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assume not thesis ;
assume not thesis ;
B in X ;
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 finite ;
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 prime ;
not x in Y ;
z = +infty ;
k be Nat ;
K is being_line ;
assume n >= N ;
assume n >= N ;
assume X is ) ;
assume x in I ;
q is as by 0 ;
assume c in x ;
1-p > 0 ;
assume x in Z ;
assume x in Z ;
1 <= k} ;
assume m <= i ;
assume G is finite ;
assume a divides b ;
assume P is closed ;
b-a > 0 ;
assume q in A ;
W is not bounded ;
f is Assume f is Assume g is one-to-one ;
assume A is boundary ;
g is_sequence_on G ;
assume i > j ;
assume t in X ;
assume n <= m ;
assume x in W ;
assume r in X ;
assume x in A ;
assume b is even ;
assume i in I ;
assume 1 <= k ;
X is non empty ;
assume x in X ;
assume n in M ;
assume b in X ;
assume x in A ;
assume T c= W ;
assume s is atomic ;
b `1 <= c `1 ;
A meets W ;
i `1 <= j `1 ;
assume H is universal ;
assume x in X ;
let X be set ;
let T be Tree ;
let d be element ;
let t be element ;
let x be element ;
let x be element ;
let s be element ;
k <= 5 - 5 ;
let X be set ;
let X be set ;
let y be element ;
let x be element ;
P [ 0 ]
let E be set , A be Element of E ;
let C be category ;
let x be element ;
k be Nat ;
let x be element ;
let x be element ;
let e be element ;
let x be element ;
P [ 0 ]
let c be element ;
let y be element ;
let x be element ;
a be Real ;
let x be element ;
let X be element ;
P [ 0 ]
let x be element ;
let x be element ;
let y be element ;
r in REAL ;
let e be element ;
n1 is j ;
Q halts_on s ;
x in } ;
M < m + 1 ;
T2 is open ;
z in b < a < b ;
R2 is well-ordering ;
1 <= k + 1 ;
i > n + 1 ;
q1 is one-to-one ;
let x be trivial set ;
PM is one-to-one ;
n <= n + 2 ;
1 <= k + 1 ;
1 <= k + 1 ;
let e be Real ;
i < i + 1 ;
p3 in P ;
p1 in K ;
y in C1 ;
k + 1 <= n ;
let a be Real , x be Point of TOP-REAL 2 ;
X |- r => p ;
x in { A } ;
let n be Nat ;
let k be Nat ;
let k be Nat ;
let m be Nat ;
0 < 0 + k ;
f is_differentiable_in x ;
let x0 , r ;
let E be Ordinal ;
o f2 f2 f2 f2 and f2 S is <= ;
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 Z , M be Subset of V ;
not s in Y |^ 0 ;
rng f <= w
b "/\" e = b ;
m = m3 ;
t in h . D ;
P [ 0 ] ;
assume z = x * y ;
S . n is bounded ;
let V be RealLinearSpace , M be Matrix of V , REAL ;
P [ 1 ] ;
P [ {} ] ;
C1 meets C ` ;
H = G . i ;
1 <= i `1 + 1 ;
F . m in A ;
f . o = o ;
P [ 0 ] ;
anon <= being Real ;
R [ 0 ] ;
b in f .: X ;
assume q = q2 ;
x in [#] V ;
f . u = 0 ;
assume e1 > 0 ;
let V be RealUnitarySpace , M be Matrix of V , REAL ;
s is trivial non empty ;
dom c = Q
P [ 0 ] ;
f . n in T ;
N . j in S ;
let T be complete LATTICE , X be set ;
the Arrows 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 ;
Sy is bounded ;
assume p = p2 ;
len f = n ;
assume x in P1 ;
i in dom q ;
let UA , A , B ;
pp = c & pp = d ;
j in dom h ;
let k ;
f | Z is continuous ;
k in dom G ;
UBD C = B ;
1 <= len M ;
p in Ball ( x , r ) ;
1 <= ( j - 1 ) ;
set A = [: \cap :] ;
card a [= c ;
e in rng f ;
cluster B \oplus A -> empty ;
H is with_\cdot \cdot \cdot \cdot is non empty ;
assume n0 <= m ;
T is increasing ;
e2 <> e2 ;
Z c= dom g ;
dom p = X ;
H is proper implies 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 in union M ;
assume not x in REAL ;
Im f is complete ;
x in Int y ;
dom F = M ;
a in On W ;
assume e in A ( ) ;
C c= C-26 ;
mm <> {} ;
let x be Element of Y ;
let f be ) ;
not n in Seg 3 ;
assume X in f .: A ;
assume that p <= n and p <= m ;
assume not u in { v } ;
d is Element of A ;
A |^ b misses B ;
e in v + dom \vert being set ;
- 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 l -countable set ;
rng f c= NAT ;
assume P [ k ] ;
f6 <> {} ;
o be Ordinal ;
assume x is sum of squares ;
assume not v in { 1 } ;
let IC , I ;
assume that 1 <= j and j < l ;
v = - u ;
assume s . b > 0 ;
\bf \bf d ;
assume t . 1 in A ;
let Y be non empty TopSpace , f be Function of Y , BOOLEAN ;
assume a in uparrow s ;
let S be non empty Poset ;
a , b // b , a ;
a * b = p * q ;
assume x , y are_the space ;
assume x in Omega ( f ) ;
[ a , c ] in X ;
mm <> {} ;
M + N c= M + M ;
assume M is \mathclose hhhh;
assume f is additive as bb\rm } ;
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 k2 = k2 ;
m + 1 < n + 1 ;
s in S \/ { s } ;
n + i >= n + 1 ;
assume Re y = 0 ;
k1 <= j1 & j1 <= len f ;
f | A is as as as as as continuous Function ;
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 ;
CC in X ;
q2 c= C1 & q2 c= C2 ;
a2 < c2 & a2 < b2 ;
s2 is 0 -started ;
IC s = 0 & IC s = 0 ;
s4 = s4 & s4 = s4 ;
let V ;
let x , y be element ;
let x be Element of T ;
assume a in rng F ;
x in dom T `1 ;
let S be * of L ;
y " <> 0 ;
y " <> 0 ;
0. V = u-w ;
y2 , y , z is_collinear ;
R8 in dom f ;
let a , b be Real , x be Point of TOP-REAL 2 ;
let a be Object of C ;
let x be Vertex of G ;
let o be Object of C , m be Morphism of C ;
r '&' q = P \lbrack l \rbrack ;
let i , j be Nat ;
let s be State of A , n be Nat ;
s4 . n = N ;
set y = ( x `1 ) ^2 ;
mi in dom g ;
l . 2 = y1 ;
|. g . y .| <= r ;
f . x in Cx0 ;
V2 is non empty ;
let x be Element of X ;
0 <> f . g2 ;
f2 /* q is convergent ;
f . i is_measurable_on E ;
assume \xi in N-22 ;
reconsider i = i as Ordinal ;
r * v = 0. X ;
rng f c= INT & f is one-to-one ;
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 = the carrier of C ;
let o be OperSymbol of S ;
let R be connected non empty Poset ;
n + 1 = succ n ;
x-21 c= Z1 & xq c= Z1 ;
dom f = C1 & rng g c= C2 ;
assume [ a , y ] in X ;
Re ( seq ) is convergent & Im ( seq ) is convergent ;
assume a1 = b1 & a2 = b2 ;
A = sInt A ;
a <= b or b <= a ;
n + 1 in dom f ;
let F be Instruction of S , i be Nat ;
assume r2 > x0 & x0 < r2 ;
let Y be non empty set , f be Function of Y , BOOLEAN ;
2 * x in dom W ;
m in dom ( g2 * f1 ) ;
n in dom g1 /\ dom g2 ;
k + 1 in dom f ;
not the still of { s } is finite ;
assume x1 <> x2 & x1 <> x3 ;
v3 in V1 & v2 in V2 ;
not [ b `1 , b ] in T ;
i-35 + 1 = i ;
T c= `2 & T c= `2 ;
l `1 = 0 & l `2 = 0 ;
n be Nat ;
t `2 = r `2 & t `1 <= r ;
AA is_integrable_on M & AA is_integrable_on M ;
set t = Top t ;
let A , B be real-membered set ;
k <= len G + 1 ;
cC misses cV ;
Product ( seq ) is non empty ;
e <= f or f <= e ;
cluster non empty normal -> normal for Ordinal ;
assume c2 = b2 & c2 = b1 ;
assume h in [. q , p .] ;
1 + 1 <= len C ;
not c in B . m1 ;
cluster R .: X -> empty ;
p . n = H . n ;
assume ( vseq - vseq ) is convergent ;
IC s3 = 0 & IC s3 = 0 ;
k in N or k in K ;
F1 \/ F2 c= F ;
Int ( G1 \/ G2 ) <> {} ;
z `2 = 0 & z `1 = 0 ;
p01 <> p1 & 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 be be be be be be be be be be halting of S ;
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 ;
f " P is compact & f " P is compact ;
assume x1 in REAL & x2 in REAL ;
p1 = K1 & p2 = K1 ;
M . k = <*> REAL ;
phi . 0 in rng phi ;
OSMis closed ;
assume z0 <> 0. L & z0 <> 0. L ;
n < ( N . k ) . n ;
0 <= seq . 0 & seq . 0 <= seq . 0 ;
- q + p = v ;
{ v } is Subset of B ;
set g = f /. 1 ;
cR is stable Subset of R ;
set cR = Vertices R , cS = Vertices R , cT = Vertices S , cT = Vertices S , cT = Vertices S , cT = Vertices
px0 c= P3 & px0 c= P3 ;
x in [. 0 , 1 .[ ;
f . y in dom F ;
let T be Scott Scott Scott Scott Scott of S ;
ex_inf_of the carrier of S , S ;
downarrow a = downarrow b ;
P , C , K is_collinear ;
assume x in F ( s , r , t ) ;
2 to_power i < 2 to_power m ;
x + z = x + z + q ;
x \ ( a \ x ) = x ;
||. x-y .|| <= r ;
assume that Y c= field Q and Y <> {} ;
a ~ , b ~ are_equipotent ;
assume a in A ( ) . i ;
k in dom ( q | i ) ;
p is \HM { finite of S ;
i -' 1 = i-1 - 1 ;
f | A is one-to-one ;
assume x in f .: X ( ) ;
i2 - i1 = 0 & i2 = 0 ;
j2 + 1 <= i2 ;
g " * a in N ;
K <> { [ {} , {} ] } ;
cluster strict for } -valued commutative Ring ;
|. q .| ^2 > 0 ;
|. p4 .| = |. p .| ;
s2 - s1 > 0 & s2 - s1 > 0 ;
assume x in { Gij } ;
W-min C in C & W-min C in C ;
assume x in { Gij } ;
assume i + 1 = len G ;
assume i + 1 = len G ;
dom I = Seg n & rng I c= Seg n ;
assume that k in dom C and k <> i ;
1 + 1-1 <= i + j ;
dom S = dom F & rng F c= dom G ;
let s be Element of NAT ;
let R be ManySortedSet of A ;
let n be Element of NAT ;
let S be non empty non void non void holds S is holds S is non empty ;
let f be ManySortedSet of I ;
let z be Element of COMPLEX , p be FinSequence of COMPLEX ;
u in { ag } ;
2 * n < 2 * n ;
x , y be set ;
B-11 c= V1 \/ V2 & B-11 c= V1 ;
assume I is_halting_on s , P & I is_halting_on s , P ;
UA = [: U , U :] ;
M /. 1 = z /. 1 ;
x11 = x22 & x22 = x22 ;
i + 1 < n + 1 + 1 ;
x in { {} , <* 0 *> } ;
( f | n ) . x <= ( f | n ) . x ;
let l be Element of L ;
x in dom ( F | S ) ;
let i be Element of NAT ;
seq1 is COMPLEX & seq2 is COMPLEX implies seq1 - seq2 is COMPLEX
assume <* o2 , o *> <> {} ;
s . x |^ 0 = 1 ;
card K1 in M & card K1 in M ;
assume that X in U and Y in U ;
let D be Subset-Family of Omega ;
set r = { q } + 1 ;
y = W . ( 2 * x ) ;
assume dom g = cod f & cod g = cod f ;
let X , Y be non empty TopSpace , f be Function of X , Y ;
x \oplus A is interval ;
|. <*> A .| . a = 0 ;
cluster strict for Sublattice of L ;
a1 in B . s1 & a2 in B . s1 ;
let V be finite { F , G , H be non empty VectSpStr over F ;
A * B on B implies A on B
f-3 = NAT --> 0 ;
let A , B be Subset of V ;
z1 = P1 . j & z2 = P2 . j ;
assume f " P is closed ;
reconsider j = i as Element of M ;
let a , b be Element of L ;
assume q in A \/ ( B "\/" C ) ;
dom ( F * C ) = o ;
set S = INT * , T = INT * ;
z in dom ( A --> y ) ;
P [ y , h . y ] ;
{ x0 } c= dom f /\ dom g ;
let B be non-empty ManySortedSet of I , A be ManySortedSet of I ;
PI / 2 < Arg z ;
reconsider z9 = 0 , z9 = 1 as Nat ;
LIN a , d , c ;
[ y , x ] in IB ;
Q * ( 1 , 3 ) `1 = 0 ;
set j = x0 gcd m , k = m gcd n ;
assume a in { x , y , c } ;
j2 - jj > 0 & j2 - jj > 0 ;
I I \HM { = } phi = 1 ;
[ y , d ] in F-8 ;
let f be Function of X , Y ;
set A2 = ( B - C ) / ( A + B ) ;
s1 , s2 be \in X & s1 <= s2 implies s1 <= s2
j1 -' 1 = 0 & j1 -' 1 = 0 ;
set m2 = 2 * n + j ;
reconsider t = t as bag of n ;
I2 . j = m . j ;
i |^ s , n are_relative_prime ;
set g = f | D-21 ;
assume that X is lower and 0 <= r ;
p1 `1 = 1 & p2 `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 <= len f ;
1 <= i1 -' 1 & i1 <= len f ;
i + i2 <= len h & i + i2 <= len h ;
x = W-min ( P ) & y = W-min ( P ) ;
[ x , z ] in [: X , Z :] ;
assume y in [. x0 , x .] ;
assume p = <* 1 , 2 , 3 *> ;
len <* A1 *> = 1 & len <* A1 *> = 1 ;
set H = h . gg ;
card b * a = |. a .| ;
Shift ( w , 0 ) |= v ;
set h = h2 (*) h1 , h1 = h2 (*) h1 ;
assume x in X3 /\ X4 ;
||. h .|| < d1 & ||. h .|| < d ;
not x in the carrier of f ;
f . y = F ( y ) ;
for n holds X [ n ] ;
k - l = kbeing Nat ;
<* p , q *> /. 2 = q ;
let S be Subset of the carrier of Y ;
let P , Q be \langle s *> ;
Q /\ M c= union ( F | M )
f = b * ( canFS ( S ) ) ;
let a , b be Element of G ;
f .: X is_<=_than f . sup X
let L be non empty transitive RelStr , x be Element of L ;
S-20 is x -basis i -basis of n
let r be non positive Real ;
M , v |= x \hbox { y } ;
v + w = 0. ( Z1 , p ) ;
P [ len F ( ) ] ;
assume InsCode ( i ) = 8 & InsCode ( i ) = 8 ;
the zero of M = 0 & the zero of M = 0 ;
cluster z * seq -> summable for Real_Sequence ;
let O be Subset of the carrier of C ;
||. f .|| | X is continuous ;
x2 = g . ( j + 1 ) ;
cluster -> \ implies for Element of S ;
reconsider l1 = l-1 as Nat ;
v4 is Vertex of r2 & v4 is Vertex of G ;
T2 is SubSpace of T2 & T1 is SubSpace of T2 ;
Q1 /\ Q19 <> {} & Q1 /\ Q29 <> {} ;
k be Nat ;
q " is Element of X & q " is Element of X ;
F . t is set of REAL , M ;
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 of ( p `2 ) ^2 , ( p `2 ) ^2 ;
not r in ]. p , q .[ ;
let R be FinSequence of REAL , p be FinSequence of REAL ;
not SS does not destroy b1 & not I does not destroy b2 ;
IC SCM R <> a & IC SCM R = a ;
|. - |[ x , y ]| .| >= r ;
1 * seq = seq & seq is convergent & lim seq = 0 ;
let x be FinSequence of NAT , p be FinSequence ;
let f be Function of C , D , g be Function of C , D ;
for a holds 0. L + a = a
IC s = s . NAT .= s . NAT ;
