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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
contradiction ;
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thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
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contradiction ;
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thesis ;
thesis ;
contradiction ;
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contradiction ;
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thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b
D c= S
let Y ;
S `1 is Cauchy ;
q ;
V ;
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 ;
x in X ;
Y `1 in Y ;
assume 0 < g ;
not c in Y ;
not v in L ;
2 in z `1 ;
assume f = g ;
N c= b ` ;
assume i < k ;
assume u = v ;
I = J ;
B `1 = b `1 ;
assume e in F ;
assume p > 0 ;
assume x in D ;
let i be element ;
assume F is onto ;
assume n <> 0 ;
let x be element ;
set k = z ;
assume o = x ;
assume b < a ;
assume x in A ;
a `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 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 `1 in B `1 ;
assume i = 1 ;
let x be element ;
x `1 = x `1 ;
let X be BCK-algebra ;
assume S is non empty ;
a in REAL ;
let p be set ;
let A be set ;
let G be _Graph , X be set ;
let G be _Graph , X be set ;
let a be Complex ;
let x be element ;
let x be element ;
let C be FormalContext , X be Subset of C ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
let y be Real ;
X c= f . a ;
let y be element ;
let x be element ;
let i be Nat ;
let x be element ;
n in NAT ;
let a be element ;
m in NAT ;
let u be element ;
i in NAT ;
let g be Function ;
Z c= NAT ;
l <= ma ;
let y be element ;
r2 in dom f ;
let x be element ;
let k1 be Integer ;
let X be set ;
let a be element ;
let x be element ;
let x be element ;
let q be element ;
let x be element ;
assume f is being_homeomorphism ;
let z be element ;
a , b // K ;
let n be Nat ;
let k be Nat ;
B `1 c= B `1 ;
set s = f as Function ;
n >= 0 + 1 ;
k c= k + 1 ;
R1 c= R ;
k + 1 >= k ;
k c= k + 1 ;
let j be Nat ;
o , a // Y ;
R c= Cl G ;
card B = card B ;
let j be Nat ;
1 <= j + 1 ;
arccot is_differentiable_on Z ;
exp ( x , a ) is_differentiable_in x ;
j < i0 ;
let j be Nat ;
n <= n + 1 ;
k = i + m ;
assume C meets S ;
n <= n + 1 ;
let n be Nat ;
h1 = {} ;
0 + 1 = 1 ;
o <> b2 ;
f2 is one-to-one ;
support p = {}
assume x in Z ;
i <= i + 1 ;
r1 <= 1 ;
let n be Nat ;
a "/\" b <= a ;
let n be Nat ;
0 <= r1 ;
let e be Real , x be set ;
not r in G . l
c1 = 0 ;
a + a = a ;
<* 0 *> in e ;
t in { t } ;
assume F is not discrete ;
m1 divides m ;
B * A <> {} ;
a + b <> {} ;
p * p > p ;
let y be ExtReal ;
let a be Int-Location , i be Integer ;
let l be Nat ;
let i be Nat ;
let r ;
1 <= i2 ;
a "\/" c = c ;
let r be Real ;
let i be Nat ;
let m be Nat ;
x = p2 ;
let i be Nat ;
y < r + 1 ;
rng c c= E
Cl R is boundary ;
let i be Nat ;
R2 ;
cluster uparrow x -> being such that x is being natural ;
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 ;
that that s >= ks ;
G . y <> 0 ;
let X be RealNormSpace , Y be Subset of X ;
a ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , X be Subset of V ;
assume x in \rbrack M ;
k < s . a ;
not t in { p } ;
let Y be 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 ;
G8 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 `1 = a * y `1 ;
rng D c= A ;
assume x in K1 ;
1 <= ii & 1 <= jj ;
1 <= ii & 1 <= jj ;
pp c= cos cos . x ;
1 <= ii & 1 <= i ;
1 <= ii & 1 <= i ;
LMP C in L ;
1 in dom f ;
let seq ;
set C = a * B ;
x in rng f ;
assume f is Lipschitzian ;
I = dom A ;
u in dom p ;
assume a < x + 1 ;
seq is bounded & lim seq = 0 ;
assume I c= P1 ;
n in dom I ;
let Q ;
B c= dom f ;
b + p _|_ a ;
x in dom g ;
F-14 is continuous ;
dom g = X ;
len q = m ;
assume A2 : A is closed ;
cluster R \ S -> real-valued ;
sup D in S ;
x << sup D ;
b1 >= X1 + X2 ;
assume w = 0. V ;
assume x in A . i ;
g in p3 X ;
y in dom t ;
i in dom g ;
assume P [ k ] ;
\rangle c= f ;
x4 is increasing ;
let e1 be element ;
- b divides b ;
F c= \tau ( F ) ;
GX is non-decreasing & GX is non-decreasing ;
GX is non-decreasing & GX is non-decreasing ;
assume v in H . m ;
assume b in [#] B ;
let S be non void ManySortedSign , X be set ;
assume P [ n ] ;
assume union S is finite ;
V is Subspace of V ;
assume P [ k ] ;
rng f c= NAT ;
assume inf X in L ;
y in rng f ;
let s , I be set , X be set ;
b `1 c= b9 `1 & b9 `2 = b9 ;
assume not x in REAL + 1 ;
A /\ B = { a } ;
assume len f > 0 ;
assume x in dom f ;
b , a // o , c ;
B in B-24 ;
cluster product p -> non empty ;
z , x // x , p ;
assume x in rng N ;
cosec is_differentiable_in x & cosec is_differentiable_in x ;
assume y in rng S ;
let x , y be element ;
i2 < i1 & i1 < i2 ;
a * h in a * H ;
p , q in Y ;
cluster sqrt I -> left ideal ;
q1 in A1 & q2 in B1 ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A2 c= A1 ;
|. \hbox { \boldmath $ m } .| < n ;
assume A c= dom f ;
Re ( f | A ) is_integrable_on M ;
let k , m ;
a , a \equiv b , b ;
j + 1 < k + 1 ;
m + 1 <= n1 ;
g is_differentiable_in x0 & g is_differentiable_in x0 ;
g is continuous & g is continuous implies g | X is continuous
assume O is symmetric & O is transitive ;
let x , y be element ;
let jj be Nat ;
[ y , x ] in R ;
let x , y be element ;
assume y in conv A ;
x in Int V ;
let v be VECTOR of V ;
P3 halts_on s ;
d , c // a , b ;
let t , u ;
let X be set ;
assume k in dom s ;
let r be non negative Real ;
assume x in F | M ;
let Y be Subset of S ;
let X be non empty TopSpace , Y be non empty SubSpace of X ;
[ a , b ] in R ;
x + w < y + w ;
{ a , b } >= c ;
let B be Subset of A , C be Subset of A ;
let S be non empty ManySortedSign ;
let x be variable , f be Function of f , g ;
let b be Element of X , c be Element of X ;
R [ x , y ] ;
x ` = x ` & x = x ;
b \ x = 0. X ;
<* d *> in D |^ 1 ;
P [ k + 1 ] ;
m in dom ( ( n + 1 ) * n ) ;
h2 . a = y ;
P [ n + 1 ] ;
cluster G * F -> prepreone-to-one ;
let R be non empty multMagma , X be Subset of R ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p `1 ;
assume f | X is lower ;
x in rng ( pion1 /. 1 ) ;
let x be Element of B ;
let t be Element of D ;
assume x in Q .last() ;
set q = s ^\ k ;
let t be VECTOR of X ;
let x be Element of A ;
assume y in rng p `1 ;
let M be be be be not empty id ;
let N be non empty Point of M ;
let R be RelStr , X be finite Subset of R ;
let n , k be Nat ;
let P , Q be RelStr ;
P = Q /\ [#] S ;
F . r in { 0 } ;
let x be Element of X ;
let x be Element of X ;
let u be VECTOR of V ;
reconsider d = x as Int-Location ;
assume I is not destroy a ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < vv ;
x <= c2 . x ;
x in F ` ` ;
cluster S --> T -> nonempty ;
assume t1 <= t2 & t2 <= t1 ;
let i , j be even Integer ;
assume F1 <> F2 & F2 <> F2 ;
c in Intersect ( union R ) ;
dom p1 = c & dom p1 = c ;
a = 0 or a = 1 ;
assume A1 : A <> A2 & B <> A1 ;
set i1 = i + 1 ;
assume a1 = b1 & a2 = b2 ;
dom g1 = A & rng g1 = A ;
i < len M + 1 ;
assume not - G in rng G ;
N c= dom f1 /\ dom f2 ;
x in dom ( sec | Z ) ;
assume [ x , y ] in R ;
set d = sqrt ( x ^2 + y ^2 ) ;
1 <= len g1 & len g1 <= len g2 ;
len ( s2 ) > 1 & len ( s2 ) > 1 ;
z in dom f1 /\ dom ( f1 + f2 ) ;
1 in dom D2 & 1 in dom D2 ;
( p `2 ) ^2 = 0 ;
j2 <= width G & j <= width G ;
len cos > 1 + 1 ;
set n1 = n + 1 ;
|. q1 .| = 1 ;
let s be SortSymbol of S ;
i |^ ( i , i ) = i ;
X1 c= dom f & X2 c= dom f ;
h . x in h . a ;
let G be mod of k ;
cluster m * n -> invertible ;
let k9 be Nat ;
i -' 1 > m - 1 ;
R is transitive & R is transitive implies R is transitive
set F = <* u , w *> ;
pp c= P3 & q c= P3 ;
I is_closed_on t , Q & I is_halting_on t , Q ;
assume [ S , x ] is real ;
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 /\ dom ( f1 + f2 ) ;
assume [ X , p ] in C ;
BX c= [: X , Y :] ;
n2 <= ( 2 |^ ( n + 1 ) ) ;
A /\ ( P ` ) c= A ` ;
cluster -> x -valued for Function ;
let Q be Subset-Family of S , P be Subset of X ;
assume n in dom ( g2 | n ) ;
let a be Element of R ;
t `1 in dom ( e | ( dom e ) ) ;
N . 1 in rng N ;
- z in A \/ B ;
let S be SigmaField of X , T be Subset-Family of X ;
i . y in rng i ;
REAL c= dom f & f | X is bounded ;
f . x in rng f ;
( |. r .| ) ^2 <= ( r ^2 ) ^2 ;
s2 in r & s2 in r ;
let z , z be complex number ;
n <= NN . m ;
LIN q , p , s ;
f . x = waybelow x /\ B ;
set L = |[ S \to T , T , f , g ;
let x be non negative Real ;
let m be Element of M ;
f in union rng F1 & f is one-to-one ;
let K be add-associative right_zeroed right_complementable associative associative distributive non empty doubleLoopStr , A be Subset of K ;
let i be Element of NAT ;
rng ( F * g ) c= Y
dom f c= dom x & f . x = x ;
n1 < n1 + 1 + 1 ;
n1 < n1 + 1 + 1 ;
cluster [: T , X :] -> *> ;
[ y2 , 2 ] `2 = z ;
let m be Element of NAT ;
let S be Subset of R ;
y in rng ( S | X ) ;
b = sup dom f & b = sup dom f ;
x in Seg len ( q | i ) ;
reconsider X = [: D , D :] as set ;
[ a , c ] in [: E , F :] ;
assume n in dom h2 ;
w + 1 = ( a1 + a2 ) * w ;
j + 1 <= j + 1 + 1 ;
k2 + 1 + 1 <= k1 + 1 ;
let i be Element of NAT ;
Support u = Support p & Support p = Support p ;
assume X is complete mod m ;
assume f = g & p = q ;
n1 <= n1 + 1 & n1 + 1 <= len f ;
let x be Element of REAL ;
assume x in rng ( s2 | X ) ;
x0 < x0 + 1 - r ;
len ( Carrier ( L ) ) = W ;
P c= Seg len ( A * ) ;
dom q = Seg n & len q = Seg n ;
j <= width M *' ;
let r8 be real-valued FinSequence of REAL ;
let k be Element of NAT ;
Integral ( P , x ) < +infty ;
let n be Element of NAT ;
assume z in C \tt not contradiction ( 0 , A ) ;
let i be set ;
n -' 1 = n - 1 ;
len ( n + m ) = n ;
`| Z c= F
assume x in X or x = X ;
x is midpoint of b , c ;
let A , B be non empty set , f be Function of A , B ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q ;
let s be Element of E -tuples_on E ;
let B1 be Basis of x , B be Subset of V ;
Carrier ( 3 ) /\ Carrier ( 4 ) = {} ;
L1 /\ L2 = {} & L1 /\ L2 = {} ;
assume ||. x .|| = ||. y .|| ;
assume b , c , b is_collinear ;
LIN q , c , c ;
x in rng ( ( f | X ) ^ ) ;
set n8 = n + j ;
let D7 be non empty set , X be non empty set ;
let K be add-associative right_zeroed right_complementable associative non empty addLoopStr , M be Matrix of K ;
assume f = f & h = f ;
R1 - R2 is total & R2 - R1 is total ;
k in NAT & 1 <= k & k <= n ;
let a be Element of G ;
assume x0 in [. a , b .] ;
K1 ` is open & ( ( TOP-REAL 2 ) | K1 ) is open ;
assume a , b ] is_a_maximal from C ;
let a , b be Element of S ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = Integral ( M , a ) ;
cluster as \vert strict strict for WFF ;
not u in { \hbox { \boldmath $ g } } ;
the carrier of f c= B ;
reconsider z = x as VECTOR of V ;
cluster the RelStr of L -> strict ;
r (#) H is \Im of X ;
s . intloc 0 = 1 ;
assume x in C & y in C ;
let U0 be strict MSAlgebra over S , x be set ;
[ x , Bottom T ] is compact ;
i + 1 + 1 + k in dom p ;
F . i is stable of M ;
r-35 in { ( y ) `1 , ( y ) `2 } ;
let x , y be Element of X ;
let A , I be J of X ;
[ y , z ] in [: O , O :] ;
that that that that that that card Macro i = 1 and card Macro i = card I ;
rng Sgm A = A & rng Sgm A = A ;
q |- All ( y , q ) ;
for n holds X [ n ] ;
x in { a } & x in d ;
for n holds P [ n ] ;
set p = |[ x , y , z ]| ;
LIN o , a , b ;
p . 2 = Z |^ Y ;
( D . x0 ) `2 = {} & ( D . x0 ) `2 = 0 ;
n + 1 + 1 + 1 <= len g ;
a in [: Al , Al :] ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
let I be Ideal of L ;
set g = f1 + f2 , h = f2 + f3 , i = f2 + f3 , j = f3 + h , n = n + 1 ;
a <= max ( a , b ) ;
i-1 < len G + 1 ;
g . 1 = f . i1 ;
x `1 , y `2 in A2 ;
( f /* s ) . k < r ;
set v = VAL g ;
i -' k + 1 <= S ;
cluster non empty multiplicative for multMagma ;
x in support ( ( support t ) | ( support t ) ) ;
assume a in [: the carrier of G , the carrier of G :] ;
i `1 <= ( y `1 ) / 2 ;
assume p divides b1 + b2 & p divides b2 + b2 ;
x0 <= sup M1 & sup M2 <= sup M2 ;
assume x in ( W /\ ( X /\ Y ) ) ;
j in dom ( z | ( n + 1 ) ) ;
let x be Element of D ;
IC Comput ( P3 , s3 , k ) = l1 + 1 ;
a = {} or a = { x } ;
set u9 = Vertices G , u9 = Vertices G ;
seq " is non-zero & lim ( seq " ) = 0 ;
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
hK c= h & h is one-to-one ;
]. a , b .[ c= Z ;
X1 , X2 are_separated implies X1 union X2 , X2 X2 X2 are_separated
a in Cl ( union F ) ;
set x1 = [ 0 , 0 ] , x2 = [ 0 , 0 ] , x3 = [ 0 , 0 ] , x4 = [ 0 , 1 ] , x4 = [ 0 , 1
k + 1 -' 1 = k - 1 ;
cluster -> real-valued for Relation of INT , INT ;
ex v st C = v + W ;
let IT be non empty addLoopStr , X be non empty set ;
assume V is Abelian add-associative right_zeroed right_complementable distributive non empty doubleLoopStr ;
X1 \/ Y in InclPoset ( L ) ;
reconsider x = x as Element of S ;
max ( a , b ) = a & max ( b , c ) = b ;
sup B is upper of B & sup B in B ;
let L be non empty reflexive RelStr , X be Subset of L ;
R is reflexive & R is transitive implies R is transitive
E , g |= All ( g , H ) ;
dom G /. y = a & G /. y = b ;
sqrt ( 1 - 4 * r ) >= - r ;
G . x0 in rng G & G . x0 in rng G ;
let x be Element of FF , y be Element of X ;
D [ P , 0 , 0 ] ;
z in dom id B & z in B ;
y in the carrier of N & y in the carrier of N ;
g in the carrier of H & g in the carrier of H ;
rng ( f | X ) c= [: X , Y :] ;
j `2 + 1 in dom s1 ;
let A , B be strict Subgroup of G ;
let C be non empty Subset of REAL ;
f . z1 in dom h & h . z1 in dom h ;
P . k1 in rng P & P . k1 in rng P ;
M = ( A +* {} ) +* ( A +* B ) ;
let p be FinSequence of REAL ;
f . n1 in rng f & f . n1 in rng f ;
M . ( F . 0 ) in REAL ;
ind [. a , b .[ = b ;
assume that the distance of V , Q is Subspace of v ;
let a be Element of ^ V ;
let s be Element of P ( ) ;
let PA be non empty RelStr ;
let n be Nat ;
the carrier of g c= B & the carrier of g c= B ;
I = halt R .= ( id the carrier of R ) ;
consider b being element such that b in B ;
set BK = BCS ( K , n ) ;
l <= ( ( -> Nat ) * ( j , k ) ) ;
assume x in ]. s , t .[ ;
( x `2 ) ^2 in ]. t `1 , t `2 .[ ;
x in JumpParts ( JumpParts JumpParts T ) ;
let h be Morphism of c , a ;
Y c= [: Y , Z :] & Y c= [: Y , Z :] ;
A2 \/ ( A1 \/ A2 ) c= Carrier ( A1 \/ A2 ) ;
assume LIN o , a , b & LIN o , a , b ;
b , c // d1 , d2 ;
x1 , x2 in Y & x1 , x2 in Y ;
dom <* y *> = Seg 1 .= Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar s .| ;
[ x , x `2 ] in X ;
for n being Nat holds 0 <= x . n
|[ a , b ]| = [. a , b .] ;
cluster -> -> -> -> -> -> -> -> -> -> -> -> -> -> closed for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 , q1 , q2 is_collinear & q2 , q2 , q2 is_collinear ;
dom ( M1 + M2 ) = Seg n ;
x = [ x1 , x2 ] & y = [ x1 , x2 ] ;
let R , Q be ManySortedSet of A ;
set d = sqrt ( 1 + n ) ;
rng ( g2 | ( dom g ) ) c= dom W ;
P . ( [#] Sigma \ B ) <> 0 ;
a in field R & a = b ;
let M be non empty Subset of V , A be Subset of M ;
let I be Program of SCM+FSA , J be Program of SCM+FSA ;
assume x in rng ( ( R ~ ) * ( R ~ ) ) ;
let b be Element of the lattice of T ;
dist ( e , z ) < rz ;
u1 + v1 in W2 + W1 & v1 in W2 + W1 ;
assume the carrier of L misses ( rng G ) ;
