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thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
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thesis ;
contradiction ;
contradiction ;
thesis ;
contradiction ;
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thesis ;
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contradiction ;
thesis ;
thesis ;
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contradiction ;
thesis ;
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thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
contradiction ;
thesis ;
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contradiction ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
B ;
a <> c
T c= S
D c= B
c 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 ;
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 ;
assume x in I ;
q is as : 0 in n ;
assume c in x ;
p > 0 ;
assume x in Z ;
assume x in Z ;
1 <= k-2 ;
assume m <= i ;
assume G is prime ;
assume a divides b ;
assume P is closed ;
\bf 2 > 0 ;
assume q in A ;
not W is bounded ;
f is IC ;
assume A is boundary ;
g is_sequence_on G ;
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 `2 <= c `2 ;
A meets W ;
i `2 <= j `2 ;
assume H is universal ;
assume x in X ;
let X be set ;
let T be Tree ;
let d be element ;
let t be element ;
let x be element ;
let x be element ;
let s be element ;
k <= 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 ;
let a be Real ;
let x be element ;
let X be element ;
P [ 0 ]
let x be element ;
let x be element ;
let y be element ;
r in REAL ;
let e be element ;
n1 is (#) ;
Q halts_on s ;
x in such that x in \in \in \in ' ;
M < m + 1 ;
T2 is open ;
z in b < a < a ;
R2 is well-ordering ;
1 <= k + 1 ;
i > n + 1 ;
q1 is one-to-one ;
let x be trivial set ;
PM is one-to-one ;
n <= n + 2 ;
1 <= k + 1 ;
1 <= k + 1 ;
let e be Real ;
i < i + 1 ;
p3 in P ;
p1 in K ;
y in C1 ;
k + 1 <= n ;
let a be Real , x be Point of REAL ;
X |- r => p ;
x in { A } ;
let n be Nat ;
let k be Nat ;
let k be Nat ;
let m be Nat ;
0 < 0 + k ;
f is_differentiable_in x ;
let x0 , x1 ;
let E be Ordinal ;
o \mathord o1 , o * ;
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 complex let M be non empty set , f be PartFunc of V , V ;
not s in Y to_power 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 , M be Subset of V ;
P [ 1 ] ;
P [ {} ] ;
C1 is component ;
H = G . i ;
1 <= i `2 + 1 ;
F . m in A ;
f . o = o ;
P [ 0 ] ;
a-0 <= being Real ;
R [ 0 ] ;
b in f .: X ;
assume q = q2 ;
x in [#] V ;
f . u = 0 ;
assume e1 > 0 ;
let V be RealUnitarySpace , M 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 , f be Function of T , T ;
the ObjectMap of F is one-to-one
sgn x = 1 ;
k in support a ;
1 in Seg 1 ;
rng f = X ;
len T in X ;
vbeing Nat st v < n ;
S.: S is bounded ;
assume p = p2 ;
len f = n ;
assume x in P1 ;
i in dom q ;
let U0 , E , f ;
pp `1 = 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 <= j9 & j <= width G ;
set A = -> non empty set ;
card a [= c ;
e in rng f ;
cluster B \oplus A -> empty ;
H has \cdot F ;
assume n0 <= m ;
T is increasing ;
e2 <> e2 & 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 is connected
assume not x in REAL ;
Im f is complete ;
x in Int y ;
dom F = M ;
a in On W ;
assume e in A ( ) ;
C c= C-26 ;
mm <> {} ;
let x be Element of Y ;
let f be let \Omega , g be 1 -element Function ;
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 \overline ( X /\ Y ) ;
- y in I ;
let A be non empty set , B be non empty set ;
P0 = 1 ;
assume r in F . k ;
assume f is simple ;
let A be TOP-REAL countable set ;
rng f c= NAT & f is one-to-one
assume P [ k ] ;
f6 <> {} ;
let o be Ordinal ;
assume x is sum of squares ;
assume not v in { 1 } ;
let IB , C ;
assume that 1 <= j and j < l ;
v = - u ;
assume s . b > 0 ;
d1 in D ;
assume t . 1 in A ;
let Y be non empty TopSpace , x be Point of Y ;
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 <> {} & mm <> {} ;
M + N c= M + M ;
assume M is connected & hhz is connected ;
assume f is additive inbr-r) ;
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 = k2 ;
m + 1 < n + 1 ;
s in S \/ { s } ;
n + i >= n + 1 ;
assume Re y = 0 ;
k1 <= j1 & j1 <= j2 ;
f | A is continuous ;
f . x being Real ;
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 , B ) ;
[ x , x ] in R ;
assume x in succ 0 ;
a in sup phi ;
Cc ;
q2 c= C1 & q2 c= C2 ;
a2 < c2 & b2 < c2 ;
s2 is 0 -started ;
IC s = 0 ;
s4 `1 = s4 `1 ;
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 function of L ;
y " <> 0 ;
y " <> 0 ;
0. V = u-w ;
y2 , y are_not zero ;
R8 ;
let a , b be Real , x be Point of REAL ;
let a be object of C ;
let x be Vertex of G ;
let o be object of C , a be object of C ;
r '&' q = P \lbrack l , r .] ;
let i , j be Nat ;
let s be State of A , x be set ;
s4 . n = N ;
set y = ( x `1 ) / ( 1 + x `2 ) ;
mi in dom g & mi in dom g ;
l . 2 = y1 ;
|. g . y .| <= r ;
f . x in Cx0 ;
not V2 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 that A is dense and 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 & xY 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 , n be Nat ;
assume that r2 > x0 and x0 < r2 ;
let Y be non empty set , f be Function of Y , BOOLEAN ;
2 * x in dom W ;
m in dom g2 & n + 1 in dom g2 ;
n in dom g1 /\ dom g2 ;
k + 1 in dom f ;
not the still of s is finite ;
assume that x1 <> x2 and x1 <> x3 ;
v1 in V1 & v2 in V2 ;
not [ b `1 , b ] in T ;
i9 + 1 = i ;
T c= \rangle ( T ) ;
( l . 1 ) `1 = 0 ;
let n be Nat ;
( t `2 ) ^2 = r ;
AA is_integrable_on M & f is_integrable_on M ;
set t = Top t ;
let A , B be real-membered set ;
k <= len G + 1 ;
[: C , C :] misses [: V , C :] ;
product ( s ) is non empty ;
e <= f or f <= e ;
cluster non empty normal for NAT -defined Function ;
assume c2 = b2 & c1 = b2 ;
assume h in [. q , p .] ;
1 + 1 <= len C ;
not c in B . m1 ;
cluster R .: X -> empty ;
p . n = H . n ;
assume that vseq is Cauchy and vseq is convergent and lim vseq = 0 ;
IC s3 = 0 & IC s3 = 1 ;
k in N or k in K ;
F1 \/ F2 c= F ;
Int ( G1 \/ G2 ) <> {} ;
( z `2 ) ^2 = 0 ;
p11 `2 <> p1 `2 ;
assume z in { y , w } ;
MaxADSet ( a ) c= F ;
ex_sup_of downarrow s , S ;
f . x <= f . y ;
let T be up-complete up-complete non empty reflexive transitive RelStr ;
q |^ m >= 1 ;
a is_>=_than X & b is_>=_than Y ;
assume <* a , c *> <> {} ;
F . c = g . c ;
G is one-to-one ;
A \/ { a } \not c= B ;
0. V = 0. Y .= 0. Y ;
let I be <= halting Instruction of S , i be Nat ;
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 ;
f " P is compact & f " P is compact ;
assume x1 in REAL & x2 in REAL ;
p1 `1 = ( K `1 ) * ( 1 , 1 ) `1 ;
M . k = <*> REAL ;
phi . 0 in rng phi ;
OSMMA is closed ;
assume z0 <> 0. L & z0 <> 0. L ;
n < N7 . k ;
0 <= seq . 0 - seq . 0 ;
- q + p = v ;
{ v } is Subset of B ;
set g = f /. 1 ;
[: R , R :] is stable Subset of R ;
set [: R , R :] = Vertices R ;
pp 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 TopLattice of S ;
ex_inf_of the carrier of S , S ;
downarrow a = downarrow b & downarrow b = { a } ;
P , C , K , L ;
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 ~ are_equipotent ;
assume a in A ( ) ;
k in dom ( q ^ <* p *> ) ;
p is FinSequence of S ;
i -' 1 = i-1 ;
f | A is one-to-one ;
assume x in f .: [: X , Y :] ;
i2 - i1 = 0 & i2 - i1 = 0 ;
j2 + 1 <= i2 ;
g " * a in N ;
K <> { [ {} , {} ] } ;
cluster strict } -valued strict for Relation ;
|. q .| ^2 > 0 ;
|. p4 .| = |. p .| ;
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 F = dom G ;
let s be Element of NAT , k be Nat ;
let R be ManySortedSet of A ;
let n be Element of NAT ;
let S be non empty non void non void non empty ManySortedSign ;
let f be ManySortedSet of I ;
let z be Element of COMPLEX , p be FinSequence of COMPLEX ;
u in { ag } ;
2 * n < 2 * ( n + 1 ) ;
let x , y be set ;
B-11 c= ( V . i ) ;
assume I is_closed_on s , P ;
U2 = U ( ) & U2 = U ( ) ;
M /. 1 = z /. 1 ;
x9 = x9 & y9 = y9 or x9 = y9 ;
i + 1 < n + 1 + 1 ;
x in { {} , <* 0 *> } ;
f7 <= ( f . i ) `1 ;
let l be Element of L ;
x in dom ( F . n ) ;
let i be Element of NAT ;
r8 is ( len ( D * ) ) -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 st D is st D is open holds D is open
set r = { Seg ( k + 1 ) } ;
y = W . ( 2 * x ) ;
assume dom g = cod f & cod g = cod f ;
let X , Y be non empty TopSpace , x be Point of X ;
x \oplus A is interval ;
|. <*> A .| . a = 0 ;
cluster strict for LATTICE ;
a1 in B . s1 & a2 in B . s1 ;
let V be finite VectSp non empty VectSpStr over F , F be FinSequence of V ;
A * B on B , A ;
f-3 = NAT --> 0 ;
let A , B be Subset of V ;
z1 = P1 . j & z2 = P2 . j ;
assume f " P is closed ;
reconsider j = i as Element of M ;
let a , b be Element of L ;
assume q in A \/ ( B "\/" C ) ;
dom ( F (#) C ) = o ;
set S = INT |^ X ;
z in dom ( A --> y ) ;
P [ y , h . y ] ;
{ x0 } c= dom f ;
let B be non-empty ManySortedSet of I , A be non empty set ;
( cos ( x ) ) ^2 < ( cos ( x ) ) ^2 ;
reconsider z9 = 0 , z9 = 1 as Nat ;
LIN a , d , c & LIN a , d , c ;
[ y , x ] in IX ;
( Q ) / ( 3 / ( 4 * ( 1 + 3 ) ) ) = 0 ;
set j = x0 div m , i = m mod m ;
assume a in { x , y , c } ;
j2 - ( j - 1 ) > 0 ;
I not ( I , phi ) = 1 ;
[ y , d ] in F-8 ;
let f be Function of X , Y ;
set A2 = ( B |^ C ) ;
s1 , s2 s2 s2 s2 , s3 be Element of R ;
j1 -' 1 = 0 or j1 -' 1 = 1 ;
set m2 = 2 * n + j ;
reconsider t = t as bag of n ;
I2 . j = m . j ;
i |^ s , n are_congruent_mod m ;
set g = f | D-21 ;
assume that X is lower and 0 <= r ;
( p1 `1 ) ^2 = 1 ;
a < ( p3 `1 ) ^2 + ( p3 `2 ) ^2 ;
L \ { m } c= UBD C ;
x in Ball ( x , 10 ) ;
not a in LSeg ( c , m ) ;
1 <= i1 -' 1 + 1 & i1 -' 1 + 1 <= len f ;
1 <= i1 -' 1 + 1 & i1 -' 1 + 1 <= len f ;
i + i2 <= len h - 1 ;
x = W-min ( P ) & y = W-min ( P ) ;
[ x , z ] in X ~ ;
assume y in [. x0 , x .] ;
assume p = <* 1 , 2 , 3 *> ;
len <* A1 *> = 1 & len <* A2 *> = 2 ;
set H = h . g , I = h . i ;
card b * a = |. a .| ;
Shift ( w , 0 ) |= v ;
set h = h2 (*) h1 ;
assume x in ( ( X /\ X1 ) \/ ( X /\ X1 ) ) ;
||. h .|| < dx0 & 0 < dx0 ;
not x in the carrier of f ;
f . y = F ( y ) ;
for n holds X [ n ] ;
k -' l = kLIN ;
<* p , q *> /. 2 = q ;
let S be Subset of the carrier 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 <= f . sup X ;
let L be non empty transitive RelStr , x be Element of L ;
S-20 is x -\leq i -basis i ;
let r be non positive Real ;
M , v |= x , y |= x ;
v + w = 0. ( Z ) ;
P [ ( len F ) - 1 ] ;
assume InsCode ( i ) = 8 & InsCode ( i ) = 8 ;
the zero Element of M = 0 & the carrier 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 -> / for Element of AllSymbolsOf S ;
reconsider l1 = l-1 as Nat ;
v4 is Vertex of r2 & v4 is \overline of r2 ;
T2 is SubSpace of T2 & T2 is SubSpace of T2 ;
Q1 /\ Q1 <> {} & Q1 /\ Q1 <> {} ;
let k be Nat ;
q " is Element of X & q is Element of Y ;
F . t is set & F . t is set ;
assume that n <> 0 and n <> 1 ;
set en = EmptyBag n , en = EmptyBag n ;
let b be Element of Bags n ;
assume for i holds b . i is commutative ;
x is root & y is root implies x <= y `1
not r in ]. p , q .[ ;
let R be FinSequence of REAL , x be set ;
S7 does not destroy b1 , b2 ;
IC SCM R <> a & IC SCM R <> a ;
|. - |[ x , y ]| .| >= r ;
1 * ( seq . n ) = seq . n ;
let x be FinSequence of NAT , k be Nat ;
let f be Function of C , D , g be Function ;
for a holds 0. L + a = a
IC s = s . NAT .= ( n + 1 ) ;
H + G = F-GG ;
CS1 . x = x2 & CS2 . x = y2 ;
f1 = f .= f2 . i .= f2 . i ;
Sum <* p . 0 *> = p . 0 ;
assume v + W = v + u + W ;
{ a1 } = { a2 } ;
a1 , b1 _|_ b , a ;
d2 , o _|_ o , a3 ;
Iz is reflexive & Iz is reflexive implies z in Cz
Iz is antisymmetric implies ( for n holds n in Cz iff n <= z )
sup rng H1 = e & sup rng H1 = e ;
x = a9 * ( a * b ) ;
|. p1 .| ^2 >= 1 ^2 ;
assume j2 -' 1 < j2 -' 1 ;
rng s c= dom f1 /\ dom f2 ;
assume support a misses support b & not a in support b ;
let L be associative commutative associative non empty doubleLoopStr , p be Polynomial of L ;
s " + 0 < n + 1 ;
p . c = ( f " ) . 1 ;
R . n <= R . ( n + 1 ) ;
Directed ( I1 ) = I1 . 0 .= i ;
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 Function ;
let X be non empty directed Subset of S ;
let S be non empty full SubRelStr of L ;
cluster <* \rangle . N -> complete for non trivial RelStr ;
( 1 - a ) " = a ;
( q . {} ) `1 = o ;
n - ( i -' 1 ) > 0 ;
assume ( 1 - t ) / 2 <= t `1 / 2 ;
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 } & v in { v } ;
let G be "\/" "\/" wgraph ;
e , v6 be set ;
c . i9 in rng c & c . i9 in rng c ;
f2 /* q is divergent_to+infty & f2 . ( lim ( f2 , x0 ) ) = 0 ;
set z1 = - z2 , z2 = - z2 , z2 = - z2 , z2 = - z2 , z1 = - z2 , z2 = - z2 , z2 = - z2 , z2 = - z2 , z2
