thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
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contradiction ;
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contradiction ;
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thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
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contradiction ;
thesis ;
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thesis ;
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thesis ;
contradiction ;
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contradiction ;
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contradiction ;
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thesis ;
contradiction ;
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contradiction ;
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contradiction ;
contradiction ;
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contradiction ;
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contradiction ;
contradiction ;
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thesis ;
thesis ;
thesis ;
contradiction ;
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contradiction ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
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thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b
D c= S
let Y ;
S ` is Cauchy
q in P ;
V ;
y in N ;
x in T ;
m < n ;
m <= n ;
n > 1 ;
let r ;
t in I ;
n <= 4 ;
M is finite ;
let X ;
Y c= Z ;
A // M ;
let U ;
a in D ;
q in Y ;
let x ;
1 <= l ;
1 <= w ;
let G ;
y in N ;
f = {} ;
let x ;
x in Z ;
let x ;
F is one-to-one ;
e <> b ;
1 <= n ;
f is special ;
S misses C
t <= 1 ;
y divides m ;
P divides M ;
let Z ;
let x ;
y c= x ;
let X ;
let C ;
x _|_ p ;
o is monotone ;
let X ;
A = B ;
1 < i ;
let x ;
let u ;
k <> 0 ;
let p ;
0 < r ;
let n ;
let y ;
f is onto ;
x < 1 ;
G c= F ;
a >= X ;
T is continuous ;
d <= a ;
p <= r ;
t < s ;
p <= t ;
t < s ;
let r ;
D <= E ;
assume e > 0 ;
assume 0 < g ;
p in X ;
x in X ;
Y `2 in Y ;
assume 0 < g ;
not c in Y ;
not v in L ;
2 in z `2 ;
assume f = g ;
N c= b ` ;
assume i < k ;
assume u = v ;
I = J ;
B `2 = b `2 ;
assume e in F ;
assume p > 0 ;
assume x in D ;
let i be element ;
assume F is onto ;
assume n <> 0 ;
let x be element ;
set k = z ;
assume o = x ;
assume b < a ;
assume x in A ;
a `2 <= b `2 ;
assume b in X ;
assume k <> 1 ;
f = Product l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is from of squares ;
assume m > 0 ;
assume A c= B ;
X is lower
assume A <> {} ;
assume X <> {} ;
assume F <> {} ;
assume G is open ;
assume f is dilatation ;
assume y in W ;
y \not <= x ;
A `2 in B `2 ;
assume i = 1 ;
let x be element ;
x `2 = x `2 ;
let X be BCK-algebra ;
assume S is non empty ;
a in REAL ;
let p be set ;
let A be set ;
let G be _Graph , x be set ;
let G be _Graph , x be set ;
let a be VECTOR of W ;
let x be element ;
let x be element ;
let C be FormalContext , D be Subset of TOP-REAL 2 ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
let y be Real ;
X c= f . a
let y be element ;
let x be element ;
let i be Nat ;
let x be element ;
n in NAT ;
let a be element ;
m in NAT ;
let u be element ;
i in NAT ;
let g be Function ;
Z c= NAT ;
l <= NAT ;
let y be element ;
r2 in r2 ;
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 = $ " ;
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 ( x , a ) is_differentiable_in x ;
j < i0 ;
let j be Nat ;
n <= n + 1 ;
k = i + m ;
assume C meets S ;
n <= n + 1 ;
let n be Nat ;
h1 = {} ;
0 + 1 = 1 ;
o <> b2 ;
f2 is one-to-one ;
support p = {}
assume x in Z ;
i <= i + 1 ;
r1 <= 1 ;
let n be Nat ;
a "/\" b <= a ;
let n be Nat ;
0 <= r1 ;
let e be Real , x be Point of TOP-REAL 2 ;
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 Nat ;
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 is open ;
cluster uparrow x -> be be be be open ;
X <> { x } ;
x in { x } ;
q , b // M ;
A . i c= Y ;
P [ k ] ;
2 to_power x in W ;
X [ 0 ] ;
P [ 0 ] ;
A = A |^ i ;
that element >= us ;
G . y <> 0 ;
let X be RealNormSpace , x be Point of X ;
a in X ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , A be Subset of V ;
assume x in - M ;
k < s . a ;
not t in { p } ;
let Y be set , f be Function of Y , BOOLEAN ;
M , L are_isomorphic ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded & L is upper-bounded ;
rng f = Y ;
Gs 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 `2 = a `2 + 1 ;
x `2 = a * y `2 ;
rng D c= A ;
assume x in K1 ;
1 <= ii ;
1 <= ii ;
pp c= PI / 2 ;
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 ;
s-7 is bounded ;
assume I c= P1 ;
n in dom I ;
let Q ;
B c= dom f ;
b + p _|_ a ;
x in dom g ;
F-14 is continuous ;
dom g = X ;
len q = m ;
assume A2 is closed ;
cluster R \ S -> real-valued ;
sup D in S ;
x << sup D ;
b1 >= Z1 & b2 >= Y2 ;
assume w = 0. V ;
assume x in A . i ;
g in PreNorms X ;
y in dom t ;
i in dom g ;
assume P [ k ] ;
Set Set Set Set = f ;
x4 is increasing ;
let e2 be element ;
- b divides b ;
F c= \mathclose { F } ;
Gseq is non-decreasing ;
Gseq is non-decreasing ;
assume v in H . m ;
assume b in [#] B ;
let S be non void ManySortedSign , X be non-empty ManySortedSet of 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 inf X in X ;
y in rng f ;
let s , I be set , s be State of SCM+FSA ;
b `2 c= ( b `2 ) ;
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 ;
|. cn .| < n ;
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_differentiable_in x0 & g is_differentiable_in x0 ;
assume O is symmetric transitive ;
let x , y be element ;
let j2 be Nat ;
[ y , x ] in R ;
let x , y be element ;
assume y in conv A ;
x in Int V ;
let v be VECTOR of V ;
P3 is_halting_on s , P ;
d , c // a , b ;
let t , u ;
let X be set ;
assume k in dom s ;
let r be non negative Real ;
assume x in F | M ;
let Y be Subset of S ;
let X be non empty TopSpace , f be Function of X , REAL ;
[ a , b ] in R ;
x + w < y + w ;
{ a , b } >= c ;
let B be Subset of A , x be Element of A ;
let S be non empty ManySortedSign ;
let x be variable , f be Function ;
let b be Element of X , a be Element of X ;
R [ x , y ] ;
x ` = x ;
b \ x = 0. X ;
<* d *> in D |^ 1 ;
P [ k + 1 ] ;
m in dom ( n (#) F ) ;
h2 . a = y ;
P [ n + 1 ] ;
cluster G * F -> pre| ;
let R be non empty multiplicative Function , a be Element of R ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p `2 ;
assume f | X is lower ;
x in rng pion1 & y in rng pion1 ;
let x be Element of B ;
let t be Element of D ;
assume x in Q .vertices() ;
set q = s ^\ k ;
let t be VECTOR of X ;
let x be Element of A ;
assume y in rng p `2 ;
let M be void maid ;
let N be non empty Subset of the \it \mathbin { 0 } ;
let R be RelStr with finite finite Anumber ;
let n , k be Nat ;
let P , Q be RelStr ;
P = Q /\ [#] S ;
F . r in { 0 } ;
let x be Element of X ;
let x be Element of X ;
let u be VECTOR of V ;
reconsider d = x as Int-Location ;
assume I does not destroy a ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < ( v - u ) ;
x <= c2 . x ;
x in F ` ;
cluster S --> T -> product is \cal ;
assume t1 <= t2 & t2 <= t1 ;
let i , j be even Nat ;
assume F1 <> F2 & F2 <> G2 ;
c in Intersect ( union R ) ;
dom p1 = c & dom p2 = c ;
a = 0 or a = 1 ;
assume A1 <> A2 & A2 <> A1 ;
set i1 = i + 1 ;
assume a1 = b1 & a2 = b2 ;
dom g1 = A & rng g2 c= B ;
i < len M + 1 ;
assume not - +infty in rng G ;
N c= dom f1 & N c= dom f2 ;
x in dom sec & y in dom sec ;
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 & 1 <= len D2 ;
( p `2 ) ^2 = 0 ;
j2 <= width G ;
len cos > 1 + 1 ;
set n1 = n + 1 ;
|. q9 .| = 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 Line of as elements of as line ;
cluster m * n -> invertible ;
let kk be Nat ;
i - 1 > m ;
R is transitive & R is transitive implies R is transitive
set F = <* u , w *> ;
pp `1 c= P3 `1 & p1 `1 <= b ;
I is_halting_on t , Q & I is_halting_on t , Q ;
assume [ S , x ] is vertical ;
i <= len f2 & j <= len f2 ;
p is FinSequence of X ;
1 + 1 in dom g ;
Sum R2 = n * r ;
cluster f . x -> complex-valued ;
x in dom f1 & y in dom f2 ;
assume [ X , p ] in C ;
BX c= XX & BX c= X ;
n2 <= ( 2 |^ ( n + 1 ) ) ;
A /\ ( P ` ) c= A ` ;
cluster x -valued -> x -valued for Function ;
let Q be Subset-Family of S , P be Subset of T ;
assume n in dom g2 & n in dom g2 ;
let a be Element of R ;
t `2 in dom e2 & t `2 in dom e2 ;
N . 1 in rng N ;
- z in A \/ B ;
let S be SetSequence of X , T be Subset of S ;
i . y in rng i ;
REAL c= dom f & rng f c= dom g ;
f . x in rng f ;
mt <= ( r / 2 ) * ( 2 * n ) ;
s2 in r-5 & s1 < s2 ;
let z , z be complex number ;
n <= NN . m ;
LIN q , p , s ;
f . x = waybelow x /\ B ;
set L = |[ S , T ]| ;
let x be non positive Real ;
let m be Element of M ;
f in union ( F1 * F ) ;
let K be add-associative right_zeroed right_complementable associative commutative associative distributive non empty doubleLoopStr , p be Polynomial 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 + 1 ;
n1 < n1 + 1 + 1 ;
cluster [: T , T :] -> transitive ;
[ y2 , 2 ] `2 = z ;
let m be Element of NAT ;
let S be Subset of R ;
y in rng S29 & y in rng S29 ;
b = sup dom f & b = sup dom f ;
x in Seg len q & y in Seg len q ;
reconsider X = [: D , D :] as set ;
[ a , c ] in E1 ;
assume n in dom h2 & n in dom h2 ;
w + 1 = ( a1 + a2 ) ;
j + 1 <= j + 1 + 1 ;
k2 + 1 <= k1 & k2 + 1 <= k2 ;
let i be Element of NAT ;
Support u = Support p & Support u = Support p ;
assume X is complete \frac of m , n ;
assume that f = g and p = q ;
n1 <= n1 + 1 & n2 + 1 <= n1 ;
let x be Element of REAL n ;
assume x in rng s2 & y in rng s2 ;
x0 < x0 + 1 / 2 ;
len ( L5 ) = W ;
P c= Seg ( len A ) ;
dom q = Seg n & rng q c= Seg n ;
j <= width M *' ;
let rr be real-valued FinSequence ;
let k be Element of NAT ;
Integral ( M , P ) < +infty ;
let n be Element of NAT ;
assume z in in c= c= c= V -] ( 0 ) ;
let i be set ;
n - 1 = n-1 - 1 ;
len ( n |-> 0 ) = n ;
Set N = F .: c ;
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 & dom q = Seg i ;
let s be Element of E |^ omega ;
let B1 be Basis of x , B ;
( ( LSeg ( L2 , j ) ) /\ L2 = {} ;
L1 /\ L2 = {} ;
assume ||. x .|| = ||. y .|| ;
assume b , c // b `1 , c `2 ;
LIN q , c , c ;
x in rng ( f-1z ) ;
set n8 = n + j ;
let D7 be non empty set , f be FinSequence of D ;
let K be add-associative right_zeroed right_complementable non empty doubleLoopStr , M be Matrix of K ;
assume that f `2 = f and h `2 = h ;
R1 - R2 is total ;
k in NAT & 1 <= k ;
let a be Element of G ;
assume x0 in [' a , b '] ;
K1 ` is open & K1 /\ K1 is open ;
assume a , b are_maximal in C ;
let a , b be Element of S ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = Integral ( M , f ) ;
cluster \vert a -> neeess\mathop ;
not u in { [: g , h :] } ;
the carrier of f c= B ;
reconsider z = x as VECTOR of V ;
cluster the RelStr of L -> -> -> -> -> -> for RelStr ;
r (#) H is being being being being being being being being being being being being PartFunc of X , REAL ;
s . intloc 0 = 1 ;
assume that x in C and y in C ;
let U0 be strict non-empty MSAlgebra over S , x be set ;
[ x , Bottom T ] is compact ;
i + 1 + k in dom p ;
F . i is stable Subset of M ;
r-35 in { { y } where y is Element of L : not contradiction } ;
let x , y be Element of X ;
let A , I be Subset of X ;
[ y , z ] in O ;
that that that card Macro i = 1 and card Macro i = 2 ;
rng Sgm A = A & card A = card A ;
q |- All ( y , q ) ;
for n holds X [ n ] ;
x in { a } & x in d ;
for n holds P [ n ] ;
set p = |[ x , y , z ]| ;
LIN o , a , b ;
p . 2 = Z |^ Y ;
( D . x0 ) `2 = {} ;
n + 1 + 1 <= len g ;
a in CQC-WFF ( Al ( ) ) ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
let I be Ideal of L ;
set g = f1 + f2 , h = f2 + f3 ;
a <= max ( a , b ) ;
i-1 < len G + 1-1 ;
g . 1 = f . i1 ;
x `1 , y `2 in A2 & y `2 in A1 ;
( f /* s ) . k < r ;
set v = VAL g ;
i -' k + 1 <= S ;
cluster non empty multiplicative -> associative for is associative ;
x in support ( ( support t ) --> x ) ;
assume a in [: G , H :] ;
i `2 <= len ( y `2 ) ;
assume p divides b1 + b2 & p divides b2 ;
x0 <= sup M1 & x0 < sup M1 ;
assume x in W-min ( X ) ;
j in dom ( z | n ) ;
let x be Element of [: D , D :] ;
IC Comput ( P1 , s , m ) = l1 ;
