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contradiction ;
contradiction ;
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thesis ;
contradiction ;
contradiction ;
contradiction ;
contradiction ;
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contradiction ;
contradiction ;
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contradiction ;
contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
B ;
a <> c
T c= S
D c= B
c in X ;
b in X ;
X ;
b in D ;
x = e ;
let m ;
h is onto ;
N in K ;
let i ;
j = 1 ;
x = u ;
let n ;
let k ;
y in A ;
let x ;
let x ;
m c= y ;
F is one-to-one ;
let q ;
m = 1 ;
1 < k ;
G is prime ;
b in A ;
d divides a ;
i < n ;
s <= b ;
b in B ;
let r ;
B is one-to-one ;
R is total ;
x = 2 ;
d in D ;
let c ;
let c ;
b = Y ;
0 < k ;
let b ;
let n ;
r <= b ;
x in X ;
i >= 8 ;
let n ;
let n ;
y in f ;
let n ;
1 < j ;
a in L ;
C is boundary ;
a in A ;
1 < x ;
S is finite ;
u in I ;
z << z ;
x in V ;
r < t ;
let t ;
x c= y ;
a <= b ;
m in NAT ;
assume f is prime ;
not x in Y ;
z = +infty ;
let k be Nat ;
K ` is being_line ;
assume n >= N ;
assume n >= N ;
assume X is 1 -element ;
assume x in I ;
q is as let ;
assume c in x ;
p > 0 ;
assume x in Z ;
assume x in Z ;
1 <= kor ;
assume m <= i ;
assume G is commutative ;
assume a divides b ;
assume P is closed ;
b-a > 0 ;
assume q in A ;
not W is bounded ;
f is elements ;
assume A is boundary ;
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 atomic ;
b `1 <= c `1 ;
A meets W ;
i `1 <= j `1 ;
assume H is universal ;
assume x in X ;
let X be set ;
let T be Tree ;
let d be element ;
let t be element ;
let x be element ;
let x be element ;
let s be element ;
k <= 5 ;
let X be set ;
let X be set ;
let y be element ;
let x be element ;
P [ 0 ]
let E be set , A be Subset of E ;
let C be Category ;
let x be element ;
let k be Nat ;
let x be element ;
let x be element ;
let e be element ;
let x be element ;
P [ 0 ]
let c be element ;
let y be element ;
let x be element ;
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 --> ;
Q halts_on s ;
x in \in \in \in \in $ ;
M < m + 1 ;
T2 is open ;
z in b < a ;
R2 is well-ordering ;
1 <= k + 1 ;
i > n + 1 ;
q1 is one-to-one ;
let x be trivial set ;
PP 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 , A be Subset of TOP-REAL n ;
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 : o <> o2 ;
O <> O2 ;
let r be Real ;
let f be FinSeq-Location ;
let i be Nat ;
let n be Nat ;
Cl A = A ;
L c= Cl L ;
A /\ M = B ;
let V be RealUnitarySpace , M be Subset of V ;
not s in Y |^ 0 ;
rng f <= w
b "/\" e = b ;
m = m3 ;
t in h . D ;
P [ 0 ] ;
assume z = x * y ;
S . n is bounded ;
let V be RealUnitarySpace , A be Subset of V ;
P [ 1 ] ;
P [ {} ] ;
C1 is component ;
H = G . i ;
1 <= i `1 + 1 ;
F . m in A ;
f . o = o ;
P [ 0 ] ;
az2 <= being Real ;
R [ 0 ] ;
b in f .: X ;
assume q = q2 ;
x in [#] V ;
f . u = 0 ;
assume e1 > 0 ;
let V be RealUnitarySpace , A be Subset of V ;
s is trivial & s is non empty ;
dom c = Q
P [ 0 ] ;
f . n in T ;
N . j in S ;
let T be complete LATTICE , S be non empty Subset of T ;
the ObjectMap of F is one-to-one
sgn x = 1 ;
k in support a ;
1 in Seg 1 ;
rng f = X ;
len T in X ;
vbeing < n ;
Sbeing set ;
assume p = p2 ;
len f = n ;
assume x in P1 ;
i in dom q ;
let U1 , U2 , E ;
p-25 = c ;
j in dom h ;
let k ;
f | Z is continuous ;
k in dom G ;
UBD C = B ;
1 <= len M ;
p in \mathbin { x } ;
1 <= jj & jj <= len f ;
set A = -> set ;
card a [= c ;
e in rng f ;
cluster B \oplus A -> empty ;
H has no means : Def2 : H is odd ;
assume n0 <= m ;
T is increasing ;
e2 <> e1 & e2 <> e2
Z c= dom g ;
dom p = X ;
H is proper ;
i + 1 <= n ;
v <> 0. V ;
A c= Affin A ;
S c= dom F ;
m in dom f ;
let X0 be set ;
c = sup N ;
R is connected implies union M = union M
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= C-26 ;
mm <> {} ;
let x be Element of Y ;
let f be Line of C , i be Nat ;
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 in dom ( X .--> E ) ;
- y in I ;
let A be non empty set , B be non empty Subset of A ;
P0 = 1 ;
assume r in F . k ;
assume f is simple ;
let A be be \cdot set ;
rng f c= NAT ;
assume P [ k ] ;
f6 <> {} ;
let o be Ordinal ;
assume x is sum of squares ;
assume not v in { 1 } ;
let II , C ;
assume that 1 <= j and j < l ;
v = - u ;
assume s . b > 0 ;
d3 in dom g ;
assume t . 1 in A ;
let Y be non empty TopSpace , f be Function of Y , TOP-REAL 2 ;
assume a in uparrow s ;
let S be non empty Poset ;
a , b // b , a ;
a * b = p * q ;
assume x , y are_the space ;
assume x in Omega ( f ) ;
[ a , c ] in X ;
mm <> {} ;
M + N c= M + M ;
assume M is connected hhthesis ;
assume f is \cap bbL~ \rangle ;
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 & k2 = k1 ;
m + 1 < n + 1 ;
s in S \/ { s } ;
n + i >= n + 1 ;
assume Re y = 0 ;
k1 <= j1 & j1 <= j2 implies j1 <= j2
f | A is non as compact ;
f . x - b <= 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 ;
Cf ;
q2 c= C1 & q2 c= C2 ;
a2 < c2 & a2 < c2 implies a2 = c2
s2 is 0 -started ;
IC s = 0 ;
s4 = s4 & s4 = s3 ;
let V ;
let x , y be element ;
let x be Element of T ;
assume a in rng F ;
x in dom T ` ;
let S be i2 of L ;
y " <> 0 ;
y " <> 0 ;
0. V = u-w ;
y2 , y , w is_collinear ;
R8 ;
let a , b be Real , f be PartFunc of REAL , REAL ;
let a be Object of C ;
let x be Vertex of G ;
let o be object of C , a be object of A ;
r '&' q = P \lbrack l , l .] ;
let i , j be Nat ;
let s be State of A , a be Int-Location ;
s4 . n = N ;
set y = ( x `1 ) / 2 ;
mi in dom g & mi in dom g ;
l . 2 = y1 ;
|. g . y .| <= r ;
f . x in CX0 ;
V1 is non empty ;
let x be Element of X ;
0 <> f . g2 ;
f2 /* q is convergent ;
f . i is_measurable_on E ;
assume \xi in N-22 ;
reconsider i = i as Ordinal ;
r * v = 0. X ;
rng f c= INT & rng f c= INT ;
G = 0 .--> goto 0 ;
let A be Subset of X ;
assume X0 is dense & A is dense ;
|. f . x .| <= r ;
let x be Element of R ;
let b be Element of L ;
assume x in W-19 ;
P [ k , a ] ;
let X be Subset of L ;
let b be Object of B ;
let A , B be category ;
set X = Vars ( C ) ;
let o be OperSymbol of S ;
let R be connected non empty Poset ;
n + 1 = succ n ;
xY c= Z1 & Z1 c= Z1 ;
dom f = [: C1 , C2 :] ;
assume [ a , y ] in X ;
Re ( seq . n ) is convergent ;
assume a1 = b1 & a2 = b2 ;
A = sInt A ;
a <= b or b <= a ;
n + 1 in dom f ;
let F be Instruction of S , a be Element of S ;
assume r2 > x0 & x0 < r2 ;
let Y be non empty set , f be Function of Y , BOOLEAN ;
2 * x in dom W ;
m in dom ( g2 | n ) ;
n in dom ( g1 | n ) ;
k + 1 in dom f ;
the still of s is finite ;
assume x1 <> x2 & x2 <> x3 ;
v1 in ( V1 \ V1 ) ;
not [ b `1 , b ] in T ;
ii + 1 = i ;
T c= \llangle T , T \rrangle ;
( l `1 ) ^2 = 0 ;
let n be Nat ;
( t `2 ) ^2 = r ;
AA is_integrable_on M & AA is integrable ;
set t = Top 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 f <= e ;
cluster non empty normal for set ;
assume c2 = b2 & c1 = c2 ;
assume h in [. q , p .] ;
1 + 1 <= len C ;
not c in B . m1 ;
cluster R .: X -> empty ;
p . n = H . n ;
assume vseq is convergent & vseq is convergent ;
IC s3 = 0 & IC s3 = 0 ;
k in N or k in K ;
F1 \/ F2 c= F & F2 c= F ;
Int ( G1 \/ G2 ) <> {} ;
( z `2 ) ^2 = 0 ;
p11 <> p1 & p11 <> p2 ;
assume z in { y , w } ;
MaxADSet ( a ) c= F ;
ex_sup_of downarrow s , S ;
f . x <= f . y ;
let T be up-complete non empty reflexive antisymmetric RelStr ;
q |^ m >= 1 ;
a >= X & b >= Y ;
assume <* a , c *> <> {} ;
F . c = g . c ;
G is one-to-one implies G is one-to-one ;
A \/ { a } \not c= B ;
0. V = 0. Y ;
let I be be be be be be be \leq Instruction of S , S ;
f-24 . x = 1 ;
assume z \ x = 0. X ;
C4 = 2 to_power n ;
let B be SetSequence of Sigma ;
assume X1 = p .: D ;
n + l2 in NAT & n + l2 in NAT ;
f " P is compact ;
assume x1 in REAL & x2 in REAL & x3 in REAL ;
p1 = ( K + 1 ) * p1 ;
M . k = <*> REAL ;
phi . 0 in rng phi ;
MMMis closed ;
assume z0 <> 0_ ( n , L ) ;
n < ( N . k ) ;
0 <= seq . 0 & seq . 0 <= seq . 0 ;
- q + p = v ;
{ v } is Subset of B ;
set g = f /. 1 ;
[: R , S :] is stable ;
set cR = Vertices R , C = Vertices R ;
p0 c= P3 & P3 c= P3 ;
x in [. 0 , 1 .[ ;
f . y in dom F ;
let T be Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott of S ;
ex_inf_of the carrier of S , S ;
sup downarrow a = downarrow b & sup downarrow a = b ;
P , C , K is_collinear ;
assume x in F ( s , r , t ) ;
2 to_power i < 2 to_power m ;
x + z = x + z + q ;
x \ ( a \ x ) = x ;
||. x-y .|| <= r ;
assume that Y c= field Q and Y <> {} ;
a ~ , b ~ \rrangle ;
assume a in A ( ) ;
k in dom ( q | i ) ;
p is Let of S , S ;
i - 1 = i-1 ;
f | A is one-to-one ;
assume x in f .: [: X , Y :] ;
i2 - i1 = 0 & i1 - i2 = 0 ;
j2 + 1 <= i2 ;
g " * a in N ;
K <> { [ {} , {} ] } ;
cluster strict for for let let let
|. q .| ^2 > 0 ;
|. p4 .| = |. p .| ;
s2 - s1 > 0 & s2 - s1 > 0 ;
assume x in { Gik } ;
W-min C in C & W-min C in C ;
assume x in { Gik } ;
assume i + 1 = len G ;
assume i + 1 = len G ;
dom I = Seg n & dom I = Seg n ;
assume that k in dom C and k <> i ;
1 + 1-1 <= i + j ;
dom S = dom F & dom T = dom F ;
let s be Element of NAT , n be Nat ;
let R be ManySortedSet of A ;
let n be Element of NAT , x be set ;
let S be non empty non void void void SCM+FSA -` SCM+FSA ;
let f be ManySortedSet of I ;
let z be Element of COMPLEX , f be Function of COMPLEX , COMPLEX ;
u in { ag } ;
2 * n < ( 2 * n ) ;
let x , y be set ;
B-11 c= ( V . n ) ;
assume I is_closed_on s , P ;
U1 = U2 & U2 = U2 implies U1 = U2
M /. 1 = z /. 1 ;
x9 = x9 & x9 = y9 implies x9 = y9
i + 1 < n + 1 + 1 ;
x in { {} , <* 0 *> } ;
( f . i ) `1 <= ( f . i ) `1 ;
let l be Element of L ;
x in dom ( F . -17 ) ;
let i be Element of NAT , k be Element of NAT ;
r8 is ( COMPLEX ) -valued ;
assume <* o2 , o *> <> {} ;
s . x |^ 0 = 1 ;
card ( K1 ) in M & card ( K1 ) in M ;
assume that X in U and Y in U ;
let D be Subset-Family of Omega ;
set r = q - { k + 1 } ;
y = W . ( 2 * x ) ;
assume dom g = cod f & cod g = cod f ;
let X , Y be non empty TopSpace , f be Function of X , Y ;
x \oplus A is interval ;
|. <*> A .| . a = 0 ;
cluster strict for non empty Poset ;
a1 in B . s1 & a2 in B . s1 ;
let V be finite < F , v be Vector of V ;
A * B on B , A ;
f-3 = NAT --> 0 ;
let A , B be Subset of V ;
z1 = P1 . j & z2 = P2 . j ;
assume f " P is closed & f " P is closed ;
reconsider j = i as Element of M ;
let a , b be Element of L ;
assume q in A \/ ( B "\/" C ) ;
dom ( F * C ) = o ;
set S = INT |^ X , T = X |^ Y ;
z in dom ( A --> y ) ;
P [ y , h . y ] ;
{ x0 } c= dom f & { x0 } c= dom g ;
let B be non-empty ManySortedSet of I , A be ManySortedSet of I ;
( PI / 2 ) < Arg z ;
reconsider z9 = 0 , z9 = 1 as Nat ;
LIN a , d , c ;
[ y , x ] in IX ;
( Q ) * ( 1 , 3 ) = 0 ;
set j = x0 div m , i = x0 mod m ;
assume a in { x , y , c } ;
j2 - jj > 0 & j - 1 > 0 ;
I \! \mathop { phi } = 1 ;
[ y , d ] in F-8 ;
let f be Function of X , Y ;
set A2 = ( B - C ) / 2 ;
s1 , s2 be Element of L ;
j1 - 1 = 0 & j - 1 = j - 1 ;
set m2 = 2 * n + j ;
reconsider t = t as bag of n ;
I2 . j = m . j ;
i |^ s , n are_relative_prime ;
set g = f | D-21 ;
assume that X is lower and 0 <= r ;
( p1 `1 ) ^2 = 1 & ( p1 `2 ) ^2 = 1 ;
a < ( p3 `1 ) ^2 & b < ( p3 `2 ) ^2 ;
L \ { m } c= UBD C ;
x in Ball ( x , 10 ) ;
not a in LSeg ( c , m ) ;
1 <= i1 -' 1 & i1 -' 1 <= i2 -' 1 ;
1 <= i1 -' 1 & i1 -' 1 <= i2 -' 1 ;
i + i2 <= len h & i + 1 <= len h ;
x = W-min ( P ) & x = W-min ( P ) ;
[ x , z ] in X ~ Z ;
assume y in [. x0 , x .] ;
assume p = <* 1 , 2 , 3 *> ;
len <* A1 *> = 1 & len <* A1 *> = 1 ;
set H = h . g , I = h . g ;
card b * a = |. a .| ;
Shift ( w , 0 ) |= v ;
set h = h2 (*) h1 , i = h2 (*) h2 ;
assume x in ( X0 /\ X1 ) ;
||. h .|| < d1 & ||. h .|| < d1 ;
not x in the carrier of f & not x in the carrier of f ;
f . y = F ( y ) ;
for n holds X [ n ] ;
k - l = kl2 - l ;
<* p , q *> /. 2 = q ;
let S be Subset of the topology of Y ;
let P , Q be succ s ;
Q /\ M c= union ( F | M )
f = b * canFS ( S ) ;
let a , b be Element of G ;
f .: X is_<=_than f . sup X
let L be non empty transitive RelStr , S be non empty Subset of L ;
S-20 is x -let i be Element of S ;
let r be non positive Real ;
M , v |= x \hbox { y } ;
v + w = 0. ( Z , n ) ;
P [ len F ( ) ] ;
assume InsCode ( i ) = 8 & InsCode ( i ) = 8 ;
the zero of M = 0 & the zero of M = 0 ;
cluster z * seq -> summable for Real_Sequence ;
let O be Subset of the carrier of C ;
||. f .|| | X is continuous ;
x2 = g . ( j + 1 ) ;
cluster non empty non empty for Element of AllSymbolsOf S ;
reconsider l1 = l-1 as Nat ;
v4 is Vertex of r2 & v4 is Vertex of r2 ;
T2 is SubSpace of T2 & T1 is SubSpace of T2 ;
Q1 /\ Q19 <> {} & Q1 /\ Q29 <> {} ;
let k be Nat ;
q " is Element of X & q is Element of X ;
F . t is set of of of M ;
assume that n <> 0 and n <> 1 ;
set e = EmptyBag n , f = EmptyBag n ;
let b be Element of Bags n ;
assume for i holds b . i is commutative ;
x is root & p `2 = ( p `2 ) ^2 ;
not r in ]. p , q .[ ;
let R be FinSequence of REAL , a be Element of R ;
S7 does not destroy b1 & S7 does not destroy b1 ;
IC SCM R <> a & IC S <> a ;
|. p - |[ x , y ]| .| >= r ;
1 * ( seq . n ) = seq . n ;
let x be FinSequence of NAT , n be Nat ;
let f be Function of C , D , g be Function of C , D ;
for a holds 0. L + a = a
IC s = s . NAT & IC s = n ;
H + G = F- ( GG ) ;
CS1 . x = x2 & CS2 . x = CS2 . x ;
f1 = f .= f2 .= ( f | n ) ^ ( f | n ) ;
Sum <* p . 0 *> = p . 0 ;
assume v + W = v + u + W ;
{ a1 } = { a2 } & { a2 } = { a2 } ;
a1 , b1 _|_ b , a & b1 , c1 _|_ b , a ;
d3 , o _|_ o , a3 ;
If is reflexive & Cf is reflexive implies If is reflexive
IO is antisymmetric implies CO is antisymmetric
sup rng H1 = e & sup rng H1 = e ;
x = a9 * ( a9 - b9 ) ;
|. p1 .| ^2 >= 1 ;
assume j2 - 1 < j2 - 1 ;
rng s c= dom ( f1 - f2 ) ;
assume support a misses support b & not a in support b ;
let L be associative distributive non empty doubleLoopStr , A be Subset of L ;
s " + 0 < n + 1 ;
p . c = ( f " ) . 1 ;
R . n <= R . ( n + 1 ) ;
Directed I1 = I1 & Directed I2 = I2 ;
set f = + ( x , y , r ) ;
cluster Ball ( x , r ) -> bounded ;
