thesis ;
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contradiction ;
contradiction ;
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thesis ;
contradiction ;
contradiction ;
contradiction ;
contradiction ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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contradiction ;
contradiction ;
thesis ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
thesis ;
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contradiction ;
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contradiction ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
B ;
a <> c
T c= S
D c= B
c in X ;
b in X ;
X c= Y ;
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 \times
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 dense ;
a in A ;
1 < x ;
S is finite ;
u in I ;
z << z ;
x in V ;
r < t ;
let t ;
x c= y ;
a <= b ;
m <= n ;
assume f is prime ;
not x in Y ;
z = +infty ;
k in NAT ;
K `2 is being_line ;
assume n >= N ;
assume n >= N ;
assume X is \equiv ;
assume x in I ;
q is \upharpoonright ;
assume c in x ;
p > 0 ;
assume x in Z ;
assume x in Z ;
1 <= k12 ;
assume m <= i ;
assume G is \kern1pt ;
assume a divides b ;
assume P is closed ;
d > 0 ;
assume q in A ;
W is bounded ;
f is weakly 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 even ;
assume i in I ;
assume 1 <= k ;
X is non empty ;
assume x in X ;
assume n in M ;
assume b in X ;
assume x in A ;
assume T c= W ;
assume s is negative ;
b `2 <= c `2 ;
A meets W ;
i `2 <= j `2 ;
assume H is universal ;
assume x in X ;
let X be set ;
let T be Tree ;
let d be element ;
let t be element ;
let x be element ;
let x be element ;
let s be element ;
k <= \bigcup 5 ;
let X be set ;
let X be set ;
let y be element ;
let x be element ;
P [ 0 ]
let E be set , f be FinSequence of E ;
let C be Category ;
let x be element ;
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 retraction ;
Q halts_on s ;
x in \rbrack 0 , 1 .[ ;
M < m + 1 ;
T2 is open ;
z in b "\/" a ;
R is well-ordering ;
1 <= k + 1 ;
i > n + 1 ;
q1 is one-to-one ;
let x be trivial set ;
P3 is one-to-one ;
n <= n + 2 ;
1 <= k + 1 ;
1 <= k + 1 ;
let e be Real ;
i < i + 1 ;
p3 in P ;
p1 in K ;
y in C1 ;
k + 1 <= n ;
let a be Real , x be Real ;
X |- r => p ;
x in { A } ;
let n be Nat ;
let k be Nat ;
let k be Nat ;
let m be Nat ;
0 < 0 + k ;
f is_differentiable_in x ;
let x0 ;
let E be Ordinal ;
o <= o2 ;
O <> x3 ;
let r be Real ;
let f be FinSequence ;
let i be Nat ;
let n be Nat ;
Cl A = A ;
L c= Cl L ;
A /\ M = B ;
let V be complex RealUnitarySpace , f be PartFunc of V , V ;
not s in Y ^ 0 ;
rng f <= w
b "/\" e = b ;
m = m2 ;
t in h . D ;
P [ 0 ] ;
assume z = x * y ;
S . n is bounded ;
let V be RealUnitarySpace , f be PartFunc of V , V ;
P [ 1 ] ;
P [ {} ]
C1 is Subset of TOP-REAL 2 ;
H = G . i ;
1 <= i `1 + 1 ;
F . m in A ;
f . o = o ;
P [ 0 ] ;
aa a <= real ;
R [ 0 ] ;
b in f .: X ;
assume q = q2 ;
x in [#] V ;
f . u = 0 ;
assume e1 > 0 ;
let V be RealUnitarySpace , W be Subspace of V ;
s is trivial & s is trivial ;
dom c = Q ;
P [ 0 ] ;
f . n in T ;
N . j in S ;
let T be complete LATTICE , f be Function of T , T ;
the ObjectMap of F is one-to-one ;
sgn x = 1 ;
k in support a ;
1 in Seg 1 ;
rng f = X ;
len T in X ;
vthesis < n ;
Smax is bounded ;
assume p = p2 ;
len f = n ;
assume x in P1 ;
i in dom q ;
let U ;
p3 = c ;
j in dom h ;
let k ;
f | Z is_differentiable_in x ;
k in dom G ;
UBD C = B ;
1 <= len M ;
p in LSeg ( x , r ) ;
1 <= j1 ;
set A = notation ;
card a [= c ;
e in rng f ;
cluster B \oplus A -> empty ;
H has has has { H } ;
assume n2 <= m ;
T is increasing ;
e1 <> e1 ;
Z c= dom g ;
dom p = X ;
H is proper ;
i + 1 <= n ;
v <> 0. V ;
A c= conv A ;
S c= dom F ;
m in dom f ;
let X2 be set ;
c = sup N ;
R is connected implies union M is connected
assume not x in REAL ;
Im f is complete ;
x in Int y ;
dom F = M ;
a in On W ;
assume e in A ( ) ;
C c= CC1 ;
m2 <> {} ;
x be Element of Y ;
let f be \mathclose Morphism , p , q be Real ;
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 , f be FinSequence of A ;
P2 = 1 ;
assume r in F . k ;
assume f is simple ;
let A be infinite set ;
rng f c= NAT ;
assume P [ k ] ;
f0 <> {} ;
let o be Ordinal ;
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 in dom f ;
assume t . 1 in A ;
let Y be non empty TopStruct , f be Function of Y , Y ;
assume a in ]. s , t .[ ;
let S be non empty RelStr ;
a , b // b , a ;
a * b = p * q ;
assume x , y are_the space ;
assume x in [#] ( f | A ) ;
[ a , c ] in X ;
m2 <> {} ;
M + N c= M + N ;
assume M is Int hhhhhhhhhhhhhhhM ;
assume f is with_n-n-n-n-n-nclosed ;
let x , y be element ;
let T be non empty TopSpace ;
b , a // b , c ;
k in dom Sum p ;
let v be Element of V ;
[ x , y ] in T ;
assume len p = 0 ;
assume C in rng f ;
k1 = k2 ;
m + 1 < n + 1 ;
s in S \/ { s } ;
n + i >= n + 1 ;
assume Re y = 0 ;
k1 <= j1 ;
f | A is continuous ;
f . x <= b ;
assume y in dom h ;
x * y in B1 ;
set X = Seg n ;
1 <= i2 + 1 ;
k + 0 <= k + 1 ;
p ^ q = p ;
j |^ y divides m ;
set m = max A ;
[ x , x ] in R ;
assume x in succ 0 ;
a in sup phi ;
CX is connected ;
q2 c= C1 ;
a2 < c2 ;
s2 is 0 -started ;
IC s = 0 ;
s6 = s5 ;
let V ;
let x , y be element ;
x be Element of T ;
assume a in rng F ;
x in dom T ` ;
let S be \frac of L ;
y " <> 0 ;
y " <> 0 ;
0. V = uw -w ;
y2 , y1 , y2 , y1 , y2 is_collinear ;
R1 is one-to-one ;
let a , b be Real , f be FinSequence of 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 \lbrack l , r \rbrack ;
let i , j be Nat ;
let s be State of A , P be State of A ;
s3 . n = N ;
set y = ( x `1 ) ^2 ;
[: i , i :] in dom g ;
l . 2 = y1 ;
|. g . y .| <= r ;
f . x in CC0 ;
V is non empty ;
let x be Element of X ;
0 <> f . g2 ;
f2 /* q is convergent ;
f . i is_measurable_on E ;
assume \xi in Nk1 ;
reconsider i = i as Ordinal of NAT ;
r (#) v = 0. X ;
rng f c= [: Z , Z , Z , Z :] ;
G = 0 .--> 1 ;
let A be Subset of X ;
assume not x0 is dense & x0 in V ;
|. f . x .| <= r ;
x be Element of R ;
let b be Element of L ;
assume x in W-19 ;
P [ k , a ] ;
let X be Subset of L ;
let b be object of B ;
let A , B be category ;
set X = Vars ( C ) ;
let o be OperSymbol of S ;
let R be connected non empty Poset ;
n + 1 = succ n ;
xk1 c= [: Z , Z , Z , Z , Z , Z , Z , Y , Z , Z , Y , Z , Z , Z , Y , Z
dom f = C1 ;
assume [ a , y ] in X ;
Re ( seq ^\ k ) is convergent ;
assume a1 = b1 ;
A = sInt A ;
a <= b or b <= a ;
n + 1 in dom f ;
let F be /* of S , f be FinSequence of S ;
assume r2 > x0 ;
let Y be non empty set , f be FinSequence of Y ;
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 & x2 <> x3 ;
v2 in [: V , V :] ;
not [ b , b ] in T ;
ii + 1 = i ;
T c= * ( T ) ;
( l - 1 ) * ( l - 1 ) = 0 ;
n be Nat ;
( t `2 ) ^2 = r ^2 ;
Ameets M ;
set t = Bottom t ;
let A , B be real-membered set ;
k <= len G + 1 ;
[: C , C :] misses [: V , C :] ;
product seq is non empty ;
e <= f or e <= f ;
cluster mode sequence of S -> non empty for set ;
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 vseq is convergent and lim seq = r ;
IC s3 = 0 ;
k in N or k in K ;
F1 \/ F2 c= F ;
Int ( G1 \/ G2 ) <> {} ;
( z - t ) ^2 = 0 ;
p1 <> p1 ;
assume z in { y , w } ;
MaxADSet ( a ) c= F ;
ex_sup_of \mathopen { \downarrow s , S ;
f . x <= f . y ;
let T be lower-bounded antisymmetric non empty reflexive transitive antisymmetric reflexive transitive antisymmetric antisymmetric RelStr ;
q1 |^ m >= 1 ;
a >= X & b >= Y ;
assume <* a , c *> <> {} ;
F . c = g . c ;
G is one-to-one & G is one-to-one ;
A \/ { a } c= B ;
0. V = 0. Y ;
let I be parahalting Program of S , a be Integer ;
f-24 . x = 1 ;
assume z \ x = 0. X ;
C4 = 2 |^ n ;
let B be sequence of Sigma ;
assume X1 = p .: D ;
n + l in NAT ;
f " P is compact ;
assume x1 in REAL + REAL ;
p1 = K & K is being_line ;
M . k = <*> REAL ;
\varphi . 0 in rng \varphi ;
MOSby A is closed ;
assume z0 <> 0. L ;
n < N . k ;
0 <= seq . 0 ;
- q + p = v ;
{ v } is Subset of B ;
set g = f /. 1 ;
R is stable & R is stable implies R is stable
set \cal R = Vertices R ;
p0 c= P4 ;
x in [. 0 , 1 .] ;
f . y in dom F ;
let T be complete TopStruct ;
inf the carrier of S in S ;
downarrow a = <* b *> ;
P , C , K is_collinear ;
assume x in LSeg ( s , r ) ;
2 |^ i < 2 |^ m ;
x + z = x + z + q ;
x \ ( a \ x ) = x ;
||. \mathopen { \Vert } x-y .|| <= r ;
assume Y c= field Q & Y <> {} ;
a , b are_isomorphic ;
assume a in A . i ;
k in dom q1 ;
p is FinSequence of S ;
i - 1 = i - 1 ;
f | A is one-to-one ;
assume x in f .: X ;
i2 - i1 = 0 ;
j2 + 1 <= i2 ;
g " * a in N ;
K <> { [ {} , {} ] } ;
cluster -> \mathclose |. -> strict for doubleLoopStr ;
|. q .| ^2 > 0 ;
|. p3 .| = |. p .| ;
s2 - s1 > 0 ;
assume x in { Gik } ;
W-min C in C ;
assume x in { Gik } ;
assume i + 1 = len G ;
assume i + 1 = len G ;
dom I = Seg n ;
assume k in dom C & 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 void for i being Nat ;
let f be ManySortedSet of I ;
let z be Element of COMPLEX , f be FinSequence of COMPLEX ;
u in { \hbox { \boldmath $ g , h } } ;
2 * n < 2 * ( n + 1 ) ;
x , y , z is_collinear ;
B-11 c= [: V , V :] ;
assume I is_closed_on s , P ;
U = U . ( i + 1 ) ;
M /. 1 = z /. 1 ;
x11 = x9 & x9 in dom ( x9 ^ {} ) ;
i + 1 < n + 1 ;
x in { {} , <* 0 *> } ;
f . ( n + 1 ) <= ( f . n ) `1 ;
l be Element of L ;
x in dom ( F . n ) ;
let i be Element of NAT ;
rr is ( the carrier of C ) -valued ;
assume <* o2 , o2 , o2 *> <> {} ;
s . x |^ 0 = 1 ;
card ( K + 1 ) in M ;
assume X in U & Y in U ;
let D be non empty Subset of Omega ;
set r = { k + 1 } ;
y = W . ( 2 * x ) ;
assume dom g = cod f ;
let X , Y be non empty TopSpace , f be Function of X , Y ;
x \oplus A is interval ;
|. <*> A .| . a = 0 ;
cluster sublattice L -> strict for non empty Subset of L ;
a1 in B . s1 ;
let V be finite VectSp of F , F be FinSequence of V ;
A * B on B ;
f-3 = NAT --> 0 ;
A , B be Subset of V ;
z1 = P1 . j ;
assume f " P is closed ;
reconsider j = i as Element of M ;
a , b be Element of L ;
assume q in A \/ ( B "\/" C ) ;
dom ( F * C ) = o ;
set S = [: X , Y :] ;
z in dom ( A --> y ) ;
P [ y , h . y ] ;
{ x0 } c= dom f ;
let B be non-empty ManySortedSet of I , f be ManySortedFunction of I , B ;
sqrt ( PI / 2 ) < Arg z ;
reconsider z0 = 0 as Nat ;
LIN a , d , c ;
[ y , x ] in [: I , I :] ;
( Q , 3 ) `1 = 0 ;
set j = x0 div m + 1 ;
assume a in { x , y , c } ;
j2 - ( j - 1 ) > 0 ;
I \! \mathop { 1 } = 1 ;
[ y , d ] in F ;
let f be Function of X , Y ;
set A2 = sqrt ( B ^2 - C ^2 ) ;
s1 , s2 , s2 , s3 , s3 be Element of L ;
j1 - 1 = 0 ;
set m2 = 2 * n + j ;
reconsider t = t as bag of n ;
I2 . j = m . j ;
i |^ s , n are_congruent_mod s ;
set g = f | [: D , D :] ;
assume X is lower & 0 <= r ;
( p1 `2 ) ^2 = 1 ^2 ;
a < ( p3 `2 ) ^2 / ( p3 `2 ) ^2 ;
L \ { m } c= UBD C ;
x in Ball ( x , 10 ) ;
not a in LSeg ( c , m ) ;
1 <= i1 -' 1 ;
1 <= 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 . ( g . x ) ;
card b * a = |. a .| * |. a .| ;
Shift ( w , 0 ) |= v ;
set h = h2 (*) h1 ;
assume x in [: X , Y :] /\ [: X , Y :] ;
||. h .|| < d ;
not x in Carrier f ;
f . y = F ( y ) ;
for n holds X [ n ] ;
k - l = kl ;
<* p , q *> /. 2 = q ;
let S be Subset of lattice the carrier of T ;
P , Q be Infor_FinSeq_of of s ;
Q /\ M c= union ( F | M ) ;
f = b * card ( S ) ;
let a , b be Element of G ;
f .: X <= f . sup X ;
let L be non empty reflexive transitive transitive reflexive transitive RelStr , X be Subset of L ;
Sw is x -to_power i , K ;
let r be non negative Real ;
M , v |= ( v , y ) \hbox { y } ;
v + w = 0. V ;
P [ len F ] ;
assume InsCode i = 8 & InsCode i = 8 ;
the zero of M = 0 ;
cluster z (#) seq -> summable ;
let O be Subset of the carrier' of C ;
||. f .|| | X is continuous ;
x2 = g . ( j + 1 ) ;
cluster -> empty for Element of S ;
reconsider ll = ll as Nat ;
ve is Vertex of ( the carrier of G ) . e ;
T is SubSpace of ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( TOP-REAL 2 ) ) ;
Q1 /\ Q <> {} ;
k in NAT ;
q " is Element of X ;
F . t is non empty ;
assume n <> 0 & n <> 1 ;
set e1 = EmptyBag n , e2 = EmptyBag n , e2 = EmptyBag n , e2 = EmptyBag n , e2 = EmptyBag n , e2 = EmptyBag n , e2 = EmptyBag n , e2 = EmptyBag n ,
let b be Element of Bags n ;
assume for i holds b . i is commutative ;
x is root of ( p . i ) . x ;
not r in ]. p , q .[ ;
let R be FinSequence of REAL ;
Sb1 does not empty ;
IC SCM R <> a ;
|. - |[ x , y ]| .| >= r ;
1 * seq = seq . ( n + 1 ) ;
let x be FinSequence of NAT ;
