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thesis ;
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thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
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thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
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thesis ;
contradiction ;
contradiction ;
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contradiction ;
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contradiction ;
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thesis ;
thesis ;
contradiction ;
thesis ;
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contradiction ;
thesis ;
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thesis ;
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 ;
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 finite ;
b in A ;
d divides a ;
i < n ;
s <= b ;
b in B ;
let r ;
B is one-to-one ;
R is total ;
x = 2 ;
d in D ;
let c ;
let c ;
b = Y ;
0 < k ;
let b ;
let n ;
r <= b ;
x in X ;
i >= 8 ;
let n ;
let n ;
y in f ;
let n ;
1 < j ;
a in L ;
C is boundary ;
a in A ;
1 < x ;
S is finite ;
u in I ;
z << z ;
x in V ;
r < t ;
let t ;
x c= y ;
a <= b ;
m in NAT ;
assume f is onto ;
not x in Y ;
z = +infty ;
k be Nat ;
K ` is being_line ;
assume n >= N ;
assume n >= N ;
assume X is .= Y ;
assume x in I ;
q is as of 0 ;
assume c in x ;
as Real ;
assume x in Z ;
assume x in Z ;
1 <= kr2 ;
assume m <= i ;
assume G is finite ;
assume a divides b ;
assume P is closed ;
b-a > 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 `1 <= c `1 ;
A meets W ;
i `2 <= j `2 ;
assume H is universal ;
assume x in X ;
let X be set ;
let T be Tree ;
let d be element ;
let t be element ;
let x be element ;
let x be element ;
let s be element ;
k <= 5 ;
let X be set ;
let X be set ;
let y be element ;
let x be element ;
P [ 0 ]
let E be set , A be Subset of E ;
let C be category ;
let x be element ;
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 \setminus ;
Q halts_on s ;
x in \in that x in \in that x ;
M < m + 1 ;
T2 is open ;
z in b seq seq ;
R2 is well-ordering ;
1 <= k + 1 ;
i > n + 1 ;
q1 is one-to-one ;
let x be trivial set ;
PM is one-to-one ;
n <= n + 2 ;
1 <= k + 1 ;
1 <= k + 1 ;
let e be Real ;
i < i + 1 ;
p3 in P ;
p1 in K ;
y in C1 ;
k + 1 <= n ;
let a be Real , b 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 , r , s ;
let E be Ordinal ;
o : o : o1 , o2 are_collinear ;
O <> O2 ;
let r be Real ;
let f be FinSeq-Location ;
let i be Nat ;
let n be Nat ;
Cl A = A ;
L c= Cl L ;
A /\ M = B ;
let V be RealUnitarySpace , M be Subset of V ;
not s in Y |^ 0 ;
rng f <= w
b "/\" e = b ;
m = m3 ;
t in h . D ;
P [ 0 ] ;
assume z = x * y ;
S . n is bounded ;
let V be RealUnitarySpace , M be Matrix of REAL ;
P [ 1 ] ;
P [ {} ] ;
C1 is component ;
H = G . i ;
1 <= i `1 + 1 ;
F . m in A ;
f . o = o ;
P [ 0 ] ;
a\vert <= being Real ;
R [ 0 ] ;
b in f .: X ;
assume q = q2 ;
x in [#] V ;
f . u = 0 ;
assume e1 > 0 ;
let V be RealUnitarySpace , M be Matrix of REAL ;
s is trivial non empty ;
dom c = Q ;
P [ 0 ] ;
f . n in T ;
N . j in S ;
let T be complete LATTICE , a be Element of 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 ;
S\HM is bounded ;
assume p = p2 ;
len f = n ;
assume x in P1 ;
i in dom q ;
let U2 , U1 , U2 , U2 , U2 , U2 ;
pp `1 = c ;
j in dom h ;
let k ;
f | Z is continuous ;
k in dom G ;
UBD C = B ;
1 <= len M ;
p in Ball ( x , r ) ;
1 <= jj & jj <= width G ;
set A = \mathclose { \rm c } ;
card a [= c ;
e in rng f ;
cluster B \oplus A -> empty ;
H is non empty set ;
assume n0 <= m ;
T is increasing ;
e2 <> e2 & e2 <> e1 ;
Z c= dom g ;
dom p = X ;
H is proper ;
i + 1 <= n ;
v <> 0. V ;
A c= Affin A ;
S c= dom F ;
m in dom f ;
X0 be set ;
c = sup N ;
R is connected ;
assume not x in REAL ;
Im f is complete ;
x in Int y ;
dom F = M ;
a in On W ;
assume e in A ( ) ;
C c= C-26 ;
mm <> {} ;
Element of Y ;
let f be ) ;
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 v + dom ex X st X in G ;
- y in I ;
let A be non empty set , B be set ;
P0 = 1 ;
assume r in F . k ;
assume f is simple ;
let A be l countable set ;
rng f c= NAT * ;
assume P [ k ] ;
f6 <> {} ;
let o be Ordinal ;
assume x is sum of squares ;
assume not v in { 1 } ;
let Is , I , J ;
assume that 1 <= j and j < l ;
v = - u ;
assume s . b > 0 ;
d1 in dom f ;
assume t . 1 in A ;
let Y be non empty TopSpace , X be Subset of Y ;
assume a in uparrow s ;
let S be non empty Poset ;
a , b // b , a ;
a * b = p * q ;
assume x , y are_the space ;
assume x in Omega ( f ) ;
[ a , c ] in X ;
mm <> {} ;
M + N c= M + M ;
assume M is connected hhfor ;
assume f is \llangle bbrrr-r) ;
let x , y be element ;
let T be non empty TopSpace ;
b , a // b , c ;
k in dom Sum p ;
let v be Element of V ;
[ x , y ] in T ;
assume len p = 0 ;
assume C in rng f ;
k1 = k2 or k1 = 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 compact ;
f . x - a <= b ;
assume y in dom h ;
x * y in B1 ;
set X = Seg n ;
1 <= i2 + 1 ;
k + 0 <= k + 1 ;
p ^ q = p ;
j |^ y divides m ;
set m = max A ;
[ x , x ] in R ;
assume x in succ 0 ;
a in sup phi ;
Cs in X ;
q2 c= C1 & q2 in C2 ;
a2 < c2 & c2 < c1 ;
s2 is 0 -started ;
IC s = 0 ;
s4 = s4 , s4 = s4 , P4 = s4 ;
let V ;
let x , y be element ;
Element of T ;
assume a in rng F ;
x in dom T ` ;
let S be as as as of L ;
y " <> 0 ;
y " <> 0 ;
0. V = u-w ;
y2 , y are_: zero ;
R8 in X ;
let a , b be Real , x be Element of REAL ;
let a be Object of C ;
let x be Vertex of G ;
let o be object of C , a be Object of C ;
r '&' q = P \lbrack l , r .] ;
let i , j be Nat ;
let s be State of A , a be Element of A ;
s4 . n = N ;
set y = ( x `1 ) ^2 ;
NAT in dom g & NAT in dom g ;
l . 2 = y1 ;
|. g . y .| <= r ;
f . x in Cx0 ;
not 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 that A is dense and A is dense ;
|. f . x .| <= r ;
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 ;
xx c= Z1 & xx c= Z1 ;
dom f = C1 & dom g = C2 ;
assume [ a , y ] in X ;
Re seq is convergent & lim ( seq ) = 0 ;
assume a1 = b1 & a2 = b2 ;
A = sInt A ;
a <= b or b <= a ;
n + 1 in dom f ;
let F be Instruction of S , a be Int-Location ;
assume r2 > x0 & x0 < r2 ;
let Y be non empty set , f be Function of Y , BOOLEAN ;
2 * x in dom W ;
m in dom g2 & n in dom g1 ;
n in dom g1 & n in dom g2 ;
k + 1 in dom f ;
the still of S is finite ;
assume that x1 <> x2 and x2 <> x3 and x3 <> x4 ;
v3 in V1 & v2 in V1 ;
not [ b `1 , b `2 ] in T ;
i-35 + 1 = i ;
T c= and T c= and T c= T ;
( l `1 ) ^2 = 0 ;
let n be Nat ;
( t `2 ) ^2 = r ^2 ;
AA is_integrable_on M & AA is integrable ;
set t = Top t ;
let A , B be real-membered set ;
k <= len G + 1 ;
cC misses [: V , V :] ;
product ( s ) is non empty ;
e <= f or f <= e ;
cluster non empty normal for set ;
assume c2 = b2 or c2 = 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 vseq is Cauchy and vseq is Cauchy ;
IC s3 = 0 & IC s3 = 0 ;
k in N or k in K ;
F1 \/ F2 c= F \/ G ;
Int G1 <> {} & Int G2 <> {} ;
( z `2 ) ^2 = 0 ;
p11 <> p1 or p11 <> p2 ;
assume z in { y , w } ;
MaxADSet ( a ) c= F ;
ex_sup_of downarrow s , S ;
f . x <= f . y ;
let T be up-complete reflexive non empty antisymmetric RelStr ;
q |^ m >= 1 ;
a >= X & b >= Y ;
assume <* a , c *> <> {} ;
F . c = g . c ;
G is one-to-one , full , full ;
A \/ { a } \not c= B ;
0. V = 0. Y .= 0. V ;
let I be non empty finite Instruction of S , S ;
f-24 . x = 1 ;
assume z \ x = 0. X ;
C4 = 2 to_power n ;
let B be SetSequence of Sigma ;
assume X1 = p .: D ;
n + l2 in NAT & n + l2 in NAT ;
f " P is compact ;
assume x1 in REAL & x2 in REAL & x1 in REAL ;
p1 = K1 & p2 = K1 & p3 = 0. TOP-REAL 2 ;
M . k = <*> REAL ;
phi . 0 in rng phi ;
thesis MMInt A is closed ;
assume z0 <> 0. L & z0 <> 0. L ;
n < N7 . k ;
0 <= seq . 0 & seq . 0 <= seq . 0 ;
- q + p = v ;
{ v } is Subset of B ;
set g = f /. 1 ;
[: R , R :] is stable ;
set cR = Vertices R , cR = Vertices R ;
p0 c= P4 & p1 c= P4 ;
x in [. 0 , 1 .[ ;
f . y in dom F ;
let T be Scott Scott Scott Scott of S ;
ex_inf_of the carrier of S , S ;
downarrow a = downarrow b ;
P , C , K is_collinear ;
assume x in F ( s , r , t ) ;
2 to_power i < 2 to_power m ;
x + z = x + z + q ;
x \ ( a \ x ) = x ;
||. x - y .|| <= r ;
assume that Y c= field Q and Y <> {} ;
a ~ , b ~ are_isomorphic ;
assume a in A ( ) ;
k in dom ( q | Seg n ) ;
p is holds p is FinSequence of S ;
i -' 1 = i-1 - 1 ;
f | A is one-to-one ;
assume x in f .: [: X , Y :] ;
i2 - i1 = 0 & i2 = 0 ;
j2 + 1 <= i2 & j2 + 1 <= width G ;
g " * a in N ;
K <> { [ {} , {} ] } ;
cluster strict for for for for } ;
|. q .| ^2 > 0 ;
|. p4 .| = |. p .| ;
s2 - s1 > 0 & s2 - s1 > 0 ;
assume x in { Gij } ;
W-min C in C & W-min C in C ;
assume x in { Gij } ;
assume i + 1 = len G ;
assume i + 1 = len G ;
dom I = Seg n & dom I = Seg n ;
assume that k in dom C and k <> i ;
1 + 1-1 <= i + j ;
dom S = dom F & 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 void non empty non void holds S is { 0 } -\rm \subseteq the carrier of S ;
let f be ManySortedSet of I ;
let z be Element of COMPLEX , x be Element of COMPLEX ;
u in { ag } ;
2 * n < 2 * n ;
let x , y be set ;
B-11 c= V1 & B-15 c= V1 ;
assume I is_halting_on s , P ;
U2 = U2 & U2 = U2 implies U2 is open
M /. 1 = z /. 1 ;
x9 = x9 & y9 = y9 ;
i + 1 < n + 1 + 1 ;
x in { {} , <* 0 *> } ;
( f | X ) . x <= ( f | X ) . x ;
let l be Element of L ;
x in dom ( F | D ) ;
let i be Element of NAT ;
r8 is COMPLEX -valued & r8 is COMPLEX -valued ;
assume <* o2 , o *> <> {} ;
s . x |^ 0 = 1 ;
card K1 in M & card K1 in card M ;
assume that X in U and Y in U ;
let D be empty Subset-Family of Omega ;
set r = Seg ( k + 1 ) ;
y = W . ( 2 * x ) ;
assume dom g = cod f & cod g = cod f ;
let X , Y be non empty TopSpace , f be Function of X , Y ;
x \oplus A is interval ;
|. <*> A .| . a = 0 ;
cluster strict for SubLattice of L ;
a1 in B . s1 & a2 in B . s2 ;
let V be finite < len V , A be Subset of V ;
A * B on B , A ;
f-3 = NAT --> 0 .= fg ;
let A , B be Subset of V ;
z1 = P1 . j & z2 = P2 . j ;
assume f " P is closed & f " P is closed ;
reconsider j = i as Element of M ;
let a , b be Element of L ;
assume q in A \/ ( B "\/" C ) ;
dom ( F * C ) = o ;
set S = INT * ( X \ Y ) ;
z in dom ( A --> y ) ;
P [ y , h . y ] ;
{ x0 } c= dom f & { x0 } c= dom f ;
let B be non-empty ManySortedSet of I , A be ManySortedSet of I ;
( PI / 2 ) * PI < Arg z ;
reconsider z9 = 0 , z9 = 1 as Nat ;
LIN a , d , c ;
[ y , x ] in IF ;
( Q ) * ( 1 , 3 ) = 0 ;
set j = x0 div m , m = x0 mod m ;
assume a in { x , y , c } ;
j2 - ( j - 1 ) > 0 ;
I \! \mathop { phi } = 1 ;
[ y , d ] in [: F-8 , F-8 :] ;
let f be Function of X , Y ;
set A2 = ( B \/ C ) ` ;
s1 , s2 are_/ 2 implies s1 , s2 are_/ 2
j1 -' 1 = 0 & j2 -' 1 = j2 ;
set m2 = 2 * n + j ;
reconsider t = t as bag of n ;
I2 . j = m . j ;
i |^ s , n are_relative_prime & i divides n ;
set g = f | D-21 ;
assume that X is lower and 0 <= r ;
( p1 `1 ) ^2 = 1 ^2 & ( p1 `2 ) ^2 = 1 ;
a < ( p3 `1 ) ^2 + ( p3 `2 ) ^2 ;
L \ { m } c= UBD C ;
x in Ball ( x , 10 ) ;
not a in LSeg ( c , m ) ;
1 <= i1 -' 1 & i1 + 1 <= len G ;
1 <= i1 -' 1 & i1 + 1 <= len G ;
i + i2 <= len h ;
x = W-min ( P ) & x in P ;
[ x , z ] in [: X , Z :] ;
assume y in [. x0 , x .] ;
assume p = <* 1 , 2 , 3 *> ;
len <* A1 *> = 1 & len <* A2 *> = 2 ;
set H = h . g , g = h . x , h = h . y , h = h . z , f = h . x , g = h . y , h
card b * a = |. a .| ;
Shift ( w , 0 ) |= v ;
set h = h2 (*) h1 , h1 = h2 (*) h2 ;
assume x in ( X0 /\ X2 ) ;
||. h .|| < d1 & ||. h .|| < d ;
not x in the carrier of f & x in the carrier of g ;
f . y = F ( y ) ;
for n holds X [ n ] ;
k -' l = k\leq ;
<* p , q *> /. 2 = q ;
let S be Subset of the carrier of Y ;
let P , Q be \langle s *> ;
Q /\ M c= union ( F | M )
f = b * canFS ( S ) ;
let a , b be Element of G ;
f .: X is_<=_than f . sup X
let L be non empty transitive RelStr , a be Element of L ;
S-20 is x -f1 -basis i ;
let r be non positive Real ;
M , v |= x \hbox { y } ;
v + w = 0. ( Z ) ;
P [ len ( F | n ) ] ;
assume InsCode ( i ) = 8 or InsCode ( i ) = 7 ;
the zero of M = 0 & the carrier of M = 0 ;
cluster z * seq -> summable for Real_Sequence ;
let O be Subset of the carrier of C ;
||. f .|| | X is continuous ;
x2 = g . ( j + 1 ) ;
cluster -> [#] for Element of S ;
reconsider l1 = l-1 as Nat ;
v4 is Vertex of r2 & v4 is Vertex of G ;
T2 is SubSpace of T2 & T2 is SubSpace of T2 ;
Q1 /\ Q19 <> {} & Q19 /\ Q29 = {} ;
k be Nat ;
q " is Element of X & q " is Element of Y ;
