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contradiction ;
contradiction ;
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thesis ;
thesis ;
contradiction ;
contradiction ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
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thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
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contradiction ;
contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
thesis ;
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contradiction ;
thesis ;
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thesis ;
contradiction ;
thesis ;
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thesis ;
contradiction ;
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contradiction ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
B ;
a <> c
T c= S
D c= B
c in X ;
b <> c ;
let X ;
b in D ;
x = e ;
let m ;
h is onto ;
N in K ;
let i ;
j = 1 ;
x = u ;
let n ;
let k ;
y in A ;
let x ;
let x ;
m c= y ;
F is one-to-one ;
let q ;
m = 1 ;
1 < k ;
G is finite ;
b in A ;
d divides a ;
i < n ;
s <= b ;
b in B ;
let r ;
B is one-to-one ;
R is total ;
x = 2 ;
d in D ;
let c ;
let c ;
b = Y ;
0 < k ;
let b ;
let n ;
r <= b ;
x in X ;
i >= 8 ;
let n ;
let n ;
y in f ;
let n ;
1 < j ;
a in L ;
C is open ;
a in A ;
1 < x ;
S is finite ;
u in I ;
z divides z ;
x in V ;
r < t ;
let t ;
x c= y ;
a <= b ;
m in NAT ;
assume f is differentiable ;
not x in Y ;
z = + \infty ;
let k be Nat ;
K is being_line ;
assume n >= N ;
assume n >= N ;
assume X is \equiv ;
assume x in I ;
q is 0 ;
assume c in x ;
1-r > 0 ;
assume x in Z ;
assume x in Z ;
1 <= k12 ;
assume m <= i ;
assume G is finite ;
assume a divides b ;
assume P is closed ;
1-r > 0 ;
assume q in A ;
W is not bounded ;
f is one-to-one ;
assume A is dense ;
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 odd ;
assume i in I ;
assume 1 <= k ;
X is not empty ;
assume x in X ;
assume n in M ;
assume b in X ;
assume x in A ;
assume T c= W ;
assume s is negative ;
b `1 <= c `1 ;
A meets W ;
i `1 <= j `1 ;
assume H is universal ;
assume x in X ;
let X be set ;
let T be tree ;
let d be element ;
let t be element ;
let x be element ;
let x be element ;
let s be element ;
k <= D-2 ;
let X be set ;
let X be set ;
let y be element ;
let x be element ;
P [ 0 ]
let E be set , f be FinSequence of E ;
let C be Category ;
let x be element ;
let 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 cod ;
Q halts_on s , P ;
x in ] ;
M < m + 1 ;
T2 is open ;
z in b relation ;
R2 is well-ordering ;
1 <= k + 1 ;
i > n + 1 ;
q1 is one-to-one ;
let x be .. ;
P3 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_on Z ;
let x0 ;
let E be Ordinal ;
o symbol of 4 ;
O <> O ;
let r be Real ;
let f be FinSequence ;
let i be Nat ;
let n be Nat ;
Cl A = A ;
L c= card L ;
A /\ M = B ;
let V be complex RealUnitarySpace , W be Subset of V ;
not s in Y |^ 0 ;
rng f <= w ;
b "/\" e = b ;
m = m1 ;
t in h . D ;
P [ 0 ] ;
assume z = x * y ;
S . n is bounded ;
let V be RealLinearSpace , W be Subset of V ;
P [ 1 ] ;
P [ {} ] ;
C1 is connected ;
H = G . i ;
1 <= i + 1 ;
F . m in A ;
f . o = o ;
P [ 0 ] ;
a-f <= b-a ;
R [ 0 ] ;
b in f .: X ;
assume q = q2 ;
x in [#] V ;
f . u = 0 ;
assume e1 > 0 ;
let V be RealUnitarySpace , W be Subspace of V ;
s is trivial & s is trivial ;
dom c = Q ;
P [ 0 ] ;
f . n in T ;
N . j in S ;
let T be complete LATTICE , f be Function of T , S ;
the object of F is one-to-one ;
sgn x = 1 ;
k in Seg len a ;
1 in Seg 1 ;
rng f = X ;
len T in X ;
\Vert < n ;
Smax is bounded ;
assume p = p2 ;
len f = n ;
assume x in P1 ;
i in dom q ;
let U ;
p1 = c ;
j in dom h ;
let k ;
f | Z is_differentiable_on Z ;
k in dom G ;
UBD C = B ;
1 <= len M ;
p in LSeg ( x , y ) ;
1 <= j1 & j1 <= width G ;
set A = <^ A , B , C , D \rbrace ;
card a [= c ;
e in rng f ;
cluster B \oplus A -> empty ;
H is_has has has \textit { H } ;
assume x0 <= m ;
T is increasing ;
e2 <> e1 ;
Z c= dom g ;
dom p = X ;
H is proper implies H is proper
i + 1 <= n ;
v <> 0. V
A c= conv A ;
S c= dom F ;
m in dom f ;
X1 be set ;
c = sup N ;
R is connected & R is connected implies R 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= { C1 } ;
-5 <> {} ;
let x be Element of Y ;
let f be Fin\infty Relation , g be \widetilde of C ;
not n in Seg 3 ;
assume X in f .: A ;
assume that p <= n and n <= m ;
assume not u in { v } ;
d is Element of A ;
A |^ b misses B ;
e in v . ( len G ) ;
- y in I ;
let A be non empty set , f be FinSequence of A ;
P0 = 1 ;
assume r in F . k ;
assume f is simple _net ;
let A be Ncountable set ;
rng f c= NAT ;
assume P [ k ] ;
{ f } <> {} ;
let o be Ordinal ;
assume x is sum of squares ;
assume not v in { 1 } ;
let I ;
assume that 1 <= j and j < l and j < l ;
v = - u ;
assume s . b > 0 ;
d in dom 4 ;
assume t . 1 in A ;
Y be non empty TopStruct , X be non empty TopSpace ;
assume a in ]. s , t .[ ;
let S be non empty RelStr ;
a , b // b , a ;
a * b = p * q ;
assume x , y // the carrier of X ;
assume x in [#] ( f ) ;
[ a , c ] in X ;
FF <> {} ;
M + N c= M + N ;
assume M is connected & M is connected ;
assume f is with_B-PCCCCCCCCCCCCCCCCCCCCCCC
let x , y ;
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 + 1 ;
m + 1 < n + 1 ;
s in S \/ { s } ;
n + i >= n + 1 ;
assume Re ( y ) = 0 ;
k1 <= j1 & k2 <= j1 ;
f | A is continuous ;
f . x <= b ;
assume y in dom h ;
x * y in B1 ;
set X = Seg n ;
1 <= i2 + 1 ;
k + 0 <= k + 1 ;
p ^ q = p ^ q ;
j |^ ( y -' 1 ) divides m ;
set m = max ( A , B ) ;
[ x , x ] in R ;
assume x in succ 0 ;
a in sup \varphi ;
{ C } is connected ;
q2 c= C1 \/ C2 ;
a2 < c2 ;
s2 is 0 -started ;
IC s = 0 ;
s3 = s2 ;
let V ;
let x , y ;
let x be Element of T ;
assume a in rng F ;
x in dom T ` ;
let S be Subset-Family of L ;
y " <> 0 ;
y " <> 0 ;
0. V = uw ;
y2 , y1 , y2 is_collinear ;
R1 is connected ;
let a , b be Real , b be Real ;
let a be Object of C ;
let x be Vertex of G ;
let o be Object of C , a , b be element ;
r '&' q = P \lbrack l , P \rbrack ;
let i , j ;
let s be State of A , A be Subset of S ;
seq . n = N . n ;
set y = ( x `1 ) ^2 ;
NAT in dom g ;
l . 2 = y1 ;
|. g . y .| <= r ;
f . x in { C } ;
V is not empty iff V is not empty
let x be Element of X ;
0 <> f . ( g . g ) ;
f2 /* q is convergent ;
f . i is_measurable_on E ;
assume \varphi in N ;
reconsider i = i as Ordinal ;
r * v = 0. X ;
rng f c= REAL ;
G = 0 .--> 0 ;
let A be Subset of X ;
assume { A } is open & A is open ;
|. f . x .| <= r ;
let x be Element of R ;
let b be Element of L ;
assume x in W1 ;
P [ k , a ] ;
let X be Subset of L ;
let b be element ;
let A , B be category ;
set X = Vars ( V , C ) ;
let o be OperSymbol of S ;
let R be connected non empty Poset ;
n + 1 = succ n ;
{ x } c= Z ;
dom f = C1 ;
assume [ a , y ] in X ;
Re ( seq ) is convergent ;
assume a1 = b1 & b1 = b2 ;
A = ssA ;
a <= b or b <= a ;
n + 1 in dom f ;
let F be sequence of S , s be sequence of S ;
assume r2 > x0 ;
Y be non empty set , f be Function of Y , BOOLEAN ;
2 * x in dom W ;
m in dom g2 ;
n in dom ( g1 * g2 ) ;
k + 1 in dom f ;
the still not bound not bound ( { s } ) is finite ;
assume x1 <> x2 ;
v1 in { V } ;
[ b `1 , b ] in T ;
x9 + 1 + 1 = i ;
T c= Hom ( T ) ;
( l ) `1 = 0 ;
let n be Nat ;
( t `2 ) ^2 = r ;
A8 : ex A st A is_integrable_on M & for n st n <= n holds ex m st n <= m holds ( ex n st n <= m holds ( ( M
set t = "/\" _ { t } ;
let A , B be real-membered set ;
k <= len G + 1 ;
{ C } misses { V } ;
product ( G ) is not empty ;
e <= f or f <= e ;
cluster -> ordinal for sequence of X ;
assume c2 = b ;
assume h in [. q , p .] ;
1 + 1 <= len C ;
not c in B . ( m1 + 1 ) ;
cluster R .: X -> empty ;
p . n = H . n ;
assume seq is Cauchy ;
IC s3 = 0 ;
k in N or k in K ;
F1 \/ F2 c= F ;
Int G1 <> {} ;
( z `1 ) ^2 = 0 ;
p1 <> p3 ;
assume z in { y , z } ;
MaxADSet ( a ) c= F ;
ex_sup_of { s } , S ;
f . x <= f . y ;
let T be continuous non empty TopSpace ;
q |^ m >= 1 ;
a >= X & b >= X ;
assume <* a , c *> <> {} ;
F . c = g . c ;
G is one-to-one & G is one-to-one ;
A \/ { a } c= B ;
0. V = 0. Y ;
I be Instruction of S , s be State of S ;
( ( reproj ( i , x ) . i ) . x = 1 ;
assume z \ x = 0. X ;
C4 = 2 |^ ( n + 1 ) ;
let B be sequence of Omega ;
assume X1 = p .: D ;
n + k2 in NAT ;
f " P is compact ;
assume x1 in REAL + ( REAL * f ) ;
p1 = K & K = ( K + L ) `2 ;
M . k = <*> REAL ;
\varphi . 0 in rng \varphi ;
OSMMMA is closed
assume z <> 0. L ;
n < N . ( k + 1 ) ;
0 <= seq . ( 0 ) ;
- q + p = v ;
{ v } is Subset of B ;
set g = f /. ( len f ) ;
{ R } is stable Subset of R ;
set \cal R = Vertices R , R = G \ { {} } ;
{ p } c= { P } ;
x in [. 0 , 1 .] ;
f . y in dom F ;
let T be continuous TopStruct ;
inf the carrier of S = S ;
downarrow a = downarrow b ;
P , C , D is_collinear ;
assume x in LSeg ( s , t ) ;
2 |^ i < 2 |^ m ;
x + z = x + z ;
x \ ( a \ x ) = x ;
||. \mathopen { \Vert } x-y .|| <= r ;
assume Y c= field Q & Y <> {} ;
a \times b = a ;
assume a in A . i ;
k in dom ( q | ( len q ) ) ;
p is FinSequence of S ;
i -' 1 = i - 1 ;
f | A is one-to-one ;
assume x in f .: { X } ;
i2 - 1 = 0 ;
j2 + 1 <= i2 ;
g " * a in N ;
K <> { [ {} , {} ] } ;
cluster -> with_D_net for Relation ;
|. q .| ^2 > 0 ;
|. p2 .| = |. p .| ;
s2 - s1 > 0 ;
assume x in { G } ;
min ( C , n ) in C ;
assume x in { G } ;
assume i + 1 = len G ;
assume i + 1 = len G ;
dom I = Seg n ;
assume that k in dom C and k <> i ;
1 + 1-1 <= i + 1-1 ;
dom S = dom F ;
let s be Element of NAT ;
let R be ManySortedSet of A ;
let n be Element of NAT ;
let S be non empty non void ManySortedSign ;
let f be ManySortedSet of I ;
let z be Element of COMPLEX , f be FinSequence of COMPLEX ;
u in { |[ g , g ]| : g < g } ;
2 * n < 2 * ( n + 1 ) ;
x , y // x , y ;
{ B } c= V ;
assume I is_closed_on s , P ;
U = { U } ;
M /. 1 = z /. 1 ;
x9 = { x9 } ;
i + 1 < n + 1 ;
x in { {} , {} } ;
{ f } <= { f . ( len f + 1 ) } ;
let l be Element of L ;
x in dom ( ( F | X ) | X ) ;
let i be Element of NAT ;
{ r } is ( { r } ) -valued ;
assume <* o , o *> <> {} ;
s . x |^ 0 = 1 ;
card ( K . ( len K ) ) in M ;
assume X in U & Y in U ;
let D be \hbox { - } ;
set r = ]. k + 1 , k + 1 .[ ;
y = W . ( 2 * x ) ;
assume dom g = cod f ;
let X , Y be non empty TopSpace , f be Function of X , Y ;
x \oplus A is interval ;
|. <*> A .| . a = 0 ;
cluster subSubSublattice of L -> strict for Subspace of L ;
a1 in B . ( s1 . a ) ;
let V be finite VectSp of F , W be Subspace of V ;
A * B on B ;
{ f } = { NAT } --> 0 ;
A , B , C is_collinear ;
z1 = P1 . j & z2 = P2 . j ;
assume f " ( P ) is closed ;
reconsider j = i as Element of M ;
a , b // a , b ;
assume q in A \/ B ;
dom ( F * G ) = o ;
set S = { { 0 } } ;
z in dom ( A --> y ) ;
P [ y , h . y ] ;
{ x0 } c= dom f ;
B be ManySortedSet of I , A be ManySortedSet of I ;
sqrt ( 2 * PI * PI ) < Arg z ;
reconsider z1 = 0 as Nat ;
LIN a , d , c ;
[ y , x ] in [: I , I :] ;
( Q ) `2 = 0 ;
set j = { x } div m ;
assume a in { x , y } ;
j1 - 1 > 0 ;
I \! \mathop { + } = 1 ;
[ y , d ] in F ;
let f be Function of X , Y ;
set A2 = ]. B , C .[ ;
s1 , s2 , s1 is_collinear & s2 , s2 is_collinear ;
j1 -' 1 = 0 ;
set k2 = 2 * n + j ;
reconsider t = t as bag of n ;
I . j = m . j ;
i |^ s , n are_relative_prime ;
set g = f | ( len f ) ;
assume X is lower & 0 <= r ;
( p1 `1 ) ^2 = 1 ;
a < ( p3 `1 ) ^2 + ( p3 `2 ) ^2 ;
L \ { m } c= UBD C ;
x in Ball ( x , 10 ) ;
not a in LSeg ( c , m ) ;
1 <= i1 -' 1 ;
1 <= i1 -' 1 ;
i + i2 <= len h ;
x = E-max ( P ) ;
[ x , z ] in X \times Z ;
assume y in [. x0 , x0 + r .[ ;
assume p = <* 1 , 2 , 3 , 4 , 5 *> ;
len <* A1 *> = 1 ;
set H = h . ( g . ( g . ) ) ;
card b * a = |. a .| ;
Shift ( w , 0 ) |= w ;
set h = h2 .: { h } ;
assume x in X1 /\ X2 ;
||. h .|| < d ;
not x in the carrier of ( f | X ) ;
f . y = F ( y ) ;
for n holds X [ n ] ;
k -' l = kl ;
<* p , q *> /. 2 = q ;
let S be Subset of Y ;
P , Q , t is_collinear ;
Q /\ M c= ( F | M ) /\ M
f = b * ( ( card S ) - 1 ) ;
let a , b , c be Element of G ;
f .: X <= f . ( sup X ) ;
let L be non empty RelStr , X be Subset of L ;
SO is x -to_power i ;
let r be non negative Real ;
M |= { v } ;
v + w = 0. V ;
P [ len F ] ;
assume InsCode ( i ) = 8 ;
the zero of M = 0 ;
cluster z * seq -> summable ;
let O be Subset of the carrier of C ;
|. f .| is continuous ;
x2 = g . ( j + 1 ) ;
cluster xx -> non empty Element of BOOLEAN ;
reconsider l1 = ll as Nat ;
{ v } is Vertex of G ;
TT is SubSpace of ( ( TOP-REAL 2 ) | P ) ;
Q /\ Q <> {} ;
let k be Nat ;
q " is Element of X ;
F . t is w.r.t. M ;
assume n <> 0 & n <> 1 ;
set d1 = EmptyBag n , d2 = EmptyBag n ;
let b be Element of Bags n ;
assume for i holds b . i is commutative ;
x is root of ( TOP-REAL 2 ) | K1 ;
not r in ]. p , q .[ ;
let R be FinSequence of REAL ;
S7 not empty & not ( b1 , b2 , b3 is_collinear ) & b1 <> b2 & b1 , b3 is_collinear & b1 <> b3 & b1 <> b3 & b1 <> b3 & b1 <> b3 & b1
IC SCM R <> a ;
|. p - [ x , y ] .| >= r ;
1 * ( s - t ) = s - t * ( s - t ) ;