assume that w is \mathop { llas of S , G ;
set f = p |-count ( t . a ) , g = p |-count ( t . a ) , h = p |-count ( t . a ) , e = p |-count ( t . a ) , f
let c be Object of C ;
assume ex a st P [ a ] ;
let x be Element of REAL m , y be Real ;
let Iy be Subset-Family of X , x be set ;
reconsider p = p as Element of NAT ;
let v , w be Point of X ;
let s be State of SCM+FSA , k be Nat ;
p is FinSequence of SCM+FSA & q is FinSequence of NAT ;
stop I ( ) c= P ( ) ;
set ci = fSet /. i , fI = fconsider i ;
w ^ t ^ s ^ t ^ s ^ t ^ s ^ t ^ s ^ s ^ t ^ s ^ t ^ s ^ t ^ s ^ t ^ s ^ t ^ s ^ t ^
seq1 /\ W = seq1 /\ W ` .= ( seq1 + seq2 ) /\ W ;
f . j is Element of J . j ;
let x , y be Element of T2 ;
ex d st a , b // b , d ;
a <> 0 & b <> 0 ;
ord x = 1 & x is positive ;
set g2 = lim ( s , x0 ) ;
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 ;
/ ( X \/ R1 ) = / ( X \/ R1 ) ;
( sin . x ) ^2 <> 0 ;
( ( exp_R * f ) `| Z ) . x > 0 ;
o1 in ( X /\ O2 ) /\ O2 ;
e , v6 be set ;
r3 > ( 1 - r ) * 0 ;
x in P .: ( F -ideal ) ;
let J be closed ideal non empty Subset of R ;
h . p1 = f2 . O .= ( f . O ) `1 ;
Index ( p , f ) + 1 <= j ;
len ( q ^ <* x *> ) = width M ;
the carrier of CK c= A & the carrier of CK c= A ;
dom f c= union rng ( F | X ) ;
k + 1 in support ( support ( n ) ) ;
let X be ManySortedSet of the carrier of S ;
[ x `1 , y `2 ] in \/ ( R ~ ) ;
i = D1 or i = D2 ;
assume a mod n = b mod n .= b mod n ;
h . x2 = g . x1 .= ( f . x2 ) `1 ;
F c= 2 -tuples_on the carrier of X ;
reconsider w = |. s1 .| as Real_Sequence ;
( 1 / ( m * m + r ) ) < p ;
dom f = dom ( I * ( 1 , 1 ) ) ;
[#] ( ( TOP-REAL 2 ) | K1 ) = [#] ( ( TOP-REAL 2 ) | K1 ) ;
cluster - x -> ExtReal -> R_eal ;
then { d1 } c= A ;
cluster ( TOP-REAL n ) | ( [#] TOP-REAL n ) -> finite-ind ;
let w1 be Element of M ;
let x be Element of dyadic ( n ) ;
u in W1 & v in W3 implies u in W2
reconsider y = y , z = z as Element of L2 ;
N is full SubRelStr of [: T , T :] ;
sup { x , y } = c "\/" c ;
g . n = n to_power 1 .= n ;
h . J = EqClass ( u , J ) ;
let seq be summable sequence of X , k be Nat ;
dist ( x `1 , y ) < ( r / 2 ) ;
reconsider mm = m , mm = n as Element of NAT ;
x- x0 < r1 - x0 & r1 < x0 + r2 ;
reconsider P ` = P ` as strict Subgroup of N ;
set g1 = p * ( idseq q `1 ) , g2 = q `2 ;
let n , m , k be non zero Nat ;
assume that 0 < e and f | A is lower ;
D2 . ( I1 ) in { x } ;
cluster subcondensed closed -> subopen for Subset of T ;
let P be compact non empty Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
Gik in LSeg ( cos , 1 ) /\ LSeg ( cos , 1 ) ;
let n be Element of NAT , x be set ;
reconsider S8 = S , S8 = T as Subset of T ;
dom ( i .--> X ' ) = { i } ;
let X be non-empty ManySortedSet of S ;
let X be non-empty ManySortedSet of S ;
op ( 1 ) c= { [ {} , {} ] } ;
reconsider m = mm - 1 as Element of NAT ;
reconsider d = x as Element of C ( ) ;
let s be 0 -started State of SCMPDS , k be Nat ;
let t be 0 -started State of SCMPDS , Q be t -started State of SCMPDS ;
b , b , b , x , y , z ;
assume that i = n \/ { n } and j = k \/ { k } ;
let f be PartFunc of X , Y ;
x0 >= ( sqrt ( c ^2 ) ) / ( sqrt ( c ^2 ) ) ;
reconsider t7 = T-1 as Point of TOP-REAL 2 ;
set q = h * p ^ <* d *> ;
z2 in U . ( y2 , y2 ) /\ Q2 . ( z2 , y2 ) ;
A |^ 0 = { <* \rangle *> } ;
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 & g1 . z in dom f ;
assume p2 `1 = ( E-max ( K ) ) `1 ;
len G + 1 <= i1 + 1 ;
f1 (#) f2 is_differentiable_in x0 & f2 (#) ( f1 (#) f2 ) . x0 = lim ( f1 , x0 ) ;
cluster seq + ( - seq ) -> summable ;
assume j in dom ( M1 /. i ) ;
let A , B , C be Subset of X ;
let x , y , z be Point of X , p be Point of X ;
b ^2 - ( 4 * a * c ) >= 0 ;
<* xy *> ^ <* xy *> \subseteq x ;
a , b in { a , b } ;
len p2 is Element of NAT & len p2 = len q2 ;
ex x being element st x in dom R ;
len q = len ( K (#) G ) ;
s1 = Initialize Initialized s , P1 = P +* I +* I +* J ;
consider w being Nat such that q = z + w ;
x ` ` is being Element of L & x ` is ` ;
k = 0 & n <> k or k > n ;
then X is discrete for A is closed ;
for x st x in L holds x is FinSequence
||. f /. c .|| <= r1 & ||. f /. c .|| <= r2 ;
c in uparrow p & not c in { p } ;
reconsider V = V as Subset of the topology of TOP-REAL n ;
let N , M be \mathbin { ts } in L ;
then z is_>=_than waybelow x & z is_>=_than compactbelow x ;
M \lbrack f , g .] = f & M \lbrack g , g .] = g ;
( ( ( ( L~ z ) --> 1 ) ) /. 1 = TRUE ;
dom g = dom f |^ X & dom g = X ;
mode : is \cal \mathfrak st G is \cal .. for W being Walk of G holds W is Walk ;
[ i , j ] in Indices ( M * ( i , j ) ) ;
reconsider s = x " , t = y " as Element of H ;
let f be Element of dom ( Subformulae p ) ;
F1 . ( a1 , - a2 ) = G1 . ( a2 , - a2 ) ;
redefine func being non empty TopSpace , a , b , r be Real ;
let a , b , c , d be Real ;
rng s c= dom ( 1 / ( f1 + f2 ) ) ;
curry ( F-19 , k ) is additive ;
set k2 = card ( dom B ) , b = card ( B ) ;
set G = DTConMSA ( X ) ;
reconsider a = [ x , s ] as object of G ;
let a , b be Element of MM , x be set ;
reconsider s1 = s , s2 = t as Element of ( S ) * ;
rng p c= the carrier of L & p . 1 = p . 2 ;
let d be Subset of the Sorts of A ;
( x .|. x ) = 0 iff x = 0. W ;
I-21 in dom stop I & I-21 in dom stop I ;
let g be continuous Function of X | B , Y ;
reconsider D = Y as Subset of ( TOP-REAL n ) | K1 ;
reconsider i0 = len p1 , j1 = len p2 as Integer ;
dom f = the carrier of S & dom g = the carrier of S ;
rng h c= union ( ( the carrier of J ) --> 1 )
cluster All ( x , H ) -> Carrier <* x *> -> reconsider reconsider p *> ;
d * N1 ^2 > N1 * 1 / ( 1 - d ) ;
]. a , b .[ c= [. a , b .] ;
set g = f " ( D1 \/ D2 ) ;
dom ( p | ( NAT ) ) = NAT ;
3 + - 2 <= k + - 2 ;
tan is_differentiable_in ( ( arccot * f ) `| Z ) . x ;
x in rng ( f /^ ( n -' p ) ) ;
let f , g be FinSequence of D ;
[: p , q :] in the carrier of [: S1 , S2 :] ;
rng f " { 0 } = dom f ;
( the Source of G ) . e = v & ( the Target of G ) . e = v ;
( G * ( i , 1 ) ) `1 < G * ( i + 1 , 1 ) `1 ;
assume v in rng ( S | E1 ) ;
assume x is root or x is root or x is root or x is root ;
assume 0 in rng ( g2 | A ) & 0 < r ;
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 carrier of C-20 ( C ) ;
i <= len G -' 1 + ( i -' 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 ) .= Q " ( Q ) ;
( ( 1 / 2 ) (#) ( 1 / 2 ) ) is summable ;
- p + I c= - p + A ;
n < LifeSpan ( P1 , s1 ) + 1 ;
CurInstr ( p1 , s1 ) = i & IC s2 = 0 ;
A /\ Cl { x } \ { x } <> {} ;
rng f c= ]. r , r + 1 .[ ;
let g be Function of S , V ;
let f be Function of L1 , L2 , g be Function ;
reconsider z = z , y = y as Element of ( 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 of C , a be object of A ;
reconsider V1 = V , V1 = V as Subset of X | B ;
attr p is valid means : Def1 : All ( x , p ) is valid ;
assume that X c= dom f and f .: X c= dom g ;
H |^ a " is Subgroup of H |^ a ;
let A1 be Let of O , E1 , E be set ;
p2 , r3 , q3 is_collinear & q2 , q2 , q3 is_collinear ;
consider x being element such that x in v ^ K ;
not x in { 0. TOP-REAL 2 } & x in { 0. TOP-REAL 2 } ;
p in [#] ( I[01] | B11 ) & p in [#] ( I[01] | B11 ) ;
0 in M . ( E8 ) ;
^ ( c , c ) / ( c , d ) = c ;
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . s2 ) & a2 in dom ( F . s2 ) ;
cluster -> *> -| for LATTICE of L ;
set i1 = the Nat , i2 = the Element of NAT ;
let s be 0 -started State of SCM+FSA , k be Nat ;
assume y in ( f1 \/ f2 ) .: A ;
f . ( len f ) = f /. ( len f ) ;
x , f . x '||' f . x , f . y ;
pred X c= Y means : Def1 : cos ( X ) c= cos ( Y ) ;
let y be upper Subset of Y , x be set ;
cluster -> non empty for non empty Nat ;
set S = <* Bags n , i9 *> , T = <* i *> , S = <* i *> , T = T ;
set T = [. 0 , 1 / 2 .] ;
1 in dom mid ( f , 1 , 1 ) ;
( 4 * PI ) / ( 2 * PI ) < ( 2 * PI ) / ( 2 * PI ) ;
x2 in dom f1 /\ dom f & x2 in dom f1 /\ dom f ;
O c= dom I & { {} } = { {} } ;
( the Source of G ) . x = v & ( the Target of G ) . x = v ;
{ HT ( f , T ) } c= Support f ;
reconsider h = R . k as Polynomial of n , L ;
ex b being Element of G st y = b * H ;
let x , y , z be Element of G ` , a , b be Element of G ;
h19 . i = f . ( h . i ) ;
( p `1 ) ^2 / ( 1 + ( p `1 / p `2 ) ^2 ) = ( p `1 ) ^2 / ( 1 + ( p `2 / p `1 ) ^2 ) ;
i + 1 <= len Cage ( C , n ) ;
len <* P *> ^ <* Q *> = len P ;
set N-26 = the max of N , N-26 = the max of N ;
len gSet + ( x + 1 ) - ( x + 1 ) <= x ;
a on B & b on B implies a on B
reconsider r-12 = r * I . v , rc = r * I . v as FinSequence ;
consider d such that x = d and a [= d ;
given u such that u in W and x = v + u ;
len f /. ( \downharpoonright n ) = len \mathbb n ;
set q2 = ( N-min C ) `1 , q2 = ( E-max C ) `1 , q2 = ( E-max C ) `2 ;
set S =
MaxADSet ( b ) c= MaxADSet ( P ) /\ Q ;
Cl ( G . q1 ) c= F . r2 ;
f " D meets h " V & f " V is open ;
reconsider D = E , E = F as non empty directed Subset of L1 ;
H = ( the_left_argument_of H ) '&' ( the_right_argument_of H ) ;
assume t is Element of ( gF ) . ( X , S ) ;
rng f c= the carrier of S2 & rng f c= the carrier of S2 ;
consider y being Element of X such that x = { y } ;
f1 . ( a1 , b1 ) = b1 . ( b2 , b2 ) ;
the carrier' of G ` = E \/ { E } ;
reconsider m = len Int k - 1 as Element of NAT ;
set S1 = LSeg ( n , LMP C ) , S2 = LSeg ( n , LMP C ) ;
[ i , j ] in Indices ( M1 + M2 ) ;
assume that P c= Seg m and M is \HM { i } 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 * L /. 1 .= p * L /. 1 ;
pp . i = pp . i .= ( f . i ) `1 ;
let PA , PA , G be a_partition of Y , a be set ;
pred 0 < r & 1 < 1 & r < 1 ;
rng ( AffineMap ( a , X ) ) = [#] X ;
reconsider x = x , y = y , z = z 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 .= card ( s ) ;
reconsider x2 = x1 , y2 = x2 , z2 = y2 , z2 = z2 as Element of L2 ;
Q in FinMeetCl ( ( the topology of X ) \ { 0 } ) ;
dom ( f . 0 ) c= dom ( u . 0 ) ;
pred n divides m & m divides n implies n = m ;
reconsider x = x , y = y as Point of I[01] , z = z ;
a in dom ( ;
not y0 in the carrier of ( f . 0 ) & not ( ex x st x in the carrier of ( f . 0 ) ) ;
Hom ( ( a ~ ) , c ) <> {} ;
consider k1 such that p " < k1 and k1 < len f and f . k1 = f . k1 ;
consider c , d such that dom f = c \ d ;
[ x , y ] in dom g ~ & [ y , x ] in dom k ;
set S1 = Let ( x , y , z ) , S2 = y , S2 = z ;
l1 = m2 & l2 = i2 & l2 = i2 & l2 = i1 & l2 = i2 implies l1 = i2
x0 in dom ( u01 /\ A ) & x0 in dom ( u01 /\ A ) ;
reconsider p = x , q = y , r = z as Point of TOP-REAL 2 ;
I[01] = R^1 | ( ( R^1 | B01 ) | B01 ) .= R^1 | B01 ;
f . p4 <= _ { P . p1 , f . p2 } ;
( F . x ) `1 <= ( x `1 ) / ( 1 + x `2 ) ;
( x `2 ) ^2 = ( seq1 . i ) ^2 + ( seq2 . i ) ^2 ;
for n being Element of NAT holds P [ n ] ;
let J , K be non empty Subset of I ;
assume 1 <= i & i <= len <* a " *> ;
0 |-> a = <*> ( the carrier of K ) ;
X . i in 2 -tuples_on A . i \ B . i ;
<* 0 *> in dom ( e --> [ 1 , 0 ] ) ;
then P [ a ] implies P [ succ a ] ;
reconsider s\overline { s } = s\overline { s } as terminal of D ;
( - i -' 1 ) <= ( len wj ) - 1 ;
[#] S c= [#] ( T ) & [#] T c= [#] ( T ) ;
for V being strict real unitary space holds V in the carrier of V implies V is open
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 , n be Nat ;
- a * ( - b ) = a * b ;
for A being Subset of GX holds A // A implies A is being_line
( for o2 being object of o2 st o2 in <^ o2 , o2 ^> holds ( id o1 ) * ( id o2 ) = id o1
then ||. x .|| = 0 & x = 0. X ;
let N1 , N2 be strict normal Subgroup of G ;
j >= len upper_volume ( g , D1 ) - len upper_volume ( g , D2 ) ;
b = Q . ( len Q - 1 + 1 ) .= Q . ( len Q - 1 ) ;
f2 * f1 /* s is divergent_to+infty & f2 * ( f1 /* s ) is divergent_to+infty ;
reconsider h = f * g as Function of N4 , G ;
assume that a <> 0 and delta ( a , b , c ) >= 0 ;
[ t , t ] in the InternalRel of A & [ t , t ] in the InternalRel of A ;
( v |-- E ) | n is Element of T7 ;
{} = the carrier of L1 + L2 & the carrier of L1 = the carrier of L2 ;
Directed I is_closed_on Initialized s , P & I is_halting_on Initialized s , P ;
Initialized p = Initialize ( p +* q ) .= p +* ( q +* ( p +* q ) ) ;
reconsider N2 = N1 , N2 = N2 as strict net of R2 ;
reconsider Y = Y as Element of \langle Ids L , \subseteq \rangle ;
"/\" ( uparrow p , L \ { p } ) <> 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 ( Q ^ R ) ;
( x1 `1 ) ^2 = ( x2 `1 ) ^2 + ( x1 `2 ) ^2 ;
InputVertices S = { x1 , x2 , x3 , x4 , x5 , x5 , 7 } ;
let x , y be Element of [: FTTTTT1 ( n ) , k :] ;
p = |[ p `1 , p `2 ]| & p `1 = p `1 ;
g * 1_ G = h " * g * h .= g * h ;
let p , q be Element of PFuncs ( V , C ) ;
x0 in dom ( x1 - x2 ) /\ dom ( x1 - x2 ) ;
( R qua Function ) " = R " ( R " ( R " ( R ) ) ) ;
n in Seg ( len ( f /^ ( i -' 1 ) ) ) ;
