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thesis ;
contradiction ;
contradiction ;
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thesis ;
thesis ;
contradiction ;
contradiction ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
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thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
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thesis ;
contradiction ;
contradiction ;
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contradiction ;
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contradiction ;
thesis ;
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contradiction ;
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contradiction ;
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thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
contradiction ;
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contradiction ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
B in X ;
a <> c
T c= S
D c= B
c in X ;
b in X ;
X in 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 \Omega ;
let q ;
m = 1 ;
1 < k ;
G is prime ;
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 :] by 0 ;
assume c in x ;
- p > 0 ;
assume x in Z ;
assume x in Z ;
1 <= kthat 1 <= kthat ;
assume m <= i ;
assume G is prime ;
assume a divides b ;
assume P is closed ;
/ 2 > 0 ;
assume q in A ;
W is not bounded ;
f is st f is one-to-one holds 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 `2 <= j `2 ;
assume H is universal ;
assume x in X ;
let X be set ;
let T be DecoratedTree ;
let d be element ;
let t be element ;
let x be element ;
let x be element ;
let s be element ;
k <= be be Nat ;
let X be set ;
let X be set ;
let y be element ;
let x be element ;
P [ 0 ]
let E be set , f be Function ;
let C be category ;
let x be element ;
k be Nat ;
let x be element ;
let x be element ;
let e be element ;
let x be element ;
P [ 0 ]
let c be element ;
let y be element ;
let x be element ;
a be Real ;
let x be element ;
let X be element ;
P [ 0 ]
let x be element ;
let x be element ;
let y be element ;
r in REAL ;
let e be element ;
n1 is \setminus { n1 } is , n1 , n2 , n3 , n3 , n3 , n1 , n2 , n3 , n3 ,
Q halts_on s ;
x in \in \in \in of -1 ;
M < m + 1 ;
T2 is open ;
z in b id a ;
R2 is well-ordering ;
1 <= k + 1 ;
i > n + 1 ;
q1 is one-to-one ;
let x be trivial set ;
PM is one-to-one
n <= n + 2 ;
1 <= k + 1 ;
1 <= k + 1 ;
let e be Real ;
i < i + 1 ;
p3 in P ;
p1 in K ;
y in C1 ;
k + 1 <= n ;
let a be Real , x be Element of REAL ;
X |- r => p ;
x in { A } ;
let n be Nat ;
let k be Nat ;
let k be Nat ;
let m be Nat ;
0 < 0 + k ;
f is_differentiable_in x ;
let x0 , x1 ;
let E be Ordinal ;
o : o ;
O <> O2 ;
let r be Real ;
let f be FinSeq-Location ;
let i be Nat ;
let n be Nat ;
Cl A = A ;
L c= Cl L ;
A /\ M = B ;
let V be RealUnitarySpace , M be Subset of V ;
not s in Y to_power 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 , M be Subset of V ;
P [ 1 ] ;
P [ {} ] ;
C1 is component & C2 is component ;
H = G . i ;
1 <= i `1 + 1 ;
F . m in A ;
f . o = o ;
P [ 0 ] ;
aLet <= non < or aSet <= r ;
R [ 0 ] ;
b in f .: X ;
assume q = q2 ;
x in [#] V ;
f . u = 0 ;
assume e1 > 0 ;
let V be RealUnitarySpace , M be Subset of V ;
s is trivial & s is non trivial implies s is non trivial
dom c = Q
P [ 0 ] ;
f . n in T ;
N . j in S ;
let T be complete LATTICE , f be Function of T , the carrier of T ;
the ObjectMap of F is one-to-one
sgn x = 1 ;
k in support a ;
1 in Seg 1 ;
rng f = X ;
len T in X ;
vbeing < n ;
Sbeing Subset of T ;
assume p = p2 ;
len f = n ;
assume x in P1 ;
i in dom q ;
let U0 , S , U ;
pp `1 = c ;
j in dom h ;
let k ;
f | Z is continuous ;
k in dom G ;
UBD C = B ;
1 <= len M ;
p in \mathbin { x } ;
1 <= j9 & j9 <= len f ;
set A = st \cap \it = { 1 , 2 } ;
card a [= c ;
e in rng f ;
cluster B \oplus A -> empty ;
H is_\cdot \cdot \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 implies H is proper
i + 1 <= n ;
v <> 0. V ;
A c= Affin A ;
S c= dom F ;
m in dom f ;
X0 be set ;
c = sup N ;
R is connected implies union M is connected
assume not x in REAL ;
Im f is complete ;
x in Int y ;
dom F = M ;
a in On W ;
assume e in A ( ) ;
C c= C-26 ;
mm <> {} ;
x be Element of Y ;
let f be ) Chain , g be Chain of f , n ;
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 + X ;
- y in I ;
let A be non empty set , f be Function ;
P0 = 1 ;
assume r in F . k ;
assume f is simple ;
let A be non empty countable set ;
rng f c= NAT ;
assume P [ k ] ;
f6 <> {} ;
o be Ordinal ;
assume x is sum of squares ;
assume not v in { 1 } ;
let II , A , B ;
assume that 1 <= j and j < l ;
v = - u ;
assume s . b > 0 ;
d1 in d2 ;
assume t . 1 in A ;
let Y be non empty TopSpace , X be Subset of Y ;
assume a in uparrow s ;
let S be non empty Poset ;
a , b // b , a ;
a * b = p * q ;
assume x , y // the carrier of V ;
assume x in Omega ( f ) ;
[ a , c ] in X ;
mm <> {} ;
M + N c= M + M ;
assume M is connected & hhz is connected ;
assume f is \llangle bbnrrbeing set ;
let x , y be element ;
let T be non empty TopSpace ;
b , a // b , c ;
k in dom ( Sum p ) ;
v be Element of V ;
[ x , y ] in T ;
assume len p = 0 ;
assume C in rng f ;
k1 = k2 & k2 = k1 ;
m + 1 < n + 1 ;
s in S \/ { s } ;
n + i >= n + 1 ;
assume Re y = 0 ;
k1 <= j1 & k1 <= len f ;
f | A is non empty ;
f . x Let x ^2 <= b ;
assume y in dom h ;
x * y in B1 ;
set X = Seg n ;
1 <= i2 + 1 ;
k + 0 <= k + 1 ;
p ^ q = p ;
j |^ y divides m ;
set m = max A ;
[ x , x ] in R ;
assume x in succ 0 ;
a in sup phi ;
Cf in X ;
q2 c= C1 & q2 c= C2 ;
a2 < c2 & a2 < c2 implies a2 = b2
s2 is 0 -started & s2 is 0 -started ;
IC s = 0 & IC s = 0 ;
s4 = s4 , s4 = s4 ;
let V ;
let x , y be element ;
x be Element of T ;
assume a in rng F ;
x in dom T `2 ;
let S be non empty as non empty as Subset of L ;
y " <> 0 ;
y " <> 0 ;
0. V = u-w ;
y2 , y , w is_collinear ;
R8 in X ;
let a , b be Real , x be Element of REAL ;
let a be Object of C ;
let x be Vertex of G ;
let o be Object of C , m be Morphism of C ;
r '&' q = P \lbrack l \rbrack ;
let i , j be Nat ;
let s be State of A , x be set ;
s4 . n = N ;
set y = ( x `1 ) / ( x `2 ) ;
mi in dom g & mi in dom g ;
l . 2 = y1 ;
|. g . y .| <= r ;
f . x in CCCV ;
V1 is non empty ;
let x be Element of X ;
0 <> f . g2 ;
f2 /* q is convergent & lim ( f2 /* q ) = 0 ;
f . i is_measurable_on E ;
assume \xi in N-22 ;
reconsider i = i as Ordinal ;
r * v = 0. X ;
rng f c= INT & rng f c= INT ;
G = 0 .--> goto 0 ;
let A be Subset of X ;
assume A0 is dense & B is dense ;
|. f . x .| <= r ;
x be Element of R ;
let b be Element of L ;
assume x in W-19 ;
P [ k , a ] ;
let X be Subset of L ;
let b be Object of B ;
let A , B be category ;
set X = Vars ( C ) ;
let o be OperSymbol of S ;
let R be connected non empty Poset ;
n + 1 = succ n ;
x-21 c= Z1 & xY c= Z1 ;
dom f = C1 & rng f c= C2 ;
assume [ a , y ] in X ;
Re ( seq ) is convergent & lim ( seq ) = 0 ;
assume a1 = b1 & a2 = b2 ;
A = Int ( A ) ;
a <= b or b <= a ;
n + 1 in dom f ;
let F be Instruction of S , i be Nat ;
assume r2 > x0 & x0 < r2 ;
Y be non empty set , f be Function of Y , BOOLEAN ;
2 * x in dom W ;
m in dom g2 & n in dom g2 ;
n in dom g1 & n in dom g2 ;
k + 1 in dom f ;
not the still of { s } is finite ;
assume x1 <> x2 & y1 <> y2 ;
v1 in V1 & v2 in V1 ;
not [ b `1 , b ] in T ;
( i + 1 ) + 1 = i ;
T c= and T c= and T c= T ;
( l - 1 ) * ( l - 1 ) = 0 ;
n be Nat ;
( t `2 ) ^2 = r ^2 ;
A=> is_integrable_on M & AA c= M ;
set t = Top t ;
let A , B be real-membered set ;
k <= len G + 1 ;
cC misses cV ;
product ( s ) is non empty ;
e <= f or f <= e ;
cluster non empty normal for normal sequence ;
assume c2 = b2 & c2 = b1 ;
assume h in [. q , p .] ;
1 + 1 <= len C ;
not c in B . m1 ;
cluster R .: X -> empty ;
p . n = H . n ;
assume vseq is Cauchy sequence of X & vseq is convergent ;
IC s3 = 0 & IC s2 = 0 ;
k in N or k in K ;
F1 \/ F2 c= F \/ F2 ;
Int ( G1 ) <> {} ;
( z `2 ) ^2 = 0 ;
p11 <> p1 & p11 <> p2 ;
assume z in { y , w } ;
MaxADSet ( a ) c= F ;
ex_sup_of downarrow s , S ;
f . x <= f . y ;
let T be up-complete non empty reflexive transitive antisymmetric RelStr ;
q |^ m >= 1 ;
a >= X & b >= Y ;
assume <* a , c *> <> {} ;
F . c = g . c ;
G is one-to-one non empty full full ;
A \/ { a } \not c= B ;
0. V = 0. Y .= 0. V ;
let I be be non empty Instruction of S , S ;
f-24 . x = 1 / x ;
assume z \ x = 0. X ;
C4 = 2 to_power n ;
let B be SetSequence of Sigma ;
assume X1 = p .: D ;
n + l2 in NAT & n + l2 in NAT ;
f " P is compact & f " P is compact ;
assume x1 in REAL & x2 in REAL ;
p1 = K1 & p2 = K1 & p3 = D2 ;
M . k = <*> REAL ;
phi . 0 in rng phi ;
OSMInt A is closed ;
assume z0 <> 0. L & 0. L in I ;
n < ( N . k ) ;
0 <= seq . 0 & seq . 0 <= seq . 0 ;
- q + p = v ;
{ v } is Subset of B ;
set g = f /. 1 ;
[: R , S :] is stable Subset of R ;
set cR = Vertices R , R = Vertices R ;
p0 c= P3 & p0 c= P3 ;
x in [. 0 , 1 .[ ;
f . y in dom F ;
let T be Scott Scott Scott Scott Scott Scott Scott Scott Scott of S ;
ex_inf_of the carrier of S , S ;
downarrow a = downarrow b & a in A ;
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 ~ , a ~ " ;
assume a in A ( ) ;
k in dom ( q ^ <* 4 *> ) ;
p is Let of S , S ;
i -' 1 = i-1 - 1 ;
f | A is one-to-one ;
assume x in f .: [: X , Y :] ;
i2 - i1 = 0 & i2 - i1 = 0 ;
j2 + 1 <= i2 & j2 + 1 <= width G ;
g " * a in N ;
K <> { [ {} , {} ] } ;
cluster strict for } : X is strict } -valued ;
|. q .| ^2 > 0 ;
|. p4 .| = |. p .| ;
s2 - s1 > 0 & s2 - s1 > 0 ;
assume x in { Gik } ;
W-min ( C ) in C & W-min ( C ) in C ;
assume x in { Gik } ;
assume i + 1 = len G ;
assume i + 1 = len G ;
dom I = Seg n & rng I c= Seg n ;
assume that k in dom C and k <> i ;
1 + 1-1 <= i + j ;
dom S = dom F & rng F = dom G ;
let s be Element of NAT , x be Element of NAT ;
let R be ManySortedSet of A ;
let n be Element of NAT , x be Element of NAT ;
let S be non empty non void non void non empty non void non empty ManySortedSign ;
let f be ManySortedSet of I ;
let z be Element of COMPLEX , x be Element of COMPLEX ;
u in { bb } ;
2 * n < ( 2 * n ) ;
x , y be set ;
B-11 c= V1 & B-15 c= V1 ;
assume I is_closed_on s , P ;
U1 = U1 & U2 = U2 implies U1 = U2
M /. 1 = z /. 1 ;
x11 = x22 & x22 = x22 ;
i + 1 < n + 1 + 1 ;
x in { {} , <* 0 *> } ;
f7 <= f6 & f7 <= f6 ;
l be Element of L ;
x in dom ( F . -17 ) ;
let i be Element of NAT , k be Element of NAT ;
r8 is COMPLEX -valued & r8 is COMPLEX -valued ;
assume <* o2 , o *> <> {} ;
s . x |^ 0 = 1 ;
card ( K1 | K1 ) in M & card ( K1 | 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 * PI ) ;
assume dom g = cod f & cod g = cod f ;
let X , Y be non empty TopSpace , f be Function of X , Y ;
x \oplus A is interval ;
|. <*> A .| . a = 0 ;
cluster strict for non empty Poset ;
a1 in B . s1 & a2 in B . s1 ;
let V be finite < n , W be Subspace of V ;
A * B on B & A on B ;
f-3 = NAT --> 0 .= fs1 ;
let A , B be Subset of V ;
z1 = P1 . j & z2 = P2 . j ;
assume f " P is closed & f " P is closed ;
reconsider j = i as Element of M ;
let a , b be Element of L ;
assume q in A \/ ( B "\/" C ) ;
dom ( F * C ) = o ;
set S = INT * , T = s2 * f ;
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 non-empty ManySortedSet of I ;
( PI / 2 ) < Arg z ;
reconsider z9 = 0 as Nat ;
LIN a , d , c ;
[ y , x ] in IE ;
( Q Q ) * ( 1 , 3 ) = 0 ;
set j = x0 gcd m , m = x0 gcd m ;
assume a in { x , y , c } ;
j2 - ( j - 1 ) > 0 ;
I \! \mathop { phi } = 1 ;
[ y , d ] in F-8 ;
let f be Function of X , Y ;
set A2 = ( B - C ) / ( A * B ) ;
s1 , s2 being Element of L st s1 , s2 ] & s1 , s2 are_card ( s1 \/ s2 ) holds s1 , s2 are_card ( s1 \/ s2 )
j1 -' 1 = 0 & j2 -' 1 = 0 ;
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 = f | D-21 ;
assume that X is lower and 0 <= r ;
( p1 `1 ) ^2 = 1 & ( p1 `2 ) ^2 = 1 ;
a < ( p3 `1 ) ^2 + ( p3 `2 ) ^2 ;
L \ { m } c= UBD C ;
x in Ball ( x , 10 ) ;
not a in LSeg ( c , m ) ;
1 <= i1 -' 1 & 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 <* A2 *> = 1 ;
set H = h . ( g . x ) ;
card b * a = |. a .| ;
Shift ( w , 0 ) |= v ;
set h = h2 ** h1 , h1 = h2 ** h2 ;
assume x in ( X2 /\ X3 ) ;
||. h .|| < dx0 & ||. h .|| < dx0 ;
not x in the carrier of f & not x in the carrier of f ;
f . y = F ( y ) ;
for n holds X [ n ] ;
k - l = k\leq k\leq ;
<* p , q *> /. 2 = q ;
let S be Subset of the lattice of Y ;
let P , Q be succ s ;
Q /\ M c= union ( F | M )
f = b * canFS ( S ) ;
let a , b be Element of G ;
f .: X <= f . sup X
let L be non empty transitive reflexive transitive RelStr , x be Element of L ;
S-20 is x -f1 of i -f1 .: S
let r be non positive Real ;
M , v |= x \hbox { y } ;
v + w = 0. Z .= v + w ;
P [ len ( F | n ) ] ;
assume InsCode ( i ) = 8 & InsCode ( i ) = 8 ;
the zero of M = 0 & the zero of M = 0 ;
cluster z * seq -> summable for sequence of X ;
let O be Subset of the carrier of C ;
||. f .|| | X is continuous ;
x2 = g . ( j + 1 ) ;
cluster -> non empty 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 & not F . t is non empty ;
assume that n <> 0 and n <> 1 ;
set en = EmptyBag n , e = EmptyBag n ;
let b be Element of Bags n , c be Element of Bags n ;
assume for i holds b . i is commutative ;
x is root & y is Element of ( the carrier of S ) * ;
not r in ]. p , q .[ ;
let R be FinSequence of REAL , x be Element of REAL ;
S7 does not destroy b1 , b2 ;
IC SCM R <> a & IC SCM R <> a ;
|. - |[ x , y ]| .| >= r ;
1 * seq = seq & 1 * seq = seq ;
let x be FinSequence of NAT , y be Element of NAT ;
let f be Function of C , D , g be Function of C , D ;
for a holds 0. L + a = a
IC s = s . NAT .= IC s .= IC s ;
H + G = F- ( G-Gas ) ;
CA1 . x = x2 & CA2 . x = y2 ;
