thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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 = b ;
0 < k ;
let b ;
let n ;
r <= b ;
x in X ;
i >= 8 ;
let n ;
let n ;
y in f ;
let n ;
1 < j ;
a in L ;
C is boundary ;
a in A ;
1 < x ;
S is finite ;
u in I ;
z << z ;
x in V ;
r < t ;
let t ;
x c= y ;
a <= b ;
m in dom f ;
assume f is max ;
not x in Y ;
z = +infty ;
k be Nat ;
K `2 is being_line ;
assume n >= N ;
assume n >= N ;
assume X is 1 -element ;
assume x in I ;
q is as let of 0 ;
assume c in x ;
as Real ;
assume x in Z ;
assume x in Z ;
1 <= k12 ;
assume m <= i ;
assume G is commutative ;
assume a divides b ;
assume P is closed ;
I > 0 ;
assume q in A ;
W is not bounded ;
f is elements ;
assume A is boundary ;
g is special ;
assume i > j ;
assume t in X ;
assume n <= m ;
assume x in W ;
assume r in X ;
assume x in A ;
assume b is even ;
assume i in I ;
assume 1 <= k ;
X is non empty ;
assume x in X ;
assume n in M ;
assume b in X ;
assume x in A ;
assume T c= W ;
assume s is atomic ;
b `2 <= c `2 ;
A meets W ;
i `2 <= j `2 ;
assume H is universal ;
assume x in X ;
let X be set ;
let T be Tree ;
let d be element ;
let t be element ;
let x be element ;
let x be element ;
let s be element ;
k <= L~ L~ f ;
let X be set ;
let X be set ;
let y be element ;
let x be element ;
P [ 0 ]
let E be set , f be FinSequence 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 ;
let a be Real ;
let x be element ;
let X be element ;
P [ 0 ]
let x be element ;
let x be element ;
let y be element ;
r in REAL ;
let e be element ;
n1 is from of n , m ;
Q halts_on s ;
x in \in \in \in \in \in \in \in $ ;
M < m + 1 ;
T2 is open ;
z in b < a < b ;
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 , f be FinSequence of 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 , x1 ;
let E be Ordinal ;
o : o : a , b are_collinear ;
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 , M be Subset 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 RealLinearSpace , M be Subset of V ;
P [ 1 ] ;
P [ {} ] ;
C1 meets L~ Cage ( C , n ) ;
H = G . i ;
1 <= i `2 + 1 ;
F . m in A ;
f . o = o ;
P [ 0 ] ;
a\not <= non < \pi ;
R [ 0 ] ;
b in f .: X ;
assume q = q2 ;
x in [#] V ;
f . u = 0 ;
assume e1 > 0 ;
let V be RealUnitarySpace , M be Subset of V ;
s is trivial & s is non empty ;
dom c = Q ;
P [ 0 ] ;
f . n in T ;
N . j in S ;
let T be complete LATTICE , 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.: S is bounded ;
assume p = p2 ;
len f = n ;
assume x in P1 ;
i in dom q ;
let 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 `2 ;
1 <= jj & jj <= len f ;
set A = Cl -> set ;
card a [= c ;
e in rng f ;
cluster B \oplus A -> empty ;
H is_\cdot \cdot the \rangle ;
assume n0 <= m ;
T is increasing implies T is increasing
e2 <> e2 & e2 <> e2 ;
Z c= dom g ;
dom p = X ;
H is proper implies G is implies H = G
i + 1 <= n ;
v <> 0. V ;
A c= Affin A ;
S c= dom F ;
m in dom f ;
let X0 be set ;
c = sup N ;
R is_connected & 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 \pi ;
not n in Seg 3 ;
assume X in f .: A ;
assume that p <= n and p <= m ;
assume not u in { v } ;
d is Element of A ;
A / b misses B ;
e in v + dom ( X .--> v ) ;
- y in I ;
let A be non empty set , f be FinSequence of A ;
P0 = 1 ;
assume r in F . k ;
assume f is simple & g in S ;
let A be Incountable set ;
rng f c= NAT ( ) ;
assume P [ k ] ;
f6 <> {} ;
let o be Ordinal ;
assume x is sum of squares ;
assume not v in { 1 } ;
let II , A ;
assume that 1 <= j and j < l ;
v = - u ;
assume s . b > 0 ;
d1 in D ;
assume t . 1 in A ;
let Y be non empty TopSpace , f be Function of Y , BOOLEAN ;
assume a in uparrow s ;
let S be non empty Poset ;
a , b // b , a ;
a * b = p * q ;
assume x , y are_the space ;
assume x in [#] ( f ) ;
[ a , c ] in X ;
mm <> {} & mm <> {} ;
M + N c= M + M ;
assume M is connected hhhz ;
assume f is bbbb-r-closed ;
let x , y be element ;
let T be non empty TopSpace ;
b , a // b , c ;
k in dom Sum p ;
let v be Element of V ;
[ x , y ] in T ;
assume len p = 0 ;
assume C in rng f ;
k1 = k2 & k2 = k1 ;
m + 1 < n + 1 ;
s in S \/ { s } ;
n + i >= n + 1 ;
assume Re y = 0 ;
k1 <= j1 & j1 <= j2 ;
f | A is as as as continuous Function ;
f . x - b <= 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 Seg n ;
q2 c= C1 & q2 c= C2 ;
a2 < c2 & a2 < c2 implies a2 = c2
s2 is 0 -started & s2 is 0 -started ;
IC s = 0 ;
s4 = s4 & s4 = s4 ;
let V ;
let x , y be element ;
let x be Element of T ;
assume a in rng F ;
x in dom T `2 ;
let S be MSAlgebra over L ;
y " <> 0 ;
y " <> 0 ;
0. V = u-w ;
y2 , y , w be Element of V ;
R8 in X ;
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 , m be Morphism of o , m ;
r '&' q = P \lbrack l \rbrack ;
let i , j be Nat ;
let s be State of A , n be Nat ;
s4 . n = N ;
set y = ( x `1 ) / ( x `2 ) ;
mi in dom g ;
l . 2 = y1 ;
|. g . y .| <= r ;
f . x in CV ;
V1 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 A0 is dense & A is dense ;
|. f . x .| <= r ;
let x be Element of R ;
let b be Element of L ;
assume x in W-19 ;
P [ k , a ] ;
let X be Subset of L ;
let b be object of B ;
let A , B be category ;
set X = Vars ( C ) ;
let o be OperSymbol of S ;
let R be connected non empty Poset ;
n + 1 = succ n ;
xx c= Z1 & Z1 c= Z1 & x in Z1 ;
dom f = C1 & dom g = C2 ;
assume [ a , y ] in X ;
Re ( seq ) is convergent & Im ( seq ) is convergent ;
assume a1 = b1 & a2 = b2 ;
A = sInt A ;
a <= b or b <= a ;
n + 1 in dom f ;
let F be Instruction of S , k be Nat ;
assume r2 > x0 & x0 < r2 ;
let Y be non empty set , f be Function of Y , BOOLEAN ;
2 * x in dom W ;
m in dom g2 & m + 1 in dom g2 ;
n in dom g1 & m in dom g2 ;
k + 1 in dom f ;
not the still of { s } is finite ;
assume x1 <> x2 & x2 <> x3 ;
v1 in V1 & v2 in V1 & v1 <> v2 ;
not [ b `1 , b ] in T ;
ii + 1 = i ;
T c= Assume T ;
( l - 1 ) * ( l - 1 ) = 0 ;
n be Nat ;
( t `2 ) ^2 = r ;
AA is integrable & f | A is bounded ;
set t = Top t ;
let A , B be real-membered set ;
k <= len G + 1 ;
C ( ) misses V ( ) ;
Product seq is non empty ;
e <= f or f <= e ;
cluster empty -> non empty normal for NAT -defined Function ;
assume c2 = b2 & c1 = c2 ;
assume h in [. q , p .] ;
1 + 1 <= len C ;
not c in B . m1 ;
cluster R .: X -> empty ;
p . n = H . n ;
assume that v-4 is Cauchy and v- is Cauchy ;
IC s3 = 0 & IC s3 = 0 ;
k in N or k in K ;
F1 \/ F2 c= F \/ G ;
Int G1 <> {} & Int G2 <> {} ;
( z `2 ) ^2 = 0 ;
p11 <> p1 & p01 <> p2 implies p1 <> p2
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 RelStr ;
q |^ m >= 1 ;
a >= X & b >= Y ;
assume <* a , c *> <> {} ;
F . c = g . c ;
G is one-to-one one-to-one implies G is one-to-one ;
A \/ { a } \not c= B ;
0. V = 0. Y & 0. V = 0. V ;
I be being being being being being the InstructionsF of S , T be Element of S ;
f-24 . x = 1 ;
assume z \ x = 0. X ;
p4 = 2 to_power n & p4 = 2 to_power n ;
let B be sequence of Sigma ;
assume X1 = p .: D ;
n + l2 in NAT & n + l2 in NAT ;
f " P is compact & f " P is compact ;
assume x1 in REAL & x2 in REAL & x3 in REAL ;
p1 `1 = ( K `1 ) * ( 1 , 1 ) `1 ;
M . k = <*> REAL ;
phi . 0 in rng phi ;
OSMMSorts A is closed
assume z0 <> 0. L & z0 <> 0. L ;
n < N7 . k ;
0 <= seq . 0 & seq . 0 <= seq . 0 ;
- q + p = v ;
{ v } is Subset of B ;
set g = f /. 1 ;
[: R , S :] is stable Subset of R ;
set cR = Vertices R , S = Vertices R ;
( p ) c= P3 & ( p ) c= P3 ;
x in [. 0 , 1 .] ;
f . y in dom F ;
let T be Scott Scott Scott Scott Scott Scott Scott Scott of S ;
ex_inf_of the carrier of S , S ;
sup downarrow a = sup downarrow b ;
P , C , K is_collinear ;
assume x in F ( s , r , t ) ;
2 to_power i < 2 to_power m ;
x + z = x + z + q ;
x \ ( a \ x ) = x ;
||. x-y .|| <= r ;
assume that Y c= field Q and Y <> {} ;
a ~ , b ~ are_equipotent ;
assume a in A ( i ) ;
k in dom ( q ^ <* k *> ) ;
p is non empty \HM { x } ;
i - 1 = i-1 - 1 ;
f | A is one-to-one ;
assume x in f .: [: X , Y :] ;
i2 - i1 = 0 & i2 - i1 = 0 ;
j2 + 1 <= i2 ;
g " * a in N ;
K <> { [ {} , {} ] } ;
cluster strict for for for } ;
|. q .| ^2 > 0 ;
|. p4 .| = |. p .| ;
s2 - s1 > 0 & s2 - s1 > 0 ;
assume x in { Gij } ;
W-min C in C & W-min C in C ;
assume x in { Gij } ;
assume i + 1 = len G ;
assume i + 1 = len G ;
dom I = Seg n & dom I = Seg n ;
assume k in dom C & k <> i ;
1 + 1-1 <= i + j ;
dom S = dom F & dom F = dom G ;
let s be Element of NAT , k be Nat ;
let R be ManySortedSet of A ;
let n be Element of NAT , x be Element of REAL ;
let S be non empty non void non void non empty non void non void for E ;
let f be ManySortedSet of I ;
let z be Element of F_Complex , f be FinSequence of COMPLEX ;
u in { ag } ;
2 * n < ( 2 * n ) ;
let x , y be set ;
B-11 c= V1 & V1 c= V1 & B-15 c= V1 ;
assume I is_halting_on s , P ;
U1 = U2 & U2 = U2 implies U1 = U2
M /. 1 = z /. 1 ;
xx = x22 & x22 = x22 ;
i + 1 < n + 1 + 1 ;
x in { {} , <* 0 *> } ;
( f . x ) `2 <= ( f . x ) `2 ;
l be Element of L ;
x in dom ( F . n ) ;
let i be Element of NAT , k be Nat ;
r8 is ( len C ) -element ;
assume <* o2 , o *> <> {} ;
s . x / 0 = 1 ;
card ( K1 ) in M & card ( K1 ) in M ;
assume that X in U and Y in U ;
let D be Subset-Family of Omega ;
set r = { q . k + 1 } ;
y = W . ( 2 * x ) ;
assume dom g = cod f & cod g = cod f ;
let X , Y be non empty TopSpace , f be Function of X , Y ;
x \oplus A is interval ;
|. <*> A .| . a = 0 ;
cluster strict for Sublattice of L ;
a1 in B . s1 & a2 in B . s1 ;
let V be finite VectSp of F , v be Element of V ;
A * B on B , A ;
f-3 = NAT --> 0 .= ( NAT --> 0 ) ;
let A , B be Subset of V ;
z1 = P1 . j & z2 = P2 . j ;
assume f " P is closed & f " P is closed ;
reconsider j = i as Element of M ;
let a , b be Element of L ;
assume q in A \/ ( B "\/" C ) ;
dom ( F * C ) = o ;
set S = ( Z |^ X ) ;
z in dom ( A --> y ) ;
P [ y , h . y ] ;
{ x0 } c= dom f & { x0 } c= dom f ;
let B be non-empty ManySortedSet of I , f be Function of I , B ;
( PI / 2 ) < Arg z ;
reconsider z9 = 0 - 1 as Nat ;
LIN a , d , c ;
[ y , x ] in IE ;
( Q ) * ( 3 , 1 ) = 0 ;
set j = x0 div m , i = x0 mod m ;
assume a in { x , y , c } ;
j2 - ( j - 1 ) > 0 ;
I ( ) = 1 & phi ( ) = 2 ;
[ y , d ] in [: F , F :] ;
let f be Function of X , Y ;
set A2 = ( B - C ) / ( A * B ) ;
s1 , s2 are_` , T & s1 , s2 are_/ 2 ;
j1 - 1 = 0 & j1 - 1 = j1 - 1 ;
set m2 = 2 * n + j ;
reconsider t = t as bag of n ;
I2 . j = m . j ;
i |^ s , n are_congruent_mod m ;
set g = f | D-21 , h = f | Dy1 ;
assume that X is lower and 0 <= r ;
( p1 `1 ) ^2 = 1 & ( p1 `2 ) ^2 = ( p1 `2 ) ^2 ;
a < ( p3 `1 ) / ( p3 `2 ) ^2 ;
L \ { m } c= UBD C ;
x in Ball ( x , 10 ) ;
not a in LSeg ( c , m ) ;
1 <= i1 -' 1 & i1 -' 1 <= i2 -' 1 ;
1 <= i1 -' 1 & i1 -' 1 <= i2 -' 1 ;
i + i2 <= len h ;
x = W-min ( P ) & y = W-min ( P ) ;
[ x , z ] in X ~ ;
assume y in [. x0 , x .] ;
assume p = <* 1 , 2 , 3 *> ;
len <* A1 *> = 1 & len <* A1 *> = 1 ;
set H = h . ( g . g1 ) ;
card b * a = |. a .| ;
Shift ( w , 0 ) |= v ;
set h = h2 (*) h1 , h1 = h2 (*) h2 ;
assume x in ( X /\ X1 ) \/ ( X /\ X2 ) ;
||. h .|| < d1 & ||. h .|| < d1 ;
not x in the carrier of f ;
f . y = F ( y ) ;
for n holds X [ n ] ;
k - l = kl2 ;
<* p , q *> /. 2 = q ;
let S be Subset of the carrier of Y ;
let P , Q be \HM { s } ;
Q /\ M c= union ( F | M )
f = b * CFS ( S ) ;
let a , b be Element of G ;
f .: X <= f . sup X ;
let L be non empty reflexive RelStr , N be net of L ;
S-20 is x -let i , K ;
let r be non positive Real ;
M , v |= x , y |= x , z ;
v + w = 0. Z & w + w = 0. Z ;
P [ len F ] implies P [ F ( len F ) ]
assume InsCode ( i ) = 8 & InsCode ( i ) = 8 ;
the zero of M = 0 & the zero of M = 0 ;
cluster z * seq -> summable for Real_Sequence ;
let O be Subset of the carrier of C ;
||. f .|| | X is continuous ;
x2 = g . ( j + 1 ) ;
cluster non empty for Element of S S ;
reconsider l1 = l-1 as Nat ;
v4 is Vertex of r2 & v4 is Vertex of r2 ;
T1 is SubSpace of T2 & T2 is SubSpace of T2 ;
Q1 /\ Q19 <> {} & Q1 /\ Q29 <> {} ;
k be Nat ;
q " is Element of X & q " is Element of X ;
F . t is set of m , M ;
assume that n <> 0 and 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 implies ( p `2 <= p `1 ) & ( p `2 <= p `2 ) ;
not r in ]. p , q .] ;
let R be FinSequence of REAL , x be Element of REAL ;
S7 does not destroy b1 & not ( b1 in dom b1 & b2 in dom b2 & b1 <> b2 ) ;
IC SCM R <> a & IC SCM R <> a ;
|. - |[ x , y ]| .| >= r ;
1 * seq = seq & 1 * seq = seq * ( seq ^\ k ) ;
let 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 .= succ IC s .= succ IC s ;
H + G = F- ( GG ) ;
CS1 . x = x2 & CS2 . x = y2 ;
f1 = f .= f .= f2 .= f1 * f ;
Sum <* p . 0 *> = p . 0 ;
assume v + W = v + u + W ;
{ a1 } = { a2 } & { a2 } = { a2 } ;
a1 , b1 _|_ b , a & a1 , b1 _|_ b , a ;
\ d1 , o _|_ o , a3 & d1 , o _|_ o , a3 ;
IC is reflexive & CC is reflexive implies C is reflexive
IO is antisymmetric implies C ( ) is antisymmetric & C ( ) = C ( )
sup rng ( H1 | n ) = e & sup rng ( H1 | n ) = e ;
x = ( a * a9 ) * ( a * b9 ) ;
|. p1 .| ^2 >= 1 ^2 / ( |. p1 .| ) ^2 ;
assume j2 -' 1 < j2 - 1 ;
rng s c= dom f1 & rng s c= dom f2 ;
assume support a misses support b & not b in support b ;
let L be associative non empty doubleLoopStr , F be FinSequence of L ;
s " + 0 < n + 1 ;
p . c = ( f " ) . 1 ;
R . n <= R . ( n + 1 ) ;
Directed ( I1 ) = I1 +* ( 1 , k ) ;
set f = + ( x , y , r ) ;
cluster Ball ( x , r ) -> bounded ;
consider r being Real such that r in A ;
cluster -> non empty for NAT -defined Function ;
let X be non empty directed Subset of S ;
let S be non empty full SubRelStr of L ;
cluster <* \rangle . ( N , \subseteq ) -> complete for non trivial set ;
( 1 - a ) " = a ;
( q . {} ) `1 = o ;
( n - i ) > 0 ;
assume ( 1 - 2 ) * t `2 <= 1 ;
card B = k + 1-1 ;
x in Union ( f | ( rng f ) ) ;
assume x in the carrier of R & y in the carrier of R ;
d in D ;
f . 1 = L . ( F . 1 ) ;
the vertices of G = { v } & not v in { v } ;
let G be : is_differentiable_on wwgraph ;
e , v6 be set , x be set ;
c . ( i - 1 ) in rng c ;
f2 /* q is divergent_to+infty & f2 /* ( f1 /* q ) is divergent_to+infty ;
set z1 = - z2 , z2 = - z2 , z2 = - z2 , z1 = - z2 , z2 = - z2 , z2 = - z2 , z2 = - z1 , z2 = - z2 , z2
assume w is_llof S , G ;
set f = p |-count ( t ) , g = p |-count ( t ) , h = p |-count ( t ) , i = p |-count ( t ) , i = p |-count ( t ) , j
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 IX be Subset-Family of X , f be Function of X , REAL ;
reconsider p = p , q = q as Element of NAT ;
let v , w be Point of X ;
let s be State of SCM+FSA , I be Program of SCM+FSA ;
p is FinSequence of SCM+FSA , f be FinSequence of the carrier of SCM+FSA ;
stop I ( ) c= P3 ( ) ;
set ci = ( f /. i ) `1 , fj = ( f /. j ) `1 ;
w ^ t ^ w ^ s ^ t ^ w ^ s ^ t ^ s ^ w ^ s ^ t ^ s ^ t ^ s ^ t ^ s ^ t ^ s ^ t ^ s ^
W1 /\ W = W1 /\ W2 ` & W1 /\ W2 = W1 /\ W2 ;
f . j is Element of J . j ;
let x , y be Element of T2 , f be Function of T2 , T2 ;
ex d st a , b // b , d ;
a <> 0 & b <> 0 & c <> 0 ;
ord x = 1 & ord x is not of G ;
set g2 = lim ( seq ^\ k ) ;
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 . ( FY ) = 0 & L1 . ( FY ) = 0 ;
/ ( X \/ R1 ) = / ( X \/ R1 ) ;
( ( ( - 1 ) (#) ( sin * cos ) ) `| Z ) . x <> 0 ;
( ( ( 1 / 2 ) (#) ( ( #Z 2 ) * ( f1 + #Z 2 ) ) ) `| Z ) = f ;
o1 in ( X /\ ( O /\ O2 ) ) /\ O2 ;
e , v6 be set , x be set ;
r3 > ( 1 - 2 ) * 0 ;
x in P .: ( F -ideal ( F ) ) ;
let J be closed Ideal of R , I be Ideal of R ;
h . p1 = f2 . O & h . O = g2 ;
Index ( p , f ) + 1 <= j ;
len ( q ^ <* M *> ) = width M & len ( q ^ <* M *> ) = len M ;
the carrier of L c= A & the carrier of L c= A ;
dom f c= union rng ( F . -10 ) ;
