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contradiction ;
contradiction ;
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contradiction ;
contradiction ;
contradiction ;
contradiction ;
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contradiction ;
contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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 commutative ;
b in A ;
d divides a ;
i < n ;
s <= b ;
b in B ;
let r ;
B is one-to-one ;
R is total ;
x = 2 ;
d in D ;
let c ;
let c ;
b = Y ;
0 < k ;
let b ;
let n ;
r <= b ;
x in X ;
i >= 8 ;
let n ;
let n ;
y in f ;
let n ;
1 < j ;
a in L ;
C is boundary ;
a in A ;
1 < x ;
S is finite ;
u in I ;
z << z ;
x in V ;
r < t ;
let t ;
x c= y ;
a <= b ;
m in dom f ;
assume f is prime ;
not x in Y ;
z = -infty ;
let k be Nat ;
K ` is being_line ;
assume n >= N ;
assume n >= N ;
assume X is 1 -element ;
assume x in I ;
q is let 0 ;
assume c in x ;
p > 0 ;
assume x in Z ;
assume x in Z ;
1 <= kr2 ;
assume m <= i ;
assume G is commutative ;
assume a divides b ;
assume P is closed ;
\bf 2 > 0 ;
assume q in A ;
W is not bounded ;
f is ' one-to-one ;
assume A is boundary ;
g is special ;
assume i > j ;
assume t in X ;
assume n <= m ;
assume x in W ;
assume r in X ;
assume x in A ;
assume b is even ;
assume i in I ;
assume 1 <= k ;
X is non empty ;
assume x in X ;
assume n in M ;
assume b in X ;
assume x in A ;
assume T c= W ;
assume s is atomic ;
b `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 non empty set ;
let C be Category ;
let x be element ;
let k be Nat ;
let x be element ;
let x be element ;
let e be element ;
let x be element ;
P [ 0 ]
let c 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 \rbrace ;
Q halts_on s ;
x in such that x in such ;
M < m + 1 ;
T2 is open ;
z in b from a ;
R2 is well-ordering ;
1 <= k + 1 ;
i > n + 1 ;
q1 is one-to-one ;
let x be trivial set ;
PP is one-to-one ;
n <= n + 2 ;
1 <= k + 1 ;
1 <= k + 1 ;
let e be Real ;
i < i + 1 ;
p3 in P ;
p1 in K ;
y in C1 ;
k + 1 <= n ;
let a be Real , x be Real ;
X |- r => p ;
x in { A } ;
let n be Nat ;
let k be Nat ;
let k be Nat ;
let m be Nat ;
0 < 0 + k ;
f is_differentiable_in x ;
let x0 ;
let E be Ordinal ;
o \rbrace <> o9 ;
O <> O2 ;
let r be Real ;
let f be FinSeq-Location ;
let i be Nat ;
let n be Nat ;
Cl A = A ;
L c= Cl L ;
A /\ M = B ;
let V be RealUnitarySpace , W be Subspace of V ;
not s in Y |^ 0 ;
rng f <= w
b "/\" e = b ;
m = m3 ;
t in h . D ;
P [ 0 ] ;
assume z = x * y ;
S . n is bounded ;
let V be RealUnitarySpace , W be Subspace of V ;
P [ 1 ] ;
P [ {} ] ;
C1 is component of C ;
H = G . i ;
1 <= i `1 + 1 ;
F . m in A ;
f . o = o ;
P [ 0 ] ;
aLet <= being Real ;
R [ 0 ] ;
b in f .: X ;
assume q = q2 ;
x in [#] V ;
f . u = 0 ;
assume e1 > 0 ;
let V be RealUnitarySpace , W be Subspace of V ;
s is trivial & s is non trivial ;
dom c = Q
P [ 0 ] ;
f . n in T ;
N . j in S ;
let T be complete LATTICE , f be Function of T , T ;
the Arrows of F is one-to-one
sgn x = 1 ;
k in support a ;
1 in Seg 1 ;
rng f = X ;
len T in X ;
vbeing < n ;
Sbeing bounded Function of S , T ;
assume p = p2 ;
len f = n ;
assume x in P1 ;
i in dom q ;
let U0 , A , B ;
p-25 = c ;
j in dom h ;
let k ;
f | Z is continuous ;
k in dom G ;
UBD C = B ;
1 <= len M ;
p in being element ;
1 <= jj ;
set A = Cl Cl i ;
card a [= c ;
e in rng f ;
cluster B \oplus A -> empty ;
H has { H } or H is \rm len } H is not empty ;
assume x0 <= m ;
T is increasing ;
e2 <> e2 & e2 <> a4 ;
Z c= dom g ;
dom p = X ;
H is proper implies 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 in union M
assume not x in REAL ;
Im f is complete ;
x in Int y ;
dom F = M ;
a in On W ;
assume e in A ( ) ;
C c= C-26 ;
mm <> {} ;
let x be Element of Y ;
let f be let of TOP-REAL 2 , x be element ;
not n in Seg 3 ;
assume X in f .: A ;
assume that p <= n and p <= m ;
assume not u in { v } ;
d is Element of A ;
A |^ b misses B ;
e in v in dom ( T | E ) ;
- y in I ;
let A be non empty set , x be Element of A ;
P0 = 1 ;
assume r in F . k ;
assume f is simple function ;
let A be \lbrack set ;
rng f c= NAT ;
assume P [ k ] ;
f6 <> {} ;
let o be Ordinal ;
assume x is sum of from squares ;
assume not v in { 1 } ;
let Ij , I ;
assume that 1 <= j and j < l ;
v = - u ;
assume s . b > 0 ;
da in dom g ;
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 RelStr ;
a , b // b , a ;
a * b = p * q ;
assume x , y are_the space ;
assume x in [#] ( f | A ) ;
[ a , c ] in X ;
mm <> {} ;
M + N c= M + M ;
assume M is connected hhz ;
assume f is additive inbsnis r-n) ;
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 <= width G ;
f | A is as as as as as continuous Function ;
f . x be 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 * m ) divides m ;
set m = max A ;
[ x , x ] in R ;
assume x in succ 0 ;
a in sup phi ;
Cj in X ;
q2 c= C1 & q2 c= C2 ;
a2 < c2 & a2 < c2 ;
s2 is 0 -started ;
IC s = 0 ;
s4 = s4 & s4 = s4 ;
let V ;
let x , y be element ;
let x be Element of T ;
assume a in rng F ;
x in dom T `2 ;
let S be cluster MSAlgebra over L ;
y " <> 0 ;
y " <> 0 ;
0. V = uw ;
y2 , y , w , y thesis ;
R8 ;
let a , b be Real , x be 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 , l .]
let i , j be Nat ;
let s be State of A , x be Element of NAT ;
s4 . n = N ;
set y = ( x `1 ) / ( x `2 ) ;
mi in dom g & mj in dom g ;
l . 2 = y1 ;
|. g . y .| <= r ;
f . x in CV ;
V1 is non empty ;
let x be Element of X ;
0 <> f . g2 ;
f2 /* q is convergent ;
f . i is_measurable_on E ;
assume \xi in N-22 ;
reconsider i = i as Ordinal ;
r * v = 0. X ;
rng f c= INT & rng f c= INT ;
G = 0 .--> goto 0 ;
let A be Subset of X ;
assume A9 is dense & A2 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 ;
x9 c= Z1 & y9 c= Z1 implies x9 , y9 c= y9
dom f = [: C1 , C2 :] ;
assume [ a , y ] in X ;
Re ( seq . n ) is convergent ;
assume a1 = b1 & a2 = b2 ;
A = Int ( A ) ;
a <= b or b <= a ;
n + 1 in dom f ;
let F be Instruction of S , a be Nat ;
assume that r2 > x0 and x0 < r2 ;
let Y be non empty set , x be Element of Y ;
2 * x in dom W ;
m in dom ( g2 * f1 ) ;
n in dom ( g1 - g2 ) ;
k + 1 in dom f ;
not the still of { s } is finite ;
assume that x1 <> x2 and x2 <> x3 ;
v2 in V1 & v2 in V1 ;
not [ b `1 , b ] in T ;
ii + 1 = i ;
T c= such that T c= D1 ;
( l - 1 ) / l = 0 ;
let n be Nat ;
( t `2 ) ^2 = r ;
AA2 : is_integrable_on M & A = { x } ;
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 normal sequence ;
assume c2 = b2 & b2 = b3 ;
assume h in [. q , p .] ;
1 + 1 <= len C ;
not c in B . m1 ;
cluster R .: X -> empty ;
p . n = H . n ;
assume that vsequence is Cauchy and lim vsequence = 0 ;
IC s3 = 0 .= 0 ;
k in N or k in K ;
F1 \/ F2 c= F \/ G ;
Int ( G1 \/ G2 ) <> {} ;
( z `2 ) ^2 = 0 ;
LSeg ( p1 , p2 ) <> p1 ;
assume z in { y , w } ;
MaxADSet ( a ) c= F ;
ex_sup_of downarrow s , S ;
f . x <= f . y ;
let T be up-complete non empty reflexive antisymmetric 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 non empty implies G is one-to-one ;
A \/ { a } c= B ;
0. V = 0. Y .= 0. V ;
let I be as <= <= <= S ;
f-24 . x = 1 / ( x + 1 ) ;
assume z \ x = 0. X ;
C4 = 2 to_power n ;
let B be sequence of Sigma ;
assume X1 = p .: D ;
n + l2 in NAT & n + l2 in NAT ;
f " P is compact ;
assume x1 in REAL & x2 in REAL ;
p1 = ( K + 1 ) * ( K + 1 ) ;
M . k = <*> ( REAL ) ;
phi . 0 in rng phi ;
of MMA ;
assume z0 <> 0. L & z0 <> 0. L ;
n < N7 . k ;
0 <= ( seq . 0 ) . ( 0 + 1 ) ;
- q + p = v ;
{ v } is Subset of B ;
set g = f /. 1 ;
[: R , R :] is stable implies R is stable
set cR = Vertices R , c = R ;
p0 c= P3 & card I = card J + 1 ;
x in [. 0 , 1 .] ;
f . y in dom F ;
let T be Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott of S ;
ex_inf_of the carrier of S , S ;
downarrow a = downarrow b & downarrow a = { a } ;
P , C , K , L , N ;
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 - x .|| <= r ;
assume that Y c= field Q and Y <> {} ;
a ~ , b ~ are_isomorphic ;
assume a in A ( ) ;
k in dom ( q ^ <* a *> ) ;
p is } -st p is FinSequence of S ;
i -' 1 = i-1 ;
f | A is one-to-one ;
assume x in f .: [: X , Y :] ;
i2 - i1 = 0 implies i2 = 1
j2 + 1 <= i2 + 1 ;
g " * a in N ;
K <> { [ {} , {} ] } ;
cluster -> strict of \HM { of } -> strict \HM { \bf \rbrace ;
|. 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 & rng I c= Seg n ;
assume that k in dom C and k <> i ;
1 + 1-1 <= i + j ;
dom S = dom F & rng S c= dom G ;
let s be Element of NAT ;
let R be ManySortedSet of A ;
let n be Element of NAT ;
let S be non empty non void non void non void holds S is |. -\subseteq the topology of S ;
let f be ManySortedSet of I ;
let z be Element of COMPLEX , x be Element of COMPLEX ;
u in { \hbox { \boldmath $ g } , f } ;
2 * n < 2 * n ;
let x , y be set ;
B-11 c= V1 & B-15 c= V1 ;
assume I is_closed_on s , P ;
U = U & U = { p } ;
M /. 1 = z /. 1 ;
x9 = x9 & y9 = y9 ;
i + 1 < n + 1 + 1 ;
x in { {} , <* 0 *> } ;
( f | n ) . x <= ( f | n ) . x ;
let l be Element of L ;
x in dom ( F . -17 ) ;
let i be Element of NAT ;
r8 is ( len ( - 1 ) ) -element ;
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 be be be be be Subset-Family of Omega ;
set r = { - 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 SubLattice of L ;
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 on B ;
f-3 = NAT --> 0 ;
let A , B be Subset of V ;
z1 = P1 . j & z2 = P1 . 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 & { x0 } c= dom g ;
let B be non-empty ManySortedSet of I , A be ManySortedSet of I ;
( PI / 2 ) < Arg z ;
reconsider z9 = 0 as Nat ;
LIN a , d , c & LIN a , d , c ;
[ y , x ] in IX ;
( Q , 3 ) `1 = 0 & ( Q , 3 ) `2 = 0 ;
set j = x0 div m ;
assume a in { x , y , c } ;
j2 - ( j + 1 ) > 0 ;
I \! \mathop { \rm \hbox { - } not } = 1 ;
[ y , d ] in F-8 ;
let f be Function of X , Y ;
set A2 = ( B * C ) @ ;
s1 , s2 , s3 , s3 , s4 , r2 , s2 , s3 , r1 , r2 , s1 , s2 , s2 , s3 ;
j1 -' 1 = 0 & j1 - 1 = 0 ;
set m2 = 2 * n + j ;
reconsider t = t as bag of n ;
I2 . j = m . j ;
i |^ s , n are_relative_prime ;
set g = f | D-21 ;
assume that X is lower and 0 <= r ;
( - ( - ( - 1 ) ) ) `1 = 1 ;
a < ( p3 `1 ) / |. p3 .| ;
L \ { m } c= UBD C ;
x in Ball ( x , 10 ) ;
not a in LSeg ( c , m ) ;
1 <= ( i1 -' 1 ) + 1 ;
1 <= ( i1 -' 1 ) + 1 ;
i + i2 <= len h ;
x = W-min ( P ) & x in L~ f ;
[ x , z ] in X ~ Z ;
assume y in [. x0 , x .] ;
assume p = <* 1 , 2 , 3 *> ;
len <* A1 *> = 1 ;
set H = h . ( g . -3 ) ;
card b * a = |. a .| ;
Shift ( w , 0 ) |= v ;
set h = h2 ** h1 ;
assume x in ( ( X2 /\ 4 ) /\ ( X1 /\ X2 ) ) ;
||. h .|| < d1 & ||. h .|| < d ;
not x in the carrier of f & not x in the carrier of f ;
f . y = F ( y ) ;
for n holds X [ n ] ;
k - l = kl2 ;
<* p , q *> /. 2 = q ;
let S be Subset of the carrier of Y ;
let P , Q be Element of >= >= >= t ;
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 -\sqcup let i be Basis of K ;
let r be non positive Real ;
M , v |= ( x , x ) \hbox { y } ;
v + w = 0. ( Z ) ;
P [ len ( F | n ) ] ;
assume InsCode ( ( i + 5 ) + 4 ) = 8 ;
the zero Element of M = 0 implies the carrier of M = 0
cluster z * seq -> summable ;
let O be Subset of the carrier of C ;
||. f .|| | X is continuous ;
x2 = g . ( j + 1 ) ;
cluster non empty non empty for Element of AllTermsOf S ;
reconsider l1 = lk - 1 as Nat ;
vI is Vertex of r2 & vI is Vertex of r2 ;
T1 is SubSpace of T2 & T2 is SubSpace of T2 ;
Q1 /\ Q19 <> {} & Q1 /\ Q19 <> {} ;
let k be Nat ;
q " is Element of X & q is Element of X ;
F . t is set of of of M ;
assume n <> 0 & n <> 1 ;
set en = EmptyBag n , e = EmptyBag n ;
let b be Element of Bags n ;
assume for i holds b . i is commutative ;
x is root implies ( p `1 ) ^2 = ( p `2 ) ^2
not r in ]. p , q .[ ;
let R be FinSequence of REAL , a be Element of REAL ;
S7 does not destroy b1 , b2 ;
IC SCM R <> a & IC SCM R <> a ;
|. - |[ x , y ]| .| >= r ;
1 * ( seq . n ) = seq . ( n + 1 ) ;
let x be FinSequence of NAT , k be Nat ;
let f be Function of C , D , g be Function of C , D ;
for a holds 0. L + a = a
IC s = s . NAT .= IC s ;
H + G = F- ( GG + G ) ;
CS1 . x = x2 & CS2 . x = y2 ;
f1 = f .= f2 .= f2 .= f1 ;
Sum <* p . 0 *> = p . 0 ;
assume v + W = v + u + W ;
{ a1 } = { a2 } & { a2 } = { a2 } ;
a1 , b1 _|_ b , a ;
d2 , o _|_ o , a3 ;
IF is reflexive implies ( for x st x in CF holds x <> 0. F )
Ij is antisymmetric implies ( for x st x in Cj holds x <= ( x + y ) / ( x - y ) )
upper_bound rng ( H1 ^ H2 ) = e & upper_bound rng ( H1 ^ H2 ) = e ;
x = ( a * a9 ) * ( a * b9 ) ;
|. p1 .| ^2 >= 1 ;
assume j2 -' 1 < 1 - 1 ;
rng s c= dom ( f1 + f2 ) ;
assume that support a misses support b and not a in support b ;
let L be associative non empty doubleLoopStr , x be Element of L ;
s " + 0 < n + 1 ;
p . c = ( f " ) . 1 .= f . c ;
R . n <= R . ( n + 1 ) ;
Directed I = I1 +* ( 1 , card I + 1 ) ;
set f = + ( x , y , r ) ;
cluster Ball ( x , r ) -> bounded ;
consider r being Real such that r in A ;
cluster non empty -> non empty for NAT -defined Function ;
let X be non empty directed Subset of S ;
let S be non empty full SubRelStr of L ;
cluster <* if . N , N *> -> complete for non trivial LATTICE ;
( 1 - a ) " = a ;
( q . {} ) `1 = o ;
n - ( i -' 1 ) > 0 ;
assume ( 1 - t ) / 2 <= t `1 ;
card B = k + 1 - 1 ;
x in union rng ( f | n ) ;
assume x in the carrier of R & y in the carrier of R ;
d in dom f ;
f . 1 = L . ( F . 1 ) ;
the carrier' of G = { v } & the carrier' of G = { v } ;
let G be : ww_Graph ;
e , v9 , v9 , v2 , w , y ;
c . ( i + 1 ) in rng c ;
f2 /* ( q ^\ k ) is divergent_to-infty & f2 /* ( q ^\ k ) is divergent_to-infty ;
set z1 = - ( - z2 ) , z2 = - ( - z2 ) ;
assume w is \mathop { las of S , G ;
set f = p \! \mathop { t } , g = p \! \mathop { t } , h = p \! \mathop { t } , h = p \! \mathop { t } , n = m \! \mathop
let c be Object of C ;
assume ex a st P [ a ] ;
let x be Element of REAL m , y be Element of REAL m ;
let IX be Subset-Family of X ;
