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thesis ;
thesis ;
contradiction ;
contradiction ;
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thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
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contradiction ;
contradiction ;
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contradiction ;
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contradiction ;
thesis ;
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contradiction ;
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contradiction ;
thesis ;
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contradiction ;
thesis ;
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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 one-to-one ;
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 NAT ;
assume f is equals g ;
not x in Y ;
z = +infty ;
k be Nat ;
K ` is being_line ;
assume n >= N ;
assume n >= N ;
assume X is 1 -element ;
assume x in I ;
q is \mathopen by 0 ;
assume c in x ;
p > 0 ;
assume x in Z ;
assume x in Z ;
1 <= kLet ;
assume m <= i ;
assume G is cyclic ;
assume a divides b ;
assume P is closed ;
\bf 2 > 0 ;
assume q in A ;
W is not bounded ;
f is IC one-to-one ;
assume A is boundary ;
g is_sequence_on G ;
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 <= \rm 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 ;
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 ;
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 iff n1 is iff n2 is 0 ;
Q halts_on s ;
x in such that x in \in \in \in \in \in \in \mathop { \rm } ;
M < m + 1 ;
T2 is open ;
z in 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 , b be Real ;
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 ;
let E be Ordinal ;
o OperSymbol 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 RealUnitarySpace , W be Subspace of V , v be VECTOR of V ;
not s in Y |^ 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 RealUnitarySpace , W be Subspace of V ;
P [ 1 ] ;
P [ {} ] ;
C1 meets L~ f ;
H = G . i ;
1 <= i `2 + 1 ;
F . m in A ;
f . o = o ;
P [ 0 ] ;
aZ <= non empty 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 trivial ;
dom c = Q
P [ 0 ] ;
f . n in T ;
N . j in S ;
let T be complete LATTICE , f be Function of T , 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 { the carrier of S } is bounded ;
assume p = p2 ;
len f = n ;
assume x in P1 ;
i in dom q ;
let U0 , U1 , U2 ;
pp `2 = c ;
j in dom h ;
let k ;
f | Z is continuous ;
k in dom G ;
UBD C = B ;
1 <= len M ;
p in \mathbin { x } ;
1 <= jj & jj <= len G ;
set A = ex / 2 st A = as set ;
card a [= c ;
e in rng f ;
cluster B \oplus A -> empty ;
H is_subformula_of H ;
assume that n0 <= m and m <= n ;
T is increasing ;
e2 <> e2 ;
Z c= dom g ;
dom p = X ;
H is_subformula_of G ;
i + 1 <= n ;
v <> 0. V ;
A c= Affin A ;
S c= dom F ;
m in dom f ;
X0 be set ;
c = sup N ;
R reduces union M , union M ;
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 <> {} ;
let x be Element of Y ;
let f be then Int of G , n be Nat ;
not n in Seg 3 ;
assume X in f .: A ;
assume p <= n & p <= m ;
assume not u in { v } ;
d is Element of A ;
A |^ b misses B ;
e in v .. ( T ) ;
- y in I ;
let A be non empty set , B be non empty set ;
P0 = 1 ;
assume r in F . k ;
assume f is simple ;
let A be Incountable set ;
rng f c= NAT & rng f c= NAT ;
assume P [ k ] ;
f6 <> {} ;
o be Ordinal ;
assume x is sum of squares ;
assume not v in { 1 } ;
let I1 , I2 , I2 ;
assume that 1 <= j and j < l ;
v = - u ;
assume s . b > 0 ;
d1 in dom d2 ;
assume t . 1 in A ;
let Y be non empty TopStruct , X be set ;
assume a in uparrow s ;
let S be non empty RelStr ;
a , b // b , a ;
a * b = p * q ;
assume x , y are_the space ;
assume x in [#] ( f ) ;
[ a , c ] in X ;
mm <> {} ;
M + N c= M + M ;
assume M is connected hh\rbrace ;
assume f is mabrnA1 ;
let x , y be element ;
let T be non empty TopStruct ;
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 or k1 = k2 ;
m + 1 < n + 1 ;
s in S \/ { s } ;
n + i >= n + 1 ;
assume Re y = 0 ;
k1 <= j1 & j1 <= j2 ;
f | A is continuous ;
f . x \not <= 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 ;
Cj in Cj ;
q2 c= C1 & q2 c= C2 ;
a2 < c2 & b2 < c1 ;
s2 is 0 -started ;
IC s = 0 ;
s4 = s4 . ( n + 1 ) ;
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 MSAlgebra over L ;
y " <> 0 ;
y " <> 0 ;
0. V = uw ;
y2 , y , y is_collinear ;
R8 c= R8 ;
let a , b be Real , f be FinSequence of REAL ;
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 , l .] ;
let i , j be Nat ;
s be State of A , a be Element of L ;
s4 . n = N ;
set y = ( x `1 ) ^2 ;
mi in dom g ;
l . 2 = y1 ;
|. g . y .| <= r ;
f . x in Cx0 ;
V2 is non 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 A9 is dense & B is dense ;
|. f . x .| <= r ;
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 ) ;
let o be OperSymbol of S ;
let R be connected non empty RelStr ;
n + 1 = succ n ;
xH c= Z1 & xH c= Z1 ;
dom f = [: C1 , C2 :] ;
assume [ a , y ] in X ;
Re ( seq ) is convergent ;
assume a1 = b1 & a2 = b2 ;
A = Int sInt A ;
a <= b or b <= a ;
n + 1 in dom f ;
let F be Instruction of S , s be State of S ;
assume r2 > x0 & x0 < r2 ;
Y be non empty set , f be Function of Y , Z ;
2 * x in dom W ;
m in dom g2 & n + 1 in dom g2 ;
n in dom ( g1 | X ) ;
k + 1 in dom f ;
not the still of { s } is finite ;
assume x1 <> x2 & x1 <> x3 ;
v2 in V1 & v2 in V2 ;
not [ b `1 , b ] in T ;
ii + 1 = i ;
T c= such that T c= elements ( T ) ;
( l . 1 ) `1 = 0 ;
n be Nat ;
( t `2 ) ^2 = r ;
AA is_integrable_on M & AA is_integrable_on M ;
set t = Top t ;
let A , B be real-membered set ;
k <= len G + 1 ;
[: C , D :] misses [: the carrier of V , the carrier of V :] ;
product seq is non empty ;
e <= f or f <= e ;
cluster non empty ordinal for sequence of NAT ;
assume c2 = b2 & b2 = 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 convergent & lim vseq = 0 ;
IC s3 = 0 & IC s3 = 0 ;
k in N or k in K ;
F1 \/ F2 c= F ;
Int ( G1 ) <> {} ;
( z `2 ) ^2 = 0 ;
p10 <> p1 & p10 <> p1 ;
assume z in { y , w } ;
MaxADSet ( a ) c= F ;
ex_sup_of downarrow s , S ;
f . x <= f . y ;
let T be up-complete up-complete non empty reflexive antisymmetric RelStr ;
q |^ m >= 1 ;
a >= X & b >= Y ;
assume <* a , c *> <> {} ;
F . c = g . c ;
G is one-to-one one-to-one one-to-one ;
A \/ { a } c= B ;
0. V = 0. Y .= 0. V ;
let I be the <= <= S , s be <= I ;
f-24 . x = 1 ;
assume z \ x = 0. X ;
C4 = 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 ;
assume x1 in REAL & x2 in REAL ;
p1 = K1 & p2 = K1 ;
M . k = <*> REAL ;
phi . 0 in rng phi ;
OSMMit is closed ;
assume z0 <> 0. L & z0 <> 0. L ;
n < N . ( k + 1 ) ;
0 <= seq . ( 0 + 1 ) ;
- q + p = v ;
{ v } is Subset of B ;
set g = f /. 1 ;
[: R , S :] is stable Subset of R ;
set cR = Vertices R , cR = Vertices R ;
pp c= P3 & j c= P3 ;
x in [. 0 , 1 .] ;
f . y in dom F ;
let T be Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott non empty TopSpace ;
ex_inf_of the carrier of S , S ;
downarrow a = downarrow b ;
P , C , K , L ;
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 ;
assume that Y c= field Q and Y <> {} ;
a ~ , b ~ \rrangle , a ~ ;
assume a in A ( ) ;
k in dom ( ( q . k ) | X ) ;
p is $ len ( p ) $ -carrier of S ;
i - 1 = i-1 ;
f | A is one-to-one ;
assume x in f .: [: X , Y :] ;
i2 - i1 = 0 & i2 - i1 = 0 ;
j2 + 1 <= i2 + 1 ;
g " * a in N ;
K <> { [ {} , {} ] } ;
cluster -> strict for of the succ a , I ;
|. q .| ^2 > 0 ;
|. p4 .| = |. p .| ;
s2 - s1 > 0 & s2 - s1 > 0 ;
assume x in { Gik } ;
W-min C in C & W-min C in C ;
assume x in { Gik } ;
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 ;
dom S = dom F ;
let s be Element of NAT ;
let R be ManySortedSet of A ;
let n be Element of NAT ;
let S be non empty non void non void holds S is non void ;
let f be ManySortedSet of I ;
let z be Element of COMPLEX , x be Element of COMPLEX ;
u in { ag } ;
2 * n < 2 * n ;
let x , y be set ;
B-11 c= V1 \/ V2 ;
assume I is_closed_on s , P ;
U1 = U2 & U2 = U1 ;
M /. 1 = z /. 1 ;
x9 = x9 & x9 = y9 ;
i + 1 < n + 1 + 1 ;
x in { {} , <* 0 *> } ;
f7 <= f7 & f7 <= len f7 ;
l be Element of L ;
x in dom ( F . d ) ;
let i be Element of NAT ;
r8 is ( len r8 ) -element ;
assume <* o2 , o *> <> {} ;
s . x |^ 0 = 1 ;
card ( K1 \/ { p } ) in M ;
assume X in U & Y in U ;
let D be \rangle of Omega ;
set r = { Seg k + 1 } ;
y = W . ( 2 * x ) ;
assume dom g = cod f ;
let X , Y be non empty TopStruct , f be Function of X , Y ;
x \oplus A is an interval ;
|. <*> A .| . a = 0 ;
cluster strict Sublattice L -> strict ;
a1 in B . s1 & a2 in B . s2 ;
let V be finite finite VectSp of F , W be Subspace of V ;
A * B on B , A ;
f-3 = NAT --> 0 ;
let A , B be Subset of V ;
z1 = P1 . j .= ( P1 ^ ) . j ;
assume f " P is closed ;
reconsider j = i as Element of M ;
a , b be Element of L ;
assume q in A \/ ( B "\/" C ) ;
dom ( F * C ) = o ;
set S = ( INT , X ) --> NAT ;
z in dom ( A --> y ) ;
P [ y , h . y ] ;
{ x0 } c= dom f ;
let B be non-empty ManySortedSet of I , A be non-empty ManySortedSet of I ;
( PI / 2 ) < Arg z ;
reconsider z9 = 0 as Nat ;
LIN a , d , c ;
[ y , x ] in IF ;
( Q Q ) * ( 1 , 1 ) = 0 ;
set j = x0 div m , n = m div n ;
assume a in { x , y , c } ;
j2 - ( j - 1 ) > 0 ;
I \! \mathop { + } phi = 1 ;
[ y , d ] in F-8 ;
let f be Function of X , Y ;
set A2 = ( B |^ C ) * ( A |^ C ) ;
s1 , s2 are_Following ( s1 , s2 ) ;
j1 -' 1 = 0 & j1 - 1 = 0 ;
set m2 = 2 * n + j ;
reconsider t = t as bag of n ;
I2 . j = m . j ;
i |^ s , n n are_relative_prime ;
set g = f | D-21 ;
assume X is lower & 0 <= r ;
( p1 `1 ) ^2 = 1 ^2 + ( p1 `2 ) ^2 ;
a `2 < ( p3 `1 ) ^2 + ( p3 `2 ) ^2 ;
L \ { m } c= UBD C ;
x in Ball ( x , 10 ) ;
not a in LSeg ( c , m ) ;
1 <= i1 -' 1 + 1 - 1 ;
1 <= i1 -' 1 + 1 - 1 ;
i + i2 <= len h ;
x = W-min ( P ) .= ( W-min ( P ) ) . x ;
[ x , z ] in X ~ Z ;
assume y in [. x0 , x .] ;
assume p = <* 1 , 2 , 3 *> ;
len <* A1 *> = 1 ;
set H = h . ( g9 . ( g . x ) ) ;
card b * a = |. a .| ;
Shift ( w , 0 ) |= v ;
set h = h2 (*) h1 , h1 = h1 (*) h2 ;
assume x in ( X /\ X1 ) ;
||. h .|| < d1 & ||. h .|| < d1 ;
not x in the carrier of f ;
f . y = F ( y ) ;
for n holds X [ n ] ;
k - l = k\leq kl2 ;
<* p , q *> /. 2 = q ;
let S be Subset of the lattice of Y ;
P , Q be >= >= >= >= m ;
Q /\ M c= union ( F | M )
f = b * canFS ( S ) ;
let a , b be Element of G ;
f .: X <= f . sup X ;
let L be non empty reflexive RelStr , D be Subset of L ;
S-20 is x -basis of i ;
let r be non positive Real ;
M , v |= x \hbox { y } ;
v + w = 0. ( Z , n ) ;
P [ len F ] implies P [ F ( len F ) ]
assume InsCode ( i ) = 8 or InsCode i = 8 ;
the zero of M = 0 & the carrier of M = 0 ;
cluster z * seq -> summable ;
let O be Subset of the carrier' of C ;
||. f .|| | X is continuous ;
x2 = g . ( j + 1 ) ;
cluster non empty non empty ex S being non empty set ;
reconsider l1 = l- 1 as Nat ;
v4 is Vertex of r2 & v4 is Vertex of r2 ;
T2 is SubSpace of T2 & T1 is SubSpace of T2 ;
Q1 /\ Q19 <> {} ;
k be Nat ;
q " is Element of X & q " is Element of X ;
F . t is set of zero & F . t = 0. ;
assume n <> 0 & n <> 1 ;
set en = EmptyBag n , em = EmptyBag n ;
let b be Element of Bags n ;
assume for i holds b . i is commutative ;
x is root of ( p `2 ) * ( p `2 ) ;
not r in ]. p , q .[ ;
let R be FinSequence of REAL , a be Element of R ;
S7 does not destroy b1 ;
IC SCM R <> a & IC R <> b ;
|. - - |[ x , y ]| .| >= r ;
1 * seq = seq . ( n + 1 ) ;
x be FinSequence of NAT , k 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 ;
H + G = F- ( GG ) ;
CS1 . x = x2 & CS2 . x = y2 ;
f1 = f . ( len f1 ) .= f2 . ( len f1 ) ;
Sum <* p . 0 *> = p . 0 ;
assume v + W = v + u + W ;
{ a1 } = { a2 } ;
a1 , b1 _|_ b , a ;
d1 , o _|_ o , a3 ;
Ix is_reflexive implies ( x in Cx ) & ( y in I implies x in I )
In is_antisymmetric implies In is_antisymmetric & In is_reflexive implies n + 1 in dom ( n + 1 )
upper_bound rng ( H1 - H2 ) = e - ( upper_bound rng H1 - lower_bound rng H2 ) ;
x = ( a * a9 ) * ( b * b9 ) ;
|. p1 .| ^2 >= 1 ^2 ;
assume j2 -' 1 < j2 + 1 - 1 ;
rng s c= dom ( f1 + f2 ) ;
assume support a misses support b & support b misses support b ;
let L be associative non empty doubleLoopStr , M be Matrix of L ;
s " + 0 < n + 1 ;
p . c = f . 1 .= p . c ;
R . n <= R . ( n + 1 ) ;
Directed I = I1 +* ( 1 , card I ) ;
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 <* *> . ( N , S ) -> complete for non trivial TopSpace ;
( 1 - a ) " = a ;
( q . {} ) `1 = o ;
n - ( i - 1 ) > 0 ;
assume ( 1 - t ) ^2 <= 1 ^2 ;
card B = k + 1-1 ;
x in union rng ( f | n ) ;
assume x in the carrier of R ;
d in D ;
f . 1 = L . ( F . 1 ) ;
the vertices of G = { v } ;
let G be : ww] ;
e , v9 be set ;
c . ( i - 1 ) in rng c ;
f2 /* q is divergent_to+infty & f2 /* ( f1 /* s ) is divergent_to+infty ;
