thesis ;
thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
contradiction ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
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thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
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thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
B ;
a <> c
T c= S
D c= B
c in X ;
b in X ;
X ;
b in D ;
x = e ;
let m ;
h is onto ;
N in K ;
let i ;
j = 1 ;
x = u ;
let n ;
let k ;
y in A ;
let x ;
let x ;
m c= y ;
F is one-to-one ;
let q ;
m = 1 ;
1 < k ;
G is rng ;
b in A ;
d divides a ;
i < n ;
s <= b ;
b in B ;
let r ;
B is one-to-one ;
R is total ;
x = 2 ;
d in D ;
let c ;
let c ;
b = Y ;
0 < k ;
let b ;
let n ;
r <= b ;
x in X ;
i >= 8 ;
let n ;
let n ;
y in f ;
let n ;
1 < j ;
a in L ;
C is boundary ;
a in A ;
1 < x ;
S is finite ;
u in I ;
z << z ;
x in V ;
r < t ;
let t ;
x c= y ;
a <= b ;
m in NAT ;
assume f is prime ;
not x in Y ;
z = +infty ;
k be Nat ;
K ` is being_line ;
assume n >= N ;
assume n >= N ;
assume X is \bf ) ;
assume x in I ;
q is as as Nat ;
assume c in x ;
p > 0 ;
assume x in Z ;
assume x in Z ;
1 <= kr2 ;
assume m <= i ;
assume G is rng ;
assume a divides b ;
assume P is closed ;
b-a > 0 ;
assume q in A ;
W is not bounded ;
f is / one-to-one ;
assume A is boundary ;
g is special ;
assume i > j ;
assume t in X ;
assume n <= m ;
assume x in W ;
assume r in X ;
assume x in A ;
assume b is even ;
assume i in I ;
assume 1 <= k ;
X is non empty ;
assume x in X ;
assume n in M ;
assume b in X ;
assume x in A ;
assume T c= W ;
assume s is atomic ;
b `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 ;
let a be Real ;
let x be element ;
let X be element ;
P [ 0 ]
let x be element ;
let x be element ;
let y be element ;
r in REAL ;
let e be element ;
n1 is , \cdot ;
Q halts_on s ;
x in that for of S holds x in \in that x in \in that x in \in S ;
M < m + 1 ;
T2 is open ;
z in b \rm \hbox { - } ;
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 : o : a ;
O <> O2 ;
let r be Real ;
let f be FinSeq-Location ;
let i be Nat ;
let n be Nat ;
Cl A = A ;
L c= Cl L ;
A /\ M = B ;
let V be complex normed space , x be Point of V ;
not s in Y |^ 0 ;
rng f <= w
b "/\" e = b ;
m = m3 ;
t in h . D ;
P [ 0 ] ;
assume z = x * y ;
S . n is bounded ;
let V be RealUnitarySpace , W be Subspace of V ;
P [ 1 ] ;
P [ {} ] ;
C1 is component ;
H = G . i ;
1 <= i `2 + 1 ;
F . m in A ;
f . o = o ;
P [ 0 ] ;
a\lbrace a - b .| <= az ;
R [ 0 ] ;
b in f .: X ;
assume q = q2 ;
x in [#] V ;
f . u = 0 ;
assume e1 > 0 ;
let V be RealUnitarySpace , W be Subspace of V ;
s is trivial & s is non empty ;
dom c = Q ;
P [ 0 ] ;
f . n in T ;
N . j in S ;
let T be complete LATTICE , x be Element of T ;
the object of F is one-to-one ;
sgn x = 1 ;
k in dom a ;
1 in Seg 1 ;
rng f = X ;
len T in X ;
vbeing < n ;
St is bounded ;
assume p = p2 ;
len f = n ;
assume x in P1 ;
i in dom q ;
let U1 , U2 , E ;
pp = c & p = c ;
j in dom h ;
let k ;
f | Z is continuous ;
k in dom G ;
UBD C = B ;
1 <= len M ;
p in Ball ( x , r ) ;
1 <= jj & jj <= width G ;
set A = L /\ L ;
card a [= c ;
e in rng f ;
cluster B \oplus A -> empty ;
H has no \cdot F ;
assume n0 <= m ;
T is increasing implies T is increasing
e2 <> e2 & e2 <> e1 ;
Z c= dom g ;
dom p = X ;
H is proper ;
i + 1 <= n ;
v <> 0. V ;
A c= Affin A ;
S c= dom F ;
m in dom f ;
let X0 be set ;
c = sup N ;
R is_connected 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 <> {} ;
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 + \HM { the } \HM { carrier } ;
- y in I ;
let A be non empty set , B be set ;
P0 = 1 ;
assume r in F . k ;
assume f is simple function of S ;
let A be l -countable set ;
rng f c= NAT ;
assume P [ k ] ;
f6 <> {} ;
let o be Ordinal ;
assume x is sum of squares ;
assume not v in { 1 } ;
let IX , I , J ;
assume that 1 <= j and j < l ;
v = - u ;
assume s . b > 0 ;
d1 in dom f ;
assume t . 1 in A ;
let Y be non empty TopSpace , f be Function of Y , TOP-REAL 2 ;
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 \times hh/. ;
assume f is additive marr-n\rm L~ f ;
let x , y be element ;
let T be non empty TopSpace ;
b , a // b , c ;
k in dom Sum p ;
let v be Element of V ;
[ x , y ] in T ;
assume len p = 0 ;
assume C in rng f ;
k1 = k2 & k2 = k2 ;
m + 1 < n + 1 ;
s in S \/ { s } ;
n + i >= n + 1 ;
assume Re y = 0 ;
k1 <= j1 & k1 <= len f ;
f | A is non empty continuous ;
f . x - a <= b ;
assume y in dom h ;
x * y in B1 ;
set X = Seg n ;
1 <= i2 + 1 ;
k + 0 <= k + 1 ;
p ^ q = p ;
j |^ y divides m ;
set m = max A ;
[ x , x ] in R ;
assume x in succ 0 ;
a in sup phi ;
CH in X ;
q2 c= C1 & not q2 in C ;
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 `2 ;
let S be { w } ;
y " <> 0 ;
y " <> 0 ;
0. V = u-w ;
y2 , y , w be Element of V ;
R8 ;
let a , b be Real , x be Point of TOP-REAL 2 ;
let a be Object of C ;
let x be Vertex of G ;
let o be object of C , a be object of C ;
r '&' q = P \lbrack l \rbrack ;
let i , j be Nat ;
let s be State of A , n be Nat ;
s4 . n = N ;
set y = ( x `1 ) * ( y `2 ) ;
mi in dom g & mi in dom g ;
l . 2 = y1 ;
|. g . y .| <= r ;
f . x in Cx0 ;
not G-19 is non empty ;
let x be Element of X ;
0 <> f . g2 ;
f2 /* q is convergent ;
f . i is_measurable_on E ;
assume \xi in N-22 ;
reconsider i = i as Ordinal ;
r * v = 0. X ;
rng f c= INT & rng f c= INT ;
G = 0 .--> goto 0 ;
let A be Subset of X ;
assume that A0 is dense and A is open ;
|. f . x .| <= r ;
Element of R ;
let b be Element of L ;
assume x in W-19 ;
P [ k , a ] ;
let X be Subset of L ;
let b be object of B ;
let A , B be category ;
set X = Vars ( C ) ;
let o be OperSymbol of S ;
let R be connected non empty Poset ;
n + 1 = succ n ;
xx c= Z1 & xx c= Z1 ;
dom f = C1 & rng f = C2 ;
assume [ a , y ] in X ;
Re ( seq ^\ k ) is convergent ;
assume that a1 = b1 and a2 = b2 ;
A = sInt A ;
a <= b or b <= a ;
n + 1 in dom f ;
let F be Instruction of S , I ;
assume that r2 > x0 and x0 < r2 ;
let Y be non empty set , f be Function of Y , BOOLEAN ;
2 * x in dom W ;
m in dom ( g2 | k ) ;
n in dom g1 & m in dom g2 ;
k + 1 in dom f ;
the still of not s in { s } ;
assume that x1 <> x2 and x2 <> x3 ;
v2 in [: the carrier of V , the carrier of V :] ;
not [ b `1 , b `2 ] in T ;
i-35 + 1 = i ;
T c= T & T c= T ;
( l `1 ) ^2 = 0 ;
let n be Nat ;
( t `2 ) ^2 = r ;
AA is_integrable_on M & AA is integrable ;
set t = Top t ;
let A , B be real-membered set ;
k <= len G + 1 ;
C ( ) misses V ( ) ;
Product seq is non empty ;
e <= f or f <= e ;
cluster non empty normal for Ordinal ;
assume c2 = b2 & c2 = b2 ;
assume h in [. q , p .] ;
1 + 1 <= len C ;
not c in B . m1 ;
cluster R .: X -> empty ;
p . n = H . n ;
assume that vq is convergent and lim vq = 0 ;
IC s3 = 0 & IC s3 = 0 ;
k in N or k in K ;
F1 \/ F2 c= F ;
Int G1 <> {} & Int G2 <> {} ;
( z `2 ) ^2 = 0 ;
p11 <> p1 & p11 <> p2 ;
assume z in { y , w } ;
MaxADSet ( a ) c= F ;
ex_sup_of downarrow s , S & ex_sup_of downarrow s , S ;
f . x <= f . y ;
let T be up-complete non empty reflexive transitive antisymmetric RelStr ;
q |^ m >= 1 ;
a >= X & b >= Y ;
assume <* a , c *> <> {} ;
F . c = g . c ;
G is one-to-one full full full full ;
A \/ { a } \not c= B ;
0. V = 0. Y ;
let I be halting Instruction of S , 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 & n + l2 in NAT ;
f " P is compact ;
assume x1 in REAL & x2 in REAL ;
p1 = K1 & p2 = K1 & p3 = D2 ;
M . k = <*> REAL ;
phi . 0 in rng phi ;
OSMMInt A is closed
assume z0 <> 0. L & z0 <> 0. L ;
n < ( N . k ) ;
0 <= seq . 0 & seq . 0 <= seq . 0 ;
- q + p = v ;
{ v } is Subset of B ;
set g = f `| 1 ;
R ( ) is stable Subset of R ;
set cR = Vertices R , R = the carrier of R ;
p0 c= P3 & p1 c= P3 ;
x in [. 0 , 1 .[ ;
f . y in dom F ;
let T be Scott Scott Scott Scott Scott Scott Scott Scott of S ;
ex_inf_of the carrier of S , S ;
\HM { a } = downarrow b ;
P , C , K is_collinear ;
assume x in F ( s , r , t ) ;
2 to_power i < 2 to_power m ;
x + z = x + z + q ;
x \ ( a \ x ) = x ;
||. x-y .|| <= r ;
assume that Y c= field Q and Y <> {} ;
a ~ , b ~ are_equipotent ;
assume a in A ( ) ;
k in dom ( q | k ) ;
p is 1 -element FinSequence of S ;
i -' 1 = i-1 - 1 ;
f | A is one-to-one ;
assume x in f .: X ( ) ;
i2 - i1 = 0 & i2 - i1 = 0 ;
j2 + 1 <= i2 & j2 + 1 <= len G ;
g " * a in N ;
K <> { [ {} , {} ] } ;
cluster strict for } commutative commutative Ring ;
|. q .| ^2 > 0 ;
|. p4 .| = |. p .| ;
s2 - s1 > 0 & s2 - s1 > 0 ;
assume x in { Gik } ;
W-min C in C & W-min C in C ;
assume x in { Gik } ;
assume i + 1 = len G ;
assume i + 1 = len G ;
dom I = Seg n & rng I = Seg n ;
assume that k in dom C and k <> i ;
1 + 1-1 <= i + j ;
dom S = dom F & rng S = dom G ;
let s be Element of NAT , k be 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 void ;
let f be ManySortedSet of I ;
let z be Element of COMPLEX , p be FinSequence of COMPLEX ;
u in { ag } ;
2 * n < ( 2 * n ) ;
let x , y be set ;
B-11 c= [: V , V :] ;
assume I is_closed_on s , P & I is_halting_on s , P ;
U2 = U2 & U2 = U2 implies U2 = U2
M /. 1 = z /. 1 ;
x11 = x22 & x22 = x22 ;
i + 1 < n + 1 + 1 ;
x in { {} , <* 0 *> } ;
ff <= ff & ff <= ff & ff <= ff ;
let l be Element of L ;
x in dom ( F . -17 ) ;
let i be Element of NAT , k be Nat ;
r8 is COMPLEX -valued Function of COMPLEX , 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 = Seg ( k + 1 ) ;
y = W . ( 2 * x ) ;
assume dom g = cod f & cod g = cod f ;
let X , Y be non empty TopSpace , f be Function of X , Y ;
x \oplus A is interval ;
|. <*> A .| . a = 0 ;
cluster strict sublattice for Sublattice of L ;
a1 in B . s1 & a2 in B . s2 ;
let V be finite over F , v be Vector of V ;
A * B on B , A ;
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 |^ X ;
z in dom ( A --> y ) ;
P [ y , h . y ] ;
{ x0 } c= dom f & { x0 } c= dom f ;
let B be non-empty ManySortedSet of I , A be ManySortedSet of I ;
PI / 2 < Arg z ;
reconsider z9 = 0 , z9 = 1 as Nat ;
LIN a , d , c ;
[ y , x ] in II ;
Q * ( Q * ( 1 , 3 ) ) = 0 ;
set j = x0 gcd m , m = x0 gcd m ;
assume a in { x , y , c } ;
j2 - jj > 0 - 1 ;
I = I I I I as Element of 1 -tuples_on U ;
[ y , d ] in F-8 ;
let f be Function of X , Y ;
set A2 = ( B - C ) / 2 ;
s1 , s2 are_) & s1 , s2 are_) implies s1 , s2 are_carrier of R
j1 -' 1 = 0 & j2 -' 1 = j2 ;
set m2 = 2 * n + j ;
reconsider t = t as bag of n ;
I2 . j = m . j ;
i |^ s , n are_relative_prime & i |^ s , n are_relative_prime ;
set g = f | D-21 , h = g | D-21 ;
assume that X is lower and 0 <= r ;
p1 `1 = 1 & p2 `2 = 1 ;
a < ( p3 `1 ) * ( p3 `2 ) ;
L \ { m } c= UBD C ;
x in Ball ( x , 10 ) ;
not a in LSeg ( c , m ) ;
1 <= i1 -' 1 & i1 + 1 <= len f ;
1 <= i1 -' 1 & i1 + 1 <= len f ;
i + i2 <= len h & i + 1 <= 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 *> = 2 ;
set H = h . gg , I = h . gg , J = h . M ;
card b * a = |. a .| ;
Shift ( w , 0 ) |= v ;
set h = h2 (*) h1 , h1 = h2 (*) h2 ;
assume x in ( X /\ 4 ) /\ X ;
||. h .|| < d1 & ||. h .|| < s ;
not x in the carrier of f & not x in the carrier of g ;
f . y = F ( y ) ;
for n holds X [ n ] ;
k - l = k\leq - l ;
<* p , q *> /. 2 = q ;
let S be Subset of the carrier of Y ;
let P , Q be \langle \rm / 2 *> ;
Q /\ M c= union ( F | M )
f = b * canFS ( S ) ;
let a , b be Element of G ;
f .: X is_<=_than f . sup X
let L be non empty transitive reflexive transitive RelStr , x be Element of L ;
S-20 is x -basis i ;
let r be non positive Real ;
M , v |= x \hbox { y } ;
v + w = 0. ( Z ) ;
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 non empty ( i -tuples_on U ) -valued for Element of S ;
reconsider l1 = l-1 as Nat ;
v4 is Vertex of r2 & v4 is Vertex of r2 ;
T2 is SubSpace of T2 & T1 is SubSpace of T2 ;
Q1 /\ Q19 <> {} & Q1 /\ Q19 <> {} ;
k be Nat ;
q " is Element of X & q " is Element of X ;
F . t is set of set with zero ;
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 ) * ( p `2 ) ;
not r in ]. p , q .] ;
let R be FinSequence of REAL , x be Element of REAL ;
not S7 does not destroy b1 & not S7 does not destroy b1 ;
IC SCM R <> a & IC SCM R <> a ;
|. - |[ x , y ]| .| >= r ;
1 * seq = seq & 1 * seq = seq ;
let x be FinSequence of NAT , k be Nat ;
let f be Function of C , D , g be Function of C , D ;
for a holds 0. L + a = a
IC s = s . NAT .= ( 0 + 1 ) ;
H + G = F- ( GG ) ;
Cx1 . x = x2 & Cy1 . x = y2 ;
f1 = f .= f2 .= ( f | X ) . x .= f . x ;
Sum <* p . 0 *> = p . 0 ;
assume v + W = v + u + W ;
{ a1 } = { a2 } & { a2 } = { a1 } ;
a1 , b1 _|_ b , a ;
d3 , o _|_ o , a3 & d1 , o _|_ o , a3 ;
II is reflexive & II is transitive ;
IO is antisymmetric & IO is antisymmetric implies [: the carrier of O , the carrier of O :] is antisymmetric
sup rng H1 = e & sup rng H2 = e ;
x = ( a * 8 ) * ( a * 8 ) ;
|. p1 .| ^2 >= 1 ;
assume that j2 -' 1 < j2 and j2 + 1 < width G ;
rng s c= dom f1 & rng s c= dom f2 ;
assume that support a misses support b and b in support b ;
let L be associative commutative non empty doubleLoopStr , p be Polynomial of L ;
s " + 0 < n + 1 ;
p . c = ( f " ) . 1 ;
R . n <= R . ( n + 1 ) ;
Directed I1 = I1 & Directed I2 = I2 ;
set f = + ( x , y , r ) ;
cluster Ball ( x , r ) -> bounded ;
consider r being Real such that r in A ;