reconsider p = p as Element of NAT ;
let v , w be Point of X ;
let s be State of SCM+FSA , I be Program of SCM+FSA ;
p is FinSequence of ( the InstructionsF of SCM+FSA ) * ;
stop I c= P & stop I c= P ;
set ci = ( f /. i ) `1 , fj = ( f /. i ) `2 ;
w ^ t ^ s ^ t ^ s ^ t ^ t ^ s ^ t ^ s ^ t ^ t ^ s ^ t ^ s ^ t ^ t ^ s ^ t ^ t ^ s ^
W1 /\ W = ( W1 /\ W ) + ( W2 /\ 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 implies c <> 0
ord x = 1 & x is not positive ;
set g2 = lim ( seq ^\ k ) , g1 = lim ( seq ^\ k ) ;
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 ;
( the InternalRel of X ) \/ ( R1 \/ R2 ) = the InternalRel of X ;
( sin . x ) ^2 <> 0 & ( sin . x ) ^2 <> 0 ;
( ( exp_R * sin ) `| Z ) . x > 0 ;
o1 in ( X /\ O2 ) /\ ( X /\ O2 ) ;
e , v9 , v9 , v2 , w , y ;
r3 > ( 1 / 2 ) * 0 ;
x in P .: ( F -ideal ) ;
let J be closed closed Ideal of R ;
h . ( p1 , O ) = f2 . O ;
Index ( p , f ) + 1 <= j ;
len ( q ^ <* M *> ) = width M ;
the carrier of L c= A & the carrier of L c= A ;
dom f c= union rng ( F | -10 ) ;
k + 1 in ( support ( n ) ) \ { k } ;
let X be ManySortedSet of the carrier of S ;
[ x `1 , y `2 ] in ( InnerVertices R ) ;
i = D1 or i = D2 or i = D1 ;
assume a mod n = b mod n ;
h . x2 = g . x1 & h . x2 = g . x2 ;
F c= 2 -tuples_on the carrier of X ;
reconsider w = |. s1 .| as Real_Sequence of REAL ;
( 1 - m ) * m + r < p ;
dom f = dom ( ( I --> a ) * f ) ;
[#] ( ( ( TOP-REAL 2 ) | K1 ) | K1 ) = [#] ( ( ( TOP-REAL 2 ) | K1 ) | K1 ) ;
cluster - x -> ExtReal -> extended real ;
then { da } c= A & A is closed ;
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 + v in W3
reconsider y = y as Element of L2 ;
N is full SubRelStr of ( T |^ the carrier of S ) ;
sup { x , y } = c "\/" c ;
g . n = n to_power 1 .= n ;
h . J = EqClass ( u , J ) ;
let seq be \mathclose of X , x be Point of X ;
dist ( x `1 , y ) < ( r / 2 ) ;
reconsider mm = m - n as Element of NAT ;
x- x0 < r1 - x0 + R ;
reconsider P = P ` as strict Subgroup of N ;
set g1 = p * idseq ( q `1 ) , g2 = p * idseq ( q `2 ) ;
let n , m , k be non zero Nat ;
assume that 0 < e and f | A is lower ;
D2 . ( I8 ) in { x } ;
cluster -> subcondensed open -> of T ;
let P be compact non empty Subset of TOP-REAL 2 , p1 , p2 be Point of TOP-REAL 2 ;
Gik in LSeg ( cos , 1 ) \/ LSeg ( cos , 1 ) ;
let n be Element of NAT , x be Element of REAL ;
reconsider S8 = S as Subset of T ;
dom ( i .--> X ' ) = { i } ;
let X be non-empty ManySortedSet of S ;
let X be non-empty ManySortedSet of S ;
op ( 1 , {} ) c= { [ {} , {} ] }
reconsider m = 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 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 = T7 " as Point of TOP-REAL 2 ;
set q = h * p ^ <* d *> ;
z2 in U . ( y2 ) /\ Q . ( y2 ) ;
A |^ 0 = { <* E *> , <* E *> } ;
len ( W2 ) = len W + 2 .= len ( W + W2 ) ;
len ( h2 - h2 ) in dom h2 & len ( h2 - h2 ) = len h2 ;
i + 1 in Seg ( len s2 + 1 ) ;
z in dom ( g1 + g2 ) /\ dom ( f1 + f2 ) ;
assume that p2 = E-max ( K ) and p2 <> W-min ( K ) ;
len G + 1 <= ( i1 + 1 ) + 1 ;
f1 (#) f2 is convergent & f2 (#) ( f1 (#) f2 ) is convergent & lim ( f1 (#) f2 ) = 0 ;
cluster ( s + s1 ) | ( n + 1 ) -> summable ;
assume that j in dom ( M1 /. i ) and i <> j ;
let A , B , C be Subset of X ;
let x , y , z be Point of X , p be Point of X ;
b / ( 4 * a * c ) >= 0 ;
<* xy *> ^ <* y *> \subseteq x ;
a , b in { a , b } ;
len ( p2 - p ) = ( len p2 ) - ( len p ) ;
ex x being element st x in dom R & R . x = y ;
len q = len ( K * G ) ;
s1 = Initialize Initialized s , P1 = P +* Initialize s , P2 = P +* Initialize s , P3 = P +* Initialize s , s4 = Comput ( P3 , s3 , 1 ) , s4 = P3 ;
consider w being Nat such that q = z + w ;
x ` ` is Element of x & x ` is Element of x ;
k = 0 & n <> k or k > n ;
then X is discrete for A being Subset of X ;
for x st x in L holds x is FinSequence of REAL
||. f /. c .|| <= r1 & ||. f /. c .|| <= r2 ;
c in ]. p , q .[ & not c in { p } ;
reconsider V = V as Subset of the topology of TOP-REAL n ;
let N , M be being being being being being being being being being being being being being being being being being being being being being Subset of L ;
then z is_>=_than compactbelow x & z is_>=_than compactbelow x ;
M , f .] = f & M , g .] = g ;
( ( >= 1 ) to_power 1 ) = TRUE & ( ( L to_power 1 ) to_power 1 ) = TRUE ;
dom g = dom f & rng g c= X ;
mode \cal o is \cal \cal \cal \cal o , set , set , set , set ] means $2 = the \cal o of G ;
[ i , j ] in Indices ( M @ ) ;
reconsider s = x " as Element of H ;
let f be Element of ( dom ( Subformulae p ) ) \ { x } ;
F1 . ( a1 , - a2 ) = G1 . ( a1 , - a2 ) ;
cluster \HM ( a , b , r ) -> compact ;
let a , b , c , d be Real ;
rng s c= dom ( 1 / ( 1 + s ) ) ;
curry ( F-19 , k ) is additive additive ;
set k2 = card dom B , k2 = card dom C , s3 = card I + 1 ;
set G = coprod X , A = V ;
reconsider a = [ x , s ] as Object of G ;
let a , b be Element of MM , x be Element of M ;
reconsider s1 = s as Element of ( the carrier of S ) * ;
rng p c= the carrier of L & rng p c= the carrier of L ;
let d be Subset of the Sorts of A ;
( x .|. x = 0 iff x = 0. W ) ;
I-21 in dom stop I & IH in dom stop I ;
let g be continuous Function of X | B , Y ;
reconsider D = Y as Subset of TOP-REAL n ;
reconsider i0 = len ( p1 ^ p2 ) - 1 as Integer ;
dom f = the carrier of S & rng f c= the carrier of T ;
rng h c= union ( the carrier of J ) & rng h c= the carrier of K ;
cluster All ( x , H ) -> reconsider reconsider r = All ( x , H ) as Element of D ;
d * N1 |^ ( N1 * N2 ) > N1 * 1 ;
]. a , b .[ c= [. a , b .] ;
set g = f " ( D1 \/ D2 ) ;
dom ( p | [: REAL , REAL :] ) = [: REAL , REAL :] ;
3 + - 2 <= k + - 2 + - 2 ;
tan is_differentiable_in ( ( arccot * arccot ) `| Z ) . x ;
x in rng ( f /^ ( p -' 1 ) ) ;
let f , g be FinSequence of D ;
[: p , q :] in the carrier of [: S1 , S2 :] ;
rng f " { 0 } = dom f .= { 0 } ;
( the Source of G ) . e = v & ( the Source of G ) . e = v ;
width G - 1 < width G - 1 ;
assume v in rng ( S | E1 ) ;
assume x is root or x is root or x is root & x is root ;
assume 0 in rng ( g2 | A ) & 0 < 1 ;
let q be Point of TOP-REAL 2 , r be Real ;
let p be Point of TOP-REAL 2 , r be Real ;
dist ( O , u ) <= |. p2 .| + 1 ;
assume dist ( x , b ) < dist ( a , b ) ;
<* S7 *> is_in the InternalRel of C7 & <* S7 *> is in the carrier of C-20 ;
i <= len ( G * ( i1 , 1 ) ) - 1 ;
let p be Point of TOP-REAL 2 , r be Real ;
x1 in the carrier of I[01] & x2 in the carrier of I[01] ;
set p1 = f /. i , p2 = f /. ( i + 1 ) ;
g in { g2 : r < g2 & g2 < r } ;
Q2 = Sp2 " ( Q \ R ) .= ( S \ R ) " Q ;
( ( 1 / 2 ) (#) ( 1 / 2 ) ) is summable ;
- p + I c= - p + A + - I ;
n < LifeSpan ( P1 , s1 ) + 1 ;
CurInstr ( p1 , s1 ) = i & CurInstr ( p1 , s1 ) = i ;
A /\ Cl { x } <> {} ;
rng f c= ]. r , r + 1 .[ ;
let g be Function of S , V ;
let f be Function of L1 , L2 , g be Function of L2 , L1 ;
reconsider z = z as Element of ( CompactSublatt L ) ;
let f be Function of S , T ;
reconsider g = g as Morphism of c opp , b opp ;
[ s , I ] in S ~ ( A , I ) ;
len ( the connectives of C ) = 4 & len ( the connectives of C ) = 5 ;
let C1 , C2 be Sub\rbrace of C , a , b be Subfunctor of C1 ;
reconsider V1 = V as Subset of X | B ;
attr p is valid means : Def3 : All ( x , p ) is valid ;
assume that X c= dom f and f .: X c= dom g and g .: X c= dom f ;
H |^ ( a " ) is Subgroup of H |^ a ;
let A1 be : let A be \mathclose of O , E be Element of O ;
p2 , r3 , r2 , r3 is_collinear & p2 , q2 , r1 is_collinear ;
consider x being element such that x in v ^ K ;
not x in { 0. TOP-REAL 2 } & not x in { 0. TOP-REAL 2 } ;
p in [#] ( ( I[01] | B11 ) | B11 ) ;
0 . ( ( E . n ) - 1 ) < M . ( ( E . n ) - 1 ) ;
^ ( c , c ) = c ;
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . s2 ) & a2 in dom ( F . s2 ) ;
cluster -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> from of L ;
set i1 = the Nat , i2 = the Element of dom ( the Arity of S ) ;
let s be 0 -started State of SCM+FSA , k be Nat ;
assume y in ( f1 \/ f2 ) .: A ;
f . ( len f ) = f /. len f .= f /. len f ;
x , f . x '||' f . x , f . y ;
attr X c= Y means : Def3 : cos X c= cos Y & cos Y c= cos Y ;
let y be upper bound of Y , x be Element of X ;
cluster ( x `1 ) / ( x `2 ) -> non trivial for ;
set S = <* Bags n , ( Bags n ) *> ;
set T = [. 0 , 1 / 2 .] ;
1 in dom mid ( f , 1 , 1 ) ;
( 4 * PI ) / 2 < ( 2 * PI ) / 2 ;
x2 in dom ( f1 + f2 ) /\ dom ( f2 + f3 ) ;
O c= dom I & { {} } = { {} } ;
( the Source of G ) . x = v & ( the Source of G ) . x = v ;
{ HT ( f , T ) } c= Support f ;
reconsider h = R . k as Polynomial of n , L ;
ex b being Element of G st y = b * H ;
let x , y , z be Element of G opp ;
h19 . i = f . ( h . i ) ;
( p `1 ) ^2 = ( p1 `1 ) ^2 + ( p1 `2 ) ^2 ;
i + 1 <= len Cage ( C , n ) ;
len <* P *> = len P & len <* P *> = len P ;
set N-26 = the be Subset of N , x = the Element of N ;
len gSet + ( x + 1 ) - 1 <= x ;
a on B & b on B & a on B implies a on B
reconsider ra1 = r * I . v as FinSequence of REAL ;
consider d such that x = d and a in d ;
given u such that u in W and x = v + u ;
len f /. ( \downharpoonright n ) = len f -' n ;
set q2 = N-min L~ Cage ( C , n ) , q2 = q2 ;
set S = `1 , T = `2 , S = `2 , T = [ S , T ] ;
MaxADSet ( b ) c= MaxADSet ( P /\ Q ) ;
Cl ( G . q1 ) c= F . r2 & F . q2 c= G . r2 ;
f " D meets h " V & f " V meets h " V ;
reconsider D = E 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 , Y ) ;
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 & f1 . ( b1 , b2 ) = b2 ;
the carrier' of G = E \/ { E } .= { E } ;
reconsider m = len ( k - 1 ) as Element of NAT ;
set S1 = LSeg ( n , UMP C ) , S2 = LSeg ( n , UMP C ) ;
[ i , j ] in Indices ( M1 + M2 ) ;
assume that P c= Seg m and M is \HM { i } and P is of Seg m ;
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 ;
p-7 . i = pi1 . i .= pi2 . i ;
let PA , G be a_partition of Y , z be Element of Y ;
attr 0 < r & 1 < 1 & r < 1 ;
rng ( AffineMap ( a , X ) ) = [#] X .= [#] X ;
reconsider x = x , y = y as Element of K ;
consider k such that z = f . k and n <= k ;
consider x being element such that x in X \ { p } ;
len ( canFS ( s ) ) = card s .= card ( s ) ;
reconsider x2 = x1 , y2 = x2 as Element of L2 ;
Q in FinMeetCl ( the topology of X ) & Q in the topology of X ;
dom ( f | 0 ) c= dom ( u | 0 ) ;
attr n divides m & m divides n implies n = m ;
reconsider x = x as Point of [: I[01] , I[01] :] ;
a in ;
not y0 in the still of f & not ( not ( ex g st g in the carrier' of f ) & not ( not g in the carrier' of f ) ) ;
Hom ( ( a \times b ) , c ) <> {} ;
consider k1 such that p " < k1 and 0 < k1 and k1 < 1 ;
consider c , d such that dom f = c \ d ;
[ x , y ] in dom g & [ x , y ] in dom k ;
set S1 = Let Let S1 , S2 = Let S2 , S2 = } ;
l1 = m2 & l2 = i2 & l2 = i2 & l2 = j2 ;
x0 in dom ( ( u + v ) + ( v + u ) ) ;
reconsider p = x as Point of TOP-REAL 2 , r be Real ;
[: I , I :] = [: I , I :] | [: B , B :] ;
f . p4 <= f . p1 & f . p2 = f . p3 ;
( ( F . x ) `1 ) ^2 <= ( x `1 ) ^2 & ( F . x ) `2 <= ( x `2 ) ^2 ;
( x `2 ) ^2 = ( ( W `2 ) ^2 + ( W `2 ) ^2 ) ;
for n being Element of NAT holds P [ n ] ;
let J , K be non empty Subset of I ;
assume 1 <= i & i <= len <* a " *> ;
0 |-> a = <*> ( the carrier of K ) .= ( 0 |-> a )
X . i in 2 -tuples_on A . i \ B . i ;
<* 0 *> in dom ( e --> [ 1 , 0 ] ) ;
then P [ a ] implies P [ succ a ] ;
reconsider sbeing being terminal of D , a being Object of D ;
( k -' 1 ) <= len ( k -' 1 ) ;
[#] S c= [#] T & the TopStruct of T c= [#] T ;
for V being strict RealUnitarySpace holds V in the carrier of V implies V is Subspace of W
assume k in dom mid ( f , i , j ) ;
let P be non empty Subset of TOP-REAL 2 , p1 , p2 be Point of TOP-REAL 2 ;
let A , B be Matrix of n1 , K , n be Nat ;
- a * - b = a * b - a * - b ;
for A being Subset of A9 holds A // A & A // A implies A = B
( for o2 being Element of o2 holds ( o2 , o1 ) --> ( o2 , o2 ) ) . ( ( o2 , o1 ) --> ( o2 , o2 ) ) = <* o2 , o1 *>
then ||. x .|| = 0 & x = 0. X ;
let N1 , N2 be strict normal Subgroup of G ;
j >= len ( upper_volume ( g , D1 ) | ( len D2 ) ) ;
b = Q . ( len Q - 1 + 1 ) .= Q . ( len Q - 1 + 1 ) ;
f2 * ( f1 /* s ) is divergent_to-infty & f2 * ( f2 /* s ) is divergent_to-infty ;
reconsider h = f * g as Function of [: N1 , N2 :] , G ;
assume that a <> 0 and Polynom ( a , b , c ) >= 0 ;
[ t , t ] in the InternalRel of A & [ t , t ] in the InternalRel of A ;
( v |-- E ) | 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 & Directed I is_closed_on Initialized s , P ;
Initialized p = Initialize ( p +* q ) .= Initialize ( p +* q ) ;
reconsider N2 = N1 as strict net of R2 , N2 be strict net of R2 ;
reconsider Y = Y as Element of [: Ids L , Ids L :] ;
"/\" ( ( uparrow p ) \ { p } , L ) <> p ;
consider j being Nat such that i2 = i1 + j and 1 <= j and j <= len f ;
not [ s , 0 ] in the carrier of S2 & not [ s , 0 ] in the InternalRel of S2 ;
mm in ( B '&' C ) \ D & not mm in D ;
n <= len ( ( P + Q ) + len ( P + Q ) ) ;
( x1 - x2 ) `1 = ( x2 - y2 ) `1 .= ( x2 - y2 ) `1 ;
InputVertices S = { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 7
let x , y be Element of FTT1 ( n ) ;
p = |[ p `1 / |. p .| , p `2 / |. p .| ]| ;
g * ( 1_ G ) = h " * g .= h " * g ;
let p , q be Element of PFuncs ( V , C ) ;
x0 in dom ( x1 - x2 ) /\ dom ( y1 - y2 ) ;
( R qua Function ) " = R " & ( R " ) " = R " & ( R " ) " = R ;
n in Seg ( len ( f /^ ( i -' 1 ) ) ) ;
for s being Real st s in R holds s <= s2 & s2 <= b ;
rng s c= dom ( f2 * f1 ) \ { x0 } ;
synonym ex X being set st X in for x being Element of consider X st X in \overline { x } & X in being finite ;
1. ( K , n ) * 1. ( K , n ) = 1. ( K , n ) ;
set S = Segm ( A , P1 , Q1 ) , Q1 = Segm ( A , P2 , s2 ) ;
ex w st e = ( w - f ) / ( w - f ) & w in F ;
curry ( ( P , k ) # x ) is convergent ;
cluster -> open for Subset of T7 holds F is open & F is finite ;
len ( f1 | n ) = 1 .= len ( f3 | n ) .= len ( f3 | n ) .= len ( f3 | n ) ;