f1 = f , f2 = f , f3 = g ;
Partial_Sums <* p . 0 *> = p . 0 ;
assume v + W = v + u + W ;
{ a1 } = { a2 } & { a2 } = { a2 } ;
a1 , b1 _|_ b , a & b1 , c1 _|_ b , a ;
d2 , o _|_ o , a3 & a3 , o _|_ a3 , a1 ;
If is reflexive & If is reflexive implies f * I is reflexive
IO is antisymmetric implies [: O , O :] is antisymmetric
sup rng ( H1 | n ) = e & sup rng ( H1 | n ) = e ;
x = ( a * ( - 1 ) ) * ( - 1 ) ;
|. p1 .| ^2 >= 1 ^2 ;
assume j2 -' 1 < 1 & j2 -' 1 < width G ;
rng s c= dom f1 & rng s c= dom f2 ;
assume not ( for a being Element of L holds a in support b ) ;
let L be associative non empty doubleLoopStr , I be non empty Subset of L ;
s " + 0 < n + 1 ;
p . c = ( f " ) . 1 .= f . c ;
R . n <= R . ( n + 1 ) ;
Directed I1 = I1 , I2 = I2 , I2 = I2 , I2 = I2 ;
set f = + ( x , y , r ) ;
cluster Ball ( x , r ) -> bounded ;
consider r being Real such that r in A ;
cluster non empty -> non empty for NAT -defined Function ;
let X be non empty directed Subset of S ;
let S be non empty full SubRelStr of L ;
cluster <* [ ] *> , <* N *> -> complete for non trivial set ;
( 1 - a ) " = a " & ( 1 - a ) " = a " ;
( q . {} ) `1 = o ;
( n - 1 ) > 0 ;
assume ( 1 - 2 ) <= t `1 / 2 ;
card B = k + 1-1 & card A = k + 1 ;
x in union rng ( f | ( len f ) ) ;
assume x in the carrier of R & y in the carrier of R ;
d in X ;
f . 1 = L . ( F . 1 ) ;
the vertices of G = { v } & { v } c= the vertices of G ;
let G be finite connected wgraph ;
e , e be set , x be set ;
c . ( i - 1 ) in rng c & c . ( i - 1 ) in rng c ;
f2 /* q is divergent_to+infty & ( f2 /* q ) . n in dom ( f2 * f1 ) ;
set z1 = - z2 , z2 = - z1 , z2 = - z2 , z1 = - z2 , z2 = - - z2 ;
assume w is_llas S , G ;
set f = p |-count ( t - p ) , g = p |-count ( t - p ) , h = p |-count ( t - p ) , f = p |-count ( t - p ) , g
let c be Object of C ;
assume ex a st P [ a ] ;
let x be Element of REAL m , y be Element of REAL m ;
let IX be Subset-Family of X , f be Function of X , X ;
reconsider p = p as Element of NAT ;
v , w be Point of X ;
let s be State of SCM+FSA , I be Program of SCM+FSA ;
p is FinSequence of SCM , k be Nat ;
stop I ( ) c= P-12 & I ( ) c= P-12 ;
set ci = fSet /. i , fi = fSet /. i , fj = fj ;
w ^ t ^ t ^ <* s *> ^ w ^ w ^ t ^ w ^ v ^ w ^ w ^ w ^ v ^ w ^ w ^ w ^ w ^ w ^ v ^ w ^
W1 /\ W = W1 /\ W2 ` .= W1 /\ W2 ` ;
f . j is Element of J . j ;
let x , y be Element of T2 , T be Subset of T2 ;
ex d st a , b // b , d ;
a <> 0 & b <> 0 & c <> 0 ;
ord x = 1 & x is prime implies x is not prime
set g2 = lim ( seq , n ) , g1 = lim ( seq , n ) ;
2 * x >= 2 * ( 1 / 2 ) ;
assume ( a 'or' c ) . z <> TRUE ;
f (*) g in Hom ( c , c ) ;
Hom ( c , c + d ) <> {} ;
assume 2 * Sum ( q | m ) > m ;
L1 . ( F-21 ) = 0 & L1 . ( F-21 ) = 0 ;
/ ( X \/ R1 ) = / ( X \/ R2 )
( sin . x ) ^2 <> 0 & ( sin . x ) ^2 <> 0 ;
( ( ( #Z 2 ) * ( exp_R + exp_R ) ) `| Z ) . x > 0 ;
o1 in [: X , Y :] /\ [: X , O :] ;
e , e be set , x be set ;
r3 > ( 1 - r2 ) * 0 ;
x in P .: ( F -ideal ) ;
let J be closed Subset of R , I be left Subset of R ;
h . p1 = f2 . O & h . O = g2 ;
Index ( p , f ) + 1 <= j ;
len ( q | M ) = width M & len ( q | M ) = len M ;
the carrier of `1 c= A & the carrier of Cmin K c= A ;
dom f c= union rng ( F | ( union rng F ) )
k + 1 in support ( support ( f ) ) ;
let X be ManySortedSet of the carrier of S ;
[ x `1 , y `2 ] in an an an an an an \mathclose of an ~ ( R ~ ) ;
i = D1 or i = D2 or i = D1 ;
assume a mod n = b mod n & b mod n = 0 ;
h . x2 = g . x1 & h . x2 = g . x2 ;
F c= 2 -tuples_on the carrier of X
reconsider w = |. s1 .| as Real_Sequence ;
( 1 - m ) * m + r < p ;
dom f = dom I-4 & dom f = dom IK ;
[#] P-17 = [#] ( ( TOP-REAL 2 ) | K1 ) .= K1 ;
cluster - x -> ExtReal -> ExtReal for ExtReal ;
then { d1 } c= A & A is closed ;
cluster TOP-REAL n -> finite-ind for non empty TopSpace ;
let w1 be Element of M , w2 be Element of S ;
x be Element of dyadic ( n ) ;
u in W1 & v in W3 implies u in W2
reconsider y = y as Element of L2 ( ) ;
N is full SubRelStr of T |^ the carrier of S ;
sup { x , y } = c "\/" c ;
g . n = n to_power 1 .= n to_power 1 .= n ;
h . J = EqClass ( u , J ) ;
let seq be -> summable sequence of X , x be Element of X ;
dist ( x `1 , y ) < ( r / 2 ) ;
reconsider mm1 = m , mm2 = n as Element of NAT ;
x- x0 < r1 - x0 & r1 - x0 < r1 - x0 ;
reconsider P = P ` as strict Subgroup of N ;
set g1 = p * ( idseq q " ) , g2 = p " ;
let n , m , k be non zero Nat ;
assume that 0 < e and f | A is lower ;
D2 . ( I8 . ( I . ( I . ( I . ( I . ( I . ( I . ( I . ( I . x ) ) ) ) ) ) ) ) ) ) )
cluster subcondensed open -> subopen ;
let P be compact non empty Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
Gik in LSeg ( cos , 1 ) /\ LSeg ( cos , 1 ) ;
n be Element of NAT , x be Element of NAT ;
reconsider S8 = S as Subset of T | S ;
dom ( i .--> X `1 ) = { i } ;
let X be non-empty ManySortedSet of S ;
let X be non-empty ManySortedSet of S ;
op ( 1 ) c= { [ {} , {} ] }
reconsider m = mm as Element of NAT ;
reconsider d = x as Element of C ( ) ;
let s be 0 -started State of SCMPDS , P be Subset of SCMPDS ;
let t be 0 -started State of SCMPDS , Q be t of SCMPDS ;
b , b , x , y , z is_collinear ;
assume that i = n \/ { n } and j = k \/ { k } ;
let f be PartFunc of X , Y ;
N1 >= ( sqrt c ) * ( sqrt c ) * ( ( 2 / ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * (
reconsider t7 = T7 as Point of TOP-REAL 2 ;
set q = h * p ^ <* d *> ;
z2 in U . ( y2 , z2 ) /\ Q2 & z2 in Q ;
A |^ 0 = { <%> E } & A |^ 0 = { <%> E } ;
len W2 = len W + 2 & len W2 = len W + 1 ;
len h2 in dom h2 & len h2 in dom h2 ;
i + 1 in Seg len s2 & i + 1 in Seg len s2 ;
z in dom g1 /\ dom f & z in dom g1 /\ dom f ;
assume p2 = E-max ( K ) & p1 in L~ f ;
len G + 1 <= i1 + 1 ;
f1 (#) f2 is convergent & lim ( f1 (#) f2 ) = x0 ;
cluster seq + seq -> summable for sequence of X , seq , n be Nat ;
assume j in dom M1 & i in dom M2 ;
let A , B , C be Subset of X ;
x , y , z be Point of X , p be Point of X ;
b ^2 - ( 4 * a * c ) >= 0 ;
<* xx , y *> ^ <* y *> \mathclose { x } << 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 q = len G ;
s1 = Initialize ( Initialized s ) , P1 = P +* I , P2 = P +* I ;
consider w being Nat such that q = z + w ;
x ` ` is Element of x ` & x ` ` is Element of x ` ;
k = 0 & n <> k or k > n & k > n ;
then X is discrete implies X is closed
for x st x in L holds x is FinSequence of REAL ;
||. f /. c .|| <= r1 & ||. f /. c .|| <= r1 ;
c in uparrow p & not c in { p } implies not c in { p }
reconsider V = V as Subset of the TopStruct of TOP-REAL n ;
N , M be Subset of L ;
then z is_>=_than compactbelow x & z >= compactbelow y ;
M [. f , g .] = f & M [. g , f .] = g ;
( ( ( Fib 1 ) * ( L /. 1 ) ) ) /. 1 = TRUE ;
dom g = dom f -tuples_on X & rng g c= dom f ;
mode : of G is : Let ;
[ i , j ] in Indices M & [ i , j ] in Indices M ;
reconsider s = x " as Element of H ;
let f be Element of ( dom ( Subformulae p ) ) -tuples_on the carrier of F ;
F1 . ( a1 , - a2 ) = G1 . ( a1 , - a2 ) ;
redefine func \mathopen { a , b , r } -> compact ;
let a , b , c , d be Real ;
rng s c= dom ( 1 / f ) & rng s c= dom f ;
curry ( F-19 , k ) is additive ;
set k2 = card ( dom B \ A ) , k1 = card ( A \ B ) ;
set G = DTConMSA ( X ) ;
reconsider a = [ x , s ] as w of G ;
let a , b be Element of MM , x be Element of M ;
reconsider s1 = s , s2 = t as Element of S0 ;
rng p c= the carrier of L & p . x in the carrier of L ;
let d be Subset of the Sorts of A ;
( x .|. x = 0 iff x = 0. W ) ;
I-21 in dom stop I & IY in dom stop I ;
g be continuous Function of X | B , Y ;
reconsider D = Y as Subset of ( TOP-REAL n ) | P ;
reconsider i0 = len p1 , j1 = len p2 as Integer ;
dom f = the carrier of S & rng f c= the carrier of S ;
rng h c= union ( the carrier of J ) & rng h c= union ( the carrier of J )
cluster All ( x , H ) -> Carrier c ;
d * N1 / ( 1 - d ) > N1 * 1 / ( 1 - d ) ;
]. a , b .[ c= [. a , b .] ;
set g = f " | D1 , f = f " | D2 ;
dom ( p | mm1 ) = mm1 & dom ( p | mm1 ) = mm1 ;
3 + - 2 <= k + - 2 & k + - 2 <= len f ;
tan + arccot is_differentiable_on Z & for x st x in Z holds ( ( arccot - arccot ) `| Z ) . x = 1 / ( x ^2 )
x in rng ( f /^ ( p -' 1 ) ) ;
f , g be FinSequence of D ;
cp1 in the carrier of S1 & cp2 in the carrier of S1 ;
rng f " = dom f & rng f = dom f ;
( the Source of G ) . e = v & ( the Source of G ) . e = v ;
width G - 1 < width G - 1 & width G - 1 < width G - 1 ;
assume v in rng ( S | E1 ) ;
assume x is root or x is root or x is root ;
assume 0 in rng ( g2 | A ) & 0 in rng ( g2 | A ) ;
let q be Point of ( TOP-REAL 2 ) | K1 , r be Real ;
let p be Point of TOP-REAL 2 , x be Point of TOP-REAL 2 ;
dist ( O , u ) <= |. p2 .| + 1 ;
assume dist ( x , b ) < dist ( a , b ) ;
<* S7 *> is_in the / of C-20 & <* S7 *> in the carrier of C-20 ;
i <= len ( G /^ 1 ) - 1 & i + 1 <= len G ;
let p be Point of TOP-REAL 2 , x be Point of TOP-REAL 2 ;
x1 in the carrier of I[01] & x2 in the carrier of I[01] & x3 in the carrier of I[01] ;
set p1 = f /. i , p2 = f /. ( i + 1 ) ;
g in { g2 : r < g2 & g2 < r } ;
Q2 = Sp2 " . ( Q " . ( P . ( P . ( Q . ( Q . ( Q . ( Q . ( Q . ( Q . ( Q . a ) ) ) ) ) ) ) ) ) )
( ( 1 / 2 ) (#) ( 1 / 2 ) ) is summable ;
- p + I c= - p + A & - p + I c= - p + I ;
n < LifeSpan ( P1 , s1 ) + 1 & n <= LifeSpan ( P2 , s2 ) ;
CurInstr ( p1 , s1 ) = i .= ( 0 + 1 ) ;
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 L2 , L1 ;
reconsider z = z as Element of CompactSublatt L , L ;
let f be Function of S , T ;
reconsider g = g as Morphism of c opp , b opp ;
[ s , I ] in S ~ ( A , I ) ;
len ( the connectives of C ) = 4 & len ( the connectives of C ) = 5 ;
let C1 , C2 be subcategory , F be subFunctor of C1 , C2 ;
reconsider V1 = V as Subset of X | B , V1 = V as Subset of X | B ;
attr p is valid means : Def2 : All ( x , p ) is valid ;
assume that X c= dom f and f .: X c= dom g and g .: X c= dom f ;
H |^ a " * ( a |^ b ) is Subgroup of H ;
let A1 be : A1 : A1 , A2 |^ A1 |^ 2 ] ;
p2 , r3 , q3 is_collinear & q2 , q3 , q3 is_collinear ;
consider x being element such that x in v ^ K ;
not x in { 0. TOP-REAL 2 } & not x in { 0. TOP-REAL 2 } ;
p in [#] ( I[01] | B11 ) & p in [#] ( I[01] | B11 ) ;
0 . 0 < M . ( E . 0 ) ;
( c / ( c / a ) ) / ( c / a ) = c ;
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . s2 ) & a2 in dom ( F . s2 ) ;
cluster -> non empty for \mathbin of the carrier of L is from of L , L ;
set i1 = the Nat , i2 = the Element of NAT , n = 1 ;
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 : Def2 : cos | X c= cos | Y ;
let y be upper Subset of Y , x be Element of X ;
cluster -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> real for non empty ;
set S = <* Bags n , ( Bags n ) *> ;
set T = [. 0 , 1 / 2 .] , S = [. 0 , 1 / 2 .] ;
1 in dom mid ( f , 1 , 1 ) & 1 in dom mid ( f , 1 , 1 ) ;
( 4 * PI ) / 2 < ( 2 * PI ) / 2 ;
x2 in dom f1 /\ dom f & x2 in dom f1 /\ dom f ;
O c= dom I & { {} } = { {} } implies O = {}
( the Source of G ) . x = v & ( the Source of G ) . x = v ;
{ HT ( f , T ) } c= Support f & Support ( f , T ) c= Support f ;
reconsider h = R . k as Polynomial of n , L ;
ex b being Element of G st y = b * H ;
let x `1 , y , z be Element of G opp ;
h19 . i = f . ( h . i ) ;
( p `1 ) ^2 = ( p1 `1 ) ^2 + ( p1 `2 ) ^2 .= ( p1 `2 ) ^2 ;
i + 1 <= len Cage ( C , n ) ;
len <* P *> @ = len P & len <* P *> = 1 ;
set N-26 = the non empty Subset of the or of N = the or of N = the or of N = the or N = the or of N = the or N = the or N = the or N is strict non empty
len gSet + ( x + 1 ) - 1 <= x ;
not a on B & b on B & not a on B implies a on B
reconsider rv = r * I . v as FinSequence of REAL ;
consider d such that x = d and a -> d ;
given u such that u in W and x = v + u ;
len f /. ( \downharpoonright n ) = len \bf n .= len f ;
set q2 = N-min C , q2 = E-max C , q2 = E-max C , q2 = E-max C , q2 = E-max C , q2 = E-max C , q2 = E-max C , q2 = E-max C , q2 = E-max C , q2 =
set S =
MaxADSet ( b ) c= MaxADSet ( P /\ Q ) ;
Cl ( G . q1 ) c= F . r2 & Cl ( G . q2 ) c= F . r2 ;
f " D meets h " ( f .: V ) & f " ( f .: V ) meets h " V ;
reconsider D = E as non empty directed Subset of L1 ;
H = ( the_left_argument_of H ) '&' ( the_left_argument_of H ) ;
assume t is Element of ( S , X ) * ;
rng f c= the carrier of S2 & rng f c= the carrier of S2 ;
consider y being Element of X such that x = { y } ;
f1 . ( a1 , b1 ) = b1 & f1 . ( b1 , b2 ) = b2 ;
the carrier' of G ` = E \/ { E } .= { E } \/ { E } ;
reconsider m = len thesis , k = len ( k - 1 ) as Element of NAT ;
set S1 = LSeg ( n , UMP C ) , S2 = LSeg ( n , UMP C ) ;
[ i , j ] in Indices M1 & ( M1 - M2 ) * ( i , j ) = M1 ;
assume that P c= Seg m and M is \HM { i , j } is not empty ;
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 ;
let PA , PA , G be a_partition of Y , z be Element of Y ;
pred 0 < r & r < 1 implies 1 < ( 1 - r ) * ( 1 - r ) ;
rng ( \mathop { a , X } ) = [#] X & rng ( \mathop { a , X } ) = [#] X ;
reconsider x = x , y = y as Element of K ;
consider k such that z = f . k and n <= k ;
consider x being element such that x in X \ { p } ;
len ( canFS ( s ) ) = card ( s ) .= card ( s ) ;
reconsider x2 = x1 , y2 = x2 as Element of L2 ;
Q in FinMeetCl ( ( the topology of X ) \/ the topology of Y ) ;
dom ( f . 0 ) c= dom ( u . 0 ) & rng ( f . 0 ) c= dom f ;
pred n divides m means : Def2 : m divides n & n = m ;
reconsider x = x as Point of [: I[01] , I[01] :] , R^1 ;
a in ) implies the carrier of let T , T be non empty set , f be Function of T , T2 ;
not y0 in the still of f & not y in the still of f implies y in the carrier of f
Hom ( ( a ~ ) , c ) <> {} ;
consider k1 such that p " < k1 and k1 < len f and f . k1 = f . k1 ;