k + 1 in support ( ( support n ) | ( support 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 & h . x2 = g . x2 ;
F c= 2 -tuples_on the carrier of X ( ) ;
reconsider w = |. s1 .| as Real_Sequence of REAL ;
( 1 / ( m * m + r ) ) < p ;
dom f = dom ( ( I - 1 ) --> ( I - 1 ) ) ;
[#] P-17 = [#] ( ( TOP-REAL 2 ) | K1 ) ;
cluster - x -> ExtReal -> ExtReal for ExtReal ;
then { d1 } c= A ;
cluster TOP-REAL n -> finite-ind for non empty TopSpace ;
let w1 be Element of M , w2 be Element of N ;
let x be Element of dyadic ( n ) ;
u in W1 & v in W3 & u in W2 implies u in W1
reconsider y = y , z = z as Element of L2 ;
N is full SubRelStr of T |^ ( the carrier of S ) ;
sup { x , y } = c "\/" c ;
g . n = n / 1 .= n ;
h . J = EqClass ( u , J ) ;
let seq be \mathclose { \Vert } , n be Nat ;
dist ( x `1 , y ) < ( r / 2 ) / 2 ;
reconsider mm = m , mn = n as Element of NAT ;
( x- x0 ) < r1 - x0 ;
reconsider P = P ` as strict Subgroup of N ;
set g1 = p * ( idseq q `1 ) , g2 = p `2 ;
let n , m , k be non zero Nat ;
assume that 0 < e and f | A is lower ;
D2 . ( I1 ) in { x } & D2 . ( I ) in { x } ;
cluster subcondensed closed -> subopen for Subset of T ;
let P be compact non empty Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
Gij in LSeg ( cos , 1 ) /\ LSeg ( cos , 1 ) ;
n be Element of NAT , x be Element of REAL n ;
reconsider S8 = S , S8 = T as Subset of T ;
dom ( i .--> X ' ) = { i } ;
let X be non-empty ManySortedSet of S ;
let X be non-empty ManySortedSet of S ;
op ( 1 ) c= { [ {} , {} ] } ;
reconsider m = mm - 1 as Element of NAT ;
reconsider d = x as Element of C ( ) ;
let s be 0 -started State of SCMPDS , I be Program of SCMPDS ;
let t be 0 -started State of SCMPDS , Q be t -started State of SCMPDS ;
b , b , x , y , z ;
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 TopSpace , t be Point of I[01] ;
set q = h * p ^ <* d *> ;
z2 in U . ( y2 ) /\ Q2 . ( z2 + 1 ) ;
A |^ 0 = { <* \rangle *> } & A |^ 1 = { {} } ;
len W2 = len W + 2 & len W2 = len W + 2 ;
len h2 in dom h2 & len h2 in dom h2 ;
i + 1 in Seg len s2 & i + 1 in Seg len s2 ;
z in dom g1 /\ dom g2 & z in dom g1 /\ dom g2 ;
assume p2 `1 = ( E-max ( K ) ) `1 & p2 `2 = ( E-max ( K ) ) `1 ;
len G + 1 <= i1 + 1 ;
f1 (#) f2 is convergent & f2 (#) ( f1 - f2 ) = lim ( f1 , x0 ) ;
cluster seq + seq + seq + ( - seq ) -> summable ;
assume j in dom M1 & i in dom M2 ;
let A , B , C be Subset of X ;
let x , y , z be Point of X , f be Function of X , Y ;
b ^2 - ( 4 * a * c ) >= 0 ;
<* x/y *> ^ <* y *> \subseteq x ;
a , b in { a , b } ;
len p2 is Element of NAT & len p2 = len q2 ;
ex x being element st x in dom R & R . x = y ;
len q = len ( K (#) G ) .= len G ;
s1 = Initialize ( ( Initialized s ) +* ( 1 , k ) ) ;
consider w being Nat such that q = z + w ;
x ` ` is X implies x ` is X & x ` is Element of X
k = 0 & n <> k or k > n ;
then X is discrete for A being Subset of X ;
for x st x in L holds x is FinSequence ;
||. f /. c .|| <= r1 & ||. f /. c .|| <= r1 ;
c in uparrow p & not c in { p } ;
reconsider V = V as Subset of the TopStruct of TOP-REAL n ;
let N , M be being being being being being being being being being being being \hbox of L ;
then z >= waybelow x & z >= compactbelow y ;
M \lbrack f , f .] = f & M \lbrack g , f .] = g ;
( ( L~ z ) /. 1 ) = TRUE ;
dom g = dom f -tuples_on X & dom g = dom f -tuples_on X ;
mode \cal o is \cal \cal \cal o of G means : Def4 : for n being Nat holds it . n = G . n ;
[ i , j ] in Indices ( M @ ) ;
reconsider s = x " , t = y " as Element of H ;
let f be Element of dom ( Subformulae p ) , g be Element of dom ( Subformulae p ) ;
F1 . ( a1 , - a2 ) = G1 & F1 . ( - a2 ) = G1 ;
cluster Sphere ( a , b , r ) -> compact ;
let a , b , c , d be Real ;
rng s c= dom ( 1 / ( f1 + f2 ) ) ;
curry ( FF , k ) is additive & curry ( FF , k ) is additive ;
set k2 = card dom B , s3 = card C , s4 = card D , s4 = card D , s4 = card D , s4 = card D , s4 = card D , s4 = card D , s4 = card D ,
set G = DTConMSA ( X ) ;
reconsider a = [ x , s ] as terminal of G ;
let a , b be Element of MM , f be FinSequence of the carrier of M ;
reconsider s1 = s , s2 = t as Element of S0 ;
rng p c= the carrier of L & rng p c= the carrier of L ;
let d be Subset of the Sorts of A ;
( x .|. x = 0 iff x = 0. W ) ;
IY in dom stop I & IY in dom stop I ;
let g be continuous Function of X | B , Y ;
reconsider D = Y as Subset of ( TOP-REAL n ) | P ;
reconsider i0 = len p1 - 1 as Integer ;
dom f = the carrier of S & dom g = the carrier of S ;
rng h c= Union ( ( Carrier J ) * ( i , 1 ) )
cluster All ( x , H ) -> reconsider reconsider H1 -LSeg ;
d * N1 ^2 > N1 * 1 / ( 2 * N ) ;
]. a , b .[ c= [. a , b .] ;
set g = f " | D1 , f = f " | D2 ;
dom ( p | ( REAL m ) ) = REAL m & dom ( p | m ) = REAL m ;
3 + - 2 <= k + - 2 ;
tan is_differentiable_in ( ( ( #Z 2 ) * ( f1 + f2 ) ) `| Z ) . x ;
x in rng ( f /^ ( len p -' 1 ) ) ;
let f , g be FinSequence of D ;
p ( ) in the carrier of S1 & p ( ) in the carrier of S2 ;
rng f " = dom f & rng f = dom f ;
( the Source of G ) . e = v & ( the Source of G ) . e = v ;
width G - 1 < width G - 1 ;
assume v in rng ( S | E1 ) & v in rng ( S | E1 ) ;
assume x is root or x is root or x is root & x is root ;
assume 0 in rng ( g2 | A ) & 0 < len ( g2 | A ) ;
let q be Point of ( TOP-REAL 2 ) | P , r be Real ;
let p be Point of ( TOP-REAL 2 ) | P , p1 , p2 be Point of TOP-REAL 2 ;
dist ( O , u ) <= |. p2 .| + 1 ;
assume dist ( x , b ) < dist ( a , b ) ;
<* S7 *> is_in the carrier of C-20 & <* 7 *> in the carrier of C-20 ;
i <= len ( G * ( i1 , j1 ) ) - 1 ;
let p be Point of ( TOP-REAL 2 ) | P , p1 , p2 be Point of TOP-REAL 2 ;
x1 in the carrier of I[01] & x2 in the carrier of I[01] & x3 in the carrier of I[01] ;
set p1 = f /. i , p2 = f /. ( i + 1 ) ;
g in { g2 : r < g2 & g2 < r } ;
Q2 = Sp2 " ( Q /\ R /\ S ) ;
( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( 1 / 2 ) ) ) ) is summable ;
- p + I c= - p + A ;
n < LifeSpan ( P1 , s1 ) + 1 & I < len Comput ( P1 , s1 , m ) ;
CurInstr ( p1 , s1 ) = i & CurInstr ( p1 , s1 ) = halt SCM+FSA ;
A /\ Cl { x } \ { x } <> {} ;
rng f c= ]. r , r + 1 .[ /\ dom f ;
let g be Function of S , V ;
let f be Function of L1 , L2 , g be Function of L1 , 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 ~ ( A , I ) ;
len ( ( the connectives of C ) . ( n + 1 ) ) = 4 ;
let C1 , C2 be subcategory of C ;
reconsider V1 = V as Subset of X | B , V1 = V | B ;
attr p is valid means : Def4 : 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 " * ( H |^ a ) is Subgroup of H ;
let A1 be Let B1 of O , E be Element of E ;
p2 , r3 , q3 is_collinear & q2 , q3 , q3 is_collinear & q2 <> q3 & q2 <> q3 & q2 <> q3 & q2 <> q3 ;
consider x being element such that x in v ^ K ;
not x in { 0. TOP-REAL 2 } \/ { 0. TOP-REAL 2 } ;
p in [#] ( I[01] | B11 ) & p in [#] ( I[01] | B11 ) ;
0 . ( 0 ) < M . ( E8 ) ;
^ ( op ( c ) , op ( c ) ) = c ;
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . s2 ) & a2 in dom ( F . s2 ) ;
cluster -> Line for *> -\cal .| for non empty Poset ;
set i1 = the Nat , i2 = the Element of NAT ;
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 : Def4 : cos ( X ) c= cos ( Y ) ;
let y be upper Subset of Y , x be Element of X ;
cluster ( x `1 ) / ( x `2 ) -> non reconsider x / ( x `2 ) -> non reconsider y / ( x `2 ) -> non zero ;
set S = <* Bags n , i9 *> , T = <* i *> , S = <* i *> , T = <* j *> , R = <* i *> , T = <* i *> , S = <* i *> , T = <* i
set T = [. 0 , 1 / 2 .] , S = [. 0 , 1 / 2 .] ;
1 in dom mid ( f , 1 , 1 ) ;
( 4 * PI ) / ( 2 * PI ) < ( 2 * PI ) / ( 2 * PI ) ;
x2 in dom ( f1 + f2 ) /\ dom ( f2 + f3 ) ;
O c= dom I & { {} } = { {} } ;
( the Source of G ) . x = v & ( 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 , y , z be Element of G opp , f be FinSequence 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 ;
set N-26 = the be let of N , x be Element of N ;
len gSet + ( x + 1 ) - 1 <= x ;
a on B & b on B & c on B ;
reconsider r-12 = r * I . v as FinSequence of REAL ;
consider d such that x = d and a [= d ;
given u such that u in W and x = v + u ;
len f /. ( \downharpoonright n ) = len f - n ;
set q2 = E-max C , q2 = E-max C , q2 = E-max C , q3 = E-max C , q3 = E-max C , q2 = E-max C , q2 = E-max C , q2 = E-max C , q2 = E-max C , q2 =
set S = S1 +* ( S1 , S2 ) ;
MaxADSet ( b ) c= MaxADSet ( P /\ Q ) ;
Cl ( G . q1 ) c= F . r2 & G . q2 c= G . q2 ;
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_left_argument_of H ) ;
assume t is Element of ( F . X ) . s ;
rng f c= the carrier of S2 & rng f c= the carrier of S2 ;
consider y being Element of X such that x = { y } ;
f1 . ( a1 , b1 ) = b1 & f1 . ( a2 , b2 ) = b2 ;
the carrier' of G `1 = E \/ { E } & E in { E } ;
reconsider m = len ( k - 1 ) as Element of NAT ;
set S1 = LSeg ( n , UMP C ) , S2 = LSeg ( n , UMP C ) ;
[ i , j ] in Indices M1 & [ i , j ] in Indices M2 ;
assume that P c= Seg m and M is \HM { 0 , 1 } ;
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 ;
( ( p . i ) `1 = ( p . i ) `1 ;
let PA , G be a_partition of Y , a be Element of Y ;
pred 0 < r & r < 1 implies 1 < ( 1 - r ) / ( 1 - r ) ;
rng ( AffineMap ( a , X ) ) = [#] X & rng ( ( AffineMap ( a , X ) ) | A ) = [#] 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 ( rng s ) ;
reconsider x2 = x1 , y2 = x2 , z2 = y2 as Element of L2 ;
Q in FinMeetCl ( ( the topology of X ) \/ ( the topology of Y ) ) ;
dom ( f . 0 ) c= dom ( ( f . 0 ) .--> ( f . 1 ) ) ;
pred n divides m & m divides n implies n = m ;
reconsider x = x , y = y , z = z as Point of I[01] ;
a in *> implies the carrier of T2 = the carrier of T2 & the carrier of T2 = the carrier of T2
not y0 in the still of f & not ( ex g st g in dom f & not g in f ) ;
Hom ( ( a ~ ) , c opp ) <> {} ;
consider k1 such that p " < k1 and k1 < len p and p . k1 = f . ( k1 + 1 ) ;
consider c , d such that dom f = c \ d ;
[ x , y ] in dom g & [ y , k ] in dom g ;
set S1 = \times \times
l2 = m2 & l2 = i2 & l2 = j2 & E = i2 & E = i2 & E = j2 ;
x0 in dom ( ( u + v ) | A ) & x0 in dom ( ( u + v ) | A ) ;
reconsider p = x as Point of ( TOP-REAL 2 ) | K1 , q = ( TOP-REAL 2 ) | K1 ;
I[01] = ( R^1 | B01 ) | B01 .= ( ( TOP-REAL 2 ) | B01 ) | B01 ;
f . p4 <= f . p1 & f . p2 <= f . p1 & f . p4 <= f . p1 ;
( ( F . x ) `1 ) ^2 / ( ( F . x ) `2 ) ^2 <= ( ( F . x ) `1 ) ^2 / ( ( F . x ) `2 ) ^2 ;
( x `2 ) ^2 = ( W `2 ) ^2 + ( W `2 ) ^2 ;
for n being Element of NAT holds P [ n ] ;
let 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 ] & P [ succ a ] ;
reconsider s\overline = seq . ( s , t ) as terminal of D ;
( k - 1 ) <= len thesis - j ;
[#] S c= [#] T & [#] T c= [#] T & [#] T c= [#] T ;
for V being strict real linear holds V in W1 implies V in W1
assume k in dom mid ( f , i , j ) ;
let P be non empty Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
let A , B be Matrix of n1 , n2 , K be Matrix of n1 , n2 , K ;
- a * - b = a * b - b * c ;
for A being Subset of A9 , B being Subset of A9 holds A // B implies A c= B
( the charact of o2 ) in <^ o2 , o2 ^> & ( the _ of o1 ) in <^ o2 , o1 ^> ;
then ||. x .|| = 0 & x = 0. X ;
let N1 , N2 be strict normal Subgroup of G , N be normal Subgroup of G ;
j >= len upper_volume ( g , D1 ) - len upper_volume ( g , D2 ) ;
b = Q . ( len Q - 1 ) + 1 .= len Q ;
f2 * f1 /* s is divergent_to+infty & f2 * ( f1 /* s ) is divergent_to+infty ;
reconsider h = f * g as Function of N4 , G ;
assume that a <> 0 and Let ( 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 = the carrier of L2 + L2 ;
Directed I is_halting_on Initialized s , P +* I & Directed I is_halting_on Initialized s , P +* I ;
Initialized p = Initialize ( p +* q +* ( i , k ) ) ;
reconsider N2 = N1 , N2 = N2 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 /\ D /\ { {} } ;
n <= len ( P + Q ) - len ( P + Q ) ;
( x1 `1 ) ^2 = ( x2 `1 ) ^2 + ( x1 `2 ) ^2 .= ( x2 `1 ) ^2 + ( x2 `2 ) ^2 ;
InputVertices S = { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 7
let x , y be Element of FTT1 ( n , k ) ;
p = |[ p `1 , p `2 ]| & p <> |[ p `1 , p `2 ]| ;
g * 1_ G = h " * g * h * h * g ;
let p , q be Element of V , a be Element of V ;
x0 in dom ( x1 - x2 ) /\ dom ( x2 - y2 ) ;
( R qua Function ) " = R " & ( R " ) * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * I ) ) ) ) ) ) = R ;
n in Seg ( len ( f /^ 1 ) ) & ( f /^ 1 ) . n = f . ( len f -' 1 ) ;
for s be Real st s in R holds s <= s2 implies s <= s2
rng s c= dom ( f2 * f1 ) /\ dom ( f2 * f1 ) ;
synonym for for for for for for for for X be Subset of \rm such X = 2 holds X is finite ;
1. ( K , n ) * 1. ( K , n ) = 1. ( K , n ) * 1. ( K , n ) ;
set S = Segm ( A , P1 , Q1 ) , T = Segm ( A , P1 , Q1 ) ;
ex w st e = ( w - f ) * w & w in F ;
curry ( P' , k ) # x is convergent & ( curry ( P' , k ) # x ) is convergent ;
cluster open -> open for Subset-Family of T7 ;
len f1 = 1 .= len f3 .= len f3 + 1 .= len f3 + 1 ;
( i * p ) / p < ( 2 * p ) / p ;
let x , y be Element of OSSub ( U0 ) ;
b1 , c1 // b9 , c9 & c1 , c2 // b , c ;
consider p being element such that c1 . j = { p } ;
assume that f " { 0 } = {} and f is total and f is total ;
assume IC Comput ( F , s , k ) = n ;
Reloc ( J , card I ) does not destroy a , I ;
( goto ( card I + 1 ) ) does not destroy c ;
set m3 = LifeSpan ( p3 , s3 ) , s4 = LifeSpan ( p3 , s3 ) , P4 = Comput ( p3 , s3 , 1 ) , P4 = P3 ;
IC SCMPDS in dom ( Initialize p +* ( l , S ) +* ( l , S ) ) ;
dom t = the carrier of SCM+FSA & dom t = the carrier of SCM+FSA ;
( ( E-max L~ f ) .. f ) .. f = 1 & ( E-max L~ f ) .. f = 1 ;
let a , b be Element of V , f be Function of V , C ;
Cl ( union Int 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 = d6 and i in N ;
then Y c= { x } or Y = {} or Y = { x } ;
M , v / ( ( y , v ) / ( x , y ) ) |= H ;
consider m being element such that m in Intersect ( FF , S ) and x = ( Intersect FF ) . m ;
reconsider A1 = support ( u1 ) , A2 = support ( v1 ) as Subset of X ;
card ( A \/ B ) = k-1 + ( 2 * 1 ) ;
assume that a1 <> a3 and a2 <> a4 and a3 <> a4 and a4 <> a5 and a4 <> a5 and a5 <> a5 ;
cluster s -\mathop { V } -> non empty for string of S ;
Carrier ( L2 /. n2 ) = Carrier ( L2 . n2 ) & Carrier ( L2 ) . n2 = Carrier ( L ) ;
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 , x be Point of TOP-REAL n ;
assume [ k , m ] in Indices ( D * ( i , m ) ) ;
0 <= ( ( 1 / 2 ) |^ p ) / p ;
( F . N | E8 ) . x = +infty ;
pred X c= Y & Z c= V & X \ V c= Y \ Z ;
( y `2 ) * ( z `2 ) <> 0. I & ( y `2 ) * ( z `2 ) <> 0. I ;
1 + card ( X-18 ) <= card ( u \/ { w } ) ;
set g = z \circlearrowleft ( E-max L~ z ) , M = z .. z , N = z .. z , S = z .. z , N = z .. z , S = z .. z , N = z .. z , S = z .. z , N = (
then k = 1 or p . k = <* x , y *> . k ;
cluster -> total for Element of C -succ ( X , D ) ;
reconsider B = A as non empty Subset of ( TOP-REAL n ) | B , C be Subset of TOP-REAL n ;
let a , b , c be Function of Y , BOOLEAN , f be Function of Y , BOOLEAN ;
L1 . i = ( i .--> g ) . i .= g . i .= g . i ;
Plane ( x1 , x2 , x3 ) c= P & Line ( x1 , x2 , x3 ) c= P ;
n <= indx ( D2 , D1 , j1 ) + 1 - 1 ;
( ( g2 . O ) `1 ) ^2 = - 1 & ( g2 . O ) `2 = 1 ;
j + p .. f - len f <= len f - len f ;
set W = W-bound C , E = E-bound C , N = E-bound C , S = Gauge ( C , n ) ;
S1 . ( a `1 , e `2 ) = a + e .= a `1 ;
1 in Seg width ( M * ( ColVec2Mx p ) ) ;
dom ( i (#) Im ( f ) ) = dom ( Im ( f ) ) ;
( Dx ) `1 = W . ( a , *' ( a , p ) ) ;
set Q = non empty contradiction , being being |= being Element of |= ( g , f ) ;
cluster -> many sorted for ManySortedSet of U1 , B be ManySortedFunction of U2 ;
