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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
thesis ;
thesis ;
assume thesis ;
assume thesis ;
B ;
a <> c ;
T c= S
D c= B ;
c ;
b <> c ;
X is non empty ;
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 injective ;
let q ;
m = 1 ;
1 < k ;
G is cyclic ;
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 closed ;
a in A ;
1 < x ;
S is finite ;
u in I ;
z \ll z1 ;
x in V ;
r < t ;
let t ;
x c= y ;
a <= b ;
m < n ;
assume f is prime ;
x in Y ;
z = - \infty ;
let k be Nat ;
K ` is width M ;
assume n >= N ;
assume n >= N ;
assume X is ./. connected ;
assume x in I ;
q is 0 ;
assume c in x ;
arccot > 0 ;
assume x in Z ;
assume x in Z ;
1 <= kG1 ;
assume m <= i ;
G is IT ;
assume a divides b ;
assume P is closed ;
arccot > 0 ;
assume q in A ;
W is bounded ;
f is one-to-one ;
assume A is closed ;
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 `2 <= c `2 ;
A meets W ;
i `2 <= j `2 ;
assume H is universal ;
assume x in X ;
let X be set ;
let T be tree ;
let d be element ;
let t be element ;
let x be element ;
let x be element ;
let s be element ;
k <= card 5 ;
let X be set ;
let X be set ;
let y y ;
let x be element ;
P [ 0 ] ;
let E be set ;
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 ;
let y y ;
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 y ;
r in REAL ;
let e be element ;
n1 is iso ;
Q is_closed_on s , P ;
x in \Omega I[01] ;
M < m + 1 ;
T2 is open ;
z in b \omega b \omega ;
L2 is well-ordering ;
1 <= k + 1 ;
i > n + 1 ;
q1 is one-to-one ;
let x be set ;
P1 is one-to-one ;
n + 2 <= n + 2 ;
1 <= k + 1 ;
1 <= k + 1 ;
let e be Real ;
i < i + 1 ;
a3 in P ;
p1 in K ;
y in C ;
k + 1 <= n ;
let a be Real ;
X |- r => p ;
x in A ;
let n be Nat ;
let k be Nat ;
let k be Nat ;
let m be Nat ;
0 < 0 + k ;
f is_differentiable_on Z ;
let x0 ;
let E be Ordinal ;
o o o , 4 are_isomorphic ;
O <> { O } ;
let r be Real ;
let f be finite FinSequence ;
let i be Nat ;
let n be Nat ;
Cl A = A ;
L c= card L ;
A /\ M = B ;
let V be complex normed space ;
s in Y |^ 0 ;
rng f <= w ;
b "/\" e = b ;
m = m1 ;
t in h . D ;
P [ 0 ] ;
assume z = x * y ;
S . n is bounded ;
let V be RealLinearSpace ;
P [ 1 ] ;
P [ {} ] ;
A1 is component ;
H = G * ( i , j ) ;
1 <= i + 1 ;
F . m in A ;
f . o = o ;
P [ 0 ] ;
ar2 <= b-a ;
R [ 0 ] ;
b in f .: X ;
assume q = q2 ;
x in [#] V ;
f . u = 0 ;
assume that e > 0 and e > 0 ;
let V be RealLinearSpace ;
s is non trivial ;
dom c = Q ;
P [ 0 ] ;
f . n in T ;
N . j in S ;
let T be LATTICE ;
the object of F is one-to-one ;
sgn x = 1 ;
k in Seg a ;
1 in Seg 1 ;
rng f = X ;
len T in X ;
v < n ;
S1 is bounded ;
assume p = { p1 : p1 `1 >= 0 & p1 `2 >= 0 & p1 `2 >= 0 & p1 `2 >= 0 & p1 `2 >=
len f = n ;
assume x in P1 ;
i in dom q ;
let U ;
p1 = c ;
j in dom h ;
let k ;
f | Z is continuous ;
k in dom G ;
UBD C = B ;
1 <= len M ;
p in Ball ( x , r ) ;
1 <= j1 ;
set A = R^1 | divset ( sec , i ) ;
card a is_less_than c ;
e in rng f ;
cluster B \oplus A -> non empty ;
H is_no no operator ;
assume that n <= m and m <= n ;
T is increasing ;
v2 <> v1 ;
Z c= dom g ;
dom p = X ;
H is proper subformula of G ;
i + 1 <= n ;
v <> 0. V ;
A c= Affin A ;
S c= dom F ;
m in dom f ;
let x0 be set ;
c = sup N ;
R is connected ;
assume x in { 0 , 1 } ;
Im f is complete ;
x in Int y ;
dom F = M ;
a in On W ;
assume e in { \cal A } ;
C c= { C1 where C1 is Subset of C : C1 c= C1 & C1 c= C2 } ;
m <> {} ;
let x be Element of Y ;
let f be non-negative chain of G ;
n in Seg 3 ;
assume X in f .: A ;
assume that p <= n and m <= n ;
assume u in { v } ;
d is Element of A ;
A |^ b misses B ;
e in v ( ) ;
- y in I ;
let A be non empty set ;
P1 = 1 ;
assume r in F . k ;
assume f is simple simple _net ;
let A be z2 z2 be element ;
rng f c= NAT ;
assume P [ k ] ;
g1 <> {} ;
let o be Ordinal ;
assume x is sum of squares ;
assume v in { 1 , 2 } ;
let I1 ;
assume that 1 <= j and j < n and n < m ;
v = - u ;
assume s . b > 0 ;
d1 , d2 are_connected ;
assume t . 1 in A ;
let Y be non empty TopSpace ;
assume a in [. s , t .] ;
let S be non empty RelStr ;
a , b // b , c ;
a * b = p * q ;
assume x , y are_connected ;
assume x in [#] ( f .: A ) ;
[ a , c ] in X ;
IT <> {} ;
M + N c= M + N ;
assume M is transitive ;
assume f is_Ubststst-st-stst-st-operator ;
let x , y ;
let T be non empty TopSpace ;
b , a // b , c ;
k in dom p ;
let v be Element of V ;
[ x , y ] in T ;
assume len p = 0 ;
assume C in rng f ;
k1 = k + 1 ;
m + 1 < n + 1 ;
s in S \/ { s } ;
i + i + 1 >= n + 1 ;
assume Re ( y ) = 0 ;
m1 <= m1 ;
f | A is continuous ;
f . x in b ;
assume y in dom h ;
x * y in { B } ;
set X = Seg n ;
1 <= i + 1 ;
k + 0 <= k + 1 ;
p ^ q ^ p ^ q ^ q ^ p = p ^ q ;
j |^ y divides m ;
set m = max A ;
[ x , y ] in R ;
assume x in succ 0 ;
a in sup \varphi ;
A1 is A3 ;
q2 c= { C , D } ;
a2 < c1 ;
T2 is 0 -started ;
IC Comput ( P3 , s3 , 0 ) = 0 ;
set q2 = s . m , q1 = s . m , q2 = s . m ;
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 MSAlgebra over L ;
y " * ( y " ) <> 0 ;
y " * ( y " ) <> 0 ;
0. V = uV -right right right right right right right complementable ;
L2 , y are_isomorphic ;
R1 , R2 are_isomorphic ;
let a , b be Real ;
let a be element ;
let x be Vertex of G ;
let o be object of C , a be object of C ;
r '&' q = P '&' l ;
let i , j be Nat ;
let s be State of A , f be State of A , g be State of A , f be State of A , g be State of A
s . n = N ;
set y = ( x + y ) / ( x + y ) ;
m1 in dom g ;
l . 2 = y1 . 2 ;
|. g . y .| <= r ;
f . x in { C } ;
V is non empty ;
let x be Element of X ;
0 <> f . ( g . ( g . n ) ) ;
f2 /* q is convergent ;
f . i is_measurable_on E ;
assume that \xi in { N where N is Real : N <= N & N <= 1 } ;
reconsider i = i ' as Element of NAT ;
r * v = 0. X ;
rng f c= { Z where Z is Subset of REAL : Z <= 1 & Z <= 1 } ;
G = 0 .--> 0 ;
let A be Subset of X ;
assume that A is dense and A is open ;
|. f . x .| <= r ;
let x be Element of R ;
let b be Element of L ;
assume x in { W1 where W1 is Element of W1 : W1 in W1 & W1 c= W1 } ;
P [ k , a ] in P ;
let X be Subset of L ;
let b be element ;
let A , B be category structure ;
set X = Vars term ( C ) ;
let o be OperSymbol of S ;
let R be connected RelStr ;
n + 1 = succ n ;
{ q1 : q1 c= Z & Z c= dom ( f1 + f2 ) & for x st x in Z holds f1 . x = ( f1 + f2 ) . x ;
dom f = { C } ;
assume [ a , b ] in X ;
Re ( seq ) is convergent ;
assume a1 = b1 ;
A = Int ( A /\ Int Int ( A /\ Int ( A /\ Int Int A ) ) ) ;
a <= b or b <= a ;
n + 1 in dom f ;
let F be sequence of the carrier of S , T be sequence of S ;
assume that 2 > x0 and x0 < r2 ;
let Y be non empty set ;
2 * x in dom W ;
m in dom g2 ;
n in dom ( g `| X ) ;
k + 1 in dom f ;
the still not bound in { s } ;
assume that x1 <> x2 and x2 <> x3 and x1 <> x3 and x1 <> x3 and x2 <> x3 ;
v1 in { 0. V } ;
[ b , b ] in T ;
intpos i + 1 = i ;
T c= exp ( T ) ;
( l ) ^2 = 0 ;
let n be Nat ;
( t `2 ) ^2 = r ^2 ;
divset ( a0 , i ) is integrable ;
set t = \top _ t ;
let A , B be real-membered set ;
k <= len G + 1 ;
A1 misses A2 & A2 misses A1 & A1 c= A2 ;
product s1 is non empty ;
e <= f or e <= f ;
cluster {} -> On yielding for set ;
assume that g2 = - g2 and g2 = - g2 ;
assume h in [. q , p .] ;
1 + 1 <= len C ;
c in B . ( m + 1 ) ;
let R ;
p . n = H . n ;
assume v is convergent and for n be Nat holds ||. v .|| is convergent ;
IC Comput ( P3 , s3 , 1 ) = 0 ;
k in N or k in K ;
F1 \/ { F } c= F \/ { G }
Int G1 <> {} ;
( |. z .| ) ^2 = 0 ;
p1 <> p2 ;
assume z in { y } ;
MaxADSet ( a ) c= F ;
sup [. s , t .] is Subset of S ;
f . x <= f . y ;
let T be non empty reflexive reflexive reflexive reflexive reflexive reflexive reflexive reflexive reflexive reflexive RelStr ;
q1 ^2 >= 1 ;
a >= X & b >= Y & b >= Y ;
assume <* a , b *> <> {} ;
F . c = g . c ;
G is one-to-one ;
A \/ { a } c= B \/ { a } ;
0. V = 0. V ;
let I be non empty Program of S ;
f1 . x = 1 ;
assume z \ x = 0. X ;
Set 4 = 2 |^ n ;
let B be sequence of \Sigma ;
assume that p = p ^ q and q ^ p in D ;
n + l in NAT ;
f " { f " } is compact ;
assume x1 in { x1 + x2 } ;
p1 = { K } ;
M . k = 0. V ;
\varphi . 0 in rng \varphi ;
MA is closed ;
assume that z1 <> 0. V and z1 <> 0. V ;
n < N . ( k + 1 ) ;
0 <= s . 0 ;
- q + p = v ;
not v in A ;
set g = f `| Z ;
A1 , A2 be stable MSAlgebra over R ;
set CVertices R = Vertices R ;
{ p } c= { p1 } ;
x in [. 0 , 1 .] ;
f . y in dom F ;
let T be Lawson complete TopAugmentation of S ;
inf ( the carrier of S ) in S ;
intpos ( a \/ b ) = intpos ( a \/ b ) ;
P , C // K , L ;
assume x in LSeg ( s , t ) ;
2 |^ i < 2 |^ m ;
x + z = x + y ;
x \ ( a \ x ) = x \ y ;
||. \mathopen { \Vert } cosec - x .|| <= r ;
assume that Y c= field Q and Y <> {} and Y <> {} and Y <> {} and Y <> {} and Y <> {} and Y <> {} and Y <> {} and Y <> {}
a \times b = id b & b is isomorphic ;
assume a in { [ i , j ] } ;
k in dom q1 ;
p is are__probability -sort ;
i -' 1 = i-1 -' 1 ;
f | A is one-to-one ;
assume x in f .: X ;
i2 - i = 0 ;
2 + 1 <= i + 1 ;
g " * a in N ;
K <> {} ;
cluster non trivial strict for _Graph ;
|. q .| ^2 > 0 ;
|. p1 .| = |. p1 .| ;
- ( 2 - PI ) > 0 ;
assume x in { G _ { -12 } where -12 is Real : 1 <= -12 & -12 <= 1 & -12 <= 1 } ;
W in C ;
assume x in { G _ { -12 } where -12 is Real : 1 <= -12 & -12 <= 1 & -12 <= 1 } ;
assume i + 1 = len G ;
assume i + 1 = len G ;
dom I = Seg n ;
assume that k in dom C and i <> k and k <> i ;
1 + 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 void non void ManySortedSign ;
let f be ManySortedSet of I ;
let z be Element of REAL ;
u in { \hbox { \boldmath $ g } } ;
2 * n < 2 * n ;
let x , y , z be set ;
Bii c= { V where V is Real : V >= 1 } ;
assume that I is_closed_on s , P and I is_halting_on s , P and I is_halting_on s , P and s , P and I is_halting_on s , P and I is_halting_on s , P and s is
U = U U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2
M /. 1 = z /. 1 ;
x1 = y1 ;
i + 1 < n + 1 ;
x in { {} , {} } ;
f1 <= f1 . m ;
let l be Element of L ;
x in dom ( F . n1 ) ;
let i be Element of NAT ;
r1 is -valued ;
assume that \langle 2 , 3 *> <> {} and <* 2 , 3 *> <> {} and 3 , 4 , 4 *> <> 3 ;
s . x = 1 ;
card { K , 1 } in M ;
assume that X in U and Y in U and X c= U ;
let D be <% non-empty net of the carrier of N ;
set r = ]. q , k .[ ;
y = W * ( x + y ) ;
assume dom g = cod f ;
let X , Y be non empty TopSpace ;
x \ A is closed ;
|. \varepsilon _ { A } .| . a = 0 ;
cluster strict for lattice ;
a1 in B . ( s . ( n + 1 ) ) ;
let V be finite over F , W be strict Subspace of V ;
A * B c= A * B ;
set f1 = { \mathbb m } --> 0 , f2 = { 0 } --> 1 ;
let A , B be Subset of V ;
z1 = z1 . j ;
assume f " { f " } 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 = { Z : Z is non empty ;
z in dom ( A --> 0 ) ;
P [ y , h . y ] ;
not x0 in dom f ;
let B be ManySortedSet of I ;
sin . 2 < Arg z ;
reconsider z1 = 0 as Nat ;
not a , b // c , d ;
[ y , x ] in { I } ;
( ( Q ) /. 3 = 0 ;
set j = { x } div m ;
assume a in { x } ;
M2 - ( j - sn ) > 0 ;
I \! \mathop { \rm \hbox { - } = 1 ;
[ y , d ] in F ;
let f be Function of X , Y ;
set S2 = \frac { B where B is Subset of C : B in B & B in C } ;
s1 is convergent & lim s1 = 0 ;
j -' 1 = 0 ;
set i2 = 2 * n + 1 ;
reconsider t = t as bag of n ;
T2 . j = m . j ;
i |^ ( s , n ) mod p = 1 & i mod p = 0 ;
set g = f | divset ( D2 , j ) ;
assume that X is lower bounded and r <= 0 ;
( p `1 ) ^2 = 1 ;
a < ( p / |. p .| ) * |. p .| ;
L \ { m } c= UBD C ;
x in Ball ( x , r ) ;
a in LSeg ( c , c ) ;
1 <= i -' 1 ;
1 <= i -' 1 ;
i + 2 <= len h ;
x = W \mathop { \rm W _ { min } ( P ) where P is Subset of W : P [ W ] } ;
[ x , z ] in [: X , Y :] ;
assume y in [. x0 , x0 .] ;
assume p = <* 1 , 2 *> ;
len <* A *> = 1 ;
set H = h . ( { g } ) ;
card b * a = |. a .| * a .| ;
Shift ( w , w ) is_monomorphism v , v ;
set h = { h . ( m + 1 ) where m is Nat : m <= n + 1 } ;
assume x in { X , Y } /\ { X , Y } ;
||. h .|| " < d ;
x in support f ;
f . y = ( F . y ) . ( f . y ) .= ( F . y ) . ( f . y ) ;
for n holds X [ n ] ;
k -' l = kl -' k ;
<* p , q *> = p ;
let S be Subset of Y ;
let P , Q , t be type of s ;
Q /\ M c= union ( F | M ) ;
f = b * ( S * ( S ) ) ;
let a , b be Element of G ;
f .: X <= f .: X ;
let L be non empty RelStr ;
S1 is \square -basis of x , i -wi ;
let r be Real ;
M , v |= _ { v } H ;
v + w = 0. V ;
P [ len F ] ;
assume InsCode i = 8 ;
the carrier of M = 0 ;
cluster z * ( |. z .| ) -> summable ;
let O be Subset of C ;
|. f .| | X is continuous ;
set z2 = g . j + ( j + 1 ) ;
cluster -> -> -> -> -> -> -> -> -> -> -> -> -> m -wff ;
reconsider m1 = lm1 as Nat ;
v1 is directed ;
3 is SubSpace of T2 ;
A1 /\ A2 <> {} ;
let k be Nat ;
q " = ( X " ) " ;
F . t is set ;
assume that n <> 0 and n <> 1 and n <> 1 ;
set e = EmptyBag ( n + 1 ) ;
let b be Element of Bags n ;
assume for i holds b . i = b . i ;
x is root of ( p ) . ( x , y ) ;
r in [. p , q .] ;
let R be finite FinSequence ;
i2 , j2 are_connected ;
IC SCM SCM <> a ;
|. [ x , y ] .| >= r ;
1 * ( s - t ) = s * ( t - s ) ;
let x be FinSequence of NAT ;