for s being Real st s in R holds s <= s2 & s2 <= 1
rng s c= dom ( f2 * f1 ) /\ dom f2 ;
synonym for for for for X is Subset of consider 2 , n st X is finite holds X is finite ;
1. K * ( 1. K , 1 ) = 1. K ;
set S = Segm ( A , P1 , Q1 ) , T = Segm ( A , Q1 , Q1 ) ;
ex w st e = ( w - f ) / ( w - f ) & w in F ;
curry ( P+* ( i , k ) ) # x is convergent ;
cluster open for Subset of TY | the carrier of T ;
len f1 = 1 .= len f3 .= len f3 + 1 .= len f3 + 1 ;
( i * p ) / p < ( 2 * p ) / p ;
let x , y be Element of ( OSSub ( U0 ) ) . s ;
b1 , c1 // b9 , c9 & b2 , c1 // b9 , c9 ;
consider p being element such that c1 . j = { p } ;
assume that f " { 0 } = {} and f is total ;
assume that IC Comput ( F , s , k ) = n and IC Comput ( F , s , k ) in dom I ;
Reloc ( J , card I ) does not destroy a ;
( goto ( card I + 1 ) ) not c in dom ( i .--> c ) ;
set m3 = LifeSpan ( p3 , s3 ) , P4 = LifeSpan ( p2 , s3 ) , P4 = LifeSpan ( p2 , s3 ) ;
IC SCMPDS in dom ( Initialize p ) & IC SCMPDS in dom ( Initialize p ) ;
dom t = the carrier of SCM & dom t = the carrier of SCM R ;
( ( ( ( f /. 1 ) ) .. f ) .. f ) .. f = 1 ;
let a , b be Element of PFuncs ( V , C ) ;
Cl ( union Int ( F ) ) c= Cl Int ( union F ) ;
the carrier of X1 union X2 misses ( A \/ B ) ;
assume not LIN a , f . a , g . a , g . b ;
consider i being Element of M such that i = d6 and i in dom f ;
then Y c= { x } or Y = { x } ;
M , v |= H1 / ( ( y , x ) / ( y , x ) ) ;
consider m being element such that m in Intersect ( F0 . i ) ;
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 a3 <> a4 ;
cluster s -\mathop { \rm \hbox { - } IC } ( V , p ) -> .| for string of S ;
Ln2 /. n2 = Ln2 . n2 .= Ln2 . n2 .= Ln2 . 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 rp2 in LSeg ( p1 , p2 ) ;
let A be non empty compact Subset of TOP-REAL n , a be Real ;
assume [ k , m ] in Indices ( D * ( i , j ) ) ;
0 <= ( ( 1 / 2 ) |^ p ) / ( 1 / 2 ) ;
( F . N ) | E8 ( ) . x = +infty ;
pred X c= Y & Z c= V & X \ V c= Y \ Z ;
( y `2 ) ^2 * ( z `2 ) ^2 <> 0. I ;
1 + card ( X-18 ) <= card ( u \/ { x } ) ;
set g = z \circlearrowleft ( ( L~ z ) .. z ) , z = ( L~ z ) .. z , z = ( L~ z ) .. z , w = ( L~ z ) .. z , i = ( len z ) .. z , j = ( len z )
then k = 1 & p . k = <* x , y *> . k ;
cluster -> total for Element of C -\mathop { \rm \hbox { - } :] ;
reconsider B = A as non empty Subset of ( TOP-REAL n ) | ( REAL n ) ;
let a , b , c be Function of Y , BOOLEAN , p be Function of Y , BOOLEAN ;
L1 . i = ( i .--> g ) . i .= g . i ;
Plane ( x1 , x2 , x3 , x4 ) c= P & Line ( x1 , x2 , x3 ) c= P ;
n <= indx ( D2 , D1 , j1 ) + 1 - 1 ;
( ( g2 ) . O ) `1 = - 1 & ( g2 ) . I = 1 ;
j + p .. f -' len f <= len f - 1 + p .. f ;
set W = W-bound C , E = E-bound C , S = E-bound C , N = E-bound C , S = E-bound C , N = E-bound C , S = E-bound C , N = E-bound C , S = E-bound C , N = E-bound C , S = E-bound
S1 . ( a , e ) = a + e .= a + e ;
1 in Seg ( width ( M * ( ColVec2Mx p ) ) ) ;
dom ( i (#) Im ( f ) ) = dom Im ( f ) ;
Dx `1 = W . ( a , *' ( a , p ) ) ;
set Q = / ( |= ( g , f , h ) ) ;
cluster -> topological for ManySortedSet of U1 , A be non-empty MSAlgebra over U1 ;
attr F = { A } means : Def1 : F is discrete ;
reconsider z9 = <* : x , y *> in product ( G ) ;
rng f c= rng f1 \/ rng f2 & f . 0 = f . 1 \/ f . 2 ;
consider x such that x in f .: A and x in f .: C ;
f = <*> ( the carrier of F_Complex ) & g = <*> ( the carrier of F_Complex ) ;
E , j |= All ( x1 , x2 , H ) / ( x1 , x2 ) ;
reconsider n1 = n , n2 = m , n2 = n as Morphism of o1 , o2 ;
assume that P is idempotent and R is idempotent and P (*) R = R ** P ;
card ( B2 \/ { x } ) = k-1 + 1 .= card B2 + 1 ;
card ( ( x \ B1 ) /\ B1 ) = 0 ;
g + R in { s : g-r < s & s < g + r } ;
set qI12 = ( q , <* s *> ) -\subseteq ( q , <* s *> ) -\subseteq ( q , <* s *> ) -\subseteq ( q , <* s *> ) -\subseteq ( q , <* s *> ) -\subseteq ( q , <*
for x being element st x in X holds x in rng f1 implies x in X
h0 /. ( i + 1 ) = h0 . ( i + 1 ) ;
set mw = max ( B , min ( NAT , { NAT } ) ) ;
t in Seg ( ( width ( I ^ J ) ) , ( len ( I ^ J ) ) ) ;
reconsider X = dom f , Y = C as Element of Fin NAT ;
IncAddr ( i , k ) = <% - l %> + k .= i ;
( ( q `1 ) / |. q .| ) ^2 <= ( q `1 ) ^2 / ( |. q .| ) ^2 ;
attr R is condensed means : Def1 : Int R is condensed & Cl R is condensed ;
pred 0 <= a & b <= 1 & a <= 1 implies a * b <= 1 ;
u in ( ( ( c /\ ( ( d /\ b ) /\ e ) ) /\ f ) /\ j ) /\ j ;
u in ( ( ( c /\ ( ( d /\ e ) /\ b ) ) /\ f ) /\ j ) /\ j ;
len C + - 2 >= 9 + - 3 + 3 ;
x , z , y is_collinear & x , z , x is_collinear implies x = y
a |^ ( n1 + 1 ) = a |^ ( n1 + 1 ) * a ;
<* \underbrace ( 0 , \dots , 0 ) *> in Line ( x , a * x ) ;
set y9 = <* y , c *> , z9 = <* c , x *> ;
FF2 /. 1 in rng Line ( D , 1 ) & FF2 /. 1 in rng Line ( D , 1 ) ;
p . m Joins r /. m , r /. ( m + 1 ) , G ;
( p `2 ) ^2 = ( f /. i1 ) ^2 .= ( f /. i1 ) ^2 ;
( W-bound X \/ Y ) `1 = W-bound X & ( W-bound X ) `1 = E-bound 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 ) = f2 . ( seq ^\ k ) ;
reconsider u2 = u , v2 = v , u2 = w as VECTOR of \llangle X , Y :] ;
p |-count ( Product ( Sgm ( X ) ) ) = 0 & p |-count ( ( Sgm ( X ) ) . p ) = 1 ;
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 ) ) ;
x in { x , y } & h . x = {} ( T . x ) ;
consider y being Element of F such that y in B and y <= x ` ;
len S = len ( the charact of ( A ) ) .= len the charact of ( A ) ;
reconsider m = M , i = I , n = N as Element of X ;
A . ( j + 1 ) = B . ( j + 1 ) \/ A . j ;
set N8 = : ( G . e ) `1 = ( G . e ) `1 ;
rng F c= the carrier of gr ( { a } , gr { a } ) ;
implies for n being Nat holds holds holds ( Q . ( K , n , r ) ) is in of D
f . k , f . ( f . ( Let n ) ) in rng f ;
h " P /\ [#] T1 = f " P /\ [#] T1 .= f " P ;
g in dom ( f2 \ f2 " { 0 } ) \ ( f2 " { 0 } ) ;
g1X /\ dom f1 = g1 " ( f " X ) .= ( f " X ) /\ ( 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 ) ;
b `2 + ( 1 - r ) < ( 1 - r ) / 2 + ( 1 - r ) / 2 ;
reconsider f1 = f , f2 = g as VECTOR of the carrier of X , Y ;
pred i <> 0 means : Def1 : i ^2 mod ( i + 1 ) = 1 ;
j2 in Seg ( len ( g2 . i2 ) ) & 1 in Seg ( len g2 ) ;
dom ( i - 1 ) = dom ( i - 2 ) .= a ;
cluster sec | ]. PI / 2 , PI / 2 .[ -> one-to-one ;
Ball ( u , e ) = Ball ( f . p , e ) ;
reconsider x1 = x0 , y1 = y0 , y2 = y1 , y2 = y2 as Function of S , T ;
reconsider R1 = x , R2 = y , R1 = z , R2 = t as Relation of L ;
consider a , b being Subset of A such that x = [ a , b ] ;
( <* 1 *> ^ p ) ^ <* n *> in Rn ;
S1 +* S2 = S2 +* ( S2 +* S1 +* S2 , S2 +* S2 ) ;
( ( exp_R * ( exp_R + cot ) ) `| Z ) = f ;
cluster -> C -valued for Function of C , REAL ;
set C7 = 1GateCircStr ( <* z , x *> , f3 ) , C8 = 1GateCircStr ( <* z , x *> , f3 ) ;
E8 . e2 = ( ( e2 . e2 ) -T . e2 ) . e2 .= T . e2 ;
( ( ( id Z ) (#) ( ln * f ) ) `| Z ) = f ;
upper_bound A = ( cos * 3 ) / 2 & lower_bound A = 0 ;
F . ( dom f , - ( f . ( cod f ) ) ) is_transformable_to F . ( cod f , - ( f . ( cod f ) ) ) ;
reconsider pp = q8 , q8 = q , q8 = r as Point of TOP-REAL 2 ;
g . W in [#] ( Y | ( the carrier of Y ) ) & [#] ( Y | ( the carrier of Y ) ) c= [#] ( Y | ( the carrier of Y ) ) ;
let C be compact non vertical non vertical non horizontal Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
LSeg ( f ^ g , j ) = LSeg ( f , j ) ;
rng s c= dom f /\ ]. - r , x0 .[ & f . x0 in dom f /\ ]. x0 , x0 + r .[ ;
assume x in { ( idseq 2 ) . ( ( idseq 2 ) . ( len ( idseq 2 ) ) ) } ;
reconsider n2 = n , m2 = m , m2 = n , m2 = m , n1 = n , n2 = m , n2 = n , n2 = m ;
for y being ExtReal st y in rng seq holds g . y <= y
for k st P [ k ] holds P [ k + 1 ]
m = m1 + m2 .= m1 + m2 .= m1 + m2 .= m1 + m2 ;
assume for n holds H1 . n = G . n -H . n ;
set B9 = f .: ( the carrier of X1 ) , B9 = f .: the carrier of X2 ;
ex d being Element of L st d in D & x << d ;
assume R ~ c= R ~ & R ~ c= R ~ & R ~ c= R ~ & R ~ c= R ~ ;
t in ]. r , s .[ or t = r or t = s ;
z + v2 in W & x = u + ( z + v2 ) ;
x2 |-- y2 iff P [ x2 , y2 ] & P [ x2 , y2 ] ;
pred x1 <> x2 means : Def1 : |. x1 - x2 .| > 0 & |. x1 - x2 .| > 0 ;
assume p2 - p1 , p3 - p1 - p1 , p2 - p1 is_collinear & p2 - p1 , p3 - p1 is_collinear ;
set q = ( -1 f ) ^ <* 'not' 'not' 'not' A *> ;
let f be PartFunc of REAL-NS 1 , REAL-NS n , x be Point of REAL-NS 1 , r be Real ;
( n mod ( 2 * k ) ) ^2 = n mod k ;
dom ( T * ( \mathbin { ^ \smallfrown } \langle t *> ) ) = dom ( \mathop { \rm \rbrack } t , dom <* t *> ) ;
consider x being element such that x in wc and x in c ;
assume ( F * G ) . v = v . x3 ;
assume that the carrier of D1 c= the carrier of D2 and for i being Nat holds [ i , j ] in the carrier of D2 iff [ i , j ] in the carrier of D2 ;
reconsider A1 = [. a , b .[ , A2 = [. a , b .] as Subset of R^1 ;
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 ) ) .. Cage ( C , n ) ;
n1 -' len f + 1 <= len - len f + 1 + 1 - 1 ;
.| ( |. q , O1 , a , b , a , b , a , b , b ) = [ u , v , a , b ] ;
set C-2 = ( \mathclose { \overline { G } } ) . ( k + 1 ) ;
Sum ( L (#) p ) = 0. R * Sum p .= 0. R ;
consider i being element such that i in dom p and t = p . i ;
defpred Q [ Nat ] means 0 = Q ( $1 ) & 1 <= $1 & $1 <= len Q ;
set s3 = Comput ( P1 , s1 , k ) , P3 = Comput ( P2 , s2 , k ) , P4 = Comput ( P2 , s2 , k ) ;
let l be variable of k , A , A1 , A2 be Subset of k ;
reconsider U2 = union ( G . n ) , G-24 = union ( G . n ) as Subset-Family of [: T , T :] ;
consider r such that r > 0 and Ball ( p `1 , r ) c= Q ` ;
( h | ( n + 2 ) ) /. ( i + 1 ) = p29 ;
reconsider B = the carrier of X1 , C = the carrier of X2 , D = the carrier of X2 , E = the carrier of X1 , F = the carrier of X2 , G = the carrier of X2 , G = the carrier of X2 , H = the carrier of X1 ;
p[#] L = <* - ( c - 1 ) , 1 , 1 *> .= <* - ( c - 1 ) , 1 *> ;
synonym f is real-valued means : Def1 : rng f c= NAT & rng f c= NAT ;
consider b being element such that b in dom F and a = F . b ;
x20 < card ( ( X0 \/ ( X0 \/ Y2 ) ) ) + card ( ( X0 \/ Y2 ) \/ ( Y1 \/ Y2 ) ) ;
pred X c= B1 means : Def1 : for B st X c= B holds \mathop { \rm len } \mathop { \rm lpst X c= B ;
then w in Ball ( x , r ) & dist ( x , w ) <= r ;
angle ( x , y , z ) = angle ( x , 0 , 0 ) ;
pred 1 <= len s means : Def1 : len ( s * s ) = 0 ;
fReconsider fK1 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 : Def1 : q '&' p in TAUT ( A ) & q in TAUT ( A ) ;
- ( t `1 / t `2 ) < ( - t `1 ) / t `2 / t `2 ;
( U . 1 ) `1 = ( U /. 1 ) `1 .= ( W /. 1 ) `1 .= ( W /. 1 ) `1 ;
f .: ( the carrier of x ) = the carrier of x & f . ( the carrier of x ) = the carrier of x ;
Indices ( O * ( i , j ) ) = [: Seg n , Seg n :] & len ( O * ( i , j ) ) = 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 \setminus & f is \setminus & f is \setminus & f is \setminus & f is \setminus implies f is \setminus & f is \setminus & f is \setminus & f is \setminus & f is \setminus & f is \setminus & f is \setminus implies f is \setminus & f is \setminus & f
[ 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 ]| `1 ^2 - ( w1 `1 ) ^2 <> 0. TOP-REAL 2 ;
reconsider t = t , s = ( t , 1 ) --> ( t , 1 ) as Element of INT ;
C \/ P c= [#] ( ( ( G | [#] ( ( ( G | A ) \ A ) ) \ A ) ) ;
f " V in ( the topology of X ) /\ D & D c= ( the topology of X ) /\ D ;
x in [#] ( ( the carrier of Y ) /\ A ) /\ ( the carrier of Y ) ;
g . x <= h1 . x & h . x <= h1 . x ;
InputVertices S = { xy , y9 , z9 , z9 } & InputVertices S = { xy , yz , z9 } ;
for n being Nat st P [ n ] holds P [ n + 1 ]
set R = ( Line ( M , i ) * ( a , i ) ) ;
assume that M1 is being_line and M2 is being_line and M3 is being_line and M3 is being_line and M3 is being_line ;
reconsider a = f4 . ( i0 -' 1 ) , b = ( f /. ( i0 -' 1 ) ) as Element of K ;
len B2 = Sum ( Len ( F1 ^ F2 ) ) .= len ( ( len F1 + len F2 ) ) ;
len ( ( the \it of n ) --> i ) = n & i in Seg n & j in Seg n implies ( ( n + 1 ) |-> i ) = n ;
dom ( f + g ) = dom ( f + g ) .= dom f /\ dom g ;
( ( the Sorts of Y ) * ( n + 1 ) ) . x = upper_bound Y1 & ( the Sorts of Y ) . x = upper_bound Y1 ;
dom ( p1 ^ p2 ) = dom ( f ^ p2 ) .= dom ( f ^ p2 ) ;
M . [ 1 / y , y ] = 1 / ( 1 / y ) .= y ;
assume that W is non trivial and W .last() c= the \frac of G2 and W .last() c= the \frac { 2 } , 2 } ;
C6 /. i1 = G1 * ( i1 , i2 ) & C6 /. ( i1 + 1 ) = G2 * ( i1 , i2 ) ;
C8 |- 'not' Ex ( x , p ) 'or' p . ( x , y ) ;
for b st b in rng g holds lower_bound rng fmmmmb <= b / 2
- ( ( q1 `1 / |. q1 .| - cn ) / ( 1 + cn ) ) = 1 ;
( LSeg ( c , m ) \/ [: l , k :] ) \/ LSeg ( l , k ) c= R ;
consider p being element such that p in { x } and p in L~ f and x = f . p ;
Indices ( X @ ) = [: Seg n , Seg n :] & len ( X @ ) = n ;
cluster s => ( q => p ) -> valid ( s => ( s => p ) ) ;
