thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
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thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
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contradiction ;
contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
thesis ;
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contradiction ;
thesis ;
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contradiction ;
thesis ;
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contradiction ;
thesis ;
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contradiction ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
B in A ;
a <> c
T c= S
D c= B
c in X ;
b in X ;
X in X ;
b in D ;
x = e ;
let m ;
h is onto ;
N in K ;
let i ;
j = 1 ;
x = u ;
let n ;
let k ;
y in A ;
let x ;
let x ;
m c= y ;
F is one-to-one ;
let q ;
m = 1 ;
1 < k ;
G is prime ;
b in A ;
d divides a ;
i < n ;
s <= b ;
b in B ;
let r ;
B is one-to-one ;
R is total ;
x = 2 ;
d in D ;
let c ;
let c ;
b = Y ;
0 < k ;
let b ;
let n ;
r <= b ;
x in X ;
i >= 8 ;
let n ;
let n ;
y in f ;
let n ;
1 < j ;
a in L ;
C is compact ;
a in A ;
1 < x ;
S is finite ;
u in I ;
z \ll z ;
x in V ;
r < t ;
let t ;
x c= y ;
a <= b ;
m <= n ;
assume f is prime ;
not x in Y ;
z = - \infty ;
k in NAT ;
K is being_line ;
assume n >= N ;
assume n >= N ;
assume X is \equiv ;
assume x in I ;
q is not 0 ;
assume c in x ;
C2 > 0 ;
assume x in Z ;
assume x in Z ;
1 <= km2 ;
assume m <= i ;
assume G is prime ;
assume a divides b ;
assume P is closed ;
d2 > 0 ;
assume q in A ;
W is not bounded ;
f is non one-to-one ;
assume A is dense ;
g is special ;
assume i > j ;
assume t in X ;
assume n <= m ;
assume x in W ;
assume r in X ;
assume x in A ;
assume b is odd ;
assume i in I ;
assume 1 <= k ;
X is non empty ;
assume x in X ;
assume n in M ;
assume b in X ;
assume x in A ;
assume T c= W ;
assume s is atomic ;
b `2 <= c ;
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 <= D-2 ;
let X be set ;
let X be set ;
let y be element ;
let x be element ;
P [ 0 ]
let E be set , x be element ;
let C be Category ;
let x be element ;
k in NAT ;
let x be element ;
let x be element ;
let e be element ;
let x be element ;
P [ 0 ]
let c be element ;
let y be element ;
let x be element ;
let a be Real ;
let x be element ;
let X be element ;
P [ 0 ]
let x be element ;
let x be element ;
let y be element ;
r in REAL ;
let e be element ;
n1 is S-Sequence_in_R2 ;
Q is_closed_on s , P ;
x in xy ;
M < m + 1 ;
T2 is open ;
z in b -Seg a ;
R2 is well-ordering ;
1 <= k + 1 ;
i > n + 1 ;
q1 is one-to-one ;
let x be non trivial set ;
P3 is one-to-one ;
n <= n + 2 ;
1 <= k + 1 ;
1 <= k + 1 ;
let e be Real ;
i < i + 1 ;
p3 in P ;
p1 in K ;
y in C1 ;
k + 1 <= n ;
let a be Real , n be Nat ;
X |- r => p ;
x in { A } ;
let n be Nat ;
let k be Nat ;
let k be Nat ;
let m be Nat ;
0 < 0 + k ;
f is_differentiable_in x ;
let x0 ;
let E be Ordinal ;
o <= o2 ;
O <> x3 ;
let r be Real ;
let f be FinSequence of NAT ;
let i be Nat ;
let n be Nat ;
Cl A = A ;
L c= Cl L ;
A /\ M = B ;
let V be RealUnitarySpace , W be Subset of V ;
not s in Y |^ 0 ;
rng f <= w
b "/\" e = b ;
m = m1 ;
t in h . D ;
P [ 0 ] ;
assume z = x * y ;
S . n is bounded ;
let V be RealUnitarySpace , W be Subset of V ;
P [ 1 ] ;
P [ {} ] ;
C1 is connected ;
H = G . i ;
1 <= i + 1 ;
F . m in A ;
f . o = o ;
P [ 0 ] ;
aa1 <= field W ;
R [ 0 ] ;
b in f .: X ;
assume q = q2 ;
x in [#] V ;
f . u = 0 ;
assume d1 > 0 ;
let V be RealUnitarySpace , W be Subset of V ;
s is non trivial & not thesis ;
dom c = Q ;
P [ 0 ] ;
f . n in T ;
N . j in S ;
let T be complete LATTICE , A be Subset of T ;
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 ;
vthesis < n ;
Sy is bounded ;
assume p = p2 ;
len f = n ;
assume x in P1 ;
i in dom q ;
let U ;
-25 = c ;
j in dom h ;
let k ;
f | Z is continuous ;
k in dom G ;
UBD C = B ;
1 <= len M ;
p in LSeg ( x , r ) ;
1 <= means : 1 <= it ;
set A = SCMPDS ;
card a [= c ;
e in rng f ;
cluster B \oplus A -> empty ;
H has no no \rbrace ;
assume x0 <= m ;
T is increasing ;
d2 <> d1 ;
Z c= dom g ;
dom p = X ;
H is proper of G ;
i + 1 <= n ;
v <> 0. V ;
A c= Affin A ;
S c= dom F ;
m in dom f ;
let X2 be set ;
c = sup N ;
R is connected implies R is connected
assume not x in REAL ;
Im f is complete
x in Int y ;
dom F = M ;
a in On W ;
assume e in [: A , B :] ;
C c= CN1 ;
mm <> {} ;
let x be Element of Y ;
let f be \mathclose extended Chain , n ;
not n in Seg 3 ;
assume X in f .: A ;
assume that p <= n and p <= m ;
assume not u in { v } ;
d is Element of A ;
A |^ b misses B ;
e in v .vertices() ;
- y in I ;
let A be non empty set , x , y be Element of A ;
P0 = 1 ;
assume r in F . k ;
assume f is simple ;
let A be infinite set ;
rng f c= NAT ;
assume P [ k ] ;
f3 <> {} ;
o in omega ;
assume x is sum of squares ;
assume not v in { 1 } ;
let I1 ;
assume that 1 <= j and j < l ;
v = - u ;
assume s . b > 0 ;
d1 <> d2 ;
assume t . 1 in A ;
Y is non empty TopSpace , X be Subset of Y ;
assume a in ]. s , t .[ ;
let S be non empty reflexive transitive RelStr ;
a , b // b , a ;
a * b = p * q ;
assume x , y // the carrier of V ;
assume x in \Omega ( f ) ;
[ a , c ] in X ;
F-14 <> {} ;
M + N c= M + N ;
assume M is with_hhhhhhhhhhhhhhhhhhhhhhhhhh
assume f is J with_BBby J ;
let x , y ;
let T be non empty TopSpace ;
b , a // b , c ;
k in dom Sum p ;
let v be Element of V ;
[ x , y ] in T ;
assume len p = 0 ;
assume C in rng f ;
k1 = k2 ;
m + 1 < n + 1 ;
s in S \/ { s } ;
n + i >= n + 1 ;
assume Re y = 0 ;
k1 <= j1 & j <= width G ;
f | A is continuous ;
f . x <= b ;
assume y in dom h ;
x * y in B1 ;
set X = Seg n ;
1 <= i2 + 1 ;
k + 0 <= k + 1 ;
p ^ q = p ^ q ;
j |^ y divides m ;
set m = max ( A , B ) ;
[ x , x ] in R ;
assume x in succ 0 ;
a in sup \varphi ;
Cj in dom h ;
q2 c= C1 * C2 ;
a2 < c2 ;
s2 is 0 -started ;
IC s = 0 ;
s2 = s2 ;
let V ;
let x , y ;
let x be Element of T ;
assume a in rng F ;
x in dom T ` ;
let S be Subset-Family of L ;
y " <> 0 ;
y " <> 0 ;
0. V = uw -xw ;
y2 , y1 , y2 is_collinear ;
R1 in X ;
let a , b be Real , x , y be Real ;
let a be object of C ;
let x be Vertex of G ;
let o be object of C , a , b be Object of C ;
r '&' q = P ;
let i , j be Nat ;
let s be State of A , P ;
s2 . n = N ;
set y = ( x - y ) / ( x - y ) ;
NAT in dom g ;
l . 2 = y1 ;
|. g . y .| <= r ;
f . x in Cx0 ;
V is non empty ;
let x be Element of X ;
0 <> f . g2 ;
f2 /* q is convergent ;
f . i is_measurable_on E ;
assume \varphi in NH ;
reconsider i = i as Ordinal ;
r * v = 0. X ;
rng f c= [: { 0 } , { 0 } :] ;
G = 0 .--> 0 ;
let A be Subset of X ;
assume that x0 is open and A is open ;
|. f . x - f . x .| <= r ;
x be Element of R ;
let b be Element of L ;
assume x in WT ;
P [ k , a ] ;
let X be Subset of L ;
let b be object of B ;
let A , B be category ;
set X = Vars ( C , n ) ;
let o be OperSymbol of S ;
let R be connected non empty Poset ;
n + 1 = succ n ;
{ x9 } c= Z ;
dom f = C1 ;
assume [ a , y ] in X ;
Re ( seq ) is convergent ;
assume a1 = b1 ;
A = sB ( ) ;
a <= b or b <= a ;
n + 1 in dom f ;
let F be sequence of S , n ;
assume r2 > x0 ;
Y is non empty set ;
2 * x in dom W ;
m in dom g2 ;
n in dom g1 ;
k + 1 in dom f ;
the still not bound in { s } ;
assume x1 <> x2 ;
v2 in V1 ;
[ b `1 , b `2 ] in T ;
-' 1 + 1 = i ;
T c= INT.Ring ( T ) ;
( l - 1 ) * l = 0 ;
n in NAT ;
( t `2 ) ^2 = r ;
AT is_integrable_on M ;
set t = Bottom T ;
let A , B be real-membered set ;
k <= len G + 1 ;
\ { C } misses { V } ;
product seq is non empty ;
e <= f or f <= g ;
cluster -> non trivial for FinSequence of REAL ;
assume c2 = b2 ;
assume h in [. q , p .] ;
1 + 1 <= len C ;
not c in B . m1 ;
cluster R .: X -> empty ;
p . n = H . n ;
assume that -4 is convergent and for n be Nat holds seq is convergent ;
IC s3 = 0 ;
k in N or k in K ;
F1 \/ F2 c= F ;
Int ( G1 ) <> {} ;
( z - 1 ) * ( z - 1 ) = 0 ;
p1 <> p1 & p1 <> p2 & p2 <> p3 & p3 <> p1 & p3 <> p4 & p3 <> p1 & p3 <> p4 & p3 <> p4 & p3 <> p4 & p3 <>
assume z in { y , w } ;
MaxADSet ( a ) c= F ;
sup \mathopen { \downarrow s } in S ;
f . x <= f . y ;
let T be non empty reflexive transitive RelStr ;
q1 |^ m >= 1 ;
a >= X & b >= Y ;
assume <* a , c *> <> {} ;
F . c = g . c ;
G is one-to-one & is one-to-one ;
A \/ { a } c= B ;
0. V = 0. Y ;
let I be non empty Instruction of S , s be State of S ;
f3 . x = 1 ;
assume z \ x = 0. X ;
C4 = 2 |^ n ;
let B be sequence of subsets ( X ) ;
assume X1 = p .: D ;
n + l2 in NAT ;
f " ( P ) is compact ;
assume x1 in [: { x1 } , { x2 } :] ;
p1 = K & K = ( len ( M * ( i , j ) ) ) ;
M . k = <*> REAL ;
\varphi . 0 in rng \varphi ;
MMMA is closed ;
assume x0 <> 0. L ;
n < log ( 2 , k ) ;
0 <= seq . 0 ;
- q + p = v ;
{ v } is Subset of B ;
set g = f /. 1 ;
Following R is stable Subset of R ;
set RR = Vertices R , R = \cal R , S = \cal S , T = \cal S , T = \cal S , T = \cal S , T = \cal
p0 c= P4 ;
x in [. 0 , 1 .] ;
f . y in dom F ;
let T be Scott non empty TopSpace ;
inf the carrier of S = S ;
intpos a = intpos b ;
P , C , K is_collinear ;
assume x in LSeg ( s , r ) ;
2 |^ i < 2 |^ m ;
x + z = x + z ;
x \ ( a \ x ) = x ;
||. \mathopen .|| <= r ;
assume that Y c= field Q and Y <> {} ;
a \times b = a & b = b ;
assume a in A . i ;
k in dom q9 ;
p is non empty FinSequence of S ;
i -' 1 = i-1 ;
f | A is one-to-one ;
assume x in f .: [: X , Y :] ;
i2 - ( i2 -' 1 ) = 0 ;
j2 + 1 <= i2 + 1 ;
g " * a * N in N ;
K <> { [ {} , {} ] } ;
cluster non trivial -> non empty for NAT ;
|. q .| ^2 > 0 ;
|. p3 .| = |. p .| ;
s2 - s1 > 0 ;
assume x in { G * ( -12 , 1 ) } ;
W-min C in C & W-min C in C ;
assume x in { G * ( -12 , 1 ) } ;
assume i + 1 = len G ;
assume i + 1 = len G ;
dom I = Seg n ;
assume that k in dom C and k <> i ;
1 + 1-1 <= i + 1 ;
dom S = dom F ;
let s be Element of NAT ;
let R be ManySortedSet of A ;
let n be Element of NAT ;
let S be non empty non void non empty non void ManySortedSign ;
let f be ManySortedSet of I ;
let z be Element of COMPLEX , x , y be Element of COMPLEX ;
u in { \hbox { \boldmath $ g } } ;
2 * n < 2 * n ;
x , y // x , y ;
B-11 c= [: [: V , W :] , V :] ;
assume I is_closed_on s , P ;
U = U . ( i , j ) ;
M /. 1 = z ;
x9 = x9 & x9 = y9 ;
i + 1 < n + 1 ;
x in { {} , {} } ;
f3 <= ( f2 | X ) . ( len f1 + 1 ) ;
l in L ;
x in dom ( F | X ) ;
let i be Element of NAT ;
min ( r , s ) is ( len s ) -element ;
assume <* o2 , o *> <> {} ;
s . x |^ 0 = 1 ;
card ( K . n ) in M ;
assume that X in U and Y in U ;
let D be SetSequence of Omega ;
set r = ]. k + 1 , k + 1 .[ ;
y = W . 2 * x ;
assume dom g = cod f ;
let X , Y be non empty TopSpace , X , Y be Subset of X ;
x \ A is interval ;
|. <*> A .| . a = 0 ;
cluster sublattice -> strict for SubSpace of L ;
a1 in B . s1 ;
let V be finite .[ , F be FinSequence of V ;
A * B on B & B * C on A ;
f-3 = NAT --> 0 ;
A , B are_equipotent implies A , B are_equipotent
z1 = P1 . j .= P1 . j ;
assume f " ( P ) is closed ;
reconsider j = i as Element of M ;
a , b // a , b ;
assume q in A \/ ( B "\/" C ) ;
dom ( F * C ) = o ;
set S = [: { 0 } , { 0 } :] ;
z in dom ( A --> y ) ;
P [ y , h . y ] ;
{ x0 } c= dom f ;
B be non-empty ManySortedSet of I , A be ManySortedSet of I ;
sqrt ( PI / 2 ) < PI / 2 ;
reconsider z1 = 0 as Nat ;
LIN a , d , b ;
[ y , x ] in [: I , I :] ;
( Q ) * ( 3 , 1 ) = 0 ;
set j = x0 div m ;
assume a in { x , y } ;
j2 - ( j + -3 ) > 0 ;
I \! 1 = 1 ;
[ y , d ] in [: F , G :] ;
let f be Function of X , Y ;
set A2 = \frac { B where B is Subset of TOP-REAL 2 : B in B } ;
s1 , s2 are_\not ;
j1 -' 1 + 1 = 0 ;
set m2 = 2 * n + j ;
reconsider t = t as bag of n ;
I2 . j = m . j ;
i |^ s , n are_relative_prime ;
set g = f | [: D2 , D1 :] ;
assume that X is lower and 0 <= r ;
( p1 `1 ) ^2 = 1 ;
a < ( p3 `1 ) ^2 ;
L \ { m } c= UBD C ;
x in Ball ( x , 10 ) ;
not a in LSeg ( c , m ) ;
1 <= i1 -' 1 ;
1 <= i1 -' 1 ;
i + i2 <= len h ;
x = W-min ( P ) ;
[ x , z ] in X [: Z , Z :] ;
assume y in [. x0 , x .] ;
assume p = <* 1 , 2 , 3 *> ;
len <* A1 *> = 1 ;
set H = h . -3 ;
card b * a = |. a .| ;
Shift ( w , 0 ) |= v ;
set h = h2 \circ h1 ;
assume x in X1 /\ X2 ;
||. h .|| < d ;
not x in the carrier of f ;
f . y = F ( y ) ;
for n holds X [ n ] ;
k -' l = kl ;
<* p , q *> /. 2 = q ;
let S be Subset of lattice Y ;
P , Q are_homotopic ;
Q /\ M c= union ( F | M )
f = b * vol ( S ) ;
let a , b be Element of G ;
f .: X <= f . sup X ;
let L be non empty reflexive transitive RelStr , X be Subset of L ;
Sw is x -to_power i ;
let r be non negative Real ;
M , v |= x ;
v + w = 0. ( X , Y ) ;
P [ len ( F | ( len F ) ) ] ;
assume InsCode ( i + 5 ) = 8 ;
the zero of M = 0 ;
cluster z * seq -> summable ;
let O be Subset of the carrier of C ;
|. f " .| is continuous ;
x2 = g . ( j + 1 ) ;
cluster non empty Element of string S ;
reconsider ll = ll as Nat ;
v2 is Vertex of G2 & v2 is Vertex of G2 ;
T is SubSpace of ( T ) | ( the carrier of T ) ;
Q1 /\ Q <> {} ;
k in NAT ;
q " is Element of X ;
F . t is set of M , M ;
assume that n <> 0 and n <> 1 ;