let L be lower-bounded non empty RelStr ;
assume [ x , y ] in [: a , b :] ;
dom ( A * e ) = NAT ;
let a , b be Vertex of G ;
let x be Element of Bool ( M ) ;
0 <= Arg a * PI & Arg a < 2 * PI ;
o , a9 // o , y & a , b // o , y ;
{ v } c= the carrier of l & v in the carrier of l ;
let x be variable ;
assume x in dom ( uncurry f ) ;
rng F c= ( product f ) |^ X ;
assume D2 . k in rng D ;
f " ( p1 ) = 0 & f " ( p1 ) = 0 ;
set x = the Element of X , y = the Element of Y ;
dom Ser G = NAT & for n holds G . n = F . n ;
let n be Element of NAT ;
assume LIN c , a , e1 ;
cluster -> finite for FinSequence of NAT ;
reconsider d = c as Element of L1 ;
( v2 |-- I ) . X <= 1 ;
assume x in the carrier of f & y in the carrier of f ;
conv ( A , B ) c= conv A & conv ( A , B ) c= conv A ;
reconsider B = b as Element of the lattice of T ;
J , v |= P ! l , v |= l ;
cluster J . i -> non empty for TopSpace ;
ex_sup_of Y1 \/ Y2 , T & for x being Element of T holds x in Y1 \/ Y2 implies x in Y1 \/ Y2
W1 is_Lin ( W1 ) & W2 is_Lin ( W2 ) implies W1 is Subspace of W2
assume x in the carrier of R & y in the carrier of R ;
dom ( n + 1 ) = Seg n & len ( n + 1 ) = Seg n ;
s4 misses ( s * ( s * ( s * ( t * ( t * ( s * ( t * ( s * ( t * ( s * ( t * ( t * ( s * ( t *
assume ( a 'imp' b ) . z = TRUE ;
assume that X is open and f = X --> d ;
assume [ a , y ] in \cup ( f | X ) ;
assume that that that that that that that that stop I c= J and card I = K ;
Im ( ( lim seq ) (#) ( Im seq ) ) = 0 ;
( ( sin * cos ) `| Z ) . x <> 0 ;
sin | Z is_differentiable_on Z & cos | A is continuous ;
t1 . n = t2 . n .= s . n ;
dom ( cos | dom F ) c= dom F ;
W1 . x = W2 . x & W2 . x = W2 . x ;
y in W .last() \/ W .last() ( x ) ;
k9 <= len ( v | ( n + 1 ) ) & k <= len ( v | ( n + 1 ) ) ;
x * a divides y * a * ( y mod m ) ;
proj2 .: S c= ( proj2 .: P ) .: P ;
h . p3 = g2 . I .= g2 . I ;
GU = U /. 1 .= G /. 1 .= G /. 1 ;
f . ( r1 + 1 ) in rng f ;
i + 1 + 1 + 1 <= len f ;
rng F = rng ( F | ( X \ Y ) ) ;
mode non empty doubleLoopStr is unital of G ;
[ x , y ] in A [: { a } , { a } :] ;
x1 . o in { L2 . o } ;
the carrier of ( m + 1 ) c= B ;
not [ y , x ] in id X & y in id X ;
1 + p .. f <= i + len f ;
seq ^\ k1 is bounded & lim ( seq ^\ k1 ) = ( lim seq ) * ( lim seq ) ;
len ( F | ( len F ) ) = len F ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be complex number , r be complex number ;
Comput ( P , s , n ) = s ;
k <= k + 1 & k + 1 <= len p ;
reconsider c = {} T as Element of T ;
let Y be with_empty \cal T ;
cluster -> directed-sups-preserving for Function of L , L ;
f . j1 in K . j1 & f . j1 in K . j1 ;
cluster J => y -> total for Function of J , the carrier of R ;
K c= 2 -tuples_on the carrier of T ;
F . b1 = F . ( b2 , b1 ) ;
x1 = x or x1 = y or x1 = z ;
pred a <> {} means : Def1 : sqrt a = 1 & sqrt a = sqrt a ;
assume that succ a c= b and b in a ;
s1 . n in rng s1 & s1 . n in rng s1 ;
{ o , b2 } on C2 & { o , b2 } on C2 ;
LIN o , b , b9 & LIN b , c , b9 ;
reconsider m = x as Element of Funcs ( V , C ) ;
let f be non empty FinSequence of D ;
let F2 be non empty <> the carrier of TOP-REAL 2 ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in F-8 ;
reconsider pp = x as Subset of m -tuples_on the carrier of L ;
let A , B , C be Element of R ;
cluster non empty strict for SubSpace of X ;
rng c `1 misses rng ( e | n ) ;
z is Element of gr ( { x } ) ;
not b in dom ( a .--> p1 ) ;
assume that k >= 2 and P [ k ] ;
Z c= dom ( ( ( ( - 1 ) (#) ( ( cot * ( arccot ) ) ) ) (#) ( ( arccot * ( arccot ) ) ) ) ) ;
the component of Q c= ( UBD A ) \/ ( UBD A ) ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( 1 / ( f ^ ) ) ;
pred f = u * f & a * f = a * u ;
for n holds P1 [ n ] ;
{ x . O : x in L } <> {} ;
let x be Element of V . s ;
let a , b be Nat ;
assume that S = S2 and p = S2 and q = S2 ;
gcd ( n1 , n2 , n1 , n2 , n3 , n2 , n3 , n4 ) = 1 ;
set o9 = ( 1 / 2 ) * ( ( 1 / 2 ) * ( ( 1 / 2 ) * ( ( 1 / 2 ) * ( ( 1 / 2 ) * ( ( 1 / 2 ) * (
seq . n < |. r1 - x0 .| & |. r1 - x0 .| < r ;
assume that seq is increasing and r < 0 and seq is increasing ;
f . ( y1 , x1 ) <= a & f . ( y1 , x1 ) <= a ;
ex c being Nat st P [ c ] ;
set g = { n / ( 1 + 1 ) where n is Element of NAT : n in n } ;
k = a or k = b or k = c ;
a9 , b9 , c9 is_collinear & b9 , c9 , a9 is_collinear & b9 , c9 , c9 is_collinear ;
assume Y = { 1 } & s = <* 1 *> ;
I1 . x = f . x .= 0 .= 0 ;
pion1 . 1 = W . ( 1 + 1 ) .= W . ( 1 + 1 ) ;
cluster -> trivial for Subgroup of G ;
reconsider u = u as Element of Bags X ;
A in B ^ implies A , B are_separated
x in { |[ 2 * n + 3 , k + 1 ]| } ;
1 >= sqrt ( ( q `2 / |. q .| - sn ) ^2 ) ;
f1 is_t` & f2 is_t` implies f1 is_\! misses f2
( f . ( q `2 / |. q .| - sn ) ) ^2 <= ( q `2 / |. q .| - sn ) ^2 ;
h is_the carrier of Cage ( C , n ) ;
( b `2 ) ^2 <= ( p `2 ) ^2 + ( p `2 ) ^2 ;
let f , g be Function of X , Y ;
S * ( k , k ) <> 0. ( K , n ) ;
x in dom ( max ( - ( f . x ) ) , ( - ( f . x ) ) ) ;
p2 in [: N1 , N2 :] & p1 in N1 /\ N2 ;
len ( ( H ) . n ) < len ( H ) ;
F [ A , F . A ] ;
consider Z such that y in Z and Z in X ;
pred 1 in C means : Def1 : A c= C & C c= D ;
assume r1 <> 0 or r2 <> 0 or r1 <> 0 ;
rng q1 c= rng ( C1 * C2 ) & rng C1 c= dom ( C2 * C2 ) ;
A1 , A2 , A3 is_collinear implies not ( A1 , A2 , A3 is_collinear ) & ( A1 , A2 , A3 is_collinear ) & ( A2 , A1 , B1 is_collinear ) & ( A1 , A2 , A3 is_collinear ) & ( A2 ,
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in ^2 ( p , S ) & b in c ;
then S is atomic implies P [ S ] ;
Cl ( [#] T ) = [#] T .= [#] T ;
( f | ( A2 \/ B2 ) ) | ( A2 \/ B2 ) = f2 | ( A2 \/ B2 ) ;
0. M in the carrier of W & 0. M in the carrier of W ;
let v , v be Element of M ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ V
let X be Subset of S ;
consider H1 such that H = 'not' H1 and H1 is Subgroup of G ;
\lbrace 1 , 1 , 0 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 1 , 0 , 1 , 1 , 1 ,
0 * a = 0. R .= a * 0. R .= a * 0. R ;
A |^ 2 = A ^^ A & A |^ 2 = A |^ 2 ;
set vX1 = vX /. n , vX = v /. n , vX = v /. n , vX = v /. n , vX = v /. n , RX = v /. n , R
r = 0. ( \langle 0. TOP-REAL n , \Vert * \Vert *> ) .= 0. TOP-REAL n ;
( f . p3 ) `1 >= 0 & ( f . p4 ) `1 >= 0 ;
len W = len ( W | ( len W ) ) .= len W ;
f /* ( s * G ) is divergent_to+infty ;
consider l being Nat such that m = F . l ;
t8 , W8 , [: b1 , b2 :] & not ex b1 , b2 st b1 , b2 , b1 , b2 is_collinear & not ( b1 , b2 , b1 , b2 is_collinear ) & ( b1 , b2 , b1 , b2 is_collinear ) & not
reconsider Y1 = X1 union X2 as SubSpace of X ;
consider w such that w in F and not x in w ;
let a , b , c be Real ;
reconsider i = i - 1 as non zero Element of NAT ;
c . x >= id the carrier of L . x ;
InclPoset the topology of T \/ \omega ( T ) is Basis of T ;
for x being element st x in X holds x in Y
cluster [ x1 , x2 ] -> pair ;
downarrow a /\ { t } is Ideal of T ;
let X be set , N be non empty set , f be Function of X , NAT ;
rng f = cluster cluster S ( ) -> non-empty ;
let p be Element of B , x be Element of S ;
max ( N1 , 2 ) >= N1 & N1 >= N1 ;
0. X <= b |^ ( m + 1 ) * ( b |^ ( m + 1 ) ) ;
assume that i in I and R0 . i = R ;
i = j1 & p1 = q1 & p1 = q2 & p2 = q2 ;
assume gR in the carrier of g & gR in the carrier of g ;
let A1 , A2 be Point of S , A be Subset of X ;
x in h " P /\ [#] ( ( TOP-REAL 2 ) | P ) ;
1 in Seg 2 & 1 in Seg 3 & 2 in Seg 3 ;
reconsider X1 = X as non empty Subset of [: T , T :] ;
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the Target of G ) -valued ;
n1 <= i2 + len g2 & i2 <= len g2 + 1 ;
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume v in the carrier of G2 & v in the carrier of G2 ;
y = Re ( y ) + ( Im ( y ) ) ;
( ( ( - 1 ) / p ) ^2 ) = 1 ;
x2 is_differentiable_on ]. a , b .[ & ( f | ]. a , b .[ ) . ( a + b ) = f . ( a + b ) ;
rng ( M * ( i , j ) ) c= rng ( D2 * ( i , j ) ) ;
for p being Real st p in Z holds p >= a
( ( Y --> f ) | X ) = ( proj1 | X ) | X .= ( proj1 | X ) | X ;
( seq ^\ m ) . k <> 0 ;
s . ( G . ( k + 1 ) ) > x0 ;
( p \! \mathop { x } ) . 2 = d ;
A \oplus ( B -- C ) = ( A \oplus B ) \ominus C
h \equiv gg . ( ( mod P ) . ( b mod P ) ) ;
reconsider i1 = i-1 as Element of NAT ;
let v1 , v2 be VECTOR of V , v be VECTOR of V ;
for V being Subspace of V holds V is Subspace of V
reconsider ii = i - 1 as Element of NAT ;
dom f c= [: C , D :] ;
x in ( the \upharpoonright of B ) . n ;
len reconsider _ 2 = len ( f2 | ( len f2 ) ) ;
( p in the topology of T ) & p in the topology of T ;
]. r , s .[ c= [. r , s .] ;
let B2 be Basis of T2 , B2 be Subset of T2 ;
G * ( B * A ) = ( id the carrier of A1 ) * ( B * A ) ;
assume that p , u , v , w is_collinear and u , v , w , y y ;
[ z , z ] in union rng ( F | ( X \/ Y ) ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S - $1 ;
LIN a1 , a3 , b1 & LIN a1 , b1 , b2 , b1 ;
f " ( f .: ( f .: ( f .: ( f .: ( f .: ( f .: ( f .: ( f .: ( f .: ( f .: ( f .: ( f .: ( f .: ( f .: ( f .: ( f .:
dom ( w2 | ( dom ( w | ( dom w ) ) ) ) = dom ( ( w | ( dom w ) ) ) ;
assume that 1 <= i and i <= n and j <= n ;
( ( g2 ) . O ) `2 <= 1 ;
p in LSeg ( E . i , F . i ) ;
I1 * ( i , j ) = 0. ( K , n ) ;
|. f . ( s . m ) - g .| < g1 ;
q1 . x in rng ( ( q | n ) | n ) ;
Carrier ( Lxy ) misses Carrier ( LR2 ) ;
consider c being element such that [ a , c ] in G ;
assume N5 = o & o = o & o = o & o = o ;
q . ( j + 1 ) = q /. j ;
rng F c= ( F-12 ) .: C .= C ;
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [. f . j , f . j .] ;
pred 0 <= x & x <= 1 & x <= 1 implies x ^2 <= x ^2 ;
p `1 <> 0. TOP-REAL 2 & q `2 <> 0. TOP-REAL 2 ;
cluster [: { 0 } , T :] -> non empty ;
let x be Element of S ~ ;
F is one-to-one of F , F . ( a , b ) is one-to-one ;
|. i - i .| <= - ( 2 |^ n ) ;
the carrier of I[01] = dom P & P . 0 = P . 1 ;
cos ( n + 1 ) > 0 * cos ( n + 1 ) ;
S c= ( A1 /\ A2 ) /\ ( A2 /\ A1 ) & ( A1 /\ A2 ) /\ ( A1 /\ A2 ) c= ( A1 /\ A2 ) /\ ( A1 /\ A2 ) ;
a3 , a4 // b2 , b2 & a3 , a4 // b2 , b3 ;
then dom A <> {} & dom A <> {} & rng A c= {} ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , Y , X , Y , Z , X , Y ;
set v2 = v1 /. ( i + 1 ) , v2 = v2 /. ( i + 1 ) ;
x = r . n .= ( r . n ) * ( r . n ) ;
f . s in the carrier of S2 & f . s in the carrier of S2 ;
dom g = the carrier of I[01] & rng g = the carrier of I[01] ;
p in Lower_Arc ( P ) /\ Lower_Arc ( P ) ;
dom ( d2 | ( A \/ B ) ) = [: A , B :] ;
0 < sqrt p + sqrt ( 1 - ( p `2 / |. p .| - sn ) ) ;
e . ( m + 1 ) <= e . ( m1 + 1 ) ;
B \ominus X \/ B \ominus Y c= B \ominus X /\ Y
- - ( Im ( g | B ) ) | B is bounded ;
cluster O \tt F -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> natural for Relation of X ;
let U1 , U2 be non-empty MSAlgebra over S , B be MSAlgebra over S ;
Proj ( i , n ) * g is_differentiable_on X ;
let x , y , z be Point of X , p be Point of X ;
reconsider pp = p . x as Subset of V ;
x in the carrier of Lin ( A ) & x in the carrier of Lin ( A ) ;
let I , J be parahalting Program of SCM+FSA ;
assume - a is lower & b is lower ;
Int Cl A c= Cl Int A & Int A c= Cl Int A ;
assume for A being Subset of X holds Cl A = A ;
assume q in Ball ( |[ x , y ]| , r ) ;
( p2 `2 ) ^2 <= ( p2 `2 ) ^2 + ( p2 `2 ) ^2 ;
Cl Q ` = [#] ( ( TOP-REAL 2 ) | P ) .= P ;
set S = the carrier of T , T = the carrier of S ;
set I1 = [: f , g :] , I2 = [: f , g :] , I2 = [: f , g :] , I2 = f , I2 = g , I2 = g , I2 = f , I2 = g , I2 = g , I2 = g ,
len p -' n = len ( q | n ) .= len p ;
A is Permutation of Funcs ( A , x , y ) ;
reconsider nn = nY. as Element of NAT ;
1 <= j + 1 & j + 1 <= len ( s | j ) ;
q9 , q9 , q be Element of M , a , b , c be Element of M ;
a9 in the carrier of S1 & b9 in the carrier of S1 & c9 in the carrier of S2 ;
c1 /. ( n1 + 1 ) = c1 . ( n1 + 1 ) .= c1 /. ( n1 + 1 ) ;
let f be FinSequence of TOP-REAL 2 , p , q be Point of TOP-REAL 2 ;
y = ( ( f * S ) . x ) . x .= f . x ;
consider x being element such that x in \mathop { \rm _ > } A ;
assume r in ( ( dist ( o ) ) .: P ) .: P ;
set i2 = ( ( n + 1 ) - ( n + 1 ) ) ;
h2 . ( j + 1 ) in rng h2 & h2 . ( j + 1 ) in rng h2 ;
Line ( M , k ) . i = M . ( i , j ) ;
reconsider m = sqrt ( x ^2 + 2 * x ^2 ) as Element of REAL n ;
let U1 , U2 be Subspace of U0 , B be Subset of U0 ;
set P = Line ( a , d ) ;
len p1 < len p2 + 1 & len p2 = len p1 + 1 ;
let T1 , T2 be b2 empty topological s ;
then x <= y & ( ex x st x in { y } ) & y in { x } ;
set M = n -to_power ( m + 1 ) , N = n -to_power m , S = n -to_power m , T = n -tuples_on X , T = n -tuples_on X , i = n -tuples_on X , i = n -tuples_on X , j = n -tuples_on X ,
reconsider i = x1 , j = x2 as Nat ;
rng ( the_arity_of ( a ) ) c= dom H & ( the_arity_of o ) . n = ( the_arity_of o ) . n ;
z1 " = z1 " * ( z1 * z2 ) .= z1 * ( z1 * z2 ) " .= z1 * ( z1 * z2 ) " ;
x0 - sqrt ( r ^2 + ( 1 - r ^2 ) ) in L /\ dom f ;
then w is \rm Lipschitzian & rng w /\ ( S /\ L ) <> {} ;
set x9 = ( x9 ^ <* Z *> ) | Z , y9 = ( x9 ^ <* Z *> ) | Z ;
len w1 in Seg len w1 & len w2 = len w2 ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of PFuncs ( V , { k } ) ;
x . n = sqrt ( |. a . n .| ) * ( |. a . n .| ) ;
( p `1 ) ^2 <= ( G * ( 1 , j ) ) ^2 ;
rng ( g | ( L~ g ) ) c= L~ ( g | ( L~ g ) ) ;
reconsider k = i-1 * l + j as Nat ;
for n being Nat holds F . n is of REAL m , REAL n
reconsider x9 = x9 , y9 = y9 as VECTOR of M ;
dom ( f | X ) = X /\ dom f /\ X .= X /\ dom f ;
p , a // p , c & b , a // c , a ;
reconsider x1 = x as Element of REAL m m -tuples_on REAL n ;
assume i in dom ( a (#) p ) ;
m . ( \hbox { \boldmath $ g } ) = p . ( \hbox { \boldmath $ g } ) ;
a |^ ( s . m ) - ( a |^ n ) <= 1 ;
S . ( n + k ) c= S . ( n + k ) ;
assume B1 \/ C2 = B2 \/ C2 \/ C2 .= ( B1 \/ B2 ) \/ ( B2 \/ C2 ) \/ ( B2 \/ C2 ) ;
X . i = { x1 , x2 } . i .= ( X --> x1 ) . i ;
r2 in dom ( h1 + h2 ) /\ dom ( h2 + h2 ) ;
||. 0. R .|| = a & b0. R = b ;
FF is closed & Q1 ( a , b ) c= Q1 & Q1 ( b , c ) c= Q ;
set T = NetUniv Inl ( X , x0 ) ;
Int ( Cl R ) c= Int R & Int ( R ) c= Int R ;
consider y being Element of L such that c . y = x ;
rng ( Fwhere x is Element of X ( ) ) = { F ( x ) where F is Element of X ( ) : P [ x ] } ;
G-23 ( { c } ) c= B \/ S \/ S ;
f9 is Relation of [: X , Y :] , X & f is Function of X , Y ;
set RP = the Point of ( ( TOP-REAL n ) | P ) | R ;
assume that n + 1 >= 1 and n + 1 <= len M ;
let k2 be Element of NAT ;
reconsider pp = u as Element of ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( j ) ) ) ;
g . x in dom f & x in dom g ;
assume that 1 <= n and n + 1 <= len f1 and f1 /. n = f1 /. n ;
reconsider T = b * N as Element of G |^ N , a , b be Element of G |^ N ;
len ( ( P +* ( j , m ) ) | ( i + 1 ) ) <= len ( P +* ( i , m ) ) ;
x in the carrier of A1 & x in the carrier of A1 & y in the carrier of A1 ;
[ i , j ] in Indices ( ( A * B ) * ( i , j ) ) ;
for m be Nat holds Re ( F . m ) is simple
f . x = a . i .= a1 . i .= f . i ;
let f be PartFunc of REAL i , REAL i , x be Element of REAL i ;
rng f = the carrier of ( ( A + B ) --> ( A + B ) ) ;
assume s1 = sqrt 2 & s2 = sqrt 2 & s1 = sqrt 2 ;
pred a > 1 & b > 0 implies a |^ b > 1 / a |^ b ;