a = {} or a = { x } ;
set uG = Vertices G , uG = Vertices G , uG = Vertices G , uG = Vertices G , uG = Vertices G , uG = Vertices G , uG
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 ( h ) ;
]. a , b .[ c= Z ;
X1 , X2 are_separated & X2 , X1 are_separated implies X1 , X2 are_separated
a in Cl ( union F \ G ) ;
set x1 = [ 0 , 0 ] ;
k + 1 -' 1 = k ;
cluster NAT -valued for NAT -defined Function ;
ex v st C = v + W ;
let IT be non empty NAT , n be Nat ;
assume V is Abelian add-associative right_zeroed right_complementable ;
XY \/ Y in \sigma ( L ) ;
reconsider x = x as Element of S ;
max ( a , b ) = a ;
sup B is upper & sup B is upper ;
let L be non empty reflexive RelStr , X be Subset of L ;
R is reflexive & X is transitive implies R \/ R is transitive
E , g |= the_left_argument_of H implies E , f |= H
dom G /. y = a ;
sqrt ( 1 - 4 ) >= - r ;
G . x0 in rng G & G . x0 in rng G ;
let x be Element of FH , y be Element of FH ;
D [ ( P-6 ) . 0 , 0 ] ;
z in dom id ( B ) & z in dom id ( B ) ;
y in the carrier of N & y in the carrier of N ;
g in the carrier of H & g in the carrier of H ;
rng f|[ 1 , 1 ]| c= NAT ;
j `2 + 1 in dom s1 ;
let A , B be strict Subgroup of G ;
let C be non empty Subset of REAL ;
f . z1 in dom h & f . z2 in dom h ;
P . k1 in rng P & P . k2 in rng P ;
M = ( A +* {} ) +* {} .= A ;
let p be FinSequence of REAL , n be Nat ;
f . n1 in rng f & f . n1 in rng f ;
M . ( F . 0 ) in REAL ;
h | [' a , b '] = b ;
assume that the distance of V , Q is v ;
let a be Element of ^ ( V ) ;
let s be Element of P ( ) ;
let PL be non empty RelStr ;
n be Nat ;
the carrier of g c= B & f .: ( g .: B ) c= B ;
I = halt SCM ( R ) .= I ;
consider b being element such that b in B ;
set BK = BCS K , BK = BCS K ;
l <= ( -> -> Real ) ;
assume x in ]. [ s , t ] , [ s , t ] } ;
( x `2 ) ^2 in ]. t `1 , t `2 .[ ;
x in JumpParts ( T . s ) ;
let h be Morphism of c , a ;
Y c= { 1_ K } & Y c= { 1_ K } ;
A2 \/ A3 c= Carrier ( L1 ) \/ Carrier ( L2 ) ;
assume LIN o `1 , a & LIN o `1 , b , c ;
b , c // d1 , d2 ;
x1 , x2 , x3 is_collinear & x2 , x3 , x4 is_collinear ;
dom <* y *> = Seg 1 .= Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar s .| ;
[ x , x `2 ] in X ~ ;
for n being Nat holds 0 <= x . n
|[ a , b ]| = [. a , b .] ;
cluster -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> closed for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 , q1 is_collinear & q2 , q2 , q1 is_collinear ;
dom M1 = Seg n & dom M2 = Seg n ;
x = [ x1 , x2 ] & y = [ x2 , x3 ] ;
let R , Q be ManySortedSet of A ;
set d = ( 1 / n ) * ( 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 , V be Subset of M ;
let I be Program of SCM+FSA , a be Int-Location ;
assume x in rng ( ( R * S ) * ( R * S ) ) ;
let b be Element of the lattice of T ;
dist ( e , z ) > r-r ;
u1 + v1 in W2 + W3 ;
assume the carrier of L misses rng G & G is finite ;
let L be lower-bounded antisymmetric non empty RelStr ;
assume [ x , y ] in [: a9 , b9 :] ;
dom ( A * e ) = NAT ;
let a , b be Vertex of G ;
let x be Element of Bool ( M , X ) ;
0 <= Arg a * PI ;
not o , a9 , y is_collinear & not o , a , y is_collinear ;
{ v } c= the carrier of l ;
let x be variable of A ;
assume x in dom ( uncurry f ) ;
rng F c= ( product f ) .: X
assume D2 . k in rng D & D . k in rng D ;
f " . p1 = 0 ;
set x = the Element of X , y = the Element of X ;
dom Ser G = NAT & dom Ser G = NAT ;
n be Element of NAT ;
assume LIN c , a , e1 ;
cluster finite for FinSequence of NAT ;
reconsider d = c as Element of L1 ;
( v2 |-- I ) . X <= 1 ;
assume x in the carrier of f .: the carrier of G ;
conv @ @ S c= conv A & conv @ S c= conv A ;
reconsider B = b as Element of the lattice of T ;
J , v |= P ! l ;
cluster the TopStruct of J . i -> non empty ;
ex_sup_of Y1 \/ Y2 , T & for x st x in Y1 holds x in Y1 implies x in Y1
W1 is_well field W1 & W2 is_\upharpoonright field W2 implies W1 is well of 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 s4 & not s4 c= s ;
assume ( a 'imp' b ) . z = TRUE ;
assume that X is open and f = X --> d ;
assume [ a , y ] in Indices ( f * g ) ;
assume that Directed I c= J and Directed I c= K ;
Im ( lim seq ) = 0 ;
( ( ( - 1 ) (#) sin ) `| Z ) . x <> 0 ;
sin is_differentiable_on Z & cos is_differentiable_on Z implies sin is_differentiable_on Z
t2 . n = t2 . n .= s . n ;
dom ( ( - F ) | Z ) c= dom F ;
W1 . x = W2 . x .= W2 . x ;
y in W .vertices() \/ W .vertices() ;
( k <= len ( vk ) ) & ( k + 1 ) <= len ( vk ) ;
x * a \equiv y * ( m mod p ) ;
proj2 .: S c= proj2 .: P & proj2 .: P c= proj2 .: P ;
h . p4 = g2 . I .= g2 . I ;
G6 = U /. 1 .= G * ( 1 , 1 ) `1 ;
f . ( r1 - r2 ) in rng f ;
i + 1 + 1-1 <= len - 1 ;
rng F = rng ( F | n ) & rng F c= rng F ;
mode upper_bound is well unital non empty multiplicative loop structure ;
[ x , y ] in A ~ ;
x1 . o in L2 . ( o . o ) ;
the carrier of m - m c= B ;
not [ y , x ] in id ( X ) ;
1 + p .. f <= i + len f ;
seq ^\ k1 is lower ;
len F-12 = len I & len Fc = len I ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be complex number , r be Real ;
Comput ( P , s , n ) = s ;
k <= k + 1 & k + 1 <= len p ;
reconsider c = {} as Element of T ;
let Y be connected Chain of T ;
cluster -> directed-sups-preserving for Function of L , L ;
f . j1 in K . j1 ;
cluster J => y -> total for Function ;
K c= 2 |^ ( the carrier of T )
F . b1 = F . b2 ;
x1 = x or x1 = y or x1 = z ;
attr a <> {} means : Def6 : for a st a in a holds ( a * x ) = 1 ;
assume that a c= b and b in a ;
s1 . n in rng s1 & s1 . n in rng s1 ;
{ o , b2 } on C2 & { o , b1 } on C2 ;
LIN o `1 , b , b9 ;
reconsider m = x as Element of Funcs ( V , V ) ;
let f be non constant FinSequence of D ;
let F2 be non empty TopSpace ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in F-8 ;
reconsider p9 = x as Subset of m m ;
A , B , C , D , E , F , J , M , N , N , M , N , N , N , M , N , N , N , M , N , N
cluster strict non empty for for us empty real number ;
rng c `1 misses rng ( e | m ) & rng e c= rng e2 ;
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 ) ;
the component of Q c= UBD A & P /\ Q c= UBD A ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( 1 / ( ( f ^ ) (#) ( g ^ ) ) ) ;
attr f = u means : Def6 : a * f = a * u ;
for n holds P1 [ prop n ] ;
{ x . O : x in L } <> {} ;
let x be Element of V . s ;
let a , b be Nat ;
assume that S = S2 and p = p2 and S = S2 ;
gcd ( n1 , n2 ) = 1 & gcd ( n1 , n2 ) = 1 ;
set o1 = a * _ 2 , o2 = b * _ 3 ;
seq . n < |. r1 .| & |. r1 - x0 .| < r ;
assume that seq is increasing and r < 0 and for n st n >= 0 holds r < 0 ;
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 is_collinear & a9 , b9 , c9 is_collinear ;
assume that Y = { 1 } and s = <* 1 *> ;
I1 . x = f . x .= 0 ;
W3 .first() = W2 . 1 .= W2 . 1 ;
cluster -> trivial for Vertex of G , finite _Graph ;
reconsider u = u as Element of Bags X ;
A in B ^ B implies A , B are_\kern1pt \kern1pt
x in { [ 2 * n + 3 , k ] } ;
1 >= ( sqrt ( ( q `1 ) ^2 ) / ( |. q .| ) ^2 ) ;
f1 is_in the carrier of ( f2 | A ) & f2 | A is ' ;
( f /. 2 ) `2 <= ( q `2 ) / ( |. q .| ) ;
h is_the carrier of Cage ( C , n ) ;
( b `2 / |. p .| - sn ) / ( 1 + sn ) <= ( p `2 / |. p .| - sn ) / ( 1 + sn ) ;
let f , g be Function of X , Y ;
S * ( k , k ) <> 0. K ;
x in dom ( max ( - f , - g ) ) ;
p2 in NN . p1 & p2 in NN . p2 ;
len ( the_left_argument_of H ) < len H & len ( H ) < len H ;
F [ A , F-14 . A ] ;
consider Z such that y in Z and Z in X ;
attr 1 in C means : Def6 : A c= C |^ A ;
assume that r1 <> 0 or r2 <> 0 ;
rng q1 c= rng C1 & rng q2 c= rng C1 ;
A1 , L , A2 is_collinear & A1 , A2 , p3 is_collinear ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in PartFunc ( p , Ss ) . ( S . x ) ;
then S is atomic means : then for n holds P-2 [ n ] ;
Cl ( Int [#] T ) = [#] T ;
f12 | ( A2 /\ A1 ) = f2 | ( A2 /\ A1 ) ;
0. M in the carrier of W & 0. M in W ;
v , v , w , y is_collinear ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ V & V \ Y c= Y \ Z ;
let X be Subset of S , T be Subset of S ;
consider H1 such that H = 'not' H1 and H1 in H ;
\bf 1 c= ( ( t * the function of A ) * ( \HM { 0 } ) ) ;
0 * a = 0. R .= a * 0. R .= a * 0. R ;
A |^ 2 = A ^^ A & A |^ 2 = A |^ 2 ;
set vY = vY /. n , vY = vY /. n ;
r = 0. ( \langle \cal E *> , \Vert \cdot \Vert \rangle ) ;
( f . p4 ) `1 >= 0 ;
len W = len ( W ^ ( m ) ) + len ( W ^ ( m ) ) ;
f /* ( s * G ) is divergent_to+infty ;
consider l being Nat such that m = F . l ;
t8 on W7 & not LIN b1 , b2 , b1 & not a2 on b2 , b1 ;
reconsider Y1 = 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 NAT -defined non empty set , f be Function of NAT , NAT ;
rng f = S1 -S1 ( S , X ) ;
let p be Element of B , x be the the ResultSort of S ;
max ( N1 , 2 ) >= N1 ;
0. X <= b |^ ( m * \mathbb m ) ;
assume that i in I and R1 . i = R . i ;
i = j1 & p1 = q1 & q1 = q2 ;
assume gR in the right & FR c= the carrier of g ;
let A1 , A2 be Subset of S , A be Subset of S ;
x in h " ( P /\ [#] ( T | P ) ) /\ [#] ( T | P ) ;
1 in Seg 2 & 1 in Seg 3 & 2 in Seg 3 ;
reconsider X-5 = X as non empty Subset of Tsuch that X = { {} } and X is non empty ;
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the Target of G ) -valued ;
n1 <= i2 + len g2 & n2 <= len g2 ;
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume v in the carrier' of G2 & v in the carrier' of G2 ;
y = Re y + ( Im y ) * i ;
( ( - 1 ) |^ p ) gcd ( p , q ) = 1 ;
x2 is_differentiable_on ]. a , b .[ & x2 in ]. a , b .[ ;
rng M5 c= rng ( D2 | D2 ) & rng ( D1 | D2 ) c= rng ( D2 | D2 ) ;
for p be Real st p in Z holds p >= a
assume that \bf X in proj1 .: ( f .: X ) and X is compact and Y is compact ;
( seq ^\ m ) . k <> 0 ;
s . ( G . ( k + 1 ) ) > x0 ;
( p |-count M ) . 2 = d ;
A \oplus ( B \ominus C ) = ( A \oplus B ) \ominus C
h \equiv gg . ( mod P ) .= g . ( mod P ) ;
reconsider i1 = i-1 - 1 as Element of NAT ;
let v1 , v2 be VECTOR of V , v be VECTOR of V ;
for V being Subspace of V holds V is Subspace of [#] V
reconsider i9 = i - 1 as Element of NAT ;
dom f c= [: C , D :] & rng f c= [: D , D :] ;
x in ( the Sorts of B ) . n ;
len - f2 in Seg len ( f2 | ( len f2 ) ) ;
p9 c= the topology of T & p9 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 ) * ( id o1 ) ;
assume that p , u are_elements and u , v // v , w ;
[ z , z ] in union rng ( F | X ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S ;
LIN a1 , a3 , a4 & LIN a1 , a3 , a4 ;
f " ( f .: x ) = { x } ;
dom w2 = dom ( r12 ) .= Seg len r12 ;
assume that 1 <= i and i <= n and j <= n ;
( ( g2 ) . O ) `2 <= 1 ;
p in LSeg ( E . i , F . i ) ;
I1 * ( i , j ) = 0. K ;
|. f . ( s . m ) - g .| < g1 ;
q9 . x in rng ( q | n ) ;
Carrier ( LF2 ) misses ( Carrier ( LF2 ) ) ` ;
consider c being element such that [ a , c ] in G ;
assume N|[ o1 , o2 ]| = o1 & o1 <> o2 ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= ( F-12 | C ) .: ( Cpion1 .: C ) ;
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [. f . j , f . j .] ;
attr 0 <= x & x ^2 <= x ;
p `2 <> 0. TOP-REAL 2 & p `2 <> 0. TOP-REAL 2 ;
cluster \cal a] ( S , T ) -> non empty ;
let x be Element of S ~ ;
( Obj F ) . ( a , b ) is one-to-one ;
|. i .| <= - ( - 2 |^ n ) / ( n + 1 ) ;
the carrier of I[01] = dom P & P . 0 = P . 1 ;
assume n * ( n + 1 ) ! > 0 * n ;
S c= ( A1 /\ A2 ) /\ ( A1 /\ A2 ) ;
a3 , a4 // a3 , a4 & a3 , a4 // a4 , a4 ;
then dom A <> {} & dom A <> {} ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , Y , G , G , j , G , k , G ;
set v2 = v4 /. ( i + 1 ) , v2 = v4 /. ( i + 1 ) ;
x = r . n .= ( rr . n ) ;
f . s in the carrier of S2 & f . s in the carrier of S2 ;
dom g = the carrier of I[01] & rng g = the carrier of I[01] ;
p in Lower_Arc ( P ) /\ Lower_Arc ( P ) ;
dom d = A2 & dom d = A2 & for x st x in A2 holds d . x = F ( x ) ;
0 < ( ||. p .|| ) / ( ||. z .|| + 1 ) ;
e . ( m + 1 ) <= e . ( m + 1 ) ;
B \ominus X \/ B \ominus Y c= B \ominus X
- - Im ( g | B ) < Integral ( M , Im g ) ;
cluster O := F -> \HM of X means : Let : F is an operation of X ;
let U1 , U2 be non-empty MSAlgebra over S , f be Function of U1 , U2 ;
Proj ( i , n ) * g is_differentiable_on X ;
x , y , z is_collinear & x , y , z is_collinear ;
reconsider p9 = p . x as Subset of V . x ;
x in the carrier of Lin ( A ) & y in the carrier of Lin ( A ) ;