consider r being Real such that r in A ;
cluster non empty NAT -defined for NAT -defined Function ;
let X be non empty directed Subset of S ;
let S be non empty full SubRelStr of L ;
cluster <* [ ] *> . N -> complete non trivial ;
( 1 - a ) " = a ;
( q . {} ) `1 = o ;
$ \bf ( i -' 1 ) > 0 ;
assume ( 1 - r ) <= t `1 ;
card B = k + 1-1 ;
x in union rng ( f | n ) ;
assume x in the carrier of R & y in the carrier of R ;
d in D ;
f . 1 = L . ( F . 1 ) ;
the vertices of G = { v } & not v in { v } ;
let G be : \cal G is : odd ;
e , v6 be set ;
c . ( i - 1 ) in rng c ;
f2 /* q is divergent_to+infty & f2 /* q is divergent_to+infty ;
set z1 = - ( z1 - z2 ) , z2 = - ( z1 - z2 ) ;
assume w is llas of S , G ;
set f = p |-count ( t - p ) ;
let c be Object of C ;
assume ex a st P [ a ] ;
let x be Element of REAL m , y be Element of REAL m ;
let IX be Subset-Family of X , Y be Subset of X ;
reconsider p = p as Element of NAT ;
let v , w be Point of X ;
let s be State of SCM+FSA , I be Program of SCM+FSA ;
p is finite implies p is empty & p is non empty
stop I ( ) c= ( P ( ) ) ;
set ci = ( f /. i ) `1 ;
w ^ t ==> w ^ s ;
W1 /\ W = W1 /\ W2 implies W1 = W2
f . j is Element of J . j ;
let x , y be \rm \rm \cdot of T2 ;
ex d st a , b // b , d ;
a <> 0 & b <> 0 & c <> 0
ord x = 1 & x is positive ;
set g2 = lim ( seq , n ) ;
2 * x >= 2 * ( 1 / 2 ) ;
assume ( a 'or' c ) . z <> TRUE ;
f (*) g in Hom ( c , c ) ;
Hom ( c , c + d ) <> {} ;
assume 2 * Sum ( q | m ) > m ;
L1 . ( F-21 ) = 0 & L1 . ( FY ) = 0 ;
the InternalRel of X \/ R1 = the InternalRel of X ;
( sin . x ) ^2 <> 0 & ( sin . x ) ^2 <> 0 ;
( ( exp_R * f ) `| Z ) . x > 0 ;
o1 in ( X /\ O ) /\ O & o2 in X /\ O ;
e , v6 be set ;
r3 > ( 1 - r ) * 0 ;
x in P .: ( F \ G ) ;
let J be closed ideal of R , I be Ideal of R ;
h . p1 = f2 . O & h . O = f . I ;
Index ( p , f ) + 1 <= j ;
len ( q @ ) = width M & width ( q @ ) = width M ;
the carrier of LK c= A & the carrier of LK c= A ;
dom f c= union rng ( F | n ) ;
k + 1 in support ( \mathop { \rm support } ( n ) ) ;
let X be ManySortedSet of the carrier of S ;
[ x `1 , y `2 ] in an \/ ( R ~ ) ;
i = D1 or i = D2 or i = D1 ;
assume a mod n = b mod n & b mod n = b mod n ;
h . x2 = g . x1 & h . x2 = f . x2 ;
F c= 2 |^ the carrier of X & F is closed ;
reconsider w = |. s1 .| as Real_Sequence ;
( 1 / m ) * m + r < p ;
dom f = dom ( I --> ( i , j ) ) ;
[#] ( ( TOP-REAL 2 ) | K1 ) = [#] ( ( TOP-REAL 2 ) | K1 ) ;
cluster - x -> R_eal for ExtReal ;
then { d1 } c= A & A is closed ;
cluster TOP-REAL n -> finite-ind for Subset of TOP-REAL n ;
let w1 be Element of M , w2 be Element of S ;
let x be Element of dyadic ( n ) ;
u in W1 & v in W2 & u in W2 implies u + v in W2
reconsider y = y as Element of L2 ;
N is full SubRelStr of T |^ ( the carrier of S ) ;
sup { x , y } = c "\/" c ;
g . n = n to_power 1 .= n ;
h . J = EqClass ( u , J ) ;
let seq be summable sequence of X , x be set ;
dist ( x `1 , y ) < ( r / 2 ) ;
reconsider mm = m - 1 as Element of NAT ;
x- x0 < r1 - x0 & x0 < r1 - x0 ;
reconsider P = P ` as strict Subgroup of N ;
set g1 = p * idseq ( q `1 , q `2 ) ;
let n , m , k be non zero Nat ;
assume that 0 < e and f | A is lower ;
D2 . ( I8 ) in { x } ;
cluster subcondensed open -> subopen ;
let P be compact non empty Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
Gik in LSeg ( Gik , 1 ) /\ LSeg ( Gik , Gij ) ;
let n be Element of NAT , x be Element of X ;
reconsider S8 = S , S8 = T as Subset of T ;
dom ( i .--> X `1 ) = { i } ;
let X be non-empty ManySortedSet of S ;
let X be non-empty ManySortedSet of S ;
op ( 1 ) c= { [ {} , {} ] } ;
reconsider m = mm - 1 as Element of NAT ;
reconsider d = x as Element of C ( ) ;
let s be 0 -started State of SCMPDS , k be Integer ;
let t be 0 -started State of SCMPDS , Q ;
b , b , x , y is_collinear & x , y , z is_collinear ;
assume that i = n \/ { n } and j = k \/ { k } ;
let f be PartFunc of X , Y ;
x0 >= ( sqrt c ) / sqrt 2 & x0 >= ( sqrt c ) / sqrt 2 ;
reconsider t7 = T-1 as Point of TOP-REAL n ;
set q = h * p ^ <* d *> ;
z2 in U . ( y2 , z2 ) /\ Q2 ;
A |^ 0 = { <* E *> } ;
len W2 = len W + 2 & len W2 = len W + 2 ;
len h2 in dom h2 & len h2 in dom h2 ;
i + 1 in Seg ( len s2 ) ;
z in dom g1 /\ dom ( f | X ) ;
assume p2 `1 = ( E-max ( K ) ) `1 & p2 `2 = ( E-max ( K ) ) `1 ;
len G + 1 <= i1 + 1 ;
f1 (#) f2 is convergent & lim ( f1 (#) f2 ) = x0 ;
cluster seq + ( s + 1 ) -> summable ;
assume j in dom ( M1 * ( i , j ) ) ;
let A , B , C be Subset of X ;
x , y , z be Point of X , p be Point of X ;
b / 2 - ( 4 * a * c ) >= 0 ;
<* x\vert y .| ^ <* y *> \geq x ;
a , b in { a , b } ;
len p2 is Element of NAT & len p1 = len p2 ;
ex x being element st x in dom R & y = R . x ;
len q = len ( K (#) G ) & len q = len G ;
s1 = Initialize ( Initialized s ) , P1 = P +* I , P2 = P +* I , P2 = P +* I ;
consider w being Nat such that q = z + w ;
x ` is Ideal of x & y is Ideal of y implies x = y
k = 0 & n <> k or k > n ;
then X is discrete for Subset of X ;
for x st x in L holds x is FinSequence of REAL
||. f /. c .|| <= r1 & ||. f /. c .|| <= r1 ;
c in uparrow p & not c in { p } ;
reconsider V = V as Subset of the TopStruct of TOP-REAL n ;
let N , M be being being being being being being being being being being being <* net of L ;
then z >= waybelow x & z >= compactbelow x ;
M \lbrack f , g .] = f & M [. g , f .] = g ;
( ( 0* 1 ) /. 1 ) = TRUE ;
dom g = dom f & dom g = dom f & rng g c= dom f ;
mode \cal holds is \cal \cal \cal \cal U } is \cal lim of G ;
[ i , j ] in Indices M & [ i , j ] in Indices M ;
reconsider s = x " as Element of H ;
let f be Element of dom ( Subformulae p ) , p be Element of dom p ;
F1 . ( a1 , - a1 ) = G1 . ( a1 , - a1 ) ;
redefine func Sphere ( a , b , r ) -> compact ;
let a , b , c , d be Real ;
rng s c= dom ( 1 / 2 ) & rng s c= dom ( 1 / 2 ) ;
curry ( F-19 , k ) is additive ;
set k2 = card dom B , s3 = card { i } ;
set G = coprod ( X ) ;
reconsider a = [ x , s ] as Object of G ;
let a , b be Element of Mf , f be Function of Mf , M ;
reconsider s1 = s , s2 = t as Element of ( S , T ) ;
rng p c= the carrier of L & rng p c= the carrier of L ;
let d be Subset of the Sorts of A ;
( x .|. x ) = 0 iff x = 0. W ;
I-21 in dom stop I & IY in dom stop I ;
let g be continuous Function of X | B , Y ;
reconsider D = Y as Subset of ( TOP-REAL n ) | D ;
reconsider i0 = len p1 , i1 = len p2 as Integer ;
dom f = the carrier of S & dom g = the carrier of S ;
rng h c= union ( the carrier of J ) ;
cluster All ( x , H ) -> \cal <* x *> ;
d * N1 / 2 > N1 * 1 / 2 ;
]. a , b .[ c= [. a , b .] ;
set g = f " | D1 , h = f " | D2 ;
dom ( p | ( ( m + 1 ) ) ) = ( m + 1 ) ;
3 + - 2 <= k + - 2 ;
tan is_differentiable_in ( ( arccot ) . x ) ;
x in rng ( f /^ n ) & y in rng ( f /^ n ) ;
let f , g be FinSequence of D ;
cp in the carrier of S1 & cp in the carrier of S2 ;
rng f " { 0 } = dom f & rng f = { 0 } ;
( the Source of G ) . e = v & ( the Source of G ) . e = v ;
width G - 1 < width G - 1 ;
assume v in rng ( S | E1 ) ;
assume x is root or x is root or x is root ;
assume 0 in rng ( ( g2 ) | A ) ;
let q be Point of ( TOP-REAL 2 ) | K1 , r be Real ;
let p be Point of ( TOP-REAL 2 ) | K1 , r be Real ;
dist ( O , u ) <= |. p2 .| + 1 ;
assume dist ( x , b ) < dist ( a , b ) ;
<* S7 *> is in the Indices of C-20 & <* C7 *> is in the carrier of C-20 ;
i <= len ( G - 1 ) - 1 + 1 ;
let p be Point of ( TOP-REAL 2 ) | K1 , r be Real ;
x1 in the carrier of [: I[01] , I[01] :] & x2 in the carrier of I[01] ;
set p1 = f /. i , p2 = f /. j ;
g in { g2 : r < g2 & g2 < r } ;
Q2 = Sp2 " ( Q /\ R ) .= Q ;
( ( 1 / 2 ) (#) ( 1 / 2 ) ) is summable ;
- p + I c= - p + A ;
n < LifeSpan ( P1 , s1 ) + 1 & I < len s1 ;
CurInstr ( p1 , s1 ) = i & CurInstr ( p1 , s1 ) = i ;
A /\ Cl { x } \ { x } <> {} ;
rng f c= ]. r - 1 , r + 1 .[ ;
let g be Function of S , V ;
let f be Function of L1 , L2 , g be Function of L1 , L2 ;
reconsider z = z as Element of ( CompactSublatt L ) ;
let f be Function of S , T ;
reconsider g = g as Morphism of c opp , b opp ;
[ s , I ] in S ~ ( A , I ) ;
len ( the connectives of C ) = 4 & len ( the connectives of C ) = 5 ;
let C1 , C2 be subcategory , a be object of C ;
reconsider V1 = V , V1 = V as Subset of X | B ;
attr p is valid means : Def4 : All ( x , p ) is valid ;
assume that X c= dom f and f .: X c= dom g and g .: X c= dom f ;
H |^ a " * H is Subgroup of H & H |^ a is Subgroup of H ;
let A1 be Let A1 of O , E , A2 be Element of E ;
p2 , r3 , q3 is_collinear & p1 , p2 , p3 is_collinear ;
consider x being element such that x in v ^ K ;
not x in { 0. TOP-REAL 2 } & not x in { 0. TOP-REAL 2 } ;
p in [#] ( ( TOP-REAL 2 ) | B11 ) ;
0 < M . ( E8 ) & M . ( E8 ) < M . ( E8 ) ;
^ ( c / c ) = c ;
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . s2 ) & a2 in dom ( F . s2 ) ;
cluster -> *> \rangle for non empty \mathbin of L ;
set i1 = the Nat , i2 = the Nat , i1 = the Nat , i2 = the Nat ;
let s be 0 -started State of SCM+FSA , I be Program of SCM+FSA ;
assume y in ( f1 \/ f2 ) .: A ;
f . ( len f ) = f /. ( len f ) ;
x , f . x '||' f . x , f . y ;
pred X c= Y means : Def4 : cos ( X ) c= cos ( Y ) ;
let y be upper Subset of Y , x be set ;
cluster -> -> natural for non empty Nat ;
set S = <* Bags n , i *> , T = <* i , j *> ;
set T = [. 0 , 1 / 2 .] ;
1 in dom mid ( f , 1 , 1 ) ;
( 4 * PI ) / 2 < ( 2 * PI ) / 2 ;
x2 in dom ( f1 - f2 ) /\ dom ( f2 - f3 ) ;
O c= dom I & { {} } = { {} } ;
( the Source of G ) . x = v & ( 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 `1 , y , z be Element of G ` ;
h19 . i = f . ( h . i ) ;
( p `1 ) ^2 = ( p1 `1 ) ^2 & ( p `2 ) ^2 = ( p1 `2 ) ^2 ;
i + 1 <= len Cage ( C , n ) ;
len <* P *> @ = len P & len <* P *> = len P ;
set N-26 = the \mathbin { of N : not contradiction } ;
len gSet + ( x + 1 ) - 1 <= x ;
a on B & b on B & not a on B ;
reconsider r-12 = r * I . v as FinSequence of REAL ;
consider d such that x = d and a "/\" d [= c ;
given u such that u in W and x = v + u ;
len f /. ( \downharpoonright n ) = len ( 9 ^ <* n *> ) ;
set q2 = ( \mathopen { - } f /. 1 ) , q2 = ( - f ) /. 1 ;
set S = { S1 , S2 , T , S , T , f , g ;
MaxADSet ( b ) c= MaxADSet ( P /\ Q ) ;
Cl ( G . q1 ) c= F . r2 & Cl ( G . q2 ) c= G . r2 ;
f " D meets h " ( V /\ W ) ;
reconsider D = E as non empty directed Subset of L1 ;
H = the_left_argument_of H '&' ( H ) '&' the_right_argument_of H ;
assume t is Element of ( F . S ) . X ;
rng f c= the carrier of S2 & rng g c= the carrier of S2 ;
consider y being Element of X such that x = { y } ;
f1 . ( a1 , b1 ) = b1 & f2 . ( a2 , b2 ) = b2 ;
the carrier' of G ` = E \/ { E } & { E } c= E ;
reconsider m = len ( k - 1 ) as Element of NAT ;
set S1 = LSeg ( n , UMP C ) , S2 = LSeg ( n , UMP C ) ;
[ i , j ] in Indices M1 & [ i , j ] in Indices M2 ;
assume that P c= Seg m and M is \HM { i } ;
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 * ( 1 / p ) ;
p-7 . i = pp1 . i & pp2 . i = pp2 . i ;
let PA , G be a_partition of Y , z be set ;
pred 0 < r & r < 1 implies 1 < ( 1 - r ) / ( 1 - r ) ;
rng ( \mathop { \rm AffineMap ( a , X ) , Y ) = [#] 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 & len canFS ( s ) = card s ;
reconsider x2 = x1 , y2 = x2 , z2 = y2 as Element of L1 ;
Q in FinMeetCl ( ( the topology of X ) . i , the topology of X ) ;
dom ( f . 0 ) c= dom ( u | ( dom f ) ) ;
redefine pred n divides m & m divides n implies n = m ;
reconsider x = x , y = y as Point of [: I[01] , I[01] :] ;
a in be
not y0 in the still of f & not ( ex g st g in dom f & not g in f ) ;
Hom ( ( a \times b ) \times c , c ) <> {} ;
consider k1 such that p " < k1 and k1 < len p and p . k1 = q . k1 ;
consider c , d such that dom f = c \ d ;
[ x , y ] in dom g & [ y , z ] in dom k ;
set S1 = \circ ( x , y , z ) ;
l2 = m2 & l1 = i2 & l2 = i2 implies l1 = i2
x0 in dom ( ( u - v ) (#) ( ( u - v ) (#) ( v - u ) ) ) ;
reconsider p = x as Point of ( TOP-REAL 2 ) | K1 , ( TOP-REAL 2 ) | K1 ;
I[01] = [: the carrier of I[01] , the carrier of I[01] :] & I = [: the carrier of I[01] , the carrier of I[01] :] ;
f . p4 <= f . p1 & f . p1 <= f . p2 ;
( ( F . n ) `1 ) ^2 <= ( x `1 ) ^2 & ( F . n ) `2 <= ( x `2 ) ^2 ;
( x `2 ) ^2 = ( W . ( len W ) ) ^2 .= ( W . ( len W ) ) ^2 ;
for n being Element of NAT holds P [ n ] ;
let J , K be non empty Subset of I ;
assume 1 <= i & i <= len <* a " *> ;
0 |-> a = <*> the carrier of K & 0 = 0. K ;
X . i in 2 -tuples_on A . i \ B . i ;
<* 0 *> in dom ( e --> [ 1 , 0 ] ) ;
then P [ a ] & P [ succ a ] ;
reconsider sbeing being such that sbeing being such that s = s\Omega D and snon empty and s is terminal of D ;
k - ( i -' 1 ) <= len thesis - j ;
[#] S c= [#] ( the TopStruct of T ) & [#] T c= [#] ( the TopStruct of T ) ;
for V being strict RealUnitarySpace holds V in the carrier of V implies V is Subspace of W
assume k in dom mid ( f , i , j ) ;
let P be non empty Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
let A , B be Matrix of n1 , K , n1 , n2 be Nat ;
- a * - b = a * b - b * a ;
for A being Subset of A9 holds A // A implies A // B
( id o2 ) in <* o2 , o2 , o2 *> & ( id o2 ) in { o2 , o2 } ;
then ||. x .|| = 0 & x = 0. X ;
let N1 , N2 be strict normal Subgroup of G , N be strict normal Subgroup of G ;
j >= len upper_volume ( g , D1 ) & len upper_volume ( g , D2 ) = len D2 ;
b = Q . ( len Qa - 1 ) + 1 ;
f2 * f1 /* s is divergent_to+infty & f2 * f1 is divergent_to+infty ;
reconsider h = f * g as Function of ( 4 * N ) , G ;
assume that a <> 0 and Polynom ( a , b , c ) >= 0 ;
[ t , t ] in the InternalRel of A & [ t , t ] in the InternalRel of A ;
( v |-- ( E , T ) ) | n is Element of T7 ;
{} = the carrier of L1 + L2 & the carrier of L1 = the carrier of L1 + L2 ;
Directed I is_closed_on Initialized s , P & Directed I is_closed_on Initialized s , P & Directed I is_halting_on Initialized s , P ;
Initialized p = Initialize ( p +* q +* I , p +* I ) ;
reconsider N2 = N1 , N2 = N2 as strict net of R1 , R2 ;
reconsider Y = Y as Element of [: Ids L , Ids L :] ;
"/\" ( uparrow p \ { p } , L ) <> p ;