cluster non empty NAT -defined NAT -defined NAT -defined Function ;
let X be non empty directed Subset of S ;
let S be non empty full SubRelStr of L ;
cluster <* \hbox { $ 1 } } -> complete for non trivial TopSpace ;
( 1 - a ) " = a " ;
( q . {} ) `1 = o ;
( n - 1 ) > 0 ;
assume ( 1 / 2 ) * t `1 <= 1 ;
card B = k + - 1 ;
x in union rng ( f | ( len f ) ) ;
assume x in the carrier of R & y in the carrier of R ;
d in dom f ;
f . 1 = L . ( F . 1 ) ;
the vertices of G = { v } & the vertices of G = { v } ;
let G be let G be let
let e , v6 be set ;
c . ii in rng c & c . ii in rng c ;
f2 /* q is divergent_to-infty & f2 /* q is divergent_to-infty ;
set z1 = - z2 , z2 = - z1 , z2 = - z2 , z2 = - z1 ;
assume w is llas of S , G ;
set f = p |-count ( t - p ) , g = p |-count ( t - p ) , h = p |-count ( t - p ) , n = p |-count ( t - p ) , m
let c be Object of C ;
assume ex a st P [ a ] ;
let x be Element of REAL m , y be Element of REAL m ;
let IX be Subset-Family of X , I be Element of X ;
reconsider p = p as Element of NAT ;
let v , w be Point of X ;
let s be State of SCM+FSA , I be Program of SCM+FSA ;
p is FinSequence of SCM & q is FinSequence of the carrier of SCM ;
stop I ( ) c= P-12 & stop I ( ) c= P-12 ;
set ci = fJ /. i , fj = fj /. j ;
w ^ t ^ s ^ t ^ s ^ t ^ t ^ s ^ t ^ s ^ t ^ t ^ s ^ t ^ t ^ t ^ s ^ t ^ t ^ s ^ t ^
W1 /\ W = W1 /\ W ` .= W1 /\ W2 ;
f . j is Element of J . j ;
let x , y be \rm \cdot of T2 , a , b be element ;
ex d st a , b // b , d ;
a <> 0 & b <> 0 & c <> 0 ;
ord x = 1 & x is positive ;
set g2 = lim ( seq ^\ k ) , g1 = ( lim seq ) ^\ k ;
2 * x >= 2 * ( 1 / 2 ) ;
assume ( a 'or' c ) . z <> TRUE ;
f (*) g in Hom ( c , c ) ;
Hom ( c , c + d ) <> {} ;
assume 2 * Sum ( q | m ) > m ;
L1 . F-21 = 0 & L1 . F-21 = 0 ;
/ ( X \/ R1 ) = / ( X \/ R1 ) ;
( sin * cos ) . x <> 0 & ( sin * cos ) . x <> 0 ;
( ( #Z n ) * ( exp_R + f ) ) . x > 0 ;
o1 in ( X /\ O2 ) /\ ( X /\ O2 ) ;
let e , v6 be set ;
r3 > ( 1 / 2 ) * 0 ;
x in P .: ( F -ideal ) ;
let J be closed Subset of R , left ideal non empty Subset of R ;
h . p1 = f2 . O & h . O = g2 . I ;
Index ( p , f ) + 1 <= j ;
len ( q | i ) = width M & width ( q | i ) = width M ;
the carrier of CL c= A & the carrier of CK c= A ;
dom f c= union rng ( F | X ) ;
k + 1 in dom ( support ( n ) ) ;
let X be ManySortedSet of the carrier of S ;
[ x `1 , y `2 ] in ( \HM { the } \HM { carrier of R ) ;
i = D1 or i = D2 or i = D1 ;
assume a mod n = b mod n & b mod n = b mod n ;
h . x2 = g . x1 & h . x2 = h . x2 ;
F c= 2 -tuples_on the carrier of X & F is one-to-one ;
reconsider w = |. s1 .| as Real_Sequence , r be Real ;
( 1 / m ) * m + r < p ;
dom f = dom ( I --> ( 0 , 1 ) ) .= { 0 } ;
[#] P-17 = [#] ( ( TOP-REAL 2 ) | K1 ) .= K1 ;
cluster - ( x - y ) -> ExtReal ;
then { d1 } c= A & A is closed ;
cluster ( TOP-REAL n ) | A -> finite-ind ;
let w1 be Element of M , w2 be Element of M ;
let x be Element of dyadic ( n ) ;
u in W1 & v in W3 & u in W3 ;
reconsider y = y as Element of L2 & x = y ;
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 Point of X ;
dist ( x `1 , y ) < r / 2 ;
reconsider mm = m , mn = n as Element of NAT ;
x- x0 < r1 - x0 & x0 < r2 - x0 ;
reconsider P = P `1 as strict Subgroup of N ;
set g1 = p * idseq ( q `2 ) , g2 = p * idseq ( q `2 ) ;
let n , m , k be non zero Nat ;
assume that 0 < e and f | A is lower ;
D2 . I8 in { x } & D2 . I8 in { x } ;
cluster subcondensed condensed -> subcondensed for Subset of T ;
let P be compact non empty Subset of TOP-REAL 2 , p , q be Point of TOP-REAL 2 ;
Gik in LSeg ( cos , 1 ) /\ LSeg ( cos , 1 ) ;
let n be Element of NAT , x be Point of TOP-REAL n ;
reconsider S8 = S , S8 = T as Subset of T ;
dom ( i .--> X `2 ) = { 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 , I be Program of SCMPDS ;
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 ( 1 + ( x ^2 ) ) ^2 ) ) ;
reconsider t7 = T-1 , T8 = T" as Point of TOP-REAL 2 ;
set q = h * p ^ <* d *> ;
z2 in U . ( y2 ) /\ Q . ( y1 + y2 ) ;
A |^ 0 = { <%> E } & A |^ 0 = { <%> E } ;
len W2 = len W + 2 & len W1 = len W2 + 2 ;
len h2 in dom h2 & len h2 in dom h2 ;
i + 1 in Seg ( len s2 ) ;
z in dom g1 /\ dom f & z in dom f1 /\ dom f2 ;
assume that p2 = E-max ( K ) and p1 <> 0. K ;
len G + 1 <= i1 + 1 ;
f1 (#) f2 is convergent & lim ( f1 (#) f2 ) = x0 ;
cluster seq + seq + seq1 -> summable for sequence of X ;
assume j in dom ( M1 /. i ) ;
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 ;
<* xxy *> ^ <* y *> ^ <* y *> ^ x iff y in dom x ;
a , b in { a , b } ;
len p2 is Element of NAT & len p1 = len p2 ;
ex x being element st x in dom R & y = R . x ;
len q = len ( K (#) G ) .= 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 is Element of L ;
k = 0 & n <> k or k > n & k > n ;
then X is discrete for A is closed Subset of X ;
for x st x in L holds x is FinSequence ;
||. f /. c .|| <= r1 & ||. f /. c .|| <= r ;
c in uparrow p & not c in { p } ;
reconsider V = V as Subset of the topology of TOP-REAL n ;
let N , M be 9 9 9 ;
then z is_>=_than waybelow x & z >= compactbelow x ;
M \lbrack f , g .] = f & M \lbrack g , g .] = g ;
( ( ( \mathop { 1 } ) /. 1 ) `1 ) = TRUE ;
dom g = dom f -tuples_on X & rng f = dom f ;
mode : of G is ^ trivial Walk of 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 ) -tuples_on the carrier of K ;
F1 . ( a1 , - a1 ) = G1 . ( a1 , - a1 ) ;
redefine func being set equals LSeg ( a , b , r ) ;
let a , b , c , d be Real ;
rng s c= dom ( 1 / ( f1 + f2 ) ) ;
curry curry ' ( F-19 , k ) is additive additive K ;
set k2 = card dom B , k1 = card dom B , k2 = card dom C ;
set G = DTConMSA X , A = the Sorts of A ;
reconsider a = [ x , s ] as 0. of G ;
let a , b be Element of [: M , M :] ;
reconsider s1 = s , s2 = t as Element of ( the carrier of S ) ;
rng p c= the carrier of L & p . ( len p ) = 0. L ;
let d be Subset of the Sorts 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 ) | D ;
reconsider i0 = len p1 , i1 = len p2 as Integer ;
dom f = the carrier of S & rng f = the carrier of T ;
rng h c= union ( the carrier of J . i ) ;
cluster All ( x , H ) -> non empty for element ;
d * N1 ^2 > N1 * 1 ;
]. a , b .[ c= [. a , b .] ;
set g = f " | D1 , h = f " | D2 ;
dom ( p | mmm1 ) = mm1 & dom ( p | mm1 ) = mm1 ;
3 + - 2 <= k + - 2 & k + - 2 <= k + - 2 ;
tan is_differentiable_in ( arccot * arccot ) . x & cot . x > 0 ;
x in rng ( f /^ ( len p -' 1 ) ) ;
let f , g be FinSequence of D ;
p ( ) in the carrier of S1 & p ( ) in the carrier of S2 ;
rng f " = dom f & rng f = dom f & f " = rng f ;
( the Target of G ) . e = v & ( the Target of G ) . e = v ;
width G -' 1 < width G - 1 & width G -' 1 <= width G ;
assume that v in rng ( S | E1 ) and v in rng ( S | E1 ) ;
assume x is root or x is root or x is root ;
assume 0 in rng ( g2 | A ) & 0 < r ;
let q be Point of ( TOP-REAL 2 ) | K1 , r be Real ;
let p be Point of ( TOP-REAL 2 ) | K1 , q be Point of ( TOP-REAL 2 ) | D ;
dist ( O , u ) <= |. p2 .| + 1 ;
assume dist ( x , b ) < dist ( a , b ) ;
<* S7 *> is_the carrier of C-20 & <* S7 *> is_\! \! \smallfrown C-20 ;
i <= len G -' 1 & j + 1 <= width G ;
let p be Point of ( TOP-REAL 2 ) | K1 , q be Point of ( TOP-REAL 2 ) | D ;
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 } ;
Q = Sthesis " ( Q ) .= Q " ( Q /\ R " ( Q /\ R ) ) ;
( ( 1 / 2 ) (#) ( 1 / 2 ) ) (#) ( 1 / 2 ) is summable ;
- p + I c= - p + A & - p + I c= - p + I ;
n < LifeSpan ( P1 , s1 ) + 1 & I <= LifeSpan ( P2 , s2 ) ;
CurInstr ( p1 , s1 ) = i .= ( the InstructionsF of SCM+FSA ) . IC 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 , x be Element of L ;
let f be Function of S , T ;
reconsider g = g opp as Morphism of c opp , b opp ;
[ s , I ] in [: S , A :] ;
len ( the connectives of C ) = 4 & len ( the connectives of C ) = 4 ;
let C1 , C2 be subcategory of C , a , b be Element of C1 ;
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 & H |^ a = H |^ a ;
let A1 be Element of O , A2 be Element of E ;
p2 , r3 , q3 is_collinear & p1 , r2 , p3 is_collinear ;
consider x being element such that x in v ^ K ;
not x in { 0. TOP-REAL 2 } & not x in { 0. TOP-REAL 2 } ;
p in [#] ( I[01] | B11 ) & p in [#] ( I[01] | B11 ) ;
0 . n < M . E8 & M . E8 < M . E8 ;
op ( c ) / ( c / ( c / ( c / ( c / ( c / ( c / ( c / ( c / ( c / ( c / ( c / ( c / ( c / ( d / ( d +
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . s2 ) & a2 in dom ( F . s2 ) ;
cluster -> Line *> -> Line \langle *> , the carrier of L ;
set i1 = the Nat , i2 = the Element of NAT , i1 = the Element of NAT ;
let s be 0 -started State of SCM+FSA , I be Program of SCM+FSA ;
assume y in ( f1 \/ f2 ) .: A ;
f . ( len f ) = f /. len f .= f /. 1 ;
x , f . x '||' f . x , f . y ;
pred X c= Y means : Def1 : cos X c= cos Y ;
let y be upper Subset of Y , x be Point of X ;
cluster the InternalRel of x `1 -> non \rm \hbox { - } NAT for non empty non empty set ;
set S = <* Bags n , il *> , T = <* Bags n , i *> ;
set T = [. 0 , 1 / 2 .] ;
1 in dom mid ( f , 1 , 1 ) ;
( 4 * PI ) / 2 < ( 2 * PI ) / 2 ;
x2 in dom f1 /\ dom f & x2 in dom f1 /\ dom f2 ;
O c= dom I & { {} } = { {} } ;
( the Source 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 ) * ( p1 `2 ) + ( p1 `2 ) * ( p1 `2 ) ;
i + 1 <= len Cage ( C , n ) ;
len <* P *> @ = len P & width <* P *> = len P ;
set N-26 = the \subseteq of the \subseteq of N , Nw = the Element of N ;
len gLet + ( x + 1 ) - 1 <= x ;
a on B & not b on B implies b on B
reconsider rr = r * I . v as FinSequence of REAL ;
consider d such that x = d and a (#) d [= c ;
given u such that u in W and x = v + u ;
len f /. ( \downharpoonright n ) = len ^2 ( n ) ;
set q2 = N-min L~ Cage ( C , n ) , q2 = W-min L~ Cage ( C , n ) ;
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 /\ h " D = f " V ;
reconsider D = E as non empty directed Subset of L1 ;
H = ( the_left_argument_of H ) '&' ( the_right_argument_of H ) & H = ( the_right_argument_of H ) '&' ( the_right_argument_of H ) ;
assume t is Element of ( the Sorts of Free ( S , X ) ) . s ;
rng f c= the carrier of S2 & f . x = f . x ;
consider y being Element of X such that x = { y } ;
f1 . ( a1 , b1 ) = b1 & f1 . ( b1 , b2 ) = b2 ;
the carrier' of G `1 = E \/ { E } .= { E } ;
reconsider m = len ( k - 1 ) as Element of NAT ;
set S1 = LSeg ( n , UMP C ) , S2 = LSeg ( n , UMP C ) ;
[ i , j ] in Indices ( M1 + M2 ) ;
assume that P c= Seg m and M is \HM { i } is non empty Subset of K ;
for k st m <= k holds z in K . k ;
consider a being set such that p in a and a in G ;
L1 . p = p * L /. 1 .= p * L /. 1 ;
p-7 . i = pp1 . i & pp2 . i = pp2 . i ;
let PA , G be a_partition of Y , a be Element of Y ;
pred 0 < r & r < 1 means : Def1 : 1 < r & r < 1 ;
rng ( ( a , X ) --> ( a , b ) ) = [#] X ;
reconsider x = x , y = y as Element of K ;
consider k such that z = f . k and n <= k ;
consider x being element such that x in X \ { p } ;
len ( canFS ( s ) ) = card ( s ) .= card ( s ) ;
reconsider x2 = x1 , y2 = x2 as Element of L2 ;
Q in FinMeetCl FinMeetCl ( the topology of X ) & Q c= the topology of X ;
dom ( f | 0 ) c= dom ( f | 0 ) & dom ( f | 0 ) c= dom ( f | 0 ) ;
pred n divides m means : Def1 : m divides n & n = m ;
reconsider x = x as Point of [: I[01] , I[01] :] ;
a in dom
not y0 in the still of f & not ( ex y st y in dom f & not y in the carrier of f ) ;
Hom ( ( a ~ ) , c ) <> {} ;
consider k1 such that p " < k1 and k1 < len p and p . k1 = q . k1 ;
consider c , d such that dom f = c \ d and c in dom f ;
[ x , y ] in [: dom g , dom k :] ;
set S1 =
l1 = m2 & l1 = l2 & l2 = i2 implies l1 + l2 = 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 & the carrier of I[01] = the carrier of I[01] & the carrier of I[01] = the carrier of I[01] ;
f . p4 <= f . ( f . p1 ) & f . p2 <= f . p1 ;
( ( F /. x ) `1 ) ^2 / ( |. x .| ) ^2 <= ( |. x .| ) ^2 / ( |. x .| ) ^2 ;
( x `2 ) ^2 = ( W `2 ) ^2 + ( W `2 ) ^2 ;
for n being Element of NAT holds P [ n ] ;
let J , K be non empty Subset of I ;
assume 1 <= i & i <= len <* a " *> ;
0 |-> a = <*> ( the carrier of K ) .= ( 0 -tuples_on 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 ] & Q [ succ a ] ;
reconsider s\mathclose = snon empty where s is Element of D : s is ' of D } as finite Subset of D ;
( i - 1 ) <= len ( thesis - j ) ;
[#] S c= [#] ( the TopStruct of T ) & the TopStruct of T = the TopStruct of T ;
for V being strict RealUnitarySpace holds V in and V in and V is Subspace of V
assume k in dom mid ( f , i , j ) ;
let P be non empty Subset of TOP-REAL 2 , p1 , p2 , q1 , q2 be Point of TOP-REAL 2 ;
let A , B be square Matrix of n1 , K , M be Matrix of n1 , K ;
- a * - b = a * b & - a * b = - b ;
for A being Subset of AS holds A // A & A is being_line implies A is being_line
( for o2 being Element of NAT holds o2 in dom <* o2 , o2 *> ) implies o2 = o1
then ||. x .|| = 0 & x = 0. X ;
let N1 , N2 be strict normal Subgroup of G , x be Element of G ;
j >= len upper_volume ( g , D1 ) & len upper_volume ( g , D2 ) = len D2 ;
b = Q . ( len Q - 1 ) + 1 .= Q . ( len Q - 1 ) ;
f2 * f1 /* s is divergent_to-infty & f2 * f1 /* s is divergent_to+infty ;
reconsider h = f * g as Function of [: N1 , N2 :] , G ;
assume that a <> 0 and Let a , b , c ;