( i * p ) / p < ( 2 * p ) / p ;
let x , y be Element of ( the carrier of U0 ) * ;
b1 , c1 // b9 , c9 & b1 , c1 // b9 , c ;
consider p being element such that c1 . j = { p } and p in F ;
assume that f " { 0 } = {} and f is total and f is total ;
assume IC Comput ( F , s , k ) = n ;
Reloc ( J , card I + 3 ) does not destroy a ;
( goto ( card I + 1 ) ) not ( card I + 1 ) does not destroy c ;
set m3 = LifeSpan ( p3 , s3 ) , P4 = LifeSpan ( p3 , s3 ) , P4 = LifeSpan ( p3 , s3 ) ;
IC Comput ( p , s , k ) in dom ( Initialize p ) ;
dom t = the carrier of ( SCM R ) \ { {} } .= { {} } ;
( ( N-min L~ f ) .. f ) .. f = 1 & ( ( E-max L~ f ) .. f ) .. f = 1 ;
let a , b be Element of PFuncs ( V , C ) ;
Cl Int ( union F ) c= Cl Int ( union F ) ;
the carrier of X1 union X2 misses the carrier of X2 ;
assume not LIN a , f . a , g . a ;
consider i being Element of M such that i = d6 and i in dom f ;
then Y c= { x } or Y = {} or Y = { x } ;
M , v / ( ( ( y , v ) / ( x , y ) ) ) |= H ;
consider m being element such that m in Intersect ( FF ) and x = [ m , m ] ;
reconsider A1 = ( support ( u1 ) ) \ { u } as Subset of X ;
card ( A \/ B ) = k-1 + ( 2 * 1 ) ;
assume that a1 <> a3 and a2 <> a4 and a3 <> a4 and a4 <> a4 and a4 <> a4 ;
cluster s -\mathop { \rm \hbox { - } IC } ( V , p ) -> .| for string string of S ;
Ln2 /. n2 = Ln2 . n2 .= Ln2 . n2 .= Ln2 . n2 ;
let P be compact non empty Subset of TOP-REAL 2 , p1 , p2 be Point of TOP-REAL 2 ;
assume that rpppppppppppppppppppppppppppppppppppppppppppppppppppppp
let A be non empty compact Subset of TOP-REAL n , a be Real ;
assume [ k , m ] in Indices ( D * ( k , m ) ) ;
0 <= ( ( 1 / 2 ) |^ p ) / 2 ;
( F . N ) | E1 = +infty & ( F . N ) . x = +infty ;
attr X c= Y means : Def3 : Z c= V & X \ V c= Y \ Z ;
( y `2 ) * ( z `2 ) <> 0. I ;
1 + card ( X-18 ) <= card ( u + card ( X ) ) ;
set g = z \circlearrowleft ( ( L~ z ) .. z ) , p = z .. z , q = z .. z , r = z .. z , s = z .. z , t = z .. z ;
then k = 1 & p . k = <* x , y *> . k ;
cluster -> total for Element of C -\mathop { - } :] ;
reconsider B = A as non empty Subset of TOP-REAL n , a be Real ;
let a , b , c be Function of Y , BOOLEAN ;
L1 . i = ( i .--> g ) . i .= g . i .= g . i ;
Plane ( x1 , x2 , x3 , x4 , x5 , x5 , 8 , 7 , 8 ) c= P ;
n <= indx ( D2 , D1 , j1 ) + 1 - 1 ;
( ( g2 ) . O ) `1 = - 1 & ( g2 ) . I = - 1 ;
j + p .. f -' len f <= len f - len f + 1 ;
set W = W-bound C , E = E-bound C , S = S-bound C , N = S-bound C , G = Gauge ( C , n ) ;
S1 . ( a `1 , e `2 ) = a + e .= a `1 ;
1 in Seg width ( M * ( ColVec2Mx p ) ) ;
dom ( i * Im f ) = dom Im f .= dom Im f ;
DI . ( x `1 , p `2 ) = W . ( a , p `2 ) ;
set Q = contradiction ( |= ( g , f , h ) ) ;
cluster -> -> many sorted for ManySortedSet of U1 ;
attr F = { A } means : Def3 : F is discrete ;
reconsider z9 = \hbox { - } \sum G as Element of product G ;
rng f c= rng f1 \/ rng f2 & rng f c= rng f1 \/ rng f2 ;
consider x such that x in f .: A and x in f .: C ;
f = <*> ( the carrier of F_Complex ) .= ( the carrier of F_Complex ) --> ( the carrier of F_Complex ) ;
E , j |= All ( x1 , x2 , H ) implies E , j |= H
reconsider n1 = 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 - 1 .= card B2 + 1 ;
card ( ( x \ B1 ) /\ B1 ) = 0 implies card ( x \ B1 ) = 1 ;
g + R in { s : g-r < s & s < g + r } ;
set q00 = ( q , <* s *> ) -\mathop { + } , q00 = ( q , <* s *> ) -\mathop { + } ;
for x being element st x in X holds x in rng ( f1 + f2 )
h0 /. ( i + 1 ) = h0 . ( i + 1 ) ;
set mw = max ( B , being Element of Bags NAT , a , b , c , d ) ;
t in Seg width ( I ^ ( n , n ) ) ;
reconsider X = dom f \ C as Element of Fin ( NAT ) ;
IncAddr ( i , k ) = <% - l , - l %> + k .= - l ;
( ( GoB f ) * ( 1 , 1 ) ) `2 <= ( ( GoB f ) * ( 1 , 1 ) ) `2 ;
attr R is condensed means : Def3 : Int R is condensed & Cl R is condensed & Cl R is condensed ;
attr 0 <= a & b <= 1 & a <= 1 implies a * b <= 1 ;
u in ( ( c /\ ( ( d /\ b ) /\ e ) ) /\ f ) /\ j ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) /\ f ) /\ j ;
len ( C + - 2 ) >= 9 + - 3 + ( - 2 ) ;
x , z , y is_collinear & x , z , y , z is_collinear implies x , y , z , x , y , z is_collinear
a |^ ( n1 + 1 ) = a |^ ( n1 + 1 ) * a ;
<* \underbrace ( 0 , \dots , 0 ) , 0 *> in Line ( x , a * x ) ;
set y9 = <* y , c *> ;
Fs2 /. 1 in rng Line ( D , 1 ) & Fs2 /. len Fs2 = Line ( D , 1 ) ;
p . m Joins r /. m , r /. ( m + 1 ) , G ;
( p `2 ) ^2 = ( f /. ( i1 + 1 ) ) `2 .= ( f /. ( i1 + 1 ) ) `2 ;
( W-bound X \/ W-bound Y ) = W-bound X + ( W-bound Y ) .= W-bound Y + ( W-bound Y ) ;
0 + ( p `2 ) ^2 <= 2 * r + ( p `2 ) ^2 ;
x in dom g & not x in g " { 0 } ;
f1 /* ( s ^\ k ) is divergent_to-infty & f2 /* ( s ^\ k ) is divergent_to-infty ;
reconsider u2 = u as VECTOR of Psqrt ( X , 1 ) , u1 = v as VECTOR of X ;
p \! \mathop { \rm \hbox { - } count } ( X ) = 0 implies p = 0
len <* x *> < i + 1 & i + 1 <= len c + 1 ;
assume that I is non empty and { x } /\ { y } = { 0. I } ;
set ii = card I + 4 .--> goto 0 , ii = ( card I + 4 ) .--> goto 0 ;
x in { x , y } & h . x = {} ( T | A ) ;
consider y being Element of F such that y in B and y <= x `1 ;
len S = len ( the charact of ( A + B ) ) .= len ( the charact of ( A + B ) ) ;
reconsider m = M , i = I as Element of X ;
A . ( j + 1 ) = B . ( j + 1 ) \/ A . j ;
set N8 = width ( ( G-15 ) \ { i } ) ;
rng F c= the carrier of gr { a } & rng F c= the carrier of gr { a } ;
<> <> <> ( ( the _ of Q , n , r ) * ( the _ of Q , n ) ) ;
f . k , f . ( Let ( Let n ) + 1 ) are_congruent_mod rng f ;
h " P /\ [#] ( T1 | P ) = f " P /\ [#] ( T1 | P ) ;
g in dom f2 \ f2 " { 0 } & f2 " { 0 } = dom f2 \ f2 " { 0 } ;
gX /\ dom ( f1 " X ) = ( g1 " X ) /\ ( g2 " X ) ;
consider n being element such that n in NAT and Z = G . n ;
set d1 = being in being dist ( x1 , y1 ) , d2 = dist ( x2 , y2 ) , d2 = dist ( x2 , y2 ) ;
b `2 + ( 1 - 2 ) < ( 1 - 2 ) * ( 1 - 2 ) ;
reconsider f1 = f as VECTOR of the carrier of X , Y ;
attr i <> 0 means : Def3 : i |^ ( i + 1 ) mod ( i + 1 ) = 1 ;
j2 in Seg len ( ( g2 ) . ( i2 + 1 ) ) & 1 <= j2 & j2 + 1 <= len ( ( g2 ) . i2 + 1 ) ;
dom ( i - j ) = dom ( i - j ) .= Seg ( len ( i - j ) ) .= dom ( i - j ) .= Seg ( len ( i - j ) ) ;
cluster sec | ]. PI / 2 , PI / 2 .[ -> one-to-one ;
Ball ( u , e ) = Ball ( f . p , e ) ;
reconsider x1 = x0 as Function of S , T ( ) , T ( ) ;
reconsider R1 = x , R2 = y , R2 = z as Relation of L ;
consider a , b being Subset of A such that x = [ a , b ] ;
( <* 1 *> ^ p ) ^ <* n *> in Rn & <* n *> ^ p in Rn ;
S1 +* S2 = S2 +* S2 & S2 +* S2 = S2 +* S2 +* S2 ;
( ( - 1 ) (#) ( ( id Z ) ^ ) ) is_differentiable_on Z ;
cluster -> non empty for Function of C , REAL ;
set C7 = 1GateCircStr ( <* z , x *> , f3 ) , C8 = 1GateCircStr ( <* z , x *> , f3 ) ;
E\langle g . e2 , T . ( e2 + T ) *> = ( E . e2 ) -T ;
( ( - 1 ) (#) ( arctan + arccot ) ) is_differentiable_on Z ;
upper_bound A = ( cos * 3 ) / 2 & lower_bound A = 0 & lower_bound A = 0 ;
F . ( dom f , - - F . ( cod f , - F . ( cod f , - F . cod f ) ) ) is \emptyset ;
reconsider p8 = ( q - p ) / ( 1 + ( q - p ) ) as Point of TOP-REAL 2 ;
g . W in [#] ( Y | 0 ) & [#] ( Y | 0 ) c= [#] ( Y | 0 ) ;
let C be compact non vertical non horizontal Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
LSeg ( f ^ g , j ) = LSeg ( f , j ) \/ LSeg ( g , i ) ;
rng s c= dom f /\ ]. x0 - r , x0 .[ & rng s c= dom f /\ ]. x0 - r , x0 + r .[ ;
assume x in { idseq ( 2 , len Rev f ) , Rev Rev f } ;
reconsider n2 = n , m2 = m as Element of NAT ;
for y being ExtReal st y in rng ( seq ^\ k ) holds g <= y
for k st P [ k ] holds P [ k + 1 ] ;
m = m1 + m2 .= m1 + m2 .= m1 + m2 .= m1 + m2 ;
assume for n holds H1 . n = G . n -H . n ;
set BB = f .: ( the carrier of X1 ) , BB = f .: the carrier of X2 ;
ex d being Element of L st d in D & x << d ;
assume R " ( a \ b ) c= R " ( a \ b ) ;
t in ]. r , s .[ or t = r or t = s ;
z + v2 in W & x = u + ( z + v2 ) ;
x2 |-- y2 iff ( x2 , y2 ) `1 = x2 & ( x2 , y2 ) `2 = y2 & ( x2 , y2 ) `2 = y2 ;
attr x1 <> x2 means : Def3 : |. x1 - x2 .| > 0 & |. x1 - x2 .| > 0 ;
assume that p2 - p1 , p3 - p2 , p3 - p1 , p2 - p3 , p3 - p1 , p2 - p3 , p3 - p1 , p2 - p3 - p3 , p3 - p1 , p2 - p3 , p3 - p1 , p2 - p3 - p3 , p3 - p1 , p2 - p3 - p3 , p3 - p1 , p2 - p3 , p3 -
set q = ( -1 f ) ^ <* 'not' A *> ^ <* 'not' A *> ;
let f be PartFunc of \langle REAL-NS 1 , REAL-NS 1 *> , REAL-NS 1 , REAL-NS 1 , REAL-NS 1 be PartFunc of REAL 1 , REAL-NS 1 ;
( n mod ( 2 * k ) ) ! = n mod k ;
dom ( T * \mathop { \rm tree } ( t ) ) = dom ( T * \mathop { \rm succ } t ) ;
consider x being element such that x in wc and x in c and x in A ;
assume ( F * G ) . ( v . x3 ) = v . x4 ;
assume that the carrier of D1 c= the carrier of D2 and the carrier of D1 c= the carrier of D2 and the carrier of D1 c= the carrier of D2 ;
reconsider A1 = [. a , b .[ as Subset of R^1 ( R^1 ( a ) ) ;
consider y being element such that y in dom F and F . y = x ;
consider s being element such that s in dom o and a = o . s ;
set p = W-min L~ Cage ( C , n ) , q = W-min L~ Cage ( C , n ) ;
n1 -' len f + 1 - 1 + 1 <= len g + 1 - 1 + 1 ;
.| ( q , O1 ) = [ u , v , a , b , a , b , c ] ;
set C-2 = ( `1 `1 ) . ( k + 1 ) , Cw2 = ( k + 1 ) ;
Sum ( L (#) p ) = 0. R * Sum p .= 0. V .= Sum ( L (#) p ) ;
consider i being element such that i in dom p and t = p . i ;
defpred Q [ Nat ] means 0 = Q ( $1 ) implies $1 in dom ( Q * R ) ;
set s3 = Comput ( P1 , s1 , k ) , P3 = Comput ( P2 , s2 , k ) , P4 = P3 ;
let l be variable of k , Al , A be non empty Subset of D ;
reconsider U2 = union ( G-24 \/ { p } ) as Subset-Family of ( T | A ) ;
consider r such that r > 0 and Ball ( p `1 , r ) c= Q ` ;
( h | ( n + 2 ) ) /. ( i + 1 ) = ( h | ( n + 2 ) ) /. ( i + 1 ) ;
reconsider B = the carrier of X1 as Subset of X , a be Real ;
p[#] L = <* - c , 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , -
synonym f is real-valued for rng f c= NAT & rng f c= NAT & rng f c= { 0 } ;
consider b being element such that b in dom F and a = F . b ;
x9 < card ( X0 \/ ( X1 \/ X2 ) ) + card ( X2 \/ X1 ) ;
attr X c= B1 means : Def3 : for B st X c= B1 holds card X c= card B & card X = card B ;
then w in Ball ( x , r ) & dist ( x , w ) <= r ;
angle ( x , y , z ) = angle ( \HM { the } \HM { carrier } \HM { of } K , n , z ) ;
attr 1 <= len s means : Def3 : len the { F , 0 } = len s ;
fY. c= f . ( k + ( n + 1 ) ) ;
the carrier of { 1_ G } = { 1_ G } & the carrier of G = { 1_ G } ;
attr p '&' q in \mathbin { \cal U } means : Def3 : q in - p & q in - p ;
- ( t `1 / t `2 ) < ( - t `1 ) / t `2 ;
U . 1 = U /. 1 .= ( U /. 1 ) `1 .= ( U /. 1 ) `1 .= ( U /. 1 ) `1 .= ( U /. 1 ) `1 ;
f .: ( the carrier of x ) = the carrier of x & f .: ( the carrier of x ) = the carrier of x ;
Indices ( O @ ) = [: Seg n , Seg n :] & Indices ( O @ ) = [: Seg n , Seg n :] ;
for n being Element of NAT holds G . n c= G . ( n + 1 )
then V in M @ ;
ex f being Element of F-9 st f is \cup of Abe Element of Abe Element of f st f = F & f is unital & f is unital ;
[ h . 0 , h . 3 ] in the InternalRel of G & [ h . 0 , h . 3 ] in the InternalRel of G ;
s +* Initialize ( ( intloc 0 ) .--> 1 ) = s3 +* Initialize ( ( intloc 0 ) .--> 1 ) ;
|[ w1 , v1 ]| `1 <> 0. TOP-REAL 2 & |[ w1 , v1 ]| `1 <> 0. TOP-REAL 2 ;
reconsider t = t as Element of ( Z , X ) * ;
C \/ P c= [#] ( ( ( ( [#] ( ( ( ( ( ( ( ( B - B ) | A ) ) | A ) ) | A ) ) ) ) ) ) ;
f " V in ( the carrier of [ X , the carrier of S ] ) /\ D ;
x in [#] ( ( the carrier of ( the carrier of ( ) ) ) /\ A ) ;
g . x <= h1 . x & h . x <= h1 . x & h . x <= h1 . x ;
InputVertices S = { xy , yz , yz , zx , zx , zx , cin } ;
for n being Nat st P [ n ] holds P [ n + 1 ] ;
set R = ( Line ( M , i , a ) ) * Line ( M , i , a ) ;
assume that M1 is being_line and M2 is being_line and M2 is being_line and M2 is being_line and M2 is being_line and M2 is being_line ;
reconsider a = f4 . ( i0 -' 1 ) as Element of K ;
len ( ( len ( F1 ^ F2 ) ) | ( i + 1 ) ) = Sum ( ( F1 ^ F2 ) | ( i + 1 ) ) ;
len ( ( the { 0 } \HM { \bf qua } \HM { FinSequence } ) * ( i , j ) ) = n ;
dom ( max ( f + g , h ) ) = dom ( f + g ) .= dom f ;
( the Sorts of ( seq ^\ k ) ) . n = upper_bound ( Y1 + ( seq ^\ k ) ) .= upper_bound ( Y1 + ( seq ^\ k ) ) ;
dom ( p1 ^ p2 ) = dom ( f ^ <* p2 *> ) .= dom ( f ^ <* p2 *> ) .= dom f ;
M . [ 1 , y ] = 1 / ( v1 * v1 ) .= 1 / ( v1 * v2 ) .= ( 1 / ( v1 * v2 ) ) * v1 ;
assume that W is not trivial and W { x } c= the \frac of G2 and not x in the carrier of G2 ;
C6 * ( i1 , i2 ) = G1 * ( i1 , i2 ) .= G1 * ( i1 , i2 ) ;
C8 |- 'not' Ex ( All ( x , p ) 'or' All ( x , p ) ) ;
for b st b in rng g holds lower_bound rng fmma <= b * ( lower_bound rng ffma )
- ( ( - ( ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ) ^2 ) = 1 ;
( LSeg ( c , m ) \/ [: NAT , NAT :] ) \/ [: { l } , NAT :] c= R ;
consider p being element such that p in { x } and p in L~ f and x in L~ f ;
Indices ( the carrier of X ) = [: Seg n , Seg n :] & [: Seg n , Seg n :] = [: Seg n , Seg n :] ;
cluster s => ( q => p ) -> valid ;
Im ( ( Partial_Sums ( F ) . m ) ) is_measurable_on E ;
cluster f . ( x1 , x2 ) -> ( f . x1 , f . x2 ) -> Element of D ;
consider g being Function such that g = F . t and Q [ t , g ] ;
p in LSeg ( ( N-min Z ) , ( ( N-min Z ) `1 ) / 2 ) ;
set R8 = R ^ ]. b , +infty .[ , R8 = R ^ <* b , +infty *> ;
IncAddr ( I , k ) = SubFrom ( da , db ) .= Exec ( I , da ) ;
seq . m <= ( ( the Sorts of A1 ) . k ) . ( ( the Sorts of A2 ) . k ) ;
a + b = ( a ` + b ) ` .= ( a ` + b ) ` .= ( a ` + b ) ` ;
id ( X /\ Y ) = id ( X /\ Y ) .= id ( X /\ Y ) ;
for x being element st x in dom h holds h . x = f . x
reconsider H = ( U1 \/ U2 ) \/ ( U2 \/ U1 ) as non empty Subset of U0 ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) /\ f ) /\ m ;
consider y being element such that y in Y and P [ y , lower_bound B ] ;
consider A being finite stable non empty Subset of R such that card A = card ( the InternalRel of R ) and card A = card A ;
