thesis ;
thesis ;
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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 ;
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thesis ;
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thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
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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 rng ;
b in A ;
d divides a ;
i < n ;
s <= b ;
b in B ;
let r ;
B is one-to-one ;
R is total ;
x = 2 ;
d in D ;
let c ;
let c ;
b = Y ;
0 < k ;
let b ;
let n ;
r <= b ;
x in X ;
i >= 8 ;
let n ;
let n ;
y in f ;
let n ;
1 < j ;
a in L ;
C is boundary ;
a in A ;
1 < x ;
S is finite ;
u in I ;
z << z ;
x in V ;
r < t ;
let t ;
x c= y ;
a <= b ;
m in NAT ;
assume f is z ;
not x in Y ;
z = +infty ;
k be Nat ;
K ` is being_line ;
assume n >= N ;
assume n >= N ;
assume X is + 1 ;
assume x in I ;
q is as Nat ;
assume c in x ;
p > 0 ;
assume x in Z ;
assume x in Z ;
1 <= k} ;
assume m <= i ;
assume G is rng ;
assume a divides b ;
assume P is closed ;
b-a > 0 ;
assume q in A ;
W is not bounded ;
f is Assume
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 DecoratedTree ;
let d be element ;
let t be element ;
let x be element ;
let x be element ;
let s be element ;
k <= 5 - -2 ;
let X be set ;
let X be set ;
let y be element ;
let x be element ;
P [ 0 ]
let E be set , f be Function of E , 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 , iff n1 is , n1 , n2 + 1 is_collinear
Q halts_on s ;
x in that x in that x in that x in that x in that x in that x in that x in that x in
M < m + 1 ;
T2 is open ;
z in b < a ;
R2 is well-ordering ;
1 <= k + 1 ;
i > n + 1 ;
q1 is one-to-one ;
let x be trivial set ;
PM is one-to-one ;
n <= n + 2 ;
1 <= k + 1 ;
1 <= k + 1 ;
let e be Real ;
i < i + 1 ;
p3 in P ;
p1 in K ;
y in C1 ;
k + 1 <= n ;
let a be Real , b be Real ;
X |- r => p ;
x in { A } ;
let n be Nat ;
let k be Nat ;
let k be Nat ;
let m be Nat ;
0 < 0 + k ;
f is_differentiable_in x ;
let x0 , r ;
let E be Ordinal ;
o on o2 ;
O <> O2 ;
let r be Real ;
let f be FinSeq-Location ;
let i be Nat ;
let n be Nat ;
Cl A = A ;
L c= Cl L ;
A /\ M = B ;
let V be RealUnitarySpace , W be strict Subspace of V ;
not s in Y |^ 0 ;
rng f is_<=_than w
b "/\" e = b ;
m = m3 ;
t in h . D ;
P [ 0 ] ;
assume z = x * y ;
S . n is bounded ;
let V be RealUnitarySpace , W be Subspace of V ;
P [ 1 ] ;
P [ {} ] ;
C1 is component ;
H = G . i ;
1 <= i `1 + 1 ;
F . m in A ;
f . o = o ;
P [ 0 ] ;
aA2 <= non [ k ] ;
R [ 0 ] ;
b in f .: X ;
assume q = q2 ;
x in [#] V ;
f . u = 0 ;
assume e1 > 0 ;
let V be RealUnitarySpace , W be Subspace of V ;
s is trivial 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 , S ;
the ObjectMap of F is one-to-one
sgn x = 1 ;
k in support a ;
1 in Seg 1 ;
rng f = X ;
len T in X ;
vbeing < n ;
S\HM is bounded ;
assume p = p2 ;
len f = n ;
assume x in P1 ;
i in dom q ;
let U0 , A , B ;
pp `1 = c ;
j in dom h ;
let k ;
f | Z is continuous ;
k in dom G ;
UBD C = B ;
1 <= len M ;
p in \mathbin { x } ;
1 <= jj & jj <= len f ;
set A = thesis ;
card a [= c ;
e in rng f ;
cluster B ++ A -> empty ;
H has no for f ;
assume n0 <= m ;
T is increasing ;
e2 <> e2 & e2 <> e1 ;
Z c= dom g ;
dom p = X ;
H is proper ;
i + 1 <= n ;
v <> 0. V ;
A c= Affin A ;
S c= dom F ;
m in dom f ;
let X0 be set ;
c = sup N ;
R is_connected implies union M is connected
assume not x in REAL ;
Im f is complete ;
x in Int y ;
dom F = M ;
a in On W ;
assume e in A ( ) ;
C c= C-26 ;
mm <> {} & mm <> {} ;
let x be Element of Y ;
let f be ) Chain , g be Chain of G ;
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 in v in dom that x in dom G ;
- y in I ;
let A be non empty set , f be Function of A , REAL ;
Px0 = 1 ;
assume r in F . k ;
assume f is simple ;
let A be w countable set ;
rng f c= NAT ;
assume P [ k ] ;
f6 <> {} ;
let o be Ordinal ;
assume x is sum of squares ;
assume not v in { 1 } ;
let IB , IB ;
assume that 1 <= j and j < l ;
v = - u ;
assume s . b > 0 ;
\bf da in dom f ;
assume t . 1 in A ;
let Y be non empty TopSpace , f be Function of Y , X ;
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 Omega ( f ) ;
[ a , c ] in X ;
mm <> {} & mm <> {} ;
M + N c= M + M ;
assume M is r1 hh) ;
assume f is
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 or k1 = k2 ;
m + 1 < n + 1 ;
s in S \/ { s } ;
n + i >= n + 1 ;
assume Re y = 0 ;
k1 <= j1 & j1 <= len G ;
f | A is continuous ;
f . x - a <= 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 ;
Cf in X ;
q2 c= C1 & q2 c= C2 ;
a2 < c2 & a2 < c1 ;
s2 is 0 -started ;
IC s = 0 & IC s = 0 ;
s4 = s4 , P4 = P3 ;
let V ;
let x , y be element ;
let x be Element of T ;
assume a in rng F ;
x in dom T `1 ;
let S be as as as of L ;
y " <> 0 ;
y " <> 0 ;
0. V = u-w ;
y2 , y , w is_collinear ;
R8 in X ;
let a , b be Real , f be Function of REAL , REAL ;
let a be Object of C ;
let x be Vertex of G ;
let o be object of C , m be Morphism of C ;
r '&' q = P \lbrack l .] ;
let i , j be Nat ;
let s be State of A , v be Element of V ;
s4 . n = N ;
set y = x `1 , z = y `2 ;
mi in dom g ;
l . 2 = y1 ;
|. g . y .| <= r ;
f . x in Cx0 ;
V-19 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 NffffF ;
reconsider i = i as Ordinal ;
r * v = 0. X ;
rng f c= INT & rng g 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 , Y = Vars C ;
let o be OperSymbol of S ;
let R be connected non empty Poset ;
n + 1 = succ n ;
x9 c= Z1 & x9 c= Z1 ;
dom f = C1 & dom g = C2 ;
assume [ a , y ] in X ;
Re ( seq ) is convergent & lim Im ( seq ) = 0 ;
assume a1 = b1 & a2 = b2 ;
A = sInt A & B = sInt B ;
a <= b or b <= a ;
n + 1 in dom f ;
let F be Instruction of S , I be Instruction of S ;
assume that r2 > x0 and x0 < r2 ;
let Y be non empty set , f be Function of Y , Z ;
2 * x in dom W ;
m in dom g2 & m + 1 in dom g2 ;
n in dom g1 /\ dom g2 ;
k + 1 in dom f ;
the still of s in { s } ;
assume x1 <> x2 & x1 <> x3 ;
v3 in Vx0 & v3 in Vx0 ;
not [ b `1 , b `2 ] in T ;
( i + 1 ) + 1 = i ;
T c= non empty TopSpace & T c= T ;
l `1 = 0 & l `2 = 0 ;
let n be Nat ;
t `2 = r `2 & t `2 = s `2 ;
Ab is_integrable_on M & Ab is_integrable_on M ;
set t = Top t ;
let A , B be real-membered set ;
k <= len G + 1 ;
C ( ) misses V ( ) ;
product ( s ) is non empty ;
e <= f or f <= e ;
cluster non empty normal for Ordinal ;
assume that c2 = b2 and 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 vseq is convergent and lim vseq = 0 ;
IC s3 = 0 & IC s3 = 0 ;
k in N or k in K ;
F1 \/ F2 c= F \/ F2 ;
Int G1 <> {} & Int G2 <> {} ;
z `2 = 0 & z `2 = 0 ;
p11 <> p1 or p11 <> p2 ;
assume z in { y , w } ;
MaxADSet ( a ) c= F ;
ex_sup_of downarrow s , S ;
f . x <= f . y ;
let T be up-complete non empty reflexive transitive antisymmetric RelStr ;
q |^ m >= 1 ;
a is_>=_than X & b is_>=_than Y ;
assume <* a , c *> <> {} ;
F . c = g . c ;
G is one-to-one one-to-one ;
A \/ { a } \not c= B ;
0. V = 0. Y ;
let I be
f-24 . x = 1 ;
assume z \ x = 0. X ;
C4 = 2 to_power n ;
let B be SetSequence of Sigma ;
assume X1 = p .: D ;
n + l2 in NAT & n + l2 in NAT ;
f " P is compact & f " P is compact ;
assume x1 in REAL & x2 in REAL ;
p1 = ( K + L ) ;
M . k = <*> REAL ;
phi . 0 in rng phi ;
OSM^ A is closed
assume z0 <> 0. L & z0 <> 0. L ;
n < ( N . k ) ;
0 <= seq . 0 & seq . 0 <= seq . 0 ;
- q + p = v ;
{ v } is Subset of B ;
set g = f /. 1 ;
R ( ) is stable set of R ;
set cR = Vertices R ;
px0 c= P3 & px0 c= P3 ;
x in [. 0 , 1 .[ ;
f . y in dom F ;
let T be Scott Scott TopAugmentation of S ;
ex_inf_of the carrier of S , S ;
\HM { a } = 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 - x .|| <= r ;
assume that Y c= field Q and Y <> {} ;
a ~ , b ~ are_isomorphic ;
assume a in A ( i ) ;
k in dom ( q | k ) ;
p is FinSequence of S ;
i -' 1 = i-1 - 1 ;
f | A is one-to-one ;
assume x in f .: X ( ) ;
i2 - i1 = 0 & i2 - i1 = 0 ;
j2 + 1 <= i2 & j2 + 1 <= i2 ;
g " * a in N ;
K <> { [ {} , {} ] } ;
cluster strict for for for \HM { of X } ;
|. q .| ^2 > 0 ;
|. p4 .| = |. p .| ;
s2 - s1 > 0 ;
assume x in { Gij } ;
W-min C in C & E-max C in C ;
assume x in { Gij } ;
assume i + 1 = len G ;
assume i + 1 = len G ;
dom I = Seg n & dom J = Seg n ;
assume that k in dom C and k <> i ;
1 + 1-1 <= i + j ;
dom S = dom F /\ dom G ;
let s be Element of NAT , k be Element of NAT ;
let R be ManySortedSet of A ;
let n be Element of NAT ;
let S be non empty non void non void holds S is - void topological structure ;
let f be ManySortedSet of I ;
let z be Element of COMPLEX , v be Element of COMPLEX ;
u in { ag } ;
2 * n < ( 2 * n ) ;
let x , y be set ;
B-11 c= V-15 \/ { x } ;
assume I is_closed_on s , P & I is_halting_on s , P ;
UA = [: U2 , U2 :] ;
M /. 1 = z /. 1 ;
x11 = x22 & x22 = x22 ;
i + 1 < n + 1 + 1 ;
x in { {} , <* 0 *> } ;
( f . x ) `1 <= ( f . y ) `1 ;
let l be Element of L ;
x in dom ( F . n ) ;
let i be Element of NAT , f be Function ;
seq1 is COMPLEX -valued & seq2 is COMPLEX -valued ;
assume <* o2 , o *> <> {} ;
s . x to_power 0 = 1 ;
card K1 in M & card K1 in M ;
assume that X in U and Y in U ;
let D be thesis in
set r = Seg ( k + 1 ) ;
y = W . ( 2 * x ) ;
assume that dom g = cod f and cod g = cod f ;
let X , Y be non empty TopSpace , f be Function of X , Y ;
x ++ A is interval ;
|. <*> A .| . a = 0 ;
cluster strict for SubLattice of L ;
a1 in B . s1 & a2 in B . s2 ;
let V be finite implies V is non empty
A * B on B & B on B ;
f-3 = NAT --> 0 ;
let A , B be Subset of V ;
z1 = P1 . j .= P1 . j ;
assume f " P is closed & f " P is closed ;
reconsider j = i as Element of M ;
let a , b be Element of L ;
assume q in A \/ ( B "\/" C ) ;
dom ( F * C ) = o ;
set S = INT |^ X ;
z in dom ( A --> y ) ;
P [ y , h . y ] ;
{ x0 } c= dom f & { x0 } c= dom g ;
let B be non-empty ManySortedSet of I , f be Function of B , B ;
PI / 2 < Arg z ;
reconsider z9 = 0 , z9 = 1 as Nat ;
LIN a , d , c & LIN a , d , c ;
[ y , x ] in If ;
Q * ( 3 , 3 ) = 0 ;
set j = x0 gcd m , m = x0 gcd m ;
assume a in { x , y , c } ;
j2 - jj > 0 & j2 - jj > 0 ;
I . phi = 1 & I . phi = 2 ;
[ y , d ] in F-8 ;
let f be Function of X , Y ;
set A2 = ( B - C ) / 2 ;
s1 , s2 are_card ( X ) ;
j1 -' 1 = 0 & j2 -' 1 = 1 ;
set m2 = 2 * n + j ;
reconsider t = t `1 as bag of n ;
I2 . j = m . j ;
i |^ s , n are_relative_prime ;
set g = f | D-21 ;
assume that X is lower and 0 <= r ;
p1 `1 = 1 or p1 `2 = - 1 ;
a < p3 `1 & p3 `1 < b ;
L \ { m } c= UBD C ;
x in Ball ( x , 10 ) ;
not a in LSeg ( c , m ) ;
1 <= i1 -' 1 & i1 + 1 <= len G ;
1 <= i1 -' 1 & i1 + 1 <= len G ;
i + i2 <= len h ;
x = W-min ( P ) or x = E-max ( P ) ;
[ x , z ] in [: X , Z :] ;
assume y in [. x0 , x .] ;
assume p = <* 1 , 2 , 3 *> ;
len <* A1 *> = 1 & len <* A2 *> = 1 ;
set H = h . gg ;
card b * a = |. a .| ;
Shift ( w , 0 ) |= v ;
set h = h2 (*) h1 ;
assume x in X3 /\ ( X3 \/ { x1 } ) ;
||. h .|| < d1 & ||. h .|| < d1 ;
not x in the carrier of f ;
f . y = F ( y ) ;
for n holds X [ n ] ;
k -' l = kbeing - kl ;
<* p , q *> /. 2 = q ;
let S be Subset of the carrier of Y ;
let P , Q be + T ;
Q /\ M c= union ( F | M )
f = b * ( canFS S ) ;
let a , b be Element of G ;
f .: X is_<=_than f . sup X
let L be non empty transitive reflexive RelStr , x , y be Element of L ;
S-20 is_be x -8 i -basis ;
let r be non positive Real ;
M , v |= x \hbox { y } ;
v + w = 0. ( Z , p ) ;
P [ len F ( ) ] ;
assume that InsCode ( i ) = 8 and InsCode ( i ) = 8 ;
the zero of M = 0 & the zero of M = 0 ;
cluster z * seq -> summable ;
let O be Subset of the carrier of C ;
||. f .|| | X is continuous ;
x2 = g . ( j + 1 ) ;
cluster -> that for Element of ( AllSymbolsOf S ) ;
reconsider l1 = l-1 as Nat ;
v4 is Vertex of r2 & v4 is Vertex of G ;
T2 is SubSpace of T2 & T2 is SubSpace of T2 ;
Q1 /\ Q19 <> {} & Q1 /\ Q1 <> {} ;
k be Nat ;
q " is Element of X & q " is Element of Y ;
F . t is set of of of non zero set ;
assume that n <> 0 and n <> 1 ;
set en = EmptyBag n , en = EmptyBag n ;
let b be Element of Bags n ;
assume for i holds b . i is commutative ;
x is root & y is root implies x * y is root
not r in ]. p , q .[ ;
let R be FinSequence of REAL , a be Element of REAL ;
( not S does not contradiction ) & not S is not empty ;
IC SCM R <> a & IC SCM R <> a ;
|. p - |[ x , y ]| .| >= r ;
1 * seq = seq & 1 * seq = seq ;
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 .= IC s1 ;
H + G = Felement ( G-GG ) ;
Cs1 . x = x2 & Cs2 . x = y2 ;
f1 = f .= f2 .= f1 + f2 ;
Sum <* p . 0 *> = p . 0 ;
assume v + W = v + u + W ;