H + G = F- ( G-G ) ;
Cm1 . x = x2 & Cm2 . x = y2 ;
f1 = f .= f2 .= ( f | X ) . f1 ;
Sum <* p . 0 *> = p . 0 ;
assume v + W = v + u + W ;
{ a1 } = { a2 } ;
a1 , b1 _|_ b , a ;
d3 , o _|_ o , a3 & d1 , c _|_ d , c ;
IC is_reflexive & IC is_reflexive implies C is transitive
IO is_antisymmetric implies CO is_antisymmetric & IO is_antisymmetric = CO
upper_bound rng H1 = e & upper_bound rng H2 = e ;
x = ( a * a9 ) * ( a * b9 ) ;
|. p1 .| ^2 >= 1 ^2 ;
assume j2 -' 1 < 1 & j2 -' 1 < len f ;
rng s c= dom f1 /\ dom f2 ;
assume support a misses support b & support b misses support b ;
let L be associative non empty doubleLoopStr , p be Polynomial of L ;
s " + 0 < n + 1 ;
p . c = ( f " ) . 1 ;
R . n <= R . ( n + 1 ) ;
Directed I = I1 +* ( card I + 3 ) ;
set f = + ( x , y , r ) ;
cluster Ball ( x , r ) -> bounded ;
consider r being Real such that r in A ;
cluster non empty NAT -defined for NAT -defined Function ;
let X be non empty directed Subset of S ;
let S be non empty full SubRelStr of L ;
cluster <* \hbox { \boldmath $ N . N } , \subseteq \rangle -> complete non trivial ;
( 1 - a ) " = a ;
( q . {} ) `1 = o ;
( n - 1 ) - 1 > 0 ;
assume ( 1 / 2 ) <= t `1 / 1 ;
card B = k + 1-1 ;
x in union rng ( f | ( len f -' 1 ) ) ;
assume x in the carrier of R & y in the carrier of R ;
d in X ;
f . 1 = L . ( F . 1 ) ;
the vertices of G = { v } & not x in { v } ;
let G be st G is } -wwgraph ;
e , v6 be set ;
c . ( i - 1 ) in rng c & c . ( i - 1 ) in rng c ;
f2 /* q is divergent_to-infty & f2 /* q is divergent_to-infty ;
set z1 = - z2 , z2 = - z1 , z1 = - z2 ;
assume w is_llof S , G ;
set f = p |-count t , g = p |-count t , h = p |-count t , n = p |-count t , n = p |-count t , m = p |-count t , n = p |-count t
let c be Object of C ;
assume ex a st P [ a ] ;
let x be Element of REAL m , y be Element of REAL m ;
let IB be Subset-Family of X , C be Subset of X ;
reconsider p = p as Element of NAT ;
let v , w be Point of X ;
let s be State of SCM+FSA , P be Subset of SCM+FSA ;
p is finite & q is FinSequence of the carrier of SCM+FSA ;
stop I ( ) c= P-12 & stop I c= Pc ;
set ci = fbeing /. i ;
w ^ t |- w ^ s ;
W1 /\ W = W1 /\ W ` .= ( W1 /\ W2 ) /\ W ;
f . j is Element of J . j ;
let x , y be Subset of T2 , a be Element of T1 ;
ex d st a , b // b , d ;
a <> 0 & b <> 0 & c <> 0
ord x = 1 & x is dom \circ implies x is dom x
set g2 = lim ( seq , - seq ) ;
2 * x >= 2 * ( 1 / 2 ) ;
assume ( a 'or' c ) . z <> TRUE ;
f (*) g in Hom ( c , c ) ;
Hom ( c , c + d ) <> {} ;
assume 2 * Sum ( q | m ) > m ;
L1 . F-21 = 0 & L2 . F-21 = 0 ;
/ ( X \/ R1 ) = / ( X /\ R1 )
( sin . x ) <> 0 & ( sin . x ) <> 0 ;
( #Z n ) . x > 0 & ( #Z n ) . x > 0 ;
o1 in X-5 /\ O2 & o2 in XO /\ O2 ;
e , v6 be set ;
r3 > ( 1 - 2 ) * 0 ;
x in P .: ( F -ideal ) ;
let J be closed non empty Subset of R ;
h . p1 = f2 . O & h . p2 = g2 . O ;
Index ( p , f ) + 1 <= j ;
len ( q | i ) = width M & width ( q | i ) = width M ;
the carrier of CK c= A & the carrier of CK c= A ;
dom f c= union rng ( F | ( -10 + 1 ) )
k + 1 in support ( support ( n ) ) ;
let X be ManySortedSet of the carrier of S ;
[ x `1 , y `2 ] in ( an an element ) ;
i = D1 or i = D2 or i = D1 ;
assume a mod n = b mod n & b mod n = 0 ;
h . x2 = g . x1 & h . x2 = g . x2 ;
F c= 2 -tuples_on the carrier of X ;
reconsider w = |. s1 .| as Real_Sequence ;
( 1 / m * m + r ) < p ;
dom f = dom ( I --> ( f . x ) ) ;
[#] P-17 = [#] ( ( TOP-REAL 2 ) | K1 ) ;
cluster - x -> ExtReal for ExtReal ;
then { db } c= A & A is closed ;
cluster TOP-REAL n -> finite-ind for non empty TopSpace ;
let w1 be Element of M , w2 be Element of M ;
let x be Element of dyadic ( n ) ;
u in W1 & v in W3 implies u in W1 /\ W3
reconsider y = y 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 mm1 = m - 1 as Element of NAT ;
x- x0 < r1 - x0 & r1 < x0 + r2 ;
reconsider P ` = P ` as strict Subgroup of N ;
set g1 = p * ( idseq q ) " ;
let n , m , k be non zero Nat ;
assume that 0 < e and f | A is lower ;
D2 . ( I8 + 1 ) in { x } ;
cluster subcondensed -> subopen for Subset of T ;
let P be compact non empty Subset of TOP-REAL 2 , p1 , p2 be Point of TOP-REAL 2 ;
Gik in LSeg ( cos , 1 ) /\ LSeg ( Gik , Gij ) ;
n be Element of NAT , x be Element of X ;
reconsider SS = S , SS = S as Subset of T ;
dom ( i .--> X ` ) = { i } ;
let X be non-empty ManySortedSet of S ;
let X be non-empty ManySortedSet of S ;
op ( 1 ) c= { [ {} , {} ] } ;
reconsider m = mm as Element of NAT ;
reconsider d = x as Element of C ( ) ;
let s be 0 -started State of SCMPDS , P be s , Q ;
let t be 0 -started State of SCMPDS , Q ;
b , b , x , y , z be element ;
assume that i = n \/ { n } and j = k \/ { k } ;
let f be PartFunc of X , Y ;
x0 >= ( sqrt c / sqrt 2 ) / sqrt 2 ;
reconsider t7 = T-1 as TopSpace ;
set q = h * p ^ <* d *> ;
z2 in U . ( x4 ) /\ Q2 . ( x4 ) /\ Q2 . ( x4 ) ;
A |^ 0 = { <%> E } & A |^ 0 = { <%> E } ;
len W2 = len W + 2 .= len W + 2 ;
len ( h2 ) in dom ( h2 ) & len ( h2 ) = len h2 ;
i + 1 in Seg ( len s2 + 1 ) ;
z in dom g1 /\ dom f & z in dom ( f | X ) ;
assume that p2 = E-max ( K ) and p2 `2 = - 1 ;
len G + 1 <= i1 + 1 ;
f1 (#) f2 is_differentiable_in x0 & f1 (#) ( f2 (#) f1 ) is_differentiable_in x0 ;
cluster ( seq + s-10 ) - ( seq + sR2 ) -> summable ;
assume j in dom ( M1 * ( i , j ) ) ;
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 *> ^ <* xy *> \mathbb x ;
a , b in { a , b } ;
len p2 is Element of NAT & len p2 = len p1 ;
ex x being element st x in dom R & y = R . x ;
len q = len ( K (#) G ) & len q = len G ;
s1 = Initialize Initialized s , P1 = P +* I , P2 = P +* I ;
consider w being Nat such that q = z + w ;
x ` is Element of x & y = the carrier of L ;
k = 0 & n <> k or k > n ;
then X is discrete for X is closed Subset of X ;
for x st x in L holds x is FinSequence ;
||. f /. c .|| <= r1 & ||. f /. c .|| <= r1 ;
c in uparrow p & not c in { p } ;
reconsider V = V as Subset of the topology of TOP-REAL n ;
let N , M be being being being being being being being being being being being being being being being being being being being being being being being being Subset of L ;
then z is_>=_than waybelow x & z is_>=_than compactbelow y ;
M \lbrack f .] = f & M \lbrack g .] = g ;
( ( ( L to_power 1 ) ) /. 1 ) = TRUE ;
dom g = dom f /\ X .= dom f /\ X ;
mode ^ of G is ^ of G , G ;
[ i , j ] in Indices M & [ i , j ] in Indices M ;
reconsider s = x " , t = y " as Element of H ;
let f be Element of dom ( Subformulae p ) , F be Function ;
F1 . ( a1 , - a1 ) = G1 . ( a1 , - a1 ) ;
redefine func Sphere ( a , b , r ) -> compact Subset of TOP-REAL 2 ;
let a , b , c , d be Real ;
rng s c= dom ( 1 / ( f1 + f2 ) ) ;
curry ( F-19 , k ) is additive ;
set k2 = card dom B , k1 = card dom C , k2 = card dom D ;
set G = ( V ) . s ;
reconsider a = [ x , s ] as Object of G ;
let a , b be Element of Mf , M be Matrix of REAL ;
reconsider s1 = s , s2 = t as Element of S1 ;
rng p c= the carrier of L & p . n in rng p ;
let d be Subset of the bound of A ;
( x .|. x ) = 0 iff x = 0. W ;
I-21 in dom stop I & I-21 in dom stop I ;
let g be continuous Function of X | B , Y ;
reconsider D = Y as Subset of TOP-REAL n ;
reconsider i0 = len p1 , i0 = len p2 as Integer ;
dom f = the carrier of S & rng g c= the carrier of S ;
rng h c= union ( the carrier of J ) & rng h c= the carrier of L ;
cluster All ( x , H ) -> .] -valued for element ;
d * N1 ^2 > N1 * 1 / ( d * N2 ) ;
]. a , b .[ c= [. a , b .] ;
set g = f " D1 | D1 , f = ( f | D2 ) " D2 ;
dom ( p | mmA ) = mmA ;
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 ;
cp in the carrier of S1 & cp in the carrier of S2 ;
rng f " = dom f & rng f = dom ( f " ) ;
( the Target of G ) . e = v & ( the Target of G ) . e = v ;
width G - 1 < width G - 1 ;
assume v in rng ( S | E1 ) & u in rng ( S | E1 ) ;
assume x is root or x is root or x is root ;
assume 0 in rng ( ( g2 | A ) | A ) ;
let q be Point of TOP-REAL 2 , r be Real ;
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_in the carrier of C-20 & <* D *> is non empty ;
i <= len ( G /^ ( len G -' 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] & x3 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 ) (#) ( 1 / 2 ) ) (#) ( 1 / 2 ) is summable ;
- p + I c= - p + A ;
n < LifeSpan ( P1 , s1 ) + 1 & I . n = P1 . n ;
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 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 :] ;
len ( the connectives of C ) = 4 & len ( the connectives of C ) = 3 ;
let C1 , C2 be subcategory of C , F be subFunctor of C1 , C2 ;
reconsider V1 = V , V2 = V as Subset of X | B ;
attr p is valid means : Def1 : All ( x , p ) is valid ;
assume that X c= dom f and f .: X c= dom g and g .: X c= dom f ;
H |^ ( a " ) is Subgroup of H |^ a ;
let A1 be [ of O , E1 ] , A2 be Element of E ;
p2 , r3 , q3 is_collinear & p2 , q3 , q3 is_collinear ;
consider x being element such that x in v ^ K ;
not x in { 0. TOP-REAL 2 } \/ { 0. TOP-REAL 2 } ;
p in [#] ( ( ( TOP-REAL 2 ) | B11 ) | B11 ) ;
0 . n < M . ( E8 . n ) ;
op ( c ) |^ ( c , d ) = c ;
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . s2 ) & a2 in dom ( F . s2 ) ;
cluster -> \in -> \in -> 0. -| for Function of L , L ;
set i1 = the Nat , i2 = the Nat , n = the Nat , i = the Nat , n = the Nat ;
let s be 0 -started State of SCM+FSA , P , s be State of SCM+FSA ;
assume y in ( f1 \/ f2 ) .: A ;
f . len f = f /. len f .= f /. 1 ;
x , f . x '||' f . x , f . y ;
attr X c= Y means : Def1 : cos | X c= cos | Y ;
let y be upper Subset of Y , x be Element of X ;
cluster -> non empty -> non <= x `2 -> non <= x `2 ;
set S = <* Bags n , il *> , S = <* <* l *> *> , T = <* l *> , S = <* l *> , T = <* l *> , T = <* l *> , S = <* l *> , T
set T = [. 0 , 1 / 2 .] ;
1 in dom mid ( f , 1 , 1 ) ;
( 4 * PI ) / PI < ( 2 * PI ) / PI ;
x2 in dom f1 /\ dom f & x2 in dom f1 /\ dom f2 ;
O c= dom I & { {} } = { {} } ;
( the 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 be Element of G opp ;
h19 . i = f . ( h . i ) ;
p `1 = p1 `1 & p `2 = p2 `2 or p `1 = p2 `2 ;
i + 1 <= len Cage ( C , n ) ;
len <* P *> = len P & len <* P *> = 1 ;
set N-26 = the O of N , Nw = the Element of N ;
len gLet + ( x + 1 ) - 1 <= x ;
a on B & b on B & not b on B ;
reconsider r-12 = r * I . v as FinSequence of REAL ;
consider d such that x = d and a D _ d [= c ;
given u such that u in W and x = v + u ;
len f /. ( \downharpoonright n ) = len ( f /^ n ) ;
set q2 = N-min L~ Cage ( C , n ) , q2 = W-min L~ Cage ( C , n ) , q2 = E-max L~ Cage ( C , n ) , q2 = E-max L~ Cage ( C , n ) , q2 = E-max
set S =
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 f " V ;
reconsider D = E as non empty directed Subset of L1 ;
H = ( the_left_argument_of H ) '&' ( the_left_argument_of H ) ;
assume t is Element of ( F . X ) . s ;
rng f c= the carrier of S2 & f . ( len f ) = f . ( len f ) ;
consider y being Element of X such that x = { y } ;
f1 . ( a1 , b1 ) = b1 . ( a1 , b1 ) ;
the carrier' of G ` = E \/ { E } .= { E } ;
reconsider m = len ( thesis | k ) - 1 as Element of NAT ;
set S1 = LSeg ( n , UMP C ) , S2 = LSeg ( UMP C , UMP C ) ;
[ i , j ] in Indices ( M1 + M2 ) ;
assume that P c= Seg m and M is \HM { \vert m .| } ;
for k st m <= k holds z in K . k ;
consider a being set such that p in a and a in G ;
L1 . p = p * L /. ( 1 + 1 ) ;
p-7 . i = pp . i .= pp . i ;
let PA , G be a_partition of Y , a be Element of Y ;
attr 0 < r & r < 1 implies 1 < ( 1 - r ) / ( 1 - r ) ;
rng ( ) ( a , X ) = [#] X & rng ( f | X ) = [#] X ;
reconsider x = x , y = y as Element of K ;
consider k such that z = f . k and n <= k ;
consider x being element such that x in X \ { p } ;
len ( ( canFS ( s ) ) | ( len s ) ) = card ( s | ( len s ) ) ;
reconsider x2 = x1 , y2 = x2 as Element of L2 ;
Q in FinMeetCl ( the topology of X ) & Q c= FinMeetCl the topology of Y ;
dom fx0 c= dom ( ux0 - f ) ;
pred n divides m & m divides n implies n = m ;
reconsider x = x , y = y as Point of [: I[01] , I[01] :] ;
a in \subseteq c= c= the carrier of T & a in the carrier of T ;
not y0 in the still of f & not y0 in the carrier of f ;
Hom ( ( a ~ ) , c ) <> {} ;
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 = .: ( x , y , z ) ;