set d1 = EmptyBag n , d2 = EmptyBag n , d2 = EmptyBag n , d2 = EmptyBag n , d2 = EmptyBag n , d2 = EmptyBag n , d2 = EmptyBag n , d2 = EmptyBag n ,
let b be Element of Bags n ;
assume for i holds b . i is commutative ;
x is root of ( p `1 ) ^2 ;
not r in ]. p , q .[ ;
let R be FinSequence of REAL ;
S7 does not empty b1 ;
IC SCM R <> a ;
|. \upupharpoons x , y ;
1 * seq = seq . ( len seq + 1 ) ;
x be FinSequence of NAT ;
let f be Function of C , D ;
for a holds 0. L + a = a
IC s = s . NAT .= n ;
H + G = FG + ( GG ) ;
Ci1 . x = x2 ;
f1 = f .= f2 .= f2 ;
Sum <* p . 0 *> = p . 0 ;
assume v + W = v + u ;
{ a1 } = { a1 } ;
a1 , b1 _|_ b , a ;
a3 , o _|_ o , a1 ;
I1 is reflexive & I1 is reflexive ;
I1 is antisymmetric & transitive in the carrier of C & [ the carrier of C , the carrier of C ] in the InternalRel of C ;
sup rng ( H1 , v ) = e ;
x = a1 * a2 + a3 * a3 ;
|. p1 .| ^2 >= 1 ;
assume j2 -' 1 < 1 ;
rng s c= dom ( f1 ^ f2 ) ;
assume that support a misses dom b and not a in dom b ;
let L be associative non empty doubleLoopStr , n be Element of NAT ;
s " + 0 < n + 1 ;
p . c = ( f " ) . 1 ;
R . n <= R . ( n + 1 ) ;
card I + ( card I + 2 ) = card I + 2 ;
set f = + ( x , y ) ;
cluster Ball ( x , r ) -> bounded ;
consider r being Real such that r in A ;
cluster -> non empty for FinSequence of NAT ;
let X be non empty Subset of S ;
let S be non empty transitive RelStr ;
cluster <* *> . N , M *> -> complete ;
sqrt ( 1 / a ) = a / sqrt ( 1 / a ) ;
( q . {} ) `1 = o ;
sqrt ( i -' 1 ) > 0 ;
assume sqrt ( 1 / 2 ) <= 1 / 2 ;
card B = k + 1 ;
x in union rng ( f1 ^ f2 ) ;
assume x in the carrier of R ;
d in dom f ;
f . 1 = L . 1 ;
the vertices of G = { v } ;
let G be finite Vwconnected non empty _Graph ;
e , v2 , f3 is_collinear ;
c . i0 in rng c ;
f2 /* q is divergent_to-infty ;
set z1 = - ( z1 - z2 ) / ( z1 - z2 ) ;
assume w is at_S of G , G ;
set f = p \! \mathop { t } ;
let c be object of C ;
assume ex a st P [ a ] ;
let x be Element of ( REAL m ) -tuples_on the carrier of ( REAL m ) ;
let I1 be Subset-Family of X ;
reconsider p = p as Element of NAT ;
v , w as Point of X ;
let s be State of SCM+FSA , P , s be State of SCM+FSA ;
p is finite Program of SCM+FSA ;
stop stop I c= card I + card J ;
set ci = fi /. ( i + 1 ) ;
w ^ t ^ w ^ t ^ w ^ w ^ t ^ w ^ w ^ t ^ w ^ w ^ w ^ t ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^
W1 /\ W2 = ( W1 + W2 ) /\ ( W2 + W1 ) ;
f . j is Element of J . j ;
let x , y be Element of T ;
ex d st a , b // b , d ;
a <> 0 & b <> 0 implies c <> 0
ord ( x ) = 1 & x is 0 ;
set g2 = lim ( s , x0 ) ;
2 * x >= 2 * x + 2 * x ;
assume ( a 'or' c ) . z <> TRUE ;
f "/\" g in Hom ( c , d ) ;
Hom ( c , c + d ) <> {} ;
assume 2 * Sum ( q | m ) > m ;
L1 . ( F . ( F . k ) ) = 0 ;
then R1 \/ R2 = id X \/ R1 \/ R2 \/ R2 = id X ;
( ( the function sin ) * ( sin * ( cos * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * (
( ( ( the function of ) ) `| Z ) = 0 ;
o1 in [: X1 /\ X2 , X2 :] ;
e , v2 , f3 is_collinear ;
s3 > sqrt ( 1 / 2 ) * 0 ;
x in P .: ( F ) ;
J be closed by \hbox { $ , 1 } , R ;
h . p1 = f2 . O ;
Index ( p , f ) + 1 <= j ;
len ( q @ ) = width M & len ( q @ ) = width M ;
the support of ( K ) c= A ;
dom f c= union rng ( F ^ ) ;
k + 1 in Seg ( k + 1 ) ;
let X be ManySortedSet of the carrier of S ;
[ x `1 , y `2 ] in Indices R & [ x `2 , y `2 ] in R ;
i = D1 or i = D2 ;
assume a mod n = b mod n ;
h . x2 = g . x1 ;
F c= 2 -tuples_on the carrier of X
reconsider w = |. s1 .| as Real_Sequence ;
sqrt ( 1 / m * r + r / m ) < p ;
dom f = dom ( I --> ( i , j ) ) ;
[#] ( Pb9 ) = [#] ( K ) ;
cluster - x -> extended real ;
then { d } c= A ;
cluster [: TOP-REAL n , TOP-REAL n :] -> finite-ind ;
let w1 be Element of M ;
x be Element of dyadic ( n ) ;
u in W1 & v in W2 ;
reconsider y = y as Element of L2 ;
N is full SubRelStr of T |^ ( n ) , T |^ ( n ) ;
sup { x , y } = c "\/" ( { x , y } ) ;
g . n = n |^ 1 .= n ;
h . J = EqClass ( u , J ) ;
let seq be complex sequence of X ;
dist ( x `1 , y `2 ) < sqrt ( r ^2 + 2 ^2 ) ;
reconsider \mathbb m = m as Element of NAT ;
x- x0 < r1 - x0 + x0 ;
reconsider P = P ` as strict Subgroup of N ;
set g1 = p * ( q `1 ) ;
let n , m , k be non zero Nat ;
assume that 0 < e and f | A is lower ;
D2 . I in { x } ;
cluster open -> subopen ;
P is compact non empty Subset of TOP-REAL 2 ;
Gik in LSeg ( \pi , 1 ) ;
n in NAT & n in NAT ;
reconsider ST = S as Subset of T ;
dom ( i .--> X ) = { i } ;
let X be non-empty ManySortedSet of S ;
let X be non-empty ManySortedSet of S ;
op ( 1 , 1 ) c= { [ {} , {} ] } ;
reconsider m = +infty as Element of NAT ;
reconsider d = x as Element of [: the carrier of C , the carrier of C :] ;
let s be 0 -started State of SCMPDS , P , s be State of SCMPDS ;
let t be 0 -started State of SCMPDS , Q ;
b , b // x , y & x , y // x , y ;
assume that i = n \/ { n } and j = k \/ { k } ;
let f be PartFunc of X , Y ;
N2 >= sqrt ( c ^2 + d ^2 ) ;
reconsider tT = T " as TopSpace ;
set q = h * p ^ <* d *> ;
z2 in U . ( y2 , z2 ) /\ Q ;
A |^ 0 = { <* E *> } ;
len W2 = len W + ( len W1 + 1 ) ;
len h2 in dom h2 & h2 . 1 in rng h2 ;
i + 1 in Seg ( len s2 ) ;
z in dom ( g1 | X ) ;
assume p2 = W-min ( K ) + W-min ( K ) ;
len G + 1 <= i1 + 1 ;
f1 (#) f2 is convergent & f2 is convergent & f2 is convergent & f2 is convergent & f2 is convergent & f2 is convergent & f2 is convergent & f2 is convergent & f2 is convergent & f2 is convergent & f2 is convergent & f1
cluster Partial_Sums seq -> summable for sequence of X ;
assume j in dom ( M1 * M2 ) ;
let A , B , C be Subset of X ;
x , y // x , y & x , y // y , z ;
b ^2 - ( 4 * a ) >= 0 ;
<* xy *> ^ <* y *> ^ <* y *> ^ <* y *> ^ <* x *> ^ <* y *> ^ <* y *> ^ <* y *> ^ <* x *> ^ <* y *> ^ <* y *> ^ <* y
a , b // { a , b } ;
len ( p2 ^ ( - 1 ) ) = len ( p2 ^ ( - 1 ) ) ;
ex x being element st x in dom R & R [ x , x ] ;
len q = len ( K * G ) ;
s1 = Initialize ( Initialized s ) .= Exec ( i , s1 ) ;
consider w being Nat such that q = z + w ;
x ` is Element of L ;
k = 0 & n > k implies k > 0
then X is discrete ;
for x st x in L holds x is FinSequence of L
||. f /. c - f /. c .|| <= r1 ;
c in ]. p , q .[ & not c in { p } ;
reconsider V = V ` as Subset of the carrier of Euclid n ;
N , M is_has with with with N ;
then z >= compactbelow x & z >= x ;
M = f & M = g implies M = g
( ( ( to_power 1 ) to_power ( 1 + 1 ) ) to_power ( 1 + 1 ) ) = TRUE ;
dom g = dom f & dom f = dom f & dom g = dom f ;
mode mode odd of G is with_\upupharpoons G , dom the InternalRel of G ;
[ i , j ] in Indices M & [ i , j ] in Indices M ;
reconsider s = x " as Element of H ;
let f be Element of dom Subformulae p ;
F1 . a1 , F1 . a2 - F1 . a2 - F1 . a2 - F1 . a2 = G1 . a1 ;
cluster circle ( a , b , r , s ) -> compact ;
let a , b , c be Real ;
rng s c= dom ( ( 1 / 2 ) (#) ( f + g ) ) ;
K is additive implies ( K + ( F , k ) ) is additive ;
set k2 = card dom B \ ( dom B ) ;
set G = DTConOSA ( X ) ;
reconsider a = [ x , s ] as Object of G ;
let a , b be Element of M , a , b be Element of M ;
reconsider s1 = s as Element of S1 ;
rng p c= the carrier of L ;
let d be Subset of the bound of A ;
( x | x ) = 0 iff x = 0. W ;
I1 in dom stop ( a , I , J ) ;
g be continuous Function of X , Y ;
reconsider D = Y as Subset of TOP-REAL n ;
reconsider i2 = len ( p1 ^ p2 ) as Integer ;
dom f = the carrier of S & dom g = the carrier of S ;
rng h c= union ( the carrier of J ) ;
cluster All ( x , H ) -> \bf \bf 1_ ;
d * N1 > N1 * N1 * N2 ;
]. a , b .[ c= [. a , b .] ;
set g = f " ( D1 , j ) ;
dom ( p | ( Seg m ) ) = [: Seg m , Seg m :] ;
3 + - 2 <= k + - 2 ;
the function tan is differentiable of ( - 1 ) (#) ( ( - 1 ) (#) ( ( #Z 2 ) * ( ( #Z 2 ) * ( #Z 2 ) ) ) ) ;
x in rng ( f /^ p ) ;
f , g be FinSequence of D ;
\ { p } in the carrier of S1 & p in the carrier of S2 ;
rng f " = dom f & rng f = dom f ;
( the Target of G ) . e = v ;
width G -' 1 + 1 < width G ;
assume v in rng ( S | ( E ) ) ;
assume x is root or x is root ;
assume 0 in rng ( g2 | A ) ;
let q be Point of TOP-REAL 2 , a , b , c , d be Real ;
let p be Point of TOP-REAL 2 ;
dist ( O , u ) <= |. p2 .| + 1 ;
assume dist ( x , b ) < dist ( a , b ) ;
<* S7 *> is in the carrier of C & <* C7 *> is Subset of the carrier of C ;
i <= len ( G /^ ( k + 1 ) ) ;
let p be Point of TOP-REAL 2 ;
x1 in the carrier of I[01] & x2 in the carrier of I[01] ;
set p1 = f /. ( i + 1 ) ;
g in { g2 : r < g2 } ;
Q = Sk2 " ( Q , Q ) ;
( sqrt ( 1 / 2 ) ) to_power ( 2 to_power ( 2 * k ) ) is summable ;
- p + I c= - p + A + - - p ;
n < LifeSpan ( P1 , s1 ) + 1 ;
CurInstr ( p1 , s1 ) = i ;
A /\ Cl { x } <> {} ;
rng f c= ]. r , s + r .[ ;
let g be Function of S , V ;
let f be Function of L1 , L2 ;
reconsider z = z as Element of \langle L , L *> ;
let f be Function of S , T ;
reconsider g = g as Morphism of c , b ;
[ s , I ] in [: S , A :] ;
len ( the connectives of C ) = 4 & len the connectives of C = 4 ;
let C1 , C2 be subsignature of C ;
reconsider V1 = V as Subset of X | B ;
attr p is valid means : Def5 : x is valid ;
assume that X c= dom f and f .: X c= dom g ;
H |^ a is Subgroup of H |^ a ;
let A1 be Element of O , B1 , B2 be Element of O ;
p2 , p3 , r is_collinear & q1 , q2 is_collinear & q2 , p1 , p2 is_collinear & q1 <> q2 & q2 <> q2 & q2 <> q2 & q2 <> q2 & q2 <> q2 & q2 <> q2 & q2 <> q2 & q2 <> q2
consider x being element such that x in v ^ K ;
not x in { 0. TOP-REAL 2 } ;
p in [#] ( ( TOP-REAL 2 ) | B ) ;
0 + ( REAL * ( E * F ) ) < M ;
^ ( c ^ ( a9 ^ c9 ) ) = c ;
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . ( s2 . ( s2 . ( s2 . s2 ) ) ) ) ;
cluster with_generated -> with_generated for SubSpace of L ;
set i1 = the Element of NAT ;
let s be 0 -started State of SCM+FSA , P , s be State of SCM+FSA ;
assume y in ( f1 \/ f2 ) .: A ;
f . len f = f /. ( len f + 1 ) ;
x , f . x '||' f . x , f . y ;
attr X c= Y means : only : \pi _ 2 ( X ) c= \pi _ 2 ( X ) ;
y in the carrier of Y & x in the carrier of X ;
cluster -> non empty for XR ;
set S = <* Bags n *> ^ ( \HM { the } \HM { connectives of K ) *> ^ ( \HM { the } \HM { connectives of K ) ;
set T = [. 0 , 1 / 2 .] ;
1 in dom mid ( f , 1 , 1 ) ;
sqrt ( 4 * PI * PI * PI + 2 * PI * PI * PI ) < sqrt 2 * PI * PI / 2 ;
x2 in dom ( f1 + f2 ) /\ dom ( f2 + g2 ) ;
O c= dom I & { {} } c= { {} } ;
( the Target of G ) . x = v ;
{ HT ( f , T ) } c= Support f ;
reconsider h = R . k as Polynomial of n , L ;
ex b being Element of G st y = b * H ;
let x , y , z be Element of G ;
h1 . i = f . ( h . i ) ;
( p `1 ) ^2 = ( p `1 ) ^2 + ( p `2 ) ^2 ;
i + 1 <= len Cage ( C , n ) ;
len <* P *> = len P & len P = len P ;
set N1 = the Element of the Element of the Element of N ;
len gy + ( x + 1 ) <= x ;
a does not lie on B & b on C ;
reconsider sum = r * I . v as FinSequence of REAL ;
consider d such that x = d and a _|_ d ;
given u such that u in W and x = v + u ;
len f /. ( \downharpoonright n ) = len fn ;
set q2 = E-max L~ Cage ( C , n ) ;
set S = \vert ( S1 , S2 ) . ( S1 , S2 ) . ( S1 , S2 ) . ( S1 , S2 ) . ( S1 , S2 ) . ( S1 , S2 ) . ( S1 , S2 ) . ( S1 , S2 )
MaxADSet ( b ) c= MaxADSet ( P ) /\ MaxADSet ( P ) ;
Cl ( G . q1 ) c= F . r2 ;
f " ( D ) meets h " ( D ) ;
reconsider D = E as non empty Subset of L1 ;
H = All ( H , F ) '&' ( H '&' ( H '&' ( H '&' F ) ) ) ;
assume t is Element of ( \mathfrak F ) . ( X , Y ) ;
rng f c= the carrier of S2 & rng f c= the carrier of S2 & rng f c= the carrier of S2 ;
consider y being Element of X such that x = { y } ;
f1 . a1 = b1 & b1 . b1 = b2 ;
the carrier of G = E \/ { E } ;
reconsider m = len wk as Element of NAT ;
set S1 = LSeg ( n , connectives ( C ) , W-min ( C ) ) ;
[ i , j ] in Indices ( M1 * M2 ) ;
assume that P c= Seg m and M is without_zero ;
for k st m <= k holds z in K . k ;
consider a being set such that p in a and a in G ;
L1 . p = p * ( i , j ) ;
-7 . i = ( p1 ^ p2 ) . i .= p1 . i ;
let PA , G be Subset of Y ;
attr 0 < r & 1 < r & r < 1 & r < 1 & r < 1 & r < 1 ;
rng proj ( a , X ) = [#] ( X ) ;
reconsider x = x , y = y as Element of K ;
consider k such that z = f . k and n <= k ;
consider x being element such that x in X \ { p } ;
len Sgm ( s | Seg len s ) = card ( s | Seg k ) ;
reconsider x2 = x1 as Element of L2 ;
Q in FinMeetCl ( the topology of X ) ;
dom ( ( f1 + f2 ) * ( f2 + g2 ) ) c= dom ( f1 + f2 ) ;
attr n divides m & m divides n ;
reconsider x = x as Point of I[01] ;
a in \bf trivial ( T , T2 ) ;
not y in the bound not bound ( f ) & not y in dom f & not x in dom f & y in dom f & not x in dom f & y in dom f ;
Hom ( a , b ) <> {} & Hom ( a , b ) <> {} ;
consider k1 such that p " < k1 and p " < k1 ;
consider c , d such that dom f = c \ d ;
[ x , y ] in dom g & [ x , y ] in dom k ;
set S1 = 1GateCircStr ( x , y , z ) ;
l = k2 & l = k2 & l = k2 & l = k2 & l = k2 & l = k2 ;
x0 in dom ( u | A ) /\ A ;
reconsider p = x as Point of TOP-REAL 2 ;
c01 = [: REAL , B :] \/ [: the carrier of B , the carrier of C :] ;
f . p3 <= _ P . p1 , f . p2 , f . p3 , f . p2 , f . p3 , f . p3 , f . p2 , f . p3 , f . p2 , f . p3 , f . p2 , f . p3 , f . p3
( F . ( x , y ) ) `1 <= ( F . x , y ) `1 ;
( x - y ) `2 = ( ( - x ) `2 ) ^2 + ( - x ) ^2 ;
