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contradiction ;
contradiction ;
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contradiction ;
contradiction ;
contradiction ;
contradiction ;
thesis ;
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thesis ;
thesis ;
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thesis ;
contradiction ;
contradiction ;
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contradiction ;
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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contradiction ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
B ;
a <> c ;
T c= S ;
D c= B ;
c ;
b ;
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 dense ;
a in A ;
1 < x ;
S is finite ;
u in I ;
z << z ;
x in V ;
r < t ;
let t ;
x c= y ;
a <= b ;
m in NAT ;
assume f is prime ;
not x in Y ;
z = +infty ;
let k be Nat ;
K is being_line ;
assume n >= N ;
assume n >= N ;
assume X is condition ;
assume x in I ;
q is \upharpoonright ;
assume c in x ;
p > 0 ;
assume x in Z ;
assume x in Z ;
1 <= k12 ;
assume m <= i ;
assume G is prime ;
assume a divides b ;
assume P is closed ;
d > 0 ;
assume q in A ;
W is bounded ;
f is Int one-to-one ;
assume A is discrete ;
g is special ;
assume i > j ;
assume t in X ;
assume n <= m ;
assume x in W ;
assume r in X ;
assume x in A ;
assume b is even ;
assume i in I ;
assume 1 <= k ;
X is non empty ;
assume x in X ;
assume n in M ;
assume b in X ;
assume x in A ;
assume T c= W ;
assume s is 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 DecoratedTree ;
let d be element ;
let t be element ;
let x be element ;
let x be element ;
let s be element ;
k <= m-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 Function of E , F ;
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 initial ;
Q halts_on s ;
x in SCMPDS ;
M < m + 1 ;
T2 is open ;
z in b +^ a ;
R2 is well-ordering ;
1 <= k + 1 ;
i > n + 1 ;
q1 is one-to-one ;
let x be trivial set ;
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_in x ;
let x0 ;
let E be Ordinal ;
o is_transformable_to o1 ;
O <> O ;
let r be Real ;
let f be FinSequence ;
let i be Nat ;
let n be Nat ;
Cl A = A ;
L c= Cl L ;
A /\ M = B ;
let V be RealUnitarySpace , W be Subspace of V ;
not s in Y |^ 0 ;
rng f <= w ;
b "/\" e = b ;
m = m2 ;
t in h . D ;
P [ 0 ] ;
assume z = x * y ;
S . n is bounded ;
let V be RealUnitarySpace , A be Subset of V ;
P [ 1 ] ;
P [ {} ] ;
C1 is component ;
H = G . i ;
1 <= i + 1 ;
F . m in A ;
f . o = o ;
P [ 0 ] ;
aa <= real ;
R [ 0 ] ;
b in f .: X ;
assume q = q2 ;
x in [#] V ;
f . u = 0 ;
assume e1 > 0 ;
let V be RealUnitarySpace , W be Subspace of V ;
s is trivial & s is trivial ;
dom c = Q ;
P [ 0 ] ;
f . n in T ;
N . j in S ;
let T be complete LATTICE , A be Subset of T ;
the ObjectMap of F is one-to-one ;
sgn x = 1 ;
k in support a ;
1 in Seg 1 ;
rng f = X ;
len T in X ;
vs < n ;
Smax is bounded ;
assume p = p2 ;
len f = n ;
assume x in P1 ;
i in dom q ;
let U ;
p-25 = c ;
j in dom h ;
let k ;
f | Z is continuous ;
k in dom G ;
UBD C = B ;
1 <= len M ;
p in right_open_halfline x ;
1 <= j1 & j1 <= j2 ;
set A = \mathclose ;
card a [= c ;
e in rng f ;
cluster B \oplus A -> empty ;
H has has has H ;
assume n <= m ;
T is increasing ;
e1 <> e1 ;
Z c= dom g ;
dom p = X ;
H is proper ;
i + 1 <= n ;
v <> 0. V ;
A c= Affin A ;
S c= dom F ;
m in dom f ;
let X be set ;
c = sup N ;
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 , A :] ;
C c= Cmax ( C , m ) ;
m1 <> {} ;
let x be Element of Y ;
let f be \mathclose extended -valued Function , x be set ;
not n in Seg 3 ;
assume X in f .: A ;
assume that p <= n and p <= m ;
assume not u in { v } ;
d is Element of A ;
A |^ b misses B ;
e in v .vertices() ;
- y in I ;
let A be non empty set , a be Element of A ;
P0 = 1 ;
assume r in F . k ;
assume f is simple ;
let A be infinite set ;
rng f c= NAT ;
assume P [ k ] ;
f6 <> {} ;
let o be Ordinal ;
assume x is sum of sum ;
assume not v in { 1 } ;
let I1 ;
assume that 1 <= j and j < l ;
v = - u ;
assume s . b > 0 ;
d1 in LSeg ( a , b ) ;
assume t . 1 in A ;
let Y be non empty TopSpace , a be Point of Y ;
assume a in ]. s , t .[ ;
let S be non empty RelStr ;
a , b // b , a ;
a * b = p * q ;
assume x , y are_the space ;
assume x in [#] ( f | A ) ;
[ a , c ] in X ;
m1 <> {} ;
M + N c= M + N ;
assume M is satisfying_Ahhhh;
assume f is with_Gsssssssssssssssssssssssss
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 & k2 = k2 ;
m + 1 < n + 1 ;
s in S \/ { s } ;
n + i >= n + 1 ;
assume Re ( y ) = 0 ;
k1 <= j1 & j1 <= j2 ;
f | A is x2 ;
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 ^ q ^ p ^ q ^ r ^ s ^ s ^ q ^ s ^ q ^ s ^ q ^ s ^ q
j |^ ( y , y ) divides m ;
set m = max A ;
[ x , x ] in R ;
assume x in succ 0 ;
a in phi ;
Cs is one-to-one ;
q2 c= C1 & q2 c= C2 ;
a2 < c2 & c2 < c2 ;
s2 is 0 -started ;
IC s = 0 ;
s3 = s3 ;
let V ;
let x , y ;
let x be Element of T ;
assume a in rng F ;
x in dom T ` ;
let S be non empty RelStr ;
y " <> 0 ;
y " <> 0 ;
0. V = uw ;
y2 , y , w , y is_collinear ;
R8 is open ;
let a , b be Real , x be Real ;
let a be object of C ;
let x be Vertex of G ;
let o be Object of C , a be Object of C ;
r '&' q = P \lbrack l , P \rbrack ;
let i , j ;
let s be State of A , a be Element of B ;
s3 . n = N ;
set y = ( x `1 ) / 2 ;
NAT in dom g ;
l . 2 = y1 ;
|. g . y .| <= r ;
f . x in C0 ;
V is non empty ;
let x be Element of X ;
0 <> f . g2 ;
f2 /* q is convergent ;
f . i is_measurable_on E ;
assume \xi in N-22 ;
reconsider i = i as Ordinal ;
r * v = 0. X ;
rng f c= INT ;
G = 0 .--> goto 0 ;
let A be Subset of X ;
assume that A9 is dense and A9 is dense ;
|. f . x .| <= r ;
let x be Element of R ;
let b be Element of L ;
assume x in W-19 ;
P [ k , a ] ;
let X be Subset of L ;
let b be object of B ;
let A , B be category ;
set X = Vars ( C , X ) ;
let o be OperSymbol of S ;
let R be connected non empty Poset ;
n + 1 = succ n ;
x-21 c= [: Z , Z :] ;
dom f = C1 & dom g = C2 ;
assume [ a , y ] in X ;
Re ( seq ) is convergent ;
assume a1 = b1 & a2 = b2 ;
A = sInt A ;
a <= b or b <= a ;
n + 1 in dom f ;
let F be instruction of S , s be State of S ;
assume r2 > x0 ;
let Y be non empty set , a be Element of Y ;
2 * x in dom W ;
m in dom ( g2 | A ) ;
n in dom ( g1 | A ) ;
k + 1 in dom f ;
the still not bound in { s } ;
assume x1 <> x2 & x2 <> x3 ;
v2 in ( V \ { 0 } ) \/ ( V \ { 0 } ) ;
not [ b `1 , b `2 ] in T ;
-' 1 + 1 = i ;
T c= Cn ( T ) ;
( l - 1 ) * ( l - 1 ) = 0 ;
let n be Nat ;
( t `2 ) ^2 = r ;
AA is integrable ;
set t = "/\" ( { t } , L ) ;
let A , B be real-membered set ;
k <= len G + 1 ;
( TOP-REAL 2 ) misses ( TOP-REAL 2 ) ;
product ( seq | A ) is non empty ;
e <= f or f <= e ;
cluster -> non empty for NAT -defined Function ;
assume c2 = b2 & c2 = b1 ;
assume h in [. q , p .] ;
1 + 1 <= len C ;
not c in B . m1 ;
cluster R .: X -> empty ;
p . n = H . n ;
assume that v-4 is convergent and lim is convergent ;
IC s3 = 0 ;
k in N or k in K ;
F1 \/ F2 c= F ;
Int ( G1 \/ G2 ) <> {} ;
( z `2 ) ^2 = 0 ;
p01 <> p1 & p2 <> p3 ;
assume z in { y , w } ;
MaxADSet ( a ) c= F ;
ex_sup_of { s } , S ;
f . x <= f . y ;
let T be complete non empty reflexive antisymmetric RelStr ;
q1 / ( m + 1 ) >= 1 ;
a >= X & b >= Y ;
assume <* a , c *> <> {} ;
F . c = g . c ;
G is one-to-one implies is one-to-one & G is one-to-one ;
A \/ { a } c= B ;
0. V = 0. ( Y | V ) ;
let I be halting Instruction of S , s be State of S ;
f-24 . x = 1 ;
assume z \ x = 0. X ;
C4 = 2 to_power n ;
let B be sequence of Sigma ;
assume X1 = p .: D ;
n + l in NAT ;
f " P is compact ;
assume x1 in REAL + ( - x2 ) ;
p1 = K & p2 = K ;
M . k = <*> ( the carrier of V ) ;
phi . 0 in rng phi ;
MMMA is closed ;
assume z <> 0. L ;
n < N . k ;
0 <= seq . 0 ;
- q + p = v ;
{ v } is Subset of B ;
set g = f /. 1 ;
[: R , S :] is stable ;
set R = Vertices R , S = Vertices R ;
p0 c= PE & PE c= PE ;
x in [. 0 , 1 .] ;
f . y in dom F ;
let T be Scott TopAugmentation of S ;
inf the carrier of S is Element of S ;
sup a = sup { b } ;
P , C , K , f , g ;
assume x in LSeg ( s , r ) ;
2 to_power i < 2 to_power m ;
x + z = x + z ;
x \ ( a \ x ) = x ;
||. \mathopen { \Vert x .|| } <= r ;
assume that Y c= field Q and Y <> {} ;
a , b are_: b , a are_isomorphic ;
assume a in A . i ;
k in dom ( ( q | n ) | ( dom q | n ) ) ;
p is FinSequence of S ;
i - 1 = i - 1 ;
f | A is one-to-one ;
assume x in f .: [: X , Y :] ;
i2 - i1 = 0 ;
j2 + 1 + 1 <= i2 ;
g " * a in N ;
K <> { [ {} , {} ] } ;
cluster strict strict strict strict for commutative Ring ;
|. q .| / |. q .| > 0 ;
|. p3 .| = |. p .| ;
s2 - s1 > 0 ;
assume x in { G * ( -12 , k ) } ;
W-min C in C & W-min C in C ;
assume x in { G * ( -12 , k ) } ;
assume i + 1 = len G ;
assume i + 1 = len G ;
dom I = Seg n & dom I = Seg n ;
assume that k in dom C and k <> i ;
1 + 1 <= i + 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 void non empty non void ManySortedSign ;
let f be ManySortedSet of I ;
let z be Element of COMPLEX , x be Element of COMPLEX ;
u in { \hbox { \boldmath $ g g } } ;
2 * n < ( 2 * n ) / ( 2 * n ) ;
let x , y ;
BW c= [: open , V :] ;
assume I is_halting_on s , P ;
U = ( U , m ) --> ( U , m ) ;
M /. 1 = z /. 1 ;
x11 = x22 ( x , y , z ) ;
i + 1 < n + 1 ;
x in { {} , <* 0 *> } ;
f . ( n + 1 ) <= f . ( n + 1 ) ;
let l be Element of L ;
x in dom ( F . n ) ;
let i be Element of NAT ;
r8 is ( the carrier of K ) -valued ;
assume <* o2 , o2 *> <> {} ;
s . x / 0 = 1 ;
card ( K . i ) in M ;
assume that X in U and Y in U ;
let D be Subset-Family of Omega ;
set r = q | { k + 1 } ;
y = W . ( 2 * x ) ;
assume dom g = cod f & cod g = cod f ;
let X , Y be non empty TopSpace , f be Function of X , Y ;
x \oplus A is interval ;
|. <*> A .| . a = 0 ;
cluster strict for non empty Poset ;
a1 in B . ( s1 . a ) ;
let V be finite VectSp of F , F be FinSequence of V ;
A * B on B , A ;
f-3 = NAT --> 0 ;
let A , B be Subset of V ;
z1 = P1 . j & z2 = P1 . j ;
assume f " P is closed ;
reconsider j = i as Element of M ;
let a , b be Element of L ;
assume q in A \/ ( B \/ C ) ;
dom ( F * C ) = o ;
set S = ( NAT , X ) --> NAT ;
z in dom ( A --> y ) ;
P [ y , h . y ] ;
{ x0 } c= dom f ;
let B be non-empty ManySortedSet of I , A be non-empty ManySortedSet of I ;
sqrt ( PI / 2 ) < Arg z ;
reconsider z9 = 0 as Nat ;
LIN a , d , c ;
[ y , x ] in [: I , I :] ;
( Q ) * ( 1 , 3 ) = 0 ;
set j = x0 div m , k = x0 mod m ;
assume a in { x , y , c } ;
j2 - ( j - 1 ) > 0 ;
I \! \mathop { + } phi = 1 ;
[ y , d ] in ( F /. d ) ;
let f be Function of X , Y ;
set A2 = sqrt ( B , C ) ;
s1 , s2 be Element of L ;
j1 -' 1 + 1 = 0 ;
set m2 = 2 * n + j ;
reconsider t = t as bag of n ;
I2 . j = m . j ;
i |^ s , n are_relative_prime ;
set g = f | ( D , i ) ;
assume that X is lower and 0 <= r ;
( p1 `1 ) ^2 = 1 ;
a < ( p3 `2 ) ^2 + ( p3 `2 ) ^2 ;
L \ { m } c= UBD C ;
x in Ball ( x , 10 ) ;
not a in LSeg ( c , m ) ;
1 <= i1 -' 1 + 1 - 1 ;
1 <= i1 -' 1 + 1 - 1 ;
i + i2 <= len h ;
x = W-min ( P ) ;
[ x , z ] in X [: Z , Z :] ;
assume y in [. x0 , x .] ;
assume p = <* 1 , 2 , 3 *> ;
len <* A1 *> = 1 ;
set H = h . ( g . x ) ;
card b * a = |. a .| ;
Shift ( w , 0 ) |= v ;
set h = h2 \circ ( h1 . x ) ;
assume x in ( 3 /\ 4 ) /\ ( 3 /\ 4 ) ;
||. h .|| < d ;
not x in the carrier of f & not x in the carrier of f ;
f . y = F ( y ) ;
for n holds X [ n ] ;
k - l = kl ;
<* p , q *> /. 2 = q ;
let S be Subset of the lattice of Y ;
P , Q be subspaces of s ;
Q /\ M c= union ( F | M )
f = b * canFS ( S ) ;
let a , b be Element of G ;
f .: X <= f . ( sup X ) ;
let L be non empty transitive RelStr , X be Subset of L ;
Sw is x -Gi -to_power i ;
let r be non positive Real ;
M , v / ( x , y ) |= y ;
v + w = 0. ( V , C ) ;
P [ len ( F | ( len F ) ) ] ;
assume InsCode i = 8 & InsCode i = 8 ;
the zero of M = 0 ;
cluster z * seq -> summable ;
let O be Subset of the carrier' of C ;
||. f .|| " X is continuous ;
x2 = g . ( j + 1 ) ;
cluster -> ( S , U ) -valued for Element of S ;
reconsider l1 = ll as Nat ;
v4 is Vertex of r2 & ( the ResultSort of C ) . ( n + 1 ) = v ;
T is SubSpace of T2 | ( the carrier of T2 ) ;
( Q /\ Q ) /\ Q <> {} ;
let k be Nat ;
q " is Element of X ;
F . t is set with the non empty ;
assume n <> 0 & n <> 1 ;
set e1 = EmptyBag n , e2 = EmptyBag n , T = EmptyBag n ;
let b be Element of Bags n ;
assume for i holds b . i is commutative ;
x is root implies ( p . x ) `1 = ( p . x ) `1
not r in ]. p , q .[ ;
let R be FinSequence of REAL ;
S7 not LIN b1 , b2 , b1 ;
IC SCM R <> a ;
|. - x , y ] >= r ;
1 * ( seq . n ) = seq . ( n + 1 ) ;
let x be FinSequence of NAT ;
let f be Function of C , D ;
for a holds 0. L + a = a
IC s = s . NAT .= s . NAT ;
H + G = F- ( GG ) ;
Cx . x = x2 . x ;
f1 = f .= ( f1 | A ) . ( f . x ) ;
Sum <* p . 0 *> = p . 0 ;
assume v + W = v + u ;
{ a1 } = { a1 } & { a1 } = { a1 } ;
a1 , b1 _|_ b , a ;
d1 , o _|_ o , a1 ;
I1 is reflexive implies ( for x being Element of C st x in the carrier' of C holds x in the carrier' of C ) & ( x in the carrier' of C ) & ( x in the carrier' of C )
I1 is antisymmetric antisymmetric antisymmetric Relation of C & ( the InternalRel of C ) . ( a , b ) is antisymmetric ;
sup ( rng ( H1 | n ) ) = e ;
x = a9 * ( a * b ) ;
|. p1 .| ^2 >= 1 ;
assume j2 - 1 < j - 1 ;
rng s c= dom ( f1 * f2 ) ;
assume support a misses support b & not a in support b ;