attr F = { A } means : Def4 : F is discrete ;
reconsider z9 = *> , \mathopen = <* x *> as Element of product G ;
rng f c= rng f1 \/ rng f2 & rng ( f1 + f2 ) c= rng f1 \/ rng f2 ;
consider x such that x in f .: A and x in f .: C ;
f = <*> ( the carrier of F_Complex ) & g = <*> ( the carrier of F_Complex ) ;
E , j |= All ( x1 , x2 ) & E , j |= All ( x2 , x3 ) ;
reconsider n1 = n , n2 = m , n3 = n , n2 = m , n3 = n , n3 = m , n1 = n , n2 = m ;
assume that P is idempotent and R is idempotent and P (*) R = R (*) P ;
card B2 \/ { x } = k-1 + 1 - 1 .= k + 1 - 1 ;
card ( ( x \ B1 ) /\ B1 ) = 0 implies ( x \ B1 ) /\ B2 = {} ;
g + R in { s : g-r < s & s < g + r } ;
set q-111 = ( q , <* s *> ) -\subseteq ( q , <* s *> ) -\subseteq ( q , <* s *> ) -\subseteq ;
for x being element st x in X holds x in rng f1 implies x in rng f1
h0 /. ( i + 1 ) = h0 . ( i + 1 ) ;
set mw = max ( B , thesis ( Bags NAT ) \ { {} } ) ;
t in Seg width ( I ^ ( n , n ) ) ;
reconsider X = dom f /\ C as Element of Fin NAT ;
IncAddr ( i , k ) = <% - l . ( k + 1 ) %> .= - ( l . k ) ;
( ( GoB f ) * ( 1 , 1 ) ) `2 <= ( ( GoB f ) * ( 1 , 1 ) ) `2 ;
attr R is condensed means : Def4 : Int R is condensed & Cl R is condensed ;
pred 0 <= a & a <= 1 & b <= 1 implies a * b <= 1 ;
u in ( ( ( c /\ ( ( ( d /\ e ) /\ f ) /\ f ) ) /\ j ) /\ e ) ;
u in ( ( ( c /\ ( ( ( d /\ e ) /\ b ) /\ f ) ) /\ j ) /\ e ) ;
len C + - 2 >= 9 + - 3 + ( - 3 ) ;
x , z , y is_collinear & x , z , y is_collinear implies x = y
a |^ ( n1 + 1 ) = a |^ n1 * a |^ n1 ;
<* \underbrace ( 0 , \dots , 0 , 0 , 0 , 0 ) *> in Line ( x , a ) ;
set y9 = <* y , c *> , z9 = <* c , x *> ;
FF /. 1 in rng Line ( D , 1 ) & len ( D * ( 1 , 1 ) ) = width D ;
p . m Joins r /. m , r /. ( m + 1 ) , G ;
( p `2 ) = ( f /. i1 ) `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 P\overline ( X ) , f be Function of X , REAL ;
p |-count ( Product ( Sgm ( X ) ) ) = 0 & p |-count ( Product ( Sgm ( X ) ) ) = 0 ;
len <* x *> < i + 1 & i + 1 <= len c + 1 ;
assume that I is non empty and { x } /\ { y } = { 0. I } ;
set ii = ( card I + 4 ) .--> ( ( card I + 3 ) + 4 ) ;
x in { x , y } & h . x = {} T & h . y = {} ;
consider y being Element of F such that y in B and y <= x `2 ;
len S = len ( the charact of ( A + B ) ) .= len the charact of ( A + B ) ;
reconsider m = M , i = I , n = N as Element of X ;
A . ( j + 1 ) = B . ( j + 1 ) \/ A . j ;
set N8 = : \HM { G : G * ( i , j ) `1 = G * ( i , j ) `1 } ;
rng F c= the carrier of gr { a } & F . ( { a } ) = F . ( { a } ) ;
implies implies for n , K , n , r being FinSequence st len Q = n & len r = n holds P [ n , r , r ]
f . k , f . ( ( p mod n ) + 1 ) are_congruent_mod n ;
h " P /\ [#] T1 = f " P & h " P = f " P ;
g in dom f2 \ f2 " { 0 } & f2 . ( g . 0 ) = f2 . ( g . 0 ) ;
gX /\ dom f1 = g1 " ( X /\ dom f1 ) .= g1 " ( X /\ dom f1 ) ;
consider n being element such that n in NAT and Z = G . n ;
set d1 = being being dist of dist ( x1 , y1 ) , d2 = dist ( x2 , y2 ) , d2 = dist ( x2 , y2 ) ;
b `2 / ( 1 + ( 1 - r ) / ( 1 + r ) ) < ( 1 - r ) / ( 1 + r ) ;
reconsider f1 = f as VECTOR of the carrier of X , g be Function of X , Y ;
pred i <> 0 means : Def4 : i ^2 mod ( i + 1 ) = 1 ;
j2 in Seg len ( g2 . i2 ) & j2 in Seg ( len g2 ) ;
dom ( i ) = dom ( i - 1 ) .= Seg ( ( len i ) - 1 ) .= Seg ( ( len i ) - 1 ) ;
cluster sec | ]. PI / 2 , PI / 2 .[ -> one-to-one ;
Ball ( u , e ) = Ball ( f . p , e ) ;
reconsider x1 = x0 , x2 = y0 as Function of S , IF = ( X --> 0 ) ;
reconsider R1 = x , R2 = y , R1 = z , R2 = t , R2 = u , R2 = v , R1 = v , R2 = w , R2 = w , R2 = w , R2 = v , R2 = w , R2 = w , R2 = v , R2 = w , R2 = w ,
consider a , b being Subset of A such that x = [ a , b ] ;
( <* 1 *> ^ p ) ^ <* n *> in RL ;
S1 +* S2 = S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2
( ( ( 1 / 2 ) (#) ( ( #Z 2 ) * ( ( #Z 2 ) * ( f1 + #Z 2 ) ) ) `| Z ) = f ;
cluster -> C -valued for Function of C , REAL ;
set C7 = 1GateCircStr ( <* z , x *> , f3 ) , C8 = 1GateCircStr ( <* x , y *> , f3 ) ;
Esuch that ( e2 . e2 ) = E8 . e2 & ( E8 . e2 ) = E8 . e2 ;
( ( ( ( 1 / 2 ) (#) ( ( #Z 2 ) * ( f1 + #Z 2 ) ) ) `| Z ) = f ;
upper_bound A = ( PI * 3 / 2 ) / 2 & lower_bound A = 0 ;
F . ( dom f , - f ) is_transformable_to F . ( cod f , - f ) ;
reconsider pbeing Point of TOP-REAL 2 , q = ( q `1 ) / |. q .| as Point of TOP-REAL 2 ;
g . W in [#] Y & [#] Y c= [#] Y & [#] Y c= [#] X & [#] X c= [#] Y ;
let C be compact connected non vertical non horizontal Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
LSeg ( f ^ g , j ) = LSeg ( f , j ) ;
rng s c= dom f /\ ]. x0 - r , x0 .[ & rng s c= dom f /\ ]. x0 - r , x0 .[ ;
assume x in { ( idseq 2 ) . ( idseq 2 ) , ( Rev ( idseq 2 ) ) . ( i + 1 ) } ;
reconsider n2 = n , m2 = m , m2 = n , m2 = m , n1 = n , n2 = m ;
for y being ExtReal st y in rng seq holds g . y <= g . 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 Bf = f .: ( the carrier of X1 ) , Bg = f .: ( the carrier of X2 ) , Bh = f .: ( the carrier of X2 ) , Bh = f .: ( the carrier of X1 ) , Bh = f .: ( the carrier of X2 ) , B
ex d being Element of L st d in D & x << d ;
assume R ~ c= R ~ & R ~ c= R ~ & R ~ c= R ~ & R ~ c= R ~ & R c= R ~ & R c= R ~ c= R ~ ;
t in ]. r , s .[ or t = r or t = s or t = s ;
z + v2 in W & x = u + ( z + v2 ) ;
x2 |-- y2 iff P [ x2 , y2 ] & P [ x2 , y2 ] & P [ y2 , x2 ] ;
pred x1 <> x2 means : Def4 : |. x1 - x2 .| > 0 & |. x1 - x2 .| > 0 ;
assume p2 - p1 , p3 - p1 - p1 - p1 , p4 - p1 - p1 , p3 - p1 is_collinear ;
set q = Ant ( f ) ^ <* 'not' 'not' A *> ;
let f be PartFunc of REAL-NS 1 , REAL-NS n , x be Point of REAL-NS m , r be Real ;
( n mod ( 2 * k ) ) + ( n mod k ) = n mod k ;
dom ( T * ( succ t ) ) = dom ( succ t ) & dom ( succ t ) = dom ( succ t ) ;
consider x being element such that x in wc and x in c ;
assume ( F * G ) . ( v . x3 ) = v . x3 ;
assume that the carrier of D1 c= the carrier of D2 and the carrier of D2 c= the carrier of D2 ;
reconsider A1 = [. a , b .[ , B1 = [. a , b .] as Subset of REAL ;
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 = E-max L~ Cage ( C , n ) , r = W-bound L~ Cage ( C , n ) ;
n1 - len f + 1 - len f + 1 <= len - ( len g - 1 ) + 1 - len f ;
.| |. |. .| ( q , O1 ) .| = [ u , v , a , b , c , d ] ;
set C-2 = ( \mathclose { \overline { \overline { \overline { \kern1pt G \kern1pt } } } ) . ( k + 1 ) ;
Sum ( L * p ) = 0. R * Sum p .= 0. V .= 0. V ;
consider i being element such that i in dom p and t = p . i ;
defpred Q [ Nat ] means 0 = Q ( $1 ) & $1 in dom Q ( ) & ( $1 in dom Q ( ) implies Q . $1 = Q ( $1 ) ) ;
set s3 = Comput ( P1 , s1 , k ) , P3 = Comput ( P2 , s2 , k ) , s4 = Comput ( P2 , s2 , k ) , P4 = Comput ( P2 , s2 , k ) , P4 = Comput ( P2 , s2 , k ) , P4 = Comput ( P2 , s2
let l be variable of k , A , B be Subset of k , x be element ;
reconsider U1 = union G-24 , U2 = union G-24 , E = union GT2 , T = union ( T \/ S ) ;
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 ;
por = <* - ( c - 1 ) , 1 , - ( c - 1 ) , - ( c - 1 ) , - ( c - 1 ) , - ( c - 1 ) , - ( c - 1 ) , - ( c - 1 ) , - ( c - 1 ) *> ;
synonym f is real-valued for rng f c= NAT & rng f c= NAT ;
consider b being element such that b in dom F and a = F . b ;
x0 < card ( X0 ) + card ( Y0 ) - 1 / ( card Y0 ) + 1 / ( card Y0 ) ;
attr X c= B1 means : Def4 : for B st X c= B holds \mathop { \rm _ lpp) ( B ) c= succ B ;
then w in Ball ( x , r ) & dist ( x , w ) <= r ;
angle ( x , y , z ) = angle ( x , y , z ) ;
pred 1 <= len s means : Def4 : len it = len s & for i being Nat st i in dom s holds it . i = s . ( i - 1 ) ;
f.: c= f . ( k + ( n + 1 ) ) ;
the carrier of { 1_ G } = { 1_ G } & the carrier of G = { 1_ G } ;
pred p '&' q in the carrier of \rm WFF means : Def4 : q '&' p in the carrier of - p ;
- ( t `1 / t `2 ) ^2 < ( t `1 / t `2 ) ^2 / t `2 ;
U1 . 1 = U1 /. 1 .= U1 /. 1 .= U1 /. ( 1 + 1 ) .= U1 . ( 1 + 1 ) ;
f .: ( the carrier of x ) = the carrier of x & f . ( the carrier of x ) = f . ( the carrier of x ) ;
Indices O = [: Seg n , Seg n :] & dom O = Seg n & rng O = Seg n ;
for n being Element of NAT holds G . n c= G . ( n + 1 ) ;
then V in M @ ;
ex f being Element of F-9 st f is \cup of Aand x = f . ( len f ) ;
[ h . 0 , h . 3 ] in the InternalRel of G & [ h . 0 , h . 3 ] in the InternalRel of G ;
s +* Initialize ( ( intloc 0 ) .--> 1 ) = s3 +* Initialize ( ( intloc 0 ) .--> 1 ) ;
|[ w1 , v1 ]| - v1 `2 <> 0. TOP-REAL 2 & |[ w1 , v1 ]| - ( v1 - v2 ) * |[ w1 , v1 ]| <> 0. TOP-REAL 2 ;
reconsider t = t as Element of ( Z |^ X ) * , s be Element of Z ;
C \/ P c= [#] ( ( G | [#] ( ( ( G | A ) \ A ) ) \ A ) ;
f " V in ( for X being Subset of [ X , f ] ) /\ D ( X , ( the carrier of S ) --> ( X , f ) ) ;
x in [#] ( ( the carrier of Y ) /\ A ) /\ ( ( the carrier of Y ) /\ A ) ;
g . x <= h1 . x & h . x <= h1 . x & h1 . x <= h1 . x ;
InputVertices S = { xy , yz , yz , yz , \emptyset , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} } ;
for n being Nat st P [ n ] holds P [ n + 1 ] & P [ n + 1 ] ;
set R = Line ( M , i , a * Line ( M , i ) ) ;
assume that M1 is being_line and M2 is being_line and M3 is being_line and M3 is being_line and M2 is being_line and M3 is being_line ;
reconsider a = f4 . ( i0 -' 1 ) , b = f4 . ( i0 -' 1 ) as Element of K ;
len B2 = Sum ( ( F1 ^ F2 ) | ( len F1 + len F2 ) ) .= len ( ( F1 ^ F2 ) | ( len F1 + len F2 ) ) ;
len ( ( ( the ` of n ) * ( i , j ) ) * ( i , j ) ) = n ;
dom ( f + g ) = dom ( f + g ) & dom ( f + g ) = dom ( f + g ) ;
( ( the Sorts of seq ) . n ) . x = upper_bound Y1 & ( ( the Sorts of seq ) . n ) . x = upper_bound Y1 ;
dom ( p1 ^ p2 ) = dom ( f ^ <* p1 *> ) .= dom ( f ^ <* p2 *> ) .= dom ( f ^ <* p2 *> ) ;
M . [ 1 / ( 1 - y ) , y ] = 1 / ( 1 - y ) * v1 .= y ;
assume that W is non trivial and W the Source of G c= the carrier of G and W is trivial ;
C6 * ( i1 , i2 ) = G1 * ( i1 , i2 ) & C6 * ( i1 , i2 ) = G1 * ( i1 , i2 ) ;
C8 |- 'not' Ex ( x , p ) 'or' p . ( x , y ) ;
for b st b in rng g holds lower_bound rng f-g <= b - a ;
- ( ( q1 `1 / |. q1 .| - cn ) / ( 1 + cn ) ) = 1 ;
( LSeg ( c , m ) \/ ( NAT \ { l } ) ) c= R ;
consider p being element such that p in such that p in { x } and p in L~ f and x = f . p ;
Indices ( X @ ) = [: Seg n , Seg n :] & [: Seg n , Seg n :] = [: Seg n , Seg n :] ;
cluster s => ( q => p ) -> valid ( s => ( s => p ) ) -> valid ;
Im ( ( Partial_Sums F ) . m ) . ( ( Partial_Sums F ) . m ) is_measurable_on E ;
cluster f . ( x1 , x2 ) -> ( f . ( x1 , x2 ) ) * ( f . ( x2 , y2 ) ) ;
consider g being Function such that g = F . t and Q [ t , g ] ;
p in LSeg ( ( N-min L~ Cage ( C , n ) ) * ( i , 1 ) , ( E-max L~ Cage ( C , n ) ) * ( i , 1 ) ) ;
set R8 = R / 1 , R8 = ]. b , +infty .[ , R8 = ]. a , b .[ ;
IncAddr ( I , k ) = AddTo ( da , db ) .= IncAddr ( da , db ) .= ( - ( - ( - ( - ( k + 1 ) ) ) ) ) ;
seq . m <= ( ( ( seq ^\ k ) ^\ n ) ^\ k ) . m ;
a + b = ( a ` *' ) ` + ( b ` *' ) .= ( a ` ` ) + ( b ` ) ;
id ( X /\ Y ) = id ( X /\ Y ) .= id X /\ id Y ;
for x being element st x in dom h holds h . x = f . x ;
reconsider H = U1 \/ U2 , U1 = U2 \/ U1 , U2 = U2 \/ U1 as non empty Subset of U0 ;
u in ( ( ( c /\ ( ( ( ( ( d /\ e ) /\ f ) ) /\ f ) ) /\ j ) ) /\ m ;
consider y being element such that y in Y and P [ y , inf B ] and P [ y , inf B ] ;
consider A being finite stable Subset of R such that card A = ( R * S ) . ( len A ) ;
p2 in rng ( f |-- p1 ) \ rng <* p1 *> & rng <* p1 *> c= rng <* p1 *> ;
len s1 - 1 > 1-1 & len s2 > 0 & len s2 - 1 > 0 ;
( ( N-min P ) `2 ) ^2 = ( ( E-max P ) `2 ) ^2 + ( ( E-max P ) `2 ) ^2 ;
Ball ( e , r ) c= LeftComp Cage ( C , k + 1 ) /\ L~ Cage ( C , k + 1 ) ;
f . a1 ` ` = f . a1 ` & f . ( a1 ` ) = f . ( a1 ` ) ;
( seq ^\ k ) . n in ]. x0 - r , x0 .[ & ( seq ^\ k ) . n in ]. x0 - r , x0 .[ ;
gg . s0 = g . s0 .= G . s0 .= G . s0 .= G . s0 .= G . s0 ;
the InternalRel of S is \lbrace the InternalRel of S , the InternalRel of S , the InternalRel of S } ;
deffunc F ( Ordinal , Ordinal ) = phi . ( $1 , $2 ) & phi . ( $2 , $2 ) = phi . ( $2 , $2 ) ;
F . ( s1 . a1 ) = F . ( s2 . a1 ) .= F . ( s2 . a1 ) .= F . ( s2 . a1 ) ;
x `1 = A . o . a .= Den ( o , A ) . a ;
Cl ( f " P1 ) c= f " P1 & f " P1 c= f " P1 & f " P1 c= f " P1 ;
FinMeetCl ( ( the topology of S ) . n ) c= the topology of T & FinMeetCl ( ( the topology of S ) . n ) c= the topology of T ;
synonym o is \hbox means : Def4 : o <> \ast & o <> * ;
assume that X |^ + = Y |^ + 1 and card X <> card Y and card Y <> card X ;
the *> ( s ) <= 1 + ( ( the *> of s ) +* ( 1 , ( the +* ( 1 , s ) ) ) ) ;
LIN a , a1 , d or b , c // b1 , c1 or a , b // b1 , c1 or a , c // c1 , c2 ;
e . 1 = 0 & e . 2 = 1 & e . 3 = 0 & e . 4 = 0 ;
EE in SE & EE in { NE } & EE in { NE } ;
set J = ( l , u ) If , K = I " ;
set A1 = Subset ( Y ) , A2 = Y +* ( A1 +* ( A2 +* ( A1 +* ( A1 +* ( m +* ( x , y , c ) ) ) ) ) ;
set c9 = [ <* cin , cin *> , and2 ] , A1 = [ <* cin , cin *> , and2 ] , A2 = [ <* cin , cin *> , and2 ] , C = [ <* cin , cin *> , and2 ] , D = [ <* cin , cin *> , and2 ] , N = [ <* cin , cin *> , and2 ] , A =
x * z `2 * x " in x * ( z * N ) * x " ;
for x being element st x in dom f holds f . x = f3 . x & f . x = f3 . x ;
Int cell ( f , 1 , G ) c= RightComp f \/ RightComp f \/ L~ f \/ L~ f ;
U2 is_an_arc_of W-min C , E-max C & W-min C <> W-min C implies W-min L~ Cage ( C , n ) in L~ Cage ( C , n )
set f-17 = f @ "/\" ( g @ ) ;
attr S1 is convergent & S2 is convergent & S1 is convergent implies S1 - S2 is convergent & lim ( S1 - S2 ) = lim ( S1 - S2 ) ;
f . ( 0 + 1 ) = ( 0 qua Ordinal ) + a .= a + ( 0 qua Ordinal ) .= a + ( 0 qua Ordinal ) ;
cluster -> Carrier -> Carrier for reflexive non empty reflexive transitive RelStr , M , N be reflexive non empty reflexive 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 ) & 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 & len ( l |^ 0 ) = len l ;
t4 is ( {} \/ rng t4 ) -valued FinSequence of ( {} \/ rng t4 ) * , D = { {} } ;
t = <* F . t *> ^ ( C . p ) ^ ( C . q ) .= ( C . q ) ^ ( C . q ) ;
set pp = W-min L~ Cage ( C , n ) , p1 = W-min L~ Cage ( C , n ) , p2 = W-min L~ Cage ( C , n ) , p3 = Cage ( C , n ) , p4 = Cage ( C , n ) , p4 = Cage ( C , n ) , p4 = Cage ( C , n ) , p4 =
( k - ( i + 1 ) ) - ( i + 1 ) = ( k - ( i + 1 ) ) - ( i + 1 ) ;
consider u being Element of L such that u = u ` *' & u in D ;
len ( ( width ( ( a - G ) * ( i , j ) ) - ( a - G * ( i , j ) ) ) ) = width ( ( a - G ) * ( i , j ) ) ;
FF . x in dom ( ( G * the_arity_of o ) . x ) ;
set cH2 = the carrier of H2 , c = the carrier of H2 ;
set cH1 = the carrier of H1 , cH2 = the carrier of H2 ;
( Comput ( P , s , 6 ) ) . intpos m = s . intpos m & ( Comput ( P , s , 6 ) ) . intpos m = s . intpos m ;
IC Comput ( Q2 , t , k ) = ( l + 1 ) - ( k + 1 ) .= ( l + 1 ) - ( k + 1 ) ;
dom ( ( ( ( 1 / 2 ) (#) ( ( #Z 2 ) * ( ( #Z 2 ) * ( f1 + #Z 2 ) ) ) ) `| REAL ) = REAL ;
cluster <* l *> ^ phi ^ phi -> ( 1 + ( not l ) ) -element for string of S ;
set b5 = [ <* ap , bp *> , and2 ] , c5 = [ <* A1 , cin *> , and2 ] , c5 = [ <* cin , cin *> , and2 ] , c5 = [ <* A1 , cin *> , and2 ] , b5 = [ <* cin , cin *> , and2 ] , b5 = [ <* cin , A1 *> , and2
Line ( Segm ( M @ , P , Q ) , x ) = L * Sgm Q ;
n in dom ( ( ( the Sorts of A ) * the_arity_of o ) * ( the Arity of S ) ) ;
cluster f1 + f2 -> continuous for PartFunc of REAL , the carrier of S , the carrier of S ;
consider y be Point of X such that a = y and ||. x - y .|| <= r ;
set x3 = t3 . DataLoc ( s2 . SBP , 2 ) , x4 = [ s2 . SBP , 2 ] , x4 = [ s2 . SBP , 2 ] , x4 = [ s2 . SBP , 2 ] , P4 = [ s2 . SBP , 2 ] , P4 = [ s2 . SBP , 2 ] , P4 = [ s2 .