for n being Element of NAT holds P [ n ] ;
J , K // J , K ;
assume 1 <= i & i <= len <* a *> ;
0 |-> a = <*> the carrier of K & a = 0. K ;
X . i in 2 -tuples_on ( A \ B ) ;
<* 0 *> in dom ( e --> <* 1 *> --> 0 ) ;
then P [ a ] ;
reconsider ss = seq . Y. as SortSymbol of D ;
Seg ( i -' 1 ) <= len fj ;
[#] S c= [#] T & [#] T c= [#] T ;
let V being strict RealUnitarySpace ;
assume k in dom mid ( f , i , j ) ;
let P be non empty Subset of TOP-REAL 2 ;
let A , B be Matrix of n , K ;
- a * b * b = a * b ;
let A being Subset of ( TOP-REAL 2 ) | A ;
id ( o2 ) in <* o2 , o2 , o1 *> ;
then ||. x .|| = 0 & x = 0. X ;
let N1 , N2 be strict Subgroup of G ;
j >= len ( g | indx ( D1 , n1 , n1 , n1 ) ) ;
b = Q . ( len Q + 1 ) .= Q . ( len Q + 1 ) ;
f2 * ( f1 /* s ) is divergent to \hbox { - \infty , - \infty , - \infty .[ ;
reconsider h = f * g as Function of I[01] , R^1 ;
assume that a <> 0 and delta ( a , b , c , d ) >= 0 ;
[ t , t ] in the InternalRel of A ;
( v |-- E ) | n is Element of ( v | E ) | n ;
{} = the support of ( L1 + L2 ) ;
Reloc ( I , s ) is_closed_on Initialized s , P +* I ;
Initialized p = Initialize ( p +* q ) .= DataPart p +* q ;
reconsider N2 = N1 as strict net of ( the carrier of G2 ) . ( the carrier of G2 ) ;
reconsider Y = Y as Element of \langle L , \subseteq \rangle ;
"/\" ( { p } , L ) \ { p } <> p ;
consider j being Nat such that i2 = i1 + j + 1 ;
not [ s , 0 ] in the carrier of S2 & not [ s , 0 ] in the carrier of S2 ;
mm in ( B \wedge C ) \wedge D '/\' E '/\' F '/\' J '/\' M '/\' M '/\' M '/\' J '/\' M '/\' M '/\' M '/\' M '/\' M '/\' N '/\' M '/\' N '/\' M '/\' N '/\' M '/\' M '/\' N '/\' M '/\' N '/\'
n <= len ( P ^ Q ) + len ( P ^ Q ) ;
( x1 - x2 ) `2 = ( x2 - y2 ) `2 ;
InputVertices S = { x1 , x2 } \/ { x1 , x2 } ;
let x , y be Element of [: F\it \it it , n :] ;
p = |[ p `1 , p `2 ]| ;
g * 1_ G = h " * g * h .= h * g ;
let p , q be Element of PFuncs ( V , C ) ;
x0 in dom ( ( f1 + f2 ) (#) ( f2 + g2 ) ) ;
( R qua Function ) " = R " * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R ) )
n in Seg len ( f /^ ( len p ) ) ;
for s be Real st s in R holds s <= s2
rng s c= dom ( ( f2 * f1 ) ^ ) ;
synonym ex r being Subset of bool X , Y st r in bool X & r in card Y ;
1_ K * ( ( - 1 ) * ( - 1 ) ) = 1_ K * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1
set S = Segm ( P1 , P1 , Q1 ) ;
ex w st e = sqrt ( f , w ) & w in F ;
curry K . k is convergent ;
cluster open -> open for Subset of T ;
len ( f1 ^ f2 ) = 1 .= len ( f1 ^ f2 ) + 1 .= len ( f1 ^ f2 ) ;
sqrt ( i * p ) < sqrt ( 2 * p ) ;
let x , y be Element of Sub ( U0 ) ;
b1 , c1 // b1 , c1 & b1 , c1 // b1 , c1 & not b1 , c1 // b1 , c1 & not b1 , c1 // b1 , c1 & not b1 , c1 // b1 , c1 & not b1 , c1 // b1 , c1 & not b1 , c1
consider p being element such that c1 . j = { p } ;
assume that f " { 0 } = {} and f is total ;
assume IC Comput ( F , s , k ) = n ;
Reloc ( J , card I + card J ) does not destroys a , I ;
Stop SCM+FSA ( card I + 1 ) does not destroys a , I ;
set s3 = LifeSpan ( p1 , s3 ) , P4 = p1 , P4 = p2 , P4 = p1 , P4 = p2 , P4 = p1 , P4 = p2 , P4 = p2 , P4 = p3 , s4 = p2 , P4 = p3 , P4 = p3 ,
IC Comput ( P , s , k + 1 ) in dom Start-At ( ( card I + 2 ) , SCMPDS ) ;
dom t = the carrier of SCM R & dom t = the carrier of SCM R ;
( E-max L~ f ) .. f = 1 ;
let a , b be Element of PFuncs ( V , C ) ;
Cl ( union F ) c= Cl ( union F ) \/ Cl ( union F ) ;
the carrier of X1 union X2 misses the carrier of X1 union X2 ;
assume not LIN a , f . a , g . b ;
consider i being Element of M such that i = d ;
then Y c= { x } or Y = { x } ;
M , v |= ( H | ( y , x ) ) ;
consider m being element such that m in Intersect ( F ) ;
reconsider A1 = support ( u | A1 ) as Subset of X ;
card ( A \/ B ) = card ( 2 * 1 ) ;
assume that a1 <> a3 and a2 <> a3 and a3 <> a4 and a1 <> a4 and a2 <> a4 and a3 <> a4 and a1 <> a4 and a2 <> a4 and a3 <> a4 and a1 <> a4 and a2 <> a4 and a3 <> a4 and a1 <> a4 and a2 <> a4
cluster s \! \mathop { V } -> .| ;
L2 /. ( n2 + 1 ) = L2 . ( n2 + 1 ) ;
let P be compact non empty Subset of TOP-REAL 2 ;
assume -7 in LSeg ( p1 , p2 ) /\ LSeg ( p2 , p1 ) ;
let A be non empty Subset of TOP-REAL n , P , Q be Subset of TOP-REAL n , p1 , p2 , p1 , p2 be Point of TOP-REAL n ;
assume [ k , m ] in Indices ( D1 * ( k , m ) ) ;
0 <= ( 1 / 2 ) * ( 1 / 2 ) ;
( F . N ) . x = + \infty ;
attr X c= Y & Z c= V implies X \ Y c= V \ Y ;
( y * z ) * ( y * z ) <> 0. I ;
1 + card ( card ( X1 \/ X2 ) ) <= card ( X1 \/ X2 ) + card ( X2 \/ X1 ) ;
set g = z .. z + ( L~ z ) .. z ;
then k = 1 & p . k = <* x *> ;
cluster -> Element of C equals ( the \mathbb of X ) . ( the carrier of X ) ;
reconsider B = A as non empty Subset of TOP-REAL n ;
let a , b , c be Function of Y , BOOLEAN ;
L1 . i = ( i .--> g ) . i .= g . i ;
indx ( x1 , x2 , x3 ) c= P ;
n <= indx ( D2 , D1 , j ) + 1 ;
( g2 . O ) `2 = - 1 ;
j + p .. f -' len f + 1 <= len f -' 1 + 1 ;
set W = E-max C , E = W-min C , W = W-min C , E = W-min C , W = W-min C , S = W-min C , S = W-min C , T = W-min C , S = W-min C , T = W-min C , T = /
S1 . a = a + e .= a + e .= a + e ;
1 in Seg width ( M * ( i , j ) ) ;
dom ( i * Im ( f , g ) ) = dom ( f , g ) ;
\upupharpoons x `1 , ( a *' p ) `2 ;
set Q = |= ( g , f ) ;
cluster -> \mathclose { \rm \mathclose { -1 } } -> \mathclose { \rm \mathclose { -1 } } ;
attr F = { A } means : ex A st F is discrete ;
reconsider -13 = i0 as Element of product \overline { G } ;
rng f c= rng ( f1 ^ f2 ) \/ rng f2 \/ rng f2 \/ rng f2 \/ rng f2 \/ rng f2 \/ rng f2 \/ rng f2 \/ rng f2 \/ rng f1 \/ rng f2 \/ rng f1 \/ rng f2 \/ rng f2 \/ rng f1 \/ rng f2 \/ rng f2
consider x such that x in f .: A and x in f .: A ;
f = <*> the carrier of C & f = id the carrier of C ;
E , j |= All ( x1 , x2 , x3 ) ;
reconsider n1 = n as Morphism of o1 , o2 ;
assume that P is associative and R is associative and P is associative and P is associative ;
card ( B2 \/ { x } ) = card ( { x } \/ { x } ) ;
card ( x \ ( B \ ( B \ ( B \ ( B \ A ) ) ) ) ) = 0 ;
g + R in { s : g-r < s & g-r < g } ;
set q9 = ( q , <* s *> ) --> ( 1 , 2 ) ;
for x being element st x in X holds x in rng ( f1 ^ f2 )
h2 /. ( i + 1 ) = h2 . ( i + 1 ) ;
set pw = max ( B , C ) , pw = max ( B , C ) , pw = max ( B , C ) , R = max ( B , C ) , R = max ( B , C ) , S = min ( B
t in Seg width ( ( 1. ( K , n ) ) * ( i , j ) ) ;
reconsider X = dom f as Element of Fin ( C ) ;
IncAddr ( i , k ) = halt SCM+FSA .= i + k ;
S in LSeg ( f , k ) /\ LSeg ( f , k ) ;
attr R is condensed means : only : for R being Subset of X st R is condensed holds it is condensed ;
attr 0 <= a & b <= 1 & a * b <= 1 & a * b <= 1 & b * c <= 1 ;
u in ( c /\ ( d /\ b ) ) /\ f ;
u in ( c /\ ( d /\ e ) ) /\ f ;
len C + ( 2 * PI * 2 + 3 ) >= 9 + 3 ;
x , y // x , y & x , y // x , y implies x , y // x , y
a |^ ( n1 + 1 ) = a |^ ( n1 + 1 ) * a ;
<* x0 , 0 *> in Line ( x , x0 ) & <* x0 , 0 *> in Line ( x , x0 ) ;
set y9 = <* y , c *> ;
F2 * ( 1 , 1 ) in rng Line ( D , 1 ) ;
p . m joins r , s ;
( p `1 ) ^2 = ( f /. ( i1 + 1 ) ) ^2 ;
W in ( the carrier of X ) \/ ( the carrier of Y ) ;
0 + ( p `2 - p `2 ) <= 2 * ( 1 + p `2 - p `2 ) ;
x in dom g & not x in { 0 } ;
f1 /* ( seq ^\ k ) is divergent to \hbox { - \infty , + \infty .[ ;
reconsider u2 = u as VECTOR of Pvector ( X , Y ) ;
p \! ( ( ( ( Sgm X ) * ( Sgm X ) ) ) ) . ( p , ( Sgm X ) * ( p | X ) ) = 0 ;
len <* x *> + 1 < i + 1 + 1 ;
assume that I is non empty and { x } /\ { y } = { 0. I } ;
set i2 = card I + 2 + 1 ;
x in { x , y } & h . x = {} implies h . x = {} & h . y = {} & h . x = {} & h . y = {} ;
consider y being Element of F such that y in B and y <= x `1 ;
len S = len ( the charact of A ) & dom the charact of A = the carrier of A ;
reconsider m = M , i = N as Element of X ;
A . ( j + 1 ) = B . j \/ A . j \/ A . j ;
set NG2 = G .order() , dG2 = G .order() , GG2 = G .order() , GG2 = G .order() , GG2 = G .order() , GG2 = G .order() , GG1 = G .order() , G2 = G .order() , GG1 = G .order() , G2 = G = G .order() , G2
rng F c= the carrier of gr ( { a } , { a } ) ;
CastNode ( Q , K , n , r ) is len p1 implies for n being Nat holds ( Q * ( n , r ) ) is len p1
f . k , f . ( Radix ( n ) ) . k // f . ( Radix ( n ) ) , f . ( Radix ( n ) ) . k ;
h " ( P ) /\ [#] ( T | P ) = f " ( P ) ;
g in dom ( f2 " ) \ ( f2 " ) ;
gX /\ X = ( gX ) " X .= ( g " ) " X ;
consider n being element such that n in NAT and Z = G . n ;
set d1 = dist ( x1 , y1 ) , d2 = dist ( y1 , y2 ) , d2 = dist ( y2 , y2 ) , d2 = dist ( y2 , y2 ) , d2 = dist ( y2 , z2 ) , d2 = dist ( y2 , z2 ) , d2 = dist ( y2
b + sqrt ( 1 / 2 ) < sqrt ( 1 / 2 ) + sqrt ( 1 / 2 ) ;
reconsider f1 = f as VECTOR of X , Y = the carrier of X ;
attr i <> 0 implies i ^2 = i ^2 + ( i ^2 + i ^2 ) ;
j2 in Seg ( len ( g2 ) ) ;
dom ( i .--> ( j + 1 ) ) = dom ( i .--> j ) .= dom ( i .--> j ) .= dom ( i .--> j ) .= dom ( i .--> j ) .= dom j ;
cluster sec | ]. - 1 , 1 .[ -> one-to-one ;
Ball ( u , e ) = Ball ( f . p , e ) ;
reconsider x1 = x0 as Function of S , T ;
reconsider R1 = x , R2 = y as Relation of L , L ;
consider a , b being Subset of A such that x = [ a , b ] ;
( <* 1 *> ^ p ) ^ <* n *> in ( len p ) -tuples_on the carrier of K ;
S1 +* S2 = S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2
( ( ( ( ( ( ( ( ( - 1 ) * ( ( #Z ) ) * ( #Z n ) ) ) ) ) ) ) `| Z ) = ( ( ( ( - 1 ) * ( ( #Z n ) * ( ( #Z n ) * ( ( #Z n ) * ( ( #Z n ) * ( ( #Z
cluster -> Function of C , REAL ;
set CM = 1GateCircStr ( <* z , x *> , f1 ) ;
Ev2 . ( v2 , v2 ) = ( the Element of ( the carrier of G ) +* ( the carrier of G ) ) +* ( the carrier of G , the carrier of G ) ;
( ( ( ( ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( ( * (
sup A = cos * PI / 2 & lower_bound A = 0 & lower_bound A = 0 ;
F ( dom f , dom g ) is Morphism of dom ( F . ( dom f , g ) ) ;
reconsider p8 = q as Point of TOP-REAL 2 ;
g . W in [#] ( Y | X ) & [#] ( Y | X ) c= [#] ( Y | X ) ;
let C be compact non vertical non horizontal Subset of TOP-REAL 2 ;
LSeg ( f ^ g , j ) = LSeg ( f , j ) ;
rng s c= dom f /\ left_open_halfline ( x0 ) ;
assume x in { idseq ( 2 ) , ( idseq 2 ) ) } ;
reconsider n2 = n , n2 = m as Element of NAT ;
for y being extended real number st y in rng seq holds g <= y
for k st P [ k ] holds P [ k + 1 ]
m = m1 + m2 .= m1 + m2 + m2 .= m1 + m2 + m2 + m2 .= m1 + m2 + m2 + m2 + m2 + g2 .= m1 + m2 + g2 ;
assume for n holds H1 . n = G . n ;
set BB = f .: ( the carrier of X1 ) , pB = f .: ( the carrier of X2 ) , C = f .: ( the carrier of X1 ) , D = the carrier of X2 ;
ex d being Element of L st d in D & x <= d ;
assume R ~ ( a , b ) c= R ~ ( a , b ) ;
t in ]. r , s .] or t = s & t in ]. r , s .] ;
z + v2 in W & x = u + ( z + v2 ) ;
x2 = y2 \rightarrow y2 iff P [ x2 , y2 ]
attr x1 <> x2 means : Def5 : |. x1 - x2 .| = 0 & |. x1 - x2 .| = 0 ;
assume p2 - p1 , p2 - p1 , p2 - p1 is_collinear & - p1 , p2 - p1 is_collinear & - p1 , p2 - p1 is_collinear ;
set q = Rotate ( f , A ) ;
let f be PartFunc of [: { 1 } , { 1 } :] , REAL ;
( n mod 2 ) = n mod 2 .= n mod 2 ;
dom ( T * succ t ) = dom T & dom ( T * succ t ) = dom T ;
consider x being element such that x in w iff x in { w } ;
assume ( F * G ) . ( x3 , x4 ) = v . x3 ;
assume the carrier of D1 c= the carrier of D2 & the carrier of D2 c= the carrier of D2 ;
reconsider A1 = [. a , b .] as Subset of REAL ;
consider y being element such that y in dom F and F . y = x ;
consider s being element such that s in dom o and a = o . s ;
set p = W-min L~ Cage ( C , n ) ;
n1 -' len f + 1 + 1 <= len f + 1 - len f + 1 ;
ConsecutiveDelta ( q , o1 , o1 ) = [ u , v ] ;
set CG = ( \mathclose { .: G . k ) `1 ;
Sum ( L * p ) = 0. R .= 0. R .= 0. R ;
consider i being element such that i in dom p and t = p . i ;
defpred Q [ Nat ] means 0 = Q . $1 & $1 = Q . $1 implies $2 = Q . $1 ;
set s3 = Comput ( P1 , s1 , k + 1 ) , P4 = P1 , P2 = P2 , s2 = P2 , P4 = P2 , P4 = P2 , P4 = P2 , P4 = P2 , P4 = Comput ( P2 , s2 , k ) , P4 = P2 , P4 = P2 , P4 =
l in k & l in { k } & l in k ;
reconsider AT = union { G where G is Subset-Family of T : G is finite } as Subset-Family of T ;
consider r such that r > 0 and Ball ( p `1 , r ) c= Q ` ;
( h | ( n + 2 ) ) /. ( i + 2 ) = p1 ;
reconsider B = the carrier of X1 as Subset of X ;
pmax = <* - ( c.. ( seq ) ) , - ( /* ( seq ) ) *> ) ;
synonym f is real-valued for rng f is FinSequence of NAT ;
consider b being element such that b in dom F and a = F . b ;
x0 < card ( X1 + X2 ) + card ( X2 + X1 ) ;
attr X c= B1 implies indx ( X , Y ) c= card ( B1 \/ B2 ) ;
then w in Cl ( x , r ) & dist ( x , w ) <= r ;
angle ( x , y , z ) = angle ( x , 0 , PI , PI ) ;
attr 1 <= len s means : DefDef: : len ( the _ of K ) = len s & len ( the _ of K ) = len s ;
LE f . x9 , f . ( n + 1 ) , P ;
the carrier of { 1_ G } = { 1_ G } \/ { 1_ G } ;
attr p '&' q in sorted \ { p } & p '&' q in the carrier of S ;
- ( ( - t ) * ( 1 + t ) ) < ( - t ) * ( 1 + t ) ;
U . 1 = U /. 1 .= U . 1 .= U . 1 .= U . 1 .= U . 1 .= U . 1 .= U . 1 .= U . 1 .= U . 1 .= U . 1 .= U . 1 .= U . 1 .= U . 1 .= U . 1 .= U . 1 .= U .