consider j being Nat such that i2 = i1 + j and j in dom f ;
not [ s , 0 ] in the carrier of S2 & not [ s , 0 ] in the carrier of S2 ;
mm in ( B '&' C ) \ D /\ { {} } ;
n <= len ( ( P + Q ) | ( len P + len Q ) ) ;
( x1 `1 ) ^2 = ( x2 `1 ) ^2 & ( x1 `2 ) ^2 = ( x2 `1 ) ^2 ;
InputVertices S = { x1 , x2 , x3 , x4 , x5 } ;
let x , y be Element of FTTF1 ( n ) ;
p = |[ p `1 , p `2 ]| & p = |[ p `1 , p `2 ]| ;
g * 1_ G = h " * g * h .= g * h * g ;
let p , q be Element of PFuncs ( V , C ) ;
x0 in dom ( x1 - x2 ) /\ dom ( x2 - x3 ) ;
( R qua Function ) " = R " & ( R " ) " = R ;
n in Seg len ( f /^ ( i -' 1 ) ) ;
for s be Real st s in R holds s <= s2 & s2 <= s2 ;
rng s c= dom ( f2 * f1 ) & rng s c= dom ( f2 * f1 ) ;
synonym for for Subset of \rm being Subset of \rm being Subset of \rm being Subset of X holds X is finite ;
1. K * ( 1. K , n ) = 1. K * ( 1. K , n ) ;
set S = Segm ( A , P1 , Q1 ) , Q = Segm ( A , B , Q1 ) ;
ex w st e = ( w - f ) & w in F & w in G ;
curry ( P+* ( i , k ) # x ) # x is convergent ;
cluster open open open -> open for Subset of T7 ;
len f1 = 1 .= len ( f3 ) .= len ( f3 ) + len ( f3 ) ;
( i * p ) / p < ( 2 * p ) / p ;
let x , y be Element of [: U0 , U0 :] ;
b1 , c1 // b9 , c9 & c1 , c1 // b9 , c9 & c1 , c1 // b9 , c ;
consider p being element such that c1 . j = { p } ;
assume that f " { 0 } = {} and f is total ;
assume IC Comput ( F , s , k ) = n & k <= n ;
Reloc ( J , card I ) does not destroy a implies J does not destroy a
card Macro ( card I + 1 ) does not destroy c ;
set m3 = LifeSpan ( p3 , s3 ) , P4 = LifeSpan ( p1 , s3 ) , P4 = Comput ( p2 , s2 , 1 ) ;
IC SCMPDS in dom Initialize ( p +* I , s ) & IC s in dom Initialize ( p +* I , s ) ;
dom t = the carrier of SCM+FSA & dom t = the carrier of SCM+FSA ;
( ( E-max L~ f ) .. f ) .. f = 1 & ( E-max L~ f ) .. f = len f ;
let a , b be Element of PFuncs ( V , C ) ;
Cl Int ( Int Cl F ) c= Cl Int ( Int Cl F ) ;
the carrier of X1 union X2 misses ( A1 \/ A2 ) & the carrier of X1 misses A1 ;
assume not LIN a , f . a , g . a ;
consider i being Element of M such that i = d6 and i in A ;
then Y c= { x } or Y = {} or Y = { x } ;
M , v |= H1 / ( ( y , x ) / ( y , x ) ) ;
consider m being element such that m in Intersect ( FF , i ) and x = Intersect m ;
reconsider A1 = support ( u1 ) , A2 = support ( v1 ) as Subset of X ;
card ( A \/ B ) = k-1 + ( 2 * 1 ) ;
assume that a1 <> a3 and a2 <> a4 and a3 <> a4 and a4 <> a5 and a5 <> a5 ;
cluster s -\mathop { V } -> non empty for string of S ;
LS2 /. n2 = LS2 . n2 & LS2 . n2 = LS2 . n2 ;
let P be compact non empty Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
assume that r-7 in LSeg ( p1 , p2 ) and rp29 in LSeg ( p1 , p2 ) ;
let A be non empty compact Subset of TOP-REAL n , a , b be Real ;
assume that [ k , m ] in Indices ( D * ( i , j ) ) ;
0 <= ( ( 1 - 2 ) |^ p ) / p ;
( F . N | E8 ) . x = +infty ;
pred X c= Y & Z c= V & X \ V c= Y \ Z ;
( y `2 ) * ( z `2 ) <> 0. I & ( y `2 ) * ( z `2 ) <> 0. I ;
1 + card ( X-18 ) <= card ( u + ( X + Y ) ) ;
set g = z \circlearrowleft ( ( L~ z ) .. z ) ;
then k = 1 implies p . k = <* x , y *> . k ;
cluster total for Element of C -bool X , S , T ;
reconsider B = A as non empty Subset of ( TOP-REAL n ) | B , C be Subset of TOP-REAL n ;
let a , b , c be Function of Y , BOOLEAN , f be Function of Y , BOOLEAN ;
L1 . i = ( i .--> g ) . i .= g . i .= g . i ;
Plane ( x1 , x2 , x3 ) c= P & Plane ( x2 , x3 , x4 ) c= P ;
n <= indx ( D2 , D1 , j1 ) & len D2 = len D1 ;
( ( g2 ) . O ) `1 = - 1 & ( ( g2 ) . I ) `1 = - 1 ;
j + p .. f - len f <= len f - len f + p .. f - len f ;
set W = W-bound C , S = S-bound C , E = E-bound C , N = S-bound C , S = S-bound C , N = S-bound C , S = S-bound C , N = S-bound C , S = S-bound C , S = S-bound C , N = S-bound
S1 . ( a `1 , e `1 ) = a + e .= a + e ;
1 in Seg width ( M * ( ColVec2Mx p ) ) ;
dom ( i (#) Im f ) = dom ( Im f ) /\ dom Im f ;
( Dx `1 ) = W . ( a , *' ( a , p ) ) ;
set Q = non |= contradiction ( g , f , h ) ;
cluster -> topological for ManySortedSet of U1 , U2 ;
attr F = { A } means : Def2 : F is discrete ;
reconsider z9 = <* ] as Element of product ( Carrier G ) ;
rng f c= rng f1 \/ rng f2 & rng f c= rng f1 \/ rng f2 ;
consider x such that x in f .: A and x in f .: C ;
f = <*> ( the carrier of F_Complex ) & f = <*> ( the carrier of F_Complex ) ;
E , j |= All ( x1 , x2 , H ) ;
reconsider n1 = n , n2 = m , n1 = n , n2 = m , n2 = n , n3 = m ;
assume that P is idempotent and R is idempotent and P ** R = R ** P ;
card ( B2 \/ { x } ) = k-1 + 1 & card ( B2 \/ { x } ) = k + 1 ;
card ( ( x \ B1 ) /\ B1 ) = 0 & card ( ( x \ B1 ) /\ B2 ) = 0 ;
g + R in { s : g-r < s & s < g + r } ;
set q-1E = ( q , <* s *> ) -\mathop { 1 } ;
for x being element st x in X holds x in rng f1 holds x in X
h0 /. ( i + 1 ) = h0 . ( i + 1 ) ;
set mw = max ( B , being Element of ) , mw = max ( B , R ) ;
t in Seg width ( I ^ ( n , n ) ) ;
reconsider X = dom f , C = [: C , D :] as Element of Fin NAT ;
IncAddr ( i , k ) = <% i , j %> + k .= i + k ;
( S-bound L~ f ) / 2 <= ( q `2 ) / 2 & ( q `2 <= ( q `2 ) / 2 ;
attr R is condensed means : Def4 : Int R is condensed & Cl R is condensed ;
pred 0 <= a & a <= 1 & b <= 1 implies a * b <= 1 ;
u in ( ( c /\ ( ( d /\ b ) /\ e ) ) /\ f ) /\ j ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) /\ f ) /\ j ) ;
len C + - 2 >= 9 + - 3 & len C - 2 >= 9 + - 3 ;
x , z , y is_collinear & x , z , y is_collinear implies x , y , z is_collinear
a |^ ( n1 + 1 ) = a |^ ( n1 + 1 ) * a ;
<* \underbrace ( 0 , \dots , 0 } , x ) in Line ( x , a * x ) ;
set yx1 = <* y , c *> ;
Fs2 /. 1 in rng Line ( D , 1 ) & Fs2 /. len Fs2 = 0. K ;
p . m Joins r /. m , r /. ( m + 1 ) , G ;
( p `2 ) ^2 = ( f /. i1 ) `2 .= ( f /. i1 ) `2 ;
W-bound ( X \/ Y ) = W-bound ( X \/ Y ) & W-bound ( X \/ Y ) = W-bound ( X \/ Y ) ;
0 + ( p `2 ) ^2 <= 2 * r + ( p `2 ) ^2 ;
x in dom g & not x in g " { 0 } ;
f1 /* ( seq ^\ k ) is divergent_to+infty & f2 /* ( seq ^\ k ) is divergent_to+infty ;
reconsider u2 = u , v2 = v , u2 = w , u1 = y as VECTOR of \mathop { \rm PInt } ( X , Y ) ;
p |-count ( Product ( Sgm ( X ) ) ) = 0 & p |-count ( Product ( Sgm ( X ) ) = 0 ;
len <* x *> < i + 1 & i + 1 <= len c + 1 ;
assume that I is non empty and { x } /\ { y } = { 0. I } ;
set ii = card I + 4 .--> ( card I + 4 ) , dom ( i + 3 ) ;
x in { x , y } & h . x = {} & h . y = {} ;
consider y being Element of F such that y in B and y <= x `1 ;
len S = len ( the charact of ( A ) ) & len S = len the charact of ( A ) ;
reconsider m = M , i = I , n = N as Element of X ;
A . ( j + 1 ) = B . ( j + 1 ) \/ A . j ;
set N8 = : ( Ge & Ge = G ) ;
rng F c= the carrier of gr { a } & rng F c= the carrier of gr { a } ;
( for K , n , r being Nat holds P . ( K , n , r ) is and Q [ K ]
f . k , f . ( Let n ) ] in rng f ;
h " P /\ [#] ( T1 | P ) = f " P /\ [#] ( T1 | P ) ;
g in dom ( f2 \ f2 " { 0 } ) /\ dom ( f2 \ f1 " { 0 } ) ;
gX /\ dom f1 = g1 " ( f .: X ) & gX /\ dom g1 = f " ( f .: X ) ;
consider n being element such that n in NAT and Z = G . n ;
set d1 = being element , d2 = dist ( x1 , y1 ) , d2 = dist ( x2 , y2 ) , d2 = dist ( x2 , y2 ) ;
b `1 + ( 1 - r ) < ( 1 - r ) + ( 1 - r ) ;
reconsider f1 = f as Vector of the carrier of X , the carrier of Y ;
pred i <> 0 implies i / ( i + 1 ) mod ( i + 1 ) = 1 ;
j2 in Seg len ( ( g2 ) . i2 ) & j in Seg len ( ( g2 ) . i2 ) ;
dom ( i - 1 ) = dom ( i - 1 ) .= dom ( i - 1 ) .= dom ( i - 1 ) ;
cluster sec | ]. PI / 2 , PI .[ -> one-to-one ;
Ball ( u , e ) = Ball ( f . p , e ) ;
reconsider x1 = x0 , y1 = x0 , z1 = x0 , z2 = x1 , z2 = x2 , z2 = x3 as Element of S ;
reconsider R1 = x , R2 = y , R1 = z , R2 = z as Relation of L , L ;
consider a , b being Subset of A such that x = [ a , b ] ;
( <* 1 *> ^ p ) ^ <* n *> in RL ;
S1 +* S2 = S2 +* ( i , S1 +* S2 ) & S2 +* ( i , S2 ) = S2 +* ( i , S2 ) ;
( ( ( 1 / 2 ) (#) ( ( #Z 2 ) * ( f1 + #Z 2 ) ) ) `| Z ) = f ;
cluster -> C -valued for Function of C , REAL , A be non empty set ;
set C7 = 1GateCircStr ( <* z , x *> , f3 ) ;
Ef . e2 = Ef . e2 -T & Ef . e2 = Ef . ( e2 -T ) ;
( ( arctan ) (#) ( arctan ) ) `| Z is_differentiable_on Z & for x st x in Z holds ( ( arctan ) (#) ( arctan ) ) `| Z = f ;
upper_bound A = ( cos * 3 / 2 ) & lower_bound A = 0 & lower_bound A = 0 ;
F . ( dom f , - F . ( cod f , - F . ( cod f , - F . ( cod f , - F . ( cod f , - F . ( cod f , - F . ( cod f ) ) ) ) ) ) ;
reconsider pbeing Point of TOP-REAL 2 = ( q `1 ) / 2 , q `2 = ( q `2 ) / 2 as Point of TOP-REAL 2 ;
g . W in [#] Y0 & [#] Y0 c= [#] Y0 & [#] Y0 c= [#] Y0 & [#] Y0 c= [#] X0 ;
let C be compact non vertical non horizontal Subset of TOP-REAL 2 , a , b , c , d be Real ;
LSeg ( f ^ g , j ) = LSeg ( f , j ) /\ LSeg ( g , i ) ;
rng s c= dom f /\ ]. x0 - r , x0 .[ & rng s c= dom f /\ ]. x0 - r , x0 .[ ;
assume x in { idseq ( 2 , len Rev Rev f ) , Rev Rev f } ;
reconsider n2 = n , m2 = m , m1 = n , m2 = m , m2 = n , n1 = m + 1 ;
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 + m2 .= m1 + m2 + m2 + m2 ;
assume for n holds H1 . n = G . n -H . 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 c= R ` & R \ a c= R ` & R \ a c= R ;
t in ]. r , s .[ or t = r or t = s & s = t ;
z + v2 in W & x = u + ( z + v2 ) implies x in W
x2 |-- y2 iff P [ x2 , y2 ] & P [ x2 , y2 , y2 , x2 , y2 , y2 , x3 ] ;
pred x1 <> x2 means : Def4 : |. x1 - x2 .| > 0 & |. x1 - x2 .| > 0 ;
assume that p2 - p1 , p3 - p1 - p1 , p3 - p1 - p1 is_collinear and p1 - p2 , p3 - p1 - p2 is_collinear ;
set q = \langle in rng f ^ <* 'not' A *> ;
let f be PartFunc of REAL 1 , REAL-NS n , x be Point of REAL-NS n , r be Real ;
( n mod ( 2 * k ) ) - 1 = n mod k - 1 ;
dom ( T * ( \mathop { 0 } ) ) = dom ( \mathop { \rm len } t ) ;
consider x being element such that x in wX and x in c and x in c ;
assume ( F * G ) . v . x3 = v . x3 ;
assume that the Sorts of D1 c= the Sorts of D2 and the Sorts of D1 c= the Sorts of D2 and the Sorts of D2 c= the Sorts of D2 ;
reconsider A1 = [. a , b .[ , A2 = [. 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 ) , r = W-bound L~ Cage ( C , n ) ;
n1 - len f + 1 - len g <= len - len g + len g - len g ;
Seg |. q , O1 , a , b , c , b , a , b , c , d , b , c , a , b , c , d , b , c , d , f , u , v , u , v , w , u , z , u , z , u , v , w , z , u
set C-2 = ( ( \mathclose { \rm c } ) | G ) . ( k + 1 ) ;
Sum ( L (#) p ) = 0. R * Sum p .= 0. V * Sum p .= 0. V ;
consider i being element such that i in dom p and t = p . i ;
defpred Q [ Nat ] means 0 = Q ( $1 ) & $1 <= len Q implies Q [ $1 ] ;
set s3 = Comput ( P1 , s1 , k ) , P3 = P1 +* I , s4 = Comput ( P2 , s2 , k ) , P4 = P2 ;
let l be variable of k , A , B be Subset of k ;
reconsider U1 = union ( G-24 | n ) , U2 = union ( G-24 | n ) as Subset-Family of ( TT | n ) ;
consider r such that r > 0 and Ball ( p `1 , r ) c= Q ` ;
( h | ( n + 2 ) ) /. i = p29 ;
reconsider B = the carrier of X1 , C = the carrier of X2 as Subset of X ;
p9 = <* - ( c - 1 ) , 1 , - ( c - 1 ) *> ;
synonym f is real-valued for f is FinSequence of NAT means : Def2 : rng f c= NAT & rng f c= NAT ;
consider b being element such that b in dom F and a = F . b ;
x9 < card X0 & x9 < card X0 implies x9 in card ( X0 \/ Y ) & x9 in card ( X0 \/ Y )
pred X c= B1 means : Def4 : X c= succ B1 & for B st X c= B holds B c= B ;
then w in Ball ( x , r ) & dist ( x , w ) <= r ;
angle ( x , y , z ) = angle ( x , y , z ) ;
pred 1 <= len s means : Def4 : for s being Element of S holds s . ( 0 + 1 ) = s . ( 0 + 1 ) ;
f/. c= f . ( k + ( n + 1 ) ) ;
the carrier of { 1_ G } = { 1_ G } & the carrier of G = { 1_ G } ;
pred p '&' q in TAUT ( A ) means : Def4 : q '&' p in TAUT ( A ) & q is valid ;
- ( t `1 / t `2 ) < ( ( t `1 / t `2 ) - ( t `2 / t `1 ) ) ;
U1 . 1 = ( U1 /. 1 ) `1 .= ( U1 . 1 ) `1 .= ( U1 . 1 ) `1 .= ( U1 . 1 ) `1 ;
f .: the carrier of x = the carrier of x & f .: the carrier of x = the carrier of x ;
Indices ( O @ ) = [: Seg n , Seg n :] & dom ( O @ ) = [: Seg n , Seg n :] ;
for n being Element of NAT holds G . n c= G . ( n + 1 ) ;
then V in M @ ;
ex f being Element of F-9 st f is \cup of A9 & f is \cup of A9 & f is \cup of A9 ;
[ h . 0 , h . 3 ] in the InternalRel of G & [ h . 2 , h . 3 ] in the InternalRel of G ;
s +* Initialize ( ( intloc 0 ) .--> 1 ) = s3 +* Initialize ( ( intloc 0 ) .--> 1 ) ;
|[ w1 , v1 ]| - ( v1 - v2 ) <> 0. TOP-REAL 2 & |[ w1 , v1 - v2 ]| - ( w1 - v2 ) = 0. TOP-REAL 2 ;
reconsider t = t as Element of ( Z , X ) * ;
C \/ P c= [#] ( ( G | [#] ( ( ( ( G | A ) \ A ) ) \ A ) ) ;
f " V in ( let X ) /\ D & D in ( the topology of S ) /\ D & f .: V in X ;
x in [#] ( the carrier of F ) /\ A & y in [#] ( F | A ) ;
g . x <= h1 . x & h . x <= h1 . x & h . x <= h1 . x ;
InputVertices S = { xy , y , z } & InputVertices S = { xy , y , z } ;
for n be Nat st P [ n ] holds P [ n + 1 ] ;
set R = Line ( M , i , a * Line ( M , i ) ) ;
assume that M1 is being_line and M2 is being_line and M3 is being_line and M2 is being_line and M1 - M2 is being_line ;
reconsider a = f4 . ( i0 -' 1 ) , b = ( f | ( i0 -' 1 ) ) as Element of K ;
len ( B2 ^ F2 ) = Sum ( Len ( F1 ^ F2 ) ) & len ( Len ( F1 ^ F2 ) ) = len ( F1 ^ F2 ) ;
len ( ( the means : Let : i in dom A ) & i <> j & j in dom A & A . i = A . j ) ;
dom max ( f , g ) = dom ( f + g ) & dom f = dom ( f + g ) ;
( the seq of seq ) . n = upper_bound Y1 & ( the seq of seq ) . n = upper_bound Y1 ;
dom ( p1 ^ p2 ) = dom ( f ^ <* 0 *> ) .= dom ( f ^ <* 1 *> ) ;
M . [ 1 / y , y ] = 1 / ( 1 - y ) * v1 .= y ;
assume that W is non trivial and W { x } c= the carrier of G2 and not x in the carrier of G2 ;
C6 * ( i1 , i2 ) `1 = G1 * ( i1 , i2 ) `1 & C6 * ( i1 , j1 ) `2 = G1 * ( i1 , j1 ) `2 ;
C8 |- 'not' Ex ( x , p ) 'or' p . ( x , y ) ;
for b st b in rng g holds lower_bound rng f-r <= b - a ;
- ( ( q1 `1 / |. q1 .| - cn ) / ( 1 - cn ) ) = 1 ;
( LSeg ( c , m ) \/ LSeg ( l , k ) ) \/ LSeg ( l , k ) c= R ;
consider p being element such that p in such that p in Ball ( x , r ) and p in L~ f ;