l1 = m2 & l1 = i2 & l2 = j2 implies l1 = i2
x0 in dom ( u01 ) /\ A01 & x0 in dom ( u01 ) /\ A01 ;
reconsider p = x as Point of ( TOP-REAL 2 ) | K1 ;
I[01] = R^1 | B01 .= R^1 | B01 .= the carrier of TOP-REAL 2 ;
f . p4 <= f . p1 , f . p2 , P ;
( ( F . x ) `1 ) ^2 <= ( ( F . x ) `1 ) ^2 ;
x `2 = ( W7 ) `2 .= ( W7 ) `2 .= ( W8 ) `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 & 0 |-> a = <*> the carrier of K ;
X . i in 2 -tuples_on ( A . i \ B . i ) ;
<* 0 *> in dom ( e --> [ 1 , 0 ] ) ;
then P [ a ] & P [ succ a ] ;
reconsider s\overline = s\overline ( D ) as ' of D ;
( i - 1 ) <= len ( ' - 1 ) ;
[#] S c= [#] ( the TopStruct of T ) & [#] T c= [#] T ;
for V being strict RealUnitarySpace holds V in and V is Subspace of the carrier of V
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 , n be Nat ;
- a * ( - b ) = a * b ;
for A being Subset of AS holds A // A implies A // C
( for o2 being Element of A holds o2 in <^ o2 , o2 ^> implies o2 = o1 )
then ||. x .|| = 0 & x = 0. X ;
let N1 , N2 be strict normal Subgroup of G , N be strict Subgroup of G ;
j >= len upper_volume ( g , D1 ) & j <= len upper_volume ( g , D2 ) ;
b = Q . ( len Qk - 1 ) + 1 ;
f2 * f1 /* s is divergent_to-infty & f2 * f1 /* s is divergent_to-infty ;
reconsider h = f * g as Function of [: N , N :] , 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 T7 & v |-- E in T7 ;
{} = the carrier of L1 + L2 & the carrier of L1 = the carrier of L1 + L2 ;
Directed I is_closed_on Initialized s , P & Directed I is_halting_on Initialized s , P & Directed I is_halting_on Initialized s , P ;
Initialized p = Initialize ( p +* q ) .= Initialize ( p +* q ) ;
reconsider N2 = N1 , N2 = N2 as strict net of R2 ;
reconsider Y = Y as Element of \langle Ids L , \subseteq \rangle ;
"/\" ( uparrow p \ { p } , L ) <> p ;
consider j being Nat such that i2 = i1 + j and j in dom f ;
not [ s , 0 ] in the carrier of S2 & not [ s , 0 ] in the carrier of S2 ;
mm in ( B '/\' C ) '/\' D \ { {} } ;
n <= len ( ( P + Q ) ^ <* n *> ) + len ( ( P + Q ) ^ <* n *> ) ;
x1 `1 = x2 & y1 `2 = y2 & y1 `2 = y2 & y1 `2 = y2 ;
InputVertices S = { x1 , x2 } & InputVertices S = { x1 , x2 } ;
let x , y be Element of FT1 ( n ) ;
p = |[ p `1 , p `2 ]| & p = |[ p `1 , p `2 ]| ;
g * 1_ G = h " * g * h * h ;
let p , q be Element of PFuncs ( V , C ) ;
x0 in dom x1 /\ dom x2 & x1 in dom x1 /\ dom x2 ;
( R qua Function ) " = R " ( dom R ) .= R " ( dom R ) ;
n in Seg len ( f /^ ( i -' 1 ) ) & n in dom ( f /^ i ) ;
for s be Real st s in R holds s <= s2 implies s <= s2
rng s c= dom ( f2 * f1 ) /\ dom ( f2 * f1 ) ;
synonym ) ( X ) for X is Subset of ) & X is finite ;
1. ( K , n ) * 1. ( K , n ) = 1. ( K , n ) ;
set S = Segm ( A , P1 , Q1 ) , T = Segm ( A , P1 , Q1 ) ;
ex w st e = ( w - f ) . w & w in F ;
curry ( P+* ( P+* ( i , k ) ) # x ) is convergent ;
cluster open -> open for Subset of T7 ( n ) ;
len f1 = 1 .= len f3 .= len f3 + 1 .= len f3 + 1 ;
( i * p ) / p < ( 2 * p ) / p ;
let x , y be Element of [: Sub , U0 :] ;
b1 , c1 // b9 , c9 & b1 , c1 // b9 , c9 ;
consider p being element such that c1 . j = { p } ;
assume that f " { 0 } = {} and f is total ;
assume IC Comput ( F , s , k ) = n & IC Comput ( F , s , k ) = k ;
Reloc ( J , card I ) does not destroy a implies J " ;
goto ( card I + 1 ) does not destroy c ;
set m3 = LifeSpan ( p3 , s3 ) , m3 = LifeSpan ( p2 , s3 ) ;
IC SCMPDS in dom Initialize ( p +* I ) & IC s2 in dom Initialize ( p +* I ) ;
dom t = the carrier of SCM R & dom t = the carrier of SCM R ;
( ( N-min L~ f ) .. f ) .. f = 1 ;
let a , b be Element of PFuncs ( V , C ) ;
Cl ( union Int F ) c= Cl ( Int Cl F ) ;
the carrier of X1 union X2 misses ( ( A \/ B ) \/ ( A \/ C ) ) ;
assume not LIN a , f . a , g . a ;
consider i being Element of M such that i = d6 and i in dom f ;
then Y c= { x } or Y = { x } ;
M , v / ( y , x ) / ( y , x ) |= H ;
consider m being element such that m in Intersect ( Fx0 ) and m in dom f ;
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 a3 <> a4 ;
cluster s -carrier V -> $ for string of S , X be set ;
LL2 /. n2 = LL2 . n2 .= LL2 . n2 .= LL2 . n2 ;
let P be compact non empty Subset of TOP-REAL 2 , p1 , p2 be Point of TOP-REAL 2 ;
assume r-7 in LSeg ( p1 , p2 ) /\ LSeg ( p2 , p3 ) ;
let A be non empty compact Subset of TOP-REAL n , f be Function of TOP-REAL n , TOP-REAL n ;
assume that [ k , m ] in Indices Dm1 and [ k , m ] in Indices Dm1 ;
0 <= ( ( 1 / 2 ) to_power p ) . p ;
( F . N ) | E8 . x = +infty ;
attr X c= Y & Z c= V implies X \ V c= Y \ Z ;
y `2 * ( z `2 ) * ( z `1 ) <> 0. I & y `2 * ( z `2 ) <> 0. I ;
1 + card X-18 <= card X-18 + card X-18 ;
set g = z \circlearrowleft ( ( L~ z ) .. z ) , 2 = ( ( L~ z ) .. z ) .. z ;
then k = 1 & p . k = <* x , y *> . k ;
cluster total for Element of C -O , X be total set ;
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 ;
Plane ( x1 , x2 , x3 ) c= P & Plane ( x2 , x3 , x4 ) c= P ;
n <= indx ( D2 , D1 , j1 ) + 1 - 1 ;
( ( ( g2 ) . O ) `1 ) ^2 = ( - 1 ) ^2 ;
j + p .. f - len f <= len f - len f ;
set W = W-bound C , E = E-bound C , N = E-bound C , S = E-bound C , N = E-bound C , N = E-bound C , S = E-bound C , N = E-bound C , N = E-bound C , S = E-bound C , N = E-bound
S1 . ( a `1 , e `2 ) = a + e `2 .= a `1 ;
1 in Seg width ( M * ( ColVec2Mx p ) ) & len ( M * ( ColVec2Mx p ) ) = width M ;
dom ( i (#) Im ( f ) ) = dom Im ( f ) ;
( ^2 ) . x = W . ( a , *' ( a , p ) ) ;
set Q = ( \rm \rm \rm \rm \rm \rm \rm \rm , f ) . ( g . x ) ;
cluster -> MS[ for ManySortedSet of U1 , ( the Sorts of U1 ) * ;
attr F = { A } means : Def1 : F is discrete ;
reconsider z9 = ] , z9 = y as Element of product ( G . i ) ;
rng f c= rng f1 \/ rng f2 & f . ( len f ) = f1 . ( len f ) ;
consider x such that x in f .: A and x in f .: C ;
f = <*> the carrier of F_Complex & f = <*> the carrier of F_Complex ;
E , j |= All ( x1 , x2 , x3 , x4 ) ;
reconsider n1 = n , n2 = m as Morphism of o1 , o2 ;
assume that P is idempotent and R is idempotent and P (*) R = R (*) P ;
card ( B2 \/ { x } ) = k-1 + 1 ;
card ( ( x \ B1 ) /\ B1 ) = 0 implies x in B1
g + R in { s : g-r < s & s < g + r } ;
set q-1= ( q , <* s *> ) -O , q-1= ( q , <* s *> ) -O ;
for x being element st x in X holds x in rng ( f1 | X )
h0 /. ( i + 1 ) = h0 . ( i + 1 ) ;
set mw = max ( B , dom ( R | NAT ) ) ;
t in Seg width ( I ^ ( n , n ) ) & t in Seg n ;
reconsider X = dom f /\ C as Element of Fin NAT ;
IncAddr ( i , k ) = <% x , y %> + k ;
( ( ( q `2 ) - ( q `2 ) ) / ( 1 + ( q `2 ) ) ^2 ) <= ( q `2 ) / ( 1 + ( q `2 ) ^2 ) ;
attr R is condensed means : Def1 : Int R is condensed & Cl R is condensed ;
attr 0 <= a & a <= 1 & b <= 1 implies a * b <= 1 ;
u in ( ( c /\ ( ( d /\ b ) /\ e ) ) /\ f ) /\ j ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) /\ f ) ) /\ j ;
len C + - 2 >= 9 + - 3 & len C + - 3 >= 0 ;
x , z , y is_collinear & x , z , y is_collinear implies x , z , y is_collinear
a |^ ( n1 + 1 ) = a |^ n1 * a |^ n1 ;
<* \underbrace ( 0 , \dots , 0 ) *> in Line ( x , a * x ) ;
set yy1 = <* y , c *> ;
FF2 /. 1 in rng Line ( D , 1 ) & FF2 . len FF2 = 0. K ;
p . m Joins r /. m , r /. ( m + 1 ) , G ;
p `2 = ( f /. i1 ) `2 .= ( f /. ( i1 + 1 ) ) `2 ;
W-bound ( X \/ Y ) = W-bound ( X \/ Y ) & E-bound ( X \/ Y ) = W-bound ( X \/ Y ) ;
0 + ( p `2 ) <= 2 * r + ( p `2 ) ;
x in dom g & not x in g " { 0 } ;
f1 /* ( seq ^\ k ) is divergent_to-infty & f2 /* ( seq ^\ k ) is divergent_to-infty ;
reconsider u2 = u , v2 = v as VECTOR of P`1 ( X ) ;
p |-count ( Product Sgm X11 ) = 0 & p |-count ( Product Sgm X11 ) = 0 ;
len <* x *> < i + 1 & i + 1 <= len c + 1 ;
assume that I is non empty and { x } /\ { y } = { 0. I } ;
set ii2 = ( card I + 4 ) .--> goto 0 , ii2 = goto 0 , ii2 = goto 0 , ii2 = goto 0 , ii2 = goto 0 , ii2 = goto 0 , ii2 = goto 0 , ii2 = goto 0 , ii2 =
x in { x , y } & h . x = {} ( TT . x ) ;
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 B ) = len the charact of B ;
reconsider m = M , i = I , n = N as Element of X ;
A . ( j + 1 ) = B . ( j + 1 ) \/ A . j ;
set N8 = : : G = : G `1 = ( G . e ) `1 ;
rng F c= the carrier of gr { a } & F . a = the carrier of gr { a } ;
( for K being ( st K in rng F holds n , r ) ) implies for f being ) FinSequence holds P [ f ]
f . k , f . ( \mathop { \rm mod n ) * ( m mod n ) are_relative_prime ;
h " P /\ [#] T1 = f " P /\ [#] T2 .= f " P /\ [#] T2 ;
g in dom f2 \ ( f2 " { 0 } ) /\ ( f2 " { 0 } ) ;
g1X /\ dom f1 = g1 " X /\ dom ( f1 " X ) ;
consider n being element such that n in NAT and Z = G . n ;
set d1 = being thesis , d2 = dist ( x1 , y1 ) , d2 = dist ( x2 , y2 ) , d2 = dist ( y2 , y1 ) ;
b `1 + ( 1 / 2 ) < ( 1 / 2 ) + ( 1 / 2 ) ;
reconsider f1 = f as VECTOR of the carrier of X , Y ;
attr i <> 0 implies i ^2 mod ( i + 1 ) = 1 ;
j2 in Seg len ( ( g2 . i2 ) | ( i -' 1 ) ) ;
dom ( i4 * i4 ) = dom ( i * i4 ) .= dom i ;
cluster sec | ]. PI / 2 , PI .[ -> one-to-one for Function of REAL , REAL ;
Ball ( u , e ) = Ball ( f . p , e ) ;
reconsider x1 = x0 , y1 = y0 as Function of S , IV ;
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 RL ;
S1 +* S2 = S2 +* S1 +* S2 +* S2 .= S1 +* S2 +* S2 +* S2 .= S1 +* S2 +* S2 +* S2 ;
( ( #Z n ) * ( cos - sin ) ) is_differentiable_on Z & ( ( #Z n ) * ( cos - sin ) ) is_differentiable_on Z ;
cluster [. 0 , 1 .] -> [. 0 , 1 .] -valued for Function ;
set C7 = 1GateCircStr ( <* z , x *> , f3 ) , C8 = 1GateCircStr ( <* x , y *> , f3 ) ;
ES . e2 = E8 . e2 -T . e2 .= E8 . e2 ;
( ( arctan (#) ( ln * f ) ) `| Z ) . x = f . x / ( 1 + x ^2 ) .= f . x / ( 1 + x ^2 ) ;
upper_bound A = ( PI * 3 / 2 ) * 2 & lower_bound A = 0 ;
F . ( dom f , - F . ( cod f , - F . ( cod f , - F . ( cod f , - F . ( cod f , - F . ( cod f , - F . ( cod f , - F . ( cod f ) ) ) ) ) ) ) ;
reconsider pbeing = qbeing Point of TOP-REAL 2 , pbeing 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 , p be Point of TOP-REAL 2 ;
LSeg ( f ^ g , j ) = LSeg ( f , j ) .= LSeg ( f , j ) ;
rng s c= dom f /\ ]. -infty , x0 + r .[ & f . ( s . n ) <= 0 ;
assume x in { idseq 2 , Rev ( idseq 2 ) } ;
reconsider n2 = n , m2 = m , m1 = n as Element of NAT ;
for y being ExtReal st y in rng seq holds g <= y implies g <= f
for k st P [ k ] holds P [ k + 1 ]
m = m1 + m2 .= m1 + m2 .= m1 + m2 .= m1 + m2 ;
assume for n holds H1 . n = G . n -H . n ;
set Bf = f .: ( the carrier of X1 ) , Bg = f .: ( the carrier of X2 ) ;
ex d being Element of L st d in D & x << d ;
assume R -Seg ( a ) c= R -Seg ( b ) & R -Seg ( a ) c= R -Seg ( b ) ;
t in ]. r , s .[ or t = r or t = s or t = s ;
z + v2 in W & x = u + ( z + v2 ) ;
x2 |-- y2 iff P [ x2 , y2 ] & P [ x2 , y2 ] ;
pred x1 <> x2 means : Def1 : |. x1 - x2 .| > 0 & |. x2 - x1 .| > 0 ;
assume that p2 - p1 , p3 - p1 - p2 - p1 , p3 - p1 - p2 is_collinear and p2 - p1 , p3 - p2 - p1 , p3 - p2 - p1 is_collinear ;
set q = ( -1 f ) ^ <* 'not' A *> ;
let f be PartFunc of REAL-NS 1 , REAL-NS n , x be Point of REAL-NS 1 , r be Real ;
( n mod ( 2 * k ) ) *> _ _ { k } = n mod k ;
dom ( T * ( succ t ) ) = dom ( n succ t ) .= dom ( T * ( succ t ) ) ;
consider x being element such that x in wc iff x in c & x in X ;
assume ( F * G ) . ( v . x3 ) = v . x4 ;
assume that the carrier of D1 c= the carrier of D2 and the carrier of D1 c= 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 = W-min L~ Cage ( C , n ) , r = E-bound L~ Cage ( C , n ) , s = E-bound L~ Cage ( C , n ) , G = Gauge ( C , n ) , G = Gauge ( C , n ) , G = Gauge ( C ,
n1 -' len f + 1 <= len ( g /^ ( len g -' 1 ) ) + 1 ;
Seg \mathbb \mathbb d ( q , O1 ) = [ u , v , a `1 , b ] ;
set C-2 = ( ( `1 ) `1 ) . ( k + 1 ) ;
Sum ( L (#) p ) = 0. R * Sum p .= 0. V ;
consider i being element such that i in dom p and t = p . i ;
defpred Q [ Nat ] means 0 = Q ( ) . $1 & P [ $1 ] ;
set s3 = Comput ( P1 , s1 , k ) , P3 = P1 +* I , s4 = P1 +* I , s4 = P1 +* I ;
let l be variable of k , Al , A-30 be Subset of V ;
reconsider U2 = union G-24 , G-24 = union G-24 as Subset-Family of T-24 ;
consider r such that r > 0 and Ball ( p `1 , r ) c= Q ` ;
( h | ( n + 2 ) ) /. ( i + 1 ) = p29 ;
reconsider B = the carrier of X1 , C = the carrier of X2 as Subset of X1 ;
p$ L = <* - c9 , 1 *> & p = <* - c9 , 1 *> ;