f .: ( the carrier of x ) = the carrier of ( TOP-REAL 2 ) | the carrier of TOP-REAL 2 ) ;
Indices ( O * ( i , j ) ) = [: Seg n , Seg n :] & len ( O * ( i , j ) ) = [: Seg n , Seg n :] ;
for n being Element of NAT holds G . n c= G . ( n + 1 )
then V in M .: { x } ;
ex f being Element of F-9 st f is w.r.t. ( A , B ) & f is w.r.t. A ;
[ h . 0 , h . 3 ] in the InternalRel of G ;
s +* Start-At ( 0 , SCM+FSA ) = Exec ( 0 , SCM+FSA ) +* Start-At ( 0 , SCM+FSA ) .= Exec ( 0 , SCM+FSA ) .= Exec ( 0 , SCM+FSA ) .= Exec ( 0 , s3 ) .= Exec ( 0 , s3 ) .= Exec ( 0 , s3 ) .= Exec ( 0 , s3 ) .=
|[ w1 `1 , v1 `2 ]| <> 0. TOP-REAL 2 ;
reconsider t = t as Element of ( len t ) -tuples_on the carrier of K ;
C \/ P c= [#] ( ( [#] ( G ) ) \ A ) ;
f " ( V ) in Hom ( X , D ) /\ Hom ( X , D ) ;
x in [#] ( A ) /\ the carrier of ( A ) ;
g . x <= h1 . x & h . x <= h1 . x & h . x <= h1 . x ;
InputVertices S = { x9 , y9 } \/ { z , dp } ;
for n being Nat st P [ n ] holds P [ n + 1 ]
set R = Line ( M , i ) * Line ( M , i ) ;
assume that M1 is being_line and M2 is being_line and M2 is being_line and M2 is being_line and M2 is being_line and M2 is being_line and M2 is being_line and M2 is being_line and M2 * M2 is being_line and M2 * M2 = M2 * M2 ;
reconsider a = f3 . i2 - 1 as Element of K ;
len ( ( Len ( F ) ) ^ ( ( Len ( F ) ) ^ ( ( Len ( F ) ) ^ ( <* F *> ) ) ) ) = Sum ( ( ( Len ( F ) ) ^ ( F ^ ( <* F *> ) ) ) ;
len ( the connectives of K ) = n & len ( the connectives of K ) = n ;
dom ( max ( f , g ) ) = dom ( f + g ) ;
( the InternalRel of seq ) . ( n + 1 ) = sup ( ( the InternalRel of seq ) * ( the InternalRel of seq ) ) ;
dom ( p1 ^ p2 ) = dom ( p1 ^ p2 ) \/ dom ( p2 ^ p2 ) ;
M . [ 1 , y ] = 1 * ( 1 , 1 ) .= y ;
assume that W is non trivial and W is non trivial and W is Subset of the carrier of G2 ;
CG1 * ( i1 , i2 ) `1 = G1 * ( i1 , i2 ) `1 ;
CJ |- 'not' All ( x , p ) => ( 'not' All ( x , p ) => ( 'not' p ) ) ;
for b st b in rng g holds inf ( rng fa ) <= b
- sqrt ( ( ( q `1 ) / |. q .| - ( q `1 / |. q .| - ( q `1 / |. q .| - ( q `1 / |. q .| - ( q `1 / |. q .| - ( q `1 / |. q .| - q `1 / |. q .| - ( q `1 / q `1
( LSeg ( c , m ) \/ LSeg ( l , m ) ) \/ LSeg ( l , m ) c= R ;
consider p being element such that p in LSeg ( x , p ) and p in L~ f and p in L~ f ;
Indices ( X ^ ) = [: Seg n , Seg n :] & len ( X ^ Y ) = [: Seg n , Seg n :] ;
cluster s => ( q => p ) -> valid ;
Im ( ( Partial_Sums ( F ) ) . m ) = ( Partial_Sums ( F ) ) . m ;
cluster f . x1 , x2 , x3 , x4 -> Element of D ;
consider g being Function such that g = F . t and Q [ t , g ] ;
p in LSeg ( ( W-min L~ Cage ( C , n ) ) , W-min L~ Cage ( C , n ) ) ;
set RR = R |^ 1 , ]. b , b .[ , ]. b , d .[ ;
IncAddr ( I , k ) = halt SCM+FSA .= ( I , k ) --> ( k + 1 ) ;
seq . m <= ( the InternalRel of seq ) . m ;
a + b = ( a *' ) *' ( a *' ) .= ( a *' ) *' ( a *' ) *' ) ;
id ( X /\ Y ) = id ( X /\ Y ) ;
for x being element st x in dom h holds h . x = f . x ;
reconsider H = U1 \/ ( U2 \ U1 ) as non empty Subset of ( the carrier of U1 ) \ the carrier of U2 ;
u in ( ( c /\ d ) /\ ( ( d /\ e ) /\ f ) ) /\ m ;
consider y being element such that y in Y and P [ y , inf B ] ;
consider A being finite Subset of R such that card A = card ( the carrier of R ) ;
p2 in rng ( f .--> p1 ) \ rng ( f , p1 ) ;
len s1 > 0 & len s2 > 0 implies len s2 = len s2 + 1 & len s2 = 0 & len s2 = 0 & len s2 = 1 & len s2 = 0 & len s2 = 1 & s2 = s2 ^ s2 ^ s2 ^ s2 ^ s2 ^ s2 ^ s2 ^ s2 ^ s2 ^ s2 ^ s2 ^ s2 ^ s2 ^ s2
( W-min ( P ) ) `2 = ( E-max ( P ) ) `2 ;
Ball ( e , r ) c= LeftComp Cage ( C , n ) \/ L~ Cage ( C , n ) ;
f . a1 ` ` = f . a1 ` ` .= f . a1 ` ` ;
( seq ^\ k ) . n in ]. x0 - r , x0 + r .[ ;
gg . x0 = g . ( x0 - g . x0 ) ;
the InternalRel of S is symmetric Relation of the carrier of S & the carrier of S = the carrier of S ;
deffunc F ( Ordinal , Ordinal ) = phi ( $2 , $2 , $2 ) ;
F . s1 = F . s2 .= F . s2 .= F . s2 ;
x = A . o .= Den ( o , A ) . o ;
Cl ( f " ) c= f " ( dom f ) ;
FinMeetCl ( the topology of S ) c= the topology of T ;
synonym o is constructor means : only : for for for o is constructor & o <> \ast & o <> \ast p & p <> {} & p <> {} & p <> {} & p <> {} & p <> {} & p <> {} & p <> {} & p <> {} & p <> {} ;
assume that X + Y = Y + ( X + Y ) and card X <> card Y ;
the carrier of s <= 1 + ( the carrier of S ) ;
LIN a , a1 , d or b , c // d , b ;
e . 1 = 0 & e . 2 = 1 & e . 3 = 0 & e . 3 = 0 & e . 3 = 0 & e . 3 = 0 & e . 3 = 0 & e . 3 = 0 & e . 3 = 0 & e . 3 = 0 & e . 1 = 0 & e . 3 = 0 & e . 3
E in S1 & E in { N } ;
set J = ( l , u ) ReassignIn I ;
set A1 = 1GateCircStr ( a , b , c ) , B1 = 1GateCircStr ( a , b , c ) , C2 = 1GateCircStr ( a , b , c ) , C1 = 1GateCircStr ( a , b , c ) , C2 = 1GateCircStr ( a , b , c ) , C2 = 1GateCircStr ( a , b , c ) , C1 = 1GateCircStr ( a , b ,
set c9 = [ <* c9 , cin *> , '&' ] , cin = [ <* cin , dp *> , '&' ] , cin = [ <* cin , dp *> , '&' ] , dp = [ <* cin , dp *> , '&' ] , cin = [ <* cin , dp *> , '&' ] , cin = [ <* cin , dp *> , '&' ] ;
x * z " * x in x * ( z * x ) ;
for x being element st x in dom f holds f . x = g . x ;
cell ( f , 1 , G ) c= cell ( GoB f , 1 , width GoB f ) \/ ( L~ f -' 1 ) \/ ( L~ f -' 1 ) ;
U is reduces E-max ( C ) , W-min ( C ) , W-min ( C ) , W-min ( C ) , W-min ( C ) , W-min ( C ) , W-min ( C ) , W-min ( C ) ;
set f9 = f ^ g ^ h ;
attr S1 is convergent means : only : S1 is convergent & for n st n >= N holds S1 . n is convergent & lim S1 = x0 ;
f . ( 0 + 1 ) = ( 0 qua Ordinal ) + a .= a ;
cluster reflexive transitive non empty for RelStr ;
consider d being element such that R reduces b , d and R reduces c , d and R reduces d , d ;
not b in dom Start-At ( ( card I + 2 ) , SCMPDS ) ;
( z + a ) + x = z + ( a + x ) .= z + x .= z + x ;
len ( l |^ ( 0 , A ) --> x ) = len l ;
t4 is ( {} , X ) -valued ;
t = <* F . t *> ^ ( C ^ D ) ^ ( C ^ D ) ;
set pmin = W-min L~ Cage ( C , n ) ;
k9 -' ( i + 1 ) = k2 - ( i + 1 ) ;
consider u being Element of L such that u = u ` and u in D ;
len ( width ( ( width ( G @ ) ) |-> a ) |-> b ) = width ( ( ( G @ ) --> b ) ) ;
F3 . x in dom ( G * the_arity_of o ) ;
set H2 = the carrier of H2 , H2 = the carrier of H2 , H2 = the carrier of H2 , H2 = the carrier of H2 , H2 = the carrier of H2 , H2 = the carrier of H2 , H2 = the carrier of H2 = the carrier of H2 , the carrier of H2 = the carrier of H2 ;
set H1 = the carrier of ( ( the carrier of ( TOP-REAL 2 ) | K1 ) ) ;
( Comput ( P , s , 6 ) ) . intpos ( m + 1 ) = s . intpos m ;
IC Comput ( P3 , s3 , k + 1 ) = l + 1 ;
dom ( ( ( ( ( ( ( 1 / 2 ) * ( ( 1 / 2 ) ) * ( ( 1 / 2 ) * ( 1 / 2 ) ) ) ) ) ) ) = [: dom ( ( 1 / 2 ) * ( 1 / 2 ) ) ) ;
cluster <* l *> ^ phi -> ( 1 + string ( S ) ) string of S ;
set ba1 = [ <* \hbox { \boldmath $ p $ } , the carrier of G *> --> TRUE ] , [ <* <* d *> , f ] *> --> [ <* d *> , f ] , [ <* d *> , f ] , f ] , g = [ <* d *> , f ] ;
Line ( M , i ) = L * ( i , j ) ;
n in dom ( the Sorts of A ) ;
cluster f1 + f2 -> continuous for PartFunc of the carrier of S , the carrier of S ;
consider y being Point of X such that a = y and ||. \mathopen \mathclose .|| <= r ;
set x3 = DataLoc ( s . SBP , 2 ) , 5 = DataLoc ( s . cin , 2 ) , 6 = s . cin , 7 = s . cin , 6 = s . cin , 6 = s . cin , 7 = s . dp , 6 = s . cin , 6 = s . cin , 6 = s .
set pI = stop I , PI = stop I , PI = P +* I , PI = P +* I , PI = P +* I , PI = +* I , PI = P +* I , PI = +* I , PI = P +* I , PI = +* I , PI = +* J , PI = P +* I , PI
consider a being Point of D2 such that a in W1 and b = g . a ;
{ A , B , C } = { A , B , C } \/ { C , D } ;
let A , B , C , D , E , F , J , M , N , N , M , N , N , M , N , N , F , M , J , M , N , N , M , N , M , N , F , M ;
|. p2 .| ^2 - ( p2 `2 ) ^2 + ( p2 `2 ) ^2 >= 0 ;
l -' 1 + 1 = l * ( l + 1 ) + 1 ;
x = v + ( a * b ) + ( b * c ) + ( b * c ) * ( a * b ) + ( b * c ) * ( a * b ) + ( b * c ) * ( a * c ) + ( b * c ) * ( a * b ) + ( b * c ) * ( a * c ) + ( b
the TopStruct of L = (# the carrier of L , the carrier of L #) ;
consider y being element such that y in dom ( H1 | y ) and x = ( H1 | y ) . y ;
f9 \ { n } = | { ( v1 \ { v } ) } ;
let Y being Subset of X , X be Subset of Y ;
2 * n in { N : 2 * Sum ( p | N ) = N & N > 0 } ;
let s being FinSequence of the carrier of G , the carrier of G , the carrier of G , the carrier of G , s be Element of G ;
for x st x in Z holds ( ( ( id Z ) ^ ) `| Z ) = ( ( id Z ) ^ ) . x
rng ( h2 * f2 ) c= the carrier of ( TOP-REAL 2 ) | K1 & rng ( h2 * f2 ) c= the carrier of ( TOP-REAL 2 ) | K1 ;
j + 1- ( len f + 1 ) <= len f + ( len f + 1 ) - len f + ( len f + 1 ) ;
reconsider R1 = R * I as PartFunc of REAL , REAL ;
Cseq . x = seq . ( x0 + 1 ) .= seq . ( x0 + 1 ) .= seq . ( x0 + 1 ) .= seq . ( x0 + 1 ) ;
1_ ( ( { \mathbb C } ) , n ) = 1 / ( n , n ) .= ( x |^ n ) * ( n + 1 ) .= x |^ ( n + 1 ) ;
t is_is_is_at ( C , s ) . I & t is_the connectives of S ;
Carrier ( f + g ) c= Carrier f \/ Carrier g \/ { v } ;
ex N st N = j1 & 2 * ( ( r - ( r - ( 4 * N ) ) ) > 0 ;
for y , p st P [ p ] holds P [ All ( y , p ) ]
{ [ x1 , x2 ] } is Subset of [: X1 , X2 :] ;
h = j \rightarrow ( j .--> h ) .= ( j .--> h ) . ( i , j ) .= ( j .--> h ) . ( i , j ) ;
ex x1 being Element of G st x1 = x & x1 * x1 = A & x1 in A & y1 in A & x1 in B ;
set X = ( ( d , q1 ) `1 ) `1 , q1 = ( d , q1 ) `2 , q2 = ( d , q1 ) `2 , q1 = ( d , q1 ) `2 , q1 = ( d , q1 ) `2 ;
b . n in { g1 : x0 - r < g1 & g1 < x0 } ;
f /* ( s1 + c ) is convergent & f /. ( x0 + c ) = lim ( f /* ( s1 + c ) ) ;
the lattice of Y = the lattice of ( Y ) "\/" ( the carrier of Y ) ;
'not' ( a . x ) '&' b . x = TRUE ;
q2 = ( len q1 ^ ( len q2 ) + len q2 ) + len q2 .= len ( ( p1 ^ p2 ) + len p2 ) + len p1 ;
sqrt ( 1 / ( ( 1 / ( ( sec * ( sec * ( sec * ( sec * ( sec * ( sec * ( sec * ( sec * ( sec * ( sec * ( sec * ( sec * ( sec * ( sec * ( sec * ( sec * ( sec * ( sec * ( sec * ( sec * ( sec * ( ( ) ) )
set K = integral ( ( lim ( ( lim ( ( lim ( ( h / ( k / n ) ) ) ) - ( lim ( h / ( k + 1 ) ) ) ) ) ) ;
assume e in { \frac { w where w is Element of F : w in F & w in G } ;
reconsider dd = dom a , dd = dom F , dd = dom G , dd = dom G , dd = dom F , d = dom G , d = dom F , d = G , e = G , e = G , e = G , f = G as Element of F ;
LSeg ( f /^ ( j + 1 ) , q ) = LSeg ( f , j ) ;
assume X in { T . ( N2 , N2 ) : h . ( N2 , N2 ) = N2 } ;
assume Hom ( d , c ) <> {} & Hom ( d , c ) * f = <* c , d *> * f & <* d , c *> * f = <* d , c *> * f ;
dom S29 = dom S /\ ( Seg n ) .= dom S /\ ( Seg n ) .= dom S /\ ( Seg n ) .= dom S /\ ( Seg n ) .= dom S /\ ( Seg n ) ;
x in H implies ex g st x = g |^ a & g in H & x in H ;
( a * ( n , 1 ) ) . a = a * ( n , 1 ) .= a * ( n , 1 ) .= a * ( n , 1 ) .= a * ( n , 1 ) .= a * ( n , 1 ) ;
D2 . D2 in { r : lower_bound A <= r & lower_bound A <= D1 . i } ;
ex p being Point of TOP-REAL 2 st p = x & P [ p ] & p <> 0. TOP-REAL 2 ;
for c holds f . c <= g . c implies f ^ g <= f ^ g ^ h ^ h ^ h ^ h ^ h ^ g ^ h ^ h ^ h ^ k ^ g ^ h ^ h ^ h ^ k ^ g ^ h ^ h ^ h ^ h ^ g ^ h ^ h ^ h ^ h ^ g ^ h ^ h ^ h ^ h ^ h
dom ( ( f1 * f2 ) | X ) /\ X c= dom ( f1 * f2 ) /\ X ;
1 = sqrt ( p * ( p * ( 1 / p ) ) ) .= p * ( 1 / p ) .= p * ( 1 / p ) ;
len g = len f + len <* x *> .= len f + len <* x *> .= len f + len <* x *> .= len f + len <* x *> ;
dom ( F | [: N1 , S :] ) = dom ( F | [: N1 , S :] ) ;
dom ( f . t ) = dom ( f . t ) ;
assume a in ( sup ( ( T |^ ( \alpha ) ) ) ) .: ( the carrier of S ) ;
assume that g is one-to-one and rng g c= dom f and rng g c= dom g ;
( x \ y ) \ z = 0. X \ ( x \ y ) .= 0. X ;
consider f such that f * f = id ( a , b ) and f = id ( b , a ) * f ;
( ( ( the function cos ) * ( 2 * PI ) ) `| Z ) is increasing ;
Index ( p , co ) <= len ( ( p .. go -' 1 ) .. co + 1 ) .. co - len co + 1 ;
t1 , t2 , t2 , t1 , t2 , t2 , t1 , t2 , t2 , t1 , t2 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 is_collinear
relational ( ( ( Frege ( L ) ) . ( h , i ) ) ) <= ( ( Frege ( L ) ) . ( h , i ) ) . ( j , i ) ;
then P [ f . i2 , f . i2 ] & F ( f . i2 , f . i2 ) < j ;
Q [ ( [ D . x , 1 ] ) `1 , F ( ) `2 ] ;
consider x being element such that x in dom ( F . s ) and y = F . x ;
l . i < r . i & [ l . i , r . i ] is Subset of G . i ;
the Sorts of A2 = ( the Sorts of A2 ) +* the Sorts of A2 +* the Sorts of A1 +* the Sorts of A2 +* the Sorts of A1 +* the Sorts of A2 +* the Sorts of A2 +* the Sorts of A1 +* the Sorts of A2 +* the Sorts of A1 +* the Sorts of A2 +* the Sorts of A1 +* the Sorts of A2 +* the Sorts of A2 +* the Sorts of
consider s being Function such that s is one-to-one and dom s = NAT and rng s = NAT and rng s = NAT and rng s c= NAT and rng s c= NAT and rng s c= NAT and rng s c= NAT and rng s c= NAT and rng s c= NAT and rng s c= NAT and rng s c= NAT and rng s c= NAT and rng s c= NAT and rng s c= NAT and
dist ( b1 , b2 ) <= dist ( ( b1 , b2 ) . ( a , b ) ) + dist ( ( b1 , b2 ) . ( a , b ) ) ;
( W-min C ) * ( len ( Cage ( C , n ) ) ) = ( W-min C ) * ( W-min C ) .= ( W-min C ) * ( W-min C ) ;
q <= ( W-min L~ Cage ( C , n ) ) `2 ;
LSeg ( f | i2 , i ) /\ LSeg ( f | i2 , j ) = {} ;
given a being ExtReal such that a <= I and A = ]. a , b .] and a = ]. a , b .] ;
consider a , b being complex number such that z = a & y = b + a & z = a + b ;
set X = { b } , Y = { b } , Z = { b } , X = { b } , Y = { b } , Z = { b } , Y = { b } , Z = { b } , X = { b } , Y = { b } , Z = { b } , X = { b } , Y = { b } , Z =
( x * y ) \ ( x * z ) = 0. X ;
set xy = [ <* xy , yz , zx *> , '&' ] , yz = [ <* xy , yz , yz *> , '&' ] , yz = [ <* yz , yz , yz *> , '&' *> , '&' ] , yz = [ <* yz , yz , yz *> , '&' ] , xy = [ <* xy , yz , yz *> , '&' ] , xy = [ <* xy , yz