Indices ( X @ ) = [: Seg n , Seg n :] & Indices ( X @ ) = [: Seg n , Seg n :] ;
cluster s => ( q => p ) -> valid ;
Im ( ( Partial_Sums F ) . m ) is measurable & Im ( ( Partial_Sums F ) . m ) is measurable ;
cluster f . ( x1 , x2 ) -> Element of D * ;
consider g being Function such that g = F . t and Q [ t , g . t ] ;
p in LSeg ( ( NW-corner L~ f ) , ( NW-corner L~ f ) `1 ) ;
set R8 = R / 1 , R8 = R / 1 , R8 = R / 2 , R8 = R / 2 ;
IncAddr ( I , k ) = SubFrom ( da , da ) .= IncAddr ( da , da ) ;
seq . m <= ( ( the seq of seq ) . k ) . ( seq . k ) ;
a + b = ( a ` *' b ` ) ` .= ( a ` ` ) ` .= ( a ` ` ) ` ;
id ( X /\ Y ) = id ( X /\ Y ) .= id ( X /\ Y ) ;
for x being element st x in dom h holds h . x = f . x ;
reconsider H = U1 \/ U2 , U1 = U2 \/ U1 , U2 = U1 \/ U2 as non empty Subset of U0 ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) /\ f ) /\ j ) /\ m ;
consider y being element such that y in Y and P [ y , lower_bound B ] ;
consider A being finite stable set of R such that card A = card ( the carrier of R ) and card A = card the carrier of R ;
p2 in rng ( f |-- p1 ) \ rng <* p1 *> & rng <* p1 *> c= rng ( f |-- p1 ) ;
len s1 - 1 > 0 & len s2 - 1 > 0 & len s2 - 1 > 0 ;
( ( NW-corner P ) `2 ) ^2 = ( S-bound P ) ^2 & ( ( NW-corner P ) `2 ) ^2 = ( S-bound P ) ^2 ;
Ball ( e , r ) c= LeftComp Cage ( C , k + 1 ) ;
f . a1 ` ` = f . a1 ` & f . a1 ` = f . a1 ` ` & f . a2 = f . ( a1 ` ) ;
( seq ^\ k ) . n in ]. x0 - r , x0 .[ & ( seq ^\ k ) . n in ]. x0 - r , x0 .[ ;
gg . s0 = g . s0 & G . s0 = g . s0 & G . s0 = g . s0 ;
the InternalRel of S is PI & the InternalRel of S is non empty implies the InternalRel of S is connected
deffunc F ( Ordinal , Ordinal ) = phi . ( $2 , $2 ) & $2 = phi . ( $2 , $2 ) ;
F . a1 = F . s2 & F . a1 = F . a1 & F . a2 = F . a2 ;
x `1 = A . o . a .= Den ( o , A . a ) . a ;
Cl ( f " P1 ) c= f " P1 & Cl ( f " P1 ) c= f " P1 & Cl ( f " P1 ) c= f " P1 ;
FinMeetCl ( ( the topology of S ) . i ) c= the topology of T & FinMeetCl ( ( the topology of S ) . i ) c= the topology of T ;
synonym o is \bf means : Def2 : o <> \ast & o <> * & o <> * ;
assume that X + Y = Y and card X <> card Y and card Y <> card Z and card X = card Z ;
the *> <= 1 + ( the *> \HM { s + 1 ) * ( the *> ^ the { s + 1 } ) ;
LIN a , a1 , d or b , c // b1 , c1 & b , c // b1 , c1 & a , b // c , c1 ;
e . 1 = 0 & e . 2 = 1 & e . 3 = 0 & e . 4 = 0 ;
EE in SS1 & EE in { NE } & EE in { NE } ;
set J = ( l , u ) If , I = l l , J = l l , T = I " ;
set A1 = .: ( ap , bm , cp , cin ) , A2 = { A1 , A2 , cin , cin } ;
set c9 = [ <* c9 , cin *> , '&' ] , s3 = [ <* c , d *> , '&' ] , f4 = [ <* d , c *> , '&' ] , f4 = [ <* c , d *> , '&' ] , A1 = [ <* d , c *> , '&' ] , A2 = [ <* c , d *> , '&' ] , A2 =
x * z `1 * x " in x * ( z * N ) * x " ;
for x being element st x in dom f holds f . x = g3 . x & f . x = g2 . x
Int cell ( f , 1 , G ) c= RightComp f \/ RightComp f & RightComp f c= RightComp f \/ RightComp f ;
U2 is_an_arc_of W-min C , E-max C , W-min C & LE E-max C , E-max C , E-max C , E-max C ;
set f-17 = f @ "/\" ( g @ ) ;
attr S1 is convergent means : Def4 : S2 is convergent & for n holds S1 . n - S2 . n is convergent & lim S1 = 0 ;
f . ( 0 + 1 ) = ( 0 qua Ordinal ) + a .= a ;
cluster be Line let RelStr which is Line be reflexive transitive ;
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 + 2 ) , SCMPDS ) & not b in dom Start-At ( ( card I + 2 ) , SCMPDS ) ;
( z + a ) + x = z + ( a + y ) .= z + a + y ;
len ( l | ( a |^ 0 ) ) = len l & len ( l | 0 ) = len l ;
t4 is {} ( {} \/ rng ( t ^ <* n *> ) ) -valued FinSequence of NAT ;
t = <* F . t *> ^ ( C . p ^ q ) .= C . ( p ^ q ) ;
set p-2 = W-min L~ Cage ( C , n ) , pi = W-min L~ Cage ( C , n ) , pi = W-min L~ Cage ( C , n ) ;
( k - i ) + ( i - 1 ) = ( k - i ) + ( i - 1 ) ;
consider u being Element of L such that u = u ` ` and u in D and x in D ;
len ( ( width ( ( ( A - B ) - a ) ) * ( A - B ) ) * ( A - B ) ) = width ( A - B ) ;
F3 . x in dom ( ( G * the_arity_of o ) . x ) ;
set cH2 = the carrier of H2 , H1 = the carrier of H1 , H2 = the carrier of H2 ;
set cH1 = the carrier of H1 , H1 = the carrier of H2 ;
( Comput ( P , s , 6 ) ) . intpos ( m + 6 ) = s . intpos ( m + 6 ) ;
IC Comput ( Q2 , t , k ) = ( l + 1 ) - ( k + 1 ) ;
dom ( ( ( 1 / 2 ) (#) ( sin * cos ) ) `| Z ) = dom f /\ dom ( sin * cos ) ;
cluster <* l *> ^ phi -> ( 1 + \ { l } ) as ( 1 + \ { l } ) string of S ;
set b5 = [ <* A1 , cin , cp *> , [ <* cin , A1 *> , [ <* cin , A1 *> , [ <* cin , A1 *> , [ <* cin , A1 *> , [ <* cin , A1 *> , [ <* cin , A1 *> , [ <* cin , A1 *> , [ <* cin , A1 *> , [ <* cin , A1 *> ] *>
Line ( Segm ( M @ , P @ , Q ) , x ) = L * Sgm Q ;
n in dom ( ( ( the Sorts of A ) * the_arity_of o ) * ( the_arity_of o ) ) ;
cluster f1 + f2 -> continuous for PartFunc of REAL , the carrier of S , the carrier of S ;
consider y be Point of X such that a = y and ||. x - y .|| <= r ;
set x3 = ( t . DataLoc ( s2 . SBP , 2 ) ) , x4 = ( t . SBP ) , 6 = ( t . SBP ) , 7 = ( t . SBP ) / ( 2 * ( 2 * ( 2 * ( 3 * ( 3 * ( 4 * ( 4 * ( 4 * ( 4 * ( 4
set p-3 = stop I , pE = stop I , pE = stop I ;
consider a being Point of D2 such that a in W1 and b = g . a and a in W2 ;
{ A , B , C , D } = { A , B , C } \/ { D , E , F , J }
let A , B , C , D , E , F , J , M , N , N , M , N , N , F , M , N , N , M , N , N , F , N , M be set ;
|. p2 .| ^2 - ( ( p2 `1 ) / |. p2 .| ) ^2 >= 0 & |. p2 .| ^2 - ( p2 `1 ) ^2 >= 0 ;
l - 1 + 1 = n-1 * ( l6 + 1 ) + ( 1 - 1 ) ;
x = v + ( a * w1 + b * w2 ) + ( c * w1 + c * w2 ) ;
the TopStruct of L = \bf \bf 1 - ( L ) & the TopStruct of L = the TopStruct of L & the TopStruct of L = the TopStruct of L ;
consider y being element such that y in dom H1 and x = H1 . y and y in H1 . x ;
ff \ { n } = Free ( All ( v1 , H ) , E ) & f in Free ( All ( v1 , H ) , E ) ;
for Y being Subset of X st Y is summable holds Y is non empty implies Y is non empty ;
2 * n in { N : 2 * Sum ( p | N ) = N & N > 0 } ;
for s being FinSequence holds len ( the { of A , A , B , C ) = len s & len ( the { of A , B , C , D ) = len s
for x st x in Z holds exp_R * f is_differentiable_in x & for x st x in Z holds exp_R * f is_differentiable_in x
rng ( h2 * ( f2 - g2 ) ) c= the carrier of ( ( TOP-REAL 2 ) | K1 ) ;
j + 1- len f <= len f + ( len f - len f ) - len f + len g - len f ;
reconsider R1 = R * I , R2 = R * I as PartFunc of REAL n , REAL-NS n , REAL-NS n ;
C8 . x = s1 . a1 .= C7 . x .= C7 . x .= C8 . x ;
power F_Complex = 1 .= x |^ ( z , n ) .= x |^ ( z , n ) .= x |^ ( z , n ) ;
t at at ( C , s ) = f . ( the connectives of S ) . t & f . t = s . ( the connectives of S ) . t ;
support ( f + g ) c= support f \/ C7 & support ( f + g ) c= support f \/ Carrier g ;
ex N st N = j1 & 2 * Sum ( r4 | N ) > N & for n st n >= N holds r4 . n > 0 ;
for y , p st P [ p ] holds P [ All ( y , p ) ]
{ [ x1 , x2 ] where x1 , x2 is Point of [: X1 , X2 :] : x1 in X } is Subset of [: X1 , X2 :]
h = ( i , j = j |-- h , id B ) . i .= H . i ;
ex x1 being Element of G st x1 = x & x1 * N c= A & N c= A & N c= A ;
set X = ( ( Seg ( q , O1 ) ) , { q } ) , Y = { [ p , q ] } ;
b . n in { g1 : x0 < g1 & g1 < x0 & g1 < x0 & g1 < x0 + r } ;
f /* s1 is convergent & f /. x0 = lim ( f /* s1 ) implies f /* s1 is convergent & lim ( f /* s1 ) = lim ( f /* s1 )
the lattice of the lattice of Y = the carrier of the lattice of Y & the carrier of Y = the carrier of Y implies the carrier of Y = the carrier of X
'not' ( a . x ) '&' b . x 'or' a . x '&' 'not' ( b . x ) '&' 'not' ( b . x ) = FALSE ;
2 = len ( ( q ^ r1 ) ^ ( p - r1 ) ) + len ( ( q ^ r1 ) ^ ( p - r1 ) ) ;
( 1 - a ) * ( sec * f1 ) - id Z is_differentiable_on Z & ( ( 1 - a ) * ( sec * f1 ) ) is_differentiable_on Z ;
set K1 = upper ( lim ( H , A ) || ( A , B ) ) , K1 = ( lim ( H , A ) || ( A , B ) ) ;
assume e in { ( w1 - w2 ) / ( w1 - w2 ) : w1 in F & w2 in G } ;
reconsider d7 = dom a `1 , d8 = dom F `1 , d8 = dom G , d8 = dom F `1 , d8 = dom G , d8 = dom F , d8 = dom G , d8 = dom F , d8 = dom G , d8 = dom F , d8 = dom G , d8 = dom F , d
LSeg ( f /^ j , q ) = LSeg ( f , j -' q .. f + q .. f -' 1 ) ;
assume X in { T . ( N2 , K1 ) : h . ( N2 , K1 ) = N2 } ;
assume that Hom ( d , c ) <> {} and <* f , g *> * f1 = <* f , g *> * f2 ;
dom S29 = dom S /\ Seg n .= dom L6 .= Seg n /\ Seg n .= dom L6 .= Seg n /\ Seg n ;
x in H implies ex g st x = g |^ a & g in H & g in H
a * ( 0. ( K , n , 1 ) ) = a `1 - ( 0 * n ) .= a `1 - ( 0 * n ) ;
D2 . j in { r : lower_bound A <= r & r <= D1 . i } ;
ex p being Point of TOP-REAL 2 st p = x & P [ p ] & p <> 0. TOP-REAL 2 ;
for c holds f . c <= g . c implies f @ c <= g @ c
dom ( f1 (#) f2 ) /\ X c= dom ( f1 (#) f2 ) /\ dom ( f2 (#) f3 ) ;
1 = ( p * p ) * p .= p * ( p * q ) .= p * ( 1 / p ) .= p * ( 1 / p ) ;
len g = len f + len <* x + y *> .= len f + 1 + 1 .= len f + 1 + 1 ;
dom ( F | ( N1 , S ) ) = dom ( F | ( N1 , S ) ) .= ( F | ( N1 , S ) ) | ( N1 , S ) ;
dom ( f . t * I . t ) = dom ( f . t * g . t ) ;
assume a in ( "\/" ( ( T |^ the carrier of S ) , F ) ) .: D ;
assume that g is one-to-one and ( the carrier' of S ) /\ rng g c= dom g and rng g c= dom f and f is one-to-one ;
( ( x \ y ) \ z ) \ ( ( x \ z ) \ ( y \ z ) ) = 0. X ;
consider f such that f * f `1 = id b and f * f `2 = id a and f * f = id b ;
( ( ( cos * cos ) | [. 0 , PI / 2 .] ) * ( cos * cos ) | [. 0 , PI / 2 .] ) is increasing ;
Index ( p , co ) <= len LS - Index ( Gij , LS ) + 1 - Index ( Gik , LS ) ;
let t1 , t2 , t2 , t be Element of ( T . s ) . ( s , t ) , s be Element of ( T . s ) . ( s , t ) ;
"/\" ( ( Frege ( Frege ( H ) ) , L ) , L ) <= "/\" ( ( Frege ( ( Frege ( G ) ) , L ) , L ) ;
then P [ f . i0 , f . i0 ] & F ( f . ( i0 + 1 ) ) < j ;
Q [ ( D . ( x , 1 ) ) `1 , F . ( D . ( 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 the 0. of G . i ;
the Sorts of A2 = ( the carrier of S2 ) --> ( the carrier of S2 ) & the Sorts of A2 = ( 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 = F and rng s c= { x } ;
dist ( b1 , b2 ) <= dist ( b1 , a ) + dist ( a , b ) & dist ( a , b ) <= dist ( a , b ) + dist ( b , a ) ;
( <* W-min C , n *> /. len <* W-min C , n *> ) = seq1 /. len ( <* W-min C , n *> ) ;
q `2 <= ( ( UMP ( C ) ) `2 ) / 2 & ( ( UMP ( C ) ) `2 <= ( ( UMP ( C ) ) ) `2 ;
LSeg ( f | i2 , i ) /\ LSeg ( f | i2 , j ) = {} & LSeg ( f | i2 , j ) = {} ;
given a being ExtReal such that a <= Ia and A = ]. a , Ia .[ and a in A and b in B ;
consider a , b be complex number such that z = a & y = b and z + y = a + b ;
set X = { b |^ n where n is Element of NAT : n <= k } , Y = { b |^ n where n is Element of NAT : n <= k } ;
( ( x * y * z \ x ) \ z ) \ ( x * y * z ) = 0. X ;
set xy = [ <* xy , y , z *> , f2 ] , yz = [ <* y , z *> , f3 ] , f4 = [ <* z , x *> , f3 ] , f4 = [ <* z , x *> , f2 ] , xy = [ <* z , x *> , f3 ] , xy = [ <* x , y *> , f3 ] , f4 = [ <* z ,
ll /. len ll = ll . ( len ll ) .= ll . ( len ll ) ;
( ( q `2 / |. q .| - sn ) / ( 1 - sn ) ) ^2 = 1 ;
( ( p `2 / |. p .| - sn ) / ( 1 - sn ) ) ^2 < 1 ;
( ( ( S \/ Y ) /\ ( X \/ Y ) ) /\ ( X \/ Y ) ) = ( ( S \/ S ) /\ ( X \/ Y ) ) /\ ( X \/ Y ) ;
( s1 - s2 ) . k = ( s1 . k - s2 . k ) / ( s1 . k - s2 . k ) ;
rng ( ( h + c ) ^\ n ) c= dom SVF1 ( 1 , f , u0 ) ;
the carrier of X = the carrier of X0 & the carrier of X0 = the carrier of X0 & the carrier of X0 = the carrier of X0 implies f is closed & f is closed
ex p4 st p3 = p4 & |. p3 - |[ a , b ]| .| = r & |. p3 - |[ a , b ]| .| = r ;
set ch = chi ( X , A ) , A5 = chi ( X , A ) , A5 = chi ( X , A ) ;
R / ( 0 * n ) = Imin ( X , X ) .= R / ( n * n ) .= R / ( n * n ) ;
( Partial_Sums ( curry ( F , n ) ) . n ) . x is nonnegative & ( Partial_Sums ( ( curry ( F , n ) ) . n ) . x is nonnegative ;
f2 = C7 . ( E7 . ( len ( V , K ) ) , len ( H ) ) ;
S1 . b = s1 . b .= s2 . b .= s2 . b .= s2 . b .= s2 . b .= s2 . b ;
p2 in LSeg ( p2 , p1 ) /\ LSeg ( p1 , p2 ) & LSeg ( p2 , p1 ) /\ LSeg ( p1 , p2 ) c= { p2 } ;
dom ( f . t ) = Seg n & dom ( I . t ) = Seg n & dom ( I . t ) = Seg n ;
assume o = ( the connectives of S ) . 11 & the ResultSort of S = ( the carrier' of S ) . 12 ;
set phi = ( l1 , l2 ) If , phi = ( l1 , l2 ) If , F = ( l1 , l2 ) If , G = ( l1 , l2 ) If , F = ( l1 , l2 ) If , G = ( l1 , l2 ) If , F = ( l1 , l2 ) ) , G = ( l1 , l2 ) <> {} ;
synonym p is invertible for p , T means : Def2 : ex p being Polynomial of n , L st p = 1. L & p <> 0. L ;
( Y1 `2 = - 1 & Y1 `1 = - 1 & Y1 `2 <> - 1 & Y1 `1 <> 0. TOP-REAL 2 & Y1 `1 <> 0. TOP-REAL 2 & Y1 <> 0. TOP-REAL 2 & Y2 <> 0. TOP-REAL 2 & Y1 <> 0. TOP-REAL 2 & Y1 <> 0. TOP-REAL 2 & Y1 <> 0. TOP-REAL 2 & Y1 <> 0. TOP-REAL 2 & Y1 <> 0. TOP-REAL 2 & Y1 <> 0. TOP-REAL 2 & Y1 <> 0. TOP-REAL 2 &
defpred X [ Nat , set , set ] means P [ $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 is_collinear ;
consider k be Nat such that for n be Nat st k <= n holds s . n < x0 + g and s . n < x0 + g ;
Det ( I |^ ( m -' n ) ) * ( m -' n ) = 1. K & Det ( I |^ ( m -' n ) ) = 1. K ;
( - b - sqrt ( b - a ^2 ) ) / ( 4 * a - b ^2 ) < 0 ;
Cf . d = C7 . ( d7 . d ) mod C7 . ( d8 . d ) .= C7 . ( d7 . d ) mod C8 . ( d7 . d ) ;
attr X1 is dense means : Def2 : X2 is dense & X1 is dense implies X1 /\ X2 is dense SubSpace of X ;
deffunc F6 ( Element of E , Element of I , Element of I ) = ( $1 * $2 ) * ( $1 * $2 ) ;
t ^ <* n *> in { t ^ <* i *> : Q [ i , T . t ] } ;
( x \ y ) \ x = ( x \ x ) \ y .= y ` \ x .= 0. X ;
for X being non empty set holds X is Basis of [: X , FinMeetCl ( Y ) :] & for Y being Subset-Family of X holds Y is Basis of [: X , FinMeetCl ( Y ) :]
synonym A , B are_separated for A , B , C , D , E , F , J , M , N , N , F , M , N , N , F , N , M , N , N , F , N , M , N , N , M , N , F , N , M , N , N , F , M be set ;
len ( M @ ) = len p & width ( M @ ) = width M & width ( M @ ) = width M & width ( M @ ) = width M & width ( M @ ) = width M ;
J . v = { x where x is Element of K : 0 < v . x & v . x < 0 } ;
( Sgm ( Seg m ) ) . d - ( Sgm ( Seg m ) . d ) <> 0 ;
lower_bound divset ( D2 , k + k2 ) = D2 . ( k + ( k2 - 1 ) ) .= D2 . ( k + ( k2 - 1 ) ) ;