let f be Function of C , D , g be Function of C , D ;
for a being Element of L holds 0. L + a = a
IC s = s . IC Comput ( P , s , 0 ) ;
H + G = FF - ( G + F ) ;
Cx . x = x2 . x ;
f1 = f . x .= f2 . x ;
Sum <* p . 0 *> = p . 0 ;
assume v + W = v + u ;
{ a1 } = { a1 , a2 } ;
a1 , b1 _|_ b , a ;
s3 , o _|_ o , a ;
I1 is reflexive implies reflexive transitive & for C being reflexive reflexive reflexive reflexive reflexive reflexive reflexive reflexive & C is reflexive
IO is antisymmetric implies [: O , O :] is antisymmetric
sup rng ( H1 ^ H2 ) = e ;
x = a1 * a2 * a3 + a3 * a1 * a3 ;
|. p1 .| ^2 >= 1 ^2 ;
assume j2 - 1 < j - 1 ;
rng s c= dom ( f1 + f2 ) ;
assume support ( a ) misses support b ;
let L be associative non empty doubleLoopStr , f be Polynomial of L , L ;
s " + 0 < n + 1 ;
p . c = f . ( 1 - c ) ;
R . n <= R . ( n + 1 ) ;
Directed ( I , 0 ) = ( card I + 1 ) -tuples_on SCM+FSA ;
set f = + ( x , y , r ) ;
cluster Ball ( x , r ) -> bounded ;
consider r being Real such that r in A ;
cluster non empty NAT -defined NAT -defined NAT ;
let X be non empty directed Subset of S ;
let S be non empty full Subset of L ;
cluster <* L1 . N , L2 *> -> complete for LATTICE ;
sqrt ( 1 + a ^2 ) = a ^2 ;
( q . {} ) `1 = o ;
( n - 1 ) - 1 > 0 ;
assume sqrt ( 1 ^2 - 2 ^2 ) <= t `2 ;
card ( B + 1 ) = k + 1 ;
x in union rng ( f | X ) ;
assume x in the carrier of R ;
d in dom f ;
f . 1 = L . ( F . 1 ) ;
the vertices of G = { v } ;
let G be -> -> -> -> -> -> -> -> -> -> -> : Let ;
e , e1 , e2 , e2 is_collinear ;
c . ( i1 + 1 ) in rng c ;
f2 /* q is divergent_to-infty ;
set z1 = - z2 , z2 = - z1 , z2 = - z2 , z1 = - z2 , z2 = - z1 , z2 = - z2 , z1 = - z2 , z2 = - z1 , z2
assume w is lllof S , G ;
set f = p \! \mathop { t } , g = p \! \mathop { t } ;
let c be Object of C ;
assume ex a st P [ a ] ;
let x be Element of REAL m ;
let I1 be Subset-Family of X ;
reconsider p = p as Element of NAT ;
v , w be Point of X ;
let s be State of SCM+FSA , i be Nat ;
p is finite for q being State of SCM+FSA , a being Int-Location ;
stop I c= P3 & I c= P3 & I c= P3 ;
set ci = f /. i , ci = f /. i , ci = f /. i , cj = f /. i , cj = f /. i , cj = f /. i
w ^ t ^ u ^ s ^ u ^ t ^ u ^ s ^ u ^ t ^ u ^ v ^ u ^ t ^ u ^ v ^ u ^ u ^ v ^ w ^ u ^
W1 /\ W = ( W1 /\ W2 ) /\ ( W2 /\ W3 ) ;
f . j is Element of J . j ;
let x , y be Element of T2 ;
ex d st a , b // b , d ;
a <> 0 & b <> 0 implies c <> 0
ord x = 1 & x is 0 ;
set g2 = lim ( seq ^\ k ) ;
2 * x >= 2 * sqrt ( 1 + ( 2 * x ) ^2 ) ;
assume ( a 'or' c ) . z <> TRUE ;
f (*) g in Hom ( c , c ) ;
Hom ( c , d + c ) <> {} ;
assume 2 * ( q | m ) > m ;
L1 . ( F . -21 ) = 0 ;
R1 \/ R1 = ( R1 \/ R2 ) \/ ( R2 \/ R2 ) ;
( ( ( - 1 ) (#) sin ) `| Z ) . x <> 0 ;
( ( the carrier of T ) . x ) ^2 > 0 ;
o1 in [: X , Y :] /\ [: X , Y :] ;
e , e1 , e2 , e2 is_collinear ;
s3 / ( 1 - r ) > sqrt ( 1 - r ^2 ) ;
x in P .: ( F .: ( F .: ( F .: ( F .: ( F .: ( F .: ( F .: ( F .: ( F .: ( F .: ( F .: ( F .: ( F .: (
let J be closed non empty doubleLoopStr , R be Subset of R ;
h . p1 = f2 . O ;
Index ( p , f ) + 1 <= j ;
len ( q ^ M ) = width M & width ( q ^ M ) = width M ;
the support of K c= A ;
dom f c= union rng ( F . n ) ;
k + 1 in support support ( support ( L ) ) ;
let X be ManySortedSet of the carrier of S ;
[ x `1 , y ] in ( ( Carrier R ) . s ) ~ ;
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 , X ;
reconsider w = |. s1 .| as convergent Real_Sequence ;
sqrt ( 1 / m * m + r / m ) < p / m ;
dom f = dom ( I . i ) ;
[#] [: P , P :] = [#] ( ( TOP-REAL 2 ) | P ) ;
cluster - x -> ExtReal for ExtReal ;
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 implies v in W3
reconsider y = y as Element of L2 ;
N is full full SubRelStr of T , T & N is full implies N is full
ex_sup_of { x , y } , c ;
g . n = n / 1 .= n / 1 ;
h . J = EqClass ( u , J ) ;
let seq be summable sequence of X ;
dist ( x `1 , y ) < sqrt ( r ^2 + r ^2 ) ;
reconsider \mathbb m = m - 1 as Element of NAT ;
- x0 < r1 - x0 & x0 - r < x0 ;
reconsider P = P ` as strict Subgroup of N ;
set g1 = p * idseq ( q `1 ) ;
let n , m , k be non zero Nat ;
assume that 0 < e and f | A is bounded ;
D2 . ( I . ( x , I ) ) in { x , y } ;
cluster -> subcondensed for Subset of T ;
let P be compact non empty Subset of TOP-REAL 2 , p , q be Point of TOP-REAL 2 ;
G * ( -13 , 1 ) in LSeg ( G * ( 1 , 1 ) , G * ( 1 , 1 ) ) ;
n be Element of NAT , x be Element of 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 , {} ) c= { [ {} , {} ] } ;
reconsider m = i-1 as Element of NAT ;
reconsider d = x as Element of C ;
let s be 0 -started State of SCMPDS , P be s of P ;
let t be 0 -started State of SCMPDS ;
b , b , x , y is_collinear & x , y , z is_collinear ;
assume i = n \/ { n } & j = k \/ { k } ;
let f be PartFunc of X , Y ;
N2 >= sqrt ( c ^2 - sqrt ( c ^2 - d ^2 ) ) ;
reconsider tT = [: T1 , T2 :] as non empty TopSpace ;
set q = h * p ^ <* d *> ;
z2 in U . ( y1 /\ y2 ) /\ Q ;
A |^ 0 = { <* E *> , E } ;
len W2 = len W + 2 ;
len ( h2 ^ h2 ) in dom h2 ;
i + 1 in Seg ( len s2 ) ;
z in dom ( g1 | X ) /\ dom f ;
assume p2 = W-min ( K ) & p1 = W-min ( K ) ;
len G + 1 <= i1 + 1 ;
f1 (#) ( f2 - f1 /* seq ) is convergent ;
cluster seq + ( seq + - is summable ;
assume j in dom ( M1 * M2 ) ;
let A , B , C be Subset of X ;
x , y , z be Point of X , f be PartFunc of X , Y ;
b ^2 - ( 4 * a * c ) >= 0 ;
<* xy *> ^ <* y *> ^ <* y *> ^ <* y *> ^ <* y *> ^ <* x *> ^ <* y *> ^ <* y *> ^ <* y *> ^ <* y *> ^ <* x *> ^ <* y
a , b in { a , b } ;
len p2 is Element of NAT ;
ex x being element st x in dom R ;
len q = len ( K * G ) ;
s1 = Initialize ( Initialized s ) +* ( Initialized s ) .= ( Initialized s ) +* ( Initialized s ) ;
consider w be Nat such that q = z + w ;
x ` ` is L ` of x ` ;
k = 0 & n <> k or k > 1 ;
then X is discrete for X being Subset of X ;
for x st x in L holds x is finite
||. f /. c .|| <= r1 ;
c in ]. p , q .[ & not c in { p } ;
reconsider V = V as Subset of the topology of [: n , n :] ;
N , M be \mathbin { ^ \smallfrown } p ;
then z >= compactbelow x ;
M , f [. f , g .] = f & M , g .] = g ;
( ( to_power 1 ) to_power ( 1 + 1 ) ) to_power ( 1 + 1 ) = TRUE ;
dom g = dom f /\ X .= dom f /\ X ;
mode subwalk of G is sub.. of G ;
[ i , j ] in Indices M ;
reconsider s = x " as Element of H ;
let f be Element of dom ( - p ) ;
F1 . a1 = G1 . a1 & F1 . a1 = G1 . a1 ;
cluster circle ( a , b , r , s ) -> compact ;
let a , b , c be Real ;
rng s c= dom ( 1 / 2 ) ;
curry ( F , k ) is additive ;
set k2 = card ( dom B ) + 1 ;
set G = coprod ( X ) ;
reconsider a = [ x , s ] as Symbol of G ;
let a , b be Element of M , f be FinSequence of M ;
reconsider s1 = s as Element of S ;
rng p c= the carrier of L ;
let d be Subset of the carrier of A ;
( x | x ) = 0 iff x = 0. W ;
I1 in dom stop I ;
let g be continuous Function of X , Y ;
reconsider D = Y as Subset of TOP-REAL n ;
reconsider i2 = len p1 - 1 as Integer ;
dom f = the carrier of S & rng f c= the carrier of S ;
rng h c= union ( the carrier of J ) ;
cluster All ( x , H ) -> \bf |. p .| ;
d * N1 > ( N1 - 1 ) * ( 1 - 1 ) ;
]. a , b .[ c= [. a , b .[ ;
set g = f " ( D1 ) , h = f " ( D2 ) ;
dom ( p | ( m + 1 ) ) = [: Seg m , Seg m :] ;
3 + ( - 2 ) <= k + - 2 ;
the function tan is differentiable in ( ( arctan + arccot ) . x ) ;
x in rng ( f /^ p ) ;
f , g be FinSequence of D ;
p in the carrier of [: S1 , S2 :] & p in the carrier of S2 ;
rng f " = dom f /\ dom f ;
( the Target of G ) . e = v ;
width G - 1 < width G - 1 ;
assume v in rng ( S | ( E , X ) ) ;
assume x is root or x is root or x is root implies x is root & x is root ;
assume 0 in rng ( ( ( ( g2 ) | A ) ^ ( ( g2 | A ) ^ ( g2 | A ) ) ) ;
let q be Point of TOP-REAL 2 , r be Real ;
let p be Point of TOP-REAL 2 , q be Point of TOP-REAL 2 ;
dist ( O , u ) <= |. p2 .| + 1 ;
assume dist ( x , b ) < dist ( a , b ) ;
<* S *> is in the carrier of CCCCCCCon ( Cbe ) ;
i <= len ( G * ( i , 1 ) , G * ( i , 1 ) ) ;
let p be Point of TOP-REAL 2 , q be Point of TOP-REAL 2 ;
x1 in the carrier of I[01] & x2 in the carrier of I[01] ;
set p1 = f /. i , p2 = f /. i ;
g in { g2 : r < g2 } ;
Q = [: [: S , T :] , Q :] ;
( ( 1 / 2 ) to_power ( n + 1 ) ) is summable ;
- p + I c= - ( p + I ) + A ;
n < LifeSpan ( P1 , s1 ) ;
CurInstr ( p1 , s1 ) = i ;
A /\ Cl { x } <> {} ;
rng f c= ]. r , s + 1 .[ ;
let g be Function of S , V ;
let f be Function of L1 , L2 ;
reconsider z = z as Element of ( Carrier L ) . i ;
let f be Function of S , T ;
reconsider g = g as Morphism of c , b , c ;
[ s , I ] in S [: A , A :] ;
len ( the connectives of C ) = 4 ;
let C1 , C2 be subcategory of C , D ;
reconsider V1 = V as Subset of X | B ;
attr p is valid means : Def3 : All ( x , p ) is valid ;
assume X c= dom f & f .: X c= dom g ;
H |^ a is Subgroup of H * , a |^ b ;
let A1 be |^ of O , A2 be Subset of E ;
p2 , q1 , q2 , q1 is_collinear & q1 , q2 , q2 is_collinear ;
consider x being element such that x in v ^ K ;
not x in { 0. TOP-REAL 2 } ;
p in [#] ( I[01] | B ) ;
0 in ( M . E ) . E ;
op ( c ) / ( op ( c ) ) = c / c ;
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . ( s . ( n + 1 ) ) ) ;
cluster with_generated empty for SubSpace of L ;
set i1 = the Nat , i2 = the Element of NAT ;
let s be 0 -started State of SCMPDS , p be Polynomial of SCM+FSA ;
assume y in ( f1 \/ f2 ) .: A ;
f . ( len f ) = f /. ( len f ) ;
x , f . x '||' f . x , f . y ;
attr X c= Y means : Def3 : for X st X c= Y holds X c= Y & Y c= Z ;
let y be upper Subset of Y , x , y be Element of X ;
cluster -> -> -> -> -> -> -> -> -> -> -> -> -> -> infor for #Z number ;
set S = <* Bags n , Y. *> ;
set T = [. 0 , 1 .] , S = [. 0 , 1 .] ;
1 in dom mid ( f , 1 , 1 ) ;
sqrt ( 4 * PI / 2 ) < sqrt ( 2 * PI / 2 * PI * 2 * PI / 2 ) ;
x2 in dom ( f1 + f2 ) /\ dom ( f2 + f1 ) ;
O c= dom I & { {} } = { {} } ;
( the Source of G ) . x = v ;
{ HT ( f , T ) } c= Support f ;
reconsider h = R . k as Polynomial of n , L ;
ex b being Element of G st y = b * H ;
let x , y , z be Element of G ;
h1 . i = f . ( h . i ) ;
( p `1 ) ^2 = ( p `2 ) ^2 + ( p `2 ) ^2 ;
i + 1 <= len Cage ( C , n ) ;
len <* P *> = len P & len P = len P ;
set N2 = the |= of N , N2 = the |= of N ;
len g\mathopen { gx + 1 } <= x ;
a on B & b on B implies b on C
reconsider rv = r * I . v as FinSequence of REAL ;
consider d such that x = d and a [= d and a [= c ;
given u such that u in W and x = v + u ;
len f /. n = len f ;
set q2 = ( N-min C ) `2 , q2 = ( E-max C ) `2 ;
set S = LSeg ( S1 , S2 ) , S2 = LSeg ( S2 , S2 ) ;
MaxADSet ( b ) c= MaxADSet ( P /\ Q ) ;
Cl ( G . q1 ) c= F . r2 ;
f " D meets h " ( V ) ;
reconsider D = E as non empty directed Subset of L1 ;
H = ( H '&' H ) '&' ( H '&' H ) ;
assume t is Element of Free ( S , X ) . s ;
rng f c= the carrier of S2 & rng f c= the carrier of S2 ;
consider y being Element of X such that x = { y } ;
f1 . a1 = b1 . a1 & f1 . b1 = b2 . b1 ;
the carrier of G = E \/ { E } ;
reconsider m = len H - k as Element of NAT ;
set S1 = LSeg ( n , ( TOP-REAL 2 ) | C ) ;
[ i , j ] in Indices ( M1 + M2 ) ;
assume P c= Seg m & M is without_independent ;
for k st m <= k holds z in K . k
consider a being set such that p in a and a in G ;
L1 . p = p * L /. 1 ;
p-7 . i = p1 . i ;
let P , Q be a_partition of Y , x be Element of Y ;
attr 0 < r & 1 < 1 & r < 1 implies 1 / 2 < r / 2 ;
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 ( canFS ( s ) ) = card ( s ) ;
reconsider x2 = x1 as Element of L ;
Q in FinMeetCl ( ( the topology of X ) \/ the topology of Y ) ;
dom ( f | [: X , Y :] ) c= dom ( u | [: X , Y :] ) ;
attr n divides m & m divides n implies n = m ;
reconsider x = x as Point of I[01] , y = x as Point of I[01] ;
a in |. D2 - ( D2 - ( D1 - D2 ) ) .| ;
not x0 in the still of f & not [ x0 , f . x0 ] in the carrier of f ;
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 = Circuit ( x , y , z ) ;
l = m2 & l = m2 & l = m2 & l = m2 & l = m2 ;
x0 in dom ( u | A ) /\ A & x0 in dom ( u | A ) /\ A ;
reconsider p = x as Point of ( TOP-REAL 2 ) | P as Point of ( TOP-REAL 2 ) | P ;
[: I , I :] = [: I , I :] ;
f . p3 <= f . p3 ;
( F . x ) `1 <= ( F . x ) `1 ;
( x `2 ) ^2 = ( ( x `2 ) ^2 + ( x `2 ) ^2 ) ;
for n being Element of NAT holds P [ n ] ;
J , K , L be non empty Subset of I ;
assume 1 <= i & i <= len <* a *> ;
0 |-> a = <*> the carrier of K ;
X . i in 2 ^ ( A . i ) \ B . i ;
<* 0 *> in dom ( e --> [ 1 , 0 ] ) ;
then P [ a ] implies P [ succ a ]
reconsider sY. = sY. as DecoratedTree of D , D ;
( k - 1 ) <= len ( H ) ;