let x be FinSequence of NAT ;
let f be Function of C , D ;
for a , b being Element of L holds 0. L + b = a
IC s = s . NAT .= ( IC s ) ;
H + G = FH + ( G ) ;
{ C . x } . x = x2 . x ;
f1 = f .= f2 .= f1 .= f2 ;
Sum <* p . 0 *> = p . 0 ;
assume v + W = v + W ;
{ a1 , a2 } = { a1 , a2 } ;
a1 , b1 // b , c ;
d , o // o , d ;
I1 is transitive & I is transitive & I is transitive & I is transitive & I is transitive & I is transitive & I is transitive & I is transitive & I is transitive & I is transitive & I is transitive &
I is antisymmetric & I is transitive & I is transitive & I is transitive & I is transitive & I is transitive & I is transitive & I is transitive & I is transitive & I is transitive & I is transitive &
upper_bound ( ( H ) .: { H } ) = e ;
x = a1 * a2 + a3 * a1 + a3 * a2 * a3 ;
|. p1 .| ^2 + 1 >= 1 ;
assume i2 -' 1 < i2 -' 1 ;
rng s c= dom ( f1 (#) f2 ) ;
assume support a misses { b } ;
let L be associative non empty multMagma , n be Element of NAT ;
s " + 0 < n + 1 ;
p . c = ( f " ) . c ;
R . n <= R . ( n + 1 ) ;
Directed ( I ) = ( card I + 2 ) ;
set f = + ( x , y ) ;
cluster Ball ( x , r ) -> bounded ;
consider r being Real such that r in A ;
cluster non empty NAT -defined NAT -defined NAT -defined NAT ;
let X be non empty directed Subset of S ;
S be non empty full relational substructure of L ;
cluster <* L1 . N , L1 . N *> -> complete ;
sqrt ( 1 - a ) ^2 = a ;
( q . {} ) `2 = o ;
( i -' 1 ) - 1 > 0 ;
assume sqrt ( 1 - 2 ) <= t `1 ;
card B = k + 1 ;
x in union rng ( f | X ) ;
assume x in the carrier of R ;
d in X ;
f . 1 = L . ( F . 1 ) ;
the vertices of G = { v } ;
let G be Go-board ;
e , f , g , h is_collinear ;
c . ( i1 + 1 ) in rng c ;
f2 /* q is divergent_to+infty & f2 /* q is convergent ;
set z1 = - ( z1 - z2 ) , z2 = - z2 + z2 - z1 - z2 ;
assume w is \mathbin { l-to to w ;
set f = p \! \mathop { t } ;
let c be Object of C ;
assume pred P [ a ] means P [ a ] ;
let x be Element of REAL m m m ;
let I1 be Subset-Family of X ;
reconsider p = p `1 as Element of NAT ;
v , w as Point of X ;
let s be State of SCM+FSA , a be Int-Location ;
p is FinSequence of NAT ;
stop I c= P & card I c= card I ;
set ci = ( f /. i ) `1 ;
w ^ t ^ w ^ w ^ w ^ t ^ w ^ w ^ t ^ w ^ t ^ w ^ w ^ t ^ w ^ t ^ w ^ w ^ t ^ w ^ w ^
W1 /\ W = W1 /\ ( W2 /\ V ) ;
f . j is Element of J . j ;
let x , y be Element of T2 ;
ex d st a , b // b , d ;
a <> 0 & b <> 0 implies c = 0
ord x = 1 & x is 0. X ;
set g2 = lim ( seq ) , g1 = ( lim seq ) * ( ( seq ) ) " ) ;
2 * x >= 2 * x ;
assume ( a 'or' c ) . z <> TRUE ;
f \circ g in Hom ( c , d ) ;
Hom ( c , c + d ) <> {} ;
assume 2 * ( Sum ( q | m ) ) > m ;
L1 . ( L1 . ( L1 . ( L1 . ( L1 . j ) ) ) = 0 ;
indx ( X \/ R1 ) \/ R1 = R1 \/ R2 ;
( ( ( ( ( - 1 ) (#) ( ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1
( ( ( ( ( ( - 1 ) (#) ( ( ( ( - 1 ) (#) ( ( ( ( ( - 1 ) (#) ( ( ( ( - 1 ) (#) ( ( ( ( ( - 1 ) /
o1 in { X where X is Subset of L : X in O & X c= O } ;
e , f , g , h is_collinear ;
s3 > sqrt ( 1 - 2 ) * ( 1 - 2 ) ;
x in P .: ( F " { x } ) ;
let J be non empty TopSpace , I be Subset of R ;
h . p1 = f2 . O ;
Index ( p , f ) + 1 <= j ;
len ( q ) = width M & len ( q @ ) = width M ;
the carrier of K c= A ;
dom f c= union rng ( F | X ) ;
k + 1 in Seg ( n + 1 ) ;
let X be ManySortedSet of the carrier of S ;
[ x `1 , y ] in \sqcup ( R ~ ) ;
i = D1 or i = D2 . i ;
assume a mod n = b mod n ;
h . x2 = g . x2 ;
F c= 2 |^ ( n + 1 )
reconsider w = |. seq .| as sequence of X ;
sqrt ( 1 / m ) < p ;
dom f = dom ( I * J ) ;
[#] ( ( TOP-REAL 2 ) | K1 ) = K1 ;
cluster - x -> -> -> -> -> -> -> zero ;
then { d } c= A ;
cluster { p1 } -> finite-ind ;
w be Element of M ;
let x be Element of REAL-NS n ;
u in W1 & v in W2 implies u in W2
reconsider y1 = y as Element of L2 ;
N is full relational structure of T ;
ex_sup_of { x , y } , L ;
g . n = n |^ ( n + 1 ) .= n ;
h . J = EqClass ( u , J ) . ( J ) ;
seq be \mathbin { \uparrow } k is convergent & seq is convergent implies seq is convergent
dist ( x , y ) < r / 2 ;
reconsider mm = m as Element of NAT ;
x0 < r1 - x0 & r1 < r2 & r2 < x0 + r2 ;
reconsider P = P ` as strict Subgroup of N ;
set g1 = p * ( q , i ) ;
let n , m , k , m be Nat ;
assume that 0 < e and f | A is bounded and f | A is bounded ;
D2 . ( I + 1 ) in { x } ;
cluster Subset of T -> subopen ;
P be compact non empty Subset of TOP-REAL 2 ;
G * ( 1 , j ) `2 in LSeg ( G * ( 1 , j ) , G * ( 1 , j ) ) ;
let n be Element of NAT , m be Nat ;
reconsider S8 = S as Subset of T ;
dom ( i .--> X ) = { i } ;
X be non-empty ManySortedSet of I ;
X be non-empty ManySortedSet of I ;
op ( {} ) c= { [ {} , {} ] } ;
reconsider m = i-1 as Element of NAT ;
reconsider d = x as Element of C ;
let s be 0 -started State of SCMPDS , P be s , Q be Initialize s , t be State of SCMPDS ;
let t be 0 -started State of SCMPDS ;
b , a // b , c ;
assume i = n \/ { n } & j = k \/ { n } ;
f be PartFunc of X , Y ;
N >= sqrt ( c * ( sqrt ( c ^2 - sqrt ( c ^2 - d ) ) ) ;
reconsider tt = T as Point of T ;
set q = h * p ^ <* d *> ;
z2 in U ( ) /\ Q ( ) ;
A |^ 0 = { <* <* E *> *> , <* E *> } ;
len W2 = len W + len W ;
len ( f2 ^ g2 ) in dom f2 ;
i + 1 in Seg ( len s2 ) ;
z in dom ( ( g1 * ( f | X ) ) | X ) ;
assume p2 = |[ - 1 , 1 ]| ;
len G + 1 <= i1 + 1 ;
f1 * f2 is convergent & f2 is convergent & f2 is convergent & f2 is convergent & f2 is convergent & f2 is convergent & f2 is convergent & f2 is convergent & f2 is convergent & f2 is convergent & f2 is convergent & f2
cluster seq + seq1 -> summable for Real_Sequence ;
assume j in dom ( ( M * ( i , j ) ) | ( i , j ) ) ;
let A , B , C , D , E , F , G , J , F , J , J , F , J , F , J , F , J , J , F , J , J , F , J ,
x , y // x , y & x , y // y , z ;
b ^2 - ( 4 * a ) >= 0 ;
<* xy *> ^ <* y *> reduces x , y ;
a , b // a , b ;
len p2 is Element of NAT ;
ex x being element st x in dom R & R [ x , y ] ;
len q = len ( K * G ) ;
s1 = Initialize s1 .= s2 ;
consider w being Nat such that q = z + w ;
x ` is Element of L ;
k = 0 & n <> 0 or k > 0 ;
then X is discrete ;
for x st x in L holds x is finite
||. f /. c .|| <= r1 * ( ||. c .|| ) ;
c in ]. p , q .[ & not c in { p } ;
reconsider V = V as Subset of the topology of T ;
N , M is_Re_of L ;
then z >= \twoheaddownarrow x ;
M = f & M = g implies M = g
( ( ( ( ( ( m , 1 ) , 1 ) , 1 ) ) `1 ) `1 = TRUE ;
dom g = dom f /\ ( dom f /\ ( dom g \ { 0 } ) ) ;
{ A } is with_constant ;
[ i , j ] in Indices M ;
reconsider s = x " as Element of H ;
let f be Element of dom Subformulae p ;
F1 . ( a1 , b1 ) = G1 . ( a1 , b1 ) ;
cluster AffineMap ( a , b , r , s ) -> compact ;
let a , b , c , d be Real ;
rng s c= dom ( ( f2 * f1 ) `| Z ) ;
[: F , G :] is additive additive ;
set k2 = card ( B \ { x } ) ;
set G = coprod ( X ) ;
reconsider a = [ x , s ] as Object of G ;
let a , b be Element of [: M , M :] , M ;
reconsider s1 = s as Element of S1 ;
rng p c= the carrier of L ;
let d be Subset of the carrier of A ;
( x | x ) | ( ( len x ) | ( len x ) ) = 0. V ;
I1 . ( len I + 1 ) in dom stop I ;
let g be continuous Function of X , Y ;
reconsider D = Y as Subset of TOP-REAL n ;
reconsider i1 = len p1 as Element of NAT ;
dom f = the carrier of S ;
rng h c= union ( the carrier of L ) ;
cluster All ( x , H ) -> All of x , H ;
d * N > N * ( N * ( N * ( i , j ) ) ) ;
]. a , b .[ c= [. a , b .] ;
set g = f " ( dom ( f | ( dom g ) ) ) ;
dom ( p | ( m + 1 ) ) = REAL ;
3 + 2 <= k + 2 ;
the carrier of ( ( TOP-REAL 2 ) | K1 ) = K1 ;
x in rng ( f /^ n ) ;
f , g be FinSequence of D ;
[ p , q ] in the InternalRel of S1 & [ p , q ] in the InternalRel of S2 ;
rng f " { 0 } = dom f ;
( the Target of G ) . e = v ;
width G -' 1 < width G ;
assume v in rng ( S | E ) ;
assume x is root or x is root or x is root or x is root or x is root or x is root ;
assume 0 in rng ( g2 | A ) ;
let q be Point of TOP-REAL 2 ;
let p be Point of TOP-REAL 2 ;
dist ( O , u ) <= |. p2 .| + 1 ;
assume dist ( x , b ) < dist ( a , b ) ;
<* S *> is in the carrier of C & <* S *> is the carrier of C ;
i <= len G -' 1 ;
let p be Point of TOP-REAL 2 ;
x1 in the carrier of I[01] & x2 in the carrier of ( TOP-REAL 2 ) | K1 ;
set p1 = f /. ( i + 1 ) ;
g in { g2 : r < g2 & g2 < x0 } ;
Q2 = Sn8 \/ ( Q /\ R ) ;
( ( ( 1 / 2 ) (#) ( ( ( ( ( 1 / 2 ) (#) ( ( ( ( ( 2 ) ) (#) ( ( ( ( ( 2 ) ) (#) ( ( ( ( ( 2 ) ) ) ) ) )
- p + I c= - p + I ;
n < LifeSpan ( P1 , s1 ) + 1 ;
CurInstr ( p1 , s1 ) = i ;
A /\ { x } <> {} ;
rng f c= ]. r , s .[ ;
let g be Function of S , V ;
let f be Function of L1 , L2 ;
reconsider z = z as Element of [: L , L :] ;
let f be Function of S , T ;
reconsider g = g as Morphism of c , d ;
[ s , I ] in S \times A ;
len ( the connectives of C ) = 4 ;
let C1 , C2 be subfunctor of C1 , C2 , C1 , C2 be subfunctor from C1 , C2 ;
reconsider V = V as Subset of X | B ;
attr p is valid means : Def3 : p in rng p ;
assume X c= dom f & X c= dom g implies f .: X c= dom g
H |^ a is Element of H |^ a ;
A1 be Element of O , A2 be Element of O ;
p2 , q2 is_collinear & p2 , p3 is_collinear & p2 , q2 is_collinear & p2 , p3 is_collinear & q , q2 is_collinear & q , q2 is_collinear & q <> q2 & q <> q2 & q <> q2 & q <> q2 & q <> q2 &
consider x being element such that x in v ^ <* x *> ;
not x in { 0. TOP-REAL 2 } ;
p in [#] ( ( TOP-REAL 2 ) | K1 ) ;
0 ( REAL ) < M . ( E ) ;
^ ( c ^ d ) = c ^ d ;
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . ( ( ( ) ) | ( dom F ) ) ) ;
cluster with_Mgenerated ' -> with_M) ;
set i1 = the Element of NAT ;
let s be 0 -started State of SCM+FSA ;
assume y in ( f1 \/ f2 ) .: A ;
f . ( len f ) = f /. ( len f + 1 ) ;
x , f . x // f . ( x , y ) ;
attr X c= Y means : only : cos X c= Y ;
y be upper Subset of Y , x , y be Element of Y ;
cluster ( x `1 ) ^2 -> non empty finite sequence of elements of D ;
set S = <* Bags n , <* an *> , <* an *> *> ;
set T = [. 0 , PI / 2 .] ;
1 in dom mid ( f , 1 , len f ) ;
sqrt ( 4 * PI * PI ) < sqrt ( 2 * PI * PI * PI ) ;
x2 in dom ( ( f1 + f2 ) `| Z ) /\ dom ( ( f1 + f2 ) `| Z ) ;
O c= dom I & { O } c= { O } ;
( the Target of G ) . ( x , y ) = v ;
{ HT ( f , T ) } c= Support f ;
reconsider h = R . k as Polynomial of n , L ;
ex b being Element of G st y = b * H ;
let x , y , z be Element of G ;
h1 . i = f . ( h . i ) ;
( p `1 ) ^2 = ( ( p `1 ) ^2 + ( p `2 ) ^2 ) ^2 ;
i + 1 <= len Cage ( C , n ) ;
len <* P *> = len P + len <* P *> ;
set NN = the Element of the CastNode ( N , v ) ;
len g\mathbin -' ( x + 1 ) + 1 <= len g-1 ;
a on B & b on B & b on C & c on D & d on D & d on D & d on D & d on D & d on D ;
reconsider rr = r * I . ( v + 1 ) as FinSequence of K ;
consider d such that x = d and a [= d and a [= d ;
given u such that u in W and x = v + u ;
len f -' n = len Rev <* n *> ;
set q2 = Cage ( C , n ) , q2 = Cage ( C , n ) /. ( i + 1 ) ;
set S = <* [ S1 , S2 ] , S2 ] , S2 = <* [ S1 , S2 ] , S2 = <* S2 , S2 , S2 ] , S2 = [ S1 , S2 ] , S2 = [ S1 , S2 ] , S2 = [
MaxADSet ( b ) c= MaxADSet ( b ) /\ MaxADSet ( b ) ;
Cl ( ( q . ( q1 . n ) ) c= F . ( ( q . n ) `1 ) ;
f " ( D ) meets h " ( D ) ;
reconsider D = E as non empty directed Subset of L1 ;
H = ( H '&' ( H ) ) => ( H '&' ( H ) ) ;
assume t is Element of [: { F } , { F } :] ;
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 . ( a1 , b1 ) ;
the carrier of G = { E } \/ { E } ;
reconsider m = len fk as Element of NAT ;
set S1 = LSeg ( n , d ) ;
[ i , j ] in Indices ( M1 @ ) ;
assume P c= Seg m & P is Seg m ;
for k st m <= k holds z in K . ( k + 1 )
consider a being set such that p in a and a in G ;
L1 . p = p * ( 1 - p ) ;
p2 . i = p1 . i .= p1 . i ;
let PA , G , PA , PA , G be Subset of Y ;
attr 0 < r & r < 1 & r < 1 ;
rng \langle a , b *> = [#] X ;
reconsider x = x , y = y as Element of K ;
consider k such that z = f . k and n <= k ;
consider x being element such that x in X \ { p } ;
len ( ( s | ( len s ) ) = card ( rng s ) ;
reconsider x2 = x1 as Element of L1 ;
Q in { the topology of X where X is Subset of X : X in F } ;
dom ( ( ( ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) (
attr n divides m & n divides m ;
reconsider x = x as Point of I[01] ;
a in \mathop { \rm Point of T2 , T2 be Point of T2 ;
not y in the still not bound ( f ) & not y in rng f implies not y in rng f
Hom ( a , b ) /\ Hom ( b , c ) <> {} ;
consider k1 such that p " < k1 and k1 < k1 & k1 < n ;
consider c , d such that dom f = c \ d ;
[ x , y ] in dom g \times ( k + 1 ) ;
set S1 = [ <* x , y *> , f1 ] ;
s3 = k2 & s3 = k2 & s3 = k2 & s3 = k2 & s3 = k2 & s3 = k2 & s3 = k2 & s3 = k2 & s3 = k2 & s3 = k2 & s3 = k2 & s3 = k2 & s3 = k2 & s3 = k2 &
x0 in dom ( u + v ) /\ dom ( v + u ) ;
reconsider p = x as Point of TOP-REAL 2 ;
reconsider \mathbb I = REAL n as Subset of REAL n ;
f . p3 <= f . p3 ;
( ( F . n ) `1 <= ( F . n ) `1 ;
( x `1 ) ^2 = ( ( ( W ) ^2 + ( W ) ^2 ) ^2 ) ^2 + ( W `1 ) ^2 ;
for n being Element of NAT holds P [ n ] ;
J , K , L , L , R be Subset of Y ;
assume 1 <= i & i <= len <* a *> ;
0 |-> a = <*> the carrier of K ;