let I , J be parahalting Program of SCM+FSA , a , b be Int-Location ;
assume that - a is lower and b is lower and a < b ;
Int ( A /\ B ) c= Cl Int ( A /\ B ) ;
assume for A being Subset of X holds Cl A = A ;
assume q in Ball ( |[ x , y ]| , r ) ;
( p2 `2 / p2 `1 ) ^2 <= ( p `2 / p2 `1 ) ^2 ;
Cl Q ` = [#] ( T | P ) ;
set S = the carrier of T , T = the carrier of S ;
set I1 = ' ( f , n ) , I2 = ' ( f , n ) , I2 = ' ( f , n ) , I2 = ' ( f , n ) , I2 = ' ( f , n ) , I2 = ' ( f , n
len p -' n = len p - n .= len p - n ;
A is Permutation of Line ( A , x ) ;
reconsider nnY = nY - 1 as Element of NAT ;
1 <= j + 1 & j + 1 <= len ( s | n ) ;
let q9 , q9 , q9 , g2 be Element of M ;
a1 in the carrier of S1 & a2 in the carrier of S2 ;
c1 /. n1 = c1 . n1 .= c2 . n1 ;
let f be FinSequence of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
y = ( f * ST ) . x .= f . x ;
consider x being element such that x in \mathop { \rm _ } A ;
assume r in ( ( dist ( o ) ) .: P ) .: P ;
set i2 = width ( h /. ( n + 1 ) ) ;
h2 . ( j + 1 ) in rng h2 ;
Line ( M29 , k ) . i = M . i ;
reconsider m = ( x - 2 ) / ( x - 2 ) as Element of ( REAL 2 ) ;
let U1 , U2 be Subspace of U0 , a be Element of U0 ;
set P = Line ( a , d ) ;
len p1 < len p2 + 1 & len p1 + 1 <= len p1 ;
let T1 , T2 be Scott Scott Scott m of L ;
then x <= y & { x } c= { y } ;
set M = n -tuples_on the carrier of G , F = n -tuples_on the carrier of G ;
reconsider i = x1 , j = x2 as Nat ;
rng ( ( the_arity_of a9 ) * ( the_arity_of a9 ) ) c= dom H ;
z1 " = z1 " * z " .= z1 " * z " " " " * z " " " " * z " " " " " " * z " " " * z " " * z " " * z " * z " " * z " * z
x0 - r in L /\ dom f & f . x0 < x0 + r ;
then w is \rm \hbox { - } that rng w /\ AllSymbolsOf S <> {} ;
set xZ = xZ ^ <* Z *> ^ <* Z *> ^ <* Z *> ^ <* Z *> ^ <* x *> ;
len w1 in Seg len w1 & len w2 = len w2 ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of PFuncs ( V , { k } ) ;
x . n = ( |. a . n .| ) / ( |. a .| ) ;
( p `1 ) ^2 / ( |. p .| ) ^2 <= ( G * ( 1 , 1 ) ) `1 / ( |. p .| ) ^2 ;
rng ( g | ( L~ g ) ) c= L~ g /\ L~ g ;
reconsider k = i-1 * ( lj + 1 ) as Nat ;
for n being Nat holds F . n is Seg ( n + 1 ) is Seg n
reconsider xM = xM as VECTOR of M ;
dom ( f | X ) = X /\ dom f /\ X .= X ;
p , a // p , c & b , a // c , d ;
reconsider x1 = x as Element of REAL m ( ) ;
assume i in dom ( a * p ^ q ) ;
m . ( |. g .| ) = p . ( |. g .| ) ;
a / ( s . m ) - ( a / ( s . n ) ) <= 1 ;
S . ( n + k ) c= S . ( n + k ) ;
assume B1 \/ C1 = B2 \/ C2 & B2 \/ C2 = B2 \/ C2 ;
X . i = { x1 , x2 } . i .= x2 ;
r2 in dom ( h1 + h2 ) /\ dom h2 ;
that that - 0. R = a and b0. R = b ;
FQ is_closed_on t1 , Q1 & PQ is_halting_on t1 , Q1 implies Q1 is_halting_on t1 , Q
set T = InInIncluster ( X , x0 ) ;
Int ( Cl R ) c= Int R & Int ( R ) c= Cl R ;
consider y being Element of L such that c . y = x ;
rng Fwhere x is Element of X ( ) st rng F: x in { F ( x ) } c= { F ( x ) } ;
G-23 ( { c } ) c= B \/ S ;
f-1 is Relation of [: X , Y :] , [: X , Y :] & X is Y ;
set RP = the Element of P , R = the Element of P , I = the carrier of P , J = the carrier of P , R = the carrier of P , T = the carrier of P , R = the carrier of P , T = the
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 ) | ( ( TOP-REAL n ) | ( j + 1 ) ) ;
g . x in dom f & x in dom g ;
assume that 1 <= n and n + 1 <= len f1 ;
reconsider T = b * N as Element of G / ( N , 1 ) ;
len ( ( P * ( 1 , k ) ) `2 ) <= len ( ( P * ( 1 , k ) ) `2 ;
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 ;
let f be PartFunc of REAL i , REAL n , x be Element of REAL n ;
rng f = the carrier of \bf SCM ( A ) & f .: A c= the carrier of \bf SCM ( A ) ;
assume s1 = sqrt ( 2 * p ) & s2 = sqrt ( 2 * p ) ;
attr a > 1 & b > 0 & a |^ b > 1 ;
let A , B , C be Subset of I1 , D be Subset of I2 ;
reconsider X0 = X , Y1 = Y as RealNormSpace ;
let f be PartFunc of REAL , REAL , x be Element of REAL m ;
r * ( v1 |-- I ) . X < r * 1 ;
assume that V is Subspace of X and X is Subspace of V and X is Subspace of V ;
t-3 , tt1 , t2 , t1 , t2 , t2 , t1 , t2 , t2 , t1 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t2 , t1 , t2 , t2 , t2
Q [ e \/ { v2 } , f ] ;
g \circlearrowleft ( W-min L~ z ) = z ;
|. |[ x , v ]| - |[ x , y ]| .| = v12 ;
- f . w = - ( L * w ) ;
z - y <= x iff z <= x + y
sqrt ( 7 + 1 ) / ( 2 * e ) > 0 ;
assume X is BCK-algebra & 0 < 0 & 0 < n ;
F . 1 = v1 & F . 2 = v2 ;
( f | X ) . x2 = f . x2 ;
( ( tan * tan ) `| Z ) . x in dom sec ;
i2 = ( f /. len f ) `2 .= ( f /. len f ) `2 ;
X1 = X2 \/ ( X1 \ X2 ) & X2 = X1 \/ X2 ;
[. a , b , 1_ G .] = 1_ G & [. a , b , c .] = 1_ G ;
let V , W be non empty VectSpStr over F_Complex , f be Function of V , W ;
dom g2 = the carrier of I[01] & rng g2 = the carrier of I[01] ;
dom f2 = the carrier of I[01] & rng f2 = the carrier of I[01] ;
( proj2 | X ) .: X = proj2 .: ( X /\ Y ) ;
f . ( x , y ) = h1 . ( x `1 , y `2 ) ;
x0 - r < a1 . n & a1 . n < x0 + r ;
|. ( f /* s ) . k - G . ( k + 1 ) .| < r ;
len Line ( A , i ) = width A & width A = width A ;
SY / S = ( S . g ) / S ;
reconsider f = v + u as Function of X , the carrier of Y ;
intloc 0 in dom Initialized p & Initialized p = Initialized s ;
i1 , i2 |= ( b , c ) & not i2 does not destroy b1 & i1 , i2 does not destroy b2 ;
( - 1 ) * ( 1 / ( r - 1 ) ) = ( sqrt ( r ) ) * ( 1 / ( r - 1 ) ) ;
for x st x in Z holds f2 is_differentiable_in x & f2 is_differentiable_in x & f2 is_differentiable_in x ;
reconsider q2 = ( q - x ) / 2 as Element of REAL ;
( 0 qua Nat ) + 1 <= i + j1 ;
assume f in the carrier of [ X , Omega ] ;
F . a = H / ( x , y ) . a ;
true T = TRUE & not ( C is Element of C implies p in T ) ;
dist ( ( a * seq ) . n , h ) < r ;
1 in the carrier of [. 0 , 1 .] & 1 <= len G ;
( p2 `1 ) ^2 - x1 > - g / ( 1 - g ) ;
|. r1 - r2 .| = |. a1 .| * |. a2 - x .| ;
reconsider S-14 = 8 as Element of Seg 8 ;
( A \/ B ) |^ b c= A |^ b \/ B |^ c
DDDW = DDW -A0 + 1 ;
i1 = ( of a + n ) * ( i , j ) & i2 = ( a + n ) * ( i , j ) ;
f . a [= f . ( f .: O1 ) ;
attr f = v & g = u + v ;
I . n = Integral ( M , F . n ) ;
[: { T } , T :] . s = 1 ;
a = VERUM ( A ) or a = VERUM ( A ) ;
reconsider k2 = s . b2 - 1 as Element of NAT ;
( Comput ( P , s , 4 ) ) . GBP = 0 ;
L~ M1 meets L~ R & L~ M2 meets L~ R implies L~ M1 misses L~ R
set h = the continuous Function of X , R , s be Real ;
set A = { L . ( ( k + 1 ) / 2 ) where k is Nat : 0 < k & k < n } ;
for H st H is atomic holds P [ H ] ;
set breal = S5 . ( i + 1 ) , S5 = S5 . ( i + 1 ) , x5 = S . ( i + 1 ) , x5 = S . ( i + 1 ) , x5 = S . ( i + 1 ) , x
Hom ( a , b ) c= Hom ( a `1 , b `2 ) ;
sqrt ( 1 + ( n + 1 ) ) < sqrt ( 1 + ( s " ) ) ;
( l /. 1 ) `1 = [ dom l , cod l ] `1 .= [ dom l , cod l ] ;
y +* ( i , y ) in dom g ;
let p be Element of CQC-WFF ( Al ( ) ) , A ( ) ;
X /\ X1 c= dom ( f1 - f2 ) /\ dom ( f2 - f3 ) ;
p2 in rng ( f /^ p1 ) & p1 in rng ( f /^ ( p1 + 1 ) ) ;
1 <= indx ( D2 , D1 , j1 ) + 1 - 1 ;
assume x in K1 /\ ( ( TOP-REAL 2 ) | K1 ) ;
- 1 <= ( ( f2 . O ) `2 / ( 1 + ( f2 . O ) `2 ) ^2 ) ;
let f , g be Function of I[01] , TOP-REAL 2 , a , b , c be Real ;
k1 -' k2 = k1 - k2 + 1 .= k1 -' k2 + 1 ;
rng seq c= ]. x0 - r , x0 + r .[ & rng seq c= ]. x0 , x0 + r .[ ;
g2 in ]. x0 - r , x0 + r .[ & g2 < x0 + r / 2 ;
sgn ( p `1 , K ) = - 1_ K .= 1_ K ;
consider u being Nat such that b = p |^ y * u ;
ex A being the the carrier of T st a = Sum A & A is limit_ordinal ;
Cl ( ( Cl H ) ` ) = union ( ( Cl H ) ` ) ;
len t = len t1 + len t2 .= len t1 + len t2 + len t2 ;
v-29 = v + w & v + A = v + A ;
v <> DataLoc ( t1 . GBP , 3 ) , X = P +* I , Y = P +* I , Y = P +* I , Y = Comput ( P , s , 3 ) , Y = P +* I , Z = Comput ( P , s , 3 ) ,
g . s = sup ( d " { s } ) ;
( \dot y ) . s = s . ( y . s ) ;
{ s : s < t } c= NAT & t = {} ;
s ` \ s = s ` \ 0. X .= ( s ` ) \ ( s ` ) ;
defpred P [ Nat ] means B + $1 in A & B + $1 in A ;
( 339 + 1 ) ! = 311139 * ( 339 + 1 ) ;
U /. succ A = ( T /. A ) `1 .= ( U /. A ) `1 ;
reconsider y = y as Element of COMPLEX ( len y , n ) ;
consider i2 being Integer such that y0 = p * i2 and i2 in dom f and x = f . i2 ;
reconsider p = Y | Seg k as FinSequence of NAT , f be FinSequence of NAT ;
set f = ( S , U ) -TruthEval 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 be Real ;
( \cal M ) . [ n + i , 'not' A ] <> 1 ;
ex r being Real st x = r & a <= r & r <= b ;
R1 , R2 be Element of ( n -tuples_on REAL ) * , R2 be Element of ( n -tuples_on REAL ) * ;
reconsider l = (0). V as Linear_Combination of A * , B ;
set r = |. e .| + |. w .| + |. s .| ;
consider y being Element of S such that z <= y and y in X ;
a 'or' ( b 'or' c ) = 'not' ( ( a 'or' b ) 'or' c ) ;
||. xg - g2 .|| < r2 & ||. x1 - g2 .|| < s ;
b9 , a9 // b9 , c9 & b9 , c9 // c9 , c9 ;
1 <= k2 -' k1 & k2 + 1 + 1 = k2 -' k1 + 1 & k2 + 1 = k2 -' k1 + 1 ;
sqrt ( ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) >= 0 ;
sqrt ( ( |. q .| - sn ) / ( 1 + sn ) ) < 0 ;
E-max C in right_cell ( R , 1 ) & E-max L~ R in rng R ;
consider e being Element of NAT such that a = 2 * e + 1 ;
Re ( lim F ) = Re ( lim G ) .= Im ( lim G ) ;
LIN b , a , c or LIN b , a , c ;
p `1 , a `2 // a `1 , b `2 ;
g . n = a * Sum ( f | n ) .= f . n ;
consider f being Subset of X such that e = f and f is \rm } ;
F | ( N2 , S ) = CircleMap * ( F * ( N , S ) ) ;
q in LSeg ( q , v ) \/ LSeg ( v , p ) ;
Ball ( m , r1 ) c= Ball ( m , s ) ;
the carrier of (0). V = { 0. V } & W is Subspace of V ;
rng ( ( ( - 1 ) (#) cos ) `| [. - 1 , 1 .] ) = [. - 1 , 1 .] ;
assume that Re ( seq ) is summable and Im ( seq ) is summable and Im ( seq ) is summable ;
||. ( ( vseq . n ) - ( vseq . m ) ) * ( vseq . n ) .|| < e / 2 ;
set g = O --> 1 ;
reconsider t2 = t11 as 0 -started string of S2 ;
reconsider xLet = seq . n as sequence of ( TOP-REAL n ) | ( the carrier of TOP-REAL n ) ;
assume that E-max L~ Cage ( C , n ) meets L~ go and L~ pion1 /\ L~ pion1 meets L~ pion1 ;
- ( 1 / ( 1 - x ) ) < F . n - x ;
set d1 = being element of being Function of REAL , REAL n , REAL n , z1 , z2 be Real ;
2 |^ ( 2 -' 1 ) = 2 |^ ( 2 - 1 ) ;
dom ( v | ( len ( v | k ) ) ) = Seg len ( ( v | k ) ^ <* d *> ) ;
set x1 = ( - ( k2 + 1 ) ) / ( k + 1 ) , x2 = ( - ( k2 + 1 ) ) / ( k + 1 ) ;
assume for n being Element of X holds 0. ( X , n ) <= F . n ;
assume that 0 <= T|^ i and T|^ ( i + 1 ) <= 1 / 2 ;
for A being Subset of X holds c . ( c . A ) = c . A
the carrier of ( LZ + L2 ) c= I & L2 c= I ;
'not' All ( x , p ) => All ( x , p ) is valid ;
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the normal normal w.r.t. over D ;
Z c= dom ( ( ( ( - 1 ) (#) ( f1 * f2 ) ) `| Z ) ;
|. 0. TOP-REAL 2 - q .| < r / ( 1 - sn ) ;
ConsecutiveSet2 ( A , succ B ) c= ConsecutiveSet2 ( A , succ ( d ) ) ;
E = dom ( L | E ) & LL is_measurable_on E & LL | E is is_measurable_on E ;
C |^ ( A + B ) = C |^ B * C |^ A ;
the carrier of W2 c= the carrier of V & the carrier of W1 c= the carrier of W2 ;
I . IC Comput ( P , s , m ) = P . IC Comput ( P , s , m ) ;
attr x > 0 means : Def6 : x = 1 / ( - x ) ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) ;
consider p being Point of T such that C = [. p , R .] and p in { p } ;
b , c are_connected & - C , - C are_not empty & - C , - C are_not Set
assume f = id the carrier of C & f is Function of the carrier of C , the carrier of C ;
consider v such that v <> 0. V and f . v = L * v ;
let l be Linear_Combination of {} ( ( the carrier of V ) \ { v } ) ;
reconsider g = f " as Function of U2 , U1 , U2 ;
A1 : x in the carrier of ( G . k ) & A2 : x in the carrier of ( G . k ) ;
|. - x .| = - ( - x ) .= - x .= - x ;
set S = \mathop { \rm ) . ( x , y , c ) ;
Fib ( n ) * ( 5 * Fib ( n ) ) >= 4 * log ( 2 , n ) ;
v3 /. ( k + 1 ) = v3 . ( k + 1 ) ;
0 mod i = - ( i * ( 0 qua Nat ) ) / i ;
Indices M1 = [: Seg n , Seg n :] & len M2 = n ;
Line ( SFinSequence , j ) = SFinSequence . j .= 0. K ;
h . ( x1 , y1 ) = [ y1 , y2 ] & h . ( y1 , y2 ) = [ y1 , y2 ] ;
|. f .| - ( Re ( |. f .| ) * h ) is nonnegative ;
assume x = ( a1 ^ <* x1 *> ) ^ <* b1 *> ^ <* b2 *> ;
MW is_halting_on IExec ( I , P , s ) , P & PI is_halting_on s , P ;
DataLoc ( t3 . a , 4 ) = intpos 0 + 4 ;
x + y < - x + y & |. x - y .| = - x + y ;
LIN c , q , b & LIN c , q , a ;