let L be unital non empty doubleLoopStr , n be Element of NAT ;
s " + 0 < n + 1 ;
p . c = ( f . 1 ) . c ;
R . n <= R . ( n + 1 ) ;
Directed ( I , x0 ) = I +* ( card I + 1 ) ;
set f = + ( x , y , r ) ;
cluster Ball ( x , r ) -> bounded ;
consider r being Real such that r in A ;
cluster non empty for Function ;
let X be non empty directed Subset of S ;
let S be non empty full RelStr ;
cluster <* L1 . N , L2 . N *> -> complete for non trivial TopSpace ;
sqrt ( 1 - a ) = a ;
( q . {} ) `1 = o ;
n - ( i - 1 ) > 0 ;
assume sqrt ( 1 - 2 ) <= t `1 ;
card ( B \ C ) = k + 1 ;
x in union rng ( f | ( rng f ) ) ;
assume x in the carrier of R ;
d in C ;
f . 1 = L . ( F . 1 ) ;
the carrier' of G = { v } \/ { v } ;
let G be finite _Graph ;
e , v2 , G2 ;
c . ( i1 + 1 ) in rng c ;
f2 /* q is divergent_to+infty ;
set z1 = - ( z1 - z2 ) , z2 = - ( z1 - z2 ) ;
assume w is SubGof S , G ;
set f = p \! \mathop { t } ;
let c be Object of C ;
assume that P [ a ] ;
let x be Element of REAL m m ;
let I1 be Subset-Family of X ;
reconsider p = p as Element of NAT ;
v , w be Point of X ;
let s be State of SCM+FSA , I be Program of SCM+FSA ;
p is FinSequence of the InstructionsF of SCM+FSA , the carrier of SCM+FSA ;
stop I c= P-12 & I c= P ;
set ci = f /. i , ci = f /. ( i + 1 ) ;
w ^ t ^ s ^ w ^ t ^ s ^ w ^ t ^ s ^ w ^ t ^ s ^ t ^ w ^ s ^ t ^ w ^ t ^ w ^ s ^ t ^
W1 /\ W = ( W1 /\ W2 ) /\ ( W2 /\ W3 ) ;
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 ;
ord x = 1 & x is \setminus ;
set g2 = lim ( s , x0 ) ;
2 * x >= 2 * sqrt ( 1 + ( 2 * x ) ^2 ) ;
assume ( a 'or' c ) . z <> TRUE ;
f (*) g in Hom ( c , c ) ;
Hom ( c , d + c ) <> {} ;
assume 2 * Sum ( q | m ) > m ;
L1 . ( F . k ) = 0 ;
( id X \/ R1 ) \/ R1 = ( id X ) \/ ( id X ) ;
( ( the Sorts of sin ) . x ) . ( ( the Sorts of sin ) . x ) <> 0 ;
( the function of exp ( C , n ) ) . x > 0 ;
o1 in [: X , Y :] /\ [: Y , Z :] ;
e , v2 , G2 ;
r3 > sqrt ( 1 - 0 ) * 0 ;
x in P .: ( F " ) ;
let J be closed non empty Subset of R ;
h . p1 = f2 . O ;
Index ( p , f ) + 1 <= j ;
len ( q ^ <* x *> ) = width M ;
the carrier of Lin K c= A ;
dom f c= union rng ( F | ( union rng F ) ) ;
k + 1 in Seg ( n + 1 ) ;
let X be ManySortedSet of the carrier of S ;
[ x `1 `1 , y `2 ] in InnerVertices R ;
i = D1 or i = D2 or i = D1 ;
assume a mod n = b mod n ;
h . x2 = g . ( x1 , x2 ) ;
F c= 2 -tuples_on the carrier of X ;
reconsider w = |. s1 .| as sequence of REAL ;
sqrt ( 1 / m * m + r ) < p ;
dom f = dom ( I | ( dom I ) ) ;
[#] ( ( ( TOP-REAL n ) | P ) | P ) = [#] ( ( TOP-REAL n ) | P ) ;
cluster - x -> real for number ;
then { d } c= A ;
cluster ( TOP-REAL n ) | A -> finite-ind ;
let w1 be Element of M ;
let x be Element of dyadic ( n ) ;
u in W1 & v in W2 & u in W2 ;
reconsider y = y as Element of L2 ;
N is full full SubRelStr of T |^ ( the carrier of S ) ;
sup { x , y } = c "\/" c ;
g . n = n / ( 1 + 1 ) .= n ;
h . J = EqClass ( u , J ) ;
let seq be sequence of X ;
dist ( x `1 , y ) < r / 2 ;
reconsider mm = m - 1 as Element of NAT ;
x- x0 < ( r1 - x0 ) * ( x0 - x0 ) ;
reconsider P = P ` as strict Subgroup of N ;
set g1 = p * idseq ( q `1 ) ;
let n , m , k be non zero Nat ;
assume 0 < e & f | A is lower ;
D2 . ( I . ( x , I ) ) in { x } ;
cluster -> closed for Subset of T ;
let P be compact non empty Subset of TOP-REAL 2 ;
G * ( -13 , 1 ) in LSeg ( \pi , 1 ) ;
let n be Element of NAT , a be Element of NAT ;
reconsider Ss = S as Subset of T ;
dom ( i .--> X ) = { i } ;
let X be non-empty ManySortedSet of S ;
let X be non-empty ManySortedSet of S ;
op ( 1 , {} ) c= { [ {} , {} ] } ;
reconsider m = min ( m , n ) as Element of NAT ;
reconsider d = x as Element of ( the carrier' of C ) . s ;
let s be 0 -started State of SCMPDS , p be Point of SCMPDS ;
let t be 0 -started State of SCMPDS ;
b , b , x , y , z is_collinear ;
assume i = n \/ { n } & j = k \/ { k } ;
let f be PartFunc of X , Y ;
N >= sqrt ( ( c + d ) / ( 2 * PI ) ) ;
reconsider tT = ( T " ) . x as TopStruct ;
set q = h * p ^ <* d *> ;
z2 in U . ( y2 , z2 ) /\ Q ;
A |^ 0 = { <* 0 *> , A } ;
len ( W2 + W1 ) = len ( W2 + W1 ) ;
len ( h2 ^ h2 ) in dom h2 & len ( h2 ^ h2 ) in dom h2 ;
i + 1 in Seg ( len s2 ) & i + 1 in Seg ( len s2 ) ;
z in dom ( g1 | A ) /\ dom f ;
assume p2 = W-min ( K ) & p2 = W-min ( K ) ;
len G + 1 <= i1 + 1 ;
f1 (#) f2 is convergent & f2 (#) ( f1 (#) f2 ) is convergent ;
cluster ( seq + seq1 ) + ( seq + seq1 ) -> summable ;
assume j in dom ( M1 * M2 ) ;
let A , B , C be Subset of X ;
let x , y , z be Point of X , f be Function of X , Y ;
b ^2 - ( 4 * a ) >= 0 ;
<* xy *> ^ <* y *> ^ <* y *> ^ <* x *> ^ <* y *> ^ <* y *> ^ <* x *> ^ <* y *> ^ <* y *> ^ <* x *> ^ <* y *> ^ <* y
a , b in { a , b } ;
len ( p2 | ( len p2 ) ) is Element of NAT ;
ex x being element st x in dom R & x = R . x ;
len q = len ( K * G ) ;
s1 = Initialize ( Initialized s ) .= Exec ( i , s1 ) .= Exec ( i , s1 ) . f ;
consider w be Nat such that q = z + w ;
x ` ` ` is Element of L ` ;
k = 0 & n <> k or k > n ;
then X is discrete implies for A being Subset of X holds A is closed ;
for x st x in L holds x is FinSequence ;
||. f /. c .|| <= r1 ;
c in ]. p , q .] & not c in { p } ;
reconsider V = V as Subset of the carrier of n n , 1 ;
let N , M be <* non empty set ;
then z is_>=_than x & z >= sup x ;
M | [. f , g .] = f & M | [. g , g .] = g ;
( ( #Z ( 1 ) ) /. 1 ) /. 1 = TRUE ;
dom g = dom f & g = ( f | X ) . x ;
mode Walk of G is Walk of G ;
[ i , j ] in Indices ( M @ ) ;
reconsider s = x " as Element of H ;
let f be Element of dom ( - p ) ;
F1 . ( a1 , a2 ) = G1 . ( a1 , a2 ) ;
cluster AffineMap ( a , b , r ) -> compact ;
let a , b , c , d be Real ;
rng s c= dom ( 1 - s ) ;
curry ( F . k , X ) is additive ;
set k2 = card ( B \/ C ) , k2 = card ( B \/ C ) ;
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 S0 -tuples_on the carrier of S ;
rng p c= the carrier of L & rng p c= the carrier of L ;
let d be Subset of the bound A ;
( x | x ) = 0 iff x = 0. W ;
I1 in dom ( stop I ) & I . ( card I + 1 ) = ( stop I ) . ( card I + 1 ) ;
let g be continuous Function of X | B , Y ;
reconsider D = Y as Subset of TOP-REAL n ;
reconsider i1 = len ( p1 , p2 ) as Integer ;
dom f = the carrier of S & rng f c= the carrier of S ;
rng h c= union ( the carrier of J ) ;
cluster All ( x , H ) -> proper ;
d * ( N1 / 2 ) > ( 1 / 2 ) * ( 1 / 2 ) ;
]. a , b .[ c= [. a , b .] ;
set g = f " ( D1 ) , h = f " ( D2 ) ;
dom ( p | ( NAT + 1 ) ) = NAT ;
3 + - 2 <= k + - 2 ;
the function tan is differentiable in ( ( arctan * arccot ) . x ) ;
x in rng ( f /^ ( p .. f ) ) ;
let f , g be FinSequence of D ;
p in the carrier of S1 & q in the carrier of S2 ;
rng ( f " ) = dom f ;
( the Target of G ) . e = v ;
width G - 1 < width G - 1 + 1 ;
assume v in rng ( S | ( E . k ) ) ;
assume x is root or x is root ;
assume 0 in rng ( ( g2 | A ) | A ) ;
let q be Point of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
let p be Point of TOP-REAL 2 , q be Point of TOP-REAL 2 ;
dist ( O , u ) <= |. p2 .| + 1 ;
assume dist ( x , b ) < dist ( a , b ) ;
<* S7 *> is in the InternalRel of ( the Sorts of C ) . ( len C7 ) ;
i <= len ( G | ( i -' 1 ) ) - 1 ;
let p be Point of TOP-REAL 2 , q be Point of TOP-REAL 2 ;
x1 in the carrier of ( ( TOP-REAL 2 ) | K1 ) ;
set p1 = f /. i , p2 = f /. i ;
g in { g2 : r < g2 } ;
Q = SnK ( Q ) .= ( Q | Q ) " ;
( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( 1 / 2 ) ) ) ) ) is summable ;
- p + I c= - p + A ;
n < LifeSpan ( P1 , s1 ) + 1 ;
CurInstr ( p1 , s1 ) = i .= i ;
A /\ Cl { x } \ { x } <> {} ;
rng f c= ]. r - 1 , r + 1 .[ ;
let g be Function of S , V ;
let f be Function of L1 , L2 ;
reconsider z = z as Element of InclPoset ( L ) ;
let f be Function of S , T ;
reconsider g = g as Morphism of c opp , b opp ;
[ s , I ] in S \/ ( the carrier' of A ) ;
len ( the connectives of C ) = 4 & len ( the connectives of C ) = 4 ;
let C1 , C2 be subsignature of C ;
reconsider V1 = V as Subset of X | B ;
attr p is valid means : Def2 : All ( x , p ) is valid ;
assume that X c= dom f and f .: X c= dom g ;
H |^ a " is Subgroup of H ;
let A1 be Element of O , A2 be Element of O ;
p2 , q2 , q1 , q2 , q2 is_collinear & q2 <> q2 ;
consider x being element such that x in v ^ K ;
not x in { 0. TOP-REAL 2 } ;
p in [#] ( ( ( TOP-REAL 2 ) | B ) | K1 ) ;
0 in ( M . E ) & ( M . E ) `2 < ( M . F ) `2 ;
op ( c , c ) / ( a , b ) = c ;
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . ( s . ( s . ( s . ( s . ( s . ( s . ( s . ( s . ( s . ( s . ( s . n ) ) ) ) ) ) ) ) ) )
cluster \mathclose { \rm \mathclose { \rm c } } -> with_\ast ;
set i1 = the Element of NAT , i2 = the Element of NAT , j1 = the Element of NAT ;
let s be 0 -started State of SCM+FSA , p be ( 0 , I ) -valued Function ;
assume y in ( f1 \/ f2 ) .: A ;
f . ( len f ) = f /. ( len f + 1 ) ;
x , f . x '||' f . y , f . y ;
attr X c= Y means : Def2 : cos ( X ) c= cos ( Y ) ;
let y be upper Subset of Y , x be Element of Y ;
cluster ( x `1 ) / 2 -> non inm for Nat ;
set S = <* Bags n , i *> ;
set T = [. 0 , 1 .] ;
1 in dom mid ( f , 1 , 1 ) ;
sqrt ( 4 * PI ) < sqrt ( 2 * PI * PI ) ;
x2 in dom ( f1 | X ) /\ dom ( f2 | X ) ;
O c= dom I & { {} } = { {} } ;
( the Target of G ) . x = v ;
{ HT ( f , T ) } c= Support f ;
reconsider h = R . k as Polynomial of n , L ;
ex b being Element of G st y = b * H ;
let x , y , z be Element of G opp ;
h19 . i = f . ( h . i ) ;
( p `1 ) ^2 = ( p `2 ) ^2 + ( p `2 ) ^2 ;
i + 1 <= len Cage ( C , n ) ;
len <* P *> = len P & len ( P ^ <* Q *> ) = len P ;
set NN = the InternalRel of N , the InternalRel of S ;
len gx + ( x + 1 ) <= x ;
a on B & b on C ;
reconsider r-12 = r * I . v as FinSequence ;
consider d such that x = d and a [= d and a [= c ;
given u such that u in W and x = v + u ;
len f /. ( \downharpoonright n ) = len f - n ;
set q2 = W-min ( C ) , q2 = W-min ( C ) , q1 = W-min ( C ) ;
set S = variables_in ( S1 , S2 , S1 ) ;
MaxADSet ( b ) c= MaxADSet ( P /\ Q ) ;
Cl ( G . q1 ) c= F . ( r1 , r2 ) ;
f " ( D ) meets h " ( V ) ;
reconsider D = E as non empty directed Subset of L1 ;
H = H '&' ( H '&' H ) ;
assume t is Element of ( \mathfrak S ) . ( X . s ) ;
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 . b2 ;
the carrier' of G = E \/ { E } ;
reconsider m = len ( p | ( k + 1 ) ) as Element of NAT ;
set S1 = LSeg ( n , W-min C ) , S2 = LSeg ( n , W-min C ) ;
[ i , j ] in Indices ( M1 @ ) ;
assume that P c= Seg m and M is Matrix of n , K ;
for k st m <= k holds z in K . k ;
consider a be set such that p in a and a in G ;
L1 . p = p * ( 1 , 0 ) ;
p-7 . i = p-7 . i .= p-7 . i ;
let PA , G be a_partition of Y ;
attr 0 < r & 1 < 1 & r < 1 ;
rng proj ( a , X ) = [#] ( X | A ) ;
reconsider x = x , y = y as Element of K ;
consider k such that z = f . k and n <= k ;
consider x being element such that x in X \ { p } ;
len ( canFS ( s ) ) = card ( s ) + 1 ;
reconsider x2 = x1 as Element of L ;
Q in FinMeetCl ( ( the topology of X ) | ( the topology of X ) ) ;
dom ( f | X ) c= dom ( u | X ) ;
attr n divides m means : Def2 : m divides n ;
reconsider x = x as Point of I[01] | P ;
a in \bf trivial ( 2 , T2 ) & b in trivial ( 2 , T2 ) ;
not y in the still of f & not y in the carrier' of f ;
Hom ( a , b ) <> {} & Hom ( a , b ) <> {} ;
consider k1 such that p " < k1 and p " < k1 ;
consider c , d such that dom f = c \ d ;
[ x , y ] in dom g & [ y , z ] in g ;
set S1 = InputVertices S2 ( x , y , z ) ;
l = m2 & l = m2 & l = m2 & l = m2 & l = m2 & l = m2 & l = m2 & l = m2 ;
x0 in dom ( u | A ) /\ ( ( u | A ) . x0 ) ;
reconsider p = x as Point of ( TOP-REAL 2 ) | K1 ;
[: I[01] , I[01] :] = [: ( TOP-REAL 2 ) | B , ( TOP-REAL 2 ) | B :] ;
f . p3 <= f . p1 & f . p2 <= f . p2 ;
( ( F . n ) . x ) `1 <= ( F . n ) `1 ;
( x `2 ) ^2 = ( ( ( y `2 ) ) ^2 + ( y `2 ) ^2 ) ;
for n being Element of NAT holds P [ n ] ;
let J , K be non empty Subset of I ;
assume 1 <= i & i <= len <* a *> ;
0 |-> a = <*> the carrier of K ;
X . i in 2 -tuples_on ( A . i \ B . i ) ;
<* 0 *> in dom ( e --> [ 1 , 0 ] ) ;
then P [ a ] ;
reconsider ss = ( smax ( D , R ) ) . s as terminal of D ;
( - i ) - 1 <= len - j ;
[#] S c= [#] ( T | S ) ;
for V being strict RealUnitarySpace holds V in (Omega). ( 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 , K ;
- a * ( - b ) = a * b ;
for A being Subset of ( TOP-REAL 2 ) holds A // A & A // C ;
id ( o2 ) in <* o2 , o2 , o1 , o2 *> ;
then ||. x .|| = 0 & x = 0. ( X ) ;
let N1 , N2 be strict normal Subgroup of G ;
j >= len ( g | indx ( g , D1 , j1 ) ) ;
b = Q . ( len Q + 1 ) .= Q . ( len Q + 1 ) ;
f2 * f1 /* s is divergent of \hbox { $ - \infty $ } , lim ( f1 /* s ) ;
reconsider h = f * g as Function of [: N , I :] , G ;
assume that a <> 0 and delta ( a , b , c ) >= 0 ;
[ t , t ] in the InternalRel of A & [ s , t ] in the InternalRel of A ;
( v |-- E ) | n is Element of ( v |-- E ) . n ;
{} = the carrier of ( L1 + L2 ) /\ ( L2 + L2 ) ;
Directed I is closed & Directed I is closed implies ;
Initialized ( p +* q ) = Initialized ( p +* q ) .= Initialized ( p +* q ) ;
reconsider N2 = N1 as strict net of ( the carrier of N ) | the carrier of N ;
reconsider Y = Y as Element of <* <* 1 *> , \subseteq *> ;
"/\" ( { p } , L ) \ { p } <> p ;
consider j being Nat such that i2 = i1 + j and j in dom f ;
not [ s , 0 ] in the InternalRel of S2 & not [ s , 0 ] in the InternalRel of S2 ;
m in ( B '&' C ) /\ D \ { {} } ;