[#] S c= [#] T & the carrier of T c= the carrier of T ;
let V being strict RealUnitarySpace ;
assume k in dom mid ( f , i , j ) ;
let P be non empty Subset of TOP-REAL 2 , p , q , r be Real ;
let A , B be Matrix of K ;
- a * b = a * b * c ;
let A being Subset of A9 , B be Subset of T ;
id o2 in <* o2 , o2 , o2 , o2 , o2 , o2 , o2 , o2 , o2 , o2 , o2 , o2 , o2 , o2 , o2 , o2 , o2 , o2 , o2 , o2 , o2 , o2 , o2 , o2 , o2
then ||. x .|| = 0 & x = 0 ;
let N1 , N2 be strict Subgroup of G , f be FinSequence of the carrier of G ;
j >= len ( ( Carrier g ) . ( len g ) ) ;
b = Q . ( len Q - 1 ) ;
f2 (#) ( f1 /* s ) is divergent ;
reconsider h = f * g as Function of [: N , N :] , G ;
assume that a <> 0 and delta ( a , b , c , d ) >= 0 ;
[ t , t ] in the InternalRel of A & [ t , t ] in the InternalRel of A ;
( v |-- E ) | n is Element of T ;
{} = the support L1 + ( Carrier ( L1 ) ) ;
I is_halting_on Initialized s , P +* I , P +* I ;
Initialized p = Initialized ( p +* q ) +* ( I +* q ) ;
reconsider N2 = N1 as strict net of ( the carrier of N ) * , N2 be net of ( the carrier of N ) ;
reconsider Y = Y as Element of <* ( Ids L ) . 1 , \subseteq \rangle ;
Bottom L \ { p } <> p ;
consider j being Nat such that i2 = i1 + j + 1 ;
[ s , 0 ] in the carrier of S2 & [ s , 0 ] in the carrier of S2 ;
m2 in ( B \wedge C ) /\ D \ { {} } ;
n <= len ( P1 + P2 ) + len ( P1 + P2 ) ;
( ( x1 - x2 ) `1 ) ^2 = ( x2 - x3 ) ^2 ;
InputVertices S = { x1 , x2 , x3 , x4 , x5 , x5 , M } ;
let x , y be Element of F\it \it T1 ;
p = |[ p `1 , p `2 ]| ;
g * 1_ G = h " * g " * h * h " * g " * h " * h * h " * g * h " * h * h * h * g * h * h * h " * h * h * h * h * h * h "
let p , q be Element of PFuncs ( V , C ) ;
x0 in dom ( f1 + f2 ) /\ dom ( f2 + f1 ) /\ dom ( f2 + f1 + f2 ) ;
( R qua Function ) " = R " ( R .: ( R .: ( R .: ( R .: ( R .: ( R .: ( R .: ( R .: ( R .: ( R .: ( R .: ( R .: ( R .: ( R .: ( R .: ( R
n in Seg len ( f /^ p ) ;
for s be Real st s in R holds s <= s2 ;
rng s c= dom ( f2 * f1 ) /\ dom ( f2 * f1 ) ;
synonym card \kern1pt X for card Y for card X + card Y ;
1_ K * 1_ K = 1_ K * 1_ K .= 1_ K * 1_ K ;
set S = Segm ( A , P1 , P2 , s2 , s3 ) ;
ex w st e = sqrt ( w , f ) & w in F ;
curry ( P1 , k ) # x is convergent ;
cluster -> open for Subset of T ;
len f1 = 1 .= len ( f1 ^ f2 ) .= len ( f1 ^ f2 ) ;
sqrt ( i * p ) < sqrt ( 2 * p ) * sqrt ( 2 * p ) ;
let x , y be Element of Sub Sub ( U1 ) ;
b1 , c1 // b1 , c2 & b1 , c1 // b2 , c2 ;
consider p being element such that c1 . j = { p } ;
assume f " { 0 } = {} & f is total ;
assume IC Comput ( F , s , k ) = n ;
Reloc ( J , card I + card J ) not exists a st I c= a ;
Stop SCM+FSA ( card I + 1 ) c= P +* I ;
set m2 = LifeSpan ( p1 , s3 ) , m2 = Comput ( p2 , s2 , m1 ) , P4 = Comput ( p2 , s2 , m1 ) , P4 = Comput ( p2 , s2 , m1 ) , P4 = Comput ( p2 , s2 , m1 ) ,
IC Comput ( p , s , i ) in dom Start-At ( IC Comput ( p , s , i ) , SCMPDS ) ;
dom t = the carrier of SCM R & dom t = the carrier of R ;
( ( E-max L~ f ) .. f + ( E-max L~ f ) .. f = 1 ;
let a , b be Element of PFuncs ( V , C ) ;
Cl ( union F ) c= Cl ( union F ) ;
the carrier of X1 union X2 misses ( the carrier of X1 union X2 ) \/ ( the carrier of X1 ) ;
assume not LIN a , f . ( a , f . a ) , f . ( a , f . a ) ;
consider i being Element of M such that i = d . i ;
then Y c= { x } or Y = { x } ;
M , v / ( ( H . ( x , y ) . v ) / ( ( H . ( y , x ) . v ) ) / ( ( H . ( y , x ) . v ) ) / ( ( H . ( y , x
consider m be element such that m in Intersect ( ( F . m ) /\ ( F . m ) ) ;
reconsider A1 = support ( u1 ) as Subset of X ;
card ( A \/ B ) = k-1 + ( 2 * 1 ) ;
assume a1 <> a3 & a2 <> a4 & a3 <> a4 & a4 <> a4 & a1 <> a4 ;
cluster s \! \mathop { V } -> S -\mathop { V } ;
L2 /. ( n + 2 ) = L2 . ( n + 2 ) ;
let P be compact non empty Subset of TOP-REAL 2 , p , q be Point of TOP-REAL 2 ;
assume that r in LSeg ( p1 , p2 ) and r in LSeg ( p1 , p2 ) ;
let A be non empty compact Subset of TOP-REAL n , f be FinSequence of TOP-REAL n ;
assume [ k , m ] in Indices ( D1 ^ D2 ) ;
0 <= ( ( 1 / 2 ) to_power p ) to_power p ;
( F . N ) . x = +infty ;
attr X c= Y & Z c= V & Z c= V implies X \ Z c= Y \ Z ;
( y - z ) * ( z - ( - ( z - y ) ) * ( z - y ) ) <> 0. I ;
1 + card ( card ( X \/ { x } ) ) <= card ( u + { x } ) ;
set g = z \circlearrowleft ( L~ z ) ;
then k = 1 & p . k = <* x , y *> ;
cluster -> Element of C -\to ( X ) for PartFunc of X , Y ;
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 ;
\uparrow ( x1 , x2 , x3 ) c= P ;
n <= indx ( D2 , D1 , j1 ) + 1 ;
( ( g2 . O ) `1 ) `1 = - 1 ;
j + p .. f - len f <= len f - len f ;
set W = E-bound ( C ) ;
S1 . ( a , e ) = a + e .= a + e ;
1 in Seg width ( M * ( ( Line ( M * ( p , 1 ) ) ) ) ;
dom ( i (#) Im ( f , h ) ) = dom ( Im ( f , h ) ) ;
\Phi ( x `1 , p `2 ) = W . ( a , p `2 ) ;
set Q = |= ( g , f , g ) ;
cluster -> qua qua Sorts of U1 for MS|. U1 .| ;
attr F = { A } means : Def3 : F is discrete ;
reconsider -13 = reproj ( i , x ) as Element of product \overline ( G ) ;
rng f c= rng ( f1 ^ f2 ) \/ rng ( f2 ^ f1 ) ;
consider x such that x in f .: A and x in f .: C ;
f = <*> ( the carrier of C ) & f is FinSequence of the carrier of C ;
E , E |= All ( x , H , E ) ;
reconsider n1 = n as Morphism of o1 , o2 , o2 be Morphism of o2 , o2 ;
assume P is associative & P is associative & P * R = P * R ;
card ( ( B2 \/ { x } ) \/ { x } ) = k-1 + 1 ;
card ( x \ ( B \/ ( B ) ) ) = 0 ;
g + R in { s : g-r < s & s < g + r } ;
set q1 = ( q , s ) := ( s , C ) , q2 = ( q , s ) := ( s , C ) ;
for x being element st x in X holds x in rng f1 & x in rng f2
h /. ( i + 1 ) = h . ( i + 1 ) ;
set \mathbb w = max ( B , max ( B , C ) ) ;
t in Seg width ( I ^ ( n , n ) ) ;
reconsider X = dom f as Element of Fin C ;
IncAddr ( i , k ) = halt SCM+FSA .= goto l + k ;
( ( S /. ( len f ) ) `2 <= ( q `2 ) `2 ;
attr R is condensed means : Def3 : R is condensed & R is condensed & R is condensed ;
pred 0 <= a & b <= 1 & a <= 1 & b <= 1 implies a * b <= 1 ;
u in ( c /\ ( d /\ b ) /\ f ) /\ j ;
u in ( c /\ ( d /\ e ) /\ f /\ j ;
len C + ( - 2 ) >= 9 + - 2 ;
x , y , z is_collinear & x , y , z is_collinear ;
a |^ ( n1 + 1 ) = a |^ ( n1 + 1 ) * a ;
<* \underbrace { 0 , \dots , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
set y9 = <* y , c *> ;
F2 /. 1 in rng Line ( D , 1 ) ;
p . m joins r /. m , r /. ( m + 1 ) ;
( p `1 ) ^2 = ( f /. ( i1 + 1 ) ) ^2 ;
( W-min ( X \/ Y ) ) \/ ( ( W-min ( X ) ) /\ Y ) = ( W-min ( X ) /\ Y ) ;
0 + ( p `2 ) ^2 <= 2 * r ^2 + ( p `2 ) ^2 ;
x in dom g & not x in g " { 0 } ;
f1 /* ( seq ^\ k ) is divergent_to-infty ;
reconsider uu = u as VECTOR of P|[ X , Y ]| ;
p \! \mathop { ( Product ( X , Y ) ) . ( len p + 1 ) = 0 ;
len <* x *> < i + 1 & i + 1 <= len c ;
assume I is non empty & { x } /\ { y } = { 0. I } ;
set i2 = card I + 4 + ( 0 + 1 ) , goto 0 = card I + 4 + 0 ;
x in { x , y } & h . x = {} implies h . x = {} ;
consider y being Element of F such that y in B and y <= x `1 ;
len S = len ( the charact of A ) & len ( the charact of A ) = len ( the charact of A ) ;
reconsider m = M , n = I as Element of X ;
A . ( j + 1 ) = B . ( j + 1 ) \/ A . ( j + 1 ) ;
set NG = LSeg ( G * ( 1 , j ) , G * ( 1 , j + 1 ) ) ;
rng F c= the carrier of gr ( { a } , { a } ) ;
Comput ( Q , Comput ( Q , s , n ) , n ) is len for p being FinSequence of D holds p . ( n + 1 ) is len -element
f . k , f . ( Radix ( n ) ) |= rng f ;
h " ( P ) /\ [#] ( T | P ) = f " ( P /\ P ) ;
g in dom ( f2 \ ( f2 " { 0 } ) ) ;
gX /\ dom f1 = ( g1 | X ) " ( { 0 } ) ;
consider n being element such that n in NAT and Z = G . n ;
set d1 = |. ( ( |. x1 - y1 .| ) ^2 - ( |. y1 - y1 .| ) ^2 ;
b `2 + sqrt ( 1 + sqrt ( 1 + ( 1 + ( 2 * 2 ) ^2 ) ) ^2 < sqrt ( 1 + ( 2 * 2 ) ^2 ) ;
reconsider f1 = f as VECTOR of the carrier of X ;
attr i <> 0 implies i ^2 mod ( i + 1 ) = 1 ;
j2 in Seg ( len ( ( ( g2 . i2 ) ) . i2 ) ;
dom i = dom ( i - 1 ) .= dom ( i - 1 ) .= dom ( i - 1 ) .= dom ( i - 1 ) ;
cluster sec | ]. 0 , PI / 2 .[ -> one-to-one ;
Ball ( u , e ) = Ball ( f . p , e / 2 ) ;
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 ) ^ ( p ^ q ) in ( R1 ^ R2 ) ^ ( R1 ^ R2 ) ;
S1 +* ( S1 +* S2 , S2 +* S2 ) = ( S2 +* S2 , S2 +* S2 ) +* ( S2 +* S2 , S2 +* S2 ) ;
( ( ( - 1 ) (#) ( ( ( ( ( - 1 ) (#) ( ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( -
cluster C -> { 0 } -valued for PartFunc of C , REAL ;
set CM = 1GateCircStr ( <* z , x *> , f2 ) , CM = 1GateCircStr ( <* z , x *> , f3 ) ;
E . ( e , e1 ) = ( ( E . e ) | ( E . e ) ^ T ) . ( e , e1 ) ;
( ( arctan * ( arctan + arccot ) ) `| Z ) = ( ( arctan * ( arctan + arccot ) ) `| Z ;
upper_bound A = sqrt ( 3 * PI * PI ) & lower_bound A = 0 ;
F . ( dom f , - F . f ) is transformable to F . ( cod f , - f . f ) ;
reconsider p8 = q1 as Point of TOP-REAL 2 ;
g . W in [#] ( Y | [#] ( Y | ( X | Y ) ) ) & [#] ( Y | Y ) c= [#] ( Y | Y ) ;
let C be compact non horizontal Subset of TOP-REAL 2 , p , q be Point of TOP-REAL 2 ;
LSeg ( f ^ g , j ) = LSeg ( f , j ) ;
rng s c= dom f /\ ]. - r , x0 .[ ;
assume x in { idseq ( 2 ) , ( idseq ( 2 ) ) . ( i + 1 ) } ;
reconsider n2 = n , m2 = m - n as Element of NAT ;
for y being ExtReal st y in rng seq holds g <= y
for k st P [ k ] holds P [ k + 1 ]
m = m1 + m2 .= m1 + m2 .= m1 + m2 .= m1 + m2 ;
assume for n holds H1 . n = G . n -( G . n ) ;
set Bf = f .: ( the carrier of X1 ) , Bf = f .: ( the carrier of X2 ) ;
ex d being Element of L st d in D & x <= d ;
assume R " ( a , b ) c= R " ( a , b ) ;
t in ]. r , s .[ or t = r or t = s ;
z + v2 in W & x = u + ( z + ( z + u ) ) ;
x2 |-- ( y1 , y2 ) = y1 & P [ y1 , y2 , y1 , y2 , z2 , y2 , z2 , z2 , z2 , z2 , z2 , z2 , z2 , z2 , z2 , z2 , z2 , z2 , z2 , z2 , z2 , z2 , z2 , z2 , z2 ,
attr x1 <> x2 & |. x1 - x2 .| > 0 implies |. x1 - x2 .| > 0 ;
assume p2 - p1 , p3 - p1 , p1 - p2 - p1 , p1 - p1 , p2 - p1 - p1 , p1 - p1 , p2 - p1 , p3 - p1 , p1 - p1 , p2 - p1 , p1 - p1 , p2 - p1 - p1 , p2 - p1 - p1 , p1 - p1 , p2 - p1 , p1 - p1 - p1
set q = 'not' f ^ <* 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not'
let f be PartFunc of \langle { 1 , 1 } , \Vert * \Vert , \Vert * \Vert \rangle ;
( n mod 2 ) = n mod 2 .= n mod 2 ;
dom ( T * succ t ) = dom succ t ;
consider x being element such that x in w1 and x in c ;
assume ( F * G ) . ( v , x3 ) = v . x3 ;
assume the carrier of D1 c= the carrier of D2 & the carrier of D2 c= the carrier of D1 & 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 ) , q = W-min L~ Cage ( C , n ) ;
n1 - len f - 1 <= len f + 1 - 1 ;
ConsecutiveSet2 ( q , O ) = [ u , v , v , w ] ;
set CG = ( \mathclose { -1 } ) . ( k + 1 ) , CG = ( G \mathclose { 1 } ) . ( k + 1 ) , CG = G . ( k + 1 ) ;
Sum ( L (#) p ) = 0. R .= 0. V .= 0. V ;
consider i being element such that i in dom p and t = p . i ;
defpred Q [ Nat ] means 0 <= $1 implies ( $1 + 1 ) <= ( $1 + 1 ) * ( $1 + 1 ) ;
set s3 = Comput ( P1 , s1 , k ) , s4 = Comput ( P1 , s1 , k ) , P4 = Comput ( P1 , s1 , k ) , P4 = Comput ( P1 , s1 , k ) , P4 = Comput ( P1 , s1 , k ) , P4 = Comput ( P1 , s1
let l be variable of k , A , A be non empty Subset of k , B be non empty Subset of l ;
reconsider U = union ( ( G . n ) /\ ( G . n ) ) as Subset-Family of T ;
consider r such that r > 0 and Ball ( p , r ) c= Q ` and Ball ( p , r ) c= Q ` ;
( h | ( n + 2 ) ) /. i = p1 /. ( n + 2 ) ;
reconsider B = the carrier of X1 as Subset of X ;
pmax ( - c , - 1 ) = <* - c , - c , - c , - c , - c , - c , - 1 , - 1 , 0 , - 1 , 0 , - 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 1 , 0
synonym f is real-valued for rng f c= NAT & rng f c= NAT ;
consider b being element such that b in dom F and a = F . b ;
x0 < card ( [: X , Y :] \/ ( card X /\ Y ) ) + card ( card Y /\ X ) ;
attr X c= B1 & X c= succ ( B \/ C ) ;
then w in Cl Ball ( x , r ) & dist ( x , w ) <= r ;
angle ( x , y , z ) = angle ( x , 0 , 0 , PI , 0 ) ;
attr 1 <= len s means : Def3 : len ( s . 0 ) = len s & len ( s . 1 ) = len s ;
f . ( k + 1 ) c= f . ( k + 1 ) ;
the carrier of G = { 1_ G } \/ { 1_ G } ;
attr p '&' q in TAUT ( A ) means : Def3 : p '&' q in TAUT ( A ) & q '&' p in TAUT ( A ) ;
- ( t `2 / t `2 - sn ) / ( 1 + sn ) < ( t `2 / t `2 ) ^2 ;
U . 1 = U /. 1 .= U . 1 .= U . ( 1 + 1 ) ;
f .: ( the carrier of x ) = the carrier of x & f .: ( the carrier of x ) = the carrier of x ;
Indices ( O * ( n , m ) ) = [: Seg n , Seg m :] ;
for n being Element of NAT holds G . n c= G . ( n + 1 )
then V in M ^ N ;
ex f being Element of F st f is having_a_unity & f is having_a_unity & f is having_a_unity ;