set z1 = - z2 , z2 = - z2 , z2 = - z2 , z2 = - z2 , z1 = - z2 , z2 = - z2 , z2 = - z2 , z2 = - z2 , z2
assume w is_llcluster S , G ;
set f = p \! \mathop { t } , g = p \! \mathop { t } , h = p \! \mathop { t } , p = p \! \mathop { t } , p = p \! \mathop
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 I1 be Subset-Family of X ;
reconsider p = p as Element of NAT ;
v , w be Point of X ;
let s be State of SCM+FSA , I be Program of SCM+FSA ;
p is FinSequence of NAT & q is FinSequence of NAT ;
stop I c= P +* stop I , P +* stop I ;
set ci = f^ ( f /. i ) ;
w ^ t seq ^ s ^ w ^ w ^ s ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w
W1 /\ W = W1 /\ W2 ;
f . j is Element of J . j ;
let x , y be Element of T2 ;
ex d st a , b // b , d ;
a <> 0 & b <> 0 & c <> 0 ;
ord x = 1 & x is positive ;
set g2 = lim ( seq , n ) ;
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 . ( F . k ) ) = 0 ;
/ ( X \/ R1 ) = / ( X \/ R2 ) ;
( ( sin - cos ) `| Z ) . x <> 0 ;
( ( exp_R * exp_R ) `| Z ) . x > 0 ;
o1 in [: X , Y :] /\ [: X , Y :] ;
e , v9 be set ;
r3 > ( 1 - 2 ) * 0 ;
x in P .: ( F -ideal ) ;
J be closed Subset of R , I be Ideal of R ;
h . p1 = f2 . O .= ( f . O ) `1 ;
Index ( p , f ) + 1 <= j ;
len ( q ^ <* x *> ) = width M ;
the carrier of LK c= A ;
dom f c= union rng ( F | X ) ;
k + 1 in support ( ( n ) | k ) ;
let X be ManySortedSet of the carrier of S ;
[ x `1 , y `2 ] in ( InnerVertices R ) ;
i = D1 or i = D2 or i = D1 ;
assume a mod n = b mod n ;
h . x2 = g . ( x1 - x2 ) ;
F c= 2 |^ the carrier of X implies F is closed
reconsider w = |. s1 .| as Real_Sequence of REAL ;
( 1 / ( m * m + r ) ) < p ;
dom f = dom ( I-4 * f ) ;
[#] ( P-17 ) = [#] ( ( TOP-REAL 2 ) | K1 ) ;
cluster - x -> ExtReal -> extended real ;
then { d1 } c= A ;
cluster TOP-REAL n -> finite-ind for non empty Subset of TOP-REAL n ;
w1 be Element of M ;
x be Element of dyadic ( n ) ;
u in W1 & v in W2 implies u in W2
reconsider y = y as Element of L2 ;
N is full full SubRelStr of ( T |^ the carrier of S ) ;
sup { x , y } = c "\/" c ;
g . n = n / 1 .= n / 1 ;
h . J = EqClass ( u , J ) ;
let seq be non sequence of X , n be Nat ;
dist ( x `1 , y ) < r / 2 ;
reconsider mm = m - n as Element of NAT ;
x- x0 < r1 - x0 + x0 ;
reconsider P = P ` as strict Subgroup of N ;
set g1 = p * ( idseq q ) ;
let n , m , k be non zero Nat ;
assume that 0 < e and f | A is lower ;
D2 . ( I2 , I2 ) in { x } ;
cluster subcondensed open -> \mathbb open for Subset of T ;
P be compact non empty Subset of TOP-REAL 2 , p1 , p2 be Point of TOP-REAL 2 ;
Gik in LSeg ( PI , 1 ) /\ LSeg ( Gik , Gij ) ;
n be Element of NAT , x be Element of NAT ;
reconsider ST = S 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 - 1 as Element of NAT ;
reconsider d = x as Element of C ( ) ;
let s be 0 -started State of SCMPDS , k be Nat ;
let t be 0 -started State of SCMPDS , Q be Subset of SCMPDS ;
b , b , x , y , z d d d ;
assume that i = n \/ { n } and j = k \/ { n } ;
let f be PartFunc of X , Y ;
x0 >= ( sqrt ( c ^2 ) ) * sqrt ( c ^2 ) ;
reconsider t7 = T" as TopStruct of TOP-REAL 2 ;
set q = h * p ^ <* d *> ;
z2 in U . ( y2 ) /\ Q2 . ( z2 . y2 ) ;
A |^ 0 = { <* E *> } .= { <* E *> } ;
len W2 = len W + 2 .= len W + 2 ;
len ( h2 ) in dom ( h2 ) & len ( h2 ) = len h2 ;
i + 1 in Seg ( len s2 ) ;
z in dom g1 /\ dom f & z in dom f1 /\ dom f2 ;
assume p2 `1 = W-bound ( K ) & p1 `2 = W-bound ( K ) ;
len G + 1 <= i1 + 1 ;
f1 (#) f2 is convergent & lim ( f1 (#) f2 ) = f1 . x0 * ( f2 . x0 ) ;
cluster seq + ( s + t ) -> summable ;
assume j in dom M1 & i in Seg n ;
let A , B , C be Subset of X ;
x , y , z be Point of X , p be Point of X ;
b ^2 - ( 4 * a * c ) >= 0 ;
<* xx *> ^ <* y *> \subseteq x ;
a , b in { a , b } ;
len p2 is Element of NAT & len p1 = len p2 ;
ex x being element st x in dom R ;
len q = len ( K * G ) ;
s1 = Initialize ( Initialized s ) , P1 = P +* I , P3 = P +* I , P3 = P +* J , P3 = P +* J , s4 = P +* J , P3 = P +* J , s4 = P
consider w being Nat such that q = z + w ;
x ` ` is X ` & x ` is a + x ;
k = 0 & n <> k or 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 .|| <= r2 ;
c in uparrow p & not c in { p } ;
reconsider V = V as Subset of the TopStruct of TOP-REAL n ;
N , M be being being being being being being being Subset of L ;
then z is_>=_than compactbelow x ;
M | [. f , g .] = f & M | [. g , f .] = g ;
( ( ( TOP-REAL 1 ) | ( n + 1 ) ) ) /. 1 = TRUE ;
dom g = dom f & rng g c= dom f ;
mode \cal o of G is \cal .. for c , d be directed Walk of G ;
[ i , j ] in Indices M & [ i , j ] in Indices M ;
reconsider s = x " as Element of H ;
let f be Element of dom ( Subformulae p ) , p be Element of Subformulae p ;
F1 . ( a1 , - a1 ) = G1 . ( a1 , - a1 ) ;
Observe ( a , b , r ) is compact ;
let a , b , c , d be Real ;
rng s c= dom ( 1 / 2 ) & rng s c= dom ( 1 / 2 ) ;
curry ( F , k ) is additive ;
set k2 = card dom B , s3 = card { x } ;
set G = Sym ( X , X ) ;
reconsider a = [ x , s ] as Object of G ;
let a , b be Element of MM , n be Nat ;
reconsider s1 = s as Element of ( the carrier of S ) * ;
rng p c= the carrier of L & p . ( len p ) = 0. L ;
d be Subset of the Sorts of A ;
( x .|. x ) = 0 iff x = 0. W
I-21 in dom stop I & IH in dom stop I ;
g be continuous Function of X | B , Y ;
reconsider D = Y as Subset of Euclid n ;
reconsider i0 = len p1 - 1 as Integer ;
dom f = the carrier of S & rng f c= the carrier of S ;
rng h c= union ( the carrier of J ) ;
cluster All ( x , H ) -> seq yielding ;
d * N1 ^2 > N1 * 1 / ( 2 * N ) ;
]. a , b .[ c= [. a , b .] ;
set g = f " [: D1 , D2 :] ;
dom ( p | [: the carrier of S , the carrier of S :] ) = [: the carrier of S , the carrier of S :] ;
3 + - 2 <= k + - 2 ;
tan is_differentiable_in ( ( arctan - arccot ) * ( f ^ ) ) . x ;
x in rng ( f /^ ( len p ) ) ;
let f , g be FinSequence of D ;
[: p , q :] in the carrier of S1 & [: p , q :] in the carrier of S2 ;
rng f " { 0 } = dom f ;
( the Source of G ) . e = v ;
width G - 1 < width G - 1 ;
assume v in rng ( S | E1 ) ;
assume x is root or x is root or x is root ;
assume 0 in rng ( g2 | A ) ;
let q be Point of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
let p be Point of TOP-REAL 2 , a be Real ;
dist ( O , u ) <= |. p2 .| + 1 ;
assume dist ( x , b ) < dist ( a , b ) ;
<* S *> . ( len S ) *> is in the carrier of C-20 ;
i <= len G -' 1 + 1 - 1 ;
let p be Point of TOP-REAL 2 , a be Real ;
x1 in the carrier of I[01] & x2 in the carrier of I[01] ;
set p1 = f /. i , p2 = f /. j ;
g in { g2 : r < g2 & g2 < x0 + r } ;
Q2 = Sp2 " . ( len Q ) .= Sp2 ;
( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( 1 / 2 ) ) ) is summable ;
- p + I c= - p + A ;
n < LifeSpan ( P1 , s1 ) + 1 ;
CurInstr ( p1 , s1 ) = i .= ( I . 0 ) ;
A /\ Cl { x } \ { x } <> {} ;
rng f c= ]. r - 1 , r + 1 .[ ;
g be Function of S , V ;
f be Function of L1 , L2 , g be Function of L2 , L2 ;
reconsider z = z as Element of ( CompactSublatt L ) ;
let f be Function of S , T ;
reconsider g = g as Morphism of c opp , b opp ;
[ s , I ] in S ~ ;
len ( the connectives of C ) = 4 & len ( the connectives of C ) = 5 ;
let C1 , C2 be Subcategory of C , a , b be Subfunctor of C1 , c , d be Subfunctor of C2 ;
reconsider V1 = V as Subset of X | B ;
attr p is valid means : Def1 : 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 |^ ( a " ) is Subgroup of H ;
let A1 be Let o of O , E , A2 be Element of E ;
p2 , r3 , q1 is_collinear & q2 , q1 , q2 is_collinear ;
consider x being element such that x in v ^ K ;
not x in { 0. TOP-REAL 2 } \/ { 0. TOP-REAL 2 } ;
p in [#] ( ( TOP-REAL 2 ) | B11 ) ;
0 . 0 < M . ( E . ( n + 1 ) ) ;
^ ( c / d ) = c / d ;
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . s2 ) ;
cluster -> \cal from from from from len } -| the carrier of L ;
set i1 = the Nat , i2 = the Nat , n = the Element of NAT , m = len f ;
let s be 0 -started State of SCM+FSA , I be Program of SCM+FSA ;
assume y in ( f1 \/ f2 ) .: A ;
f . ( len f ) = f /. len f ;
x , f . x '||' f . x , f . y ;
pred X c= Y means : Def2 : cos ( X ) c= cos ( Y ) ;
y be upper Subset of Y , x be Element of X ;
cluster ( x `1 ) -> non cluster -> non cluster -> non cluster -> non cluster -> non cluster -> non cluster non Set for non trivial FinSequence of NAT ;
set S = <* Bags n , i9 *> , T = <* <* i *> , i *> , S = <* i *> , T = <* i *> , S = <* i *> , T = <* i *> , S = <* i *> ,
set T = [. 0 , 1 / 2 .] , G = [. 0 , 1 .] ;
1 in dom mid ( f , 1 , 1 ) ;
( sqrt ( 4 * PI ) ^2 ) < ( 2 * PI ) ^2 ;
x2 in dom ( f1 + f2 ) /\ dom ( f2 + f3 ) ;
O c= dom I & { {} } = { {} } ;
( the Source of G ) . x = v ;
{ HT ( f , T ) } c= Support f ;
reconsider h = R . k as Polynomial of n , L ;
ex b being Element of G st y = b * H ;
let x `1 , y , z be Element of G ;
h19 . i = f . ( h . i ) ;
( p `1 ) ^2 = ( p1 `1 ) ^2 + ( p1 `2 ) ^2 ;
i + 1 <= len Cage ( C , n ) ;
len <* P *> |^ len P = len P & len P = len P ;
set No = the MSAlgebra over N , No = the MSAlgebra over N ;
len g: x + ( x + 1 ) - 1 <= x ;
a on B & b on B ;
reconsider rj = r * I . v as FinSequence of REAL ;
consider d such that x = d and a is_less_than d ;
given u such that u in W and x = v + u ;
len f /. ( \downharpoonright n ) = len \mathbb n - n ;
set q2 = q2 , p = q2 , q = q2 , r = q2 ;
set S = ex S1 , S2 being RelStr st S = ( S1 , S2 ) --> S2 ;
MaxADSet ( b ) c= MaxADSet ( P /\ Q ) ;
Cl ( G . q1 ) c= F . r2 ;
f " D meets h " V & f " V meets h " V ;
reconsider D = E as non empty directed Subset of L1 ;
H = ( the_left_argument_of H ) '&' ( the_right_argument_of H ) ;
assume t is Element of ( F . X ) . ( t . a ) ;
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 . ( b1 , b2 ) ;
the carrier' of G = E \/ { E } ;
reconsider m = len ( p - k ) as Element of NAT ;
set S1 = LSeg ( n , UMP C ) , S2 = LSeg ( n , UMP C ) ;
[ i , j ] in Indices M1 & [ i , j ] in Indices M1 ;
assume that P c= Seg m and M is non empty and P is of 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 .= p * L /. 1 ;
pp . i = pp . i .= p . i ;
let PA , PA , G be a_partition of Y ;
pred 0 < r & 1 < r & r < 1 & r < 1 ;
rng ( AffineMap ( a , X ) ) = [#] X ;
reconsider x = x , y = y 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 ( 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 ) (#) ( v . 1 ) ) ;
pred n divides m & m divides n & n divides m ;
reconsider x = x as Point of [: I[01] , I[01] :] ;
a in ;
not y0 in the still of f & not ( not ( y in dom f & not y in dom f ) ;
Hom ( ( a \times b ) , c ) <> {} ;
consider k1 such that p " < k1 and k1 < len p and p . k1 = p . k1 ;
consider c , d such that dom f = c \ d ;
[ x , y ] in dom g & [ y , k ] in g ;
set S1 = Let S1 , S2 = \rm a1 , S2 = a2 , z = a3 ;
l1 = m2 & l1 = m2 & l1 = m2 & l1 = y2 ;
x0 in dom ( ( u + v ) + ( v + u ) ) ;
reconsider p = x as Point of ( TOP-REAL 2 ) | K1 ;
I[01] = [: R^1 , R^1 :] | B01 .= [: the carrier of I[01] , the carrier of I[01] :] ;
f . p4 <= f . p1 & f . p1 <= f . p2 ;
( F . ( x - y ) ) `1 <= ( F . ( x - y ) ) `1 ;
( x `2 ) ^2 = ( W . ( len W ) ) ^2 + ( W . ( len W ) ) ^2 ;
for n being Element of NAT holds P [ n ] ;
J , K be non empty Subset of I ;
assume 1 <= i & i <= len <* a " *> ;
0 |-> a = <*> the carrier of K ;
X . i in 2 -tuples_on A . i \ B . i ;
<* 0 *> in dom ( e --> [ 1 , 0 ] ) ;
then P [ a ] implies P [ succ a ] ;
reconsider sY. = seq . ( s + t ) as \rangle of D ;
( - ( i -' 1 ) ) <= len ( - ( i - 1 ) ) ;
[#] S c= [#] T & [#] T c= [#] T ;
for V being strict real linear holds V in the carrier of V implies V is RealUnitarySpace
assume k in dom mid ( f , i , j ) ;
let P be non empty Subset of TOP-REAL 2 , p1 , p2 be Point of TOP-REAL 2 ;
let A , B be Matrix of n1 , K ;
- a * ( - b ) = a * b ;
for A being being_line , P being Subset of A holds A // P & P is being_line
( ( o2 o2 ) . ( o1 , o2 ) ) in <^ o2 , o2 ^> ;
then ||. x .|| = 0 & x = 0. ( X ) ;
let N1 , N2 be strict normal Subgroup of G ;
j >= len upper_volume ( g , D1 ) - len upper_volume ( g , D2 ) ;
b = Q . ( len Q - 1 + 1 ) .= Q . ( len Q - 1 ) ;
f2 * ( f1 /* s ) is divergent_to+infty & f2 * ( f1 /* s ) is divergent_to+infty ;
reconsider h = f * g as Function of N4 , G ;
assume a <> 0 & Polynom ( a , b , c ) >= 0 ;
[ t , t ] in the InternalRel of A & [ t , t ] in the InternalRel of A ;
( v |-- E ) | n is Element of T7 ;
{} = the carrier of ( L1 + L2 ) .= the carrier of ( L1 + L2 ) ;
Directed I is_closed_on Initialized s , P & Directed I is_halting_on Initialized s , P ;
Initialized p = Initialize ( p +* q ) .= p +* q ;
reconsider N2 = N1 as strict net of R2 ;
reconsider Y = Y as Element of \langle Ids L , \subseteq \rangle ;
"/\" ( uparrow p \ { p } , L ) <> p ;
consider j being 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 + Q ) ^ ( len P + 1 ) ) ;
( x1 - x2 ) `1 = ( x2 - x1 ) `1 .= ( x2 - x1 ) `1 - ( x1 - x2 ) `1 ;
