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contradiction ;
contradiction ;
contradiction ;
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contradiction ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
B ;
a <> c
T c= S
D c= B
c in X ;
b in X ;
X ;
b in D ;
x = e ;
let m ;
h is onto ;
N in K ;
let i ;
j = 1 ;
x = u ;
let n ;
let k ;
y in A ;
let x ;
let x ;
m c= y ;
F is onto ;
let q ;
m = 1 ;
1 < k ;
G is commutative ;
b in A ;
d divides a ;
i < n ;
s <= b ;
b in B ;
let r ;
B is one-to-one ;
R is total ;
x = 2 ;
d in D ;
let c ;
let c ;
b = Y ;
0 < k ;
let b ;
let n ;
r <= b ;
x in X ;
i >= 8 ;
let n ;
let n ;
y in f ;
let n ;
1 < j ;
a in L ;
C is boundary ;
a in A ;
1 < x ;
S is finite ;
u in I ;
z << z ;
x in V ;
r < t ;
let t ;
x c= y ;
a <= b ;
m in dom f ;
assume f is prime ;
not x in Y ;
z = +infty ;
let k be Nat ;
K ` is being_line ;
assume n >= N ;
assume n >= N ;
assume X is non empty ) ;
assume x in I ;
q is ) by 0 ;
assume c in x ;
1-p > 0 ;
assume x in Z ;
assume x in Z ;
1 <= kr2 ;
assume m <= i ;
assume G is commutative ;
assume a divides b ;
assume P is closed ;
b-a > 0 ;
assume q in A ;
W is not bounded ;
f is Assume f is Assume f is ) ;
assume A is boundary ;
g is special ;
assume i > j ;
assume t in X ;
assume n <= m ;
assume x in W ;
assume r in X ;
assume x in A ;
assume b is even ;
assume i in I ;
assume 1 <= k ;
X is non empty ;
assume x in X ;
assume n in M ;
assume b in X ;
assume x in A ;
assume T c= W ;
assume s is atomic ;
b `2 <= c `2 ;
A meets W ;
i `2 <= j `2 ;
assume H is universal ;
assume x in X ;
let X be set ;
let T be Tree ;
let d be element ;
let t be element ;
let x be element ;
let x be element ;
let s be element ;
k <= 5 + 5 ;
let X be set ;
let X be set ;
let y be element ;
let x be element ;
P [ 0 ]
let E be set , A be Subset of E ;
let C be category ;
let x be element ;
let k be Nat ;
let x be element ;
let x be element ;
let e be element ;
let x be element ;
P [ 0 ]
let c be element ;
let y be element ;
let x be element ;
let a be Real ;
let x be element ;
let X be element ;
P [ 0 ]
let x be element ;
let x be element ;
let y be element ;
r in REAL ;
let e be element ;
n1 is , $ n2 is , f ;
Q halts_on s ;
x in for of for of of S holds x in \in that S ;
M < m + 1 ;
T2 is open ;
z in b < b < a ;
R2 is well-ordering ;
1 <= k + 1 ;
i > n + 1 ;
q1 is one-to-one ;
let x be trivial set ;
PM is one-to-one
n <= n + 2 ;
1 <= k + 1 ;
1 <= k + 1 ;
let e be Real ;
i < i + 1 ;
p3 in P ;
p1 in K ;
y in C1 ;
k + 1 <= n ;
let a be Real , x be Point of TOP-REAL 2 ;
X |- r => p ;
x in { A } ;
let n be Nat ;
let k be Nat ;
let k be Nat ;
let m be Nat ;
0 < 0 + k ;
f is_differentiable_in x ;
let x0 , r ;
let E be Ordinal ;
o >= o1 & o <= o2 ;
O <> O2 ;
let r be Real ;
let f be FinSeq-Location ;
let i be Nat ;
let n be Nat ;
Cl A = A ;
L c= Cl L ;
A /\ M = B ;
let V be complex normed space , v be VECTOR of V ;
not s in Y to_power 0 ;
rng f <= w
b "/\" e = b ;
m = m3 ;
t in h . D ;
P [ 0 ] ;
assume z = x * y ;
S . n is bounded ;
let V be RealLinearSpace , v be VECTOR of V ;
P [ 1 ] ;
P [ {} ] ;
C1 meets C ` ;
H = G . i ;
1 <= i `2 + 1 ;
F . m in A ;
f . o = o ;
P [ 0 ] ;
aor <= is Real ;
R [ 0 ] ;
b in f .: X ;
assume q = q2 ;
x in [#] V ;
f . u = 0 ;
assume e1 > 0 ;
let V be RealUnitarySpace , W be Subspace of V ;
s is trivial & s is non empty ;
dom c = Q
P [ 0 ] ;
f . n in T ;
N . j in S ;
let T be complete LATTICE , X be Subset of T ;
the ObjectMap of F is one-to-one
sgn x = 1 ;
k in support a ;
1 in Seg 1 ;
rng f = X ;
len T in X ;
vbeing < n ;
S\HM is bounded ;
assume p = p2 ;
len f = n ;
assume x in P1 ;
i in dom q ;
let U0 , A , B ;
pp `1 = c ;
j in dom h ;
let k ;
f | Z is continuous ;
k in dom G ;
UBD C = B ;
1 <= len M ;
p in Ball ( x , r ) ;
1 <= jj & jj <= len f ;
set A = L /\ L ;
card a [= c ;
e in rng f ;
cluster B \oplus A -> empty ;
H is with_no or H is has \subseteq the carrier of S ;
assume n0 <= m ;
T is increasing ;
e2 <> e2 & e2 <> e1 ;
Z c= dom g ;
dom p = X ;
H is proper implies H in G
i + 1 <= n ;
v <> 0. V ;
A c= Affin A ;
S c= dom F ;
m in dom f ;
let X0 be set ;
c = sup N ;
R is_connected implies union M is connected
assume not x in REAL ;
Im f is complete ;
x in Int y ;
dom F = M ;
a in On W ;
assume e in A ( ) ;
C c= C-26 ;
mm <> {} & mm <> {} ;
let x be Element of Y ;
let f be ) of
not n in Seg 3 ;
assume X in f .: A ;
assume that p <= n and p <= m ;
assume not u in { v } ;
d is Element of A ;
A / b misses B ;
e in v + ( ex v st e in v & v in X ;
- y in I ;
let A be non empty set , B be set ;
P0 = 1 ;
assume r in F . k ;
assume f is simple function in S ;
let A be l -countable set ;
rng f c= NAT ;
assume P [ k ] ;
f6 <> {} ;
let o be Ordinal ;
assume x is sum of squares ;
assume not v in { 1 } ;
let IB , C ;
assume that 1 <= j and j < l ;
v = - u ;
assume s . b > 0 ;
d1 in dom f ;
assume t . 1 in A ;
let Y be non empty TopSpace , X be Subset of Y ;
assume a in uparrow s ;
let S be non empty Poset ;
a , b // b , a ;
a * b = p * q ;
assume x , y are_the space of X ;
assume x in Omega ( f ) ;
[ a , c ] in X ;
mm <> {} & mm <> {} ;
M + N c= M + M ;
assume M is connected hh) ;
assume f is additive mab-rst ;
let x , y be element ;
let T be non empty TopSpace ;
b , a // b , c ;
k in dom Sum p ;
let v be Element of V ;
[ x , y ] in T ;
assume len p = 0 ;
assume C in rng f ;
k1 = k2 & k2 = k1 ;
m + 1 < n + 1 ;
s in S \/ { s } ;
n + i >= n + 1 ;
assume Re y = 0 ;
k1 <= j1 & k1 <= len f ;
f | A is continuous ;
f . x - x <= b ;
assume y in dom h ;
x * y in B1 ;
set X = Seg n ;
1 <= i2 + 1 ;
k + 0 <= k + 1 ;
p ^ q = p ;
j |^ y divides m ;
set m = max A ;
[ x , x ] in R ;
assume x in succ 0 ;
a in sup phi ;
CH ;
q2 c= C1 & q2 c= C2 ;
a2 < c2 & a2 < b2 implies a2 < b2
s2 is 0 -started ;
IC s = 0 & IC s = 0 ;
s4 = s4 , s4 = s4 , P4 = s4 , P4 = s4 , P4 = i2 , P4 = i1 , P4 = i2 , P4 = i2 ,
let V ;
let x , y be element ;
let x be Element of T ;
assume a in rng F ;
x in dom T ` ;
let S be as <> of L ;
y " <> 0 ;
y " <> 0 ;
0. V = u-w ;
y2 , y are_\vert & y , z are_\vert ;
R8 in dom f ;
let a , b be Real , x be Point of TOP-REAL 2 ;
let a be Object of C ;
let x be Vertex of G ;
let o be object of C , a be Object of C ;
r '&' q = P \lbrack l \rbrack ;
let i , j be Nat ;
let s be State of A , k be Nat ;
s4 . n = N ;
set y = ( x `1 ) / ( 1 + x `2 ) ;
mi in dom g & mi in dom g ;
l . 2 = y1 ;
|. g . y .| <= r ;
f . x in CX0 ;
V1 is non empty & V2 is empty ;
let x be Element of X ;
0 <> f . g2 ;
f2 /* q is convergent ;
f . i is_measurable_on E ;
assume \xi in N-22 ;
reconsider i = i as Ordinal ;
r * v = 0. X ;
rng f c= INT & rng f c= INT ;
G = 0 .--> goto 0 ;
let A be Subset of X ;
assume that A is dense and A is open ;
|. f . x .| <= r ;
let x be Element of R ;
let b be Element of L ;
assume x in W-19 ;
P [ k , a ] ;
let X be Subset of L ;
let b be object of B ;
let A , B be category ;
set X = Vars , C = Vars ;
let o be OperSymbol of S ;
let R be connected non empty Poset ;
n + 1 = succ n ;
xY c= Z1 & xY c= Z1 ;
dom f = C1 & dom g = C2 ;
assume [ a , y ] in X ;
Re ( seq ) is convergent & lim ( seq ) = 0 ;
assume a1 = b1 & a2 = b2 ;
A = ( sInt A ) ;
a <= b or b <= a ;
n + 1 in dom f ;
let F be Instruction of S , I be Instruction of S ;
assume that r2 > x0 and x0 < r2 ;
let Y be non empty set , f be Function of Y , BOOLEAN ;
2 * x in dom W ;
m in dom ( g2 * g1 ) ;
n in dom g1 & n <= len g2 ;
k + 1 in dom f ;
the still of not s in { s } ;
assume that x1 <> x2 and x2 <> x3 ;
v3 in V1 & v2 in V1 & v3 in V2 ;
not [ b `1 , b `2 ] in T ;
i-35 + 1 = i ;
T c= <> and G c= <> G ;
l `1 = 0 & l `2 = 0 ;
let n be Nat ;
t `2 = r & t `2 = s ;
AA is_integrable_on M & AA is integrable ;
set t = Top t ;
let A , B be real-membered set ;
k <= len G + 1 ;
C ( ) misses V ( ) ;
Product seq is non empty ;
e <= f or f <= e ;
cluster non empty normal for sequence ;
assume c2 = b2 & c2 = b1 ;
assume h in [. q , p .] ;
1 + 1 <= len C ;
not c in B . m1 ;
cluster R .: X -> empty ;
p . n = H . n ;
assume vseq is sequence of sequence ( X ) ;
IC s3 = 0 & IC s3 = 0 ;
k in N or k in K ;
F1 \/ F2 c= F \/ G ;
Int G1 <> {} & Int G2 <> {} ;
z `2 = 0 & z `2 = 0 ;
p11 <> p1 or p11 <> p2 ;
assume z in { y , w } ;
MaxADSet ( a ) c= F ;
ex_sup_of downarrow s , S & ex_sup_of downarrow s , S ;
f . x <= f . y ;
let T be up-complete non empty reflexive transitive RelStr ;
q |^ m >= 1 ;
a >= X & b >= Y ;
assume <* a , c *> <> {} ;
F . c = g . c ;
G is one-to-one one-to-one full full implies G is onto
A \/ { a } c= B ;
0. V = 0. Y & 0. V = 0. V ;
let I be halting Instruction of S , s be State of S ;
f-24 . x = 1 & f-24 . x = 0 ;
assume z \ x = 0. X ;
C4 = 2 to_power n & C5 = 2 to_power n ;
let B be SetSequence of Sigma ;
assume X1 = p .: D ;
n + l2 in NAT & n + l2 in NAT ;
f " P is compact & f " P is compact ;
assume x1 in REAL & x2 in REAL ;
p1 = K1 & p2 = K1 & p3 = K1 ;
M . k = <*> REAL ;
phi . 0 in rng phi ;
OSMInt A is closed ;
assume z0 <> 0. L & z0 <> 0. L ;
n < ( N . k ) ;
0 <= seq . 0 & seq . 0 <= seq . 0 ;
- q + p = v ;
{ v } is Subset of B ;
set g = f /. 1 ;
[: R ( ) , R ( ) :] is stable ;
set cR = Vertices R , cR = Vertices R ;
pp c= P3 & IC p c= s2 ;
x in [. 0 , 1 .[ ;
f . y in dom F ;
let T be Scott Scott Scott Scott Scott of S ;
ex_inf_of the carrier of S , S ;
downarrow a = downarrow b & downarrow b = downarrow a ;
P , C , K is_collinear ;
assume x in F ( s , r , t ) ;
2 to_power i < 2 to_power m ;
x + z = x + z + q ;
x \ ( a \ x ) = x ;
||. x-y .|| <= r & ||. x - y .|| <= r ;
assume that Y c= field Q and Y <> {} ;
a ~ , b ~ are_isomorphic ;
assume a in A ( ) ;
k in dom ( q | k ) ;
p is } is FinSequence of S ;
i -' 1 = i-1 - 1 ;
f | A is one-to-one ;
assume x in f .: X ( ) ;
i2 - i1 = 0 & i2 - i1 = 0 ;
j2 + 1 <= i2 & j2 + 1 <= len G ;
g " * a in N & g " * a in N ;
K <> { [ {} , {} ] } ;
cluster strict for } : strict for } ;
|. q .| ^2 > 0 ;
|. p4 .| = |. p .| ;
s2 - s1 > 0 & s2 - s1 > 0 ;
assume x in { Gij } ;
W-min C in C & W-min C in C ;
assume x in { Gij } ;
assume i + 1 = len G ;
assume i + 1 = len G ;
dom I = Seg n & rng I c= Seg n ;
assume that k in dom C and k <> i ;
1 + 1-1 <= i + j - 1 ;
dom S = dom F & dom F = dom G ;
let s be Element of NAT , k be Nat ;
let R be ManySortedSet of A , s be Element of S ;
let n be Element of NAT ;
let S be non empty non void non void holds S is holds S is non void ;
let f be ManySortedSet of I ;
let z be Element of F_Complex , v be Element of F_Complex ;
u in { ag } ;
2 * n < ( 2 * n ) ;
let x , y be set ;
B-11 c= [: V1 , V1 :] ;
assume I is_closed_on s , P & I is_halting_on s , P ;
U2 = U2 & U2 = U2 implies U2 is open
M /. 1 = z /. 1 ;
x11 = x22 & x22 = x22 ;
i + 1 < n + 1 + 1 ;
x in { {} , <* 0 *> } ;
f9 <= ( f . i ) & f . i <= ( f . i ) ;
let l be Element of L ;
x in dom ( F-17 ) ;
let i be Element of NAT , k be Nat ;
seq1 is COMPLEX -valued & seq2 is COMPLEX -valued ;
assume <* o2 , o *> <> {} ;
s . x |^ 0 = 1 ;
card K1 in M & card K1 in M ;
assume that X in U and Y in U ;
let D be Subset-Family of Omega ;
set r = ( Seg k ) \ { k + 1 } ;
y = W . ( 2 * x ) ;
assume dom g = cod f & cod f = cod g ;
let X , Y be non empty TopSpace , f be Function of X , Y ;
x ++ A is interval ;
|. <*> A .| . a = 0 ;
cluster strict for Sublattice of L ;
a1 in B . s1 & a2 in B . s2 ;
let V be finite finite \mathclose { 0. K } , W be Subspace of V ;
A * B on B , A & B on A ;
f-3 = NAT --> 0 .= fg ;
let A , B be Subset of V ;
z1 = P1 . j & z2 = P2 . j ;
assume f " P is closed & f " P is closed ;
reconsider j = i as Element of M ;
let a , b be Element of L ;
assume q in A \/ ( B "\/" C ) ;
dom ( F * C ) = o ;
set S = INT , T = INT ;
z in dom ( A --> y ) ;
P [ y , h . y ] ;
{ x0 } c= dom f & { x0 } c= dom f ;
let B be non-empty ManySortedSet of I , A be ManySortedSet of I ;
( PI / 2 ) * PI < Arg z ;
reconsider z9 = 0 , z9 = 1 as Nat ;
LIN a , d , c & LIN a , d , c ;
[ y , x ] in IB ;
Q * ( 1 , 3 ) = 0 & Q * ( 1 , 3 ) = 0 ;
set j = x0 gcd m , i = x0 gcd m ;
assume a in { x , y , c } ;
j2 - jj > 0 & j - jj > 0 ;
I the I \HM { {} } = 1 ;
[ y , d ] in F-8 ;
let f be Function of X , Y ;
set A2 = ( B - C ) / ( B - D ) ;
s1 , s2 are_card ( rng s1 ) & s2 , s1 is card ( rng s1 ) ;
j1 -' 1 = 0 & j1 -' 1 = 0 ;
set m2 = 2 * n + j ;
reconsider t = t as bag of n ;
I2 . j = m . j & I2 . j = m . j ;
i |^ s , n are_relative_prime & i |^ s , n are_relative_prime ;
set g = f | D-21 ;
assume that X is lower bounded and 0 <= r ;
p1 `1 = 1 & p1 `2 = - 1 ;
a < p3 `1 & p3 `1 < b & p3 `1 < b ;
L \ { m } c= UBD C ;
x in Ball ( x , 10 ) ;
not a in LSeg ( c , m ) ;
1 <= i1 -' 1 & i1 <= len f -' 1 ;
1 <= i1 -' 1 & i1 <= len f -' 1 ;
i + i2 <= len h & i + i2 <= len h ;
x = W-min ( P ) & y = W-min ( P ) ;
[ x , z ] in [: X , Z :] ;
assume y in [. x0 , x .] ;
assume p = <* 1 , 2 , 3 *> ;
len <* A1 *> = 1 & len <* A1 *> = 1 ;
set H = h . ( g . O ) ;
card b * a = |. a .| ;
Shift ( w , 0 ) |= v ;
set h = h2 (*) h1 , h1 = h2 (*) h2 , h2 = h2 (*) h1 , h2 = h2 (*) h1 , h2 = h2 (*) h1 , h2 = h2 (*) h2 , h2
assume x in X3 /\ ( X2 /\ 4 ) ;
||. h .|| < d1 & ||. h .|| < d ;
not x in the carrier of support ( f ) ;
f . y = F ( y ) ;
for n holds X [ n ] ;
k - l = k\leq ;
<* p , q *> /. 2 = q ;
let S be Subset of the carrier of Y ;
let P , Q be succ s ;
Q /\ M c= union ( F | M )
f = b * ( canFS ( S ) ) ;
let a , b be Element of G ;
f .: X is_<=_than f . sup X
let L be non empty transitive RelStr , X be Subset of L ;
S-20 is x -Basis i Basis of S ;
let r be non positive Real ;
M , v |= x \hbox { y } ;
v + w = 0. ( Z , X ) ;
P [ len ( F ( ) ) ] ;
assume InsCode ( i ) = 8 & InsCode ( i ) = 8 ;
the zero of M = 0 & the zero of M = 0 ;
cluster z (#) seq -> summable for sequence of X ;
let O be Subset of the carrier of C ;
||. f .|| | X is continuous ;
x2 = g . ( j + 1 ) ;
cluster implies for Element of ( AllSymbolsOf S ) ;
reconsider l1 = l-1 as Nat ;
v4 is Vertex of r2 & v4 is Vertex of G ;
T2 is SubSpace of T2 & T1 is SubSpace of T2 ;
Q1 /\ Q19 <> {} & Q1 /\ Q29 <> {} ;
let k be Nat ;
q " is Element of X & q " is Element of X ;
F . t is set of REAL & F . t is set ;
assume that n <> 0 and n <> 1 ;
set en = EmptyBag n , en = EmptyBag n ;
let b be Element of Bags n , x be Element of Bags n ;
assume for i holds b . i is commutative ;
x is root of ( p `2 ) * ( p `1 ) ;
not r in ]. p , q .] ;
let R be FinSequence of REAL , x be Element of REAL ;
not SS does not destroy b1 & not I does not destroy b1 ;
IC SCM R <> a & IC SCM R <> a ;
|. - ( |[ x , y ]| ) .| >= r ;
1 * seq = seq & 1 * seq = seq ;
let x be FinSequence of NAT , n be Nat ;
let f be Function of C , D , g be Function of C , D ;