n <= len ( ( P | ( len P ) ) | ( len P ) ) ;
( x1 - x2 ) `1 = ( x2 - y2 ) `1 .= ( x2 - y2 ) `1 ;
InputVertices S = { x , y , c } & InputVertices S = { x , y , c } ;
let x , y be Element of FIA1 ( n ) ;
p = |[ ( p `1 ) / ( 1 + ( p `2 ) / ( 1 + ( p `2 ) ) ^2 ) , ( p `2 ) / ( 1 + ( p `2 ) ) ^2 ) ]| ;
g * 1_ G = h " * g * h ;
let p , q be Element of PFuncs ( V , C ) ;
x0 in dom ( f1 | X ) /\ dom ( f2 | X ) ;
( R qua Function ) " = R " ( dom R ) ;
n in Seg ( len ( f /^ n ) ) & n in dom ( f /^ n ) ;
for s being Real st s in R holds s <= s2 ;
rng s c= dom ( f2 * f1 ) /\ dom ( f2 * f1 ) ;
synonym for for Subset of ( TOP-REAL n ) | ( Seg n ) for x being element st x in Seg n holds x in Seg n ;
1_ ( K , n , m ) * ( 0. ( K , n , m ) ) = 0. ( K , n , m ) ;
set S = Segm ( A , P1 , Q1 ) , P1 = Segm ( A , P1 , Q1 ) ;
ex w st e = sqrt ( w , f ) & w in F ;
( curry ( Pk , k ) ) # x is convergent ;
cluster -> open for Subset of T | P ;
len f1 = 1 .= len ( f1 ^ f2 ) .= len ( f1 ^ f2 ) .= len ( f1 + f2 ) ;
sqrt ( i * p ) < sqrt ( 2 * p ) ;
let x , y be Element of ( U1 ) * ;
b1 , c1 // b1 , c1 & b1 , c1 // b1 , c2 ;
consider p being element such that c1 . j = { p } ;
assume that f " { 0 } = {} and f " { 0 } = {} ;
assume IC Comput ( F , s , k ) = n ;
Reloc ( J , card I + card J ) not a in dom J ;
not ( card I + 1 ) does not destroy c ;
set m1 = LifeSpan ( p , s ) , m2 = LifeSpan ( p , s ) , m2 = Comput ( p , s , m1 ) , m2 = Comput ( p , s , m1 ) ;
IC Comput ( p , s , k ) in dom Initialize ( p +* I ) ;
dom t = the carrier of SCM ( ) & dom t = the carrier of R ( ) ;
( W-min L~ f ) .. f = 1 & ( W-min L~ f ) .. f = 1 ;
let a , b be Element of PFuncs ( V , C ) ;
Cl ( union F ) c= Cl ( union F ) ;
the carrier of X1 union X2 misses ( the carrier of X1 ) \/ ( the carrier of X2 ) ;
assume not LIN a , f . a , g . a ;
consider i being Element of M such that i = d and i in dom ( d | i ) ;
then Y c= { x } or Y = { x } ;
M , v / ( y , x ) / ( y , x ) / ( y , x ) / ( y , x ) / ( y , x ) / ( y , x ) / ( y , x ) / ( y , x ) / (
consider m being element such that m in Intersect ( ( F . m ) . x ) ;
reconsider A1 = support ( u | A1 ) as Subset of X ;
card ( A \/ B ) = k-1 + ( 2 * 1 ) ;
assume that a1 <> a3 and a2 <> a4 and a1 <> a4 and a1 <> a4 and a2 <> a4 and a1 <> a4 ;
cluster s \! \mathop { \rm \hbox { - } -> ( S , X ) -valued string of S ;
L2 /. ( n + 2 ) = L . ( n + 2 ) ;
let P be compact non empty Subset of TOP-REAL 2 ;
assume that r-7 in LSeg ( p1 , p2 ) and P-7 in LSeg ( p2 , p3 ) ;
let A be non empty compact Subset of TOP-REAL n , a be Real ;
assume [ k , m ] in Indices ( ( D1 | j1 ) | j1 ) ;
0 <= ( 1 / 2 ) to_power ( p . n ) ;
( F . N ) . x = +infty ;
attr X c= Y means : Def2 : Z c= V \ Y ;
( y * ( z * ( y * z ) ) ) * ( ( y * z ) * ( z * x ) ) <> 0. I ;
1 + card ( ( card ( X /\ Y ) ) ) <= card ( u /\ ( X /\ Y ) ) ;
set g = z \circlearrowleft ( L~ z ) , h = z /. ( len z + 1 ) ;
then p = 1 & p . k = <* x , y *> ;
cluster -> ( C , D ) -for Element of D ;
reconsider B = A as non empty Subset of TOP-REAL n ;
let a , b , c be Function of Y , BOOLEAN ;
L1 . i = ( i .--> g ) . i .= g . i ;
( for x1 , x2 , x3 being Element of D holds P [ x1 , x2 , x3 ] ) implies P [ x1 , x2 , x3 ]
n <= indx ( D2 , D1 , j1 ) + 1 ;
( g2 . O ) `1 = ( - 1 ) * ( ( g2 . O ) `1 ) `1 ;
j + p .. f - ( len f - 1 ) <= len f - ( len f - 1 ) ;
set W = W-min ( C ) ;
S1 . ( a , e ) = a + e .= a + e ;
1 in Seg width ( M * ( p , q ) ) ;
dom ( i * Im ( f ) ) = dom ( Im ( f ) ) ;
W . ( x `1 , p ) = W . ( a , p ) ;
set Q = |= ( g , f , h ) ;
cluster -> \mathclose for Relation of U1 ;
attr F = { A } means : Def2 : F is discrete ;
reconsider z9 = H as Element of product ( G . i ) ;
rng f c= rng ( f1 + f2 ) \/ rng ( f2 + g2 ) ;
consider x such that x in f .: A and x in f .: C ;
f = <*> ( the carrier of C ) & f is one-to-one ;
E , j |= All ( x , y , H ) ;
reconsider n1 = n as Morphism of o1 , o2 ;
assume that P is commutative and R is commutative and P = P * R ;
card ( ( B \/ { x } ) \/ { x } ) = k-1 + 1 ;
card ( x \ ( x \ ( B \ C ) ) ) = 0 ;
g + R in { s : g-r < s + r } ;
set q9 = ( q , <* s *> ) := ( <* s *> , <* s *> ) , <* s *> ) := ( s , <* s *> ) ;
for x being element st x in X holds x in rng ( f1 | X ) ;
h1 /. ( i + 1 ) = h2 . ( i + 1 ) ;
set mw = max ( B , Funcs ( B , C ) ) ;
t in Seg width ( I ^ ( n , n ) ) & t in dom ( I ^ ( n , n ) ) ;
reconsider X = dom f as Element of ( the carrier of V ) -tuples_on the carrier of V ;
IncAddr ( i , k ) = halt S + k .= ( i + k ) + k ;
( W-min L~ f ) `2 <= ( q `2 ) / ( |. q .| ) ;
attr R is condensed means : Def2 : for x being Element of R st x in R holds Cl ( R . x ) is condensed ;
attr 0 <= a & b <= 1 & a * b <= 1 ;
u in ( c /\ ( ( d /\ b ) /\ e ) /\ f ) /\ f ;
u in ( c /\ ( ( d /\ e ) /\ f ) /\ f ) /\ j ;
len C + ( - 2 ) >= 9 + - ( - 2 ) ;
x , z , y , z is_collinear & x , y , z is_collinear ;
( a |^ ( n + 1 ) ) * a = ( a |^ ( n + 1 ) ) * a ;
<* \underbrace 0 , \dots 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 y9 = <* y , c *> ;
( F /. 1 ) /. 1 in rng ( Line ( D , 1 ) ) ;
p . m joins r /. m , r /. ( m + 1 ) ;
( p `1 ) `2 = ( ( f /. ( i1 + 1 ) ) `2 ) `2 ;
W-min ( X \/ Y ) = W-min ( X ) & W-min ( X ) = W-min ( X ) ;
0 + ( p `2 ) <= 2 * r + ( p `2 ) ^2 ;
x in dom g & not x in { 0 } ;
f1 /* ( s ^\ k ) is divergent_to+infty ;
reconsider u2 = u as VECTOR of Lin ( X ) , \bf 0. ( X ) ;
p \! ( Product ( Sgm ( X ) ) ) = 0 ;
len <* x *> < i + 1 + 1 ;
assume that I is non empty and { x } /\ { y } = { 0. I } ;
set i = card ( I + 4 ) + card ( I + 4 ) ;
x in { x , y } & h . x = {} & h . y = {} ;
consider y being Element of F such that y in B and y <= x `1 ;
len S = len ( the charact of ( A , 0 ) ) .= len ( the charact of ( A , 0 ) ) ;
reconsider m = M , i = N , n = n as Element of X ;
A . ( j + 1 ) = B . ( j + 1 ) \/ A . ( j + 1 ) ;
set NG = |. ( ( G . n ) `1 ) `1 , PG . ( n + 1 ) `2 ]| ;
rng F c= the carrier of gr { a } & rng F c= the carrier of gr { a } ;
( for n holds ( for K , n holds F . ( n , n ) , F . ( n , n ) ) is a complex-valued FinSequence ;
f . k , f . ( Radix ( n ) ) in rng f ;
h " ( P ) /\ [#] ( ( TOP-REAL 2 ) | P ) = f " ( P ) ;
g in dom ( f2 \ ( f " ) ) \ ( f " { 0 } ) ;
gX /\ dom ( f1 | X ) = ( g1 | X ) " X ;
consider n being element such that n in NAT and Z = G . n ;
set d1 = \mathopen ( x1 , y1 , y2 ) , d2 = dist ( y1 , y2 , y1 ) ;
b `1 + sqrt ( 1 + ( 1 + ( 1 + ( 1 + ( 1 + ( 1 + ( 1 + ( 1 + 1 ) ) ) ) ^2 ) ) ) < sqrt ( 1 + ( 1 + ( 1 + ( 1 + ( 1 + ( 1 + ( 1 +
reconsider f1 = f as VECTOR of the carrier of X ;
attr i <> 0 implies i |^ ( i + 1 ) mod ( i + 1 ) = 1 ;
j2 in Seg ( ( g2 . ( i2 + 1 ) ) ) & ( g2 . ( i2 + 1 ) ) = ( g2 . ( i2 + 1 ) ) ;
dom ( i , ( n + 1 ) --> ( n + 1 ) ) = dom ( i , ( n + 1 ) --> ( n + 1 ) ) .= a ;
cluster sec | ]. 0 , PI / 2 .[ -> one-to-one ;
Ball ( u , e ) = Ball ( f . p , e ) ;
reconsider x1 = x0 as Function of [: S , T :] , T ;
reconsider R1 = x , R2 = y as Relation of L , the carrier of L ;
consider a , b being Subset of A such that x = [ a , b ] ;
( <* 1 *> ^ p ) ^ <* n *> ^ <* n *> in R1 ;
S1 +* S2 = S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2
( ( ( id Z ) (#) ( ( exp_R + exp_R ) `| Z ) `| Z ) = ( ( exp_R * ( exp_R + exp_R ) `| Z ) `| Z ) ;
cluster -> Function for Function of C , REAL ;
set Cx = 1GateCircStr ( <* z , x *> , f ) , Cy = 1GateCircStr ( <* x , y *> , f ) ;
E . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 .
( ( ( arctan * ( arctan + arccot ) ) `| Z ) `| Z ) = ( ( arctan * ( exp_R + arccot ) ) `| Z ) | A ;
lower_bound A = sqrt ( cos * PI ) & lower_bound A = 0 ;
F ( dom f , - ( F . a ) ) is transformable to F ( cod f , - ( F . a ) ) ;
reconsider p8 = ( q `2 / |. q .| - sn ) / ( 1 + sn ) as Point of TOP-REAL 2 ;
g . W in [#] ( Y | W ) & [#] ( Y | W ) c= [#] ( Y | W ) ;
let C be compact connected non vertical non horizontal Subset of TOP-REAL 2 ;
LSeg ( f ^ g , j ) = LSeg ( f , j ) ;
rng s c= dom f /\ ]. - r , x0 .[ ;
assume x in { idseq ( 2 ) , ( idseq ( 2 ) ) . ( ( idseq 2 ) . ( len ( idseq 2 ) ) ) } ;
reconsider n2 = n , m2 = m as Element of NAT ;
for y being ExtReal st y in rng ( seq | n ) holds g <= y
for k st P [ k ] holds P [ k + 1 ]
m = m1 + m2 .= m1 + m2 .= m1 + m2 + m2 ;
assume for n holds H1 . n = G . n -H . n ;
set BX = f .: ( the carrier of X1 ) , pX = f .: ( the carrier of X2 ) ;
ex d being Element of L st d in D & x << d ;
assume R " ( a , b ) c= R " ( a , b ) ;
t in ]. r , s .] or t = r ;
z + v2 in W & x = u + ( z + v2 ) ;
x2 |-- ( y2 , y2 ) iff P [ x2 , y2 ] ;
attr x1 <> x2 means : Def2 : |. x1 - x2 .| > 0 ;
assume p2 - p1 , p1 - p2 , p2 - p1 , p3 - p1 , p2 - p1 , p3 - p1 , p2 - p1 , p3 - p1 , p2 - p1 , p3 - p1 , p2 - p1 , p3 - p1 , p2 - p1 , p3 - p1 , p2 - p1 , - p1 , p3 - p1 , p2 - p1 , - p1
set q = ( 'not' f ) ^ <* 'not' A *> ;
let f be PartFunc of \langle 1 , 1 , 2 , 3 *> , ||. f /. 1 , f /. 3 .|| ;
( n mod ( 2 * k ) ) = n mod ( 2 * k ) ;
dom ( T * succ t ) = dom T .= dom T ;
consider x being element such that x in w and x in c ;
assume ( F * G ) . ( v , w ) = v . ( ( F * G ) . ( v , w ) ) ;
assume the carrier' of ( D1 ) c= the carrier' of ( D2 ) \/ the carrier' of D2 ;
reconsider A1 = [. a , b .] as Subset of R^1 | A1 ;
consider y being element such that y in dom F and F . y = x ;
consider s being element such that s in dom o and a = o . s ;
set p = W-min L~ Cage ( C , n ) , q = W-min L~ Cage ( C , n ) ;
n1 - len f + 1 + 1 <= len f - 1 + 1 ;
ConsecutiveSet ( q , O ) = [ u , v ] ;
set C-2 = ( \mathclose { -1 } ) . ( k + 1 ) ;
Sum ( L * p ) = 0. ( V * p ) .= 0. ( V * p ) ;
consider i being element such that i in dom p and t = p . i ;
defpred Q [ Nat ] means 0 = ( $1 + 1 ) * ( $1 + 1 ) ;
set s3 = Comput ( P1 , s1 , k + 1 ) , s4 = Comput ( P2 , s2 , k + 1 ) , P4 = Comput ( P2 , s2 , k + 1 ) , P4 = P3 ;
let l be variable of k , A , l be Nat ;
reconsider U = union ( ( G | n ) | ( G | n ) ) as Subset-Family of T ;
consider r such that r > 0 and Ball ( p `1 , r ) c= Q ` ;
( h | ( n + 2 ) ) /. ( i + 1 ) = p29 ;
reconsider B = the carrier of X1 as Subset of X ( ) ;
pmax = <* - ( ( - ( c1 + c2 ) ) * ( 1 , 0 ) *> ;
synonym f -> real-valued means : Def2 : rng f c= NAT & rng f c= NAT ;
consider b being element such that b in dom F and a = F . b ;
x0 < card ( X | ( Y | ( X | Y ) ) ) + card ( Y | ( X | Y ) ) ;
attr X c= B1 means : Def2 : for x being Element of X holds X . x c= succ ( B1 . x ) ;
then w in Cl ( x , r ) & dist ( x , w ) <= r ;
angle ( x , y , z ) = angle ( x , 0 , PI , PI ) ;
attr 1 <= len s means : Def2 : for s being Element of NAT holds ( the _ of s ) . ( s . 0 ) = s . 0 ;
fx9 c= f . ( k + n ) ;
the carrier of { 1_ G } = { 1_ G } ;
attr p '&' q in TAUT ( Al ) means : Def2 : q '&' p in TAUT ( Al ) ;
- ( t `2 / |. t .| - sn ) < ( ( t `2 / |. t .| - sn ) ) / ( 1 - sn ) ;
U . 1 = U /. 1 .= U /. 1 .= U . 1 .= U . 1 .= U . 1 ;
f .: ( the carrier of x ) = the carrier of x & f .: ( the carrier of x ) = the carrier of x ;
Indices ( O * ( n , m ) ) = [: Seg n , Seg n :] & [: Seg n , Seg n :] c= Indices ( O * ( n , m ) ) ;
for n being Element of NAT holds G . n c= G . ( n + 1 )
then V in M .: { x } ;
ex f being Element of ( F . ( len A ) ) st f is \emptyset & ( for i being Element of NAT st i in dom A holds f . i = F ( i ) ) ;
[ h . 0 , h . 3 ] in the InternalRel of G ;
s +* ( intloc 0 , 1 ) . ( intloc 0 ) = s3 . ( 0 ) .= s . ( 0 + 1 ) ;
|[ w1 , v1 ]| + |[ w1 , v1 ]| <> 0. TOP-REAL 2 ;
reconsider t = t as Element of ( the carrier of V ) -tuples_on the carrier of V ;
C \/ P c= [#] ( ( G | A ) \ A ) ;
f " ( V ) in ( ( the carrier of X ) /\ D ) /\ ( the carrier of X ) ;
x in [#] ( ( A /\ A ) /\ ( A /\ B ) ;
g . x <= h1 . x & h . x <= 1 ;
InputVertices S = { x , y , z } & { x , y , z } = { x , y , z } ;
for n being Nat st P [ n ] holds P [ n + 1 ]
set R = Line ( M , i ) * Line ( M , i ) ;
assume that M1 is being_line and M2 is being_line and M2 is being_line and M2 is being_line ;
reconsider a = ( f . ( i1 -' 1 ) ) - 1 as Element of K ;
len ( ( ( ( F1 ^ F2 ) ^ ( F1 ^ F2 ) ) ^ ( F2 ^ F2 ) ) ) = Sum ( ( F1 ^ F2 ) ^ ( F2 ^ F2 ) ) ;
len ( the addF of n , m ) = n & len ( the multF of n , m ) = n ;
dom max ( f + g , f ) = dom ( f + g ) ;
( the Sorts of ( seq + 8 ) ) . n = sup ( ( the Sorts of ( seq + 8 ) ) . n ) ;
dom ( p1 ^ p2 ) = dom ( p1 ^ p2 ) .= dom ( p1 ^ p2 ) .= dom ( p1 ^ p2 ) ;
M . ( [ 1 , y ] , [ y , z ] ) = 1 * ( v . ( y , z ) ) .= y ;
assume that W is non trivial and W \cap ( the carrier of G2 ) c= the carrier' of G2 and W \cap ( the carrier of G2 ) c= the carrier' of G2 ;
CG * ( i1 , j1 ) `1 = G * ( i1 , j1 ) `1 .= G * ( i2 , j1 ) `1 ;
C8 |- 'not' All ( x , p ) 'or' ( 'not' p => q ) ;
for b st b in rng g holds lower_bound rng fb <= b implies lower_bound fb <= b
- ( ( - ( ( q `1 / |. q .| - sn ) ) / ( 1 + sn ) ) / ( 1 + sn ) ) = 1 ;
( LSeg ( c , m ) \/ ( l , k ) \/ ( l , k ) \/ ( l , k ) \/ ( l , k ) c= R ;
consider p being element such that p in LSeg ( x , p ) and p in L~ f and p in L~ f ;
Indices X = [: Seg n , Seg n :] & [: Seg n , Seg n :] = [: Seg n , Seg n :] ;
cluster s => ( q => p ) -> valid ;
Im ( ( Partial_Sums ( F ) ) . m ) is_measurable_on E ;
cluster f . ( x1 , x2 ) -> Element of D ;
consider g being Function such that g = F . t and Q [ g , t ] ;
p in LSeg ( ( W-min ( C , n ) ) , W-min ( C , n ) ) ;