[ t , t ] in the InternalRel of A & [ t , t ] in the InternalRel of A ;
( v |-- E ) | n is Element of ( T . n ) -tuples_on ( T . n ) ;
{} = 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 ) .= p +* q ;
reconsider N2 = N1 , N2 = N2 as strict net of R2 , the carrier of R2 ;
reconsider Y = Y as Element of ( Ids L ) , \subseteq the carrier of L ;
"/\" ( 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 & not contradiction ;
n <= len ( P + ( len P ) ) + len ( P + ( len P ) ) ;
( x1 `1 ) ^2 = ( x2 `2 ) ^2 + ( x2 `2 ) ^2 ;
InputVertices S = { x1 , x2 } & InputVertices S = { x1 , x2 } ;
let x , y be Element of FTT1 ( n ) ;
p = |[ p `1 , p `2 ]| & q = |[ p `1 , q `2 ]| ;
g * 1. G = h " * g * h .= h " * g .= h ;
let p , q be Element of V , a , b , c ;
x0 in dom ( x1 + x2 ) /\ dom ( y1 + y2 ) & x0 in dom ( y1 + y2 ) ;
( R qua Function ) " = R " * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ) ) ) ) ) ) )
n in Seg len ( f /^ ( len p -' 1 ) ) ;
for s being Real st s in R holds s <= s2 implies s <= s1
rng s c= dom ( f2 * f1 ) /\ dom ( f2 * f1 ) ;
synonym ) for for for for for for ( for X holds X in R ) ;
1. K * 1. K = 1. K & 1. K * 1. K = 1. K ;
set S = Segm ( A , P1 , Q1 ) , T = Segm ( A , P1 , Q1 ) ;
ex w st e = ( w / f ) & w in F ;
curry ' ( P+* ( x , k ) ) # x is convergent ;
cluster open -> open for Subset of T<* S , T *> ;
len f1 = 1 .= len f3 .= len f3 + 1 .= len f3 + 1 .= len f3 + 1 ;
( i * p ) / p < ( 2 * p ) / p ;
let x , y be Element of OSSub ( U0 ) ;
b1 , c1 // b9 , c9 & b1 , c1 // b9 , c ;
consider p being element such that c1 . j = { p } ;
assume that f " { 0 } = {} and f is total and f is total ;
assume IC Comput ( F , s , k ) = n & IC Comput ( F , s , k ) = n ;
Reloc ( J , card I ) does not destroy a & Reloc ( J , card I ) does not destroy a ;
( goto ( card I + 1 ) ) does not destroy c ;
set m3 = LifeSpan ( p3 , s3 ) , P4 = p +* I , P4 = p +* I ;
IC SCMPDS in dom Initialize ( p +* I ) & IC SCMPDS in dom Initialize ( p +* I ) ;
dom t = the carrier of SCM & dom t = the carrier of SCM & t . a = s . a ;
( E-max L~ f ) .. f = 1 & ( E-max L~ f ) .. f = ( E-max L~ f ) .. f ;
let a , b be Element of V , C be Element of V ;
Cl Int ( union F ) c= Cl Int ( union F ) ;
the carrier of X1 union X2 misses ( the carrier of X1 union X2 ) ;
assume not LIN a , f . a , g . a , g . b ;
consider i being Element of M such that i = d6 and i in dom f ;
then Y c= { x } or Y = { x } ;
M , v |= H1 / ( y , x ) & M , v |= H2 ;
consider m being element such that m in Intersect ( FF . 0 ) and m in dom f ;
reconsider A1 = support u1 , A2 = support v1 as Subset of X ;
card ( A \/ B ) = k-1 + ( 2 * 1 ) ;
assume that a1 <> a3 and a2 <> a4 and a3 <> a4 and a1 <> a5 ;
cluster s -\bf carrier of S -> $ string of S , D be string of S ;
Lf2 /. n2 = Lf2 . n2 & Lf2 /. n2 = Lf2 . n2 ;
let P be compact non empty Subset of TOP-REAL 2 , p , q be Point of TOP-REAL 2 ;
assume that r-7 in LSeg ( p1 , p2 ) and r-7 in LSeg ( p1 , p2 ) ;
let A be non empty compact Subset of TOP-REAL n , a , b , c , d be Real ;
assume [ k , m ] in Indices ( D | [: Seg n , Seg n :] ) ;
0 <= ( 1 / 2 ) |^ p & ( 1 / 2 ) |^ p <= 1 ;
( F . N | E8 ) . x = +infty & ( F . N ) . x = +infty ;
pred X c= Y means : Def1 : Z c= V & X \ V c= Y \ Z ;
( y `2 ) * ( z `2 ) <> 0. I & ( y `2 ) * ( z `2 ) <> 0. I ;
1 + card ( X-18 ) <= card u & card ( X-18 ) <= card ( X \/ Y ) ;
set g = z :- ( E-max L~ z ) , M = z .. z , N = .. z , S = L~ z , N = .. z , N = .. z , S = .. z , N = .. z , N = .. z , S = .. z
then k = 1 & p . k = <* x , y *> . k ;
cluster total for Element of C -O , the carrier of S ;
reconsider B = A as non empty Subset of ( TOP-REAL n ) | B ;
let a , b , c be Function of Y , BOOLEAN , p be Function of Y , BOOLEAN ;
L1 . i = ( i .--> g ) . i .= g . i .= g . i ;
Plane ( x1 , x2 , x3 , x4 ) c= P & Plane ( x2 , x3 , x4 ) c= P ;
n <= indx ( D2 , D1 , j1 ) + 1 & indx ( D2 , D1 , j1 ) + 1 <= len D2 ;
( ( g2 . O ) `1 ) ^2 = - 1 & ( ( g2 . O ) `2 ) ^2 = 1 ;
j + p .. f -' len f <= len f - len f + 1 ;
set W = W-bound C , S = S-bound C , E = E-bound C , N = N-bound C , N = N-bound C , S = S-bound C , N = E-bound C , S = E-bound C , N = E-bound C , N = E-bound C , S = E-bound
S1 . ( a `1 , e `2 ) = a + e `2 .= a `2 ;
1 in Seg width ( M * ( ColVec2Mx p ) ) ;
dom ( i (#) Im ( f , x ) ) = dom Im ( f , x ) ;
( ^2 x ) = W . ( a , *' ( a , p ) ) ;
set Q = non empty ( \rm \rm \rm \rm I ( ) } ) ;
cluster Carrier ( U1 ) -> MS[ U1 , U2 ] ] -> MS[ U1 , U2 ] ;
attr A means : Def1 : ex A st F = { A } ;
reconsider z9 = \hbox { z } , z9 = z as Element of product G ;
rng f c= rng f1 \/ rng f2 & f . 1 = f1 . 1 \/ f2 . 2 ;
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 , H ) ;
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 .= k + 1 ;
card ( ( x \ B1 ) /\ B1 ) = 0 implies x in ( x \ B1 ) /\ ( x \ B1 )
g + R in { s : g-r < s & s < g + r } ;
set q-19 = ( q , <* s *> ) : not contradiction } , qv1 = ( q , <* s *> ) : not contradiction } ;
for x being element st x in X holds x in rng f1 implies x in X
h0 /. ( i + 1 ) = h0 . ( i + 1 ) ;
set mw = max ( B , dom ( R | NAT ) ) ;
t in Seg width ( I ^ ( n , n ) ) ;
reconsider X = dom f , C = C as Element of Fin ( NAT ) ;
IncAddr ( i , k ) = <% - l . k , k %> + k .= succ k ;
S-bound L~ f <= q `2 & q `2 <= ( q `2 ) * ( 1 + ( q `2 / q `1 ) ^2 ) ;
attr R is condensed means : Def1 : Int R is condensed & Cl R is condensed ;
pred 0 <= a & a <= 1 & b <= 1 implies a * b <= 1 ;
u in ( ( c /\ ( ( d /\ b ) /\ e ) ) /\ f ) /\ j ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) /\ f ) ) /\ j ;
len C + - 2 >= 9 + - 3 & len C + - 2 >= 9 + - 2 ;
x , z , y is_collinear & x , z , y is_collinear & x , z , y is_collinear ;
a |^ ( n1 + 1 ) = a |^ n1 * a .= a |^ n1 * a ;
<* \underbrace ( 0 , \dots , 0 *> , x ) in Line ( x , a * x ) ;
set y9 = <* y , c *> ;
FF2 /. 1 in rng Line ( D , 1 ) & FF2 . 1 = Line ( D , 1 ) ;
p . m joins r /. m , r /. ( m + 1 ) ;
p `2 = ( f /. i1 ) `2 & ( f /. i1 ) `2 = ( f /. i1 ) `2 ;
W-bound ( X \/ Y ) = W-bound ( X \/ Y ) & W-bound ( X \/ Y ) = W-bound ( X \/ Y ) ;
0 + ( p `2 ) ^2 <= 2 * r + ( p `2 ) ^2 ;
x in dom g & not x in g " { 0 } implies x in dom g
f1 /* ( seq ^\ k ) is divergent_to+infty & f2 /* ( seq ^\ k ) is divergent_to+infty ;
reconsider u2 = u , v2 = v as VECTOR of ( the carrier of V ) --> ( u , v ) ;
p |-count Product ( Sgm ( X11 ) ) = 0 & p |-count ( Sgm ( X11 ) ) = 1 ;
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 ;
x in { x , y } & h . x = {} ( Th . x , h . y ) ;
consider y being Element of F such that y in B and y <= x `1 ;
len S = len ( the charact of ( A . 0 ) ) .= len ( the charact of ( A . 0 ) ) ;
reconsider m = M , i = I , n = N as Element of X ;
A . ( j + 1 ) = B . ( j + 1 ) \/ A . j ;
set N8 = : ( G is : n <= len G-15 & n <= len G-15 ) ;
rng F c= the carrier of gr { a } & F . a = ( gr { a } ) . a ;
g is ) of |. Q . ( K , n , r ) .| is a |. f .| ;
f . k , f . ( mod n ) in rng f & f . ( mod n ) in rng f ;
h " P /\ [#] T1 = f " P /\ [#] T1 .= [#] T1 /\ [#] T2 .= [#] T1 ;
g in dom f2 \ f2 " { 0 } & ( f2 " { 0 } ) . g in dom f2 ;
gX /\ dom f1 = g1 " { 0 } & gX /\ dom f2 = g2 " { 0 } ;
consider n being element such that n in NAT and Z = G . n ;
set d1 = dist ( x1 , y1 ) , d2 = dist ( x2 , y2 ) , d2 = dist ( x2 , y2 ) ;
b `2 + ( 1 - r ) < ( 1 - r ) + ( 1 - r ) ;
reconsider f1 = f , g1 = g as VECTOR of the carrier of X , Y ;
pred i <> 0 means : Def1 : i ^2 mod ( i + 1 ) = 1 ;
j2 in Seg len ( g2 . i2 ) & ( g2 . i2 ) . j2 = ( g2 . i2 ) . j2 ;
dom ii = dom ( i - 1 ) .= dom ( i - 1 ) .= dom ( i - 1 ) .= dom ( i - 1 ) ;
cluster sec | ]. PI / 2 , PI / 2 .[ -> one-to-one ;
Ball ( u , e ) = Ball ( f . p , e ) ;
reconsider x1 = x0 , y1 = x0 , y2 = x0 as Point of [: S , T :] ;
reconsider R1 = x , R2 = y , R2 = z as Relation of L , the carrier of K ;
consider a , b being Subset of A such that x = [ a , b ] ;
( <* 1 *> ^ p ) ^ <* n *> in RX ;
S1 +* S2 = S2 +* S1 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2
( ( #Z n ) * ( cos * ( cos - cos ) ) ) `| Z ) = f ;
cluster -> continuous for Function of C , REAL * , a be Real ;
set C7 = 1GateCircStr ( <* z , x *> , f3 ) , C8 = 1GateCircStr ( <* x , y *> , f3 ) ;
Ea1 . e2 = E8 . e2 + T . e2 .= T . e2 + T . e2 ;
( ( ( arctan * ln ) `| Z ) . ( x + x0 ) ) (#) ( ln * ln ) is_differentiable_on Z ;
upper_bound A = PI * ( 3 / 2 ) & lower_bound A = 0 ;
F . ( dom f , - F . ( cod f ) ) is_transformable_to F . ( cod f , - F . ( cod f ) ) ;
reconsider pbeing being Point of TOP-REAL 2 , p8 = ( |. q .| ) as Point of TOP-REAL 2 ;
g . W in [#] Y0 & [#] Y0 c= [#] ( Y | X ) ;
let C be compact non vertical non vertical non horizontal Subset of TOP-REAL 2 ;
LSeg ( f ^ g , j ) = LSeg ( f , j ) .= LSeg ( f , j ) ;
rng s c= dom f /\ ]. - r , x0 .[ & ( for n holds s . n < x0 ) implies f | ]. x0 , x0 + r .[ is convergent
assume x in { idseq ( 2 ) , Rev Rev idseq ( 2 ) } ;
reconsider n2 = n , m2 = m , n1 = n as Element of NAT ;
for y being ExtReal st y in rng seq holds g <= y & y <= g
for k st P [ k ] holds P [ k + 1 ] ;
m = m1 + m2 .= m1 + m2 .= m1 + m2 .= m1 + m2 .= m1 + m2 ;
assume for n holds H1 . n = G . n -H . n ;
set B" = f .: ( the carrier of X1 ) , B" = 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 ( b ) c= R -Seg ( a ) ;
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 ] or P [ x2 , y2 ] ;
pred x1 <> x2 means : Def1 : |. x1 - x2 .| > 0 & |. x2 - x1 .| > 0 ;
assume that p2 - p1 , p3 - p1 - p1 , p2 - p1 - p1 , p3 - p1 - p1 is_collinear ;
set q = ( -1 f ) ^ <* 'not' 'not' A *> ;
let f be PartFunc of REAL-NS 1 , REAL-NS 1 , x0 be Point of REAL-NS 1 , x be Point of REAL-NS 1 ;
( n mod ( 2 * k ) ) + 1 = n mod k ;
dom ( T * succ t ) = dom ( \mathop { 0 } --> 1 ) .= dom ( T * succ t ) ;
consider x being element such that x in wX and x in c and x in X ;
assume ( F * G ) . v . x3 = v . x3 & ( F * G ) . x3 = v . x3 ;
assume that the Sorts of D1 c= the Sorts of D2 and the Sorts of D1 c= the Sorts of D2 and the Sorts of D1 c= the Sorts 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 = q ;
n1 -' len f + 1 <= len ( - 1 ) + 1 + 1 - len f + 1 ;
\lbrace \lbrace q , O1 , a , b , a , b , b } = { [ u , v , a , b ] } ;
set C-2 = ( ( n + 1 ) .--> G ) . ( k + 1 ) ;
Sum ( L (#) p ) = 0. R * Sum p .= 0. V * 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 ) & ( $1 + 1 ) <= n ;
set s3 = Comput ( P1 , s1 , k ) , P3 = P1 +* I , s4 = P2 +* I ;
let l be -> -> -> [: of k , Al ( ) , D ( ) :] , A ;
reconsider U2 = union G-24 , G-24 = union G-24 , G-24 = union G-24 as Subset-Family of TL ;
consider r such that r > 0 and Ball ( p `1 , r ) c= Q ` ;
( h | ( n + 2 ) ) /. ( i + 1 ) = p2 . ( i + 2 ) ;
reconsider B = the carrier of X1 , C = the carrier of X2 as Subset of X ;
p$ 9 = <* - vs , 1 , - 1 , - 1 , 1 *> .= <* - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 *> ;
synonym f is real-valued means : Def1 : rng f c= NAT & for n being Nat holds f . n = f . n ;
consider b being element such that b in dom F and a = F . b ;
x10 < card X0 + card Y0 & card ( X0 \/ Y0 ) <= card ( X0 \/ Y0 ) ;
pred X c= B1 means : Def1 : for B1 st X c= B1 holds X c= B1 & not X c= B1 ;
then w in Ball ( x , r ) & dist ( x , w ) <= r ;
angle ( x , y , z ) = angle ( x-y , 0 , p2 ) ;
pred 1 <= len s means : Def1 : len ( the { of S } --> 0 ) = s & for i being Nat holds s . i = 0 ;
fP1 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 : Def1 : q '&' p in TAUT ( A ) ;
- ( ( t `1 ) / |. t .| ) < ( ( t `2 ) / |. t .| ) ;
( ( U . 1 ) = U2 /. 1 ) .= ( ( U . 1 ) * ( U2 /. 1 ) ) .= ( ( U . 1 ) * ( U2 /. 1 ) ) ;
f .: ( the carrier of x ) = the carrier of x & f .: ( the carrier of x ) = the carrier of x ;
Indices O = [: Seg n , Seg n :] & Indices O = [: Seg n , Seg n :] ;
for n being Element of NAT holds G . n c= G . ( n + 1 ) ;
then V in M @ ;
ex f being Element of F-9 st f is unital & ( f is unital of A-29 ) & ( f is the carrier of A-29 ) ;
[ 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 ]| - |[ w , v1 ]| <> 0. TOP-REAL 2 & |[ w1 , v1 ]| - |[ w , v1 ]| = |[ w , v1 ]| ;
reconsider t = t as Element of INT * , s be Element of INT ;
C \/ P c= [#] ( GX | ( [#] ( GX \ A ) ) ) ;
f " V in ( the carrier of X ) /\ D & f " ( the carrier of X ) = D /\ ( the carrier of X ) ;
x in [#] ( the carrier of ( the carrier of ( F . 0 ) ) /\ A ) ;
g . x <= h1 . x & h . x <= h1 . x ;
InputVertices S = { xy , y , z } & InputVertices S = { xy , y , z } ;
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 M2 is being_line and M3 is being_line ;
reconsider a = f4 . ( i0 -' 1 ) , b = f4 . ( i0 -' 1 ) as Element of K ;
len B2 = Sum ( Len F1 ^ F2 ) .= len ( Len F1 ) + len ( Len F2 ) .= len ( Len F1 ) + len ( Len F2 ) ;
len ( ( the S of n ) --> i ) = n & len ( ( the Sorts of n ) --> i ) = n ;
dom max ( - ( f + g ) , ( - f ) ) = dom ( f + g ) ;
( the Sorts of seq ) . n = upper_bound Y1 & ( the Sorts of seq ) . n = upper_bound Y2 ;
dom ( p1 ^ p2 ) = dom ( f12 ) & dom ( f12 ) = dom ( f12 ) ;
M . [ 1 , y ] = 1 / ( 1 - M ) * v1 .= 1 / ( 1 - M ) .= 1 / ( 1 - M ) ;