f^ . ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + z ) .= x1 + ( y1 + z ) ;
f_ { let a } . a = f[ a , f ] & v in InputVertices S & v in InputVertices S & u in InputVertices S ;
( p `1 ) ^2 / ( ( E-max C ) / ( p `1 ) ^2 <= ( ( E-max C ) / ( p `1 ) ) ^2 / ( p `1 ) ^2 ;
set R8 = Cage ( C , n ) \circlearrowleft E , E8 = Cage ( C , n ) /. E , E8 = Cage ( C , n ) /. i , E8 = Cage ( C , n ) /. i , E8 = Cage ( C , n ) /. ( i + 1 ) , E
( p `1 ) ^2 >= ( E-max C ) ^2 / ( ( E-max C ) / ( p `2 ) ^2 ) ;
consider p such that p = p9 and s1 < p 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 ;
f1 /* ( s1 /* s1 ) is convergent & f2 /* ( s1 ^\ k ) is convergent ;
f . x1 = x1 & f . y1 = y1 & f . y2 = y2 ;
len f <= len f + 1 & len f + 1 <> 0 ;
dom ( Proj ( i , n ) * s ) = REAL m .= Seg m ;
n = k * ( 2 * t ) + ( n mod ( 2 * k ) ) ;
dom B = 2 \ { {} } .= the carrier of V ;
consider r such that r _|_ a and r _|_ x and r _|_ y ;
reconsider B1 = the carrier of Y1 , B2 = the carrier of Y2 as Subset of X ;
1 in the carrier of [. 1 / 2 , 1 .] & 1 / 2 in [. 1 / 2 , 1 .] ;
for L being complete LATTICE holds <* <* <* a *> *> , L *> is isomorphic iff L is isomorphic
[ gi , gj ] in I \ ( I \ { i } ) ;
set S2 = 1GateCircStr ( <* x , y *> , '&' ) ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and for r st x0 in dom f1 /\ dom f2 holds f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2
reconsider y = ( a " ) / ( Fq " ) as Element of L ;
dom s = { 1 , 2 , 3 } & s . 1 = d1 & s . 2 = d2 ;
( min ( g , ( 1 - f ) ) . c ) <= h . c ;
set G2 = the subgraph of G , s3 = the Vertex of G , s3 = the Vertex of G , s3 = the Vertex of G , s3 = the Vertex of G , s3 = the Vertex of G , s3 = the Vertex of G , s3 = the Vertex of G , s3 = the Vertex of G , s3
reconsider g = f as PartFunc of REAL , REAL-NS n , REAL-NS n ;
|. s1 . m .| / |. p .| < d / ( p . m ) / ( |. p .| / ( |. p .| ) ) ;
for x being element st x in for u being element st x in B ( ) holds x in B ( ) iff x in B ( )
P = the carrier of ( ( TOP-REAL n ) | K1 ) | K1 .= K1 ;
assume that p1 in LSeg ( p1 , p2 ) /\ LSeg ( p01 , p2 ) and p2 in LSeg ( p1 , p2 ) /\ LSeg ( p01 , p2 ) ;
( 0. X \ x ) |^ ( m + 1 ) = 0. X ;
let g be Element of Hom ( cod f , cod f ) ;
2 * a * b + ( 2 * c ) <= 2 * C1 * C2 + ( 2 * C2 ) ;
let f , g , h be PartFunc of X , Y ;
set h = Hom ( a , g ) ;
then idseq ( n ) | Seg m = idseq ( m ) & m <= n ;
H * ( g " * a ) in the right of H & H * ( g " * a ) in the carrier of H ;
x in dom ( ( ( - 1 ) (#) ( ( #Z 2 ) * ( ( #Z 2 ) * ( #Z 2 ) ) ) ) `| REAL ) ;
cell ( G , i1 , j2 -' 1 ) misses C & LSeg ( G * ( i1 , j2 ) , 1 ) misses C ;
LE q2 , q2 , P , p1 , p2 & LE q2 , q1 , P , p1 , p2 & LE q2 , q2 , P , p1 , p2 ;
attr B is an component means : Def6 : B c= BDD A & B c= BDD A ;
deffunc D ( set , set ) = union rng $2 & $2 = union rng $2 ;
n + - n < len p + ( - n ) + ( - n ) ;
attr a <> 0. K means : Let : for M st M <> 0. K holds width M = width M & len M = width M ;
consider j such that j in dom \mathbb J and I = len |^ j + j ;
consider x1 such that z in x1 and x1 in ( P * f ) . x ;
for n ex r being Element of REAL st X [ n , r ]
set C1 = Comput ( P2 , s2 , i + 1 ) , C1 = Comput ( P2 , s2 , i + 1 ) , C1 = Comput ( P2 , s2 , i + 1 ) , C1 = Comput ( P2 , s2 , i + 1 ) , C1 = Comput ( P2 , s2
set \cal v = 3 / ( { a , b } \/ { c } ) , w = 3 / ( { a , b } \/ { c } ) , e = 3 / ( { a , b } \/ { c } ) , f = 3 / ( { a , b } \/ { c } )
conv @ ( @ W ) c= union ( F .: ( E " ) ) ;
1 in [. - 1 , 1 .] /\ dom ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * (
r3 <= s1 + ( ( r - s2 ) / 2 ) * ( ( r - s1 ) / 2 ) ;
dom ( f * ( f0 * f1 ) ) = dom f /\ dom ( f2 * f1 ) .= dom f /\ dom f1 ;
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 ) | ( ( TOP-REAL n ) | K1 ) | K1 ;
( T * h . s ) . x = T . ( h . s ) ;
I . ( J . x ) = ( I * L ) . J ;
y in dom an an an implies ( Frege ( A . ( ( Frege A ) . o ) ) . y = ( Frege ( A . ( ( Frege A ) . o ) ) ) . y ;
for I being non degenerated commutative commutative Ring holds I is commutative
set s2 = s +* Initialize ( ( intloc 0 ) .--> 1 ) , P2 = P +* Initialize ( ( intloc 0 ) .--> 1 ) , s2 = P +* Initialize ( ( intloc 0 ) .--> 1 ) , P2 = P +* Initialize ( ( intloc 0 ) .--> 1 ) , s2 = P +* S2 , s2 = P +* S2 ,
P1 /. IC s1 = P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 ;
lim S1 in the carrier of [' a , b '] & lim S1 in the carrier of [' a , b '] ;
v . ( lpp . i ) = ( v *' ) . i .= ( v *' ) . i ;
consider n being element such that n in NAT and x = seq . n ;
consider x being Element of c such that F1 . x <> F2 . x and F2 . x <> F2 . x ;
card ( X , 0 , x1 , x2 , x3 , x4 , x1 , x2 , x3 , x4 , x1 , x2 , x3 , x4 , x1 , x2 , x3 , x4 , x1 , x2 , x3 , x4 , x1 , x2 , x3 , x4 , x1 , x2 , x3 , x4 , x1 , x2 , x3 ,
j + ( 2 * ( k + 1 ) ) + m1 > j + ( 2 * ( k + 1 ) ) ;
{ s , t } on A3 & { s , t } on B2 & { s , t } on B2 ;
n1 > len crossover ( p2 , p1 , n1 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n4 , n4 , n4 , n4 , n3 , n3 , n4 , n3 , n4 , n4 , n4 , n4 , n3 , n4 , n4
( ( for m being Element of NAT st ( HT ( f2 , T ) ) + T ) = 0. L holds ( ( g2 ) + T ) . m = 0. L ;
then H1 , H2 are_: for H st H = H1 & H is normal holds H is be be be Subgroup of G ;
( E-max L~ f ) .. f > 1 & ( E-max L~ f ) .. f > 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 , T be PartFunc of S , T ;
DigA ( t5 , z ) is Element of k -tuples_on k -tuples_on k ;
I /. 122z = d1 & I /. IC I = k2 & I /. IC I = k2 ;
uu = { [ a , u ] } & u = { [ a , v ] } ;
( w | p ) | ( w | w ) = p ;
consider u2 such that u2 in W2 and x = v + u2 and u = v + u2 ;
for y st y in rng F ex n st y = a |^ n & y in a |^ n
dom ( ( g * ( :] --> C ) ) | K ) = K ;
ex x being element st x in ( ( [#] U0 ) \/ A ) . s & x in ( the Sorts of U0 ) . s ;
ex x being element st x in ( ( ( ( ( ( ( ( ( L ) . O ) ) . s ) ) . s ) . s ) . s ) . x ;
f . x in the carrier of [. - r , 1 .] & f . x in [. - r , 1 .] ;
( the carrier of X1 ) /\ ( the carrier of X2 ) <> {} & ( the carrier of X1 ) /\ ( the carrier of X2 ) <> {} ;
L1 /\ LSeg ( p01 , p2 ) c= { p01 } /\ LSeg ( p01 , p2 ) ;
sqrt ( b + ( b `1 ) ^2 ) in { r : a < r & r < b & b < b } ;
ex_sup_of { x , y } , L & x "\/" y = sup { x , y } ;
for x being element st x in X ex u being element st P [ x , u ]
consider z being Point of G such that z = y and P [ z ] and P [ z ] ;
( the sequence of \overline ( M ) ) . n <= e ;
len ( w ^ w2 ) + 1 = len w + 2 + 1 .= len w + 1 ;
assume q in the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 & q in the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 ;
f | E-4 = g | E-4 .= ( g | E-4 ) | E-4 .= ( g | E-4 ) | E-4 ;
reconsider i1 = x1 , i2 = x2 , j2 = x3 as Element of NAT ;
( a * A ) ` = ( a * ( A * B ) ) ` ;
assume ex x0 being Element of NAT st f .: ( n + 1 ) is <= x0 & x0 < x0 ;
Seg len ( ( ( ( f ^ f2 ) | i ) ^ ( ( f ^ f2 ) | i ) ) = dom ( ( f ^ f2 ) | i ) ;
( Complement A1 ) . m c= ( Complement A1 ) . n & ( Complement A1 ) . m c= ( Complement A1 ) . n ;
f1 . p = p9 & g1 . ( p . p ) = d & g1 . ( p . p ) = d ;
FinS ( F , Y ) = FinS ( F , dom ( F | Y ) ) ;
( x | y ) | z = z | ( y | x ) ;
sqrt ( |. x .| ^2 ) <= sqrt ( ( r2 ^2 ) / ( 2 * n ) ) ;
Sum ( F-12 ) = Sum f & dom ( F-12 ) = dom g ;
assume for x , y being set st x in Y & y in Y holds x /\ y in Y ;
assume that W1 is Subspace of W2 and W2 is Subspace of W3 and W1 is Subspace of W2 and W2 is Subspace of W3 ;
||. t-15 . x - lim ( x - y ) .|| = lim ||. ( x - y ) .|| .= ||. x - y .|| ;
assume that i in dom D and f | A is lower and g | A is bounded and g | A is bounded ;
sqrt ( ( p `2 ) ^2 + ( p `2 ) ^2 ) <= sqrt ( 1 + ( p `2 ) ^2 ) ;
g | Sphere ( p , r ) = id ( Ball ( p , r ) ) ;
set N8 = E-max L~ Cage ( C , n ) , N8 = W-min L~ Cage ( C , n ) ;
for T being non empty TopSpace holds T is countable iff T is countable
width B |-> 0. K = Line ( B , i ) .= B * ( i , j ) .= B * ( i , j ) ;
attr a <> 0 means : Def6 : ( A ^^ B ) Y. = ( A Y. ) Y. ;
then f is_differentiable 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 b > 0 and c > 0 and d > 0 and d > 0 ;
w1 , w2 in Lin { w1 , w2 } & w2 , w1 , w2 , w1 , w2 , w1 , w2 , w1 , w2 } is Element of the carrier of V ;
p2 /. IC Comput ( p2 , s , k ) = p2 . IC Comput ( p1 , s , k ) ;
ind ( T-10 | b ) = ind B - ind B .= ind B - ind B .= ind B - 1 ;
[ a , A ] in the Points of Line ( D , k ) & [ a , A ] in the P of Line ( D , k ) ;
m in ( the Arrows of C ) . ( o1 , o2 ) & ( the Arrows of C ) . ( o1 , o2 ) = ( the Arrows of C ) . ( o1 , o2 ) ;
( ( a , CompF ( PA , G ) ) . z ) . z = FALSE ;
reconsider phi = phi /. 11 , phi = phi /. 2 as Element of from from from from ^ 11 , ^ <* y *> ;
len s1 - ( len s2 - 1 ) > 0 + 1 - 1 ;
\delta ( D * ( f . sup A ) - f . ( lower_bound A ) ) < r ;
[ f21 , f22 ] in the carrier' of A & [ f22 , f22 ] in the carrier' of A ;
the carrier of ( ( TOP-REAL 2 ) | K1 ) = K1 & ( ( TOP-REAL 2 ) | K1 ) | K1 = K1 ;
consider z being element such that z in dom g2 and p = g2 . z and g2 . z = g2 . z ;
[#] V1 = { 0. V } .= the carrier of V1 .= { 0. V } ;
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 that x1 in dom ( f | X ) and ||. x1 - x0 .|| < s and ||. x1 - x0 .|| < s ;
h1 = f ^ ( <* p3 *> ^ <* p *> ) .= h ^ ( <* p3 *> ^ <* p *> ) .= h ;
c /. [ b , c ] = c /. ( a , c ) .= c /. ( a , c ) ;
reconsider t1 = p1 , t2 = p2 , t2 = p2 , t1 = p2 as Term of C , V , s be State of C , a , b be Element of C , s be Element of C , t be Element of C , a , b be Element of C , s be Element of C , r be Element of C , r be
sqrt ( 1 - 2 ) in the carrier of [. 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 `1 ) ) + D * ( p1 `1 + D * ( p1 `1 ) ) ;
R . b real 2 = 2 * - b .= 2 * b ;
consider e1 such that B = 1- `1 * C + ( 0 * A ) and 0 <= e1 and e1 <= 1 ;
dom g = dom ( ( the Sorts of A ) * ( the Sorts of A ) * ( the Arity of S ) ) ;
[ P . ( l2 , P . ( l2 , y ) ) , P . ( l2 , y ) ] in => ( T . ( l2 , y ) ) ;
set s2 = Initialize s , P2 = P +* I , P2 = P +* I , s2 = P +* I , P2 = P +* I , P3 = P +* I , s4 = P +* I , s2 = Comput ( P3 , s3 , 1 ) , P4 = P3 ;
reconsider M = mid ( z , i2 , i1 ) as non empty FinSequence of TOP-REAL 2 ;
y in product ( ( the Sorts of J ) +* ( { 1 } --> { 1 } ) ) ;
1 / ( 0 , 1 ) = 1 / ( 0 , 1 ) & 0 / ( 0 , 1 ) = 0 ;
assume x in the left .[ or x in the right of g or x in the right of g ;
consider M be strict Subspace of A such that a = M and T is elements of M and for A being Subset of M holds T . A = A ;
for x st x in Z holds ( ( ( ( exp_R * f ) `| Z ) `| Z ) . x <> 0
len W1 + len W2 + m = 1 + len W2 + len W1 + len W2 + m ;
reconsider h1 = ( ( vseq . n ) - tt ) * t-16 as Lipschitzian LinearOperator of X , Y ;
( ( len p + len q ) + 1 ) in dom ( p + q ) ;
assume that s2 is negative and F in the |= of s2 and F is the |= of s2 and F is the |= of s2 ;
( ( ( ( ( ( x , y ) ) / 3 ) ) / 3 ) / 3 ) / 3 = ( ( x , y ) / 3 ) / 3 ) / 3 ;
for u being element st u in Bags n holds ( p `2 + m ) . u = p . u
for B being Subset of u-5 st B in E holds A = B or A = B or A misses B ;
ex a being Point of X st a in A & A /\ Cl { y } = { a } ;
set W2 = from [: p , q :] \/ [: q , p :] ;
x in { X where X is Ideal of L : X is Ideal of L & X is Ideal of L } ;
the carrier of W1 /\ W2 c= the carrier of W2 & the carrier of W1 /\ W2 c= the carrier of W1 /\ W2 ;
( ( 1 / a ) * id ( a + b ) ) * id ( a + b ) = ( 1 / a ) * 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 => ( p => q ) ) in TAUT ( A ) ;
set cos = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
set cos = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
- 1 + 1 <= ( ( - 1 ) |^ ( n -' m ) ) + 1 ;
( reproj ( 1 , z0 ) ) . x in dom ( f1 * f2 ) /\ dom f2 ;
assume that b1 . r = { c1 } and b2 . r = c2 . r 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 & a1 on P & a2 on P & a1 on P & a2 on P & a1 on P & a2 on P & a1 on P & a2 on P
reconsider gf = g `2 * f `2 , hg = h `2 * g `2 as strict Element of X ;
consider v1 being Element of T such that Q = ( downarrow v1 ) ` and v1 in ( { v1 } ) ` and v2 in ( { v1 } ) ` ;
n in { i where i is Nat : i < n1 + 1 & n1 < n + 1 } ;
( F /. i , j ) `2 >= ( F /. m ) `2 ;
assume K1 = { p : ( p `1 >= cn & p `2 >= cn ) & p <> 0. TOP-REAL 2 } ;