{ a1 } = { a2 } ;
a1 , b1 _|_ b , a ;
d1 , o _|_ o , a3 , a4 ;
If is_reflexive implies f is_reflexive & g is_reflexive
IB is antisymmetric & IB is antisymmetric implies IB is antisymmetric
sup rng H1 = e & sup rng H1 = e ;
x = ( a * ( - 1 ) ) * ( - 1 ) ;
|. p1 .| ^2 >= 1 ^2 ;
assume j2 -' 1 < 1 & j2 -' 1 < len G ;
rng s c= dom f1 /\ dom f2 ;
assume that support a misses support b and support b misses support b ;
let L be associative well-unital non empty doubleLoopStr , p be Polynomial of L ;
s " + 0 < n + 1 ;
p . c = ( f " ) . 1 ;
R . n <= R . ( n + 1 ) ;
Directed ( I1 , I2 ) = I1 +* Directed ( I2 , 1 ) ;
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 <* [ ] , N . N *> -> complete non trivial ;
( 1 - a " ) * a = a ;
( q . {} ) `1 = o ;
( n - 1 ) > 0 ;
assume 1 / 2 <= t `1 & t `2 <= 1 ;
card B = k + 1-1 ;
x in union rng ( f | k ) ;
assume x in the carrier of R & y in the carrier of R ;
d in dom f ;
f . 1 = L . ( F . 1 ) ;
the vertices of G = { v } & not v in { x } ;
let G be Q be \rm : ] ;
e , v9 , x , y be set ;
c . ( i9 - 1 ) in rng c ;
f2 /* q is divergent_to-infty & f2 /* q is divergent_to-infty ;
set z1 = - z2 , z2 = - z1 ;
assume w is_llas of S , G ;
set f = p |-count t , g = p |-count t , h = p |-count t , p = p |-count t , n = p |-count t , m = p |-count t , n = p |-count t
let c be Object of C ;
assume ex a st P [ a ] ;
let x be Element of REAL m , y be Element of REAL m ;
let IB be Subset-Family of X , IB be Subset-Family of X ;
reconsider p = p 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 , k be Nat ;
stop I ( ) c= P-12 & stop I ( ) c= Pc ;
set ci = fci /. i ;
w ^ t ^ s ^ t ^ s ^ t ^ t ^ t ^ s ^ t ^ t ^ t ^ t ^ t ^ t ^ t ^ t ^ t ^ t ^ t ^ t ^
W1 /\ W = W1 /\ W ` ;
f . j is Element of J . j ;
let x , y be Subset of T2 , a be Real ;
ex d st a , b // b , d ;
a <> 0 & b <> 0 & c <> 0
ord x = 1 & x is positive implies x is positive
set g2 = lim ( seq ) , g1 = lim ( seq ) ;
2 * x >= 2 * 1 / 2 ;
assume ( a 'or' c ) . z <> TRUE ;
f (*) g in Hom ( c , c ) ;
Hom ( c , c + d ) <> {} ;
assume 2 * Sum ( q | m ) > m ;
L1 . ( FY . k ) = 0 ;
/ ( X \/ R1 ) = h / ( X \/ R1 ) ;
( ( sin - cos ) `| Z ) . x <> 0 ;
( ( #Z 2 ) * ( f1 + f2 ) ) . x > 0 ;
o1 in X-5 /\ ( XO2 /\ O2 ) ;
e , v9 , x , y be set ;
r3 > ( 1 / 2 ) * 0 ;
x in P .: ( F -ideal of L ) ;
let J be closed Ideal of R , left ideal non empty Subset of R ;
h . p1 = f2 . O & h . p2 = g2 . I ;
Index ( p , f ) + 1 <= j ;
len ( q | i ) = width M ;
the carrier of CK c= A ;
dom f c= union rng F-10 & rng F-10 c= union rng F-10 ;
k + 1 in ( support n ) \/ ( support n ) ;
let X be ManySortedSet of the carrier of S ;
[ x `1 , y `2 ] in ( ( \HM { x } ) \/ ( the carrier of R ) ) ;
i = D1 or i = D2 or i = D1 ;
assume a mod n = b mod n ;
h . x2 = g . x1 & h . x1 = g . x2 ;
F c= 2 -tuples_on the carrier of X ;
reconsider w = |. s1 .| as Real_Sequence ;
1 / ( m * m + r ) < p ;
dom f = dom IK1 & dom g = dom IK1 ;
[#] ( P-17 ) = [#] ( ( TOP-REAL 2 ) | K1 ) ;
cluster - x -> ExtReal & x <= - y ;
then { da } c= A & A is closed ;
cluster TOP-REAL n -> finite-ind for Subset of TOP-REAL n ;
let w1 be Element of M , w2 be Element of M ;
let x be Element of dyadic ( n ) ;
u in W1 & v in W3 implies u + v in W3
reconsider y = y , z = z as Element of L2 ;
N is full SubRelStr of ( T |^ the carrier of S ) ;
sup { x , y } = c "\/" c ;
g . n = n to_power 1 .= n ;
h . J = EqClass ( u , J ) ;
let seq be summable sequence of X , x be Element of X ;
dist ( x `1 , y ) < r / 2 ;
reconsider mm = m - 1 as Element of NAT ;
x0 - r < r1 - x0 & r1 < r2 - x0 ;
reconsider P ` = P ` as strict Subgroup of N ;
set g1 = p * ( idseq q `1 ) ;
let n , m , k be non zero Nat ;
assume that 0 < e and f | A is lower ;
D2 . ( I8 + 1 ) in { x } ;
cluster -> subcondensed for Subset of T ;
let P be compact non empty Subset of TOP-REAL 2 , p1 , p2 be Point of TOP-REAL 2 ;
Gij in LSeg ( cos , 1 ) /\ LSeg ( Gik , Gij ) ;
let n be Element of NAT , x be Element of X ;
reconsider SS = S , SS = T as Subset of T ;
dom ( i .--> X ` ) = { i } ;
let X be non-empty ManySortedSet of S ;
let X be non-empty ManySortedSet of S ;
op ( 1 ) c= { [ {} , {} ] } ;
reconsider m = mm as Element of NAT ;
reconsider d = x `2 as Element of C ( ) ;
let s be 0 -started State of SCMPDS , k be Nat ;
let t be 0 -started State of SCMPDS , Q ;
b , b , b , x , y is_collinear ;
assume that i = n \/ { n } and j = k \/ { k } ;
let f be PartFunc of X , Y ;
N2 >= ( sqrt c / sqrt 2 ) / 2 ;
reconsider [: T , T :] = [: T , T :] as TopSpace ;
set q = h * p ^ <* d *> ;
z2 in U . ( y2 ) /\ Q . ( y2 ) ;
A |^ 0 = { <%> E } & A |^ 1 = A ;
len W2 = len W + 2 & len W + 1 = len W ;
len h2 in dom h2 & len h2 = len h2 ;
i + 1 in Seg len s2 & i + 1 in dom s2 ;
z in dom g1 /\ dom f & z in dom g1 ;
assume that p2 = E-max ( K ) and p1 `2 >= 0 ;
len G + 1 <= i1 + 1 ;
f1 (#) f2 is_differentiable_in x0 & f1 (#) ( f2 (#) f1 ) is_differentiable_in x0 ;
cluster s-10 + sp -> summable for Real_Sequence ;
assume j in dom M1 & i <= len M2 ;
let A , B , C be Subset of X ;
let x , y , z be Point of X , p be Point of X ;
b ^2 - ( 4 * a * c ) >= 0 ;
<* x/y *> ^ <* y *> ^ <* y *> ^ <* x *> ^ <* y *> ;
a , b in { a , b } ;
len p2 is Element of NAT & len p1 = len p2 ;
ex x being element st x in dom R & y = R . x ;
len q = len ( K (#) G ) ;
s1 = Initialize Initialized s , P1 = P +* I ;
consider w being Nat such that q = z + w ;
x ` is Element of L & x ` is Element of L ;
k = 0 & n <> k or k > n ;
then X is discrete for A is closed Subset of X ;
for x st x in L holds x is FinSequence ;
||. f /. c .|| <= r1 & ||. f /. c .|| <= r2 ;
c in uparrow p & not c in { p } ;
reconsider V = V as Subset of the topology of TOP-REAL n ;
let N , M be being being being being being being being 0 Element of L ;
then z is_>=_than waybelow x & z is_>=_than compactbelow x ;
M \lbrack f , f .] = f & M \lbrack g , f .] = g ;
( ( >= 1 ) to_power ( 1 + 1 ) ) = TRUE ;
dom g = dom f .: X .= dom f ;
mode ) Walk of G is ^ Assume W is * * -valued ;
[ i , j ] in Indices ( M @ ) ;
reconsider s = x " , t = y " as Element of H ;
let f be Element of dom ( Subformulae p ) ;
F1 . ( a1 , - a1 ) = G1 . ( a1 , - a1 ) ;
redefine func being Real equals LSeg ( a , b , r ) ;
let a , b , c , d be Real ;
rng s c= dom ( 1 / 2 ) ;
curry ( ( F . -19 ) . k ) is additive ;
set k2 = card dom B , k2 = card dom C , k1 = card D , k2 = card C ;
set G = ( X ) --> NAT ;
reconsider a = [ x , s ] as Object of G ;
let a , b be Element of Mf , c be Element of Mf ;
reconsider s1 = s , s2 = t as Element of ( the carrier of S ) ;
rng p c= the carrier of L & rng p c= the carrier of L ;
let d be Subset of the bound of A ;
( x .|. x = 0 iff x = 0. W )
I-21 in dom stop I & Ik = stop I ;
let g be continuous Function of X | B , Y ;
reconsider D = Y as Subset of TOP-REAL n ;
reconsider i0 = len p1 , i2 = len p2 as Integer ;
dom f = the carrier of S & rng g c= the carrier of S ;
rng h c= union ( ( Carrier J ) . i ) ;
cluster All ( x , H ) -> carrier of and x in Carrier p ;
d * N1 ^2 > N1 * 1 / 2 ;
]. a , b .[ c= [. a , b .] ;
set g = f " | D1 , h = f " | D2 ;
dom ( p | mm1 ) = mm1 ;
3 + - 2 <= k + - 2 ;
tan is_differentiable_in ( arccot - arccot ) . x & tan . x > 0 ;
x in rng ( f /^ n ) /\ rng ( f /^ n ) ;
let f , g be FinSequence of D ;
p ( ) in the carrier of S1 & p ( ) in the carrier of S1 ;
rng f " = dom f & rng f = rng g ;
( the Target of G ) . e = v & ( the Target of G ) . e = v ;
width G -' 1 < width G -' 1 + 1 ;
assume v in rng ( S | E1 ) /\ rng ( S | E1 ) ;
assume x is root or x is root or x is root ;
assume that 0 in rng ( g2 | A ) and 0 < r ;
let q be Point of TOP-REAL 2 , r be Point of TOP-REAL 2 ;
let p be Point of TOP-REAL 2 , r be Real ;
dist ( O , u ) <= |. p2 .| + 1 ;
assume dist ( x , b ) < dist ( a , b ) ;
<* SS *> is_the carrier of C-20 & <* N *> is_<* the carrier of C *> ;
i <= len Ga -' 1 & j + 1 <= width Ga -' 1 ;
let p be Point of TOP-REAL 2 , r be Real ;
x1 in the carrier of I[01] & x2 in the carrier of I[01] ;
set p1 = f /. i , p2 = f /. ( i + 1 ) ;
g in { g2 : r < g2 & g2 < r } ;
Q2 = SL " Q .= SL " Q ;
( ( 1 / 2 ) to_power ( 1 / 2 ) ) is summable ;
- p + I c= - p + A ;
n < LifeSpan ( P1 , s1 ) + 1 + 1 ;
CurInstr ( p1 , s1 ) = i .= halt SCM+FSA ;
A /\ Cl { x } \ { x } <> {} ;
rng f c= ]. r - 1 , r + 1 .[ ;
let g be Function of S , V ;
let f be Function of L1 , L2 , g be Function of L1 , L2 ;
reconsider z = z , t = t 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 , B :] :] ;
len ( the connectives of C ) = 4 & len ( the connectives of C ) = 3 ;
let C1 , C2 be subFunctor of C , D ;
reconsider V1 = V , V2 = V as Subset of X | B ;
attr p is valid means : Def3 : All ( x , p ) is valid ;
assume that X c= dom f and f .: X c= dom g and X c= dom f ;
H |^ ( a " ) is Subgroup of H & H |^ a is Subgroup of H ;
let A1 be $1 ) & A1 on E1 & A2 on E1 ;
p2 , r3 , q2 is_collinear & q2 , r2 , q3 is_collinear ;
consider x being element such that x in v ^ K ;
not x in { 0. TOP-REAL 2 } & not x in { 0. TOP-REAL 2 } ;
p in [#] ( ( I[01] | B11 ) | B11 ) ;
0 in M . ( E . n ) ;
op ( c ) , op ( c ) are_isomorphic ;
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . s2 ) /\ dom ( F . s2 ) ;
cluster -> Nat for w w of L ;
set i1 = the Nat , i2 = the carrier of S ;
let s be 0 -started State of SCM+FSA , I be Program of SCM+FSA ;
assume y in ( f1 union f2 ) .: A ;
f . len f = f /. len f .= p ;
x , f . x '||' f . x , f . y ;
pred X c= Y means : Def3 : cos | X c= cos | Y ;
let y be upper Subset of Y , x be Element of X ;
cluster -> -> as as Element of >= + i + 1 ;
set S = <* Bags n , ( Bags n ) , ( Bags n ) *> ;
set T = [. 0 , 1 / 2 .] ;
1 in dom mid ( f , 1 , 1 ) ;
( 4 * PI ) / 4 < ( 2 * PI ) / 4 ;
x2 in dom f1 /\ dom f & x2 in dom f1 /\ dom f ;
O c= dom I & { {} } = { {} } ;
( the Target of G ) . x = v & ( the Target 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 , t be Element of G opp ;
h19 . i = f . ( h . i ) ;
p `1 = p1 `1 & p `2 = p2 `2 or p `2 = p2 `2 ;
i + 1 <= len Cage ( C , n ) ;
len <* P *> @ = len P & len <* P *> = 1 ;
set N-26 = the non empty Subset of N ;
len gLet gLet + ( x + 1 ) - 1 <= x ;
a on B & b on B & not a 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 |^ n ;
set q2 = N-min L~ Cage ( C , n ) ;
set S = len S1 , T = len S2 , S = len S1 , T = len S2 ;
MaxADSet ( b ) c= MaxADSet ( P /\ Q ) ;
Cl ( G . q1 ) c= F . r2 & Cl ( G . q2 ) c= F . r2 ;
f " D meets h " V & f " D meets h " V ;
reconsider D = E as non empty directed Subset of L1 ;
H = ( the_left_argument_of H ) '&' ( the_right_argument_of H ) ;
assume t is Element of ( the carrier of S ) * ;
rng f c= the carrier of S2 & rng g c= the carrier of S2 ;
consider y being Element of X such that x = { y } ;
f1 . ( a1 , b1 ) = b1 & f1 . ( b1 , b2 ) = b2 ;
the carrier' of G ` = E \/ { E } ;
reconsider m = len thesis - k as Element of NAT ;
set S1 = LSeg ( n , UMP C ) , S2 = LSeg ( n , UMP C ) ;
[ i , j ] in Indices M1 & [ i , j ] in Indices M1 ;
assume that P c= Seg m and M is \HM { an } ;
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-7 . i = pp1 . i .= pp1 . i ;
let PA , G be a_partition of Y , z be Element of Y ;
pred 0 < r & r < 1 implies 1 < 1 / r ;
rng ( g1 . a , X ) = [#] X .= X ;
reconsider x = x , y = y , z = z as Element of K ;
consider k such that z = f . k and n <= k ;
consider x being element such that x in X \ { p } ;
len ( canFS s ) = card s .= card ( rng s ) ;
reconsider x2 = x1 , y2 = x2 as Element of L2 ;
Q in FinMeetCl ( the topology of X ) & Q c= the topology of X ;
dom fx0 c= dom ( u + v ) & dom fx0 = dom u ;
pred n divides m means : Def2 : m divides n & n = m ;
reconsider x = x , y = y as Point of [: I[01] , I[01] :] ;
a in ;
not y0 in the carrier of f & not y0 in the carrier of f ;
Hom ( ( a [: b , c :] , c ) , ( a , b ) ) <> {} ;
consider k1 such that p " < k1 and k1 < len f ;
consider c , d such that dom f = c \ d ;
[ x , y ] in [: dom g , dom k :] ;
set S1 = a1 +* ( y , z ) , S2 = a2 +* ( z , x ) ;
l2 = m2 & l1 = i2 & l2 = j2 implies l1 = i2
x0 in dom ( u01 ) /\ ( dom ( h + c ) ) ;
reconsider p = x , q = y as Point of TOP-REAL 2 ;
I[01] = R^1 | B01 .= ( TOP-REAL 2 ) | B01 ;
f . p4 <= f . f . p1 , f . p2 , P ;
( F `1 ) ^2 <= ( x `1 ) ^2 + ( x `2 ) ^2 ;
x `2 = ( W . ( len W ) ) `2 .= ( W . ( len W ) ) `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 ] and P [ succ a ] & P [ succ a ] ;
reconsider s, s, snon = s, snon empty where of D ;
( k - 1 ) <= len thesis - j ;