F . t is set of empty set ;
assume that n <> 0 and n <> 1 ;
set en = EmptyBag n , en = EmptyBag n ;
let b be Element of Bags n ;
assume for i holds b . i is commutative ;
x is root & x `2 = ( p `2 ) ;
not r in ]. p , q .[ ;
let R be FinSequence of REAL , x be Element of REAL ;
S7 does not destroy b1 or S7 does not destroy b1 ;
IC SCM R <> a & IC SCM R <> a ;
|. - |[ x , y ]| .| >= r ;
1 * ( s * a ) = ss * a ;
let x be FinSequence of NAT , n be Element of 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 .= IC s ;
H + G = F- ( GG ) ;
Cx1 . x = x2 & Cx1 . x = y2 ;
f1 = f .= f2 .= f2 * f1 .= f2 * f1 ;
Sum <* p . 0 *> = p . 0 ;
assume v + W = v + u + W ;
{ a1 } = { a2 } & { a2 } = { a2 } ;
a1 , b1 _|_ b , a ;
d1 , o _|_ o , a3 ;
IF is reflexive & IF is reflexive implies IF is reflexive
Iy is antisymmetric implies [: Iy , Iy :] is antisymmetric
upper_bound rng H1 = e & upper_bound rng H1 = e ;
x = ( a * ( b * a ) ) * ( b * a ) ;
|. p1 .| ^2 >= 1 ^2 ;
assume j2 -' 1 < 1 & j2 -' 1 < width G ;
rng s c= dom f1 & rng s c= dom f2 ;
assume support a misses support b & support b misses support a ;
let L be associative commutative non empty doubleLoopStr , a be Element of L ;
s " + 0 < n + 1 ;
p . c = ( f " ) . 1 ;
R . n <= R . ( n + 1 ) ;
Directed IV = IV .= IV ;
set f = + ( x , y , r ) ;
cluster Ball ( x , r ) -> bounded ;
consider r being Real such that r in A ;
cluster non empty NAT -defined NAT -defined NAT -defined NAT -defined Function ;
let X be non empty directed Subset of S ;
let S be non empty full SubRelStr of L ;
cluster <* [ ] , ] *> -> complete for non trivial TopSpace ;
( 1 - a ) " = a ;
( q . {} ) `1 = o ;
( n - 1 ) - 1 > 0 ;
assume ( 1 / 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 X ;
f . 1 = L . F . 1 ;
the vertices of G = { v } & the vertices of G = { v } ;
let G be } -wgraph ;
e , v6 be set , v be set ;
c . ( i - 1 ) in rng c ;
f2 /* q is divergent_to-infty & f2 /* q is divergent_to-infty ;
set z1 = - z2 , z2 = - z1 , z2 = - z2 , z1 = - z2 , z2 = - z2 , z2 = - z1 , z2 = - z2 , z1 = - z2 , z2
assume w is Element of las ( S , G ) ;
set f = p |-count ( t ) , g = p |-count ( t ) , h = p |-count ( t ) , t = p |-count ( t ) , f = p |-count ( t ) , g
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 Is be Subset-Family of X , P be Subset of X ;
reconsider p = p as Element of NAT ;
v , w be Point of X ;
let s be State of SCM+FSA , a be Int-Location ;
p is FinSequence of ( the carrier of SCM ) * ;
stop I ( ) c= P-12 & I ( ) c= P-12 ;
set ci = f^ /. i , fi = fSet /. i , cj = fn ;
w ^ t ^ t ^ <* s *> ^ w ^ t ^ w ^ t ^ w ^ t ^ w ^ t ^ w ^ t ^ t ^ w ^ t ^ w ^ t ^ w ^
W1 /\ W = W1 /\ W ` .= W1 /\ W2 ;
f . j is Element of J . j ;
let x , y be \rm \rm \rm \rm \hbox { - } of T ;
ex d st a , b // b , d ;
a <> 0 & b <> 0 ;
ord x = 1 & x is positive implies x is where x is Element of NAT : x is not of a & x is not L
set g2 = lim ( seq ) , g1 = lim ( seq ) ;
2 * x >= 2 * ( 1 / 2 ) ;
assume ( a 'or' c ) . z <> TRUE ;
f (*) g in Hom ( c , c ) ;
Hom ( c , c + d ) <> {} ;
assume 2 * Sum ( q | m ) > m ;
L1 . F-21 = 0 & L1 . Fk1 = 0 ;
( / X ) \/ R1 = ( / X ) \/ ( id X ) ;
( sin . x ) ^2 <> 0 & ( sin . x ) ^2 <> 0 ;
( ( exp_R * exp_R ) `| Z ) . x > 0 ;
o1 in [: X , Y :] /\ [: Y , X :] ;
e , v6 be set , v be set ;
r3 > ( 1 - 2 ) * 0 ;
x in P .: ( F -ideal ( F ) ) ;
let J be closed Subset of R , I be non empty Subset of R ;
h . p1 = f2 . O & h . O = g2 ;
Index ( p , f ) + 1 <= j ;
len ( q | i ) = width M .= width ( q | i ) ;
the carrier of CK c= A ;
dom f c= union rng ( F | n ) ;
k + 1 in support ( support ( n ) ) ;
let X be ManySortedSet of the carrier of S ;
[ x `1 , y `2 ] in ( an \/ R ) ;
i = D1 or i = D2 or i = D1 ;
assume a mod n = b mod n ;
h . x2 = g . x1 & h . x2 = g . x2 ;
F c= 2 -tuples_on the carrier of X
reconsider w = |. s1 .| as Real_Sequence ;
( 1 / m * m + r ) < p ;
dom f = dom I-4 & dom IK = dom IK ;
[#] P-17 = [#] ( ( TOP-REAL 2 ) | K1 ) .= K1 ;
cluster - x -> ExtReal for ExtReal ;
then { d1 } c= A & A is closed ;
cluster ( TOP-REAL n ) | ( [#] TOP-REAL n ) -> finite-ind ;
let w1 be Element of M , a be Element of M ;
let x be Element of dyadic ( n ) ;
u in W1 & v in W3 implies u in W2
reconsider y = y , z = z as Element of L2 ;
N is full SubRelStr of ( T |^ the carrier of S ) ;
sup { x , y } = c "\/" c ;
g . n = n to_power 1 .= n ;
h . J = EqClass ( u , J ) ;
let seq be summable sequence of X , x be Element of X ;
dist ( x `1 , y ) < ( r / 2 ) ;
reconsider mm = m , mn = n as Element of NAT ;
x- x0 < r1 - x0 & r1 < x0 + r2 ;
reconsider P ` = P ` as strict Subgroup of N ;
set g1 = p * idseq ( q `1 ) , g2 = q `2 ;
let n , m , k be non zero Nat ;
assume that 0 < e and f | A is lower ;
D2 . I8 in { x } & D2 . I8 in { x } ;
cluster subcondensed -> subcondensed for Subset of T ;
let P be compact non empty Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
Gik in LSeg ( cos , 1 ) /\ LSeg ( cos , 1 ) ;
let n be Element of NAT , x be Element of NAT ;
reconsider S8 = S , S8 = T as Subset of T ;
dom ( i .--> X ` ) = { i } ;
let X be non-empty ManySortedSet of S ;
let X be non-empty ManySortedSet of S ;
op ( 1 ) c= { [ {} , {} ] } ;
reconsider m = mm - 1 as Element of NAT ;
reconsider d = x as Element of C ( ) ;
let s be 0 -started State of SCMPDS , a be Int-Location ;
let t be 0 -started State of SCMPDS , Q ;
b , b , x , y is_collinear ;
assume that i = n \/ { n } and j = k \/ { k } ;
let f be PartFunc of X , Y ;
Nx0 >= ( sqrt c ) ^2 - ( sqrt c ) ^2 ;
reconsider t7 = T7 as TopSpace , T7 = T6 as TopSpace ;
set q = h * p ^ <* d *> ;
z2 in U . ( y2 ) /\ Q2 & z2 in Q ;
A |^ 0 = { <%> E } .= { <%> E } ;
len W2 = len W + 2 .= len W + 1 ;
len h2 in dom h2 & len h2 in dom h2 ;
i + 1 in Seg len s2 & i + 1 in Seg len s2 ;
z in dom g1 /\ dom f & z in dom f1 /\ dom f2 ;
assume p2 = E-max ( K ) & p3 = E-max ( K ) ;
len G + 1 <= i1 + 1 & i1 + 1 <= len G ;
f1 (#) f2 is convergent & lim ( f1 (#) f2 ) = x0 ;
cluster s-10 + sRRRRRRRRRRRp -> summable ;
assume j in dom M1 & i in dom M2 ;
let A , B , C be Subset of X ;
x , y , z be Point of X , p be Point of X ;
b ^2 - ( 4 * a * c ) >= 0 ;
<* xy *> ^ <* y *> ^ <* y *> \subseteq x ;
a , b in { a , b } ;
len p2 is Element of ( len p2 ) -tuples_on NAT ;
ex x being element st x in dom R & R . x = y ;
len q = len ( K (#) G ) .= len G ;
s1 = Initialize Initialized s , P1 = P +* I , P2 = P +* I ;
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 is closed Subset of X ;
for x st x in L holds x is FinSequence ;
||. f /. c .|| <= r1 & ||. f /. c .|| <= r1 ;
c in uparrow p & not c in { p } ;
reconsider V = V as Subset of the TopStruct of TOP-REAL n ;
N , M be being being being being being \hbox of L ;
then z is_>=_than waybelow x & z is_>=_than compactbelow y ;
M [. f , g .] = f & M | [. g , f .] = g ;
( ( L /. 1 ) `1 ) ^2 = TRUE ;
dom g = dom f -tuples_on X & dom g = dom f ;
mode : of G is : Let is .| ;
[ i , j ] in Indices M & [ i , j ] in Indices M ;
reconsider s = x " as Element of H ;
let f be Element of ( dom Subformulae p ) -tuples_on the carrier of K ;
F1 . ( a1 , - a2 ) = G1 & F1 . ( a2 , - a2 ) = G1 ;
redefine func Sphere ( a , b , r ) -> compact Subset of TOP-REAL 2 ;
let a , b , c , d be Real ;
rng s c= dom ( 1 / 2 ) & rng s c= dom ( 1 / 2 ) ;
curry ( F-19 , k ) is additive ;
set k2 = card dom B , k2 = card dom C , k1 = card dom C , k2 = card dom D ;
set G = DTConMSA ( X ) ;
reconsider a = [ x , s ] as w of G ;
let a , b be Element of ML , M be Matrix of n , K ;
reconsider s1 = s , s2 = t as Element of SS ;
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 & Ik in dom stop I ;
let g be continuous Function of X | B , Y ;
reconsider D = Y as Subset of ( TOP-REAL n ) | P ;
reconsider i0 = len p1 , i2 = len p2 as Integer ;
dom f = the carrier of S & rng f = the carrier of T ;
rng h c= union ( the carrier of J ) & rng h c= the carrier of L ;
cluster All ( x , H ) -> non empty for element ;
d * N1 / 2 > N1 * 1 / 2 * 1 ;
]. a , b .[ c= [. a , b .] ;
set g = f " D1 , g1 = f " D2 ;
dom ( p | ( Seg m ) ) = ( m + 1 ) ;
3 + - 2 <= k + - 2 & k + - 2 <= k + - 2 ;
tan is_differentiable_in ( ( arccot * arccot ) `| Z ) . x ;
x in rng ( f /^ ( n -' 1 ) ) ;
let f , g be FinSequence of D ;
cp in the carrier of S1 & cp in the carrier of S2 ;
rng f " = dom f & rng f = dom f ;
( the Target of G ) . e = v & ( the Target of G ) . e = v ;
width G - 1 < width G - 1 & width G - 1 < width G ;
assume v in rng ( S | E1 ) & u in rng ( S | E1 ) ;
assume x is root or x is root or x is root ;
assume 0 in rng ( g2 | A ) & 0 in rng ( g2 | A ) ;
let q be Point of ( TOP-REAL 2 ) | K1 , r be Real ;
let p be Point of ( TOP-REAL 2 ) | K1 , q be Point of ( TOP-REAL 2 ) | K1 ;
dist ( O , u ) <= |. p2 .| + 1 ;
assume dist ( x , b ) < dist ( a , b ) ;
<* S7 *> is_in the carrier of C-20 & S7 in the carrier of C-20 ;
i <= len G -' 1 & G * ( i1 , j1 ) `2 <= s ;
let p be Point of ( TOP-REAL 2 ) | K1 , q be Point of ( TOP-REAL 2 ) | K1 ;
x1 in the carrier of I[01] & x2 in the carrier of I[01] & x = f . x1 ;
set p1 = f /. i , p2 = f /. j , p3 = f /. ( i + 1 ) , p4 = f /. ( i + 1 ) , p4 = f /. ( i + 1 ) , p4 = f /. (
g in { g2 : r < g2 & g2 < r } ;
Q2 = Sp2 " ( Q /\ R ) .= Sp2 " ( Q /\ R ) ;
( 1 / 2 ) |^ ( n + 1 ) is summable ;
- p + I c= - p + A ;
n < LifeSpan ( P1 , s1 ) & n <= card I ;
CurInstr ( p1 , s1 ) = i .= ( - 1 ) ;
A /\ Cl { x } \ { x } <> {} ;
rng f c= ]. r , r + 1 .[ ;
let g be Function of S , V ;
let f be Function of L1 , L2 , g be Function of L2 , L2 ;
reconsider z = z as Element of CompactSublatt L , x be Element of L ;
let f be Function of S , T ;
reconsider g = g as Morphism of c opp , b opp ;
[ s , I ] in S ~ & [ s , I ] in S ;
len ( the connectives of C ) = 4 & len ( the connectives of C ) = 5 ;
let C1 , C2 be subcategory , F be subFunctor of C1 , C2 ;
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 " * ( H |^ a ) is Subgroup of H ;
let A1 be /. of O , E , A2 be Element of E ;
p2 , r3 , q3 is_collinear & q2 , q3 , q3 is_collinear & p1 , r2 , p3 is_collinear ;
consider x being element such that x in v ^ K ;
not x in { 0. TOP-REAL 2 } or x in { 0. TOP-REAL 2 } ;
p in [#] ( I[01] | B11 ) & p in the carrier of ( I[01] ) ;
0 . 0 < M . E8 & E8 < M . E8 ;
^ ( c , c ) @ = c ;
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . s2 ) & a2 in dom ( F . s2 ) ;
cluster -> *> for Sub| | the carrier of L is ) ;
set i1 = the Nat , i2 = the Element of NAT ;
let s be 0 -started State of SCM+FSA , a be Int-Location ;
assume y in ( f1 \/ f2 ) .: A ;
f . ( len f ) = f /. len f .= f /. 1 ;
x , f . x '||' f . x , f . y ;
attr X c= Y means : Def3 : cos ( X ) c= cos ( Y ) ;
let y be upper Subset of Y , x be Element of X ;
cluster ( x `1 ) `1 -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> of \rm \rm thesis ;
set S = <* Bags n , i\mathopen *> , S = <* <* i *> *> ;
set T = [. 0 , 1 / 2 .] ;
1 in dom mid ( f , 1 , 1 ) ;
( 4 * PI ) / 2 < ( 2 * PI ) / 2 ;
x2 in dom f1 /\ dom f & x2 in dom f1 /\ dom f ;
O c= dom I & { {} } = { {} } ;
( the Target of G ) . x = v & ( the Target of G ) . x = v ;
{ HT ( f , T ) } c= Support f ;
reconsider h = R . k as Polynomial of n , L ;
ex b being Element of G st y = b * H ;
let x , y , z be Element of G 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 *> = 1 ;
set N-26 = the \subseteq of the or N-26 = the InternalRel of N ;
len g: + ( x + 1 ) - 1 <= x ;
a on B & b on B implies a on B
reconsider rv = r * I . v as FinSequence of REAL ;
consider d such that x = d and a [= d ;
given u such that u in W and x = v + u ;
len f /. ( \downharpoonright n ) = len for p st p in rng f holds p = f /. n
set q2 = ( N-min C ) `2 , q2 = ( E-max C ) `2 ;
set S = \leq ( S1 , S2 ) `1 , T = S2 `2 ;
MaxADSet ( b ) c= MaxADSet ( P /\ Q ) ;
Cl ( G . q1 ) c= F . r2 & Cl ( G . q2 ) c= F . r2 ;
f " D meets h " V & f " D meets f " 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 & t is Element of ( F . X ) . s ;
rng f c= the carrier of S2 & rng f c= the carrier of S2 ;
consider y being Element of X such that x = { y } ;
f1 . ( a1 , b1 ) = b1 & 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 & [ i , j ] in Indices M2 ;
assume that P c= Seg m and M is \HM { i , j } is Seg of n ;
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 = p1 . i .= p1 . i ;
let PA , PA , G be a_partition of Y , a be Element of Y ;
pred 0 < r & r < 1 & 1 < r & r < 1 ;
rng ( \mathop { 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 ) \/ the topology of Y ) ;