assume that W is non trivial and W .vertices() c= the carrier' of G2 and not W is trivial and not W is trivial ;
C6 * ( i1 , i2 ) = G1 * ( i1 , i2 ) .= G1 * ( i1 , i2 ) ;
C8 |- 'not' Ex ( x , p ) 'or' p . ( x , y ) ;
for b st b in rng g holds lower_bound rng f\rbrace <= b & b <= upper_bound rng f\rbrace
- ( ( q1 `1 / |. q1 .| - cn ) / ( 1 + cn ) ) = 1 ;
( LSeg ( c , m ) \/ LSeg ( l , k ) ) \/ LSeg ( l , k ) c= R ;
consider p being element such that p in { x } and p in L~ f and x in L~ f ;
Indices ( ( X @ ) * ( i , j ) ) = [: Seg n , Seg n :] ;
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 * ;
consider g being Function such that g = F . t and Q [ t , g ] ;
p in LSeg ( NW-corner Z , NW-corner L~ f ) /\ LSeg ( NW-corner Z , NW-corner L~ f ) ;
set R8 = R / ( 1 - R ) , R8 = R / ( 1 - R ) ;
IncAddr ( I , k ) = SubFrom ( da , da ) .= IncAddr ( da , da ) .= ( - - - n ) ;
seq . m <= ( the Sorts of seq ) . k & ( the Sorts of seq ) . k <= ( the Sorts of seq ) . k ;
a + b = ( a ` *' b ) ` + ( a ` *' b ) ` .= ( a ` *' b ) ` ;
id ( X /\ Y ) = id ( X /\ id Y ) .= id ( X /\ Y ) ;
for x being element st x in dom h holds h . x = f . x ;
reconsider H = U1 \/ U2 , U2 = U2 \/ U2 , U1 = U1 /\ U2 as non empty Subset of U0 ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) /\ f ) /\ j ) /\ m /\ n ;
consider y being element such that y in Y and P [ y , lower_bound B ] ;
consider A being finite stable set of R such that card A = len R and card A = len R ;
p2 in rng ( f |-- p1 ) \ rng <* p1 *> & p2 in rng ( f |-- p1 ) \ rng <* p1 *> ;
len s1 - 1 > 1-1 & len s2 - 1 > 0 & len s1 - 1 > 0 ;
( N-min P ) `2 = N-bound P & ( N-min P ) `2 = N-bound P & ( E-max P ) `2 = N-bound P ;
Ball ( e , r ) c= LeftComp Cage ( C , k + 1 ) & Ball ( e , r ) c= LeftComp Cage ( C , k + 1 ) ;
f . a1 ` = f . a1 ` & f . a2 = f . a2 ` & f . a2 = f . a2 ;
( seq ^\ k ) . n in ]. -infty , x0 + r .[ & ( seq ^\ k ) . n in dom f ;
gg . s0 = g . s0 | G . s0 .= g . s0 .= g . s0 ;
the InternalRel of S is \lbrace field ( the InternalRel of S ) , the InternalRel of S ( ) } ;
deffunc F ( Ordinal , Ordinal ) = phi . ( $2 , $1 ) & phi . ( $2 , $2 ) = phi . ( $2 , $2 ) ;
F . a1 = F . ( s2 . a1 ) & F . a1 = F . ( s2 . a1 ) ;
x `2 = A . o .= Den ( o , A . a ) .= Den ( o , A . a ) ;
Cl ( f " P1 ) c= f " ( Cl P1 ) & Cl ( f " P1 ) c= Cl ( f " P1 ) ;
FinMeetCl ( ( the topology of S ) \/ ( the topology of T ) ) c= the topology of T & the topology of T = the topology of T ;
synonym o is \bf means : Def1 : o <> {} & o <> {} & o <> {} ;
assume that X = Y + Z and card X <> card Y and card Y <> card Z and card X = card Z ;
the *> of s <= 1 + ( the +* ( s +* ( x +* ( s +* ( x +* ( x +* ( x , y ) ) ) ) ) ) ;
LIN a , a1 , d or b , c // b1 , c1 & a , c // a1 , c1 or b , c // b1 , c1 ;
e / 2 . 1 = 0 & e / 2 . 2 = 1 & e / 2 . 3 = 0 ;
E in SS1 & not ES1 in { NS1 } & not ES1 in SS2 ;
set J = ( l , u ) If ;
set A1 = } , A2 = ( a , b , c , d ) , A2 = ( a , b , c , d ) ;
set vs = [ <* cin , cin *> , '&' ] , xy = [ <* A1 , cin *> , '&' ] , i2 = [ <* cin , cin *> , '&' ] , a3 = [ <* A1 , cin *> , '&' ] ;
x * z `2 * x `2 * x `2 in x * ( z * N ) * x `2 ;
for x being element st x in dom f holds f . x = g3 . x & f . x = g3 . x ;
Int cell ( f , 1 , G ) c= RightComp f \/ RightComp f \/ L~ f \/ L~ f \/ L~ f ;
U2 is_an_arc_of W-min C , W-min C & P = LSeg ( W-min C , E-max C ) implies P = L~ f
set f-17 = f @ "/\" ( g @ ) ;
attr S1 is convergent means : Def1 : S is convergent & lim ( S - x ) = 0 & ( lim S ) = 0 ;
f . ( 0 + 1 ) = ( 0 qua Ordinal ) + a .= a + ( 0 qua Ordinal ) .= a + ( 0 qua Nat ) .= a + ( 0 qua Nat ) ;
cluster -> \llangle -> \in \lbrace reflexive transitive transitive transitive non empty RelStr , F , G , F , G , F , G , H , F , G , H , G , H , F , G , H , H , F , G , H , G , F , G , H , H , F , G , H , F
consider d being element such that R reduces b , d and R reduces c , d and R reduces d , b ;
not b in dom Start-At ( ( card I + 2 ) , SCMPDS ) & not b in dom Start-At ( ( card I + 2 ) , SCMPDS ) ;
( z + a ) + x = z + ( a + y ) .= z + a + y ;
len ( l \lbrack a \rbrack ) = len l & len ( l (#) x ) = len l ;
t4 } is ( {} \/ rng t4 ) -valued ( {} \/ rng t4 ) -valued FinSequence of NAT ;
t = <* F . t *> ^ ( C . p ) .= ( C . p ) ^ ( C . q ) ;
set p-2 = W-min L~ Cage ( C , n ) , pw2 = Cage ( C , n ) , pw2 = Cage ( C , n ) ;
( k -' ( i + 1 ) ) = ( k - ( i + 1 ) ) + ( i - ( i + 1 ) ) ;
consider u being Element of L such that u = u `1 ^ ( u `2 ^ v `2 ) and u in D ;
len ( ( width ( a |-> b ) ) |-> a ) = width ( ( a --> b ) * ( a --> b ) ) .= len ( a --> b ) ;
FM . x in dom ( ( G * the_arity_of o ) . x ) & ( G * the_arity_of o ) . x = ( G * the_arity_of o ) . x ;
set cH2 = the carrier of H2 , cH1 = the carrier of H2 , cH2 = the carrier of H1 , cH2 = the carrier of H2 ;
set cH1 = the carrier of H1 , cH2 = the carrier of H2 , cH2 = the carrier of H2 ;
( Comput ( P , s , 6 ) ) . intpos ( m + 6 ) = s . intpos ( m + 6 ) ;
IC Comput ( Q , t , k ) = ( l + 1 ) .= ( IC t ) + 1 ;
dom ( ( cos * sin ) `| REAL ) = REAL & dom ( ( cos * sin ) `| REAL ) = REAL & dom ( ( cos * sin ) `| REAL ) = REAL ;
cluster <* l *> ^ phi -> ( 1 + 1 ) -element for string of S ;
set b5 = [ <* A1 , cin *> , <* cin , cin *> ] , b5 = [ <* A1 , cin *> , '&' ] ;
Line ( Segm ( M @ , P , Q ) , x ) = L * Sgm Q .= Line ( M , x ) ;
n in dom ( ( the Sorts of A ) * the_arity_of o ) & ( ( the Sorts of A ) * the_arity_of o ) . n = ( the Sorts of A ) . n ;
cluster f1 + f2 -> continuous for PartFunc of REAL , the carrier of S , the carrier of S ;
consider y being Point of X such that a = y and ||. x-y .|| <= r ;
set x3 = being Element of NAT , Q = Q . DataLoc ( s4 . SBP , 2 ) , x4 = Q . DataLoc ( s3 . SBP , 2 ) , P4 = Q . DataLoc ( s3 . SBP , 2 ) , P4 = Q . DataLoc ( s3 . SBP , 2 ) , P4 = Q . DataLoc ( s3 . SBP
set p-3 = stop I ( ) , p-3 = stop I ( ) , p-3 = stop I ( ) , p-3 = stop I ( ) , ps1 = stop I ( ) , ps1 = stop I ( ) , ps1 = stop I ( ) , ps1 = stop I ( ) , ps1 = stop
consider a being Point of D2 such that a in W1 and b = g . a and a in W1 ;
{ A , B , C , D , E } = { A , B } \/ { C , D , E }
let A , B , C , D , E , F , J , M , N , M , N , N , A , M , N , M , N , A , N , M , N , A , N , M , N , A , N , M be set ;
|. p2 .| ^2 - ( ( p2 `2 ) ^2 - ( p2 `2 ) ^2 ) >= 0 ;
l -' 1 + 1 = n-1 * ( l + ( 1 + 1 ) ) + 1 .= ( 1 + 1 ) + 1 ;
x = v + ( a * w1 + b * w2 ) + ( c * w2 ) + ( c * w2 ) ;
the TopStruct of L = , , , the TopStruct of L = [: the topology of L , the topology of L :] ;
consider y being element such that y in dom H1 and x = H1 . y and y in H . x ;
fH \ { n } = Free ( All ( v1 , H ) ) & not f . ( n + 1 ) in Free ( H ) ;
for Y being Subset of X st Y is summable & Y is summable holds Y is not summable
2 * n in { N : 2 * Sum ( p | N ) = N & N > 0 } ;
for s being FinSequence holds len ( the { - } \rm \rm \rm \rm ' } Shift ( s ) ) = 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 > 0 ;
rng ( h2 * f2 ) c= the carrier of ( ( TOP-REAL 2 ) | ( the carrier of TOP-REAL 2 ) ) | ( the carrier of ( TOP-REAL 2 ) | K1 ) ;
j + ( len f ) - len f <= len f + ( len g - len f ) - len f ;
reconsider R1 = R * I , R2 = R * I as PartFunc of REAL , REAL-NS n , REAL-NS n ;
C8 . x = s1 . x0 .= C8 . x .= C8 . x .= C8 . x ;
power F_Complex . ( z , n ) = 1 .= x |^ n .= x |^ n .= x |^ n ;
t at ( C , s ) = f . ( the connectives of S ) . t .= f . ( the connectives of S ) . t ;
support ( f + g ) c= support f \/ ( support g ) & support ( f + g ) c= support f \/ support g ;
ex N st N = j1 & 2 * Sum ( ( r4 | N ) | N ) > N ;
for y , p st P [ p ] holds P [ All ( y , p ) ] ;
{ [ x1 , x2 ] where x1 is Point of [: X1 , X2 :] : x1 in X1 & x2 in X2 } c= the carrier of [: X1 , X2 :]
h = ( i = j |-- h , id B . i ) .= H . i .= H . i ;
ex x1 being Element of G st x1 = x & x1 * N c= A & N c= A ;
set X = ( ( \lbrace q , O1 } ) . ( q , 4 ) ) `1 , Y = ( ( { q , O1 } ) . ( q , 4 ) ) `1 , Z = { q , p } ;
b . n in { g1 : x0 < g1 & g1 < x0 & g1 < x0 } ;
f /* s1 is convergent & f /. x0 = lim ( f /* s1 ) & f /. x0 = lim ( f /* s1 ) ;
the lattice of Y = the lattice of the lattice of Y & the carrier of Y = the carrier of the carrier of Y & the carrier of Y = the carrier of Y ;
( 'not' ( a . x ) '&' b . x ) 'or' a . x '&' ( b . x ) = FALSE ;
2 = len ( q0 ^ r1 ) + len ( q1 ^ q2 ) .= len ( q1 ^ q2 ) + len ( q2 ^ r2 ) .= len ( q1 ^ q2 ) + len ( q2 ^ r2 ) ;
( 1 / a ) (#) ( sec * f1 ) - id Z ) is_differentiable_on Z & ( ( 1 / a ) (#) ( sec * f1 ) ) `| Z = f ;
set K1 = integral ( ( lim ( lim ( H , A ) ) || ( A , B ) ) , D2 = integral ( ( lim ( H , A ) || ( A , B ) ) , D ) ;
assume e in { ( w1 - w2 ) : w1 in F & w2 in G } ;
reconsider d7 = dom a `1 , d6 = dom F `1 , d6 = dom F `1 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 SA2 = dom S /\ Seg n .= dom ( L | Seg n ) .= dom ( L | Seg n ) .= dom ( L | Seg n ) .= dom ( L | Seg n ) .= dom ( L | Seg n ) ;
x in H |^ a implies ex g st x = g |^ a & g in H & g in H
a * 0. ( INT , n ) . ( a , 1 ) = a `2 - ( 0 * n ) .= a `2 ;
D2 . j in { r : lower_bound A <= r & r <= D1 . i } ;
ex p being Point of TOP-REAL 2 st p = x & P [ p ] & p <> 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 c= dom ( f1 (#) f2 ) ;
1 = ( p * p ) * p .= p * ( p * p ) .= p * p .= p ;
len g = len f + len <* x + y *> .= len f + 1 .= len f + 1 .= len f + 1 ;
dom F-11 = dom ( F | [: N1 , S-23 :] ) .= [: N1 , S-23 :] .= [: N1 , S-23 :] ;
dom ( f . t ) * I . t = dom ( f . t ) * g . t .= ( f . t ) * g . t ;
assume a in ( "\/" ( ( T |^ the carrier of S ) , F ) ) .: D & b in ( the carrier of S ) ;
assume that g is one-to-one and ( the carrier' of S ) /\ rng g c= dom g and g is one-to-one and rng g c= dom f ;
( ( x \ y ) \ z ) \ ( ( x \ z ) \ ( y \ z ) ) = 0. X ;
consider f such that f * f `1 = id b and f * f `2 = id a and f is one-to-one and f is one-to-one ;
( cos | [. 2 * PI , 0 + ( 2 * PI * 0 ) .] ) | [. 2 * PI , 0 + ( 2 * PI * 0 ) .] is increasing ;
Index ( p , co ) <= len LS - LS .. LS - LS .. LS + 1 .= len LS - LS .. LS + 1 ;
let t1 , t2 , Z be Element of [: T , S :] , s , t be Element of S ;
( Frege ( ( Frege ( curry H ) ) . h ) ) . h <= "/\" ( rng ( ( Frege ( ( Frege G ) . h ) . ( i + 1 ) ) ) ;
then P [ f . i0 , f . ( i0 + 1 ) ] & F ( f . i0 , f . ( i0 + 1 ) ) < j ;
Q [ [ D . x , 1 ] , F . [ D . x , 1 ] ] ;
consider x being element such that x in dom ( F . s ) and y = ( F . s ) . x ;
l . i < r . i & [ l . i , r . i ] is Element of G . i ;
the Sorts of A2 = ( the carrier of S2 ) --> ( the carrier of S2 ) .= the Sorts of A1 +* A2 .= the Sorts of A2 ;
consider s being Function such that s is one-to-one and dom s = NAT & rng s = F and rng s = the carrier of S ;
dist ( b1 , b2 ) <= dist ( b1 , a ) + dist ( a , b2 ) + dist ( a , b2 ) ;
( Upper_Seq ( C , n ) ) /. len ( Upper_Seq ( C , n ) ) = WL~ Cage ( C , n ) ;
q `2 <= ( UMP Upper_Arc L~ Cage ( C , 1 ) ) `2 & ( UMP L~ Cage ( C , 1 ) ) `2 <= ( E-max L~ Cage ( C , 1 ) ) `2 ;
LSeg ( f | i2 , i ) /\ LSeg ( f | i2 , j ) = {} & LSeg ( f | i2 , j ) = {} ;
given a being ExtReal such that a <= II and A = ]. a , II .] and a <= II ;
consider a , b being complex number such that z = a & y = b & z + y = a + b ;
set X = { b |^ n where n is Element of NAT : n <= k } , Y = { b |^ n where b is Element of NAT : b <= n } ;
( ( x * y * z \ x ) \ z ) \ ( x * y \ x ) = 0. X ;
set xy = [ <* xy , y *> , f1 ] , yz = [ <* y , z *> , f2 ] , yz = [ <* z , x *> , f3 ] , yz = [ <* x , y *> , f2 ] , yz = [ <* z , x *> , f3 ] ;
lq /. len lq = lq . len ( lq | ( len q -' 1 ) ) .= ( q | ( len q -' 1 ) ) . len ( q | ( len q -' 1 ) ) ;
( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 = 1 ;
( ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) * ( 1 + sn ) < 1 ;
( ( E-max X ) `2 ) ^2 = ( ( E-max X ) `2 ) ^2 + ( ( E-max X ) `2 ) ^2 .= ( ( E-max X ) `2 ) ^2 + ( ( E-max X ) `2 ) ^2 ;
( seq - seq ) . k = seq . k - seq . k .= ( seq ^\ k ) . k - seq . k .= ( seq ^\ k ) . k - 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 X & the carrier of X = the carrier of X ;
ex p4 st p3 = p4 & |. p4 - |[ a , b ]| .| = r & |. p4 - |[ a , b ]| .| = r ;
set ch = chi ( X , A5 ) , c5 = chi ( X , A5 ) ;
R to_power ( 0 * n ) = I\HM ( X , X ) .= R to_power n .= R to_power ( 0 * n ) ;
( Partial_Sums ( curry ( F-19 , n ) ) ) . n is nonnegative & ( ( curry ( F-19 , n ) ) . n ) . m = ( ( ( curry ( F-19 , n ) ) . m ) . n ) . m ;
f2 = C7 . ( len E7 , K , len H ) .= C8 . ( len E7 + len H ) .= ( the V of V ) . ( len H + 1 ) ;
S1 . b = s1 . b .= S2 . b .= S2 . b .= S2 . b .= S2 . b .= S2 . b ;
p2 in LSeg ( p2 , p1 ) /\ LSeg ( p1 , p01 ) /\ LSeg ( p1 , p01 ) & p2 in LSeg ( p1 , p01 ) /\ LSeg ( p1 , p01 ) ;
dom ( f . t ) = Seg n & dom ( I . t ) = Seg n & rng ( I . t ) = Seg n ;
assume o = ( the connectives of S ) . 11 & the connectives of S = ( the connectives of S ) . 11 & the connectives of S = the carrier' of S ;
set phi = ( l1 , l2 ) \mathop { {} } , phi = ( l1 , l2 ) \mathop { {} } , F = ( S , l2 ) \mathop { {} } , G = ( S , l2 ) \mathop { {} } , F = ( S , l2 ) \mathop { {} } , G = ( S , l2 ) \mathop { {} } , F = ( S , l2 ) \mathop { {} } , G =
synonym p is invertible for p is invertible & HT ( p , T ) = 1. L & p is invertible ;