[ h . 0 , h . 3 ] in the InternalRel of G & [ h . 3 , h . 3 ] in the InternalRel of G ;
s +* ( Initialize ( ( intloc 0 ) .--> 1 ) ) = s3 +* ( ( intloc 0 ) .--> 1 ) ;
|[ w1 , v1 ]| `1 <> 0. TOP-REAL 2 & |[ w1 , v1 ]| `2 <> 0 ;
reconsider t = t as Element of Z -tuples_on ( X , Y ) ;
C \/ P c= [#] ( ( G \ A ) \ ( G \ A ) ;
f " ( V , V ) in exists ( X /\ D ) /\ D & f .: ( V , V ) in exists ;
x in [#] ( ( the carrier of F ) /\ A ) /\ ( the carrier of F ) ;
g . x <= h1 . x & h . x <= h1 . x ;
InputVertices S = { x , y , z } & { x , y , z } = { x , y , z } ;
for n be Nat st P [ n ] holds P [ n + 1 ]
set R = Line ( M , i ) * Line ( M , i ) ;
assume that M1 is being_line and M2 is being_line and M1 is being_line and M2 is being_line ;
reconsider a = f . ( i1 -' 1 ) as Element of K ;
len ( ( Len ( F ^ G ) ) ^ ( ( ( Len F ) ^ ( ( len G ) + len G ) ) ) = Sum ( ( ( Len F ) ^ ( len G ) ) ) ;
len ( the addF of n , i ) = n & len ( the _ of n , i ) = n ;
dom max ( f , g ) = dom ( f + g ) /\ dom g ;
( the Sorts of s1 ) . n = sup ( ( the Sorts of s1 ) . n ) ;
dom ( p1 ^ p2 ) = dom ( p1 ^ p2 ) /\ dom ( p1 ^ p2 ) ;
M . [ 1 , y ] = 1 * v .= 1 * v .= v . 1 ;
assume W is non trivial & W .vertices() c= the carrier of ( G ) \ the carrier of ( G ) ;
C6 * ( i1 , i2 ) `1 = G * ( i2 , j2 ) `1 ;
C8 |- 'not' All ( x , p => 'not' p ) => 'not' ( 'not' All ( x , p => 'not' p ) ) ;
for b st b in rng g holds lower_bound rng gb <= b ;
- sqrt ( 1 + ( - ( q `2 / |. q .| - sn ) ) ^2 ) = 1 ^2 ;
LSeg ( c , m ) \/ LSeg ( l , k ) c= R ;
consider p being element such that p in LSeg ( x , p ) and p in L~ f and p in L~ f ;
Indices X = [: Seg n , Seg n :] & [: Seg n , Seg n :] = [: Seg n , Seg n :] ;
cluster s => ( q => p ) -> valid ;
( Partial_Sums ( ( Partial_Sums F ) . n ) . m is_measurable_on E ;
cluster f . ( x1 , x2 ) -> Element of D ;
consider g being Function such that g = F . t & Q [ g , t ] ;
p in LSeg ( |[ ( ( N-min Z ) . i , ( TOP-REAL 2 ) . i ) `1 , ( ( TOP-REAL 2 ) . i ) `1 ]| ;
set Rf = R ^ ]. b , + \infty .[ ;
IncAddr ( I , k ) = ( goto ( d + k ) ) . k ;
seq . m <= ( ( ( ( the Sorts of A1 ) . k ) . k ) . m ;
a + b = ( a ` *' ) *' ( b ` + a ` ) .= ( a ` + b ` ) *' ;
id X /\ Y = id X /\ Y & ( id Y ) /\ Y = {} ;
for x being element st x in dom h holds h . x = f . x
reconsider H = U \/ ( U1 \/ U2 ) as non empty Subset of [: U1 , U2 :] ;
u in ( c /\ ( d /\ e ) /\ f /\ j /\ m ;
consider y being element such that y in Y and P [ y , inf B ] ;
consider A being finite non empty set such that card ( A \/ B ) = card ( ( A \/ B ) \/ B ) ;
p2 in rng ( f |-- p1 ) \ rng ( p1 |-- p2 ) ;
len s1 - 1 > 1 & len s1 - 1 > 0 ;
( ( ( N-min ( P ) ) | ( P ) ) `2 = ( ( E-max ( P ) ) `2 ) `2 ;
Ball ( e , r + ( Cage ( C , n + 1 ) ) ) c= LeftComp Cage ( C , n + 1 ) ;
f . a1 ` ` ` = f . a1 ` ` ` ` ` ` ` .= f . a1 ` ` ` ` ` ;
( seq ^\ k ) . n in ]. - r , x0 .[ ;
gg . ( s . ( n + k ) ) = g . ( s . ( n + k ) ;
the InternalRel of S is transitive & ( the InternalRel of S ) \/ ( the InternalRel of S ) c= ( the InternalRel of S ) \/ ( the InternalRel of S ) ;
deffunc F ( Ordinal , Ordinal , set ) = phi . ( $1 + 1 ) ;
F . ( s1 . a1 ) = F . ( s2 . a1 ) ;
x `2 = A . a .= Den ( o , A ) . a .= Den ( o , A ) . a ;
Cl ( f " ) c= f " ( Cl ( Cl ( P1 /\ P2 ) ) ) ;
FinMeetCl ( the topology of S ) c= the topology of T ;
synonym o is constructor means : Def3 : o <> \ast ;
assume X ^ Y = Y ^ Z & card ( X ^ Y ) <> card ( Y ^ Z ) ;
the consider the consider \/ s *> <= 1 + ( the \hbox { - } tree s ) . ( 1 + s ) ;
LIN a , a1 , d or b , c // b1 , c1 & b , c // b1 , c1 ;
e . 1 = 0 & e . 2 = 1 & e . 3 = 0 ;
E in [: S , T :] & not E in { [: S , T :] ;
set J = ( l , u ) ReassignIn I ;
set A1 = 1GateCircStr ( a , a9 , b9 , c9 ) , A2 = and2 ( a , b , c , c9 ) ;
set c9 = [ <* c , d *> , '&' ] , C2 = [ <* d , c *> , '&' ] , C2 = [ <* c , d *> , '&' ] , C2 = [ <* d , c *> , '&' ] , C2 = [ <* c , d *> , '&' ] , C2 = [ <* c , d *> , '&' ] , C2 =
x * z * x " in x * ( z * N ) * x " ;
for x being element st x in dom f holds f . x = g2 . x ;
Int cell ( GoB f , 1 , G ) c= RightComp f \/ RightComp f \/ RightComp f ;
U is_an_arc_of W-min ( C ) , W-min ( C ) , W-min ( C ) , W-min ( C ) ;
set f3 = f ^ ( g ^ f ) ;
attr S1 is convergent means : Def3 : S1 is convergent & S2 is convergent & lim ( S1 - S2 ) implies S1 - S2 is convergent & lim ( S1 - S2 ) = 0 ;
f . ( 0 qua Ordinal ) = ( 0 qua Ordinal ) + a .= a ;
cluster reflexive transitive for reflexive transitive non empty reflexive transitive transitive transitive transitive transitive transitive transitive transitive transitive transitive transitive transitive non empty reflexive transitive transitive transitive transitive transitive transitive reflexive transitive transitive transitive transitive transitive non empty RelStr ;
consider d being element such that R reduces b , d and R reduces c , d and R reduces c , d ;
not b in dom Start-At ( ( card I + card J + 2 ) , SCMPDS ) ;
( z + a ) + x = z + ( a + a ) .= z + ( a + a ) .= z + a ;
len ( l (#) ( a , A ) +* x ) = len l & len ( l , A ) = len l ;
t4 is ( {} , rng t ) -valued ;
t = <* F . t *> ^ ( C ^ q1 ^ q2 ) ;
set pa = W-min ( C ) , pa = W-min ( C ) , cb = W-min ( C ) , cb = W-min ( C ) , cb = W-min ( C ) ;
k9 - ( i + 1 ) = ( k - 1 ) - ( i + 1 ) ;
consider u being Element of L such that u = u ` and u in D and u in D ;
len ( ( width \hbox { \hbox { $ ( ( ( G , G ) ) --> a , b ) ) --> ( len G ) ) = width ( ( ( G , b ) --> b ) ;
F . x in dom ( G * the_arity_of o ) ;
set H2 = the carrier of H2 , H = the carrier of H2 , I = the carrier of H , I = I /\ H ;
set H = the carrier of H1 , G = the carrier of H2 , I = the carrier of H2 , H = the carrier of H2 , I = the carrier of H , G = the carrier of H , H = the carrier of H , I = the carrier of H , H = the carrier of H ;
( Comput ( P , s , 6 ) ) . intpos ( m + 6 ) = s . intpos ( m + 6 ) ;
IC Comput ( P3 , Comput ( P3 , t , k + 1 ) , Comput ( P3 , s3 , k ) ) = l + 1 ;
dom ( ( ( ( ( the carrier of TOP-REAL 2 ) ) * ( ( the carrier of TOP-REAL 2 ) | K1 ) ) | K1 ) = the carrier of ( ( TOP-REAL 2 ) | K1 ) ;
cluster <* l *> ^ phi -> ( 1 + \rbrace , S ) -valued for string of S ;
set bb = [ <* \hbox { \boldmath $ p $ } , { p , m } , { p , m } , { p , m } , { p , m } , { p , m } , { p , m } } , bb = [ <* p , m *> , { p , m } , { p , m } , { p , m
Line ( Segm ( M , P , Q ) , x ) = L * Sgm Q * Sgm Q ;
n in dom ( ( the Sorts of A ) * ( the_arity_of o ) ) ;
cluster f1 + f2 -> continuous for PartFunc of S , T ;
consider y being Point of X such that a = y and ||. \mathopen { \Vert y .|| } <= r ;
set x3 = ( t . intpos ( m + 1 ) ) . intpos ( m + 1 ) , y1 = t . intpos ( m + 1 ) , y2 = t . intpos ( m + 1 ) , y1 = t . intpos ( m + 1 ) , y2 = t . intpos ( m + 1 ) , y1 = t .
set pI = stop I , PI = P +* I , PI = P +* I , PI = P +* I ;
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 be set ;
|. p2 .| ^2 - ( - ( p2 `2 / |. p2 .| - sn ) ) ^2 >= 0 ;
l - 1 + 1 = l * ( l - 1 ) + ( l - 1 ) ;
x = v + ( a * w1 + b * w2 ) + ( c * w2 ) * w2 + ( c * w2 ) * w2 ;
the topological L = k1 (# the carrier 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 ;
f \ { n } = ( ( Free ( ( { v } , H ) ) \/ { n } ) \/ { n } ) \/ { n } ;
let Y be Subset of X ;
2 * n in { N : 2 * Sum ( p | N ) = N & N > 0 } ;
let s be FinSequence of D ;
for x st x in Z holds ( ( exp_R * f ) `| Z ) . x = exp_R . x / ( cos . x ) ^2
rng ( ( h * ( f2 * f1 ) - ( f2 * f1 ) - ( f2 * f2 ) ) c= the carrier of ( ( TOP-REAL 2 ) ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 )
j + ( len f - len f ) <= len f + ( len f - len g ) - len f ;
reconsider R1 = R * I as PartFunc of REAL , REAL n , REAL ;
Cx . ( x - x0 ) = s1 . ( x - x0 ) .= s1 . ( x - x0 ) .= s1 . ( x - x0 ) ;
power ( L , n , z ) . ( z , n ) = 1 / ( x |^ n ) .= x |^ n ;
t at ( C , s ) = f . ( ( the connectives of S ) . ( s , C ) ) ;
support ( f + g ) c= ( support f ) \/ ( support g ) \/ ( support g ) ;
ex N st N = j1 & 2 * Sum ( ( r (#) ( ( r (#) ( ( ( ( r (#) ( ( r - N ) - N ) ) - ( r (#) ( ( r - N ) - N ) ) ) ) ) > N ;
for y , p st P [ p ] holds P [ 'not' p ]
{ [ x1 , x2 , x3 , x4 ] where x1 , x2 is Point of [: X , Y :] : x1 in [: X , Y :] & x2 in [: X , Y :] } ;
h = ( j , h ) +* ( h , h ) .= H . i ;
ex x1 being Element of G st x1 = x & x1 * N c= A & N c= A ;
set X = ( |. q .| ) * ( |. q1 .| ) , Y = |. q1 .| * ( |. q1 .| ) , Z = |. q1 .| * ( |. q1 .| ) , Y = |. q1 .| * ( |. q1 .| ) , Z = |. q1 .| * ( |. q1 .| ) , Y = |. q1 .| * ( |. q1 .| ) ;
b . n in { g1 : x0 - r < g1 & g1 < x0 + r } ;
f /* ( s1 ^\ k ) is convergent & f /. x0 = lim ( f /* s1 ) ;
the lattice of lattice ( Y ) = the lattice of ( the lattice of F ) |^ the carrier of F ;
'not' ( a . x ) '&' b . x = TRUE ;
2 = len ( ( q ^ <* 0 *> ^ ( ( q ^ <* 1 *> ^ ( q ^ <* 1 *> ) ) ) ) + len ( ( q ^ <* 1 *> ^ ( q ^ <* 1 *> ) ) ) ;
sqrt ( 1 / a * ( arctan * f1 + arctan * f2 ) ^2 ) = 1 / ( ( arctan * f1 + arctan * f2 ) ^2 ) ;
set K = upper upper upper upper \ ( lim ( H , A ) ) , H = ( lim ( H , A ) ) \/ ( lim ( H , A ) ) ;
assume e in { \frac w + ( w1 + w2 ) where w1 is Element of F : w1 in G & w2 in F } ;
reconsider d = dom a , e = dom F , d = dom F as finite set ;
LSeg ( f , j ) = LSeg ( f , j ) \/ LSeg ( f , j ) ;
assume X in { T . ( N2 , K1 ) : h . ( N2 , K1 ) = { N . ( N2 , K1 ) } ;
assume Hom ( d , c ) <> {} & <* f , g *> * f = <* f , g *> * f ;
dom Sb = dom S /\ Seg n .= dom ( L | Seg n ) .= dom L /\ Seg n .= dom L /\ Seg n .= dom L /\ Seg n .= dom L /\ Seg n ;
x in H ^ a implies ex g st x = g |^ a & g in H
a * ( n , 1 ) . a = a * ( n , 1 ) . a .= a * ( n , 1 ) . a ;
D2 . j in { r : lower_bound A <= r & r <= upper_bound A } ;
ex p being Point of TOP-REAL 2 st p = x & P [ p ] ;
for c holds f . c <= g . c implies f ^ g . c ^ f = g ^ f ;
dom ( f1 (#) f2 ) /\ X c= dom ( f1 (#) f2 ) /\ X /\ X ;
1 = sqrt ( p * p ) .= p * sqrt ( p * q ) .= p * sqrt ( p * q ) ;
len g = len f + len <* x *> .= len f + len <* y *> .= len f + 1 ;
dom ( F | [: { 1 } , { 1 } :] ) = [: [: { 1 } , { 1 } :] , { 1 } :] ;
dom ( f . t ) = dom ( f . t ) /\ dom ( g . t ) ;
assume a in ( ( ( T , ( \alpha , the carrier of S ) ) .: D ) .: D ) ;
assume that g is one-to-one and ( the carrier of S ) /\ rng g c= dom g ;
( x \ y ) \ ( x \ z ) = 0. X ;
consider f such that f * f = id ( b , a ) and f * f = id ( b , a ) ;
( ( ( ( - 1 / 2 ) * ( cos + cos ) ) `| Z ) = ( ( - 1 ) (#) ( cos + cos ) ) `| Z ) ;
Index ( p , co ) <= len ( ( p , co ) - 1 ) ;
t1 , t2 , t2 be Element of ( T . s ) . ( t , s ) = t . ( s , t ) .= t . ( s , t ) ;
Inf ( ( Frege ( K , L ) ) . h ) <= Inf ( ( ( Frege ( K , L ) ) . h ) ;
then P [ f . ( i2 + 1 ) , f . ( i2 + 1 ) ] ;
Q [ ( [ D . x , 1 ] , F . ( x + 1 ) ] ;
consider x being element such that x in dom ( F . s ) and y = F . s . x ;
l . i < r . i & [ l . i , r . i ] is r of G . i ;
the Sorts of ( ( the carrier of S2 ) --> ( the carrier of S2 ) ) = ( the carrier of S2 ) --> ( the carrier of S2 ) ;
consider s being Function such that s is one-to-one and dom s = NAT and rng s = NAT and rng s c= NAT and rng s c= NAT ;
dist ( b1 , b2 ) <= dist ( b1 , b2 ) + dist ( b2 , b2 ) ;
( ^\ ( C , n ) ) /. ( len ( Cage ( C , n ) ) = ( W-min L~ Cage ( C , n ) ) /. ( len ( Cage ( C , n ) ) ) ;
q <= ( ( ( w + ( TOP-REAL 2 ) ) | ( L~ Cage ( C , n ) ) ) `2 ;
LSeg ( f | i2 , i ) /\ LSeg ( f , i2 ) = {} ;
given a being ExtReal such that a <= I and A = ]. a , b .[ and A = ]. a , b .[ ;
consider a , b be complex number such that z = a & y = b + a and z = a + b ;
set X = { b } where b is Element of NAT : b in X } ;
( x * y + z * x ) \ ( y * z ) = 0. X ;
set xx9 = [ <* x , y , z *> , f2 ] , xz = [ <* z , x , y *> , f3 ] , xz = [ <* z , x , z *> , f3 ] , xz = [ <* z , x , z *> , f3 ] , xz = [ <* x , y , z *> , f3 ] , xz = [ <*
( l /. len l ) `1 = ( l /. len l ) `1 .= ( l /. len l ) `1 ;
sqrt ( ( q `2 / |. q .| - sn ) ^2 ) = 1 ^2 * ( ( q `2 / |. q .| - sn ) ^2 ) ;
sqrt ( ( p `2 / |. p .| - sn ) ^2 ) < 1 ^2 / sqrt ( 1 + sn ^2 ) ^2 ;
( ( ( ( X \/ Y ) \/ Y ) \/ X ) \/ Y = ( ( X \/ Y ) \/ Y ) \/ Y ) \/ Y ;
( seq - ( seq ^\ k ) ) . n = ( seq . ( k + 1 ) - ( seq ^\ k ) ) . n - ( seq . k ) . n ;
rng ( h + c ) c= dom SVF1 ( 1 , f , x0 ) /\ dom ( f + c ) ;
the carrier of X = the carrier of X & the carrier of X = the carrier of Y implies X is SubSpace of Y
ex p3 , p4 st p3 = p3 & |. p3 - p4 .| = r & |. p3 - p4 .| = r ;