X . i in 2 |^ ( i + 1 ) \ B . i ;
<* 0 *> in dom ( e --> [ 1 , 0 ] ) ;
then P [ a ] ;
reconsider cin = |[ |[ ]| , |[ ]| , |[ ]| , |[ ]| ]| ]| `1 ]| `1 ;
( i -' 1 ) - 1 <= len w- 1 ;
[#] S c= [#] T ;
let V being strict Subspace of V ;
assume k in dom mid ( f , i , j ) ;
let P be non empty Subset of TOP-REAL 2 ;
let A , B be Matrix of n , m , k , m , n , k , Nat ;
- a * b = a * b ;
let A be Subset of [: A , B :] ;
id ( o ) in <* o , o *> ;
then ||. x .|| = 0 & x = 0. X ;
let N , H be strict Subgroup of G ;
j >= len ( g , D1 ) ;
b = Q . ( len Q + 1 ) .= Q . ( len Q + 1 ) ;
f2 * f1 /* seq is convergent ;
reconsider h = f * g as Function of I[01] , TOP-REAL 2 ;
assume that a <> 0 and delta ( a , b , c , d ) >= 0 ;
[ t , t ] in the InternalRel of A ;
( v |-- E ) | ( n + 1 ) is Element of T ;
{} = the carrier of L1 + ( the carrier of L1 ) .= the carrier of L1 ;
Directed I is_closed_on s , P , s ) ;
Initialized p = Initialize ( p +* q ) ;
reconsider N2 = N as strict net of [: N , N :] ;
reconsider Y = Y as Element of \langle \mathop { \rm to } ( L ) , \subseteq \rangle ;
"/\" ( { p } , L ) <> p ;
consider j being Nat such that i2 = i1 + j + 1 and j <= len z ;
[ s , 0 ] in the InternalRel of S2 & [ s , 0 ] in the InternalRel of S2 ;
m in ( B /\ C ) \ { {} } ;
n <= len ( ( P . ( len P ) + 1 ) + 1 ) ;
( x1 , x2 ) `2 = ( x2 , x3 ) `2 ;
InputVertices S = { x , y , z } ;
let x , y be Element of FFFFFFFFFFFFare } ;
p = [ p `1 , p `2 ] ;
g * h = h " * g ;
let p , q be Element of PFuncs ( V , C ) ;
x0 in dom ( ( x1 - x2 ) / ( n + 1 ) ) /\ dom ( ( x1 - x2 ) / ( n + 1 ) ) ;
( R qua Function ) " = R " * ( R qua Function ) ;
n in Seg len ( ( f /^ ( n + 1 ) ) | ( i + 1 ) ) ;
for s be Real st s in R holds s <= s
rng s c= dom ( ( f2 * f1 ) `| Z ) ;
synonym subsets X for non empty Subset of X ;
1_ K * ( ( 1_ K ) * ( ( 1_ K ) * ( ( 1_ K ) * ( ( 1_ K ) ) ) ) = 0. K ;
set S = Segm ( A , P , Q ) ;
ex w st e = sqrt ( w , f ) & w in F & w in F ;
curry k . ( n + k ) is convergent ;
cluster open -> open for Subset of T ;
len ( f1 ^ f2 ) = 1 .= len f1 + len f2 .= len f1 + len f2 ;
sqrt ( i * p ) < sqrt ( 2 * p ) ;
let x , y be Element of [: U1 , U2 :] ;
b1 , c1 // b1 , c1 ;
consider p being element such that c1 . j = { p } ;
assume f " { 0 } = {} & f " { 0 } = {} ;
assume IC Comput ( F , s , k ) = n ;
Reloc ( J , card I + card J ) not a ;
goto ( card I + 1 ) not a ;
set m1 = LifeSpan ( p1 , s ) + 1 ;
IC Comput ( P , s , k ) in dom Initialize s ;
dom t = the carrier of R ;
( E-max L~ f ) .. f = 1 ;
let a , b , c be Element of PFuncs ( V , C ) ;
Cl F c= Cl F ;
the carrier of X1 \/ X2 misses the carrier of X1 & the carrier of X2 = the carrier of X2 ;
assume not LIN a , f . ( a , b ) , f . ( a , b ) ;
consider i being Element of M such that i = d and i <= n ;
then Y c= { x } or Y = { x } ;
M , v |= ( ( { y } \leftarrow { x } ) ) . ( ( { y } \leftarrow { x } ) ) . ( ( { y } \leftarrow { x } ) ) . ( { y } ) ) |= H . ( { x }
consider m being element such that m in Intersect ( F ) ;
reconsider A1 = ( support u ) as Subset of X ;
card A \/ B = card ( 2 * ( 2 * ( 1 + 1 ) ) ;
assume a1 <> a3 & a2 <> a3 & a3 <> a4 & a1 <> a4 & a1 <> a4 & a1 <> a4 & a2 <> a4 & a1 <> a4 & a1 <> a4 & a1 <> a4 & a1 <> a4 & a1 <> a4 & a1 <> a4 & a1 <> a4 &
cluster s \! \mathop { - V } -> ( S , V ) -provable ;
L1 /. ( n + 1 ) = L1 . ( n + 1 ) ;
let P be compact non empty Subset of TOP-REAL 2 ;
assume |[ p1 , p2 ]| in LSeg ( |[ - 1 , 1 ]| , |[ 1 , 1 ]| ) ;
let A be non empty Subset of TOP-REAL n , B being Subset of TOP-REAL n , C being Subset of TOP-REAL n st C = { A } & C is compact & C is compact & C is compact & C is compact holds C is compact
assume [ k , m ] in Indices ( D1 ^ D2 ) ;
0 <= ( ( 1 - 2 ) |^ ( n + 1 ) ) * ( ( 1 - 2 ) |^ ( n + 1 ) ) ) ;
( F . N ) . x = + \infty . x ;
attr X c= Y & Z c= Y implies X \ Z c= Y \ Z
( y - z ) * ( ( - z ) * ( y - z ) ) <> 0. F ;
1 + card ( X1 /\ X2 ) <= card X1 + card X2 ;
set g = z /^ ( len z -' 1 ) ;
then k = 1 implies p . k = <* x , y *> . k
cluster C -\mathbin { - } functor -> total for Relation of X , Y ;
reconsider B = A as non empty Subset of TOP-REAL n ;
let a , b , c , d , e , f , g , h be Function of Y , BOOLEAN ;
L1 . i = ( i .--> g ) . i .= g . i ;
Plane ( x1 , x2 , x3 ) c= P ;
n <= indx ( D2 , D1 , j1 ) + 1 ;
( g2 . O ) `2 = - 1 ;
j + 1 <= len f -' 1 ;
set W = \langle W , W , E , F , G , F *> ;
S1 . a = a + e .= a + e ;
1 in Seg width ( M * ( ( ( M , j ) , i ) ) ) ;
dom ( i * ( ( f . i ) * ( ( ( f . i ) * ( ( f . i ) * ( ( f . i ) * ( ( f . i ) * ( ( f . i ) * ( ( f .
\varphi . x = W . ( a , p ) ;
set Q = |= ( g , f ) ;
cluster -> MSsorted for Relation of U1 ;
attr F = { A } means : ex B st F = { A } ;
reconsider reproj = reproj ( i , z ) as Element of product G ;
rng f c= rng ( f1 ^ f2 ) \/ rng ( f2 ^ g2 ) ;
consider x such that x in f .: A and x in f .: A ;
f = <*> the carrier of V & f is one-to-one implies f is one-to-one
E |= All ( x , H ) ;
reconsider n1 = n as Morphism of o1 , o2 ;
assume that P is commutative and R is commutative and R is commutative & P is commutative ;
card ( B \/ { x } ) = card ( B \/ { x } ) ;
card ( x \ B ) = 0 ;
g + R in { s : g-r < s & s < g } ;
set q1 = ( q , <* s *> ) . ( n + 1 ) , q2 = ( q , n ) `2 ;
for x being element st x in X holds x in rng ( f1 | X )
h /. ( i + 1 ) = h . ( i + 1 ) ;
set Gw = max ( B , A ) , R = max ( B , A ) , S = max ( B , A ) , S = max ( A , B ) , T = max ( A , B ) , T = max ( A , B
t in Seg width ( I ^ <* n *> ) ;
reconsider X = dom f as Element of Fin ( V ) ;
IncAddr ( i , k ) = goto l + k ;
( S - ( f /. ( n + 1 ) ) `1 <= ( q `1 ) `1 ) `1 ;
attr R is condensed means : Def3 : for n st n in dom R holds R . n is condensed ;
attr 0 <= a & a <= 1 & b <= 1 & a <= 1 & b <= 1 & a <= 1 & b <= 1 & a <= 1 & b <= 1 & a <= 1 & b <= 1 & a <= 1 & b <= 1 & a <= 1 & b <= 1 & b
u in ( c /\ d ) /\ ( e /\ f ) ;
u in ( c /\ d ) /\ ( b /\ e ) ;
len C + 1 - 2 >= 9 + 2 - 2 ;
x , y // x , y & x , y // y , z ;
a |^ ( n1 + 1 ) = a |^ ( n1 + 1 ) ;
<* \underbrace 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0
set y1 = <* y , c , d *> ;
{ F /. 1 } in rng ( Line ( D , 1 ) ) ;
p . m joins r , s . ( m + 1 ) ;
( ( GoB f ) * ( i , j ) `2 = ( GoB f ) * ( i , j ) `2 ;
W in ( the carrier of X ) \/ ( the carrier of X ) ;
0 + ( p `2 ) <= 2 * ( r `2 ) `2 + ( r - p `2 ) `2 ;
x in dom g & not x in dom g implies x in dom g
f1 /* seq is divergent_to+infty & f2 /* seq is convergent & f2 /* seq is convergent & f2 /* seq is convergent & f2 /* seq is convergent & f2 /* seq is convergent & f2 /* seq is convergent & f2 /* seq is convergent & f2 /* seq is convergent & f2 /* seq is convergent
reconsider v2 = u as VECTOR of [: X , Y :] , [: X , Y :] ;
p \! \mathop { \rm \hbox { - } count ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( - - p ) ) ) /. /. 1 ) ) ) ) ) ) ) ) ) ) ) )
len <* x *> < i + 1 ;
assume I is not empty & I /\ { x } = { x } ;
set i2 = card I + 4 + 0 ;
x in { x , y } & h . x = {} ;
consider y being Element of F such that y in B and y <= x and x <= y ;
len S = len ( the connectives of A ) .= len the connectives of A ;
reconsider m = M , n = I as Element of X ;
A . ( j + 1 ) = B . ( j + 1 ) \/ A . ( j + 1 ) ;
set N8 = // // // // // // // // // // // // // // // // , // // // // // // // // // // // // // // // // // // // // // // // // // // // // // // // // // // // // // // // // //
rng F c= the carrier of gr { a } ;
NQ \ { F ( n ) } is rng -valued ;
f . k , f . ( prime ( n + 1 ) ) are_relative_prime ;
h " ( P /\ Q ) /\ ( P /\ Q ) = f " ( P /\ Q ) ;
g in dom ( f2 \ ( dom f2 ) ) ;
gX /\ dom ( ( g1 | X ) | X ) = g1 /\ g2 .= g1 /\ g2 ;
consider n being element such that n in NAT and Z = G . n ;
set d = dist ( x1 , x2 ) , e = dist ( x2 , x3 ) , f = dist ( x2 , x3 ) , f = dist ( x2 , x3 ) , g = dist ( x2 , x3 ) ;
b `1 + sqrt 5 < sqrt 5 + ( sqrt 5 ) ^2 + ( sqrt 5 ) ^2 + ( sqrt 5 ) ^2 + ( sqrt 5 ) ^2 + ( sqrt 5 ) ^2 + ( sqrt 5 ) ^2 + ( sqrt 5 ) ^2 + ( sqrt 5 ) ^2 ;
reconsider f1 = f as VECTOR of X , f2 = g as VECTOR of X ;
attr i <> 0 implies i mod ( i mod ( i mod ( i mod ( i mod ( i mod ( i mod ( i mod ( n ) ) ) ) ) ) = 1
j1 in Seg ( len g2 + 1 ) ;
dom ( ( i + 4 ) |-> ( i + 4 ) ) = dom ( i + 4 ) .= dom ( i + 4 ) ;
cluster sec | ]. - 1 , 1 .[ -> one-to-one ;
Ball ( u , e ) = Ball ( f . ( p , e ) , e ) ;
reconsider x1 = x1 as Function of S , T ;
reconsider R1 = x , R2 = y as Relation of L ;
consider a , b being Subset of A such that x = [ a , b ] and b in A ;
( <* 1 *> ^ <* p *> ) ^ <* 1 *> in { R } ;
S1 +* S2 = S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 ;
( ( ( ( ( ( - 1 ) (#) ( ( ( ( ( ( ( ( - 1 ) (#) ( ( ( ( ( ( ( - 1 ) / 2 ) ) * ( ( ( ( ( ( ( 1 / 2 ) / 2 ) * ( ( ( ( ( - 1 ) / 2 ) * ( ( ( (
cluster -> one-to-one for Function of C , D ;
set cp = 1GateCircStr ( <* z , x *> , '&' ) , dp = 1GateCircStr ( <* z , x *> , '&' ) ;
E . ( 8 ) = ( E . ( 8 + 1 ) ) . ( n + 1 ) .= ( E . ( n + 1 ) ) . ( n + 1 ) ;
( ( ( ( ( ( - 1 ) * ( ( ( ( ( ( ( ( ( 1 / 2 ) * ( ( ( ( ( ( 2 ) ) * ( ( ( ( ) ) / 2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) is_differentiable_on Z ;
upper_bound A = cos * ( 3 * ( 2 * ( 1 + 1 ) ) `1 & lower_bound A = cos * ( 3 * ( 1 + 1 ) `1 ;
F . ( dom f ) is Morphism of dom f , cod f ;
reconsider p8 = q as Point of TOP-REAL 2 ;
g . W in [#] Y & g . W in [#] Y ;
let C be compact non empty Subset of TOP-REAL 2 ;
LSeg ( f , j ) = LSeg ( f , j ) ;
rng s c= dom f /\ ]. - r , r .[ ;
assume x in { idseq ( 2 ) , <* ( 2 *> ) . ( <* 2 *> ) *> ;
reconsider n2 = n , n2 = m as Element of NAT ;
for y being ExtReal st y in rng seq holds g . y <= g . y
for k st P [ k ] holds P [ k + 1 ]
m = m1 + ( m + 1 ) .= m + ( n + 1 ) .= m + ( n + 1 ) ;
assume for n holds ( G . n ) . ( n + 1 ) = G . ( n + 1 ) . ( n + 1 ) ;
set BX = f .: ( the carrier of X ) ;
ex d being Element of L st d in D & x <= d ;
assume R " { a } c= R " { b } ;
t in ]. r , s .[ or t in ]. r , s .[ ;
z + v2 in W & x + y = u + ( z + y ) ;
x2 = x2 iff P [ x2 , y2 ]
attr x1 <> x2 means : Def3 : |. x1 - x2 .| > 0 ;
assume p2 - p1 = p2 - p1 & - p1 = p2 - p1 ;
set q = \mathbin { ^ \smallfrown } <* 'not' 'not' p *> ;
f be PartFunc of REAL , REAL n ;
( n mod 2 ) -' ( k + 1 ) -' 1 = n - 2 ;
dom ( T * ( <* t *> ) ) = dom <* T *> ;
consider x being element such that x in w iff x in { w } ;
assume ( F * G ) . ( ( " ) * ( F " ) ) . ( ( " ) * ( F " ) ) . ( ( " ) * ( F " ) ) . ( ( " ) ) = v ;
assume the carrier of D1 c= the carrier of D2 & the carrier of D2 c= the carrier of D2 ;
reconsider A1 = [. a , b .] as Subset of REAL ;
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 = Cage ( C , n ) /. ( i + 1 ) ;
n1 -' len f + 1 <= len f + 1 ;
ConsecutiveDelta ( q , O ) . ( O , O ) = [ u , v ] ;
set SG = ( the_Vertices_of G ) . ( k + 1 ) ;
Sum ( L * p ) = 0. V .= Sum ( L * 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 ) ;
set s3 = Comput ( P1 , s1 , k ) , P3 = Comput ( P1 , s1 , k ) , s4 = Comput ( P2 , s2 , k ) , P4 = Comput ( P2 , s2 , k ) , P4 = Comput ( P2 , s2 , k ) , P4 = Comput ( P2 , s2
let l be Nat , A be Matrix of k , Al , A , B be Matrix of k , Al ;
reconsider U = union { G where G is Subset of T : G c= F } as Subset of T ;
consider r such that r > 0 and Ball ( p , r ) c= Q ;
( h | ( n + 2 ) ) /. ( i + 2 ) = p1 `1 ;
reconsider B = the carrier of X1 as Subset of X ;
{ p } = <* - c , - c , d *> ;
synonym f is one-to-one for rng f c= { 0 } ;
consider b being element such that b in dom F and a = F . b ;
x0 < card ( X \/ { x0 } ) + ( card { x0 } ) ;
attr X c= { x1 , x2 , x3 , x4 } ;
then w in Cl Ball ( x , r ) ;
angle ( x , y , z ) = angle ( x , y , z ) ;
attr 1 <= len s means : Def3 : for n st n in dom s holds ( ( the InternalRel of G ) * ( s , n ) ) . ( n ) = s . n ;
{ f } c= f . ( k + 1 ) ;
the carrier of ( ( TOP-REAL 2 ) | K1 ) = { [ { 0. TOP-REAL 2 , { 0. } } , { 0. TOP-REAL 2 } } ;
pred p '&' q in TAUT ( Y ) ;
( - t ) `1 < ( - t ) `1 ;
U . 1 = U /. ( len U + 1 ) .= U /. ( len U + 1 ) ;
f .: ( the carrier of S ) = the carrier of S ;
the carrier of ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL
for n being Element of NAT holds G . n c= G . ( n + 1 )
then V in M .: { x } ;
ex f being Element of [: { F } , { F } :] st f is [: { F } , { F } :] & f is [: { F } , { F } :] ;
[ h . 0 , h . 3 ] in the InternalRel of G ;
s +* ( intloc 0 ) = s1 +* ( intloc 0 ) ;
[ w , v ] <> 0. TOP-REAL 2 & [ w , v ] <> 0. TOP-REAL 2 ;
reconsider t = t as Element of REAL n ;
C \/ P c= [#] ( ( ( ( ( ) \ { x } ) ) \/ P ) ) ;
f " ( [#] X ) /\ ( X /\ X ) in Hom ( X , X ) /\ Hom ( X , X ) ;
x in [#] ( ( TOP-REAL n ) | A ) ;
g . x <= h1 . x & h . x <= h1 . x ;
InputVertices S = { x , y , z } \/ { x , y , z } ;
for n be Nat st P [ n ] holds P [ n + 1 ]
set R = Line ( M , i ) , a = Line ( M , i ) ;