( Y1 `2 ) ^2 = - 1 & 0. ( TOP-REAL 2 ) <> 0. ( TOP-REAL 2 ) & ( the carrier of ( TOP-REAL 2 ) | K1 ) = the carrier of ( TOP-REAL 2 ) | K1 ;
defpred X [ Nat , set , set , set ] means P [ $2 , $2 , $2 , $2 , $2 ] ;
consider k being Nat such that for n being Nat st k <= n holds s . n < x0 + g and s . n < x0 + g ;
Det ( I |^ ( m -' n ) ) * ( m - n ) = 1. K & Det ( I |^ ( m -' n ) ) = 0. K ;
( - b - sqrt ( b - a ^2 ) ) / ( 4 * a * c ) < 0 ;
Cd . d = Cd . d mod Cd . ( dd + 1 ) .= Cd . d mod Cd . ( dd + 1 ) .= Cd . d mod Cd . ( Cd + 1 ) ;
attr X1 is dense means : Def1 : X2 is dense dense & X1 is dense SubSpace of X & X2 is dense SubSpace of X ;
deffunc F6 ( Element of E , Element of I ) = ( $1 * $2 ) * ( $1 * $2 ) ;
t ^ <* n *> in { t ^ <* i *> : Q [ i , T . t ] } ;
( x \ y ) \ x = ( x \ x ) \ y .= y ` .= 0. X ;
for X being non empty set holds for F being Subset-Family of X holds F is Basis of [: X , UniCl ( Y ) :]
synonym A , B are_separated means : Def1 : Cl A misses Cl B & Cl B misses Cl Cl Cl A & Cl Cl B = Cl Cl A ;
len ( M @ ) = len p & width ( M @ ) = width ( M @ ) & width ( M @ ) = width ( M @ ) ;
J . v = { x where x is Element of K : 0 < v . x & v . x = 1 } ;
( Sgm ( seq ( m ) ) . d - ( Sgm ( seq ( m ) ) . e ) ) . e <> 0 ;
lower_bound divset ( D2 , k + k2 ) = D2 . ( k + k2 - 1 ) .= D2 . ( k + k2 - 1 ) ;
g . r1 = - 2 * r1 + 1 & dom h = [. 0 , 1 .] & rng h = [. 0 , 1 .] ;
|. a .| * ||. f .|| = 0 * ||. f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| ;
f . x = ( h . x ) `1 & g . x = ( h . x ) `2 & h . x = ( h . x ) `2 ;
ex w st w in dom B1 & <* 1 *> ^ s = <* 1 *> ^ w & len w = len B1 + 1 ;
[ 1 , {} , <* d1 *> ] in ( { [ 0 , {} , {} ] } \/ S1 ) \/ S2 ;
IC Exec ( i , s1 ) + n = IC Exec ( i , s2 ) .= IC Exec ( i , s2 ) .= IC Exec ( i , s2 ) ;
IC Comput ( P , s , 1 ) = IC Comput ( P , s , 9 ) .= 5 + 9 .= ( 5 + 9 ) ;
( IExec ( W6 , Q , t ) ) . intpos ( \mathbb d + 3 ) = t . intpos ( 0 + 3 ) .= t . intpos ( 0 + 3 ) ;
LSeg ( f /^ q , i ) misses LSeg ( f /^ q , j ) \/ LSeg ( f /^ q , j ) ;
assume for x , y being Element of L st x in C & y in C holds x <= y or y <= x ;
integral ( integral ( f , C ) , A ) = f . ( upper_bound C ) - lower_bound ( rng ( f | C ) ) ;
for F , G being one-to-one FinSequence st rng F misses rng G holds F ^ G is one-to-one & F ^ G is one-to-one
||. R /. ( L . h ) .|| < e1 * ( K + 1 * ||. h .|| ) ;
assume a in { q where q is Element of M : dist ( z , q ) <= r } ;
set p4 = [ 2 , 1 ] .--> [ 2 , 0 , 1 ] ;
consider x , y being Subset of X such that [ x , y ] in F and x c= d and y c= d ;
for y , x being Element of REAL st y `1 in Y & x in X ` holds y `1 <= x `1 + x `2
func |. Sum ( p \bullet q ) .| -> -> [: of A , A :] equals min ( NBI , p ) .= Sum ( NNI ) ;
consider t being Element of S such that x `1 , y `2 '||' z `1 , t `2 and x `2 '||' y `2 , t `2 ;
dom x1 = Seg ( len x1 ) & len x1 = len l1 & for i st i in dom x1 holds x1 . i = ( x1 . i ) * ( x2 . i ) ;
consider y2 being Real such that x2 = y2 and 0 <= y2 and y2 <= 1 / 2 and |. y2 - x0 .| < r ;
||. f | X .|| /* s1 = ||. f .|| | X & ||. f .|| /* s1 = ||. f .|| /* s1 .= ( f | X ) /* s1 ;
( the InternalRel of A ) ` /\ ( the InternalRel of A ) = {} \/ {} .= {} \/ {} .= {} .= {} ;
assume that i in dom p and for j be Nat st j in dom q holds P [ i , j ] and i + 1 in dom p and p . i = q . j and j + 1 in dom p and p . j = p . j ;
reconsider h = f | X ( ) as Function of X ( ) , rng f , rng f :] , rng f ;
u1 in the carrier of W1 & u2 in the carrier of W2 & v in the carrier of W1 & v = [ v1 , v2 ] ;
defpred P [ Element of L ] means M <= f . $1 & f . $1 <= f . $1 & f . $1 <= f . $1 ;
l . ( u , a , v ) = s * x + ( - ( s * x ) + y ) .= b ;
- ( x-y - x ) = - x + - ( - y ) .= - x + - y .= x + y .= x ;
given a being Point of GX such that for x being Point of GX holds a , x , a , x , y is_collinear and a , x , y is_collinear ;
fJ = [ dom @ f2 , cod @ g2 ] & [ dom @ f2 , cod @ g2 ] in dom @ ( @ f2 ) ;
for k , n being Nat st k <> 0 & k < n & n is prime holds k , n are_relative_prime & k , n are_relative_prime
for x being element holds x in A |^ d iff x in ( ( A ` ) |^ d ) ` & x in ( A ` ) |^ d
consider u , v being Element of R , a being Element of A such that l /. i = u * a * v and a in I ;
( ( - ( p `1 / |. p .| - cn ) ) / ( 1 + cn ) ) ^2 > 0 ;
L-13 . k = L9 . ( F . k ) & F . k in dom ( L9 | dom F ) & F . k = L9 . ( F . k ) ;
set i2 = AddTo ( a , i , - n ) , i1 = goto - ( card I + 1 ) ;
attr B is thesis means : Def1 : for S being SubSubSub of B holds S = ( B `1 ) \/ ( B `2 ) ;
a9 "/\" D = { a "/\" d where d is Element of N : d in D } & ( for x being Element of N holds d in D iff x in D ) ;
|( \square , q29 )| * |( q , q29 )| + |( \square , q29 )| * |( q , q29 )| >= |( \square , q29 )| ;
( - f ) . sup A = ( ( - f ) | A ) . 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 .= G * ( len G , k ) `1 ;
( Proj ( i , n ) ) . LM = <* ( proj ( i , n ) ) . LM *> .= ( proj ( i , n ) ) . LM ;
f1 + f2 * reproj ( i , x ) is_differentiable_in ( ( reproj ( i , x ) ) . i ) & f2 + ( ( reproj ( i , x ) ) . x ) = ( f2 + ( reproj ( i , x ) ) . i ) ;
pred ( cos * tan ) . x <> 0 means : Def1 : ( tan * tan ) . x = tan . x * tan . x ;
ex t being SortSymbol of S st t = s & h1 . t . x = h2 . t . x & ( h1 . t ) . x = h2 . x ;
defpred C [ Nat ] means ( P . $1 ) is non empty and ( P . $1 is non empty ) & ( not P [ $1 ] ) & ( not P [ $1 ] ) ;
consider y being element such that y in dom ( p | 8 ) and ( p | 8 ) . y = ( p | 8 ) . y ;
reconsider L = product ( { x1 } +* ( index B , l ) ) as Subset of ( Carrier A ) . l ;
for c being Element of C ex d being Element of D st T . ( id c ) = id d & for a being Element of C holds T . ( id c ) = id d
N = ( f | n ) ^ <* p *> .= ( f | n ) ^ <* p *> .= f | n ^ <* p *> ;
( f * g ) . x = f . ( g . x ) & ( f * h ) . x = f . ( h . x ) ;
p in { 1 / 2 * ( G * ( i + 1 , j + 1 ) + G * ( i + 1 , j + 1 ) ) } ;
f `2 - cp = ( f | ( n , L ) ) *' - ( f | ( n , L ) ) .= ( f - ( f | ( n , L ) ) ) *' ;
consider r being Real such that r in rng ( f | divset ( D , j ) ) and r < m + s ;
f1 . ( |[ ( r `1 ) / ( 1 + ( r `2 / r `1 ) ^2 ) , ( r `2 ) / ( 1 + ( r `2 / r `1 ) ^2 ) ]| ) in f1 .: W1 ;
eval ( a | ( n , L ) , x ) = eval ( a | ( n , L ) , x ) .= a . ( x * x ) .= a . ( x * x ) ;
z = DigA ( tk , xx ) .= DigA ( tk , ( k + 1 ) ) .= DigA ( tk , ( k + 1 ) ) .= DigA ( tk , ( k + 1 ) ) ;
set H = { Intersect S where S is Subset-Family of X : S c= G } , G = { meet S where S is Subset of X : S c= G } , F = the carrier of X ;
consider S19 being Element of D * , d being Element of ( the carrier of K ) * 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 and f . x2 = f . x2 ;
- 1 <= ( q `2 / |. q .| - sn ) / ( 1 + sn ) & q `2 / |. q .| - sn <= ( q `2 / |. q .| - sn ) / ( 1 + sn ) ;
0. ( V ) is Linear_Combination of A & Sum ( 1. ( V ) ) = 0. V & Sum ( 0. ( V ) ) = 0. V ;
let k1 , k2 , k1 , k2 , k2 , k2 , 6 be Instruction of SCM+FSA , a , b , c , k2 be Int-Location ;
consider j being element such that j in dom a and j in g " { k `2 } and x = a . j and y = a . j ;
H1 . x1 c= H1 . x2 or H1 . x2 c= H1 . x1 or H1 . x2 c= H1 . x2 or H1 . x2 c= H1 . x2 & H1 . x2 c= H1 . x2 ;
consider a being Real such that p = or p = a * p1 + ( a * p2 ) and 0 <= a and a <= 1 ;
assume that a <= c & c <= d & [' a , b '] c= dom f and [' a , b '] c= dom g and g . a = g . b ;
cell ( Gauge ( C , m ) , len Gauge ( C , m ) -' 1 , 0 ) is non empty ;
A, AX in { ( S . i ) `1 where i is Element of NAT : not contradiction } ;
( T * b1 ) . y = L * b2 /. y .= ( F /. y ) . y .= ( F /. y ) . y .= ( F /. y ) . y ;
g . ( s , I ) . x = s . y & g . ( s , I ) . y = |. s . x - s . y .| ;
( log ( 2 , k + 1 ) ) ^2 >= ( log ( 2 , k + 1 ) ) ^2 ;
then p => q in S & not x in the still of p & not x in the carrier of p & not x in the carrier of p & not x in the carrier of p ;
dom ( the InitS of r-10 ) misses dom ( the InitS of r-11 ) & dom ( the InitS of r-11 ) = the carrier of r-11 & dom ( the InitS of r-11 ) = the carrier of r-11 ;
synonym f is integer means : Def1 : for x being set st x in rng f holds x is Integer & f . 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 p1 + len <* x *> .= len p1 + len <* x *> .= len p1 + 1 .= len p1 + 1 ;
l ( ) = g /. ( ( - 1 ) * k ) + ( k ( ) * k ) - ( e /. ( - 1 ) * k ) .= ( e /. ( - 1 ) ) + ( e /. ( - 1 ) ) ;
CurInstr ( P2 , Comput ( P2 , s2 , l ) ) = halt SCM+FSA .= CurInstr ( P2 , Comput ( P2 , s2 , l ) ) .= CurInstr ( P2 , Comput ( P2 , s2 , l ) ) ;
assume for n be Nat holds ||. seq .|| . n <= ( ||. seq .|| ) . n & ( ||. seq .|| ) . n <= ( ||. seq .|| ) . n ;
sin ( 0. K ) = sin r * cos ( - r ) .= cos ( - r ) * sin ( - r ) .= cos r * sin ( - r ) .= - sin r * sin r ;
set q = |[ g1 . t0 `1 , g2 . t0 `2 , g1 . t0 `2 ]| , r = |[ r , g2 . t0 `2 ]| , s = |[ r , g2 . t0 `2 ;
consider G being sequence of S such that for n being Element of NAT holds G . n in G and G . n in implies G . n in
consider G such that F = G and ex G1 st G1 in SM & G = [: X , G1 :] & G = [: X , G1 :] ;
the root of [ x , s ] in ( the Sorts of Free ( C , X ) ) . s & ( the Sorts of Free ( C , X ) ) . s = ( the Sorts of Free ( C , X ) ) . s ;
Z c= dom ( exp_R * ( f + ( #Z 3 ) * ( f + ( #Z 3 ) ) ) ) ;
for k being Element of NAT holds seq1 . k = ( sum ( Im ( f , S-3 ) ) ) . k + ( Im ( f , S-3 ) ) . k
assume that - 1 < n and ( q `2 / |. q .| - sn ) > 0 and ( q `2 / |. q .| - sn ) < 0 and ( q `2 / |. q .| - sn ) < 0 ;
assume that f is continuous one-to-one and a < b and a < c and f = g and f . a = c and f . b = d and f . c = d ;
consider r being Element of NAT such that s-> Element of NAT such that s-> = Comput ( P1 , s1 , r ) and r <= q and r <= q ;
LE f /. ( i + 1 ) , f /. j , L~ f , f /. ( len f ) , f /. ( len f ) ;
assume that x in the carrier of K and y in the carrier of K and ex_inf_of { x , y } , L and inf { x , y } = inf { x , y } ;
assume f +* ( i1 , \xi ) . 1 in ( proj ( F , i2 ) ) " ( A . 1 ) & f . ( i1 + 1 ) = ( proj ( F , i2 ) ) . ( A . 1 ) ;
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 the carrier of T } ;
consider l be Nat such that for m be Nat st l <= m holds ||. ( s1 . m - x0 ) .|| < g / 2 ;
consider t being 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 v ^ <* 1 *> in dom p ;
consider a being Element of the carrier of [ X , A ] , A being Element of the lines of [ X , A ] such that not a on A and not b on A ;
( - x ) |^ ( k + 1 ) * ( ( - x ) |^ ( k + 1 ) ) " = 1 ;
for D being set st for i st i in dom p holds p . i in D holds p . i is FinSequence of D & p . i = p . i
defpred R [ element ] means ex x , y st [ x , y ] = $1 & P [ x , y ] & P [ y , x ] ;
L~ f2 = union { LSeg ( p0 , p01 ) , LSeg ( p01 , p2 ) } .= { LSeg ( p01 , p2 ) } \/ { LSeg ( p01 , p2 ) } ;
i -' len h11 + 2 - 1 < i -' len h11 + 2 - 1 + 1 + 1 - 1 + 1 + 1 - 1 + 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 ;
for n being Element of NAT st n in dom F holds F . n = |. ( F . n ) . ( n -' 1 ) .| ;
for r , s1 , s2 , r holds r in [. s1 , s2 .] iff r <= s1 & s1 <= s2 & s1 <= s2 & s1 <= s2 & s2 <= s2
assume v in { G where G is Subset of T2 : G in B2 & G c= z1 & G c= z2 } ;
let g be \vert \vert is \vert non-empty Function of A , INT , b be Element of INT ;
min ( g . [ x , y ] , k . [ y , z ] ) = ( min ( g . k , k , x ) ) . y ;
consider q1 being sequence of CNS such that for n holds P [ n , q1 . n ] & P [ n , q1 . n ] ;
consider f being Function such that dom f = NAT & for n being Element of NAT holds f . n = F ( n ) and for n being Element of NAT holds f . n = F ( n ) ;
reconsider B-6 = B /\ O , OO = O /\ Z , Z = { O } , Z = { O , O } , S = { O , O } , T = { O , O } , O = { O , O } , O = { O , O } , Q = { O , O } , S = { O , O } , T = { O , O } , Q = { O , O } , S = {
consider j being Element of NAT such that x = the ` of n and j <= n and 1 <= j and j <= n and j <= n ;
consider x such that z = x and card ( x . O2 ) in card ( x . O2 ) and x in L1 . O2 and x . O2 in L1 . O2 ;
( C * _ T4 ( k , n2 ) ) . 0 = C . ( ( _ T4 ( k , n2 ) ) . 0 ) .= C . ( ( T . k ) . 0 ) ;
dom ( X --> rng f ) = X & dom ( X --> f ) = dom ( X --> f ) & dom ( X --> f ) = dom ( X --> f ) ;
S-bound L~ SpStSeq L~ f <= ( ( SpStSeq L~ f ) /. 1 ) `2 & ( ( 0. L~ f ) `2 <= ( ( 0. L~ f ) `2 ) `2 ;
synonym x , y are_collinear means : Def1 : x = y or ex l being Nat st { x , y } c= l & x in l ;
consider X being element such that X in dom ( f | ( n + 1 ) ) and ( f | ( n + 1 ) ) . X = Y ;
assume that Im k is continuous and for x , y being Element of L , a , b being Element of Im k st a = x & b = y holds x << y & x << y & y << a ;
( 1 / 2 * ( ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( 1 / 2 ) ) ) ) ) ) `| REAL = ( 1 / 2 * ( ( #Z 2 ) * ( 1 / 2 ) ) ) ;
defpred P [ Element of omega ] means ( the Sorts of A1 ) . $1 = A1 . $1 & ( the Sorts of A2 ) . $1 = A2 . $1 ;
IC Comput ( P , s , 2 ) = succ IC Comput ( P , s , 1 ) .= IC Comput ( P , s , 1 ) .= 6 + 1 .= 6 ;
f . x = f . g1 * f . g2 .= f . g1 * 1_ H .= f . ( g1 . g2 ) .= ( f . g2 ) * ( f . g2 ) .= ( f . g1 ) * ( f . g2 ) .= ( f . g2 ) * ( f . g2 ) .= ( f . g1 ) * ( f . 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 + L2 c= the carrier of L1 & the carrier of L1 + L2 = the carrier of L1 + L2 ;
