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contradiction ;
contradiction ;
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contradiction ;
contradiction ;
contradiction ;
contradiction ;
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contradiction ;
contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
B in F ;
a <> c
T c= S
D c= B
c in X ;
b in X ;
X ;
b in D ;
x = e ;
let m ;
h is onto ;
N in K ;
let i ;
j = 1 ;
x = u ;
let n ;
let k ;
y in A ;
let x ;
let x ;
m c= y ;
F is one-to-one ;
let q ;
m = 1 ;
1 < k ;
G is commutative ;
b in A ;
d divides a ;
i < n ;
s <= b ;
b in B ;
let r ;
B is one-to-one ;
R is total ;
x = 2 ;
d in D ;
let c ;
let c ;
b = 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 commutative ;
assume x in I ;
q is let of 0 ;
assume c in x ;
p > 0 ;
assume x in Z ;
assume x in Z ;
1 <= k-2 ;
assume m <= i ;
assume G is commutative ;
assume a divides b ;
assume P is closed ;
\bf 2 > 0 ;
assume q in A ;
W is non bounded ;
f is IC ;
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 Tree ;
let d be element ;
let t be element ;
let x be element ;
let x be element ;
let s be element ;
k <= 5 ;
let X be set ;
let X be set ;
let y be element ;
let x be element ;
P [ 0 ]
let E be set , A be Subset of E ;
let C be Category ;
let x be element ;
k be Nat ;
let x be element ;
let x be element ;
let e be element ;
let x be element ;
P [ 0 ]
let c be element ;
let y be element ;
let x be element ;
let a be Real ;
let x be element ;
let X be element ;
P [ 0 ]
let x be element ;
let x be element ;
let y be element ;
r in REAL ;
let e be element ;
n1 is (#) f ;
Q halts_on s ;
x in such that y in \in \in \in \in \in \in \in h ;
M < m + 1 ;
T2 is open ;
z in b Y. ;
R2 is well-ordering ;
1 <= k + 1 ;
i > n + 1 ;
q1 is one-to-one ;
let x be non 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 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 , y0 ;
let E be Ordinal ;
o OperSymbol o2 ;
O <> O2 ;
let r be Real ;
let f be FinSeq-Location ;
let i be Nat ;
let n be Nat ;
Cl A = A ;
L c= Cl L ;
A /\ M = B ;
let V be RealUnitarySpace , W be Subspace of V ;
not s in Y / 0 ;
rng f <= w ;
b "/\" e = b ;
m = m3 ;
t in h . D ;
P [ 0 ] ;
assume z = x * y ;
S . n is bounded ;
let V be RealUnitarySpace , W be Subspace of V ;
P [ 1 ] ;
P [ {} ] ;
C1 ` is component ;
H = G . i ;
1 <= i `1 + 1 ;
F . m in A ;
f . o = o ;
P [ 0 ] ;
aZ <= being Real ;
R [ 0 ] ;
b in f .: X ;
assume q = q2 ;
x in [#] V ;
f . u = 0 ;
assume e1 > 0 ;
let V be RealUnitarySpace , W be Subspace of V ;
s is non trivial & s is non empty ;
dom c = Q
P [ 0 ] ;
f . n in T ;
N . j in S ;
let T be complete LATTICE , f be Function of T , T ;
the Arrows of F is one-to-one ;
sgn x = 1 ;
k in support a ;
1 in Seg 1 ;
rng f = X ;
len T in X ;
vbeing Nat ;
Sbeing bounded non empty set ;
assume p = p2 ;
len f = n ;
assume x in P1 ;
i in dom q ;
let U0 , E , F , G ;
pp `1 = c ;
j in dom h ;
let k ;
f | Z is continuous ;
k in dom G ;
UBD C = B ;
1 <= len M ;
p in \mathbin { x } ;
1 <= jj & jj <= len G ;
set A = -> set ;
card a [= c ;
e in rng f ;
cluster B \oplus A -> empty ;
H is_\rrangle H1 ;
assume n0 <= m ;
T is increasing ;
e2 <> e2 ;
Z c= dom g ;
dom p = X ;
H is proper ;
i + 1 <= n ;
v <> 0. V ;
A c= Affin A ;
S c= dom F ;
m in dom f ;
let X0 be set ;
c = sup N ;
R is_connected implies union M is connected ;
assume not x in REAL+ ;
Im f is complete ;
x in Int y ;
dom F = M ;
a in On W ;
assume e in A ( ) ;
C c= C-26 ;
mm <> {} ;
let x be Element of Y ;
let f be let ;
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 and v in dom ( E `2 ) ;
- y in I ;
let A be non empty set , F be Function of A , A ;
P0 = 1 ;
assume r in F . k ;
assume f is simple ;
let A be Y. -countable set ;
rng f c= NAT ;
assume P [ k ] ;
f6 <> {} ;
let o be Ordinal ;
assume x is sum of from squares ;
assume not v in { 1 } ;
let Ix , Iy be Element of REAL ;
assume that 1 <= j and j < l ;
v = - u ;
assume s . b > 0 ;
d1 in dom d2 ;
assume t . 1 in A ;
let Y be non empty TopSpace , f be Function of Y , Z ;
assume a in uparrow s ;
let S be non empty Poset ;
a , b // b , a ;
a * b = p * q ;
assume x , y are_the space ;
assume x in [#] ( f ) ;
[ a , c ] in X ;
mm <> {} ;
M + N c= M + M ;
assume M is connected connected hh) ;
assume f is additive inbr-r) ;
let x , y be element ;
let T be non empty TopSpace ;
b , a // b , c ;
k in dom Sum p ;
let v be Element of V ;
[ x , y ] in T ;
assume len p = 0 ;
assume C in rng f ;
k1 = k2 & k2 = k2 ;
m + 1 < n + 1 ;
s in S \/ { s } ;
n + i >= n + 1 ;
assume Re y = 0 ;
k1 <= j1 & j1 <= j2 ;
f | A is continuous ;
f . 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 ;
Cx in C ;
q2 c= C1 & q2 c= C2 ;
a2 < c2 & a2 < c2 ;
s2 is 0 -started ;
IC s = 0 & IC s = 0 ;
s4 = s4 & s4 = s3 ;
let V ;
let x , y be element ;
let x be Element of T ;
assume a in rng F ;
x in dom T ` ;
let S be function of L ;
y " <> 0 ;
y " <> 0 ;
0. V = uw ;
y2 , y , w as u1 of V ;
R8 in dom R ;
let a , b be Real , x be Real ;
let a be Object of C ;
let x be Vertex of G ;
let o be Object of C , a be Object of C ;
r '&' q = P \lbrack l , l .] ;
let i , j be Nat ;
s be State of A , a be Element of A ;
s4 . n = N ;
set y = ( x `1 ) / ( x `2 ) ;
mi in dom g ;
l . 2 = y1 ;
|. g . y .| <= r ;
f . x in CM ;
V1 is non empty ;
let x be Element of X ;
0 <> f . g2 ;
f2 /* q is convergent ;
f . i is_measurable_on E ;
assume \xi in N-22 ;
reconsider i = i as Ordinal ;
r * v = 0. X ;
rng f c= INT & rng f c= INT ;
G = 0 .--> goto 0 ;
let A be Subset of X ;
assume that X0 is dense and A is dense ;
|. f . x .| <= r ;
let x be Element of R ;
let b be Element of L ;
assume x in W-19 ;
P [ k , a ] ;
let X be Subset of L ;
let b be Object of B ;
let A , B be category ;
set X = Vars ( C ) ;
let o be OperSymbol of S ;
let R be connected non empty Poset ;
n + 1 = succ n ;
xY c= Z1 & xY c= Z1 ;
dom f = [: C1 , C2 :] ;
assume [ a , y ] in X ;
Re ( seq . n ) is convergent ;
assume a1 = b1 & a2 = b2 ;
A = Int ( A ) ;
a <= b or b <= a ;
n + 1 in dom f ;
let F be Instruction of S , k be Nat ;
assume that r2 > x0 and x0 < r2 ;
let Y be non empty set , f be Function of Y , Z ;
2 * x in dom W ;
m in dom g2 & m + 1 in dom g2 ;
n in dom g1 & n in dom g2 ;
k + 1 in dom f ;
not the still of { s } is finite ;
assume x1 <> x2 & x1 <> x3 ;
v1 in V1 & v2 in V1 & v1 in V1 ;
not [ b `1 , b `2 ] in T ;
ii + 1 = i ;
T c= and T c= and T is TopSpace ;
( l . 1 ) `1 = 0 ;
n be Nat ;
( t `2 ) ^2 = r ;
AA is_integrable_on M & AA is integrable ;
set t = Top t ;
let A , B be real-membered set ;
k <= len G + 1 ;
[: C , D :] misses [: V , C :] ;
product ( s ) is non empty ;
e <= f or f <= e ;
cluster non empty -> non empty normal for NAT ;
assume c2 = b2 & c1 = b2 ;
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 convergent & vseq is convergent ;
IC s3 = 0 & IC s2 = 0 ;
k in N or k in K ;
F1 \/ F2 c= F ;
Int ( G1 \/ G2 ) <> {} ;
( 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 antisymmetric RelStr ;
q |^ m >= 1 ;
a is_>=_than X & b is_>=_than Y ;
assume <* a , c *> <> {} ;
F . c = g . c ;
G is one-to-one non empty full ;
A \/ { a } c= B ;
0. V = 0. Y .= 0. Y ;
let I be non empty halting Instruction of S , S be Nat ;
f-24 . x = 1 ;
assume z \ x = 0. X ;
C4 = 2 to_power n ;
let B be SetSequence of Sigma ;
assume X1 = p .: D ;
n + l2 in NAT & n + l2 in NAT ;
f " P is compact ;
assume x1 in REAL & x2 in REAL ;
p1 `1 = ( K `1 ) * ( 1 + 1 ) `1 ;
M . k = <*> REAL ;
phi . 0 in rng phi ;
OSMbeing closed non empty set ;
assume z0 <> 0. L & z0 <> 0. L ;
n < NT . k ;
0 <= ( seq . 0 ) ^2 ;
- q + p = v ;
{ v } is Subset of B ;
set g = f /. ( \restriction 1 ) ;
[: R , R :] is stable ;
set cR = Vertices R , cR = Vertices R ;
pp c= P3 & P3 c= P3 ;
x in [. 0 , 1 .[ ;
f . y in dom F ;
let T be Scott Scott Scott Scott Scott Scott Scott TopLattice of S ;
inf the carrier of S = S ;
downarrow a = downarrow b & downarrow a c= downarrow b ;
P , C , K , L is_collinear ;
assume x in F ( s , r , t ) ;
2 / i < 2 / m ;
x + z = x + z + q ;
x \ ( a \ x ) = x ;
||. x-y .|| <= r ;
assume that Y c= field Q and Y <> {} ;
a ~ , b ~ are_isomorphic ;
assume a in A ( ) ;
k in dom ( q4 ^ <* x *> ) ;
p is holds holds p is FinSequence of 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 ;
g " * a in N ;
K <> { [ {} , {} ] } ;
cluster -> strict for succ \rm \mathbb } ;
|. q .| ^2 > 0 ;
|. p4 .| = |. p .| ;
s2 - s1 > 0 & s2 - s1 > 0 ;
assume x in { Gij } ;
W-min C in C & W-min C in C ;
assume x in { Gij } ;
assume i + 1 = len G ;
assume i + 1 = len G ;
dom I = Seg n & dom I = Seg n ;
assume that k in dom C and k <> i ;
1 + 1-1 <= i + --> j ;
dom S = dom F & rng S c= dom F ;
let s be Element of NAT , a be Element of NAT ;
let R be ManySortedSet of A ;
let n be Element of NAT , x be set ;
let S be non empty non void non void holds S is non void ;
let f be ManySortedSet of I ;
let z be Element of F_Complex , p be FinSequence of COMPLEX ;
u in { \hbox { \boldmath $ g } } ;
2 * n < 2 * ( n + 1 ) ;
let x , y be set ;
B-11 c= V1 & B-15 c= V1 ;
assume I is_halting_on s , P ;
U2 = U2 & U2 = U2 & U2 = U2 ;
M /. 1 = z /. 1 ;
x11 = x22 & x22 = x22 ;
i + 1 < n + 1 + 1 ;
x in { {} , <* 0 *> } ;
fT <= fT & fT <= fT ;
l be Element of L ;
x in dom ( F . -17 ) ;
let i be Element of NAT , a be Element of REAL ;
r8 is ( len p ) -element ;
assume <* o2 , o *> <> {} ;
s . x |^ 0 = 1 ;
card ( K1 ) in M & card ( K1 ) in M ;
assume that X in U and Y in U ;
let D be st D is st D is \frac of Omega D ;
set r = { q } + 1 ;
y = W . ( 2 * x ) ;
assume dom g = cod f & cod g = cod g ;
let X , Y be non empty TopSpace , f be Function of X , Y ;
x \oplus A is interval ;
|. <*> A .| . a = 0 ;
cluster -> strict for Sublattice of L ;
a1 in B . s1 & a2 in B . s1 ;
let V be finite VectSp of F , W be Subspace of V ;
A * B on B & A * B on A ;
fg = NAT --> 0 ;
let A , B be Subset of V ;
z1 = P1 . j & z2 = P1 . j ;
assume f " P is closed ;
reconsider j = i as Element of M ;
let a , b be Element of L ;
assume q in A \/ ( B "\/" C ) ;
dom ( F (#) C ) = o ;
set S = INT / ( X , Y ) ;
z in dom ( A --> y ) ;
P [ y , h . y ] ;
{ x0 } c= dom f & { x0 } c= dom f ;
let B be non-empty ManySortedSet of I , A be ManySortedSet of I ;
( PI / 2 ) < Arg z ;
reconsider z9 = 0 - z as Nat ;
LIN a , d , c ;
[ y , x ] in II ;
( Q ) `1 = 0 & ( Q `1 ) = 0 ;
set j = x0 div m , m = x0 mod m ;
assume a in { x , y , c } ;
j2 - jj > 0 & j2 - 1 > 0 ;
I ( ) = 1 & I ( ) = 1 ;
[ y , d ] in FF ;
let f be Function of X , Y ;
set A2 = ( B |^ C ) * ( A |^ C ) ;
s1 , s2 are_/ 2 implies s1 , s2 |= s2 , T
j1 -' 1 = 0 & j1 -' 1 = 0 ;
set m2 = 2 * n + j ;
reconsider t = t as bag of n ;
I2 . j = m . j ;
i |^ s , n are_relative_prime ;
set g = f | D-21 , h = f | D-21 ;
assume that X is lower and 0 <= r ;
( p1 `1 ) ^2 = 1 ;
a < ( p3 `1 ) / ( p3 `2 ) ;
L \ { m } c= UBD C ;
x in Ball ( x , 10 ) ;
not a in LSeg ( c , m ) ;
1 <= i1 -' 1 & i1 -' 1 <= len G ;
1 <= i1 -' 1 & i1 -' 1 <= len G ;
i + i2 <= len h ;
x = W-min ( P ) & x = E-max ( P ) ;
[ x , z ] in X ~ ;
assume y in [. x0 , x .] ;
assume p = <* 1 , 2 , 3 *> ;
len <* A1 *> = 1 & len <* A1 *> = 1 ;
set H = h . gg , I = h . W ;
card b * a = |. a .| ;
Shift ( w , 0 ) |= v ;
set h = h2 ** h1 , h1 = h2 ** h2 ;
assume x in ( X2 /\ X1 ) /\ ( X2 /\ X2 ) ;
||. h .|| < d1 & ||. h .|| < d1 ;
not x in the carrier of f & x in the carrier of f ;
f . y = F ( y ) ;
for n holds X [ n ] ;
k - l = kl2 ;
<* p , q *> /. 2 = q ;
let S be Subset of the topology of Y ;
, P 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 reflexive transitive RelStr , f be Function of L , L ;
Sbe x -basis i is x -basis of i ;
let r be non positive Real ;
M , v |= All ( x , y ) ;
v + w = 0. ( Z , n ) ;
P [ len F ] implies P [ F ( ) ]
assume InsCode ( i , 5 ) = 8 & InsCode ( i , 5 ) = 8 ;
the zero Element of M = 0 & the carrier of M = 0 ;
cluster z * seq -> summable ;
let O be Subset of the carrier of C ;
||. f .|| | X is continuous ;
x2 = g . ( j + 1 ) ;
cluster -> / for Element of AllTermsOf S ;
reconsider l1 = l- 1 as Nat ;
v4 is Vertex of r2 & v4 is Vertex of r2 ;
T1 is SubSpace of T2 & T2 is SubSpace of T2 ;
Q1 /\ Q19 <> {} & Q1 /\ Q19 c= Q1 /\ Q19 ;
k be Nat ;
q " is Element of X & q is Element of Y ;
F . t is set of zero non zero set ;
assume that n <> 0 and n <> 1 ;
set en = EmptyBag n , e = EmptyBag n ;
let b be Element of Bags n ;
assume for i holds b . i is commutative ;
x is root implies ( p . x ) `2 = ( p . x ) `2
not r in ]. p , q .[ ;
let R be FinSequence of REAL , a be Real ;
S7 does not destroy b1 , T ;
IC SCM R <> a & IC SCM R <> a ;
|. - |[ x , y ]| .| >= r ;
1 * ( seq . m ) = seq . m ;
let x be FinSequence of NAT , k be Nat ;
let f be Function of C , D , g be Function ;
for a holds 0. L + a = a
IC s = s . NAT .= succ IC s .= succ IC s ;
H + G = F- ( GG ) ;
Cp1 . x = x2 & Cp1 . x = y2 ;
f1 = f .= f2 .= f2 .= f1 * f2 ;
Sum <* p . 0 *> = p . 0 ;
assume v + W = v + u + W ;
{ a1 } = { a2 } ;
a1 , b1 _|_ b , a ;
d1 , o _|_ o , a3 ;
I is_reflexive & I is_reflexive implies I is_\circ ( I , J )
IT is antisymmetric implies ( for C being Element of I holds C is antisymmetric iff C is antisymmetric )
sup rng ( H1 | n ) = e & sup rng ( H1 | n ) = e ;
x = a9 * a9 & y = b9 * c9 ;
|. p1 .| ^2 >= 1 ^2 ;
assume j2 -' 1 < 1 & j2 -' 1 < len G ;
rng s c= dom f1 & rng s c= dom f2 ;