g . r1 = - 2 * r1 + 1 & dom h = [. 0 , 1 .] & rng g = [. 0 , 1 .] ;
|. a .| * ||. f .|| = 0 * ||. f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| ;
f . x = ( h . x ) `1 & g . x = ( h . x ) `2 & h . x = ( h . x ) `2 ;
ex w st w in dom B1 & <* 1 *> ^ s = <* 1 *> ^ w & for i st i in dom B1 holds w . i = <* 1 *> ^ w ;
[ 1 , {} , <* d1 *> ] in ( { [ 0 , {} , {} ] } \/ ( { [ 0 , {} ] } \/ { [ 1 , {} ] } ) ) ;
IC Exec ( i , s1 ) + n = IC Exec ( i , s2 ) + n .= ( i + n ) + n ;
IC Comput ( P , s , 1 ) = succ IC s .= ( 5 + 9 ) .= 5 + 9 .= 5 ;
( IExec ( W6 , Q , t ) ) . intpos i = t . intpos i & ( IExec ( W6 , Q , t ) ) . intpos i = t . intpos i ;
LSeg ( f /^ q , i ) misses LSeg ( f /^ i , j ) & LSeg ( f /^ i , j ) misses LSeg ( f /^ i , j ) ;
assume for x , y being Element of L st x in C & y in C holds x <= y or y <= x ;
integral ( integral ( f , C ) , f ) = f . ( upper_bound C ) - lower_bound C .= f . ( upper_bound C ) - lower_bound C ;
for F , G being one-to-one FinSequence st rng F misses rng G & rng G misses rng F holds F ^ G is one-to-one
||. R /. L - R /. h .|| < e1 * ( K + 1 - K ) ;
assume a in { q where q is Element of M : dist ( z , q ) <= r & r <= q & q in { r where r is Real : r <= r & r <= s } ;
set p4 = [ 2 , 1 ] .--> [ 2 , 0 , 1 ] ;
consider x , y being Subset of X such that [ x , y ] in F and x c= d and y c= d ;
for y , x being Element of REAL st y `1 in Y & x `1 in X & x `2 in Y holds y `1 <= x `1 & x `2 <= x `2
func |. p \bullet |. p .| -> variable means : Def4 : for i holds it . i = min ( NBBBBBBIIIIBBBIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII
consider t being Element of S such that x `1 , y `2 '||' z `1 , t `2 and x `1 , z `2 '||' y `1 , t `2 ;
dom x1 = Seg len x1 & len x2 = len y1 & len y2 = len x2 & len y1 = len y2 & len y2 = len y2 & len y1 = len y2 & len y2 = len y2 & len y2 = len y2 & len y1 = len y2 & len y2 = len y2 & len y1 = len y2 & len y2 = len y2 & len y2 = len y2 & len y1 = len y2 & len y2 = len y2 & len y2 = len y2 & len y1
consider y2 being Real such that x2 = y2 and 0 <= y2 and y2 <= 1 and y2 <= 1 and y2 <= 1 ;
||. f | X /* s1 .|| = ||. f .|| | X & ||. f /. s1 .|| = ||. f /. s1 .|| & ||. f /. s1 .|| = ||. f /. s1 .|| ;
( the InternalRel of A ) \ ( x ` ) = {} \/ {} .= {} \/ {} .= {} .= {} ;
assume that i in dom p and for j be Nat st j in dom q holds P [ i , j ] and for i , j be Nat st i in dom q & j in dom q & i + 1 in dom q & i + 1 in dom q & i + 1 in dom q holds q . i = p . j ;
reconsider h = f | [: X , Y :] as Function of [: X , Y :] , rng ( f | [: X , Y :] ) ;
u1 in the carrier of W1 & u2 in the carrier of W2 & v in the carrier of W1 & u in the carrier of W2 implies v + u1 in the carrier of W1
defpred P [ Element of L ] means M <= f . $1 & f . $1 <= $1 & f . $1 <= f . $1 ;
l . ( u , a , v ) = s * x + ( - ( s * x ) + y ) .= b ;
- ( x - y ) = - x + - y .= - x + ( - y ) .= - x + y .= - x + y ;
given a being Point of GX such that for x being Point of GX holds a , x are_ed ed a , b and a , x are_\HM { a } ;
fbe Function of [: dom ( @ f2 ) , cod ( @ f2 ) :] , cod ( @ f2 ) :] , cod ( f ) ;
for k , n being Nat st k <> 0 & k < n & k < n & n , n are_relative_prime holds k divides n & n divides k
for x being element holds x in A |^ d iff x in ( ( A ` ) |^ d ) ` & x in ( ( A ` ) |^ d ) `
consider u , v being Element of R , a being Element of A such that l /. i = u * a * v and a in A ;
- ( ( p `1 / |. p .| - cn ) / ( 1 - cn ) ) ^2 > 0 ;
L-13 . k = LLlet . ( F . k ) & F . k in dom Lw & F . k in dom Lw ;
set i2 = AddTo ( a , i , - n ) , i1 = a := ( - n ) ;
attr B is max means : Def2 : for S being SubSubuniversal of Al holds S is ( B , S ) `1 & S is ( B , S ) `1 ;
a9 "/\" D = { a "/\" d where d is Element of N : d in D } & a "/\" ( d "/\" D ) = a "/\" ( d "/\" ( d "/\" D ) ) ;
|( \square , q29 - q )| * |( - 1 , q )| * |( - 1 , q )| * |( - 1 , q )| * |. q .| * |. q .| * |. q .| * |. q .| >= 0 ;
( - f ) . sup A = ( ( - f ) | A ) . sup A .= ( ( - f ) | A ) . sup A .= ( - f ) . sup A ;
( G * ( len G , k ) ) `1 = ( G * ( len G , k ) ) `1 .= G * ( 1 , k ) `1 .= G * ( 1 , k ) `1 ;
( Proj ( i , n ) ) . LM = <* ( proj ( i , n ) ) . LM *> .= ( Proj ( i , n ) ) . LM ;
f1 + f2 * reproj ( i , x ) is_differentiable_in ( ( f1 + f2 ) * reproj ( i , x ) ) . x ;
pred ( for x st x in Z holds ( ( tan * tan ) `| Z ) . x <> 0 & ( tan * tan ) `| Z ) . x = tan . x ;
ex t being SortSymbol of S st t = s & h1 . t = h2 . ( t . x ) & for x being set st x in dom h1 holds h1 . x = ( h2 . x ) . x ;
defpred C [ Nat ] means ( P8 . $1 is as as as as as non empty & ( A is as as non empty & A is non empty or A is non empty ) ;
consider y being element such that y in dom ( p | i ) and q9 . i = ( p | i ) . y and y in dom ( p | i ) ;
reconsider L = product ( { x1 } +* ( index B , l ) ) as Basis of \bf A , { x1 } , { x2 } ;
for c being Element of C ex d being Element of D st T . ( id c ) = id d & for d being Element of D holds T . ( id c ) = id d
be mid ( f , n , p ) = ( f | n ) ^ <* p *> .= f ^ <* p *> .= f ^ <* p *> ;
( f (#) g ) . x = f . ( g . x ) & ( f (#) h ) . x = f . ( h . x ) ;
p in { ( 1 - 2 ) * ( G * ( i + 1 , j ) + G * ( i + 1 , j + 1 ) } ;
f `1 - p `2 = ( f | ( n , L ) ) *' ( - ( f | ( n , L ) ) .= ( - ( c - f ) ) * ( - ( f | ( n , L ) ) ) ;
consider r be Real such that r in rng ( f | divset ( D , j ) ) and r < m + s ;
f1 . ( |[ ( 8 - r ) / 2 , ( 8 - r ) / 2 ]| ) in f1 .: W1 & f2 .: W2 c= W2 ;
eval ( a | ( n , L ) , x ) = eval ( a | ( n , L ) , x ) .= a * ( x | ( n , L ) ) .= a * ( x | ( n , L ) ) ;
z = DigA ( tz , x9 ) .= DigA ( tz , x9 ) .= DigA ( tz , x9 ) .= DigA ( tz , x9 ) ;
set H = { Intersect S where S is Subset-Family of X : S c= G & S c= G } , F = { Intersect S where S is Subset-Family of X : S c= G } ;
consider S19 being Element of D such that S `1 = S19 ^ <* d *> and S `2 = S19 ^ <* d *> ;
assume that x1 in dom f and x2 in dom f and f . x1 = f . x2 and f . x2 = f . x2 and f . x3 = f . x3 ;
- 1 <= ( ( q `2 / |. q .| - sn ) / ( 1 - sn ) ) ^2 & - 1 <= ( q `2 / |. q .| - sn ) ^2 ;
0. ( V , C\ ( v + 1 ) ) is Linear_Combination of A & Sum ( L ) = 0. V implies Sum ( L ) = 0. V
let k1 , k2 , k1 , k2 , k2 , x4 , 6 , 8 be Element of NAT , a , b , c , d be Int-Location , b be Int-Location of SCM+FSA ;
consider j being element such that j in dom a and j in g " { k } and x = a . j and x = a . j ;
H1 . x1 c= H1 . x2 or H1 . x2 c= H1 . x2 or H1 . x2 c= H1 . x2 & H1 . x2 c= H1 . x2 implies H1 is open & H2 is open
consider a being Real such that p = *> * p1 + ( a * p2 ) and 0 <= a and a <= 1 ;
assume that a <= c & c <= d and [' a , b '] c= dom f and [' a , b '] c= dom g and f | [' a , b '] is bounded ;
cell ( Gauge ( C , m ) , i , width Gauge ( C , m ) -' 1 , 0 ) is non empty ;
A9 in { ( S . i ) `1 where i is Element of NAT : i <= n & not contradiction } ;
( T * b1 ) . y = L * ( b2 /. y ) .= ( F * b1 ) . y .= ( F * b1 ) . y .= ( F * b1 ) . y ;
g . ( s , I ) . x = s . y & g . ( s , I ) . y = |. s . x - s . y .| ;
( log ( 2 , k + 1 ) ) / 2 >= ( log ( 2 , k + 1 ) ) / 2 ;
then p => q in S & not x in the still of p & not x in S & not p => All ( x , p ) in S ;
dom ( the InitS of r-10 ) misses dom ( the InitS of rM ) & dom ( the InitS of rM ) misses dom ( the InitS of rM ) ;
synonym f is extended as or means : Def2 : for x being set st x in rng f holds x is ExtReal & f . x is ExtReal ;
assume for a being Element of D holds f . { a } = a & for X being Subset-Family of D holds f . ( f .: X ) = f . union X ;
i = len p1 .= len p3 + len <* x *> .= len p1 + len <* x *> .= len p1 + len <* x *> .= len p1 + len <* x *> .= len p1 + len <* x *> .= len p1 + len <* x *> .= len p1 + len <* x *> ;
( l , 3 ) `1 = ( g . ( 1 , 3 ) ) `1 + ( k , 3 ) `1 - ( e . ( 1 , 3 ) ) `1 .= ( g . ( 1 , 3 ) ) `1 + ( e . ( 1 , 3 ) ) `1 ;
CurInstr ( P2 , Comput ( P2 , s2 , l ) ) = halt SCM+FSA & CurInstr ( P2 , Comput ( P2 , s2 , l ) ) = halt SCM+FSA ;
assume for n be Nat holds ||. seq . n .|| <= ( R . n ) & ( R . n ) is summable & ( R . n ) is summable & R . n <= ( R . n ) * ( R . n ) ) ;
sin . ( Let ) = sin . ( sin . ( cos . ( - 1 ) ) * cos . ( sin . ( - 1 ) ) ) .= 0 ;
set q = |[ g1 `1 / ( ( t `1 ) ^2 ) , g2 `2 / ( t `2 ) ^2 ]| , r = |[ r , s ]| , s = |[ r , t ]| , t = |[ r , s ]| , t = |[ r , t ]| , r ]| ;
consider G be sequence of S such that for n being Element of NAT holds G . n in let holds G . n in let
consider G such that F = G and ex G1 , G2 st G1 in SM & G2 in SM & G = ( the carrier of G1 ) \/ ( the carrier of G2 ) ;
the root of [ x , s ] in ( ( the Sorts of Free ( C , X ) ) * ( the Arity of S ) ) . s ;
Z c= dom ( exp_R * ( f + ( ( #Z 3 ) * ( f1 + #Z 3 ) ) ) ) ;
for k being Element of NAT holds r0 . k = ( ( upper ( Im ( f ) ) ) . k ) * ( ( Im ( f ) ) . k ) )
assume that - 1 < n and q `2 > 0 and ( q `1 / |. q .| - cn ) < 0 and ( q `1 / |. q .| - cn ) < 0 ;
assume that f is continuous and a < b and a < b and c < d and f . a = c and f . b = d and f . c = d ;
consider r being Element of NAT such that syy1 = Comput ( P1 , s1 , r ) and r <= q and r <= q ;
LE f /. ( i + 1 ) , f /. j , L~ f & LE f /. 1 , f /. ( len f ) , f /. ( len f ) , f /. ( len f ) ;
assume that x in the carrier of K and y in the carrier of K and inf { x , y } = x and inf { x , y } = y ;
assume f +* ( i1 , \xi ) in ( proj ( F , i2 ) ) " ( ( proj ( F , i2 ) ) " ( ( Carrier ( F , i2 ) ) " ( A ) ) ;
rng ( ( ( Flow M ) | ( the carrier of M ) ) | ( the carrier' of M ) ) c= the carrier' of M & rng ( ( ( Flow M ) | ( the carrier' of M ) ) ) c= the carrier' of M ;
assume z in { ( the carrier of G ) \ { t where t is Element of T : t in A } ;
consider l be Nat such that for m be Nat st l <= m holds ||. s1 . m - f /. x0 .|| < g ;
consider t be VECTOR of product G such that mt = ||. D5 . t .|| and ||. t .|| <= 1 ;
assume that the carrier of v = 2 and v ^ <* 0 *> , v ^ <* 1 *> ] in dom p and v ^ <* 1 *> in dom p and p . ( len p + 1 ) = v and p . ( len p + 1 ) = v ;
consider a being Element of the Points of X , A being Element of the lines of X such that a on A and b on A and a on A ;
( - x ) |^ ( k + 1 ) * ( ( - x ) |^ ( k + 1 ) ) = 1 ;
for D being set st for i st i in dom p holds p . i in D holds p . i in D & p . i in D
defpred R [ element ] means ex x , y st [ x , y ] = $1 & P [ x , y ] & P [ y , x ] ;
L~ ( f2 ) = union { LSeg ( p0 , p2 ) , LSeg ( p1 , p2 ) , LSeg ( p1 , p2 ) } .= { LSeg ( p1 , p2 ) , LSeg ( p1 , p2 ) } ;
i - len h11 + 2 - 1 < i - len h11 + 2 - 1 + 2 - 1 + 1 - 1 + 2 - 1 + 1 ;
for n be Element of NAT st n in dom F holds F . n = |. ( n\leq . ( n -' 1 ) ) .| ;
for r , s1 , s2 , s3 being Real holds r in [. s1 , s2 .] iff s1 <= r & r <= s2 & s1 <= s2 & s2 <= s2
assume v in { G where G is Subset of [: T2 , T2 :] : G in B2 & G c= z & z in G & G c= z } ;
let g be then :] Function of A , INT , X , Y be Subset of INT , b be Element of INT , f be Function of A , the carrier of b , X ;
min ( g . [ x , y ] , k . [ y , z ] ) = ( min ( g . [ y , z ] , k . [ y , z ] ) ) . y ;
consider q1 being sequence of Cf such that for n holds P [ n , q1 . n , q1 . n ] and for n holds P [ n , q1 . n ] ;
consider f being Function such that dom f = NAT and for n being Element of NAT holds f . n = F ( n ) and for n being Element of NAT holds f . n = F ( n ) ;
reconsider B-6 = B /\ B , Ed = O /\ ( A , O ) , Bd = O /\ ( A , O ) as Subset of B ;
consider j be Element of NAT such that x = the the \rm the is \ of n and 1 <= j and j <= n and 1 <= n and n <= len f and f . j = f . ( j + 1 ) ;
consider x such that z = x and card ( x . O2 ) in card ( x . O2 ) and x in L1 and x in L1 and x in L2 ;
( C * ( _ T4 . k , n2 ) ) . 0 = C . ( ( ( _ T4 . k , n2 ) ) . 0 ) ;
dom ( X --> rng f ) = X & dom ( X --> f ) = X & rng ( X --> f ) = dom ( X --> f ) & rng ( X --> f ) = X ;
S-bound L~ SpStSeq C <= ( ( L~ SpStSeq C ) * ( i , j ) ) `2 & ( S-bound L~ SpStSeq C ) * ( i , j ) `2 <= ( S-bound L~ SpStSeq C ) * ( i , j ) `2 ;
synonym x , y means : Def2 : x = y or ex l being Subset of S st { x , y } c= l & ex l being Subset of S st { x , y } c= l ;
consider X being element such that X in dom ( f | ( n + 1 ) ) and ( f | ( n + 1 ) ) . X = Y ;
assume that Im k is continuous and for x , y being Element of L , a , b being Element of Image k st a = x & b = y & a = b holds a << b ;
( 1 / 2 * ( ( - ( ( #Z 2 ) * ( ( - 1 ) / 2 ) ) * ( ( #Z 2 ) * ( ( - 1 ) / 2 ) ) ) ) ) is_differentiable_on REAL ;
defpred P [ Element of omega ] means ( ( for k st k <= $1 holds ( for n holds n <= k ) implies ( for n holds n <= $1 ) implies ( for n holds ( n <= k ) implies ( for n holds ( n <= k ) implies ( n <= k ) implies ( n <= k ) ) implies ( n <= k ) ) ;
IC Comput ( P , s , 2 ) = succ IC Comput ( P , s , 1 ) .= 6 + 1 .= 6 + 1 .= 6 ;
f . x = f . g1 * f . g2 .= f . g1 * f . g2 .= f . g1 * 1_ H .= f . g1 * 1_ H * 1_ H .= f . g1 * 1_ H ;
( M * ( F . n ) ) . n = M . ( ( F . n ) . n ) .= M . ( { ( canFS ( Omega ) ) . n } ) .= M . ( { ( canFS ( Omega ) ) . n } ) ;
the carrier of L1 + L2 c= ( the carrier of L1 ) \/ ( the carrier of L2 ) & the carrier of L1 + L2 c= the carrier of L1 & the carrier of L1 + L2 c= the carrier of L2 implies L1 + L2 c= the carrier of L1
pred a , b , c , x , y , a , b , c , d , x , y , z be element means : Def2 : a , b , c , x is_collinear & a , c , x is_collinear & b , c , y is_collinear ;
( the partial of s ) . n <= ( ( the partial of s ) . n ) * ( ( the Sorts of s ) . n ) & ( the Sorts of s ) . n <= ( the Sorts of s ) . n ;
pred - 1 <= r & r <= 1 implies ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) / ( r - 1 ) ) ) ) `| Z ) = - 1 / ( r - 1 ) ;
seq in { p ^ <* n *> where n is Nat : p ^ <* n *> in T1 & p ^ <* n *> in T1 } ;
|[ x1 , x2 , x3 ]| . 2 - |[ y1 , y2 , x3 ]| . 2 - |[ y1 , y2 , x3 ]| . 2 = x2 - y2 - y2 - x2 - x3 ;
attr for m being Nat holds F . m is nonnegative means : Def4 : for n being Nat holds ( Partial_Sums F ) . n is nonnegative & ( Partial_Sums F ) . n is nonnegative ;