consider c , d such that dom f = c \ d and rng f = d ;
[ x , y ] in dom g & [ y , x ] in dom g ;
set S1 = Let L , S2 = L +* ( x , y , z ) ;
l2 = m2 & l2 = i2 & E = C & E = F . i2 implies E = F
x0 in dom ( u01 /\ dom ( f | A ) ) & x0 in dom ( f | A ) ;
reconsider p = x as Point of ( TOP-REAL 2 ) | K1 , ( TOP-REAL 2 ) | K1 ;
I[01] = R^1 | B01 & dom ( ( TOP-REAL 2 ) | B01 ) = B01 ;
f . p4 `1 <= f . p1 `1 & f . p2 `2 <= f . p1 `2 ;
( ( F . x ) `1 ) ^2 <= ( ( F . x ) `1 ) ^2 ;
( x `2 ) ^2 = ( W7 ) ^2 + ( W8 ) ^2 .= ( W8 ) ^2 + ( x `2 ) ^2 ;
for n being Element of NAT holds P [ n ] implies P [ n + 1 ]
let J , K be non empty Subset of I ;
assume 1 <= i & i <= len <* a " *> ;
0 |-> a = <*> the carrier of K & 0 |-> a = {} implies a = {}
X . i in 2 -tuples_on A . i \ B . i ;
<* 0 *> in dom ( e --> [ 1 , 0 ] ) ;
then P [ a ] & P [ succ a ] ;
reconsider sp2 = seq as \rangle of D , the carrier of D ;
( k - 1 ) <= len thesis - j ;
[#] S c= [#] the TopStruct of T & [#] T is open & [#] T is open ;
for V being strict RealUnitarySpace holds V in the carrier of V implies V in the carrier of V
assume k in dom mid ( f , i , j ) ;
let P be non empty Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
let A , B be Matrix of n1 , n2 , n3 be Matrix of n1 , n2 , n3 be Nat ;
- a * - b * a = a * b - b * a
for A being Subset of AS holds A // A & A // A implies A = B
( for o2 being Element of o2 holds ( o1 , o2 ) in <^ o2 , o1 ^> ) implies o1 = o2
then ||. x .|| = 0 & x = 0. X ;
let N1 , N2 be strict normal Subgroup of G , N2 be strict normal Subgroup of G ;
j >= len ( upper_volume ( g , D1 ) | j ) ;
b = Q . ( len Q - 1 + 1 ) ;
f2 * f1 /* s is convergent & lim ( f2 * f1 ) = x0 ;
reconsider h = f * g as Function of N1 , G ;
assume that a <> 0 and Let a , b , c be Real ;
[ t , t ] in the InternalRel of A & [ t , t ] in the InternalRel of A ;
( v |-- E ) | n is Element of ( T | ( n + 1 ) ) -tuples_on REAL ;
{} = the carrier of L1 + L2 & the carrier of L1 + L2 = the carrier of L2 + L2 ;
Directed I is_closed_on Initialized s , P & Directed I is_halting_on Initialized s , P ;
Initialized p = Initialize ( p +* q ) , p1 = p +* I , p2 = p +* I ;
reconsider N2 = N1 , N2 = N2 as strict net of R1 , R2 ;
reconsider Y = Y as Element of \langle Ids L , \subseteq \rangle ;
"/\" ( uparrow p \ { p } , L ) <> p ;
consider j being Nat such that i2 = i1 + j and j in dom f ;
not [ s , 0 ] in the carrier of S2 & not [ s , 0 ] in the carrier of S2 ;
mm in ( B '/\' C ) '/\' D \ { {} } ;
n <= len ( ( P + Q ) ^ <* a *> ) + len ( P ^ <* b *> ) ;
( x1 - x2 ) ^2 = ( x2 - y2 ) ^2 + ( y2 - z2 ) ^2 ;
InputVertices S = { x1 , x2 , x3 , x4 , x5 } ;
let x , y be Element of FTTT1 ( n ) ;
p = |[ p `1 , p `2 ]| & p = |[ p `1 , p `2 ]| ;
g * 1_ G = h " * g * h * h .= g " * h ;
let p , q be Element of being Element of being Element of being Element of being Element of being Element of being set ;
x0 in dom ( x1 - x2 ) /\ dom ( x2 - y2 ) ;
( R qua Function ) " = R " * ( R " ) .= R " * ( R " ) ;
n in Seg len ( f /^ ( len f -' 1 ) ) ;
for s being Real st s in R holds s <= s2 & s2 <= 1 ;
rng s c= dom ( ( f2 * f1 ) ^ ) & rng s c= dom ( ( f2 * f1 ) ^ ) ;
synonym for for for for for for the carrier of consider X , the carrier of X ;
1. ( K , n ) * 1. ( K , n ) = 1. ( K , n ) ;
set S = Segm ( A , P1 , Q1 ) , S = Segm ( A , P1 , Q1 ) ;
ex w st e = ( w - f ) / ( w - f ) & w in F ;
curry ( P+* ( i , k ) ) # x is convergent & curry ( P+* ( i , k ) ) # x is convergent ;
cluster open open -> open for Subset of [: T , T :] ;
len f1 = 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 & c1 , c2 // b9 , c ;
consider p being element such that c1 . j = { p } and p in { 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 implies Reloc ( J , card I ) not " ;
goto ( card I + 1 ) not contradiction & i in dom ( i .--> a ) ;
set m3 = LifeSpan ( p3 , s3 ) , P4 = LifeSpan ( p2 , s2 ) ;
IC SCMPDS in dom ( Initialize p ) & IC SCMPDS in dom ( Initialize p ) ;
dom t = the carrier of SCM & dom t = the carrier of SCM & rng t c= the carrier of SCM ;
( ( E-max L~ f ) .. f ) .. f = 1 & ( ( E-max L~ f ) .. f ) .. f = 1 ;
let a , b be Element of being Element of being Element of being Element of being Element of being Element of V ;
Cl ( union F ) c= Cl ( Int union F ) ;
the carrier of X1 union X2 misses ( A1 \/ A2 ) & the carrier of X1 union X2 misses ( A1 \/ A2 ) ;
assume not LIN a , f . a , g . a ;
consider i being Element of M such that i = d6 and i in A ;
then Y c= { x } or Y = {} or Y = { x } ;
M , v |= H1 / ( ( y , x ) / ( y , x ) ) ;
consider m being element such that m in Intersect ( FF . m ) and x = ( Intersect ( FF . m ) ) . m ;
reconsider A1 = support u1 , A2 = support v1 as Subset of X ;
card ( A \/ B ) = k-1 + ( 2 * 1 ) ;
assume that a1 <> a3 and a2 <> a4 and a3 <> a4 and a4 <> a5 ;
cluster s -\mathop { V } -> non empty for string of S ;
LG2 /. n2 = LG2 . n2 & LG2 /. n2 = LG2 . n2 ;
let P be compact non empty Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
assume that r-7 in LSeg ( p1 , p2 ) and r29 in LSeg ( p1 , p2 ) ;
let A be non empty compact Subset of TOP-REAL n , B be Subset of TOP-REAL n ;
assume [ k , m ] in Indices ( ( - 1 ) * ( i , j ) ) ;
0 <= ( ( 1 / 2 ) |^ p ) * ( ( 1 / 2 ) |^ p ) ;
( F . N | E8 ) . x = +infty ;
pred X c= Y means : Def2 : Z c= V & X \ V c= Y \ Z ;
( y - z ) * ( z - w ) <> 0. I & ( - z ) * ( w - y ) <> 0. I ;
1 + card ( X-18 ) <= card ( u \/ { u } ) ;
set g = z \circlearrowleft ( L~ z ) , 2 = ( L~ z ) .. z , M = ( L~ z ) .. z , N = ( L~ z ) .. z , S = ( L~ z ) .. z , N = ( L~ z ) .. z , S
then k = 1 & p . k = <* x , y *> . k ;
cluster -> total for Element of C -\mathop { X } , the carrier of S ;
reconsider B = A as non empty Subset of ( TOP-REAL n ) | ( A ` ) ;
let a , b , c be Function of Y , BOOLEAN , f be Function of Y , BOOLEAN ;
L1 . i = ( i .--> g ) . i .= g . i .= g . i ;
Plane ( x1 , x2 , x3 , x4 , x5 , x5 , x5 ) c= P ;
n <= indx ( D2 , D1 , j1 ) + 1 - 1 ;
( ( g2 ) . O ) `1 = - 1 & ( ( g2 ) . O ) `2 = 1 ;
j + p .. f - len f <= len f - len f + 1 - len f ;
set W = W-bound C , S = S-bound C , E = E-bound C , N = E-bound C , N = E-bound C , S = S-bound C , N = E-bound C , S = S-bound C , N = W-bound C , S = W-bound C , S = W-bound
S1 . ( a `1 , e `2 ) = a + e .= a `1 + e ;
1 in Seg width ( M * ( ColVec2Mx p ) ) & 1 in Seg len ( M * ( ColVec2Mx p ) ) ;
dom ( i * Im ( f , Im ( f , Im ( f , Im ( f , Im ( f , Im ( f , Im ( f , Im ( f , Im ( f , Im ( f , Im ( f , Im ( f
that that .: ( means x `1 = W . ( a , *' ( a , p ) ) ) ;
set Q = means : every Element of ) \ \mathopen { f . g , h . g } ;
cluster -> non-empty for ManySortedSet of U1 , ( the Sorts of U1 ) * ;
attr ex A st F = { A } & F is discrete ;
reconsider z9 = \hbox { y where y is Element of product \overline G : y in G } as non empty set ;
rng f c= rng f1 \/ rng f2 & rng f c= rng f1 \/ rng f2 ;
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 implies f = id the carrier of F_Complex
E , j |= All ( x1 , x2 , x3 , x4 ) ;
reconsider n1 = n , n2 = m as Morphism of o1 , o2 ;
assume that P is idempotent and R is idempotent and P ** R = R ** P ;
card ( B2 \/ { x } ) = k-1 + 1 .= card B2 + 1 ;
card ( x \ B1 ) /\ ( x \ B1 ) = 0 implies x in B1
g + R in { s : g-r < s & s < g + r } ;
set q-1r = ( q , <* s *> ) \mathop { 1 } , qr = ( q , <* s *> ) \mathop { 1 } ;
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 , being = \overline ( B , NAT ) ) ;
t in Seg width ( ( I ^ ( n , n ) ) @ ) ;
reconsider X = dom f , Y = [: the carrier of NAT , the carrier of NAT :] as Element of Fin NAT ;
IncAddr ( i , k ) = <% i %> + k .= ( - i ) + k ;
( S-bound L~ f ) ^2 <= ( q `2 ) ^2 & ( q `2 ) ^2 <= ( q `2 ) ^2 ;
attr R is condensed means : Def2 : Int R is condensed & Cl R is condensed & Cl R is condensed ;
pred 0 <= a & a <= 1 & b <= 1 implies a * b <= 1 ;
u in ( ( c /\ ( d /\ b ) ) /\ f ) /\ j ;
u in ( ( c /\ ( d /\ e ) ) /\ f ) /\ j ;
len C + - 2 >= 9 + - 3 & len C - 3 >= 9 + - 3 ;
x , z , y is_collinear & x , z , y is_collinear implies x = y
a |^ ( n1 + 1 ) = a |^ n1 * a |^ n1 ;
<* \underbrace ( 0 , \dots , 0 ) *> in Line ( x , a * x ) ;
set yx1 = <* y , c *> , yx2 = <* c , x *> ;
FG2 /. 1 in rng Line ( D , 1 ) & FG2 /. len FG2 = D ;
p . m Joins r /. m , r /. ( m + 1 ) , G ;
( p `2 ) ^2 = ( f /. i1 ) ^2 + ( f /. i2 ) ^2 .= ( f /. i1 ) ^2 + ( f /. i2 ) ^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 & ( f1 /* ( seq ^\ k ) ) . n in dom ( f1 /* ( seq ^\ k ) ) ;
reconsider u2 = u as VECTOR of \overline { X , Y } , f = u + v as VECTOR of X ;
p |-count ( Product Sgm ( X11 ) ) = 0 & p |-count ( Product Sgm ( X11 ) ) = 0 ;
len <* x *> < i + 1 & i + 1 <= len c + 1 ;
assume that I is non empty and { x } /\ { y } = { 0. I } ;
set ii = card I + 4 .--> goto 0 , goto 0 = goto 0 , goto 0 = goto 0 , goto 0 , goto 0 = goto 0 ;
x in { x , y } & h . x = {} T & h . y = {} T ;
consider y being Element of F such that y in B and y <= x `1 ;
len S = len ( the charact of ( A ) ) & len ( the charact of ( A ) ) = len the charact of ( A ) ;
reconsider m = M , i = I , n = N as Element of X ;
A . ( j + 1 ) = B . ( j + 1 ) \/ A . ( j + 1 ) ;
set N8 = : ( G-15 ) . e = ( G-15 ) . e ;
rng F c= the carrier of gr { a } & rng F c= the carrier of gr { a } ;
( for K being ( ) , n , r being Nat holds K is a .| -valued implies ( K is a -sequence of K )
f . k , f . ( Let ( p |^ n ) ) ] in rng f ;
h " P /\ [#] ( T1 | P ) = f " P /\ [#] ( T1 | P ) ;
g in dom f2 \ ( f2 " { 0 } ) & ( f2 " { 0 } ) " { 0 } = dom f2 ;
gfinite X /\ dom f1 = g1 " X & gX /\ dom f1 = dom g1 /\ dom f1 ;
consider n being element such that n in NAT and Z = G . n ;
set d1 = being thesis , d2 = dist ( x1 , y1 ) , d2 = dist ( x2 , y2 ) ;
b `2 + ( 1 - r ) < ( 1 - r ) * ( 1 - r ) ;
reconsider f1 = f as VECTOR of the carrier of X , the carrier of Y ;
pred i <> 0 means : Def2 : i ^2 mod ( i + 1 ) = 1 ;
j2 in Seg len ( g2 . i2 ) & 1 <= j2 & j2 <= len ( g2 . i2 ) ;
dom ( i * ( i - 1 ) ) = dom ( i * ( i - 1 ) ) .= a ;
cluster sec | ]. PI / 2 , PI / 2 .[ -> one-to-one ;
Ball ( u , e ) = Ball ( f . p , e ) & Ball ( u , r ) c= Ball ( u , r ) ;
reconsider x1 = x0 , y1 = x1 as Function of S , IF ;
reconsider R1 = x , R2 = y , R1 = z as Relation of L , L ;
consider a , b being Subset of A such that x = [ a , b ] ;
( <* 1 *> ^ p ) ^ <* n *> in ( R ^ p ) ;
S1 +* S2 = S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2
( ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * (
cluster -> [. 0 , 1 .] -valued for Function of C , REAL ;
set C7 = 1GateCircStr ( <* z , x *> , f3 ) , C8 = 1GateCircStr ( <* z , x *> , f3 ) ;
Ea1 . e2 = E8 . e2 -T & E8 . e2 = ( ( e | e2 ) | T ) . e2 ;
( ( ( arctan + arccot ) (#) ( ( arccot + arccot ) * ( f ^ ) ) ) `| Z ) = f ;
upper_bound A = ( PI * 3 ) / 2 & lower_bound A = 0 ;
F . ( dom f , - F . ( cod f , - F . ( cod f , - F . ( cod f , - F . ( cod f , - F . ( cod f , - F . ( cod f , - F . ( cod f , - F . ( cod f , - F . ( cod f
reconsider p8 = ( q `1 ) / |. q .| as Point of Euclid 2 ;
g . W in [#] Y0 & [#] Y0 c= [#] Y0 & [#] Y0 c= [#] Y0 & [#] Y0 c= [#] Y0 ;
let C be compact connected non vertical non horizontal Subset of TOP-REAL 2 , p , q be Point of TOP-REAL 2 ;
LSeg ( f ^ g , j ) = LSeg ( f , j ) \/ LSeg ( g , i ) ;
rng s c= dom f /\ ]. x0 - r , x0 .[ & rng s c= dom f /\ ]. x0 , x0 + r .[ ;
assume x in { idseq ( 2 ) , Rev ( idseq ( 2 ) ) } ;
reconsider n2 = n , m1 = m , m2 = n as Element of NAT ;
for y being ExtReal st y in rng seq holds g <= y & g <= x
for k st P [ k ] holds P [ k + 1 ] ;
m = m1 + m2 .= m1 + m2 .= m1 + m2 + m2 .= m1 + m2 + m2 ;
assume for n holds H1 . n = G . n -H . n ;
set Bf = f .: ( the carrier of X1 ) , Bf = f .: ( the carrier of X2 ) ;
ex d being Element of L st d in D & x << d ;
assume R ~ ( a ) c= R ~ ( b ) & R ~ ( a ) c= R ~ ( b ) ;
t in ]. r , s .[ or t = r or t = s or t = s ;
z + v2 in W & x = u + ( z - v2 ) ;
x2 -tree y2 in X iff P [ x2 , y2 ] & P [ y2 , x2 ] ;
pred x1 <> x2 means : Def2 : |. x1 - x2 .| > 0 & |. x1 - x2 .| > 0 ;
assume that p2 - p1 , p3 - p1 - p2 , p3 - p1 - p2 - p1 is_collinear and p2 - p1 , p3 - p2 - p3 - p1 , p3 - p1 - p2 - p3 - p1 - p2 .| ;
set q = ( Ant ( f ) ^ <* 'not' A *> ) ^ <* 'not' A *> ;
let f be PartFunc of REAL-NS 1 , REAL-NS n , g be PartFunc of REAL 1 , REAL-NS n , x0 be Point of REAL-NS n ;
( n mod ( 2 * k ) ) + 1 = n mod ( 2 * k ) ;
dom ( T * ( succ t ) ) = dom ( succ t ) & dom ( T * ( succ t ) ) = dom ( succ t ) ;
consider x being element such that x in wc iff x in c & x in c ;
assume ( F * G ) . ( v . 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 = W-bound L~ Cage ( C , n ) ;
n1 - len f + 1 <= len - len f + 1 - len f + 1 ;
Seg ( q , O1 ) = [ u , v , a , b , b , a , b , c ] ;
set C-2 = ( dom ( { v : not G . ( k + 1 ) } ) ) ;
Partial_Sums ( L * p ) . 0 = 0. R * Sum p .= 0. V .= 0. V ;
consider i being element such that i in dom p and t = p . i and p . i = x ;
defpred Q [ Nat ] means 0 = Q ( $1 ) & $1 <= len Q & Q [ $1 ] ;
set s3 = Comput ( P1 , s1 , k ) , P3 = Comput ( P2 , s2 , k ) , P4 = P2 , P3 = P3 ;
let l be variable of k , A , P be Subset of A , x be element ;
reconsider U1 = union ( Gf1 , T ) as Subset-Family of [: T , T :] ;
consider r such that r > 0 and Ball ( p `1 , r ) c= Q ` ;
( h | ( n + 2 ) ) /. ( i + 1 ) = p29 ;
reconsider B = the carrier of X1 , C = the carrier of X2 as Subset of X ;
por = <* - ( c - 1 ) , 1 , - ( c - 1 ) *> ;
synonym f is real-valued means : Def2 : rng f c= NAT & rng f c= NAT & f is one-to-one ;
consider b being element such that b in dom F and a = F . b and b in dom F ;
x10 < card X0 + card Y0 & ( card Y0 + 1 ) < card Y0 + card Y0 ;