assume support a misses support b & support b misses support c ;
let L be associative commutative non empty doubleLoopStr , a be Element of L ;
s " + 0 < n + 1 ;
p . c = ( f " ) . 1 ;
R . n <= R . ( n + 1 ) ;
Directed I1 = I1 +* ( 1 , card I ) ;
set f = + ( x , y , r ) ;
cluster Ball ( x , r ) -> bounded ;
consider r being Real such that r in A ;
cluster non empty for NAT -defined Function ;
let X be non empty directed Subset of S ;
let S be non empty full SubRelStr of L ;
cluster <* *> . ( N , N ) -> complete for non trivial set ;
( 1 - a ) " = a ;
( q . {} ) `1 = o ;
n - ( i -' 1 ) > 0 ;
assume ( 1 - t `1 ) ^2 <= t `1 ;
card B = k + - 1 ;
x in union rng ( f | ( len f ) ) ;
assume x in the carrier of R & y in the carrier of R ;
d in D ;
f . 1 = L . ( F . 1 ) ;
the carrier' of G = { v } & the carrier' of G = { v } ;
let G be *> : G is : Let ;
e , v6 , v6 , v7 be set ;
c . ( i - 1 ) in rng c ;
f2 /* q is divergent_to+infty & f2 /* q is divergent_to+infty ;
set z1 = - z2 , z2 = - z2 , z2 = - z2 , z2 = - z2 , z1 = - z2 , z2 = - z2 , z2 = - z2 , z2 = - z2 , z2
assume w is_llof S , G ;
set f = p |-count ( t , n ) ;
let c be Object of C ;
assume ex a st P [ a ] ;
let x be Element of REAL m , y be Real ;
let Ix be Subset-Family of X , y be set ;
reconsider p = p as Element of NAT ;
let v , w be Point of X ;
let s be State of SCM+FSA , k be Nat ;
p is FinSequence of ( the InstructionsF of SCM+FSA ) ;
stop I c= ( card I + 1 ) ;
set ci = fSet /. i , fj = f22 /. i ;
w ^ t -tree s , w ^ t ^ s ^ t ^ t ^ s ^ t ^ t ^ s ^ t ^ s ^ t ^ t ^ t ^ s ^ t ^ t ^ s ^
W1 /\ W = W1 /\ W2 ` .= W1 /\ W2 ;
f . j is Element of J . j ;
let x , y be Element of T2 , a be Element of T2 ;
ex d st a , b // b , d ;
a <> 0 & b <> 0 & c <> 0 ;
ord x = 1 & x is positive ;
set g2 = lim ( seq ^\ k ) , g1 = lim ( seq ^\ k ) ;
2 * x >= 2 * ( 1 / 2 ) ;
assume ( a 'or' c ) . z <> TRUE ;
f (*) g in Hom ( c , c ) ;
Hom ( c , c + d ) <> {} ;
assume 2 * Sum ( q | m ) > m ;
L1 . ( F . ( F . k ) ) = 0 ;
/ ( X \/ R1 ) = / ( X \/ R1 ) ;
( sin . x ) ^2 <> 0 & ( sin . x ) ^2 <> 0 ;
( ( exp_R * sin ) `| Z ) . x > 0 ;
o1 in ( ( X /\ O ) /\ O2 ) /\ O2 ;
e , v6 , v6 , v7 be set ;
r3 > ( 1 - 2 ) * 0 ;
x in P .: ( F -ideal L ) ;
let J be closed Subset of R , I be Ideal of R ;
h . p1 = f2 . O & h . p2 = g2 . O ;
Index ( p , f ) + 1 <= j ;
len ( q | ( len M ) ) = width M ;
the carrier of LK c= A & the carrier of LK c= A ;
dom f c= union rng ( F | m ) ;
k + 1 in support ( support ( n ) ) ;
let X be ManySortedSet of the carrier of S ;
[ x `1 , y `2 ] in InnerVertices ( R ~ ) ;
i = D1 or i = D2 or i = D1 ;
assume a mod n = b mod n ;
h . x2 = g . x1 & h . x2 = g . x2 ;
F c= 2 |^ the carrier of X ;
reconsider w = |. s1 .| as Real_Sequence ;
( 1 / ( m * m + r ) ) < p ;
dom f = dom ( I . ( len I ) ) ;
[#] ( ( TOP-REAL 2 ) | K1 ) = [#] ( ( ( TOP-REAL 2 ) | K1 ) | K1 ) ;
cluster - x -> R_eal -> R_eal ;
then { d1 } c= A & A is closed ;
cluster TOP-REAL n -> finite-ind for non empty Subset of TOP-REAL n ;
let w1 be Element of M ;
x be Element of dyadic ( n ) ;
u in W1 & v in W2 & u 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 / 1 .= n ;
h . J = EqClass ( u , J ) ;
let seq be summable sequence of X , m be Nat ;
dist ( x `1 , y ) < ( r / 2 ) / 2 ;
reconsider mm = m - 1 as Element of NAT ;
x- x0 < r1 - x0 & r1 < x0 - x0 ;
reconsider P ` = P ` as strict Subgroup of N ;
set g1 = p * idseq ( q `1 ) , g2 = p * idseq ( q `2 ) ;
let n , m , k be non zero Nat ;
assume that 0 < e and f | A is lower ;
D2 . ( I . ( len I ) ) in { x } ;
cluster -> subcondensed for Subset of T ;
let P be compact non empty Subset of TOP-REAL 2 , p1 , p2 be Point of TOP-REAL 2 ;
G * ( len G , 1 ) in LSeg ( cos , 1 ) ;
n be Element of NAT , x be Element of REAL n ;
reconsider S8 = S as Subset of T ;
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 - 1 as Element of NAT ;
reconsider d = x as Element of C ( ) ;
let s be 0 -started State of SCMPDS , k be Nat ;
let t be 0 -started State of SCMPDS , Q be Subset of SCMPDS ;
b , b , x , y , a is_collinear ;
assume that i = n \/ { n } and j = k \/ { k } ;
let f be PartFunc of X , Y ;
x0 >= ( sqrt c ) ^2 + ( sqrt c ) ^2 ;
reconsider t7 = T7 as Point of TOP-REAL 2 ;
set q = h * p ^ <* d *> ;
z2 in U . ( y2 ) /\ Q2 . ( z2 + 1 ) ;
A |^ 0 = { <* E *> } ;
len W2 = len W + 2 & len W2 = len W1 + 1 ;
len h2 in dom h2 & len h2 in dom h2 ;
i + 1 in Seg ( len s2 + 1 ) ;
z in dom g1 /\ dom f & z in dom f1 /\ dom f2 ;
assume p2 `1 = E-bound ( K ) & p2 `1 <= E-bound ( K ) ;
len G + 1 <= i1 + 1 ;
f1 (#) f2 is convergent & lim ( f1 (#) f2 ) = x0 ;
cluster ( seq + seq ) ^\ k -> summable ;
assume j in dom ( M1 * ( i , j ) ) ;
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 ;
<* xy *> ^ <* y *> Y. iff <* xy *> Y.
a , b in { a , b } ;
len p2 is Element of NAT & len p1 = len p2 ;
ex x being element st x in dom R & R . x = y ;
len q = len ( K (#) G ) ;
s1 = Initialize ( Initialized s ) , P1 = P +* Initialize s , P2 = P +* Initialize s , P3 = P +* P3 ;
consider w being Nat such that q = z + w ;
x ` is Element of L & x is Element of L ;
k = 0 & n <> k or k > n ;
then X is discrete for A is closed ;
for x st x in L holds x is FinSequence ;
||. f /. c .|| <= r1 & ||. f /. c .|| <= r ;
c in uparrow p & not c in { p } ;
reconsider V = V as Subset of the topology of TOP-REAL n ;
N , M be being being being being being being being being being being being being being being being being being being being being being <* of L ;
then z is_>=_than waybelow x & z is_>=_than compactbelow y ;
M \lbrack f , f .] = f & M \lbrack g , f .] = g ;
( ( ( ( ( to_power 1 ) ) to_power 1 ) ) to_power 1 ) = TRUE ;
dom g = dom f & rng g c= dom f ;
mode \cal il of G is \cal ppp.. of G ;
[ i , j ] in Indices M & M * ( i , j ) = M * ( i , j ) ;
reconsider s = x " as Element of H ;
let f be Element of dom ( Subformulae p ) , F be Element of dom ( Subformulae p ) ;
F1 . ( a1 , - a2 ) = G1 . ( a1 , a2 ) ;
cluster -> compact ( a , b , r ) -> compact ;
let a , b , c , d be Real ;
rng s c= dom ( 1 / 2 ) & rng s c= dom ( 1 / 2 ) ;
curry ( ( F . -19 ) . k ) is additive ;
set k2 = card dom B , s3 = card { x } ;
set G = DTConMSA ( X ) ;
reconsider a = [ x , s ] as terminal of G ;
let a , b be Element of MM , M be Matrix of REAL ;
reconsider s1 = s , s2 = t as Element of ( S , X ) ;
rng p c= the carrier of L & rng p c= the carrier of L ;
let d be Subset of the Sorts of A ;
( x .|. x = 0 iff x = 0. W ) ;
I-21 in dom stop I & Ik in dom stop I ;
let g be continuous Function of X | B , Y ;
reconsider D = Y as Subset of TOP-REAL n ;
reconsider i0 = len p1 - 1 as Integer ;
dom f = the carrier of S & rng f c= the carrier of S ;
rng h c= union ( the carrier of J ) ;
cluster All ( x , H ) -> reconsider All ( x , G ) ;
d * N1 ^2 > N1 * 1 & d * N2 ^2 > N2 * 1 ;
]. a , b .[ c= [. a , b .] ;
set g = f " ( D1 \/ D2 ) ;
dom ( p | mm1 ) = mm1 & dom ( p | mm2 ) = mm2 ;
3 + - 2 <= k + - 2 + - 2 ;
tan is_differentiable_in ( ( arccot * ( arccot ) ) * ( f ^ ) ) . x ;
x in rng ( f /^ ( p .. f ) ) ;
let f , g be FinSequence of D ;
[: p , q :] in the carrier of [: S1 , S2 :] ;
rng f " { x } = dom f & rng f = { x } ;
( the Source of G ) . e = v & ( the Source of G ) . e = v ;
width G -' 1 < width G - 1 & width G - 1 < width G ;
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 < len g2 ;
let q be Point of TOP-REAL 2 , a , b be Real ;
let p be Point of TOP-REAL 2 , r be Real ;
dist ( O , u ) <= |. p2 .| + 1 ;
assume dist ( x , b ) < dist ( a , b ) ;
<* S7 *> is_in the InternalRel of C-20 ( C7 ) ;
i <= len G -' 1 + ( len G -' 1 ) ;
let p be Point of TOP-REAL 2 , r be Real ;
x1 in the carrier of I[01] & x2 in the carrier of I[01] ;
set p1 = f /. i , p2 = f /. j ;
g in { g2 : r < g2 & g2 < r } ;
Q2 = Sp2 " ( Q , P ) ;
( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 )
- p + I c= - p + A ;
n < LifeSpan ( P1 , s1 ) + 1 ;
CurInstr ( p1 , s1 ) = i & CurInstr ( p2 , s2 ) = halt S ;
A /\ Cl { x } \ { x } <> {} ;
rng f c= ]. r , r + 1 .[ ;
let g be Function of S , V ;
let f be Function of L1 , L2 , g be Function of L1 , L2 ;
reconsider z = z as Element of ( ( L ) | the carrier of L ) ;
let f be Function of S , T ;
reconsider g = g as Morphism of c opp , b opp ;
[ s , I ] in S ~ ;
len ( the connectives of C ) = 4 & len ( the connectives of C ) = 5 ;
let C1 , C2 be Subcategory of C , a , b be object of C1 ;
reconsider V1 = V as Subset of X | B ;
attr p is valid means : Def3 : All ( x , p ) is valid ;
assume that X c= dom f and f .: X c= dom g and f .: X c= dom g ;
H |^ a |^ ( a |^ b ) is Subgroup of H ;
let A1 be Let of O , E1 , A2 be Element of E ;
p2 , r3 , p2 is_collinear & q2 , r2 , p3 is_collinear ;
consider x being element such that x in v ^ K ;
not x in { 0. TOP-REAL 2 } & x in { 0. TOP-REAL 2 } ;
p in [#] ( I[01] | B11 ) & p in [#] ( I[01] | B11 ) ;
0 . m < M . ( E8 . m ) ;
^ ( c ^ ( c ^ ( a ^ b ) ) ) = c ;
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . s2 ) & a2 in dom ( F . s2 ) ;
cluster -> Line for Subas non empty as thesis of L ;
set i1 = the Nat , i2 = the Element of NAT ;
let s be 0 -started State of SCM+FSA , P be Subset of P ;
assume y in ( f1 \/ f2 ) .: A ;
f . ( len f ) = f /. ( len f ) ;
x , f . x '||' f . x , f . y ;
attr X c= Y means : Def4 : cos ( X ) c= cos ( Y ) ;
let y be upper Subset of Y , x be Element of Y ;
cluster ( x `1 ) / ( x `2 ) -> non trivial for non empty non trivial NAT ;
set S = <* Bags n , <* i9 *> *> ;
set T = [. 0 , PI / 2 .] ;
1 in dom mid ( f , 1 , 1 ) ;
( 4 * PI * PI ) / 2 < ( 2 * PI * PI ) / 2 ;
x2 in dom ( f1 | X ) /\ dom ( f2 | X ) ;
O c= dom I & { {} } = { {} } ;
( the Source of G ) . x = v & ( the Source of G ) . x = v ;
{ HT ( f , T ) } c= Support f ;
reconsider h = R . k as Polynomial of n , L ;
ex b being Element of G st y = b * H ;
let x `1 , y , z `2 , x , y , z `2 be Element of G ;
h19 . i = f . ( h . i ) ;
( p `1 ) ^2 = ( p1 `1 ) ^2 + ( p1 `2 ) ^2 ;
i + 1 <= len Cage ( C , n ) ;
len <* P *> = len P & len <* P *> = len P ;
set N-26 = the be Element of the set of N , x = the Element of N ;
len gy0 + ( x + 1 ) - 1 <= x ;
a on B & b on B & a on B implies a on b
reconsider rc = r * I . v as FinSequence of REAL ;
consider d such that x = d and a is_less_than d ;
given u such that u in W and x = v + u ;
len f /. ( \downharpoonright n ) = len ( 9 /. n ) ;
set q2 = N-min L~ Cage ( C , n ) , q2 = E-max L~ Cage ( C , n ) ;
set S =
MaxADSet ( b ) c= MaxADSet ( P ) /\ Q ;
Cl ( G . q1 ) c= F . r2 ;
f " D meets h " ( V ) /\ h " ( V ) ;
reconsider D = E as non empty directed Subset of L1 ;
H = ( the_left_argument_of H ) '&' ( the_right_argument_of H ) ;
assume t is Element of ( gF ) . ( X , Y ) ;
rng f c= the carrier of S2 & rng f c= the carrier of S2 ;
consider y being Element of X such that x = { y } ;
f1 . ( a1 , b1 ) = b1 & f1 . ( a2 , b2 ) = b2 ;
the carrier' of G = E \/ { E } .= { E } ;
reconsider m = len wk - 1 as Element of NAT ;
set S1 = LSeg ( n , UMP C ) , S2 = LSeg ( n , LMP C ) ;
[ i , j ] in Indices M1 & M1 * ( i , j ) = M1 * ( i , j ) ;
assume that P c= Seg m and M is \HM { of Seg m } ;
for k st m <= k holds z in K . k ;
consider a being set such that p in a and a in G ;
L1 . p = p * L /. 1 .= p * L /. 1 ;
p-7 . i = pi1 . i .= pi2 . i ;
let PA , PA , G be a_partition of Y , a be Element of Y ;
attr 0 < r & r < 1 implies 1 < ( 1 - r ) / ( r - s ) ;
rng ( AffineMap ( 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 ) \ { x } ) ;
dom ( f . 0 ) c= dom ( u . 0 ) ;
redefine pred n divides m & m divides n & n = m ;
reconsider x = x as Point of I[01] , I[01] ;
a in ;
not y0 in the still of f & not ( f . y in the still of f ) ;
Hom ( ( a , b ) --> c ) <> {} ;
consider k1 such that p " < k1 and k1 < len p and p . k1 = x ;
consider c , d such that dom f = c \ d ;
[ x , y ] in dom g & [ y , x ] in dom g ;
set S1 = Let ( x , y , z ) , S2 = y , S2 = z ;
l = m2 & l2 = i2 & l2 = j2 & l2 = i2 & l2 = j2 ;
x0 in dom ( ( u + v ) | A ) /\ A ;
reconsider p = x as Point of TOP-REAL 2 , r be Real ;
I[01] = R^1 | B01 & R^1 | B01 = ( R^1 | B01 ) | B01 ;
f . p4 `1 <= f . p1 `1 & f . p2 `1 <= f . p3 `1 ;
( F . ( x , y ) ) `1 <= ( F . ( x , y ) ) `1 ;
( x `2 ) ^2 = ( W `2 ) ^2 + ( W `2 ) ^2 ;
for n being Element of NAT holds P [ n ] ;
let J , K be non empty Subset of I ;
assume 1 <= i & i <= len <* a " *> ;
0 |-> a = <*> the carrier of K ;
X . i in 2 to_power ( A . i \ B . i ) ;
<* 0 *> in dom ( e --> [ 1 , 0 ] ) ;
then P [ a ] & P [ succ a ] ;
reconsider sbeing object of D , sbeing object of D ;
( Seg i -' 1 ) <= len wj ;
[#] S c= [#] T & [#] T c= [#] T ;
for V being strict RealUnitarySpace holds V in the carrier of V implies V in : V in the carrier of V
assume k in dom mid ( f , i , j ) ;
let P be non empty Subset of TOP-REAL 2 , p1 , p2 be Point of TOP-REAL 2 ;
let A , B be Matrix of n1 , n2 , K ;
- a * - b = a * b - a * c ;
for A being Subset of GX holds A // A implies A is being_line
( the Arrows of o2 ) . ( o1 , o2 ) in <^ o2 , o2 ^> ;
then ||. x .|| = 0 & x = 0. X ;
let N1 , N2 be strict normal normal Subgroup of G ;
j >= len upper_volume ( g , D1 ) & j <= len upper_volume ( g , D1 ) ;
b = Q . ( len Q- 1 + 1 ) ;
f2 * ( f1 /* s ) is divergent_to+infty & f2 * ( f1 /* s ) is divergent_to+infty ;
reconsider h = f * g as Function of N4 , G ;
assume that a <> 0 and delta ( a , b , c ) >= 0 ;