assume M1 is being_line & M2 is being_line & M2 is being_line & M2 is being_line & M2 is being_line & M2 is being_line & M2 is being_line & M2 is being_line & M2 is being_line & M2 is being_line & M2 is being_line & M2 is being_line & M2 is being_line & M2 is being_line & M2 is being_line & M2 is being_line & M2
reconsider a = f . ( i1 -' 1 ) as Element of K ;
len ( ( Len ( F ) ) = len ( ( ( ( ( F ^ <* d *> ) ^ <* d *> ) ) ^ ( ( <* d *> ^ <* d *> ) ) ) ) ) ;
len ( ( the InternalRel of K ) * ( i , j ) ) = n ;
dom ( ( f + g ) `| Z ) = dom ( f + g ) ;
( the InternalRel of Y ) . ( n + 1 ) = ( ( the InternalRel of Y ) . ( n + 1 ) ) . ( ( n + 1 ) ) ;
dom ( p1 ^ p2 ) = dom ( p1 ^ p2 ) ;
M . [ 1 , y ] = 1 * ( ( 1 - y ) * ( M . 1 ) ) .= 1 ;
assume W is not trivial & W is not trivial & W is not trivial & W is not trivial & W is not trivial & W is not trivial & W is not trivial & W is not trivial & W is not trivial ;
{ C _ { 6 } } /. 1 = G * ( 1 , 1 ) `1 ;
{ C } |- 'not' p => ( 'not' p ) => ( 'not' p ) ;
for b st b in rng g holds b <= b
- sqrt ( ( ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 ) = 1 ;
( LSeg ( c , m ) \/ LSeg ( c , m ) ) \/ LSeg ( c , m ) c= R ;
consider p being element such that p in LSeg ( x , p ) and p in LSeg ( f , p ) ;
the carrier of X |^ ( n + 1 ) = [: the carrier of X , the carrier of X :] ;
cluster s => ( q => r ) -> valid ;
Im ( F ) is convergent & ( Partial_Sums ( F ) ) . m is convergent implies ( ( F ) . n ) to_power ( k + 1 ) <= ( Partial_Sums ( F ) ) . m
cluster f . ( x1 , x2 ) -> Element of D ;
consider g being Function such that g = F . t and g is one-to-one ;
p in LSeg ( |[ - 1 , 1 ]| , |[ 1 , 1 ]| ) ;
set R1 = R |^ ( b + 1 ) , R2 = R |^ ( b + 1 ) ;
IncAddr ( I , k ) = Exec ( I , s ) . ( k + 1 ) ;
seq . m <= ( ( ( n + 1 ) * ( ( ( n + 1 ) / ( m + 1 ) ) * ( ( ( n + 1 ) / ( m + 1 ) ) ) ) / ( ( ( n + 1 ) / ( ( n + 1 ) * ( ( n + 1 ) * ( (
a + b = ( a ` ` ) ` .= ( a ` ` ) ` ;
id X = id X & id X = id X ;
for x being element st x in dom h holds h . x = f . x
reconsider H = U \/ { {} } as non empty Subset of U ;
u in ( ( c /\ d ) /\ ( b /\ d ) /\ ( b /\ d ) ) /\ ( b /\ d ) /\ ( b /\ d ) /\ ( b /\ d ) /\ ( b /\ d ) ) ;
consider y being element such that y in Y and P [ y , x ] ;
consider A being finite non empty set such that card A = card ( A /\ B ) ;
p2 in rng ( f ^ <* p1 *> ) \ { p1 } ;
len s1 > 0 & len s2 > 0 & len s2 > 0 implies len s2 = len s2
( ( N-min L~ Cage ( C , n ) ) `2 = ( E-max L~ Cage ( C , n ) ) `2 ;
Ball ( e , r ) c= LeftComp Cage ( C , n ) ;
f . a1 ` ` = f . a1 ` ` .= f . a1 ;
( seq ^\ k ) . n in ]. x0 - r , x0 + r .[ ;
g1 . ( s . ( n + k ) ) = g . ( s . ( n + k ) ) ;
the InternalRel of S is transitive ;
deffunc F ( Ordinal , Ordinal ) = L . ( $1 + 1 ) ;
F . s1 = F . s2 .= F . s2 ;
x ` = A . ( o , a ) .= Den ( o , A ) . ( a , a ) ;
Cl ( f " { p } ) c= f " { p } ;
the topology of ( T ) c= the topology of T ;
synonym o is \ast means : Def3 : o <> {} ;
assume X + + 1 = Y ^ <* X *> & X <> Y implies X /\ ( X \/ 1 ) <> Y
the carrier of s <= 1 + ( the carrier of S ) ;
LIN a , d , b or b , c // b , c ;
e . 1 = 0 & e . 2 = 1 & e . 3 = 0 ;
{ E } in { S } & { E } in { E } ;
set J = ( l , u ) -TruthEval ;
set A1 = \llangle a , b , c , d \rrangle , cin = [ <* a , b , c , dp , cin , dp , cin , cin , dp , cin , dp , cin , cin , dp , cin , dp , cin , cin *> , cin , dp , cin , dp , cin *> , cin *> ;
set c9 = [ <* c , 8 *> , <* d *> , f1 ] , cin = [ <* d , c *> , f2 ] , dp = [ <* d , c *> , f2 ] , cin = [ <* d , c *> , f2 ] , cin = [ <* d , 8 *> , f2 ] , *> , *> ;
x * z " * x in x * ( z * ( x * z ) ) ;
for x being element st x in dom f holds f . x = g . x
right cell ( GoB f , 1 , 1 ) c= RightComp f \/ { f . ( 1 + 1 ) } ;
U is reduces of E-max L~ Cage ( C , n ) , E-max L~ Cage ( C , n ) ;
set f9 = f ^ <* d *> ^ <* d *> ;
attr S1 is convergent means : only : for n holds S1 . n is convergent & ( for n holds S1 . n = ( lim ( S1 ) ) * ( ( lim S2 ) ) ) & ( lim ( S1 ) ) = ( lim S1 ) * ( ( lim S2 ) * ( lim S2 ) ) ;
f . ( 0 qua Ordinal ) = ( 0 qua Ordinal ) + 1 .= a ;
cluster reflexive transitive for transitive RelStr ;
consider d being element such that R reduces b , d and R reduces b , d and R reduces b , c and R reduces b , d and R reduces b , c and R reduces b , d and R reduces b , c and R reduces b , d and R reduces b , c and R reduces b , d ;
not b in dom Start-At ( card I + 2 , SCMPDS ) ;
( z + a ) + x = z + ( a + x ) .= z + x .= z + x ;
len ( l (#) ( l ) ) = len ( l (#) ( l (#) ( F ) ) ) ;
tt is ( X \/ Y ) -valued & ( X \/ Y ) \ Y is ( X \/ Y ) \ Y is finite ;
t = <* F . t *> ^ ( C ^ D ) .= <* F . t *> ^ ( C ^ D ) ;
set p1 = ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ;
k1 -' ( i + 1 ) = k1 - ( i + 1 ) ;
consider u being Element of L such that u = u ` and u in D and u in u ` ;
len ( ( width G ) |-> ( a , b ) ) = width G ;
{ F ( o ) } . x in dom ( ( G * the_arity_of o ) . x ) ;
set H = the carrier of ( TOP-REAL 2 ) | K1 , G = the carrier of ( ( TOP-REAL 2 ) | K1 , G | K1 ) ;
set H = the carrier of ( ( TOP-REAL n ) | K1 ) | K1 ;
( Comput ( P , s , 6 ) ) . intpos ( m + 6 ) = s . intpos ( m + 6 ) ;
IC Comput ( Q , t , k ) = { l } + 1 ;
dom ( ( ( ( ( ( ( ( ( - 1 ) (#) ( ( ( ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( - 1
cluster <* l *> -> ( 1 + 2 ) -element ;
set b = [ <* \hbox { \boldmath $ p $ } , { p } , { p } , { p } , { p } } ] ;
Line ( M , i ) = L * ( Sgm Seg len M , i ) ;
n in dom ( ( the Sorts of A ) * ( ( the ResultSort of S ) * ( the ResultSort of S ) ) ) ;
cluster f1 + f2 -> continuous for PartFunc of REAL , REAL ;
consider y being Point of X such that a = y and ||. \mathopen { \Vert } \mathclose { \Vert } <= r ;
set x8 = t . ( DataLoc ( ( m + 2 ) + 3 ) , 3 ) ;
set p1 = stop I , p2 = stop I , s2 = stop I , s2 = Comput ( P2 , s2 , 1 ) , P2 = P2 , s2 = P2 ;
consider a being Point of D2 such that a in W1 and b = g . a ;
{ A , B , C } = { A , B , C } \/ { D , E , F , J , M } ;
let A , B , C , D , E , F , J , J , M , N , N , F , J , J , M , N , F , J , F , J , J , F , J , J , F , J , M ;
|. p2 .| ^2 - ( |. p2 .| ) ^2 >= 0 ;
l -' 1 + 1 = l * ( m + 1 ) + 1 ;
x = v + ( a * b ) + ( b * c ) .= ( a * b ) * ( b * c ) + ( b * c ) * ( b * c ) ;
the TopStruct of L = ( the TopStruct of L ) | ( the carrier of L ) ;
consider y being element such that y in dom ( ( H . n ) | ( ( ) ) | ( ( { y } \ { y } ) ) ) and x = H . ( y ) ;
{ f } \ { n } = { ( ( 'not' 'not' 'not' 'not' f ) . n ) } ;
let Y being Subset of X , X be non empty Subset of Y ;
2 * n in { N : N * ( n + 1 ) = N * ( n + 1 ) } ;
let s being FinSequence of D ;
for x st x in Z holds ( ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1
rng ( f2 * f1 ) c= the carrier of ( TOP-REAL 2 ) | K1 ) ;
j + 1 <= len f + ( len f -' 1 ) ;
reconsider R1 = R * I as PartFunc of REAL , REAL n ;
seq . ( n + k ) = seq . ( n + k ) .= seq . ( n + k ) ;
<% ( z , z ) . ( n + 1 ) %> . ( n + 1 ) = 1 .= x ;
t is_not at ( C , s ) & t is_not at the connectives of S ;
( support f ) \/ { x } c= ( support f ) \/ { x }
ex N st N = j1 & 2 * ( ( - N ) * ( ( - N ) * ( ( - N ) * ( ( - N ) * ( ( - N ) * ( ( - N ) * ( ( - N ) * ( ( - N ) * ( ( - N ) * ( ( - N ) * ( ( - N ) *
for y , p , q being Element of Y st P [ p , q ] holds P [ p , q ]
{ [ x1 , x2 ] } is Subset of [: X , Y :] ;
h = ( i |-> h ) . ( i , j ) .= H . ( i , j ) .= H . ( i , j ) ;
ex x1 be Element of G st x1 = x & x1 in A & x1 in A & y1 in B ;
set X = ( ( d , O ) `1 , O ) `2 , O = ( d , O ) `2 , I = ( d , O ) `2 , I = ( d , O ) `2 , I ) `2 , I = ( d , O ) `2 , I = ( d , O ) `2 , O = I ;
b . n in { g1 : x0 < g1 & g1 < x0 } ;
f /* s1 is convergent & f /* s1 is convergent & f /* s1 is convergent implies f /* s1 is convergent & f /* s1 is convergent & lim s1 = lim s1
the lattice of Y = the lattice of Y & the carrier of Y = the carrier of X ;
'not' a . x 'or' b . x = TRUE ;
k2 = len ( ( q ^ <* 0 *> ) ^ <* 1 *> ) + len <* 1 *> .= len ( q ^ <* 1 *> ) + len <* 1 *> ;
( ( ( ( ( ( 1 / 2 ) * ( ( ( ( ( ( ( 2 ) * ( ( ( ( ( 2 ) ) * ( ( ( ( 2 ) * ( ( ( ( 2 ) ) * ( ( ( ( 2 ) ) * ( ( ( ( 2 ) ) * ( ( ( ( ( 2 ) ) * ( ( ( ) ) ) )
set K = lim ( ( ( H ) | A ) | A ) , H = ( lim ( H | A ) | A ) | A ) | A ;
assume e in { \frac w + ( 2 * e ) + ( 2 * e ) / ( 2 * ( 2 * e ) + ( 2 * e ) / ( 2 * ( 2 * e ) ) } ;
reconsider d = dom a , e = dom F , e = F . ( len F ) , f = F . ( len F ) , g = F . ( len F ) , g = F . ( len F ) , h = F . ( len F ) , h = F . ( len F ) , f = F . ( len F ) , g = F
LSeg ( f , j ) = LSeg ( f , j ) ;
assume X in { T . ( N , T ) : h . ( N , T ) = N . ( N , T ) } ;
assume Hom ( d , c ) <> {} & Hom ( d , c ) * f = <* c , d *> * f ;
dom Sseq = dom S /\ ( dom ( S | ( n + 1 ) ) ) .= dom ( S | ( n + 1 ) ) /\ ( ( n + 1 ) \ { n } ) .= ( ( S | ( n + 1 ) ) /\ ( ( n + 1 ) \ { n } ) /\ ( n + 1 ) ) ;
x in { H } implies ex g st x = g |^ k & g in H
a * ( a , n ) = a * ( n , n ) .= a * ( n , n ) ;
D2 . j in { r : r <= D1 & D1 <= len D2 } ;
ex p being Point of TOP-REAL 2 st p = x & |. p .| <= 1 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <>
for c holds f . c <= g . c implies f ^ ) ^ g ^ g ^ h ^ h ^ h ^ h ^ h
dom ( ( ( ( ( ( ( ( ( - 1 ) (#) ( ( ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( 1 / 2 ) / 2 ) * ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( - 1 ) / 2 ) * ( ( ( ( 1 / 2 ) / 2 ) * ( (
1 = sqrt ( p * ( ( 1 - p `2 ) * ( 1 - p `2 ) ) .= p `1 ;
len g = len f + len <* x *> .= len <* x *> + len <* y *> .= len <* x *> ;
dom ( F | ( N \/ { N } ) = dom ( F | ( N \/ { N } ) ) ;
dom ( f . t ) = dom ( f . t ) ;
assume a in ( ( sup ( { T } ) .: D ) .: D ) .: D ;
assume g is one-to-one & ( the carrier of S ) /\ ( the carrier of T ) c= ( the carrier of T ) /\ ( the carrier of S ) ;
( x \ ( x \ z ) ) \ ( x \ z ) = 0. X ;
consider f such that f * f = id Z and f is one-to-one ;
( ( ( ( ( 2 * ( ( ( ( 2 * ( ( 2 * ) ) ) ) * ( ( ( ( 2 * ( ( 2 * ( ( 2 * ( 2 ) ) ) ) ) ) ) ) ) ) ) ) `| Z ) ) ) is differentiable ;
Index ( p , co ) <= len ( L (#) ( G , 1 ) ) - len ( L (#) ( G , 1 ) ) ;
t1 , t2 , t2 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t1 , t2 ,
( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) . ( ( ( ( ( ( ( ( ) ) . ( ( ( ( ( ( ) ) ) . ( ( ( ( ( ( ) ) . ( ( ( ( ( ( ) ) . ( i , ) ) ) ) ) ) )
then P [ f . ( i1 + 1 ) ] & F ( i1 + 1 ) . ( i1 + 1 ) < j + 1 ;
Q [ ( [ D , {} ] , <* {} *> ] ) `1 = [ D , {} ] `1 , D ] ;
consider x being element such that x in dom ( F . s ) and y = F . x ;
l . i < r . i & l . i < r . i ;
the Sorts of A = ( the Sorts of ( the Sorts of A ) ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A )
consider s being one-to-one Function such that s is one-to-one & dom s = NAT & rng s = { s } ;
dist ( b1 , c1 ) <= dist ( b1 , c1 ) + dist ( b1 , c1 ) ;
( <* C *> /. ( n + 1 ) ) `1 = W-bound L~ Cage ( C , n ) `1 ;
q <= ( ( E-max L~ Cage ( C , n ) ) `1 ;
LSeg ( f | ( i + 1 ) , f /. ( i + 1 ) ) /\ LSeg ( f , i ) = {} ;
given a being ExtReal such that a <= I and A = { a } ;
consider a , b being complex number such that z = a & y = b and a = a + b ;
set X = { b } , Y = { b } , Z = { b } , Y = { b } , Z = { b } , Y = { b } , Z = { b } , { b } , { b } } ;
( x * z ) \ ( x \ z ) = 0. X ;
set x9 = [ <* x9 , cin *> , '&' ] , y9 = [ <* cin , dp *> , '&' ] , c9 = [ <* cin , dp *> , '&' ] , cin , dp ] ;
l1 /. ( len l + 1 ) = l1 . ( len l + 1 ) .= l /. ( len l + 1 ) ;
sqrt ( ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 ) = 1 ;
sqrt ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( 1 + 2 ) / 2 ) * ( ( 1 + 2 ) / 2 ) ) ) ) / ( ( 1 + 2 ) * ( ( 1 + 2 ) / 2 ) ) ) ) < 1 ;
( ( ( ( ( ( X \/ Y ) \/ Y ) \/ Y ) \/ Y ) \/ ( ( X \/ Y ) \/ Y ) ) \/ Y ) ) /\ ( X \/ Y ) ) /\ ( X \/ Y ) ) = ( X \/ Y ) /\ ( X \/ Y ) ;
( seq - ( k + 1 ) ) . ( n + 1 ) = seq . ( n + 1 ) - seq . ( n + 1 ) ;
rng ( h + c ) c= dom ( ( h + c ) ^\ n ) ;
the carrier of ( X ) = the carrier of X & the carrier of X = the carrier of X ;
ex p3 st p3 = p1 & |. p3 .| = r & |. p3 .| <= 1 & |. p3 .| <= 1 & |. p3 .| <= 1 & |. p3 .| <= 1 & |. p3 .| = 1 & |. p3 .| = 1 & |. p3 .| = 1 & |. p3 .| = 1 & |. p3 .| = 1 & |. p3 .| = 1 & |. p3 .| = 1 & |. p3 .| =
set h = \raise .4ex ( X , Y ) , A , B , C , D , E , F , J , M , N , N , F , J , M , N , N , F F J J , J , M , F F N , J , J , M , N , F F F N , J , J N N , F F J J J M M , M ,
R |^ ( 0 * n ) = R |^ ( 0 * n ) .= R |^ ( 0 * n ) ;
( Partial_Sums ( F ) ) . ( n + 1 ) is n1 + 1 <= len ( ( F ) . ( n + 1 ) ) ;
f2 = { C ( ) : for n , m st n >= 8 ( ) holds ( for n st n >= 8 ( ) holds ( ( for n st n >= >= 8 ( ) holds ( ( the Eof G ) . ( n + 1 ) ) . ( m + 1 ) = - ( n + 1 ) )
S1 . b = s1 . b .= s2 . b ;
p2 in LSeg ( p2 , p2 ) /\ LSeg ( p2 , p2 ) ;
dom ( f . t ) = Seg n & dom ( f . t ) = Seg n ;
assume o = ( the connectives of S ) . 11 & ( the connectives of S ) . 11 in ( the carrier' of S ) . 11 ;
set \varphi = ( l1 , {} ) , l1 = ( l1 , {} ) , r2 = ( l1 , {} ) , r2 = ( l1 , {} ) , r2 = ( l1 , {} ) , r2 = ( l1 , {} ) , r2 = ( l1 , {} ) , r2 = ( l1 , {} ) , r2 = ( l1 , {} ) , r2 = ( l1 , {} ) , r2 = ( l1 , {}