ConsecutiveSet ( A , succ O1 ) = ( ConsecutiveSet ( A , O1 ) ) .: ( A , O1 ) ;
set I1 = Macro ( a , intloc 0 ) , I2 = P +* while>0 ( a , intloc 0 ) , I2 = P +* while>0 ( a , intloc 0 ) , I2 = P +* while>0 ( a , intloc 0 ) , I2 = P +* while>0 ( a , intloc 0 ) , I2 = P +* while>0 ( a , intloc 0 ) , I1 = P +* while>0 ( a , intloc 0 ) , I1 = P +* while>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 L2 c= the carrier of L2 ;
consider xp being Element of GF ( p ) such that xp |^ 2 = a and xp = x ;
reconsider e1 = e1 , fe2 = f . ( x , y ) , ffff:] as Element of D ( ) ;
ex O being set st O in S & C1 c= O & M . O = 0. ( Cl O ) ;
consider n be Nat such that for m being Nat st n <= m holds S . m in U1 ;
f * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) * reproj ( i , x ) ) . x ;
defpred P [ Nat ] means A + succ $1 = succ $1 & A = succ $1 & B = succ $1 ;
the left One of - ( - g ) = the left st - ( - g ) = the left ' of g ;
reconsider p9 = x , p9 = y , q9 = z as Point of ( TOP-REAL 2 ) | K1 ;
consider g2 such that g2 = y and x <= g2 and g2 <= x0 and x0 < g2 & g2 < x0 and g2 < x0 and g2 < x0 and g2 < x0 and g2 in dom f ;
for n being Element of NAT holds X [ n , r ] implies X [ n , r ]
len ( x2 ^ 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 the set of K & x in the set of K & y = the set of K . n ;
LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = {} ;
func " ( X ) -> set means : Def6 : for x being set holds it . x = ( the carrier of X ) \ ( id X ) ;
len ( ( Cage ( C , n ) /. len ( Cage ( C , n ) ) ) ) <= len ( ( Cage ( C , n ) /. len ( Cage ( C , n ) ) ) ;
attr K is a L means : Def6 : a <> 0. K & v . ( a |^ i ) = i * v . ( a |^ i ) ;
consider o being OperSymbol of S such that t . {} = [ o , the carrier of S ] and o <> the carrier of S ;
for x st x in X ex y st x c= y & y in X & x is a holds y is a holds f . x is a fixpoint of f
IC Comput ( P1 , s1 , k ) in dom ( Comput ( P1 , s1 , k ) ) ;
attr q < s & r < s & s < q & p < q & q < s implies ]. p , q .[ c= ]. p , s .[ ;
consider c being Element of Class ( f . c , x ) such that Y = ( F . c ) `1 and c in X ;
func the ResultSort of S2 -> ResultSort means : Def6 : for x being Element of S2 holds it . x = id the carrier of S2 & for y being Element of S2 holds it . y = id the carrier of S2 . y ;
set y9 = [ <* y , z *> , f2 ] , z9 = [ <* z , x *> , f2 ] , y9 = [ <* y , z *> , f3 ] , z9 = [ <* z , x *> , f3 ] , z9 = [ <* z , x *> , f3 ] , zx = [ <* z , x *> , f3 ] ;
assume x in dom ( ( ( ( ( #Z 2 ) * ( ( #Z 2 ) * ( #Z 2 ) ) ) `| Z ) ;
r-7 in Int cell ( GoB f , i , width GoB f ) \ { LSeg ( GoB f , i + 1 , width GoB f ) where f is FinSequence of TOP-REAL 2 : 1 <= i & i + 1 <= len GoB f } ;
( q `2 / |. q .| - sn ) >= ( ( Cage ( C , n ) ) /. ( i + 1 ) ) `2 ;
set Y = { a "/\" a ` : a in X } ;
i -' len f <= len f + ( len f -' 1 ) - len f + 1 - len f + 1 ;
for n ex x st x in N & x in N1 & h . n = x0 + r / ( n + 1 )
set s0 = ( \mathop { \it true } ( a , I ) ) . i , p = ( \mathop { \it true } ( a , I ) ) . i , q = ( \mathop { \it true } ( a , I ) ) . i , s = ( \mathop { \it true } ( a , I ) ) . i , s = ( \mathop { \it true } ( a , I )
p . k = 1 or p . k = - 1 or p . k = - 1 ;
u + Sum ( L-18 ) in ( U \ { u } ) \/ { u + Sum ( L-18 ) } ;
consider xm being set such that x in xm and xm in V1 and xm c= V and xm c= V and xm c= V ;
( p ^ q ) . m = ( q | k ) . ( Seg len p ) .= p . ( len p ) ;
g + h = gg + h + 1 & for x holds -SVF1 ( g , X , h ) = g + h + x
L1 is distributive & L2 is distributive implies L1 * L2 is distributive & L2 * L2 is distributive
attr x in rng f & y in rng ( f | x ) & f | x = f | y ;
assume that 1 < p and p + 1 / ( 2 * q ) = 1 and 0 <= p and p <= 1 / ( 2 * q ) ;
FM * ( fM *' ) = rpoly ( 1 , the carrier of L ) *' .= 0. L ;
for X being set , A being Subset of X holds A ` = {} implies A = {} & A = {}
( ( ( E-max X ) `2 ) / ( 1 + ( ( E-max X ) `2 ) / ( 1 + ( ( E-max X ) `2 ) / ( 1 + ( S-bound X ) `2 ) ^2 ) ) <= ( ( E-max X ) / ( 1 + ( E-max X ) / ( 1 + ( S-bound X ) / ( 1 + ( S-bound X ) / ( 1 + ( q `2 ) / ( 1 + ( q
for c being Element of the Sorts of A , a being Element of the Sorts of A holds c <> a implies c <> a
s1 . GBP = ( Exec ( i2 , s2 ) ) . intpos ( card I + 3 ) .= 0 ;
for a , b being Real holds [ a , b ] in ( y >= 0 iff b >= 0 & a >= 0 ) implies b = 0
for x , y being Element of X holds x ` \ y = ( x \ y ) `
mode BCK-algebra of i , j , m , n , m , n , m , k being Nat holds m * n , j , k * m * n , n * m * n , m * n + k * n , m * n + k * n , m * n + k * n , m * n + k * n is commutative
set x2 = ( Re ( y ) ) * ( Im ( y ) ) ;
[ y , x ] in dom u & [ u , x ] in g . y ;
]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .] c= A & [. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .] c= A ;
0 <= delta ( S2 . n , S2 . n ) & |. delta ( S2 . n , S2 . n ) - 0 .| < e / 2 ;
( - ( - ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 <= ( - ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 ;
set A = ( 2 / 2 ) * ( - 1 ) ;
for x , y being set st x in R" holds x , y are_\hbox { x }
deffunc F2 ( Nat ) = b . ( $1 * M . ( $1 + 1 ) ) * ( M . ( $1 + 1 ) ) ;
for s being element holds s in ( { f } \/ { g } ) iff s in ( { f } \/ { g } )
for S being non void holds S is connected holds S is connected iff S is connected
max ( ( |. z .| ) / ( |. z .| ) ) >= 0 ;
consider n1 be Nat such that for k holds seq . ( n1 + k ) < r + s / 2 ;
Lin ( A /\ B ) is Subspace of Lin ( A /\ B ) & Lin ( B /\ A ) is Subspace of Lin ( B ) ;
set n-15 = nnv '&' ( M . x qua Element of BOOLEAN ) , nv = ( n -tuples_on BOOLEAN ) . ( n qua Element of BOOLEAN ) ;
f " V in ( ( f " ) .: X ) & f " V in D & f " ( ( f " ) .: X ) in D ;
rng ( ( a \HM { c } ) +* ( 1 , b ) ) c= { a , c } ;
consider y being Wsubgraph of G1 such that y `1 = y and dom y `1 = WG1 and dom y `1 = WG2 and y `2 = WG2 ;
dom ( 1 / f ) /\ ]. - 1 , 1 .[ c= ]. - 1 , 1 .[ & dom ( 1 / f ) /\ ]. - 1 , 1 .[ c= ]. - 1 , 1 .[ ;
in in in , j , n , - ( i , j ) , - ( j , n ) , - ( i , j ) , - ( j , n ) , - ( i , j ) , - ( j , n ) ) ;
v ^ ( ( n |-> 0 ) ^ ( n |-> 0 ) ) in Lin ( ( ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B | ( B
ex a , k1 , k2 st i = a := k1 & k2 = b := k2 & i = c := k2 ;
t . NAT = ( NAT .--> ( i1 , m ) ) . NAT .= ( NAT .--> ( i1 , m ) ) . NAT .= ( NAT .--> ( i1 , m ) ) . NAT .= ( NAT .--> ( i1 , m ) ) . NAT ;
assume that F is b\times and rng p = Seg ( n + 1 ) and dom p = Seg ( n + 1 ) and rng p = Seg ( n + 1 ) and rng p = Seg ( n + 1 ) ;
not LIN b `1 , b9 , c9 & not LIN a , b , c & not LIN a , c , c9
( L1 there L2 ) | O c= ( L1 | O ) | O & ( L2 | O ) | O c= ( L1 | O ) | O ;
consider F be ManySortedSet of E such that for d being Element of E holds F . d = F ( d ) ;
consider a , b such that a * ( v + w ) = b * ( -w ) and 0 < a and a < b and b < 0 ;
defpred P [ FinSequence of D ] means |. Sum ( $1 ) .| <= Sum ( |. $1 .| ) * |. $1 .| ;
u = cos . ( x , y ) * x + cos . ( x , y ) * y .= v . ( x , y ) * y ;
dist ( ( seq . n ) + x , g + x ) <= dist ( ( seq . n ) , g ) + 0 ;
P [ p , |. p .| : p <> {} & ( id the carrier of A ) . ( id the carrier of A ) = id the carrier of A
consider X being Subset of CQC-WFF ( Al ) such that X c= Y and X is finite and X is finite and X is non empty ;
|. 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 <= ( h . l1 ) `1 & ( g . l1 ) `2 <= ( h . l ) `2 } ;
( ( Partial_Sums ( G . n ) ) . m ) / ( ( G . n ) to_power k ) <= ( Partial_Sums ( G . n ) ) . m / ( ( G . n ) to_power k ) ;
f . y = x .= x * 1_ L .= ( 1_ L ) * ( 1_ L ) .= x * ( 1_ L ) .= x * ( 1_ L ) ;
NIC ( ( \bf if i1 , i2 ) , ( 0 , k ) ) = { i1 , i2 , k } & { i1 , i2 , k , k } in { i1 , i2 , k , k } ;
LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = { p1 } /\ LSeg ( p1 , p2 ) ;
Product ( ( the Sorts of I1 ) +* ( i , { 1 } ) ) in Z1 ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) | ( the carrier of S2 ) .= Following ( s1 , n ) | ( the carrier of S2 ) ;
W-min ( Q ) <= ( q1 `1 ) / ( |. q1 .| ) & ( q1 `1 ) / ( |. q1 .| ) <= ( q1 `1 ) / ( |. q1 .| ) ;
f /. i2 <> f /. ( ( i1 + len g -' 1 ) + 1 ) ;
M , v / ( x. 3 , m ) / ( x. 4 , m ) / ( x. 4 , m ) |= H ;
len ( ( ( P ^ ) ^ ( P ^ ) ) ^ ) in dom ( ( P ^ ) ^ ( P ^ ) ) ;
A |^ ( m , n ) c= A |^ ( m , n ) & A |^ ( k , n ) 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 y = p1 . n1 ;
consider X be set such that X in Q and for Z being set st Z in Q & Z <> {} holds X in Z ;
CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA & CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA ;
for v be VECTOR of l1 holds ||. v .|| = upper_bound rng |. ( ( ( ||. v .|| ) . v ) .| + |. ( ( v .| ) . v ) .|
for phi , phi st phi in X holds phi . phi in X & phi . phi in X & phi . phi in Y ;
rng ( Sgm dom ( f | ( dom f ) ) ) c= dom ( f | ( dom f /\ dom ( f | ( dom f ) ) ) ) ;
ex c being FinSequence of D ( ) st len c = k & P [ c ] & P [ c ] ;
( the_arity_of a , b ) <> <* ( <* b *> , <* c *> *> , <* d *> *> , <* e *> *> , <* f *> *> , <* g *> , <* h *> *> , <* g *> *> , <* h *> *> , <* h *> , <* g *> , <* h *> *> , <* h *> , <* h *> , <* h *> *> , <* h *> , <* h *> *> , <* h *> *> , <* h *>
consider f1 be Function of the carrier of X , REAL such that f1 = |. f .| and f1 is continuous one-to-one and for x be Element of X , r be Real st x in X & r > 0 holds f1 . x <> r ;
a1 = b1 & a2 = b2 & a3 = b3 & a4 = b3 & a4 = b3 & a4 = b3 & a5 = b3 & a4 = 6 & a5 = 6 & 8 = 7 & 8 = 8 & 8 = 7 & 8 = 8 & 8 = 8 & 7 = 8 & 8 = 7 & 8 = 8 & 7 = 8 & 8 = 8 & 8 = 8 & 8 = 7 & 8 = 7 & 8 = 8 & 7 = 7
D2 . indx ( D2 , D1 , n1 ) = D1 . ( indx ( D2 , D1 , n1 ) + 1 ) .= D1 . ( indx ( D2 , D1 , n1 ) + 1 ) ;
f . ( ||. r .|| ) = ||. ||. r .|| * ||. r .|| .= ||. r .|| * ||. r .|| .= ||. r .|| * ||. r .|| .= ||. r .|| * ||. r .|| ;
consider n be Nat such that for m being Nat st n <= m holds Cseq . m = Cseq . m ;
consider d be Real such that for a , b be Real st a in X & b in Y holds a <= b & b <= d & a <= b ;
||. L /. h - ( K * |. h .| ) + ( K * |. h .| ) .|| <= x0 + ( K * |. h .| ) ;
attr F is commutative means : Def6 : for b being Element of X holds F -|^ { b } = f . b ;
p = 2 * ( p1 + 0. TOP-REAL 2 ) .= 1 * ( p1 + 0. TOP-REAL 2 ) .= 1 * ( p1 + 0. TOP-REAL 2 ) .= p1 `1 ;
consider z1 such that b , x3 , z1 is_collinear and o , x1 , x2 is_collinear and o <> x1 and o <> x2 and o <> x2 and o <> x1 and o <> x2 and o <> x1 and o <> x2 and o <> x1 and o <> x2 ;
consider i such that Arg ( ( Rotate ( s , q ) ) . q ) = s + Arg ( q , p ) and 2 * PI < ( 2 * PI ) * ( i , j ) ;
consider g such that g is one-to-one and dom g = card f . x and rng g = { f . x } and rng g = { f . x } and rng g = { g . x } ;
assume that A = P2 \/ Q2 and P2 <> {} and P /\ P2 = {} and P /\ P2 = {} and P /\ Q = {} and P /\ Q = {} and P /\ Q = {} and P /\ Q = {} ;
attr F is associative means : Def6 : F .: ( F .: ( f , g ) , h ) = F .: ( f , g ) ;
ex x being Element of NAT st m = x `1 & x `1 < z `1 & z `2 < i & m < len z `2 ;
consider k2 be Nat such that k2 in dom ( Pk . ( k1 + k2 ) ) and l in dom ( Pk . ( k2 + k2 ) ) ;
seq = r * seq implies for n holds seq . n = r * seq . n & seq . n = r * seq . ( n + 1 )
F1 . [ ( id a ) , ( id a ) . [ a , a ] ] = [ f * ( id a ) , f * ( id 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 ) | ( dom F ) ) and ( dom F ) = y and ( dom F ) . z = y ;
for x , y being element st x in dom f & y in dom f & f . x = y holds x = y
Int cell ( G , i , j ) = { |[ r , s ]| : r <= G * ( 0 + 1 , 1 ) `1 & s <= G * ( 0 , 1 ) `2 } ;
consider e being element such that e in dom ( T | E1 ) and ( T | E1 ) . e = v ;
( F `2 * b1 ) . x = ( ( Mx2Tran J ) . x ) * ( ( Mx2Tran J ) . j ) .= ( ( Mx2Tran J ) . j ) * ( ( Mx2Tran J ) . j ) ;
- 1 _ { \mathbb R } = ( ( m (#) D ) | n ) | D .= ( ( m (#) D ) | n ) | ( ( m (#) D ) | n ) .= ( ( Det M ) | n ) | ( ( m (#) D ) | n ) .= ( Det M ) | ( ( m * D ) | n ) ;
attr for x being set st x in dom f /\ dom g holds g . x <= f . x + g . x ;
len ( f1 . j ) = len ( f2 /. j ) .= len ( f2 /. j ) .= len ( f2 /. j ) ;
All ( 'not' All ( a , A , G ) , B , G ) |= All ( 'not' All ( 'not' a , B , G ) , A , G ) ;
LSeg ( E . k1 , F . k2 ) c= Cl RightComp Cage ( C , k + 1 ) & LSeg ( Cage ( C , k + 1 ) , F /. ( k + 1 ) ) c= Cl RightComp Cage ( C , k + 1 ) ;