[ t , t ] in the InternalRel of A & [ t , t ] in the InternalRel of A ;
( v |-- E ) | n is Element of TT & v . n in TT ;
{} = the carrier of L1 + L2 & the carrier of L1 = the carrier of L2 ;
Directed I is_halting_on Initialized s , P & Directed I is_halting_on Initialized s , P ;
Initialized p = Initialize ( p +* q ) , p2 = p +* q ;
reconsider N2 = N1 , N2 = N2 as strict net of R2 ;
reconsider Y = Y as Element of [: Ids L , Ids L :] ;
"/\" ( uparrow p , L ) <> p ;
consider j being Nat such that i2 = i1 + j and j in dom G ;
not [ s , 0 ] in the carrier of S2 & not [ s , 0 ] in the carrier of S2 ;
mm in ( B '&' C ) '/\' D \ { {} } ;
n <= len ( ( P + Q ) . ( n + 1 ) ) ;
( x1 `1 ) ^2 = ( x2 `1 ) ^2 + ( x1 `2 ) ^2 ;
InputVertices S = { x1 , x2 , x3 , x4 , x5 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 } ;
let x , y be Element of let F_T1 ( n ) ;
p = |[ p `1 / |. p .| - cn , p `2 / |. p .| - cn ]| ;
g * 1_ G = h " * g * h .= g " * g ;
let p , q be Element of let V , C , D be Element of V ;
x0 in dom x1 /\ dom x2 & x0 in dom x1 /\ dom x2 /\ dom x3 ;
( R qua Function ) " = R " & ( R " ) " = R " ;
n in Seg len ( f /^ ( len f -' 1 ) ) ;
for s being Real st s in R holds s <= s2 & s2 <= s2 ;
rng s c= dom ( f2 * f1 ) /\ dom ( f2 * f1 ) ;
synonym for for for for for for for for for the carrier of X , the carrier of Y ;
1. ( K , n ) * 1. ( K , n ) = 1. ( K , n ) ;
set S = Segm ( A , P1 , Q1 ) , T = Segm ( A , P2 , s2 ) ;
ex w st e = ( w / f ) * f & w in F ;
curry ( Pbe ( Pholds x , k ) # x is convergent ;
cluster -> open for Subset of TK ;
len f1 = 1 .= len f3 .= len f3 .= len f3 + len f3 .= len f3 + len f3 ;
( i * p ) / p < ( 2 * p ) / p ;
let x , y be Element of OSSub ( U0 ) ;
b1 , c1 // b9 , c9 & b1 , c1 // b9 , c9 ;
consider p being element such that c1 . j = { p } ;
assume that f " { 0 } = {} and f is total ;
assume IC Comput ( F , s , k ) = n ;
Reloc ( J , card I ) does not destroy a ;
( goto ( card I + 1 ) ) not ( card I + 1 ) does not destroy c ;
set m3 = LifeSpan ( p3 , s3 ) , P4 = P +* I , P4 = P +* I , P4 = P +* I , P4 = Comput ( P3 , s3 , 1 ) , P4 = Comput ( P3 , s3 , 1 ) , P4 = P3 ;
IC Comput ( P , s , k ) in dom ( Initialize p ) ;
dom t = the carrier of SCM R & dom t = the carrier of SCM R ;
( ( N-min L~ f ) .. f ) .. f = 1 & ( ( E-max L~ f ) .. f ) .. f = 1 ;
let a , b be Element of Let ( V , C ) ;
Cl ( union Int Cl ( Int Cl F ) ) c= Cl ( Int Cl ( union F ) ) ;
the carrier of X1 union X2 misses ( ( X1 union X2 ) \/ ( X1 union X2 ) ) ;
assume not LIN a , f . a , g . b ;
consider i being Element of M such that i = d6 and i in dom f ;
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 ) ;
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 <> a4 ;
cluster s -\mathop { V } -> .| for string of S ;
Ln2 /. n2 = Ln2 . n2 .= Ln2 . n2 .= Ln2 . n2 ;
let P be compact non empty Subset of TOP-REAL 2 , p1 , p2 be Point of TOP-REAL 2 ;
assume that r-7 in LSeg ( p1 , p2 ) and rp2 in LSeg ( p1 , p2 ) ;
let A be non empty compact Subset of TOP-REAL n , a be Real ;
assume [ k , m ] in Indices ( D1 | Seg n ) ;
0 <= ( 1 - ( 2 |^ p ) ) / ( 2 |^ p ) ;
( F . N ) | E1 = +infty ;
attr X c= Y means : Def3 : Z c= V & X \ V c= Y \ Z ;
( y `2 ) * ( z `2 ) <> 0. I & ( y `2 ) * ( y `2 ) <> 0. I ;
1 + card ( X-18 ) <= card ( u \/ { x } ) ;
set g = z \circlearrowleft ( ( L~ z ) .. z ) , z = z .. z , p = z .. z , q = z .. z , r = z .. z , s = z .. z , s = z .. z , s = z .. z ,
then k = 1 & p . k = <* x , y *> . k ;
cluster -> total for Element of C -succ X ;
reconsider B = A as non empty Subset of TOP-REAL n , C be Subset of TOP-REAL n ;
let a , b , c be Function of Y , BOOLEAN ;
L1 . i = ( i .--> g ) . i .= g . i .= g . i ;
Plane ( x1 , x2 , x3 , x4 , x5 ) c= P & Plane ( x1 , x2 , x3 , x4 , x5 ) c= P ;
n <= indx ( D2 , D1 , j1 ) + 1 - 1 ;
( ( g2 ) . O ) `1 = - 1 & ( g2 ) . I `2 <= 1 ;
j + p .. f -' len f <= len f - len f ;
set W = W-bound C , S = S-bound C , E = E-bound C , N = E-bound C , S = E-bound C , N = E-bound C , N = E-bound C , S = E-bound C , N = E-bound C , S = E-bound C , N = E-bound
S1 . ( a `1 , e ) = a + e .= a `1 ;
1 in Seg width ( ( M * ColVec2Mx p ) * ( ColVec2Mx p ) ) ;
dom ( i (#) Im ( f | X ) ) = dom Im ( f | X ) ;
DW . ( x `1 ) = W . ( a , \ast ( a , p ) ) ;
set Q = non empty contradiction , f = g +* f , h = h +* f , f = h +* f , g = h +* f , h = f +* g ;
cluster -> topological for ManySortedSet of U1 , U2 ;
attr ex A st F = { A } ;
reconsider z9 = \hbox { y0 , y } as Element of product G ;
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 V ) & g = <*> ( the carrier of V ) ;
E , j |= All ( x1 , x2 ) implies E , j |= All ( x1 , x2 )
reconsider n1 = n as Morphism of o1 , o2 ;
assume that P is idempotent and R is idempotent and P ** R = R ** P ;
card ( B2 \/ { x } ) = k-1 + 1 ;
card ( ( x \ B1 ) /\ B1 ) = 0 implies card ( ( x \ B1 ) /\ B2 ) = 0 ;
g + R in { s : g-r < s & s < g + r } ;
set q14 = ( q , <* s *> ) -\subseteq ( q , <* s *> ) -\subseteq q ;
for x being element st x in X holds x in rng f1 ;
h0 /. ( i + 1 ) = h0 . ( i + 1 ) ;
set mw = max ( B , support ( R | NAT ) ) ;
t in Seg width ( I ^ ( n , n ) ) ;
reconsider X = dom f /\ C as Element of Fin NAT ;
IncAddr ( i , k ) = <% halt SCM+FSA %> + k .= ( i + k ) ;
( ( W-bound L~ f ) / 2 ) * ( ( GoB f ) / 2 ) <= ( q `2 ) * ( ( GoB f ) / 2 ) ;
attr R is condensed means : Def3 : 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 ) /\ e ) ) /\ f ) /\ j ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) /\ f ) /\ j ) /\ f /\ j ;
len C + - 2 >= 9 + - 3 + 3 - 2 ;
x , z , y is_collinear & x , z , y is_collinear implies x , y , z is_collinear
a |^ ( n1 + 1 ) = a |^ n1 * a ;
<* \underbrace ( 0 , \dots , 0 ) *> in Line ( x , a * x ) ;
set y9 = <* y , c *> ;
FG2 /. 1 in rng Line ( D , 1 ) ;
p . m Joins r /. m , r /. ( m + 1 ) , G ;
( p `2 ) ^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 / |. p .| - cn ) <= 2 * r + ( p `2 / |. p .| - cn ) ;
x in dom g & not x in g " { 0 } ;
f1 /* ( seq ^\ k ) is divergent_to+infty & f2 /* ( seq ^\ k ) is divergent_to+infty ;
reconsider u2 = u as VECTOR of P\rm Space ( X ) ;
p |-count ( Product ( Sgm ( X ) ) ) = 0 & p |-count ( Sgm ( X ) ) = 0 ;
len <* x *> < i + 1 & i + 1 <= len c + 1 ;
assume that I is non empty and { x } /\ { y } = 0. I ;
set ii = ( card I + 4 ) .--> ( goto 0 + 4 ) ;
x in { x , y } & h . x = {} T & h . y = {} ;
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 ) ) ;
reconsider m = M , i = I , n = N as Element of X ;
A . ( j + 1 ) = B . ( j + 1 ) \/ A . j ;
set N8 = : : x0 in : x0 in x0 & x0 in dom ( G . n ) ;
rng F c= the carrier of gr { a } & rng F c= the carrier of gr { a } ;
for K being st NQ ( K , n , r ) is in Indices Q holds Q is in rng P ;
f . k , f . ( Let n ) . k are_congruent_mod n are_congruent_mod n ;
h " P /\ [#] T1 = f " P & h " P = f " P ;
g in dom f2 \ f2 " { 0 } & f2 " { 0 } c= dom f2 \ f2 " { 0 } ;
gX /\ dom f1 = g1 " ( X /\ dom g1 ) .= g1 " ( X /\ dom g1 ) ;
consider n being element such that n in NAT and Z = G . n ;
set d1 = being element , d2 = dist ( x1 , y1 ) , d2 = dist ( x2 , y2 ) ;
b `1 + ( 1 - r ) < ( 1 - r ) + ( 1 - r ) ;
reconsider f1 = f as VECTOR of the carrier of X , the carrier of Y ;
attr i <> 0 means : Def3 : i |^ ( i + 1 ) mod ( i + 1 ) = 1 ;
j2 in Seg len ( g2 . i2 ) & 1 <= len ( g2 . i2 ) ;
dom ( i , ( i , j ) --> ( a , b ) ) = dom ( a , b ) .= a ;
cluster sec | ]. PI / 2 , PI / 2 .[ -> one-to-one ;
Ball ( u , e ) = Ball ( f . p , e ) ;
reconsider x1 = x0 as Function of S , I * , ( the Sorts of U1 ) * ;
reconsider R1 = x , R2 = y as Relation of L , L ;
consider a , b being Subset of A such that x = [ a , b ] ;
( <* 1 *> ^ p ) ^ <* n *> in RL ;
S1 +* S2 = S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2
( ( ( - 1 / 2 ) (#) ( ( #Z 2 ) * ( ( #Z 2 ) * ( f1 + f2 ) ) ) ) `| Z ) = f ;
cluster -> C -valued for Function of C , REAL ;
set C7 = 1GateCircStr ( <* z , x *> , f3 ) , C8 = 1GateCircStr ( <* z , x *> , f3 ) ;
E8 . e2 = E8 . e2 & E8 . e2 = E8 . e2 & E8 . e2 = E8 . e2 ;
( ( ( arctan * ( arctan + arccot ) ) (#) ( ( arctan * ( f1 + f2 ) ) ) ) `| Z ) = f ;
upper_bound A = ( PI * 3 ) / 2 & lower_bound A = 0 ;
F . ( dom f , - - f ) is Functor of F . ( cod f , - f ) ;
reconsider p8 = q8 as Point of TOP-REAL 2 , q be Point of TOP-REAL 2 ;
g . W in [#] Y & [#] Y c= [#] Y & g . W in [#] Y ;
let C be compact non vertical non horizontal Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
LSeg ( f ^ g , j ) = LSeg ( f , j ) \/ LSeg ( f , i ) ;
rng s c= dom f /\ ]. - r , x0 .[ & rng s c= ]. - r , x0 .[ ;
assume x in { idseq ( 2 ) , Rev ( idseq 2 ) } ;
reconsider n2 = n , m2 = m , m2 = n as Element of NAT ;
for y being R_eal st y in rng ( seq ^\ k ) holds g <= y
for k st P [ k ] holds P [ k + 1 ]
m = m1 + m2 .= m1 + m2 .= m1 + m2 .= m1 + m2 ;
assume for n holds H1 . n = G . n -H . n ;
set BB = f .: the carrier of X1 , BB = f .: the carrier of X2 ;
ex d being Element of L st d in D & x << d ;
assume R ~ c= R ~ & R ~ c= R ~ & R ~ c= R ~ & R ~ c= R ~ ;
t in ]. r , s .[ or t = r or t = s ;
z + v2 in W & x = u + ( z + v2 ) ;
x2 |-- y2 iff P [ x2 , y2 ] & P [ x2 , y2 ] ;
attr x1 <> x2 means x1 <> x2 & |. x1 - x2 .| > 0 & x1 <> x2 ;
assume p2 - p1 , p3 - p2 - p1 , p3 - p2 - p1 is_collinear ;
set q = ( -1 f ) ^ <* 'not' 'not' A *> ;
let f be PartFunc of REAL-NS 1 , REAL-NS n , x be Point of REAL-NS n ;
( n mod ( 2 * k ) ) = n mod k ;
dom ( T * ( succ t ) ) = dom ( ( succ t ) . ( T . t ) ) ;
consider x being element such that x in wc and x in c ;
assume ( F * G ) . v = v . x3 & ( F * G ) . x3 = v . x4 ;
assume that the carrier of D1 c= the carrier of D2 and the carrier of D1 c= the carrier of D2 ;
reconsider A1 = [. a , b .[ as Subset of R^1 ( R^1 ( A ) ) ;
consider y being element such that y in dom F and F . y = x ;
consider s being element such that s in dom o and a = o . s ;
set p = W-min L~ Cage ( C , n ) , q = E-max L~ Cage ( C , n ) , r = W-bound L~ Cage ( C , n ) , s = E-bound L~ Cage ( C , n ) , s = E-bound L~ Cage ( C , n ) , w = E-bound L~ Cage ( C , n ) , G =
n1 - len f + 1 <= len ( - f + 1 ) + len ( - f + 1 ) ;
|. .| ( q , O1 ) = [ u , v , a , b ] ;
set C-2 = ( `1 ) . ( k + 1 ) , C-2 = ( `1 ) . ( k + 1 ) ;
Sum ( L (#) p ) = 0. R * Sum p .= 0. V * p .= 0. V ;
consider i being element such that i in dom p and t = p . i ;
defpred Q [ Nat ] means 0 = Q ( $1 ) & $1 <= len Q ( ) ;
set s3 = Comput ( P1 , s1 , k ) , P3 = P1 +* stop I , s4 = P2 +* stop I , P4 = P2 +* stop I , P4 = P2 +* stop I , P4 = Comput ( P2 , s2 , k ) , P4 = P2 +* stop I , P4 = P2 +* stop
l be variable of k , Al , x be set , y be element ;
reconsider U2 = union ( G . n ) as Subset-Family of ( T . n ) ;
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 ;
pS = <* - c , 1 , - 1 *> & pS = <* - c , 1 *> ;
synonym f is real-valued for rng f c= NAT & rng f c= NAT ;
consider b being element such that b in dom F and a = F . b ;
x20 < card X0 + card Y2 & card X0 + card Y2 < card X0 + card Y2 + card Y2 ;
attr X c= B1 means : Def4 : for B being set holds \mathop { \rm Bseq X , B is non empty ;
then w in Ball ( x , r ) & dist ( x , w ) <= r ;
angle ( x , y , z ) = angle ( x , y , z ) ;
attr 1 <= len s means : Def3 : len it = len s & for i st i in dom s holds it . i = s . i ;
fbeing set c= f . ( k + ( n + 1 ) ) ;
the carrier of { 1_ G } = { 1_ G } & the carrier of G = { 1_ G } ;
attr p '&' q in \mathbin { \rm TAUT ( A ) : q in TAUT ( A ) } ;
- ( t `1 / t `2 ) < ( t `1 / t `2 ) / t `2 ;
( U . 1 ) = U /. 1 .= ( U /. 1 ) `1 .= ( W . 1 ) `1 .= ( W . 1 ) `1 ;
f .: ( the carrier of x ) = the carrier of x & f .: ( the carrier of x ) = the carrier of x ;
Indices ( 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 \setminus & f is \setminus closed & f is commutative ;
[ h . 0 , h . 3 ] in the InternalRel of G & [ h . 0 , h . 3 ] in the InternalRel of G ;
s +* Initialize ( ( intloc 0 ) .--> 1 ) = s3 +* Initialize ( ( intloc 0 ) .--> 1 ) ;
|[ w1 , v1 ]| `1 <> 0. TOP-REAL 2 & |[ w1 , v1 ]| `1 <> 0. TOP-REAL 2 ;
reconsider t = t as Element of ( the carrier of X ) * ;
C \/ P c= [#] ( ( ( ( the carrier of ( ( ( ( the carrier of ( 2 ) ) \ A ) ) \ A ) ) ) ) ) ;
f " V in ( for X holds V in ( the topology of X ) /\ D ) & V in ( the topology of Y ) /\ D ;
x in [#] ( ( the carrier of T ) /\ A ) /\ the carrier of T ;
g . x <= h1 . x & h . x <= h1 . x & h . x <= h . x ;
InputVertices S = { xy , y9 , z9 , \emptyset , {} , {} , {} , {} , {} , {} , {} , {} , {} } ;
for n being Nat st P [ n ] holds P [ n + 1 ]
set R = Line ( M , i ) * Line ( M , i ) ;
assume that M1 is being_line and M2 is being_line and M1 is being_line and M2 is being_line and M1 is being_line ;
reconsider a = f4 . i0 - 1 as Element of K ;
len B2 = Sum ( Len ( F1 ^ F2 ) ) & len ( Len ( F1 ^ F2 ) ) = len ( Len ( F1 ^ F2 ) ) ;
len ( ( the every finite sequence of elements of n , m ) * ( i , j ) ) = n & len ( ( the _ of n , m ) * ( i , j ) ) = n ;
dom max ( f , g ) = dom ( f + g ) .= dom f /\ dom g ;
( the Sorts of Y ) . n = upper_bound Y1 & ( the Sorts of Y ) . n = upper_bound Y2 ;
dom ( p1 ^ p2 ) = dom ( f ^ p2 ) .= dom ( f ^ p1 ) \/ dom p2 .= dom f \/ dom g ;
M . [ 1 / y , y ] = 1 / ( v1 * v1 ) .= y ;
assume that W is non trivial and W in the carrier of G2 and W c= the carrier of G2 ;
C6 * ( i1 , i2 ) `1 = G1 * ( i1 , i2 ) `1 .= G1 * ( i1 , i2 ) `1 ;