let A , B , C be Subset of [: I , J :] ;
reconsider X0 = X , Y1 = Y as RealNormSpace , Y2 = X as Subset of X ;
let f be PartFunc of REAL , REAL , g be PartFunc of REAL , REAL ;
r * ( v1 |-- I ) . X < r * 1 ;
assume that V is Subspace of X and X is Subspace of V ;
t-3 , tt1 , tt1 , tt1 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , 6 , 6 , 6 , 6 , 8 , 8 , 8 , 8 , 8
Q [ e1 , v \/ { v } ] & f . v = [ v , v ] ;
g \circlearrowleft ( ( L~ z ) .. z ) = z ;
|. |[ x , v ]| - |[ x , v ]| .| = vv1 ;
- f . w = - ( L * w ) .= - ( L * w ) ;
z -' y <= x + y iff z <= x + y & z <= y + x ;
sqrt 7 + ( 1 + ( 1 - e ) * ( 1 - e ) ) > 0 ;
assume X is BCK-algebra & 0 < 0 implies X is BCK-algebra of 0 , X ;
F . 1 = v1 & F . 2 = v2 & F . 1 = v2 ;
( f | X ) . x2 = f . ( x2 - y2 ) ;
( ( ( tan * ( cos + cos * ( cos * ( cos * ( ( cos * ( ( cos * ( ( cos * ( ( cos * ( ( f + ( cos * ( ( f * ( f * ( f * ( f * ( ( f * (
i2 = ( f /. len f ) /. ( len f -' 1 ) .= f /. ( len f -' 1 ) ;
X1 = X2 \/ ( X1 union X2 ) .= X1 union X2 \/ X2 .= X1 union X2 ;
[. a , b , c .] = 1_ G & G * ( i , j ) = 1_ G ;
let V , W be non empty VectSpStr over K ;
dom ( g | the carrier of I[01] ) = the carrier of I[01] .= the carrier of I[01] ;
dom ( f2 | the carrier of I[01] ) = the carrier of I[01] .= the carrier of I[01] ;
( proj2 | X ) .: X = proj2 .: ( X /\ Y ) .= ( proj2 | X ) .: Y ;
f . ( x , y ) = h1 . ( x , y ) ;
x0 - a1 < a1 . n & x0 < a1 . n ;
|. ( f /* s ) . k - ( f /* s ) . k .| < r ;
len Line ( A , i ) = width A & width A = width B ;
SY. ^2 = ( S . g ) ^2 .= ( S . g ) ^2 ;
reconsider f = v + u as Function of X , the carrier of Y ;
( intloc 0 ) in dom Initialized ( p ) & ( Initialized ( p ) ) . 0 = p . 0 ;
i1 <> i2 & i2 <> j2 & j2 <> j2 & i1 <> i2 & i1 <> i2 & i2 <> j2 & j2 <> j2 & i1 <> i2 & i2 <> j2 & i1 <> i2 & i1 <> i2 & i2 <> j2 & i1 , i2 , j2 , j1 implies i1 , i2 , j1 , j2
sqrt r + ( ( cos r ) ^2 + ( cos r ) ^2 ) = sqrt ( ( cos r ) ^2 + ( cos r ) ^2 ) ;
for x st x in Z holds f2 * ( f1 + f2 ) is_differentiable_in x
reconsider q2 = sqrt ( q `2 ) / ( q `2 ) ^2 as Element of REAL ;
( 0 qua Nat ) + 1 <= i + j1 + 1 ;
assume f in the carrier of [ X , [#] Y :] ;
F . a = H / ( x , y ) . a ;
( true T ) div ( C , u ) = TRUE ;
dist ( ( a * seq ) . n , ( a * seq ) . n ) < r ;
1 in the carrier of [. 0 , 1 .] & 1 <= j & j <= 1 ;
( p2 - p1 ) `1 - p1 `1 > - p1 `1 + p1 `1 - p1 `1 ;
|. r1 - r1 .| = |. a1 - a2 .| * |. q1 - q2 .| ;
reconsider S-14 = 8 as Element of Seg 8 , X be non empty set ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b
DkW = DW .let ( n + 1 ) + 1 ;
i1 = ( ( a + n ) + n ) & i2 = ( a + n ) + ( n + 1 ) ;
f . a [= f . ( f .: O1 "\/" f .: ( f .: O1 ) ) ;
pred f = v & g = u u + v & f + g = v + u ;
I . n = Integral ( M , F | E ) | E ;
[: [: T , S :] . s = 1 & [: T , S :] . s = 1 ;
a = VERUM ( A ) or a = VERUM ( A ) or a = VERUM ( A ) ;
reconsider k2 = s . b2 - 1 as Element of NAT ;
( Comput ( P , s , 4 ) ) . DataLoc ( ( s + 4 ) + 2 ) = 0 ;
L~ M1 meets L~ M2 & L~ M2 /\ L~ M2 = { M1 } ;
set h = ( the continuous Function of X , R ) | X , f = ( the carrier of X ) | X ;
set A = { L . ( k9 . n ) where k is Element of NAT : P [ k ] } ;
for H st H is atomic holds P [ H ] ;
set b = [: S , i :] , c = [: S , i :] , d = [: S , i :] , e = [: S , i :] , f = [: S , i :] , g = [: S , j :] , g = [: S , j :] , g =
Hom ( a , b ) c= Hom ( a , b ) ;
sqrt ( 1 + ( n + 1 ) ) ^2 < sqrt ( 1 + ( n + 1 ) ) ^2 ;
( l , 1 ) `1 = [ dom l , cod l ] `1 .= [ dom l , cod l ] `1 ;
y +* ( i , y ) in dom g ;
let p be Element of CQC-WFF ( Al ) ;
X /\ X1 c= dom ( f1 - f2 ) /\ ( dom ( f2 - f1 ) ) ;
p2 in rng ( f /^ ( len f -' 1 ) ) ;
1 <= indx ( D2 , D1 , j1 ) + 1 ;
assume x in ( ( ( ( TOP-REAL 2 ) | K1 ) \/ ( ( TOP-REAL 2 ) | K1 ) \/ ( ( TOP-REAL 2 ) | K1 ) ) ;
- 1 <= ( ( f2 . O ) . O ) `2 & ( ( f2 . O ) . O ) `2 <= ( ( f2 . O ) `2 ) `2 ;
let f , g be Function of I[01] , TOP-REAL 2 , a , b , c , d be Real ;
k1 -' k2 = k1 - k2 + 1 .= k1 - k2 + 1 .= k1 - k2 + 1 ;
rng seq c= ]. x0 - r , x0 .[ & x0 < x0 + r ;
g2 in ]. x0 - r , x0 + r .[ & g2 < x0 + r ;
sgn ( p `1 , K ) = - ( - ( p `1 ) ) .= - ( - p `1 ) ;
consider u being Nat such that b = p |^ y * u ;
ex A being \rbrack or a = Sum A ;
card ( union the carrier of X ) = union ( ( the carrier of X ) \/ { x } ) .= card ( the carrier of X ) ;
len t = len t1 + len t2 .= len t1 + len t2 .= len t1 + len t2 ;
vx0 = v + w & Av + Aw = v + ( - w ) ;
v <> DataLoc ( t1 . DataLoc ( 0 , 3 ) , 2 ) ;
g . s = sup ( d " { s } ) .= s ;
( \dot y ) . s = s . ( \dot y ) ;
{ s : s < t & t < s } = [: Q , Q :] & t = {} ;
s ` \ s = s ` \ ( 0. X ) .= 0. X ;
defpred P [ Nat ] means B + $1 in A & $1 in B + $1 ;
( 329 + 1 ) ! = 3129 * ( 329 + 1 ) ;
U = [: A , A :] & U = [: A , A :] ;
reconsider y = y as Element of COMPLEX n -tuples_on COMPLEX ;
consider i2 being Integer such that y0 = p * i2 and i2 in A * ( i2 , j2 ) ;
reconsider p = Y | Seg k as FinSequence of ( the carrier of X ) \ { i } as FinSequence of NAT ;
set f = ( S , U ) \! \mathop { x } , f = S \! \mathop { x } , f = S \! \mathop { x } , f = S \! \mathop { x } , f = S \! \mathop { x } , f = S \! \mathop { x
consider Z being set such that lim s in Z and Z in F ;
let f be Function of I[01] , TOP-REAL n , R^1 , q be Point of TOP-REAL n ;
( ( M + i ) . [ n + i , 'not' A ] ) <> 1 ;
ex r being Real st x = r & a <= r & r <= b ;
R1 , R2 be Element of REAL n , a , b be Element of REAL n ;
reconsider l = 0. ( V ) as Linear_Combination of A ;
set r = |. e .| + |. w + v .| + a * v + a * v ;
consider y being Element of S such that z <= y and y in X ;
a 'imp' ( b 'or' c ) = 'not' ( ( a 'or' b ) 'or' c ) '&' ( a 'or' b ) '&' ( a 'or' c ) )
||. ( x9 - y9 ) * ( g - g2 ) .|| < r2 * ( g - g2 ) ;
b9 , a9 // b9 , c9 & b9 , c9 // b9 , c9 & b9 , c9 // a9 , c9 & b9 , c9 // c9 , a9 ;
1 <= k2 -' k1 & k2 + 1 + 1 = k2 -' k1 & k2 + 1 = k2 -' k1 + 1 ;
sqrt ( ( p `2 / |. p .| - sn ) / ( 1 - sn ) ) ^2 >= 0 ;
sqrt ( ( ( q `2 / |. q .| - sn ) / ( 1 - sn ) ) ^2 ) < 0 ;
E-max C in cell ( RR , 1 , 1 ) & W c= cell ( RR , 1 , 1 ) ;
consider e being Element of NAT such that a = 2 * e + 1 ;
Re ( ( lim F ) | D ) = Re ( ( lim G ) | D ) .= Re ( ( lim G ) | D ) ;
LIN b , a , c or LIN b , c , a ;
p `1 , a // a `1 , b `1 or p `1 , a // b , c ;
g . n = a * Sum ( f | n ) .= f . n * ( f | n ) ;
consider f being Subset of X such that e = f and f is elements of X ;
F | ( N2 , S ) = CircleMap * ( F | [: N2 , S :] ) .= ( F | [: N2 , S :] ) | [: N2 , S :] ;
q in LSeg ( q , v ) \/ LSeg ( v , p ) ;
Ball ( m , r ) c= Ball ( m , s ) ;
the carrier of V = { 0. V } .= { 0. V } ;
rng ( ( cos * ( cos ) ) `| REAL ) = [. - 1 , 1 .] ;
assume Re ( seq ) is summable & Im ( seq ) is summable ;
||. ( v . n - t ) * ( ( v . n - t ) * ( v . n ) ) .|| < e ;
set g = O --> 1 ;
reconsider t2 = t . {} as 0 -started string of S2 ;
reconsider x9 = seq . n as sequence of ( TOP-REAL n ) | ( the carrier of TOP-REAL n ) ;
assume that Index ( E-max C , n ) meets L~ go \/ L~ pion1 and p in L~ pion1 and p in L~ pion1 and p in L~ pion1 ;
- ( 1 / ( n + 1 ) ) < F . ( n + 1 ) - F . x ;
set d1 = \bf \bf ( ( x1 , y1 ) , z1 ) , d2 = ( x1 , y1 ) , d2 = ( x1 , y2 ) , d2 = ( x1 , y1 ) , d2 = ( x2 , y2 ) , d2 = ( x2 , y2 ) , d2 = ( x1 , y2 ) ,
2 |^ ( ( q -' 1 ) / ( q -' 1 ) ) = ( 2 |^ ( q -' 1 ) ) ;
dom ( v | ( Seg len v ) ) = Seg len ( v | ( Seg len v ) ) ;
set x1 = - ( k2 + 1 ) + ( k2 + 1 ) * ( k2 + 1 ) , x2 = ( k2 + 1 ) * ( k2 + 1 ) + ( k2 + 1 ) * ( k1 + 1 ) ;
assume for n being Element of X holds 0. <= F . n & 0. <= F . n ;
assume that 0 <= T-32 . i and T-32 . ( i + 1 ) <= 1 ;
for A being Subset of X holds c . ( c . A ) = c . A
the carrier of ( ( Carrier ( Lq ) + Carrier ( Lq ) ) ) c= [: I , { 0 } :] ;
'not' All ( x , p ) => All ( x , p ) is valid ;
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the normal Subgroup of A ;
Z c= dom ( ( ( ( ( ( ( ( ( exp_R * f ) ^ ) ) * ( ( exp_R * f ) ^ ) ) ) `| Z ) ) ) ;
|. 0. TOP-REAL 2 - ( 0. TOP-REAL 2 ) - ( 0. TOP-REAL 2 ) .| < r ;
ConsecutiveSet2 ( A , succ B ) c= ConsecutiveSet2 ( A , succ ( succ ( d , succ ( d , C ) ) ) ;
E = dom ( Carrier ( L ) ) & ( for x st x in E holds ( r (#) L ) . x = ( r (#) L ) . x ) ;
C |^ ( A + B ) = C |^ B * C |^ ( A + B ) ;
the carrier of W2 c= the carrier of V & the carrier of W1 c= the carrier of V ;
I . IC Comput ( P , s , 2 ) = P . IC Comput ( P , s , 2 ) ;
pred x > 0 means : Def1 : sqrt ( 1 - x ^2 ) = x ^2 - x ^2 ;
LSeg ( f ^ g , i ) = LSeg ( f , i ) \/ LSeg ( g , i ) ;
consider p being Point of T such that C = [. p , q .] and p in C ;
b , c are_connected & - C , - C are_connected & - C , - C are_connected ;
assume f = id ( the carrier of Y ) & f is continuous Function of Y , the carrier of Y ;
consider v such that v <> 0. V and f . v = L * v ;
let l be Linear_Combination of {} ( ( the carrier of V ) \ { 0. V } ) ;
reconsider g = f " ( ( f | U1 ) | U2 ) as Function of U1 , U2 ;
A1 : x in the carrier of ( ( G . k ) | X ) & ( G . k ) | X = G . ( k + 1 ) ;
|. - x .| = - ( x + - x ) .= - x + - x .= x + - x ;
set S = 1GateCircStr ( x , y , c ) ;
Fib ( n ) * ( 5 * ( 5 + 1 ) ) >= 4 * sqrt 5 ;
vv /. ( k + 1 ) = vv . ( k + 1 ) ;
0 mod i = sqrt ( ( i * 0 ) * ( i mod j ) ) ;
Indices M1 = [: Seg n , Seg n :] & width M1 = [: Seg n , Seg n :] ;
Line ( S , j ) . j = S . ( j , i ) .= S . j ;
h . ( x1 , y1 ) = [ y1 , y2 ] & h . ( y1 , y2 ) = [ y2 , y1 ] ;
|. f .| (#) Re ( ( |. b .| (#) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( b - b ) / ( b - a ) ) ) * h ) ) * h ) ) (#) h ) ) | A ) ) ) is nonnegative ;
assume x = ( a1 ^ <* b1 *> ) ^ ( a2 ^ <* b2 *> ) ^ ( b1 ^ <* b1 *> ) ;
MM is_closed_on IExec ( I , P , s ) , P ;
DataLoc ( t3 . a , 4 ) = intpos ( 0 + 4 ) .= 0 ;
x + y < - x + y & |. x - y .| = - x + y ;
LIN c , q `1 , b & LIN c , b , q ;
f\rangle . ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + y2 ) .= ( x1 + y1 ) + ( y2 + y2 ) ;
f[ f . a , v ] = [: f . a , f . b :] .= f . a ;
( p `1 ) ^2 <= ( E-max C ) ^2 + ( E-max C ) ^2 ;
set R8 = Cage ( C , n ) \circlearrowleft E-max L~ Cage ( C , n ) ;
( p `1 ) ^2 >= ( E-max C ) ^2 + ( E-max C ) ^2 ;
consider p such that p = p and s1 < p and p < s2 and p < s2 ;
|. ( f /* ( s * F ) ) . l - ( f /* ( s * F ) ) . l .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len Line ( N , k + 1 ) = width N & width N = width N ;
f1 /* s1 is convergent & f2 /* s1 is convergent & lim ( f1 /* s1 ) = lim ( f2 /* s1 ) ;
f . x1 = x1 & f . y1 = y1 & f . y2 = y2 ;
len f <= len f + 1 & len f + 1 <> 0 & len f + 1 <> 0 ;
dom ( Proj ( i , n ) * s ) = REAL m .= REAL m ;
n = k * ( 2 * t + ( n mod ( 2 * t ) ) ) + ( n mod ( 2 * t ) ) ;
dom B = 2 -tuples_on the carrier of V \ { {} } .= the carrier of V ;
consider r such that r , r _|_ a , x and r , r _|_ x , y ;
reconsider B1 = the carrier of X1 , B2 = the carrier of X2 as Subset of X ;
1 in the carrier of [. 1 / 2 , 1 .] & 1 / 2 <= 1 / 2 * ( 1 / 2 ) ;
for L being complete LATTICE holds lattice (# C , L #) , L #) is isomorphic
[ gi , gj ] in [: I , I :] \ { i } ;
set S2 = 1GateCircStr ( x , y , c ) ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and for x st x in dom f1 /\ dom f2 holds f2 . x = f2 . x ;
reconsider y = ( a ` ) / ( F ` ) as Element of L ;
dom s = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8
( min ( g , ( f , g ) ) . c ) <= h . c ;
set G2 = the subgraph of G , v = the Vertex of G , e = the Vertex of G , x = the Vertex of G ;
reconsider g = f as PartFunc of REAL , REAL-NS n , REAL-NS m , r ;
|. s1 . m - p . ( n + 1 ) .| < d / ( p . m ) ;
for x being element st x in B ( ) holds x in B ( ) & x in B ( )
P = the carrier of ( TOP-REAL n ) | P .= P ;
assume p1 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ;
( 0. X \ x ) * ( m + k ) = 0. X ;
let g be Element of Hom ( cod f , dom g ) ;
2 * a * b + ( 2 * c ) * ( a + b ) <= 2 * C1 + ( 2 * C2 ) * ( a + b ) ;
let f , g , h be PartFunc of X , Y , x be Point of X , Y ;
set h = Hom ( a , g ) , f = Hom ( a , b ) ;
then Seg ( n + 1 ) = Seg ( m + 1 ) & m <= n + 1 ;
H * ( g " * a ) in the carrier of H & ( g * a ) * ( g * a ) in the carrier of H ;
x in dom ( ( cos * ( sin + cos * ( cos * ( cos * ( cos * ( ( sin * ( ( cos * ( f + cos * ( f + cos * ( f + cos * ( ( f + cos * ( f + cos * ( f + ( f * ( f + ( f * ( f
cell ( G , i1 , j1 -' 1 ) misses C & C misses C & C misses L~ f ;
LE q2 , q2 , P , p1 , p2 & LE q2 , q2 , P , p1 , p2 ;
pred B is component means : Def1 : B c= BDD A & B c= BDD A ;
deffunc D ( set , set ) = union rng $2 & $2 = union rng $2 ;
n + - n < len ( p | ( n + 1 ) ) + n - n + 1 ;
pred a <> 0. K means : Def1 : width ( M * ( i , j ) ) = width ( M * ( i , j ) ) ;
consider j such that j in dom Seg len |^ J and I = Seg ( len J + 1 ) + j ;
consider x1 such that z in x1 and x1 in P and x2 in P and x1 in P . x1 ;
for n being Element of REAL ex r being Element of REAL st X [ n , r ]
set C1 = Comput ( P2 , s2 , i + 1 ) , C1 = Comput ( P2 , s2 , i + 1 ) , C1 = Comput ( P2 , s2 , i + 1 ) ;
set v = 3 / ( a , b ) , w = 3 / ( { a , b } , c ) , y = 2 / ( a , b ) , z = 2 / ( a , b ) , w = 2 / ( a , b ) , t = 2 / ( a , b ) ,
conv ( F .: W ) c= union ( F .: ( E .: W ) ) ;
1 in [. - 1 , 1 .] /\ dom ( ( ( #Z 2 ) * ( ( arccot ) * ( arccot ) ) ) ) ;
r3 <= s3 + ( r2 - ( r2 - ( r2 - ( r2 - r2 ) ) / 2 ) ) ;
dom ( f (#) ( ( f ^ ) | X ) ) = dom f /\ X .= dom ( f | X ) ;
dom ( f * G ) = dom ( l (#) F ) /\ Seg k .= Seg k ;
rng ( s ^\ k ) c= dom f1 /\ ( { x0 } \ { x0 } ) ;
reconsider g2 = gp as Point of ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( L~ g ) ) | ( ( L~ g ) /\ ( L~ g ) ) ;
( T * h ) . x = T . ( h . ( s . x ) ) ;
I . ( J . ( J . x ) ) = ( I * L ) . ( J . x ) ;
y in dom ( ( Frege ( A * ( ( Frege ( A * ( ( Frege ( A * ( ( Frege ( A * ( ( A * ( A * ( o * ( A * ( A * ( A * ( A * ( A * ( A * ( A * ( A * ( A * ( A * ( A
for I being non degenerated commutative Ring holds I is commutative
set s2 = s +* Initialize ( ( intloc 0 ) .--> 1 ) , P2 = P +* Initialize ( ( intloc 0 ) .--> 1 ) ;
P1 /. IC Comput ( P1 , s1 , i + 1 ) = P1 . IC Comput ( P2 , s2 , i + 1 ) .= P1 . IC Comput ( P2 , s2 , i + 1 ) ;
lim S1 in the carrier of \lbrack a , b .] & lim S1 = a & lim S1 = b ;
v . ( l . i ) = ( v *' l ) . i .= ( v *' l ) . i ;
consider n being element such that n in NAT and x = seq . n and n in NAT and x = seq . n ;
consider x being Element of c such that F1 . x <> F2 . x and F1 . x <> F2 . x ;
cluster cluster cluster cluster cluster cluster cluster cluster cluster cluster cluster cluster cluster cluster 0 , 0 , 1 , 2 , 1 , 2 , 3 , 4 , 5 , 7 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 9 , 8 , 7 , 8 , 8 , 8 ,
j + ( 2 * ( k2 + 1 ) ) > j + ( 2 * ( k2 + 1 ) ) ;
{ s , s } on A2 & { s , t } on B2 & { s , t } on B2 ;
n1 > len crossover ( p2 , p1 , p2 , p1 , p2 , p1 , p2 , p1 , p2 , p1 , p2 , p1 , p2 , p1 , p2 , p1 , p2 , p1 , p2 , p1 , p2 , p2 , p1 , p2 , p1 , p2 , p2 , p1 , p2 , p1 , p2 , p1
( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
then H1 , H2 are_Subgroup of H1 & ( the carrier of H1 ) , ( the carrier of H2 ) , ( the carrier of H2 ) , ( the carrier of H1 ) is Subgroup of H1 ;