pred a , b , c , x , y , c , d , x , y , z , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , z , x , y , z , x , y , z , x , z , y , z , x , z , y , z , x , y , z , x , y , z , x , y , z , z , x , y , z
( the partial of s ) . n <= ( the Sorts of s ) . n * ( the Sorts of s ) . ( n + 1 ) & ( the Sorts of s ) . n = ( the Sorts of s ) . n ;
pred - 1 <= r & r <= 1 & ( arccot - 1 ) * ( arccot - 1 ) = - 1 / ( 1 + r ^2 ) ;
seq in { p ^ <* n *> where n is Nat : p ^ <* n *> in T1 & p ^ <* n *> in T1 } ;
|[ x1 , x2 , x3 , x4 ]| . 2 - |[ y1 , y2 , x3 , x4 ]| . 2 = x2 - y2 & |[ y1 , y2 , x3 , x4 ]| . 2 = x2 - y2 ;
attr m be Nat means : Def1 : F . m is nonnegative & ( Partial_Sums F ) . m is nonnegative & ( Partial_Sums F ) . m is nonnegative ;
len ( ( G . z ) . x ) = len ( ( ( G . ( x , y ) ) + ( G . ( y , z ) ) ) . x ) .= len ( G . ( x , y ) ) .= len ( G . ( y , z ) ) ;
consider u , v being VECTOR of V such that x = u + v and u in W1 /\ W2 and v in W2 /\ W3 and v in W3 /\ W3 ;
given F be finite FinSequence of NAT such that F = x and dom F = n & rng F c= { 0 , 1 } and Sum F = k and Sum F = k ;
0 = 1 * over \llangle 0 , 0 * u] iff 1 = ( 1 - ( 1 - ( 1 - ( 1 - 0 ) ) * ( 1 - 0 ) ) ) / ( 1 - 0 )
consider n being Nat such that for m being Nat st n <= m holds |. ( f # x ) . m - lim ( f # x ) .| < e ;
cluster -> \mathclose means : Def1 : ( the carrier of \mathop { \rm o } ) is Boolean non empty & ( the carrier of \mathop { \rm o } ) is Boolean & ( the carrier of \mathop { \rm o } ) is Boolean & ( the carrier of \mathop { \rm o } ) is Boolean ;
"/\" ( BB , L ) = Top BB .= Top ( S , L ) .= "/\" ( [#] I , L ) .= "/\" ( I , L ) .= "/\" ( I , L ) .= "/\" ( I , L ) ;
( r / 2 ) ^2 + ( r / 2 ) ^2 <= ( r / 2 ) ^2 + ( r / 2 ) ^2 + ( r / 2 ) ^2 ;
for x being element st x in A /\ dom ( f `| X ) holds ( f `| X ) . x >= r2 & ( f `| X ) . x >= r2
2 * r1 - ( 2 * r1 - ( 2 * r1 - 1 ) ) * ( b - a ) = 0. TOP-REAL 2 .= ( 2 * r1 - ( 2 * r2 - 1 ) ) * ( b - a ) ;
reconsider p = P * ( \square , 1 ) , q = a " * ( ( - a ) * ( ( - a ) * ( - b ) ) ) 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 = ( upper_volume ( g , M7 ) ) . n & ( the Sorts of M7 ) . n = ( the Sorts of M7 ) . n ;
consider y , z being element such that y in the carrier of A and z in the carrier of A and i = [ y , z ] and i = [ y , z ] ;
given H1 , H2 being strict Subgroup of G such that x = H1 & y = H2 and H1 is Subgroup of H2 and H2 is Subgroup of H1 and the carrier of H1 = the carrier of H2 and H2 is Subgroup of H2 ;
for S , T being non empty < , d being Function of T , S st T is complete holds d is directed-sups-preserving & d is monotone & d is monotone & d is monotone
[ a + 0. F_Complex , b2 ] in ( the carrier of F_Complex ) \/ ( the carrier of F_Complex ) & [ a + 0. F_Complex , b2 + 0. F_Complex ] in ( the carrier of F_Complex ) \/ ( the carrier of V ) ;
reconsider mm = max ( len F1 , len ( p . n ) * ( x |^ n ) ) , mm = max ( len F1 , len ( p . n ) * ( x |^ n ) ) as Element of NAT ;
I <= width GoB ( ( GoB h ) * ( len GoB h , width GoB h ) ) & ( GoB ( ( GoB h ) * ( len GoB h , width GoB h ) ) `2 <= ( GoB h ) * ( 1 , width GoB h ) `2 ;
f2 /* q = ( f2 /* ( f1 /* s ) ) ^\ k .= ( f2 * f1 ) /* s .= ( ( f2 * f1 ) /* s ) ^\ k .= ( ( f2 * f1 ) /* s ) ^\ k ;
attr A1 \/ A2 is linearly-independent means : Def1 : A1 is linearly-independent & A2 is linearly-independent & Lin ( A1 \/ A2 ) = Lin ( A1 \/ A2 ) & Lin ( A1 \/ A2 ) = Lin ( A1 \/ A2 ) ;
func A -carrier C -> set equals union { A . s where s is Element of R : s in C } & for s being Element of R holds s in C . s ;
dom ( Line ( v , i + 1 ) ) = dom ( ( F . ( p + m ) ) ^ ( ( F . ( p + m ) ) . ( p + m ) ) ) .= dom ( F . ( p + m ) ) ;
cluster [ x `1 , 4 ] , [ x `2 , 4 ] , [ x `1 , 4 ] ] -> to [ x `1 , 4 ] , [ x `1 , 4 ] , [ x `1 , 4 ] ] -> to ;
E , f |= All ( x2 , ( x2 } \hbox ( x1 , x2 ) ) '&' ( x1 '&' x2 ) '&' ( x2 '&' x3 ) '&' ( x1 '&' x2 ) '&' ( x2 '&' x3 ) '&' ( x1 '&' x2 ) '&' ( x2 '&' x3 ) '&' ( x2 '&' x3 ) '&' ( x1 '&' x2 ) '&' ( x2 '&' x3 ) '&' ( x2 '&' x3 ) ;
F .: ( id X , g ) . x = F . ( id X , g . x ) .= F . ( x , g . x ) .= F . ( x , g . x ) .= F . ( x , g . x ) ;
R . ( h . m ) = F . x0 + h . ( m + 1 ) - ( h . x0 ) + ( h . x0 - h . x0 ) .= ( F . x0 - h . x0 ) + ( F . x0 - h . x0 ) ;
cell ( G , ( X -' 1 ) + ( Y -' 1 ) , ( X + 1 ) ) \ L~ f meets ( L~ f ) \/ ( L~ f ) & ( L~ f ) \/ ( L~ f ) = ( L~ f \/ L~ f ) \/ ( L~ f ) ;
IC Result ( P2 , s2 ) = IC IExec ( I , P , Initialize s ) .= IC IExec ( I , P , Initialize s ) .= ( card I + 1 ) .= ( card I + 1 ) .= ( card I + 1 ) .= ( card I + 1 ) ;
sqrt ( ( - ( ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ) ^2 ) ) > 0 ;
consider x0 being element such that x0 in dom a & x0 in g " { k `2 } and y0 = a . x0 and a . x0 = b . x0 and b . x0 = a . x0 and a . x0 = b . x0 ;
dom ( r1 (#) chi ( A , A ) ) = dom chi ( A , A ) .= dom ( chi ( A , A ) ) /\ dom ( chi ( A , A ) ) .= dom ( ( r1 (#) chi ( A , A ) ) | A ) .= dom ( ( r1 (#) chi ( A , A ) ) | A ) .= dom ( ( r1 (#) chi ( A , A ) ) | A ) .= dom ( ( r1 (#) chi ( A , A ) ) ;
d-7 . [ y , z ] = ( y `2 ) - ( y `2 ) * ( y `2 ) .= ( y `2 ) - ( y `2 ) * ( y `2 ) .= ( y `2 ) - ( y `2 ) * ( y `2 ) ;
pred for i being Nat holds C . i = A . i /\ B . i means : Def1 : C . i c= C . i /\ C . i ;
assume that x0 in dom f and f is_continuous_in x0 and ||. f .|| is_continuous_in x0 and ||. f .|| is_continuous_in x0 and ||. f .|| is_continuous_in x0 and ||. f .|| is_continuous_in x0 and ||. f .|| is_continuous_in x0 and ||. f .|| is_continuous_in x0 ;
p in Cl A implies for K being Basis of p , Q being Subset of T st Q in K & A is open & Q is open holds A meets Q
for x being Element of REAL n st x in Line ( x1 , x2 ) holds |. y1 - y2 .| <= |. y1 - y2 .| & |. y1 - y2 .| <= |. y1 - y2 .|
func /. <*> a -> Ordinal means : Def1 : a in it & for b being Ordinal st a in b holds it . b c= b & it . b = 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 ) & [ a1 , a2 , a3 ] in ( the carrier of A ) \/ ( the carrier of A ) ;
ex a , b being element st a in the carrier of S1 & b in the carrier of S2 & x = [ a , b ] & x = [ a , b ] ;
||. ( vseq . n ) - ( vseq . m ) .|| * ||. x .|| < ( e / ||. ( ||. x .|| + ||. y .|| ) * ||. x .|| ) * ||. x .|| ;
then for Z being set st Z in { Y where Y is Element of I7 : F c= Y & Y in Z } holds z in Z & z in Z ;
sup compactbelow [ s , t ] = [ sup ( { s } . ( s , t ) ) , sup ( { s } . ( s , t ) ) ] .= [ sup ( { s } . ( s , t ) ) , t ] ;
consider i , j being Element of NAT such that i < j and [ y , f . j ] in IX and [ f . i , z ] in IX and [ f . i , f . j ] in IX ;
for D being non empty set , p , q being FinSequence of D st p c= q holds ex p being FinSequence of D st p ^ q = q & p ^ q = q ^ p
consider e39 being Element of the carrier of X such that c9 , a9 // a9 , e39 and a9 <> c9 and a9 <> b9 and b9 <> c9 and a9 <> c9 and a9 <> b9 and b9 <> c9 and a9 <> c9 and a9 <> b9 and a9 <> c9 & b9 <> c9 ;
set U2 = I \! \mathop { - } , U2 = I \! \mathop { - } , E = I \! \mathop { - } , F = S S S S S S S S S S S S S ;
|. q3 .| ^2 = ( |. q3 .| ) ^2 + ( |. q2 .| ) ^2 .= |. q .| ^2 + ( |. q2 .| ) ^2 .= |. q .| ^2 + ( |. q2 .| ) ^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 = y /\ x implies x "/\" y = 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 ) & 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 ) | X ) .= dom ( ( ||. h .|| | X ) | X ) .= dom ( ( ||. h .|| | X ) | X ) ;
for N1 , N2 being Element of G8 holds dom ( h . K1 ) = N & rng ( h . K1 ) = N1 & rng ( h . K1 ) = N2 & rng ( h . K1 ) = N1 & rng ( h . K1 ) c= N2
( mod ( u , m ) + mod ( v , m ) ) . i = ( mod ( u , m ) ) . i + ( mod ( v , m ) . i ) .= ( mod ( v , m ) ) . i ;
- ( q `1 ) ^2 / ( |. q .| ) ^2 < - ( q `2 ) ^2 or ( q `2 ) ^2 / ( |. q .| ) ^2 >= - ( q `2 / |. q .| ) ^2 / ( |. q .| ) ^2 ;
pred r1 = ff & r2 = ff & for x st x in dom ff holds r1 * ( x - y ) = ff . x * ( x - y ) ;
vseq . m is bounded Function of X , the carrier of Y & x9 . m = ( vseq . m ) . x & ( vseq . m ) . x = ( vseq . m ) . x ;
pred a <> b & b <> c & b <> c & angle ( a , b , c ) = PI & angle ( b , c , a ) = 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 q1 = q1 ^ q1 and q1 = q1 ^ q2 and q1 = q2 ^ q1 and q1 = q2 ^ q1 and q1 = q2 ^ q1 ;
( 1. ( A , B ) ) . ( r1 , r2 , s1 , s2 , s2 , Amp ) = ( s2 . ( r2 , s2 ) ) * ( r2 , s2 ) .= ( s2 . ( r2 , s2 ) ) * ( r2 , s2 ) ;
( ( LMP A ) `2 ) ^2 = lower_bound ( proj2 .: ( A /\ from ( w + 1 ) ) /\ ( proj2 .: ( A /\ from ( w + 1 ) ) ) ) & proj2 .: ( A /\ from ( w + 1 ) ) is non empty ;
s |= ( H , H1 ) |= H2 iff s |= ( H1 , H2 ) & s |= ( H , H1 ) & ( H , H1 |= H2 iff s |= ( H , H1 ) ) & ( s |= ( H , H2 ) ) ;
len s5 + 1 = card ( support b1 ) + 1 .= card ( support b2 ) + 1 .= card ( support b2 ) + 1 .= card ( support b2 ) + 1 .= len ( b1 + 1 ) + 1 .= len ( b1 + 1 ) + 1 .= len ( b1 + 1 ) + 1 .= len ( b1 + 1 ) ;
consider z being Element of L1 such that z >= x and z >= y and for z being Element of L1 st z >= x & z >= y holds z `1 >= y `1 and z `2 >= y `2 and z `2 >= y `2 ;
LSeg ( UMP D , |[ W-bound D , ( W-bound D ) / 2 ) * ( ( W-bound D ) / 2 ) + ( ( E-bound D ) / 2 ) * ( ( W-bound D ) / 2 ) ) = { UMP D } ;
lim ( ( ( f `| N ) / ( g `| N ) ) /* b ) = lim ( ( f `| N ) / ( g `| N ) ) .= ( ( f `| N ) / ( g `| N ) ) /* b ;
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 . 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 n ) /\ ( dom ( ( #Z n ) * f ) ) \ ( ( #Z n ) * f ) " { 0 } ) & Z c= dom ( ( #Z n ) * f ) \ ( ( #Z n ) * f ) " { 0 } ) ;
ex j being Nat st j in dom ( l ^ <* x *> ) & j < i & y = ( l ^ <* x *> ) . j & i = 1 + len l + j & z = ( l ^ <* x *> ) . j & i = 1 + len l + j ;
for u , v being VECTOR of V , r being Real st 0 < r & r < 1 & u in dom _ N holds r * u + ( 1-r ( V ) ) . v in L~ N
A , Int A , Int Cl A , Cl Int A , Cl Int Cl A , Cl Int Cl A , Cl Int Cl Int Cl A , Cl Int Cl Int Cl A , Cl Int Cl Int Cl A , Cl Int Cl Int Cl A , Cl Int Cl A , Cl Int Cl Int Cl A , Cl Int Cl Int Cl A , Cl Int Cl Int Cl A , Cl Int Cl Int Cl A , Cl Int Cl Cl Int Cl A , Cl Cl Cl Cl Cl Int Cl A , Cl Cl Int Cl A , Cl Int Cl A , Cl Int Cl A , Cl Cl A , Cl Cl A , Cl Int Cl A , Cl
- Sum <* v , u , w *> = - ( v + u + w ) .= - ( v + u ) + w .= - ( v + u ) + u .= - v + u ;
( Exec ( a := b , s ) ) . IC SCM R = ( Exec ( a := b , s ) ) . IC SCM R .= Exec ( ( a := b ) , s ) . IC SCM R .= succ IC s .= succ IC s .= succ IC s .= ( succ IC s ) .= ( succ IC s ) ;
consider h being Function such that f . a = h and dom h = I and for x being element st x in I holds h . x in ( the carrier of J ) . x and h . x in ( the carrier of J ) . x ;
for S1 , S2 , S2 , D being non empty reflexive RelStr , D being non empty Subset of [: S1 , S2 :] holds cos ( D ) is directed & cos ( D ) is directed & cos ( D ) is directed
card X = 2 implies ex x , y st x in X & y in X & x <> y & x <> y or x = y & x = y or x = y & y = x or x = y & y = x or y = x & x = y
E-max L~ Cage ( C , n ) in rng ( Cage ( C , n ) \circlearrowleft W-min L~ Cage ( C , n ) ) & W-min L~ Cage ( C , n ) in rng ( Cage ( C , n ) \circlearrowleft W-min L~ Cage ( C , n ) ) ;
for T , t being tree , p being Element of dom T , q being Element of dom T st p element q in dom T holds ( T -tree ( p , T ) ) . 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 ) & ( k divides ( k gcd n ) ) & ( k divides ( k gcd n ) & ( k divides ( k gcd n ) ) & ( k divides ( k gcd n ) ) & ( k divides ( k gcd n ) ) implies k divides n ) ;
dom F " = the carrier of X2 & rng F " = the carrier of X1 & F " = the carrier of X2 & F " = the carrier of X2 & F " = the carrier of X2 & F " = the carrier of X1 & F " = 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 ( BM \/ C ) and C = Lin ( BM \/ C ) and C = Lin ( BM \/ B ) ;
V is prime implies for X , Y being Element of [: the topology of T , the topology of T :] st X /\ Y c= V & X c= Y holds X c= Y or Y c= V
set X = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } , Y = { F ( v1 ) : P [ v1 ] } , Z = { F ( v1 ) : P [ v1 ] } ;
angle ( p1 , p3 , p4 ) = 0 .= angle ( p2 , p3 , p4 ) .= angle ( p2 , p3 , p4 ) .= angle ( p , p3 , p2 ) .= angle ( p , p3 , p2 ) .= angle ( p , p3 , p2 ) ;
- sqrt ( ( - ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 ) = - sqrt ( ( - ( ( q `2 / |. q .| - cn ) / ( 1 + cn ) ) ^2 ) ) .= - ( ( - ( q `2 / |. q .| - cn ) / ( 1 + cn ) ) ) .= - ( - ( q `2 / |. q .| - cn ) ) ;
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 . 0 = p2 & f . 1 = p3 & f . 1 = p4 & f . - 1 = p4 ;
attr f is partial differentiable of REAL , u0 means : Def1 : SVF1 ( 2 , pdiff1 ( f , 1 ) , u0 ) . u = SVF1 ( 2 , pdiff1 ( f , 1 ) , u0 ) . u ;
ex r , s st x = |[ r , s ]| & G * ( len G , 1 ) `1 < r & r < G * ( 1 , 1 ) `2 & s < G * ( 1 , 1 ) `2 ;
assume that f is_sequence_on G and 1 <= t and t <= len G and G * ( t , width G ) `2 >= N-bound L~ f and G * ( t , width G ) `2 >= N-bound L~ f and f /. 1 = ( f /. len f ) `2 and f /. len f = ( f /. len f ) `2 ;