Carrier ( l ) = { l . ( len l ) + 1 } ;
sqrt ( ( q `1 ) ^2 + ( q `2 ) ^2 ) = 1 ;
sqrt ( ( p `1 ) ^2 + ( p `2 ) ^2 + ( p `2 ) ^2 ) < 1 ;
( ( E-max X ) `1 ) = ( E-max X ) `1 .= ( E-max X ) `1 .= ( E-max X ) `1 ;
( seq - seq ) . k = seq . ( ( - seq ) . k ) .= seq . ( - seq . k ) ;
rng ( h + c ) c= dom SVF1 ( 1 , f , x0 ) ;
the carrier of X = the carrier of ( X union the carrier of Y ) & the carrier of X = the carrier of Y ;
ex p3 st p3 = p3 & |. p3 - p3 .| = r & |. p3 - p3 .| = |. p3 - p3 .| ;
set h = \raise .4ex ( X , A ) , A , B = ( X , A ) --> ( X , A ) ;
R |^ ( 0 * n ) = Ireal ( X , n ) .= R |^ ( 0 ) .= R |^ ( 0 ) ;
( Partial_Sums ( ( ||. ( lim ( F ) ) ) . n ) ) . m ) + ( ( Partial_Sums ( ( lim ( F ) ) . n ) ) ) . m ) + ( ( Partial_Sums ( ( lim ( F ) ) . n ) ) ) ) . m + ( ( ( Partial_Sums ( F ) ) . n ) . m ) ) ) + ( ( Partial_Sums ( F ) . n
f2 = C7 . ( E7 , len ( V , the carrier of G ) ) ;
S1 . b = s1 . b .= s2 . b .= s2 . b .= s2 . b .= s2 . b .= s2 . b .= s2 . b .= s2 . b .= s2 . b .= s2 . b .= s2 . b .= s2 . b .= s2 . b ;
p2 in LSeg ( p2 , p1 ) /\ LSeg ( p1 , p2 ) ;
dom ( f . t ) = Seg n & dom ( f . t ) = Seg n ;
assume o = ( the connectives of S ) . 11 & the connectives of S = ( the connectives of S ) . 11 ;
set phi = ( l , l ) -TruthEval ( l , 2 ) , phi = l , l = l , l = l , phi = l , l = l , l = l , l = l , l = l , l = ( l , 2 ) -wff = ( l , 2 ) -wff ;
synonym p is T means : Def5 : HT ( p , T ) = 1. ( L ) ;
( Y1 - p1 ) `2 = - 1 & ( 0. ( TOP-REAL 2 ) | P ) `2 = - 1 & 0. TOP-REAL 2 = - 1 & 0. TOP-REAL 2 = 0. TOP-REAL 2 & 0. TOP-REAL 2 = 0. TOP-REAL 2 ;
defpred X [ Nat , set ] means P [ $2 , $2 , $2 ] means P [ $2 , $2 , $2 ] ;
consider k being Nat such that for n being Nat st k <= n holds s . n < x0 + g ;
Det ( I ^ ( m -' n ) ) = ( 1_ K ) ^ ( m , n ) ;
sqrt ( - b ^2 - b ^2 + c ^2 ) < 0 ;
CC . d = ( C . d ) mod ( C . d ) .= ( C . d ) mod ( C . d ) mod ( C . d ) mod ( C . d ) mod ( C . d ) mod ( C . d ) mod ( C . d ) mod ( C . d ) mod ( C . d ) mod ( C . d ) mod ( C . d ) mod ( C . d
attr X1 is dense means : only : X1 is dense means : only : X1 is dense of X & X1 is dense & X2 is dense implies X1 is dense of X , Y ;
deffunc F6 ( Element of E , Element of E ) = ( the Element of $1 ) * ( ( the Element of $1 ) * ( the Element of $1 ) ;
t ^ <* n *> in { t ^ <* i *> *> ^ <* i *> ^ ( T ^ <* i *> ^ t ) ;
( x \ y ) \ x = ( x \ y ) \ x .= 0. X .= 0. X ;
let X being non empty Subset of X , Y being Subset of X , T being Subset-Family of X , i being Subset of X st i in dom <* Y *> holds i is basis of X
synonym A , B , C means : Def5 : for : for A , B being Subset of X st A = B & B = C holds A misses B & B is open & B is open & A is open & B is open & B is open & A c= C & B is open & A c= C & B c= C ;
len ( M @ ) = len p & len ( M @ ) = width ( M @ ) & len ( M @ ) = width ( M @ ) ;
J = { x where x is Element of K : 0 < x & x < 1 } ;
( Sgm Y ) . d - ( Sgm Y ) . d - ( Sgm Y ) . d - ( Sgm Y ) . d - ( Sgm Y ) . d ) <> 0 ;
lower_bound divset ( D2 , k + 1 ) = D2 . ( k + 1 ) .= D2 . ( k + 1 ) ;
g . r1 = ( 2 * r1 ) * ( 2 * r1 ) & dom h = [. 0 , 1 .] & dom h = [. 0 , 1 .] & dom h = [. 0 , 1 .] & dom g = [. 0 , 1 .] & dom h = [. 0 , 1 .] ;
|. a * ||. f .|| .| = 0 * ||. f /. ( a * b ) .|| .= ||. a * ( f /. b ) .|| ;
f . x = ( h . x ) `1 & g . x = ( h . x ) `1 ;
ex w st w in dom B1 & <* 1 *> ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w
[ 1 , {} , <* d1 *> ] in ( { <* 0 *> , {} ) \/ ( { 0 } , {} ) \/ ( { 0 } , {} ) ) \/ ( { 0 } , {} ) \/ ( { 0 } , {} ) ) ;
IC Exec ( i , s1 ) + n = IC Exec ( i , s1 ) + n .= IC Exec ( i , s2 ) + n ;
IC Comput ( P , s , 1 ) = IC Comput ( P , s , 1 ) .= card I + 1 .= card I + 1 .= card I + 1 .= card I + 1 ;
( IExec ( W6 , Q , t ) ) . intpos ( 0 + 1 ) = t . intpos ( 0 + 1 ) ;
LSeg ( f , i ) misses LSeg ( f , i ) \/ LSeg ( f , j ) ;
assume for x , y being Element of L st x in C holds x <= y or y <= x & x <= y ;
integral ( f , C ) . x = f . ( sup C ) - ( lower_bound C ) . x ) ;
let F being one-to-one FinSequence , G be FinSequence of the carrier of G , the carrier of G , the carrier of G , the carrier of G , the carrier of G , the carrier of G , the carrier of G , the carrier of G , the carrier of G , the carrier of G , the carrier of G , the carrier of G , the carrier of G , the carrier of G , carrier of G , the carrier of
||. R /. ( h + c ) - R /. ( h + c ) .|| < e * ( K * ( c + d ) ) ;
assume a in { q where q is Element of M : dist ( z , q ) <= r } ;
set p3 = [ 2 , 1 ] .--> |[ 2 , 1 ]| ;
consider x , y being Subset of X such that [ x , y ] in F and x in d and y in G and x in G ;
for y being Element of REAL st y in Y & x in X & y in Y holds x <= y
redefine func |. p ^ <* p *> *> -> Element of A equals ( p ^ ( @ @ ( @ ( @ ( p ^ @ ) ) ) ) . ( p ^ @ ( @ ( @ ( p ^ @ @ @ @ ( p @ ) ) ) ) ;
consider t being Element of S such that x `1 , y `2 // z `1 , t `2 and x `2 <= y `2 ;
dom x1 = Seg len ( x1 ^ x2 ) & len x1 = len x1 & len x2 = len x2 & len x1 = len x2 & len x1 = len x2 implies len x1 = len x1 + len x2 & len x2 = len x1 + len x3 ;
consider y2 being Real such that x2 = y2 & 0 <= y2 & y2 <= 1 / 2 ;
||. f .|| .|| | ( X \ dom ( f | X ) ) = ||. f | X .|| ;
( the InternalRel of A ) ~ = ( the InternalRel of A ) ~ .= {} .= ( the InternalRel of A ) "\/" ( the InternalRel of A ) .= ( the InternalRel of A ) "\/" ( the InternalRel of A ) "\/" ( the InternalRel of A ) .= ( the InternalRel of A ) "\/" ( the InternalRel of A ) "\/" ( the InternalRel of A ) ;
assume that i in dom p and for j being Nat st j in dom p holds P [ j , p . j ] ;
reconsider h = f | [: X , Y :] as Function of X , Y ;
u1 in the carrier of W1 & u2 in the carrier of W2 & u1 in the carrier of W1 & u2 in the carrier of W2 & u1 in the carrier of W2 & u2 in the carrier of W2 & u1 in the carrier of W1 & u2 in the carrier of W2 & u1 in the carrier of W2 & u1 in the carrier of W2 ;
defpred P [ Element of L ] means M <= f . $1 & f . $1 <= f . $1 & f . $1 <= f . $1 ;
T ( u , a , v , w ) . ( x , y ) = s * x + ( - s ) * y .= b * y + ( - s ) * y .= b * x + ( - s ) * y .= b * y + ( - s ) * y .= b * x + ( - s ) * y ;
- ( - ( cosec * y ) ) = - ( - ( - ( - ( - ( - ( - ( x + y ) ) ) ) ) ) .= - ( - ( - ( - ( x + y ) ) ) ) .= - ( - ( - ( x + y ) ) ) .= - ( - ( x + y ) ) ;
given a being Point of G1 such that for x being Point of G2 holds a in the carrier of G1 & x in the carrier of G2 ;
f9 = [ dom ( ( @ ( f , g ) ) , ( @ ( f , g ) ) . ( ( @ ( g , h ) ) . ( f , g ) ) , ( @ ( f , g ) . ( f , h ) ) . ( f , g ) . ( f , h ) . ( f , h ) . ( f , g ) = h ;
let k being Nat , n be Nat , m be Nat , k be Nat , n be Nat ;
for x being element holds x in A |^ ( d , f ) iff x in A |^ ( d , f )
consider u , v being Element of R such that l /. i = u * a ;
1- sqrt ( ( p `1 ) ^2 + ( p `2 ) ^2 + ( p `2 ) ^2 ) > 0 ;
L-13 . k = ( L . k ) `1 & F . k = ( L . k ) `1 ;
set i2 = SubFrom ( a , i , n ) , i2 = i2 + ( i + n ) ;
attr B is atomic means : only : : : : : : : : : ( ( B @ ) | ( B @ ) ) = ( B @ ) | ( B @ ) ;
a9 "/\" D = { a "/\" d where d is Element of N : d in D } ;
( \square , \square ) * ( ( \square , v2 ) * ( ( \square , v2 ) ) ) . ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
( - f ) . sup A = ( - f ) . sup A .= - f . sup A ;
( G * ( len G , 1 ) ) `1 = ( G * ( len G , 1 ) `1 ) ;
( proj ( i , n ) ) . ( proj ( i , n ) ) . ( proj ( i , n ) . ( proj ( i , n ) . ( proj ( i , n ) . ( proj ( i , n ) . ( proj ( i , n ) . ( proj ( i , n ) . ( proj ( i , n ) . ( proj ( i , n ) . ( proj ( i , n ) ) ) ) )
f1 + f2 * reproj ( i , x ) is_differentiable_in x0 & reproj ( i , x ) . x0 = ( reproj ( i , x ) ) . x0 ;
attr ( ( ( ( ( ( ( ( ( ( ( ( ( ( 1 / ) ) ) ) ) ) ) ) ) `| Z ) = ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) )
ex t being SortSymbol of S st t = s & dom ( h . x ) = ( h . x ) . t ;
defpred C [ Nat ] means ( for n being Nat st n <= $1 holds ( for i being Nat st i in dom $1 holds ( P . i ) is thesis ) & ( ( P . i ) } is thesis ) ;
consider y being element such that y in dom ( p ^ q ) and q = ( p ^ q ) . y ;
reconsider L = product ( { x1 , B } ) as Subset of product ( B ) ;
for c being Element of C holds T . ( id c ) = id ( c ) & T . ( c , d ) = T . ( c , d ) ;
LIN f , n , p ^ <* p *> ^ <* p *> ^ <* p *> .= f ^ <* p *> ^ <* p *> ^ <* p *> ^ <* p *> ^ <* p *> ^ <* p *> ^ <* p *> ^ <* p *> ^ <* p *> ^ <* p *> ^ <* p *> ^ <* p *> ^ <* p *> ^ <* p *> ^ <* p *> ^ <* p *> ^ <* p
( f * g ) . x = f . x & ( f * g ) . x = f . x ;
p in { 1 / 2 * ( i + 1 ) } ;
f - p = ( - p ) *' ( - p ) .= ( - p ) *' + ( - p ) *' .= - ( - p ) *' + ( - p ) *' ) .= - ( - p ) *' + ( - p ) *' ) .= - ( - p ) *' + ( - p ) *' + ( - p ) *' ) .= - ( - p ) *' + ( - p *' ) *' ;
consider r being Real such that r in rng ( f | divset ( D , j ) ) and r < m + r ;
( f1 . r2 ) in [: ( ( f2 . r2 ) , ( f2 . r2 ) ) , ( ( f2 . r2 ) . r2 :] ) ;
eval ( a | ( n , L ) , x ) = eval ( a | ( n , L ) ) .= ( a | ( n , L ) ) . x ;
z = \llangle ( \llangle \rrangle , x \rrangle ) `1 .= ( \llangle x , y \rrangle ) `1 .= ( ( \llangle x , y \rrangle ) `1 ) `1 .= ( ( x , y ) `1 ) `1 .= ( ( x , y ) `1 ) `1 ;
set H = { Intersect ( S , T ) where S is Subset-Family of X : S in G } , G = { {} } ;
consider S19 being Element of D such that S = ( S ^ <* d *> ) ^ <* d *> ;
assume that x1 in dom f and x2 in dom f and f . x1 = f . x2 and f . x2 = f . x1 ;
- 1 <= sqrt ( ( q `1 ) / |. q .| - cn ) / ( 1 + cn ) ;
0. ( ( TOP-REAL 2 ) | A ) is Linear_Combination of A & Sum ( ( TOP-REAL 2 ) | A ) = 0. ( TOP-REAL 2 ) | A ;
let k1 , k2 , k1 , k2 be Nat ;
consider j being element such that j in dom a and j in dom a and a = g . j and x = a . j ;
H1 . x1 c= ( H1 . x1 ) & ( H2 . x1 ) . x2 = ( H1 . x2 ) . x1 ;
consider a being Real such that p = a * p1 + ( a * p2 ) and a <= 1 and a <= 1 and a <= 1 and a <= 1 and a <= 1 and a <= 1 and a <= 1 and b <= 1 and a <= 1 and a <= 1 and b <= 1 and a <= 1 and b <= 1 and a <= 1 and b <= 1 and a <= 1 and a <= 1 and b <= 1 and a <= 1 and b
assume that a <= c and [ a , b ] <= [ a , b ] and [ a , b ] in dom f and [ a , b ] in dom f and [ a , b ] in dom g and [ a , b ] in dom g and [ a , b ] in dom g and [ a , b ] in dom g and [ a , b ] in dom g and [ a , b ] in dom g and [ a , b ] in dom g
cell ( Gauge ( C , m ) , m ) /\ cell ( Gauge ( C , m ) , 1 , 0 ) is non empty ;
AA2 in { ( S . i ) `1 where i is Element of NAT : i in dom S } ;
( T * b1 ) . y = L * ( ( F * b1 ) . y ) .= ( F * b1 ) . y .= ( F * b1 ) . y ;
g . ( s , I ) . x = s . y & g . ( s , I ) . y = s . x ;
( log ( 2 , k ) ) ^2 + ( log ( 2 , k ) ) ^2 + ( ( log ( 2 , k ) ) ^2 + ( ( k + 1 ) ^2 ) ) >= ( ( k + 1 ) ^2 + ( k + 1 ) ^2 ) ;
then p => q in the carrier of S & not x in the carrier of S & not x in the carrier of S ;
dom ( the succ ( the succ of InsCode ( the connectives of C ) ) --> ( the carrier' of C ) ) misses dom ( the connectives of C ) ;
synonym f is extended real-valued for for for for for x is extended real-valued of f , x , y ;
assume for a being Element of D holds f . { a } = a & f . { a } = f . a ;
i = len ( p1 ^ p2 ) + len ( p1 ^ p2 ) .= len ( p1 ^ p2 ) + len ( p1 ^ p2 ) .= len ( p1 ^ p2 ) + len p2 .= len p1 + len p2 + 1 .= len p1 + 1 ;
( l - 3 ) * ( l - 3 ) = ( g - 3 ) * ( l - 3 ) + ( l - 3 ) * ( l - 3 ) * ( l - 3 ) ;
CurInstr ( P2 , Comput ( P2 , s2 , i + 1 ) ) = halt SCMPDS .= halt SCMPDS ;
assume for n be Nat holds ||. seq . n - seq . n .|| <= ||. seq . n - seq . n .|| ;
sin . r2 = sin . ( cos . r2 ) .= sin . ( cos . r2 ) .= sin . ( cos . r2 ) .= sin . ( cos . r2 ) .= sin . ( cos . r2 ) .= sin . ( cos . r2 ) .= sin . ( sin . r2 ) .= 0 ;
set q = |[ g1 `1 , g2 `2 ]| , g2 `2 ]| ;
consider G being sequence of S such that for n being Element of NAT holds G . n in qua Element of G ;
consider G such that F = G and ex G1 , G2 being Subset of [: the carrier of G , the carrier of G :] st G1 = G & G2 = ( the carrier of G ) \/ the carrier of G ;
the root of \llangle x , s \rrangle in ( the Sorts of Free ( F ) ) . s ;
Z c= dom ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
for k being Element of NAT holds --Index ( f , S ) . k = ( \Im ( f , S ) ) . k
assume that - 1 < n and ( - 1 ) * ( q `1 - 1 ) < 0 and sqrt ( ( q `1 - 1 ) * ( q `1 - 1 ) ) < 0 and sqrt ( ( q `1 - 1 ) * ( q `2 - 1 ) ) < 0 ;
assume that f is continuous and a < b and c < d and f . a = c and f . b = d and f . c = d and f . d = d and f . d = c and f . c = d and f . d = d and f . d = c and f . c = d and f . d = d and f . d = d and f . c = d and f . d = d and f
consider r being Element of NAT such that s2 = Comput ( P1 , s1 , i ) and r <= s2 and r <= s2 ;
LE f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 )
assume that x in the carrier of K and y in the carrier of K and x in the carrier of K and y in the carrier of K and x in the carrier of K and y in the carrier of K ;
assume f +* ( i1 , i2 ) in ( proj ( F , i1 ) ) . ( ( proj ( F , i2 ) ) . ( ( proj ( F , i2 ) ) . ( ( proj ( F , i2 ) ) . ( ( proj ( F , i1 ) ) . ( ( proj ( F , i2 ) ) . ( ( proj ( F , i2 ) ) . ( ( proj ( F , i2 ) ) . ( ( proj ( F , i2 )
rng ( ( Flow M ) ~ ) c= the carrier of M & rng ( ( Flow M ) ~ ) c= the carrier of M ;
assume z in { ( the carrier of G ) \/ { t } } ;
consider l being Nat such that for m be Nat st l <= m holds ||. ( s1 . m ) - g .|| < g ;
consider t being VECTOR of product G such that for t being Element of product G holds ||. t . t - x .|| <= 1 / 2 ;
assume that the carrier of v = 2 and v ^ <* 0 *> ^ ( <* 1 *> ^ ( <* 0 *> ^ ( <* 1 *> ^ ( <* 1 *> ^ ( 2 *> ^ ( 2 *> ^ ( 2 ^ ( 2 ^ ( 2 ^ ( 2 ^ 1 ) ) ) ) ) ) ;