[#] S c= [#] the TopStruct of T & [#] S c= [#] T ;
for V being strict RealUnitarySpace holds V in and V in the carrier of W
assume k in dom mid ( f , i , j ) ;
let P be non empty Subset of TOP-REAL 2 , p1 , p2 be Point of TOP-REAL 2 ;
let A , B be square Matrix of n1 , K ;
- a * - b = a * b - a ;
for A being Subset of AS holds A // A implies A // C
( for o2 being Element of A st o2 in dom o2 holds o2 . o2 = <* o2 , o2 *> ) implies o1 = o2
then ||. x .|| = 0 & x = 0. X ;
let N1 , N2 be strict normal Subgroup of G , a be Element of G ;
j >= len ( upper_volume ( g , D1 ) ) + len ( upper_volume ( g , D1 ) ) ;
b = Q . ( len Qk - 1 ) ;
f2 (#) f1 /* s is divergent_to-infty & lim ( f2 (#) f1 ) = 0 ;
reconsider h = f * g as Function of N2 , G ;
assume that a <> 0 and delta ( a , b , c ) >= 0 ;
[ t , t ] in the InternalRel of A & [ t , t ] in the InternalRel of A ;
( v |-- E ) | n is Element of ( T . n ) * ;
{} = the carrier of L1 + L2 & {} = the carrier of L1 + L2 ;
Directed I is_closed_on Initialized s , P & Directed I is_halting_on Initialized s , P ;
Initialized p = Initialize ( p +* q ) , p = p +* q ;
reconsider N2 = N1 , N2 = N2 as strict net of R2 ;
reconsider Y ` = Y as Element of <* Ids L , \subseteq \rangle ;
"/\" ( ( uparrow p ) \ { p } , L ) <> p ;
consider j being Nat such that i2 = i1 + j and j in dom f ;
not [ s , 0 ] in the carrier of S2 & not [ s , 0 ] in the carrier of S2 ;
mm in ( B '/\' C ) '/\' D \ { {} } ;
n <= len ( ( P + Q ) ^ <* n *> ) ;
x1 `1 = x2 `1 & x1 `2 = x3 `2 or x1 `2 = x4 `2 ;
InputVertices S = { x1 , x2 } & InputVertices S = { x1 , x2 } ;
let x , y be Element of FTT1 ( n ) ;
p = |[ p `1 , p `2 ]| & p `2 = |[ p `2 , p `2 ]| ;
g * 1_ G = h " * g * h .= h " * g ;
let p , q be Element of is Element of PFuncs ( V , C ) ;
x0 in dom x1 /\ dom x2 & x0 in dom x1 /\ dom x2 ;
( R qua Function ) " = R " & ( R " ) " = R " ;
n in Seg len ( f /^ k ) & n in dom ( f /^ k ) ;
for s being 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 for for for for for for Seg n ;
1_ K * 1. ( K , n ) = 1_ K * 1. ( K , n ) ;
set S = Segm ( A , P1 , Q1 ) , T = Segm ( A , P1 , Q1 ) ;
ex w st e = ( w - f ) / 2 & w in F ;
curry ( ( P+* ( i , k ) ) # x ) is convergent ;
cluster open -> open for Subset of ( T | \sigma ) ;
len f1 = 1 .= len f3 + 1 .= len f3 + 1 .= len f3 + 1 ;
( i * p ) / p < ( 2 * p ) / p ;
let x , y be Element of OSSub U0 ;
b1 , c1 // b9 , c9 & o , c1 // o , c ;
consider p be element such that c1 . j = { p } ;
assume that f " { 0 } = {} and f is total ;
assume that IC Comput ( F , s , k ) = n and F . IC Comput ( F , s , k ) = 0 ;
Reloc ( J , card I ) does not ` not h " ;
( goto ( card I + 1 ) ) does not ` not h ;
set m3 = LifeSpan ( p3 , s3 ) , m3 = LifeSpan ( p3 , s3 ) ;
IC SCMPDS in dom Initialize p & IC SCMPDS in dom Initialize p ;
dom t = the carrier of SCM R & dom t = the carrier of SCM R ;
( E-max L~ f ) .. f = 1 & ( E-max L~ f ) .. f = 1 ;
let a , b be Element of thesis of thesis , f , g be Element of V ;
Cl ( union Int F ) c= Cl Int Cl ( union F ) ;
the carrier of X1 union X2 misses ( ( the carrier of X1 ) \/ the carrier of X2 ) ;
assume not LIN a , f . a , g . a , f . b ;
consider i being Element of M such that i = d6 and i in M ;
then Y c= { x } or Y = {} or Y = { x } ;
M , v / ( y , x ) |= H1 / ( y , x ) ;
consider m being element such that m in Intersect ( Fx0 ) and m in Y ;
reconsider A1 = support u1 , A2 = support u2 as Subset of X ;
card ( A \/ B ) = k-1 + ( 2 * 1 ) ;
assume that a1 <> a3 and a2 <> a4 and a3 <> a4 and a4 <> a5 ;
cluster s -\mathop { t } -> string of S ;
LW2 /. n2 = LW2 . n2 .= LW2 /. n2 .= LW2 /. n2 ;
let P be compact non empty Subset of TOP-REAL 2 , p1 , p2 be Point of TOP-REAL 2 ;
assume that r-7 in LSeg ( p1 , p2 ) and rp2 in LSeg ( p1 , p2 ) ;
let A be non empty compact Subset of TOP-REAL n , a be Real ;
assume that [ k , m ] in Indices DD1 and k + 1 <= len DD1 ;
0 <= ( ( 1 / 2 ) to_power p ) . ( p / 2 ) ;
( F . N ) | E8 . x = +infty ;
pred X c= Y & Z c= V implies X \ V c= Y \ Z ;
y `2 * ( z `2 ) * ( y `2 ) * ( z `2 ) <> 0. I ;
1 + card X-18 <= card u + card X-18 ;
set g = z \circlearrowleft E-max L~ z , 2 = ( E-max L~ z ) .. z ;
then k = 1 & p . k = <* x , y *> . k ;
cluster total for Element of C -carrier ( X ) ;
reconsider B = A as non empty Subset of TOP-REAL n , a be Real ;
let a , b , c be Function of Y , BOOLEAN , p be Function of Y , BOOLEAN ;
L1 . i = ( i .--> g ) . i .= g . i .= g . i ;
Plane ( x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 ) c= P ;
n <= indx ( D2 , D1 , j1 ) + 1 - 1 ;
( ( g2 ) . O ) `1 = - 1 & ( ( g2 ) . I ) `2 = 1 ;
j + p .. f - len f <= len f - len f ;
set W = W-bound C , S = E-bound C ;
S1 . ( a `1 , e `2 ) = a + e `2 .= a `2 ;
1 in Seg width ( M * ( ColVec2Mx p ) ) ;
dom ( i (#) Im f ) = dom Im f /\ dom Im f ;
( \mathbb x ) `2 = W . ( a , *' ( a , p ) ) ;
set Q = non empty set , g = ( \models g , f , h ) ;
cluster -> MS} for ManySortedSet of U1 * ( MS' A ) ;
attr F = { A } means : Def3 : F is discrete ;
reconsider z9 = \hbox { z } as Element of product \overline G ;
rng f c= rng f1 \/ rng f2 & rng f1 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 , H ) implies E , j |= H
reconsider n1 = n , n2 = m , n1 = n as Morphism of o1 , o2 ;
assume P is idempotent & R is idempotent & P ** R = R ** P ;
card ( B2 \/ { x } ) = k-1 + 1 ;
card ( ( x \ B1 ) /\ ( x \ B2 ) ) = 0 ;
g + R in { s : g-r < s & s < g + r } ;
set q-112 = ( q , <* s *> ) -\subseteq ( q , <* s *> ) -\subseteq ;
for x being element st x in X holds x in rng f1 implies x in X
h0 /. ( i + 1 ) = h0 . ( i + 1 ) ;
set mw = max ( B , ( } , 1 ) --> 0 ) ;
t in Seg width ( I ^ ( n , n ) ) ;
reconsider X = dom f , Y = rng f as Element of Fin NAT ;
IncAddr ( i , k ) = <% l , k %> + k ;
( for q being Point of TOP-REAL 2 st q in L~ f holds q `2 <= ( q `2 ) * ( 1 + 1 ) `2 ) implies q `2 = ( q `2 ) * ( 1 + 1 ) `2
attr R is condensed means : Def3 : for A being Subset of R st A is condensed holds Cl A is condensed ;
pred 0 <= a & a <= 1 & b <= 1 implies a * b <= 1 ;
u in ( ( c /\ ( d /\ b ) ) /\ e ) /\ f /\ j ;
u in ( ( c /\ ( d /\ e ) ) /\ f ) /\ j ;
len C + - 2 >= 9 + - 3 ;
x , z , y is_collinear & x , z , x is_collinear implies x = y
a |^ ( n1 + 1 ) = a |^ n1 * a .= a |^ n1 * a ;
<* \underbrace ( 0 , \dots 0 , 0 *> *> in Line ( x , a * x ) ;
set yy1 = <* y , c *> ;
FF2 /. 1 in rng Line ( D , 1 ) ;
p . m Joins r /. m , r /. ( m + 1 ) , G ;
p `2 = ( f /. i1 ) `2 .= ( f /. i1 ) `2 .= ( f /. i1 ) `2 ;
W-bound ( X \/ Y ) = W-bound ( X \/ Y ) .= W-bound ( X \/ Y ) ;
0 + p `2 <= 2 * r + p `2 + p `2 - p `2 ;
x in dom g & not x in g " { 0 } ;
f1 /* ( seq ^\ k ) is divergent_to-infty & ( f1 /* ( seq ^\ k ) ) . n = lim ( f1 /* ( seq ^\ k ) ) ;
reconsider u2 = u , v2 = v as VECTOR of P`1 , X ;
p |-count ( Product Sgm X11 ) = 0 & p |-count ( Sgm X11 ) = 0 ;
len <* x *> < i + 1 & i <= len c + 1 ;
assume that I is non empty and { x } /\ { y } = { 0. I } ;
set ii2 = ( card I + 4 ) .--> goto 0 ;
x in { x , y } & h . x = {} ( Tx , y ) ;
consider y being Element of F such that y in B and y <= x ` ;
len S = len ( the charact of A0 ) & len ( the charact of A0 ) = len the charact of A0 ;
reconsider m = M , i = I , n = N as Element of X ;
A . ( j + 1 ) = B . ( j + 1 ) \/ A . j ;
set N8 = : G = : G = ( G . e ) `1 ;
rng F c= the carrier of gr { a } & F is one-to-one ;
Comput ( Q , t , n ) is FinSequence & Q is o implies Q is o \bf
f . k , f . ( \mathop { \rm mod n ) ] in rng f ;
h " P /\ [#] T1 = f " P /\ [#] T2 .= f " P /\ [#] T1 ;
g in dom f2 \ f2 " { 0 } & g in dom f2 \ f2 " { 0 } ;
g1X /\ dom f1 = g1 " X /\ dom f1 .= dom g1 ;
consider n being element such that n in NAT and Z = G . n ;
set d1 = \bf dist ( x1 , y1 ) , d2 = dist ( x2 , y2 ) ;
b `2 `2 + 1 / 2 < 1 / 2 + 1 / 2 ;
reconsider f1 = f as VECTOR of the carrier of X , Y ;
pred i <> 0 means : Def2 : i ^2 mod ( i + 1 ) = 1 ;
j2 in Seg len ( g2 . i2 ) & j2 + 1 in Seg len ( g2 . i2 ) ;
dom ( i ) = dom ( i ) .= dom ( i ) .= dom ( i ) ;
cluster sec | ]. PI / 2 , PI / 2 .[ -> one-to-one ;
Ball ( u , e ) = Ball ( f . p , e ) ;
reconsider x1 = x0 , y1 = x0 , y2 = x0 , y1 = x0 , y2 = x0 ;
reconsider R1 = x , R2 = y , R1 = z as Relation of L ;
consider a , b being Subset of A such that x = [ a , b ] ;
( <* 1 *> ^ p ) ^ <* n *> in Rn ;
S1 +* S2 = S2 +* ( S1 +* S2 ) .= S2 +* ( S1 +* S2 ) ;
( ( ( 1 / 2 ) (#) ( cos * f1 ) ) `| Z ) = f ;
cluster -> continuous for Function of C , REAL , f be Function of C , REAL ;
set C7 = 1GateCircStr ( <* z , x *> , f3 ) , C7 = 1GateCircStr ( <* z , x *> , f3 ) ;
Eex e be Element of E8 , T be ( e2 , e ) -carrier of G ;
( ( arctan (#) ln ) `| Z ) = ( ( arctan (#) ln ) `| Z ) ;
upper_bound A = PI * 3 / 2 & lower_bound A = 0 ;
F . ( dom f , - g ) is_transformable_to F . ( cod f , - g ) ;
reconsider pNAT = q`2 , pNAT = q`2 as Point of TOP-REAL 2 ;
g . W in [#] Y0 & [#] Y0 c= [#] Y0 & g . W in [#] Y0 ;
let C be compact non vertical non horizontal Subset of TOP-REAL 2 ;
LSeg ( f ^ g , j ) = LSeg ( f , j ) /\ LSeg ( g , j ) ;
rng s c= dom f /\ ]. x0 - r , x0 .[ & rng s c= dom f /\ ]. x0 , x0 + r .[ ;
assume x in { idseq 2 , Rev ( idseq 2 ) } ;
reconsider n2 = n , m2 = m , m2 = n + 1 as Element of NAT ;
for y being ExtReal st y in rng seq holds g <= y implies g <= y
for k st P [ k ] holds P [ k + 1 ] ;
m = m1 + m2 .= m1 + m2 .= m1 + m2 .= m1 + m1 + m2 ;
assume for n holds H1 . n = G . n -H . n ;
set Bf = f .: the carrier of X1 , Bf = f .: the carrier of X2 ;
ex d being Element of L st d in D & x << d ;
assume that R -Seg ( a ) c= R -Seg ( b ) and R -Seg ( a ) c= R -Seg ( b ) ;
t in ]. r , s .[ or t = r or t = s or t = s ;
z + v2 in W & x = u + ( z + v2 ) ;
x2 |-- y2 iff P [ x2 , y2 ] & not ( x2 = y2 or x2 = y2 ) ;
pred x1 <> x2 means : Def2 : |. x1 - x2 .| > 0 & |. x1 - x2 .| > 0 ;
assume that p2 - p1 , p3 - p1 - p3 - p1 , p3 - p1 - p3 is_collinear and p2 - p3 , p3 - p1 - p3 + p1 + p3 - p1 is_collinear ;
set q = ( x0 , f ) ^ <* 'not' A *> ;
let f be PartFunc of REAL-NS 1 , REAL-NS 1 , g be PartFunc of REAL-NS 1 , REAL-NS 1 , REAL-NS 1 ;
( n mod ( 2 * k ) ) + 1 = n mod k ;
dom ( T * ( succ t ) ) = dom ( T * ( succ t ) ) ;
consider x being element such that x in wf iff x in c & x in f ;
assume ( F * G ) . ( v . x3 ) = v . x4 ;
assume that the carrier of D1 c= the carrier of D2 and for x being Element of D1 holds x in the carrier of D2 ;
reconsider A1 = [. a , b .[ , A2 = [. a , b .] as Subset of R^1 ;
consider y being element such that y in dom F and F . y = x ;
consider s being element such that s in dom o and a = o . s ;
set p = W-min L~ Cage ( C , n ) , q = E-max L~ Cage ( C , n ) ;
n1 -' len f + 1 <= len ( len ( g | 1 ) ) + 1 ;
Seg ( q , O1 ) = [ u , v , a , b , b , c , d ] ;
set C-2 = ( ( ( n + 1 ) `1 ) `1 ) ;
Sum ( L (#) p ) = 0. R * Sum p .= 0. V * p .= p ;
consider i being element such that i in dom p and t = p . i ;
defpred Q [ Nat ] means 0 = Q ( $1 ) & P [ $1 ] ;
set s3 = Comput ( P1 , s1 , k ) , P3 = P1 , s4 = P1 , P4 = P1 , P4 = P1 , P4 = P2 ;
let l be variable of k , A , A1 be Subset of A ;
reconsider U2 = union G-24 , GN = union GN as Subset-Family of ( T | A ) ;
consider r such that r > 0 and Ball ( p `1 , r ) c= Q ` ;
( h | ( n + 2 ) ) /. ( i + 1 ) = p29 ;
reconsider B = the carrier of X1 , C = the carrier of X2 as Subset of X ;
p$ c = <* - vs , 1 , - 1 *> .= <* - vs , - 1 , - 1 *> ;
synonym f is real-valued means : Def2 : rng f c= NAT & for x being element st x in NAT holds f . x = f . x ;
consider b being element such that b in dom F and a = F . b ;
x9 < card X0 + card Y0 & x9 in card Y0 + card Y0 + card Y0 + 1 ;
pred X c= B1 means : Def3 : for A st A c= B holds ( X c= succ B1 ) & ( X c= B implies X c= A ) ;
then w in Ball ( x , r ) & dist ( x , w ) <= r ;
angle ( x , y , z ) = angle ( x-y , 0 , PI ) ;
pred 1 <= len s means : Def3 : for i being Element of NAT st i in dom s holds ( the mapping of s ) . i = s ;
fz c= f . ( k + ( n + 1 ) ) ;
the carrier of { 1_ G } = { 1_ G } & the carrier of { 1_ G } = { 1_ G } ;
pred p '&' q in \cdot ( p '&' q ) means : Def3 : q '&' p in * ( p '&' q ) ;
- ( t `1 ) < ( t `1 ) / 2 & - ( t `2 ) < - ( t `1 ) / 2 ;
UA . 1 = U2 /. 1 .= ( W /. 1 ) `1 .= ( W /. 1 ) `1 .= W /. 1 ;
f .: the carrier of x = the carrier of x & f .: the carrier of x = the carrier of x ;
Indices OO = [: Seg n , Seg n :] & Indices OO = [: Seg n , 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_' Aand f . x = F ( f , x ) ;