pred i in dom G means : Def1 : r (#) ( f * reproj ( i , x ) ) = r (#) f * reproj ( i , x ) & for i st i in dom G holds f . i = r * reproj ( i , x ) ;
consider c1 , c2 being bag of o1 + o2 such that ( decomp c ) /. k = <* c1 , c2 *> and c = ( decomp c1 ) /. k and ( decomp c ) /. k = ( decomp c1 ) . ( c1 + c2 ) and ( decomp c ) /. k = ( decomp c1 ) . ( c2 + c1 ) ;
u0 in { |[ r1 , s1 ]| : r1 < G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `2 < s1 & s1 < G * ( 1 , 1 ) `2 } ;
( X ^ Y ) . k = the carrier of X . k2 .= ( C . k2 ) . k2 .= ( C . k2 ) . k2 .= ( C . k2 ) . k2 .= ( C . k2 ) . k2 .= ( C . k2 ) . k2 .= ( C . k2 ) . k2 ;
pred len M1 = len M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & len M1 = width M2 ;
consider g2 being Real such that 0 < g2 and { y where y is Point of S : ||. y - x0 .|| < g2 & y in dom f & f . y = ( f | N ) . ( y - x0 ) + g2 . ( y - x0 ) ;
assume x < ( - b + sqrt ( delta ( a , b , c ) ) ) / ( 2 * a ) or x > ( - b - sqrt ( delta ( a , b , c ) ) / ( 2 * a ) ) ;
( 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 ) + M1 * ( i , j )
for f being FinSequence of NAT , i being Element of NAT st i in dom f & for j being Element of NAT st j in dom f & j <> i holds i divides f /. j holds i divides Sum f & f /. j = Sum f
assume F = { [ a , b ] where a , b is set : for c being set st c in B39 & c in B39 & a c= c holds b c= c } & F c= c & a c= c & b c= c ;
b2 * q2 + ( b3 * q3 ) + ( - ( a * q2 ) + ( - ( a * q3 ) ) ) = 0. TOP-REAL n + ( - ( a * q2 ) + ( a * q3 ) ) .= 0. TOP-REAL n + ( a * q2 ) .= ( a * ( - a * q2 ) ) + ( a * ( - a * q2 ) ) .= 0. TOP-REAL n + ( a * ( - a * q2 ) ) .= 0. TOP-REAL n ;
Cl ( Cl F ) = { D where D is Subset of T : ex B being Subset of T st D = Cl B & B in F & Cl ( Cl ( Cl F ) ) = Cl ( Cl ( Cl ( Cl F ) ) ) & Cl ( Cl ( Cl ( Cl F ) ) ) = Cl ( Cl ( Cl ( Cl F ) ) ) ) ;
attr seq is summable means : Def1 : seq is summable & seq is summable & Partial_Sums ( seq ) is summable & Partial_Sums ( seq ) is summable & Partial_Sums ( seq ) is convergent & Partial_Sums ( seq ) is convergent & Partial_Sums ( seq ) is convergent & lim ( seq ) = Sum ( seq ) + Partial_Sums ( seq ) ;
dom ( ( ( ( cn max ) | D ) | D ) ) = ( the carrier of ( ( TOP-REAL 2 ) | D ) ) /\ D .= the carrier of ( ( TOP-REAL 2 ) | D ) .= the carrier of ( ( TOP-REAL 2 ) | D ) .= the carrier of ( ( TOP-REAL 2 ) | D ) ;
X is full full full full full full SubRelStr of ( [#] Z ) |^ the carrier of Z & [ X \to Y ] is full full SubRelStr of ( [#] Z ) |^ the carrier of Z ;
G * ( 1 , j ) `2 = G * ( i , j ) `2 & G * ( 1 , j ) `2 <= G * ( 1 , j ) `2 & G * ( 1 , j ) `2 <= G * ( 1 , j + 1 ) `2 ;
synonym m1 c= m2 means : Def1 : for p being set st p in P holds the } of p in the carrier of ( m + 1 ) & the carrier of ( m + 1 ) in the carrier of ( m + 1 ) -tuples_on the carrier of ( m + 1 ) -tuples_on the carrier of L ;
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 mode multiplicative loop over R means : : : : the carrier of it is multiplicative loop of R means : Def1 : the carrier of it = the carrier of R & the multF of it = the multF of R & the multF of it = the multF of R ;
L ( a , b , 1 ) + L ( c , d ) = b + L ( c , d ) .= b + L ( c , d ) .= b + ( c + d ) .= the carrier of L ( a + c , b + d ) ;
cluster + _ ( \mathbb Z ) -> } means : Def1 : for i1 , i2 being Element of INT holds it . ( i1 , i2 ) = + ( i1 , i2 ) & it . ( i1 , i2 ) = + ( i2 , i1 ) ;
( ( - s2 ) * p1 + ( s2 * p2 - ( s2 * p2 - ( s2 * p2 ) ) ) * p1 + ( s2 * p2 - ( s2 * p2 ) ) * p1 - ( s2 * p2 - ( s2 * p2 ) ) * p1 ) = ( ( - s2 ) * p1 + ( s2 * p2 ) ) * p1 + ( s2 * p2 - ( s2 * p2 ) ) * p1 - ( s2 * p2 - ( s2 * p2 ) ) * p1 + ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - (
eval ( ( a | ( n , L ) ) *' p , x ) = eval ( a | ( n , L ) , x ) * eval ( p , x ) .= eval ( a | ( n , L ) , x ) * eval ( p , x ) .= eval ( a , x ) ;
assume that the TopStruct of S = the TopStruct of T and for D being non empty directed Subset of S , V being Subset of Omega S st V in D & V is open & V is open & for V being Subset of S st V in V holds V is open & V is open & V is open & V is open & V is open & V is open & V is open ;
assume that 1 <= k & k <= len w + 1 and T-7 . ( ( len ( q , w ) -O ) . k ) = ( T-7 . ( len ( q , w ) -O ) ) . k and T-7 . ( ( len ( q , w ) -O ) . k ) = ( T-7 . ( len ( q , w ) -O ) ) . k ;
2 * a |^ ( n + 1 ) + ( 2 * b |^ ( n + 1 ) ) >= a |^ n + ( b |^ n ) + ( b |^ n ) + ( b |^ n ) + ( b |^ n ) + ( a |^ n ) + ( b |^ n ) + ( b |^ n ) + ( b |^ n ) ;
M , v2 |= All ( x. 3 , All ( x. 0 , All ( x. 4 , All ( x. 0 , All ( x. 4 , H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H ( x. 0 ) ) ) ) ) ;
assume that f is_differentiable_on l and for x0 st x0 in l holds 0 < f . x0 & for x0 st x0 in l holds 0 < f . x0 & f . x0 < 0 ;
for G1 being _Graph , W being Walk of G1 , e being set , W being Walk of G1 , e being Vertex of G2 , x being set st e in W & not e in W holds W is Walk of G1 & W is Walk of G2
not c9 is not empty iff iff iff not ( y1 is not empty & not y2 is not empty & not y2 is not empty & not y1 is not empty & not y2 is not empty & not y2 is not empty & not q1 is not empty & not q1 is not empty & not q2 is not empty ) & not q1 is not empty & not q2 is not empty & not q2 is not empty & not q2 is not empty ;
Indices GoB f = [: dom GoB f , Seg width GoB f :] & i1 + 1 in dom GoB f & i2 + 1 in Seg width GoB f & j + 1 in Seg width GoB f & f /. ( i1 + 1 ) = GoB f & f /. ( i1 + 1 ) = GoB f & f /. ( i1 + 1 ) = GoB f ;
for G1 , G2 , G3 being Group , O being Subgroup of G1 , G2 being Subgroup of G2 , G1 being Subgroup of G2 st G1 is stable & G2 is stable & G1 is stable & G2 is stable holds G1 * G2 is stable Subgroup of G2 * & G1 * G2 is stable Subgroup of G1 * & G2 * G1 is stable Subgroup of G2 * G1
UsedIntLoc ( in4 ( f , intloc 0 ) ) = { intloc 0 , intloc 1 , intloc 0 , intloc 0 , intloc 0 , intloc 0 , intloc 0 , intloc 0 , intloc 0 , intloc 0 , intloc 0 , intloc 0 , intloc 0 , 1 , 1 ) , 1 } \/ UsedIntLoc ( in4 ( f , intloc 0 , intloc 0 , intloc 0 , 1 ) ) ;
for f1 , f2 being FinSequence of F st f1 ^ f2 is p -element & Q [ f1 ^ f2 ] & Q [ f1 ^ f2 ] & Q [ f1 ^ f2 ] holds Q [ f1 ^ f2 ]
( 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 ) , ( x2 - x3 ) )| = |( x1 , x2 - x3 )| & |( x1 - x2 , x2 - x3 )| = |( x2 , x3 )| + |( x2 , x3 )| + |( x3 , x2 - x3 )| + |( x2 , x3 )| + |( x3 , x3 )|
for x st x in dom ( ( F | A ) | A ) holds ( ( ( F | A ) | A ) . ( - x ) = - ( ( F | A ) . x ) * ( - ( F | A ) )
for T being non empty TopSpace , P being Subset-Family of T , x being Point of T , B being Subset of T st P c= the topology of T & x in P holds ex B being Basis of T st B c= P & B is Basis of x
( a 'or' b 'imp' c ) . x = 'not' ( ( a 'or' b ) . x ) 'or' c . x .= 'not' ( ( a . x ) 'or' b . x ) 'or' c . x .= TRUE 'or' TRUE .= TRUE 'or' TRUE .= TRUE ;
for e being set st e in [: A , Y :] ex X1 being Subset of [: X , Y :] , Y1 being Subset of Y st e = [: X1 , Y1 :] & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open holds Y1 is open
for i be set st i in the carrier of S for f be Function of [: S . i , S1 . i :] , S1 . i st f = H . i & F . i = f | ( the carrier of S1 ) holds F . i = f | ( the carrier of S2 )
for v , w st for y st x <> y holds w . y = v . y holds Valid ( VERUM ( Al , J ) , J ) . v = Valid ( VERUM ( Al , J ) , J ) . v
card D = card D1 + card D2 - card { i , j } .= ( c1 + c2 ) + ( { i , j } - 1 ) .= c1 + ( c2 + ( { j } - 1 ) ) - 1 .= c1 + ( c2 + ( { j } - 1 ) ) - 1 .= 2 * c1 + ( c2 + ( { j } - 1 ) ) - 1 .= 2 * c1 + ( c2 + ( { j } - 1 ) ) ;
IC Exec ( i , s ) = ( s +* ( 0 .--> succ ( 0 , s ) ) ) . 0 .= ( 0 .--> succ ( 0 , s ) ) . 0 .= ( 0 .--> succ ( 0 , s ) ) . 0 .= succ IC s .= succ IC s .= succ IC s .= ( IC s ) ;
len f /. ( len f -' 1 ) + 1 = len f -' 1 + 1 .= len f -' 1 + 1 .= len f -' 1 + 1 .= len f -' 1 + 1 .= len f -' 1 + 1 .= len f -' 1 + 1 .= len f -' 1 + 1 .= len f -' 1 + 1 ;
for a , b , c being Element of NAT st 1 <= a & 2 <= b & k <= len b holds a <= b + b-2 or a = b + b-2 or b = a + b-2 or b = a + b-2 or b = a + b-2 or b = a + b-2 or b = a + 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 ) & i <= len f & i <= len f & f /. i = f /. ( i + 1 ) holds Index ( p , f ) <= Index ( p , f )
( curry curry ( P+* ( k , n + 1 ) ) ) # x = ( ( curry curry ( F+* ( k , n ) ) ) # x ) + ( ( curry curry ( F+* ( k , n + 1 ) ) # x ) ) . x ;
z2 = g /. ( len g -' ( n1 + 1 ) + 1 ) .= g . ( i -' ( n2 + 1 ) + 1 ) .= g . ( i -' ( n2 + 1 ) + 1 ) .= g . ( i -' ( n2 + 1 ) + 1 ) .= g . ( i -' ( n2 + 1 ) + 1 ) .= g . ( i + ( n2 + 1 ) + 1 ) ;
[ f . 0 , f . 3 ] in id ( the carrier of G ) \/ ( the InternalRel of G ) or [ f . 0 , f . 3 ] in the InternalRel of C6 & [ f . 0 , f . 3 ] in the InternalRel of C6 ;
for G being Subset-Family of B st G = { R [ X ] where R is Subset of A , B is Subset of B : R in F6 & R in F6 } holds ( Intersect ( F , B ) ) . [ X , B ] = Intersect ( G , B )
CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 ) ) = CurInstr ( P1 , Comput ( P1 , s1 , m1 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m1 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m1 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m1 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m1 ) ) .= halt SCMPDS ;
assume that a on M and b on M and c on N and d on N and p on M and p on N and a on M and c on N and p on M and a on N and c on N and p on M and c on N and p on N and p on M and a on N and c on M and p on N and p on M and a on N and b on N and c on N and p on N and a on M and b on N and c on N and c on N and p on N and p on M and p on N and p on N and p on N and p on M and p on M and p on N and p on M and p on N and p on N and p on M and p on N and p on N and p on N and p on N and p on M and a on M and a on M and a on M
assume that T is \hbox { T _ 4 } and F is closed of T and ex F being Subset-Family of T st F is closed & F is finite-ind & ind F <= 0 & ind F <= 0 and ind T <= 0 and ind T <= 0 ;
for g1 , g2 st g1 in ]. ( r - g ) / 2 , r + g .[ & |. f . g1 - f . g2 .| <= ( g1 - g ) / 2 holds |. ( f - g ) / 2 .| <= ( f - g ) / 2
( ( - ( 1 / 2 ) ) * ( z + z2 ) ) = ( ( - ( 1 / 2 ) ) * ( z + z2 ) ) * ( z + w ) .= ( ( - ( 1 / 2 ) ) * ( z + w ) ) * ( z + w ) ;
F . i = F /. i .= 0. R + r2 .= ( b |^ n + r2 ) |^ n .= <* ( n + 1 ) |^ 0 , ( n + 1 ) |^ ( n + 1 ) , \dots , ( n + 1 ) |^ n , ( n + 1 ) |^ 0 , ( n + 1 ) |^ n , ( n + 1 ) |^ 0 *> ;
ex y being set , f being Function st y = f . n & dom f = NAT & f . 0 = A ( ) & for n holds f . ( n + 1 ) = R ( n , f . n ) & f . ( n + 1 ) = R ( ) . ( n + 1 ) .= f . ( n + 1 ) ;
func f (#) F -> FinSequence of V means : Def1 : len it = len F & for i be Nat st i in dom it holds it . i = F /. i * F /. i & for i be Nat st i in dom it holds it . i = f /. i * F /. i ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 } = { x1 , x2 , x3 , x4 } \/ { x2 , x3 , x4 , x5 , x5 , x5 } \/ { x5 , x5 , x5 , x5 , x5 } \/ { x5 , x5 , x5 , x5 , x5 } \/ { x5 , x5 , x5 , x5 , x5 } \/ { x5 , x5 , x5 } \/ { x5 , x5 , x5 , x5 , x5 , x5 } \/ { x5 , x5
for n being Nat for x being set st x = h . n holds h . ( n + 1 ) = o ( x , n ) & x in InputVertices S ( x , n ) & o . ( x , n ) in InnerVertices S ( x , n ) & o . ( x , n ) in InnerVertices S ( x , n )
ex S1 being Element of CQC-WFF ( Al ) st SubP ( P , l , e ) = S1 & ( S , e ) `1 = e & ( S , e ) `2 = e & ( S , e ) `2 = e & ( S , e ) `2 = e & ( S , e ) `2 = e & ( S , e ) `2 = e ) ;
consider P being FinSequence of G8 such that p8 = Product P and for i st i in dom P ex t being Element of the carrier of G st P . i = t & t is Element of the carrier of G & t is i , t is Element of the carrier of G & t is i , t is Element of the carrier of G ;
for T1 , T2 being non empty TopSpace , P being Basis of T1 , Q being Basis of T2 st the carrier of T1 = the carrier of T2 & P = the carrier of T2 & P = the topology of T2 & P = the topology of T1 & Q = the topology of T2 holds P is Basis of T1 & Q is Basis of T2
assume that f is_is_\cal U and r (#) pdiff1 ( f , 3 ) is_partial_differentiable_in u0 , 2 and partdiff ( r (#) pdiff1 ( f , 3 ) , u0 , 2 ) = r (#) pdiff1 ( f , 3 ) . u0 and partdiff ( r (#) pdiff1 ( f , 3 ) , u0 , 2 ) = r * pdiff1 ( f , u0 ) . u0 ;
defpred P [ Nat ] means for F , G being FinSequence of bool ( the carrier of V ) , G be Permutation of V st len F = $1 & G = F * s & not G = F * s & not F = G * s & not G = F * s & not G in rng F & not G in rng F & not G in rng F & not G in rng F & not F in rng F ;
ex j st 1 <= j & j < width GoB f & ( GoB f ) * ( 1 , j ) `2 <= s & s <= ( GoB f ) * ( 1 , j + 1 ) `2 & ( GoB f ) * ( 1 , j + 1 ) `2 <= s `2 & s <= ( GoB f ) * ( 1 , j + 1 ) `2 ;
defpred U [ set , set ] means ex F-23 be Subset-Family of T st $2 = F-23 & $2 is open & union F\times F-23 is open & union F\times F\times F-23 is Subset-Family of T & union F\times F\times F\times F\times Fmax ( F , G ) = union the topology of T & union F\times F\times F = union the topology of T & union F\times F = union the topology of T ;