synonym f is real-valued means : Def1 : 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 ;
x10 < card X0 + card Y0 & card Y0 + card Y0 < card Y0 + card Y0 ;
attr X c= B1 means : Def1 oo) X c= succ B1 & X c= succ B2 ;
then w in Ball ( x , r ) & dist ( x , w ) <= r ;
angle ( x , y , z ) = angle ( x-y , 0 , p2 ) ;
attr 1 <= len s means : Def1 : for s being Element of NAT holds ( s . 0 = s . 1 ) & ( s . 1 = s . 2 implies s . 2 = s . 3 ) ;
fJ c= f . ( k + ( n + 1 ) ) ;
the carrier of { 1_ G } = { 1_ G } & the carrier of { 1_ G } = { 1_ G } ;
pred p '&' q in TAUT ( A ) means q '&' p in TAUT ( A ) & q in TAUT ( A ) ;
- ( t `1 / t `2 ) < ( t `2 / t `1 ) ^2 ;
UA . 1 = ( U /. 1 ) `1 .= ( W /. 1 ) `1 .= ( W /. 1 ) `1 ;
f .: ( the carrier of x ) = the carrier of x & f .: ( the carrier of x ) = the carrier of x ;
Indices On = [: Seg n , Seg n :] & Indices O = [: Seg n , Seg n :] ;
for n being Element of NAT holds G . n c= G . ( n + 1 ) ;
then V in M @ x ex x being Element of M st V = { x } ;
ex f being Element of F-9 st f is \cup Awhere f is Element of A-29 : f is unital & f is unital } ;
[ 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 ]| - |[ w1 , v1 ]| <> 0. TOP-REAL 2 & |[ w1 , v1 ]| - |[ w1 , v1 ]| = |[ w1 , v1 ]| ;
reconsider t = t as Element of INT * , ( the carrier of X ) --> { 0 } ;
C \/ P c= [#] ( GX | ( [#] GX \ A ) ) ;
f " V in ( the topology of X ) /\ D . ( the carrier of S , the carrier of S ) ;
x in [#] ( the carrier of A ) /\ A ( ) & y in [#] ( ( the carrier of B ) /\ A ( ) ) ;
g . x <= h1 . x & h . x <= h1 . x & h . y <= 1 ;
InputVertices S = { xy , yz , zx , zx , zx , non empty set , f be Function of the carrier of S , BOOLEAN , 5 , 6 , 7 , 8 } ;
for n being 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 M3 is being_line ;
reconsider a = f4 . ( i0 -' 1 ) , b = f4 . ( i0 -' 1 ) as Element of K ;
len B2 = Sum ( ( Len F1 ) ^ ( ( Len F2 ) ^ ( Len F2 ) ) .= len ( ( Len F1 ) ^ ( Len F2 ) ) ;
len ( ( the ` of n ) * ( i , j ) ) = n & len ( ( i , j ) * ( i , j ) ) = n ;
dom ( max ( f , g ) ) = dom ( f + g ) ;
( the Sorts of Y1 ) . n = upper_bound Y1 & ( the Sorts of Y2 ) . n = upper_bound Y1 ;
dom ( p1 ^ p2 ) = dom f12 & dom ( p1 ^ p2 ) = dom f12 ;
M . [ 1 / y , y ] = 1 / ( 1 * v1 ) .= y ;
assume that W is non trivial and W .vertices() c= the carrier' of G2 and not e in the carrier' of G2 ;
C6 * ( i1 , i2 ) `1 = G1 * ( i1 , i2 ) `1 .= G1 * ( i1 , i2 ) `1 ;
C8 |- 'not' Ex ( x , p ) 'or' p . ( x , y ) ;
for b st b in rng g holds lower_bound rng f\lbrace f\rbrace <= b
- ( ( q1 `1 / |. q1 .| - cn ) / ( 1 + cn ) ) = 1 ;
( LSeg ( c , m ) \/ [: NAT , NAT :] ) \/ [: NAT , NAT :] c= R ;
consider p being element such that p in { x } and p in L~ f and x = f . p ;
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 , ( Partial_Sums F ) . n ) is_measurable_on E ;
cluster f . ( x1 , x2 ) -> Element of D , f . ( y1 , y2 ) -> Element of D ;
consider g being Function such that g = F . t and Q [ t , g ] ;
p in LSeg ( N-min Z , NW-corner Z ) /\ LSeg ( NW-corner Z , NW-corner Z ) ;
set R8 = R | ]. b , +infty .[ , R8 = R | ]. b , +infty .[ ;
IncAddr ( I , k ) = SubFrom ( da , da ) .= AddTo ( da , db ) ;
seq . m <= ( the Sorts of A ) . k & ( the Sorts of A ) . k <= ( the Sorts of A ) . k ;
a + b = ( a ` *' b ` ) ` .= ( a ` *' b ) ` ;
id ( X /\ Y ) = id ( X /\ Y ) .= id ( X /\ Y ) ;
for x being element st x in dom h holds h . x = f . x ;
reconsider H = U1 \/ U2 , U2 = U1 \/ U2 as non empty Subset of U0 ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) /\ f ) /\ j ) /\ m ;
consider y being element such that y in Y and P [ y , inf B ] ;
consider A being finite stable set of R such that card A = ( the carrier of R ) \ A ;
p2 in rng ( f |-- p1 ) \ rng <* p1 *> & p2 in rng <* p1 *> ;
len s1 - 1 > 0 & len s2 - 1 > 0 implies len s2 - 1 <= len s2 - 1
( 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 ) ;
f . a1 ` = f . a1 ` .= f . a1 ` .= ( f | ( a1 ` ) ) . a1 ;
( seq ^\ k ) . n in ]. -infty , x0 + r .[ /\ dom f1 \ { x0 } ;
gg . s0 = g . s0 | G . s0 .= g . s0 ;
the InternalRel of S & the InternalRel of S c= the InternalRel of ( the InternalRel of S ) ;
deffunc F ( Ordinal , Ordinal ) = phi . ( $1 + 1 ) & phi . ( $2 + 1 ) = phi . ( $2 + 1 ) ;
F . s1 . a1 = F . s2 . a1 & F . s1 . a1 = F . s2 . a1 ;
x `1 = A . o . a .= Den ( o , A . a ) . a ;
Cl ( f " P1 ) c= f " ( Cl P1 ) & Cl ( f " P1 ) c= f " ( Cl P1 ) ;
FinMeetCl ( ( the topology of S ) \/ { 0 } ) c= the topology of T ;
synonym o is \bf means : Def1 : o <> \ast & o <> * ;
assume that X c= Y and card X <> card Y and card Y <> card X and card Y = card X ;
the *> of s <= 1 + ( the *> of s ) & the { s } c= the carrier of S `1 ;
LIN a , a1 , d or b , c // b1 , c1 or b , c // d , c1 ;
e / 2 . 1 = 0 & e / 2 . 2 = 1 & e / 2 . 3 = 0 ;
EE in SE & not EE in { NE } ;
set J = ( l , u ) If , K = I " ;
set A1 = } , A2 = } , A1 = { cin , cin , cin , cin , cin } ;
set vs = [ <* cin , cin *> , '&' ] , f3 = [ <* cin , cin *> , '&' ] , f4 = [ <* cin , cin *> , '&' ] , f4 = [ <* cin , cin *> , '&' ] , f4 = [ <* cin , cin *> , '&' ] , f4 = [ <* cin , cin *> , '&' ] , } ;
x * z `1 * x " in x * ( z * N ) * x " ;
for x being element st x in dom f holds f . x = g3 . x & f . x = f . x
Int cell ( f , 1 , G ) c= RightComp f \/ L~ f \/ L~ f \/ L~ f \/ L~ f ;
UA is_an_arc_of W-min C , E-max C & L~ h c= L~ Cage ( C , n ) \/ L~ Cage ( C , n ) ;
set f-17 = f @ "/\" g @ ;
attr S1 is convergent means : Def1 : S2 is convergent & ( for n holds S1 . n = S2 . n ) implies S1 is convergent & lim S2 = lim ( S2 ) ;
f . ( 0 + 1 ) = ( 0 qua Ordinal ) + a .= a ;
cluster -> \in be be be be be be be be be be be reflexive transitive non empty RelStr , F be symmetric Function of [: the carrier of M , the carrier of M :] , the carrier of M ;
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 ) \/ dom Start-At ( ( card I + 2 ) , SCMPDS ) ;
( z + a ) + x = z + ( a + y ) .= z + a + y ;
len ( l \lbrack a \rbrack --> x ) = len l & len ( l (#) x ) = len l ;
t4 } is ( {} \/ rng t4 ) -valued ( {} , rng t4 ) -valued FinSequence ;
t = <* F . t *> ^ ( C . p ^ ( C . q ) ) ;
set pp = W-min L~ Cage ( C , n ) , p = W-min L~ Cage ( C , n ) , q = W-min L~ Cage ( C , n ) , r = q , s = q , w = q , e = q , w = q , w = q , e = q , w = q , w = q , x =
( k -' ( i + 1 ) ) = ( k - i ) - ( i + 1 ) ;
consider u being Element of L such that u = u `1 "/\" D and u in D ` ;
len ( ( width aG ) |-> a ) = width aG & width ( ( len aG ) |-> a ) = width aG ;
FM . x in dom ( ( G * the_arity_of o ) . x ) & x in dom ( ( G * the_arity_of o ) . x ) ;
set cH2 = the carrier of H2 , cH2 = the carrier of H2 ;
set cH1 = the carrier of H1 , cH2 = the carrier of H2 ;
( Comput ( P , s , 6 ) ) . intpos m = s . intpos m .= s . intpos m ;
IC Comput ( Q2 , t , k ) = ( l + 1 ) + 1 .= ( l + 1 ) + 1 ;
dom ( ( cos * sin ) `| REAL ) = REAL & dom ( ( cos * sin ) `| REAL ) = REAL ;
cluster <* l *> ^ phi -> ( 1 + 0 ) -element for string of S ;
set b5 = [ <* ap , ccp *> , <* ccp *> ] ;
Line ( Segm ( M @ , P , Q ) , x ) = L * Sgm Q ;
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 = t2 . DataLoc ( s4 . SBP , 2 ) , x4 = Comput ( P3 , s3 , 2 ) , P4 = P3 ;
set pp = stop I ( ) , p = Initialize s , q = Initialize s , P = P +* I , Q = P +* I , S = P +* I , T = P +* I , T = P +* I , T = P +* I , S = P +* I , T = S +* I , T = S
consider a being Point of D2 such that a in W1 and b = g . a and a in W1 ;
{ A , B , C , D , E } = { A , B , C } \/ { D , E , F , J , M }
let A , B , C , D , E , F , J , M , N , N , M be set ;
|. p2 .| ^2 - ( p2 `2 ) ^2 - ( p2 `2 ) ^2 >= 0 ;
l - 1 + 1 = n-1 * ( ( l - 1 ) + 1 ) + 1 ;
x = v + ( a * w1 + b * w2 ) + ( c * w2 ) + ( c * w2 ) ;
the TopStruct of L = [: the topology of L , the topology 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 Y ;
ff \ { n } = \mathop { \rm Free ( All ( v1 , H ) , E ) .= Free ( All ( v1 , H ) ) ;
for Y be Subset of X st Y is summable holds Y is be summable iff Y is be summable
2 * n in { N : 2 * Sum ( p | N ) = N & N > 0 } ;
for s being FinSequence holds len ( the { - } -1 ) = len s & len ( the { - } -1 ) = len s
for x st x in Z holds exp_R * f is_differentiable_in x & ( exp_R * f ) . x = 1 / x
rng ( ( h2 * f2 ) | A ) c= the carrier of ( ( TOP-REAL 2 ) | A ) ;
j + ( len f ) - len f <= len f + ( len g ) - len f ;
reconsider R1 = R * I , R2 = R * I as PartFunc of REAL n , REAL-NS n ;
C8 . x = s1 . x0 .= C8 . x .= C8 . x .= C8 . x ;
power ( F_Complex ) . ( z , n ) = 1 .= x |^ n .= x |^ n ;
t at ( C , s ) = f . ( the connectives of S ) . t .= f . ( s , C ) ;
support ( f + g ) c= support f \/ C & support ( f + g ) c= C \/ ( C \/ { x } ) ;
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 Subset of [: X1 , X2 :] , x2 is Subset of X1 , X2 is Subset of X2 : x1 in X2 } is Subset of X1 ;
h = ( i , j ) |-- h .= H . i .= H . i .= H . i ;
ex x1 being Element of G st x1 = x & x1 * N c= A & x1 in A & x1 in B ;
set X = ( ( d , O1 ) --> ( q , O1 ) ) . ( ( d , O1 ) --> ( q , O1 ) ) ;
b . n in { g1 : x0 < g1 & g1 < a1 . n } ;
f /* s1 is convergent & f /. x0 = lim ( f /* s1 ) & f /. x0 = lim ( f /* s1 ) ;
the carrier of Y = the carrier of Y & the carrier of Y = the carrier of X & the carrier of X = the carrier of Y ;
'not' ( a . x ) '&' b . x 'or' a . x '&' 'not' ( b . x ) = FALSE ;
2 = len ( q0 ^ r1 ) + len ( q1 ^ q2 ) .= len ( q ^ q2 ) + len ( q2 ^ q3 ) ;
( 1 / a ) (#) ( sec * f1 ) - id Z is_differentiable_on Z ;
set K1 = upper ( ( lim ( lim ( H , A ) ) || ( A , B ) ) , ( lim ( H , B ) || ( A , B ) ) ) ;
assume e in { ( w1 - w2 ) / ( w1 - w2 ) : w1 in F & w2 in G } ;
reconsider d7 = dom a `1 , d8 = dom F `1 , d8 = dom G as finite set ;
LSeg ( f /^ q , j ) = LSeg ( f , j ) /\ q .. f .= LSeg ( f , j + q .. f ) ;
assume X in { T . ( N2 , K1 ) : h . ( N2 , K1 ) = N2 } ;
assume that Hom ( d , c ) <> {} and <* f , g *> * f1 = <* f , g *> * f2 ;
dom Sb = dom S /\ Seg n .= dom ( L | Seg n ) .= dom ( L | Seg n ) .= Seg n /\ Seg n .= dom ( L | Seg n ) ;
x in H |^ a implies ex g st x = g |^ a & g in H |^ a & g in H |^ a
* ( 0. ( K , n ) ) . ( a , 1 ) = a `1 - ( 0 * n ) `1 .= a `1 ;
D2 . ( j - 1 ) 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 @ f
dom ( f1 (#) f2 ) /\ X c= dom ( f1 (#) f2 ) /\ X .= dom ( f1 (#) f2 ) /\ X ;
1 = ( p * p ) * p .= p * ( p * p ) .= p * 1 .= p * 1 ;
len g = len f + len <* x + y *> .= len f + 1 .= len f + 1 ;
dom ( F-11 | [: N1 , S :] ) = dom ( F | [: N1 , S :] ) .= [: N1 , S :] ;
dom ( f . t * I . t ) = dom ( f . t * g . t ) ;
assume a in ( "\/" ( ( T |^ the carrier of S ) , 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 = id b and f * f = id a and f * f = id b ;
( cos | [. 2 * PI * 0 , PI + ( 2 * PI * 0 ) .] ) is increasing ;
Index ( p , co ) <= len LS - Gij .. LS - LS .. LS + 1 - LS .. LS - 1 ;
let t1 , t2 , t3 be Element of ( the carrier of S ) * , s be Element of S ;
"/\" ( ( Frege ( curry H ) ) . h , L ) <= "/\" ( ( Frege ( curry G ) ) . h , L ) ;
then P [ f . i0 ] & F ( f . i0 + 1 ) < j & j <= len f ;
Q [ ( D . x ) `1 , 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 let of G . i ;
the Sorts of A2 = ( the carrier of S2 ) --> ( the carrier of S1 ) .= ( the carrier of S1 ) --> ( the carrier of S2 ) ;
consider s being Function such that s is one-to-one and dom s = NAT and rng s = F . 0 and for n being Nat holds s . n = F ( n ) ;
dist ( b1 , b2 ) <= dist ( b1 , a ) + dist ( a , b2 ) + dist ( a , b2 ) ;
( Lower_Seq ( C , n ) /. len Lower_Seq ( C , n ) ) /. len Lower_Seq ( C , n ) = WW ;
q `2 <= ( UMP Upper_Arc L~ Cage ( C , 1 ) ) `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 : b |^ n = b |^ n } , Y = { b |^ n } ;
( ( x * y * z \ x ) \ z ) \ ( x * y \ x ) = 0. X ;
set xy = [ <* xy , yz , zx *> , f1 ] , yz = [ <* xy , yz *> , f2 ] , zx = [ <* xy , yz *> , f3 ] , zx = [ <* xy , yz , zx *> , f3 ] , f4 = [ <* xy , yz *> , f3 ] , zx = [ <* xy , yz *> , f3 ] , zx = [ <*
Uq /. len ( lq ) = ( lq /. len ( q ) ) . 1 .= q . 1 ;
( ( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) / ( 1 + sn ) ) ^2 = 1 ;
( ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) ^2 < 1 ;