InputVertices S = { x1 , x2 , x3 , x4 , x5 , x5 , 7 , 8 } ;
let x , y be Element of FTTTTTTTT1 ( n ) ;
p = |[ p `1 , p `2 ]| .= |[ p `1 , p `2 ]| ;
g * 1_ G = h " * g * h .= h " * g ;
let p , q be Element of PFuncs ( V , C ) ;
x0 in dom ( x1 - x2 ) /\ dom ( y1 - y2 ) ;
( R qua Function ) " = R " * ( R * ( f | X ) ) ;
n in Seg len ( f /^ ( len p ) ) ;
for s being Real st s in R holds s <= s2 & s2 <= s ;
rng s c= dom ( f2 * f1 ) /\ dom ( f2 * f1 ) ;
synonym for for for for for for X \/ Y for X c= Y holds X c= Y ;
1_ K * ( 1_ K ) = 1_ K * 1_ K .= 1_ K ;
set S = Segm ( A , P1 , Q1 ) ;
ex w st e = ( w - f ) . w & w in F ;
curry ( P+* ( I , k ) # x ) is convergent ;
cluster -> open for Subset of TK ;
len f1 = 1 .= len f3 .= len f3 .= len f3 + len f3 .= len f3 + len f3 ;
( i * p ) / p < ( 2 * p ) / p ;
let x , y be Element of ( the Sorts of U1 ) . 0 ;
b1 , c1 // b9 , c9 & b1 , c1 // b9 , c9 ;
consider p being element such that c1 . j = { p } ;
assume that f " { 0 } = {} and f is total ;
assume IC Comput ( F , s , k ) = n ;
Reloc ( J , card I + 3 ) does not destroy a ;
Macro ( card I + 1 ) not c does not destroy c ;
set m3 = LifeSpan ( p3 , s3 ) , P4 = LifeSpan ( p3 , s3 ) , P4 = P +* I ;
IC Comput ( p , s , k ) in dom ( Initialize p ) ;
dom t = the carrier of ( SCM R ) \ { 0 } .= { 0 } ;
( ( E-max L~ f ) .. f ) .. f = 1 & ( E-max L~ f ) .. f = 1 ;
let a , b be Element of PFuncs ( V , C ) ;
Cl ( union F ) c= Cl Int ( union F ) ;
the carrier of X1 union X2 misses ( A1 \/ A2 ) ;
assume not LIN a , f . a , g . b ;
consider i being Element of M such that i = dI . i ;
then Y c= { x } or Y = { x } ;
M , v / ( y , x ) |= H1 / ( y , x ) ;
consider m being element such that m in Intersect ( F . 0 ) ;
reconsider A1 = support ( u1 ) as Subset of X ;
card ( A \/ B ) = k-1 + ( 2 * 1 ) ;
assume that a1 <> a3 and a2 <> a4 and a3 <> a4 and a4 <> a5 and a4 <> 6 and 8 <> 7 and 8 <> 7 ;
cluster s \! \mathop { \rm \hbox { - } -> ( S , V ) \rm \hbox { - } \mathclose { ^ { -1 } } -> 1 -element for string of S ;
Ln2 /. n2 = Ln2 . n2 .= Ln2 . n2 .= Ln2 . n2 ;
let P be compact non empty Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
assume that r-7 in LSeg ( p1 , p2 ) and r-7 in LSeg ( p1 , p2 ) ;
let A be non empty compact Subset of TOP-REAL n , B be Subset of TOP-REAL n ;
assume [ k , m ] in Indices ( D1 * D2 ) ;
0 <= ( ( 1 / 2 ) |^ p ) / ( p |^ q ) ;
( F . N | E1 ) . x = +infty ;
pred X c= Y & Z c= V & X \ V c= Y \ Z ;
( y `2 ) * ( z `2 ) <> 0. I ;
1 + card ( X \/ Y ) <= card ( u \/ v ) ;
set g = z \circlearrowleft ( ( L~ z ) .. z ) ;
then k = 1 + p . k ;
cluster -> total for Element of C -one-to-one ( X , Y ) ;
reconsider B = A as non empty Subset of TOP-REAL n , C be non empty Subset of TOP-REAL n ;
let a , b , c be Function of Y , BOOLEAN ;
L1 . i = ( i .--> g ) . i .= g . i .= g . i ;
Plane ( x1 , x2 , x3 , x4 , x5 ) c= P ;
n <= indx ( D2 , D1 , j1 ) + 1 - 1 ;
( ( g2 ) . O ) `1 = - 1 & ( g2 ) . I = 1 ;
j + p .. f - len f <= len f - len f ;
set W = W-bound C , E = E-bound C , S = Gauge ( C , n ) ;
S1 . ( a `1 , e ) = a + e .= a ;
1 in Seg width ( M * ( ColVec2Mx p ) ) ;
dom ( i (#) Im f ) = dom ( Im f ) ;
Dx . ( x `1 , p ) = W . ( a , p ) ;
set Q = non empty contradiction , v = |= ( g , f ) ;
cluster -> many sorted for ManySortedSet of U1 ;
attr F = { A } means : Def2 : F is discrete ;
reconsider zj = <* *> as Element of product G ;
rng f c= rng ( f1 ^ f2 ) \/ rng ( f2 ^ f3 ) ;
consider x such that x in f .: A and x in f .: C ;
f = <*> ( the carrier of F_Complex ) .= ( <*> the carrier of F_Complex ) ;
E , j |= All ( x1 , x2 ) & E , f |= H ;
reconsider n1 = n as Morphism of o1 , o2 ;
assume P is idempotent & R is idempotent & P (*) R = R ** P ;
card ( B2 \/ { x } ) = k-1 + 1 - 1 ;
card ( ( x \ B1 ) /\ ( ( x \ A1 ) \ A1 ) ) = 0 ;
g + R in { s : g-r < s & s < g + r } ;
set q00 = ( q , <* s *> ) \mathop { \rm \hbox { - } bag } ;
for x being element st x in X holds x in rng f1 ;
h0 /. ( i + 1 ) = h0 . ( i + 1 ) ;
set mw = max ( B , support ( R ) ) ;
t in Seg width ( I ^ ( n , n ) ) ;
reconsider X = dom f , Y = C as Element of Fin NAT ;
IncAddr ( i , k ) = ( \bf if l + k ) ;
( ( q `2 ) / ( |. q .| ) <= ( |. q .| ) / ( |. q .| ) ;
attr R is condensed means : Def2 : Int R is condensed & Cl R is condensed ;
pred 0 <= a & a <= 1 & b <= 1 , 1 ;
u in ( ( c /\ ( ( d /\ b ) /\ f ) ) /\ j ;
u in ( ( c /\ ( ( d /\ e ) /\ f ) /\ j ) /\ f ) /\ j ;
len C + ( - 2 ) >= 9 + ( - 3 ) ;
x , z , y is_collinear & x , y , z is_collinear ;
a |^ ( n1 + 1 ) = a |^ ( n1 + 1 ) * a ;
<* \underbrace ( 0 , \dots , 0 *> , x ) in Line ( x , a * x ) ;
set y9 = <* y , c *> ;
FG2 /. 1 in rng Line ( D , 1 ) ;
p . m joins r /. m , r /. ( m + 1 ) ;
( p `2 ) = ( f /. ( i1 + 1 ) ) `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 } ;
f1 /* ( seq ^\ k ) is divergent_to+infty & f2 /* ( seq ^\ k ) is divergent_to+infty ;
reconsider u2 = u as VECTOR of \mathop { \rm P\mathbb R } ( X ) ;
p \! \mathop { Product ( Sgm X ) } = 0 implies p = 0 & p = ( Sgm X ) . p
len <* x *> < i + 1 & i + 1 <= len c ;
assume I is non empty & { x } /\ { y } = { 0. I } ;
set ii = card I + 4 , ii = goto ( 0 + 4 ) , P4 = goto ( 0 + 4 ) , P4 = goto ( 0 + 4 ) , P4 = goto ( 0 + 4 ) , P4 = goto ( 0 + 4 ) , P4 = goto ( 0 +
x in { x , y } & h . x = {} ( Ty ) ;
consider y being Element of F such that y in B and y <= x ` ;
len S = len ( the charact of ( A ) ) .= len ( the charact of ( A ) ) ;
reconsider m = M , i = I , n = N as Element of X ;
A . ( j + 1 ) = B . ( j + 1 ) \/ A . j ;
set N8 = : : |. G8 .| = |. G8 .| ;
rng F c= the carrier of gr { a } & rng F c= the carrier of gr { a } ;
( C , P ) . ( K , n , r ) is in rng p ;
f . k , f . ( Let ( n ) ) . k ] in rng f ;
h " ( P ) /\ [#] T1 = f " ( P ) /\ f " ( P ) ;
g in dom ( f2 \ f2 " { 0 } ) \ f2 " { 0 } ;
gN /\ dom f1 = g1 " ( f1 " X ) .= g1 " ( f1 " X ) ;
consider n being element such that n in NAT and Z = G . n ;
set d1 = \bf dist ( x1 , y1 ) , d2 = dist ( x2 , y2 ) , d2 = dist ( x2 , y2 ) ;
b `2 + ( 1 - 2 ) < ( 1 - 2 ) * ( 1 - 2 ) ;
reconsider f1 = f as VECTOR of the carrier of X , Y be bounded Subset of X ;
pred i <> 0 implies i ^2 mod ( i + 1 ) = 1 mod ( i + 1 ) ;
j2 in Seg len ( ( g2 . ( i2 + 1 ) ) * ( ( g2 . ( i2 + 1 ) ) ) ;
dom ( i - 1 ) = dom ( i - 1 ) .= Seg ( i - 1 ) .= Seg ( i - 1 ) ;
cluster sec | ]. PI / 2 , PI / 2 .[ -> one-to-one ;
Ball ( u , e ) = Ball ( f . p , e ) ;
reconsider x1 = x0 as Function of S , IF ( ) . x0 , IF ( ) . x0 ;
reconsider R1 = x , R2 = y , R2 = z as Relation of L ;
consider a , b being Subset of A such that x = [ a , b ] ;
( <* 1 *> ^ p ) ^ <* n *> in Rn & ( <* n *> ^ p ) ^ q in Rn ;
S1 +* S2 = S2 +* S2 +* S2 +* S2 & S2 +* S2 = S1 +* S2 +* S2 +* S2 ;
( ( ( - 1 ) (#) ( ( #Z 2 ) * ( cos * f ) ) ) `| Z ) = f ;
cluster -> continuous for Function of C , REAL ;
set C7 = 1GateCircStr ( <* z , x *> , f3 ) ;
E\langle e2 *> . ( e2 + T . ( e2 + T . ( e2 + T . ( e2 + T . ( e2 + T . ( e2 + T . ( e2 + T . ( e2 + T . ( $1 + T ) ) ) ) ) *> ) ) = EE ;
( ( arctan ) (#) ( ( arctan ) `| Z ) ) . x = ( ( arctan ) . x ) * ( ( ( arctan ) . x ) ^2 ) ;
upper_bound A = ( PI * 3 ) / 2 & lower_bound A = 0 ;
F . ( dom f , - f ) is Functor of F . ( cod f , - f ) , F . ( cod f , - f ) ;
reconsider pp = qelement . ( q . ( len q ) ) as Point of Euclid 2 ;
g . W in [#] Y & [#] Y c= [#] Y & g . W c= [#] Y ;
let C be compact non vertical non vertical non horizontal Subset of TOP-REAL 2 ;
LSeg ( f ^ g , j ) = LSeg ( f , j ) ;
rng s c= dom f /\ ]. x0 - r , x0 .[ & f | ]. x0 - r , x0 .[ is bounded ;
assume x in { idseq ( 2 , len ( idseq 2 ) ) , ( Rev ( idseq 2 ) ) } ;
reconsider n2 = n , m2 = m , m2 = n as Element of NAT ;
for y being ExtReal st y in rng seq holds g . y <= f . y ;
for k st P [ k ] holds P [ k + 1 ] ;
m = m1 + m2 .= m1 + m2 .= m1 + m2 .= m1 + m2 + m2 .= m1 + m2 + m2 ;
assume for n holds H1 . n = G . n -H . n ;
set BX1 = f .: ( the carrier of X1 ) , BX2 = f .: ( the carrier of X2 ) ;
ex d being Element of L st d in D & x << d ;
assume R ~ c= R ~ & ( R ~ ) c= R ~ & ( R ~ ) c= R ~ ;
t in ]. r , s .[ or t = r or t = s ;
z + v2 in W & x = u + ( z + v2 ) ;
x2 |-- y2 iff P [ x2 , y2 ] or P [ y2 , y2 ] ;
pred x1 <> x2 & |. x1 - x2 .| > 0 & |. x1 - x2 .| > 0 ;
assume that p2 - p1 , p3 - p1 , p3 - p1 , p1 - p2 is_collinear and p2 - p1 , p3 - p1 , p3 - p1 is_collinear ;
set q = ( \mathbb f ) ^ <* 'not' 'not' A *> ;
let f be PartFunc of \langle REAL-NS 1 , REAL-NS 1 *> , REAL-NS 1 , REAL-NS 1 ;
( n mod ( 2 * k ) ) = n mod k ;
dom ( T * ( \mathop { \rm \lbrace 0 } , 1 } ) ) = dom ( \mathop { \rm succ 0 ) ;
consider x being element such that x in wc and x in c ;
assume ( F * G ) . ( v . x3 ) = v . x4 ;
assume that the Sorts of D1 c= the carrier' of D2 and the Sorts of D2 c= the carrier' of D2 and the Sorts of D1 c= the carrier' of D2 ;
reconsider A1 = [. a , b .] as Subset of R^1 | [. a , b .] ;
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 ) ;
n1 - len f + 1 <= len - ( len g + 1 ) + 1 - 1 ;
|. |. |. q , O1 , a , b , a , b , c , d ) .| = [ u , v , a , b , c , d ] ;
set C-2 = ( `1 ) . ( k + 1 ) , C-2 = ( n + 1 ) - 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 ) & $1 <> 0 implies ( $1 = 1 implies $1 = 2 ) ;
set s3 = Comput ( P1 , s1 , k ) , P3 = Comput ( P3 , s3 , k ) , P4 = P3 ;
l be variable of k , A , B be Subset of A ;
reconsider U1 = union ( Gf1 \/ Gf2 ) as Subset-Family of ( T | A ) ;
consider r such that r > 0 and Ball ( p `1 , r ) c= Q ` ;
( h | ( n + 2 ) ) /. ( i + 1 ) = p29 ;
reconsider B = the carrier of X1 , C = the carrier of X2 as Subset of X ;
p9 = <* - ( c . i ) , - ( c . i ) *> .= - ( c . i ) ;
synonym f is real-valued for rng f c= NAT & rng f c= NAT & f is one-to-one ;
consider b being element such that b in dom F and a = F . b ;
x10 < card ( X0 \/ ( X0 \/ Y2 ) ) + card ( Y1 \/ Y2 ) ;
attr X c= B1 means : Let for B1 , B2 being Subset of X holds \mathop { \rm _ c } ( X , B1 ) c= succ B2 ;
then w in Ball ( x , r ) & dist ( x , w ) <= r ;
angle ( x , y , z ) = angle ( an , 0 , 2 ) ;
pred 1 <= len s means : Let : len ( the { - } Shift ( s , 0 ) ) = len s ;
f/. c= f . ( k + ( n + 1 ) ) ;
the carrier of { 1_ G } = { 1_ G } .= { 1_ G } ;
pred p '&' q in \mathbin { \rm mod } A , p '&' q in TAUT ( A ) ;
- ( t `1 ) ^2 < ( - t `1 ) ^2 + ( - t `2 ) ^2 ;
U1 . 1 = U /. 1 .= ( U1 /. 1 ) `1 .= ( U1 /. 1 ) `1 .= ( U1 /. 1 ) `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 :] & [: Seg n , Seg n :] c= Indices ( O ) ;
for n being Element of NAT holds G . n c= G . ( n + 1 ) ;
then V in M .: \square ex x being Element of M st V = { x } ;
ex f being Element of F st f is \cup ( A , B ) & f in F ;
[ h . 0 , h . 3 ] in the InternalRel of G & [ h . 2 , h . 3 ] in the InternalRel of G ;
s +* Initialize ( ( intloc 0 ) .--> 1 ) = s3 +* Initialize ( ( intloc 0 ) .--> 1 ) ;
|[ w1 , v1 ]| \not <> 0. TOP-REAL 2 & |[ w1 , v1 ]| <> 0. TOP-REAL 2 ;
reconsider t = t as Element of ( the carrier of Z ) * ;
C \/ P c= [#] ( ( ( the carrier of ( ( ( ( ( A ) ) | A ) | A ) ) | A ) ) ;
f " V in ( the carrier of m ) /\ D ( the carrier of C ) ;
x in [#] ( ( the carrier of F ) /\ A ) ;
g . x <= h1 . x & h . x <= h1 . x ;
InputVertices S = { xy , yz , \mathopen { xy , yz , z1 } , { xy , yz , z1 } } ;
for n be Nat st P [ n ] holds P [ n + 1 ] ;
set R = Line ( M , i ) * Line ( M , i ) ;
assume that M1 is being_line and M2 is being_line and M1 is being_line and M2 is being_line ;
reconsider a = f3 . ( i0 -' 1 ) as Element of K ;
len ( ( Len F1 ) ^ ( ( Len F2 ) ^ ( ( len F1 ) + len F2 ) ) = len ( ( len F1 ) + len ( len F2 ) ) ;
len ( ( the _ of K ) * ( i , j ) ) = n & len ( ( n |-> 0. K ) * ( i , j ) ) = n ;
dom max ( f + g , h ) = dom ( f + g ) ;
( the Sorts of seq ) . n = upper_bound Y1 & ( the Sorts of seq ) . n = upper_bound Y1 ;
dom ( p1 ^ p2 ) = dom ( ( f ^ p2 ) ^ <* p2 *> ) .= dom ( f ^ <* p2 *> ) ;
M . [ 1 , y ] = 1 / ( v - y ) .= y * v1 - y * v .= y * v1 - y * v ;
assume that W is non trivial and W .vertices() c= the carrier of G2 and W c= the carrier of G2 ;
C6 /. i1 = G1 * ( i1 , i2 ) .= G1 * ( i1 , i2 ) ;
C8 |- 'not' Ex ( x , p ) 'or' p . ( x , y ) ;
for b st b in rng g holds lower_bound rng f\lbrace b \rbrace <= b * ( upper_bound rng f\lbrace b \rbrace )
- ( ( q1 `1 ) / |. q1 .| - cn ) = 1 - ( ( q1 `1 ) / |. q1 .| - cn ) ;
( LSeg ( c , m ) \/ [: l , k :] ) \/ LSeg ( l , k ) c= R ;
consider p being element such that p in { x } and p in L~ f and x in L~ f ;
Indices ( X @ ) = [: Seg n , Seg n :] & [: Seg n , Seg n :] = [: Seg n , Seg n :] ;
cluster s => ( q => p ) => ( q => ( s => p ) ) -> valid ;
Im ( ( Partial_Sums ( F ) ) . m ) is_measurable_on E ;
cluster f . ( x1 , x2 ) -> ( f . x1 , f . x2 ) -> Element of D ;
consider g being Function such that g = F . t and Q [ g , f . t ] ;