( ( ( GoB f ) * ( 1 , 1 ) ) `1 ) `1 > 1 & ( ( GoB f ) * ( 1 , 1 ) ) `1 > 1 ;
]. s , 1 .[ = ]. s , 2 .[ /\ [. 0 , 1 .] .= [. s , 1 .] ;
x1 in [#] ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( L~ g ) ) | ( ( L~ g ) | ( L~ g ) ) ) ;
let f1 , f2 be PartFunc of REAL , REAL , x0 be Point of S , r be Real ;
DigA ( ti1 , z ) is Element of k -tuples_on ( k -tuples_on the carrier of X ) ;
I = d1 & I = d2 & I = d2 & I = d2 & I = d2 & I = d2 ;
u9 `1 = { [ a , u ] , [ b , v ] } .= [ a , b ] `1 ;
( w | p ) | ( p | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w
consider u2 such that u2 in W2 and x = v + u and u1 in W2 and u2 in W1 + W2 ;
for y st y in rng F ex n st y = a |^ n & n in dom F ;
dom ( ( g * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( M M of R ) ) ) ) ) | K ) ) ) | K ) ) ) ) ) | K ) ) ) = K ;
ex x being element st x in ( [#] U0 ) \/ A . s & x in ( the carrier of U0 ) ;
ex x being element st x in ( ( ( the Sorts of U1 ) * A ) . s ) . s & ( the Sorts of U1 ) . s = ( the Sorts of U1 ) . s ;
f . x in the carrier of [. - r , r .[ & f . x in [. - r , - r .[ ;
( the carrier of X1 union X2 ) /\ ( the carrier of X2 ) <> {} implies X1 union X2 /\ ( the carrier of X2 ) <> {}
L1 /\ LSeg ( p1 , p2 ) c= { p1 : p1 in LSeg ( p1 , p2 ) } ;
sqrt ( b + ( bs ) ^2 ) in { r : a < r & r < b } ;
ex_sup_of { x , y } , L & x "\/" y = sup { x , y } ;
for x being element st x in X ex u being element st P [ x , u ]
consider z being Point of G8 such that z = y and P [ z ] and P [ z ] ;
( the Lipschitzian of ( bounded ) ) . ( ( bounded ) * ( ( bounded ) * ( ( bounded ) * ( ( bounded ) * ( ( bounded ) * ( ( bounded ) * ( ( bounded ) ) * ( ( bounded ) * ( ( bounded ) * ( ( bounded ) * ( ( bounded ) * ( ( bounded
len ( w ^ ( w2 ^ ( w1 ^ w2 ) ) ) = len w + ( len w1 + 1 ) ;
assume q in the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 & q = ( ( TOP-REAL 2 ) | K1 ) | K1 ;
f | E-4 = g | E-4 .= g | EK .= g | EK .= g | EK ;
reconsider i1 = x1 , i2 = x2 , z = y2 as Element of ( the carrier of X ) * ;
( a * A ) * B = ( a * ( A * B ) ) * ( a * B ) ;
assume ex n1 being Element of NAT st f |^ n1 is sequence of X & f . n1 is convergent & lim f = x0 ;
Seg len ( ( ( f | ( len f ) ) ) | ( len f ) ) = dom ( ( f | ( Seg len f ) ) ) .= Seg len ( ( f | ( Seg len f ) ) ) ;
( Complement ( A1 ) ) . m c= ( Complement ( A1 ) ) . n ;
f1 . p = p1 & g1 . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . (
FinS ( F , Y ) = FinS ( F , Y ) .= FinS ( F , Y ) ;
( x | y ) | z = z | ( y | x ) ;
sqrt ( ( |. x .| ^2 ) ^2 + ( |. x .| ^2 ) <= sqrt ( ( |. x .| ^2 ) ^2 ) ;
Sum ( F ) = Sum f & dom ( F | ( len F ) ) = dom ( F | ( len F ) ) ;
assume for x , y being set st x in Y & y in Y holds x /\ y in Y ;
assume that W1 is Subspace of W2 and W2 is Subspace of W1 and W1 is Subspace of W2 ;
||. ( t . x ) - ( t . x ) .|| = lim ( ( t . x ) - ( t . x ) ) + ( t . x ) ;
assume that i in dom D and f | A is bounded and g | A is bounded ;
sqrt ( ( p `2 ) ^2 + ( p `2 ) ^2 ) <= sqrt ( ( p `2 ) ^2 + ( p `2 ) ^2 ) ;
g | Ball ( p , r ) = id ( Ball ( p , r ) ) .= id ( Ball ( p , r ) ) ;
set Nmin = ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ;
for T being non empty TopSpace holds T is with_countable implies the TopStruct of T is countable
width B |-> 0. ( K , i ) = Line ( B , i ) .= width B ;
pred a <> 0 means : Def1 : ( A -- B ) = ( A -- B ) -- ( A -- a ) ;
then f is_partial u0 , u & pdiff1 ( f , u ) is_differentiable_in u & pdiff1 ( f , u ) is_differentiable_in u ;
assume that a > 0 and a <> 1 and b <> 0 and c <> 0 and a <> 0 and c <> 0 ;
w1 , w2 , w1 , w2 , w1 , w2 , w1 , w2 , w1 , w2 , w1 , w1 , w2 , w1 , w2 , w1 , w2 , w1 , w1 , w2 , w1 , w1 , w2 , w1 , w1 , w2 , w1 , w1 , w2 , w1 , w1 , w2 , w1 , w1 , w1 , w2 , w1 , w1
p2 /. IC Comput ( p2 , s2 , k ) = p2 . IC Comput ( p2 , s2 , k ) .= Exec ( I , Comput ( p2 , s2 , k ) , Comput ( p2 , s2 , k ) ) ;
ind ( T | b ) = ind b .= ind B .= ind B .= ind B ;
[ a , A ] in the topology of ( ( ( 2 ) * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 *
m in ( the Arrows of C ) . ( o1 , o2 ) & m in ( the Arrows of C ) . ( o1 , o2 ) ;
( ( Y. ( PA , CompF ( PA , G ) , G ) ) . z ) . z = TRUE ;
reconsider phi = phi /. 11 , phi = phi /. ( 11 + 1 ) as Element of phi ;
len s1 - ( len s2 - 1 ) > 0 + 1 - 1 ;
\delta ( D * ( f . ( upper_bound A ) - f . ( lower_bound A ) ) ) < r ;
[ f9 , f9 ] in the carrier of A & f9 , f ] in the carrier of A ;
the carrier of ( ( TOP-REAL 2 ) | K1 ) = ( ( TOP-REAL 2 ) | K1 ) | K1 .= K1 .= K1 ;
consider z being element such that z in dom g2 and p = g2 . z and q = g2 . z ;
[#] ( V1 ) = { 0. ( V1 ) } .= the carrier of ( V1 ) /\ V1 .= the carrier of ( V1 ) ;
consider P2 be FinSequence of NAT such that rng P2 = M and P2 is one-to-one and P2 is one-to-one and P2 is one-to-one and P2 is one-to-one ;
assume that x1 in dom ( f | X ) and ||. x1 - x0 .|| < s and ||. x1 - x0 .|| < s ;
h1 = f ^ ( <* p3 *> ^ <* p3 *> ) .= h ^ ( <* p3 *> ^ <* p3 *> ) .= h ;
c /. |[ b , c ]| = c /. |[ a , c ]| .= c /. ( |[ a , c ]| ) .= c /. ( |[ b , c ]| ) ;
reconsider t1 = p1 , t2 = p2 as term of C , a = p . ( i + 1 ) as term of C ;
sqrt ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 / 2 ) ) ) ^2 ) ) ^2 ) in the carrier of ( ( ( 1 - ( 1 / 2 ) ) ^2 ) ) ;
ex W being Subset of X st p in W & W is open & h .: W c= V & h .: W c= V ;
( h . p1 ) `2 = C * ( p1 `2 ) + D * ( p1 `2 ) + D * ( p1 `2 ) ;
R . b ` = 2 * Re b .= 2 * PI + 2 * PI .= 2 * PI ;
consider y2 such that B = ( - 1 ) * C + ( - 1 ) * C and 0 <= y2 & y2 <= 1 ;
dom g = dom ( ( the Sorts of A ) * ( ( the Sorts of A ) * ( the_arity_of o ) ) ) ;
[ P . l , P . l ] in => ( T . l , T . l ) ;
set s2 = Initialize s , P2 = P +* I ;
reconsider M = mid ( z , i2 , i1 ) , i1 = mid ( z , i1 , j1 ) , i2 = ( i1 + i2 ) + ( i1 + i2 ) , i2 = ( i1 + i2 ) + ( i1 + i2 ) as Element of REAL n ;
y in product ( ( Carrier J ) +* ( V , { 1 } ) ) ;
1 / ( |[ 0 , 1 ]| ) = 1 & 0 <= ( |[ 0 , 1 ]| ) / ( |[ 0 , 1 ]| ) = 0 ;
assume x in the left of g or x in the carrier of ( g * f ) \/ the carrier of ( g * f ) ;
consider M being strict Subgroup of A such that a = M and T is Subspace of M and M is Subspace of M ;
for x st x in Z holds ( ( ( ( ( ( exp_R + ( exp_R + exp_R ) ) * f ) `| Z ) ) . x ) <> 0
len W1 + len W2 = 1 + len W2 + len W1 + len W2 .= len W1 + len W2 + len W1 + len W2 ;
reconsider h1 = ( v - t ) . n - t . n as VECTOR of X , Y ;
( Y. + len ( p + q ) ) + 1 in dom ( p + q ) ;
assume that s2 is conjunctive and F in the carrier of ( ( TOP-REAL 2 ) | ( the carrier of 2 ) ) and F is U and F is U ;
( ( ( gcd ( x , y ) ) ) * ( ( gcd ( x , y ) ) * ( ( x , y ) * ( ( x , y ) * ( ( x , y ) * ( ( x , y ) * ( ( x , y ) * ( ( y , z ) * ( ( x , z ) * ( ( y ,
for u being element st u in Bags n holds ( p + m ) . u = p . u + m . u
for B being Subset of u st B in E holds A = B or A misses B or B misses C or B misses C ;
ex a being Point of X st a in A & A /\ Cl { y } = { a } ;
set W2 = [: p , W1 :] \/ [: { p } , { q } :] ;
x in { X where X is Ideal of L : X in I & X in J } ;
the carrier of W1 /\ W2 c= the carrier of W1 + W2 & the carrier of W1 c= the carrier of W2 + W2
( ( 1 + b ) * id a ) * id a = ( 1 - b ) * id a .= ( 1 - b ) * id a ;
( dom ( X --> f ) ) . x = ( X --> f ) . x .= ( X --> f ) . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) , y = LSeg ( g , n ) /\ LSeg ( g , m ) ;
p => ( q => r ) => ( p => ( q => r ) ) in TAUT ( A ) ;
set cos = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
set cos = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
- 1 + 1 <= sqrt ( ( E-bound / 2 ) ^2 + ( m + 1 ) ^2 ) + 1 ;
( reproj ( 1 , z0 ) ) . x in dom ( ( f1 (#) ( f2 * ( f1 + f2 ) ) ) ;
assume that b1 . r = { c1 . r } and b2 . r = { c2 . r } and b1 . r = c2 . r ;
ex P st a1 on P & a2 on P & a1 on P & a2 on P & a1 on P & a2 on P & a1 on P & a2 on P & a1 on P & a2 on P & a1 on P & a2 on P & a2 on P & a1 , a2 on P & a2 , c on P & a1 , a2 on P & a1
reconsider gf = g `1 * f as strict Subgroup of X , Y ;
consider v1 being Element of T such that Q = ( downarrow v1 ) ` and v1 in ( downarrow v2 ) ` and v1 in ( downarrow v2 ) ` ;
n in { i where i is Nat : i < n + 1 & n < i + 1 } ;
( F /. ( i , j ) ) `2 >= ( F /. ( m , k ) ) `2 ;
assume K1 = { p : ( p `1 / |. p .| - sn ) / ( 1 + sn ) >= sn & ( p `2 / |. p .| - sn ) / ( 1 + sn ) <= sn & ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) / ( 1 + sn ) <= sn ;
ConsecutiveSet2 ( A , succ O1 ) = ( ConsecutiveSet2 ( A , O1 ) ) .: ( ( ConsecutiveSet2 ( A , O1 ) ) .: ( ( ConsecutiveSet2 ( A , O1 ) ) .: ( ( ConsecutiveSet2 ( A , O1 ) ) .: ( ( ConsecutiveSet2 ( A , O1 ) ) ) .: ( ( ConsecutiveSet2 ( A , O1 ) ) .: ( ( ConsecutiveSet2 ( A , O1 ) ) .: ( ( ConsecutiveSet2 ( A , O1 ) ) .: (
set I1 = Macro ( a , intloc 0 ) , I2 = Macro ( a , intloc 0 ) , I2 = Macro ( a , intloc 0 ) , I2 = P +* stop I , I2 = P +* stop J , I2 = P +* stop J , P3 = P +* stop J , s3 = P +* stop 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 ) & X c= the carrier of L1 & X c= the carrier of L1 & X c= the carrier of L1 ;
consider x9 being Element of GF ( p ) such that x9 |^ 2 = a |^ ( p |^ 2 ) and x9 in X ;
reconsider e1 = e1 , e2 = f . ( e , f . ( e , f . ( e , f . ( e , f . ( e , f . ( e , f . ( e , f . ( e , f . ( e , f . ( e , f . ( e , f . ( e , f . ( e , f . ( e , f . ( e , f . ( e , f .
ex O being set st O in S & C1 c= O & M . O = 0. ( X , Y ) ;
consider n be Nat such that for m be Nat st n <= m holds S . m in U1 ( n , m ) ;
f * g * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) * f ) . x ;
defpred P [ Nat ] means A + succ $1 = succ $1 + A & A = succ $1 + A ;
the left of ( - g ) = the left of ( - g ) .= ( - g ) * ( ( - g ) * ( ( - g ) * ( ( - g ) * ( ( - g ) * ( ( - g ) * ( ( - g ) * ( ( - g ) * ( ( - g ) * ( ( - g ) * ( ( - g ) * ( ( - g )
reconsider pp = x , pp = y as Point of TOP-REAL 2 , p = p , q = q , r = p , s = q , t = p , t = q , s = q , t = p , s = q , t = p , s = q , t = q , s = q , t = p , s = q , t = q , s = q , t = q ,
consider g2 such that g2 = y and x <= g2 and g2 <= x0 and x0 < g2 and g2 < g2 and g2 < x0 and g2 in dom f ;
for n being Element of NAT ex r being Element of REAL st X [ n , r ]
len ( x2 ^ y2 ) = len ( x2 ^ y2 ) + len ( y2 ^ y1 ) .= len ( x2 ^ y2 ) + len y2 ;
for x being element st x in X holds x in the set of ( n , m ) --> ( n , m )
LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = {} .= { p1 } /\ LSeg ( p1 , p2 ) .= { p1 } ;
func product ( X , Y ) -> set equals [: X , Y :] & [: X , Y :] c= [: X , Y :] ;
len ( ( Cage ( C , n ) /. 1 ) | ( len ( Cage ( C , n ) ) ) ) <= len ( ( Cage ( C , n ) /. ( len ( Cage ( C , n ) ) ) ) ;
pred K is every Field means a <> 0. K & v . ( a |^ i ) = i * v . ( a |^ i ) ;
consider o being OperSymbol of S such that t . {} = [ o , the carrier of S ] and o in the carrier' of S ;
for x st x in X ex y st x c= y & y in X & x in X & y in Y holds y in f . x
IC Comput ( P1 , s , k ) in dom ( Comput ( P2 , s2 , k ) , ( k + 1 ) ) ;
pred q < s & r < s implies ]. r , s .[ c= ]. p , q .[ \/ ]. p , q .[ ;
consider c being Element of Class ( f , c ) such that Y = ( F . c ) . ( ( F . c ) . ( c , c ) ) ;
func ( the ResultSort of S2 ) * ( id the carrier of S2 ) -> Function of the carrier of S2 , the carrier of S2 ;
set y9 = [ <* y , z *> , f2 ] , y2 = [ <* y , z *> , f2 ] , z2 = [ <* z , x *> , f3 ] ;
assume x in dom ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
r2 in Int cell ( f , i , j ) \ ( ( GoB f ) * ( i , j ) + ( GoB f ) * ( i , j ) ) ;
( q `2 ) ^2 >= ( ( Cage ( C , n ) /. ( i + 1 ) ) `2 ;
set Y = { a "/\" a : a in X } , Z = { a "/\" b : b in X } ;
i -' len f <= len f + ( len f -' 1 ) - len f + 1 - len f ;
for n holds x in N & x in N1 & x in N & h . n = - ( x0 + h )
set s0 = ( > ( a , I , p ) , s ) . i , s1 = ( a , I ) . i , s2 = ( a , I ) . i , s2 = ( a , I ) . i , s2 = ( a , I ) . i , s2 = ( a , I ) . i , s2 = ( a , I ) . i , s2 = ( a
( p . k ) . 0 = 1 or ( p . k ) . 0 = 1 or ( p . k ) . 0 = 1 or ( p . k ) . 0 = 1 ;
u + Sum ( L-18 ) in ( U \ { u + Sum ( L-18 ) } ) \/ { u + Sum ( L-18 ) } ;
consider x9 being set such that x in x9 and x9 in V and x9 in V and x = [ x9 , y9 ] ;
( p ^ q ) . m = ( q | k ) . ( ( q | k ) . ( m + 1 ) ) .= ( q | ( m + 1 ) ) . ( ( q | ( m + 1 ) ) . ( m + 1 ) ) ;
g + h = gg + h + { h } & f + g = g + h + { h + c } ;
L1 is distributive & L2 is distributive implies L1 "\/" L2 is distributive & L1 "\/" L2 is distributive & L1 "\/" L2 = L1 & L2 "\/" L2 = L2
pred x in rng f & y in rng ( f | x ) & f | ( x .. f ) = f | ( x .. f ) ;
assume that 1 < p and p + sqrt ( 1 - q ) = 1 and 0 <= a and a <= b and b <= c ;
F*' ( f , <* H *> *' t ) = rpoly ( 1 , H ) *' t + t *' ( <* H *> *' t ) .= 1 ;
for X being set , A being Subset of X holds A ` = {} implies A = {} implies A = {} & A = {}
( ( ( ( ( ( ( ( ( ( ( ( ( X X X ) ) ) ) ) ) ) ) `1 ) / ( ( ( ( ( ( ( ( X ) ) `1 ) / ( ( X ) ) `1 ) ^2 ) ) ^2 ) ) ) ^2 ) ) ^2 <= ( ( ( ( ( X ) / ( ( Y ) / ( X ) ) ^2 ) ) ^2 ;
for c being Element of the bound of A , a being Element of the bound of A holds c <> a
s1 . DataLoc ( s2 . a , i ) = ( Exec ( i2 , s2 ) ) . intpos ( i + 1 ) .= Exec ( i2 , s2 ) . intpos ( i + 1 ) .= Exec ( i2 , s2 ) . intpos ( i + 1 ) ;
for a , b being Real holds [ a , b ] in ( y iff b >= 0 ) implies b >= 0
for x , y being Element of X holds x ` \ y = ( x \ y ) ` & y = ( x \ y ) ` \ ( y \ x ) `
mode BCK-algebra of i , j , m , n , m , k be Nat holds m * ( i , j ) = m * ( i , j )
set x2 = ( Re ( y ) ) | ( ( Re ( y ) ) | ( ( Im ( y ) ) | ( ( Im ( y ) ) ) | ( ( Im ( y ) ) ) | ( ( Im ( y ) ) | ( ( Im ( y ) ) | ( ( Im ( y ) ) ) ) ) ;
[ y , x ] in dom ( u | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | (
]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .] c= A ;
0 <= \delta ( S2 . n ) & |. \delta ( S2 . n ) - 0 .| < e / 2 ;
( - ( q `2 / |. q .| - sn ) ) ^2 <= ( ( q `2 / |. q .| - sn ) ) ^2 * ( ( q `2 / |. q .| - sn ) ) ^2 ;
set A = sqrt 2 , B = sqrt 2 , C = sqrt 2 , D = sqrt 2 , E = sqrt 2 , A = sqrt 3 , B = sqrt 3 , C = sqrt 5 , D = / 2 , E = sqrt 5 , N = / 2 , N = sqrt 5 , N = / 2 , N = / 2 , N = / 2 , N = / 2 , N = 2 / 2
for x , y being set st x , y ] in RO holds x , y are_\hbox { x , y }
deffunc F ( Nat ) = b . ( $1 * M ) * ( M * G ) . $1 * ( M * G ) . $1 * ( M * G ) . $1 * ( M * G ) . $1 * ( M * G ) . $1 * ( M * G ) . $1 * ( M * G ) . $1 * ( M * G ) . $1 * ( M * G ) . $1 *
for s being element holds s in ( ( f 'or' g ) | ( X \/ Y ) ) iff s in ( f \/ g ) | ( X \/ Y )
for S being non empty non void non void non empty non void ManySortedSign st S is connected holds S is connected
max ( ( degree ( z ) ) , ( degree ( z ) ) ) / ( ( degree ( z ) ) ^2 + ( degree ( z ) ) ^2 ) >= 0 ;
consider n1 be Nat such that for k holds seq . ( n + k ) < r + s / 2 ;
Lin ( A /\ B ) is Subspace of Lin ( A ) & Lin ( B ) is Subspace of Lin ( B ) ;
set nnw = nnv , M = ( M . x ) `2 , nv = ( M . x ) `2 , nv = ( M . x ) `2 , nv = ( M . x ) `2 , nv = ( M . x ) `2 , nv = ( M . x ) `2 , nv = ( M . x ) `2 , nv = ( M .