pred X c= B1 means : Def2 : for X st X c= succ B1 holds X in succ B & X is non empty ;
then w in Ball ( x , r ) & dist ( x , w ) <= r ;
angle ( x , y , z ) = angle ( x-y , 0 , PI ) ;
pred 1 <= len s means : Def2 : for s being 2 -w holds s . s = s . ( 2 * s ) ;
f/. i c= f . ( k + ( n + 1 ) ) ;
the carrier of { 1_ G } = { 1_ G } & the carrier of G = { 1_ G } ;
pred p '&' q in - ( p '&' q ) means : Def2 : q '&' p in - ( p '&' q ) ;
- ( t `1 ) < ( t `1 ) ^2 + ( t `2 ) ^2 ;
( U . 1 ) = ( U /. 1 ) `1 .= ( W /. 1 ) `1 .= W . 1 .= W . 1 ;
f .: ( the carrier of x ) = the carrier of x & f .: ( the carrier of x ) = the carrier of x ;
Indices ( O @ ) = [: Seg n , Seg n :] & [: Seg n , Seg n :] = [: 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 \cup ( f * F ) & f is \setminus & f is \setminus implies f is FinSequence of the carrier of [: A , A :]
[ h . 0 , h . 3 ] in the InternalRel of G & [ h . 2 , h . 3 ] in the InternalRel of G ;
s +* Initialize ( ( intloc 0 ) .--> 1 ) = s3 +* Initialize ( ( intloc 0 ) .--> 1 ) ;
|[ w1 , v1 ]| - |[ w1 , v1 ]| <> 0. TOP-REAL 2 & |[ w1 , v1 ]| - |[ w1 , v1 ]| = 0. TOP-REAL 2 ;
reconsider t = t as Element of INT * , X be non empty set ;
C \/ P c= [#] ( ( G \ A ) \ ( [#] ( G \ A ) ) ) ;
f " V in ( the carrier of X ) /\ D & ( the carrier of X ) /\ D = { the carrier of X } ;
x in [#] ( the carrier of ( the carrier of ( F . m ) ) ) /\ the carrier of ( ( F . m ) /\ the carrier of ( F . m ) ) ;
g . x <= h1 . x & h . x <= h1 . x implies g . x <= h1 . x
InputVertices S = { xy , yz , yz , yz } & InputVertices S = { xy , yz , yz } ;
for n being Nat st P [ n ] holds P [ n + 1 ] implies P [ n + 1 ]
set R = Line ( M , i ) , a = Line ( M , i ) , N = Line ( M , i ) ;
assume that M1 is being_line and M2 is being_line and M3 is being_line and M3 is being_line and M2 is being_line ;
reconsider a = f4 . ( i0 -' 1 ) , b = f4 . ( i0 -' 1 ) as Element of K ;
len B2 = Sum ( Len F1 ^ F2 ) .= len ( Len F1 ^ ( Len F2 ) ) ;
len ( ( the _ of n ) * ( i , j ) ) = n & len ( ( f ^ ) ) = n ;
dom ( ( f + g ) (#) f ) = dom ( f + g ) /\ dom f ;
( for n holds seq . n = upper_bound Y1 ) implies ( seq is convergent & lim seq = ( seq ^\ n ) * ( seq ^\ n ) )
dom ( p1 ^ p2 ) = dom f12 & dom ( p1 ^ p2 ) = dom f12 ;
M . [ 1 / y , y ] = 1 / ( 1 * v1 ) * v1 .= 1 / ( 1 * v1 ) .= 1 / ( 1 * v1 ) ;
assume that W is non trivial and W { v } c= the carrier' of G2 and W is trivial ;
C6 /. i1 = G1 * ( i1 , i2 ) & C6 /. i1 = G2 * ( i1 , i2 ) ;
C8 |- 'not' Ex ( x , p ) 'or' p . ( x , y ) ;
for b st b in rng g holds lower_bound rng fwhere fwhere fmeans is Element of REAL n : 0 <= fLet & fover n , b } <= b
- ( ( q1 `1 ) ^2 - ( q1 `2 ) ^2 ) = 1 - ( ( q1 `2 ) ^2 ) .= ( ( q1 `2 ) ^2 - ( q1 `2 ) ^2 ) ;
( LSeg ( c , m ) \/ { l } ) \/ ( LSeg ( l , k ) ) c= R ;
consider p being element such that p in Ball ( x , r ) and p in L~ f and x = f . p ;
Indices ( X @ ) = [: Seg n , Seg n :] & [: Seg n , Seg n :] = [: Seg n , Seg n :] ;
cluster s => ( q => p ) -> valid & ( q => ( s => p ) ) => ( s => q ) is valid ;
Im ( ( Partial_Sums ( F ) ) . m ) is_measurable_on E & ( Im ( ( Partial_Sums ( F ) ) . m ) ) is_measurable_on E ;
cluster f . ( x1 , x2 ) -> Element of D * , f . ( x1 , x2 ) -> Element of D * ;
consider g being Function such that g = F . t and Q [ t , g . t ] ;
p in LSeg ( ( N-min Z ) /. 1 , ( E-max Z ) /. 2 ) & p in LSeg ( ( func \hbox { - } ) , ( E-max Z ) /. 2 ) ;
set R8 = R / ( ]. b , +infty .[ ) , R8 = R / ( ]. a , +infty .[ ) ;
IncAddr ( I , k ) = SubFrom ( d8 , ( d + n ) + k ) .= goto ( ( d + n ) + k ) ;
seq . m <= ( ( the seq of seq ) ^\ k ) . m & ( seq ^\ k ) . m <= ( ( seq ^\ k ) * ( seq ^\ k ) ) . m ;
a + b = ( a *' *' ) *' *' .= ( a *' *' ) *' *' .= a *' *' *' *' .= a *' *' *' ;
id X /\ Y = id X /\ id Y .= id Y /\ id Y .= id Y ;
for x being element st x in dom h holds h . x = f . x & h . x = f . x ;
reconsider H = U1 \/ U2 , U1 = U1 \/ U2 as non empty Subset of U0 ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) /\ f ) /\ j ) /\ m ;
consider y being element such that y in Y and P [ y , inf B ] and y in B ;
consider A being finite stable Subset of R such that card A = ( the carrier of R ) and card A = 1 ;
p2 in rng ( f |-- p1 ) \ rng <* p1 *> & p2 in rng ( f |-- p1 ) \ { p2 } ;
len s1 - 1 > 1-1 & len s2 - 1 > 0 & len s2 - 1 > 0 ;
( ( N-min ( P ) ) `2 ) ^2 = ( ( E-max ( P ) ) ) ^2 .= ( ( E-max ( P ) ) ) ^2 ;
Ball ( e , r ) c= LeftComp Cage ( C , k + 1 ) & L~ Cage ( C , k + 1 ) c= LeftComp Cage ( C , k + 1 ) ;
f . a1 ` = f . a1 ` .= f . a1 ` .= f . a1 ` .= f . a1 ;
( seq ^\ k ) . n in ]. x0 - r , x0 .[ & ( seq ^\ k ) . n in ]. x0 - r , x0 .[ ;
gg . s0 = g . s0 | G . s0 .= g . s0 .= g . s0 ;
the InternalRel of S is \lbrace the carrier of S , the carrier of S , the carrier of S , the carrier of S } ;
deffunc F ( Ordinal , Ordinal ) = phi . $1 & phi . $1 = phi . $1 ;
F . ( s1 . a1 ) = F . ( s2 . a1 ) .= F . ( s2 . a1 ) ;
x `2 = A . ( o . a ) .= Den ( o , A . a ) ;
Cl f " P1 c= f " ( Cl P1 ) & Cl f " ( Cl P2 ) c= f " ( Cl P1 ) ;
FinMeetCl ( ( the topology of S ) \/ the topology of T ) c= the topology of T & the topology of T c= the topology of T ;
synonym o is \bf means : Def2 : o <> o & o <> * ;
assume that X c= Y and card X <> card Y and Y <> {} and Y <> {} ;
the finite the { F of s <= 1 + ( the } \HM { the } \HM { \upharpoonright } s ) . ( len s + 1 ) ;
LIN a , a1 , d or b , c // b1 , c1 or a , c // b1 , c1 ;
e . 1 = 0 & e . 2 = 1 & e . 3 = 0 & e . 4 = 0 ;
EE in SE & not EE in { NE } & not EE in { NE } ;
set J = ( l , u ) If , K = I " ;
set A1 = EqClass ( z , ap , cin ) , A2 = \mathop { A1 , cin } ;
set xy = [ <* xy , cin *> , '&' ] , yz = [ <* cin , cin *> , '&' ] , yz = [ <* cin , cin *> , '&' ] ;
x * z `1 * x " in x * ( z * N ) * x " ;
for x being element st x in dom f holds f . x = g2 . x & f . x = g1 . x ;
Int cell ( f , 1 , G ) c= RightComp f \/ RightComp f & ( L~ f ) misses ( L~ f \/ L~ f ) ;
that U1 is_an_arc_of W-min C , E-max C and L~ ( f ) c= L~ ( f ) and L~ ( f ) = L~ f and L~ ( f ) = L~ f and L~ ( f ) = L~ f and L~ ( f ) = L~ f and L~ ( f ) = L~ f and L~ ( f ) = L~ f and L~ ( f ) = L~ f and
set f-17 = f @ "/\" g @ ;
attr S1 is convergent means : Def2 : S2 is convergent & ( for n holds S2 . n = S2 . n ) implies S2 is convergent & lim S2 = lim S2 ;
f . ( 0 + 1 ) = ( 0 qua Ordinal ) + a .= a + ( 0 qua Ordinal ) .= a + ( 0 qua Nat ) .= a ;
cluster -> \llangle -> \mathclose be in , reflexive transitive non empty transitive strict non empty for non empty reflexive transitive RelStr ;
consider d being element such that R reduces b , d and R reduces c , d and R reduces d , d ;
not b in dom Start-At ( ( card I + 2 ) , SCMPDS ) & not b in dom Start-At ( ( card I + 2 ) , SCMPDS ) ;
( z + a ) + x = z + ( a + y ) .= z + a + y ;
len ( l \lbrack a \rbrack ) = len l & len ( l ) = len l ;
t4 ^ {} is ( {} \/ rng t4 ) -valued FinSequence of ( {} \/ rng t4 ) * * , ( {} \/ rng t4 ) * * , ( {} \/ rng t4 ) * , ( {} , rng t4 ) * ) * ) -valued Function ;
t = <* F . t *> ^ ( C . ( p ^ q ) ) .= ( C ^ ( p ^ q ) ) ^ ( C ^ q ) ;
set p-2 = W-min L~ Cage ( C , n ) , p`2 = W-bound L~ Cage ( C , n ) ;
( k -' ( i + 1 ) ) = ( k - ( i + 1 ) ) + ( i - ( i + 1 ) ) ;
consider u being Element of L such that u = u ` and u in D ` and u in D ` ;
len ( ( width ( ( ( B - A ) |-> a ) * ( A @ ) ) ) @ ) = width ( ( B - A ) |-> a ) ;
FM . x in dom ( ( G * the_arity_of o ) . x ) ;
set cH2 = the carrier of H2 , cH2 = the carrier of H2 ;
set cH1 = the carrier of H1 , cH2 = the carrier of H2 ;
( Comput ( P , s , 6 ) ) . intpos ( m + 6 ) = s . intpos ( m + 6 ) ;
IC Comput ( Q2 , t , k ) = ( l + 1 ) + 1 .= ( l + 1 ) ;
dom ( ( ( cos * sin ) `| Z ) ) = REAL & dom ( ( cos * cos ) `| Z ) = REAL ;
cluster <* l *> ^ phi -> ( 1 + 1 ) -element for string of S ;
set b9 = [ <* ap , bm *> , [ <* A1 , cin *> , [ <* cin , cin *> , [ <* cin , cin *> , [ <* cin , cin *> , [ <* cin , cin *> , [ <* cin , cin *> ] ] ] , [ <* cin , y2 *> , [ <* cin , y2 *> , [ <* cin , y2 *> ] ]
Line ( Segm ( M @ , P @ , x ) , x ) = L * ( Sgm Q ) ;
n in dom ( ( the Sorts of A ) * ( the_arity_of o ) ) & ( the Sorts of A ) . n in dom ( ( the Sorts of A ) * ( the_arity_of o ) ) ;
cluster f1 + f2 -> continuous for PartFunc of REAL , the carrier of S , the carrier of S ;
consider y be Point of X such that a = y and ||. x-y .|| <= r and ||. y - x .|| <= r ;
set x3 = t8 . DataLoc ( s . SBP , 2 ) , x4 = s . SBP , x5 = s . SBP , P4 = s . SBP , P4 = s . SBP , P4 = s . SBP , P4 = s . SBP , \mathbin { IC } } ;
set p-3 = stop I , pI = stop I , pI = stop I , pI = stop I , pI = stop I , pI = stop I , pI = stop I , pI = stop I , II = stop I , pI = stop I , II = stop I ;
consider a being Point of D2 such that a in W1 and b = g . a and a in W2 ;
{ A , B , C , D , E } = { A , B , C } \/ { D , E , F , J }
let A , B , C , D , E , F , J , M , N , N , F , J , M , N , N , F , J , M , N , N , F , J , M , N , N , F , N , N , F , J , M , N , N , F ,
|. p2 .| ^2 - ( p2 `1 ) ^2 - ( p2 `2 ) ^2 >= 0 ;
l - 1 + 1 = n-1 * ( ( m + 1 ) + 1 ) + 1 ;
x = v + ( a * w1 + ( b * w2 ) ) + ( c * w2 ) ;
the TopStruct of L = reconsider the TopStruct of L , the TopStruct of L = the TopStruct of L , the TopStruct of L = the TopStruct of L ;
consider y being element such that y in dom H1 and x = H1 . y and y in H1 . x ;
fv \ { n } = Free ( All ( x , H ) ) \ { n } & fv \ { n } = Free ( H ) ;
for Y being Subset of X st Y is summable & Y is iff Y is \overline of Y
2 * n in { N : 2 * Sum ( p | N ) = N & N > 0 } ;
for s being FinSequence holds len ( the { of s } * the { - } ) = len s & len s = len s
for x st x in Z holds exp_R * f is_differentiable_in x & ( exp_R * f ) . x = 1
rng ( h2 * ( f ^ ) ) c= the carrier of ( ( TOP-REAL 2 ) | ( the carrier of TOP-REAL 2 ) ) ;
j + ( len f ) - len f <= len f + ( len f - len f ) - len f ;
reconsider R1 = R * I , R2 = R * I as PartFunc of REAL , REAL-NS n , REAL-NS n ;
C8 . x = s1 . ( a - 0 ) .= C8 . x - ( s1 - s2 ) . x .= ( s1 - s2 ) . x ;
power F_Complex = 1 / ( z , n ) .= 1 / ( x |^ n ) .= x |^ n .= x |^ n ;
t at ( C , s ) = f . ( the that ( 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 Element of [: X1 , X2 :] : x1 in X } is Subset of [: X1 , X2 :] ;
h = ( i , j ) -tree h , id B = H . i , id B = H . i ;
ex x1 being Element of G st x1 = x & x1 * N c= A & x1 in A & x1 in B ;
set X = ( ( Seg ( q , O1 ) ) , ( a , b ) ) `1 , Y = ( a , b ) `1 , Z = ( a , b ) `2 , Y = ( a , b ) `1 , Z = ( a , b ) `1 , Z = ( a , b ) `1 , Z = ( a , b ) `1 , Z = ( a , b ) `1 , Z =
b . n in { g1 : x0 < g1 & g1 < x0 & g1 < x0 + r } ;
f /* s1 is convergent & f /. x0 = lim ( f /* s1 ) implies f /* s1 is convergent & lim ( f /* s1 ) = lim ( f /* s1 )
the lattice of Y = the lattice of the lattice of Y & the lattice of Y = the lattice of the open of Y implies Y is open
'not' ( a . x ) '&' b . x 'or' a . x '&' 'not' ( b . x ) = FALSE ;
s2 = ( len ( q1 ^ r1 ) ) + len ( q1 ^ r2 ) .= len ( q1 ^ r2 ) + len ( q2 ^ r2 ) ;
( 1 / a ) (#) ( sec * f1 ) - ( ( sec * f1 ) - ( sec * f2 ) ) is_differentiable_on Z ;
set K1 = upper_volume ( f , ( lim H ) || A ) , K1 = ( lim H ) || A , K1 = ( lim H ) || A ;
assume e in { ( w1 - w2 ) / 2 : w1 in F & w2 in G } ;
reconsider d7 = dom a `1 , d8 = dom F as finite non empty set ;
LSeg ( f /^ ( j -' 1 ) , j ) = LSeg ( f , j + q .. f ) ;
assume X in { T . ( N2 , N2 ) : h . ( N2 , N2 ) = N2 } ;
assume that Hom ( d , c ) <> {} and <* f , g *> * f1 = <* f , g *> * f1 ;
dom S29 = dom S /\ Seg n .= dom ( L * ( S * ( S * ( S * ( S * ( S * ( S * ( S * ( S * ( S * ( S * ( S * ( S * ( S * ( S * ( S * ( S * ( S * ( S * ( S * ( S * ( S * ( S *
x in H |^ a implies ex g st x = g |^ a & g in H & a in H
a * ( - n ) . ( a , 1 ) = a `1 - ( 0 * n ) .= a `1 - ( 0 * n ) .= a `1 - ( 0 * n ) ;
D2 . ( j - 1 ) in { r : lower_bound A <= r & r <= D1 . i } ;
ex p being Point of TOP-REAL 2 st p = x & P [ p ] & ( for p being Point of TOP-REAL 2 st p in P holds p `1 <= 0 ) ;
for c holds f . c <= g . c implies f ^ @ c < g ^ @ c
dom ( f1 (#) f2 ) /\ X c= dom ( f1 (#) f2 ) /\ dom ( f1 (#) f3 ) ;
1 = ( p * p ) * ( 1 - p ) .= p * 1 - p * ( 1 - p ) .= p * 1 - p * 1 ;
len g = len f + len <* x + y *> .= len f + 1 .= len f + 1 .= len f + 1 ;
dom F-11 = dom ( F | ( N1 ~ ) ) & rng ( F | ( N1 ~ ) ) = ( N1 ~ ) ;
dom ( f . t * I . t ) = dom ( f . t * g . t ) ;
assume a in ( "\/" ( ( T |^ the carrier of S ) , F ) ) .: D ;
assume that g is one-to-one and ( the carrier' of S ) /\ rng g c= dom g and g 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 " = id b and f * f = id a and f is one-to-one ;
( ( cos | [. 2 * PI , 0 .] ) | [. 2 * PI , 0 + PI .] ) | [. 2 , PI .] is increasing ;
Index ( p , co ) <= len LS - Index ( Gij , LS ) + 1 - Index ( Gik , LS ) ;
t1 , t2 , Z be Element of ( S , s ) * , s be Element of ( S , s ) * ;
"/\" ( ( Frege ( curry H ) ) . h , L ) <= "/\" ( rng ( Frege ( curry H ) ) , L ) ;
then P [ f . i0 ] & F ( f . ( i0 + 1 ) ) < j & j < len f ;
Q [ [ D . ( x , 1 ) , F . ( D . ( x , 1 ) ) ] ;
consider x being element such that x in dom ( F . s ) and y = ( F . s ) . x ;
l . i < r . i & [ l . i , r . i ] is for i holds r . i is a < G . i ;
the Sorts of A2 = ( the carrier of S2 ) --> ( the carrier of S2 ) .= ( the carrier of S2 ) --> ( the carrier of S2 ) ;
consider s being Function such that s is one-to-one and dom s = NAT and rng s = F . s and rng s c= dom F ;
dist ( b1 , b2 ) <= dist ( b1 , a ) + dist ( a , b2 ) + dist ( a , b2 ) ;
( Lower_Seq ( C , n ) ) /. len ( Lower_Seq ( C , n ) ) = WU . 1 .= W ;
q <= ( UMP Upper_Arc C ) `2 & ( UMP C ) `2 <= ( UMP C ) `2 ;
LSeg ( f | i2 , i ) /\ LSeg ( f | i2 , j ) = {} implies LSeg ( f | i2 , i ) = {}