p in LSeg ( ( N-min Z ) /. ( len ( \mathopen { - } Z ) ) , ( ( \mathopen { - Z } ) /. ( len ( p - Z ) ) ) ) ;
set R8 = R |^ 1 , R8 = R |^ ( b + a ) , R8 = R |^ ( b + a ) , R7 = R |^ ( b + a ) , R8 = R |^ ( b + a ) , R8 = R |^ ( b + a ) , R8 = R |^
IncAddr ( I , k ) = IncAddr ( da , da ) .= I . ( da + k ) ;
seq . m <= ( ( the Sorts of seq ) . k ) . ( ( seq ^\ k ) . n ) ;
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 as non empty Subset of ( U1 \/ U2 ) ;
u in ( ( c /\ ( ( d /\ e ) /\ f ) /\ j ) /\ m ;
consider y being element such that y in Y and P [ y , inf B ] ;
consider A being finite stable Subset of R such that card A = card ( R * A ) ;
p2 in rng ( f |-- p1 ) \ rng <* p1 *> & p1 in rng ( f |-- p2 ) \ { p1 } ;
len s1 - 1 > 0 & len s2 - 1 > 0 & len s1 - 1 > 0 ;
( ( N-min L~ z ) `2 ) = ( E-max L~ z ) `2 .= ( E-max L~ z ) `2 ;
Ball ( e , r ) c= LeftComp Cage ( C , k + 1 ) \/ LeftComp Cage ( C , k + 1 ) ;
f . a1 ` = f . a1 ` .= f . a1 .= ( f . a1 ) ` ;
( seq ^\ k ) . n in ]. - ( x0 - r ) , x0 + r .[ ;
gg . s0 = g . ( s . s0 ) | G . ( s . f0 ) ;
the InternalRel of S is If & the InternalRel of S is PI & the InternalRel of S is transitive ;
deffunc F ( Ordinal , Ordinal ) = phi . ( $2 , $2 ) ;
F . ( s1 . a1 ) = F . ( s2 . a1 ) .= F . ( s2 . a1 ) ;
x `2 = A ( o ) . a .= Den ( o , A ( ) ) . a ;
Cl ( f " P1 ) c= f " ( Cl P1 ) \/ f " ( Cl P2 ) ;
FinMeetCl ( ( the topology of S ) . i ) c= the topology of T ;
synonym o is " for o <> \ast & o <> * & o <> * ;
assume that X c= Y and card X <> card Y and card X <> card Y and card Y <> card X ;
the \rangle <= 1 + ( the \rangle of ( the consider of S ) * ( the \rangle ) ;
LIN a , a1 , d or b , c // b1 , c1 or b , c // b1 , c1 ;
e2 . 1 = 0 & e2 . 2 = 1 & e2 . 3 = 0 ;
E in S1 & not E in S1 & not E in S1 & not E in S1 ;
set J = ( l , u ) If ;
set A1 = Let ( ap , bm , cp ) , A2 = non-empty ( ap , bm , cp ) , A1 = non-empty ( ap , bm , cp ) ;
set c9 = [ <* cin , d *> , and2 ] , A2 = [ <* cin , d *> , and2 ] , every set ;
x * z `1 * x " in x * ( z * N ) ;
for x being element st x in dom f holds f . x = f3 . x ;
Int cell ( f , 1 , G ) c= RightComp f \/ RightComp f \/ L~ f ;
U2 is_an_arc_of W-min ( C ) , W-min ( C ) , W-min ( C ) , W-min ( C ) ;
set f-17 = f .: @ g ;
attr S1 is convergent & S2 is convergent & lim ( S1 - S2 ) = 0 & for n holds S1 . n = S2 . n - S2 . n ;
f . ( 0 + 1 ) = ( 0 qua Ordinal ) + a .= a + ( 0 qua Ordinal ) .= a + ( 0 qua Ordinal ) .= a + b ;
cluster -> \in be \in be \in be in , reflexive transitive transitive non empty RelStr , F , G be symmetric non empty RelStr ;
consider d being element such that R reduces b , d and R reduces c , d ;
not b in dom Start-At ( ( card I + 2 ) , SCMPDS ) ;
( z + a ) + x = z + ( a + y ) .= z + ( a + y ) ;
len ( l \lbrack ( a |^ 0 ) * x ) = len l ;
t4 ^ {} is ( {} \/ rng t4 ) -valued ( {} \/ rng t4 ) -valued FinSequence ;
t = <* F . t *> ^ ( C . p ^ q ) .= C . p ^ q . ( p ^ q ) ;
set pp = W-min L~ Cage ( C , n ) , p = W-min L~ Cage ( C , n ) , q = W-min L~ Cage ( C , n ) ;
( ( k -' 1 ) + ( i - 1 ) ) = ( k - 1 ) + ( i - 1 ) ;
consider u being Element of L such that u = u ` and u in D and u in D ;
len ( ( width ( ( b ) |-> a ) ^ ( b * c ) ) ) = width ( ( b * c ) |-> ( a * c ) ) ;
FM . x in dom ( ( G * the_arity_of o ) . x ) ;
set cH2 = the carrier of H2 , cH = the carrier of H , cH = the carrier of H ;
set H1 = the carrier of H1 , H2 = the carrier of H2 , E = the carrier of H1 , f = the carrier of H2 , g = the carrier of H2 ;
( Comput ( P , s , 6 ) . intpos ( m + 1 ) ) = s . intpos ( m + 1 ) ;
IC Comput ( Q2 , t , k ) = ( l + 1 ) - ( ( card I + 1 ) + ( card I + 2 ) ) ;
dom ( ( - cos ) * ( ( sin * sin ) ) `| Z ) = REAL & dom ( ( - cos ) * ( ( sin * cos ) ) `| Z ) = REAL ;
cluster <* l *> ^ phi -> ( 1 + 1 ) -element for string of S ;
set b9 = [ <* ap , bm *> , and2 ] , c9 = [ <* A1 , cin *> , and2 ] , A2 = [ <* cin , cin *> , and2 ] , it = [ <* A1 , cin *> , and2 ] , it = [ <* A1 , cin *> , and2 ] , it = [ <* cin , ap *> , and2 ] ;
Line ( Segm ( M @ , P @ , x ) , x ) = L * Sgm Q ;
n in dom ( ( the Sorts of A ) * ( the_arity_of o ) ) ;
cluster f1 + f2 -> continuous for PartFunc of REAL , the carrier of S ;
consider y be Point of X such that a = y and ||. y - x .|| <= r ;
set x3 = t3 . DataLoc ( s2 . SBP , 2 ) , x4 = Comput ( P3 , s3 , 2 ) , P4 = P3 ;
set pp = stop I , pp = P +* stop I , P3 = P +* stop I ;
consider a being Point of D2 such that a in W1 and b = g . a ;
{ A , B , C , D } = { A , B , C } \/ { D , E , F , J , M , N , N , N , N , F , M , N , N , N , I , N , M , N , N , N , N , I , J , M , N ,
let A , B , C , D , E , F , J , M , N , N , N , F , M , N , N , N , F , J , M , N , N , N , F , J , M , N , N , N , F , N , M , N , N , N ,
|. p2 .| ^2 - ( ( p2 `2 ) ^2 - ( p2 `2 ) ^2 ) >= 0 ;
l - 1 + 1 = n-1 * ( l + 1 ) + ( 1 - 1 ) ;
x = v + ( a * w1 + b * w2 ) + ( c * w2 ) ;
the TopStruct of L = TopSpaceMetr ( ( the Scott Scott Scott Scott Scott Scott of L ) ) ;
consider y being element such that y in dom H1 and x = H1 . y ;
f9 \ { n } = Free ( All ( v1 , H ) ) \/ Free ( H ) ;
for Y being Subset of X st Y is summable holds Y is non empty iff X is non empty ;
2 * n in { N : 2 * Sum ( p | N ) = N & N > 0 } ;
for s being FinSequence holds len ( the { - } Shift ( s ) ) = len s & len ( the { - } Shift ( s ) ) = len s ;
for x st x in Z holds exp_R * f is_differentiable_in x & exp_R * f is_differentiable_in x ;
rng ( ( h2 * f2 ) | X ) c= the carrier of ( ( TOP-REAL 2 ) | X ) ;
j + ( len f - len f ) <= len f + ( len f - len f ) - len f ;
reconsider R1 = R * I as PartFunc of REAL , REAL-NS n , REAL-NS n , r be Real ;
C8 . x = s1 . ( a - 1 ) .= C8 . x - C8 . x .= C8 . x - C8 . x ;
power F_Complex = 1 .= ( x |^ n ) * ( z |^ n ) .= ( x |^ n ) * ( x |^ n ) ;
t at ( C , s ) = f . ( ( the connectives of S ) . o ) ;
support ( f + g ) c= support f \/ support g \/ support g ;
ex N st N = j1 & 2 * Sum ( ( r | N ) | N ) > 0 ;
for y , p st P [ p ] holds P [ All ( y , p ) ] ;
{ [ x1 , x2 ] where x1 , x2 is Point of [: X , Y :] : x1 in X & x2 in Y } c= [: X , Y :] ;
h = ( i , j ) |-- ( id B , id B ) . i .= H . i ;
ex x1 being Element of G st x1 = x & x1 * N c= A & x1 * N c= A ;
set X = ( ( |. q .| ) * ( x , O ) ) `1 , Y = ( ( |. q .| ) * ( x , O ) ) `1 , Z = { ( q `1 ) * ( x , O ) } , O = { ( q `1 ) * ( x , O ) } ;
b . n in { g1 : x0 - g1 < g1 & g1 < x0 & g1 < x0 + g2 } ;
f /* s1 is convergent & f /. ( lim s1 ) = lim ( f /* s1 ) ;
the lattice of the lattice of Y = the lattice of the carrier of Y & the carrier of X = the carrier of Y ;
'not' ( a . x ) '&' b . x 'or' a . x = FALSE ;
2 = len ( ( q ^ r1 ) ^ ( ( q ^ r1 ) ^ ( ( q ^ r1 ) ^ ( ( q ^ r1 ) ^ ( ( q ^ r1 ) ^ ( ( q ^ r1 ) ^ ( q ^ r1 ) ) ) ) ) ) ;
( ( 1 / a ) (#) ( sec * f1 ) - ( ( 1 / a ) (#) ( sec * f1 ) ) is_differentiable_on Z ) ;
set K1 = upper ( ( H | A ) || ( A /\ B ) ) , D2 = ( H | A ) || ( B /\ A ) ;
assume e in { ( ( w1 - w2 ) / ( w1 - w2 ) ) * ( w1 - w2 ) : w1 in F & w2 in G } ;
reconsider d7 = dom a `1 , d8 = dom F , d8 = dom F , d7 = dom G as finite set ;
LSeg ( f /^ ( j + 1 ) , j ) = LSeg ( f , j + q .. f ) ;
assume X in { T . ( N2 , K ) : h . ( N2 , K ) = N2 } ;
assume Hom ( d , c ) <> {} & <* f , g *> * f1 = <* f , g *> * f2 ;
dom ( S . n ) = dom S /\ Seg n .= dom ( L . n ) .= dom ( L . n ) /\ Seg n .= dom ( L . n ) ;
x in H |^ a implies ex g st x = g |^ a & g in H |^ b
a * ( 0. ( K , n ) ) = a `1 - ( 0 * n ) .= a `1 - ( 0 * n ) ;
D2 . ( j - 1 ) in { r : lower_bound A <= r & r <= upper_bound A & upper_bound A <= upper_bound A } ;
ex p being Point of TOP-REAL 2 st p = x & P [ p ] & p <> 0. TOP-REAL 2 ;
for c holds f . c <= g . c implies f ^ @ c <= g @ c ;
dom ( ( f1 (#) f2 ) | X ) /\ X c= dom ( ( f1 (#) f2 ) | X ) ;
1 = ( p * p ) * ( p * q ) .= p * ( 1 / p ) .= p * ( 1 / p ) ;
len g = len f + len <* x + y *> .= len f + 1 .= len f + 1 + 1 .= len f + 1 + 1 ;
dom ( F | [: N1 , S :] ) = dom ( F | [: N1 , S :] ) .= [: N1 , S :] ;
dom ( f . t ) * I . t = dom ( f . t ) * g . t ;
assume a in ( "\/" ( ( T |^ the carrier of S ) , F ) ) .: D ;
assume that g is one-to-one and ( the carrier' of S ) /\ rng g c= dom g and g . ( len g ) = f . ( len g ) ;
( ( ( x \ y ) \ z ) \ ( ( x \ y ) \ z ) ) \ ( ( x \ y ) \ z ) = 0. X ;
consider f such that f * f = id b and f * f = id a and f * f = id b ;
( ( ( cos * cos ) | [. 0 , PI * 0 .] ) | [. 0 , PI * 0 .] is increasing ;
Index ( p , co ) <= len LS - Index ( Gij , LS ) + 1 - Index ( Gij , LS ) ;
t1 , t2 , p3 be Element of ( the carrier of T ) * , s be Element of ( the carrier of S ) * ;
( -> "/\" ( ( Frege ( Frege H ) ) . h ) <= ( -> Element of ( ( Frege H ) . h ) . ( j , i ) ) ;
then P [ f . ( i0 + 1 ) , F ( f . ( i0 + 1 ) ) ] & F ( f . ( i0 + 1 ) , F ( f . ( i0 + 1 ) ) ) < j ;
Q [ ( D . [ D . x , 1 ] ) , F ( D . [ D . x , 1 ] ) ] ;
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 a \HM { of G . i : not contradiction } ;
the Sorts of A2 = ( the carrier of S2 ) --> TRUE .= ( the Sorts of S2 ) +* ( the Sorts of S2 ) ;
consider s being Function such that s is one-to-one and dom s = NAT and rng s = F . s and rng s c= F . s ;
dist ( b1 , b2 ) <= dist ( b1 , a ) + dist ( a , b ) ;
( ( <* Cage ( C , n ) /. len ( Cage ( C , n ) ) *> /. len ( Cage ( C , n ) ) ) = W /. len ( Cage ( C , n ) /. len ( Cage ( C , n ) ) ) ;
q `2 <= ( ( UMP Upper_Arc L~ Cage ( C , 1 ) ) ) `2 & ( ( UMP L~ Cage ( C , 1 ) ) `2 <= ( ( UMP L~ Cage ( C , 1 ) ) `2 ) ;
LSeg ( f | i2 , i ) /\ LSeg ( f | i2 , j ) = {} ;
given a being ExtReal such that a <= Ij and A = ]. a , I .[ and a in A and b in A ;
consider a , b being complex number such that z = a & y = b and z + y = a + b ;
set X = { b |^ n where b is Element of NAT : b <= n & b <= n } , Y = { b } ;
( ( x * y * z ) \ ( x * y ) ) \ ( x * y ) = 0. X ;
set xy = [ <* xy , \mathopen { x } , \mathopen { x } *> ] , yz = [ <* y , z *> , f3 ] , xy = [ <* z , x *> , f4 ] , yz = [ <* x , y *> , f3 ] , xy = [ <* z , x *> , f4 ] , xy = [ <* x , y *> , f3 ] ;
ll /. len ( l ) = ( l . len ( l ) ) * ( l /. len l ) ;
( ( ( q `2 ) - sn ) / ( 1 + sn ) ) ^2 = 1 - ( ( q `2 ) - sn ) / ( 1 + sn ) ^2 ;
( ( ( p `2 ) - sn ) / ( 1 + sn ) ) ^2 < 1 ^2 + ( ( p `2 ) - sn ) ^2 ;
( ( ( ( X \/ Y ) \/ Y ) \/ X ) `2 = ( ( X \/ Y ) \/ Y ) `2 ;
( seq . ( k - s ) ) . k = seq . ( k - s ) .= seq . ( k - s ) ;
rng ( ( h + c ) ^\ n ) c= dom SVF1 ( 1 , f , u0 ) ;
the carrier of X = the carrier of X & the carrier of X = the carrier of X & the carrier of X = the carrier of X ;
ex p3 st p3 = p4 & |. p3 - |[ a , b ]| .| = r & |. p3 - |[ a , b ]| .| = r ;
set ch = chi ( X , the carrier of X ) , Aj = ( the Sorts of A ) . j ;
R |^ ( 0 * n ) = Il ( X , X ) .= R |^ n * R |^ ( 0 * n ) .= R |^ n * R |^ ( 0 * n ) ;
( Partial_Sums ( curry ( F ) ) ) . ( n + 1 ) is nonnegative & ( Partial_Sums ( F ) ) . ( n + 1 ) is nonnegative ;
f2 = C7 . ( ET . ( ET . ( V , len V ) ) ) .= 0. K ;
S1 . b = s1 . b .= s2 . b .= s2 . b .= S2 . b .= S2 . b .= S2 . b ;
p2 in LSeg ( p2 , p1 ) /\ LSeg ( p1 , p10 ) \/ LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ;
dom ( f . t ) = Seg n & dom ( I . t ) = Seg n & rng ( I . t ) = Seg n ;
assume o = ( the connectives of S ) . 11 & ( the connectives of S ) . 12 in ( the carrier' of S ) . 12 ;
set phi = ( l1 , l2 ) If , phi = ( l1 , l2 ) If , phi = ( l1 , l2 ) If , F = ( l1 , l2 ) ) . ( l ) ;
synonym p is_\mathop { m , T } for ( p , T ) . ( p , q ) = 1 / ( p , T ) ;
( Y1 `2 = - 1 ) & ( 0. TOP-REAL 2 ) <> 0. TOP-REAL 2 & ( 0. TOP-REAL 2 ) <> 0. TOP-REAL 2 & ( 0. TOP-REAL 2 ) <> 0. TOP-REAL 2 & 0. TOP-REAL 2 <> 0. TOP-REAL 2 & 0. TOP-REAL 2 <> 0. TOP-REAL 2 & 0. TOP-REAL 2 <> 0. TOP-REAL 2 ;
defpred X [ Nat , set , set ] means P [ $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , P , N , N , f , f , f , g , h , i , i , f , g , i , i , j
consider k be Nat such that for n be Nat st k <= n holds s . n < x0 + g ;
Det ( I |^ ( m -' n ) , ( m - n ) * ( m - n ) ) = 1_ K ;
( - b ) / sqrt ( b ^2 - ( 4 * a * c ) ) < 0 ;
Cj . d = Cj . d mod Cj . d .= Cj . d mod Cj . d .= Cj . d mod Cj . d ;
attr X1 is dense dense means : Def3 : X1 is dense dense & X2 is dense dense SubSpace of X ;
deffunc F ( Element of E , Element of I , Element of I ) = $1 * $2 + ( $2 * $2 ) * $2 ;
t ^ <* n *> in { t ^ <* i *> : Q [ i , T ( ) ] } ;
( x \ y ) \ x = ( x \ x ) \ y .= y \ 0. X .= 0. X ;
for X being non empty set for X being Subset-Family of X holds X is Basis of [: X , product <* X *> :]
synonym A , B , C for Cl ( A \/ B ) misses Cl ( B \/ C ) ;
len ( M @ ) = len p & width ( M @ ) = width M & width ( M @ ) = width M & width ( M @ ) = width M & width ( M @ ) = width M ;
J . v = { x where x is Element of K : 0 < v . x & v . x < 0 } ;
( ( Sgm ( Seg m ) ) . d - ( Sgm ( Seg m ) ) . e <> 0 ;