consider a being Element of the Points of X such that a on the Points of X and a on A and a on A ;
( - x ) |^ ( k + 1 ) = 1 / ( ( - x ) |^ ( k + 1 ) ) .= 1 / ( ( - x ) |^ k ) ;
let D being set such that for i being Nat st i in dom p holds p . i in D . i ;
defpred R [ element , element ] means ex x , y st [ x , y ] = [ x , y ] & [ x , y ] in R & [ x , y ] in R ;
L~ ( f2 ^ g2 ) = union { LSeg ( p1 , p2 ) , p1 , p2 ) } ;
i -' len ( h1 ^ h2 ) + 2 - 2 + 2 < i + ( len h2 + 2 ) + 2 ;
for n be Element of NAT st n in dom F holds F . n = |. F . n - F . ( n -' 1 ) .|
for r , s being Real holds r <= s iff r <= s & s <= t & t <= t & s <= s & t <= s & s <= t & t <= s & s <= t & t <= s & s <= t & t <= s & s <= t & t <= s & s <= t & t <= s & s <= t & t <= t & s <= t & t <= s & s <= t & t <= s & s <= t
assume v in { G where G is Subset of T : G in B & G c= B } ;
let g be Element of A , X , Y , Z , X , Y , Z , X , Y , Z , x , y , z be Element of A , x , y , z be Element of A , x , y , z be Element of Z , x , y , z be Element of Z , x , y , z be Element of Z ;
min ( g . [ x , y ] , k ) = ( min ( g . ( y , z ) , k ) , k ) . ( y , z ) ;
consider q1 being sequence of elements of ( the carrier of G ) * such that for n being Nat holds P [ n , q1 . n ] ;
consider f being Function such that dom f = NAT and for n being Element of NAT holds f . n = F ( n ) ;
reconsider BO = B /\ B as Subset of T ;
consider j being Element of NAT such that x = the Element of ( n -tuples_on the carrier of K ) and 1 <= j and j + 1 <= n and 1 <= n and n <= len p ;
consider x being element such that z = x & card ( O . ( O , o2 ) ) in dom ( O . ( O , o2 ) ) ;
( C * T4 ) . ( k + 2 ) = C ( ( S4 ) . ( k + 2 ) ) ;
dom ( X --> rng f ) = X & dom ( X --> f ) = X ;
( S - ( TOP-REAL 2 ) ) `2 <= ( ( TOP-REAL 2 ) | ( L~ f ) ) `2 ;
synonym x , y , x means : Def5 : { x , y } is collinear ;
consider X being element such that X in dom ( f | ( n + 1 ) ) and ( f | X ) . X = Y ;
assume that Im k is continuous and for x , y being Element of L st x = y & y in the carrier of L holds x <= y iff x <= y ;
sqrt ( 1 / 2 * ( cos * ( #Z n ) ) ) + ( ( ( #Z n ) * ( #Z n ) ) ) * ( ( #Z n ) * ( #Z n ) ) ) is_differentiable_on REAL ;
defpred P [ Element of omega ] means ( the Element of ( A \ ( A \ { i } ) ) . $1 = ( A \ { i } ) . $1 ;
IC Comput ( P , s , 2 + 1 ) = succ IC Comput ( P , s , 2 + 1 ) .= IC Comput ( P , s , 2 + 1 ) .= 6 + 1 .= 6 + 1 .= 6 + 1 ;
f . x = f . ( g1 * ( g1 * ( g2 * ( g2 * ( g1 * ( g2 * ( g2 * ( g1 * ( g2 * ( g1 * ( g2 * ( g1 * ( g2 * ( g1 * ( g2 * ( g1 * ( f * ( g * ( g * ( ) ) ) ) ) ) ) ) ) ) ) .= f * ( ( ( ( g * ( g * ( g * ( g * ( g * ( g * ( g * ( g * ( g * ( g
( M * ( - ( - ( M * ( - ( M * ( - ( M * ( - ( M * ( - ( M * ( - M ) ) ) ) ) ) ) ) ) ) ) ) . n = M * ( ( - ( M * ( - ( M * ( - ( M * ( - ( M * ( - ( M * ( - M ) ) ) ) ) ) ) .= M * ( ( - ( M * ( - ( M * ( - ( M * ( - ( - (
the support ( L1 + L2 ) c= ( Carrier ( L1 ) ) \/ Carrier ( L2 ) \/ Carrier ( L2 ) ;
attr a , b , c , d means : Def5 : for x , y , z being Element of X st x in X & y in Y holds x = y & y = z & z = x & x = y ;
( the partial of product s ) . n <= ( the Sorts of product ( A * ) ) . n ;
attr - 1 <= r & 1 <= r & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & 1 <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 & r <= 1 &
seq ^ <* n *> in { p where p is Element of NAT : p ^ <* n *> *> in T } ;
[ x1 , x2 ] . ( x2 , y2 ) = [ x1 , x2 ] . ( x2 , y2 ) . ( x2 , y2 ) .= [ x1 , x2 ] ;
attr for m being Nat holds F . m is non-negative means : Def5 : for n be Nat holds ( Partial_Sums ( F ) ) . n = ( Partial_Sums ( F ) ) . n ;
len ( \mathopen { ^ ( ( G , y ) . ( z , y ) ) ) = len ( ( G , y ) . ( z , y ) ) + 1 ;
consider u , v being VECTOR of V such that x = u + v and u in the carrier of W1 and v in the carrier of W2 and u in the carrier of W1 and v in the carrier of W2 ;
given F being FinSequence of NAT such that F = x & dom F = { 0 , 1 } and dom F = { 0 , 1 } and Sum F = { 0 , 1 } and Sum ( F | ( 0 , 1 ) ) = 1 and Sum F = 0 and Sum F = 0 ;
0 = 1- If * uq iff 1 = ( 1- px ) * ( 1- px `1 ) + ( ( px `1 ) * ( px `1 ) ) ;
consider n be Nat such that for m be Nat st n <= m holds |. ( ( ||. x .|| # x ) - ( lim ( f # x ) ) ) .| < e ;
cluster non empty for 19 non empty 19 for RelStr ;
"/\" ( B , B ) = {} "/\" ( B , C ) .= ( the carrier of S ) "/\" ( the carrier of S ) .= ( the carrier of S ) "\/" ( the carrier of S ) .= ( the carrier of S ) "\/" ( the carrier of S ) .= ( the carrier of S ) "\/" ( the carrier of S ) ;
sqrt ( r ^2 + ( r ^2 + ( r ^2 + r ^2 ) ) ^2 ) + ( r ^2 + ( r ^2 + r ^2 ) ) + ( r ^2 + ( r ^2 + r ^2 ) ) + ( r ^2 + r ^2 ) ) <= sqrt ( r ^2 + ( r ^2 + r ^2 ) + ( r ^2 + r ^2 ) ) + ( r ^2 + r ^2 ) ;
for x being element st x in A /\ dom ( f `| A ) holds ( f `| A ) . x >= ( f `| A ) . x
2 * ( a * c ) - ( a * b ) * ( b * c ) = 0. TOP-REAL 2 - ( a * b ) * ( b * c ) ;
reconsider p = P /. ( \square , 1 ) , q = P " * ( ( ( ( the 0. K ) ^ ) + ( the carrier of K ) ) * ( ( the carrier of K ) ) * ( ( the carrier of K ) ) * ( the carrier of K ) ) ;
consider x1 , x2 being element such that x1 in uparrow s and x2 in downarrow s and x = [ x1 , x2 ] ;
for n being Nat st 1 <= n & n <= len q1 holds q1 . n = ( g /. ( len q1 ) ) . n
consider y , z being element such that y in the carrier of A and z in the carrier of A and y = [ y , z ] ;
given H1 , H2 being strict Subgroup of G such that x = H1 & y = H2 & ( ex H being Subset of G st x = H & ( H is Subgroup of G ) & ( H is Subgroup of H ) & ( H is Subgroup of G ) implies H is Subgroup of G ) ;
let S being non empty RelStr , T being non empty Poset , d being Function of S , T , S , T , T ;
[ a + 0 , i ] in ( the carrier of ( ( the carrier of ( TOP-REAL 2 ) ) | the carrier of ( TOP-REAL 2 ) | the carrier of ( TOP-REAL 2 ) | the carrier of ( TOP-REAL 2 ) | the carrier of ( TOP-REAL 2 ) | the carrier of ( TOP-REAL 2 ) | the carrier of ( TOP-REAL 2 ) | the carrier of ( TOP-REAL 2 ) | the carrier of ( TOP-REAL 2 ) | the carrier of ( TOP-REAL 2 ) | the carrier of ( TOP-REAL 2 ) | the carrier of ( TOP-REAL 2 ) | the carrier of ( TOP-REAL 2 ) |
reconsider mm = max ( ( ( p . n ) * ( ( p . n ) * ( p . n ) ) ) ) as Element of NAT ;
I <= width ( ( the Go-board of f ) * ( len GoB f , j ) ) & ( ( ( the Go-board of f ) * ( i , j ) ) * ( i , j ) ) * ( i , j ) = ( ( the Go-board of f ) * ( i , j ) ) * ( i , j ) ;
f2 /* q = ( f2 /* ( f1 /* q ) ) ^\ k .= ( f2 /* ( f1 /* q ) ) ^\ k ;
attr A1 \/ A2 is linearly independent means : Def5 : for A , B st A c= B holds ( A \/ B ) \ ( A \/ B ) = { 0. V } ;
redefine func A -the carrier of C -> Element of the carrier of R means : Def5 : for s being Element of C holds it . s = { s . ( s , C ) } ;
dom ( Line ( v , i + 1 ) ) = dom ( ( Line ( p , i ) ) * ( ( Line ( p , i ) ) ) ;
cluster [ x , x ] -> LSeg ( x , x ) , [ x , x ] , [ x , x ] , [ x , x ] , [ x , x ] , [ x , x ] , [ x , x ] , [ x , x ] ) ;
E , x1 |= ( x2 x1 x1 x1 ) => ( x2 , x1 ) => ( x2 , x1 ) => ( x2 , x1 ) ) => ( x2 , x1 ) => ( x2 , x1 ) => ( x2 , x1 ) => ( x2 , x1 ) ) => ( ( x1 , x1 ) => ( x2 , x1 ) ) => ( x2 , x1 ) => ( x2 , x1 ) ) ) ;
F .: ( id X ) . x = F . ( id X ) .= F . x .= F . x .= F . x .= F . x .= F . x .= F . x .= F . x .= F . x .= F . x .= F . x ;
R . m = F . x0 + h . x0 + h . x0 .= ( R + h ) . x0 + ( R + h ) . x0 ;
cell ( G , ( X1 -' 1 , width G ) -' 1 , j2 ) \ ( Y \ { Y } ) c= UBD ( L~ f ) \/ ( L~ f -' 1 ) ;
IC Comput ( P2 , s2 , i + 1 ) = IC Comput ( P2 , s2 , i + 1 ) .= card I + 2 ;
sqrt ( ( - ( - ( q `1 / |. q .| - q .| ) ) ) ^2 + ( - ( - ( q `1 / |. q .| - q `2 ) ) ) ^2 ) ) > 0 ;
consider x0 being element such that x0 in dom a and x0 in dom a and x0 = a . ( k + 1 ) and x0 = a . ( k + 1 ) ;
dom ( ( 1 / ( m + 1 ) ) * ( ( m + 1 ) / ( m + 1 ) ) ) = dom ( ( ( ( m + 1 ) / ( m + 1 ) ) * ( m + 1 ) ) ) .= dom ( ( ( ( ( m + 1 ) / ( m + 1 ) ) / ( m + 1 ) ) / ( m + 1 ) ) .= ( ( ( m + 1 ) / ( m + 1 ) ) / ( m + 1 ) ) / ( m + 1 ) ) ;
d-7 . [ y , z ] = ( ( y + z ) * ( y + z ) ) * ( z + y ) ;
attr for i being Nat holds C . i = A . i /\ B . i ;
assume that x0 in dom f and f | X is bounded and for x st x in X holds ||. f /. x - f /. x .|| <= r ;
p in Cl A implies for i being Element of T st i in A holds p meets A & p meets B & p meets B holds p meets B
for x be Element of REAL st x in Line ( x1 , y1 ) holds |. ( x1 - y1 ) - ( x2 - y2 ) .| <= |. x1 - y1 .|
func mode Ex{ <*> a , b , c -> Ordinal means : Def5 : for a being Ordinal st a in it holds it . a = c & it . a = b & it . a = c & it . b = c ;
[ a1 , a2 ] in ( the carrier of A ) \/ the carrier of B ;
ex a , b being element st a in the carrier of S1 & b in the carrier of S2 & x = [ a , b ] ;
||. ( vseq . n ) - ( vseq . m ) .|| < sqrt ( ( vseq . n ) - ( vseq . m ) ) * ( vseq . n ) ;
then for Z being set st Z in { Y where Y is Element of I : not contradiction } holds z in { Y } ;
sup compactbelow ( s , t ) = [ sup compactbelow ( s , t ) , sup compactbelow ( s , t ) ] .= [ sup compactbelow ( s , t ) , t ) , sup sup ( s , t ) ] ;
consider i , j being Element of NAT such that i < j and [ y , z ] in [: the carrier of G , of G :] and [ y , z ] in [: the carrier of G , the carrier of G :] and [ y , z ] in [: the carrier of G , the carrier of G , { z } :] ;
let D being non empty set , p , q , r being Element of D st p ^ q = r ^ r holds p ^ r ^ r ^ r ^ r ^ r ^ s ^ s ^ s ^ s ^ s ^ s ^ s ^ s ^ r ^ s ^ s ^ s ^ s ^ s ^ r ^ s ^ s ^ r ^ s ^ s ^ s ^ r ^ s ^ s ^ s ^ r ^ s ^ s ^ s ^ r ^ s ^ s ^ s ^ s ^ s ^ s ^ s ^ s ^ s ^ s ^ s ^ s ^ s ^
consider d1 being Element of the carrier of X such that c , d1 // b , d and a , b // c , d and b , c // d , d ;
set Fm2 = I \! \mathop { + } , U = I \! \mathop { + } ;
|. q1 .| ^2 = ( ( |. q1 .| ) ^2 + ( |. q2 .| ) ^2 ) + ( ( |. q2 .| ) ^2 ) .= ( |. q2 .| ) ^2 + ( |. q2 .| ) ^2 ) + ( |. q2 .| ) ^2 ) ;
let T being non empty TopSpace , x , y being Element of T , x , y being Element of T st x "/\" y = x "/\" y & y "/\" x = x "\/" y & x "/\" y = x "\/" y ;
dom signature ( U1 ) = dom ( the Arity of U1 ) & dom ( the Arity of U1 ) = dom ( the Arity of U1 ) & dom ( the Arity of U1 ) = the carrier' of U1 ;
dom ( h | X ) = dom h /\ X .= X /\ ( dom h ) .= X /\ X ;
for N1 , N2 being Element of ( the carrier of G1 ) * holds ( ( h * ( ( h * ( f * g ) ) ) . ( h * ( f * g ) ) ) . ( h * ( f * g ) ) . ( h * ( f * g ) ) . ( h * ( f * g ) ) . ( h * ( f * g ) ) . ( h * ( f * g ) ) ) = ( h * ( f * g ) . ( h * ( f * g ) ) . ( h * ( f * g ) ) . ( h * g ) ) . ( h *
( mod ( u , m ) ) + mod ( u , m ) = ( u mod m ) + ( v mod m ) ;
- ( ( - 1 ) * ( q `1 / |. q .| - ( q `1 / |. q .| - ( q `1 / q `1 ) / |. q .| - ( q `1 / q `1 - ( q `1 / q `1 ) / |. q .| - ( q `2 / q `1 - ( q `1 / q `1 / q `1 ) / |. q .| - ( q `1 / q `1 ) ) ) ) ) ) = - ( - ( q `1 / q `1 / q `1 ) / |. q `1 / q `1 - ( q `1 / q `1 / q `1 / q `1 - q `1 / q `1 / q `1 /
attr r1 = ( f | X ) . ( a , b ) & ( f | X ) . ( a , b ) = ( f | X ) . ( a , b ) ;
vseq . m is bounded & ( for x be Element of X , Y holds ||. ( vseq . x ) - ( vseq . x ) ) . m = ( vseq . x ) - ( vseq . x ) ) . m ;
attr a <> b & b <> c & c <> d & d <> b & c <> d & d <> d & d <> b & c <> d & d <> d & d <> b & c <> d & d <> d & c <> d & d <> d & d <> b & d <> b & c <> d & d <> d & d <> b & c <> d & d <> d & d <> b & c <> d & d <> d & c <> d & d <> d & d <> d & d <> d & d <> b & d <> d & d <> d & d
consider i , j being Nat such that p1 = [ i , j ] and i = [ i , j ] and j = [ i , j ] ;
|. p ^2 - ( 2 * ( p `1 - p `2 ) ) ^2 + ( p `2 - p `2 ) ^2 + ( p `2 - p `2 ) ^2 ) = |. p `1 - p `2 ) + ( p `2 - p `2 ) ^2 + ( p `2 - p `2 ) ^2 ;
consider p1 , q1 being Element of ( X \ ( Y \ { p1 } ) ) such that y = p1 ^ q1 and ( p1 ^ q1 ) ^ q1 = ( p1 ^ q1 ) ^ ( p1 ^ q1 ) ^ q1 ^ q2 ) ;
gcd ( ( 1 / 2 ) * ( 1 / 2 ) , ( 1 / 2 ) * ( 1 / 2 ) ) = ( 1 / 2 ) * ( 1 / 2 ) ;
( ( proj2 | ( ( TOP-REAL 2 ) | D ) ) | K1 ) . q = ( proj2 | K1 ) . q & ( proj2 | K1 ) . q = ( proj2 | K1 ) . q & ( proj2 | K1 ) . q = ( proj2 | K1 ) . q & ( proj2 | K1 ) . q = ( proj2 | K1 ) . q ;
s , -4 |= ( ( H , ( H , i ) ) |= ( ( H , i ) ) . ( i , j ) iff s |= ( H , i ) . ( i , j ) ) ;
len ( f5 + 1 ) + 1 = card ( ( support b1 ) + 1 ) .= card ( support b1 ) + 1 .= card ( support b1 ) + 1 .= ( support b1 ) + ( card ( support b1 ) ) + 1 ) .= card ( support b1 ) + ( card ( support b1 ) + 1 ) .= card ( support b1 ) + card ( support b1 ) .= ( support b1 ) + card ( support b1 ) + 1 .= ( support b1 ) + card ( support b1 ) + card ( support b1 ) + card ( support b1 ) + card ( support b1 ) + card ( support b1 ) +
consider z being Element of L1 such that z >= x and z >= y and z >= x and z >= y ;
LSeg ( Gauge ( D , n ) , len Gauge ( D , n ) ) /\ LSeg ( |[ ( ( D , n ) , ( D , n ) ) `1 ) = { w + ( ( D , n ) + 1 ) / 2 ) ;
lim ( ( ( ( ( f `| Z ) ^ ) (#) g ) `| Z ) = diff ( ( f `| Z ) ^ ) . x0 ;
P [ i , pr1 ( f , g ) . i ] ;
for r be Real st 0 < r ex m be Nat st for k be Nat st k <= m holds ||. ( seq . k ) - ( lim seq ) .|| < r
let X being set , P , Q being Subset of X , a , b , c , d being Element of X st a in P & b in Q & c in Q & d in Q & a = b holds a = b & b = c & c = d & a = d & a = c & b = c & a = c & c = d & a = d & b = d & c = d & a = d & b = d & a = d & a = d & b = d & a = d & c = d & d = d & a = d & d = d
Z c= dom ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ;
ex j being Nat st j in dom ( l ^ <* x *> ^ <* y *> ) & ( len l ) = i + j & j + 1 = i + j + 1 & i + j = i + j + 1 ;
for u , v being VECTOR of V st 0 < r & for u being VECTOR of V st u < r holds r * ( u + v ) + r * ( v + u ) in [. r , s .] holds r * ( v + u ) + r * ( v + u ) ) in [. r , s .]