let f be Function of C , D ;
for a , b being Element of L holds 0. L = a * b
IC Comput ( P3 , s3 , 0 ) = IC Comput ( P3 , s3 , 1 ) ;
H + G = F-F-operator ( G-operator ) ;
C . x = { x } ;
f1 = f . ( f1 . ( len f1 ) ) .= f1 . ( len f1 + 1 ) .= f1 . ( f1 . ( len f1 + 1 ) ) .= f1 . ( f1 . ( len f1 +
Sum ( p ) = p . 0 ;
assume v + W = v + W ;
not a in { a , b } ;
a1 , b // b , c ;
a3 , o // o , a ;
reflexive transitive transitive transitive RelStr ;
Ia9 is antisymmetric ;
sup { H , 1 } = e ;
x = { a } * ( b * ( a * b ) ) ;
|. p1 .| >= 1 ;
assume that j -' 1 < j -' 1 and j + 1 < len G ;
rng s c= dom f2 /\ dom f2 ;
assume support a misses support b ;
let L be non empty doubleLoopStr ;
s " + 0 < n + 1 ;
p . c = { f . c where c is Real : c <= d & d <= 1 & c <= 1 } ;
R . n <= R . ( n + 1 ) ;
Directed ( I ) = I " ;
set f = x + y ;
cluster Ball ( x , r ) -> bounded ;
consider r being Real such that r in A ;
cluster non empty for Relation ;
let X be non empty directed Subset of S ;
let S be full SubRelStr of L ;
cluster \langle L1 ( N , N ) , L2 \rangle -> complete ;
\frac 1 " * a " = a " ;
( q . {} ) . {} = o ;
( i -' i -' 1 ) mod ( i -' 1 ) > 0 ;
assume that 1 - t <= 1 and t `2 <= 1 ;
card ( B \ card B ) = k + card k ;
x in union rng ( f1 + f2 ) ;
assume x in the carrier of R ;
d ;
f . 1 = L . ( F . 1 ) ;
the vertices of G = { v } ;
let G be G1 G1 , wwwwwwwG1 be _Graph ;
let e , v2 be set ;
c . i in rng c ;
f2 /* q is divergent to \hbox { - \infty $ } ;
set z1 = - ( z1 - z2 ) , z2 = z1 - z2 ;
assume w is Element of atllFuncs ( X , Y ) ;
set f = p \! \mathop { \rm \hbox { - } ;
let c be object of C ;
assume ex a st P [ a ] ;
let x be Element of REAL m ;
let S1 be Subset-Family of X ;
reconsider p = p as Element of NAT ;
let v , w be Point of X ;
let s be State of SCM+FSA ;
p be finite finite sequence of elements of the carrier of SCM R ;
stop I c= card I ;
set cU = { f /. i where i is Nat : i <= j & j + 1 <= i + 1 } ;
w ^ t ^ w ^ w ^ t ^ w ^ w ^ w ^ t ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^
W1 /\ W = W /\ ( W + V ) ;
f . j is Element of J . j ;
let x , y be Element of T2 ;
ex d , b st a , b // d , c ;
a <> 0 & c <> 0 ;
ord ( x * ( x * y ) ) = 1 & x * y = 1 ;
set g2 = lim ( ( ( ( ( ( ( ( s - n ) - ( 1 - n ) ) (#) ( ( ( s - n ) - ( ( s - n ) (#) ( ( s - n
2 * x >= 2 * x ;
assume ( a 'or' c ) '&' ( a 'or' c ) <> TRUE ;
f \circ g in Hom ( c , d ) ;
Hom ( c , c ) <> {} ;
assume 2 * ( q | m ) > m ;
L1 . ( F . b1 ) = 0 ;
\times X = id X \/ Y ;
( ( ( ( the function sin ) * ( ( id Z ) * ( ( id Z ) * ( ( id Z ) * ( id Z ) ) ) ) ) ) ) . x <> 0 ;
( the function of exp ( n ) ) . x > 0 ;
o in { X where X is Subset of [: X , Y :] : X in F & Y in F } ;
let e , v2 be set ;
3 > \frac { 1 } * ( 2 * PI ) ;
x in P .: ( ( F " ) " ) ;
let J be closed Subset of \Vert \Vert \Vert \Vert \Vert \Vert \Vert \Vert \Vert \Vert * \Vert .|| ;
h . ( p1 , p2 ) = f2 . ( p1 , p2 ) .= p1 . ( p1 , p2 ) ;
Index ( p , f ) + 1 <= j ;
len ( M ) = width M ;
the carrier of K c= A ;
dom f c= union rng ( F | union { -10 } ) ;
k + 1 in Seg ( n + 1 ) ;
let X be ManySortedSet of the carrier of S ;
[ x , y ] in InnerVertices R ;
i = { D1 : i <= len D1 & j <= len D2 & i <= len D2 & i <= len D2 } ;
assume a mod n = b mod n ;
h . ( { x } , y ) = g . ( y , z ) .= g . ( y , z ) ;
F c= 2 |^ ( X ) ;
reconsider w = |. seq .| as sequence of real numbers ;
\frac 1 - m * m + r * m < p ;
dom f = [: dom ( I , Seg n ) , Seg n :] ;
[#] TOP-REAL 2 = [#] [#] K ;
cluster the functor of - \infty -> \Vert ;
then not { d } c= A ;
cluster -> ind for TopSpace ;
let i1 be Element of M ;
let x be Element of dyadic ( n ) ;
u in { W , v } & v in { v , w } ;
reconsider y = y as Element of L ;
N is full full SubRelStr of T & N is full
sup { x , y } = c "\/" c .= c ;
g . n = n ^ <* 1 *> .= n ^ ( n |-> 0 ) .= n ;
h . J = EqClass ( u , J ) ;
||. seq .|| .|| is summable ;
dist ( x , y ) < r / 2 ;
reconsider m1 = m as Element of NAT ;
x0 - x0 < r1 ;
reconsider P = P ` as strict Subgroup of N ;
set g = p * ( q " ) ;
let n , m be Nat ;
assume that 0 < e and e < 1 and f | A is bounded ;
T2 . ( { I } , {} } , T2 . ( I , {} ) ) in { I } ;
cluster non empty Subset of T -> condensed ;
let P be compact Subset of TOP-REAL 2 , f be Function of TOP-REAL 2 , TOP-REAL 2 ;
Y1 in LSeg ( \pi , 1 ) ;
let n be Element of NAT , i be Element of NAT ;
reconsider S8 = S as Subset of T ;
dom ( i .--> ( i .--> X ) ) = { i } ;
let X be ManySortedSet of the carrier of S ;
let X be ManySortedSet of the carrier of S ;
op ( 1 , 1 ) c= { {} } ;
reconsider m = i-1 as Element of NAT ;
reconsider d = x as Element of C ;
let s be 0 -started State of SCMPDS , i be Nat ;
let t be 0 -started State of SCMPDS , Q be t be 0 -started State of SCMPDS ;
b , c are_connected ;
assume that i = n \/ k and j = k \/ ( n \/ k ) and i = k \/ ( n + 1 ) ;
let f be PartFunc of X , Y ;
N >= \frac { c where c is Real : c <= c & c <= 1 & c <= 1 & c <= 1 & c <= 1 } ;
reconsider t1 = T " as Point of T ;
set q = h * ( p ^ <* d *> ) ^ <* d *> ;
z2 in U /\ ( U2 /\ U2 ) ;
A |^ 0 = { <* 0 *> , 0 *> ;
len W2 = len W + len W ;
len M2 in dom ( M2 + M2 ) ;
i + 1 in Seg ( len s + 1 ) ;
z in dom ( g `| X ) ;
assume that M2 = 0. K and M2 = 0. K and M2 = 0. K ;
len G + 1 <= i + 1 ;
f1 * ( ( ( ( f1 (#) f2 ) (#) ( f1 + f2 ) ) ) ) (#) ( f1 (#) f2 ) ) ) ) is_differentiable_on Z ;
cluster |. seq .| + |. seq .| -> summable ;
assume j in dom ( M1 * M2 ) ;
let A , B be Subset of X ;
let x , y be Point of X , x be Point of X ;
b - ( 4 * ( a - b ) ) >= 0 ;
<* xxy *> ^ <* x *> ^ <* y *> ^ <* y *> ^ <* x *> ^ <* y *> ^ <* y *> ^ <* y *> ^ <* x *> ^ <* y *> ^ <* y *> ^ <*
a , b // a , b ;
len M2 = n ;
ex x being element st x in dom R & x in R ;
len q = len ( K * G ) ;
set s1 = Initialize ( s ) , s2 = Initialize s , s3 = Initialize s , s3 = Initialize s , s3 = Initialize s , P4 = Initialize s , P4 = P +* stop I , P4 = P +* I ;
consider w being Nat such that q = z + w ;
x ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` = x ` ` ` ` ` x .= x ` ` ` ` ` ` x ` ` `
k = 0 & k <> n ;
attr X is discrete means : Def5 : for X being discrete SubSpace of X holds X is closed ;
for x being set st x in L holds x in L
||. f .|| . c - f . c .|| <= r ;
c in \mathopen { \uparrow } p : p < p & p <= g } ;
reconsider V = V as Subset of the TopStruct TopStruct TopStruct TopStruct TopStruct TopStruct TopStruct TopStruct TopStruct TopStruct TopStruct TopStruct TopStruct TopStruct TopStruct TopStruct ;
let N , M be Boolean net over L ;
then z >= compactbelow x ;
M , f are_isomorphic & f , g are_connected ;
( ( to_power ( 1 ) ) to_power ( 1 + 1 ) ) to_power ( 1 + 1 ) = ( ( 1 / ( 1 + 1 ) ) to_power ( 1 + 1 ) ) to_power ( ( 1 + 1 ) ) ;
dom g = dom ( f ^ g ) ;
mode traktraktraktratratratratra\smallfrown G -> _Graph means : Def14 : for W being Walk of G holds W is D\smallfrown W ;
[ i , j ] in the indices of M ;
reconsider s = x " * y as Element of H ;
let f be Element of On Seg n ;
F . ( a , b ) = G . ( a , b ) .= G . ( a , b ) ;
cluster strict strict strict strict strict strict for TopSpace ;
let a , b be Real ;
rng s c= dom f2 /\ dom f2 ;
curry ' ( F , -19 ) is additive ;
set i2 = card ( B \ { card ( B \ { i } ) where i is Element of NAT : i in dom B } ) ;
set G = DTConMSA X ;
reconsider a = [ x , s ] as terminal of G ;
let a , b be Element of M ;
reconsider s = s as Element of { S } ;
rng p c= the carrier of L ;
let d be Subset of the bound of A ;
( x | ( x | n ) ) | ( x | n ) = 0 implies x = 0
I-21 in dom stop I ;
let g be Function of X , Y ;
reconsider D = Y as Subset of TOP-REAL 2 ;
reconsider x9 = len { p1 where p1 is Element of NAT : len p1 = len p1 & p1 = len p1 & p1 = p1 . 1 & p1 . 1 = p1 . 1 ;
dom f = the carrier of S ;
rng h c= union ( the carrier of J ) ;
cluster All ( x , H ) -> WFF ;
d * N > 1 * N ;
[. a , b .] c= [. a , b .] ;
set g = f " (#) ( f " ) ;
dom ( p | ( m + 1 ) ) = Seg m ;
3 + 3 + 2 <= k + 2 ;
the function arccot is differentiable ;
x in rng ( f ^ <* p *> ) ;
let f , g , h be FinSequence ;
p in the carrier of S ;
rng f " { f " } = dom f ;
( the target of G ) . e = v . e ;
width G -' 1 < width G ;
assume v in rng ( S | ( ( ( S | ( ( E ) | E ) ) | E ) ) | E ) ) ;
assume x is root of g or x = t or x = t . x ;
assume 0 in rng ( ( ( ( ( |. g .| | A ) | A ) | A ) | A ) ) ;
let q be Point of TOP-REAL 2 ;
let p be Point of TOP-REAL 2 ;
dist ( O , u ) <= |. ( |. ( { p .| - u .| ) .| + |. ( |. p .| ) .| ) .| + |. ( |. p .| - u .| ) .| ;
assume dist ( x , y ) < dist ( x , y ) ;
<* S1 , S2 *> is Sconnectives ;
i <= len G -' len G -' 1 ;
let p be Point of TOP-REAL 2 ;
x1 in the carrier of I[01] ;
set p1 = f /. i ;
g in { g : g < r & g in { r } ;
q2 = { S : S in F & Q c= F . ( Q . ( Q . ( Q . ( Q . ( Q . ( Q . ( Q . ( Q . ( Q . ( Q . ( Q . ( Q
( ( 2 * ( 2 |^ ( n + 1 ) ) ) (#) ( 2 |^ ( n + 1 ) ) ) ) . 0 is convergent ;
- p + I c= - A + A ;
n < LifeSpan ( P1 , s1 ) ;
CurInstr ( p1 , s1 ) = i ;
A /\ Cl ( { x } \ { y } ) <> {} ;
rng f c= [. r , s .] ;
let g be Function of S , T ;
let f be Function of the carrier of L , the carrier of L ;
reconsider z = z as Element of InclPoset Ids L ;
let f be Function of S , T ;
reconsider g = g as Morphism of c , b ;
[ s , I ] in [: [: [: A , B :] , A :] , A :] ;
len ( the connectives of C ) = 4 ;
let A1 , A2 be A1 , C be strict signature ;
reconsider V1 = V as Subset of X ;
attr p is valid means : for x st x in still_not-bound_in p holds p . x = x ;
assume that X c= dom f and dom f = X and for r st r in X holds f . r = r ;
H " { a } is normal Subgroup of H " { a } ;
A1 , B1 be ManySortedFunction of the Sorts of A1 , A2 ;
a2 , b2 // b3 , b3 & b3 , b3 , b3 // b3 , b3 , b3 ;
consider x being element such that x in v ^ K ;
x in { 0. TOP-REAL 2 } ;
p in [#] ( { \mathbb I } | { { B } } ) ;
0 in M . ( E . ( n + 1 ) ) ;
op c ^ op c = op c ;
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . 0 ) ;
cluster Boolean Boolean Boolean for lattice ;
set i1 = the Element of NAT ;
let s be 0 -started State of SCM+FSA , i be Nat ;
assume y in ( f1 \/ f2 ) \/ ( f1 \/ f2 ) ;
f . ( len f -' 1 ) = f . ( len f -' 1 ) .= f . ( len f -' 1 ) ;
x , f . x \bfparallel f . x , f . x ;
attr X c= Y means : for Y being Subset of X st Y in Y holds Y c= Y ;
let y be upper Subset of Y ;
cluster the functor of ( x , y ) -> Xyyus sequence ;
set S = \langle n , m *> ^ <* i *> ;
set T = [. 0 , 1 .] ;
1 in dom ( ( ( mid ( f , 2 , 1 ) ) ) * ( i + 1 ) ) ;
sin * PI < PI * PI ;
{ x } in dom ( f1 + f2 ) ;
O c= dom I & dom I = I & for i being Element of I st i in I holds I . i = I . i ;
( the source of G ) . x = v . x ;
not HT ( p , T ) c= HT ( p , T ) ;
reconsider h = R . k as Function of n , L ;
ex b being Element of G st y = b * H ;
let x , y , z be Element of G ;
h . i = f . ( h . i ) ;
( p `1 ) ^2 = ( p `2 ) ^2 ;
i + 1 <= len Cage ( C , n ) ;
len <* P *> = len P ;
set N = the the carrier of N , N = the carrier of N ;
len gW + ( x + ( ( x + ( y + 1 ) ) ) ^2 ) + ( y + 1 ) ^2 ) <= x ;
a is_not lie between B , C & b is_not lie between C , B ;
reconsider -12 = v * ( v + ( v + u ) ) as FinSequence ;
consider d such that x = d and a [= d ;
consider u being element such that u in W and v = u + v ;
len f = len f-n ;
set q2 = E-max C , q1 = E-max C ;
set S = [ S1 , S2 ] , S2 = [ S1 , S2 ] , f3 = [ S1 , S2 ] , f3 = [ S1 , S2 ] , f3 = [ S1 , S2 ] , f3 = [ S1 , S2 ] , f3 =
MaxADSet ( b ) c= MaxADSet ( b ) ;
Cl ( ( G . ( ( ( ( ( F . ( ( ( ( ( ( ( ( ( ( ( ) ) , len G ) ) ) , ( ( F . ( n + 1 ) ) ) ) ) ) ) ) ) ) ,
f " D /\ D meets h " ( D ) ;
reconsider D = E as non empty directed directed Subset of L ;
H = ( ( the ObjectMap of H ) . ( ( the ObjectMap of H ) . i ) ) . i ;
assume t is Element of ( \mathfrak F ) . ( X , Y ) ;
rng f c= the carrier of S ;
consider y being Element of X such that x = y and y in X ;
f1 . ( a , b ) = f1 . ( a , b ) .= f1 . ( a , b ) ;
the carrier of G = E \/ { E } .= E \/ { E } ;
reconsider m = len |^ k as Element of NAT ;
set S1 = LSeg ( Gauge ( C , n ) * ( i , j ) , Gauge ( C , n ) * ( i , j ) ) ;
[ i , j ] in the indices of M ;
assume that P c= Seg m and m is mod mod mod mod mod ( m mod n ) ;
for k st m <= k & k <= m holds m . k in K ;
consider a being set such that p in a and a in G ;
L1 . ( p + 1 ) = p . ( p + 1 ) .= L1 . ( p + 1 ) ;
p2 . i = p1 . i .= p2 . i ;
let q1 , q2 be a_partition of Y ;
attr 0 < 1 & 1 < r & r < 1 ;
rng proj ( a , X ) = [#] X ;
reconsider x = x , y = y as Element of K ;
consider z such that z = f . k and k <= n ;
consider x being element such that x in X \ { p } and x in X ;
len ( s | Seg ( len s ) ) = card ( Seg ( len s ) - 1 ) ) ;
reconsider z2 = x1 as Element of L ;
Q in FinMeetCl ( the topology of X ) ;
dom ( f1 | ]. x0 , x0 + r .[ ) c= dom f2 ;
attr n divides m & m divides n & n = m ;
reconsider x = x as Point of I[01] ;
a in from ( TOP-REAL 2 ) | ( TOP-REAL 2 ) ;
{ y , z } misses the still of f , x , y , z } ;
Hom ( a , b ) <> {} ;
consider k such that p " < k and k < n ;
consider c , d such that dom f = c \ d ;