set R8 = R / ]. b , a .[ ;
IncAddr ( I , k ) = IncAddr ( goto ( d + 1 ) , k ) .= Exec ( goto ( d + 1 ) , k ) ;
seq . m <= ( the Sorts of ( s . m ) ) . k ;
a + b = ( a + b ) *' ( a *' ) .= ( a *' ) *' ;
id ( X /\ Y ) = id ( X /\ Y ) /\ id ( X /\ Y ) ;
for x being element st x in dom h holds h . x = f . x ;
reconsider H = U \/ ( U1 \/ U2 ) as non empty Subset of ( TOP-REAL 2 ) | U1 ;
u in ( ( ( c /\ ( d /\ e ) ) /\ f ) /\ j ) /\ m /\ m ;
consider y being element such that y in Y and P [ y , inf B ] ;
consider A being finite stable set such that card ( A ) = card ( A ) ;
p2 in rng ( f |-- ( p1 , p2 , p3 ) ) \ rng <* p1 , p2 *> ;
len ( s1 - s2 ) > 0 & len ( s2 - s2 ) > 0 ;
( W-min ( P ) ) `2 = ( W-min ( P ) ) `2 .= ( E-max ( P ) ) `2 ;
Ball ( e , r + 1 ) c= LeftComp ( Cage ( C , k + 1 ) ) ;
f . a1 ` = f . ( a1 ` + b1 ` ) .= f . ( a1 ` + b1 ) ;
( seq ^\ k ) . n in ]. - \infty , x0 .[ ;
( gg . ( s . ( s . ( s . ( s . ( s . ( s . ( s . ( s . ( s . ( s . ( s . ( s . ( s . ( s . ( s . ( s . ( s . ( s . ( s . ( s . ( s
the InternalRel of S is \rrangle & ( the InternalRel of S ) \/ ( the InternalRel of S ) is non empty ;
deffunc F ( Ordinal , set , set ) = phi . ( $2 , $2 ) ;
F . ( s1 . a ) = F . ( s1 . a ) .= F . ( s1 . a ) ;
x `2 = A . ( o . a ) .= Den ( o , A . a ) ;
Cl ( f " ) c= f " ( ( Cl ( P1 \/ P2 ) ) " ) ;
FinMeetCl ( ( the topology of S ) | ( the topology of S ) ) c= the topology of T ;
synonym o is OperSymbol means : Def2 : o <> * & o <> * ;
assume that X = Y + ( card X ) and card X <> card Y + 1 ;
the \frac of s <= 1 + ( the \frac of s , s ) * ( the ResultSort of s ) ;
LIN a , a1 , d or b , c // a1 , b1 ;
e /. 1 = 0 & e . 2 = 1 & e . 3 = 0 ;
E in SN & not E in { N } ;
set J = ( l , u ) := u ;
set A1 = 1GateCircStr ( <* a , b , c *> , and2 ) , cin = \vert a , b , c , d .| ;
set c9 = [ <* c , 8 *> , <* d , c *> ] , [ <* d , c *> , and2 ] , [ <* c , 8 *> , and2 ] , [ <* d , c *> , and2 ] , [ <* c , 8 *> , and2 ] *> , [ <* d , 8 *> , and2 ] ;
x * z * x " in ( x * N ) * x ;
for x being element st x in dom f holds f . x = g2 . x ;
Int cell ( GoB f , 1 , len GoB f ) c= RightComp f \/ RightComp f \/ RightComp f \/ L~ f ;
U is_an_arc_of W-min C , W-min C & W-min C in L~ Cage ( C , n ) ;
set f-17 = f .: @ @ @ ;
attr S1 is convergent means : Def2 : for x st x in dom S1 holds S1 . x - S2 . x ;
f . ( 0 + 1 ) = ( 0 qua Ordinal ) + a .= a ;
cluster -> with_with_with_symmetric for non empty RelStr ;
consider d being element such that R reduces b , d and R reduces c , d ;
not b in dom Start-At ( card I + 2 , SCMPDS ) ;
( z + a ) + x = z + ( a + y ) .= z + ( a + y ) ;
len ( l | A ) = len l .= len l ;
t4 is ( {} \/ rng ( t ^ <* n *> ) ) -valued FinSequence ;
t = <* F . t *> ^ ( C . p ^ q ) ^ ( C . ( p ^ q ) ) ;
set p-2 = W-min ( C , n ) , pmin = W-min ( C , n ) , pmin = W-min ( C , n ) ;
( ( k + 1 ) - ( i + 1 ) ) = ( k + 1 ) - ( k + 1 ) ;
consider u being Element of L such that u = u *' ( p *' ) and u in D ;
len ( ( width ( ( b |-> a ) |-> ( len b ) ) ) |-> ( len ( b --> a ) ) ) = width ( ( b --> a ) ) ;
FF . x in dom ( ( G * the_arity_of o ) ) ;
set H2 H2 = the carrier of H2 , H2 = the InternalRel of H2 ;
set H1 = the carrier of H1 , H2 = the carrier of H2 , H1 = the InternalRel of H2 ;
( Comput ( P , s , 6 ) ) . intpos ( m + 6 ) = s . intpos ( m + 6 ) ;
IC Comput ( Q , t , k ) = ( l + 1 ) + 1 .= ( l + 1 ) + 1 ;
dom ( ( ( ( id ( the Sorts of A ) * ( ( the Sorts of A ) * ( ( the Sorts of A ) * ( the Sorts of A ) ) ) ) | ( ( the Sorts of A ) * ( the Arity of B ) ) ) | ( ( the Arity of B ) * ( ( the Arity of B ) * ( ( the Arity of
cluster <* l *> ^ phi -> ( 1 + 0 ) string string of S ;
set b5 = [ <* [ <* A1 , cin *> , and2 ] , [ <* cin , dp *> , and2 ] , [ <* cin , dp *> , and2 ] , [ <* cin , dp *> , and2 ] , [ cin , c9 ] , [ m , .. ] ] , [ m , width ] ] , [ m , width ] ] *> ;
Line ( Segm ( M @ , P , Q ) , x ) = L * Sgm ( Q @ ) ;
n in dom ( ( the Sorts of A ) * ( the_arity_of o ) ) ;
cluster f1 + f2 -> continuous for PartFunc of the carrier of S , the carrier of S ;
consider y be Point of X such that a = y and ||. y .|| <= r ;
set xs = ( ( the connectives of S ) . ( \mathop { \rm SBP of S ) , the carrier' of S ) ;
set pr = stop ( I , P , s ) ;
consider a being Point of D2 such that a in W1 and b = g . a ;
{ A , B , C , D } = { A , B , C } \/ { D , E } ;
let A , B , C , D , E , F , J , M be set ;
|. p2 .| ^2 - ( p2 `2 ) ^2 >= 0 ;
l - 1 + 1 = l * ( l + 1 ) + ( l - 1 ) ;
x = v + ( a * w + ( b * w ) ) + ( c * w ) + ( c * w ) ;
the TopStruct of L = k1 ( the TopStruct of L ) & the TopStruct of L = the TopStruct of L ;
consider y being element such that y in dom ( H1 . y ) and x = ( H1 . y ) . y ;
f \ { n } = card ( ( the Sorts of Free ( v1 , v2 ) ) . n ) ;
for Y being Subset of X st Y is not empty holds Y is not not empty implies Y is not empty
2 * n in { N : 2 * Sum ( p | N ) = N } ;
for s being FinSequence holds len ( the _ of A ) = len the _ { s }
for x st x in Z holds ( exp_R * f ) . x = exp_R . x * x + 1 / ( x + 1 )
rng ( h2 * f2 ) c= the carrier of ( TOP-REAL 2 ) | K1 ;
j + ( len f - 1 ) <= len f + ( len f - 1 ) - ( len f - 1 ) ;
reconsider R1 = R * I as PartFunc of REAL n , REAL n ;
Cx . ( x , a ) = ( Cx . ( a , b ) ) . ( x , a ) .= Cx . ( a , b ) ;
power ( F_Complex , z ) = 1 .= ( x |^ ( n + 1 ) ) * ( x |^ ( n + 1 ) ) .= ( x |^ ( n + 1 ) ) * ( x |^ ( n + 1 ) ) ;
t in dom ( the connectives of S , X ) . ( ( the connectives of S ) . ( t . {} ) ) ;
support ( f + g ) c= support f \/ ( support ( f + g ) ) ;
ex N st N = j1 & 2 * Sum ( ( r | N ) | N ) > 0 ;
for y , p , q st P [ p ] holds P [ 'not' p ]
{ [ x1 , x2 ] , [ x2 , y2 ] } is Subset of [: X1 , X2 :] ;
h = ( i , j ) |-> ( id B ) . ( id B ) .= H . ( i , j ) .= H . ( i , j ) ;
ex x1 being Element of G st x1 = x & x1 * N c= A ;
set X = ( ConsecutiveSet ( q , O ) ) . ( ( ConsecutiveSet ( q , O ) ) . ( O , 4 ) ) ;
b . n in { g1 : x0 < g1 & g1 < x0 + r } ;
f /* ( f1 /* s ) is convergent & f /. ( lim s ) = lim ( f /* s ) ;
the lattice of Y = the lattice of ( the lattice of Y ) & the lattice of ( the lattice of X ) = the lattice of ( the lattice of Y ) | ( the carrier of Y ) ;
( 'not' a . x ) '&' b . x = TRUE ;
2 = ( ( ( q ^ <* 0 *> ) ^ ( ( q ^ <* 0 *> ) ^ ( ( q ^ <* 0 *> ) ^ ( ( q ^ <* 0 *> ) ) ) ) ) + len ( ( q ^ <* 0 *> ) ^ ( ( q ^ <* 0 *> ) ) ) ;
( 1 / a ) (#) ( ( sec * f1 ) - id Z ) is_differentiable_on Z ;
set K = upper ( lim ( ( lim ( h , A ) ) , lim ( c , A ) ) , lim ( h , A ) ) , lim ( ( h , A ) , lim ( c , A ) ) ) ;
assume e in { ( \frac { w where w is Element of : w in F & w in G } ) where w is Element of D : ( for x being Element of D st x in F holds w in G } ;
reconsider d1 = dom a `1 , d2 = dom F , d2 = cod F as finite set ;
LSeg ( f , j ) -' ( q .. f ) = LSeg ( f , j ) + q .. f ;
assume X in { T . ( N , 2 ) : h . ( N , 2 ) = N . ( N , 2 ) } ;
assume that Hom ( d , c ) <> {} and <* f , g *> * <* f *> = <* f , g *> ;
dom ( Ss ) = dom S /\ Seg n .= Seg n .= Seg n .= Seg n .= Seg n .= Seg n .= Seg n .= Seg n .= Seg n .= Seg n .= Seg n .= Seg n ;
x in H |^ a implies ex g st x = g |^ a & g in H |^ a ;
a * ( ( 0. ( K , n ) ) . ( a , 1 ) ) = a * ( 0 , n ) .= a * ( 0 , n ) .= a * ( 0 , n ) ;
D2 . j in { r : lower_bound A <= r & r <= upper_bound A } ;
ex p being Point of TOP-REAL 2 st p = x & p `1 <= - 1 & p `2 <= - 1 ;
for c holds f . c <= g . c implies f ^ <* a *> ^ g ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h
dom ( f1 (#) f2 ) /\ X /\ X c= dom ( f1 (#) f2 ) /\ X ;
1 = sqrt ( p * p ) .= p * sqrt ( p * q ) .= p * sqrt ( q * p ) ;
len g = len f + len <* x *> .= len f + len <* y *> .= len f + 1 ;
dom ( F | [: ( ( N | ( S1 | S1 ) ) , ( F | S1 ) | S1 ) ) = dom ( F | S1 ) | S1 ) ;
dom ( f . t ) = dom ( f . t ) .= dom ( f . t ) ;
assume a in ( "\/" ( ( T |^ ( \alpha ) ) ) .: D ) .: ( ( id the carrier of S ) .: D ) ;
assume that g is one-to-one and ( the carrier' of S ) /\ rng g c= dom g ;
( ( x \ y ) \ ( ( x \ z ) \ ( y \ z ) ) ) \ ( ( x \ z ) \ ( y \ z ) ) = 0. X ;
consider f such that f * f = id ( b ) and f * f = id ( b ) ;
( ( ( cos | [. 0 , PI / 2 .] ) | [. 0 , PI .] ) | [. 0 , PI .] ) is increasing ;
Index ( p , co ) <= len ( co - LS ) - Index ( Gij , co ) ;
t1 , t2 , t2 , t1 , t2 , t2 , t1 , t2 , t2 , t1 , t2 , t2 , t2 , t1 , t2 , t2 , t1 , t2 , t2 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 8 ,
( ( ( the _ of L ) * ( the _ of L ) ) . h ) . ( ( the _ of L ) . ( h . ( a , b ) ) ) ) <= ( the _ of L ) . ( ( the _ of L ) . ( a , b ) ) ;
then P [ f . ( i1 + 1 ) , F . ( i1 + 1 ) ] ;
Q [ ( [ D . ( x , 1 ) , F . ( x , 1 ) ] , [ D . ( x , 1 ) ] , [ D . ( x , 1 ) , F . ( x , 1 ) ] ] ) ;
consider x being element such that x in dom ( F . s ) and y = F . s ;
l . i < r . i & [ l . i , r . i ] is Element of G . i ;
the Sorts of ( ( the Sorts of A2 ) * ( the Arity of S ) ) = ( the Sorts of A2 ) * ( the Arity of S ) ;
consider s being Function such that s is one-to-one and dom s = NAT and rng s = { 1 } and rng s c= { 1 } ;
dist ( b1 , b2 ) <= dist ( b1 , a ) + dist ( a , b ) ;
( for n holds ( proj ( C , n ) /. ( len C ) ) /. ( len C ) = W-min ( C , n )
q <= ( W-min ( C , 1 ) ) `1 & ( W-min ( C , 1 ) ) `2 <= ( W-min ( C , 1 ) ) `2 ;
LSeg ( f | i2 , i ) /\ LSeg ( f | i2 , j ) = {} ;
given a being ExtReal such that a <= I and A = ]. a , b .] and A = ]. a , b .] ;
consider a , b be complex number such that z = a + b and z = a + b ;
set X = { b / n where b is Element of NAT : not contradiction } ;
( ( x * y ) * z \ ( x * y ) ) \ ( x * z ) = 0. X ;
set xxy = [ <* xy , yz , yz *> , [ <* yz , yz *> , and2 ] , [ <* yz , yz *> , [ xy , yz ] , [ xy , yz ] , [ xy , yz ] , [ ] , [ ] , [ ] , [ ] , [ xy , yz ] ] , [ xy , yz ] ] , [ xy , yz ] ] ,
( l /. len ( l /. len l ) ) = ( l /. len ( l /. len l ) ) .= ( l /. len l ) ;
sqrt ( ( q `1 / |. q .| - sn ) / ( 1 + sn ) ) ^2 ) = 1 ;
sqrt ( ( p `1 / |. p .| - sn ) / ( 1 + sn ) ) < 1 ;
( ( ( ( ( X \/ Y ) \/ Y ) \/ Z ) \/ Z ) \/ ( ( X \/ Y ) \/ Z ) \/ Z ) = ( X \/ Y ) \/ Z ;
( ( seq - s ) - ( seq - s ) ) . k = ( seq - s ) . k + ( - s ) . k ;
rng ( ( h + c ) ^\ n ) c= dom SVF1 ( 1 , f , x0 ) ;
the carrier of ( the addF of X ) = the carrier of X & the addF of ( X ) = the addF of X ;
ex p3 st p3 = p3 & |. p3 - p4 .| = r & |. p3 .| = 1 ;
set h = ( \raise .4ex ( X , X ) , ( the Sorts of A ) * ( the Arity of S ) ) . ( ( the Arity of S ) . ( o , X ) ) ;
R |^ ( 0 * n ) = I--Element ( X , X ) |^ 0 .= R |^ ( 0 * n ) ;
( Partial_Sums ( ( F ) ) . n ) . n + ( ( ( F . n ) . m ) ) . n is nonnegative ;
f2 = C7 ( V , - ( - 1 , - 1 , - 1 ) ) ;
S1 . b = ( S2 . b ) . b .= S2 . b .= S2 . b ;
p2 in LSeg ( p2 , p1 ) /\ LSeg ( p1 , p2 ) ;
dom ( f . t ) = Seg n & dom ( I . t ) = Seg n & dom ( I . t ) = Seg n ;
assume o = ( the connectives of S ) . 11 & ( the connectives of S ) . 12 in ( the carrier' of S ) . 12 ;
set phi = ( l , l ) \HM ( l , l ) \HM { + l ( ) \HM { l ( ) } ;
synonym p is T T T means : Def2 : HT ( p , T ) = 1 ;
( ( Y1 | ( Y | ( X | ( X | ( X | ( X | ( X | ( X | ( X ) ) ) ) ) ) ) ) ) | ( X | ( X | ( X | ( X | ( X | ( X | ( X | ( X | ( X | ( X | ( X ) ) ) ) ) ) ) ) ) ) | ( X /\ ( X | (
defpred X [ Nat , set , set , set , set , set , set ) = { $2 , $2 } ;
consider k be Nat such that for n being Nat st n <= k holds s . n < x0 + g ;
Det ( I |^ ( ( m -' n ) -' n ) ) = 1_ ( K , n ) ;
sqrt ( b - b ) - sqrt ( b - a ) < 0 ;
Cd . d = CC . ( d , e ) .= CC . ( d , e ) .= CC . ( d , e ) ;
attr X1 is dense means : Def2 : X1 is dense & X2 is dense implies X1 /\ X2 is dense SubSpace of X ;
deffunc F ( Element of E , Element of I ) = ( $1 , $2 ) * ( $1 , $2 ) ;
t ^ <* n *> in { t ^ <* i *> where i is Element of NAT : Q [ i , T . i ] } ;
( x \ y ) \ x = ( x \ x ) \ ( y \ x ) .= ( y \ x ) \ ( y \ x ) .= 0. X ;
for X being non empty set holds ( for Y being Subset-Family of X holds Y is Basis of <* X , Y *> ) implies ( for Y being Subset-Family of X st Y in F holds Y is Basis of X )
synonym A , B , C , D , E , F , G , N , F , G , G , H , H , G ) for A , B , H , F , G , H , G , H , G , H , F , G , H , G , H , H , G , F , G , H , G , H , H , G , F , G , H , G ,
len ( ( M @ ) ) = len p & width ( M @ ) = width ( M @ ) ;
v = { x where x is Element of K : 0 < v . x } ;
( ( ( Sgm ( Seg m ) ) . d - ( Sgm ( Seg m ) ) . e ) ) . e <> 0 ;
lower_bound divset ( D2 , k + 1 ) = D2 . ( k + 1 ) - D2 . ( k + 1 ) ;
g . r1 = ( 2 * r1 + 1 ) * r1 & dom h = [. 0 , 1 .] & dom h = [. 0 , 1 .] ;
|. a .| * ||. f .|| = 0 * ||. f .|| .= ||. a * f .|| .= ||. f .|| ;
f . x = ( h . x ) `1 & g . x = ( h . x ) `1 ;
ex w st w in dom ( B ^ <* w *> ) & <* 1 *> ^ s = <* 1 *> ^ s ;
[ 1 , {} , <* d1 *> ] in ( { [ 0 , 0 ] } \/ ( { 0 , 0 } ) \/ ( { 0 , 0 } ) \/ { 0 , 0 } ) \/ ( { 0 , 0 } ) \/ { 0 , 0 } ) ;
IC Exec ( i , s1 ) + n = IC Exec ( i , s1 ) + n .= ( IC Exec ( i , s1 ) + n ) ;
IC Comput ( P , s , 1 ) = IC Comput ( P , s , 1 ) .= 5 + ( card I + 1 ) .= 5 + 1 ;
( IExec ( W6 , Q , t ) ) . intpos ( i + 1 ) = t . intpos ( i + 1 ) ;
LSeg ( f , i -' 1 ) misses LSeg ( f , j ) /\ LSeg ( f , j ) ;
assume for x , y being Element of L st x in C holds x <= y or x <= y ;
integral ( f , X ) = f . ( upper_bound ( C ) ) .= f . ( upper_bound ( C ) ) ;
for F , G being one-to-one FinSequence st rng F misses rng G holds F ^ G is one-to-one
||. R /. h . ( h + c ) .|| < e * ( K + c ) ;