set pp = stop I , p1 = P +* I , p2 = P +* I , p3 = P +* I , p4 = P +* I , p4 = P +* I , p4 = P +* I , P4 = P +* I , P4 = Comput ( P3 , s3 , 1 ) , P4 = P3 ;
consider a being Point of D2 such that a in W1 and b = g . a and a in W2 ;
{ A , B , C , D } = { A , B , C } \/ { D , E , F , J , M , N , N , M , N , N , N , F , M , N , N , N , F , M , N , N , M , N , N , F , M ,
let A , B , C , D , E , F , J , M , N , N , F , M , N , N , F , J , M , N , N , F , N , M , N , N , F , N , M , N , N , F , N , M , N , N ,
|. p2 .| ^2 - ( ( p2 `2 ) / |. p2 .| ) ^2 >= 0 & ( p2 `2 ) / |. p2 .| ) ^2 >= 0 ;
l - 1 + 1 = ( n * ( l6 + 1 ) ) + ( ( n + 1 ) + 1 ) ;
x = v + ( a * w1 + ( b * w2 ) ) + ( c * w2 ) + ( c * w2 ) + ( c * w2 ) + ( c * w2 ) ;
the TopStruct of L = TopSpaceMetr ( ( the Scott of L ) | the carrier of L ) .= TopSpaceMetr ( ( the Scott of L ) | the TopStruct of L ) ;
consider y being element such that y in dom H1 and x = H1 . y and y in dom H1 and x = H1 . y ;
f9 \ { n } = Free ( All ( v1 , H ) ) & not f in Free ( All ( v1 , H ) ) ;
for Y being Subset of X st Y is summable holds Y is iff X is iff Y is iff X is iff Y is iff X is iff Y is iff X = Y
2 * n in { N : 2 * Sum ( p | N ) = N & N > 0 } ;
for s being FinSequence holds len ( the { of } the { - } \rm <* s *> ) = len s & len ( the { - } -\rm Shift s ) = len s
for x st x in Z holds ( exp_R * f ) is_differentiable_in x & ( exp_R * f ) . x > 0
rng ( h2 * f2 ) c= the carrier of ( ( TOP-REAL 2 ) | P ) | the carrier of ( ( TOP-REAL 2 ) | P ) ;
j + ( len f - len f ) <= len f + ( len f - len f ) - len f ;
reconsider R1 = R * I as PartFunc of REAL n , REAL-NS n , x be Point of REAL-NS m ;
C8 . x = s1 . ( a - 1 ) .= C8 . ( a - 1 ) .= C8 . ( a - 1 ) ;
( power F_Complex ) . ( z , n ) = 1 .= ( x |^ n ) * ( z |^ n ) .= ( x |^ n ) * ( z |^ n ) ;
t at ( C , s ) = f . ( ( the connectives of S ) . t ) & t = s . ( ( the connectives of S ) . t ) ;
support ( f + g ) c= support f \/ C /\ support g \/ support f /\ support g ;
ex N st that N = j1 & 2 * Sum ( ( r | N ) | N ) > N & N > 0 ;
for y , p st P [ p ] holds P [ All ( y , p ) ]
{ [ x1 , x2 ] , [ x2 , x3 ] } is Subset of [: X1 , X2 :] ;
h = ( i = j = i |-- h , id B . i ) .= H . i ;
ex x1 being Element of G st x1 = x & x1 * N c= A & x1 in N ;
set X = ( |. ( |. |. ( q .| ) ) .| ) * ( ( |. q .| ) .| ) , Y = ( |. q .| ) * ( ( |. q .| ) ) * ( ( |. q .| ) ) ;
b . n in { g1 : x0 < g1 & g1 < x0 } & ( for n holds g1 . n < x0 ) implies f . ( n + 1 ) < f . ( n + 1 )
f /* s1 is convergent & f /. x0 = lim ( f /* s1 ) implies f /* s1 is convergent & lim ( f /* s1 ) = lim ( f /* s1 )
the lattice of the lattice of Y = the lattice of the lattice of the lattice of Y & the carrier of X = the carrier of Y implies X = Y
'not' ( a . x ) '&' b . x 'or' a . x '&' 'not' ( b . x ) '&' 'not' ( b . x ) = FALSE ;
( 2 = len ( ( q1 ^ <* r1 *> ) + len ( q1 ^ <* r1 *> ) ) + len ( q1 ^ <* r1 *> ) ) ;
( ( 1 / a ) (#) ( sec * f1 ) - id Z ) is_differentiable_on Z ;
set K1 = upper ( lim ( H , A ) || ( A , B ) ) , D2 = ( lim ( H , A ) || ( A , B ) ) ;
assume e in { ( w1 - w2 ) : w1 in F & w2 in G & w2 in G } ;
reconsider d7 = dom a `1 , d8 = dom F `1 , d8 = dom F `1 , d8 = dom G `1 , d8 = dom F `1 , d8 = dom G `1 , d8 = dom F `1 , d8 = dom G `1 , d8 = dom G `1 , d8 = dom F `1 , d8 = dom G `1
LSeg ( f /^ ( j + 1 ) , q ) = LSeg ( f , j ) + q .. f ;
assume X in { T . ( N2 , L2 ) : h . ( N2 , L2 ) = N2 } ;
assume that Hom ( d , c ) <> {} and <* f , g *> * f1 = <* f , g *> * f2 ;
dom S-34 = dom S /\ Seg n .= dom L /\ Seg n .= dom ( L | Seg n ) .= dom ( L | Seg n ) .= Seg n /\ Seg n ;
x in H |^ a implies ex g st x = g |^ a & g in H |^ a & g in H
a * ( ( a , n ) * ( a , 1 ) ) = a `2 - ( 0 * n ) .= a `2 - ( 0 * n ) ;
D2 . ( j - j ) in { r : lower_bound A <= r & r <= upper_bound A } ;
ex p being Point of TOP-REAL 2 st p = x & P [ p ] & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 ;
for c holds f . c <= g . c implies f ^ @ g ^ @ @ f ;
dom ( f1 (#) f2 ) /\ X c= dom ( f1 (#) f2 ) /\ dom ( f2 (#) f3 ) ;
1 = ( p * p ) * p .= p * ( p * q ) .= p * ( q * p ) ;
len g = len f + len <* x + y *> .= len f + 1 + 1 .= len f + 1 + 1 .= len f + 1 ;
dom ( ( F | ( N1 /\ S-23 ) ) | ( N1 /\ S-23 ) ) = dom ( F | ( N1 /\ S-23 ) ) ;
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 ) = g . ( len g ) ;
( ( x \ y ) \ z ) \ ( ( ( x \ z ) \ ( y \ z ) ) \ ( z \ x ) ) = 0. X ;
consider f such that f * f " = id b and f * f = id a and f * f = id b ;
( ( ( ( ( 2 ) * cos ) * cos ) `| [. 0 , PI / 2 .] ) ) . ( ( ( 2 * cos ) * cos ) `| [. 0 , PI / 2 .] ) is increasing ;
Index ( p , co ) <= len LS - Gij .. LS + 1 - Index ( Gij , LS ) - 1 ;
let t1 , t2 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2
( the mapping of ( ( Frege ( H ) ) . h ) ) . h <= ( the mapping of ( ( Frege ( G ) ) . h ) ) . ( h . ( h . ( h . ( k + 1 ) ) ) ) ;
then P [ f . i0 ] & F ( f . ( i0 + 1 ) ) < j & F ( f . i0 ) < j ;
Q [ ( D . ( [ D . x , 1 ] ) , F . [ 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 \HM { l . i } ;
the Sorts of A2 = ( the carrier of S2 ) --> ( the carrier of S2 ) .= ( the carrier of S2 ) --> ( the carrier of S2 ) ;
consider s being Function such that s is one-to-one and dom s = NAT and rng s = F and rng s c= F ;
dist ( b1 , b2 ) <= dist ( b1 , a ) + dist ( a , b2 ) + dist ( a , b2 ) ;
( Upper_Seq ( C , n ) /. len Upper_Seq ( C , n ) ) `1 = W-bound L~ Cage ( C , n ) ;
q `2 <= ( ( UMP C ) * ( ( UMP C ) * ( 1 , 1 ) ) `2 ) ;
LSeg ( f | i2 , i ) /\ LSeg ( f | i2 , j ) = {} & LSeg ( f | i2 , i ) /\ LSeg ( f | i2 , j ) = {} ;
given a being ExtReal such that a <= Ia and A = ]. a , Ia .] and a < I and I = ]. a , I .] ;
consider a , b being complex number such that z = a & y = b and z + y = a + b ;
set X = { b |^ n where n is Element of NAT : n in dom b } , Y = { b |^ n where n is Element of NAT : n in dom b } ;
( ( x * y * z ) \ x ) \ ( ( x * y * z ) \ x ) = 0. X ;
set xy = [ <* xy , y *> , f1 ] , yz = [ <* y , z *> , f2 ] , yz = [ <* z , x *> , f3 ] , yz = [ <* x , y *> , f3 ] , \mathopen = [ <* y , z *> , f3 ] , f4 = [ <* z , x *> , f3 ] , f4 = [ <* x , y *>
ll /. len ll = ll . ( len ll ) .= ll . ( len ll ) ;
( ( q `2 / |. q .| - cn ) / ( 1 + cn ) ) ^2 = 1 ;
( ( p `2 / |. p .| - cn ) / ( 1 + cn ) ) ^2 < 1 ;
( ( ( ( ( S \/ S ) \/ S ) ) \/ S ) `2 = ( ( ( S \/ S ) \/ S ) \/ S ) `2 ;
( s1 - s2 ) . k = s1 . k - s2 . k .= s1 . k - s2 . k .= s1 . k - s2 . k .= s1 . k - s2 . k .= s1 . k - s2 . k ;
rng ( ( h + c ) ^\ n ) c= dom SVF1 ( 1 , f , u0 ) /\ dom SVF1 ( 1 , f , u0 ) ;
the carrier of ( the carrier of X ) = the carrier of X & the carrier of ( X ) = the carrier of X ;
ex p4 st p4 = p4 & |. p4 - |[ a , b ]| .| = r & |. p4 - |[ a , b ]| .| = r ;
set ch = chi ( X , A ) , AA = chi ( X , A ) , AB = chi ( X , A ) ;
R / ( 0 * n ) = I---> ( X , X ) * ( R / n ) .= R / ( n * n ) ;
( Partial_Sums ( ( curry ( F , I ) ) . n ) . ( ( curry ( F , I ) . n ) ) . ( ( Partial_Sums ( ( curry ( F , I ) ) . n ) ) . ( ( Partial_Sums ( ( curry ( F , I ) ) . n ) ) . ( ( Partial_Sums ( ( curry ( F , I ) ) . n ) ) ) . ( ( Partial_Sums
f2 = C7 . ( E7 , len ( V . ( K . ( K . ( K . ( L . ( K . ( L . ( K . ( L . ( K . m ) ) ) ) ) ) ) ) ) ) ;
S1 . b = s1 . b .= s2 . b .= s2 . b .= s2 . b .= s2 . b .= s2 . b .= s2 . b ;
p2 in LSeg ( p2 , p1 ) /\ LSeg ( p1 , p01 ) \/ LSeg ( p1 , p01 ) /\ LSeg ( p1 , p01 ) ;
dom ( f . t ) = Seg n & dom ( I . t ) = Seg n & dom ( I . t ) = Seg n ;
assume o = ( the connectives of S ) . 11 & ( the connectives of S ) . 12 in ( the carrier' of S ) . 11 ;
set phi = ( l1 , l2 ) If , phi = ( l1 , l2 ) If , phi = ( l1 , l2 ) If , phi = ( l1 , l2 ) If , phi = ( l1 , l2 ) <> {} ;
synonym p is invertible for p , T means : Def4 : HT ( p , T ) = 1 & HT ( p , T ) = 0. L ;
( Y1 `2 = - 1 & ( Y1 `2 ) - cn ) <> 0 & ( Y1 `2 / Y1 `1 <> - cn ) & ( Y1 `2 / Y1 `1 <> cn ) & ( Y1 `2 / Y1 `1 <> - cn ) & ( Y1 `2 / Y1 `1 <> - cn ) & ( Y1 `2 / Y1 `1 <> - cn ) & ( Y1 `2 / Y1 `1 <> - cn ) & ( Y1 `2 / Y1
defpred X [ Nat , set , set , set ] means P [ $2 , $2 , $1 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2
consider k being Nat such that for n being Nat st k <= n holds s . n < x0 + g ;
Det ( I |^ ( m -' n ) ) * ( m -' n ) = 1. K * ( ( m - n ) * ( m - n ) ) ;
( - b - sqrt ( b ^2 - 4 * a * c ) ) / ( 2 * a * c ) < 0 ;
Cf . d = C7 . ( d7 . d ) mod C7 . ( d8 . d ) .= C7 . ( ( d - 1 ) mod C7 . d ) mod C7 . ( ( d - 1 ) mod C7 . d ) ;
attr X1 is dense means : Def4 : X2 is dense dense & X1 /\ X2 is dense implies X1 union X2 is dense SubSpace of X ;
deffunc FF ( Element of E , Element of I , Element of I ) = ( $1 * $2 ) * ( $2 * $1 ) ;
t ^ <* n *> in { t ^ <* i *> : Q [ i , T . t ] } ;
( x \ y ) \ x = ( x \ x ) \ y .= y ` \ ( x \ y ) .= 0. X ;
for X being non empty set holds X is Basis of <* X , \subseteq FinMeetCl ( X , FinMeetCl Y ) *> iff X is Basis of <* X , \subseteq \subseteq X , FinMeetCl Y *>
synonym A , B are_separated for Cl A , B , C , D , E , F , J , M , N , N , F , M , N , N , F , J , M , N , N , F , N , M , N , N , F , M , N , N , F , N , M , N , N , N , F , N , M , N , N , N
len ( M8 ) = len p & width ( M8 ) = width ( p * ( i , j ) ) & width ( p * ( i , j ) ) = width ( p * ( i , j ) ) ;
J . v = { x where x is Element of K : 0 < v . ( x + 1 ) } ;
( ( ( Sgm ( support m ) ) . d - ( Sgm ( support m ) ) . e ) <> 0 ;
lower_bound divset ( D2 , k + k2 ) = D2 . ( k + k2 - 1 ) .= D2 . ( k + k2 - 1 ) ;
g . r1 = - 2 * r1 + 1 & dom h = [. 0 , 1 .] & dom g = [. 0 , 1 .] & rng h c= [. 0 , 1 .] ;
|. a .| * ||. f .|| = 0 * ||. f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| ;
f . x = ( h . x ) `1 & g . x = ( h . x ) `2 & h . x = ( h . x ) `2 ;
ex w st w in dom B1 & <* 1 *> ^ s = <* 1 *> ^ w & len w = len <* 1 *> & len w = len w + 1 ;
[ 1 , {} , <* d1 *> ] in ( { [ 0 , {} ] } \/ S1 ) \/ S2 ;
IC Exec ( i , s1 ) + n = IC Exec ( i , s2 ) .= ( IC Exec ( i , s2 ) ) + n .= ( IC Exec ( i , s2 ) ) ;
IC Comput ( P , s , 1 ) = succ IC s .= ( ( 0 + 1 ) + 1 ) .= 5 .= ( 5 + 1 ) ;
( IExec ( W6 , Q , t ) ) . intpos ( e + 1 ) = t . intpos ( e + 1 ) ;
LSeg ( f /^ q , i ) misses LSeg ( f /^ q , j ) & LSeg ( f /^ q , i ) misses LSeg ( f /^ q , j ) ;
assume for x , y being Element of L st x in C & y in C holds x <= y or y <= x ;
integral ( integral ( f , C ) , A ) = f . ( upper_bound C ) - f . ( lower_bound C ) .= f . ( upper_bound C ) - f . ( lower_bound C ) ;
for F , G being one-to-one FinSequence st rng F misses rng G holds F ^ G is one-to-one & F ^ G is one-to-one
||. R /. L - L /. h .|| < e1 * ( K + 1 ) * ( K + 1 ) ;
assume a in { q where q is Element of M : dist ( z , q ) <= r } ;
set p4 = [ 2 , 1 ] .--> [ 2 , 1 ] , p4 = [ 3 , 1 , 1 ] ;
consider x , y being Subset of X such that [ x , y ] in F and x c= d and y in F and x \not c= F ;
for y , x being Element of REAL m st y in Y ` & x in X holds y <= x ` & y <= x ;
func |. p \bullet q .| -> variable of A means : Def4 : for p being FinSequence of A holds it . p = min ( NBBI ( p ) , q ) ;
consider t being Element of S such that x `1 , y `2 '||' z `1 , t `2 & x , y '||' z `1 , t `2 ;
dom x1 = Seg len x1 & len x2 = len x2 & len y2 = len x2 & len ( x1 ^ x2 ) = len ( x1 ^ x2 ) ;
consider y2 being Real such that x2 = y2 and 0 <= y2 and y2 < 1 / 2 and y2 <= 1 / 2 ;
||. f | X /. s1 .|| = ||. f /. ( X + s1 ) .|| .= ||. f /. ( X + s1 ) .|| .= ||. f /. ( X + s1 ) .|| ;
( ( the InternalRel of A ) ~ ) /\ ( ( the InternalRel of A ) ` /\ Y ) = {} \/ {} .= {} ;
assume that i in dom p and for j being Nat st j in dom q holds P [ i , j ] and for i , j being Nat st i in dom q & j in dom q & q . i = q . j holds p . ( i + 1 ) = p . ( j + 1 ) ;
reconsider h = f | [: X , Y :] as Function of [: X , Y :] , rng ( f | [: X , Y :] ) ;
u1 in the carrier of W1 & u2 in the carrier of W2 implies ( u + u1 ) + ( u + u2 ) in the carrier of W1 & ( u + u1 ) + ( u + u2 ) in W2
defpred P [ Element of L ] means M <= f . $1 & f . $1 <= f . ( $1 + 1 ) & f . ( $1 + 1 ) <= f . ( $1 + 1 ) ;
l . ( u , a , v ) = s * x + ( - ( s * x ) + y ) .= b ;
- ( x - y ) = - x + ( - y ) .= - x + ( - y ) .= - x + y .= - x + y ;
given a being Point of GX such that for x being Point of GX holds a , x are_ed ed & a , x are_elements of G ;
fconsider fconsider A2 = [ [ dom ( f2 * f2 ) , cod ( f2 * g2 ) ] , h2 = [ cod ( f2 * g2 ) , cod ( f2 * g2 ) ] , h2 = [ cod ( f2 * g2 ) , cod ( f2 * g2 ) ] ;
for k , n being Nat st k <> 0 & k < n & k <> n holds k , n are_congruent_mod k , n * ( k * n )
for x being element holds x in A |^ d iff x in ( ( A ` ) |^ d ) ` & ( ( A ` ) |^ d ) ` = ( ( 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 ) ) ^2 > 0 ;
Carrier ( L-13 ) . k = Carrier ( L ) . ( F . k ) & F . k in dom ( L ) ;
set i2 = AddTo ( a , i , - n ) , i1 = a - ( - n ) ;
attr B is max means : Def4 : for S being Subsqrt of B holds ( Let S , B ) `1 = ( B `1 ) `1 ;
a9 " D = { a "/\" d where d is Element of N : d in D } & N = { a "/\" d where a is Element of N : a in D } ;
|( \square , ( - q ) * ( - q ) , ( - q ) * ( - q ) )| >= |( - q , - q )| ;
( - f ) . ( sup A ) = ( ( - f ) | A ) . ( sup A ) .= - f . ( sup A ) ;
( G * ( i , k ) ) `1 = G * ( len G , k ) `1 .= G * ( 1 , k ) `1 .= G * ( 1 , k ) `1 .= G * ( 1 , k ) `1 ;
( Proj ( i , n ) . LM ) . LM = <* ( proj ( i , n ) . LM ) . LM *> .= <* ( proj ( i , n ) . LM ) . LM *> ;
f1 + f2 * reproj ( i , x ) is_differentiable_in ( ( reproj ( i , x ) ) . ( x0 - x ) ) ;