p2 in rng ( f |-- p1 ) \ rng <* p1 *> & p2 in rng <* p1 *> ;
len s1 - 1 > 0 & len s2 - 1 > 0 ;
( ( N-min L~ f ) `2 ) `2 = ( ( N-min L~ f ) `2 ) `2 .= ( ( E-max L~ f ) `2 ) `2 ;
Ball ( e , r ) c= LeftComp Cage ( C , k + 1 ) \/ LeftComp Cage ( C , k + 1 ) ;
f . a1 ` = f . a1 ` .= f . a1 .= ( f | ( a ` ) ) . a1 .= ( f | ( a ` ) ) . a1 ;
( seq ^\ k ) . n in ]. - \infty , x0 + r .[ & ( seq ^\ k ) . n in ]. - r , x0 + r .[ ;
gg . s0 = g . ( s0 | G ) . ( s . m ) .= g . ( s . m ) ;
the InternalRel of S is If & 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 . ( s1 . a1 ) = F . ( s2 . a1 ) .= F . ( s2 . a1 ) ;
x `1 = A . o . a .= Den ( o , A . a ) ;
Cl ( f " P1 ) c= f " ( ( f " P1 ) \/ f " P1 ) ;
FinMeetCl ( the topology of S ) c= the topology of T & the topology of T c= the topology of T ;
synonym o is " means : Def3 : o <> \ast p & o <> 0. S ;
assume that X |^ + = Y |^ + and card X <> card Y and card Y <> card X ;
the non empty the real number <= 1 + ( the { x } \HM { \bf qua } \HM { Real } ) ;
LIN a , a1 , d or b , c // b1 , c1 or b , c // b1 , c1 ;
e . 1 = 0 & e . 2 = 1 & e . 3 = 0 & e . 4 = 0 ;
( for x st x in SS1 holds not x in { NS1 } ) & not x in { NS1 } ;
set J = ( l , u ) If , K = I " ;
set A1 = Let ( ap , bm , cp , cp , cin ) , A2 = Boolean , C = Boolean , d = cin ;
set c9 = [ <* cin , d *> , '&' ] , A2 = [ <* cin , d *> , '&' ] , C = [ <* cin , d *> , '&' ] , D = [ <* cin , d *> , '&' ] , N = [ <* cin , d *> , '&' ] , E = [ <* cin , c *> , '&' ] , N =
x * z `1 * x " in x * ( z * N ) " * x ;
for x being element st x in dom f holds f . x = f3 . x & f . x = f3 . x ;
Int cell ( f , 1 , G ) c= RightComp f \/ L~ f \/ L~ f \/ L~ f ;
U is_an_arc_of W-min C , W-min C & L~ Cage ( C , n ) c= L~ Cage ( C , n ) implies L~ Cage ( C , n ) = L~ Cage ( C , n )
set f-17 = f @ "/\" ( g @ ) ;
attr S1 is convergent means : Def3 : S2 is convergent & for x st x in dom S1 holds S1 - S2 is convergent & lim ( S1 - S2 ) = 0 ;
f . ( 0 + 1 ) = ( 0 qua Ordinal ) + a .= a + ( 0 qua Ordinal ) .= a ;
cluster -> in -> of such that -> of -> reflexive transitive transitive transitive and the InternalRel of ( the InternalRel of N ) -symmetric ;
consider d being element such that R reduces b , d and R reduces c , d and R reduces c , d ;
not b in dom Start-At ( ( card I + 2 ) , SCMPDS ) \/ dom Start-At ( ( card I + 2 ) , SCMPDS ) ;
( z + a ) + x = z + ( a + y ) .= z + a + y ;
len ( l (#) ( a (#) ( 0 .--> x ) ) ) = len l .= len l ;
t4 is ( {} \/ rng t4 ) -valued FinSequence of ( {} \/ rng t4 ) * * * ( 1 , len t5 ) ) -valued FinSequence ;
t = <* F . t *> ^ ( C . p ^ ( C . q ) ) .= <* F . t *> ^ ( C . q ^ ( C . p ^ q ) ) ;
set p-2 = W-min L~ Cage ( C , n ) , p-2 = W-min L~ Cage ( C , n ) , p-2 = W-bound L~ Cage ( C , n ) ;
( k -' 1 ) + ( i -' 1 ) = ( k + 1 ) - ( i + 1 ) ;
consider u being Element of L such that u = u `1 and u in D and u in D ;
len ( ( width ( ( ( ( ( ( a , i ) --> a ) ^ <* a *> ) ^ <* b *> ) ^ <* b *> ) ) |-> a ) ) = width ( ( a , b ) --> ( <* b , c *> ^ <* c , d *> ) ) ;
FM . x in dom ( ( G * the_arity_of o ) . x ) ;
set H2 = the carrier of H2 , y2 = the carrier of H2 ;
set H1 = the carrier of ( H1 , H2 ) , H2 = the carrier of ( H1 , H2 ) ;
( Comput ( P , s , 6 ) ) . intpos ( m + 1 ) = s . intpos ( m + 1 ) ;
IC Comput ( Q2 , t , k ) = ( l + 1 ) + 1 .= ( l + 1 ) + 1 ;
dom ( ( - 1 ) (#) ( ( - 1 ) (#) ( sin + cos ) ) ) = REAL ;
cluster <* l *> ^ phi -> ( 1 + 1 ) -element for string of S ;
set b9 = [ <* xy , that ] , [ <* A1 , cin *> , '&' ] ] , Z = [ <* cin , A1 *> , '&' ] ;
Line ( Line ( M , P ) , x ) = L * ( Sgm Q ) ;
n in dom ( ( the Sorts of A ) * the_arity_of o ) & dom ( ( the Sorts of A ) * the_arity_of o ) = dom ( the Sorts of A ) ;
cluster f1 + f2 -> continuous for PartFunc of REAL , REAL , x be Point of S ;
consider y be Point of X such that a = y and ||. - y .|| <= r ;
set x3 = ( Y . SBP , 2 = ( Y . SBP , 2 ) , x4 = ( Y . SBP , 2 ) , 7 = ( Y . SBP , 2 ) , 8 = ( Y . SBP , 3 ) , 6 = ( Y . SBP , 2 ) , 6 = ( Y . SBP , 3 ) , 8
set p-3 = stop I , ps2 = stop I , ps3 = stop I ;
consider a being Point of D2 such that a in W1 and b = g . a and a in W2 ;
{ A , B , C , D } = { A , B , C , D } ;
let A , B , C , D , E , F , J , M , N , N , N , F , M , N , N , A , N , M , N , N , A , N , A , N , A , N , M , N , N , A , N , N , A , N ,
|. p2 .| ^2 - ( - ( p2 `1 / |. p2 .| - cn ) ) ^2 >= 0 ;
l -' 1 + 1 = n-1 * ( l + 1 ) + ( ( l1 + 1 ) + 1 ) ;
x = v + ( a * w1 + ( b * w2 ) ) + ( c * w2 ) + ( c * w2 ) + ( c * w2 ) + ( c * w2 ) + ( c * w2 ) + ( c * w2 ) + ( c * w2 ) + ( c * w2 ) + ( c * w2 ) = x ;
the TopStruct of L = TopSpaceMetr (# the topology of L , the topology of L #) & the TopStruct of L = the TopStruct of L ;
consider y being element such that y in dom ( H1 . y ) and x = ( H1 . y ) `1 ;
f9 \ { n } = Free ( All ( v1 , H ) ) & f in Free ( All ( v1 , H ) ) ;
for Y being Subset of X st Y is summable holds Y is summable iff Y is iff Y is \overline Y
2 * n in { N : 2 * Sum ( p | N ) = N & N > 0 } ;
for s being FinSequence holds len ( the { - } \rm <* - *> ) = len s & len ( the { - } -\rm <* - *> ) = len s
for x st x in Z holds exp_R * f is_differentiable_in x & exp_R * f is_differentiable_in x
rng ( h2 * ( f2 * f1 ) ) c= the carrier of ( ( len f2 ) -tuples_on the carrier of K ) ;
j + ( len f - len f ) <= len f + ( len f - len f ) - len f ;
reconsider R1 = R * I as PartFunc of REAL , REAL-NS n , a be Real ;
C8 . x = s1 . ( a - 1 ) .= C8 . ( a - 1 ) .= C8 . ( a - 1 ) .= ( a - 1 ) * ( a - 1 ) ;
power ( F_Complex , n , z ) . ( z , n ) = 1 / ( n + 1 ) .= ( x |^ n ) |^ ( z , n ) .= ( x |^ n ) |^ ( z , n ) ;
t at ( C , s ) = f . ( the connectives of S ) . t .= f . ( s , C ) ;
( support f + g ) c= ( support f ) \/ ( support g ) ;
ex N st N = j1 & 2 * Sum ( ( r | N ) | N ) > N ;
for y , p st P [ p ] holds P [ All ( y , p ) ]
{ [ x1 , x2 ] , [ y1 , y2 ] } is Subset of [: X1 , X2 :] ;
h . i = ( j |-- h , id B . i ) .= H . i .= H . i ;
ex x1 being Element of G st x1 = x & ( x1 * N ) c= A & ( x1 * N ) c= A ;
set X = ( ( |. q .| ) |^ ( O1 , O2 ) ) , Y = ( |. q .| ) |^ ( O1 , O2 ) , Z = { ( |. q .| ) |^ ( O2 , O2 ) } ;
b . n in { g1 : x0 < g1 & g1 < x0 & g1 < x0 & g1 < x0 } ;
f /* ( s1 + s2 ) is convergent & f /. x0 = lim ( f /* ( s1 + s2 ) ) ;
the carrier of Y = the carrier of the lattice of Y & the carrier of Y = the carrier of X & the carrier of Y = the carrier of Y ;
'not' ( a . x ) '&' b . x 'or' a . x = TRUE ;
( 2 to_power ( ( q ^ r1 ) + 1 ) ) = ( len ( ( q ^ r1 ) + 1 ) + len ( q ^ r1 ) ) + len ( ( q ^ r1 ) + 1 ) ;
( 1 / a ) (#) ( sec * ( f1 - f2 ) - ( 1 / a ) (#) ( ( 1 / a ) (#) ( ( 1 / a ) (#) ( ( 1 / a ) (#) ( f1 - f2 ) ) ) ) ) is_differentiable_on Z ) ;
set K1 = upper ( H , lim ( H , lim ( I , H ) ) ) , K1 = lim ( H , lim ( I , lim ( H , lim ( I , lim ( H , lim ( I , lim ( H , lim ( I , lim ( H , lim ( H , lim ( I , lim ( H , lim ( H , lim ( H
assume that e in { ( w1 + w2 ) : w1 in F & w2 in G & w1 in G } and w1 in F ;
reconsider d7 = dom a `1 , d8 = dom F , d8 = dom G as finite set ;
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 .= Seg n .= Seg n .= Seg n .= Seg n .= Seg n .= Seg n .= Seg n .= Seg n ;
x in H |^ a implies ex g st x = g |^ a & g in H |^ a
a * ( 0. ( K , n , 1 ) ) = a `1 - ( 0 , 1 ) .= a `2 - ( 0 , 1 ) ;
D2 . ( j - 1 ) in { r : lower_bound A <= r & r <= D1 . i } ;
ex p being Point of TOP-REAL 2 st p = x & ( for p being Point of TOP-REAL 2 st p in P holds p `1 >= 0 ) & ( for p being Point of TOP-REAL 2 st p in P holds p <> 0. TOP-REAL 2 ) ;
for c holds f . c <= g . c implies f @ c <= g @ c
dom ( ( f1 (#) f2 ) | X ) /\ X c= dom ( ( f1 (#) f2 ) | X ) /\ X ;
1 = ( p * p ) |^ ( 1 + 1 ) .= p * 1 + 1 .= p * 1 + 1 ;
len g = len f + len <* x + y *> .= len f + len <* y + x *> .= len f + 1 + 1 .= len f + 1 + 1 ;
dom ( F-11 | [: N1 , S :] ) = dom ( ( F | [: N1 , S :] ) | [: N1 , S :] ) ;
dom ( f . t * I . t ) = dom ( f . t * g . t ) ;
assume a in ( "\/" ( ( T |^ the carrier of S ) , F ) ) .: D ;
assume that g is one-to-one and ( the carrier' of S ) /\ rng g c= dom g and rng g c= dom g ;
( ( x \ y ) \ z ) \ ( ( x \ z ) \ ( y \ z ) ) = 0. X ;
consider f such that f * f `1 = id b and f * f = id a and f * f = id b ;
( cos | [. 2 * PI , 0 + 1 .] ) | [. 2 * PI , 0 + 1 .] is increasing ;
Index ( p , co ) <= len LS - Index ( Gij , LS ) + Index ( Gij , LS ) ;
t1 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t2 , t1 , 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 ,
( Frege ( ( Frege ( Frege H ) ) . h ) ) . x <= ( Frege ( ( Frege H ) . h ) ) . ( y , x ) ;
then ( for i0 st i0 in dom f holds F . i0 < G . i0 ) & ( for i0 st i0 < i0 holds F . i0 < G . i0 ) ;
Q [ ( D . [ D . x , 1 ] ) `1 , F ( D . [ D . x , 1 ] ) `2 ] ;
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 Sorts of A2 ;
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 ) ;
( ( <* Gauge ( C , n ) * ( len Gauge ( C , n ) , 1 ) ) `1 , ( <* Gauge ( C , n ) * ( len Gauge ( C , n ) , 1 ) *> ) `2 , ( <* Gauge ( C , n ) * ( len Gauge ( C , n ) , 1 ) *> ) `2 *> ) `2 = ( ( <* Gauge (
q <= ( UMP ( Upper_Arc ( L~ Cage ( C , 1 ) ) ) ) `2 & ( UMP ( L~ Cage ( C , 1 ) ) ) `2 <= ( N-bound ( L~ Cage ( C , 1 ) ) ) `2 ;
LSeg ( f | i2 , i ) /\ LSeg ( f | i2 , j ) = {} ;
given a being ExtReal such that a <= II and A = ]. a , I .[ and a < I ;
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 <= len b } , Y = { b |^ n where n is Element of NAT : n <= len b } ;
( ( x * y ) \ z ) \ ( x * z ) = 0. X ;
set xy = [ <* xy , yz , zx *> , [ <* xy , yz , yz ] , [ <* xy , yz , yz *> , [ ] , f4 ] ] ;
( l /. len l ) = ( l . len l ) .= ( l . len l ) * ( l . ( len l ) ) ;
( ( - ( q `2 / |. q .| - sn ) ) / ( 1 + sn ) ) ^2 = 1 ;
( ( - ( p `2 / |. p .| - sn ) ) / ( 1 + sn ) ) ^2 < 1 ;
( ( ( ( ( ( S \/ X ) \ { p } ) \ { p } ) \ { p } ) ) \ { p } ) ;
( ( - 1 ) |^ ( k + 1 ) ) . i = ( - 1 ) |^ ( k + 1 ) .= ( - 1 ) |^ ( k + 1 ) ;
rng ( ( h + c ) ^\ n ) c= dom SVF1 ( 1 , f , u0 ) ;
the carrier of ( the carrier of X ) = the carrier of ( the carrier of X ) \ { 0 } .= the carrier of X ;
ex p4 st p3 = p4 & |. p4 - |[ a , b ]| .| = r & |. p4 - |[ a , b ]| .| = r ;
set ch = chi ( X , A ) , Aj = chi ( X , A ) , Aj = card X ;
R |^ ( 0 * n ) = Ireal ( X , 0 ) .= R |^ ( n * n ) .= R |^ ( n * n ) .= R |^ ( n * n ) ;
( Partial_Sums ( ( ( - ( F . -19 ) ) ) ) . n ) + ( ( - ( F . -19 ) ) . n ) ) is nonnegative ;
f2 = C7 . ( ( the EE8 of V , len ( the N8 of V , len ( the N8 of V ) ) ) ) ;
S1 . b = s1 . b .= s2 . b .= s2 . b .= s2 . b .= ( s2 * s1 ) . b .= ( s2 * s2 ) . b .= ( s2 * s2 ) . b .= ( s2 * s2 ) . b .= ( s2 * 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 & ( the connectives of S ) . 11 in ( the carrier' of S ) ;
set phi = ( l1 , l2 ) If , phi = ( l1 , l2 ) If , phi = ( l1 , l2 ) If , phi = ( l1 , l2 ) <> {} ;
synonym p is invertible for p is invertible for ( p , T ) *' = 1 & ( p , T ) *' = 1 ;
( ( Y1 `2 ) ^2 = - 1 & ( Y1 `1 ) ^2 + ( Y1 `2 ) ^2 <> 0 ) implies ( Y1 `1 ) ^2 + ( Y1 `2 ) ^2 = ( Y1 `1 ) ^2 + ( Y1 `2 ) ^2
defpred X [ Nat , set , set , set , set , set , set , set , set , set ) = $1 \ { $2 , $2 , $1 , $2 , $1 , $2 , $2 , $2 , $1 , $2 , $2 , $2 , $1 , $2 , $2 , $2 , $2 , $2 , $2 , $1 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 ,
consider k being Nat such that for n being Nat st k <= n holds s . n < x0 + g ;
Det ( I |^ ( m -' n ) ) * ( m -' n ) = 1. ( K , m -' n ) * ( m -' n ) ;
( - b ) / sqrt ( b ^2 - ( - 4 * a * c ) ) < 0 ;
CF . d = CF . ( d mod ( n + 1 ) ) .= CF . ( d mod ( n + 1 ) ) .= CF . ( d mod ( n + 1 ) ) ;
attr X1 is dense dense means : Def3 : X2 is dense dense & X1 is dense SubSpace of X2 & X2 is dense SubSpace of X1 ;
deffunc F6 ( Element of E , Element of I ) = ( $1 * $2 ) * ( $1 * $2 ) ;
t ^ <* n *> in { t ^ <* i *> : Q [ i , T . t ] } ;
( x \ y ) \ x = ( x \ x ) \ y .= y \ ( x \ y ) .= 0. X ;
for X being non empty set for Y being Subset-Family of X holds X is Basis of [: X , Y :] iff X is Basis of [: X , Y :]
synonym A , B , C , D , A , B , C , D , E F F , G , G , H , G , I , J , N , N , N , G , N , G , N , G , N , G , N , G , N , G , N , G , N , G , G , G , N , G , N , G , N , G , N
len ( M @ ) = len p & width ( M @ ) = width M & width ( M @ ) = width M & width ( M @ ) = width M ;
J = { v where x is Element of K : 0 < v . x & 0 < v . x } ;
( ( Sgm ( Seg m ) ) . d - ( Sgm ( Seg m ) ) . e ) <> 0 ;
lower_bound divset ( D2 , k + 1 ) = D2 . ( k + 1 - 1 ) .= D2 . ( k + 1 - 1 ) ;
g . r1 = - 2 * r1 + 1 & dom h = [. 0 , 1 .] & dom h = [. 0 , 1 .] & rng h c= [. 0 , 1 .] ;
|. a .| * ||. f .|| = 0 * ||. f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| ;
f . x = ( h . x ) `1 & g . x = ( h . x ) `2 & h . x = ( h . x ) `2 ;
ex w st w in dom B1 & <* 1 *> ^ s = <* 1 *> ^ w & <* 1 *> ^ s = <* 1 *> ^ w ^ w ;
[ 1 , {} , <* d1 *> ] in ( { 0 , {} } \/ ( { 1 , {} } \/ { {} } ) ) \/ ( { 1 , {} } \/ { {} } ) ;
IC Exec ( i , s1 ) + n = IC Exec ( i , s2 ) .= IC Exec ( i , s2 ) .= IC Exec ( i , s2 ) ;
IC Comput ( P , s , 1 ) = IC Comput ( P , s , 9 ) .= 5 + 9 .= 5 + 9 .= 5 + 9 .= 5 + 9 .= 5 + 9 ;
( IExec ( W6 , Q , t ) ) . intpos ( 4 + 1 ) = t . intpos ( 4 + 1 ) ;
LSeg ( f /^ q , i ) misses LSeg ( f /^ q , j ) \/ LSeg ( f /^ q , j ) ;
assume for x , y being Element of L st x in C & y in C holds x <= y or y <= x ;
||. integral ( f , C ) . x - f /. x0 .|| = f . ( upper_bound C - lower_bound C ) .= ||. lower_bound C - lower_bound C .|| ;
for F , G being one-to-one FinSequence st rng F misses rng G & rng F misses rng G holds F ^ G is one-to-one
||. R /. ( L . h ) - R /. ( L . h ) .|| < e1 * ( K + 1 ) ;
assume a in { q where q is Element of M : dist ( z , q ) <= r } ;
set p4 = [ 2 , 1 ] .--> [ 2 , 1 , 1 ] ;
consider x , y being Subset of X such that [ x , y ] in F and x c= d and y c= d and x c= y ;
