thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b
D c= S
let Y ;
S ` is Cauchy
q in P ;
V in F ;
y in N ;
x in T ;
m < n ;
m <= n ;
n > 1 ;
let r ;
t in I ;
n <= 4 ;
M is finite ;
let X ;
Y c= Z ;
A // M ;
let U ;
a in D ;
q in Y ;
let x ;
1 <= l ;
1 <= w ;
let G ;
y in N ;
f = {} ;
let x ;
x in Z ;
let x ;
F is one-to-one ;
e <> b ;
1 <= n ;
f is special ;
S misses C
t <= 1 ;
y divides m ;
P divides M ;
let Z ;
let x ;
y c= x ;
let X ;
let C ;
x _|_ p ;
o is monotone ;
let X ;
A = B ;
1 < i ;
let x ;
let u ;
k <> 0 ;
let p ;
0 < r ;
let n ;
let y ;
f is onto ;
x < 1 ;
G c= F ;
a >= X ;
T is continuous ;
d <= a ;
p <= r ;
t < s ;
p <= t ;
t < s ;
let r ;
D <= E ;
assume e > 0 ;
assume 0 < g ;
p in X ;
x in X ;
Y `1 in Y ;
assume 0 < g ;
not c in Y ;
not v in L ;
2 in z `1 ;
assume f = g ;
N c= b ` ;
assume i < k ;
assume u = v ;
I = J ;
B `1 = b `1 ;
assume e in F ;
assume p > 0 ;
assume x in D ;
let i be element ;
assume F is onto ;
assume n <> 0 ;
x be element ;
set k = z ;
assume o = x ;
assume b < a ;
assume x in A ;
a `1 <= b `1 ;
assume b in X ;
assume k <> 1 ;
f = product l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is generated ;
assume m > 0 ;
assume A c= B ;
X is lower bounded
assume A <> {} ;
assume X <> {} ;
assume F <> {} ;
assume G is open ;
assume f is dilatation ;
assume y in W ;
y \not <= x ;
A ` in B ` ;
assume i = 1 ;
let x be element ;
x `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 , A , B be Subset of G ;
let G be _Graph , A , B be Subset of G ;
let a be UNKNOWN of F ;
let x be element ;
let x be element ;
let C be FormalContext , A , B 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 ;
y be Real ;
X c= f . a
let y be element ;
let x be element ;
i be Nat ;
let x be element ;
n in NAT ;
let a be element ;
m in NAT ;
let u be element ;
i in NAT ;
let g be Function ;
Z c= NAT ;
l <= NAT ;
let y be element ;
r2 in X ;
let x be element ;
k1 be Integer ;
let X be set ;
let a be element ;
let x be element ;
let x be element ;
let q be element ;
let x be element ;
assume f is being_homeomorphism ;
let z be element ;
a , b // K ;
let n be Nat ;
let k be Nat ;
B ` c= B ` ;
set s = 1 ;
n >= 0 + 1 ;
k c= k + 1 ;
R1 c= R ;
k + 1 >= k ;
k c= k + 1 ;
let j be Nat ;
o , a // Y ;
R c= Cl G ;
Cl B = B ;
let j be Nat ;
1 <= j + 1 ;
arccot is_differentiable_on Z ;
exp 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 <> b3 ;
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 <= r0 ;
let e be Real , x be Real ;
not r in G . l
c1 = 0 ;
a + a = a ;
<* 0 *> in e ;
t in { t } ;
assume F is non 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 downarrow x -> being being and x is closed ;
X <> { x } ;
x in { x } ;
q , b // M ;
A . i c= Y ;
P [ k ] ;
2 |^ x in W ;
X [ 0 ] ;
P [ 0 ] ;
A = A |^ i ;
2 >= s ;
G . y <> 0 ;
let X be RealNormSpace , A be Subset of X ;
a in X ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , A , B be Subset of V ;
assume x in - M ;
k < s . a ;
not t in { p } ;
let Y be such that Y is such that X c= Y and Y c= X ;
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 <= ii ;
p10 c= cos .: A ;
1 <= ii ;
1 <= ii ;
UMP 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 ;
assume I c= P1 ;
n in dom I ;
let Q ;
B c= dom f ;
b + p \not _|_ 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 >= Z1 & b2 in Z1 ;
assume w = 0. V ;
assume x in A . i ;
g in the carrier of _ X ;
y in dom t ;
i in dom g ;
assume P [ k ] ;
if C c= dom f holds f in C
x4 is increasing ;
let e2 be element ;
- b divides b ;
F c= \tau ( F ) ;
IT is non-decreasing ;
IT is non-decreasing ;
assume v in H . m ;
assume b in [#] B ;
let S be non void ManySortedSign , A , B be non-empty MSAlgebra over S ;
assume P [ n ] ;
assume union S is independent & card S is finite ;
V is Subspace of V ;
assume P [ k ] ;
rng f c= NAT ;
assume ex_inf_of X , L ;
y in rng f ;
let s , I be set , F be Function of s , I ;
b `1 c= b9 & b `1 c= b9 ;
assume not x in NAT + 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 & i2 < i2 ;
a * h in a * H ;
p , q in Y ;
cluster sqrt I -> left ideal ;
q1 in A1 & q2 in A2 ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A2 c= A1 ;
n < n & n < m ;
assume A c= dom f ;
Re ( f ) is_integrable_on M ;
let k , m be element ;
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_in x0 & g is_differentiable_in x0 ;
assume O is symmetric 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 ;
v be Vector of V ;
P3 halts_on s , P2 = P +* stop I ;
d , c // a , b ;
let t , u ;
let X be set with a non-empty ManySortedSet of X ;
assume k in dom s ;
let r be non negative Real ;
assume x in F | M ;
Y be Subset of S ;
let X be non empty TopSpace , A be Subset of X ;
[ a , b ] in R ;
x + w < y + w ;
{ a , b } >= c ;
let B be Subset of A , a , b be Element of B ;
let S be non empty ManySortedSign ;
let x be variable , f , g be Function ;
let b be Element of X , a , b be Element of X ;
R [ x , y ] ;
x ` ` = x ;
b \ x = 0. X ;
<* d *> in D |^ 1 ;
P [ k + 1 ] ;
m in dom ( n - 1 ) ;
h2 . a = y ;
P [ n + 1 ] ;
cluster G * F -> Int W ;
let R be non empty multiplicative loop over R , S be Subset of R ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p `1 ;
assume f | X is lower bounded ;
x in rng co ;
let x be Element of B ;
let t be Element of D ;
assume x in Q .first() ;
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 void maid ;
let N be non empty // \cal \cal \in ;
let R be RelStr with finite and R is finite ;
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 not I does not destroy a ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < v9 ;
x <= c2 . x ;
x in F ` ;
cluster S --> T -> \in such that S --> is \cal ;
assume t1 <= t2 & t2 <= t1 ;
let i , j be even ;
assume F1 <> F2 & F2 <> F1 ;
c in Intersect ( union R ) ;
dom p1 = c & dom p2 = c ;
a = 0 or a = 1 ;
assume A1 <> A2 & A2 <> A3 ;
set i1 = i + 1 ;
assume a1 = b1 & a2 = b2 ;
dom g1 = A & rng g2 = A ;
i < len M + 1 ;
assume not - r in rng G ;
N c= dom f1 & N c= dom f2 ;
x in dom ( sec | Z ) ;
assume [ x , y ] in R ;
set d = ( x - y ) / 2 ;
1 <= len g1 & len g2 <= len g2 ;
len s2 > 1 & len s2 > 1 ;
z in dom f1 & z in dom f2 ;
1 in dom D2 & D2 . 1 = D2 . 1 ;
( p `2 ) ^2 = 0 ;
j2 <= width G & j2 <= width G ;
len cos > 1 + 1 ;
set n1 = n + 1 ;
|. q2 .| = 1 ;
let s be SortSymbol of S ;
( i , i ) *' = i ;
X1 c= dom f & X2 c= dom f ;
h . x in h . a ;
let G be upper_bound as line ;
cluster m * n -> square ;
let k9 be Nat ;
i -' 1 > m ;
R is transitive implies R is transitive
set F = <* u , w *> ;
p0 c= P3 & p4 c= P3 ;
I is_halting_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 f2 ;
assume [ X , p ] in C ;
BX c= XX & BX c= BX ;
n2 <= ( 2 |^ 4 ) * ( 2 |^ 4 ) ;
A /\ c9 c= A ` ;
cluster -> x -valued for Function ;
Q be Subset-Family of S , B be Subset of T ;
assume n in dom g2 ;
let a be Element of R ;
t `1 in dom e2 & t `2 in dom e2 ;
N . 1 in rng N ;
- z in A \/ B ;
let S be SigmaField of X , T be Subset of X ;
i . y in rng i ;
REAL c= dom f & rng f c= dom f ;
f . x in rng f ;
NAT <= ( r - 1 ) / 2 ;
s2 in r-5 & s1 in r-5 ;
let z , z be complex number ;
n <= NN . m ;
LIN q , p , s ;
f . x = waybelow x /\ B ;
set L = [ S \to T ] ;
let x be non positive Real ;
m be Element of M ;
f in union rng F1 ;
let K be add-associative right_zeroed right_complementable non empty doubleLoopStr , M , N be Matrix of K ;
let i be Element of NAT ;
rng ( F * g ) c= Y
dom f c= dom x & rng f c= dom x ;
n1 < n1 + 1 & n2 + 1 < n1 ;
n1 < n1 + 1 & n2 + 1 < n1 ;
cluster {} T -> \rbrace for set ;
[ y2 , 2 ] `2 = z ;
let m be Element of NAT ;
let S be Subset of R ;
y in rng Sit ;
b = sup ( dom f ) ;
x in Seg ( len q ) ;
reconsider X = [: D , D :] as set ;
[ a , c ] in E1 ;
assume n in dom h2 ;
w + 1 = ma1 ;
j + 1 <= j + 1 + 1 ;
k2 + 1 <= k1 & k2 + 1 <= k2 ;
i be Element of NAT ;
Support u = Support p & Support u = Support p ;
assume X is complete complete ] ;
assume f = g & p = q ;
n1 <= n1 + 1 & n2 + 1 <= n1 ;
let x be Element of REAL ;
assume x in rng s2 & y in rng s2 ;
x0 < x0 + 1 & x0 < x0 + 1 ;
len ( L - K ) = W ;
P c= Seg ( len A ) ;
dom q = Seg n & rng q c= Seg n ;
j <= width M *' ;
let r8 be real-valued FinSequence ;
let k be Element of NAT ;
Integral ( M , P ) < +infty ;
let n be Element of NAT ;
assume z in 0 := f . 0 ;
let i be set ;
n -' 1 = n-1 ;
len ( n - m ) = n ;
cell ( Z , c , F ) 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 |^ omega ;
let B1 be Basis of x , B ;
L2 /\ L2 = {} ;
L1 /\ L2 = {} ;
assume downarrow x = downarrow y ;
assume b , c '||' b , c ;
LIN q , c , c ;
x in rng ( f-1J ) ;
set n8 = n + j ;
let D7 be non empty set , A , B be Matrix of n , K ;
let K be add-associative right_zeroed right_complementable non empty doubleLoopStr , M be Matrix of K ;
assume f `1 = f & h `1 = h ;
R1 - R2 is total & R2 is total ;
k in NAT & 1 <= k & k <= n ;
let a be Element of G ;
assume x0 in [. a , b .] ;
K1 ` is open Subset of ( TOP-REAL 2 ) | K1 ;
assume a , b ] in maximal ;
a , b be Element of S ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = Integral ( M , f ) ;
cluster as nreal for \vert -nA1 ;
not u in { is_differentiable_on g } ;
the carrier of f c= B ;
reconsider z = x as VECTOR of V ;
cluster the RelStr of L -> to ;
r (#) H is partial ;
s . intloc 0 = 1 ;
assume that x in C and y in C ;
let U0 be strict non-empty MSAlgebra over S , A be non-empty MSAlgebra over S ;
[ x , Bottom T ] is compact ;
i + 1 + k in dom p ;
F . i is stable Subset of M ;
r-35 in { ( r . y ) `1 : not contradiction } ;
let x , y be Element of X ;
A , I be \cal _ empty set ;
[ y , z ] in O ;
card Macro i = 1 + 1 ;
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 o9 , a , b ;
p . 2 = Z |^ Y ;
( D . 0 ) `2 = {} ;
n + 1 + 1 <= len g ;
a in VERUM ( Al ) ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
I be Ideal of L ;
set g = f1 + f2 , h = f2 + f3 ;
a <= max ( a , b ) ;
i-1 < len G + 1-1 ;
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 multiplicative multMagma ;
x in support ( ( support t ) | ( support t ) ) ;
assume a in [: the carrier of G , the carrier of G :] ;
i `1 <= len ( y-5 ) ;
assume p divides b1 + b2 ;
M1 <= sup M1 & sup M1 <= sup M1 ;
assume x in ( W-min X ) .: X ;
j in dom ( z | n ) ;
let x be Element of D ( ) ;
IC Comput ( P3 , s3 , l ) = l1 .= 0 ;
a = {} or a = { x } ;
set Wmin = Vertices G , u9 = Vertices G ;
seq " is non-zero implies seq " is non-zero
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
h-4 c= h-14 .: { x } ;
]. a , b .[ c= Z ;
X1 , X2 are_\Vert implies X1 , X2 are_\Vert
a in Cl ( union F \ G ) ;
set x1 = [ 0 , 0 ] ;
k + 1 -' 1 = k ;
cluster -> real-valued for Relation ;
ex v st C = v + W ;
let IT be non empty zero structure , a , b be Element of IT ;
assume V is Abelian add-associative right_zeroed right_complementable distributive non empty doubleLoopStr ;
XY \/ Y in \sigma ( L ) ;
reconsider x = x as Element of S ;
max ( a , b ) = a ;
sup B is upper & sup B in B ;
let L be non empty reflexive antisymmetric RelStr , X be Subset of L ;
R is reflexive & X is transitive implies R is transitive
E , g |= the_right_argument_of H ;
dom G /. y = a ;
sqrt ( 1 - 4 ) >= - r / 2 ;
G . p4 in rng G ;
let x be Element of FF , y be Element of FF ;
D [ P-6 , 0 ] ;
z in dom id ( B ) & z in B ;
y in the carrier of N & z in the carrier of N ;
g in the carrier of H & g in the carrier of H ;
rng fA1 c= [: { x } , NAT :] ;
j ' + 1 in dom s1 ;
A , B be strict Subgroup of G ;
C be non empty Subset of REAL ;
f . z1 in dom h & h . z2 in dom h ;
P . k1 in rng P ;
M = ( A +* {} ) +* {} ;
let p be FinSequence of REAL , r be Real ;
f . n1 in rng f & f . n2 in rng f ;
M . ( F . 0 ) in REAL ;
h | [. a , b .] = b ;
assume the distance of V , Q is_to v , w ;
let a be Element of ^ ( V ) ;
let s be Element of PP ( ) ;
let PP be non empty RelStr ;
n be Nat ;
the carrier of g c= B ;
I = halt SCM R & I = ( l , R ) --> ( l , R ) ;
consider b be element such that b in B ;
set BK = BCS K , BK = BCS K ;
l <= ( ( -> Real ) . j ;
assume x in downarrow [ s , t ] ;
( x - t ) / 2 in ]. t - t , s .[ ;
x in ( JumpParts T ) . a ;
let h be Morphism of c , a ;
Y c= { 1_ R } implies the_rank_of ( Y ) c= A
A2 \/ A3 c= L2 \/ L2 ;
assume LIN o , a , b ;
b , c // d1 , e2 ;
x1 , x2 in Y & x2 in Y ;
dom <* y *> = Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar s .| ;
[ x , x `1 ] in X ~ ;
for n be Nat holds 0 <= x . n
|[ a , b ]| = [. a , b .] ;
cluster -> -> -> -> -> -> sqrt closed for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 , q1 is_collinear & q2 , q2 , q2 is_collinear ;
dom M1 = Seg n & dom M2 = Seg n ;
x = [ x1 , x2 ] & y = [ x2 , x3 ] ;
R , Q be ManySortedSet of A ;
set d = ( 1 / n + 1 ) * ( 1 / n ) ;
rng g2 c= dom W & rng g2 c= dom W ;
P . ( [#] Sigma \ B ) <> 0 ;
a in field R & a = b ;
let M be non empty Subset of V , A , B be Subset of V ;
let I be Program of SCM+FSA , a , b be Int-Location ;