f " V in ( ( X , p ) --> ( X , p ) ) & f " V in ( ( X , p ) --> ( X , p ) ) . ( X , p ) ;
rng ( ( a , c ) +* ( 1 , b ) ) c= { a , c } \/ { a , b } ;
consider y being Vertex of G1 such that y `1 = y and dom y = { y } and y `2 = WG1 ;
dom ( ( 1 - ( f . x ) ) (#) ( f . x ) ) c= ]. - ( f . x ) , ( f . x ) * ( f . x ) * ( f . x ) ) ;
( for i , j , n , m , n , m , n , m ) is Element of TOP-REAL n & m <> 0 implies ( m * n , n ) * ( m , n ) = ( m * n ) * ( m , n )
v ^ ( n |-> 0 ) in Lin ( ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B ) ) ) ) ) )
ex a , k1 , k2 st i = a := k1 & ( i = a := k1 ) & ( i = a := k1 ) & ( i = a := k1 ) & ( i = a := k1 ) & i = b implies i = b ) ;
t . ( [: NAT , NAT :] ) = ( ( ( ( the carrier of A ) --> ( the carrier of A ) --> ( the carrier of A ) ) ) . ( t . ( t . ( t . ( t . ( t . ( t . ( t ) ) ) ) ) ) .= ( t . ( t . ( t . ( t . ( t . ( t . ( t . ( t .
assume that F is bbSubset-Family and rng p = F and rng p = Seg ( n + 1 ) and p is one-to-one ;
not LIN b , b9 , a & not LIN b , c , a & not LIN c , a , b , c
( L1 or L2 ) . O c= ( L1 L1 L1 ) . O & ( L2 L1 ) . O = ( L1 L1 ) . O ;
consider F be ManySortedSet of E such that for d being Element of E holds F . d = F ( d ) ;
consider a , b such that a * ( u + b ) = b * ( -w ) and 0 < a and b < b and 0 < a and a < b ;
defpred P [ FinSequence of D ] means |. Sum ( $1 ) .| <= Sum ( |. $1 .| ) + Sum ( |. $1 .| ) ;
u = cos ( x , y ) * x + cos ( x , y ) * y .= v . ( x , y ) * y + cos ( y , v ) * y .= v . ( x , y ) * v ;
dist ( seq . n + x , g + x ) <= dist ( seq . n , g ) + 0 ;
P [ p , |. p .| : p in [: the carrier of A , the carrier of A :] & q in the carrier of A } ;
consider X being Subset of CQC-WFF ( Al ) such that X c= Y and X is finite and X is ininof X ;
|. b .| * |. eval ( f , z ) .| >= |. b .| * |. eval ( f , z ) .| ;
1 < ( ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ) .. Cage ( C , n ) ;
l in { l1 where l1 is Real : g <= l1 & l <= h . l1 & l <= h . l1 & l <= h . l } ;
( Partial_Sums ( G . n ) ) . ( ( G . n ) . ( ( G . n ) . ( n + 1 ) ) ) <= ( Partial_Sums ( G ) ) . ( ( G . n ) . ( ( G . n ) . ( ( G . n ) . ( n + 1 ) ) ) ;
f . y = x * ( x * y ) .= x * ( ( 0. L ) * y ) .= x * ( ( 0. L ) * y ) .= x * ( y * y ) ;
NIC ( i1 , i1 ) = { i1 , i2 , j2 } & { i1 , i2 , j2 } = { i1 , i2 , j2 } & { i1 , i2 , j2 } = { i1 , i2 , j2 } ;
LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = { p1 , p2 } .= { p1 , p2 } ;
Product ( ( ( Carrier ( I1 ) ) +* ( i , { 1 } ) ) ) in Z & ( ( Carrier ( I1 ) ) +* ( i , { 1 } ) ) . x in Z ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) +* ( the carrier of S2 ) .= Following ( s1 , n ) ;
( W-min ( Q ) ) `1 <= ( ( q `2 ) / ( |. q .| ) ) ^2 & ( q `2 ) ^2 <= ( ( q `2 ) / ( |. q .| ) ) ^2 ;
f /. i2 <> f /. ( ( i1 + len f -' 1 ) -' 1 ) & f /. ( ( i1 + len f -' 1 ) -' 1 ) = f /. ( ( i1 + len f -' 1 ) -' 1 ) ;
M , f / ( ( ( x , y ) / ( ( x , y ) / ( x , y ) ) / ( ( x , y ) / ( x , y ) ) / ( x , y ) ) / ( x , y ) / ( x , y ) ) / ( x , y ) / ( x , y ) / ( x , y ) ) |= H ;
len ( ( P ^ Q ) ^ ( P ^ Q ) ) in dom ( ( P ^ Q ) ^ ( P ^ Q ) ) ;
A |^ ( m , n ) c= A |^ ( m , n ) & A |^ ( m , n ) c= A |^ ( m , n ) ;
( ( TOP-REAL n ) \ { q : |. q .| < a } ) c= { q1 : |. q1 .| >= a }
consider n1 being element such that n1 in dom p1 and y1 = p1 . n1 and p1 . n1 = p2 . n1 and p1 . n1 = p2 . n1 and p1 . n1 = p2 . n1 ;
consider X being set such that X in Q and for Z being set st Z in Q & Z <> X holds X c= Z ;
CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA .= halt SCM+FSA .= halt SCM+FSA ;
for v being VECTOR of l1 holds ||. v .|| = upper_bound ( ( ||. v .|| ) | ( ||. v .|| ) ) & ||. v .|| = ||. v .|| * ||. v .||
for phi holds phi in X implies phi in X & phi in X & phi in X & phi in X & phi in X
rng ( ( Sgm dom ( ( f | ( dom ( g | ( dom g ) ) ) ) | ( dom ( ( g | ( dom ( g | ( dom ( g | ( dom ( g | ( dom ( g | ( dom g ) ) ) ) ) ) ) ) ) ) ) c= dom ( ( g | ( dom ( g | ( dom ( g | ( dom ( g | ( dom ( g | ( dom g ) ) ) )
ex c being FinSequence of D st len c = k & for n st n in dom c holds P [ n , c . n ] ;
the_arity_of ( a , b ) = <* Hom ( b , c ) , Hom ( c , d ) *> .= <* Hom ( b , c ) , Hom ( c , d ) *> ;
consider f1 be Function of the carrier of X , REAL such that f1 = |. f .| and f1 | X is continuous and f1 | X is continuous ;
a1 = b1 & a2 = b2 & a1 = b2 & a2 = b1 & b2 = b2 & b1 = b2 & b2 = b3 implies a1 = b1 & a2 = b2 & b2 = b3 & b1 = b3 & b2 = b3 & b1 = b3 & b2 = b3 & b2 = b3 & b1 = b3 & b2 = b3
D2 . indx ( D2 , D1 , n1 ) = D1 . indx ( D2 , D1 , n1 ) .= D1 . indx ( D2 , D1 , n1 ) ;
f . ( ||. r .|| ) = ||. ( r - r ) .|| .= ||. r .|| .= ||. r .|| .= ||. r .|| ;
consider n be Nat such that for m be Nat st n <= m holds Cseq . m = Cseq . ( m + n ) ;
consider d being Real such that for a , b being Real st a in X & b in Y holds a <= b + d ;
||. L /. h - ( K * |. h .| ) + ( K * |. h .| ) + ( K * |. h .| ) + ( K * |. h .| ) <= x0 + ( K * |. h .| ) + ( K * |. h .| ) ;
attr F is commutative means : Def1 : F \hbox { b } = f . b & F is commutative & F is commutative & F is commutative ;
p = *> * ( p1 + 0. TOP-REAL 2 ) .= 1 * p1 + 0. TOP-REAL 2 .= ( p1 + p2 ) * p1 + 0. TOP-REAL 2 .= ( p1 + p2 ) * p1 + 0. TOP-REAL 2 * p2 + 0. TOP-REAL 2 * p2 .= ( p1 + p2 ) * p2 + 0. TOP-REAL 2 ;
consider z1 such that b , x1 , x3 , x4 , z1 , z2 is_collinear and ( o , z1 , z2 is_collinear ) & ( o , z1 , z2 is_collinear & ( o , z1 , z2 is_collinear & ( o , z1 , z2 is_collinear & ( o , z1 , z2 is_collinear & ( o , z1 , z2 is_collinear & ( o , z1 , z2 is_collinear & o , z1 , z2 is_collinear ) & ( o , z1 , z2 is_collinear ) & ( o
consider i such that Arg ( ( Rotate ( s ) ) . q ) = s + ( 2 * PI * i ) + ( 2 * PI * i ) ;
consider g such that g is one-to-one and dom g = card f and rng g c= f . x and g is one-to-one and g is one-to-one and g is one-to-one and g is one-to-one and g is one-to-one and g is one-to-one and g is one-to-one and g is one-to-one and g is one-to-one and g is one-to-one and g is one-to-one ;
assume that A = P2 \/ P2 and ( for i st i in dom P2 holds P2 . i misses P2 . i ) and ( for i st i in dom P2 holds P2 . i misses P2 . i ) & ( for i st i in dom P2 holds P2 . i misses P2 . i ) ;
attr F is associative means : Def1 : F .: ( F .: ( f , g ) , h ) = F .: ( f , g ) ;
ex x being Element of NAT st m = x `1 & x in z `1 & m < n & n < m & m < n + 1 & n + 1 <= m + 1 & m + 1 <= n + 1 ;
consider k2 being Nat such that k2 in dom ( P . ( k2 + 1 ) ) and l in P . ( k2 + 1 ) and l = P . ( k2 + 1 ) ;
seq = r * seq implies for n holds seq . n = r * seq . ( n + 1 ) & seq is convergent & lim seq = r * seq . ( n + 1 )
F1 . [ id a , id a ] = f * ( id a , id a ) .= f * ( id a ) .= f * ( id a ) ;
{ p } "\/" D2 = { p "\/" y where y is Element of L : y in D & p in D } ;
consider z being element such that z in dom ( ( dom F ) | ( dom F ) ) and ( ( F | ( dom F ) ) ) . z = y ;
for x , y being element st x in dom f & y in dom f & f . x = f . y holds x = y
cell ( G , i , 1 ) = { |[ r , s ]| : r <= G * ( 0 + 1 , 1 ) `1 & G * ( 1 , 1 ) `2 <= s } ;
consider e being element such that e in dom ( T | ( E , X ) ) and ( T | ( E , X ) ) . e = v ;
( F `1 * b1 ) . x = ( ( ( Mx2Tran ( J , b2 ) ) . x ) * ( ( ( ( J , b2 ) . J ) . J ) * ( ( ( J , b2 ) . J ) * ( ( B , b2 ) . j ) ) ) ;
- 1 / ( 1 / D ) = ( ( 1 / D ) (#) D ) (#) ( ( 1 / D ) (#) ( ( 1 / D ) (#) ( ( 1 / D ) (#) ( ( 1 / D ) (#) ( ( 1 / D ) (#) ( ( 1 / D ) (#) ( ( 1 / D ) (#) ( ( 1 / D ) (#) ( ( 1 / D ) (#) ( ( 1 / D ) (#) ( ( 1 /
pred x in dom f /\ dom g & g . x <= f . x & g . x <= f . x ;
len ( f1 . j ) = len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) ;
All ( All ( 'not' a , A , G ) , B , G ) '<' All ( All ( 'not' a , B , G ) , A , G ) ;
LSeg ( E . x0 , F . ( x0 + 1 ) ) c= Cl ( ( Cage ( C , n + 1 ) ) . x0 ) ;
x \ a = x \ ( ( a |^ m ) * a ) .= ( ( ( x |^ k ) * a ) * a ) * a .= ( ( x |^ k ) * a ) * a ;
k -{ \it true ( I ) } = ( ( commute ( I1 ) ) | ( ( commute ( I2 ) ) | ( ( commute ( I2 ) ) | ( ( commute ( I2 ) ) | ( ( Y ) ) | ( Y ) ) ) ) ) .= ( ( commute ( I1 ) ) | ( Y ) ) . k ;
for s being State of A holds Following ( s , n ) . ( n + 1 ) + ( n + 1 ) * ( n + 1 ) is stable
for x st x in Z holds f1 . x = a ^2 & ( f1 - f2 ) . x <> 0 & ( f1 - f2 ) . x <> 0 ) implies ( f1 - f2 ) . x = ( f1 - f2 ) . x
support ( ( support ( n ) ) \/ support ( ( support ( m ) ) ) c= support ( ( support ( n ) ) ) \/ support ( ( support ( m ) ) ) ;
reconsider t = u as Function of ( the carrier of A ) , the carrier of B ;
- ( a * sqrt ( 1 + b ^2 ) ) <= - ( b * sqrt ( 1 + b ^2 ) ) ;
phi /. ( succ a ) = g . a & phi . ( b , a ) = f . ( g . ( a , b ) ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( F ^ <* p *> ) and i = len ( F ^ <* p *> ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 , 7 , 8 , 8 , 7 , 8 , 8 , 7 , 8 } = { x1 , x2 , x3 , x4 , x5 , 7 , 8 , 8 , 7 } \/ { x2 , x3 , x4 , 8 , 7 , 8 , 7 } .= { x1 , x2 , x3 , x4 } \/ { x2 , x4 , 8 } ;
the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U1 & the Sorts of U1 c= the Sorts of U2 /\ ( U2 "\/" U2 ) ;
( - ( 2 * a ) + ( 2 * a ) + b ) / ( 2 * a ) + ( 2 * a ) / ( 2 * a ) > 0 ;
consider seq1 such that for z being element holds z in seq1 & z in seq2 & ( ex y being element st y in seq1 & ( ex z being element st z in seq2 & y = seq1 . z ) & ( z in seq1 ) & ( z in seq2 ) implies ( z in seq2 ) implies z = ( seq1 union seq2 ) . z ) ;
assume that ( the Arity of S ) . o = <* a *> and ( the Arity of S ) . o = r and ( the Arity of S ) . o = r and ( the Arity of S ) . o = r and ( the Arity of S ) . o = r ;
Z = dom ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 1 / / ) ) ) ) ) ) ) ) ) ) ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
integral ( f , S , T ) is convergent & lim ( f , S ) = integral ( f , S , T ) - integral ( g , T , i ) ;
( ( a . ( f . ( x ) ) ) => ( ( f . ( x ) ) => ( f . ( x ) ) ) ) => ( ( f . ( x ) ) => ( f . ( x ) ) ) ) in TAUT ( Y ) ;
len ( M2 * M2 ) = n & width ( M2 * M2 ) = n & width ( M2 * M2 ) = n & width ( M2 * M2 ) = n ;
attr X1 union X2 is open means : Def1 : X1 union X2 = the carrier of X1 & X1 union X2 = the carrier of X2 & X1 union X2 = the carrier of X2 ;
for L being lower-bounded antisymmetric non empty RelStr for X being non empty Subset of L holds X "\/" { Bottom L } = { Bottom L }
reconsider f9 = ( F . b ) . ( ( F . b ) . ( F . b ) ) , f9 = F . ( ( F . b ) . ( F . b ) ) as Function of ( F . b ) . ( F . b ) , ( F . b ) . ( F . b ) ;
consider w being FinSequence of I such that the carrier of M is_q , w , I and w ^ <* w *> ^ w , I ^ w , I ^ w ^ w ^ w ^ w ^ w , I ^ w ^ w ^ w ^ w , I ^ w ^ w ^ w , I ^ w ^ w , I ^ w ^ w , I ^ w ^ w , I ^ w , I ^ w ^ w ^ w , I ^ w ^ w ^ w ,
g . ( a |^ 0 ) = g . ( 1_ G ) .= ( g |^ ( a |^ 0 ) ) |^ ( a |^ 0 ) .= ( g |^ ( a |^ 0 ) ) |^ ( a |^ 0 ) .= ( g |^ ( a |^ 0 ) ) |^ ( a |^ 0 ) .= g |^ ( a |^ 0 ) .= g |^ ( a |^ 0 ) ;
assume for i being Nat st i in dom f ex z being Element of L st f . i = rpoly ( 1 , z ) ;
ex L being Subset of X st Carrier ( L ) = C & for K being Subset of X st K in C holds L /\ K <> {} & L /\ K <> {} ;
( the carrier of C1 ) /\ ( the carrier of C2 ) c= the carrier of C1 & the carrier of C2 = the carrier of C2 & the carrier of C2 = the carrier of C2 & the carrier of C2 = the carrier of C2 ;
reconsider o9 = o `1 , y9 = o `2 as Element of TS ( ( the Sorts of A ) . o ) ;
1 * x1 + ( 0 * x2 ) + ( 0 * x2 ) = x1 + ( 0 * x2 ) .= ( 0 * x2 ) + ( 0 * x2 ) .= ( 0 * x2 ) + ( 0 * x2 ) + ( 0 * x2 ) .= ( 0 * x2 ) + ( 0 * x2 ) + ( 0 * x2 ) ;
Ef " . 1 = ( Ef ) " . 1 .= ( ( E . 1 ) qua Function ) . 1 .= ( E . 1 ) " .= ( E . 1 ) " .= ( E . 1 ) " .= E . 1 ;
reconsider u1 = the carrier of U1 /\ ( U1 "\/" U2 ) , u2 = the carrier of U2 /\ ( U1 "\/" U2 ) as non empty Subset of U0 ;
( ( x "/\" z ) "/\" ( x "/\" y ) ) "\/" ( ( x "/\" y ) "/\" ( x "/\" z ) ) <= ( x "/\" ( y "/\" z ) ) "\/" ( x "/\" ( y "\/" z ) ) ;
|. f . ( s1 . ( l1 + 1 ) ) - f . ( s1 . ( l + 1 ) ) .| < r / ( M * ( M * ( l , 1 ) ) ) ;
LSeg ( ( Cage ( C , n ) /. i ) , ( Cage ( C , n ) /. i ) ) is vertical & ( Cage ( C , n ) /. i ) `2 is vertical ;
( f | Z ) /. x - ( f | Z ) /. ( x - x0 ) = L /. ( x - x0 ) + R /. ( x - x0 ) ;
g . c * ( ( g . c ) * ( f . c ) + ( f . c ) * ( f . c ) ) <= h . c * ( ( f . c ) + ( f . c ) * ( f . c ) ) + ( f . c ) * ( f . c ) + ( f . c ) * ( f . c ) * ( f . c ) ) ;
( f + g ) | divset ( D , i ) = f | divset ( D , i ) + ( g | divset ( D , i ) ) | divset ( D , i ) ;
assume that ( width f ) in the carrier of A and width ( f * ( b , n ) ) = width A and width ( f * ( b , n ) ) = width A ;
len ( - ( M1 + M2 ) ) = len M1 & width ( - M1 ) = width M1 & width ( - M1 ) = width M1 & width ( - M1 ) = width M1 ;
for n , i being Nat st i + 1 < n & i + 1 < n holds not ( i + 1 ) in the InternalRel of ( ( TOP-REAL n ) | the carrier of ( TOP-REAL n ) | P )
pdiff1 ( f1 , 2 ) is_differentiable_in x0 & pdiff1 ( f2 , 2 ) is_differentiable_in y0 & pdiff1 ( f2 , 2 ) = ( f1 + f2 ) . y0 + ( f2 + f3 ) . y0 ;
pred a <> 0 & b <> 0 & Arg a = Arg b & Arg a = Arg b & Arg b = Arg a & Arg a = Arg b & Arg b = Arg a & Arg a = Arg b & Arg b = Arg a & Arg a = Arg b ;
for c being set st c in [. a , b .] holds not c in Intersection ( ( the topology of X ) | [. a , b .] )
assume that V1 is closed and V1 = { v + u : v in V1 & u in V1 & v in V1 } and V1 = { v + u : v in V1 & u in V1 & v in V1 } ;
z * ( x1 + ( 1 - ( z * x2 ) ) * ( x1 + ( z * x2 ) ) ) in M & ( z * ( x1 + ( z * x2 ) ) * ( x1 + ( z * x2 ) ) ) in N ;
rng ( ( ( P qua Function ) * ( ( P * ( P * ( R * ( R * ( R * ( R * ( R * ( P * ( R * ( R * S ) ) ) ) ) ) ) ) ) ) ) ) = Seg ( card ( ( P * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R
