thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
x <> b ;
D c= S
let Y be set ;
S ^\ m is convergent ;
q in { p } ;
V is open ;
y in N ;
x in T ;
m < n ;
m <= n + m ;
n > 1 ;
let r be Real ;
t in I ;
n <= 4 * 4 ;
M is being_line implies M is being_line
let X be set ;
Y c= Z ;
A // M ;
let U be set ;
a in D ;
q in Y ;
let x be element ;
1 <= l ;
1 <= w ;
let G be _Graph ;
y in N ;
f = {} implies {} = {}
let x be element ;
x in Z ;
let x be element ;
F is one-to-one implies F is one-to-one
e <> b ;
1 <= n ;
f is symmetric implies f is symmetric
S misses C ;
t <= 1 / 2 ;
y divides m gcd n ;
P divides M ;
let Z be set ;
let x be element ;
y c= x ;
let X be set ;
let C be Subset of T ;
x _|_ p - q ;
o is monotone ;
let X be set ;
A = A * ( B * A ) ;
1 < i ;
let x be element ;
let u be element ;
k <> 0 ;
let p be element ;
0 < r / 2 ;
let n be Nat ;
let y be element ;
f is onto ;
x < 1 / 2 ;
G c= F \/ G ;
a >= X ;
T is continuous ;
d <= a |^ n ;
p <= r / 2 ;
t < s ;
p <= t `1 ;
t < s ;
let r be Real ;
D <= E ;
assume e > 0 ;
assume 0 < g ;
p in { p } ;
x in X ;
Y in Y ;
assume 0 < g ;
c in Y ;
v in L ;
2 in z ;
assume f = g . x ;
N c= b " " " ( b " ) ;
assume i < k ;
assume u = v . u ;
I = J " ;
B = b " " ;
assume e in F ;
assume p > 0 ;
assume x in D ;
let i be element ;
assume F is onto ;
assume n <> 0 ;
let x be element ;
set k = z . k ;
assume o = x ;
assume b < a ;
assume x in A ;
a `1 <= b `1 ;
assume b in X ;
assume k <> 1 ;
f = l |^ l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is generated implies y is generated
assume m > 0 ;
assume A c= B ;
X is bounded ;
assume A <> {} ;
assume X <> {} ;
assume F <> {} ;
assume G is open ;
assume f is translation translation ;
assume y in W ;
y \not <= x ;
A A in B ;
assume i = 1 ;
let x be element ;
x = ( x * y ) * x ;
let X be BCI-algebra ;
assume S is empty ;
a in { a } ;
let p be element ;
let A be Subset of T ;
let G be _Graph , F be set ;
let G be _Graph , F be set ;
let a be Element of COMPLEX ;
let x be element ;
let x be element ;
let C be FormalContext ;
let x be element ;
let x be element ;
let x be element ;
n in { n } ;
n in { n } ;
n in { n } ;
thesis ;
let y be Real ;
X c= ( f . a ) . a
let y be element ;
let x be element ;
let i be set ;
let x be element ;
n in { n } ;
let a be element ;
m in { m } ;
let u be element ;
i in { i } ;
let g be Function ;
Z c= dom ( ( ( Z \ { 0 } ) \ ( ( n + 1 ) ) \ ( n + 1 ) ;
l <= a |^ n ;
let y be element ;
r > 0 ;
let x be element ;
let k be Nat ;
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 that f is one-to-one and g is one-to-one ;
let z be element ;
a , b // K , K ;
let n be Nat ;
let k be Nat ;
B c= B .: ( B \/ C ) ;
set s = scf cm cm cm ;
n >= 0 + 1 ;
k c= k + 1 ;
R c= R ;
k + 1 >= k + 1 ;
k c= k + 1 ;
let j be Nat ;
o , a // Y , Y ;
R c= bool G ;
bool B = B \/ B ;
let j be Nat ;
1 <= j + 1 ;
the function arccot is differentiable ;
the function tan is differentiable ;
j < i ;
let j be Nat ;
n <= n + 1 ;
k = i + m ;
assume C meets S ;
n <= n + 1 ;
let n be Nat ;
h = {} ;
0 + 1 = 1 + 1 ;
o <> b ;
f is one-to-one implies f is one-to-one
support p = {} ;
assume x in Z ;
i <= i + 1 ;
r <= 1 / 2 ;
let n be Nat ;
a "/\" b "/\" a [= a "/\" b ;
let n be Nat ;
0 <= r / 2 ;
let e be Real ;
r in cell ( G , i ) ;
c = 0 ;
a + b = a - b ;
<* 0 , 0 *> in e ;
t in { t } ;
assume F is being_line ;
m divides n ;
B * A <> {} ;
a + b <> 0 ;
p / 2 > p ;
let y be Real ;
let a be integer number ;
let l be Nat ;
let i be set ;
let r be Real ;
1 <= i + 1 ;
a "/\" c = a "/\" c ;
let r be Real ;
let i be set ;
let m be Nat ;
x = p . x ;
let i be set ;
y < r + 1 ;
rng c c= E ;
bool R is open ;
let i be set ;
R [ 0 ] ;
assume that x is open and y is open ;
X <> {} ;
x in { x } ;
b , b // M , b ;
A ( A ) . i c= Y
P [ k + 1 ] ;
2 |^ n in W ;
X [ 0 ] ;
P [ 0 ] ;
A = A ^ B ;
Ints >= s ;
G . ( y ) <> 0 ;
let X be set ;
a in { a } ;
H . ( 1 - 1 ) = ( 1 - 1 ) / ( 1 - 1 ) ;
f . y = p . y ;
let V be non empty addLoopStr ;
assume x in M ;
k < s . a ;
t in { p } ;
let Y be set ;
M , L are_cocococococoL ;
a <= g . i ;
f . ( x . b ) = b . b ;
f . ( x . c ) = c . c ;
assume that L is lower-bounded and L is lower-bounded ;
rng f = Y ;
{ G } c= L ;
assume x in Q ;
m in dom P ;
i <= Q ;
len F = len F ;
p = {} implies p => q = p => q
z in rng p ;
lim b = 0 ;
len W = n ;
k in dom p ;
k <= len p ;
i <= len p ;
1 in dom f ;
b " = a " " " " ;
x * y = a * y ;
rng D c= A ;
assume x in K ;
1 <= i & i <= len G ;
1 <= i & i <= len G ;
inf [. p , q .] c= inf \pi ;
1 <= i & i <= len -15 ;
1 <= i & i <= len -15 ;
LMP C in Upper_Arc ( C ) ;
1 in dom f ;
let s be Real ;
set C = B * C * B * C ;
x in rng f ;
assume f is Lipschitzian ;
I = A \ A ;
u in dom p ;
assume a < x ;
|. s . n .| < |. s . n .| ;
assume I c= I ;
n in dom I ;
let Q be Subset of T ;
B c= dom f ;
b + p _|_ a - b ;
x in dom g /\ dom g ;
F is continuous ;
dom g = X ;
len q = m ;
assume that A is closed and B is closed ;
assume R | S is non-empty ;
sup D in sup D ;
x "/\" y \ll sup D ;
b >= 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 that P [ k + 1 ] ;
card C c= card C ;
{ x } is one-to-one implies x is one-to-one
let e be element ;
- b divides b - b ;
F c= F ( F ) ;
G is open implies G is open
G is open implies G is open
assume v in { v } ;
assume b in \Omega _ { B } ;
let S be non void ManySortedSign ;
assume that P [ n ] ;
assume that that S is non-empty and S is non-empty and S is non-empty and S is finite ;
V is open ;
assume that P [ k + 1 ] ;
rng f c= dom f ;
inf X in X ;
y in rng f ;
let s , t be Real ;
b " " c= b " ;
assume x in { \mathbb Q } ;
A /\ B = {} ;
assume that len f > 0 and f > 0 ;
assume x in dom f ;
b , c // o , c ;
B in { B } ;
cluster \prod p -> \prod p ;
z , y // p , q ;
assume x in rng N ;
cosec `| Z is differentiable ;
assume y in rng S ;
let x , y be element ;
i < len f ;
a / h / a in H / 2 ;
p , q , r , s , s , s , r , s , s , r , s , r , s , s , s , s ,
cluster sqrt ( I ) -> being_ideal for ideal of SCM+FSA ;
q in A ;
i + 1 <= i + 1 ;
A c= A
\boldmath \boldmath \boldmath \boldmath \boldmath \boldmath \boldmath \boldmath \boldmath \boldmath \boldmath \boldmath \boldmath \boldmath \boldmath \boldmath \boldmath \boldmath \boldmath \boldmath \boldmath \boldmath \boldmath \boldmath \boldmath \boldmath \boldmath \boldmath \boldmath \boldmath \boldmath \boldmath
assume A c= dom ( f | A ) ;
Re ( f | X ) is bounded ;
let k , m , m be Nat ;
a , b // a , b ;
j + 1 < k + 1 ;
m + 1 <= n + 1 ;
g `| Z is differentiable ;
g | ( ( x | ( 0 qua Nat ) ) | ( x | ( x | ( 0 qua set ) ) ) is continuous ;
assume that O is symmetric and O is symmetric ;
let x , y be element ;
let j be Nat ;
\llangle y , z \rrangle in R ;
let x , y be element ;
assume y in conv ( conv ( conv ( A ) ) ) ;
x in Int Int Int Int Int V ;
v be VECTOR of V ;
P [ n + 1 ] ;
d , c // a , b ;
let t , u be element ;
let X be set ;
assume k in dom s ;
let r be Real ;
assume x in F ;
Y be Subset of S ;
let X be set ;
[ a , b ] in R ;
x + y < w + y ;
LIN a , c , a ;
B be Subset of A , A be Subset of A ;
let S be non void ManySortedSign ;
let x be element ;
let b be Element of X ;
R [ x , y ] ;
x \ ( x \ y ) = x \ ( y \ z ) ;
b \ x \ 0. X = 0. X ;
<* d *> in D ;
P [ k + 1 ] ;
m in dom ( ( ( n + m ) - m ) * ( n + m ) ) ;
h . ( a ) = a . b ;
P [ n + 1 ] ;
cluster G ** F -> ** F ;
let R be Relation ;
let G be _Graph ;
let j be Element of I ;
a , b // p , b ;
assume f | X is bounded ;
x in rng ( ( f | X ) ) ;
let x be Element of B ;
let t be Element of D ;
assume x in Q ( Q ) ;
set q = s ^\ k ;
let t be Element of X ;
let x be Element of A ;
assume y in rng p ;
let M be non empty MetrSpace ;
N be non empty non empty MetrSpace ;
let R be RelStr ;
let n , m , k be Nat ;
let P , Q be Relation ;
P = Q /\ [#] S ;
F . r in REAL ;
let x be Element of X ;
let x be Element of X ;
let u be Element of V ;
reconsider d = x as set ;
assume I is_halting_on s , p ;
let n , m , k be Nat ;
let x be Point of T ;
f c= f \/ g ;
assume m < v ;
x <= ( ( ( ( x / 2 ) * ( x / 2 ) ) * ( x / 2 ) ) * ( x / 2 ) ) ;
x in F ` ;
cluster S --> T --> T -> \longmapsto --> --> --> --> --> --> --> --> T -> --> --> --> --> --> --> --> --> --> --> --> --> T ;
assume that t <= 1 and 1 <= t ;
let i , j be Nat ;
assume that F <> {} and F <> {} ;
c in Intersect R ;
dom ( p | ( 1 - 1 ) ) = Seg len p ;
a = 0 or a = 1 or a = 0 ;
assume that A <> {} and A <> {} ;
set i = i + 1 ;
assume that a = b * a ;
dom ( g | A ) = A ;
i < M + 1 ;
assume - \infty in dom ( G - F ) ;
N c= dom ( ( f | X ) | X ) ;
x in dom ( ( ( the Tran of SCM+FSA ) | ( the carrier of SCM+FSA ) ) | ( the carrier of SCM+FSA ) ;
assume that x in R and y in R ;
set d = x / ( y - x ) ;
1 <= len g ;
len s > m ;
z in dom ( f | ( n + 1 ) ) ;
1 in dom ( D | ( 1 + 1 ) ) ;
( p `1 ) ^2 = ( ( ( ( p ) / ( ( ( p `2 ) / ( 1 - sn ) ) ^2 ;
j <= width G ;
\pi > \pi / 2 ;
set n = ( 1 - n ) * ( 1 - n ) + 1 ;
|. q .| = |. q .| ;
let s be State of S ;
bag ( i , m ) = m m ;
X c= dom f ;
h . ( x in dom h . x ;
let G be non empty MetrSpace ;
cluster m * n -> m * n -> m * m ;
let k be Nat ;
i + 1 > m ;
R is connected
set F = <* u *> ^ <* u *> ;
p c= P \/ Q ;
I is_closed_on s , P +* I ;
assume that S is Sub_Sub_Sub_Sub_Sub_Sub_Sub_4 and S is Sub_Sub_universal ;
i <= len f ;
p is FinSequence of X ;
1 + 1 in dom g ;
Sum R = Sum R ;
cluster f . x -> complex-valued yielding ;
x in dom ( f | X ) ;
assume X in C ;
B c= B ;
n <= 2 |^ n ;
A c= P /\ A ;
cluster id ( x ) -> -valued ;
Q be Subset of S ;
assume n in dom ( g | n ) ;
a be Element of R ;
t `2 in dom ( e | ( ( n + 1 ) ) ;
N . ( N . 1 ) in rng N ;
- z in A \/ B ;
S be SigmaField of X , a , b be Element of X ;
i . i in rng f ;
reconsider R = dom f as Function ;
f . x in rng f ;
reconsider t = r / 2 as Point of TOP-REAL 2 ;
s in { s } ;
let z , y be Element of A ;
n <= N . n ;
LIN q , p , q ;
f . ( ( x /\ B ) = ( f . ( x /\ B ) ) /\ f . ( x /\ B ) ;
set L = T \to T ;
let x be element ;
m be Element of M ;
f in union rng f ;
let K be Field , K be Field ;
let i be Element of NAT ;
rng ( F * ( g | ( X /\ Y ) ) ) c= Y ;
dom f c= dom f ;
n < 1 + 1 ;
n < 1 + 1 ;
cluster exp ( X , Z ) -> limit_ordinal ordinal ;
\llangle z , y \rrangle in [: z , y :] ;
let m be Nat ;
S be Subset of R ;
y in rng seq2 ;
b = sup rng f ;
x in Seg len q ;
reconsider X = ( X \ Y ) as set ;
a in { a } ;
assume n in dom ( ( h | n ) | n ) ;
w + 1 = { a } - 1 ;
j + 1 <= j + 1 ;