for y , x being Element of REAL st y in Y ` & x in X holds y in Y ` & y in Y ;
func |. |. p ^ <* p *> .| -> variable of A equals : Def3 : ( for i st i in dom NBI holds ( p ^ NBI ) . i = ( - NBI ) . i ) ;
consider t being Element of S such that x `1 , y `2 '||' z `1 , t `2 and x , z `1 '||' y , t `2 ;
dom ( x1 - x2 ) = Seg len ( x1 - x2 ) & len ( x1 - x2 ) = len ( x1 - x2 ) & len ( x1 - x2 ) = len ( x1 - x2 ) ;
consider y2 being Real such that x2 = y2 and 0 <= y2 and y2 < 1 and y2 <= 1 and y2 <= 1 ;
||. f | X /* s1 .|| = ||. ( f | X ) /* s1 .|| .= ||. ( f | X ) /. ( lim s1 ) .|| .= ||. ( f | X ) /. ( lim s1 ) .|| ;
( the InternalRel of A ) \ ( x ` \ Y ) = {} .= {} \/ {} .= {} .= {} .= {} ;
assume that i in dom p and for j being Nat st j in dom q holds P [ i , j ] and for i being Nat st i in dom p holds P [ i , p . j ] ;
reconsider h = f | [: X , Y :] as Function of X , Y , X , Y ;
u1 in the carrier of W1 & u2 in the carrier of W2 implies ( the carrier of W1 ) + ( the carrier of W2 ) = the carrier of ( ( the carrier of W1 ) + ( the carrier of W2 ) ) + ( the carrier of V )
defpred P [ Element of L ] means M <= f . $1 & f . $1 <= f . $1 & f . $1 <= f . $1 ;
\mathbin { + } ( u , a , v ) = s * x + ( - ( s * x ) + y ) .= b ;
- ( - ( - y ) ) = - x + - ( - y ) .= - x + - y .= - x + - y .= - x + y .= - x + y ;
given a being Point of IT such that for x being Point of IT holds a , x , a , x , y , z is_collinear ;
fA2 = [ [ dom ( ( [ f , g ] , ( [ f , g ] , [ g , h ] ) ) , [ g , h ] ] , [ g , h ] ] ] ;
for k , n being Nat st k <> 0 & k < n & n < k holds k , n are_relative_prime & k , n are_relative_prime implies k , n are_relative_prime
for x being element holds x in A |^ d iff x in ( ( A ` ) |^ d ) ` & x in ( A ` ) |^ d
consider u , v being Element of R , a being Element of A such that l /. i = u * a * v ;
- ( ( - ( p `1 / |. p .| - cn ) ) / ( 1 + cn ) ) ^2 > 0 ;
L-13 . k = Lz . ( F . k ) & F . ( k + 1 ) in dom ( L * F ) ;
set i2 = AddTo ( a , i , - n ) , i1 = a := ( a , i , - n ) , i2 = a := ( a , i , - n ) , i2 = - ( - n ) ;
attr B is \overline means : Def3 : for S being SubSubInt of B holds ( .= ( B , S ) --> ( x , y ) ) `1 = ( B , S ) `1 ;
a9 "/\" D = { a "/\" d where d is Element of N : d in D } ;
|( \square , REAL . ( ( q - q ) * ( q - p ) ) , REAL . ( q - p ) )| >= |( \square , REAL . ( q - p ) , REAL . ( q - p ) )| ;
( - f ) . ( upper_bound A ) = ( - f ) . ( upper_bound A ) .= - ( f . ( lower_bound A ) ) .= - ( f . ( lower_bound A ) ) ;
( G * ( len G , k ) ) `1 = ( G * ( len G , k ) ) `1 .= G * ( 1 , k ) `1 .= G * ( 1 , k + 1 ) `1 ;
( Proj ( i , n ) * ( g - h ) ) . ( ( Proj ( i , n ) * ( g - h ) ) . ( h - i ) ) = <* ( proj ( i , n ) * ( g - i ) ) . ( h - i ) *> ;
f1 + f2 * reproj ( i , x ) is_differentiable_in ( ( - 1 ) (#) reproj ( i , x ) ) . ( x + x0 ) ;
attr ( for x st x in Z holds ( ( - 1 / 2 ) (#) ( cos * sin ) ) `| Z ) . x <> 0 ;
ex t being SortSymbol of S st t = s & h1 . t = h2 . ( x , t ) & h1 . ( x , t ) = t ;
defpred C [ Nat ] means ( ( for n being Nat st n < $1 holds ( n + 1 ) is as $1 ) & ( ( n + 1 ) < $1 implies n is $1 ) implies ( n + 1 ) in $1 ;
consider y being element such that y in dom ( ( p | i ) | ( Seg n ) ) and ( ( p | i ) | ( Seg n ) ) . y = ( p | ( Seg n ) ) . y ;
reconsider L = product ( { x1 } +* ( index B , l ) ) as Basis of product A ;
for c being Element of C ex d being Element of D st T . ( id c ) = id d & for d being Element of D st d in dom T holds T . ( id c ) = id d
be Element of ( f | n ) ^ <* p *> = ( f | n ) ^ <* p *> .= f /^ n ^ <* p *> .= f /^ n ;
( f * g ) . x = f . ( g . x ) & ( f * h ) . x = f . ( h . x ) ;
p in { 1 / 2 * ( G * ( i + 1 , j + 1 ) + G * ( i + 1 , j + 1 ) + G * ( i + 1 , j + 1 ) + G * ( i , j + 1 ) + G * ( i , j + 1 ) + p / 2 * ( i , j + 1 ) + 1 ) } ;
f `2 - cp = ( f | ( n , L ) ) *' .= f - ( c * ( f | ( n , L ) ) ) *' .= f - ( f - ( f | ( n , L ) ) ) *' .= f - ( f - ( g - ( g - p ) ) ) *' ) ;
consider r being Real such that r in rng ( f | divset ( D , j ) ) and r < m + s ;
f1 . ( |[ ( 8 - r ) / ( 8 - 1 ) , ( 8 - 1 ) / ( 8 - 1 ) ]| ) in [: `1 , `2 :] ;
eval ( a | ( n , L ) , x ) = eval ( a | ( n , L ) , x ) .= a * eval ( b , x ) .= a * eval ( b , x ) ;
z = DigA ( tk , x9 ) .= DigA ( tk , x ) .= DigA ( tk , x ) .= DigA ( tk , x ) .= 0 ;
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 finite } ;
consider S19 being Element of D ( ) , d being Element of D ( ) such that S `1 = S19 ^ <* d *> and S . ( i + 1 ) = d ;
assume that x1 in dom f and x2 in dom f and f . x1 = f . x2 and f . x2 = f . x2 and f . y2 = f . y2 ;
- 1 <= ( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 ;
0. V is Linear_Combination of A & Sum ( - L ) = 0. V implies Sum ( - L ) = Sum ( - L )
let k1 , k2 , x4 , k2 , 6 , 7 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 ,
consider j being element such that j in dom a and j in g " { k } and x = a . j ;
H1 . ( x1 , x2 ) c= H1 . ( x2 , y2 ) or H1 . ( x1 , x2 ) c= H1 . ( x2 , y2 ) ;
consider a being Real such that p = \rbrace * p1 + ( a * p2 ) and 0 <= a and a <= 1 ;
assume that a <= c and c <= d and [ a , b ] c= dom f and [ a , b ] in dom g and g . a = g . b ;
cell ( Gauge ( C , m ) , len Gauge ( C , m ) -' 1 , 0 ) is non empty ;
A-1 in { ( S . i ) `1 where i is Element of NAT : not contradiction } ;
( T * b1 ) . y = L * ( b2 /. y ) .= ( F * b2 ) . ( y , x ) .= ( F * b2 ) . ( y , x ) .= ( F * b2 ) . ( y , x ) ;
g . ( s , I ) . x = s . y & g . ( s , I ) . y = |. s . x - s . y .| ;
( log ( 2 , k + 1 ) ) / ( 2 * k + 1 ) >= ( log ( 2 , k + 1 ) ) / ( 2 * k + 1 ) ;
then that not p => q in S and not x in the still of p and not p => All ( x , p ) in S ;
dom ( the { of r-10 } --> ( the { 0 } , the carrier of r-11 ) ) misses dom ( the { 0 } --> ( the { 0 } , the carrier of r-11 ) ) ;
synonym f is extended real means : Def3 : for x being set st x in rng f holds x is extended real ;
assume for a being Element of D holds f . { a } = a & for X being Subset-Family of D holds f . ( a , X ) = f . union X ;
i = len p1 + len <* x *> .= len p1 + len <* x *> .= len p1 + len <* x *> .= len p1 + len <* x *> .= len p1 + len <* x *> .= len p1 + len <* x *> .= len ( p1 + x ) + len <* x *> .= len ( p1 + x ) + len <* x *> .= len ( p1 + x ) + len <* x *> .= len ( p1 + x ) + 1 ;
( l , 3 ) `1 = ( g /. 1 ) `1 + ( k , 3 ) `1 .= ( g /. 1 ) `1 + ( k , 3 ) `2 .= ( g /. 1 ) `1 + ( k , 3 ) `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 ) is summable ;
sin . ( upper_bound ( - r ) ) = sin . ( - r ) * cos . ( upper_bound ( - r ) ) .= 0 ;
set q = |[ g1 . a , g2 . a ]| , r = |[ g1 . a , g2 . a ]| , s = |[ g1 . a , g2 ]| , t = |[ g1 . a , g2 ]| , s = |[ g1 . a , g2 ]| , r = |[ g1 . a , g2 ]| , s = |[ g1 , g2 ]| ;
consider G being sequence of S such that for n being Element of NAT holds G . n in GG5 ( F . n ) and G is W5 ;
consider G such that F = G and ex G1 st G1 in SX & G in SX & ( for x st x in SX holds G . x = F ( x ) ) ;
the root of [ x , s ] in ( the Sorts of Free ( C , s ) ) . s & ( the Sorts of Free ( C , s ) ) . s = ( the Sorts of Free ( C , s ) ) . s ;
Z c= dom ( exp_R * ( f + ( exp_R * ( f + g ) ) ) ) ;
for k be Element of NAT holds ( ( Im ( Im f ) ) | ( S . k ) ) . i = ( ( Im f ) | ( S . k ) ) . i
assume that - 1 < n and ( q `2 / |. q .| - sn ) < 0 and ( q `1 / |. q .| - sn ) < 0 ;
assume that f is continuous and a < b and a < b and c < d and f . a = c and f . b = d and g . a = d and g . c = d ;
consider r being Element of NAT such that s\mathopen { + } r = Comput ( P1 , s1 , r ) and r <= q ;
LE f /. ( i + 1 ) , f /. j , L~ f & LE f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) ;
assume that x in the carrier of K and y in the carrier of K and inf { x , y } in the carrier of K and inf { x , y } in the carrier of K ;
assume that f +* ( i1 , \xi ) in ( proj ( F , i2 ) ) " ( A /\ B ) and f " ( A /\ B ) in ( proj ( F , i2 ) ) " ( A /\ B ) ;
rng ( ( ( ( ( ( ( ( ( the carrier of M ) \/ the carrier of S ) ) \/ the carrier of S ) \ the carrier of S ) ) \/ the carrier of S ) ) \ the carrier of S ) c= the carrier of M ;
assume z in { ( the carrier of G ) \ { t where t is Element of T : t in A } ;
consider l being Nat such that for m be Nat st l <= m holds ||. s1 . m - lim s1 .|| < g ;
consider t be VECTOR of product G such that mt = ||. Dt . t .|| and ||. t .|| <= 1 ;
assume that the carrier of v = 2 and the carrier of v = 2 and v ^ <* 0 *> ^ <* 1 *> ^ ( v ^ <* 1 *> ) in dom p and v ^ <* 1 *> ^ <* 1 *> in dom p ;
consider a being Element of the Points of Xbe Element of the Points of [ X , A ] , A being Element of the Points of [ X , A ] ;
( - x ) |^ ( k + 1 ) * ( ( - x ) |^ k ) " = 1 ;
for D being set st for i st i in dom p holds p . i in D holds p . i in D & for i st i in dom p holds 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 ( p2 , p1 ) ;
i -' len h11 + 2 - 1 < i -' len h11 + 2 - 1 + 2 - 1 + 1 + 1 ;
for n being Element of NAT st n in dom F holds F . n = |. ( ( n -' 1 ) + 1 ) * ( n -' 1 ) .| ;
for r , s1 , s2 , s3 , s3 , r1 , r2 , s1 , s2 , r2 , s2 , r2 , s1 , s2 , r2 , s2 , r2 , r2 , s2 , r2 , r2 , r2 , s2 , r2 , r2 , r2 , s2 , r2 , r2 , r2 , r2 , s2 , r2 , r2 , r2 , r2 , r2 , r2 , r2 , r2 , s2 , r2 , r2 , r2 , r2 , r2 , r2 , r2 , r2
assume that v in { G where G is Subset of T2 : G in B2 & G c= B1 & G c= B1 & G c= B1 & F c= G & G c= { G where G is Subset of T2 : G in B2 & G c= B1 } ;
let g be Element of A , f be Element of INT , X be set , Y be set , f be Function of [: X , Y :] , INT , b be Element of [: X , Y :] , Z :] ;
min ( g . [ x , y ] , k ) . [ y , z ] = ( min ( g , k ) ) . [ y , z ] ;
consider q1 being sequence of Ck such that for n holds P [ n , q1 . n , q1 . ( n + 1 ) ] ;
consider f being Function such that dom f = NAT and for n being Element of NAT holds f . n = F ( n ) and for x being Element of NAT holds f . x = F ( n ) ;
reconsider B-6 = B /\ B , Gq2 = O /\ ( { A } \/ { B } ) as Subset of [: B , C :] ;
consider j being Element of NAT such that x = the { the } \HM { for i being Nat st 1 <= i & i < n holds 1 <= j & j <= n ;
consider x such that z = x and card ( x . O2 ) in card ( x . O2 ) and x in L1 and x in L2 and x in L1 . ( x . O2 ) ;
( C * _ T4 ( k , n2 ) ) . 0 = C . ( ( ( C * _ ( k , n2 ) ) . 0 ) ) .= C . ( ( ( ( C * _ 1 , n2 ) ) . 0 ) ) ;
dom ( X --> rng f ) = X & dom ( X --> f ) = X & dom ( X --> f ) = X & rng ( X --> f ) = X ;
( ( ( GoB f ) * ( len ( Gauge ( C , n ) , 1 ) ) `1 ) / ( ( ( GoB f ) * ( 1 , 1 ) ) `1 ) <= ( ( GoB f ) * ( 1 , 1 ) ) `1 ) / ( ( ( GoB f ) * ( 1 , 1 ) ) `2 ) ;
synonym x , y , z means : Def3 : x = y or ex l being Nat st { x , y } c= l or ex l being Nat st { x , l } c= l ;
consider X being element such that X in dom ( f | ( n + 1 ) ) and ( f | ( n + 1 ) ) . X = Y ;
assume that $ \mathop { \rm Im } k is continuous and for x , y being Element of L st a = x & b = y & x << y holds a << b iff a << b ;
( 1 / 2 * ( ( - ( m + 1 ) ) * ( ( - m ) / 2 ) ) ) (#) ( ( - m ) / 2 * ( ( - m ) / 2 ) (#) ( ( - m ) / 2 ) ) ) is_differentiable_on REAL ;
defpred P [ Element of omega ] means ( the partial of A1 ) . $1 = ( the partial of A2 ) . $1 & ( the partial of A2 ) . $1 = ( the partial of A2 ) . $1 ;
IC Comput ( P , s , 2 ) = succ IC Comput ( P , s , 1 ) .= 6 .= 6 + 1 .= 6 + 1 .= 6 + 1 .= 6 + 1 .= 6 + 1 .= 6 + 1 ;
f . x = f . ( g1 * f . g2 ) .= f . ( g1 . g2 ) .= f . ( g1 . g2 ) .= ( f . ( g1 . g2 ) ) * ( f . g2 ) .= ( f . ( g2 ) ) * ( f . g2 ) .= ( f . g2 ) * ( f . g2 ) .= ( f . g2 ) * ( f . g2 ) ;
( M * ( F-4 ) ) . n = M . ( ( canFS ( Omega ) ) . n ) .= M . ( ( canFS ( Omega ) ) . 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 , a , b , c , d , x , y , z , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , y , z
( the partial of s ) . n <= ( the partial of s ) . ( n + 1 ) * ( ( the partial of s ) . n ) ;
attr - 1 <= r & r <= 1 implies ( - 1 ) * ( ( - 1 ) * ( r - 1 ) ) = - ( r * ( r - 1 ) ) * ( r - 1 ) ;
seq in { p ^ <* n *> where n is Nat : p ^ <* n *> in T1 & p ^ <* n *> in T1 } ;
|[ x1 , x2 , x3 ]| . 2 - |[ y1 , y2 ]| . 2 - |[ y1 , y2 , x4 ]| . 3 = x2 - |[ y2 , z2 ]| . 2 - |[ y1 , y2 , x4 ]| . 3 - |[ y1 , y2 ]| . 3 - |[ y1 , y2 , x4 ]| . 2 - |[ y1 , y2 , x4 ]| . 3 - |[ y1 , y2 , x4 ]| . 2 = x2 - |[ y1 , y2 , x4 ]| . 3 ;
attr for m being Nat holds F . m is nonnegative & ( Partial_Sums F ) . m is nonnegative & ( Partial_Sums F ) . m is nonnegative ;
len ( ( G . ( y , z ) ) * ( ( G . ( y , z ) ) * ( G . ( y , z ) ) ) ) = len ( ( G . ( y , z ) ) * ( G . ( y , z ) ) ) ;
consider u , v being VECTOR of V such that x = u + v and u in W1 /\ W2 and v in W2 and u in W2 and u in W1 and v in W2 ;
given F being 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 and Sum ( F ) = k ;
0 = ( 1 * a2 ) , 1 = ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - { 1 - { 1 - { 1 / ( 1 - 0 ) ) ) ) ) ) ) ) ) ) / ( 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 ;
cluster -> Boolean -> Boolean for non empty { q , r , s , r , s , r , s , r , s , q , r , s , p , q , r , s , s , r , s , q , r , s , s , r , s , s , r , s , s , r , s , q , r , s , s , s , r , s , q , r , s , s , s , r , s , q ,
"/\" ( B , {} ) = Top ( B , {} ) .= "/\" ( [#] S , {} ) .= "/\" ( [#] S , {} ) .= "/\" ( [#] S , {} ) .= "/\" ( [#] S , {} ) .= "/\" ( {} S , {} ) .= "/\" ( {} , {} ) .= "/\" ( {} , {} ) .= "/\" ( {} , {} ) .= "/\" ( {} , {} ) .= "/\" ( {} , {} ) .= "/\" ( {} , {} ) ;
( 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 ) . x >= r2
2 * r1 - ( 2 * |[ a , c ]| - ( 2 * |[ a , c ]| - ( 2 * |[ a , c ]| - |[ b , c ]| ) ) = 0. TOP-REAL 2 ;
reconsider p = P * ( \square , 1 ) , q = a " * ( ( - - 1_ K ) * ( ( - 1_ K ) * ( 1 , 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 y = [ x1 , x2 ] ;
for n be Nat st 1 <= n & n <= len q1 holds q1 . n = ( ( lower ( g , M7 ) ) | ( i + 1 ) ) . n