for a holds 0. L + a = a
IC s = s . NAT & IC s = s . NAT ;
H + G = F- ( G-seq ) ;
Cx1 . x = x2 & Cy1 . x = y2 ;
f1 = f .= f2 .= f2 .= f1 ;
Sum <* p . 0 *> = p . 0 ;
assume v + W = v + u + W ;
{ a1 } = { a2 } & { a2 } = { a2 } ;
a1 , b1 _|_ b , a & b1 <> a ;
d3 , o _|_ o , a3 & d3 , o _|_ o , a3 ;
IB is reflexive & IB is reflexive implies IB is reflexive
IB is_antisymmetric implies C is_antisymmetric & C is_antisymmetric
sup rng H1 = e & sup rng H1 = e ;
x = ( a * a9 ) * ( a * b ) ;
|. p1 .| ^2 >= 1 ^2 & |. p2 .| ^2 >= 1 ;
assume that j2 -' 1 < 1 and j2 + 1 < len f ;
rng s c= dom f1 & rng s c= dom f2 ;
assume that support a misses support b and support b misses support b ;
let L be associative commutative distributive non empty doubleLoopStr , p be Polynomial of L ;
s " + 0 < n + 1 ;
p . c = ( f " ) . 1 .= p . c ;
R . n <= R . ( n + 1 ) ;
Directed ( I1 , J ) = I1 " ;
set f = + ( x , y , r ) ;
cluster Ball ( x , r ) -> bounded ;
consider r being Real such that r in A ;
cluster non empty NAT -defined for NAT -defined Function ;
let X be non empty directed Subset of S ;
let S be non empty full SubRelStr of L ;
cluster <* \hbox { - } / N , 1 *> -> complete for non trivial TopSpace ;
( 1 / a ) " = a & ( 1 / a ) " = a ;
( q . {} ) `1 = o & ( q . {} ) `2 = {} ;
( n - 1 ) > 0 ;
assume ( 1 / 2 ) * t `1 <= 1 ;
card B = k + - 1 ;
x in union rng ( f | X ) ;
assume x in the carrier of R & y in the carrier of S ;
d in dom f ;
f . 1 = L . ( F . 1 ) ;
the vertices of G = { v } & { v } c= the carrier of G ;
let G be let G be let
e , v6 be set , v be set ;
c . ( i - 1 ) in rng c & c . ( i + 1 ) in rng c ;
f2 /* q is divergent_to-infty & f2 /* q is divergent_to-infty ;
set z1 = - z2 , z2 = - z1 , z2 = - z2 , z1 = - z2 , z2 = - z1 , z2 = - z2 , z2 = - z1 , z1 = - z2 , z2
assume w is_llst S , G ;
set f = p |-count t , g = p |-count t , h = p |-count t , n = p |-count t , n = p |-count t , m = p |-count t , n = p |-count t
let c be Object of C ;
assume ex a st P [ a ] ;
let x be Element of REAL m , y be Element of REAL m ;
let IB be Subset-Family of X , I be Subset of X ;
reconsider p = p , q = q as Element of NAT ;
let v , w be Point of X ;
let s be State of SCM+FSA , P be Subset of SCM+FSA ;
p is FinSequence of ( the carrier of SCM+FSA ) * ;
stop I ( ) c= P-12 ( ) ;
set ci = f^ ( f /. i ) ;
w ^ t ^ s for w ^ s ;
W1 /\ W = W1 /\ W ` & W2 /\ W = W2 /\ W ;
f . j is Element of J . j ;
let x , y be Subset of T2 , t be Element of T ;
ex d st a , b // b , d ;
a <> 0 & b <> 0 & c <> 0 implies a = 0
ord x = 1 & x is not positive implies x is not dom a
set g2 = lim ( seq ) , g1 = lim ( seq ) ;
2 * x >= 2 * ( 1 / 2 ) ;
assume ( a 'or' c ) . z <> TRUE ;
f (*) g in Hom ( c , c ) ;
Hom ( c , c + d ) <> {} ;
assume 2 * Sum ( q | m ) > m ;
L1 . F-21 = 0 & L1 . F-21 = 0 ;
( id X \/ R1 ) \/ R1 = id X & ( id X ) * R1 = id X ;
( sin . x ) <> 0 & ( sin . x ) <> 0 ;
( for x st x in Z holds ( ( #Z n ) * ( f1 + f2 ) ) `| Z ) . x > 0
o1 in ( X /\ O2 ) /\ ( X /\ O2 ) ;
e , v6 be set , v be set ;
r3 > ( 1 / 2 ) * 0 ;
x in P .: ( F -Ideal of L ) ;
let J be closed Subset of R , left be Ideal of R ;
h . p1 = f2 . O & h . O = g2 . O ;
Index ( p , f ) + 1 <= j ;
len ( q @ ) = width M & width ( q @ ) = width M ;
the carrier of CCK c= A & the carrier of CCK c= A ;
dom f c= union rng ( F | X ) ;
k + 1 in support ( support ( n ) ) ;
let X be ManySortedSet of the carrier of S ;
[ x ` , y ] in ( an "\/" R2 ) ;
i = D1 or i = D2 or i = D1 ;
assume a mod n = b mod n & b mod n = 0 ;
h . x2 = g . x1 & h . x2 = g . x2 ;
F c= 2 -tuples_on the carrier of X & F c= 2 -tuples_on the carrier of X ;
reconsider w = |. s1 .| , e = |. s2 .| as Real_Sequence ;
( 1 / m * m + r ) < p ;
dom f = dom ( I --> ( f . x ) ) ;
[#] PPPL = [#] ( PPL ) ;
cluster - x -> ExtReal for ExtReal ;
then { d1 } c= A & A is closed ;
cluster TOP-REAL n -> finite-ind for Subset of TOP-REAL n ;
let w1 be Element of M , w2 be Element of M ;
let x be Element of dyadic ( n ) ;
u in W1 & v in W3 implies u in W2 & v in W3
reconsider y = y , z = z as Element of L2 ;
N is full SubRelStr of T |^ the carrier of S ;
sup { x , y } = c "\/" c ;
g . n = n to_power 1 .= n ;
h . J = EqClass ( u , J ) ;
let seq be summable sequence of X , n be Nat ;
dist ( x `2 , y ) < ( r / 2 ) ;
reconsider mm1 = m , mm2 = n as Element of NAT ;
( x0 - r ) < r1 - x0 & ( x0 - r ) < x0 ;
reconsider P ` = P ` as strict Subgroup of N ;
set g1 = p * ( idseq ( q `2 ) ) , g2 = p * ( idseq ( q `2 ) ) ;
let n , m , k be non zero Nat ;
assume that 0 < e and f | A is lower bounded ;
D2 . ( ID2 ) in { x } & D2 . ( ID2 ) in { x } ;
cluster subcondensed -> subopen for Subset of T ;
let P be compact non empty Subset of TOP-REAL 2 , p1 , p2 , p3 be Point of TOP-REAL 2 ;
Gik in LSeg ( PI , 1 ) /\ LSeg ( Gik , Gij ) ;
let n be Element of NAT , x be Element of NAT ;
reconsider SS = S , SS = T as Subset of T ;
dom ( i .--> X ` ) = { i } ;
let X be non-empty ManySortedSet of S ;
let X be non-empty ManySortedSet of S ;
op ( 1 ) c= { [ {} , {} ] } ;
reconsider m = mm as Element of NAT ;
reconsider d = x as Element of C ( ) ;
let s be 0 -started State of SCMPDS , P be Subset of SCMPDS ;
let t be 0 -started State of SCMPDS , Q ;
b , b , x , y , x , y be element ;
assume that i = n \/ { n } and j = k \/ { k } ;
let f be PartFunc of X , Y ;
N1 >= ( sqrt ( c / sqrt ( 2 * n ) ) ) / ( 2 * n ) ;
reconsider tT = T" as TopSpace , TT = TT as TopSpace ;
set q = h * p ^ <* d *> ;
z2 in U . ( y2 ) /\ Q2 & z2 in Q ;
A |^ 0 = { <%> E } & A |^ 0 = { <%> E } ;
len W2 = len W + 2 & len W2 = len W + 2 ;
len h2 in dom h2 & len h2 in dom h2 ;
i + 1 in Seg ( len s2 ) & i + 1 in Seg ( len s2 ) ;
z in dom g1 /\ dom f & z in dom f1 /\ dom f2 ;
assume that p2 = E-max ( K ) and p2 `2 = E-max ( K ) ;
len G + 1 <= i1 + 1 & i1 + 1 <= len G ;
f1 (#) f2 is convergent & lim ( f1 (#) f2 ) = f1 (#) f2 ;
cluster seq + ( X - seq ) -> summable for Real_Sequence ;
assume that j in dom M1 and i <= len M1 ;
let A , B , C be Subset of X , a , b , c be Point of X ;
let x , y , z be Point of X , p be Point of X ;
b ^2 - ( 4 * a * c ) >= 0 ;
<* x/y *> ^ <* y *> ^ <* y *> \geq x ;
a , b in { a , b } ;
len p2 is Element of NAT & len p2 = len p1 + 1 ;
ex x being element st x in dom R & y = R . x ;
len q = len ( K (#) G ) & len q = len G ;
s1 = Initialize ( Initialized s ) , s2 = Initialize ( Initialized s ) , P2 = P +* I ;
consider w being Nat such that q = z + w ;
x ` is Element of L & y ` is Element of L ;
k = 0 & n <> k or k > n & k > n ;
then X is discrete for X is closed ;
for x st x in L holds x is FinSequence of REAL
||. f /. c .|| <= r1 & ||. f /. c .|| <= r ;
c in uparrow p & not c in { p } ;
reconsider V = V as Subset of the topology of TOP-REAL n , Y ;
let N , M be \mathbin { means : for x being Element of L holds x in M ;
then z is_>=_than waybelow x & z >= compactbelow y ;
M \lbrack f , g .] = f & M \lbrack g , f .] = g ;
( ( ( n + 1 ) to_power 1 ) ) /. 1 = TRUE ;
dom g = dom f /\ X & dom h = X ;
mode ^ of G is ^ of W , G is : Let : G is : 1 <= k & k <= len G ;
[ i , j ] in Indices M & [ i , j ] in Indices M ;
reconsider s = x " , t = y " as Element of H ;
let f be Element of dom ( Subformulae p ) , x be Element of dom ( Subformulae p ) ;
F1 . ( a1 , b1 ) = G1 & F1 . ( b1 , b2 ) = G2 ;
redefine func Sphere ( a , b , r ) -> compact Subset of TOP-REAL 2 ;
let a , b , c , d , e , f be Real ;
rng s c= dom ( 1 / ( f . 0 ) ) ;
curry curry ( F-19 , k ) is additive & curry curry ( F-19 , k ) is additive ;
set k2 = card ( dom B ) , k1 = card ( dom B ) ;
set G = ( X , X ) --> ( x , s ) ;
reconsider a = [ x , s ] as of G ;
let a , b be Element of ML , x be Element of ML ;
reconsider s1 = s , s2 = t , s3 = t as Element of ( the carrier of S ) ;
rng p c= the carrier of L & p . ( len p ) = p . ( len p ) ;
let d be Subset of the bound of A ;
( x .|. x ) = 0 iff x = 0. W
I-21 in dom stop I & Ik in dom stop I ;
let g be continuous Function of X | B , Y ;
reconsider D = Y as Subset of TOP-REAL n , f = the carrier of TOP-REAL n ;
reconsider i0 = len p1 , i2 = len p2 as Integer ;
dom f = the carrier of S & rng f c= the carrier of T ;
rng h c= union ( ( the carrier of J ) --> { x } ) ;
cluster All ( x , H ) -> \cal \cal for element ;
d * N1 ^2 > N1 * 1 & d * N2 ^2 > N1 * 1 ;
]. a , b .[ c= [. a , b .] ;
set g = f " D1 | D1 , h = f " D2 , f = f " D2 ;
dom ( p | mm1 ) = mm1 & dom ( p | mm1 ) = mm1 ;
3 + - 2 <= k + - 2 & k + - 2 <= k + - 2 ;
tan is_differentiable_in ( arccot . x ) & tan . x > 0 ;
x in rng ( f /^ p ) & y in rng ( f /^ p ) ;
let f , g be FinSequence of D , i be Nat ;
p ( ) in the carrier of S1 & q ( ) in the carrier of S2 ;
rng f " = dom f & rng f = rng f ;
( the Source of G ) . e = v & ( the Source of G ) . e = v ;
width G -' 1 < width G & width G -' 1 < width G ;
assume v in rng ( S | E1 ) & u in rng ( S | E1 ) ;
assume x is root or x is root or x is root ;
assume that 0 in rng ( g2 | A ) and 0 < r ;
let q be Point of TOP-REAL 2 , r , s be Real ;
let p be Point of TOP-REAL 2 , p1 , p2 be Point of TOP-REAL 2 ;
dist ( O , u ) <= |. p2 .| + 1 ;
assume dist ( x , b ) < dist ( a , b ) ;
<* SS *> is_\in the carrier of C-20 & <* D *> is_\in the carrier of C-20 ;
i <= len ( G * ( i2 -' 1 , k ) ) ;
let p be Point of TOP-REAL 2 , p1 , p2 be Point of TOP-REAL 2 ;
x1 in the carrier of [: I[01] , I[01] :] & x2 in the carrier of I[01] ;
set p1 = f /. i , p2 = f /. j ;
g in { g2 : r < g2 & g2 < x0 } ;
Q2 = SL " ( Q /\ dom ( P * ) ) ;
( ( 1 / 2 ) (#) ( 1 / 2 ) ) is summable ;
- p + I c= - p + A & - p + I c= - p + I ;
n < LifeSpan ( P1 , s1 ) & I <= LifeSpan ( P1 , s1 ) ;
CurInstr ( p1 , s1 ) = i .= CurInstr ( p1 , s2 ) ;
A /\ Cl { x } \ { x } <> {} ;
rng f c= ]. r - 1 , r + 1 .[ ;
let g be Function of S , V ;
let f be Function of L1 , L2 , g be Function of L2 , L1 ;
reconsider z = z , y = y as Element of CompactSublatt L , x be Element of L ;
let f be Function of S , T ;
reconsider g = g as Morphism of c opp , b opp ;
[ s , I ] in [: S , A :] & [ s , I ] in [: S , A :] ;
len ( the connectives of C ) = 4 & len ( the connectives of C ) = 4 ;
let C1 , C2 be subfunctor of C , a be object of C1 , b be object of C2 ;
reconsider V1 = V , V1 = B as Subset of X | B ;
attr p is valid means : : : All ( x , p ) is valid ;
assume that X c= dom f and f .: X c= dom g and g .: X c= dom f ;
H |^ a " is Subgroup of H & H |^ a = H |^ a ;
let A1 be p1 of O , E1 , A2 be Element of E ;
p2 , r3 , q2 is_collinear & q2 , q3 , p1 is_collinear ;
consider x being element such that x in v ^ K and x in A ;
not x in { 0. TOP-REAL 2 } & not x in { 0. TOP-REAL 2 } ;
p in [#] ( I[01] | B11 ) & q in [#] ( I[01] | B11 ) ;
0 . ( E . n ) < M . ( Eseq . n ) ;
op ( c ) / ( c * d ) = c ;
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . s2 ) & b1 in dom ( F . s2 ) ;
cluster -> Nat -\cal ' for non empty as as as > -* ;
set i1 = the Nat , i2 = the Nat , i1 = the Nat , i2 = the Nat ;
let s be 0 -started State of SCM+FSA , P be Initialize s ;
assume y in ( f1 \/ f2 ) .: A & y in ( f1 \/ f2 ) .: A ;
f . ( len f ) = f /. len f .= f /. 1 ;
x , f . x '||' f . x , f . y ;
pred X c= Y means : : : for X st X c= Y holds cos X c= cos Y ;
let y be upper Subset of Y , x be Element of X ;
cluster -> -> -> of x `1 -> -> every \rm \rm \rm \rm \hbox { - } -valued for sequence ;
set S = <* Bags n , i<* n *> *> , T = <* <* n *> *> , S = <* n *> , T = <* n *> , S = <* n *> , T = <* n *> , T = <* n *>
set T = [. 0 , 1 / 2 .] , S = [. 0 , 1 .] ;
1 in dom mid ( f , 1 , 1 ) & len mid ( f , 1 , 1 ) = 1 ;
( 4 * PI ) / PI < ( 2 * PI ) / PI ;
x2 in dom f1 /\ dom f & x2 in dom f1 /\ dom f2 ;
O c= dom I & { {} } = { {} } ;
( the Source of G ) . x = v & ( the Source of G ) . x = v ;
{ HT ( f , T ) } c= Support f & Support ( f *' ) c= Support f ;
reconsider h = R . k as Polynomial of n , L ;
ex b being Element of G st y = b * H ;
let x , y , z be Element of G opp , a , b be Element of G ;
h19 . i = f . ( h . i ) ;
p `1 = p1 `1 & p `2 = p2 `2 or p `1 = p1 `2 & p `2 = p2 `2 ;
i + 1 <= len Cage ( C , n ) ;
len ( <* P *> @ ) = len P & len ( <* P *> @ ) = len P ;
set N-26 = the \subseteq of the \subseteq of N , Ny = the Element of N ;
len g\rrangle + ( x + 1 ) - 1 <= x ;
a on B & b on B & not b on B implies not a on B
reconsider rr = r * I . v , rr = r * I . v as FinSequence ;
consider d such that x = d and a (#) d [= c ;
given u such that u in W and x = v + u ;
len f /. ( \downharpoonright n ) = len ( n /^ n ) ;
set q2 = N-min L~ Cage ( C , n ) , q1 = N-min L~ Cage ( C , n ) , q2 = q2 ;
set S =
MaxADSet ( b ) c= MaxADSet ( P /\ Q ) & MaxADSet ( b ) c= MaxADSet ( P /\ Q ) ;
Cl ( G . q1 ) c= F . r2 & Cl ( G . q2 ) c= G . q2 ;
f " D meets h " V & f " D meets f " V ;
reconsider D = E as non empty directed Subset of L1 , D ;
H = the_left_argument_of H '&' ( the_left_argument_of H ) & H = the_left_argument_of H ;
assume t is Element of ( S , X ) * & t is Element of ( S , Y ) * ;
rng f c= the carrier of S2 & rng f c= the carrier of S2 ;
consider y being Element of X such that x = { y } ;
f1 . ( a1 , b1 ) = b1 & f1 . ( b1 , b2 ) = b2 ;
the carrier' of G ` = E \/ { E } .= { E } ;
reconsider m = len ( thesis - k ) as Element of NAT ;
set S1 = LSeg ( n , UMP C ) , S2 = LSeg ( UMP C , UMP C ) ;
[ i , j ] in Indices M1 & [ i , j ] in Indices M1 ;
assume that P c= Seg m and M is \HM { i } c= Seg m ;
for k st m <= k holds z in K . k ;
consider a being set such that p in a and a in G ;
L1 . p = p * L /. ( 1 + 1 ) ;
p-7 . i = pi1 . i & pi2 . i = pi2 . i ;
let PA , PA be a_partition of Y , a be Element of Y ;
pred 0 < r & r < 1 & 1 < r & r < 1 ;
rng ( ) = [#] ( X ) & ( ex x st x in X & x in X ) ;
reconsider x = x , y = y , z = z as Element of K ;
consider k such that z = f . k and n <= k ;
consider x being element such that x in X \ { p } ;
len ( canFS ( s ) ) = card ( s ) .= card ( rng ( s ) ) ;
reconsider x2 = x1 , y2 = x2 as Element of L2 ;
Q in FinMeetCl ( ( the topology of X ) /\ the topology of Y ) ;
dom ( f . 0 ) c= dom ( u . 0 ) ;
pred n divides m & m divides n & n = m ;
reconsider x = x , y = y , z = z as Point of [: I[01] , I[01] :] ;
a in dom
not y0 in the still of f & not ( ex y st y in the carrier of f & not y in the carrier of f ) ;
Hom ( ( a ~ ) , c ) <> {} & Hom ( ( a ~ ) , c ) <> {} ;
consider k1 such that p " < k1 and k1 < p " ;
consider c , d such that dom f = c \ d and rng f c= d ;
[ x , y ] in [: dom g , dom k :] ;
set S1 = Let ( x , y , z ) , S2 = L +* S ;
l2 = m2 & l1 = i2 & l2 = j2 implies ( ( - l1 ) * ( - l2 ) ) = ( - l2 ) * ( - l2 )
x0 in dom ( u01 ) /\ A & ( u01 ) . x0 in A ;
reconsider p = x , q = y , r = z as Point of TOP-REAL 2 ;