assume x in rng ( ( R * R ) * ( S * R ) ) ;
let b be Element of the lattice of T ;
dist ( e , z ) - r > r-r ;
u1 + v1 in W2 & v1 in W2 ;
assume func support L -> Subset of L equals rng G ;
let L be lower-bounded antisymmetric transitive antisymmetric transitive RelStr with L ;
assume [ x , y ] in a9 ;
dom ( A * e ) = NAT ;
let a , b be Vertex of G ;
let x be Element of Bool ( M ) ;
0 <= 2 * PI ;
o `1 , a9 // o `1 , y `2 ;
{ v } c= the carrier of l & { v } c= the carrier of l ;
x be bound of A ;
assume x in dom ( ( curry f ) . x ) ;
rng F c= ( product f ) |^ X
assume D2 . k in rng D ;
f " . p1 = 0 & f " . p2 = 1 ;
set x = the Element of X , y = the Element of X ;
dom Ser ( G ) = NAT & rng Ser ( G ) = NAT ;
n be Element of NAT ;
assume LIN c , a , e1 ;
cluster non empty 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 @ S c= conv @ A ;
reconsider B = b as Element of the lattice of T ;
J , v |= P \lbrack l , P \lbrack ;
cluster the carrier of J . i -> non empty for TopSpace ;
ex_sup_of Y1 \/ Y2 , T & ex_sup_of Y1 , T ;
W1 is_well field W1 & W2 is_\upharpoonright field W1 implies W1 is well field W2
assume x in the carrier of R & y in the carrier of R ;
dom ( n |-> 0 ) = Seg n & rng ( n --> 0 ) = Seg n ;
s4 misses s2 & not ( not x in dom ( s ) ) ;
assume ( a 'imp' b ) . z = TRUE ;
assume X is open & f = X --> d ;
assume [ a , y ] in Indices ( a * f ) ;
assume that that that that that that that that that card I c= J and card Shift J c= K ;
Im ( ( lim seq ) (#) ( Im seq ) ) = 0 ;
( sin * sin ) . x <> 0 & ( sin * cos ) . x <> 0 ;
sin * cos is_differentiable_on Z & cos * cos is_differentiable_on Z ;
t1 . n = t2 . n & t1 . n = t2 . n ;
dom ( ( - F ) | A ) c= dom F ;
W1 . x = W2 . x & W1 . x = W2 . x ;
y in W .first() \/ W .first() \/ W .first() ;
k9 <= len ( v | ( len v + 1 ) ) ;
x * a \equiv y * a . ( mod m ) ;
proj2 .: S c= proj2 .: P ;
h . p4 = g2 . I . I .= g2 . I ;
Gij `1 = U * ( 1 , 1 ) `1 .= Gij `1 ;
f . ( r - 1 ) in rng f ;
i + 1 + 1 <= len f ;
rng F = rng ( F | 2 ) ;
mode Subset of \in is well unital non empty multiplicative loop structure ;
[ x , y ] in A ~ ;
x1 . o in L2 . o ;
the carrier of support _ m c= B ;
not [ y , x ] in id X ;
1 + p .. f <= i + len f ;
seq ^\ k1 is lower bounded & seq ^\ k1 is lower bounded ;
len ( F - I ) = len I - len I ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be Complex , a , b be complex number ;
Comput ( P , s , n ) . x = s . x ;
k <= k + 1 & k <= len p ;
reconsider c = {} T as Element of L ;
let Y be \rbrace is an \rbrace \cal \cal \cal T ;
cluster -> directed-sups-preserving for Function of L , L ;
f . j1 in K . j1 ;
cluster J => y -> total for Function ;
K c= 2 -tuples_on the carrier of T ;
F . b1 = F . b2 ;
x1 = x or x1 = y or x1 = z or x1 = z ;
attr a <> {} means a / ( a * a ) = 1 ;
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 o9 , b , b9 & LIN o9 , b9 , b9 ;
reconsider m = x as Element of Funcs ( V , C ) ;
let f be non constant FinSequence of D ;
let F2 be non empty for F be non empty set ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in F-8 ;
reconsider p9 = x as Subset of m , n = m as Element of NAT ;
A , B , C be Element of R ;
cluster non empty for b9 for // -order ;
rng c `1 misses rng e & rng e c= rng e1 ;
z is Element of gr ( { x } ) ;
not b in dom ( a .--> p1 ) ;
assume that k >= 2 and P [ k ] ;
Z c= dom ( cot * cot ) & Z c= dom ( cot * cot ) ;
the component of Q c= UBD A & ( not x in A ) ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( 1 - ( x - a ) ) ;
pred f = u means a * f = a * u ;
for n holds P1 [ n ] implies P [ n + 1 ]
{ x . O : x in L } <> {} ;
x be Element of V . s ;
a , b be Nat ;
assume that S = S2 and p = p2 and p = p2 and q = p1 ;
gcd ( n1 , n2 ) = 1 & gcd ( n1 , n2 ) = 1 ;
set o9 = ( a * _ of INT ) . ( a , b ) ;
seq . n < |. r1 .| & |. r1 .| < r ;
assume that seq is increasing and r < 0 and seq is increasing ;
f . ( y1 , x1 ) <= a ;
ex c being Nat st P [ c ] ;
set g = { n |^ 1 : n in NAT } ;
k = a or k = b or k = c ;
a9 , b9 , c9 , b9 is_collinear & b9 , c9 , c9 is_collinear ;
assume that Y = { 1 } and s = <* 1 *> ;
I1 . x = f . x .= 0 .= 0 ;
W3 .first() = W2 . 1 & W . 1 = W2 . 1 ;
cluster -> trivial for subgraph of G , finite _Graph ;
reconsider u = u as Element of Bags X ;
A in B ^ implies A , B are_are \lbrack
x in { [ 2 * n + 3 , k ] } ;
1 >= sqrt ( ( q `1 ) ^2 ) / |. q .| ;
f1 is_] ] ] ] ] ] ] ] ] then implies f1 is_] not iff f1 = f2
( f . I ) `2 <= ( q `2 ) / |. q .| ;
h is_the carrier of Cage ( C , n ) ;
( b - a ) / 2 <= ( p - a ) / 2 ;
let f , g be Function of X , Y ;
S * ( k , k ) <> 0. K ;
x in dom ( max ( - f ) ) ;
p2 in NO . p1 & p2 in NO . p2 ;
len ( the_right_argument_of H ) < len H & len ( the_right_argument_of H ) < len H ;
F [ A , F-14 . A ] ;
consider Z such that y in Z and Z in X ;
attr 1 in C means A c= C |^ A ;
assume r1 <> 0 or r2 <> 0 or r1 <> 0 ;
rng q1 c= rng C1 & rng C1 c= rng C1 ;
A1 , L , A2 , A3 is_collinear implies A1 , A2 , A3 is_collinear
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in ^2 ( p , Sp ) & c in dom ( p , Sp ) ;
then S is atomic implies P-2 [ S ] ;
Cl Int [#] T = [#] T & Int [#] T = [#] T ;
f12 | A2 = f2 | A2 & f12 | A2 = f2 | A2 ;
0. M in the carrier of W & 0. M in the carrier of W ;
v , v `1 , v `2 be Element of M ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ V ;
let X be Subset of S , T be Subset of S ;
consider H1 such that H = 'not' H1 and H1 in H ;
1 c= ( 0 * p2 ) * ( \HM { the } \HM { the } \HM { function } ) ;
0 * a = 0. R .= a * 0 .= 0 ;
A |^ 2 = A ^^ ( A |^ 2 ) ;
set vY = v4 /. n , vY = v4 /. n ;
r = 0. ( \langle \cal E *> , \Vert \cdot \Vert \rangle ) ;
( f . p4 ) `1 >= 0 & ( f . p4 ) `2 >= 0 ;
len W = len ( W | ( W . n ) ) + 1 .= n ;
f /* ( s * G ) is divergent_to+infty ;
consider l being Nat such that m = F . l ;
t8 in W7 or W7 does not let b1 , b2 ;
reconsider Y1 = X1 as SubSpace of X | X1 as SubSpace of X ;
consider w such that w in F and not x in w ;
let a , b , c , d be Real ;
reconsider i = i as non zero Element of NAT ;
c . x >= id ( L . x ) ;
\sigma ( T ) \/ omega is Basis of T ;
for x being element st x in X holds x in Y
cluster [ x1 , x2 ] -> pair for set ;
sup downarrow a /\ downarrow t is Ideal of T ;
let X be with with NAT , F be non empty set ;
rng f = S1 -*> & f is c -iff f is one-to-one
let p be Element of B , x be the the sort of S ;
max ( N1 , 2 ) >= N1 & max ( N1 , 2 ) >= N1 ;
0. X <= b |^ ( m * NAT ) ;
assume that i in I and R1 . i = R ;
i = j1 & p1 = q1 & p2 = q2 & q1 = q2 ;
assume gR in the right & g in the right of g ;
let A1 , A2 be Point of S , A be Subset of T ;
x in h " P /\ [#] ( T1 | P ) ;
1 in Seg 2 & 1 in Seg 3 & 1 in Seg 3 ;
reconsider X-5 = X as non empty Subset of Tsuch that X = { x } and X is non empty and X is non empty ;
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the Source of G ) -valued ;
n1 <= i2 + len g2 & i2 + len g2 <= i2 + len g2 ;
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume v in the carrier' of G2 & v in the carrier' of G2 ;
y = Re ( y ) + ( Im ( y ) ) ;
( exists ( - 1 ) ) \equiv 1 ;
x2 is_differentiable_on ]. a , b .[ & ]. a , b .[ c= dom ( a (#) ( b (#) ( b `| Z ) ) ) ;
rng M5 c= rng D2 & rng D2 c= rng D1 ;
for p be Real st p in Z holds p >= a
( \bf X ) --> ( f . x ) = proj1 . x ;
( seq ^\ m ) . k <> 0 ;
s . ( G . ( k + 1 ) ) > x0 ;
( p -Path M ) . 2 = d ;
A \oplus ( B \ominus C ) = ( A \oplus B ) \ominus C ;
h \equiv gg . ( mod P ) ;
reconsider i1 = i-1 - 1 as Element of NAT ;
let v1 , v2 be VECTOR of V , a , b be Real ;
for V being Subspace of V holds V is Subspace of [#] V
reconsider i9 = i as Element of NAT , s be Element of NAT ;
dom f c= [: C , D :] ;
x in ( the Sorts of B ) . n /\ ( the Sorts of B ) . n ;
len [ len f1 , f2 ] in Seg len ( f1 ^ f2 ) ;
p1 c= the topology of T & P c= the topology of T ;
]. r , s .[ c= [. r , s .] ;
B2 be Basis of T2 , B be Basis of T2 ;
G * ( B * A ) = ( id o1 ) * ( B * A ) ;
assume that p , u are_\! to u and u , v are_collinear ;
[ z , z ] in union rng ( F | 5 ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S , G = $1 .. S ;
LIN a1 , b3 , b3 & LIN a1 , b3 , b3 ;
f " ( f .: x ) = { x } ;
dom ( w2 ) = dom r12 .= dom r12 .= dom r12 ;
assume that 1 <= i and i <= n and j <= n ;
( ( g2 ) . O ) `2 <= 1 ;
p in LSeg ( E . i , F . i ) ;
Ix * ( i , j ) = 0. K ;
|. f . ( s . m ) -g .| < g1 ;
q7 . x in rng ( q | Seg n ) ;
Carrier ( L2 ) misses Carrier ( L2 ) ` ;
consider c being element such that [ a , c ] in G ;
assume N\lbrack o1 , o2 .] = o9 & for o being object of S holds o . ( o1 , o2 ) = o1 . ( o2 , o1 ) ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= ( F |^ CZ ) " C ;
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [. f . j , f . j .] ;
attr 0 <= x & x ^2 <= 1 & x ^2 <= 1 ;
p `1 - q `2 <> 0. TOP-REAL 2 & p `2 <> 0. TOP-REAL 2 ;
cluster ( _ S ) --> T -> non empty ;
x be Element of S , T ;
cluster cluster cluster cluster cluster cluster there F , b ) -> one-to-one ;
|. i .| <= - 2 |^ n & |. i - n .| < r ;
the carrier of I[01] = dom P & the carrier of I[01] = dom P ;
n * ( n + 1 ) ! > 0 * n ;
S c= ( A1 /\ A2 ) /\ ( A2 /\ A3 ) ;
a3 , a4 // b3 , b3 & a3 , a4 // b3 , b3 ;
then dom A <> {} & dom A <> {} & rng A <> {} ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , Y , G , z , F , G , H , G ;
set v2 = v4 /. ( i + 1 ) , v1 = v4 /. ( i + 1 ) ;
x = r . n .= ( r . n ) / 2 ;
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 Upper_Arc ( P ) /\ Lower_Arc ( P ) ;
dom d2 = A2 & dom d2 = A2 & dom d2 = A2 ;
0 < ( p / 2 ) * ( ||. z .|| + 1 ) ;
e . ( m + 1 ) <= e . ( m + 1 ) ;
B \ominus X \/ B \ominus Y c= B \ominus X
- g < Integral ( M , Im ( g | B ) ) ;
cluster O := F -> \HM { additive } for operation of X ;
let U1 , U2 be non-empty MSAlgebra over S , B be non-empty MSAlgebra over S ;
Proj ( i , n ) * g is_differentiable_on X ;
x , y , z be Point of X , a , b , c be Real ;
reconsider p9 = 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 , a , b be Int-Location ;
assume - a is lower of - X & b is lower ;
Int Cl Int A c= Cl Int Cl Int A & Int Cl Int A c= Cl Int 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 `1 ) ^2 / ( p2 `2 ) ^2 ;
Cl Q ` = [#] ( T | P ) .= P ;
set S = the carrier of T , T = the carrier of T ;
set I8 = ' ( f , n ) , I8 = ' ( f , n ) , I8 = ' ( f , n ) ;
len p -' n = len q - n .= len p - n ;
A is Permutation of Seg len ( A , x , y ) ;
reconsider n6 = n6 - 1 as Element of NAT ;
1 <= j + 1 & j + 1 <= len ( s | [. 1 , 1 .] ) ;
let q9 , q9 be State of M , a , b be Element of M ;
a1 in the carrier of S1 & a2 in the carrier of S2 & a3 in the carrier of S2 ;
c1 /. n1 = c1 . n1 & c2 /. n1 = c1 . n1 ;
let f be FinSequence of TOP-REAL 2 , a , b be Real ;
y = ( f * S8 ) . x .= ( f * S8 ) . x ;
consider x being element such that x in = an " A ;
assume r in ( dist ( o ) ) .: P ;
set i2 = ( ( n + 1 ) -tuples_on h ) . i ;
h2 . ( j + 1 ) in rng h2 & h2 . ( j + 1 ) in rng h2 ;
Line ( Mit , k ) . i = M . i ;
reconsider m = ( x - 2 ) / 2 as Element of ( - 1 ) / 2 ;
let U1 , U2 be U2 of U0 , U2 be Subspace of U0 ;
set P = Line ( a , d ) ;
len p1 < len p2 + 1 & len p2 + 1 <= len p1 + 1 ;
let T1 , T2 be Scott Scott Scott Scott of L , T be Scott Scott topological of L ;
then x <= y & ( { x } c= { y } ) ;
set M = n -tuples_on the carrier of K ;
reconsider i = x1 , j = x2 as Nat ;
rng ( ( the_arity_of a ) | ( dom the_arity_of o ) ) c= dom H ;
z1 " = z1 " * ( z1 " ) .= z1 " * ( z1 " ) ;
x0 - sqrt ( r ^2 ) in L /\ dom f ;
then w is strict {} , \mathop { 0 } :] /\ \mathop { 1 } <> {} ;
set x-10 = xZ ^ <* Z *> ^ <* Z *> ^ <* Z *> ^ <* Z *> ;
len w1 in Seg len w1 & len w2 = len w2 & len w2 = width w1 ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of PFuncs ( V , { k } ) ;
x . n = ( |. a . n .| ) * ( |. a . n .| ) ;
( p `1 ) ^2 / ( |. p .| ) ^2 <= ( |. p .| ) ^2 / ( |. p .| ) ^2 ;
rng ( g | X ) c= L~ ( g | X ) ;
reconsider k = i-1 * ( l + j ) as Nat ;
for n be Nat holds F . n is \HM { 0 } ;
reconsider x9 = x9 as Vector of M , the carrier of M ;
dom ( f | X ) = X /\ dom f /\ X .= X ;
p , a // p , c & b , a // c , c ;
reconsider x1 = x as Element of REAL m , x2 be Element of REAL m ;
assume i in dom ( a * p ) ;
m . b9 = p . ( b9 , c ) .= ( m + 1 ) * ( m + 1 ) ;
a / ( s . m ) -f . n <= 1 ;
S . ( n + k + 1 ) c= S . ( n + k ) ;
assume B1 \/ C1 = B2 \/ C2 & C2 \/ C1 = B2 \/ C2 ;
X . i = { x1 , x2 } . i .= x2 ;
r2 in dom ( h1 + h2 ) & r2 in dom ( h1 + h2 ) ;
that ||. 0. R .|| = a and b-0 = b ;
F8 is_closed_on t1 , Q1 & Q1 is_halting_on t1 , Q1 ;
set T = For Incluster 1 ( X , x0 ) ;
Int ( Int R ) c= Int R & Int ( R ) c= Int R ;
consider y being Element of L such that c . y = x ;
rng ( F | _ { x } ) = { F . x } ;
G-23 " { c } c= B \/ S " S ;
f9 is Relation of [: X , X :] , X & X is Subset of [: X , Y :] ;
set RQ = the Point of P , RQ = the Point of Q , R = the Point of Q ;