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 Subgroup the carrier of G and H1 , H2 / ( x , y ) / ( x , y ) / ( x , y ) / ( x , y ) / ( x , y ) / ( x , y ) ;
for S , T being non empty RelStr , d being Function of T , S st T is complete holds d is monotone iff d is monotone
[ a + 0. F_Complex , b + 0. F_Complex ] in ( the carrier of COMPLEX ) \ ( the carrier of COMPLEX ) & [ a + 0. F_Complex , b ] in the carrier of COMPLEX ;
reconsider mm = max ( len ( F1 . n ) , len ( F1 . n ) * ( p . n ) ) as Element of NAT ;
I <= width GoB ( ( GoB f ) * ( 1 , width GoB f ) + ( GoB f ) * ( 1 , width GoB f ) + ( GoB f ) * ( 1 , width GoB f ) ) ;
f2 /* q = ( f2 /* ( f1 /* s ) ) ^\ k .= ( f2 * ( f1 /* s ) ) ^\ k .= ( f2 * ( f1 /* s ) ) ^\ k .= ( f2 * ( f1 /* s ) ) ^\ k .= ( f2 * ( f1 /* s ) ) ^\ k ;
attr A1 \/ A2 is linearly-independent means : Def3 : A1 is linearly-independent & A2 is linearly-independent & ( for x being Element of A1 st x in A2 holds not x in A1 & x in A2 & not x in A2 & x in A1 & x in A2 & not x in A2 ;
func A -carrier C -> set equals union { A . s where s is Element of R : s in C & s in A } ;
dom ( ( Line ( v , i + 1 ) ) ^ ( ( Line ( p , i + 1 ) ) ^ ( ( Line ( p , i + 1 ) ) ^ ( ( Line ( p , i + 1 ) ) ^ ( Line ( p , i + 1 ) ) ) ) ) = dom ( F ^ ( Line ( p , i + 1 ) ) ) ;
cluster [ ( x `1 ) / 4 , ( x `2 ) / 4 ] -> to |[ x `1 , x `2 ]| & [ x `1 , x `2 ] , [ x `1 , x `2 ] , [ x `2 , x `2 ] ) ;
E , All ( x2 , All ( x2 , All ( x2 , All ( x2 , x2 , x3 , x4 ) ) ) |= All ( x2 , All ( x2 , x2 , x3 , x4 ) ) => ( x2 , Ex ( x2 , x3 , x4 , x4 ) ) ;
F .: ( id X , g ) . x = F . ( id X , g . x ) .= F . ( id X , g . x ) .= F . ( id X , g . x ) .= F . ( id X , g . x ) .= F . ( id X , g . x ) ;
R . ( h . m ) = F . ( x0 + h . m ) - ( h . m ) + ( h . m ) + ( h . m ) - ( h . m ) ;
cell ( G , ( X -' 1 , t + ( t + 1 ) ) , ( X -' 1 ) ) \ L~ f meets ( L~ f ) \ L~ f ;
IC Comput ( P2 , s2 , k ) = IC Comput ( P2 , s2 , k ) .= ( card I + ( card I + 1 ) ) .= ( card I + ( card I + 1 ) ) + ( card I + 1 ) .= ( card I + ( card I + 1 ) ) + ( card I + 1 ) .= card I + ( card I + 1 ) .= card I + ( card I + 1 ) ;
sqrt ( ( - ( - ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ) ^2 ) ) > 0 ;
consider x0 being element such that x0 in dom a and x0 in dom a and y = ( a " { k } ) . x0 and x0 in dom ( a " { k } ) and y = a . x0 and x0 in N ;
dom ( r1 (#) ( chi ( A , A ) ) ) = dom ( ( r1 (#) ( chi ( A , A ) ) ) | ( A /\ B ) ) .= dom ( ( r1 (#) ( chi ( A , B ) ) | ( A /\ B ) ) .= C ;
d-7 . [ y , z ] = ( ( y , z ) `2 ) `2 - ( ( y , z ) `2 ) - ( ( y , z ) `2 ) ;
attr for i being Nat holds C . i = A . i /\ B . i & C . i c= ( A /\ B ) . i ;
assume that x0 in dom f and f is continuous and for x st x in dom f holds ||. f /. x - f /. x0 .|| < r ;
p in Cl A implies for K being Basis of p , Q being Basis of T st K 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 .|
func the \emptyset { <*> a } -> Ordinal means : Def3 : a in it & for b being Ordinal st a in it holds it . b = b & for a being Ordinal st a in it holds it . a = b ;
[ a1 , a2 , a3 ] in ( the carrier of A ) \ ( the carrier of B ) & [ a1 , a2 , a3 ] in ( the carrier of A ) \ ( the carrier of B ) ;
ex a , b being element st a in the carrier of S1 & b in the carrier of S2 & x = [ a , b ] & [ a , b ] in the InternalRel of S2 ;
||. ( ( vseq . n ) - ( vseq . m ) ) * ( ( vseq . n ) - ( vseq . m ) ) .|| < ( e / ( ( ||. vseq . n .|| - ( vseq . m ) ) ) * ||. ( vseq . m ) - ( vseq . m ) .|| ) ;
then for Z being set st Z in { Y where Y is Element of I : for x being Element of I st x in Y holds F . x = x & Z in { x } ;
sup compactbelow [ s , t ] = [ sup ( { s , t } ) , sup ( { s , t } ) ] .= [ sup ( { s , t } ) , sup ( { s , t } ) ] ;
consider i , j being Element of NAT such that i < j and [ y , f . j ] in Ij and [ f . i , f . j ] in Ij and [ f . i , f . j ] in Ij ;
for D being non empty set , p , q being FinSequence of D st p c= q & p ^ q = q holds ex p being FinSequence st p = q & p = p ^ q
consider e1 being Element of the affine of X such that c9 , a9 // a9 , c9 and not LIN a9 , b9 , c9 and not LIN a9 , c9 , a9 and not LIN a9 , b9 , c9 and not a9 , c9 // a9 , b9 and not a9 , c9 // a9 , b9 and not a9 , c9 // a9 , b9 and not a9 , c9 // b9 , c9 , c9 ;
set U2 = I \! \mathop { \vert F . a .| , U2 = I \! \mathop { \vert a .| } , d = I \! \mathop { |. a .| } , F = { a } , d = { a } , e = { a } , d = { a } , e = { a } , d = { a } , e = { a } , d = { a } , e = { a } , d = { a } , e = { a } , d = { a } , e = { a } , d = { a } , d = { a } , e = { a } , d
|. q2 .| ^2 = ( ( q2 `1 ) ^2 + ( q2 `2 ) ^2 ) ^2 .= |. q2 .| ^2 + ( q2 `2 ) ^2 .= |. q2 .| ^2 + ( q2 `1 ) ^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 ) & dom ( the charact of U1 ) = dom ( the charact of U1 ) & dom ( the charact of U1 ) = the carrier of U1 & dom ( the charact of U1 ) = the carrier of U1 ;
dom ( h | X ) = dom h /\ X .= dom ( ( h | X ) | X ) .= dom ( ( h | X ) | X ) .= dom ( ( h | X ) | X ) .= dom ( ( h | X ) | X ) .= dom ( ( h | X ) | X ) .= dom ( ( h | X ) | X ) .= dom ( ( h | X ) | X ) .= dom ( h | X ) .= dom ( h | X ) .= dom ( h | X ) /\ X .= dom ( h | X ) /\ X .= dom ( h | X ) /\ X .= dom (
for N1 , N2 being Element of [: N1 , N1 :] , N1 , N2 , N1 , N2 , N1 , N2 , N1 , N2 , N1 , N2 , N1 , N2 , N1 , N2 , N2 , N1 , N2 , N1 , N2 , N2 , N1 , N2 , N2 , N1 , N2 , N2 , N1 , N2 , N2 , N2 , N2 , N1 , N2 , N2 , N2 , N1 , N2 , N2 , N1 , N2 , N2 , N1 , N2 , N1 , N2 , N1 , N2 , N2 , N1 , N2 , N1 , N2 , N2 , N2 , N2 , N2 , N2 , N1 , N2 , N2
( ( mod ( u , m ) ) + mod ( v , m ) ) . i = ( mod ( u , m ) ) . i + ( mod ( v , m ) ) . i .= ( mod ( v , m ) ) . i + ( mod ( v , m ) ) . i ;
- ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) < - ( q `1 / |. q .| - cn ) / ( 1 + cn ) & - ( q `1 / |. q .| - cn ) >= 0 ;
attr r1 = f9 & r2 = f9 & s1 = q9 & s2 = f9 & s1 = q9 & s2 = q9 & s1 = q9 & s2 = q9 & s1 = s2 & s1 = s2 & s2 = s3 & s1 = s2 & s2 = s3 & s1 = s3 & s2 = s3 & s1 = s2 & s1 = s2 & s1 = s3 & s2 = s3 & s1 = s2 & s1 = s3 & s1 = s3 & s2 = s3 & s1 = s3 & s2 = s3 & s1 = s3 & s2 = s3 & s1 = s3 & s2 = s3 & s1 = s3 & s2 = s3 & s1
vseq . m is bounded Function of X , the carrier of Y & x9 . m = ( vseq . m ) * ( ( vseq . m ) * ( vseq . m ) ) & x9 . m = ( vseq . m ) * ( vseq . m ) ;
attr a <> b & b <> c & c <> 0 & angle ( a , b , c ) = PI & angle ( b , c , a ) = 0 & angle ( b , c , a ) = 0 ;
consider i , j being Nat , r being Real such that p1 = [ i , r ] and p2 = [ j , r ] and r < s and s < 1 ;
|. p .| ^2 - ( 2 * |( ( p , q ) , q )| ) ^2 + |. ( p , q ) .| ^2 = |. p .| ^2 + |. q .| ^2 ;
consider p1 , q1 being Element of [: X , Y :] such that y = p1 ^ q1 and q1 ^ <* q1 *> = p1 ^ q1 and p1 ^ q1 = q1 ^ q1 and q1 ^ q2 = q1 ^ q2 ;
( ( the multF of A ) . ( r1 , r2 ) , ( the _ of A ) . ( r2 , s2 ) ) = ( s2 - s2 ) * ( ( r2 - s2 ) * ( r2 - s2 ) ) ;
( ( LMP A ) `2 = lower_bound ( proj2 .: ( A /\ \mathop { \rm Ball } ( w , r ) ) ) & ( proj2 .: ( A /\ \mathop { \rm Ball } ( w , r ) ) ) is non empty ) implies ( proj2 .: ( A /\ \mathop { w } ) ) is non empty ;
s , ( ( H , 1 ) |= H1 '&' H2 iff s , ( H , 1 ) |= H2 ) & s , ( ( H , 1 ) |= H2 '&' ( H1 , 1 ) ) ;
len ( s + t ) = card ( ( support b1 ) \/ { t } ) .= card ( ( support b2 ) \/ { t } ) .= card ( ( support b1 ) \/ { t } ) .= card ( ( support b2 ) \/ { t } ) .= card ( ( support b2 ) \/ { t } ) .= card ( ( support b1 ) \/ { t } ) .= ( len b1 ) + ( len b2 ) + 1 ) .= ( len b2 + 1 ) + 1 .= len b2 + 1 ;
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 ( UMP D , |[ ( E-bound D ) / 2 , ( E-bound D ) / 2 ) / 2 ) = { UMP D , ( ( UMP D ) / 2 ) / 2 , ( ( UMP D ) / 2 ) / 2 }
lim ( ( f `| N ) /* ( g `| N ) ) = lim ( ( f `| N ) /* ( g `| N ) ) .= lim ( ( f `| N ) /* ( g `| N ) ) .= lim ( ( f `| N ) /* ( g `| N ) ) ;
P [ i , pr1 ( f ) . i , pr1 ( f ) . ( i + 1 ) ] & [ f . i , pr1 ( f ) . ( i + 1 ) ] in the InternalRel of A ;
for r be Real st 0 < r ex m be Nat st for k be Nat st m <= k holds ||. ( ( seq . k ) - ( R /. m ) ) - ( R /. k ) .|| < r
for X being set , P being a_partition of X , x , a , b , c being Element of X st x in a & a in P & b in P & c in P & a in P & b in P & c in P & a in P & b in P & c in P & a in P & b in P & c in P & a in P & b in P & c in P & a in P & b in P & c in P & c in P & a in P & c in P & c in P & c in P & c in P & c in P & c
Z c= dom ( ( - 1 / 2 ) (#) ( ( - 1 ) (#) f ) ) \ ( ( - 1 ) (#) ( ( - 1 ) (#) f ) " { 0 } ) ;
ex j being Nat st j in dom ( l ^ <* x *> ) & j < i & ( l ^ <* x *> ) . j = ( l ^ <* x *> ) . j & ( l ^ <* x *> ) . j = 1 & ( l ^ <* x *> ) . j = 1 ;
for u , v being VECTOR of V , r being Real st 0 < r & u in N & v in N holds r * u + ( r * v ) in c= c= Seg N
A , Int Cl A , B , C , D , E , F , G , G , H , G , I , G , H , G , I , G , H , G , H , I , G , G , H , G , I , G , H , G , H , I , G , H , G ) = H , I , G , H , G , I , G , H , G , I , G , H , G , I , G , H , G , G , G , H , G , G , H , G , H , G , G , H , G , H ,
- Sum <* v , u , w *> = - ( v + u + u ) .= - ( v + u + u ) .= - ( v + u ) + ( u + u ) .= - ( v + u ) + ( u + u ) .= - ( v + u ) + ( u + u ) .= - ( v + u ) + u .= - ( v + u ) ;
( Exec ( a := b , s ) ) . IC SCM = ( Exec ( a := b , s ) ) . IC SCM .= ( Exec ( a := b , s ) ) . IC SCM .= succ IC s .= ( IC s ) .= ( IC s ) .= ( IC s ) .= ( IC s ) .= ( IC s ) .= ( IC s ) + ( IC s ) ;
consider h being Function such that f . a = h and dom h = I and for x being element st x in I holds h . x in ( the carrier of J ) . x ;
for S1 , S2 being non empty RelStr , D being non empty Subset of S1 , X being non empty Subset of S2 , Y being non empty Subset of S1 , x being Element of S2 , y being Element of S1 , z being Element of S2 , x being Element of S2 , y being Element of S2 st x = [ x , y ] & y = [ x , z ] holds x is directed
card X = 2 implies ex x , y st x in X & y in X & x <> y & not ( ex z st z in X & z in X & x = z ) or ex x st x = z & y = x ) ;
E-max L~ Cage ( C , n ) in rng ( Cage ( C , n ) \circlearrowleft E-max L~ Cage ( C , n ) ) & E-max L~ Cage ( C , n ) /. len ( Cage ( C , n ) \circlearrowleft E-max L~ Cage ( C , n ) ) = Cage ( C , n ) /. len ( Cage ( C , n ) \circlearrowleft E-max L~ Cage ( C , n ) ) ;
for T , T , p , q , r being tree , p , q being Element of dom T st p ^ q ^ r = T . q & ( T -tree ( p , q ) ) . q = T . q holds ( T -tree ( p , r ) ) . q = T . q
[ 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 , n ) divides ( k , n ) & ( k divides m ) & ( for m being Nat st k divides m & m divides n & k <= n ) & ( for m being Nat st m divides n & k <= m holds k divides m ) implies k divides n ) ;
dom F " = the carrier of X2 & rng F = the carrier of X2 & rng F = the carrier of X2 & rng F = the carrier of X2 & rng F = the carrier of X2 & rng F = the carrier of X2 & rng F = the carrier of X2 & rng F = the carrier of X2 & rng F = the carrier of X2 & rng F = the carrier of X2 & rng F = the carrier of X2 ;
consider C being finite Subset of V such that C c= A and card C = m and the carrier of C = Lin ( B \/ C ) and the carrier of C = Lin ( B \/ C ) and the carrier of C = Lin ( A \/ B ) ;
V is prime implies for X , Y being Element of \langle the topology of T , the topology of T *> st X /\ Y c= V holds X c= Y or Y c= V
set X = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } , Y = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } ;
angle ( p1 , p3 , p4 ) = 0 .= angle ( p2 , p3 , p4 ) .= angle ( p2 , p3 , p4 ) .= angle ( p2 , p3 , p4 ) .= angle ( p2 , p3 , p4 ) .= angle ( p2 , p3 , p4 ) .= angle ( p2 , p3 , p4 ) ;
- sqrt ( ( - ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ) ^2 ) = - ( - ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) .= - ( - ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) .= - ( - ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ;
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 . 1 = p3 & f . 0 = p4 & f . 1 = p3 & f . 1 = p4 & f . 1 = p4 & f . 1 = p4 & f . 1 = p2 & f . 1 = p4 & f . 1 = p4 & f . 1 = p4 ;
attr f is partial differentiable of REAL means : Def3 : for u , v , y0 being Element of REAL holds SVF1 ( 2 , pdiff1 ( f , 1 ) , u ) . ( 3 + 1 ) = ( proj ( 2 , pdiff1 ( f , 1 ) ) ) . ( 3 + 1 ) ;
ex r , s st x = |[ r , s ]| & G * ( len G , 1 ) `1 < r & r < G * ( 1 , 1 ) `1 & s < G * ( 1 , 1 ) `2 & s < G * ( 1 , 1 ) `2 & s < G * ( 1 , 1 ) `2 ;
assume that f is_sequence_on G and 1 <= t and t <= len G and G * ( t , width G ) `2 >= S-bound L~ f and G * ( t , width G ) `2 >= S-bound L~ f and G * ( t , width G ) `2 >= S-bound L~ f ;
attr i in dom G means : Def3 : r * ( f (#) ( reproj ( i , x ) ) ) = r * ( reproj ( i , x ) ) . i ;
consider c1 , c2 being bag of o1 + o2 such that ( decomp c ) /. k = <* c1 , c2 *> and ( decomp c ) /. k = <* c1 , c2 *> and ( decomp c ) /. k = <* c1 , c2 *> and ( decomp c ) /. k = c1 + c2 ;
u0 in { |[ r1 , s1 ]| : r1 < G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `2 < G * ( 1 , 1 ) `2 } ;
Cl ( X ^ Y ) = the carrier of X .= ( ( X ^ Y ) \ ( X ^ Y ) ) \ ( X ^ Y ) .= ( ( X ^ Y ) \ ( X ^ Y ) ) \ ( X ^ Y ) .= ( X \ Y ) \ ( X \/ Y ) .= ( X \ Y ) \ ( X \/ Y ) .= ( X \ Y ) \ ( X \ Y ) ) \ ( X \ Y ) ;
attr M1 = len M2 & M2 = len M2 & M1 = M2 & M2 = M2 & M1 = M2 & M2 = M2 & M1 = M2 & M2 = M2 implies M1 + M2 = M2 + M1 & M1 = M2 + M2 & M1 = M2 + M2 & M2 = M2 + M2 ;
consider g2 being Real such that 0 < g2 and { y where y is Point of S : ||. ( y - x0 ) - ( x0 - r ) .|| < g2 & g2 in N } c= N2 ;
assume x < ( - b + sqrt ( a , b , c ) ) / ( 2 * a ) or x > - b + sqrt ( a , b , c ) / ( 2 * a ) ;
( G1 '&' G2 ) . i = ( <* 3 *> ^ ( G1 ^ G2 ) ) . i & ( G1 ^ G2 ) . i = ( <* 3 *> ^ ( G1 ^ G2 ) ) . i & ( G1 ^ G2 ) . i = ( <* 3 *> ^ ( G1 ^ G2 ) ) . i ;
for i , j st [ i , j ] in Indices ( ( M2 + M1 ) * ( i , j ) ) holds ( ( M2 + M1 ) * ( i , j ) ) `1 < ( M2 + M2 ) * ( i , j ) `1