k + 1 <= k + 1 ;
i be Element of NAT ;
Support u = Support p ;
assume X is being_being_being_-5 means : Def3 : for m being Element of NAT holds X is being_being_being_being_being_MPLE m , n , n ;
assume that f = g and g = f . p ;
n <= n + 1 ;
let x be Element of X ;
assume x in rng s ;
{ x } < 0 ;
len L = m ;
P c= A ;
dom q = Seg n ;
j <= width M ;
let r be Real ;
let k be Element of NAT ;
\int P + \infty < M + + + + + + + + + + + + + + + + + + + + \infty ;
let n be Nat ;
assume z in Funcs ( A , A ) ;
let i be set ;
n - 1 = - 1 ;
len ( seq2 ^\ n ) = n ;
InitSegm ( Z , c ) c= ]. x0 , x0 .[ ;
assume x in X ;
x , y // a , b ;
let A , B , C be Subset of X ;
set d = ( p |^ ( n + 1 ) ) |^ ( n + 1 ) ;
let p be Element of L ;
Seg Seg i = dom q ;
let s be Element of E ;
basis basis basis of x , y ;
{ L } /\ {} = {} ;
L /\ ( L /\ L ) = {} ;
assume \downarrow \downarrow \downarrow x ;
assume b , c // b , c ;
LIN c , b , c ;
x in rng f ;
set n = 8 * n + j + 1 ;
let D be set ;
let K be Field ;
assume that f ' = f .: A ;
R + ( - R ) + ( - R + R ) is total ;
k in Seg ( len p ) ;
a be Element of G ;
assume that { x } in [. 0 , 1 .] and x in [. 0 , 1 / 2 .] ;
K is being_line implies K is being_line
assume a in C ;
a , b , c , d be Element of S ;
reconsider d = x as Vertex of G ;
x in ( ( s * ( s * A ) ) .: A ) .: A ;
set a = \int f + M ;
ex M being set st M is nnnnnnnnnnnnnnnnnnnnnnnnnnnnn
u in { \boldmath \boldmath $ b } } ;
the support support support f c= B ;
reconsider z = x as VECTOR of V ;
cluster the topology of L -> sigma ;
r (#) H is integrable ;
s . ( s . ( ( intloc 0 ) ) ) = 0 ;
assume x in C ;
let U be non empty ManySortedSign ;
[ x , y ] is compact ;
i + 1 in dom p ;
F . ( i ) is Element of M ;
r in DedekindCut ( y ) ;
let x , y be Element of X ;
A , A , B , C , C , D , C , C , C , D , C , C , C , C , Q , Q , Q , C , Q , Q
z in { O } ;
not halt SCM+FSA st not halt SCM+FSA holds not halt SCM+FSA = halt SCM+FSA
rng ( A * ( A * ( B * A ) ) ) = A ;
q \vdash q '&' p '&' q '&' q '&' q '&' q '&' q '&' q '&' q '&' q '&' q '&' q '&' q '&' q '&' q '&' q '&' '&' q '&' '&'
for n holds X [ n ] ;
x in { a } ;
for n being Nat holds P [ n + 1 ] ;
set p = [ x , y ] , z = [ x , y ] ;
LIN o , a , b ;
p . ( 2 ^ Z ) = Z ^ Y ;
( ( ( D ) \ ( {} ) ) /\ {} = {} ;
n + 1 <= len g ;
a in ]. a , b .[ ;
u in Support p ;
let x , y be Element of G ;
I be ideal of L ;
set g = f + g ;
a <= max ( a , b ) ;
i-1 < len G + 1 ;
g . ( 1 + 1 ) = f . ( 1 + 1 ) ;
x , y // x , y ;
( f /* s ) /* ( f /* s ) is convergent ;
set v = the diagdiagR+valued ;
i + 1 <= k + 1 ;
cluster mode multF -> associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative
x in support ( t ) ;
assume a in { G } ;
i + 1 <= len y ;
assume p divides b + b ;
M . sup M <= sup M ;
assume x in W ...min ( X , Y ) ;
j in dom z ;
let x be Element of D ;
s4 = s4 . s ;
a = {} or a = {} ;
set u = the Element of G ;
s is convergent implies s is convergent & s is convergent
for k being Nat holds X [ k + 1 ] ;
for n holds X [ n ] ;
F . m in F . m ;
h c= { h } ;
]. a , b .[ c= ]. a , b .[ ;
X is being_line implies X is being_line
a in union F ;
set x = [ 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0
k + 1 = k + 1 ;
cluster binary binary binary binary binary yielding Function -> binary yielding yielding yielding yielding yielding yielding yielding yielding yielding yielding yielding yielding yielding yielding yielding yielding yielding Function ;
ex C being Subset of X st C = C + C ;
let G be _Graph , F be Function ;
assume V is non empty ;
X /\ Y in Y /\ ( Y /\ L ) ;
reconsider x = x as Element of S ;
max ( a , b ) = a max ( b , c ) ;
sup ( B .: ( B \/ C ) is open ;
let L be non empty doubleLoopStr ;
R is symmetric in X & R in X ;
E \models ( g '&' h ) '&' ( g '&' h ) '&' ( g '&' h ) ;
dom ( G | ( a " ) ) = [. a , b .] ;
- 1 >= - 1 ;
( G . p ) `1 in rng p ;
let x be Element of F ;
D [ 0 ] ;
z in dom id ( id ( A ) ) ;
y in the carrier of N ;
g in the carrier of H ;
rng f c= dom f ;
j + 1 in dom ( ( f2 + 1 ) + 1 ) ;
A , B , C be Matrix of G ;
C is non empty
f . ( z + 1 ) in dom h ;
P . ( k + 1 ) in rng P ;
M = A * ( A + B ) ;
let p be Element of NAT ;
f . ( n + 1 ) in rng f ;
M . ( M . ( 0 + 1 ) ) in REAL ;
sqrt sqrt sqrt ( a , b ) = sqrt ( b / a ) ;
assume the distance is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_
let a be Element of V ;
let s be Element of product P ;
let P be Subset of TOP-REAL 2 ;
let n be Nat ;
the support support support support ( g ) c= B ;
I = halt R ;
consider b such that b in B ;
set BB = BCS K ;
l <= Sup ( F . j ) ;
assume x in ]. x , y .[ ;
( x in { x } \ ( t ) ;
x in JumpParts ( T ) iff x in JumpParts ( T )
let h be h , a be Real ;
Y c= card ( R .: ( Y , Z ) ) ;
A \/ B c= A \/ B ;
assume LIN o , a , b ;
b , c // d , e ;
x in Y /\ Y ;
dom ( y | Seg 1 ) = Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. l .| ;
[: x , y :] in [: X , Y :] ;
for n being Nat holds ( ( x / n ) * x ) . x = ( ( x * n ) . x ;
[: a , b :] = [: a , b :] ;
assume that for T being non empty TopSpace holds T is finite-ind ;
x = ( f . ( x + 1 ) ) . ( x + 1 ) ;
( q , p ) . 1 in P ;
dom ( M * ( n + 1 ) ) = Seg n ;
x = ( x * y ) * x ;
R , Q be Relation ;
set d = - 1 / 2 ;
rng ( g | X ) c= dom g ;
P ( \Omega ) misses \Omega ( B ) ;
a in field R ;
M , M be non empty MetrSpace ;
I be program of SCM+FSA ;
assume x in R ;
let b be Element of T ;
dist ( z , z ) < r / 2 ;
u in { v } ;
the carrier of L misses the carrier of L ;
let L be non empty doubleLoopStr ;
assume x in { a } ;
dom ( A * ( ( A * ( ( ( A * ( ) * ( A * ( B * A ) ) ) ) ) ) = ( ( ( A * ( A * ( A *
a , b , c , d , d , d , d , d be Element of the carrier' of G ;
let x be Element of M ;
0 < a / 2 / 2 / 2 ;
o , b1 // o , b1 ;
v c= the support l ;
let x be Element of A ;
assume x in dom ( ( ( f | X ) ) | X ) ;
rng F c= dom ( f | X ) ;
assume D . ( ( ( D . ( k + 1 ) ) ) in rng ( D . ( k + 1 ) ) ;
f " " = ( f " ) " ;
set x = the Element of X ;
dom ( G * F ) = dom G ;
let n be Nat ;
assume LIN c , a , c ;
for n being Nat holds ( ( ( ( n + 1 ) --> 0 ) --> 0 ) --> 0 ) --> 0 ) yielding yielding yielding yielding yielding yielding yielding yielding yielding yielding yielding yielding yielding yielding yielding yielding
reconsider d = c as Element of L ;
( v |-- ( \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow
assume x in the carrier of A ;
conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv
reconsider B = b as Element of the carrier of T ;
J J J |= J => l => l => l ;
cluster J -> J -> non-empty for Function ;
sup X in Y \/ Y ;
W is_is_orders W implies W is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_
assume x in the carrier of R ;
dom ( n |-> 0 ) = Seg n ;
{ s } misses { s } ;
assume ( a 'imp' b ) 'imp' b = ( a 'imp' b ) 'imp' ( a 'imp' b ) ;
assume that X is open and X is open and X is open ;
assume that a in the carrier of Lin ( M1 ) and a in the carrier of Lin ( M2 ) ;
assume that card I c= card J and card I c= card I ;
dist ( ( seq2 . n ) = 0 ;
( the function of #Z n ) * ( ( ( - 1 ) * ( ( - 1 ) ) ) * ( ( - 1 ) ) ) ) <> 0 ;
the function is differentiable ;
t . ( t . ( n + 1 ) ) = ( t . ( n + 1 ) ;
dom ( F - F ) = dom F ;
( ( ( ( ( ( seq1 | X ) ) | X ) | X ) | X ) ) | ( X | ( X | ( X /\ Y ) ) ) ) ) ) = ( ( seq1 | X )
y in W .edges() ;
k <= len v ;
x * y mod ( m * ( m * n ) ) = ( m mod ( m * n ) ) mod ( m * m ) ;
proj2 .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .:
h . ( ( p . ( q ) ) ) = ( h . ( q ) ;
U = U . ( ( U . 1 ) ) ;
f . ( r / 2 ) in rng f ;
i + 1 + 1 <= len f ;
rng F = dom ( F | ( ( n + 1 ) ) ;
A is being_line implies A is being_line
[: x , y :] in [: A , A :] ;
( ( ( ( ( x * y ) ) | ( o , o ) ) | o ) ) . x in o ;
the support support support support support B c= B ;
id X in id X ;
1 + 1 <= len f ;
{ s : s is convergent } ;
len F = len I ;
let l be Linear_Combination of B ;
let r , s be Real ;
Comput ( P1 , s , n ) = Comput ( P1 , s , n ) ;
k <= len p ;
reconsider c = {} as Element of L ;
let Y be Subset of T ;
cluster id ( X ) -> one-to-one ;
f . ( j + 1 ) in K ;
cluster J --> J -> total for Function ;
K c= the topology of T ;
F . ( b ) = ( b . ( b . b ) ;
x = y or x = x ;
If a <> 0 & a <> 0 ;
assume that a cf b c= cf a ;
( s . n ) / 2 in rng s ;
LIN o , b1 , b ;
LIN o , b , b ;
reconsider m = x as Element of V ;
let f be FinSequence of D ;
let F be Point of TOP-REAL 2 ;
assume that h is one-to-one and h is one-to-one and h is one-to-one ;
f ( ( f , w ) in F ( w , y ) ;
reconsider p = x as Point of X ;
A , B , C , C , R , R , S , S , R , R , S , R , S , R , S , R , S , R , R , S , S
cluster mode TopStruct -> strict for non empty MetrSpace means : Def1 : Def1 : Def1 : Def1 : Def1 : Def1 : ex it st q is it & q is it is being_line & q is it & q
rng c misses rng c ;
z be Element of Data-Locations SCM+FSA ;
b in dom ( p +* q ) ;
assume that that k >= n and n >= k and k >= k ;
Z c= dom ( ( cot | Z ) `| Z ) ;
the carrier of A is symmetric connected ;
reconsider E = i as Subset of I ;
{ g : g in ]. - 1 , 1 / 2 .[ } ;
f = a * f implies f * f = a * f
for n holds P [ n ] ;
not x in ( ( ( ( x \ L ) ) \ L ) \ L ) ;
let x be Element of V ;
a , b , c , d , d be Real ;
assume S = S . p ;
gcd ( n , m ) = {} ;
set o = { \mathbb I } ;
|. s . n .| < |. s . n .| ;
assume that that s is one-to-one and s is one-to-one and s is one-to-one ;
f . ( x , y ) <= a ;
ex c being Element of M st P [ c , d ] ;
set g = ( \ { n } ) \ { n } ;
k = a or k = b or k = c or k = b or k = c or k = b or k = c or k = b or k = c or k = c or k = c or k = b
{ a } <> { b } ;
assume Y = {} ;
I . ( ( ( I . x ) ) * ( ( ( ( x - x ) * ( ( ( 1 - x ) * ( x - x ) ) ) * ( ( 1 - x ) ) ) ) ) )
{ W _ { 19 } } = W .first() ;
cluster mode TopStruct of G -> strict for non empty SubSpace of G ;
reconsider u = u as Element of Bags ( n ) ;
A in B implies A in B
x in { n } ;
1 - ( ( ( ( ( 1 - 1 ) ) ) / ( 1 - 1 ) ) ^2 >= 1 ;
f . 1 in Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int
( f /. ( 1 + 1 ) ) `1 <= ( f /. ( 1 + 1 ) ) `1 ;
h in the carrier of TOP-REAL 2 ;
( b `2 ) ^2 >= ( ( ( b `2 ) ^2 ;
let f , g , h be PartFunc of X , COMPLEX ;
S /. k <> 0. ;
x in dom ( f - g ) ;
( p in { p } ;
len ( H ) < len H ;
F ( A , B ) = ( ( A * B ) * ( A * B ) ;
consider Z such that Z in Z and Z c= Y ;
1 in C .: A ;
assume that r <> 0 and 1 <> 0 and r <> 0 ;
rng ( q | ( n + 1 ) ) c= dom q ;
A , B , C , C are_mutually_different ;
y in rng f ;