set h = IExec ( X , A , C ) , A = +* ( X , C ) , B = ( the Sorts of A ) +* ( X , C ) ;
R |^ ( 0 * n ) = ( IX * ( R |^ n ) ) |^ ( 0 * n ) .= R |^ ( 0 * n ) ;
( Partial_Sums ( ( ( ( F . \alpha ) ) . n ) ) . n + ( ( ( ( ( F . 0 ) ) . n ) ) . n ) is nonnegative ;
f2 = C7 . ( - ( - ( V , - ( - ( - ( V , - ( - ( V , - ( - ( V , - ( - ( V , - ( V , - ( - ( V , ( - ( V , ( - ( V , ( - ( V , ( V , ( - ( V , ( - ( V , ( V , ( - ( V
S1 . b = s1 . 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 carrier' of S ) . 11 in ( the carrier' of S ) . 11 ;
set phi = ( l , ( l , l ) ) ReassignIn phi , l = l , l = l , l = ( l , u ) \HM u = l } ;
synonym p is is_distributive_wrt of T means : Def3 : HT ( p , T ) = 1 ;
( Y1 `1 ) ^2 = ( 1 - ( Y1 `2 ) ^2 & ( Y1 `2 ) ^2 = 1 & ( Y1 `2 ) ^2 = 0 ;
defpred X [ Nat , set , set , set , set , set , set , set , set = $2 , $2 = $2 , $2 = $2 , $2 = $2 , $2 = $2 , $2 = 1 / ( $2 + 1 ) ;
consider k be Nat such that for n be Nat st k <= n holds s . n < x0 + g ;
Det ( I ^ ( m - n ) ) = 1_ K ;
sqrt ( - b ^2 - sqrt ( b ^2 - a ^2 ) * sqrt ( b ^2 - b ^2 ) < 0 ;
Cd . d = Cd . d .= ( Cd . d ) . d .= ( Cd . d ) . d .= ( Cd . d ) . d ;
attr X1 is dense means : Def3 : X1 is dense & X1 is dense & X2 is dense & X1 is dense implies X1 union X2 is dense & X1 union X2 is dense implies X1 union X2 is dense & X1 union X2 is dense & X1 is dense implies X1 union X2 is dense ;
deffunc F ( Element of E , Element of I , Element of I , Element of E . ( $1 , $2 ) = ( 2 * $1 ) * ( 2 * $1 + 1 ) ;
t ^ <* n *> in { t ^ <* i *> where i is Nat : Q [ i ] } ;
( x \ y ) \ x = ( x \ y ) \ y .= y \ 0. X .= y \ 0. X ;
let X being non empty set ;
synonym A , B , C , D , E , F , G , G , N , f , g , h , h be Function of A , B , g be Function of A , B , h be Function of A , B , g be Function of A , B , h be Function of A , B , g be Function of A , B , h be Function of A , B , g be Function of A
len ( M ^ p ) = len p & width ( M ^ q ) = width M & width ( M ^ q ) = width M ;
v2 = { v where v is Element of K : 0 < v & v < 1 } ;
( ( Sgm ( \mathbb m ) ) . d - ( Sgm ( \mathbb m ) ) . d ) <> 0 ;
lower_bound divset ( D2 , k + 1 ) = D2 . ( k + 1 ) - D2 . ( k + 1 ) ;
g . r1 = - ( 2 * r1 + 1 ) & dom h = [. 0 , 1 .] & dom h = [. 0 , 1 .] ;
|. a .| * ||. f .|| = 0 * ||. f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| ;
f . x = ( h . x ) `1 & g . x = ( h . x ) `1 ;
ex w st w in dom ( B ^ w ) & <* 1 *> ^ s = B ^ w ^ w ;
[ 1 , {} , <* d1 , d2 *> ] in ( { [ 0 , {} , {} ] } \/ [: { {} , {} } , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {} , {}
IC Exec ( i , s1 ) + n = IC Comput ( i , s1 , n ) + n .= IC Comput ( i , s1 , n ) ;
IC Comput ( P , s , 1 ) = IC Comput ( P , s , 1 ) .= IC Comput ( P , s , 1 ) .= IC Comput ( P , s , 1 ) .= IC Comput ( P , s , 1 ) ;
( IExec ( W6 , Q , t ) ) . intpos ( card I + 3 ) = t . intpos ( card I + 3 ) ;
LSeg ( f , i ) misses LSeg ( f , i ) \/ LSeg ( f , j ) \/ LSeg ( f , j ) \/ LSeg ( f , j ) = LSeg ( f , j ) \/ LSeg ( f , j ) ;
assume for x , y being Element of L st x in C holds x <= y or y <= x ;
integral ( f ' , C ) = f ' . ( upper_bound C ) - f ' . ( upper_bound C ) ;
let F , G be one-to-one FinSequence , G be FinSequence of the carrier of G ;
||. R /. ( h + c ) .|| < e * ( K * ( 1 + c ) - R /. ( h + c ) ) ;
assume a in { q where q is Element of M : dist ( z , q ) <= r } ;
set p3 = [ 2 , 1 , 0 , 1 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 6 , 8 , 8 , 8 , 8 , 6 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8
consider x , y being Subset of X such that [ x , y ] in F and x in d and y in d and x in d and y in d and y in G and x in G and y in G and x in G and y in G ;
for y being Element of REAL , x being Element of REAL st y in Y & x in X holds x <= y & y <= x ;
func |. p ^ <* p *> -> ^ of A means : Def3 : for x being Element of A holds it . x = ( p ^ <* x *> ) . ( len p + 1 ) ;
consider t being Element of S such that x , y '||' z , t and x , z '||' z , t and x , z '||' z , t ;
dom ( x1 . ( len x1 ) ) = Seg len x1 & len ( x1 . ( len x1 ) ) = len x1 & len ( x1 . ( len x1 ) ) = len x1 & len ( x1 . ( len x1 ) ) = len x1 ;
consider y2 be Real such that x2 = y2 & 0 <= y2 & y2 < 1 and y2 < 1 / 2 ;
||. f /* s1 .|| = ||. f /* s1 .|| & ||. f /* s1 .|| = ||. f /* s1 .|| ;
( 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 q holds P [ j ] holds i + 1 <= len q & i + 1 <= len p + 1 ;
reconsider h = f | [: X , Y :] as Function of [: X , Y :] , Y , Y :] ;
u1 in the carrier of W1 & v1 in the carrier of W2 & v2 in the carrier of W1 & v1 in the carrier of W2 implies v1 + v2 in the carrier of W1 & v2 in the carrier of W2
defpred P [ Element of L ] means M <= f . $1 & f . $1 <= f . $1 ;
T . ( u , a ) = s . x * ( - ( s . x + s . y ) ) .= b * ( s . x + ( s . y ) * y ) .= b * b ;
- ( - ( \mathopen { - x } ) ) = - x + ( - x ) .= - x + ( - x ) .= - x ;
given a being Point of G1 such that for x being Point of G1 holds x in G iff x in G ;
fthesis = [ [ dom ( f , g ) , cod ( f , h ) ] , fh = [ f , h ] , fh = [ f , h ] , h1 = [ f , h ] , h1 = [ f , h ] , h1 = [ f , h ] , h1 = [ f , h ] , h1 = [ f , h ] , h1 = [ f ,
let k , n be Nat , n be Nat , k be Nat , n be Nat ;
for x being element holds x in A ^ ( ( A \/ B ) ` ) iff x in ( A ` ) `
consider u , v being Element of R such that l /. i = u * a * v ;
sqrt ( ( p `2 / |. p .| - sn ) ^2 ) > 0 ;
L-13 . k = LL . ( F . k ) & F . ( k + 1 ) in dom ( L . k ) ;
set i2 = AddTo ( a , i , n ) , i2 = AddTo ( a , i , n ) , i1 = goto ( - 1 ) ;
attr B is bound means : Def3 : ( for S being non empty set holds S is \sum ( All ( x , S ) ) ;
a9 "/\" D = { a "/\" d where d is Element of N : d in D } ;
( ( \square - PI / 2 ) * ( ( q - PI / 2 ) ) * ( ( q - PI / 2 ) ) >= ( ( q - PI / 2 ) * ( q - PI / 2 ) ) * ( q - PI / 2 ) ;
( - f ) . ( upper_bound A ) = ( - f ) . ( upper_bound A ) .= - f . ( upper_bound A ) ;
( G * ( len G , k ) , G * ( len G , k ) ) `1 = ( G * ( len G , k ) ) `1 ;
( Proj ( i , n ) ) . ( L . i ) = <* ( proj ( i , n ) ) . ( L . i ) *> ;
f1 + ( ( ( ( \HM { i } , x ) * reproj ( i , x ) ) * reproj ( i , x ) ) `| Z ) . x = ( ( ( ( \HM { i } , x ) * reproj ( i , x ) ) (#) ( reproj ( i , x ) ) `| Z ) . x ;
( ( ( - tan ) `| Z ) . x = 0 ;
ex t being SortSymbol of S st t = s & ( for x being Element of X holds ( h . x ) . x = ( h . x ) . x ;
defpred C [ Nat ] means ( $1 is $1 -Subset of NAT ) & ( $1 is $1 -NAT implies $1 is consistent implies ( $1 is consistent implies $1 is consistent & ( $1 is consistent implies $1 is consistent implies $1 is consistent & ( $1 is consistent implies $1 is consistent implies $1 is consistent ) ) ;
consider y being element such that y in dom ( p . i ) and q . i = p . i ;
reconsider L = product ( { x1 , x2 , x3 , x3 , x4 } ) as Basis of product A ;
for c being Element of C ex d being Element of D st T . ( id C ) = id d & c in d ;
Comput ( f , n , p ) . n = ( f | n ) . n .= f . n ;
( f * g ) . x = f . ( g . x ) & ( f * h ) . x = f . ( g . x ) ;
p in { sqrt ( 1 - ( G * ( i + 1 , j + 1 ) + 1 ) where j is Nat : j < 1 } ;
f ' - p = ( f - ( c + 1 ) ) \ast ( f - p ) .= ( f - ( f - p ) ) \ast ( f - p ) .= ( f - ( f - p ) ) * ( f - p ) .= ( f - f ) * ( f - p ) ;
consider r be Real such that r in rng ( f | divset ( D , j ) ) and r < m + r ;
f1 . ( |[ r , s ]| ) in ( f1 .: ( [. r , s .] ) /\ ( f2 .: ( [. r , s .] ) ) ;
eval ( a | n , L ) = eval ( a , ( a | n ) ) .= ( a | n ) . ( a . n ) ;
z = DigA ( [: { t , x } , { t , x } :] , t = ( [: { t , x } , { t , x } :] ) . z ;
set H = { Intersect ( S ) where S is Subset-Family of X : S is finite } , G = G \ { {} } ;
consider S19 being Element of D ( ) , d being Element of D ( ) such that S = S ^ <* d *> and d in d ( ) and d in d ( ) and d in d ( ) ;
assume that x1 in dom f and x2 in dom f and f . x1 = f . x2 and f . x2 = f . x2 ;
- 1 <= sqrt ( ( q `2 / |. q .| - sn ) / sqrt ( 1 + sn ) ^2 ) ;
0. V is Linear_Combination of A & Sum ( L ) = 0. V implies Sum ( L ) = 0. V
let k1 , k2 be Nat , k1 , k2 be Nat , a be k1 , b be bag of SCMPDS , k1 be Integer ;
consider j being element such that j in dom a and j in dom a and a . j = g . j ;
H1 . ( x1 , x2 ) c= H1 . ( x1 , x2 ) or H1 . ( x1 , x2 ) c= H1 . ( x2 , x3 ) ;
consider a be Real such that p = a * p1 + ( a * p2 ) and 0 <= a and a <= 1 and a <= 1 ;
assume that a <= c and d <= b and [ a , b ] c= dom f and [ a , b ] in dom g and [ a , b ] in dom g and [ a , b ] in dom f and [ a , b ] in dom g and [ a , b ] in dom g and [ a , b ] in dom g ;
cell ( Gauge ( C , m ) , m -' 1 , G * ( m , 1 ) -' 1 ) is non empty ;
ADef : { ( S . i ) `1 - ( S . i ) `2 <= ( S . i ) `2 - ( S . i ) `2 ;
( T * b1 ) . y = L * ( b1 /. y ) .= ( F * b1 ) . y .= ( F * b1 ) . y ;
g . ( s , I ) . x = s . y & g . ( s , I ) . y = |. s . x .| ;
( ( ( ( log ( 2 , k + 1 ) ) to_power ( k + 1 ) ) ) to_power ( k + 1 ) >= ( ( ( ( ( k + 1 ) to_power ( k + 1 ) ) to_power ( k + 1 ) ) ) to_power ( k + 1 ) ) ;
then p => q in S & not x in the carrier of ( p => q ) & not p => ( p => q ) in S ;
dom ( the - of ( the - of ( rreal , the carrier of ( ( the carrier of ( ( the carrier of ( ( the carrier of ( ( ( the carrier of ( ( the carrier of ( ( the carrier of ( ( the carrier of ( ( ( the carrier of ( ( the carrier of ( ( the carrier of ( ( the carrier of ( ( the carrier of ( ( the carrier of ( ( the carrier of ( the carrier of ( the carrier of ( the
synonym f is extended extended real-valued for for for for for for for for for x being set st x in rng f holds x is extended real ;
assume for a being Element of D holds f . { a } = a & for X being finite Subset of D holds f . ( a , X ) = f . X ;
i = len p1 + len <* x *> .= len p1 + len <* x *> .= len p1 + len <* x *> .= len p1 + len <* x *> .= len p1 + len <* x *> ;
( l /. 3 ) `1 = ( g /. 3 ) `1 + ( g /. 3 ) `1 .= ( g /. 3 ) `1 + ( g /. 3 ) `1 ;
CurInstr ( P2 , Comput ( P2 , s2 , l ) ) = halt SCMPDS .= halt SCMPDS ;
assume for n be Nat holds ||. seq . n .|| <= ( ||. seq .|| ) . n & ||. seq .|| . n .|| < r ;
sin . r2 = sin r * sin ( ( - ( cos . s ) ) * sin ( - ( sin . s ) ) ) .= 0 ;
set q = |[ g1 `1 , g2 `2 ]| , q2 = |[ 0 , 0 ]| , q1 = |[ 0 , 1 ]| , q2 = |[ 0 , 1 ]| , q2 = |[ 0 , 1 ]| , q2 = |[ 0 , 1 ]| , q2 = |[ 0 , 1 ]| , q2 = |[ 0 , 1 ]| , q2 = |[ 0 , 1 ]| ;
consider G being sequence of S such that for n being Element of NAT holds G . n in GGGJ ( F . n ) ;
consider G such that F = G and ex G1 , G2 being Subset of [: X , Y :] st G1 in [: X , Y :] & G = [: X , Y :] ;
the root tree of \llangle x , s . i ] in ( the Sorts of Free ( C , X ) ) . s ;
Z c= dom ( ( ( ( exp_R + ( exp_R * f ) + ( exp_R * f ) ) `| Z ) ;
for k be Element of NAT holds Sum ( ( Im ( f , S ) . k ) = ( Sum ( ( Im ( f , S ) . k ) ) ) . k
assume that - 1 < n and n < 1 and n < 1 and q `1 / |. q .| < 1 / ( n + 1 ) and q `1 / |. q .| < 1 / ( n + 1 ) and q `1 / |. q .| < 1 / ( n + 1 ) and q `1 / |. q .| < 1 / ( n + 1 ) and q `1 / |. q .| < 1 / ( n + 1 ) ;
assume that f is continuous and a < b and f . a = c and f . b = d and f . a = c and f . b = d and f . c = d ;
consider r being Element of NAT such that s1 = Comput ( P1 , s1 , 1 ) and r <= len s1 and r <= q and r <= q ;
LE 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 inf { x , y } in K and inf { x , y } in K and inf { x , y } in K ;
assume f +* ( i1 , i2 ) in ( proj ( F , i2 ) ) " ( the carrier of ( F . i2 ) ) ;
rng ( ( ( Flow M ) ~ ) | ( the carrier of M ) ) c= the carrier of M ;
assume z in ( the carrier of G ) \times { t } ;
consider l be Nat such that for m be Nat st l <= m holds ||. ( ( s1 . m ) - ( s2 . m ) ) .|| < g ;
consider t being VECTOR of product G such that t = ||. ( D . t ) - ( D . t ) .|| and ||. t .|| <= 1 ;
assume that the carrier of v = 2 and v ^ <* 0 *> in dom p and v ^ <* 1 *> in dom p ;
consider a being Element of the Points of XX such that a on A and not a on A and not a on A ;
( - x ) |^ ( k + 1 ) * ( ( - x ) |^ ( k + 1 ) ) = 1 ;
let D being set such that for i being Nat st i in dom p holds p . i in D ( ) holds p . i in D ( ) ;
defpred R [ element , element ] means ex x , y being element st [ x , y ] = $1 & P [ x , y ] ;
L~ ( f2 , p1 ) = union { LSeg ( p1 , p2 ) , LSeg ( p2 , p1 ) , p1 = LSeg ( p2 , p1 ) , p2 = LSeg ( p1 , p2 ) , q1 = LSeg ( p2 , p1 ) , q2 = LSeg ( p1 , p2 ) , q2 = LSeg ( p2 , p1 ) , q2 = LSeg ( p1 , p2 ) ;
i - len ( h1 - ( i + 1 ) ) - 2 < i - ( len ( h1 - ( i + 1 ) ) - 2 ;
for n be Element of NAT st n in dom F holds F . n = |. ( ( n -' 1 ) - 1 ) .|