Im ( ( Partial_Sums ( F ) ) . m ) is_measurable_on E ;
cluster f . ( x1 , x2 ) -> Element of D ( ) ;
consider g being Function such that g = F . t and Q [ t , g ] ;
p in LSeg ( ( NW-corner L~ f ) , ( E-max L~ f ) .. f ) /\ LSeg ( ( NW-corner L~ f ) , ( E-max L~ f ) .. f ) ;
set R8 = R / ( 1 / ( b + r ) ) ;
IncAddr ( I , k ) = SubFrom ( da , db ) .= ( - ( d + k ) ) ;
seq . m <= ( ( the Sorts of seq ) . k ) . ( ( seq . k ) + ( seq . k ) ) ;
a + b = ( a ` *' ) ` + ( b ` *' ) ` .= ( a ` ) ` + ( b ` *' ) ;
id ( X /\ Y ) = id ( X /\ Y ) .= id X ;
for x being element st x in dom h holds h . x = f . x ;
reconsider H = U1 \/ U2 , U1 = ( U1 \/ U2 ) /\ ( U2 /\ U1 ) 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 , inf B ] ;
consider A being finite stable Subset of R such that card A = ( R * ) + 1 and A is finite ;
p2 in rng ( f |-- p1 ) \ rng <* p1 *> & p2 in rng f \ { p1 } ;
len s1 - 1 > 0 & len s2 - 1 > 0 & len s2 - 1 > 0 ;
( ( ( ( NW-corner ( P ) ) ) `2 ) ^2 = ( ( E-max ( P ) ) ) ^2 ;
Ball ( e , r ) c= LeftComp Cage ( C , k + 1 ) ;
f . a1 ` ` ` = f . a1 ` ` .= ( f . a1 ` ) ` .= ( f . a1 ) ` ;
( seq ^\ k ) . n in ]. - r , x0 .[ & ( seq ^\ k ) . n in ]. x0 , x0 + r .[ ;
gg . s0 = g . s0 .= G . ( G . s0 ) .= G . ( s . m ) ;
the InternalRel of S is symmetric & the InternalRel of S is transitive implies the InternalRel of S is transitive
deffunc F ( Ordinal , Ordinal ) = phi ( $1 , $2 ) & phi ( $1 , $2 ) = phi ( $1 , $2 ) ;
F . a1 = F . a2 & F . a2 = F . a1 & F . a2 = F . a2 ;
x `2 = A . o . a .= Den ( o , A . a ) ;
Cl ( f " P1 ) c= f " ( Cl P1 ) & f " ( Cl P1 ) c= f " ( Cl P1 ) ;
FinMeetCl ( ( the topology of S ) . i ) c= the topology of T & the topology of S c= the topology of T ;
synonym o is OperSymbol means : Def1 : o <> *' & o <> * ;
assume that X |^ n = Y |^ n + card X and card X <> card Y ;
the finite Subset of S <= 1 + ( the *> ^ the *> ) + ( the *> ^ the *> ^ the card of F ) ;
LIN a , a1 , d or b , c // b1 , c1 or b , c // b1 , c1 ;
e / 2 . 1 = 0 & e / 2 . 2 = 1 & e / 2 . 3 = 0 ;
EE in SE & EE in { N } implies not EE in { N }
set J = ( l , u ) If , K = I ;
set A1 = ( a1 , a2 , a3 , a4 , a5 , a5 , 8 , 9 , 8 , 8 , 9 } ) ;
set vs = [ <* c , d *> , '&' ] , xy = [ <* d , c *> , '&' ] , yz = [ <* c , d *> , '&' ] , \mathopen = [ <* d , e *> , '&' ] , e = [ <* c , e *> , '&' ] , ] , ] = [ <* d , e *> , '&' ] ,
x * z `2 * x " in x * ( z * N ) * x " ;
for x being element st x in dom f holds f . x = g2 . x & f . x = g2 . x ;
Int cell ( f , 1 , G ) c= RightComp f \/ RightComp f \/ L~ f \/ L~ f \/ L~ f ;
U2 is_an_arc_of W-min C , W-min C , W-min C , W-min C , W-min C , W-min C , W-min C , W-min C ;
set f-17 = f @ "/\" ( g @ ) ;
attr S1 is convergent & S2 is convergent & S2 is convergent implies S1 - S2 is convergent & lim ( S1 - S2 ) = 0 ;
f . ( 0 + 1 ) = ( 0 qua Ordinal ) + a .= a + a .= a ;
cluster -> } -be reflexive transitive transitive transitive non empty transitive for RelStr ;
consider d being element such that R reduces b , d and R reduces c , d ;
not b in dom Start-At ( ( card I + 2 ) , SCMPDS ) \/ dom Start-At ( ( card I + 2 ) , SCMPDS ) ;
( z + a ) + x = z + ( a + y ) .= z + a + y ;
len ( l \lbrack ( a |^ 0 ) * x , x \rbrack ) = len l ;
t4 is ( {} \/ rng t4 ) -valued finite Function of ( {} \/ rng t4 ) , ( {} \/ rng t4 ) * ) * \ ( {} , { {} } ) * \ { {} } is empty * \ { {} } ;
t = <* F . t *> ^ ( C . p ^ q ) .= C . ( p ^ q ) ;
set pp = ( W-min L~ Cage ( C , n ) ) .. Cage ( C , n ) ;
( k -' ( i + 1 ) ) = ( k - ( i + 1 ) ) + ( i + 1 ) ;
consider u being Element of L such that u = u ` ` and u in D ` and u in D ` ;
len ( ( ( ( ( G . ( i + 1 ) ) ) |-> a ) ^ ( ( G . ( i + 1 ) ) --> a ) ) ) = len ( ( G . i ) --> a ) ;
FM . x in dom ( ( G * the_arity_of o ) . x ) ;
set cH2 = the carrier of H2 , cH2 = the carrier of H2 ;
set cH1 = the carrier of H1 , cH2 = the carrier of H2 ;
( Comput ( P , s , 6 ) ) . intpos ( m + 6 ) = s . intpos ( m + 6 ) ;
IC Comput ( Q2 , t , k ) = ( l + 1 ) + 1 .= ( l + 1 ) ;
dom ( ( ( id Z ) (#) ( ( id Z ) ^ ) ) `| Z ) = REAL & dom ( ( id Z ) ^ ) = REAL ;
cluster <* l *> ^ phi -> ( 1 + ( not not not not ( not ( not l is ( not m ) ) ) ) string of S ) for string of S ;
set b9 = [ <* ap , bm *> , and2 ] , c9 = [ <* A1 , cin *> , and2 ] , z2 = [ <* cin , cin *> , and2 ] , e = [ <* A1 , cin *> , and2 ] , in = [ <* cin , cin *> , and2 ] , w = [ <* A1 , cin *> , and2 ] , e = [ <*
Line ( Line ( M , P ) , x ) = L * Sgm Q .= ( Sgm Q ) . x ;
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 T ;
consider y being Point of X such that a = y and ||. x-y .|| <= r ;
set x3 = t2 . DataLoc ( s2 . SBP , 2 ) , x4 = Comput ( s2 , s2 , 2 ) , x4 = Comput ( s2 , s2 , 2 ) , P4 = Comput ( P2 , s2 , 2 ) , P4 = P3 ;
set pp = stop I , p1 = P +* I , p2 = P +* 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 } ;
let A , B , C , D , E , F , J , M , N , M , N , N , F , M , N , N , F , M , N , N , F , M , N , N , M , N , N , F , M , N , N , F , M , N ,
|. p2 .| ^2 - ( p2 `1 ) ^2 / ( 1 + ( p2 `2 / p2 `1 ) ^2 ) >= 0 ;
l -' 1 + 1 = n-1 * ( ( mm + 1 ) + 1 ) ;
x = v + ( a * w1 + ( b * w2 ) ) + ( c * w2 ) ;
the TopStruct of L = reconsider the TopStruct of L , C = the Scott Scott Scott Scott Scott Scott Function of L , L ;
consider y being element such that y in dom H1 and x = H1 . y and y in dom H1 and x = H1 . y ;
fv \ { n } = Free ( All ( x , H ) ) & not n in Free ( All ( x , H ) ) ;
for Y being Subset of X st Y is summable & Y is summable holds Y is iff X is iff Y is iff X is iff Y is not empty
2 * n in { N : 2 * Sum ( p | N ) = N & N > 0 } ;
for s being FinSequence holds len ( the multF of A ) = len s & len ( the { F } ^ s ) = len s
for x st x in Z holds exp_R * f is_differentiable_in x & exp_R * f . x > 0
rng ( h2 * f2 ) c= the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 & rng ( h2 * f2 ) c= the carrier of ( ( TOP-REAL 2 ) | K1 ) ;
j + ( - len f ) <= len f + ( len f - len f ) - len f + ( len f - len f ) ;
reconsider R1 = R * I , R2 = R * I as PartFunc of REAL , REAL-NS n , REAL-NS n ;
C8 . x = s1 . ( a , a ) .= C8 . ( a , a ) .= C8 . ( a , a ) ;
power F_Complex . ( z , n ) = 1 .= x |^ n .= x |^ n .= x |^ n ;
t at ( C , s ) = f . ( ( the connectives of S ) . t ) & t . ( C , s ) = s . ( C , s ) ;
support ( f + g ) c= support f \/ C /\ ( support g ) & support ( f + g ) c= support f ;
ex N st N = j1 & 2 * Sum ( ( r | N ) | N ) > N & 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 & x2 in Y } is Subset of [: X1 , X2 :] ;
h = ( i , j ) |-- ( id B , id B ) . i .= H . i ;
ex x1 being Element of G st x1 = x & x1 * N c= A & N c= A ;
set X = ( ( |. ( .| ( O1 , O2 ) ) ) . ( O1 , 4 ) ) , Y = { [ ( |. O1 .| ) . ( O2 , 4 ) ] } ;
b . n in { g1 : x0 - r < g1 & g1 < x0 + r } ;
f /* s1 is convergent & f /. x0 = lim ( f /* s1 ) implies f /. x0 = lim ( f /* s1 )
the lattice of Y = the lattice of the lattice of Y & the carrier of Y = the carrier of the carrier of Y & the carrier of Y = the carrier of Y ;
'not' ( a . x ) '&' b . x 'or' a . x '&' 'not' ( b . x ) = FALSE ;
2 = ( len ( ( q ^ r1 ) ^ ( q ^ r1 ) ) ) + len ( ( q ^ r1 ) ^ ( q ^ r1 ) ) .= len ( q ^ r1 ) + len ( q ^ r1 ) ;
( 1 / a ) (#) ( ( sec * f1 ) - id Z ) is_differentiable_on Z ;
set K1 = ( upper ( lim ( H , A ) ) ) || ( H , A ) , D2 = ( lim ( H , A ) ) || ( H , A ) ;
assume that e in { ( w1 /. i ) `1 : w1 in F & w2 in G } and w1 in G and w2 in G ;
reconsider d7 = dom a `1 , d8 = dom F `1 , d8 = dom F `1 , d8 = dom G `1 , d8 = dom G `1 , d8 = dom G `1 , d8 = dom G `1 , d8 = G `1 , d8 = G `1 , d8 = G . ( i , j ) , d8 = G
LSeg ( f /^ j , j ) = LSeg ( f , j ) /\ LSeg ( f , j + q .. f -' 1 ) ;
assume X in { T . ( N2 , K ) : h . ( N2 , K ) = N2 } ;
assume that Hom ( d , c ) <> {} and <* f , g *> * f1 = <* f , g *> * f2 ;
dom SA2 = dom S /\ Seg n .= dom ( LA2 * ( S . n ) ) .= dom ( LA2 * ( S . n ) ) ;
x in H |^ a implies ex g st x = g |^ a & g in H |^ a & g in H |^ a
a * ( ( a , 1 ) to_power n ) = a `1 - ( 0 * n ) .= a `1 ;
D2 . j in { r : lower_bound A <= r & r <= ( f . i ) `1 } ;
ex p being Point of TOP-REAL 2 st p = x & P [ p ] & p `2 >= 0 ;
for c holds f . c <= g . c implies f ^ @ c c= g @ c
dom ( f1 (#) f2 ) /\ X c= dom ( f1 (#) f2 ) /\ X ;
1 = ( p * p ) / p .= p * ( 1 / p ) .= p * 1 / p .= 1 / p ;
len g = len f + len <* x + y *> .= len f + 1 .= len f + 1 + 1 .= len f + 1 ;
dom F-11 = dom ( F | ( N1 , S-23 ) ) .= [: N1 , { 0 } :] ;
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 g . 0 = g . 1 ;
( ( x \ y ) \ z ) \ ( ( x \ z ) \ ( y \ z ) ) = 0. X ;
consider f such that f * f `1 = id b and f `2 = id b and f `2 = id b ;
( ( ( id Z ) (#) cos ) | [. 2 , PI / 2 .] ) is increasing ;
Index ( p , co ) <= len LS - Gij .. LS + 1 .= len LS - Gij .. LS + 1 .= len LS -' 1 + 1 ;
let t1 , t2 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2
"/\" ( ( Frege ( curry G ) ) . h , h ) <= "/\" ( ( Frege ( curry G ) ) . h , h ) ;
then P [ f . ( i0 + 1 ) , f . ( i0 + 1 ) ] & F ( f . ( i0 + 1 ) , f . ( i0 + 1 ) ) < j ;
Q [ ( D . [ D . x , 1 ] ) , F . [ D . ( 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 Element of G . i ;
the Sorts of A2 = ( the carrier of S2 ) --> TRUE .= ( the carrier of S2 ) --> TRUE .= 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= F ;
dist ( b1 , b2 ) <= dist ( b1 , a ) + dist ( a , b2 ) + dist ( a , b2 ) ;
( ( Cage ( C , n ) ) /. len ( Cage ( C , n ) ) = ( W-min L~ Cage ( C , n ) ) ;
q `2 <= ( LMP ( Lower_Arc ( C ) ) ) `2 & ( LMP ( Lower_Arc ( C ) ) ) `2 <= ( LMP ( C ) ) `2 ;
LSeg ( f | i2 , i ) /\ LSeg ( f | i2 , j ) = {} ;
given a being ExtReal such that a <= IB and A = ]. a , IB .[ and B = ]. a , IB .] ;
consider a , b being complex number such that z = a & y = b and z + y = a + b ;
set X = { b |^ n where n is Element of NAT : n in dom b } , Y = { b |^ n where n is Element of NAT : n in dom b } ;
( ( x * y * z \ x ) \ z ) \ ( x * y \ x ) = 0. X ;
set xy = [ <* xy , yz , \mathopen { x } , { y } ] , yz = [ <* z , x *> , f1 ] , yz = [ <* y , z *> , f2 ] , yz = [ <* z , x *> , f3 ] , zx = [ <* x , y *> , f3 ] , zx = [ <* z , x *> , f3 ] , ] ;
lz /. len lz = ( lz /. len ( lz ^ ( i + 1 ) ) ) * ( ( i + 1 ) + 1 ) ;
( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 = 1 ;
( ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) ^2 < 1 ;
( ( ( ( ( ( X \/ Y ) \/ Y ) ) \/ Y ) /\ X ) ) `2 = ( ( ( X \/ Y ) \/ Y ) /\ X ) `2 ;
( s1 - s ) . k = ( s1 . k - s . k ) / ( 1 - s . k ) .= ( s1 . k - s . k ) / ( 1 - s . k ) ;
rng ( ( h + c ) ^\ n ) c= dom SVF1 ( 1 , f , u0 ) /\ dom SVF1 ( 1 , f , u0 ) ;
the carrier of X is the carrier of X & the carrier of X is the carrier of X implies X is non empty & X is non empty
ex p4 st p3 = p4 & |. p4 - |[ a , b ]| .| = r & |. p4 - |[ a , b ]| .| = r ;
set ch = chi ( X , A ) , A5 = chi ( X , A ) , A5 = ( X * ) ;
R to_power ( 0 * n ) = I----> Element of X * , X * ( 0 , 0 ) ;
( Partial_Sums ( curry ( f , n ) ) . 0 ) . n is nonnegative & ( Partial_Sums ( curry ( f , n ) ) . 0 = ( Partial_Sums ( curry ( f , n ) ) . 0 ) . n ;
f2 = C7 . ( ( E7 , K , len ( V , len ( V , K , len ( V , len V , len V ) ) ) ) ;
S1 . b = s1 . b .= s2 . b .= s2 . b .= s2 . b .= s2 . b .= ( s1 * s2 ) . b ;
p2 in LSeg ( p2 , p1 ) /\ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ;
dom ( f . t ) = Seg n & dom ( I . t ) = Seg n & dom ( I . t ) = Seg n ;
assume o = ( the connectives of S ) . 11 & o in ( the carrier' of S ) . 11 ;
set phi = ( l1 , l2 ) support phi , phi = ( l1 , l2 ) . ( l + 1 ) , N = ( l1 , l2 ) . ( l + 1 ) ;
synonym p is \cup g for ( p , T ) `1 , p `2 = 1 & ( p , T ) `2 = 1 ;
( Y1 `2 ) ^2 = - 1 & ( Y1 `2 ) ^2 + ( Y1 `2 ) ^2 <> {} or ( Y1 `1 ) ^2 = 1 ;
defpred X [ Nat , set , set , set ] means P [ $1 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2
consider k be Nat such that for n be Nat st k <= n holds s . n < x0 + g / 2 ;
Det ( I |^ ( ( m -' n ) -' n ) ) = 1. K & len ( I |^ ( m -' n ) ) = len ( I |^ ( m -' n ) ) ;
( - b ) / sqrt ( b ^2 - ( 4 * a * c ) ) < 0 ;
Cz . d = Cz . d mod ( C . d ) .= Cz . d mod ( C . d ) .= ( C . d ) mod ( C . d ) ;
attr X1 is dense means : Def1 : 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 ) * ( $2 , $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 for Y being Subset-Family of X holds X is Basis of <* X , product ( Y ) *>
synonym A , B are_separated means : Def1 : Cl ( A \/ B ) misses Cl ( B \/ C ) & Cl ( A \/ B ) misses Cl ( B \/ C ) ;
len ( M @ ) = len p & width ( M @ ) = width M & width ( M @ ) = width M & width ( M @ ) = width M ;
J . v = { x where x is Element of K : 0 < x & x < 1 } ;
( ( Sgm ( Seg m ) ) . d - ( Sgm ( Seg m ) ) . e ) <> 0 ;