given a being ExtReal such that a <= II and A = ]. a , I .[ and a in A and b in B ;
consider a , b being complex number such that z = a & y = b and z + y = a + b ;
set X = { b |^ n where n is Element of NAT : n <= len b } , Y = { b |^ n where n is Element of NAT : n <= len b } ;
( ( x * y * z ) \ x ) \ z = 0. X ;
set xy = [ <* xy , yz , yz *> , f1 ] , yz = [ <* xy , yz *> , f2 ] , yz = [ <* xy , yz *> , f3 ] , xy = [ <* xy , yz *> , f3 ] ;
ll /. len ll = ( l . len ( l | i ) ) * ( l | i ) ;
( ( q `2 ) ^2 - ( |. q .| ) ^2 ) * ( ( q `2 ) ^2 ) = 1 ;
( ( ( p `1 ) / |. p .| - cn ) / ( 1 + cn ) ) * ( 1 + cn ) < 1 ;
( ( ( S \/ Y ) --> ( x , y ) ) ) `2 = ( ( S \/ Y ) --> ( x , y ) ) `2 ;
( seq - seq ) . k = seq . k - seq . ( k + 1 ) .= seq . k - seq . ( k + 1 ) ;
rng ( ( h + c ) ^\ n ) c= dom SVF1 ( 1 , f , u0 ) /\ dom SVF1 ( 1 , f , u0 ) ;
the carrier of X = the carrier of X & the carrier of Y = the carrier of X & the carrier of Y = the carrier of Y implies X = Y
ex p4 st p3 = p4 & |. p3 - |[ a , b ]| .| = r & |. p3 - |[ a , b ]| .| = r ;
set ch = chi ( X , A5 ) , A5 = chi ( X , A5 ) ;
R / ( 0 * n ) = Ilet ( X , X ) to_power 0 .= R / ( 0 * n ) .= R / ( 0 * n ) ;
( Partial_Sums ( curry ( F1 , n ) ) . n ) . x is nonnegative & ( Partial_Sums ( F1 , n ) ) . x <= ( Partial_Sums ( F1 , n ) ) . x ;
f2 = C7 . ( ( the EEof V , len ( V , K ) ) . ( len ( V , K ) ) ) ;
S1 . b = s1 . b .= s2 . b .= S2 . b .= S2 . b .= S2 . b .= S2 . b ;
p2 in LSeg ( p2 , p1 ) /\ LSeg ( p2 , p1 ) & p2 in LSeg ( p2 , p1 ) /\ LSeg ( p1 , p2 ) ;
dom ( f . t ) = Seg n & dom ( I . t ) = Seg n & rng ( I . t ) = Seg n ;
assume o = ( the connectives of S ) . 11 & the carrier' of S = ( the carrier' of S ) . 11 ;
set phi = ( l1 , l2 ) \mathop { l1 } , phi = ( l2 , l2 ) \mathop { l2 } , C = ( l2 , l2 ) \mathop { l2 } , D = ( l2 , l2 ) \mathop { l2 } , E = ( X , l2 ) . { l2 } , F = ( X , l2 ) . { l2 } , F = ( X , l2 ) . { l2 } , D =
synonym p is invertible for ( p , T ) " , ( p , T ) " , ( p , T ) " ;
( Y1 `2 = - 1 & ( Y1 `2 = 1 & Y2 `2 = 1 or Y1 `1 = 1 & Y1 `2 = 1 ) & ( Y1 `2 = 1 implies Y1 `2 = 1 ) ;
defpred X [ Nat , set , set ] means P [ $1 , $2 , , , , , ] & P [ $1 , $2 , $2 , $2 , $2 ] ;
consider k be Nat such that for n be Nat st k <= n holds s . n < x0 + g and x0 < g ;
Det ( I @ ) * ( m - n ) = 1. ( K , n ) & ( I @ ) * ( m - n ) = 1. ( K , n ) ;
( - b - sqrt ( b - a ^2 - ( 4 * a * c ) ) / ( 2 * a * c ) ) < 0 ;
Ci . d = C7 . d mod ( d7 . d ) .= C7 . d mod ( C7 . d ) .= C8 . d mod ( C7 . d ) ;
attr X1 is dense means : Def2 : X2 is dense & X2 is dense implies X1 /\ X2 is dense SubSpace of X & X2 /\ X1 is dense SubSpace of X ;
deffunc F6 ( Element of E , Element of I , Element of I ) = $1 * $2 & $2 = ( $1 * $2 ) * ( $2 * $2 ) ;
t ^ <* n *> in { t ^ <* i *> : Q [ i , T . ( t `1 ) ] } ;
( x \ y ) \ x = ( x \ x ) \ y .= y ` \ 0. X .= 0. X .= 0. X ;
for X being non empty set for T being Subset-Family of X holds the topology of [: X , Y :] is Basis of [: X , Y :]
synonym A , B are_separated for Cl ( A \/ B ) , Cl ( A \/ B ) , Cl ( A \/ B ) ` , Cl ( A \/ B ) ` ;
len ( ( - 1 ) * ( - 1 ) ) = len p & len ( - 1 ) * ( - 1 ) = len p & len ( - 1 ) * ( - 1 ) = len p ;
J = { x where x is Element of K : 0 < v . x & x < v . x } ;
( Sgm ( Seg m ) ) . d - ( Sgm ( Seg m ) ) . e <> 0 ;
lower_bound divset ( D2 , k + k2 ) = D2 . ( k + k2 - 1 ) .= D2 . ( k + k2 - 1 ) ;
g . r1 = - 2 * r1 + 1 & dom h = [. 0 , 1 .] & rng h c= [. 0 , 1 .] ;
|. a .| * ||. f .|| = 0 * ||. f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| ;
f . x = ( h . x ) `1 & g . x = ( h . x ) `2 ;
ex w st w in dom B1 & <* 1 *> ^ s = <* 1 *> ^ w & len w = len b1 + len b2 ;
[ 1 , {} , <* d1 *> ] in ( { [ 0 , {} , {} ] } \/ S1 ) \/ S2 ;
IC Exec ( i , s1 ) + n = IC Exec ( i , s2 ) .= IC Exec ( i , s2 ) ;
IC Comput ( P , s , 1 ) = IC ( s , 9 ) .= 5 + 9 .= 5 + 9 .= ( card I + 1 ) .= ( card I + 1 ) ;
( IExec ( W6 , Q , t ) ) . intpos i = t . intpos i .= t . intpos i .= t . intpos i ;
LSeg ( f /^ ( i -' 1 ) , i ) misses LSeg ( f /^ ( i -' 1 ) , j ) ;
assume for x , y being Element of L st x in C & y in C holds x <= y or y <= x ;
integral ( f , C ) . x = f . ( upper_bound C ) - f . ( lower_bound C ) .= f . ( lower_bound C ) - f . ( lower_bound C ) ;
for F , G being one-to-one FinSequence st rng F misses rng G holds F ^ G is one-to-one & F ^ G is one-to-one
||. R /. ( L . h ) - R /. ( K + 1 ) .|| < e1 * ( K + 1 ) ;
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 ` in Y ` & x in X ` holds y ` <= x ` ;
func |. p \bullet |. p .| -> variable equals : Def2 : ( p . 1 ) . ( ( - NBI ) . ( p . 1 ) ) ;
consider t being Element of S such that x `1 , y `2 '||' z `1 , t `2 and x `1 , y `2 '||' y `1 , t `2 ;
dom x1 = Seg len x1 & len x1 = len x2 & len y1 = len x2 & len y1 = len x2 & len y1 = len y2 ;
consider y2 being Real such that x2 = y2 and 0 <= y2 and y2 <= 1 and y2 <= 1 and y2 <= 1 ;
||. f | X /* s1 .|| = ||. f .|| | X & ||. f .|| | X = ( 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 i + 1 in dom q and j + 1 in dom q ;
reconsider h = f | X ( ) , g = f | X ( ) as Function of X ( ) , Y ( ) ;
u1 in the carrier of W1 & u2 in the carrier of W2 & u2 in the carrier of W1 implies u1 + u2 in the carrier of W1 + W2
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 - ( s * x ) + y .= b - ( s * x ) ;
- ( x-y ) = - x + - y .= - x + - y .= - x + y .= - x + y .= - x + y .= x + y ;
given a being Point of GX such that for x being Point of GX holds a , x , a , x is_collinear and a , x , x , y is_collinear ;
f\lbrace x , y , cod ( f * f2 ) , y , cod ( f * g2 ) , y , cod ( f * g2 ) , y , cod ( f * g2 ) ) = [ x , y , f . x ] ;
for k , n being Nat st k <> 0 & k < n & n is prime & k , n are_relative_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 ) ` & ( ( A ` ) |^ d ) ` = ( A ` ) `
consider u , v being Element of R , a being Element of A such that l /. i = u * a * v ;
( - ( ( p `1 / |. p .| - cn ) / ( 1 + cn ) ) ^2 ) > 0 ;
L-13 . k = Lw . ( F . k ) & F . k in dom ( L * F ) ;
set i2 = AddTo ( a , i , - n ) , i1 = goto - ( n + 1 ) ;
attr B is \frac means : Def2 : ( for B holds B in Subrng ( B , Sv ) ) & ( B is non empty or B is non empty ) ;
a9 "/\" D = { a "/\" d where d is Element of N : d in D } & a "/\" d in D ;
|( \square , q29 )| * |( q1 , q2 )| * |( q2 , q1 )| >= |( \square , 1 )| * |( q2 , q2 )| ;
( - f ) . ( upper_bound A ) = ( ( - f ) | A ) . ( upper_bound A ) .= f . ( upper_bound 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 . LM ) *> .= ( Proj ( i , n ) ) . ( LM . LM ) ;
f1 + f2 * reproj ( i , x ) - ( reproj ( i , x ) ) . i = ( ( ( ( - 1 ) (#) reproj ( i , x ) ) ^ ) ) . i ;
pred ( ( tan (#) tan ) `| Z ) . x <> 0 & ( tan (#) tan ) . x = 1 & ( tan (#) tan ) . x = 1 ;
ex t being SortSymbol of S st t = s & h1 . t = h2 . t & ( h1 . t ) . x = h2 . x ;
defpred C [ Nat ] means P8 . $1 is non empty & A8 is non empty & A8 is non empty & A8 is non empty ;
consider y being element such that y in dom ( p9 | i ) and q9 . i = p9 . y and y in dom ( p9 | i ) and x = ( p9 | i ) . y ;
reconsider L = product ( { x1 } +* ( index B , l ) ) as Basis of A . ( index A ) ;
for c being Element of C ex d being Element of D st T . ( id c ) = id d & for d being Element of D holds T . ( id d ) = id d
be Element of ( f | n ) ^ , p be Element of ( f | n ) ^ , q be Element of ( f | n ) * ;
( f * g ) . x = f . ( g . x ) & ( f * h ) . x = f . ( h . x ) ;
p in { ( 1 - 2 ) * ( G * ( i + 1 , j ) + G * ( i + 1 , j + 1 ) ) } ;
f `2 - p `2 = ( f | ( n , L ) ) *' - ( f *' ) . ( - p ) .= ( f - ( - ( - ( f . p ) ) ) ) ) . p ;
consider r being Real such that r in rng ( f | divset ( D , j ) ) and r < m + s ;
f1 . ( |[ r2 , s2 ]| ) in f1 .: W1 & f2 . ( |[ r2 , s2 ]| ) in f1 .: W2 & ( f1 .: W2 ) c= f1 .: W2 ;
eval ( a | ( n , L ) , x ) = ( a | ( n , L ) ) . x .= a * ( x * x ) .= a * x ;
z = DigA ( tz , x ) .= DigA ( tz , x ) .= DigA ( tz , x ) .= DigA ( tz , x ) ;
set H = { Intersect S where S is Subset-Family of X : S c= G & S c= G } , F = { Intersect S where S is Subset-Family of X : S c= G } , G = { Intersect S where S is Subset of X : S c= G } ;
consider S19 being Element of D * , d being Element of D * such that S `1 = S19 ^ <* d *> and S29 in 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 `1 ) / |. q .| - cn ) / ( 1 + cn ) ;
( ( for v be VECTOR of V holds v in A ) & ( for v be VECTOR of V holds v in A iff v in A ) implies Sum ( L (#) F ) = Sum ( L (#) F )
let k1 , k2 , k2 , x4 , x5 , 7 , 8 , 7 , 8 , 8 , 7 , 8 be Element of NAT ;
consider j being element such that j in dom a and j in g " { k `2 } and x = a . j and a . j = b . j ;
H1 . x1 c= H1 . x2 or H1 . x2 c= H1 . x2 or H1 . x2 c= H1 . x2 & H1 . x2 c= H1 . x2 & H1 . x2 c= H1 . x2 ;
consider a being Real such that p = \hbox { a * p1 + ( a * p2 ) * p2 and 0 <= a and a <= 1 and a <= 1 ;
assume that a <= c & c <= d and [' a , b '] c= dom f and [' a , b '] c= dom g and f | [' a , b '] is bounded ;
cell ( Gauge ( C , m ) , 1 , ( X -' 1 ) -' 1 ) is non empty & cell ( Gauge ( C , m ) , 1 , 0 ) is non empty ;
A\HM { A where i is Element of NAT : i in { S . i where i is Element of NAT : not contradiction } c= { S . i where i is Element of NAT : not contradiction }
( T * b1 ) . y = L * b2 /. y .= ( F * b1 ) . y .= ( F * b1 ) . y .= ( F * b1 ) . y ;
g . ( s , I ) . x = s . y & g . ( s , I ) . y = |. s . x - s . y .| ;
( log ( 2 , k + 1 ) ) to_power ( k + 1 ) >= ( log ( 2 , k + 1 ) ) to_power ( k + 1 ) ;
then p => q in S & not x in the still of p & not x in S & not x in S & x in S ;
dom ( the InitS of r-10 ) misses dom ( the InitS of rM ) & dom ( the InitS of rM ) misses dom ( the InitS of rM ) ;
synonym f is ExtReal means : Def2 : for for x being set st x in rng f holds x is ExtReal ;
assume for a being Element of D holds f . { a } = a & for X being Subset-Family of D holds f . ( f .: X ) = f . union X ;
i = len p1 .= len p3 + len <* x *> .= len p3 + len <* x *> .= len p3 + 1 .= len p3 + 1 .= len p3 + 1 ;
( l - 1 ) * ( k - 3 ) = ( g - k ) * ( k - 3 ) + ( k - 3 ) * ( k - 3 ) .= ( g - k ) * ( k - 3 ) ;
CurInstr ( P2 , Comput ( P2 , s2 , l ) ) = halt SCM+FSA .= halt SCM+FSA .= ( halt SCM+FSA ) . l .= ( CurInstr ( P2 , s2 ) ) ;
assume for n be Nat holds ||. seq .|| . n <= ( ||. seq .|| ) . n & ( ||. seq .|| ) . n <= ( ||. seq .|| ) . n ;
sin . ( 0. X ) = sin . ( cos . ( - PI ) * cos . ( - PI ) ) .= sin . ( - PI ) * cos . ( - PI ) .= 0 ;
set q = |[ g1 `1 / ( t `2 ) ^2 , g2 `2 / ( t `1 ) ^2 ]| , r = |[ g1 `1 / ( t `2 ) ^2 , g2 `2 / ( t `2 ) ^2 ]| ;
consider G being sequence of S such that for n being Element of NAT holds G . n in WAsubsets ( F . n ) and G . n = B . n ;
consider G such that F = G and ex G1 st G1 in SM & G = [: G1 , G2 :] & G1 in SM & G2 in SM ;
the root of [ x , s ] in ( the Sorts of Free ( C , X ) ) . s & ( the Sorts of Free ( C , X ) ) . s in ( the Sorts of Free ( C , X ) ) . s ;
Z c= dom ( exp_R * ( f + ( #Z 3 ) * ( f + #Z 3 ) ) ) ;
for k be Element of NAT holds r0 . k = ( ( Im ( Im ( f , S ) ) ) . k ) * ( ( Im ( f , S ) ) . k )
assume that - 1 < n and n > 0 and ( q `2 <= 1 or q `1 < 1 & q `2 <= 1 or q `1 < 1 & q `1 <= 1 ) ;
assume that f is continuous and a < b and f is continuous and c < d and f . a = c and f . b = d and f . c = d ;
consider r being Element of NAT such that s\mathopen { + } ( P1 , s1 ) , r ) = Comput ( P1 , s1 , r ) and r <= q ;
LE f /. ( i + 1 ) , f /. ( j + 1 ) , L~ f , f /. ( i + 1 ) , L~ f ;
assume that x in the carrier of K and y in the carrier of K and ex_inf_of { x , y } , L and x in the carrier of K and y in the carrier of K ;
assume f +* ( i1 , \xi ) in ( proj ( F , i2 ) ) " ( ( proj ( F , i2 ) ) " ( A7 ) ) . ( ( f . ( A . ( A . ( B . ( A . ( B . ( A . ( B . ( A . ( A . ( A . ( A . ( A . ( A . ( A . ( A . ( A . ( A . ( A . ( A . (
rng ( ( ( Flow M ) | ( the carrier of M ) ) | ( the carrier' of M ) ) c= the carrier' of M & ( ( Flow M ) | ( the carrier' of M ) ) | ( the carrier' of M ) c= the carrier' of M ;
assume z in { ( the carrier of G ) \ { t } where t is Element of T : t in 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 be VECTOR of product G such that mt = ||. Dt . t .|| and ||. t .|| <= 1 ;
assume that the carrier of v = 2 and v ^ <* 0 *> in dom p and v ^ <* 1 *> in dom p and p . 1 = 0 ;
consider a being Element of the Points of Xbe that not a on ( the Points of X29 ) and not a on ( the Points of X29 ) ;
( - x ) |^ ( k + 1 ) * ( ( - x ) |^ ( k + 1 ) ) " = 1 ;
for D being set for i st i in dom p holds p . i in D & p . i in D implies p . i in D
defpred R [ element ] means ex x , y st [ x , y ] = $1 & P [ x , y ] & P [ y , x ] ;
L~ f2 = union { LSeg ( p0 , p10 ) , LSeg ( p1 , p10 ) } .= { LSeg ( p10 , p2 ) , LSeg ( p1 , p10 ) } ;
i - len h11 + 2 - 1 < i - len h11 + 2 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 < i - len h11 + 1 - 1 ;
for n being Element of NAT st n in dom F holds F . n = |. ( F . ( n -' 1 ) ) . n .| ;
for r , s1 , s2 , s2 , s3 being Real holds s1 in [. s1 , s2 .] iff s1 <= s2 & s2 <= s2 & s1 <= s2 & s2 <= s2 & s2 <= s1 & s1 <= s2
assume v in { G where G is Subset of T2 : G in B2 & G c= B1 & G c= B2 & A c= G & B c= G } ;
let g be ) :] non-empty Element of A , X , Y be Element of INT , b be Element of INT , f be Function of X , Y ;
min ( g . [ x , y ] , k ) . [ y , z ] = ( min ( g , k , x ) ) . y ;
consider q1 being sequence of CP such that for n holds P [ n , q1 . n ] and for n holds q1 . n = F ( 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 /\ O ) , Z = O /\ ( O /\ O ) as Subset of B ;
consider j being Element of NAT such that x = ( the ` of n ) * ( j , i ) and 1 <= j and j <= n and i <= n ;