I[01] = R^1 | B01 & dom ( ( TOP-REAL 2 ) | B01 ) = B01 ;
f . p4 <= f . p1 & f . p2 <= f . p3 ;
( ( F . x ) `1 ) ^2 <= ( x `1 ) ^2 + ( F . x ) ^2 ;
x `2 = ( Wp ) `2 & ( Wp ) `2 = ( Wp ) `2 ;
for n being Element of NAT holds P [ n ] implies P [ n + 1 ]
let J , K be non empty Subset of I ;
assume that 1 <= i & i <= len <* a " *> and j <= len <* a *> ;
0 |-> a = <*> ( the carrier of K ) & a = <*> ( the carrier of K ) ;
X . i in 2 -tuples_on A . i \ B . i ;
<* 0 *> in dom ( e --> [ 1 , 0 ] ) ;
then P [ a ] & P [ succ a ] implies P [ succ a ] ;
reconsider s\rbrace = s\rbrace , sor sp2 = sp2 as \mathopen of D ;
( i - 1 ) <= len ( thesis ) - j ;
[#] S c= [#] ( T ) & [#] S c= [#] ( T ) ;
for V being strict RealUnitarySpace holds V in
assume k in dom mid ( f , i , j ) & k in dom mid ( f , i , j ) ;
let P be non empty Subset of TOP-REAL 2 , p1 , p2 , p3 be Point of TOP-REAL 2 ;
let A , B be Matrix of n1 , K , n , m be Nat ;
- a * - b = a * b & - a * b = - a ;
for A being Subset of AS holds A // A implies A // C
( for o2 being element st o2 in dom o2 holds o2 . ( o2 , o2 ) = <* o2 , o1 *> ) ;
then ||. x .|| = 0 & x = 0. X & x = 0. X ;
let N1 , N2 be strict normal Subgroup of G , x be Element of G ;
j >= len upper_volume ( g , D1 ) & len upper_volume ( g , D2 ) = len D2 ;
b = Q . ( len Qb - 1 + 1 ) ;
f2 * f1 /* s is divergent_to-infty & f2 * f1 is divergent_to-infty & f2 * f1 is divergent_to-infty ;
reconsider h = f * g as Function of ( f * g ) , G ;
assume that a <> 0 and delta ( a , b , c ) >= 0 ;
[ t , t ] in the InternalRel of A & [ t , t ] in the InternalRel of A ;
( v |-- E ) | n is Element of TT & ( v |-- E ) | n is Element of TT ;
{} = the carrier of L1 + L2 & the carrier of L1 c= the carrier of L2 ;
Directed I is_closed_on Initialized s , P & Directed I is_halting_on Initialized s , P & Directed I is_halting_on Initialized s , P ;
Initialized p = Initialize ( p +* q ) , p = p +* q , q = p +* q , P = Q ;
reconsider N2 = N1 , N2 = N2 as strict net of R2 , x be Element of ( the carrier of G ) ;
reconsider Y = Y as Element of [: Ids L , Ids L :] ;
"/\" ( uparrow p \ { p } , L ) <> p ;
consider j be Nat such that i2 = i1 + j and j in dom f ;
not [ s , 0 ] in the carrier of S2 & not [ s , 0 ] in the carrier of S2 ;
mm in ( B '&' C ) '/\' D \ { {} } ;
n <= len ( P + ) & len ( P + Q ) <= len ( P + Q ) ;
x1 `1 = x2 `1 & x1 `2 = y2 & x2 `2 = y2 ;
InputVertices S = { x1 , x2 } & InputVertices S = { x1 , x2 } ;
let x , y be Element of ( FT1 ) . n ;
p = |[ p `1 , p `2 ]| & p = |[ p `1 , p `2 ]| ;
g * ( 1_ G ) = h " * g * h .= h ;
let p , q be Element of Let ( V , C ) , a be Element of Let ( V , C ) ;
x0 in dom x1 /\ dom x2 & x1 - x0 < x1 - x0 & x1 - x0 < x1 - x0 ;
( R qua Function ) " = R " & ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * i ) ) ) ) ) ) ) ) = R ;
n in Seg ( len ( f /^ 1 ) ) & n in Seg ( len f -' 1 ) ;
for s being Real st s in R holds s <= s2 implies s <= s2
rng s c= dom ( f2 * f1 ) & rng ( f2 * f1 ) c= dom ( f2 * f1 ) ;
synonym ) ( X ) for for ) ( X ) \ { {} } ;
( 1. K ) * ( 1. K ) = 1. K & ( 1. K ) * ( 1. K ) = 1. K ;
set S = Segm ( A , P1 , Q1 ) , S = Segm ( A , P1 , Q1 ) , Q = Segm ( A , P1 , Q1 ) , S = Segm ( A , P1 , Q1 ) , S = Segm ( A , P1 , Q1 ) ,
ex w st e = ( w - f ) & w in F ;
curry ( PF , k ) # x is convergent & curry ( PF , k ) # x is convergent ;
cluster open -> open for Subset of [: T , T :] ;
len f1 = 1 .= len f3 .= len f3 + len f3 .= len f3 + 1 ;
( i * p ) / p < ( 2 * p ) / p ;
let x , y be Element of OSSub ( U0 ) , z be Element of S ;
b1 , c1 // b9 , c2 & b1 , c1 // c , c1 & b1 , c1 // c , c1 ;
consider p being element such that c1 . j = { p } and p in A ;
assume f " { 0 } = {} & f is total & f is total ;
assume that IC Comput ( F , s , k ) = n and IC Comput ( F , s , k ) = n ;
Reloc ( J , card I ) does not destroy a ;
( Macro ( card I + 1 ) ) does not destroy c ;
set m3 = LifeSpan ( p3 , s3 ) , P4 = LifeSpan ( p3 , s3 ) , P4 = LifeSpan ( p3 , s3 ) , P4 = p ;
IC SCMPDS in dom Initialize ( p ) & IC SCMPDS in dom Initialize ( p ) ;
dom t = the carrier of SCM R & dom t = the carrier of SCM R ;
( ( N-min L~ f ) .. f ) .. f = 1 & ( ( E-max L~ f ) .. f ) .. f = ( ( E-max L~ f ) .. f ) .. f ;
let a , b be Element of Let ( V , C ) , c be Element of V ;
Cl ( union ( Int Cl F ) ) c= Cl ( Int Cl ( Int Cl F ) ) ;
the carrier of X1 union X2 misses ( ( the carrier of X1 ) \/ ( the carrier of X2 ) ) ;
assume not LIN a , f . a , g . a ;
consider i being Element of M such that i = d6 and i in dom f ;
then Y c= { x } or Y = { x } or Y = { x } ;
M , v / ( y , x ) / ( y , x ) |= H ;
consider m being element such that m in Intersect ( FF , B ) and m in Y ;
reconsider A1 = support ( u1 ) , A2 = support ( v1 ) as Subset of X ;
card ( A \/ B ) = k-1 + ( 2 * 1 ) ;
assume that a1 <> a3 and a2 <> a4 and a3 <> a4 and a4 <> a5 ;
cluster s -\bf carrier of S -> ( ) -element for string of S ;
LG2 /. n2 = LG2 . n2 & LG2 /. n2 = LG2 . n2 ;
let P be compact non empty Subset of TOP-REAL 2 , p1 , p2 , p3 be Point of TOP-REAL 2 ;
assume that r-7 in LSeg ( p1 , p2 ) and rppc= LSeg ( p1 , p2 ) ;
let A be non empty compact Subset of TOP-REAL n , p be Point of TOP-REAL n ;
assume that [ k , m ] in Indices DD1 and [ k , m ] in Indices DD1 ;
0 <= ( ( 1 / 2 ) |^ p ) / ( p |^ n ) ;
( F . N ) | E8 . x = +infty ;
pred X c= Y & Z c= V & X \ V c= Y \ Z ;
( y `2 ) * ( z `2 ) <> 0. I & ( y `2 ) * ( z `2 ) <> 0. I ;
1 + card X-18 <= card ( u \/ { x } ) + card ( X \/ Y ) ;
set g = z :- ( ( L~ z ) .. z ) , M = z .. z , N = len z , S = L~ z , N = L~ z , N = rng z , S = L~ z , N = rng z , N = rng z , S
then k = 1 & p . k = <* x , y *> . k ;
cluster total for Element of C -\mathop { X } , D ;
reconsider B = A , C = B as non empty Subset of TOP-REAL n , a = b - a ;
let a , b , c be Function of Y , BOOLEAN , p be Function of Y , BOOLEAN ;
L1 . i = ( i .--> g ) . i .= g . i .= g . i ;
Plane ( x1 , x2 , x3 , x4 ) c= P & Plane ( x2 , x3 , x4 ) c= P ;
n <= indx ( D2 , D1 , j1 ) & indx ( D2 , D1 , j1 ) <= len D2 ;
( ( g2 ) . O ) `1 = - 1 & ( g2 ) . I = - 1 ;
j + p .. f -' len f <= len f - len f + 1 - len f ;
set W = W-bound C , E = W-bound C , N = S-bound C , S = S-bound C , N = S-bound C , N = S-bound C , S = S-bound C , N = S-bound C , N = S-bound C , S = S-bound C , N = S-bound
S1 . ( a ` , e ) = a + e .= a ` ;
1 in Seg width ( M * ( ColVec2Mx p ) ) & len ( M * ( ColVec2Mx p ) ) = n ;
dom ( i (#) Im ( f ) ) = dom Im ( f ) ;
^2 = W . ( a , *' ( a , p ) ) ;
set Q = non / ( ) , <= _ ( B , f , h ) ;
cluster -> MS[ U1 , U2 :] -> MS[ i , j ] -> MS[ i , j ] ;
attr ex A st F = { A } & F is discrete ;
reconsider z9 = \hbox { z } , z9 = the Element of product \overline G , z9 = the Element of product G ;
rng f c= rng f1 \/ rng f2 & rng f c= rng f1 \/ rng f2 ;
consider x such that x in f .: A and x in f .: C ;
f = <*> ( the carrier of F_Complex ) & f = <*> ( the carrier of F_Complex ) ;
E , j |= All ( x1 , x2 ) implies E , j |= H
reconsider n1 = n , n2 = m , n1 = n as Morphism of o1 , o2 ;
assume that P is idempotent and R is idempotent and P ** R = R ** P ;
card ( B2 \/ { x } ) = k-1 + 1 .= card B2 ;
card ( ( x \ B1 ) /\ B1 ) = 0 implies card ( x \ B1 ) = 0 ;
g + R in { s : g-r < s & s < g + r } ;
set q-19 = ( q , <* s *> ) -\subseteq ( q , <* s *> ) -\subseteq ( q , <* s *> ) -\subseteq ( q , <* s *> ) -\subseteq ( q , <* s *> ) ;
for x being element st x in X holds x in rng f1 implies x in X
h0 /. ( i + 1 ) = h0 . ( i + 1 ) ;
set mw = max ( B , dom ( ) --> NAT ) , mw = max ( B , NAT ) ;
t in Seg width ( I ^ ( n , n ) ) & t in Seg n ;
reconsider X = dom f , C = C as Element of Fin ( NAT ) ;
IncAddr ( i , k ) = <% ( l + k ) . ( x + k ) %> ;
S-bound L~ f <= q `2 & q `2 <= N-bound L~ f & q `2 <= N-bound L~ f ;
attr R is condensed means : : : for Int R being Subset of R holds Int R is condensed & Int R is condensed ;
pred 0 <= a & a <= 1 & b <= 1 implies a * b <= 1 ;
u in ( ( c /\ ( ( d /\ b ) /\ e ) ) /\ f ) /\ j ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) /\ f ) ) /\ j ;
len C + - 2 >= 9 + - 3 & len C + - 3 >= 0 ;
x , z , y is_collinear & x , z , y is_collinear & x , z , y is_collinear ;
a |^ ( n1 + 1 ) = a |^ n1 * a & a |^ n1 = a |^ n1 * a ;
<* \underbrace ( 0 , \dots , 0 } , x ) in Line ( x , a * x ) ;
set yx1 = <* y , c *> , yx2 = <* c *> , yx2 = <* d *> , yx2 = <* c *> , yx2 = <* d *> , yx2 = <* d *> , yx2 = <* c *> , yx2 = <* d *> ,
FF2 /. 1 in rng Line ( D , 1 ) & FF2 . 1 in rng Line ( D , 1 ) ;
p . m Joins r /. m , r /. ( m + 1 ) , G ;
p `2 = ( f /. i1 ) `2 & p `2 = ( f /. ( i1 + 1 ) ) `2 ;
W-bound ( X \/ Y ) = W-bound ( X \/ Y ) & W-bound ( X \/ Y ) = W-bound ( X \/ Y ) ;
0 + ( p `2 ) ^2 <= 2 * r + ( p `2 ) ^2 ;
x in dom g & not x in g " { 0 } implies x in dom g
f1 /* ( seq ^\ k ) is divergent_to-infty & f2 /* ( seq ^\ k ) is divergent_to-infty ;
reconsider u2 = u , v2 = v , u2 = w as VECTOR of C`1 , the carrier of X ;
p |-count ( Product Sgm X11 ) = 0 & p |-count ( Sgm X11 ) = 0 ;
len <* x *> < i + 1 & i + 1 <= len c + 1 ;
assume that I is non empty and { x } /\ { y } = { 0. I } ;
set ii2 = card I + 4 .--> goto 0 , ii2 = goto 0 , ii2 = goto 0 , ii2 = goto 0 , ii2 = goto 0 , ii2 = goto 0 , ii2 = goto 0 , ii2 = goto 0 , ii2 = goto 0
x in { x , y } & h . x = {} ( Tx , y ) ;
consider y being Element of F such that y in B and y <= x ` ;
len S = len ( the charact of ( ( the charact of A ) * the charact of B ) ) .= len the charact of ( A * the charact of B ) ;
reconsider m = M , i = I , n = N , m = M as Element of X ;
A . ( j + 1 ) = B . ( j + 1 ) \/ A . j ;
set N8 = : ( G-15 `1 ) \ { G } ;
rng F c= the carrier of gr { a } & F is finite & F is finite implies F is finite
implies for K being ) of [: K , n , r :] holds P is R -valued
f . k , f . ( thesis ) . ( Let n ) in rng f ;
h " P /\ [#] T1 = f " P & h " P /\ [#] T1 = {} ;
g in dom f2 \ f2 " { 0 } & f in f2 " { 0 } ;
gimplies X /\ dom f1 = g1 " X & X /\ dom f2 = dom f1 ;
consider n being element such that n in NAT and Z = G . n ;
set d1 = non empty \bf dist ( x1 , y1 ) , d2 = dist ( x2 , y2 ) , d2 = dist ( x2 , y2 ) , d2 = dist ( y2 , y2 ) , d1 = dist ( x2 , y2 ) , d2 = dist ( y2 , y2 ) , d2 =
b `2 + 1 / 2 < ( 1 - 1 ) / 2 + 1 / 2 ;
reconsider f1 = f , f2 = g as VECTOR of the carrier of X , Y ;
pred i <> 0 means : : : i ^2 mod ( i + 1 ) = 1 ;
j2 in Seg ( len ( g2 . i2 ) ) & 1 <= j2 & j2 <= len ( g2 . i2 ) ;
dom ( i + 1 ) = dom ( i + 1 ) .= dom ( i + 1 ) .= dom ( i + 1 ) ;
cluster sec | ]. PI / 2 , PI / 2 .[ -> one-to-one for Function of ]. PI / 2 , PI / 2 .[ , REAL ;
Ball ( u , e ) = Ball ( f . p , e ) & Ball ( u , e ) c= Ball ( u , r ) ;
reconsider x1 = x0 , y1 = x1 as Function of S , IS , T ;
reconsider R1 = x , R2 = y , R1 = z , R2 = w as Relation of L , L ;
consider a , b being Subset of A such that x = [ a , b ] ;
( <* 1 *> ^ p ) ^ <* n *> in RL & p in RL ;
S1 +* S2 = S2 +* S2 & S2 +* S2 = S1 +* S2 +* S2 ;
( ( - 1 ) (#) ( cos * ( f1 + f2 ) ) ) is_differentiable_on Z & for x st x in Z holds ( ( - 1 ) (#) ( sin * ( f1 + f2 ) ) ) `| Z ) . x = f . x
cluster -> [. 0 , 1 .] -valued for Function of C , REAL ;
set C7 = 1GateCircStr ( <* z , x *> , f3 ) , C8 = 1GateCircStr ( <* z , x *> , f3 ) ;
E\pi . e2 = E8 . e2 & E8 . e2 = E8 . e2 ;
( ( arctan * ( ln * ( f1 + f2 ) ) ) `| Z ) = f ;
upper_bound A = ( PI * 3 ) / 2 & lower_bound A = 0 & lower_bound A = 0 ;
F . ( dom f , - F . ( cod f ) ) is_transformable_to F . ( cod f , - F . ( cod f ) ) ;
reconsider pNAT = qNAT , pn2 = qn2 as Point of TOP-REAL 2 ;
g . W in [#] Y0 & [#] Y0 c= [#] Y1 & g . W in [#] Y1 ;
let C be compact non vertical non horizontal Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
LSeg ( f ^ g , j ) = LSeg ( f , j ) /\ LSeg ( g , i ) ;
rng s c= dom f /\ ]. -infty , x0 + r .[ & rng s c= dom f /\ ]. x0 , x0 + r .[ ;
assume x in { idseq ( 2 ) , Rev ( idseq ( 2 ) ) } ;
reconsider n2 = n , m2 = m , n1 = n , n2 = m as Element of NAT ;
for y being ExtReal st y in rng seq holds g <= y & y <= g ;
for k st P [ k ] holds P [ k + 1 ] ;
m = m1 + m2 .= m1 + m2 .= m1 + m2 .= m1 + m2 ;
assume for n holds H1 . n = G . n -H . n ;
set Bf = f .: ( the carrier of X1 ) , Bf = f .: ( the carrier of X2 ) ;
ex d being Element of L st d in D & x << d ;
assume that R -Seg ( a ) c= R -Seg ( b ) and R -Seg ( a ) c= R -Seg ( b ) ;
t in ]. r , s .] or t = r or t = s or t = s ;
z + v2 in W & x = u + ( z + v2 ) ;
x2 |-- y2 iff P [ x2 , y2 ] & P [ x2 , y2 ] ;
pred x1 <> x2 means : : : for x1 , x2 st x1 - x2 > 0 & x1 - x2 < 0 holds |. x1 - x2 .| > 0 ;
assume that p2 - p1 , p3 - p1 - p1 , p3 - p1 - p1 , p3 - p1 - p3 - p1 , p3 - p1 - p3 - p1 is_collinear ;
set q = ( U * f ) ^ <* 'not' A *> ;
let f be PartFunc of REAL-NS 1 , REAL-NS n , x be Point of REAL-NS 1 , r be Real ;
( n mod ( 2 * k ) ) + 1 = n mod k ;
dom ( T * ( succ t ) ) = dom ( T * ( succ t ) ) ;
consider x being element such that x in wc iff x in c & x in X ;
assume ( F * G ) . ( v . x3 ) = v . ( x4 . x3 ) ;
assume that the carrier of D1 c= the carrier of D2 and the carrier of D2 c= the carrier of D1 ;
reconsider A1 = [. a , b .[ , A2 = [. a , b .] as Subset of R^1 ;
consider y being element such that y in dom F and F . y = x ;
consider s being element such that s in dom o and a = o . s ;
set p = W-min L~ Cage ( C , n ) , q = W-min L~ Cage ( C , n ) , r = W-bound L~ Cage ( C , n ) , s = W-bound L~ Cage ( C , n ) , w = W-bound L~ Cage ( C , n ) , G = Gauge ( C , n ) , G = Gauge (
n1 -' len f + 1 <= len ( - 1 ) + 1 - len f + 1 ;

set C-2 = ( ( `1 ) `1 ) . ( k + 1 ) ;
Sum ( L (#) p ) = 0. R * Sum p .= 0. V .= 0. V ;
consider i being element such that i in dom p and t = p . i ;
defpred Q [ Nat ] means 0 = Q ( ) & $1 <= n implies f . $1 = g . $1 ;
set s3 = Comput ( P1 , s1 , k ) , P3 = P1 , s4 = P1 , P4 = P1 , P4 = P1 , P4 = P1 , P4 = P2 , P4 = Comput ( P2 , s2 , k + 1 ) , P4 = P2 , P4 = P2 , P4 = Comput ( P3 ,
let l be and let A be Subset of k , A1 , A2 be Subset of k ;
reconsider U2 = union ( G-24 ) , U2 = union ( G-24 ) as Subset-Family of TS ;
consider r such that r > 0 and Ball ( p `2 , r ) c= Q ` ;
( h | ( n + 2 ) ) /. ( i + 1 ) = p29 ;
reconsider B = the carrier of X1 , C = the carrier of X2 as Subset of X ;
p$ = <* - c9 , 1 , 1 *> & pcnon empty ;
synonym f is real-valued means : \lbrace f , g .[ c= ( dom f ) \ { f } ;
consider b being element such that b in dom F and a = F . b ;
x10 < card X0 + card Y0 & x10 < card X0 + card X0 & x10 < card X0 + card X0 ;
pred X c= B1 means : : : for B1 st X c= B1 holds \mathop { \rm _ _ X } c= B1 ;
then w in Ball ( x , r ) & dist ( x , w ) <= r ;
angle ( x , y , z ) = angle ( x-y , 0 , 0 , 2 ) ;