synonym p is T means : Def3 : ex q being Polynomial of n , L st q is w.r.t. I & q <> 0. L ;
( ( Y ) | ( ( X \/ Y ) | ( X \/ Y ) ) ) | ( X \/ Y ) = ( Y | ( X \/ Y ) ) | ( X \/ Y ) ) | ( X \/ Y )
defpred X [ Nat , set ] means for n being Nat st $1 in dom $2 holds $2 = F ( n ) ;
consider k being Nat such that for n being Nat st n <= k holds s . n < x0 + r ;
Det ( I |^ ( m -' n ) ) = 0. K ;
sqrt ( b - sqrt b ) - sqrt ( b - sqrt ( b - sqrt 5 ) ) < 0 ;
{ C . d } = { C . d } ;
attr X1 is open means : only : ( X /\ Y ) /\ Y is open & X /\ Y is open implies X /\ Y is open ;
deffunc F ( Element of E , Element of E ) = ( the Element of E ) . ( $2 , $2 ) ;
t ^ <* n *> in { t ^ <* n *> *> ;
( x \ y ) \ ( x \ y ) = ( x \ y ) \ ( x \ y ) .= 0. X ;
let X being non empty Subset-Family of [: X , Y :] ;
synonym A , B , C , D , E , F , G , G , F , G , G , F } ;
len ( M * ( i , j ) ) = len ( M * ( i , j ) ) & len ( M * ( i , j ) ) = len ( M * ( i , j ) ) ;
J . v = { x where x is Element of K : 0 < x & x < 1 } ;
( ( Sgm m ) . d ) . ( ( Sgm m ) . d ) - ( ( Sgm m ) . d ) ) <> 0 ;
lower_bound divset ( D2 , k ) + ( lower_bound divset ( D2 , k ) - ( lower_bound A ) - ( lower_bound A ) <= ( upper_bound A ) - ( lower_bound A ) - ( lower_bound A ) ) ;
g . r1 = ( 2 * ( 1 + 2 ) ) * ( ( 1 + 2 ) * ( ( 1 + 2 ) * ( ( 1 + 2 ) * ( 1 + 2 ) ) ) ) & dom h = [. 0 , 1 .] ;
|. a .| * ||. f .|| = 0 * ||. f .|| .= 0 ;
f . x = ( h . x ) `1 & g . x = ( h . x ) `1 ;
ex w st w in dom B & <* w *> ^ B = <* w *> ^ B ;
[ 1 , {} , {} ] in ( { [ {} , {} ] } \/ { {} } ) \/ { {} } ;
IC Exec ( i , s1 ) + n = IC Exec ( i , s1 ) + n .= ( IC Exec ( i , s1 ) + n ) ;
IC Comput ( P , s , 1 ) = succ IC Comput ( P , s , 1 ) .= ( card I + 1 ) .= 0 ;
( IExec ( W6 , Q , t ) ) . intpos ( m + 1 ) = t . intpos ( m + 1 ) ;
LSeg ( f , i ) misses LSeg ( f , i ) ;
assume for x , y being Element of L st x in C & y in C holds x <= y or y <= x & x <= y ;
]. f /. ( n + 1 ) - f /. ( n + 1 ) .[ = f . ( n + 1 ) - f /. ( n + 1 ) ;
let F being one-to-one FinSequence of D , G being FinSequence of D , F being FinSequence of D st F misses G & F is one-to-one & G is one-to-one & G is one-to-one & F is one-to-one & G is one-to-one & G is one-to-one & F is one-to-one & G is one-to-one & G is one-to-one & G is one-to-one holds F is one-to-one
||. R /. ( h . ( h . ( k + 1 ) ) - R /. ( h . ( k + 1 ) ) .|| < e * ( K . ( k + 1 ) ) ;
assume a in { q where q is Element of M : dist ( z , q ) <= r } ;
set p2 = [ 2 , 0 ] .--> ( 2 * 0 ) ;
consider x , y being Subset of X such that [ x , y ] in F and [ x , y ] in F and [ x , y ] in F ;
for y , z being Element of REAL m , x being Element of REAL m st y in Y & x in X & y in Y holds x + z in X
func |. p .| -> ManySortedSet of NAT equals |. p .| ;
consider t being Element of S such that x = z and x , y // z , t ;
dom x1 = Seg len x1 & len x1 = len x1 & len x1 = len x2 & len x1 = len x2 & len x1 = len x2 & len x1 = len x2 & len x1 = len x2 & len x1 = len x2 & len x1 = len x2 & len x2 = len x2 & len x1 = len x2 & len x1 = len x2 & len x1 = len x2 & len x1 = len x2 & len x1 = len x2 implies x1 = x2
consider x2 being Real such that x2 = x2 and 0 <= x2 and x2 <= 1 and x2 <= 1 and x2 <= 1 and x2 <= 1 and x2 <= 1 and x1 <= 1 and x2 <= 1 & x2 <= 1 & x2 <= 1 & x2 <= 1 & x1 <= 1 & x2 <= 1 & x2 <= 1 & x2 <= 1 & x2 <= 1 & x1 <= 1 & x2 <= 1 & x2 <= 1 & x1 <= 1 & x2 <= 1 & x2 <= 1 & x2
||. f .|| = ||. ( f | X ) . ( n + 1 ) - f /. ( n + 1 ) .|| .= ||. f /. ( n + 1 ) - f /. ( n + 1 ) .|| ;
( the InternalRel of A ) \/ ( the InternalRel of A ) /\ ( the InternalRel of A ) = {} .= {} ;
assume i in dom p implies for j st j in dom p holds p . j = q . j
reconsider h = f | X as Function of X , Y , ( ) | X ) , ( Y | X ) | X ;
u1 in the carrier of W1 & u1 in the carrier of W2 & u in the carrier of W1 & u1 in the carrier of W2 & u in the carrier of W1 ;
defpred P [ Element of L ] means M <= f . ( $1 + 1 ) & f . ( $1 + 1 ) <= f . ( $1 + 1 ) ;
T . ( u , a ) = s * ( x , a ) .= s * ( x , a ) ;
( - ( a2 - a3 ) ) . ( x - y ) = - ( a2 - a3 ) . ( x - y ) .= - ( a1 - a2 ) ;
given a being Point of G such that for x being Point of G , y being Point of G st [ x , y ] in the InternalRel of G holds [ x , y ] in [: the carrier of G , { x } :] ;
f = [ <* dom ( f ^ <* f *> ) *> , <* f *> ] ;
let k , n be Nat ;
for x being element holds x in A |^ ( n + 1 ) iff x in A |^ ( n + 1 )
consider u , v being Element of R such that l /. i = u * v and u in I and v in I ;
( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( - 1 ) / 2 ) * ( ( - 1 ) / 2 ) ) ) ) / ( ( - 1 ) * ( ( - 1 ) / 2 ) ) ) ) ) ) ^2 ) > 0 ;
L1 . ( k + 1 ) = L1 . ( k + 1 ) & L1 . ( k + 1 ) = L1 . ( k + 1 ) ;
set i2 = if>0 ( a , i , n , k ) ;
If B is universal & B is universal & C is universal & D is universal & B is universal & C is being_line & D is being_line & C is being_line & D is being_line & D is being_line & D is being_line & B is being_line & D is being_line & D is being_line & B is being_line & D is being_line & D is being_line & D is being_line & D is being_line & D is being_line & D is being_line & D is being_line & D is being_line &
{ a } "/\" D = { a "/\" b where a is Element of N : a in D & b in D } ;
( \square ) * ( ( \square , len ( ( ( n ) ) * ( ( \square , len ( ( n ) ) ) ) ) ) ) + ( ( ( n + 1 ) * ( ( \square , len ( ( n ) ) ) ) ) ) ) ) * ( ( ( \square , len ( ( ( n ) ) * ( ( \square , len ( ( n ) ) ) ) ) ) ) ) ) ) >= ( n ) *
( - f ) . ( upper_bound A ) = ( - f ) . ( upper_bound A ) .= ( - f ) . ( lower_bound A ) .= ( - f ) . ( lower_bound A ) ;
( G * ( i , j ) `1 = ( G * ( i , j ) `1 ) `1 ;
( Proj ( i , n ) . ( Proj ( i , n ) . ( ( Proj ( i , n ) . ( ( Proj ( i , n ) . ( ( Proj ( i , n ) * ( ( Proj ( i , n ) * ( ( Proj ( i , n ) * ( ( Proj ( i , n ) * ( ( Proj ( i , n ) * ( ( Proj ( i , n ) * ( ( Proj ( i
f1 + f2 * ( ( ( ( i - f2 ) * f1 ) ) `| Z ) . x = ( ( i - f2 ) * f1 ) `| Z ) . x ;
attr ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( - 1 ) / 2 ) * ( ( ( - 1 ) / 2 ) * ( ( ( - 1 ) / 2 ) * ( ( ( - 1 ) / 2 ) * ( ( ( ( - 1 ) / 2 ) ) ) ) ) ) ) ) )
ex t being SortSymbol of S st t = s & ( for n being Nat holds ( h . n ) . ( x , n ) = h . ( x , n ) ;
defpred C [ Nat ] means ( for n being Nat st n >= $1 holds P [ n ] implies ( for n st n >= $1 & n <= $1 holds P [ n ] implies P [ n + 1 ] ) & ( for n st n >= $1 & n <= $1 & n <= $1 holds P [ n ] implies P [ n + 1 ] ) & P [ n + 1 ] ) implies P [ n +
consider y being element such that y in dom ( p ^ <* y *> ) and q = ( p ^ <* y *> ) . i ;
reconsider L = product ( { x1 , x2 } ) as Subset of TOP-REAL n ;
for c being Element of C , d being Element of D st T . ( c , d ) = id d holds T . ( c , d ) = id d
LIN f , n , p . ( n + 1 ) .= f . ( n + 1 ) ;
( f * g ) . x = f . ( x , y ) & ( f * g ) . x = f . ( x , y ) ;
p in { 1 / 2 * ( ( G * ( i + 1 , j ) `1 ) + G * ( i , j + 1 ) `1 } ;
f ` - ( c ` ) = ( - ( c ` ) ) ` .= ( - ( c ` ) ) ` .= ( - ( c ` ) ) ` ;
consider r being Real such that r in rng ( f | divset ( D , j ) ) and r < m + 1 ;
f1 ( ( ( ( ( ( ( ( ( ( ( ( ( G ) /. i ) , 1 ) , 1 ) , 1 ) , 1 ) ) ) ) ) `1 in { ( ( ( GoB f ) * ( ( GoB f ) * ( 1 , 1 ) ) ) `1 } ) ;
eval ( a , ( n + 1 ) , x ) = ( a , x ) . ( n + 1 ) .= ( a , x ) . ( n + 1 ) ;
z = \llangle `1 , z `2 , z `2 , z ] .= [ `1 , z `2 , z `2 , z `2 , z ] ;
set H = { Intersect ( S ) where S is Subset of X : S is open } ;
consider S19 being Element of D such that S = { S19 } and S = { S19 } and { 19 } = { 19 } ;
assume that x1 in dom f and x2 in dom f and f . x1 = f . x2 ;
- 1 <= sqrt ( ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 ) ;
{ 0. V } is Carrier of A & { 0. V } is Carrier of V
let k1 , k2 , k2 , k1 , k2 , k2 , k2 , k1 , k2 , k2 , k2 , k2 , k2 , k1 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k1 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 ,
consider j being element such that j in dom a and j in dom g and x = g . j ;
p1 . ( x + y ) c= p1 . ( x + y ) or p1 . ( x + y ) c= p1 . ( x + y ) ;
consider a being Real such that p = a * ( ( 1 - a ) * ( ( 1 - a ) * ( ( 1 - a ) * ( ( 1 - a ) * ( 1 - a ) ) ) ) + ( 1 - a ) * ( 1 - a ) ) ;
assume a <= c & b <= d & c <= d & d <= b & a <= b & b <= c & d <= c & c <= d & d <= b & d <= b & b <= c & c <= d & d <= b & d <= b & b <= c & c <= d & d <= d & d <= b & c <= d & d <= b & d <= b & c <= d & d <= b & d <= b & c
cell ( Gauge ( C , m , k ) * ( i , j ) , G * ( i , j ) ) is not empty ;
A5 in { ( S . ( i + 1 ) ) `1 where i is Element of NAT : i <= n & n <= len S } ;
( T * ( b1 /. ( len b1 ) ) . ( len b1 ) = L . ( b1 /. ( len b1 ) ) .= ( F /. len b1 ) . ( len b1 ) ;
g . ( s , I ) . ( y , I ) = s . ( y , I ) . ( y , I ) . ( y , I ) .= s . ( I , I ) . ( y , I ) ;
( log ( 2 , k ) ) ^2 + ( ( 2 * ( ( ( ( ( 2 * ( ( 2 * ( ( 2 * ( ( 2 * ( ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * (
then p => q in S & not p in the carrier of S & p in the carrier of S & q in the carrier of S & p => q in the carrier of S ;
dom ( ( the Target of G1 ) +* ( the carrier' of G2 ) ) misses ( the carrier' of G1 ) \/ ( the carrier' of G2 ) ;
synonym f is -> extended real-valued for for f is extended real-valued Function of X , Y ;
assume for a being Element of D holds f . ( a , b ) = a & f . ( a , b ) = f . ( a , b ) ;
i = len ( ( p ^ <* x *> ) + len <* x *> ) .= len <* x *> + len <* x *> .= len <* x *> + len <* y *> .= len <* y *> + len <* y *> ;
( l ) `1 = ( g /. ( k + 1 ) ) `1 + ( g /. ( k + 1 ) ) `1 .= ( g /. ( k + 1 ) ) `1 + ( g /. ( k + 1 ) ) `1 ;
CurInstr ( P2 , Comput ( P2 , s2 , i ) ) = halt SCMPDS ;
assume for n be Nat holds ||. ( seq . n ) . n .|| <= ( ||. seq .|| . n ) . ( ( seq . n ) . n ) & ( seq . n ) . ( ( seq . n ) ) . ( ( seq . n ) . ( ( seq . n ) . n ) ) <= ( seq . n ) . ( ( seq . n ) . ( ( seq . n ) . ( ( seq . n ) ) ) ;
sin . r2 = sin . ( cos . r2 ) .= cos . ( cos . r2 ) .= cos . ( cos . r2 ) .= cos . ( cos . r2 ) .= cos . ( cos . r2 ) ;
set q = [ g1 , g2 ] `2 , r = [ r , g2 ] `2 ;
consider G being sequence of S such that for n being Element of NAT holds G . n in F . n ;
consider G such that F = G and G in { S } and G in { S } ;
the carrier of ( ( the Sorts of Free ( C , X ) ) . s ) . ( ( the Sorts of Free ( C , X ) ) . s ) = ( ( the Sorts of Free ( C , X ) ) . s ) . ( ( ( the Sorts of Free ( C , X ) ) . s ) ;
Z c= dom ( ( ( ( ( ( ( ( - 1 ) (#) ( ( ( ( ( ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) * ( ( ( ( ) ) ) * ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ) ) ) * ( ( ( ( ( ) / 2 ) ) * ( ( ( (
for k being Element of NAT holds ( ( ( Im ( f ) ) . k ) . ( n + 1 ) ) . k = ( ( Im ( f ) ) . ( n + 1 ) ) . k
assume that 1 < n and n < len ( ( f | n ) | n ) and ( f | n ) | n = ( f | n ) | n ;
assume that f is continuous and a < b and f is one-to-one and f | [. a , b '] is continuous and f | [' a , b '] is bounded and f | [' a , b '] is bounded and f | [' a , b '] is bounded and f | [' a , b '] is bounded and f | [' a , b '] is bounded and f | [' a , b '] is bounded and f | [' a , b '] is bounded
consider r being Element of NAT such that s = Comput ( P1 , s1 , i ) and r <= ( Comput ( P1 , s1 , i ) ) . ( k + 1 ) ;
LE f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) ) , f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) ) ;
assume that x in the carrier of K and y in the carrier of K and x in the carrier of K and y in the carrier of K and y in the carrier of K and y in the carrier of K and x in the carrier of K and y in the carrier of K and y in the carrier of K and y in the carrier of K ;
assume f +* ( i1 , i2 ) . ( i1 + 1 ) in ( ( ( ( ) -' 1 ) + 1 ) ) . ( i1 + 1 ) ) . ( i1 + 1 ) ;
rng ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( \ ) \ ) ) ) | ( ( ( ( \ ) ) | ( ( ( ( \ \ ( ) | ( ( ( \ \ ( ) \ ( ( \ \ ( ) \ ( ( \ \ ( ) ) | ( ( \ ( ) \ ( ( \ \ ( \ \ ( ) ) ) | ( ( \ ( \ ( ) ) ) | ( ( ( \
assume z in { ( the carrier of G ) \/ { t } where t is Element of G : t in X } ;
consider l being Nat such that for m be Nat st l <= m holds ||. ( ( ( G . m ) - ( G . m ) ) - ( G . m ) ) - ( G . m ) ) .|| < g ;
consider t being VECTOR of product G such that for n being Nat holds ( ex m be Element of NAT st n <= m & m <= n & n <= m holds ( G . n ) . ( m + 1 ) <= 1 ) ;
assume that the topology of v = 2 and v in dom ( <* 0 *> ^ ( <* 1 *> ) ) and v in dom <* 1 *> ^ ( <* 1 *> ^ <* 0 *> ) ;
consider a being Element of the Points of X such that a on X and A on X ;
( - x ) |^ ( k + 1 ) = 1 ;
let D being set such that for i being Nat st i in dom p holds p . i in D . i ;
defpred R [ element ] means ex x , y being element st [ x , y ] in $1 & [ y , x ] in R & [ x , y ] in R ;
L~ f2 = union { LSeg ( f2 , len f2 ) where f2 is Point of TOP-REAL 2 : len f2 = 2 } ;
i -' len ( h ) + 1 - len ( h ) + 1 - len ( h ) + 1 - len ( h ) + 1 - len ( h ) + 1 ) < i - len ( h ) + 1 - len ( h ) ;
for n being Element of NAT st n in dom F holds F . n = |. ( F . n ) .|