dom ( f | u ) c= dom ( u | v ) & dom ( f | v ) = dom u ;
pred n divides m & m divides n & n = m ;
reconsider x = x as Point of [: I[01] , I[01] :] ;
a in dom the
not y0 in the still of f & not y0 in the carrier of f ;
Hom ( ( a , b ) `1 , c ) <> {} ;
consider k1 such that p " < k1 and k1 < len f and f . k1 = f . k1 ;
consider c , d such that dom f = c \ d and rng f c= A ;
[ x , y ] in [: dom g , dom k :] ;
set S1 = Let ( x , y , z ) , S2 = y , S1 = z ;
l1 = m2 & l1 = i2 & l2 = j2 implies l1 = i2 & l2 = j2
x0 in dom ( u + v ) /\ dom ( v + u ) ;
reconsider p = x , q = y as Point of ( TOP-REAL 2 ) | K1 ;
I[01] = R^1 | B01 .= R^1 | B01 .= the carrier of ( TOP-REAL 2 ) | B01 ;
f . p4 <= f . p1 & f . p1 in P ;
( ( F `1 ) ^2 + ( F `2 ) ^2 ) <= ( x `1 ) ^2 + ( F `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 .= <* a *> ;
X . i in 2 -tuples_on A . i \ B . i ;
<* 0 *> in dom ( e --> [ 1 , 0 ] ) ;
then P [ a ] & P [ succ a ] & P [ succ a ] ;
reconsider s\mathclose = s/. i as w of D ;
( k -' 1 ) <= len ( ' - 1 ) ;
[#] S c= [#] the TopStruct of T & [#] T c= the TopStruct of T ;
for V being strict RealUnitarySpace holds V in the carrier of V implies V is Subspace of V
assume k in dom mid ( f , i , j ) ;
let P be non empty Subset of TOP-REAL 2 , p , q be Point of TOP-REAL 2 ;
let A , B be Matrix of n1 , K , n , m be Nat ;
- a * - b = a * b - a * b ;
for A being Subset of AS holds A // A & A // A implies A = B
for o2 being object of o2 st o2 in <^ o2 , o2 ^> holds <^ o2 , o2 ^> <> {} ;
then ||. x .|| = 0 & x = 0. X ;
let N1 , N2 be strict normal Subgroup of G , N be strict Subgroup of G ;
j >= len upper_volume ( g , D1 ) & len upper_volume ( g , D2 ) = len D2 ;
b = Q . ( len Qc - 1 + 1 ) ;
f2 * f1 /* s is divergent_to-infty & f2 * f1 is convergent & lim ( f2 * f1 ) = x0 ;
reconsider h = f * g as Function of N4 , G ;
assume that a <> 0 and Let a , b , c , d ;
[ t , t ] in the Relation of A & [ t , t ] in the InternalRel of A ;
( v |-- E ) | n is Element of T7 & ( v |-- E ) | n is Element of T7 ;
{} = the carrier of L1 + L2 .= the carrier of L1 + ( the carrier of L2 ) .= the carrier of L1 ;
Directed I is_halting_on Initialized s , P & Directed I is_halting_on Initialized s , P ;
Initialized p = Initialize ( p +* q ) .= Initialize ( p +* q ) ;
reconsider N2 = N1 , N2 = N2 as strict net of R2 , N be net of R2 ;
reconsider Y = Y as Element of \langle Ids L , \subseteq \rangle ;
"/\" ( uparrow p , L ) <> p ;
consider j being Nat such that i2 = i1 + j and j in dom f ;
not [ s , 0 ] in the carrier of S2 & not [ s , 0 ] in the carrier of S2 ;
mm in ( B '/\' C ) '/\' D \ { {} } ;
n <= len ( P + Q ) & n <= len ( P + Q ) ;
( x1 `1 ) ^2 = ( x2 `1 ) ^2 + ( x1 `2 ) ^2 .= ( x2 `1 ) ^2 + ( x1 `2 ) ^2 ;
InputVertices S = { x1 , x2 , x3 , x4 , x5 , x5 , 7 , 8 } ;
let x , y be Element of FTT1 ( n ) ;
p = |[ p `1 / p `2 , p `2 / p `1 ]| .= |[ p `1 , p `2 ]| ;
g * 1_ G = h " * g * h .= h " * g * h .= h ;
let p , q be Element of PFuncs ( V , C ) ;
x0 in dom x1 /\ dom x2 & x0 in dom x1 /\ dom x2 ;
( R qua Function ) " = R " & ( R " ) " = R ;
n in Seg len ( f /^ ( len p -' 1 ) ) ;
for s being Real st s in R holds s <= s2 ;
rng s c= dom ( f2 * f1 ) & rng s c= dom ( f2 * f1 ) ;
synonym ex \mathop { X } , 1 } for X is Subset of iff X is finite ;
1. ( K , n ) * 1. ( K , n ) = 1. ( K , n ) ;
set S = Segm ( A , P1 , Q1 ) , Q1 = Segm ( A , P1 , Q1 ) ;
ex w st e = ( w / f ) * f & w in F ;
curry ( P+* ( i , k ) ) # x is convergent ;
cluster open -> open for Subset of [: T , Y :] ;
len f1 = 1 .= len f3 .= len f3 .= len f3 .= len f3 ;
( i * p ) / p < ( 2 * p ) / p ;
let x , y be Element of [: \HM { the carrier of U0 } , the carrier of U0 :] ;
b1 , c1 // b9 , c9 or b1 , c1 // c , a ;
consider p being element such that c1 . j = { p } ;
assume f " { 0 } = {} & f is total ;
assume IC Comput ( F , s , k ) = n & IC Comput ( F , s , k ) = n ;
Reloc ( J , card I ) does not destroy a ;
goto ( card I + 1 ) does not destroy c ;
set m3 = LifeSpan ( p3 , s3 ) , P4 = Comput ( p3 , s3 , 1 ) , P4 = P3 ;
IC SCMPDS in dom Initialize p & IC SCMPDS in dom Initialize p & IC SCMPDS in dom p ;
dom t = the carrier of SCM R & dom t = the carrier of SCM R ;
( ( E-max 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 & Cl Int Cl F c= Cl Int Cl F ;
the carrier of X1 union X2 misses ( A \/ B ) ;
assume not LIN a , f . a , g . a , g . b ;
consider i being Element of M such that i = d6 and i in dom f ;
then Y c= { x } or Y = {} or Y = { x } ;
M , v / ( y , x ) / ( y , x ) |= H1 ;
consider m being element such that m in Intersect ( FF ) and m in Intersect ( FF ) ;
reconsider A1 = support u1 , A2 = support v1 as Subset of X ;
card ( A \/ B ) = k-1 + ( 2 * 1 ) ;
assume that a1 <> a3 and a2 <> a4 and a3 <> a4 and a3 <> a4 and a3 <> a4 and a3 <> a4 ;
cluster s -\bf as Element of S , V , s be non empty Subset of S ;
LG2 /. n2 = LG2 . n2 .= LG2 . n2 .= LG2 . n2 ;
let P be compact non empty Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
assume r-7 in LSeg ( p1 , p2 ) /\ LSeg ( p2 , p3 ) ;
let A be non empty compact Subset of TOP-REAL n , a be Real ;
assume [ k , m ] in Indices ( D * ( i , j ) ) ;
0 <= ( 1 / 2 ) |^ p / 2 & 1 <= ( 1 / 2 ) |^ p ;
( F . N | E8 ) . x = +infty ;
attr X c= Y means : Def3 : Z c= V & X \ V c= Y \ Z ;
( y `2 ) * ( z `2 ) * ( z `2 ) <> 0. I ;
1 + card X-18 <= card u & card X-18 <= card u ;
set g = z \circlearrowleft ( ( E-max L~ z ) .. z ) , M = z .. z , N = L~ z , S = L~ z , N = L~ z , S = L~ z , N = L~ z , N = L~ z , S = L~ z ,
then k = 1 & p . k = <* x , y *> . k ;
cluster -> total for Element of C -\mathop { X } ;
reconsider B = A as non empty Subset of ( TOP-REAL n ) | A as Subset of ( TOP-REAL n ) | A ;
let a , b , c be Function of Y , BOOLEAN ;
L1 . i = ( i .--> g ) . i .= g . i .= g ;
Plane ( x1 , x2 , x3 , x4 ) c= P & Plane ( x1 , x2 , x3 , x4 ) c= P ;
n <= indx ( D2 , D1 , j1 ) & 1 <= j1 & j1 + 1 <= len ( D2 ) ;
( ( g2 . O ) `1 ) ^2 = - 1 & ( g2 . O ) `2 = 1 ;
j + p .. f - len f <= len f - len f - len f ;
set W = W-bound C , E = E-bound C , S = S-bound C , N = N-bound C , S = S-bound C , N = S-bound C , S = S-bound C , N = S-bound C , S = S-bound C , N = S-bound C , S = S-bound
S1 . ( a , e ) = a + e .= a ` ;
1 in Seg width ( M * ( ColVec2Mx p ) ) ;
dom ( i (#) Im f ) = dom Im f .= dom ( i (#) Im f ) ;
( that ^2 ) . x = W . ( a , *' ( a , p ) ) ;
set Q = \mathop { \rm _ _ 0 } ( g , f , h ) ;
cluster -> e -] for ManySortedSet of U1 , U2 ;
attr ex A st F = { A } ;
reconsider z9 = \hbox { y : y in product G } as Element of product \overline G ;
rng f c= rng f1 \/ rng f2 & rng f c= rng f1 \/ rng f2 ;
consider x such that x in f .: A and x in f .: C ;
f = <*> the carrier of F_Complex & f is FinSequence of the carrier of F_Complex ;
E , j |= All ( x1 , x2 , x3 , x4 ) ;
reconsider n1 = n , n2 = m 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 .= k + 1 ;
card ( ( x \ B1 ) /\ B1 ) = 0 implies card ( x \ B1 ) = 0
g + R in { s : g-r < s & s < g + r } ;
set q0S = ( q , <* s *> ) -\mathop { <* s *> } ;
for x being element st x in X holds x in rng f1 implies x in Y
h0 /. ( i + 1 ) = h0 . ( i + 1 ) ;
set mw = max ( B , o ) , mw = max ( B , o ) ;
t in Seg width ( I ^ ( n , n ) ) ;
reconsider X = dom f /\ C as Element of Fin NAT ;
IncAddr ( i , k ) = <% IC SCM R + k %> .= succ IC S ;
( ( S-bound L~ f ) / 2 ) <= ( q `2 ) / 2 & ( ( q `2 ) / 2 ) <= ( ( q `2 ) / 2 ) ;
attr R is condensed means : Def3 : Int R is condensed & Cl R is condensed & R is condensed ;
pred 0 <= a & b <= 1 & a * b <= 1 ;
u in ( ( c /\ ( ( d /\ b ) /\ e ) ) /\ f ) /\ j ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) /\ f ) ) /\ j ;
len C + - 2 >= 9 + - 3 & len C + - 2 >= 0 ;
x , z , y is_collinear & x , z , x is_collinear implies x = y
a |^ ( n1 + 1 ) = a |^ n1 * a .= a |^ n1 * a ;
<* \underbrace ( 0 , \dots , 0 ) *> in Line ( x , a * x ) ;
set y9 = <* y , c *> ;
FG2 /. 1 in rng Line ( D , 1 ) & FG2 /. len FG2 in rng Line ( D , 1 ) ;
p . m Joins r /. m , r /. ( m + 1 ) , G ;
( p `2 ) ^2 = ( f /. i1 ) ^2 & ( f /. i1 ) ^2 = ( f /. i1 ) ^2 ;
W-bound ( X \/ Y ) = W-bound ( X \/ Y ) .= W-bound ( X \/ Y ) ;
0 + ( p `2 ) ^2 <= 2 * r + ( p `2 ) ^2 ;
x in dom g & not x in g " { 0 } ;
f1 /* ( seq ^\ k ) is divergent_to-infty & f2 /* ( seq ^\ k ) is divergent_to-infty ;
reconsider u2 = u as Vector of ( the carrier of \langle V , \subseteq *> ) , REAL ;
p |-count ( Product Sgm X ) = 0 & p |-count ( Product Sgm X ) = 0 ;
len <* x *> < i + 1 & i + 1 <= len c + 1 ;
assume that I is non empty and { x } /\ { y } = { 0. I } ;
set ii2 = card I + 4 .--> goto 0 , ii2 = goto 0 , ii2 = goto 0 , ii2 = goto 0 , ii2 = goto 0 , ii2 = goto 0 , ii2 = goto 0 , ii2 = goto 0 , ii2 = goto 0
x in { x , y } & h . x = {} ( TF ) ;
consider y being Element of F such that y in B and y <= x `1 ;
len S = len ( the charact of ( ( the charact of ( A ) ) * the charact of B ) ) ;
reconsider m = M , i = I , n = N as Element of X ;
A . ( j + 1 ) = B . ( j + 1 ) \/ A . j ;
set N8 = : G : G = : G c= dom ( G-15 | G ) ;
rng F c= the carrier of gr { a } & F is finite & G is finite implies F is finite
P is reconsider being reconsider being reconsider K of Q as FinSequence of ( the carrier of K ) * , r = Q as FinSequence of K ;
f . k , f . ( Let ( Let n ) + 1 ) ) in rng f ;
h " P /\ [#] T1 = f " P /\ [#] T1 .= [#] T2 /\ [#] T2 .= [#] T2 ;
g in dom f2 \ f2 " { 0 } & f2 " { 0 } = f2 " { 0 } ;
g+ X /\ dom f1 = g1 " X .= dom g1 /\ dom ( f1 | X ) .= X ;
consider n being element such that n in NAT and Z = G . n ;
set d1 = being Subset of dist ( x1 , y1 ) , d2 = dist ( x2 , y2 ) , d2 = dist ( y2 , y2 ) ;
b `1 + ( 1 / 2 ) < ( 1 + ( 1 + 1 ) ) / 2 ;
reconsider f1 = f as VECTOR of the carrier of X , Y ;
pred i <> 0 means : Def3 : i ^2 mod ( i + 1 ) = 1 ;
j2 in Seg len ( g2 . i2 ) & j2 in Seg width ( g2 . i2 ) ;
dom ( i ) = dom ( i ) .= dom ( i ) .= dom ( i ) .= dom ( i ) ;
cluster sec | ]. PI / 2 , PI / 2 .[ -> one-to-one ;
Ball ( u , e ) = Ball ( f . p , e ) .= Ball ( u , e ) ;
reconsider x1 = x0 , y1 = x1 as Function of S , IV , IV = x1 as Function of S , IV ;
reconsider R1 = x , R2 = y , R1 = z as Relation of L , R ;
consider a , b being Subset of A such that x = [ a , b ] ;
( <* 1 *> ^ p ) ^ <* n *> in Rv ;
S1 +* S2 = S2 +* S2 +* S2 .= S2 +* S2 +* S2 .= S1 +* S2 +* S2 +* S2 +* S2 ;
( ( ( #Z 2 ) * ( cos + arccot ) ) `| Z ) = f ;
cluster -> C -valued for Function of C , REAL * , REAL ;
set C7 = 1GateCircStr ( <* z , x *> , f3 ) , C8 = 1GateCircStr ( <* z , x *> , f3 ) ;
EEE8 . e2 = E8 . e2 .= E8 . e2 .= E8 . e2 ;
( ( arctan * ( arctan + arccot ) ) `| Z ) = ( arctan * ( arctan + arccot ) ) `| Z ;
upper_bound A = ( PI * 3 / 2 ) * 2 & lower_bound A = 0 ;
F . ( dom f , - F . ( cod f , - F . ( cod f , - F . ( cod f , - F . ( cod f , - F . ( cod f , - F . ( cod f , - F . ( cod f , - F . ( cod f , - F . ( cod f
reconsider p8 = q8 , p8 = q8 as Point of Euclid 2 ;
g . W in [#] Y0 & [#] Y0 c= [#] Y0 & [#] Y0 c= [#] Y0 ;
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 ( f , i ) ;
rng s c= dom f /\ ]. - r , x0 .[ & rng s c= dom f /\ ]. x0 , x0 .[ ;
assume x in { idseq 2 , Rev idseq 2 } /\ { idseq 2 , <* 1 *> } ;
reconsider n2 = n , m2 = m , m1 = n as Element of NAT ;
for y being ExtReal st y in rng seq holds g <= y
for k st P [ k ] holds P [ k + 1 ] ;
m = m1 + m2 .= m1 + m2 .= m1 + m2 .= m1 + m2 + m2 .= m1 + m2 + m2 ;
assume for n holds H1 . n = G . n -H . n ;
set B-1 = f .: ( the carrier of X1 ) , B29 = f .: the carrier of X2 ;
ex d being Element of L st d in D & x << d ;
assume R ~ ( a ) c= R ~ ( b ) & R ~ ( b ) c= R ~ ( a ) ;
t in ]. r , s .[ or t = r or t = s & s < t ;
z + v2 in W & x = u + ( z + v2 ) ;
x2 |-- y2 iff P [ x2 , y2 ] & P [ y2 , y2 ] or P [ y2 , y2 ] & P [ y2 , y2 ] ;
pred x1 <> x2 & |. x1 - x2 .| > 0 & x1 - x2 > 0 & x1 - x2 < 0 ;
assume p2 - p1 , p3 - p1 , p3 - p1 , p2 - p3 as linearly-independent non empty Subset of TOP-REAL 2 ;
set q = ( be \upupharpoons f ^ <* 'not' A *> ) ;
let f be PartFunc of REAL-NS 1 , REAL-NS n , x be Point of REAL-NS n , r be Real ;
( n mod ( 2 * k ) ) + 1 = n mod k ;
dom ( T * succ t ) = dom succ t & dom ( T * succ t ) = dom succ t ;
consider x being element such that x in wc iff x in c ;
assume ( F * G ) . v . x3 = v . x4 ;
assume that the Sorts of D1 c= the Sorts of D2 and the Sorts of D1 c= the Sorts of D2 and the Sorts of D2 c= the Sorts of D1 and the Sorts of D1 c= the Sorts of D2 ;