C8 |- 'not' Ex ( x , p ) 'or' p . x ;
for b st b in rng g holds lower_bound rng fmmmmb <= b * r
- ( ( q1 `1 / |. q1 .| - cn ) / ( 1 + cn ) ) = 1 ;
( LSeg ( c , m ) \/ [: NAT , NAT :] ) \/ [: { l } , { l } :] c= R ;
consider p being element such that p in such that p in LSeg ( x , f . p ) and p in L~ f ;
Indices ( X @ ) = [: Seg n , Seg n :] & [: Seg n , Seg n :] = [: Seg n , Seg n :] ;
cluster s => ( q => p ) -> valid ;
Im ( ( Partial_Sums F ) . m ) is measurable of E , ( ( Partial_Sums F ) . m ) . ( ( Partial_Sums F ) . m ) ;
cluster f . ( x1 , x2 ) -> ( x1 , x2 ) -valued for Element of D ;
consider g being Function such that g = F . t and Q [ t , g ] ;
p in LSeg ( ( NW-corner Z ) * ( i , 1 ) , ( E-max Z ) * ( i + 1 ) ) ;
set R8 = R / ( ]. b , +infty .[ ) , R8 = R / ( ]. a , +infty .[ ) ;
IncAddr ( I , k ) = SubFrom ( da , da ) .= ( - ( d + k ) ) ;
( seq . m ) . m <= ( ( seq ^\ k ) ^\ ( m + 1 ) ) . n ;
a + b = ( a ` + b ) ` + ( a ` + b ) ` ;
id ( X /\ Y ) = id ( X /\ Y ) .= id ( X /\ Y ) ;
for x being element st x in dom h holds h . x = f . x ;
reconsider H = U1 \/ U2 , U = 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 , lower_bound B ] ;
consider A being finite stable Subset of R such that card A = card ( R ) and card A = card ( R ) ;
p2 in rng ( f |-- p1 ) \ rng <* p1 *> & p2 in rng ( f |-- p1 ) \ rng <* p2 *> ;
len s1 - 1 > 0 & len s2 - 1 > 0 & len s2 - 1 > 0 ;
( ( N-min L~ f ) .. f ) .. f = ( ( N-min L~ f ) .. f ) .. f & ( N-min L~ f ) .. f = ( ( E-max L~ f ) .. f ) .. f ;
Ball ( e , r ) c= LeftComp Cage ( C , k + 1 ) \/ LeftComp Cage ( C , k + 1 ) ;
f . a1 ` = f . a1 ` & f . a2 = f . a2 ` & f . a1 = f . a2 ` ;
( seq ^\ k ) . n in ]. - r , x0 .[ & ( seq ^\ k ) . n in ]. - r , x0 .[ ;
gg . s0 = g . s0 | G . ( s . s0 ) .= g . ( s . s0 ) ;
the InternalRel of S is symmetric implies the InternalRel of S is symmetric
deffunc F ( Ordinal , Ordinal ) = phi . ( $2 , $2 ) ;
F . a1 = F . a2 & F . a1 = F . a1 & F . a2 = F . a2 ;
x `2 = A . ( o . a ) .= Den ( o , A . a ) ;
Cl ( f " P1 ) c= f " P1 & f " P1 c= f " P2 implies f " P1 is closed
FinMeetCl ( ( the topology of S ) . i ) c= the topology of T & the topology of S is finite ;
synonym o is " means : Def4 : o <> *' & o <> 0. C ;
assume that X = Y and card X = card Y and card Y <> card Y and card Y = card X ;
the finite non empty Subset of s <= 1 + ( the finite Subset of dom s ) * ( the finite Subset of dom s ) ;
LIN a , a1 , d or b , c // b1 , c1 or b , c // b1 , c1 ;
e . 1 = 0 & e . 2 = 1 & e . 3 = 0 & e . 4 = 0 ;
Ex in Sx & Ex in Sx & Ex in { Nx } ;
set J = ( l , u ) \mathop { I } ;
set A1 = Let ( ap , bm , cp ) , A2 = non-empty ( ( ap , bm , cp ) | ( { x , y } ) ) ;
set vs = [ <* vs , dA1 *> , and2a ] , xy = [ <* cin , A1 *> , and2b ] , is non empty ;
x * z `1 * x " in x * ( z * N ) " * x ;
for x being element st x in dom f holds f . x = g2 . x ;
Int cell ( f , 1 , G ) c= RightComp f \/ RightComp f \/ RightComp f \/ RightComp f \/ RightComp f \/ RightComp f ;
U2 is_an_arc_of W-min C , E-max C , E-max C , E-max C , E-max C , E-max C ;
set f-17 = f ^ @ g ;
attr S1 is convergent & S2 is convergent & ( for n holds ( S1 . n ) - ( S2 . n ) ) implies ( S1 . n - S2 . n ) is convergent ;
f . ( 0 + 1 ) = ( 0 qua Ordinal ) + a .= a ;
cluster -> in -> in be in in the carrier of reflexive transitive transitive non empty transitive strict for non empty reflexive transitive RelStr ;
consider d being element such that R reduces b , d and R reduces c , d and R reduces c , d ;
not b in dom Start-At ( ( card I + 2 ) , SCMPDS ) \/ dom Start-At ( ( card I + 2 ) , SCMPDS ) ;
( z + a ) + x = z + ( a + y ) .= z + a + y ;
len ( l \lbrack ( 0. ( A , A ) ) ) = len l & len ( l (#) ( l (#) ( l (#) ( l (#) ( l (#) ( l (#) ( l (#) ( m , A ) ) ) ) ) ) ) = len l ;
t4 is ( {} \/ rng t4 ) -valued ( {} \/ rng t4 ) -valued ;
t = <* F . t *> ^ ( C . p ) ^ ( C . q ) ;
set pp = W-min L~ Cage ( C , n ) , p1 = W-min L~ Cage ( C , n ) , p2 = Cage ( C , n ) , p3 = Cage ( C , n ) , p1 = Cage ( C , n ) , p2 = Cage ( C , n ) , p3 = Cage ( C , n ) , p4 = Cage (
( ( k -' 1 ) + 1 ) - ( i + 1 ) = ( k - 1 ) + ( i + 1 ) ;
consider u being Element of L such that u = u ` and u in D and u in D ;
len ( ( width ( ( ( ( G , i ) |-> a ) + ( G , j ) ) + ( ( G , j ) |-> a ) ) + ( ( G , j ) |-> a ) ) ) = width ( ( ( G , j ) |-> a ) + ( ( G , j ) --> a ) ) ;
FF . 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 ( ( ( - 1 / 2 ) (#) ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * (
cluster <* l *> ^ phi -> ( 1 + ( \mathop ( 1 + 2 ) ) ) -element for string of S ;
set b9 = [ <* ap , bm *> , <* bm , cp *> ] , a9 = [ <* ap , bm *> , <* cin , that a , b *> ] ;
Line ( Line ( M , P ) , x ) = L * ( Sgm Q ) ;
n in dom ( ( the Sorts of A ) * ( the_arity_of o ) ) & ( ( the Sorts of A ) * ( the_arity_of o ) ) . n = ( the Sorts of A ) . ( the_arity_of o ) ;
cluster f1 + f2 -> continuous for PartFunc of REAL , REAL n ;
consider y be Point of X such that a = y and ||. x-y - x .|| <= r ;
set x3 = t3 . DataLoc ( s2 . SBP , 2 ) , x4 = Comput ( P1 , s2 , 2 ) , P4 = P3 ;
set pg = stop I , pg = stop I , pg = stop I ;
consider a being Point of D2 such that a in W1 and b = g . a ;
{ A , B , C , D } = { A , B , C } \/ { C , D , E , F , J , M } ;
let A , B , C , D , E , F , J , M , N , N , M , N , N , N , F , M , N , N , N , M , N , N , N , F , M , N , N , M , N , N , N , N , M , N ,
|. p2 .| ^2 - ( p2 `1 / p2 `2 ) ^2 >= 0 & ( p2 `1 / p2 `2 ) ^2 - ( p2 `1 / p2 `2 ) ^2 >= 0 ;
l -' 1 + 1 = n-1 * ( ( mm + 1 ) + 1 ) + 1 ;
x = v + ( a * w1 + ( b * w2 ) ) + ( c * w2 ) ;
the TopStruct of L = TopSpaceMetr ( 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 x = H1 . y ;
fv \ { n } = Free ( All ( v1 , H ) ) & fv = Free ( All ( v1 , H ) ) ;
for Y being Subset of X st Y is summable holds Y is not summable & Y is not summable implies Y is not iff X is not empty
2 * n in { N : 2 * Sum ( p | N ) = N & N > 0 } ;
for s being FinSequence holds len ( the { - } Shift Shift Shift s ) = len s & len s = len s ;
for x st x in Z holds exp_R * f is_differentiable_in x & exp_R * f is_differentiable_in x ;
rng ( h2 * f2 ) c= the carrier of ( ( TOP-REAL 2 ) | K1 ) & rng ( h2 * g2 ) c= the carrier of ( ( TOP-REAL 2 ) | K1 ) ;
j + 1- len f <= len f + ( len f - len f ) - len f + len f ;
reconsider R1 = R * I as PartFunc of REAL , REAL-NS n , REAL-NS n ;
C8 . x = s1 . a .= C8 . x .= C8 . x .= C8 . x ;
power ( F_Complex ) . ( z , n ) = 1 .= x |^ n .= x |^ n ;
t at at at at ( C , s ) = f . ( the connectives of S ) . t ;
support ( f + g ) c= support f \/ C & support ( f + g ) c= C \/ { f } ;
ex N st N = j1 & 2 * Sum ( ( r | N ) | N ) > N & N . ( r | N ) > 0 ;
for y , p st P [ p ] holds P [ All ( y , p ) ]
{ [ x1 , x2 ] where x1 , x2 is Point of [: X1 , X2 :] : x1 in X & x2 in Y } c= [: X1 , X2 :]
h . ( i , j ) = ( j , h = id B ) . i .= H . i ;
ex x1 being Element of G st x1 = x & x1 * N c= A & x1 in N ;
set X = ( |. ( |. q .| ) . O ) . ( ( |. q .| ) . O ) . ( ( |. q .| ) . O ) . ( ( |. q .| ) . O ) . O ) ;
b . n in { g1 : x0 < g1 & g1 < x0 & g1 < x0 } ;
f /* s1 is convergent & f /. x0 = lim ( f /* s1 ) implies f /* s1 is convergent & lim ( f /* s1 ) = lim ( f /* s1 )
the carrier of Y = the carrier of the lattice of Y & the carrier of Y = the carrier of the topology of Y implies the carrier of Y = the carrier of the topology of X
'not' ( a . x ) '&' b . x 'or' a . x = FALSE ;
2 = len ( ( q0 ^ r1 ) ^ ( q1 ^ q2 ) ) + len ( ( q1 ^ q2 ) ^ ( q1 ^ q2 ) ) ;
( 1 - a ) (#) ( ( sec * f1 ) - ( sec * f2 ) ) is_differentiable_on Z ) ;
set K1 = upper ( ( lim ( H , A ) || ( A , B ) ) , ( lim ( H , B ) || ( A , B ) ) , ( lim ( H , B ) || ( A , C ) ) ) ;
assume e in { ( ( w1 w1 , w2 ) `1 : w1 in F & w2 in G } ;
reconsider da = dom a `1 , dd = dom F `1 , dd = dom F `1 , dd = dom G as finite set ;
LSeg ( f /^ j , j ) = LSeg ( f , j ) \/ LSeg ( f , j + q .. f ) ;
assume X in { T . ( N2 , L2 ) : h . ( N2 , K ) = N2 } ;
assume that Hom ( d , c ) <> {} and <* f , g *> * f1 = <* f , g *> * f1 and f = g ;
dom S29 = dom S /\ Seg n .= Seg n /\ Seg n .= Seg n /\ Seg n .= Seg n /\ Seg n .= Seg n ;
x in H |^ a implies ex g st x = g |^ a & g in H |^ b
a * ( - n ) = a `1 - ( 0 * n ) .= a `1 - ( 0 * n ) .= a `1 - 1 ;
D2 . j in { r : lower_bound A <= r & r <= D1 . i } ;
ex p being Point of TOP-REAL 2 st p = x & P [ p ] & p `2 >= 0 & p <> 0. TOP-REAL 2 ;
for c holds f . c <= g . c implies f ^ g . c <= g . c
dom ( f1 (#) f2 ) /\ X c= dom ( f1 (#) f2 ) /\ dom ( f1 (#) f3 ) ;
1 = ( p * p ) * ( p * q ) .= p * 1 .= p * 1 ;
len g = len f + len <* x + y *> .= len f + 1 + 1 .= len f + 1 ;
dom ( F | [: N1 , N2 :] ) = dom ( F | [: N1 , N2 :] ) .= [: N1 , N2 :] ;
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 ;
( ( x \ y ) \ z ) \ ( ( x \ z ) \ ( y \ z ) ) = 0. X ;
consider f such that f * f `1 = id b and f * f `2 = id a and f * g `2 = id b ;
( ( ( cos | [. - PI / 2 , 0 .[ ) | [. - PI / 2 , 0 .[ ) | [. - PI / 2 , 0 .[ ) is increasing ;
Index ( p , co ) <= len LS - Gij .. LS + 1 & Index ( Gij , LS ) + 1 <= len LS - Gij .. LS ;
t1 , t2 , t2 , t1 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t2 , t2 , t1 , t2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8
( "/\" ( ( Frege ( ( Frege ( ( curry G ) . i ) ) ) . h ) ) . h ) . x <= "/\" ( ( Frege ( ( Frege ( ( Frege G ) . i ) ) . h ) ) . x ) . x ;
then P [ f . i0 , f . i0 ] & F ( f . i0 , f . i0 ) < j ;
Q [ ( D . [ D , x ] ) `1 , F ( ) . [ D . [ D , x ] ) `2 ] ;
consider x being element such that x in dom ( F . s ) and y = F . s . x ;
l . i < r . i & [ l . i , r . i ] is Element of G . i ;
the Sorts of A2 = ( the Sorts of S2 ) +* ( the Sorts of S2 ) +* ( the Sorts of S2 ) ;
consider s being Function such that s is one-to-one and dom s = NAT and rng s = F and rng s c= F ;
dist ( b1 , b2 ) <= dist ( b1 , a ) + dist ( a , b2 ) ;
( ( for n holds ( for i st i in Seg n holds ( i in Seg n ) ) implies ( i in Seg n ) ) & ( i in Seg n implies ( i in Seg n implies ( i in Seg n ) )
q `2 <= ( UMP Lower_Arc L~ Cage ( C , 1 ) ) `2 & q `2 <= ( UMP L~ Cage ( C , 1 ) ) `2 ;
LSeg ( f | i2 , i ) /\ LSeg ( f | i2 , j ) = {} & LSeg ( f | i2 , j ) /\ LSeg ( f | i2 , j ) = {} ;
given a being R_eal such that a <= Ia and A = ]. a , I .[ and a < I ;
consider a , b being complex number such that z = a & y = b and z + y = a + b ;
set X = { b |^ n where n is Element of NAT : n <= len b } ;
( ( x * y ) * z \ x ) \ ( x * y ) = 0. X ;
set xy = [ <* xy , y9 , z9 *> , [ <* xy , z9 , \upharpoonright ] ] , yz = [ <* xy , yz , z9 ] , yz = [ <* xy , yz , z9 *> , [ <* xy , yz *> , [ <* xy , yz *> , <* xy , z0 *> ] ] ;
lx /. ( len lx ) = ( lx . ( len ( l . ( len ( l . ( len l ) ) ) ) ) ) . ( len ( l . ( len l ) ) ) ) ;
( ( - ( q `2 / |. q .| - sn ) ) / ( 1 + sn ) ) ^2 = 1 ;
( ( - ( p `2 / |. p .| - sn ) ) / ( 1 + sn ) ) ^2 < 1 ;
( ( ( ( ( X \/ Y ) \ { x } ) \/ ( Y \/ { x } ) ) ) /\ ( ( X \/ Y ) \ { x } ) ) = ( ( ( X \/ Y ) \ { x } ) \/ ( Y \/ { x } ) ) /\ ( ( X \/ Y ) \ { x } ) ;
( ( s1 - seq ) ^\ k ) . k = ( s1 . k - seq . ( k + 1 ) ) .= ( s1 . k - seq . ( k + 1 ) ) ;
rng ( ( h + c ) ^\ n ) c= dom SVF1 ( 1 , f , u0 ) ;
the carrier of X = the carrier of X & the carrier of X = the carrier of X & the carrier of X = the carrier of Y ;
ex p3 st p3 = p4 & |. p3 - |[ a , b ]| .| = r & |. p3 - p3 .| = r ;
set ch = chi ( X , [ x , A ] ) , Ah = chi ( X , [ x , A ] ) ;
R / ( ( 0 * n ) * n ) = I---> Element of I-set ( X * n ) * n ;
( Partial_Sums ( ( curry ( F . -19 ) ) ) . n ) . m is nonnegative & ( ( ( curry ( F . -19 ) ) . n ) . m ) . m is nonnegative ;
f2 = CK . ( EK , ( EK . ( V , len ( V , len ( V , len ( V , f ) ) ) ) ) ) ;
S1 . b = s1 . b .= s2 . b .= s2 . b .= s2 . b .= s2 . b ;
p2 in LSeg ( p2 , p1 ) /\ LSeg ( p1 , p11 ) \/ LSeg ( p1 , p11 ) /\ LSeg ( p1 , p11 ) \/ LSeg ( p11 , p2 ) /\ LSeg ( p1 , p10 ) c= { p2 } ;
dom ( f . t ) = Seg n & dom ( I . t ) = Seg n & rng ( I . t ) = Seg n ;
assume o = ( the connectives of S ) . 11 & ( the connectives of S ) . 11 = ( the connectives of S ) . 11 ;
set phi = ( l1 , l2 ) \mathop ( l1 , l2 ) , phi = ( l1 , l2 ) \mathop ( l1 , l2 ) ;
synonym p is \cup g for ( 1 , T ) *' p & ( p is invertible implies p is invertible ) ;
( Y1 `2 = - 1 & Y1 `1 = - 1 & Y1 `2 <= 1 implies Y1 `1 = - 1 & Y1 `2 <= 1 ) & ( Y1 `2 <= 1 implies Y1 `1 <= 1 & Y1 `2 <= 1 )
defpred X [ Nat , set , set , set ] means ( $1 in $2 implies $2 = $2 ) & ( $1 in $2 implies $2 = $2 ) ;
consider k be Nat such that for n be Nat st k <= n holds s . n < x0 + g ;
Det I |^ ( ( m -' n ) mod n ) = 1. ( K , n ) & Det I = 1. ( K , n ) ;
( - b ) / sqrt ( b ^2 - 4 * a * c ) < 0 ;
CT . d = CT . d mod CT . d & CT . d = CT . d mod CT . d & CT . d = CT . d mod C . d ;
attr X1 is dense means : Def3 : X2 is dense & X1 is dense implies X1 /\ X2 is dense SubSpace of X1 & X2 is dense implies X1 is dense SubSpace of X2 ;
deffunc FF ( Element of E , Element of I ) = ( $1 * $2 ) * ( $2 , $2 ) ;
t ^ <* n *> in { t ^ <* i *> : Q [ i , T ( ) . t ] } ;
( x \ y ) \ x = ( x \ x ) \ y .= y \ 0. X .= 0. X ;
for X being non empty set for Y being Subset-Family of X holds X is Basis of <* X , Y *> iff Y is Basis of X
synonym A , B are_separated for Cl ( A \/ B ) for Cl ( B \/ C ) for Cl ( B \/ C ) ;