for r , s being Real holds ( for n being Nat holds ( for n holds s . n ) `1 <= r & ( for n st n >= 1 holds r <= s . n ) `1 ) implies ex m st m <= n & m <= m & m <= n & n <= len s ) implies m <= n
assume v in { G where G is Subset of T : G in B & G * ( i , j ) `1 <= G * ( i , j ) `1 } ;
let g be as as as as as as as as as as as as as as as as as as as as as as as as as as differentiable differentiable differentiable differentiable non empty Element of A , D ;
min ( g . ( [ x , y ] , k ] , k ) = ( min ( g . ( k + 1 ) , k ) ) . ( ( k + 1 ) ) ;
consider q1 being sequence of NAT 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 BO = B /\ B , GO = O /\ ( B /\ { O } ) as Subset of T ;
consider j being Element of NAT such that x = ( the InternalRel of K ) * ( ( n + 1 ) , j ) ) `1 and 1 <= j & j <= n & j <= n & n <= len f & f /. j = f /. j ;
consider x such that z = x and card { O where O is Subset of L : O in F & O c= F } and x in { O where O is Subset of L : O in F } ;
( C * ( ( G . ( n + 2 ) ) ) . 0 = ( ( C . ( n + 2 ) ) . 0 ) . 0 ;
dom ( X --> ( n + 1 ) ) = X & dom ( X --> ( n + 1 ) ) = X ;
( S - ( T ) ) . ( ( - S ) . ( T . ( T . n ) ) <= ( ( S - T ) . ( T . n ) ) `1 ;
synonym x is collinear means : Def3 : { x , y } = { x , y } ;
consider X being element such that X in dom ( f | X ) and ( f | X ) . X = ( f | X ) . X ;
assume that Im k is continuous and for x , y being Element of L st x in X & y in X & x <= y holds x + y <= y + ( x + y ) ;
( 1 / 2 * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( ( ( ( n ) * ( ( ( n + 2 ) * ( ( ( ( n + 2 ) ) * ( ( ( ( 1 + 2 ) * ( ( ( 1 + 2 ) * ( ( ( ( 2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) is_differentiable_on Z ) )
defpred P [ Element of omega ] means ( ( the InternalRel of $1 ) . ( len $1 ) = ( the InternalRel of $1 ) . ( len $1 ) & ( the InternalRel of $1 ) . ( len $1 ) = ( the InternalRel of $1 ) . ( len $1 ) ;
IC Comput ( P , s , 2 ) = succ IC Comput ( P , s , 2 ) .= ( card I + 2 ) .= ( card I + 2 ) .= ( card I + 2 ) + 2 ;
f . x = f . ( ( g . x ) * f . x ) .= f . ( ( g . x ) * f . x ) .= f . ( g . x ) ;
( M * ( F . n ) ) . ( ( M . n ) . ( j + 1 ) ) = M * ( ( ( ( ( M . n ) . ( j + 1 ) ) . ( j + 1 ) ) . ( j + 1 ) ) .= M * ( ( M . n ) . ( j + 1 ) ) ;
the carrier of ( ( TOP-REAL n ) | K1 ) \/ ( ( TOP-REAL n ) | K1 ) \/ ( ( TOP-REAL n ) | K1 ) ) \/ ( ( TOP-REAL n ) | K1 ) ) /\ K1 = K1 ;
pred a , b , c , d is_collinear means : Def3 : for x , y , z being Element of Y st x in X & y in X & z in Y holds x , y // z , x ;
( ( the InternalRel of product G ) . ( n + 1 ) ) . ( ( n + 1 ) ) <= ( ( the InternalRel of G ) . ( n + 1 ) ) . ( ( n + 1 ) ) . ( ( n + 1 ) ) ;
attr 1 <= r & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= ( r <= 1 & r <= 1 & r <=
seq . ( n + 1 ) in { p where p is Element of NAT : p in dom <* p *> & p in T } ;
[ x1 , x2 ] . ( 2 * ( 2 * ( ( 2 * ( ( 2 * ( ( ( ( ( - 1 ) / 2 ) ) * ( ( ( - 1 ) / 2 ) ) ) ) ) ) ) ) . ( 2 * ( 2 * ( ( ( - 1 ) / 2 ) * ( ( ( - 1 ) / 2 ) * ( ( ( - 1 ) / 2 ) * ( ( ( - 1 ) / 2 ) ) ) ) ) ) )
hence for m be Nat holds ( ( Partial_Sums F ) . m ) . ( n + 1 ) is non-negative
len ( <* G *> ^ <* y *> ) = len ( <* G *> ^ <* y *> ) + len <* y *> .= len <* G *> + len <* y *> ;
consider u , v being VECTOR of V such that x = u + v and u in W and v in W and u in W and v in W and u in W and v in W and W /\ W = { u } ;
given F being FinSequence of NAT such that F = x & dom F = n & dom F = n ;
0 = r1 * cos . If iff 1 = r1 * cos . If
consider n being Nat such that for m be Nat st n <= m holds |. ( ( f # x ) . m - g .| < e ;
cluster non empty transitive for non empty RelStr ;
"/\" ( { B } , L ) = "/\" ( { B } , L ) .= "/\" ( { B } , L ) .= "/\" ( { B } , L ) .= "/\" ( { B } , L ) .= "/\" ( { B } , L ) .= "/\" ( { B } , L ) .= "/\" ( { B } , L ) ;
sqrt ( r ^2 + ( r ^2 + ( r ^2 ) ) ^2 ) + ( r ^2 ) ^2 <= r ^2 + ( r ^2 ) ^2 + ( r ^2 ) ^2 + ( r ^2 ) ^2 + ( r ^2 ) ^2 + ( r ^2 ) ^2 + ( r ^2 ) ^2 + ( r ^2 ) ^2 ) ;
for x being element st x in A /\ B holds ( ( f `| A ) `| A ) . x >= r
2 * ( a - b ) * ( b - c ) = 0. TOP-REAL n ;
reconsider p = P /. ( \square , j ) , q = P /. ( j , j ) as FinSequence of K ;
consider x1 , y1 , y2 being element such that x1 in uparrow s and y1 in uparrow s and x = [ x1 , y1 ] and y = [ x1 , y1 ] and [ x1 , y1 ] in uparrow [ x1 , y1 ] ;
for n be Nat st 1 <= n & n <= len q holds ( ( ( ( ( ) /. n ) ) `1 <= ( ( ( ( ( ( G /. n ) ) `1 ) ) `1 ) ^2 )
consider y , z being element such that y in the carrier of A and z in the carrier of A and y = [ y , z ] and z = [ y , z ] ;
given H1 , H2 being strict Subgroup of G such that x = H1 & ( ex g being Element of G st g = H1 & g in H2 & g in H ) & g in H ) & f = g . ( g . x ) ;
let S being non empty RelStr , T be non empty Poset , S be full non empty Subset of T ;
[ a + 0 , b ] in ( the carrier of V ) \/ { a } ;
reconsider mF = max ( ( len F - 1 ) , ( len F - 1 ) - 1 ) as Element of NAT ;
I <= width ( ( GoB f ) * ( i , j ) , ( GoB f ) * ( i , j ) ) `1 ;
f2 /. ( len f2 ) = ( f2 /* ( f1 /* ( seq + k ) ) ) /. ( ( f2 /* ( seq ^\ k ) ) ) /. ( ( ( f2 /* seq ) ^\ k ) ) .= ( f2 /* ( seq ^\ k ) ) /* ( seq ^\ k ) ) .= ( f2 /* ( seq ^\ k ) ) /* ( seq ^\ k ) ) ^\ k ) . ( ( ( ( ( n + k ) - k ) ) ;
attr A1 \/ A2 is linearly of V means : Def3 : for n , m being Nat st n <= m & m <= len ( ( m + n ) |-> ( m + n ) ) holds ( ( m , n ) --> ( m , n ) ) . ( m + 1 ) = ( ( m , n ) --> ( m , n ) ) . ( m + 1 ) ;
func A -ManySortedSet of I means : Def3 : for n being Element of I holds it . n = { A . n } ;
dom ( ( Line ( ( ( ( ( ( ( ( ( ( ( ( ( ( , m , , , width ( ) ) ) , ( ( ( ( ( ( ( ( , , m ) , ( ( ( ( , ) , ) , ( ( ( ( ( ( ( ( , ) , ) , ( ( ( ( ( ( , ) ) , ( ( ( ( ( ( ) ) , ( ( ( ) ) ) , ( ( ( ( ( ) ) ) , ( ( ( ( ( ) ) ) , ( ( ( ( ( ( ( ( ( ) ) )
cluster [ x , y ] -> Point of TOP-REAL 2 equals [ x , y ] `1 , x ] ;
E |= All ( x , H ) => All ( x , H ) => All ( x , H ) => All ( x , H ) => All ( x , H ) ) => All ( x , H ) = All ( x , H ) => All ( x , H ) ;
F .: ( { x } \/ { y } ) = F .: ( { x } \/ { y } ) .= F .: { x } ;
R . ( h . m ) = F . ( ( h . m ) . ( n + 1 ) ) + R . ( h . ( n + 1 ) ) .= F . ( ( h . m ) . ( n + 1 ) ) + F . ( n + 1 ) ;
cell ( G , i1 , j1 -' 1 , j1 -' 1 ) meets L~ f ;
IC Comput ( P2 , s2 , k ) = IC Comput ( P2 , s2 , k ) .= ( card I + 2 ) + 2 ;
sqrt ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) / 2 ) ) * ( ( ( ( 1 + 2 ) / 2 ) ) ) ) ) ) ) ) ) ^2 ) ) ) ) ) ) > 0 ;
consider x0 being element such that x0 in dom a and x0 in dom g and x0 = g . ( n + k ) ;
dom ( ( r (#) ( ( ( ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( 1 / 2 ) / 2 ) ) ) ) ) ) ) ) ) ) ) ) ) = dom ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( - 1 )
d . ( y , z ) = ( ( y , z ) . ( x , y ) ) . ( y , z ) .= ( ( y , z ) . ( y , z ) ) . ( y , z ) ;
hence for i being Nat , C being Subset of X , D being non empty Subset of X st C . i = A /\ D holds C is open
assume that x0 in dom f and f | X is continuous and f | X is continuous ;
p in A implies for K st K in A & K is open & p in K & K c= A
for x being Element of REAL n st x in Line ( x1 , x ) holds |. ( x1 - x2 ) - ( y1 - y2 ) .| <= |. x1 - x2 .|
func <* a *> -> e of a , b , c *> means : Def3 : for b being Ordinal of n , b , c being Ordinal of n , L st b in b & c in b & b in b holds it . ( b , c ) = b * c + ( b * c ) * ( b , c )
[ a1 , a2 ] in ( the InternalRel of A ) \/ ( the InternalRel of A ) \/ ( the InternalRel 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 ;
||. ( ||. seq .|| . n ) - ( ( seq . n ) . m ) .|| < r / 2 ;
then Z in { Y where Y is Subset of [: I , I :] : Y in F } ;
ex_sup_of { s ( ) where s is Element of L : s in X ( ) & s in X ( ) } , L ( ) is "\/" Subset of L ( )
consider i , j being Element of NAT such that i < j and [ i , j ] in Indices GoB f and [ i , j ] in Indices GoB f and [ i , j ] in Indices GoB f ;
let D being non empty set , p , q , r being FinSequence of D , s being FinSequence of D st p = q & r = s & s = r & p ^ s = s holds p ^ s = q
consider d1 being Element of the carrier of X such that c , d1 // a , b and c , d // b , c and b , c // c , d ;
set U = I \! \mathop { + } , SS = I \! \mathop { + } ;
|. q .| ^2 = ( ( ( ( q `1 ) / |. q .| ) ^2 + ( ( q .| ) ^2 ) ) ^2 ) ^2 + ( ( q `2 ) ^2 ) ^2 + ( ( q `2 ) ^2 ) ^2 ) .= ( ( q `2 ) ^2 + ( q `2 ) ^2 ) ^2 + ( q `2 ) ^2 ;
let T being non empty TopStruct , x , y being Element of T , a , b being Element of T st x = a "\/" b & y in { x } & x <= y & y <= b holds x is Subset of T
dom ( ( the InternalRel of U1 ) * ( the Arity of S ) ) = dom ( the ResultSort of S ) & ( the ResultSort of S ) . ( ( the ResultSort of S ) . ( the carrier' of S ) = ( the ResultSort of S ) . ( the carrier' of S ) ;
dom ( h | X ) = dom ( h | X ) .= ( h | X ) /\ X .= ( h | X ) /\ X .= ( h | X ) /\ X .= ( h | X ) /\ X .= ( h | X ) /\ X .= ( h | X ) /\ X ;
for N , K being non empty TopSpace , f being Function of [: N , N :] , [: N , N :] , [: N , N :] , [: N , N :] , { f } :] , [: N , N :] = [: N , N :] ;
( mod ( u , m ) ) . ( i + 1 ) = ( ( mod m ) . ( i + 1 ) ) . ( ( m + 1 ) mod m ) . ( i + 1 ) ) ;
- ( q `1 ) / ( 1 + ( q `1 / |. q .| - cn ) ) <= - ( q .| / ( 1 + cn ) ) / ( 1 + cn ) or - ( q `1 / |. q .| - cn ) <= - ( q `1 / |. q .| - cn ) / ( 1 + cn ) ;
attr r1 = f & r2 = g & r1 = h & r2 = h implies r1 = g & r2 = h & r1 = h & r2 = h & h = f & h = h & h = g & h = h & h = f & h = h & h = h & h = g & h = h & h = h * ( h , h ) implies h = h * ( h , h , h )
Partial_Sums ( seq ) . m is bounded & ( for n be Nat holds seq . n = ( ( ||. seq .|| . m ) * ( ( ||. seq .|| . n ) * ( ||. seq .|| . m ) ) ) . m
attr a <> b & b <> c & a <> b & b , c is_collinear & a , b is_collinear & b , c is_collinear & a , c is_collinear & b , c is_collinear & a , c is_collinear & b , c is_collinear & a , b , c is_collinear & b , c is_collinear & a , c is_collinear & b , c is_collinear & a , c is_collinear & b , c is_collinear & a , c is_collinear & b , c is_collinear & a , c is_collinear & b , c is_collinear & b , c is_collinear & a , c is_collinear & b , c is_collinear & b , c is_collinear & a , c is_collinear & a
consider i , j being Nat such that p1 = [ i , j ] and i = [ i , j ] and i = [ i , j ] and j = [ i , j ] ;
|. p .| ^2 - ( 2 * ( ( 2 * ( ( ( ( ( ( ( - p ) / |. p .| - cn ) / ( ( 1 + cn ) / ( ( 1 + cn ) ) ^2 ) ) ) ) ) ) .| = |. p .| - ( ( ( ( - 1 ) / ( |. p .| - cn ) ) ) ) ^2 ) .| ;
consider p1 , p2 being Element of [: X , Y :] such that y = p1 ^ p2 and ( for n being Element of NAT holds p1 . n = p1 ^ p2 ) & ( for n being Element of NAT holds p1 . n = p2 . n ) implies ex k being Element of X st k <= n & k <= len p1 & k <= len p1 & k <= len p2 & k <= len p2 & k <= len p2 & k <= len p2 & k <= len p2 & k <= len p2 & k <= len p2 & k <= len p2 & k <= len p2 & k <= len p2 & k
<* A , B , C , D , E , F , G , G , D , F , G , J , J , F , J , M , J , F , J , M , N , N , F , J , J , M , F , J , M , N , F , J , J , M , N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N
( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 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 ) )
s |= All ( H , H ) implies s |= All ( H , H ) . ( len H )
len ( ( b + d ) + 1 ) = card ( ( b + d ) + d ) .= len ( b + d ) + d .= len ( b + d ) + d .= len ( b + d ) + d .= len ( b + d ) + d .= len ( b + d ) + d .= len ( b + d ) + d .= len ( b + d ) + d ;
consider z being Element of L1 such that z >= x and z >= y and z >= y ;
LSeg ( |[ ( ( ( TOP-REAL 2 ) | K1 ) | K1 ) | K1 , ( ( TOP-REAL 2 ) | K1 ) | K1 ) | K1 ) = { |[ ( ( ( ( TOP-REAL 2 ) | K1 ) | K1 ) ) : 1 <= cn & cn >= 0 & cn >= 0 & cn <= 0 & ( ( TOP-REAL 2 ) | K1 ) ^2 <= 0 & ( ( ( TOP-REAL 2 ) | K1 ) | K1 ) ^2 <= 0 & ( ( ( TOP-REAL 2 ) | K1 ) ^2 <= 0 & ( ( ( TOP-REAL 2 ) | K1 ) ^2 ) & ( ( ( TOP-REAL 2 ) | K1 )
lim ( ( ( f `| N ) `| N ) - ( ( f `| N ) `| N ) /* ( h ^\ N ) ) ) = ( ( f `| N ) - ( ( f `| N ) `| N ) `| N ) . ( ( ( f `| N ) `| N ) . ( ( f `| N ) `| N ) . ( ( ( f `| N ) `| N ) . ( ( ( f `| N ) `| N ) - ( ( f `| N ) `| N ) ) . ( ( ( h ) - ( ( f `| N ) `| N ) - ( ( f `| N ) `| N
P [ i , ( ( the InternalRel of A ) . ( i + 1 ) ) ] ;
for r be Real st 0 < r ex m be Nat st for n be Nat st n <= m holds |. ( seq . n ) - ( seq . n ) .| < r
let X being non empty set , P , Q being Subset of X , Q being Subset of X , Q being Subset of X st Q = P & Q is open & P is open & Q is open & Q is open holds P is open
Z c= dom ( ( ( ( ( ( ( - 1 ) (#) ( ( ( ( ( ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( ( ( 1 ) ) * ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( ) ) / 2 ) ) * ( ( ( ( ( ( ) ) ) * ( ( ( ( ( ( ) ) / 2 ) * ( ( ( ( ( ( ( ( ( ) ) ) * ( ( ( ( ( (
ex j being Nat st j in dom ( l ^ <* x *> ) & ( i = j ) & ( j = 1 implies j = 1 ) & ( i = j implies i = j ) & ( i = j ) implies i = j ) & ( i = j implies i = j ) implies i = j ) & ( i = j ) implies i = j ) & i = j
for u , v being VECTOR of V , w being VECTOR of V st 0 < u & u in W holds ( r * u + ( r * v ) ) * ( ( r * w ) ) in W