reconsider A1 = [. a , b .[ , A2 = [. a , b .] as Subset of R^1 ;
consider y being element such that y in dom F and F . y = x ;
consider s being element such that s in dom o and a = o . s ;
set p = W-min L~ Cage ( C , n ) , q = W-min L~ Cage ( C , n ) , r = W-bound L~ Cage ( C , n ) , s = E-bound L~ Cage ( C , n ) , w = E-bound L~ Cage ( C , n ) , G = Gauge ( C , n ) , G = Gauge (
n1 - len f + 1 <= len ( - len f + 1 ) - len f + 1 ;
Seg |. ( q , O1 ) , a , b , a , b , c *> = { u , v , a , b , c } ;
set C-2 = ( ( `1 ) `1 ) . ( k + 1 ) ;
Sum ( L * p ) = 0. R * Sum p .= 0. V .= 0. V ;
consider i being element such that i in dom p and t = p . i ;
defpred Q [ Nat ] means 0 = Q ( $1 ) & $1 in dom Q & Q [ $1 ] ;
set s3 = Comput ( P1 , s1 , k ) , P3 = P1 +* I , s4 = P1 +* I , P4 = P1 +* I , P4 = P1 +* I , P4 = P2 +* I , P4 = P2 +* I , P4 = P1 +* I , P4 = P2 +* I , P4 = P1
let l be [: of k , A , A-30 , A-30 , Al be Nat ;
reconsider U2 = union G-24 , Gs = union Gs as Subset-Family of [: T , T :] ;
consider r such that r > 0 and Ball ( p `1 , r ) c= Q ` ;
( h | ( n + 2 ) ) /. ( i + 1 ) = p29 ;
reconsider B = the carrier of X1 , B = the carrier of X2 as Subset of X ;
pQ = <* - vs , 1 , 1 *> .= <* - vs , 1 , 1 *> ;
synonym f is real-valued means : Def3 : rng f c= NAT & rng f c= NAT & f is one-to-one ;
consider b being element such that b in dom F and a = F . b ;
x9 < card X0 & x9 in card Y0 & y9 in card Y0 & x9 in card Y0 implies x9 in card ( X \/ Y )
attr X c= B1 means : Def3 : for B being Subset of X st B c= succ B1 holds X c= succ B & X c= B ;
then w in Ball ( x , r ) & dist ( x , w ) <= r ;
angle ( x , y , z ) = angle ( x , y , z ) ;
pred 1 <= len s means : Def3 : len ( the m of s ) = len s & for i being Nat st i in dom s holds s . i = s . i ;
f-47 c= f . ( k + ( n + 1 ) ) ;
the carrier of { 1_ G } = { 1_ G } .= { 1_ G } .= { 1_ G } ;
pred p '&' q in the carrier of \rm WFF means : Def3 : q '&' p in the carrier of assume ( A ) ;
- ( t `1 ) ^2 < ( t `1 ) ^2 & t `2 < - ( t `2 ) ^2 ;
U . 1 = U /. 1 .= ( W /. 1 ) `1 .= ( W /. 1 ) `1 .= W /. 1 .= W /. 1 ;
f .: ( the carrier of x ) = the carrier of x & f .: ( the carrier of x ) = the carrier of x ;
Indices [: O , O :] = [: Seg n , Seg n :] & [: O , 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 the carrier of A-9 & f is \cup the carrier of A-9 = the carrier of A-9 ;
[ h . 0 , h . 3 ] in the InternalRel of G & [ h . 2 , h . 3 ] in the InternalRel of G ;
s +* Initialize ( ( intloc 0 ) .--> 1 ) = s3 +* Initialize ( ( intloc 0 ) .--> 1 ) ;
|[ w1 , v1 ]| - |[ w1 , v1 ]| <> 0. TOP-REAL 2 & |[ w1 , v1 ]| - |[ w1 , v1 ]| = 0. TOP-REAL 2 ;
reconsider t = t as Element of INT -tuples_on INT , ( the carrier of INT ) * ;
C \/ P c= [#] ( ( G | A ) \ A ) & C /\ P = [#] ( ( G | A ) \ A ) ;
f " V in the topology of [: X , D :] /\ D & f " V in the topology of [: X , Y :] ;
x in [#] ( the carrier of A ) /\ A & x in the carrier of ( the carrier of B ) /\ A ;
g . x <= h1 . x & h . x <= h1 . x ;
InputVertices S = { xy , y , z } \/ { xy , y , z } .= { xy , y , z } ;
for n being Nat st P [ n ] holds P [ n + 1 ] ;
set R = being Matrix of M , a , a * Line ( M , i ) ;
assume that M1 is being_line and M2 is being_line and M3 is being_line and M3 is being_line and M3 is being_line and M3 is being_line ;
reconsider a = f4 . i0 -' 1 as Element of K , i = len f - 1 as Element of K ;
len B2 = Sum Len ( F1 ^ F2 ) .= len ( F1 ^ F2 ) + len ( F2 ^ F1 ) .= len F1 + len F2 ;
len ( ( the ` of n ) * ( i , j ) ) = n & len ( ( i , j ) * ( i , j ) ) = n ;
dom max ( - ( f + g ) , f + g ) = dom ( f + g ) ;
( the Sorts of seq ) . n = upper_bound Y1 & ( the Sorts of seq ) . n = upper_bound Y1 ;
dom ( p1 ^ p2 ) = dom ( f12 ) .= dom ( f12 ) .= dom ( f12 ) .= dom ( f12 ) ;
M . [ 1 / y , y ] = 1 / y * v1 .= 1 / y * v1 .= 1 ;
assume that W is non trivial and W .vertices() c= the carrier' of G2 and W .vertices() c= the carrier' of G2 and W is not empty ;
C6 * ( i1 , i2 ) = G1 * ( i1 , i2 ) .= G1 * ( i1 , i2 ) .= G1 * ( i1 , i2 ) ;
C8 |- 'not' Ex ( x , p ) 'or' p . ( x , y ) ;
for b st b in rng g holds lower_bound rng f\lbrace fwhere b is Real : b <= a & b <= b } <= b
- ( ( q1 `1 ) / |. q1 .| - cn ) = 1 - cn & ( q1 `1 ) / |. q1 .| - cn = 1 ;
( LSeg ( c , m ) \/ NAT ) \/ LSeg ( l , k ) c= R ;
consider p being element such that p in Ball ( x , r ) and p in L~ f and x = f . p ;
Indices ( X @ ) = [: Seg n , Seg n :] & Indices ( X @ ) = [: Seg n , Seg n :] ;
cluster s => ( q => p ) => ( q => ( s => p ) ) -> valid ;
Im ( ( Partial_Sums F ) . m ) . m is_measurable_on E & Im ( ( Partial_Sums F ) . m ) . n is_measurable_on E ;
cluster f . ( x1 , x2 ) -> Element of D & f . ( x1 , x2 ) -> Element of D ;
consider g being Function such that g = F . t and Q [ t , g ] ;
p in LSeg ( ( N-min Z ) /. 1 , ( \hbox { \boldmath $ p $ } ) /. 2 ) ;
set R8 = R .: ]. b , +infty .[ ;
IncAddr ( I , k ) = SubFrom ( da , da ) .= SubFrom ( da , da ) .= goto ( ( card I + k ) + k ) ;
seq . m <= ( the Sorts of seq ) . k & ( the Sorts of seq ) . m <= ( the Sorts of seq ) . 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 = U2 as non empty Subset of ( the carrier of U0 ) * ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) /\ f ) /\ m ) /\ n ;
consider y being element such that y in Y and P [ y , lower_bound B ] ;
consider A being finite stable set such that card A = card ( the carrier of R ) and A is finite ;
p2 in rng ( f |-- p1 ) \ rng <* p1 *> & p2 in rng ( f |-- p1 ) \ rng <* p1 *> ;
len s1 - 1 > 1-1 & len s2 - 1 > 0 & len s2 - 1 > 0 ;
( ( ( N-min P ) `2 ) / ( 1 + ( ( q `2 ) ) / ( 1 + ( q `2 ) ) ) ) ) = ( ( ( ( q `2 ) ) ) / ( 1 + ( q `2 ) / ( 1 + ( q `2 ) ) ) ) ;
Ball ( e , r ) c= LeftComp Cage ( C , k + 1 ) & Ball ( e , r ) c= LeftComp Cage ( C , k + 1 ) ;
f . a1 ` = f . a1 ` .= f . a1 ` .= f . a1 ` .= f . a1 ;
( seq ^\ k ) . n in ]. -infty , x0 .[ & ( seq ^\ k ) . n in ]. x0 , x0 + r .[ ;
gg . s0 = g . s0 | G . s0 .= g . s0 .= ( g | G ) . s0 ;
the InternalRel of S is \lbrace the carrier of S , the carrier of S , the carrier of S , the carrier of S } ;
deffunc F ( Ordinal , Ordinal ) = phi . ( $2 , $1 ) & $2 = phi . ( $2 , $2 ) ;
F . a1 = F . s2 & F . a1 = F . a1 & F . a1 = F . a1 ;
x `2 = A . o .= Den ( o , A . a ) .= Den ( o , A . a ) ;
Cl ( f " P1 ) c= f " P1 & Cl ( f " P1 ) c= f " P1 ;
FinMeetCl ( ( the topology of S ) \/ the topology of T ) c= the topology of T & FinMeetCl ( ( the topology of T ) \/ the topology of T ) c= the topology of T ;
synonym o is \bf means : Def3 : o <> *' & o <> {} & o <> {} ;
assume that X |^ + = Y |^ + 1 and card X <> card Y and Y is finite and X is finite and Y is finite ;
the *> s <= 1 + ( the *> \HM { the } \HM { is Element of ( the Sorts of A ) * the Arity of S ) ;
LIN a , a1 , d or b , c // b1 , c1 or b , c // a1 , c1 or b , c // a1 , d ;
e / 2 . 1 = 0 & e / 2 . 2 = 1 & e / 2 . 3 = 0 ;
EE in SE & EE in { NE } implies EE in NE & EE in FE
set J = ( l , u ) If , K = I " ;
set A1 = } , A2 = [ <* ap , set *> , '&' ] , A2 = [ <* cin , set *> , '&' ] , A2 = [ <* cin , set *> , '&' ] , A1 = [ <* cin , A1 *> , '&' ] , A2 = [ <* cin , A1 *> , '&' ] , A2 = [ <* cin , set *> , '&'
set vs = [ <* xy , *> , {} ] , xy = [ <* xy , yz *> , {} ] , yz = [ <* xy , yz *> , {} ] , yz = [ <* xy , yz *> , {} ] , yz = [ <* xy , yz *> , {} ] , xy = [ <* xy , yz *> , {} ] , xy = [
x * z * x " in x * ( z * N ) * x * x * x * z " ;
for x being element st x in dom f holds f . x = g3 . x & f . x = g2 . x ;
Int cell ( f , 1 , G ) c= RightComp f \/ L~ f & Int cell ( f , 1 , G ) c= RightComp f \/ L~ f ;
U2 is_an_arc_of W-min C , E-max C & P /\ L~ f = { E-max C , E-max C } ;
set f-17 = f @ "/\" @ g ;
attr S1 is convergent means : Def3 : S2 is convergent & ( for n st n >= k holds ( S1 - S2 ) . n = ( S1 . n ) - ( S2 . n ) ) ;
f . ( 0 + 1 ) = ( 0 qua Ordinal ) + a .= a ;
cluster -> } -be -> \mathclose reflexive transitive transitive transitive for reflexive -symmetric , symmetric , reflexive -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 ) & not b in dom Start-At ( ( card I + 2 ) , SCMPDS ) ;
( z + a ) + x = z + ( a + y ) .= z + a + y ;
len ( l \lbrack a |^ 0 , x .] ) = len l .= len ( l |^ 0 ) .= 0 ;
t4 being {} -valued FinSequence of ( {} \/ rng t4 ) * * -valued Function of ( {} \/ rng t4 ) , ( {} \/ rng t4 ) * \ { {} } ;
t = <* F . t *> ^ ( C . p ^ q ) .= <* F . t *> ^ ( C . q ^ q ) ;
set p-2 = W-min L~ Cage ( C , n ) , p-2 = W-min L~ Cage ( C , n ) , p`1 = W-bound L~ Cage ( C , n ) , p`1 = W-bound L~ Cage ( C , n ) , p`1 = W-bound L~ Cage ( C , n ) , p`1 = W-bound L~ Cage ( C , n )
( k -' 1 ) + ( i + 1 ) = ( k - 1 ) + ( i + 1 ) .= ( k - 1 ) + ( i + 1 ) ;
consider u being Element of L such that u = u ` and u in D and u in D ;
len ( ( width E ) |-> a ) = width ( ( len E ) |-> a ) .= len ( ( len E ) |-> a ) .= len ( ( len E ) |-> a ) ;
FF . x in dom ( ( G * the_arity_of o ) . x ) & ( G * the_arity_of o ) . x in dom ( ( G * the_arity_of o ) * the_arity_of o ) ;
set cH2 = the carrier of H2 , cH2 = the carrier of H1 , cH2 = the carrier of H2 ;
set cH1 = the carrier of H1 , cH2 = the carrier of H2 ;
( Comput ( P , s , 6 ) ) . intpos m = s . intpos m .= s . intpos m ;
IC Comput ( Q2 , t , k ) = ( l + 1 ) .= ( l + 1 ) + 1 ;
dom ( ( ( - 1 ) (#) ( sin * cos ) ) `| REAL ) = REAL & dom ( ( - 1 ) (#) ( cos * cos ) ) `| REAL ) = REAL ;
cluster <* l *> ^ phi -> ( 1 + 2 ) -element for string of S ;
set b5 = [ <* thesis , A1 *> , <* y *> ] , b5 = [ <* z , x *> , <* y *> ] , b5 = [ <* z , x *> , <* y *> ] ;
Line ( Segm ( M @ , P , Q ) , x ) = L * Sgm Q .= Line ( M , x ) ;
n in dom ( ( the Sorts of A ) * the_arity_of o ) & n in dom ( ( the Sorts of A ) * the_arity_of o ) ;
cluster f1 + f2 -> continuous for PartFunc of REAL , the carrier of S , the carrier of T ;
consider y be Point of X such that a = y and ||. x - y .|| <= r ;
set x3 = t2 . DataLoc ( s2 . SBP , 2 ) , x4 = s2 . SBP , P4 = s2 . SBP , P4 = s2 . SBP , P4 = s2 . SBP , P4 = s2 . SBP , P4 = s2 . SBP , P4 = s2 . SBP , P4 = s2 . SBP , P4 = s2 . SBP , P4
set p-3 = stop I ( ) , pE = stop I ( ) , pE = stop I ( ) , pE = stop I ( ) , pE = stop I ( ) , pE = stop I ( ) , pE = stop I ( ) , pE = stop I ( ) , pE = stop
consider a being Point of D2 such that a in W1 and b = g . a and a in W2 ;
{ A , B , C , D , E , F , J } = { A , B , C , D , E , F , J }
let A , B , C , D , E , F , J , M , N , N , M , N , N , M , N , N , M , N , N , M , N , N , N , M , N , N , M , N , N , N , M be set ;
|. p2 .| ^2 - ( p2 `2 ) ^2 - ( p2 `2 ) ^2 >= 0 ;
l -' 1 + 1 = n-1 * ( ( x + 1 ) + ( x + 1 ) ) + 1 ;
x = v + ( a * w1 + ( b * w2 ) ) + ( c * w2 ) ;
the TopStruct of L = reconsider the TopStruct of L , the TopStruct of L = [: the topology of L , the topology of L :] as Scott ;
consider y being element such that y in dom H1 and x = H1 . y and y in the carrier of H1 and x = H1 . y ;
fv \ { n } = \mathop ( Free ( v1 , H ) , E ) .= Free ( v1 , H ) .= E ;
for Y being Subset of X st Y is summable & Y is non empty holds Y is iff Y is number
2 * n in { N : 2 * Sum ( p | N ) = N & N > 0 } ;
for s being FinSequence holds len ( the { 0 } --> \rm Shift ( s ) ) = len s & ( the { 0 } --> an = s )
for x st x in Z holds exp_R * f is_differentiable_in x & exp_R * f . x > 0 ;
rng ( h2 * f2 ) c= the carrier of ( ( TOP-REAL 2 ) | K1 ) .= the carrier of ( ( TOP-REAL 2 ) | K1 ) .= K1 ;
j + ( len f ) - len f <= len f + ( len g - len f ) - len f ;
reconsider R1 = R * I , R2 = R * I as PartFunc of REAL n , REAL-NS n ;
C8 . x = s1 . x0 .= C7 . x .= C7 . x .= C8 . x .= C8 . x ;
power F_Complex . ( z , n ) = 1 .= x |^ n .= x |^ n .= x |^ n ;
t at ( C , s ) = f . ( the connectives of S ) . t & t in dom ( the connectives of S ) ;
support ( f + g ) c= support f \/ C & support ( f + g ) c= dom f /\ dom g ;
ex N st N = j1 & 2 * Sum ( ( r | N ) | N ) > N & N > 0 ;
for y , p st P [ p ] holds P [ All ( y , p ) ]
{ [ x1 , x2 ] , [ x2 , x3 ] } 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 in N & x1 in A ;
set X = ( ( Seg ( q , O1 ) ) `1 , ( q , 4 ) `2 ) ;
b . n in { g1 : x0 < g1 & g1 < x0 & g1 < x0 + r } ;
f /* s1 is convergent & f /. x0 = lim ( f /* s1 ) implies f /* s1 is convergent & lim ( f /* s1 ) = 0
the lattice of the lattice of Y = the lattice of the topology of Y & the topology of Y = the topology of X & the topology of Y = the topology of Y & the topology of Y = the topology of X ;
'not' ( a . x ) '&' b . x 'or' a . x '&' 'not' ( b . x ) '&' 'not' ( a . x ) '&' b . x = FALSE ;