pred ( ( ( ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 )
ex t being SortSymbol of S st t = s & h1 . t = h2 . x & t = h2 . x & t = h2 . x ;
defpred C [ Nat ] means P8 . $1 is non empty & A8 : ( A is $1 is non empty & A is $1 & A is $1 is non empty or A is non empty ) ;
consider y being element such that y in dom ( ( p - q ) | i ) and ( ( p - q ) | i ) . y = ( ( p - q ) | i ) . y ;
reconsider L = Product ( { x1 } +* ( index B , l ) ) as Subset of product Carrier A ;
for c being Element of C ex d being Element of D st T . ( id c ) = id d & ( T . ( id c ) ) . ( id c ) = id d
( ( f , n ) --> p ) = ( f | n ) ^ <* p *> .= f ^ <* p *> .= f ^ <* 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 `2 = ( ( c | n ) *' ) . ( - ( f . ( - g ) ) ) .= ( - c ) * ( ( - g ) . ( - ( - g ) ) ) ;
consider r be Real such that r in rng ( f | divset ( D , j ) ) and r < m + s ;
f1 . ( |[ ( 8 - r1 ) / 2 , ( 8 - r1 ) / 2 ]| ) in f1 .: W1 /\ W2 ;
eval ( a | ( n , L ) , x ) = eval ( a | ( n , L ) ) .= a * ( ( a | ( n , L ) ) . x ) .= a * ( ( a | ( n , L ) ) . x ) ;
z = DigA ( tl , x9 ) .= DigA ( tl , x9 ) .= DigA ( tl , x9 ) .= DigA ( tl , x9 ) .= DigA ( tl , x9 ) ;
set H = { Intersect S where S is Subset-Family of X : S c= G & S c= G } , G = { Intersect S where S is Subset-Family of X : S c= G } , F = { Intersect S where S is Subset-Family of X : S c= G } , G = { Intersect S : S c= G } , H = { Intersect S where S is Subset of X : S c= G } ;
consider S19 being Element of D such that S `1 = S19 ^ <* d *> and S . ( i + 1 ) = S . ( d + 1 ) ;
assume that x1 in dom f and x2 in dom f and f . x1 = f . x2 and f . x2 = f . x2 and f . x1 = f . x2 ;
- 1 <= ( ( q `2 / |. q .| - cn ) / ( 1 + cn ) ) ^2 / ( 1 + cn ) ^2 / ( 1 + cn ) ^2 ;
( 0. ( V ) ) is Linear_Combination of A & Sum ( ( 0. V ) (#) ( ( 0. V ) (#) ( ( 0. V ) (#) ( ( 0. V ) (#) ( ( 0. V ) (#) ( ( 0. V ) (#) ( 0. V ) ) ) ) ) = 0. V ) ;
let k1 , k2 , k1 , k2 , k2 , x4 , 6 , 7 , 8 , 8 , 8 , 9 , 8 , 7 , 8 , 8 , 8 , 9 , 8 , 8 , 8 , 9 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 9 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 ,
consider j being element such that j in dom a and j in g " { k `2 } and x = a . j and a . j = b . j ;
H1 . x1 c= H1 . x2 or H1 . x2 c= H1 . x2 or H1 . x1 c= H1 . x2 & H1 . x2 c= H1 . x2 or H1 . x2 c= H1 . x2 & H1 . x2 c= H1 . x2 ;
consider a being Real such that p = *> * p1 + ( a * p2 ) and 0 <= a and a <= 1 and a <= 1 ;
assume that a <= c and c <= d and |[ a , b ]| c= dom f and |[ a , b ]| c= dom g and g . a = g . b ;
cell ( Gauge ( C , m ) , ( X , 1 ) -' 1 , 0 ) is non empty ;
Ain { ( S . i ) `1 where i is Element of NAT : i in dom ( S . i ) `1 } ;
( T * b1 ) . y = L * b2 /. y .= L * ( b1 /. y ) .= ( F /. y ) * ( G * ( b1 /. y ) ) .= ( F * ( b1 /. y ) ) * ( G * ( b1 /. y ) ) ;
g . ( s , I ) . x = s . y & g . ( s , I ) . y = |. s . x - s . y .| ;
( log ( 2 , k + 1 ) ) / ( 2 * ( k + 1 ) ) >= ( log ( 2 , k + 1 ) ) / ( 2 * ( k + 1 ) ) ;
then p => q in S & not x in the still of p & not p in S & p => q in S ;
dom ( the InitS of r-10 ) misses dom ( the InitS of rM ) & dom ( the InitS of rM ) misses dom ( the InitS of rM ) ;
synonym f is extended real means : Def4 : for x being set st x in rng f holds x is ExtReal & f . x is ExtReal ;
assume for a being Element of D holds f . { a } = a & for X being Subset-Family of D holds f . ( f .: X ) = f . union X ;
i = len p1 .= len p3 + len <* x *> .= len p3 + len <* x *> .= len p3 + 1 .= len p3 + 1 + 1 .= len p3 + 1 + 1 .= len p3 + 1 + 1 ;
( l ) * ( 1 , 3 ) = ( g . ( 1 , 3 ) + ( k + 1 ) * ( 1 , 3 ) - ( k + 1 ) * ( 1 , 3 ) ) ;
CurInstr ( P2 , Comput ( P2 , s2 , l ) ) = halt SCM+FSA .= halt SCM+FSA .= ( halt SCM+FSA ) . l .= ( halt SCM+FSA ) . l .= ( halt SCM+FSA ) . l .= ( halt SCM+FSA ) . l .= ( halt SCM+FSA ) . l .= ( ( halt SCM+FSA ) . l ) ;
assume for n be Nat holds ||. seq .|| . n <= ( R . n ) * ( R . n ) & ( R . n ) * ( R . n ) <= ( R . n ) * ( R . n ) ;
sin ( st Let \mathclose { -1 } ) = sin r * cos ( ( - 1 ) * ( sin ( r ) ) ) .= 0 ;
set q = |[ g1 . ( 0 ) , g2 . ( 0 + 1 ) ]| , r = |[ g1 . ( 0 + 1 ) , g2 . ( 0 + 1 ) ]| , s = |[ r , g2 ]| ;
consider G being sequence of S such that for n being Element of NAT holds G . n in <* W\sqcup ( F . n ) , G . n *> ;
consider G such that F = G and ex G1 st G1 in SM & G = ( X \/ G1 ) . ( n + 1 ) and G . ( n + 1 ) = G . ( n + 1 ) ;
( the root of [ x , s ] ) . [ x , s ] in ( ( the Sorts of Free ( C ) ) . s ) . [ x , s ] ;
Z c= dom ( ( exp_R * f ) + ( ( exp_R * f ) ) `| Z ) & f = ( ( exp_R * f ) + ( exp_R * f ) ) . ( x + x0 ) ;
for k being Element of NAT holds r0 . k = ( upper_volume ( Im ( f ) , S ) ) . k - ( ( Im ( f ) ) . k ) * ( ( Im ( f ) ) . k )
assume that - 1 < n and q `2 > 0 and ( q `2 / |. q .| - cn ) * ( 1 + cn ) < 0 and q `2 <= 0 ;
assume that f is continuous and a < b and a < d and f . a = c and f . b = d and f . a = d and f . b = c ;
consider r being Element of NAT such that s-> Element of ( P1 + P2 ) * ( i , r ) & r <= q ;
LE f /. ( i + 1 ) , f /. j , L~ f , f /. ( len f ) , f /. ( len f ) , f /. ( len f ) , f /. ( len f ) ;
assume that x in the carrier of K and y in the carrier of K and inf { x , y } = x and x <> y and y <> z ;
assume f +* ( i1 , \xi ) in ( proj ( F , i2 ) ) " ( ( proj ( F , i2 ) ) . ( A . ( A . ( i + 1 ) ) ) ) ;
rng ( ( ( ( Flow M ) | ( the carrier of M ) ) | ( the carrier' of M ) ) ) c= the carrier' of M ;
assume z in { ( the carrier of G ) --> { t } where t is Element of T : t in { t } } ;
consider l being Nat such that for m being Nat st l <= m holds ||. s1 . m - ( lim s1 ) .|| < g / 2 ;
consider t be VECTOR of product G such that mt = ||. Dt . t .|| and ||. t .|| <= 1 / 2 ;
assume that the carrier of v = 2 and v ^ <* 0 *> , v ^ <* 1 *> , v ^ <* 1 *> *> in dom p and v ^ <* 1 *> in dom p ;
consider a being Element of the \rm lines of X , A being Element of the \rm lines of X such that a on A and b on A and a on A ;
( - x ) |^ ( k + 1 ) * ( ( - x ) |^ ( k + 1 ) ) " = 1 ;
for D being set st for i st i in dom p holds p . i in D holds p . i in D & p . ( i + 1 ) in D
defpred R [ element ] means ex x , y st [ x , y ] = $1 & P [ y , x ] & P [ x , y ] ;
L~ f2 = union { LSeg ( p0 , p1 ) , LSeg ( p1 , p01 ) } .= { LSeg ( p1 , p01 ) , LSeg ( p1 , p01 ) } \/ { LSeg ( p1 , p01 ) } ;
i - len h11 + 2 - 1 < i - len h11 + 2 - 1 + 2 - 1 + 2 - 1 + 2 - 1 + 2 - 1 ;
for n being Element of NAT st n in dom F holds F . n = |. ( nbeing ) . ( n -' 1 ) .| & F . ( n -' 1 ) = |. ( nbeing ) . ( n -' 1 ) .| ;
for r , s1 , s2 , s3 being Element of REAL holds r in [. s1 , s2 .] iff s1 <= r & r <= s2 & s1 <= s2 & s2 <= s2 & s2 <= r & r <= s2 & s2 <= s2 implies s1 <= s2
assume v in { G where G is Subset of T2 : G in B2 & G c= F & F c= G & G c= F & G c= F & G c= F & G c= F & G c= F & F c= G & G c= F ;
let g be then then of A |^ X , Z , f , g , h , i , j , z be Element of Z ;
min ( g . [ x , y ] , k . [ y , z ] ) = ( min ( g , k , x ) ) . y ;
consider q1 being sequence of CP such that for n holds P [ n , q1 . n ] and q1 . ( n + 1 ) = F ( n ) ;
consider f being Function such that dom f = NAT & for n being Element of NAT holds f . n = F ( n ) and f . ( n + 1 ) = F ( n ) ;
reconsider B9 = B /\ B , O1 = O /\ O , Z1 = O /\ O , Z1 = O /\ O , I = O /\ O , I = O /\ O , E = I /\ O , I = I /\ O , I = I /\ O , E = I /\ I , I = I /\ I , I = I /\ I , E = I /\ I , I = I /\ I , I = I /\ I , E = I /\ I
consider j being Element of NAT such that x = ( the ` of n ) \ ( j - 1 ) and 1 <= j and j <= n and j <= n ;
consider x such that z = x and card ( x . O2 ) in card ( x . O2 ) and x in L1 . O2 and x in L1 . O2 ;
( C * \vert \vert \vert : T4 ( k , n2 ) .| ) . 0 = C . ( ( \vert \rm : .| ( k , n2 ) ) . 0 ) ;
dom ( X --> rng f ) = X & dom ( X --> f ) = dom ( X --> f ) & dom ( X --> f ) = dom ( X --> f ) ;
( ( N-bound L~ SpStSeq C ) `2 ) / 2 <= ( ( E-max C ) `2 ) / 2 & ( ( E-max C ) `2 ) / 2 <= ( ( E-max C ) `2 ) / 2 ;
synonym x , y are_collinear means : Def4 : x = y or ex l being Subset of S st { x , y } c= l & { x , y } c= l ;
consider X being element such that X in dom ( f | ( n + 1 ) ) and ( f | ( n + 1 ) ) . X = Y ;
assume that \mathop { \rm Im k is continuous and for x , y being Element of L st a = x & b = y holds x << y iff x << y ;
( 1 / 2 * ( ( ( - ( n + 1 ) ) (#) ( ( #Z n ) * ( ( #Z n ) * ( ( #Z n ) * ( 1 + 1 ) ) ) ) ) `| REAL ) = f ;
defpred P [ Element of omega ] means ( ( ( the partial of A1 ) * ( $1 + 1 ) ) | ( ( ( the Sorts of A1 ) * ( $1 + 1 ) ) /\ ( ( ( the Sorts of A1 ) * ( $1 + 1 ) ) | ( ( ( the Sorts of A1 ) * ( $1 + 1 ) ) /\ ( ( ( the Sorts of A1 ) * ( $1 + 1 ) ) | ( ( $1 + 1 ) * ( $1 + 1 ) ) ) ) ) ;
IC Comput ( P , s , 2 ) = succ IC Comput ( P , s , 1 ) .= ( 6 + 1 ) .= ( 6 + 1 ) .= 6 + 1 ;
f . x = f . g1 * f . g2 .= f . g1 * f . g2 .= f . g1 * 1_ H .= f . g1 * 1_ H .= f . g1 * 1_ H .= f . g2 * f . g2 ;
( M * ( ( F . n ) . n ) ) . n = M . ( ( ( canFS ( Omega ) ) . n ) ) .= M . ( { ( ( canFS ( Omega ) ) . n } ) . n ) ;
the carrier of L1 + L2 c= ( the carrier of L1 ) \/ ( the carrier of L2 ) & the carrier of L1 + L2 c= ( the carrier of L1 ) \/ ( the carrier of L2 ) ;
pred a , b , c , x , y , z , w , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , z , z , x
( ( the partial of s ) . n ) . n <= ( ( the partial of s ) . n ) . ( ( n + 1 ) * s . n ) ;
pred - 1 <= r & r <= 1 implies ( ( ( 1 - r ) (#) ( ( 1 - r ) (#) ( ( 1 - r ) (#) ( ( 1 - r ) / ( 1 - r ) ) ) ) `| Z ) = - r / ( 1 - r ) ;
seq in { p ^ <* n *> where n is Nat : p ^ <* n *> in T1 } & p in T1 & p in T1 & q in T2 & p in T1 & q in T2 & p in T1 & q in T2 & p in T1 & q in T2 } ;
|[ x1 , x2 , x3 ]| . 2 - |[ x1 , x2 , x3 ]| . 2 - |[ x2 , x3 , x4 ]| . 3 = x2 - x3 - x3 .= x2 - x3 ;
attr m is nonnegative means : Def4 : for n be Nat holds ( Partial_Sums ( F ) . n ) . m is nonnegative & ( Partial_Sums ( F ) . n ) . m <= ( ( Partial_Sums ( F ) ) . n ) . m ;
len ( ( G . ( z - x ) ) * ( ( G . ( x - y ) ) * ( ( G . ( y - z ) ) ) ) ) = len ( ( G . ( y - z ) ) * ( ( G . ( y - z ) ) * ( ( G . ( y - z ) ) * ( ( G . ( y - z ) ) * ( ( G . ( y - z ) ) ) ) ) ;
consider u , v being VECTOR of V such that x = u + v and u in W1 /\ W2 and v in W2 and u in W1 /\ W2 and v in W2 /\ W3 ;
given F being FinSequence of NAT such that F = x and dom F = n and rng F c= { 0 , 1 } and Sum F = k and Sum ( F ) = k ;
0 = 0 * ( 'not' ( - u ) * uH ) iff 1 = ( ( - 1 ) * ( - ( - ( - ( - ( 1 - ( - ( - ( 1 - ( 1 - 2 ) ) ) ) ) ) ) * ( ( - ( 1 - ( 1 - ( 1 - 2 ) ) ) ) ) ;
consider n be Nat such that for m be Nat st n <= m holds |. ( f # x ) . m - ( lim ( f # x ) ) .| < e ;
cluster -> being being being being being being being non empty for \frac for \frac of ( Y ) , ( ( Y ) ` ) ` , ( Y ) ` ) is Boolean & ( ( Y ) ` ) ` = ( Y ` ) ` ;
"/\" ( BB , L ) = Top BB .= Top S .= "/\" ( [#] S , L ) .= "/\" ( [#] S , L ) .= "/\" ( ( [#] S ) \ { {} } , L ) .= "/\" ( ( [#] S ) \ { {} } , L ) ;
( r / 2 ) ^2 + ( r / 2 ) ^2 / 2 <= ( r / 2 ) ^2 + ( r / 2 ) ^2 ;
for x being element st x in A /\ dom ( f `| X ) holds ( ( f `| X ) `| X ) . x >= r2
2 * r1 " * |[ a , c ]| - ( 2 * r1 - 1 ) * |[ b , c ]| = 0. TOP-REAL 2 & ( 2 * r1 - 1 ) * |[ b , c ]| = 0. TOP-REAL 2 ;
reconsider p = P * ( \square , 1 ) as a " * ( ( - ( - ( - ( 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 uparrow t and x = [ x1 , x2 ] and y = [ x1 , x2 ] ;
for n being Nat st 1 <= n & n <= len q1 holds q1 . n = ( upper_volume ( g , M7 ) ) . ( n + 1 ) & ( len q1 ) = len ( ( g | ( i + 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 Subgroup of G such that x = H1 and y = H2 and H1 is Subgroup of H2 and H2 is Subgroup of H1 and H2 is Subgroup of H2 ;
for S , T being non empty RelStr , d being Function of T , S st T is complete for x being Element of T holds d is monotone & x is monotone & d is monotone & x is monotone
[ a + 0 , b + i , b2 + ( - 1 ) * ( - 1 ) ] in ( the carrier of ( ( ( n + 1 ) + 1 ) * ( - 1 ) ) * ( - 1 ) ) ;
reconsider mm = max ( len F1 , len ( p . n ) * ( p . n ) ) as Element of NAT ;
I <= width GoB ( ( GoB ( GoB ( h , n ) ) * ( i , 1 ) ) , ( GoB ( h , n ) ) * ( i , 1 ) ) ;
f2 /* q = ( f2 /* ( f1 /* s ) ) ^\ k .= ( f2 * ( f1 /* s ) ) ^\ k .= ( f2 * ( f1 /* s ) ) ^\ k .= ( f2 * ( f1 /* s ) ) ^\ k .= ( f2 * ( f1 /* s ) ) ^\ k ;
attr A1 \/ A2 is linearly-independent means : Def4 : A1 misses A2 & A2 misses A1 & ( A1 \/ A2 ) /\ ( A2 \/ A1 ) = ( ( A1 \/ A2 ) \/ ( A2 \/ A1 ) ) /\ ( A1 \/ A2 ) ;
func A -_ C -> set means : Def4 : union it = union { A . s where s is Element of R : s in C } & for x being Element of R st x in A holds it . x = A ( x ) ;
dom ( ( Line ( v , i + 1 ) ) ^ ( ( ( Line ( p , m ) ) * ( ( Gauge ( p , m ) * ( i , 1 ) ) * ( i , 1 ) ) ) ) = dom ( F ^ ( ( Line ( F , m ) ) * ( i , 1 ) ) ) ;
cluster [ ( x `1 ) / 4 , ( x `2 ) / 4 ] -> [ x `1 , ( x `2 ) / 4 ] -> [ x `1 , ( x `2 ) / 4 ] -> [ x `1 , ( x `2 ) / 4 ] -> [ x `1 , ( x `2 ) / 4 ] ;
E , All ( x2 , All ( x2 , x2 ) ) |= All ( x2 , All ( x2 , x2 ) ) => ( x2 '&' ( x2 '&' ( x2 '&' x3 ) ) ) ;
F .: ( id X , g ) . x = F . ( id X , g . x ) .= F . ( id X , g . x ) .= F . ( x , g ) .= F . ( x , g ) ;
R . ( h . m ) = F . x0 + h . ( m + 1 ) - h . x0 .= ( F . x0 - h . x0 ) - ( F . x0 ) ;
cell ( G , ( X -' 1 , Y ) -' ( t + 1 ) , G ) meets UBD L~ f & ( L~ f ) meets L~ f implies ( L~ f ) meets ( L~ f ) \/ ( L~ f ) ;