assume that n + 1 >= 1 and n + 1 <= len M ;
k2 be Element of NAT , x be Element of NAT ;
reconsider p9 = u as Element of ( ( TOP-REAL n ) | ( i + 1 ) ) | ( i + 1 ) ;
g . x in dom f & x in dom g implies x in dom g
assume that 1 <= n and n + 1 <= len f1 and n + 1 <= len f1 ;
reconsider T = b * N as Element of G / ( N , G ) ;
len ( P\mathopen { - } P } ) <= len ( P\mathopen { - } P } ) ;
x " in the carrier of A1 & x " in the carrier of A1 & x " in the carrier of A1 ;
[ i , j ] in Indices ( A * ( i , j ) ) ;
for m be Nat holds Re ( F . m ) is simple
f . x = a . i .= a1 . k .= a1 . k ;
let f be PartFunc of REAL i , REAL n , x be Real ;
rng f = the carrier of ( ( Carrier A ) | ( the carrier of ( Carrier A ) | ( the carrier of ( Carrier A ) | ( the carrier of ( Carrier A ) | ( i + 1 ) ) ) ;
assume s1 = sqrt ( 2 * p ) ^2 + ( 2 * p ) ^2 ;
attr a > 1 & b > 0 & a |^ b > 1 ;
let A , B , C be Subset of IX ;
reconsider X0 = X , Y = Y as RealNormSpace ;
f be PartFunc of REAL , REAL , r be Real ;
r (#) ( v1 |-- I ) . X < r * 1 ;
assume that V is Subspace of X and X is Subspace of V ;
let t-3 , t-4 be binary relation ;
Q [ e1 \/ { v } ] \/ { v } [ e1 , f ] ;
g \circlearrowleft ( E-max L~ z ) = z & ( E-max L~ z ) .. z = z ;
|. |[ x , v ]| - |[ x , y ]| .| = v-0 ;
- f . w = - ( L * w ) ;
z -' y <= x iff z <= x + y & z <= x + y
sqrt ( 7 * p1 ) ^2 > 0 ;
assume X is BCK-algebra , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ;
F . 1 = v1 & F . 2 = v2 & F . 3 = v2 ;
( f | X ) . x2 = f . x2 ;
( tan * tan ) . x in dom ( sec * tan ) ;
i2 = ( f /. len f ) /. ( len f -' 1 ) & i2 = ( f /. len f ) . ( len f -' 1 ) ;
X1 = X2 \/ ( X1 \ X2 ) & X2 \ X1 = X2 ;
[. a , b , \bf 1_ G .] = 1_ G & [. a , b .] = 1_ G ;
let V , W be non empty VectSpStr over F_Complex , a , b be Element of K ;
dom g2 = the carrier of I[01] & rng g2 = 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 .= proj2 .: X ;
f . ( x , y ) = h1 . ( x `1 , y `2 ) ;
x0 - r < a1 . n & x0 < x0 + r ;
|. ( f /* s ) . k - G . 3 .| < r ;
len Line ( A , i ) = width A & width Line ( A , i ) = width A ;
SY ^2 = S . ( g . x ) ^2 .= S . ( g . x ) ^2 ;
reconsider f = v + u as Function of X , the carrier of Y ;
intloc 0 in dom Initialized ( p ) & Initialized ( p ) . intloc 0 in dom ( p ) ;
i1 , i2 , 3 , 4 , 5 , 6 , 7 } not exists exists a , b , c st i = b & ( a , b , c ) := c ;
( r + sqrt ( r ^2 + 1 ) ) / 2 = ( cos . x ) ^2 + 0 ;
for x st x in Z holds f2 is_differentiable_in x & f2 . x > 0 ;
reconsider q2 = ( q - x ) / 2 as Element of REAL ;
( 0 qua Nat ) + 1 <= i + j1 & j + 1 <= j ;
assume f in the carrier of [ X \to [#] Y ] ;
F . a = H / ( x , y ) . a ;
( TRUE _ { T } ) . ( u , u ) = TRUE ;
dist ( ( a * seq ) . n , h ) < r ;
1 in the carrier of [. 0 , 1 .] & [. 0 , 1 .] c= dom G ;
( ( p2 `2 ) - x1 ) - x1 > - g & ( p2 `2 ) - x1 < - g ;
|. r1 - r2 .| = |. a1 .| * |. p2 - p1 .| ;
reconsider S-14 = 8 as Element of Seg 8 ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b
DppW .first() = DW .first() + 1 .= DW . ( k + 1 ) ;
i1 = [: NAT + n , K :] & i2 = [: K , K :] & k = [: K , K :] ;
f . a [= f . ( f .: O1 "\/" a ) ;
pred f = v & g = u , h = v + u ;
I . n = Integral ( M , F . n ) ;
( chi ( S , T1 ) . s ) . s = 1 ;
a = VERUM ( A ) or a = VERUM ( A ) ;
reconsider k2 = s . b3 as Element of NAT ;
( Comput ( P , s , 4 ) ) . GBP = 0 ;
L~ M1 meets L~ ( R4 * R ) & L~ ( R4 * R ) meets L~ ( R4 * R ) ;
set h = the continuous Function of X , R | X ;
set A = { L . ( k9 . n ) : not contradiction } ;
for H st H is atomic holds P7 [ H ] ;
set b9 = S5 \ S5 , I5 = S5 \ S5 ;
Hom ( a , b ) c= Hom ( a opp , b opp ) ;
sqrt ( 1 + n ) < sqrt ( 1 - s ^2 ) ;
( l . 1 ) `1 = [ l . 1 , cod l ] `1 .= [ l . 1 , cod l ] ;
y +* ( i , y /. i ) in dom g ;
let p be Element of QC-WFF ( Al ) , A be Subset of QC-WFF ( Al ) ;
X /\ X1 c= dom ( f1 - f2 ) /\ ( X /\ X1 ) ;
p2 in rng ( f /^ p1 ) & p1 in rng ( f /^ ( p1 + 1 ) ) ;
1 <= indx ( D2 , D1 , j1 ) + 1 & indx ( D2 , D1 , j1 ) + 1 <= len D2 ;
assume x in K1 /\ K0 \/ ( 3 * K ) /\ K0 ;
- 1 <= ( ( f2 . O ) . O ) `2 & - 1 <= ( ( f2 . I ) . I ) `2 ;
f , g be Function of I[01] , TOP-REAL 2 , a , b , c be Real ;
k1 -' k2 = k1 - k2 + 1 .= k2 -' 1 + 1 .= k2 -' 1 + 1 ;
rng ( seq ^\ k ) c= ]. x0 , x0 + r .[ & rng ( seq ^\ k ) c= dom f ;
g2 in ]. x0 - r , x0 + r .[ & g2 < x0 + r ;
sgn ( p `1 , K ) = - ( - 1_ K ) .= - 1_ K ;
consider u be Nat such that b = p |^ y * u ;
ex A being the normal normal sequence of X st a = Sum A & A is limit_ordinal ;
Cl ( union H ) = union ( ( Cl H ) \ H ) ;
len t = len t1 + len t2 & len t1 = len t2 + len t2 ;
v = v + w |-- v + w |-- A ;
v <> DataLoc ( t1 . GBP , 3 ) & v <> DataLoc ( t1 . GBP , 3 ) ;
g . s = sup ( d " { s } ) ;
( \dot y ) . s = s . ( y , s . s ) ;
{ s : s < t } in NAT & t = {} + 1 } ;
s ` \ s = s ` \ 0. X .= 0. X \ 0. X ;
defpred P [ Nat ] means B + $1 in A & not $1 in B ;
( 3such + 1 ) ! = 111111111111111! * ( 1\mathopen + 1 ) ;
U . succ A = T . ( 1_ U , A ) .= ( U , A ) . ( IC U , A ) ;
reconsider y = y as Element of ( len y ) -tuples_on COMPLEX ;
consider i2 being Integer such that y0 = p * i2 and i2 <> 0 and i1 <> 0 ;
reconsider p = Y | Seg k as FinSequence of NAT ;
set f = ( S , U ) \mathop \mathop { z } ;
consider Z be set such that ( lim s ) in Z and Z in F ;
let f be Function of I[01] , TOP-REAL n , a , b , c be Real ;
( ( the M of M ) . [ n + i , 'not' A ] <> 1 ;
ex r being Real st x = r & a <= r & r <= b ;
R1 , R2 be Element of ( len R ) -tuples_on the carrier of K ;
reconsider l = 0. ( V , m ) as Linear_Combination of A ;
set r = |. e .| + |. w .| + |. w .| + a ;
consider y being Element of S such that z <= y and y in X ;
a ' "\/" ( b ' "\/" c ) = 'not' ( ( a 'or' b ) 'or' c ) ;
||. x9 - g .|| < r2 & ||. g - g .|| < r2 ;
b9 , a9 // b9 , c9 & b9 , c9 // b9 , c9 & b9 , c9 // c9 , c9 ;
1 <= k2 -' k1 & k2 + 1 - k1 + 1 = k2 - k1 + 1 & k2 + 1 = k2 - k1 ;
sqrt ( ( p `1 ) ^2 - ( p `2 ) ^2 ) >= 0 ;
sqrt ( ( q `1 ) ^2 - ( q `2 ) ^2 ) < 0 ;
E-max C in cell ( RR , 1 , 1 ) & E-max L~ R in L~ R ;
consider e being Element of NAT such that a = 2 * e + 1 ;
Re ( ( lim F ) | D ) = Re ( ( lim G ) | D ) ;
LIN b , a , c or LIN b , a , c ;
p `1 , a // a `1 , b `2 or p `1 , a // b `1 , b `2 ;
g . n = a * Sum ( f | n ) .= f . n ;
consider f being Subset of X such that e = f and f is being that f is \rm that f is \rm f ;
F | ( N2 , S ) = CircleMap * ( F | [: N2 , S :] ) ;
q in LSeg ( q , v ) \/ LSeg ( v , p ) ;
Ball ( m , r0 ) c= Ball ( m , s ) ;
the carrier of (0). V = { 0. V } & the carrier of V = { 0. V } ;
rng ( ( - 1 ) (#) cos ) = [. - 1 , 1 .] ;
assume Re ( seq ) is summable & Im ( seq ) is summable ;
||. ( vseq . n ) - ( vseq . m ) .|| < e ;
set g = O --> 1 ;
reconsider t2 = t11 as 0 -started string of S2 , x be 0 -started string of S2 ;
reconsider x9 = seq . n as sequence of REAL-NS n , r be Real ;
assume that E-max C meets L~ go and L~ pion1 meets L~ pion1 and L~ pion1 /\ L~ pion1 meets L~ pion1 ;
- ( - 1 / 2 ) < F . n - F . x ;
set d1 = ]. \bf 1 , 0 .[ , d2 = ]. 1 , 0 .[ , d2 = ]. 1 , 0 .[ , d2 = ]. 1 , 0 .[ , d2 = ]. 1 , 0 .[ , d2 = ]. 1 , 0 .[ , d2 = ]. 1 , 0 .[ , d2 = ]. 1 ,
2 |^ ( q -' 1 ) - 1 = 2 |^ ( q -' 1 ) - 1 ;
dom ( v | 2 ) = Seg len ( d | 2 ) .= Seg len d ;
set x1 = - ( k2 + 1 ) + |. k2 + 1 .| ;
assume for n being Element of X holds 0. ( X , n ) <= F . n ;
assume that 0 <= T-32 . i and T-32 . ( i + 1 ) <= 1 and Tc . i <= 1 ;
for A being Subset of X holds c . ( c . A ) = c . A
the carrier of ( Carrier ( L2 ) + L2 ) c= I & L2 is finite ;
'not' All ( x , p ) => All ( x , p ) is valid ;
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the normal w.r.t. of A ;
Z c= dom ( ( sin (#) sin ) `| Z ) ;
|. 0. TOP-REAL 2 - ( q `2 / |. q .| - cn ) .| < r / 2 ;
ConsecutiveSet2 ( B , succ B ) c= ConsecutiveSet2 ( A , succ ( d ) ) ;
E = dom L & L is_measurable_on E & L is_measurable_on E implies L + K is_measurable_on E
C |^ ( A + B ) = C |^ B * C |^ A ;
the carrier of W2 c= the carrier of W1 & the carrier of W1 c= the carrier of V ;
I . IC Comput ( P2 , s2 , k ) = P . IC Comput ( P2 , s2 , k ) ;
pred x > 0 means : Def1 : sqrt ( 1 + x ^2 ) = x ^2 ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) ;
consider p being Point of T such that C = [. p , R .] and p in P ;
b , c connected connected connected connected connected connected & - C , - C are_homotopic implies - C , - C \mathclose { \lbrack } \upharpoonright C is continuous
assume f = id the carrier of O & f is Function of O , O ;
consider v such that v <> 0. V and f . v = L * v ;
let l be Linear_Combination of {} ( the carrier of V ) ;
reconsider g = f " as Function of U1 , U2 * , U2 * ;
A1 in the carrier of ( G . k ) | ( X . k ) ;
|. - x .| = - ( - x ) .= - x .= - x ;
set S = \mathop { \rm ) } ( x , y , c ) ;
( Fib n ) * ( 5 * ( 5 * ( 5 * ( 5 * n ) ) ) >= 4 * log ;
v9 /. ( k + 1 ) = vk . ( k + 1 ) ;
0 mod i = - ( i * ( 0 qua Nat ) ) .= - ( i * ( 0 qua Nat ) ) ;
Indices M1 = [: Seg n , Seg n :] & width M1 = [: Seg n , Seg n :] ;
Line ( S\mathopen { - } j } , j ) . j = Sj . j ;
h . ( x1 , y1 ) = [ y1 , x1 ] & h . ( y1 , y2 ) = [ y2 , x2 ] ;
|. f .| - ( Re ( f (#) ( b (#) h ) ) ) is nonnegative ;
assume x = ( a1 ^ <* b1 *> ) ^ <* b1 *> ^ ( a2 ^ <* b2 *> ) ;
MW is_halting_on IExec ( I , P , s ) , P & M is_halting_on s , P ;
DataLoc ( t3 . a , 4 ) = intpos ( 0 + 4 ) ;
x + y < - x + y & |. x .| = - x + y ;
LIN c , q , b & LIN c , q , c ;
f9 . ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + z1 ) .= x + ( y1 + z1 ) ;
f9 . a = f9 . a & v in InputVertices S & v in InputVertices S & v in InputVertices S ;
( p `1 ) ^2 <= ( E-max C ) `1 & ( E-max C ) `1 <= ( E-max C ) `1 ;
set R8 = Cage ( C , n ) \circlearrowleft E8 , E7 = Cage ( C , n ) ;
( p `1 ) ^2 >= ( E-max C ) `1 & ( E-max C ) `1 <= ( E-max C ) `1 ;
consider p such that p = p2 and s1 < p and p < s2 and p < s2 and p < s2 ;
|. ( f /* ( s (#) F ) ) . l - G . l .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len Line ( N , k + 1 ) = width N & width Line ( N , k + 1 ) = width N ;
f1 /* s1 is convergent & f2 /* s1 is convergent & lim ( f1 /* s1 ) = x0 ;
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 ) ;
dom B = 2 -tuples_on the carrier of V \ { {} } ;
consider r such that r _|_ a and r _|_ x and r _|_ y ;
reconsider B1 = the carrier of Y1 as Subset of X | X1 , B2 = the carrier of X1 | X2 as Subset of X ;
1 in the carrier of [. ( 1 / 2 ) / 2 , 1 .] & 1 / 2 < 1 ;
for L being complete LATTICE holds <* <* \mathbb L *> , L *> is isomorphic ;
[ gi , gj ] in Ii \ Ij " ;
set S2 = 1GateCircStr ( x , y , c ) , S2 = 1GateCircStr ( y , c ) ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and for x st x in dom f1 holds f1 . x <> 0 ;
reconsider y = ( a ` ) / ( F ` ) , z = ( a ` ) / ( F ` ) as Element of L ;
dom s = { 1 , 2 , 3 } & s . 1 = d1 & s . 2 = d2 ;
( min ( g , ( g . c ) ) . c <= h . c ;
set G2 = the subgraph of G , s3 = the subgraph of G , s3 = the subgraph of G , s3 = the subgraph of G , 4 = the consider of G , 5 = the consider of G , 6 being Element of G such that 5 = 6 and 6 = 6 and 6 = 5 ;
reconsider g = f as PartFunc of REAL , REAL-NS n , REAL-NS n ;
|. s1 . m .| / |. p .| < d / p |^ m ;
for x being element st x in ( 1 - u ) holds x in ( 1 - u ) * ( x - u )
P = the carrier of ( TOP-REAL n ) | K1 .= ( TOP-REAL n ) | K1 .= ( TOP-REAL n ) | K1 ;
assume that p10 in LSeg ( p1 , p2 ) /\ L2 and LSeg ( p1 , p2 ) /\ L2 = {} ;
( 0. X \ x ) |^ ( m * ( k + 1 ) ) = 0. X ;
let g be Element of Hom ( cod f , dom g ) ;
2 * a * b + ( 2 * c ) * d <= 2 * C1 * C2 ;
f , g , h be Point of complex X , g be Point of complex X ;
set h = Hom ( a , g ) (*) f ;
then idseq n | Seg m = ( idseq m ) | Seg m & m <= n ;
H * ( g " * a ) in the carrier of H & ( g " * a ) * ( g " * a ) in the carrier of H ;
x in dom ( ( - 1 ) (#) ( sin * cos ) ) & ( ( - 1 ) (#) ( cos * cos ) ) `| Z ) . x = ( - 1 ) (#) ( cos * cos ) . x ;
cell ( G , i1 , j2 -' 1 ) misses C & cell ( G , i1 , j2 -' 1 ) misses C ;
LE q2 , q2 , P , p1 , p2 or LE q2 , q2 , P , p1 , p2 ;
attr B is component means : Def1 : B c= BDD A & B c= BDD A ;
deffunc D ( set , set ) = union rng $2 & $2 in rng $2 ;
n + - n < len ( p + - n ) + n - n ;
attr a <> 0. K means $1 in dom ( a * M ) & the_rank_of ( M * M ) = a ;
consider j such that j in dom |^ Z and I = len |^ j + j and I = len |^ j ;
consider x1 such that z in x1 and x1 in ( P . n ) and x2 in ( P . n ) ;
for n ex r being Element of REAL st X [ n , r ]
set C1 = Comput ( P2 , s2 , i + 1 ) , C2 = Comput ( P2 , s2 , i + 1 ) , C2 = Comput ( P2 , s2 , i ) ;
set v = 3 -tuples_on BOOLEAN , w = 2 -tuples_on { a , b , c } , w = 1 -tuples_on BOOLEAN , y = 2 -tuples_on BOOLEAN , z = 3 -tuples_on BOOLEAN , w = 1 -tuples_on BOOLEAN , z = 2 -tuples_on BOOLEAN , w = 3 -tuples_on BOOLEAN , z = 1 -tuples_on BOOLEAN , w = 2 -tuples_on
conv @ W c= union ( F .: ( E " W ) ) ;
1 in [. - 1 , 1 .] /\ dom ( ( #Z 2 ) * ( arccot ) ) ;