2 = len ( ( q ^ r1 ) ^ ( q ^ r2 ) ) + len ( ( q ^ r1 ) ^ ( q ^ r2 ) ) .= len ( q ^ r1 ) + len ( q ^ r2 ) ;
( 1 / a ) * ( sec * f1 ) - ( 1 / a ) * ( ( 1 / a ) * ( ( 1 / a ) * ( ( 1 / a ) * ( 1 / a ) ) ) ) is_differentiable_on Z ;
set K1 = integral ( ( lim ( H ) || A ) ) , D2 = ( ( lim H ) || A ) , g2 = ( lim H ) || A ;
assume e in { ( w1 - w2 ) `1 : w1 in F & w2 in G } ;
reconsider d7 = dom a `1 , d7 = dom F , d7 = dom G as finite set ;
LSeg ( f /^ j , q ) = LSeg ( f , j ) /\ LSeg ( f , j + q .. f ) .= LSeg ( f , j + q .. f ) ;
assume X in { T . ( N2 , K1 ) : h . ( N2 , K1 ) = N2 } ;
assume that Hom ( d , c ) <> {} and <* f , g *> * f1 = <* f , g *> * f2 and f = <* f , g *> * f1 ;
dom S29 = dom S /\ Seg n .= dom L6 .= 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 in H
* ( 0. ( INT , n ) ) . ( a , 1 ) = a `1 - ( 0 * n ) `1 .= a `1 ;
D2 . ( j - 1 ) in { r : lower_bound A <= r & r <= D1 . i } ;
ex p being Point of TOP-REAL 2 st p = x & P [ p ] & p `1 <= 0 & p <> 0. TOP-REAL 2 ;
for c holds f . c <= g . c implies f ^ @ = g @ c
dom ( f1 (#) f2 ) /\ X c= dom ( f1 (#) f2 ) /\ X .= dom ( f1 (#) f2 ) /\ X .= dom ( f1 (#) f2 ) ;
1 = ( p * p ) * p .= p * ( 1 / p ) .= p * 1 / p .= 1 ;
len g = len f + len <* x + y *> .= len f + 1 .= len f + 1 .= len f + 1 ;
dom F-11 = dom ( F | [: N1 , S :] ) .= [: N1 , S :] .= [: N1 , S :] ;
dom ( f . t * I . t ) = dom ( f . t * g . t ) .= dom ( f . t * g . t ) .= dom ( f . t * g . t ) ;
assume a in ( "\/" ( ( T |^ the carrier of S ) , F ) ) .: D & b in ( the carrier of S ) ;
assume that g is one-to-one and ( the carrier' of S ) /\ rng g c= dom g and g is one-to-one and rng g = rng g ;
( ( x \ y ) \ z ) \ ( ( x \ z ) \ ( y \ z ) ) = 0. X ;
consider f such that f * f opp = id b and f * f = id b and f = id b and f = id b ;
( cos | [. 2 * PI * 0 , PI .] ) | [. 0 , PI .] is increasing ;
Index ( p , co ) <= len LS - Gij .. LS & Index ( Gij , LS ) <= len LS - Gij .. LS ;
t1 , t2 , t2 , t be Element of Funcs ( NAT , NAT ) , s be Element of ( the carrier of S ) * ;
( Frege ( Frege curry H ) ) . h <= ( Frege ( curry G ) ) . h & ( Frege ( curry G ) ) . h <= ( Frege ( curry G ) ) . h ;
then P [ f . i0 , f . i0 ] & F ( f . i0 , f . i0 ) < j & j < len f ;
Q [ [ D . x , 1 ] , F . [ D . x , 1 ] ] ;
consider x being element such that x in dom ( F . s ) and y = F . s . x ;
l . i < r . i & [ l . i , r . i ] is an Element of G . i ;
the Sorts of A2 = ( the carrier of S2 ) --> ( the carrier of S2 ) .= the Sorts of A2 ;
consider s being Function such that s is one-to-one and dom s = NAT & rng s = F and rng s = { x } and rng s = { x } ;
dist ( b1 , b2 ) <= dist ( b1 , a ) + dist ( a , b2 ) & dist ( a , b2 ) <= dist ( a , b2 ) + dist ( b , a ) ;
( <* Cage ( C , n ) /. len Upper_Seq ( C , n ) *> /. 1 ) `1 = W-bound L~ Cage ( C , n ) .= W-bound L~ Cage ( C , n ) ;
q `2 <= ( UMP ( Upper_Arc ( C ) ) ) `2 & ( UMP ( C ) ) `2 <= ( UMP ( C ) ) `2 ;
LSeg ( f | i2 , i ) /\ LSeg ( f | i2 , j ) = {} or LSeg ( f | i2 , j ) /\ LSeg ( f | i2 , i ) = {} ;
given a being ExtReal such that a <= Is and A = ]. a , Is .[ and a < Is and for x being element st x in dom Is holds x in [. a , Is .] ;
consider a , b being Complex such that z = a & y = b & z + y = a + b ;
set X = { b |^ n where n is Element of NAT : n in dom b & b . n = 0 } ;
( ( x * y * z ) \ x ) \ z = 0. X ;
set xy = [ <* xy , y *> , f1 ] , yz = [ <* y , z *> , f2 ] , yz = [ <* z , x *> , f3 ] , yz = [ <* z , x *> , f3 ] , yz = [ <* y , z *> , f3 ] , xy = [ <* z , x *> , f3 ] , xy = [ <* z , x *>
( l /. len l ) `1 = ( l /. len l ) `1 .= ( l /. len l ) `1 .= ( l /. len l ) `1 ;
( ( q `2 ) / |. q .| - sn ) / ( 1 + sn ) ) ^2 = 1 ;
( ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) ^2 < 1 ^2 ;
( ( ( ( ( ( S ) \/ Y ) \/ X ) \/ Y ) \/ X ) ) `2 = ( ( ( ( S ) \/ Y ) \/ X ) \/ Y ) `2 .= ( ( ( S ) \/ X ) \/ Y ) `2 ) `2 ;
( seq - seq ) . k = seq . k - seq . k .= seq . k - seq . k .= seq . k - seq . k .= seq . k ;
rng ( ( h + c ) ^\ n ) c= dom SVF1 ( 1 , f , u0 ) /\ dom SVF1 ( 1 , f , u0 ) ;
the carrier of X is the carrier of the carrier of X & the carrier of Y = the carrier of X & the carrier of Y = the carrier of Y ;
ex p4 st p3 = p4 & |. p4 - |[ a , b ]| .| = r & |. p4 - |[ a , b ]| .| = r ;
set ch = chi ( X , A5 ) , A5 = chi ( X , A5 ) ;
R |^ ( 0 * n ) = Iseq ( X , X ) .= R |^ n * 0 .= R |^ n * 0 .= R |^ n ;
( Partial_Sums curry ( F1 , n ) ) . n is nonnegative & ( Partial_Sums curry ( F1 , n ) ) . n is nonnegative ;
f2 = C7 . ( E7 , len ( V , len ( K , len ( H , len ( K , len ( H , len ( K , len ( H , len ( H , len ( H , len ( H , len ( H ) ) ) ) ) ) ) ) ) ) ;
S1 . b = s1 . b .= s2 . b .= s2 . b .= ( s2 * s2 ) . b .= ( s2 * s2 ) . b ;
p2 in LSeg ( p2 , p1 ) /\ LSeg ( p2 , p1 ) \/ LSeg ( p1 , p11 ) /\ LSeg ( p2 , p1 ) .= { p2 } ;
dom ( f . t ) = Seg n & dom ( I . t ) = Seg n & rng ( I . t ) = Seg n ;
assume o = ( the connectives of S ) . 11 & 11 in ( the carrier' of S ) . 11 ;
set phi = ( l1 , l2 ) \mathop { l1 } , phi = ( l1 , l2 ) \mathop { l1 } , C = ( l1 , l2 ) \mathop { l2 } , D = ( l1 , l2 ) \mathop { l1 } , D = ( l1 , l2 ) \mathop { l2 } , E = ( l1 , l2 ) \mathop { l2 } , C = ( l1 , l2 ) \mathop { l2 } , D =
synonym p is is invertible means : Def3 : HT ( p , T ) = 1 & HT ( p , T ) = 1 ;
( Y1 ) `2 = - 1 & ( Y1 ) `2 <> 0. TOP-REAL 2 or Y1 = ( Y1 ) `2 & ( Y1 = ( Y1 ) `2 or Y1 = ( Y1 ) `2 ) & ( Y1 = ( Y1 ) `2 ) ;
defpred X [ Nat , set , set , set ] means P [ $1 , $2 , $2 , $2 , $2 , $2 , $2 ] & P [ $1 , $2 , $2 , $2 , $2 ] ;
consider k being Nat such that for n being Nat st k <= n holds s . n < x0 + g and x0 < g ;
Det I = ( ( m -' n ) * ( m - n ) ) * ( m - n ) .= 1. K .= ( ( m -' n ) * ( m - n ) ) * ( m - n ) ;
( - b - sqrt ( b ^2 - 4 * a * c ) ) / ( 2 * a * c ) < 0 ;
Cs . d = C7 . d mod C7 . d .= C7 . d mod C7 . d .= C7 . d mod C7 . d .= C8 . d ;
attr X1 is dense means : Def3 : X2 is dense & X1 is dense & X2 is dense & X1 is dense & X2 is dense & X1 is dense ;
deffunc F6 ( Element of E , Element of I , Element of I ) = $1 * $2 & $2 = $1 * $2 & $2 = $1 * $2 ;
t ^ <* n *> in { t ^ <* i *> : Q [ i , T . t ] } ;
( x \ y ) \ x = ( x \ x ) \ y .= y ` \ 0. X .= 0. X ;
for X being non empty set for Y being Subset-Family of X holds for F being Subset-Family of [: X , product <* Y *> :] holds F is Basis of [: X , product <* Y *> :]
synonym A , B are_separated means : Def3 : Cl ( A ) misses Cl B & Cl ( A ) misses Cl Cl ( B ) ;
len M8 = len p & width M8 = width M & width M8 = width M8 & width M8 = width M8 & width M8 = width M8 ;
*> = { x where x is Element of K : 0 < v . x & x < 1 } ;
( Sgm Seg m ) . d - ( Sgm Seg m ) . e <> 0 ;
lower_bound divset ( D2 , k + k2 ) = D2 . ( k + k2 - 1 ) .= D2 . ( k + k2 - 1 ) .= D2 . ( k + 1 - 1 ) ;
g . r1 = - 2 * r1 + 1 & dom h = [. 0 , 1 .] & rng h = [. 0 , 1 .] ;
|. a .| * ||. f .|| = 0 * ||. f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| ;
f . x = ( h . x ) `1 & g . x = ( h . x ) `2 & g . x = ( h . x ) `2 ;
ex w st w in dom B1 & <* 1 *> ^ s = <* 1 *> ^ w & len w = len <* 1 *> & len w = len <* 1 *> + len w ;
[ 1 , {} , <* d1 *> ] in ( { [ 0 , {} , {} ] } \/ S1 ) \/ S2 .= [ 0 , {} , {} ] ;
IC Exec ( i , s1 ) + n = IC Exec ( i , s2 ) .= IC Exec ( i , s2 ) .= IC Exec ( i , s2 ) ;
IC Comput ( P , s , 1 ) = IC ( s , 9 ) .= 5 .= 5 + 9 .= ( IC s ) .= 5 ;
( IExec ( W6 , Q , t ) ) . intpos ( e + 1 ) = t . intpos ( e + 1 ) .= t . intpos ( e + 1 ) ;
LSeg ( f /^ i , q ) misses LSeg ( f /^ i , j ) \/ LSeg ( f /^ j , q ) .= LSeg ( f /^ j , q ) \/ LSeg ( f /^ j , q ) ;
assume for x , y being Element of L st x in C & y in C holds x <= y or y <= x ;
integral ( integral ( f , C ) , f ) = f . ( upper_bound C ) - f . ( lower_bound C ) .= f . ( lower_bound C ) - f . ( lower_bound C ) ;
for F , G being one-to-one FinSequence st rng F misses rng G holds F ^ G is one-to-one & F ^ G is one-to-one
||. R /. ( L . h ) .|| < e1 * ( K + 1 * ||. h .|| ) .= e1 * ( K + 1 * ||. h .|| ) ;
assume a in { q where q is Element of M : dist ( z , q ) <= r } ;
set p4 = [ 2 , 1 ] .--> [ 2 , 0 , 1 ] ;
consider x , y being Subset of X such that [ x , y ] in F and x c= d and y in F ;
for y , x being Element of REAL st y ` in Y & x in X ` holds y ` <= x ` & y ` <= x ` ;
func |. p \bullet |. p .| -> variable of A means : Def3 : for x being element st x in it holds it . x = min ( NBI ( p ) , p . x ) ;
consider t being Element of S such that x `1 , y `2 '||' z `1 , t `2 and x `1 , z `2 '||' y `1 , t `2 ;
dom x1 = Seg len x1 & len x1 = len l1 & len y1 = len l1 & len y1 = len l1 & len y1 = len l1 & len y1 = width l1 & len y1 = width l1 & len y1 = width l1 & len y1 = width l1 & len y1 = width l1 ;
consider y2 being Real such that x2 = y2 and 0 <= y2 and y2 < 1 and y2 <= 1 and y2 <= 1 ;
||. f | X /* s1 .|| = ||. f .|| | X .= ||. f /. s1 .|| .= ||. f /. s1 .|| .= ||. f /. s1 .|| .= ||. f /. s1 .|| ;
( the InternalRel of A ) ` ` ` /\ Y = {} ( the carrier of A ) \/ {} .= {} ( the carrier of A ) .= {} ;
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 q & i + 1 in dom q holds P [ i , j ] and P [ i , j ] ;
reconsider h = f | [: X , Y :] as Function of [: X , Y :] , rng ( f | [: X , Y :] ) ;
u1 in the carrier of W1 & u2 in the carrier of W2 & u1 in the carrier of W2 implies u1 + u2 in the carrier of W1 & u2 in the carrier of W2
defpred P [ Element of L ] means M <= f . $1 & f . $1 <= f . $1 & f . $1 <= f . $1 ;
^ ( u , a , v ) = s * x + ( - ( s * x ) + y ) .= b ;
- ( - ( x - y ) ) = - x + - ( - y ) .= - x + - y .= - x + y .= x + y ;
given a being Point of GX such that for x being Point of GX holds a , x are_ed ;
fT = [ [ dom @ f2 , cod f2 ] , [ cod f2 , cod f2 ] ] , fT = [ cod f2 , cod f2 ] , fT = [ cod f2 , cod f2 ] , fT = [ cod f2 , cod f2 ] , fT = [ cod f2 , cod f2 ] , fT = [ cod f2 , cod g2 ] ;
for k , n being Nat st k <> 0 & k < n & n is prime & k <> 0 holds k , n are_relative_prime implies k , n are_relative_prime
for x being element holds x in A |^ d iff x in ( ( A ` ) ` ) ` & ( A ` ) ` = ( A ` ) `
consider u , v being Element of R , a being Element of A such that l /. i = u * a and u in A ;
( - ( p `1 / |. p .| - cn ) ) / ( 1 + cn ) > 0 ;
L-13 . k = Ls . ( F . k ) & F . k in dom ( Ls * Ls ) & F . k in dom ( Ls * Ls ) ;
set i2 = SubFrom ( a , i , - n ) , i1 = goto ( n + 1 ) ;
attr B is thesis means : Def3 : for S being Sub\mathop of B holds S = ( B `1 ) `1 & S = ( B `1 ) `1 ;
a9 "/\" D = { a "/\" d where d is Element of N : d in D & d in D } ;
|( exp_R , q2 )| * |( exp_R , q2 )| * |( exp_R , q2 )| >= |( exp_R , exp_R )| * |( exp_R , q2 )| .= |( exp_R , q2 )| * |( exp_R , q2 )| ;
( - f ) . ( upper_bound A ) = ( ( - f ) | A ) . ( upper_bound A ) .= f . ( upper_bound A ) .= f . ( lower_bound A ) ;
G * ( 1 , k ) `1 = G * ( len G , k ) `1 .= G * ( 1 , k ) `1 .= G * ( 1 , k ) `1 .= G * ( 1 , k ) `1 ;
( Proj ( i , n ) ) . LM = <* ( proj ( i , n ) ) . LM *> .= <* ( proj ( i , n ) ) . LM *> ;
f1 + f2 * reproj ( i , x ) is_differentiable_in ( ( reproj ( i , x ) ) * reproj ( i , x ) ) . x0 & f2 * reproj ( i , x ) is_differentiable_in ( ( reproj ( i , x ) ) * reproj ( i , x ) ) . x0 ;
pred ( ( - 1 ) (#) ( tan * tan ) ) . x <> 0 & ( tan * tan ) . x = tan . x ;
ex t being SortSymbol of S st t = s & h1 . t = h2 . x & t in ( the Sorts of A ) . x ;
defpred C [ Nat ] means P8 . $1 is non empty & A8 is as n -Subset of NAT & A8 is n -V & A8 is non empty ;
consider y being element such that y in dom ( p | i ) and ( q | i ) . y = ( p | i ) . y ;
reconsider L = product ( { x1 } +* ( index B , l ) ) as Subset of ( \bf T ) ;
for c being Element of C ex d being Element of D st T . ( id c ) = id d & for c being Element of C st c in dom T holds T . ( id c ) = id d
not ( for n , p being FinSequence st n in dom f holds f . n = ( f | n ) ^ <* p *> ) . n .= f . n ;
( f * g ) . x = f . ( g . x ) & ( f * h ) . x = f . ( h . x ) ;
p in { 1 / 2 * ( G * ( i + 1 , j ) + G * ( i + 1 , j + 1 ) ) } ;
f `2 - cp = ( f | ( n , L ) ) *' ( - ( f | ( n , L ) ) ) .= ( f - ( f | ( n , L ) ) ) *' ( - ( f | ( n , L ) ) ) ;
consider r being Real such that r in rng ( f | divset ( D , j ) ) and r < m + s and m < n + s ;
f1 . |[ ( r2 `1 ) ^2 , ( r2 `2 ) ^2 ]| in f1 .: W1 & f2 . |[ r2 `1 , ( r2 `2 ) ^2 ]| in f2 .: W2 ;
eval ( a | ( n , L ) , x ) = eval ( a | ( n , L ) ) .= a * eval ( b , x ) .= a * eval ( b , x ) ;
z = DigA ( tz , xx ) .= DigA ( tz , xx ) .= DigA ( tz , xx ) .= DigA ( tz , xx ) .= DigA ( tz , x ) ;