IC Comput ( P2 , s2 , 2 ) = IC Comput ( P2 , s2 , 1 ) .= ( card I + ( card I + 2 ) ) .= ( card I + 2 ) .= ( card I + 2 ) + ( card I + 2 ) .= ( card I + 2 ) + ( card I + 2 ) .= card I + 2 ;
sqrt ( ( - ( ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ) ^2 ) > 0 ;
consider x0 being element such that x0 in dom a and x0 in dom g and x0 in g " { k } and y = a . ( k + 1 ) and x0 in dom g and g . ( k + 1 ) = a . ( k + 1 ) ;
dom ( r1 (#) chi ( A , C ) ) = dom chi ( A , C ) /\ dom chi ( A , C ) .= dom ( ( r1 (#) chi ( A , C ) ) | A ) .= dom ( ( r1 (#) chi ( A , C ) ) | A ) .= dom ( ( r1 (#) chi ( A , C ) ) | A ) .= dom ( ( r1 (#) chi ( A , C ) ) | A ) .= dom ( ( r1 (#) chi ( A , C ) ) ;
d-7 . [ y , z ] = ( ( y - z ) * ( x - z ) ) * ( ( y - z ) * ( x - z ) ) ;
attr i being Nat means : Def4 : C . i = A . i /\ B . i & C . i c= A . i /\ B . i ;
assume that x0 in dom f and f is_continuous_in x0 and for x be Element of REAL st x in dom f holds f is_differentiable_in x0 & f is_differentiable_in x0 holds ( f | X ) . x = - ( f | X ) . x0 ;
p in Cl A implies for K being Basis of p , Q being Subset of T st Q in K holds A meets Q & A meets Q implies A meets Q
for x being Element of REAL n st x in Line ( x1 , x2 ) holds |. y1 - y2 .| <= |. y1 - y2 .| & |. y1 - y2 .| <= |. y1 - y2 .|
func Sum ( <*> a ) -> Ordinal means : Def4 : a in it & for b being Ordinal st a in it holds it . b c= a & it . b c= b ;
[ a1 , a2 , a3 , a4 ] in ( the carrier of A ) /\ ( the carrier of B ) & [ a1 , a2 , a3 ] in ( the carrier of A ) /\ ( the carrier of B ) ;
ex a , b being element st a in the carrier of S1 & b in the carrier of S2 & x = [ a , b ] & [ b , a ] = [ a , b ] ;
||. ( ( vseq . n ) - ( vseq . m ) ) - ( vseq . n ) .|| < ( e / ( ||. x .|| + ||. x .|| ) ) * ||. x .|| ;
then for Z being set st Z in { Y where Y is Element of I7 : F . Y c= Z & Z in { Y } holds z in Z ;
sup compactbelow [ s , t ] = [ sup ( compactbelow s ) . ( sup compactbelow s ) , sup ( compactbelow s ) . ( sup compactbelow s ) ] .= sup ( compactbelow s ) . ( sup compactbelow s ) ;
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 holds p ^ q = q ^ p & q ^ p = p ^ q
consider e1 being Element of the affine of X such that c9 , a9 // a2 , e1 and a <> e1 and a <> e1 and b <> e1 and a <> e1 and b <> e1 and c <> e1 and a <> e1 and b <> e1 and c <> e1 and a <> e1 and b <> e2 and c , d // b , e2 ;
set U2 = I \! \mathop { + } , E = I \! \mathop { + } , F = I \! \mathop { + } , G = I \! \mathop { + } , E = { F } , N = { F } , N = { F } , N = { F } , N = { F } , N = { F } , N = { F . N } , N = { F . N } , N = { F . N } , N = { F . N } , N = { F . N } , N = { F . N , N = { F . N , N = { N .
|. q3 .| ^2 = ( ( q3 `1 ) ^2 + ( q3 `2 ) ^2 ) + ( ( q3 `2 ) ^2 + ( q3 `2 ) ^2 ) .= ( q `1 ) ^2 + ( q `2 ) ^2 .= ( q `1 ) ^2 + ( q `2 ) ^2 ;
for T being non empty TopSpace , x , y being Element of [: the topology of T , the topology of T :] holds x "\/" y = x \/ y & x "/\" y = x /\ y implies x "/\" y = 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 ) | X ) .= dom ( ( h | X ) | X ) .= dom ( ( h | X ) | X ) .= dom ( ( h | X ) | X ) .= dom ( ( h | X ) | X ) .= dom ( ( h | X ) | X ) .= dom ( h | X ) ;
for N1 , N1 , N2 be Element of G1 holds dom ( h . K1 ) = N & rng ( h . N1 ) = N1 & rng ( h . N1 ) c= N1 & rng ( h . N1 ) c= N1 & rng ( h . N1 ) c= N1 & rng ( h . N1 ) c= N1 & rng ( h . N1 ) c= N2
( 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 ) >= - cn & - ( q `1 / |. q .| - cn ) / ( 1 + cn ) >= cn & - cn < q `2 / ( 1 + cn ) / ( 1 + cn ) ;
attr r1 = f9 , r2 = f9 , s1 = f2 , s2 = f9 , s3 = q9 , s3 = f9 , s3 = q9 , s3 = Comput ( f1 , f2 , i ) , s3 = Comput ( f2 , f3 , i ) , s4 = Comput ( f1 , f3 , i ) , P4 = Comput ( f2 , f3 , i ) , P4 = Comput ( f2 , f3 , i ) , P4 = Comput ( f2 , f3 , i ) , P4 = Comput ( f2 , f3 , i ) , P4 = Comput ( f2 , f3 ) , P4 = f1 , s4 = f2 , s4 = f3
( ( vseq . m ) is bounded & xx . m = ( seq_id ( ( vseq . m ) , X ) ) . x & ( vseq . m ) ) . x = ( ( seq_id ( ( vseq . m ) , X ) ) . x ) . x ;
pred a <> b & b <> c & angle ( a , b , c , a ) = PI & angle ( b , c , a ) = 0 & angle ( b , c , a ) = 0 ;
consider i , j being Nat , r being Real such that p1 = [ i , r ] and p2 = [ j , s ] and r < j and s < 1 ;
|. p .| ^2 - ( 2 * |( p , q )| ) ^2 + ( 2 * |( p , q )| ) ^2 = |. p .| ^2 + ( 2 * |( p , q )| ) ^2 ;
consider p1 , q1 being Element of X ( ) such that y = p1 ^ q1 and q1 ^ q2 = p1 ^ q1 and p1 ^ q1 = q1 ^ q2 and q1 = q1 ^ q2 and q1 = q2 and q2 = q2 and q1 = q2 and q2 = q2 and q2 = q2 ;
1. ( A , ( 1 , r1 , s1 ) ) * ( 1 , s1 ) = ( s2 * ( s1 , s2 ) ) * ( s1 , s2 ) .= ( s2 * s1 ) * ( s1 , s2 ) ;
( ( LMP A ) `2 = lower_bound ( proj2 .: ( A /\ Vertical_Line w ) ) & ( proj2 .: ( A /\ Vertical_Line w ) ) is non empty & ( proj2 .: ( A /\ Vertical_Line w ) ) c= ( proj2 .: ( A /\ Vertical_Line w ) ) ;
s , ( k + 1 ) |= H1 '&' H2 iff s |= All ( H1 , H2 ) & s |= All ( H2 , H1 ) & s |= H2 implies s |= All ( H2 , H2 ) & s |= H2 ;
len ( s + 1 ) = card ( support b1 ) + 1 .= card ( support b2 ) + ( len b2 ) .= card ( support b2 ) + ( len b2 ) .= len ( s ) + ( len b2 ) .= len ( s ) + len ( b1 ) .= len ( s ) + len ( b1 ) ;
consider z being Element of L1 such that z >= x and z >= y and for z being Element of L1 st z >= x & z >= y holds z >= x ;
LSeg ( ( UMP D ) * ( ( ( ( ( ( ( D ) ) * ( 1 , 1 ) ) + ( ( ( D ) * ( 1 , 1 ) ) + ( D * ( 1 , 1 ) ) ) ) / 2 ) , ( ( D ) * ( 1 , 1 ) ) / 2 ) ) /\ D = { ( ( D ) * ( 1 , 1 ) ) / 2 } ;
lim ( ( ( f `| N ) / ( g `| N ) ) /* b ) = lim ( ( f `| N ) / ( g `| N ) ) .= lim ( ( f `| N ) / ( g `| N ) ) ;
P [ i , pr1 ( f ) . ( i , pr1 ( f ) . ( i + 1 ) , pr1 ( f ) . ( i + 1 ) ) ] & pr1 ( f ) . ( i + 1 ) = pr1 ( f ) . ( i + 1 ) ;
for r be Real st 0 < r ex m be Nat st for k be Nat st m <= k holds ||. ( seq . k ) - ( seq . k ) .|| < r
for X being set , P being a_partition of X , x , a , b being set st x in P & a in P & b in P & x in P & a in P & b in P & x in P holds a = b
Z c= dom ( ( ( 1 / 2 ) (#) f ) `| Z ) \ ( ( ( 1 / 2 ) (#) f ) `| Z ) " { 0 } \/ dom ( ( 1 / 2 ) (#) f ) " { 0 } ;
ex j being Nat st j in dom ( l ^ <* x *> ) & j < i & y = ( l ^ <* x *> ) . j & i = 1 + j & j = len l + 1 & i = len l + 1 + j ;
for u , v being VECTOR of V , r being Real st 0 < r & u in N & v in N holds r * u + ( r * v ) + ( r * v ) in N
A , Int A , Cl Int A , Cl Int A , Cl Int A , Cl Int A , Cl Int A , Cl Int A , Cl Int A , Cl Cl ( Cl ( A \/ B ) ) , Cl ( A \/ B ) , Cl ( A \/ B ) , Cl ( A \/ B ) , Cl ( A \/ B ) , Cl ( A \/ B ) , Cl ( A \/ B ) , Cl ( A \/ B ) , Cl ( A \/ B ) , Cl ( A , B ) , Cl ( A , B ) , Cl ( A \/ B ) , Cl ( A \/ B ) , Cl ( A
- Sum <* v , u , w *> = - ( v + u + w ) .= - ( v + u + w ) .= - ( v + u ) + ( - u ) .= - ( v + u ) ;
( Exec ( a := b , s ) ) . IC SCM = ( Exec ( a := b , s ) ) . IC SCM .= ( Exec ( a := b , s ) ) . IC SCM .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ
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 ( Carrier J ) . x ;
for S1 , S2 being non empty reflexive RelStr , D being non empty Subset of S1 , f being Function of S1 , S2 , g being Function of S2 , S2 st f = g holds ( f * g ) * ( f * g ) is directed & ( f * g ) * ( f * g ) is directed
card X = 2 implies ex x , y st x in X & y in X & x <> y & x <> y or x = y or x = y or x = z or y = z or x = z or x = z or y = z or x = z or x = z or y = z or x = z or y = z or x = z or y = z
E-max L~ Cage ( C , n ) in rng Cage ( C , n ) implies ( Cage ( C , n ) \circlearrowleft E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) < ( Cage ( C , n ) ) .. Cage ( C , n )
for T , T being decorated tree , p , q being Element of dom T st p ^ q in dom T holds ( T -tree ( p , q ) ) . q = T . q & ( T tree ( p , q ) ) . q = T . q
[ i2 + 1 , j2 ] in Indices G & [ i2 + 1 , j2 ] in Indices G & f /. k = G * ( i2 + 1 , j2 ) & f /. k = G * ( i2 + 1 , j2 ) & f /. k = G * ( i2 + 1 , j2 ) ;
cluster ( k gcd n ) divides k & k divides ( k gcd n ) & ( k divides m ) & ( k divides n implies k divides m ) & ( k divides n ) & ( k divides m ) implies k divides m ) & ( k divides m ) & ( k divides m ) implies k divides m ) ;
dom F " = the carrier of X2 & rng F = the carrier of X1 & F " { 0 } = the carrier of X2 & rng F = the carrier of X2 & rng F = the carrier of X2 & rng F = the carrier of X1 & rng F = the carrier of X2 & rng F = the carrier of X2 & F is one-to-one implies F " { 0 } = F " { 0 }
consider C being finite Subset of V such that C c= A and card C = n and the carrier of V = n and C = Lin ( B \/ C ) and C = Lin ( B \/ C ) and C = Lin ( B \/ C ) ;
V is prime implies for X , Y being Element of \langle the topology of T , \subseteq \rangle st X /\ Y c= V & X c= V or Y c= V or X c= V & Y c= V & V c= V implies X 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 ) + angle ( p4 , p4 ) .= angle ( p2 , p4 ) + angle ( p3 , p4 ) .= angle ( p4 , p4 ) + angle ( p4 , p4 ) .= angle ( p2 , p4 ) + angle ( p4 , p4 ) ;
- sqrt ( ( - ( q `1 / |. q .| - cn ) ) ^2 / ( 1 + cn ) ^2 ) = - sqrt ( ( q `1 / |. q .| - cn ) ^2 / ( 1 + cn ) ^2 ) .= - ( q `1 / |. q .| - cn ) / ( 1 + cn ) ^2 / ( 1 + cn ) ^2 ) .= - ( q `1 / |. q .| - cn ) ;
ex f being Function of I[01] , TOP-REAL 2 st f is continuous one-to-one & rng f = P & f . 0 = p1 & f . 1 = p2 & f . 0 = p4 & f . 1 = p4 & f . 1 = p4 & f . 0 = p4 & f . 1 = p4 & f . 1 = p4 & f . 1 = p4 ;
attr f is partial means : Def4 : SVF1 ( 2 , pdiff1 ( f , 1 ) , u0 ) . ( 2 * ( u - x0 ) ) = ( proj ( 2 , 3 ) ) . ( ( u - x0 ) * ( u - x0 ) ) ;
ex r , s st x = |[ r , s ]| & G * ( len G , 1 ) `1 < r & r < G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `2 < s & s < G * ( 1 , 1 ) `2 ;
assume that f is_sequence_on G and 1 <= t and t <= len G and G * ( t , width G ) `2 >= G * ( t , width G ) `2 and G * ( t , width G ) `2 >= G * ( t , width G ) `2 ;
pred i in dom G means : Def4 : r (#) ( f * reproj ( G , i ) ) = r * ( f * reproj ( G , i ) ) . x ;
consider c1 , c2 being bag of o1 + o2 such that ( <* c1 , c2 *> ) /. k = <* c1 , c2 *> and ( <* c1 , c2 *> ) /. k = <* c1 , c2 *> and ( <* c1 , c2 *> ) /. k = <* c1 , c2 *> ;
u0 in { |[ r1 , s1 ]| : r1 < G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `2 < G * ( 1 , 1 ) `2 } & G * ( 1 , 1 ) `2 < G * ( 1 , 1 ) `2 } ;
Cl ( X ^ Y ) . k = the carrier of X . ( k2 + 1 ) .= ( C . ( k2 + 1 ) ) .= ( C . ( k2 + 1 ) ) . k .= ( C . ( k2 + 1 ) ) . k .= ( C . ( k2 + 1 ) ) . k ;
attr M1 = len M2 & width M1 = width M2 & width M2 = width M2 & width M2 = width M2 & width M1 = width M2 & width M2 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 ;
consider g2 be Real such that 0 < g2 and { y where y is Point of S : ||. ( y - x0 ) - x0 .|| < g2 } c= N2 & N c= dom ( ( ||. y .|| ) | N ) ;
assume x < ( - b + sqrt ( delta ( a , b , c , d ) ) / ( 2 * a ) ) or x > ( - b + sqrt ( 2 * a ) ) / ( 2 * a ) ;
( G1 '&' G2 ) . i = ( <* 3 *> ^ G1 ) . i & ( H1 '&' H2 ) . i = ( <* 3 *> ^ ( H1 ^ H2 ) ) . i & ( H1 ^ H2 ) . i = ( <* 3 *> ^ ( H1 ^ H2 ) ) . i ;
for i , j st [ i , j ] in Indices ( M3 + M1 ) holds ( M3 + M2 ) * ( 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 <> j holds i divides f /. ( j + 1 ) & i divides len f & j divides len f holds i divides f /. ( j + 1 ) & i divides len f
assume F = { [ a , b ] where a , b is Subset of X : for c being set st c in B\mathopen the carrier of X & c in B\mathopen the carrier of X holds a c= b & b c= c & c c= a & a c= b ;
b2 * q2 + ( b3 * q3 + ( b3 * q3 ) ) + ( ( a * q2 ) + ( - ( a * q3 ) ) + ( - ( a * q3 ) ) ) = 0. TOP-REAL n + ( - ( a * q2 ) ) .= 0. TOP-REAL n + ( - ( a * q2 ) ) .= 0. TOP-REAL n + ( - ( a * q2 ) ) .= 0. TOP-REAL n + ( - ( a * q2 ) ) .= 0. TOP-REAL n ;
Cl ( Cl F ) = { D where D is Subset of T : ex B being Subset of T st D = Cl B & B in F & A c= B & B c= D & A c= B } ;
attr seq is summable means : Def4 : seq is summable & seq is summable & Sum ( seq ) = Sum ( seq ) & Sum ( seq ) = Sum ( seq ) & Sum ( seq ) = Sum ( seq ) & Sum ( seq ) = Sum ( seq ) + Sum ( seq ) ;
dom ( ( ( ( ( ( ( sn ) | D ) ) | D ) ) | D ) = ( the carrier of ( ( TOP-REAL 2 ) | D ) | D ) .= the carrier of ( ( TOP-REAL 2 ) | D ) | D .= D ;
[ X \to Z ] is full full full SubRelStr of ( [#] Z ) |^ the carrier of X & [ X \to Y ] is full SubRelStr of ( [#] Z ) |^ the carrier of X ;
( G * ( 1 , j ) ) `2 = ( G * ( i , j ) ) `2 & ( G * ( 1 , j ) ) `2 <= ( G * ( i , j ) ) `2 ;
synonym m1 c= m2 means : Def4 : for p be set st p in P & ( for m be set st p in P & m <> 1 holds ( m in P & p in P & p <> m ) holds p is__ P & p is__ 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 ] ;
synonym IT is multiplicative non empty multMagma means : Def4 : the carrier of it = [: the carrier of IT , the carrier of IT :] & the carrier of it = [: the carrier of IT , the carrier of IT :] ;
directed ( a , b , c ) + l ( c , d ) = b + l ( c , d ) .= b + l ( c , d ) .= the l of the carrier of thesis + ( a + c ) .= the l of the carrier of thesis + ( a + c ) + d ( b + c ) ;
cluster + ( i1 , i2 ) -> Z for Element of INT , i1 , i2 , j1 , j2 be Element of INT , i1 , i2 be Element of NAT , i2 be Element of NAT st i1 = i2 & i2 = i1 & j1 = i2 holds i1 = i2 & j1 = i2 implies i1 = i2 or j1 = j2 & j2 = i2 or i1 = i2 & i2 = i2 & i2 = j2 implies i1 = i2 + i2 + j2
( - s2 ) * p1 + ( s2 * p2 ) - ( s2 * p2 ) = ( - r2 ) * p1 + ( s2 * p2 ) - ( s2 * p2 ) .= ( - s2 ) * p2 + ( s2 * p2 ) - ( s2 * p2 ) ;