len ( M @ ) = len p & width ( M @ ) = width M & width ( M @ ) = width M & width ( M @ ) = width M & width ( M @ ) = width M & width ( M @ ) = width M & width ( M @ ) = width M & width ( M @ ) = width M & width ( M @ ) = width M ;
J . v = { x where x is Element of K : 0 < v . ( x + 1 ) & x in rng ( v . ( x + 1 ) ) } ;
( ( Sgm [: Seg m , Seg m :] ) . d - ( Sgm Sgm [: Seg m , Seg m :] ) . e ) <> 0 ;
lower_bound divset ( D2 , k + k2 ) = D2 . ( k + k2 - 1 ) ;
g . r1 = - 2 * r1 + 1 & dom h = [. 0 , 1 .] & rng h c= [. 0 , 1 .] ;
|. a .| * ||. f .|| = 0 * ||. f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| ;
f . x = ( h . x ) `1 & g . x = ( h . x ) `2 & g . x = ( h . x ) `2 ;
ex w st w in dom B1 & <* 1 *> ^ s = <* 1 *> ^ w & <* 1 *> ^ s = <* 1 *> ^ w ;
[ 1 , {} , <* d1 *> ] in ( { 0 , {} } \/ S1 ) \/ S2 \/ S2 ) \/ S2 \/ S2 \/ S2 \/ S2 \/ S2 \/ S2 \/ S2 \/ S2 \/ S \/ S \/ S \/ S \/ S \/ S \/ S \/ S \/ S \/ S \/ T \/ S \/ T \/ S \/ S \/ S \/ S \/ S \/ S \/ S \/
IC Exec ( i , s1 ) + n = IC Exec ( i , s2 ) + n .= IC Exec ( i , s2 ) ;
IC Comput ( P , s , 1 ) = DataLoc ( s , 9 ) .= 5 + 9 .= 5 + 9 .= 5 ;
( IExec ( W6 , Q , t ) ) . intpos ( e + 1 ) = t . intpos ( e + 1 ) ;
LSeg ( f /^ i , i ) misses LSeg ( f /^ j , j ) \/ LSeg ( f /^ j , j ) ;
assume for x , y being Element of L st x in C & y in C holds x <= y or y <= x ;
integral ( integral ( f , C ) , x ) = f . ( upper_bound C ) - ( lower_bound C ) * ( 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 /. ( L . h ) .|| < 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 `1 , x being Element of REAL st y `1 in Y & x `2 in X holds y `2 <= x `2 & x `2 <= x `2 ;
func |. p \bullet q -> variable equals min ( NBI , p , q ) . ( i + 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 x1 & for i st i in Seg len x1 holds x1 . i = x1 . i ) implies x1 = x2
consider y2 being Real such that x2 = y2 and 0 <= y2 and y2 < 1 / 2 and y2 <= 1 / 2 ;
||. f | X /* s1 .|| = ||. f .|| | X & ||. f .|| | X is convergent & ||. f .|| /. ( lim s1 ) = ||. f .|| /. ( lim s1 ) ;
( the InternalRel of A ) ` /\ Y = {} \/ {} .= {} \/ {} .= {} \/ {} .= {} .= {} ;
assume that i in dom p and for j being Nat st j in dom q holds P [ i , j ] and for i be Nat st i in dom p holds P [ i , j ] ;
reconsider h = f | [: X , Y :] as Function of [: X , Y :] , [: Y , X :] ;
u1 in the carrier of W1 & u2 in the carrier of W2 implies v1 + u1 in the carrier of W1 & v1 + u1 in the carrier of W1 & v1 + u1 in the carrier of 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 ;
- ( - x ) = - x + - y .= - x + - y .= - x + - y .= - x + y ;
given a being Point of IT such that for x being Point of IT holds a , x are_ed ed implies a , x are_ed ;
fbeing Function of [: dom ( ( dom f ) | [: dom f2 , cod g2 :] ) , cod ( f | [: cod f , cod g2 :] ) :] , cod ( f | [: cod f , cod f :] ) :] ;
for k , n being Nat st k <> 0 & k < n & k is prime holds k , n are_relative_prime & k , n are_relative_prime & k , n are_relative_prime holds k , n are_relative_prime
for x being element holds x in A |^ d iff x in ( ( A ` ) |^ d ) ` & x in ( ( A ` ) |^ d ) `
consider u , v being Element of R , a being Element of A such that l /. i = u * a and a * v = a * v ;
- ( ( - ( p `1 / |. p .| - cn ) ) / ( 1 + cn ) ) ^2 > 0 ;
L-13 . k = LL . ( F . k ) & F . k in dom ( L * F ) ;
set i2 = AddTo ( a , i , - n ) , i1 = a , i2 = b , i2 = c , n = - n , n = - ( - n ) , m = - ( - n ) , m = - ( - n ) , n = - ( - n ) , m = - ( - n ) , m = - ( - n ) , m = - ( - n ) , n = - ( - m )
attr B is \frac means : Def4 : for S being Subuniversal non empty Subset of \mathopen ( B , S-13 ) holds S is non empty ;
a9 "/\" D = { a "/\" d where d is Element of N : d in D } ;
|( \square , q29 )| * |( q29 , q29 )| , |( q29 )| >= |( ( 1 - 2 ) * |( q29 , q29 )| ;
( - f ) . ( upper_bound A ) = ( - f ) . ( upper_bound A ) .= - f . ( lower_bound A ) ;
( G * ( len G , k ) `1 ) `1 = ( G * ( len G , k ) `1 ) `1 .= ( G * ( len G , k ) `1 ) `1 ;
( Proj ( i , n ) ) . ( ( proj ( i , n ) ) . ( ( proj ( i , n ) ) . ( x - x0 ) ) ) = <* ( proj ( i , n ) ) . ( x - x0 ) *> ;
f1 + f2 * reproj ( i , x ) is_differentiable_in ( ( reproj ( i , x ) * reproj ( i , x ) ) ) . x ;
attr ( for x st x in Z holds ( ( tan (#) f ) `| Z ) . x = ( tan . x ) * ( cos . x ) ^2 ) ;
ex t being SortSymbol of S st t = s & h1 . t = h2 . t & h2 . x = h2 . t ;
defpred C [ Nat ] means P8 . $1 is non empty & ( for n st n <= $1 holds A8 . n is non empty & A8 . n is non empty ) ;
consider y being element such that y in dom ( p9 . i ) and qp . i = p9 . y and qp . i = x . i ;
reconsider L = product ( { x1 } +* ( index B ) ) as Basis of product ( A . ( index B ) ) ;
for c being Element of C ex d being Element of D st T . ( id c ) = id d & T . ( id c ) = id d ;
( f , n ) = ( f | n ) ^ <* p *> .= f | n ^ <* p *> .= 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 - \cal p = ( f | ( n , L ) ) *' ( f | ( n , L ) ) .= ( f - c ) *' ( f | ( n , L ) ) ;
consider r being Real such that r in rng ( f | divset ( D , j ) ) and r < m + s ;
f1 . ( |[ r2 `1 , s2 ]| ) in f1 .: ( W1 /\ W2 ) & f2 . ( r2 ) in f1 .: ( W2 /\ W3 ) ;
eval ( a | ( n , L ) ) = eval ( a | ( n , L ) ) .= a . ( a | ( n , L ) ) .= a . ( a . ( n , L ) ) ;
z = DigA ( t9 , x ) .= DigA ( t9 , x ) .= DigA ( t9 , x ) .= DigA ( t9 , x ) ;
set H = { Intersect S where S is Subset-Family of X : S c= G & S is finite } , G = { Intersect S where S is Subset of X : S is finite } ;
consider S19 being Element of D * , d being Element of D * such that S `1 = S19 ^ <* d *> and S `2 = d ;
assume that x1 in dom f and x2 in dom f and f . x1 = f . x2 and f . x2 = f . x1 and f . x2 = f . x2 ;
- 1 <= ( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 / ( 1 + sn ) ^2 ;
0. ( V ) is Linear_Combination of A & Sum ( l ) = 0. V implies Sum ( l ) = 0. V
for k1 , k2 being Nat , k1 being Element of NAT holds k1 in dom ( the InstructionsF of SCM+FSA ) & k2 in dom ( the InstructionsF of SCM+FSA ) implies k1 is Element of NAT
consider j being element such that j in dom a and j in dom a and x = a " . j and a . j = a . j ;
H1 . x1 c= H1 . x2 or H1 . x1 c= H1 . x2 or H1 . x1 c= H1 . x2 & H1 . x2 c= H1 . x1 or H1 . x2 c= H1 . x2 ;
consider a being Real such that p = *> * p1 + ( a * p2 ) and 0 <= a and a <= 1 and a <= 1 ;
assume that a <= c & c <= d and |[ a , b ]| c= dom f and f . a = g . b ;
cell ( Gauge ( C , m ) , ( Gauge ( C , m ) ) * ( i , 1 ) + Gauge ( C , m ) * ( i + 1 , 1 ) + Gauge ( C , m ) * ( i + 1 , 1 ) ) ) is non empty ;
Ain { ( S . i ) `1 where i is Element of NAT : i in dom S & i in dom S } ;
( T * b1 ) . y = L * b2 /. y .= ( F * b1 ) . y .= ( F * b2 ) . y .= ( F * b2 ) . y ;
g . ( s , I ) . x = s . y & g . ( s , I ) . y = |. s . x - s . y .| ;
( log ( 2 , k + 1 ) ) ^2 >= ( log ( 2 , k + 1 ) ) ^2 ;
then that p => q in S and not x in the still of p and not x in S and not x in S ;
dom ( the InitS of r-10 ) misses dom ( the InitS of r-11 ) & dom ( the InitS of r-11 ) = dom ( the InitS of r-11 ) & dom ( the InitS of r-11 ) = the carrier of r-11 ;
synonym f is extended real for for for for for for x is Real st x in rng f holds f is R_eal ;
assume that for a being Element of D holds f . { a } = a and for X being Subset-Family of D holds f . { a } = f . union X ;
i = len p1 + len <* x *> .= len p1 + len <* x *> .= len p1 + 1 + 1 .= len p1 + 1 + 1 ;
( l . ( 1 , 3 ) ) `1 = ( g . ( 1 , 3 ) ) `1 + ( g . ( 1 , 3 ) ) `1 - ( g . ( 1 , 3 ) ) `1 ) / ( 2 * ( 1 + 3 ) ) `1 ) ;
CurInstr ( P2 , Comput ( P2 , s2 , l ) ) = halt SCM+FSA .= ( halt SCM+FSA ) . l .= halt SCM+FSA .= ( l l ) . l ;
assume for n be Nat holds ||. ( seq . n ) - ( seq . m ) .|| <= ( seq . n ) - ( seq . m ) & ( seq . m ) is summable ;
sin . ( PI / 2 ) = sin . ( PI / 2 ) * cos . ( PI / 2 ) .= 0 ;
set q = |[ g1 `1 / ( a - b ) , g2 `2 / ( a - b ) ]| , G = |[ a , b ]| ;
consider G be sequence of S such that for n being Element of NAT holds G . n in implies G . n in
consider G such that F = G and ex G1 st G1 in SM & G in SM & F . ( len G ) = G . ( len G1 ) ;
the root of [ x , s ] in ( the Sorts of Free ( C , s ) ) . s & ( the Sorts of Free ( C , s ) ) . s c= ( the Sorts of Free ( C , s ) ) . s ;
Z c= dom ( exp_R * ( f + ( 3 / 2 ) ) ) /\ dom ( ( exp_R * ( f + g ) ) ) ;
for k be Element of NAT holds ( ( Im ( Im f ) ) . k ) . x = ( ( Im ( Im f ) ) . x ) . x ;
assume that - 1 < n and q `2 > 0 and q `2 <= 1 and q `2 <= 1 and q `2 <= 1 and q `2 <= 1 and q `2 <= 1 and q `2 <= 1 and q `2 <= 1 and q `2 <= 1 and q `2 <= 1 and q `2 <= 1 and q `2 <= 1 and q `2 <= 1 and q `2 <= 1 ;
assume that f is continuous and a < b and a < d and c < d and f . a = g and f . b = g and g . a = f . c and f . c = g . d ;
consider r being Element of NAT such that sy0 = Comput ( P1 , s1 , r ) and r <= q and r <= q ;
LE f /. ( i + 1 ) , f /. ( j + 1 ) , L~ f , f /. ( j + 1 ) , f /. ( len f ) , f /. ( len f ) , f /. ( len f ) ) , f /. ( len f ) , f /. ( len f ) , f /. ( len f ) , f /. ( len f ) , f /. ( len f ) , f /. ( len f ) , f /. ( len
assume that x in the carrier of K and y in the carrier of K and inf { x , y } in the carrier of K and x <> y ;
assume f +* ( i1 , \xi ) in ( proj ( F , i2 ) ) " ( ( proj ( F , i1 ) ) . ( i2 , j2 ) ) " ( ( proj ( F , i2 ) ) . ( i1 , j2 ) ) ;
rng ( ( ( ( ( ( ( ( ( the carrier of M ) ) | the carrier of M ) ) | the carrier of M ) ) | the carrier of M ) ) | the carrier of M ) ) = the carrier of M ;
assume z in { ( the carrier of G ) \ { t where t is Element of T : t in F } ;
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 .|| & ||. t .|| <= 1 ;
assume that the carrier of v = 2 and v ^ <* 0 *> in dom p and v ^ <* 1 *> in dom p and p ^ <* 1 *> in dom p ;
consider a being Element of the Points of Ximplies a on A & a on B & a on B & a on B ;
( - x ) |^ ( k + 1 ) * ( ( - x ) |^ ( k + 1 ) ) " = 1 ;
for D being set st for i st i in dom p holds p . i in D holds p . i in D & p . i in D
defpred R [ element ] means ex x , y st [ x , y ] = $1 & P [ x , y ] & P [ y , x ] ;
L~ f2 = union { LSeg ( p0 , p2 ) , LSeg ( p1 , p2 ) } .= LSeg ( p1 , p2 ) \/ LSeg ( p11 , p2 ) ;
i -' len h11 + 2 - 1 < i - len h11 + 2 - 1 + 2 - 1 + 1 + 1 - 1 + 1 ;
for n being Element of NAT st n in dom F holds F . n = |. ( nbeing ) . ( n -' 1 ) .| ;
for r , s1 , s2 being Real holds r in [. s1 , s2 .] iff r <= s1 & s1 <= s2 & s1 <= s2 & s2 <= 1 & s1 <= 1 & s1 <= s2 & s2 <= 1 implies r <= r & r <= s2 & s1 <= 1
assume v in { G where G is Subset of T2 : G in B2 & G c= B1 & G c= H & H c= G & G c= H } ;
let g be then succ of A , ( X , Z ) .: ( b , ( X , Z ) .: ( b , ( X , Z ) .: ( b , c ) ) ) ) -> Element of S ;
min ( g . [ x , y ] , k ) . ( y , z ) = ( min ( g , k ) ) . ( y , z ) ;
consider q1 being sequence of CL such that for n holds P [ n , q1 . n ] ;
consider f being Function such that dom f = NAT and for n being Element of NAT holds f . n = F ( n ) ;
reconsider B-6 = B /\ B , OO = O /\ Z , $ O = O /\ Z as Subset of B ;
consider j being Element of NAT such that x = the \it the ` of n and 1 <= j and j <= n and n <= len f ;
consider x such that z = x and card ( x . O2 ) in card ( x . O2 ) and x in L1 . O2 and x in L2 . O2 ;
( C * _ T4 ( k , n2 ) ) . 0 = C . ( ( _ T4 ( k , n2 ) ) . 0 ) .= C . ( ( _ 4 ( k , n2 ) ) . 0 ) ) ;
dom ( X --> rng f ) = X & dom ( ( X --> f ) | X ) = dom ( X --> f ) ;
( ( N-bound L~ Cage ( C , n ) ) / 2 ) * ( ( \upharpoonright L~ Cage ( C , n ) ) / 2 ) <= ( ( ( E-max L~ Cage ( C , n ) ) ) / 2 ) * ( ( \upharpoonright L~ Cage ( C , n ) ) / 2 ) ;
synonym x , y are_collinear means : Def3 : x = y or ex l being Nat st { x , y } c= l or x c= l ;
consider X being element such that X in dom ( f | ( n + 1 ) ) and ( f | ( n + 1 ) ) . X = Y ;
assume that $ \mathop { \rm Im k is continuous and for x , y being Element of L st x = x & y = y holds x << y iff x << y ;
( 1 / 2 ) (#) ( ( ( #Z 2 ) * ( ( AffineMap ( n , 0 ) ) * ( ( AffineMap ( n , 0 ) ) * ( ( AffineMap ( n , 0 ) ) * ( ( AffineMap ( n , 0 ) ) * ( ( AffineMap ( n , 0 ) ) * ( ( AffineMap ( n , 0 ) ) * ( ( AffineMap ( n , 0 ) ) * ( ( AffineMap ( n , 0 ) ) * ( ( AffineMap ( n , 0 ) ) * ( ( AffineMap ( n , 0 ) ) * ( ( AffineMap ( n ,
defpred P [ Element of omega ] means ( ( the partial of A1 ) . $1 = A1 . $1 ) & ( for n holds A1 . n = A2 . n ) implies A1 is convergent & A2 is convergent & A1 is convergent & A2 is convergent & lim A1 = A2 . $1 ;
IC Comput ( P , s , 2 ) = succ IC Comput ( P , s , 1 ) .= 6 + 1 .= 6 + 1 .= 6 + 1 ;
f . x = f . g1 * f . g2 .= f . g1 * ( g . g2 ) .= f . g2 * ( g . g2 ) .= f . g2 * ( g . g2 ) .= f . g2 * ( g . 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 c= L2 + L2
pred a , b , c , x , y , x , y , z , x , y , z , x , y , z is_collinear means : Def3 : a , b , c , x is_collinear & x , y , z is_collinear & x , y , z is_collinear ;
( ( the partial of s ) . n ) . n <= ( ( the Sorts of s ) . n ) . ( ( the Sorts of s ) . n ) . ( ( the Sorts of s ) . n ) . x ) ;
attr - 1 <= r & r <= 1 implies ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) / ( r - 1 ) ) ) ) `| Z ) = - ( 1 / ( r - 1 ) ) ;
seq in { p ^ <* n *> where p is Nat : p ^ <* n *> in T1 & p in T1 & p in T2 } ;
|[ x1 , x2 , x3 ]| . 2 - |[ x1 , x2 ]| . 2 = x2 - |[ x1 , x3 ]| . 2 - |[ x2 , x3 ]| . 3 - |[ x1 , x2 ]| . 3 ;
attr for m being Nat holds F . m is nonnegative & ( Partial_Sums F ) . m is nonnegative ;
len ( ( ( G . ( x , y ) ) ) . ( ( ( G . ( x , y ) ) . ( y , z ) ) ) ) = len ( ( G . ( x , y ) ) . ( x , y ) ) ) ;