A , B , C is_collinear & A c= B implies A , B , C is_collinear & B , C is_collinear & C , D is_collinear & C , D is_collinear & C , D is_collinear & C , D is_collinear & A , D is_collinear & C <> D implies B , D , C is_collinear & A , D , D is_collinear & C , D , D is_collinear & A , D , D is_collinear & A , D , D is_collinear & B , D , E is_collinear & A , D , D is_collinear & B , D is_collinear & B , D , E is_collinear & A , D , E is_collinear & B , D , E is_collinear & B , D is_collinear
- Sum <* v , u *> = - ( v + u ) .= - ( v + u ) + ( - u ) .= - ( v + u ) + ( - u ) .= - ( v + u ) + ( - u ) .= - ( v + u ) + ( - u ) + ( - u ) .= ( - u ) + ( - u ) + ( - u ) + ( - u ) ;
( Exec ( a := b , s ) ) . IC SCM SCM R = ( Exec ( a := b , s ) ) . IC SCM R .= succ IC s .= IC s .= IC s ;
consider h being Function such that f . a = h and dom h = I and for x being element st x in I holds h . x in the carrier of J and h . x in the carrier of J ;
let S1 , S2 be non empty RelStr , S1 , S2 be non empty Subset of S1 , S2 , S2 being non empty Subset of S2 , S1 being Subset of S2 , S2 being non empty Subset of S2 st S1 = S1 & S2 = S2 & S2 is non empty & S1 is non empty & S2 is directed & S2 is directed holds S1 is directed
card X = 2 implies ex x , y st x in X & y in Y & x in X & y in Y & x in Y & y in X & x in Y ;
E-max ( C ) in rng ( Cage ( C , n ) ) & W-min ( C , n ) in rng Cage ( C , n ) ;
let T being decorated tree , p , q , r be Element of dom T , T , x , y be Element of dom T , T , y be Element of dom T ;
[ i2 + 1 , j2 ] in Indices G & [ i2 , j2 ] in Indices G & [ i2 , j2 ] in Indices G implies f /. ( i2 + 1 , j2 ) = G * ( i2 , j2 ) ;
cluster INT.Ring ( k , n ) -> natural & k > 0 implies k divides ( k -' n ) * ( k -' n ) ;
dom ( F " ) = the carrier of ( ( the carrier of X1 ) \/ the carrier of X2 ) & rng ( F " ) = the carrier of ( the carrier of X2 ) & rng ( F " ) = the carrier of ( the carrier of X1 union X2 ) & rng ( F " ) = the carrier of X1 union the carrier of X2 ;
consider C being finite Subset of V such that C c= A and card C = card C + 1 and card C = card C + 1 ;
V is prime implies for X , Y being Subset of T st X /\ Y c= the carrier of T holds X c= Y or Y c= X
set X = { F ( ) where v1 is Element of B : P [ v1 ] } , Y = { F ( ) where v1 is Element of B : P [ v1 ] } ;
angle ( p1 , p2 , p3 ) = 0 & angle ( p1 , p2 , p3 ) = 0 & angle ( p1 , p2 , p3 ) = 0 & angle ( p1 , p2 , p3 ) = 0 & angle ( p1 , p2 , p3 ) = 0 & angle ( p1 , p3 , p2 , p3 ) = 0 ;
- sqrt ( ( q `1 ) ^2 + ( q `2 ) ^2 ) = - ( q `1 ) ^2 + ( q `2 ) ^2 ) .= - ( - 1 ) ^2 + ( q `2 ) ^2 ;
ex f being Function of I[01] , TOP-REAL 2 st f is continuous one-to-one & f . 0 = p1 & f . 1 = p2 & f . 0 = p1 & f . 1 = p1 & f . 1 = p2 & f . 0 = p1 & f . 1 = p3 & f . 1 = p1 & f . 1 = p2 & f . 0 = p1 & f . 1 = p2 & f . 1 = p1 & f . 1 = p1 & f . 1 = p3 = p3 & f . 1 = p3 = p4 & f . 1 = p3 & f . 1 = p4 & f . 1 = p4 & f . 1 =
attr f is partial & u in dom ( f , u ) & f | [' x0 , y0 '] & u in dom ( f , u ) ;
ex r , s st x = [ r , s ] & s < G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `1 < r & r < G * ( 1 , 1 ) `1 ;
assume that f is FinSequence and 1 <= t and t <= G * ( t , 1 ) and 1 <= ( G * ( t , 1 ) ) `1 ;
attr i in dom G means : where : r * ( reproj ( i , x ) ) = r * reproj ( i , x ) ;
consider c1 , c2 being bag of o1 , L such that ( /* c1 ) /. k = <* c1 , c2 *> and ( /* c1 ) /. k = <* c1 , c2 *> ;
x0 in { |[ r1 , s1 ]| : r1 < r1 & s1 < s2 & s1 < s2 } ;
Cl ( X ^ Y ) . k = the carrier of ( X \/ Y ) . k .= ( the carrier of Y ) . k .= ( the carrier of X ) . k .= ( the carrier of Y ) . k ;
attr len M1 = len M1 & width M1 = width M2 & width M2 = width M2 & width M1 = width M2 implies M1 * M2 = M2 * M1 + M2 * M2 + M2 * M2 + M1 * M2 + M1 * M2 + M1 * M2 + M1 * M2 + M1 * M2 + M1 * M2 + M1 * M2 + M1 * M2 + M1 * M2 + M1 * M2 + M1 * M2 + M1 * M2 + M1 * M2 + M1 * M2 + M1 * M2 + M1 * M2 * M2 * M2 * M2 + M1 * M2 + M1 * M2 + M1 * M2 * M2 + M1 * M2 + M1 * M2 + M1
consider g2 be Real such that 0 < g2 and for y be Point of S st y in { g2 where g2 is Point of TOP-REAL 2 : ||. g2 - g2 .|| < g2 } holds g2 . y - g2 . y < g2 . y - g2 . y ;
assume x < sqrt ( - b ) + sqrt ( ( - a ) * sqrt ( b ) + sqrt ( b ) ) ;
( G1 '&' G2 ) . i = ( <* 3 *> ^ ( G1 ^ G2 ) ) . i & ( G1 ^ G2 ) . i = ( <* 3 *> ^ ( G1 ^ G2 ) . i ;
for i , j st [ i , j ] in Indices ( M1 + M2 ) & [ i , j ] in Indices M1 & [ i , j ] in Indices M1 + M2 implies M1 * ( i , j ) + M2 * ( i , j ) < M1 * ( i , j ) + M2 * ( i , j )
let f being FinSequence of NAT , i be Element of NAT , j be Element of NAT st i in dom f & j <= len f & i <= len f holds i divides j & j divides f /. ( i + 1 ) ;
assume F = { [ a , b ] where a , b is Element of X : for c being Element of X st c in B holds a c= c } ;
b2 * ( q2 - q2 ) + ( - ( q2 - q2 ) * ( q2 - q2 ) ) + ( - ( q2 - q2 ) * ( q2 - q2 ) ) * ( q2 - q2 ) + ( - ( q2 - q2 ) * ( q2 - q2 ) ) * ( q2 - q2 ) + ( - q2 ) * ( q2 - q2 ) * ( q2 - q2 ) ) = 0. TOP-REAL 2 + ( - q2 ) * ( q2 - q2 ) ;
Cl D = { D where D is Subset of T : D is open & D is open } ;
attr seq is summable means : Def5 : for n st n >= 1 holds seq . n = seq . n & seq . n = seq . n ;
dom ( ( ( TOP-REAL 2 ) | D ) | K1 ) = ( the carrier of ( TOP-REAL 2 ) | D ) /\ K1 .= the carrier of ( ( TOP-REAL 2 ) | D ) .= the carrier of ( ( TOP-REAL 2 ) | D ) ;
[ X \to Z , Y ] is full SubRelStr of ( X --> Z ) |^ ( the carrier of Y ) & [ X , Y ] is full SubRelStr of ( X --> Z ) |^ ( the carrier of Y ) ;
( G * ( 1 , j ) ) `2 = ( G * ( 1 , j ) ) `2 & ( G * ( 1 , j ) ) `2 = ( G * ( 1 , j ) `2 ) `2 ;
synonym m1 c= ( m1 + m2 ) & for for ( m1 + m2 ) * ( m + k ) = ( m1 + m2 ) * ( m + k ) ;
consider a being Element of [: B ( ) , C ( ) :] such that x = { F ( ) where b is Element of B ( ) : a in A ( ) } ;
We say that { multiplicative loop s , w -> Element of the carrier of G , s -> Element of the carrier of G , s -> Element of the carrier of G means : where the carrier of it = the carrier of it & the carrier of it = the carrier of it & the carrier of it = the carrier of it & the carrier of it = the carrier of it iff the carrier of it is set ;
indx ( a , b , c ) + indx ( a , b , c ) + indx ( a , b , c ) = b + indx ( a , b , c ) .= b + indx ( a , b , c ) + indx ( a , b , c ) .= b + indx ( a , b , c ) + indx ( a , b , c ) ;
cluster redefine func redefine func redefine func redefine func ( 1 , 1 ) --> ( 1 , 1 ) -> Element of ( 1 , 1 ) -tuples_on ( the carrier of K ) ;
1- ( ( 1 - ( ( 1 - ( 1 - ( 1 / 2 ) ) ) * ( 1 + ( 1 / 2 ) ) ) + ( 1 / 2 ) * ( 1 / 2 ) ) = ( - 1 / 2 ) * ( ( 1 / 2 ) * ( 1 / 2 ) ) + ( - 1 / 2 ) * ( 1 / 2 ) * ( 1 / 2 ) ) .= ( - 1 / 2 ) * ( 1 / 2 ) ;
eval ( a | ( n , L ) , x ) = eval ( a | ( n , L ) ) * eval ( a , x ) .= a * eval ( a , x ) .= a * 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 ) * eval ( a , x ) * eval ( a , x ) .= a * eval ( a , x ) * eval ( a , x ) * eval ( a , x ) * eval ( a , x ) * eval ( a , x ) * eval (
assume that the TopStruct of S = the TopStruct of T and for D being Subset of S st D = the carrier of T holds D is open & D is open and for V being Subset of T st V in V holds V is open & V is open & V is open & V is open ;
assume that 1 <= k + 1 and k + 1 <= len w & T . ( k + 1 ) = T . ( k + 1 ) & T . ( k + 1 ) = T . ( k + 1 ) ;
2 * ( a |^ ( n + 1 ) ) + ( 2 * ( b |^ ( n + 1 ) ) + b |^ ( n + 1 ) ) >= ( a |^ ( n + 1 ) + b |^ ( n + 1 ) ) + ( 2 * ( b |^ ( n + 1 ) ) + ( 2 * ( b |^ ( n + 1 ) ) + b |^ ( n + 1 ) ) ) ;
M , v2 / ( x , y ) |= ( ( x , y ) / ( x , y ) ) / ( x , y ) ) & ( ( x , y ) / ( x , y ) ) / ( x , y ) = ( x , y ) / ( x , y ) / ( x , y ) / ( x , y ) ) ;
assume that f is_differentiable_on l and for x0 st x0 in dom f holds 0 < ( f `| Z ) . x0 - f . x0 ;
let G1 being finite _Graph , W being Subset of G , W being Subset of G , e being set st e in W & W is open & W is open & W is open holds W is Subset of G & W is open & W is open & e is open & W is open & W is open ;
not \it it is not empty iff is not empty & not is not empty & not is not empty & not is not empty & not is not empty & not is not empty & not is not empty & not is not empty & not empty & not is not empty & not is not empty & not empty & not is not empty & not is not empty & not is not empty & not is not empty & not empty & not is not empty & not is not empty & not empty is not empty & not is not empty & not empty & not is not empty & not empty & not empty & not empty & not empty & not is not empty & not empty & not empty & not empty is not empty & not is not empty & not empty is not empty & not empty & not empty & not
Indices ( ( - 1 ) * ( i , j ) ) = [: dom ( - 1 ) , Seg n :] & len ( - 1 ) = [: Seg n , Seg n :] & len ( - 1 ) * ( i , j ) = [: Seg n , Seg n :] ;
let G1 , G2 be Subgroup of O , G , H , G be Subset of O , H being Subset of ( the carrier of G ) | ( the carrier of H ) , the carrier of G , H ;
UsedIntLoc ( f , 3 ) = { ( f . intloc 0 ) , f . intloc 0 , f . intloc 0 , f . intloc 0 , f . intloc 0 , 1 , 1 , f . intloc 0 , 1 , f . intloc 0 , 1 , f . intloc 0 , 1 , f . intloc 0 , 1 , f . intloc 0 , 1 , 1 , f . intloc 0 , 1 , 1 , 1 , 5 ) ;
for f1 , f2 being FinSequence of F st f1 ^ f2 is Element of the carrier of ( len f1 ) -tuples_on the carrier of ( len f1 ) -tuples_on the carrier of ( len f1 ) -tuples_on the carrier of ( len f1 ) -tuples_on the carrier of ( len f1 ) -tuples_on the carrier of ( len f1 ) -tuples_on the carrier of ( len f1 ) -tuples_on the carrier of ( len f1 ) -tuples_on the carrier of ( len f1 ) -tuples_on the carrier of ( len f1 ) -tuples_on the carrier of ( len f1 ) -tuples_on the carrier of ( len f1 ) -tuples_on the carrier of ( len f1 ) -tuples_on the carrier of ( len f1 ) -tuples_on the carrier of ( len f1 ) -tuples_on the carrier of ( len f1 ) -tuples_on the carrier of ( len f1 ) -tuples_on the carrier of ( len
sqrt ( ( p `1 ) ^2 + ( p `2 ) ^2 + ( p `2 ) ^2 ) = sqrt ( ( p `2 ) ^2 + ( p `2 ) ^2 + ( p `2 ) ^2 ) ;
let x1 , x2 be Element of REAL n , x1 , x2 be Element of REAL n , x2 be Element of REAL n , y1 be Element of REAL n , y2 be Element of REAL n , y2 be Element of REAL n , y2 be Element of REAL n ;
for x , y st x in dom ( cos | A ) holds ( cos | A ) . x = ( cos | A ) . y
let T being non empty TopSpace , P , Q being Subset of T , x being Point of T st P c= the topology of T & x in Q & P is open holds P is open
( a 'or' b ) . x = 'not' ( a 'or' b ) . x .= TRUE ;
for e being set st e in A & e in X & ex X1 being Subset of X st X1 = X & X1 is open & X2 is open & X1 is open & X1 is open & X1 is open & X2 is open & X1 is open & X1 is open & X1 is open & X1 is open & X2 is open & X1 is open & X1 is open & X2 is open & X1 is open & X2 is open & X1 is open & X1 is open & open implies X1 is open & open is open & open implies X1 is open & open is open & open is open & open is open & open is open & open is open & open is open & open is open & open is open & open is open & open implies open is open & open is
for i be set st i in the carrier of S for f being Function of ( the carrier of S ) . i , ( the carrier of S ) . i holds F ( i ) = f . i & F ( i ) = f . i & F ( i ) = f . i ;
for v , w st for x , y st x <> y holds J . ( x , y ) = J . ( y , x ) holds J . ( x , y ) = J . ( x , y )
card D = card ( D1 + D2 ) + card ( D1 + D2 ) .= 2 * ( D1 + D2 ) + card ( D1 + D2 ) .= 2 * ( D1 + D2 ) + card ( D1 + D2 ) + card ( D1 + D2 ) .= 2 * ( D1 + D2 ) + ( D1 + D2 ) + ( D1 + D2 ) ) + ( D1 + D2 ) + ( D1 + D2 ) + ( D1 + D2 ) ) .= 2 * ( D1 + D2 ) + ( D1 + D2 ) .= 2 * ( D1 + D2 ) + ( D1 + D2 ) + ( D1 + D2 ) + ( D1 + D2 ) .= 2 * ( D1 + D2 ) .= 2 * ( D1 + D2 ) + ( D1 +
IC Exec ( i , s ) = ( 0 .--> ( s . 0 ) ) .= ( 0 .--> ( s . 0 ) ) .= ( 0 .--> ( s . 0 ) ) .= ( 0 .--> ( s . 0 ) ) .= ( 0 .--> ( s . 0 ) ) .= ( 0 .--> ( s . 0 ) ) .= ( 0 .--> ( s . 0 ) ) .= ( 0 .--> ( s . 0 ) ) .= ( 0 .--> ( s . 0 ) ) .= ( 0 .--> ( s . 0 ) .= ( s . 0 ) .= ( s . 0 ) .= ( s . 0 ) .= ( s . 0 ) .= ( s . 0 ) .= ( s . 0 ) .= ( s . 0 ) .= ( s . 0
len ( f /^ ( i1 -' 1 ) ) + 1 = len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) + 1 ;
for a , b , c being Element of NAT st 1 <= a & a <= b & b <= c & c <= d holds a + b + c = b + c
let f being FinSequence of TOP-REAL 2 , p , q be Point of TOP-REAL 2 , r be Real st r = ( r * ( 1 / 2 ) ) * ( 1 / 2 ) & p <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & r = ( r / 2 ) * ( 1 / 2 ) + ( r / 2 ) * ( 1 / 2 ) ) holds p = r / 2 ;
lim ( ( ( ( curry ( k ) ) + 1 ) to_power ( k + 1 ) ) to_power ( k + 1 ) ) = ( lim ( ( abs ( k + 1 ) ) to_power ( k + 1 ) ) to_power ( k + 1 ) ) ;
z2 = 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 ) ;
[ f . 0 , f . 3 ] in id the carrier of G & [ f . 0 , f . 3 ] in the InternalRel of G & [ f . 0 , f . 3 ] in the InternalRel of G ;
let G being Subset-Family of B , R being Subset of A , B being Subset of B st G = { [ X , B ] } & ( ( Intersect ( F ) ) . ( X , B ) ) . ( X , B ) = ( Intersect ( F ) ) . ( X , B ) ;
CurInstr ( P1 , Comput ( P1 , s1 , i + 1 ) ) = halt SCMPDS .= halt SCMPDS .= ( halt SCMPDS ) .= halt SCMPDS .= IC SCMPDS .= IC SCMPDS .= IC SCMPDS .= IC SCMPDS .= IC SCMPDS .= IC SCMPDS .= IC SCMPDS .= IC SCMPDS .= IC SCMPDS + IC SCMPDS .= IC SCMPDS + IC SCMPDS .= IC SCMPDS + IC SCMPDS .= IC SCMPDS + IC SCMPDS .= IC SCMPDS + IC SCMPDS ;
assume that a on M and b on M and c on N and d on N and p <> q and p <> q and p <> q and q <> q and p <> q and p <> q and q <> q and p <> q & p <> q & q <> q & p <> q & p <> q & p <> q & p <> q & q <> q & p <> q & p <> q & p <> q & q <> q & p <> q & p <> q & p <> q & p <> q & p <> q & p <> q & p <> q & q <> q & p <> q & p <> q & p <> q & q <> q & p <> q & p <> q & p <> q & p <> q & q & p <> q & p <> q & q & p <> q & q <> q & p <> q & p <> q & q <> q
assume that T is \hbox { $ T , 4 , T and for F being Subset-Family of T st F is closed & F is closed holds card F <= 0 & card F <= 2 ;
for g1 , g2 st g1 in ]. x0 - r , x0 + r .[ & |. ( f | X ) . g1 - r .| <= ( f | X ) . g2 holds |. ( f | X ) . g2 - r .| <= r * ( f | X ) . g2 - r * ( f | X ) . g2 )
( ( signature ( - ( 1 / 2 ) ) + ( - ( 1 / 2 ) ) * ( ( - ( 1 / 2 ) ) + ( - ( 1 / 2 ) ) * ( - ( 1 / 2 ) ) * ( - ( 1 / 2 ) ) * ( - ( 1 / 2 ) ) * ( - ( 1 / 2 ) ) * ( - ( 1 / 2 ) ) * ( - ( 1 / 2 ) ) * ( - ( 1 / 2 ) ) = ( - ( 1 / 2 ) * ( - ( 1 / 2 ) * ( - ( 1 / 2 ) * ( - ( 1 / 2 ) * ( - ( 1 / 2 ) ) * ( - ( 1 / 2 ) ) * ( - ( 1 / 2 ) * ( - ( 1 / 2 ) * ( - ( 1 / 2 ) ) ) * ( - ( 1 / 2 ) ) * (
F . i = F /. i .= 0. ( R ^ ( n + 1 ) ) .= 0. ( R ^ ( n + 1 ) ) .= ( ( ( n + 1 ) |-> 0. ( n + 1 ) --> 0. ( n + 1 ) ) ^ ( n + 1 ) ) .= <* ( ( n + 1 ) --> 0. ( n + 1 ) ) *> ;
ex y being set st y = f . n & dom f = NAT & for x being element st x in NAT holds f . x = F ( x ) ;