[ x , y ] in dom g ;
set S1 = over over F_Complex ;
3 = 2 & 4 = 2 * ( i + 1 ) ;
x0 in dom ( u | A ) /\ ( ( u | A ) | A ) ;
reconsider p = x as Point of TOP-REAL 2 ;
reconsider c01 = { 0. TOP-REAL 2 } as Subset of TOP-REAL 2 ;
f . p1 <= f . p1 & f . p1 <= p1 ;
( F . ( ( F . ( ( F . n ) ) ) ) ) ) ) . x <= ( F . ( n + 1 ) ) . x ;
( x + ( z1 + z2 ) ) ^2 = ( z1 + z2 ) ^2 ;
for n be Element of NAT st n <= m holds |. ( ( P ) . n ) . m - ( P . n ) .| < r ;
let J , K be non empty subsets of I ;
assume i <= i & i + 1 <= len <* a *> ;
0 |-> 0. K = 0. K ;
X . i in 2 |^ ( i + 1 ) ;
<* 0 , 1 *> in dom ( e .--> ( e .--> 1 ) ) ;
attr P [ a ] means ex a , b being Element of R st a = [ b , b ] ;
reconsider s4 = { s _ { .|| } as Element of D ;
( |^ i -' j + 1 ) -' j <= len ( |^ j -' j ) ;
[#] S c= [#] T ;
let V be strict Subspace of V ;
assume k in dom ( mid ( f , i , j ) ) ;
let P be non empty Subset of TOP-REAL 2 ;
let A , B , C be Matrix of n , K ;
- a * b = a * b ;
let A be Subset of T9 ;
id 2 in <* 2 , 3 *> ;
then ||. x - x .|| = 0 ;
let N , normal normal normal Subgroup of G , N be normal Subgroup of G ;
j >= len ( ( ( ( ( ( ( ( ( ( ( g , len g ) , len g ) ) -' 1 ) ) ) ) ) ) ) ) * ( ( ( ( g , len g ) -' 1 ) ) ) ) ) ) )
b = Q ( len Q + 1 ) + ( Q + 1 ) ;
f2 * ( f1 + f2 ) is_differentiable_on Z ;
reconsider h = f * g as Function of N , G * g as Function of N , G ;
assume that a <> 0 and a <= b and b <= 0 and a <= 0 and c <= 0 ;
[ t , t ] in A ;
( v .--> E ) | n is Element of ( v | E ) | n ;
{} = the carrier of L1 + L2 ;
Directed I , Initialized s , k ) is halting ;
Initialized p = Initialize ( s +* p ) ;
reconsider N = { N where N is strict net of over over R : N in F & N c= F . N ;
reconsider Y = Y as Element of InclPoset Ids ( Ids L ) ;
inf ( p "/\" q ) <> p ;
consider j being Nat such that 2 = j + 1 and j + 1 = i + j ;
[ s , 0 ] in the carrier of S ;
m in ( B \wedge C ) \wedge ( B \wedge C ) ;
n + 1 <= len ( ( len g1 ) + 1 ) + ( len g1 - 1 ) ;
( x1 - x2 ) * ( x1 - x2 ) = x1 - x2 * y2 ;
InputVertices S = { x , y , z } ;
let x , y , z be Element of FTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
p = [ p , q ] ;
g * h = h " * g " ;
let p , q be Element of CQC-WFF ( V ) ;
x0 in dom ( f1 + f2 ) /\ dom ( f2 + g2 ) ;
( R qua Function ) " = R " ;
n in Seg len ( f ^ <* p *> ) ;
for s be Real st s in R holds ( s - ( s - t ) ) * ( s - t ) ) * ( s - t ) = t ;
rng s c= dom ( ( f2 (#) f1 ) (#) f2 ) ;
synonym the topology of InclPoset X -> topology of InclPoset ( X ) ;
- 1 = - ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( / ( ) ) ) )
set S = Segm ( A , B , C ) ;
ex w st e = \frac { w where w is Real : w <= w & w <= 1 & w <= 1 & w <= 1 } ;
curry ' ( -48 ' , k ) . x is convergent ;
cluster -> Subset of T topology of L1 -> open ;
len ( f1 + f2 ) = len ( f1 + f2 ) .= len ( f1 + f2 ) ;
i * p < \frac { 2 * p where p is Real : p <= 2 & p <= 2 * p & p <= 2 * p & p <= 2 * p & p <= 2 * p & p <= 2 * p `2 & p <= 2 * p `2 &
let x , y , z be Element of OSSub U1 U1 ;
b , c // b , c ;
consider p being element such that j in { p } and p . j = { p } ;
assume that f " { 0 } = {} and f .: { 0 } = {} ;
assume IC Comput ( F , s , k ) = n ;
if=0 ( J , card I , card J ) = a " ;
P3 is_closed_on SCM+FSA , P , Initialize s ;
set m1 = LifeSpan ( p1 , s1 ) , m2 = LifeSpan ( p1 , s1 ) ;
IC Comput ( P3 , s3 , 1 ) in dom I ;
dom t = the carrier of SCM R ;
( L~ f ) .. f = 1 ;
let a , b be Element of PFuncs ( V , C ) ;
union ( union F ) c= union ( union F ) ;
the carrier of X1 union X2 \/ X1 misses X2 \/ X1 union X2 ;
assume not a in { a , b } ;
consider i being Element of M such that i = { d where d is Element of M : d <= i & d <= i & i <= len d } ;
then Y c= { x } & Y = { x } ;
M , v |= _ { v } ( { y } , v ) } ;
consider m being element such that m in Intersect ( F ) and m in meet ( F . m ) ;
reconsider A1 = support u as Subset of X ;
card A \/ card A = card ( 2 * ( 2 * ( 2 * 1 ) ) ) + 1 ;
assume that a <> <> <> <> b and a <> b and b <> c and a <> c ;
cluster s \! \mathop { \rm \hbox { - } -> ( V , V ) -> ( V , V ) -wff for Element of S ;
L1 /. ( n + 1 ) = L1 . ( n + 1 ) .= L1 /. ( n + 1 ) ;
let P be compact Subset of TOP-REAL 2 ;
assume that p2 in LSeg ( p1 , p2 ) and p1 = p2 and p2 = p1 ;
let A be non empty Subset of TOP-REAL 2 ;
assume [ k , m ] in the InternalRel of D ;
0 <= ( ( 1 / 2 ) |^ ( n + 1 ) ) * ( n + 1 ) ;
( F . N ) . x = - ( F . x ) . x ;
attr X c= Y & X c= Y implies X c= Y
( y * z ) * ( y * z ) <> 0. V ;
1 + card { card { \kern1pt { \kern1pt } where where where where where where where where where where is is is Element of NAT : 0 <= card { \kern1pt card { \kern1pt X \kern1pt } } c= card { \kern1pt X \kern1pt } ;
set g = z \circlearrowleft ( ( p ) ) , z = ( z ) .. ( z .. ( z .. z ) ) ;
then p = <* x *> . k ;
cluster -> Element of C -\hbox { - } -> total ;
reconsider B = A as Subset of TOP-REAL 2 ;
let a , b , c be set ;
L1 . i = ( L1 . i ) . i .= L1 . i .= L1 . i ;
space , x1 , x2 , x3 , x4 , x5 , x6 ( x1 , x2 , x4 ) c= P ;
n <= indx ( D2 , D1 , j ) ;
( g . O ) `2 = - ( g . O ) `2 ;
j + p .. f + len f - len f + len f - len f - len f + len f - len f - len f - len f ;
set W = \mathop { \rm W \hbox { - } ;
S . a = a + b .= a + b .= a + b ;
1 in Seg width ( M * ( len M , len M ) ) ;
dom ( i * ( i * ( i * ( i * ( i , j ) ) ) ) ) = dom ( i * ( i * ( i * ( i , j ) ) ) ;
( W ' ) . ( a , p ' ) = W . ( a , p ' ) ;
set Q = Rmax ( g , h ) ;
cluster strict for ManySortedSet of the Sorts of U1 , U2 ;
attr A = { A where A is discrete Subset of X : A = F . A } ;
reconsider Y1 = reproj ( i , y ) as Element of product G ;
rng f c= rng ( f1 \/ f2 ) \/ rng ( f2 \/ g2 ) ;
consider x such that x in dom f and f .: A = f .: A ;
f = \varepsilon _ { \alpha } & f = the carrier of G ;
E , j |= All ( x1 , x2 ) ;
reconsider n1 = n as Morphism of o , U2 ;
assume that P is NonZero and P is NonZero and R is NonZero ;
card { \kern1pt { \kern1pt { 2 } \/ { x } } = card { card { x } } + 1 ;
card ( ( x \ y ) \ { x } ) = 0 ;
g + R in { s : s < g & g < g & g < g + R } ;
set q1 = ( q , s ) -... ;
for x being element st x in X holds x in X
h /. ( i + 1 ) = h . ( i + 1 ) .= h /. ( i + 1 ) .= h /. ( i + 1 ) ;
set mw = max ( B , m ) ;
t in Seg n & t = Seg n ;
reconsider X = dom f as Element of Fin C ;
IncAddr ( i , k ) = goto l + k ;
S is non empty ;
attr R is condensed means : Def5 : for A being Subset of R st A = Int A holds R is condensed ;
attr 0 <= a & a <= 1 & 1 <= a & a <= 1 ;
u in ( c /\ d ) /\ ( e /\ d ) ;
u in ( c /\ d ) /\ ( b /\ d ) ;
len C + ( 2 * ( 1 - 1 ) ) + ( 2 * ( 1 - 1 ) ) + ( 2 * ( 2 * ( 1 - 1 ) ) ) + ( 2 * ( 1 - 1 ) ) ) + ( 2 * ( 1 - 1 ) ) ) + ( 2 *
x , y // z , x & y , z // z , y ;
a |^ ( n + 1 ) = a |^ ( n + 1 ) .= a |^ ( n + 1 ) ;
<* 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
set y1 = <* y *> ^ <* c *> ;
F in rng Line ( D , 1 ) ;
p . m joins joins joins joins r , r ;
( p `1 ) ^2 = ( ( ( ( ( ( f /. i ) `1 ) `1 ) ^2 ) ^2 ;
W is_SubSubSubSublattice ( X ) ;
0 + ( p + r ) <= 2 * ( p + r ) + 2 * ( p + r ) ;
x in dom g & x in dom g & y in dom g ;
f1 /* ( s ^\ k ) is divergent_to+infty ;
reconsider u2 = u as VECTOR of PBoundedFunctions ( X , Y ) ;
p \! \mathop { \rm \hbox { - } = 0 ;
len <* x *> + i + 1 <= i + 1 ;
assume that I is non empty and for x being Element of I holds x in I implies x in I & x in I ;
set i = card I + 4 + 4 + 4 = 0 ;
x in { x where x is Element of T : x in A & x in A & x in A & y in A & y in A } ;
consider y being Element of F such that y in B and y <= x ;
len S = len ( the charact of A ) .= len ( the charact of A ) ;
reconsider m = M as Element of NAT ;
A . ( j + 1 ) = B . ( j + 1 ) \/ A . ( j + 1 ) ;
set N = \rm AP ( ) , P4 = ( G , -15 ) . ( -15 , -15 ) , P4 = G . ( -15 , -15 ) , P4 = G . ( -15 , -15 ) , P4 = G . ( -15 , -15 ) , P4 = G . ( -15 , -15 ) ,
rng F c= the carrier of gr { H } ;
Q1 ( Q1 , Q1 , n ) is binary tree ;
f . k , f . ( m + 1 ) are_relative_prime ;
h " ( P " ) = f " ( P " ) " ;
g in dom ( f2 \ { 0 } ) \ { 0 } ;
gX /\ LX = ( g | X ) .: X ;
consider n being element such that n in { \mathbb N } and n = G . n ;
set d = \rho ( \rho ( numbers , ^2 ) , dist ( ^2 ) ) , \rho ( ^2 ) , 0 = dist ( 0 , 0 ) ^2 ) ;
b + ( 1 - ( 1 - ( 2 * ( 2 * ( 1 - ( 2 * ( 1 - ( 2 * ( 1 - ( 2 * ( 1 - ( 2 * ( 1 - ( 2 * ( ( 1 / 2 ) ) ) ) ) ) ) ) ) ) )
reconsider f1 = f as VECTOR of X ;
attr i <> 0 & i mod ( i + 1 ) = 1 ;
j in Seg ( ( len ( g . ( i + 1 ) ) ) - ( i + 1 ) ) ) ;
dom ( { i _ 4 } ) = dom ( { i _ 4 } --> { i _ 4 } ) .= { i _ 4 } ;
cluster sec | [. - 1 , 1 .] -> continuous ;
Ball ( u , p ) = Ball ( u , p ) ;
reconsider x1 = { x1 , x2 } as Function of the carrier of S , the carrier of T ;
reconsider R1 = x as Relation of L , the carrier of L ;
consider a , b being subsets of A such that x = a \/ b and a in A ;
( ( 1 + p ) ^ <* n *> ) ^ <* n *> ) ^ ( n + 1 ) in { n + 1 } ;
{ S , { S , { S , T } } \/ { S , T } = { S , T } \/ { S , T } ;
( the function of ( ( the function sin ) * ( id Z ) ) ) `| Z ) = ( ( id Z ) * ( id Z ) ;
cluster -> non empty for PartFunc of C , REAL ;
set X0 = 1GateCircStr ( <* z , x *> , f ) ;
v1 . ( Ev2 . ( Ev2 ) = Ev2 . ( len v2 + 1 ) + ( len v2 ) - ( len v2 - len v1 - len v1 - len v2 ) .= len v2 - ( len v2 - len v1 - len v2 ) ;
( the function of ln * ( id Z ) ) `| Z ) . x = ( ( id Z ) * ( id Z ) . x ;
sup A = \frac { 2 * PI * PI * PI } & A = 0 * PI * PI * PI + 2 * PI * PI * PI * PI * PI * PI * PI * PI * PI * PI * PI * PI * PI * PI * PI * PI * PI ;
F ( dom f ) = F ( f , g ) ;
reconsider pq2 = { q1 where q1 is Point of TOP-REAL 2 : q1 `1 >= 0 & q2 `2 >= 0 & q2 `2 >= 0 & q2 `2 >= 0 & q1 `2 >= 0 & q1 `2 >= 0 & q1 `2 >= 0 & q1 `2 >= 0 & q1 `2 >= 0 & q2 `2 >= 0 & q1 `2 >= 0
g . W in [#] Y & g . W = [#] Y ;
let C be compact non empty compact Subset of TOP-REAL 2 ;
LSeg ( f , j ) = LSeg ( f , j ) ;
rng s c= dom f /\ ]. x0 - r , x0 + r .[ ;
assume x in { idseq ( 2 ) where 2 is Element of NAT : 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * (
reconsider T2 = n as Element of NAT ;
for y being ExtReal st y in rng seq holds y <= g . y
for k st k < n holds P [ k ] ;
m = ( m + 1 ) + ( m + 1 ) .= m + ( m + 1 ) .= m + ( m + 1 ) ;
assume for n holds H . n = G . n ;
set B" = f .: ( the carrier of X ) ;
ex d being Element of L st d in D & x <= d ;
assume that R \ ( a \ b ) c= R \ ( a \ b ) and R \ ( a \ b ) = R \ ( a \ b ) ;
t in [. r , s .] or t = s ;
z + u = v + u & v + u = v + u + v ;
{ 2 , 1 } --> 2 in { 2 , 3 } implies for n , m being Element of NAT st n in dom ( ( 2 .--> 0 ) --> 2 holds ( 2 -tuples_on 3 ) * ( 2 -tuples_on 3 ) = 2 * ( 2 -tuples_on 3 )
attr x1 <> x0 & x1 <> x0 & x0 - ( x1 - x0 ) < r ;
assume that { p1 - p2 } = |. p1 .| and |. p2 .| = 1 and |. p2 .| <= 1 and |. p2 .| <= 1 and |. p2 .| <= 1 and |. p2 .| <= 1 and |. p2 .| <= 1 and |. p2 .| <= 1 ;
set q = \over \over { f } ( A ) ;
let f be PartFunc of REAL , REAL ;
( n -' 2 ) mod ( n -' 2 ) = n mod ( n -' 2 ) ;
dom ( T * t1 ) = dom ( t * t1 ) ;
consider x being element such that x in { w where w is element : w in c } ;
assume F * v = v * ( ( ( |. v .| * |. v .| ) .| ) ) * v ;
assume the carrier of D1 c= the carrier of D2 & the carrier of D2 = the carrier of D2 ;
reconsider A1 = [. a , b .] as Subset of R^1 ;
consider y being element such that y in dom F and y in F and y in F ;
consider s being element such that s in dom o and o = o . s and o = o . s ;
set p = \mathop { \rm W _ { min } } ( \widetilde { \cal L } ( Cage ( C , n ) ) ) ;
n1 -' len f -' len f -' len f + 1 + 1 <= len f - 1 ;
ConsecutiveDelta ( q , u , v ) = [ u , v , u , v , v , w , v , w , u , v , v , w , u , v , v , w , u , v , w , v , w , u , v , v , w , w , u , v , v , v
set B1 = ( Partial_Sums G ) . k + B2 . k ;
Sum ( p * p ) = 0. V * p .= 0. V * p .= 0. V * p ;
consider i being element such that i in dom p and i = p . i ;
defpred Q [ Nat ] means $1 = ( $1 + 1 ) * ( $1 + 1 ) ;
set s3 = Comput ( P1 , s1 , k ) , P4 = Comput ( P1 , s1 , k ) , P4 = P3 ;
let l be Nat , k be Nat , n1 be Nat ;
reconsider ii = union { union { ii where ii is Subset of T : ii <= ii & ii <= len } as Subset of T ;
consider r such that r > 0 and r > 0 ;
( h | ( n + 1 ) ) . i = ( h | ( i + 1 ) ) . i ;
reconsider B = the carrier of X as Subset of X ;
p = |[ - 1 , 0 ]| .= |[ 1 , 0 ]| ;
synonym f -> real-valued for NAT for NAT \ { 0 } ;
consider b being element such that b in dom F and a = b . b ;
x0 < card { X where X is Subset of \overline { X : X in card { X where X is Subset of \overline { X where X is Subset of X : X is finite } ;
attr X c= { { B : B is Subset of X : B is infinite } , X } ;
then w in bool the carrier of M ;
angle ( x , y , z ) = angle ( x , y , 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
attr 1 <= len s means : means : len it = len s & for n st n <= len s holds it . n = s . n ;
{ f /. ( k + 1 ) } c= f . ( k + 1 ) ;
the carrier of ( |[ - 1 , 1 ]| ) = { |[ 1 , 1 ]| where 1 is Real : 1 <= 1 & 1 <= 1 & 1 <= 1 & 1 <= 1 & 1 <= 1 & 1 <= len ( |[ 1 , 1 ]| ) `1 } ;