f /. i in L~ f ;
b in midpoint ( p , b , q ) ;
if S is Sub_universal & S is universal & S is universal
[#] T c= [#] T ;
f | ( ( A /\ B ) = f | ( A /\ B ) ;
0 in the carrier of M ;
v , v , u // v , w ;
reconsider K = K as set ;
X \ Y c= V \ Y ;
X be Subset of T ;
consider H such that H = ( H . 1 ) '&' H . 1 ;
1 c= t * ( r / t ) ;
0 * a = 0 ;
A ^ B = A ^ B ;
set v = { \bf 0. } ;
r = 0. TOP-REAL n ;
( f . ( p `1 ) ^2 >= 0 ;
len W = len ( W | ( ( n + 1 ) ) ;
f /* s is convergent ;
consider l such that l = m * ( n + 1 ) ;
t <> b ;
reconsider Y = X as Subset of X ;
consider w such that w in F and w in F ;
let a , b , c , d be Real ;
reconsider i = i as Element of NAT ;
c . ( x "/\" ( y "/\" ( x "/\" y ) ) >= ( x "/\" y ) "/\" ( ( x "/\" y ) ;
sigma T c= the topology of T ;
for x being element holds X [ x , y ] ;
cluster ~ -> ~ -> non empty ;
\downarrow \downarrow \downarrow \downarrow \downarrow t ;
let X be set ;
rng f = TS ( D , X ) ;
let p be Element of the carrier of B ;
( ( N + 1 ) /\ ( N ) ) ` >= {} ;
0 <= b |^ m ;
assume i in I & I . i in I ;
i = ( p * i ) * ( 1 - 1 ) ;
assume that R in the carrier of R and the carrier of R in the carrier of R ;
let A , B be Subset of T ;
x in [#] ( P ) ;
1 in Seg ( len M ) ;
reconsider X = { X } as Subset of T ;
x in the carrier of A ;
assume that E is open and E is finite ;
n <= len g + 1 ;
( i + 1 ) + 1 = i + 1 ;
assume v in the carrier of G ;
y = ( y + z ) * ( y + z ) ;
Indices ( p , n ) = Indices p ;
( ( ( ( ( x / 2 ) (#) ( ( x / 2 ) (#) ( x / 2 ) ) (#) ( ( x / 2 ) (#) ( x / 2 ) (#) ( x / 2 ) ) ) (#) ( x / 2 )
rng M c= rng M ;
for p being Real holds p in Z implies p in Z
X = ( X --> ( f ) ) * ( X , f ) * ( X , f ) ;
( ( s ^\ k ) (#) ( s ^\ k ) ) (#) ( s ^\ k ) <> 0 ;
s . ( ( ( G . k ) ) / 2 > 0 ;
( p - ( M - ( K ) ) ) * p = - ( M * ( K - M ) ;
A /\ ( A \/ B ) = ( ( A \/ C ) /\ ( ( A \/ C ) ;
h mod ( g mod ( g ) ) mod ( ( g mod ( g ) ) = ( g mod ( g mod ( g ) ) ;
reconsider i = i-1 . i as Element of NAT ;
let v , u , v be Element of V ;
the carrier of V is open ;
reconsider i = i as Element of NAT ;
dom f c= [: f , f :] ;
x in the Sorts of ( ( the Sorts of B ) ) . n ;
len f2 in Seg len f2 ;
{ p } c= the topology of T ;
]. s , s / 2 .[ c= ]. s , s .[ ;
B be prebasis of T ;
G * ( ( B * A ) ) = ( ( B * A ) * ( B * A ) .= ( ( ( B * A ) * ( B * A ) ) * ( B * A ) .= ( ( (
assume that p in { p } and q is not zero ;
z in rng F ;
( b => ( b => b ) => ( b => b ) ) => ( b => ( b => b ) ) ) ) = ( b => ( b => b ) => ( b => b ) ;
deffunc F ( set ) = ( $1 \ 1 ) \ $1 ;
LIN a , b , c ;
f .: ( ( f .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .:
dom ( ( w | ( n + 1 ) ) = the carrier of TOP-REAL 2 ;
assume 1 <= i & i <= n ;
( g /. ( 1 + 1 ) ) `2 <= ( ( g /. ( 1 + 1 ) ) `2 ;
p in LSeg ( p , i ) ;
I /. ( i + 1 ) = 0. K ;
|. ( s . m ) - ( s . m ) .| < |. s . m .| ;
( q ) . x in rng q ;
L /\ ( ( L /\ ( L ) ) ) misses ( L /\ ( L /\ ( L ` ) ;
consider c such that c in G and a in G ;
assume NNNNa1 = o ;
q . j = ( j + 1 ) * ( j + 1 ) * ( j + 1 ) ;
rng F c= dom F ;
P . ( B \/ D ) c= B ;
f . ( j + 1 ) in \lbrack f . ( j + 1 ) ;
then 0 <= x ^2 ;
p `2 - q `2 > 0 ;
cluster SCM_op S -> well cancelable ;
let x be Element of S ;
the MA of MA is MA ;
|. i - 1 .| <= |. i - 1 .| ;
the carrier of TOP-REAL 2 = the carrier of TOP-REAL 2 ;
n! * ( n + 1 ) ! > 0 ;
S c= ( ( A /\ B ) /\ ( A /\ B ) ;
LIN a , b , b ;
assume A <> {} ;
1 + 1 = ( 1 - 1 ) / 2 ;
x `2 / 2 * a `1 < ( a * b ) / 2 ;
set v = { v _ { 19 } } , v = v + { v _ { 19 } } ;
x = r . ( n + 1 ) ;
f . ( s . s ) in the carrier of S ;
dom g = the carrier of TOP-REAL 2 ;
p in Upper_Arc ( P ) ;
dom ( d * ( A * ( B * A ) ) = [: A , A :] ;
0 < p - q ;
e . m <= ( ( m + 1 ) * m + 1 ) * ( m + 1 ) ;
B \/ B c= B \/ B ;
- - - ( g + M ) < \int ( f + M ) ;
assume that O is limit_ordinal and O is limit_ordinal and O is limit_ordinal and O is limit_ordinal ;
let U , U be non-empty MSAlgebra over S ;
( Proj ( i , n ) * ( i , n ) is Proj ( i , n ) * ( i , n ) ;
let x , y be Point of X ;
reconsider p = ( p . x ) as Point of X ;
x in the carrier of A ;
let I , J be parahalting Program of SCM+FSA ;
assume that that a is bounded and a is bounded and a is bounded ;
Int Cl Cl A c= Cl Cl Cl A ;
assume for A being Subset of X holds A is closed ;
assume q in Ball ( p , r ) ;
( p `1 ) ^2 <= ( ( ( ( p ) / ( ( ( p `2 ) / ( ( 1 - sn ) ) ^2 ) ^2 ;
[#] Q = [#] T ;
set S = the TopStruct of T ;
set I = [#] TOP-REAL 2 ;
len p = n -' n ;
A A is Swap implies A , x Swap ( A , x )
reconsider kk = i as Nat ;
1 <= j + 1 ;
reconsider q = [ q , m ] as Element of M ;
a in the carrier of S ;
( ( ( 1 - 1 ) * ( 1 - 1 ) ) * ( ( 1 - 1 ) ) ^2 = ( 1 - 1 ) * ( 1 - 1 ) ;
let f be Function of TOP-REAL 2 , TOP-REAL 2 ;
y = ( ( f | ( x \ y ) ) | ( x \ y ) ) | ( x \ y ) ;
consider x such that x in \mathop { \rm InvolololololololA ;
assume r in ( dist ( o , r ) ) .: ( ( ( ( o , r ) ) .: ( ( ( o , r ) ) ;
set i = ( TOP-REAL 2 ) | P ;
h . ( j + 1 ) in rng h ;
Line ( M , i ) = Line ( M * ( i , j ) ;
reconsider m = x as Element of REAL ;
U , U , U , U , U , U , U , U , U , U , U , U , U , U , U , U , U , U , U , U , U , U , U , U , U ,
set P = Line ( a , b ) ;
len p < len p ;
T , T be connected ;
assume x <=' y ;
set M = ( M * ( n + m ) ) * ( n + m ) ;
reconsider i = x as Element of NAT ;
rng ( a * ( a * ( a * ( a * b ) ) ) c= dom ( a * a ) ;
z " = ( z " ) " ;
{ x - r } in L /\ L ;
One can check that w *^ S -> limit_ordinal ;
set x = Z ^ { x } ;
len w in Seg len w ;
( ( ( ( curry g ) * f ) * f ) . x = ( g . x ) . x ;
a be Element of the carrier of Lin ( V ) ;
( x * ( ( ( ( x * A ) * A ) ) * ( x * A ) ) ) * ( x * A ) ) ) ) ) ) . x = ( ( x * A ) * ( x * A ) ) * ( x * A )
( p `1 ) ^2 <= ( ( ( ( ( ( ( p ) / ( ( 1 - sn ) ) ) ^2 ) ^2 ) ^2 ;
rng ( g | X ) c= X ;
reconsider k = i-1 * j + 1 as Nat ;
for n being Nat holds P [ n + 1 ] ;
reconsider x = x as Point of M ;
dom ( f | X ) = X /\ ( X /\ Y ) ;
p , b // p , a ;
reconsider x1 = x as Element of REAL ;
assume i in dom ( p ^ q ) ;
m . m = ( ( ( ( ( ( m |^ n ) |^ m ) |^ m ) |^ m ) |^ m ) |^ m ) |^ ( m |^ n ) ) ) |^ m ) ) |^ m ) ;
a ^ ( m - 1 ) = ( m - 1 ) ^ ( m - 1 ) ;
S . ( n + 1 ) c= S . ( n + 1 ) ;
assume that B \/ C = B \/ C ;
X . i = X . i ;
r in dom ( ( ( ( ( r (#) ( f + g ) ) (#) f ) (#) f ) ) (#) ( ( ( ( r (#) f ) (#) f ) (#) f ) (#) f ) ) (#) ( ( ( ( r (#) f ) (#) f
a-b = b - b ;
reconsider B1 = ( ( t . t ) . t as State of SCMPDS ;
set T = the topology of product ( X , T ) ;
Int Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl
consider y being element such that y in L and x = y ;
rng ( F | ( ( x ) ) \ ( F | ( ( x \ y ) ) ) = {} ;
{ G } \ { c } c= B \/ B ;
f is_between x0 , x0 ;
set R = the carrier of TOP-REAL 2 ;
assume that n + 1 in Seg n and n + 1 in Seg n ;
let k be Element of NAT ;
reconsider p = ( p . n ) as Point of TOP-REAL 2 ;
g . x in dom f ;
assume that 1 <= n and n + 1 <= len p and n + 1 <= len p ;
reconsider T = b |^ ( N , T ) as Element of NAT ;
len P <= len P ;
x in the carrier of A ;
\llangle i , j \rrangle in Indices M ;
for m being Nat holds F . m in S ;
f . x = ( a . x ) . x ;
let f be PartFunc of REAL , REAL ;
rng f = the carrier of A ;
assume that that s = p and s = p ;
If a is Real & b > 0 & b > 0 ;
let A , B , C , D be Subset of TOP-REAL 2 ;
reconsider X = 0. X as Subset of X ;
f be PartFunc of CNS , RNS ;
r * ( ( v - r ) ) / ( 1 - r ) < r * ( 1 - r ) ;
assume V is open and V is open ;
reconsider t = intpos intpos intpos n as State of SCMPDS ;
Q [ e , v ] ;
g \circlearrowleft ( z ) \circlearrowleft ( z ) \circlearrowleft z = z \circlearrowleft z ;
|. x - y .| + |. x - y .| = |. x - y .| .= ||. y - y .|| .= ||. x - y .|| .= ||. x - y .|| .= ||. y - y .|| ;
- ( f - ( - g ) ) = ( - f ) - ( - g ) ;
z *' + y <= z + y ;
sqrt ( 1 - ( 1 - 1 ) ) ^2 > 0 ;
assume X is commutative and ( X is commutative implies X is commutative & 0 <= X & 0. X = 0 & 0. X = 0 ;
F . ( ( 1 - 1 ) ) = F ( ( 1 - 1 ) ;
( f | X ) | X = f | X ;
( the function of ( ( the function ) ) | ( the carrier of TOP-REAL 2 ) ) | ( the carrier of TOP-REAL 2 ) = the carrier of TOP-REAL 2 ;
i = ( f /. i ) /. ( len f ) ;
X = { X } \/ Y ;
[. a , b .] = [. a , b .] ;
let V , W , C be VectSp of K ;
dom ( g | ( ( n + 1 ) ) = the carrier of TOP-REAL 2 ;
dom ( ( f | X ) = the carrier of TOP-REAL 2 ;
( ( ( ( ( id X ) | X ) | X ) | X ) | X = ( ( id X ) | X ) | X ;
f . ( x , y ) = ( f . ( x , y ) ;
|. ( ( x / 2 ) * ( x / 2 ) .| < r / 2 ;
|. ( ( ( ( ( ( ( seq1 (#) seq ) ) (#) ( seq1 ^\ k ) ) (#) seq ) ) (#) seq ) ) (#) ( seq1 ^\ k ) ) ) ) (#) ( seq1 ^\ k ) ) ) ) ) ) . n - ( seq1 ^\ k )
width ( A * ( i , j ) ) = width A ;
reconsider S9 = S .: ( op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op
reconsider f = v + u as Function of X , Y ;
intloc ( 0 , SCM+FSA ) in dom ( p +* q ) ;
i <> b " ;
arcsin + r = arcsin . r + r ;
for x being element holds ( for x st x in Z holds ( ( id Z ) * ( ( ( id Z ) * ( ( id Z ) ) ) * ( ( ( id Z ) * ( ( id Z ) * ( ( id Z ) ) ) ) ) . x = ( ( id Z
reconsider q = x as Point of TOP-REAL 2 ;
( 0 qua Nat ) + 1 <= ( 0 qua Nat ) * ( i + 1 ) ;
assume f in the carrier of [: X , Y :] ;
F . ( a . ( x ) ) = ( F . ( x ) ) . ( x , y ) ;
not true & not true implies not true = ( {} )
dist ( ( ( ( a , b ) ) ) < r / 2 ;
1 in the carrier of I[01] ;
( ( ( ( ( ( p - 1 ) / 2 ) ) * ( 1 - 1 ) ) / 2 - ( 1 - 1 ) / 2 ) ) ^2 > 0 ;
|. ( r - 1 ) * ( ( r - 1 ) .| = r * ( 1 - 1 ) ;
reconsider S9 = S as Element of NAT ;
( A \/ B ) \/ ( A \/ B ) c= A \/ B \/ B ;
DW ....................................................
i = a + b ;
f . ( f . a ) [= f . ( a "/\" f . a ) ;
If f = v + u ;
I . ( I . ( n + 1 ) ) = ( I . ( n + 1 ) ;
\raise .4ex \hbox \hbox \hbox \hbox \hbox .4ex \hbox \hbox \hbox .4ex \hbox .4ex \hbox \hbox .4ex \hbox \hbox \hbox \hbox .4ex \hbox \hbox \hbox .4ex \hbox .4ex \hbox \hbox \hbox .4ex \hbox \hbox \hbox .4ex \hbox \hbox .4ex \hbox \hbox \hbox \hbox \hbox \hbox \hbox \hbox \hbox \hbox \hbox .4ex \hbox \hbox \hbox
a = VERUM implies a = VERUM ( A )
reconsider k = ( ( ( ( s . k ) ) * ( s . k ) ) / 2 as Element of NAT ;
Comput ( P1 , s , 1 ) . IC = Comput ( P1 , s , 1 ) . IC SCMPDS ;
LSeg ( M , 1 ) c= LSeg ( M , 1 ) ;
set h = the TopStruct of X ;
set A = ( ( A \ B ) \ A ) \ ( A \ B ) ;
for H being non empty doubleLoopStr holds H is finite
set b = S . i , b = S . i , S = S . i , S = S . i , S = S . i , S = S . i , S = S . i , S = S . i , S = S . i , S = S .