len ( 2 * ( G . ( x , y ) ) ) = len ( ( 2 * ( G . ( x , y ) ) ) + ( 2 * ( G . ( y , z ) ) ) ) .= len ( G . ( y , z ) ) ;
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 and v in W1 /\ W2 ;
given F be finite FinSequence of NAT such that F = x and dom F = n & rng F c= { 0 , 1 } and Sum F = k and Sum ( F ) = k ;
0 = 1 * \upupharpoons \upupharpoons 0 * umid iff 1 = ( 1 - ( 1 - ( 1 - 0 ) ) * ( 1 - ( 0 - 0 ) ) * ( 1 - 0 ) ) ;
consider n be Nat such that for m be Nat st n <= m holds |. ( f # x ) . m - ( lim ( f # x ) ) .| < e ;
cluster being being being being being being non empty \frac for non empty \frac , \hbox { 0 } , { 1 } , { 2 } , { 3 } } is Boolean & ( for p being Element of L holds p in { 3 } iff p is Boolean )
"/\" ( BB , L ) = Top ( BB , L ) .= "/\" ( ( [#] S ) , L ) .= "/\" ( [#] S , L ) .= "/\" ( ( [#] S ) , L ) ;
( r / 2 ) ^2 + ( r / 2 ) ^2 <= ( r / 2 ) ^2 + ( r / 2 ) ^2 ;
for x being element st x in A /\ dom ( f `| X ) holds ( f `| X ) . x >= r2 & ( f `| X ) . x >= r2
2 * r1 - ( 2 * |[ a , c ]| - ( 2 * r1 - ( 2 * r2 - ( 2 * r2 - ( 2 * r1 - 1 ) ) ) ) ) = 0. TOP-REAL 2 ;
reconsider p = P * ( \square , 1 ) , q = a " * ( ( - ( - ( K , n , 1 ) ) * ( 1 , 1 ) ) ) as FinSequence of K ;
consider x1 , x2 being element such that x1 in uparrow s and x2 in < t and x = [ x1 , x2 ] and x = [ x1 , x2 ] ;
for n be Nat st 1 <= n & n <= len q1 holds q1 . n = ( ( upper_volume ( g , M ) ) . n ) * ( Sum ( g , M ) ) ;
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 and y = H2 and H1 is Subgroup of H2 and H2 is Subgroup of H1 and H1 is Subgroup of H2 ;
for S , T being non empty RelStr , d being Function of T , S st T is complete & d is directed-sups-preserving holds d is monotone & d is monotone
[ a + 0. F_Complex , b + ( - b ) ] in ( the carrier of COMPLEX ) /\ ( the carrier of COMPLEX ) & [ a , b ] in the carrier of COMPLEX ;
reconsider mm = max ( len F1 , len ( p . n ) * ( p . n ) / ( n + 1 ) ) as Element of NAT ;
I <= width GoB ( ( GoB h ) * ( 1 , j ) , ( GoB h ) * ( 1 , j ) ) & I <= width GoB ( ( GoB h ) * ( 1 , j ) , ( GoB h ) * ( 1 , j ) ) = I ;
f2 /* q = ( f2 /* ( f1 /* s ) ) ^\ k .= ( ( f2 * f1 ) /* s ) ^\ k .= ( ( f2 * f1 ) /* s ) ^\ k .= ( ( f2 * f1 ) /* s ) ^\ k ;
attr A1 \/ A2 is linearly-independent means : Def4 : A1 is linearly-independent & A2 misses A1 & ( for A being Subset of V st A misses A1 holds Lin ( A1 ) /\ Lin ( A2 ) = (0). ( A1 ) & Lin ( A1 /\ A2 ) = (0). ( A1 ) ;
func A -carrier C -> set equals union { A . s where s is Element of R : s in C & s in C } ;
dom ( Line ( v , i + 1 ) ^ ( ( Line ( p , m ) ) * ( Line ( p , i ) ) ) ) = dom ( F ^ ( Line ( p , m ) ) ) ;
cluster [ ( x `1 ) , ( x `2 ) , ( x `2 ) ] -> R & [ x `1 , ( x `2 ) , ( x `2 ) ] in R ;
E , All ( x2 , All ( x2 , H ) '&' ( All ( x3 , x2 ) '&' ( All ( x3 , x3 ) '&' ( All ( x4 , x4 ) '&' ( All ( x4 , x2 ) '&' ( All ( x4 , x3 ) ) '&' ( All ( x4 , x2 ) '&' ( All ( x4 , x3 ) ) '&' ( All ( x4 , x2 ) ) ) ) ) ) |= ;
F .: ( id X , g ) . x = F . ( id X , g . x ) .= F . ( id X , g . x ) .= F . ( id X , g . x ) .= F . ( id X , g . x ) ;
R . ( h . m ) = F . x0 + h . ( m + 1 ) - h . ( h . m ) .= L . ( h . m ) - R . ( h . m ) ;
cell ( G , ( X -' 1 , Y -' 1 ) , ( X -' 1 ) + ( t + 1 ) ) meets UBD L~ f & ( for n holds f . n in ( L~ f ) ` implies f . n meets ( L~ f ) ` )
IC Result ( P2 , s2 ) = IC Comput ( P2 , s2 , i ) .= ( card I + 1 ) .= ( card I + 1 ) .= ( card I + 1 ) .= ( card I + 1 ) .= ( card I + 1 ) .= card I + 1 ;
sqrt ( ( - ( ( q `1 / |. q .| - cn ) ) / ( 1 - cn ) ) ^2 ) > 0 ;
consider x0 being element such that x0 in dom a and x0 in g and x0 in g " { k } and y = a . x0 and x0 in g " { k } ;
dom ( r1 (#) chi ( A , C ) ) = dom ( chi ( A , A ) ) .= dom ( chi ( A , A ) ) .= dom ( ( r1 (#) chi ( A , A ) ) | ( A /\ A ) ) .= dom ( r1 (#) ( chi ( A , A ) ) ) .= A ;
d-7 . [ y , z ] = ( ( ( y `1 ) - z `1 ) - ( ( y `2 ) - z `2 ) ) - ( ( y `2 ) - z `2 ) ;
redefine for i being Nat holds C . i = A . i /\ B . i & C . i c= A . i /\ B . i ;
assume that x0 in dom f and f is_continuous_in x0 and f is_continuous_in x0 and for r st r in dom f & r > 0 ex g st g < r & g in dom f & for g st g in dom f & g in dom f & g in dom f holds f . g <= ( f /* x0 ) . g ;
p in Cl A implies for K being Basis of p , Q being Subset of T st Q in K & A meets Q holds A meets Q
for x be Element of REAL n st x in Line ( x1 , x2 ) holds |. y1 - y2 .| <= |. y1 - y2 .| & |. y1 - y2 .| <= |. y1 - y2 .|
func Sum <*> { a } -> Ordinal means : Def4 : a in it & for b being Ordinal st a in it holds it c= b & for b being Ordinal st b in it holds it c= b ;
[ a1 , a2 , a3 , a4 , a5 , a5 , a6 ] in ( the carrier of A ) /\ ( the carrier of A ) & [ a1 , a2 , a3 , a4 , a5 ] in ( the carrier of A ) /\ ( the carrier of A ) ;
ex a , b being element st a in the carrier of S1 & b in the carrier of S2 & x = [ a , b ] & x = [ a , b ] ;
||. ( vseq . n ) - ( vseq . m ) .|| * ||. x - y .|| < ( e / ( ||. x .|| + ||. y .|| ) ) * ||. x - y .|| ;
then for Z be set st Z in { Y where Y is Element of I7 : F c= Y & Y c= Z } holds z in Z & z in Z ;
sup compactbelow [ s , t ] = [ sup ( [: the carrier of S , the carrier of T :] ) , sup ( ( compactbelow s ) . s ) ] .= sup ( ( compactbelow s ) . s ) ;
consider i , j being Element of NAT such that i < j and [ y , f . j ] in If and [ f . i , z ] in If and [ y , z ] in If ;
for D being non empty set , p , q being FinSequence of D st p c= q & p is FinSequence of D ex n being Element of NAT st p ^ q = q & for k being Element of NAT st k < n holds p ^ q = q
consider e19 being Element of the affine of X such that c9 , a9 // a9 , e and a9 , e // e , e and not a9 , c9 // e , e and not b9 , c9 // e , e and not a9 , c9 // e , e ;
set U2 = I \! \mathop { {} } , U2 = I \! \mathop { {} } , E = I \! \mathop { {} } , F = I \! \mathop { {} } , N = I \! \mathop { {} } , E = I \! \mathop { {} } , F = I \! \mathop { {} } , E = I \! \mathop { {} } , F = I \! \mathop { {} } , E = I \! \mathop { {} } , F = I -\mathop { {} } , N = I \! \mathop { {} } , E = I -\mathop { {} } , F = I -\mathop { {} } , A = I -\mathop
|. q3 .| ^2 = ( ( |. q3 .| ) ^2 + ( |. q .| ) ^2 ) + ( ( |. q .| ) ^2 ) .= |. q .| ^2 + ( |. q .| ) ^2 .= |. q .| ^2 + ( |. q .| ) ^2 ;
for T being non empty TopSpace , x , y being Element of [: the topology of T , the topology of T :] holds x "\/" y = x \/ y & x "/\" y = x /\ y implies x "/\" y = x /\ y
dom signature U1 = dom ( the charact of U1 ) & Args ( o , MSAlg U1 ) = dom ( the charact of U1 ) & Args ( o , MSAlg U1 ) = dom ( the charact of U1 ) & Args ( o , MSAlg U1 ) = dom ( the charact of U1 ) ;
dom ( h | X ) = dom h /\ X .= dom ( ||. h .|| | X ) .= dom ( ||. h .|| | X ) .= dom ( ||. h .|| | X ) .= dom ( ||. h .|| | X ) .= dom ( ||. h .|| | X ) .= dom ( ||. h .|| | X ) .= dom ( ||. h .|| | X ) .= dom ( ||. h .|| | X ) ;
for N1 , N1 , N2 be Element of ( the carrier of G ) * , N1 , N2 be non empty Subset of ( the carrier of G ) * , N1 , N2 be Subset of ( the carrier of G ) * , N1 , N2 be Subset of ( the carrier of G ) * ;
( mod ( u , m ) + mod ( v , m ) ) . i = ( mod ( u , m ) ) . i + ( mod ( v , m ) ) . i ;
- ( q `1 ) ^2 < - 1 or - ( q `2 ) ^2 / ( |. q .| ) ^2 & - ( q `1 ) ^2 / ( |. q .| ) ^2 <= - ( q `1 ) ^2 / ( |. q .| ) ^2 ;
pred r1 = f & r2 = f & for r , s st r = f & s = f & r in dom f & s in dom f & r in dom f & s in dom f & r in dom f & s in dom f & s in dom f & r in dom f & s in dom f & f . s in f . s ;
vseq . m is bounded Function of X , the carrier of Y & x9 . m = ( seq_id ( vseq . m , X ) ) . x & x9 . m = ( seq_id ( vseq . m , X ) ) . x ;
pred a <> b & b <> c & a <> c & angle ( a , b , c ) = PI & angle ( b , c , a ) = 0 & angle ( b , c , a ) = 0 ;
consider i , j being Nat , r being Real such that p1 = [ i , r ] and p2 = [ j , s ] and i < j and r < s and s < j ;
|. p .| ^2 - ( 2 * |( p , q )| ) ^2 + |. q .| ^2 = |. p .| ^2 + |. q .| ^2 ;
consider p1 , q1 being Element of X ( ) such that y = p1 ^ q1 and q1 ^ p1 = p1 ^ q1 and p1 ^ q1 = p2 and p1 ^ q1 = p2 and q1 ^ q1 = p1 ^ q1 and p1 ^ q1 = p2 ^ q2 ;
1. ( K , A ) . ( r1 , r2 ) = ( s2 - s1 ) * ( s2 - s1 ) .= ( s2 - s2 ) * ( s2 - s1 ) .= ( s2 - s2 ) * ( s2 - s1 ) .= ( s2 - s2 ) * ( s2 - s1 ) ;
( ex w being Real st w = lower_bound ( proj2 .: A ) & ( for w being Real st w in A holds w <= ( proj2 .: A ) ) & ( for w being Real st w in A holds w <= ( proj2 .: A ) ) & ( for w being Real st w in A holds w <= ( proj2 .: A ) ) implies w is non empty )
s , ( k , 1 ) |= H1 '&' H2 iff s , ( k , 1 ) |= H1 & s , ( k , 1 ) |= H2 & ( s , ( k , 1 ) |= H1 ) & ( s , ( k , 1 ) |= H2 ) ;
len ( s + t ) = card ( support b1 ) + 1 .= card ( support b1 ) + ( support b2 ) .= card ( support b1 ) + ( support b2 ) .= card ( support b1 ) + ( support b2 ) .= card ( support b1 ) + ( support b2 ) .= card ( support b1 ) + ( support b2 ) .= card ( support b1 ) ;
consider z being Element of L1 such that z >= x and z >= y and for z being Element of L1 st z >= x & z >= y holds z >= y ;
LSeg ( ( UMP D ) , ( ( ( E-bound D ) + E-bound D ) / 2 ) ) /\ D = { ( ( UMP D ) + ( ( E-bound D ) / 2 ) / 2 } /\ D .= { ( LMP D ) / 2 } ;
lim ( ( f `| N ) / g ) = lim ( ( f `| N ) / g ) .= lim ( ( f `| N ) / g ) .= lim ( ( f `| N ) / g ) ;
P [ i , pr1 ( f ) . i , pr1 ( f ) . ( i + 1 ) , pr1 ( f ) . ( i + 1 ) ] ;
for r be Real st 0 < r ex m be Nat st for k be Nat st m <= k holds ||. ( seq . k ) - ( seq . k ) .|| < r
for X being set , P being a_partition of X , x , a , b being set st x in a & a in P & x in P & b in P & a in P & b in P & a in P & b in P holds a = b
Z c= dom ( ( ( 1 / 2 ) (#) f ) `| Z ) /\ dom ( ( ( 1 / 2 ) (#) f ) `| Z ) & Z c= dom ( ( 1 / 2 ) (#) f ) /\ dom f " { 0 } implies f | Z is continuous
ex j being Nat st j in dom ( l ^ <* x *> ) & j < i & y = ( l ^ <* x *> ) . j & z = ( l ^ <* x *> ) . j & i = 1 + j & j = len l + 1 & z = ( l ^ <* x *> ) . j & i = 1 + j & j = len l + 1 ;
for u , v being VECTOR of V , r being Real st 0 < r & for u , v being VECTOR of V st 0 < r & u in N & v in N holds r * u + ( 1-r r ) * v in N
A , Int Cl A , Cl Int Cl Int Cl A , Cl Int Cl A , Cl Int Cl A ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` = Cl ( Cl ( Cl ( A /\ Cl ( A /\ B ) ) ` ) ;
- Sum <* v , u , w *> = - ( v + u + u ) .= - ( v + u ) + u .= - ( v + u ) + u .= - ( v + u ) + u .= - ( v + u ) ;
( Exec ( a := b , s ) ) . IC SCM R = ( Exec ( a := b , s ) ) . NAT .= ( Exec ( a := b , s ) ) . NAT .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s ;
consider h being Function such that f . a = h and dom h = I and for x being element st x in I holds h . x in ( the carrier of J ) . x and for x being element st x in I holds h . x in ( the carrier of J ) . x ;
for S1 , S2 being non empty reflexive RelStr , D being non empty Subset of [: S1 , S2 :] , f being Function of [: S1 , S2 :] , D holds ( cos * f ) . D is directed & ( cos * f ) . D is directed & ( cos * f ) . D = f . D
card X = 2 implies ex x , y st x in X & y in X & x <> y & x <> y & not ( ex z st z in X & z in X & not z in Y ) & not ( ex x st x in X & x in X & y in X ) & not ( ex y st y in X & not y in X & y in X )
( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) in rng ( Cage ( C , n ) \circlearrowleft W-min L~ Cage ( C , n ) ) ;
for T , T being decorated tree , p , q being Element of dom T , p being Element of dom T st p divides q & p in T holds ( T -\hbox { p } ) . q = T . q
[ i2 + 1 , j2 ] in Indices G & [ i2 + 1 , j2 ] in Indices G & f /. k = G * ( i2 + 1 , j2 ) implies f /. k = G * ( i2 + 1 , j2 )
cluster that k divides ( k , n ) and k divides ( k , n ) and n divides ( k , n ) and ( k divides n implies k divides ( k , n ) ) ;
dom F " = the carrier of X2 & rng F = the carrier of X1 & rng F = the carrier of X2 & F " { 0 } = the carrier of X2 & rng F = the carrier of X2 & rng F = the carrier of X2 & rng F = the carrier of X2 & rng F = the carrier of X2 & F is one-to-one implies F " { 0 } = F " { 0 }
consider C being finite Subset of V such that C c= A and card C = n and the carrier of V = n and the carrier of W = Lin ( BM \/ ( BM ) ) and C is linearly-independent and C is linearly-independent ;
V is prime implies for X , Y being Element of [: the topology of T , the topology of T :] st X /\ Y c= V & X c= V holds X c= V or Y c= V
set X = { F ( v1 ) where v1 is Element of B ( ) , v2 is Element of B ( ) : P [ v1 ] } , Y = { F ( v2 ) where v2 is Element of B ( ) : P [ v2 ] } ;
angle ( p1 , p3 , p4 ) = 0 .= angle ( p2 , p3 , p4 ) .= angle ( p2 , p3 , p4 ) .= angle ( p3 , p2 , p4 ) .= angle ( p3 , p2 , p4 ) ;
- sqrt ( ( - ( q `1 / |. q .| - cn ) ) / ( 1 - cn ) ) ^2 = - ( ( q `1 / |. q .| - cn ) / ( 1 - cn ) ) ^2 .= - ( q `1 / |. q .| - cn ) / ( 1 - cn ) ;
ex f being Function of I[01] , ( TOP-REAL 2 ) | P st f is continuous one-to-one & rng f = P & f . 0 = p1 & f . 1 = p2 & f . 1 = p2 & f . 1 = p3 & f . 0 = p4 & f . 1 = p4 ;
attr f is partial differentiable of partial , u0 means : Def4 : for u holds SVF1 ( 2 , pdiff1 ( f , 1 ) , u0 ) . u = ( proj ( 2 , 3 ) ) . u + ( proj ( 2 , 3 ) ) . u ;
ex r , s st x = |[ r , s ]| & G * ( len G , 1 ) `1 < r & r < G * ( 1 , 1 ) `1 & s < G * ( 1 , 1 ) `2 & s < G * ( 1 , 1 ) `2 ;
assume that f is special and 1 <= t and t <= len G and G * ( t , width G ) `2 >= S-bound L~ f and G * ( t , width G ) `2 >= S-bound L~ f and p `2 >= S-bound L~ f and p `2 <= N-bound L~ f and p `2 <= N-bound L~ f and p `2 <= N-bound L~ f and p `2 <= N-bound L~ f ;
pred i in dom G means : Def4 : r * ( f * reproj ( i , x ) ) = r * f * reproj ( i , x ) & for i be Nat holds r * ( f * reproj ( i , x ) ) = r * f * reproj ( i , x ) ;
consider c1 , c2 being bag of o1 + o2 such that ( decomp c ) /. k = <* c1 , c2 *> and c = c1 + c2 and ( decomp c ) /. k = c1 + c2 and ( decomp c ) /. k = c1 + c2 and ( decomp c ) /. k = c2 + c2 ;
u0 in { |[ r1 , s1 ]| : r1 < G * ( 1 , 1 ) `1 & G * ( 1 , j ) `2 < s1 & s1 < G * ( 1 , j + 1 ) `2 } ;
Cl ( X ^ Y ) . k = the carrier of X . ( k2 + 1 ) .= C4 . ( k + 1 ) .= C4 . ( k + 1 ) .= C4 . ( k + 1 ) .= C4 . ( k + 1 ) ;