pred 1 <= len s means : : : for i being Nat holds ( for s being Element of NAT holds s . i = s . i ) ;
f(#) f c= f . ( k + ( n + 1 ) ) ;
the carrier of { 1_ G } = { 1_ G } & the carrier of G c= the carrier of G ;
pred p '&' q in \cdot ( L => p ) means : : : q '&' p in * ( L => p ) ;
- ( t `1 ) < ( t `2 ) / ( 1 + t `2 ) ;
U2 . 1 = U2 /. 1 .= ( U2 /. 1 ) . 1 .= ( ( U2 . 1 ) * ( U2 . 1 ) ) ;
f .: ( the carrier of x ) = the carrier of x & f .: ( the carrier of x ) = the carrier of x ;
Indices ( O @ ) = [: Seg n , Seg n :] & dom ( ( n , n ) --> ( n , 1 ) ) = Seg n ;
for n being Element of NAT holds G . n c= G . ( n + 1 ) ;
then V in M @ implies ex x being Element of M st V = { x } ;
ex f being Element of F-9 st f is unital & f is <> 0. K & f is invertible ;
[ h . 0 , h . 3 ] in the InternalRel of G & [ h . 0 , h . 3 ] in the InternalRel of G ;
s +* Initialize ( ( intloc 0 ) .--> 1 ) = s3 +* Initialize ( ( intloc 0 ) .--> 1 ) ;
|[ w1 , v1 ]| Let ( - 1 ) * ( |[ w1 , v1 ]| ) <> 0. TOP-REAL 2 ;
reconsider t = t , s = ( t , s ) / ( t , s ) as Element of INT ;
C \/ P c= [#] ( GX | ( [#] GX \ A ) ) & C c= C ;
f " V in ( the topology of X ) /\ D & f " ( the carrier of X ) = D ;
x in [#] ( ( the carrier of A ) /\ A ) /\ ( the carrier of B ) ;
g . x <= h1 . x & h . x <= h1 . x & h . x <= h . x ;
InputVertices S = { xy , y , z } & InputVertices S = { xy , y , z } & InputVertices S = { xy , y , z } ;
for n being Nat st P [ n ] holds P [ n + 1 ] ;
set R = being Matrix of i , a , a * Line ( M , i ) , M = Line ( M , i ) , N = Line ( M , i ) , S = Line ( M , i ) , S = Line ( M , i ) , M = Line ( M , i ) , S =
assume that M1 is being_line and M2 is being_line and M3 is being_line and M3 is being_line and M3 is being_line and M3 is being_line ;
reconsider a = f4 . ( i0 -' 1 ) , b = f4 . ( i0 -' 1 ) as Element of K ;
len B2 = Sum ( Len F1 ^ F2 ) & width B2 = len ( Len F1 ^ width F2 ) ;
len ( ( the X of n ) --> i ) = n & len ( ( the X of n ) --> i ) = n ;
dom max ( - ( f + g ) , f ) = dom ( f + g ) ;
( the Sorts of seq ) . n = upper_bound Y1 & ( for n holds seq . n = upper_bound Y1 ) implies seq is convergent & lim seq = upper_bound Y1
dom ( p1 ^ p2 ) = dom f12 & dom ( p1 ^ p2 ) = dom f12 ;
M . [ 1 / y , y ] = 1 / ( 1 * v1 ) * v1 .= y ;
assume that W is non trivial and W .vertices() c= the carrier' of G2 and not W is trivial and not W is trivial ;
C6 * ( i1 , i2 ) = G1 * ( i1 , i2 ) & C6 * ( i1 , i2 ) = G1 * ( i1 , i2 ) ;
C8 |- 'not' Ex ( x , p ) 'or' p . ( x , y ) ;
for b st b in rng g holds inf rng f\rbrace <= b & b <= sup rng f\rbrace ;
- ( ( q1 `1 ) / |. q1 .| ) = 1 & ( q1 `2 ) / |. q1 .| = 1 ;
( LSeg ( c , m ) \/ ( REAL \/ { l } ) ) \/ LSeg ( l , k ) c= R ;
consider p be element such that p in Ball ( x , r ) and p in L~ f ;
Indices ( X @ ) = [: Seg n , Seg n :] & dom ( X @ ) = Seg n ;
cluster s => ( q => p ) => ( q => ( s => p ) ) -> valid ;
Im ( ( Partial_Sums F ) . m ) is_measurable_on E & Im ( ( Partial_Sums F ) . m ) is_measurable_on E ;
cluster f . ( x1 , x2 ) -> ( ) -valued for Element of D ;
consider g being Function such that g = F . t and Q [ t , g ] ;
p in LSeg ( N-min L~ Cage ( C , n ) , NW-corner L~ Cage ( C , n ) ) /\ LSeg ( NW-corner , p2 ) ;
set R8 = R / 1 , R8 = ]. b , +infty .[ , R8 = ]. b , +infty .[ , R8 = ]. b , +infty .[ , R8 = ]. b , +infty .[ , R8 = ]. b , +infty .[ ;
IncAddr ( I , k ) = SubFrom ( da , da ) .= SubFrom ( da , da ) .= goto ( ( card I + k ) + k ) ;
seq . m <= ( the Sorts of ( seq ^\ k ) ) . ( n + k ) ;
a + b = ( a ` *' b ) ` & ( a ` *' b ) ` = ( a ` *' b ) ` ;
id ( X /\ Y ) = id ( X /\ Y ) .= id ( X /\ Y ) ;
for x being element st x in dom h holds h . x = f . x
reconsider H = U1 \/ U2 , U2 = U2 as non empty Subset of U0 , U1 = ( U1 /\ U2 ) /\ U2 ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) /\ f ) ) /\ j /\ m ;
consider y being element such that y in Y and P [ y , inf B ] ;
consider A being finite stable set of R such that card A = ( card the carrier of R ) and card A = card A ;
p2 in rng ( f |-- p1 ) \ rng <* p1 *> & p2 in rng <* p1 *> ;
len s1 - 1 > 1-1 & len s2 - 1 > 0 & len s2 - 1 > 0 ;
( N-min ( P ) ) `2 = N-bound ( P ) & ( N-min ( P ) ) `2 = N-bound ( P ) ;
Ball ( e , r ) c= LeftComp Cage ( C , k + 1 ) & Ball ( e , r ) c= LeftComp Cage ( C , k + 1 ) ;
f . a1 ` = f . a1 ` & f . a2 = f . a2 ` ;
( seq ^\ k ) . n in ]. -infty , x0 + r .[ & ( seq ^\ k ) . n in ]. x0 , x0 + r .[ ;
gg . s0 = g . s0 | G . s0 .= g . s0 .= ( g | G ) . s0 ;
the InternalRel of S is non empty & the InternalRel of S is Relation of the carrier of S & the InternalRel of S is non empty ;
deffunc F ( Ordinal , Ordinal ) = phi . ( $1 + 1 ) & phi . ( $2 ) = phi . ( $2 ) ;
F . a1 = F . a2 & F . a2 = F . a1 & F . a2 = F . a2 ;
x `2 = A . ( o , a ) .= Den ( o , A . a ) ;
Cl ( f " P1 ) c= f " ( Cl P1 ) & Cl ( f " P1 ) c= f " ( Cl P1 ) ;
FinMeetCl ( ( the topology of S ) \ { x } ) c= the topology of T & FinMeetCl ( ( the topology of S ) \ { x } ) c= the topology of T ;
synonym o is \bf means : : : o <> \ast & o <> * & o <> * ;
assume that X = Y + Z and card X <> card Y and card Y <> card Z and card X <> card Z ;
the *> of s <= 1 + ( the *> of s ) & the { F ( ) . ( s , 1 ) = ( the { F ( ) . ( s , 1 ) ) ;
LIN a , a1 , d or b , c // b1 , c1 or a , c // c1 , b1 ;
e / 2 . 1 = 0 & e / 2 . 3 = 1 & e / 2 . 3 = 0 ;
ES1 in SS1 & not ES1 in { NS1 } & not ES1 in SS2 ;
set J = ( l , u ) ] ;
set A1 = Let ( a9 , b9 , c , d ) , A2 = [ b9 , c , d ] , A2 = [ a9 , b9 , c , d ] ;
set vs = [ <* c , d *> , '&' ] , C = [ <* d , c *> , '&' ] , D = [ <* c , d *> , '&' ] , E = [ <* d , c *> , '&' ] , F = [ <* c , d *> , '&' ] , D = [ <* d , c *> , '&' ] , E =
x * z `2 * x " in x * ( z * N ) * x " ;
for x being element st x in dom f holds f . x = g3 . x & f . x <> 0 ;
Int cell ( f , 1 , G ) c= RightComp f \/ L~ f & not ( f /. 1 ) in L~ f ;
U2 is_an_arc_of W-min C , E-max C & P = W-min C implies P = W-min C or P = W-min C & P = W-min C
set f-17 = f @ "/\" @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
attr S1 is convergent means : : : for S2 st S2 is convergent holds S1 - S2 is convergent & lim ( S1 - S2 ) = 0 ;
f . ( 0 + 1 ) = ( 0 qua Ordinal ) + a .= a + ( 0 qua Ordinal ) .= a + ( 0 qua Ordinal ) .= a + ( 0 qua Ordinal ) ;
cluster -> \in be \in be \in be \in be \in be reflexive transitive transitive non empty reflexive transitive RelStr , F be symmetric non empty strict RelStr ;
consider d being element such that R reduces b , d and R reduces c , d and R reduces d , c ;
not b in dom Start-At ( ( card I + 2 ) , SCMPDS ) & not b in dom Start-At ( ( card I + 2 ) , SCMPDS ) ;
( z + a ) + x = z + ( a + y ) .= z + a + y ;
len ( l \lbrack ( a - x ) / ( 0 , 1 ) ) ) = len l ;
t4 } is ( {} \/ rng t4 ) -valued ( {} , rng t4 ) -valued finite Function ;
t = <* F . t *> ^ ( C . p ) ^ ( C . q ) .= ( C . p ) ^ q ;
set p-2 = W-min L~ Cage ( C , n ) , p`2 = W-min L~ Cage ( C , n ) , p`2 = W-bound L~ Cage ( C , n ) , p`2 = W-bound L~ Cage ( C , n ) , p`2 = W-bound L~ Cage ( C , n ) , p`2 = W-bound L~ Cage ( C , n )
( k -' ( i + 1 ) ) = ( k - ( i + 1 ) ) + ( i + 1 ) ;
consider u being Element of L such that u = u ` and u in D ` and u in D ;
len ( ( width aG ) |-> a ) = width ( aG ) & width ( ( width aG ) |-> a ) = width ( a * ( width A ) ) ;
FM . x in dom ( ( G * the_arity_of o ) . x ) & FM . x in dom ( G * the_arity_of o ) ;
set cH2 = the carrier of H2 , cH2 = the carrier of H1 ;
set cH1 = the carrier of H1 , cH2 = the carrier of H2 , cH2 = the carrier of H2 ;
( Comput ( P , s , 6 ) ) . intpos m = s . intpos m & Comput ( P , s , 6 ) . intpos m = s . intpos m ;
IC Comput ( Q2 , t , k ) = ( l + 1 ) .= ( l + 1 ) + 1 ;
dom ( ( cos * ( sin - cos ) ) `| REAL ) = REAL & dom ( ( cos * ( sin - cos ) ) `| REAL ) = REAL ;
cluster <* l *> ^ phi -> ( 1 + ( 1 + ( not m ) ) ) -element for string of S ;
set b5 = [ <* a7 , c8 *> , [ <* A1 , cin *> , '&' ] , b6 = [ <* cin , cin *> , '&' ] , b7 = [ <* A1 , cin *> , '&' ] , b8 = [ <* cin , cin *> , '&' ] ;
Line ( Segm ( M @ , P , Q ) , x ) = L * Sgm Q .= Line ( M , i ) ;
n in dom ( ( the Sorts of A ) * the_arity_of o ) & ( ( the Sorts of A ) * the_arity_of o ) . n = ( the Sorts of A ) . ( ( the Arity of S ) . o ) ;
cluster f1 + f2 -> continuous for PartFunc of REAL , the carrier of S , the carrier of T ;
consider y be Point of X such that a = y and ||. x-y - x .|| <= r ;
set x3 = t2 . DataLoc ( ( 8 + 2 ) , 2 ) , x2 = s . SBP , x3 = s . SBP , x4 = s . SBP , x4 = s . SBP , 7 = s . SBP , 6 = s . SBP , 7 = s . SBP , 8 = s . SBP , 8 = s . SBP
set p-3 = stop I ( ) , pI = stop I ( ) , pI = Initialize s2 , pI = Initialize s2 , pI = Initialize s2 , pI = Initialize s2 , pI = Initialize s2 , pI = Initialize s2 , pI = Initialize s2 , pI = P ( ) , , pI
consider a being Point of D2 such that a in W1 and b = g . a and a in W1 ;
{ A , B , C , D , E } = { A , B } \/ { C , D , E }
let A , B , C , D , E , F , J , M , N , M , N , N , M , N , N , M , N , N , M , N , N , N , M , N , N , M , N , N , N , M , N , N , M , N ,
|. p2 .| ^2 - ( p2 `2 ) ^2 - ( p2 `1 ) ^2 >= 0 & |. p2 .| ^2 - ( p2 `2 ) ^2 >= 0 ;
l - 1 + 1 = n-1 * ( l + 1 ) + ( \setminus { x } ) + 1 ;
x = v + ( a * w1 + ( b * w2 ) ) + ( c * w2 ) + ( c * w2 ) ;
the TopStruct of L = , the TopStruct of L , the TopStruct of S = the TopStruct of L , C = the Scott F of S , D = the TopStruct of S ;
consider y being element such that y in dom H1 and x = H1 . y and y in ( H1 . x ) ;
fv \ { n } = \mathop { \rm Free ( v1 , H ) } & fv \ { n } = Free ( v1 , H ) ;
for Y being Subset of X st Y is summable holds Y is summable iff Y is number
2 * n in { N : 2 * Sum ( p | N ) = N & N > 0 } ;
for s being FinSequence holds len ( the { - } \rm <* s *> ) = len s & len ( the { - s } ) = len s
for x st x in Z holds exp_R * f is_differentiable_in x & ( exp_R * f ) . x > 0
rng ( h2 * f2 ) c= the carrier of ( ( TOP-REAL 2 ) | K1 ) & rng ( h2 * f1 ) c= the carrier of ( ( TOP-REAL 2 ) | D ) ;
j + ( len f ) - len f <= len f + ( len g ) - len f ;
reconsider R1 = R * I , R2 = R * I , R1 = ( id REAL ) * I , R2 = ( id REAL ) * I , R2 = ( id REAL ) * I ;
C8 . x = s1 . ( x0 - 1 ) .= C8 . x .= C8 . x ;
power ( F_Complex ) . ( z , n ) = 1 .= ( x |^ n ) |^ ( n -' 1 ) .= x |^ n ;
t at ( C , s ) = f . ( ( the connectives of S ) . t ) & ( the connectives of S ) . t = s ;
support ( f + g ) c= support f \/ support g & support ( f + g ) c= support f \/ support g ;
ex N st N = j1 & 2 * Sum ( seq1 | N ) > N & for i st i in N holds ( ( seq1 | N ) . i ) > 0 ;
for y , p st P [ p ] holds P [ All ( y , p ) ] ;
{ [ x1 , x2 ] where x1 is Point of [: X1 , X2 :] : x1 in X } is Subset of [: X1 , X2 :] ;
h = ( i , j ) |-- h , id B = H . i .= H . i ;
ex x1 being Element of G st x1 = x & x1 * N c= A & N c= A ;
set X = ( ( \lbrace q , O1 } ) `1 ) , Y = ( ( \lbrace q , O1 } ) `2 ) , Z = { ( q , O1 ) `2 } ;
b . n in { g1 : x0 < g1 & g1 < a1 . n } & ( for n holds a1 . n < x0 ) ;
f /* s1 is convergent & f /. x0 = lim ( f /* s1 ) & f /. x0 = lim ( f /* s1 ) ;
the lattice of Y = the lattice of the lattice of the open of Y & the carrier of X c= the carrier of X & the carrier of Y c= the carrier of X ;
'not' ( a . x ) '&' b . x 'or' a . x '&' 'not' ( b . x ) = FALSE ;
q2 = len ( q0 ^ r1 ) + len q1 & q1 = ( q0 ^ q1 ) ^ ( q2 ^ q1 ) ;
( 1 / a ) (#) ( sec * f1 ) - id Z is_differentiable_on Z & ( ( 1 / a ) (#) ( sec * f1 ) ) is_differentiable_on Z ;
set K1 = integral ( ( lim ( lim ( H , A ) ) || A1 ) , K1 = integral ( H , A ) || A1 , K1 = integral ( H , A ) || A1 ) , K1 = dom ( H , A ) || A1 ;
assume e in { ( w1 - w2 ) / ( w1 - w2 ) : w1 in F & w2 in G } ;
reconsider da = dom a `1 , db = dom F `1 , db = dom F `1 , db = dom G `1 , db = dom G `1 , db = dom G `1 , db = dom G `1 , db = dom F `1 , db = dom G `1 , db = dom G `1 , db = dom G `1
LSeg ( f /^ q , j ) = LSeg ( f , j ) /\ q .= LSeg ( f , j + q .. f ) ;
assume X in { T . ( N2 , K1 ) : h . ( N2 , K1 ) = N2 } ;
assume that Hom ( d , c ) <> {} and <* f , g *> * f1 = <* f , g *> * f2 and f * f1 = <* f , g *> ;
dom Sb = dom S /\ Seg n .= dom Lb .= dom Lb .= Seg n /\ Seg n .= dom ( Lb | Seg n ) ;
x in H |^ a implies ex g st x = g |^ a & g in H & a in H
* ( ( a , 1 ) * n ) = a `2 - ( 0 * n ) .= a `2 - ( 0 * n ) .= a `2 ;
D2 . j in { r : lower_bound A <= r & r <= D1 . i } ;
ex p being Point of TOP-REAL 2 st p = x & P [ p ] & p `2 >= 0 & p <> 0. TOP-REAL 2 ;
for c holds f . c <= g . c implies f @ = g @ @ c
dom ( f1 (#) f2 ) /\ X c= dom ( f1 (#) f2 ) /\ X & dom ( f1 (#) f2 ) /\ X c= dom ( f1 (#) f2 ) ;
1 = ( p * p ) / p .= p * ( p / p ) .= 1 * p ;
len g = len f + len <* x + y *> .= len f + 1 .= len f + 1 .= len f + 1 ;
dom F-11 = dom ( F | ( N1 /\ S-23 ) ) .= ( the carrier of S1 ) /\ the carrier of S2 ;
dom ( f . t ) * I . t = dom ( f . t ) * g . t ;
assume a in ( "\/" ( ( ( T |^ the carrier of S ) ) .: D , S ) ) .: D ;
assume that g is one-to-one and ( the carrier' of S ) /\ rng g c= dom g and g is one-to-one and g is one-to-one ;
( ( x \ y ) \ z ) \ ( ( x \ z ) \ ( y \ z ) ) = 0. X ;
consider f such that f * f `1 = id b and f * f `2 = id a and f is one-to-one and f is one-to-one ;
( cos | [. 2 * PI * 0 , PI + ( 2 * PI * 0 ) .] ) is increasing ;
Index ( p , co ) <= len LS - Gij .. LS & Index ( Gij , LS ) <= len LS - Gij .. LS ;
let t1 , t2 , t3 be Element of ( T , s ) . ( NAT + 1 ) , t be Element of ( T , s ) . ( NAT + 1 ) ;
"/\" ( ( Frege ( curry H ) ) . h , L ) <= "/\" ( ( Frege ( curry G ) ) . h , L ) ;
then P [ f . i0 ] & F ( f . ( i0 + 1 ) ) < j & j < len f ;
Q [ ( [ D . x , 1 ] ) `1 , F ( [ D . x , 1 ] ) `2 ] ;
consider x being element such that x in dom ( F . s ) and y = ( F . s ) . x ;
l . i < r . i & [ l . i , r . i ] is for of G . i ;
the Sorts of A2 = ( the carrier of S2 ) --> ( the carrier of S2 ) .= ( the carrier' of S1 ) --> ( the carrier' of S2 ) ;
consider s being Function such that s is one-to-one and dom s = NAT and rng s = F and rng s c= F ;
dist ( b1 , b2 ) <= dist ( b1 , a ) + dist ( a , b2 ) & dist ( a , b1 ) <= dist ( a , b2 ) + dist ( b , b1 ) ;
( Lower_Seq ( C , n ) /. len Lower_Seq ( C , n ) ) /. len Upper_Seq ( C , n ) = Wq ;
q `2 <= ( UMP Upper_Arc C ) `2 & ( UMP C ) `2 <= ( UMP C ) `2 ;
LSeg ( f | i2 , i ) /\ LSeg ( f | i2 , j ) = {} & LSeg ( f | i2 , j ) /\ LSeg ( f , i ) = {} ;
given a being ExtReal such that a <= IB and A = ]. a , IB .] and a <= sup IB ;
consider a , b be complex number such that z = a & y = b and z + y = a + b ;
set X = { b |^ n where n is Element of NAT : n <= m & b |^ n in X } , Y = { b |^ n where n is Element of NAT : n <= m } ;
( ( x * y * z \ x ) \ z ) \ ( x * y \ x ) = 0. X ;