consider u , v being VECTOR of V such that x = u + v and u in W1 /\ W2 and v in W2 /\ W3 and u in W2 /\ W3 and v in W2 /\ W3 ;
given F being finite FinSequence of NAT such that F = x and dom F = n and rng F c= { 0 , 1 } and Sum F = k and Sum F = k ;
0 = 1 * ( D1 , D2 ) , 1 = ( - 1 ) * ( ( - 1 ) * ( D1 , D2 ) ) ;
consider n be Nat such that for m be Nat st n <= m holds |. ( f # x ) . m - ( lim ( f # x ) ) . m .| < e ;
cluster non empty -> Boolean for \hbox { ( st ( ( st ( implies ( ( implies ( ( implies ( } } _ 1 ) ) ) ) ) ) ) , ( ( st ( ( st ( st ( st ( ( } ( st ( } ( ( } ) ) ) ) ) ) ) ) ) ) ) ) is Boolean ) ) ;
"/\" ( BB , L ) = Bottom ( B , L ) .= Bottom S .= "/\" ( ( the carrier of S ) \ {} ) .= "/\" ( ( the carrier of S ) \ {} , L ) .= "/\" ( I , L ) ;
( 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 - c * |[ a , c ]| - ( 2 * r1 - c ) * |[ b , c ]| = 0. TOP-REAL 2 ;
reconsider p = P * ( \square , 1 ) , q = a " * ( ( - - ( - 1 ) |^ n ) * ( 1 , 1 ) ) as FinSequence of K ;
consider x1 , x2 being element such that x1 in uparrow s and x2 in downarrow t and x = [ x1 , x2 ] and x = [ x1 , x2 ] ;
for n be Nat st 1 <= n & n <= len q1 holds q1 . n = ( upper_volume ( g , M7 ) ) . ( n + 1 )
consider y , z being element such that y in the carrier of A and z in the carrier of A and i = [ y , z ] and i = [ y , z ] ;
given H1 , H2 being strict Subgroup of G such that x = H1 and y = H2 and H1 is Subgroup of H2 and H2 is Subgroup of H1 and H2 is Subgroup of H2 ;
for S , T being non empty RelStr , d being Function of T , S st T is complete for d being Function of T , S st d is monotone holds d is monotone & d is monotone
[ a + 0. ( F_Complex , b2 ) , b2 ] in ( the carrier of ( COMPLEX ) ) \ ( the carrier of ( COMPLEX ) ) ;
reconsider mm = max ( ( len F1 ) * ( ( p . n ) * ( x |^ n ) ) ) as Element of NAT ;
I <= width GoB ( ( GoB f ) * ( 1 , width GoB f ) + ( GoB f ) * ( 1 , width GoB f ) ) & I <= width GoB f & f /. len f = ( GoB f ) * ( 1 , width GoB f ) + ( GoB f ) * ( 1 , width GoB f ) ;
f2 /* q = ( f2 /* ( f1 /* s ) ) ^\ k .= ( f2 * ( f1 /* s ) ) ^\ k .= ( f2 * ( f1 /* s ) ) ^\ k .= ( f2 * ( f1 /* s ) ) ^\ k ;
attr A1 \/ A2 is linearly-independent means : Def3 : A1 is linearly-independent & A2 is linearly-independent & ( for A being Subset of V st A is Subset of A1 holds ( A /\ A2 ) /\ ( A1 /\ A2 ) = { 0. V } ) ;
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 ) ) ) ) ) = dom ( ( F ^ <* p *> ) (#) ( G ^ <* p *> ) ) ;
cluster [ x , ( x , 4 ) `1 , ( x , 4 ) `2 , ( x , 4 ) `2 ] -> Morphism of x , 4 & x <> 0. TOP-REAL 2 ;
E , All ( x1 , x2 ) |= All ( x2 , All ( x1 , x2 ) '&' ( x1 '&' ( x1 '&' ( x1 '&' ( x1 '&' ( x1 '&' x2 ) ) ) ) ) ) '&' ( x1 '&' ( x1 '&' ( x1 '&' ( x1 '&' ( x1 '&' ( x1 '&' ( x1 '&' x2 ) ) ) ) ) ) ;
F .: ( id X , g ) . x = F . ( id X , g . x ) .= F . ( id X , g . x ) .= F . ( x , g . x ) ;
R . ( h . m ) = F . ( x0 + h . m ) - h . ( x0 + h . m ) ;
cell ( G , ( X -' 1 ) + ( Y -' 1 ) ) \ ( ( X -' 1 ) + ( Y -' 1 ) ) meets ( UBD L~ f ) ;
IC Comput ( P2 , s2 , 2 ) = IC Comput ( P2 , s2 , 1 ) .= ( card I + 2 ) .= card I + ( card I + 2 ) .= card I + ( card I + 2 ) .= card I + ( card I + 2 ) .= card I + ( card I + 2 ) ;
sqrt ( ( - ( ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ) ^2 ) ) ^2 / ( 1 + cn ) ^2 ) > 0 ;
consider x0 being element such that x0 in dom a and x0 in g and x0 in g " { k } and y0 = a . x0 and x0 in g " { k } ;
dom ( r1 (#) chi ( A , A ) ) = dom ( ( chi ( A , A ) ) | ( Seg m ) ) .= dom ( ( ( chi ( A , A ) ) | ( Seg m ) ) ) .= C ;
d-7 . [ y , z ] = ( [ y , z ] `1 - ( y `1 ) ) * ( [ y , z ] `1 - ( y `2 ) * ( [ y , z ] `1 ) ) ;
attr for i being Nat holds C . i = A . i /\ B . i & C . i c= C . i /\ C . i ;
assume that x0 in dom f and f is continuous and f is continuous and for x st x in dom f holds f . x - f . x <> 0 and f . x <> 0 and f . x <> 0 and f . x <> 0 ;
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 & Q meets Q implies A meets Q
for x be Element of REAL n st x in Line ( x1 , x2 ) holds |. y1 - y0 .| <= |. y1 - y0 .| & |. y1 - y0 .| <= |. y1 - y0 .| ;
func Sum ( <*> a ) -> Ordinal means : Def3 : a in it & for b being Ordinal st a in it holds it . b c= b & for a being Ordinal Ordinal st a in it holds it . a c= b ;
[ a1 , a2 , a3 ] in ( the carrier of A ) \ ( the carrier of B ) & [ a1 , a2 , a3 ] in 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 ] & [ a , b ] in the InternalRel of S2 ;
||. ( vseq . n ) - ( vseq . m ) .|| * ||. x .|| < ( e / ( 2 |^ n ) ) * ||. x .|| ;
then for Z being set st Z in { Y where Y is Element of I : F . Y in Z & Z in F } holds z in Z ;
sup compactbelow [ s , t ] = [ sup ( compactbelow [ s , t ] ) , sup ( compactbelow s ) ] .= [ sup ( compactbelow s ) , sup ( compactbelow t ) ] ;
consider i , j being Element of NAT such that i < j and [ y , f . j ] in I and [ f . i , f . j ] in I and [ f . i , f . j ] in I ;
for D being non empty set , p , q being FinSequence of D st p c= q & p ^ q = q holds ex p being FinSequence of D st p ^ q = q & p ^ q = p ^ q
consider e19 being Element of the affine of X such that c9 , a9 // a9 , b9 and not a9 , b9 // a9 , b9 and not b9 , c9 // a9 , b9 and not a9 , b9 // c9 , b9 & not b9 , c9 // a9 , b9 & not b9 , c9 // b9 , c9 ;
set U2 = I \! \mathop { {} } , E = I \! \mathop { {} } , F = S \! \mathop { {} } , G = S \! \mathop { {} } , F = S \! \mathop { {} } , G = S \! \mathop { {} } , F = S \! \mathop { {} } , G = S \! \mathop { {} } , F = S \! \mathop { {} } , G = S \! \mathop { {} } , E = S \! \mathop { {} } , E = S \! \mathop { {} } , A = S \! \mathop { {} } , F = S \! \mathop { {} } , A = S \! \mathop
|. q2 .| ^2 = ( |. q2 .| ) ^2 + ( |. q2 .| ) ^2 .= |. q2 .| ^2 + ( |. q2 .| ) ^2 .= |. q2 .| ^2 + ( |. q2 .| ) ^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 & x "/\" y = x /\ y & x "/\" y = x /\ y & x "/\" x = x /\ y & x "/\" y = x "/\" y & x "/\" y = x "/\" y & x "/\" y = x "/\" y
dom signature ( U1 ) = dom ( the charact of U1 ) & ( for o be Element of O holds o in ( the Sorts of U1 ) . o iff ( the_arity_of o ) . o = ( the Sorts of U1 ) . o ) . o ) & ( the_arity_of o ) . o = ( the Sorts of U1 ) . o ;
dom ( h | X ) = dom h /\ X .= dom h /\ X .= dom ( ( |. h .| ) | X ) .= dom ( ( |. h .| ) | X ) .= dom ( ( |. h .| ) | X ) .= dom ( ( |. h .| ) | X ) .= dom ( ( |. h .| ) | X ) .= dom ( ( |. h .| ) | X ) ;
for N1 , N2 , N1 , N1 , N2 being Element of ( the carrier of G ) * holds ( h . N1 ) . ( h . N2 ) = N1 & ( h . N1 ) . ( h . N2 ) = N2 & ( h . N1 ) . ( h . N1 ) = N1
( mod ( u , m ) + mod ( v , m ) ) . i = ( mod ( u , m ) ) . i + ( mod ( v , m ) ) . i ;
- ( q `1 / |. q .| - cn ) < - 1 or - ( q `1 / |. q .| - cn ) >= - ( q `1 / |. q .| - cn ) & - ( q `1 / |. q .| - cn ) <= - ( q `1 / |. q .| - cn ) ;
attr r1 = f9 & r2 = f9 & s1 = f9 & s2 = f9 & s1 = f9 * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * g ) ) ) ) ) ) ) ) & s2 = f9 * ( f * ( f * ( f * ( f * ( f * ( f * g ) ) ) ) ) ;
vseq . m is bounded Function of X , the carrier of Y & vseq . m = ( seq_id ( vseq . m ) ) . x & vseq . m = ( seq_id ( vseq . m ) ) . x ;
attr a <> b & b <> c & c <> 0 & angle ( a , b , c ) = PI & angle ( b , c , a ) = 0 & angle ( c , a , b ) = 0 ;
consider i , j being Nat , r being Real such that p1 = [ i , r ] and p2 = [ j , s ] and r < s and s < p2 and p2 < r ;
|. p .| ^2 - ( 2 * |( p , q )| ) ^2 + |. q .| ^2 = |. p .| ^2 + |. q .| ^2 - ( 2 * |( p , q )| ) ^2 ;
consider p1 , q1 being Element of [: X , Y :] such that y = p1 ^ q1 and q1 in A and p1 ^ q1 = p2 ^ q2 and p1 ^ q1 = p2 ^ q2 and p2 ^ q2 = p2 ^ q2 ;
Y. ( , r1 , r2 , s1 , s2 , s1 , s2 , t2 ) = ( s2 s2 s2 , t2 ) / 2 & s1 , s2 , t2 , t2 is_collinear & s1 , s2 , t2 is_collinear & s1 , s2 , t2 is_collinear & s1 , s2 , t2 is_collinear implies s1 , s2 , t2 is_collinear
( ( LMP A ) `2 = lower_bound ( proj2 .: ( A /\ holds w `2 <= ( proj2 .: ( A /\ holds w `2 <= w `2 ) ) ) & ( proj2 .: ( A /\ holds w `2 <= w `2 ) ) ) is non empty ;
s |= ( ( k , H1 ) |= H2 iff s |= ( H1 , H2 ) |= ( ( H1 , H2 ) |= ( H1 , H2 ) |= ( H1 , H2 ) '&' ( H2 , k ) ) ;
len ( s + 1 ) = card ( support b1 ) + 1 .= card ( support b2 ) + card ( support b2 ) .= card ( support b2 ) + card ( support b2 ) .= card ( support b2 ) + card ( support b2 ) .= card ( support b2 ) + card ( support b2 ) ;
consider z being Element of L1 such that z >= x and z >= y and for z being Element of L1 st z >= x holds z `1 >= y `1 & z `2 >= y `2 & z `2 >= x `2 ;
LSeg ( UMP D , |[ ( W-bound D ) / 2 ) / 2 , ( E-bound D ) / 2 ]| ) /\ D = { UMP D , ( E-bound D ) / 2 } /\ D .= { UMP D } ;
lim ( ( f `| N ) /* ( g `| N ) /* b ) = ( lim ( ( f `| N ) /* ( g `| N ) ) ) / ( ( g `| N ) /* b ) ;
P [ i , pr1 ( f ) . i , pr1 ( f ) . ( i + 1 ) ] implies pr1 ( f ) . ( i + 1 ) = pr1 ( f ) . ( i + 1 )
for r be Real st 0 < r ex m be Nat st for k be Nat st m <= k holds ||. ( ( seq . k ) - ( seq . m ) ) - ( seq . k ) .|| < r
for X being set , P being a_partition of X , a , b being set st x in a & a in P & b in P & x in P & x in P & b in P holds a = b
Z c= dom ( ( ( 1 / 2 ) (#) f ) `| Z ) \ ( ( ( 1 / 2 ) (#) f ) `| Z ) & f | Z is continuous implies ( ( ( 1 / 2 ) (#) f ) `| Z ) . ( 0 + 1 ) = ( ( 1 / 2 ) (#) f ) `| Z ) . ( 0 + 1 )
ex j being Nat st j in dom ( l ^ <* x *> ) & j < i & i = ( l ^ <* x *> ) . j & ( l ^ <* x *> ) . j = 1 + ( l ^ <* x *> ) . j & z = 1 + ( l ^ <* x *> ) . j ;
for u , v being VECTOR of V , r being Real st 0 < r & r < 1 holds r * u + ( r * v ) in c= c= c= N * ( - r ) + ( r * v )
A , Int Cl A , Cl Int Cl A , Cl Cl Int Cl A , Cl Cl Int Cl A , Cl Cl Cl Cl A , Cl Cl A ` ` ` ` , Cl Cl A ` ` ` ` ` ` ` ` ` ` ` , Cl ( Cl ( Cl A , Cl A , Cl A ) ` ) ` ` ` ` ) ` ` ` ` ` ` ` ` ` ;
- Sum <* v , u , w *> = - ( v + u + w ) .= - ( v + u ) + u .= - ( v + u ) ;
( Exec ( a := b , s ) ) . IC SCM R = ( Exec ( a := b , s ) ) . IC SCM .= ( Exec ( a := b , s ) ) . IC SCM .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s ;
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 , D being non empty Subset of [: S1 , S2 :] , f being Function of S1 , S2 holds cos ( f ) is directed & cos ( f ) is directed & cos ( f ) is directed implies f is directed
card X = 2 implies ex x , y st x in X & y in X & x <> y or x = y & y = x or x = y or x = y or x = y or x = z or x = z or x = x or x = y or x = z or x = x or x = y or x = z or x = y or x = z or x = x or x = y or x = z & y = z or x = x or x = z or x = x or x = z or x = z or x = x or x = z or x = z or x =
( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) in rng ( Cage ( C , n ) \circlearrowleft Cage ( C , n ) ) & ( Cage ( C , n ) \circlearrowleft Cage ( C , n ) ) .. Cage ( C , n ) in rng Cage ( C , n ) implies ( Cage ( C , n ) \circlearrowleft Cage ( C , n ) ) /. len Cage ( C , n ) = E-max L~ Cage ( C , n )
for T , T being decorated tree , p , q being Element of dom T , p being Element of dom T st p divides q holds ( T -tree p ) . q = T . q & ( T -tree p ) . q = T . q ;
[ i2 + 1 , j2 ] in Indices G & [ i2 + 1 , j2 ] in Indices G & f /. k = G * ( i2 + 1 , j2 ) & f /. k = G * ( i2 + 1 , j2 ) & f /. k = G * ( i2 + 1 , j2 ) ;
cluster -> commutative means : Def3 : k divides it & k divides it & ( for m being Nat st k divides m & m divides k holds it divides m ) & ( for m being Nat st m divides k & m divides m holds it divides m ) & ( for m being Nat st m divides k holds it divides m ) & ( m divides k ) & ( m divides k implies m divides k ) ;
dom F " = the carrier of X2 & rng F = the carrier of X1 & F " { x } = the carrier of X2 & F " { x } = the carrier of X1 & F " { x } = the carrier of X2 & F " { x } = the carrier of X2 & F " { x } = the carrier of X1 & F " { x } = the carrier of X2 & x is Function of X1 , X2 ;
consider C being finite Subset of V such that C c= A and card C = omega and the carrier of V = Lin ( B \/ C ) and the carrier of C = Lin ( B \/ C ) and the carrier of C = Lin ( B \/ C ) ;
V is prime implies for X , Y being Element of [: the topology of T , the topology of T :] st X /\ Y c= V holds X c= Y or Y c= V or X c= Y or Y c= V
set X = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } , Y = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } ;
angle ( p1 , p3 , p2 ) = 0 .= angle ( p2 , p3 , p2 ) .= angle ( p2 , p3 , p2 ) .= angle ( p2 , p3 , p2 ) .= angle ( p2 , p3 , p2 ) ;
- sqrt ( ( - ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 ) ) = - sqrt ( ( - ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 ) .= - 1 ;
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 = p4 & f . 1 = p2 & f . 0 = p3 & f . 1 = p4 & f . 1 = p4 & f . 1 = p4 & f . 0 = p4 ;
attr f is partial means : Def3 : for u0 be Point of pdiff1 ( f , 1 ) holds SVF1 ( 2 , pdiff1 ( f , 1 ) , u0 ) . u0 = ( proj ( 2 , 3 ) ) . u0 & SVF1 ( 2 , 3 ) . u0 = ( proj ( 2 , 3 ) ) . u0 ;
ex r , s st x = |[ r , s ]| & G * ( len G , 1 ) `1 < r & r < G * ( 1 , 1 ) `1 & s < G * ( 1 , 1 ) `2 & G * ( 1 , 1 ) `2 < s & s < G * ( 1 , 1 ) `2 } ;
consider f being FinSequence such that f is_sequence_on G and 1 <= t and t <= len G and G * ( t , width G ) `2 >= N-bound L~ f and f /. ( t + 1 ) `2 >= N-bound L~ f ;
attr i in dom G means : Def3 : r (#) ( f (#) reproj ( i , x ) ) = r * f * reproj ( i , x ) + r (#) reproj ( i , x ) ;
consider c1 , c2 being bag of o1 + o2 such that ( decomp c ) /. k = <* c1 , c2 *> and ( decomp c ) /. k = c1 + c2 and ( decomp c ) /. k = c1 + c2 and c2 /. k = c2 + c2 ;