for p4 being Point of TOP-REAL 2 st LE p4 , p4 , P , p1 , p2 & LE p4 , p , P , p1 , p2 & LE p4 , p , P , p1 , p2 & LE p , p1 , P , p1 , p2 & LE p , p1 , P , p1 , p2 & LE p , p1 , P , p2 & LE p , p1 , P , p1 , p2 & LE p , p1 , P , p2 , p1 , p2 & LE p , p2 , P , p1 , p2 , p2 , p2 , p2 & LE p1 , p2 , P , p1 , p2 , p2 , p2 , p2 , p2 , p2 , p2 , p2 , P , p1 , p2 , p2 & LE p2 , p2 , P , p1 , p2 & LE p2 , p2 , P , p1 , p2 & LE p1 , p2 , P , p1 , p2 & LE p1 , p2 , P , p1 , p2 , p1 ,
f in D ( ) & for g st g in D ( ) & ( for y st y <> x holds g . y = f ( y ) ) holds x in Free ( H ( ) ) & f . y = f ( y ) ) implies f in Free ( H ( ) ) & f . ( x , y ) = f . ( x , y )
ex 8 being Point of TOP-REAL 2 st x = 8 & ( ( ( - 1 ) * ( |. 8 .| ) ) / ( 1 + ( |. 8 .| ) ) ^2 ) >= 8 & ( ( - 1 ) * ( |. 8 .| ) ) / ( 1 + ( |. 8 .| ) ^2 ) ) >= 0 ;
assume for d7 being Element of NAT st d7 <= d7 holds ( ( ( n + 1 ) -d7 ) . d7 = ( ( n + 1 ) -d7 ) . d7 ) & ( ( n + 1 ) -d7 = ( ( n + 1 ) -d7 ) . d7 ) & ( ( n + 1 ) -d7 = ( n + 1 ) -d8 ) . 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 E st { e } = Ball ( x , r ) /\ Sphere ( y , r ) and e = Ball ( x , r ) /\ Sphere ( y , r ) ;
given r such that 0 < r and for s st 0 < s holds 0 < s or ex x1 be Point of CNS st x1 in dom f & ||. x1 - x0 .|| < s & |. f /. x1 - f /. x0 .|| < r & |. f /. x0 - f /. x0 .| < r ;
( p | x ) | ( p | ( x | x ) ) = ( ( x | x ) | ( x | x ) ) | ( ( x | x ) | p ) .= ( ( x | x ) | p ) | p ;
assume that x , x + h / 2 in dom sec and ( for x st x in dom sec holds sec . x = ( 4 * sec . x + h / 2 ) * sin . x ) / ( cos . x + h / 2 ) * sin . x ) ^2 and sin | A is bounded ;
assume that i in dom A and len A > 1 and for B st B > 1 & B c= dom A & A . i = B holds ( A * B ) * ( i , j ) = ( A * ( i , j ) ) * ( B * ( i , j ) ) ;
for i be non zero Element of NAT st i in Seg n holds ( i divides n or i = n or i = n ) & ( i <> n & i <> n implies h . i = 1. F_Complex ) & ( i divides n implies h . i = 1. F_Complex ) & ( i divides n implies h . i = 1. F_Complex )
( ( b1 'imp' b2 ) '&' ( c1 'imp' c2 ) '&' ( ( b1 'or' b2 ) '&' ( c1 'or' c2 ) '&' ( a1 'or' b1 ) '&' ( b1 'or' b2 ) '&' ( a1 'or' b1 ) '&' ( b1 'or' b2 ) '&' ( a1 'or' b1 ) '&' ( b1 'or' b2 ) '&' ( a1 'or' b1 ) '&' ( b1 'or' b2 ) '&' ( a1 'or' b1 ) '&' ( b1 'or' b2 ) '&' ( b1 'or' b2 ) '&' ( b1 'or' b2 ) '&' ( a1 'or' b2 ) '&' ( a1 'or' b2 ) '&' ( b1 'or' b2 ) '&' ( b1 'or' b2 ) '&' ( a1 'or' b2 ) '&' ( a1 'or' b2 ) '&' ( b1 'or' b2 ) '&' ( b1 'or' b2 ) '&' ( b1 'or' b2 ) '&' ( b1 'or' b2 ) '&' ( b1 'or' b2 ) '&' ( a1 'or' b2 ) '&' ( a1 'or' b2 ) '&' ( a1 'or' b2 ) '&' ( a1 'or' b2 ) '&' ( a1 'or' b2 ) '&'
assume that for x holds f . x = ( ( cot * cot ) `| Z ) . x and x in dom ( cot * cot ) and for x st x in Z holds ( ( cot * cot ) `| Z ) . x = cos . ( x - x0 ) and ( ( cot * cot ) `| Z ) . x = cos . ( x - x0 ) ;
consider R8 , I-8 being Real such that R8 = Integral ( M , Re ( F . n ) ) and I8 = Integral ( M , Im ( F . n ) ) and Integral ( M , Im ( F . n ) ) = Integral ( M , Im ( F . n ) ) + Integral ( M , Im ( F . n ) ) ;
ex k being Element of NAT st k = k & 0 < d & for q be Element of product G st q in X & ||. q- f /. x .|| < d holds ||. partdiff ( f , q , k ) - partdiff ( f , x , k ) .|| < r ;
x in { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 } \/ { x5 , x5 , x5 } \/ { x5 , x5 } \/ { x5 , x5 , x5 } \/ { x5 , x5 , x5 , x5 , x5 } \/ { A , x5 , x5 , x5 } \/ { A , x2 } \/ { x5 , x5 , x5 } \/ { x5 , x5 } \/ { x5 , x5 } \/ { A , x2 } \/ { A , x2 } \/ { x5 , x5 } \/ { x5 , x5 } \/ { x5 , x5 } \/ { x5 , x5 } \/ { x5 , x5 }
G * ( j , ii ) `2 = G * ( 1 , ii ) `2 .= G * ( 1 , ii ) `2 .= G * ( 1 , jj ) `2 .= G * ( 1 , jj ) `2 .= G * ( 1 , jj ) `2 .= G * ( 1 , jj ) `2 .= G * ( 1 , jj ) `2 .= G * ( 1 , jj ) `2 .= G * ( 1 , jj ) `2 .= G * ( 1 , jj ) `2 .= G * ( 1 , jj ) `2 .= G * ( 1 , jj ) `2 .= G * ( 1 , jj ) `2 .= G * ( 1 , jj ) `2 .= G * ( 1 , jj ) `2 .= G * ( 1 , jj ) `2 .= G * ( 1 , jj ) `2 .= G * ( 1 , jj ) `2 .= G * ( 1 , jj ) `2 .= G * ( 1 , jj ) `2 .= G * ( 1 , jj
f1 * p = p .= ( the Arity of S1 ) +* ( the Arity of S2 ) . o .= ( the Arity of S1 ) +* ( the Arity of S2 ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o ;
func tree ( T , P , T1 ) -> FinSequence of T means : Def1 : q in it iff q in T & for p st p in P holds p in T or ex p st p in P & p in T1 & q = T . p or p in T1 & q in T1 & p in T1 or p in T1 & q in T1 & p in T1 & q in T1 or p in T1 & q in T1 & p in T1 & q in T1 ;
F /. ( k + 1 ) = F . ( k + 1 -' 1 ) .= F\circ ( p . ( k + 1 -' 1 ) , k + 1 -' 1 ) .= F9 . ( p . ( k + 1 -' 1 ) , k + 1 -' 1 ) .= F9 . ( p . ( k + 1 -' 1 ) , k + 1 ) .= F9 . ( p . ( k + 1 -' 1 ) , k + 1 ) .= F9 . ( p . ( k + 1 ) ;
for A , B , C being Matrix of K st len B = len C & width B = width C & width B = width C & len B = width C & len B > 0 & len A > 0 & len B > 0 & width A = width B & len B > 0 & width B = width C holds A * ( B * C ) = B * ( C * B )
seq . ( k + 1 ) = 0. 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 ) .= ( 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 [ x , y ] in the InternalRel of CQ and [ y , x ] in the InternalRel of CQ ;
defpred P [ Element of NAT ] means for f st len f = $1 & ( for k st k in dom f holds ( for i st i in dom f holds f . ( f . ( k + 1 ) ) = ( VAL g ) . ( f . ( k + 1 ) ) ) '&' ( ( VAL g ) . ( f . ( k + 1 ) ) ) ;
assume that 1 <= k and k + 1 <= len f and f is_sequence_on G and [ i , j ] in Indices G and [ i + 1 , j ] in Indices G and f /. k = G * ( i , j ) and f /. ( k + 1 ) = G * ( i , j ) and f /. ( k + 1 ) = G * ( i + 1 , j ) ;
assume that sn < 1 and q `1 > 0 and ( q `2 / |. q .| - sn ) / ( 1 + sn ) >= sn and ( q `2 / |. q .| - sn ) / ( 1 + sn ) >= sn and ( q `2 / |. q .| - sn ) / ( 1 + sn ) >= sn and ( q `2 / |. q .| - sn ) / ( 1 + sn ) >= sn ;
for M being non empty dist , x being Point of M , f being Function of M , M st x = x `1 holds ex x being Point of M st f . x = Ball ( x `1 , 1 / ( 2 |^ n ) ) & for n being Element of NAT holds f . n = Ball ( x `1 , 1 / ( 2 |^ 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 . x = 1 / ( x + 1 / ( x + 1 ) ) ) & ( f1 - f2 ) `| Z = ( f1 `| Z ) . x - ( f2 `| Z ) . x ) ;
defpred P1 [ Nat , Point of CNS ] means $2 in Y & ||. f /. $2 - f /. x0 .|| < r & $2 in Y & ||. f /. $2 - f /. x0 .|| < r & ||. f /. $2 - f /. x0 .|| < r ;
( f ^ mid ( g , 2 , len g ) ) . i = ( mid ( g , 2 , len g ) ) . ( i -' len f + 1 ) .= g . ( i -' len f + 1 ) .= g . ( i -' len f + 1 ) .= g . ( i -' len f + 1 ) .= g . ( i -' len f + 1 ) .= g . ( i -' len f + 1 ) ;
( 1 / 2 * n0 + 2 * ( 2 * n0 + 2 * ( 2 * n0 + 1 ) ) ) * ( 2 * n0 + 2 * ( 2 * n0 + 1 ) ) = ( 1 / 2 * ( 2 * n0 + 1 ) ) * ( 2 * n0 + 1 ) .= 1 / 2 * ( 2 * n0 + 1 ) * ( 2 * n0 + 1 ) .= 1 / 2 * ( 2 * n0 + 1 ) ;
defpred P [ Nat ] means for G being non empty finite strict finite RelStr , R being strict symmetric RelStr st G is space & card the carrier of G = $1 & the carrier of R = the carrier of G & the carrier of R = the carrier of R & the InternalRel of R = the InternalRel of R & the InternalRel of R = the InternalRel of R & the InternalRel of R = the InternalRel of R ;
not f /. 1 in Ball ( u , r ) & 1 <= m & m <= len ( - 1 ) & not 1 <= m & m <= len ( - 1 ) & LSeg ( f , i ) /\ Ball ( u , r ) <> {} & not m in Ball ( u , r ) & not m in Ball ( - 1 ) & not m in Ball ( - 1 , r ) & not m in Ball ( - 1 , r ) ;
defpred P [ Element of NAT ] means ( Partial_Sums ( cos ( $1 ) ) ) . ( 2 * $1 ) = ( cos ( $1 ) ) . ( 2 * $1 ) & ( Partial_Sums ( cos ( $1 ) ) ) . ( 2 * $1 + 1 ) = ( cos ( $1 ) ) . ( 2 * $1 + 1 ) ;
for x being Element of product F holds x is FinSequence of G & dom x = I & x . i = ( the carrier of F ) . i & for i being set st i in dom F holds x . i = ( the carrier of F ) . i & x . i = ( the carrier of F ) . i & x . i = ( the carrier of F ) . i
( x " ) |^ ( n + 1 ) = ( x " ) |^ n * x .= ( x |^ n ) |^ n * x .= ( x |^ n ) |^ n * x .= ( x |^ n ) |^ n * x .= ( x |^ n ) |^ n * x .= ( x |^ n ) |^ n .= ( x |^ n ) |^ n ;
DataPart Comput ( P +* ( a , I ) , LifeSpan ( P +* I , Initialized s ) + 3 ) = DataPart Comput ( P +* I , Initialize s , LifeSpan ( P +* I , LifeSpan ( P +* I , s ) + 3 ) ) .= DataPart Comput ( P +* I , s , LifeSpan ( P +* I , s ) + 3 ) .= DataPart Comput ( P +* I , s , LifeSpan ( P +* I , s ) + 3 ) ;
given r such that 0 < r and ]. x0 , x0 + r .[ c= dom ( f1 + f2 ) /\ dom ( f2 + f1 ) and for g st g in ]. x0 , x0 + r .[ /\ dom ( f1 + f2 ) and g in ]. x0 , x0 + r .[ /\ dom ( f1 + f2 ) and for g st g in ]. x0 , x0 + r .[ /\ dom ( f1 + f2 ) holds f1 . g <= f2 . g ;
assume that X c= dom f1 /\ dom f2 and f1 | X is continuous and f2 | X is continuous and ( f1 - f2 ) | X is continuous and ( f1 - f2 ) | X is continuous and ( f1 - f2 ) | X is continuous and ( f1 - f2 ) | X is continuous and ( 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 Subset of L & for x being Element of L st x in X holds x is directed & x is complete & x is complete & x is complete & x is complete holds x = sup ( waybelow l /\ waybelow ( x , L ) )
Support e8 in { m *' p where m is Nat : i in dom ( m *' p ) & ex i being Nat st i in dom ( m *' p ) & ex p being Polynomial of n , L st p in Support ( m *' p ) & i <= len ( m *' p ) & p . i = ( m *' p ) . i ;
( f1 - f2 ) /* s1 = lim ( f1 /* s1 ) - lim ( f2 /* s1 ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) .= lim ( ( f1 - f2 ) /* s1 ) .= lim ( ( f1 - f2 ) /* s1 ) ;
ex p1 being Element of CQC-WFF ( Al ) st F . p1 = g `1 & for g being Function of [: CQC-WFF ( Al ) , D ( ) :] , D ( ) st P [ g , ( len F ) --> 1 ] holds P [ g , ( len F ) --> 1 ] ;
( mid ( f , i , len f -' 1 ) ^ <* f /. ( len f -' 1 ) *> ) /. j = ( mid ( f , i , len f -' 1 ) ) . j .= ( mid ( f , i , len f -' 1 ) ) . j .= ( mid ( f , i , len f -' 1 ) ) . j .= ( mid ( f , i , len f -' 1 ) ) . j ;
( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) = ( ( p ^ q ) . ( len p + k ) ) . ( len q + k ) .= ( ( p ^ q ) . ( len p + k ) ) . ( len q + k ) .= ( ( p ^ q ) . ( len q + k ) ) . ( len q + k ) .= ( p ^ q ) . ( len p + k ) .= ( p . ( len p + k ) .= ( p . ( len p + k ) .= ( p . ( len p + k ) . ( len p + k ) .= ( p . ( len p + k ) .= ( p . ( len p + k ) .= ( p . ( len p + k ) .= ( p . ( len p + k ) .= ( p . ( len p + k ) + ( q . ( len p + k ) .= ( p . ( len p + k ) .= ( p . ( len p + k ) .= ( p . ( len p + k ) .= ( p . ( len
len mid ( upper_volume ( D2 , D1 ) , indx ( D2 , D1 , j1 ) + 1 , indx ( D2 , D1 , j ) ) = indx ( D2 , D1 , j1 ) - indx ( D2 , D1 , j1 ) + 1 .= indx ( D2 , D1 , j1 ) + 1 + 1 .= indx ( D2 , D1 , j1 ) + 1 ;
x * y * z = MQ . ( x * y , z ) .= x * ( y * z ) .= ( x * ( y * z ) ) * ( x * ( y * z ) ) .= ( x * ( y * z ) ) * ( x * ( y * z ) ) .= x * ( y * z ) .= x * ( y * z ) ;
v . <* x , y *> + ( <* x0 , y0 *> ) . i = partdiff ( v , ( x - x0 ) * ( x - x0 ) + ( y - x0 ) * ( x - x0 ) + ( y - x0 ) * ( x - x0 ) + ( y - x0 ) * ( x - x0 ) ) + ( proj ( 1 , 1 ) * ( x - x0 ) + ( y - x0 ) * ( x - x0 ) ) ;
i * i = <* 0 * ( - 1 ) - ( 0 * 0 ) - ( 0 * 0 ) + 0 * 0 , 0 * 0 + 0 * 0 + 0 * 0 , 0 * 0 + 0 * 0 + 0 * 0 , 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 , 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 * 0 + 0 * 0 , 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 , 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 * 0 * 1 , 0 * 1 * 1 , 0 * 1 , 0 * 1 , 0 * 1 + 0 * 1 * 1 * 1 + 0 * 1 + 0 * 1 + 0 * 1 + 0 * 1 + 0 * 1 * 1 + 0 * 1 * 1 + 0 * 1 + 0
Sum ( L (#) F ) = Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( L (#) ( L (#) ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( L (#) ( L (#) ( L (#) F2 ) ) .= Sum ( L (#) ( L (#) F2 ) ) .= Sum ( L (#) ( L (#) ( L (#) F2 ) ) + Sum ( F1 ^ F2 ) + Sum ( F1 ^ F2 ) + Sum ( L (#) F2 ) + Sum ( L (#) ( L (#) F2 ) + Sum ( L (#) F2 ) ) + Sum ( L (#) ( L (#) F2 ) + Sum ( L (#) F2 ) + Sum ( L (#) F2 )
ex r be Real st for e be Real st 0 < e ex Y be finite Subset of X st 0 < e & Y is non empty & for Y1 be finite Subset of X st Y1 is non empty & Y1 c= Y holds |. ( - lower_bound ( Y1 ) ) . e - lower_bound ( Y1 ) ) .| < r ;
( GoB f ) * ( i , j + 1 ) = f /. ( k + 2 ) & ( GoB f ) * ( i , j + 1 ) = f /. ( k + 2 ) or ( GoB f ) * ( i , j + 1 ) = f /. ( k + 2 ) & ( GoB f ) * ( i , j + 1 ) = f /. ( k + 2 ) ;
( cos ( x ) ) ^2 = ( - sin ( x ) ) ^2 + ( sin ( x ) ) ^2 .= ( - sin ( x ) ) ^2 + ( cos ( x ) ) ^2 .= ( - sin ( x ) ) ^2 + ( cos ( x ) ) ^2 .= ( - sin ( x ) ) ^2 + ( cos ( x ) ) ^2 .= ( - sin ( x ) ) ^2 + ( cos ( x ) ) ^2 .= ( - cos ( x ) ) ^2 ;
( - b + sqrt ( delta ( a , b , c ) ) / ( 2 * a ) ) < 0 & ( - b - sqrt ( delta ( a , b , c ) ) / ( 2 * a ) ) < 0 or ( - b - sqrt ( delta ( a , b , c ) ) / ( 2 * a ) ) < 0 ;
assume that ex_inf_of uparrow X /\ C , L and ex_sup_of X /\ C , L and ex_sup_of ( uparrow X /\ C ) , L and "\/" ( uparrow X , L ) = "/\" ( uparrow "\/" ( uparrow X /\ L ) , L ) and not "\/" ( uparrow X , L ) = "/\" ( uparrow "\/" ( uparrow X /\ L ) , L ) and "\/" ( uparrow X , L ) = "/\" ( uparrow "\/" ( uparrow X /\ L ) , L ) ;
( ( for j holds j in the Sorts of U1 ) . ( j , i ) = ( j |-- ( B , i ) ) ** id ( the Sorts of U1 , j ) ) & ( j = i |-- ( B , i ) ) ** id ( the Sorts of U1 , j ) ) = ( j |-- ( B , i ) ) ** id ( the Sorts of U1 , j ) )