eval ( ( a | ( n , L ) ) *' , p ) = eval ( a | ( n , L ) ) * eval ( p , x ) .= a * eval ( p , x ) * eval ( p , x ) .= a * eval ( p , x ) * eval ( q , x ) ;
assume that the TopStruct of S = the TopStruct of T and for D being non empty directed Subset of S , V being open Subset of S st D in V holds V meets V and V meets V and V meets V and V meets V and V meets V and V meets V and V meets V and V meets V and V meets V and V meets V and V meets V and V meets V and V meets V and V meets V and V meets V and V meets V and V meets V and V meets V and V meets V and V meets V and V meets V and V meets V and V meets V and V meets V and V meets V and V meets V and V meets V and V meets V and V meets V and V meets V and V meets V and V
assume that 1 <= k and k <= len w + 1 and T-7 . ( ( ( q11 , w ) U ) . k ) = ( T-7 . ( ( q11 , w ) . k ) ) . k and T-7 . ( ( ( q11 , w ) U . k ) . k ) and not ( ( ( T . k ) , w ) . k = ( ( T . ( q11 , w ) ) . k ) . k ) ;
2 * a |^ ( n + 1 ) + ( 2 * b |^ ( n + 1 ) ) >= ( a |^ ( n + 1 ) + ( b |^ ( n + 1 ) ) + ( ( a |^ ( n + 1 ) ) + ( b |^ ( n + 1 ) ) ) + ( ( a |^ ( n + 1 ) ) + b |^ ( n + 1 ) ) ;
M , v / ( x. 3 , x. 0 ) / ( x. 0 , x. 4 ) / ( x. 0 , x. 0 ) / ( x. 4 , x. 0 ) / ( x. 0 , x. 4 ) / ( x. 0 , x. 0 ) / ( x. 0 , x. 4 ) / ( x. 0 , x. 4 ) / ( x. 0 , x. 4 ) ) / ( x. 0 , x. 4 ) / ( x. 0 , x. 4 ) ) ;
assume that f is_differentiable_on l and for x0 st x0 in l holds 0 < f . x0 - f . x0 and for x0 st x0 in l holds f . x0 < 0 and f . x0 < 0 ;
for G1 being _Graph , W being Walk of G1 , e being Vertex of G1 , x being Vertex of G2 st e in W holds not ( W is Walk of G & e in W implies W is Walk of G ) implies W is Walk of G
c01 is not empty iff q1 is not empty & not q1 is not empty & not q2 is not empty & not q1 is not empty & not q2 is not empty & not q1 is not empty & not q2 is not empty & not q1 is not empty & not q2 is not empty & not q2 is not empty & not q1 is not empty & not q2 is not empty & not q2 is not empty & not q1 is not empty & not q2 is not empty & not q2 is not empty & not q2 is not empty & not q2 is not empty & not q2 is not empty & not q2 is not empty & not q2 is not empty & not q2 is not q1 is not empty & not q2 is not empty & not q2 is not empty & not q2 is not empty & not q2 is
Indices GoB f = dom GoB f & ( GoB f ) * ( i1 , j1 ) in dom GoB f & ( GoB f ) * ( i1 , j1 ) in dom GoB f or ( GoB f ) * ( i1 , j1 ) = GoB f & ( GoB f ) * ( i2 , j1 ) = GoB f & ( GoB f ) * ( i2 , j1 ) = GoB f ) implies ( GoB f ) * ( i2 , j1 ) = ( GoB f ) * ( i2 , j1 )
for G1 , G2 , G3 , G1 , G2 , G3 being stable Subgroup of O st G1 is_stable in the carrier of G2 & G2 is_stable in the carrier of G2 & G1 is_stable in the carrier of G2 holds G1 is stable & G2 is stable & G2 is stable & G1 is stable & G2 is stable
UsedIntLoc ( int ) = { intloc 0 , intloc 1 , intloc 1 , intloc 2 , intloc 3 , intloc 4 , intloc 0 , intloc 3 , intloc 4 , intloc 5 , intloc 5 , 6 , 7 , 8 , 8 , 6 , 8 , 7 , 8 , 8 , 9 , 8 , 9 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 9 , 8 , 8 , 8 , 9 , 8 , 8 , 7 , 8 , 8 , 9 , 8 , 8 , 8 , 8 , 8 , 9 , 8 , 8 , 9 , 8 , 8 , 9 , 9 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 9 , 8 , 8 , 9 , 8 , 9 , 9 ,
for f1 , f2 be FinSequence of F st f1 ^ f2 is p -element & Q [ p ] & P [ q ] holds Q [ p ^ f1 , p ^ f2 ] & Q [ q ^ f1 , p ^ f2 ] & Q [ p ^ f1 , q ^ f2 ]
( p `1 / sqrt ( 1 + ( p `2 / p `1 ) ^2 ) ) ^2 = ( q `1 / sqrt ( 1 + ( q `2 / q `1 ) ^2 ) ) ^2 / ( 1 + ( q `2 / q `1 ) ^2 ) ;
for x1 , x2 , x3 , x4 , x1 , x2 , x3 , x4 , x1 , x2 , x3 , x4 , x1 , x2 , x3 , x4 , x1 , x2 , x3 , x4 , x1 , x2 , x3 , x4 , x1 , x2 , x3 , x4 , x1 , x2 , x3 , x4 , x1 , x2 , x3 , x4 , x1 , x2 , x3 , x4 , x1 , x2 , x3 , x2 , x3 , x4 , x1 , x2 , x3 , x2 , x3 , x4 , x1 , x2 , x3 , x4 , x1 , x2 , x3 , x4 , x2 , x4 , x1 , x4 , x2 , x2 , x4 , x1 , x2 , x3 , x4 , x1 , x2 , x3 , x4 , x4 , x2 , x3
for x st x in dom ( ( ( ( h | A ) - x ) | A ) ) holds ( ( ( ( h | A ) - x ) | A ) . ( - x ) = - ( ( ( h | A ) - x ) | A ) . x )
for T being non empty TopSpace , P being Subset-Family of T , B being Basis of T st P c= the topology of T for x being Point of T st x in P & B c= P holds 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 .= TRUE 'or' 'not' ( a 'or' b ) . x .= TRUE ;
for e being set st e in A8 ex X1 being Subset of X , Y1 being Subset of X st e = X1 & 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 & Y1 is open implies Y1 meets X1
for i be set st i in the carrier of S for f be Function of [: S , T :] , S1 , S2 be Function of [: S , T :] , S2 be Function of [: S , T :] , S2 st f = H . i & g = H . i holds F . i = f | [: F , F :]
for v , w st for x st x <> y holds w . y = v . y holds Valid ( VERUM ( Al ( ) ) , J ) . v = Valid ( VERUM ( Al ( ) ) , J ) . w holds Valid ( VERUM ( Al ( ) ) , J ) . v = Valid ( VERUM ( Al ( ) ) , J ) . w
card D = card D1 + card D2 - card { i , j } .= ( 1 + 1 ) - ( 1 + 1 ) .= ( 1 + 1 ) - ( 1 + 1 ) .= 2 * ( i - 1 ) - ( 1 + 1 ) .= 2 * ( i - 1 ) - ( 1 + 1 ) .= 2 * ( i - 1 ) - ( 1 + 1 ) ;
IC Exec ( i , s ) = ( s +* ( 0 .--> ( s . 0 ) ) ) . 0 .= ( ( s +* ( 0 .--> 1 ) ) +* ( 0 .--> 1 ) ) . 0 .= ( ( s +* ( 0 .--> 1 ) ) +* ( 0 .--> 1 ) ) . 0 .= ( ( s +* ( 0 .--> 1 ) ) +* ( 0 .--> 1 ) ) . 0 .= succ IC s ;
len f /. ( i1 -' 1 ) - 1 + 1 = len f -' ( i1 -' 1 ) + 1 - 1 .= len f -' ( i1 -' 1 ) + 1 - 1 .= len f - ( i1 -' 1 ) + 1 - 1 .= len f - ( i1 -' 1 ) + 1 - 1 ;
for a , b , c being Element of NAT st 1 <= a & 2 <= b holds k < a + b-2 or k = a + b-2 or k = a + b-2 or k = b + b-2 or k = a + b-2 or k = b + b-2 or k = a + b-2 or k = b + b-2 or k = a + b-2 or k = b + b-2 or k = a + b ;
for f being FinSequence of TOP-REAL 2 , p being Point of TOP-REAL 2 st p in LSeg ( f , i ) & p <> f /. ( i + 1 ) holds Index ( p , f ) <= Index ( p , f ) & Index ( p , f ) <= Index ( p , f )
lim ( ( curry ' ( I , k + 1 ) ) # x ) = lim ( ( curry ' ( I , k ) ) # x ) + lim ( ( curry ' ( I , k ) ) # x ) .= lim ( ( curry ' ( I , k ) ) # x ) + lim ( ( curry ' ( I , k ) ) # x ) ;
z2 = g /. ( ( i -' n1 + 1 ) + 1 ) .= g /. ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i - n1 + 1 ) .= g . ( i - n1 + 1 ) .= g . ( i - n1 + 1 ) .= g . ( i - n1 + 1 ) .= ( g /. ( i - n1 + 1 ) ) + ( z - n1 ) ;
[ f . 0 , f . 3 ] in id ( the carrier of G ) \/ ( the InternalRel of G ) or [ f . 0 , f . 3 ] in the InternalRel of C & [ f . 0 , f . 3 ] in the InternalRel of G ;
for G being Subset-Family of B st G = { R [ X ] where R is Subset of A ( ) , Y ( ) st R in F ( ) & Y in G ( ) holds ( Intersect ( R , G ) ) . ( X , Y ) = Intersect ( G , F ) . ( X , Y ) )
CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 ) ) = CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 ) ) .= halt SCMPDS .= halt SCMPDS .= halt SCMPDS .= ( halt SCMPDS ) ;
assume that a on M and b on M and c on N and d on N and p on M and p on N and a on M and p on M and p on M and a on M and p on M and p on M and a on M and p on M and p on M and p on M and a on M and p on M and p on M and a on M and p on M and a on M and p on M and a on M and p on M and p on M and p on M and a on M and p on M and p on M and p on M and p on M and p on M and p on M and a on M and p on M and a on M and p on M and p on M and p on M and a on M and a on M and a on M and p on M and a on M and p on M and p on M
assume that T is \hbox { T _ 4 4 , T , B be closed Subset of T , F be Subset-Family of T , n , m , m , n st F is closed & m <= n & n <= m holds F . ( n + 1 ) <= 0 and F . ( n + 1 ) <= 0 ;
for g1 , g2 st g1 in ]. r - r , r .[ & g2 in ]. r - r , r .[ & |. f . g1 - r .| <= ( |. f - g .| ) . ( g1 - g2 ) holds |. f . g1 - f . g2 .| <= ( |. f - g .| ) / ( |. f - g .| ) / ( |. f - g .| )
( ( - 1 ) / ( z1 + z2 ) ) / ( z2 + z2 ) = ( - ( - 1 ) / ( z1 + z2 ) ) / ( z2 + z2 ) .= ( - ( - 1 ) / ( z1 + z2 ) ) / ( z2 + z2 ) ;
F . i = F /. i .= 0. R + r2 .= ( b |^ n ) * r2 .= <* ( n + 1 ) -tuples_on the carrier of K ) .= <* ( n + 1 ) -tuples_on the carrier of K , ( n + 1 ) -tuples_on the carrier of K , ( n + 1 ) -tuples_on the carrier of K , 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 : Def4 : len it = len F & for i be Nat st i in dom F holds it . i = f /. i * F /. ( i - 1 ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , 6 , 7 , 8 , 8 , 7 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 7
for n being Nat , x being set st x = h . n holds h . ( n + 1 ) = o ( x , n ) & x in InputVertices S ( x , n ) & o . ( x , n ) in InnerVertices S ( x , n ) & o . ( x , n ) in InnerVertices S ( x , n ) ;
ex S1 being Element of QC-WFF ( Al ( ) ) , e being Element of D ( ) st [ P , e , S1 . ( k + 1 ) ] = S1 & ( for n being Nat holds P [ n , S1 . n , S1 . ( k + 1 ) ] ) & ( for n being Nat holds P [ n , S1 . n ] ) & ( for n being Nat holds P [ n , S1 . n ] ) ;
consider P being FinSequence of GS2 such that pS2 = Product P and for i being Element of dom t st i in dom P ex t being Element of Seg k st t . i = ( t . i ) * ( t . i ) & t . i = ( t . i ) * ( t . i ) ;
for T1 , T2 being non empty TopSpace , P being Basis of T1 , Q being Basis of T2 st the carrier of T1 = the carrier of T2 & P is Basis of T2 holds P is Basis of T1 & P is Basis of T2 implies P is Basis of T2
assume that f is_is_\cal \bf 2 2 } (#) 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 ) , 3 ) = r * pdiff1 ( f , 3 ) . ( 2 * pdiff1 ( f , 3 ) ) ;
defpred P [ Nat ] means for F , G being FinSequence of bool ( Seg $1 ) , s be FinSequence of REAL st len F = $1 & rng s = Seg $1 & rng s = Seg $1 holds Sum ( F , s ) = Sum ( G , s ) * ( s , G ) ;
ex j st 1 <= j & j < width GoB f & ( GoB f ) * ( 1 , j ) `2 <= s & s <= ( GoB f ) * ( 1 , j + 1 ) `2 & s <= ( GoB f ) * ( 1 , j + 1 ) `2 ;
defpred U [ set , set ] means ex FF being Subset-Family of T st $1 = FF & $2 = ( union FF ) . $1 & ( union FF ) . $1 = ( union FF ) . $1 & ( union FF ) . $1 = ( union FF ) . $1 & ( union FF ) . $1 = ( union FF ) . $1 & ( union FF ) . $1 = ( union FF ) . $1 ;
for p4 being Point of TOP-REAL 2 st LE p4 , p4 , P , p1 , p2 & LE p4 , p , P , p1 , p2 holds LE p4 , p , P , p1 , p2 & LE p4 , p , P , p1 , p2 & LE p , p , P , p1 , p2 & LE p , p1 , P , p1 , p2 & LE p , p1 , P , p1 , p2 & LE p , p1 , P , p2 & LE p , p1 , P , p1 , p2 , p2 , p2 , K & LE p , p1 , P , p1 , p2 , K & LE p , p1 , P , p1 , p2 , K & LE p , p1 , P , p1 , p2 , K & LE p , p1 , P , K & LE p , p1 , P , p1 , K & LE p , p1 , P , p1 , p2 , p1 , K & LE p , p1 , P ,
f in St ( E , H ) & for g st g . y <> f . y holds x = g . y iff for y st y <> x holds g . y = f . ( g . y ) & f . ( g . y ) = f . ( All ( x , H ) . y ) ;
ex 8 being Point of TOP-REAL 2 st x = 8 & ( ( for p being Point of TOP-REAL 2 st p in 8 holds p `2 / |. p .| >= cn ) & ( p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 ) & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 implies p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 implies p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL
assume for d7 being Element of NAT st d7 <= d7 holds ( for t being Element of NAT st d7 <= t & t <= ( |. 7 .| ) * ( t + |. 7 .| ) ) & ( t + |. 7 .| ) * ( t + |. 7 .| ) = ( ( |. 7 .| ) * ( t + |. 7 .| ) ) ;
assume that s <> t and s is Point of Sphere ( x , r ) and s is Point of Sphere ( x , r ) and not ex e being Point of E st e = Ball ( x , r ) /\ Ball ( s , r ) & e in Sphere ( x , r ) /\ Ball ( x , r ) ;
given r such that 0 < r and for s holds 0 < s or ex x1 be Point of CNS st x1 in dom f & |. x1 - x0 .| < s & |. x1 - x0 .| < s & |. x1 - x0 .| < r & |. x1 - x0 .| < s ;
( p | x ) | ( p | ( x | ( x | x ) ) ) = ( ( ( x | x ) | ( x | ( x | x ) ) ) | ( p | ( x | ( x | x ) ) ) ) | ( p | ( x | ( x | x ) ) ) ;
assume that x , x + h in dom sec and ( for x st x in dom sec holds ( ( ( 2 * sec ) `| Z ) . x ) = ( 4 * ( ( 2 * ( x + h ) ) / ( 2 * x ) ) ) / ( 2 * x + h / ( 2 * x ) ) ) and ( ( 2 * sec ) `| Z ) . x = ( 4 * ( x + h ) / ( 2 * x ) ) / ( 2 * x ) ) / ( 2 * x ) ;
assume that i in dom A and len A > 1 and len B > 0 and len A = len B and width A = len B and width A = width B and width A = width B and width A = width B and len A = width B and width A = width B and len A = width B ;
for i being non zero Element of NAT st i in Seg n holds i divides n or i = <* 1. F_Complex *> or i = <* 1. F_Complex *> & ( i divides n or i divides n & i <> n & n <> 0 & n <> 0 & n <> 0 & n <> 0 & n <> 0 ) implies h . ( n * i ) = <* 1. F_Complex * n , n * n *>
( ( ( b1 'imp' b2 ) '&' ( c1 '&' c2 ) ) '&' ( ( ( b1 'or' b2 ) '&' ( c1 '&' c2 ) ) '&' ( ( a1 'or' a2 ) '&' ( b1 'or' c2 ) ) '&' ( ( a2 'or' b2 ) '&' ( ( a1 'or' a2 ) '&' ( a1 'or' a2 ) ) ) ) ) '&' ( ( a1 'or' a2 ) '&' ( a1 'or' a2 ) ) '&' ( ( a1 'or' a2 ) '&' ( a1 'or' a2 ) '&' ( a1 'or' a2 ) ) '&' ( a2 'or' a2 ) '&' ( a1 'or' a2 ) '&' ( a2 'or' a2 ) '&' ( a1 'or' a2 ) '&' ( a1 'or' a2 ) '&' ( a2 'or' c2 ) '&' ( a1 'or' a2 ) '&' ( a1 'or' a2 ) '&' ( a1 'or' a2 ) '&' ( a1 'or' a2 ) '&' ( a1 'or' a2 ) '&' ( ( a1 'or' a2 ) '&' ( ( ( a1 'or' a2 ) '&' ( ( a1 'or' a2 ) '&' ( a2 ) )
assume that for x holds f . x = ( ( ( - 1 ) (#) ( ( cot ) * ( ( cot ) * ( sin ) ) ) `| Z ) . x and for x st x in Z holds ( ( ( - 1 ) (#) ( cot ) ) `| Z ) . x = - ( sin . x ) / ( sin . x ) ^2 ) and for x st x in Z holds ( ( ( - 1 ) (#) ( cot ) ) `| Z ) . x = - ( sin . x ) / ( sin . 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 / ( sin . x ) ^2 + ( sin . x ) ^2 + ( sin . x ) ^2 + ( sin .