attr p => q in WFF means : for p st p in WFF holds p => q in WFF ;
- ( t - ( 1 - t ) ) < t ;
U . 1 = U . ( ( U . 1 ) , U . 1 ) .= U . ( U . 1 ) .= U . ( ( U . 1 ) , U . 1 ) .= U . ( ( U . 1 ) , U . 1 ) .= U . ( U . 1 ) .= U . ( U
f .: the carrier of ( the carrier of ( TOP-REAL 2 ) | K1 ) = the carrier of ( TOP-REAL 2 ) | K1 ;
the InternalRel of [: the carrier of G , the carrier of G :] = [: the carrier of G , the carrier of G :] ;
for n being Element of NAT st n <= m holds G . n = G . n
then V in M ^ \square ;
ex f being Element of F st f is_monomorphism U1 , U2 ;
[ h . 0 , h . 1 ] in the InternalRel of G ;
s +* ( intloc 0 ) = s +* ( intloc 0 , 1 ) .= s +* ( intloc 0 , 1 ) ;
[ w , v ] <> 0. TOP-REAL 2 ;
reconsider t = t as Element of [: Z , Z :] , Z = t (#) ( t (#) ( id Z ) ) as Element of dom ( t (#) ( id Z ) ) ;
C \/ P \/ P c= [#] ( G \ A ) ;
f " ( V " ) in from ( V .: X ) , V .: ( V .: X ) ;
x in [#] ( \alpha ) ;
g . x <= h . x & g . x <= h . x ;
InputVertices S = { x , y , z } ;
for n being Nat st n < m holds P [ n ] ;
set R = Line ( M , i ) ;
assume that M1 is line and M2 is width M2 and M2 is width M2 and M2 = width M2 and M2 = M2 * ( M2 , width M2 ) ;
reconsider a = { f . i - ( { f . i } ) where i is Element of NAT : i <= len f } as Element of NAT ;
len ( Len ( Len ( ( Len ( F ) ) ) ) + ( ( Len ( F ) ) ) ) ) ) = len ( ( ( ( ( ( ( F ) ^ ) ^ ( ( ( F ) ^ ( ( F ^ <* F ) ) ^ <* F *> ) ) ) ) ) .= len ( ( (
len ( ( the InternalRel of n ) \mapsto 0. K ) = n ;
dom ( - ( f + g ) (#) f ) = dom ( ( - g ) (#) f ) ;
( the dist of seq ) . n = ( the dist ( seq , n ) ) . n ;
dom ( p1 ^ p2 ) = dom ( p1 ^ p2 ) ;
M . ( 1 , y ) = 1 * ( y , z ) .= 1 * ( y , z ) .= 1 * ( y , z ) .= y * ( y , z ) .= y * ( y , z ) .= y * ( z , z ) .= y * ( y , z ) .= y * ( y
assume that W is trivial and W is trivial and W is trivial and for n being Nat st n <= m holds W . n = W . n and W . n = W . n and W . n = W . n ;
C /. ( i + 1 ) = G * ( i + 1 , j ) .= G * ( i + 1 , j ) .= G * ( i + 1 ) .= G * ( i + 1 , j + 1 ) ;
C |- 'not' p '&' ( 'not' p ) '&' ( 'not' p ) '&' ( 'not' p ) '&' ( 'not' p ) ) ;
for b being set st b in rng g holds b <= b
- ( ( - ( ( ( ( ( ( |. q .| / |. q .| ) .| ) .| ) ) ^2 ) ) ^2 = 1 ;
LSeg ( c , c ) \/ LSeg ( c , m ) c= R ;
consider p being element such that p in LSeg ( x , p ) and p in LSeg ( x , p ) ;
the carrier of X = Seg n & the carrier of X = Seg n ;
let s ;
Im ( ( ( Im F ) . m ) ) . m <= ( Im F ) . m ;
cluster the functor func f . ( x1 , y1 ) -> Element of D ;
consider g being Function such that g = F . ( t , 1 ) and g . ( t , 1 ) = g . t ;
p in LSeg ( ( ( ( ( Gauge ( C , n ) * ( i , 1 ) , 1 ) ) , ( ( Gauge ( C , n ) * ( i , 1 ) ) , ( i , 1 ) ) ) `1 ;
set R8 = R ^ <* b *> , R8 = [. b , c *> ;
IncAddr ( I , k ) = Exec ( I , Comput ( I , s2 , k ) ) , k ) .= Exec ( I , s2 ) ;
s . m <= ( the dist ( m , n ) ) . m ;
a + b = ( a " ) * ( a " ) .= a " ;
id X = id X /\ Y .= id Y ;
for x being element st x in dom h holds h . x = f . x
reconsider H = U \/ { U } as non empty Subset of U ;
u in ( c /\ ( d /\ e ) ) /\ m ;
consider y being element such that y in Y and y in Y and y in Y ;
consider A being finite stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable stable
{ p : p in rng f & p in rng f & f . p = ( f . p ) . ( len f ) & p in rng f & p in rng f & p in rng f & f . ( len f ) = ( f . 1 ) . ( len f ) ;
len ( s " ) > 0 & len s = n ;
( ( N-min L~ f ) * ( i + 1 ) ) `2 = ( N-min L~ f ) /. ( i + 1 ) `2 .= ( N-min L~ f ) /. ( i + 1 ) `2 .= ( N-min L~ f ) /. ( i + 1 ) `2 .= ( N-min L~ f ) /. ( i + 1 ) `2 ;
Ball ( e , r ) c= LeftComp Cage ( C , n ) ;
f . ( a , b ) = f . ( a , b ) .= f . ( a , b ) ;
( s ^\ k ) . n in [. x0 , x0 + r .] ;
g1 . ( ( g . ( m + 1 ) ) ) = g . ( m + 1 ) .= g . ( m + 1 ) ;
the InternalRel of S is InternalRel of S ;
deffunc F ( Ordinal ) = \varphi ( $1 , $1 ) ;
F . ( a , b ) = F . ( a , b ) .= F . ( a , b ) ;
x = A . ( o , a ) .= ( ( the Sorts of A ) . ( o , a ) ) . ( o , a ) .= ( the Sorts of A ) . ( o , a ) ;
card ( f " { f " } ) c= card ( f " { f " { 0 } ) " { 0 } ;
FinMeetCl ( the topology of S ) c= the topology of S ;
attr o is \ast means : Def5 : o <> \ast it & o <> \ast p ;
assume that X + card Y = card Y + card Y and card Y = card Y and card Y = card Y + card Y and card Y = card Y ;
the still of s is stable & the Following of s is stable ;
not a , b // c , d or a , b // c , d ;
e . 1 = 0 & e = 0 ;
E in { { { S where S is Subset of S : S <= N & N <= N & N <= N & N <= N & N <= N & N c= S & S <= N & N <= N & N c= S & N c= S & N c= S & S <= N & N c= S ;
set J = ( l , u ) -TruthEval , J = ( l , u ) -TruthEval , I = ( l , u ) -TruthEval , J = ( l , u ) , J = ( l , u ) , J = ( l , u ) , J = ( l , u ) , J = ( l , u ) -TruthEval , J = ( l , u
set A1 = 1GateCircStr ( <* <* a *> , b *> , cp ) ;
set IT = [ <* { , IT *> , f1 ] , IT = [ <* IT , IT *> , IT ] , IT = [ IT , IT *> , IT ] , IT ] , IT = [ IT , IT ] , IT ] , IT = [ IT , IT ] , IT ] , IT = [ IT , IT ] , IT = [ IT , IT
x * z " * x in ( N " ) * ( N " ) ;
for x being element st x in dom f holds f . x = g . x
RightComp f c= RightComp f \/ RightComp f \/ RightComp f ;
U is arc of W1 , W2 ;
set L1 = f ^ g ;
attr attr attr attr attr attr : : : for S st S is convergent & lim S = x0 holds ex n st S = ( for n st n <= m holds ( for n st n >= m holds ( ( for n st n >= m holds ( ( for m st n >= m holds ( ( for m st n >= m holds
f . ( 0 qua Nat ) = ( a qua Nat ) . ( 0 qua Nat ) .= a ;
cluster the InternalRel of M -> reflexive reflexive ;
consider d being element such that R reduces b , d and d , d // b , d ;
b in dom Start-At ( card I + 2 ) ;
( z + a ) * x = z + ( a * x ) * x .= z * x + y * y ;
len ( ( <% a %> --> 0 ) --> 0 ) = len ( <% a %> --> 0 ) ;
t1 . ( t . ( t . ( t . ( t . ( t ) ) ) ) ) ) = ( t . ( t . ( t . ( t . ( t . ( t . ( t . ( t . t ) ) ) ) ) ) . ( t . t ) .= ( t . t ) . ( t
t = <* F ( t ) *> ^ <* p ( t ) *> ^ <* q ( t ) *> ^ <* p ( t ) *> ;
set p1 = \mathop { \rm W _ { min } ( C ) , p2 = \mathop { \rm E _ { max } } , p3 = \mathop { \rm E _ { max } } ;
k1 -' ( i + 1 ) = ( i + 1 ) - ( i + 1 ) ;
consider u being Element of L such that u = u ^ <* x *> and u in D and u in D ;
len ( ( ( width G , ( width G , width G , width G , width G , width G , width G , width G , width G , width G ) ) ) = width G ;
F . x in dom ( G . x ) ;
set z2 = the carrier of H , z1 = the carrier of H , z2 = the carrier of H , z1 = the carrier of H , z2 = the carrier of H , z2 = the carrier of H , z1 = the carrier of H , z2 = the carrier of H , z1 = the carrier of H , z2 = the carrier of H , z1
set V1 = the carrier of H , V2 = the carrier of G , V2 = the carrier of H , V2 = the carrier of H , V2 = the carrier of G , V2 = the carrier of H , V2 = the carrier of H , V2 = the carrier of G , V2 = the carrier of G , E = the carrier of H , E
Comput ( P1 , s , m ) . intpos m = s . intpos m ;
IC Comput ( P3 , s3 , k ) = l + 1 + 1 ;
dom ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) | ( dom ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) )
cluster <* l *> ^ ( 1 , l ) -> ( ( 1 , m ) -wff string of S ) * ( ( ( ( ( ( m , m ) ) .| ) ) ) -> m -> ( ( ( ( ( m , m ) ) ) ) ) -> m -wff ;
set b = [ <* \hbox { \boldmath $ p $ } , <* a *> , b *> , c = [ <* b , c *> , <* c *> , c ] , d ] ;
Line ( M , i ) = L * ( M , i ) ;
n in dom ( ( the Sorts of A ) * ( ( the Sorts of A ) * ( the Sorts of A ) * ( the Sorts of A ) ) ) . o ;
cluster f1 + f2 -> Lipschitzian for PartFunc of S , T ;
consider y being Point of X such that y = y & y <= r ;
set x3 = t . ( DataLoc ( s . a , k1 ) , k1 ) , k2 = t . b , k2 = t . b , k2 = t . b , k2 = t . b , k2 = t . b , k2 = t . b , k2 = t . b , Element of SCMPDS ;
set p-3 = stop I , PJ = stop J , PJ = Initialize s , PJ = Initialize s , P3 = Initialize s , PJ = Initialize s , PJ = P3 , PJ = P3 = P3 , P3 = P3 , s3 = P3 = P3 , s3 = P3 = P3 , P3 = P3 , s3
consider a being Point of T2 such that a in g and g . a = g . a ;
not A in { A , B } \/ { A , B } ;
let A , B be non empty set ;
|. p2 .| - |. p2 .| >= 0 ;
l -' 1 = n-1 * ( n-1 + 1 ) + ( n + 1 ) ;
x = v + ( a * ( b * ( b * ( a * ( b * ( a * ( b * ( b * ( b * ( b * ( a * ( b * ( b * ( a * ( b * ( b * ( b * ( ( - - a ) ) ) ) ) ) ) ) ) ) ) ) ) ) .= ( a *
the TopStruct of L = ( the TopStruct of L ) | ( the TopStruct of L ) .= the TopStruct of L ;
consider y being element such that y in dom H and y in H and y in H ;
{ f . n } \ { v . ( n + 1 ) where v is Element of Free ( { v } ) : v . ( n + 1 ) = ( f . ( n + 1 ) ) . ( n + 1 ) ;
let Y be Subset of X ;
2 * n in { N : N > 0 & N > 0 & N > 0 } ;
let s be FinSequence of the carrier of G ;
for x st x in Z holds ( ( - 1 / 2 ) `| Z ) . x = - 1 / 2
rng ( f2 * f1 ) c= the carrier of V ;
j + 1- len f + len f - len f + len f - len f - len f + len f - len f - len f - len f + len f - len f - len f + len f - len f -' len f -' len f -' len f -' len f -' len f -' len f -' len f -' len f -' len f -'
reconsider R1 = R * ( I , i ) as PartFunc of REAL ;
C . x = { s . x } .= { s . x } .= { s . x } ;
0. ( ( 0. ( ( ( ( z ) ) * |. z .| ) .| ) ) ) ) ) ) = 1 * ( 1 * ( |. z .| ) ) .= 1 * ( |. z .| * |. z .| ) .= |. z .| * |. z .| .= |. z .| * |. z .| * |. z .| .= |. z .| * |. z .|
t --connectives ( C , s ) = f ( the connectives of C , s ) ;
support ( f + g ) c= support f \/ { 0 } ;
ex N st N = { j : j >= N & j >= N & N >= N & N > N } ;
for y st p [ y , p ] & p = [ y , p ] holds y = [ p , q ]
\ { x , y } is Subset of [: X , Y :] \ { y , x } ;
h = ( j .--> ( i .--> ( j .--> ( i .--> ( i .--> j ) ) ) ) ) . i .= ( j .--> ( i .--> ( i .--> j ) ) . i ) . i .= ( j .--> ( i .--> j ) ) . i ;
ex x1 being Element of G st x = x1 & x1 * N c= A * N ;
set X = ( ( ( ( ( ( q , m ) ) * ( ( q , m ) * ( ( q , m ) ) * ( q , m ) ) ) ) , Y = ( ( q , m ) * ( q , m ) ) ) * ( q , m ) ) , Y = ( q , m ) * ( q , m ) ) * ( q , m ) ;
b . n in { g . n where g is Real : 0 <= g . n & g . n < g . n } ;
f /* ( s \mathbin { \uparrow } k ) is convergent & lim ( ( f /* s ) \mathbin { \uparrow } ) = ( f /* s ) . k ;
the carrier of Y = the carrier of Y & the carrier of Y = the carrier of Y ;
( a '&' b ) '&' ( a '&' b ) = ( a '&' b ) '&' ( a '&' b ) ;
set S2 = ( len { q } ^ <* r *> ) + ( len q ) + ( len q ) + 1 ) ;
( a * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( , , , , , , , , , , , , ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
set V1 = upper upper upper upper upper upper \ _ integral ( f , T ) , integral ( f , T ) , integral ( f , T ) , integral ( f , T ) , integral ( f , T ) , integral ( f , T ) = integral ( f , T ) ;
assume e in { w where w is Element of : w in F & w in F & w in G & w in F } ;
reconsider a9 = a as set ;
LSeg ( f , j -' 1 ) = LSeg ( f , j ) ;
assume X in { T where { N is Nat : N <= N & N <= N & N <= N & N <= N & N <= N & N <= N & N <= N & N <= N & N <= N & N <= N & N <= N & N <= N & N <= N & N <= N & N <= N ;
assume that Hom ( d , c ) = {} and dom ( d , c ) = {} and dom ( d , c ) = {} and dom ( d , c ) = {} ;
dom U2 = [: the carrier of U1 , the carrier of U2 :] .= the carrier of U2 .= the carrier of U2 ;
x in { g : g <= g & g in A & g <= a & a <= b } ;
a * ( a * ( a * b ) ) = a * ( a * b ) .= a * ( a * b ) .= a * ( a * b ) .= a * ( a * b ) ;
D2 . mm in { r where r is Real : r <= s & s <= s & s <= 1 & s <= 1 } ;
ex p being Point of TOP-REAL 2 st p = x & p in P & p in P & p in P & p in Q & p in Q ;
for c , f st f . c <= g . c holds g . c <= f . c
dom ( ( f1 (#) f2 ) `| Z ) = dom ( f1 (#) ( f1 (#) f2 ) `| Z ) ;
1 = p * ( 1 - p ) .= p * ( 1 - p ) .= p * ( 1 - p ) .= p * ( 1 - p ) .= p * ( 1 - p ) ;
len g = len f + len <* x *> .= len f + len f ;
dom ( F | ( n1 + 1 ) ) = ( F | ( n1 + 1 ) ) . n1 ;
dom ( ( f * ( t ) ) ) = dom ( f * ( t ) ) ) ;
assume a in ( bool the carrier of ( T ) .: the carrier of T ;
assume that g is one-to-one and rng g c= dom ( f | X ) and rng g c= dom ( f | X ) ;
( x \ y ) \ y = 0. X ;
consider f being Function of b , a such that f = id b and f = id a ;
( the function of cos ) | [. 0 , 1 .] is continuous ;
Index ( p , co ) + 1 <= len G - ( len G -' 1 ) + 1 ;
t1 , t2 are_connected , t1 , t2 , t1 , t2 , t1 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t1 , t2 , t1 , t2 , t1 , t1 , t2 , t1 , t1 , t2 , t1 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t1 , t2 , t1 , t2 , t1 , t2 , t1
relational ( ( ( relational ( F ) . n ) ) . m ) <= ( ( relational ( F . m ) ) . n ) ;
then for f st f . ( ( f . ( i + 1 ) ) ) < j holds f . ( i + 1 ) < j ;
[ Q , D ] in F ;
consider x being element such that x in dom ( F . s ) and y = F . x ;
l . i < r . i & r . i < G . i ;
the Sorts of S2 = ( the Sorts of S2 ) . ( the Sorts of U2 ) . ( the Sorts of U2 ) .= ( the Sorts of U2 ) . ( the Sorts of U2 ) . ( the Sorts of U2 ) . ( the Sorts of U2 ) . ( the Sorts of U2 ) . ( the Sorts of U2 ) . ( the Sorts of U2 ) .= ( the Sorts of U2
consider s being Function of the carrier of G , the carrier of G such that rng s = the carrier of G and rng s c= the carrier of G ;