lower_bound divset ( D2 , k + k2 ) = D2 . ( k + k2 - 1 ) .= D2 . ( k + k2 - 1 ) ;
g . r1 = - 2 * r1 + 1 & dom h = [. 0 , 1 .] & dom h = [. 0 , 1 .] ;
|. a .| * ||. f .|| = 0 * ||. f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| ;
f . x = ( h . x ) `1 & g . x = ( h . x ) `2 & h . x = ( h . x ) `2 ;
ex w st w in dom B1 & <* 1 *> ^ s = <* 1 *> ^ w & <* 1 *> ^ w = <* 1 *> ^ w ;
[ 1 , {} , <* d1 *> ] in ( { [ 0 , {} , {} ] } \/ S1 ) \/ S2 ;
IC Exec ( i , s1 ) + n = IC Exec ( i , s2 ) + n .= IC Exec ( i , s2 ) + n .= 0 ;
IC Comput ( P , s , 1 ) = IC Comput ( P , s , 9 ) .= 5 + 9 .= 5 ;
( IExec ( W6 , Q , t ) ) . intpos ( i + 1 ) = t . intpos ( i + 1 ) ;
LSeg ( f /^ i , q /. i ) misses LSeg ( f /^ i , j ) \/ 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 | X ) . x ) = f . ( upper_bound C ) - ( f . ( lower_bound C ) ) ;
for F , G being one-to-one FinSequence st rng F misses rng G holds F ^ G is one-to-one & F ^ G is one-to-one
||. R /. L - L /. h .|| < e1 * ( K + 1 ) * ( K + 1 ) ;
assume a in { q where q is Element of M : dist ( z , q ) <= r } ;
set p4 = [ 2 , 1 ] .--> [ 2 , 1 , 0 ] ;
consider x , y being Subset of X such that [ x , y ] in F and x c= d and y c= d ;
for y , x being Element of REAL st y in Y & x in X & x in X holds y <= x & x <= y implies x <= y
func |. p ^ <* 0 *> -> variable of A equals min ( p , NBI ) . ( p ^ <* 0 *> ) ;
consider t being Element of S such that x ` , y ` '||' z , t and x , z '||' y , t ;
dom x1 = Seg ( len x1 ) & len x1 = len y1 & for i st i in Seg ( len x1 ) holds x1 . i = ( x1 /. i ) * ( y1 /. i )
consider y2 being Real such that x2 = y2 and 0 <= y2 and y2 <= 1 / 2 and y2 <= 1 / 2 and y2 <= 1 / 2 ;
||. f | X /. s1 .|| = ||. f .|| /. ( ( ||. f .|| /. s1 ) - f /. ( lim s1 ) ) .= ||. f /. s1 .|| ;
( the InternalRel of A ) ` \ ( x ` \ Y ) = {} \/ ( the InternalRel of A ) ` .= {} \/ {} .= {} ;
assume that i in dom p and for j be Nat st j in dom q holds P [ i , j ] and for i be Nat st i in dom q holds P [ i , j ] and P [ i , 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 & u2 in the carrier of W1 & v2 in the carrier of W2 implies ( ( the carrier of W1 ) /\ ( the carrier of W2 ) ) c= the carrier of W2
defpred P [ Element of L ] means M <= f . $1 & f . $1 <= f . $1 & f . $1 <= f . $1 ;
\bf T . ( u , a , v ) = s * x + ( - ( s * x ) + y ) .= b ;
- ( x - y ) = - x + ( - y ) .= - x + ( - y ) .= - x + y .= - x ;
given a being Point of GX such that for x being Point of GX holds a , x / ( a , x ) / ( a , x ) / ( a , x ) / ( a , x ) / ( a , x ) / ( a , x ) = a ;
f\mathopen { [ dom , cod ( f2 , g2 ) ] , [ cod ( f2 , g2 ) , cod ( f2 , g2 ) ] } = [ cod ( f2 , g2 ) , cod ( f2 , g2 ) ] ;
for k , n being Nat st k <> 0 & k < n & n < k holds k , n are_relative_prime implies k divides n
for x being element holds x in A |^ d implies x in ( ( A ` ) |^ d ) ` & f . x = ( ( A ` ) |^ d ) `
consider u , v being Element of R , a being Element of A such that l /. i = u * a * v ;
( - ( ( p `1 / |. p .| - cn ) ) / ( 1 + cn ) ) ^2 > 0 ;
Carrier ( Z ) . k = Carrier ( L ) . ( F . k ) & F . k in dom Carrier ( L ) ;
set i2 = AddTo ( a , i , - n ) , i1 = - ( n + 1 ) , i2 = - ( n + 1 ) ;
attr B is atomic means : Def1 : for S being Subsqrt of ( B , Sz ) holds ( S is non empty implies S is non empty ;
a9 "/\" D = { a "/\" d where d is Element of N : d in D & d in D } .= D ;
|( \square , q29 )| * |( q29 , q29 )| , ( 2 * q ) * ( b ^2 ) )| >= |( \square , ( 2 * q ) * ( b ^2 ) )| ;
( - f ) . ( upper_bound A ) = ( ( - f ) | A ) . ( upper_bound A ) .= ( - f ) . ( upper_bound 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 ) ) . ( ( proj ( i , n ) ) . LM ) *> ;
f1 + f2 * reproj ( i , x ) is_differentiable_in ( ( reproj ( i , x ) * reproj ( i , x ) ) . x ) & f2 + ( ( reproj ( i , x ) * reproj ( i , x ) ) . x ) = ( ( reproj ( i , x ) * reproj ( i , x ) ) . x ) ;
pred ( for x st x <> 0 holds ( ( x / 2 ) * ( x / 2 ) ) ^2 = ( x / 2 ) * ( x / 2 ) ;
ex t being sort symbol of S st t = s & h1 . t = h2 . t & h1 . x = h2 . x & h2 . x = t ;
defpred C [ Nat ] means P8 . $1 is non empty & A8 : A is non empty & A is n -element implies A is n -element ;
consider y being element such that y in dom ( p9 . i ) and q9 . i = p9 . y and 1 <= y and y <= 1 ;
reconsider L = product ( { x1 } +* ( index ( B ) , l ) ) as Subset of ( \bf SCM } ( A ) ) ;
for c being Element of C ex d being Element of D st T . ( id c ) = id d & ( id d ) . ( id c ) = id d
every f , n , p = ( f | n ) ^ <* p *> .= f ^ <* p *> ^ <* p *> .= f ^ <* p *> ;
( f * g ) . x = f . g . x & ( f * h ) . x = f . ( h . x ) ;
p in { ( 1 - 2 ) / 2 * ( G * ( i + 1 , j ) + G * ( i + 1 , j + 1 ) ) } ;
f `2 - cp = ( f | ( n , L ) ) *' ( - ( f . 0 ) ) .= ( f . ( - ( f . 0 ) ) ) ;
consider r being Real such that r in rng ( f | divset ( D , j ) ) and r < m + s ;
f1 . ( |[ ( r . 8 ) , ( r . 8 ) ]| ) in f1 .: ( W1 /\ W2 ) ;
eval ( a | ( n , L ) , x ) = eval ( a | ( n , L ) ) .= a * eval ( b , x ) .= a * eval ( b , x ) ;
z = DigA ( tz , xz ) .= DigA ( tz , xz ) .= DigA ( tz , xz ) .= DigA ( tz , xz ) .= DigA ( tz , xz ) ;
set H = { Intersect S where S is Subset-Family of X : S c= G & S c= G } , G = { Intersect S where S is Subset-Family of X : S is open & S is open } , H = { Intersect S where S is Subset of X : S is open } ;
consider S19 being Element of D ( ) , d being Element of D ( ) such that S `1 = S19 ^ <* d *> and d = 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 . x1 = f . x2 ;
- 1 <= ( ( q `2 / |. q .| - sn ) ) / ( 1 + sn ) ;
( ( 0. K ) * ( a * b ) ) is Linear_Combination of A & Sum ( ( 0. K ) * ( b * a ) ) = 0. K ;
let k1 , k2 , k2 , x4 , k2 , x4 , 8 , 7 , 8 , 8 , 8 , 8 , 9 , 8 , 9 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 9 be Element of NAT ;
consider j being element such that j in dom a and j in g " { k `2 } and x = a . j and x = a . j ;
H1 . x1 c= H1 . x2 or H1 . x1 c= H1 . x2 or H1 . x2 c= H1 . x2 & H1 . x2 c= H1 . x2 & H1 . x2 c= H1 . x2 ;
consider a being Real such that p = ( a * p1 + ( a * p2 ) ) + ( a * p2 ) and 0 <= a and a <= 1 ;
assume that a <= c & d <= b & [ a , b ] c= dom f and [ a , b ] c= dom g and g . a = g . b ;
cell ( Gauge ( C , m ) , ( X , width Gauge ( C , m ) -' 1 ) , 0 ) is non empty ;
Aid in { ( S . i ) `1 where i is Element of NAT : i in dom S & i in dom S } ;
( T * b1 ) . y = L * b2 /. y .= ( F /. b1 ) * ( G /. b2 ) .= ( F /. b2 ) * ( G /. b2 ) ;
g . ( s , I ) . x = s . y & g . ( s , I ) . y = |. s . x - s . y .| ;
( log ( 2 , k + 1 ) ) ^2 >= ( log ( 2 , k + 1 ) ) ^2 / ( 2 * k ) ^2 ;
then that p => q in S and not x in the still of p and not p in S and p => All ( x , p ) in S ;
dom ( the InitS of r-10 ) misses dom ( the InitS of r-10 ) & dom ( the InitS of r-10 ) = dom ( the InitS of r-10 ) ;
synonym f is extended real means : Def1 : for x being set st x in rng f holds 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 + 1 .= len p1 + 1 + 1 .= len p2 + 1 ;
( l ) ^2 = ( g . ( 1 , 3 ) ) ^2 + ( k + 1 ) ^2 .= ( g . ( 1 , 3 ) ) ^2 + ( k + 1 ) ^2 .= ( k + 1 ) ^2 ;
CurInstr ( P2 , Comput ( P2 , s2 , l ) ) = halt SCM+FSA .= halt SCM+FSA .= ( halt SCM+FSA ) ;
assume for n be Nat holds ||. seq .|| . n <= ( ||. seq .|| ) . n & ( ||. seq .|| ) . n <= ( ||. seq .|| ) . n & ( ||. seq .|| ) . n <= ( ||. seq .|| ) . n ;
sin . ( \kern1pt \kern1pt ) = sin . r * cos ( ( - cos . ( sin . ( - r ) ) * sin ( ( - r ) * sin ( ( - r ) * sin ( ( - r ) * sin ( ( - r ) / sin ( ( - r ) ) * sin ( ( - r ) / sin ( ( - r ) / sin ( ( - r ) ) * sin ( ( - r ) ) * sin ( ( - r ) ) ) ) ) ) )
set q = |[ g1 . ( t1 . t ) , g2 . ( t2 . a ) ]| , r = |[ 0 , 0 ]| , s = |[ 0 , 1 ]| , s = |[ 0 , 1 ]| , t = |[ 0 , 1 ]| , s = |[ 0 , 1 ]| , s = |[ 0 , 1 ]| , t = |[ 0 , 1 ]| , s = |[ 0 , 1 ]| , s = |[ 0 , 1 ]| ;
consider G being sequence of S such that for n being Element of NAT holds G . n in <* /\ ( F . n ) , F . n *> ;
consider G such that F = G and ex G1 st G1 in SM & G = ( the carrier of G1 ) --> ( G1 , G2 ) ;
the root of [ x , s ] in ( the Sorts of Free ( C , X ) ) . s & ( the Sorts of Free ( C , X ) ) . s c= ( the Sorts of Free ( C , X ) ) . s ;
Z c= dom ( exp_R * ( ( exp_R + f1 ) (#) ( exp_R + f1 ) ) ) ;
for k be Element of NAT holds ( ( Im ( Im ( f ) ) ) . k ) = ( ( Im ( Im ( f ) ) ) . k ) * ( ( Im ( f ) ) . k )
assume that - 1 < n and q `2 > 0 and q `2 / |. q .| < 1 and q `1 <= 0 and q `1 <= 0 and q `1 <= 0 and q `1 <= 0 and q `1 <= 0 and q `1 <= 0 and q `1 <= 0 and q `1 <= 0 and q `1 <= 0 and q `1 <= 0 and q `1 <= 0 and q `1 <= 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 . a = d ;
consider r being Element of NAT such that s-> Element of NAT and r <= Comput ( P1 , s1 , r ) and r <= q and r <= q ;
LE f /. ( i + 1 ) , f /. j , f /. ( j + 1 ) , f /. ( len f + 1 ) , f /. ( len f + 1 ) , f /. ( len f + 1 ) ;
assume that x in the carrier of K and y in the carrier of K and inf { x , y } in the carrier of K and inf { x , y } in L ;
assume f +* ( i1 , \xi ) . ( i1 , \xi ) in ( proj ( F , ( i2 + 1 ) ) " ( A . ( i1 , i2 ) ) ) " ( A . ( i1 , i2 ) ) ;
rng ( ( ( ( ( ( ( the carrier' of M ) ) * ) | ( the carrier' of 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 { ( the carrier of G ) --> t } ;
consider l be Nat such that for m be Nat st l <= m holds ||. s1 . m - g .|| < g / 2 * ( ||. x .|| + ||. x .|| ) ;
consider t be VECTOR of product G such that m5 = ||. D5 . t .|| & ||. t .|| <= 1 / 2 and ||. t .|| <= 1 / 2 ;
assume that the carrier of v = 2 and v ^ <* 0 *> , v ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ v ^ <* 1 *> in dom p and v ^ <* 1 *> ^ <* 1 *> in dom p ;
consider a being Element of the Points of X\mathopen { a } , A , B being Element of the lines of X\mathopen { a } , B , C such that a on A & b on B ;
( - 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 , p1 ) , 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 + 2 + 1 + 1 + 1 + 1 + 1 + 1 + 1 ;
for n being Element of NAT st n in dom F holds F . n = |. ( ( n -' 1 ) + 1 ) .| ;
for r , s1 , s2 , r , s , t being Real holds r in [. s1 , s2 .] iff r <= s & s <= t & t <= q & q <= r & r <= s implies r <= s
assume v in { G where G is Subset of T2 : G in B2 & G c= z & G c= z & z in G & G c= z } ;
let g be :] :] , A , h be Element of INT , b , c , d , b be Element of INT , b , c , d be Element of INT ;
min ( g . [ x , y ] , k . [ y , z ] ) = ( min ( g , k , x ) ) . y ;
consider q1 being sequence of CNS such that for n holds P [ n , q1 . n , q1 . n ] and 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 B9 = B /\ O , O1 = O /\ O , Z1 = O , Z1 = I /\ O , Z1 = I /\ O , I = I /\ O , I = I /\ O , I = I /\ O , I = I /\ O , I = I /\ O , J = I /\ O , I = I /\ O , I = I /\ O , I = I /\ I , J = I /\ I , I = I /\ I , I
consider j being Element of NAT such that x = ( the \it n ) --> ( j + 1 ) and j + 1 <= n and j + 1 <= n ;
consider x such that z = x and card ( x . O2 ) in card ( x . O2 ) and x in L1 . O2 and x in L2 . O2 ;
( C * of _ T4 ( k , n2 ) ) . 0 = C . ( ( of of T4 ( k , n2 ) ) . 0 ) ;
dom ( X --> rng f ) = X & dom ( X --> f ) = dom ( X --> f ) & dom ( X --> f ) = X --> f ;
( ( ( ( TOP-REAL 2 ) ) * ( ( SpStSeq L~ Cage ( C , n ) ) ) ) `1 <= ( ( ( ( TOP-REAL 2 ) ) * ( ( SpStSeq L~ Cage ( C , n ) ) ) ) `1 ;
synonym x , y are_collinear means : Def1 : x = y or 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 that L is continuous and for x , y being Element of L for a , b being Element of L st a = x & b = y & a << b holds a << b iff a << b ;
( 1 / 2 ) (#) ( ( ( #Z 2 ) * ( ( ( #Z 2 ) * ( ( #Z 2 ) * ( 1 + 0 ) ) ) ) ) is_differentiable_on REAL ;
defpred P [ Element of omega ] means ( ( the partial of A1 ) . $1 ) . ( ( the partial of A1 ) . $1 ) = A1 . ( ( the partial of A2 ) . $1 ) ;
IC Comput ( P , s , 2 ) = succ IC Comput ( P , s , 1 ) .= ( 6 + 1 ) .= 6 + 1 .= 6 + 1 ;
f . x = f . g1 * f . g2 .= f . g1 * f . g2 .= f . g1 * ( f . g2 ) .= f . g1 * ( f . g2 ) .= f . g2 * ( f . g2 ) .= f . g2 ;
( M * F-4 ) . 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 L2 ;
pred a , b , c , x , y , c , a , b , c , x , y , y , z , x , y , z , y , z , x , y , z , y , x , y , z , x , y , z , y , z , x , y , z , y , z , x , y , z , x , y , z , y , z , x , y , z , x , y , y , z , x , y , z , z , y
( ( the partial of s ) . n ) . n <= ( ( the partial of s ) . n ) * ( ( the Sorts of s ) . n ) . ( ( the Sorts of s ) . n ) ;
pred - 1 <= r & r <= 1 implies ( ( - 1 ) / r ) ^2 = - ( ( 1 - r ) / r ) ^2 ;
seq in { p ^ <* n *> where n is Nat : p ^ <* n *> in T1 & n in T1 } \/ { p ^ <* n *> : p ^ <* n *> in T1 } ;
|[ x1 , x2 , x3 , x4 ]| . 2 - |[ x1 , x2 , x3 ]| . 2 = ( x1 - x2 ) * ( y1 - y2 ) .= ( x1 - x2 ) * ( y1 - y2 ) ;