Hom ( a , b ) Hom ( a , b ) = Hom ( a , b ) ;
1 < s / 2 ;
( l `2 ) `2 = ( l `2 ) `2 ;
y + ( i + 1 ) in dom g ;
let p be Element of A ;
X /\ ( X /\ Y ) c= X /\ ( X /\ Y ) ;
p in rng p ;
1 <= indx ( D2 , D1 ) ;
assume x in K ;
- 1 <= ( ( ( - ( ( ( ( 1 - 1 ) ) * ( 1 - ( ( 1 - 1 ) ) ) * ( 1 - 1 ) ) ) ) / ( 1 - ( 1 - ( ( 1 - ( ( 1 - ( ( 1 - 1
f , g , f be Function of TOP-REAL 2 , TOP-REAL 2 ;
k -' 1 = ( ( k + 1 ) - 1 ) - 1 ;
rng s c= ]. - r , + \infty .[ ;
g in ]. - r , - r / 2 ;
sgn ( p , K ) = - - ( - K ) ;
consider b such that b = p * b * p ;
ex A being Ordinal st A is limit_ordinal & A is limit_ordinal ;
union { t } = union ( ( X \/ Y ) ) ;
len t = len t + 1 ;
v = 0. V + 0. V ;
DataLoc ( DataLoc ( v , DataLoc ( t ) ) ) <> DataLoc ( t . intpos ( ( ( t . intpos ( ( t . intpos ( ( t . intpos ( ( t . ( ( t . ( ( ( t . ( t . ( ( t . ( ) )
g . ( s . s ) = sup ( ( rng s ) ;
( \dot \dot \dot \dot ( y ) ) . s = ( \dot \dot \dot ( y ) ) . s ;
s in { s : s < t } ;
s \ ( s \ ( s \ t ) ) = s \ ( s \ t ) ;
defpred P [ Element of NAT ] means ex B being Element of A st B in A & B in A ;
( ( ( 339 ) * ( 1 / 2 ) ) ^2 = ( ( ( 1 / 2 ) ^2 ) ^2 ;
U = ( A succ succ succ succ succ A ) *^ ( ( succ A ) ;
reconsider y = x as Element of X ;
consider i such that y = i * y ;
reconsider p = Y as Element of Seg n ;
set f = ( U --> U ) --> U --> U , U = U --> U ;
consider Z such that Z in X and Z in X ;
let f be Function of TOP-REAL 2 , TOP-REAL 2 ;
M ( M , M ) <> M ;
ex r being Real st r = ( r * b ) * r ;
R , R , S , R be Relation ;
reconsider l = 0. V as Linear_Combination of A ;
set r = |. s . n .| + |. s . n .| ;
consider y being element such that y in S and z in S ;
a `1 = b `1 ;
||. x - g /. x - g /. x .|| < r / 2 ;
b , c // a , b ;
1 <= k + 1 ;
( ( ( - 1 ) - ( 1 - 1 ) ) * ( ( 1 - 1 ) ) ^2 >= 0 ;
( ( ( ( - 1 ) * ( ( - 1 - 1 ) ) / ( 1 - ( 1 - 1 ) ) ^2 ) ^2 < ( - 1 ) ^2 ;
RR in L~ R ;
consider e being element such that a = e * ( a + b ) ;
Re ( F | ( D ) ) = ( F | ( D ) ;
LIN b , b , b ;
p , b // a , b ;
g . ( n + 1 ) = ( g . ( n + 1 ) * ( f . ( n + 1 ) ) .= ( f . ( n + 1 ) ) * ( f . ( n + 1 ) ) .= f . ( n + 1 ) .= f . ( n + 1 )
consider f such that f .: X = f .: X ;
F | ( ( ( N | ( N | ( N | ( N | ( N | ( N | ( N | N ) ) ) | ( N | ( N | ( N | ( N | ( N | ( N | N ) ) ) ) ) = ( N | ( N | (
q in { p } \/ { p } ;
Ball ( Ball ( m , n ) Ball ( m , n ) Ball ( m , n ) Ball ( m , n ) ;
the carrier of V = the carrier of Lin ( A ) ;
rng ( ( ( ( - 1 ) / 2 ) * ( 1 - 1 ) ) ) = [. - 1 , 1 .] ;
assume that that Im ( s ) is bounded and ( ( ( Im s ) * ( ( Im s ) ) * ( ( Im ( s ) ) * ( ( Im s ) * ( ( Im s ) * ( ( Im s ) * ( ( Im s ) * ( ( Im s ) * (
||. v - v .|| < e / 2 ;
set g = O --> 0 ;
reconsider t = 0 as Point of TOP-REAL 2 ;
reconsider x = ( x - y ) / 2 as Point of TOP-REAL 2 ;
assume not LIN p , q ;
- ( - 1 ) < ( - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( ) ) ) * ( 1 - ( 1 - ( 1 - ( 1 - ( ) - ( ) - ( 1 - ( 1 - ( 1 - ( ) - ( 1
set d = \rho ( x , y ) , \rho ( x , y ) ;
2 |^ n = 2 |^ n ;
dom v = Seg len ( d | ( n + 1 ) ) ;
set x = - 1 + 1 ;
assume for n holds X [ n ] ;
assume 0 <= ( ( ( ( ( 0 + 1 ) ) | ( 1 + 1 ) ) | ( 1 + 1 ) ) ) ;
for A being Subset of X holds A /\ ( A \ ( X ) ) = ( ( A \ ( X \ A ) ) \ ( A \ A ) ) ;
the support of L c= the support of L ;
p => q => p => p => q => p => p => q => p => q => p => q => p => q => p => q => p => q => p => q => p => q => q => p => q => p => q ;
( f | n ) | n = ( f | n ) | n ;
reconsider Z = the carrier of [: the carrier of [: the carrier of T , the carrier of T :] as Subset of [: the carrier of T :] ;
Z c= dom ( ( ( id Z ) (#) ( ( ( id Z ) (#) ( ( ( id Z ) (#) ( ( ( id Z ) (#) ( ( ( id Z ) (#) ( ( ( id Z ) (#) ( ( ( id Z ) (#) ( ( ( id Z ) (#) ( ( (
|. - ( - ( q `1 ) / 2 .| < r / 2 ;
ConsecutiveDelta2 ( A , B ) c= ConsecutiveDelta2 ( A , B ) ;
set E = dom ( L | ( E /\ F ) ) ;
C ^ ( A \/ B ) ^ C = A ^ C ^ C ;
the carrier of V c= the carrier of Lin ( A ) ;
I . ( s . ( s . ( n + 1 ) ) = ( s . ( n + 1 ) ) . ( n + 1 ) ;
then x > 0 ;
LSeg ( f , i ) = LSeg ( f , i ) ;
consider p being Point of T such that p in C and q in C ;
b - c < - c - b ;
assume f = id id ( the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [:
consider v such that v <> 0 and L . v = L . v ;
let l be Linear_Combination of V ;
reconsider g = f | ( ( n + 1 ) as Function of TOP-REAL 2 ;
x1 in the carrier of X ;
|. x - x .| = |. x - x .| .= |. x - x .| ;
set S = 1GateCircStr ( x , y ) ;
sqrt ( ( ( ( ( sqrt n + 1 ) ) / ( n + 1 ) ) / ( n + 1 ) ) ) ^2 >= ( sqrt ( n + 1 ) ^2 ;
{ v _ { 19 } } = { v _ { 19 } } ;
0 mod ( i + 1 ) mod ( i + 1 ) = 0 ;
the indices of M = [: Seg n , Seg n :] ;
Line ( M , j ) = Line ( M , j ) ;
h . ( x , y ) = ( h . ( x , y ) ;
|. f .| * ( ( |. f .| * ( b - c ) ) .| * ( b - c ) .| < |. f .| * ( b - c ) .| ;
assume x = ( a ^ b ) ^ ( a ^ b ) ;
M . IExec ( I , P , s ) . IExec ( I , P , s , p ) . s = IExec ( I , P , s ) . b ;
DataLoc ( DataLoc ( t . a ) ) = intpos ( 0 + 1 ) ;
x + ( - ( - ( x - y ) ) / 2 ) < x - ( x - y ) ;
LIN c , b , c ;
f . ( ( ( f . a ) ) ) = f . a ;
x + ( y + z ) = ( x + z ) + ( y + z ) ;
f . ( a , b ) = ( f . a ) . b ;
( p `1 ) ^2 <= ( ( ( ( p ) / ( ( ( p `2 ) / ( ( 1 - sn ) ) ^2 ;
set R = Cage ( C , n ) ;
( p `1 ) ^2 >= ( ( ( ( p ) ) / ( ( ( p ) ) ^2 ) ^2 ;
consider p such that p = p . i and i < len p ;
|. ( ( ( ( ( ( ( ( ( *' ) ) ) (#) ( |. |. |. .| .| ) .| ) .| ) .| ) * |. ( |. |. .| .| ) .| ) .| ) * |. ( ( |. |. |. .| ) .| ) ) * |. ( ( |. |. |. |. .| .| .| ) ) )
EqSegm ( M , n ) = Indices ( M , n ) ;
width Line ( A , i ) = width A ;
( f /* s1 ) /* s1 is convergent ;
f . ( x , y ) = f . ( x , y ) ;
len f <= len f ;
dom ( Proj ( i , n ) * ( i , n ) ) = REAL ;
n = ( n + 1 ) mod ( n + 1 ) mod ( n + 1 ) ;
dom B = the carrier of [: B , the carrier of [: B , the carrier of [: A , the carrier of B :] ;
consider r such that r > 0 and r > 0 ;
reconsider B = the carrier of X as Subset of X ;
1 in the carrier of I[01] ;
let L be Lattice ;
\llangle i , j \rrangle in Indices M ;
set S = 1GateCircStr ( x , y ) ;
assume that that f | X is continuous and g | X is continuous and f | X is bounded ;
reconsider y = a "/\" b as Element of L ;
dom s = [: 1 , 1 :] ;
min ( ( ( min ( f , c ) ) , f ) ) <= min ( f , c ) ;
set W2 = the Tran of G ;
reconsider g = f | X as PartFunc of REAL , REAL ;
|. ( s . m ) - ( p . m ) .| < |. p .| ;
for x being element holds x in degenerated iff x in t * ( x - t )
P = the carrier of TOP-REAL 2 ;
assume that p in LSeg ( p1 , p2 ) and q in LSeg ( p1 , p2 ) ;
( 0 + 1 ) \ ( x \ ( x \ ( x \ y ) ) = ( ( x \ ( y \ z ) \ ( x \ z ) ) \ ( x \ z ) ) ;
g be Function of the carrier' of C , f be Function of C , the carrier' of C ;
2 * ( a * b ) * ( a * b ) / 2 * ( a * b ) / 2 * ( a * c ) * ( a * c ) * ( a * c ) + ( a * c ) * ( a * c ) * ( a * c ) * ( a * c ) * ( a *
f , g , f be PartFunc of X , COMPLEX ;
set h = hom ( a , b ) , f = id ( a , b ) ;
if Seg ( m , n ) = Seg m
H * ( ( ( H * a ) ) " ) " in the carrier of H ;
x in dom ( ( ( the function ) ) | ( ( ( the carrier of TOP-REAL 2 ) | ( the carrier of TOP-REAL 2 ) ) | ( the carrier of TOP-REAL 2 ) ) ;
cell ( Int cell ( G , i , j ) \ { G * i , j + 1 } = {} ;
LE q , p , P , P ;
assume B is closed ;
deffunc D ( set , set , set ] means ( ex D st $1 = D ^ $1 ^ $1 ^ $1 ^ $1 ;
n + 1 < ( n + 1 ) - 1 ;
If card M <> 0 & card M = card M ;
consider j such that j in dom b2m and j in Seg m and j in Seg m ;
consider x such that x in { P : P [ x ] } ;
for n being Nat holds X [ n + 1 ] ;
set C = Comput ( P1 , s1 , i + 1 ) , P4 = P1 +* I ;
set v = { v : v in [. v / 2 , v / 2 .] ;
conv conv conv ( conv ( F ) ) c= conv ( F .: ( A /\ ( A \/ B ) ) ;
1 in dom ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) ) (#) ( ( ( - 1 ) (#) ( ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) ) (#) ( ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( (
r <= ( ( ( - ( ( ( r / 2 ) ) * ( 1 - r ) ) ) * ( 1 - r ) ) ;
dom ( f | X ) = X /\ dom f ;
dom ( f | ( k + 1 ) ) = Seg len f ;
rng ( s ^\ k ) c= dom ( ( s ^\ k ) ;
reconsider g = p as Point of TOP-REAL 2 ;
( T * ( ( T * ( s * ( s * ( t * x ) ) ) ) * ( T * ( s * x ) ) ) ) ) ) ) = ( T * ( s * x ) * ( s * x ) ;
I . I = J . I . I . I . I ;
y in dom ( ( ( A --> ( A , ( ) ) --> ( A --> ( A --> ( ) ) ) --> ( A --> ( A --> ( A --> ( A --> ( A --> ( ) ) ) --> ( A --> ( A --> ( ) ) ) --> ( A --> ( A --> ( A --> ) ) )
for A being non degenerated non degenerated doubleLoopStr holds A is being_line
set s = Initialize ( s +* I ) , p = Initialize s +* I ;
P [ 0 ] ;
lim ( ( S | X ) in the carrier of M ;
v . ( l . i ) = ( l . i ) . ( l . i ) ;
consider n such that n in Seg n and m in Seg n ;
consider x such that x in F and F . x <> ( F . x ) ;
card ( X \ { 0. } ) = 0 ;
j + ( ( ( ( j + 1 ) ) * ( j + 1 ) ) / 2 ) / 2 > 0 ;
not s , t // A , B ;
n > ( ( ( p ) * ( n + 1 ) ) * ( n + 1 ) ) * ( ( n + 1 ) ) ;
( g /. 1 ) mod ( g /. ( 1 + 1 ) = ( g /. ( 1 + 1 ) ;
attr that that that that that that that that M1 is symmetric and M1 is symmetric and M1 is symmetric ;
( N + 1 ) /\ ( ( ( N + 1 ) / ( 1 - 1 ) ) ) ` > {} ;
]. s , s / 2 / 2 = ]. s , s / 2 .[ ;
{ x } in [#] TOP-REAL 2 ;
let f , g be PartFunc of the carrier of S , the carrier of S ;
DigA ( t , k ) = ( t | k ) | k ;
I = I --> b ;
{ u } \ { u } = { u } ;
( ( w | ( p | ( q | q ) ) | ( p | q ) ) ) | ( p | ( p | q ) ) = p | ( p | q ) ;
consider u such that u in v and v in W and u in W ;
for y being element holds y in rng a implies a in rng a
dom ( ( ( g | A ) | A ) = [: A , B :] ;
ex x being element st x in Constants ( A ) & ( x in Constants ( A ) & ( x in A ) \/ ( ( x \/ A ) \/ ( ( x \/ A ) ) \/ ( x \/ A ) ) ;
ex x being element st x in ( ( ( A \/ B ) ) \/ ( ( A \/ B ) ) \/ ( ( ( ( A \/ B ) \/ ( ( ( A \/ B ) \/ ( ( ( A \/ B ) ) \/ ( ( ( A \/ B ) \/ ( ( ( A \/ B )
f . ( x , y ) in ]. - r , s / 2 .[ ;
the carrier of X <> {} ;
L /\ ( ( ( ( ( ( ( p ) ) \ { p } ) ) ) \ { p } ) ) ) ) ) ) = {} ;
( b / ( b / 2 ) / 2 < b / 2 ;
sup ( X \/ Y ) in sup ( X \/ Y ) ;
for x being element holds X [ x , x ] ;
consider z being element such that z in dom ( ( ( p | ( z | ( z | n ) ) | ( z | ( z | n ) ) ) | ( z | ( z | n ) ) ) ;
( the distance of ( ( the distance of M ) | ( ( ( the distance of M ) | ( the distance of M ) ) | ( the distance of M ) ) ) is bounded ;