x \ ( a |^ m ) = x \ ( ( a |^ k ) * a ) .= ( x \ ( a |^ k ) ) \ a ;
k -inininin0 ( I ) = ( commute I ) . k .= ( ( commute I ) . k ) .= ( ( commute I ) . k ) .= ( ( commute I ) . k ) . ( ( commute I ) . k ) .= ( ( commute I ) . k ) . ( ( commute I ) . k ) ;
for s being State of A holds Following ( s , n ) . ( 0 + 1 ) is stable implies Following ( s , n ) . ( 2 * n + 1 ) is stable
for x st x in Z holds f1 . x = a / ( x - a ) & ( f1 . x ) <> 0 & ( f1 . x ) <> 0 implies f1 . x = a / ( x - a ) )
support ( ( support ( m ) ) \/ support ( ( support ( m ) ) ) c= support ( ( support ( m ) ) ) \/ support ( ( support ( m ) ) ) ) ;
reconsider t = u as Function of ( the carrier of A ) , the carrier of B ;
- ( a * sqrt ( 1 + b ^2 ) ) <= - ( b * sqrt ( 1 + b ^2 ) ) / sqrt ( 1 + b ^2 ) ;
phi /. ( succ b1 ) = g . a & phi /. ( b + a ) = f . ( g . ( a , b ) ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( F ^ <* p *> ) and i <> j ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x6 , x6 , x6 , x5 , x6 , x6 , x6 , x5 , x6 , x6 , x5 , x6 , x6 , x6 , x6 , x5 , x6 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x6 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , a9 , x5 , x6 , x5 , x5 , x5 ,
the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U2 & the Sorts of U2 /\ ( U1 "\/" U2 ) c= the Sorts of U1 /\ ( U2 "\/" U2 ) ;
( - ( 2 * a ) + b ) / ( 2 * a ) > 0 ;
consider W00 such that for z being element holds z in W00 iff z in N ~ & P [ z ] ;
assume that ( the Arity of S ) . o = <* a *> and ( the Arity of S ) . o = r and ( the Arity of S ) . o = r ;
Z = dom ( ( exp_R ^ ) (#) ( ( exp_R ^ ) (#) ( ( exp_R ^ ) (#) ( ( exp_R ^ ) ) ) ) `| Z ) ;
lim sum ( f , S1 ) is convergent & lim lim lim lim sum ( f , S1 ) = 0 ;
( X . a9 ) => ( ( f . a9 ) => ( g . b9 ) ) in [: the carrier of L , the carrier of L :] ;
len ( M2 * M1 ) = n & width ( M2 * M1 ) = n & width ( M2 * M1 ) = n ;
attr X1 \/ X2 is open means : Def6 : X1 , X2 are_separated & X1 , X2 are_separated & X2 , X1 are_separated ;
for L being lower-bounded antisymmetric RelStr for X being Subset of L holds X "\/" { Top L } = { Top L } implies X "\/" { Top L } = { Top L }
reconsider f-1be Function of [: M . ( 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 q , w ^ <* s *> , q ^ <* s *> , q ^ <* s *> , q ^ <* s *> , q ^ w , q ^ <* s *> , q ^ <* s *> , w ^ <* s *> , q ^ <* s *> , w ^ <* s *> , q ^ <* s *> , w ^ <* s *> , w ^ <* s *> , q ^ <* s *>
g . ( a |^ 0 ) = g . ( 1_ G ) .= ( 1_ G ) * g . ( a * a ) .= ( 1_ G ) * g . ( a * a ) .= ( 1_ G ) * g . a ;
assume for i being 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 C1 ;
reconsider o9 = o `2 , carrier' = o `2 as Element of TS ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . o ) ) ;
1 * x1 + ( 0 * x2 ) + ( 0 * x3 ) = x1 + ( 0 * x2 ) .= x1 + ( 0 * x2 ) .= x1 + ( 0 * x2 ) ;
EP " . 1 = ( ( E qua Function ) " ) . 1 .= ( E qua Function ) . 1 .= E " . 1 .= E " . 1 .= E " . 1 ;
reconsider u1 = the carrier of U1 /\ ( U1 "\/" U2 ) as non empty Subset of U0 ;
( ( x "/\" z ) "\/" ( x "/\" y ) ) "\/" ( z "/\" y ) <= ( x "/\" z ) "\/" ( x "/\" y ) ;
|. f . ( s1 . ( m1 + 1 ) ) - f . ( s1 . ( m1 + 1 ) ) .| < r / ( 1 - M ) ;
LSeg ( ( Cage ( C , n ) /. i , ( Cage ( C , n ) /. ( i + 1 ) ) ) , ( L~ Cage ( C , n ) ) /. ( i + 1 ) ) is vertical ;
( f | Z ) /. x - ( f | Z ) /. x0 = L /. ( x- - x0 ) + R /. ( a1 - x0 ) ;
g . c * ( g . c ) + f . c <= h . c * ( ( - 1 ) * f . c ) + f . c ;
( f + g ) | divset ( D , i ) = f | divset ( D , i ) + g | divset ( D , i ) ;
assume that width ( f ) in the carrier of ( A ) and len ( f ) = width ( A * B ) and width ( f * B ) = width ( A * B ) and width ( f * B ) = width ( A * B ) ;
len ( - M1 ) = len M1 & width ( - M2 ) = width M1 & width ( - M2 ) = width M1 ;
for n , i being Nat st i + 1 < n holds [ i , i ] in the InternalRel of [: n , m :]
pdiff1 ( f1 , 2 ) is_partial_differentiable_in z , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z , 1 & pdiff1 ( f2 , 2 ) . 1 = z * pdiff1 ( f2 , 1 ) . 1 ;
attr a <> 0 & b <> 0 & Arg a = Arg b & Arg b = Arg - Arg a ;
for c being set st not c in [. a , b .[ holds not c in Intersection ( the open m of a , b )
assume that V1 is linearly closed and V2 is linearly closed and V1 is closed and V1 /\ V2 = { v + u : v in V1 & u in V1 & v in V1 & u in V1 & v in V1 & u in V1 & v in V1 & u + v in V1 & v + u in V1 & v + u in V1 and v + u in V1 ;
z * x1 + ( 1 - z ) * x2 in M & z * ( y + z ) + ( 1 - z ) * x2 in N ;
rng ( ( ( ( ( ( ( ( ( ( P qua Function ) qua Function ) ) " ) * S ) * S ) * ( ( ( ( P * R ) * S ) * S ) * f ) * ( ( ( P * R ) * S ) * f ) ) ) = Seg card ( ( ( ( P * R ) * S ) * f ) ) ;
consider s2 being Integer such that s2 is convergent and b = lim s2 and for n holds s2 . n = lim s2 and s2 . n <> 0 ;
h2 " . n = h2 . n " * ( ( - 1 / 2 ) * ( ( - 1 / 2 ) * ( ( - 1 / 2 ) * ( ( - 1 / 2 ) * ( ( - 1 / 2 ) * ( ( - 1 / 2 ) * ( ( - 1 / 2 ) * ( ( - 1 / 2 ) * ( ( - 1 / 2 ) * ( ( - 1 / 2 ) * ( ( - 1 ) / 2 ) ) ) ) ) ) ;
( Partial_Sums ||. ( r (#) ( seq ^\ k ) ) . m ) . m = ||. ( r (#) ( seq ^\ k ) ) . m .|| .= 0 ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= Comput ( P2 , s2 , 1 ) . b .= Comput ( P2 , s2 , 1 ) . b ;
- v = ( - 1_ G ) * v & - w = ( - 1_ G ) * v & - w = - w * v ;
sup ( ( k .: D ) .: D ) = sup ( ( k .: D ) .: D ) .= sup ( ( k .: D ) .: D ) .= sup ( ( k .: D ) .: D ) .= sup ( ( k .: D ) .: D ) ;
A |^ ( k , l ) = ( A |^ n ) |^ ( k , l ) .= ( A |^ n ) |^ ( 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 = ( I + J ) \ ( I + J )
( 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 holds ( for n being Nat holds Fib ( a * b ) = ( Fib ( a ) ) . n + ( Fib ( b ) ) . n
consider A5 being countable set such that r is countable and A5 is non empty and ( A is closed & B is closed & A /\ B is closed ) & ( for n being Nat holds A c= A iff A is closed ) ;
for X being non empty additive Subset of X , M being Subset of X st M in M holds x + M in M + M
{ [ x1 , x2 ] , [ y1 , y2 ] } c= { x1 , x2 } & { y1 , y2 } in { x1 } ;
h . O = |[ A * ( f . O ) + B * ( f . I ) , C * ( f . O ) + D * ( f . I ) + D * ( f . I ) + D * ( f . I ) + D * ( f . I ) + D * ( f . I ) ]| ;
( Gauge ( C , n ) * ( k , i ) ) `2 in L~ Cage ( C , n ) /\ L~ Cage ( C , n ) ;
cluster m , n ) mod ( 2 , n ) -> prime implies for Nat holds p divides m & p divides n implies 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 \ c <= c holds a "/\" b <= c
consider b being element such that b in dom ( H / ( ( x , y ) . b ) ) and z = H . ( ( x , y ) . b ) ;
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 . 2 , G & e Joins W . 3 , G . 3 , G ;
( exists h st x0 = ( h * f ) . ( 2 * n ) ) . x = ( h * f ) . ( 2 * n ) ;
j + 1 = j + ( len h2 -' 1 ) .= i + 1 -' len h11 + 1 .= i + 1 -' len h11 + 1 ;
( *' S ) . f = S *' . ( f ^ ) .= S . ( ( S ^ ) . f ) .= S . ( ( S ^ ) . f ) .= S . ( ( S ^ ) . f ) .= S . ( ( S ^ ) . f ) ;
consider H such that H is one-to-one and rng H = the carrier of L2 and Sum ( L2 * H ) = Sum ( L2 * H ) and Sum ( L2 * H ) = Sum ( L2 * H ) ;
attr R is in R means : Def6 : p <> q & p <> q & q <> p & p <> q & p <> q & q <> r & r < p ;
dom ( Product ( X --> f ) ) = meet ( dom ( X --> f ) ) .= meet ( dom ( X --> f ) ) .= dom ( X --> f ) .= dom ( X --> f ) .= dom ( X --> f ) .= dom ( X --> f ) .= dom ( X --> f ) ;
sup ( ( proj2 .: ( Lower_Arc C ) /\ Vertical_Line w ) ) <= sup ( ( proj2 .: ( Lower_Arc C ) /\ Vertical_Line w ) ) & sup ( ( proj2 .: ( w ) /\ Vertical_Line w ) /\ Vertical_Line w ) <= sup ( ( proj2 .: ( 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 - g /. x0 .| < r
i * fN - fN = i * ( f . ( i + 1 ) ) .= i * ( f . ( i + 1 ) ) - f . ( i + 1 ) ;
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 g = [ g1 , g2 ] and [ g1 , g2 ] in union C and [ g2 , g2 ] in [: the carrier of Y , the carrier of X :] ;
func d gcd n -> Nat means : Def6 : d divides n & ( d |^ n ) divides ( d |^ n ) & ( d |^ n ) divides ( d |^ n ) ;
f\rbrack . [ 0 , t ] = f . [ 0 , t ] .= ( - P * ( x , t ) ) . [ 0 , t ] .= a * ( - P * ( x , t ) ) .= a ;
t = h . D or t = h . C or t = h . D or t = h . E or t = h . F ;
consider m1 be Nat such that for n st n >= m1 holds dist ( ( ( seq . n ) , ( seq . n ) ) , ( lim seq ) ) < 1 / ( ( lim seq ) to_power k ) ;
sqrt ( ( q `1 / |. q .| - sn ) / ( 1 + sn ) ^2 ) <= sqrt ( ( q `1 / |. q .| - sn ) ^2 ) ;
h1 . ( i + 1 ) = h2 . ( i + 1 + 1 ) .= h2 . ( i + 1 + 1 ) .= h2 . ( i + 1 ) ;
consider o being Element of the carrier' of S , x2 being Element of { the carrier of S } such that a = [ o , x2 ] and a = [ o , x2 ] ;
for L being RelStr , a , b being Element of L holds a <= { b } iff a >= b & b >= a & a >= b & b <= a
||. 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 ) - ( exp_R * exp_R ) . x ;
attr r = F .: ( p , q ) means : Def6 : len r = len p & for i st i in dom p holds r . i = F . ( p . i , q . i ) ;
sqrt ( ( r / 2 ) ^2 + ( r / 2 ) ^2 ) <= sqrt ( ( r / 2 ) ^2 + ( 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 ( ( Line ( M , i ) ) * ( i , j ) )
then a " <> 0. R & a " * ( a * v ) = 1 / ( a * v ) & a " * ( a * v ) = 1 / ( a * v ) ;
p . ( j -' 1 ) * ( q *' r ) . ( i + 1 ) = Sum ( p . ( j -' 1 ) * r ) ;
deffunc F ( Nat ) = L . 1 + ( R /* ( h ^\ n ) ) . ( h . n ) " ;
assume that the carrier of H2 = f .: ( the carrier of H1 ) and the carrier of H2 = f .: ( the carrier of H2 ) and the carrier of H1 = f .: ( the carrier of H2 ) and the carrier of H2 = f .: ( the carrier of H1 ) and the carrier of H2 = the carrier of H2 ;
Args ( o , Free ( X , Free ( X ) ) ) = ( ( the Sorts of Free ( S , X ) ) * the Arity of S ) . o ;
H1 = n + 1 / ( |. 2 .| + h ) .= n + 1 / ( |. N .| + h ) .= n + 1 / ( |. N .| + h ) ;
( O /. 1 ) `1 = 0 & ( O /. 1 ) `2 = 1 & ( O /. 1 ) `2 = 1 & ( O /. 1 ) `2 = 1 ;
F1 .: ( dom F1 /\ dom F2 ) = F1 .: ( dom F1 /\ dom F2 ) .= { f /. ( n + 1 ) } .= { f /. ( n + 1 ) } ;
attr b <> 0 & d <> 0 & b <> d & a = ( - e ) / 2 implies ( a = ( - e ) / 2 ) / 2 & ( b = ( - e ) / 2 ) / 2 ;
dom ( ( f +* g ) | D ) = dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom f \/ dom g /\ D .= dom f /\ D .= dom ( f +* g ) /\ D .= dom f /\ D ;
for i being set st i in dom g ex u being Element of L st g /. i = u * a & ex u being Element of B st g /. i = u * a * v
g `2 * P `2 = g `2 * ( g `2 ) * ( g `2 ) .= g `2 * ( g `2 ) * ( g `2 ) ;
consider i , s1 such that f . i = s1 and not ( ( s1 . i ) in s1 & ( not s1 . i in s1 & s1 . i in s1 & not s1 . i in s1 ) & not s1 . i in s1 . i ) ;
h5 | ]. a , b .[ = ( g | ]. a , b .[ ) | ( ]. a , b .[ \/ ( g | ]. a , b .[ ) ) | ( ]. a , b .[ \/ ( g | ]. a , b .[ ) ) .= g | ]. a , b .[ ;
[ s1 , t1 ] in R & [ s2 , t2 ] in R & [ s1 , t1 ] in R implies [ s1 , t2 ] in R & [ s2 , t2 ] in R
then H is negative & H is non negative & H is non negative & H is non negative implies H is non negative & H is non negative & H is non negative ;
attr f1 is total means : Def6 : 1 / f2 is total & for c st c in dom f1 holds ( ( f1 (#) f2 ) . c ) (#) ( f2 * f1 ) . c = ( f1 . c ) * ( f2 . c ) ;
z1 in W2 " ( W2 " ( { z } ) ) & ( z in W2 " ( { z } ) ) & ( z in W2 " ( { z } ) ) & ( z in W2 " ( { z } ) implies z in W2 " ( { z } ) ) ;
p = 1 * p " * a * p .= a " * p " * ( b " * q ) .= a " * p " * ( b " * q " ) .= a " * p " * p " * q " * q " ;
for rr be Real for K be Real st for n be Nat st n <= K holds rr . n <= K . ( n + 1 ) holds upper_bound rng ( ( r * K ) . n ) <= K . ( n + 1 )
\hbox { E-max C } meets L~ go \/ L~ pion1 or not ( L~ Cage ( C , n ) meets L~ pion1 ) & ( not ( not W-min C in L~ pion1 ) & ( not W-min C in L~ pion1 ) & ( not W-min C in L~ pion1 ) & ( not W-min C in L~ pion1 ) & not ( W-min C in L~ pion1 ) & not ( W-min C in L~ pion1 ) & not W-min C in L~ pion1 ) ;
||. f . ( g . ( k + 1 ) ) - g . ( g . ( k + 1 ) ) .|| <= ||. g . ( g . ( k + 1 ) ) - g . ( k + 1 ) .|| * ( K * ( K to_power k ) ) ;
assume h = ( ( B .--> C ) +* ( D .--> E ) +* ( F .--> J ) +* ( M .--> N ) +* ( M .--> N ) +* ( M .--> N ) +* ( M .--> N ) +* ( M .--> N ) +* ( M .--> N ) +* ( M .--> N ) ) +* ( M .--> N ) assume h = ( B .--> N ) +* ( M .--> N ) +* ( M .--> N ) assume assume h = h ;