dist ( b , a ) <= dist ( b , a ) + dist ( b , a ) ;
( LowerSeq ( C , n ) ) . ( len Gauge ( C , n ) , 1 ) ) = ( ( vseq . n ) * ( 1 , 1 ) ) * ( 1 , 1 ) .= ( ( vseq . n ) * ( 1 , 1 ) ) * ( 1 , 1 ) ;
q <= ( W-min L~ Cage ( C , n ) ) `1 ;
LSeg ( f , i ) /\ LSeg ( f , i ) = {} ;
consider a being Real such that a <= a and a <= b & b <= 1 ;
consider a , b being complex number such that z = a * b + b * c + c * c and a * c = b * c + c * c ;
set X = { b where b is Element of NAT : b <= n & n <= m & m <= m } ;
( x * y ) * z = 0. X ;
set L1 = <* L1 , L2 *> , R1 = <* L1 , L2 *> , R1 = <* L1 , L2 *> , R1 = <* L1 , L2 *> , R1 = <* L2 , R1 *> , R1 = <* L1 , R1 *> , R1 = L2 , R1 = L2 ;
l = ( ( len l ) + 1 ) * ( len l ) .= l ;
( - ( ( ( ( ( |. q .| / |. q .| - sn ) ) ) sn ) ) sn ) ^2 = 1 ;
\frac ( p - ( |. p .| - sn ) ) ^2 < 1 ;
( Partial_Sums ( S \/ { 0 } ) ) . n = ( Partial_Sums ( S \/ { 1 } ) ) . n .= ( Partial_Sums ( S \/ { 1 } ) ) . n ;
( - 1 ) * ( k - 1 ) = - ( ( - 1 ) * k ) * ( k - 1 ) .= ( - 1 ) * ( k - 1 ) ;
rng ( h + c ) c= dom ( h + c ) ;
the carrier of X = the carrier of X ;
ex p3 , p2 st p3 = |. p3 - p2 .| & |. p2 .| = |. p2 .| ;
set h = .4ex .4ex .4ex .4ex .4ex .4ex .4ex .4ex \hbox \chi \chi , \chi , A , A , A , B , B , C ) ;
R |^ ( 0 , n ) = ( R |^ ( 0 , n ) ) |^ ( 0 , n ) .= R |^ ( 0 , n ) .= R |^ ( 0 , n ) .= R |^ ( 0 , n ) .= R |^ ( 0 , n ) .= R |^ ( 0 , n ) .= R |^ ( 0 , n ) .= R |^ ( 0 , n ) .= R |^ (
( Partial_Sums ( ( Partial_Sums ( ( ( ( ( ( ( ( ( ( ( ( ( ExpSeq ) # ) # ) ) ) # ( \alpha ) ) ) ) # ( \alpha ) ) ) ) ) ) ) ) . n ) ) . n is non-negative ;
set f2 = EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEV = EEEV ;
S1 . b = S1 . b .= S2 . b .= S2 . b .= S2 . b ;
p2 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ;
dom ( f . ( t . ( t . n ) ) ) = Seg n & dom ( f . t ) = Seg n ;
assume o = ( the connectives of S ) . ( ( the connectives of S ) . 11 ) ;
set \varphi = ( l , ( l , n ) ) * ( l , n ) ;
attr p is top means : Def5 : for q being polynomial of L , L holds p . q = 1. L ;
( Y - ( Y - 1 ) ) * ( Y - 1 ) = ( Y - 1 ) * ( Y - 1 ) & Y - ( Y - 1 ) * ( Y - 1 ) = Y ;
defpred X [ Nat ] means $1 in dom ( ( ( $1 , $1 ) --> ( $1 , $1 ) --> ( $1 , $1 ) ) ) ;
consider k being Nat such that for n being Nat st n <= k holds s . n < x0 + g . n ;
Det ( m + ( m + n ) ) = - ( m + n ) ;
sqrt ( b - sqrt ( b ^2 ) ) ^2 < 0 ;
C . d = C . d .= C . d .= C . d ;
attr X is dense means : Def5 : for A being Subset of X st A in A holds A is dense & A is dense & A is dense & A is dense ;
defpred F ( Element of E , Element of E ) = ( $1 , $1 ) * ( $1 , $1 ) ;
t ^ <* n *> in { t ^ <* n *> where n is Nat : n in dom t & t in dom t & t ^ <* n *> in T } ;
( x \ y ) \ y = ( x \ y ) \ y .= x \ y .= y \ y .= x \ y ;
let X be non empty Subset-Family of X ;
synonym A \ A -> non empty ;
len ( M1 * M2 ) = len ( M1 * M2 ) & width M2 = width ( M2 * M2 ) ;
vp = { x where x is Element of K : x <= v & v <= y & y <= v & v <= y & not x in v . x } ;
( Sgm Seg ( m + 1 ) ) . ( m + 1 ) = ( ( Sgm Seg ( m + 1 ) ) . ( m + 1 ) ;
inf divset ( D2 , k + 1 ) = D2 . ( k + 1 ) ;
g . ( 2 * ( 2 * ( 2 * n + 1 ) ) ) = - ( 2 * n ) * ( 2 * n ) ;
|. a .| * ||. f .|| = 0 * ||. f .|| .= 0 * ||. f .|| .= 0 ;
f . x = ( h . x ) . ( h . x ) & f . x = ( h . x ) . ( h . x ) ;
ex w st w in dom ( B ^ <* w *> ) & w in dom ( B ^ <* w *> ) ;
[ 1 , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} ,
IC Comput ( P3 , s3 , n ) = n + 1 ;
IC Comput ( P3 , s3 , 1 ) = IC Comput ( P3 , s3 , 1 ) .= IC Comput ( P3 , s3 , 1 ) .= IC Comput ( P3 , s3 , 1 ) .= 0 ;
( IExec ( W6 , Q , t ) ) . intpos >= 0 ;
LSeg ( f , i -' 1 ) misses LSeg ( f , i ) ;
assume for x , y being Element of L st x <= y & y <= x & x <= y holds y <= x
integral ( f , C ) = f . ( f . x ) .= f . x ;
let F be FinSequence of the carrier of G ;
||. R /. ( h + 1 ) - R /. ( h + 1 ) .|| < R * ( h + 1 ) ;
assume a in { q where q is Element of M : q <= r & r <= q & q <= 1 } ;
set q1 = [ 2 , 0 ] , q2 = [ 2 , 1 ] , q1 = [ 2 , 1 ] , q1 = [ 2 , 0 ] , q1 = [ 2 , 1 ] , q1 = [ 2 , 1 ] , q1 = [ 2 , 1 ] , q1 = [ 2 , 1 ] , q1 = [ 2 , 1 ] , q1 = [ 2 , 1 ] , q1 = [ 2 , 1
consider x , y being element such that x in F and y in F and x in F and y in F ;
for y being Element of REAL st y in Y & y in Y holds y in Y
func |. p .| ^ |. p .| -> complex number equals Sum ( p ^ |. p .| ) ;
consider t being Element of S such that x , y // t , t and t , u // t , u , t , t , t , t , t , u , t , t , t , u , t , t , t , u , t , t , t , t , u , t , t , t , t , u , t , t , t , t , u , t , t , t
dom ( x1 * x2 ) = Seg len ( x1 * x2 ) & dom ( x1 * x2 ) = Seg len ( x1 * x2 ) ;
consider z2 being Real such that z2 = 0 and z2 < z1 and z1 < z2 & z2 < z1 & z1 < z2 ;
||. f .|| | X = ||. f .|| ;
( the InternalRel of A ) \/ ( the InternalRel of A ) = {} \/ ( the InternalRel of A ) .= {} \/ ( the InternalRel of A ) .= {} ;
assume that i in dom p and i + 1 <= len p and i + 1 <= len p ;
reconsider h = f | X as Function of X , Y ;
u in the carrier of W1 & v in the carrier of W1 & u in the carrier of W2 ;
defpred P [ Element of L ] means $1 <= len f & f . $1 = ( f . $1 ) . ( $1 + 1 ) ;
T . ( u , v ) = s . ( u , v ) .= s . ( v , v ) .= s . ( v , v ) .= s . ( v , v ) .= s . ( v , v ) + s . ( v , v ) .= s . ( v , v ) .= s . ( v , v ) .= s . ( v , v ) ;
- ( ( - sin ) . x ) = - ( sin . x ) .= ( - sin . x ) * ( sin . x ) .= ( - sin . x ) * ( sin . x ) .= ( - sin . x ) * ( sin . x ) .= ( sin . x ) * ( sin . x ) .= ( sin . x ) * ( sin . x ) .= ( sin .
consider a being Point of G1 | the carrier of G1 , the carrier of G1 ;
f1 = [ f1 , f2 ] ^ <* f1 , f2 *> , f2 ] ;
let k be Nat ;
for x being element st x in A holds ( A ` ) ` = ( A ` ) `
consider u , v being Element of R such that v = u * v + v * v ;
1- ( p - ( |. p .| - sn ) ) ^2 > 0 ;
L1 . k = L1 . k & L2 . k = L1 . k ;
set i2 = AddTo ( a , i , n ) , i2 = AddTo ( a , i , n ) ;
attr B is universal means : Def14 : for S being full SubRelStr of ( B ) . ( ( B . ( ( B . ( m + 1 ) ) ) , S . ( m + 1 ) ) ) ) . ( m + 1 ) = ( B . ( m + 1 ) ) . ( m + 1 ) ;
z1 "/\" D = { d where d is Element of N : d in D } ;
| ( ( \square \square , n ) * ( b , n ) ) . ( b , n ) = ( \square | n ) . ( b , n ) .= ( \square , n ) . ( b , n ) ;
( - f ) . ( ( - f . ( - f . ( - f . n ) ) ) ) = ( - f . ( - f . n ) ) ;
( G * ( 1 , 1 ) ) * ( 1 , j ) = G * ( 1 , j ) ;
( proj ( i , n ) ) . i = ( proj ( i , n ) ) . i .= ( proj ( i , n ) . i ) . i ;
f1 + ( ( ( ( ( ( reproj ( i , m ) ) * ( i , m ) ) * ( ( reproj ( i , m ) ) * ( ( reproj ( i , m ) ) * ( ( ( ( ( i , m ) * ( ( ( ( ( ( i , m ) * ( ( ( ( ( i , m ) * ( ( ( i , m ) * ( ( ( ( i
attr the function of ( the carrier of X ) , ( the carrier of X ) , ( the carrier of Y ) , the carrier of Y be Subset of X ;
ex t being symbol of S st t = s & t = s . t & for x being Element of S holds x . x = t . x ;
defpred C [ Nat ] means $1 . $1 = ( $1 + 1 ) . $1 & $1 = ( $1 + 1 ) . $1 ;
consider y being element such that y in dom ( p * q ) and y in X and y in Y ;
reconsider L = product ( { { x } } ^ ( Carrier ( B ) ) ) ^ ( Carrier ( A ) ) ) as Subset of \prod ( A ^ ( B ^ ( B ^ ( B ) ) ) ) ;
for c being Element of C st c in C holds ( id id C ) . c = id C
Line ( f , n ) = ( f | n ) | ( p | n ) .= p ;
( f * g ) . x = f . x & ( f * g ) . x = f . x ;
p in { ( ( 2 * ( i + 1 ) ) + ( 2 * ( i + 1 ) ) ) ) + ( 2 * ( i + 1 ) ) ) ;
f - ( ( - ( ( ( ( - ( c - ( c - sn ) ) ) (#) ( f - g ) ) ) (#) ( f - g ) ) ) ) = ( ( - ( c - sn ) (#) ( ( - sn ) (#) ( ( - ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
consider r being Real such that r in rng ( f | divset ( D , j ) ) and for m be Nat st m <= m holds |. ( f | divset ( D , j ) ) - ( f | divset ( D , j ) ) .| < r ;
f1 . ( ( |. ( ( |. ( ( |. ( ( ( ( |. .| .| ) .| ) ) ) .| ) .| ) ) .| ) ) ^2 in { f . ( |. f .| ) .| ;
eval ( a , x ) = eval ( a , x ) .= eval ( a , x ) .= eval ( a , x ) .= eval ( a , x ) .= eval ( a , x ) .= eval ( a , x ) .= eval ( a , x ) .= eval ( a , x ) ;
z = DigA ( t , 1 ) .= ( ( t , 1 ) * ( z , 1 ) ) * ( z , 1 ) .= ( t * ( z , 1 ) ) * ( z , 1 ) ;
set H = { meet ( S , S ) where S is Subset-Family of X : S in F } ;
consider W1 being Element of D such that S = { W1 ^ W2 where W2 is Element of D : len W2 = len W2 } ;
assume that for x1 st x1 in dom f & x1 in dom f and x1 in dom f and x1 in dom f and x1 in dom f and x1 in dom f and x1 in dom f ;
- 1 <= ( ( - ( q `1 / |. q .| - sn ) ) / ( 1 + sn ) ) ^2 ;
0. V = 0. V & 0. V = 0. V ;
let i1 , i2 be Nat ;
consider j being element such that j in dom g and j in g . j and j in g . j ;
H . ( x1 , y1 ) c= H . ( x1 , y1 ) or H . ( x1 , y1 ) = H . ( x1 , y1 ) ;
consider a such that p = exp_R * ( a + 1 ) and a <= 1 and a <= 1 ;
assume that a <= c and c <= d and d <= b and c <= d ;
cell ( Gauge ( C , m ) , width Gauge ( C , m ) -' 1 , width Gauge ( C , m ) -' 1 ) ) c= cell ( Gauge ( C , m ) -' 1 , width Gauge ( C , m ) -' 1 ) ;
A1 in { ( A2 . \alpha ) where \alpha is is Element of NAT : ( ( S . \alpha ) . i ) `1 <= ( ( S . i ) `1 ) ^2 } ;
( T * b ) . y = ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( - ( ( ( ( ( ( ( ( ( ( ( o , , o ) , , ( ) ) ) * ( ( o ) ) ) * b ) ) ) ) ) ) ) ) ) ) ) . y ) ) ) ) . y .= ( ( ( ( ( ( o ) ) * ( ( o ) ) ) )
g . ( s , x ) = |. s . ( y , x ) .| & g . ( y , x ) = |. s . ( y , x ) .| ;
( log ( 2 , k ) ) ^2 >= ( 2 * k ) ^2 ;
attr p => q in S & p => q in S & p => q in S ;
dom ( the Tran of ( the carrier of Euclid n ) ) misses dom ( the Tran of n ) ;
synonym f -> ExtReal means : for for for for for for for x being set st x in dom f holds f . x = x
assume for a being Element of D holds f . a = a * f . a ;
i = len ( <* p *> ^ <* x *> ) .= len ( <* y *> ^ <* x *> ) + len ( <* y *> ^ ( <* y *> ^ ( <* x *> ^ ( y ^ <* y *> ) ) .= len ( <* y *> ^ ( y ^ <* x *> ) .= len ( <* y *> ^ ( y ^ ( y ^ <* y *> ) .= len ( y ^ ( y ^ <* y *> ) + 1 ) .= len ( y ^ ( y ^ ( y
( l + 1 ) * ( l + 1 ) = ( l - 1 ) * ( l - 1 ) .= ( l - 1 ) * ( l - 1 ) ;
CurInstr ( P3 , Comput ( P3 , s3 , 1 ) ) = P3 . IC Comput ( P3 , s3 , 1 ) .= P3 . IC Comput ( P3 , s3 , 1 ) ;
assume for n be Nat holds ||. seq . n - seq . n .|| < r ;
sin . r2 = sin . r2 .= sin . r2 .= sin . r2 .= sin . r2 .= sin . r2 ;
set q = [ g , t ] , r = [ t , r ] , t = [ t , r ] , t = [ t , t ] , t = [ t , t ] , t = [ t , t ] , t = [ t , t ] , t = [ t , t ] , t = [ t , t ] , t = [ t , t ] , t = [ t , t ] , t = [ t , t ] , t = [ t , t ]
consider G being sequence of S such that for n being Element of NAT holds G . n in G . n ;
consider G such that F = G and G in { G * ( 1 , 1 ) where G * ( 1 , 1 ) `1 } ;
the root of ( the carrier of Free F ) . s in ( the Sorts of F ) . s ;
Z c= dom ( ( ( ( ( ( ( ( ( ( 1 / ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( / ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) (#) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
for k be Element of NAT st k = ( Sum ( ( Im ( Im ( Im ( Im ( Im ( Im ( Im ( ( Im ( Im ( ( Im ( ( ( Im ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) * ( ( Im ( ( ( ( ( ( Im ( ( ( (
assume that 1 < ( |. q .| ) ^2 and |. q .| = |. q .| and |. q .| = |. q .| ;
assume that f is continuous and for a being Real st a < b & a <= b holds f . a = a * b ;
consider r being Element of NAT such that s = Comput ( P1 , s1 , 1 ) and r <= len Comput ( P1 , s1 , 1 ) ;
LE f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) ;
assume that x in the carrier of K and y in the carrier of K and y in the carrier of K and x in the carrier of K and y in the carrier of K and y in the carrier of K and y in the carrier of K and y in the carrier of K ;
assume f +* ( i , i1 ) in ( the InternalRel of A ) * ( i , i1 ) ;
rng ( ( ( ( ( M ~ ) | the carrier of K ) ) | the carrier of K ) ) | the carrier of K ) ) = the carrier of K ;
assume z in { the carrier of G where G is Element of G : G * ( len G , 1 ) `1 } ;
consider l being Nat such that for m be Nat st m <= n holds |. ( ( s . m ) - ( s . n ) ) .| < g ;
consider t be VECTOR of product G such that t = ||. t .|| & ||. t .|| < 1 ;
attr the topology of v = 2 ^ <* v *> & v = 2 ^ <* v *> ;
consider a being Point of the TopStruct of X2 such that A = the TopStruct of X2 and A c= the TopStruct of X2 ;
( - ( x - ( ( ( - ( ( x - ( k - n ) ) ) / ( k + 1 ) ) ) ^2 ) ) ) ^2 = 1 ;
let D be set , i be set ;
defpred R [ element ] means $1 in dom ( f | $1 ) & $1 in dom ( f | $1 ) ;
LSeg ( f1 , p2 ) = union LSeg ( f1 , p2 ) ;
i -' ( len h -' ( i + 1 ) + 1 ) + 1 + 1 - ( i + 1 ) - 1 + 1 + 1 < i - 1 ;
for n be Element of NAT st n in dom ( F . n ) holds |. ( F . n ) - ( F . n ) .| < |. ( F . n ) - ( F . n ) .|
for r , s being Real , f being Function of the carrier of X , the carrier of Y st r in the carrier of Y & s in the carrier of Y holds f .: ( r , s ) = f .: ( r , s )
assume v in { G where G is Subset of T : G * ( 1 , 1 ) `1 <= G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `2 <= G * ( 1 , 1 ) `2 & G * ( 1 , 1 ) `2 <= G * ( 1 , 1 ) `2 & G * ( 1 , 1 ) `2 <= G * ( 1 , 1 ) `2 ;