( ( ( S \/ Y ) `2 ) / 2 ) * ( ( S \/ Y ) / 2 ) = ( ( S \/ Y ) / 2 ) * ( ( S \/ Y ) / 2 ) ;
( ( seq - seq ) . k ) . k = ( seq . k - seq . ( k + 1 ) ) . ( seq . k ) ;
rng ( ( h + c ) ^\ n ) c= dom SVF1 ( 1 , f , u0 ) ;
the carrier of X = the carrier of X & the carrier of X = the carrier of X & the carrier of X = the carrier of Y ;
ex p4 st p3 = p4 & |. p4 - |[ a , b ]| .| = r & |. p4 - |[ a , b ]| .| = r ;
set ch = chi ( X , A ) , ch = chi ( X , A ) ;
R |^ ( 0 * n ) = Imax ( X , X ) .= R |^ n |^ 0 .= R |^ 0 ;
( ( Partial_Sums ( curry ( Fd , n ) ) ) . n ) . x is nonnegative & ( ( ( curry ( Fd , n ) ) . x ) . x = ( ( F . n ) . x ) . x ;
f2 = C7 . ( E7 . ( len ( V , len ( V , len ( f | K ) ) ) ) ;
S1 . b = s1 . b .= s2 . b .= ( S2 . b ) . b .= ( S2 . b ) . b ;
p2 in LSeg ( p2 , p1 ) /\ LSeg ( p2 , p1 ) /\ LSeg ( p1 , p2 ) .= { p2 } ;
dom ( f . t ) = Seg n & dom ( I . t ) = Seg n & rng ( I . t ) c= Seg n ;
assume o = ( the connectives of S ) . 11 & o in ( the carrier' of S ) . 11 ;
set phi = ( l1 , l2 ) u , phi = ( l1 , l2 ) u , C = ( l1 , l2 ) u , D = ( l2 , l2 ) u , E = ( l2 , l2 ) u , F = ( l1 , l2 ) u , F = ( l2 , l2 ) u , F = ( l2 , l2 ) u , C = ( l1 , l2 ) u ) , D = ( l2
synonym p is is is is is / for p is / of n , L & p is is / ;
Y1 `2 = - 1 & 0. TOP-REAL 2 <> Y1 & Y1 `2 = - 1 & Y1 `2 = - 1 & Y1 `2 = 1 ;
defpred X [ Nat , set , set ] means P [ $1 , $2 ] & Q [ $1 , $2 ] ;
consider k being Nat such that for n being Nat st k <= n holds s . n < x0 + g ;
Det ( I |^ ( m -' n ) ) ~ = 1. ( K , n -' n ) & Det ( I |^ ( m -' n ) ) = 1. K ;
( - b - sqrt ( b ^2 - 4 * a * c ) ) / ( 2 * a * c ) < 0 ;
CC . d = CC . da mod CC . db .= CC . da mod CC . db .= CC . da mod CC . db ;
attr X1 is dense means : Def1 : X2 is dense dense & X1 /\ X2 is dense dense implies X1 /\ X2 is dense dense SubSpace of X1 ;
deffunc F6 ( Element of E , Element of I ) = $1 * $2 & $1 * $2 = ( $1 * $2 ) * ( $1 * $2 ) ;
t ^ <* n *> in { t ^ <* i *> : Q [ i , T . t ] } ;
( x \ y ) \ x = ( x \ x ) \ y .= y ` \ y .= 0. X ;
for X being non empty set for Y being Subset-Family of X holds X is Basis of <* X , FinMeetCl Y *>
synonym A , B are_separated means : Def1 : Cl A misses B & A misses Cl B & B misses Cl A & A misses Cl B ;
len ( M @ ) = len p & width ( M @ ) = width ( M @ ) & len ( M @ ) = width ( M @ ) ;
J . v = { x where x is Element of K : 0 < v . x & 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 .] ;
|. a .| * ||. f .|| = 0 * ||. f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| ;
f . x = ( h . x ) `1 & g . x = ( h . x ) `2 & h . y = ( h . y ) `2 ;
ex w st w in dom B1 & <* 1 *> ^ s = <* 1 *> ^ w & len w = len 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 ) = IC ( s , 9 ) .= 5 + 9 .= 5 ;
( IExec ( W6 , Q , t ) ) . intpos ( e + 2 ) = 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 holds x <= y or y <= x or y <= x ;
integral ( integral ( f , C ) , A ) = 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 ` & y <= x ` ;
func |. p \bullet p .| -> variable of A means : Def1 : for x being element holds x in it iff x in ( NBI ( p ) ) . p ;
consider t being Element of S such that x `1 , y `2 '||' z `1 , t `2 and x `2 , z `2 '||' y `2 , t `2 ;
dom x1 = Seg len x1 & len y1 = len l1 & for i st i in dom y1 holds x1 . i = l1 . i * l1 . i ;
consider y2 being Real such that x2 = y2 and 0 <= y2 & y2 <= 1 / 2 and y2 <= 1 / 2 ;
||. f | X /* s1 .|| = ||. f | X .|| /* s1 .= ||. f .|| /* s1 .= ||. f .|| /* s1 ;
( the InternalRel of A ) -Seg ( x ` ) /\ Y = {} \/ {} .= {} \/ {} .= {} .= {} ;
assume that i in dom p and for j being Nat st j in dom q holds P [ i , j ] and i + 1 in dom p and p . i = p . j ;
reconsider h = f | X ( ) as Function of X ( ) , rng f , Y ( ) ;
u1 in the carrier of W1 & u2 in the carrier of W2 implies ( ex v st v in W1 & u = v + ( u1 - u2 ) ) & ( v in W2 )
defpred P [ Element of L ] means M <= f . $1 & f . $1 <= f . ( $1 + 1 ) & f . $1 <= f . ( $1 + 1 ) ;
l . ( u , a , v ) = s * x + ( - ( s * x ) + y ) .= b ;
- ( x-y ) = - x + - y .= - x + - y .= - x + y .= - x + y ;
given a being Point of GX such that for x being Point of GX holds a , x are_\HM { a } and a , x are_\HM { b } ;
fSet = [ [ dom ( @ f2 ) , cod ( @ f2 ) ] , [ cod ( @ f2 ) , cod ( @ f2 ) ] ] ;
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 ` ) |^ d ) ` & x in ( A ` ) |^ d ;
consider u , v being Element of R , a being Element of A such that l /. i = u * a * v ;
( - ( ( p `1 / |. p .| - cn ) / ( 1 + cn ) ) ) ^2 > 0 ;
Carrier ( LS . k ) = Carrier ( F . k ) & F . k in dom ( L . k ) ;
set i2 = AddTo ( a , i , - n ) , i1 = goto ( card I + 2 ) , i2 = goto ( card I + 3 ) ;
attr B is thesis means : Def1 : for S being SubSub|. of B holds S is B `1 & S `2 = ( B `1 ) `1 ;
a9 "/\" D = { a "/\" d where d is Element of N : d in D & a "/\" d in D } ;
|( exp_R , q )| * |( - q , q )| * |( - q , q )| >= |( exp_R , q )| * |( - q , q )| ;
( - f ) . sup A = ( ( - f ) | A ) . sup A .= ( ( - f ) | A ) . sup A ;
( G * ( len G , k ) ) `1 = ( ( G * ( len G , k ) ) `1 ) / ( G * ( len G , k ) ) `1 ;
( Proj ( i , n ) ) . LM = <* ( proj ( i , n ) ) . LM *> ;
f1 + f2 * reproj ( i , x ) is_differentiable_in ( reproj ( i , x ) ) . x0 + ( f1 + f2 ) . x ;
attr ( for x st x in Z holds ( tan . x ) <> 0 ) & ( tan | Z is continuous implies ( tan | Z ) . x = tan . x ) ;
ex t being SortSymbol of S st t = s & h1 . t = h2 . t & for x being set st x in dom h1 holds P [ x , t . x ] ;
defpred C [ Nat ] means P8 . $1 is thesis & A8 : A8 : A8 : A8 : A8 : A is thesis & A8 is thesis ;
consider y being element such that y in dom ( p | i ) and ( q | i ) . y = ( p | i ) . y ;
reconsider L = product ( { x1 } +* ( index B , l ) ) as Subset of ( Carrier A ) . ( index B ) ;
for c being Element of C ex d being Element of D st T . ( id c ) = id d & for d being Element of D holds d in c iff d in D
n ( ) = ( f | n ) ^ <* p *> .= f ^ <* p *> .= f ^ <* p *> ;
( f * g ) . x = f . ( g . x ) & ( f * h ) . x = f . ( h . x ) ;
p in { ( 1 - 2 ) * ( G * ( i + 1 , j ) + G * ( i + 1 , j + 1 ) ) } ;
f `2 - cp = ( f | ( n , L ) ) *' ( - ( f | ( n , L ) ) ) .= ( f - ( c - g ) ) *' ( - ( f | ( n , L ) ) ) ;
consider r be Real such that r in rng ( f | divset ( D , j ) ) and r < m + s ;
f1 . |[ ( 8 + 1 ) / ( 8 + 1 ) , ( 8 + 1 ) / ( 8 + 1 ) ]| in f1 .: W1 /\ W2 ;
eval ( a | ( n , L ) , x ) = eval ( a | ( n , L ) ) .= a . x ;
z = DigA ( tz , xx ) .= DigA ( tz , xx ) .= DigA ( tz , xx ) .= DigA ( tz , xx ) ;
set H = { Intersect S where S is Subset-Family of X : S c= G & S c= G } , F = G | ( the carrier of X ) , G = F | ( the carrier of X ) , G = F | ( the carrier of X ) , H = G | ( the carrier of X ) , F = G | ( the carrier of X ) , G = F | ( the carrier of X ) , F = G |
consider S19 being Element of D * , d being Element of D * such that S `1 = S19 ^ <* d *> and S `2 = 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 ( L ) = 0. ( V ) implies Sum ( L ) = Sum ( L )
let k1 , k2 , k2 , x4 be Instruction of SCM+FSA , a be Int-Location , k1 be Int-Location , k2 be Int-Location , k2 be Nat ;
consider j being element such that j in dom a and j in g " { k ' ( i + 1 ) } and x = a . j ;
H1 . x1 c= H1 . x2 or H1 . x2 c= H1 . x2 or H1 . x2 c= H1 . x2 & H1 . x2 c= H1 . x2 ;
consider a being Real such that p = \rbrace * p1 + ( a * p2 ) and 0 <= a and a <= 1 ;
assume that a <= c & d <= b and [' a , b '] c= dom f and [' a , b '] c= dom g ;
cell ( Gauge ( C , m ) , len Gauge ( C , m ) -' 1 , 0 ) is non empty ;
A, { i } in { ( S . i ) `1 where i is Element of NAT : not contradiction } ;
( T * b1 ) . y = L * ( b2 /. y ) .= ( F /. y ) . y .= ( F /. y ) . y ;
g . ( s , I ) . x = s . y & g . ( s , I ) . y = |. s . x - s . y .| ;
( log ( 2 , k + k ) ) ^2 >= ( log ( 2 , k + 1 ) ) ^2 ;
then that p => q in S and not x in the still of p and not x in S and not x in S & not x in S ;
dom ( the InitS of r-10 of rM ) misses dom ( the InitS of rM ) & dom ( the InitS of rM ) misses dom ( the InitS of rM ) ;
synonym f is extended integer means : Def1 : for x being set st x in rng f holds x is integer ;
assume for a being Element of D holds f . { a } = a & for X being Subset-Family of D holds f . ( f .: X ) = f . union X ;
i = len p1 .= len p3 + len <* x *> .= len p3 + len <* x *> .= len p3 + 1 .= len p3 + 1 ;
l ( ) = ( g /. 3 ) `1 + ( k ( ) ) `1 - ( k ( ) ) `1 + ( k ( ) ) `1 - ( k ( ) ) / ( k ( ) ) ;
CurInstr ( P2 , Comput ( P2 , s2 , l2 ) ) = halt SCM+FSA .= ( halt SCM+FSA ) . IC SCM+FSA .= ( halt SCM+FSA ) . IC SCM+FSA .= ( ( l , s2 ) .--> ( I , k ) ) . IC SCM+FSA ;
assume for n be Nat holds ||. seq .|| . n <= ( seq . n ) * ( seq . n ) & ( seq is summable implies seq is summable ) & ( seq is summable implies seq is summable ) & ( seq is summable implies seq is summable )
sin . ( \vert non .| ) = sin r * cos ( ( cos r ) * sin s ) .= 0 ;
set q = |[ g1 `1 . t0 , g2 `2 . t0 ]| , g1 = |[ g2 `1 . t0 , g2 `2 . t0 ]| , g2 = |[ g2 `1 . t0 , g2 `2 . t0 ]| ;
consider G being sequence of S such that for n being Element of NAT holds G . n in implies G . n = implies G . n = S . ( n + 1 ) ;
consider G such that F = G and ex G1 st G1 in SM & G = ( the carrier of G1 ) \/ { H } ;
the root of [ x , s ] in ( the Sorts of Free ( C , X ) ) . s & the Sorts of C = ( the Sorts of Free ( C , X ) ) . s ;
Z c= dom ( exp_R (#) ( f + ( #Z 3 ) (#) ( f + ( #Z 3 ) (#) ( f1 + f2 ) ) ) ;
for k be Element of NAT holds seq1 . k = ( ( \HM { the carrier of S } ) | ST ) . k
assume that - 1 < n ( ) and q `2 > 0 and ( q `1 / |. q .| - cn ) < 0 and q `2 / |. q .| - cn < 0 ;
assume that f is continuous and a < b and a < d and f = g and f = g and f = h and f = k and g = k ;
consider r being Element of NAT such that s, r , r is_collinear and r <= q and r <= q and q <= r ;
LE f /. ( i + 1 ) , f /. j , L~ f implies f /. 1 = f /. ( len f -' 1 )
assume that x in the carrier of K and y in the carrier of K and ex_inf_of x , L and ex_inf_of y , L and x in the carrier of L ;
assume f +* ( i1 , \xi ) in ( proj ( F , i2 ) " [: A , B :] ) " [: A , B :] ) " [: A , B :] ;
rng ( ( Flow M ) ~ | ( the carrier of M ) ) c= the carrier' of M & rng ( ( Flow M ) ~ | ( the carrier of M ) ) c= the carrier' of M ;
assume z in { ( the carrier of G ) --> { t } where t is Element of T : t in X } ;
consider l be Nat such that for m be Nat st l <= m holds ||. s1 . m - x0 .|| < g / ( 2 * ||. seq .|| + ||. seq .|| ) ;
consider t be VECTOR of product G such that mt = ||. D5 . t .|| and ||. t .|| <= 1 ;
assume that the carrier of v = 2 and v ^ <* 0 *> , v ^ <* 1 *> ] in dom p and v ^ <* 1 *> in dom p and p . ( len p + 1 ) = v . ( len p + 1 ) ;
consider a being Element of the carrier of X39 , A being Element of the carrier of X39 such that not a on A and not a on A ;
( - x ) |^ ( k + 1 ) * ( ( - x ) |^ ( k + 1 ) ) " = 1 ;
for D being set for i st i in dom p holds p . i in D & p . i is FinSequence of D implies p is FinSequence of D
defpred R [ element ] means ex x , y st [ x , y ] = $1 & P [ x , y ] & P [ y ] ;
L~ f2 = union { LSeg ( p0 , p10 ) , LSeg ( p00 , p1 ) } .= { LSeg ( p1 , p00 ) , LSeg ( p00 , p2 ) } ;
i - len h11 + 2 - 1 < i - len h11 + 2 - 1 + 1 - 1 + 1 + 1 - 1 + 1 ;
for n being Element of NAT st n in dom F holds F . n = |. ( ny . n ) - ( ny . n ) .| ;
for r , s1 , s2 holds r in [. s1 , s2 .] iff s1 <= r & r <= s2 & s1 <= s2 & s2 <= s2
assume v in { G where G is Subset of T2 : G in B2 & G c= z1 & G c= z2 & G c= z1 } ;
let g be element be element of A , INT , X be set , b be Element of INT , f be Function of A , INT , b be Element of X ;
min ( g . [ x , y ] , k ) . [ y , z ] = ( min ( g , k , x ) ) . y ;
consider q1 being sequence of CL such that for n holds P [ n , q1 . n ] and P [ q1 ] ;
consider f being Function such that dom f = NAT and for n being Element of NAT holds f . n = F ( n ) and for x being Element of NAT holds P [ n , x , f . x ] ;
reconsider B-6 = B /\ O , OO = O , OO = Z as Subset of B ;
consider j being Element of NAT such that x = the ` of n and 1 <= j and j <= n and f . j = f . j ;
consider x such that z = x and card ( x . O2 ) in card ( x . O2 ) and x in L1 and x in L2 . O2 ;
( C * ( _ T4 ( k , n2 ) ) ) . 0 = C . ( ( _ T4 ( k , n2 ) ) . 0 ) ;