|. ( ( Let ( H . n ) `| ( A . n ) ) . k ) - ( ( H . n ) `| ( A . m ) ) . k .| <= e * ( b * ( b * ( a + 0 ) ) ) ) . k ;
( ( a holds ( the Sorts of U1 ) . e ) . v = [ ( the Sorts of U1 ) . ( ( the Sorts of U2 ) . v ) , ( the Sorts of U1 ) . ( ( the Sorts of U2 ) . v ) ] ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x6 , x6 , x6 , x5 , x6 , x6 , x6 , x5 , x6 , x6 , x5 , x6 , x5 , x6 , x6 , x6 , x5 , x6 , x5 , x5 , x5 , x5 , x5 , x5 , a9 , b9 , c9 , a9 , c9 , a9 , b9 , c9 , a9 , b9 , c9 , a9 , c9 , a9 , b9 , c9 , a9 , c9 , a9 , b9 , c9 , c9 , a9 , b9 , c9 , c9 , a9 , b9 , c9 , c9 , a9
assume that A = [. 0 , 2 * PI .] and [' a , b '] = 0 and [' a , b '] = |. a .| * |. b .| and |. a .| = |. b .| * |. b .| ;
p `2 is Permutation of dom f1 & p `2 = ( Sgm Y ) . i " * p " * ( Sgm Y ) . ( Sgm Y ) " = ( Sgm Y ) . ( Sgm Y ) " * p " * ( Sgm Y ) . ( Sgm Y ) " ;
for x , y st x in A holds |. ( 1 / 2 ) * ( f . x ) - ( 1 / 2 ) * ( f . y ) .| <= 1 * |. f . x - ( 1 / 2 ) * ( f . y ) .|
( p2 `2 / |. p2 .| - sn ) / ( 1 + sn ) = |. q2 .| * sqrt ( 1 + sn ^2 ) / sqrt ( 1 + sn ^2 ) ;
for f be PartFunc of the carrier of C , REAL st dom f is compact & rng f c= dom f & f | X is bounded holds rng f c= dom f & rng f c= dom ( f | X )
assume for x being Element of Y st x in EqClass ( z , CompF ( B , G ) ) holds ( Ex ( a , CompF ( B , G ) ) ) . x = TRUE ;
consider F3 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 , v , u1 , v1 , u1 , v1 , v1 , u1 , v1 , v1 , u1 , v1 , u1 , v1 , v1 , v2 , v1 , u1 , v1 , v2 , v1 , u1 , v1 , v2 , v1 , u1 , v1 , u1 , v1 , u1 , v1 , v1 , u1 , v1 , u1 , v1 , u1 , v1 , u1 , v1 , u1 , v1 , u1 , v1 , v1 , u1 , v1 , v2 , v1 , u1 , v1 , u1 , v1 , u1 , v1 , u1 , u1 , v1 , u1
for G being Group , A , B being strict Subgroup of G holds ( N ` A ) * ( N ` A ) = N ` A * ( N ` A )
for s be Real st s in dom F holds F . s = ( ( R to_power 0 ) (#) ( ( f + g ) (#) ( e to_power 0 ) ) ) . x
width ( AutMt ( f1 , b1 , b2 ) ) = len ( ( f2 * ( f1 , b2 ) ) * ( ( f2 * ( f1 , b2 ) ) * ( ( f2 * ( f1 , b2 ) ) * ( ( f2 * ( f2 * ( f1 , b2 ) ) ) ) ) .= len ( ( f2 * ( f1 , b1 ) ) * ( ( f2 * ( f2 * ( f1 * ( f2 * ( f2 * ( f1 , b2 ) ) ) ) ) ) ) ) ;
f | ]. - PI / 2 , PI / 2 .[ = f & f " ( ]. - PI / 2 , PI / 2 .[ ) = f " ( ]. - PI / 2 , PI / 2 .[ ) & f " ( ]. - PI / 2 , PI / 2 .[ ) = f " ( ]. - PI / 2 , PI / 2 .[ )
assume that X is closed and a in X and a in X and y in X and x in X and y in X and x in X and y in X and x in X and y in X ;
Z = dom ( ( ( ( ( exp_R * arctan ) `| Z ) / ( exp_R * arccot ) ) `| Z ) /\ dom ( ( exp_R * arccot ) `| Z ) ;
func VERUM ( V ) -> Subset of V means : Def6 : 1 <= it & it . k = l . k & for k st 1 <= k & k <= len l holds it . k = V . k ;
for L being non empty TopSpace , N being net of L , M being net of L st c is net of N for c being Element of N st c is cluster cluster /. 1 ) holds c is cluster ;
for s being Element of NAT holds ( ( ( ( 0. X ) . v ) + ( 0. X ) . v ) + ( ( 0. X ) . v ) ) . s = ( ( ( 0. X ) . v ) + ( 0. X ) . v ) ;
then z /. 1 = ( E-max L~ z ) .. z & ( E-max L~ z ) .. z < ( E-max L~ z ) .. z ;
len ( p ^ <* 0 *> ) = len p + len <* 0 *> .= len p + 1 + 1 .= len p + 1 + 1 .= len p + 1 ;
assume that Z c= dom ( ( ( - ln * f ) `| Z ) and for x st x in Z holds f . x = a - x / ( a - x ) ) and f . x = a - x / ( a - x ) ) and for x st x in Z holds f . x = a - x / ( a - x ) ;
for R being add-associative right_zeroed right_complementable distributive non empty doubleLoopStr , I being non empty Subset of R , J being Subset of R holds ( I + J ) *' ( I /\ J ) c= I /\ J
consider f being Function of B1 , B2 such that for x being Element of B1 holds f . x = F ( x ) and f . x = F ( x ) ;
dom ( x2 + y2 ) = Seg len x .= dom ( x (#) z ) .= Seg len ( x (#) z ) .= Seg len ( x (#) z ) .= Seg len x ;
for S being -1 functor of C , B being Morphism of B for c being Object of C holds ( S . id c ) . ( id c ) = id ( ( Obj S ) . c ) * ( S . c )
ex a st a = a2 & a in dom f /\ ( dom ( f | X ) ) & for x st x in dom f /\ ( dom ( f | X ) ) holds f . x = F ( f5 , x ) ) ;
a in Free ( H2 , ( x. 4 ) ) '&' ( ( x. 4 ) '&' ( ( x. 4 ) '&' ( ( x. 4 ) '&' ( ( x. 4 ) '&' ( ( x. 4 ) '&' ( ( x. 4 ) '&' ( ( x. 4 ) '&' ( ( x. 4 ) '&' ( ( x. 4 ) '&' ( ( x. 4 ) '&' ( ( x. 4 ) '&' ( ( x. 4 ) '&' ( ( x. 4 ) '&' ( ( x. 4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ;
for C1 , C2 being set , f being stable Function of C1 , C2 st f is stable holds f = g iff f = g
( W-min L~ go ) `1 = ( W-min L~ pion1 ) `1 .= ( W-bound L~ pion1 ) `1 .= ( W-bound L~ pion1 ) `1 .= ( W-bound L~ pion1 ) `1 ;
assume that u = <* x0 , y0 , z0 *> and f is PartFunc of 3 , u & u = SVF1 ( 3 , f , u ) . 3 and SVF1 ( 3 , f , u ) . 3 = SVF1 ( 3 , f , u ) . 3 ;
then ( t . {} in Vars & ex x being Element of Vars st x = ( t . {} ) `1 & ( t . {} = ( t . {} ) `1 & ( t . {} ) `2 = ( t . {} ) `2 ;
Valid ( p '&' J , 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 & y = f . x holds a >= f . y & b >= f . x & a >= b ;
func Class R -> Subset of R means : Def6 : for A being Subset of R holds A in it iff ex a being Element of R st a = Class ( R , a ) ;
defpred P [ Nat ] means ( ( ( ( ( ( ( the Target of G ) . $1 ) ) `1 ) `1 ) `1 ) `1 c= G * ( ( the Element of G ) . $1 ) `1 ;
assume that dim W1 = 0 and dim W2 = 0 and dim W2 = 0 and dim W1 = 0 and dim W1 = 0 and dim W2 = 1 and dim W1 = 0 and dim W1 = 1 and dim W1 = 0 and dim W1 = 1 and dim W1 = 0 ;
mam ( t ) = ( m . t ) `1 .= ( m . t ) `1 .= ( m . t ) `1 .= ( m . t ) `1 .= ( m . t ) `1 ;
d11 = x11 ^ ( ( y , d ) --> ( x , y ) ) .= f . ( ( y , d ) --> ( x , y ) ) .= ( f ^ <* ( y , z ) *> ^ ( f ^ <* x , z *> ) ) .= ( f ^ <* y *> ) . ( ( y , z ) --> d ) .= ( f ^ <* y , z *> ) . ( y , z ) ;
consider g such that x = g and dom g = dom f and for x being element st x in dom f holds g . x = f . x ) & for x being element st x in dom f holds g . x = f . x ;
x + 0. F_Complex = x + ( len x ) .= x + ( len x ) .= x + ( len x ) .= x + ( len x ) .= x + ( len x ) .= x + ( len x ) ;
( ( f /^ ( ( len f -' 1 ) + 1 ) ) + 1 ) in dom ( f /^ ( ( len f -' 1 ) + 1 ) ) /\ dom ( f /^ ( ( len f -' 1 ) + 1 ) ) ;
assume that P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 /\ P2 = { p1 } and P1 /\ P2 = { p1 , p2 } and P1 /\ P2 = { p1 , p2 } and P1 /\ P2 = { p2 , p3 } and P1 /\ P2 = { p2 , p3 } and P1 /\ P2 = { p1 , p3 } and P1 /\ P2 = { p2 , p3 } and P1 /\ P2 = { p1 , p3 } and P1 /\ P2 = { p2 , p3 } = { p1 , p3 } and p2 /\ LSeg ( p1 , p2 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p3 , p4 , p4 , p4 , p4 , p2 , p4 , p4 , p4 , p4 , p4
reconsider a1 = a , b1 = b , c1 = c , c2 = d , c2 = c , c2 = d , c1 = d , c2 = c , c2 = d , c2 = c , c1 = d , c2 = d , c2 = c , c2 = d , c2 = d , c1 = d , c2 = c , c2 = d , c2 = d , c2 = d , c2 = d , c2 = c , c1 = d , c2 = d , c2 = d , c2 = c , c2 = d , c2 = d , c2 = d , 7 = d , 7 = d , 7 = d , 8 = d , 7 = d , 8 = c , 7 = d , 8 = d , 7 = c = d , 8 = d ,
reconsider GFFFFFFFf = G1 . t * F1 . a 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 ( M , P . m ) | dom ( P . n ) <= Integral ( M , P . n ) | dom ( P . n ) ;
assume that dom f1 = dom f2 and for x , y being element st x in dom f2 holds f1 . ( x , y ) = f2 . ( x , y ) and f2 . ( x , y ) = f2 . ( x , y ) ;
consider v such that v = y and dist ( u , v ) < min ( ( r - ( G * ( i , 1 ) ) `1 ) / 2 , ( G * ( i , 1 ) `2 ) / 2 ) ;
for G being Group , H being Subgroup of G , a being Element of G , b being Element of G st a = b holds a |^ b = b |^ a * b
consider B be 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 = { p2 where p2 is Point of TOP-REAL 2 : P [ p1 ] & p2 `2 <= 0 & p1 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 } as Subset of TOP-REAL 2 ;
sqrt ( ( ( W-bound C ) / 2 ) ^2 + ( ( W-bound C ) / 2 ) ^2 ) <= ( ( W-bound C ) / 2 ) / 2 + ( ( W-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 ( @ @ ) = len ( ( @ @ p ) ^ <* 0 *> ) + len <* 1 *> .= len ( @ p ) + 1 .= len ( @ p ) + 1 ;
v / ( x. 3 , m1 ) / ( x. 4 , m ) . ( x. 4 , m ) . ( x. 4 , m ) = m1 / ( x. 4 , m ) . ( x. 4 , m ) . ( x. 4 , m ) . ( x. 4 , m ) . ( x. 4 , m ) . ( x. 4 , m ) . ( x. 4 , m ) . ( x. 4 , m ) = m1 / ( x. 4 , m ) ;
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 ) ) |= r ;
func U1 \ w2 -> Element of Union ( G , R ) equals ( the Sorts of G ) . ( ( the Sorts of R ) * ( the Arity of S ) . ( ( the Arity of S ) . ( ( the Arity of S ) . ( ( the Arity of S ) . ( ( the Arity of S ) . ( ( the Arity of S ) . ( ( the Arity of S ) . ( o ) ) ) ) ) ;
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= Exec ( n2 , s2 ) . b2 .= Exec ( n2 , s2 ) . b2 .= Exec ( i2 , s2 ) . b2 .= Exec ( i2 , s2 ) . b2 .= Exec ( i2 , s2 ) . b2 .= s2 . b2 ;
for n , k be Nat holds 0 <= ( Partial_Sums |. seq .| ) . ( n + k ) - ( Partial_Sums |. seq .| ) . ( n + k )
set F = S \! \mathop { \vert F .| , G = S \! \mathop { \vert F .| } ;
( Partial_Sums ( seq ) ) . ( K + 1 ) + Partial_Sums ( seq ) . ( K + 1 ) >= ( Partial_Sums ( seq ) ) . ( K + 1 ) + Partial_Sums ( seq ) . ( K + 1 ) ;
consider L , R such that for x st x in N holds ( f | Z ) . x - f . x0 = L . ( x- ( 1 / 2 ) ) * R . ( x- ( 1 / 2 ) ) ;
func the closed non empty Subset of TOP-REAL 2 means : Def6 : for a , b , c being Real st a in the carrier of \HM { a , b } holds it . ( a , c ) = ( the distance of \HM { a } ) . ( b , c ) ;
a * b ^2 + ( a * c ) + ( b * d ) >= 6 * a * c + ( b * d ) ;
v / ( x1 , m1 ) / ( x2 , m1 ) . ( x2 , m ) = v / ( x2 , m1 ) / ( x2 , m ) . ( x2 , m ) . ( x2 , m ) . ( x2 , m ) . ( x2 , m ) . ( x2 , m ) . ( x2 , m ) . ( x2 , m ) . ( x2 , m ) . ( x2 , m ) ) = v / ( x2 , m ) . ( x2 , m ) . ( x2 , m ) ;
( ( Q ^ <* x *> ) . ( M ^ <* y *> ) = ( ( ( Q ^ <* x *> ) +* ( M ^ <* y *> ) ) +* ( M ^ <* x *> ) ) . ( ( M ^ <* y *> --> FALSE ) ) . ( M ^ <* x *> ) ;
Sum ( F3 = ( r to_power n1 ) * Sum ( C3 ) .= ( C to_power n1 ) * ( ( r to_power n1 ) * ( r to_power n1 ) ) .= C . ( ( r to_power n1 ) * ( r to_power n1 ) ) .= C . ( ( r to_power n1 ) * ( r to_power n1 ) ) ;
( GoB f ) * ( len GoB f , 1 ) `1 = ( ( GoB f ) * ( len GoB f , 1 ) ) `1 .= ( ( GoB f ) * ( 1 , 1 ) ) `1 .= ( GoB f ) * ( 1 , 1 ) `1 ;
defpred X [ Element of NAT ] means ( Partial_Sums s ) . $1 = ( Partial_Sums ( a * $1 ) ) . ( 2 * $1 + 1 ) * b ;
( the_arity_of g ) . ( ( the Arity of S ) . g ) = ( the Arity of S ) . ( ( the Arity of S ) . g ) .= ( ( the Arity of S ) . g ) . ( ( the Arity of S ) . g ) .= ( the Arity of S ) . g ;
( X \times Y ) Z tolerates X ^ Y & ( X \/ Y ) c= ( X \/ Y ) \/ ( X \/ Y ) implies ( X \/ Y ) \/ ( X \/ Y ) = ( X \/ Y ) \/ ( X \/ Y )
for a , b being Element of S for s being Element of NAT st s = F . n & a = F . n & b = N . n holds b = N . ( n + 1 ) \ G . ( n + 1 )
E , f |= All ( x , H ) => ( ( All ( x , H ) '&' ( All ( x , H ) ) '&' ( ( All ( x , H ) '&' ( ( All ( x , H ) '&' ( ( All ( x , H ) ) '&' ( ( All ( x , H ) '&' ( ( All ( x , H ) ) '&' ( ( All ( x , H ) ) '&' ( ( x , H ) ) ) ) ) ) ) ) ) ;
ex R2 be 1-sorted st R2 = ( p | ( n2 -' 1 ) ) . i & ( the carrier of p ) . i = the carrier of R & ( the carrier of p ) . i = the carrier of R ;
[. a , b + sqrt ( 1 - ( a * b ) ) / ( 1 - ( b * f ) ) , b + sqrt ( 1 - ( b * f ) ) / ( 1 - ( b * f ) ) .] is Element of the W of the W of X ;
Comput ( P , s , 2 + 1 ) = Exec ( a3 , Comput ( P , s , 2 ) ) . IC SCM+FSA .= Exec ( a3 , s ) . IC SCM+FSA .= s . IC SCM+FSA .= s . IC SCM+FSA ;
card ( h1 *' ) . k = ( power ( F_Complex , n ) ) . ( ( - 1_ F_Complex ) * u ) .= ( ( ( - 1_ F_Complex ) * u ) ) . k .= ( ( ( - 1_ F_Complex ) * u ) . k ) * u .= ( ( ( - 1_ F_Complex ) * u ) . k ) * u ;
sqrt ( ( f * g ) /. c ) = f /. c * ( g /. c ) " .= ( f * g ) /. c * ( g /. c ) " .= ( f * g ) /. c * ( g /. c ) " .= ( f * g ) /. c ;
len Cv - len ( ( ( ( ( ( ( ( ( ( ( ( ( B /. 1 ) ) ) /. len ( ( ( ( ( ( C /. 1 ) ) ) /. len ( ( ( ( C /. 1 ) ) /. len ( ( ( ( C /. 1 ) ) /. len ( ( ( C /. 1 ) ) /. len ( ( D /. 1 ) ) /. len ( ( D /. 1 ) ) ) ) ) ) ) ) ) = len ( ( ( ( ( ( ( ( ( ( ( D /. len ( ( ( D /. len ( ( D /. len ( ( D /. len ( ( D /. len ( ( D /. len ( ( ( D /. len ( ( D /. len ( ( ( D /. len ( ( D /. len ( ( D /. len ( ( (
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 ) ;
defpred P [ Nat ] means 2 * Fib ( n + 1 ) = Fib ( n ) * Fib ( $1 ) + Fib ( n ) * Fib ( $1 ) * Fib ( $1 ) * Fib ( $1 ) ;