for f being FinSequence of NAT , i being Element of NAT st i in dom f & i in dom f & ( for j being Element of NAT st j in dom f & j < len f holds i divides j ) holds i divides Sum f
assume that F = { [ a , b ] where a , b is Element of X : for c being set st c in B\mathopen { \rbrack a , b .[ & c in B\mathopen { \rbrack a , b .[ } holds a c= b ;
b2 * q2 + ( b3 * q2 ) + ( b3 * q2 ) + ( ( ( a , b ) + ( - ( a , b ) + ( - ( a , b ) + ( - ( a , c ) + ( - ( a , b ) + ( - ( a , c ) + - ( a , b ) + ( - ( a , c ) + ( - ( a , b ) + c ) ) ) ) ) ) = 0. TOP-REAL n ;
Cl ( F ) = { D where D is Subset of T : ex B being Subset of T st D = Cl B & B is open & x in D & B in F } ;
attr seq is summable means : Def3 : for n holds seq is summable & ( for m st m + n < k holds seq . m = Sum ( seq ) ) & ( for m holds seq . m = Sum ( seq ) ) & ( for m st m <= k holds seq . m = Sum ( seq ) ) ;
dom ( ( ( cn -FanMorphN ) | D ) | D ) = ( the carrier of ( ( TOP-REAL 2 ) | D ) | D ) .= the carrier of ( ( ( TOP-REAL 2 ) | D ) | D ) .= the carrier of ( ( ( TOP-REAL 2 ) | D ) | D ) .= the carrier of ( ( TOP-REAL 2 ) | D ) | D ) .= the carrier of ( ( TOP-REAL 2 ) | D ) | D ) ;
[ X \to Z ] is full full non empty SubRelStr of ( [#] Z ) |^ the carrier of Z & [ X \to Y ] is full SubRelStr of ( [#] Y ) |^ the carrier of Z ;
( G * ( 1 , j ) `2 = ( G * ( 1 , j ) `2 ) `2 & ( G * ( 1 , j + 1 ) `2 <= ( G * ( 1 , j + 1 ) `2 ) `2 ) ;
synonym m1 c= m2 means : Def3 : for for p , q being set st p in P & q in P & ( for p being set st p in P holds ( m1 , m2 ) `1 <= ( m , m2 ) `2 ) & ( m , m2 ) `2 <= ( m , m2 ) `2 ;
consider a being Element of B ( ) such that x = F ( a ) and a in { G ( b ) where b is Element of B ( ) : P [ b ] } and for x being Element of A ( ) st P [ x , a , b ] holds a = b ;
attr IT is multiplicative means : Def3 : for s being Element of IT holds mas (# the carrier of IT , the carrier of IT #) #) (# carrier -> Element of the carrier of IT , the carrier of IT #) : the multF of it = the multF of s ;
( the carrier of \HM { a , b , c } + 1 ) + v = b + c + d .= b + d .= ( the carrier of X + c ) + d .= ( the carrier of X + c ) + d .= ( the carrier of X + d ) + ( b + d ) .= ( the carrier of X + c ) + ( d + c ) .= ( the carrier of X + d ) + ( d + c ) .= ( the carrier of X + d ) + ( b + d ) .= ( the carrier of X + d ) + ( b + d ) .= ( the carrier of X + d ) + ( b + d ) .= ( the carrier of X + d ) + ( b + d ) + ( c +
cluster ( i + 1 ) -tuples_on INT -> ( for Element of INT , i , j be Element of INT , k be Element of INT , i , j be Element of INT , i , j be Element of INT , k be Element of INT st i = j + 1 & j = i + 1 holds k = j + 1 ;
- ( s2 * p1 + ( s2 * p2 ) ) = - ( r2 * p1 + s2 * p2 ) .= - ( r2 * p1 + s2 * p2 ) .= - ( r2 * p1 + s2 * p2 ) .= - ( r2 * p1 + s2 * p2 ) ;
eval ( ( a | ( n , L ) ) *' p , x ) = eval ( a | ( n , L ) ) * eval ( p , x ) .= a * 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 , A being Subset of T st A = the carrier of S & for V being open Subset of S st V in the topology of T holds V 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 ;
assume that 1 <= k and k <= len w + 1 and T-7 . ( ( ( ( ( q , w ) -\mathop { \rm \hbox { - } F ) -\mathop { - } F ) . k ) = ( ( ( q , w ) -\mathop { - } F . k ) ) \hbox { - } F . k } ) `1 and ( ( ( q , w ) -\mathop { - } F . k ) ) `1 = ( ( q , w ) -\mathop { - } , w ) `1 , ( ( q , w ) -\mathop { - 1 , w } ) `1 , ( q , w ) `1 , ( q , w ) `1 , ( q , w ) `1 , ( q , w ) `1 , ( q , w ) `1 ,
2 * a |^ ( n + 1 ) + ( 2 * b |^ ( n + 1 ) ) >= ( a |^ n + b |^ ( n + 1 ) ) + ( 2 * a |^ ( n + 1 ) ) + ( 2 * b |^ ( n + 1 ) ) ;
M , v2 / ( x. 3 , x ) / ( x. 0 , x ) / ( x. 4 , x ) / ( x. 0 , x ) / ( x. 0 , x ) / ( x. 4 , x ) / ( x. 0 , x ) / ( x. 0 , x ) ) / ( x. 0 , x ) / ( x. 4 , x ) / ( x. 0 , x ) / ( x. 0 , x ) ) / ( x. 0 , x ) / ( x. 4 , x ) / ( x. 0 , x ) ) / ( x. 0 , x ) / ( x. 0 , x ) / ( x. 0 , x ) / ( x. 0 , x ) / ( x. 0 , x ) / ( x. 0 , x ) / ( x. 0 , x ) /
assume that f is_differentiable_on l and for x0 st x0 in l holds 0 < f . x0 and for x0 st x0 in l holds 0 < f . x0 - f . x0 and for x0 st x0 in l holds f . x0 < f . x0 and f . x0 < 0 ;
for G1 being _Graph , W being Walk of G1 , e being Vertex of G2 , x being Vertex of G1 , y being Vertex of G2 , e being Vertex of G2 st e in W & x in W holds not e in ( the carrier' of G1 ) \ ( the carrier' of G2 ) \ ( { x } ) ;
not ( not ( ( iff ( not ( not ( not ( ( not ( ( not empty lim lim _ 1 is not true is not empty & not ( not .| is not empty & not ( not .| is not empty ) & not ( not .| is not empty ) & not ( not .| is not empty ) ) ) & not ( not empty is not empty ) ) ) ;
Indices ( GoB f ) = [: dom f , Seg width GoB f :] & dom ( GoB f ) = [: Seg len GoB f , Seg width GoB f :] & dom ( f | [: Seg len GoB f , Seg width GoB f :] ) = [: Seg len GoB f , Seg width GoB f :] & dom ( f | [: Seg width GoB f , Seg width GoB f :] ) = Seg width GoB f ;
for G1 , G2 , G1 , G2 being Group st G1 is_stable _of O & G2 is_stable $ O & G1 is_stable $ O & G2 is_stable $ O & G2 is_stable $ O holds G1 is stable Subgroup of G1 & G2 is stable
UsedIntLoc ( int ( f , 1 ) ) = { [ ( int ( ) , 1 , 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , 1 ) , 1 ) } ;
for f1 , f2 being FinSequence of F st f1 is p -element & f2 is p -element & ( for i st i in dom f1 holds f1 . i = f2 . i ) & ( for i st i in dom f1 holds f1 . i = f2 . i ) holds f1 is ( p ^ f2 ) | ( i + 1 ) = f2 | ( i + 1 )
( ( p `1 ) / sqrt ( 1 + ( p `2 ) ^2 ) ) ^2 = ( ( q `1 ) / sqrt ( 1 + ( q `2 ) ^2 ) ) / sqrt ( 1 + ( q `1 ) ^2 ) ;
for x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 7 , 8
for x , - x st - x in dom ( ( - F ) | A ) holds ( ( - F ) | A ) . ( - x ) = - ( ( - F ) | A ) . ( - x )
for T being non empty TopSpace , P being Subset-Family of T , B being Basis of T st P c= the topology of T & for x being Point of T st x in P & B c= P holds ex B being Basis of T st B c= P & B is Basis of T
( a 'or' b ) 'imp' c . x = 'not' ( ( a 'or' b ) . x 'or' c . x ) 'or' c . x .= 'not' ( ( a 'or' b ) . x 'or' c . x ) 'or' 'not' ( ( a 'or' b ) . x 'or' c . x ) .= TRUE ;
for e being set st e in A9 ex X1 being Subset of X , Y1 being Subset of Y 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 & Y2 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open implies Y1 is open ;
for i be set st i in the carrier of S for f being Function of [: S , T :] , [: S , T :] , [: S , T :] st f = H . i & F . i = f | [: S , T :] holds F . i = f | [: S , T :]
for v , w st for y st x <> y holds w . y = v . y holds J . ( v . y ) = J . ( ( VERUM ( Al , J ) . v ) . w )
card D = card ( D1 + card D2 - 1 ) .= card ( D1 + 1 - 1 ) - card ( { i + 1 } - 1 ) .= 2 * card ( D1 + 1 - 1 ) - card ( { i + 1 } - 1 ) .= 2 * card ( D1 + 1 - 1 ) - card ( { i + 1 } - 1 ) .= 2 * card ( D1 + 1 - 1 ) ;
IC Exec ( i , s ) = ( s +* ( 0 .--> ( 1 , 0 ) ) ) . 0 .= ( 0 .--> ( 1 , 0 ) ) . 0 .= ( 0 .--> ( 1 , 0 ) ) . 0 .= ( 0 .--> ( 1 , 0 ) ) . 0 .= ( 0 .--> ( 1 , 0 ) ) . 0 .= ( 0 .--> ( 1 , 0 ) ) . 0 .= ( 0 .--> ( 1 , 0 ) ) . 0 .= ( 0 .--> ( 1 , 0 ) ) . 0 .= ( 0 .--> ( 1 , 0 ) .= ( 0 .--> ( 1 , 0 ) .= ( 0 .--> ( 1 , 0 ) .= ( 0 .--> ( 1 , 1 ) .= ( 0 .--> ( 1 , 1 ) )
len f /. ( \downharpoonright i1 -' 1 + 1 ) = len f -' ( len f -' 1 + 1 ) .= len f -' ( len f -' 1 + 1 ) + 1 .= len f -' ( len f -' 1 + 1 ) + 1 .= len f -' ( len f -' 1 + 1 ) + 1 .= len f -' ( len f -' 1 + 1 ) + 1 .= len f -' ( len f -' 1 + 1 ) + 1 .= len f -' 1 + 1 + 1 + 1 + 1 + 1 .= len f -' 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 .= len f -' 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 +
for a , b , c being Element of NAT st 1 <= a & 2 <= b & k < a holds a <= b + b-2 or a = b + b-2 or a = b + b-2 or a = b + b-2 or a = b + b-2 or a = b + b-2 or a = b-2 or a = b-2 or a = b-2 or a = b-2 or a = b-2 or a = b-2 or a = b-2 or b = b-2 or b = b-2 or b = b-2 or b = b`1 ;
for f being FinSequence of TOP-REAL 2 , p , q being Point of TOP-REAL 2 , r being Real st p in LSeg ( f , i ) & q in LSeg ( f , i ) & Index ( p , f /. 1 ) = r & Index ( p , f /. len f ) = Index ( p , f /. 1 ) holds Index ( p , f /. len f ) <= Index ( p , f /. len f )
lim ( ( curry ( ( F , k + 1 ) # x ) ) . 0 ) = lim ( ( ( F , k + 1 ) # x ) . 0 ) + lim ( ( ( F , k + 1 ) # x ) . 0 ) ;
z2 = g /. ( \downharpoonright n1 -' 1 + 1 ) .= g . ( i -' n1 + 1 + 1 ) .= g . ( i -' n1 + 1 + 1 ) .= g . ( i -' n1 + 1 + 1 ) .= g . ( i -' n1 + 1 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) ;
[ f . 0 , f . 3 ] in id ( the carrier of G ) \/ ( the InternalRel of G ) or [ f . 0 , f . 3 ] in the InternalRel of ( G ) \/ the InternalRel of G ;
for G being Subset-Family of B st G = { [ X , R ] where X is Subset of A ( ) , Y is Subset of B ( ) , R is Subset of A ( ) st X = { [ X , Y ] where Y is Subset of A ( ) : Y in F ( ) & Y in G ( ) } holds Y is finite
CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 ) ) = CurInstr ( P1 , Comput ( P2 , s2 , m1 + m2 ) ) .= CurInstr ( P2 , Comput ( P2 , s2 , m1 + m2 ) ) .= CurInstr ( P2 , Comput ( P2 , s2 , m1 + m2 ) ) .= CurInstr ( P2 , Comput ( P2 , s2 , m1 + 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 c on N and p on M and a on M and p on N and c on N and p on M and p on N and p on M and p on N and p on M and p on N and p on M and p on N and p on N and p on M and p on N and p on N and p on M and p on N and p on N and p on N and p on M and p on N and p on N and p on N and p on M and p on N and p on N and p on N and p on N and p on M and p on M and p on N and p on M and p on M and p on N and p on N and p on N and p on M and p on N and p on N and p on M
assume that T is \hbox of 4 and ( ex F being Subset-Family of T st F is closed & for F being Subset-Family of T st F is closed & F is finite-ind & for n being Nat st n <= 0 holds F . n <= 0 ) and ind F <= 0 and ind F <= 0 ;
for g1 , g2 st g1 in ]. r - g , r + g .[ & g2 in ]. r - g , r + g .[ & |. f . g1 - g .| <= ( r - g ) / ( r - g ) holds |. f . g2 - f . g2 .| <= ( r - g ) / ( r - g )
( ( - 1 ) / ( |. z .| + |. z .| ) ) / ( |. z .| + |. z .| ) = ( - 1 ) / ( |. z .| + |. z .| ) * ( |. z .| + |. z .| ) .= ( - 1 ) / ( |. z .| + |. z .| ) * ( |. z .| + |. z .| ) ;
F . i = F /. i .= 0. R + r2 .= ( b |^ n ) * ( r |^ ( i + 1 ) ) .= <* ( n + 1 ) -tuples_on ( the carrier of K ) ) * ( r |^ ( i + 1 ) , ( n + 1 ) -tuples_on ( the carrier of K ) ) .= <* ( n + 1 ) -tuples_on ( the carrier of K ) *> ;
ex y being set , f being Function st y = f . n & dom f = NAT & for n being Nat holds f . ( n + 1 ) = R ( n , f . n ) & for x being element st x in NAT holds f . ( n + 1 ) = R ( n , x ) ;
func f (#) F -> FinSequence of V means : Def3 : len it = len F & for i be Nat st i in dom it holds it . i = F . i * F . ( i , j ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8
for n being Nat for x , n being Nat st x = h . n & h . ( n + 1 ) = o ( x , n ) & x in InputVertices S ( x , n ) & o . ( n + 1 ) in InputVertices S ( x , n ) & o . ( n + 1 ) in InputVertices S ( x , n ) holds o . ( n + 1 ) = S . ( n + 1 )
ex S1 being Element of QC-WFF ( Al ( ) ) st ( for e being Element of Al ( ) , P being Element of D ( ) , l being Element of D ( ) st P [ e , S1 , l ] ) & ( for k being Element of NAT ( ) holds P [ k , l , k ] ) & ( for k being Element of NAT st P [ k ] ) & P [ k , l , k ] ) holds P [ k , l , P [ k , l , P , P [ k , l , P , P , P [ k , l , P , P [ k , P , l , P , P , l , P , P , l , P , l , P , l , P , P , l , P , P , P , P , P , P , P , P ,
consider P being FinSequence of Gs2 such that ps2 = product P and for i being Element of dom t st i in dom P ex t7 being Element of the carrier of K st P . i = t . i & P . i = t . i & P . i = t . i ;
for T1 , T2 being non empty TopSpace , P being Basis of T1 , P1 being Basis of T2 st the topology of T1 = the topology of T2 & the topology of T1 = the topology of T2 & the topology of T2 = the topology of T2 & the topology of T1 = the topology of T2 & the topology of T2 = the topology of T2 & the topology of T1 = the topology of T2 & the topology of T2 = the topology of T2 & the topology of T2 = the topology of T2 & the topology of T2 & the topology of T2 is open of T2 & the topology of T2 is open & the topology of T2 is open of T2 is open of T2 & the topology of T2 & the topology of T2 is open & the topology of T2 & the topology of T2 = the topology of T2 & the topology of T1 = the topology of T2 = the topology
assume that f is_partial_differentiable_in u0 , u0 and r (#) pdiff1 ( f , 3 ) is_partial_differentiable_in u0 , 2 and partdiff ( r (#) pdiff1 ( f , 3 ) , 3 ) = r * pdiff1 ( f , 3 ) and partdiff ( r (#) pdiff1 ( f , 3 ) , 3 ) = r * pdiff1 ( f , 3 ) ;
defpred P [ Nat ] means for F , G being FinSequence of bool REAL st len F = $1 & for s being FinSequence of REAL st len s = $1 & len s = $1 & for i being Nat st i in Seg $1 holds F . i = G . i & G . ( i + 1 ) = F . ( s . i ) holds Sum ( F , G ) = Sum ( F , G ) ;
ex j st 1 <= j & j < width GoB f & ( GoB f ) * ( 1 , j + 1 ) `2 <= s & s <= ( GoB f ) * ( 1 , j + 1 ) `2 & ( GoB f ) * ( 1 , j + 1 ) `2 <= s ) ;
defpred U [ set , set ] means ex Fi1 being Subset-Family of T st $1 = Fi1 & $2 is open & ( for x being Element of T st x in Fi1 holds ( x in F . x ) & ( x in F . x implies x in F . x ) & ( x in F . x implies x in F . x ) & ( x in F . x implies x in F . x ) ;
for p4 being Point of TOP-REAL 2 st LE p4 , p2 , P & LE p4 , p1 , P & LE p4 , p1 , P & LE p3 , p1 , P & LE p4 , p1 , P & LE p4 , p1 , P & LE p3 , p1 , P & LE p4 , p1 , P & LE p4 , p1 , P & LE p2 , p1 , P & LE p3 , p1 , P & LE p4 , p1 , P & LE p4 , p1 , P & LE p4 , p1 , P & LE p4 , p1 , P & LE p4 , p1 , P & LE p4 , p1 , P & LE p4 , p1 , P & LE p4 , p1 , P & LE p4 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2
f in the carrier of E ( ) & for g st g in f . y holds for x st g in f . x holds x in g . ( x , g . x ) implies f in the carrier of ( the carrier of E ( ) ) & g in the carrier of ( the carrier of E ( ) )