for r , s1 , s2 being Real , s1 , s2 being State of TOP-REAL 2 holds s1 in [. r , s .] iff s1 <= s2 & s2 <= s2 & s2 <= 1
assume v in { G where G is Subset of T : G in ( B \/ C ) & G c= ( B \/ C ) ` ;
let g be non-empty Element of A , f be Function of A , ( X --> 0 ) , ( X --> 1 ) --> 1 , ( X --> 1 ) --> 1 ;
min ( g . [ x , y ] , k , h . ( [ y , z ] , h . ( y , z ) ) , k ) = ( min ( g . ( y , z ) , h . ( y , z ) ) , k ) . ( k + 1 ) ;
consider q1 be sequence of CC such that for n holds P [ n , q1 . 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 /\ ( O \/ { O } ) as Subset of B ;
consider j being Element of NAT such that x = the carrier of ( n -tuples_on the carrier of K ) and 1 <= j and j <= n and 1 <= n and j <= n and 1 <= n and n <= len f and 1 <= j and 1 <= n and j <= len f ;
consider x such that z = x and card ( x . O ) in card ( x . O ) and x in ( L . O ) . O and x in ( L . O ) . O ;
( C * ( k , n2 ) ) . 0 = ( C * ( ( ( ( k + 1 ) + 1 ) ) . 0 ) ;
dom ( X --> rng f ) = X & dom ( X --> f ) = X & dom ( X --> f ) = X ;
( ( TOP-REAL 2 ) | ( L~ Cage ( C , n ) ) ) `2 <= ( ( ( TOP-REAL 2 ) | ( L~ Cage ( C , n ) ) ) `2 ;
synonym x , y , l means : Def3 : { x , y } = y or { x , y } c= l ;
consider X be element such that X in dom ( f | ( n + 1 ) ) and ( f | ( n + 1 ) ) . X = Y ;
assume that k is continuous and for x , y being Element of L st x = y & y in compact holds x <= y & y <= k & x <= y ;
sqrt ( 1 / ( cos . ( tan . x ) ) ^2 ) * ( ( ( ( sin . x ) ^2 ) ) is_differentiable_on REAL ;
defpred P [ Element of omega ] means ( the partial PartFunc of omega , A ) . $1 = ( the partial of A1 ) . $1 & ( the partial of A1 ) . $1 = ( the partial of A1 ) . $1 ;
IC Comput ( P , s , 2 ) = succ IC Comput ( P , Comput ( P , s , 2 ) , 1 ) .= 6 + 1 .= ( card I + 1 ) + 1 .= 1 + 1 .= 1 + 1 .= 1 + 1 .= 1 + 1 ;
f . x = f . ( g1 . ( g1 . ( g1 . ( g1 . ( g1 . ( g1 . ( g1 . ( g1 . ( g1 . ( g1 . ( g1 . x ) ) ) ) ) ) ) .= f . ( f . ( g1 . ( g1 . x ) ) ;
( M * ( F . n ) ) . n = M . ( ( ( M . n ) . n ) .= M . ( ( M . n ) . n ) ;
the support of L1 + ( Carrier ( L1 ) ) c= ( Carrier ( L1 ) \/ Carrier ( L2 ) ) \/ ( Carrier ( L2 ) ) \/ Carrier ( L2 ) ;
pred a , b , c , x , y , z be Element of o means : Def3 : for a , b being Element of o holds a , b , c is_collinear & a , b , z is_collinear & b , c , x is_collinear ;
( the partial of product s ) . n <= ( ( the partial of product s ) . n ) * ( ( the carrier of product s ) . n ) ;
attr - 1 <= r & r <= 1 & r <= 1 implies ( ( - 1 ) * ( - 1 ) ) * ( ( - 1 ) * ( - 1 ) = ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) ;
seq in { p ^ <* n *> where p is Nat : p ^ <* n *> in T } ;
|[ x1 , x2 , x3 , x4 , x4 , x5 , x5 , x5 , x5 , 8 , 8 , 8 , x5 , 8 , x5 , 8 , 8 , x5 , M , N , M , N , N , M , N , N , M be Element of [: the carrier of S , the carrier of S , f , g , h being Function of S , T ;
attr for m be Nat holds F . m is nonnegative & ( Partial_Sums ( F . m ) . m <= +infty ;
len ( ( the addF of G , z ) ^ <* ( ( the addF of G ) . ( x , y ) ) *> ) = len ( ( the addF of G ) . ( y , z ) ) + 1 ;
consider u , v being VECTOR of V such that x = u + v and u in W1 /\ W2 and v in W2 and u in W2 /\ W3 and v in W2 /\ W3 ;
given F be FinSequence of NAT such that F = x and dom F = n and dom F = n and for k being Nat st k in dom F holds F . k = { 0 , 1 } and Sum F = 1 ;
0 = r1 * uwhere where where where where where where where \HM \HM { is Real : 1 <= |. 1- sn .| * ( 1 + sn ) = 1 } * ( 1 - sn ) ;
consider n be Nat such that for m be Nat st n <= m holds |. ( f # x ) . m - lim ( f # x ) .| < e ;
cluster strict for non empty additive for RelStr ;
Bottom ( ( B \/ { {} } ) ) = Bottom S .= Bottom S .= ( Bottom S ) \/ ( [#] S ) .= ( Bottom S ) \/ ( [#] S ) ;
sqrt ( r ^2 + ( r ^2 + ( r ^2 + ( r ^2 ) ) ^2 ) <= sqrt ( r ^2 + ( r ^2 ) ^2 ) ;
for x being element st x in A /\ ( f `| X ) holds ( f `| X ) . x >= r2
2 * r1 - ( a * c ) * ( b - a ) = 0. TOP-REAL 2 + ( a * c ) * ( b - c ) * ( b - c ) * ( b - c ) * ( b - c ) * ( b - c ) * ( b - c ) * ( b - c ) = 0. TOP-REAL 2 ;
reconsider p = P /. 1 , q = P " * ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - 1 ) / n ) ) ) / ( 1 + 1 ) ) ) ) ) as FinSequence of K ;
consider x1 , x2 being element such that x1 in uparrow s and x2 in uparrow t and x = [ x1 , x2 , x3 , x4 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , \Vert , x5 , \Vert , \Vert , \Vert , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8
for n be Nat st 1 <= n & n <= len q1 holds q1 . n = ( ( ( Carrier ( g , n ) ) . i ) * ( ( Carrier ( g , n ) ) . i ) * ( ( Carrier ( g , n ) ) . i )
consider y , z being element such that y in the carrier of A and z in the carrier of A and i = [ y , z ] and i = [ y , z ] ;
given H1 , H2 being strict Subgroup of G such that x = H1 & y = H2 & the carrier of H1 = the carrier of H2 and H2 is Subgroup of H1 and H1 is Subgroup of H2 and H2 is Subgroup of H2 and H1 is Subgroup of H2 ;
let S be non empty RelStr , T be complete Function of S , T , f be Function of S , T ;
[ a + i , b ] in ( the carrier of V ) /\ ( the carrier of V ) & [ a , b ] in ( the carrier of V ) /\ ( the carrier of V ) ;
reconsider m1 = max ( len F - len ( p - n ) ) as Element of NAT ;
I <= width GoB ( h , i ) & I * ( h , i ) = [ [ h , h ] , [ h , i ] , [ h , i ] , [ h , i ] , [ h , i ] ] ) ;
f2 /* q = ( f2 /* ( f1 /* s ) ) ^\ k .= ( ( f2 /* s ) ^\ k .= ( ( f1 /* s ) ^\ k ) ^\ k ;
attr A1 \/ A2 is linearly-independent means : Def3 : ( for i being Nat holds ( A1 . i ) /\ ( A1 . i ) /\ ( A1 . i ) = { 0. V } ;
func A -|^ C -> set equals union { A . s where s is Element of R : s in C } ;
dom ( Line ( v , i + 1 ) (#) ( ( \mathopen { - v } ) (#) ( <* v , m *> ) ) ) = dom ( F ^ <* v *> ) ;
cluster [ x , x , y ] -> LSeg ( x , y ) & [ x , y ] in [: { x , y } , { y } :] ;
E , E |= All ( x2 , H , E ) => ( ( x1 , x2 ) |= All ( x1 , H ) => ( x1 , x2 ) => ( x1 , x2 ) => ( x1 , x3 ) => ( x1 , x2 ) => ( x1 , x3 ) ;
F .: ( ( id X ) . x , g ) = F . ( ( id X ) . x , g . x ) .= F . ( ( id X ) . x , g . x ) ;
R . ( h . m ) = F . ( x0 + h . m ) - ( R . m ) . x0 ;
cell ( G , ( X , t ) -' 1 , t ) \ ( X \/ Y ) c= UBD ( ( L~ f ) \ { t } ) ;
IC Comput ( P2 , Comput ( P2 , s , i + 1 ) , Comput ( P2 , s , i + 1 ) ) = IC Comput ( P2 , Comput ( P2 , s , i + 1 ) , Comput ( P2 , s , i + 1 ) ) .= card I + 2 .= card I + 2 ;
sqrt ( ( - ( ( - ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 ) ^2 > 0 ;
consider x0 being element such that x0 in dom a and a in dom a and x0 in dom a and a . ( k + 1 ) = g . ( k + 1 ) ;
dom ( ( 1 / ( A . m ) ) (#) ( ( ( ( A . m ) --> ( A . m ) ) --> ( ( A . m ) --> ( A . m ) ) --> ( ( A . m ) --> ( ( A . m ) --> ( A . m ) ) ) = dom ( ( A . m ) --> ( A . m ) --> ( A . m ) ) .= ( A . m ) --> ( A . m ) --> ( ( A . m ) --> ( ( A . m ) --> ( ( A . m ) --> ( ( A .
d . [ y , z ] = ( ( [ y , z ] ) . [ y , z ] ) . [ y , z ] ;
attr for i being Nat holds C . i = A . i /\ B . i & C . i c= C . i /\ B . i ;
consider x0 such that x0 in dom f and f . x0 - f . x0 = ( f . x0 ) . x0 - f . x0 and for r st r in dom f holds ||. f /. r - f /. x0 .|| < r ;
p in Cl A implies for K being Basis of T st K in F & K is Basis & A c= K holds A meets K
for x being Element of REAL n st x in Line ( x1 , x2 ) holds |. y1 - y2 .| <= |. y1 - y2 .|
func mode sigma_# a -> Ordinal means : Def1 : for b being Ordinal st a in it holds it . b = b ;
[ a1 , a2 , a3 ] 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 ( e * ||. x .|| - ( vseq . m ) - ( vseq . n ) ) ^2 * sqrt ( e * ||. x .|| ) ;
then for Z being set st Z in { Y where Y is Element of I : Y in F } holds z in Z ;
upper_bound compactbelow ( s , t ) = [ sup { s . t where t is Element of L : not contradiction } & sup { s . t } = sup { s . t where t is Element of L : t in X } ;
consider i , j being Element of NAT such that i < j and [ y , f . i ] in [: I , I :] and [ y , f . i , f . j ] in [: I , I :] and [ y , f . i ] in [: I , I :] ;
let D being non empty set , p , q being FinSequence of D , r being FinSequence of D ;
consider e being Element of the carrier of X such that e , e // e , b and e , b // e , c and e , c // e , b and e , b // e , c and e , c // e , d ;
set U = I \! \mathop { {} } , u = I \! \mathop { {} } , v = I \! \mathop { {} } ;
|. q1 .| ^2 = ( ( |. q1 .| ) ^2 + ( |. q2 .| ) ^2 ) ^2 .= ( |. q1 .| ) ^2 + ( |. q2 .| ) ^2 .= ( |. q1 .| ) ^2 + ( |. q2 .| ) ^2 ;
let T being non empty TopStruct , x , y be Element of T , f being Function of the topology of T , the carrier of T ;
dom ( ( the charact of U1 ) * the Arity of U2 ) = dom ( ( the charact of U1 ) * the Arity of U2 ) & dom ( ( the Arity of U1 ) * the Arity of U2 ) = dom ( the Arity of U1 ) /\ ( the carrier' of U2 ) ;
dom ( h | X ) = dom h /\ X .= dom ( h | X ) /\ X .= dom ( h | X ) /\ X .= dom ( h | X ) /\ X .= dom ( h | X ) /\ ( ( h | X ) /\ X ) .= dom ( h | X ) /\ X .= dom ( ( h | X ) /\ X ) /\ X /\ X .= dom ( ( h | X ) /\ X ) /\ X /\ X .= dom ( ( h | X ) /\ X /\ X /\ dom ( ( h | X ) /\ X /\ dom ( ( h | X ) /\ X /\ dom ( ( h |
for N1 , N2 being Element of [: the carrier of G1 , the carrier of G1 , the carrier of G1 :] holds ( h . ( h . ( h . ( f . f ) ) ) . ( h . f ) = N & ( h . ( h . f ) ) . ( f . f ) = N ;
( mod ( u , m ) ) . i = ( mod ( u , m ) ) . i + ( mod ( v , m ) ) . i .= ( mod ( u , m ) ) . i + ( mod ( v , m ) ) . i ;
- ( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 < - ( ( q `2 / |. q .| - sn ) ^2 / ( 1 + sn ) ^2 & ( q `2 / |. q .| - sn ) ^2 >= 0 ;
attr r1 = f1 & r2 = f2 & f1 = g2 implies f1 + f2 = g1 + g2 & g1 + g2 = g2 + g2 & g1 + g2 = g2 + g2 & g1 + g2 = g1 + g2 & g1 + g2 = g2 + g2 & g1 + g2 = g1 + g2 + g2 & g1 + g2 = g1 + g2 implies g1 + g2 = g1 + g2 + g2 & g1 + g2 = g1 + g2 + g2
( for v be bounded Function of X , Y holds ( for m be Nat holds ||. ( vseq . m ) - vseq . m ) .|| . m = ( vseq . m - vseq . m ) ) . m
attr a <> b & b <> c & a <> c & b <> c & c <> d & d <> d & a <> b & c <> d & d <> d & d <> c & c <> d implies angle ( b , c , d ) = angle ( b , c , d ) ;
consider i , j being Nat , r being Real such that p1 = [ i , j ] and r < j and j < n and i < j and j < n and r < n and s . i < j and s . j < n ;
|. p .| ^2 - ( 2 * ( p `2 / |. p .| - sn ) ) ^2 = |. p .| ^2 + ( p `2 / |. p .| - sn ) ^2 ;
consider p1 , q1 , q1 , q2 being Element of ( X , Y ) * such that y = p1 ^ q1 and q1 ^ q2 = q1 ^ q2 and q1 ^ q2 = q2 ^ q2 and q1 ^ q2 = q2 ^ q2 and q1 ^ q2 = q2 ^ q2 ;
<* ( <* A , B *> ) . ( 1 , 1 ) = sqrt ( ( 1 - B ) * ( 1 - B ) ) * ( 1 - B ) ;
( ( TOP-REAL 2 ) | A ) `1 = lower_bound ( ( proj2 .: A ) /\ ( ( TOP-REAL 2 ) | ( TOP-REAL 2 ) ) ^2 ) & ( ( proj2 .: A ) `2 ) `2 = ( ( proj2 .: A ) `2 ) ^2 ;
s , ( ( H / ( 1 + 1 ) ) |= ( ( H / ( 1 + 1 ) ) , ( ( H / ( 1 + 1 ) ) ) / ( ( ( H / ( 1 + 1 ) ) ) , ( ( H / ( 1 + 1 ) ) ) / ( ( ( H / ( 1 + 1 ) ) ) ) / ( ( ( H / ( 1 + 1 ) ) ) to_power ( ( ( ( ( H / ( 1 + 1 ) ) ) ) ) ;
len ( ( ( ( support b1 ) + 1 ) + 1 ) = card ( support b1 ) + 1 .= card ( support b1 ) + 1 .= card ( support b1 ) + 1 .= card ( support b1 ) + 1 .= card ( support b1 ) + 1 .= card ( support b1 ) + 1 .= card ( support b1 ) + 1 .= card ( support b1 ) ;
consider z being Element of L1 such that z >= x and for z being Element of L1 st z >= y holds z >= x & z >= y ;
LSeg ( ( ( proj2 D ) | ( ( TOP-REAL 2 ) | D ) ) /\ D ) = { ( ( TOP-REAL 2 ) | D ) /\ D .= { ( ( TOP-REAL 2 ) | D ) /\ D .= { ( ( TOP-REAL 2 ) | D ) /\ D } ;
lim ( ( f ' ' - g ' /* ( h ^\ k ) - f /* ( h ^\ k ) - f /* ( h ^\ k ) - f /* ( h ^\ k ) - f /* ( h ^\ k ) ) = lim ( ( f ' - g ' - f ' /* ( h ^\ k ) - f ' /* ( h ^\ k ) ) ;
P [ i , pr1 ( f , i ) , pr1 ( f , i ) ] ;
for r be Real st 0 < r ex m be Nat st for n be Nat st n <= m holds ||. ( seq . n ) - ( seq . m ) .|| < r
let X be set , P be a_partition of X , a , b , c , d being Element of P , P , P be non empty Subset of X ;
Z c= dom ( ( ( ( ( ( ( \HM { the } \HM { carrier } ) ) ^ ) - ( ( \HM { the } \HM { carrier } ) ^ ) - ( ( ( \HM { the } \HM { carrier } \HM { of } ) ^ ) ) - ( ( ( \HM { the } \HM { carrier } \HM { 0 } ) ^ ) - ( ( ( \HM { the } \HM { 0 } ) ^ ) ) - ( ( ( \HM { the } \HM { 0 } ) - ( ( \HM { 0 } ) ^ ( ( \HM { 0 } ) ^ ( ( \HM { 0 } ) - ( (
ex j being Nat st j in dom ( l ^ <* x *> ) & j < i + 1 & i < len ( l ^ <* x *> ) & j = 1 + j & j < len ( l ^ <* x *> ) & j = i + j + 1 ;
for u , v being VECTOR of V , r being Real st 0 < r & r < 1 holds r * ( 1 - r ) in [. r , s .]