redefine func f * F -> FinSequence of V means : Def5 : for x being Element of the carrier of V st x in dom it holds it . x = F . x * ( x | ( i -' 1 ) ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 *> \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 , x5 \/ x5 , x5 ,
for n being Nat , x being set st x = h . n & h . n = o & x in InputVertices S & h . n = o & x = h . n & h . n = o & x = o & h . n = x & x = o & h . n = o & x = o & h . n = o & x = h . n & x = o ;
ex S1 be Element of QC-WFF ( Al ( ) ) st ( for e being Element of NAT holds ( P [ e ] ) implies ( P [ e ] ) & ( P [ e ] ) & ( P [ e ] implies P [ e ] ) & ( P [ e ] ) implies ( P [ e ] ) & ( P [ e ] ) & ( P [ e ] ) & ( P [ e ] ) implies P [ e ] ) implies ( P [ e , d , d ] ) & ( P [ e , d ] ) & ( P [ e , d ] ) & ( P [ e , d ] ) implies ( P [ e , d ] ) & ( P [ e , d ] ) & ( P [ e , d ] ) & ( P [ e , d , d ] ) implies ( P [ e
consider P being FinSequence of ( the carrier of G2 ) * , P being Element of the carrier of G2 such that p = P & P . i = P . i & P . i = Q . i ;
let T1 , T2 be non empty TopSpace , T1 , T2 be Function of the carrier of T1 , the carrier of T2 | the carrier of T2 , the carrier of T2 | the carrier of T2 ;
assume that f is partial & r (#) u = r * u + r * v and r (#) ( f (#) u ) = r * u + r * v + r * u ;
defpred P [ Nat ] means for F being FinSequence of the carrier of K st len F = $1 & for n being Nat st n in dom F holds F . n = F . n holds $1 = Sum ( F | n ) ;
ex j st 1 <= j & j < width GoB f & ( GoB f ) * ( 1 , j ) `1 <= ( GoB f ) * ( 1 , j ) `1 ;
defpred U [ set , set ] means ex F be Subset-Family of T st F is $2 & ( for n being Nat st n in dom F holds F . n = F . n ) & ( for n being Nat st n in dom F holds F . n = G . n ) implies for n being Element of NAT st n in dom F holds F . n is union of T & for n being Element of NAT st n in dom F holds F . n is union of the carrier of T holds F is union of T holds F is union of T & for n being Element of NAT st n in dom F holds F is union of T & for n being Element of NAT st n in dom F holds F is union of T holds F is union of T holds F is union of T holds F is union of T holds F is connected & for n being Element of T st n in the carrier of T st n in dom F holds F
for p2 being Point of TOP-REAL 2 st LE p1 , p2 , P & LE p1 , p2 , P & LE p1 , p2 , P holds LE p1 , p2 , P , p1 , p2
f in Funcs ( E , H ) iff for x , y st x <> y & y <> x & x in y holds f . x = f . y & g . y = f . x & f . y = g . y & f . x = h . y ;
ex p2 being Point of TOP-REAL 2 st x = p2 & |. p2 .| = |. p2 .| & |. p2 .| = |. p2 .| & |. p2 .| = |. p2 .| & p2 <> p2 & p2 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <>
assume for d being Element of NAT st d <= ( ( d + ( d + 1 ) ) / ( d + 1 ) ) holds d <= ( d + ( d + 1 ) / ( d + 1 ) / ( d + 1 ) ) / ( d + 1 ) ;
assume that s <> t and not s in LSeg ( x , r ) and not x in LSeg ( x , r ) and not x in LSeg ( x , r ) and y in LSeg ( x , r ) and y in LSeg ( x , r ) and y in LSeg ( x , r ) ;
given r such that 0 < r and for x1 , x2 st x1 in dom f & x2 in dom f & f . x1 - f . x2 < r holds |. f . x1 - f . x2 .| < r ;
( p | x ) | ( x | ( x | x ) ) = ( ( p | x ) | ( x | x ) ) | ( x | x ) ) | ( x | ( x | x ) ) ;
assume that x in dom sec and h + h in dom sec and h + c = ( 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 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2
assume that i in dom A and i > 1 and j in dom B and B * ( i , j ) = B * ( i , j ) ;
for i be non zero Element of NAT st i in Seg n & i in Seg n holds h . i = <* 1_ F_Complex *> ^ ( i , n ) & h . i = h . i & h . i = h . i & h . i = h . i & h . j = i & h . i = j ;
( ( ( ( ( ( ( b1 ) ) '&' ( b2 ) ) ) '&' ( ( ( ( b1 ) '&' ( ( b2 ) ) '&' ( ( ( ( b1 ) ) '&' ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
assume for x holds f . x = ( \HM { the } \HM { function } ) . x & ( \HM { the } \HM { function } ) . x = ( \HM { the } \HM { function } ) . x ) & ( \HM { the } \HM { function } ) . x = ( \HM { the } \HM { function } ) . x ) & ( \HM { the } \HM { function } \HM { sin } ) . x = ( \HM { the } \HM { function } ) . x ) & ( for x st x in Z holds ( \HM { the } \HM { function } ) . x = ( \HM { the } \HM { function } ) . x = ( \HM { the } \HM { x ) & ( for x ) . x = ( \HM { the } ) . x ) & ( for x ) . x ) & ( \HM { the } ) . x = ( \HM { the } \HM { function } )
consider R1 , R2 be Real such that R1 = ( ( Re ( F . n ) ) * ( ( Im ( F . n ) ) * ( ( Im ( F . n ) ) * ( ( F . n ) ) * ( ( F . n ) ) * ( ( F . n ) ) * ( ( F . n ) * ( ( F . n ) ) * ( ( F . n ) * ( ( F . n ) ) * ( ( F . n ) ) ) ) ) ) ;
ex k be Element of NAT st x0 = k & 0 < k & for x be Element of product G st x in X holds ||. ( f , x ) - ( f , x ) .|| < r ;
x in { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 *> implies x in { x1 , x2 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 ,
( G * ( j , i ) ) `2 = ( G * ( 1 , i ) ) `2 .= ( G * ( 1 , j ) ) `2 .= ( G * ( 1 , j ) ) `2 .= ( G * ( 1 , j ) ) `2 .= ( G * ( 1 , j ) ) `2 .= ( G * ( 1 , j ) ) `2 ;
f1 * p = p * ( the Arity of S1 ) .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o ;
redefine func <* T , P , T *> -> Function of T , P means : where : for p , q st p in T & q in T & p in T & q in T & p in T & q in T & p in T & p in T } ;
F /. ( k + 1 ) = F . ( k + 1 ) .= F . ( k + 1 ) .= F . ( k + 1 ) .= F . ( k + 1 ) .= F . ( k + 1 ) .= F . ( k + 1 ) .= F . ( k + 1 ) .= F . ( k + 1 ) .= F . ( k + 1 ) .= F . ( k + 1 ) .= F . ( k + 1 ) .= F . ( k + 1 ) .= F . ( k + 1 ) .= F . ( k + 1 ) .= F . ( k + 1 ) .= F . ( k + 1 ) .= F . ( k + 1 ) .= F . ( k + 1 ) .= F . ( k + 1 ) .= F . ( k + 1 ) .= F . ( k + 1 ) .= F . ( k + 1 ) .= F . ( k + 1 ) .= F . (
let A , B , C be Matrix of n , K , len B , width B , width B , width B , width B , C , D , D , E , F , J , M , N , M , N , N , M , N , N , M , N , N , M , N , N , M , N , M , N , M , N , M , N , N , M , N , N , M , N , M , N , M , N , N , M , N , N , N , M , N , M , N , N , M , N , N , M , M , M , N , M , N , N , N , N , N , M , N , M , N , N , N , N , M , M , M , M , M , N , M , N , M , N , M , M , N , M , M , M
seq . ( k + 1 ) = 0. ( ( seq ^ ) . ( k + 1 ) ) .= ( seq ^ ) . ( k + 1 ) .= ( seq ^ ) . ( k + 1 ) .= ( seq ^ ) . ( k + 1 ) .= ( seq ^ ) . ( k + 1 ) ;
assume that x in ( the carrier of ( the carrier of C ) ) & [ x , y ] in the InternalRel of ( the carrier of C ) & [ x , y ] in the InternalRel of the carrier of C ;
defpred P [ Element of NAT ] means for f being Element of NAT st len f = $1 holds ( ( VAL ^ <* f *> ) ^ <* f /. ( $1 + 1 ) *> ) . $1 = ( ( VAL ^ ( g ^ <* f /. ( $1 + 1 ) ) ) ^ ( ( ( VAL ^ ( g /^ ( $1 ) ) ) . ( ( g /^ ( $1 ) ) . ( len f ) ) ) ;
assume that 1 <= k and k + 1 <= len f and f /. k = G * ( i , j ) and f /. k = G * ( i , j ) ;
assume that -4 < 1 and ( - 1 ) * ( 1 + ( 1 + ( 1 + ( 1 + ( 1 / 2 ) ) * ( 1 + ( 1 / 2 ) ) ) ) * ( 1 / 2 ) ) * ( 1 / 2 ) ) + ( 1 / 2 ) * ( 1 / 2 ) * ( 1 / 2 ) ) * ( 1 / 2 ) ) * ( 1 / 2 ) * ( 1 / 2 ) ) * ( 1 / 2 ) * ( 1 / 2 ) * ( 1 / 2 ) * ( 1 / 2 ) * ( 1 / 2 ) * ( 1 / 2 ) * ( 1 / 2 ) * ( 1 / 2 ) * ( 1 / 2 ) * ( 1 / 2 ) * ( 1 / 2 ) * ( 1 / 2 ) * ( 1 / 2 ) * ( 1 / 2 ) * ( 1 / 2 ) * ( 1 / 2 ) * ( 1 / 2 ) * (
let M be non empty TopSpace , x be Point of M , y be Point of M , x be Point of M , r be Point of M , x be Point of M , y be Point of M , x be Point of M , y be Point of M , r be Point of M , x be Point of M , y be Point of M , r be Point of M , x , Point of M , y be Point of M , r be Point of M , s be Point of M , s be Point of M , s be Point of M , s be Point of M , s , s , t be Point of M , t be Point of M , t be Point of M , t be Point of M , t be Point of M , t , s , t , s , s , t be Point of M , x be Point of M , x , s be Point of M , x , y be Point of M , x be Point of M
defpred P [ Element of omega ] means for f being Element of omega st f . $1 in Z holds ( f | Z ) . $1 = f . $1 & ( f | Z ) . $1 = 0 ) implies ( ( f | Z ) . $1 = 0 ) & ( ( f | Z ) . $1 = 0 ) & ( ( f | Z ) . $1 = 0 ) ;
defpred P1 [ Nat , Element of C ] means ||. ( ( f . $1 ) - ( f . $1 ) ) - ( f . $1 ) .|| < \frac ( 1 / 2 ) * ( ( f . $1 ) - ( f . $1 ) ) & ( ||. f . $1 - f . $1 .|| ) * ( ( f . $1 - f . $1 ) ) < sqrt 2 ;
( f ^ mid ( g , 2 , len g + 1 ) ) . i = ( g ^ mid ( g , 2 , len g + 1 ) ) . i .= ( g ^ mid ( g , 2 , len g + 1 ) ) . i .= ( g ^ mid ( g , 2 , len g + 1 ) ) . i .= ( g ^ mid ( g , 2 , len g + 1 ) ) . i ;
sqrt ( 1 / 2 * ( n + 1 ) ) = ( 1 / 2 ) * ( n + 1 / 2 ) .= ( 1 / 2 ) * ( n + 1 / 2 ) .= ( 1 / 2 ) * ( n + 1 / 2 ) .= ( 1 / 2 ) * ( n + 1 / 2 ) ;
defpred P [ non empty RelStr ] means for G being non empty RelStr , A being Subset of G st the carrier of G in the carrier of G & the carrier of G = the carrier of G & the carrier of G = the carrier of G & the carrier of G = the carrier of G & the carrier of G = the carrier of G & the carrier of G = the carrier of G & the carrier of G = the carrier of G & the carrier of G or the carrier of G = the carrier of G or the carrier of G or the carrier of G or the carrier of G or the carrier of G or the carrier of G = the carrier of G or the carrier of G or the carrier of G = the carrier of G or the carrier of G or the carrier of G = G or the carrier of G = the carrier of G or the carrier of G or the carrier of G or the carrier of G = the carrier of G or the carrier of G = the carrier of G or the carrier of G = the carrier of G
assume not f in Ball ( u , r ) & 1 <= m & m <= len u & u <> 0. TOP-REAL 2 & f . m = u & f . ( m + 1 ) = v & f . ( m + 1 ) = v & f . ( m + 1 ) = v & f . ( m + 1 ) = v & f . ( m + 1 ) = v & f . ( m + 1 ) = v & f . ( m + 1 ) = v & f . ( m + 1 ) = v & f . ( m + 1 ) = v & f . ( m + 1 ) = v & f . ( m + 1 ) = v & f . ( m + 1 ) = v & f . ( m + 1 ) = v & f . ( m + 1 ) = v & f . ( m + 1 ) = v & f . ( m + 1 ) = u & f . ( m + 1 ) = v & f . ( m + 1 ) = v &
defpred P [ Element of NAT ] means ( Sum ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . $1 ;
for x being Element of product F holds x in ( the carrier of F ) \/ the carrier of G & x in the carrier of F implies x in the carrier of G
( x " ) |^ ( n + 1 ) = ( x |^ ( n + 1 ) ) * ( x |^ ( n + 1 ) ) .= ( x |^ ( n + 1 ) ) * ( x |^ ( n + 1 ) ) .= ( x |^ ( n + 1 ) ) * ( x |^ ( n + 1 ) ) .= ( x |^ ( n + 1 ) ) * ( x |^ ( n + 1 ) ) ;
DataPart Comput ( P +* I , s , LifeSpan ( P +* I , s ) + 3 ) = DataPart Comput ( P +* I , s , k + 1 ) .= DataPart Comput ( P +* I , s , k + 1 ) ;
given r such that 0 < r and ]. x0 - r , x0 + r .[ c= dom ( f1 + f2 ) /\ dom f2 ;
assume that X c= dom ( f1 (#) f2 ) and ( f1 (#) f2 ) | X is bounded and ( f1 (#) f2 ) | X is bounded and ( f1 (#) f2 ) | X is bounded and ( f1 (#) f2 ) | X is bounded and ( f1 (#) f2 ) | X is bounded and ( f1 (#) f2 ) | X is bounded and ( f1 (#) f2 ) | X is bounded & ( f1 (#) f2 ) | X is bounded & ( f1 (#) f2 ) | X is bounded & ( ( f1 (#) f2 ) | X is bounded & ( ( ( f1 (#) f2 ) | X is bounded & ( f1 (#) f2 ) | X is bounded & ( ( f1 (#) f2 ) | X is bounded & ( ( f1 (#) f2 ) | X is bounded & & ( ( f1 (#) f2 ) | X is bounded & ( f1 (#) f2 ) | X is bounded & ( ( f1 (#) f2 ) | X is bounded & ( ( f1 (#) f2 ) | X is bounded & ( ( f1 (#) f2 ) | X is bounded & ( ( ( f2 ) | X is bounded &
let L being complete LATTICE , X being Subset of L st X = sup ( X ) holds X is directed Subset of L & for x being Element of L st x in X holds x is Subset of L holds x is "/\" of X , L & x is Element of L ;
consider i being Element of NAT such that i in dom A and A /. i = m *' ( p *' ( p *' ( p *' ( p *' ) ) ) ) ;
( f1 - f2 ) /* ( h + c ) = ( f1 - f2 ) /* ( h + c ) .= ( f1 - f2 ) /* ( h + c ) .= ( f1 - f2 ) /* ( h + c ) ;
ex p1 being Element of QC-WFF ( Al ( ) ) st F . ( p , p1 ) = g & for g being Element of QC-WFF ( Al ( ) ) st g in D ( ) holds g . ( p , g ) = F ( g . ( p , g ) ) ;
( mid ( f , i , len f -' 1 ) , i ) = ( mid ( f , i , len f -' 1 ) ) ^ mid ( f , i , len f -' 1 ) .= f ^ mid ( f , i , len f -' 1 ) ;
( p ^ q ) . ( len p + k ) = ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) ;
len ( indx ( D2 , D1 , j ) ) + 1 = indx ( D2 , D1 , j ) + 1 .= indx ( D2 , D1 , j ) + 1 ;
x * y = ( ( - ( ( - ( x - y ) ) * y ) ) * ( ( - ( x - y ) * y ) * ( ( - ( x - y ) * y ) * ( - ( x - y ) * y ) ) ) .= ( - ( x - y ) * ( x - y ) ) * ( x - y ) .= ( - ( - ( x - y ) * ( x - y ) ) * ( x - y ) * ( x - y ) * ( x - y ) ) * ( x - y ) * ( x - y ) * ( x - y ) * ( x - y ) * ( x - y ) ) * ( x - y ) * ( x - y ) * ( x - y ) ) * ( x - y ) * ( x - y ) * ( x - y ) .= ( - ( x - y ) * ( x - y ) * ( x - y ) * ( x - y ) ) * ( x - y ) * ( x - y ) * ( x - y ) * ( x - y )
v . <* x0 , y0 *> = ( <* x0 , y0 *> ) . ( x0 , y0 ) + ( x0 , y0 ) * ( x0 , y0 ) + ( x0 , y0 ) * ( x0 , y0 ) ) + ( x0 , y0 ) * ( x0 , y0 ) + ( x0 , y0 ) * ( x0 , y0 ) + ( x0 , y0 ) * ( x0 , y0 ) ) ;
i * i = <* 0 , 0 , 0 , 0 *> .= <* 0 , 0 , 0 , 0 *> ;
Sum ( L * F ) = Sum ( L * F ) + Sum ( ( L * F ) ) .= Sum ( L * F ) + Sum ( ( L * F ) ) .= Sum ( L * F ) + Sum ( L * F ) .= Sum ( L * F ) + Sum ( L * F ) .= Sum ( L * F ) + Sum ( L * F ) .= Sum ( L * F ) + Sum ( L * F ) .= Sum ( L * F ) + Sum ( L * F ) + Sum ( L * F ) + Sum ( L * F ) + Sum ( L * F ) .= Sum ( L * F ) + Sum ( L * F ) + Sum ( L * F ) .= Sum ( L * F ) + Sum ( L * F ) .= Sum ( L * F ) + Sum ( L * F ) + Sum ( L * F ) + Sum ( L * F ) + Sum ( L * F ) + Sum ( L * F ) + Sum ( L * F ) .= Sum ( L * F ) + Sum ( L * F ) + Sum ( L * F
ex r be Real st for Y be Subset of X st Y in the carrier of X ex r1 being Real st 0 < r1 & r1 < r2 & r1 < r2 & r1 < r2 & r2 in the carrier of X & r1 < r2 & r2 in the carrier of X & r2 in the carrier of X ;
( GoB f ) * ( i , j ) = f /. ( k + 1 ) & ( GoB f ) * ( i , j + 1 ) = f /. ( k + 1 ) & ( GoB f ) * ( i + 1 , j + 1 ) = f /. ( k + 1 ) ;
( ( ( - 1 ) * ( ( - 1 ) * ( - 1 ) ) ) / ( 1 + ( - 1 ) * ( - 1 ) ) ) = ( - 1 ) / ( 1 + ( - 1 ) * ( - 1 ) ) .= ( - 1 ) / ( 1 + ( - 1 ) * ( - 1 ) ) .= ( - 1 / ( 1 + ( - 1 ) * ( - 1 ) ) / ( 1 + 1 ) ) .= ( - 1 / ( 1 + ( - 1 ) / ( 1 + ( - 1 ) / ( 1 + 1 ) / ( 1 + 1 ) / ( 1 + 1 ) / ( 1 + ( 1 + 1 ) / ( 1 + 1 ) / ( 1 + ( 1 + 1 ) / ( 1 + ( 1 + ( 1 + ( 1 + ( 1 + ( 1 + ( 1 + ( 1 + ( 1 + ( 1 + ( - 1 ) ) ) / ( 1 + ( 1 + ( 1 + ( 1 + ( 1 + ( 1 + ( 1 + ( 1 + ( 1 + ( 1 + ( 1 + ( 1 +
x- ( a - b ) + sqrt ( - b * sqrt ( a * b ) + sqrt ( - b * sqrt ( a * sqrt ( a * b ) + sqrt ( b * sqrt ( a * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * ( b * sqrt ( b * ( b * ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * sqrt ( b * ( b * ( b * ( b * sqrt ( b * sqrt ( b * sqrt ( b *
assume that inf ( X /\ Y ) in X and sup ( X /\ Y ) in X and sup ( X /\ Y ) in Y ;
( B BB ) . j = ( B --> ( i , j ) ) . j & ( B --> ( i , j ) ) . j = ( B --> ( i , j ) ) . j ;