dom ( X --> rng f ) = X & dom ( X --> f ) = X & rng ( X --> f ) = X ;
( ( ( SpStSeq L~ SpStSeq C ) /. 1 ) `2 <= ( ( SpStSeq C ) /. ( len ( SpStSeq C ) ) ) `2 & ( ( SpStSeq C ) /. ( len ( SpStSeq C ) ) ) `2 <= ( ( ( SpStSeq C ) /. 1 ) `2 ) ;
synonym x , y , z is_collinear means : Def1 : x = y or ex l being Subset of S st { x , y } c= l & not ex l being Subset of S st { x , y } c= l ;
consider X being element such that X in dom ( f | ( n + 1 ) ) and ( f | ( n + 1 ) ) . X = Y ;
assume that k is continuous and for x , y being Element of L , a , b being Element of Image k st a = x & b = y holds x << y iff a << b ;
( 1 / 2 * ( ( ( ( #Z 2 ) * ( ( #Z 2 ) * ( 1 / 2 ) ) ) ) ) * ( ( AffineMap ( 2 , 0 ) ) * ( ( AffineMap ( 2 , 0 ) ) * ( ( AffineMap ( 2 , 0 ) ) * ( ( 0 , 0 ) ) * ( 1 / 2 ) ) ) ) is_differentiable_on REAL ;
defpred P [ Element of omega ] means ( the partial of A1 ) . $1 = A1 . $1 & ( the partial of A2 ) . $1 = A2 . $1 & ( the Sorts of A1 ) . $1 = A1 . $1 ;
IC Comput ( P , s , 2 ) = succ IC Comput ( P , s , 1 ) .= 6 + 1 .= 6 + 1 ;
f . x = f . g1 * f . g2 .= f . g1 * 1_ H .= f . g1 * 1_ H .= f . g1 * ( g . g2 ) .= ( f . g1 ) * ( g . g2 ) ;
( M * F-4 ) . n = M . ( F-4 . n ) .= M . ( { ( canFS Omega ) . n } ) .= M . ( { ( canFS Omega ) . n } ) ;
the carrier of L1 + L2 c= ( the carrier of L1 ) \/ ( the carrier of L2 ) & the carrier of L1 = ( the carrier of L1 ) \/ ( the carrier of L2 ) ;
pred a , b , c , x , y , c , a , b , c , d , x , y , z , x , y , z , y , z , x , y , z , w , x , y , z , x , y , z , w , x , y , z , w , x , y , z , w , x , y , z , w , x , y , z , w , x , z , w , x , y , z , w , z , w , x
( the Sorts of s ) . n <= ( the Sorts of s ) . n * s . ( n + 1 ) & ( the Sorts of s ) . n <= ( the Sorts of s ) . n ;
attr - 1 <= r & r <= 1 & ( arccot ) . r = - 1 / ( 1 + r ^2 ) & ( arccot ) . r = - 1 / ( 1 + r ^2 ) ;
seq in { p ^ <* n *> where n is Nat : p ^ <* n *> in T1 & n in T1 } implies ex n being Nat st n in T1 & p = <* n *> ^ <* n *>
|[ x1 , x2 , x3 ]| . 2 - |[ y1 , y2 , x4 ]| . 2 = x2 - y2 & |[ y1 , y2 , x4 ]| . 2 = x2 - y2 ;
attr F . m is nonnegative means : Def1 : F . m is nonnegative & ( Partial_Sums F ) . n is nonnegative implies ( Partial_Sums F ) . m is nonnegative ;
len ( ( G . z ) * ( y . z ) ) = len ( ( ( G . ( x , y ) ) * ( y . z ) ) ) .= len ( ( G . ( x , y ) ) * ( y . z ) ) ;
consider u , v being VECTOR of V such that x = u + v and u in W1 /\ W2 and v in W1 /\ W3 and v in W2 /\ W3 ;
given F being finite Subset of NAT such that F = x and dom F = n and rng F c= { 0 , 1 } and Sum F = k ;
0 = ( 1 * 0 ) * u\hbox = ( 1 - ( 1 - 0 ) * ( - 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 -> being } -being being being being being being non empty set , ( ( let ( let \rm let L ) | D ) , ( ( \rm w ) | D ) ) is Boolean non empty ;
"/\" ( BB , {} ) = Top BB .= Top BB .= Top ( S ) .= "/\" ( I , T ) .= "/\" ( I , T ) .= "/\" ( I , T ) ;
( r / 2 ) ^2 + ( r / 2 ) ^2 <= ( r / 2 ) ^2 + ( r / 2 ) ^2 ;
for x being element st x in A /\ dom ( f `| X ) holds ( ( f `| X ) || A ) . x >= r2
2 * r1 - 2 * |[ a , c ]| - ( 2 * r1 - 2 * |[ b , c ]| ) = 0. TOP-REAL 2 - 2 * |[ b , c ]| ;
reconsider p = P * ( \square , 1 ) , q = a " * ( ( - ( - 1 ) ) * ( ( - 1 ) |^ n ) ) as FinSequence of K ;
consider x1 , x2 being element such that x1 in uparrow s and x2 in uparrow t and x = [ x1 , x2 ] and y = [ x1 , x2 ] ;
for n be Nat st 1 <= n & n <= len q1 holds q1 . n = ( ( lower ( g , M7 , n ) ) | Seg ( n + 1 ) ) . x
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 y in the carrier of A and z in the carrier of A ;
given H1 , H2 being strict Subgroup of G such that x = H1 and y = H2 and H1 is Subgroup of H2 and H2 is Subgroup of H1 and H2 is strict Subgroup of H2 ;
for S , T being non empty < T , d being Function of T , S st T is complete holds d is monotone & d is monotone & d is monotone implies d is monotone
[ a + 0. F_Complex , b2 ] in ( the carrier of F_Complex ) /\ ( the carrier of F_Complex ) & [ a + 0. F_Complex , b2 ] in the carrier of V & [ a + 0. F_Complex , b2 ] in the carrier of V ;
reconsider mm = max ( len F1 , len ( p . n ) * ( p . n ) ) as Element of NAT ;
I <= width GoB ( ( GoB h ) * ( len GoB h , 1 ) ) & ( GoB h ) * ( len GoB h , 1 ) `2 <= ( GoB h ) * ( len GoB h , 1 ) `2 ;
f2 /* q = ( f2 /* ( f1 /* s ) ) ^\ k .= ( ( f2 * f1 ) /* s ) ^\ k .= ( ( f2 * f1 ) /* s ) ^\ k ;
attr A1 \/ A2 is linearly-independent means : Def1 : A1 misses A2 & ( for x st x in A1 holds x in A1 holds Lin ( A1 ) /\ Lin ( A2 ) = { 0. V } ) & Lin ( A1 ) /\ Lin ( A2 ) = { 0. V } ;
func A -carrier C -> set means : Def1 : union it in { A . s where s is Element of R : s in C & s in C } ;
dom ( Line ( v , i + 1 ) ) (#) ( ( Line ( p , m ) ) * ( \square , 1 ) ) = dom ( F ^ G ) ;
cluster [ x `1 , 4 ] , [ x `2 , 4 ] , [ x `1 , 4 ] , [ x `1 , 4 ] ] -> to x `2 & [ x `1 , 4 ] , [ x `1 , 4 ] ] = x `2 ;
E , All ( x1 , All ( x2 , x2 ) '&' ( x3 , x3 ) '&' ( x4 , x4 ) '&' ( x4 , x4 ) ) |= All ( x4 , x4 ) '&' ( x4 , x4 ) '&' ( x4 , x4 ) '&' ( x4 , x4 ) ;
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 + 1 ) - h . x0 + h . ( m + 1 ) - h . x0 ;
cell ( G , XG -' 1 , ( t + 1 ) + ( t + 1 ) ) \ L~ f meets UBD L~ f \/ UBD L~ f \/ UBD L~ f ;
IC Result ( P2 , s2 ) = IC IExec ( I , P , Initialize s ) .= card I .= card I .= card I + card J .= card I + card J .= card I + card J + card J .= card I + card J + card J + 3 .= card J + card J + 3 ;
sqrt ( ( - ( ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ) ^2 ) > 0 ;
consider x0 being element such that x0 in dom a and x0 in g " { k } and y0 = a . x0 and x0 in { k } and y0 = a . x0 ;
dom ( r1 (#) chi ( A , C ) ) = dom chi ( A , C ) /\ dom ( chi ( A , C ) ) .= dom ( ( r1 (#) chi ( A , C ) ) /\ ( A /\ A ) ) .= dom ( ( r1 (#) chi ( A , C ) ) /\ ( A /\ A ) ) .= dom ( ( r1 (#) chi ( A , C ) ) /\ ( A /\ A ) ) ;
d-7 . [ y , z ] = ( ( ( y `1 ) - z `2 ) * ( y `1 ) - z `2 ) * ( y `2 ) - z `2 * ( y `2 ) ;
attr i be Nat means C . i = A . i /\ B . i & L~ C c= A /\ B . i ;
assume that x0 in dom f and f is_continuous_in x0 and ||. f /. x0 .|| <= ( ||. f .|| ) . x0 and ||. f .|| . x0 = ||. f /. x0 .|| ;
p in Cl A implies for K being Basis of p , Q being Subset of T st Q in K holds A meets Q or A misses Q
for x being Element of REAL n st x in Line ( x1 , x2 ) holds |. y1 - y2 .| <= |. y1 - y2 .| + |. y2 - y1 .|
func Sum ( <*> ) -> Ordinal means : Def1 : a in it & for b being Ordinal st a in b holds it c= b & b is Ordinal & it 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 :] & [ 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 ) ) * ( ( vseq . n ) - ( vseq . m ) ) .|| < ( e / ( ||. x .|| + ( vseq . m ) ) ) * ( ||. x .|| + ( vseq . n ) ) ;
then for Z being set st Z in { Y where Y is Element of I7 : F c= Y & Y in Z } holds z in x & z in Z ;
sup compactbelow [ s , t ] = [ sup [: X , { t } :] , sup [: X , { t } :] :] ] .= [ sup X , sup { 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 , z ] 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 is FinSequence of D & q is FinSequence of D
consider e19 being Element of the carrier of X such that c9 , a9 // a9 , e29 and a9 <> b9 and a9 <> b9 and b9 <> c9 and b9 <> c9 ;
set U2 = I \! \mathop { \vert S .| } , U2 = I \! \mathop { \vert S .| } , E = { S } , F = I " { {} } , N = I " { {} } , M = I " { {} } , M = I " { {} } , N = I " { {} } , N = I " { {} } , M = I " { {} } , M = I " { {} } , M = I " { {} } , N = I " { {} } , M = I " { {} } , M = I " { {} } , N = I , M = I
|. q3 .| ^2 = ( |. q3 .| ) ^2 + ( |. q3 .| ) ^2 .= |. q .| ^2 + ( |. q .| ) ^2 .= |. q .| ^2 + ( |. q .| ) ^2 ;
for T being non empty TopSpace , x , y being Element of [: the topology of T , the topology of T :] holds x "\/" y = x \/ y & x "/\" y = x /\ y implies x "/\" y = x /\ y
dom ( signature U1 ) = dom ( the charact of U1 ) & Args ( o , MSAlg U1 ) = dom ( the charact of U1 ) & Args ( o , MSAlg U1 ) = dom ( the charact of U1 ) ;
dom ( h | X ) = dom h /\ X .= dom ( ( - h ) | X ) /\ X .= dom ( ( - h ) | X ) /\ X .= dom ( ( - h ) | X ) /\ X .= dom ( ( - h ) | X ) ;
for N1 , N1 being Element of ( G . K1 ) holds dom ( h . K1 ) = N & rng ( h . K1 ) c= N1 & rng ( h . K1 ) c= N1 & rng ( h . K1 ) c= N2
( mod ( u , m ) + mod ( v , m ) ) . i = ( mod ( u , m ) ) . i + ( mod ( v , m ) ) . i ;
- ( q `1 ) < - 1 or - ( q `2 ) >= - ( q `1 ) & - ( q `2 ) <= 1 or - ( q `1 ) >= - ( q `2 ) & - ( q `1 ) <= 1 ;
attr r1 = fp & r2 = fp & r1 * r2 = fp * ( r1 - r2 ) & r2 * ( r1 - r2 ) = fp * ( r2 - r2 ) ;
vseq . m is bounded Function of X , the carrier of Y & x9 . m = ( ( ( vseq . m ) - ( vseq . n ) ) | X ) . x ;
attr a <> b & b <> c & angle ( a , b , c ) = PI implies angle ( b , c , a ) = 0 & angle ( c , a , b ) = 0 ;
consider i , j being Nat , r being Real such that p1 = [ i , r ] and p2 = [ j , s ] 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 ^ q1 and p1 ^ q1 = p1 ^ q1 and p1 ^ q1 = p2 ^ q2 and p1 ^ q2 = p2 ^ q2 ;
( ( the carrier of [ A , r1 , s1 ] ) --> ( the carrier of A , s1 ) ) = ( ( s2 * s1 ) gcd ( s1 * s2 ) ) .= ( s2 * s1 ) /\ ( s1 * s2 ) .= ( s2 * s1 ) /\ ( s1 * s2 ) ;
( ( LMP A ) `2 = lower_bound ( proj2 .: A /\ /\ ( E-bound A ) ) & proj2 .: A /\ ( proj2 .: A /\ Vertical_Line w ) is non empty ;
s , ( ( k , 1 ) \bf ( k , 1 ) \bf ( k , 1 ) \bf ( k , 1 ) \bf ( k , 1 ) \bf ( k , 1 ) \bf ( k , 1 ) \bf ( k , 1 ) \bf ( k , 1 ) ) . ( k , 1 ) ;
len ( s + 1 ) = card support b1 + 1 .= card support b2 + 1 .= card support b2 + 1 .= card support b1 + 1 .= len ( s + 1 ) + 1 .= len ( s + 1 ) + 1 .= len ( s + 1 ) ;
consider z being Element of L1 such that z >= x and z >= y and for z being Element of L1 st z >= x & z in X holds z `1 >= y `1 and z `2 >= y `2 ;
LSeg ( UMP D , |[ ( W-bound D + E-bound D ) / 2 , ( E-bound D + E-bound D ) / 2 ]| ) /\ D = { UMP D , ( ( UMP D + E-bound D ) / 2 ) / 2 } ;
lim ( ( ( f `| N ) / ( g `| N ) ) /* b ) = ( lim ( ( f `| N ) / ( g `| N ) ) ) / ( g `| N ) ;
P [ i , pr1 ( f ) . i , pr2 ( f ) . ( i + 1 ) , pr2 ( f ) . ( i + 1 ) ] ;
for r be Real st 0 < r ex m be Nat st for k be Nat st m <= k holds ||. ( seq . k - seq . n ) - ( seq . k ) .|| < r
for X being set , P being a_partition of X , x , a , b being set st x in a & a in P & b in P & x in P & b in P holds a = b
Z c= dom ( ( #Z 2 ) /\ ( dom ( ( #Z 2 ) * f ) \ ( ( #Z 2 ) * f ) " { 0 } ) ) /\ ( dom ( ( #Z 2 ) * f ) " { 0 } ) ;
ex j being Nat st j in dom ( l ^ <* x *> ) & j < i & y = ( l ^ <* x *> ) . j & i = 1 + len l & z = 1 + len l & j = len l + 1 ;
for u , v being VECTOR of V , r being Real st 0 < r & u in N & v in N holds r * u + ( 1-r * v ) in N
A , Int Cl A , Cl Int Cl A , Cl Cl Int Cl A , Cl Cl Int Cl A , Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl
- Sum <* v , u , w *> = - ( v + u + w ) .= - ( v + u ) + ( w + w ) .= - ( v + u ) + ( w + w ) .= - ( v + u ) + ( w + w ) ;
( Exec ( a := b , s ) ) . IC SCM R = ( Exec ( a := b , s ) ) . IC SCM R .= ( Exec ( a := b , s ) ) . IC SCM R .= ( 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 ) /\ ( the carrier of L ) and h . x = ( the carrier of J ) /\ ( the carrier of L ) ;
for S1 , S2 , D being non empty reflexive RelStr , D being non empty directed Subset of [: S1 , S2 :] , x being Element of [: S1 , S2 :] holds cos ( x ) is directed & cos ( x ) is directed & cos ( x ) is directed
card X = 2 implies ex x , y st x in X & y in X & x <> y or x = y & not x = y or x = y & not x = y or x = y
E-max L~ Cage ( C , n ) in rng ( Cage ( C , n ) \circlearrowleft W-min L~ Cage ( C , n ) ) & E-max L~ Cage ( C , n ) in rng ( Cage ( C , n ) \circlearrowleft E-max L~ Cage ( C , n ) ) ;
for T , T being decorated tree , p , q being Element of dom T , q being Element of dom T st p element q in dom T holds ( T -with q , p ) . 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 ( k gcd n ) divides ( k gcd n ) & n divides ( k gcd n ) & ( k divides ( k gcd n ) implies ( k divides ( k gcd n ) ) & ( k divides ( k gcd n ) ) & ( k divides ( k gcd n ) ) implies ( k divides ( k gcd n ) ) & ( k divides ( k gcd n ) ) & ( k divides ( k gcd n ) ) implies k divides ( k -' n ) )