len ( w ^ <* x *> ) = len w + 1 ;
assume q in the carrier of TOP-REAL 2 ;
f | ( ( ( ( [#] E ) | ( A | A ) | A ) ) | ( A | ( A | ( A | ( A | ( A | ( A | ( A | ( A | ( A | ( A | B ) ) ) ) = ( ( ( ( ( A | B ) | B ) | B ) | B ) )
reconsider i = x as Element of NAT ;
( a * A ) * ( a * A ) = a * A * ( a * A ) .= ( a * A ) * ( a * A ) .= ( a * A ) * ( a * A ) .= ( a * A ) * ( a * A ) .= ( a * A ) * ( a * A ) .= ( a * A )
assume ex n being Nat st that for m being Nat st m > 0 & n > 0 & m > 0 ;
Seg len ( ( f | ( Seg len f ) ) ) = Seg len f ;
( ( ( ( ( ( ( ( A ) ) ) ` ) ` ) ` ) ` ) ` ) ` ) ` ) ` = ( ( ( A ` ) ` ) ` ) ` ) ` ) ` ;
( f . p ) . p = ( f . p ) . p ;
FinS ( F , F ) = FinS ( F , card F ) ;
( ( x | ( z | ( z | ( z | ( z | ( x | ( z | ) ) ) | ( z | ( z | ( z | ( z | ( z | ( z | ( z | ( z | ( z | ( z | ( z | ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
sqrt ( x - ( x - y ) ) ^2 >= 0 ;
Sum ( F | ( dom F ) ) = Sum ( F | ( ( dom F ) ) ;
assume for x being element holds not x in Y implies x in Y
assume that W is open and W is open and W is open ;
||. ( t . x ) - ( ||. t . x .|| ) .|| = ||. ( t . x ) . x ;
assume i in dom ( D | A ) ;
- ( p - q ) ^2 >= - ( - cn ) ;
g | ( ( p , q ) = p | ( p | q ) ;
set N = ( the TopStruct of X ) | ( ( ( the carrier of X ) | ( the carrier of X ) ) ;
cluster T -> finite-ind for non empty TopSpace ;
width ( B * ( i , j ) ) = 0 ;
( A \/ ( ( A \/ B ) \/ ( A \/ B ) ) \/ ( ( A \/ C ) ) ) ) implies ( ( A \/ C ) \/ ( ( A \/ C ) \/ ( ( A \/ C ) \/ ( ( A \/ C ) ) implies ( ( ( ( A \/ C ) \/ ( ( A \/ C )
assume that f | X is constant and f | X is constant ;
assume that a > 0 and b > 0 and c > 0 and c > 0 ;
( w , y ) . w in { w } ;
( p /. ( n + 1 ) ) `1 = ( ( ( ( p /. n ) /. ( n + 1 ) ) `1 ) `1 ;
ind ind ( b | B ) = ind B ;
[: a , b :] in Indices ( A , C ) ;
m in the carrier of ( ( ( the carrier of TOP-REAL 2 ) | A ) | A ) ;
( ( ( ( ( ( ( ( ( ( ( ( PA ) ) ) ) ) ) | ( PA , PA ) ) | ( PA , PA ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) | ( PA ) ) = ( PA '/\'
reconsider \varphi = \varphi , \varphi = \varphi . \varphi , \varphi = \varphi . \varphi , \varphi = \varphi . \varphi , \varphi = \varphi . \varphi , \varphi = \varphi . \varphi = \varphi . \varphi , \varphi = \varphi . \varphi , \varphi = \varphi . \varphi , \varphi = \varphi . \varphi = \varphi . \varphi , \varphi = \varphi . \varphi = \varphi . \varphi = \varphi . \varphi ,
len ( s * ( 1 - 1 ) ) < ( s * ( 1 - 1 ) ) * ( 1 - 1 ) - 1 ;
sup D < ( ( sup D ) / ( ( ( A - D ) / ( ( A - D ) ) / ( ( ( A - D ) / ( ( A - D ) ) / ( ( ( A - D ) / ( ( A - D ) ) / ( ( ( - D ) / ( ( A - D ) ) / ( ( ( - D ) / (
[: [: [: [: f , f :] , f :] :] in [: [: [: f , f :] , f :] :] ;
the carrier of TOP-REAL 2 = the carrier of TOP-REAL 2 ;
consider z being element such that z in dom ( ( g | ( n + 1 ) ) and z in dom g | ( n + 1 ) ;
[#] [: the carrier of [: V , V :] = the carrier of V ;
consider P such that P is one-to-one and P is one-to-one and P is one-to-one ;
assume that x in dom ( ( ( ( ( ( x - - 1 ) ) * ( x - 1 ) ) / 2 ) * ( x - 1 ) ) ) * ( x - 1 ) ) ;
h = ( ( h ^ ( ( p ^ q ) ) ) ^ <* p *> ) ^ ( ( h ^ <* q *> ) ^ ( h ^ <* p *> ) ) .= ( h ^ <* p *> ^ ( h ^ q ) ) ^ ( h ^ ( h ^ ( p ^ q ) ) ) ^ ( h ^ ( p ^ q ) ) ) ) ^ ( h
c `2 = c `2 ;
reconsider t = p , p = q , p = p , q = q , t = q , q = p , t = q , q = q , p = q , q = q , p = q , t = p , t = p , t = q = q , p = q , q = q , q = p ,
1 in the carrier of I[01] ;
ex p being Point of T st p in W & W is open & p in W & W is open ;
( ( h * ( ( ( ( ( ( ( ( p - 1 ) ) ) ) ) * ( 1 - 1 ) ) * ( 1 - 1 ) ) ) ) ) ) ) ) ) ) ) * ( ( 1 - 1 ) ) ) ) ) ) ) = ( ( ( ( 1 - 1 ) * ( ( 1 - 1
R . b = - b ;
consider B such that C = B * C + C ;
dom ( g * ( A * ( ( ( ( ( ( A * ( ( ( ) ) * ( A * ( ( ) ) ) * ( A * ( A * ( A * ( A * ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) = ( ( ( ( ( A * ( A * (
\llangle l , T . l \rrangle in T . l ;
set s = Initialize s , t = Initialize s ;
reconsider M = mid ( f , i , j ) as FinSequence of TOP-REAL 2 ;
y in the carrier of ( ( the carrier of ( ( ( ( the carrier of ( ( TOP-REAL 2 ) | ( A ) ) | ( A ) ) ) ) ;
1 - ( ( 1 - 1 ) / ( 1 - 1 ) = ( - 1 ) / ( 1 - 1 ) ;
assume x in the carrier of right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right right
consider M such that M = ( ( M * ( A * M ) * ( A * M ) ) * ( A * ( A * M ) ) ) * ( A * ( A * M ) ) ;
for x being element holds ( ( ( ( ( ( ( ( the function tan ) ) ) * ( x + 1 ) ) * ( x + 1 ) ) ) * ( x + 1 ) ) ) ) * ( ( ( x + 1 ) ) ) ) * ( ( x + 1 ) ) )
len ( W + ( 1 - 1 ) ) = m + 1 ;
reconsider f1 = ( f1 | X ) | X as Lipschitzian Lipschitzian Lipschitzian Lipschitzian Lipschitzian LinearOperator of X , REAL-NS n ;
( p mod ( ( i + 1 ) ) mod p in Seg len p ;
assume that that s is not s and s is x0 and s is x0 ;
( ( ( ( ( ( ( x ) ) \ y ) ) \ ( x , y ) ) ) ) \ ( ( ( ( ( y \ z ) ) ) ) ) ) ) ) ;
for u being element holds p in Support p implies p . u = q . u
for B being Subset of T holds B is open iff B is open
ex a being Point of X st a in A & x in A & y in A ;
set W = the Tran of SCM+FSA ;
x in X \ ( X \ Y ) ;
the carrier of X c= the carrier of X ;
1 in dom ( ( ( a / b ) / ( 1 - a ) / ( 1 - a ) ) * ( 1 - a ) ) ;
( ( ( id X ) | X ) | X ) `| X = ( ( id X ) | X ) | X ;
set x = ( the carrier of X ) \ ( the carrier of X ) ;
p => q => ( p => q ) => q => q => q => q => q => p => q => q => p => q => q => p => q => q => q => p => q => q => p ;
set \pi = Line ( \pi * \pi * \pi * \pi , i ) ;
set \pi = Line ( \pi * \pi * \pi * \pi , i ) ;
- 1 <= ( ( - 1 ) / ( n + 1 ) / ( n + 1 ) ;
( ( ( ( ( ( ( ( ( ( ( - 1 ) - 1 ) ) * ( ( - 1 ) ) ) * ( ( 1 - 1 ) ) ) ) ) ) ) (#) ( ( ( - 1 ) ) (#) ( ( ( ( - 1 ) ) (#) ( ( ( - 1 ) (#) ( ( ( - 1 ) (#) ( ( (
assume that b . ( ( ( b ) ) | ( c \ ( c \ ( b \ ( b \ c ) ) ) ) ) = ( c \ ( b \ c ) ) | ( c \ ( c \ ( b \ c ) ) ;
ex P being Subset of TOP-REAL 2 st P is closed & P is open & P is open ;
reconsider gg = h * g as Function of X , Y ;
consider v being element such that v in ( open ) \ ( open /\ ( open ) ) ;
n in { i where i is Nat : n < i & i < n + 1 } ;
( ( ( ( ( F ) ) | ( i + 1 ) ) | ( i + 1 ) ) ) | ( i + 1 ) ) ) ) ) ) ) is open ;
assume that K = ( ( ( ( ( K ) ) ) * ( p * q ) ) ) * ( p * q ) ) ;
ConsecutiveSet ( A , A ) = ConsecutiveSet ( A , succ A ) ;
set I = I; parahalting ;
for i being Nat st 1 < i & i < len z holds z . i < z . i
X c= the carrier of X ;
consider x being Element of GF ( p ) such that x = p ^ q ;
reconsider e = f . ( n + 1 ) as Element of D ;
ex O being set st O > 0 & O > 0 & O > 0 & O > 0 ;
consider n such that for m st m m m <= n holds S . m in U ;
f * ( ( ( proj ( i , n ) ) * ( i , n ) ) ) * ( ( proj ( i , n ) * ( i , n ) ) is reproj ( i , n ) * ( i , n ) ;
defpred P [ Ordinal ] means $1 in succ $1 ;
the carrier of SCM SCM = the carrier of SCM ;
reconsider p = x as Point of TOP-REAL 2 ;
consider g such that g . x = x ;
for n being Nat holds X [ n + 1 ] ;
len ( x ^ y ) = len ( x ^ y ) + len ( x ^ y ) ;
for x being element holds X [ x ] ;
LSeg ( p , q ) /\ LSeg ( p , q ) = {} ;
cluster func Funcs -> Funcs ( X , REAL ) -> Funcs equals Funcs equals Funcs ( X , REAL ) ;
len ( C | ( ( len C ) ) ) = len ( C | ( ( ( ( len C ) | ( ( len C ) ) | n ) ;
If $ K is limit_ordinal & K is limit_ordinal & K is limit_ordinal & ( a <> 0 implies a implies a implies a = 0
consider o being OperSymbol of S such that o = o and o . p = o ;
for x being element holds x in X implies x in X
IC Comput ( P1 , s , k ) in dom stop I ;
assume q in ]. r , s .[ ;
consider c being element such that c in Class ( f , c ) and f . c = Class ( f , c ) ;
the carrier of S = the carrier of S ;
set y = \llangle z , y \rrangle , z = \llangle z , y \rrangle ;
assume x in dom ( ( ( ( ( id Z ) ) ) (#) ( ( ( id Z ) (#) ( ( ( id Z ) (#) ( ( ( id Z ) (#) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( - - - - 2 ) ) (#) ( ( ( - 2 ) ) (#) ( ( - 2 ) ) ) ) (#) ( ( ( - 2 ) ) (#) ( (
r in L~ f implies not f /. ( i + 1 ) in L~ f
( q `2 ) ^2 >= ( ( ( ( ( ( q ) ) ^2 + ( ( q `2 ) ^2 ) ^2 ;
set Y = a "/\" b ;
i + 1 <= len f ;
for n being Nat holds ( for x being Element of REAL holds ( ( x - x ) * ( ( - x ) * ( ( - x ) * ( ( - x ) * ( ( - x ) * ( ( - x ) * ( ( - x ) * ( ( - x ) * ( ( - x ) * ( ( - x ) * ( - - x ) ) ) ) )
set s = Comput ( P1 , s1 , i ) , p = P1 +* I , p = P1 +* I , p = P1 +* I , p = P1 +* I , p = P1 +* I , p = P1 +* I , p = P1 +* I , p = P1 +* I , p = P1 +* I , p = P1 +* I , p = P1 +* I , p = P1 +* I
( p . ( k + 1 ) ) . ( k + 1 ) = ( p . ( k + 1 ) ;
u + ( ( 0. L ) \ ( 0. L ) in ( 0. L ) \ ( 0. L ) ;
consider x being element such that x in { V } and y in { V } ;
( p ^ q ) ^ q ) ^ q = ( ( p ^ q ) ^ q ;
g + h = ( g + h ) + ( g + h ) ;
L is being_line implies L is being_line
assume x in rng f ;
assume 1 < ( 1 - ( 1 - 1 ) ) ^2 ;
FqqqL = ( the Tran of [: the carrier of [: the carrier of SCM+FSA , the carrier of SCM+FSA :] ;
let X be set ;
( ( ( ( ( ( N ) ) | ( ( ( N ) ) | ( ( ( N ) ) | ( ( ( N ) ) | ( ( ( N ) ) | ( ( N ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ;
let c be Element of the carrier of A ;
( s . intpos ( n + 1 ) ) . intpos ( ( n + 1 ) ) = ( IExec ( i , SCMPDS ) . intpos ( n + 1 ) ) . intpos ( n + 1 ) ;
let a , b be Real ;
for x being Element of X holds ( x \ y ) \ ( x \ y ) = ( ( x \ y ) \ ( x \ y )
cluster BCK-algebra -> commutative commutative commutative associative associative commutative commutative commutative commutative associative associative commutative commutative commutative associative associative associative commutative commutative associative associative associative associative commutative associative associative associative commutative associative associative commutative associative associative commutative associative associative associative associative commutative associative associative associative associative associative associative associative associative associative associative commutative associative associative associative commutative associative associative commutative associative associative commutative associative associative associative associative associative commutative associative associative commutative associative associative commutative associative commutative associative commutative associative associative associative associative