lower_bound divset ( D2 , k + ( k + 1 ) ) = D2 . ( k + ( k + 1 ) - 1 ) ;
g . r1 = - 2 * r1 + 1 & dom h = [. 0 , 1 .] & rng h = [. 0 , 1 .] ;
|. a .| * ||. f .|| = 0 * ||. f .|| .= ||. f .|| * ||. f .|| .= ||. f .|| * ||. f .|| .= ||. f .|| * ||. f .|| ;
f . x = ( h . x ) `1 & g . x = ( h . x ) `2 ;
ex w st w in dom B1 & <* 1 *> ^ s = <* 1 *> ^ w & w ^ s = <* 1 *> ^ w ;
[ 1 , {} , <* d1 *> ] in ( { [ 0 , {} , {} ] } \/ ( { [ 0 , {} ] } \/ { [ 0 , {} ] } ) ;
IC Exec ( i , s1 ) + n = IC Exec ( i , s2 ) .= IC Exec ( i , s2 ) .= ( 0 + n ) ;
IC Comput ( P , s , 1 ) = IC Comput ( P , s , 9 ) .= 5 .= 5 + 9 .= 5 ;
( IExec ( W6 , Q , t ) ) . intpos ( 8 + 3 ) = t . intpos ( 8 + 3 ) ;
LSeg ( f /^ q , i ) misses LSeg ( f /^ q , j ) \/ LSeg ( f /^ q , j ) ;
assume for x , y being Element of L st x in C holds x <= y or y <= x or x <= y ;
integral ( integral ( f , C ) , ( f `| X ) . x ) = f . ( upper_bound C ) - ( lower_bound C ) . ( lower_bound C ) ;
for F , G being one-to-one FinSequence st rng F misses rng G & F ^ G is one-to-one holds F ^ G is one-to-one
||. R /. ( L /. h ) .|| < e1 * ( K + 1 ) * ||. K /. 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 `1 in Y & x `2 <= y `2 holds y `1 <= x `1 ;
func |. p ^ <* p *> -> \rbrack of A equals ( p ^ <* @ p *> ) . ( len @ p ) ;
consider t being Element of S such that x `1 , y `2 '||' z `1 , t `2 and x `1 , y `2 '||' y `1 , t `2 ;
dom ( x1 ) = Seg len ( l1 ^ <* x1 *> ) & len ( x1 ^ <* x1 *> ) = len ( l1 ^ <* x1 *> ) & len ( x1 ^ <* x1 *> ) = len ( l1 ^ <* x1 *> ) ;
consider y2 being Real such that x2 = y2 and 0 <= y2 and y2 < 1 and y2 < 1 and y2 <= 1 and y2 <= 1 ;
||. f | X /* s1 .|| = ||. f | X .|| . ( lim s1 ) .= ||. f /. x0 .|| .= ||. f /. x0 .|| .= ||. f /. x0 .|| ;
( the InternalRel of A ) ` /\ ( the InternalRel of A ) ` = {} \/ {} .= {} \/ {} .= {} ;
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 i + 1 in dom p ;
reconsider h = f | [: X , Y :] as Function of [: X , Y :] , Y ;
u1 in the carrier of W1 & u2 in the carrier of W2 implies ( ( the carrier of W1 ) + ( the carrier of W2 ) + ( the carrier of V ) = the carrier of V
defpred P [ Element of L ] means M <= f . $1 & f . $1 <= f . $1 & f . $1 <= f . $1 ;
l . ( u , a , v ) = s * x + ( - ( s * x ) + y ) .= b ;
- ( ( x-y ) . x ) = - x + ( - y ) .= - x + y .= - x + y .= - x + y ;
given a being Point of IT such that for x being Point of IT holds a , x , a , x is_collinear and a , x , y is_collinear ;
f\rbrace = [ [ [ dom ( ( @ f2 ) , cod ( @ f2 ) ] , cod ( @ f2 ) ] , cod ( ( @ f2 ) ) ] ] ;
for k , n be Nat st k <> 0 & k < n & n <= k holds k divides n implies k divides n & k divides n
for x being element holds x in A |^ d iff x in ( ( A ` ) |^ d ) `
consider u , v being Element of R , a being Element of A such that l /. i = u * a * v ;
- ( ( p `1 ) / |. p .| - cn ) / ( 1 + cn ) > 0 ;
Carrier ( Lj . k ) = Carrier ( Lj . k ) & F . k in dom ( Lj . k ) ;
set i2 = AddTo ( a , i , - n ) , i1 = a := ( - n ) , i2 = - ( - n ) ;
attr B is + sqrt ( ( Al ) `1 ) = ( ( ( ( ( ( ( ( B ) ) ) | ( S ) ) | ( S ) ) ) | ( S ) ) ) ;
a9 " "/\" D = { a "/\" d where d is Element of N : d in D } .= { a "/\" d where d is Element of N : d in D } ;
|( \square , ( - q ) * |( \square , q )| >= |( \square , q )| * |( - q )| ;
( - f ) . ( upper_bound A ) = ( ( - f ) | A ) . ( upper_bound A ) .= - ( f | A ) . ( upper_bound A ) ;
( G * ( len G , k ) ) `1 = ( G * ( len G , k ) ) `1 .= G * ( len G , k ) `1 ;
( Proj ( i , n ) ) . ( LM . ( h . 3 ) ) = <* ( proj ( i , n ) ) . ( h . 3 ) *> ;
f1 + f2 * reproj ( i , x ) is_differentiable_in ( ( reproj ( i , x ) ) . x ) & diff ( f1 , x ) = diff ( ( reproj ( i , x ) ) . x ) ;
pred ( ( ( tan * cos ) `| Z ) . x ) <> 0 & ( ( tan * cos ) `| Z ) . x = ( tan . x ) ^2 * ( cos . x ) ^2 ;
ex t being SortSymbol of S st t = s & h1 . t = h2 . ( x . t ) ;
defpred C [ Nat ] means ( P8 . $1 is n + 1 ) & ( not $1 is A & $1 is n + 1 ) & ( not $1 is A & not $1 is n + 1 ) & ( not $1 is A & not $1 is S & not $1 is S ) ;
consider y being element such that y in dom ( p9 . i ) and q9 . i = p9 . y ;
reconsider L = product ( { x1 } +* ( index B , l ) ) as Subset of product A ;
for c being Element of C ex d being Element of D st T . ( id c ) = id d & ( id d ) . ( id d ) = id d
( f , n ) = ( f | n ) ^ <* p *> .= f | n ^ <* p *> .= f | n ^ <* p *> ;
( f * g ) . x = f . ( g . x ) & ( f * h ) . x = f . ( h . x ) ;
p in { ( 1 - 2 ) * ( G * ( i + 1 , j + 1 ) + G * ( i + 1 , j + 1 ) } ;
f `2 - p = ( f | ( n , L ) ) *' .= ( f * ( - ( p ) ) ) *' .= ( f * ( - ( p ) ) ) *' ;
consider r being Real such that r in rng ( f | divset ( D , j ) ) and r < m + s ;
f1 . [ |[ ( ( ( ( ( ( ( ( ( ( - r ) ) ) * ( f2 ) ) * ( f2 ) ) * ( f1 ) ) , ( ( ( - r ) * ( f2 ) ) * ( f3 ) ) * ( f3 ) ) , ( ( ( - r ) * ( f3 ) ) * ( f3 ) ) ) ] in f1 .: W1 ;
eval ( a | ( n , L ) , x ) = eval ( a | ( n , L ) , x ) .= a . x * 0. L .= 0. L ;
z = DigA ( tn , x9 ) .= DigA ( tn , x ) .= DigA ( tn , x ) .= DigA ( tn , x ) .= 0 ;
set H = { Intersect S where S is Subset-Family of X : S c= G & S c= G } , F = G \ { F where G is Subset-Family of X : G c= F & G c= F } , G = G \ { G where G is Subset-Family of X : G c= F & G c= F } , H = G \ { G where G is Subset-Family of X : G c= F } , H = G \ { G where G
consider S9 being Element of D * , d being Element of D such that S `1 = ( S ^ <* d *> ) . ( i + 1 ) and d = ( S ^ <* d *> ) . ( i + 1 ) ;
assume that x1 in dom f and x2 in dom f and f . x1 = f . x2 and f . x2 = f . x1 ;
- 1 <= ( ( q `2 ) / |. q .| - cn ) / ( 1 + cn ) ;
( for V be Linear_Combination of A holds Sum ( l ) = 0. V ) & Sum ( l ) = 0. V & Sum ( l ) = 0. V ;
let k1 , k2 , k2 , x4 , x5 , 7 be Element of NAT , a be Int-Location , b be Int-Location ;
consider j being element such that j in dom a and j in g " { k } and x = a . j ;
H1 . ( x1 ) c= H1 . ( x2 ) or H1 . ( x2 ) c= H1 . ( x2 ) or H1 . ( x2 ) c= H1 . ( x2 ) ;
consider a being Real such that p = len ( p * p1 + ( a * p2 ) ) and 0 <= a and a <= 1 and a <= 1 ;
assume that a <= c & c <= d and [ a , b ] c= dom f and [ a , b ] c= dom g ;
cell ( Gauge ( C , m ) , len Gauge ( C , m ) -' 1 , 0 ) is non empty ;
A\HM : x in { ( S . i ) `1 where i is Element of NAT : not contradiction } ;
( T * b1 ) . y = L * b2 /. y .= ( F * b2 ) . y .= ( F * b2 ) . y .= ( F * b2 ) . y ;
g . ( s , I ) . x = s . y & g . ( s , I ) . y = |. s . x - s . y .| ;
( log ( 2 , k + 1 ) ) / 2 >= ( log ( 2 , k + 1 ) ) / 2 ;
then p => q in S & not x in the still of p & not x in S & not p => All ( x , p ) in S ;
dom ( the InitS of r-10 ) misses dom ( the InitS of ri1 ) & dom ( the InitS of ri1 ) misses dom ( the InitS of ri1 ) ;
synonym f is extended real for for for for for for x being set st x in rng f holds x is ExtReal ;
assume for a being Element of D holds f . { a } = a & for X being finite Subset of D holds f .: ( f .: X ) = f . union X ;
i = len p1 .= len p3 + len <* x *> .= len p3 + len <* x *> .= len p3 + len <* x *> .= len p3 + len <* x *> .= len p3 + len <* x *> .= len p3 + len <* x *> .= len p3 + len <* x *> .= len p3 + len <* x *> ;
( l /. ( 1 + 3 ) ) = ( g /. ( 1 + 3 ) ) * ( ( k + 3 ) - ( k + 3 ) ) ;
CurInstr ( P2 , Comput ( P2 , s2 , l ) ) = halt SCM+FSA .= ( halt SCM+FSA ) . IC SCM+FSA .= ( halt SCM+FSA ) . IC SCM+FSA .= ( IC I ) .= ( IC I ) .= ( IC I ) + ( card I ) ;
assume for n be Nat holds ||. seq . n .|| <= ( seq . n ) & ( seq . n ) is summable & ( seq . n ) <> 0 ;
sin . ( Let Let Let *> = sin . ( sin . ( - s ) ) * cos . ( sin . ( - s ) ) .= 0 ;
set q = |[ g1 . ( ( g1 . ( t - 1 ) ) , g2 . ( t - 1 ) ]| , g2 = |[ g2 . ( t - 1 ) , g2 ]| ;
consider G being sequence of S such that for n being Element of NAT holds G . n in implies G . n in \mathopen { J . n where n is Element of NAT : n <= len G } ;
consider G such that F = G and ex G1 , G2 st G1 in S & G2 in S & G = ( S . i ) `1 & G2 in S & F = ( S . i ) `1 ;
the root of [ x , s ] in ( the Sorts of Free ( C ) ) . s & ( the Sorts of Free ( C ) ) . s c= ( the Sorts of Free ( C ) ) . s ;
Z c= dom ( ( exp_R * ( f + exp_R ) ) `| Z ) ;
for k being Element of NAT holds ( ( Im ( Im f ) ) . k ) . k = ( ( Im ( Im f ) ) . k ) . k ;
assume that - 1 < n and q `2 > 0 and ( q `2 <= - 1 ) and ( q `2 <= - 1 ) and ( q `2 <= 1 or q `1 >= 0 & - 1 <= q `1 or - 1 <= q `1 & q `2 <= 1 or - 1 <= q `1 & q `2 <= 1 or - 1 <= q `1 & q `1 <= 1 & q `2 <= 1 ;
assume that f is continuous and a < b and a < b and f . a = c and f . b = d and f . a = d and f . c = d ;
consider r being Element of NAT such that s8 = Comput ( P1 , s1 , r ) and r <= q and r <= q ;
LE f /. ( i + 1 ) , f /. ( j + 1 ) , L~ f , f /. ( j + 1 ) , f /. ( j + 1 ) , f /. ( j + 1 ) , f /. ( j + 1 ) , f /. ( j + 1 ) ;
assume that x in the carrier of K and y in the carrier of K and ex_inf_of x , y , L ;
assume f +* ( i1 , \xi ) . ( i1 + 1 ) in ( proj ( F , i2 ) ) . ( i1 + 1 ) ;
rng ( ( ( ( ( ( ( ( ( ( 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 ) \times { t } where t is Element of T : t in A & t in B } ;
consider l be Nat such that for m be Nat st l <= m holds ||. s1 . m - lim ( f /* s1 ) .|| < g / 2 ;
consider t being VECTOR of product G such that mt = ||. D . t .|| and ||. t .|| <= 1 ;
assume that the carrier of v = 2 and v ^ <* 0 *> , <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ v ^ <* 1 *> ^ w in dom p and p ^ <* 1 *> in dom p and p ^ <* 1 *> in dom p ;
consider a being Element of the Points of X29 , A being Element of the lines of X29 such that a on A and a on A ;
( - x ) |^ ( k + 1 ) * ( ( - x ) |^ ( k + 1 ) ) = 1 * ( ( - x ) |^ k ) ;
for D being set st for i st i in dom p holds p . i in D holds p . i in D . i
defpred R [ element ] means ex x , y st [ x , y ] = $1 & P [ x , y ] & P [ y , x ] ;
L~ ( f2 ) = union { LSeg ( p0 , p2 ) , LSeg ( p1 , p2 ) } .= LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ;
i - len h11 + 2 - 1 < i - len h11 + 2 - 1 + 2 - 1 + 2 - 1 + 2 - 1 + 2 - 1 + 1 - 1 + 2 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 < i - 1 + 1 - 1 + 1 - 1 ;
for n being Element of NAT st n in dom F holds F . n = |. ( nthesis ) . ( n -' 1 ) .| ;
for r , s1 , s2 , r being Real holds r in [. s1 , s2 .] iff s1 <= r & r <= s2 & s1 <= s2 & s2 <= r & s1 <= s2 & s2 <= s2 & s1 <= s2 implies s1 <= s2
assume v in { G where G is Subset of T2 : G in B2 & G c= z & z in G & G c= F & G c= F } ;
g be Element of A , X be Element of INT , f be Element of ( the carrier of A ) | ( X , X ) | ( b , X ) | ( b , X ) <> 0 ;
min ( g . [ x , y ] , k . [ y , z ] ) = ( min ( g , k ) ) . y ;
consider q1 being sequence of Cj such that for n holds P [ n , q1 . n , q1 . n ] ;
consider f being Function such that dom f = NAT & 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 , O = O /\ ( A /\ B ) as Subset of B ;
consider j being Element of NAT such that x = ( the X of n ) * ( j , i ) and 1 <= j and j <= n and i <= n ;
consider x such that z = x and card ( x . O2 ) in card ( x . O1 ) and x in card ( x . O2 ) and x in card ( x . O2 ) ;
( C * _ T4 ( k , n2 ) ) . 0 = C . ( ( _ ( ( len T4 ( k ) ) ) . 0 ) .= C . ( ( ( ( k + 1 ) + 1 ) ) . 0 ) ;
dom ( X --> rng f ) = X & dom ( X --> f ) = X & rng ( X --> f ) = dom ( X --> f ) ;
( ( E-max L~ Cage ( C , n ) ) `2 <= ( ( E-max L~ Cage ( C , n ) ) `2 ) / 2 & ( ( E-max L~ Cage ( C , n ) ) `2 <= ( ( E-max L~ Cage ( C , n ) ) `2 ) / 2 ;
synonym x , y for x = y or ex l being \mathopen of S st { x , y } c= l & x <> y
consider X being element such that X in dom ( f | ( n + 1 ) ) and ( f | ( n + 1 ) ) . X = Y ;
assume that \cdot k is continuous and for x , y being Element of L st x = y & for a , b being Element of L st a = b & b = y holds x << y ;
( ( 1 / 2 ) (#) ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( #Z 2 ) ) ) ) ) ) ) is_differentiable_on REAL ) ;
defpred P [ Element of omega ] means ( the partial of A1 ) . ( $1 + 1 ) = A1 . ( $1 + 1 ) & ( for n holds P [ n ] implies P [ n + 1 ] ) ;
IC Comput ( P , s , 2 ) = succ IC Comput ( P , s , 1 ) .= ( ( card I + 1 ) + 1 ) .= 6 + 1 .= ( card I + 1 ) .= ( card I + 1 ) + 1 .= ( card I + 1 ) + 1 .= ( card I + 1 ) ;
f . x = f . g1 * f . g2 .= f . g1 * ( f . g2 ) .= f . g2 * ( f . g2 ) .= ( f . g2 ) * ( f . g2 ) .= ( f . g2 ) * ( f . g2 ) .= ( f . g2 ) * ( f . g2 ) ;
( M * ( F . n ) ) . n = M . ( ( canFS ( Omega ) ) . n ) .= M . ( ( canFS ( Omega ) ) . n ) .= M . ( ( canFS ( Omega ) ) . n ) ;
the carrier of ( L1 + L2 ) c= ( the carrier of L1 ) \/ ( the carrier of L2 ) ;
pred a , b , c , x , y , a , b , c , d , x , y , z be element means : L~ x = { a , b , c , d } ;
( the PartFunc of s , X ) . n <= ( the PartFunc of s , X ) . ( n + 1 ) * ( ( the PartFunc of s , X ) . n ) ;
pred - 1 <= r & r <= 1 & ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1
seq in { p ^ <* n *> where n is Nat : p ^ <* n *> in T1 & p ^ <* n *> in T2 } ;