r3 <= s3 + ( r - r2 ) / 2 * ( 1 - r2 ) / 2 * ( 1 - r2 ) / 2 * ( 1 - r2 ) / 2 * ( 1 - r2 ) / 2 * ( 1 - r2 ) ;
dom ( f (#) ( 4 (#) f3 ) ) = dom f /\ dom ( 4 (#) f3 ) .= dom f /\ dom f3 ;
dom ( f (#) G ) = dom ( l (#) F ) /\ Seg k .= Seg k ;
rng ( s ^\ k ) c= dom f1 \ { x0 } & rng ( s ^\ k ) c= dom f2 \ { x0 } ;
reconsider g2 = gp as Point of ( TOP-REAL n ) | K1 , ( TOP-REAL n ) | K1 ;
( T * h . s ) . x = T . ( h . s ) . x ;
I . ( J . x ) = ( I * L ) . x ;
y in dom \mathopen { *> \! \! *> \! ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( A A A ) A ) ) ) ) ) . o ) ) ) . o ) ) ) ) . x ) ) ;
for I being non degenerated commutative commutative Ring holds I is commutative commutative commutative commutative associative non empty doubleLoopStr ;
set s2 = s +* Initialize ( ( intloc 0 ) .--> 1 ) , P2 = Initialize ( ( intloc 0 ) .--> 1 ) ;
P1 /. IC s1 = P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 ;
lim S1 in the carrier of [. a , b .] & lim S2 in [. a , b .] ;
v . ( l-13 ) = ( v *' ( lpp ) ) . i ;
consider n be element such that n in NAT and x = seq . n and x = seq . n ;
consider x being Element of c such that F1 . x <> F2 . x and F1 . x <> 0 ;
card ( X , 0 , x1 , x2 , x3 , x4 ) = { E } \/ { F } ;
j + ( 2 * k ) + m1 > j + ( 2 * k ) ;
{ s , t } on A2 & { s , t } on A2 & { s , t } on A2 ;
n1 > len crossover ( p2 , p1 , p2 , n1 , n2 , n3 , n3 ) & n2 <= len crossover ( p2 , p1 , n1 , n2 , n3 ) ;
( ( for b being bag of g2 ) , T being HT ( f2 , T ) ) = 0. L holds ( b * T ) = ( b * T ) *' ( b * T )
then H1 , H2 are_isomorphic , H are_isomorphic is are ` & ( H , H1 ) / ( H , H1 ) is are isomorphic ;
( E-max L~ f ) .. ( f /. ( len f -' 1 ) ) > 1 & ( E-max L~ f ) .. ( f /. ( len f -' 1 ) ) > 1 ;
]. s , 1 .[ = ]. s , 2 .[ /\ [. 0 , 1 .] ;
x1 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) | ( L~ g ) ) ;
let f1 , f2 be continuous PartFunc of REAL , the carrier of S , r be Real ;
DigA ( t-23 , z1 ) is Element of k -tuples_on k -tuples_on k -tuples_on k ;
I \mathop { \rm I2211111z } = k2 & I \mathop { \rm \hbox { - } \rm @ } = k2 ;
Wmin ~ = { [ a , u9 ] , [ b , u9 ] } ;
( w | p ) | ( w | w ) = p | ( w | w ) ;
consider u2 such that u2 in W2 and x = v + u2 and u = v + u2 and v in W and u in W ;
for y st y in rng F ex n st y = a |^ n & n >= 1
dom ( ( g * ( f , K ) ) | K ) = K ;
ex x being element st x in ( ( [#] U0 ) \/ A ) . s ;
ex x being element st x in ( ( ( ( ( ( ( O ) \/ A ) . s ) ) . s ) . x ;
f . x in the carrier of [. - r , r .] & f . x in [. - r , - r .] ;
( the carrier of X1 union X2 ) /\ ( the carrier of X2 ) <> {} implies ( the carrier of X1 union X2 ) /\ ( the carrier of X1 union X2 ) <> {}
L1 /\ LSeg ( p10 , p2 ) c= { p10 } /\ LSeg ( p10 , p2 ) ;
sqrt ( b + ( b-0 ) ) / 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 (#) of ( ( the \overline of ( the carrier of X ) ) ) . an <= e ;
len ( w ^ w2 ) + 1 = len w + 2 + 1 .= len w + 2 ;
assume q in the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 & q in the carrier of ( TOP-REAL 2 ) | K1 ;
f | E-4 ` = g | E-4 ` .= g | E-4 ` ` ` ` .= g | E-4 ` ` ` ;
reconsider i1 = x1 , i2 = x2 , j1 = x3 as Element of NAT ;
( a * A ) ` = ( a * ( A * B ) ) ` ;
assume ex n1 being Element of NAT st f |^ n1 is seq & f |^ n1 is seq ;
Seg len ( ( Sum ( f2 ) ) | ( len ( ( Sum ( f2 ) ) ) ) = dom ( ( Sum ( f2 ) ) | ( dom ( ( Sum ( f2 ) ) ) ) ;
( Complement A1 ) . m c= ( Complement A1 ) . n & ( Complement A1 ) . n c= ( Complement A1 ) . n ;
f1 . p = p0 & g1 . p = d & g2 . p = d & g2 . p = c ;
FinS ( F , Y ) = FinS ( F , dom ( F | Y ) ) ;
( x | y ) | z = z | ( y | x ) ;
sqrt ( ( |. x .| ^2 ) ^2 - ( x ^2 ) ^2 <= sqrt ( ( r ^2 ) ^2 ) ;
Sum ( F ) = Sum f & dom ( F - g ) = dom g & rng ( F - g ) c= dom g ;
assume for x , y being set st x in Y & y in Y holds x /\ y in Y ;
assume W1 is Subspace of W2 & W2 is Subspace of W1 & W1 is Subspace of W2 implies W1 + W2 is Subspace of W1
||. ( ( t . x ) - ( t . x ) ) .|| = lim ||. ( x - x0 ) .|| .= 0 ;
assume that i in dom D and f | A is bounded and g | A is bounded ;
sqrt ( ( p `1 ) ^2 + d ^2 ) <= sqrt ( ( p `1 ) ^2 ) ;
g | Sphere ( p , r ) = id Ball ( p , r ) & g | Ball ( p , r ) = id Ball ( p , r ) ;
set N8 = ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ;
for T being non empty TopSpace holds T is countable countable implies the TopStruct of T is countable countable
width B |-> 0. K = Line ( B , i ) .= Line ( B , i ) .= B * Line ( B , i ) ;
attr a <> 0 means A implies ( A \+\ B ) \subseteq ( A Y. ) \ ( B Y. ) ;
then f is_differentiable 3 , u & pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 & pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 ;
assume that a > 0 and a <> 1 and b > 0 and c > 0 and d > 0 and c > 0 ;
w1 , w2 in Lin { w1 , w2 } & w2 in Lin { w1 , w2 } ;
p2 /. IC Comput ( p2 , s2 , k ) = p2 . IC Comput ( p2 , s2 , k ) .= ( IC Comput ( p2 , s2 , k ) ) ;
ind ( T-10 | b ) = ind b - ind b .= ind B - ind B .= ind B - ind B .= ind B - ind B .= ind B - ind B ;
[ a , A ] in the Line of Line ( A , len Line ( A9 , i ) ) & [ a , A ] in the Line of Line ( A , i ) ;
m in ( the Arrows of \HM { o } ) . ( o1 , o2 ) & m in ( the Arrows of C ) . ( o1 , o2 ) ;
( ( a , CompF ( PA , G ) ) . z = FALSE ;
reconsider phi = phi /. 11 , phi = phi /. 2 as Element of phi ;
len s1 - ( len s2 - 1 ) + 1 > 0 + 1 - 1 ;
( \delta D ) * ( f . ( upper_bound A ) - f . ( lower_bound A ) ) < r ;
[ f21 , f22 ] in the InternalRel of A & [ f22 , f22 ] in the InternalRel of A ;
the carrier of ( ( TOP-REAL 2 ) | K1 ) = K1 & the carrier of ( ( TOP-REAL 2 ) | K1 ) = K1 ;
consider z being element such that z in dom g2 and p = g2 . z and p = g2 . z ;
[#] V1 = { 0. V1 } .= the carrier of V1 .= the carrier of V1 .= the carrier of V1 ;
consider P2 be FinSequence such that rng P2 = M and P2 is one-to-one and P2 is one-to-one and P [ P2 ] ;
assume x1 in dom ( f | X ) & ||. x1 - x0 .|| < s & ||. x1 - x0 .|| < s ;
h1 = f ^ ( <* p3 *> ^ <* p *> ) .= h ^ <* p *> .= h ;
c /. [ b , c ] `2 = c /. [ a , c ] .= c /. ( a , c ) ;
reconsider t1 = p1 , t2 = p2 , t2 = p2 as Term of C , V , a , b be Element of C ;
sqrt ( 1 + 2 ) in the carrier of [. 1 / 2 , 1 .] & sqrt ( 1 + 2 ) < 1 ;
ex W being Subset of X st p in W & W is open & h .: W c= V ;
( h . p1 ) `2 = C * ( p1 `1 ) + D * ( p1 `2 ) ;
R . b ||. a .|| = 2 * PI - b .= 2 * b ;
consider I1 such that B = 1- I1 * C + ( 0 * A ) and 0 <= I1 and I1 <= 1 ;
dom g = dom ( ( the Sorts of A ) * ( a , 6 ) ) .= dom ( ( the Sorts of A ) * ( a , 6 ) ) ;
[ P . ( l1 + 1 ) , P . ( l1 + 1 ) ] in => => ( T . ( l1 + 1 ) ) ;
set s2 = Initialize s , P2 = Initialize s , P2 = P +* stop I ;
reconsider M = mid ( z , i2 , i1 ) as Matrix of 2 , K , len z , K ;
y in product ( ( Carrier J ) +* ( V , { 1 } ) ) ;
1 / ( 0 , 1 ) = 1 & 0 / ( 0 , 1 ) = 0 & 1 / ( 0 , 1 ) = 1 ;
assume x in the left st x in the left of g or x in the left st not x in the right of g & x in the right of g ;
consider M be strict Subgroup of A such that a = M and T is Subspace of M and for A being Subset of M holds T . A is Subspace of M ;
for x st x in Z holds ( ( ( exp_R * f ) + ( exp_R * f ) ) `| Z ) . x <> 0
len W1 + len W2 + m = 1 + len W2 + m .= 1 + len W2 + m .= len W1 + m + 1 ;
reconsider h1 = vseq . n - t1 as Lipschitzian LinearOperator of X , Y ;
( i mod len ( p + q ) ) + 1 in dom ( p + q ) ;
assume that s2 is_or F in the |= of s1 and F in the |= of s2 and F . ( len F ) = s2 ;
( ( for x , y holds ( x , y ) ) 3 , ( x , y ) ) 3 = ( x , y ) / 3 , ( x , y ) / 3 = ( x , y ) / 3
for u being element st u in Bags n holds ( p *' m ) . u = p . u
for B being Subset of u st B in E holds A = B or A misses B or A misses B ;
ex a being Point of X st a in A & A /\ Cl { y } = { a } ;
set W2 = [: p , q :] \/ [: p , q :] ;
x in { X where X is Ideal of L : X is Ideal of L |^ n } ;
the carrier of W1 /\ W2 c= the carrier of W1 & the carrier of W1 /\ W2 c= the carrier of W1 /\ W2 ;
( id ( a + b ) ) * id ( a + b ) = id ( a + b ) * id ( a + b ) ;
( dom ( X --> f ) ) . x = ( X --> dom f ) . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) ;
p => ( q => r ) => ( p => ( q => r ) ) in TAUT ( A ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
- 1 + 1 <= ( - 2 |^ n ) * ( n -' m ) + 1 ;
( reproj ( 1 , z0 ) ) . x in dom ( f1 (#) f2 ) & ( f1 (#) f2 ) . x in dom ( f1 (#) f2 ) ;
assume that b1 . r = { c1 } and b2 . r = { c2 } and b2 . 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 & a2 on P & a1 on P & a2 on P & a1 on P & a2 on P & a2 on P & a1 on P & a2 on P
reconsider gf = g opp * f , hh = h opp * g as strict Subgroup of X ;
consider v1 being Element of T such that Q = ( downarrow v1 ) ` and v1 in ( downarrow v1 ) ` and v1 in ( downarrow v1 ) ` ;
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 .| - cn ) / ( 1 + cn ) >= cn } ;
ConsecutiveSet ( A , succ O1 ) = ( ConsecutiveSet ( A , O1 ) ) * ( ConsecutiveSet ( A , O1 ) ) ;
set I1 = Macro ( a , intloc 0 ) , I2 = AddTo ( a , intloc 0 ) , I2 = AddTo ( a , intloc 0 ) , I3 = AddTo ( a , intloc 0 ) , I2 = AddTo ( a , intloc 0 ) , I3 = [ a , intloc 0 ] ;
for i be Nat st 1 < i & i < len z holds z /. i <> z /. 1 ;
X c= ( the carrier of L1 ) \/ ( the carrier of L2 ) & the carrier of L1 c= the carrier of L2 ;
consider x9 being Element of GF ( p ) such that x9 |^ 2 = a and x9 |^ 2 = a ;
reconsider e1 = e1 , e2 = e2 , e2 = e2 as Element of D * , e2 = e2 as Element of D * ;
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 ;
f * g * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) . x ) ;
defpred P [ Nat ] means A + succ $1 = succ A & A + $1 = succ $1 ;
the left ]| of ( - g ) * ( - g ) = the left st ( - g ) * ( - g ) = the left st ( - g ) * ( - g ) = ( - g ) * ( - g ) ;
reconsider p9 = x , q9 = y , p9 = z as Point of ( TOP-REAL 2 ) | K1 , e = z as Point of ( TOP-REAL 2 ) | K1 ;
consider g2 such that g2 = y and x <= g2 and g2 <= x0 and x0 <= g2 and g2 <= x0 and g2 <= x0 and g2 <= x0 and g2 <= x0 ;
for n being Element of NAT ex r being Element of REAL st X [ n , r ]
len ( x2 ^ y2 ) = len x2 + len y2 .= len x2 + len y2 .= len x2 + len y2 .= len x2 + len y2 ;
for x being element st x in X holds x in the set of K iff x in the set of K & not x in K & not x in K
LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = {} or LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = {} ;
func ;
len ( ( ( \mathbb C ) | ( len ( ( \mathbb C ) | ( len C ) ) ) ) ) <= len ( ( C | ( len C ) ) ) ;
attr K is L means a <> 0. K & v . ( a |^ i ) = i * v . a ;
consider o being OperSymbol of S such that t . {} = [ o , the carrier of S ] and o . {} = [ o , the carrier of S ] ;
for x st x in X ex y st x c= y & y in X & y in X & x in X holds f . x in a
IC Comput ( P3 , s3 , k ) in dom ( Comput ( P3 , s3 , k ) ) ;
attr q < s & r < s implies ]. r , s .[ c= ]. p , q .[ ;
consider c being Element of Class ( f . c , 3 ) such that Y = ( F . c ) . ( 1 , 3 ) and c in X ;
func the ResultSort of S2 -> Function means : Def1 : for x being Element of the carrier' of S2 holds it . x = id the carrier' of S2 & x in the carrier' of S2 iff x is Function of the carrier' of S2 , the carrier' of S2 ;
set y-13 = [ <* y , z *> , f2 ] , y] = [ <* z , x *> , f3 ] ;
assume x in dom ( ( ( exp_R * ( arctan * arccot ) ) `| Z ) ;
r-7 in Int cell ( GoB f , i , GoB f ) \ L~ f & ( GoB f ) * ( i + 1 , width GoB f ) c= LeftComp f ;
( q `2 ) ^2 >= ( ( Cage ( C , n ) ) /. ( i + 1 ) ) `2 ;
set Y = { a "/\" a ` : a in X } ;
i -' len f <= len f + 1 - len f + 1 & i + 1 <= len f + 1 - len f ;
for n ex x st x in N & x in N1 & h . n = x0 & h . n = x0 + r
set s0 = ( IExec ( 0 , p , s ) , p , s ) . i , s = s . b ;
( p . k ) . 0 = 1 or ( p . k ) . 0 = 1 or ( p . k ) . 1 = 0 ;
u + Sum L-18 in ( U \ { u } ) \/ { u + Sum L } ;
consider x9 be set such that x in x9 and x9 in V1 and x9 in V1 and x9 in V1 and x9 in V1 and x = [ x9 , y9 ] ;
( p ^ ( q | k ) ) . m = ( q | k ) . ( len p - 1 ) .= p . ( len p - 1 ) ;
g + h = gg + h + h1 & x + h = g + h + h ;
L1 is distributive & L2 is distributive & L2 is distributive implies L1 [: L1 , L2 :] is distributive
pred x in rng f & y in rng ( f | x ) & f . x = f . y ;
assume that 1 < p and ( sqrt ( 1 + ( p `1 / p `2 ) ^2 ) = 1 and 0 <= a and a <= b and b <= 1 and a <= b ;
F* ( f , <* *> ) = rpoly ( 1 , the carrier of F_Complex ) *' *' t + t *' t .= 0. F_Complex ;
for X being set , A being Subset of X holds A ` = {} implies A = {} or A = {}
( ( E-max X ) `1 <= ( E-max X ) `1 & ( E-max X ) `2 <= ( E-max X ) `2 ;
for c being Element of the Sorts of A , a being Element of the Sorts of A holds c <> a ;
s1 . GBP = ( Exec ( i2 , s2 ) ) . DataLoc ( s . GBP , 2 ) .= s . DataLoc ( s . GBP , 2 ) .= s . DataLoc ( s . GBP , 2 ) .= s . DataLoc ( s . GBP , 2 ) ;
for a , b being Real holds [ a , b ] in ( y iff b >= 0 ) & a >= 0 ) implies a >= 0
for x , y being Element of X holds x ` \ y ` = ( x \ y ) ` \ ( x \ y ) `
mode BCK-algebra of i , j , m , n , m , n , m , n , m , k be Nat holds m + n , n , m , k , n + 1 ;