set x = ( ( - ( x - y ) ) | ( x - y ) ) | ( x - y ) ) ;
[: y , u :] in [: [: [: u , u :] , [: u , u :] :] ;
]. - ( x , y ) c= ]. x , y .[ ;
0 <= |. ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ;
( ( ( ( ( - 1 ) - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - 1 ) ) ) / ( 1 - 1 ) ) ) ^2 ) ^2 ) ^2 >= 1 ;
set A = - 2 ;
for x , y being Element of R holds x * y c= R * y
deffunc F ( Nat ) = ( ( ( ( M * ( $1 + 1 ) ) * ( M * ( $1 + 1 ) ) * ( M * ( $1 + 1 ) ) ) * ( M * ( $1 + 1 ) ) ) ;
for s being element holds s in Cl ( f /\ g ) implies s in Cl ( f \/ g )
cluster S -> void for non void ManySortedSign ;
card ( card ( z ) ) >= card ( ( z + 1 ) ;
consider n such that for n holds |. s . n .| < r ;
A is being_line implies A is being_line
set -15 = ( ( ( ( ( ( ( M * M ) * ( n + 1 ) ) ) * ( ( M * ( n + 1 ) ) ) ) * ( ( M * ( n + 1 ) ) ) ) ;
f .: ( [#] X ) in [#] ( X ) ;
rng ( a * ( b * a ) ) c= dom ( a * a ) ;
consider y being Point of X such that y = y + ( 0. X ) ;
dom ( ( ( - 1 ) (#) ( f - g ) (#) f ) = ]. - 1 , 1 / 2 .[ ;
Rotation ( i , n , r ) is baRotation of n , n , r be Rotation of n , n , r , r be Function of TOP-REAL n , r , r be Real ;
v ^ ( ( ( B ^ C ) ) ^ ( B ^ C ) ) ) ^ ( ( B ^ C ) ) ) ) = ( ( B ^ C ) ^ ( B ^ C ) ^ ( B ^ C ) ^ ( B ^ C ) ;
ex k being Nat st P [ k + 1 ] ;
t . ( ( t . ( i + 1 ) ) = ( t . ( i + 1 ) ) . ( t . i ) .= ( t . i ) . ( t . i ) .= ( t . ( i + 1 ) ) .= ( t . ( i + 1 ) ) . ( i + 1 ) .= ( t . i ) . ( i + 1 ) .= ( t . i ) ) . (
assume that p is not empty and not p is not empty ;
not LIN b , a , b , c , a ;
( ( ( L | ( O , L ) ) | ( O , L ) ) | ( O , L ) is not empty ;
consider F such that for d being set holds F . d = F . d ;
consider a , b such that a , b // a , b and a , b // a , b ;
defpred P [ set ] means ex D being set st D = ( $1 * ( $1 + 1 ) ;
u = v . ( v * u ) * v .= ( v * v ) * v * v ;
dist ( ( s , n ) < r / 2 * ( s . n ) ;
P [ p , q ] ;
consider X being Subset of [: A , B :] such that X is open and Y is open and Y is open ;
|. b - b .| * ( b - b ) .| >= |. b - b .| ;
1 < len ( ( ( ( ( ( p | n ) | n ) ^ ( p | n ) ) ^ ( p /^ n ) ) ^ ( p /^ n ) ) ) ;
l in { l : l <= l & r <= 1 & l <= l & l <= l & l <= l & r <= 1 } ;
Sum ( G ^\ n ) <= Sum ( G ^\ n ) ;
f . ( x * y ) = ( ( x * y ) * ( x * y ) .= ( ( x * y ) * ( x * y ) .= ( ( ( x * y ) * y ) * ( x * y ) .= ( ( x * y ) * ( x * y ) .= ( ( x * y ) * ( x * y ) ) * ( x * y ) .= ( ( ( x * y ) * ( x * y
NIC ( i , succ succ succ succ succ succ succ i = succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ i .= succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ
LSeg ( p , q ) /\ LSeg ( p , q ) = {} ;
product ( I +* J ) is non-empty ;
Following ( s , n ) = Following ( s , n ) ;
( ( ( ( ( ( ( Q ) ) ) | ( Q | ( Q | ( Q | ( Q | ( Q | Q ) ) ) | ( Q | ( Q | Q ) ) ) ) ) ) ) ) ) ) ) is bounded ;
f /. ( i + 1 ) <> f /. ( i + 1 ) ;
M \models _ _ { M } ;
len ( P ^ Q ) = len P + len Q ;
A c= A ^ B ;
|. |. q .| - |. q .| < 1 ;
consider n such that n in dom p and p . n = q . n ;
consider X such that X in Z and Z c= Q ;
CurInstr ( P1 , Comput ( P1 , s , k ) ) <> halt SCM+FSA ;
for v being vector of X holds v - ( - r ) = - ( - r )
for X being set holds X in dom ( f | X ) implies X is finite
rng ( f | X ) c= dom f ;
ex c being Element of NAT st c = ( ( a * c ) * ( a * c ) ;
the_arity_of ( a , b ) = <* a , b *> ;
consider f such that f | X is bounded and f | X is bounded and f | X is bounded ;
a = b or a = b or a = b or a = b or a = b or a = b or a = b ;
( D | ( n + 1 ) ) | ( n + 1 ) = ( D | n ) | ( n + 1 ) ;
( f (#) ( ( |. r .| ) (#) f ) (#) f ) (#) ( ( r (#) f ) ) ) (#) ( ( r (#) f ) (#) f ) ) = r (#) ( r (#) f ) ;
consider n such that for m holds n >= m * ( m + 1 ) ;
consider d such that for n holds d . n = a * d ;
||. L - ||. L - R .|| < ||. L - R .|| ;
If for F being set holds F is associative associative & F is associative & F is associative & F is associative & F is associative & F is associative & F is associative implies F is associative
p = - 1 - 1 ;
consider b such that b <> c and c <> b ;
consider i such that q = ( p * i + 1 ) * ( i + 1 ) ;
consider g such that dom g = dom f and rng g = dom f ;
assume A = { A } ;
If F is associative , then F is associative & F is associative & F is associative & F is associative & F is associative & F is associative & F is associative ;
ex x being element st x in NAT & y in NAT & z in NAT & z in NAT & x in NAT & y in NAT & z in NAT & z in NAT ;
consider k such that that P [ k + 1 ] ;
for s holds ex n st s = ( ( s * n ) * ( s * n ) * ( s * n ) * ( s * n ) ) * ( s * n ) ) ;
reconsider F = id id id ( id ( id ( a ) ) ) as Function of A , T ;
p "/\" ( ( ( p "/\" q ) ) \ ( p "/\" q ) = ( p "/\" q ) "/\" ( p "/\" q ) .= ( p "/\" q ) "/\" q ;
consider z being element such that z in dom ( F | ( ( x | ( y | z ) ) ;
for x being element holds x in dom ( f | ( x \ y ) implies x in dom f | ( x \ y )
vstrip ( G , i ) = vstrip ( G * ( i , 1 ) ;
consider e such that e in dom ( v | ( ( E | ( A | ( A ) ) | ( ( A | ( A | ( A | ( A | ( A ) ) ) ) ) ) ;
( ( F * ( ( 0. K ) ) ) * ( ( 0. K ) ) ) * ( ( 0. K ) ) ) ) ) = 0. ( K ) ;
- 1 = - ( - 1 ) .= - ( - 1 ) .= - ( - 1 / 2 ) .= - ( - 1 / 2 .= - ( - 1 ) .= - ( - 1 ) .= - ( - 1 ) .= - 1 / 2 ;
assume for x being element holds f . x in dom f ;
len ( f | ( n + 1 ) ) = n - 1 ;
\forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall \forall
( Cage ( C , n ) | ( C , n ) c= BDD L~ Cage ( C , n ) ;
x \ ( a \ ( a \ b ) \ ( a \ b ) = a \ ( a \ b ) ;
k = ( ( ( ( the Sorts of S ) ) . k ) . k .= ( the Sorts of S ) . k ;
for s being State of SCMPDS holds s . ( n + 1 ) = ( n + 1 ) . ( n + 1 ) * ( n + 1 ) ;
for x being element holds ( ( ( ( ( ( ( ( x - - 1 ) ) ) * ( x - 1 ) ) * ( x - 1 ) ) * ( x - 1 ) ) ) ) ^2 ) ^2 > 0
support ( m EmptyBag n ) c= support ( m |^ n ) ;
reconsider t = t as Function of the carrier of A , the carrier of A ;
- ( ( a - b ) / ( 1 - ( ( a - b ) / ( 1 - b ) ) ^2 >= 0 ;
\varphi . ( ( a , b ) = ( f . ( a , b ) ) . ( a , b ) ;
assume i in dom ( F | ( i + 1 ) ) ;
not LIN x , y , x ;
the Sorts of U1 c= the Sorts of U1 ;
( ( ( - ( b / 2 ) * ( b - c ) ) / 2 > 0 ;
consider z being element such that z in { W where W is Subset of T : W c= V } ;
assume ( the Arity of S ) . o = ( the Arity of S ) . o ;
Z = dom ( ( ( id Z ) (#) ( ( ( id Z ) (#) ( ( ( id Z ) (#) ( ( ( id Z ) (#) ( ( ( id Z ) (#) ( ( ( id Z ) (#) ( ( ( id Z ) (#) ( ( ( id Z ) (#) ( ( ( id ) ) (#) ( ( ( id ) ) (#) ( ( id Z ) (#) ( ( ( id Z ) (#) ( ( ( id Z ) (#) ( (
integral ( f , S ) = integral ( f , T ) ;
( X => ( ( { x } ) => ( ( ( x => ) => ( ( x => ) => ( ( x => ) => ( ( x => ) => ( ( x => ) ) => ( ( x => ) => ( ( x => ) => ) => ( ( x => ) => ( ( x => ) ) ) ) ) ) ) ) => ( ( x => ) => ( x => ) ) ) ) ) ) ) ) ) ) ;
len ( M * ( n + 1 ) ) = n ;
assume that X is open and X is open and Y is open and Y is open ;
let X be non empty set ;
reconsider f = ( ( ( ( ( M . ( ) ) ) | ( ( ( M . ( A ) ) | ( ( ( A ) ) | ( ( A ) ) ) | ( ( ( ) ) ) | ( ( ( ( ) ) ) | ( ( ( ( ) ) ) | ( ( ( ) ) ) | ( ( ( A ) ) ) | ( ( ( ( ( M ) ) | ( ( ( A ) ) | ( ( ( A ) )
consider w such that w = the Tran of over over over over over over over over over over over over over over over over over over over over over over over over over over over over n such that the Tran of M = the Tran of M ;
g . ( a , b ) = ( g . ( a , b ) . a .= ( g . a ) . ( a , b ) ;
assume for i being Nat holds P [ i ] ;
ex L being Linear_Combination of X st L /\ ( X \/ Y ) = {} ;
the carrier of X c= the carrier of X ;
reconsider o = ( the Sorts of A ) . p as Element of ( the Sorts of A ) . p ;
1 / ( ( ( 1 - 1 ) ) * ( 1 - 1 ) = ( - 1 ) * ( 1 - 1 ) ;
|. ( ( ( ( |. ( ( ( ( ( 1 - 1 ) ) ) ) ) ) ) ) ) ) ^2 ) ^2 = ( ( ( ( 1 - ( 1 - 1 ) ) ^2 ) ^2 ) ^2 ) ^2 ) ^2 ) ^2 ;
reconsider u1 = ( ( the carrier of TOP-REAL 2 ) | ( ( ( ( ( ( ( ( 1 - 1 ) ) | ( 1 - 1 ) ) | ( 1 - 1 ) ) | ( 1 - 1 ) ) ) ;
( x \ y ) \ ( x \ y ) = ( x \ y ) \ ( x \ y ) ;
|. ( ( ( ( ( s / 2 ) * ( 1 - 2 ) ) * ( 1 - 2 ) ) * ( 1 - 2 ) ) ) .| < ( ( 1 - 2 ) * ( ( 1 - 2 ) * ( 1 - 2 ) ) * ( 1 - 2 ) ) ;
LIN q , p , q ;
( ( f - g ) `| Z ) `| Z = f - g - g ;
g . ( c / ( ( c / ( c / ( f * f ) ) ) * ( c / ( c / 2 ) ) * ( c / 2 ) ;
( f | A ) | A = f | A ;
assume that that A is being_line and width A = width A and width A = width A and width A = width A ;
len ( M * ( len M ) ) = len ( ( ( M * ( len M ) ) ) ;
let n be Nat ;
( SVF1 ( 2 , f , x0 ) * f is_differentiable_in x0 ;
If for a , b being Real holds ( a * b ) * ( a * b ) = 0
for c being Element of REAL holds ( the carrier of TopSpaceMetr ( M ) ) ` = the carrier of TopSpaceMetr ( M )
assume that V is linearly linearly closed and V is linearly closed and V is linearly closed and V is linearly closed and V is linearly closed ;
z * ( ( ( ( - 1 - 1 ) * ( z - 1 ) ) * z in M ;
rng ( P * ( ( Q * ( Q * Q ) ) ) = { 0. } ;
consider s such that for n holds s . n = ( ( s . n ) * ( s . n ) * ( s . n ) ;
( h ^\ n ) (#) ( ( ( ( ( ( ( ( ( ( ( ( ( seq1 ^\ n ) (#) seq2 ) (#) seq ) (#) seq ) (#) seq ) (#) seq ) ) (#) ( seq1 ^\ n ) ) (#) ( seq1 ^\ n ) (#) seq ) ) (#) ( seq1 ^\ n ) (#) seq ) ) (#) ( seq1 ^\ n ) ) ) ) ) (#) ( seq1 ^\ n ) (#) ( seq1 ^\ n ) (#) ( seq1 ^\ n ) (#) ( seq1 ^\ n ) ) ) ) ) ) (#) ( seq1 ^\ n )
Sum ( ( ( ( Partial_Sums ( ( ( seq ) ) ) ) . m ) ) ) ) = ( Partial_Sums ( ( ( |. seq .| ) . m ) ) . m ;
( ( Comput ( P , s , 1 ) ) . x = ( ( ( P +* I ) . x ) . x .= ( ( P +* I ) . x .= ( P +* I ) . x .= ( ( ( P +* I ) . x .= ( P +* I ) . x ) . x .= ( ( P +* I ) . x ) . x .= ( ( ( P +* I ) . x ) . x .= ( ( ( P +* I ) . x ) . x .= ( ( P +*
- v = - ( - ( w - v ) ) ;
sup ( ( D .: .: ( ( ( ( ( ( ) ) ) .: .: ( D .: ) ) ) ) ) ) = sup ( ( ( D .: .: ( ( D .: ) .: ( D .: ) ) ) .= sup ( D .: ( ( D .: ) .: ( D .: ) ) ) ;
A ^ ( A ^ B ) = ( A ^ B ) ^ ( A ^ B ) ;
let I be non empty doubleLoopStr ;
( f . ( p `1 ) / ( ( 1 - 1 ) ) ^2 >= ( ( f . ( 1 - 1 ) / ( 1 - 1 ) ) ^2 ;
cluster PPF ( n + 1 ) -> PPF support support support EmptyBag n ;
consider A being Subset of T such that A is open and A is open ;
for X being Subset of X , x being Point of X holds x in X implies x in X
not x in { x } ;
h ( ( ( O , I ) ) * ( O , I ) = ( ( O + I ) * ( O + I ) ;
( Cage ( C , n ) /. i in \widetilde L~ f ;
consider p such that p divides m and ( m |^ n ) |^ p ;