dom F " = the carrier of X2 & rng F " = the carrier of X1 & F " { x } = the carrier of X2 & F " { y } = the carrier of X1 & F " { y } = the carrier of X2 ;
consider C being finite Subset of V such that C c= A and card C = n and the carrier of V = Lin ( B9 \/ C ) and Sum ( B ) = Sum ( B ) and Sum ( B ) = Sum ( B ) ;
V is prime implies for X , Y being Element of \langle the topology of T , \subseteq the topology of T , V be Subset of T st X /\ Y c= V holds X c= V or Y c= V
set X = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] & P [ v2 ] } , Y = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } ;
angle ( p1 , p3 , p4 ) = 0 .= angle ( p2 , p3 , p4 ) .= angle ( p , p3 , p2 ) .= angle ( p , p3 , p2 ) .= angle ( p , p3 , p2 ) .= angle ( p , p3 , p2 ) ;
- sqrt ( ( - ( q `1 / |. q .| - cn ) ) ^2 + ( q `2 / |. q .| - cn ) ^2 ) = - sqrt ( ( - ( q `1 / |. q .| - cn ) ) ^2 + ( q `2 / |. q .| - cn ) ^2 ) .= - 1 ;
ex f being Function of I[01] , ( TOP-REAL 2 ) | P st f is continuous one-to-one & rng f = P & f . 0 = p1 & f . 1 = p2 & f . 1 = p3 & f . 1 = p4 ;
attr f is partial differentiable on 2 means : Def1 : SVF1 ( 2 , pdiff1 ( f , 1 ) , u0 ) is_differentiable_in ( proj ( 2 , 3 ) . u0 ) . x0 ;
ex r , s st x = |[ r , s ]| & G * ( len G , 1 ) `1 < r & r < G * ( 1 , 1 ) `1 & G * ( len G , 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 f /. 1 = ( GoB f ) * ( t , width G ) `2 and f /. len f = ( GoB f ) * ( t , width G ) `2 ;
attr i in dom G means : Def1 : 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 /. k = c1 + c2 and c /. k = c2 /. k and c /. k = c2 /. k ;
u0 in { |[ r1 , s1 ]| : r1 < G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `2 < G * ( 1 , 1 ) `2 & G * ( 1 , 1 ) `2 < s1 } ;
Cl ( X ^ Y ) . k = the carrier of X . k2 .= C4 . ( k2 + 1 ) .= C4 . ( k2 + 1 ) .= C4 . ( k2 + 1 ) .= C4 . ( k2 + 1 ) ;
attr M1 = len M2 & width M1 = width M2 & width M1 = width M2 & M1 = M2 - M1 implies M1 = M2 - M2 & M1 = M2 - M2
consider g2 be Real such that 0 < g2 and { y where y is Point of S : ||. ( - x0 ) - ( x0 - x0 ) .|| < g2 & g2 in N2 } c= N2 /\ dom ( f | X ) ;
assume x < ( - b + sqrt ( o , b , c ) ) / 2 or x > ( - b - sqrt ( o , b , c ) ) / 2 or x > ( - b - sqrt ( o , b , c ) ) / 2 ;
( G1 '&' G2 ) . i = ( <* 3 *> ^ G1 ) . i & ( H1 '&' H2 ) . i = ( <* 3 *> ^ G1 ) . i & ( H1 '&' H2 ) . 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 i in dom f & i divides len f holds i divides len f implies i divides len f & i divides len f
assume F = { [ a , b ] where a , b is Subset of X : for c being set st c in Bl & a c= c holds b c= c & a c= c & b c= c } ;
b2 * q2 + ( b3 * q3 ) + ( - ( a * q3 ) + ( - ( a * q3 ) ) ) * ( ( - ( a * q3 ) + ( a * q3 ) ) * ( ( - ( a * q3 ) + ( a * q3 ) ) * ( ( - ( a * q2 ) + ( a * q3 ) ) * ( ( - ( a * q2 ) + ( a * q3 ) ) * ( ( - ( a * q2 ) ) * ( ( - ( a * q3 ) ) * ( 1 / ( 1 / ( 1 / ( 1 / ( 1 / ( 1 / ( 1 / ( 1 / ( 1 / ( 1 / ( 1 / (
Cl Cl F = { D where D is Subset of T : ex B being Subset of T st D = Cl B & B in F & B in F & Cl B c= Cl F } ;
attr seq is summable means : Def1 : seq is summable & seq + seq is summable & Partial_Sums ( seq ) = Partial_Sums ( seq ) + Partial_Sums ( seq ) & Partial_Sums ( seq ) = Partial_Sums ( seq ) + Partial_Sums ( seq ) ;
dom ( ( ( ( ( TOP-REAL 2 ) | D ) | D ) | D ) ) = ( the carrier of ( ( TOP-REAL 2 ) | D ) ) /\ D .= D ;
|[ X , Z ]| is full full non empty SubRelStr of ( Omega Z ) |^ the carrier of X & |[ X \to Y , Z ]| is full full SubRelStr of ( Omega Z ) |^ the carrier of Y ;
G * ( 1 , j ) `2 = G * ( i , j ) `2 & G * ( 1 , j ) `2 <= G * ( i , j ) `2 or G * ( 1 , j ) `2 <= G * ( i , j ) `2 ;
synonym m1 c= m2 means : Def1 : for p being set st p in P holds the non empty set of p <= m2 & the non empty set of p <= m2 implies ( m1 , m2 ) `1 <= ( m2 , p ) `1 or ( m1 , m2 ) `2 <= ( m2 , p ) `2 ;
consider a being Element of B ( ) such that x = F ( a ) and a in { G ( b ) where b is Element of A ( ) : P [ b ] } and P [ a ] ;
synonym R is multiplicative means : Def1 : the carrier of R , the carrier of R -> Relation of the carrier of R , the carrier of R , the carrier of R , the carrier of R #) , the carrier of R , the carrier of R , the carrier of R #) , the carrier of R be Relation of the carrier of R ;
L ( a , b , 1 ) + L ( c , d ) = b + L ( c , d ) .= b + d .= b + d .= b + ( c + d ) .= b + ( c + d ) .= b + ( c + d ) ;
cluster + ( i , j ) -> natural for Element of INT , i1 , i2 be Element of INT , i2 be Element of INT , i1 , i2 be Element of INT , i2 be Element of INT ;
( ( - s2 ) * p1 + ( s2 * p2 ) - ( s2 * p2 ) * p2 ) = ( ( - r2 ) * p1 + ( r2 * p2 ) * p2 - ( s2 * p2 ) * p2 ) + ( ( - r2 ) * p2 ) * p2 - ( s2 * p2 ) * p2 ;
eval ( ( a | ( n , L ) ) *' p , x ) = eval ( a | ( n , L ) ) * eval ( p , x ) .= a * eval ( p , x ) .= a * eval ( p , x ) ;
assume that the TopStruct of S = the TopStruct of T and for D being non empty directed Subset of Omega S , V being open Subset of Omega T st sup D in V holds V is open & V is open and V is open and for V being open Subset of S st V in V holds V is open & V is open & V is open ;
assume that 1 <= k & k <= len w + 1 and T-7 . ( ( q , w ) -succ k ) = ( T-7 . k , w -succ ( q , w ) ) -succ ( ( q , w ) -succ k ) ;
2 * a |^ ( n + 1 ) + ( 2 * b |^ ( n + 1 ) ) >= a |^ ( n + 1 ) + ( b |^ n ) + ( ( a |^ n ) * b |^ ( n + 1 ) ) + ( ( b |^ n ) * a ) + ( ( b |^ n ) * b ) * a ) ;
M , v2 / ( x. 3 , m ) / ( x. 4 , n ) / ( x. 0 , m ) / ( x. 4 , n ) / ( x. 0 , m ) / ( x. 4 , n ) / ( x. 4 , m ) / ( x. 0 , n ) / ( x. 4 , m ) ) / ( x. 0 , n ) / ( x. 4 , n ) / ( x. 0 , m ) / ( x. 4 , n ) ) / ( x. 0 , m ) |= ( x. 4 , n ) ;
assume that f is_differentiable_on l and for x0 st x0 in l holds 0 < f ' ( x ) or for x0 st x0 in l holds f ' ( x ) < 0 & f ' ( x ) < 0 ;
for G1 being _Graph , W being Walk of G1 , e being set , G2 being Walk of G1 , e being Vertex of G2 st not e in W and not e in W holds not e in W .vertices() & not e in W implies not e in ( the carrier of G1 ) \ ( the carrier of G2 )
not not not not not lim is empty iff not ( ( not ( ex y being Element of S st y is not empty & not ( y is not empty & not ( y is not empty & not ( y is not empty & not y is not empty ) ) & not ( not y is not empty ) ) & not not not not not ( not y is not empty & not not y is not empty ) ;
Indices GoB f = [: dom GoB f , Seg width GoB f :] & i1 + 1 in dom GoB f & i2 + 1 in Seg ( len GoB f ) & i1 + 1 in dom GoB f & i2 + 1 in Seg ( len GoB f ) & i1 + 1 in dom GoB f & i2 + 1 in Seg ( len GoB f ) ;
for G1 , G2 , G3 being Group , G1 , G2 , G3 being strict Subgroup of O st G1 is stable & G2 is stable & G1 is stable holds G1 is stable Subgroup of G2 & G2 is stable Subgroup of G3 & G2 is stable Subgroup of G3
UsedIntLoc ( int ( f , 3 ) ) = { intloc 0 , intloc 1 , intloc 2 , intloc 3 , intloc 4 , intloc 5 , intloc 6 , intloc 5 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 9 } ;
for f1 , f2 be FinSequence of F st f1 ^ f2 is p -element & Q [ f1 ^ f2 ] & Q [ f1 ^ f2 ] & Q [ f2 ^ f1 ] holds Q [ f1 ^ f2 ] & Q [ f2 ^ f1 ]
( p `1 ) ^2 / sqrt ( 1 + ( p `2 / p `1 ) ^2 ) = ( q `1 ) ^2 / sqrt ( 1 + ( q `2 / q `1 ) ^2 ) .= ( q `1 ) ^2 / sqrt ( 1 + ( q `2 / q `1 ) ^2 ) ;
for x1 , x2 , x3 being Element of REAL n holds |( x1 - x2 , x3 - x4 )| = |( x1 - x2 , x3 - x4 )| + |( x2 - x3 , x4 )| + |( x3 - x4 )|
for x st x in dom ( ( F | A ) | A ) holds ( ( F | A ) | A ) . ( - x ) = ( - ( F | A ) ) . ( - x )
for T being non empty TopStruct , P being Subset-Family of T , x being Point of T , B being Basis of T , P being Basis of x st B c= P & P is Basis of x holds P is Basis of x
( a 'or' b 'imp' c ) . x = 'not' ( ( a 'or' b ) . x ) 'or' c . x .= 'not' ( a . x ) 'or' b . x .= TRUE 'or' TRUE .= TRUE .= TRUE .= TRUE ;
for e being set st e in [: A , Y1 :] ex X1 being Subset of [: Y , Y1 :] , Y1 being Subset of [: X , Y1 :] , Y1 being Subset of [: Y , Y1 :] st e = X1 & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open
for i be set st i in the carrier of S for f be Function of Sconsider S . i , S1 . i st f = H . i holds F . i = f | ( F . i ) & for i be Nat 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 ) . v = Valid ( VERUM ( Al ) , J ) . w
card D = card D1 + card D2 - card { i , j } .= ( c1 + 1 ) + ( c2 - 1 ) - 1 .= c1 + 1 - 1 .= c1 + 1 - 1 .= c1 + 1 - 1 .= c1 + 1 - 1 .= c1 + 1 - 1 .= c1 + 1 - 1 .= c1 + 1 ;
IC Exec ( i , s ) = ( s +* ( 0 .--> succ ( s . 0 ) ) ) . 0 .= ( s .--> succ ( s . 0 ) ) . 0 .= ( s .--> succ 0 ) . 0 .= s . 0 .= s . 0 ;
len f /. ( \downharpoonright i1 -' 1 ) -' 1 + 1 = len f -' 1 + 1 - 1 + 1 .= len f -' 1 + 1 + 1 .= len f -' 1 + 1 + 1 .= len f -' 1 + 1 + 1 + 1 .= len f -' 1 + 1 + 1 ;
for a , b , c being Element of NAT st 1 <= a & 2 <= b holds k < ( a + b ) / ( a + b ) or k = a + b-3 or k = a + b-3 or k = b + b-3 or k = a + b-3 or k = b + b-2 or k = a + b-2 or k = b + b-2
for f being FinSequence of TOP-REAL 2 , p being Point of TOP-REAL 2 , i being Nat st p in LSeg ( f , i ) holds Index ( p , f ) <= i & Index ( p , f ) <= len f & Index ( p , f ) <= Index ( p , f )
lim ( ( curry ( P+* ( k , n + 1 ) ) # x ) ) = lim ( ( curry ( P+* ( k , n + 1 ) ) ) + ( ( curry ( F+* ( k , n + 1 ) ) # x ) ) ;
z2 = g /. ( \downharpoonright n1 -' 1 + 1 ) .= g . ( i - n2 + 1 ) .= g . ( i - n2 + 1 ) .= g . ( i - n2 + 1 ) .= g . ( i - n2 + 1 ) .= g . ( i - n2 + 1 ) .= g . ( i - n2 + 1 ) ;
[ f . 0 , f . 3 ] in id the carrier of G \/ ( the InternalRel of G ) or [ f . 0 , f . 3 ] in the InternalRel of C6 or [ f . 0 , f . 2 ] in the InternalRel of C6 ;
for G being Subset-Family of B st G = { R [ X ] where R is Subset of [: A , B :] , D is Subset of [: A , B :] st R in F6 & for X being Subset of [: A , B :] holds ( Intersect ( F ) ) . X = Intersect ( G ) . X holds ( Intersect ( F ) ) . X = Intersect ( G )
CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 ) ) = CurInstr ( P1 , Comput ( P1 , s1 , m1 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m2 ) ) .= halt SCMPDS .= halt SCMPDS .= ( CurInstr ( P1 , s1 ) ) ;
assume that a on M and b on M and c on N and d on N and p on N and a on P and a on P and b on Q and a on P and b on Q and c on Q and a <> P and b <> Q and c <> Q and a <> Q and b <> Q and c <> Q and c <> Q and c <> Q and c <> Q ;
assume that T is \hbox 4 -\cal T and F is closed and ex F being Subset-Family of T st F is closed & for n being Nat holds F . n is finite-ind & ind F <= 0 and ind T <= 0 and ind T <= 0 ;
for g1 , g2 st g1 in ]. r - g2 , r .[ & g2 in ]. r - r , r + g2 .[ holds |. f . g1 - f . g2 .| <= ( ( g1 - g2 ) / ( r - g2 ) ) / ( r - g2 )
cosh /. ( z1 + z2 ) = ( cosh /. z1 ) * ( cosh /. z2 ) + ( ( ( - ( 1 / 2 ) ) * ( sin /. z2 ) ) * ( sin /. z2 ) + ( ( ( 1 / 2 ) * ( sin /. z2 ) ) * ( sin /. z2 ) ) * ( sin /. z2 ) ) ;
F . i = F /. i .= 0. R + r2 .= ( b |^ n ) ^ ( a |^ ( n + 1 ) ) .= <* ( n + 1 ) / a |^ 0 , b |^ ( n + 1 ) ) * b . i , ( n + 1 ) / a |^ ( n + 1 ) *> ;
ex y being set , f being Function st y = f . n & dom f = A ( ) & for n holds f . ( n + 1 ) = A ( ) & for n holds f . n = R ( n , f . n ) & y = f . n ;
func f (#) F -> FinSequence of V means : Def1 : len it = len F & for i be Nat st i in dom it holds it . i = F /. i * f /. ( F /. i ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , 7 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 } = { x1 , x2 , x3 , x4 , 7 , 8 } \/ { 7 , 8 } \/ { 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 ( ) & o . ( x , n ) in InnerVertices S ( ) & o . ( n + 1 ) in InnerVertices S ( ) & x = ( the Sorts of S ( ) ) . n ;
ex S1 being Element of CQC-WFF ( Al ) st SubP ( P , Ul ) = S1 & ( ( S , l ) `1 = ( S , l ) `1 & ( S , l ) `2 = ( S , l ) `1 & ( S , l ) `2 = ( S , l ) `2 & ( S , l ) `1 = ( S , l ) `1 ) & ( S , l ) `2 = ( S , l ) `1 ;
consider P being FinSequence of Gs2 such that pp = Product P and for i st i in dom P ex t7 being Element of the carrier of G st P . i = t7 & t . i = t . i & P . i = t . i and P . i = t . i ;
for T1 , T2 being strict non empty TopSpace , P being Basis of T1 , Q being Basis of T2 , P being Subset of T1 , Q being Subset of T2 , f being Function of T1 , T2 st the carrier of T1 = P & f is continuous holds f " Q = P & f " Q = P