set x2 = |( Re ( y - x ) , ( Im ( y - x ) ) )| ;
[ y , x ] in dom u & u2 . ( y , x ) = g . y ;
]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .] c= A ;
0 <= \delta & |. \delta .| . n < ( 0 - 2 ) / 2 ;
( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) <= ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ;
set A = ( 2 * b ) / 2 ;
for x , y being set st x in R" holds x , y are_x , T
deffunc F2 ( Nat ) = b . $1 * ( M . $1 ) * ( M . $1 ) ;
for s being element holds s in ( { f } \/ g ) iff s in ( ( { f } \/ g ) \/ ( ( { g } \/ { f } ) )
for S being non empty non void holds S is connected holds S is connected implies S is connected
max ( ( degree ( K , z ) ) / ( 1 + ( d / ( K + 1 ) ) ) ) >= 0 ;
consider n1 be Nat such that for k holds seq . ( n1 + k ) < r + s and for n holds seq . ( n1 + k ) < r + s ;
Lin ( A /\ B ) is Subspace of Lin ( A ) & Lin ( B ) is Subspace of Lin ( B ) ;
set n-15 = nw '&' ( M . x qua Element of BOOLEAN ) , nw = M . ( x qua Element of BOOLEAN ) , nw = M . ( x qua Element of BOOLEAN ) ;
f " V in ( the carrier of X ) & f " V in D & f " V in D & f " V in D ;
rng ( ( a ^\ c ) \mathbin { + } \cdot ( 1 , b ) ) c= { a , c } ;
consider y being Wsubgraph of G1 such that y `1 = y and dom y `1 = WWG2 and y `2 = WWWmod G ;
dom ( 1 / 2 ) /\ ]. - 1 , 1 .[ c= ]. - 1 , 1 .[ & dom ( 1 / 2 ) /\ ]. - 1 , 1 .[ c= dom ( 1 / 2 , 1 .[ ;
i , j , n , r , n , r be Element of Matrix ( i , j , n , r ) ;
v ^ ( n |-> 0 ) in Lin ( ( B | ( B -' 1 ) ) ) & v ^ ( ( B | ( B -' 1 ) ) ) in Lin ( ( B | ( B -' 1 ) ) ;
ex a , k1 , k2 st i = a := k1 & i = b := k2 & i <> k2 & i <> k2 ;
t . NAT = ( NAT .--> ( i1 , i2 ) ) . NAT .= ( NAT .--> ( i1 , i2 ) ) . NAT .= ( NAT .--> ( i1 , i2 ) ) . NAT .= ( NAT --> i1 ) . NAT .= ( NAT --> i1 ) . NAT ;
assume F is bbbfamily & rng p = F & dom p = Seg ( n + 1 ) & rng p c= Seg ( n + 1 ) ;
( not b in Line ( b , a , b ) & not LIN a , b , c ) & not LIN a , b , c
( L1 \HM { L1 } \HM { or } L2 c= ( L1 \HM { L2 } ) \HM { and ( L2 \HM { L2 } ) \HM { is } } is \HM { \HM { 0 } } -and ( L2 of L2 ) consider O being set such that O c= ( L1 \HM { L } ) there } ;
consider F being ManySortedSet of E such that for d being Element of E holds F . d = F ( d ) ;
consider a , b such that a * ( u - w ) = b * ( -w ) and 0 < a and a < b and b < 1 ;
defpred P [ FinSequence of D ] means |. Sum $1 .| <= Sum ( $1 ) & Sum ( $1 ) <= Sum ( $1 ) ;
u = cos / ( x , y ) . v + ( cos / ( x , y ) ) . v .= v + u . v .= v ;
dist ( ( seq . n ) + x , g + x ) <= dist ( ( seq . n ) + g , x ) + 0 ;
P [ p , |. p .| : p (#) ( p (#) id NAT ) = ( id the carrier of A ) . ( id the carrier of A )
consider X be Subset of CQC-WFF ( Al ) such that X c= Y and X is finite and X is finite and inininininP ;
|. b .| * |. eval ( f , z ) .| >= |. b .| * |. eval ( f , z ) .| ;
1 < ( E-max L~ Cage ( C , n ) ) .. ( Cage ( C , n ) ) ;
l in { l1 where l1 is Real : g <= l1 & l1 <= h & g <= h & l1 <= h } ;
( ( Partial_Sums ( G . n ) ) . i <= ( Partial_Sums ( G . n ) ) . i & ( Partial_Sums ( G . n ) ) . i <= ( Partial_Sums ( G . n ) ) . i ;
f . y = x .= x * 1. L .= x * 1. L .= x * 1. L .= x * 1. L .= x ;
NIC ( ( IC SCM+FSA , { i1 } ) , i ) = { ( a , i ) , ( i + 1 ) } .= { ( a , i ) } ;
LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = { p1 } /\ LSeg ( p1 , p2 ) .= { p1 } ;
Product ( ( the support of I-15 ) +* ( i , { 1 } ) ) in ( Z . i ) ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) | ( the carrier of S2 ) .= Following ( s1 , n ) | ( the carrier of S2 ) ;
( W-min ( Q ) ) `1 <= ( q `1 ) / 2 & ( q `2 ) / 2 <= ( q `1 ) / 2 ;
f /. i2 <> f /. ( len f -' 1 ) & f /. ( len g -' 1 ) = f /. ( len f -' 1 ) ;
M , f / ( x. 3 , m ) / ( x. 3 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 0 , m ) / ( x. 4 , m ) |= H ;
len ( ( P ^ ) ^ ( P ^ ) ) in dom ( ( P ^ ) ^ ( P ^ ) ) ;
A |^ ( m , n ) c= A |^ ( m , n ) & A |^ ( k , l ) c= A |^ ( k , l ) ;
R |^ 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 not p1 . n1 = p2 . n1 ;
consider X be set such that X in Q and for Z being set st Z in Q & Z <> X holds X \not c= Z ;
CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA & CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA ;
for v be VECTOR of l1 holds ||. v .|| . v = upper_bound ( |. v .| ) & ||. v .|| . v = upper_bound ( |. v .| )
for phi holds phi in X implies phi phi in X & phi phi in X & phi phi in X
rng ( ( Sgm dom f-6 ) | ( dom ( Sgm dom ( f-6 ) ) ) c= dom ( Sgm dom ( f-6 ) | ( dom ( Sgm dom ( f-6 ) ) ) ;
ex c being FinSequence of D st len c = k & P [ c ] & a = c & b = c & a = b ;
( the_arity_of ( a , b , c ) ) . <* b , c *> = <* Hom ( b , c ) , Hom ( a , b ) *> . <* b , c *> ;
consider f1 be Function of the carrier of X , REAL such that f1 = |. f .| and f1 is continuous and for x be Element of X holds f1 . x = f . x ;
a1 = b1 & a2 = b2 or a1 = b3 or a2 = b3 or a1 = b3 or a2 = b3 & a3 = b3 ;
D2 . indx ( D2 , D1 , n1 ) + 1 = D1 . indx ( D2 , D1 , n1 ) + 1 .= D1 . indx ( D2 , D1 , n1 ) ;
f . ( ||. r .|| ) = ||. r .|| /. 1 .= <* r *> /. 1 .= r .= r . 1 .= x ;
consider n be Nat such that for m be Nat st n <= m holds Cseq . m = Cseq . m ;
consider d be Real such that for a , b being Real st a in X & b in Y holds a <= d and a <= b ;
||. L /. h .|| - ( K * |. h .| ) + ( K * |. h .| ) <= x0 + ( K * |. h .| ) ;
attr F is commutative associative means : Def1 : for b being Element of X holds F \hbox { b } -\hbox { b } . b = f . b ;
p = ( - 1 ) * ( 0. TOP-REAL 2 + 0 ) .= 1 * ( 0. TOP-REAL 2 ) .= 1 * ( 0. TOP-REAL 2 + 0 ) .= 1 * ( 0. TOP-REAL 2 + 0 ) .= 1 * ( 0. TOP-REAL 2 + 0 ) .= 1 * ( 0. TOP-REAL 2 + 0 ) .= 1 * ( 0. TOP-REAL 2 + 0 ) .= 1 * ( 0. TOP-REAL 2 + 0 ) ;
consider z1 such that b `1 , x3 , x3 , x4 is_collinear and o , x1 , x2 is_collinear and o <> x2 and o <> x1 and o <> x2 and o <> x2 and o <> x1 and o <> x2 ;
consider i such that Arg ( ( Rotate ( s ) ) . q ) = s + ( 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 . x = f . x ;
assume that A = P2 \/ Q2 and Q1 <> {} and for i , j st i in dom P2 & j in dom P2 & i <> j holds P2 . i misses P2 . j and P2 . j misses P2 . i ;
attr F is associative means : Def1 : F .: ( f , g ) = F .: ( f , g ) ;
ex x being Element of NAT st m = x `1 & x `1 < z & z `2 < x `2 or m < i & z `2 < i `2 & z `2 < i `2 ;
consider k2 be Nat such that k2 in dom P-2 and l in P-2 and k = ( P-2 . k2 ) `1 + ( k - 1 ) * ( k - 1 ) ;
seq = r (#) seq implies for n holds seq . n = r * seq . n & seq . n = r * seq . n
F1 . [ id a , [ a , a ] ] = [ f * ( id a , a ] ) , f * ( id a , a ) ] ;
{ p } "\/" D2 = { p "\/" y where y is Element of L : y in D2 & p in D1 } ;
consider z being element such that z in dom ( ( dom F ) | F ) and ( ( dom F ) | 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 , s ) = { |[ r , s ]| : r <= G * ( 0 + 1 , 1 ) `1 } ;
consider e being element such that e in dom ( T | E1 ) and ( T | E1 ) . e = v and ( T | E1 ) . e = v ;
( F ' * b1 ) . x = ( Mx2Tran J ) . ( b1 , b2 ) .= ( ( Mx2Tran J ) . ( b2 , b3 ) ) . ( b2 , b3 ) ;
- 1 _ { \mathbb R } = ( m (#) D ) | n .= ( m (#) D ) | n .= ( m (#) D ) | n .= ( m (#) D ) | n .= ( m (#) D ) | n .= ( m (#) D ) | n .= ( m (#) D ) | n .= ( m (#) D ) | n ;
attr for x being set st x in dom f /\ dom g holds g . x <= f . x ;
len ( f1 . j ) = len ( f2 /. j ) .= len ( f2 /. j ) .= len ( f2 /. j ) .= len ( f2 /. j ) ;
All ( 'not' All ( a , A , G ) , B , G ) |= All ( 'not' All ( a , B , G ) , A , G ) ;
LSeg ( E . k , F . ( k + 1 ) ) c= Cl RightComp Cage ( C , k + 1 ) \/ RightComp Cage ( C , k + 1 ) ;
x \ a |^ m = x \ ( a |^ k ) .= ( x \ a ) |^ k .= x \ a ;
k -func func func ( I ) -> Element of ( ( commute I ) . k ) . k = ( ( ( commute I ) . k ) . k .= ( ( ( commute I ) . k ) . k .= ( ( ( commute I ) . k ) . k ;
for s being State of A2 holds Following ( s , n ) . ( n + 2 * n ) is stable ;
for x st x in Z holds f1 . x = a ^2 & ( f1 - f2 ) . x <> 0 & ( f1 - f2 ) . x <> 0 ;
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 ) * the carrier of C ;
- ( a * sqrt ( 1 + b ^2 ) ) <= - ( b * sqrt ( 1 + b ^2 ) ) ;
phi /. ( succ b1 ) = g . a & phi /. ( a + 1 ) = f . ( g . a ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( ( F ^ <* p *> ) | i ) and i <> j ;
{ x1 , x2 , x3 , x4 } = { x1 } \/ { x2 , x3 , x4 } .= { x1 } \/ { x2 , x3 , x4 } ;
the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U1 /\ ( U2 "\/" U2 ) ;
( - 2 * ( a * ( b * ( b - a ) ) + b ) / 2 > 0 ;
consider W00 such that for z being element holds z in W00 iff z in N ~ & P [ z ] ;
assume ( the Arity of S ) . o = <* a *> & ( the ResultSort of S ) . o = r & ( the ResultSort of S ) . o = r ;
Z = dom ( ( exp_R * ( arccot * arccot ) ) `| Z ) & for x st x in Z holds ( ( exp_R * arccot ) `| Z ) . x = exp_R . x / ( 1 + x ^2 )
integral ( f , S1 ) is convergent & lim ( integral ( f , S2 ) ) = integral ( f , S2 ) - integral ( g , S2 ) . x0 ;
( for a9 holds ( a . f ) => ( g . x ) => ( ( x . f ) => ( x . f ) ) in 1 iff ( x . f ) => ( x . f ) in
len ( M2 * M1 ) = n & width ( M2 * M1 ) = n & width ( M2 * M1 ) = n & width ( M2 * M1 ) = n ;
attr X1 union X2 is open means : Def1 : X1 , X2 are_separated & X2 , X1 are_separated & X1 , X2 are_separated & X2 , X2 are_separated ;
for L being lower-bounded antisymmetric antisymmetric antisymmetric RelStr holds X "\/" Y = { Bottom L }
reconsider f9 = F2 . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . (
consider w being FinSequence of I such that the initial of M , the initial q -{ s } ^ w ^ <* s *> ^ w ^ w ^ w ^ w ^ ( <* s *> ^ w ) and q ^ w ^ w ^ w ^ w ^ w ^ w ^ w ;
g . ( a |^ 0 ) = g . ( 1_ G ) .= g . ( 1_ G ) .= g . ( 1_ G ) .= g . ( 1_ G ) .= g . ( 1_ G ) ;
assume for i be Nat st i in dom f ex z being Element of L st f . i = rpoly ( 1 , z ) & 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 <> {} ;
( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C1 & ( the carrier' of C2 ) /\ ( the carrier' of C2 ) c= the carrier' of C2 & ( the carrier' of C2 ) /\ ( the carrier' of C2 ) c= ( the carrier' of C2 ) /\ ( the carrier' of C2 ) ;
reconsider o9 = o `1 as Element of TS ( the Sorts of A ) , p = ( the Sorts of A ) . o as Element of TS ( ( the Sorts of A ) . v ;
1 * x1 + ( 0 * x2 + 0 * x3 ) + ( 0 * x3 ) = x1 + <* \underbrace ( 0 , \dots , 0 ]| , 0 , 0 ) .= x1 + 0 ;
Eg " . 1 = ( Eg " ) . 1 .= ( ( E qua Function ) " ) . 1 .= ( E " ) . 1 .= ( E " ) . 1 .= ( E " ) . 1 .= ( E " ) . 1 .= ( E " ) . 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 U2 ;
( ( x "/\" z ) "\/" ( x "/\" y ) ) "\/" ( z "/\" y ) <= ( x "/\" z ) "\/" ( x "/\" y ) ;
|. f . ( s1 . l1 + 1 ) - f . ( s1 . l1 ) .| < ( 1 - M ) / ( 1 - M ) ;
LSeg ( ( Cage ( C , n ) /. i , ( Gauge ( C , n ) * ( i + 1 , j ) ) /. ( i + 1 ) ) is vertical ;
( f | Z ) /. x - ( f | Z ) /. x0 = L /. ( x- + R ) /. ( x- + R ) ;
g . c * ( g . c ) * f . c + f . c <= h . c * ( f . c ) + f . c ;
( f + g ) | divset ( D , i ) = f | divset ( D , i ) + g | divset ( D , i ) ;
assume that ColVec2Mx f in the carrier of [ A , b ] and ColVec2Mx f = the carrier of [ A , b ] and len f = width A and width f = width A and width f = width A and width f = width A and width f = width A and width f = width A and width f = width A and width f = width A ;
len ( - M1 ) = len M1 & width ( - M2 ) = width M1 & width ( - M2 ) = width M1 & width ( - M2 ) = width M1 ;
for n , i being Nat st i + 1 < n holds [ i , i + 1 ] in the InternalRel of ( \cal E of TOP-REAL n ) \ ( { i } \/ { i } )
pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 2 ) is_differentiable_in z0 implies pdiff1 ( f2 , 2 ) is_differentiable_in z0
attr a <> 0 & b <> 0 & Arg a = Arg b & Arg ( - b ) = Arg ( - b ) & Arg ( - b ) = Arg ( - b ) ;
for c being set st not c in [. a , b .] holds not c in Intersection ( the open m of a , b )
assume V1 is linearly-independent & V2 is closed & V2 is closed & V1 = { v + u : v in V1 & u in V1 } ;
z * x1 + ( 1 - z ) * x2 in M & z * x1 + ( 1 - z ) * x2 in N ;
rng ( ( ( P qua Function ) " ) * S1 ) = Seg card ( ( P " ) " ) .= Seg card ( ( P " ) " ) .= Seg card ( ( P " ) " ) ;
consider s2 being Integer such that s2 is convergent and b = lim s2 and for n holds s2 . n <= b and s2 . n <= b ;
h2 " . n = h2 . n " & 0 < ( - 1 ) / ( 2 * ( - 1 ) ) & 0 < ( - 1 ) / ( 2 * ( - 1 ) ) ;
( Partial_Sums ||. seq .|| ) . m = ||. ( seq . m ) - ( seq . n ) .|| .= ||. ( seq . m ) - ( seq . n ) .|| .= 0 ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= Comput ( P2 , s2 , 1 ) . b .= Comput ( P2 , s2 , 1 ) . b .= Comput ( P2 , s2 , 1 ) . b ;
- v = ( - 1 ) * v & - w = ( - 1 ) * v & - w = ( - 1 ) * v & - w = ( - 1 ) * v ;
sup ( k .: D ) = sup ( k .: D ) .= sup ( k .: D ) .= sup ( k .: D ) .= sup ( k .: D ) .= sup ( k .: D ) .= sup ( k .: D ) ;
A |^ k , l l l l l , k + ( n , l ) .. ( A |^ k , l ) = ( A |^ n , l + ( n , l ) .. ( A |^ k , l ) ) ;