let g be non-empty non-empty MSAlgebra over A , B be non-empty MSAlgebra over A ;
min ( g , k ) = ( k + 1 ) * ( k + 1 ) ;
consider q1 being sequence of the carrier of C , r being Real such that for n holds q1 . n = r * ( n + 1 ) ;
consider f being Function of NAT , NAT such that for n being Element of NAT holds f . n = F ( n ) ;
reconsider -6 = B /\ Z as Subset of [: B , B :] ;
consider j being Element of NAT such that x = [ j , n ] and j <= n and n <= len f and n <= len f ;
consider z such that z = x and card ( z + 1 ) = card ( z + 1 ) ;
( C * ( k + 1 ) ) * ( k + 1 ) = C * ( k + 1 ) ;
dom ( X --> 0 ) = X & dom ( X --> 0 ) = X & dom ( X --> 0 ) = X ;
S is non empty implies ( for n being Nat holds ( for n being Nat holds ( ( S . n ) . n ) . ( S . n ) ) . ( n + 1 ) = ( S . n ) . ( n + 1 )
synonym x \ y -> collinear means : for for x , y being Element of S st x = y holds x \ y = y ;
consider X being element such that X in dom ( f | X ) and X c= Y ;
assume that for k being Nat st k is compact and k <= n holds ||. ( x - y ) - ( y - z ) .|| < r ;
( ( 2 * PI ) * ( ( ( 2 * PI ) * ( ( 2 * PI ) ) ) ) ) ) ) * ( ( 2 * PI ) * ( ( 2 * PI ) ) ) ) = ( 2 * PI ) * ( ( 2 * PI ) ) * ( 2 * PI ) ;
defpred P [ Element of \omega ] means ( for n holds ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( $1 . . . . ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . $1 ;
IC Comput ( P3 , s3 , 1 ) = IC Comput ( P3 , s3 , 1 ) .= ( P3 , s3 ) + 1 .= ( P3 , s3 ) . IC SCM+FSA .= ( P3 , s3 ) . IC SCM+FSA .= ( P3 , s3 ) . IC SCM+FSA .= ( P3 , s3 ) . IC SCM+FSA .= ( P3 , s3 ) . IC SCM+FSA .= ( P3 , s3 ) . IC SCM+FSA .= ( P3 , s3 ) . IC SCM+FSA ;
f . x = f . x .= f . x .= f . x .= f . x ;
( M * F ) . n = M . ( ( M * F ) . n ) .= M . ( ( M * F ) . n ) .= M . ( ( M * F ) . n ) .= M . ( ( M * F ) . n ) .= M . ( ( M * F ) . n ) ;
the carrier of L1 + ( the carrier of L1 ) c= ( the carrier of L1 ) + ( the carrier of L2 ) ;
attr a , b , c , d , c , d , x , y , y , c , d , x , y , y , c , d , x , y , c , y , x , y , c , d , x , y , c , y , x , y , c , y , x , y , c , y , c , d , x , y , c , y , d , x , y , c , y , x , y , c , y , x , y , c , x
( the partial PartFunc of X , REAL ) . n <= ( the partial PartFunc of X , REAL ) . n ;
attr 1 - ( 1 - ( ( 1 - ( ( ( ( ( ( ( ( ( - ( ( ( ( ( ( `1 / |. p .| .| ) ) ) ) ) ) ) / ( 1 + sn ) ) ) ^2 ) ) ^2 ) ) ^2 ) ) ^2 = ( 1 - sn ) ^2 ;
s in { p where p is FinSequence of T : p in A & len p = n & len p = n + 1 & len p = n + 1 & len p = n ;
[ x1 , x2 ] - ( x1 - x2 ) = x1 - x2 - y2 .= x1 - x2 - y2 ;
attr F is nonnegative means : Def5 : for m being Nat st m <= n holds F . m <= ( F . m ) . ( n + 1 ) ;
len ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) , , ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ^ ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ^ ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ^ ( ( (
consider u , v being VECTOR of V such that x = u + v and v in W and u in W and v in W and v in W and u in W ;
consider F be finite FinSequence of NAT such that dom F = Seg ( 0 qua FinSequence st len F = k & len F = 0 & len F = 0 ;
0 = exp_R * exp_R * exp_R + exp_R * exp_R * exp_R * exp_R . \alpha + 0 * <i> * <i> * <i> ) & 0 = exp_R * exp_R . \alpha ;
consider n be Nat such that for m be Nat st n <= m holds |. ( f /* ( ( f /* ( ( ( ( ( ( ( ( ( f # x ) - ( f # x ) ) ) ) ) ) ) ) ) ) ) ) ) . m - ( f # x ) ) ) . m .| < e ;
cluster non empty strict for TopSpace ;
inf { B where B is Subset of S : B = "/\" ( B , S ) } .= "/\" ( B , S ) ;
sqrt ( 2 * ( 1 + r ) ^2 ) <= 2 * ( 2 * ( 1 + r ) ^2 ) ^2 ;
for x being element st x in A holds ( f `| X ) . x >= ( f `| X ) . x
2 * ( a - b ) + ( 2 * ( a - b ) ) + ( 2 * ( a - b ) ) ) = 0 ;
reconsider p = a * ( \square , 1 ) as FinSequence of K ;
consider x1 , y1 being element such that x1 in uparrow s and y1 in uparrow s and y2 in uparrow t and y1 in uparrow t and y2 in uparrow t ;
for n be Nat st 1 <= n & n <= len ( ( ( ( ( ( |. ( ( ( ( ( ( ( ( ( ( ( ( - ( vseq .| ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( vseq ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) * ( ( ( ( ( ( ( ( ( vseq ) )
consider y being element such that y in the carrier of A and y in the carrier of A and y in the carrier of A ;
consider H1 being strict Subgroup of G , H being strict Subgroup of H such that x = H1 "\/" ( H1 "\/" H ) and for a being Element of G st a = "\/" ( H1 , H ) & a <= a "\/" ( H1 "\/" H ) ;
let T be non empty TopSpace ;
[ a , b ] in ( the carrier of V ) /\ the carrier of V ;
reconsider m1 = max ( ( ( p * ( n + 1 ) ) ) ) as Element of NAT ;
I <= width ( the InternalRel of G ) & I = width G ;
f2 /* q = ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( - ( ( ( ( - ( ( ( ( ( - ( ( ( ( ( ( ( - ( ( ( ( ( - ( ( ( ( - ( ( ( ( - ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . k .= ( ( ( ( ( ( ( ( ( ( ( ( ( ( - ( ( ( ( ( ( (
attr A1 : : : for A , B being Subset of V st A in B holds A /\ B = 0. V ;
func A -#Z ( A , B ) -> set equals union ( A , B ) ;
dom ( Line ( p , i ) ) = dom ( ( Line ( p , i ) ) ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( i , , i ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ;
cluster |[ x , y ]| -> Real ;
E \models All ( x1 , x2 ) '&' All ( x2 , y2 ) '&' All ( x2 , y2 ) '&' All ( x2 , y2 ) '&' All ( x1 , y2 ) '&' All ( x2 , y2 ) '&' All ( x2 , y2 ) '&' All ( x2 , y2 ) '&' All ( x2 , y2 ) = All ( x2 , y2 ) '&' All ( x2 , y2 ) '&' All ( x2 , y2 ) '&' All ( x2 , y2 ) '&' All ( x2 , y2 ) '&' All ( x2 , y2 ) '&' All ( x2 , y2 ) '&' All ( x2 , y2 )
F .: ( id X ) = F . ( id X ) .= F . ( id X ) .= F . ( id X ) .= F . ( id X ) .= F . ( id X ) .= F . ( id X ) ;
( h . m ) . ( ( h . m ) . ( ( h . m ) . ( ( h . m ) . ( ( h . m ) . ( ( h . n ) . ( ( h . n ) . ( ( h . n ) . ( ( h . n ) . ( ( h . n ) . ( ( h . n ) ) ) ) ) ) ) ) ) ) ) . ( ( h . m ) . ( ( h . m ) ) ) ) in ( ( ( h . m ) . ( ( h . n ) ) . (
cell ( G , i1 , j1 -' 1 ) /\ cell ( G , i1 , j1 -' 1 ) c= cell ( G , i1 , j1 -' 1 ) ;
IC Comput ( P3 , s3 , 1 ) = card I .= card I ;
sqrt ( 1 - ( ( ( ( ( ( q `1 / |. q .| ) / |. q .| - sn ) ) ^2 ) ) ^2 ) ) ^2 > 0 ;
consider x0 being element such that dom a = g & for k being element st k in dom g holds g . k = a * ( k + 1 ) ;
dom ( ( ( ( ( ( ( ( f1 * ( f1 * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( , , , , , , , , ( , , , , ) , ( ) ) ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( , , , , , , , ) ) , * ( ( ( ( ( ( ( ( ( ( ( ( , , , , , , , , ) ) , ( ( ( ( ( ( ( ( , , * ( ( ( ( (
d . ( y , z ) = ( y . ( y , z ) ) . ( y , z ) .= ( y . ( y , z ) ) . ( y , z ) .= ( y . ( y , z ) ) . ( y , z ) .= ( y . ( y , z ) ) . ( y , z ) .= ( y . ( y , z ) ) . ( y , z ) ;
attr attr means : for i being Nat st i in dom A holds A . i = A . i ;
attr x0 in dom ( f `| X ) & for x0 be Point of REAL m st x0 in dom ( f `| X ) holds ( f `| X ) . x0 = ( f `| X ) . x0 ;
p in Int A implies for A being Subset of T st A in Int A holds A is open
for x being Element of REAL st x in Line ( x1 , y1 ) holds |. ( x1 - y1 ) - ( y1 - y2 ) .| < r
func \omega ( a \omega ) -> set means : for a being Ordinal , b being Element of NAT st b in it holds it . b = b . b & b in b . b ;
[ a , b ] in ( the InternalRel of A ) \/ ( the InternalRel of A ) \/ ( the InternalRel of A ) ;
ex a being element st a in the carrier of S & a in the carrier of S & a in the carrier of S & b in the carrier of S ;
||. v .|| . m - ||. v .|| < e * ||. v .|| ;
then Z in { Y where Y is set : Y in Z & Y in Z & Z c= { Y where Y is Element of Z : Y in Z } ;
sup ( D , s ) = sup ( D , s ) .= sup ( D , s ) ;
consider i , j such that i < j and j < i and i + 1 <= i and i + 1 <= j ;
let D be non empty set , p be FinSequence of D ;
consider a9 being Element of the carrier of X such that a9 , b9 // a9 , b9 and a9 , b9 // b9 , c9 and c9 , b9 // c9 , c9 and c9 , b9 // c9 , c9 ;
set T1 = I \! \hbox { - } , T2 = I \! \hbox { - } ;
|. q1 .| = ( |. q1 .| ) ^2 .= |. q1 .| ^2 + |. q1 .| ^2 .= |. q1 .| ^2 ;
let T be non empty TopSpace , A be Subset of T ;
dom ( the charact of U1 ) = the carrier of U2 & dom ( the charact of U2 ) = the carrier of U2 ;
dom ( h | X ) = dom ( h | X ) /\ ( dom ( h | X ) /\ X ) .= X /\ ( dom ( h | X ) /\ Y ) .= X /\ ( dom ( h | X ) /\ Y ) .= X /\ Y ;
for N , K being Element of NAT st N = dom ( ( ( ( ( ( ( |. ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( .| .| .| ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) | N holds N = ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) | N ) ) | N ) | N ) | N
( m mod ( m + 1 ) ) mod ( m + 1 ) = ( m mod ( m + 1 ) ) mod ( m + 1 ) .= ( m mod ( m + 1 ) ) mod ( m + 1 ) ;
- ( ( - ( q `1 / |. q .| - sn ) ) ) ^2 >= - sn & ( - 1 ) ^2 >= 0 ;
attr r1 = r1 & r2 = r2 & r1 = r2 implies r1 = r2 & r2 = r2 & r1 = r2 & r2 = r2 & r1 < r2 & r2 < r2 & r2 < r2 & r2 < r2 & r2 < r2 & r2 < r2 & g2 < r2 & g2 < r2 & g2 < r2 & g2 < r2 & r2 < r2 & g2 < r2 & g2 < r2 & r2 < r2 & g2 < r2 & g2 < r2 & r2 < r2 & g2 < r2 & g2 < r2 & g2 < r2 & g2 < r2 & g2 < r2 & g2 < r2 & g2 < r2 & g2
v . m = ( |. v .| ) . m & v . m = ( |. v .| ) . m ;
attr a <> b & a <> 0 implies a = 0 & a = 0 & b = 0 & a = 0 & a = 0 & b = 0 & a = 0 & a = 0 & b = 0 ;
consider i , j such that i = [ i , j ] and i < j and j < i and i < j and i < j ;
|. p - ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( ( 2 * ( 2 * ( 2 * ( 2 * ( 1 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 1 ) ( 1 ) ) ) ) ) ) ) ) ) ) ) ) .| ) ) ) .| ) .| ) .| ) .| )
consider y such that y = [ y1 , y2 ] and y1 in X and y2 in Y and y1 in Y and y2 in Y ;
<* 1 *> , 2 *> = \frac { 1 , 2 *> * ( 1 , 2 ) .= ( 1 , 2 ) * ( 1 , 2 ) ;
( ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | K1 ) ) ) ) ^2 = ( ( TOP-REAL 2 ) | K1 ) ^2 & ( ( TOP-REAL 2 ) | K1 ) ^2 = ( ( TOP-REAL 2 ) | K1 ) ^2 ;
s |= All ( 'not' All ( 'not' All ( 'not' All ( 'not' All ( 'not' 'not' All ( 'not' All ( 'not' 'not' All ( 'not' 'not' All ( 'not' 'not' All ( 'not' 'not' All ( 'not' 'not' All ( 'not' 'not' All ( 'not' 'not' 'not' 'not' All ( 'not' 'not' 'not' 'not' All ( 'not' 'not' 'not' All ( 'not' 'not' All ( 'not' 'not' All ( 'not' 'not' All ( 'not' 'not' 'not' All ( 'not' 'not' 'not' 'not' All ( 'not' 'not' 'not' All ( 'not' 'not' 'not' All ( 'not' 'not' All ( 'not' 'not' All ( 'not' 'not' 'not' All ( 'not' 'not' 'not' 'not' All ( 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' All
len ( ( \sum ( { b } ^ <* a *> ) ) + ( len ( ( b ^ <* b *> ) ) ) ) ) = len ( ( b ^ <* a *> ) ) .= len ( ( b ^ <* b *> ) + ( b ^ <* b *> ) .= len ( ( b ^ <* a *> ) + ( b ^ <* b *> ) .= len ( b ^ <* b *> ) + ( b ^ <* b *> ) .= len ( b ^ <* a *> ) + ( b ^ <* b *> ) .= len ( b ^ <* b *> ) + ( b ^ <* b *> ) .=
consider z being Element of L such that z >= x & for y being Element of L st y in X & y >= x holds y >= x ;
LSeg ( |[ - ( - ( |[ `1 / 2 , 0 ]| ) , 0 ]| ) , |[ 1 , 0 ]| ) = |[ 0 , 1 ]| ;
lim ( ( ( ( ( ( ( ( ( ( ( f `| N ) `| N ) ) `| N ) `| N ) `| N ) ) `| N ) `| N ) `| N ) ) `| N = lim ( ( ( ( ( ( ( ( f `| N ) `| N ) `| N ) `| N ) `| N ) ) ) `| N ;
P [ i , i ] in dom ( ( ( ( f * ( i , j ) ) * ( i , j ) ) * ( i , j ) ) * ( i , j ) ) ) ;
for r be Real st r < r holds ||. ( seq . m - seq . m ) - ( seq . m ) - ( seq . m ) .|| < r ;
let X be set , a be set , b be Element of X ;
Z c= dom ( ( ( ( ( ( ( ( ( ( ( - ( ( ( ( ( ( ( - ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) (#) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) (#) ( ( ( ( ( ( ( ( ( ( ( ) ) ) (#) ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
ex j being Nat st j in dom ( l ^ <* x *> ) & j + 1 = i + j & j + 1 = i + j ;
for u , v being VECTOR of V st 0 < r & v in M holds u + v in M
A , A , B , C , D , E , F , J , J , E , F , J , J , E , F , J , J , E , F , J , J , E , F , J , J , E , F , J , F , J , E , F , J , F , J , E , F , J , J , E , F , J , E , F , J , F , J , E , F , J , J , F , J , J , J , J , E , F , J , J , J , E , F , J , J , E
- v = - ( v - u ) .= - ( v - u ) .= - ( v - u ) .= - ( v - u ) .= - ( v - u ) .= - ( v - u ) .= - ( v - u ) .= - ( v - u ) .= - ( v - u ) .= - ( v - u ) ;
( Exec ( a , k1 , k2 ) ) . IC SCM+FSA = succ IC SCM+FSA .= succ IC SCM+FSA .= succ IC SCM+FSA .= succ IC SCM+FSA .= succ IC SCM+FSA ;
consider h being Function such that f . a = h . ( ( the carrier of J ) . a ) and for x being element st x in dom ( the InternalRel of J ) . x holds h . x = ( the InternalRel of J ) . x ;
let S be non empty ManySortedSign , f be Function of the carrier of S , the carrier of S ;
card X = card X & for x being set st x in X holds x in X & x in X & x in X & x in X & x in X & x in X
L~ Cage ( C , n ) = rng Cage ( C , n ) ;
let T be tree , p , q be FinSequence of the carrier of T , r be Real ;
[ i2 , j2 ] in Indices G & [ i2 , j2 ] in Indices G ;
cluster the functor of k -> natural ;
dom F = the carrier of X & for x being Element of X holds ( F " ) . x = ( F " ) . x ;