[ 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 ]| - b <> 0. TOP-REAL 2 & |[ w1 , v1 ]| - b = 0. TOP-REAL 2 ;
reconsider t = t as Element of INT * , ( the carrier of X ) * ;
C \/ P c= [#] ( GX | ( [#] GX \ A ) ) & C /\ A = {} ( GX | A ) ;
f " V in the topology of X /\ D . ( the carrier of C , the carrier of C ) ;
x in [#] ( ( the carrier of A ) /\ ( the carrier of B ) ) ;
g . x <= h1 . x & h . x <= h1 . x ;
InputVertices S = { xy , y , z } & InputVertices S = { xy , y , z } ;
for n be Nat st P [ n ] holds P [ n + 1 ] ;
set R = Line ( M , i ) * Line ( M , i ) ;
assume that M1 is being_line and M2 is being_line and M3 is being_line and M3 is being_line and M2 is being_line ;
reconsider a = f4 . ( i0 -' 1 ) , b = f4 . ( i0 -' 1 ) as Element of K ;
len B2 = Sum Len ( F1 ^ F2 ) .= len ( Len F1 ^ width F2 ) + len ( Len F2 ) ;
len ( ( the FinSequence of n ) * ( i , j ) ) = n & len ( ( the FinSequence of n ) * ( i , j ) ) = n ;
dom max ( - ( f + g ) , f + g ) = dom ( f + g ) ;
( the + H ) . n = upper_bound Y1 & ( the + H ) . n = upper_bound Y1 ;
dom ( p1 ^ p2 ) = dom f12 & dom ( p1 ^ p2 ) = dom f12 ;
M . [ 1 , y ] = 1 / ( 1 - y ) * v1 .= y ;
assume that W is non trivial and W .vertices() c= the carrier' of G2 and not W is Vertex of G2 ;
godo /. i1 = G1 * ( i1 , i2 ) & card L~ godo = 1 & card L~ godo = 1 ;
C8 |- 'not' Ex ( x , p ) 'or' p . ( x , y ) ;
for b st b in rng g holds lower_bound rng f\lbrace b , b } <= b
- ( ( q1 `1 / |. q1 .| - sn ) / ( 1 + sn ) ) = 1 ;
( LSeg ( c , m ) \/ [: l , k :] ) \/ [: l , k :] c= R ;
consider p be element such that p in x and p in L~ f and p in L~ f ;
Indices ( X @ ) = [: Seg n , Seg 1 :] & Indices ( X @ ) = [: Seg n , Seg 1 :] ;
cluster s => ( q => p ) => ( q => ( s => p ) ) -> valid ;
Im ( ( Partial_Sums F ) . m ) is_measurable_on E & Im ( ( Partial_Sums F ) . m ) is_measurable_on E ;
cluster f . ( x1 , x2 ) -> Element of D * ;
consider g being Function such that g = F . t and Q [ t , g ] ;
p in LSeg ( N-min Z , \mathop { \rm \hbox { - } corner Z } , p1 ) /\ LSeg ( p1 , p2 ) ;
set R8 = R .: ]. b , +infty .[ ;
IncAddr ( I , k ) = SubFrom ( da , da ) .= SubFrom ( da , db ) ;
seq . m <= ( the + seq ) . k & ( the + seq ) . m <= ( the + seq ) . k ;
a + b = ( a ` *' b ) ` .= ( a ` *' b ) ` .= a ` ;
id ( X /\ Y ) = id ( X /\ id Y ) .= id ( X /\ Y ) ;
for x being element st x in dom h holds h . x = f . x ;
reconsider H = U1 \/ U2 , H = U2 \/ U1 as non empty Subset of U0 ;
u in ( c /\ ( ( d /\ e ) /\ b ) ) /\ m /\ j ;
consider y being element such that y in Y and P [ y , lower_bound B ] ;
consider A being finite stable set of R such that card A = ( the carrier of R ) and card A = card A ;
p2 in rng ( f |-- p1 ) \ rng <* p1 *> & p2 in rng <* p1 *> ;
len s1 - 1 > 1-1 & len s2 - 1 > 0 or len s2 - 1 > 0 ;
( N-min P ) `2 = N-bound P & ( N-min P ) `2 = N-bound P ;
Ball ( e , r ) c= LeftComp Cage ( C , k + 1 ) & LeftComp Cage ( C , k + 1 ) c= LeftComp Cage ( C , k + 1 ) ;
f . a1 ` = f . a1 ` .= f . a1 ` .= ( f | ( a1 ` ) ) . a1 ;
( seq ^\ k ) . n in ]. -infty , x0 + r .[ /\ dom ( f1 (#) f2 ) ;
gg . s0 = g . s0 | G . s0 .= g . s0 ;
the InternalRel of S is symmetric & the InternalRel of S is transitive implies the InternalRel of S is transitive
deffunc F ( Ordinal , Ordinal ) = phi . $2 & phi . $2 = phi . $2 ;
F . s1 . a1 = F . s2 . a1 .= F . a1 .= s . a1 ;
x `2 = A . o . a .= Den ( o , A . a ) ;
Cl ( f " P1 ) c= f " ( Cl P1 ) & f " P1 c= f " ( Cl P1 ) ;
FinMeetCl ( the topology of S ) c= the topology of T & the topology of S c= the topology of T ;
synonym o is \bf means : Def2 : o <> \ast & o <> * & o <> * ;
assume that X = Y |^ + and card X <> card Y and X <> Y and Y <> {} ;
the such that the { s } <= 1 + ( the { s } ) & s <= 1 + ( the { s } ) ;
LIN a , a1 , d or b , c // b1 , c1 or LIN a , c , d ;
e /. 1 = 0 & e /. 2 = 1 & e /. 3 = 0 ;
Ef in SS1 & not Ef in { Nf } ;
set J = ( l , u ) If , K = ( l , u ) If , L = ( l , u ) If , L = ( l , u ) If ;
set A1 = Y , A2 = Y , A1 = A1 +* A2 , A2 = A2 +* A1 , A2 = A1 +* A2 ;
set vs = [ <* vs , c *> , '&' ] , f3 = [ <* V , m *> , '&' ] , f4 = [ <* c , m *> , '&' ] ;
x * z `2 * x " in x * ( z * N ) * x " ;
for x being element st x in dom f holds f . x = g3 . x & f . x <> g3 . x
Int cell ( f , 1 , G ) c= RightComp f \/ L~ f \/ RightComp f \/ RightComp f ;
UA is_an_arc_of W-min C , E-max C & \subseteq L~ Cage ( C , n ) /\ L~ Cage ( C , n ) = { E-max C } ;
set f-17 = f @ g "/\" g @ @ @ f ;
attr S1 is convergent means : Def2 : S2 is convergent & lim ( S1 - S2 ) = 0 ;
f . ( 0 + 1 ) = ( 0 qua Ordinal ) + a .= a ;
cluster -> \in -> \in -> \in reflexive transitive transitive reflexive transitive reflexive transitive for non empty RelStr , the carrier of L , the carrier of L ;
consider d being element such that R reduces b , d and R reduces c , d and R reduces d , c ;
not b in dom Start-At ( ( card I + 2 ) , SCMPDS ) & not b in dom Start-At ( ( card I + 2 ) , SCMPDS ) ;
( z + a ) + x = z + ( a + y ) .= z + a + y ;
len ( l (#) ( a |^ 0 ) ) = len l & len ( l (#) ( a |^ 0 ) ) = len l ;
t4 ^ {} is ( {} \/ rng t4 ) -valued FinSequence ;
t = <* F . t *> ^ ( C . p ^ q ) .= <* F . t *> ^ q ;
set p-2 = W-min L~ Cage ( C , n ) , pi = W-min L~ Cage ( C , n ) , pi = W-min L~ Cage ( C , n ) ;
( k -' ( i + 1 ) ) = ( k - ( i + 1 ) ) - ( i + 1 ) ;
consider u being Element of L such that u = u ` ` and u in D ` ;
len ( ( width aG ) |-> a ) = width aG & width ( ( width aG ) |-> a ) = width aG ;
FM . x in dom ( ( G * the_arity_of o ) . x ) & FM . x = dom ( G * the_arity_of o ) ;
set cH2 = the carrier of H2 , cH1 = the carrier of H1 ;
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 ;
IC Comput ( P3 , t , k ) = ( l + 1 ) + 1 .= ( l + 1 ) ;
dom ( ( cos * sin ) `| Z ) = REAL & dom ( ( cos * sin ) `| Z ) = dom f ;
cluster <* l *> ^ phi -> ( 1 + 1 ) -element for string of S ;
set b5 = [ <* that p , { 1 } , f1 ] , b5 = [ <* p , f1 *> , f2 ] ;
Line ( Segm ( M @ , P , Q ) , x ) = L * Sgm Q .= L ;
n in dom ( ( the Sorts of A ) * the_arity_of o ) & dom ( ( the Sorts of A ) * the_arity_of o ) = dom the_arity_of o ;
cluster f1 + f2 -> continuous for PartFunc of REAL , the carrier of S ;
consider y be Point of X such that a = y and ||. x-y .|| <= r ;
set x3 = being Element of Q . DataLoc ( s2 . SBP , 2 ) , x4 = Comput ( P3 , s3 , 2 ) , x4 = P3 ;
set p-3 = stop I ( ) , p-3 = stop I ( ) ;
consider a being Point of D2 such that a in W1 and b = g . a and a <= b ;
{ A , B , C , D , E } = { A , B } \/ { C , D , E } ;
let A , B , C , D , E , F , J , M , N , M be set ;
|. p2 .| ^2 - ( p2 `2 ) ^2 >= 0 & |. p2 .| ^2 - ( p2 `2 ) ^2 >= 0 ;
l -' 1 + 1 = n-1 * ( l + 1 ) + ( mm + 1 ) ;
x = v + ( a * w1 + b * w2 ) + ( c * w2 + c * w2 ) ;
the TopStruct of L = , the TopStruct of L = [: the topology of L , the topology of L :] ;
consider y being element such that y in dom H1 and x = H1 . y and y in H1 . x ;
ff \ { n } = ( Free All ( v1 , H ) ) \/ ( Free All ( v1 , H ) ) ;
for Y being Subset of X st Y is summable & Y is summable holds Y is iff Y is Sum
2 * n in { N : 2 * Sum ( p | N ) = N & N > 0 } ;
for s being FinSequence holds len ( the { + } * } ) = len s & len ( the { + } * s ) = len s
for x st x in Z holds exp_R * f is_differentiable_in x & ( exp_R * f ) . x > 0
rng ( h2 (#) f2 ) c= the carrier of ( ( TOP-REAL 2 ) | ( the carrier of TOP-REAL 2 ) ) ;
j + ( len f ) <= len f + ( len len g - len f ) - len f ;
reconsider R1 = R * I as PartFunc of REAL , REAL-NS n , REAL-NS n , REAL-NS n ;
C8 . x = s1 . x0 .= C8 . x .= C8 . x .= ( C * f ) . x ;
power ( F_Complex ) . ( z , n ) = 1 .= x |^ n .= x |^ n ;
t at at ( C , s ) = f . ( the connectives of S ) . t .= s ;
support ( f + g ) c= support f \/ ( C \/ support g ) & support ( f + g ) = support f \/ support g ;
ex N st N = j1 & 2 * Sum ( seq1 | N ) > N & N > 0 ;
for y , p st P [ p ] holds P [ All ( y , p ) ] ;
{ [ x1 , x2 ] where x1 is Point of [: X1 , X2 :] : x1 in X } c= [: X1 , X2 :]
h = ( i = j |-- h , id B . i ) .= H . i .= H . i ;
ex x1 being Element of G st x1 = x & x1 * N c= A & x1 in N ;
set X = ( ( \lbrace q , O1 , L ) `1 , { q , 4 } } ) `1 , Y = { q , p } ;
b . n in { g1 : x0 < g1 & g1 < a1 . n & g1 < x0 + r } ;
f /* s1 is convergent & f /. x0 = lim ( f /* s1 ) implies f /* s1 is convergent & lim ( f /* s1 ) = lim ( f /* s1 )
the carrier of the lattice of Y = the carrier of the topology of Y & the carrier of Y = the carrier of the topology of Y ;
'not' ( a . x ) '&' b . x 'or' a . x '&' 'not' ( b . x ) = FALSE ;
2 = len ( q0 ^ r1 ) + len q1 .= len ( p ^ q ) + len q1 .= len p + len q ;
( 1 / a ) (#) ( sec * f1 ) - id Z is_differentiable_on Z ;
set K1 = upper_volume ( lim ( H , H ) , x0 ) , K1 = integral ( H , x0 ) , K1 = lim ( H , x0 ) ;
assume e in { ( w1 - w2 ) `1 : w1 in F & w2 in G } ;
reconsider d7 = dom a `1 , d6 = dom F `1 , d6 = dom G `1 as finite set ;
LSeg ( f /^ q , j ) = LSeg ( f , j ) \/ LSeg ( f , j + q .. f ) ;
assume X in { T . ( N2 , N2 ) : h . N2 = N2 } ;
assume that Hom ( d , c ) <> {} and <* f , g *> * f1 = <* f , g *> * f2 ;
dom S\cdot = dom S /\ Seg n .= dom L6 .= Seg n /\ Seg n .= dom L6 .= Seg n /\ Seg n .= dom L6 ;
x in H |^ a implies ex g st x = g |^ a & g in H & g in H
a * ( 0. ( Z , n ) ) = a `2 - ( 0 * n ) .= a `2 - ( 0 * n ) .= a `2 ;
D2 . j in { r : lower_bound A <= r & r <= D1 . i } ;
ex p being Point of TOP-REAL 2 st p = x & P [ p ] & p `2 >= 0 & p <> 0. TOP-REAL 2 ;
for c holds f . c <= g . c implies f @ @ g ;
dom ( f1 (#) f2 ) /\ X c= dom ( f1 (#) f2 ) /\ X & dom ( f1 (#) f2 ) /\ X c= dom f1 /\ dom f2 ;
1 = ( p * p ) / p .= p * ( p / p ) .= p * 1 .= p ;
len g = len f + len <* x + y *> .= len f + 1 .= len f + 1 + 1 ;
dom F-11 = dom ( F | ( N1 , S-23 ) ) .= [: the carrier of S , the carrier of S :] ;
dom ( f . t * I . t ) = dom ( f . t * g . t ) ;
assume a in ( "\/" ( ( T |^ the carrier of S ) , T ) ) .: D ;
assume that g is one-to-one and ( the carrier' of S ) /\ rng g c= dom g and g is one-to-one ;
( ( x \ y ) \ z ) \ ( ( x \ z ) \ ( y \ z ) ) = 0. X ;
consider f such that f * f `1 = id b and f * f `2 = id a and f * g = id b ;
( cos | [. 2 * PI * 0 , PI + 2 * PI * 0 .] ) is increasing ;
Index ( p , co ) <= len LS - Gij .. LS - Gij .. LS + 1 - LS .. LS ;
let t1 , t2 , t3 , t2 , t2 , t1 , t2 , t2 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2
( the mapping of ( ( curry H ) . h ) ) . h <= ( the mapping of ( ( curry G ) . h ) ) . ( ( ( Frege H ) . h ) . ( ( Frege H ) . h ) ) ;
then P [ f . i0 ] & F ( f . ( i0 + 1 ) ) < j ;
Q [ ( 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 { of G . i , G . i } ;
the Sorts of A2 = ( the carrier of S2 ) --> BOOLEAN .= ( the carrier of S1 ) --> TRUE .= the Sorts of A1 ;
consider s being Function such that s is one-to-one & dom s = NAT & rng s = F and for n being Nat st n in NAT holds P [ n , s . n ] ;
dist ( b1 , b2 ) <= dist ( b1 , a ) + dist ( a , b2 ) + dist ( a , b2 ) ;
( Lower_Seq ( C , n ) /. len Lower_Seq ( C , n ) ) /. 1 = ( W /. 1 ) `1 ;
q `2 <= ( UMP C ) `2 & ( UMP C ) `2 <= ( UMP C ) `2 ;
LSeg ( f | i2 , i ) /\ LSeg ( f | i2 , j ) = {} ;
given a being ExtReal such that a <= IB and A = ]. a , IB .[ and a <= IB ;
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 <= b } , Y = { b |^ n where n is Element of NAT : n <= b } ;
( ( x * y * z \ x ) \ z ) \ ( x * y \ x ) = 0. X ;
set xy = [ <* xy , y , z *> , f1 ] , yz = [ <* y , z *> , f2 ] , xy = [ <* z , x *> , f3 ] , yz = [ <* z , x *> , f3 ] ;
Ul /. len ll = ll . len ll .= ll /. len ll .= ll /. len ll ;
( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 = 1 ;
( ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) ^2 < 1 ;
( ( for x st x in X \/ Y holds x `2 <= ( ( for x st x in X holds x in Y ) ) implies x in Y
( ss1 - ss2 ) . k = ss1 . k - ss2 . k .= ss2 . k - ss2 . k ;
rng ( ( h + c ) ^\ n ) c= dom SVF1 ( 1 , f , u0 ) /\ dom SVF1 ( 1 , f , u0 ) ;
the carrier of X = the carrier of X & the carrier of X = the carrier of X implies X is non empty
ex p4 st p3 = p4 & |. p4 - |[ a , b ]| .| = r & p4 `2 <= b & p4 `2 <= d ;
set ch = chi ( X , A5 ) , A5 = chi ( X , A5 ) ;
R |^ ( 0 * n ) = I\HM ( X , X ) .= R |^ n |^ 0 .= R |^ n |^ 0 ;
( ( ( curry ( F-19 , n ) ) . 0 ) . x ) . x is nonnegative ;
f2 = C7 . ( E7 , len ( V ) - 1 ) .= C7 . ( len ( V ) - 1 ) ;
S1 . b = s1 . b .= s2 . b .= s2 . b .= s2 . b .= s2 . b .= s2 . b ;
p2 in LSeg ( p2 , p1 ) /\ LSeg ( p2 , p1 ) or p2 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ;