for R being add-associative right_zeroed right_complementable associative distributive non empty doubleLoopStr , I , J being Subset of R holds I + ( J + K ) = ( I + J ) + K
( f . p ) `1 = sqrt ( ( p `1 ) ^2 + ( p `2 ) ^2 ) .= sqrt ( ( p `1 ) ^2 + ( p `2 ) ^2 ) ;
for a , b being non zero Nat st a , b are_relative_prime & b , a are_relative_prime holds ( for n being Nat holds ( a * b ) = ( ( a * b ) + ( b * n ) ) + ( ( a * n ) + ( b * n ) )
consider A9 being countable set such that r is countable and r is countable and A9 is countable and for i being Nat holds A9 is closed & not i in A and not i in A ;
for X being non empty addLoopStr , M , N being Subset of X st y in M & for x , y being Point of X st x in M holds x + y in M + N holds x + y in M + N
{ [ x1 , x2 ] , [ y1 , y2 ] } c= { [ x1 , x2 ] } & { [ y1 , y2 ] } c= { [ y1 , y2 ] } ;
h . ( f . O ) = |[ A * ( f . O ) + B , C * ( f . O ) + D ]| ;
( Gauge ( C , n ) * ( k , i ) ) `1 in L~ Upper_Seq ( C , n ) /\ L~ Upper_Seq ( C , n ) ;
cluster m , n -> prime means : Def1 : for Nat holds ( for p being prime Nat holds p divides n iff p divides n ) & ( not p divides n ) & not p divides n ;
( f (#) F ) . x1 = f . ( F . x1 ) & ( f (#) F ) . x2 = f . ( F . x2 ) ;
for L being Lattice , a , b , c being Element of L st a \ b <= c & b \ a <= c holds a "/\" b <= c
consider b be element such that b in dom ( H / ( x , y ) ) and z = H / ( x , y ) and z = H / ( 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 , G & e Joins W . 5 , G . 6 , G ;
( ( exists h h h ) [ f ] ) . n = ( ( h h h ) . n ) * ( h . n ) .= ( h * ( n + 1 ) ) . x ;
j + 1 = j - len h11 + 1 .= i + 1 - len h11 + 1 .= i + 1 - len h11 + 1 - len h11 + 1 ;
( *' S ) . f = S *' . ( f , ( *' S ) . f ) .= S *' . ( f , f ) .= S *' . ( f , f ) .= S *' . ( f , f ) ;
consider H such that H is one-to-one and rng H = the carrier of L2 and Sum ( L2 * H ) = Sum ( L2 ) and Sum ( L2 * H ) = Sum ( L2 ) and Sum ( L2 * H ) = Sum ( L2 ) ;
attr R is element means : Def1 : p in R & q <> q & p <> q & q <> r & r < s & s in R & s in R ;
dom ( product ( X --> f ) ) = meet ( X --> f ) .= meet ( X --> f ) .= meet ( X --> f ) .= meet ( X --> f ) .= dom f ;
sup ( proj2 .: ( Upper_Arc ( C ) /\ Vertical_Line w ) ) <= sup ( proj2 .: ( Upper_Arc ( C ) /\ Vertical_Line w ) ) & sup ( proj2 .: ( Upper_Arc w ) /\ Vertical_Line w ) <= sup ( proj2 .: ( Vertical_Line w ) /\ Vertical_Line w ) ;
for r be Real st 0 < r ex n be Nat st for m be Nat st n <= m holds |. S . m - S . n .| < r
i * fN - f . j = i * ( f - g ) .= i * ( f - g ) .= i * ( f - g ) .= i * ( f - g ) ;
consider f being Function such that dom f = 2 -tuples_on X ( ) and for Y being set st Y in 2 -tuples_on X ( ) holds f . Y = F ( Y ) ;
consider g1 , g2 being element such that g1 in [#] Y and g2 in union C and g2 in union C and g = [ g1 , g2 ] and g1 in C and g2 in C and g2 in C and g2 in C and g2 in C and g2 in C and g2 in C and g2 in C and g2 in C and g2 in C and g2 in C and g2 in C ;
func d -count ( n ) -> Nat means : Def1 : d |^ n divides n & d |^ ( n + 1 ) divides n & d |^ ( n + 1 ) divides n ;
f9 . [ 0 , t ] = f . [ 0 , t ] .= ( - P ) . [ 2 * x , t ] .= ( - P ) . [ 2 * x , t ] .= a ;
t = h . D or t = h . C or t = h . E or t = h . F or t = h . J or t = M . M ;
consider m1 be Nat such that for n st n >= m1 holds dist ( ( seq . n ) , ( seq . n ) ) < 1 / ( 2 |^ ( n + 1 ) ) ;
sqrt ( ( q `1 ) ^2 + ( q `2 ) ^2 ) <= sqrt ( ( q `1 ) ^2 ) + sqrt ( ( q `2 ) ^2 ) ;
h1 . ( i + 1 + 1 ) = h2 . ( i + 1 + 1 -' len h11 ) .= h2 . ( i + 1 -' len h11 ) ;
consider o being Element of the carrier' of S , x2 being Element of { [ o , x2 ] } such that a = [ o , x2 ] and [ o , x2 ] in dom p and [ o , x2 ] in dom p ;
for L being RelStr , a , b being Element of L holds a <= b iff a <= b & b <= a & a <= b
||. h1 .|| . n = ||. h1 . n .|| .= ||. h1 . n .|| .= ||. h1 . n .|| .= ||. h1 . n .|| .= ||. h1 . n .|| .= ||. h1 .|| . n ;
( ( - ( exp_R * exp_R ) ) . x = f . x - ( exp_R * exp_R ) . x .= ( - ( exp_R * exp_R ) ) . x .= ( - ( exp_R * exp_R ) . x ) .= ( - ( exp_R * exp_R ) ) . x ;
attr r = F .: ( p , q ) means : Def1 : len r = len p & for i holds r . i = min ( p , q ) ;
sqrt ( ( r ^2 ) ^2 + ( r ^2 ) ^2 ) + sqrt ( ( r ^2 ) ^2 + ( r ^2 ) ^2 ) <= sqrt ( ( r ^2 ) ^2 + ( r ^2 ) ^2 ) + sqrt ( ( r ^2 ) ^2 ) ;
for i being Nat , M being Matrix of n , K st i in Seg n & i in Seg n holds Det ( M , i ) = Sum ( Det ( M , i ) ) + ( Det ( M , i ) )
then a " * ( a * v ) = 1 * v & a " * ( a * v ) = 1 * v & a " * ( a * v ) = 1 * v ;
p . ( j -' 1 ) * ( q *' r ) . ( i + 1 -' j ) = Sum ( p . j - q . ( i + 1 -' j ) ) * ( q . j - r ) ;
deffunc F ( Nat ) = L . 1 + ( R /* ( h ^\ n ) ) . $1 * ( ( R /* ( h ^\ n ) ) . $1 - ( R /* ( h ^\ n ) ) . $1 ;
assume that the carrier of H2 = f .: ( the carrier of H1 ) and the carrier of H1 = f .: ( the carrier of H2 ) and the carrier of H1 = f .: ( the carrier of H1 ) and the carrier of H1 = f .: ( the carrier of H2 ) and the carrier of H1 = f .: ( the carrier of H1 ) ;
Args ( o , Free ( X , Free ( X ) ) = ( ( the Sorts of Free ( S , X ) ) * ( the Arity of S ) ) # * ( the Arity of S ) ;
H1 = n + 1 -] & ( |. 2 to_power ( n + 1 ) .| + h = n + 1 -\hbox { - } cluster N .| .= n + 1 -\hbox { - } cluster N } ;
( O ( ) ) `1 = 0 & ( O ( ) ) `1 = 1 & ( O ( ) ) `1 = 0 & ( O ( ) ) `1 = 0 & ( O ( ) ) `1 = 1 & ( O ( ) ) `1 = 0 & O ( ) ) `2 = 0 ;
F1 .: ( dom F1 /\ dom F2 ) = F1 .: ( dom F1 /\ dom F2 ) .= F1 .: ( dom F1 /\ dom F2 ) .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } ;
attr b <> 0 & d <> 0 & b <> d & c <> d & d = b & a = b & b = c & c = d ;
dom ( ( f +* g ) | D ) = dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom f \/ dom g ;
for i be set st i in dom g ex u , v being Element of L st g /. i = u * v & v in B * a & u in B * a
g `1 * P " * g " = g `1 * ( g " * P " * g " ) .= g " * ( g " * P " * g " ) .= g " * ( g " * g " * g " ) .= g " * ( g " * g " ) .= g " * ( g " * g " ) ;
consider i , s1 such that f . i = s1 and not ( f . i = s1 & not i in s1 & s1 . ( i + 1 ) <> s1 ) & f . ( i + 1 ) <> s1 and f . ( i + 1 ) <> s1 and f . ( i + 1 ) <> s2 ;
h5 | ]. a , b .[ = ( g | Z ) | Z .= g | Z .= g | Z .= g | Z .= g | Z ;
[ s1 , t1 ] , [ s2 , t2 ] ] in R & [ s2 , t2 ] in R & [ s2 , t2 ] in R & [ s2 , t2 ] in R & [ s2 , t2 ] in R ;
then H is negative & H is negative & H is non negative implies H is non --+ -g\mathopen { x } -g\mathclose { x } -g\mathopen { x } }
attr f1 is total means : Def1 : 1 / 2 is total & ( 1 / 2 ) (#) ( f1 - f2 ) = f1 . c * ( f2 . c ) " ;
z1 in W2 " ( { z2 } ) or z1 = z2 " ( { z2 } ) or z1 = z2 " ( { z2 } ) & z2 in W2 " ( { z2 } ) ;
p = 1 * p .= a " * a * p .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * b " * ( b " * q ) .= b " * ( b " * q ) ;
for r be sequence of REAL , K be Real st for n be Nat holds ( for n be Nat holds r . n <= K . n ) holds upper_bound ( rng ( r - seq ) <= K . n
( E-max C ) meets ( L~ go \/ L~ pion1 ) \/ ( L~ pion1 \/ L~ pion1 ) \/ ( L~ pion1 \/ L~ pion1 ) meets ( L~ pion1 \/ L~ pion1 ) \/ ( L~ pion1 \/ L~ pion1 ) meets ( L~ pion1 \/ L~ pion1 \/ L~ co ) ;
||. f . ( g . ( k + 1 ) ) - g . ( k + 1 ) .|| <= ||. g . ( g . ( k + 1 ) ) .|| * ( K / 2 ) ;
assume h = ( ( B .--> B ) +* ( C .--> D ) +* ( E .--> F ) +* ( F .--> J ) ) +* ( E .--> F ) +* ( F .--> J ) +* ( E .--> E ) +* ( F .--> J ) +* ( F .--> E ) +* ( F .--> E ) ;
|. ( ( lower H ) . n ) - ( ( lower H ) . n ) .| <= e * ( ( lower H ) . n ) - ( ( lower H ) . n ) ;
( ( the Sorts of Free ( C , v ) ) . e = [ the Sorts of C , the carrier of C ] -tree ( the Sorts of C ) . e , ( the Sorts of C ) . e ] .= [ ( the Sorts of C ) . e , ( the Sorts of C ) . e ] ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 } = { x1 , x2 , x3 , x4 } .= { x1 , x2 , x3 } .= { x2 , x3 } \/ { x3 } .= { x1 , x2 } ;
assume that A = [. 0 , 2 * PI .] and integral ( A , A ) = ( ( #Z n ) * sin ) . x and ( A = #Z n ) * sin . x ;
p `1 is Permutation of dom f1 & p `2 = ( Sgm Y ) . i & p `2 = ( Sgm Y ) . i & p `2 = ( Sgm Y ) . i ;
for x , y st x in A holds |. ( 1 - x ) (#) ( 1 - x ) .| <= 1 * |. f . x - x .|
( ( p2 `2 ) * sqrt ( 1 + ( p2 `1 ) ^2 ) = |. p2 .| * sqrt ( 1 + ( p2 `2 ) ^2 ) .= |. p2 .| * sqrt ( 1 + ( p2 `2 ) ^2 ) ;
for f be PartFunc of the carrier of C , REAL st dom f is compact & f | X is compact holds f | X is compact iff f | X is compact
assume for x being Element of Y st x in EqClass ( z , CompF ( B , G ) ) holds ( All ( a , CompF ( B , G ) ) . x = TRUE ;
consider F3 be Function such that dom F3 = n1 and for k be Nat st k in n1 holds Q [ k , F3 . k ] ;
ex u , u1 st u <> u1 & u , u1 / v / v / ( u , u1 ) / ( u , v1 ) / ( u , u1 ) / ( u , v1 ) / ( u , u1 ) > 0 & u , v / ( u , u1 ) / ( u , v / ( u , v1 ) / ( u , v / v ) / ( u , u1 ) / ( u , v1 / v ) > 0 ;
for G being Group , A , B being non empty Subgroup of G holds ( N : N : N = ( N , A ) ` ) * ( N , B ) = N ` A
for s be Real st s in dom F holds F . s = integral ( R ^2 > ( R ^2 ) (#) ( f ^2 ) ) . s - ( R ^2 ) . s
width ( AutMt ( f1 , b1 , b2 ) ) = len ( ( f2 , b1 , b2 ) * ( f1 , b2 ) ) .= len ( ( f2 , b1 ) * ( f1 , b2 ) ) .= width ( ( f2 , b1 ) * ( f1 , b2 ) ) .= width ( ( f2 , b1 ) * ( f1 , b2 ) ) ;
f | ]. - PI / 2 , 0 .[ = f | ]. - PI / 2 , 0 .[ & f | ]. - PI / 2 , 0 .[ = f | ]. - PI / 2 , 0 .[ ;
assume that X is closed and a in X and a in X and y in X and not x in X and y in { [ n , x ] : not contradiction } ;
Z = dom ( ( ( exp_R * ( arctan * ( arctan * ( arctan + arccot ) ) ) `| Z ) /\ dom ( ( exp_R * ( arctan + arccot ) ) ) " { 0 } ;
func VERUM ( V ) -> Subset of V means : Def1 : for k holds 1 <= k & k <= len l iff l . k in V & l . k in V ;
for L being non empty RelStr , N being net of L , M being net of L st c is net of N for c being Point of N st c is Point of M holds c is Point of N holds c in N
for s being Element of NAT holds ( ( ( id the carrier of V ) + ( id the carrier of V ) ) + ( id the carrier of V ) ) . s = ( ( id the carrier of V ) + ( id the carrier of V ) ) . s
then z /. 1 = ( E-max L~ z ) .. z & ( E-max L~ z ) .. z < ( E-max L~ z ) .. z & ( E-max L~ z ) .. z < ( E-max L~ z ) .. z ;
len ( p ^ <* ( 0 qua Real ) *> ) = len p + len <* ( 0 qua Real ) *> .= len p + 1 .= len p + 1 .= len p + 1 ;
assume that Z c= dom ( ( - ln * f ) `| Z ) and for x st x in Z holds f . x = a and f . x > 0 ;
for R being add-associative right_zeroed right_complementable distributive non empty doubleLoopStr , I , J being Subset of R holds ( I + J ) *' c= I /\ J
consider f being Function of [: B1 , B2 :] , B2 such that for x being Element of B1 holds f . x = F ( x ) and f . x = F ( x ) ;
dom ( x2 + y2 ) = Seg len x .= Seg len x .= dom ( x2 (#) z ) .= dom ( x2 (#) z ) .= dom ( x2 (#) z ) .= dom ( x2 (#) z ) .= dom ( x2 (#) z ) .= dom ( x2 (#) z ) .= dom ( x2 (#) z ) ;
for S being -1 functor of C , B being functor of B for c being Object of C holds id S . ( id c ) = id ( ( the carrier' of C ) . c ) & id ( ( the carrier' of C ) . c ) = id ( ( the carrier' of C ) . c )
ex a st a = a2 & a in [: S , T :] /\ [: S , T :] & for x being Element of S holds ( f . x ) = \/ ( f . x , f . x ) ;
a in Free ( H / ( x. 4 , x. 4 ) ) \/ ( ( x. 4 , x. 0 ) '&' ( x. 4 , x. 0 ) ) ;
for C1 , C2 being C2 , f , g being stable Function of C1 , C2 st @ f = g holds f = g iff f = g
( W-min L~ go \/ W-min L~ pion1 ) `1 = ( W-min L~ pion1 ) `1 .= ( W-min L~ pion1 ) `1 .= ( W-min L~ pion1 ) `1 .= ( W-min L~ pion1 ) `1 .= ( W-min L~ pion1 ) `1 .= ( W-min L~ pion1 ) `1 .= ( W-min L~ pion1 ) `1 .= ( W-min L~ pion1 ) `1 .= ( W-min L~ pion1 ) `1 ;
assume that u = <* x0 , y0 , z0 *> and f is_continuous_in x0 and u = y0 and SVF1 ( 3 , f , u ) = SVF1 ( 3 , f , u ) . 3 and SVF1 ( 3 , f , u ) . 3 = SVF1 ( 3 , f , u ) . 3 ;
then ( t . {} ) `1 in Vars & ex x being Element of Vars st x = ( t . {} ) `1 & t . {} = x & t . {} = x & t . {} = y & t . {} = y ;
Valid ( p '&' p , J ) . v = Valid ( p , J ) . v '&' Valid ( p , J ) . v .= Valid ( p , J ) . v .= Valid ( p , J ) . v ;
assume for x , y being Element of S st x <= y for a , b being Element of T st a = f . x & b = f . y holds a >= b ;
func Class R -> Subset-Family of R means : Def1 : for A being Subset of R holds A in it iff ex a being Element of R st a in it & A c= a & a in A ;
defpred P [ Nat ] means ( ( ( the Target of G ) . ( $1 + 1 ) ) `1 c= G . ( ( the Target of G ) . ( $1 + 1 ) ) `1 & ( ( the Target of G ) . ( $1 + 1 ) ) `2 c= G . ( ( the Element of G ) . ( $1 + 1 ) ) `2 ;
assume that dim W1 = 0 and dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W1 ) = 0 ;
mat . m = ( m . t ) . {} .= ( m . t ) . {} .= ( m . t ) . {} .= ( m . t ) . {} .= ( m . t ) . {} .= m . t ;
d11 = x9 ^ d .= f . ( y9 , d ) .= f . ( y9 , d ) .= f . ( y9 , d ) .= f . ( y9 , d ) .= f . ( y9 , d ) .= f . ( y9 , d ) .= ( f . ( y9 , d ) ) . ( y9 , d ) .= ( f . ( y9 , d ) ) . ( y9 , d ) .= ( f . ( y9 , d ) . ( y9 , d ) .= ( f . ( y9 , d ) . ( y9 , d ) . ( y9 , d ) .= ( f . ( y9 , d ) . ( y9 , d ) . ( y9 , d ) . ( y9 , d ) .= ( f . ( y9 , d ) .