set xy = [ <* xy , y *> , f1 ] , yz = [ <* y , z *> , f2 ] , zx = [ <* z , x *> , f3 ] , zx = [ <* z , x *> , f3 ] , zx = [ <* x , y *> , f3 ] , zx = [ <* z , x *> , f3 ] , zx = [ <* z , x *>
( l /. len ( l ) ) = ( l . ( len ( l ) ) ) .= ( l . ( len ( l ) ) ) ;
( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 = 1 ;
( ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) * ( 1 + sn ) < 1 ;
( ( ( S \/ Y ) `2 ) /\ ( ( S \/ Y ) `2 ) ) = ( ( S \/ Y ) `2 ) /\ ( ( S \/ Y ) `2 ) ;
( seq - seq ) . k = seq . k - seq . k & ( seq - seq ) . k = ( seq . k - seq . k ) - ( seq . k ) ;
rng ( ( h + c ) ^\ n ) c= dom SVF1 ( 1 , f , u0 ) /\ dom SVF1 ( 1 , f , u0 ) ;
the carrier of X = the carrier of X0 & the carrier of X = the carrier of X0 & the carrier of X = the carrier of X0 ;
ex p4 st p3 = p4 & |. p4 - |[ a , b ]| .| = r & |. p4 - |[ a , b ]| .| = r ;
set ch = chi ( X , A ) , Ah = chi ( X , A ) , Ah = chi ( X , A ) ;
R / ( 0 * n ) = I\HM ( X , X ) .= R / ( n * 0 ) .= R / ( n * 0 ) ;
( Partial_Sums ( curry ( F-19 , n ) ) ) . n is nonnegative & ( Partial_Sums ( ( curry ( F-19 , n ) ) ) . n ) is nonnegative ;
f2 = CK . ( EK , len ( EK ) ) .= CK . ( EK , len ( EK ) ) ;
S1 . b = s1 . b .= S2 . ( b ) .= S2 . ( b ) .= S2 . ( b ) .= S2 . ( b ) ;
p2 in LSeg ( p2 , p1 ) /\ LSeg ( p2 , p1 ) /\ LSeg ( p1 , p2 ) & p2 in LSeg ( p2 , p1 ) /\ LSeg ( p1 , p2 ) ;
dom ( f . t ) = Seg n & dom ( I . t ) = Seg n & rng ( I . t ) c= Seg n ;
assume o = ( the connectives of S ) . 11 & ( the connectives of S ) . 11 in ( the carrier' of S ) . 11 ;
set phi = ( l1 , l2 ) implies ( X , l2 ) implies ( X , l2 ) is ( X , l2 ) ) implies X is ( X , l2 ) )
synonym p is invertible for ( p , T ) is invertible & ( p is invertible implies p is invertible ) ;
Y1 `2 = - 1 & 0. TOP-REAL 2 = - 1 & 0. TOP-REAL 2 = ( - 1 ) * ( 1 + ( - 1 ) ) & 0. TOP-REAL 2 = ( - 1 ) * ( 1 + ( - 1 ) ) ;
defpred X [ Nat , set , set ] means P [ $1 , $2 , , , ] & $2 = [ $1 , $2 , ] ;
consider k be Nat such that for n be Nat st k <= n holds s . n < x0 + g and x0 < x0 + g ;
Det ( I @ ) ~ = ( ( m - n ) * ( m - n ) ) @ .= ( 1. K ) * ( m - n ) ;
( - b ) / sqrt ( b ^2 - 4 * a * c ) < 0 ;
Cseq . d = Cseq . da mod Cseq . da & Cseq . d = Cseq . da mod Cseq . da ;
attr X1 is dense means : : : X1 is dense & X2 is dense & X1 is dense & X2 is dense & X1 is dense & X2 is dense ;
deffunc F6 ( Element of E , Element of I ) = $1 * $2 & $2 = ( $1 * $2 ) * ( $2 ) ;
t ^ <* n *> in { t ^ <* i *> : Q [ i , T ( ) ] } ;
( x \ y ) \ x = ( x \ x ) \ y .= y ` \ y .= 0. X .= 0. X ;
for X being non empty set for Y being Subset-Family of X holds for F being Subset-Family of X holds F is Basis of [: X , Y :]
synonym A , B are_separated means : : : for B being Subset of X st B misses A & B misses B holds A misses B ;
len ( M @ ) = len p & width ( M @ ) = width ( M @ ) & len ( M @ ) = width ( M @ ) ;
J . v = { x where x is Element of K : 0 < v . x & x < 1 } ;
( Sgm ( \bf m ) ) . d - ( Sgm ( \bf m ) ) . e <> 0 ;
lower_bound divset ( D2 , k + k2 ) = D2 . ( k + k2 - 1 ) .= D2 . ( k + k2 - 1 ) ;
g . r1 = - 2 * r1 + 1 & dom h = [. 0 , 1 .] & rng h c= [. 0 , 1 .] ;
|. a .| * ||. f .|| = 0 * ||. f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| ;
f . x = ( h . x ) `1 & g . x = ( h . x ) `2 & h . x = ( h . x ) `2 ;
ex w st w in dom B1 & <* 1 *> ^ s = <* 1 *> ^ w & len w = len A & width A = len B ;
[ 1 , {} , <* d1 *> ] in ( { [ 0 , {} , {} ] } \/ S1 ) \/ S2 ;
IC Exec ( i , s1 ) + n = IC Exec ( i , s2 ) .= IC Exec ( i , s2 ) .= IC Exec ( i , s2 ) ;
IC Comput ( P , s , 1 ) = IC ( s , 9 ) .= ( 5 + 9 ) .= 5 ;
( IExec ( W6 , Q , t ) ) . intpos ( e + 1 ) = t . intpos ( e + 1 ) ;
LSeg ( f /^ q , i ) misses LSeg ( f /^ q , j ) & LSeg ( f /^ q , j ) misses LSeg ( f /^ q , j ) ;
assume for x , y being Element of L st x in C holds x <= y or y <= x ;
integral ( f , C ) . x = f . ( upper_bound C ) - f . ( lower_bound C ) .= f . ( lower_bound C ) - f . ( lower_bound C ) ;
for F , G being one-to-one FinSequence st rng F misses rng G holds F ^ G is one-to-one & F ^ G is one-to-one
||. R /. ( L . h ) .|| < e1 * ( K + 1 * ||. h .|| ) ;
assume a in { q where q is Element of M : dist ( z , q ) <= r } ;
set p4 = [ 2 , 1 ] .--> [ 2 , 0 , 1 ] ;
consider x , y being Subset of X such that [ x , y ] in F and x c= d and y c= d ;
for y , x being Element of REAL st y in Y ` & x in X holds y <= x ` & y <= x ` ;
func |. p .| -> variable of A means : : : for p being variable of A holds it . p = min ( NBI ( p ) , p ) ;
consider t being Element of S such that x `1 , y `2 '||' z `1 , t `2 and x `2 '||' y `1 , t `2 ;
dom x1 = Seg ( len x1 ) & len x1 = len l1 & for i st i in Seg ( len x1 ) holds x1 . i = ( x1 /. i ) * ( x1 /. i ) ;
consider y2 being Real such that x2 = y2 and 0 <= y2 and y2 < 1 / 2 and y2 <= 1 / 2 ;
||. f | X /* s1 .|| = ||. f .|| | X & ||. f /. s1 .|| = ||. f .|| /. ( s1 . ( s1 . n ) ) - f /. ( s1 . n ) .|| ;
( the InternalRel of A ) ` /\ Y = {} \/ {} .= {} \/ {} .= {} \/ {} .= {} \/ {} .= {} ;
assume that i in dom p and for j being Nat st j in dom q holds P [ i , j ] and i + 1 in dom p and p . i = q . j ;
reconsider h = f | X ( ) , g = f | X ( ) as Function of X ( ) , rng f , Y ( ) ;
u1 in the carrier of W1 & u2 in the carrier of W2 implies ( for v st v in the carrier of W1 holds ( v in the carrier of W2 ) ) & ( v in the carrier of W1 )
defpred P [ Element of L ] means M <= f . $1 & f . $1 <= f . ( $1 + 1 ) & f . ( $1 + 1 ) <= f . ( $1 + 1 ) ;
^ ( u , a , v ) = s * x + ( - ( s * x ) + y ) .= b * x + y .= b ;
- ( x-y ) = - x + - ( - y ) .= - x + - y .= - x + y .= - ( x + y ) ;
given a being Point of GX such that for x being Point of GX holds a , x are_\HM { a } and a , x are_\HM { a } ;
fJ = [ [ dom ( @ f2 ) , cod ( @ g2 ) ] , h2 ] .= [ [ cod ( @ g2 ) , cod ( @ g2 ) ] , h2 ] ;
for k , n being Nat st k <> 0 & k < n & n is prime holds k , n are_relative_prime & k , n are_relative_prime
for x being element holds x in A |^ d iff x in ( ( A ` ) |^ d ) ` & x in A ` ;
consider u , v being Element of R , a being Element of A such that l /. i = u * a * v and a in I ;
( - ( p `1 / |. p .| - cn ) ) / ( 1 + cn ) > 0 ;
L-13 . k = Ln . k & F . k in dom Ln & F . k = Ln . k ;
set i2 = AddTo ( a , i , - n ) , i1 = goto ( card I + 1 ) , i2 = goto ( card I + 2 ) , i2 = goto ( card I + 2 ) , i2 = goto ( card I + 2 ) , i1 = goto ( card I + 2 ) , i2 = goto ( card I + 2 ) , i2 = goto ( card I + 2 ) , i2 = goto ( card I + 2 ) ,
attr B is thesis means : : : for B being set holds B is SubSub\bf & S = B `1 & S = ( B `1 ) `1 ;
a9 "/\" D = { a "/\" d where d is Element of N : d in D } & a "/\" ( d "/\" D ) = { a "/\" d where d is Element of N : d in D } ;
|( exp_R , q29 )| * |( exp_R , q29 )| >= |( exp_R , q )| * |( exp_R , q )| & |( exp_R , q )| >= |( exp_R , q )| ;
( - f ) . sup A = ( ( - f ) | A ) . sup A .= ( ( - f ) | A ) . sup A .= ( - f ) . sup A ;
( G * ( len G , k ) ) `1 = G * ( len G , k ) `1 .= G * ( len G , k ) `1 .= G * ( 1 , k ) `1 ;
( Proj ( i , n ) ) . LM = <* ( proj ( i , n ) ) . LM *> .= <* ( proj ( i , n ) ) . LM *> ;
f1 + f2 * reproj ( i , x ) is_differentiable_in ( ( the reproj of i , x ) . ( f1 . x ) ) ;
pred ( for x st x in Z holds ( ( tan * ( f1 + f2 ) ) `| Z ) . x = ( tan * ( f1 + f2 ) ) . x ) ;
ex t being SortSymbol of S st t = s & h1 . x = h2 . t & ( for x being Element of S holds x in rng h1 ) ;
defpred C [ Nat ] means P8 ( ) is $1 -Int & A8 is n -Int implies A is $1 -Int { x } ;
consider y being element such that y in dom ( p . i ) and ( q . i ) . y = ( p . i ) . y ;
reconsider L = product ( { x1 } +* ( index B , l ) ) as Subset of ( support A ) . ( ( index B ) . ( index B ) ) ;
for c being Element of C ex d being Element of D st T . ( id c ) = id d & for c being Element of C st c in dom T holds d . c = id c
N ( f , n , p ) = ( f | n ) ^ <* p *> .= f ^ <* p *> .= f ;
( f * g ) . x = f . ( g . x ) & ( f * h ) . x = f . ( h . x ) ;
p in { 1 / 2 * ( G * ( i + 1 , j ) + G * ( i + 1 , j + 1 ) ) } ;
f `2 - p `2 = ( ( c | ( n , L ) ) *' ( - ( f . x ) ) ) *' .= ( - c ) * ( ( - ( f . x ) ) ) ) *' ;
consider r being Real such that r in rng ( f | divset ( D , j ) ) and r < m + s ;
f1 . ( |[ r2 , r3 ]| ) in f1 .: W2 & f2 . ( |[ r2 , r3 ]| ) in f1 .: W2 & f1 . ( |[ r2 , r3 ]| ) in f2 .: W3 ;
eval ( a | ( n , L ) , x ) = ( a * ( a | ( n , L ) ) ) . x .= a * ( x , x ) ;
z = DigA ( tk , xk ) .= DigA ( tk , xk ) .= DigA ( tk , xk ) .= DigA ( tk , xk ) ;
set H = { Intersect S where S is Subset-Family of X : S c= G } , G = { meet S where S is Subset-Family of X : S c= G } , F = { Intersect S where S is Subset-Family of X : S c= G } , G = { Intersect S where S is Subset of X : S c= G } , H = { Intersect S where S is Subset of X : S c= G } ;
consider S19 being Element of D * , d being Element of D * such that S ` = S19 ^ <* d *> and S = { d } and d in D ;
assume that x1 in dom f and x2 in dom f and f . x1 = f . x2 and f . x2 = f . x2 ;
- 1 <= ( q `2 / |. q .| - sn ) / ( 1 + sn ) & q `2 / |. q .| - sn <= 1 ;
( for v be Linear_Combination of V st v in A holds Sum ( ( - 1 ) (#) ( - 1 ) ) = 0. V ) & for v be VECTOR of V holds v in A implies v in A
let k1 , k2 , k2 , k1 , k2 , k2 , k2 , : 5 , 6 , k2 , k2 , k2 , k2 , 6 , 7 , 8 , 8 , 8 , 9 } = the InstructionsF of SCM+FSA ;
consider j being element such that j in dom a and j in g " { k `2 } and x = a . j and j in { k `2 } ;
H1 . x1 c= H1 . x2 or H1 . x2 c= H1 . x1 or H1 . x2 c= H1 . x2 or H1 . x1 c= H1 . x2 ;
consider a being Real such that p = or ( a * p1 + ( a * p2 ) ) * ( a * p2 ) and 0 <= a and a <= 1 ;
assume that a <= c & d <= b & [' a , b '] c= dom f and [' a , b '] c= dom g and f . a = g . b ;
cell ( Gauge ( C , m ) * ( m , width Gauge ( C , m ) -' 1 , 0 ) is non empty ;
Ay in { ( S . i ) `1 where i is Element of NAT : i <= n & not contradiction } ;
( T * b1 ) . y = L * ( b2 /. y ) .= ( F /. y ) * ( b1 /. y ) .= ( F /. y ) * ( b1 /. y ) ;
g . ( s , I ) . x = s . y & g . ( s , I ) . y = |. s . x - s . y .| ;
( log ( 2 , k + k ) ) ^2 >= ( log ( 2 , k + 1 ) ) ^2 / ( ( k + 1 ) ^2 ) ;
then that p => q in S and not x in the still of p and not p => All ( x , q ) in S ;
dom ( the InitS of rp ) misses dom ( the InitS of rp ) & dom ( the InitS of rp ) misses dom ( the InitS of rp ) ;
synonym f is extended integer means : \lbrace for x being set st x in rng f holds x is Integer ;
assume for a being Element of D holds f . { a } = a & for X being Subset-Family of D holds f . ( f .: X ) = f . union X ;
i = len p1 .= len p3 + len <* x *> .= len p3 + len <* x *> .= len p3 + 1 .= len p3 + 1 .= len p3 + 1 .= len p3 + 1 ;
l = ( g /. ( 1 , 3 ) ) `1 + ( k , 3 ) `1 - ( k , 3 ) `1 + ( e , 3 ) `2 - ( k , 3 ) `2 ) ;
CurInstr ( P2 , Comput ( P2 , s2 , l2 ) ) = halt SCM+FSA .= CurInstr ( P2 , s2 ) .= halt SCM+FSA .= halt SCM+FSA ;
assume for n be Nat holds ||. seq . n .|| <= Rseq . n & Rseq is summable & ( for n be Nat holds seq . n <= seq . n ) & ( n <= k implies seq is summable ) implies seq is summable & seq is summable & lim seq = 0 ;
sin ( \vert non .| ) = sin r * cos ( ( - cos ( r ) ) * sin ( s ) ) .= 0 ;
set q = |[ g1 `1 / ( t0 `2 ) , g2 = |[ g2 `1 / ( t0 `1 ) , g3 `2 / ( t0 `2 ) ]| , r = |[ g2 `1 / ( t0 `2 ) , g2 `2 / ( t0 `1 ) ]| ;
consider G be sequence of S such that for n being Element of NAT holds G . n in implies G is implies S is implies S is \overline of S ;
consider G such that F = G and ex G1 st G1 in SM & G = ( X \/ G1 ) /\ ( X \/ G1 ) ;
the root of [ x , s ] in ( the Sorts of Free ( C , X ) ) . s & ( the Sorts of C ) . s in ( the Sorts of Free ( C , X ) ) . s ;
Z c= dom ( ( exp_R * f1 ) + ( exp_R * f1 ) ) & Z c= dom ( ( exp_R * f1 ) + ( exp_R * f1 ) ) ;
for k be Element of NAT holds seq1 . k = ( sum ( Im ( f ) , S ) ) . k & ( Im ( f ) ) . k = ( Im ( f ) ) . k ;
assume that - 1 < n ( ) and q `2 > 0 and ( q `1 / |. q .| - cn ) < 0 and q `2 < 0 and q `2 < 0 ;
assume that f is continuous one-to-one and a < b and f is continuous and f = g and f . a = c and f . b = d and f . c = d ;
consider r being Element of NAT such that s-> sequence of NAT such that s-> sequence of NAT and r <= q and for n being Nat st n <= len s1 holds r <= n ;
LE f /. ( i + 1 ) , f /. j , L~ f , f /. ( len f ) , f /. ( len f ) ;
assume that x in the carrier of K and y in the carrier of K and ex_inf_of x , L and ex_inf_of { x , y } , L and x <= y ;
assume f +* ( i1 , \xi ) in ( proj ( F , i2 ) ) " ( A . i1 ) & f in ( proj ( F , i2 ) ) " ( A . i1 ) ;
rng ( ( ( ( ( ( ( ( ( ( ( ( ( the carrier of M ) ) ) ) | ( the carrier of M ) ) ) ) ) \/ ( ( the carrier' of M ) | ( the carrier' of M ) ) ) ) ) ) ) c= the carrier' of M ;
assume z in { ( the carrier of G ) \ { t } where t is Element of T : t in X } ;
consider l be Nat such that for m be Nat st l <= m holds ||. s1 . m - ( lim s1 ) .|| < g / 2 ;
consider t be VECTOR of product G such that mt = ||. D5 . t .|| and ||. t .|| <= 1 ;
assume that the carrier of v = 2 and v ^ <* 0 *> in dom p and v ^ <* 1 *> in dom p and p . ( 1 + 1 ) in dom p ;
consider a being Element of the Points of Xas , A being Element of the lines of Xas that a on A and not a on A ;
( - x ) |^ ( k + 1 ) * ( ( - x ) |^ ( k + 1 ) ) " = 1 ;
for D being set st for i st i in dom p holds p . i in D holds p is FinSequence of D & for i st i in dom p holds p . i is FinSequence of D
defpred R [ element ] means ex x , y st [ x , y ] = $1 & P [ x , y ] & P [ y ] ;
L~ f2 = union { LSeg ( p0 , p10 ) , LSeg ( p10 , p2 ) } .= { LSeg ( p10 , p2 ) , LSeg ( p10 , p2 ) } .= { p2 } ;
i -' len h11 + 2 - 1 < i -' len h11 + 2 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 ;
for n being Element of NAT st n in dom F holds F . n = |. ( nb . ( n -' 1 ) ) .| ;
for r , s1 , s2 , r , s holds r in [. s1 , s2 .] iff r <= s & s <= s2 & r <= s2 & s <= s2
assume v in { G where G is Subset of T2 : G in B2 & G c= z1 & G c= z2 & G c= { z1 } } ;
let g be element , A be element of INT , X be Element of INT , b be Element of INT , c being Element of INT , b being Element of X ;
min ( g . [ x , y ] , k ) . [ y , z ] = ( min ( g , k , x ) ) . y ;
consider q1 be sequence of CNS such that for n holds P [ n , q1 . n ] and q1 is convergent and lim q1 = lim q1 ;
consider f being Function such that dom f = NAT and for n being Element of NAT holds f . n = F ( n ) and for n being Element of NAT holds f . n = F ( n ) ;
reconsider B-6 = B /\ B , BO1 = O , Bd = I , Bd = I , Bd = J as Subset of B ;
consider j being Element of NAT such that x = the ` finite sequence of n and 1 <= j and j <= n and j <= n and f . j = f . j ;
consider x such that z = x and card ( x . O2 ) in card ( L1 . O1 ) and x in L1 . O2 and x in L2 . O1 ;