set H = { Intersect S where S is Subset-Family of X : S c= G & S c= G } , G = { Intersect S where S is Subset of X : S c= G } , F = { Intersect S where S is Subset of X : S c= G } ;
consider S19 being Element of D * , d being Element of D * such that S `1 = S19 ^ <* d *> and S19 in dom S19 and S19 in dom S19 and S19 in dom S29 ;
assume that x1 in dom f and x2 in dom f and f . x1 = f . x2 and f . x2 = f . x2 and f . x1 = f . x2 ;
- 1 <= ( q `2 / |. q .| - sn ) / ( 1 + sn ) & q `1 / |. q .| - sn <= ( q `2 / |. q .| - sn ) / ( 1 + sn ) ;
0. ( V ) is Linear_Combination of A & Sum ( L ) = 0. V implies Sum ( L ) = Sum ( L ) & Sum ( L ) = Sum ( L )
let k1 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 be Element of NAT , a be Int-Location , b be Int-Location , k1 , k2 , k2 be Int-Location , k2 be Int-Location , k2 be Element of NAT ;
consider j being element such that j in dom a and j in dom g and x = a " . { k } and x = g . j ;
H1 . x1 c= H1 . x2 or H1 . x2 c= H1 . x1 or H1 . x2 c= H1 . x2 or H1 . x1 c= H1 . x2 & H1 . x2 c= H1 . x1 & H1 . x2 c= H1 . x2 ;
consider a being Real such that p = L~ ( Let * p1 + ( a * p2 ) * p2 ) and 0 <= a and a <= 1 ;
assume that a <= c & c <= d & [' a , b '] c= dom f and [' a , b '] c= dom g and g = f | [' a , b '] ;
cell ( Gauge ( C , m ) , len Gauge ( C , m ) -' 1 , 0 ) is non empty & cell ( Gauge ( C , m ) , len Gauge ( C , m ) -' 1 , 0 ) is non empty ;
Aand in { ( S . i ) `1 where i is Element of NAT : i in dom S & i < n } ;
( T * b1 ) . y = L * b2 /. y .= ( F /. y ) * ( F /. y ) .= ( F /. y ) * ( F /. y ) ;
g . ( s , I ) . x = s . y & g . ( s , I ) . y = |. s . x - s . y .| ;
( log ( 2 , k + k ) ) ^2 >= ( log ( 2 , k + 1 ) ) ^2 + ( log ( 2 , k + 1 ) ) ^2 ;
then p => q in S & not x in the still of p & not x in S & p => All ( x , q ) in S ;
dom ( the InitS of r-10 ) misses dom ( the InitS of r-10 ) & dom ( the InitS of r-11 ) = dom ( the InitS of rM ) & dom ( the InitS of rM ) = the carrier of rM ;
synonym f is integer means : Def3 : for x being set st x in rng f holds x is an integer number & f . x = 0 ;
assume for a being Element of D holds f . { a } = a & for X being Subset-Family of D holds f . ( f .: X ) = f . union X ;
i = len p1 .= len p3 + len <* x *> .= len p3 + len <* x *> .= len p3 + 1 .= len p3 + 1 .= len p3 + 1 .= len p3 + 1 ;
( l ) `1 = ( g ) `1 ) ^2 + ( k ) ^2 .= ( g ) ^2 + ( k ) ^2 + ( e ) ^2 .= ( g ) ^2 + ( e ) ^2 + ( e ) ^2 ;
CurInstr ( P2 , Comput ( P2 , s2 , l ) ) = halt SCM+FSA .= CurInstr ( P2 , Comput ( P2 , s2 , l ) ) .= halt SCM+FSA ;
assume for n be Nat holds ||. seq . n .|| <= ( ||. seq .|| ) . n & ( ||. seq .|| ) . n <= ( ||. seq .|| ) . n ;
sin . ( M - i ) = sin r * cos ( ( - cos . ( r - i ) ) / ( 2 * PI ) ) .= 0 ;
set q = |[ g1 . t0 , g2 . t0 ]| , r = |[ s , t ]| , s = |[ s , t ]| , t = |[ s , t ]| , s = |[ s , t ]| , t = |[ s , t ]| , s = |[ s , t ]| , t = |[ s , t ]| , s = |[ s , t ]| , t = |[ s , t ]| , s = |[ s , t ]| , t ]| , s = |[ s , t ]| ,
consider G being sequence of S such that for n being Element of NAT holds G . n in implies G in WAAAAAS ( F . n ) ;
consider G such that F = G and ex G1 st G1 in SM & G = ( the carrier of G1 ) \/ the carrier of G2 & G1 is finite & G is finite ;
the root of [ x , s ] in ( the Sorts of Free ( C , X ) ) . s & the Sorts of C = ( the Sorts of Free ( C , X ) ) . s ;
Z c= dom ( exp_R * ( f + ( #Z 2 ) * f1 ) + ( exp_R * ( f + ( #Z 2 ) * f1 ) ) ) ;
for k being Element of NAT holds rV . k = ( upper_volume ( Im f , S-3 ) ) . k & rV . k = ( upper_volume ( Im f , S-3 ) ) . k ;
assume that - 1 < n and n `2 / |. q .| > 0 and ( q `2 / |. q .| - sn ) < 0 and q `2 < 0 ;
assume that f is continuous one-to-one and a < b and c < d and f . a = c and f . b = d and f . a = d ;
consider r being Element of NAT such that s\mathclose = Comput ( P1 , s1 , r ) and r <= q and q <= r and r <= s ;
LE f /. ( i + 1 ) , f /. j , L~ f & LSeg ( f , i ) /\ LSeg ( f , j ) c= { 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 and inf { x , y } in the carrier of K ;
assume f +* ( i1 , \xi /. 1 ) in ( proj ( F , i2 ) ) " ( A ) & f . ( i1 + 1 ) in ( proj ( F , i2 ) ) " ( A ) ;
rng ( ( ( ( ( ( ( Flow M ) ) ~ ) | ( the carrier of M ) ) \/ the carrier of M ) ) \/ the carrier of M ) ) c= the carrier' of M ;
assume z in { ( the carrier of G ) \/ { t } where t is Element of T : t in the carrier of T } ;
consider l be Nat such that for m be Nat st l <= m holds ||. s1 . m - x0 .|| < g / 2 and g < x0 + g ;
consider t be VECTOR of product G such that mt = ||. D5 . t .|| and ||. t .|| <= 1 and t <= 1 ;
assume that the carrier of v = 2 and v ^ <* 0 *> in dom p and v ^ <* 1 *> in dom p and p . 1 = <* 1 *> and p . 2 = <* 0 *> ;
consider a being Element of the Points of X29 , A being Element of the Points of X29 such that a on A and a on A and a on A ;
( - x ) |^ ( k + 1 ) * ( ( - x ) |^ ( k + 1 ) ) " = 1 ;
for D being set for i st i in dom p holds p . i in D & p . i in D & p . i in D & p . i in D
defpred R [ element ] means ex x , y st [ x , y ] = $1 & P [ x , y ] & P [ y , x ] & P [ x , y ] ;
L~ f2 = union { LSeg ( p0 , p2 ) , LSeg ( p1 , p11 ) } .= { LSeg ( p1 , p11 ) , LSeg ( p1 , p11 ) } .= { p1 , p2 } ;
i -' len h11 + 2 - 1 < i - len h11 + 2 - 1 + 1 & i - len h11 + 2 - 1 < i - len h11 + 2 - 1 + 1 ;
for n being Element of NAT st n in dom F holds F . n = |. ( nbeing . ( n -' 1 ) ) .| * |. ( F . n ) .| ;
for r , s1 , s2 , s3 being Real holds r in [. s1 , s2 .] iff s1 <= s2 & s2 <= 1 & s1 <= 1 & s2 <= 1 & s1 <= 1 & s2 <= 1 & s1 <= 1
assume v in { G where G is Subset of T2 : G in B2 & G c= z & z in A & G c= z } ;
let g be Z :] , X be INT -valued Element of INT , b be Element of INT , c be Element of INT , b be Element of INT ;
min ( g . [ x , y ] , k ) . [ y , z ] = ( min ( g , k , x ) ) . y ;
consider q1 being sequence of CV such that for n holds P [ n , q1 . n ] and q1 is convergent and for n holds q1 . n = U ( n ) ;
consider f being Function such that dom f = NAT & for n being Element of NAT holds f . n = F ( n ) and for n being Element of NAT holds f . n = F ( n ) ;
reconsider B-6 = B /\ O , OI = O , TI = O , TI = O , Z = { Z } , Z = { Z } , T = { Z } , T = { Z } , S = { Z , Z , Z } as Subset of [: B , C :] ;
consider j being Element of NAT such that x = the j j j and j <= n and 1 <= j and j <= n and n <= len f and f . j = f . j ;
consider x such that z = x and card ( x . O2 ) in card ( x . O1 ) and x in L1 and x in ( x . O2 ) and x in ( x . O2 ) ;
( C * : T . ( k , n2 ) ) . 0 = C . ( ( of of T4 ) . ( k , n2 ) ) .= C . ( ( T * T ) . ( k , n2 ) ) ;
dom ( X --> rng f ) = X & dom ( X --> f ) = dom ( X --> f ) & dom ( X --> f ) = dom ( X --> f ) ;
( N-bound L~ SpStSeq C ) `2 <= ( N-bound L~ SpStSeq C ) `2 & ( N-bound L~ SpStSeq C ) `2 <= ( N-bound L~ SpStSeq C ) `2 or ( S-bound L~ SpStSeq C ) `2 <= ( N-bound L~ SpStSeq C ) `2 ;
synonym x , y are_collinear means : Def3 : x = y or ex l being Nat st { x , y } c= l & x in l ;
consider X being element such that X in dom ( f | ( n + 1 ) ) and ( f | ( n + 1 ) ) . X = Y ;
assume that Im k is continuous and for x , y being Element of L for a , b being Element of Im k st a = x & b = y & x << y holds a << b & b << a ;
( 1 / 2 * ( ( ( ( ( ( ( ( ( ( 1 / 2 ) * ( ( ( 1 / 2 ) ) * ( ( ( 1 / 2 ) * ( ( 1 / 2 ) ) * ( ( ( 1 / 2 ) * ( ( 1 / 2 ) ) * ( 1 + ( 1 / 2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) `| REAL = f ;
defpred P [ Element of omega ] means ( the partial of A1 ) . $1 = A1 . $1 & ( the partial of A2 ) . $1 = A2 . $1 & ( the Sorts of A1 ) . $1 = A2 . $1 ;
IC Comput ( P , s , 2 ) = succ IC Comput ( P , s , 1 ) .= 6 + 1 .= ( card I + 1 ) .= 6 + 1 .= ( card I + 1 ) .= 6 + 1 ;
f . x = f . g1 * f . g2 .= f . g1 * 1_ H .= f . g1 * 1_ H .= f . g1 * f . g2 .= f . g1 * f . g2 .= f . g1 * f . g2 .= f . g1 * f . g2 .= f . g1 * f . g2 ;
( M * F-4 ) . n = M . ( F-4 . 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 L1 & the carrier of ( L1 + L2 ) \/ the carrier of L2 = the carrier of L1 ;
pred a , b , c , x , y , c , d , x , y , z , y , z , x , y , z , y , x , y , z , y , z , x , y , z , x , y , z , z , y , x , z , y , z , x , y , z , z , x , y , z , x , y , z , z , x , y , z , x , y , z , z , y , z , z , x
( the PartFunc of s ) . n <= ( the PartFunc of s ) . n * s . n & ( the Sorts of s ) . ( n + 1 ) = ( the Sorts of s ) . n * s . n ;
pred - 1 <= r & r <= 1 & ( arccot | [. - 1 , r .[ ) . r = - 1 & ( arccot | [. - 1 , r .[ ) . r = - 1 ;
s8 in { p ^ <* n *> where n is Nat : p ^ <* n *> in T1 & p ^ <* n *> in T1 } ;
|[ x1 , x2 , x3 ]| . 2 - |[ y1 , y2 ]| . 2 - |[ y1 , y2 ]| . 2 = |[ x2 , y2 ]| . 2 - |[ y2 , y2 ]| . 2 - |[ y1 , y2 ]| . 2 .= |[ y2 , y2 ]| . 2 - |[ y2 , y2 ]| . 2 - |[ y2 , y2 ]| . 2 - |[ y2 , y2 ]| . 2 ]| ;
attr for m being Nat holds F . m is nonnegative means : Def3 : ( Partial_Sums F ) . m is nonnegative & ( Partial_Sums F ) . m is nonnegative ) & ( Partial_Sums F ) . m is nonnegative ;
len ( Element of ( G . z ) ) = len ( ( G . x9 ) + ( G . y ) ) .= len ( G . x9 ) .= len ( G . x9 ) .= len ( G . x9 ) ;
consider u , v being VECTOR of V such that x = u + v and u in W1 /\ W2 and v in W2 and u in W3 and v in W3 and u in W3 ;
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 = 1 ;
0 = 1 * ( 1- @ ) * uon + 1 * ( - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - 0 ) ) ) ) ) * ( - ( 1 - ( 1 - 0 ) ) ) * ( 1 - ( 1 - 0 ) ) ) ;
consider n be Nat such that for m be Nat st n <= m holds |. ( f # x ) . m - ( lim ( f # x ) ) ) . n .| < e ;
cluster -> } -\mathbin Boolean for non empty _ of { ( let Y ) , ( Y ) , ( Y ) , ( Y ) , ( Y ) , ( Y ) , ( Y ) , ( Y ) , ( Y ) , ( Y ) } is Boolean
"/\" ( Bs , {} ( S , T ) ) = Top ( S , T ) .= "/\" ( [#] ( S , T ) , {} ( S , T ) ) .= "/\" ( Is , T ) .= "/\" ( Is , T ) ;
( r / 2 ) ^2 + ( r / 2 ) ^2 <= ( r / 2 ) ^2 + ( r / 2 ) ^2 + ( r / 2 ) ^2 ;
for x being element st x in A /\ dom ( f `| X ) holds ( f `| X ) . x >= r2 & ( f `| X ) . x >= r2
2 * r1 - 2 * |[ a , c ]| - ( 2 * r1 - ( 2 * r1 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - 1 ) ) ) ) ) ) ) / ( 2 * r1 ) ) ]| ) = 0. TOP-REAL
reconsider p = P * ( 1 , 1 ) , q = a " * ( ( - ( - ( K , n , 1 ) ) ) * ( ( - ( K , n , 1 ) ) * ( 1 , 1 ) ) ) as FinSequence of K ;
consider x1 , x2 being element such that x1 in uparrow s and x2 in < t and x = [ x1 , x2 ] and x = [ x1 , x2 ] and y = [ x1 , x2 ] ;
for n being Nat st 1 <= n & n <= len q1 holds q1 . n = ( upper_volume ( g , M7 ) ) . n & ( upper_volume ( g , M7 ) ) . n = ( upper_volume ( g , M7 ) ) . n
consider y , z being element such that y in the carrier of A & z in the carrier of A and i = [ y , z ] and i = [ y , z ] ;
given H1 , H2 being strict Subgroup of G such that x = H1 and y = H2 and H1 is Subgroup of H2 and H2 is Subgroup of H1 and H1 is Subgroup of H2 and H2 is Subgroup of H2 ;
for S , T being non empty RelStr , d being Function of T , S st T is complete holds d is directed-sups-preserving iff d is monotone & d is monotone
[ a + 0 , i + b2 ] in ( the carrier of F_Complex ) /\ ( the carrier of F_Complex ) .= [: the carrier of F_Complex , the carrier of F_Complex :] .= [: the carrier of F_Complex , the carrier of F_Complex :] .= [: the carrier of F_Complex , the carrier of F_Complex :] ;
reconsider mm = max ( len F1 , len ( p . n ) * ( <* x *> |^ n ) ) as Element of NAT ;
I <= width GoB ( ( GoB ( ( GoB ( ( GoB h ) ) * ( 1 , 1 ) ) `1 ) , ( GoB ( ( GoB ( ( GoB ( ( GoB ( h ) ) * ( 1 , 1 ) ) ) ) ) ) ) ) ) ) ;
f2 /* q = ( f2 /* ( f1 /* s ) ) ^\ k .= ( f2 * f1 ) /* s .= ( ( 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 st x in A1 holds x in A1 & x in A2 holds x in A2 ) & ( for x st x in A1 holds x in A1 ) implies x in A2 or x in A1 & x in A2 ;
func A -carrier of 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 , m ) ) ) * ( \square , 1 ) ) = dom ( F ^ <* G * ( i , 1 ) *> ) .= dom ( F ^ <* G * ( i , 1 ) *> ) .= dom ( F ^ <* G * ( i , 1 ) *> ) ;
cluster [ x `1 , 4 ] , [ x `2 , 4 ] , [ x `1 , 4 ] ] -> reduces x `1 , x `2 & x `1 = x `1 & x `2 = 4 or x `1 = x `2 & x `2 = 4 ;
E , f |= All ( x1 , All ( x2 , x2 , x3 , x4 ) ) => ( All ( x3 , x1 , x2 , x3 ) ) ;
F .: ( id X , g ) . x = F . ( id X , g . x ) .= F . ( id X , g . x ) .= F . ( x , g ) .= F . ( x , g ) ;
R . m = F . x0 + h . m - h . x0 + h . m - h . x0 .= ( F - G ) . x0 + ( F - G ) . x0 .= ( F - G ) . x0 ;
cell ( G , Xs -' 1 , ( Y + 1 ) + ( t + 1 ) ) \ L~ f meets ( UBD L~ f ) \/ ( L~ f ) ;