pred len M1 = len M2 & width M1 = width M2 & width M2 = width M2 & width M2 = width M2 & width M2 = width M2 & width M2 = width M2 & width M2 = width M2 & width M2 = width M2 & width M2 = width M2 & width M1 = width M2 & width M2 = width M2 & width M1 = width M2 & width M1 = width M2 & width M2 = width M2 & width M1 = width M2 & width M2 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M2 = width M2 & width
consider g2 be Real such that 0 < g2 and { y where y is Point of S : ||. y - x0 .|| < g2 & ||. y - x0 .|| < g2 } c= N2 & for x be Point of S st x in N holds ||. ( f | X ) /. x - ( f | X ) /. x0 .|| < g2 } c= N2 ;
assume x < ( - b + sqrt ( delta ( a , b , c ) ) / 2 ) or x > ( - b - sqrt ( a - b ) ) / 2 ;
( G1 '&' G2 ) . i = ( <* 3 *> ^ G1 ) . i & ( H1 '&' G1 ) . i = ( <* 3 *> ^ H1 ) . i & ( H1 '&' G1 ) . i = ( <* 3 *> ^ H1 ) . i ;
for i , j st [ i , j ] in Indices ( M3 + M1 ) & ( M2 + M1 ) * ( i , j ) < M2 holds ( M2 + M1 ) * ( i , j ) < M2 * ( i , j )
for f being FinSequence of NAT , i being Element of NAT st i in dom f & i in dom f & i <> j holds i divides f /. i & i divides f /. j implies i divides f /. ( i + 1 )
assume F = { [ a , b ] where a , b is set : for c st c in B\mathopen { a , b } & a c= c & b c= c & a c= c } ;
b2 * q2 + ( b3 * q3 ) + ( - ( a * q3 ) + - ( a * q3 ) ) * ( ( - ( a * q2 ) + ( - ( a * q3 ) ) + ( - ( a * q3 ) ) ) * ( ( - ( a * q2 ) ) + ( - ( a * q2 ) ) * ( - ( a * q2 ) ) * ( - ( b * q2 ) ) * ( - ( b * q2 ) ) ) = b2 * ( - ( a * q2 ) ) ;
Cl ( Cl F ) = { D where D is Subset of T : ex B being Subset of T st D = Cl B & B is open & A c= B & B c= Cl F } & F is closed & A c= Cl ( Cl F ) & B c= Cl ( Cl F ) ;
attr seq is summable means : Def4 : for n holds seq . n is summable & seq is summable & ( for k st k <= n holds seq . k = Sum ( seq ^\ k ) ) & ( for k be Nat holds seq . k = Sum ( seq ^\ k ) ) & ( for k be Nat holds seq . k = Sum ( seq ^\ k ) ) implies seq is summable & lim seq = Sum ( seq ^\ k )
dom ( ( ( cn - cn ) | D ) | D ) = ( the carrier of ( ( TOP-REAL 2 ) | D ) /\ D ) .= the carrier of ( ( TOP-REAL 2 ) | D ) .= the carrier of ( ( TOP-REAL 2 ) | D ) .= the carrier of ( ( TOP-REAL 2 ) | D ) .= D ;
X [ X \to Z ] is full full full non empty SubRelStr of [: Omega , Omega :] & [ X \to Y , X ] is full full SubRelStr of [: Omega , Omega :] implies X is full non empty & Y is full non empty
G * ( 1 , j ) `2 = ( G * ( i , j ) ) `2 & G * ( 1 , j ) `2 <= ( G * ( i , j ) ) `2 ;
synonym m1 c= m2 for P c= P & ( for p be set st p in P holds ( for p be set st p in P holds m1 . p <= ( m + 1 ) * p ) & ( for n be set st n in P holds not ( n , m ) in P & n <= m ) implies not n <= m ;
consider a being Element of B ( ) such that x = F ( a ) and a in { G ( b ) where b is Element of A ( ) : P [ b ] } and for a being Element of A ( ) st P [ a ] holds a c= b ;
mode multiplicative Str over S , s -> multiplicative non empty multMagma over S means : Def4 : it = [ the carrier of S , the carrier of S ] & for a being Element of the carrier of S holds it . a = [ a , the carrier of S ] & for b being set holds [ b , the carrier of S ] in it iff ex f being Function of the carrier of S , the carrier of S st f = f & f is multiplicative ;
the carrier of the carrier of \HM { a , b } + the subset of the carrier of T = b + the carrier of the carrier of T .= b + the carrier of T .= the carrier of T + the carrier of T .= the carrier of T + the carrier of T .= the carrier of T + the carrier of T ;
cluster ( i + 1 ) * ( i1 , i2 ) -> natural for Element of INT , i , j be Element of INT , i1 , i2 , j1 , j2 be Element of INT , i2 , j2 be Element of NAT ;
- s2 * p1 + ( s2 * p2 - s1 * p2 ) = ( ( - s2 ) * p1 + ( s2 * p2 - s1 * p2 ) ) * p1 + ( ( - s2 ) * p2 - s1 * p2 ) * p2 + ( ( - s2 ) * p2 - s1 * p2 ) * p1 ;
eval ( ( a | ( n , L ) ) *' p , x ) = eval ( a | ( n , L ) , x ) * eval ( p , x ) .= a * eval ( p , x ) * eval ( p , x ) .= a * eval ( p , x ) ;
assume that the TopStruct of S = the TopStruct of T and for D being non empty directed Subset of S , D being non empty directed Subset of S st D = the carrier of S & D = [#] S & for V being open Subset of S st V in V holds V is open & V is open and V c= V and V is open and V c= V ;
assume that 1 <= k and k <= len w + 1 and TU . ( ( ( q , w ) \mathclose { \rm c } } ) . k = ( ( ( T . ( ( q , w ) ) \mathclose { \rm c } } ) | ( i + 1 ) ) ) . k ;
2 * a |^ ( n + 1 ) + ( 2 * b |^ ( n + 1 ) ) >= a |^ ( n + 1 ) + ( a |^ ( n + 1 ) ) + ( a |^ ( n + 1 ) ) ;
M , v / ( All ( x. 3 , All ( x. 4 , All ( x. 4 , x. 4 , x. 0 ) ) , ( All ( x. 4 , All ( x. 4 , x. 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 ) ) ) ) ;
assume that f is_differentiable_on l and for x0 st x0 in l holds 0 < f . x0 and for x1 st x1 in l holds 0 < f . x1 and for x1 st x1 in l holds f . x1 - f . x0 < f . x1 ;
for G1 being _Graph , W being Walk of G1 , W being Walk of G1 , e being Vertex of G2 , G2 being Walk of G1 st e in W & not ( ex G2 being Walk of G1 st e in W & not ( W is Walk & not ( W is Walk & not y in W & y in W & not y in W ) ) & not ( ex x being Vertex of G1 st x in W & not x in W ) )
not c9 is not empty iff ( ( ex y1 , y2 st y1 is not empty & y2 is not empty & not ( ex x1 , x2 , x3 st y1 is not empty & y2 is not empty & not ( x1 is not empty & x2 is not empty & y1 is not empty & y2 is not empty ) ) & not ( not y1 is not empty & not y2 is not empty ) & not ( y1 is not empty & not y2 is not empty ) ;
Indices GoB f = [: dom GoB f , Seg width GoB f :] & ( for i st i in dom GoB f & i + 1 in dom GoB f holds ( GoB f ) * ( i , j ) ) `1 = ( GoB f ) * ( i , j ) `1 & ( GoB f ) * ( i , j ) `2 = ( GoB f ) * ( i + 1 , j ) `2
for G1 , G2 , G3 being strict Subgroup of O st G1 is stable & G2 is stable & G1 is stable & G2 is stable & G2 is stable & G1 is stable & G2 is stable & G2 is stable holds G1 is stable & G2 is stable & G2 is stable & G1 is stable & G2 is stable & G2 is stable & G1 is stable & G2 is stable & G2 is stable & G1 is stable & G2 is stable implies G1 is stable
UsedIntLoc ( inint f ) = { intloc 0 , intloc 1 , intloc 2 , intloc 3 , 1 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 8 , 9 } ;
for f1 , f2 be FinSequence of F st f1 is p -element & f2 is p & Q [ f1 ^ f2 , p ] & Q [ f2 , p ] & Q [ f1 ^ f2 , p ^ f2 , p ^ q ] holds Q [ f1 ^ f2 , p ^ q , p ^ q ]
( p `1 / sqrt ( 1 + ( p `2 / p `1 ) ^2 ) ) ^2 = ( q `1 / sqrt ( 1 + ( q `2 / q `1 ) ^2 ) ) ^2 .= ( q `1 / sqrt ( 1 + ( q `2 / q `1 ) ^2 ) ) ^2 ;
for x1 , x2 , x3 , x4 being Element of REAL n holds |( x1 - x2 , x3 - x4 )| = |( x1 , x3 - x4 , x1 - x2 - x3 )| & |( x1 - x2 , x3 - x4 )| = |( x1 , x2 - x3 )| - |( x1 , x2 - x3 )| , x1 - x3 )|
for x st x in dom ( ( let x - h ) | A ) holds ( ( ( - 1 ) (#) ( F | A ) ) | A ) . x = - ( ( - 1 ) (#) ( F | A ) ) . x
for T being non empty TopSpace , P being Subset-Family of T , P being Subset of T st P c= the topology of T & for x being Point of T holds P is open & x in P holds ex B being Basis of T st B c= P & B c= P & B c= P
( a 'or' b 'imp' c ) . x = 'not' ( ( a 'or' b ) . x 'or' c . x ) 'or' c . x .= 'not' ( ( a 'or' b ) . x 'or' c . x ) 'or' c . x .= TRUE ;
for e being set st e in A9 ex X1 being Subset of X st e = [: X1 , Y1 :] & for X1 being Subset of Y st X1 = [: X1 , Y1 :] & Y1 is open & X1 is open & Y1 is open & Y1 is open & X1 is open & Y1 is open & Y1 is open & X1 c= Y1 & Y1 c= Y1 holds X1 meets Y1
for i be set st i in the carrier of S for f being Function of [: S , T :] , S1 , S2 being Function of [: S , T :] , S2 st f = H . i & for i being set st i in dom H holds F . i = f | ( F . i ) holds F . i = f | ( F . i )
for v , w st for y st x <> y holds w . y = v . y holds Valid ( VERUM ( Al ( ) , J ) , v . w ) = Valid ( VERUM ( Al ( ) , J ) , v . w )
card D = card D1 + card D2 - card { i , j } - card { i , j } - ( i - 1 ) .= ( i + 1 - 1 ) - ( i - 1 ) + ( i - 1 ) .= ( i + 1 - 1 ) + ( i - 1 ) - ( i - 1 ) .= ( i + 1 - 1 ) - ( i - 1 ) ;
IC Exec ( i , s ) = ( s +* ( 0 .--> ( s . 0 ) ) ) . 0 .= ( 0 .--> ( s . 0 ) ) . 0 .= ( 0 .--> ( s . 0 ) ) . 0 .= ( 0 .--> ( s . 0 ) ) . 0 .= ( 0 .--> ( s . 0 ) ) . 0 .= ( 0 .--> ( s . 0 ) ) . 0 .= ( 0 .--> ( s . 0 ) ) . 0 .= ( 0 .--> ( s . 0 ) .= ( 0 .--> ( s . 0 ) ) . 0 .= ( 0 .--> ( s . 0 ) .= ( 0 .--> ( s . 0 ) .= ( 0 .--> ( s . 0 ) .= ( 0 .--> ( s . 0 ) .=
len f -' i1 -' 1 + 1 = len f - i1 + 1 - 1 + 1 .= len f - i1 + 1 - 1 + 1 .= len f - i1 + 1 - 1 + 1 .= len f - i1 + 1 - 1 + 1 - 1 + 1 .= len f - i1 + 1 - 1 + 1 ;
for a , b , c being Element of NAT st 1 <= a & 2 <= b & k <= a holds k <= a + b or k = a + b-2 or k = a + b-2 or k = b + b-2 or k = a + b-2 or k = b + b-2 ;
for f being FinSequence of TOP-REAL 2 , p being Point of TOP-REAL 2 , i being Nat st i in LSeg ( f , i ) & p in LSeg ( f , i ) & Index ( p , f ) <= i holds Index ( p , f ) <= i & Index ( p , f ) + 1 <= Index ( p , f )
lim ( ( curry ' ( f , k + 1 ) ) # x ) = lim ( ( curry ' ( f , k ) ) # x ) + lim ( ( curry ' ( f , k ) ) # x ) .= lim ( ( curry ' ( f , k ) ) # x ) + lim ( ( curry ' ( f , k ) ) # x ) ;
z2 = g /. ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) .= ( g | n1 + 1 ) ;
[ f . 0 , f . 3 ] in id ( the carrier of G ) \/ ( the InternalRel of G ) or [ f . 0 , f . 3 ] in the InternalRel of C6 & [ f . 0 , f . 3 ] in the InternalRel of C6 ;
for G being Subset-Family of B st G = { R [ X ] where R is Subset of A , Y is Subset of A ( ) : R in F & Y in F } holds ( ( Intersect ( R , A ) ) . ( X , Y ) ) = Intersect ( G , A )
CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 ) ) = CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m1 ) ) .= halt SCMPDS .= ( halt SCMPDS ) ;
assume that a on M and b on M and c on N and d on N and p on M and c on N and p on M and a on M and b on N and c on N and d on M and p on M and a on M and b on M and c on N and p on M and a on M and b on M and a on M and b on M and a on M and b on M and c on N and a on M and b on M and c on M and a on M and b on M and c on M and b on M and a on M and b on M and b on M and b on M and c on M and c on M and a on M and c on M and a on M and b on M and c on M and c on M and c on M and a on M and c on M and c on M and c on M
assume that T is \hbox { T _ 4 } and F is closed and ex F being Subset-Family of T st F is closed & for A being Subset of T st A is closed & A is closed & F is finite-ind & F is finite-ind holds ind ( T ) <= 0 and ind ( T ) <= 0 ;
for g1 , g2 st g1 in ]. r - r , r .[ & g2 in ]. r - r , r + r .[ & |. f . g1 - f . g2 .| <= ( ( g1 - f ) / ( r - r ) ) / ( r - r ) & |. f . g2 - f . x0 .| <= ( ( g1 - f ) / ( r - r ) ) / ( r - r )
( ( - 1 ) * ( ( - 1 ) * ( - 1 ) ) ) = ( ( - 1 ) * ( ( - 1 ) * ( - 1 ) ) ) .= ( ( - 1 ) * ( ( - 1 ) * ( - 1 ) ) .= ( - 1 ) * ( ( - 1 ) * ( - 1 ) ) .= ( - 1 ) * ( ( - 1 ) * ( - 1 ) ) ;
F . i = F /. i .= 0. R + r2 .= ( b |^ ( n + 2 ) ) * ( a |^ ( n + 1 ) ) .= ( b |^ ( n + 1 ) ) * ( a |^ ( n + 1 ) ) .= b |^ ( n + 1 ) * ( a |^ ( n + 1 ) ) .= b |^ ( n + 1 ) * ( a |^ ( n + 1 ) ) ;
ex y being set , f being Function st y = f . n & dom f = NAT & f . 0 = A & for n holds f . ( n + 1 ) = Rf . ( n + 1 ) & for n holds f . ( n + 1 ) = Rf . ( n + 1 ) & f . ( n + 1 ) = y ;
func f (#) F -> FinSequence of V means : Def4 : len it = len F & for i be Nat st i in dom it holds it . i = F . i * ( F . i ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , 6 , 7 , 8 , 7 , 8 } = { x1 , x2 , x3 , x4 , 6 , 7 , 8 } \/ { x1 , x2 , x3 , x4 , 6 , 7 , 8 } .= { x1 , x2 , x3 , x4 , 6 , 7 } ;
for n being Nat , x being set st x = h . n holds h . ( n + 1 ) = o . ( x , n ) & x in InputVertices S & o <> x & o <> x & o <> x & o <> x & o <> x & o <> x & o <> x & o <> x & o <> x & o <> x & o <> x & o <> x & o <> x & o <> x & o <> x & o <> x & o <> x & o <> x & o <> x & o <> y & o <> y & o <> y & o <> y implies o <> y & o <> y & o <> y & o <> y & o <> y & o <> y & o <> y & o <> y & o <> y & o <> y & o <> y & o <>
ex S1 being Element of CQC-WFF ( Al ( ) ) , e being Element of CQC-WFF ( Al ( ) ) st S = S1 & ( for S being non empty Subset of Al ( ) holds S is non empty & ( S is non empty & S is non empty & ( S is non empty & S is non empty & A is non empty & S is non empty & S is non empty & A is non empty ) & A is non empty ) ;
consider P being FinSequence of G7 such that p7 = Product P and for i being Element of dom P ex t7 being Element of the carrier of K st P . i = t & t7 = t & t7 = t & t7 = t & t7 = t ;
for T1 , T2 being strict non empty TopSpace , P being Subset of T1 , P being Subset of T1 , p1 , p2 being Subset of T2 st the carrier of T1 = the carrier of T2 & P = the carrier of T2 & P = the carrier of T1 & p1 = p2 holds P is Basis of T1 & P is Basis of T2 & P is Basis of T2 & P is Basis of T1 & P is closed & P is closed & P is closed & P is closed & P is closed & Q is closed & P is closed & P is closed & P is closed & Q is closed & P is closed & Q is closed & R is closed & R is closed & R is closed & R is closed & R is closed & R is closed & R is closed & R is closed & R is closed & R is closed & R is closed &
assume that f is partial differentiable on \rbrace and r (#) pdiff1 ( f , 3 ) is_partial_differentiable_in u0 and r (#) pdiff1 ( f , 3 ) is_partial_differentiable_in u0 and for u st u in dom pdiff1 ( f , 3 ) holds partdiff ( r (#) pdiff1 ( f , 3 ) , u ) = r * partdiff ( f , u ) ;
defpred P [ Nat ] means for F , G being FinSequence of bool ( Seg $1 ) , G st len F = $1 & for s being Permutation of Seg $1 st len s = $1 & for i being Nat st i in Seg $1 holds G . i = F . i holds Sum ( F ) = Sum ( G ) ;
ex j st 1 <= j & j < width GoB f & ( ( GoB f ) * ( 1 , j ) ) `2 <= s & s <= ( ( GoB f ) * ( 1 , j ) ) `2 & s <= ( ( GoB f ) * ( 1 , j + 1 ) ) `2 ;
defpred U [ set , set ] means ex Fmax be Subset-Family of T st $1 = Fmax & $2 = Fmax & for F be Subset-Family of T st F is open & F is open & F is open & F is open & F is open & F is open & F is open & F is open & F is open & F is open & F is open & F is open & F is open & F is open & F is open & F is open & F is open & ( ( ex S is open & ( ( ex S be Subset of S & S is open & ( S is open & S is open & S is open & S is open & S is open & S is open & S is open & S is open & S is open & S is open & S is open & S is open & S is open & ( S is open & S is open & S is open & S is open & S