|[ x1 , x2 , x3 ]| . 2 - |[ y1 , y2 ]| . 2 - |[ y1 , y2 ]| . 3 = |[ x2 - y2 , y2 ]| . 2 - |[ y1 , y2 ]| . 3 - |[ y1 , y2 ]| . 2 - |[ y1 , y2 ]| . 3 - |[ y2 , y2 ]| . 3 - |[ y1 , y2 ]| . 2 = x2 - y2 ;
attr for m be Nat holds F . m is nonnegative & ( Partial_Sums F ) . m is nonnegative & ( Partial_Sums F ) . m is nonnegative ;
len ( ( ( G . ( y , z ) ) ) . ( y , z ) ) = len ( ( ( G . ( y , z ) ) ) . ( y , z ) ) .= len ( ( G . ( y , z ) ) . ( y , z ) ) ;
consider u , v being VECTOR of V such that x = u + v and u in W1 /\ W2 and v in W2 /\ W3 ;
given F be FinSequence of NAT such that F = x and dom F = n and rng F c= { 0 , 1 } and Sum F = k and Sum ( F ) = 1 ;
0 = 1 * ( 0 - 1 ) * u] iff 1 = ( ( 1 - 1 ) * ( ( - 1 ) * ( 1 - 1 ) ) ) * ( 1 - ( ( 1 - 1 ) * ( 1 - 1 ) ) ) ;
consider n be Nat such that for m be Nat st n <= m holds |. ( f # x ) . m - lim ( f # x ) .| < e ;
cluster -> Boolean for non empty _ of ( ( ( ( ( ( ( Y ) ) ) | ( D , 1 ) ) | ( D , 2 ) ) ) , ( ( ( ( ( ( ( ( ( ( ( ( ( Y ) ) | D ) ) | ( D , 3 ) ) | ( D , 1 ) ) ) ) ) ) ) ) , L ;
"/\" ( B , {} ) = Top ( B , {} ) .= "/\" ( [#] S , {} ) .= "/\" ( [#] S , {} ) .= "/\" ( [#] S , {} ) .= "/\" ( {} , {} , {} ) .= "/\" ( {} , {} , {} ) .= "/\" ( {} , {} , {} ) ;
( ( r / 2 ) ^2 + ( r / 2 ) ^2 ) <= ( r / 2 ) ^2 + ( r / 2 ) ^2 ;
for x being element st x in A /\ dom ( ( f `| X ) `| A ) holds ( ( f `| X ) `| A ) . x >= r2
2 * r1 - ( 2 * |[ a , c ]| - ( 2 * |[ b , c ]| ) ) = 0. TOP-REAL 2 + ( 2 * |[ b , c ]| - ( 2 * |[ b , c ]| ) ;
reconsider p = P * ( \square , 1 ) , q = a " * ( ( - ( - ( - ( K , n , 1 ) ) ) * ( ( - ( K , n , 1 ) ) ) * ( ( - ( K , n , 1 ) ) ) * ( ( - ( K , n , 1 ) ) * ( ( - ( K , n , 1 ) ) * ( ( - ( K , n , 1 ) ) ) ) as FinSequence of K ;
consider x1 , x2 being element such that x1 in uparrow s and x2 in downarrow t and x = [ x1 , x2 ] and x = [ x1 , x2 ] ;
for n be Nat st 1 <= n & n <= len q1 holds q1 . n = ( upper_volume ( g , M1 ) ) . ( 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 ] and i = [ y , z ] ;
given H1 , H2 being strict Subgroup of G such that x = H1 and y = H2 and H1 , H2 // H2 , H and H1 , H2 // H2 , H ;
for S , T being non empty RelStr , d being Function of T , S st T is complete holds d is monotone & d is monotone & d is monotone
[ a + ( 0. F_Complex , b ) , b ] in ( the carrier of V ) \/ ( the carrier of V ) ;
reconsider mm = max ( len F1 , len ( p . n ) * ( <* x *> |^ n ) ) as Element of NAT ;
I <= width GoB ( ( GoB h ) * ( len GoB h , 1 ) , 1 ) & I <= len GoB ( ( GoB h ) * ( len GoB h , 1 ) ) ;
f2 /* q = ( f2 /* ( f1 /* ( s ^\ k ) ) ) ^\ k .= ( f2 * f1 ) /* ( s ^\ k ) .= ( f2 * f1 ) /* ( s ^\ k ) .= ( f2 * f1 ) /* ( s ^\ k ) ;
attr A1 \/ A2 is linearly-independent means : Def3 : A1 misses A2 & A2 misses A1 & ( not A1 c= A2 & A2 c= A1 & ( not A2 c= A1 & not A2 c= A1 & not A2 c= A1 & not A2 c= A1 & not A1 c= A2 & not A2 c= A1 & not A2 c= A1 & not A2 c= A1 & not A2 c= A1 & not A2 c= A1 & A1 c= A2 & A2 c= A1 & not A2 c= A1 & A1 c= A2 ;
func A -the carrier of C -> set equals union { A ( s ) where s is Element of R ( ) : s in A ( ) & s in C ( ) } ;
dom ( Line ( v , i + 1 ) ^ ( ( Line ( p , m ) ) /. ( \square , 1 ) ) = dom ( F ^ <* p . ( i + 1 ) *> ) ;
cluster [ ( x `1 ) , ( x `2 ) ] -> [ x `1 , ( x `2 ) ] , [ x `2 , ( x `2 ) ] , [ x `2 , ( x `2 ) ) ] , [ x `1 , ( x `2 ) `2 ] , [ x `2 , ( x `2 ) `2 ] , [ x `1 , ( x `2 ) , ( x `2 ) ] , [ x `2 , ( x `2 ) ] , [ x `2 , ( x `2 ] , [ x `2 , ( x `2 , x `2 ] , [ x `2 ] , [ x `2
E , f |= All ( x2 , All ( x2 , x1 ) ) => ( All ( x2 , x2 ) '&' ( x1 'in' x2 ) ) ;
F .: ( id X , g ) . x = F . ( id X , g . x ) .= F . ( id X , g . x ) .= F . ( id X , g . x ) .= F . ( id X , g . x ) ;
R . ( h . m ) = F . ( x0 + h . m ) - R . ( h . m ) + R . ( h . m ) - R . ( h . m ) ;
cell ( G , ( X -' 1 ) , ( Y -' 1 ) ) \ ( L~ f ) meets ( UBD L~ f ) \ ( ( X - 1 ) - ( Y - 1 ) ) ;
IC Comput ( P2 , s2 , 2 ) = IC Comput ( P2 , s2 , 2 ) .= ( card I + 2 ) .= ( card I + 2 ) .= ( card I + 2 ) + 2 .= card I + 2 ;
sqrt ( ( - ( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 ) ) > 0 ;
consider x0 being element such that x0 in dom a and x0 in g " { k } and y = a . x0 and x0 = a . x0 and x0 = a . x0 ;
dom ( ( r1 * chi ( A , A ) ) | ( A . m ) ) = dom ( ( chi ( A , A ) ) | ( A . m ) ) .= dom ( ( r1 * chi ( A , A ) ) | ( A . m ) ) .= C ;
d-7 . [ y , z ] = ( ( y - z ) * ( y - z ) ) * ( ( y - z ) * ( y - z ) ) ;
attr for i being Nat holds C . i = A . i /\ B . i ;
assume that x0 in dom f and f is_continuous_in x0 and f is_continuous_in x0 and for r , s st r in dom f & s in dom f & r < s & s < x0 holds ||. f /. r - f /. s .|| < r ;
p in Cl A implies for K being Basis of p , Q being Basis of T st K in A & Q c= K holds A meets Q
for x be Element of REAL n st x in Line ( x1 , x2 ) holds |. y1 - y2 .| <= |. y1 - y2 .| + |. y2 - x .| ;
func the \rangle -> <= a means : Def2 : a in it & for b being Ordinal Ordinal st a in it holds b c= it iff for b being Ordinal Ordinal st b in it holds b c= a & b is Ordinal ;
[ 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 ] & x = [ a , b ] ;
||. ( vseq . n ) - ( vseq . m ) .|| * ||. x .|| < ( e * ||. x .|| ) * ||. x .|| + ||. x .|| * ||. x .|| ;
then for Z being set st Z in { Y where Y is Element of I7 : F c= Y & Y in Z } holds z in Z ;
sup compactbelow [ s , t ] = [ sup { 1 , t } , sup { 1 , t } ] .= sup { { 1 , t } , t } ;
consider i , j being Element of NAT such that i < j and [ y , f . j ] in Ij and [ f . i , f . j ] in Ij and [ f . i , f . j ] in Ij ;
for D being non empty set , p , q being FinSequence of D st p c= q & p ^ q = p ^ q holds ex p being FinSequence of D st p = p ^ q & p ^ q = p ^ q
consider e2 being Element of the affine of X such that c9 , a9 // c9 , b9 and c9 <> a9 and c9 <> b9 and c9 , b9 // c9 , c9 and c9 , b9 // a9 , b9 and c9 , c9 // a9 , b9 and c9 , b9 // c9 , c9 ;
set U2 = I \! \mathop { \rm \hbox { - } F } , A = { I \! \mathop { - } F } , B = { I \! \mathop { - } F } , C = { I \! \mathop { - } F } , D = { I \! \mathop { - } F } , D = { I \! \mathop { - } E } , E = { I \! \mathop { - F } } , E = { I \! \mathop { - } E } , A = { I \! \mathop { - F } , C = { I \! \mathop { - F } , D = { E } , D = { E }
|. q2 .| ^2 = ( ( |. q2 .| ) ^2 + ( |. q2 .| ) ^2 ) * ( ( |. q2 .| ) ^2 + ( |. q2 .| ) ^2 ) .= |. q .| ^2 * ( |. q2 .| ) ^2 + ( |. q2 .| ) ^2 .= |. q .| ^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
dom signature U1 = dom ( ( the charact of U1 ) * ( the charact of U2 ) ) & Args ( o , ( ( the charact of U1 ) * ( the charact of U2 ) ) ) = dom ( ( the charact of U1 ) * ( the charact of U2 ) ) ;
dom ( h | X ) = dom h /\ X .= dom ( ( |. h .| ) | X ) .= dom ( ( |. h .| ) | X ) .= dom ( ( |. h .| ) | X ) .= dom ( ( |. h .| ) | X ) .= dom ( |. h .| ) /\ X .= dom ( |. h .| ) /\ X .= dom ( |. h .| ) /\ X .= dom ( |. h .| ) /\ X .= X ;
for N1 , N2 , K being Element of [: the carrier of G , the carrier of G :] holds ( h . K ) `1 = N & ( h . K ) `2 = ( h . K ) `2
( mod ( u , m ) + mod ( v , m ) ) . i = ( mod ( u , m ) ) . i + ( mod ( v , m ) ) . i .= ( mod ( v , m ) ) . i + ( mod ( v , m ) ) . i ;
- ( ( q `1 ) / |. q .| - cn ) / ( 1 + cn ) < - ( ( q `1 ) / ( 1 + cn ) ) / ( 1 + cn ) & - ( ( q `1 ) / ( 1 + cn ) ) / ( 1 + cn ) <= - ( ( q `1 ) / ( 1 + cn ) ) / ( 1 + cn ) ;
attr r1 = f9 & r2 = f9 & for x , y st x in dom ( u * v ) & y in dom ( u * v ) holds ( ( u * v ) . x = ( u * v ) . y ;
vseq . m is bounded Function of X , Y & x9 . m = ( seq_id ( vseq . m , X ) ) . x & x9 . m = ( seq_id ( vseq . m , X ) ) . x ;
pred a <> b & b <> c & angle ( a , b , c ) = PI & angle ( b , c , a ) = 0 & angle ( b , c , a ) = 0 & angle ( a , b , c ) = 0 ;
consider i , j being Nat , r being Real such that p1 = [ i , r ] and p2 = [ j , s ] and r < s and s < 0 and r < s and s < 0 ;
|. p .| ^2 - ( 2 * |( p , q )| ) ^2 + |. q .| ^2 = |. p .| ^2 + ( |. q .| ) ^2 ;
consider p1 , q1 being Element of [: X ( ) , Y ( ) :] such that y = p1 ^ q1 and p1 ^ q1 = p1 ^ q1 and q1 ^ q1 = p1 ^ q1 and p1 ^ q1 = p1 ^ q1 and p1 ^ q1 = p1 ^ q1 ;
( ( the carrier of A ) . ( r1 , r2 ) , s1 ) = ( ( s2 ) . ( s1 , s2 ) ) * ( ( s2 - s1 ) * ( s1 - s2 ) ) .= ( s2 - s1 ) * ( s1 - s2 ) ;
( ( ( UMP A ) . w ) `2 = lower_bound ( ( proj2 .: ( A /\ \mathop { \rm Ball } ( w , r ) ) ) ) & ( proj2 .: ( A /\ \mathop { \rm Ball } ( w , r ) ) ) is non empty ;
s , k |= ( H1 , H2 ) |= ( H iff s |= H ) iff s |= ( H , k ) |= ( H , ( k + 1 ) ) |= ( H , ( k + 1 ) ) |= ( H , ( k + 1 ) ) |= ( H , ( k + 1 ) ) ;
len ( s + 1 ) = card ( support b1 ) + 1 .= card ( support b2 ) + 1 .= card ( support b2 ) + 1 .= card ( support b2 ) + 1 .= card ( support b2 ) + 1 .= card ( support b2 ) + 1 .= card ( support b2 ) + 1 .= card ( support b2 ) + 1 .= card ( support b2 ) + 1 .= card ( support b2 ) + 1 ;
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 >= x ;
LSeg ( ( UMP D ) . |[ ( W-bound D ) / 2 , ( ( E-bound D ) / 2 ) / 2 ]| ) /\ D = { UMP D } /\ D ;
lim ( ( f `| N ) /* ( g `| N ) /* b ) = lim ( ( f `| N ) /* ( g `| N ) ) .= lim ( ( f `| N ) /* b ) ;
P [ i , pr1 ( f . i , pr1 ( f . i , pr1 ( f . i , pr1 ( f . i , pr1 ( f . i , pr1 ( f . i , F . i ) ) ) ) ] ;
for r be Real st 0 < r ex m be Nat st for k be Nat st m <= k holds ||. ( seq . k ) - ( seq . m ) .|| < r
for X being set , P being a_partition of X , x , y being set st x in a & y in P & x in P & y in P & x <> y holds a = b
Z c= dom ( ( ( - 1 ) (#) f ) `| Z ) \ ( ( - 1 ) (#) f ) " { 0 } implies f is_differentiable_on Z & for x st x in Z holds ( ( - 1 ) (#) f ) . x = - 1 / ( x + a ) * f . x
ex j being Nat st j in dom ( l ^ <* x *> ) & j < i & y = ( l ^ <* x *> ) . j & z = ( l ^ <* x *> ) . ( j + 1 ) & z = ( l ^ <* x *> ) . ( j + 1 ) & z = ( l ^ <* x *> ) . ( j + 1 ) ;
for u , v being VECTOR of V , r being Real st 0 < r & u in N & v in Seg 1 holds r * u + ( r * v ) in N
A , Int A , Cl ( A , B ) , B , C , D , E , F , J J , M , N , N , N , F , J J M M , N , N , F , J J M , N , N , M , N , N , F F M , J , N , N , M , N , N , N , N , F , J , M , N , N , N , N , F being Element of A , A , J being Element of A , K being Element of A , K being Element of A , K , J being Element of A , M being Element
- Sum <* v , u , w *> = - ( v + u ) .= - ( v + u ) + ( u + w ) .= - ( v + u ) + ( u + w ) .= - ( v + u ) + ( v + w ) .= - ( v + u ) + ( v + w ) ;
( Exec ( a := b , s ) ) . IC SCM R = ( Exec ( a := b , s ) ) . IC SCM R .= ( Exec ( I , s ) ) . IC SCM .= succ IC s .= IC s .= IC s .= IC s .= IC s .= IC s .= IC s .= IC s .= IC s .= IC s .= IC s .= IC s .= IC s .= IC s .= IC s .= 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 support of J ) . x ;
for S1 , S2 being non empty reflexive RelStr , D being non empty directed Subset of S1 , f being Function of S1 , S2 for g being Function of S2 , S2 holds cos ( g ) is directed & cos ( f ) is directed & cos ( g ) is directed
card X = 2 implies ex x , y st x in X & y in X & not ( ex z st z in X & x = [ z , y ] or x = [ z , y ] ) & ( for x st x in X holds x <> y or x = z or x = x ) ;
E-max L~ Cage ( C , n ) in rng Cage ( C , n ) implies ( Cage ( C , n ) \circlearrowleft E-max L~ Cage ( C , n ) ) /. ( len Cage ( C , n ) + 1 ) in rng Cage ( C , n )
for T being decorated tree , T , p , q being Element of dom T st p element q & p divides q holds ( T -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 = G * ( i2 + 1 , j2 ) ;
cluster ( k gcd n ) divides ( k * n ) & ( k divides ( k * n ) ) & ( k divides ( k * n ) ) & ( k divides ( k * n ) ) & ( k divides ( k * n ) ) & ( k divides ( k * n ) ) & ( k divides ( k * n ) ) implies k divides ( k * n ) * ( k divides ( k * n ) )
dom F " = the carrier of X2 & rng F = the carrier of X1 & F " is one-to-one & rng F = the carrier of X2 & F " is one-to-one & rng F = the carrier of X1 & F " is one-to-one ;
consider C being finite Subset of V such that C c= A and card C = n and the carrier of V = A and the carrier of C = A \/ B and the carrier of V = A \/ 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= Y or Y c= V
set X = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } , Y = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } , Z = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } ;
angle ( p1 , p3 , p4 ) = 0 .= angle ( p2 , p3 , p4 ) .= angle ( p3 , p4 , p2 ) .= angle ( p3 , p4 , p2 ) .= angle ( p3 , p4 , p2 ) .= angle ( p3 , p4 ) + angle ( p3 , p4 , p2 ) ;
- sqrt ( ( - ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 ) = - ( ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ) .= - ( - ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 ) .= - 1 ;
ex f being Function of I[01] , TOP-REAL 2 st f is continuous & rng f = P & f . 0 = p1 & f . 1 = p2 & f . 0 = p3 & f . 1 = p4 & f . 1 = p4 & f . 0 = p4 ;