assume a in { q where q is Element of M : dist ( z , q ) <= r } ;
set p3 = [ 2 , 0 , 1 , 0 ] .--> [ 2 , 0 ] ;
consider x , y being Subset of X such that [ x , y ] in F and x c= d and y c= d ;
for y , x being Element of REAL st y in Y & x in X & y in Y holds x <= y
func |. p ^ |. p .| .| -> variable of A , ( the Sorts of A ) . ( p ^ <* p *> ) ;
consider t being Element of S such that x `1 , y `2 , z `2 and x , y `2 '||' z `1 , t `2 ;
dom ( x1 | ( len x1 ) ) = Seg ( len x1 ) & len ( x1 | ( len x1 ) ) = len ( x1 | ( len x1 ) ) ;
consider y2 being Real such that x2 = y2 & 0 <= y2 & y2 <= 1 and 0 <= y2 & y2 <= 1 ;
||. f | X .|| = ||. f /. ( lim s1 ) .|| .= ||. f /. ( lim s1 ) .|| ;
( the InternalRel of A ) ` /\ ( the InternalRel of B ) /\ ( the InternalRel of C ) = {} \/ ( the InternalRel of C ) /\ ( the InternalRel of C ) .= {} ;
assume that i in dom p and for j being Nat st j in dom q holds P [ i , j ] and P [ i , j ] ;
reconsider h = f | [: X , Y :] as Function of [: X , Y :] , the carrier of Y ;
u1 in the carrier of W1 & ( the InternalRel of W1 ) /\ the carrier of W2 = the InternalRel of W2 & ( the InternalRel of W1 ) /\ ( the InternalRel of W2 ) = the InternalRel of W2 ;
defpred P [ Element of L ] means $1 <= f . $1 & f . $1 <= f . $1 ;
T . ( u , a , v ) = s * ( x + ( - x ) * y ) .= b * ( ( s * x ) + ( - x ) * y ) .= b ;
- ( - ( R1 + R2 ) ) = - ( x + y ) .= - ( x + y ) + ( y + x ) .= - ( x + y ) + ( y + x ) .= - ( x + y ) ;
given a being Point of ( G | A ) such that for x being Point of G holds x in A iff x , a ] in A ;
f = [ [ [ dom ( f2 ) , ( cod f2 ) ] , [ cod f2 ] , [ cod f2 ] ] , [ cod f2 , cod f2 ] ] , [ cod f2 ] , [ cod f2 ] ] , [ cod f2 ] ] ;
for k , n being Nat holds k < 0 & k < n implies k * ( n + 1 ) is prime
for x being element holds x in A |^ d iff x in ( A ` ) |^ d
consider u , v being Element of R such that l /. i = u * a ;
( - sqrt ( ( p `1 / |. p .| - sn ) / ( 1 + sn ) ) ^2 ) > 0 ;
L-13 . k = LU . ( F . k ) & F . k in rng ( L . k ) ;
set i2 = AddTo ( a , i , n ) , i2 = AddTo ( a , i , n ) ;
attr B is Al means : Def2 : for S being non empty ManySortedSign holds S . ( S , x ) = ( B . S ) . ( S . x , x ) ;
a9 "/\" D = { a "/\" d where d is Element of N : d in D & a "/\" d in D } ;
( ( ( \square | ( len ( q | ( len q ) ) ) ) | ( len ( q | ( len q ) ) ) ) ) | ( len ( q | ( len q ) ) ) ) >= ( ( q | ( len q ) ) ) | ( len q ) ) | ( len q ) ;
( - f ) . ( upper_bound A ) = ( - f ) . ( upper_bound A ) .= ( - f ) . ( upper_bound A ) ;
( G * ( len G , 1 ) ) `1 = ( G * ( len G , 1 ) ) `1 .= ( G * ( len G , 1 ) ) `1 ;
( Proj ( i , n ) * ( L . n ) ) . ( L . n ) = <* proj ( i , n ) . ( L . n ) *> ;
f1 + f2 * reproj ( i , x ) is_differentiable_in ( ( the PartFunc of G , x ) * reproj ( i , x ) ) . ( f1 . ( x - x ) ) ;
attr ( ( ( id Z ) . x ) (#) ( ( id Z ) . x ) ) <> 0 ;
ex t being Term of S , X st ( h1 . x ) . ( x . s ) = ( h2 . s ) . ( x . s ) ;
defpred C [ Nat ] means ( P8 . $1 ) is countable & ( ( n + $1 ) -tuples_on the carrier of S ) is countable & ( n + $1 ) -A\ A ) is countable ;
consider y being element such that y in dom ( ( p | i ) | ( Seg n ) ) and ( q | ( Seg n ) ) . y = ( p | ( Seg n ) ) . y ;
reconsider L = product ( { x1 } , ( indx ( B ) ) -' l ) as Basis of A ;
for c being Element of C holds T . ( id c ) = id ( the carrier' of C ) . ( id c )
diff ( f , n , p ) = ( f | n ) ^ <* p . n *> .= f | n ^ <* p . n *> ;
( f * g ) . x = f . ( g . x ) & ( f * h ) . x = f . ( h . x ) ;
p in { 1 / 2 * ( G * ( i + 1 , j ) + G * ( i + 1 , j ) } ;
f `2 = ( c | ( n , L ) ) *' ( g - f ) *' .= ( - ( c | n ) ) *' ;
consider r be Real such that r in rng ( f | divset ( D , j ) ) and r < m + r ;
f1 . ( ( ( ( ( ( ( ( ( ( 8 8 8 8 ) ) ) ) | ( ( 8 8 8 ) | ( ( the carrier of S ) ) | ( the carrier of S ) ) ) | ( the carrier of S ) ) ) ) ) ) in ( ( the carrier of S ) | ( the carrier of S ) ) | ( the carrier of S ) ) ;
eval ( a | n , L ) = eval ( a | n , x ) .= ( a | n ) . ( x . x ) .= a . x ;
z = ( DigA ( tl , x ) ) . ( ( n + 1 ) + 1 ) .= ( DigA ( tl , x ) ) . ( ( n + 1 ) + 1 ) .= ( DigA ( tl , x ) ) . ( ( n + 1 ) + 1 ) .= ( n + 1 ) + 1 ;
set H = { ( Intersect S ) . ( ( Intersect S ) . ( i , j ) ) where S is Subset-Family of X : S is open } ;
consider S19 being Element of D .: ( j , d ) , d being Element of D .: ( j , d ) such that S = S19 ^ <* d *> ;
assume that x1 in dom f and x2 in dom f and f . x1 = f . x2 ;
- 1 <= sqrt ( ( q `1 / |. q .| - sn ) / ( 1 + sn ) ) ;
Sum ( L (#) ( - id V ) ) = 0. ( V , m ) & Sum ( - ( 0. ( V , m ) ) ) = 0. ( V , m ) ;
let k1 , k2 , k2 , k2 , k1 , k2 , k2 , k2 be Element of NAT , a , k1 , k2 , k2 be Element of D ;
consider j being element such that j in dom a and j in dom a and x = a . j and x = a . j ;
H1 . ( x1 , x2 ) c= H1 . ( x2 , y2 ) or H1 . ( x1 , x2 ) c= H1 . ( x2 , y2 ) ;
consider a be Real such that p = 1 * ( p1 + p2 ) + ( 0 * ( p1 + p2 ) ) and 0 <= a and a <= 1 ;
assume that a <= c and b <= d and [ a , b ] c= [: f , g :] ;
cell ( Gauge ( C , m ) , len Gauge ( C , m ) -' 1 , 0 ) is non empty ;
AZ in { ( S . i ) `1 where i is Element of NAT : i < n } ;
( T * b1 ) . y = L * ( ( F * b1 ) . y ) .= ( F * b1 ) . y .= ( F * b1 ) . y ;
g . ( s , I ) . ( s , I ) = s . ( s , I ) . ( s , I ) & g . ( s , I ) . ( s , I ) = |. s . I . I .| ;
( ( log ( 2 , k + 1 ) ) / ( ( log ( 2 , k + 1 ) ) ) ) to_power ( ( ( log ( 2 , k + 1 ) ) / ( ( log ( 2 , k + 1 ) ) ) ) to_power ( ( ( log ( 2 , k + 1 ) ) ) to_power ( ( ( log ( 2 , k + 1 ) ) ) to_power ( ( k + 1 ) ) ) ) ) ) >= ( ( ( log (
then that p => q in S and not x in the carrier of S and not x in the carrier of S ;
dom ( the multF of ( ( the multF of ( ( the InstructionsF of ( ( the Arity of ( ( the connectives of ( ( union ( ( the Arity of n ) ) ) ) ) ) ) ) ) ) ) ) ) misses dom ( the multF of ( ( the connectives of ( ( the connectives of ( n + 1 ) ) ) ) ) ;
synonym f is \kern1pt -> \kern1pt means : Def2 : for x being set st x in rng f holds x is integer ;
assume for a being Element of D holds f . { a } = a & f .: { a } = f . ( union rng f ) ;
i = len ( p1 + p2 ) .= len ( p1 + p2 ) .= len ( p1 + p2 ) + len ( p2 + p3 ) .= len ( p1 + p3 ) + len ( p3 + p3 ) .= len ( p3 + p3 ) + len ( p3 + p3 ) .= len ( p3 + p3 ) ;
( l + 1 ) * ( k + 1 ) = ( g /. ( k + 1 ) ) * ( ( l + 1 ) * ( k + 1 ) ) ;
CurInstr ( P2 , Comput ( P2 , s2 , l ) ) = halt S .= halt S ;
assume for n being Nat holds ||. ( seq . n ) - ( seq . n ) .|| <= ||. ( seq . n ) - ( seq . n ) .|| ;
sin ( w ) = sin ( r * ( cos ( w ) ) ) .= sin ( r ) * sin ( s ) .= 0 ;
set q = |[ ( g1 `1 ) * ( g2 `2 ) , ( g2 `2 ) * ( g2 `2 ) ]| ;
consider G being sequence of S such that for n being Element of NAT holds G . n in GGs ( F . n ) ;
consider G such that F = G and ex G1 being Element of S st G1 in ( the carrier of G ) & G = ( the InternalRel of G ) . ( len G ) ;
the root of [ x , s ] in ( the Sorts of Free ( C , s ) ) . ( ( the Sorts of Free ( C , s ) ) . ( ( the Sorts of Free ( C , s ) ) . ( ( the Sorts of Free ( C , s ) ) . ( ( the Sorts of Free ( C , s ) ) . ( ( the Sorts of Free ( C , s ) ) . ( ( the Sorts of Free (
Z c= dom ( ( exp_R * ( exp_R + ( exp_R * f1 ) ) ) ) /\ dom ( ( exp_R * ( exp_R * f1 ) ) ) ;
for k being Element of NAT holds ( Im ( f ) ) . k = ( lim ( Im ( f ) ) ) . k
assume that - 1 < ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) ) * ( ( - 1 ) * ( ( -
assume that f is continuous and a < b and f . a = c and f . b = d ;
consider r being Element of NAT such that ( ex s being Element of NAT st s = Comput ( P1 , s1 , 1 ) & r <= s & s <= 1 ) & r <= 1 ;
LE f /. ( i + 1 ) , f /. ( j + 1 ) , f /. ( j + 1 ) , f /. ( j + 1 ) , f /. ( j + 1 ) ;
assume that x in the carrier of K and y in the carrier of K and inf { x , y } in K and inf { x , y } in K ;
assume f +* ( i1 , R ) in ( proj ( F , i1 ) * ( i2 , R ) ) " ;
rng ( ( Flow M ) | ( the carrier of M ) ) c= the carrier of M ;
assume z in { ( the carrier of G ) \times { t } where t is Element of T : not contradiction } ;
consider l be Nat such that for m be Nat st l <= m holds ||. ( s1 . m ) - ( lim s1 ) .|| < g ;
consider t be VECTOR of product ( G . t ) such that ( ex m being Element of dom ( G . t ) st ( for t being Element of dom ( G . t ) holds ||. t .|| <= 1 ) ;
cluster the carrier of v = 2 * card ( { 0 } ) , v ^ <* 1 *> , v ^ <* 1 *> , v ^ <* 1 *> ) -> Element of dom p ;
consider a being Element of the points of ( X ) , A being Element of the carrier' of ( X ) such that a on A , B and not a on A ;
( - x ) |^ ( k + 1 ) * ( ( - x ) |^ ( k + 1 ) ) = 1 ;
for D being set st i in dom p holds p . i in D & p . i in D & p . i in D ;
defpred R [ element , element ] means ex x , y being element st $1 = [ x , y ] & $2 = [ x , y ] ;
L~ ( f2 | ( L~ ( f1 | ( L~ ( f1 | ( L~ ( f1 | ( f1 | ( ( f1 | ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( L~ f1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) | ( ( ( ( ( f2 | ( L~ ( f1 |
i - ( len ( h - 2 ) + 1 ) + 2 - ( i - 1 ) + 1 < i - ( len h - 2 + 1 ) + 2 - ( i - 1 ) + 1 ;
for n being Element of NAT st n in dom F holds F . n = |. ( n -' 1 ) . ( n -' 1 ) .| ;
for r , s1 , s2 , s3 , s3 , s3 , s2 , s3 , s3 , s3 , s3 , s3 , s2 , s3 , s3 , s3 , s2 , s3 , s3 , s2 , s3 , s3 , s2 , s3 , s3 , s2 , s3 , s2 , s3 , s3 , s2 , t2 , s2 , t2 , t2 , s2 , t2 ) is Real ;
assume v in { G where G is Subset of ( TOP-REAL 2 ) | B : G * ( i , j ) `1 <= G * ( i , j ) `1 } ;
let g be non-empty Function of A , ( the Sorts of A ) * , ( the Sorts of A ) * , ( the Sorts of A ) * ;
min ( g . [ x , y ] , k ) . ( [ y , z ] , [ x , z ] ) = ( min ( g . ( y , z ) , [ x , z ] ) , [ y , z ] ) ;
consider q1 be sequence of CC such that for n holds P [ n , q1 . n , q1 . ( n + 1 ) ] ;
consider f being Function such that dom f = NAT and for n being Element of NAT holds f . n = F ( n ) ;
reconsider BZ = B /\ ( O /\ B ) , CZ = O /\ ( O /\ ( O /\ Z ) as Subset of B ;
consider j being Element of NAT such that x = the Element of ( n -tuples_on the carrier of K ) and 1 <= j and j <= n and 1 <= n and j <= n and n <= n ;
consider x such that z = x and card ( x . O ) in card ( x . O ) and x . O in ( x . O ) . O ;
( C * ( k , n2 ) ) . 0 = C ( ( k , n2 ) . 0 ) .= C ( ( k , n2 ) . 0 ) ;
dom ( X --> f ) = X & dom ( X --> f ) = X & rng ( X --> f ) = dom ( X --> f ) ;
( W-min L~ Cage ( C , n ) ) `2 <= ( E-max L~ Cage ( C , n ) ) `2 & ( W-min L~ Cage ( C , n ) ) `2 <= N-bound L~ Cage ( C , n ) `2 ;
synonym x , y , x or { x , y } = l or { x , y } on l ;
consider X be element such that X in dom ( f | ( n + 1 ) ) and ( f | ( n + 1 ) ) . X = Y ;
assume that Im k is continuous and for x , y being Element of L st x = x & y = y holds x << y iff x << y ;
( 1 / 2 * ( AffineMap ( 0 , 0 ) ) ) (#) ( ( AffineMap ( 0 , 0 , 0 ) ) (#) ( ( AffineMap ( 0 , 0 , 0 ) ) (#) ( ( AffineMap ( 0 , 0 , 0 ) ) (#) ( ( AffineMap ( 0 , 0 , 0 ) ) ) ) ) ) is_differentiable_on REAL ;
defpred P [ Element of omega ] means ( the partial of A1 ) . $1 = ( the partial of A2 ) . $1 & ( the partial of A2 ) . $1 = ( the partial of A2 ) . $1 ;
IC Comput ( P , s , 2 ) = succ IC Comput ( P , s , 2 ) .= 6 + 1 .= 6 + 1 ;
f . x = f . ( g1 . x ) * f . ( g2 . x ) .= f . ( g1 . x ) * f . ( g2 . x ) .= f . ( ( f1 . x ) * f . x ) .= f . ( ( f1 . x ) * f . x ) ;
( M * ( F . n ) ) . ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( s ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( F
the carrier of ( L1 + L2 ) c= ( the carrier of L2 ) \/ ( the carrier of L2 ) \/ ( the carrier of L1 ) \/ ( the carrier of L2 ) ;
pred a , b , c , x , y , z ) means : Def2 : for a , b , c being Element of o , x , y being Element of o , a being Element of o , b being Element of o , a being Element of o , b st x = a & y , z is_collinear & x , y // b , z ;
( the partial of product s ) . n <= ( the partial of product s ) . ( n + 1 ) ;
attr - 1 <= r & ( - 1 ) * ( - 1 ) = ( - 1 ) * ( - 1 ) ;
seq in { p ^ <* n *> where n is Nat : p ^ <* n *> in dom ( p ^ <* n *> ) } ;
[ x1 , x2 , x3 ] . ( 2 * PI ) = [ x1 , x2 ] . ( 2 * PI ) ;
attr F . m is nonnegative means : Def2 : ( Partial_Sums F ) . m is nonnegative ;
len ( the addF of G , z ) = len ( ( the multF of G ) . ( ( the _ of G ) . ( ( the Arity of G ) . ( ( the Arity of G ) . ( ( the Arity of G ) . ( o , z ) ) ) ) ) ;
consider u , v being VECTOR of V such that x = u + v and u in W /\ ( W1 /\ W2 ) ;
given F being FinSequence of NAT such that F = x and dom F = n and rng F c= { 0 , 1 } and Sum F = 0 and Sum F = 1 ;
0 = 1- ( 1- @ ) iff 1 = ( - ( 1- ( 1- @ ) ) * ( 1 - ( - ( 0 ) ) * ( 1 - ( 0 ) ) * ( 1 - ( 0 ) ) * ( 1 - ( 0 ) ) * ( 1 - 0 ) ) ) ;
consider n be Nat such that for m be Nat st n <= m holds |. ( f # x ) . m - lim ( f # x ) .| < e ;
cluster \mathclose { \rm \mathclose { \rm c } } -> \mathclose { \rm c } & ( for c being Element of L holds ( ( \mathclose { \rm c } ) | ( { c } ) ) . ( { c } ) ) is Boolean & ( ( \mathclose { \rm c } ) | ( { c } ) ) . ( { c } ) is Boolean
"/\" ( B , {} ) = "/\" ( ( the carrier of S ) , ( the InternalRel of S ) . ( the InternalRel of S ) ) .= "/\" ( ( the InternalRel of S ) . ( the InternalRel of S ) . ( the InternalRel of S ) . ( the InternalRel of S ) . ( the InternalRel of S ) ) .= "/\" ( ( the InternalRel of S ) . ( the InternalRel of S ) . ( the InternalRel of S ) ) .= "/\" ( ( the InternalRel of S ) .