ex 8 being Point of TOP-REAL 2 st x = 8 & ( for p being Point of TOP-REAL 2 st p in 8 holds |. p .| <= 8 ) & ( for q being Point of TOP-REAL 2 st q in 8 holds |. q .| <= 8 ) & ( for q being Point of TOP-REAL 2 st q in 8 holds |. q .| <= 8 ) implies |. q .| <= |. q .| ) & ( |. q .| <= |. q .| <= |. q .| ) ;
assume for d7 being Element of NAT st d7 <= ( n , 8 ) holds s1 . ( n + 1 ) = s2 . ( n + 1 ) & s2 . ( n + 1 ) = s2 . ( n + 1 ) & s1 . ( n + 1 ) = s2 . ( n + 1 ) & s2 . ( n + 1 ) = s2 . ( n + 1 ) ;
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 TOP-REAL 2 st e in Ball ( x , r ) & s = Ball ( e , r ) and s = Ball ( e , r ) and s = Ball ( e , r ) ;
given r such that 0 < r and for s holds 0 < s or ex x1 , x2 being Point of CNS st x1 in dom f & x2 in dom f & ||. x1 - x2 .|| < s & ||. x1 - x2 .|| < s & ||. x1 - x2 .|| < s ;
( p | x ) | ( ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | R ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) | ( ( 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 , x , h st x + h in dom sec & x in dom sec & h in dom sec & x + h in dom sec & x + h in dom sec & x + h in dom sec & x + h in dom sec & x + h in dom sec & x + h in dom sec & x + h in dom sec & x + h in dom sec & x + h in dom sec & x + h in dom sec & x + h in dom sec & x + h in dom sec & x + h in dom sec & x + h in dom sec & x + h in dom sec & x + h in dom sec & x + h in dom sec & x + h in dom sec & x + h in dom sec & x + h in dom sec & x + h in dom sec ( 2 * cos & x + h in dom sec & x + h + h in dom sec & x + h
assume that i in dom A and len A > 1 and len B > 1 and len B = width A and width B = width B and width B = width B and width B = width B and width B = width B and len A = width B and width A = width B and len A = width B and width A = width B and len B = width B ;
for i be non zero Element of NAT st i in Seg n holds ( i divides n or i = n or i = n & ( i divides n implies h . i = <* 1. F_Complex , 1_ F_Complex , 0. L *> ) & ( i divides n implies h . i = <* 1. L , 0. L *> ) & ( i divides n implies h . i = 1. L ) ) & ( i divides n implies h . i = 1. L )
( ( ( b1 => b2 ) '&' ( ( b1 'or' b2 ) '&' ( ( b1 'or' b2 ) '&' ( ( b1 'or' b2 ) '&' ( ( b1 'or' b2 ) '&' ( ( b1 'or' b2 ) '&' ( ( b1 'or' b2 ) '&' ( ( b1 'or' b2 ) '&' ( ( b1 'or' b2 ) '&' ( ( b1 'or' b2 ) '&' ( ( b1 'or' b2 ) '&' ( b1 'or' b2 ) '&' ( b1 'or' b2 ) ) ) ) ) ) ) ) '&' ( ( b1 'or' b2 ) '&' ( ( b1 'or' b2 ) '&' ( ( b1 'or' b2 ) '&' ( ( b1 'or' b2 ) '&' ( b2 'or' b2 ) '&' ( ( b1 'or' b2 ) '&' ( ( b1 'or' b2 ) '&' ( b1 'or' b2 ) '&' ( b1 'or' b2 ) '&' ( b1 'or' b2 ) ) ) ) ) / ( ( ( b2 ) '&' ( ( b1 'or' b2 ) ) ) / ( ( ( b1 'or' b2 ) '&' ( ( b1 'or'
assume that for x holds f . x = ( ( - 1 ) (#) ( ( - 1 ) (#) ( sin - cos ) ) `| Z ) and for x st x in Z holds ( ( - 1 ) (#) ( sin - cos ) ) `| Z ) . x = - cos . ( x- x ) and for x st x in Z holds ( ( - 1 ) (#) ( sin - cos ) ) `| Z ) . x = - cos . ( x - x0 ) ;
consider Rd , I-8 be Real such that Rd = Integral ( M , Re ( F . n ) ) and I-8 = Integral ( M , Im ( F . n ) ) and Integral ( M , Im ( F . n ) ) = Integral ( M , Im ( F . n ) ) ;
ex k being Element of NAT st ' = k & 0 < d & 0 < d & for q be Element of product G st q in X & ||. q - x .|| < r holds ||. partdiff ( f , q ) - partdiff ( f , x , k ) .|| < r ;
x in { x1 , x2 , x3 , x4 , x5 , x5 , x5 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 7 , 7 , 8 , 7 , 8 , 7 ,
( G * ( j , 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 , 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 ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 .= ( G * (
f1 * p = p .= ( the Arity of S1 ) . ( ( the Arity of S2 ) . o ) .= ( the Arity of S1 ) . ( ( the Arity of S1 ) . o ) .= ( the Arity of S1 ) . ( ( the Arity of S1 ) . o ) .= ( the Arity of S1 ) . ( ( the Arity of S1 ) . o ) .= ( the Arity of S1 ) . ( ( the Arity of S1 ) . o ) .= ( the Arity of S1 ) . ( ( the Arity of S1 ) . o ) ;
func tree ( T , P , T1 ) -> Tree means : Def3 : for q st q in P & q in T holds it . q = T . ( q , p ) or for p st p in P & p in T holds it . p = T . ( q , p ) ;
F /. ( k + 1 ) = F . ( k + 1 -' 1 ) .= Fx0 .= Fx0 .= Fx0 .= Fx0 .= Fx0 .= Fx0 .= Fx0 .= ( ( p . k ) -' 1 ) * ( p . k ) .= Fx0 ;
for A , B , C being Matrix of len C , K st len B = len C & len C = width C & len B = width C & len C > 0 & len A > 0 & len B > 0 & len C > 0 & len A > 0 & len C > 0 & len A > 0 & width A > 0 & width A = width B & width A = width B holds A * ( B + C , i ) = B * ( i , j )
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 ) .= ( Partial_Sums ( seq ) ) . ( k + 1 ) ;
assume that x in ( the carrier of CF ) and y in ( the carrier of CF ) and z in ( the carrier of CF ) and x = [ x , y ] ;
defpred P [ Element of NAT ] means for f st len f = $1 & ( for k st k in dom f holds f . k = ( the Element of D ) . ( k , f . k ) ) holds ( the Element of D ) . ( k , f . k ) = ( the Element of D ) . ( k , f . k ) ;
assume that 1 <= k and k + 1 <= len f and f is_sequence_on G and [ i , j ] in Indices G and [ i , j ] in Indices G and [ i , j ] in Indices G and [ i , j ] in Indices G and [ i , j ] in Indices G and [ i , j ] in Indices G and f /. k = G * ( i , j ) ;
assume that sn < 1 and ( q `1 / |. q .| - sn ) >= 0 and ( q `1 / |. q .| - sn ) >= 0 and ( q `1 / |. q .| - sn ) >= 0 and ( q `1 / |. q .| - sn ) >= 0 and ( q `2 / |. q .| - sn ) >= 0 ;
for M being non empty metric space , x being Point of M , f being Function of M , M st x = x & ex x being Point of M st f = x & for p being Point of M st p in B holds f . p = dist ( x , f . p ) holds x = f . p
defpred P [ Element of omega ] means ( for x st x in Z holds f1 . x > 0 ) & ( for x st x in Z holds f1 . x = 1 ) & ( for x st x in Z holds f1 . x = 0 ) implies ( f1 - f2 ) is_differentiable_on Z & for x st x in Z holds f1 . x = 1 ) & f2 is_differentiable_on Z & for x st x in Z holds f1 . x > 0 ) ;
defpred P1 [ Nat , Point of Cj ] means ( for r be Real st r in Y & $1 < $2 holds ||. ( f /. $1 ) - ( f /. $2 ) .|| < r ) & ||. ( f /. $1 ) - ( f /. $2 ) .|| < r ;
( f ^ mid ( g , 2 , len g ) ) . i = ( mid ( g , 2 , len g ) ) . ( i + 1 ) .= ( g . i ) . ( i + 1 ) .= g . ( i + 1 ) .= g . ( i + 1 ) .= g . ( i + 1 ) .= g . ( i + 1 ) .= g . ( i + 1 ) .= g . ( i + 1 ) ;
( 1 - 2 * ( n + 2 ) ) * ( 2 * ( n + 1 ) ) = ( 1 - 2 * ( n + 1 ) ) * ( 2 * ( n + 1 ) ) .= 1 * ( 2 * ( n + 1 ) ) .= 1 * ( 2 * ( n + 1 ) ) .= 1 * ( 2 * ( n + 1 ) ) ;
defpred P [ Nat ] means for G being non empty RelStr st G is of \times the carrier of G , the carrier of G st G is the such that ( for x being Element of G holds x in the carrier of G ) & ( for x being Element of G holds x in the carrier of G ) & ( for x being Element of G st x in the carrier of G holds x = the InternalRel of G ) holds x = the RelStr of G ) ;
( not f /. 1 in Ball ( u , r ) & not 1 <= m & m <= len f & for i st 1 <= i & i <= len f holds LSeg ( f , i ) /\ LSeg ( f , m ) <> {} ) implies not LSeg ( f , i ) /\ LSeg ( f , m ) = { f /. i }
defpred P [ Element of NAT ] means ( Partial_Sums ( cos ) . $1 = ( Partial_Sums ( cos ) . $1 ) * ( ( cos . $1 ) |^ ( $1 + 1 ) ) & ( Partial_Sums ( cos ) . $1 = ( Partial_Sums ( cos ) . $1 ) * ( cos . $1 ) |^ ( $1 + 1 ) ) ;
for x being Element of product F holds x is FinSequence of ( the carrier of G ) & ( for i being set st i in dom x holds x . i = ( the |^ i ) . ( x . i ) ) & ( for i being set st i in dom x holds x . i = ( the |^ i ) . ( x . i ) ) & ( for i be set st i in dom x holds x . i = ( the |^ i ) . i ) & ( for i be set st i in dom x ) holds x . i in ( the carrier of G ) . i = ( the carrier of G ) . i ) & ( for i be set st i in dom x ) & ( for i be set st i in dom x ) & ( x . i in ( the carrier of G ) . i in ( the carrier of G ) & ( for i be set st i in dom x holds x . i in ( the carrier of G . i ) & ( for i be set st i in dom
( x " ) |^ ( n + 1 ) = ( x " ) |^ ( n + 1 ) .= ( x |^ n ) |^ ( n + 1 ) .= ( x |^ n ) |^ ( n + 1 ) .= ( x |^ n ) |^ ( n + 1 ) .= ( x |^ n ) |^ ( n + 1 ) .= ( x |^ n ) |^ ( n + 1 ) .= ( x |^ n ) |^ ( n + 1 ) ;
DataPart ( ( P +* I ) +* ( 1 , ( ( card I + 1 ) + 1 ) ) + 1 ) = DataPart Comput ( P +* I , s , k ) .= DataPart Comput ( P +* I , s , k ) .= DataPart Comput ( P +* I , s , k ) ;
given r such that 0 < r and ]. x0 - r , x0 + r .[ c= dom ( f1 + f2 ) and for g st g in ]. x0 - r , x0 + r .[ holds f1 . g <= ( f1 + f2 ) . g and for g st g in ]. x0 - r , x0 + r .[ & g in ]. x0 - r , x0 + r .[ holds f1 . g <= 0 ;
assume that X c= dom f1 /\ dom f2 and f1 | X is continuous and f2 | X is continuous and ( for x st x in X /\ X holds f1 . x = 1 ) and ( for x st x in X /\ X holds f1 . x = 1 ) and ( for x st x in X holds f1 . x = 1 ) implies f2 | X is continuous & f2 | X is continuous & f2 | X is continuous ;
for L being continuous complete LATTICE for l being Element of L ex X being Subset of L st l = sup X & for x being Element of L st x in X holds x is Element of L & for x being Element of L st x in X holds x is Element of L & x is ` } is `
Support ( e *' p ) in { m *' ( p *' q ) where m is Nat : m in dom ( p *' q ) & p = ( p *' q ) *' q } ;
( f1 - f2 ) /. ( lim s1 ) = lim ( f1 /* s1 ) - ( f2 /* s1 ) .= lim ( f1 /* s1 ) - ( f2 /* s1 ) .= lim ( f1 /* s1 ) - ( f2 /* s1 ) .= lim ( f1 /* s1 ) - ( f2 /* s1 ) .= lim ( f1 /* s1 ) - ( f2 /* s1 ) .= lim ( f2 /* s1 ) - ( f2 /* s1 ) .= lim ( f2 /* s1 ) ;
ex p1 being Element of QC-WFF ( Al ( ) ) st F . p1 = g . ( p1 ) & for g being Function of [: Al ( ) , D ( ) :] , D ( ) st ( for g being Function of D ( ) , D ( ) holds P [ g , f . g ] ) & for g being Function of D ( ) , D ( ) st P [ g , f . g ] holds F [ g , f . g ] ) ;
( mid ( f , i , len f -' 1 ) ) /. j = ( mid ( f , i , len f -' 1 ) ) /. ( j + 1 ) .= ( mid ( f , i , len f -' 1 ) ) /. j .= ( mid ( f , i , len f -' 1 ) ) /. j .= f /. ( j + 1 ) .= f /. ( j + 1 ) ;
( ( p ^ q ) . ( len p + k ) ) = ( ( p ^ q ) . ( len p + k ) ) .= ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( ( ( p ^ q ) + ( k + k ) ) . ( len p + k ) . ( len p + k ) .= ( ( p ^ q ) . ( k + k ) . ( k + k ) .= ( ( p ^ q ) + ( k + k ) . ( k + k ) .= ( ( p ^ q ) . ( len p + k ) . ( len p + ( k + 1 ) .= ( ( k + 1 ) . ( len p + k ) . ( len p + k ) .= ( ( p ^ q ) . ( k + 1 ) .= ( ( p ^ q ) . ( k + 1 )
len mid ( D2 , D1 , indx ( D2 , D1 , j1 ) + 1 ) = len ( D2 , indx ( D2 , D1 , j1 ) + 1 ) .= len ( D2 , indx ( D2 , D1 , j1 ) + 1 ) .= len ( D2 , indx ( D2 , D1 , j1 ) + 1 ) .= len D2 + 1 + 1 .= len D2 + 1 + 1 .= len D2 + 1 + 1 .= len D2 + 1 + 1 .= len D2 + 1 + 1 .= len D2 + 1 + 1 + 1 .= len D2 + 1 + 1 + 1 + 1 + 1 .= len D2 + 1 + 1 + 1 + 1 + 1 .= len D2 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 .= len D2 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 .= len D2 + 1 + 1 + 1 + 1 + 1 + 1
x * y * z = Mj . ( ( x * ( y * z ) ) * ( y * z ) ) .= ( x * ( y * z ) ) * ( x * z ) .= ( x * ( y * z ) ) * ( x * z ) .= ( x * ( y * z ) ) * ( x * z ) .= ( x * ( y * z ) ) * ( x * z ) .= ( x * ( y * z ) * ( x * z ) .= ( x * z ) * ( x * z ) * ( x * z ) .= ( x * ( y * z ) * ( x * z ) .= ( x * z ) * ( x * z ) * ( x * z ) * ( x * z ) * ( x * z ) .= ( x * z ) * ( x * z ) * ( x * z ) * ( x * z ) * ( x * z ) * ( x * z ) .= ( x * ( x * z ) * ( x * z ) * ( x * z ) .= ( x * ( x * z
v . <* x , y *> + ( <* x0 , y0 *> ) . i * ( <* x0 , y0 *> . i + ( <* x0 , y0 *> . ( y - x0 ) ) + ( <* x0 , y0 *> . ( y - x0 ) ) + ( ( <* x0 , y0 *> . ( y - x0 ) ) * ( y - x0 ) ) + ( ( <* x0 , y0 *> . ( y - x0 ) ) *> ) ;
i * i = <* 0 * ( 1 - i ) - ( 1 - i ) * ( 1 - i ) + ( 1 - i ) * i .= <* - 1 * ( 1 - i ) + ( 1 - i ) * i + ( 1 - i ) * i .= <* - 1 * i + 1 * i + 1 * i - i * i + 1 * i + 1 * i - 1 * i .= 1 * i + 1 * i ;
Sum ( L (#) F ) = Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( ( L (#) F2 ) ^ ( F2 ^ F2 ) ) .= Sum ( ( L (#) F2 ) ^ ( F2 ^ F2 ) ) .= Sum ( ( L (#) F2 ) + Sum ( F2 ^ F2 ) ) .= Sum ( ( L (#) ( F2 ^ F2 ^ F2 ) + Sum ( F2 ^ F2 ) + Sum ( F2 ^ F2 ) + Sum ( F2 ^ F2 ) + Sum ( F2 ^ F2 ) + Sum ( F2 ^ F2 ) + Sum ( F2 ^ F2 ) .= Sum ( F2 ^ F2 ) + Sum ( F2 ^ ( F2 ^ F2 ^ ( F2 ^ F2 ) .= Sum ( F2 ^ F2 ) + Sum ( F2 ^ F2 ) + Sum ( F2 ^ F2 ) + Sum ( F2 ^ F2 ) + Sum ( F2 ^ F2 ) + Sum ( F2 ^ F2 ) + Sum ( F2 ^ F2 )
ex r be Real st for e be Real st 0 < e ex Y1 be finite Subset of REAL st Y1 in Y & for Y1 be finite Subset of REAL st Y1 c= Y & Y1 c= Y & Y1 c= Y & Y1 c= Y & Y is finite & Y is finite & Y is finite & Y is finite ;
( GoB f ) * ( i , j + 1 ) = f /. ( k + 2 ) & ( GoB f ) * ( i + 1 , j + 1 ) = f /. ( k + 2 ) or ( GoB f ) * ( i + 1 , j + 1 ) = f /. ( k + 1 ) & ( GoB f ) * ( i + 2 , j + 1 ) = f /. ( k + 2 ) ;
( ( - 1 ) (#) cos ) . x = - ( ( - 1 ) * cos . x ) / ( sin . x ) .= - ( ( - 1 ) * cos . x ) / ( sin . x ) .= - ( ( - 1 ) * cos . x ) / ( sin . x ) .= - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - x ) ) * cos . x ) ) ) ^2 ) .= - ( 1 - ( 1 - ( 1 - x ) * cos . x ) ) / ( cos . x ) ) .= - ( 1 - x ) * cos . x ) / ( 1 - ( 1 - x ) * cos . x ) .= - ( 1 - x ) ^2 .= - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - x ) / ( 1 - ( 1 - x ) * cos . x ) / ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - x ) * cos . x )
- ( - b + sqrt ( a , b ) ) / sqrt ( a , b ) > 0 & - ( - b + sqrt ( a , b ) ) / sqrt ( a , b ) > 0 implies - ( - b + sqrt ( a , b ) ) / sqrt ( a , b ) + sqrt ( a , b ) / sqrt ( a , b ) ) > 0
assume that inf ( \mathopen { \uparrow } X /\ C ) in L and sup ( \mathopen { \uparrow } X ) = L and for X st X in { x } holds not X in ( the carrier of subrelstr X ) /\ ( the carrier of L ) and not X in ( the carrier of subrelstr X ) /\ ( the carrier of L ) ;
( for j being Element of I holds ( j = i implies j = i ) implies ( i = j = j implies j = i ) & ( j = i implies j = i ) & ( j = i implies j = i implies j = i ) ) & ( j = i implies j = i implies j = i ) implies j = i ) & j = i implies j = i implies j = i )