attr for m being Nat holds F . m is nonnegative & ( Partial_Sums F ) . m is nonnegative & ( Partial_Sums F ) . m is nonnegative ;
len ( ( G . ( x , z ) ) ) = len ( ( ( G . ( x , y ) ) + ( G . ( y , z ) ) ) .= len ( G . ( x , y ) ) ;
consider u , v being VECTOR of V such that x = u + v and u in W1 /\ W2 and v in W2 and u in W2 /\ W3 ;
given F be finite FinSequence of NAT such that F = x and dom F = n and rng F c= { 0 , 1 } and Sum F = k and Sum F = k ;
0 = ( 1 * a2 ) * u] iff 1 = ( ( 1 - ( 1 - ( 1 - ( 1 - ( ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 / 2 ) ) ) ) ) ) ) ) ) ) ) ) * ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 -
consider n be Nat such that for m be Nat st n <= m holds |. ( f # x ) . m - ( lim ( f # x ) ) .| < e / 2 ;
cluster -> real for non empty implies L is \in ( ( { \rm o } _ 2 ) ) --> ( ( { o } _ 2 ) ) & ( ( { o } _ 3 ) --> ( ( { o } _ 3 ) ) ) is Boolean
"/\" ( BB , L ) = "/\" ( BB , L ) .= "/\" ( ( the carrier of 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 - c * |[ a , c ]| - ( 2 * r1 - c ) = 0. TOP-REAL 2 - ( 2 * r1 - c ) ;
reconsider p = P * ( 1 , 1 ) , q = a " * ( ( - ( - ( K , n , 1 ) ) ) * ( ( - ( K , n , 1 ) ) ) as FinSequence of K ;
consider x1 , x2 being element such that x1 in uparrow s and x2 in downarrow t and x = [ x1 , x2 ] and x = [ x1 , x2 ] and x = [ x1 , x2 ] ;
for n be Nat st 1 <= n & n <= len q1 holds q1 . n = ( upper_volume ( g , M7 ) ) . ( n + 1 ) & q2 . n = ( upper_volume ( g , M7 ) ) . ( n + 1 ) ;
consider y , z being element such that y in the carrier of A and z in the carrier of A and i = [ y , z ] and i = [ y , z ] ;
given H1 , H2 being strict Subgroup of G such that x = H1 and y = H2 and H1 , H2 / ( x , y ) / ( x , y ) / ( x , y ) / ( x , y ) / ( x , y ) / ( x , y ) / ( x , y ) / ( x , y ) / ( x , y ) / ( x , y ) / ( x , y ) ) ;
for S , T , 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 , b + b2 ] in ( the carrier of COMPLEX ) \/ ( the carrier of COMPLEX ) & [ a + b2 , b2 + b2 ] in the carrier of COMPLEX ;
reconsider mm = max ( ( len F1 . n ) * ( p . n ) * ( x . n ) ) , m5 = max ( ( len F1 . n ) * ( x . n ) * ( x . n ) ) as Element of NAT ;
I <= width GoB ( ( GoB ( h ) ) * ( 1 , 1 ) , ( GoB ( h ) ) * ( 1 , 1 ) ) & ( GoB ( h ) ) * ( 1 , 1 ) = ( GoB ( h ) ) * ( 1 , 1 ) ;
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 : Def1 : A1 is linearly-independent & A2 is linearly-independent & ( ( 1 - 1 ) * A1 ) * A2 = ( 1 - 1 ) * A2 & ( 1 - 1 ) * A2 = ( 1 - 1 ) * A1 ;
func A -_ C -> set equals union { A . s where s is Element of R : s in C & s in C } ;
dom ( ( Line ( v , i + 1 ) ) (#) ( ( ( Line ( v , m ) ) (#) ( ( ( v , m ) * ( x , 1 ) ) ) ) ) ) = dom ( F ^ <* x , m *> ) ;
cluster [ x , ( x , 4 ) ] -> to & [ x , ( x , 4 ) ] in R & [ x , ( x , 4 ) ] in R ;
E , f |= All ( x2 , All ( x2 , ( x2 , x1 ) ) '&' ( x2 , x1 ) ) '&' ( x2 , ( x2 , x1 ) ) '&' ( x2 , ( x2 , x2 ) ) '&' ( x2 , x1 ) ) ;
F .: ( id X , g ) . x = F . ( id X , g . x ) .= F . ( id X , g . x ) .= F . ( x , g . x ) .= F . ( x , g . x ) ;
R . ( h . m ) = F . x0 + h . ( m + 1 ) - h . x0 .= ( h . x0 + h . x0 ) - ( h . x0 ) ;
cell ( G , ( XX -' 1 , Y1 + ( t + 1 ) ) , ( t + 1 ) ) \ L~ f meets ( UBD L~ f ) ` & ( t + 1 ) in ( UBD L~ f ) ` ;
IC Comput ( P2 , s2 , 2 ) = IC Comput ( P2 , s2 , 2 ) .= IC Comput ( P2 , s2 , 2 ) .= ( card I + 1 ) .= ( card I + 2 ) .= ( card I + 2 ) .= ( card I + 2 ) .= ( card I + 2 ) ;
sqrt ( ( - ( ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ) ^2 ) > 0 ;
consider x0 being element such that x0 in dom a and x0 in g " { k } and y0 = a . ( k + 1 ) and x0 = a . ( k + 1 ) and y0 = a . ( k + 1 ) ;
dom ( r1 (#) chi ( A , A ) ) = dom ( chi ( A , A ) ) /\ dom ( chi ( A , A ) ) .= dom ( chi ( A , A ) ) /\ dom ( chi ( A , A ) ) .= dom ( ( r1 (#) chi ( A , A ) ) ) ;
d-7 . [ y , z ] = ( ( [ y , z ] `1 ) ^2 - ( ( y , z ) `2 ) ^2 + ( z `2 ) ^2 .= ( ( y , z ) `2 ) ^2 - ( z `2 ) ^2 ;
pred for i being Nat holds C . i = A . i /\ B . i & C . i c= A . i /\ B . i ;
for x0 st x0 in dom f & f is_differentiable_in x0 & f is_differentiable_in x0 & for r st r < x0 & x0 < r ex g st g in dom f & f . g = r & g in dom ( f /* x0 ) & ( for r st r in dom f & g . r in dom ( f /* x0 ) holds ( f /* x0 ) . r = ( f /* x0 ) . r
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 : Def1 : a in it & for b being Ordinal st a in it holds it . b c= b & for a being Ordinal st a in it holds it . a c= b ;
[ a1 , a2 , a3 ] in ( the carrier of A ) ~ & [ a1 , a2 , a3 ] in ( the carrier of A ) ~ & [ a2 , a3 ] in ( 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 .|| < ( e / ( ||. x .|| + ||. x .|| ) * ||. x .|| ) * ||. x .|| ;
then for Z being set st Z in { Y where Y is Element of I : F ( Y ) c= Z & Z c= x & z in Z & z in Z } implies z in Z ;
sup compactbelow [ s , t ] = [ sup compactbelow ( [ s , t ] ) , sup compactbelow ( [ s , t ] ) ] .= sup { sup { s . ( s , t ) } where s is Element of L : s in compactbelow ( s , t ) } ;
consider i , j being Element of NAT such that i < j and [ y , f . j ] in Ij and [ f . i , z ] in Ij and [ f . i , z ] in Ij ;
for D being non empty set , p , q being FinSequence of D st p c= q & q c= p holds ex p being FinSequence st p = q & q = p ^ q & p = q
consider e19 being Element of the affine \bf of X such that c9 , a9 // a9 , e29 and not a9 , b9 // a9 , e29 and not a9 , c9 // b9 , e29 and not b9 , c9 // a9 , e29 and not c9 , e // b9 , e ;
set U2 = I \! \mathop { + } , E = S \! \mathop { + } ;
|. q1 .| ^2 = ( q1 `1 ) ^2 + ( q2 `2 ) ^2 .= ( q1 `1 ) ^2 + ( q2 `2 ) ^2 .= ( q2 `1 ) ^2 + ( q2 `2 ) ^2 .= ( q2 `1 ) ^2 + ( q2 `2 ) ^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 & x "/\" y = x /\ y
dom ( signature U1 ) = dom ( ( the charact of U1 ) * ( the charact of U2 ) ) & Args ( o , ( ( the charact of U1 ) * ( the charact of U2 ) ) ) = dom ( ( the charact of U1 ) * ( the charact of U2 ) ) ;
dom ( h | X ) = dom h /\ X .= dom ( ( |. h .| ) | X ) .= dom ( ( |. h .| ) | X ) .= dom ( ( |. h .| ) | X ) .= X /\ dom ( ( |. h .| ) | X ) .= X ;
for N1 , N1 , N2 being Element of [: the carrier of G , the carrier of G :] holds dom ( h . K1 ) = N & rng ( h . K1 ) = N1 & rng ( h . K1 ) c= N1 & rng ( h . K1 ) c= N2
( ( mod ( u , m ) ) + mod ( v , m ) ) . i = ( ( mod ( u , m ) ) . i + ( mod ( v , m ) ) . i ;
- ( q `1 / |. q .| - cn ) ^2 / ( 1 + cn ) ^2 < - ( q `1 / |. q .| - cn ) ^2 / ( 1 + cn ) ^2 / ( 1 + cn ) ^2 & - ( q `1 / |. q .| - cn ) / ( 1 + cn ) ^2 <= - ( q `1 / |. q .| - cn ) / ( 1 + cn ) ^2 ;
pred r1 = f9 & r2 = g9 & r3 = g9 & r2 = f9 & r3 = r2 & r3 = r2 & r2 = ^2 & r3 = ^2 & r2 = ^2 ;
vseq . m is bounded Function of X , the carrier of Y & x9 . m = ( seq_id ( vseq . m ) , X . m ) & x9 . m = ( seq_id ( vseq . m , X . m ) ) . x ;
pred a <> b & b <> c & angle ( a , b , c ) = PI & angle ( b , c , a ) = 0 ;
consider i , j being Nat , r being Real such that p1 = [ i , r ] and p2 = [ j , s ] and r < j and i < j and j <= len s ;
|. p .| ^2 - ( 2 * |( p , q )| ) ^2 + |. q .| ^2 = |. p .| ^2 + |. q .| ^2 + |. q .| ^2 ;
consider p1 , q1 being Element of [: X , Y :] such that y = p1 ^ q1 and q1 = p1 ^ q1 and q2 = q1 ^ q2 and q1 = q2 ^ q2 and q2 = q2 ^ q2 and q2 = q2 ^ q2 ;
, r2 = ( s2 . r1 , s2 . r2 , s2 . r2 , s2 . r2 , s2 . r2 , s2 . r2 , s2 . r2 , s2 . r2 , s2 . r2 , s2 . r2 , s2 . r2 , s2 . r2 , s2 . r2 , s2 . r2 , s2 . r2 , s2 . r2 , s2 . r2 , s2 . r2 , s2 . r2 , s2 . r2 , s2 . r2 , s2 . r2 , s2 . r2 , s2 . r2 , s2 . r2 , s2 . r2 r2 , s2 . r2 , s2 . r2 , s2 . r2 , s2 . r2 r2 ,
( ( LMP A ) `2 ) ^2 = lower_bound ( ( proj2 .: ( A /\ ( A /\ ( w /\ ( w /\ ( w /\ k ) ) ) ) ) ) ) & ( ( proj2 .: ( A /\ ( w /\ k ) ) ) ) ^2 is non empty ;
s |= ( ( H , ( 1 , k ) ) |= H1 ) iff s |= ( ( H1 , ( k , k ) ) |= H2 ) & s |= ( H1 , ( k , k ) ) |= H2 ;
( len s5 + 1 ) = card ( support b1 ) + 1 .= card ( support b2 ) + ( 1 + 1 ) .= card ( support b2 ) + ( 1 + 1 ) .= card ( support b2 ) + ( 1 + 1 ) .= card ( support b2 ) + ( 1 + 1 ) .= card ( support b2 ) ;
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 >= x ;
LSeg ( LMP D , |[ ( W-bound D ) / 2 , ( E-bound D ) / 2 ]| ) /\ D = { LMP D , ( E-bound D ) / 2 } /\ D .= { LMP D , ( E-bound D ) / 2 } ;
lim ( ( ( f `| N ) / g ) /* b ) = ( lim ( ( f `| N ) / g ) .= ( ( f `| N ) / g ) . 0 .= ( ( f `| N ) / g ) . 0 ;
P [ i , pr1 ( f ) . i , pr1 ( f ) . ( i + 1 ) ] & 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 ) - ( R /. k ) ) - ( R /. k ) .|| < r
for X being set , P , a , b , x being set st x in a & a in P & x in P & b in P & x in P holds a = b
Z c= dom ( ( ( id Z ) ^ ) (#) f ) /\ ( ( id Z ) ^ ) \ ( ( id Z ) ^ ) " { 0 } implies f . x = ( ( id Z ) ^ ) (#) f . x - ( id Z ) ^ { 0 }
ex j being Nat st j in dom ( l ^ <* x *> ) & j < i & j < i implies ( l ^ <* x *> ) . j = 1 & ( i + 1 ) = i & ( i + 1 ) = i & ( i + 1 ) = i & ( i + 1 ) = i & ( i + 1 ) = i & ( i + 1 ) = j & ( i + 1 ) = j implies i = 1 ;
for u , v being VECTOR of V , r being Real st 0 < r & r < 1 & u in dom ( - r ) & v in N holds r * u + ( r * v ) in N
A , Int A , Cl ( A , B ) \ ( Cl ( A , B ) ) \ ( Cl ( A , B ) ) \ ( Cl ( A , B ) ) \ ( Cl ( A , B ) ) , Cl ( A , B ) \ ( Cl ( A , B ) ) \ ( Cl ( A , B ) ) \ ( Cl ( A , B ) ) \ ( Cl ( A , B ) ) ) \ ( Cl ( A , B ) ) , Cl ( A , B ) \ ( Cl ( A , B ) \ ( Cl ( A , B ) ) \ ( Cl ( A , B
- Sum <* v , u , w *> = - ( v + u + u ) .= - ( v + u + u ) .= - ( v + u + u ) .= - ( v + u ) ;
( Exec ( a := b , s ) ) . IC SCM R = ( Exec ( ( a := b , s ) , s ) ) . IC SCM R .= 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 support of J ) . x ;
for S1 , S2 , S2 , D , D , E , F , G being non empty directed Subset of [: S1 , S2 :] , f being Function of S1 , S2 , g being Function of S1 , S2 st f . D = g & g . D = f . D holds cos ( f , D ) is directed
card X = 2 implies ex x , y st x in X & y in X & x <> y or x = y or x = z or y = x or x = y or x = z or x = y or x = z or x = y or x = z or x = y or x = z or x = y or x = z or x = y ;
( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) in rng ( Cage ( C , n ) \circlearrowleft ( Cage ( C , n ) ) ) & ( W-min L~ Cage ( C , n ) ) .. Cage ( C , n ) in rng Cage ( C , n ) ;
for T , T , p , q , q , r , s , q , r , s , p , q , r , s , r , s , q , r , s , s , r , q , s , r , s , r , s , s , r , s , q , r , s , s , r , s , s , r , s , s , r , s , q , s , r , s , s , r , s , q , s , s , s , q , r , s , s , s , s , r , s , s , q , r , s , s
[ i2 + 1 , j2 ] in Indices G & [ i2 + 1 , j2 ] in Indices G & f /. k = G * ( i2 + 1 , j2 ) & f /. k = G * ( i2 + 1 , j2 ) ;
cluster ( k gcd n ) -> natural & k divides ( k -' n ) & n divides ( k -' n ) & ( k -' n ) divides ( k -' n ) implies ( k divides ( k -' n ) ) & ( k divides ( k -' n ) ) & ( k divides ( k -' n ) ) & ( k divides ( k -' n ) ) & ( k divides ( k -' n ) ) & ( k divides ( k -' n ) ) ) ;
dom F " = the carrier of X2 & rng F = the carrier of X1 & F " { 0 } = the carrier of X2 & F " { 0 } = the carrier of X1 & F " { 0 } = the carrier of X2 & F " { 1 } is one-to-one ;
consider C being finite Subset of V such that C c= A and card C = thesis and the carrier of V = C and the carrier of V = Lin ( B9 \/ C ) and the carrier of V = Lin ( B9 \/ C ) ;
V is prime implies for X , Y being Element of \langle the topology of T , \subseteq the topology of T , \subseteq the topology of T , the topology of T , the topology of T , the topology of T *> st X = V holds X is open or Y is open
set X = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } , Y = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } , Z = { F ( v2 ) where v2 is Element of B ( ) : P [ v2 ] } ;
angle ( p1 , p3 , p4 ) = 0 .= angle ( p2 , p3 , p4 ) .= angle ( p2 , p3 , p4 ) .= angle ( p2 , p3 , p4 ) .= angle ( p2 , p3 , p4 ) ;
- sqrt ( ( - ( q `1 / |. q .| - cn ) ) ^2 ) = - sqrt ( ( q `1 / |. q .| - cn ) ^2 ) .= - sqrt ( ( q `1 / |. q .| - cn ) ^2 ) .= - sqrt ( ( q `1 / |. q .| - cn ) ^2 ) .= - sqrt ( 1 + ( q `2 / q `2 ) ^2 ) .= - 1 ;
ex f being Function of I[01] , ( TOP-REAL 2 ) st f is continuous one-to-one & rng f = P & f . 0 = p1 & f . 1 = p2 & f . 0 = p2 & f . 1 = p3 & f . 1 = p4 ;
attr f is partial means : Def1 : SVF1 ( 2 , pdiff1 ( f , 1 ) , u0 ) . ( u0 + 1 ) = ( proj ( 2 , pdiff1 ( f , 3 ) ) . ( u0 + 1 ) ;
ex r , s st x = |[ r , s ]| & G * ( len G , 1 ) `1 < r & G * ( 1 , 1 ) `2 < s & s < G * ( 1 , 1 ) `2 ;
for f being FinSequence of TOP-REAL 2 st 1 <= t & t <= len G & 1 <= t & t <= len G holds ( G * ( t , width G ) ) `1 >= ( ( G * ( t , width G ) ) `1 ) / 2
pred i in dom G & r (#) ( f (#) reproj ( i , x ) ) = r (#) ( f (#) reproj ( i , x ) ) & f . ( r (#) reproj ( i , x ) ) = r (#) ( reproj ( i , x ) ) ;