sqrt ( r / 2 + ( r / 2 ) / 2 ) <= sqrt ( r / 2 + ( r / 2 ) / 2 ) ;
for x being element st x in A /\ dom ( f `| X ) holds ( f `| X ) . x >= r2
2 * ( r1 - c ) - ( 2 * ( a - c ) ) = 0. TOP-REAL 2 ;
reconsider p = P * ( \square , 1 ) , q = a " * ( ( the multF of K ) * ( 1- ( n , 1 ) ) ) as FinSequence of K ;
consider x1 , x2 being element such that x1 in uparrow s and x2 in uparrow s and x = [ x1 , x2 ] ;
for n being Nat st 1 <= n & n <= len ( q1 ^ q2 ) holds q1 . n = ( ( the Sorts of A1 ) . ( n + 1 ) ) . ( n + 1 )
consider y , z being element such that y in the carrier of A and z in the carrier of A and i = [ y , z ] ;
given H1 , H2 being strict Subgroup of G such that x = H1 & y = H2 & H1 = H2 and H2 = H ;
for S , T being non empty RelStr , d being Function of S , T holds d is complete implies d is complete & d is complete & d is complete & d is complete & d is complete
[ a + 0 , i ] in ( the addF of V ) /\ [: the carrier of V , the carrier of V :] ;
reconsider m1 = max ( ( ( p . n ) * ( <* x *> ) ) * ( <* x *> ) ) as Element of NAT ;
I <= width ( ( the multF of K ) * ( i , j ) ) & ( the addF of K ) * ( i , j ) = ( ( the multF of K ) * ( i , j ) ) * ( i , j ) ;
f2 /* q = ( f2 /* ( f1 /* s ) ) /* ( ( f1 /* s ) ^\ k ) .= ( ( f1 /* s ) /* s ) ^\ k .= ( ( f1 /* s ) /* s ) ^\ k .= ( f1 /* s ) ^\ k ;
attr A1 \/ A2 means : Def2 : for A , B being Element of V st A misses B & A misses C & B <> C & C <> D holds A /\ ( B \/ C ) = C /\ D ;
func A -\to C -> set equals union { A . s where s is Element of C : s in A & s in C } ;
dom ( Line ( v , i + 1 ) (#) ( ( \mathopen ( p , m ) ) (#) ( ( \mathopen ( p , m ) ) (#) ( ( q , m ) (#) ( <* p *> ) ) ) ) ) = dom ( F ^ ( m , n ) ) ;
cluster [ ( x `1 ) , ( x `2 ) ] -> LSeg ( x `2 , ( x `2 ) `2 ) -> empty ;
E , ( All ( x , y , G ) ) . ( ( All ( x , y , G ) . ( x , y , G ) ) . ( x , y , G ) ) |= All ( x , y , G ) . ( x , y , G ) . ( x , y , G . ( x , y , G ) ) ;
F .: ( id X , g ) . x = F . ( id X , g . x ) .= F . ( id X , g . x ) .= F . ( id X , g . x ) ;
R . ( h . m ) = F . ( x0 + h . m ) + ( h . m ) , ( h . m ) . x0 + ( h . m ) . x0 ;
cell ( G , ( X -' 1 , Y + 1 ) , ( X -' 1 ) \ ( X \/ Y ) \ ( X \/ Y ) \ ( X \/ Y ) ) meets ( the carrier of TOP-REAL 2 ) \ ( X \/ Y ) ;
IC Comput ( P2 , s2 , k ) = IC Comput ( P2 , s2 , k ) .= IC Comput ( P2 , s2 , k ) .= IC Comput ( P2 , s2 , k ) .= IC Comput ( P2 , s2 , k ) ;
sqrt ( ( - ( ( ( q `1 / |. q .| - sn ) / ( 1 + sn ) ) / ( 1 + sn ) ) ^2 ) ) > 0 ;
consider x0 being element such that x0 in dom a and x0 in dom a and g . x0 = a . x0 and g . x0 = a . x0 ;
dom ( ( r1 (#) ( A * ( m , C ) ) ) | A ) = dom ( ( A * ( m , C ) ) | A ) .= A /\ C .= C ;
d . ( y , z ) = ( ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y
attr i , C . i = A . i /\ B . i means : Def2 : for C being Nat holds it . C = A . i /\ B . i ;
assume that x0 in dom f and f is continuous and for x st x in dom f holds f . x = ( f . x ) / ( x + x0 ) ;
p in Cl A implies for K being Basis of p st K in K & A c= K holds A meets K
for x being Element of REAL n st x in Line ( x1 , x2 ) holds |. ( x1 - x2 ) . x - ( x2 - y2 ) . x .| <= |. ( x1 - x2 ) . x - ( x2 - y2 ) . x .|
func <* a *> -> Ordinal means : Def2 : for b being Ordinal st a in it holds it . b = b & for a being Ordinal st a in it holds it . a = a & for b being Ordinal st b in it holds it . b = b ;
[ a1 , a2 ] in ( the InternalRel of A ) \/ ( the InternalRel of B ) & [ a1 , a2 ] in ( the InternalRel of B ) \/ ( the InternalRel of C ) ;
ex a , b being element st a in the carrier of S1 & b in the carrier of S2 & x = [ a , b ] & x = [ a , b ] ;
||. ( v . n - v . m ) - ( v . n - v . m ) .|| < r / 2 * ||. x .|| ;
then for Z being set st Z in { Y where Y is Element of I : for x being Element of I st x in Z holds F . x = { x } } ;
sup { [ s , t ] where s , t is Element of S : P [ s , t ] } = { [ s , t ] where s is Element of S ( ) , t is Element of S ( ) : P [ s , t ] } ;
consider i , j such that i < j and [ y , f . i ] in [: I , I :] and [ i , f . j ] in [: I , I :] ;
for D being non empty set , p being FinSequence of D st p c= q & ex q being FinSequence st q c= p & q is FinSequence of D & q is FinSequence of D & p is FinSequence of D & q is FinSequence of D
consider e1 being Element of the carrier of X such that not c , e1 // a , b and not a , b // c , d and not c , d // a , b and a , b // c , d ;
set U = I \! \mathop { N } , m = I \! \mathop { N } ;
|. ( q `1 / |. q .| - sn ) / ( 1 + sn ) .| = ( |. q .| ) / ( 1 + sn ) .= |. q .| / ( 1 + sn ) ;
for T being non empty TopSpace , x , y being Element of [: the topology of T , the topology of T :] holds x "/\" y = x "\/" y & x "/\" y = x "\/" ( y "/\" ( x "/\" y ) )
dom ( ( the charact of U1 ) * ( the Arity of S ) ) = dom ( the Arity of S ) & ( the Arity of S ) * ( the Arity of S ) = ( the Arity of S ) * ( the Arity of S ) ;
dom ( h | X ) = dom h /\ X .= X /\ ( dom ( h | X ) ) .= X /\ dom ( h | X ) .= X /\ X .= X /\ ( dom ( h | X ) ) .= X /\ X ;
for N1 , N2 being Element of ( the carrier of G ) , h being Element of G holds ( h . ( h . ( x , y ) ) ) = N & ( h . ( x , y ) ) = ( h . ( x , y ) ) ) & ( h . ( x , y ) ) = ( h . ( x , y ) ) & ( h . ( x , y ) ) = ( h . ( x , y ) ) & ( h . ( x , y ) ) & ( h . ( x , y ) = ( h . ( x , y ) ) . ( x , y ) )
( ( mod ( u , m ) ) + mod ( v , m ) ) . i = ( mod ( u , m ) ) ) . i + ( mod ( v , m ) ) . i ;
- ( q `1 / |. q .| - sn ) < - ( - ( q `1 / |. q .| - sn ) ) & - ( - ( q `1 / |. q .| - sn ) ) / ( 1 - sn ) <= - ( ( q `1 / |. q .| - sn ) ) / ( 1 - sn ) ;
attr r1 = f means : Def2 : for x st x in dom f holds f . x = ( f . x ) * ( f . x ) ;
( for m being bounded Function holds ||. ( seq . m ) - ( seq . m ) .|| = ( ( seq . m ) - ( seq . m ) ) ) . x
attr a <> b & b <> c & a , b , c is_collinear & not ex a , b st a , b , c is_collinear & angle ( a , b , c ) = 0 & angle ( a , b , c ) = 0 & angle ( a , b , c , d ) = 0 ;
consider i , j , s being Real such that p1 = [ i , j ] and p2 = [ i , j ] and p1 = [ i , j ] and p2 = [ i , j ] ;
|. p .| ^2 + ( 2 * ( p , q ) ) ^2 + ( 2 * ( p , q ) ) ^2 + ( 2 * ( p , q ) ) ^2 ) = |. p .| ^2 + ( 2 * ( p , q ) ) ^2 ;
consider p1 , q1 , q2 being Element of ( X , p1 ) , q1 , q2 being Element of ( X , p1 ) , q2 being Element of ( X , p2 ) , q1 , q2 be Element of ( X , p1 ) , q2 be Element of X ;
<* ( r / ( r1 + r2 ) ) / ( s1 + s2 ) *> = <* ( r / ( s1 + s2 ) ) / ( s1 + s2 ) *> ;
( ( for w being Element of A , v being Element of B st w = inf ( A /\ ( B /\ C ) ) holds ( ( A /\ ( C /\ D ) ) ) . ( w , v ) ) is non empty & ( ( A /\ ( C /\ D ) ) . ( w , v ) ) is not empty ) implies not ( ( A /\ ( C /\ D ) ) . ( w , v ) is not empty & not ( for w being Element of B st w in C ) & ( for w being Element of C ) . ( w , v ) is not empty ) &
s , ( H / ( H / ( H / ( H / ( H ) ) ) ) ) / ( H / ( H / ( H / ( H ) ) ) ) ) / ( H / ( H / ( H ) ) ) ) / ( H / ( H / ( H / ( H / ( H ) ) ) ) ) / ( H / ( H . ( H / ( H ) ) ) ) ) ) / ( H / ( H . ( H . ( H / ( H ) ) ) ) ) ) / ( H . ( H . ( H . ( H . ( H . ( H
len ( ( b + 1 ) + 1 ) = card ( ( b + 1 ) + 1 ) .= card ( b + 1 ) + 1 .= ( b + 1 ) + 1 .= ( b + 1 ) + 1 .= ( b + 1 ) + 1 .= ( b + 1 ) + 1 ;
consider z being Element of L1 such that z is_>=_than x and for x being Element of L1 st x >= y holds z >= x & z >= y ;
LSeg ( W-min ( D , |[ a , b ]| ) , |[ b , c ]| ) /\ LSeg ( |[ b , d ]| , |[ b , c ]| ) = { W-min ( D , a ) + |[ b , c ]| } ;
lim ( ( f / g ) /* ( h + c ) ) = lim ( ( f / g ) /* ( h + c ) ) .= lim ( ( f / g ) /* ( h + c ) ) ;
P [ i , ( pr1 ( f ) ) . ( 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 . m ) .|| < r ;
for X being set , P being a_partition of X , a being Element of X , b being Element of X st x in P & b in P & a in P & b in P & a in P & b in P & a in P & b in P & a in P & b in P & b in P & a in P & b in P & a in P & b in P & a in P & b in P & a in P & b in P & a in P & b in P & b in P & a in P & b in P & a in P & b in P & a
Z c= dom ( ( ( exp_R * f ) `| Z ) /\ ( ( exp_R * f ) `| Z ) \ ( ( exp_R * f ) `| Z ) \ ( exp_R * f ) " { 0 } ) ;
ex j being Nat st j in dom ( l ^ <* x *> ) & j = ( l ^ <* x *> ) . j & i = ( l + 1 ) + 1 ;
for u , v being VECTOR of V , r being Real st 0 < r & for u being VECTOR of V st u in N holds r * u + ( 1 - r ) * ( 1 - r ) in N
A , Int A , Int ( A \/ B ) , Int ( A \/ B ) , Int ( A \/ C ) , Int ( B \/ C ) , Int ( A \/ C ) , B ) , Int ( A \/ C ) , B ) , C , D ) ;
- Sum <* v , u , w *> = - ( v + u ) .= - ( v + u ) + ( u + u ) .= ( - ( u + u ) + ( - u ) ) + ( - u ) .= ( - ( v + u ) ) + ( - u ) .= ( - ( u + u ) ) + ( - u ) ;
( Exec ( a := b , s ) ) . IC S = ( Exec ( a := b , s ) ) . IC S .= succ IC S .= succ IC S .= succ IC S .= succ IC S ;
consider h being Function such that f . a = h and dom h = I and for x being element st x in I holds h . x = ( the Sorts of J ) . x ;
for S1 , S2 being non empty RelStr , D being non empty Subset of S1 , f being Function of S1 , S2 st f is directed & g is directed holds f is directed & g is directed & for x being Element of S1 , y being Element of S2 st x in D & y in D holds f . ( x , y ) is directed & f . ( x , y ) is directed & f . ( y , x ) is directed & f . ( y , x ) is directed & f . ( y , x ) is directed & f . ( y , x ) is directed & f . ( y , x ) is directed
card X = 2 implies ex x st x in X & ex y st y in X & not x in X & y in X & not x in X & y in X & not x in X & y in X & not x in X & y in X & not x in X & x = y or x = y ) & not x in X & y in X & y in X & x = y ) & x = y ) & y = y ;
W-min ( C , n ) in rng ( W-min ( C , n ) \circlearrowleft W-min ( C , n ) ) ;
for T , S being DecoratedTree , T being Element of dom T , p being Element of dom T st p in dom T & q in dom T holds ( T , p ) . ( q , p ) = T . ( p , q )
[ i2 + 1 , j2 ] in Indices G & [ i2 + 1 , j2 ] in Indices G & f /. ( i2 + 1 ) = G * ( i2 + 1 , j2 ) ;
cluster ( for Nat , n being Nat holds ( k divides n ) divides ( k |^ n ) & ( k divides n ) & ( k divides n implies ( k divides n ) mod ( k -' n ) ) implies ( k divides n ) mod ( k -' n ) ) ;
dom F = the carrier of ( X | A ) & rng F = the carrier of ( X | A ) & rng F = the carrier of ( X | A ) & rng F = the carrier of ( X | A ) ;
consider C being finite Subset of V such that C c= A and card C = card C and card C = card C and card C = card C = card C ;
V is prime implies for X being Subset of [: the topology of T , the topology of T :] st X /\ Y c= V holds X c= V or X c= V
set X = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } , Y ( ) ;
angle ( p1 , p2 , p3 ) = 0 or angle ( p2 , p3 , p4 ) = 0 & angle ( p2 , p3 , p4 , p4 ) = angle ( p2 , p4 , p4 , p4 ) ;
- sqrt ( ( q `1 / |. q .| - sn ) / ( 1 + sn ) ) / ( 1 + sn ) = - ( ( q `2 / |. q .| - sn ) ) / ( 1 + sn ) ) / ( 1 + sn ) .= - ( ( q `2 / |. q .| - sn ) ) / ( 1 + sn ) ) / ( 1 + sn ) ;
ex f being Function of I[01] , TOP-REAL 2 st f is continuous one-to-one & rng f = P & rng f = Q & rng f = Q & rng f = Q & rng f = Q & rng f = Q & rng f = Q & rng ( f ) = Q & rng ( f ) = Q & rng ( f ) = Q & rng ( f ) = Q & rng ( f ) = Q & rng ( f ) = Q & rng ( f ) c= Q & rng ( f ) = Q & rng ( f ) c= Q & rng ( f ) = Q & rng ( f ) c= Q & rng ( f )
attr f is partial differentiable of RNS means : Def2 : for x0 st x0 in dom f holds SVF1 ( 2 , f , x0 ) . x0 - f . x0 ;
ex r , s st x = |[ r , s ]| & G * ( 1 , 1 ) `1 < r & r < G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `2 < s & s < G * ( 1 , 1 ) `2 } ;
assume that f is FinSequence and 1 <= t and t <= len G and ( G * ( t , width G ) ) `1 <= ( G * ( t , width G ) `1 ;
attr i in dom G means : Def2 : r * ( f * reproj ( i , x ) ) = r * diff ( f , x ) ;
consider c1 , c2 being bag of o1 + o2 such that ( <* c1 , c2 *> /. k ) /. ( k + 1 ) = <* c1 , c2 *> /. ( k + 1 ) and ( <* b1 , c1 *> /. k ) = <* c1 , c2 *> /. ( k + 1 ) *> ;
y0 in { |[ r1 , s1 ]| : r1 < s1 & s1 < 1 } ;
Cl ( X ^ Y ) . k = the carrier of X & ( for k being Element of NAT holds ( X . k ) . k = ( X . k ) . k ) * ( X . k ) ;
attr len M1 = len M2 means : Def2 : width ( M1 @ ) = width ( M2 @ ) & width ( M2 @ ) = width ( M2 @ ) ;
consider g2 be Real such that 0 < g2 and { y where y is Point of S : ||. y - x0 .|| < g2 } c= dom f2 & ||. ( y - x0 ) /. ( y - x0 ) .|| < g2 /. ( y - x0 ) ;