consider s2 being Real such that s2 is convergent and b = lim s2 and for n holds s2 . n <= ( lim s2 ) * ( ( lim s2 ) * ( ( lim s2 ) * ( lim s2 ) ) ) ;
h2 " . n = h2 . ( ( h . n ) " ) * ( ( h . n ) " ) * ( ( h . n ) " ) * ( ( h . n ) " ) * ( ( h . n ) " ) * ( ( h . n ) " ) * ( ( h . n ) " ) * ( ( h . n ) " ) * ( ( h . n ) " ) * ( ( h . n ) " ) ) ;
( Partial_Sums ( |. ( r ) .| ) ) . m = ( |. ( r ) .| ) . m .= ( |. ( r ) .| ) . m .= ( |. r .| ) . m .= ( |. r .| ) . m ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b ;
- v = ( - ( - ( v + u ) ) * v ) & - ( - ( v + u ) * w ) = ( - ( v + u ) * w ) * w & - ( v + u ) * w = ( - ( v + u ) ) * w ;
sup ( ( k .: D ) .: D ) = sup ( ( k .: D ) .: D ) .= ( k .: D ) .: D .= ( k .: D ) .: D .= ( k .: D ) .: D .= ( k .: D ) .: D .= ( k .: D ) .: D .= ( k .: D ) .: D ;
A |^ ( k , l ) = ( A |^ ( n , l ) ) |^ ( n , l ) .= ( A |^ ( n , l ) ) |^ ( n , l ) ;
for R being add-associative right_zeroed right_complementable associative associative distributive non empty doubleLoopStr , I , J being Subset of R holds I + J + K = ( I + J ) + K
( ( f . p ) `1 ) ^2 = sqrt ( ( p `1 ) ^2 + ( p `2 ) ^2 ) .= ( p `1 ) ^2 + ( p `2 ) ^2 ;
for a , b being non zero Nat st a , b are_relative_prime & a * b = ( a * b ) + ( b * a ) * ( a * b ) + ( a * b ) * ( a * b ) = ( a * b ) * ( a * b ) + ( a * b ) * ( a * b )
consider A9 being countable Subset of Al such that r is countable & A9 is countable and A9 is countable and A9 is countable and A9 is countable and A9 is countable and A9 is countable and A9 is countable ;
for X being non empty addLoopStr , M being Subset of X , x being Point of X , y being Point of X st x in M holds x + y in M + M
{ [ x1 , x2 ] , [ y1 , y2 ] } c= { x1 , y1 , y2 } & { x1 , x2 } c= { x1 , y1 , y2 } ;
h . ( f . O ) = [ A * ( f . O ) + B * ( f . O ) + C * ( f . O ) + D * ( f . O ) + D * ( f . O ) + D * ( f . O ) + D * ( f . O ) + D * ( f . O ) + D * ( f . O ) + D * ( f . O ) + D * ( f . O ) ) ;
( Gauge ( C , n ) * ( k , i ) ) in L~ Upper_Seq ( C , n ) /\ L~ Cage ( C , n ) ;
cluster m , n -> prime means : Def1 : for Nat of n , m be Nat st p divides m & p divides m holds p divides m & p divides n ;
( f * F ) . x1 = f . ( F . ( F . x1 ) ) & ( f * F ) . x2 = f . ( F . ( F . x2 ) ) ;
for L being LATTICE , a , b , c being Element of L st a \ b <= c holds a \ b <= c \ a & b \ c <= c \ a
consider b being element such that b in dom ( H / ( ( x , y ) / ( ( x , y ) / ( ( x , y ) / ( x , y ) ) ) ) and z = H / ( ( x , y ) / ( x , y ) ) ;
assume that x in dom ( F (#) g ) and y in dom ( F (#) g ) and ( F (#) g ) . x = ( F (#) g ) . y ;
assume ex e being element st e Joins W . 1 , W . 5 , W . 6 , W . 7 , W . 6 , W . 6 , W . 7 , W . 8 , W . 6 , W . 8 , W . 8 , W . 6 , W . 8 , W ;
( ( ' ' ) * f ) . x = ( ( ' ' ) * f ) . x .= ( f ' ) . x .= ( f ' ) . x ;
j + 1 = j + ( len ( h | ( len h -' 1 ) ) ) .= i + ( len h -' 1 ) .= i + ( len h -' 1 ) ;
( S *' ) . f = S *' ( S *' ) .= S *' ( S *' ) .= S *' ( S *' ) .= S *' ( S *' ) .= S *' ( S *' ) ;
consider H such that H is one-to-one and rng H = the carrier of ( ( ( Carrier ( L2 ) * H ) ) * ( Sum ( L2 ) ) ) and Sum ( ( L2 ) * ( Sum ( L2 ) ) ) = Sum ( ( L2 ) * ( Sum ( L2 ) ) ) ;
attr R is Rev means : Def1 : for p , q st p in R & q in R & p <> q & p <> q holds ex P st P [ p , q ] & P [ q , p ] ;
dom ( product ( X --> f ) ) = meet ( ( dom X --> dom f ) --> ( f . x ) ) .= meet ( ( X --> dom f ) --> ( f . x ) ) .= meet ( X --> f . x ) .= meet ( X --> f . x ) ;
sup ( ( proj2 .: ( Upper_Arc ( C ) /\ Vertical_Line w ) ) /\ ( ( proj2 .: ( C /\ Vertical_Line w ) ) /\ ( ( proj2 .: ( C /\ Vertical_Line w ) ) /\ ( ( proj2 .: ( C /\ Vertical_Line w ) ) /\ ( ( proj2 .: ( C /\ Vertical_Line w ) ) /\ ( ( proj2 .: ( C /\ Vertical_Line w ) ) /\ ( ( proj2 .: ( C /\ Vertical_Line w ) ) /\ ( ( proj2 .: ( C /\ Vertical_Line w ) ) /\ ( ( proj2 .: ( C /\ Vertical_Line w )
for r be Real st 0 < r ex n be Nat st for m be Nat st n <= m holds |. S . m - 0 .| < r
i * ( ( f - g ) * ( ( f - g ) * ( ( f - g ) * ( ( f - g ) * ( ( f - g ) * ( ( f - g ) * ( ( f - g ) * ( ( f - g ) * ( ( f - g ) * ( ( f - g ) * ( f - g ) * ( ( f - g ) * ( f - g ) * ( f - g ) ) ) ) ) ) = i * ( ( f - g )
consider f being Function such that dom f = 2 -tuples_on X ( ) & for Y being set st Y in 2 -tuples_on X ( ) holds f . Y = F ( Y ) ;
consider g1 , g2 being element such that g1 in [#] Y & g2 in [#] Y & g1 in C & g2 in C and g1 in C and g2 in C and g2 in C and g1 in C and g2 in C and g2 in C and g2 in C and g1 , g2 ] ;
func d \! \hbox { n } -> Nat means : Def1 : d |^ ( n + 1 ) divides d |^ ( n + 1 ) & d |^ ( n + 1 ) divides d |^ ( n + 1 ) & d |^ ( n + 1 ) divides d |^ ( n + 1 ) ;
f9 . [ 0 , t ] = f . [ 0 , 0 ] .= ( - P ) . [ 0 , t ] .= ( - P ) . [ 0 , t ] .= a ;
t = h . D or t = h . B or t = h . C or t = h . D or t = h . E or t = h . J or t = h . M or t = h . J or t = h . M or t = h . M ;
consider m1 be Nat such that for n st n >= m1 holds dist ( ( seq . n ) . m , ( seq . n ) . m ) < 1 / ( ( seq . m ) . n ) ;
sqrt ( ( ( q `2 / |. q .| - sn ) / ( 1 - sn ) ) ^2 + ( q `2 / |. q .| - sn ) ^2 <= ( ( q `2 / |. q .| - sn ) ) ^2 + ( q `2 / |. q .| - sn ) ^2 ;
h1 . ( i + 1 + 1 ) = h1 . ( i + 1 + 1 ) .= h1 . ( i + 1 + 1 ) .= h1 . ( i + 1 ) ;
consider o being Element of the carrier' of S such that a = [ o , x2 ] and o in { [ o , x2 ] } and o <> {} ;
for L being RelStr for a , b being Element of L holds a <= b iff a <= b & b <= a & a <= b & b <= a & a <= b implies a <= b & b <= a & a <= b
||. h1 .|| . n = ||. h1 . n .|| .= ||. h1 . n .|| .= ||. h1 . n .|| .= ||. h1 .|| . n .= ||. h1 .|| . n .= ||. h1 .|| . n ;
( ( ( ( - ( ( ( ( #Z ( 1 / 2 ) ) ) ) * ( ( #Z ( 2 * x ) ) ) ) ) ) `| Z ) . x = f . x - ( ( #Z ( 2 * x ) ) * ( ( #Z ( 2 * x ) ) ) ) . x .= ( ( #Z ( 2 * x ) ) ) . x ;
pred r = F .: ( p , q ) means : Def1 : len r = len p & for i st i in dom p holds p . i = F ( p . i , q . i ) ;
sqrt ( ( r ^2 + ( r ^2 + ( r ^2 + ( r ^2 + ( r ^2 + 1 ) ) ^2 ) ) ^2 ) <= sqrt ( ( r ^2 + ( r ^2 + ( r ^2 + 1 ) ^2 ) ) ^2 ) + sqrt ( ( r ^2 + ( r ^2 + 1 ) ^2 ) ) ;
for i being Nat , M being Matrix of n , K st i in Seg n & i in Seg n holds ( Det ( M , i ) ) . ( i , j ) = Sum ( Line ( M , i ) )
then a <> 0. R & a " * ( a * v ) = 1 * v .= a * v .= a * v ;
p . ( j -' 1 ) * ( q *' r ) . ( i -' 1 ) = Sum ( p *' r ) * ( q *' r ) . ( j -' 1 ) * ( q *' r ) . ( j -' 1 ) ;
deffunc F ( Nat ) = L . 1 + ( ( R /* h ) * ( ( R /* h ) ^\ n ) * ( ( R /* h ) ^\ n ) * ( ( h ^\ n ) /* ( h ^\ n ) ) ;
assume that the carrier of H = f .: ( the carrier of H ) and the carrier of H = f .: ( the carrier of H ) and the carrier of H = f .: ( the carrier of H ) and the carrier of H = f .: ( the carrier of H ) ;
Args ( o , Free ( S , X ) ) = ( ( ( the Sorts of Free ( S , X ) ) * ( the_arity_of o ) ) . o .= ( ( the Sorts of Free ( S , X ) ) * ( the_arity_of o ) ) . o ;
H1 = n + 1 / ( 2 to_power ( 2 to_power ( n + 1 ) ) ) .= n + 1 / ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 to_power ( 2 / ( 2 to_power ( 2 to_power ( 2 to_power
( O O O O ) . ( O , O ) = 0 & ( O O ) . ( O , O ) = 0 & ( O O O O ) . ( O , O ) = 0 & ( O O ) . ( O , O ) = 0 ;
F1 .: ( dom F1 /\ dom F2 ) = F1 .: ( dom F1 /\ dom F2 ) .= ( f | ( ( n + 1 ) ) | ( ( n + 1 ) -tuples_on ( ( n + 1 ) -tuples_on ( ( n + 1 ) -tuples_on ( ( n + 1 ) -tuples_on ( ( n + 1 ) -tuples_on ( ( n + 1 ) -tuples_on ( ( n + 1 ) -tuples_on ( ( n + 1 ) -tuples_on ( ( n + 1 ) -tuples_on ( ( ( n + 1 ) ) ) ) ) ) ) ) ) ) .= f .: ( ( n + 1 ) ) ) .= f
pred b <> 0 & d <> 0 & b <> 0 implies b = ( id the carrier of X ) --> ( b , a ) & ( a = ( id the carrier of X ) --> ( b , a ) ) = ( ( id the carrier of X ) --> ( b , a ) )
dom ( ( f +* g ) | D ) = dom ( ( f +* g ) | D ) /\ D .= ( f +* g ) | D .= ( f +* g ) | D .= ( f +* g ) | D .= ( f +* g ) | D .= f +* ( g +* ( g +* ( g +* ( g +* ( g +* ( g +* ( g +* ( g +* ( g +* ( g +* ( g +* f ) ) ) ) ) ) ) ) ;
for i being set st i in dom g ex u being Element of B st g /. i = u * v & u in B * v
g `2 * P `2 * ( g * P ) `2 = g `2 * ( g * P ) `2 .= g `2 * ( g * P ) `2 .= g `2 * ( g * P ) `2 ;
consider i , s1 such that f . i = s1 & ( not ex i , j st i in dom s1 & j in dom s1 & not ( s1 . i = s1 . i ) & ( not s1 . i = s1 . j ) & ( not s1 . i = s2 . j ) ;
h5 | ]. a , b .[ = ( g | ]. a , b .[ ) | ]. a , b .[ .= ( g | ]. a , b .[ ) | ]. a , b .[ .= ( g | ]. a , b .[ ) | ]. a , b .[ .= ( g | [. a , b .[ ) | [. a , b .[ ;
[ s1 , t1 ] , [ s2 , t2 ] ] in R & [ s2 , t2 ] in R & [ s1 , t2 ] in R & [ s2 , t2 ] in R ;
then H is negative means : Def1 : H is non negative & H is non empty implies H is non empty ;
attr f1 is total means : Def1 : ( 1 - f1 (#) f2 ) | X is total & ( f1 (#) f2 ) | X = ( f1 (#) f2 ) | X & ( f1 (#) f2 ) | X = ( f1 (#) f2 ) | X ) | X ;
z1 in W2 ` & z2 in W2 ` & z1 in W2 ` implies ( z1 in W2 ) & ( z1 in W2 ) & ( z1 in W2 ) & ( z1 in W2 ) & ( z1 in W1 ) & ( z1 in W2 ) & ( z1 in W2 ) & ( z1 in W2 ) implies z1 = z2 ) & ( z1 in W2 ) & ( z1 in W2 ) implies z1 = z2 )
p = 1 * p * a .= a " * p * a .= a * ( p * a ) .= a * ( p * a ) .= a * ( p * a ) .= a * ( p * a ) .= a * ( p * a ) .= a * ( p * a ) .= a * ( p * a ) ;
for K be Real , K be Real st K <= K & K <= n holds upper_bound ( K * L ) <= K * ( K * L )
( E-max C ) meets ( L~ go \/ L~ pion1 \/ L~ pion1 ) or ( E-max C ) /\ ( L~ pion1 \/ L~ pion1 ) meets ( L~ pion1 \/ L~ pion1 ) or ( E-max C ) meets ( L~ pion1 \/ L~ pion1 \/ L~ pion1 ) \/ ( L~ pion1 \/ L~ pion1 ) & ( E-max C ) \/ L~ pion1 c= ( L~ go C \/ L~ pion1 \/ L~ pion1 ) \/ L~ pion1 \/ L~ pion1 \/ L~ pion1 ) ;
||. f . ( g . ( k + 1 ) ) - f . ( g . ( k + 1 ) ) .|| <= ||. g . ( g . ( k + 1 ) ) .|| * ( K * ( K + 1 ) ) ;
assume h = ( ( B .--> C ) +* ( D .--> E ) +* ( E .--> F ) +* ( J .--> M ) +* ( M .--> N ) +* ( N .--> F ) +* ( M .--> N ) +* ( N .--> N ) +* ( M .--> N ) +* ( N .--> N ) +* ( M .--> N ) +* ( N .--> N ) ) +* ( M .--> N ) +* ( N .--> N ) ) ;
|. ( ( Gauge ( C , n ) . ( i , j ) ) - ( ( Cage ( C , n ) . ( i , j ) ) ) .| <= e * ( 2 * ( i , j ) ) ;
( ( is_{ i } -tree v ) ) . e = [ ( the Sorts of U1 ) . e , ( the Sorts of U2 ) . e ] .= ( the Sorts of U1 ) . e ;
{ x1 , y1 , y2 , y1 , y2 , y2 , y1 , y2 , y2 , y1 , y2 , y2 , y1 , y2 , y2 , y1 , y2 , y2 , y1 , y2 , y2 , y2 , y1 , y2 , y2 , y1 , y2 , y2 , y1 , y2 , y2 , y2 , y1 , y2 , y2 , y2 , y2 , y1 , y2 , y2 , y2 , y1 , y2 , y2 , y2 , y2 , y1 , y2 , y2 , y2 , y2 , y2 , y2 , y1 , y2 , y2 , y2 , y2 , y2 , y2 , y2
assume that A = [. 0 , PI * cos | A , PI * sin | A .] and integral ( sin , A ) = 0 ;
p `1 is Permutation of dom f1 & p `2 `2 = ( Sgm Y ) . i & p `2 `2 = ( Sgm Y ) . i & p `2 = ( Sgm Y ) . i ;
for x , y st x in A holds |. ( 1 - f ) . x - f . y .| <= 1 * |. ( f . x ) . y - f . y .|
( p2 `2 ) ^2 = |. q2 .| * ( ( p2 `2 / |. q2 .| - sn ) ) ^2 + ( p2 `2 / |. q2 .| - sn ) ^2 * ( ( p2 `2 / |. q2 .| - sn ) ) ^2 + ( p2 `2 / |. q2 .| - sn ) ^2 * ( ( p2 `2 / |. q2 .| - sn ) ) ^2 ) ;
for f being PartFunc of the carrier of C , REAL , g be PartFunc of C , REAL st dom f = the carrier of C & f | C is bounded & g | C is bounded holds f | C is bounded
assume for x being Element of Y st x in EqClass ( z , CompF ( PA , G ) ) holds ( All ( a , PA ) ) . x = TRUE ;
consider F3 such that dom F3 = n1 & for k be Nat st k in n1 holds Q [ k , F . k ] ;
ex u , u1 st u <> u1 & u , u1 , v1 , u1 , v1 , v2 , v1 , v2 , v2 , v1 , v2 , v1 , v2 , v2 , v1 , v2 , v2 , v1 , v2 , v1 , v2 , v2 , v1 , v2 , v2 , v1 , v2 , v2 , v1 , v2 , v2 , v2 , v1 , v2 , v2 , v2 , v1 , v2 , v2 , v1 , v2 , v2 , v1 , v2 , v2 , v1 , v2 , v2 , v2 , v2 , v1 , v2 , v2 , v2 , v2 , v2 , v2 , v2 , v2
for G being Group , N being non empty Subgroup of G , N being normal Subgroup of G holds ( N , N ) * ( N , N ) = N " N * ( N , N )
for s be Real st s in dom F holds F . s = integral ( R ^2 ) * ( ( R ^2 + ( R ^2 ) (#) ( f ^2 ) ) ^2 )
width ( ( ( f1 ^ f2 ) * ( ( f2 ^ g2 ) * ( f1 + g2 ) ) ) | ( len ( ( f2 ^ g2 ) ) ) ) = len ( ( ( f2 ^ g2 ) * ( f1 + g2 ) ) .= len ( ( f2 ^ g2 ) * ( ( f2 ^ g2 ) * ( f1 + g2 ) ) ) .= len ( ( f2 ^ g2 ) (#) ( ( f2 ^ g2 ) * ( f1 + g2 ) ) ) ;
f | ]. - PI / 2 , PI / 2 .[ = f & f | [. - PI / 2 , PI / 2 .[ = f | [. - PI / 2 , PI / 2 .[ & f | [. - PI / 2 , PI / 2 .[ = f | [. - PI / 2 , PI / 2 .[ ;
assume that X is closed and a in X and a in X and y in a & x in a \/ { [ n , x ] } and x in X \/ { [ n , x ] } ;
Z = dom ( ( ( ( ( ( ( ( ( ( ( ( arctan + arctan ) ) * ( arctan + arccot ) ) * ( ( arctan + arccot ) ) * ( ( arctan + arccot ) * ( ( arctan + arccot ) * ( ( arctan + arccot ) * ( ( arctan + arccot ) * ( ( arctan + arccot ) * ( ( arctan + arccot ) * ( ( arctan + arccot ) * ( ( arctan + arccot ) * ( ( arctan + f1 ) ) * ( ( arctan + f1 ) ) * ( ( arctan + f1 ) * ( ( arctan + f1 ) * ( ( arctan + f1 ) *
func [: l , l :] -> Subset of V means : Def1 : 1 <= l & l <= len l & l <= len l & l . k = l . k ;
for L being non empty TopSpace , N being net of L , M being net of L , N being net of L , c being Element of L st c is Element of N holds M . ( c , N ) is convergent & N is convergent & lim c = c
for s being Element of NAT holds ( ( ( id the carrier of X ) + ( id the carrier of Y ) ) + ( id the carrier of Y ) ) . s = ( ( id the carrier of X ) + ( id the carrier of Y ) ) . s
then z /. 1 = ( E-max L~ z ) .. z & ( E-max L~ z ) .. z < ( E-max L~ z ) .. z ;
len ( p ^ <* 0 *> ) = len p + len <* 0 *> .= len p + len <* 0 *> .= len p + 1 ;
assume that Z c= dom ( ( ( - 1 ) (#) ( ( ( #Z 2 ) * ( f + ( #Z 2 ) * ( f + ( #Z 2 ) * ( f + #Z 2 ) ) ) ) ) and for x st x in Z holds f . x = exp_R ( x ) / ( x + b ) ) ^2 and f . x = b ;
for R being add-associative right_zeroed right_complementable associative associative distributive non empty doubleLoopStr , I being Subset of R holds ( I + J ) *' ( I + J ) c= I /\ J