dom ( f . t ) = Seg n & dom ( I . t ) = Seg n & dom ( I . t ) = Seg n ;
assume o = ( the connectives of S ) . 11 & o = ( the connectives of S ) . 12 ;
set phi = ( l1 , l2 ) \mathop { l1 } , phi = ( l1 , l2 ) \mathop { l2 } ;
synonym p is is is is is / for p , q , T ;
Y1 `2 = - 1 & 0. TOP-REAL 2 <> 0. TOP-REAL 2 implies Y1 `2 = - 1 & Y1 `2 = - 1 & Y1 `2 = - 1
defpred X [ Nat , set , set ] means P [ $2 , $2 , $2 ] & $2 = $2 ;
consider k be Nat such that for n be Nat st k <= n holds s . n < x0 + g ;
Det ( I |^ ( m -' n ) ) = 1. ( K , m ) * ( 1. ( K , n ) ) .= 1. ( K , m ) ;
( - b - sqrt ( b ^2 - 4 * a * c ) ) / 2 < 0 ;
Cd . d = Cd . da mod Cd . db .= Cd . da mod Cd . db ;
attr X1 is dense means : Def3 : X2 is dense dense & X1 is dense SubSpace of X & X2 is dense SubSpace of X ;
deffunc F6 ( Element of E , Element of I ) = $1 * $2 & $2 = ( $1 * $2 ) * $2 ;
t ^ <* n *> in { t ^ <* i *> : Q [ i , T . t ] } ;
( x \ y ) \ x = ( x \ x ) \ y .= y ` .= 0. X ;
for X being non empty set for Y being Subset-Family of X holds for X being Subset-Family of [: X , \mathop { the carrier of Y } :] holds X is Basis of Y
synonym A , B are_separated means : Def2 : Cl A misses Cl B & A misses Cl B & B misses Cl A ;
len ( M @ ) = len p & width ( M @ ) = width ( M @ ) & width ( M @ ) = width ( M @ ) ;
J . v = { x where x is Element of K : 0 < v . x & v . x < 1 } ;
( Sgm ( Seg m ) ) . d - ( Sgm ( Seg m ) ) . e <> 0 ;
lower_bound divset ( D2 , k + k2 ) = D2 . ( k + k2 - 1 ) .= D2 . ( k + k2 - 1 ) ;
g . r1 = - 2 * r1 + 1 & dom h = [. 0 , 1 .] & rng h c= [. 0 , 1 .] ;
|. a .| * ||. f .|| = 0 * ||. f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| ;
f . x = ( h . x ) `1 & g . x = ( h . x ) `2 ;
ex w st w in dom B1 & <* 1 *> ^ s = <* 1 *> ^ w & <* 1 *> ^ s = <* 1 *> ^ w ;
[ 1 , {} , <* d1 *> ] in ( { [ 0 , {} , {} ] } \/ S1 ) \/ S2 ;
IC Exec ( i , s1 ) + n = IC Exec ( i , s2 ) .= IC Exec ( i , s2 ) ;
IC Comput ( P , s , 1 ) = succ IC s .= 5 + 9 .= 5 + 9 .= 5 ;
( IExec ( W6 , Q , t ) ) . intpos ( e + 2 ) = t . intpos ( e + 2 ) ;
LSeg ( f /^ q , i ) misses LSeg ( f /^ q , j ) \/ LSeg ( f /^ q , j ) ;
assume for x , y being Element of L st x in C & y in C holds x <= y or y <= x ;
integral ( f , C , f ) = 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 . h ) .|| < e1 * ( K + 1 * ||. h .|| ) ;
assume a in { q where q is Element of M : dist ( z , q ) <= r } ;
set p4 = [ 2 , 1 ] .--> [ 2 , 0 , 1 ] ;
consider x , y being Subset of X such that [ x , y ] in F and x c= d and y \not c= d ;
for y , x being Element of REAL st y ` in Y ` & x in X ` holds y <= x ` & x <= x ` ;
func |. \bullet p .| -> variable of A equals min ( NBI ( p ) , p ) ;
consider t being Element of S such that x `1 , y `2 '||' z `1 , t `2 and x `1 , z `2 '||' y `1 , t `2 ;
dom x1 = Seg len x1 & len x1 = len l1 & for i st i in Seg len x1 holds x1 /. i = x1 . i * ( x1 /. i ) ;
consider y2 being Real such that x2 = y2 and 0 <= y2 and y2 < 1 / 2 and y2 <= 1 / 2 ;
||. f | X /* s1 .|| = ||. f .|| | X & ||. f .|| | X is convergent & lim ( ||. f .|| | X ) = ||. f .|| ;
( the InternalRel of A ) -Seg ( x ` ) /\ Y = {} \/ {} .= {} \/ {} .= {} \/ {} .= {} ;
assume i in dom p & for j be Nat st j in dom q holds P [ i , j ] & i + 1 in dom p & j + 1 in dom p ;
reconsider h = f | X ( ) as Function of X ( ) , rng f ( ) ;
u1 in the carrier of W1 & u2 in the carrier of W2 implies ( W1 + W2 ) + ( W2 + W3 ) = W1 + W2 + W3
defpred P [ Element of L ] means M <= f . $1 & f . $1 <= $1 & f . $1 <= f . $1 ;
^ ( u , a , v ) = s * x + ( - ( s * x + y ) ) .= b - s + y .= b ;
- ( x-y ) = - x + - y .= - x + y .= - x + y .= - x + y ;
given a being Point of GX such that for x being Point of GX holds a , x are_\HM { x } and a , x are_\HM { x } ;
fSet = [ [ dom @ f2 , cod @ f2 ] , h2 ] .= [ [ cod @ f2 , cod @ g2 ] , h2 ] ;
for k , n be Nat st k <> 0 & k < n & n is prime holds k , n are_relative_prime & k , n are_relative_prime
for x being element holds x in A |^ d iff x in ( ( A ` ) |^ f ) ` & x in ( A ` ) `
consider u , v being Element of R , a being Element of A such that l /. i = u * a * v ;
( - ( p `1 / |. p .| - cn ) ) / ( 1 + cn ) > 0 ;
L-13 . k = Lk . ( F . k ) & F . k in dom ( Lk | dom F ) ;
set i2 = SubFrom ( a , i , - n ) , i2 = SubFrom ( a , i , - n ) , i1 = i2 ;
attr B is thesis means : Def3 : for S being SubSub of B holds S = B `1 & S = B `2 ;
a9 "/\" D = { a "/\" d where d is Element of N : d in D } & a "/\" b in D ;
|( exp_R , q )| * |( exp_R , q )| * |( exp_R , q )| >= |( exp_R , q )| * |( exp_R , q )| .= |( exp_R , q )| * |( exp_R , q )| ;
( - f ) . ( sup A ) = ( ( - f ) | A ) . sup A .= - ( f | A ) . sup A ;
GG2 /. k = ( ( G * ( len G , k ) ) `1 ) `1 .= G * ( len G , k ) `1 .= G * ( 1 , k ) `1 ;
( Proj ( i , n ) * ( L . x ) ) = <* ( proj ( i , n ) ) . ( L . x ) *> ;
f1 + f2 (#) reproj ( i , x ) is_differentiable_in ( the reproj of i , x ) . x & f2 + f1 is_differentiable_in x ;
pred ( for x st x in Z holds ( tan . x ) <> 0 ) & ( for x st x in Z holds ( tan . x ) <> 0 ) ;
ex t being SortSymbol of S st t = s & h1 . t . x = h2 . t . x & t . x = h2 . t ;
defpred C [ Nat ] means ( ( P . $1 ) is n -\hbox { x } & ( P . $1 ) is n -\hbox { x } ) ;
consider y being element such that y in dom p9 and q9 . i = p9 . y and p . i = p9 . y ;
reconsider L = product ( { x1 } +* ( index B , l ) ) as Subset of ( Carrier A ) . ( i + 1 ) ;
for c being Element of C ex d being Element of D st T . ( id c ) = id d & for x being Element of C st x in c holds d <= c
not ( for f , n , p holds f = ( f | n ) ^ <* p *> ) .= f ^ <* p *> ^ <* p *> ;
( f (#) g ) . x = f . ( g . x ) & ( f (#) h ) . x = f . ( h . x ) ;
p in { 1 / 2 * ( G * ( i + 1 , j ) + G * ( i + 1 , j + 1 ) ) } ;
f `2 `2 - p `2 = ( ( c | n ) *' ( f - g ) ) *' - ( ( c (#) ( f - g ) ) *' ) .= ( ( c (#) ( f - g ) ) *' ) *' - ( ( f - g ) ) *' ;
consider r be Real such that r in rng ( f | divset ( D , j ) ) and r < m + s ;
f1 . ( |[ r2 , r2 ]| ) in f1 .: W2 & f2 . ( |[ r2 , r2 ]| ) in f1 .: W3 ;
eval ( a | ( n , L ) , x ) = ( a | ( n , L ) ) . x .= a . x ;
z = DigA ( tk , x9 ) .= DigA ( tk , x9 ) .= DigA ( tk , x9 ) .= DigA ( tk , x9 ) ;
set H = { Intersect S where S is Subset-Family of X : S c= G } , G = { Intersect S where S is Subset-Family of X : S c= G } ;
consider S19 being Element of D .: j , d being Element of D .: j such that S ` = S19 ^ <* d *> and S = S19 ^ <* d *> ;
assume that x1 in dom f and x2 in dom f and f . x1 = f . x2 and f . x2 = f . x2 ;
- 1 <= ( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 / ( 1 + sn ) ^2 ;
(0). V is Linear_Combination of A & Sum ( (0). V ) = 0. V & Sum ( L ) = 0. V implies Sum ( L ) = Sum ( L )
let k1 , k2 , k1 , k2 , k2 , k2 , k1 , k2 , k2 , k2 , k2 , k2 , k1 , k2 , k2 , k2 , k1 , k2 , k2 , k2 be Instruction of SCM+FSA ;
consider j being element such that j in dom a and j in g " { k `2 } and x = a . j and y = a . j ;
H1 . x1 c= H1 . x2 or H1 . x2 c= H1 . x1 or H1 . x1 c= H1 . x2 or H1 . x1 c= H1 . x2 ;
consider a being Real such that p = -' a * p1 + ( a * p2 ) and 0 <= a and a <= 1 ;
assume that a <= c & c <= d & [' a , b '] c= dom f and [' a , b '] c= dom g and [' a , b '] c= dom g ;
cell ( Gauge ( C , m ) , i , X ) -' 1 is non empty or cell ( Gauge ( C , m ) , i , 0 ) is non empty ;
A5 in { ( S . i ) `1 where i is Element of NAT : i in dom S } ;
( T * b1 ) . y = L * b2 /. y .= ( F `1 * b1 ) . y .= ( F `1 * b1 ) . y ;
g . ( s , I ) . x = s . y & g . ( s , I ) . y = |. s . x - s . y .| ;
( log ( 2 , k + 1 ) ) ^2 >= ( log ( 2 , k + 1 ) ) ^2 ;
then p => q in S & not x in the carrier of p & not p => All ( x , 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 integer ;
assume for a being Element of D holds f . { a } = a & for X being Subset-Family of D holds f . ( f .: X ) = f . union X ;
i = len p1 .= len p3 + len <* x *> .= len p3 + 1 .= len p3 + 1 .= len p3 + 1 + 1 .= len p3 + 1 + 1 ;
l /. ( 1 , 3 ) = g /. ( g /. 1 , 3 ) + k * ( e /. 3 , e /. 1 ) - e /. 3 + e /. 3 - e /. 3 + e /. 1 ;
CurInstr ( P2 , Comput ( P2 , s2 , l2 ) ) = halt SCM+FSA .= halt SCM+FSA .= halt SCM+FSA .= halt SCM+FSA .= halt SCM+FSA ;
assume for n be Nat holds ||. seq .|| . n <= Rseq . n & Rseq is summable & Rseq is summable & Rseq is summable ;
sin . ( n + 1 ) = sin . r * cos . ( - cos . s ) .= 0 * ( sin . s ) .= 0 ;
set q = |[ g1 `1 . t0 , g2 `2 . t0 , f3 `2 . t0 ]| , r1 = g1 `1 . t0 , r2 = g2 . t0 ;
consider G being sequence of S such that for n being Element of NAT holds G . n in implies G in implies G in implies for n being Element of NAT holds G . n in S ;
consider G such that F = G and ex G1 st G1 in SM & G = ( the carrier of G1 ) & G is one-to-one & G is one-to-one ;
the root of [ x , s ] in ( the Sorts of Free ( C , X ) ) . s & ( the Sorts of Free ( C , X ) ) . s = s ;
Z c= dom ( exp_R * ( f + ( #Z 2 ) * f1 ) ) /\ dom ( exp_R * f1 + exp_R * f1 ) ;
for k be Element of NAT holds seq1 . k = ( \HM { Im ( f , S ) . k , S . ( k + 1 ) ) . x
assume that - 1 < n ( ) and q `2 > 0 and ( q `1 / |. q .| - cn ) < 0 and ( - 1 ) < 0 ;
assume that f is continuous one-to-one and a < b and c < d and f = g and f . a = c and f . b = d ;
consider r being Element of NAT such that sLet = Comput ( P1 , s1 , r ) and r <= q and r <= q ;
LE f /. ( i + 1 ) , f /. j , L~ f , f /. 1 , 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 ex_inf_of { x , y } , L and x <> y and y <> x ;
assume that f +* ( i1 , \xi ) . 1 in ( proj ( F , i2 ) ) " ( A . ( i1 + 1 ) ) and f . ( i1 + 1 ) = f " A ;
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 A } ;
consider l be Nat such that for m be Nat st l <= m holds ||. s1 . m - x0 .|| < g / 2 and g in dom ( ||. f .|| ) ;
consider t be VECTOR of product G such that mt = ||. D5 . t .|| and ||. t .|| <= 1 ;
assume that the degree degree of v = 2 and v ^ <* 0 *> , v ^ <* 1 *> ^ p ^ <* 1 *> ^ q ^ <* 1 *> ^ q ^ <* 1 *> ^ q ^ <* 1 *> ^ q ^ <* 1 *> ^ q ^ <* 1 *> ^ q ^ <* 1 *> ^ q ^ <* 1 *> ^ q ^ q ;
consider a being Element of the carrier of X39 , A being Element of the lines of X39 such that a on A and not a on A ;
( - x ) |^ ( k + 1 ) * ( ( - x ) |^ ( k + 1 ) ) " = 1 ;
for D being set st for i st i in dom p holds p . i in D holds p is FinSequence of D & for i st i in dom p holds p . i is FinSequence of D
defpred R [ element ] means ex x , y st [ x , y ] = $1 & P [ x , y ] & P [ y ] ;
L~ f2 = union { LSeg ( p0 , p1 ) , LSeg ( p1 , p01 ) } .= { p1 , p2 } \/ { p1 , p01 } ;
i -' len h11 + 2 - 1 < i -' len h11 + 2 - 1 + 1 + 1 - 1 + 1 + 1 - 1 + 1 ;
for n being Element of NAT st n in dom F holds F . n = |. ( nthesis . ( n -' 1 ) ) .| ;
for r , s1 , s2 holds r in [. s1 , s2 .] iff s1 <= r & r <= s2 & s1 <= s2 & r <= s2 & s1 <= s2
assume v in { G where G is Subset of T2 : G in B2 & G c= z1 & G c= z2 & G c= z2 } ;
let g be .| -:] Function of A , INT , ( INT |^ X ) | b , ( ( 0 , 1 ) --> 1 ) , A ;
min ( g . [ x , y ] , k ) . [ y , z ] = ( min ( g , k , x , z ) ) . y ;
consider q1 being sequence of CNS such that for n holds P [ n , q1 . n ] and q1 is convergent and lim q1 = lim q1 ;
consider f being Function such that dom f = NAT and for n being Element of NAT holds f . n = F ( n ) and for n being Element of NAT holds P [ n , f . n ] ;
reconsider B-6 = B /\ B , Od = O , sequence = I /\ Z , Subset of B ;
consider j being Element of NAT such that x = the b \HM { x where x is FinSequence of NAT : 1 <= j & j <= n } and j = len f ;
consider x such that z = x and card ( x . O2 ) in card ( x . O ) and x in L1 . O and x <> O ;
( C * \mathop { \rm T4 } ( k , n2 ) ) . 0 = C . ( ( \mathop { \rm T4 ( k , n2 ) ) . 0 ) ;
dom ( X --> rng f ) = X & dom ( ( X --> f ) . x ) = dom ( X --> f . x ) ;
( for x being Element of L~ SpStSeq C st x in b holds ( for x being Element of C st x in b holds x <= ( ( L~ SpStSeq C ) * ( i , j ) ) `2 ) implies ( L~ SpStSeq C ) * ( i , j ) `2 <= ( ( L~ SpStSeq C ) * ( i , j ) ) `2
synonym x , y are_collinear means : Def2 : x = y or ex l being that { 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 ( for k , x being Element of L , a , b being Element of L st a = x & b = y holds x << y iff a << b ) & ( for x , y being Element of L st x in X & y in X holds a << b ) ;
( 1 / 2 * ( ( ( - 1 ) * ( ( #Z n ) * ( AffineMap ( n , 0 ) ) ) ) ) is_differentiable_on REAL ;
defpred P [ Element of omega ] means ( the partial of A1 ) . $1 = A1 . $1 & ( the partial of A1 ) . $1 = A2 . $1 ;
IC Comput ( P , s , 2 ) = succ IC Comput ( P , s , 1 ) .= 6 + 1 .= 6 + 1 .= 6 ;