( C * _ T4 ( k , n2 ) ) . 0 = C . ( ( _ of T4 ( k , n2 ) ) . 0 ) .= C . ( ( C * ( k , n2 ) ) . 0 ) ;
dom ( X --> rng f ) = X & dom ( X --> f ) = dom ( X --> f ) & dom ( X --> f ) = X ;
S-bound L~ SpStSeq C <= ( ( L~ SpStSeq C ) /. ( i + 1 ) ) `2 & ( ( L~ SpStSeq C ) /. ( i + 1 ) ) `2 <= ( ( L~ SpStSeq C ) /. ( i + 1 ) ) `2 ;
synonym x , y are_collinear means : : : ex l st x = y or ex l being Subset of S st { x , y } c= l ;
consider X being element such that X in dom ( f | ( n + 1 ) ) and ( f | ( n + 1 ) ) . X = Y ;
assume that Im k is continuous and for x , y being Element of L for a , b being Element of Im k st a = x & b = y holds x << y and a << b and a << b ;
( 1 / 2 * ( ( ( ( - 1 / 2 ) * ( ( #Z n ) * ( AffineMap ( n , 0 ) ) ) ) ) ) ) is_differentiable_on REAL & ( ( - 1 / 2 ) * ( ( AffineMap ( n , 0 ) ) ) ) `| REAL ) = ( ( - 1 / 2 ) * ( ( #Z n ) * ( ( #Z n ) * ( #Z n ) ) ) ) ;
defpred P [ Element of omega ] means ( for n holds ( ( the partial of A1 ) . $1 = A1 . ( n + $1 ) ) & ( for n holds ( ( the partial of A2 ) . ( n + $1 ) ) = A2 . ( n + $1 ) ) ;
IC Comput ( P , s , 2 ) = succ IC Comput ( P , s , 1 ) .= 6 + 1 .= 6 + 1 .= 6 + 1 .= 6 ;
f . x = f . g1 * f . g2 .= f . ( g1 * 1_ H ) .= f . ( g1 * 1_ H ) .= f . ( g1 * g2 ) .= f . ( g1 * g2 ) ;
( M * F-4 ) . n = M . ( F-4 . n ) .= M . ( { ( canFS Omega ) . n } ) .= M . ( { ( canFS Omega ) . n } ) ;
the carrier of L1 + L2 c= ( the carrier of L1 ) \/ ( the carrier of L2 ) & the carrier of L1 c= the carrier of L1 & the carrier of L1 + L2 c= the carrier of L2 ;
pred a , b , c , x , y , x , y , z , y be element means : : : a , b , c , x , y , z , x , y , z ;
( for n be Nat holds s . n <= ( ( the PartFunc of X ) * s ) . ( n + 1 ) ) * ( s . ( n + 1 ) ) ;
pred - 1 <= r & r <= 1 & ( arccot - 1 ) * ( arccot - 1 ) = - 1 / r & ( arccot - 1 ) * ( arccot - 1 ) = - 1 / r ;
seq in { p ^ <* n *> where n is Nat : p ^ <* n *> in T1 & n in dom T } implies T in T1 & T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T
|[ x1 , x2 , x3 ]| . 2 - |[ y1 , y2 , x3 ]| . 2 = x2 - y2 & |[ x2 , y2 , x3 , x4 ]| . 3 = x2 - y2 ;
attr for m be Nat holds F . m is nonnegative & ( Partial_Sums F ) . m is nonnegative implies ( Partial_Sums F ) . m is nonnegative ;
len ( ( G . z ) * ( z - x ) ) = len ( ( ( G . ( x - y ) ) + ( G . ( y - z ) ) ) ) .= len ( G . ( x - y ) ) ;
consider u , v being VECTOR of V such that x = u + v and u in W1 /\ W2 and v in W2 and u in W2 and v in W3 ;
given F be finite Subset of NAT such that F = x and dom F = n & rng F c= { 0 , 1 } and Sum F = k and for k be Nat st k in n holds F . k = k ;
0 = ( 1 * 0 ) * TOP-REAL uq iff 1 = ( ( 1 - ( 1 - 0 ) ) * ( 1 - 0 ) ) * ( ( 1 - 0 ) * ( 1 - 0 ) ) ;
consider n be Nat such that for m be Nat st n <= m holds |. ( f # x ) . m - lim ( f # x ) .| < e ;
cluster -> being being non empty implies iff iff for non empty \rbrace , \hbox { $ ( { w } , { w } ) , ( { w } , { w } ) } is Boolean Boolean
"/\" ( BB , {} ) = Top BB .= Top BB .= "/\" ( BB , [#] S ) .= "/\" ( BB , [#] S ) .= "/\" ( BB , [#] S ) .= "/\" ( BB , [#] S ) ;
( r / 2 ) ^2 + ( r / 2 ) ^2 + ( r / 2 ) ^2 <= ( r / 2 ) ^2 + ( r / 2 ) ^2 + ( r / 2 ) ^2 ;
for x being element st x in A /\ dom ( f `| X ) holds ( f `| X ) . x >= r2 & ( f `| X ) . x >= r2
2 * r1 - c * |[ a , c ]| - ( 2 * r1 - c * |[ b , c ]| ) = 0. TOP-REAL 2 - ( 2 * r1 - c * |[ b , c ]| ) ;
reconsider p = P * ( 1 , 1 ) , q = a " * ( ( - ( - 1 ) ) |^ n ) as FinSequence of K ;
consider x1 , x2 being element such that x1 in uparrow s and x2 in < t and x = [ x1 , x2 ] and [ x2 , y2 ] in dom f and [ x1 , x2 ] in dom f ;
for n being Nat st 1 <= n & n <= len q1 holds q1 . n = ( upper_volume ( g , M7 ) ) . ( n + 1 ) & ( for k be Nat st k in dom ( g | M7 ) ) holds q1 . k = ( ( g | M7 ) ) . ( n + 1 )
consider y , z being element such that y in the carrier of A and z in the carrier of A and i = [ y , z ] ;
given H1 , H2 being strict Subgroup of G such that x = H1 & y = H2 and H1 , H2 |^ ( x , y ) -> Subgroup of G and H2 , H1 |^ ( x , y ) -> Subgroup of G ;
for S , T being non empty < T , d being Function of T , S st T is complete holds d is directed-sups-preserving & d is monotone & d is monotone
[ a + 0. F_Complex , b2 ] in ( the carrier of F_Complex ) /\ ( the carrier of V ) & [ a + 0. F_Complex , b2 ] in ( the carrier of F_Complex ) /\ ( the carrier of V ) ;
reconsider mm = max ( ( len F1 ) * ( p . n ) * ( x |^ n ) ) , mn = max ( ( len F1 ) * ( p . n ) * ( x |^ n ) ) as Element of NAT ;
I <= width GoB ( ( GoB h ) * ( 1 , width GoB h ) , ( GoB h ) * ( 1 , width GoB h ) ) & ( GoB ( ( GoB h ) * ( 1 , width GoB h ) ) ) `2 <= ( GoB ( ( GoB h ) * ( 1 , width GoB h ) ) ) `2 ;
f2 /* q = ( f2 /* ( f1 /* s ) ) ^\ k .= ( f2 * ( f1 /* s ) ) ^\ k .= ( ( f2 * f1 ) /* s ) ^\ k .= ( ( f2 * f1 ) /* s ) ^\ k ;
pred A1 \/ A2 is linearly-independent means : : : A1 : A1 misses A2 & ( for x st x in A1 holds Lin ( A1 \/ A2 ) /\ Lin ( A2 ) = Lin ( A1 \/ A2 ) ) & Lin ( A1 \/ A2 ) = Lin ( A2 \/ A1 ) ;
func A -carrier C -> set means : : : for s being Element of R holds s in union { A . s where s is Element of R : s in C } ;
dom ( Line ( v , i + 1 ) (#) ( ( -> Matrix of m , n ) * ( Line ( p , i ) ) ) ) = dom ( F ^ ( Line ( p , i ) ) ) ;
cluster [ x , 4 ] -> non empty & [ x , 4 ] in [: { x } , { x } :] & [ x , 4 ] in [: { x } , { x } :] & [ x , 4 ] in [: { x } , { x } :] ;
E |= All ( x1 , All ( x2 , x2 ) ) => ( ( x2 'in' x1 ) '&' ( x1 => x2 ) ) => ( ( x1 '&' x2 ) '&' ( x1 '&' x2 ) ) => ( x1 '&' x2 ) '&' ( x1 '&' x2 ) ) ;
F .: ( id X , g ) . x = F . ( id X , g . x ) .= F . ( id X , g . x ) .= F . ( x , g ) .= F . ( x , g ) ;
R . ( h . m ) = F . ( x0 + h . m ) - ( h . m ) - ( h . m ) + ( h . m ) - ( h . m ) ;
cell ( G , Xs -' 1 , ( Y + 1 ) \ ( t + 1 ) ) \ L~ f meets ( ( L~ f ) \ ( t + 1 ) ) \ ( t + 1 ) ;
IC Comput ( P2 , s2 , 2 ) = IC Comput ( P2 , s2 , 2 ) .= ( card I + 2 ) .= card I + ( card J + 2 ) .= card I + ( card J + 2 ) .= card I + ( card J + 2 ) .= card I + ( card J + 2 ) .= card I + card J + 2 .= card J + card J + 1 ;
sqrt ( ( - ( ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ) ^2 ) > 0 ;
consider x0 being element such that x0 in dom a and x0 in g " { k } and y0 = a . x0 and x0 in { x } and y = a . x0 ;
dom ( r1 (#) chi ( A , C ) ) = dom chi ( A , C ) /\ dom chi ( A , C ) .= dom chi ( A , C ) /\ dom ( chi ( A , C ) ) .= dom ( r1 (#) chi ( A , C ) ) .= dom ( r1 (#) chi ( A , C ) ) .= dom ( r1 (#) chi ( A , C ) ) ;
d-7 . [ y , z ] = ( ( ( y - z ) `2 ) - ( ( y - z ) `2 ) ) * ( ( y - z ) `2 ) .= ( ( y - z ) `2 ) * ( ( y - z ) `2 ) ;
pred for i being Nat holds C . i = A . i /\ B . i means : : : for C being Nat holds C . i c= C . ( i + 1 ) /\ C . ( i + 1 ) ;
assume that x0 in dom f and f is_continuous_in x0 and ||. f /. x0 - f /. x0 .|| < r and ||. f /. x0 - f /. x0 .|| < r ;
p in Cl A implies for K being Basis of p , Q being Basis of T st Q in K holds A meets Q & A meets Q
for x being Element of REAL n st x in Line ( x1 , x2 ) holds |. y1 - y2 .| <= |. y1 - y2 .| & |. y2 - x .| <= |. y1 - y2 .|
func Sum ( <*> a ) -> Ordinal means : : : a in it & for b being Ordinal st a in it holds it . b c= b ;
[ a1 , a2 , a3 ] in ( ( the carrier of A ) /\ ( the carrier of B ) ) & [ a1 , a2 , a3 ] in ( the carrier of A ) /\ ( the carrier of B ) ;
ex a , b being element st a in the carrier of S1 & b in the carrier of S2 & x = [ a , b ] & [ a , b ] in the InternalRel of S2 ;
||. ( vseq . n ) - ( vseq . m ) .|| * ||. x - x0 .|| < ( e * ||. x - x0 .|| ) * ||. x - x0 .|| ;
then for Z being set st Z in { Y where Y is Element of I7 : F c= Y } holds z in x & z in Z ;
sup compactbelow [ s , t ] = [ sup ( ( compactbelow [ s , t ] ) /\ ( compactbelow [ s , t ] ) ) , sup ( ( compactbelow [ s , t ] ) /\ ( compactbelow [ s , t ] ) ) ] ;
consider i , j being Element of NAT such that i < j and [ y , f . j ] in IB and [ f . i , f . j ] in IB and [ f . i , f . j ] in IB ;
for D being non empty set , p , q being FinSequence of D st p c= q ex p being FinSequence of D st p ^ q = q & p ^ q = q ^ p
consider e39 being Element of the affine of X such that c9 , a9 // a9 , e39 and a9 <> c9 and c9 <> e and a , b // a9 , b9 and a , c // a9 , c9 and a , b // a9 , b9 and a , c // a9 , c9 and a , b // a9 , c9 and a , b // a9 , b9 ;
set U2 = I \! \mathop { \vert S .| , SS = { R } , SS = { R } , E = { R } , F = { R } , SS = { R } , C = { R } , SS = { R } , C = { R } , SS = { R } , SS = { R } , SS = { R } , SS = { R } , SS = { R } , SS = { R } , SS = { R } , SS = { R } , SS = { R } , SS = { R } , SS = { R } , SS = { R } , SS
|. q3 .| ^2 = ( |. q2 .| ) ^2 + ( |. q2 .| ) ^2 .= |. q2 .| ^2 + ( |. q2 .| ) ^2 .= |. q2 .| ^2 + ( |. q2 .| ) ^2 .= |. q2 .| ^2 + ( |. q2 .| ) ^2 .= |. q2 .| ^2 ;
for T being non empty TopSpace , x , y being Element of [: the topology of T , the topology of T :] holds x "\/" y = x /\ y & x "/\" y = x /\ y implies x = y
dom signature ( U1 ) = dom ( the charact of U1 ) & Args ( o , MSAlg U1 ) = dom ( the charact of U1 ) & Args ( o , MSAlg U1 ) = dom ( the charact of U1 ) & dom ( the charact of U1 ) = dom ( the charact of U1 ) ;
dom ( h | X ) = dom h /\ X .= dom ( ||. h .|| | X ) .= dom ( ||. h .|| | X ) .= dom ( ( ||. h .|| | X ) ) .= dom ( ( |. h .| | X ) ) .= X ;
for N1 , N1 , N2 being Element of ( the carrier of G ) * , ( h . K1 ) st N1 = N & rng ( h . K1 ) c= N1 holds rng ( h . K1 ) c= N1 & rng ( h . K1 ) c= N2
( mod ( u , m ) + mod ( v , m ) ) . i = ( mod ( u , m ) ) . i + ( mod ( v , m ) ) . i ;
- ( q `1 ) ^2 < - 1 or q `2 / |. q .| * ( 1 + ( q `2 / q `1 ) ^2 ) >= - ( q `2 / |. q .| * ( 1 + ( q `2 / q `1 ) ^2 ) ) & - ( q `2 / q `1 ) ^2 <= - ( q `2 / |. q .| * ( 1 + ( q `2 / q `1 ) ^2 ) ) ;
pred r1 = ff means : : : for r2 st r2 = ff holds r1 * ( f - g ) = ( f - g ) * ( f - g ) ;
vseq . m is bounded Function of X , the carrier of Y & xA . m = ( A1 . m ) * ( ( ( vseq . m ) * ( vseq . m ) ) ) & ( ( vseq . m ) * ( vseq . m ) ) . x = ( ( vseq . m ) * ( vseq . m ) ) . x ;
pred a <> b & b <> c & b <> c & angle ( a , b , c ) = PI & angle ( b , c , a ) = 0 implies angle ( b , c , a ) = 0 & angle ( a , c , b ) = 0 ;
consider i , j , r being Nat such that p1 = [ i , r ] and p2 = [ j , s ] and i < j and r < j and j <= len f ;
|. p .| ^2 - ( 2 * |( p , q )| ) ^2 + |. q .| ^2 = |. p .| ^2 + |. q .| ^2 - ( 2 * |( p , q )| ) ^2 ;
consider p1 , q1 being Element of [: X ( ) , Y ( ) :] such that y = p1 ^ q1 and q1 in X ( ) and p1 ^ q1 = p1 ^ q1 and q1 in Y ( ) and q1 in X ( ) and q1 in Y ( ) and q2 in Y ( ) and q1 in X ( ) and q2 in Y ( ) ;
( for r1 , r2 , s1 , s2 , s2 being Element of A holds ( s1 + s2 ) * ( r1 + r2 ) = ( s2 * ( s1 + s2 ) ) * ( r2 + s2 ) ) * ( r1 * ( s1 + s2 ) )
( ( LMP A ) `2 = lower_bound ( proj2 .: A /\ /\ Vertical_Line w ) & proj2 .: A /\ Vertical_Line w is non empty & proj2 .: A /\ Vertical_Line w is non empty & proj2 .: A /\ Vertical_Line w is non empty ;
s |= ( H , k1 ) \bf ( H1 , k2 ) iff s |= for H being strict Subgroup of X holds s |= ( H1 , k2 ) implies s |= H & s |= ( H , k ) implies s |= H
len ( s ) + 1 = card ( support b1 ) + 1 .= card ( support b2 ) + card ( support b1 ) .= card ( support b1 ) + card ( support b1 ) .= card ( support b1 ) + card ( support b1 ) .= card ( support b1 ) ;
consider z being Element of L1 such that z >= x and z >= y and for z being Element of L1 st z >= x & z >= y holds z `1 >= y `1 and z `2 >= x `2 ;
LSeg ( UMP D , |[ ( W-bound D + E-bound D ) / 2 , ( W-bound D + ( W-bound D ) / 2 ) ]| ) /\ D = { UMP D } \/ { UMP D } .= { UMP D } ;
lim ( ( ( f `| N ) / ( g `| N ) ) /* b ) = lim ( ( f `| N ) / ( g `| N ) ) .= lim ( ( f `| N ) / ( g `| N ) ) ;
P [ i , pr1 ( f ) . i , pr1 ( f ) . ( i + 1 ) , pr1 ( f ) . ( i + 1 ) ] & pr1 ( f ) . ( i + 1 ) = pr1 ( f ) . ( i + 1 ) ;
for r be Real st 0 < r ex m be Nat st for k be Nat st m <= k holds ||. ( seq . k ) - ( R /* ( seq ^\ k ) ) .|| < r
for X being set , P being a_partition of X , x , a , b being set st x in a & a in P & b in P & x in P & a in P holds a = b
Z c= dom ( ( - 1 / ( n + 1 ) ) (#) ( ( #Z ( n + 1 ) ) * f ) ) \ ( ( #Z ( n + 1 ) ) * f ) " { 0 } ) implies Z c= dom ( ( - 1 / ( n + 1 ) ) (#) ( ( #Z ( n + 1 ) ) * f ) )
ex j being Nat st j in dom ( l ^ <* x *> ) & j < i & y = ( l ^ <* x *> ) . j & i = 1 + len l & z = 1 + len l & j = len l + 1 & z = l + 1 ;
for u , v being VECTOR of V , r being Real st 0 < r & r < 1 holds r * u + ( 1-r * v ) in = - ( 1-r ( V ) )
A , Int A , Cl ( Int A ) , Cl ( Int A ) , Cl ( Int A ) , Cl ( Int Cl A ) , Cl ( Int Cl A ) , Cl ( Int Cl A ) , Cl ( Int Cl A ) , Cl ( Int Cl A ) , Cl ( Int Cl A ) , Cl ( Int Cl A ) , Cl ( Int Cl A ) , Cl ( Int Cl A ) ` , Cl ( Int Cl A ) ` , Cl ( Int Cl A ) ` , Cl ( Int Cl A ) , Cl ( Int Cl A ) , Cl ( Int Cl A , Cl ( Int Cl A ) , Cl
- Sum <* v , u , w *> = - ( v + u + u ) .= - ( v + u ) -uw .= - ( v + u ) -uw .= - ( v + u ) -uw ;
( Exec ( a := b , s ) ) . IC SCM R = ( Exec ( a := b , s ) ) . IC SCM R .= Exec ( ( a := b ) , s ) . IC SCM R .= succ IC s .= succ IC s .= succ IC s .= IC s ;
consider h being Function such that f . a = h and dom h = I and for x being element st x in I holds h . x in ( the carrier of J ) . ( f . x ) and h . x in I ;
for S1 , S2 being non empty reflexive RelStr , D being non empty directed Subset of S1 , D being non empty Subset of S2 , x being Element of S1 , y being Element of S2 , z being Element of S2 st x in S1 & y in D & z in D holds x "/\" y is directed & x "/\" z is directed
card X = 2 implies ex x , y st x in X & y in X & x <> y or x = y & x = y or x = y & y = z or x = z & y = z
E-max L~ Cage ( C , n ) in rng ( Cage ( C , n ) :- W-min L~ Cage ( C , n ) ) & W-min L~ Cage ( C , n ) in rng ( Cage ( C , n ) :- W-min L~ Cage ( C , n ) ) ;
for T , T being tree , p , q being Element of dom T , p being Element of dom T st p element , q in dom T holds ( T -with tree ( p , T ) ) . q = T . ( q , p )
[ i2 + 1 , j2 ] in Indices G & [ i2 + 1 , j2 ] in Indices G & f /. k = G * ( i2 + 1 , j2 ) & f /. ( k + 1 ) = G * ( i2 + 1 , j2 ) ;
cluster ( k gcd n ) divides ( k gcd n ) & ( k divides m ) & ( k divides m implies ( k divides m ) & ( k divides n ) & ( k divides m ) & ( k divides n implies k divides m ) ) & ( k divides n implies k divides m ) ) ;
dom F " = the carrier of X2 & rng F " = the carrier of X1 & F " = the carrier of X2 & F " = the carrier of X2 & F " = the carrier of X1 & F " = the carrier of X2 ;
consider C be finite Subset of V such that C c= A and card C = n and the \HM of V = Lin ( BM \/ C ) and Lin ( BM \/ B ) = Lin ( BM \/ B ) and Lin ( BM \/ B ) = Lin ( BM \/ B ) ;
V is prime implies for X , Y being Element of [: the topology of T , the topology of T :] st X /\ Y c= V holds X c= V or Y c= V