consider x such that z = x and card ( x . O2 ) in card ( x . O2 ) and x in ( x . O2 ) . x and x in ( x . O2 ) . x ;
( C * dom _ 4 ( k , n2 ) ) . 0 = C . ( ( dom ( T . k ) ) . 0 ) .= C . ( ( T . 0 ) . 0 ) ;
dom ( X --> rng f ) = X & dom ( X --> f ) = dom ( X --> f ) & rng ( X --> f ) = X ;
( N-bound L~ SpStSeq C ) <= ( ( SpStSeq L~ SpStSeq C ) * ( ( SpStSeq L~ SpStSeq C ) * ( ( SpStSeq L~ SpStSeq C ) * ( ( SpStSeq L~ SpStSeq C ) * ( ( SpStSeq L~ SpStSeq C ) * ( ( SpStSeq L~ Cage C ) * ( i , 1 ) ) + ( S-bound L~ Cage ( C , n ) ) * ( i , 1 ) ) ) ) ;
synonym x , y , z means : Def2 : x = y or ex l being Subset of S st { x , y } c= l & x in l & y in l ;
consider X be 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 for a , b being Element of Im k st a = x & b = y holds x << y iff a << b & b << a ;
( 1 - ( 2 * ( ( ( - ( m - n ) ) * ( ( m - n ) ) * ( ( m - n ) ) * ( ( m - n ) ) * ( ( m - n ) * ( ( m - n ) * ( ( m - n ) * ( ( m - n ) * ( ( m - n ) * ( ( m - n ) * ( m - n ) ) * ( ( m - n ) ) * ( ( m - n ) ) ) ) ) ) ) ) ) ) ) ) / ( m -
defpred P [ Element of omega ] means ( the partial of A1 ) . $1 = A1 . $1 & ( the partial of A2 ) . $1 = A2 . $1 & ( the partial of A1 ) . $1 = A2 . $1 ;
IC Comput ( P , s , 2 ) = succ IC Comput ( P , s , 1 ) .= 6 + 1 .= ( card I + 1 ) .= 6 ;
f . x = f . g1 * f . g2 .= f . g1 * 1_ H .= f . g1 * 1_ H .= f . g1 * ( f . g2 ) .= f . g1 * ( f . g2 ) ;
( M * ( F . n ) ) . n = M . ( ( ( canFS ( Omega ) ) . n ) ) .= M . ( ( ( canFS ( Omega ) ) . n ) ) .= M . ( ( ( canFS ( Omega ) ) . n ) ) ;
the carrier of L1 + L2 c= ( the carrier of L1 ) \/ ( the carrier of L2 ) & the carrier of L1 + L2 c= the carrier of L2 implies L1 + L2 = L2 + L1
pred a , b , c , x , y , z , x , y , z , x , y , z be set ;
( the partial of s ) . n <= ( the partial of s ) . n * s . n & ( the partial of s ) . n <= ( the partial of s ) . n * s . n ;
pred - 1 <= r & r <= 1 & ( ( - 1 ) (#) ( arccot ) ) `| Z = - ( 1 / r ) / ( 1 + r ^2 ) ;
seq in { p ^ <* n *> where p is Nat : p ^ <* n *> in T1 & p ^ <* n *> in T1 } ;
|[ x1 , x2 , x3 , x4 ]| . 2 - |[ y1 , y2 , x4 ]| . 2 = x2 - y2 & |[ x1 , x2 , x3 , x4 ]| . 2 = x2 - y2 ;
attr for m be Nat holds F . m is nonnegative means : Def2 : ( Partial_Sums F ) . m is nonnegative & ( Partial_Sums F ) . m is nonnegative ;
len ( ( ( G . z ) + ( G . ( x , y ) ) ) ) = len ( ( G . ( x , y ) ) + ( G . ( y , z ) ) ) .= len ( ( G . ( x , y ) ) + ( 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 u in W1 /\ W2 ;
given F be FinSequence of NAT such that F = x and dom F = n and rng F c= { 0 , 1 } and Sum F = k and Sum ( F ) = k ;
0 = ( 1 * or 0 = ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - 0 ) ) ) ) ) ) * ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - 0 ) ) ) ) ) ) ) ;
consider n be Nat such that for m be Nat st n <= m holds |. ( f # x ) . m - ( f # x ) . n .| < e ;
cluster -> Boolean non empty for \hbox { $ ( ( implies ( ( u ) ) , ( u ) ) *> , ( ( u ) ) --> ( ( u ) ) , ( ( u ) ) --> ( ( u ) ) , ( ( u ) ) --> ( ( u ) ) , ( ( u ) ) --> ( ( u ) ) ) ;
"/\" ( BB , {} ) = Top ( B ) .= Top S .= "/\" ( [#] S , {} ) .= "/\" ( [#] S , {} ) .= "/\" ( [#] S , {} ) .= "/\" ( {} , {} , {} ) ;
( r / 2 ) ^2 + ( r / 2 ) ^2 <= ( r / 2 ) ^2 + ( r / 2 ) ^2 ;
for x being element st x in A /\ dom ( f `| X ) holds ( f `| X ) . x >= r2 & ( f `| X ) . x >= r2
2 * r1 - ( 2 * r1 - ( 2 * r1 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - r2 ) ) ) ) ) ) ) ) ) ) ) )
reconsider p = P * ( 1 , 1 ) , q = a " * ( ( - ( - ( - ( K , n , 1 ) ) ) ) * ( ( - ( K , n , 1 ) ) ) ) as FinSequence of K ;
consider x1 , x2 being element such that x1 in uparrow s and x2 in uparrow t and x = [ x1 , x2 ] and x2 in uparrow t and t . x1 = [ x1 , x2 ] ;
for n be Nat st 1 <= n & n <= len q1 holds q1 . n = ( ( upper_volume ( g , M7 ) ) . n ) * ( ( upper_volume ( g , 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 and y = H2 and H1 , H2 |^ g |^ f |^ g |^ f = H2 |^ f and H1 , H2 |^ g |^ f = H2 |^ f ;
for S , T being non empty RelStr , d being Function of T , S st d is complete holds d is monotone iff d is monotone & d is monotone
[ a + 0 , b + i ] in ( the carrier of COMPLEX ) /\ ( the carrier of V ) & [ a , b ] in ( the carrier of F_Complex ) /\ the carrier of V ;
reconsider mm = max ( len F1 , len ( p . n ) * ( <* x *> |^ n ) ) as Element of NAT ;
I <= width ( ( GoB ( ( GoB ( ( h ) ) * ( 1 , 1 ) ) ) ) ) & ( ( GoB ( ( GoB ( ( h ) ) * ( 1 , 1 ) ) ) ) ) * ( ( GoB ( ( h ) ) * ( 1 , 1 ) ) ) ) <= ( ( GoB ( ( h ) ) * ( 1 , 1 ) ) ) ;
f2 /* q = ( f2 /* ( f1 /* s ) ) ^\ k .= ( ( f2 * f1 ) /* s ) ^\ k .= ( ( ( f2 * f1 ) /* s ) ^\ k ) ^\ k .= ( ( ( f2 * f1 ) /* s ) ^\ k ) ^\ k ;
attr A1 \/ A2 is linearly-independent means : Def2 : A1 is linearly-independent & A2 misses A1 & ( for x being Element of V holds x in A1 & x in A2 implies x in A1 & x in A2 ;
func A -carrier C -> set equals union { A . s where s is Element of R : s in C & s in C } ;
dom ( ( Line ( v , i + 1 ) ) ^ ( ( Line ( p , m ) ) * ( Line ( p , 1 ) ) ) ) = dom ( F ^ ) .= dom ( F ^ ) ;
cluster [ x , 4 ] -> non empty & [ x , 4 ] in [: { x } , { x } :] & [ x , 4 ] in [: { x } , { x } :] & [ x , 4 ] in [: { x } , { x } :] ;
E , All ( x2 , All ( x2 , All ( x2 , x2 ) ) ) |= All ( x2 , All ( x2 , x2 ) ) => ( x2 , All ( x2 , x2 ) ) ;
F .: ( id X , g ) . x = F . ( id X , g . x ) .= F . ( id X , g . x ) .= F . ( id X , g . x ) .= F . ( id X , g . x ) ;
R . ( h . m ) = F . ( x0 + h . m ) - ( h . m ) + ( h . m ) - ( h . m ) .= ( ( - 1 ) * ( h . m ) ) * ( h . m ) ;
cell ( G , ( X -' 1 ) , ( Y + 1 ) + ( t + 1 ) ) \ ( ( X + 1 ) \ ( X + 1 ) ) meets ( ( X + 1 ) \ ( X + 1 ) ) ;
IC Result ( P2 , s2 ) = IC IExec ( I , P , Initialize s ) .= ( card I + card J + 3 ) .= ( card I + card J + 3 ) .= ( card I + card J + 3 ) .= ( card I + card J + 3 ) ;
sqrt ( ( - ( - ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ) ^2 ) ) > 0 ;
consider x0 being element such that x0 in dom a and x0 in g " { k `2 } and y0 = a . x0 and x0 in g " { k } and x0 in { k } ;
dom ( ( r1 (#) chi ( A , A ) ) | ( A . m ) ) = dom ( chi ( A , A ) ) /\ ( A . m ) .= C /\ ( A . m ) .= C /\ ( A . m ) .= C /\ ( A . m ) .= C /\ ( ( A . m ) /\ ( A . m ) /\ ( A . m ) ) .= C ;
d-7 . [ y , z ] = ( ( [ y , z ] ) `1 - ( ( [ y , z ] ) `2 ) * ( ( [ y , z ] `2 - ( z , y ) `2 ) ) * ( ( [ y , z ] `2 - ( z , y ) `2 ) ) * ( ( z , y ) `2 - ( z , y ) `2 ) * ( ( z , y ) `2 - ( z , y ) `2 ) * ( ( z , y ) `2 - ( z , y ) `2 ) ) * ( z , y ) `2 )
pred for i being Nat holds C . i = A . i /\ B . i & C . i c= d /\ ( A /\ B ) ;
consider x0 such that x0 in dom f and f is continuous and for x be Element of REAL m holds ||. f /. x - f /. x0 .|| <= ( f /. x - f /. x0 ) * ( f /. x0 ) ;
p in Cl A implies for K being Basis of p , Q being Subset of T st Q in K holds A meets Q & A meets Q
for x being Element of REAL n st x in Line ( x1 , x2 ) holds |. y1 - y2 .| <= |. y1 - y2 .| & |. y2 - x2 .| <= |. y1 - y2 .|
func the \times <*> of a -> Ordinal means : Def2 : a in it & for b being Ordinal st a in it holds it . a c= b & it . b = a ;
[ a1 , a2 , a3 ] in ( the carrier of A ) /\ ( the carrier of A ) & ( the carrier of A ) /\ ( the carrier of A ) = ( the carrier of A ) /\ ( the carrier of A ) ;
ex a , b being element st a in the carrier of S1 & b in the carrier of S2 & x = [ a , b ] & x = [ a , b ] ;
||. ( ( vseq . n ) - ( vseq . m ) ) * ( ( vseq . m ) - ( vseq . n ) ) .|| < ( e / ( ||. x .|| ) * ||. x .|| ) * ||. x .|| ;
then for Z being set st Z in { Y where Y is Element of I7 : F . Y in Z & z in Z } holds z in x & z in Z ;
sup compactbelow [ s , t ] = [ sup ( { [ s , t ] } ) , sup ( compactbelow s ) ] , sup ( compactbelow t ) ] ;
consider i , j being Element of NAT such that i < j and [ y , f . j ] in Ii and [ f . i , f . j ] in Ii and [ f . i , f . j ] in Ii ;
for D be 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 & p ^ q = q ^ p
consider e19 being Element of the affine of X such that c9 , a9 // a9 , e and a9 , c9 // a9 , e and a9 , c9 // a9 , e and a9 , c9 // a9 , e and a9 , c9 // a9 , e and a9 , c9 // a9 , e ;
set U1 = I \! \mathop { \vert I .| , U2 = I \! \mathop { \vert I .| } , E = { I .| , F = { I } , N = { I } , F = { I } , N = { I } , E = { I } , N = { I } , F = { I } , N = { I } , F = { I } , E = { I } , F = { I } , E = { I } , F = { I } , F = { I } , F = { I } , F = { I } , F = { I } , F =
|. q3 .| ^2 = ( ( q3 `1 ) ^2 + ( ( q3 `2 ) ^2 ) * ( ( q3 `1 ) ^2 ) ) ^2 .= ( ( q `1 ) ^2 + ( q `2 ) ^2 ) * ( ( q `2 ) ^2 ) .= ( q `1 ) ^2 + ( q `2 ) ^2 ;
for T being non empty TopSpace , x , y being Element of [: the topology of T , the topology of T :] holds x "\/" y = x \/ y & x "/\" y = x /\ y implies x "/\" y = x /\ y
dom signature U1 = dom ( the charact of U1 ) & Args ( o , MSAlg U1 ) = dom ( the charact of U1 ) & Args ( o , MSAlg U1 ) = dom ( ( the charact of U1 ) * the Arity of U1 ) ;
dom ( h | X ) = dom h /\ X .= dom ( ( h | X ) | X ) .= dom ( ( h | X ) | X ) .= dom ( h | X ) .= X ;
for N1 , N2 being Element of N1 , N1 , N2 being Element of N1 holds dom ( h . K1 ) = N & rng ( h . K1 ) = N1 & rng ( h . K1 ) = N2 & rng ( h . K1 ) c= N1 & rng ( h . K1 ) c= N1 & rng ( h . K1 ) c= N2
( mod ( u , m ) + mod ( v , m ) ) . i = ( mod ( u , m ) ) . i + ( mod ( v , m ) ) . i ;
- ( - ( q `1 ) ) < - 1 or - ( q `2 ) >= - ( q `1 ) & - ( q `2 ) <= - ( q `1 ) & - ( q `1 ) <= - ( q `2 ) ;
pred r1 = f9 & r2 = f9 & s1 = g9 & s2 = f9 & s1 = g9 & s2 = f9 & s1 = g9 & s2 = f9 & s1 = g9 & s1 = s2 & s1 = s2 & s1 = s2 & s2 = s1 & s1 = s2 & s1 = s2 & s1 = s2 & s1 = s2 & s1 = s2 & s1 = s2 & s1 = s2 & s1 = s2 ;
vseq . m is bounded Function of X , the carrier of Y & x9 . m = ( ( vseq . m ) * ( vseq . m ) ) . x & ( ( vseq . m ) * ( vseq . m ) ) . x = ( ( vseq . m ) * ( vseq . m ) ) . x ;
pred a <> b & b <> c & angle ( a , b , c ) = PI & angle ( b , c , a ) = 0 & angle ( b , c , a ) = PI ;
consider i , j being Nat , r , s being Real such that p1 = [ i , r ] and p2 = [ j , s ] and i < j and j < len s and r < s ;
|. p .| ^2 - ( 2 * |( p , q )| ) ^2 + |. q .| ^2 = |. p .| ^2 + |. q .| ^2 - ( 2 * |. p .| ) ^2 ;
consider p1 , q1 being Element of [: X , Y :] such that y = p1 ^ q1 and q1 ^ q2 = p1 ^ q1 and p1 ^ q1 = p2 ^ q2 and p1 ^ q1 = p2 ^ q2 ;
( of of A , r1 , r2 , s2 , s1 , s2 , s1 , s2 , s2 , s2 , t2 , t2 , t1 , t2 , t2 , t1 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2
( ( LMP A ) `2 = lower_bound ( proj2 .: ( A /\ holds ( w - 1 ) ) / 2 ) ) & ( proj2 .: ( A /\ ( w - 1 ) ) ) is non empty ;
s , ( k , 1 ) |= H1 iff s |= ( H1 , H2 ) iff s |= ( H1 , H2 ) & s |= ( H2 , k ) ) & s |= ( H2 , k )
len ( s + t ) = card ( support b1 ) + 1 .= card ( support b1 ) + 1 .= card ( support b1 ) + 1 .= card ( support b1 ) + 1 .= card ( support b1 ) + 1 .= card ( support b1 ) ;
consider z being Element of L1 such that z >= x and z >= y and for z being Element of L1 st z >= x & z >= y holds z `1 >= y `1 & z `2 >= x `1 ;
LSeg ( UMP D , |[ ( W-bound D - E-bound D ) / 2 , ( E-bound D - S-bound D ) / 2 ]| ) /\ D = { UMP D , ( UMP D - E-bound D ) / 2 } ;
lim ( ( ( f `| N ) / ( g `| N ) ) /* b ) = ( ( f `| N ) / ( g `| N ) ) . b .= ( ( f `| N ) / ( g `| N ) ) . b ;
P [ i , pr1 ( f ) . ( i + 1 ) , pr1 ( f ) . ( i + 1 ) ] & pr1 ( f , i ) . ( i + 1 ) = pr1 ( f , i ) . ( 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 & x in P & b in P & a in P & b in P holds a = b
Z c= dom ( ( ( #Z 2 ) * f ) ^ ) /\ ( ( #Z 2 ) * f ) " { 0 } ) & Z c= dom ( ( #Z 2 ) * f ) /\ ( ( #Z 2 ) * f ) " { 0 } ) ;
ex j be Nat st j in dom ( l ^ <* x *> ) & j < i & y = ( l ^ <* x *> ) . j & i = 1 + j & j = len ( l ^ <* x *> ) & z = 1 + j & j = 1 + 1 ;
for u , v being VECTOR of V , r being Real st 0 < r & u in N & v in c= dom _ \kappa N holds r * u + ( 1-r * v ) in c= c= c= c= c= , N
A , Int Cl A , Cl Int Cl A , Cl Int Cl A , Cl Int Cl A , Cl Cl Int Cl A , Cl Cl Int Cl A , Cl Cl Int Cl A , Cl Cl Int Cl A , Cl Cl Cl Cl A ` ` ` ` ` ` ` ` ` ` ` ` ` ` , Cl Cl ( Cl Cl ( Cl Int Cl ( Cl ( Cl ( Cl ( Cl ( Cl ( Cl ( A ` ) ) ` ) , Cl Cl ( Cl ( Cl ( A , A ` ) ) , Cl ( Cl ( A , A ` ) , Cl ( A , A ` ) , Cl ( A , A
- Sum <* v , u , w *> = - ( v + u + w ) .= - ( v + u ) + - ( u + w ) .= - ( v + u ) + - ( u + w ) .= - ( v + u ) ;
( 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 .= 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 support of J ) . x ;
for S1 , S2 being non empty reflexive RelStr for D being non empty directed Subset of S1 , D being non empty directed Subset of S2 for x being Element of S1 , y being Element of S2 holds x in D iff x is directed & y in D & x in D
card X = 2 implies ex x , y st x in X & y in X & x <> y & x in X or x = y & x = y or x = y & y = x or x = y
E-max L~ Cage ( C , n ) in rng ( Cage ( C , n ) \circlearrowleft W-min L~ Cage ( C , n ) ) & E-max L~ Cage ( C , n ) in rng Cage ( C , n ) ;
for T , T being DecoratedTree , p , q being Element of dom T , p being Element of dom T st p divides q & p in dom T holds ( T -with p , q ) . q = T . q
[ i2 + 1 , j2 ] in Indices G & f /. k , j2 ] in Indices G & f /. k = G * ( i2 + 1 , j2 ) & f /. k = G * ( i2 + 1 , j2 ) ;
cluster ( k gcd ( k , n ) ) divides ( ( k gcd ( k , n ) ) ) & ( k divides ( k gcd ( k , n ) ) ) & ( k divides ( k gcd ( k , n ) ) ) implies ( k divides ( k gcd ( k gcd ( k , n ) ) ) ) & ( k divides ( k gcd ( k gcd ( k gcd ( k , n ) ) ) ) ) ;
dom F " = the carrier of X2 & rng F = the carrier of X1 & F " = ( the carrier of X2 ) \/ the carrier of X2 & F " = ( the carrier of X1 ) \/ the carrier of X2 & F " = ( the carrier of X2 ) \/ 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 is linearly-independent of A , B and C is linearly-independent ;