consider c1 , c2 being bag of o1 + o2 such that ( decomp c ) /. k = <* c1 , c2 *> and c /. k = <* c1 , c2 *> and c1 /. k = c1 + c2 and c2 /. k = c2 /. k and c1 /. k = c2 /. k * c2 /. k ;
u0 in { |[ r1 , s1 ]| : r1 < G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `2 < G * ( 1 , 1 ) `2 } ;
Cl ( X ^ Y ) . k = the carrier of X . ( k2 + 1 ) .= ( C . ( k2 + 1 ) ) . ( k + 1 ) .= C . ( k2 + 1 ) .= C . ( k2 + 1 ) ;
pred that len M1 = len M2 and width M1 = width M2 and width M2 = width M2 and width M1 = width M2 and width M1 = width M2 and width M1 = width M2 and width M2 = width M2 and width M1 = width M2 and width M1 = width M2 ;
consider g2 being Real such that 0 < g2 and { y where y is Point of S : ||. ( y - x0 ) - ( x0 - r ) .|| < g2 & g2 . y = ( g2 . y - r ) * ( x0 - r ) + R . ( y - x0 ) ;
assume x < ( - b + sqrt ( delta ( a , b , c ) ) ) / ( 2 * a ) or x > ( - b + sqrt ( 2 * a ) ) / ( 2 * a ) ;
( G1 '&' G2 ) . i = ( <* 3 *> ^ G1 ) . i & ( H1 '&' H2 ) . i = ( <* 3 *> ^ ( H1 ^ H2 ) ) . i & ( H1 ^ H2 ) . i = ( <* 3 *> ^ ( H1 ^ H2 ) ) . i ;
for i , j st [ i , j ] in Indices ( M3 + M1 ) holds ( M3 + M2 ) * ( i , j ) < M2 * ( 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 f /. i divides f /. j & f /. j divides f /. j
assume that F = { [ a , b ] where a , b is Subset of X : for c st c in B\mathopen { a , b } holds a c= c } and for c st c in B\mathopen { a , b } holds a c= c ;
b2 * q2 + ( b3 * q2 ) + ( b3 * q2 ) + ( a * q2 ) + ( a * q2 ) + ( a * q2 ) + ( a * q2 ) + ( a * q2 ) = 0. TOP-REAL n + ( a * q2 ) .= 0. TOP-REAL n + ( 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 & B in F & B c= Cl ( Cl F ) } ;
attr seq is summable & seq is summable & ( for n holds seq . n = Partial_Sums ( seq ) . n ) implies Partial_Sums ( seq ) . n = Partial_Sums ( seq ) . ( Partial_Sums ( seq ) . n ) & Partial_Sums ( seq ) . ( Partial_Sums ( seq ) ) . n = Partial_Sums ( seq ) . ( Partial_Sums ( seq ) ) . n ;
dom ( ( ( 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 ) ;
|[ X \to Z ]| is full full SubRelStr of ( Omega Z ) |^ the carrier of Z & [ X \to Y , Z ] is full full SubRelStr of ( Omega Z ) |^ the carrier of Z ;
( G * ( 1 , j ) ) `2 = ( G * ( i , j ) ) `2 & ( G * ( 1 , j ) ) `2 <= ( G * ( i , j ) ) `2 ;
synonym m1 c= m2 & ( for p be set st p in P & ( for p be set st p in P holds ( p in P implies p in P ) & ( for m be set st m in P holds m <= m holds p . m = ( m + 1 ) ) & ( m <= k implies p . m = ( m + 1 ) ) ;
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 P [ a ] ;
synonym the multMagma of K is \vert means : Def1 : the multF of K is multiplicative & the multF of K is associative & the multF of K is associative & the multF of K is associative & the multF of K is associative & the multF of K is associative & the multF of K is associative & the multF of K is associative & the multF of K is associative & the multF of K is associative & the multF of K is associative ;
directed ( a , b , 1 ) + \mathop ( c , d ) = b + ( c + d ) .= b + ( c + d ) .= b + ( c + d ) .= sequence ( a + c + d ) .= sequence ( a + c + d ) .= sequence ( a + c + d ) ;
cluster + _ + -> strict for Element of INT means : Def1 : for i , j being Element of INT holds it . ( i , j ) = + ( i , j ) & j = j implies it . ( i , j ) = + ( i , j ) ;
( - s2 ) * p1 + ( s2 * p2 ) - ( s2 * p2 ) = ( ( - s2 ) * p1 + ( s2 * p2 ) ) * p2 + ( ( - s2 ) * p2 ) .= ( ( - s2 ) * p2 ) + ( ( - s2 ) * p2 ) ;
eval ( ( a | ( n , L ) ) *' p , x ) = eval ( a | ( n , L ) ) * eval ( p , x ) .= a * eval ( p , x ) .= a * eval ( p , x ) ;
assume that the TopStruct of S = the TopStruct of T and for D being non empty Subset of S , D being non empty directed Subset of S st D = the carrier of S & D is open holds D is open & for V being open Subset of S st V in V holds V is open & V is open & V is open & V is open & V is open ;
assume that 1 <= k & k <= len w + 1 and T-7 . ( ( ( q11 , w ) -w , w ) ) = ( ( T11 . k ) -w ) . ( ( ( q11 , w ) -w ) . k ) and ( ( T . k ) -w ) . k = ( ( T . k ) -w ) . k ;
2 * a |^ ( n + 1 ) + ( 2 * b |^ ( n + 1 ) ) >= ( a |^ n + 1 ) + ( b |^ ( n + 1 ) ) + ( b |^ ( n + 1 ) ) + ( a |^ ( n + 1 ) ) + ( b |^ ( n + 1 ) ) ;
M , v / ( x. 0 , x. 4 ) / ( x. 0 , x. 0 , x. 0 ) / ( x. 4 , x. 0 ) / ( x. 0 , x. 4 , x. 0 ) / ( x. 0 , x. 0 ) / ( x. 4 , x. 0 , x. 0 ) / ( x. 4 , x. 0 , x. 0 ) / ( x. 0 , x. 0 ) ) / ( x. 4 , x. 0 ) |= ( x. 0 , x. 4 , x. 0 ) ) |= ( x. 0 , x. 0 , x. 0 , x. 0 , x. 0 , x. 4 , x. 0 , x. 0 ) / ( x. 4 , x. 0 , x. 0 ) ;
assume that f is_differentiable_on l and for x0 st x0 in l holds 0 < f . x0 or for x0 st x0 in l holds f . x0 < ( f . x0 ) ^2 and for x0 st x0 in l holds f . x0 < ( f . x0 ) ^2 and for x0 st x0 in l holds f . x0 < ( f . x0 ) ^2 ;
for G1 being _Graph , W being Walk of G1 , e being Vertex of G2 , G2 being Walk of G2 , e being Vertex of G2 st e in W holds not ( e in W implies e in G & not ( e in G implies e in G ) implies e in G & e in G & e in G implies ( e in G ) implies ( e in G ) implies ( e in G )
not c01 is non empty iff ( not ( ( ex y st y is not empty & not ( y is not empty & not ( y is not empty & not ( not ( y is not empty & not ( y is not empty ) ) ) ) ) & not ( not not ( not not ( not ( y is not empty ) & not ( not ( y is not empty ) ) ) ) ) ;
Indices GoB f = [: dom GoB f , Seg ( len GoB f ) :] & ( for i st i in dom GoB f holds f /. i = ( GoB f ) * ( i , j ) ) implies ( f /. i = ( GoB f ) * ( i , j ) ) & ( f /. i = ( GoB f ) * ( i , j ) )
for G1 , G2 , G2 , G3 being Group st G1 is stable & G2 is stable & G2 is stable & G1 is stable & G2 is stable & G2 is stable & G2 is stable holds G1 * ( G1 * G2 ) is stable & G2 * ( G2 * G2 ) is stable
UsedIntLoc ( int ) = { intloc 0 , intloc 1 , intloc 2 , intloc 3 } \/ { intloc 0 , intloc 4 , intloc 0 } .= { intloc 0 , intloc 4 , intloc 5 } \/ { intloc 0 , intloc 4 , intloc 5 } .= { intloc 0 , intloc 4 , intloc 5 } ;
for f1 , f2 being FinSequence of F st f1 is p -element & f2 is p -element & Q [ f1 ^ f2 , f2 . p ] & Q [ f1 ^ f2 , f2 . p ] holds Q [ f1 ^ f2 , f2 . p ]
( p `1 / sqrt ( 1 + ( p `1 / p `2 ) ^2 ) ) ^2 = ( q `1 / sqrt ( 1 + ( q `1 / q `2 ) ^2 ) ) ^2 .= ( q `1 / sqrt ( 1 + ( q `1 / q `2 ) ^2 ) ) ^2 ;
for x1 , x2 , x3 , x4 being Element of REAL n holds |( x1 , x2 , x3 )| = |( x1 , x2 , x3 )| & |( x1 , x2 , x3 )| = |( x1 , x2 , x3 )| + |( x2 , x3 )| + |( x2 , x3 )| + |( x2 , x3 )| + |( x2 , x3 )| + |( x2 , x3 )| + |( x2 , x3 )| + |( x2 , x3 )|
for x st x in dom ( ( ( ( ( ( ( ( ( - x ) / 2 ) ) (#) ( ( id Z ) ^ 2 ) ) (#) ( ( id Z ) ^ 2 ) ) ) `| Z ) holds ( ( ( ( ( ( ( ( ( ( - x ) / 2 ) (#) ( ( id Z ) / 2 ) ) (#) ( ( id Z ) ^ 2 ) ) (#) ( ( id Z ) ^ 2 ) ) ) ) `| Z ) ) . x = - ( x + x ) (#) ( ( ( ( Z ) ^ 2 ) (#) ( ( id Z ) ^ 2 ) ) ) . x )
for T being non empty TopSpace , P being Subset-Family of T , x being Point of T st P c= the topology of T & x in P ex B being Basis of T st B c= P & x in B
( a 'or' b 'imp' c ) . x = 'not' ( ( a 'or' b ) . x ) 'or' c . x .= ( ( a 'or' b ) . x ) 'or' c . x .= TRUE 'or' TRUE .= TRUE ;
for e being set st e in A8 ex X1 , Y1 being Subset of X st e = [: X1 , Y1 :] & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open ;
for i be set st i in the carrier of S for f being Function of [: S , T :] , S1 . i st f = H . i & f is Function of [: S , T :] , S1 . i holds F . i = f | [: F . i , F . i :]
for v , w st for x st x <> y holds w . y = v . y holds J . ( ( VERUM ( Al , J ) ) . v ) = J . ( ( VERUM ( Al , J ) ) . w )
card D = card D1 + card D2 - card D1 .= card D1 + card D2 - card D2 .= ( i + 1 ) - card D1 + ( i + 1 ) .= ( i + 1 ) - ( i + 1 ) .= ( i + 1 ) - ( i + 1 ) .= ( i + 1 ) + ( i + 1 ) - ( i + 1 ) .= ( i + 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 ;
len f /. ( \downharpoonright i1 -' 1 ) + 1 -' 1 = 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 ;
for a , b , c being Element of NAT st 1 <= a & 2 <= b & k <= m holds a <= a + b or k = a + b-2 or k = a + b-2 or k = a + b-2 or k = a + 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 ) holds Index ( p , f ) <= Index ( p , f ) + 1 & Index ( p , f ) + 1 <= Index ( p , f ) + 1
lim ( ( curry ( I , k + 1 ) ) # x ) = ( lim ( curry ( I , k ) ) ) + ( lim ( ( curry ( I , k ) ) # x ) .= lim ( ( curry ( I , k ) ) # x ) ;
z2 = g /. ( \downharpoonright n1 -' n2 + 1 ) .= g /. ( i -' n2 + 1 ) .= g . ( i -' n2 + 1 ) .= g . ( i -' n2 + 1 ) .= g /. ( i -' n2 + 1 ) .= g /. ( i -' n2 + 1 ) .= g /. ( i -' n2 + 1 ) ;
[ f . 0 , f . 3 ] in id ( the carrier of G ) \/ ( the InternalRel of G ) or [ f . 0 , f . 3 ] in the InternalRel of G & [ f . 0 , f . 3 ] in the InternalRel of C ;
for G being Subset-Family of B st G = { R [ X ] where R is Subset of A ( ) , R is Subset of B ( ) , R is Subset of A ( ) st R is open & R is open & R is open & R is open holds ( Intersect R ) . ( ( Intersect R ) . ( ( Intersect R ) . ( x , y ) ) = Intersect ( G . ( x , y ) ) )
CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 + m2 ) ) = CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 + m2 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 + m2 ) ) .= halt SCMPDS ;
assume that a on M and b on M and c on N and d on N and p on N and p on M and a on M and p on N and d on N and p <> d and a <> b and p <> d and a <> b and p <> d and a <> d and a <> b and p <> d and a <> d and a <> d and a <> b ;
Suppose T is \hbox of 4 , T _ 4 & B is closed and ex F being Subset-Family of T st F is closed & F is closed & for n st n <= 0 holds F . n <= 0 & ind F <= 0 and ind F <= 0 ; Then T <= 0 & ind T <= 0 ;
for g1 , g2 st g1 in ]. r - g , r .[ & g2 in ]. r - g , r + g .[ & |. f . g1 - g .| <= ( f . g1 - f . g2 ) / ( r - g ) / ( r - g ) & |. f . g2 - f . g2 .| <= ( f . g1 - f . g2 ) / ( r - g ) / ( r - g ) / ( r - g ) .|
( ( - 1 ) / ( z1 + z2 ) ) / ( z1 + z2 ) = ( - ( 1 / z1 ) ) / ( z1 + z2 ) .= ( - 1 / z1 ) / ( z1 + z2 ) .= ( - 1 / z1 ) / ( z1 + z2 ) .= ( - 1 / z1 ) / ( z1 + z2 ) .= ( - 1 / z1 ) / ( z1 + z2 ) ;
F . i = F /. i .= 0. R + r2 .= ( b |^ n ) * r2 .= ( ( n + 1 ) -tuples_on REAL ) .= ( ( n + 1 ) -tuples_on REAL ) .= ( ( n + 1 ) -tuples_on REAL ) ^ ( ( n + 1 ) -tuples_on REAL ) .= ( n + 1 ) -tuples_on REAL ;
ex y being set , f being Function st y = f . n & dom f = NAT & f . 0 = A ( ) & for n holds f . ( n + 1 ) = R ( n , f . n ) & for n holds f . ( n + 1 ) = R ( n , f . n ) ;
func f (#) F -> FinSequence of V means : Def1 : len it = len F & for i be Nat st i in dom it holds it . i = F /. i * F /. i & for i be Nat st i in dom it holds it . i = f /. i * F /. i ;
{ x1 , x2 , x3 , x4 , x4 , x5 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 } = { x1 , x2 , x3 , x4 , 8 } \/ { x4 , 8 , 7 , 8 } \/ { 8 , 8 , 7 } \/ { 8 , 8 , 7 } ;
for n being Nat for x , n being set st x = h . n & h . ( n + 1 ) = o ( x , n ) & x in InputVertices S ( x , n ) & o ( x , n ) in InnerVertices S ( x , n ) holds o ( x , n ) in InnerVertices S ( x , n )
ex S1 being Element of QC-WFF ( Al ( ) ) st [ P , l , e ] = S1 & ( for k being Element of NAT holds P [ k , l , e , e ] ) & ( for k being Element of NAT st P [ k , l , e ] ) & ( for k being Nat st P [ k , l , e ] holds P [ k , l , e ] ) ;
consider P being FinSequence of G2 such that p9 = Product P and for i st i in dom P ex t7 , t7 st P . i = t & t7 = t . i & t7 = t . i & t7 = t . i & t7 = t . i & t7 = t . i ;
for T1 , T2 being non empty TopSpace , P being Basis of T1 , Q being Basis of T2 st the topology of T1 = the topology of T2 & P is Basis of T2 & P is Basis of T2 holds P is Basis of T1 & Q is Basis of T2
consider f being PartFunc of REAL , REAL such that f is_partial_differentiable_in u0 , u0 & r (#) pdiff1 ( f , 3 ) is_partial_differentiable_in u0 , 2 and partdiff ( r (#) pdiff1 ( f , 3 ) , u0 , 2 ) = r (#) pdiff1 ( f , 3 ) & partdiff ( r (#) pdiff1 ( f , 3 ) , u0 , 2 ) = r (#) pdiff1 ( f , 3 ) ;
defpred P [ Nat ] means for F , G be FinSequence of bool ( Seg ( $1 + 1 ) ) , G be FinSequence of REAL st len F = $1 & G = F * s holds Sum ( F ^ G ) = Sum ( F ^ G ) & Sum ( F ^ G ) = Sum ( F ^ G ) ;
ex j st 1 <= j & j < width GoB f & ( GoB f ) * ( 1 , j ) `2 <= s & s * ( 1 , j ) `2 <= ( GoB f ) * ( 1 , j + 1 ) `2 & s * ( 1 , j + 1 ) `2 <= ( GoB f ) * ( 1 , j + 1 ) `2 ;
defpred U [ set , set ] means ex Fa1 be Subset-Family of T st $1 = Fa1 & $2 is open & union ( Fa1 , Fa1 ) is open & ( for n st n <= $1 holds Fa1 . n is open & ( for n st n <= $1 holds Fa1 . n is open implies $2 is open ) & ( for n st n <= $1 holds F . n is discrete implies F . n is discrete ) ;
for p4 being Point of TOP-REAL 2 st LE p4 , p4 , P , p1 , p2 & LE p4 , p , P , p1 , p2 & LE p4 , p , P , p1 , p2 holds LE p4 , p , P , p1 , p2 & LE p , q , P , p1 , p2