IC Result ( P2 , s2 ) = IC IExec ( I , P , Initialize s ) .= IC IExec ( I , P , Initialize s ) .= IC s .= IC IExec ( I , P , Initialize s ) .= IC s .= IC s .= IC s .= IC s .= IC s .= IC s .= IC s .= IC s ;
sqrt ( ( - ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ) ^2 ) > 0 ;
consider x0 being element such that x0 in dom a and x0 in dom g and x0 in g " { k } and y = a . x0 and x0 in { k } and g . x0 = a . x0 ;
dom ( r1 (#) chi ( A , A ) ) = dom chi ( A , A ) .= dom ( chi ( A , A ) ) .= dom ( r1 * chi ( A , A ) ) .= dom ( r1 * chi ( A , A ) ) .= dom ( r1 * chi ( A , A ) ) .= dom ( r1 * chi ( A , A ) ) .= dom ( r1 * chi ( A , A ) ) .= dom ( r1 * chi ( A , A ) ) ;
d-7 . [ y , z ] = ( ( y `1 ) ^2 - ( y `2 ) ^2 ) * ( ( y `2 ) ^2 - ( y `2 ) ^2 ) .= ( y `2 ) ^2 - ( y `2 ) ^2 ;
attr for i being Nat holds C . i = A . i /\ B . i & C . i = LSeg ( x , i ) /\ LSeg ( x , i ) ;
assume that x0 in dom f and f is_continuous_in x0 and f is_continuous_in x0 and for r st r in dom f holds ||. f /. r - f /. x0 .|| < r & f /. x0 - f /. x0 .|| < r ;
p in Cl A implies for K being Basis of p , Q being Subset of T st Q in K & K is open & Q is open holds A meets Q
for x being Element of REAL n st x in Line ( x1 , x2 ) holds |. y1 - y2 .| <= |. y1 - y2 .| & |. y2 - y2 .| <= |. y1 - y2 .|
func Sum <*> a -> w -\mathop { a } means : Def3 : a in it & for b being Ordinal st b in a holds it . b c= a & it . b = b ;
[ a1 , a2 , a3 ] in ( the carrier of A ) /\ ( the carrier of A ) & [ a1 , a2 , a3 ] in [: the carrier of A , the carrier of A :] ;
ex a , b being element st a in the carrier of S1 & b in the carrier of S2 & x = [ a , b ] & x = [ a , b ] ;
||. ( vseq . n ) - ( vseq . m ) .|| * ||. x .|| < ( e / ||. x .|| ) * ||. x .|| + ||. x .|| * ||. x .|| .= e * ||. x .|| + ||. x .|| .= e * ||. x .|| + e * ||. x .|| .= 0 ;
then for Z being set st Z in { Y where Y is Element of I7 : F c= Y & Y in Z & z in Z } holds z in x & z in Z ;
sup compactbelow [ s , t ] = [ sup ( { 1 } /\ compactbelow [ s , t ] ) , sup ( { 1 } /\ compactbelow [ s , t ] ) ] .= [ s , t ] .= t ;
consider i , j being Element of NAT such that i < j and [ y , f . j ] in IF and [ f . i , z ] in IF and [ f . i , f . j ] in IF and [ f . i , f . j ] in IF ;
for D being non empty set , p , q being FinSequence of D st p c= q holds ex p being FinSequence of D st p ^ q = q & len p = len q & p = q ^ p
consider e19 being Element of the affine of X such that c9 , a9 // a9 , e29 and a9 <> c9 & c9 <> a9 & c9 <> a9 & c9 <> a9 & c9 <> a9 & c9 <> a9 & c9 <> a9 & c9 <> a9 & c9 <> a9 & c9 <> a9 & c9 <> a9 & c9 <> a9 & c9 <> a9 & c9 <> a9 & c9 <> a9 & c9 <> a9 & c9 <> a9 & c9 <> a9 & c9 <> a9 & a9 , c9 // c9 & a9 , a9 // c9 & c9 , a9 // c9 & a9 , a9 // c9 & c9 ,
set U = I \! \mathop { \vert x .| : not contradiction } , * = { I : I in { 1 } } , * = { 1 } , * = { 1 } , * = { 1 } , * = { 1 } , * = { 2 } , * = { 2 } , * = { 2 } , * = { 2 } , * = { 2 } , * = { 2 } , * = { 1 } , * = { 2 } , * = { 2 } , * = { 2 } , * = { 2 } , * = { 2 } , * = { 2 } , * = {
|. q2 .| ^2 = ( q2 `1 ) ^2 + ( q2 `2 ) ^2 .= |. q2 .| ^2 + ( q2 `2 ) ^2 .= |. q2 .| ^2 + ( q2 `2 ) ^2 .= |. q2 .| ^2 + ( q2 `2 ) ^2 ;
for T being non empty TopSpace , x , y being Element of [: the topology of T , the topology of T :] holds x "\/" y = x \/ y & x "/\" y = x /\ y implies x "/\" y = x /\ y
dom signature U1 = dom the charact of U1 & Args ( o , MSAlg U1 ) = dom the charact of U2 & Args ( o , MSAlg U1 ) = dom the charact of U1 & Args ( o , MSAlg U1 ) = dom the charact of U2 & dom ( the charact of U1 ) = dom the charact of U1 & dom ( the charact of U1 ) = dom the charact of U2 ;
dom ( h | X ) = dom h /\ X .= dom ( ||. h .|| | X ) /\ X .= dom ( ||. h .|| | X ) .= dom ( ||. h .|| | X ) .= dom ( ||. h .|| | X ) .= dom ( ||. h .|| | X ) .= dom ( ||. h .|| | X ) .= X ;
for N1 , N2 being Element of [: G , G :] , K st dom ( h . K1 ) = N & rng ( h . K1 ) = N & rng ( h . K1 ) = N & rng ( h . K1 ) c= N & rng ( h . K1 ) c= N & ( h . N1 ) c= N & ( h . N1 ) c= N & ( h . N1 ) c= N
( mod ( u , m ) + mod ( v , m ) ) . i = ( mod ( u , m ) ) . i + ( mod ( v , m ) ) . i .= m . i + m . i ;
- ( q `1 ) ^2 < - 1 or q `2 / |. q .| * ( 1 + cn ) <= - ( q `1 / q `2 ) ^2 & - ( q `1 / q `2 ) ^2 <= - ( q `1 / q `2 ) ^2 ;
attr r1 = ff & r2 = ff & for x st x in dom f holds r1 * f . x = ff . x * ( f . x ) & ( for x st x in dom f holds f . x = f . x ) ;
vseq . m is bounded Function of X , the carrier of Y & x9 . m = ( vseq . m ) . x & ( vseq . m ) . x = ( vseq . m ) . x ;
pred a <> b & b <> c & angle ( a , b , c ) = PI & angle ( b , c , a ) = PI & angle ( c , a , b ) = PI ;
consider i , j being Nat , r , s being Real such that p1 = [ i , r ] and p2 = [ j , s ] and i < j and j < len s and s < n ;
|. p .| ^2 - ( 2 * |( p , q )| ) ^2 + |. q .| ^2 = |. p .| ^2 + |. q .| ^2 - ( |. q .| ^2 + |. q .| ^2 ) ;
consider p1 , q1 being Element of [: X , Y :] such that y = p1 ^ q1 and q1 in dom ( p ^ q ) and p1 ^ q1 = p ^ q1 and q1 ^ q2 = p ^ q2 and len p1 = len q1 + len q2 ;
Assume that } ( A , r1 , r2 , s1 , s2 , s1 , s2 , s2 , s3 ) = ( s2 , r2 , s2 , s3 ) * ( r1 , r2 , s2 , s1 ) and r2 = s2 and r2 = s2 and s1 = s2 and s2 = s1 and s1 = s2 and r2 = s2 ;
( ( UMP A ) `2 ) ^2 = lower_bound ( proj2 .: ( A /\ Vertical_Line ( w ) ) ) & ( proj2 .: ( A /\ Vertical_Line ( w ) ) ) `2 = ( proj2 .: ( A /\ Vertical_Line ( w ) ) ) `2 & ( proj2 .: ( A /\ Vertical_Line ( w ) ) ) `2 = ( proj2 .: ( A /\ Vertical_Line ( w ) ) ) `2 ;
s , k1 |= ( H1 , k1 ) |= H2 iff s , k1 |= ( H1 , k2 ) . ( len ( H1 , k1 ) ) & s , k1 |= ( H1 , k1 ) . ( len ( H1 , k1 ) ) ;
len s5 + 1 = card ( support b1 ) + 1 .= card ( support b2 ) + 1 .= card ( support b1 ) + 1 .= card ( support b2 ) + 1 .= card ( support b1 ) + 1 .= card ( support b1 ) + 1 .= card ( support b1 ) + 1 .= card ( support b1 ) + 1 .= card ( support b1 ) + 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 `1 >= y holds z `1 >= y & z `2 >= x ;
LSeg ( UMP D , |[ ( W-bound D ) / 2 , ( ( E-bound D ) / 2 ) / 2 ) ]| , ( ( ( E-bound D ) / 2 ) / 2 ) / 2 ) /\ D = { UMP D } ;
lim ( ( ( f `| N ) / g ) `| N ) = ( ( f `| N ) / g ) `| N .= ( ( f `| N ) / g ) `| N .= ( ( f `| N ) / g ) `| N .= ( ( f `| N ) / g ) `| N .= ( ( f `| N ) / g ) `| N ;
P [ i , pr1 ( f ) . i , pr1 ( f ) . ( i + 1 ) ] & pr1 ( f ) . ( i + 1 ) = pr1 ( f ) . ( i + 1 ) & pr1 ( f ) . ( i + 1 ) = pr1 ( f ) . ( i + 1 ) ;
for r be Real st 0 < r ex m be Nat st for k be Nat st m <= k holds ||. ( seq . k ) - ( R /. ( k + 1 ) ) .|| < r
for X being set , P being a_partition of X , x , a , b being set st x in a & a in P & b in P & x in P & b in P holds a = b
Z c= dom ( ( ( - 1 ) (#) ( ( #Z 2 ) * f ) ) ^ ) \ ( ( #Z 2 ) * f ) " { 0 } ) .= dom ( ( - 1 ) (#) ( ( #Z 2 ) * f ) ) ;
ex j being Nat st j in dom ( l ^ <* x *> ) & j < i & i = 1 & y = ( l ^ <* x *> ) . j & z = ( l ^ <* x *> ) . j & z = ( l ^ <* x *> ) . j ;
for u , v being VECTOR of V for r being Real st 0 < r & u in N & v in N holds r * u + ( 1-r * v ) in c= c= c= c= c= ( - 1 ) * N
A , Int Cl A , Cl Cl Int Cl A , Cl Cl Int Cl A , Cl Cl Cl Int Cl A , Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl
- Sum <* v , u , w *> = - ( v + u + w ) .= - ( v + u ) + u .= - ( v + u ) + u .= - ( v + u ) + u .= - ( v + u ) ;
( Exec ( a := b , s ) ) . IC SCM = ( Exec ( a := b , s ) ) . IC SCM .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= IC s .= IC s .= IC s .= IC s .= IC s .= IC s .= IC s .= IC s .= IC s .= IC s .= 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 and h . x in ( the carrier of J ) . x ;
for S1 , S2 being non empty reflexive RelStr , D being non empty directed Subset of [: S1 , S2 :] , f being Function of [: S1 , S2 :] , [: S2 , S2 :] , [: S2 , S2 :] st f is directed & f . 0 = sup D holds ( f * f ) . 1 is directed & ( f * f ) . 2 = ( f * g ) . 2
card X = 2 implies ex x , y st x in X & y in X & x <> y & x <> y or x = y & x = y or x = y & x = z or x = y or x = z or x = x or x = y or x = z or x = z or x = y or x = z or x = z or x = x or x = y ;
E-max L~ Cage ( C , n ) in rng ( Cage ( C , n ) \circlearrowleft W-min L~ Cage ( C , n ) ) & E-max L~ Cage ( C , n ) in rng ( Cage ( C , n ) \circlearrowleft W-min L~ Cage ( C , n ) ) ;
for T , T being DecoratedTree , p , q being Element of dom T st p in dom T holds ( T -with tree ( p , q ) ) . q = T . q & ( T -with tree ( p , q ) ) . q = T . q
[ i2 + 1 , j2 ] in Indices G & [ i2 , j2 ] in Indices G & f /. k = G * ( i2 + 1 , j2 ) & f /. k = G * ( i2 + 1 , j2 ) & f /. k = G * ( i2 + 1 , j2 ) & f /. k = G * ( i2 + 1 , j2 ) ;
cluster ( k -' n ) divides ( k -' n ) & ( k -' n ) divides ( k -' n ) & ( k -' n ) divides ( k -' n ) & ( k -' n ) divides ( k -' n ) implies k divides ( k -' n ) & ( k -' n ) divides ( k -' n ) ) ;
dom F " = the carrier of X2 & rng F = the carrier of X1 & rng F = the carrier of X2 & F " = F " * ( F " * ( F " * ( F * ( the carrier of X1 ) ) ) ) & F " * ( F * ( the carrier of X2 ) ) = F " * ( F * ( the carrier of X1 ) ) ;
consider C being finite Subset of V such that C c= A and card C = n and the carrier of V = Lin ( BM \/ C ) and C is linearly-independent and C is linearly-independent and C is linearly-independent and C is linearly-independent ;
V is prime implies for X , Y being Element of [: the topology of T , the topology of T :] st X /\ Y c= V holds X c= Y or Y c= V or X c= Y
set X = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] & v1 in A ( ) & P [ v1 ] } ;
angle ( p1 , p3 , p4 ) = 0 .= angle ( p2 , p3 , p4 ) .= angle ( p3 , p4 ) .= angle ( p3 , p4 ) .= angle ( p3 , p4 ) .= angle ( p3 , p4 ) .= angle ( p3 , p4 ) ;
- sqrt ( ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ) ^2 = - ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 .= - ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ) ^2 .= - ( - ( q `1 / |. q .| - cn ) ) ^2 ;
ex f being Function of I[01] , TOP-REAL 2 st f is continuous one-to-one & rng f = P & f . 0 = p1 & f . 1 = p2 & f . 0 = p3 & f . 1 = p4 & f . 1 = p4 & f . 0 = p4 & f . 1 = p4 & f . 1 = p4 ;
attr f is partial u , u0 means : Def3 : SVF1 ( 2 , pdiff1 ( f , 1 ) , u0 ) is continuous & SVF1 ( 2 , pdiff1 ( f , 3 ) , u0 ) . u0 = ( proj ( 2 , 3 ) ) . u0 ;
ex r , s st x = |[ r , s ]| & G * ( len G , 1 ) `1 < r & r < G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `2 < s & s < G * ( 1 , 1 ) `2 ;
assume that f is sequence which and 1 <= t and t <= len G and G * ( t , width G ) `2 >= s and s <= width G and G * ( t , width G ) `2 >= s and G * ( t , width G ) `2 >= s and s <= G * ( t , width G ) `2 ;
pred i in dom G means : Def3 : r * ( f * reproj ( G , i ) ) = r * f * reproj ( G , i ) & f . i = r * ( reproj ( G , i ) ) ;
consider c1 , c2 being bag of o1 + o2 such that ( decomp c ) /. k = <* c1 , c2 *> and c = c1 + c2 and c1 in dom ( <* c1 , c2 *> ) and c2 in dom ( <* c1 , c2 *> ) and c2 in dom <* c1 , c2 *> and c1 in dom <* c2 , c2 *> ;
u0 in { |[ r1 , s1 ]| : r1 < G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `2 < s1 & s1 < G * ( 1 , 1 ) `2 } ;
Cl ( X ^ Y ) . k = the carrier of X . k2 .= C4 . k .= C4 . k .= C4 . k .= C4 . k .= C4 . k .= C4 . k .= C4 . k ;
attr M1 = len M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 ;
consider g2 be Real such that 0 < g2 and { y where y is Point of S : ||. y - x0 .|| < g2 & y in dom ( ||. y .|| * ( ||. y .|| ) ) & ||. y - x0 .|| < g2 & g2 in N } c= dom ( ||. y .|| * ( ||. y .|| ) ) ;
assume x < ( - b + sqrt ( integral ( a , b , c ) ) / ( 2 * a ) ) or x > - b & x > 0 ;
( G1 '&' G2 ) . i = ( <* 3 *> ^ H1 ) . i & ( H1 '&' H2 ) . i = ( <* 3 *> ^ H1 ) . i & ( H1 '&' H2 ) . i = ( <* 3 *> ^ H1 ) . i & ( H1 '&' H2 ) . i = ( <* 3 *> ^ H1 ) . i ;
for i , j st [ i , j ] in Indices M3 holds ( M3 + M1 ) * ( i , j ) < M2 * ( i , j ) & M2 * ( i , j ) < M2 * ( i , j )
for f being FinSequence of NAT , i being Element of NAT st i in dom f & i in dom f & i <> j holds i divides f /. i & ( for j being Element of NAT st j in dom f & j < i holds f /. j = Sum ( f /. j ) ) implies f /. i = Sum ( f /. j )
assume F = { [ a , b ] where a , b is Subset of X : for c being set st c in B\mathopen the carrier of Y & c c= b holds a c= c } ;
b2 * q2 + ( b3 * q3 ) + - ( ( a * q2 ) * q2 ) + ( ( a * q2 ) * q2 ) = 0. TOP-REAL n + ( ( a * q2 ) * q2 ) .= 0. TOP-REAL n + ( ( a * q2 ) * q2 ) .= 0. TOP-REAL n + ( a * q2 ) * q2 .= 0. TOP-REAL n + ( a * q2 ) * q2 .= 0. TOP-REAL n ;