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= V holds X \/ Y is finite
set X = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } , Y = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } ;
angle ( p1 , p3 , p4 ) = 0 .= angle ( p2 , p3 , p4 ) .= angle ( p2 , p3 , p4 ) .= angle ( p3 , p3 , p4 ) .= angle ( p2 , p3 , p4 ) .= angle ( p3 , p4 , p4 ) ;
- sqrt ( ( - ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ) ^2 ) = - ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) .= - ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) .= - ( q `2 / |. q .| - cn ) ;
ex f being Function of I[01] , TOP-REAL 2 st f is continuous one-to-one & rng f = P & f . 0 = p1 & f . 1 = p2 & f . 0 = p3 & f . 1 = p2 & f . 1 = p4 & f . 1 = p2 & f . 2 = p4 ;
attr f is partial differentiable of REAL , u0 means : Def2 : SVF1 ( 2 , pdiff1 ( f , 1 ) , u0 ) . u = ( proj ( 2 , 3 ) * ( pdiff1 ( f , 1 ) , u0 ) ) . u ;
ex r , s st x = |[ r , s ]| & G * ( len G , 1 ) `1 < r & r < G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `2 < s & s < G * ( 1 , 1 ) `2 ;
assume that f is FinSequence and 1 <= t and t <= len G and G * ( t , width G ) `2 >= S-bound L~ f and G * ( t , width G ) `2 >= S-bound L~ f and G * ( t , width G ) `2 <= N-bound L~ f ;
pred i in dom G means : Def2 : r * ( f * reproj ( i , x ) ) = r * f * reproj ( i , x ) ;
consider c1 , c2 being bag of o1 + o2 such that ( decomp c ) /. k = <* c1 , c2 *> and c1 in dom ( <* c1 , c2 *> ) and c2 in dom ( <* c1 , c2 *> ) and c2 in dom ( <* c1 , c2 *> ) ;
u0 in { |[ r1 , s1 ]| : r1 < s1 & s1 < s1 & s1 < s2 & s2 < s1 } & ( s1 + s2 ) * ( s1 + s2 ) = ( s1 + s2 ) * ( s1 + s2 )
Cl ( X ^ Y ) . k = the carrier of X . ( k2 + 1 ) .= C4 . ( k2 + 1 ) .= C4 . ( k2 + 1 ) .= C4 . ( k2 + 1 ) .= C4 . ( k2 + 1 ) ;
pred len M1 = len M2 & width M1 = width M2 & width 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 & M1 = M2 ;
consider g2 be Real such that 0 < g2 and { y where y is Point of S : ||. y - x0 .|| < g2 & y in dom f & ||. y - x0 .|| < g2 & g2 /. y - x0 .|| < g2 } c= N2 ;
assume x < ( - b + sqrt ( Let ( a , b , c ) ) ) / ( 2 * a ) or x > ( - b - sqrt ( a , b , c ) ) / ( 2 * a ) ;
( G1 '&' G2 ) . i = ( <* 3 *> ^ G1 ) . i & ( H1 ^ G1 ) . i = ( <* 3 *> ^ G1 ) . i & ( H1 ^ G1 ) . i = ( <* 3 *> ^ G1 ) . i ;
for i , j st [ i , j ] in Indices ( M3 + M1 ) holds ( M3 + M2 ) * ( i , j ) < ( M3 + M2 ) * ( i , j )
for f being FinSequence of NAT , i being Element of NAT st i in dom f & i in dom f holds i divides f /. i & i divides len f & i divides len f & i divides len f & i divides len f implies f divides f
assume F = { [ a , b ] where a , b is Subset of X : for c being set st c in B\mathopen the carrier of X & a c= c holds b c= c } ;
b2 * q2 + ( b3 * q3 ) + ( - ( ( a * q2 ) + ( - ( a * q2 ) ) ) ) * ( ( a * q2 ) + ( - ( a * q2 ) ) ) * ( ( a * q2 ) + ( - ( a * q2 ) ) ) * ( ( a * q2 ) + ( - ( a * q2 ) ) * ( ( a * q2 ) + ( a * q2 ) ) * ( ( b * 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 D c= Cl B } & F is closed & Cl D = Cl B & Cl D = Cl B & Cl D = Cl B & Cl D = Cl B ;
attr seq is summable means : Def2 : seq is summable & seq is summable & ( for n holds seq . n = Partial_Sums ( seq ) . n ) implies seq is convergent & lim seq = Partial_Sums ( seq ) . n ;
dom ( ( ( cn ) | D ) | D ) = ( the carrier of ( TOP-REAL 2 ) ) /\ D .= the carrier of ( ( TOP-REAL 2 ) | D ) .= D ;
X [ X \to Z ] is full full non empty SubRelStr of ( Omega Z ) |^ the carrier of Z & [ X \to Y ] is full non empty full SubRelStr of ( Omega Z ) |^ the carrier of Z ;
G * ( 1 , j ) `2 = G * ( i , j ) `2 & G * ( 1 , j ) `2 <= G * ( i , j ) `2 ;
synonym m1 c= m2 means : Def2 : for for p being set st p in P holds the non empty set of p <= ( m2 + p ) is_pLet ( m2 + p ) , ( m2 + p ) is_pLet ( m2 + p ) , ( m1 + p ) -that p in the carrier of p ;
consider a being Element of B ( ) such that x = F ( a ) and a in { G ( b ) where b is Element of A ( ) : P [ b ] } and a in A ( ) ;
synonym empty empty multMagma (# carrier -> set means : Def2 : the multMagma of it = [ the carrier of it , the multF of it , the multF of it , the multF of it , the multF of it #) is Relation of the carrier , the carrier of it ;
( the carrier of not ( a , b , 1 ) + ( the InternalRel of c ) = b + ( the InternalRel of a , d ) .= b + ( a + c ) .= b + ( a + c ) .= b + ( a + c ) .= b + ( a + c ) ;
cluster ( 1 + 2 ) * -> Element of INT means : Def2 : for i1 , i2 being Element of INT holds it . ( i1 , i2 ) = ( 1 + 2 ) * ( i1 , i2 ) + ( 1 + 2 ) * ( i1 , i2 ) ;
( - s2 * p1 + ( s2 * p2 ) - ( s2 * p2 ) ) * p1 = ( ( - s2 ) * p1 + ( ( s2 * p2 ) - ( s2 * p2 ) ) * p2 .= ( ( - s2 ) * p1 + ( ( s2 * p2 ) - ( s2 * p2 ) ) * p2 ) ;
eval ( ( a | ( n , L ) ) *' , x ) = eval ( a | ( n , L ) ) * eval ( p , x ) .= a * eval ( p , x ) * eval ( x , x ) .= a * eval ( p , x ) * eval ( x , x ) ;
assume that the TopStruct of S = the TopStruct of T and for D being non empty set , D being non empty directed Subset of S , V being Subset of Omega S , S being Subset of Omega T st D in V holds V meets V ;
assume that 1 <= k and k <= len w + 1 and TU . k = ( TU . ( q11 , w ) ) . k and TU . ( q11 , w ) = ( TU . ( q11 , w ) ) . k and TU . ( q11 , w ) = ( ( U . ( q11 , w ) ) | k ) ;
2 * a |^ ( n + 1 ) + ( 2 * b |^ ( n + 1 ) ) >= ( a |^ ( n + 1 ) ) + ( ( b |^ ( n + 1 ) ) * a ) + ( ( b |^ ( n + 1 ) ) * a ) ;
M , v2 |= All ( x. 3 , All ( x. 0 , All ( x. 4 , All ( x. 0 , All ( x. 0 , All ( x. 0 , All ( x. 0 , x. 0 ) ) ) ) ) ) ) ;
assume that f is_differentiable_on l and for x0 st x0 in l holds 0 < f . x0 and for x0 st x0 in l holds f . x0 - f . x0 < 0 and for x1 st x1 in l holds f . x1 - f . x0 < f . x1 ;
for G1 , G2 being _Graph , W being Walk of G1 , e being set st e in W and not e in W holds e in W & not e in W implies e in W & e in W
not c9 is empty iff ( ex m1 , m2 st m1 is not empty & not ( m1 is not empty & not m2 is not empty & not ( m1 is not empty & not m2 is not empty ) & not ( m1 is not empty & not m2 is not empty ) & not R is not empty ) & not R is not empty & not R is not empty & not R is not empty & not R is not empty & not R is not empty & not R is not empty & not R is not empty & not R is not empty & not R is not empty & not R is not empty & not R is not empty & not R is not empty & not R is not empty & not R is not empty & not R is not empty & not R is not empty & not
Indices GoB f = [: dom GoB f , Seg width GoB f :] & ( GoB f ) * ( i1 , j1 ) in Indices GoB f & ( GoB f ) * ( i1 , j1 ) in Indices GoB f & ( GoB f ) * ( i1 , j1 ) in Indices GoB f implies ( GoB f ) * ( i1 , j1 ) in Indices GoB f
for G1 , G2 , G2 , G3 being strict Subgroup of O , O being stable Subgroup of O st G1 is_stable & G2 is_stable & G1 is_stable holds G1 , G2 is_stable & G2 is_stable & G1 , G2 is_stable implies G1 , G2 is_stable
UsedIntLoc ( int -> Element of UsedIntLoc ( 1 ) ) = { intloc 0 , intloc 1 , intloc 2 , intloc 3 , intloc 4 , intloc 5 , intloc 5 , intloc 6 , intloc 5 , 7 , 8 , 8 , 9 } \/ UsedIntLoc ( where 5 is Element of { 0 } , { 0 } , { 1 , 1 } , { 2 , 3 , 4 , 5 , 6 , 7 , 8 } ) ;
for f1 , f2 being FinSequence of F st f1 ^ f2 is p -element & Q [ f1 ^ f2 ] & Q [ f2 ^ <* f1 ^ f2 *> ] holds Q [ f1 ^ f2 ^ <* f1 ^ f1 *> ]
( ( p `1 ) ^2 + ( p `2 ) ^2 ) = ( q `1 ) ^2 + ( p `2 ) ^2 .= ( q `1 ) ^2 + ( q `2 ) ^2 .= ( q `1 ) ^2 + ( q `2 ) ^2 ;
for x1 , x2 , x3 , x4 being Element of REAL n holds |( x1 - x2 , x3 - x4 )| = |( x1 , x2 - x3 )| + |( x2 , x3 - x4 )| + |( x1 , x2 - x3 )| , ||. x2 - x3 )| + |( x2 , x3 )| + |( x2 , x3 )| + |( x2 , x3 )| + |( x2 , x3 )|
for x st x in dom ( ( ( ( F - G ) | A ) | A ) ) holds ( ( ( ( F - G ) | A ) | A ) . ( - x ) ) = - ( ( ( F - G ) | A ) . x )
for T being non empty TopSpace , P being Subset-Family of T , B being Basis of T st P c= the topology of T for x being Point of T st x in P holds P is Basis of x
( a 'or' b 'imp' c ) . x = 'not' ( ( a 'or' b ) . x 'or' c . x ) .= 'not' ( ( a 'or' b ) . x 'or' ( b 'or' c ) . x ) .= TRUE 'or' TRUE .= TRUE .= TRUE ;
for e being set st e in A8 ex X1 being Subset of X , Y1 being Subset of Y st e = X1 & Y1 is open & 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 & Y2 is open
for i be set st i in the carrier of S for f being Function of Sconsider i , S1 . i st f = H . i & F . i = f . i holds F = f | ( F . i )
for v , w st for y st x <> y holds w . y = v . y holds Valid ( VERUM ( Al , J ) , J ) . v = Valid ( VERUM ( Al , J ) , J ) . ( v . y )
card D = card D1 + card D2 - 1 .= ( { i , j } - 1 ) * ( { i , j } - 1 ) .= ( ( i + 1 ) - 1 ) * ( ( i + 1 ) - 1 ) * ( ( i + 1 ) - 1 ) .= ( ( i + 1 ) - 1 ) * ( ( i + 1 ) - 1 ) * ( ( i + 1 ) - 1 ) * ( i + 1 ) ) .= 2 * ( ( i + 1 ) * ( ( i + 1 ) - 1 ) * ( ( i + 1 ) * ( ( i + 1 ) * ( i + 1 ) * ( i + 1 ) * ( i + 1 ) * ( i + 1
IC Exec ( i , s ) = ( s +* ( 0 .--> succ ( s . 0 ) ) ) . 0 .= ( 0 .--> ( s . 0 ) ) . 0 .= ( 0 .--> ( s . 0 ) ) . 0 .= ( 0 .--> ( s . 0 ) ) . 0 .= ( 0 .--> ( s . 0 ) ) . 0 .= ( 0 .--> ( s . 0 ) ) . 0 .= 0 ;
len f /. ( ( i1 -' 1 ) + 1 ) = len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) + 1 ;
for a , b , c being Element of NAT st 1 <= a & a <= b & b < a holds a <= a + b-2 or a = b + b-2 or a = b + b-2 or a = b + b-2 or a = b + b-2 or a = b + b-2 or a = b + b-2 or a = b + b-2
for f being FinSequence of TOP-REAL 2 , p being Point of TOP-REAL 2 , i being Nat st i in LSeg ( f , i ) & i <= len f holds Index ( p , f ) <= i & Index ( p , f ) <= len f
( ( curry ' ( [: 0 , k + 1 :] , n ) # x ) # x ) . x = ( ( curry ' ( [: 0 , k :] , n ) ) # x ) . x + ( ( curry ' ( 0 , k ) , n ) # x ) . x ;
z2 = g /. ( ( i -' n1 + 1 ) + 1 ) .= g . ( ( i -' n1 + 1 ) + 1 ) .= g . ( ( i -' n1 + 1 ) + 1 ) .= g . ( ( i -' n1 + 1 ) + 1 ) .= g . ( ( i -' n1 + 1 ) + 1 ) .= g . ( ( i -' n1 + 1 ) + 1 ) ;
[ f . 0 , f . 3 ] in id the carrier of ( the carrier of G ) \/ ( the InternalRel of G ) or [ f . 0 , f . 3 ] in the InternalRel of ( the carrier of G ) \/ the InternalRel of G ;
for G being Subset-Family of B for R being Subset of A st G = { [ R , X ] where X is Subset of A , Y is Subset of A ( ) st X in F6 & Y in F6 } holds ( ( Intersect ( F , G ) ) . X = Intersect ( G , G ) . Y )
CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 ) ) = CurInstr ( P1 , Comput ( P1 , s1 , m1 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m1 ) ) .= halt SCMPDS .= CurInstr ( P1 , Comput ( P2 , s2 , m1 ) ) .= halt SCMPDS ;
assume that not a on M and b on M and c on N and d on N and p on M and a on N and p on M and a on N and p on M and a on M and p on N and p on M and a on M and p on N and a on M and b on N and a on M and b on N and a on M and b on N and a on N and b on N and a on M and b on N and a on N and b on N and b on N and b on N and a on N and b on N and a on N and b on N and b on M and b on N and a on N and b on N and b on N and b on M and b on N and b on M and a on N and a on N and c on N and a on N and a on N and c on N and c on
assume that T is \hbox { T _ 4 } and F is closed and ex F being Subset-Family of T st F is closed & for n being Nat st n <= 0 holds F . n is finite-ind and ind F <= n and ind F <= n and ind F <= n and ind F <= n ;
for g1 , g2 st g1 in ]. r - g2 , r .[ & g2 in ]. r - g2 .[ & |. f . g1 - f . g2 .| <= ( g1 - g2 ) / ( |. g1 .| - |. g2 .| ) holds g1 <= g2
( ( - ( 1 / 2 ) ) * ( z1 + z2 ) ) = ( ( - ( 1 / 2 ) ) * ( z1 + z2 ) ) * ( z2 + z2 ) .= ( ( - ( 1 / 2 ) ) * ( z1 + z2 ) ) * ( z2 + z2 ) ;
F . i = F /. i .= 0. R + r2 .= ( b |^ ( n + 1 ) ) * ( a |^ ( n + 1 ) ) .= ( ( n + 1 ) -tuples_on a ) * ( b |^ ( n + 1 ) ) .= ( ( n + 1 ) -tuples_on a ) * ( b |^ ( n + 1 ) ) .= ( ( n + 1 ) -tuples_on a ) * ( b |^ ( n + 1 ) ) ;
ex y being set , f being Function st y = f . n & dom f = NAT & f . 0 = A ( ) & for n holds f . ( n + 1 ) = R ( n , f . n ) & for n holds f . ( n + 1 ) = R ( n , f . n ) ;
func f * F -> FinSequence of V means : Def2 : 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 } = { x1 , x2 , x3 , x4 , x5 } \/ { x5 , x5 , x5 } .= { x1 , x2 , x3 , x4 } \/ { x4 , x5 , x5 } .= { x1 , x2 , x3 , x4 } \/ { x2 , x4 } ;
for n being Nat for x being set st x = h . n holds h . ( n + 1 ) = o . ( x , n ) & o . ( x , n ) in InnerVertices S & o . ( x , n ) in InnerVertices S ;
ex S1 being Element of CQC-WFF ( Al ( ) ) st SubP ( P , l , e ) = S1 & ( for i being Nat holds ( S . i = F ( i ) ) & ( S . i = F ( i ) ) & ( S . i = F ( i ) ) & ( S . i = F ( i ) ) & ( S . i = F ( i ) ) ;
consider P being FinSequence of GL2 such that p9 = product P and for i being Element of dom P ex t being Element of the carrier of K st P . i = t & ex i being Element of NAT st P . i = t & ex i being Element of NAT st i in dom P & t . i = t . i & P . i = t . i ;
for T1 , T2 being strict non empty TopSpace for P being Basis of T1 , Q being Basis of T2 st the carrier of T1 = the carrier of T2 & P = the topology of T2 & P = the topology of T2 holds P is Basis of T1 & P is Basis of T2
assume that f is_partial differentiable on u0 , u0 and r (#) pdiff1 ( f , 3 ) is_partial_differentiable_in u0 , 2 and partdiff ( r (#) pdiff1 ( f , 3 ) , u0 , 2 ) , u0 , 2 & partdiff ( r (#) pdiff1 ( f , 3 ) , u0 , 2 ) = r * pdiff1 ( f , 3 ) . 2 and partdiff ( r (#) pdiff1 ( f , 3 ) , u0 , 2 ) = r * pdiff1 ( f , 3 ) . 2 ;
defpred P [ Nat ] means for F , G being FinSequence of bool the carrier of X for s being Permutation of Seg $1 , G being Permutation of Seg $1 st len F = $1 & rng s = rng G & for i being Nat st i in Seg $1 holds F . i = G . i & G . i = F . i holds Sum ( F , s ) = Sum ( F , s ) ;
ex j st 1 <= j & j < width GoB f & ( GoB f ) * ( 1 , j ) `2 <= s & s <= ( GoB f ) * ( 1 , j + 1 ) `2 & s <= ( GoB f ) * ( 1 , j + 1 ) `2 ;