A , Int Cl A , Int Cl Int ( A \/ B ) ` ` , Int ( A \/ B ) ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ;
- Sum <* v , u , v *> = - ( v + u + u ) .= - ( v + u ) .= - ( v + u ) .= - ( v + u ) .= - ( v + u ) .= ( v + u ) + ( u + u ) .= ( v + u ) + ( u + u ) .= ( v + u ) + ( u + u ) ;
( Exec ( a := b , s ) ) . IC SCM R = ( Exec ( a , s ) ) . IC SCM R .= succ IC s .= succ IC s .= IC s .= IC s .= IC s .= IC s .= IC s .= IC s .= IC s .= 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 = ( the Sorts of J ) . x ;
let S1 , S2 be non empty reflexive RelStr , D be non empty Subset of [: S1 , S2 :] , S be non empty Subset of [: S2 , S2 :] ;
card X = 2 implies ex x , y st x in X & y in X & x in Y & y in Y & x in Y & y in Z & z in Z ;
( E-max L~ Cage ( C , n ) ) .. ( Cage ( C , n ) ) in rng ( Cage ( C , n ) ) ;
let T be DecoratedTree , p , q be Element of dom T , r be Element of dom T holds ( T . ( p ^ q ) ) . ( len T + 1 ) = T . ( len T + 1 ) ;
[ i2 + 1 , j2 ] in Indices G & [ i2 + 1 , j2 ] in Indices G & f /. ( i2 + 1 ) = G * ( i2 + 1 , j2 ) ;
cluster \frac ( k , n ) -> natural for Nat , m be Nat , n be Nat , k be Nat st n <= k & k <= m holds n <= m & k <= m & m <= n ;
dom F " ( the carrier of X1 union X2 ) = the carrier of X1 & rng F c= the carrier of X2 & F " ( the carrier of X1 union X2 ) is one-to-one & rng F c= the carrier of X2 & rng F c= the carrier of X1 & rng F c= the carrier of X2 ;
consider C being finite Subset of V such that C c= A and card C = n and card C = n and card C = n and card C = n and card C = n and card C = n and card C = n and card C = n and card C = n and card C = n and card C = n and card C = n ;
V is prime implies for X being finite Subset of the topology of T st X /\ Y c= V holds X is open or V is open & V is open
set X = { F ( v1 ) where v1 is Element of B : P [ v1 ] } , Y = { F ( v1 ) where v1 is Element of C : P [ v1 ] } ;
angle ( p1 , p3 , p4 , p4 , p4 , p4 , p4 , p1 , p2 , p4 , p1 , p4 , p4 , p1 , p2 , p4 , p1 , p4 , p4 , p4 , p1 , p2 , p4 , p4 , p1 , p3 , p4 , p4 , p1 , p2 be Point of TOP-REAL 2 ;
- sqrt ( ( 1 - ( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 ) = - sqrt ( 1 - sn ) ^2 ) .= - 1 - sn ^2 .= 1 - sn ^2 ;
ex f being Function of I[01] , ( TOP-REAL 2 ) | P st f is continuous & f . 0 = P & f . 1 = P & f . 0 = P & f . 1 = Q & f . 1 = Q & f . 1 = Q ;
attr f is partial differentiable means : Def3 : f . ( 2 + 1 ) = ( proj ( 2 , 3 ) ) . ( 2 + 1 ) ;
ex r , s st x = |[ r , s ]| & G * ( 1 , 1 ) `1 < r & r < G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `1 < r & G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `2 < s & s < G * ( 1 , 1 ) `2 } c= { G * ( 1 , 1 ) `1 } ;
consider f such that f is special and 1 <= t and t <= len G and G * ( t , width G ) `1 <= ( ( G * ( t , width G ) `1 ) `1 and G * ( t , width G ) `2 <= ( ( G * ( t , width G ) `1 ) `1 ;
attr i in dom G means : Def3 : r (#) ( reproj ( i , x ) ) = r (#) reproj ( i , x ) ;
consider c1 , c2 being bag of o1 , b2 being bag of o2 such that ( that c1 + c2 ) /. k = <* c1 , c2 , c2 *> and c1 = <* c1 , c2 , c2 , c1 , c2 , c2 , c2 , c2 , c1 , c2 , c2 , c2 , c2 , c1 , c2 , c2 , c2 , c2 , c2 , c2 , c2 , c2 , c1 , c2 , c2 , c2 , c2 , c2 , c2 , c2 , c2 , c2 , c1 , c2 , c1 , c2 , c1 , c2 , c1 , c2 , c2 , c1 , c1 , c2 , c2 , c1 , c2 , c2 , c1 ,
x0 in { |[ r1 , s1 ]| : r1 < r1 & r1 < s1 & s1 < s2 & s1 < s2 } ;
Cl ( X ^ Y ) . k = the carrier of ( X ^ Y ) . k .= ( ( the carrier of Y ) \/ ( the carrier of Y ) ) . k .= ( the carrier of Y ) . k ;
attr len M1 = len M2 & width M1 = width M2 & width M2 = width M2 implies M1 - M2 = - - - - M1 + - M2 + - - M2 + - - M1 + - - M1 + - - - M1 + - - - M1 + - - M1
consider g2 be Real such that 0 < g2 and for y being Point of S , g being Function of T , T st y in { g . y where y is Element of T : ||. y - g .|| < g2 } holds ||. ( g . y - g . y - g . y ) .|| < g2 . y ;
assume x < sqrt ( ( - a ) * sqrt ( a + b ) ^2 ) or x > sqrt ( - a ^2 + b ^2 ) ;
( G1 '&' G2 ) . i = ( <* 3 *> ^ ( G1 ^ G2 ) . i & ( G1 ^ G2 ) . i = ( <* 3 *> ^ ( G1 ^ G2 ) ) . i & ( G1 ^ G2 ) . i = ( <* 3 *> ^ ( G1 ^ G2 ) . i ) ^ ( G1 ^ G2 ) . i ;
for i , j st [ i , j ] in Indices ( M + ( - 1 , j ] ) & j < len ( M + ( - 1 , j ) ) holds ( M + ( - 1 , j ) ) * ( - j , j ) < ( M + ( - 1 ) ) * ( - j )
let f be FinSequence of NAT , i be Element of NAT , j be Element of NAT , f be FinSequence of NAT , i be Element of NAT , f be FinSequence of NAT , i be Nat st i in dom f & j in dom f holds f . i = f /. j ;
assume F = { [ a , b ] where a , b is Element of X : for c being Element of X st c in B holds a <= b } ;
b2 * ( b2 - b1 ) + ( b2 - b2 ) * ( b2 - b1 ) = 0. TOP-REAL n + ( b2 - b1 ) * ( b2 - b1 ) .= ( - b2 ) * ( b2 - b1 ) * ( b2 - b1 ) .= ( - b2 ) * ( b2 - b1 ) ;
Cl { D where D is Subset of T : D in F & ex B being Subset of T st B in F & B c= F & B c= F & B is open } ;
attr seq is summable means : Def3 : seq is summable & seq is summable & seq is summable & seq is summable & seq is summable & seq is summable implies Sum ( seq + seq ) = Sum ( seq + seq ) ;
dom ( ( TOP-REAL 2 ) | D ) = ( ( TOP-REAL 2 ) | D ) /\ D .= ( ( TOP-REAL 2 ) | D ) /\ D .= ( TOP-REAL 2 ) /\ D .= ( TOP-REAL 2 ) /\ D .= ( TOP-REAL 2 ) /\ D .= D ;
[ X \to Z ] is full full full SubRelStr of ( [#] Z ) |^ the carrier of Z & [ X , Z ] is full Subset of Z ;
( G * ( 1 , j ) `1 ) `1 = ( G * ( 1 , j ) `1 ) `1 & ( G * ( 1 , j ) ) `1 <= ( G * ( 1 , j + 1 ) `1 ) `1 ;
synonym m c= m2 , m1 , m2 , p be ( m2 , m ) -tuples_on the carrier of p means : Def3 : for p being Polynomial of n , L st p in P & m <= n holds it . m <= ( the \it M+* ( m , n ) ) . m ;
consider a being Element of B ( ) such that x = F ( a ) and a in { G ( b ) where b is Element of B ( ) : P [ b ] } ;
We say that the multiplicative loop over S is multiplicative where where where \alpha is carrier of S , a is Element of the carrier of S : a <> {} & b is thesis } is set ;
Morphism ( a , b , 1 , 1 ) + ( b + 1 ) = b + 1 + 1 .= b + 1 .= b + 1 + 1 .= b + 1 ;
cluster -> strict for non empty Subset of REAL , i , j be Nat , a , b be Element of REAL , c be Real , i be Element of NAT , a , b be Real st a = b + c & b = a + b holds a = b + c + d
( ( 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 ) = eval ( a *' ( p , x ) , x ) .= eval ( a *' ( p , x ) , x ) * eval ( p , x ) .= eval ( p , x ) * eval ( p , x ) .= eval ( p , x ) ;
assume the TopStruct of S = the TopStruct of T & for D being non empty Subset of S st D in the topology of T holds D is open & D is open & D is open ;
assume that 1 <= k and k <= len w + 1 and ( ( ( ( the carrier of w ) + w ) --> ( k + 1 ) ) . k = ( ( the carrier of w ) --> ( k + 1 ) ) . k ;
2 * ( a |^ n + 1 ) + ( 2 * ( a |^ n + 1 ) ) >= ( a |^ n + 1 ) * ( a |^ n + 1 ) + ( a |^ n ) * ( a |^ n + 1 ) ;
M , v / ( ( ( v , v / ( x , m ) ) / ( x , m ) ) / ( x , m ) / ( x , m ) ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x
assume that f is_differentiable_on l and for x0 st x0 in l holds 0 < f . ( x0 - x0 ) and 0 < g . ( x0 - x0 ) ;
let G1 be _Graph , W be walk of G1 , e be Vertex of G , x be Vertex of G , y be Vertex of G , e be Vertex of G , x being Vertex of G , y being Vertex of G , e being Vertex of G ;
c01 is not empty iff not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty & not empty is not empty & not empty is not empty & not empty & not empty is not empty & not empty & not empty & not empty
Indices ( GoB f ) = [: dom f , Seg ( len f ) :] & Indices ( f | [: f , f :] ) = [: dom f , Seg ( len f ) :] ;
let G1 , G2 be Group , G1 , G2 be Subgroup of G , f being FinSequence of the carrier of G1 , g being Element of G2 , h being Element of G , f being Element of G , g being Element of G st h = f & g is Subgroup of G2 & h is Subgroup of G holds f is normal & g is Subgroup of G
UsedIntLoc ( ( the Sorts of t ) +* f ) = { ( ( the Sorts of t ) +* ( f . ( intloc 0 ) ) . f , ( the Sorts of t ) +* ( f . ( intloc 0 ) ) . f , ( the Sorts of t ) +* ( f . ( intloc 0 ) ) . f , ( the Sorts of t ) +* ( f . ( intloc 0 ) ) . f , ( the Sorts of t ) +* ( ( the Sorts of t ) +* ( ( the Sorts of t ) +* ( ( the Sorts of t ) . f ) . f ) . f , ( the Sorts of t ) . f ) . f ) . f ) . f , ( the Sorts of t ) . f , ( the Sorts of
for f1 , f2 being FinSequence of F , p , q being FinSequence of F st f1 ^ f2 is p -element & p ^ q is p -element & q ^ <* p *> in F holds f1 ^ f2 is p -element
sqrt ( ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + 1 + 1 ) ^2 ) ^2 ) ) ^2 ) ) ^2 ) ^2 ) ^2 ) ^2 ) = sqrt ( ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2
let x1 , x2 , x3 , x4 , x5 , x2 , x4 , x5 , x5 , x5 , x5 , 8 , x5 , 6 , x5 , 8 , 8 , 6 , x5 , 8 , 8 , x5 , 6 , x5 , 6 , x5 , 6 , x5 , \Vert , \Vert , x5 , \Vert , \Vert , \Vert *> = <* x1 , x2 , x3 , x4 , x5 , \Vert , \Vert , \Vert , \Vert *> ;
for x , y st x in dom ( ( - ( F . x ) ) | A ) holds ( - ( F . y ) ) . x = - ( F . x ) . y
let T be non empty TopSpace , P be Basis of T , x be Point of T , T be Basis of T , P be Basis of T , x be Point of T ;
( a 'or' b ) 'or' c = 'not' ( a 'or' b ) 'or' ( 'not' a 'or' b ) 'or' ( 'not' a 'or' b ) 'or' ( 'not' a 'or' b ) .= 'not' ( a 'or' b ) 'or' ( 'not' a 'or' b ) 'or' ( 'not' a ) 'or' ( 'not' a 'or' b ) 'or' ( 'not' b ) 'or' ( 'not' a ) 'or' ( 'not' b ) 'or' ( 'not' a ) 'or' ( 'not' b ) 'or' ( 'not' b ) 'or' ( 'not' a ) 'or' ( 'not' a ) 'or' ( 'not' b ) 'or' ( 'not' b ) 'or' ( 'not' b ) 'or' ( 'not' b ) 'or' ( 'not' b ) 'or' ( 'not' b ) 'or' ( 'not' b ) 'or' ( 'not' b ) 'or' ( 'not' b ) 'or' ( 'not' b ) 'or' ( 'not' b )
for e being set st e in [: X1 , X2 :] ex X1 being Subset of [: Y1 , Y2 :] st e = [: X1 , X2 :] & [: X1 , X2 :] is open & [: X1 , X2 :] is open & [: X1 , X2 :] is open & [: X1 , X2 :] is open & [: X1 , X2 :] is open & [: X1 , X2 :] is open & [: Y1 , X2 :] is open & [: Y1 , X2 :] is open & [: X1 , X2 :] is open & [: X1 , X2 :] is open & [: X1 , X2 :] is open & [: X1 , X2 :] is open & [: X1 , X2 :] is open & [: X1 , X2 :] is open & [: X1 , X2 :] is open & [: X1 , X2 :]
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 | ( the carrier of S ) . i
for v , w st for x , y holds w <> y holds w . ( v , y ) = v . ( v , w )
card D = card ( D1 + D2 ) + card ( D1 + D2 ) .= 1 + ( card ( D1 + D2 ) + D2 ) .= 1 + ( card ( D1 + D2 ) + D2 ) .= 1 + ( card ( D1 + D2 ) + D2 ) .= 1 + ( card ( D1 + D2 ) + D2 ) .= 1 + ( card ( D1 + D2 ) + D2 ) .= 1 + ( 2 + D2 ) ;
IC Exec ( i , s ) = ( s +* ( i .--> 0 ) ) . 0 .= ( s +* ( i .--> 0 ) ) . 0 .= ( s +* ( i .--> 0 ) ) . 0 .= ( s +* ( i .--> 0 ) ) . 0 .= ( ( s +* ( i .--> 0 ) ) . 0 .= ( ( s +* ( i .--> 0 ) ) . 0 .= ( ( s +* ( i .--> 0 ) ) . 0 .= ( ( s +* ( i .--> 0 ) ) . 0 .= ( ( s +* ( i .--> 0 ) ) . 0 .= ( ( s +* ( i --> 0 ) .= ( ( s +* ( i .--> 0 ) ) . 0 .= ( s +* ( i --> 0 ) )
len f /. ( i1 -' 1 ) = len f - ( i1 - 1 ) .= len f - ( i1 - 1 ) .= len f - ( i1 - 1 ) .= len f - ( i1 - 1 ) .= len f - ( i1 - 1 ) .= len f - ( i1 - 1 ) .= len f - ( i1 - 1 ) ;
for a , b , c being Element of NAT st 1 <= a & a <= b & b <= c holds a + b < a + b or a + c < b + c
let f be FinSequence of TOP-REAL 2 , p be Point of TOP-REAL 2 , i be Nat , f be FinSequence of TOP-REAL 2 , p be Point of TOP-REAL 2 , r being Real st f . 1 = p & f . ( len f ) = q & f . ( len f ) = r holds Index ( p , f ) <= r ;
lim ( ( curry ( P , k + 1 ) ) # x ) = lim ( ( ( ( curry ( P , k + 1 ) ) # x ) + lim ( ( ( curry ( P , k + 1 ) ) # x ) ) ;
z2 /. ( n + 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 ) \/ ( the InternalRel of G ) or [ f , f . 3 ] in the InternalRel of G ;
let G being Subset-Family of B , X be non empty set , F be Subset-Family of G , G be Subset-Family of B , X be set , Y being Subset of G , x being Element of G st X = { [ X , Y ] where Y is Element of B : Y in F } ;
CurInstr ( P1 , Comput ( P1 , s1 , m + 1 ) ) = CurInstr ( P1 , Comput ( P1 , s1 , m + 1 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m ) ) ;
assume that a on M and b on M and c on N and d on N and p on M and q on N and p on M and q on N and p on M and q on N and p on M and q on N and p on M and q on N and p on M and q on N and q on N and r , s , s and s , q , r and s , r , s and p , s , q is_collinear ;
assume that T is \hbox { 4 , 4 , F , G , F , G be finite Subset-Family of T , F be Subset-Family of T , G be Subset-Family of T , f be Subset-Family of T , g be Subset-Family of T , g be Subset-Family of T , h be Subset-Family of T ;
for g1 , g2 st g1 in ]. r , s .[ & g1 in ]. r , s .[ holds |. f . g1 - f . g2 .| < ( r - f ) . g2 - f . g2
( ( - 1 / ( the carrier of V ) ) + ( - 1 / ( the carrier of V ) ) * ( - 1 / ( the carrier of V ) ) = - 1 / ( the carrier of V ) * ( - 1 / ( the carrier of V ) ) ;
F . i = F /. i .= F /. ( n + 1 ) .= F /. ( n + 1 ) .= F /. ( n + 1 ) .= F /. ( n + 1 ) .= F /. ( n + 1 ) .= F /. ( n + 1 ) .= F /. ( n + 1 ) ;
ex y being set st y = f . n & dom f = NAT & for n being Nat holds f . n = A ( ) & f . n = R ( n ) ;