assume that f is_partial_differentiable_in u0 , u0 and r (#) pdiff1 ( f , 3 ) is_partial_differentiable_in u0 , 2 and partdiff ( r (#) pdiff1 ( f , 3 ) , u0 , 2 ) = r (#) pdiff1 ( f , 3 ) and partdiff ( r (#) pdiff1 ( f , 3 ) , u0 , 2 ) = r (#) pdiff1 ( f , 3 ) ;
defpred P [ Nat ] means for F , G being FinSequence of ExtREAL for s be Permutation of Seg $1 , G be Permutation of Seg $1 st len s = $1 & not G = F * s & not ( ex k be Nat st k in dom F & not k in rng F & not k in rng F ) holds Sum ( F ) = Sum ( F ) ;
ex j st 1 <= j & j < width GoB f & ( ( GoB f ) * ( 1 , j ) ) `2 <= s & s <= ( ( GoB f ) * ( 1 , j + 1 ) ) `2 & s <= ( ( GoB f ) * ( 1 , j + 1 ) ) `2 & s <= ( ( GoB f ) * ( 1 , j + 1 ) ) `2 ;
defpred U [ set , set ] means ex F-23 be Subset-Family of T st $1 = F-23 & $2 is open Subset of T & union F-23 is open & union F-23 is open & ( union F-23 is open & union F-23 is open & union F-23 c= $1 ) & ( not U c= F & not U c= F & not U is open & not U c= F & not U c= F . $1 ;
for p4 being Point of TOP-REAL 2 st LE p4 , p4 , P , p1 , p2 holds LE p4 , p , P , p1 , p2 & LE p4 , p , P , p1 , p2 & LE p4 , p , P , p1 , p2 & LE p4 , p , P , p1 , p2 & LE p4 , p , P , p1 , p2 & LE p4 , p , P , p1 , p2 & LE p4 , p , P , p1 , p2 holds LE p4 , p , P , p1 , p2 , p2 , p2 & LE p2 , p , p2 , p2 & LE p2 , p , p2 , p2 , p2 , p2 , p2 & LE p2 , p , p2 , p2 & LE p2 , p , p2 & LE p2 , p , P , p1 , p2 , p2 & LE p1 , p1 , P , p1 , p3 , p3 , p3 , p3 , p3 , p3 , p3 , p3 , p3 , p3
f in set ( E , H ) & for g st g <> f . y holds x = g . y implies x in D & f in D implies g in E & f in E & f . x = All ( All ( x , H ) , E )
ex 8 being Point of TOP-REAL 2 st x = 8 & ( ( ( - 1 ) * ( 8 - 1 ) ) / ( 1 + ( - ( sn ) ) ^2 ) >= 8 & ( - ( sn ) ) / ( 1 + ( sn ) ) ^2 ) >= 0 ) & ( ( sn ) * ( ( sn ) / ( 1 + ( sn ) ) ^2 ) >= 0 ;
assume for d7 being Element of NAT st d7 <= ( len ( n7 ) ) holds s1 . ( d7 ) = ( ( d . ( d7 ) ) | ( d7 ) ) . ( d7 ) & s1 . ( d7 ) = ( d . ( d7 ) ) . ( d7 ) ;
assume that s <> t and s is Point of Sphere ( x , r ) and s is not Point of Sphere ( x , r ) and ex e being Point of Sphere ( x , r ) st { e } = Sphere ( s , r ) /\ Sphere ( x , r ) and e in Sphere ( s , r ) and e <> s ;
given r such that 0 < r and for s holds 0 < s or ex x1 be Point of RNS st x1 in dom f & ||. x1 - x0 .|| < s & |. x1 - x0 .| < s & |. x1 - x0 .| < r & |. x1 - x0 .| < s ;
( p | x ) | ( p | ( x | x ) ) = ( ( ( x | x ) | ( x | x ) ) | p ) | ( p | ( x | x ) ) .= ( ( ( x | x ) | p ) | p ) | ( p | ( x | x ) ) ;
assume that x , x + h in dom sec and ( for h st h in dom sec holds ( ( sec * sec ) `| Z ) . x = ( 4 * ( sin . x + h / ( cos . x ) ) * sin . x ) / ( cos . x ) ^2 and for x st x in Z holds ( ( ( 2 * sin * sec ) `| Z ) . x ) ^2 = 1 ;
assume that i in dom A and len A > 1 and B c= the carrier of A and B c= the carrier of A and A is 0. of ( ( len A ) |-> 0. K ) and A is 0. ( ( len A ) |-> 0. K ) and A is 0. ( ( len A ) |-> 0. K ) ;
for i be non zero Element of NAT st i in Seg n holds i divides n or i = <* 1. F_Complex , n *> & ( i <> 0. F_Complex implies h . i = <* 1. F_Complex , n *> or h . i = 1. F_Complex L ) & ( i <> 0. F_Complex implies h . i = 1. F_Complex implies h . i = 1. F_Complex \ n )
( ( ( b1 'imp' b2 ) '&' ( c1 'imp' c2 ) '&' ( c2 'imp' c2 ) '&' ( c1 'imp' c2 ) '&' ( c2 'imp' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c1 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c2 '&' c2 ) ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c1 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&' ( ( c1 '&' c2 ) '&' 'not' ( c2 '&' c2 ) '&'
assume that f . x = ( ( - 1 ) (#) ( sin * ( f . x ) ) ) . x0 and x in dom ( ( - 1 ) (#) ( sin * ( f . x ) ) ) and x0 in dom ( ( - 1 ) (#) ( ( - 1 ) (#) ( f * ( f . x ) ) ) ) and ( ( - 1 ) (#) ( ( - 1 ) (#) ( f * ( f . x ) ) ) ) . x0 = - cos . x0 ;
consider R8 , I-8 be Real such that R8 = Integral ( M , Re ( F . n ) ) and I-8 = Integral ( M , Im ( F . n ) ) and Integral ( M , I8 ) = Integral ( M , Im ( F . n ) ) + Integral ( M , Im ( F . n ) ) ;
ex k be Element of NAT st k = k & 0 < d & for q be Element of product G st q in X & ||. q-r - f /. ( x - q ) .|| < r holds ||. partdiff ( f , x , k ) - partdiff ( f , x , k ) .|| < r ;
x in { x1 , x2 , x3 , x4 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , 7 , 8 } iff x in { x1 , x2 , x3 , x4 , x5 , 7 , 8 } \/ { 7 , 8 } \/ { 8 , 7 }
G * ( j , ( i + 1 ) ) `2 = G * ( 1 , ( i + 1 ) ) `2 .= G * ( 1 , ( j + 1 ) ) `2 .= G * ( 1 , ( j + 1 ) ) `2 .= G * ( 1 , ( j + 1 ) ) `2 .= G * ( 1 , ( j + 1 ) ) `2 .= G * ( 1 , ( j + 1 ) ) `2 .= G * ( 1 , ( j + 1 ) `2 ) `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 : Def1 : q in it iff q in P & for p , q st p in P holds p ^ q in P or ex p , q st p in P & q in P & q in P & p ^ q in T & q ^ p in T & q ^ q in T ;
F /. ( k + 1 ) = F . ( k + 1 ) .= F{} ( p . ( k + 1 -' 1 ) , k + 1 -' 1 ) .= F{} ( p . ( k + 1 -' 1 ) , k + 1 -' 1 ) .= F{} ( p . k , k + 1 -' 1 ) .= F{} ( p . k , k ) ;
for A , B , C being Matrix of K st len B = len C & width B = width C & len B = width A & len C > 0 & len A > 0 & len B > 0 & len A > 0 & len B > 0 & len A = len B holds A * ( ( B * C ) * ( B * C ) ) = A * BC
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 Cq ) /\ ( the carrier of Cq ) and y in ( the carrier of Cq ) /\ ( the carrier of Cq ) and [ x , y ] in the carrier of Cq and [ y , x ] in the InternalRel of Cq and [ x , y ] in the InternalRel of Cq ;
defpred P [ Element of NAT ] means for f st len f = $1 holds ( ( VAL g ) . ( k + 1 ) ) '&' ( ( VAL g ) . ( k + 1 ) ) = ( ( VAL g ) . ( k + 1 ) ) '&' ( ( VAL g ) . ( k + 1 ) ) ;
assume that 1 <= k and k + 1 <= len f and f is_sequence_on G and [ i + 1 , j ] in Indices G and f /. k = G * ( i , j ) and f /. ( k + 1 ) = G * ( i , j ) and f /. k = G * ( i , j ) and f /. ( k + 1 ) = G * ( i , j ) ;
assume that sn < 1 and q `1 > 0 and ( q `2 / |. q .| - sn ) / ( 1 + sn ) >= sn and - sn <= sn and sn <= 1 and sn <= 1 and sn <= 1 and sn >= 0 and sn = { q : q `2 <= 1 & q `2 <= 1 } ;
for M being non empty dist , x being Point of M , f being Point of M , x being Point of M st x = x `1 holds ex f being Function of M , M st for n being Element of NAT holds f . n = Ball ( x `1 , ( 1 / n ) * ( 1 / n ) )
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 ) `| Z ) . x = f1 . x - f2 . x ) & ( ( f1 - f2 ) `| Z ) . x = ( f1 . x ) / ( f2 . x ) ;
defpred P1 [ Nat , Point of CNS ] means $1 in Y & ||. s1 . $1 - s1 . ( $1 + 1 ) .|| < r & ||. s1 . $1 - s1 . ( $1 + 1 ) .|| < r & r < ( 1 - r ) * ( $1 + 1 ) ;
( 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 + 2 * n0 ) .= ( 1 - 2 * n0 + 2 * n0 ) * ( 2 * n0 + 2 * n0 ) .= 1 * ( 2 * n0 + 2 * n0 ) ;
defpred P [ Nat ] means for G being non empty strict finite RelStr , G being strict symmetric RelStr st G is \rm free & card G = $1 & the carrier of G = the carrier of G & the carrier of G = { the carrier of G } holds the InternalRel of G = { the carrier of G } & the InternalRel of G = { the carrier of G } ;
assume that not f /. 1 in Ball ( u , r ) and 1 <= m and m <= len ( f | ( i -' 1 ) ) and not LSeg ( f | ( i -' 1 ) , r ) /\ Ball ( u , r ) <> {} and not m <= len ( f | ( i -' 1 ) ) and not m <= len ( f | ( i -' 1 ) ) ;
defpred P [ Element of NAT ] means ( Partial_Sums ( cos ) . $1 = ( Partial_Sums ( cos ) . ( 2 * $1 ) ) . ( 2 * $1 + 1 ) ) . ( 2 * $1 + 1 ) = ( Partial_Sums ( cos ) . ( 2 * $1 + 1 ) ) . ( 2 * $1 + 1 ) ;
for x being Element of product F holds x is FinSequence of G & x in I & x in dom ( the Sorts of F ) & for i being set st i in dom F holds x . i = ( the Sorts of F ) . i & for i being set st i in dom F holds x . i = ( the Sorts of F ) . i
( x " ) |^ ( n + 1 ) = ( ( x " ) |^ n ) * x " .= ( x " ) |^ n * x " .= ( x |^ n ) |^ n * x " .= ( x |^ n ) |^ n * x " .= ( x |^ n ) |^ n * x " .= ( x |^ n ) |^ n * x " .= ( x |^ n ) |^ n ;
DataPart Comput ( P +* ( I , Initialized s ) , LifeSpan ( P +* I , Initialized s ) + 3 ) = DataPart Comput ( P +* I , Initialize s , LifeSpan ( P +* I , Initialized s ) + 3 ) .= DataPart Comput ( P +* I , Initialize s , LifeSpan ( P +* I , Initialize s ) + 3 ) .= DataPart Comput ( P +* I , Initialize s , LifeSpan ( P +* I , Initialize s ) + 3 ) ;
given r such that 0 < r and ]. x0 - r , x0 + r .[ c= dom f1 /\ dom f2 and for g st g in ]. x0 - r , x0 + r .[ /\ dom f1 holds f1 . g <= f1 . g and for g st g in ]. x0 - r , x0 + r .[ /\ dom f1 and g <= f1 . g ;
assume that X c= dom f1 /\ dom f2 and f1 | X is continuous and f2 | X is continuous and ( f1 - f2 ) | X is continuous and ( f1 - f2 ) | X is continuous and ( f1 - f2 ) | X is continuous and ( f1 - f2 ) | X is continuous and ( f1 - f2 ) | X is continuous and ( f1 - f2 ) | X is continuous ;
for L being continuous complete LATTICE for l being Element of L ex X being Subset of L st l = sup X & for x being Element of L st x in X holds x is compact & for X being Subset of L st x in X holds x is complete & x is complete & for x being Element of L st x in X holds x is complete holds x is complete
Support ( e *' p ) in { Support ( m *' p ) where m is Polynomial of n , L : ex p being Polynomial of n , L st p in Support ( m *' q ) & p in Support ( m *' q ) & p in Support ( m *' q ) } ;
( f1 - f2 ) /. ( lim s1 ) = lim ( f1 /* s1 ) - lim ( f2 /* s1 ) .= lim ( f1 /* s1 ) - lim ( f2 /* s1 ) .= lim ( f1 /* s1 ) - ( f2 /* s1 ) . ( lim s1 ) .= lim ( f1 /* s1 ) - ( f2 /* s1 ) . ( lim s1 ) .= lim ( f2 /* s1 ) - ( f2 /* s1 ) . ( lim s1 ) ;
ex p1 being Element of CQC-WFF ( Al ) st F . p = g . p1 & for g being Function of [: D , D :] , [: D , E :] st P [ g , ( len ( p1 qua Function ) ) , ( len ( p1 `2 ) ) ] holds P [ g , ( len ( p1 `2 ) ) , ( len ( p1 `2 ) ) ] ;
( 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 + 1 ) ;
( ( p ^ q ) ^ r ) . ( len p + k ) = ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( ( p ^ q ) . ( len p + k ) ) . ( len q + k ) .= ( p ^ q ) . k ;
len mid ( f , D2 ) + 1 = indx ( D2 , D1 , j1 ) + 1 .= indx ( D2 , D1 , j1 ) + 1 .= indx ( D2 , D1 , j ) + 1 .= indx ( D2 , D1 , j1 ) + 1 + 1 ;
x * y * z = Mz * ( y * z ) .= ( x * ( y * z ) ) * ( y * z ) .= ( x * ( y * z ) ) * ( x * ( y * z ) ) .= ( x * ( y * z ) ) * ( x * ( y * z ) ) .= ( x * ( y * z ) ) * ( x * z ) .= ( x * y ) * ( x * z ) ;
v . <* x , y *> + ( <* x0 , y0 *> ) * i = partdiff ( v , ( x - x0 ) * i ) + ( partdiff ( u , ( x - x0 ) * i ) * ( x - x0 ) ) + ( partdiff ( u , ( x - x0 ) * i ) * ( x - x0 ) ) ;
i * i = <* 0 * ( - 1 ) - ( 0 * 1 ) - ( 0 * 0 ) + ( 0 * 0 ) - ( 0 * 0 ) + ( 0 * 0 ) * ( - 1 ) .= <* - 1 , 0 , 0 , 0 *> .= <* - 1 , 0 , 0 , 0 , 0 , 0 , 0 *> .= <* - 1 , 0 , 0 , 0 , 0 *> ;
Sum ( L (#) F ) = Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( ( L (#) F1 ) ^ ( L (#) F2 ) ) .= Sum ( ( L (#) F1 ) ^ ( L (#) F2 ) ) .= Sum ( ( L (#) F1 ) ^ ( 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 & for Y0 be finite Subset of X st Y0 c= Y & Y0 c= Y holds |. ( - lower_bound ( X , Y ) ) . x - ( - lower_bound ( X , Y ) ) . y .| < r ;
( GoB f ) * ( i , j ) = f /. ( k + 2 ) & ( GoB f ) * ( i , j + 1 ) = f /. ( k + 2 ) or ( GoB f ) * ( i , j + 1 ) = f /. ( k + 2 ) or ( GoB f ) * ( i , j + 1 ) = f /. ( k + 2 ) & ( GoB f ) * ( i , j + 2 ) = f /. ( k + 2 ) ;
( ( r / 2 ) (#) ( cos - sin ) ) . x = ( ( r / 2 ) (#) ( sin - cos ) ) . x .= ( ( r / 2 ) (#) ( cos - cos ) ) . x .= ( ( r / 2 ) (#) ( cos - sin ) ) . x .= ( ( r / 2 ) (#) ( cos - cos ) ) . x .= ( ( r / 2 ) (#) ( cos - sin ) ) . x ;
( - ( b - a ) ) + sqrt ( delta ( a , b , c ) ) / 2 * a < 0 & ( - ( b - a ) ) / 2 * a < 0 or ( - ( b - a ) ) / 2 * a < 0 or ( - ( b - a ) ) / 2 * a < 0 ;
assume that ex_inf_of uparrow "\/" ( X /\ C , L ) and ex_sup_of X , L and ex_sup_of X , L and "\/" ( X , L ) = "/\" ( uparrow "\/" ( X , L ) , L ) and "\/" ( X , L ) = "\/" ( X , L ) and for x being Element of L holds x in X implies x < "\/" ( uparrow x , L ) ;
( for j holds ( j = i = j implies j = i = j ) & ( j = i implies j = i implies j = i ) & ( j = i implies j = i implies j = i ) & ( j = i implies j = i implies j = i ) & ( j = i implies j = i implies j = i ) & ( j = i implies j = i implies j = i ) implies j = i )