attr f is_partial_differentiable_in u0 , u0 means : Let for u , v holds SVF1 ( 2 , pdiff1 ( f , 1 ) , u0 ) . ( u + v ) = ( proj ( 2 , pdiff1 ( f , 3 ) ) . ( u + v ) ;
ex r , s st x = |[ r , s ]| & G * ( len G , 1 ) `1 < r & r < G * ( 1 , 1 ) `1 & s < G * ( 1 , 1 ) `2 & s < G * ( 1 , 1 ) `2 & G * ( 1 , 1 ) `2 < s & s < G * ( 1 , 1 ) `2 ;
assume that f is_sequence_on G and 1 <= t and t <= len G and G * ( t , width G ) `2 >= ( GoB f ) * ( t , width G ) `2 and G * ( t , width G ) `2 >= ( GoB f ) * ( t , width G ) `2 ;
pred i in dom G means : Def2 : r * ( f * reproj ( i , x ) ) = r * ( reproj ( i , x ) . i ) ;
consider c1 , c2 being bag of o1 + o2 such that ( decomp c ) /. k = <* c1 , c2 *> and ( ( decomp c ) /. k = <* c1 , c2 *> and ( ( decomp c ) /. k ) = c1 + c2 ;
u0 in { |[ r1 , s1 ]| : r1 < s1 & s1 < s1 & s1 < s2 & s1 < s2 & s1 < s2 & s2 < s2 & s2 < s1 & s1 < s1 & s1 < s2 & s1 < s2 & s1 < s2 & s2 < s2 & t1 < s2 & t1 < t2 } ;
Cl ( X ^ Y ) . k = the carrier of X . ( k + 1 ) .= CX . ( k + 1 ) .= CX . ( k + 1 ) .= CX . k .= CX . k .= CX . k ;
attr M1 = len M2 & width M1 = width M2 & len M2 = len M2 & width M1 = len M2 & width M1 = len M2 & width M1 = width M2 & M1 = M2 * M1 & M1 = M2 * M1 ;
consider g2 be Real such that 0 < g2 and { y where y is Point of S : ||. y - x0 .|| < g2 & y in N & ||. y - x0 .|| < g2 & g2 . ( y - x0 ) < g2 . ( y - x0 ) ;
assume x < ( - b + sqrt ( delta ( a , b , c ) ) * sqrt ( C ) or x > - sqrt ( b - a ) * sqrt ( C ) ;
( G1 '&' G2 ) . i = ( <* 3 *> ^ ( G1 ^ G2 ) ) . i & ( G1 ^ G2 ) . i = ( <* 3 *> ^ ( G1 ^ G2 ) ) . i & ( G1 ^ G2 ) . i = ( G1 ^ ( G2 ^ H ) ) . i ;
for i , j st [ i , j ] in Indices ( M3 + M1 ) holds ( M2 + M1 ) * ( i , j ) < M2 * ( i , j ) + M2 * ( i , j )
for f being FinSequence of NAT , i being Element of NAT st i in dom f & i <= len f holds i divides f /. ( j + 1 ) & i divides f /. ( j + 1 )
assume F = { [ a , b ] where a , b is Subset of X : for c st c in B\mathopen { \rbrack a , b .[ & c c= b & a c= c & b c= c } ;
b2 * q2 + ( b2 * q2 ) + ( - ( ( - ( b2 * q2 ) ) * q2 ) ) = 0. TOP-REAL n + ( - ( ( - ( ( b2 * q2 ) ) * q2 ) ) * q2 ) .= 0. TOP-REAL n + ( - ( ( - ( ( b2 * q2 ) ) * q2 ) ) * q2 ) ;
Cl ( Cl F ) = { D where D is Subset of T : ex B being Subset of T st D = Cl B & B in F & B c= Cl ( Cl B ) } ;
attr seq is summable & seq is summable means : Def2 : seq is summable & ( for n holds seq . n = Sum ( seq ) & seq is summable & seq is summable & Sum ( seq ) = Sum ( seq ) ;
dom ( ( cn " ) | D ) = ( the carrier of ( TOP-REAL 2 ) | D ) /\ D .= the carrier of ( ( TOP-REAL 2 ) | D ) .= D ;
[ X \to Z ] is full full full full SubRelStr of ( [#] Z ) |^ the carrier of Z , [ X \to Y ] ] is full full SubRelStr of ( [#] Z ) |^ the carrier of Z ;
( G * ( 1 , j ) `2 = ( G * ( i , j ) ) `2 & ( G * ( i , j ) ) `2 <= ( G * ( i , j ) ) `2 ;
synonym m1 c= m2 & for for for for for p , q being set st p in P & q in P & p in P & p in P & q in P & p in P & p in P holds p , q is_the functor empty 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 P [ a ] ;
for R being multiplicative non empty multMagma , s being multiplicative multMagma over R , a , b being Element of R holds [ a , b ] in [: the carrier of R , the carrier of R :] & [ b , a ] in [: the carrier of R , the carrier of R :] & [ a , b ] in the carrier of R
>= ( n , b , 1 ) + ( n , d ) .= b + ( n , d ) .= b + ( n , d ) .= b + ( n , d ) .= b + ( n + 1 ) .= n + 1 + 1 ;
cluster ( i1 + i2 ) -> Z for Element of INT , i , j be Element of INT , k be Element of NAT , n be Nat , x be Element of INT , y be Element of INT ;
( ( - s2 ) * p1 + ( ( - s1 ) * p2 ) * p2 ) = ( - ( ( - s1 ) * p1 ) + ( ( - s1 ) * p2 ) * p2 .= ( - ( ( - s1 ) * p2 ) ) * p1 + ( ( - s1 ) * p2 ) * p2 ;
eval ( ( a | ( n , L ) *' p ) , x ) = eval ( a | ( n , L ) , x ) * eval ( p , x ) .= a * eval ( p , x ) * eval ( p , x ) .= a * eval ( p , x ) * eval ( p , x ) ;
assume that the TopStruct of S = the TopStruct of T and for D being non empty Subset of S , V being open Subset of T st D in V & V is open holds V meets V and V is open and V is open and for V being open Subset of S st V in V holds V meets V ;
assume that 1 <= k and k <= len w + 1 and T-7 . ( ( ( q , w ) div 2 ) * ( k + 1 ) ) = ( T-7 . ( ( q , w ) div 2 ) * ( k + 1 ) ) and T-7 . ( ( q , w ) div 2 ) = ( T-7 . ( ( q , w ) div 2 ) * ( k + 1 ) ) ;
2 * ( a |^ ( n + 1 ) + ( 2 * b ) ) >= ( a |^ n + ( 2 * a ) * b ) + ( 2 * a ) * ( b |^ ( n + 1 ) + ( 2 * a ) * b ) ;
M , v / ( x. 3 , x ) / ( x. 0 , x ) / ( x. 4 , x ) / ( x. 0 , x ) / ( x. 0 , x ) / ( x. 0 , x ) / ( x. 0 , x ) / ( x. 4 , x ) / ( x. 0 , x ) / ( x. 0 , x ) ) |= H ;
assume that f is_differentiable_on l and for x0 st x0 in l holds 0 < f . ( x - x0 ) or for x0 st x0 in l holds f /. x0 < f /. ( x - x0 ) ;
for G1 being _Graph , W being Walk of G1 , e being Vertex of G2 , x being Vertex of G1 , y being Vertex of G2 st e in W & x in W holds not x in W .vertices() & e in W .vertices() implies x in y
not ( not ( ( ( not ( ( ( ( not ( ( ( ( y is y ) ) ) & not ( ( not ( ( not ( ( y is not ( ( ( y ) ) ) & not ( not ( not ( ( not y is not ( ( not ( ( ( not y is not ( not ( ( not ( ( not y is not ( not ( not ( not ( not y is not ( not ( not y ) & not ( not ( not ( not ( ( not ( ( not y & not ( ( ( ( ( not y & not ( ( ( ( ( ( ( not y ) & not ( ( ( ( not y ) & not ( ( ( not y is not ( ( not ( ( not ( not ( ( not ( not not (
Indices GoB f = [: dom GoB f , Seg width GoB f :] & ( GoB f ) * ( i1 , 1 ) = [: dom GoB f , Seg width GoB f ) & ( GoB f ) * ( i1 , 1 ) = ( GoB f ) * ( i1 , 1 ) ;
for G1 , G2 , G3 being Group , O being stable Subgroup of G1 , A being stable Subgroup of G2 st G1 is_stable in O & G2 is_stable & G1 is_stable & G2 is_stable , O holds ( G1 * ( G * ( i , j ) ) is stable of G2 * ( G * ( i , j ) )
UsedIntLoc ( ( int ) := ( 1 , ( intloc 0 ) ) ) = { intloc 0 , ( ( intloc 0 ) .--> 1 ) , ( ( intloc 0 ) .--> 1 ) , ( ( intloc 0 ) .--> 1 ) , ( ( intloc 0 ) .--> 1 ) ) , ( ( intloc 0 ) .--> 1 ) ;
for f1 , f2 being FinSequence of F st f1 is p -element & f2 is p -element & Q [ f1 ^ f2 ] & Q [ f1 ^ f2 ] holds Q [ f1 ^ f2 ]
( ( p `1 ) ^2 + ( p `2 ) ^2 ) = ( ( q `1 ) ^2 + ( q `2 ) ^2 * ( ( q `2 ) ^2 ) .= ( ( q `1 ) ^2 + ( q `2 ) ^2 * ( ( q `2 ) ^2 ) ;
for x1 , x2 , x3 , x4 being Element of REAL n holds |( x1 - x2 , x3 - x4 )| = |( x1 , x3 , x4 )| & |( x1 , x2 , x3 )| = |( x1 , x3 , x4 )| - |( x2 , x3 )|
for x st x in dom ( ( ( - ( x ) ) | A ) ) holds ( ( ( - ( x ) ) | A ) . x ) = - ( ( ( - ( x ) ) | A ) . x )
for T being non empty TopSpace , P being Subset of T , A being Subset of T st P c= the topology of T & for x being Point of T st x c= A ex B being Basis of x st B c= A & B c= B & P is Basis of x
( a 'or' b 'imp' c ) . x = 'not' ( ( a 'or' b ) . x ) 'or' c . x .= 'not' ( ( a 'or' b ) . x ) 'or' c . x .= ( 'not' ( a 'or' b ) . x ) 'or' c . x .= TRUE ;
for e being set st e in A8 ex X1 being Subset of X , Y1 being Subset of Y st e = X1 & Y1 is open & ex Y1 being Subset of X st e = Y1 & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open ;
for i be set st i in the carrier of S for f being Function of S . i , S1 . i st f = H . i & f . i = f | ( F . i ) holds F . i = f | ( F . i )
for v , w st for x , y st x <> y holds w . ( y ) = v . y holds Valid ( VERUM ( Al , J ) , J ) . ( v . ( y ) ) = Valid ( VERUM ( Al ) , J ) . ( y )
card D = card D1 + card D2 - card { i + 1 } .= card D1 + card { i + 1 } - card { i + 1 } .= 2 * ( i + 1 ) - card { i + 1 } .= 2 * ( i + 1 ) - card { i + 1 } .= 2 * ( i + 1 ) - card { i + 1 } .= 2 * ( i + 1 ) - card { i + 1 + 1 } .= 2 * ( i + 1 ) - card D - 1 - 1 - 1 - 1 .= 2 * ( i + 1 - 1 - 1 + 1 .= 2 * ( i + 1 ) - 1 + 1 - 1 - 1 + 1 .= 2 * ( i + 1 - 1
IC Exec ( i , s ) = ( s +* ( 0 , s ) ) . 0 .= ( 0 .--> ( s . 0 ) ) . 0 .= ( 0 .--> ( s . 0 ) ) . 0 .= ( 0 .--> ( s . 0 ) ) . 0 .= ( 0 .--> ( s . 0 ) ) . 0 .= ( ( 0 .--> s ) . 0 .= ( 0 .--> s ) . 0 .= ( ( 0 .--> s ) . 0 ) . 0 .= ( ( 0 .--> s ) . 0 .= ( ( 0 .--> s ) . 0 .= ( 0 .--> s . 0 .= ( 0 .--> s ) . 0 .= ( 0 .--> s ) . 0 .= ( 0 .--> ( s . 0 ) .= ( 0 .--> ( s .
len f /. ( len f -' 1 ) + 1 - 1 = len f -' ( len f -' 1 ) + 1 - 1 .= len f - ( len f -' 1 ) + 1 - 1 .= len f - ( len f -' 1 ) + 1 - 1 .= len f - ( len f -' 1 ) + 1 .= len f - 1 + 1 - 1 .= len f - 1 + 1 .= len f - 1 + 1 .= len f - 1 + 1 ;
for a , b , c being Element of NAT st 1 <= a & 2 <= b & k < a holds a <= b + b- c or a = b + b- c or a = b + b- c
for f being FinSequence of TOP-REAL 2 , p being Point of TOP-REAL 2 , i being Nat st i in LSeg ( f , i ) & i <= len f & p = f /. i holds Index ( p , f /. i ) = Index ( p , f /. i )
lim ( ( curry ( F , k + 1 ) # x ) ) = lim ( ( curry ( F , k ) # x ) + ( ( curry ( F , k ) # x ) # x ) .= lim ( ( curry ( F , k ) # x ) ) + lim ( ( curry ( F , k ) # x ) # x ) ;
z2 = g /. ( len g -' ( n + 1 ) + 1 ) .= g . ( i + 1 + ( n + 1 ) - 1 ) .= g . ( i + 1 + ( n + 1 ) - 1 ) .= g . ( i + 1 + ( n + 1 ) - 1 ) .= g . ( i + 1 + ( n + 1 ) - 1 ) .= g . ( i + 1 ) ;
[ f . 0 , f . 3 ] in id ( the carrier of G ) or [ f . 0 , f . 3 ] in ( the InternalRel of G ) \/ ( the InternalRel of G ) ;
for G being Subset-Family of B st G = { R [ X ] where R is Subset of A ( ) , Y is Subset of B ( ) st R in F & Y in G ( ) & X c= Y ( ) & Y c= G ( ) holds ( Intersect F ) . Y = Intersect G
CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 ) ) = CurInstr ( P1 , Comput ( P1 , s1 , m1 ) ) .= CurInstr ( P1 , Comput ( P2 , s2 , m1 ) ) .= CurInstr ( P1 , Comput ( P2 , s2 , m1 ) ) .= CurInstr ( P2 , Comput ( P2 , s2 , m1 ) ) .= halt SCMPDS ;
assume that a on M and b on M and c on N and d on N and p on N and p on N and a on b and p on N and p on N and p on N and p on N and p on N and p on N and p on N and p on N and p on N and p on N and p on N and p on N and p on N and p on N and p on N and p on N and p on N and p on N and p on N and p on N and p on N and p on N and p on N and p on N and p on N and p on N and p on N and p on N and p on N and p on N and p on N and p on N and p on N and p on N and p on M and p on M and p on M and p on M and p on M and p on M
assume that T is \hbox { 4 , T _ 4 } and B is of T and ex F being Subset-Family of T st F is closed & for p being Point of T st p in F ex F being Subset-Family of T st F is finite-ind & p = F . p & F is finite-ind & ind F <= 0 ;
for g1 , g2 st g1 in ]. r - g , r + g .[ & g2 in ]. r - g , r + g .[ holds |. ( f - g ) . g1 - ( f - g ) . g2 .| <= ( ( f - g ) . g2 ) / ( r - g )
( ( exp_R ) /. ( z1 + z2 ) ) = ( exp_R /. z1 ) * ( ( exp_R /. z2 ) * ( ( exp_R /. z2 ) * ( ( exp_R /. z2 ) * ( ( exp_R /. z2 ) ) * ( ( exp_R /. z2 ) * ( ( exp_R /. z2 ) * ( ( exp_R /. z2 ) * ( ( exp_R /. z2 ) * ( ( exp_R /. z2 ) * ( ( exp_R /. z2 ) * ( ( exp_R /. z2 ) ) ) ) ) ) ) ;
F . i = F /. i .= 0. R + r2 .= ( b |^ n ) * ( a |^ ( n + 1 ) ) .= ( b |^ n ) * ( a |^ ( n + 1 ) ) .= ( b |^ n ) * ( a |^ ( n + 1 ) ) .= ( b |^ n ) * ( a |^ ( n + 1 ) ) ;
ex y being set , f being Function st y = f . n & dom f = NAT & f . 0 = A & for n holds f . ( n + 1 ) = R ( n , f . n ) & for n holds f . ( n + 1 ) = R ( n , f . n ) ;
func f * F -> FinSequence of V means : Def1 : len it = len F & for i be Nat st i in dom it holds it . i = F /. i * 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 , x5 , x5 , x5 } = { 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 , x5 , 7 , 7 , 7 , 7 , 7 , 7 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5
for n being Nat , x being set st x = h . n holds h . ( n + 1 ) = o ( 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 [ S , e , e ] = S1 & ( for i , l being Element of NAT st [ S , l ] = S1 . i & ( S . i ) `1 = ( S . i ) `1 & ( S . i ) `1 = ( S . l ) `1 ) & ( S . l = ( S . l ) `1 ) `1 ) ;
consider P being FinSequence of Gj such that p9 = Product P and for i st i in dom P ex t being Element of the carrier of K st P . i = t & t . i = i & t . i = j ;
for T1 , T2 being strict non empty TopStruct , P being Subset of T1 , p1 being Subset of T2 st the topology of T1 = the topology of T2 & P = the topology of T2 & P is Basis of T2 holds P is Basis of T1
assume that f is_partial_differentiable_in u0 , u0 and r (#) pdiff1 ( f , 3 ) is_partial_differentiable_in u0 , 2 and r (#) pdiff1 ( f , 3 ) is_partial_differentiable_in u0 , 2 and partdiff ( r (#) pdiff1 ( f , 3 ) , u0 ) . 3 = r * pdiff1 ( f , 3 ) . 3 and partdiff ( r (#) pdiff1 ( f , 3 ) , u0 , 3 ) . 3 = r * pdiff1 ( f , 3 ) . 3 ;
defpred P [ Nat ] means for F , G being FinSequence of bool REAL , s being Permutation of Seg $1 st len F = $1 & len G = $1 & for s being Permutation of Seg $1 st len s = $1 holds Sum F = Sum G & Sum s = 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 & ( GoB f ) * ( 1 , j + 1 ) `2 <= s * ( 1 , j + 1 ) `2 ;