func f * F -> FinSequence of V means : Def1 : for i being Nat st i in dom it holds it . i = F . i * F . ( i + 1 ) & for i being Nat st i in dom it holds it . i = F . ( i + 1 ) * F . ( i + 1 ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , 8 , x5 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 7 } } = { x1 , x2 , x3 , x4 , x5 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 ,
for n being Nat , x being set st x = h . n holds h . n = o ( x , n , n ) & h . n in InnerVertices S ( x , n , n ) & h . n = o ( x , n , n ) & h . n = o ( x , n , n ) ;
ex S1 being Element of Al ( ) , e being Element of Al ( ) , l being Element of l ( ) , A ( ) , e being Element of l ( ) , f being Function of [: Al ( ) , D ( ) , D ( ) :] , D ( ) , l ( ) , l ( ) , l ( ) , l ( ) , l ( ) , l ( ) ) , l ( ) , l ( ) , l ( ) ) is Element of D ( ) ;
consider P being FinSequence of ( the carrier of G ) * such that p = product P and for i being Nat st i in dom P holds P . i = t . i and P . i = t . i ;
let T1 being strict non empty TopSpace , T1 , T2 be Basis of T2 , T2 be Basis of T2 , T2 be Basis of T2 , T2 be Basis of T2 , T2 be Basis of T2 ;
assume that f is_differentiable_in z0 and r (#) ( f (#) ( 3 (#) f ) ) is_differentiable_in z0 and r (#) ( f (#) ( 3 (#) f ) ) is_differentiable_in z0 & r (#) ( f (#) f ) is_differentiable_in z0 & r (#) ( f (#) f ) is_differentiable_in z0 ;
defpred P [ Nat ] means for F being FinSequence of Seg ( $1 ) , G being FinSequence of REAL , s being FinSequence of REAL , r being FinSequence of REAL st len F = $1 & len G = $1 & len G = $1 holds r ^ s = Sum ( F ^ G ) ^ ( r ^ s ) ) ;
ex j st 1 <= j & j < width GoB f & ( ( GoB f ) * ( 1 , j ) + 1 ) `2 <= s * ( ( GoB f ) * ( 1 , j + 1 ) + ( GoB f ) * ( 1 , j + 1 ) ) `2 ;
defpred U [ set , set , set ] means ex F being Subset-Family of T st F = ( F . $1 ) & ( for n being Nat holds F . n = union F & ( for n being Nat holds F . n = union F ) & ( for n being Nat holds F . n = union F ) implies union F is open & union F is open & union F is open & union F is open & union F is open & union F is open & union F is open & union F is open & union F is open & union F is open & union F is open & union F is open & union F is open & union F is open & union F is open & union F is open & union F is open & union F is open & union F is open & union F is open & union F is open & union F is open & union F is open & union F is open & union F is open & union
for p2 being Point of TOP-REAL 2 , e being Real st LE e , p1 , p2 , p1 , p2 holds LE e , p1 , P & LE e , p1 , P & LE e , p1 , P
f in St ( E , H ) & for g st g in E holds f . g = f . g implies for x st x in E holds g . x = f . x & g . x = f . x & f . x = g . x & f . x = h . x & f . x = h . x
ex p2 being Point of TOP-REAL 2 st x = p2 & |. p2 .| >= sn & p2 `1 <= 0 & p2 `2 <= 0 & p2 `2 <= 0 & p2 `2 <= 0 & p2 `2 <= 0 & p2 `2 <= 0 & p2 `2 <= 0 & p1 `2 <= 0 & p2 `2 <= 0 & p1 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p1 `1 <= 0 & p2 `2 = 0 & p2 `2 = 1 & p2 `2 = 1 & p2 `2 = 1 & p2 `2 = 1 & p2 `2 = 1 & p2 `2 = 1 & p2 `2 = 1 & p2 `2 = 1 & p2 `2 = 1 & p2 `2 = 1 & p2 `2 = 1 & p2 `2 = 1 & p2 `2 = 1 & p2 `2 = 1 & p2 `2 = 1 & p2 `2 = 1 & p1 `2 = 1 & p1 `2 = 1 & p1 `2 = 1 & p1 `2 = 1 & p1 `2 = 1
assume for d being Element of NAT st d <= ( ( n + 1 ) -tuples_on ( the carrier of K ) ) . d holds d = ( n + 1 ) -tuples_on ( the carrier of K ) . d & ( n + 1 ) -tuples_on ( the carrier of K ) . d = ( n + 1 ) -tuples_on ( the carrier of K ) . d ;
consider s such that s <> t and s is Point of Closed-Interval-TSpace ( x , r ) and not ex e being Point of TOP-REAL n st e in Ball ( x , r ) & e in Ball ( x , r ) & not e in Ball ( x , r ) ;
given r such that 0 < r and for s st 0 < s ex x1 st x1 in dom f & |. x1 - x0 .| < s & |. x1 - x0 .| < s & |. x1 - x0 .| < s & |. x1 - x0 .| < r ;
( p | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x
consider x , h such that x + h in dom sec and h . x = sqrt ( ( 1 + ( 2 * x ) ^2 ) * ( 2 * h ^2 + h ^2 ) ) and h . x = sqrt ( 2 * x ^2 + ( 2 * h ^2 ) ^2 ) ;
assume that i in dom A and i > 1 and i < 1 and j in Seg n and i in dom A and j in dom B and A * ( i , j ) = B * ( i , j ) and B * ( i , j ) = B * ( i , j ) ;
for i be non zero Element of NAT st i in Seg n holds h . i = <* 1_ L *> or h . i = <* 1_ L *> & h . i = <* 1_ L *> ^ h . i & h . i = <* 1_ L *> ^ h ;
( ( ( b1 'imp' b2 ) 'imp' ( b2 'imp' b1 ) ) '&' ( ( b1 'imp' b2 ) 'imp' ( b2 'imp' b1 ) ) '&' ( b2 'imp' b1 ) '&' ( b2 'imp' b1 ) '&' ( b2 'imp' b1 ) '&' ( b2 'imp' b1 ) '&' ( b2 'imp' b1 ) '&' ( b2 'imp' b1 ) '&' ( b2 'imp' b1 ) '&' ( b2 'imp' b1 ) '&' ( b2 'imp' b1 ) '&' ( b2 'imp' b1 ) '&' ( b2 'imp' b1 ) '&' ( b2 'imp' b1 ) '&' ( b2 'imp' b1 ) '&' ( b2 'imp' b2 ) '&' ( b2 'imp' b1 ) '&' ( b2 'imp' b1 ) '&' ( b2 'imp' b2 ) '&' ( b2 'imp' b1 ) '&' ( b2 'imp' b1 ) '&' ( b2 'imp' b1 ) '&' ( b2 'imp' b2 ) '&' ( b2 'imp' b1 ) '&' ( b2 'imp' b2 ) '&' ( b2 'imp' b2 ) '&' ( b2 'imp' b1 ) '&' ( b2 'imp' b2 ) '&' ( b2 'imp' b1 ) '&' ( b2 'imp'
assume that for x holds f . x = ( ( - cot ) * ( \HM { x } ) ) . x and for x st x in dom ( ( - cot ) * ( \HM { x } ) ) holds f . x = - ( - ( - cot ) * ( \HM { x } ) ) . x ;
consider Rbe Real , I be Element of REAL such that RI = \int ( F . n ) + I and for i be Nat st i in dom I holds I . i = \int ( F . i , F . i , F . i ) + I . i and I . i = \int ( F . i , F . i , F . i ) + I . i ;
ex k be Element of NAT st k = k & 0 < k & for q be Element of product G st q in X & q in X holds Ball ( f , q ) c= r ;
x in { x1 , x2 , x3 , x4 , x5 , x5 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 8 , 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 in iff x in { x1 , x2 , x3 , x4 , 8 , 8 , 8 , 7 , 8 , 7 , 7 , 8 , 7 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 7 , 7 , 8 , 7 , 7 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 7 , 8 , 8 , 7 , 8 ,
( G * ( j , i ) `1 ) `1 = ( G * ( 1 , j ) `1 ) `1 .= ( G * ( 1 , j + 1 ) `1 ) `1 .= G * ( 1 , j + 1 ) `1 .= G * ( 1 , j + 1 ) `1 ;
f1 * p = p +* ( the Arity of S ) .= ( the Arity of S ) . o .= ( the Arity of S ) . o ;
func <* T , P , T , T , T , P , T , T , T *> -> Tree means : Def3 : for p , q being Element of T st p in P & q in P & p in T & q in T holds it . p = T . q ;
F /. ( k + 1 ) = F . ( k + 1 ) .= FH . ( p . ( k + 1 ) ) .= FH . ( p . ( k + 1 ) ) .= FH . ( p . ( k + 1 ) ) ;
let A , B , C be Matrix of K , K , f , g be FinSequence of K , a , b , c be Real ;
seq . ( k + 1 ) = 0. ( ( the carrier of V ) + ( k + 1 ) ) .= ( ( ( ( k + 1 ) + 1 ) + 1 ) * ( ( k + 1 ) + 1 ) .= ( ( ( k + 1 ) + 1 ) * ( ( k + 1 ) + 1 ) * ( ( k + 1 ) + 1 ) .= ( ( ( k + 1 ) + 1 ) * ( k + 1 ) ) * ( ( ( k + 1 ) * ( ( k + 1 ) * ( k + 1 ) * ( k + 1 ) * ( k + 1 ) * ( k + 1 ) * ( k + 1 ) * ( k + 1 ) * ( k + 1 ) * ( k + 1 ) * ( k + 1 ) * ( k + 1 ) * ( ( k + 1 ) + 1 ) * ( k + 1 ) * ( k + 1 ) * ( k +
assume that x in ( the addF of CCC ) & y in ( the carrier of CC ) /\ the carrier of CC and z in the carrier of CC and x in the carrier of CC and y in the carrier of CC and z in the carrier of CC and x in the carrier of CC and y in the carrier of CC and z in the carrier of CC and z in the carrier of CC and z in the carrier of CC and y in the carrier of CC and z in the carrier of CC and z in the carrier of CC and z in the carrier of CC and [ x "\/" y ;
defpred P [ Element of NAT ] means for f being ( Element of NAT ) , g being ( Element of D ) . ( $1 + 1 ) , h being Element of D st len f = $1 & g . ( ( ( ex f being Function of D , D ) . ( $1 + 1 ) ) . ( ( ( ( VAL g ) . ( $1 + 1 ) ) . ( ( ( VAL g ) . ( $1 + 1 ) ) ) . ( ( ( ( ( ( ( n + 1 ) ) . ( n + 1 ) ) . ( n + 1 ) ) . ( n + 1 ) ) = ( ( ( ( n + 1 ) ) . ( ( ( ( ( ( n + 1 ) ) . ( n + 1 ) ) . ( n + 1 ) ) . ( ( n + 1 ) ) . ( ( ( ( n + 1 ) ) . ( ( n + 1 ) ) . ( ( n + 1 ) ) .
assume that 1 <= k and k + 1 <= len f and f /. k = G * ( i + 1 , j ) and [ i + 1 , j + 1 ] in Indices G and [ i + 1 , j + 1 ] in Indices G and f /. k = G * ( i + 1 , j + 1 ) ;
consider sn such that sn < 1 and sn < 1 and sn >= 0 and ( sn TOP-REAL 2 ) | K1 = ( TOP-REAL 2 ) | K1 and ( ( TOP-REAL 2 ) | D ) . O = ( sn TOP-REAL 2 ) | D and ( ( TOP-REAL 2 ) | D ) . O = ( sn TOP-REAL 2 ) | D and ( TOP-REAL 2 ) . O = ( TOP-REAL 2 ) | D and ( TOP-REAL 2 ) | D = ( TOP-REAL 2 ) | D ) | D ;
let M be non empty dist , x be Point of M , f be Function of M , M , x be Point of M , r be Real ;
defpred P [ Element of omega ] means for f being PartFunc of omega , REAL , Z st f is_differentiable_on Z holds f . ( $1 + 1 ) = f . ( $1 + 1 ) - f . ( $1 + 1 ) * f . ( $1 + 1 ) - f . ( $1 + 1 ) & f . ( $1 + 1 ) > 0 ;
defpred P1 [ Nat , Nat ] means ex f being Function of [: ( the carrier of C ) , the carrier of C :] st f . $1 = r & f . ( $1 + 1 ) = ( f . $1 ) . ( $1 + 1 ) & f . ( $1 + 1 ) < r & f . ( $1 + 1 ) < r ;
( f ^ mid ( g , 2 , len g ) ) . i = ( g ^ mid ( g , 2 , len g ) ) . i .= g . ( i + 1 ) .= g . ( i + 1 ) ;
sqrt ( 1 - ( 2 * n + 1 ) * ( 2 * n + 1 ) ) = ( 1 - ( 2 * n + 1 ) * ( 2 * n + 1 ) ) * sqrt ( 1 - ( 2 * n + 1 ) * sqrt ( 2 * n + 1 ) ) .= 1 * sqrt ( 2 * n + 1 ) .= 1 * sqrt ( 2 * n + 1 ) ;
defpred P [ Nat ] means G in FinGG\ ( G . $1 , G . $1 ) & ( for n being Nat st n in dom G holds G . n = ( the InternalRel of G ) . $1 ) & ( the InternalRel of G ) . n = ( the InternalRel of G ) . ( n + $1 ) ;
assume that f /. 1 in Ball ( u , r ) and 1 <= m and m <= len f and m <= len f and f /. ( m + 1 ) = LSeg ( f /. m , r ) and f /. ( m + 1 ) = LSeg ( f /. m , r ) and f /. ( m + 1 ) = LSeg ( f , r ) ;
defpred P [ Element of NAT ] means ( ( ( ( for n holds ( ( ( ( ( ( ( ( ( ( ( 1 / 2 ) * ( 1 / 2 ) ) ) * ( 1 / 2 ) ) ) ) - ( ( 1 / 2 ) * ( ( 1 / 2 ) * ( ( 1 / 2 ) * ( ( 1 / 2 ) * ( ( 1 / 2 ) * ( 1 / 2 ) ) ) ) ) ) . $1 = ( ( ( 1 / 2 ) ) * ( ( 1 / 2 ) ) ) . $1 ;
for x being Element of product F holds x in ( the carrier of F ) . i & for i being set st i in dom ( the carrier of F ) holds x = ( the carrier of F ) . i & for i being set st i in dom ( the carrier of F ) holds x = ( the carrier of F ) . i
( x " + ( x " ) * ( x " ) ) |^ n = ( ( x " ) * ( x " ) ) |^ n .= ( x " ) |^ n .= ( x " ) |^ n .= ( x " ) |^ n .= ( x " ) |^ n ;
DataPart IExec ( P +* I , s ) = DataPart Comput ( P +* I , Comput ( P +* I , s , LifeSpan ( P +* I , s ) + 3 ) .= DataPart Comput ( P +* I , s , LifeSpan ( P +* I , s ) + 3 ) ;
given r such that 0 < r and ]. g , x0 + r .[ c= dom ( f1 + f2 ) and for g st g in ]. x0 - r , x0 + r .[ holds f1 . g <= ( f1 + f2 ) . g ;
assume that X c= dom ( f1 + f2 ) /\ X and f | X is continuous and f | X is continuous and f | X is continuous ;
let L being complete continuous continuous continuous LATTICE , X be Subset of L , x being Element of L st x = "\/" ( X , L ) holds x is directed & x is directed & x is directed ;
consider e being Element of dom A such that e in dom ( m *' p ) and p = m *' ( p *' q ) and ex m being Nat st m in dom ( m *' p ) & p . m = ( m *' p ) . m ;
( f1 - ( f1 - f2 ) /* ( h + c ) - f1 /* ( h + c ) = lim ( f1 - f2 /* ( h + c ) - f2 /* c ) .= lim ( f1 - f2 /* c ) - f2 /* c .= lim ( f1 - f2 /* c ) - f2 /* c ;
ex p1 being Element of Al ( ) , q1 being Element of Al ( ) st F ( p ) = g ( p ) & for g being Function of D ( ) , D ( ) st g in D ( ) holds g . ( g . ( len g ) ) = f . ( g . ( len g ) ) ;
( mid ( f , i , len f -' 1 ) ) /. ( j + 1 ) = ( mid ( f , i , len f -' 1 ) ) . ( j + 1 ) .= ( mid ( f , i , len f -' 1 ) . j ;
( p ^ q ) . ( n + 1 ) = ( p ^ q ) . ( n + 1 ) .= ( p ^ q ) . ( n + 1 ) .= ( p ^ q ) . ( n + 1 ) ;
len mid ( ( f , indx ( D2 , D1 , j ) , 1 ) , indx ( D2 , D1 , j ) ) = len ( ( D2 , D1 , j ) ) + 1 .= len ( ( D2 . n1 ) + 1 ) ;
x * y = ( ( x * y ) * z ) * ( ( y * z ) * z ) .= ( x * ( y * z ) * z .= x * ( y * z ) * z .= x * ( y * z ) .= x * ( y * z ) .= x * ( y * z ) ;
v . ( <* x , y *> ) = proj ( 1 , 1 ) . ( <* x0 , y0 *> ) + proj ( 1 , 1 ) . ( <* x0 , y0 *> ) * ( <* x0 , y0 *> ) ;
i * ( 0 * i + 1 ) = <* 0 * ( 1 - i ) + 0 * ( 1 - i ) .= 0 * ( 1 - i * i ) + 0 * ( 1 - i * i ) .= 0 * ( 1 - i * i ) + 0 * ( 1 - i * i ) .= 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 ) ;
ex r be Real st 0 < r & for Y be finite Subset of X st Y in rng r holds ||. Y - r .|| < r & ||. Y - r .|| < r ;
( GoB f ) * ( i , j + 1 ) = f /. ( k + 1 ) & ( GoB f ) * ( i + 1 , j + 1 ) = f /. ( k + 1 ) ;
( ( 1 / 2 ) * ( 1 - r ) ) / 2 = ( 1 / 2 ) * ( 1 - r ) .= ( 1 / 2 ) / 2 .= ( 1 / 2 ) * 2 * 2 * 2 ;
sqrt ( - ( a , b ) * sqrt ( a , b ) + sqrt ( - sqrt ( a , b ) * sqrt ( a , b ) ) ^2 ) > 0 & sqrt ( - sqrt ( a , b ) * sqrt ( a , b ) + sqrt ( - sqrt ( a , b ) ^2 ) < 0 ;
assume that inf ( X /\ Y ) in X and for X being non empty set holds X in ( ( the carrier of L ) /\ Y ) /\ Y & X is directed implies X is directed & Y is directed & X is directed & Y is directed ;
( ( Normalizdd ) . i , j ) (*) ( ( i , j ) --> ( i , j ) ) = ( i --> ( j , j ) --> ( i , j ) ) (*) ( i , j ) & ( i = j implies j = i implies j = i implies j = i & j = j implies j = i & j = j )