u0 in { |[ r1 , s1 ]| : r1 < G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `2 < s1 & s1 < G * ( 1 , 1 ) `2 } ;
Cl ( X ^ Y ) = the carrier of X . ( k2 + 1 ) .= ( C . ( k2 + 1 ) ) . ( k2 + 1 ) .= C . ( k2 + 1 ) .= C . ( k2 + 1 ) .= C . ( k2 + 1 ) ;
attr len M1 = len M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & M1 = M2 implies M1 - M2 = M2 - M1 & M1 - M2 = M2 - M2 ;
consider g2 being Real such that 0 < g2 and { y where y is Point of S : ||. ( - x0 ) - ( x - x0 ) .|| < g2 & g2 < r2 } c= N2 & N2 c= N2 & g2 in N2 & g2 in N2 & g2 in N2 } c= N2 & g2 in N2 & g2 in N2 & g2 in N2 & g2 in N2 & g2 in N2 } c= N2 ;
assume x < ( - b + sqrt ( delta ( a , b , c ) ) ) * ( 2 * a ) or x > - b + sqrt ( delta ( a , b , c ) ) * ( 2 * a ) ) ;
( G1 '&' G2 ) . i = ( <* 3 *> ^ G1 ) . i & ( H1 '&' H2 ) . i = ( <* 3 *> ^ ( H1 ^ H2 ) ) . i & ( H1 ^ H2 ) . i = ( <* 3 *> ^ ( H1 ^ H2 ) ) . i ;
for i , j st [ i , j ] in Indices ( M3 + M1 ) holds ( M3 + M2 ) * ( i , j ) < M2 * ( i , j ) + M2 * ( i , j )
for f being FinSequence of NAT , i being Element of NAT st i in dom f & i in dom f holds f divides f /. i & f /. i = Sum f & f /. ( i + 1 ) = Sum f ;
assume that F = { [ a , b ] where a , b is Element of X : for c being set st c in B\mathopen { a , b } & c in B\mathopen { b : a in B } } and a c= c } ;
b2 * q2 + ( b3 * q2 ) + ( b3 * q2 ) + ( b3 * q2 ) + ( b3 * q2 ) = 0. TOP-REAL n + ( b1 * q2 ) + ( b2 * q2 ) .= 0. TOP-REAL n + ( b2 * q2 ) ;
Cl ( Cl F ) = { D where D is Subset of T : ex B being Subset of T st B = Cl ( F . D ) & B in F & B in F } ;
attr seq is summable means : Def4 : seq is summable & ( for n holds seq . n is summable & seq . n is summable ) & ( for n holds seq . n = Partial_Sums ( seq ) . n ) & ( for n holds seq . n = Partial_Sums ( seq ) . n ) implies seq is summable & ( seq is summable ) & ( seq is summable implies seq is summable ) & ( seq is summable ) & ( seq is summable ) implies seq is summable ) ;
dom ( ( ( ( cn ) | D ) | D ) ) = ( the carrier of ( ( TOP-REAL 2 ) | D ) | D ) /\ D ) .= the carrier of ( ( ( TOP-REAL 2 ) | D ) | D ) .= the carrier of ( ( ( TOP-REAL 2 ) | D ) | D ) .= the carrier of ( ( TOP-REAL 2 ) | D ) | D ) .= the carrier of ( ( TOP-REAL 2 ) | D ) | D ) ;
[: X , Z :] is full full SubRelStr of ( Omega Y ) |^ the carrier of Z & [ X , Y ] is full SubRelStr of ( Omega Y ) |^ the carrier of Z ;
( G * ( 1 , j ) `2 = G * ( i , j ) `2 & ( G * ( 1 , j ) `2 <= G * ( 1 , j ) `2 or G * ( 1 , j ) `2 <= G * ( 1 , j ) `2 ) & ( G * ( 1 , j ) `2 <= ( G * ( 1 , j ) `2 ) ) ;
synonym m1 c= m2 for for for for for for for ( m1 , m2 ) `1 , m2 = m2 & m2 <= m2 & m1 <= m2 & m2 <= m2 holds ( m1 , m2 ) `2 <= m2 `2 & m2 <= m2 `2 & m2 <= m2 `2 & m2 <= m2 `2 & m2 <= m2 `2 & m2 <= m2 `2 & m2 <= m2 `2 ) ;
consider a being Element of B ( ) such that x = F ( a ) and a in { G ( ) where b is Element of B ( ) : P [ b ] } ;
func multiplicative loop loop Str (# carrier -> multiplicative loop over R means : Def3 : for a being Element of it holds a , a are_in it iff a , b are_if a , b are_commutative in it & b , a are_if a , b are_commutative in it , b ) ;
the carrier of : ( the carrier of a , b ) + 1 = b + ( c + d ) .= b + d .= b + ( c + d ) .= b + ( c + d ) .= b + ( c + d ) .= b + ( c + d ) .= b + ( c + d ) ;
cluster -> add-associative right_zeroed right_complementable for Element of INT , F , G be Element of INT , i , j be Element of NAT holds ( i = j implies F . ( i1 , i2 ) = + F . ( i2 , i1 ) ) & ( i = i2 implies F . ( i1 , i2 ) = ( i1 + i2 ) + ( i1 - i2 ) )
( - s2 ) * p1 + ( s2 * p2 ) - ( s2 * p2 ) = ( r2 * p1 + s2 * p2 ) + ( s2 * p2 ) - ( s2 * p2 ) .= ( r2 * p1 + s2 * p2 ) + s2 * p2 ;
eval ( ( a | ( n , L ) ) *' p , x ) = eval ( a | ( n , L ) ) * eval ( p , x ) .= a * eval ( p , x ) .= a * eval ( p , x ) ;
assume that the TopStruct of S = the TopStruct of T and for D being non empty Subset of S , V being open Subset of S st V in V holds V is open & for V being open Subset of S st V in V holds V is open & V is open & V is open & V is open & V is open & V is open & V is open & V is open ;
assume that 1 <= k & k <= len w + 1 and T-7 ( k , w ) = ( T11 . ( q11 , w ) ) -\mathop { \rm \hbox { - } m } and T11 . ( k + 1 ) = ( T11 . ( q11 , w ) ) -\mathop { - m } ;
2 * a |^ ( n + 1 ) + ( 2 * b |^ ( n + 1 ) ) >= ( a |^ n + 1 ) + ( b |^ ( n + 1 ) ) + ( a |^ ( n + 1 ) ) + ( b |^ ( n + 1 ) ) ;
M , v / ( x. 3 , x ) / ( x. 0 , x ) / ( x. 4 , x ) / ( x. 0 , x ) / ( x. 4 , x ) / ( x. 0 , x ) ) / ( x. 4 , x ) / ( x. 0 , x ) / ( x. 0 , x ) / ( x. 4 , x ) / ( x. 0 , x ) / ( x. 4 , x ) / ( x. 0 , x ) ) = x ;
assume that f is_differentiable_on l and for x0 st x0 in l holds 0 < f ' Z & for x0 st x0 in l holds f . x0 < 0 & f . x0 < 0 & f . x0 < 0 ;
for G1 being _Graph , W being Walk of G1 , e being Vertex of G1 , x being set st e in W & not x in W holds W is Walk of ( G ) st e in W & x in W holds W is Walk of ( G )
not vs is non empty iff not ( not <= <= <= & & & <= <= & & & not <= <= & & & not <= <= & & & not <= <= & & & not <= <= & & & & & not <= <= & & & & & not <= <= & & & & not <= <= & & & & & & & & not not & & & & & & not not not <= & & & & not not <= & & & & not not <= & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & = & & & & & & = & & & & = & & = = = & & & & & =
Indices GoB f = [: dom GoB f , Seg width GoB f :] & [: dom GoB f , Seg width GoB f :] = [: Seg len GoB f , Seg width GoB f :] & [: dom GoB f , Seg width GoB f :] = [: Seg len GoB f , Seg width GoB f :] & [: dom GoB f , Seg width GoB f :] = [: Seg len GoB f , Seg width GoB f :] ;
for G1 , G2 , G3 being stable Subgroup of O st G1 , G2 are_: for a being stable Element of G1 , b being stable Element of G2 st a in G1 & b in G2 holds G1 , G2 are_commutative & G2 is stable holds G1 , G2 are_stable
UsedIntLoc ( int ) = { ( intloc 0 ) , ( ( intloc 1 ) .--> ( ( intloc 1 ) .--> ( ( intloc 1 ) .--> ( ( intloc 1 ) .--> ( ( intloc 0 ) .--> ( ( intloc 0 ) .--> ( ( intloc 1 ) .--> ( ( intloc 0 ) .--> ( ( intloc 1 ) .--> ( ( intloc 0 ) .--> ( ( intloc 0 ) .--> ( ( intloc 1 ) .--> ( ( intloc 0 ) ) intloc ( ( intloc 1 ) ) 1 ) ) ) ) ) ) ) ) ) ) ) ) , ( ( intloc 1 ) .--> ( ( intloc 1 ) ) ) ) , ( ( intloc 1 ) ) ) ) , ( ( intloc 1 ) ) ) , ( ( intloc 1 ) .--> ( ( intloc 1 ) .--> (
for f1 , f2 being FinSequence of F st f1 is p -element & f2 is p & Q [ f1 , f2 ] holds Q [ f2 , f1 ^ f2 ] & Q [ f1 ^ f2 , f2 ^ g2 ]
( ( p `1 ) ^2 + ( p `2 ) ^2 + ( p `1 ) ^2 ) = ( q `1 ) ^2 + ( q `2 ) ^2 + ( q `1 ) ^2 + ( q `2 ) ^2 ;
for x1 , x2 , x3 being Element of REAL n , x , y being Real holds |( x1 - x2 , x1 - x2 )| = |( x1 , x1 - x2 )| & |( x1 , x2 )| = |( x1 , x2 )| - |( x2 , x1 )| + |( x2 , x1 )| + |( x1 , x2 )|
for x st x in dom ( ( F - x ) | A ) holds ( ( F - x ) | A ) . ( - x ) = - ( ( F - x ) | A ) . ( - x ) )
for T being non empty TopSpace , P being Subset-Family of T , B being Basis of T st P c= the topology of T for x being Point of T st x in P holds P is Basis of x
( a 'or' b ) . x = 'not' ( ( a 'or' b ) . x ) 'or' c . x .= ( 'not' a . x 'or' b . x ) 'or' ( 'not' a . x ) .= 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 [: S , T :] , [: S , T :] st f = H . i & f is Function of [: S , T :] , T . i holds F . i = f | ( F . i )
for v , w st for x st x <> y holds w . y = v . y holds Valid ( VERUM ( Al ) , J ) . ( v . x ) = Valid ( VERUM ( Al ) , J ) . ( v . x )
card D = card D1 + card D1 - card { { i , j } } - card { i , j } .= 2 * ( i + 1 ) - card { i , j } .= 2 * ( i + 1 ) - card { i + 1 } .= 2 * ( i + 1 ) - card { i + 1 } .= 2 * ( i + 1 ) - card { i + 1 } .= 2 * ( i + 1 ) ;
IC Exec ( i , s ) = ( s +* ( 0 .--> ( s . 0 ) ) ) . 0 .= ( ( 0 .--> ( s . 0 ) ) . 0 ) . 0 .= ( ( 0 .--> ( s . 0 ) ) . 0 ) . 0 .= ( ( 0 .--> ( s . 0 ) ) . 0 ) . 0 .= ( ( 0 .--> ( s . 0 ) ) . 0 ) . 0 .= ( ( 0 .--> ( s . 0 ) ) . 0 ) . 0 .= ( ( ( 0 .--> ( s . 0 ) ) . 0 ) . 0 ) . 0 .= ( ( 0 .--> ( s . 0 ) ) . 0 .= ( ( ( 0 .--> ( s . 0 ) ) . 0 ) . 0
len f /. ( \downharpoonright i1 -' 1 + 1 ) -' 1 + 1 = len f -' ( i1 -' 1 + 1 ) + 1 .= len f -' ( i1 -' 1 + 1 ) + 1 .= len f -' ( i1 -' 1 + 1 ) + 1 ;
for a , b , c being Element of NAT st 1 <= a & 2 <= b & k < b holds a <= a + b-2 or a = b + b-2 or a = a + b-2 or a = b-2 or a = b-2 or a = b-2 or a = b-2 or a = b-2 or a = b-2 or a = b-2 or a = b-2 ;
for f being FinSequence of TOP-REAL 2 , p being Point of TOP-REAL 2 , i being Element of NAT st i in LSeg ( f , i ) & p in LSeg ( f , i ) holds Index ( p , f ) <= i & Index ( p , f ) <= i & Index ( p , f ) <= Index ( p , f ) + 1
lim ( ( curry ( P , k + 1 ) ) # x ) = lim ( ( curry ( P , k ) ) # x ) + ( ( curry ( P , k ) ) # x ) # x ) ;
z2 = g /. ( \downharpoonright n1 -' n2 + 1 ) .= g /. ( i -' n2 + 1 ) .= g . ( i -' n2 + 1 ) .= g . ( i -' n2 + 1 ) .= g . ( i -' n2 + 1 ) .= g . ( i -' n2 + 1 ) ;
[ f . 0 , f . 3 ] in id the carrier of G or [ f . 0 , f . 3 ] in id the carrier of G & [ f . 0 , f . 3 ] in the InternalRel of G & [ f . 2 , f . 3 ] in the InternalRel of G ;
for G being Subset-Family of B st G = { R [ X ] where R is Subset of A , Y is Subset of B : R in F & Y in F } holds ( Intersect ( F ) ) . ( X , Y ) = Intersect ( G ) . ( X , Y ) )
CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 ) ) = CurInstr ( P1 , Comput ( P1 , s2 , m2 ) ) .= CurInstr ( P1 , Comput ( P1 , s2 , m2 ) ) .= halt SCMPDS ;
assume that a on M and b on M and c on N and d on N and c on N and p on N and p on M and a on M and c on N and p on M and a on M and p on M and a on M and c on M and p on M and a <> b and p <> c & a <> b & p <> c & p <> c & a <> b & p <> c & a <> c & p <> c & a <> b & p <> c & a <> c & p <> c & a <> c & p <> d & p <> d & p <> d & p <> d & p <> d & p <> d & p <> d & p <> d & p <> d & p <> d & p <> d & p <> d & p <> d & p <> d & p <> d & p <> d & p <> d & p <> d
assume that T is \hbox of T , T1 and ex F be Subset-Family of T st F is closed & F is closed & F is finite-ind & for n being Nat st n <= len F holds F . n <= 0 & ind F <= 0 ;
for g1 , g2 st g1 in ]. ( r1 - r ) / 2 , ( r1 - r ) / 2 .[ & |. ( f . g1 ) - ( g . g2 ) .| <= ( r1 - r ) / 2 holds |. ( f . g1 ) - ( g . g2 ) .| <= ( r1 - r ) / 2 * ( ( r1 - r ) / 2 )
( ( > ( > 0 ) / ( z1 + z2 ) ) / ( z1 + z2 ) ) = ( ( > 0 ) / ( z1 + z2 ) ) * ( z1 + z2 ) + ( ( 1 / 2 ) * ( z1 + z2 ) ) * ( z1 + z2 ) ) ;
F . i = F /. i .= 0. R + r2 .= ( b |^ ( n + 1 ) ) * ( a |^ ( n + 1 ) ) .= ( b |^ ( n + 1 ) ) * ( a |^ ( n + 1 ) ) .= ( b |^ ( n + 1 ) ) * ( a |^ ( n + 1 ) ) ;
ex y being set , f being Function st y = f . n & dom f = NAT & for n holds f . ( n + 1 ) = RF ( n , f . n ) & for n holds f . ( n + 1 ) = R ( n , f . n ) ;
func f (#) F -> FinSequence of V means : Def1 : len it = len F & for i be Nat st i in dom it holds it . i = F /. i * ( F /. i ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 , 6 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 } = { x1 , x2 , x3 , 6 , 7 , 7 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 } \/ { x1 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7
for n being Nat , x being set st x = h . n holds h . ( n + 1 ) = o ( x , n ) & ( x in InputVertices S ( ) & x in InputVertices S ( ) & x in InputVertices S ( ) ) ) & ( x in InputVertices S ( ) implies x in InnerVertices S ( ) ) & x in InputVertices S ( ) ) ;
ex S1 being Element of QC-WFF ( Al ) st SubP ( P , l , e ) = S1 & ( for n being Element of NAT holds ( ( P . n ) . n = ( P . n ) . ( l + 1 ) ) & ( P . n ) . ( l + 1 ) = ( P . n ) . ( l + 1 ) ) ;
consider P being FinSequence of IT such that pthat pI = product P and for i being Element of NAT st i in dom P ex tI being Element of the carrier of K st P . i = t1 & P . i = t1 & P . i = t2 ;
for T1 , T2 being strict non empty TopSpace , P being Basis of T1 , Q being Basis of T2 st the topology of T1 = the topology of T2 & P is Basis of T2 & P is Basis of T2 holds P is Basis of T1 & Q is Basis of T2 & P is Basis of T2 implies P is Basis of T2
assume that f is partial u0 on u0 and r (#) pdiff1 ( f , 3 ) is_partial_differentiable_in u0 , 2 and partdiff ( r (#) pdiff1 ( f , 3 ) , u0 ) = r * pdiff1 ( f , 3 ) and partdiff ( r (#) pdiff1 ( f , 3 ) , u0 ) = r * pdiff1 ( f , 3 ) and partdiff ( r (#) pdiff1 ( f , 3 ) , u0 ( f , 3 ) , u0 ) = r * pdiff1 ( f , 3 ) ;
defpred P [ Nat ] means for F , G being FinSequence of REAL , s being FinSequence of REAL st len F = $1 & for i being Element of NAT st i in $1 holds G = F * s & ( for i being Element of NAT st i in $1 holds F . i = F . i ) holds F = G . i ) ;
ex j st 1 <= j & j < width GoB f & ( GoB f ) * ( 1 , j ) `2 <= s & s <= ( GoB f ) * ( 1 , j ) `2 & s <= ( GoB f ) * ( 1 , j + 1 ) `2 & s <= ( GoB f ) * ( 1 , j + 1 ) `2 & s <= ( GoB f ) * ( 1 , j + 1 ) `2 }
defpred U [ set , set ] means ex FF be Subset-Family of T st $1 = FF & $2 = F . $1 & ( for n being Nat holds $2 . n is open & F . n is open & F . n is open & F . n is open & F . n is open & F . n is open & F . ( $1 + 1 ) is open ) & F is discrete ;
for p4 being Point of TOP-REAL 2 st LE p4 , p2 , P , p1 , p2 & LE p4 , p2 , P , p1 , p2 & LE p2 , p3 , P , p1 , p2 holds LE p4 , p2 , P , p1 , p2 & LE p4 , p2 , P , p1 , p2 & LE p4 , p2 , P , p1 , p2 & LE p4 , p2 , P , p2 & LE p4 , p2 , P , p1 , p2 & LE p4 , p2 , P , p1 , p2 & LE p4 , p2 , P , p2 & LE p4 , p2 , P , p1 , p2 & LE p4 , p2 , P , p1 , p2 & LE p2 , p1 , P , p1 , p2 & LE p2 , p1 , P , p3 & LE p4 , p1 , P , p1 , p2 & LE p2 , p1 , P , p3 , p3 , K & LE p2 , p1 , K , p3 , p3 , p3