defpred U [ set , set ] means ex Fa1 being Subset-Family of T st $1 = F & $2 is discrete & ( union F is discrete & ( union F is discrete & $1 is discrete & $2 is connected & $2 is connected & union F is discrete ) & ( union F is discrete & union F is discrete & union F is discrete ) ;
for p4 being Point of TOP-REAL 2 st LE p4 , p1 , P & LE p1 , p2 , P & LE p1 , p2 , P & LE p2 , p1 , P & LE p1 , p2 , P & LE p2 , p1 , P & LE p1 , p2 , P & LE p2 , p1 , P & LE p1 , p2 , P & LE p2 , p1 , P & LE p1 , p2 , P & LE p2 , p1 , P & LE p1 , p2 , P & LE p1 , p2 , P & LE p1 , p2 , P & LE p2 , p2 , P & LE p1 , p2 , P & LE p1 , p2 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p1 , p2 , P & LE p1 , p2 , P & LE p1 , p2 , P & LE p1 , p2 , P & LE p2 , p2 , P & LE p2 , p2 , P & LE p2
f in \rbrace & for y st g . y <> f . y & for x st x <> y holds x = y holds g . x = f . ( All ( x , H ) . y ) implies f in for x st x in dom ( ( the Sorts of H ) . ( y , x ) ) holds f . x in rng ( ( the Sorts of H ) . ( y , x ) )
ex 8 being Point of TOP-REAL 2 st x = 8 & ( ( |. 8 .| ) * ( |. 8 .| ) >= 0 & ( |. 8 .| ) * ( |. 8 .| ) >= 0 & ( |. 8 .| ) * ( |. 8 .| ) >= 0 & ( |. 8 .| ) * ( |. 8 .| ) >= 0 & ( |. 8 .| ) * ( |. 8 .| ) >= 0 ;
assume for d7 being Element of NAT st d7 <= 8 holds s1 . ( |. 7 - 2 .| + 1 ) = s2 . ( |. 7 - 2 .| + 1 ) & s1 . ( |. 7 - 2 .| + 1 ) = s2 . ( |. 7 - 2 .| + 1 ) & s1 . ( |. 7 - 2 + 1 ) = s2 . ( |. 7 - 2 + 1 ) ;
assume that s <> t and s is Point of Sphere ( x , r ) and s is not Point of Sphere ( x , r ) and ex e being Point of E st e = Ball ( x , r ) /\ Sphere ( x , r ) ;
given r such that 0 < r and for s holds 0 < s or ex x1 , x2 being Point of CNS st x1 in dom f & x2 in dom f & ||. x1 - x2 .|| < s & ||. x1 - x2 .|| < s & ||. x1 - x2 .|| < s & ||. x2 - x1 .|| < s & ||. x2 - x1 .|| < s ;
( p | x ) | ( p | ( x | ( x | ( x | ( x | ( x | ( x | ( x | p ) ) ) ) ) ) ) | ( p | ( x | ( x | ( x | ( x | ( x | p ) ) ) ) ) ) = ( ( ( x | ( x | ( x | ( x | p ) ) ) ) | ( p | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | p ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) | ( ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | (
for x , h , x st x + h in dom sec holds ( ( ( sec * sec ) * sec ) `| Z ) . x = ( ( ( ( 2 * sec ) * sec ) `| Z ) . x + ( ( ( 2 * sec ) * sec ) `| Z ) . x ) / ( ( ( 2 * sec ) * sec ) . x ) ^2 )
assume that i in dom A and len A > 1 and len B > 1 and B c= rng A and B * ( i , j ) = A * ( i , j ) and B * ( i , j ) = A * ( i , j ) and A * ( i , j ) = A * ( i , j ) and B * ( i , j ) = A * ( i , j ) ;
for i be non zero Element of NAT st i in Seg n holds i divides n or i = n or i = n & h . i = <* 1_ F_Complex *> & ( i divides n implies h . i = ( 1. L ) * h . i ) & ( i divides n implies h . i = ( 1. L ) * h . i
( ( ( b1 'imp' b2 ) '&' ( ( b1 'or' b2 ) '&' ( ( b1 'or' b2 ) '&' ( ( b2 'or' b2 ) '&' ( b2 'or' b3 ) ) ) ) '&' ( ( ( b1 'or' b2 ) '&' ( ( b1 'or' b2 ) '&' ( ( b1 'or' b2 ) '&' ( b2 'or' b3 ) ) '&' ( ( b1 'or' b2 ) '&' ( ( b1 'or' b2 ) '&' ( b2 'or' b3 ) ) '&' ( ( b1 'or' b2 ) '&' ( b2 'or' b3 ) ) ) ) ) ) ) ) ) 'imp' ( ( ( b1 'or' b2 ) '&' ( ( b1 'or' b2 ) '&' ( ( b1 'or' b2 ) '&' ( ( b2 ) '&' ( ( ( b2 ) '&' ( ( ( b2 'or' b3 ) '&' ( ( b2 'or' b3 ) '&' ( ( b2 'or' b3 ) '&' ( ( ( b2 'or' b3 ) '&' ( ( b2 ) '&' ( ( b2 'or' b3 ) '&' ( ( b2 ) '&' ( ( b2 ) '&'
assume that for x holds f . x = ( ( - cot ) * ( sin - sin ) ) . x and for x st x in dom ( ( - cot ) * ( sin - sin ) ) holds ( ( - cot ) * ( sin - cos ) ) . x = - sin . x and ( ( - cot ) * ( sin - cos ) ) . x = - cos . x ;
consider R8 , I-8 be Real such that R8 = Integral ( M , Re ( F . n ) ) and R-8 = Integral ( M , Im ( F . n ) ) and I = Integral ( M , Im ( F . n ) ) and for n be Nat holds I . 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 & ||. q - p .|| < r holds ||. partdiff ( f , x ) - partdiff ( f , x ) .|| < r ;
x in { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , 7 , 8 , 7 , 8 , 7 , 8 , 7 , 8 , 7 , 8 , 7 , 8 , 7 , 8 , 7 , 8 , 7 , 8 , 7 , 8 , 7 , 8 , 7 , 8 , 7 , 8 , 8 , 7 , 8 , 7 , 8 , 8 , 7 , 8 , 7 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 7 , 7 , 7 , 8 , 8 , 8 , 8 , 8 , 7 , 7 , 8 , 8 , 7 , 7 , 7 , 7 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 7 , 7 , 7 , 8 , 8 , 7 , 8 , 8 ,
( G * ( j , i ) ) `2 = G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 ;
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 ) . o .= ( the Arity of S1 ) . o ;
func tree ( T , P , T1 ) -> tree of T means : Def1 : q in P iff for p st p in P & p in P & p in P & p in P holds p ^ q in P ;
F /. ( k + 1 ) = F . ( p . ( k + 1 ) , k + 1 ) .= F . ( p . ( k + 1 ) , p . ( k + 1 ) ) .= F . ( p . ( k + 1 ) , p . ( k + 1 ) ) .= F . ( p . ( k + 1 ) , p . ( k + 1 ) .= F /. ( k + 1 ) ;
for A , B , C being Matrix of n , K st len B = len C & len C = len B & len B = len C & len C = len A & len C > 0 & len B > 0 & len C > 0 & len A > 0 & len B > 0 & len A > 0 & len B > 0 & len A > 0 & width B = 0 holds A * ( B * C ) = B * ( A * C )
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 ) + Partial_Sums ( seq ) . ( k + 1 ) .= Partial_Sums ( seq ) . ( k + 1 ) + Partial_Sums ( seq ) . ( k + 1 ) ;
assume that x in ( the carrier of CP ) and y in ( the carrier of CP ) and [ x , y ] in ( the carrier of CP ) and [ y , z ] in ( the carrier of CP ) ;
defpred P [ Element of NAT ] means for f st len f = $1 & ( for k st k in dom f holds ( VAL g ) . k = ( VAL g ) . k ) holds ( ( VAL g ) . ( k + 1 ) ) . ( ( VAL g ) . k ) = ( VAL g ) . ( k + 1 ) ;
assume that 1 <= k and k + 1 <= len f and f /. k = G * ( i , j ) and [ i + 1 , j ] in Indices G and f /. k = G * ( i + 1 , j ) and f /. ( k + 1 ) = G * ( i + 1 , j ) and f /. ( k + 1 ) = G * ( i + 1 , j ) ;
assume that cn < 1 and ( q `1 ) ^2 > 0 and ( q `2 <= 1 or q `2 >= 0 & q `2 <= 1 or q `1 >= 0 & q `2 <= 1 & q `2 <= 1 or q `1 >= 0 & q `2 <= 1 & q `2 <= 1 or q `1 >= 0 & q `1 <= 1 & q `2 <= 1 & q `2 <= 1 & q `2 <= 1 & q `2 <= 1 & q `2 <= 1 & q `1 <= 1 & q `2 <= 1 & q `2 <= 1 & q `2 <= 1 or q `1 <= 1 & q `2 <= 1 & q `2 <= 1 & q `2 <= 1 & q `2 <= 1 & q `2 <= 1 & q `2 <= 1 or q `2 <= 1 & q `2 <= 1 & q `2 <= 1 & q `2 <= 1 & q `2 <= 1 & q `1 <= 1 & q `2 <= 1 & q `1 <= 1 & q `1 <= 1 & q `2 <=
for M being non empty metric Space , x being Point of M , p being Point of M st x = x & ex f being Function of M , TOP-REAL n st f is continuous & for n being Element of NAT holds f . n = Ball ( x , f . n ) holds f . x = Ball ( x , r )
defpred P [ Element of omega ] means f1 is_differentiable_on Z & f2 is_differentiable_on Z & Z = ]. - 1 , 1 .[ & for x st x in Z holds ( ( f1 - f2 ) `| Z ) . x = ( - 1 ) * ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( f1 - f2 ) ) ) . x ;
defpred P1 [ Nat , Point of Cj ] means ( $1 in Y & $2 in Y & ||. $2 - $2 .|| < r & ||. $2 - $2 .|| < r & ||. $2 - $2 .|| < r / 2 ;
( f ^ mid ( g , 2 , len g ) ) . i = ( mid ( g , 2 , len g ) ) . ( i + 1 ) .= g . ( i + 1 ) .= g . ( i + 1 ) .= ( g . i ) .= ( g . i ) . ( len g ) .= ( g . i ) . ( len g ) .= ( g . i ) . ( len g ) .= ( g . i ) . ( len g + 1 ) .= ( g . ( len g ) ;
( 1 / 2 * ( n + 2 ) ) * ( 2 * ( n + 1 ) ) = ( ( 1 / 2 ) * ( n + 1 ) ) * ( 2 * ( n + 1 ) ) .= ( 1 / 2 ) * ( 2 * ( n + 1 ) ) .= 1 / 2 * ( 2 * ( n + 1 ) ) .= 1 / 2 * ( 2 * n ) ;
defpred P [ Nat ] means for G being non empty set , A being non empty finite RelStr , P , Q being non empty finite non empty RelStr st G is the carrier of A & P is the carrier of Q & Q is the RelStr of P & P is the RelStr of Q holds P is strict ;
assume that not f /. 1 in Ball ( u , r ) and not 1 <= m and m <= len ( - f ) and not ( for i st 1 <= i & i <= len f holds LSeg ( f , i ) /\ Ball ( u , r ) <> {} ) and not ( ex i st 1 <= i & i <= len f & f /. i <> 0 ) ;
defpred P [ Element of NAT ] means ( Partial_Sums ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) * ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . $1 = ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( r ) ) * ) ) * ( ( ( ( ( ( ( ( ( ( ( ( ( r ) ) * ) ) ) ) ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( r ) ) * ) ) ) ) ) * ) * ( ) ) * ( ) ) * ( ( ( ( ( ( ( ( ( ( (
for x being Element of product F holds x is FinSequence of product F & ( for i being set st i in dom x holds x . i = I . i ) & ( for i being set st i in dom x holds x . i = F . i ) implies x is FinSequence of product F
( x " ) |^ ( n + 1 ) = ( x " ) |^ n * x .= ( x |^ n ) * x .= ( x |^ n ) * x .= ( x |^ n ) * x .= ( x |^ n ) * x .= ( x |^ n ) * x .= ( x |^ n ) * x .= ( x |^ n ) * x .= x |^ n * x |^ n ;
DataPart Comput ( P +* I , ( LifeSpan ( P +* I , s ) ) + 3 ) = DataPart Comput ( P +* I , s +* I , LifeSpan ( P +* I , s ) ) .= DataPart Comput ( P +* I , s +* I ) ;
given r such that 0 < r and ]. x0 - r , x0 + r .[ c= dom ( f1 + f2 ) /\ dom ( f2 + f3 ) and for g st g in ]. x0 - r , x0 + r .[ holds f1 . g <= ( f1 + f2 ) . g + ( f2 + f3 ) . g ;
assume that X c= dom f1 /\ dom f2 and f1 | X is continuous and ( for x st x in X /\ dom f2 holds f1 . x = f1 . x ) and ( for x st x in X /\ dom f2 holds f1 . x = f2 . x ) and ( for x st x in X /\ dom f2 holds f1 . x = f1 . x ) and ( f1 | X ) . x = f2 . x ;
for L being continuous complete LATTICE st 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 ` & x is prime & x is prime holds x is prime
Support ( e *' p ) in { m *' ( p , T ) where m is Element of NAT : ex p being Polynomial of n , L st p = { m *' p , T & p . ( m + 1 ) = p . ( m + 1 ) & p . ( m + 1 ) = p . ( m + 1 ) & p . ( m + 1 ) = p . ( m + 1 ) ;
( f1 - f2 ) /. ( lim ( f1 /* ( seq ^\ k ) ) ) = lim ( ( f1 /* ( seq ^\ k ) ) ) .= lim ( ( f1 /* ( seq ^\ k ) ) ) .= lim ( ( f1 /* ( seq ^\ k ) ) ) .= lim ( ( f1 /* ( seq ^\ k ) ) ) .= lim ( ( f1 /* ( seq ^\ k ) ) /* ( f2 /* ( seq ^\ k ) ) ) ;
ex p1 being Element of CQC-WFF ( Al ) st F . p1 = g . ( p `1 ) & for g being Function of Al ( ) , D ( ) st P [ g , p ] holds P [ g , p1 , g ] ;
( mid ( f , i , len f -' 1 ) ) /. ( j + 1 ) = ( mid ( f , i , len f -' 1 ) ) /. ( j + 1 ) .= ( mid ( f , i , len f -' 1 ) ) /. ( j + 1 ) .= ( mid ( f , i , len f -' 1 ) ) /. ( j + 1 ) .= ( mid ( f , i , len f -' 1 ) ) /. j .= ( mid ( f , i , len f -' 1 ) . ( j + 1 ) ;
( ( p ^ q ) . ( len p + k ) ) = ( ( p ^ q ) . ( len p + k ) ) .= ( ( p ^ q ) . ( len p + k ) ) .= ( ( p ^ q ) . ( len p + k ) ) .= ( ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p
len mid ( ( upper_volume ( f , D1 ) , indx ( D2 , D1 , j1 ) + 1 ) , indx ( D2 , D1 , j1 ) ) = indx ( D2 , D1 , indx ( D2 , D1 , j1 ) + 1 ) - indx ( D2 , D1 , j1 ) + 1 .= indx ( D2 , D1 , j1 ) - 1 ;
x * y * z = Mj . ( x * y , z ) .= x * ( y * z ) .= ( x * y ) * ( y * z ) .= x * ( y * z ) .= x * ( y * z ) .= x * y * z ;
v . ( <* x , y *> ) + ( <* y , z *> ) . i = partdiff ( v , y , i ) * ( ( reproj ( 1 , y ) ) . ( ( reproj ( 1 , y ) ) . ( y - x0 ) ) + ( proj ( 1 , y ) ) . ( ( reproj ( 1 , y ) ) . ( y - x0 ) ) + ( proj ( 1 , y ) . ( y - x0 ) ) ;
i * i = <* 0 * ( 1 - i ) * ( 1 - i ) *> .= <* - 1 * ( i - i ) * ( i - i ) *> .= <* - 1 * ( i - i ) * ( i - i ) *> .= <* - 1 * ( i - i ) * ( i - i ) *> .= <* - 1 * ( i - i ) * ( i - i ) *> .= <* - 1 * ( i - i ) * ( i - i ) ;
Sum ( L * F ) = Sum ( L * ( F1 ^ F2 ) ) .= Sum ( L * F ) + Sum ( L * F ) .= Sum ( L * F ) + Sum ( L * F ) .= Sum ( L * F ) + Sum ( L * F ) .= Sum ( L * F ) + Sum ( L * F ) .= Sum ( L * F ) + Sum ( L * F ) .= Sum ( L * F ) + Sum ( L * F ) .= Sum ( L * F ) + Sum ( L * F ) + Sum ( L * F ) .= Sum ( L * F ) + Sum ( L * F ) .= Sum ( L * F ) + Sum ( L * F ) + Sum ( L * F ) .= Sum ( L * F ) + Sum ( L * F ) + Sum ( L * F ) + Sum ( L * F ) .= Sum ( L * F ) + Sum ( L * F ) .= Sum ( L * F ) .= Sum ( L * F ) + Sum ( L * F ) .= Sum ( L * F ) + Sum ( L * F
ex r be Real st for e be Real st 0 < e ex Y1 be finite Subset of X st Y1 c= Y & for x be Element of X st Y1 c= Y & x in Y1 holds |. ( - r ) . x - ( r ) . x .| < r ;
( GoB f ) * ( i , j + 1 ) = f /. ( k + 2 ) & ( GoB f ) * ( i + 1 , j ) = f /. ( k + 2 ) or ( GoB f ) * ( i + 1 , j + 1 ) = f /. ( k + 2 ) or ( GoB f ) * ( i + 1 , j + 1 ) = f /. ( k + 2 ) ;
( ( - cos ) . x ) ^2 = ( - ( sin . x ) ) ^2 - ( sin . x ) ^2 .= ( - ( sin . x ) ) ^2 - ( sin . x ) ^2 .= ( - ( sin . x ) ) ^2 - ( sin . x ) ^2 .= ( - ( sin . x ) ) ^2 - ( sin . x ) ^2 ;
- ( ( - b ) + sqrt ( delta ( a , b , c ) ) / 2 ) * a + ( - b ) / sqrt ( ( - b ) ^2 ) > 0 & - ( ( - b ) / sqrt ( ( - a ) ^2 ) ) / 2 + ( - b ) / sqrt ( ( - a ) ^2 ) > 0 ;
Suppose ex_inf_of downarrow L /\ X , L and ex_sup_of X , L and ex_sup_of X , L and for x being Element of L holds x < "\/" ( X , L ) and not x < "\/" ( X , L ) and not x < "\/" ( X , L ) and not x < "\/" ( X , L ) ;
( ( ( - B ) . ( i , j ) ) (*) ( i , j ) ) = ( j |-> ( id O ) . ( i , j ) ) (*) ( i , j ) & ( j = i implies j = i & j = j & i = j implies j = i & j = j & i = j & j = j