consider R8 , I8 be Real such that R8 = Integral ( M , Re ( n + 1 ) ) and I8 = Integral ( M , ( Im ( n + 1 ) ) ) and I = Integral ( M , ( Im ( n + 1 ) ) ) and I = Integral ( M , ( Im ( n + 1 ) ) ) and I = Integral ( M , ( Im ( n ) ) ) ;
ex k being Element of NAT st ( for q be Element of NAT st q in X & q <> 0 holds ||. ( qx0 - f . ( q - 1 ) ) - ( f . ( q - 1 ) ) .|| < r ) & ||. partdiff ( f , q , k ) - partdiff ( f , q , k ) .|| < r ;
x in { x1 , x2 , x3 , x4 , x5 , 6 , 7 , 8 , 7 , 8 , 8 , 6 , 7 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 7 , 8 , 8 , 7 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 7 , 7 , 8 , 7 , 6 , 8 , 8 , 6 , 7 , 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 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1
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 .= ( the Arity of S1 ) . o ;
func tree ( T , P , T1 , T2 ) -> Tree means : Def4 : q in it iff q in P & p in P & q in P & p in P & q in P & p in P & q in P & p in P & q in P & p in P & q in P & p in P & q in P & p in P & q in P & p in P & q in P & p in P & q in P implies p in P or p in P & q in P & p in P & q in P & p in P & p in P & q in P & p in P & p in P & p in P & p in P & q in P & p in P & q in P & q in P & q in P & p in P & p in P & p in P & p in P & p in P & p in P & q in P & p in
F /. ( k + 1 ) = F . ( k + 1 ) .= Fpp . ( p . ( k + 1 -' 1 ) , k + 1 -' 1 ) .= FIm ( p . ( k + 1 -' 1 ) , k ) .= FIm ( p . k , k ) .= FIm ( p . k , k ) ;
for A , B , C being Matrix of K st len B = len C & len C = len C & len A = len B & len B > 0 & len C > 0 & len A > 0 & len B > 0 & len A > 0 & len B > 0 & len A > 0 & len B > 0 & width B > 0 & len A > 0 & width B > 0 & width B > 0 & width B > 0 holds A * B = B * C
seq . ( k + 1 ) = 0. F_Complex + seq . ( k + 1 ) .= ( Partial_Sums ( seq ) ) . ( k + 1 ) .= ( Partial_Sums ( seq ) ) . ( k + 1 ) + seq . ( k + 1 ) .= ( Partial_Sums ( seq ) ) . ( k + 1 ) + seq . ( k + 1 ) .= ( Partial_Sums ( seq ) ) . ( k + 1 ) + seq . ( k + 1 ) ;
assume that x in ( the carrier of C1 ) and y in ( the carrier of C1 ) and z in ( the carrier of C1 ) and x = [ y , z ] ;
defpred P [ Element of NAT ] means for f st len f = $1 & ( for k st k in dom f holds f . k = ( ( VAL g ) . ( k + 1 ) ) * ( ( VAL g ) . ( k + 1 ) ) holds ( ( VAL g ) . ( k + 1 ) ) . f = ( ( VAL g ) . ( k + 1 ) ) * ( ( VAL g ) . k ) ;
assume that 1 <= k and k + 1 <= len f and [ i , j ] in Indices G and [ i + 1 , j ] in Indices G and f /. k = G * ( i , j ) and f /. k = G * ( i + 1 , j ) and f /. k = G * ( i , j ) ;
assume that sn < 1 and q `1 > 0 and ( q `2 / |. q .| - cn ) * ( 1 + cn ) >= 0 and ( q `2 / |. q .| - cn ) * ( 1 + cn ) >= 0 and ( q `2 / |. q .| - cn ) * ( 1 + cn ) >= 0 and ( q `2 / |. q .| - cn ) * ( 1 + cn ) >= 0 ;
for M being non empty metric space , x being Point of M , f being Function of M , M st x = x `1 & f . ( n + 1 ) = Ball ( x `1 , 1 / ( n + 1 ) ) holds f . ( n + 1 ) = Ball ( x `1 , 1 / ( n + 1 ) )
defpred P [ Element of omega ] means $1 is differentiable & f1 is_differentiable_on Z & f2 is_differentiable_on Z & ( for x st x in Z holds f1 . x = - 1 & f2 is_differentiable_on Z holds ( ( f1 - f2 ) `| Z ) . x = - ( f1 . x ) / ( ( f1 . x ) ^2 + ( f2 . x ) ^2 ) ;
defpred P1 [ Nat , Point of CNS ] means ( $1 in Y & $2 in Y & $2 in X & $2 in Y & $2 in X & $2 in Y & $2 in Y & $2 in X & $2 in Y & $2 in X & $2 in Y & $2 in X & $2 in Y & $2 in X & $2 in X & $2 in X & $2 in X & $2 in X & $2 in X & $2 in X & $2 in X ;
( f ^ mid ( g , 2 , len g ) ) . i = ( mid ( g , 2 , len g ) ) . ( i - len f + 1 ) .= g . ( i - len f + 1 ) .= g . ( i - len f + 1 ) .= g . ( i - len f + 1 ) .= g . ( i - len f + 1 ) .= g . ( i - len f + 1 ) ;
( 1 - ( 2 * n0 + 2 * ( n0 + 2 * n ) ) ) * ( ( 2 * n0 + 2 * n ) ) = ( ( 1 - 2 * ( n + 2 * n ) ) * ( ( n + 2 * n ) ) * ( ( n + 2 * n ) ) .= ( ( 1 - 2 ) * ( n + 2 * n ) ) * ( ( n + 2 * n ) ) .= ( ( n + 2 ) * ( n + 2 ) ) * ( ( n + 2 ) ) * ( ( n + 2 ) * ( ( n + 2 ) * ( ( n + 2 ) * ( ( n + 2 ) ) * ( ( n + 2 ) ) * ( ( n + 2 ) * ( ( n + 2 ) * ( n + 1 ) ) * ( ( n + 1 ) ) * ( ( n + 1 ) * ( ( n + 1 ) * ( ( n + 1 ) * ( n + 1 ) ) * ( ( n
defpred P [ Nat ] means for G being non empty RelStr , N being non empty finite RelStr st G is $1 -A\ for x being Element of G st x in N & not ( the carrier of G = the carrier of N & x in N & not ( the carrier of G = the carrier of N ) & ( the carrier of G = the carrier of N ) holds x in the carrier of N ) ;
assume that f /. 1 in Ball ( u , r ) and 1 <= m and m <= len - ( f /. ( 1 + 1 ) ) and LSeg ( f , m ) /\ LSeg ( f , m ) <> {} and LSeg ( f , m ) /\ LSeg ( f , m ) <> {} and LSeg ( f , m ) <> {} and m in Ball ( f , m ) and m in Ball ( f , m ) ;
defpred P [ Element of NAT ] means ( Partial_Sums ( ( ( cos * ( ( cos * ( ( 1 / 2 ) ) * ( ( cos * ( 1 / 2 ) ) ) ) ) ) `| Z ) . ( ( ( cos * ( ( 1 / 2 ) * ( ( cos * ( ( 1 / 2 ) * ( ( cos * ( 1 / 2 ) ) ) ) ) . ( 2 * $1 ) ) ) ;
for x being Element of product F holds x is FinSequence of product F & dom x = I & for i being set st i in I holds x . i = ( ( Carrier F ) . i ) . x & for i be set st i in I holds x . i = ( Carrier F ) . i ) . x
( x " ) |^ ( n + 1 ) = ( x " ) |^ n * x " .= ( ( x " ) |^ n ) * ( x " ) .= ( ( x " ) |^ n ) * ( x " ) .= ( ( x " ) |^ n ) * ( x " ) .= ( ( x " ) |^ n ) * ( x " ) .= ( ( x " ) |^ n ) * ( x " ) .= ( x * ( x * ( x * ( x * ( x * ( x * ( x * ( x * ( x * ( x * ( x * ( x * ( x * ( x * ( x * ( x * ( x * ( x * ( x * ( x * ( x * ( x * ( x * ( x * ( x * ( x * y ) ) ) ) ) ) * ( x * y ) ) * ( x * y ) ) * ( x * y ) ) * ( x * y ) ) * ( x * y ) ) * ( x * y ) ) * ( x
DataPart Comput ( P +* I , ( ( intloc 0 ) .--> ( ( intloc 0 ) .--> 1 ) ) , ( ( intloc 0 ) .--> 1 ) ) = DataPart Comput ( P +* I , ( ( intloc 0 ) .--> 1 ) +* I , ( ( intloc 0 ) .--> 1 ) +* I ) ;
given r such that 0 < r and ]. x0 - r , x0 .[ c= dom ( f1 + f2 ) /\ dom ( f2 + g2 ) and for g st g in ]. x0 - r , x0 .[ /\ dom ( f1 + f2 ) holds ( f1 + f2 ) . g <= ( f1 + f2 ) . g ;
assume that X c= dom f1 /\ dom f2 and f1 | X is continuous and f2 | X is continuous and ( f1 - f2 ) | X is continuous and ( f1 - f2 ) | X is continuous and ( f1 - f2 ) | X is continuous and ( f1 - f2 ) | X is continuous and f2 | X is continuous and ( f1 - f2 ) | X = f1 | X ) ;
for L being continuous LATTICE for l being Element of L ex X being Subset of L st l = sup X & for x being Element of L st x in X holds x is Element of L & x is Element of L & x is prime & l is prime & x is prime & l is prime & x in X holds x = sup ( L /\ L )
Support ( e8 ) in { m *' p where m is Polynomial of n , L : i in dom ( m *' p ) & p . ( i + 1 ) = m . ( p . ( i + 1 ) ) & p . ( i + 1 ) = m . ( p . ( i + 1 ) ) } ;
( f1 - f2 ) /. ( lim s1 ) = lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) ;
ex p1 being Element of QC-WFF ( Al ( ) ) st F . p1 = g . p1 & for g being Function of [: Al ( ) , D ( ) :] , D ( ) st P [ g , f , g ] holds P [ g , f , g , h ] ;
( mid ( f , i , len f -' 1 ) ) /. j = ( mid ( f , i , len f -' 1 ) ) /. ( j + 1 ) .= ( mid ( f , i , len f -' 1 ) ) /. ( j + 1 ) .= f /. ( j + 1 ) .= f /. ( j + 1 ) ;
( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) = ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) . ( 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 + k ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) . ( len p + k ) . ( len p + k ) .= ( p ^ q )
len ( mid ( f , D2 , indx ( D2 , D1 , j ) ) + 1 ) = indx ( D2 , D1 , indx ( D2 , D1 , j ) ) - indx ( D2 , D1 , j ) .= indx ( D2 , D1 , j ) - indx ( D2 , D1 , j ) + 1 ;
x * y * z = MM . ( ( x * y ) * z ) .= x * ( ( y * z ) * z ) .= ( x * ( y * z ) ) * ( z * x ) .= ( x * ( y * z ) ) * ( z * x ) .= ( x * ( y * z ) ) * ( z * x ) .= ( x * ( y * z ) ) * ( z * x ) ;
v . ( <* x , y *> ) * ( <* x0 , y0 *> ) = partdiff ( v , ( x - x0 ) * ( <* x0 , y0 *> ) + ( ( ( proj ( 1 , 1 ) ) * ( ( reproj ( 1 , 1 ) ) * ( ( reproj ( 1 , 1 ) ) * ( ( reproj ( 1 , 1 ) ) * ( ( reproj ( 1 , 1 ) ) * ( ( reproj ( 1 , 1 ) ) * ( ( reproj ( 1 , 1 ) ) ) ) ) ) ) ) + ( ( reproj ( 1 , 1 ) ) . ( ( reproj ( 1 , 1 ) ) ) . ( ( reproj ( 1 , 1 ) ) . ( ( reproj ( 1 , 1 ) ) . ( ( reproj ( 1 , 1 ) ) . ( ( reproj ( 1 , ( h ) ) . ( ( reproj ( 1 , 1 ) ) . ( ( reproj ( 1 , 1 ) ) . ( ( reproj ( 1 , 1 ) ) . ( ( x- ( 1 , 1 ) ) . ( ( x- ( 1 , 1 ) ) . ( ( x- ( 1 , 1 )
i * i = <* 0 * ( - 1 ) - ( 0 * 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) .= <* - 1 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 1 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 1 , 0 , 0 *> .= 1 ;
Sum ( L (#) F ) = Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( ( L (#) F1 ) ^ ( L (#) F2 ) ) .= Sum ( ( L (#) F1 ) ^ ( L (#) F2 ) ) .= Sum ( ( L (#) F1 ) ^ ( L (#) F2 ) ) + Sum ( ( L (#) F2 ) ^ ( L (#) F2 ) ) .= Sum ( L (#) F1 ) + Sum ( ( L (#) F2 ) ^ ( L (#) F2 ) ) .= Sum ( L (#) F1 ) + Sum ( ( L (#) F2 ) ) .= Sum ( L (#) F1 ) + Sum ( ( L (#) F2 ) ^ ( L (#) F2 ) ) .= Sum ( ( L (#) F2 ) ) + Sum ( ( L (#) F2 ) ) .= Sum ( L (#) F2 ) ) .= Sum ( L (#) F2 ) + Sum ( ( L (#) F2 ) + Sum ( ( L (#) F1 ) + Sum ( ( L (#) F2 ) + Sum ( ( L (#) F2 ) ^ ( L (#) F2 ) .= Sum ( L (#) F1 ) + Sum ( ( L (#) F2 ) + Sum ( ( L (#) F2 ) ) .=
ex r be Real st for e be Real st 0 < e ex Y1 be Subset of X st Y1 is non empty & ( for Y1 be Subset of X st Y1 is non empty & Y1 c= Y holds |. ( - r ) * ( Y1 ) - ( r * Y2 ) ) .| < r ;
( GoB f ) * ( i , j + 1 ) = f /. ( k + 2 ) & ( GoB f ) * ( i , j + 1 ) = f /. ( k + 1 ) or ( GoB f ) * ( i , j + 1 ) = f /. ( k + 1 ) & ( GoB f ) * ( i , j + 1 ) = f /. ( k + 2 ) or ( GoB f ) * ( i , j ) = f /. ( k + 1 ) ;
( ( ( - 1 ) (#) ( ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( - 1 ) ) ) ) ) ) `| Z ) = ( - 1 / ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( - 1 ) (#) ( - 1 ) (#) ( - 1 ) ) ) ) / ( 1 - 1 ) ) ) .= - ( ( - 1 ) (#) ( ( - 1 ) (#) ( - 1 ) (#) ( - 1 ) (#) ( - 1 ) ) / ( 1 - 1 ) ) / ( 1 - 1 ) ) .= - ( 1 - 1 ) ) / ( 1 - 1 ) ) / ( 1 - ( 1 - 1 ) ) / ( 1 - ( 1 - 1 ) ) / ( 1 - ( 1 - 1 ) (#) ( 1 - 1 ) ) .= - ( 1 - 1 ) ) / ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1
( - b + sqrt ( delta ( a , b , c ) ) / ( 2 * a ) ) > 0 & ( - b - sqrt ( 2 * a ) ) / ( 2 * a ) > 0 implies ( - b - sqrt ( 2 * a ) / ( 2 * a ) ) / ( 2 * a ) > 0 or ( - b - sqrt ( 2 * a ) ) / ( 2 * a ) > 0
Suppose inf ( uparrow L /\ X ) and sup X in X and sup ( ( subrelstr L ) /\ C ) = "/\" ( ( subrelstr L ) /\ ( ( subrelstr L ) /\ X ) ) and "\/" ( ( subrelstr L ) /\ X ) = "/\" ( ( subrelstr L ) /\ ( ( subrelstr L ) /\ X ) , L ) ; Then "\/" ( ( subrelstr L ) /\ X , L ) = "/\" ( ( subrelstr L ) /\ X , L ) ;
( for j being Element of I holds ( ( j = i ) --> ( j , i ) ) & ( j = i implies i = j ) & ( j = i implies j = i ) & ( j = i implies j = i ) & ( j = i implies j = i ) & ( j = i implies j = i ) implies j = i ) & j = i & j = i implies j = i & j = i )