consider C being finite finite finite finite finite finite finite finite finite finite finite finite finite finite finite finite finite Subset of V such that card C = card C and card C = card C and card C = card C + card C and card C = card C + card C + card C ;
V is prime implies the topology of X = the topology of X & V c= the topology of Y
set X = { { v where v is Element of V : v in A & v in A & v in A } ;
angle ( p1 , p2 , p3 ) = 0 * PI * PI + 0 * PI * PI * PI * PI * PI , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
|. ( - ( ( ( ( ( ( ( |. q .| / |. q .| ) .| ) .| ) ) ^2 ) ^2 ) .| = |. ( ( |. q .| ) .| ) ^2 ) .| ^2 .= |. ( |. q .| ) ^2 .| ^2 .= |. ( |. q .| ) ^2 .| ^2 .= |. q .| ^2 .| ^2 .= |. q .| ^2 ;
ex f being Function of the carrier of TOP-REAL 2 st f = f & for x being Point of TOP-REAL 2 st x in P & x in P holds f . x = f . x ;
attr f is partial differentiable means : Def5 : for u , v being Point of REAL st u in dom ( SVF1 ( 2 , 2 , 2 ) ) holds u in dom ( SVF1 ( 2 , 2 , 2 ) ) ;
ex r st x = r & for n st n >= m & n >= m holds |. ( G . n ) . ( G . n ) - ( G . n ) .| < r ;
attr f is special means : Def5 : for n , m being Nat st n <= m & m <= len G holds ( ( G * ( i , j ) ) `1 <= ( G * ( i , j ) `1 ) `1 ;
attr i in dom ( r * reproj ( i , n ) ) ;
consider c1 , c2 being bag of o1 , c2 being bag of o2 , c being Function of o1 , o2 such that ( ( ( ) + 1 ) * ( c , c2 ) = c1 * ( c , c2 ) and ( c * c2 ) = c2 * c2 ;
u in { r where r is Real : 0 <= r & r <= 1 & r <= 1 & s <= 1 & 0 <= 1 & r <= 1 & s <= 1 & 1 <= 1 & r <= 1 & r <= 1 & s <= 1 & 1 <= 1 & r <= 1 & 1 <= 1 & 1 <= 1 & r <= 1 & 1 <= 1 & r <= 1 & 1 <= 1 & 1 <= 1 & 1 <= 1 & 1 <= 1 & 1 <= 1 & 1 <= 1 & 1 <= 1 & 1 <= 1 & 1 <= 1 & 1 <= 1 & 1 <= 1 & 1 <= 1 & 1 <= 1 & 1
card X \/ Y = ( the carrier of X ) \/ Y .= ( the carrier of Y ) \/ Y .= the carrier of Y ;
attr len M1 = len M2 & width M2 = width M2 & width M2 = width M2 & width M2 = width M2 & width M2 = width M2 & width M2 = width M2 & width M2 = width M2 & width M2 = width M2 & width M2 = width M2 ;
consider g2 being Real such that 0 < g2 and g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 in { x0 where x0 is Point of TOP-REAL 2 : x0 - g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < x0 & g2 < x0 & g2 < x0 & g2 < x0 & g2 < x0 & g2 <
assume x < - sqrt ( a * b ) & x + ( a * b ) > 0 ;
( H1 '&' ( 3 '&' i ) ) '&' ( ( H1 '&' ( i , j ) ) '&' ( ( H1 '&' ( i , j ) ) '&' ( ( H1 '&' ( i , j ) ) '&' ( ( i , j ) ) '&' ( ( i , j ) '&' ( ( i , j ) '&' ( ( i , j ) '&' ( ( i , j ) '&' ( i , j ) ) ) ) ) = ( ( H1 '&' ( i , j ) ) '&' ( ( ( i , j ) '&' ( i , j ) ) ) '&' ( ( i , j ) ) ) '&' ( ( ( i , j ) '&' (
for i , j being Nat st i in dom ( M + ( i + 1 ) ) & j in dom ( M + ( i + 1 ) ) holds ( M + ( i + 1 ) ) . j = ( M + ( i + 1 ) ) . j
let f be FinSequence of NAT , i be Element of NAT ;
assume F = { [ a , b ] where a is set : a in { b } & b in X } ;
2 * ( ( ( ( |. p2 .| - sn ) ) .| ) ^2 + ( |. p2 .| ) ^2 ) ^2 = ( |. p2 .| ) ^2 + ( |. p2 .| ) ^2 ) ^2 ;
card F = card { D where D is Subset of T : D in F & D c= F & D c= F & D c= F & D c= F & D c= F & D c= F ;
attr attr attr attr attr attr : : : for IT , s , r , s being Real holds ( for n , m being Nat st n <= m holds ( for n be Nat st n >= m holds ||. ( ( s . n ) - ( s . m ) ) - ( s . n ) .|| < r ;
dom ( ( ( proj2 " ) | D ) ) = ( the carrier of ( TOP-REAL 2 ) | D ) .= ( the carrier of ( TOP-REAL 2 ) | D ) .= ( the carrier of ( TOP-REAL 2 ) | D ) | D .= ( the carrier of 2 ) | D .= ( ( TOP-REAL 2 ) | D ) | D .= ( ( TOP-REAL 2 ) | D ) | D .= ( ( TOP-REAL 2 ) | D ) | D ;
[ X \to Z , Y ] is full & X is full
( G * ( i , j ) ) * ( i , j ) = G * ( i , j ) ;
attr m c= means : for n , m being set st n in dom ( m + 1 ) holds ( for n , m being Nat st n <= m holds ( for n st n <= m holds ( ( m + 1 ) mod ( m + 1 ) ) mod ( m + 1 ) ) mod ( m + 1 ) = ( m + 1 ) mod ( m + 1 ) ;
consider a being Element of ( bool the carrier of A ) . i such that x = [ a , b ] and a in A and b in B ;
redefine func strict for non empty additive loop (# carrier -> non empty additive loop (# carrier , carrier -> set , carrier -> set , carrier , carrier -> set , carrier , carrier -> set , carrier -> set , -> set , carrier , carrier -> set , -> set , -> set , -> set , -> set , -> set , -> set , -> set , -> set , -> set , -> set , -> set , -> set , -> set , -> set , -> set , -> set , -> set , -> set , -> set , -> set , -> set , -> set , -> set , -> set , -> set , -> set , -> set , -> set , -> set , -> set , -> set , -> set , -> set , -> set , -> set , -> set
from ( a , b ) * ( c , d ) = b * ( c , d ) .= b * ( c , d ) .= b * ( c , d ) .= b * ( d , d ) .= b * ( d , d ) .= b * ( d , d ) .= b * ( d , d ) .= b * ( d , d ) .= b * ( d , d ) .= d * ( d , d ) .= d * ( d , d ) .= d * ( d , d ) .= d * ( d , d ) .= d * ( d , d ) .= d * ( d , d ) .= d * ( d , d ) .= d * ( d , d ) .= d * ( d , d ) .= d
cluster the functor of ( i , j ) -> Element of NAT ;
1- ( 1 - ( ( 1 - ( ( ( ( ( ( ( ( p / |. p .| ) ) ) ) ) ) ) ) ) ) ) ) ^2 = ( 1 - ( ( ( p `2 / |. p .| ) .| ) ^2 ) ^2 ) ^2 ) ^2 ;
eval ( a , 0. L ) = eval ( a , 0. L ) .= eval ( a , 0. L ) * eval ( a , 0. L ) .= eval ( a , 0. L ) * eval ( a , 0. L ) .= eval ( a , 0. L ) * eval ( a , 0. L ) .= eval ( a , 0. L ) * eval ( a , 0. L ) .= eval ( a , 0. L ) * eval ( a , 0. L ) ;
assume the TopStruct of ( S , T ) = the TopStruct of ( S , T ) ;
attr 1 <= k & k <= len ( w + ( len w ) - ( k + 1 ) ) - ( k + 1 ) ) ;
2 * ( a + b ) + ( 2 * ( a + b ) ) + ( 2 * ( a + b ) ) + ( 2 * ( a + b ) ) ) + ( 2 * ( a + b ) ) ) + ( 2 * ( a + b ) ) ) + ( 2 * ( a + b ) ) ) + ( 2 * ( a + b ) ) ) + ( 2 * ( b + b ) ) ) ) + ( 2 * ( a * b ) ) + ( 2 * ( a * b ) ) + ( 2 * ( b + b ) ) + ( 2 * ( b * b ) ) + ( 2 * ( b * b ) ) >= 2 * ( a * b ) + ( 2 * b ) ) + (
M , { v , w } |= All ( x , y , z ) '&' All ( y , z ) '&' All ( x , y , z ) '&' All ( x , y , z ) '&' All ( x , y , z ) '&' All ( y , z ) '&' All ( x , y , All ( x , y , All ( x , z ) ) '&' All ( y , z ) '&' All ( y , z ) '&' All ( y , z ) '&' All ( y , z ) '&' All ( x , z ) '&' All ( y , z ) '&' All ( x , z ) '&' All ( y , z ) '&' All ( y , z ) '&' All ( y , z ) '&' All ( y , z ) '&' All ( y , z )
assume that f is_differentiable_on Z and for n st n in dom f & n < m holds |. f . n - f . n .| < r ;
let G be _Graph , W be Walk of G , v be Vertex of G ;
deffunc IT [ non empty set ] means $1 is non empty & $1 is non empty & $2 is not empty & $1 is not empty & $2 is not empty & $2 is not empty ;
the carrier of ( the carrier of ( TOP-REAL 2 ) | ( the carrier of 2 ) ) = the carrier of ( 2 ) ;
let H1 , H2 be strict Subgroup of G , a , b be Element of G ;
UsedIntLoc ( IExec ( i , intloc 0 , 1 ) , 1 ) = { intloc 0 } ;
for f1 , f2 being FinSequence , f1 , f2 being FinSequence of the carrier of G st len f1 = len ( f1 ^ f2 ) & len ( f1 ^ ( f1 ^ f2 ) ) = len ( f1 ^ ( ( f1 ^ f2 ) ) ) & len ( ( f1 ^ f2 ) ^ ( ( f1 ^ f2 ) ^ ( ( f1 ^ f2 ) ) ) ) = len ( f1 ^ ( ( f1 ^ f2 ) ^ ( ( f1 ^ f2 ) ) ) ;
sqrt ( ( p + q ) ^2 ) = ( |. p .| ^2 ) ^2 + ( |. q .| ) ^2 ;
let x1 , x2 be Element of REAL ;
for x st x in dom ( - ( sin `| A ) `| A ) holds ( ( sin `| A ) `| A ) `| A is bounded
let P be topology of T ;
( a 'or' b ) '&' ( a 'or' b ) = ( a 'or' b ) '&' ( a 'or' b ) .= ( a 'or' b ) '&' ( a 'or' b ) ) '&' ( a 'or' b ) .= ( a 'or' b ) '&' ( a 'or' c ) .= ( a 'or' b ) '&' ( a 'or' c ) .= ( a 'or' c ) '&' ( a 'or' c ) ) '&' ( a 'or' c ) .= ( a 'or' b ) '&' ( a 'or' c ) ) '&' ( a 'or' c ) .= ( a 'or' c ) '&' ( a 'or' c ) .= ( a 'or' c ) '&' ( a 'or' c ) '&' ( a 'or' c ) '&' ( a 'or' c ) '&' ( a 'or' c ) '&' ( a 'or' c ) .= ( a 'or'
for e being set , A being set , B being Subset of [: { X , Y } , { X } :] st A = { X where Y is Subset of Y : X <= Y & Y <= Y } holds A is closed
for i be set st i in dom ( ( the Sorts of U1 ) * ( i , j ) ) holds ( ( the Sorts of U1 ) * ( i , j ) ) . i = ( ( the Sorts of U1 ) . i ) . i
for v , w being Element of V st v <> w & w = v + w holds ( for u , v being Element of V st u in v holds u + v = v + w
card ( card { D } + 1 ) = card ( { D } + 1 ) + 1 ) .= card ( { D } + 1 ) + 1 + 1 .= ( { D } + 1 ) + 1 ) + 1 .= ( { D } + { D } ) + ( { D } + 1 ) + 1 ) ;
IC Comput ( P3 , s3 , 0 ) = ( P3 +* I ) . IC SCM+FSA .= ( ( P3 , s3 ) . IC SCM+FSA .= ( ( P3 , s3 ) . IC SCM+FSA ) .= ( ( ( P3 , s3 ) . IC SCM+FSA ) .= ( ( ( ( ( ( ( s , s3 ) . IC SCM+FSA ) ) ) . IC SCM+FSA .= ( ( ( ( s , s3 ) . IC SCM+FSA ) ) ) ) . IC SCM+FSA .= ( ( ( s , 0 ) ) . IC SCM+FSA ) . IC SCM+FSA ) . IC SCM+FSA .= ( ( ( s , 0 ) . IC SCM+FSA ) . IC SCM+FSA ) . IC SCM+FSA ) . IC SCM+FSA ) . IC SCM+FSA .= ( ( ( s , 0 ) . IC SCM+FSA ) . IC SCM+FSA
len ( f /^ ( i + 1 ) ) = len ( f /^ ( i + 1 ) ) .= len ( f | ( i + 1 ) ) .= ( f | ( i + 1 ) ) + ( f | ( i + 1 ) ) .= ( f | ( i + 1 ) ) .= ( f | ( i + 1 ) ) ;
for a , b being Element of NAT st a <= b & b <= len a holds a <= b + b
let f be FinSequence of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
lim ( ( Partial_Sums ( ( seq1 # # ( \alpha ) ) # \alpha ) ) . k ) = ( Partial_Sums ( seq1 # ( \alpha ) ) . k ) . k + ( ( seq1 # x ) # x ) . k .= ( ( ( seq1 # x ) # x ) . k + ( ( seq1 # x ) # x ) . k ;
deffunc z1 = g /. ( i + 1 ) .= g /. ( i + 1 ) .= g /. ( i + 1 ) .= g /. ( i + 1 ) .= g /. ( i + 1 ) .= g /. ( i + 1 ) .= g /. ( i + 1 ) .= g /. ( i + 1 ) .= g /. ( i + 1 ) .= g /. ( i + 1 ) .= g /. ( i + 1 ) .= g /. ( i + 1 ) .= g /. ( i + 1 ) .= g /. ( i + 1 ) .= g /. ( i + 1 ) .= g /. ( i + 1 ) .= g /. ( i + 1 ) .= g /. ( i + 1 ) .= g /. ( i + 1 ) .= g /. ( i + 1 ) .= g /. ( i + 1 ) .= g /. ( i + 1 ) .= g /. ( i + 1 ) .= g /.
[ f . 0 , f . 1 ] in the InternalRel of G or [ f . 0 , f . 1 ] in the InternalRel of G ;
let G be bool bool bool the carrier of G ;
CurInstr ( P1 , Comput ( P1 , s1 , m ) ) = CurInstr ( P1 , s1 , m ) .= CurInstr ( P1 , s1 , m ) .= i ;
assume that a on M and b on M and a on M and a , b // M ;
attr T is second-countable means : Def2 : for F being family of T st F is closed & for A being Subset of T holds A is open & A in F & A c= F implies A is open & A is open & A is open & A is open & A is open & A is open & A is open & A is open & A is open & A is open & A is open & A is open & A is open & A is open & A is open & A is open & A is open ;
for g1 , g2 being Real st g1 in ]. g1 , g2 .[ & g2 < g1 & g1 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g1 holds g1 <= g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 < g2 & g2 <
cosh ( z , u ) = exp_R . ( z , u ) .= 0 ;
F . i = ( F . i ) . ( i + 1 ) .= ( F . i ) . ( i + 1 ) .= ( F . i ) . ( i + 1 ) .= ( F . i ) . ( i + 1 ) .= ( F . i ) . ( i + 1 ) .= ( F . ( i + 1 ) ) . ( i + 1 ) .= ( F . i ) . ( i + 1 ) .= ( F . ( i + 1 ) .= ( F . ( i + 1 ) .= ( F . ( i + 1 ) .= ( F . ( i + 1 ) .= ( F . ( i + 1 ) .= ( F . ( i + 1 ) . ( i + 1 ) .= ( F . ( i + 1 ) .= ( F . ( i + 1 ) . ( i + 1
ex y being set st y = f . y & for n being Nat st n in dom f & n <= m holds |. f . n - f . n .| < r ;
func f * F -> FinSequence of the carrier of V , the carrier of V ;
not x1 in { x1 , x2 } \/ { x1 , x2 } ;
for n being Nat , x being set , y being set st x in y & y in S & x in S & y in S & y in S & x in S & y in S & y in S & y in S & x in S & y in S & y in S & y in S & y in S & y in S ;
ex n1 being Element of NAT st ( ( ( |. n1 .| ) | n1 ) ) . n1 = |. n1 .| & n1 <= len ( |. n1 .| ) & n1 <= len ( |. n1 .| ) ;
consider P being FinSequence of the carrier of [: { 2 } , the carrier of G :] such that P = \prod ( P ^ <* <* 1 *> ) and for i being Element of NAT st i in dom ( P ^ <* 1 *> ) holds P [ i , j ] ;
let T1 be non empty TopSpace , A be Subset of T ;
attr f is partial differentiable means : Def5 : for r st r in dom ( f * ( f , 2 ) ) holds ( ( f * ( f , 2 ) ) (#) ( f * ( ( f , 2 ) ) (#) ( f , 2 ) ) ) (#) ( ( f * ( f , 2 ) ) ) (#) ( f (#) ( f , 2 ) ) ) = ( ( f (#) ( f (#) ( f , 2 ) ) (#) ( f (#) ( f , 2 ) ) ) (#) ( ( ( f , 2 ) (#) ( f , 2 ) ) (#) ( ( f (#) ( f , 2 ) ) ;
defpred P [ Nat ] means for n being Nat st n <= $1 holds ( for m be Nat st n <= m holds ( for n be Nat st n <= m holds |. ( ( seq . m ) . n ) - ( ( seq . n ) . m ) - ( ( seq . n ) . m ) ) .| < r ;
ex j st j < width G & ( ( GoB f ) * ( i , j ) + ( 1 , j ) `2 ) `2 & ( ( GoB f ) * ( i , j ) `2 = ( ( GoB f ) * ( i , j ) `2 ) `2 ;
defpred U ( set , set , set ) = union ( $1 , $1 , $1 ) ;
for p1 , p2 being Point of TOP-REAL 2 st p1 = p2 & p2 = p1 & p1 `2 >= p2 `2 holds p2 `2 >= p3 `2
f in St & for n holds f . n = [ n , f . n ] ;
ex p1 being Point of TOP-REAL 2 st x = p1 & |. p1 .| >= sn & |. p1 .| >= sn & |. p1 .| >= sn & |. p1 .| >= sn & |. p1 .| >= sn & |. p1 .| >= sn & |. p1 .| >= sn & |. p1 .| >= sn & |. p1 .| >= sn & |. p1 .| >= sn & |. p1 .| >= sn & |. p1 .| >= sn & |. p1 .| >= sn & |. p1 .| >= sn & |. p1 .| >= sn & |. p1 .| >= sn & |. p1 .| >= sn & |. p1 .| >= sn & |. p1 .| >= sn & |. p1 .| >= sn & |. p1 .| >= sn & |. p1 .| >= sn & |. p1 .| >= sn & p2 .| >= sn & |. p1 .| >= sn & |. p1 .| >= sn & |. p1 .| >= sn & |. p1 .| >= sn & |. p1 .| >= sn & |. p1 .| >= sn & |.