for p4 being Point of TOP-REAL 2 st LE p4 , p1 , P & LE p1 , p2 , P & LE p2 , p1 , P & LE p1 , p2 , P & LE p2 , p1 , P & LE p1 , p2 , P & LE p2 , p1 , P & LE p1 , p2 , P & LE p2 , p1 , P & LE p2 , p1 , P holds LE p4 , p1 , P
f in St ( E , H ) & for y st g in f . y & x = f . y holds for y st y in dom f & y in S holds f . y = f . ( All ( x , H ) . y ) ) implies f in St ( All ( x , H ) , E )
ex 8 being Point of TOP-REAL 2 st x = 8 & ( ( |. 8 .| ) * ( 1 - cn ) ) / ( 1 - cn ) >= 8 & ( |. 8 .| ) * ( 1 - cn ) >= 8 & ( |. 8 .| ) * ( 1 - cn ) >= 8 & ( |. 8 .| ) * ( 1 - cn ) >= 8 & ( |. 8 .| ) * ( 1 - cn ) >= 8 ;
assume for d7 being Element of NAT st d7 <= 8 holds ( for t being Element of NAT st d7 <= t holds ( for d being Element of NAT st d <= ( ( n - 1 ) div t ) holds ( ( n - 1 ) div t ) * ( ( n - 1 ) div t ) ) = s2 * ( ( n - 1 ) div t ) ;
assume that s <> t and s is Point of Sphere ( x , r ) and s is Point of Sphere ( x , r ) and ex e being Point of E st e = Ball ( x , r ) & e in Ball ( x , r ) and e in Ball ( x , r ) and e in Ball ( x , r ) and s in Ball ( x , r ) ;
given r such that 0 < r and for s holds 0 < s or ex x1 , x2 being Point of CNS st x1 in dom f & x2 in dom f & ||. x1 - x2 .|| < s & ||. x1 - x2 .|| < s & ||. f /. x1 - f /. x2 .|| < r ;
( p | x ) | ( p | ( x | x ) ) = ( ( ( x | x ) | x ) | ( x | x ) ) .= ( ( ( x | x ) | x ) | ( x | x ) ) | ( x | x ) ;
assume that x , x + h in dom sec and ( for x holds h . x = ( 4 * ( sin . x ) + sin . x ) / ( sin . x ) ^2 ) and for x holds h . x = ( 4 * ( sin . x ) + sin . x ) / ( sin . x ) ^2 and for x holds h . x = 1 ;
assume that i in dom A and len A > 1 and len B > 1 and B c= dom A and for i , j st i in dom B & j in dom B & i <> j & i <> j holds A * ( i , j ) = A * ( i , j ) and A * ( i , j ) = A * ( i , j ) and A * ( i , j ) = A * ( i , j ) and A * ( i , j ) = A * ( i , j ) and B * ( i , j ) = A * ( i , j ) and B * ( i , j ) = B * ( i , j ) and B * ( i , j ) and B * ( i , j ) and B * ( i , j ) = B * ( i , j ) = B * ( i , j ) = B * ( i , j ) = B * ( i ,
for i be non zero Element of NAT st i in Seg n holds i divides n or i = <* 1. F_Complex *> or i = <* 1. F_Complex *> & ( i divides n & i divides n & i divides n implies h . i = <* 1. F_Complex *> & ( i divides n implies h . i = 1. F_Complex ) & ( i divides n implies h . i = 1. F_Complex & h . i = 1. F_Complex )
( ( ( b1 'imp' b2 ) '&' ( c1 'imp' c2 ) ) '&' ( ( b1 'or' b2 ) '&' ( c1 'or' c2 ) ) '&' ( ( b1 'or' b2 ) '&' ( c1 'or' c2 ) ) '&' ( ( b1 'or' b2 ) '&' ( c1 'or' c2 ) ) '&' ( ( b1 'or' b2 ) '&' ( c1 'or' c2 ) ) '&' ( ( b1 'or' b2 ) '&' ( c1 'or' c2 ) ) '&' ( c1 'or' c2 ) ) '&' ( c1 'or' c2 ) ;
assume that for x holds f . x = ( ( ( - 1 / 2 ) (#) ( cot * sin ) ) `| Z ) . x and for x st x in Z holds ( ( - 1 / 2 ) (#) ( cot * sin ) ) `| Z ) . x = - 1 / ( sin . x ) ^2 and for x st x in Z holds ( ( - 1 / 2 ) (#) ( cot * sin ) ) . x = - 1 / ( sin . x ) ^2 and for x holds x = 1 / ( sin . x ) ^2 and for x st x = 1 / ( sin . x ) ^2 and for x holds x = 1 / ( sin . x ) ^2 and f . x = 1 / ( sin . x ) ^2 and f . x = 1 / ( sin . x ) ^2 and for x holds x = 1 / ( sin . x ) ^2 and for x = 1 / ( sin . x ) ^2
consider Rd , Id be Real such that Rd = Integral ( M , Re ( F . n ) ) and Id = Integral ( M , Im ( F . n ) ) and Id = Integral ( M , Im ( F . n ) ) and for i be Nat st i in dom Rd holds I . i = Integral ( M , Im ( F . n ) ) ;
ex k be Element of NAT st k = k & 0 < d & for q be Element of product G st q in X & 0 < d & ||. q-r - f /. ( q - x ) .|| < r holds ||. partdiff ( f , q , k ) .|| < r ;
x in { x1 , x2 , x3 , x4 , x5 , 6 , 7 , 8 , 8 , 7 , 8 , 6 , 7 , 8 } implies x in { x1 , x2 , x3 , x4 , 6 , 7 , 8 } \/ { x1 , x2 , x3 , x4 , 6 , 7 , 8 }
G * ( j , i ) `2 = G * ( 1 , i ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , k ) `2 .= G * ( 1 , k ) `2 .= G * ( 1 , k ) `2 .= G * ( 1 , k ) `2 .= G * ( 1 , k ) `2 .= G * ( 1 , k ) `2 .= G * ( 1 , k ) `2 .= G * ( 1 , k ) `2 .= G * ( 1 , k ) `2 .= G * ( 1 , k ) `2 .= G * ( 1 , k ) `2 .= G * ( 1 , k ) `2 .= G * ( 1 , k ) `2 .= G * ( 1 , k ) `2 .= G * ( 1 , k ) `2 .= G * ( 1 , k
f1 * p = p .= ( ( the Arity of S1 ) +* ( the Arity of S2 ) ) . o .= ( ( the Arity of S1 ) +* ( the Arity of S2 ) ) . o .= ( ( the Arity of S1 ) +* ( the Arity of S2 ) ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o ;
func tree ( T , P , T1 , T2 ) -> Tree means : Def4 : q in it iff ex p , q st p in P & q in P & p ^ q in T & p ^ q in T & p ^ q in T & p ^ q in T & q ^ p in T & p ^ q in T & p ^ q in T & q ^ p in T ;
F /. ( k + 1 ) = F . ( k + 1 -' 1 ) .= F9 . ( p . ( k + 1 -' 1 ) , p . ( k + 1 -' 1 ) ) .= F9 . ( p . k , p . ( k + 1 -' 1 ) ) .= F9 . ( p . k , p . ( k + 1 -' 1 ) ) .= F9 . ( p . k , p . k ) ;
for A , B , C being Matrix of K st len B = len C & len C = len C & len B = width C & len C = len B & len A > 0 & len B > 0 & len C > 0 & len A > 0 & len B > 0 & len A > 0 & len B > 0 & len B > 0 & len A = len B & width B = 0 & width B = 0 & width B = 0 & width B = 0 & width B = 0 & width B = 0 & width B = 0 & width B = 0 & width B = 0 & width B = 0 & width B = 0 & width B = 0 & width B = 0 & width B = 0 & width B = 0 & width B = 0 & width B = 0 & width B = 0 & len C = 0 & len C = 0 & len C = 0 & len C = 0 & width B = 0
seq . ( k + 1 ) = 0. F_Complex + seq . ( k + 1 ) .= ( Partial_Sums seq ) . ( k + 1 ) .= ( Partial_Sums seq ) . ( k + 1 ) + seq . ( k + 1 ) .= ( Partial_Sums seq ) . ( k + 1 ) + seq . ( k + 1 ) .= ( Partial_Sums seq ) . ( k + 1 ) ;
assume that x in ( the carrier of Cf ) and y in ( the carrier of Cf ) and z in ( the carrier of Cf ) and x in ( the carrier of Cf ) and y in ( the carrier of Cf ) and z in ( the carrier of Cf ) and x in ( the carrier of Cf ) ;
defpred P [ Element of NAT ] means for f st len f = $1 & for k st k < $1 holds ( for i st i < $1 holds ( for k st k < $1 holds ( for k st k < $1 holds ( for k st k < $1 holds ( for k st k < $1 holds ( for k st k < n holds ( v . k ) < ( v . k ) ) ) & k < len ( v . k ) ) holds P [ k , f . k ] ;
assume that 1 <= k and k + 1 <= len f and f is special and [ i , j ] in Indices G and [ i + 1 , j ] in Indices G and f /. k = G * ( i , j ) and f /. k = G * ( i + 1 , j ) and f /. k = G * ( i + 1 , j ) ;
assume that sn < 1 and q `1 > 0 and ( q `1 / |. q .| - cn ) / ( 1 - cn ) >= 0 and ( q `1 / |. q .| - cn ) / ( 1 - cn ) >= 0 and ( q `1 / |. q .| - cn ) / ( 1 - cn ) >= 0 ;
for M being non empty metric space , x being Point of M , f being Point of M , x being Point of M st x = x holds ex f being Function of M , M st for n being Element of NAT holds f . n = Ball ( x , ( 1 / n ) ) & f . n = Ball ( x , ( 1 / n ) )
defpred P [ Element of omega ] means ( f1 is_differentiable_on Z & f2 is_differentiable_on Z & for x st x in Z holds f1 . x = - ( f1 . x ) / ( f1 . x ) ^2 ) & for x st x in Z holds ( f1 - f2 ) `| Z is_differentiable_in x & ( f1 - f2 ) `| Z = ( f1 - f2 ) `| Z ) . x ;
defpred P1 [ Nat , Point of Cseq ] means ( $1 in Y & $2 in Y & ||. $2 - $1 .|| < r & ||. $2 - $1 .|| < r ) & ||. f /. $2 - f /. ( $1 + 1 ) .|| < r ) implies ||. f /. $2 - f /. ( $1 + 1 ) .|| < r ;
( f ^ mid ( g , 2 , len g ) ) . i = ( mid ( g , 2 , len g ) ) . i .= ( mid ( g , 2 , len g ) ) . i .= ( mid ( g , 2 , len g ) ) . i .= ( mid ( g , 2 , len g ) ) . i .= ( mid ( g , 2 , len g ) ) . i ;
( 1 - 2 * ( n + 2 * ( n + 2 * ( n + 1 ) ) ) * ( 2 * ( n + 2 * ( n + 1 ) ) ) = ( ( 1 - 2 * ( n + 2 * ( n + 1 ) ) ) * ( 2 * ( n + 1 * ( n + 1 ) ) ) .= ( 1 - 2 * ( n + 1 * ( n + 1 ) ) ) * ( 2 * ( n + 1 ) ) .= ( 1 - 2 * ( n + 1 * ( n + 1 ) ) * ( 2 * ( n + 1 ) ) .= ( 1 - 2 * ( n + 1 ) ) * ( 2 * ( n + 1 ) ) * ( 2 * ( n + 1 ) ) * ( 2 * ( n + 1 ) ) * ( n + 1 ) .= ( ( n + 1 ) * ( n + 1 ) * ( n + 1 ) ) .= ( ( 1 - 2 * ( n + 1 ) ) * ( n
defpred P [ Nat ] means for G being non empty strict RelStr , F being non empty finite Subset of the carrier of G , A being non empty set st G is space & card F = $1 & for n being Element of NAT st n in dom F & n < $1 holds the RelStr of G = ( the RelStr of F ) . n & the RelStr of G = ( the RelStr of F ) . n ;
assume that f /. 1 in Ball ( u , r ) and not 1 <= m and m <= len - ( f /. 1 ) and not ( ex i st i <= len f & LSeg ( f , i ) /\ LSeg ( f , i ) <> {} ) and not m in Ball ( u , r ) and not m in Ball ( u , r ) and not m in Ball ( u , r ) ;
defpred P [ Element of NAT ] means ( Partial_Sums ( ( ( cos - cos ) * ( $1 - r ) ) ) . $1 = ( ( cos - cos ) * ( $1 - r ) ) / ( $1 - r ) & ( ( cos - cos ) * ( $1 - r ) ) / ( $1 - r ) = ( ( cos - cos ) * ( $1 - r ) ) / ( $1 - r ) ;
for x being Element of product F holds x is FinSequence of product F & dom x = I & for i being set st i in I holds x . i = I . i & for i being set st i in I holds x . i = ( the Sorts of F ) . i & x . i = ( the Sorts of A ) . i & x . i = ( the Sorts of A ) . i
( x " ) |^ ( n + 1 ) = ( x " ) |^ n * x .= ( x |^ n ) * x .= ( x |^ n ) * x .= ( x |^ n ) * x .= ( x |^ n ) * x .= ( x |^ n ) * x .= ( x |^ n ) * x .= ( x |^ n ) * x .= x |^ n * x .= x |^ n * x ;
DataPart ( Comput ( P +* I , s +* I , k ) +* I , Initialized s ) = DataPart Comput ( P +* I , s +* I , LifeSpan ( P +* I , s +* I , k ) ) .= DataPart Comput ( P +* I , s +* I , Initialize s ) ;
given r such that 0 < r and ]. x0 - r , x0 .[ c= dom ( f1 (#) f2 ) /\ ]. x0 - r , x0 .[ and for g st g in ]. x0 - r , x0 .[ & g in ]. x0 - r , x0 .[ holds f1 . g <= ( f1 (#) f2 ) . g ;
assume that X c= dom f1 /\ dom f2 and f1 | X is continuous and f2 | X is continuous and for x st x in X holds ( f1 - f2 ) | X is continuous and ( f1 - f2 ) | X is continuous and ( f1 - f2 ) | X is continuous and ( f1 - f2 ) | X is continuous and f2 | X is continuous ;
for L being continuous complete LATTICE for l being Element of L ex X being Subset of L st l = sup X & for x being Element of L st x in X holds x is finite & for x being Element of L st x in X holds x is finite & x is finite holds x is finite & x is finite
Support ( e *' p ) in { m *' p where m is Polynomial of n , L : ex m being Nat st m in Support ( m *' p ) & p . m = ( m *' p ) . ( i + 1 ) & ex n being Element of NAT st n in dom p & p . n = ( m *' p ) . ( i + 1 ) & p . n = ( m *' q ) . ( i + 1 ) ;
( f1 - f2 ) /* s1 = lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) ;
ex p1 being Element of [: Al ( ) , NAT ( ) :] st F . p1 = g . p1 & for g being Function of [: Al ( ) , NAT ( ) , NAT ( ) , NAT ( ) , NAT ( ) , NAT ( ) , NAT ( ) st P [ g , p1 , g , h , h , h , i ( ) , 0 ( ) , 0 ( ) ] ;
( mid ( f , i , len f -' 1 ) ) /. j = ( mid ( f , i , len f -' 1 ) ) . j .= ( mid ( f , i , len f -' 1 ) ) . j .= ( mid ( f , i , len f -' 1 ) ) . j .= ( mid ( f , i , len f -' 1 ) ) . j ;
( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) = ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + ( q ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q
len mid ( ( f , D2 ) , indx ( D2 , D1 , j1 ) + 1 , indx ( D2 , D1 , j1 ) + 1 ) = indx ( D2 , D1 , j1 ) - 1 + 1 .= indx ( D2 , D1 , j1 ) - 1 + 1 ;
x * y * z = MM . ( x * ( y * z ) , z * ( y * z ) ) .= x * ( ( x * y ) * z ) .= x * ( y * z ) .= x * ( y * z ) .= x * ( y * z ) .= x * ( y * z ) .= x * ( y * z ) ;
v . <* x , y *> - ( <* x0 , y0 *> ) * i = partdiff ( v , ( x - x0 ) * i + ( y - x0 ) * ( ( y - x0 ) * i ) + ( ( y - x0 ) * i ) + ( ( y - x0 ) * ( ( y - x0 ) * i ) ) ;
i * i = <* 0 * ( - 1 ) - ( 0 * ( - 1 ) ) * ( - 1 ) .= <* - 1 , 0 , 0 , 0 *> .= <* - 1 , 0 , 0 , 0 *> .= <* - 1 , 0 , 0 , 0 *> ;
Sum ( L (#) F ) = Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) .= Sum ( F1 ^ F2 ) + Sum ( F1 ^ F2 ) + Sum ( F1 ^ F2 ) .= Sum ( F1 ^ F2 ) + Sum ( F1 ^ F2 ) .= Sum ( F1 ^ F2 ) .= Sum ( F1 ^ F2 ) + Sum ( F2 ) .= Sum ( F1 ^ F2 ) + Sum ( F1 ^ F2 ) + Sum ( F1 ^ F2 ) + Sum ( F1 ^ F2 ) + Sum ( F2 ^ F2 ) + Sum ( F2 ^ F2 ) + Sum ( F1 ^ F2 ) + Sum ( F2 ) + Sum ( F2 ^ F2 ) .= Sum (
ex r be Real st for e be Real st 0 < e ex Y1 be Subset of X st 0 < e & Y1 c= Y & for Y1 be Subset of X st Y1 is non empty & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y2 c= Y & Y1 c= Y & Y1 c= Y implies Y1 c= Y implies ex Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y2 c= Y & Y1 c= Y & Y1 c= Y & Y2 c= Y & Y1 c= Y & Y1 c= Y implies ex Y1 c= Y
( GoB f ) * ( i , j + 1 ) `1 = f /. ( k + 2 ) & ( GoB f ) * ( i , j + 1 ) `1 = f /. ( k + 2 ) or ( GoB f ) * ( i , j + 1 ) `1 = f /. ( k + 2 ) & ( GoB f ) * ( i + 1 , j ) `2 = f /. ( k + 2 ) implies ( GoB f ) * ( i + 1 , j ) `2 = f /. ( k + 1 ) `2
( ( - 1 ) (#) ( sin - cos ) ) `| Z = ( - 1 ) * ( sin - cos ) .= ( - 1 ) * ( sin - cos ) .= ( - 1 ) * ( sin - cos ) .= ( - 1 ) * ( sin - cos ) .= ( - 1 ) * ( sin - cos ) .= ( - 1 ) * ( sin - sin ) ;
( - b - sqrt delta ( a , b , c ) ) / 2 + ( - b - sqrt delta ( a , b , c ) ) / 2 > 0 & ( - b - sqrt delta ( a , b , c ) ) / 2 + ( - b - sqrt delta ( a , b , c ) ) / 2 + ( - b - sqrt delta ( a , b , c ) ) / 2 > 0 ;
Suppose ex_inf_of "/\" ( uparrow X , C ) , L and ex_sup_of X , C and sup X , C and for X st X in C & X in C holds "\/" ( ( uparrow L ) .: ( ( sub L ) .: ( ( sub L ) .: ( C ) ) , L ) = "/\" ( ( uparrow L ) .: ( C /\ ( C /\ L ) ) , L ) ;
( for j holds j . ( i , i ) = ( j , j ) --> id the Sorts of B ) . ( i , j ) & ( j = i implies ( j = i implies j = i ) & ( j = i implies j = i ) ) & ( j = i implies j = i implies j = i ) & ( j = i implies j = i ) & ( j = i implies j = i ) ) & ( j = i implies i = i implies j = i implies j = i implies j = i implies j = i implies j = i ) & ( j = i implies j = i implies j = i implies j = i implies j = i ) & ( j = i implies j = i implies j = i implies j = i & i = i & i = i implies j = i & i = i & i = i & i = i implies j = i & i = i implies i = i & i = i implies j = i implies i = i & i = i & i = i & i = i & i = i implies j = i & i = i implies