f in St ( E , H ) & for g st g in S holds g . y = f . y iff for x st x in S holds g . x = f . ( All ( x , H ) . x ) ) & f in for x st x in S holds g . x = f . ( All ( x , H ) . x ) ) ;
ex p9 being Point of TOP-REAL 2 st x = p9 & ( ( for q being Point of TOP-REAL 2 st q in the carrier of TOP-REAL 2 holds |. q .| >= p9 ) & ( ( for q being Point of TOP-REAL 2 st q in the carrier of TOP-REAL 2 holds |. q .| >= sn ) & ( ( for q being Point of TOP-REAL 2 st q in the carrier of TOP-REAL 2 holds q `2 >= 0 ) ) & ( ( for q being Point of TOP-REAL 2 st q in the carrier of TOP-REAL 2 ) holds q `2 <= 0 ) & q `2 <= 0 ) & ( ( q `2 <= q `1 <= 0 ) & q `2 <= 0 ) & ( ( q `2 <= q `1 <= 0 & q `2 <= 0 ) & ( ( q `2 <= q `1 <= 0 & q `2 <= 0 & q `2 <= 0 & q `2 <= 0 & q `2 <= q `1 <= 0 & q `1 <= q `1 <= q `1
assume for d7 being Element of NAT st d7 <= d7 & d7 <= d7 holds s1 . ( d7 ) = s2 . ( d7 ) & s1 . ( d7 ) = s2 . ( d7 ) & s1 . ( d7 ) = s2 . ( d7 ) ;
assume that s <> t and s is Point of Sphere ( x , r ) and s is Point of Sphere ( x , r ) and ex e being Point of E st e = Ball ( x , r ) /\ Ball ( x , r ) & e in Sphere ( x , r ) /\ Sphere ( x , r ) ;
given r such that 0 < r and for s st 0 < s ex x1 , x2 being Point of CNS st x1 in dom f & x2 in dom f & ||. x1 - x2 .|| < s & ||. x1 - x2 .|| < s & ||. x1 - x2 .|| < r & ||. x1 - x2 .|| < s ;
( p | x ) | ( p | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x
assume that x , x + h / 2 in dom sec and ( ex h st h = ( 4 * ( sec * ( ( 2 * x + h ) ) / 2 ) ) * ( sin * ( 2 * x + h / 2 ) ) ) / 2 ) ) and ( h = 4 * ( ( 2 * x + h / 2 ) / 2 ) ;
assume that i in dom A and len A > 1 and len B > 1 and B * ( i , j ) = ( ( A * ( i , j ) ) * ( i , j ) ) and ( A * ( i , j ) ) * ( i , j ) = ( A * ( i , j ) ) * ( i , j ) ;
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 or i = 1. F_Complex & i <> 0. F_Complex & i <> 0. F_Complex & i <> 0. F_Complex & i <> 1. F_Complex & i <> 0. F_Complex & i <> 0. F_Complex & i <> 0. F_Complex & i <> 0. F_Complex & i <> 0. F_Complex holds h . i = <* 1. F_Complex , 1. F_Complex *>
( ( ( b1 'imp' b2 ) '&' ( ( b1 'or' b2 ) '&' ( ( b1 'or' b2 ) '&' ( ( b1 'or' b2 ) '&' ( ( b1 'or' b2 ) '&' ( b2 'or' b3 ) ) ) ) ) ) '&' ( ( ( b1 'or' b2 ) '&' ( b1 'or' b3 ) ) '&' ( ( b1 'or' b2 ) '&' ( b2 'or' b3 ) ) ) ) '&' ( ( ( b1 'or' b2 ) '&' ( b1 'or' b2 ) '&' ( b2 'or' b3 ) ) ) ) ) '&' ( ( ( b1 'or' b2 ) '&' ( ( b1 'or' b2 ) '&' ( ( b1 'or' b2 ) '&' ( b2 'or' b3 ) '&' ( b2 'or' b3 ) ) ) ) ) '&' ( ( b1 'or' b2 ) '&' ( ( b1 'or' b2 ) '&' ( ( b1 'or' b2 ) '&' ( b2 'or' b3 ) ) '&' ( ( b2 'or' b2 ) ) '&' ( ( ( b2 'or' b3 ) ) '&' ( ( b2 'or' b3 ) ) '&' (
assume that for x holds f . x = ( ( - 1 ) (#) ( ( cot * ( sin - x ) ) / ( sin . x ) ) ) and for x st x in dom ( ( - 1 ) (#) ( sin * ( sin - x ) ) ) holds ( ( - 1 ) (#) ( sin * ( sin - x ) ) ) `| Z ) . x = - ( sin . x ) / ( sin . x ) ;
consider R8 , I7 be Real such that R8 = Integral ( M , Re ( F . n ) ) and R7 = Integral ( M , Im ( F . n ) ) and I = ( Im ( F . n ) ) . ( n + 1 ) and I = ( Im ( F . n ) ) . ( n + 1 ) ;
ex k be Element of NAT st k = k & 0 < d & d < d & for q be Element of product G st q in X & ||. qx0 - f /. x .|| < r holds ||. partdiff ( f , x , i ) - partdiff ( f , x , i ) .|| < r ;
x in { x1 , x2 , x3 , x4 , x5 , 6 , 7 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 } \/ { 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 ,
( G * ( j , i ) `2 ) `2 = G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * (
f1 * p = p .= ( the Arity of S1 ) . ( ( the Arity of S2 ) . ( g . o ) ) .= ( the Arity of S1 ) . ( g . o ) .= ( the Arity of S1 ) . ( g . o ) .= ( the Arity of S1 ) . ( g . o ) ;
func tree ( T , P , T1 ) -> Tree means : Def1 : for q being FinSequence st q in T holds q in T iff ex p , q st p in T & q in T & q in T & p in T & q in T & p in T & q in T & q in T & p in T & q in T & q in T ;
F /. ( k + 1 ) = F . ( k + 1 -' 1 ) .= Fx0 . ( p . ( k + 1 -' 1 ) ) , F . ( p . ( k + 1 ) ) ) .= Fx0 . ( p . ( k + 1 -' 1 ) ) .= Fx0 . ( p . ( k + 1 -' 1 ) ) .= Fx0 . ( p . ( k + 1 -' 1 ) ) ;
for A , B , C being Matrix of len C , K st len B = len C & len B = width C & len C > 0 & len C > 0 & len A > 0 & len B > 0 & len A > 0 holds A * ( being being being Matrix of len A , width A ) = B * C & len A > 0 & len B > 0 & width A > 0 holds A * ( BC ) = B * C
( ( Partial_Sums ( seq ) ) . ( k + 1 ) ) . ( n + 1 ) = 0. ( X , ( seq . ( k + 1 ) ) ) . ( ( Partial_Sums ( seq ) ) . ( k + 1 ) ) . ( n + 1 ) ) .= ( Partial_Sums ( seq ) ) . ( n + 1 ) . ( n + 1 ) ;
assume that x in ( the carrier of Cs ) and y in ( the carrier of Cs ) and x in the carrier of the carrier of Cs and y in the carrier of Cs and x in the carrier of s ;
defpred P [ Element of NAT ] means for f st len f = $1 & ( for k st k < $1 holds f . k = ( VAL g ) . ( f /. k ) ) holds ( ( VAL g ) . ( f /. k ) ) . ( f /. ( k + 1 ) ) = ( ( VAL g ) . ( f /. k ) ) . ( f /. ( k + 1 ) ) ;
assume that 1 <= k and k + 1 <= len f and f is_sequence_on G and [ i , j ] in Indices G and f /. k = G * ( i + 1 , j ) and f /. ( k + 1 ) = G * ( i + 1 , j ) and f /. ( k + 1 ) = G * ( i + 1 , j ) and f /. ( k + 1 ) = G * ( i + 1 , j ) ;
assume that sn < 1 and ( q `1 / |. q .| - cn ) >= 0 and ( q `2 / |. q .| - cn ) >= 0 and ( q `2 / |. q .| - cn ) >= 0 and ( q `2 / |. q .| - cn ) >= 0 and ( q `2 / |. q .| - cn ) >= 0 ;
for M being non empty metric space , x being Point of M , f being Point of M st x = x `1 holds ex n being Nat st for m being Nat holds f . m = Ball ( x `1 , 1 / n ) & f . n = Ball ( x `1 , 1 / n )
defpred P [ Element of omega ] means f1 is_differentiable_on Z & f2 is_differentiable_on Z & for x st x in Z holds f2 . x > 0 & f2 . x > 0 & f2 . x > 0 & f2 . x < 1 & f2 . x < 1 & f2 . x < 1 & f2 . x < 1 & f2 . x < 1 & f2 . x < 1 & f2 . x < 1 & f2 . x < 1 ;
defpred P1 [ Nat , Point of ( C ) ] means ( for r be Point of ( C ) st r in Y & ( for s be Point of ( C ) st s in Y holds ||. ( s1 . s ) - ( s2 . s ) ) .|| < r ) & ( ||. f /. ( s ) - ( s2 . s ) .|| < r ) implies f /. ( s - ( s - s1 ) ) < r ) ;
( f ^ mid ( g , 2 , len g ) ) . i = ( mid ( g , 2 , len g ) ) . ( i -' len f + 1 ) .= g . ( i -' len f + 1 ) .= g . ( i -' len f + 1 ) ;
( 1 - 2 * n0 + 2 * n0 ) * ( 2 * n0 + 2 * n0 ) = ( 1 - 2 * n0 + 2 * n0 ) * ( 2 * n0 + 1 ) .= 1 * ( 2 * n0 + 2 * n0 ) .= 1 * ( 2 * n0 + 2 * n0 ) .= 1 * ( 2 * n0 + 2 * n0 ) ;
defpred P [ Nat ] means for G being non empty strict strict finite RelStr , F being Function of the carrier of G , the carrier of G st F is non empty holds the carrier of F = the carrier of G & the carrier of F = the carrier of G & the carrier of F = the carrier of G & the InternalRel of F = the InternalRel of G ;
assume that f /. 1 in Ball ( u , r ) and not 1 <= m and m <= len - ( f /. 1 ) and for i st 1 <= i & i <= len f holds LSeg ( f , i ) /\ LSeg ( f , m ) = { f /. i } and LSeg ( f , i ) /\ LSeg ( f , m ) = { f /. i } and LSeg ( f , m ) /\ LSeg ( f , m ) = { f /. m , f /. ( m + 1 ) /\ LSeg ( f , m ) = { f /. m , f /. ( m + 1 ) = { f /. ( m + 1 ) /\ LSeg ( f , m ) = { f /. ( m + 1 ) } and LSeg ( f , m ) = { f /. ( m + 1 ) /\ LSeg ( f , m ) /\ LSeg ( f , f /. ( m + 1 ) /\ LSeg ( f , f /. ( m + 1 ) /\ LSeg ( f , m ) = { f /. m ,
defpred P [ Element of NAT ] means ( Partial_Sums ( ( ( ( ( ( 1 / 2 ) (#) ( ( n + 1 ) / ( 2 * n ) ) ) ) ) ) ) . ( 2 * $1 ) ) . ( 2 * $1 ) ) . ( 2 * $1 ) ) . ( 2 * $1 ) = ( ( ( ( ( n / 2 ) * ( ( n + 1 ) / ( 2 * $1 ) ) ) ) / ( 2 * $1 ) ) . ( 2 * $1 ) ) . ( 2 * $1 ) ) . ( 2 * $1 ) ) . ( 2 * $1 ) ) . ( 2 * $1 ) ) . ( 2 * $1 ) ) . ( 2 * $1 ) . ( 2 * $1 ) ) . ( 2 * $1 ) . ( 2 * $1 + ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( n + $1 ) ) . ( ( ( ( ( ( ( (
for x being Element of product F holds x is FinSequence of dom x & ( for i being set st i in dom x holds x . i = I . i ) & ( for i being set st i in dom x holds x . i in I . i ) implies x . i = I . i )
( x " ) |^ ( n + 1 ) = ( x " ) |^ n * x .= ( x |^ n ) |^ ( n + 1 ) .= ( x |^ n ) |^ ( n + 1 ) .= ( x |^ n ) |^ ( n + 1 ) .= ( x |^ n ) |^ ( n + 1 ) .= ( x |^ n ) |^ ( n + 1 ) ;
DataPart Comput ( P +* I , ( LifeSpan ( P +* I , s ) ) + ( LifeSpan ( P +* I , s ) ) + 3 ) = DataPart Comput ( P +* I , s +* I , ( LifeSpan ( P +* I , s ) ) + 3 ) ) ;
given r such that 0 < r and ]. x0 - r , x0 .[ c= dom f1 and for g st g in ]. x0 - r , x0 .[ holds f1 . g <= ( f1 . g ) . g & f2 . g <= 0 ;
assume that X c= dom f1 /\ dom f2 and f1 | X is continuous and f2 | X is continuous and ( for x be Element of X holds f1 . x = ( f1 . x ) - f2 . x ) and ( f1 | X ) . x = ( f1 | X ) . x ;
for L being continuous complete LATTICE for l being Element of L ex X being Subset of L st l = sup X & for x being Element of L st x in X holds x is Element of L & x is ` & l is ` & x is ` ) implies for x being Element of L st x in { l where l is Element of L : l is prime }
Support ( e ) in { m where m is Element of NAT : A /. i = Support ( m *' p ) & ex p being Polynomial of n , L st p = m & p . i = m & p . i = m & p . i = m . i & p . i = m . i & p . i = m . i & p . i = m . i & p . i = m . i & p . i = m . i & p . i = m . i & p . i = m . i & p . i = m . i & p . i = m . i & p . i = m . i & p . i = m . i & p . i = m . i & p . i = m . i & p . i = m . i & p . i = m . i & p . i = m . i & p . i = m . i & p . i = m . i & p . i = m . i & p . i = m . i & p . i
( f1 - f2 ) /* ( seq ^\ k ) = lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) .= ( lim ( ( f1 /* s1 ) - ( f2 /* s2 ) ) ) ) * ( ( lim s1 ) - ( f2 /* s2 ) ) .= ( lim ( ( f1 /* s1 ) - ( f2 /* s2 ) ) ) * ( lim ( ( f1 /* s1 ) - ( f2 /* s2 ) ) ) .= ( lim ( ( f1 /* s1 ) - ( f2 /* s2 ) ) ) ) ;
ex p1 being Element of QC-WFF ( Al ) st F . p1 = g . p1 & for g being Function of QC-WFF ( Al ) , D st g = g . ( g . ( len g ) ) holds p1 = g . ( len g ) ) ;
( mid ( f , i , len f -' 1 ) ) /. ( j + 1 ) = ( mid ( f , i , len f -' 1 ) ) /. ( j + 1 ) .= f /. ( j + 1 ) .= f /. ( j + 1 ) ;
( ( p ^ q ) . ( len p ) ) . ( ( len p ) + k ) = ( ( p ^ q ) . ( len p ) ) . ( ( len p ) + ( len q ) ) ) . ( ( len p ) + ( len q ) + ( len p ) ) .= ( ( p ^ q ) . ( len p ) ) . ( ( len p + k ) ) .= ( ( p ^ q ) . ( len p + k ) ) . ( ( len p + k ) ) . ( ( len p + k ) ) .= ( ( p ^ q ) . ( ( len p + k ) ) . ( ( len p + k ) ) + ( ( len p + k ) ) . ( ( len p + k ) . ( ( len p + ( len p + ( len p + ( len p + ( len p + ( len p + ( len p + ( len p + ( len p + ( len p + k ) ) ) .= ( ( len p + k ) ) + ( len p + ( len p + k ) ) + ( len p +
len mid ( upper_volume ( f , D1 ) , 1 ) + ( indx ( D2 , D1 , j ) + 1 ) - ( indx ( D2 , D1 , j ) + 1 ) = indx ( D2 , D1 , j ) - ( indx ( D2 , D1 , j ) + 1 ) + ( indx ( D2 , D1 , j ) + 1 ) - ( indx ( D2 , D1 , j ) + 1 ) ;
x * y * z = ( x * ( y * z ) ) * ( ( x * ( y * z ) ) * ( x * ( y * z ) ) ) .= ( x * ( y * z ) ) * ( x * ( y * z ) ) .= ( x * ( y * z ) ) * ( x * ( y * z ) ) .= ( x * ( y * z ) ) * x ;
v . ( <* x , y *> ) * ( <* x0 , y0 *> ) . i = partdiff ( v , ( x - x0 ) * ( <* x0 , y0 *> ) ) + ( ( proj ( 1 , 1 ) * ( ( reproj ( 1 , 1 ) ) * ( ( reproj ( 1 , 1 ) ) * ( ( reproj ( 1 , 1 ) ) * ( ( reproj ( 1 , 1 ) ) * ( ( reproj ( 1 , 1 ) ) ) ) ) ) ) ) . ( ( reproj ( 1 , 1 ) ) ) . ( ( reproj ( 1 , 1 ) ) ) . ( ( reproj ( 1 , 1 ) ) . ( ( reproj ( 1 , 1 ) ) . ( ( reproj ( 1 , 1 ) ) . ( ( reproj ( 1 , 1 ) ) . ( ( ( reproj ( 1 , y ) ) . ( ( x , y ) ) ) ) * ( ( reproj ( 1 , y ) ) . ( ( reproj ( 1 , y ) ) . ( ( reproj ( 1 , y ) ) . ( ( reproj ( 1 , y ) ) . ( ( reproj ( 1 ,
i * i = <* 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 ) * ( - 1 , 1 ) * ( - 1 , 1 ) * ( - 1 , 1 ) * ( - 1 , 1 ) * ( - 1 , 1 ) * ( - 1 , 1 , 1 ,
Sum ( L (#) F ) = Sum ( L (#) ( F1 ^ F2 ) ) + Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) + Sum ( ( L (#) F2 ) (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) + Sum ( ( L (#) F2 ) (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) ;
ex r be Real st for e be Real st 0 < e ex Y1 be finite Subset of REAL st Y1 is non empty & for Y1 be finite Subset of X st Y1 c= Y & Y1 c= Y holds |. Sum ( Y1 ) - Sum ( Y2 ) .| < r & Sum ( Y1 ) <= Sum ( Y1 ) ;
( GoB f ) * ( i , j + 1 ) `1 = f /. ( k + 2 ) & ( GoB f ) * ( i + 1 , j ) `2 = f /. ( k + 2 ) or ( GoB f ) * ( i + 1 , j + 1 ) `2 = f /. ( k + 2 ) `2 ) or ( GoB f ) * ( i + 1 , j + 1 ) `2 = f /. ( k + 2 ) `2 ;
( ( - 1 ) (#) ( cos | [. - 1 , 1 .[ ) ) / ( sin | [. - 1 , 1 .] ) = ( - 1 ) / ( cos . ( - 1 ) ) .= ( - 1 ) / ( cos . ( - 1 ) ) .= ( - 1 ) / ( cos . ( - 1 ) ) .= ( - 1 ) / ( cos . ( - 1 ) ) .= ( - 1 ) / ( sin . ( - 1 ) ) .= ( - 1 ) / ( sin . ( - 1 ) ) .= ( - 1 ) / ( sin . ( - 1 ) ) / ( sin . ( - 1 ) ;
( - ( - b + sqrt ( a , b ) ) / ( 2 * a ) ) < 0 & ( - ( - b ) / ( 2 * a ) ) / ( 2 * a ) < 0 implies ( - ( - b ) / ( 2 * a ) ) / ( 2 * a ) < 0 ) & ( - ( - b ) / ( 2 * a ) < 0 )
assume that inf ( uparrow X /\ C ) in L and ex X st X in the carrier of L & for x st x in X holds x in X & x in X & x in sup ( ( ( downarrow X ) /\ ( downarrow x ) ) /\ ( ( downarrow x ) /\ ( downarrow x ) ) ) ) & x in X ;
( ( for j holds ( j = i = j ) implies ( i = j = j ) implies i = j ) & ( j = j implies i = j ) & ( j = j implies i = j ) ) & ( j = j implies i = j ) ) & ( j = j implies i = j ) ) & ( j = j implies i = j ) ) & ( j = j implies i = j ) ) implies i = j )