consider f being Function of B1 , B2 such that for x being Element of B1 holds f . x = F ( x ) & f . x = F ( x ) ;
dom ( x2 + y2 ) = Seg len ( x + y ) .= Seg len ( x + y ) .= Seg len ( x + y ) .= Seg len ( x + y ) .= Seg len ( x + y ) .= Seg len ( x + y ) ;
for S being transitive Functor of C , B for c being Object of C holds ( S . ( id B ) ) . ( id B ) = id ( ( the carrier of C ) . c ) . ( id B )
ex a st a = a2 & a in [: f , g :] /\ ( f , g ) & ( for a , b st a in f /\ g holds f . a = ( f , g ) . b ) & ( f , g ) . a = ( f , g ) . b ) & ( f , g ) . b = ( f , g ) . b ) & ( f , g ) . b = f , g ) ;
a in Free ( H / ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( H ( ( H ( ( ( ( H ( ( H ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
for C1 , C2 being set , f being Function of C1 , C2 holds ( for g being Function of C2 , C2 holds f = g ) iff f = g
( W-min L~ go \/ L~ pion1 ) `1 = ( W-min L~ go \/ L~ pion1 \/ L~ pion1 ) \/ ( L~ pion1 \/ L~ pion1 ) \/ L~ pion1 .= ( W-min L~ go \/ L~ pion1 ) \/ L~ pion1 \/ L~ pion1 \/ L~ pion1 \/ L~ pion1 \/ L~ pion1 \/ L~ pion1 \/ L~ pion1 \/ L~ pion1 \/ L~ pion1 \/ L~ pion1 \/ L~ pion1 c= L~ pion1 \/ L~ pion1 \/ L~ pion1 \/ L~ pion1 \/ L~ pion1 \/ L~ pion1 \/ L~ pion1 \/ L~ pion1 L~ pion1 /\ L~ pion1 \/ L~ pion1 L~ pion1 L~ pion1 \/ L~ pion1 /\ L~ pion1 L~ pion1 \/ L~ pion1 L~ pion1 \/ L~ pion1 \/ L~ pion1 \/ L~ pion1 L~ pion1 L~ pion1 L~
assume that u = <* x0 , y0 , z0 , z0 , N , z0 , N , z0 , z0 , N , z0 , z0 , z0 , N , z0 , z0 , N , z0 , N , z0 , N , z0 , N , z0 , N , z0 , N , z0 , N , z0 , N , z0 , N , z0 , N , z0 , z0 , z0 , N , z0 , z0 , z0 , z0 , N , z0 , z0 , N , z0 , N , z0 , z0 , N , N , z0 , N , z0 , N , z0 , z0 , N , N , z0 , z0 , z0 , N , N , z0 , N , N , N , z0 , N , z0 , z0 , z0 , z0 , z0 , z0 , z0
then ex x being Element of Vars ( C ) st x = ( t . {} ) `1 & ( t . {} ) `1 = ( t . {} ) `1 & ( t . {} ) `2 = ( t . {} ) `1 ;
Valid ( p '&' q , J ) . v = Valid ( p , J ) . v '&' Valid ( q , J ) . v .= Valid ( p , J ) . v '&' Valid ( q , J ) . v .= Valid ( p , J ) . v ;
assume for x , y being Element of S st x <= y & y = f . x holds a >= f . y & b >= f . y ;
func Class R -> Subset-Family of R means : Def1 : for A being Subset of R holds it = Class ( it , A ) ;
defpred P [ Nat ] means ( ( ( j ) + 1 ) ) `1 c= G . ( ( j + 1 ) ) `1 & ( ( j + 1 ) ) `1 c= G . ( ( j + 1 ) + 1 ) ) `1 ;
assume that dim ( W1 ) = 0 and dim ( W2 ) = 0 and dim ( W1 ) = 0 & dim ( W1 ) = 0 & dim ( W1 ) = 0 & dim ( W2 ) = 0 ;
mam ( m ) . t = ( m . ( m . t ) ) . {} .= ( m . ( m . t ) ) . {} .= ( m . ( m . t ) ) . t .= ( m . ( m . t ) ) . t .= ( m . ( m . t ) ) . t .= ( m . ( m . t ) ) . t ;
d1 = ( x9 ^ <* d *> ) ^ ( ( y ^ <* d *> ) | ( y ^ <* d *> ) ) .= ( f | ( y ^ <* d *> ) ) | ( y ^ <* d *> ) .= ( f | ( y ^ <* d *> ) ) | ( y ^ <* d *> ) .= ( f | ( y ^ <* d *> ) ) | ( y ^ <* d *> ) .= ( f | ( y ^ <* d *> ) | ( y ^ <* d *> ) .= ( f | ( y ^ <* d *> ) ) | ( y ^ <* d *> ) | ( y ^ <* d *> ) | ( y ^ <* d *> ) .= ( f | ( y ^ <* d *> ) | ( y ^ <* d
consider g such that x = g and dom g = dom ( f | X ) and for x being element st x in dom ( f | X ) holds g . x = ( f | X ) . x ;
x + 0. ( ( len x ) |-> 0. ( K , len x ) ) = x + 0. ( K , len x ) .= x + 0. ( K , len x ) .= x + 0. ( K , len x ) .= x + 0. ( K , len x ) .= x + 0. ( K , len x ) ;
( ( k -' 1 ) + 1 ) + ( ( k -' 1 ) + 1 ) + ( ( k -' 1 ) + ( ( k -' 1 ) + 1 ) ) + ( ( k -' 1 ) + ( ( k -' 1 ) + 1 ) + ( ( k -' 1 ) + 1 ) ) + ( ( k -' 1 ) + 1 ) + ( ( k -' 1 ) + 1 ) + ( ( k -' 1 ) + 1 ) ) = ( ( k -' 1 ) + ( ( k -' 1 ) + 1 ) ) + ( ( k -' 1 ) + ( ( k -' 1 ) + ( ( k -' 1 ) + ( ( k -' 1 ) + ( ( k -' 1 ) + 1 ) ) + ( ( k -' 1
assume that P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p2 , p1 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 c= P1 and P1 c= P2 and P1 \/ P2 = { p1 , p2 } and P1 \/ P2 = { p1 , p2 } and P1 \/ P2 = { p1 , p2 } and P1 = { p1 , p2 } and P1 = { p1 , p2 } and P1 = { p1 , p2 } ;
reconsider a1 = a , b1 = b , c1 = c , c2 = d , c2 = c , c1 = d , c2 = d , c2 = d , c2 = c , c1 = d , c2 = d , c2 = d , c2 = c , c1 = d , c2 = d , c2 = d , c2 = c , c2 = d , c2 = d , c2 = d , c2 = d , c1 = d , c2 = d , c2 = c , c2 = d , c2 = d , c2 = d , c2 = d , c2 = d , c2 = d , c2 = d , c2 = d , c2 = d , c2 = d , c2 = d , c2 = d , c2 = d , c2 = c = d , c2 = d ,
reconsider F2 = G1 . ( t , b ) * ( t , b ) as Morphism of ( G1 * F1 ) . ( t , b ) * ( t , b ) * ( t , b ) as Morphism of ( G1 * F2 ) . ( t , b ) * ( t , b ) ;
LSeg ( f , i + 1 ) = LSeg ( f /. ( i + 1 ) , f /. ( i + 1 ) ) .= LSeg ( f /. ( i + 1 ) , f /. ( i + 1 ) ) ;
Integral ( P . m , P . ( n + m ) ) <= Integral ( M , P . ( n + m ) ) + Integral ( M , P . ( n + m ) ) ;
assume that dom f1 = dom f2 and for x , y being element st x in dom f1 /\ dom f2 holds f1 . ( x , y ) = f2 . ( x , y ) and f2 . ( x , y ) = f2 . ( x , y ) ;
consider v such that v = y and dist ( u , v ) < min ( ( r - ( G * ( i , 1 ) ) ) / ( 2 * ( ( G * ( i , 1 ) + G * ( i , 1 ) ) ) / ( 2 * ( ( G * ( i , 1 ) + G * ( i , 1 ) ) ) ) ;
for G being Group , H being Subgroup of G , a being Element of H holds a = b implies a |^ H = b |^ H & a |^ H = b |^ H
consider B being Function of Seg ( S + L ) , the carrier of V such that for x being element st x in Seg ( S + L ) holds P [ x , B . x ] ;
reconsider K1 = { p2 where p2 is Point of TOP-REAL 2 : P [ p2 ] & p2 `1 <= 0 & p2 <> 0. TOP-REAL 2 } as Subset of TOP-REAL 2 ;
sqrt ( ( ( ( E-max C ) `1 ) / 2 ) ^2 + ( ( E-max C ) / 2 ) ^2 ) <= sqrt ( ( E-max C ) / 2 ) ^2 + ( ( E-max C ) / 2 ) ^2 + ( E-max C ) ^2 ;
for x be Element of X , n be Nat st x in E holds |. Re ( F . n ) .| <= P . x & Im ( F . n ) <= P . x
len ( @ ( @ ( <* 2 *> ^ q ) ) ) = len ( @ ( <* 2 *> ^ q ) ) + len <* 2 *> .= len ( @ ( <* 2 *> ^ q ) ) + len q .= len ( <* 2 *> ^ q ) + len q .= len ( <* 2 *> ^ q ) + len q ;
v / ( ( ( ( x , m ) / ( ( x , m ) / ( x , m ) ) / ( x , m ) ) ) / ( ( x , m ) / ( x , m ) ) / ( x , m ) ) ) = m1 / ( ( x , m ) / ( x , m ) ) / ( x , m ) ) ;
consider r being Element of M such that M , v / ( ( ( x , m ) / ( ( x , m ) / ( x , m ) ) / ( x , m ) ) / ( x , m ) ) / ( x , m ) / ( x , m ) / ( x , m ) ) / ( x , m ) / ( x , m ) = r ;
func ( ( the carrier of G ) \ ( the carrier of G ) ) \ ( ( the carrier of G ) \ ( the carrier of G ) \ ( the carrier of G ) ) \ ( the carrier of G ) ;
s2 . b = ( Exec ( n2 , s1 ) ) . b .= Exec ( n2 , s2 ) . b .= Exec ( n2 , s2 ) . b .= Exec ( i2 , s2 ) . b .= Exec ( i2 , s2 ) . b .= Exec ( i2 , s2 ) . b .= Exec ( i2 , s2 ) . b .= Exec ( i2 , s2 ) . b ;
for n , k be Nat holds 0 <= ( Partial_Sums ( |. seq .| ) ) . ( n + k ) - ( Partial_Sums ( |. seq .| ) ) . ( n + k ) - ( Partial_Sums ( |. seq .| ) ) . ( n + k ) ) + ( Partial_Sums ( |. seq .| ) ) . ( n + k ) - ( Partial_Sums ( |. seq .| ) ) . n <= ( Partial_Sums ( |. seq .| ) ) . ( n + k )
set F = S \! \mathop { N } , G = S \! \mathop { N } , F = S \! \mathop { N } , F = S \! \mathop { N } , F = S \! \mathop { N } , G = S \! \mathop { N } , C = S \! \mathop { N } , F = S \! \mathop { N } , G = S \! \mathop { N } , F = S \! \mathop { N } , G = S \! \mathop { N } , F = S \! \mathop { N } , F = S \! \mathop { N } , G = S \! \mathop { N } , F = S \! \mathop { N } , F = S \! \mathop { N }
( Partial_Sums seq ) . ( n + 1 ) + Partial_Sums ( seq ) . ( n + 1 ) >= ( Partial_Sums ( seq ) ) . ( n + 1 ) + ( Partial_Sums ( seq ) ) . ( n + 1 ) + ( Partial_Sums ( seq ) ) . ( n + 1 ) + ( Partial_Sums ( seq ) ) . ( n + 1 ) >= ( Partial_Sums ( seq ) ) . ( n + 1 ) + ( Partial_Sums ( seq ) ) . ( n + 1 ) ;
consider L , R such that for x st x in N holds ( f | Z ) . x - ( f | Z ) . ( f . x ) = L . ( x- ( x ) ) + R . ( x - x0 ) ;
the closed of \HM { a , b , c , d \HM { a , b } = ( the distance of \HM { a , b } ) | ( the carrier of TOP-REAL n ) .= ( the distance of TOP-REAL n ) | ( the carrier of TOP-REAL n ) .= ( the distance of TOP-REAL n ) | ( the carrier of n ) ;
a * b ^2 + ( a * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 >= 6 * ( a * c ) + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b *
v / ( x1 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) = v / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) = v / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 )
( ^ <* x *> ^ <* M *> ) . ( ( 1 , 0 ) --> ( M . ( x , 0 ) ) ) = ( ( 1 , 0 ) --> ( M . ( x , 0 ) ) ) +* ( ( ( M --> ( x , 0 ) ) --> ( x , 0 ) ) .= ( ( 1 , 0 ) --> ( x , 0 ) ) --> ( ( x , 0 ) --> ( x , 0 ) ) ) ;
Sum ( F |^ ( n1 + 1 ) ) = ( r |^ n1 ) * ( F |^ n1 ) .= ( ( r |^ n1 ) * ( F |^ n1 ) ) * ( F |^ n1 ) .= ( ( r |^ n1 ) * ( F |^ n1 ) ) * ( F |^ n1 ) .= ( ( r |^ n1 ) * ( r |^ n1 ) ) * ( r |^ n1 ) ;
( ( GoB f ) * ( len GoB f , 1 ) , 1 ) `1 = ( ( GoB f ) * ( 1 , 1 ) ) `1 .= ( ( GoB f ) * ( 1 , 1 ) ) `1 ;
defpred X [ Element of NAT ] means ( Partial_Sums s ) . $1 = ( Partial_Sums ( s ) ) . $1 + ( Partial_Sums ( s ) ) . $1 * ( Partial_Sums ( s ) ) . $1 * ( Partial_Sums ( s ) ) . $1 + ( Partial_Sums ( s ) ) . $1 * ( Partial_Sums ( s ) ) . $1 * ( Partial_Sums ( s ) ) . $1 * ( Partial_Sums ( s ) ) . $1 * ( Partial_Sums ( s ) ) . $1 * ( $1 * ( 1 / ( 1 / ( 1 / ( 1 / ( 1 / ( 1 / ( 1 / ( 1 / ( 1 / ( 1 / ( 1 / ( 1 / ( 1 / ( 1 / ( 1 / ( 1 / ( 1
( the_arity_of g ) . g = ( ( the Arity of S ) . g ) . g .= ( ( the Arity of S ) . g ) . g .= ( ( the Arity of S ) . g ) . g .= ( ( the Arity of S ) . g ) . g .= ( ( the Arity of S ) . g ) . g ;
( X \times Y ) c= X [: Z , Y :] & ( X c= Y ) & Y c= X implies ( X c= Y ) & ( Y c= X implies Y c= X & Z c= Y ) implies X c= Y & Y c= X & Z c= Y & Y c= Y & Z c= X implies X c= Y & Y c= X & Y c= Y
for a , b being Element of S , s being Element of NAT st s = F . n & a = F . n & b = F . n holds b = F . ( n + 1 ) \ G . ( n + 1 )
E , f / ( ( ( ( ( ( ( ( x , y ) / ( ( x , y ) / ( ( x , y ) / ( ( x , y ) / ( x , y ) ) ) ) / ( ( x , y ) / ( ( x , y ) / ( ( x , y ) / ( ( x , y ) / ( x , y ) / ( x , z ) ) ) ) ) ) ) |= H ;
ex R2 being 1-sorted st R2 = ( p | ( n + 1 ) ) & ( ( the carrier of R ) | ( n + 1 ) ) = ( the carrier of R ) /\ ( the carrier of R ) & ( the carrier of R ) /\ ( the carrier of R ) = the carrier of R ;
[. a , b + sqrt ( 1 + ( b + ( a + b ) ) ) / ( a + b ) ) is Element of the topology of Y & ( a + b ) / ( b + ( a + b ) ) is Element of REAL ;
Comput ( P , s , 2 + 1 ) = Exec ( P , Comput ( P , s , 2 ) , Comput ( P , s , 2 ) ) .= Exec ( P , Comput ( P , s , 2 ) , 2 ) .= Exec ( P , s ) , Exec ( P , s ) ) ;
card h1 = ( power ( F_Complex ) ) . k .= ( ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( 1 ) / 2 ) ) ) ) ) ) ) ) ) ) ) * Sum u ) .= ( ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( 1 ) ) ) ) ) ) ) ) ) ) ) ) * Sum u ) ) ) .= ( ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - (
sqrt ( ( f /. c ) * ( g /. c ) ) = f /. c * ( g /. c ) .= ( f /. c ) * ( g /. c ) .= ( f /. c ) * ( g /. c ) .= ( f /. c ) * ( g /. c ) .= ( f /. c ) * ( f /. c ) .= ( f /. c ) * ( f /. c ) * ( f /. c ) ;
len CC - len ( ( C | ( len C -' 1 ) ) | ( len C -' 1 ) ) = len ( C | ( len C -' 1 ) ) .= len ( C | ( len C -' 1 ) ) .= len ( C | ( len C -' 1 ) ) ;
dom ( ( r (#) f ) | X ) = dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) .= X /\ ( ( r (#) f ) | X ) .= X /\ ( ( r (#) f ) | X ) .= X /\ ( ( r (#) f ) | X ) .= X /\ ( ( r (#) f ) | X ) .= X /\ ( ( r (#) f ) | X ) ;
defpred P [ Nat ] means 2 * $1 = Fib ( n + $1 ) * Fib ( $1 ) + ( Fib ( $1 ) ) * Fib ( $1 ) * Fib ( $1 ) * Fib ( $1 ) * Fib ( $1 ) + ( Fib ( $1 ) ) * Fib ( $1 ) * Fib ( $1 ) * Fib ( $1 ) * Fib ( $1 ) * Fib ( $1 ) * Fib ( $1 ) * Fib ( $1 ) * Fib ( $1 ) * Fib ( $1 ) + ( Fib ( $1 ) * Fib ( $1 ) * Fib ( $1 ) * Fib ( $1 ) * Fib ( $1 ) * Fib ( $1 ) * Fib ( $1 ) * Fib ( $1 ) * Fib ( $1 ) * Fib ( $1 ) * Fib ( $1 ) * Fib ( $1 ) * Fib ( $1 * Fib
consider f being Function of REAL , REAL n such that f = f and f is onto and f is onto and rng f = { n } and f is onto and f is onto and g is onto and g is onto and g is onto and g is onto and g is onto and g is onto and g is onto and g " { n } = f " { n } ;
consider c9 being Function of S , BOOLEAN such that c9 = IExec ( A \/ B , C ) and ( for A , B being Element of S holds ( c . A \/ B ) . ( A \/ B ) = Prob ( c , B ) . ( A \/ B ) ) & ( c . B \/ c ) . ( A \/ B ) = Prob ( c , C ) . ( c . ( A \/ B ) ) ;
consider y being Element of [: Y , Y :] such that a = "\/" ( { F ( x , y ) where y is Element of Y : P [ x , y ] } , y ) & y in Y & P [ y , x ] ;
assume that A c= Z and f = ( ( ( id Z ) (#) ( ( exp_R + ( exp_R + ( exp_R + ( exp_R + exp_R ) ) ) * ( ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + exp_R ) ) ) ) ) ) ) ) `| Z ) ;
( ( GoB f ) * ( i , j ) ) `2 = ( ( GoB f ) * ( 1 , j ) ) `2 .= ( ( GoB f ) * ( 1 , j ) ) `2 .= ( ( GoB f ) * ( 1 , j ) ) `2 .= ( ( GoB f ) * ( 1 , j ) ) `2 ;
dom ( ( Seq ( q2 , len ( q | i ) ) ) | ( dom ( ( q | i ) ^ ( ( q | i ) | ( Seg len q ) ) ) ) = { j + ( q | i ) where j is Nat : j in dom ( ( q | i ) ^ ( q | i ) ) } ;
consider G1 , G2 being Element of V such that G1 <= G2 & G2 <= G1 & G2 <= G2 and G1 = G2 and G2 = G2 and G1 = G2 & G2 = G2 & G2 = G2 & G1 = G2 & G2 = G2 & G2 = G2 & G1 = G2 & G2 = G2 & G2 = G2 & G2 = G2 ;
func - f -> PartFunc of C , V means : Def1 : dom it = dom f & for c st c in dom it holds it . c = - f /. c & f /. c = ( f | X ) /. c ) * ( f /. c ) + ( f /. c ) * ( f /. c ) * ( f /. c ) ) ;
consider phi such that phi is increasing and for a st phi . a = a & phi is continuous holds union L |= ( union L ) & union ( union L ) |= ( union L ) \/ ( union L ) ;
consider i1 , j1 such that [ i1 , j1 ] in Indices GoB f and f /. ( i1 + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i1 + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i1 + 1 ) = ( GoB f ) * ( i1 , j1 ) ;