f in St ( E , H ) & for g st g . y <> f . y holds for x st x <> y holds g . x = f . ( g . x ) iff for x st x in dom ( f . x ) holds f . x = All ( x , H . ( x , y ) )
ex p9 being Point of TOP-REAL 2 st x = p9 & ( for q being Point of TOP-REAL 2 st q in the carrier of ( TOP-REAL 2 ) holds ( ( q `2 / |. q .| - sn ) ) ^2 >= 0 & ( ( q `2 / |. q .| - sn ) ) ^2 >= 0 & ( q `2 / |. q .| - sn ) ^2 >= 0 ;
assume for d7 being Element of NAT st d7 <= ( ( n + 1 ) -tuples_on NAT ) & d7 <= ( ( n + 1 ) -tuples_on NAT ) holds s1 . ( ( n + 1 ) -tuples_on NAT ) = s2 . ( ( n + 1 ) -tuples_on NAT ) & s2 . ( ( n + 1 ) -tuples_on NAT ) = s2 . ( ( n + 1 ) -tuples_on NAT ) ;
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 ) /\ Ball ( x , r ) & e in Ball ( x , r ) /\ Ball ( x , r ) ;
given r such that 0 < r and for s st 0 < s for x1 , x2 being Point of CNS st x1 in dom f & x2 in dom f & f /. x1 = f /. ( x1 - x2 ) holds |. f /. x1 - f /. x2 .| < r / ( 1 + 1 ) / ( 1 + 1 ) ;
( p | x ) | ( p | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x
for x , h , x , h st x + h in dom sec & x + h in dom sec & x - h / 2 = ( 4 * ( ( 2 * x + h / 2 ) * ( x + h / 2 ) ) ) / ( x + h / 2 ) holds h / ( x + h / 2 ) = ( 4 * x + h / 2 ) / ( x + h / 2 )
for i st i in dom A & len A > 1 & i > 1 holds ( ( A * B ) * ( i , j ) ) = ( A * ( i , j ) ) * ( i , j ) & ( A * ( i , j ) ) = A * ( i , j ) & ( A * ( i , j ) ) = A * ( i , j )
for i be non zero Element of NAT st i in Seg n holds i divides n or ( i = n or i = n or i = n or i = n & i <> n implies h . i = <* 1. F_Complex , n * ( i , n ) *> & h . i = 1. F_Complex ( F_Complex , n )
( ( ( b1 'imp' b2 ) '&' ( c1 '&' c2 ) ) '&' ( ( b1 'or' b2 ) '&' ( c1 '&' c2 ) ) ) '&' ( ( b1 'or' b2 ) '&' ( c1 '&' c2 ) ) '&' ( ( b1 'or' b2 ) '&' ( c1 '&' c2 ) ) '&' ( ( b1 'or' b2 ) '&' ( c1 '&' c2 ) ) '&' ( ( b1 'or' b2 ) '&' ( b1 'or' b2 ) '&' ( c1 '&' c2 ) ) ) '&' ( ( b1 'or' b2 ) '&' ( b2 '&' c2 ) ) '&' ( b2 'or' c2 ) ) '&' ( b1 'or' b2 ) ) '&' ( b2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( b2 '&' c2 ) '&' ( a2 '&' c2 ) '&' ( a2 '&' c2 ) '&' ( a2 '&' c2 ) '&' ( b2 '&' c2 ) '&' ( a2 '&' c2 ) '&' ( a2 '&' c2 ) '&' ( a2 '&' c2 ) '&' ( a2 '&' c2 ) '&' ( a2 '&' c2 ) '&' (
assume that for x holds f . x = ( ( - 1 ) (#) ( cot * ( sin . x ) ) ) and for x st x in dom f holds f . x = ( - 1 ) * ( sin . x ) and for x st x in dom f holds f . x = - 1 / ( sin . x ) and f . x = - 1 / ( sin . x ) and f . x = - 1 / ( sin . x ) and f . x = 1 / ( sin . x - 1 / ( sin . x ) and f . x - 1 / ( sin . x - 1 / ( sin . x - 1 / ( sin . x ) and f . x = - 1 / ( sin . x ) and f . x = 1 / ( sin . x - 1 / ( sin . x - 1 / ( sin . x ) and for x - 1 / ( sin . x ) and f . x
consider R8 , I8 be Real such that R8 = Integral ( M , Re ( F . n ) ) and I8 = Integral ( M , Im ( F . n ) ) and I = dom ( ( Im F ) . n ) and I = dom ( ( Im F ) . n ) and for n be Nat holds I . n = ( Im F ) . n ;
ex k being Element of NAT st ( ex q being Element of NAT st ( for d be Element of product G st q in X & 0 < d & for q be Element of product G st q in X holds ||. ( qx0 , q ) - partdiff ( f , x , k ) .|| < r ) & ||. partdiff ( f , x , k ) - partdiff ( f , x , k ) .|| < r ;
x in { x1 , x2 , x3 , x4 , x5 , 6 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 } \/ { 8 , 8 , 8 , 7 , 8 } \/ { 8 , 8 , 8 } \/ { 8 , 8 , 7 } \/ { 8 , 8 , 7 } \/ { 8 , 8 , 7 } \/ { 8 , 8 , 7 } \/ { 8 } \/ { 8 } \/ { 8 } \/ { 8 , 8 } \/ { 8 } \/ { 8 , 8 } \/ { 8 , 8 } \/ { 8 } \/ { 8 } \/ { 8 } \/ { 8 } \/ { 8 } \/ { 8 } \/ { 8 } \/ { 8 } \/ { 8 } \/ { 8 } \/ { 8 }
( G * ( j , i9 ) ) `2 = G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 ;
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 ) . o .= ( the Arity of S1 ) . o ;
func tree ( T , P , T1 ) -> Tree means : Def1 : q in it iff q in T & q in T or ex p , q st p in P & q in T & p ^ q in T & q in T & p ^ q in T & q in T & p ^ q in T ;
F /. ( k + 1 ) = F . ( k + 1 -' 1 ) .= Fcontradiction . ( p . ( k + 1 -' 1 ) , p . ( k + 1 -' 1 ) ) .= Fcontradiction . ( p . k , p . ( k + 1 -' 1 ) ) .= Fcontradiction . ( p . k , p . ( k + 1 -' 1 ) ) .= FF] . k ;
for A , B , C , D , E , F , G , G , C , D st len B = len C & len C = len C & len D = len C & len D = len C & len C > 0 & len D > 0 & for i st i in dom C holds A * ( B * C , C * D ) = A * ( B * C , C * D )
seq . ( k + 1 ) = 0. F_Complex .= ( Partial_Sums ( seq ) ) . ( k + 1 ) .= Partial_Sums ( seq ) . ( k + 1 ) .= Partial_Sums ( seq ) . ( k + 1 ) .= Partial_Sums ( seq ) . ( k + 1 ) .= Partial_Sums ( seq ) . ( k + 1 ) ;
assume that x in ( the carrier of Cq ) ~ and y in ( the carrier of Cq ) and z in ( the carrier of Cq ) and x = [ x , y ] and y = [ z , x ] ;
defpred P [ Element of NAT ] means for f st len f = $1 & ( for k st k in $1 holds f . k = ( ( the \it false ) --> ( f . k ) ) . ( k + 1 ) ) holds ( ( the \it false of K ) +* ( k + 1 ) ) . ( f . k ) = ( ( the \it false of K ) --> ( k + 1 ) ) . ( f . k ) ;
assume that 1 <= k and k + 1 <= len f and f /. k = G * ( i , j ) and [ i , j ] in Indices G and f /. ( k + 1 ) = G * ( i + 1 , j ) and f /. ( k + 1 ) = G * ( i + 1 , j ) and f /. ( k + 1 ) = G * ( i + 1 , j ) ;
( for q st q < 1 & q `1 > 0 & q `2 > 0 & q `2 <= 0 holds ( q `1 / |. q .| - cn ) / ( 1 + cn ) >= 0 & ( q `1 / |. q .| - cn ) / ( 1 + cn ) >= 0 & ( q `1 / |. q .| - cn ) / ( 1 + cn ) >= 0 ) implies q `1 / |. q .| = q `1 / |. q .| / ( 1 + cn ) & q `1 / ( 1 + cn ) / ( 1 + cn ) / ( 1 + cn ) / ( 1 + cn ) / ( 1 + cn ) / ( 1 + cn ) / ( 1 + cn ) / ( 1 + cn ) / ( 1 + cn ) / ( 1 + cn ) / ( 1 + cn ) / ( 1 + cn ) / ( 1 + cn ) / ( 1 + cn ) / ( 1 + cn ) / ( 1 + cn
for M being non empty metric space , x being Point of M , f being Point of M st x = x `1 & ex x being Point of M st f . n = Ball ( x `1 , ( 1 - x ) / ( 1 - x ) ) & for n being Nat holds f . n = Ball ( x `1 , ( 1 - x ) / ( 1 - x ) )
defpred P [ Element of omega ] means f1 is_differentiable_on Z & f2 is_differentiable_on Z & Z c= dom ( f1 - f2 ) & for x st x in Z holds ( ( f1 - f2 ) `| Z ) . x = f1 . x - f2 . x & ( f1 - f2 ) `| Z ) . x = ( f1 . x - f2 . x ) / ( f2 . x ) ;
defpred P1 [ Nat , Point of CNS ] means ( ( for r be Real st r in Y & $1 < $2 holds ||. ( f /. $1 ) - ( f /. $1 ) .|| < r ) & ||. ( f /. $1 ) - ( f /. $1 ) .|| < r ;
( f ^ mid ( g , 2 , len g ) ) . i = ( mid ( g , 2 , len g ) ) . ( i + 1 ) .= ( mid ( g , 2 , len g ) ) . ( i + 1 ) .= g . ( i + 1 ) .= g . ( i + 1 ) .= ( f /. ( i + 1 ) ) .= ( f /. i ) + ( f /. ( i + 1 ) ) ;
( 1 / 2 * ( n0 + 2 ) ) * ( 2 / ( n0 + 2 ) ) = ( ( 1 / 2 ) * ( ( 2 / ( n0 + 2 ) ) ) * ( 2 / ( n0 + 2 ) ) ) * ( 2 / ( n0 + 2 ) ) ) .= ( 1 / 2 ) * ( 2 / ( n0 + 2 ) ) .= 1 / 2 * ( 2 / ( n0 + 2 ) ) .= 1 / 2 ;
defpred P [ Nat ] means for G being non empty finite strict finite RelStr st G is non empty finite & card G = $1 & for n being Element of NAT st n in $1 holds the carrier of G = ( the carrier of G ) \/ ( the carrier of G ) & the carrier of G = ( the carrier of G ) \/ the carrier of G ;
assume that f /. 1 in Ball ( u , r ) and not 1 <= m and m <= len - 1 and for i st 1 <= i & i <= len - 1 holds LSeg ( f , i ) /\ LSeg ( f , m ) <> {} & LSeg ( f , i ) /\ LSeg ( f , m ) <> {} & LSeg ( f , m ) <> {} ;
defpred P [ Element of NAT ] means ( Partial_Sums ( cos , Z ) . ( n + 1 ) ) . ( ( Partial_Sums ( cos , Z ) . ( n + 1 ) ) . ( x + 1 ) ) = ( Partial_Sums ( cos , Z ) . ( n + 1 ) ) . ( x + 1 ) ;
for x being Element of product F holds x is FinSequence of G & dom x = I & for i be set st i in I holds x . i = ( ( the Sorts of F ) * ( i , x ) ) . i & for i be set st i in I holds x . i = ( ( the Sorts of A ) * ( i , x ) ) . 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 ;
DataPart Comput ( P +* I , s ( ) , LifeSpan ( P +* I ( ) + 1 ) ) = DataPart Comput ( P +* I ( ) +* I ( ) , Comput ( P +* I ( ) , s ( ) +* I ( ) , LifeSpan ( P +* I ( ) ) + 3 ) ) ;
given r such that 0 < r and ]. x0 - r , x0 .[ c= ( dom f1 /\ dom f2 ) and for g st g in dom f1 /\ dom f2 holds f1 . g <= ( f2 . g ) . g and for g st g in dom ( f2 * f1 ) & g in dom ( f2 * f1 ) holds ( f2 * f1 ) . g <= ( f2 . g ) . g ;
for X st X c= dom f1 /\ dom f2 & f1 | X is continuous & f2 | X is continuous & ( for x st x in X holds f1 . x = f1 . x ) & f2 | X is continuous & ( for x st x in X holds f1 . x = - 1 & f2 . x = 1 ) implies f1 + f2 is continuous & f2 | X is continuous & ( f1 + f2 ) | X is continuous & f2 | X is continuous & ( f1 + f2 ) | X is continuous & ( f1 + f2 ) | X is continuous & ( f1 + f2 ) | X is continuous & ( f1 + f2 ) | X is continuous & ( f1 + f2 ) | X is continuous & ( f1 + f2 ) | X is continuous & ( f1 + f2 ) | X is continuous & ( f1 - f2 is continuous & ( f1 - f2 ) | X is continuous & ( f1 - f2 ) | X is continuous & ( f1 - f2 ) | X is continuous & ( f1 - f2 ) | X is continuous & ( f1 - f2 is continuous & (
for L being continuous LATTICE for l being Element of L st l = sup X ex X being Subset of L st l = sup X & for x being Element of L st x in X holds x is directed & x is directed & x is directed & x is directed & x is directed & x is directed
Support ( ee ) in { ( m *' p ) where m is Polynomial of n , L : m in Support ( m *' p ) & p . i = m . ( p /. i ) & p /. i = m . ( p /. i ) } ;
( f1 - f2 ) /. ( lim s1 ) = lim ( f1 /* s1 ) - lim ( f2 /* s1 ) .= lim ( f1 /* s1 ) - lim ( f2 /* s1 ) .= lim ( f1 /* s1 ) - lim ( f2 /* s1 ) .= lim ( f1 /* s1 ) .= lim ( f2 /* s1 ) ;
ex p1 being Element of QC-WFF ( A ) st F . p = g . p1 & for g being Function of [: Al ( ) , D ( ) :] , D ( ) st P [ g , p1 , g . ( len g ) ] & for g being Function st P [ g , p1 , g . ( len g ) ] holds P [ g , p1 ]
( 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 .= f /. j ;
( ( p ^ q ) . ( len p + k ) ) = ( ( p ^ q ) . ( k + 1 ) ) . ( len p + k ) .= ( ( p ^ q ) . ( k + 1 ) ) . ( k + 1 ) .= ( ( p ^ q ) . ( k + 1 ) ) . ( k + 1 ) .= ( ( p ^ q ) . ( k + 1 ) ) . ( k + 1 ) ;
len mid ( upper_volume ( f , D2 , indx ( D2 , D1 , j1 ) + 1 ) , indx ( D2 , D1 , j ) ) + indx ( D2 , D1 , j1 ) = indx ( D2 , D1 , j1 ) + 1 .= indx ( D2 , D1 , j1 ) + 1 .= indx ( D2 , D1 , j1 ) + 1 .= indx ( D2 , D1 , j1 ) + 1 ;
x * y * z = ( x * ( y * z ) ) * ( x * z ) .= ( x * ( y * z ) ) * ( x * z ) .= ( x * ( y * z ) ) * ( x * z ) .= ( x * y ) * ( x * z ) .= x * ( y * z ) .= x * ( y * z ) ;
v . <* x , y *> * ( <* x0 , y0 *> ) . i = partdiff ( v , ( x - y ) * ( x - x0 ) ) + ( partdiff ( u , ( x - x0 ) * ( x - x0 ) ) ) + ( partdiff ( u , ( x - x0 ) * ( x - x0 ) ) ) + ( partdiff ( u , ( x - x0 ) * ( x - x0 ) ) ) ;
i * i = <* 0 * ( - 1 ) * ( 1 / ( 2 * 1 ) ) - ( 1 / ( 2 * 1 ) ) * ( 1 / ( 2 * 1 ) ) + ( 1 / ( 2 * 1 ) ) * ( 1 / ( 2 * 1 ) ) * ( 1 / ( 2 * 1 ) ) + ( 1 / ( 2 * 1 ) ) * ( 1 / ( 2 * 1 ) ) ) .= ( - 1 / ( 2 * 1 ) ) * ( 1 / ( 2 * 1 ) ) ;
Sum ( L (#) F ) = Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) + Sum ( ( L (#) F1 ) ^ ( ( L (#) F2 ) ^ ( ( L (#) F2 ) ^ ( ( L (#) F2 ) ^ ( ( L (#) F1 ) ^ ( ( L (#) F1 ) ^ ( ( L (#) F1 ) ^ ( ( L (#) F2 ) ^ ( ( L (#) F1 ) ^ ( ( L (#) F1 ) ) ) ) ) ) ) ) ) .= Sum ( ( ( L (#) F1 ) ^ ( ( L (#) F1 ) ) ) ) + ( Sum ( ( L (#) F1 ) ^ ( ( L (#) F1 ) ) ) ) .= Sum ( ( L (#) F1 ) ^ ( ( L (#) F1 ) ) ) .= Sum ( ( ( L (#) F1 ) ) + Sum ( ( L (#) F1 ) ) + Sum ( ( ( L (#) F1 ) ) .= Sum ( ( L (#) F1 ) ^ ( ( L (#) F1 ) ) ) .= Sum ( ( L (#) F1 ) ) .= Sum ( L (#) F1 ) ) + Sum ( L (#) F2 ) )
ex r be Real st for e be Real st 0 < e ex Y1 be finite Subset of REAL st Y1 is non empty & for Y1 be finite Subset of X st Y1 is non empty & Y1 c= Y & Y1 is non empty & Y is non empty & for Y1 be finite Subset of X st Y1 is non empty & Y1 is non empty holds |. ( Y1 - Y2 ) .| < r / 2
( GoB f ) * ( i , j + 1 ) `1 = f /. ( k + 2 ) & ( GoB f ) * ( i + 1 , j + 1 ) `2 = f /. ( k + 2 ) or ( GoB f ) * ( i + 1 , j + 1 ) `2 = f /. ( k + 2 ) or ( GoB f ) * ( i + 1 , j + 1 ) `2 = f /. ( k + 2 ) ;
( ( - 1 ) (#) ( cos . x ) ) ^2 = ( - 1 ) * ( sin . x ) ^2 .= ( - 1 ) * ( sin . x ) ^2 .= ( - 1 ) * ( sin . x ) ^2 .= ( - 1 ) * ( sin . x ) ^2 .= ( - 1 ) * ( sin . x ) ^2 .= ( - 1 ) * ( sin . x ) ^2 ;
( - ( b - sqrt ( a , b ) ) + sqrt ( a , c ) ) / ( 2 * a ) > 0 & ( - b ) / ( 2 * a ) + sqrt ( b - sqrt ( a , b ) ) / ( 2 * a ) > 0 ;
Suppose inf ( \mathopen { \uparrow } L ) /\ X = X and ex_sup_of X , L and for x st x in X holds x is maximal & x is maximal & x is ) and for x st x in X holds "\/" ( ( ( subrelstr X ) /\ ( subrelstr X ) ) , L ) = "/\" ( ( subrelstr X ) /\ ( subrelstr X ) , L ) ;
( ( for j holds ( ( j , i ) --> ( j , j ) ) ) . ( j , i ) = ( j , i ) --> ( j , j ) ) . ( j , i ) & ( j , i ) = j implies ( j , i ) = ( j , i ) --> ( j , i ) ) . ( j , i )