f . x = f . g1 * f . g2 .= f . g1 * 1_ H .= f . g1 * 1_ H .= f . g1 * ( g . g2 ) .= f . g1 * ( g . g2 ) .= f . g1 ;
( M * FK1 ) . n = M . ( FK1 . n ) .= M . { ( ( canFS ( Omega ) ) . n ) } .= M . { ( canFS ( Omega ) ) . n } ;
the carrier of L1 + L2 c= ( the carrier of L1 ) \/ ( the carrier of L2 ) & the carrier of L1 + L2 c= the carrier of L1 \/ the carrier of L2 ;
pred a , b , c , x , y , c , d d d of o , o , y , c , d , x , y ;
( the PartFunc of s , X ) . n <= ( the PartFunc of s , X ) . n * s . ( n + 1 ) ;
pred - 1 <= r & r <= 1 means : Def3 : for r st r in Z holds arccot . r = - 1 / r + r / r ;
seq in { p ^ <* n *> where n is Nat : p ^ <* n *> in T1 } & p ^ <* n *> in T1 ;
|[ x1 , x2 , x3 ]| . 2 - |[ y1 , y2 , x3 , x4 ]| . 2 = x2 - y2 & |[ x1 , x2 , x3 , x4 , x4 ]| . 2 = x2 - y2 ;
attr for m be Nat holds F . m is nonnegative & ( Partial_Sums ( F ) ) . m is nonnegative & ( Partial_Sums ( F ) ) . m is nonnegative ;
len ( ( G . z ) * ( G . ( x9 ) ) ) = len ( ( G . x9 ) * ( G . ( y9 ) ) ) .= len ( G . ( y9 ) ) ;
consider u , v being VECTOR of V such that x = u + v and u in W1 /\ W2 and v in W2 /\ W3 and u in W2 /\ W3 ;
given F being finite FinSequence of NAT such that F = x and dom F = n and rng F c= { 0 , 1 } and Sum F = k ;
0 = 0 * ( - \hbox { - 1 } ) * ( 1 - ( - 1 ) * ( - ( - 1 ) * ( - ( - 1 ) ) ) ;
consider n be Nat such that for m be Nat st n <= m holds |. ( f # x ) . m - lim ( f # x ) .| < e ;
cluster -> } for } -`1 of ( ( the carrier of L ) --> ( ( the carrier of L ) --> ( the carrier of L ) ) , L ;
"/\" ( BB , {} ) = Top BB .= the carrier of S .= Top ( S , T ) .= "/\" ( IB , {} ) .= "/\" ( IB , {} ) .= "/\" ( BB , {} ) ;
( r / 2 ) ^2 + ( r\mathopen ( 2 , r ) ) ^2 <= ( r / 2 ) ^2 + ( r / 2 ) ^2 + ( r / 2 ) ^2 ;
for x being element st x in A /\ dom ( f `| X ) holds ( f `| X ) . x >= r2 & ( f `| X ) . x >= r2
2 * r1 - c * |[ a , c ]| - ( 2 * r1 - b ) * |[ b , c ]| = 0. TOP-REAL 2 ;
reconsider p = P * ( \square , 1 ) , q = a " * ( ( - ( - ( 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 x = [ x1 , x2 ] ;
for n be Nat st 1 <= n & n <= len q1 holds q1 . n = ( ( upper_volume g , M7 ) * ( len ( D2 , D1 ) ) ) . n
consider y , z being element such that y in the carrier of A and z in the carrier of A and i = [ y , z ] and i = [ y , z ] ;
given H1 , H2 being strict Subgroup of G such that x = H1 & y = H2 and H1 is Subgroup of H2 and H2 is Subgroup of H1 and H1 is Subgroup of H2 ;
for S , T being non empty that T is complete and T is complete holds d is directed-sups-preserving iff d is monotone & d is monotone & d is monotone
[ a + 0. F_Complex , b2 ] in ( the carrier of F_Complex ) /\ ( the carrier of V ) & [ a + 0. F_Complex , b2 ] in ( the carrier of V ) /\ ( the carrier of V ) ;
reconsider mm = max ( len F1 , len ( p . n ) * <* x *> ) as Element of NAT , x be Element of NAT ;
I <= width GoB ( ( GoB h ) * ( len GoB h , width GoB h ) , ( GoB h ) * ( len GoB h , width GoB h ) ) ;
f2 /* q = ( f2 /* ( f1 /* s ) ) ^\ k .= ( f2 * f1 ) /* s .= ( ( f2 * f1 ) /* s ) ^\ k .= ( ( f2 * f1 ) /* s ) ^\ k ;
attr A1 \/ A2 is linearly-independent means : Def2 : A1 misses A2 & for x st x in A1 holds x in A1 & x <> 0. V holds ( x in A1 & x <> 0. V ) ;
func A -carrier C -> set equals union { A . s where s is Element of R : s in C } ;
dom ( Line ( v , i + 1 ) ) = dom ( F ^ ( a , m ) ) & dom ( Line ( p , i ) ) = dom ( F ^ ( a , m ) ) ;
cluster [ x `1 , 4 , x `2 , 4 , 5 ] -> non empty & [ x `1 , 4 , 5 ] `1 = x `1 & [ x `1 , 4 ] `2 = 4 ;
E , f |= All ( x1 , All ( x2 , x2 ) '&' ( x1 , x2 ) '&' ( x1 , x2 ) '&' ( x1 , x2 ) ) => All ( x1 , x2 ) '&' ( x1 , x2 ) '&' ( x1 , x2 ) '&' ( x1 , x2 ) '&' ( x1 , x2 ) '&' ( x1 , x2 ) '&' ( x1 , x2 ) '&' ( x1 , x2 ) '&' ( x1 , x2 ) '&' ( x1 , x2 ) ;
F .: ( id X , g ) . x = F . ( id X , g . x ) .= F . ( x , g . x ) .= F . ( x , g . x ) ;
R . ( h . m ) = F . x0 + h . m - h . x0 + h . m - h . x0 + h . m - h . x0 ;
cell ( G , ( X -' 1 , Y + 1 ) \ ( t + 1 ) ) meets ( UBD L~ f ) & ( for k st k in dom f holds f /. k meets ( L~ f ) \ ( t + 1 ) ) implies ( L~ f ) meets ( L~ f )
IC Result ( P2 , s2 ) = IC IExec ( I , P , Initialize s ) .= ( card I + card J + 3 ) .= ( card I + 3 ) .= ( card I + 3 ) ;
sqrt ( ( - ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) / ( 1 + cn ) ) ^2 ) > 0 ;
consider x0 being element such that x0 in dom a and x0 in g " { k `2 } and y0 = a . x0 and x0 in g " { k } ;
dom ( r1 (#) chi ( A , C ) , C ) = dom chi ( A , C ) /\ dom ( chi ( A , C ) , C ) .= C /\ dom ( ( r1 (#) chi ( A , C ) ) , C ) .= C /\ dom ( ( r1 (#) chi ( A , C ) ) , C ) .= C /\ dom ( ( r1 (#) chi ( A , C ) ) , C ) ;
d-7 . [ y , z ] = ( ( [ y , z ] `1 ) `2 - ( [ y , z ] `2 ) `2 - ( [ y , z ] `2 ) `2 - ( [ y , z ] `2 - ( [ y , z ] `2 ) `2 ) ;
attr for i being Nat holds 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 ||. f .|| is_continuous_in x0 and for r st r in dom f & 0 < r ex g st g < r & g in dom f & g in dom f ;
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
for x being Element of REAL n st x in Line ( x1 , x2 ) holds |. y1 - y2 .| <= |. y1 - y2 .| & |. y1 - y2 .| <= |. y1 - y2 .|
func /. ( <*> a ) -> w Ordinal means : Def3 : a in it & for b being l of a st a in b holds it . b c= b ;
[ a1 , a2 , a3 ] in ( [: the carrier of A ) /\ ( the carrier of A ) & [ a1 , a2 , a3 ] in [: the carrier of A , the carrier of A :] ;
ex a , b being element st a in the carrier of S1 & b in the carrier of S2 & x = [ a , b ] & x = [ a , b ] ;
||. ( vseq . n ) - ( vseq . m ) .|| * ||. x - y .|| < e / ( ||. x .|| + ||. y .|| ) * ||. x - y .|| ;
then for Z being set st Z in { Y where Y is Element of I7 : F c= Y } holds z in Z & z in Z & z in Z ;
sup compactbelow [ s , t ] = [ sup ( \pi , t ) , sup ( compactbelow s ) ] .= [ sup ( compactbelow s ) , sup ( compactbelow t ) ] .= [ sup ( compactbelow s ) , t ] ;
consider i , j being Element of NAT such that i < j and [ y , f . j ] in If and [ f . i , z ] in If and [ y , f . i ] in If ;
for D being non empty set , p , q being FinSequence of D st p c= q holds ex p being FinSequence of D st p ^ q = q & p ^ q = p ^ q
consider e19 being Element of the carrier of X such that c9 , a9 // a9 , e39 and a9 <> c9 and a9 <> c9 and a , b // a9 , e and a , c // a9 , b9 ;
set U2 = I \! \mathop { x } , U2 = I -\mathop { y } ;
|. q3 .| ^2 = ( ( q3 `1 ) ^2 + ( q3 `2 ) ^2 ) + ( ( q3 `2 ) ^2 + ( q3 `1 ) ^2 .= |. q .| ^2 + ( q `2 ) ^2 .= |. q .| ^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
dom signature U1 = dom ( the charact of U1 ) & Args ( o , U1 ) = dom ( the charact of U1 ) & Args ( o , U1 ) = dom ( the charact of U1 ) ;
dom ( h | X ) = dom h /\ X .= dom ( ||. h .|| ) /\ X .= dom ( ||. h .|| ) /\ X .= dom ( ||. h .|| ) /\ X .= dom ( ||. h .|| ) /\ X .= dom ( ||. h .|| ) /\ X .= dom ( ||. h .|| ) ;
for N1 , N1 being Element of GX holds dom ( h . K1 ) = N & rng ( h . K1 ) = N1 & rng ( h . K1 ) = N1 & rng ( h . K1 ) c= N2
( mod ( u , m ) + mod ( v , m ) ) . i = ( mod ( u , m ) ) . i + ( mod ( v , m ) ) . i ;
- ( q `1 ) ^2 < - 1 or q `2 >= - ( q `1 ) ^2 & - ( q `2 ) ^2 <= - ( q `1 ) ^2 or - ( q `1 ) ^2 >= - ( q `1 ) ^2 ;
pred r1 = fp & r2 = fp & r1 * r2 = fp * ( r1 + r2 ) & r2 * r1 = fp * ( r1 + r2 ) ;
vseq . m is bounded Function of X , the carrier of Y & x9 . m = ( ( vseq . m ) - ( vseq . n ) ) . x & ( vseq . m ) - ( vseq . n ) ) . x = ( ( vseq . m ) - ( vseq . n ) ) . x ;
pred a <> b & b <> c & angle ( a , b , c ) = PI & angle ( b , c , a ) = 0 & angle ( c , a , b ) = PI ;
consider i , j , r being Nat such that p1 = [ i , r ] and p2 = [ j , s ] and i < j and i < j and r < s ;
|. p .| ^2 - ( 2 * |( p , q )| ) ^2 + |. q .| ^2 = |. p .| ^2 + |. q .| ^2 - ( 2 * |( p , q )| ) ^2 ;
consider p1 , q1 being Element of X ( ) such that y = p1 ^ q1 and q1 ^ p1 = p1 ^ q1 and p1 ^ q1 = p1 ^ q1 and p1 ^ q1 = q1 ^ q1 and p1 ^ q1 = q1 ^ q2 ;
, 1. ( K , r1 , r2 , s2 , s1 , s2 , s2 , s2 , t2 ) = ( s2 - s1 ) * ( s1 - s2 ) + ( s2 - s1 ) * ( s2 - s2 ) ;
( LMP A ) `2 = lower_bound ( proj2 .: A /\ E-bound ( w ) ) & proj2 .: A is non empty & proj2 .: ( A /\ E-bound ( w ) ) is non empty ;
s |= ( k , H1 ) \bf ( H , k ) iff s |= ( H , k ) implies s |= ( H , k ) '&' ( H , k ) & s |= ( H , k ) '&' ( H , k )
len s5 + 1 = card ( support b1 ) + 1 .= card ( support b2 ) + card ( support b2 ) .= card ( support b2 ) + card ( support b2 ) .= card ( support b2 ) + card ( support b2 ) .= card ( support b2 ) + card ( support b2 ) ;
consider z being Element of L1 such that z >= x and z >= y and for z being Element of L1 st z >= x & z `1 >= y holds z `1 >= y and z `2 >= x ;
LSeg ( UMP D , |[ ( W-bound D + E-bound D ) / 2 , ( E-bound D + E-bound D ) / 2 ]| ) /\ D = { UMP D } /\ D .= { UMP D } ;
lim ( ( ( f `| N ) / g ) /* b ) = lim ( ( f `| N ) / g ) .= lim ( ( f `| N ) / g ) .= lim ( ( f `| N ) / g ) ;
P [ i , pr1 ( f ) . i , pr1 ( f ) . i , 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 ) - Rx ) - Rx .|| < r
for X being set , P being a_partition of X , x , a , b being set st x in a & a in P & x in P & b in P & a <> b holds a = b
Z c= dom ( ( #Z 2 ) * f ) /\ ( dom ( ( #Z 2 ) * f ) \ ( ( #Z 2 ) * f ) " { 0 } ) & Z c= dom f /\ ( dom f \ f ) " { 0 } ;
ex j be Nat st j in dom ( l ^ <* x *> ) & j < i & y = ( l ^ <* x *> ) . j & i = 1 + len l + j & z = x + len l + j & i = len l + 1 ;
for u , v being VECTOR of V , r being Real st 0 < r & r < 1 & u in dom ( r (#) u ) holds r * u + ( 1-r (#) v ) in <* u *>
A , Int A , Cl A , Cl Int A , Cl Int A , Cl Int A , Cl Int A , Cl Int A , Cl Int A , Cl Int A , Cl Int A , Cl 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 , Cl Cl A , Cl Cl A , Cl Cl A , Cl Cl A , Cl Cl A , Cl Cl A , Cl Cl A , Cl Cl A , Cl Cl A , Cl Cl A , Cl Cl A , Cl Cl A , Cl Cl A , Cl Cl
- Sum <* v , u , w *> = - ( v + u + w ) .= - ( v + u ) + w .= - ( v + u ) + w .= - v + u + w ;
Exec ( a := b , s ) . IC SCM R = ( Exec ( a := b , s ) ) . NAT .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= IC s ;
consider h being Function such that f . a = h and dom h = I and for x being element st x in I holds h . x in ( the carrier of J ) . x and h . x = ( the carrier of J ) . x ;
for S1 , S2 , D being non empty reflexive RelStr , D being non empty Subset of [: S1 , S2 :] , f being Function of S1 , S2 , g being Function of S2 , S2 st f is directed & g is directed holds g is directed & g is directed
card X = 2 implies ex x , y st x in X & y in X & x <> y & x <> y or z = x or z = y or z = x or z = y or z = x or z = x or z = y
E-max L~ Cage ( C , n ) in rng ( Cage ( C , n ) \circlearrowleft E-max L~ Cage ( C , n ) ) & E-max L~ Cage ( C , n ) in rng Cage ( C , n ) ;
for T , T being DecoratedTree , p , q being Element of dom T st p for q being Element of dom T holds ( T , p ) . q = T . q & ( T , q ) . q = T . q
[ i2 + 1 , j2 ] in Indices G & [ i2 , j2 ] in Indices G & f /. k = G * ( i2 + 1 , j2 ) & f /. k = G * ( i2 + 1 , j2 ) ;
cluster -> commutative means : Def2 : k divides it & n divides it & for m being Nat st k divides m & m divides m & m divides n holds it divides m & for k being Nat st k divides m & k divides m holds it divides k ;
dom F " = the carrier of X2 & rng F " = the carrier of X1 & F " = the carrier of X2 & F " = F " * F " & F " = F " * F " ;
consider C being finite Subset of V such that C c= A and card C = \cdot and the carrier of V = Lin ( B9 \/ C ) and card C = card ( B \/ C ) and card C = card ( B \/ C ) ;
V is prime implies for X , Y being Element of [: the topology of T , the topology of T :] st X /\ Y c= V holds X c= V or Y c= V or Y c= V
set X = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } , Y = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } ;
angle ( p1 , p3 , p4 ) = 0 .= angle ( p2 , p3 , p2 ) .= angle ( p , p3 , p2 ) .= angle ( p , p3 , p2 ) .= angle ( p , p3 , p2 ) .= angle ( p , p3 , p2 ) ;
- sqrt ( ( - ( q `1 / |. q .| - cn ) / ( 1 - cn ) ) ^2 ) = - sqrt ( ( - ( q `1 / |. q .| - cn ) / ( 1 - cn ) ) ^2 ) .= - ( - ( - ( q `1 / |. q .| - cn ) / ( 1 - cn ) ) ;
ex f being Function of I[01] , TOP-REAL 2 st f is continuous one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 & f . 0 = p4 & f . 1 = p4 ;
attr f is_partial u0 means : Def2 : SVF1 ( 2 , pdiff1 ( f , 1 ) , u0 , 1 ) is_differentiable_in ( proj ( 2 , 3 ) . u0 ) & SVF1 ( 2 , 3 ) . u0 = ( proj ( 2 , 3 ) ) . u0 ;
ex r , s st x = |[ r , s ]| & G * ( len G , 1 ) `1 < r & s < G * ( 1 , 1 ) `2 & G * ( 1 , 1 ) `2 < s & s < G * ( 1 , 1 ) `2 ;