assume x < sqrt ( - b + c ) + sqrt ( - a * b ) or x > - sqrt ( - a * b ) ;
( G1 '&' G2 ) . i = ( <* 3 *> ^ ( G1 ^ G2 ) ) . i & ( G1 ^ G2 ) . i = ( <* 3 *> ^ ( G1 ^ G2 ) . i ) & ( G1 ^ G2 ) . i = ( <* 3 *> ^ ( G1 ^ G2 ) ) . i ;
for i , j st [ i , j ] in Indices ( ( M1 + M2 ) * ( i , j ) ) & ( M2 + M2 ) * ( i , j ) < ( M2 + M2 ) * ( i , j ) ;
for f being FinSequence of NAT , i being Element of NAT st i in dom f & for j being Element of NAT st j in dom f holds f . j = Sum ( f | i ) & f . ( j + 1 ) = Sum ( f | ( j + 1 ) ) ;
assume F = { [ a , b ] where a is Element of X : for c being Element of X st c in B holds a c= c } ;
b2 * ( q + q2 ) + ( b2 * ( q + q2 ) ) + ( b1 * ( q + q2 ) ) = 0. TOP-REAL n + ( b1 * ( q + q2 ) ) ;
Cl { D where D is Subset of T : ex A being Subset of T st D = { D where D is Subset of T : ex a being Point of T st a in A & A c= D & a c= D & b c= D } c= Cl ( F " { a } )
attr seq is summable means : Def2 : for n holds seq . n = Sum ( seq ) + Sum ( seq ) ;
dom ( ( TOP-REAL 2 ) | D ) = ( ( TOP-REAL 2 ) | D ) | K1 .= ( ( TOP-REAL 2 ) | K1 ) | K1 .= K1 .= K1 ;
[ X \to Z , Z ] is full full full SubRelStr of ( ( X --> Z ) |^ ( X , Z ) , Z ) |^ ( X , Z ) |^ ( X , Z ) |^ ( X , Z ) |^ ( X , Z ) |^ ( X , Z ) |^ ( X , Z ) |^ ( X , Z ) |^ ( X , Z ) is full full SubRelStr of ( ( X --> Z ) |^ ( X , Z ) )
( G * ( 1 , j ) ) `2 = ( G * ( 1 , j ) ) `2 & ( G * ( 1 , j ) ) `2 <= ( G * ( 1 , j ) ) `2 ;
synonym m1 c= ( m1 + m2 ) \ ( m2 + g2 ) for p being Element of ( m1 + m2 ) \ ( m2 + g2 ) & ( m1 + m2 ) \ ( m2 + g2 ) <= ( m1 + m2 ) \ ( m2 + g2 ) ;
consider a being Element of B ( ) such that x = F ( a ) and a in { G ( b ) where b is Element of B ( ) : P [ b ] } ;
cluster multiplicative loop s -> multiplicative for non empty RelStr , s , t be Element of R ;
Morphism ( a , b , c ) + Morphism ( c , d , b ) = b + Morphism ( a , b , c ) .= Morphism ( a , b , c ) + Morphism ( c , d , b ) .= Morphism ( a , b , c ) + Morphism ( b , d , c ) .= Morphism ( a , b , c ) + Morphism ( c , d , d ) ;
cluster -> ( i + 1 ) -tuples_on ( the carrier of K ) -> ( i , j ) -tuples_on ( the carrier of K ) ;
( ( 2 * p1 + ( 2 * p2 ) + ( 2 * p2 ) ) / ( 2 * p1 + ( 2 * p2 ) / ( 2 * p2 ) ) ) = ( ( 2 * p1 + ( 2 * p2 ) / ( 2 * p2 ) ) ) / ( 2 * p2 ) ;
eval ( a | ( n , L ) *' x ) = eval ( a | ( n , L ) *' x ) * eval ( p , x ) .= ( a | ( n , L ) *' x ) * eval ( p , x ) .= ( a | ( n , L ) *' x ) * eval ( p , x ) ;
assume that the TopStruct of S = the TopStruct of T and for x being Point of S st x in the topology of T holds x in the topology of S and not x in the topology of T ;
assume that 1 <= k + 1 and ( ( ( ( q + 1 ) / 2 ) * w ) / ( k + 1 ) ) = ( ( ( q + 1 ) / 2 ) * w ) / ( k + 1 ) ) and ( ( ( q + 1 ) / 2 ) * w ) / ( k + 1 ) = ( ( q + 1 ) / 2 ) / ( k + 1 ) ;
2 * ( a |^ ( n + 1 ) ) + ( 2 * ( b |^ ( n + 1 ) ) + ( 2 * ( b |^ ( n + 1 ) ) + 1 ) ) >= ( 2 * ( a |^ ( n + 1 ) + b |^ ( n + 1 ) ) + ( 2 * ( b |^ n + 1 ) ) + ( 2 * ( b |^ n + 1 ) ) + ( 2 * ( b |^ n ) ) ) ;
M , v / ( x. 3 , x ) / ( x. 4 , x ) / ( x. 0 , x ) / ( x. 4 , x ) / ( x. 4 , x ) / ( x. 4 , x ) / ( x. 0 , x ) / ( x. 4 , x ) / ( x. 4 , x ) / ( x. 4 , x ) ) / ( x. 4 , x ) / ( x. 0 , x ) ) / ( x. 4 , x ) ) / ( x. 4 , x ) / ( x. 4 , x ) / ( x. 4 , x ) / ( x. 0 , x ) ) / ( x. 4 , x ) / ( x. 4 , x ) / ( x. 0 , x ) / ( x. 4 , x ) / ( x. 4 , x )
assume that f is_differentiable_on l and for x0 st x0 in l holds 0 < f . x0 - f . x0 ;
for G1 being _Graph , W being Walk of G1 , e being Vertex of G2 , x being Vertex of G1 , y being Vertex of G2 st e in dom W & x in W holds W . ( e ) = W . ( e ) & W . ( e ) = W . ( e ) ;
( not ( not ( ex x1 st x1 in the carrier of S ) & not x1 in the carrier of S ) & not ( x1 in the carrier of S ) & not x1 in the carrier of S ) & not ( x1 , x2 ) is not empty & ( x1 , x2 ) is not empty & ( x1 , x2 ) is not empty ) & ( x1 , x2 , x3 is_collinear ) & ( x1 , x2 , x3 is_collinear ) & ( x1 , x2 , x3 is_collinear & x2 , x3 , x4 is_collinear & ( x1 , x2 , x3 is_collinear & x2 , x3 , x4 is_collinear & x2 , x3 , x4 is_collinear & x3 , x3 , x4 is_collinear & x3 , x3 , x4 is_collinear & x3 , x3 , x4 is_collinear & x3 , x4 , x4 is_collinear & x3 , x4 , x4
Indices ( GoB f ) = [: Seg n , Seg n :] & [: Seg n , Seg n :] c= Indices GoB f & [: Seg n , Seg n :] c= Indices GoB f & [: Seg n , Seg n :] c= Indices GoB f implies ( GoB f ) * ( i , j ) = ( GoB f ) * ( i , j )
for G1 , G2 being Group , O being Element of O st G1 is normal & G2 is normal holds the carrier of G1 = the carrier of G2 & the carrier of G1 = the carrier of G2 & the carrier of G2 = the carrier of G1 & the carrier of G2 = the carrier of G2 & the carrier of G1 = the carrier of G2
( card ( the Sorts of t ) +* ( the Sorts of t ) ) = { ( the Sorts of t ) . ( ( the Sorts of t ) . ( the Sorts of t ) . ( o + 1 ) ) } ;
for f1 , f2 being FinSequence of F st f1 ^ f2 is p -element & ( for i being Nat st i in dom f1 holds f1 . i = p . i ) & ( for i being Nat st i in dom f1 holds f1 . i = p . ( i + 1 ) ) holds f1 . ( i + 1 ) = p . ( i + 1 )
sqrt ( ( p `1 / |. p .| - ( p `2 / |. p .| - sn ) ) / ( 1 + sn ) ) ^2 ) = sqrt ( ( p `2 / |. p .| - sn ) ) ^2 ) ;
for x1 , x2 , x3 , x4 being Element of REAL n holds |. ( x1 - x2 ) - x3 .| = |. ( x2 - x3 ) - x3 .|
for x st x in dom ( ( - ( x | A ) | A ) ) holds ( ( - x | A ) | A ) . x = - ( x | A ) . x
for T being non empty TopSpace , P being Subset-Family of T , x being Point of T st P c= the topology of T & P is Basis of T holds P is Basis of T
( a 'or' b ) . x = ( 'not' ( a 'or' b ) . x ) 'or' ( 'not' ( a 'or' b ) . x ) .= ( 'not' ( a 'or' b ) . x ) 'or' ( 'not' ( a '&' b ) . x ) .= ( 'not' ( a '&' b ) . x ) 'or' ( 'not' ( a '&' b ) . x ) .= ( 'not' ( a '&' b ) . x ) 'or' ( 'not' ( a '&' b ) . x ) .= ( 'not' ( a '&' b ) . x 'or' ( 'not' ( a '&' b ) . x 'or' ( 'not' ( a '&' b ) . x ) 'or' ( 'not' ( a '&' b ) . x ) 'or' ( 'not' ( a '&' b ) . x ) 'or' ( 'not' ( a '&' b )
for e being set st e in ( A \/ B ) ex X1 being Subset of X st ( for X1 being Subset of X st X1 = X1 & X2 is open & X1 is open & X1 is open & X2 is open & X1 is open & X2 is open & X1 is open & X2 is open holds ex X1 being Subset of X st X1 = X1 & X1 is open & X2 is open & X1 is open & X2 is open & X1 is open & X1 is open & X2 is open & X2 is open & X1 is open & X2 is open & X1 is open & X2 is open & X1 meets X2 is open & X1 meets X2 is open & X2 is open & X2 is open & X2 is open & X1 meets X2 is open & X2 is open &
for i being set st i in the carrier of S for f being Function of ( the carrier of S ) . i , the carrier of S st f = H . i & f is one-to-one holds F . i = f | ( the carrier of S ) . ( i , f . ( i , f . ( i , f . ( i , f . ( i , f . ( i , f . ( i , f . ( i , f ) ) ) ) ) )
for v , w st for y st x <> y holds ( for y st y in y holds ( J . y ) . ( y , w ) = Valid ( v , y , w ) ) . ( y , w ) = Valid ( v , y , w )
card ( D * ( i , j ) ) = card ( ( i + 1 ) * ( i , j ) ) + card ( ( i + 1 ) * ( i , j ) ) .= 2 * ( i + j ) + 1 .= 2 * ( i + j ) + 1 ;
IC Exec ( i , s ) = ( s +* ( 0 .--> 1 ) ) . ( IC S ) .= ( s +* ( 0 .--> 1 ) ) . ( IC S ) .= ( s +* ( 0 .--> 1 ) ) . ( IC S ) .= ( s +* ( 0 .--> 1 ) ) . ( IC S ) .= ( s +* ( 0 .--> 1 ) ) . ( IC S ) .= ( s +* ( 0 .--> 1 ) ) . ( IC S ) .= ( s +* ( 0 .--> 1 ) ) . ( IC S ) .= ( s . ( 0 .--> 1 ) ) . ( IC S ) .= ( s . ( IC S ) .= ( s . ( 0 .--> 1 ) . ( IC S ) .= ( s .
len f /. ( i1 -' 1 ) + 1 - 1 = len f - ( i1 -' 1 ) + 1 .= len f - ( i1 -' 1 ) + 1 .= ( f /. ( i1 -' 1 ) ) + 1 - 1 .= ( f /. ( i1 -' 1 ) ) + 1 ;
for a , b , c being Element of NAT st 1 <= a & 2 <= b & k < a & k < a holds b + c = a + b or k = a + b or k = a + b or k = a + b or k = a + b or k = a + b + c or k = a + b + c or k = a + b ;
for f being FinSequence of TOP-REAL 2 , p being Point of TOP-REAL 2 , f being FinSequence of TOP-REAL 2 , i being Element of NAT st i in Seg n & f . i = p . ( i + 1 ) & f . ( i + 1 ) = q . ( i + 1 ) holds f . ( i + 1 ) = p . ( i + 1 )
lim ( ( ( ( P +* ( k + 1 , n ) ) # x ) # x ) ) = lim ( ( ( P +* ( k + 1 , n ) # x ) ) ) + lim ( ( ( P +* ( k + 1 , n ) # x ) # x ) ) ;
z2 = g /. ( i -' ( n + 1 ) ) .= g /. ( i -' ( n + 1 ) ) .= g /. ( i -' ( n + 1 ) ) .= g /. ( i + ( n + 1 ) ) .= g /. ( i + ( n + 1 ) ) .= g /. ( i + 1 ) ;
[ f . 0 , f . 3 ] in id ( the carrier of G ) \/ ( the InternalRel of G ) \/ ( the InternalRel of G ) \/ ( the InternalRel of G ) or [ 0 , f ] in the InternalRel of G ;
for G being Subset-Family of B st G = { [ X , Y ] where X is Subset of A ( ) , Y is Subset of B ( ) : X [ Y ] } holds ( ( ( ( Intersect F ) | X ) | Y ) . ( X , Y ) ) . ( X , Y ) = ( ( ( ( ( Intersect F ) | Y ) | Y ) | Y ) . ( X , Y ) )
CurInstr ( P1 , Comput ( P1 , s1 , m ) ) = CurInstr ( P2 , Comput ( P2 , s2 , m ) ) .= CurInstr ( P2 , Comput ( P2 , s2 , m ) ) .= CurInstr ( P2 , Comput ( P2 , s2 , m ) ) .= CurInstr ( P2 , Comput ( P2 , s2 , m ) ) .= CurInstr ( P2 , Comput ( P2 , s2 , m ) , m ) .= CurInstr ( P2 , Comput ( P2 , s2 , m ) ) .= CurInstr ( P2 , Comput ( P2 , s2 , m ) , Comput ( P2 , s2 , m ) , Comput ( P2 , s2 , m ) , Comput ( P2 , s2 , m ) ) .= CurInstr ( P2 , Comput ( P2 , s2 , m ) ) .= CurInstr ( P2 , Comput ( P2 , s2 , m ) .= CurInstr ( P2 , Comput ( P2 , s2 , m ) .= CurInstr ( P2 , m ) .= CurInstr
assume that a on M and b on N and c on N and a on N and b on N and c on N and a on N and b on N and c on N and a on N and b on N and c on N and a on N and b on N and c on N and a <> b and b <> c ;
cluster T is \hbox { T T T T , T is \hbox { T T T is closed & T is closed implies for F being Subset-Family of T st F is closed & for T being Subset-Family of T st F is closed & T is closed holds ind F <= 0 & ind F <= 0 holds ind T <= 0 ;
for g1 , g2 st g1 in ]. r , s .] & g2 in ]. r , s .] holds |. ( f . g1 ) . g1 - r , s . g2 .| <= ( f . g2 ) . g2 - ( f . g2 ) . g2
( ( exp_R * ( exp_R + z ) ) + ( ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 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 * ( q * ( q * (
F . i = F /. i .= 0. R + ( F /. ( n + 1 ) ) .= <* b *> ^ ( F /. ( n + 1 ) ) .= <* b *> ^ ( F /. ( n + 1 ) ) *> ;
ex y being set , f being Function st y = f . n & dom f = NAT & rng f = { 0 , 1 } & rng f c= { 0 , 1 } & rng f c= { 0 , 1 } & rng f c= { 0 , 1 } & rng f c= { 0 , 1 } & rng f c= { 0 , 1 } ;
func f * F -> FinSequence of V means : Def2 : for i being Nat st i in dom F holds it . i = F ( i ) * F ( i ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8
for n being Nat , x being set for n being Nat st x = h . n & h . n = o ( x , n ) & h . n = o ( x , n ) & h . n = o ( x , n ) & h . n = o ( x , n ) & h . n = o ( x , n ) & h . n = o ( x , n ) ;
ex S1 being Element of QC-WFF ( Al ( ) ) st ( for e being Element of Al ( ) ) holds ( for e being Element of D ( ) st e in S ( ) & e in S ( ) & ( for n being Nat st n < 1 holds P [ n , e . n ] ) & P [ n ] holds P [ n ] ) implies P [ n + 1 ]
consider P being FinSequence of ( the carrier of G ) such that ( ex i being Element of NAT st P . i = product P & ( ex t being Element of the carrier of G st t = ( the |^ n ) * ( i , t ) ) ;
for T1 , T2 being non empty TopSpace , T being Subset-Family of T1 , P being Basis of T2 , T holds the topology of T = the topology of T2 & the topology of T = the TopStruct of T2 & the TopStruct of T = the TopStruct of T2
attr f is partial differentiable means : Def2 : r (#) ( f /* ( h + c ) ) is partially of r , x0 ;
defpred P [ Nat ] means for F being FinSequence of ( the carrier of V ) , s being FinSequence of the carrier of V st len F = $1 & for i being Element of NAT st i in dom F ex s being FinSequence of the carrier of V st P [ i , s ] & for i being Element of NAT st i in dom F holds P [ i , s . i ] ;
ex j st 1 <= j & j < width GoB f & ( GoB f ) * ( 1 , j ) `2 <= s & ( GoB f ) * ( 1 , j ) `2 <= s & ( GoB f ) * ( 1 , j ) `2 <= s & ( GoB f ) * ( 1 , j ) `2 <= s & ( GoB f ) * ( 1 , j ) `2 <= s & ( GoB f ) * ( 1 , j ) `2 & ( GoB f ) * ( 1 , j ) `2 <= s & ( GoB f ) * ( 1 , j ) `2 <= s & ( GoB f ) * ( 1 , j ) `2 <= s & ( GoB f ) * ( 1 , j ) `2 <= s & ( GoB f ) * ( 1 , j ) `2 <= ( GoB f ) * ( 1 , j ) `2 <= ( GoB f ) * ( 1 , j ) `2 <= ( GoB f ) *