A , B , C , D is_collinear ;
- Sum <* v , u , w *> = - ( - u ) .= - ( - u ) .= - ( - u ) .= - ( - u ) .= - ( - u ) .= - ( - u ) .= - ( - u ) .= - ( - u ) .= - ( - u ) .= - u .= - u .= - u ;
( a := b ) . ( IC SCM R R ) = ( a := b ) . ( IC SCM R ) .= ( a := b ) . ( IC R ) . ( IC R R R ) ;
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 = I . x ;
let S being non empty RelStr , S1 , S2 being non empty Subset of S , S2 being non empty Subset of S1 , S2 being non empty Subset of S2 st S1 = S2 & S2 = S2 & S2 is open & S2 is open holds S1 is open
card X = 2 implies for x , y being Element of X st x in X & y in X & x in X & y in X holds x = y
( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) in rng Cage ( C , n ) ;
let T being decorated with T , p , q , r be Element of dom T , s be Element of dom T , t being Element of dom T , r be Element of T ;
[ i2 + 1 , j1 + 1 ] in Indices G & [ i2 + 1 , j1 + 1 ] in Indices G * ( i2 , j1 ) = G * ( i2 , j1 ) ;
cluster the InternalRel of k -> natural for Nat ;
dom F " { 0 } = the carrier of X & rng F = the carrier of X & rng F = the carrier of X & rng F = the carrier of X ;
consider C being finite Subset of V such that C c= A and card C = n and card C = n and card C = n and card C = n and card C = n ;
V is prime implies for X , Y being Subset of T st X in { {} } & Y c= X holds V c= Y
set X = { { F ( ) where F is Element of B ( ) : P [ F ] } , { F ( ) where F is Element of B ( ) : P [ F ( ) , F ( ) ] } ;
angle ( p1 , p2 , p3 ) = 0 .= 0 * ( 2 * ( 2 * ( ( 2 * ( ( ( 2 * ( ( 1 , width f ) ) ) + ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 *
- sqrt ( ( ( ( ( ( ( ( ( ( - ( q `1 / |. q .| - cn ) / |. q .| - cn ) / ( 1 + cn ) ) / ( ( 1 + cn ) ) ^2 ) ) ) ) = ( ( ( ( ( ( ( ( - ( ( q `1 / |. q .| - cn ) ) / ( ( 1 + cn ) ) ^2 ) ) ) ^2 ) ) ) .= ( ( ( ( ( ( ( ( - cn ) / ( ( 1 + cn ) / ( ( 1 + cn ) ) ^2 ) ) ^2 ) ) ^2 ) ) ^2 ) ^2 ) .= ( (
ex f being Function of I[01] , TOP-REAL 2 st f is continuous & f is continuous & rng f = P & f is one-to-one & f is one-to-one & f . 0 = p1 & f . 1 = p2 & f . 1 = p2 & f . 1 = p2 & f . 0 = p2 & f . 1 = p2 & f . 1 = p2 & f . 1 = p2 & f . 0 = p2 & f . 1 = p2 & f . 1 = p2 & f . 1 = p2 & f . 1 = p2 & f . 1 = p2 & f . 1 = p2 & f . 1 & f . 1 = p2 &
attr f is PartFunc of REAL , REAL means : Def3 : for r be Real st r in dom f holds f /. r = r * ( r * f /. r ) ;
ex r st x = |[ r , s ]| & r < s & s < 1 & G * ( 1 , 1 ) `2 < s & s < 1 } c= { p }
assume that f is FinSequence which elements which <= len G and t <= len G and t <= len G and G * ( 1 , j ) `1 <= G * ( 1 , j ) `1 ;
attr i in dom G means : Def3 : r * ( ( f . i ) - ( f . i ) ) = r * ( ( f . i ) - ( f . i ) ) ;
consider c1 , c2 being bag of o1 such that ( ( EmptyBag n ) /. ( k + 1 ) ) = <* c1 . ( k + 1 ) *> and c1 . ( k + 1 ) = <* c1 . ( k + 1 ) *> and c1 . ( k + 1 ) = <* c1 . ( k + 1 ) *> ;
|[ r1 , r2 ]| in { |[ r1 , r2 ]| : |[ r1 , r2 ]| in Indices G & |[ r1 , r2 ]| in Indices G & G * ( 1 , 1 ) `1 <= G * ( 1 , 1 ) `1 } ;
Cl ( X ^ Y ) = the carrier of ( X \/ Y ) .= ( the carrier of X ) \/ ( the carrier of Y ) .= ( the carrier of X ) \/ ( the carrier of Y ) .= ( the carrier of X ) \/ ( the carrier of Y ) .= ( the carrier of X ) \/ ( the carrier of Y ) ;
attr M1 = len ( M2 @ ) & width ( M2 @ ) = width ( M2 @ ) & width ( M2 @ ) = width ( M2 @ ) ;
consider g2 being Real such that 0 < g2 & g2 < g2 & g2 < x0 & g2 < g2 & g2 g2 g2 g2 g2 g2 g2 g2 g2 g2 g2 g2 g2 g2 g2 g2 g2 g2 g2 g2 is convergent & lim g2 = g2 g2 g2 g2 g2 g2 g2 g2 g2 g2 g2 g2 g2 & g2 is convergent & lim ( g2 , g2 ) = g2 g2 g2 g2 ) ;
assume x < sqrt ( a * b ) + sqrt ( a * b ) * sqrt ( a * b ) * sqrt ( a * b ) * sqrt ( a * b ) * sqrt ( a * b ) * sqrt ( a * b ) ;
( ( ( ( ( 3 ) --> ( 0 ) ) +* ( ( 3 , 1 ) .--> ( ( 3 , 1 ) ) ) ) ) . ( ( 3 , 1 ) ) . ( ( 3 , 1 ) ) = ( ( ( 3 , 1 ) --> ( 3 , 1 ) ) . ( ( 3 , 1 ) --> ( 3 , 1 ) ) . ( ( 3 , 1 ) ) . ( ( 3 , 1 ) ) ) & ( ( ( ( 3 , 1 ) ) . ( ( 3 , 1 ) ) . ( ( 3 , 1 ) ) . ( ( 3 , 1 ) ) . ( ( 3 , 1 ) )
for i , j st [ i , j ] in Indices ( M * ( i , j ) , ( i , j ) ) & [ i , j ] in Indices ( M * ( i , j ) ) & ( ( M * ( i , j ) ) * ( i , j ) ) * ( i , j ) ) * ( i , j ) = ( M * ( i , j ) ) * ( i , j ) ) * ( i , j )
let f being FinSequence of NAT , i being Element of NAT , j being Nat st i in dom f & j <= n & i <= len f holds f . i = ( Sum f ) . ( i + 1 )
assume F = { [ a , b ] : a in B & b in B & a in C } ;
b2 * ( q - ( q - ( q - ( q - ( q - q ) ) ) ) = 0. TOP-REAL n + ( q - q ) * ( ( q - ( q - q ) ) * ( ( q - q ) - ( q - q ) ) * ( ( q - q ) - ( q - q ) ) * ( ( q - q ) - ( q - q ) ) * ( ( q - q ) - ( q - q ) ) .= ( ( q - q ) - ( ( q - q ) - ( ( q - q ) ) * ( ( q - q ) - ( q - q ) ) * ( ( ( q - q ) - ( q
card F = card ( D /\ { x } ) ;
attr attr attr attr attr : : : for n be Nat holds ( for m be Nat holds ( for n be Nat holds ( n >= 1 ) * ( ( n + 1 ) * ( ( n + 1 ) * ( ( n + 1 ) * ( ( n + 1 ) ) ) ) ) & ( n >= 1 ) * ( ( n + 1 ) * ( n + 1 ) ) ) = ( n + 1 ) * ( ( n + 1 ) * ( ( n + 1 ) ) & ( n + 1 ) * ( n + 1 ) ) * ( ( n + 1 ) * ( n + 1 ) ) * ( n + 1 ) * ( n + 1 ) ) * (
dom ( ( ( ( TOP-REAL 2 ) | K1 ) | K1 ) | K1 ) = ( ( ( TOP-REAL 2 ) | K1 ) | K1 ) | K1 ) | K1 ) .= K1 ;
[ X \to Z ] is full & Z is full implies X is full
( G * ( i , j ) `2 <= ( G * ( i , j ) `2 ) `2 & ( G * ( i , j ) `2 <= ( G * ( i , j ) `2 ) `2 ;
synonym m1 c= m + ( n + 1 ) for p , q being Element of n + 1 st p in P & q in P & ( for k being Nat st k in P & k <= n holds ( for k being Nat st k in P & k <= m holds ( ( P [ k ] implies P [ k + 1 ] ) implies ( P [ k + 1 ] ) & ( for n being Nat st n <= m holds ( P [ n ] implies P [ n ] implies P [ n + 1 ] implies P [ n + 1 ] implies P [ n + 1 ] ) implies P [ n + 1 ] ) implies P [ n + 1 ] ) implies P [ n + 1 ] ) & ( for n + 1 ]
consider a being Element of B such that x = F ( a ) and a in { [ a , b ] } ;
synonym -multiplicative loop s for for for for carrier of [: the carrier of L , the carrier of L , the carrier of L , the carrier of L :] is [: the carrier of L , the carrier of L , the carrier of L :] & the carrier of L is [: the carrier of L , the carrier of L , the carrier of L , the carrier of L :] is [: the carrier of L , the carrier of L , the carrier of L , the carrier of L , the carrier of L ;
AffineMap ( a , b , c , d ) . ( b + c ) = b + c .= b + c .= b + d ;
cluster strict for Subspace of V , W , L , L be Subset of V ;
( ( ( ( - 1 ) (#) ( ( ( ( ( ( - 1 ) (#) ( ( ( ( ( ( - 1 ) (#) ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( - 1 ) / 2 ) * ( ( ( - 1 ) / 2 ) * ( ( ( - 1 ) / 2 ) * ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( - 1 ) / 2 ) ) ) ) ) ) ) ) ) ) ) ) ) )
eval ( a , x ) = eval ( a , x ) * eval ( a , x ) .= a * eval ( x , x ) ;
assume the TopStruct of S = the TopStruct of T & the TopStruct of S = the TopStruct of T ;
assume that 1 <= k + 1 and k <= len ( w ^ <* q *> ) and ( w ^ <* q *> ) . ( k + 1 ) = ( w ^ <* q *> ) . ( k + 1 ) ;
2 * ( ( a |^ ( n + 1 ) ) + ( b |^ ( n + 1 ) ) * ( a |^ ( n + 1 ) ) + ( b |^ ( n + 1 ) ) * ( a |^ ( n + 1 ) ) + ( b |^ ( n + 1 ) ) * ( b |^ ( n + 1 ) ) ) >= a |^ ( n + 1 ) + b |^ ( n + 1 ) ;
M , v |= All ( x , H ) => All ( x , H ) ;
assume that f is_differentiable_on Z and for x st x in Z holds f . x = f . x - r / 2 ;
let G being _Graph , W being Subset of G , W being Subset of G , a , b being Point of G st a in W & b in W & b in W & a in W & b in W & b in W & W is open & W is there being Subset of G st W is open & W is open & W is open & W is open & W is open & W is open & W is open & W is open & ( for W being Subset of G st W is open & ( for W being Subset of G st W is open & ( for W is open & ( for W being Subset of G st W is open & ( ex W being Subset of G st W is open & ( for W is open & ( for W being Subset of G st
empty 01 is not empty iff ex y being Element of Y st y is not empty & y is not empty & not empty & not y is not empty & not empty or ex z being Element of Y st z is not empty & not empty & not empty & not empty & not empty iff ex y being Element of Y st y is not empty & not empty & not empty & ex z being Element of Y st z is not empty & z is not empty & z is not empty & not empty & not empty iff iff ex z is not empty & z is not empty & not empty iff iff iff ex y is not empty & z is not empty & z is not empty & z is not empty & z is not empty & not empty iff iff iff ex z is not empty & z is
the carrier of ( TOP-REAL 2 ) | K1 = K1 & ( ( GoB f ) | K1 ) | K1 = K1 & ( GoB f ) | K1 or ( GoB f ) | K1 = K1 or ( GoB f ) | K1 = K1 or ( GoB f ) | K1 = K1 & ( GoB f ) | K1 = K1 & ( GoB f ) | K1 = K1 & ( GoB f ) | K1 = K1 & ( GoB f ) | K1 = K1 & ( GoB f ) | K1 = K1 & ( GoB f ) | K1 or ( GoB f ) | K1 = K1 & ( GoB f ) | K1 or ( GoB f ) | K1 = K1 & ( GoB f ) | K1 = K1 & ( GoB f ) | K1 = K1 or ( GoB f ) | K1 =
let G1 , G2 being Group , F , G be Element of [: the carrier of G , the carrier of G :] , the carrier of G , the carrier of H :] ;
UsedIntLoc ( f , 2 ) = { ( f , 2 ) . ( 3 + 1 ) } ;
for f1 , f2 being FinSequence of F st f1 is FinSequence of F & f2 is FinSequence of F & f1 is FinSequence of F & f2 is FinSequence of F & f1 is FinSequence of F & f2 is FinSequence of F & f1 is FinSequence of F & f2 is FinSequence of F & f2 is FinSequence of F & f1 is FinSequence of F
sqrt ( ( ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( - 1 ) / 2 ) / 2 ) ) ) ) ) ) ) ) ) ) ) ) ^2 ) = ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( - 1 ) / 2 ) * ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( - 1 ) / 2 ) / 2 ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
let x1 , x2 , x3 , x4 , x5 , x5 , %> = <* x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x6 , x5 , x5 , x5 , x5 , x6 , x5 , x5 , x5 , x6 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 ,
for x st x in dom ( ( ( G . n ) * ( ( G . n ) * ( ( G . n ) * ( ( G . n ) * ( ( G . n ) * ( ( G . n ) * ( ( G . n ) * ( ( G . n ) * ( ( G . n ) * ( ( G . n ) ) ) ) ) ) ) ) ) ) = ( ( G . n ) * ( ( G . n ) * ( ( G . n ) * ( ( G . n ) ) ) ) ) ) ;
let T being non empty TopStruct , P be Subset-Family of T , Q being Subset-Family of T , Q being Subset-Family of T st Q c= P & Q is open & Q is open holds Q is open
( a 'or' b ) . x = 'not' a . x 'or' ( b 'or' c ) . x .= TRUE 'or' ( c ) . x .= TRUE ;
for e being set st e in A & e in B & e in A & e in B & e in A & e in B & e in A & e in A & e in B & e in A & e in B & e in A & f in A & f in A & f in A & f in A & f in A & f in A & f is one-to-one & f is one-to-one & f is one-to-one & f is one-to-one & f is one-to-one & f is one-to-one & f is one-to-one & f is one-to-one & g is one-to-one & g is one-to-one & g is one-to-one & g is one-to-one & g is one-to-one & g is one-to-one & g is one-to-one & g is one-to-one & g is one-to-one & g is one-to-one & g is one-to-one & g is one-to-one
for i be set st i in the carrier of S & i in the carrier of S & ( for n being Nat st n in dom F holds F . n = f . ( n + 1 ) ) holds F . ( n + 1 ) = f . ( n + 1 )
for v , w being Element of l st for y , z being Element of l st for y , z being Element of l holds ( J . ( y , z ) ) . ( y , z ) = J . ( y , z )
card D = card ( ( ( ( i + 1 ) - 1 ) / 2 ) ) .= 2 * ( ( i - 1 ) - 1 ) / 2 ) .= 2 * ( ( i - 1 ) - 1 ) / 2 ) .= 2 * ( i - 1 ) ;
IC Exec ( i , s ) = ( s +* ( i , s ) ) . ( IC s ) .= ( s +* ( i , s ) ) . IC s .= ( s +* ( i , s ) ) . IC s .= ( s +* ( i , s ) ) . IC s ;
len f -' 1 = len f -' 1 .= len f -' 1 .= len f -' 1 ;
for a , b , c being Element of NAT st 1 <= a & b <= c & c <= b holds a + c <= b + c
let f being FinSequence of TOP-REAL 2 , p , q , r being Real st p in LSeg ( f , i ) & q = f /. ( i + 1 ) holds f /. ( i + 1 ) = f /. ( i + 1 )
lim ( ( ( curry ( k ) ) # x ) = ( lim ( ( ( ( ( vseq ^\ k ) # x ) ) ) ) ) . n + ( ( ( vseq # x ) ) . n ) ) . n ;
z2 = g /. ( i + 1 ) .= g /. ( i + 1 ) .= g /. ( i + 1 ) ;
[ f . 0 , f . ( 0 + 1 ) ] in [: the carrier of G , { the carrier of G } , { the carrier of G } ;
let G being Subset-Family of B , F being Subset-Family of A , G being Subset-Family of B st G = { [ F , G ] where F , G ] where F is Element of A ( ) , G is Subset of B ( ) : F is finite } ;
CurInstr ( P1 , Comput ( P1 , s1 , m ) ) = ( CurInstr ( P1 , s1 ) ) . ( IC Comput ( P1 , s1 , m ) ) .= ( halt SCMPDS ) . IC Comput ( P2 , s2 , m ) ) . IC Comput ( P2 , s2 , m ) ) ;
assume a on M & b on M & b on M & c on M & d on M & a on M & b on M & a on M & b on M & b on M & d on M & a on M & b on M & b on M & c <> M & b <> M & b <> d & b <> d & b <> d & b <> d & b <> d & b <> d & b <> c & b <> d & b <> d & b <> c & b <> c & b <> c & b <> c & b <> c & b <> c & b <> d & b <> c & b <> c & b <> d & b <> c & d <> d & b <> d & b <> c & d <> d & d <> d & d <> d & d <> d & d <> d & d <> d & d <> d & d <> d &