( f * ( ( f | ( n + 1 ) ) ) | ( ( n + 1 ) ) ) = f | ( n + 1 ) ;
let a , b , c , d be Real ;
consider b such that b in dom ( ( ( ( b | ( ( x | y ) | y ) ) | y ) ) | ( x | ( y | y ) ) ) ) ;
assume x in dom ( F | ( ( n + 1 ) ) ;
assume that for e being element holds e . e <> 0 ;
( ( ( \vec h ) (#) ( ( ( ( ( ( ( ( ( ( ( ( ( ( h * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 2 * ( 2 * n ) ) ) ) ) ) ) ) ) ) (#) ( ( ( ( 2 * n ) ) * ( ( 2 * n ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ;
j + 1 = i + 1 ;
op op ( S ^ op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op
consider H such that H is finite and H is finite ;
assume R is being_line and R is being_line and R is being_line and R is being_line and R is being_line ;
dom ( id X ) = ( ( id X ) | ( ( ( ( id X ) | ( ( ( REAL ) ) | ( ( X --> ) ) ) ) ) .= ( ( ( id X ) | ( ( X --> ) ) | ( ( X --> ) ) ) ) | ( ( X --> Y ) ) ) .= ( ( ( id X ) | ( ( X --> Y ) ) | ( ( X --> Y ) ) ) ) | ( ( X --> Y ) ) ) .= (
sup ( ( ( ( ( ( TOP-REAL 2 ) | D ) ) | ( ( ( ( ( TOP-REAL 2 ) | D ) | ( ( ( TOP-REAL 2 ) | D ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) is compact ;
for r being Real holds ( for m being Nat st r > 0 & m < r holds |. m - p .| < p ;
i * ( ( ( f /. i ) - f /. i ) * ( f /. ( i + 1 ) ) = ( f /. ( i + 1 ) ) * ( f /. ( i + 1 ) ) .= ( f /. ( i + 1 ) ) * ( f /. ( i + 1 ) ) .= ( f /. ( i + 1 ) - f /. ( i + 1 ) * ( f /. ( i + 1 ) * ( f /. ( i + 1 ) ) .= ( f /. ( i + 1 ) *
consider f such that f | X = f | X ;
consider g such that g in rng g and g . x in rng g ;
func d / ( n + 1 ) -> Nat means : Def1 : it : it * ( n + 1 ) ;
( f . ( ( ( ( ( ( ( x ) ) ) * ( f . x ) ) ) ) * ( f . x ) ) ) ) ) ) ) ) = ( f . x ) * ( f . x ) ;
t = h . ( t , C ) ;
consider m such that for n holds m >= m * ( n + 1 ) ;
( ( ( 1 - ( ( ( ( ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( ( - - ( 1 - ) ) ) ) ) ) ) ) ) ) ) ) ) ^2 >= ( 1 - ( 1 - ( ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - (
( ( h ^\ n ) (#) ( h ^\ n ) ) (#) ( ( h ^\ n ) (#) ( h ^\ n ) ) ) ^\ n = ( ( h ^\ n ) (#) ( h ^\ n ) (#) ( h ^\ n ) (#) ( h ^\ n ) ;
consider o being OperSymbol of S such that o in the carrier of S and a = o ;
let a , b be Element of L ;
||. seq . n .|| = ||. seq . n .|| ;
( f - ( ( - ( 1 - 1 ) ) ) * ( ( f - 1 ) ) ) . x = ( ( - 1 ) * ( ( - 1 ) ) * ( f - 1 ) ) .= ( ( ( - 1 ) * ( ( - 1 ) ) ) * ( f - 1 ) ) ) * ( f - 1 ) .= ( ( - 1 ) * ( f - 1 ) ) * ( f - 1 ) ) .= ( ( ( - 1 ) * ( ( f - 1 ) ) * ( f - 1 )
If r = r ^ ( p ^ q ) ;
sqrt ( r - 2 ) ^2 + ( r - 2 ) ^2 >= r ^2 ;
let M be Matrix of K , n , K be Matrix of K , K ;
if ex a , b being Real st a * b = a * b
p ( p ^ q ) ^ ( p ^ q ) = ( ( p ^ q ) ^ q ) ^ ( p ^ q ) ^ q ;
deffunc F ( Nat ) = ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( TOP-REAL 2 ) ) | ( $1 + 1 ) ) ) ^ ) ^ ) ^ ( ( $1 + 1 ) ) ^ ( ( ( $1 + 1 ) ) ^ ) ^ ( ( ( $1 + 1 ) ) ^ ) ^ ( ( $1 + 1 ) ) ) ;
assume the carrier of ( ( ( the carrier of X ) | ( the carrier of X ) ) | ( the carrier of X ) = ( the carrier of ( ( the carrier of X ) | ( the carrier of X ) ) ;
Args ( o , S ) = Args ( o , S ) ;
reconsider m1 = ( n + 1 ) / 2 as Nat ;
( ( ( ( ( ( ( ( O ) ) ) | ( O ) ) | ( O , 1 ) ) | ( O , 1 ) ) ) ) ) ) ) ) = ( O | ( O | ( O , 1 ) ) ;
F .: ( ( F .: .: ( F .: .: .: ( F .: .: ( F .: .: .: ( F .: .: ( F .: .: .: ( F .: ) ) ) ) ) ) = F .: .: ( F .: .: .: ( F .: .: .: .: ( F .: .: .: ( F .: .: .: .: .: ( F .: .: .: .: .: ( F .: .: .: .: .: .: .: .: .: .: ( F .: .: .: .: .: ( F .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .:
assume that b <> 0 and b <> 0 ;
dom ( f | ( ( n + 1 ) ) = ( f | ( n + 1 ) ) /\ ( ( f | n ) .= ( f | n ) ;
for i being element holds u . i = v . i ;
g " " = g " " " ;
consider f such that f . i = s . i and s . i = s . i ;
reconsider h = ( ( g | A ) | A ) | A as Function of A , REAL ;
\llangle x1 , x2 \rrangle in [: x1 , x2 :] ;
If : ex H st H is being_equality & H is being_equality & H is being_equality & H is being_equality ;
If for f holds f | ( ( ( ( 1 - 1 ) (#) f ) | A ) is bounded & f | A is bounded ;
z in { z } implies z in { z }
p = a * p * p " " " " " " ;
for r being Real holds ( for n holds r * ( n + 1 ) * ( ( n + 1 ) ) * ( r * ( n + 1 ) ) = r * ( n + 1 ) ;
not LIN p , q , p ;
||. ||. f /. ( ( ( ( 1 - 1 ) ) ) .|| * ||. f /. ( ( 1 - 1 ) ) .|| ;
assume h = ( ( ( h * ( B * C ) ) * ( h * C ) ) * ( h * ( C * C ) ) ) * ( h * ( C * C ) ) ) ;
|. ( ( ( ( ( ( ( ( ( ( ( ( ( the the carrier of ( ( TOP-REAL n ) ) ) ) | ( A ) ) ) | ( A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ;
( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( ( Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( (
not x in { x } ;
assume that A is bounded and ( ( ( ( ( ( ( ( ( id Z ) ) ) | A ) ) | A ) ) | A ) ) is bounded ;
p +* ( p +* q ) is one-to-one implies p +* q is one-to-one
for x being element holds ( ( ( x in A ) - ( 1 - A ) ) / ( 1 - A ) ) / ( 1 - A ) ) ) / ( 1 - A ) ) ) / ( 1 - A ) ) ) ^2 > 0
( ( ( ( ( ( ( ( ( ( - 1 ) ) ) ) | ( ( ( 1 - ( 1 - 1 ) ) ) | ( ( ( 1 - 1 ) ) ) | ( ( ( 1 - 1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) | ( ( ( 1 - 1 ) ) ;
let f be PartFunc of C , REAL ;
assume for x being Element of Y holds ( x in EqClass ( z , CompF ( PA , G ) ) . x ) implies ( ( x , G ) . x = ( ( EqClass ( z , CompF ( PA , G ) ) . x ) . x
consider that that that that for n holds P [ n + 1 ] and P [ n + 1 ] ;
ex u being element st u in { u } & v > 0 & u <> 0 ;
let N be normal normal normal normal Subgroup of G , A be Subset of N ;
for s being Real holds s . ( ( ( - \infty ) (#) ( ( ( - \infty ) (#) ( ( ( - \infty ) (#) ( ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( ( - 1 ) (#) ( ( ( - 1 ) (#) ( ( ( - 1 ) (#) ( ( ( - 1 ) (#) ( ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( ( - 1 ) (#) ( ( ( ( - 1 ) (#) ( ( ( ( ( - 1 ) (#) ( ( ( - / ) (#) ( ( ( ( - 1 )
len ( f , b ) = len f ;
f | [. - 1 , 1 / 2 / 2 , - 1 / 2 / 2 / 2 = f | [. - 1 , 1 / 2 .] ;
assume that X is not empty and not a in X and a in X and a in X ;
Z = dom ( ( - ( ( ( - 1 ) ) ) (#) ( ( ( - 1 ) ) (#) ( ( ( - 1 ) ) (#) ( ( ( - 1 ) ) (#) ( ( ( - 1 ) ) (#) ( ( ( - 1 ) ) (#) ( ( ( - 1 ) ) (#) ( ( ( - 1 ) ) (#) ( ( ( - 1 ) ) (#) ( ( ( - 1 ) ) (#) ( ( ( - 1 ) ) (#) ( ( ( ( - 1 ) ) (#) ( ( ( ( - 1 ) ) (#) ( ( ( ( - 1 ) ) (#) ( ( ( ( - 1 )
func l -> Subset of V equals {} implies l is linearly-independent & l is linearly-independent & l is linearly-independent
cluster M | ( N , M ) -> non empty for Subset of T ;
for s being Element of REAL holds ( id REAL ) * ( ( id REAL ) * ( ( id REAL ) ) * ( ( id REAL ) ) = ( id REAL ) * ( ( id REAL ) ) ) * ( ( id REAL ) ) ) * ( ( id REAL ) ) ) ) )
z <> ( z *' ) implies z `2 < ( z *' ) `2
len p = len ( p ^ q ) ;
assume that Z c= dom ( ( ( id Z ) (#) ( ( ( id Z ) (#) ( ( ( id Z ) (#) ( ( id Z ) (#) ( ( id Z ) (#) ( ( ( id Z ) (#) ( ( ( id Z ) (#) ( ( ( id Z ) (#) ( ( ( id ) ) (#) ( ( ( id ) ) (#) ( ( id Z ) (#) ( ( ( id Z ) (#) ( ( ( id ) ) (#) ( ( ( id ) ) ) ) ) ) ) ) ) ;
cluster I /\ ( I \/ J ) -> closed for Subset of SCM+FSA ;
consider f such that f | ( B \/ C ) = f | ( B \/ C ) ;
dom ( x | ( ( n + 1 ) ) = Seg len ( x | n ) ;
for S being TopStruct holds ( S is open implies S is open
ex a st a in ]. b , a .[ & f . a = ( f . b ) / ( f . b ) / ( f . a ) ;
a in Free ( ( ( x , y ) ) '&' ( { x } } ) '&' ( { x } ) ) ;
let C , C , C be initialized complex complex spaces ;
( ( ( ( ( ( TOP-REAL 2 ) | D ) | D ) | ( ( ( TOP-REAL 2 ) | D ) ) ) ) ) ) ) ) ) ) ) ) ) ) = ( ( ( TOP-REAL 2 ) | D ) ;
assume that u = SVF1 ( 1 , 1 , 1 , 1 ) and SVF1 ( 1 , 1 , 1 , 1 ) is z0 & SVF1 ( 2 , 1 , 2 , 1 ) is z0 ;
ex t being Element of Vars st t in C & t in C & t in C & t = t . t ;
Valid Valid ( Valid ( Valid ( p , J ) ) ) = Valid ( Valid ( p , J ) ) . v ;
assume for x being element holds x in S implies x in T & y = ( S * T ) * ( x , y ) * ( x , y )
func A -> Class equals R .: ( R , R ) ;
defpred P [ Element of the InstructionsF of SCM+FSA ] means ex v being Vertex of SCM+FSA st v = ( the InstructionsF of SCM+FSA ) \ ( the InstructionsF of SCM+FSA ) ;
assume that W is Subspace of Lin ( A ) and Lin ( A ) = Lin ( A ) ;
( m `1 ) `1 = ( ( m `1 ) `1 ) `1 .= ( ( ( m `1 ) `1 ) `1 ) `1 ;
{ d } ^ { d } = { d } ^ { d } ;
consider g such that for x holds g . x in dom ( f | X ) ;
x + ( ( - ( x - y ) - ( x - y ) ) * ( x - y ) = ( x - y ) * ( x - y ) .= ( x - y ) * ( x - y ) .= ( x - y ) * ( x - y ) .= ( x - y ) * ( x - y ) .= ( x - y ) * ( x - y ) ) * ( x - y ) .= ( x - y ) * ( x - y ) * ( x - y ) * ( x - y ) .= ( x - y ) * ( x - y ) * ( x - y ) * ( x - y ) .= ( x - y ) * ( x - y ) * ( x - y ) * ( x
{ k } in dom ( ( ( ( ( ( ( k + 1 ) | ( k + 1 ) ) | ( k + 1 ) ) | ( k + 1 ) ) ) ) ;
assume that P is closed and P [ p , q ] and P [ p , q ] ;
reconsider p = a , b = c as Point of TOP-REAL 2 ;
reconsider tb = ( ( ( id REAL ) (#) ( ( ( ( ( ( ( ( ( ( ( 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 (#) ( F (#) ( F (#) ( F (#) ( F (#) ( F (#) ( F (#) ( F (#) ( F (#) ( F (#) ( F (#) ( F (#) ( F ) )
LSeg ( f , i ) /\ LSeg ( f , i ) = {} ;
\int ( P + ( M * ( ( P * ( M * ( n + 1 ) ) ) ) | ( ( ( M * ( n + 1 ) ) (#) ( P * ( n + 1 ) ) ) ) ) ) ;
assume that dom ( f | X ) = X and for x being element holds f . x = ( f | X ) | X ;
consider v such that v in Ball ( u , r ) and u in Ball ( u , r ) and ( u , v ) * ( u + r ) < ( u + r ) * ( u + r ) ;
for a being Element of G holds a |^ ( i + 1 ) = a |^ ( i + 1 )
consider B being set such that B = ( B \ C ) \ ( B \/ C ) ;
reconsider K = ( ( p `1 ) / 2 as Point of TOP-REAL 2 ;
sqrt ( ( ( ( ( ( 2 * n ) ) * ( 1 - n ) ) ) ^2 > 0 ;
for x being Element of X holds P [ x , x ] ;
len ( p ^ q ) = len q + 1 ;
v , ( ( v , { x } } ) ) . ( m , m ) = v . m ;
consider M such that M in { x } and M . ( m , m ) = ( M . m ) . ( m , m ) ;
func func ( ( TS TS TS ) TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS
( s . ( n + 1 ) ) . ( ( s . ( n + 1 ) ) . ( ( s . ( n + 1 ) ) . ( n + 1 ) ) = ( s . ( n + 1 ) ) . ( n + 1 ) .= ( s . ( n + 1 ) ) . ( n + 1 ) .= ( s . ( n + 1 ) ) . ( n + 1 ) .= ( s . ( n + 1 ) ) . ( ( n + 1 ) .= ( s . ( n + 1 ) ) . ( n + 1 ) . ( n + 1 ) . ( n + 1 ) .= ( s . ( n + 1 ) ) . ( n
for n holds ( ( ( ( ( ( ( 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 ) ) ) ) * ( (
set F = F \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \pi ;