assume for d1 , d2 being Element of NAT st d1 <= d2 & d2 <= d1 & d1 <= d2 holds d1 <= d2 + d2 * d2
assume that s <> t and for t being Point of TOP-REAL 2 st t <> 0 holds Ball ( t , r ) c= Ball ( t , r ) ;
consider r such that 0 < r and for n st n in dom ( f /* s1 ) & ||. ( f /* s1 ) . n - f /. n .|| < r ;
( p | ( ( p | ( Seg n ) ) ) ) | ( Seg n ) ) = p | ( Seg n ) ;
attr x + h + h in dom ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( - ( ( ( ( ( ( ( - ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( , , , , , , , , , , .| ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ;
assume that i in dom A and i in dom A and j in width A and i + 1 <= width A and i + 1 <= len A and j + 1 <= width A ;
for i being Element of NAT st i in Seg n & i in dom ( <* c *> ^ <* d *> ) holds ( ( ( <* d *> ^ <* d *> ) ^ <* d *> ) ^ <* d *> ) ^ ( <* d *> ) ^ ( <* d *> ^ <* d *> ) = ( ( ( <* d *> ^ <* d *> ^ <* d *> ) ^ <* d *> ) ^ <* d *> ) ^ ( <* d *> )
( ( b '&' c ) '&' ( a '&' c ) ) '&' ( b '&' c ) '&' ( b '&' c ) '&' ( b '&' c ) ) '&' ( b '&' c ) '&' ( b '&' c ) '&' ( b '&' c ) ) '&' ( b '&' c ) '&' ( b '&' c ) '&' ( b '&' c ) '&' ( b '&' c ) '&' ( b '&' c ) '&' ( b '&' c ) ) '&' ( b '&' c ) ) '&' ( b '&' c ) ) '&' ( b '&' ( b '&' c ) ) '&' ( b '&' c ) ) '&' ( b '&' ( b '&' c ) ) '&' ( b '&' c ) ) '&' ( b '&' ( b '&' c ) ) '&' ( b '&' c ) '&' ( b '&' c ) ) '&' ( b '&' c ) '&' ( b '&' ( b '&' ( b '&' c ) '&' ( b '&' c ) '&' ( b '&' ( b '&' c ) '&' (
attr x , y means : means : for x0 st x0 in dom ( ( for x st x in N holds ( ( for x st x in N holds ( ( f `| N ) `| N ) . x0 - ( f `| N ) . x0 ) = ( f `| N ) . x0 ;
consider R2 , R1 such that RR1 = RR1 + RR1 * ( RR1 , RR1 ) + RR1 * ( RR1 , RR1 ) and RR1 = RR1 * ( RR1 , RR1 ) ;
ex k be Element of NAT st q = k & for q be Element of REAL st q in X & |. q - ( k - q ) .| < r holds |. q - q .| < r
x in { x1 , x2 } \/ { x2 , x3 , x4 } & x in { x1 , x2 , x3 , x4 } ;
( G * ( j , 1 ) ) `1 = G * ( i , j ) `1 .= G * ( i , j ) `1 .= G * ( i , j ) `1 .= G * ( i , j ) `1 .= G * ( i , j ) `1 .= G * ( i , j ) `1 .= G * ( i , j ) `1 .= G * ( i , j ) `1 .= G * ( i , j ) `1 .= G * ( i , j ) `1 .= G * ( i , j ) `1 .= G * ( i , j ) `1 .= G * ( i , j ) `1 .= G * ( i , j ) `1 .= G * ( i , j ) `1 .= G * ( i , j ) `1 .= G * ( i , j ) `1 .= G * ( i , j ) `1 .= G * ( i , j ) `1 .= G * ( i , j ) `1 .= G * ( i
f1 * ( the Arity of S ) . o = ( the Arity of S ) . o .= ( the connectives of S ) . o .= ( the connectives of S ) . o .= ( the connectives of S ) . o ;
func tree ( T , p ) -> FinSequence of the carrier of T , the carrier of T , the carrier of T , the carrier of T , the carrier of T , the carrier of T ;
F . ( k + 1 ) = FF1 . ( k + 1 ) .= FF1 . ( k + 1 ) .= FF1 . ( k + 1 ) .= FF1 . ( k + 1 ) .= FF1 . ( k + 1 ) .= FF1 . ( k + 1 ) .= FF1 . ( k + 1 ) .= FF1 . ( k + 1 ) .= FF1 . ( k + 1 ) .= FF1 . ( k + 1 ) .= FF1 . ( k + 1 ) .= FF1 . ( k + 1 ) .= FF1 . ( k + 1 ) .= FF1 . ( k + 1 ) .= FF1 . ( k + 1 ) .= FF1 . ( k + 1 ) .= FF1 . ( k + 1 ) .= FF1 . ( k + 1 ) .= FF1 . ( k + 1 ) .= FF1 . ( k + 1 ) .= FF1 . ( k + 1
let A be Matrix of len A , REAL ;
( Partial_Sums ( seq + k ) ) . n = ( Partial_Sums ( seq + k ) ) . n .= ( Partial_Sums ( seq + k ) ) . n .= ( ( Partial_Sums ( seq + k ) ) . n ) . n .= ( ( Partial_Sums ( seq + k ) ) . n .= ( ( Partial_Sums ( seq + k ) ) . n ) . n .= ( ( ( seq + k ) ) . n ) . n .= ( ( ( ( seq + k ) ) . n ) . n .= ( ( ( seq + k ) ) . n .= ( ( ( seq + k ) . n ) . n .= ( ( ( seq + k ) . n ) . n .= ( ( ( seq + k ) ) . n ) . n ) . n .= ( ( ( seq + k ) . n ) . n .= ( ( ( seq + k ) . n ) . n .= ( ( ( ( seq + k ) . n ) . n .=
assume that x in the carrier of C and y in the carrier of C and x in the carrier of C and y in the carrier of C and y in the carrier of C and y in the carrier of C ;
defpred P [ Nat ] means ( for k being Element of NAT st k < $1 holds ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( $1 , /. $1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . $1 ) ) ) ) . k ) ) . k ) ) . ( = ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) * ( ( ( ( ( ( ( ( ) ) ) ) * ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) * ( ( ( ( ( (
assume that 1 <= k and k + 1 <= len G and k + 1 <= width G and 1 <= width G and 1 <= width G and 1 <= width G and 1 <= width G and 1 <= width G and 1 <= width G and 1 <= width G and k + 1 <= width G and G * ( 1 , 1 ) `1 = G * ( 1 , 1 ) `1 and G * ( 1 , 1 + 1 ) `1 = G * ( 1 , 1 ) `1 and G * ( 1 , 1 ) `1 and G * ( 1 , 1 ) `1 and G * ( 1 , 1 ) `1 and G * ( 1 , 1 ) `1 and G * ( 1 , 1 ) `1 and G * ( 1 , 1 ) `1 = G * ( 1 , 1 , 1 , 1 , 1 ) `1 and G * ( 1 , 1 ) `1 and G * ( 1 , 1 ) `1 and G * ( 1 , 1 , 1 ) `1 and
attr that for s st s < 0 & s < 1 holds |. ( ( ( s - s ) * ( 1 - s ) ) ) ^2 < r ;
let M be non empty MetrSpace , f be Function of M , M ;
defpred P [ Element of \omega ] means $1 in Z & for n st n in Z holds ( ( ( for n st n in Z holds ( ( for n st n in Z holds f1 . n ) . n = ( f1 . n ) . n ) * ( f1 . n ) ) * ( f1 . n ) ) * ( f1 . n ) ) ) * ( f1 . n ) ) * ( f1 . n ) ) ) * ( f1 . n ) ) * ( f1 . n ) * ( f1 . n ) ) * ( f1 . n ) ) * ( f1 . n ) ) = f1 . n ;
defpred P [ Nat ] means $1 < ( |. ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) - ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . $1 ;
( f ^ <* ( f ^ <* i *> ) *> ) /. ( i + 1 ) = ( f ^ <* i *> ) /. ( i + 1 ) .= ( f ^ <* i + 1 *> ) /. ( i + 1 ) .= ( f ^ <* i + 1 *> ) /. ( i + 1 ) .= ( f ^ <* i + 1 *> ) /. ( i + 1 ) .= ( f ^ <* i + 1 *> ) /. ( i + 1 ) .= ( f /. ( i + 1 ) .= ( f /. ( i + 1 ) .= ( f /. ( i + 1 ) .= ( f /. ( i + 1 ) .= ( f /. ( i + 1 ) .= ( f /. ( i + 1 ) .= ( f /. ( i + 1 ) .= ( f /. ( i + 1 ) /. ( i + 1 ) ;
\frac { 1 + ( 2 * ( 2 * ( 2 * ( 2 * ( 1 + 1 ) ) ) ) ) where 2 is Element of NAT : 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( ( ( ( ( ( ( ( ( ( ( ( ( 1 * ( 2 * ( 2 * ( 2 * ( 2 * ( ( 1 ) ( 1 ) ) )
defpred P [ Nat ] means $1 in the carrier of G & $1 in the carrier of G & $1 in the carrier of G & $1 in the carrier of G ;
assume that 1 <= i and i <= len Cage ( C , n ) and i <= len Cage ( C , n ) and 1 <= i and i + 1 <= len Cage ( C , n ) ;
defpred P [ Element of NAT ] means ( ( for n holds ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . $1 ;
for x being Element of product F st x in dom ( F . i ) holds ( for i be Element of NAT st i in dom ( F . i ) holds ( F . i ) . i = ( F . i ) . i
( x " ) * ( x " ) = ( x " ) * ( x " ) .= x " * ( x " ) .= x " * ( x " ) .= x * ( x " ) .= x * ( x " ;
DataPart Comput ( P3 , s3 , LifeSpan ( P3 , s3 ) + 1 ) = DataPart Comput ( P3 , s3 , 1 ) .= DataPart ( P3 , s3 ) ;
consider r such that 0 < r and for g st g in dom ( f `| X ) holds |. ( f `| X ) . g - ( f `| X ) . g .| < r ;
attr X c= dom ( f1 (#) f2 ) & for x0 be Real st x0 in X holds ( f1 (#) f2 ) . x0 = ( f1 (#) f2 ) . x0 holds ( f1 (#) ( f1 (#) f2 ) . x0 = ( f1 (#) ( f2 (#) g2 ) . x0 ) * ( ( f2 (#) g2 ) . x0 ) * ( ( f2 (#) g2 ) . x0 ) * ( ( f2 (#) g2 ) . x0 ) * ( ( f2 (#) g2 ) . x0 ) * ( ( ( f2 (#) g2 ) . x0 ) * ( ( ( (#) g2 ) . x0 ) * ( ( ( ( ( ( ( ( ( ( g2 (#) g2 ) . x0 ) (#) ( ( ( ( ( ( ( ( ( ( ( ( (#) g2 ) (#) g2 ) . x0 ) ) * ( ( ( ( g2 (#) g2 ) . x0 ) ) ) * ( ( ( g2 (#) g2 ) . x0 ) . x0 ) ) * ( ( ( ( g2 ) (#) g2 ) . x0 ) * ( ( ( ( ( ( g1 (#) g2 ) . x0 ) (#) g2 ) . x0 )
let l be continuous continuous continuous Function of L , L ;
consider i being Element of NAT such that i in dom A and i = m . i and p . i = m . i ;
( ( f1 - f2 ) /* ( s + c ) ) . n = ( f1 /* ( s + c ) ) . n .= ( f1 /* ( s + c ) ) . n .= ( f1 /* ( s + c ) ) . n .= ( f1 /* ( s + c ) ) . n .= ( f1 /* ( s + c ) ) . n .= ( f1 /* ( s + c ) ) . n .= ( f1 /* ( s + c ) ) . n .= ( f1 /* ( ( f1 + c ) ) . n .= ( f1 /* ( s + c ) . n .= ( f1 /* ( ( ( s + c ) ) . n .= ( f1 /* ( ( f2 /* c ) . n ) .= ( f1 /* ( s + c ) ) . n .= ( ( ( s + c ) . n .= ( f1 /* ( s /* c ) . n .= ( f1 /* c ) . n .= ( ( f1 /* c ) . n .= ( f1 /* c ) . n .= ( ( f2 /* c ) . n ) . n .=
ex p1 being Element of QC-WFF ( A ) st ( ( ( ( ( ( ( ( ( ( ( ( ( ( the_arity_of o ) . n ) ) . ( i + 1 ) ) ) . i ) ) . i ) ) . i = p1 . ( i + 1 ) ) . i ;
( mid ( f , i , j ) ) . ( i + 1 ) = ( ( ( mid ( f , i , j ) ) ) . ( i + 1 ) .= ( ( ( f , i2 ) ) . ( i + 1 ) ) . ( i + 1 ) .= ( f | ( i + 1 ) ) . ( i + 1 ) .= ( f | ( i + 1 ) ) . ( i + 1 ) .= ( f | ( i + 1 ) ) . ( i + 1 ) .= ( f | ( i + 1 ) ) . ( i + 1 ) .= ( f | ( i + 1 ) . ( i + 1 ) . ( i + 1 ) . ( i + 1 ) ) . ( i + 1 ) .= ( f /. ( i + 1 ) . ( i + 1 ) . ( i + 1 ) . ( i + 1 ) .= ( f /. ( i + 1 ) . ( i + 1 ) .= ( f /. ( i + 1 ) . ( i + 1 ) . ( i + 1 ) . ( i + 1 ) .= ( f /. ( i + 1
( p ^ q ) . k = ( p ^ q ) . k .= ( p ^ q ) . k .= ( ( p ^ q ) ^ r ) . k .= ( ( p ^ r ) ^ r ) . k .= ( ( ( ( ( ( ( ( len r ) ^ r ) ^ r ) ^ r ) . k .= ( ( ( ( len r ) ^ r ) ^ r ) ^ r ) . k ;
len ( D2 + ( len D2 ) - 1 ) = len ( D2 + 1 ) - ( len D2 - 1 ) ;
x * y = x * y .= x * y .= x * y .= y * y ;
v . ( ( \langle \underbrace ( 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
i * i = 0 * ( i - 1 ) .= 0 ;
Sum ( L * F ) = 0. V .= Sum ( L * F ) .= 0. V * F .= 0. V * F . n .= 0. V * F . n .= 0. V * F . n .= 0. V * F . n .= 0. V * F . n .= 0. V * F . n .= 0. V * F . n .= 0. V * F . n .= 0. V ;
ex r be Real st ( for r be Real st r in X holds |. ( ( ( ( ( ( ( ( ( ( ( ( ( - r ) / 2 ) ) ) ) ) ) ) ) ) ) ) ) ^2 < r ;
( the Go-board of f ) * ( i1 , j1 + 1 ) = ( GoB f ) * ( i1 , j1 ) or ( ( GoB f ) * ( i1 , j1 ) `2 = ( GoB f ) * ( i1 , j1 ) `1 or ( ( GoB f ) * ( i1 , j1 ) `2 ) `2 = ( GoB f ) * ( i1 , j1 + 1 ) `2 or ( f /. ( i1 + 1 ) `2 = ( GoB f ) * ( i1 , j1 ) `2 or ( ( ( f /. ( i1 + 1 ) `2 ) `2 = ( f /. ( i1 + 1 ) `2 or ( ( f /. ( i1 + 1 ) `2 ) `2 or ( ( f /. ( i1 + 1 ) `2 ) `2 ) `2 ) `2 ) `2 = ( f /. ( i1 + 1 ) `2 or ( ( f /. ( i1 + 1 ) `2 or ( ( f /. ( i1 + 1 , j1 + 1 ) `2 ) `2 ) `2 ) `2 ;
( the carrier of ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | K1 ) ) ) . x = ( sn sn ) . x .= ( sn -FanMorphE ) . x .= ( sn -FanMorphE ) . x .= ( sn sn ) . x .= ( sn -FanMorphE ) . x .= ( sn sn -FanMorphE ) . x .= ( sn sn ) . x .= ( sn sn ) . x .= ( sn -FanMorphE ) . x ) ^2 .= ( sn -FanMorphE ) . x ;
sin . ( a + b ) < 0 & sin . ( a + b ) < 0 ;
attr inf X = "/\" ( X , L ) means : Def2 : for x being Element of L st x in X holds x in X & x in X & x in X ;
( ( Exec ( i , B ) ) . j = ( Exec ( i , B ) ) . j ) . j .= ( Exec ( i , B ) . j ) . j .= ( Exec ( i , B ) . j ) . j ;