assume that T is with_with_BB) and for F being Subset-Family of T st F is open & F is open & F is open & F is open & F is open & F is open & F is open & F is open & F is open & F is open & F is open & F is open & F is open & F is open & ( for n st n >= 1 & n >= 1 holds F is open implies implies ( for n st n >= 1 & n >= 1 holds F is open & ( for n st n >= 1 & n >= 1 & n <= n & n >= 1 implies implies implies implies implies ( for n >= 1 & n >= 1 implies implies implies for n >= 1 implies implies for n >= 1 & n >= 1 implies for n >= 1 & n >= 1 & n >= 1 & n <= n & n >= 1 & n >= 1 & n >= 1 & n >= 1
for g1 , g2 being Real st g1 in ]. x0 - r , x0 + r .[ & g1 in ]. x0 - r , x0 + r .[ holds g1 in ]. x0 - r , x0 + r .[
exp_R /. ( ( - 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
F . i = F /. ( i + 1 ) .= <* b /. ( i + 1 ) *> ^ <* b *> . ( i + 1 ) .= <* b *> /. ( i + 1 ) .= <* b *> /. ( i + 1 ) ;
ex y being set st y = f . n & for n being Nat holds y in dom f & f . n = R ( n ) & f . n = R ( n ) ;
func f * F -> FinSequence of V means : Def3 : for i , j being Nat st i in dom F holds it . ( i + 1 ) = F . ( i + 1 ) * F . ( i + 1 ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 } = { x1 , x2 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 } \/ { x5 , x5 , x5 , x5 , x5 , x5 } \/ { x5 , x5 , x5 , x5 , x5 } ;
for n being Nat , x , y being Element of NAT , n being Nat st x = h . ( n + 1 ) & y in rng h & n <= len h holds h . n = o ( x , y , n )
ex S1 being Element of [: A , B :] st [ S1 , S2 ] is [: A , B :] & [ S1 , S2 ] is ^ & [ S1 , S2 ] is ^ & [ S1 , S2 ] is ^ ^ <* S1 , S2 *> ;
consider P being FinSequence of [: { \mathbb R } , { \mathbb R } :] such that P . ( i + 1 ) = P . ( i + 1 ) and P . ( i + 1 ) = Q * ( i + 1 ) ;
let T being strict TopStruct , R be Subset-Family of T , S be Subset of T ;
assume that f is PartFunc of dom f , dom ( r (#) f ) and r (#) f is_differentiable_on Z ;
defpred P [ Nat ] means for F being FinSequence of REAL , G being Matrix of n , REAL st len F = $1 & G is one-to-one & G is one-to-one & G is one-to-one & G is one-to-one holds Sum ( F ) = Sum ( G )
ex j st 1 <= j & j < len GoB f & ( GoB f ) * ( i , j ) `1 <= s & s * ( i , j ) `1 <= s & s * ( i , j ) `2 <= s * ( i , j ) `2 & s * ( i , j ) `2 <= s * ( i , j ) `2 & s * ( i , j ) `2 <= s * ( i , j ) `2 & s * ( i , j ) `2 <= s * ( i , j ) `2 & s * ( i , j ) `2 <= s * ( i , j ) `2 & s * ( i , j ) `2 & s * ( i , j ) `2 <= s * ( i , j ) `2 & s * ( i , j ) `2 <= s * ( i , j ) `2 <= s * ( i , j ) `2 & s * ( i , j + 1 ) `2
defpred U ( set , set ) = { F ( ) where F is Subset-Family of T : for n being Nat st n >= $1 & n <= $1 holds F . n is open } ;
for p1 being Point of TOP-REAL 2 st p1 `1 <= p2 & p2 `1 <= 1 holds LE p1 , p2 , P
f in St ( H ) & for y st y in { x } holds g . y in { y } & g . y = f . y
ex p2 being Point of TOP-REAL 2 st x = p2 & p2 `1 <= p2 `1 & p2 `2 <= p2 `2 & p2 `2 <= - 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2
assume for d being Element of NAT st d <= ( d + 1 ) * ( ( d + 1 ) * ( ( d + 1 ) * ( d + 1 ) ) ) & ( ( d + 1 ) * ( d + 1 ) ) * ( d + 1 ) ) * ( d + 1 ) ) = ( d + 1 ) * ( d + 1 ) * ( d + 1 ) ;
assume that s <> t and not s is Point of Closed-Interval-TSpace ( x , r ) and not ex e being Point of Closed-Interval-TSpace ( x , r ) st e in [. x , r .[ & e in [. x , r .[ ;
given r such that 0 < r and for n st n in dom r holds |. ( f /. n ) - f /. ( n + 1 ) .| < r ;
( p | ( ( len p ) | ( len p ) ) ) | ( ( len p ) | ( len p ) ) = ( ( len p ) | ( len p ) ) | ( ( len p ) ) ;
assume that x + h in dom ( ( cos * ( ( ( ( ( ( ( ( - h ) / 2 ) * ( ( ( ( ( - h ) / 2 ) * ( ( ( - h ) / 2 ) * ( ( ( - h ) / 2 ) * ( ( ( - h ) / 2 ) * ( ( ( - h ) / 2 ) * ( ( ( ( - h ) / 2 ) * ( ( ( - h ) / 2 ) * ( ( ( ( ( ( ( ( ( ( 2 ) ) * ( ( ( ( - h ) / 2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) = ( ( ( ( ( - h ) / 2 ) * ( ( ( ( ( ( ( ( ( ( ( 2 * ( ( ( ( ( ( ( ( 2 ) * ( ( ( ( 2 * ( ( - h ) * (
assume that i in dom A and len A > 0 and A is Matrix of n , len A , len A , n , D , k , Nat , Nat , Nat , Nat st k > 0 & A is \times ( A , n ) & A * ( i , j ) = A * ( i , j ) ;
for i being non zero Element of NAT st i in Seg n holds h . i = <* <* 1_ L *> . i , h . ( i + 1 ) *>
( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) /. L~ f ) ) ) /. ( m + 1 ) ) ) /. ( m + 1 ) ) ) ) ) ) ) ) ) ) ) /. ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( m m ) /. 1 ) ) /. ( m + 1 ) ) /. ( m + 1 ) ) /. ( m + 1 ) ) /. ( m + 1 ) -' 1 ) -' 1 ) -' 1 ) -' 1 ) -' 1 ) -' 1 ) -' 1 ) -' 1 ) -' 1 ) ) ) ) ) /. ( ( m + 1 ) -' 1 ) ) ) ) /. ( ( m ) -' 1 ) -' 1 ) ) /. ( ( ( 1 ) -' 1 ) -' 1 ) ) ) /. ( ( (
cluster ( ( for PartFunc of REAL , REAL , REAL ) . x ) `| Z ) . x = ( ( ( \HM { the } \HM { function } ) * ( ( \HM { the } \HM { function } \HM { cos } ) `| Z ) . x ) & ( ( \HM { the } \HM { f } ) `| Z ) . x = ( ( \HM { the } \HM { function } ) `| Z ) . x ) . x ;
consider R1 , R2 being Real such that R1 = ( ( |. F .| * ( i + 1 ) ) | ( i + 1 ) ) and ( ( ( |. F .| * ( i + 1 ) ) | ( i + 1 ) ) | ( i + 1 ) ) = ( ( ( ( F ) | ( i + 1 ) ) | ( i + 1 ) ) | ( i + 1 ) ) ;
ex k being Element of NAT st k = k & 0 < r & r < 1 & r < 1 & r < 1 & r < 1 & r < 1 & r in dom f & f /. k = r & f /. k = r & f /. k = r & f /. k = r & f /. k = r & r = r & r = r & r = r & f /. k = r & r = r & r = r & f /. k = r & f /. k = r & ex r be Real st r = r & f /. k = r & f /. k = r & f /. k = r & f /. k = r & f /. k = r & f /. k = r & f /. k = r & f /. k = r & f /. k = r & f /. k = r & f /. k = r & f /. k = r & f /. k = r &
x in { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 } iff x in { x1 , x2 , x3 , x4 , x5 , x5 }
( G * ( i , j ) `2 = ( G * ( i , j ) `2 ) `2 .= ( G * ( i , j ) `2 ) `2 .= ( G * ( i , j ) `2 ) `2 ;
f1 * ( ( the Arity of S ) * ( the Arity of S ) ) = ( ( the Arity of S ) * ( the Arity of S ) ) . o .= ( the ResultSort of S ) . o ;
func from { [ T , P ] : p in P & P [ p ] } -> Subset of T ;
F /. ( k + 1 ) = F . ( k + 1 ) .= Fq . ( k + 1 ) .= Fq . ( k + 1 ) ;
let A being Matrix of n , K , D , k , D , k , m , n , m , k , m , n , m , k , m , k , n , m , k , m , n , m , k , n , m , k , m , k be Nat st n > k & k <= m & k <= n & k <= n & k <= n & k <= n & k <= n & n <= m holds ( ( ( n , m ) * ( ( n , m ) * ( ( n , m ) ) * ( ( n , m ) ) * ( ( n , m ) * ( ( n , m ) ) * ( ( n , m ) ) = ( n , m ) * ( ( n , m ) ) * ( ( ( n , m ) ) * ( ( n , m ) * ( ( n , k ) * ( ( n , m ) ) ) *
seq . ( k + 1 ) = 0. V + ( seq . ( k + 1 ) ) .= ( seq . ( k + 1 ) ) . ( k + 1 ) .= ( seq . ( k + 1 ) ) . ( k + 1 ) ;
assume x in ( the InternalRel of C ) . ( the carrier of C ) & y in the carrier of C & z in the carrier of C & z in the carrier of C ;
defpred P [ Element of NAT ] means ( for k st k <= $1 holds ( ( ( f /. k ) /. ( $1 + 1 ) ) /. ( $1 + 1 ) ) `1 = ( ( ( ( ( f /. k ) /. ( $1 + 1 ) ) ) `1 ) `1 ;
assume that 1 <= k and k + 1 <= len f and f /. k = G * ( i , j ) `1 and f /. k = G * ( i , j ) `1 and f /. k = G * ( i , j ) `1 ;
assume that s < 1 and p `1 >= 0 and p <> 0. TOP-REAL 2 and p <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q
let M being non empty TopSpace , x , y being Point of M , r being Real st x = r & y in M & x = r & y in M & x in M & y in M & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open & M is open
defpred P [ Element of omega ] means ( ( for n st n >= 1 holds ( ( ( n + 1 ) / ( n + 1 ) ) * ( ( ( n + 1 ) * ( ( n + 1 ) ) ) ) `| Z ) . $1 = ( n + 1 ) * ( ( n + 1 ) * ( ( n + 1 ) * ( ( n + 1 ) * ( ( n + 1 ) * ( ( n + 1 ) ) ) ) ) . $1 ;
defpred P [ Nat , Real ] means ( for n st n in dom f holds ( f /. n ) `1 <= ( f /. $1 ) `1 ) `1 & ( f /. ( n + 1 ) `1 <= ( f /. n ) `1 ) `1 & ( f /. ( n + 1 ) ) `1 <= ( f /. n ) `1 ) `1 ;
( f ^ mid ( g , 2 , len g ) ) . ( len g + 1 ) = ( g ^ mid ( g , 2 , len g + 1 ) ) . ( len g + 1 ) .= ( g ^ mid ( g , 2 , len g + 1 ) ) . ( len g + 1 ) .= ( g ^ mid ( g , 2 , len g + 1 ) ) . ( len g + 1 ) ;
sqrt ( 1 - ( 2 * ( ( 2 * ( ( ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( - 1 / 2 ) / 2 ) * ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( - 1 ) / 2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) = ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
defpred P [ Nat ] means ( the carrier of G ) \/ the carrier of G ) \/ the carrier of G = the carrier of G & the carrier of G = the carrier of G implies the carrier of G = the carrier of G ) & G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict or G is strict G is strict or G is strict or G is strict
assume that f . ( u + 1 ) in Ball ( u , r ) and f . ( u + 1 ) = ( u + 1 ) * ( u + 1 ) ;
defpred P [ Element of NAT ] means ( ( Partial_Sums ( ( F ) . $1 ) . ( n + 1 ) ) . ( ( n + 1 ) ) . ( ( n + 1 ) ) . ( ( n + 1 ) ) = ( Partial_Sums ( F ) ) . ( n + 1 ) ) . ( n + 1 ) ;
for x being Element of product F st x in dom ( the InternalRel of F ) & x in dom ( the InternalRel of F ) & x in dom ( the InternalRel of F ) & ( for i being Nat st i in dom ( the InternalRel of F ) . i holds x in dom ( the InternalRel of F ) & ( for i be Nat st i in dom ( the InternalRel of F ) . i ) holds x in dom ( the InternalRel of F )
( x " ) * ( ( x " ) * ( x " ) ) = ( ( x " ) * ( x " ) ) * ( x " ) .= ( x " ) * ( x " ) * ( x " ) .= ( x " ) * ( x " ) * ( x " ) .= ( x " ) * ( x " ) ;
DataPart Comput ( P +* I , s , k ) = DataPart Comput ( P +* I , s , k ) .= DataPart Comput ( P +* I , s , k ) ;
given r such that 0 < r and for g st g in dom f & g in dom f holds g in dom ( f | X ) ;
assume X c= dom ( ( f1 (#) f2 ) `| Z ) & ( f1 (#) f2 ) `| Z ) . x = ( f1 (#) f2 ) . x & ( f1 (#) f2 ) . x = ( f1 (#) f2 ) . x & ( f1 (#) f2 ) . x = ( f1 (#) f2 ) . x & ( f1 (#) f2 ) . x = ( f1 (#) f2 ) . x & ( f1 (#) f2 ) . x = ( f1 (#) f2 ) . x & ( f1 (#) f2 ) . x = ( f1 (#) f2 ) . x = ( f1 (#) f2 ) . x & ( f1 (#) f2 ) . x & ( f1 (#) f2 ) . x & ( f1 (#) f2 ) . x = ( f1 (#) f2 ) . x & ( f1 (#) f2 ) . x & ( f1 (#) f2 ) . x = ( f1 (#) f2 ) . x & ( f1 (#) f2 ) . x = ( f1 (#) f2 ) . x = ( f1 (#) f2 ) . x = ( f1 (#) f2 ) . x = ( f1 (#) f2 ) . x = ( f1 (#) f2
let L being continuous continuous continuous continuous continuous non empty Poset , X , Y being Subset of L , a , b being Element of L st X = { b } & b in X & a in X holds b is open
consider i being Element of NAT such that i in dom A and A /. i = m \ast ( p *' ( q *' p ) ) ;
( ( ( f1 - f2 ) /* seq ) ^\ k ) . n = ( f1 /* seq ) . n ) - ( f1 /* seq ) . n ) .= ( f1 /* seq ) . n - ( f1 /* seq ) . n ;
ex p1 being Element of [: A ( ) , A ( ) :] st F ( p ) = g ( p ) & for n being Nat holds F ( n ) = p ( n ) & F ( n ) = g ( n ) ;
( mid ( f , i , len f ) ) /. ( i + 1 ) = ( f /. ( i + 1 ) ) /. ( i + 1 ) .= ( f /. ( i + 1 ) ) /. ( i + 1 ) .= ( f /. ( i + 1 ) ) `1 ;
( p ^ q ) . ( n + 1 ) = ( ( p ^ q ) . ( n + 1 ) ) . ( ( len p + 1 ) + 1 ) .= ( len p + 1 ) + ( len q ) ;
len ( D2 , indx ( D2 , D1 , j1 ) + 1 ) = len D2 + 1 ;
x * z = ( x * ( ( ( ( ( - x ) * z ) * z ) ) * ( ( - x ) * z ) ) .= ( x * ( - x ) * z ) * ( ( - x ) * z ) .= x * ( - x ) * ( - x ) .= x * ( - x ) .= x * ( - x ) * ( - x ) .= x * ( - x * z ) .= x * ( - x * ( - x * z ) .= x * ( - x * z ) .= x * ( - x * z ) .= x * ( - x * z ) .= x * ( - x * z ) .= x * ( - x * z ) .= x * ( - x * z ) .= x * ( - x * ( - x * ( - x * z ) * ( - x * z ) .= x * ( - x * z ) .= x * ( - x * ( - x * z ) .= x * ( - x * z ) .= x * ( - x * ( - x * z ) .= x * ( - x
v . ( <* x , y *> . ( i + 1 ) ) = <* <* y *> . ( i + 1 ) *> . ( ( ( <* y *> . ( i + 1 ) ) - ( <* z *> . ( i + 1 ) ) ) ) ) . ( ( <* y *> . ( i + 1 ) ) ) . ( ( i + 1 ) ) - ( ( i + 1 ) ) ) . ( ( i + 1 ) ) ) ;
i * i = <* 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
Sum ( L * F ) = Sum ( L * F ) .= Sum ( L * F ) + Sum ( ( L * F ) ) .= Sum ( L * F ) + Sum ( ( L * F ) * F ) .= Sum ( L * F ) + Sum ( L * F ) .= Sum ( L * F ) + Sum ( L * F ) .= Sum ( L * F ) + Sum ( L * F ) .= Sum ( L * F ) + Sum ( L * F ) + Sum ( L * F ) .= Sum ( L * F ) + Sum ( L * F ) .= Sum ( L * F ) + Sum ( L * F ) .= Sum ( L * F ) + Sum ( L * F ) .= Sum ( L * F ) ;
ex r be Real st for Y be Subset of X , Y being Subset of X st Y in Y & Y c= X holds r in Y
( ( GoB f ) * ( i , j ) `2 = f /. ( k + 1 ) `2 & ( GoB f ) * ( i , j ) `2 = f /. k & ( GoB f ) * ( i , j + 1 ) `2 = f /. k ) `2 ;
( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( 1 / 2 ) / 2 ) * ( ( ( ( - 1 ) / 2 ) ) ) ) ) ) / ( ( ( ( - 1 ) / 2 ) * ( ( ( ( - 1 ) / 2 ) * ( ( ( - 1 ) / 2 ) * ( ( - 1 ) / 2 ) ) ) ) ) ) / ( ( ( - 1 ) / 2 ) * ( ( ( ( ( 1 - 1 ) / 2 ) * ( ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( 1 - 1 ) / 2 ) ) ) ) ) ) ) / ( ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( - 1 ) / 2 ) ) ) ) ) ) ) ) ) ) / ( ( ( ( ( ( ( - 1 ) / 2 ) / 2 ) ) ) ) ) ) = ( ( ( ( ( ( ( ( ( ( ( (
x- ( a * b ) + ( - b * c ) * ( - b * c ) < 0 & - b * c + b * c < 0 ;
cluster inf { X where X is Subset of L : X in F & X is directed } -> directed Subset of L ;
( ( ( ( B , i ) --> ( j , i ) ) . ( j , i ) ) . ( j , i ) = ( i |-> ( j , i ) ) . ( j , i ) ) . ( j , i ) .= ( i |-> ( j , i ) ) . ( j , i ) ) . ( j , i ) .= ( i |-> ( j , i ) ) . ( j , i ) ) . ( j , i ) ;