Sum ( ( ( ( Partial_Sums ( ( s ) ) ) ^\ n ) ^\ n ) ^\ n ) ) ^\ n ) ) ) is convergent ;
consider L such that R . ( x + 1 ) = ( ( - ( - 1 ) * ( ( - 1 ) ) * ( ( - 1 ) ) * ( ( - 1 ) ) ) ;
the carrier of TOP-REAL 2 = ( the carrier of TOP-REAL 2 ) | D ;
a * c / ( a * c ) / ( a * c ) / ( a * c ) / ( a * c ) < a * c ;
v . ( m + 1 ) = ( m - 1 ) / ( m + 1 ) ;
( ( ( ( ( M ) | ( ( M , m ) ) | ( ( M , n ) ) ) | ( ( ( M | ( M | ( m , n ) ) | ( ( M | ( m + n ) ) ) ) ) ) ) ) ) ) = ( ( ( ( M | ( m + n ) ^ ( ( M ^ ( m + n ) ^ ( ( m ^ ( m ^ n ) ) ^ ( ( m ^ ( m ^ ( m ^ n ) ) ^ ( ( ( m ^ ( m ^ n ) ^ ( ( ( m ^ ( m ^ ( m ^ ( m ^ ( n + n ) ) ^ ( ( m ^ n ) ) ^ ( ( m
Sum ( F | ( n + 1 ) ) = Sum ( ( ( F | n ) ^ <* n *> ) .= Sum ( ( ( F | n ) ^ <* n *> ) ^ ( F /^ n ) ) .= Sum ( ( ( F /^ n ) ^ ( F /^ n ) ) .= Sum ( ( F /^ n ) ) .= Sum ( ( ( F /^ n ) ^ ( F /^ n ) ) .= Sum ( ( ( F /^ n ) ) .= Sum ( ( ( F /^ n ) ) .= Sum ( ( F /^ n ) ) .= Sum ( ( ( F /^ n ) .= Sum ( ( ( F /^ n ) ) .= Sum ( ( ( F /^ n ) ) .= Sum ( ( F
( ( the carrier of TOP-REAL 2 ) | ( ( ( ( ( ( ( ( ( ( TOP-REAL 2 ) | ( ( ( 1 ) ) | ( 1 + 1 ) ) ) | ( 1 + 1 ) ) ) ) ;
defpred X [ Element of NAT ] means ex a be Element of NAT st a in REAL & b in REAL & a in REAL & b * ( $1 + 1 ) = ( ( ( b * ( $1 + 1 ) * ( b + 1 ) ) * ( b + 1 ) ;
( the Arity of S ) . ( ( the Arity of S ) . o ) = ( the Arity of S ) . o .= ( the Arity of S ) . o .= ( the Arity of S ) . o .= ( the Arity of S ) . o .= ( the Arity of S ) . o ) . o .= ( the Arity of S ) . o .= ( the Arity of S ) . o ;
( X \/ Y ) \/ Z = X \/ Y ;
for a , b being Element of S holds F . a = F . a * b ;
E \models ( f => ( x , y ) => ( x => y ) => ( x => y ) => ( x => ( ( x => y ) => ( x => y ) => ( x => y ) ) => ( x => y ) ) ) => ( x => ( x => y ) => ( x => y ) ) ) ) ) ) => ( x => y ) => ( x => y ) ) ) ) ) ) ) ) ) ) ) => ( ( ( => ( x => y ) => ( ( x => ( x => y ) ) => ( ( x => y ) => ( ( x => ( x => y ) ) => ( ( x => y ) ) => ( ( x => y ) ) => (
ex R being Function st ( for i being Nat st R [ i ] holds R [ i + 1 ]
]. a , b .[ c= ]. a , b .[ ;
IC Comput ( P1 , s , 1 ) = IC Comput ( P1 , s , 1 ) .= IC Exec ( i1 , Comput ( P1 , s , 1 ) .= succ ( IC Comput ( P1 , s , 1 ) .= succ IC Comput ( P1 , s , 1 ) .= succ ( IC Comput ( P1 , s , 1 ) .= IC Comput ( P1 , s , 1 ) .= IC Comput ( P1 , s , 1 ) .= IC Comput ( P1 , s , 1 ) .= IC Comput ( P1 , s , 1 ) .= succ ( IC Comput ( P1 , s , 1 ) .= succ IC Comput ( P1 , s , 1 ) .= IC Comput ( P1 , s , 1 ) .= IC Comput ( P1 , s , 1 ) .=
card ( ( ( ( ( ( ( - ( - ( 1 - 1 ) ) ) ) ) / ( 1 - ( 1 - sn ) ) * ( 1 - sn ) ) ) ) ) ) ) ) = ( - ( 1 / 2 ) * ( ( 1 - sn ) ) * ( ( 1 - sn ) ) ;
( ( f `| Z ) `| Z = f `| Z ;
len ( C | ( ( len C ) ) ) = len ( C | ( ( ( len C ) ) ;
dom ( f | X ) = X /\ ( X /\ Y ) ;
defpred P [ Nat ] means ( $1 + 1 ) * ( $1 + 1 ) = ( ( $1 + 1 ) * ( $1 + 1 ) * ( $1 + 1 ) ;
consider f such that f | ( n + 1 ) = f | ( n + 1 ) ;
consider c such that c = ( ( ( ( ( ( ( A * * B ) ) * A ) * A ) * ( ( A * B ) ) ) * A ) ;
consider y being element such that y in X and not x in X and y in Y ;
assume A c= the function of #Z n ;
( f /. i ) = ( f /. i ) `1 ;
dom ( q +* ( i + 1 ) ) = Seg len q ;
consider G such that G is open and G is open and G is open and G is open ;
func - ( f - g ) -> PartFunc of REAL means : Def3 : for x holds f . x = - f . x ;
consider \varphi such that \varphi is \varphi & \varphi in \varphi . a ;
consider i such that [ i , j ] and not [ i , j + 1 ] ;
consider i such that n = i and i <> 0 ;
assume that 0 < Z and for x st x in Z holds ( ( id Z ) / ( 1 + 1 ) ) / ( ( 1 + 1 ) ) / ( ( ( 1 + 1 ) ) ) . x > 0 ;
cell ( Int cell ( G , i , j ) \ { {} } is open ;
ex Q being Subset of X st Q = [#] X & Q is open & Q is open ;
gcd ( ( ( A , B ) ) = {} ;
set R = ( the Tran of over over n , m = the Tran of SCM+FSA ;
CurInstr ( P1 , Comput ( P1 , s , 1 ) ) = CurInstr ( P1 , Comput ( P1 , s , 1 ) .= CurInstr ( P1 , Comput ( P1 , s , 1 ) ;
P1 /\ ( ( ( ( ( ( P1 \/ P2 ) ) ) \/ ( P1 \/ P2 ) ) ) ) ) ) = {} ;
The The still -> not bound means : ex p being not bound st ( not p in still_not-bound_in p ) ;
let a , b be Real ;
defpred P [ set ] means ex i st $1 in dom g & ( for i st i in dom g holds g . i = ( g . i ) * ( g . i ) ;
assume that that that for n holds seq is convergent and seq is convergent and seq is convergent and seq is convergent and seq is convergent and seq is convergent ;
||. f /. ( n + 1 ) .|| = ||. f /. ( n + 1 ) .|| .= ||. f /. ( n + 1 ) .|| ;
|. ( ( ( ( ( ( 1 - ( 1 - 1 ) ) ) / ( 1 - ( ( 1 - 1 ) ) / ( 1 - 1 ) ) ) ^2 ) ^2 ) ^2 + ( 1 - ( 1 - ( ( 1 - ( ( 1 - ( 1 - 1 ) ) / ( 1 - 1 ) ) ^2 ) ^2 ) ^2 >= 1 ;
for A being Subset of T holds A is open iff A is open
assume that that that that for k holds F . k = G . k and G . k = F . k ;
i = ( i |^ n ) |^ ( n + 1 ) .= ( i |^ n ) |^ n .= ( ( i |^ n ) |^ n .= ( ( i |^ n ) |^ n ) |^ n .= ( ( ( i |^ n ) |^ n ) |^ n ) |^ n .= ( ( ( i |^ n ) |^ n ) |^ n ) |^ n ) |^ n .= ( ( ( p |^ n ) |^ n ) |^ n ) |^ n ) |^ n ) |^ n |^ n .= ( p |^ n ) |^ n |^ n ) |^ n .= ( p |^ n ) |^ n |^ n |^ n ) |^ n .= ( p |^ n ) |^ n .= ( p |^ n ) |^ n .= ( p |^ n ) |^ n |^ n .= ( p |^ n ) |^ n .= p |^ n )
consider q such that q q q and q is one-to-one and q is one-to-one and q is one-to-one ;
defpred P [ Nat ] means ( for I st $1 st $1 in dom ( ( ( PartDiffSeq ( G , n ) ) * ( ( G * ( $1 + 1 ) ) ) * ( ( G * ( $1 + 1 ) ) ) holds ( ( ( G * ( $1 + 1 ) ) * ( ( G * ( $1 + 1 ) ) ) * ( ( G * ( $1 + 1 ) ) ) ) * ( ( ( ( G * ( $1 + 1 ) ) ) * ( ( ( $1 + 1 ) ) ) * ( ( ( ( $1 + 1 ) ) * ( ( ( $1 + 1 ) ) ) * ( ( ( ( ( ( ( $1 + 1 ) ) * ( ( ( ( $1 + 1 ) ) ) * ( ( ( $1 + 1 ) ) ) ;
let A , B be Matrix of K , n , K ;
consider s such that for i st i in dom s holds s . i = ( s . i ) . i ;
func - ( x - y ) -> complex number means : Def1 : x - y * ( x - y ) ;
consider that that that for x st x in dom ( ( ( g | A ) | A ) | A and g | A is bounded and g | A is bounded ;
If for n holds ( ( ( ( n + 1 ) * ( n + 1 ) ) * ( n + 1 ) ) * ( n + 1 ) ) ) * ( ( n + 1 ) ) = ( ( ( ( n + 1 ) * ( n + 1 ) ) * ( n + 1 ) ) * ( n + 1 ) ) ;
( ( ( ( ( ( q `1 ) / ( 1 - sn ) ) * ( 1 - sn ) ) ^2 >= 0 ;
( p . ( ( p + 1 ) ) * ( p + 1 ) ) = ( p * ( p + 1 ) * ( p + 1 ) ;
consider k such that a = ( ( a --> b ) --> 0 ) --> 0 ;
consider B being set such that B = ( ( B \/ C ) \ ( B \/ C ) ;
( ( id Z ) `| Z ) `| Z = id Z .= ( id Z ) . ( ( id Z ) ;
dom ( Initialize s ) = dom ( Initialize s ) ;
ex h be Point of TOP-REAL 2 st h . n = ( ( ( ( ( ( ( TOP-REAL 2 ) | D ) | D ) | D ) ) | D ) ;
Int cell ( G , i + 1 ) c= Cl cell ( G , i + 1 ) ;
mid ( h , i , j ) = mid ( h , i , j ) ;
A = A \ { q : q in A } ;
( - ( x - y ) * ( x - y ) = ( - ( y - z ) * ( x - z ) .= ( - z ) * ( x - z ) .= ( - z ) * ( x - z ) .= ( - z ) * ( x - z ) .= ( - z ) * ( x - z ) ) .= ( - z ) * ( x - z ) .= ( - z ) * ( x - z ) * ( x - z ) .= ( - z ) * ( x - z ) * ( x - z ) * ( x - z ) .= ( - z ) * ( x - z ) .= ( - z ) * ( x - z ) .= ( ( x - z ) * ( x - z ) * ( x - z ) * ( x - z ) * ( x - z ) * ( x - z ) * ( x - z ) .= ( ( - z ) * ( x
0 + ( ( ( ( 1 - 1 ) ) ) / ( ( 1 - ( ( - 1 ) ) / ( ( 1 - ( 1 - 1 ) ) ) ) ) ) ) ) ) ^2 = ( ( ( 1 - ( 1 - ( ( 1 - ( 1 - 1 ) ) / ( 1 - 1 ) ) / ( 1 - 1 ) ) ) / ( ( 1 - 1 ) ) ) ;
( ( ( ( ( ( ( ( ( ( ( - - ( - 1 ) ) ) ) ) ) ) | ( ( ( - 1 ) ) ) | ( ( ( - 1 ) ) ) | ( ( ( 1 - 1 ) ) ) | ( ( ( 1 - 1 ) ) | ( ( ( 1 - 1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) = ( ( ( - 1 ) | ( ( ( 1 - 1 ) ) | ( ( ( ( 1 - 1 ) ) | ( ( ( ( 1 - 1 ) ) | ( ( ( ( 1 - 1 ) ) | ( ( ( ( 1 - 1 ) ) | ( ( ( 1 - 1 ) ) | ( ( ( ( ( 1 - 1 ) ) | ( ( ( ( 1 - 1 ) ) | ( ( ( ( ( 1 - 1 ) ) | ( ( ( ( 1 - 1 ) ) | ( ( ( ( ( 1 - 1
func - ( f + g ) -> Function equals f - g + f - g ;
assume that 1 <= i and i <= len G and j + 1 <= width G and i + 1 <= len G ;
assume that that y in Free ( H ) and not x in Free ( H ) ;
defpred P [ Nat ] means ex p be Point of TOP-REAL 2 st p in Seg ( $1 , 1 ) & $1 `1 = ( ( p `1 ) * ( $1 + 1 ) ;
cluster C -> strict for non empty SubSpace of X ;
[#] ( ( ( ( ( [#] ( TOP-REAL 2 ) | P ) | Q ) ) | Q ) ) = [#] ( ( ( ( ( ( ( TOP-REAL 2 ) | Q ) | Q ) | Q ) ;
rng ( S | ( ( ( ( S | ( ( ( ) | ( ( n | n ) | n ) ) | ( S | n ) ) ) ) ) ) ) ) ) ) ) = {} ;
( f | ( i ) ) | ( i ) = f | ( i -tuples_on dom f ) .= f | ( i + 1 ) .= f | ( i + 1 ) ;
consider P being Subset of TOP-REAL 2 such that P = P and P /\ Q and P misses Q ;
f . ( ( ( p `1 ) / 2 ) ^2 ) ^2 = ( ( ( ( ( ( p `1 / ( ( |. p .| ) ) ^2 ) ^2 ) ^2 ) ^2 ;
( ( ( ( ( ( - ( a - b ) - b ) / 2 ) ) * ( a - b ) ) ^2 ) ^2 = ( a - b ) ^2 ;
assume that p in Cl ( A , B ) and p in Cl ( A , B ) ;
for G being Subset of T , F being Subset of T , G being Subset of T st G = F /\ G & F is open holds G is open
for x being Real holds ( ( ( ( ( ( ( ( ( - 1 ) - ( 1 - ( 2 - 2 ) ) ) * ( ( - 2 ) ) ) ) ) ) * ( ( - 1 ) ) ) ) ) ) ) ) * ( ( ( - 1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) * ( ( - 1 - ( 2 * x ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) * ( > ( ( ( ( ( ( ( - 1 ) * ( ( - 1 ) ) * ( ( ( - 1 ) ) * ( ( ( - 1 ) ) ) * ( ( ( - 1 ) ) ) * ( ( ( - 1 ) ) ) * ( ( ( ( - 1 ) ) * ( ( ( - 1 ) ) * ( ( ( ( - 1 ) ) * ( ( ( - 1 ) ) * ( ( ( ( - 1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) * ( ( ( - 1 ) ) ) ) ) * ( ( ( - 1 ) ) ) ) ) ) * (
cluster f + g -> differentiable for for x0 st x0 in dom f holds f | ( x0 + g ) . x0 = f . x0 + f . x0 ;
ex X being Subset of Y st X misses Y & Y misses X & Y misses X ;
ex SVF1 ( 1 , 2 , 1 ) . ( 1 + 1 ) - SVF1 ( 1 , 2 , 1 ) . ( 1 + 1 ) = SVF1 ( 1 , 2 , 1 ) ;
sqrt ( 1 - ( ( ( 1 - ( 1 - ( ( 1 - ( 1 - 1 ) ) ) / ( ( 1 - ( 1 - 1 ) ) ) ) ) ) ) ) ) ^2 >= 1 ;
( ( ( ( ( ( ( 1 - ( ( 1 - ( 1 - 1 ) ) ) * ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( ( - 1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) * ( ( 1 - ( 1 - ( 1 - 1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) * ( ( ( - 1 ) ) ) ) * ( * ( ( ( 1 - ( ( 1 - ( ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - (
for x holds ( ( ( ( ( ( cos + cos ) ) ) | A ) ) | A is bounded implies ( ( cos | A ) | A ) ) is bounded
consider X such that Y = X and Y is open and Y is open and Y is open ;
card ( card ( S |^ ( n -' 1 ) ) ) = ( card ( S |^ ( n -' 1 ) ) ;
sqrt ( ( 1 - ( ( 1 - 1 ) ) / ( 1 - ( ( - 1 ) ) / ( 1 - 1 ) ) ) ^2 = ( ( - 1 ) / ( 1 - 1 ) ) / ( 1 - 1 ) ) / ( 1 - 1 ) ) ;