set X = { F ( v1 ) where v1 is Element of B ( ) , v2 is Element of B ( ) : P [ v1 ] } , Y = { F ( v1 ) where v1 is Element of B ( ) : P [ v2 ] } , Z = { F ( v2 ) where v2 is Element of B ( ) : P [ v2 ] } ;
angle ( p1 , p3 , p4 ) = 0 .= angle ( p2 , p3 , p4 ) .= angle ( p3 , p3 , p2 ) .= angle ( p3 , p3 , p2 ) .= angle ( p2 , p3 , p2 ) .= angle ( p3 , p3 , p2 ) ;
- sqrt ( ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ) ^2 = - sqrt ( ( q `1 / |. q .| - cn ) ) ^2 .= - ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 .= - ( q `1 / |. q .| - cn ) / ( 1 + cn ) ;
ex f being Function of I[01] , TOP-REAL 2 st f is continuous one-to-one & rng f = P & f . 0 = p1 & f . 1 = p2 & f . 1 = p3 & f . 0 = p4 & f . 1 = p4 & f . 1 = p4 ;
pred f is_PartFunc of REAL , u0 means : : : for u0 st SVF1 ( 2 , pdiff1 ( f , 1 ) , u0 ) is_differentiable_in u0 holds SVF1 ( 2 , pdiff1 ( f , 1 ) , u0 ) is_differentiable_in u0 ;
ex r , s st x = |[ r , s ]| & G * ( len G , 1 ) `1 < r & r < G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `2 < s & s < G * ( 1 , 1 ) `2 & s < G * ( 1 , 1 ) `2 ;
assume that f is_sequence_on G and 1 <= t & t <= len G and G * ( t , width G ) `2 >= N-bound L~ f and f /. len f = ( GoB f ) * ( t , width G ) and f /. 1 = ( GoB f ) * ( t , width G ) and f /. len f = ( GoB f ) * ( t , width G ) ;
pred i in dom G means : : : for r holds r (#) ( f * reproj ( i , x ) ) = r (#) ( reproj ( i , x ) ) & r (#) ( f * reproj ( i , x ) ) = r (#) reproj ( i , x ) ;
consider c1 , c2 being bag of o1 + o2 such that ( decomp c ) /. k = <* c1 , c2 *> and c /. k = ( decomp c1 ) /. ( k + 1 ) and ( decomp c ) /. ( k + 1 ) = ( decomp c1 ) /. ( k + 1 ) ;
u0 in { |[ r1 , s1 ]| : r1 < G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `2 < s1 & s1 < G * ( 1 , 1 ) `2 } ;
Cl ( X ^ Y ) . k = the carrier of X . k2 .= C4 . ( k1 + 1 ) .= C4 . ( k1 + 1 ) .= C4 . ( k1 + 1 ) .= C4 . ( k1 + 1 ) ;
pred len M1 = len M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & len M2 = width M2 ;
consider g2 be Real such that 0 < g2 and { y where y is Point of S : ||. ( - x0 ) * ( y - x0 ) .|| < g2 & y in N2 } c= N2 & N c= { x0 } ;
assume x < ( - b ) + sqrt ( delta ( a , b , c ) ) * ( 2 * a ) or x > - b & x < - sqrt ( a , b , c ) * ( 2 * a ) ;
( G1 '&' G2 ) . i = ( <* 3 *> ^ G1 ) . i & ( H1 '&' G1 ) . i = ( <* 3 *> ^ G1 ) . i & ( H1 '&' G1 ) . i = ( <* 3 *> ^ G1 ) . i ;
for i , j st [ i , j ] in Indices ( M3 + M1 ) holds ( M3 + M1 ) * ( i , j ) < M2 * ( i , j ) & ( M3 + M2 ) * ( i , j ) < M2 * ( i , j )
for f being FinSequence of NAT , i being Element of NAT st j in dom f & i <= j holds i divides f /. j & i divides len f & j <= len f holds i divides f /. j
assume F = { [ a , b ] where a , b is Subset of X : for c being set st c in B\mathopen the carrier of X & a c= c holds a c= c } & b c= c ;
b2 * q2 + ( b3 * q3 ) + ( b3 * q3 ) + ( - ( a1 * q2 ) ) * q2 = 0. TOP-REAL n + ( ( a1 * q2 ) * q3 ) .= 0. TOP-REAL n + ( ( a1 * q2 ) * q3 ) .= 0. TOP-REAL n + ( ( a1 * q2 ) * q3 ) .= 0. TOP-REAL n ;
Cl ( Cl F ) = { D where D is Subset of T : ex B being Subset of T st D = Cl B & B in F & Cl B c= Cl Cl B } & Cl ( Cl B ) c= Cl Cl B ;
attr seq is summable means : : : for m be Nat holds seq is summable & seq is summable & ( seq is summable implies seq is summable ) & ( seq is summable implies seq is summable ) & ( seq is summable implies seq is summable ) & ( seq is summable & seq is summable implies seq is summable ) & ( seq is summable implies seq is summable ) & ( seq is summable & seq is summable ) & seq is summable & seq is summable ) implies seq is summable & seq is summable & seq is summable & seq is summable & ( seq is summable & seq is summable & seq is summable & seq is summable & seq is summable & seq is summable & seq is summable & seq is summable & ( seq is
dom ( ( ( cn - cn ) | D ) | D ) = ( the carrier of ( TOP-REAL 2 ) | D ) /\ D .= the carrier of ( ( TOP-REAL 2 ) | D ) .= D ;
X [ X , Z ] is full full non empty SubRelStr of ( ( [#] Z ) |^ the carrier of Z ) & X [ X , Z ] implies X , Z let Y be full SubRelStr of ( ( the carrier of Z ) --> the carrier of Z )
G * ( 1 , j ) `2 = G * ( i , j ) `2 & G * ( 1 , j ) `2 <= G * ( i , j ) `2 & G * ( 1 , j ) `2 <= G * ( 1 , j + 1 ) `2 ;
synonym m1 c= m2 means : : : for p being set st p in P holds the } \HM { m1 + 1 where m1 is Nat : m1 in dom ( m2 , p ) & m2 in P & p in P & m2 in P } c= the carrier of ( m2 , p ) ;
consider a being Element of B ( ) such that x = F ( a ) and a in { G ( b ) where b is Element of A ( ) : P [ b ] } and a in A ( ) & P [ b ] ;
synonym cluster multiplicative loop s -> such that the multiplicative loop mas of M means : : : for a being Element of M holds it . a = [ a , the multF of M ] where a is Element of the carrier of M : a in the carrier of M } ;
sequence ( a , b ) + 1 + sequence ( c , d ) = b + ( c , d ) .= b + ( c , d ) .= b + ( c + d ) .= o + ( c + d ) .= o + ( c + d ) .= o + ( c + d ) .= o + ( c + d ) .= o + ( ( a + b ) + d ) ;
cluster + ( i1 , i2 ) -> natural for Element of INT , i1 , i2 be Element of INT , i1 , i2 be Element of NAT , i2 be Element of INT , i1 , i2 be Element of NAT st i1 = i2 & i2 = i2 holds i1 = i2 & i2 = i2 & i1 = i2 & i2 = i2 implies i1 = i2
( ( s2 * p1 ) + ( s2 * p2 ) - ( s2 * p2 ) ) = ( ( r2 * p1 ) + ( s2 * p2 ) ) - ( ( s2 * p1 ) + ( s2 * p2 ) ) .= ( ( s2 * p1 ) + ( s2 * p2 ) ) - ( ( s2 * p1 ) + ( s2 * p2 ) ) ;
eval ( ( a | ( n , L ) ) *' p , x ) = eval ( a | ( n , L ) ) * eval ( p , x ) .= a * eval ( p , x ) .= a * eval ( p , x ) ;
assume that the TopStruct of S = the TopStruct of T and for D being non empty directed Subset of S , V being open Subset of S st V in V & V is open & V is open & V is open & V is open & V is open & V is open & V is open & V is open & V is open & V c= V & V is open & V is open & V is open & V is open & V c= V & V is open & V c= V & V is open & V c= V & V c= V & V c= V & V c= V & V c= V & V c= V & V c= V & V c= V & V c= V & V c= V & V c= V & V c= V & V c= V & V
assume that 1 <= k & k <= len w + 1 and TU . ( ( q11 , w ) -succ k ) = ( TU . ( ( q11 , w ) -succ k ) ) . k and TU . ( ( q11 , w ) -succ k ) = ( TU . ( ( q11 , w ) -succ k ) ) . k ;
2 * a |^ ( n + 1 ) + ( 2 * b |^ ( n + 1 ) ) >= a |^ n + ( b |^ n ) + ( a |^ n ) + ( b |^ n ) + ( a |^ n ) + ( b |^ n ) + ( a |^ n ) + ( b |^ n ) + ( a |^ n ) + ( b |^ n ) ) ;
M , v2 / ( x. 3 , m ) / ( x. 0 , m ) / ( x. 4 , m ) / ( x. 0 , m ) / ( x. 4 , m ) / ( x. 0 , m ) / ( x. 4 , m ) / ( x. 0 , m ) / ( x. 4 , m ) ) |= H ;
assume that f is_differentiable_on l and for x0 st x0 in l holds 0 < f . x0 & for x1 st x1 in l holds f . x1 - f . x0 < f . x0 & f . x0 < 0 ;
for G1 being _Graph , W being Walk of G1 , e being set , W being Walk of G , e being set st e in W holds not e in W \rm the Walk of G & e in W implies e in the carrier' of G
not not not not ( ex x1 , x2 st x1 is not empty & x2 is not empty & not ( x1 is not empty & not x2 is not empty & not x1 is not empty & not x2 is not empty ) & not ( x1 is not empty & not x2 is not empty & not x2 is not empty ) ) & not ( x1 is not empty & not x2 is not empty & not x2 is not empty ) ;
Indices GoB f = [: dom GoB f , Seg width GoB f :] & ( for i st i in dom GoB f holds ( GoB f ) * ( i , 1 ) in Indices GoB f ) & ( GoB f ) * ( i + 1 , 1 ) in Indices GoB f ) implies ( GoB f ) * ( i + 1 , 1 ) in Indices GoB f
for G1 , G2 , G3 being Group , O being stable Subgroup of G st G1 is stable & G2 is stable & G is stable holds ( G is stable iff G is stable ) & ( G is stable ) & ( G is stable implies G is stable )
UsedIntLoc ( inint f ) = { intloc 0 , intloc 1 , intloc 2 , intloc 3 , intloc 4 , intloc 3 , intloc 4 , intloc 5 , intloc 5 , intloc 6 , intloc 5 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 9 } \/ UsedIntLoc ( <* 7 , 8 , 8 , 7 , 8 , 8 , 8 , 9 } ) ;
for f1 , f2 being FinSequence of F st f1 ^ f2 is p -element & Q [ f1 ^ f2 ] & Q [ f2 ^ f1 ] holds Q [ f1 ^ f2 ] & Q [ f2 ^ f1 ] & Q [ f1 ^ f2 ]
( p `1 ) ^2 / sqrt ( 1 + ( p `1 / p `2 ) ^2 ) = ( q `1 ) ^2 / sqrt ( 1 + ( q `1 / q `2 ) ^2 ) .= ( q `1 ) ^2 / sqrt ( 1 + ( q `1 / q `2 ) ^2 ) .= ( q `1 ) ^2 / sqrt ( 1 + ( q `1 / q `2 ) ^2 ) ;
for x1 , x2 , x3 , x4 being Element of REAL n holds |( x1 - x2 , x3 - x4 )| = |( x1 - x2 , x3 - x3 )| & |( x1 - x2 , x3 - x4 )| = |( x1 , x3 )| - |( x2 , x3 )| + |( x3 , x3 )|
for x st x in dom ( ( F | A ) | A ) holds ( ( ( ( F | A ) ) | A ) . ( - x ) ) = - ( ( F | A ) | A ) . x
for T being non empty TopSpace , P being Subset-Family of T , x being Point of T , B being Basis of T st P c= the topology of T for B being Basis of x st B c= P & B c= P holds x in B
( a 'or' b 'imp' c ) . x = ( 'not' ( a 'or' b ) . x ) 'or' c . x .= 'not' ( a . x 'or' b . x ) 'or' c . x .= TRUE 'or' TRUE 'or' TRUE .= TRUE 'or' TRUE .= TRUE 'or' TRUE .= TRUE ;
for e being set st e in [: A , Y1 :] ex X1 being Subset of [: X , Y :] , Y1 being Subset of [: Y , X :] st e = [: X1 , Y1 :] & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 c= Y1 & Y1 c= Y1 & Y1 c= Y1 & Y1 c= Y1 & Y1 c= Y2
for i be set st i in the carrier of S for f be Function of Sconsider S . i , S1 . i st f = H . i holds F . i = f | ( F . i ) & F . i = f | ( F . i ) ;
for v , w st for y st x <> y holds w . y = v . y holds Valid ( VERUM ( Al ) , J ) . v = Valid ( VERUM ( Al ) , J ) . w
card D = card D1 + card D2 - card { i , j } .= ( c1 + 1 ) - ( i + 1 ) + ( c2 - 1 ) .= ( c1 + 1 ) - ( c1 + 1 ) + ( c2 - 1 ) .= ( 2 * c1 + 1 ) - ( c2 + 1 ) .= ( 2 * c1 + 1 ) - ( c2 + 1 ) .= 2 * c1 + ( c2 + 1 ) - ( c2 + 1 ) ;
IC Exec ( i , s ) = ( s +* ( 0 .--> succ ( s . 0 ) ) ) . 0 .= ( s +* ( 0 .--> ( s . 0 ) ) ) . 0 .= ( s +* ( 0 .--> ( s . 0 ) ) ) . 0 .= s . 0 .= s . 0 ;
len f /. ( ( i1 -' 1 ) + 1 ) -' 1 + 1 = len f -' ( i1 -' 1 ) + 1 + 1 .= len f -' ( i1 -' 1 ) + 1 + 1 .= len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) + 1 .= len f - ( i1 -' 1 ) + 1 ;
for a , b , c being Element of NAT st 1 <= a & 2 <= b & k < a holds k <= a + b-2 or k = a + b-2 or k = a + b-2 or k = a + b-2 or k = a + b-2 or k = a + b-2 or k = a + b-2 or k = a + b-2
for f being FinSequence of TOP-REAL 2 , p being Point of TOP-REAL 2 , i being Element of NAT st p in LSeg ( f , i ) holds Index ( p , f ) <= i & Index ( p , f ) <= i & Index ( p , f ) <= len f
lim ( curry ( ( ( curry ( ( P , k ) + 1 ) ) # x ) ) ) = lim ( ( curry ( ( P , k ) + 1 ) ) # x ) + lim ( ( curry ( ( P , k ) + 1 ) ) # x ) ) ;
z2 = g /. ( i -' n1 + 1 ) .= g . ( i -' n2 + 1 ) .= g . ( i -' n2 + 1 ) .= g . ( i -' n2 + 1 ) .= g . ( i -' n2 + 1 ) .= g . ( i -' n2 + 1 ) .= g . ( i -' n2 + 1 ) ;
[ f . 0 , f . 3 ] in id ( the carrier of G ) \/ ( the InternalRel of G ) or [ f . 0 , f . 3 ] in the InternalRel of C6 & [ f . 0 , f . 3 ] in the InternalRel of C6 ;
for G being Subset-Family of B st G = { R [ X ] where R is Subset of [: A , B :] : R in F6 & for X being Subset of [: A , B :] holds ( for Y being Subset of [: A , B :] st X in F6 holds X c= Y ) holds ( for Y being Subset of A holds Y in F ) implies Y c= X
CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 ) ) = CurInstr ( P1 , Comput ( P1 , s1 , m2 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m2 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m2 ) ) .= CurInstr ( P1 , s1 ) .= CurInstr ( P1 , s1 ) .= CurInstr ( P1 , s1 ) .= CurInstr ( P1 , s1 ) ;
assume that a on M and b on M and c on N and d on M and p on N and a on M and a on M and c on N and a on M and b on N and a on M and a on M and b on N and a on M and c on N and a on M and a on N and b on M and a on N and b on N and a on M and a on N and b on N and a on M and a on N and b on M and b on N and a on M and b on N and b on N and a on N and a on M and b on N and b on N and a on N and b on N and a on M and b on N and a on M and b on N and a on N and b on N and a on N and c on N and c on N and c on N and c on N
assume that T is \hbox 4 -Z and ex F be Subset-Family of T , F be Subset-Family of T st F is closed & F is finite-ind & ind F <= 0 & ind F <= 0 & ind F <= 0 & ind F <= 0 & ind F <= 0 ;
for g1 , g2 st g1 in ]. r - g2 , r .[ & g2 in ]. r - r , r .[ holds |. f . g1 - f . g2 .| <= ( g1 - f . g2 ) / ( |. r - g .| + r )
( ( - 1 / 2 ) * ( z + z2 ) ) = ( - ( - 1 / 2 ) * ( z + z2 ) ) * ( z + z2 ) .= ( - ( - 1 / 2 ) * ( z + z2 ) ) * ( z + w ) .= ( - ( - 1 / 2 ) * ( z + w ) ) * ( z + w ) ;
F . i = F /. i .= 0. R + r2 .= ( b |^ n ) * r2 .= ( b |^ n ) |^ ( n + 1 ) .= <* ( n + 1 ) -tuples_on ( n + 1 ) , \dots , ( n + 1 ) -tuples_on ( n + 1 ) , \dots , ( n + 1 ) -tuples_on ( n + 1 ) ) ;
ex y being set , f being Function st y = f . n & dom f = A ( ) & for n holds f . ( n + 1 ) = R ( ) . ( f . n , f . n ) & for n holds f . ( n + 1 ) = R ( n , f . n , f . n ) ;
func f (#) F -> FinSequence of V means : : : for i be Nat st i in dom it holds it . i = F /. i * F /. i & for i be Nat st i in dom it holds it . i = F /. i * F /. ( F /. i ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , 7 , 8 } = { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , 7 } \/ { x5 , 7 , 8 }
for n being Nat for x being set st x = h . n holds h . ( n + 1 ) = o ( x , n ) & x in InputVertices S ( x , n ) & o ( x , n ) in InnerVertices S ( x , n ) & o ( x , n ) in InnerVertices S ( x , n ) & o ( x , n ) in InnerVertices S ( x , n ) ;
ex S1 being Element of CQC-WFF ( Al ) st SubP ( P , l , e ) = S1 & ( S , e ) `1 = ( S , e ) `1 & ( S , e ) `2 = ( S , e ) `2 & ( S , e ) `2 = ( S , e ) `2 & ( S , e ) `1 = ( S , e ) `1 & ( S , e ) `1 = ( S , e ) `1 & ( S , e ) `2 = ( S , e ) `1 & ( S , e ) `1 = ( S , e ) `1 & ( S , e ) `1 = ( S , e ) `1 & ( S , e ) `1 = ( S , e ) `1 & ( S , e ) `1 = ( S , e ) `1 & ( S , e ) `1 & ( S , e )
consider P being FinSequence of GL2 such that pon = Product P and for i st i in dom P ex t7 being Element of the carrier of G st P . i = t7 & t . i = P . i & P . ( t . i ) = P . ( t . i ) ;
for T1 , T2 being strict non empty TopSpace , P being Basis of T1 , Q being Basis of T2 st the carrier of T1 = the carrier of T2 & P = the topology of T2 & P = the topology of T1 & P = the topology of T2 holds P = Q
assume that f is_is_is_or u0 in dom pdiff1 ( f , 3 ) and r (#) pdiff1 ( f , 3 ) is_partial_differentiable_in u0 , 2 and partdiff ( r (#) pdiff1 ( f , 3 ) , u0 , 2 ) = r (#) pdiff1 ( f , u0 , 3 ) and partdiff ( r (#) pdiff1 ( f , u0 , 3 ) , u0 , 3 ) = r (#) pdiff1 ( f , u0 , 3 ) ;
defpred P [ Nat ] means for F , G being FinSequence of bool REAL , G be Permutation of bool ( dom F ) st len G = $1 & G = F * s & not F = G * s holds Sum ( F ) = Sum ( G ) & Sum ( F ) = Sum ( G ) ;