consider f being Function of [: { n + 1 } , NAT :] , NAT such that f = f and f is onto and for n being Nat st n < 1 holds f " . n = n / ( n + 1 ) and f is onto ;
consider c2 being Function of S , BOOLEAN such that c = IExec ( A , B , D ) and E . ( A \/ B ) = Prob ( c , D , E ) and E . ( A \/ B ) = Prob ( c , C , D ) ;
consider y being Element of [: Y , Z :] such that a = "\/" ( { F ( x , y ) where x is Element of Y : P [ x , y ] } , Z ) and P [ y , x ] ;
assume that A c= dom f and f = ( ( - 1 ) (#) ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( #Z 2 ) * ( ( #Z 2 ) * ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2
( f /. i ) `2 = ( ( GoB f ) * ( 1 , j2 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 + 1 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 + 1 ) ) `2 ;
dom Shift ( q2 , len Seq q1 ) = { j + len Seq q1 where j is Nat : j in dom Seq q1 & len Seq q1 = j + 1 } ;
consider G1 , G2 being Morphism of V such that G1 <= G2 and G2 <= G2 and for f being Morphism of G1 , G2 st f in G1 & f in G2 holds f * ( G1 * f ) = G1 * ( G2 * f ) ;
func - f -> PartFunc of C , V means : Def6 : dom it = dom f & for c st c in dom it holds it /. c = - f /. c + f /. c ;
consider phi such that phi is increasing and for a st phi . a = a & {} <> a holds L . ( union L ) = a & for v st v in union L holds L . ( v , v ) = a ;
consider i1 , j1 such that [ i1 , j1 ] in Indices GoB f and f /. ( i1 + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( k + 1 ) = ( GoB f ) * ( i1 , j1 ) ;
consider i , n such that n <> 0 and sqrt p = sqrt ( i + n ) and for n1 being Nat st n1 <> 0 & n1 < n holds sqrt p = ( i + n ) / ( n + 1 ) ;
assume that 0 in Z and Z c= dom ( ( ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) * ( ( 1 / 2 ) * ( ( 1 / 2 ) * ( ( 1 / 2 ) * ( 1 / 2 ) * ( ( 1 / 2 ) * ( 1 / 2 ) ) ) ) ) ) ) ) ) ) ) and for x ) ) ) ) ) ) and for x ) )
cell ( G1 , i1 -' 1 , j1 -' 1 ) \ ( ( L~ f1 ) \ ( L~ f1 ) \ ( L~ f1 ) ) c= BDD L~ f1 \/ ( L~ f1 ) \ ( L~ f1 ) ;
ex Q1 being open Subset of X st s = Q1 & ex Q being Subset of Y st Q c= F & ( for x being Point of Y st x in Q ex y being Point of Y st y in Q & P [ x , y ] ) & ( for x being Point of Y st x in Q holds Q [ x , y ] ) implies P [ x , y ]
gcd ( A , ( 1. ( K , n ) ) , ( 1. ( K , n ) ) , ( 1. ( K , n ) ) ) = 1 / ( 1. ( K , n ) ) .= 1. ( K , n ) ;
R8 = ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( s2 s2 ) ) ) ) ) ) ) ) ) ) ) ) . ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( s2 ) ) ) ) . ( m ) ) ) ) ) ) ) ) . ( x + 1 ) ) ) ) ) ) . ( 2 , 2 ) ) ) ) . ( 2 , 2 ) ) . ( 2 , 2 ) ) . ( 2 , 2 ) ) . ( 2 , 2 ) ) . ( 2 , 2 , 2 ) ) . ( 2 , 2 , 1 ) ) . ( 2 , 2 , 1 ) ) . ( 2 , 2 , 2 ) ) . ( 2 , 2 )
CurInstr ( P3 , Comput ( P3 , s3 , m ) ) = CurInstr ( P3 , Comput ( P3 , s3 , m ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m ) ) .= halt SCMPDS ;
P1 /\ P2 = ( { p1 } \/ LSeg ( p1 , p2 ) ) /\ LSeg ( p1 , p2 ) \/ ( LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) ) /\ LSeg ( p1 , p2 ) \/ ( LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) ) /\ LSeg ( p1 , p2 ) ;
func -> Subset of the Sorts of A means : Def6 : a in it iff ex i st i in dom f & not ( ex p st p in dom f & f . i = p ) & ( for i st i in dom f holds f . i = p ) & ( not f . i in rng f ) ;
for a , b being Element of F_Complex st |. a .| > |. b .| & |. a .| >= 1 holds a * ( f | ( len f ) ) is ]
defpred P [ Nat ] means 1 <= $1 & $1 + 1 <= len g & for i , j st [ i , j ] in Indices G & G * ( i , j ) = g * ( $1 , j ) holds G * ( i , j ) = g * ( $1 , j ) ;
assume that C1 , C2 are_Following f and g is stable and for s being State of C1 , f being Function of C1 , C2 st f = s holds f . s = ( s * f ) . s and for t being State of C1 , s being State of C2 st t = s holds t . t = s . t ;
( ||. f .|| | X ) . c = ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| ;
|. q .| ^2 = ( |. q .| ) ^2 + ( |. q .| ) ^2 + ( |. q .| ) ^2 / ( |. q .| ) ^2 + ( |. q .| ) ^2 / ( |. q .| ) ^2 ) ;
for F being Subset-Family of T st F is open & {} in F & for A , B being Subset of T st A in F & B in F & A <> B holds card A = card B & card A = k & card B = k + 1 holds card A = k + 1
assume that len F >= 1 and len F = k + 1 and len F = k and for n st n in dom F holds H . n = F . ( F . n , G . n ) and for k st k in dom F holds H . k = F . ( k , n ) ;
i |^ ( mod n ) = i |^ ( s + k ) .= i |^ ( s + k ) * i .= i |^ ( s + k ) * i .= i |^ ( s + k ) * i ;
consider q being oriented oriented oriented Chain of G such that r = q and q <> {} and F . ( q . 1 ) = v and rng q c= rng ( p ^ q ) and rng q c= rng p and rng q c= rng p and not q in rng p and not q in rng p and q in rng p ;
defpred P [ Element of NAT ] means ( ( g , Z ) . ( len g ) ) . ( len g + $1 ) = ( ( ( f , Z ) ^ <* x *> ) . ( len f + $1 ) ) . ( len f + $1 ) ;
for A , B being Matrix of n , K holds len ( A * B ) = len A & width ( A * B ) = n & width ( A * B ) = n implies len ( A * B ) = n & width ( A * B ) = n
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 * c ;
func ( Re x , y ) .|. ( ( Re x ) * ( Re y ) ) -> Element of COMPLEX equals ( Re x ) * ( Re y ) + ( Im y ) * ( Im y ) ;
consider g2 be FinSequence of Fq such that g2 is continuous and rng g2 = A and g2 . 0 = x0 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 ;
then n1 >= len p1 & n2 >= len p1 & n3 = len p1 + len p2 & n3 = len p1 + len p2 & n3 = len p1 + len p2 ;
( q `1 ) * a <= ( q `1 ) * ( q `1 ) & ( - q `1 ) * ( q `1 ) * ( q `1 ) >= ( - q `1 ) * ( q `1 ) ;
Fv . ( len ( p . len p ) ) = Fv . ( len p ) .= ( ( v . len p ) ) * v . ( len p ) .= v . ( len p ) ;
consider k1 being Nat such that k1 + k = 1 and a := k = ( ( a := intloc 0 ) .--> 1 ) . ( ( intloc 0 ) .--> 1 ) . ( ( intloc 0 ) .--> 1 ) . ( ( intloc 0 ) .--> 1 ) . ( ( intloc 0 ) .--> 1 ) . ( ( intloc 0 ) .--> 1 ) . ( ( intloc 0 ) .--> 1 ) . ( ( intloc 0 ) .--> 1 ) . ( ( intloc 0 ) .--> 1 ) . ( ( intloc 0 ) .--> 1 ) = 1 ) . ( ( intloc 0 ) .--> 1 ) . ( ( intloc 0 ) . ( ( intloc 0 ) .--> 1 ) . ( ( intloc 0 ) . intloc 0 ) . ( ( intloc 0 ) . intloc 0 ) . intloc 0 ) . intloc 0 ) . intloc 0 ) ;
consider BY being Subset of B1 , DY being Subset of B2 such that BY is finite and BY is finite and DY = \mathop ( card A1 , card B1 ) and BY = indx ( BY , card B1 ) and LY = indx ( BY , card B1 ) ;
v2 . b2 = ( curry F2 ) . ( ( ( ( ( F . b2 ) * ( F . b2 ) ) * ( ( F . b2 ) * ( F . b2 ) ) ) ) .= ( ( ( ( F . b2 ) * ( F . b2 ) ) * ( ( F . b2 ) * ( F . b2 ) ) ) ) . b2 ;
dom IExec ( I , P , Initialize s ) = the carrier of SCMPDS .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) +* Start-At ( ( card I + 2 ) , SCMPDS ) .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ;
ex d|^ be Real st d|^ > 0 & |. h .| < d & |. h .| " * ||. ( R /. ( L + 1 ) - R /. ( L + 1 ) ) .|| < e / 2 ;
LSeg ( G * ( len G , 1 ) , G * ( len G , 1 ) ) /\ LSeg ( G * ( len G , 1 ) , G * ( len G , 1 ) ) c= Int cell ( G , len G , width G ) ;
LSeg ( mid ( h , i1 , i2 ) , i ) = LSeg ( h /. ( i1 + 1 ) , h /. ( i1 + 1 ) ) .= LSeg ( h /. ( i1 + 1 ) , h /. ( i1 + 1 ) ) , h /. ( i1 + 1 ) ) ;
A = { q where q is Point of TOP-REAL 2 : LE q , p , P , p1 , p2 & LE q , p , P , p1 , p2 & LE q , p , P , p1 , p2 & LE q , p , P , p1 , p2 & LE q , p , P , p1 , p2 & LE q , p , P , P , p1 , p2 & LE q , p , P , p2 & LE q , p , P , p1 , P , p1 , p2 & LE p , q , P , P , p1 , p2 & LE p , q , P , P , p1 , P , p2 , P , p1 , p2 & LE p , q , P , p1 , p2 & LE p , q , P , P , p2 , P , P , p1 , p2 & LE p , q , P , p1 , P , p1 , P , p1 , P , P , p1 , P , p1 , p2 ,
( ( - x ) .|. y ) = ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( 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 `1 / p `2 ) ^2 ) ;
sqrt ( ( U * ( W * ( W * ( W * ( W * ( W * ( x , y ) ) ) ) ) / ( 2 * ( W * ( x , y ) ) ) ) ^2 ) ) = ( ( U * ( W * ( x , y ) ) ) / ( 2 * ( W * ( x , y ) ) ) ) / ( 2 * ( x , y ) ) ) ;
func Shift ( f , h ) -> PartFunc of REAL , REAL means : Def6 : for x be Element of REAL m holds it . x = ( - h ) . x * ( - h ) . x ) & for x be Element of REAL m st x in dom it holds it . x = ( - h ) . x * ( - h ) . x ;
assume that 1 <= k and k + 1 <= len f and [ i , j ] in Indices GoB f and [ i + 1 , j ] in Indices GoB f and [ i + 1 , j ] in Indices GoB f and f /. k = G * ( i , j ) and f /. k = G * ( i , j ) ;
assume that 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 y in Free H and x in Free H and y in Free H and y in Free H and x in Free H and y in Free H ;
defpred P [ Element of NAT , Element of NAT ] means ( p |^ $1 ) |^ ( p |^ $1 ) = p |^ $1 & ( for n st n in $1 holds $2 = ( p |^ $1 ) |^ ( p |^ $1 ) ) |^ ( p |^ n ) ) ;
func \sigma ( C ) -> non empty Subset-Family of X means : Def6 : for A , B being Subset of X holds it . ( A \/ B ) <= it . ( A \/ B ) & for W being Subset of X holds W . ( W . ( A \/ B ) ) <= W . ( W . ( A \/ B ) ) ;
[#] ( ( ( ( ( ( TOP-REAL 2 ) | Q ) .: P ) .: Q ) .: Q ) = ( ( ( ( ( ( TOP-REAL 2 ) | P ) .: Q ) .: Q ) .: P ) .: Q ) .: Q ) .: ( [#] ( ( ( ( TOP-REAL 2 ) | P ) .: Q ) ) ;
rng ( F | ( [: S , T :] ) = { 1 } or rng ( F | ( S , T ) ) = { 1 } or rng ( F | ( S , T ) ) = { 1 } ;
( f " ) . i = f . i " .= ( f " ) . i .= ( ( f " ) . i ) " .= ( f " ) . i .= ( f " ) . i .= ( f " ) . i ;
consider P1 , P2 being Subset of TOP-REAL 2 such that P1 /\ P2 = { p1 , p2 } and P1 /\ P2 = { p1 , p2 } and P1 /\ P2 = { p1 , p2 } and P1 /\ P2 = { p2 } and P1 /\ P2 = { p2 } and P1 /\ P2 = { p1 , p2 } and P1 /\ P2 = { p2 , p1 } and P1 /\ P2 = { p2 , p1 } ;
f . p2 = |[ ( p2 `1 ) / sqrt ( 1 + ( p2 `1 ) ^2 ) , ( p2 `1 ) / sqrt ( 1 + ( p2 `1 ) ^2 ) ]| ;
( ( for a , X be non empty TopSpace holds ( for x being Point of X st x in the carrier of X holds ( ( a , X ) qua Function ) " ) . x = - a * x + b * x .= 0. X + b * x .= 0. X ;
for T being non empty normal TopSpace , A being Subset of T , B being Subset of T , A being Subset of T , B being Subset of T st A <> {} & A misses B holds ( Initialized G ) . p = ( .[ / A ) . p
for i , [#] G st i + 1 in dom F for G1 , G2 being strict Subgroup of G st G1 = F . i & G2 = F . ( i + 1 ) holds G1 is strict Subgroup of G
for x st x in Z holds ( ( arctan * arccot ) `| Z ) . x = ( ( ( arctan * arccot ) `| Z ) . x / ( 1 + x ^2 ) ) / ( 1 + x ^2 ) / ( 1 + x ^2 )
synonym f /* a is right convergent means : by : for x0 st x0 in dom f & x0 in dom f holds f . x0 < ( f /* a ) . x0 + r / ( f . x0 ) ;
then X1 , X2 are_separated & X1 , X2 are_separated & X2 , Y2 are_separated & X1 , X2 are_separated & X2 , Y2 are_separated & X1 , X2 are_separated & X2 , X1 misses X2 & X1 , X2 are_separated implies X1 , X2 are_separated & X2 , X1 are_separated & X1 , X2 are_separated & X2 , X1 are_separated & X2 , X1 are_separated & X1 , X2 are_separated & X1 , X2 are_separated & X2 , X1 are_separated implies X1 , X2 are_separated
ex N be Neighbourhood of x0 st N c= dom SVF1 ( 1 , f , u ) & for x st x in N holds ( SVF1 ( 1 , f , u ) ) . x - SVF1 ( 1 , f , u ) . x = L . ( u - x ) ;
sqrt ( ( p2 `1 ) ^2 + ( p2 `2 ) ^2 ) >= sqrt ( ( p2 `1 ) ^2 + ( p2 `2 ) ^2 ) ;
( ( ( 1 / t ) (#) ( ( ( 1 / t ) (#) ( ( 1 / t ) (#) ( ( 1 / t ) (#) ( ( 1 / t ) (#) ( ( 1 / t ) (#) ( ( 1 / t ) (#) ( ( 1 / t ) (#) ( ( 1 / t ) (#) ( ( 1 / t ) (#) ( ( 1 / t ) (#) ( ( 1 / t ) * ( 1 / t ) ) ) ) ) ) ) ) `| REAL ) . x = ( ( 1 / t ) ) `| REAL ) . x ) ) . x ) . x ) . x ) . x ) . x ) . x ) . x / t . x ) . x / t . x ) . x / t . x ) . x ) . x = ( ( 1 / t ) . x / t . x ) . x / t . x ) = ( ( ( ( 1 / t ) * ( ( 1 / t ) * ( ( 1 / t ) * ( ( 1 / t ) * ( ( 1 / t ) * ( ( 1 / t ) * ( ( 1 / t
assume that for x holds f . x = ( ( - 1 / 2 ) (#) ( sin * cos ) ) . x and for x st x in Z holds ( ( ( - 1 / 2 ) (#) ( sin * cos ) ) `| Z ) . x = ( ( - 1 / 2 ) (#) ( cos * sin ) ) . x ;
consider Xj1 being Subset of Y , Y1 being Subset of X such that t = Y1 and Y1 = Y1 /\ Y2 and Y1 = Y2 and Y1 /\ Y2 = Y1 /\ Y2 and Y1 /\ Y2 = Y1 /\ Y2 and Y1 /\ Y2 = Y1 /\ Y2 and Y1 /\ Y2 = Y1 /\ Y2 ;
card S . ( n + 1 ) = card { [ d , c ] where d is Element of GF ( p ) : [ d , c ] in Indices ( p * a ) } .= card ( p * a ) ;
sqrt ( ( ( W-bound D ) / 2 ) * ( ( i1 - 1 ) / 2 ) ) = ( ( W-bound D ) / 2 ) * ( ( i1 - 1 ) / 2 ) ) / 2 .= ( ( W-bound D ) / 2 ) * ( ( i1 - 1 ) / 2 ) ;