assume that f is_sequence_on G and 1 <= t & t <= len G and G * ( t , width G ) `2 >= N-bound L~ f and G * ( t , width G ) `2 >= N-bound L~ f and t <= width G ;
pred i in dom G means : Def3 : r * ( f * reproj ( i , x ) ) = r * f * reproj ( i , x ) ;
consider c1 , c2 being bag of o1 + o2 such that ( decomp c ) /. k = <* c1 , c2 *> and c = c1 + c2 and c1 = c2 + c2 and c1 = c2 + c2 ;
u0 in { |[ r1 , s1 ]| : r1 < G * ( 1 , 1 ) `1 & s1 < G * ( 1 , 1 ) `2 & G * ( 1 , 1 ) `2 < s1 & s1 < G * ( 1 , 1 ) `2 } ;
( X ^ Y ) . k = the carrier of X . k2 .= C4 . k .= C4 . k .= ( C4 ^ C3 ) . k .= ( C4 ^ C3 ) . k .= ( C4 ^ C3 ) . k ;
attr M1 = len M2 means : Def2 : len M1 = width M2 & width M1 = width M2 & for i st i in dom M1 holds M1 . i = M1 . i - M2 . i ;
consider g2 be Real such that 0 < g2 and { y where y is Point of S : ||. ( y - x0 ) - x0 .|| < g2 & y in N2 } c= N2 and for x be Point of S st x in N holds |. ( x - x0 ) - x0 .| < g2 . x ;
assume x < ( - b + sqrt ( a , b , c ) ) / 2 or x > ( - b - sqrt ( a , b , c ) ) / 2 or x > - b ;
( G1 '&' G2 ) . i = ( <* 3 *> ^ G1 ) . i & ( H1 '&' H2 ) . i = ( <* 3 *> ^ G1 ) . i & ( H1 '&' G1 ) . i = ( <* 3 *> ^ G1 ) . i ;
for i , j st [ i , j ] in Indices ( M3 + M1 ) holds ( M3 + M1 ) * ( i , j ) < M2 * ( i , j ) + M2 * ( i , j )
for f being FinSequence of NAT , i being Element of NAT st for j being Element of NAT st j in dom f & j <> i holds i divides f /. j holds i divides ( f /. j ) & i divides ( f /. j )
assume F = { [ a , b ] where a , b is Subset of X : for c being set st c in B\mathopen } & a c= c & b c= c holds b c= c ;
b2 * q2 + ( b3 * q3 ) + - ( ( a1 * q2 ) + - ( a2 * q3 ) ) = 0. TOP-REAL n + ( ( a1 * q2 ) + - ( a2 * q3 ) ) .= 0. TOP-REAL n + ( ( a1 * q2 ) + - ( a2 * q3 ) ) .= 0. TOP-REAL n + ( a2 * q2 ) ;
Cl Cl F = { D where D is Subset of T : ex B being Subset of T st D = Cl B & B in F & A in Cl B & B in Cl F } ;
attr seq is summable means : Def2 : seq is summable & Partial_Sums ( seq ) is summable & Partial_Sums ( seq ) is summable & Partial_Sums ( seq ) is summable & Partial_Sums ( seq ) is summable & Partial_Sums ( seq ) is summable & Partial_Sums ( seq ) is summable & Partial_Sums ( seq ) is summable ;
dom ( ( cn " ) | D ) = ( the carrier of ( TOP-REAL 2 ) ) /\ D .= the carrier of ( ( TOP-REAL 2 ) | D ) .= the carrier of ( ( TOP-REAL 2 ) | D ) .= D ;
|[ X , Z ]| is full full SubRelStr of ( Omega Z ) |^ the carrier of X & |[ X , Y ] is full SubRelStr of ( Omega Z ) |^ the carrier of Y implies X is full SubRelStr of ( Omega Z ) |^ the carrier of Y
G * ( 1 , j ) `2 = G * ( i , j ) `2 & G * ( 1 , j ) `2 <= G * ( i + 1 , j ) `2 & G * ( 1 , j ) `2 <= G * ( i + 1 , j ) `2 ;
synonym m1 c= m2 means : Def2 : for p being set st p in P holds the set of m1 <= p & p is_pw & for n being Nat st n in dom m1 holds m1 . n <= ( m2 . n ) `1 & m1 . n = m2 . n ;
consider a being Element of B ( ) such that x = F ( a ) and a in { G ( b ) where b is Element of A ( ) : P [ b ] } ;
mode multiplicative loop structure means : Def3 : the carrier of it = [ ( a , a ) * the multF of S , ( a , b ) * the multF of S ] & the carrier of it = [: the carrier of S , the carrier of S :] ;
sequence ( a , b , 1 ) + sequence ( c , d , 1 ) = b + sequence ( c , d , 1 ) .= b + d + c .= sequence ( a + c , b + d ) ;
cluster -> + _ { \mathbb Z } means : Def2 : for i1 , i2 being Element of INT holds it . ( i1 , i2 ) = + ( i1 , i2 ) & for i being Element of NAT holds i1 . ( i1 , i2 ) = ( i1 + i2 ) . ( i1 , i2 ) ;
( - s2 ) * p1 + ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - s1 ) ) ) ) ) ) ) = ( - r2 ) * p1 + ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 -
eval ( ( a | ( n , L ) ) *' p , x ) = eval ( a | ( n , L ) ) * eval ( p , x ) .= a * eval ( p , x ) .= a * eval ( p , x ) ;
assume that the TopStruct of S = the TopStruct of T and for D being non empty Subset of S , V being open Subset of Omega S st V in V & V is open & V c= V & V c= V holds V meets V and V is open and for W being open Subset of S st W in V & W c= V holds W meets V ;
assume that 1 <= k & k <= len w + 1 and T-7 . ( ( q11 , w ) -succ k ) = ( T11 . k , w ) -succ ( ( q11 , w ) -succ k ) ;
2 * a |^ ( n + 1 ) + ( 2 * b |^ ( n + 1 ) ) >= a |^ ( n + 1 ) + ( ( a |^ n ) * b + ( b |^ n ) * a + b |^ ( n + 1 ) ) ;
M , v2 |= All ( x. 3 , All ( x. 0 , All ( x. 4 , H ) ) '&' ( All ( x. 4 , H ) '&' ( x. 4 , All ( x. 0 , H ) ) ) ) implies M , v2 |= All ( x. 4 , H ) '&' ( x. 0 , All ( x. 4 , H ) )
assume that f is_differentiable_on l and for x0 st x0 in l holds 0 < f ' ( x0 ) or for x0 st x0 in l holds f ' ( x0 ) < 0 and for x0 st x0 in l holds f ' ( x0 ) < 0 ;
for G1 being _Graph , W being Walk of G1 , e being set st not e in W and not e in W & not e in W & not e in W & e in W & not e in W & e in W & not e in W & e in W holds W is Walk of G2
not vs is not empty iff that not .| is not empty & not is not empty or not q1 is not empty & not q2 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 q3 q2 is not empty & not q1 is not empty ;
Indices GoB f = [: dom GoB f , Seg width GoB f :] & i1 + 1 in dom GoB f & i2 + 1 in Seg width GoB f & 1 <= i1 & i1 + 1 in dom GoB f & 1 <= i2 & i1 + 1 in dom GoB f & 1 <= i2 & i2 + 1 in Seg width GoB f & 1 <= i2 & i2 + 1 <= len GoB f & 1 <= i2 & i2 + 1 <= len GoB f implies ( GoB f ) * ( i1 , i2 ) = ( GoB f ) * ( i1 + 1 , j2 + 1 , j2 + 1 , j2 + 1 ) & ( GoB f ) * ( i1 + 1 , j2 + 1 , j2 + 1 , j2 + 1 = ( GoB f ) * ( i1 + 1 = j2 +
for G1 , G2 , G3 being strict Subgroup of O st G1 is stable & G2 is stable & G2 is stable & G2 is stable & G1 is stable & G2 is stable & G2 is stable & G2 is stable & G2 is stable holds G1 is stable of G2 & G2 is stable of G3
UsedIntLoc ( inint f ) = { intloc 0 , intloc 1 , intloc 2 , intloc 3 , intloc 4 , intloc 5 , intloc 4 , intloc 5 , intloc 5 , intloc 6 , intloc 5 , intloc 5 , intloc 6 , intloc 5 } .= UsedIntLoc ( f , s ) \/ UsedIntLoc ( f , s ) ;
for f1 , f2 be FinSequence of F st f1 ^ f2 is p -element & Q [ p ^ <* q *> ] & Q [ q ^ <* p *> ] holds Q [ f1 ^ <* p *> ^ f2 ]
( p `1 / sqrt ( 1 + ( p `1 / p `2 ) ^2 ) ) ^2 = ( q `1 / sqrt ( 1 + ( q `1 / q `2 ) ^2 ) ) ^2 .= ( q `1 / sqrt ( 1 + ( q `1 / q `2 ) ^2 ) ) ^2 ;
for x1 , x2 , x3 , x4 being Element of REAL n holds |( x1 - x2 , x3 )| = |( x1 , x3 )| + |( x2 , x3 )| + |( x3 , x4 )| + |( x2 , x3 )| + |( x3 , x4 )| + |( x3 , x4 )| + |( x2 , x3 )| + |( x3 , x4 )| + |( x3 , x2 )| + |( x3 , x3 )| + |( x3 , x3 )|
for x st x in dom ( ( F - G ) | A ) holds ( ( F - G ) | A ) . ( - x ) = - ( ( F - G ) | A ) . x
for T being non empty TopStruct , P being Subset-Family of T st P c= the topology of T for x being Point of T , B being Basis of x st B c= P & x in B holds P is Basis of T
( a 'or' b 'imp' c ) . x = 'not' ( ( a 'or' b ) . x ) 'or' c . x .= 'not' ( a . x ) 'or' c . x .= TRUE '&' TRUE .= TRUE .= TRUE .= TRUE ;
for e being set st e in [: A , Y1 :] ex X1 being Subset of Y , Y1 being Subset of Y st e = [: X1 , Y1 :] & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open
for i be set st i in the carrier of S for f be Function of Sfor i be Element of S1 . i , S1 . i st f = H . i holds F . i = f | ( F . i ) & for i be set st i in dom F holds F . i = f | ( F . i ) ;
for v , w st for y st x <> y holds w . y = v . y holds Valid ( VERUM ( Al , J ) , J ) . v = Valid ( VERUM ( Al , J ) , J ) . w
card D = card D1 + card D2 - card { i , j } .= c1 + 1 - 1 + 1 .= c1 + 1 - 1 + 1 .= c1 + 1 - 1 + 1 .= c1 + 1 - 1 + 1 .= c1 + 1 - 1 + 1 - 1 .= c1 + 1 - 1 ;
IC Exec ( i , s ) = ( s +* ( 0 .--> succ ( s . 0 ) ) ) . 0 .= ( 0 .--> succ ( s . 0 ) ) . 0 .= ( 0 .--> ( s . 0 ) ) . 0 .= ( ( 0 .--> ( s . 0 ) ) . 0 .= ( 0 .--> ( s . 0 ) ) . 0 .= ( 0 .--> ( s . 0 ) ) . 0 .= ( 0 .--> ( s . 0 ) ) . 0 .= ( 0 .--> ( s . 0 ) .= ( 0 .--> ( s . 0 ) ) .= ( 0 .--> ( s . 0 ) .= ( 0 .--> 1 ) .= ( 0 .--> 1 ) .= ( 0 .--> 1 ) .= ( 0 .--> 1 ) .= ( 1
len f /. ( len f -' 1 ) -' 1 + 1 = len f /. ( len f -' 1 ) + 1 - 1 + 1 .= len f -' 1 + 1 - 1 + 1 .= len f -' 1 + 1 - 1 + 1 ;
for a , b , c being Element of NAT st 1 <= a & a <= b & b <= b holds a <= a + b-2 or a = a + b-2 or b = a + b-2 or a = b + b-2 or a = a + b-2 or b = a + b-2 or a = b
for f being FinSequence of TOP-REAL 2 , p being Point of TOP-REAL 2 st p in LSeg ( f , i ) & p in LSeg ( f , i ) holds Index ( p , f ) <= i & Index ( p , f ) <= len f
lim ( ( curry ( P+* ( i , k + 1 ) ) # x ) = lim ( ( curry ( P+* ( i , k ) ) # x ) + lim ( ( curry ( F+* ( i , k + 1 ) ) # x ) ) ;
z2 = g /. ( len g -' n1 + 1 ) .= g . ( i -' n2 + 1 + n1 ) .= g . ( i -' n2 + 1 ) .= g . ( i -' n2 + 1 ) .= g . ( i -' n2 + 1 ) .= g . ( i -' n2 + 1 ) ;
[ f . 0 , f . 3 ] in id ( the carrier of G ) \/ ( the InternalRel of G ) or [ f . 0 , f . 3 ] in the InternalRel of C6 & [ f . 0 , f . 3 ] in the InternalRel of C6 ;
for G being Subset-Family of B st G = { R [ X ] where R is Subset of [: A , B :] : R in F6 & R in F6 } holds ( for X being Subset of A , Y being Subset of B st X in F6 holds R [ X ] ) & ( for X being Subset of A st X in F6 holds R [ X ] ) implies for X being Subset of A , Y being Subset of B st X in F6 holds X c= Y
CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 ) ) = CurInstr ( P1 , Comput ( P1 , s1 , m1 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m1 ) ) .= halt SCMPDS .= halt SCMPDS .= halt SCMPDS .= ( CurInstr ( P1 , s2 ) ) .= halt SCMPDS ;
assume that a on M and b on M and c on N and d on M and p on P and a on P and c on P and d on P and a on Q and b on Q and a on P and b on Q and a on Q and b on Q and a on Q and b on Q and a on Q and a on Q and b on Q and a on Q and a on Q and b on Q and a on Q and a on Q and b on Q and a on Q and b on Q and b on Q and b on Q and a on Q and b on Q and b on Q and b on Q and b on Q and a on Q and b on Q and b on Q and a on Q and a on Q and a on Q and a on Q and b on Q and b on Q and b on Q and b on Q and c on Q and b on Q
assume that T is \hbox { T _ 4 } and ex F being Subset-Family of T st F is closed , countable , T & ( for n st n >= 0 holds F . n <= 0 ) & ( for n st n >= 0 holds F . n <= 0 ) & ( for n st n >= 0 holds F . n <= 0 ) implies T is finite-ind ) ;
for g1 , g2 st g1 in ]. r1 - r2 , r .[ & g2 in ]. r1 - r2 , r .[ holds |. f . g1 - f . g2 .| <= ( g1 - g ) / ( |. r1 - r2 .| + |. r2 .| ) & |. f . g2 - f . g2 .| <= ( g1 - g ) / ( |. r1 - r2 .| + |. r2 .| )
cosh /. ( z1 + z2 ) = ( cosh /. z1 ) * ( cosh /. z2 ) + ( ( cosh /. z1 ) * ( cosh /. z2 ) + ( ( cosh /. z1 ) * ( cosh /. z2 ) ) .= ( ( cosh /. z1 ) ) * ( ( cosh /. z2 ) * ( cosh /. z2 ) ) + ( ( cosh /. z1 ) * ( cosh /. z2 ) ) ;
F . i = F /. i .= 0. R + r2 .= b |^ ( n + 1 ) .= <* ( n + 1 ) |^ 0 , b |^ ( n + 1 ) , a , b , c *> .= <* ( n + 1 ) |^ ( n + 1 ) , b , c *> .= <* ( n + 1 ) |^ ( n + 1 ) , b , a *> ;
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 : Def3 : len it = len F & for i be Nat st i in dom it holds it . i = F /. i * f /. i ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 } = { x1 , x2 } \/ { x3 , x4 , 8 , 7 , 8 } \/ { 8 } \/ { 7 , 8 } ;
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 CQC-WFF ( Al ) st SubP ( P , l , e ) = S1 & ( for k st k in dom S1 holds ( ( k <= l implies S1 . k = e ) ) & ( not k <= l implies S1 . k = e ) & ( not k <= l implies S1 . k = e ) ) ;
consider P being FinSequence of Gs2 such that pB = product P and for i st i in dom P ex t7 being Element of the carrier of G st P . i = t7 & t7 is prime & ex t7 being Element of the carrier of G st P . i = t7 & t7 is i & t7 is prime ;
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 T1 & P = the topology of T2 & P = the topology of T2 & P = the topology of T1 & P = the topology of T2 holds P = Q
assume that f is_partial_differentiable_in u0 , u0 and r (#) pdiff1 ( f , 3 ) is_partial_differentiable_in u0 , 2 and partdiff u0 , 2 and partdiff ( r (#) pdiff1 ( f , 3 ) , u0 , 2 ) = r * pdiff1 ( f , u0 ) and partdiff ( r (#) pdiff1 ( f , 3 ) , u0 , 2 ) = r * pdiff1 ( f , u0 , 2 ) ;
defpred P [ Nat ] means for F , G being FinSequence of ExtREAL for s being Permutation of Seg $1 , G st len F = $1 & G = F * s & not F = G * s & not F = F * s holds Sum F = Sum G ;
ex j st 1 <= j & j < width GoB f & ( ( GoB f ) * ( 1 , j ) `2 <= s & s < ( GoB f ) * ( 1 , j + 1 ) `2 or s < j & j < width GoB f ) & ( GoB f ) * ( 1 , j + 1 ) `2 <= s ) ;