defpred U [ set , set ] means ex Fn1 being Subset-Family of T st $1 = Fn1 & $2 is open & ( union FF ) is open & ( union Fn1 is open implies $2 is open ) & ( union Fn1 is discrete implies union Fn1 is discrete ) & ( union Fn1 is discrete implies union Fn1 is discrete ) ;
for p4 being Point of TOP-REAL 2 st LE p4 , p1 , P & LE p4 , p1 , P & LE p1 , p2 , P & LE p2 , p3 , P holds LE p4 , p1 , P & LE p1 , p2 , P & LE p2 , p3 , P & LE p1 , p2 , P & LE p2 , p3 , P & LE p2 , p3 , P & LE p2 , p3 , P & p1 , p2 , P & p2 in P & p1 in P & p2 in P & p2 in P & p1 in P & p2 in P & p2 in P & p2 in P & p2 in P & p2 in P & p2 in P & p2 in P & p2 in P & p2 in P & p2 in P & p1 in P & p2 in P & p2 in P & p1 in P & p1 in P & p1 in P & p1 in P & p1 in P & p1 in P & p1 in P & p1
f in \rbrace implies for g st g in \rbrace & for y st y <> f . y holds x in dom f & g in dom ( All ( x , H ) ) implies f in as Function of E , E & f in the carrier of E & g in the carrier of E & f in the carrier of E & g in the carrier of E
ex 8 being Point of TOP-REAL 2 st x = 8 & ( ( ( ( 8 - a ) / ( 8 - a ) ) * ( 8 - a ) ) / ( 8 - a ) ) >= 8 & ( ( 8 - a ) / ( 8 - a ) ) * ( 8 - a ) <= 8 * ( 8 - a ) & ( 8 - a ) / ( 8 - a ) <= 8 * ( 8 - a ) ;
assume for d7 being Element of NAT st d7 <= ( n -d7 ) . ( ( n -\hbox { t } ) . ( n -\hbox { t } ) ) holds s1 . ( ( n -\hbox { t } ) . ( n -\hbox { t } ) ) = s2 . ( ( n -\hbox { t } ) . ( n -\hbox { t } ) ) ;
consider s such that s <> t and s is Point of Sphere ( x , r ) and not s is Point of Sphere ( x , r ) and ex e being Point of E st e = Ball ( x , r ) /\ Ball ( x , r ) & e in Ball ( x , r ) /\ Ball ( y , r ) ;
given r such that 0 < r and for s st 0 < s for x1 , x2 be Point of CNS st x1 in dom f & x2 in dom f & ||. x1 - x2 .|| < s holds |. f /. x1 - f /. x2 .| < r & |. f /. x2 - f /. x2 .| < r ;
( p | x ) | ( ( p | x ) | ( ( x | x ) | ( x | x ) ) ) = ( ( ( x | x ) | ( x | x ) ) | ( ( x | x ) | ( x | x ) ) ) | ( ( x | x ) | ( x | x ) ) ) ;
assume that x , x + h in dom sec and ( for x st x in dom sec holds ( ( sin * sec ) `| Z ) . x = ( 4 * sin . x + cos . x ) * sin . x ) / ( cos . x ) ^2 and ( cos * sec ) . x = ( 4 * sin . x + cos . x ) / ( cos . x ) ^2 ;
assume that i in dom A and len B > 1 and i in dom A and B = ( A @ ) . i and ( A @ ) * ( B @ ) = ( A @ ) * ( B @ ) and ( A @ ) * ( B @ ) = ( A @ ) * ( B @ ) and ( A @ ) * ( B @ ) = ( A @ ) * ( B @ ) ;
for i be non zero Element of NAT st i in Seg n holds i divides n or i = <* 1. F_Complex *> or i = <* 1. F_Complex *> & ( i divides n implies h . i = 1. F_Complex ) & ( i divides 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 'or' c2 ) ) '&' ( ( b1 'or' c2 ) '&' ( a2 'or' c2 ) ) '&' 'not' ( ( b1 'or' c2 ) '&' 'not' ( a2 'or' c2 ) '&' 'not' ( a2 'or' c2 ) ) '&' 'not' ( ( a2 'or' c2 ) '&' 'not' ( a2 'or' c2 ) '&' 'not' ( a2 'or' c2 ) '&' 'not' ( a2 'or' c2 ) '&' 'not' ( a2 'or' c2 ) '&' 'not' ( a2 'or' c2 ) '&' 'not' ( a2 'or' c2 ) '&' 'not' ( a2 'or' c2 ) '&' 'not' ( a2 'or' c2 ) '&' 'not' ( a2 'or' c2 ) '&' 'not' ( a2 'or' c2 ) '&' 'not' ( a2 'or' c2 ) '&' 'not' ( a2 'or' c2 ) '&' 'not' ( a2 'or' c2 ) '&' 'not' ( a2 'or' c2 ) '&' 'not' ( a2 'or' c2 ) '&' 'not' ( a2 'or' c2 ) '&' 'not' ( a2 'or' c2 ) '&' 'not' ( a2 'or' c2 ) '&' 'not' ( a2 'or' c2 ) '&' 'not' (
assume that for x holds f . x = ( ( - cot * sin ) `| Z ) . x and for x st x in dom ( ( - cot * sin ) `| Z ) holds ( ( - cot * sin ) `| Z ) . x = - cos . ( x - h . x ) / sin . ( x - h . x ) .= cos . ( x - h . x ) / cos . x ;
consider R8 , I8 be 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 ) ) and Integral ( M , Im ( F . n ) ) = ( Im ( F . n ) ) * i ;
ex k be Element of NAT st k9 = k & 0 < d & for q be Element of product G st q in X & ||. q-x - f /. ( q - x ) .|| < r holds ||. partdiff ( f , q ) . k - partdiff ( f , x ) . ( q - x ) .|| < r
x in { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 } iff x in { x1 , x2 , x3 , x4 , x5 , x5 } \/ { x5 , x5 , x5 , x5 } \/ { x5 , x5 , x5 , x5 } \/ { x5 , x5 , x5 } \/ { x5 , x5 , x5 }
G * ( j , i ) `2 = G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j
f1 * p = p .= ( ( the Arity of S1 ) * the Arity of S2 ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o ;
func tree ( T , P , T1 ) -> DecoratedTree means : : : for q st q in it holds q in P or ex p , q st p in P & q in P & p in P & q in P & p in P & q in P & p in P & q in P & p in P & q in P & p in P & q in P & p in P ;
F /. ( k + 1 ) = F . ( p . ( k + 1 -' 1 ) , k + 1 -' 1 ) .= Fx0 . ( p . ( k + 1 -' 1 ) ) .= Fx0 . ( p . ( k + 1 -' 1 ) ) .= Fx0 . ( p . ( k + 1 -' 1 ) ) .= Fx0 . ( p . ( k + 1 -' 1 ) ) .= Fx0 . ( p . ( k + 1 -' 1 ) .= F /. ( k + 1 ) ;
for A , B , C being Matrix of len C , K st len B = len C & len C = len A & len B = len C & len C = len A & len B > 0 & len A > 0 & len B > 0 & len C = len B & len C > 0 & len A > 0 & len C = len B & len C = len C & len C > 0 holds A * C = C * C
seq . ( k + 1 ) = 0. F_Complex + seq . ( k + 1 ) .= ( Partial_Sums ( seq ) ) . ( k + 1 ) .= ( Partial_Sums ( seq ) ) . ( k + 1 ) .= ( Partial_Sums ( seq ) ) . ( k + 1 ) .= ( Partial_Sums ( seq ) ) . ( k + 1 ) ;
assume that x in ( the carrier of Cs ) and y in ( the carrier of Cs ) and x in ( the carrier of Cs ) and y in ( the carrier of Cs ) and z in ( the carrier of Cs ) and x in ( the carrier of Cs ) and y in ( the carrier of Cs ) ;
defpred P [ Element of NAT ] means for f st len f = $1 holds ( for k st k in $1 holds f . k = ( ( VAL g ) . ( k + 1 ) ) '&' ( ( VAL g ) . ( k + 1 ) ) ) '&' ( ( VAL g ) . ( k + 1 ) ) ;
assume that 1 <= k and k + 1 <= len f and f is FinSequence of TOP-REAL 2 and [ i + 1 , j ] in Indices G and f /. k = G * ( i , j ) and f /. ( k + 1 ) = G * ( i + 1 , j ) and f /. ( k + 1 ) = G * ( i , j ) ;
assume that sn < 1 and ( q `1 > 0 & q `2 <= 0 or q `1 >= 0 & q `2 >= 0 & q `1 <= 0 & q `2 <= 1 or q `1 = 1 & q `2 = 1 or q `1 = 1 & q `2 = 1 or q `1 = 1 & q `2 = 1 & q `2 = 1 or q `1 = 1 & q `2 = 1 & q `2 = 1 & q `2 = 1 or q `1 = 1 & q `1 = 1 & q `2 = 1 & q `2 = 1 & q `2 = 1 & q `2 = 1 & q `2 <= 1 or q `1 = 1 & q `2 = 1 & q `2 <= 1 & q `2 = 1 & q `2 = 1 & q `2 = 1 & q `2 = 1 & q `2 = 1 & q `2 = 1 & q `2 = 1 & q `2 = 1 & q `2 = 1 or q `1 = 1 & q `2 = 1 & q
for M be non empty metric , x be Point of M , f be Point of M , x be Point of M st x = x ` holds ex f being sequence of M st f is sequence of M & f . 0 = Ball ( x , 1 / ( 2 |^ n ) ) & f . 1 = Ball ( x , 1 / ( 2 |^ n ) )
defpred P [ Element of omega ] means f1 is__ _ $1 & ( for x st x in Z holds ( ( f1 - f2 ) `| Z ) . x = ( ( f1 - f2 ) `| Z ) . $1 - ( ( f1 - f2 ) `| Z ) . $1 ) & ( ( f1 - f2 ) `| Z ) . $1 = ( ( f1 - f2 ) `| Z ) . $1 - ( ( f1 - f2 ) `| Z ) . $1 ;
defpred P1 [ Nat , Point of CNS ] means ( $1 < $2 & $2 in Y & ||. $2 - x0 .|| < r & $2 in Y & ||. $2 - x0 .|| < r & ||. $2 - x0 .|| < r & ||. $2 - x0 .|| < r & ||. $2 - x0 .|| < r ;
( f ^ mid ( g , 2 , len g ) ) . i = ( mid ( g , 2 , len g ) ) . ( i - 1 ) .= g . ( i - 1 ) .= g . ( i - 1 ) .= g . ( i - 1 ) .= ( g . ( i - 1 ) ) .= ( g . ( i - 1 ) ) .= ( g . ( i - 1 ) ) ;
( 1 - 2 * ( n + 2 ) ) * ( 2 * ( n + 1 ) ) = ( ( 1 - 2 * ( n + 1 ) ) * ( n + 1 ) ) * ( 2 * ( n + 1 ) ) .= 1 * ( ( 1 - 2 * ( n + 1 ) ) * ( n + 1 ) ) * ( 2 * ( n + 1 ) ) .= 1 * ( ( 1 - 2 * ( n + 1 ) ) * ( n + 1 ) ) .= 1 * ( ( n + 1 ) * ( n + 1 ) ) * ( n + 1 ) * ( n + 1 ) ) .= 1 * ( ( n + 1 ) * ( n + 1 ) ) .= 1 * ( ( n + 1 ) * ( n + 1 ) * ( n + ( n + 1 ) .= ( ( n + 1 ) * ( n + ( n + 1 ) * ( n + 1 ) ) * ( n + 1 ) .= 1 * ( n + 1 ) .=
defpred P [ Nat ] means for G being non empty finite strict strict finite strict non empty strict strict non empty finite strict strict non empty RelStr for S being non empty strict strict non empty strict non empty strict non empty RelStr for G being non empty strict non empty RelStr , F being Function of the carrier of G , the carrier of S st F is d holds F is strict non empty ;
assume that f /. 1 in Ball ( u , r ) and not 1 <= m and m <= len f and not ( f . 1 in Ball ( u , r ) & not ( f . len f ) /\ Ball ( u , r ) <> {} & not ( f . len f in Ball ( u , r ) & not ( f . len f in Ball ( u , r ) & not ex m st m in Ball ( u , r ) & m in Ball ( u , r ) ) ;
defpred P [ Element of NAT ] means ( Partial_Sums ( cos ) . $1 = ( Partial_Sums ( cos ) ) . $1 - ( cos . $1 ) * ( cos . $1 ) * ( cos . $1 - cos . $1 ) / ( cos . $1 - cos . $1 ) / ( cos . $1 - cos . ( $1 + 1 ) * ( cos . $1 - cos . ( $1 + 1 ) ) ) ;
for x being Element of product F holds x is FinSequence & dom x = I & for i being set st i in dom F holds x . i in ( the Sorts of F ) . i & for i being set st i in dom F holds x . i in ( the Sorts of F ) . i
( x " ) |^ ( n + 1 ) = ( ( x " ) * x ) " .= ( x " ) * x .= ( x " ) * x .= ( x " ) * x .= x " * x .= x " * x .= x " * x .= x " * x .= x " * x .= x " * x .= x " * x .= x " * x ;
DataPart Comput ( P +* ( I , s ) , LifeSpan ( P +* I , Initialized s ) + 3 ) = DataPart Comput ( P +* I , s , LifeSpan ( P +* I , Initialized s ) + 3 ) , LifeSpan ( P +* I , Initialized s ) + 3 ) ;
given r such that 0 < r and ]. x0 - r , x0 .[ c= ( dom f1 /\ dom f2 ) /\ dom ( f1 - f2 ) and for g st g in ]. x0 - r , x0 .[ /\ dom ( f1 - f2 ) and g in ]. x0 - r , x0 .[ /\ dom ( f1 - f2 ) and g in ]. x0 - r , x0 .[ /\ dom ( f1 - f2 ) ;
assume that X c= dom f1 /\ dom f2 and f1 | X is continuous and ( f1 + f2 ) | X is continuous and ( f1 + f2 ) | X is continuous and ( f1 + f2 ) | X is continuous and ( f1 + f2 ) | X is continuous and ( f1 + f2 ) | X is continuous and ( f1 + f2 ) | X is continuous ;
for L being continuous complete LATTICE for l being Element of L for X being Subset of L ex x being Element of L st x = sup X & for x being Element of L st x in X holds x is directed & x is directed & x is directed & x is directed
Support ( e *' ) in { m *' where m is Polynomial of n , L : ex p being Polynomial of n , L st p in Support ( m *' ) & ex m being Polynomial of n , L st m in Support ( m *' ) & p in Support ( m *' ) & p in Support ( m *' ) & p in Support ( m *' ) } ;
( f1 - f2 ) /* s1 = lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) ;
ex p1 being Element of CQC-WFF ( Al ) st F . p1 = g . p1 & for g being Function of [: Carrier ( Al ) , the carrier of Al ) , the carrier of Al st P [ g , p1 , g ] holds P [ g , p1 , g , h , h , h , i ] ;
( mid ( f , i , len f -' 1 ) ^ <* f /. ( len f -' 1 ) *> ) /. ( j + 1 ) = ( mid ( f , i , len f -' 1 ) ) /. ( j + 1 ) .= f /. ( j + 1 ) .= f /. ( j + 1 ) .= f /. ( j + 1 ) ;
( ( p ^ q ) . ( len p + k ) ) . ( len p + 1 ) = ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( ( p ^ q ) . ( len p + k ) . ( len p + k ) .= ( ( p ^ q ) . ( len p + k ) . ( len p + k ) . ( len p + k ) .= ( ( p ^ q ) . ( len p + k ) .= ( ( p ^ q ) . ( len p + k ) . ( len p + k ) .= ( ( p ^ q ) . ( len p + ( len q + k ) .= ( ( ( len q + k ) . ( len q + k ) .= ( ( p ^ q ) + ( ( p ^ q ) . ( len q + k ) ) . ( len q + k
len ( mid ( D2 , D1 , j1 ) + 1 ) = indx ( D2 , D1 , j1 ) + 1 .= indx ( D2 , D1 , j1 ) + 1 .= indx ( D2 , D1 , j1 ) + 1 ;
x * y * z = ( x * ( y * z ) ) * ( y * z ) .= ( x * ( y * z ) ) * ( x * z ) .= ( x * ( y * z ) ) * ( x * z ) .= ( x * ( y * z ) ) * ( x * z ) .= ( x * ( y * z ) ) * ( x * z ) .= x * ( y * z ) ;
v . ( <* x , y *> ) + ( <* x0 , y0 *> ) . i = partdiff ( v , ( x - y ) ) * ( ( reproj ( 1 , 1 ) ) . ( ( reproj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . (
i * i = <* 0 * ( - 1 ) - ( 0 * ( - 1 ) ) * ( - 1 ) + ( - 1 * ( - 1 ) ) * ( - 1 ) * ( - 1 ) * ( - 1 * ( - 1 ) ) * ( - 1 * ( - 1 ) ) .= <* - 1 * ( - 1 * ( - 1 ) * ( - 1 * ( - 1 ) ) * ( - 1 * ( - 1 ) ) * ( - 1 * ( - 1 * ( - 1 * ( - 1 * ( - 1 * ( - 1 * ( - 1 * ( - 1 * ( - 1 * ( - 1 * ( - 1 ) ) ) ) * ( - 1 * ( - 1 * ( - 1 * ( - 1 * ( - 1 * ( - 1 ) ) * ( - 1 * ( - 1 * ( - 1 * ( - 1 * ( - 1 * ( - 1 * ( - 1 * ( - 1 * ( - 1 * ( - 1 * ( - 1 * ( - 1 * ( - 1 * ( - 1 * ( - 1 *
Partial_Sums ( L * F ) = Partial_Sums ( L * F1 ) + ( L * F2 ) .= Partial_Sums ( L * F1 ) + ( L * F2 ) .= ( Sum ( L * F1 ) ) + ( Sum ( L * F2 ) ) .= ( Sum ( L * F1 ) ) + ( Sum ( L * F2 ) ) .= ( Sum ( L * F1 ) ) + ( Sum ( L * F2 ) ) .= ( Sum ( L * F1 ) ) + ( Sum ( L * F2 ) .= ( Sum ( L * F1 ) + ( Sum ( L * F2 ) .= ( Sum ( L * F1 ) + ( Sum ( L * F2 ) .= ( Sum ( L * F1 ) + Sum ( L * F2 ) .= ( Sum ( L * F1 ) .= ( Sum ( L * F2 ) + ( Sum ( L * F2 ) + ( Sum ( L * F2 ) + ( Sum ( L * F2 ) + ( Sum ( L * F2 ) + ( Sum ( L * F2 ) ) .= ( Sum ( L * F1 ) + ( Sum ( L * F2 ) + ( Sum ( L * F2 ) .= (
ex r be Real st for e be Real st 0 < e ex Y1 be Subset of X st Y1 is non empty & ( for Y1 be Subset of X st Y1 is non empty & Y1 c= Y holds |. ( lower_bound ( Y1 , Y2 ) ) . Y1 - ( lower_bound ( Y1 , Y2 ) ) . Y1 .| < r ) ;
( GoB f ) * ( i , j + 1 ) = f /. ( k + 2 ) & ( GoB f ) * ( i , j + 1 ) = f /. ( k + 1 ) or ( GoB f ) * ( i , j ) = f /. ( k + 1 ) & ( GoB f ) * ( i , j ) = f /. ( k + 1 ) ;
( ( ( - 1 ) / 2 ) * ( cos . x ) ) ^2 = ( ( - 1 ) / 2 ) * ( cos . x ) ^2 .= ( ( 1 / 2 ) * ( cos . x ) ) ^2 .= ( ( 1 / 2 ) * ( cos . x ) ) ^2 .= ( ( 1 / 2 ) * ( cos . x ) ) ^2 .= ( ( 1 / 2 ) * ( cos . x ) ) ^2 .= ( 1 / 2 ) ^2 ;
( - ( - b - sqrt ( a , b ) ) + sqrt ( a , c ) ) / ( 2 * a ) < 0 & ( - b - sqrt ( a , b ) ) / ( 2 * a ) < 0 or ( - b - sqrt ( a , c ) ) / ( 2 * a ) < 0 ;
assume that ex_inf_of uparrow "\/" ( X , L ) , L and ex X st X is maximal & for Y st Y in X holds "\/" ( X , L ) = "/\" ( Y , L ) and for Y st Y in X holds Y < "/\" ( Y , L ) ;
( ( for i being Element of NAT holds i = j implies ( i = j = j ) ) implies ( i = j implies i = j or i = j ) & ( j = j implies i = j ) & ( j = j implies i = j implies j = j ) )