defpred U [ set , set , set , set , set ] means ex F being Subset-Family of T st ( ( for x being Point of T st x in $1 holds F . x = union ( F . $1 , $2 ) ) & ( for x being Point of T st x in $1 holds F . x = union ( F . x , $2 ) ) ;
for p2 being Point of TOP-REAL 2 st LE p2 , p3 , P , p1 , p2 , p2 holds LE LE p2 , p3 , P , p1 , p2 & LE p2 , p3 , P , p2 , p1 , p2 & LE p2 , p4 , P , p2 , p1 , p2 , p2 & LE p2 , p4 , P , p1 , p2 , p2 , P & LE p2 , p4 , P , p2 , p1 , p2 , p2 & LE p4 , p4 , P , p2 , p1 , p2 , p2 , p2 , p1 , p2 , P , p1 , p2 , p2 , P , p1 , p2 , P , p1 , p2 , P , p1 , p2 , P , p1 , p2 , p1 , p2 , p1 , p2 , P , p1 , p2 , P , p1 , p2 , P , p1 , p2 , p1 , p2 , p1 , p2 , p1 , p2 , p2 , p1 , P , p1 , p2 ,
f in Funcs ( E , H ) & for y st y in E holds f . y = f . ( y , x ) implies f . ( y , x ) = f . ( y , x )
ex 8 being Point of TOP-REAL 2 st x = p2 & |. 8 .| = 1 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <= 8 & 8 <=
assume for d being Element of NAT st d <= ( ( n + 1 ) -tuples_on NAT ) & ( for t being Element of NAT st t in { ( n + 1 ) -tuples_on NAT ) holds ( n + 1 ) -tuples_on NAT = ( n + 1 ) -tuples_on NAT ) & ( n + 1 ) -tuples_on NAT = ( n + 1 ) -tuples_on NAT ) ;
assume that s <> t and s is Point of Closed-Interval-TSpace ( x , r ) and not ex e being Point of TOP-REAL n st e = s & not ex e being Point of TOP-REAL n st e in Ball ( x , r ) & not e in Ball ( x , r ) ;
given r such that 0 < r and for s st 0 < s ex x1 , x2 being Point of TOP-REAL 2 st x1 < s & x2 < s & ||. x1 - x2 .|| < s & ||. f /. ( x1 - x2 ) - f /. ( x1 - x2 ) .|| < s ;
( p | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x
for x , h , x st x + h in dom sec holds ( ( ( x + h ) (#) ( 2 * x + h ) ) `| Z ) . x = ( ( 2 * x ) (#) ( 2 * x + h * x ) ) `| Z ) . x + ( ( 2 * x ) (#) ( 2 * x + h * x ) ) ^2
assume that i in dom A and i > 1 and i > 1 and j in dom A and i in dom B and j in dom B and i in dom B and j in dom B and i = j and j in dom B and i in dom B and j in dom B and i in dom B and i = j and j in dom B ;
for i being non zero Element of NAT st i in Seg n holds h . i = <* 1_ F_Complex *> & h . i = <* 1_ F_Complex *> & h . i = <* 1_ F_Complex *> & h . i = 1_ F_Complex ;
( ( ( b1 '&' b2 ) '&' ( b2 '&' c2 ) ) '&' ( ( b1 '&' b2 ) '&' ( b2 '&' c2 ) ) '&' ( ( b1 '&' b2 ) '&' ( b2 '&' c2 ) ) '&' ( ( b1 '&' b2 ) '&' ( b1 '&' c2 ) ) '&' ( ( b1 '&' b2 ) '&' ( b2 '&' c2 ) ) '&' ( ( b1 '&' b2 ) '&' ( b1 '&' c2 ) ) '&' ( ( b1 '&' b2 ) '&' ( b1 '&' b2 ) ) ) '&' ( ( b1 '&' b2 ) ) '&' ( ( b2 '&' c2 ) ) '&' ( ( b1 '&' b2 ) '&' ( ( b1 '&' b2 ) '&' ( ( b1 '&' b2 ) '&' ( ( b1 '&' b2 ) ) '&' ( b2 '&' c2 ) ) '&' ( ( b1 '&' c2 ) '&' ( b2 '&' c2 ) ) '&' ( ( b1 '&' c2 ) ) '&' ( ( b2 '&' c2 ) '&' ( ( b1 '&' c2 ) ) '&' ( ( b2 ) )
assume that f . x = ( ( the Sorts of A ) * ( the Arity of S ) ) . ( ( the Arity of S ) . ( ( the Arity of S ) . ( o , x ) ) ) and ( the Arity of S ) . ( o , x ) = ( the Arity of S ) . ( o , x ) ;
consider RZ , RZ be Real such that RZ = \int ( F . n , Z ) and ( for i be Nat st i in dom ( F . n ) holds ( I . i ) . i = ( I . i ) . ( n + i ) ;
ex k being Element of NAT st ( for q being Element of product G st q in X & 0 < k ex n being Element of NAT st for q being Element of product G st q in X & q in X holds ||. ( f , x ) - ( f /. ( q - x ) ) .|| < r ) & ||. ( f , x ) - ( f /. ( q - x ) ) .|| < r ;
x in { x1 , x2 , x3 , x4 , x5 , x5 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 6 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 ,
( G * ( j , i ) ) `2 = ( G * ( 1 , i ) ) `2 .= ( G * ( 1 , i ) ) `2 .= ( G * ( 1 , i ) ) `2 .= ( G * ( 1 , i ) ) `2 ;
f1 * p = p .= ( the Arity of S1 ) . ( ( the Arity of S2 ) . ( o , f ) ) .= ( the Arity of S2 ) . ( o , f ) .= ( the Arity of S2 ) . ( o , f ) .= ( the Arity of S2 ) . ( o , f ) .= ( the Arity of S2 ) . ( o , f ) ;
func tree ( T , P , T ) -> DecoratedTree means : Def2 : for p , q being Element of T st p in P & q in Q & p in Q & q in Q holds it . p = T . ( p , q ) ;
F /. ( k + 1 ) = F . ( k + 1 ) .= F . ( ( p . ( k + 1 ) ) + F /. ( k + 1 ) ) .= F . ( k + 1 ) .= F . ( k + 1 ) + F /. ( k + 1 ) .= F . ( k + 1 ) ;
for A , B , C , D , E , F , G being Matrix of K st len A = len B & len B = width C & len B = width C & len A = width C & width B = width C & width B = width C & width B = width C & width B = width C & width B = width C & width B = width C & width B = width C & width B = width C & width B = width C & width B = width C = width C & width B = width C = width C & width B = width C = width C & width B = width C = width C & width B = width C & width B = width C & width B = width C & width B = width C & width B = width C = width C & width B = width C = width C & width B = width C & width B = width C = width C & width B = width C
seq . ( k + 1 ) = 0. ( X . ( k + 1 ) ) .= ( Partial_Sums ( seq ) ) . ( k + 1 ) .= ( Partial_Sums ( seq ) ) . ( k + 1 ) ) . ( k + 1 ) .= ( Partial_Sums ( seq ) ) . ( k + 1 ) ;
assume that x in ( the InternalRel of C1 ) & y in ( the InternalRel of C2 ) /\ ( the InternalRel of C2 ) and x in ( the InternalRel of C2 ) /\ ( the InternalRel of C2 ) ;
defpred P [ Element of NAT ] means for f being Element of NAT st len f = $1 & for k being Element of NAT st k in dom f holds ( ex g being Element of D st g . ( k + 1 ) = ( for f being Element of D st f . ( k + 1 ) = ( for k being Element of D st k < n holds ( for n being Element of NAT holds f . ( k + 1 ) = ( n + 1 ) -tuples_on D ) . ( k + 1 ) ) & ( ( n + 1 ) /. ( k + 1 ) ) = ( n + 1 ) /. ( k + 1 ) /. ( k + 1 ) ) & ( ( n + 1 ) /. ( k + 1 ) ) & ( ( n + 1 ) /. ( k + 1 ) = ( ( n + 1 ) /. ( k + 1 ) ) & ( ( n + 1 ) = ( n + 1 ) /. ( k +
assume that 1 <= k and k + 1 <= len f and f /. k = G * ( i , j ) and f /. ( i + 1 ) = G * ( i + 1 , j ) and f /. ( i + 1 ) = G * ( i + 1 , j ) ;
assume that s < 1 and ( for q st q in X holds ( q . q ) `1 = ( |. q .| ) * ( |. q .| ) + ( |. q .| ) * ( |. q .| ) ) and ( for q st q in X holds ( |. q .| ) * ( |. q .| ) ) ^2 >= 0 ;
for M being non empty dist , x being Point of M , f being Function of M , ( TOP-REAL n ) | ( M , f ) st x = f & ex x being Point of TOP-REAL n st x = f . x & ex n being Element of TOP-REAL n st x = f . n & f . n = ( TOP-REAL n ) . ( x , f . n )
defpred P [ Element of omega ] means ( f1 - f2 ) . $1 = ( f1 - f2 ) . $1 & ( f1 - f2 ) . $1 = ( f1 - f2 ) . $1 ;
defpred P1 [ Nat , Element of C , set ] means ( ex c being Element of C st $1 < $2 & $2 in Y & ( ex n being Element of NAT st for x being Element of C st x in Y holds ||. ( f . $1 , f . $1 ) - f /. ( $1 + 1 ) .|| < r ) & ||. ( f . $1 , f . $1 ) - f /. ( $1 + 1 ) .|| < r ) ;
( f ^ mid ( g , 2 , len g ) ) . i = ( f ^ mid ( g , 2 , len g ) ) . i .= ( f ^ mid ( g , 2 , len g ) ) . i .= ( f ^ mid ( g , 2 , len g ) ) . i .= ( f ^ mid ( g , 2 , len g ) ) . i .= ( f ^ mid ( g , 2 , len g ) ) . i .= ( f ^ mid ( g , 2 , len g ) ) . i .= ( f ^ mid ( g , 2 , len g ) ) . i .= ( f ^ mid ( g , 2 , len g ) ) . i .= ( f ^ mid ( g , 2 , len g , len g , len g ) . i .= ( f ^ mid ( g , 2 , len g ) . ( i + 1 , len g ) . ( i + 1 , len g ) . ( i + 2 ) . ( i + 2 ) . ( i + 2
sqrt ( 1 - ( 2 * n + 1 ) * ( 2 * n + 1 ) ) = ( 2 * n + 1 ) * ( 2 * n + 1 ) .= ( 2 * n + 1 ) * ( 2 * n + 1 ) .= ( 2 * n + 1 ) * ( 2 * n + 1 ) ;
defpred P [ Nat ] means ( the InternalRel of G ) . ( $1 + 1 ) = the InternalRel of G & ( the InternalRel of G ) . ( $1 + 1 ) = the InternalRel of G ;
assume that f /. 1 in Ball ( u , r ) and 1 <= m and m <= len f and f /. ( 1 + 1 ) = f /. ( 1 + 1 ) ;
defpred P [ Element of NAT ] means ( Partial_Sums ( ( x | $1 ) (#) ( x | $1 ) ) ) . $1 = ( Partial_Sums ( ( x | $1 ) * ( x | $1 ) ) ) . $1 ;
for x being Element of product F holds x in ( the Sorts of F ) . i & x in ( the Sorts of F ) . i & ( the Sorts of F ) . i = ( the Sorts of F ) . i & ( the Sorts of F ) . i = ( the Sorts of F ) . i ) . i ;
( x " ) |^ ( n + 1 ) = ( x " ) |^ ( n + 1 ) .= ( x " ) |^ ( n + 1 ) .= ( x " ) |^ ( n + 1 ) .= ( x " ) |^ ( n + 1 ) ;
DataPart Comput ( P +* I , LifeSpan ( P +* I , s ) + 3 ) = DataPart Comput ( P +* I , Comput ( P +* I , s , LifeSpan ( P +* I , s ) + 3 ) ) .= DataPart Comput ( P +* I , s , LifeSpan ( P +* I , s ) + 3 ) ;
given r such that 0 < r and ]. x0 - r , x0 + r .[ c= dom ( f1 | ]. x0 - r , x0 + r .[ ) and for g st g in ]. x0 - r , x0 + r .[ holds f1 . g <= ( f1 | ]. x0 - r , x0 + r .[ ) . g ;
assume that X c= dom ( f1 | X ) /\ X and ( f1 | X ) | X is continuous and ( f1 | X ) | X is continuous ;
for L being continuous complete LATTICE , X being Subset of L st X = sup ( ( { X } |^ ( L , L ) ) holds X is directed & for x being Element of L st x in X holds x is prime implies X is complete
consider i being Element of NAT such that i in dom A and A * ( i , p ) = ( ( m *' p ) *' ( i , p ) ) *' ( i , p ) ;
( f1 - f2 ) /* ( f1 - f2 ) = lim ( ( f1 - f2 ) /* ( f1 /* ( f1 /* ( f2 /* ( f1 /* ( ( f1 + f2 ) /* ( f1 /* ( ( f1 + f2 ) /* ( f1 /* ( ( f1 + f2 ) ) ) ) ) ) ) ) .= ( ( f1 - f2 ) /* ( f1 /* ( ( f1 + f2 ) /* ( f1 /* ( f1 + f2 ) ) ) ) ) ) ;
ex p1 being Element of QC-WFF ( Al ( ) ) st F . ( p ( ) ) = g ( p ( ) ) & for p being Element of QC-WFF ( Al ( ) ) st p ( ) = g ( p ( ) ) holds P [ p ( ) ] ;
( mid ( f , i , len f -' 1 ) ) /. ( i + 1 ) = ( mid ( f , i , len f -' 1 ) ) /. ( i + 1 ) .= ( mid ( f , i , len f -' 1 ) ) /. ( i + 1 ) .= ( mid ( f , i , len f -' 1 ) ) /. ( i + 1 ) .= ( mid ( f , i , len f -' 1 ) ) /. ( i + 1 ) ;
( ( p ^ q ) ^ r ) . ( len p + k ) = ( ( p ^ q ) ^ r ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) ;
len mid ( ( ( upper_volume ( f , D1 , j1 ) ) + 1 , indx ( f , D1 , j1 ) ) + 1 ) ) + 1 = indx ( f , D1 , j1 ) + 1 ;
x * y = ( ( x * y ) * z ) * ( ( y * z ) * ( x * z ) ) .= ( x * ( y * z ) ) * ( x * z ) .= ( x * ( y * z ) ) * ( x * z ) .= ( x * ( y * z ) ) * ( x * z ) ;
v . ( <* x , y *> , v ) = ( <* x , y *> , v ) * ( <* y , z *> , v ) + ( ( proj ( 1 , 1 , 0 ) * ( u , z ) ) ) * ( proj ( 1 , 0 , 0 ) ) + ( proj ( 1 , 0 , 0 ) ) * ( ( proj ( 1 , 0 , 0 ) ) ) * ( proj ( 1 , 0 , 0 ) ) ) ;
i * i = <* 0 * ( 1 - 0 ) , 0 * ( i - 0 ) *> .= <* 0 *> * ( i - 0 ) + 0 * ( i - 0 ) .= 0 ;
Sum ( L * F ) = Sum ( ( L * F ) + ( 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 ) ) .= Sum ( L * F ) .= Sum ( L ) + Sum ( L * F ) .= Sum ( L * F ) .= Sum ( ( ( F ) ) + Sum ( L * F ) + Sum ( L * F ) ) .= Sum ( L * F ) .= Sum ( L * F ) + Sum ( L ) + Sum ( L * F ) .=
ex r be Real st 0 < e & for Y be finite Subset of X st Y in the carrier of X ex a being Real st a < r & for Y being Subset of X st Y in the carrier of X & Y c= Y & a <= b holds r <= a & for Y being Subset of X st Y in Y holds Y is finite & Y is finite & for Y being Subset of X st Y in Y holds Y is finite & for Y being Subset of X st Y in Y holds Y is finite & Y is finite & Y is finite & Y is finite & Y is finite & Y is finite & Y is finite & Y is finite & Y is finite & Y is finite & Y is finite & Y is finite & Y is finite & Y is finite & Y is finite & Y is finite & Y is finite & Y is finite & Y is finite & Y is finite & Y is finite & Y is finite & Y is finite & Y is finite & Y is finite & Y is finite & Y is finite & Y is finite & Y is finite & Y is finite & Y is finite & Y is finite & Y is
( GoB f ) * ( i , j + 1 ) = f /. ( k + 1 ) & ( GoB f ) * ( i + 1 , j ) = f /. ( k + 1 ) ;
( ( - 1 ) (#) ( ( ( 1 - 1 ) (#) ( ( 1 - 1 ) (#) ( ( 1 - 1 ) (#) ( ( 1 - 1 ) (#) ( ( 1 - 1 ) (#) ( ( 1 - 1 ) (#) ( ( 1 - 1 ) (#) ( ( 1 - 1 ) (#) ( 1 - 1 ) ) ) ) ) ) ) ) ) = ( ( - 1 ) (#) ( ( 1 - 1 ) (#) ( 1 - 1 ) ) ) ) * ( ( 1 - 1 ) (#) ( ( 1 - 1 ) (#) ( ( 1 - 1 ) (#) ( ( 1 - 1 ) (#) ( 1 - 1 ) ) ) ) ) .= ( ( 1 - 1 ) (#) ( ( 1 - 1 ) (#) ( ( 1 - 1 ) (#) ( ( 1 - 1 ) ) ) ) ) ) * ( ( 1 - 1 ) ) ) .= ( ( 1 - 1 ) ) * ( ( 1 - 1 ) (#) ( ( 1 - 1 ) (#) ( ( 1 - 1 ) ) ) ) * ( ( 1 - 1 ) (#) ( ( 1 - 1 ) (#) ( ( 1 - 1 ) ) )
( - b + sqrt ( a , b ) + sqrt ( a , b ) ) / 2 > 0 & sqrt ( - a , b ) / 2 < 0 & sqrt ( - b , c ) / 2 < 0 ;
assume that inf ( { X } /\ C ) in L and for X st X in C holds X is maximal & for Y st Y in C holds Y in L & Y in L ;
( ( B ) . i , j ) = ( i |-> ( j , i ) ) & ( i --> ( j , i ) ) = ( i |-> ( j , i ) ) & ( i --> ( j , i ) ) = ( i |-> ( j , i ) ) . ( i , j ) ;