defpred U [ set , set ] means ex Fi1 being Subset-Family of T st $1 = Fi1 & union Fi1 is open & union Fi1 is open & union Fi1 is open & union Fi1 is open & union Fi1 is open & union Fi1 is open & union Fi1 is open & union Fi1 = union Fi1 & union Fi1 is 8 & union Fi1 is 8 & union Fi1 is discrete ;
for p4 being Point of TOP-REAL 2 st LE p4 , p4 , P , p1 , p2 & LE p4 , p , P , p1 , p2 & LE p4 , p , P , p1 , p2 & LE p4 , p , P , p1 , p2 holds LE p4 , p , P , p1 , p2 & LE p4 , p , P , p1 , p2
f in set ( E , H ) & for g st g . y <> f . y holds x in g implies g in set ( E , H ) & f in set ( E , H ) implies f in set ( E , H ) & f in set ( E , H )
ex 8 being Point of TOP-REAL 2 st x = 8 & ( ( ( 8 `2 / |. 8 .| - sn ) / ( 1 + sn ) ) / ( 1 + sn ) <= sn & ( ( 8 + sn ) / ( 1 + sn ) ) / ( 1 + sn ) <= 0 & ( ( 8 + sn ) / ( 1 + sn ) ) / ( 1 + sn ) <= 0 ) ;
assume for d7 being Element of NAT st d7 <= ( ( ( ( ( ( ( ( ( ( ( ( ( 1 1 , 1 ) , 1 ) \ t ) } , 1 ) --> 1 ) ^ <* 1 *> ) ^ ( t , ( ( 1 , 1 ) \ t ) ) , ( ( 1 , 1 ) \ t ) ) ) holds s1 . d = s2 . d ;
assume that s <> t and s is Point of Sphere ( x , r ) and s is Point of Sphere ( x , r ) and ex e being Point of E st { e } = Sphere ( s , t ) /\ Sphere ( x , r ) and e = Sphere ( s , t ) /\ Sphere ( 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 & |. f /. x1 - f /. x0 .| < r & |. f /. x1 - f /. x0 .| < r ;
( p | x ) | ( p | ( x | x ) ) = ( ( ( x | x ) | x ) | p ) | ( ( ( x | x ) | x ) | p ) ;
assume that x , x + h in dom sec and ( for x st x in dom sec holds ( h . x = 4 * sin . x + h . x ) / ( sin . x ) ^2 and ( h . x ) ^2 / ( cos . x ) ^2 = ( 4 * sin . x + h . x ) / ( cos . x ) ^2 / ( cos . x ) ^2 / ( cos . x ) ^2 and for x st x in dom sec holds ( h . x = 4 * sin . x = 4 * sin . x ) ^2 / ( cos . x ) ^2 / ( cos . x ) ^2 / ( cos . x ) ^2 / ( cos . x ) ^2 / ( cos . x ) ^2 / ( sin . x ) ^2 / ( cos . x ) ^2 / ( cos . x ) ^2 / ( cos . x ) ^2 / ( cos . x ) ^2 / ( cos . x ) ^2 /
assume that i in dom A and len A > 1 and B c= the set of \HM { i , j } and A c= the set of \HM { i , j } and B = ( i , j ) |-> ( i , j ) and ( i = j implies A = B ) and ( i = j implies A = B ) ;
for i be non zero Element of NAT st i in Seg n holds i divides n or i = <* 1. F_Complex , n *> & ( i <> n implies h . i = <* 1_ F_Complex , n *> ) & ( i <> n implies h . i = - 1_ F_Complex ) & ( i <> n implies h . i = - 1_ F_Complex implies h . i = - 1_ F_Complex )
( ( b1 'imp' b2 ) '&' ( c1 'imp' c2 ) '&' ( a1 'or' b1 'or' c1 ) '&' ( a2 'or' b2 ) '&' 'not' ( a2 '&' b2 ) '&' 'not' ( a2 '&' b2 ) '&' 'not' ( b2 '&' c2 ) '&' 'not' ( a2 '&' b2 ) '&' 'not' ( a2 '&' b2 ) '&' 'not' ( b2 '&' c2 ) '&' 'not' ( a2 '&' b2 ) '&' 'not' ( b2 '&' c2 ) '&' 'not' ( a2 '&' b2 ) '&' 'not' ( a2 '&' b2 ) '&' 'not' ( b2 '&' c2 ) '&' 'not' ( a2 '&' b2 ) '&' 'not' ( b2 '&' c2 ) '&' 'not' ( b2 '&' c2 ) '&' 'not' ( b2 ) '&' 'not' ( b2 '&' c2 ) '&' 'not' ( b2 '&' c2 ) '&' 'not' ( b2 '&' c2 ) '&' 'not' ( b2 '&' c2 ) '&' 'not' ( b2 '&' c2 ) '&' 'not' ( a2 '&' c2 ) '&' 'not' ( a2 '&' c2 ) '&' 'not' ( a2 '&' c2 ) '&' 'not' ( a2 '&' c2 ) '&' 'not'
assume that for x holds f . x = ( ( cot * ( sin + cos ) ) / ( sin . x ) ^2 and x , h / ( sin . x ) ^2 / ( sin . x ) ^2 and for x st x in Z holds ( ( ( cot * ( sin + cos ) ) / ( sin . x ) ^2 ) = cos . ( x- ) ;
consider Rd , I-8 be Real such that Rd = Integral ( M , F . n ) and I-8 = Integral ( M , F . n ) and Integral ( M , F . n ) = Integral ( M , F . n ) + Integral ( M , F . n ) and Integral ( M , F . n ) = Rd + ( I . n ) * i ;
ex k be Element of NAT st k = k & 0 < d & for q be Element of product G st q in X & ||. qLet ( f , q , k ) - partdiff ( f , x , k ) .|| < r holds ||. partdiff ( f , q , k ) - partdiff ( f , x , k ) .|| < r
x in { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 } iff x in { x1 , x2 , x3 , x4 , 8 , 8 , 7 , 8 } \/ { 8 , 8 , 7 , 8 }
G * ( j , i ) `2 = G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= p `2 .= G * ( 1 , j ) `2 .= p `2 .= G * ( 1 , j ) `2 .= p `2 .= p `2 .= p `2 .= p `2 .= p `2 .= p `2 .= p `2 .= p `2 ;
f1 * p = p .= ( ( the Arity of S1 ) +* ( the Arity of S2 ) ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o ;
func tree ( T , P , T1 ) -> DecoratedTree means : Def3 : q in it iff q in T & for p , q st p in P & q in T holds p ^ q in T1 or ex r , s st r in P & s in T & p ^ r = p ^ r ;
F /. ( k + 1 ) = F . ( k + 1 -' 1 ) .= Fq . ( k + 1 -' 1 ) .= Fq . ( k + 1 -' 1 ) .= Fq . ( k + 1 -' 1 ) .= Fq . k .= Fq . k .= Fq . k ;
for A , B , C being Matrix of K st len B = len C & width B = width C & len B = width C & len A > 0 & len B > 0 & len A > 0 & len B > 0 & len A > 0 & len B > 0 & len A > 0 & len B > 0 holds A * ( B * C ) = A * B- B * ( B * C )
seq . ( k + 1 ) = 0. F_Complex + 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 Cy ) /\ ( the carrier of Cy ) and y in ( the carrier of Cy ) /\ ( the carrier of Cy ) and [ x , y ] in the InternalRel of Cy and [ x , y ] in the InternalRel of Cy and [ x , y ] in the InternalRel of Cy ;
defpred P [ Element of NAT ] means for f st len f = $1 holds ( VAL g ) . ( k + 1 ) = ( VAL g ) . ( k + 1 ) '&' ( VAL g ) . ( k + 1 ) '&' ( VAL g ) . ( k + 1 ) ;
assume that 1 <= k and k + 1 <= len f and f is_sequence_on G and [ i , j ] in Indices G and f /. k = G * ( i , j ) and f /. ( k + 1 ) = G * ( i , j ) and f /. ( k + 1 ) = G * ( i , j ) and f /. ( k + 1 ) = G * ( i + 1 , j ) ;
assume that cn < 1 and q `1 > 0 and q `2 / |. q .| >= sn and p = ( sn and q `2 / |. q .| >= sn and p = ( sn and q `2 / |. q .| ) and p = ( sn ) / |. q .| and p = ( sn ) / |. q .| and q = ( sn ) / |. q .| and p = ( sn ) / |. q .| ;
for M being non empty TopSpace , x being Point of M , f being Point of M st x = x `1 holds ex f being sequence of B`2 st for n being Element of NAT holds f . n = Ball ( x `1 , 1 / ( n + 1 ) ) & f . 0 = Ball ( x `1 , 1 / ( n + 1 ) )
defpred P [ Element of omega ] means f1 is_differentiable_on Z & f2 is_differentiable_on Z & for x st x in Z holds ( f1 - f2 ) . x = f1 . x - f2 . x / ( f1 . x ) ^2 / ( f1 . x ) ^2 / ( f1 . x ) ^2 / ( f1 . x ) ^2 / ( f1 . x ) ^2 / ( f1 . x ) ^2 / ( f1 . x ) ^2 / ( f1 . x ) ^2 ;
defpred P1 [ Nat , Point of CNS ] means $2 in Y & ||. s1 . $1 - ( f /. $2 ) .|| < r & ||. f /. $1 - ( f /. ( $1 + 1 ) ) - ( f /. ( $1 + 1 ) ) .|| < r / 2 ;
( 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 ) ;
( 1 / 2 * n0 + 2 * n0 ) * ( 2 * n0 + 2 * n0 ) = ( ( 1 / 2 * n0 + 2 * n0 ) * ( 2 * n0 + 2 * n0 ) ) * ( 2 * n0 + 2 * n0 + 2 * n0 ) .= ( 1 / 2 * n0 + 2 * n0 + 2 * n0 ) * ( 2 * n0 + 2 * n0 ) .= 1 * ( 2 * n0 + 2 * n0 ) ;
defpred P [ Nat ] means for G being non empty strict finite symmetric RelStr st G is as free for S being non empty RelStr st S is as non empty & the carrier of G = { the carrier of G } & the carrier of G = { the carrier of G } holds the RelStr of G = the RelStr of G ;
assume that not f /. 1 in Ball ( u , r ) and 1 <= m & m <= len f and for i st 1 <= i & i <= len f & LSeg ( f , i ) /\ Ball ( u , r ) <> {} and not f /. i in Ball ( u , r ) and not m <= i & i <= len f holds not f /. i in Ball ( u , r ) ;
defpred P [ Element of NAT ] means ( Partial_Sums ( cos ) . $1 ) . ( x - r ) = ( Partial_Sums ( cos ) . ( x - r ) ) . ( x - r ) + ( ( cos ) . ( x - r ) ) * ( x - r ) ;
for x being Element of product F holds x is FinSequence of G & dom x = I & x = ( the carrier of F ) & for i being set st i in dom F holds x . i in ( the carrier of F ) & for i being set st i in dom F holds x . i in ( the carrier of F ) . i
( x " ) |^ ( n + 1 ) = ( x " ) |^ n * x " .= ( x * x ) " .= ( x * x ) " .= ( x * x ) " .= ( x * x ) " .= ( x * x ) " .= ( x * x ) " .= ( x * x ) " .= ( x * x ) " .= ( x * x ) " ;
DataPart Comput ( P +* ( a , I ) , Initialized s , LifeSpan ( P +* I , s ) + 3 ) = DataPart Comput ( P +* I , s , LifeSpan ( P +* I , s ) + 3 ) .= DataPart Comput ( P +* I , s , LifeSpan ( P +* I , s ) + 3 ) ;
given r such that 0 < r and ]. x0 , x0 + r .[ c= ( dom f1 /\ dom f2 ) and for g st g in ]. x0 , x0 + r .[ /\ dom f1 and g <= x0 and for g st g in ]. x0 , x0 + r .[ /\ dom f1 holds f1 . g <= f1 . g & f1 . g <= f1 . g ;
assume that X c= dom f1 /\ dom f2 and f1 | X is continuous and f2 | X is continuous and ( for r st r in X /\ dom f2 holds f1 + f2 | X is continuous & ( for r st r in X /\ dom f2 holds f1 . r = r ) and ( for r st r in X /\ dom f2 holds f1 . r = r ) implies f1 + f2 is continuous & ( for r st r in X /\ dom f2 holds f1 + f2 | X is continuous & f1 + f2 is continuous & ( for r st r in X & r in X & r < r ex r ex r ex r st r in X & r <= r ) & ( r - f2 ) implies f1 + f2 is continuous & ( r - f2 ) implies f1 + f2 is continuous & ( r - f2 ) implies f1 + f2 is continuous & ( r - f2 ) & ( r - f2 | X is continuous & ( r < r implies f1 + f2 is continuous & ( r - f2 ) | X is continuous & ( r - f2 ) | X is continuous &
for L being continuous complete LATTICE st for l being Element of L ex X being Subset of L st l = sup X & for x being Element of L st x in X holds x is an & for x being Element of L st x in X holds x is an & x is an & x is an & x is Element of L holds x = "\/" ( waybelow l , L )
Support e8 in { Support ( m *' p ) where m is Polynomial of n , L : m in dom ( m *' p ) & ex i being Element of NAT st i in dom ( m *' p ) & ( m /. i ) = p . i & ( m /. i ) = p . i & ( m /. i ) = p . i ) ;
( f1 - f2 ) /. ( lim s1 ) = lim ( f1 /* s1 ) - lim ( f2 /* s1 ) .= lim ( f1 /* s1 ) - lim ( f2 /* s1 ) .= lim ( f1 /* s1 ) - lim ( f2 /* s1 ) .= lim ( f1 /* s1 ) - lim ( f2 /* s1 ) .= lim ( f2 /* s1 ) - lim ( f2 /* s1 ) ;
ex p1 being Element of CQC-WFF ( Al ) st p1 = g . p1 & for g being Function of [: V , ( len p1 ) , D ( ) :] , D ( ) st P [ g , p1 , ( len p1 ) + 1 ] holds P [ p1 , ( len p1 ) + 1 ] ;
( mid ( f , i , len f -' 1 ) ^ <* f /. j *> ) /. j = ( mid ( f , i , len f -' 1 ) ) /. j .= ( mid ( f , i , len f -' 1 ) ) /. j .= ( mid ( f , i , len f -' 1 ) ) /. j .= f /. ( j + i -' 1 ) ;
( ( p ^ q ) ^ r ) . ( len p + k ) = ( ( p ^ q ) . k ) ^ r . ( len p + k ) .= ( ( p ^ q ) . k ) ^ r . ( len p + k ) .= ( p ^ r ) . k .= ( p ^ r ) . k .= ( p ^ r ) . k .= p . k ;
len mid ( upper_volume ( f , D2 ) , indx ( D2 , D1 , j1 ) + 1 , indx ( D2 , D1 , j1 ) ) = indx ( D2 , D1 , j1 ) + ( indx ( D2 , D1 , j1 ) + 1 ) - ( indx ( D2 , D1 , j1 ) + 1 ) ;
x * y * z = ( x * y ) * ( z * w ) .= ( x9 * y9 ) * ( y9 * z9 ) .= x9 * ( y9 * z9 ) .= x9 * ( y9 * z9 ) .= x9 * ( y9 * z9 ) .= x9 * ( y9 * z9 ) .= x9 * ( y9 * z9 ) .= x9 * ( y9 * z9 ) .= x9 * y9 ;
v . <* x , y *> + ( <* x0 , y0 *> ) * i = partdiff ( v , ( x - x0 ) * i ) + partdiff ( u , ( x - x0 ) * i ) + ( proj ( 1 , 1 ) * i ) + proj ( 1 , 1 ) * ( ( x - x0 ) * ( ( x - x0 ) * i ) + proj ( 1 , 1 ) * ( ( x - x0 ) * i ) ;
i * i = <* 0 * ( - 1 ) - ( 0 * 0 ) , 0 * 0 + ( 0 * 0 ) , 0 * 0 + ( 0 * 0 ) + 0 * 0 + ( 0 * 0 ) - ( 0 * 0 ) , 0 * 0 + ( 0 * 0 ) - 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 .= <* - 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0
Sum ( L (#) F ) = Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) F1 ) + Sum ( L (#) F2 ) .= Sum ( L (#) F1 ) + Sum ( L (#) F2 ) .= Sum ( L (#) F1 ) + Sum ( L (#) F2 ) .= Sum ( L (#) F1 ) + Sum ( L (#) F2 ) .= Sum ( L (#) F1 ) + Sum ( L (#) F2 ) .= Sum ( L (#) F1 ) ;
ex r be Real st for e be Real st 0 < e ex Y0 be finite Subset of X st Y0 is non empty & Y0 c= Y & for Y1 be finite Subset of X st Y1 c= Y & Y1 c= Y holds |. ( union Y1 ) - ( lower_bound Y1 ) .| < r
( GoB f ) * ( i , j ) = f /. ( k + 2 ) & ( GoB f ) * ( i , j + 1 ) = f /. ( k + 1 ) or ( GoB f ) * ( i , j + 1 ) = f /. ( k + 1 ) & ( GoB f ) * ( i , j ) = f /. ( k + 1 ) or ( GoB f ) * ( i , j ) = f /. ( k + 1 ) ;
( ( - cos ) . x ) ^2 = ( ( r ^2 - 1 ) * ( sin . x ) ^2 + ( - 1 ) * ( sin . x ) ^2 .= ( ( r ^2 - 1 ) * ( sin . x ) ) ^2 + ( ( r ^2 - 1 ) * ( sin . x ) ^2 ) .= ( r ^2 - 1 ) * ( cos . x ) ^2 .= ( r ^2 - 1 ) * ( cos . x ) ^2 ;
x- ( b + sqrt ( a , b , c ) / 2 * a ) < 0 & - ( b - sqrt ( a , b , c ) / 2 * a ) / 2 < 0 or - ( b - sqrt ( a , b , c ) / 2 * a ) / 2 > 0 & - ( - ( a , b , c ) / 2 * a ) / 2 > 0 ;
assume that ex_inf_of uparrow X /\ C and ex_sup_of X , L and ex_sup_of X , L and "\/" ( ( subrelstr X ) /\ C , L ) = "/\" ( ( subrelstr X ) /\ C , L ) and not "\/" ( ( subrelstr X ) /\ C , L ) = "/\" ( ( subrelstr X ) /\ C , L ) and not "\/" ( ( subrelstr X ) /\ C , L ) = "/\" ( ( uparrow X ) /\ C , L ) ;
( for j being Element of OL ) . ( j , i ) = ( j |-- id the carrier of B ) . ( i , j ) .= ( i |-- id B ) . ( i , j ) .= ( i |-- id B ) . ( i , j ) .= ( i |-- id B ) . ( i , j ) .= ( i |-> id B ) . ( i , j ) ;