consider g such that x = g and dom g = dom f and for x being element st x in dom f holds g . x in f . x and g . x in f . x ;
x + 0. F_Complex = x + ( len x ) .= x + x .= x + 0. F_Complex .= x + 0. F_Complex .= x + 0. F_Complex .= x + 0. F_Complex .= x + 0. F_Complex .= x + 0. F_Complex .= x + 0. F_Complex .= x + 0. F_Complex .= x + 0. F_Complex .= x + x ;
( k -' ( k + 1 ) ) + 1 in dom ( f | ( len f -' 1 ) ) & ( f | ( k + 1 ) ) . ( k + 1 ) = ( f | ( k + 1 ) ) . ( k + 1 ) ;
assume that P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 is_an_arc_of p2 , p1 and P1 is_an_arc_of p1 , p2 and P1 = { p1 } and P2 = { p2 } and P1 /\ P2 = { p1 } and P1 /\ P2 = { p2 } and P1 /\ P2 = { p2 } and P1 /\ P2 = { p2 } and P1 /\ P2 = { p1 } and P1 /\ P2 = { p2 } and P1 /\ P2 = { p2 } and P1 /\ P2 = { p1 } and P1 /\ P2 = { p2 } and P1 /\ P2 = { p2 } and P1 /\ P2 = { p2 } and P1 /\ P2 = { p2 } /\ P2 = { p1 } and P1 /\ P2 = { p2 } and P1 /\ P2 = { p2 } /\ P2 /\ P2
reconsider a1 = a , b1 = b , c1 = c , c2 = d , c1 = d , c2 = c , c2 = d , c2 = d , b3 = c , c1 = d as Element of [: the carrier of A , the carrier of A :] ;
reconsider Itt11f = G1 . t * F1 . b as Morphism of ( G1 * F1 ) . a , ( G1 * F2 ) . b ;
LSeg ( f , i + i1 -' 1 ) = LSeg ( f /. ( i + 1 -' 1 ) , f /. ( i + 1 -' 1 ) ) .= LSeg ( f /. ( i + 1 -' 1 ) , f /. ( i + 1 -' 1 ) ) ;
Integral ( P . m , P . n ) | dom ( P . n ) <= Integral ( M , P . m ) | dom ( P . n ) ;
assume that dom f1 = dom f2 and for x , y being element st [ x , y ] in dom f1 & [ x , y ] in dom f2 holds f1 . ( x , y ) = f2 . ( x , y ) ;
consider v such that v = y and dist ( u , v ) < min ( ( r - ( ( G * ( i , 1 ) `1 ) / 2 ) , ( G * ( i + 1 , 1 ) `2 ) ) ;
for G being Group , H being Subgroup of G , a being Element of H st a = b holds a |^ H = b |^ a * H
consider B being Function of Seg ( S + L ) , the carrier of V1 such that for x being element st x in Seg ( S + L ) holds P [ x , B . x ] ;
reconsider K1 = { p0 where 7 is Point of TOP-REAL 2 : P [ 7 ] & for p being Point of TOP-REAL 2 holds P [ p ] } as Subset of TOP-REAL 2 ;
sqrt ( ( ( E-max C ) `1 - ( W-bound C ) / 2 ) / 2 ) <= sqrt ( ( W-bound C ) / 2 ) ^2 + ( ( E-bound C ) / 2 ) ^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 @ @ All ( 2 , 0 ) = len ( @ @ @ @ @ @ @ @ q ) + len <* q *> .= len ( @ @ q ) + len ( @ q ) .= len ( @ q ) + 1 .= len ( @ q ) + 1 ;
v / ( x. 3 , m1 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , x. 0 ) / ( x. 0 , x. 4 ) . ( x. 0 , x. 4 ) . ( x. 4 , x. 0 ) . ( x. 4 , x. 0 ) . ( x. 4 , x. 0 ) . ( x. 4 , x. 0 ) . ( x. 4 , x. 0 ) . ( x. 4 , x. 4 ) . ( x. 4 , x. 0 ) = x. 4 , x. 0 ) . ( x. 0 , x. 0 ) . ( x. 0 ) . ( x. 4 , x. 0 ) . ( x. 0 ) . (
consider r being Element of M such that M , v / ( x. 3 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m )
func w1 \ w2 -> Element of Union ( G , R ) equals ( ( the Sorts of G ) * ( the Arity of S ) ) . ( w1 , w2 ) ;
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= Exec ( n2 , s2 ) . b2 .= Exec ( n2 , s2 ) . b2 .= Exec ( n2 , s2 ) . b2 .= Exec ( n2 , s2 ) . b2 .= s . b1 .= s . b1 .= s . b1 ;
for n , k be Nat holds 0 <= ( Partial_Sums ( |. seq .| ) ) . ( n + k ) - ( Partial_Sums ( |. seq .| ) ) . ( n + k )
set F = S \! \mathop { 0 } ;
( Partial_Sums ( seq ) ) . K + ( Partial_Sums ( seq ) ) . K + ( Partial_Sums ( seq ) ) . K >= ( Partial_Sums ( seq ) ) . K + ( Partial_Sums ( seq ) ) . K + ( Partial_Sums ( seq ) ) . K ;
consider L , R such that for x st x in N holds ( f | Z ) . ( x - x0 ) = L . ( x - x0 ) + R . ( x - x0 ) ;
func the closed of \HM { a , b , c } -> Subset of rectangle ( a , b , c ) equals ( the Element of rectangle ( a , b , c ) ) \/ ( the Element of Closed-Interval-TSpace ( a , b , c ) ) ;
a * b ^2 + ( a * c ) + ( b * c ) + ( b * c ) + ( b * c ) + ( b * c ) + ( b * c ) >= 6 * a * b ;
v / ( x1 , m1 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) = v / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) = v / ( x2 , y2 ) ;
( ( Q ^ <* x *> ) . M = ( ( ( Q ^ <* x *> ) ) . ( M ^ <* x *> ) ) . ( M . ( M ^ <* x *> ) ) .= ( ( Q ^ <* x *> ) . ( M ^ <* x *> ) .= ( Q ^ <* x *> ) . ( M ^ <* x *> ) .= ( Q ^ <* x *> ) . ( M ^ <* x *> ) ;
Sum ( F ) = r |^ ( n1 + 1 ) * Sum ( C |^ n1 ) .= C |^ ( n1 + 1 ) * ( C |^ n1 ) .= ( C |^ n1 ) * ( C |^ n1 ) .= ( C |^ n1 ) * ( C |^ n1 ) .= ( C |^ n1 ) * ( C |^ n1 ) .= ( C |^ n1 ) * ( C |^ n1 ) .= ( C |^ n1 ) * ( C |^ n1 ) .= ( C |^ n1 ) * ( C |^ n1 ) * ( C |^ n1 ) .= ( C |^ n1 ) * ( C |^ n1 ) .= ( C |^ n1 ) * ( C |^ n1 ) * ( C |^ n1 ) .= ( C |^ n1 ) * (
( GoB f ) * ( len GoB f , 2 ) `1 = ( GoB f ) * ( len GoB f , 1 ) `1 .= ( GoB f ) * ( 1 , 1 ) `1 .= ( 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 ;
( the Arity of g ) . g = ( the Arity of S ) . g .= [ ( the Arity of S ) . g , ( the Arity of S ) . g ] .= [ g , ( the Arity of S ) . g ] .= [ g , ( the Arity of S ) . g ] .= [ g , ( the Arity of S ) . g ] ;
( X \times Y ) ^ w tolerates X ^ Y & card ( X , Y ) = card X + card Y & card ( X , Y ) = card X + card Y ;
for a , b being Element of S , s being Element of NAT st s = n & a = F . n & b = F . n & s = G . ( n + 1 ) holds b = G . ( n + 1 ) \ G . ( n + 1 )
E , f |= All ( x , ( ( x. 2 ) --> ( x. 2 , x ) ) '&' ( x. 2 , x ) ) '&' ( x. 2 , x ) '&' ( x. 2 , x ) '&' ( x. 2 , x ) '&' ( x. 2 , x ) '&' ( x. 2 , x ) '&' ( x. 2 , x ) '&' ( x. 2 , x ) '&' ( x. 2 , x ) '&' ( x. 2 , x ) '&' ( x. 2 , x ) '&' ( x. 2 , x ) '&' ( x. 2 , x ) '&' ( x. 2 , x ) '&' ( x. 2 , x ) '&' ( x. 2 , x ) '&' ( x. 2 , x ) '&' ( x. 2 , x ) '&' ( x. 2
ex R2 be 1-sorted st R2 = ( p | ( n + 1 ) ) . i & ( the carrier of p ) . i = the carrier of ( p | ( n + 1 ) ) & ( the carrier of p ) . i = the carrier of ( p | ( n + 1 ) ) ;
[. a , b + sqrt ( 1 + ( k + 1 ) ) / ( k + 1 ) .] is Element of the partial } & ( the partial } \HM { f . k : f . k = f . ( k + 1 ) } is Element of the partial } ;
Comput ( P , s , 2 + 1 ) . IC SCM+FSA = Exec ( P . 2 , Comput ( P , s , 2 ) . IC SCM+FSA .= Exec ( a , s1 ) . IC SCM+FSA .= Exec ( a , s2 ) . IC SCM+FSA .= Exec ( a , s2 ) . IC SCM+FSA .= Exec ( a , s2 ) . IC SCM+FSA ;
card ( h1 " ) . k = ( power F_Complex ) . ( - 1_ F_Complex , k ) .= ( - 1_ F_Complex ) . ( - 1_ F_Complex ) . k .= ( - 1_ F_Complex ) . k * Sum u .= ( - 1_ F_Complex ) . k * Sum u .= ( - 1_ F_Complex ) . k * Sum u .= ( - 1_ F_Complex ) . k * Sum u .= ( - g ) . k * Sum u .= ( - g ) . k ;
( ( f (#) g ) /. c = f /. c * ( g /. c ) " .= ( f /. c ) * ( g /. c ) .= ( f /. c ) * ( g /. c ) .= ( f /. c ) * ( g /. c ) .= ( f /. c ) * ( g /. c ) .= ( f (#) g ) /. c ;
len Cseq - len ( ( len ( ( len ( ( len ( ( the R ) ) ) | ( len ( ( the carrier of ( ( len ( ( the carrier of ( ( len ( the carrier of ( ( carrier of ( carrier of ( carrier of ( carrier of ( C ) ) ) ) ) ) ) ) ) ) ) ) ) = len ( ( the carrier of ( ( carrier of ( C ) ) ) ) ) ;
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\
defpred P [ Nat ] means for n holds 2 * Fib ( n + $1 ) = Fib ( n ) * Fib ( n + $1 ) + Fib ( n + $1 ) * Fib ( n + $1 ) * Fib ( n + $1 ) * Fib ( n + $1 ) * Fib ( n + $1 ) + Fib ( n + $1 ) * Fib ( n + $1 ) * Fib ( n + $1 ) * Fib ( n + $1 ) * Fib ( n + $1 ) * Fib ( n + $1 ) * Fib ( n + $1 ) * Fib ( n + $1 ) * Fib ( n + $1 ) * Fib ( n + $1 ) * Fib ( n + $1 ) * Fib ( n ) * Fib ( n ) + Fib ( n + $1 ) * Fib ( n + $1 * Fib (
consider f being Function of INT , INT such that f = f ' and f is onto and f is onto and for n being Nat holds f " . n = n and f " { f . n } = n and f " { f . n } = n ;
consider c9 be Function of S , BOOLEAN such that c9 = IExec ( A \/ B , S ) and E1 . ( A \/ B ) = Prob . ( A \/ B ) and E1 . ( A \/ B ) = Prob . ( A \/ B ) and E1 . ( A \/ B ) = Prob . ( A \/ B ) ;
consider y being Element of [: Y , { x } :] such that a = "\/" ( { F ( x , y ) where x is Element of [: Y , { x } :] : P [ x , y ] } , [: Y , { y } :] ) and Q [ y , x ] ;
assume A c= Z & f = f (#) ( 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 * ( 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 * f ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) `| 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
( f /. i ) `2 = ( ( GoB f ) * ( 1 , j2 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 + 1 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 + 1 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 + 1 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 + 1 ) ) `2 ;
dom Shift ( q , len q ) = { j + len q where q is Element of NAT : len q = j + 1 & q in dom Seq ( q , len q ) } ;
consider G1 , G2 , G2 being Morphism of V such that G1 <= G2 and G2 <= G2 and f . 1 = G1 & g . 2 = G2 & g . 3 = G2 & f . 1 = G2 and g . 2 = G2 and f . 3 = G2 and f . 1 = G2 and f . 2 = G2 and f . 3 = G2 and f . 1 = G2 and f . 3 = G2 and f . 1 = G2 and f . 2 = G2 and f . 3 = G2 and f . 3 = G2 and f . 4 = G2 and f . 4 = G2 and f . 4 = G2 and f . 4 = G2 and f . 4 = G2 and f . 4 = G2 and f . 4 = G2 and f . 4 ;
func - f -> PartFunc of C , V means : Def1 : dom it = dom f & for c st c in dom it holds it /. c = - f /. c & for c be element st c in dom it holds it /. c = - f /. c ;
consider phi such that phi is increasing and for a st phi . a = a & phi is increasing holds for v st v in union L holds L . v = union ( L | union L ) and L . a = v ;
consider i1 , j1 such that [ i1 , j1 ] in Indices GoB ( GoB f ) and f /. ( i1 + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i1 + 1 ) = ( GoB f ) * ( i1 + 1 , j1 ) and f /. ( i1 + 1 ) = ( GoB f ) * ( i1 , j1 ) ;
consider i , n such that n <> 0 and sqrt p = sqrt n and sqrt p = sqrt n and for n1 being Integer number st n1 <> 0 & n <> 0 holds sqrt p = sqrt ( n ^2 ) and sqrt p = sqrt n ;
assume that not 0 in Z and Z c= dom ( ( id Z ) (#) ( arccot * arccot ) ) and for x st x in Z holds ( ( ( id Z ) (#) ( arccot * arccot ) ) `| Z ) . x > - 1 and for x st x in Z holds ( ( ( id Z ) (#) ( arccot * arccot ) ) . x > - 1 ;
cell ( G1 , i1 -' 1 , j1 -' 1 ) \ cell ( G1 , i2 , j1 -' 1 ) c= BDD L~ f1 \/ ( ( L~ f1 ) \ { p } \/ ( L~ f2 ) \ { p } ) ;
ex Q1 being open Subset of X st s = Q1 & ex Q1 being Subset of Y st Q1 c= F & for a being Point of Y st a in F holds Q1 is finite & a in Q & b in Q holds [#] ( Y | a ) c= union ( F | a )
gcd ( A1 , r2 ) . ( ( 1. ( K , n ) ) . ( r1 , r2 ) , r2 ) . ( ( 1. ( K , n ) ) . ( r2 , s2 ) ) = 1 / 2 * ( ( 1. ( K , n ) ) . ( r1 , r2 ) ) .= 1 / 2 * ( ( 1. ( K , n ) ) . ( r2 , s2 ) ) ;
R8 = ( ( ( ( ( ( ( ( ( ( ( ( ( , s2 ) ) ) . 1 ) ) ) . ( m2 + 1 ) ) . ( m2 + 1 ) ) ) . ( m2 + 1 ) ) .= ( ( ( ( ( ( the Sorts of A ) . 2 ) . ( m2 + 1 ) ) . ( A2 + 1 ) ) . ( A2 + 1 ) .= [ 3 , 4 ] ;
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 .= halt SCMPDS ;