consider i , n such that n <> 0 and sqrt p = sqrt ( i + n ) and for i1 , i2 being Integer st i1 <> i2 & i1 <> i2 & i1 <> i2 & i1 <> i2 & i2 <> 0 & i1 <> 0 & i2 <> 0 & i1 < i2 & i1 < i2 holds ( ( i + 1 ) * ( ( i + 1 ) * ( ( i + 1 ) * ( i + 1 ) ) ) = ( i + 1 ) * ( ( i + 1 ) * ( ( i + 1 ) * ( ( i + 1 ) * ( ( i + 1 ) ) ) ;
assume that 0 in Z and Z c= dom ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
cell ( G1 , i1 -' 1 , j1 -' 1 ) \ ( ( L~ f ) \ ( L~ f ) \ ( L~ f ) \/ ( L~ f ) \/ ( L~ f ) ) c= ( ( L~ f ) \ ( L~ f ) \/ ( L~ f ) \/ ( L~ f ) ) \/ ( ( L~ f ) \ ( L~ f ) \/ ( L~ f ) /\ ( ( L~ f ) /\ ( L~ f ) ) ) ;
ex Q1 being open Subset of X st s = Q1 & ( for n being Nat holds F . n = F ( n ) ) & ( for n being Nat holds F . n = union ( F ( n ) ) ) & ( F . n ) is finite implies F is finite ) & for n being Nat holds F . n = union ( F . n )
gcd ( A , B , C , D ) = 1 / ( ( the carrier of A ) , the carrier of B ) & ( the carrier of A ) , ( the carrier of A ) , ( the carrier of B ) , ( the carrier of A ) , ( the carrier of B ) , the carrier of C = the carrier of A ;
R8 = ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
CurInstr ( P3 , Comput ( P3 , s3 , m1 + 1 ) ) = CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= halt SCMPDS ;
P1 /\ P2 = ( { p1 } \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) ) \/ ( LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) ) .= { p1 } \/ LSeg ( p1 , p2 ) \/ { p2 } .= { p1 } \/ { p1 } .= { p1 } \/ { p2 } ;
func -> Subset of the carrier of Al means : Def1 : ex p st p in it & p in it & not p in it & p in it & p in it & p = q ;
for a , b being Element of COMPLEX st |. a .| > |. b .| & |. b .| > 1 & a * ( - b ) >= - 1 holds ( a * ( - b ) ) / ( - ( - b ) ) / ( - ( - b ) ) / ( - ( - b ) ) / ( - ( - b ) ) / ( - ( - b ) ) / ( - ( - b ) ) / ( - ( - ( - b ) ) ) / ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( b ) ) ) ) ) ) ) ) ) ) / ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - (
defpred P [ Nat ] means 1 <= $1 & $1 <= len g implies for i , j st [ i , j ] in Indices G & G * ( i , j ) = G * ( i , j ) & G * ( i , j ) = G * ( i , j ) ;
assume that C1 , C2 , f , g , h being Function of C1 , C2 and f = g and h = h and g = h and h = i and g = h and h = i and h = i and g = j ;
( ||. f .|| | X ) . c = ||. f .|| /. c .= ( f | X ) /. c .= ( f | X ) /. c .= ( f | X ) /. c .= ( f | X ) /. c .= ( f | X ) /. c .= ( f | X ) /. c .= ( f | X ) /. c .= ( f | X ) /. c .= ( f | X ) /. c ) /. c ;
|. q .| ^2 = ( ( q `2 / |. q .| - sn ) / ( 1 - sn ) ) ^2 + ( ( q `2 / |. q .| - sn ) / ( 1 - sn ) ) ^2 + ( ( q `2 / |. q .| - sn ) / ( 1 - sn ) ) ^2 + ( q `2 / |. q .| - sn ) ^2 + ( q `2 / |. q .| ) ^2 ) ;
for F being Subset-Family of T st F is open & not {} in F & not ex A being Subset of T st A in F & A in F & A c= F & A c= F & A c= F & A c= F & A c= F & A c= F & A c= F & A c= F & F c= F & F c= G & F c= G & G c= G implies F c= G
assume that len F >= 1 + 1 and len F = k + 1 and for k st k in dom F holds F . k = G . ( F . k ) and for k st k in dom F holds F . k = G . ( F . k ) ;
i |^ ( ( order ( n , m ) - i ) ) |^ ( ( n + m ) - i ) = i |^ ( ( n + m ) - i ) .= i |^ ( ( n + m ) - i ) * ( i |^ ( n + m ) ) .= i |^ ( n + m ) - i |^ ( n + m ) ;
consider q being oriented oriented Chain of G such that r = q and q <> {} and ( for q st q in G holds q . ( q . ( q . ( q . ( q . ( q . ( q . ( q . ( q . ( q . ( q . ( q . ( q . ( q . ( q . ( q . ( q . ( q . ( q . ( q . ( q . ( q . ( q ) q ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) & ( q . ( q . ( q . ( q . ( q . ( q . ( q . ( q . ( q . ( q . ( q . ( q . ( q . ( q . ( q . ( q . ( q . ( q . (
defpred P [ Element of NAT ] means ( ( X , Z ) --> ( f , Z ) ) . ( $1 + 1 ) = ( ( X , Z ) +* ( f , Z ) ) . ( $1 + 1 ) ;
for A , B being Matrix of n , REAL holds len ( A * B ) = len A & width ( B * A ) = width B & width ( B * A ) = width B & width ( B * A ) = width B & width ( B * A ) = width B & width ( B * A ) = width B & width ( B * A ) = width B
consider s being FinSequence of the carrier of R such that Sum s = u & for i being Element of NAT st i in dom s holds s . i = a * s . i & s . i = b * s . i ;
func Re ( x , y ) -> Element of COMPLEX equals ( Re ( x , y ) ) * ( Re ( x , y ) ) + ( Im ( x , y ) ) * ( Im ( x , y ) ) + ( Im ( x , y ) ) * ( Im ( x , y ) ) ;
consider g2 be FinSequence of ( the carrier of Y ) | A such that g2 is continuous and rng g2 c= A and g2 . 1 = ( f | A ) . ( g2 . len g2 ) and g2 is continuous and g2 is continuous and rng g2 c= A and g2 is continuous and rng g2 c= A and g2 is continuous and rng g2 c= A and g2 is continuous and rng g2 c= A ;
then n1 >= len p1 & n2 >= len p2 & p1 = crossover ( p1 , p2 , p3 , p4 , p2 , p1 , p2 , p3 , p4 , p1 , p2 , p4 , p1 , p2 , p1 , p2 , p4 , p2 , p1 , p2 , p3 , p4 , p1 , p2 , p4 , p1 , p2 , p2 , p1 , p2 , p4 , p2 , p1 , p2 , p2 , p1 , p2 , p1 , p2 , p3 , p1 , p2 , p2 , p1 , p2 , p4 , p1 , p2 , p1 , p2 , p4 , p1 , p2 , p2 , p4 , p1 , p2 , p4 , p1 , p2 , p1 , p2 , p2 , p1 , p2 , p4 , p1 , p2 , p4 , p1 , p2 , p4 , p4 , p1 , p2 , p1 , p1 , p2 , p2 , p1 , p3 , p2 , p1 , p2 , p3 ,
( q `1 ) * a <= ( q `1 ) * ( ( q `1 ) * ( q `2 ) ) & ( q `1 ) * ( ( q `2 ) * ( q `2 ) ) <= ( q `1 ) * ( q `2 ) ) * ( ( q `2 ) * ( q `2 ) ) & ( q `1 ) * ( q `2 ) >= 0 ;
( F . ( p . ( len p ) ) ) = F . ( p . ( len p ) ) .= F . ( p . ( len p ) ) .= v . ( p . ( len p ) ) .= v . ( p . ( len p ) ) .= v . ( p . ( len p ) ) ;
consider k1 being Nat such that k1 + k = 1 and a := k1 = ( <* a *> --> ( k + 1 ) ) ^ ( <* a *> --> ( k + 1 ) ) ^ ( <* a *> --> ( k + 1 ) ) ;
consider B1 being Subset of [: B1 , B2 :] , B2 being finite Subset of [: B1 , B2 :] such that B1 is finite and B2 is finite and B1 c= the carrier of [: B1 , B2 :] and B1 c= the carrier of [: B1 , B2 :] and B2 c= the carrier of B1 and B2 c= the carrier of B1 ;
v2 . ( b2 , g ) = ( curry ( F2 , g ) ) . ( ( curry ( F2 , g ) ) . ( ( curry ( F2 , g ) ) . ( ( curry ( F2 , g ) ) . ( ( curry ( F2 , g ) ) . ( ( curry ( F2 , g ) ) . ( ( curry ( F2 , g ) ) . ( ( curry ( F2 , g ) ) . ( ( curry ( F2 , g ) ) . ( ( curry ( F2 , g ) ) ) ) ) ) ) .= ( ( ( F2 , g ) . ( ( curry ( F2 , g ) ) . ( ( curry ( F2 , g ) ) . ( ( F2 , g ) ) ) ) .= ( ( ( F2 , g ) ) ) ) . ( ( F2 , ( F2 , ( F2 , ( F2 , ( F2 , ( F2 , ( F2 ,
dom IExec ( I1 , P , s ) = the carrier of SCMPDS .= ( ( card I + card J + card J + card J + card I + card J + card J + card I + card J + card J + card I + card J + card I + card J + card J + card I + card J + card J + card I + card J + card J + card I + card J + card I + card J + card J + card I + card I + card I + card I + card I + card I + card I + card I + card I + card I + card I + card I + card J + card J + card J + card J + card J + card J + card J + card J + card J + card J + card J + card J + card J + card I + card J + card I + card J + card
ex d1 be Real st d1 > 0 & for h be Real st h in [. 0 , PI / 2 * ( ( R + R ) * ( L + R ) ) * ( h + R ) ) & ( h + R ) * ( L + R ) ) * ( h + R ) < ( R + ( R + R ) ) * ( h + R ) ) * ( h + R ) ;
LSeg ( G * ( len G , 1 ) , G * ( len G , 1 ) ) c= Int cell ( G , len G , 1 ) \/ { G * ( len G , 1 ) } \/ { G * ( len G , 1 ) } ;
LSeg ( mid ( h , i1 , i2 ) , i ) = LSeg ( h /. ( i + 1 ) , h /. ( i + 1 ) ) .= LSeg ( h /. ( i + 1 ) , h /. ( i + 1 ) ) .= LSeg ( h /. ( i + 1 ) , h /. ( i + 1 ) ) ;
A = { q where q is Point of TOP-REAL 2 : LE q , q , P , p1 , p2 & LE q , q , P , p1 , p2 & LE q , p1 , P , p1 , p2 } ;
( ( - x ) | ( - x ) ) | ( ( - x ) | ( - x ) ) = ( - x ) | ( - x ) .= ( ( - x ) | ( - x ) ) | ( - x ) .= ( - x ) | ( - x ) .= ( - x ) | ( - x ) .= ( - x ) | ( - x ) ;
0 * sqrt ( 1 + ( p `2 / p `1 ) ^2 ) = sqrt ( ( p `1 / p `1 ) ^2 + ( p `2 / p `1 ) ^2 ) * sqrt ( 1 + ( p `2 / p `1 ) ^2 ) .= ( p `1 / p `1 ) ^2 + ( p `2 / p `1 ) ^2 ;
sqrt ( ( ( U1 + U2 ) * ( ( U1 + U2 ) * ( q + j ) ) ) ^2 = ( ( U1 + U2 ) * ( q + j ) ) ^2 .= ( ( U1 + U2 ) * ( q + j ) ) ^2 .= ( U1 + U2 ) * ( q + j ) ) ^2 .= ( U1 + U2 ) * ( q + j ) ;
func Shift ( f , h ) -> PartFunc of REAL , REAL n means : Def1 : for x be Element of REAL m holds it . x = ( f + h ) . x & ( f + h ) . x = ( f + h ) . x + ( f + g ) . x = ( f + g ) . x + ( f + h ) . x ;
assume that 1 <= k and k + 1 <= len f and [ i , j ] in Indices G and f /. k = G * ( i , j ) and f /. k = G * ( i , j ) and f /. ( k + 1 ) = G * ( i , j ) ;
assume that not y in Free H and not ( x in Free ( H ) or y in Free ( H ) ) and not ( x in Free ( H ) ) & not ( x in Free ( H ) ) & ( not x in Free ( H ) ) or x in Free ( H ) ) ;
defpred P [ Element of NAT ] means ( p |^ $1 ) |^ $1 = p |^ $1 & ( p |^ $1 ) |^ $1 = p |^ $1 |^ $1 & ( p |^ $1 ) |^ $1 = p |^ $1 |^ $1 & ( p |^ $1 ) |^ $1 = p |^ $1 |^ $1 implies p |^ $1 = p |^ $1 * ( p |^ $1 ) |^ $1 * ( p |^ $1 ) |^ $1 * ( p |^ $1 ) * ( p |^ $1 ) ) ;
func natural number ( C ) -> non empty Subset-Family of X means : Def1 : for A being Subset of X holds A in it iff A in it iff C in it & A c= it & for W being Subset of X st W in it holds C . W = C . ( W . W ) ;
[#] ( ( ( ( dist ( P ) ) .: Q ) ) .: Q ) = ( ( dist ( P ) ) .: Q ) .: Q & ( ( dist ( P ) ) .: Q ) .: Q = ( ( dist ( P ) ) .: Q ) .: Q ) .: Q & ( dist ( P ) ) .: Q = ( dist ( P ) ) .: Q ;
rng ( F | ( [: S , T :] , { 0 } :] ) = { 1 } or rng ( F | ( [: S , T :] , { 0 } :] ) = { 1 } or rng ( F | [: S , T :] , { 0 } :] ) = { 1 } or rng ( F | [: S , T :] ) = { 1 } ;
( f " ( ( ( f " ) (#) ( f " ) ) * ( f " ) ) ) . i = ( f " ) . i .= ( f " ) . i .= ( f " ) . i .= ( f " ) . i .= ( f " ) . i .= ( f " ) . i .= ( f " ) . i .= ( f " ) . i ;
consider P1 , P2 being non empty Subset of TOP-REAL 2 such that P1 /\ P2 = { p1 , p2 } and P1 \/ P2 = { p1 , p2 } and P1 \/ P2 = { p1 , p2 } and P1 \/ P2 = { p1 , p2 } and P1 \/ P2 = { p1 , p2 } and P1 \/ P2 = { p1 , p2 } and P1 = { p1 , p2 } and P1 = { p1 , p2 } ;
f . p2 = |[ ( ( p2 `2 ) / |. p2 .| ) ^2 + ( p2 `2 / |. p2 .| ) ^2 + ( p2 `2 / |. p2 .| ) ^2 + ( p2 `2 / |. p2 .| ) ^2 ;
( \HM { the } \HM { carrier } \HM { of X , X ) " . x = ( \HM { the } \HM { carrier } \HM { of X , Y ) . x .= ( the TopStruct of X , Y ) . x .= 0. X + ( Y + X ) . x .= 0. X + 0. Y .= 0. Y + 0. Y .= 0. Y + 0. Y + 0. Y ;
for T being non empty normal TopSpace , A being Subset of T , B being Subset of T , r being Real st A <> {} & B <> {} & r in A holds r in B & r in A & for p being Point of T st p in B holds p in B & p in A & p in B holds r <= p ;
for i , j being strict Subgroup of G st i + 1 in dom F & j in dom F & i + 1 in dom F holds F . i = F . ( i + 1 ) & G . ( i + 1 ) = F . ( i + 1 ) & G . ( i + 1 ) = G . ( i + 1 )
for x st x in Z holds ( ( ( arctan * arctan ) + ( arccot * arccot ) ) `| Z ) . x = ( ( arctan * arccot ) . x ) / ( 1 + ( arctan * arccot ) . x ) ^2 ) / ( 1 + ( arctan * arccot ) . x ) ^2
pred f /* a is convergent means : Def1 : for x0 st x0 in dom ( f /* a ) & x0 in dom ( f /* a ) & for r st x0 in dom ( f /* a ) holds ( f /* a ) . ( r / a ) <= ( f /* a ) . ( r / a ) ;
then X1 , X2 are_separated & X1 , X2 are_separated implies X1 union X2 , X1 union X2 , X2 union X1 , X2 union X2 , X1 union X2 , X2 union X1 , X2 union X2 , X1 union X2 , X2 union X2 , X2 union X2 , X1 union X2 , X2 union X2 , X1 union X2 , X2 union X1 , X2 union X2 , X1 union X2 , X2 union X2 , X1 union X2 , X2 union X2 , X1 union X2 , X2 union X2 , X1 union X2 , X2 union X2 , X1 union X2 , X1 union X2 , X1 union X2 , X2 union X2 , X2 union X2 , X1 union X2 , X1 union X2 , X1 union X2 , X1 union X2 , X1 union X2 , X2 union X2 , X1 union X2 , X2 union X2 , X2 union X2 , X2 union X2 , X1 union X2 , X1 union X2 , X1 union X2 , X1 union X2 , X1 union X2 union X2 union X2 union X2 , X1 union X2 , X1 union X2 , X1 union X2 , X2 union X2 union X2 union X2 union X2 union X2 union X2 union X2 union X2 , X1 union X2 union X2 union X2 , X2 union X2 union X2 union
ex N be Neighbourhood of x0 st N c= dom ( SVF1 ( 1 , f , u ) ) & ( SVF1 ( 1 , f , u ) ) . ( x - x0 ) = ( SVF1 ( 1 , f , u ) ) . ( x - x0 ) ;
sqrt ( ( p2 `1 ) ^2 + ( p2 `2 ) ^2 + ( p2 `2 ) ^2 + ( p2 `2 ) ^2 ) >= sqrt ( ( p2 `1 ) ^2 + ( p2 `2 ) ^2 + ( p2 `2 ) ^2 ) ;
( sqrt ( 1 - ( t * ( 1 / t ) ) ^2 ) / ( 1 - t * ( 1 / t ) ) ^2 ) = ( ( 1 - t ) / ( 1 - t ) ) / ( 1 - t ) ) / ( 1 - t ) .= ( 1 - t ) / ( 1 - t ) ) / ( 1 - t ) ;
assume that for x holds f . ( x + h ) = ( ( - 1 ) (#) ( ( sin + cos ) (#) ( sin + cos ) ) `| Z ) & ( f + ( cos + sin ) (#) ( cos + cos ) ) `| Z ) . x = ( ( sin + cos ) (#) ( cos + cos ) ) `| Z ) . x ;
consider X1 being Subset of Y , Y1 being Subset of X such that t = X1 & Y1 in X1 & Y1 in X2 and X1 c= X1 and Y1 c= X1 and Y1 c= X2 and Y1 c= X1 and Y1 c= X2 and Y1 c= X1 and Y1 c= X1 and Y1 c= X1 and Y1 c= X2 and Y1 c= X1 and Y1 c= X1 and Y1 c= X1 and Y1 c= X1 and Y1 c= X1 and Y1 c= X1 and Y1 c= X1 and X1 c= X2 and X1 c= X1 and Y1 union Y1 union Y1 & X1 union Y1 union Y1 is Subset of X1 union X2 and X1 c= X1 union X2 and X1 union X2 and X1 union Y1 is open X1 union X2 and X1 union Y1 & Y1 = X1 union X2 and X1 union X2 and X1 c= X1 union X2 and X1 c= X1 union X2 and X1 union X2 is open implies X1 union X2 = X1 union X2 and X1 union X2 implies X1 union X2 = X1 union X2 & X1 union X2 is open X1 union X2 is open implies X1 union X2 implies X1 union X2 = X1 union X2 & X1 union X2 is open implies X1 union X2 = X1 union X2 & X1 union X2 = X1 union X2 is open
card S = card { ( d |^ n ) + ( d |^ ( n + 1 ) ) * b |^ ( n + 1 ) where d is Element of GF ( p ) : d in { ( d |^ ( n + 1 ) ) * b |^ ( n + 1 ) } .= ( d |^ ( n + 1 ) ) * ( ( d |^ ( n + 1 ) ) * b |^ ( n + 1 ) ) ;
sqrt ( ( ( ( W-bound D ) - ( E-bound D ) / 2 ) / 2 ) ^2 + ( ( E-bound D ) / 2 ) ^2 = ( ( E-bound D ) / 2 ) ^2 + ( ( E-bound D ) / 2 ) ^2 .= ( ( E-bound D ) / 2 ) ^2 + ( ( E-bound D ) / 2 ) ^2 ;