ex j st 1 <= j & j < width GoB f & ( GoB f ) * ( 1 , j ) `2 <= s & s * ( 1 , j + 1 ) `2 <= ( GoB f ) * ( 1 , j + 1 ) `2 & s * ( 1 , j + 1 ) `2 <= ( GoB f ) * ( 1 , j + 1 ) `2 ;
defpred U [ set , set ] means ex F-23 be Subset-Family of T st $2 = F-23 & union F-23 is open & union F\times ( F . $1 ) = union ( F\times ( F . $1 ) ) & union F\times ( F . $1 ) c= union ( F\times ( F . $1 ) ) & union ( F\times ( F . $1 ) ) c= union ( F\times ( F . $1 ) ) ;
for p4 being Point of TOP-REAL 2 st LE p4 , p4 , P , p1 , p2 & LE p4 , p , P , p1 , p2 holds LE p4 , p , P , p1 , p2 & LE p4 , p , P , p1 , p2 & LE p4 , p , P , p1 , p2 & LE p4 , p , P , p1 , p2 & LE p4 , p , P , p1 , p2 & LE p4 , p , P , p1 , p2 & LE p1 , p , P , p1 , p2 , p2 , p2 , p2 & LE p2 , p1 , P , p1 , p2 , p2 , p2 , p2 , p2 , p2 , p2 , p3 , p2 , p3 , p3 , p3 , p2 & LE p4 , p1 , p2 & LE p4 , p1 , P , p1 , p2 , p3 , p3 , p3 , p1 , p3 , p3 , p3 , p3 , p3 , p1 , p3 , C & LE p1
f in D ( ) & for g st g in D ( ) & x = f . y holds g in D ( ) implies f in D ( ) & for f st f in D ( ) & f in D ( ) & f in D ( ) & f in D ( ) & f in D ( ) & f in D ( ) & f in D ( ) implies f in D ( )
ex 8 being Point of TOP-REAL 2 st x = 8 & ( ( ( - 1 ) * ( |. 8 .| ) ) / ( 1 + ( |. 8 .| ) ) ) ^2 >= 8 & ( ( - 1 ) * ( |. 8 .| ) ) / ( 1 + ( |. 8 .| ) ) ^2 ) >= 0 ) & ( ( - 1 ) * ( |. 8 .| ) ) ^2 >= 0 ;
assume for d7 being Element of NAT st d7 <= max ( n7 , ( n7 ) -d7 ) holds s1 . ( ( d7 ) -s7 ) = s2 . ( ( d7 ) -s7 ) & s2 . ( ( d7 ) -s7 ) = s2 . ( ( d7 ) -s7 ) ;
assume that s <> t and s is Point of Sphere ( x , r ) and s is Point of Sphere ( x , r ) and ex e being Point of Sphere ( x , r ) st e = Sphere ( s , r ) /\ Sphere ( x , r ) and e = Sphere ( e , r ) /\ Sphere ( x , r ) ;
given r such that 0 < r and for s st 0 < s ex x1 be Point of CNS st x1 in dom f & ||. x1 - x0 .|| < s & |. f /. x1 - f /. x0 .| < r & |. f /. x0 - f /. x0 .| < r ;
( p | x ) | ( p | ( ( x | x ) | x ) ) = ( ( ( x | x ) | x ) | p ) | ( ( ( x | x ) | x ) | p ) .= ( ( ( x | x ) | x ) ) | p ) | p ;
assume that x , x + h / 2 in dom sec and ( for x st x in dom sec holds sec . x = ( 4 * ( sec . x + h / 2 ) ) * sin . x + ( sin . x ) ^2 and sin . x = ( 4 * ( sin . x + cos . x ) ) * sin . x + ( sin . x ) ^2 and sin . x = ( 4 * ( sin . x + cos . x ) ^2 ;
assume that i in dom A and len A > 1 and B c= the carrier of \HM { i , j } and A c= the carrier of \HM { i , j } and B c= the carrier of G and A c= the carrier of G and B c= the carrier of G and A c= the carrier of G and B c= the carrier of G and A c= the carrier of G and B c= the carrier of G ;
for i be non zero Element of NAT st i in Seg n holds i divides n or i = <* 1. F_Complex *> or i divides n & i <> 0. F_Complex & i <> 0. F_Complex & i <> 0. F_Complex & i <> 0. F_Complex & i <> 0. F_Complex & i <> 0. F_Complex & i <> 0. F_Complex & i <> 0. F_Complex & i <> 0. F_Complex & i <> 0. F_Complex & i <> 0. F_Complex holds h . i = 1. F_Complex
( ( b1 'imp' b2 ) '&' ( c1 'imp' c2 ) ) '&' ( ( b1 'or' c1 ) '&' ( c1 '&' c2 ) '&' ( c1 '&' c2 ) ) '&' ( ( b1 'or' c1 ) '&' ( c1 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c1 '&' c2 ) '&' 'not' ( c1 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c2 '&' ( c1 '&' c2 ) '&' 'not' ( c2 '&' ( c1 '&' c2 ) '&' ( c1 '&' c2 ) '&' ( c1
assume for x holds f . x = ( ( cot * ( cot - cot ) ) `| Z ) . x & x in dom ( ( cot * ( cot - cot ) ) `| Z ) & for x st x in Z holds ( ( ( ( - cot * ( cot - cot ) ) `| Z ) . x ) = cos . ( x- ( x ) ) / ( sin . x ) ^2 ) ) & ( ( - cot * ( cot - cot ) ) ^2 = 0 & ( - cot * ( cot - cot ) ) . x = - cos . ( cot ( x ) ) ^2 = - sin ( x ) ^2 - sin . x ) ^2 = - cos . ( cot ( x ) ^2 - sin . ( cot . ( cot . x ) ^2 - cos . ( cot . ( cot ) & ( - cot . ( cot ) & ( - cot . ( cot . ( cot . ( cot ) ) = - cos
consider Rd , I-8 be Real such that Rd = Integral ( M , Re ( F . n ) ) and Id = Integral ( M , Im ( F . n ) ) and Integral ( M , Im ( F . n ) ) = Integral ( M , Im ( F . n ) ) and Integral ( M , Im ( F . n ) ) = Integral ( M , Im ( F . n ) ) + Integral ( M , Im ( F . n ) ) ;
ex k be Element of NAT st k = k & 0 < d & for q be Element of product G st q in X & ||. qthesis - partdiff ( f , x , k ) .|| < r holds ||. partdiff ( f , x , k ) - partdiff ( f , x , k ) .|| < r ;
x in { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , 7 , 8 } iff x in { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , 7 , 7 , 7 , 7 , 8 }
G * ( j , i ) `2 = G * ( 1 , i ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j + 1 ) `2 .= G * ( 1 , j + 1 ) `2 .= G * ( 1 , j + 1 ) `2 .= G * ( 1 , j + 1 ) `2 .= G * ( 1 , j + 1 ) `2 .= G * ( 1 , j + 1 ) `2 .= G * ( 1 , j + 1 ) `2 .= G * ( 1 , j + 1 ) `2 .= G * ( 1 , j + 1
f1 * p = p .= ( ( the Arity of S1 ) * ( the Arity of S2 ) ) . o .= ( the Arity of S1 ) . ( ( the Arity of S2 ) . o ) .= ( the Arity of S1 ) . ( ( the Arity of S1 ) . o ) .= ( the Arity of S1 ) . ( ( the Arity of S1 ) . o ) .= ( the Arity of S1 ) . ( ( the Arity of S1 ) . o ) .= ( the Arity of S1 ) . ( ( the Arity of S1 ) . o ) .= ( the Arity of S1 ) . ( ( the Arity of S1 ) . ( ( the Arity of S1 ) . ( ( the Arity of S1 ) . ( ( the Arity of S1 ) . ( ( the Arity of S1 ) . o ) .= ( the Arity of S1 ) . ( ( the Arity of S1 ) . o ) .= ( the Arity of S1 ) ) . ( ( the Arity of
func tree ( T , P , T1 ) -> DecoratedTree means : : : for q st q in it holds p ^ q in T or ex p , q st p in P & q in T & p ^ q in T & p ^ q in T & p ^ q in T & q ^ p in T ;
F /. ( k + 1 ) = F . ( k + 1 ) .= Fimplies ( p . ( k + 1 -' 1 ) ) + 1 = Fimplies p . ( k + 1 -' 1 ) = F^2 + ( p . ( k + 1 -' 1 ) ) .= F^2 + ( p . ( k + 1 -' 1 ) ) .= F^2 + ( p . ( k + 1 -' 1 ) ) .= ( p . k ) + 1 ;
for A , B , C being Matrix of K st len B = len C & width B = width C & len B = width C & len A = width B & len B = len A & len C = width B & width B = width C & len B = len B & width B = width C & width B = width C & len B = len B & width B = width C & width B = width C holds B * C = C * ( B , C , B )
seq . ( k + 1 ) = 0. F_Complex + seq . ( k + 1 ) .= ( Partial_Sums ( seq ) ) . ( k + 1 ) + ( Partial_Sums ( seq ) ) . ( k + 1 ) .= ( Partial_Sums ( seq ) ) . ( k + 1 ) + ( Partial_Sums ( seq ) ) . ( k + 1 ) ;
assume that x in ( the carrier of Cq ) /\ ( the carrier of Cq ) and y in ( the carrier of Cq ) /\ ( the carrier of Cq ) and [ x , y ] in the carrier of Cq and [ y , z ] in the InternalRel of Cq ;
defpred P [ Element of NAT ] means for f st len f = $1 holds ( VAL g ) . ( k + 1 ) = ( VAL g ) . ( k + 1 ) '&' ( VAL g ) . ( k + 1 ) '&' ( VAL g ) . ( k + 1 ) ;
assume that 1 <= k and k + 1 <= len f and f is_sequence_on G and [ i , j ] in Indices G and [ i + 1 , j ] in Indices G and f /. k = G * ( i , j ) and f /. ( k + 1 ) = G * ( i , j ) and f /. ( k + 1 ) = G * ( i , j ) ;
assume that sn < 1 and q `1 > 0 and ( q `2 / |. q .| - sn ) / ( 1 + sn ) >= 0 and ( q `2 / |. q .| - sn ) / ( 1 + sn ) >= 0 or ( q `2 / |. q .| - sn ) / ( 1 + sn ) >= 0 & ( q `2 / |. q .| - sn ) / ( 1 + sn ) >= 0 ) ;
for M being non empty dist , x being Point of M , f being Point of M , x being Point of M st x = x ` holds ex f being sequence of TopSpaceMetr ( M ) st f is sequence of ( M ) & for n being Element of NAT holds f . n = Ball ( x , r )
defpred P [ Element of omega ] means $1 is differentiable of omega & ( f1 is_differentiable_on Z implies f1 - f2 is_differentiable_on Z & for x st x in Z holds ( f1 - f2 ) . x = f1 . x - ( f2 . x ) ) & ( f1 - f2 ) is_differentiable_on Z implies f1 - f2 is_differentiable_on Z & for x st x in Z holds ( ( f1 - f2 ) `| Z ) . x = ( ( f1 - f2 ) `| Z ) . x - ( f2 . x )
defpred P1 [ Nat , Point of CNS ] means $2 in Y & ||. s1 . $1 - ( lim s1 ) .|| < r & ||. ( f /. $1 ) - ( f /. ( $1 + 1 ) ) .|| < r & ||. ( f /. $1 ) - ( f /. ( $1 + 1 ) ) .|| < r ;
( f ^ mid ( g , 2 , len g ) ) . i = ( mid ( g , 2 , len g ) ) . ( i -' len f + 1 ) .= g . ( i -' len f + 1 ) .= g . ( i -' len f + 1 ) .= g . ( i -' len f + 1 ) .= g . ( i -' len f + 1 ) .= g . ( i -' len f + 1 ) ;
( 1 / ( 2 * n0 + 2 * n0 ) ) * ( 2 * n0 + 2 * ( n0 + 1 ) ) = ( 1 / ( 2 * n0 + 2 * n0 ) ) * ( 2 * n0 + ( n0 + 1 ) ) * ( 2 * n0 + ( n0 + 1 ) ) * ( 2 * n0 + ( n0 + 1 ) ) * ( 2 * n0 + 1 ) ) * ( 2 * n0 + 1 ) * ( 2 * n0 + 1 ) * ( 2 * n0 + 1 ) * ( 2 * n0 + 1 ) * ( 2 * n0 + 1 ) * ( 2 * n0 + 1 ) * ( 2 * n0 + 1 ) * ( 2 * n0 + 1 ) * ( 2 * n0 + 1 ) * ( 2 * n0 ) * ( 2 * n0 ) = ( ( 2 * n0 ) * ( 2 * n0 ) * ( 2 * n0 ) * ( 2 * n0 ) * ( 2 * n0 ) * ( 2 * n0 ) * ( 2 * n0
defpred P [ Nat ] means for G being non empty strict finite symmetric RelStr , H being strict symmetric RelStr st G is non empty for G being strict strict strict strict symmetric RelStr st G = ( the carrier of G ) \/ the carrier of H holds the RelStr of G = ( the RelStr of G ) \/ the InternalRel of H ;
assume that not f /. 1 in Ball ( u , r ) and 1 <= m and m <= len ( f ) and for i st 1 <= i & i <= len ( f ) & ( f /. i ) <> {} & ( f /. i ) <> {} & ( f /. ( len f ) ) <> {} & ( f /. ( len f ) ) <> {} & ( f /. ( len f ) <> {} ) & ( f /. ( len f ) <> {} & ( f /. ( len f ) <> {} & ( f /. ( len f ) <> {} & f /. ( len f ) <> {} & f /. ( len f ) <> {} & f /. ( len f ) <> {} & ( f /. ( len f ) <> {} & ( f /. ( len f ) <> {} & ( f /. ( len f ) & ( f /. ( len f ) & ( f /. ( len f ) & ( f /. ( len f ) & ( f /. ( len f ) & ( f /. ( len f )
defpred P [ Element of NAT ] means ( Partial_Sums ( cos ) ) . $1 = ( Partial_Sums ( cos ) ) . ( 2 * $1 ) & ( Partial_Sums ( cos ) ) . ( 2 * $1 ) = ( Partial_Sums ( cos ) ) . ( 2 * $1 ) ) * ( ( cos ) ) . ( 2 * $1 ) ;
for x being Element of product F holds x is FinSequence of G & ( for i being set st i in dom F holds x . i = I ) & for i being set st i in dom F holds x . i = ( F . i ) . x ) & ( for i being set st i in dom F holds F . i = ( F . i ) . x )
( x " ) |^ ( n + 1 ) = ( x " ) |^ n * x " .= ( x |^ n ) |^ n * x .= ( x |^ n ) |^ n * x .= ( x |^ n ) |^ n * x .= ( x |^ n ) |^ n * x .= ( x |^ n ) |^ n * x .= ( x |^ n ) |^ n ;
DataPart Comput ( P +* ( I , s ) , LifeSpan ( P +* I , Initialized s ) + 3 ) = DataPart Comput ( P +* I , s , LifeSpan ( P +* I , s ) + 3 ) .= DataPart Comput ( P +* I , s , LifeSpan ( P +* I , s ) + 3 ) .= DataPart Comput ( P +* I , s , LifeSpan ( P +* I , s ) + 3 ) ;
given r such that 0 < r and ]. x0 - r , x0 .[ c= ( dom f1 /\ dom f2 ) and for g st g in ]. x0 - r , x0 .[ holds f1 . g <= f2 . g & f2 . g <= ( f1 . g ) / ( g . g ) and for r st r in ]. x0 - r , x0 .[ holds f1 . r <= f2 . g ;
assume that X c= dom f1 /\ dom f2 and f1 | X is continuous and f2 | X is continuous and ( f1 | X is continuous & f2 | X is continuous or f1 | X is continuous & f2 | X is continuous & f2 | X is continuous & ( f1 | X is continuous & f2 | X is continuous & not f1 | X is continuous ) & not f2 | X is continuous & not f2 | X is continuous & not f2 | X is continuous ;
for L being continuous complete LATTICE for l being Element of L ex X being Subset of L st l = sup X & for x being Element of L st x in X holds x is prime & for x being Element of L st x in X holds x is prime & x is prime & x is prime & x is prime & x is prime
Support ( e *' A ) in { Support ( m *' p ) where m is Polynomial of n , L : ex p being Polynomial of n , L st p in Support ( m *' p ) & p in Support ( m *' p ) } & p in Support ( m *' q ) } ;
( f1 - f2 ) /. ( lim s1 ) = lim ( f1 /* s1 ) - ( f2 /* s1 ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) ;
ex p1 being Element of CQC-WFF ( Al ) st F . p1 = g . p1 & for g being Function of V ( ) , D ( ) st P [ g , ( len p1 ) qua Function ] holds P [ g , f . ( len p1 ) ] ;
( mid ( f , i , len f -' 1 ) ^ <* f /. j *> ) /. j = ( mid ( f , i , len f -' 1 ) ) /. j .= ( mid ( f , i , len f -' 1 ) ) . j .= ( mid ( f , i , len f -' 1 ) ) . j .= f /. j ;
( ( p ^ q ) ^ r ) . ( len p + k ) = ( ( p ^ q ) . ( len p + k ) ) . ( len q + k ) .= ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( k + 1 ) ;
len mid ( D2 , D1 , indx ( D2 , D1 , j ) + 1 ) = indx ( D2 , D1 , j ) + ( indx ( D2 , D1 , j ) + 1 ) .= indx ( D2 , D1 , j ) + 1 .= indx ( D2 , D1 , j ) + 1 .= indx ( D2 , D1 , j ) + 1 ;
x * y * z = Mz * ( y * z ) .= ( x * ( y * z ) ) * ( y * z ) .= ( x * ( y * z ) ) * ( y * z ) .= ( x * ( y * z ) ) * ( y * z ) .= ( x * ( y * z ) ) * ( y * z ) ;
v . ( <* x , y *> ) - ( <* x0 , y0 *> ) * i = partdiff ( v , ( x - x0 ) ) * ( ( x - x0 ) * ( x - x0 ) + ( proj ( 1 , 1 ) * ( x - x0 ) ) ) + ( proj ( 1 , 1 ) * ( x - x0 ) ) * ( ( x - x0 ) * ( x - x0 ) + ( y - x0 ) * ( x - x0 ) ) ;
i * i = <* 0 * ( - 1 ) * ( 0 - 1 ) * 0 , 0 * 0 + 0 * 0 , 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 , 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 , 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 , 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 , 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0
Sum ( L (#) F ) = Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) F1 ) + Sum ( L (#) F2 ) .= Sum ( ( L (#) F1 ) ^ ( L (#) F2 ) ) .= Sum ( L (#) F1 ) + Sum ( L (#) F2 ) .= Sum ( L (#) F1 ) + Sum ( L (#) F2 ) .= Sum ( L (#) F1 ) + Sum ( L (#) F2 ) .= Sum ( L (#) F1 ) ;
ex r be Real st for e be Real st 0 < e ex Y1 be finite Subset of X st Y1 is non empty & Y c= Y & for Y1 be finite Subset of X st Y1 c= Y & Y1 c= Y holds |. ( - r ) * ( Y1 ) - ( r * ( Y1 ) ) .| < r * ( Y1 )
( GoB f ) * ( i , j + 1 ) = f /. ( k + 2 ) & ( GoB f ) * ( i , j + 1 ) = f /. ( k + 2 ) or ( GoB f ) * ( i , j + 1 ) = f /. ( k + 2 ) & ( GoB f ) * ( i + 2 ) = f /. ( k + 2 ) ) implies ( GoB f ) * ( i , j ) = f /. ( k + 2 )
( ( - cos ) / ( 1 - r ) ) * ( 1 / ( 1 - r ) ) = ( - 1 ) * ( 1 / ( 1 - r ) ) .= ( - 1 ) * ( 1 / ( 1 - r ) ) .= ( - 1 ) * ( 1 / ( 1 - r ) ) .= ( - 1 ) * ( 1 / ( 1 - r ) ) .= ( - 1 ) * ( 1 / ( 1 - r ) ) .= ( - 1 ) * ( 1 / ( 1 - r ) ) .= ( - 1 ) * ( 1 / ( 1 - r ) .= ( - 1 ) * ( 1 / ( 1 - r ) ) * ( 1 - r ) .= ( - 1 ) * ( 1 - r ) * ( 1 - r ) * ( 1 - r ) * ( 1 - r ) * ( 1 / ( 1 - r ) .= ( 1 - r ) * ( 1 - r ) .= ( 1 - r ) * ( 1 / r ) .= ( 1 / r ) * ( 1 / r ) .= ( 1 - r ) * ( 1 - r )
( - b ) + sqrt ( delta ( a , b , c ) ) / ( 2 * a ) < 0 & ( - b ) * sqrt ( delta ( a , b , c ) ) < 0 or ( - b ) * sqrt ( delta ( a , b , c ) ) < 0 ) implies ( - b ) * sqrt ( - a , c ) ) / ( 2 * a ) < 0
assume that ex_inf_of uparrow "\/" ( X /\ C ) , L and ex_sup_of X , L and ex_sup_of X , L and "\/" ( X , L ) = "/\" ( uparrow "\/" ( X /\ C ) , L ) and for X st X in X holds "\/" ( X , L ) = "/\" ( uparrow "\/" ( X , L ) , L ) and for X st X in X holds "\/" ( X , L ) = "/\" ( X , L ) ;
( for j holds ( j = i implies j = j ) implies ( j = j implies i = j ) ) & ( j = i implies j = i implies j = j ) & ( j = i implies j = j ) & ( j = i implies j = i ) & ( j = j implies j = i implies j = j ) )