P1 /\ P2 = ( { p1 } \/ LSeg ( p1 , p2 ) ) \/ ( LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) ) \/ ( LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) ) ;
func not f is Subset of the Sorts of A2 means : Def1 : a in dom f iff ex p st p in dom f & a = f . p & not p in dom f & not p in dom f & not p in dom f & not p in dom f ;
for a , b being Element of F_Complex st |. a .| > b & |. b .| > 1 holds a * ( f . b ) >= 1 & b * ( f . b ) >= 0 implies a * ( f . b ) = 1
defpred P [ Nat ] means $1 <= len g implies 1 <= i & i < j & j <= len G & G * ( i , j ) `2 < g & G * ( i , j ) `2 < g & g < G * ( i , j ) `2 ;
attr C1 , C2 , f , g being \cdot f , g being \cdot g being \cdot f , h being Function of C1 , C2 , a , b being Real st f = g & g = h holds f = g * f & g = h * f ;
( ||. f .|| ) . c = ||. f .|| . c .= ||. f .|| . c .= ||. f .|| . c .= ||. f .|| . c .= ||. f .|| . c .= ||. f .|| . c .= ||. f .|| . c .= ||. f .|| . c ;
|. q .| ^2 = ( q `1 ) ^2 + ( q `2 ) ^2 & 0 <= ( q `2 ) ^2 & 0 <= ( q `1 ) ^2 & 0 <= ( q `2 ) ^2 & 0 <= ( q `2 ) ^2 ;
for F being Subset-Family of T7 st F is open & not {} in F & for A , B being Subset of T7 st A in F & B in F & A misses B holds card A = card B & card B = card ( A \/ B )
assume that len F >= 1 and len F = k + 1 and len F = len G and for k st k in dom F holds H . k = g . k and for k st k in dom F holds H . k = g . k and H . k = g . k ;
i |^ ( ( mod n ) |^ s ) - i |^ s = i |^ s |^ s - i |^ s .= i |^ s * ( i |^ s ) - i |^ s .= i |^ s * ( i |^ s ) - i |^ s .= i |^ s * ( i |^ s ) - i |^ s .= i |^ s * ( i |^ s ) - i |^ s .= i |^ s * ( i |^ s ) ;
consider q being oriented oriented oriented Chain of G such that r = q and q <> {} and q . 1 = v and ( F . ( q . len q ) = v and ( F . ( q . len q ) = v and ( F . ( q . len q ) ) . 1 = v and ( F . ( q . len q ) ) . 1 = v and ( F . ( q . len q ) = v ;
defpred P [ Element of NAT ] means $1 <= len ' implies ( ( ( g , Z ) . ( len g ) ) . ( len g + $1 ) = ( ( ( g , Z ) . ( len g + $1 ) ) . ( len g + $1 ) ;
for A , B being Matrix of n , K holds len ( A * B ) = len A & width ( A * B ) = width A & width ( A * B ) = width A & width ( A * B ) = width B & width ( A * B ) = width A & width ( A * B ) = width B
consider s being FinSequence of the carrier of R such that Sum s = u and for i being Element of NAT st 1 <= i & i <= len s ex a , b being Element of R st s . i = a * b & a * b = b * a and s * a = b * b ;
func |( x , y )| -> Element of COMPLEX equals |( ( Re x , Re y ) , ( Re y ) , ( Im x ) , ( Im y ) )| + ( ( Im x ) ^2 + ( Im y ) ^2 ) ;
consider g2 be FinSequence of F such that g2 is continuous and rng g2 c= A and g2 . 1 = x0 and g2 . ( len g2 ) = x0 and g2 . ( len g2 ) = x0 and g2 . ( len g2 ) = x0 and g2 . ( len g2 ) = x0 and g2 . ( len g2 ) = x0 and g2 . ( len g2 ) = x0 and g2 . ( len g2 ) = x0 and g2 . ( len g2 ) = x0 and g2 . ( len g2 ) = x0 and g2 . ( len g2 ) = x0 and g2 . ( len g2 ) = x0 and g2 . ( len g2 ) = x0 and g2 . ( len g2 ) = x0 and g2 . ( len g2 ) = x0 and g2 . ( len g2 ) = x0 and g2 . ( len g2 ) = x0 and g2 . ( len g2 ) = x0 and g2 . ( len g2 ) = x0 and
then n1 >= len p1 & crossover ( p1 , p2 , n1 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n4 , n4 , n4 , n3 , n4 , n4 ) = crossover ( p1 , p2 , n1 , n2 , n3 , n3 , n3 , n3 , n4 , n4 , n4 , n4 , n4 ) ;
( q * a ) * a <= ( q * a ) * ( q * b ) & ( - q ) * a <= ( - q ) * ( - q ) & ( - q ) * ( - q ) <= ( - q ) * ( - q ) ;
( F . ( p . len p ) ) . ( len p + 1 ) = ( F . ( p . len p ) ) . ( len p + 1 ) .= ( F . ( p . len p ) ) . ( len p + 1 ) .= ( F . ( p . 1 ) ) . ( len p + 1 ) .= ( F . ( p . 1 ) ) . ( len p + 1 ) .= ( F . ( p . 1 ) ) . ( len p + 1 ) .= ( F . 1 ) . ( len p + 1 ) .= ( F . 1 ) . ( p . 1 ) . ( len p + 1 ) .= ( F . 1 ) . ( len p + 1 ) .= ( F . 1 ) . ( len p + 1 ) . ( len p + 1 ) .= ( F . 1 ) . ( len p + 1 ) .=
consider k1 being Nat such that k1 + 1 = k and a := k = ( <* a *> := ( k , a ) ) := ( k , a ) and a := ( k , a ) = ( <* a *> := ( k , a ) ) := ( k , a ) ;
consider B8 being Subset of [: B1 , B2 :] , y1 being Subset of [: B1 , B2 :] such that B8 : y1 is finite and [: B1 , y2 :] is finite and for a being Element of B1 holds B1 . a = \mathop { x where x is Element of [: B1 , B2 :] : x in B1 & y in B2 } c= B1 and y2 in B2 } ;
v2 . b2 = ( curry F2 , g ) * ( ( curry F2 ) . b2 ) .= ( ( curry F2 ) . b2 ) * ( ( curry F2 ) . b2 ) .= ( ( curry F2 ) . b2 ) * ( ( curry F2 ) . b2 ) .= ( ( curry F2 ) . b2 ) * ( ( curry F2 ) . b2 ) .= ( ( curry F2 ) . b2 ) * ( ( curry F2 ) . b2 ) .= ( ( curry F2 ) . b2 ) * ( id B ) . b2 ) * ( id B ) . b2 ) * ( id B ) . b2 ) .= ( ( ( id B ) . b2 ) * ( id B ) . b2 .= ( ( ( B ) . b2 ) . b2 ;
dom IExec ( I , P , Initialize s ) = the carrier of SCMPDS .= dom IExec ( I , P , Initialize s ) .= dom IExec ( I , P , Initialize s ) .= dom IExec ( I , P , Initialize s ) .= dom IExec ( I , P , Initialize s ) .= dom IExec ( I , P , Initialize s ) .= dom IExec ( I , P , Initialize s ) ;
ex d-32 be Real st d-32 > 0 & for h be Real st h <> 0 & |. h .| < d & |. h .| < d holds |. h .| " * ||. ( R + R1 ) /. h .|| < e / 2 * ||. ( R2 + R1 ) /. h .||
LSeg ( G * ( len G , 1 ) + |[ 1 , 0 ]| , |[ 1 , 0 ]| ) c= Int cell ( G , len G , 1 ) \/ { |[ 1 , 0 ]| } \/ Int cell ( G , len G , 1 ) \/ { |[ 1 , 0 ]| } ;
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 ) ) .= LSeg ( h /. ( i + 1 ) , h /. ( i + 1 ) ) .= LSeg ( h /. ( i + 1 ) ;
A = { q where q is Point of TOP-REAL 2 : LE q , p1 , P , p1 , p2 & LE q , p1 , P , p1 , p2 & LE q , p1 , P , p1 , p2 } , P } ;
( ( - x ) .|. y ) = ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .=
0 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) = sqrt ( ( p `1 / p `2 ) ^2 ) * sqrt ( 1 + ( p `2 / p `2 ) ^2 ) .= sqrt ( ( p `1 / p `2 ) ^2 ) * sqrt ( 1 + ( p `2 / p `2 ) ^2 ) ;
sqrt ( ( U * ( - 1 ) ) * ( ( W * ( - 1 ) ) / ( - 1 ) ) = ( ( U * ( - 1 ) ) ) * ( - 1 ) .= ( U * ( - 1 ) ) * ( - 1 ) .= ( U * ( - 1 ) ) * ( - 1 ) .= ( U * ( - 1 ) ) * ( - 1 ) ;
func Shift ( f , h ) -> PartFunc of REAL m , REAL means : Def1 : for x be Real st x in dom h holds it . x = ( - h ) * ( x - h ) & for x be Real st x in dom h holds it . x = ( - h ) * ( x - h ) ;
assume that 1 <= k and k + 1 <= len f and [ i + 1 , j ] in Indices G and [ i + 1 , j ] in Indices G and [ i + 1 , j ] in Indices G and f /. k = G * ( i + 1 , j ) and f /. k = G * ( i + 1 , j ) ;
assume that not y in Free H and x in Free H and not x in Free H and not y in Free ( H ) and not x in Free ( H ) and not x in Free ( H ) and not y in Free ( H ) and not x in Free ( H ) and not y in Free ( H ) and not x in Free ( H ) and x in Free ( H ) and y in Free ( H ) ;
defpred P11 [ Element of NAT , Element of NAT ] means P [ p ] implies ( $1 |^ p ) |^ ( $1 + 1 ) = ( p |^ ( $1 + 1 ) ) |^ ( $1 + 1 ) & ( $1 |^ ( $1 + 1 ) = ( p |^ $1 ) |^ ( $1 + 1 ) |^ ( $1 + 1 ) ;
func \sigma ( C ) -> non empty Subset-Family of X means : Def1 : for A , B being Subset of X holds A c= it iff for W being Subset of X st W c= A \ B & W c= B \ C holds W c= A \ B ;
[#] ( ( dist ( \in ( Euclid 2 ) ) | Q ) = ( dist ( ( dist ( ( dist ( ( Euclid 2 ) ) | Q ) ) ) .: Q ) & lower_bound ( ( dist ( ( Euclid 2 ) ) | Q ) = inf ( ( Euclid 2 ) | Q ) ;
rng ( F | ( [: S , 2 :] :] ) = {} or rng ( F | ( [: S , 2 :] ) ) = { 1 } or rng ( F | ( [: S , 2 :] ) ) = { 1 } or rng ( F | ( [: S , 2 :] :] ) = { 2 } or rng ( F | ( [: S , 2 :] ) ) = { 1 } ;
( f " ) . i = f . i " " .= ( f " ) . i .= ( f " ) . i .= ( f " ) . i .= ( f " ) . i .= ( f " ) . i .= ( f " ) . i .= ( f " ) . i .= ( f " ) . i .= ( f " ) . i .= ( f " ) . i ;
consider P1 , P2 being non empty Subset of TOP-REAL 2 such that P1 /\ P2 = { p1 , p2 } and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p2 , p1 and P1 is_an_arc_of p2 , p1 and P1 is_an_arc_of p2 , p1 and P1 is_an_arc_of p2 , p1 and P1 is_an_arc_of p2 , p1 and P1 is_an_arc_of p2 , p1 and P1 is_an_arc_of p2 , p1 and P1 is_an_arc_of p2 , p1 and P1 is_an_arc_of p2 , p1 and P1 is_an_arc_of p2 , p1 and P1 is_an_arc_of p2 , p1 and P1 is_an_arc_of p2 , p1 , p2 and P1 is_an_arc_of p2 , p1 , p2 , p2 and P1 is_an_arc_of p2 , p1 , p2 , p2 , p2 , p1 , p2 ;
f . p2 = |[ ( p2 `1 ) ^2 / sqrt ( 1 + ( p2 `2 ) ^2 ) , ( p2 `2 ) ^2 / sqrt ( 1 + ( p2 `1 ) ^2 ) ]| .= |[ 1 , 0 ]| ;
( \lbrace a , X , Y ) " . x = ( \HM { the } \HM { carrier } \HM { of } X } ) . x .= ( ( \lbrace a , X } qua } \HM { Function } ) . x .= 0. X + 0 .= 0 + 0 .= 0 ;
for T being non empty normal TopSpace , A , B being closed Subset of T , r being Real st A <> {} & A misses B & A misses B holds A misses B or A misses B or B misses C or A /\ C = B /\ C
for i , [#] G1 , G being strict Subgroup of G st i + 1 in dom F for G1 , G2 being strict Subgroup of G st G1 = F . i & G2 = F . ( i + 1 ) & G2 = F . ( i + 1 ) holds G1 is strict Subgroup of G2
for x st x in Z holds ( ( arctan * arctan ) `| Z ) . x = ( ( arctan * arccot ) . x + ( arctan * arccot ) . x / ( 1 + x ^2 ) / ( 1 + x ^2 )
synonym f /* ( a ^\ k ) = lim ( f /* a ) & f . x0 = ( f /* a ) . ( a + k ) & for n st n in dom f holds f . ( a + k ) < ( f /* a ) . n ;
then X1 , X2 are_separated or X1 , X2 are_separated implies ex Y1 , Y2 being non empty SubSpace of X st Y1 meet Y2 misses ( X1 union X2 ) & ( X1 , X2 meet X2 ) meet ( X1 union X2 ) = ( X1 union X2 ) union ( X2 meet X2 ) ;
ex N be Neighbourhood of x0 st N c= dom SVF1 ( 1 , f , u ) & ex L be Real st for x st x in N holds ( SVF1 ( 1 , f , u ) ) . x - SVF1 ( 1 , f , u ) . x0 = L . ( x - x0 ) + R . ( x - x0 ) ;
sqrt ( ( p2 `1 ) ^2 + ( p2 `2 ) ^2 ) >= sqrt ( 1 + ( p2 `1 ) ^2 ) * sqrt ( 1 + ( p2 `2 ) ^2 ) ;
( ( sqrt ( 1 - ( t1 * t2 ) ) / 2 ) to_power ( n + 1 ) = ( ( 1 - ( t1 * t2 ) ) / 2 ) to_power ( n + 1 ) & ( ( sqrt ( 1 - ( t1 * t2 ) ) / 2 ) to_power ( n + 1 ) = ( 1 - ( t1 * t2 ) ) to_power ( n + 1 ) ;
assume that for x holds f . x = ( sin + cos ) . x and for x st x in Z holds ( ( sin + cos ) `| Z ) . x = sin . x - sin . x and for x st x in Z holds ( ( sin + cos ) `| Z ) . x = sin . x - cos . x ;
consider X1 being Subset of Y , Y1 being Subset of X such that t = X1 and Y1 in A and X1 in B and Y1 in A and X1 /\ Y1 = { X1 } and Y1 /\ Y1 = { X2 } and X1 is open and Y1 is open and X1 /\ Y1 <> {} and Y1 /\ Y1 <> {} and X1 <> {} and Y1 <> {} and X1 <> {} and Y1 <> {} ;
card S . ( n + 3 ) = card { [ d , c ] } + b .= ( d * ( [: d , c :] + b ) ) * b .= ( d * ( [: d , c :] + b ) ) * b .= ( d * ( a , b ) ) * b .= ( d * ( a , b ) ) * b ;
sqrt ( ( ( W-bound D ) / 2 ) * ( ( W-bound D ) / 2 ) - ( ( W-bound D ) / 2 ) * ( ( W-bound D ) / 2 ) * ( ( W-bound D ) / 2 ) ) = ( ( W-bound D ) / 2 ) * ( ( E-bound D ) / 2 ) - ( E-bound D ) / 2 * ( ( W-bound D ) / 2 ) * ( ( W-bound D ) / 2 ) .= ( ( W-bound D ) / 2 ) / 2 ;