Cl ( Cl F ) = { D where D is Subset of T : ex B being Subset of T st D = Cl B & B in F & Cl ( Cl ( Cl ( Cl F ) ) ` ) = Cl Cl B & Cl ( Cl ( Cl ( Cl ( Cl F ) ) ` ) ) = Cl Cl Cl Cl B & Cl ( Cl ( Cl ( Cl ( Cl ( Cl F ) ) ` ) ) = Cl Cl Cl Cl B } ;
attr seq is summable means : Def3 : seq is summable & seq is summable & Sum ( seq ) = Sum ( seq ) & Sum ( seq ) = Sum ( seq ) & Sum ( seq ) = Sum ( seq ) & Sum ( seq ) = Sum ( seq ) & Sum ( seq ) = Sum ( seq ) & Sum ( seq ) = Sum ( seq ) ;
dom ( ( cn " ) | 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 , Z :] is full full non empty SubRelStr of [: Omega , Z :] & [ X , Y ] is full full SubRelStr of [: Omega , Z :] & [: X , Y :] is full SubRelStr of [: X , Y :] ;
G * ( 1 , j ) `2 = G * ( i , j ) `2 & G * ( 1 , j ) `2 <= G * ( 1 , j ) `2 & G * ( 1 , j ) `2 <= G * ( 1 , j + 1 ) `2 ;
synonym m1 c= m2 means : Def3 : for p being set st p in P holds the non empty set of ( m2 ) <= ( m2 ) & ( for p being set st p in P holds p is__ X ) & p is__ X implies p is__ X & p is__ X ;
consider a being Element of B ( ) such that x = F ( a ) and a in { G ( b ) where b is Element of A ( ) : P [ b ] } and a in A ( ) & P [ b ] ;
attr IT is $1 -loop Str means : Def3 : the carrier of IT = { [ a , the carrier of IT ] , [ a , the carrier of IT ] } , [ the carrier of IT ] } , the carrier of IT = [: the carrier of IT , the carrier of IT :] , the carrier of IT = [: the carrier of IT , the carrier of IT :] ;
sequence ( a , b , 1 ) + sequence ( c , d ) = b + sequence ( c , d ) .= b + d .= b + ( c + d ) .= b + ( c + d ) .= b + ( c + d ) .= b + ( c + d ) .= b + ( c + d ) .= b + ( c + d ) ;
cluster + _ { \mathbb Z } -> $ for Element of INT , i1 , i2 , j1 , j2 be Element of INT , i1 , i2 , j1 be Element of NAT st i1 = i2 & j1 = i2 holds ( i1 + i2 ) + ( j1 + 1 ) = ( i1 + i2 ) + ( j1 + 1 ) .= ( i1 + i2 ) + ( j1 + 1 ) ;
( - s2 ) * p1 + ( s2 * p2 ) - ( s2 * p2 ) * p2 = ( ( - r2 ) * p1 + ( s2 * p2 ) ) * p2 - ( ( - s2 ) * p2 ) * p2 .= ( ( - r2 ) * p1 ) + ( ( - r2 ) * p2 ) * p2 ;
eval ( ( a | ( n , L ) ) *' p , x ) = eval ( a | ( n , L ) ) * eval ( p , x ) .= a * eval ( p , x ) * eval ( p , x ) .= a * eval ( p , x ) * eval ( q , x ) .= a * eval ( p , x ) * eval ( q , x ) ;
assume that the TopStruct of S = the TopStruct of T and for D being non empty directed Subset of Omega S , V being Subset of Omega S st V in the topology of S & V is open & V is open holds for V being Subset of S st V in the topology of T & V is open holds V meets V ;
assume that 1 <= k & k <= len w + 1 and T-7 . ( ( q11 , w ) -/. k ) = ( T-7 . ( q11 , w ) ) . k and T21 . ( ( q11 , w ) -/. k ) = ( T-7 . ( q11 , w ) ) . k ;
2 * a |^ ( n + 1 ) + ( 2 * b |^ ( n + 1 ) ) >= a |^ ( n + 1 ) + ( b |^ ( n + 1 ) ) + ( b |^ ( n + 1 ) ) + ( b |^ ( n + 1 ) ) ;
M , v2 / ( x. 3 , m ) / ( x. 0 , ( x. 4 , ( x. 0 ) / ( x. 4 , ( x. 0 ) / ( x. 4 , ( x. 0 ) / ( x. 0 , x. 0 ) ) ) ) ) |= H ;
assume that f is_differentiable_on l and for x0 st x0 in l holds 0 < f . x0 and for x st x in l holds f . x < f . x0 and for x st x in l holds f . x < f . x ;
for G1 being _Graph , W being Walk of G1 , e being set , G2 being Walk of G1 , e being set st e in W holds not e in W implies ( W is Walk of G2 & e in W & W is Walk of G2 )
not c9 is empty iff ( not ( ex y1 , y2 being Element of REAL st y1 is not empty & y2 is not empty ) & not ( not y2 is not empty & not y2 is not empty ) & not ( not y2 is not empty ) & not ( not y1 is not empty & not y2 is not empty ) & not ( y2 is not empty ) & not ( y2 is not empty ) ;
Indices GoB f = [: dom GoB f , Seg width GoB f :] & ( for i st i in dom GoB f holds ( GoB f ) * ( i , 1 ) ) `1 = ( GoB f ) * ( i , 1 ) `1 & ( GoB f ) * ( i + 1 , 1 ) `2 = ( GoB f ) * ( i + 1 , 1 ) `2
for G1 , G2 , G3 being finite Group , G being stable Subgroup of O st G1 is stable & G2 is stable & G2 is stable & G1 is stable & G2 is stable & G2 is stable holds G1 * G is stable
UsedIntLoc ( int ) = { intloc 0 , intloc 1 , intloc 2 , intloc 3 , intloc 4 , intloc 5 , intloc 5 , x. 5 , x. 6 , 7 , 8 , 8 , 9 } .= { intloc 0 , intloc 5 , 6 , 7 , 8 } \/ { intloc 5 , 6 , 7 , 8 } .= UsedIntLoc ( i ) \/ { intloc 5 , intloc 6 , 8 } ;
for f1 , f2 being FinSequence of F st f1 ^ f2 is p -element & Q [ f1 ^ f2 ] & ( for i st i in dom f1 holds Q [ i , f2 . i ] implies f1 . i = f2 . ( f1 . i ) ) & ( for i st i in dom f1 holds Q [ i , f2 . i ] implies f1 . i = f2 . ( f1 . i ) )
( p `1 ) ^2 / ( sqrt ( 1 + ( p `2 / p `1 ) ^2 ) ) ^2 = ( q `1 ) ^2 / ( sqrt ( 1 + ( q `2 / q `1 ) ^2 ) ) ^2 .= ( q `1 ) ^2 / ( sqrt ( 1 + ( q `2 / q `1 ) ^2 ) ) ^2 ;
for x1 , x2 , x3 , x4 being Element of REAL n holds |( x1 - x2 , x3 )| = |( x1 , x3 )| + |( x2 , x3 )| + |( x1 , x3 )| + |( x2 , x3 )| + |( x2 , x3 )| + |( x1 , x3 )| + |( x2 , x3 )| + |( x2 , x3 )| + |( x2 , x3 )| + |( x2 , x3 )| + |( x2 , x3 )|
for x st x in dom ( ( ( ( ( ( ( ( ( x | A ) | A ) ) | A ) ) | A ) ) ) | A ) holds ( ( ( ( ( ( ( F | A ) | A ) ) | A ) | A ) ) | A ) . x = - ( ( ( ( ( F | A ) | A ) | A ) | A ) | 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 & P = B holds P is Basis of T
( a 'or' b 'imp' c ) . x = 'not' ( ( a 'or' b ) . x ) 'or' c . x .= ( 'not' ( a . x ) 'or' b . x ) 'or' c . x .= ( 'not' ( a . x ) 'or' b . x ) 'or' c . x .= ( ( 'not' a ) 'or' b . x ) 'or' c . x .= TRUE ;
for e being set st e in [: A , Y :] ex X1 being Subset of Y , 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 & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open
for i being set st i in the carrier of S for f being Function of [: S . i , S1 . i :] , S1 . i st f = H . i & f is Function of [: S . i , S1 . i :] , S1 . i holds F . i = f | [: S . i , S1 . i :]
for v , w st for y st x <> y holds w . y = v . y holds Valid ( VERUM ( Al , J ) , J ) . v = Valid ( VERUM ( Al , J ) , J ) . w
card D = card D1 + card D2 - 1 .= ( c1 + 1 ) - ( c2 + 1 ) .= c1 + ( c2 + 1 ) - ( c1 + 1 ) .= c1 + ( c2 + 1 ) - ( c1 + 1 ) .= 2 * c1 + ( c1 + 1 ) - ( c1 + 1 ) .= 2 * c1 + ( c1 + 1 ) - ( c1 + 1 ) .= 2 * c1 + ( c1 + 1 ) - ( c1 + 1 ) ;
IC Exec ( i , s ) = ( s +* ( 0 .--> succ ( s . 0 ) ) ) . 0 .= ( 0 .--> succ ( s . 0 ) ) . 0 .= succ IC s .= IC Exec ( i .--> succ ( s . 0 ) ) .= IC Exec ( i , s ) .= IC Exec ( i , s ) .= IC Exec ( i , s ) .= IC Exec ( i , s ) ;
len f /. ( \downharpoonright i1 -' 1 ) -' 1 + 1 = len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) + 1 ;
for a , b , c being Element of NAT st 1 <= a & b <= 2 & c <= a holds a <= b or a <= b & b <= a + b-2 or a = b + b-2 or b = a + b or a = a + b + c
for f being FinSequence of TOP-REAL 2 , p being Point of TOP-REAL 2 , i being Element of NAT st i in LSeg ( f , i ) & p in LSeg ( f , i ) holds Index ( p , f ) <= i & Index ( p , f ) <= len f
lim ( curry ( P-19 , k + 1 ) # x ) = lim ( curry ( P-19 , k ) # x ) + lim ( curry ( P-19 , k ) # x ) .= lim ( curry ( F-19 , k ) # x ) + lim ( curry ( F-19 , k ) # x ) .= lim ( curry ( F-19 , k ) # x ) ;
z2 = g /. ( \downharpoonright n1 -' 1 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) ;
[ f . 0 , f . 3 ] in id ( the carrier of G ) \/ ( the InternalRel of G ) or [ f . 0 , f . 2 ] in the InternalRel of C6 & [ f . 0 , f . 2 ] in the InternalRel of C6 & [ f . 2 , f . 3 ] in the InternalRel of C6 ;
for G being Subset-Family of B st G = { R [ X ] where X is Subset of A , Y is Subset of B : R in F & Y in F } holds ( Intersect ( F , G ) ) . X = Intersect ( G , F )
CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 ) ) = CurInstr ( P1 , Comput ( P1 , s1 , m1 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m1 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m1 ) ) .= halt SCMPDS .= halt SCMPDS ;
assume that a on M and b on M and c on N and d on N and p on M and a on N and p on N and a on M and b on N and p on N and p on N and a on M and b on N and p on N and p on N and a on M and b on N and p on N and p on N and a on M and b on N and a on N and b on N and a on M and b on N and b on N and b on N and b on N and b on N and b on N and b on N and b on N and b on N and b on N and b on N and b on N and b on N and b on N and b on N and b on N and b on N and c on M and c on M and c on M and c on M and c on M and c on M
assume that T is \hbox { T _ 4 } and F is closed and ex F being Subset-Family of T st F is closed & ( for n being Nat st n <= len F holds F . n is finite-ind & ( for n being Nat st n <= len F holds F . n is finite-ind ) & ( for n st n <= len F holds F . n <= 0 ) ) ;
for g1 , g2 st g1 in ]. r - s , r .[ & g2 in ]. r - s .[ & g1 < g2 & g2 in ]. r - s , r + s .[ holds |. f . g1 - f . g2 .| <= ( f - g ) / ( r - s )
( ( - ( - 1 ) ) * ( z1 + z2 ) ) * ( z2 + z2 ) = ( - ( - ( - ( - 1 ) ) * ( z2 + z2 ) ) ) * ( z2 + z2 ) .= ( - ( - ( - ( - ( - ( - 1 ) ) * ( z2 + z2 ) ) ) * ( z2 + z2 ) ) ) * ( z2 + z2 ) ;
F . i = F /. i .= 0. R + r2 .= ( b |^ n + 1 ) * ( a |^ n ) .= <* ( ( n + 1 ) -' a ) * ( b |^ n ) * ( a |^ n ) *> .= <* ( ( n + 1 ) -' a ) * ( b |^ n ) ) * a * b |^ n *> .= <* ( n + 1 ) * a |^ n * b |^ n *> .= <* a * b * b * b * a * b * b * a * b * a * b * a * b * b * b * b * a * b * a * b * a * b * b * b * a * b * a * b * a * a * b * a * a * a * a * b * b * b * b * b * b
ex y being set , f being Function st y = f . n & dom f = NAT & f . 0 = A ( ) & for n holds f . ( n + 1 ) = RF ( n , f . n ) & for n holds f . ( n + 1 ) = RF ( n , f . n ) ;
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 /. ( F /. i ) & for i be Nat st i in dom it holds it . i = F /. i * F /. ( F /. i ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , 7 , 8 } = { x1 , x2 , x3 , x4 , x5 , 7 , 8 } \/ { x5 , 6 , 7 , 7 } .= { x1 , x2 , x3 , x4 } \/ { x4 , 7 , 8 } .= { x1 , x2 , x3 , x4 , 7 } \/ { x4 , 7 , 8 } ;
for n being Nat for x being set st x = h . n holds h . ( n + 1 ) = o ( x , n ) & x in InputVertices S ( x , n ) & o . ( x , n ) in InnerVertices S ( x , n ) & o . ( x , n ) in InnerVertices S ( x , n )
ex S1 being Element of CQC-WFF ( Al ) st SubP ( P , l , e ) = S1 & ( S `1 = P `1 ) & ( S `1 = P `1 ) & ( S `2 = P `2 ) & ( S `2 = e & S `1 = P `2 ) & ( S `1 = P `1 ) & ( S `2 = P `2 ) & ( S `1 = P `1 ) & ( S `2 = P `2 ) & ( S `1 = P `1 ) & ( S `2 = P `2 = P `2 ) & ( S `2 = P `2 = P `2 = P `2 ) & ( S `2 = P `2 = P `2 = P `2 ) & ( S `2 = P `2 = P `2 ) & ( S `2 = P `2 = P `2 ) & ( S `2 = P `2 ) & ( S `2 = P `2
consider P being FinSequence of GL2 such that p7 = product P and for i st i in dom P ex t7 being Element of the carrier of K st P . i = t7 & t7 = t & t7 = ex t7 being FinSequence of the carrier of K st t = t7 & t7 = t & t7 = t & t7 = t & t7 = t & t7 = t ;
for T1 , T2 being strict non empty TopSpace , P being Basis of T1 , T2 being Basis of T2 st the carrier of T1 = the carrier of T2 & P = the carrier of T2 & P = the topology of T2 & P = the topology of T2 holds P is Basis of T1 & P is Basis of T2 & P is Basis of T2
assume that f is_is_is_(#) , u0 and r (#) pdiff1 ( f , 3 ) is_partial_differentiable_in u0 , 2 and partdiff ( r (#) pdiff1 ( f , 3 ) , u0 , 3 ) = r (#) pdiff1 ( f , 3 ) and partdiff ( r (#) pdiff1 ( f , 3 ) , u0 , 3 ) = r (#) pdiff1 ( f , 3 ) and partdiff ( r (#) pdiff1 ( f , 3 ) , u0 , 3 ) = r (#) pdiff1 ( f , 3 ) ;
defpred P [ Nat ] means for F , G being FinSequence of bool ( the carrier of K ) , G be Permutation of the carrier of K st len G = $1 & G = F * s & G = F * s holds Sum ( F ) = Sum ( G ) & Sum ( G ) = Sum ( G ) & Sum ( G ) = Sum ( G ) ;
ex j st 1 <= j & j < width GoB f & ( GoB f ) * ( 1 , j ) `2 < s & s < ( GoB f ) * ( 1 , j + 1 ) `2 & ( GoB f ) * ( 1 , j + 1 ) `2 < s ) `2 ;
defpred U [ set , set ] means ex Fn1 be Subset-Family of T st $1 = Fn1 & $2 is open & union Fn1 is open & union Fn1 is open & union Fn1 is empty & union Fn1 is empty & union Fn1 c= $1 & union Fn1 c= $1 & $2 in union Fn1 & union Fn1 c= $1 & union Fn1 c= $2 & union Fn1 c= $2 ;
for p4 being Point of TOP-REAL 2 st LE p4 , p , P & LE p4 , p , P & LE p4 , p , P & LE p4 , p , P holds LE p4 , p , P & LE p , p , P & LE p , p , P & LE p , p , P & LE p , p , P & LE p , p , P & LE p , p , P
f in the carrier of ( E , H ) & for g st g . y <> f . y holds x in the carrier of ( E , H ) & g in the carrier of ( E , H ) & f in the carrier of ( E , H ) & g in the carrier of ( E , H ) & f . x = f . x implies f . x = g . x
ex 8 being Point of TOP-REAL 2 st x = 8 & ( ( ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) * ( 1 + sn ) ) / ( 1 + sn ) <= cn & ( ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) * ( 1 + sn ) <= 0 ) & ( ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) / ( 1 + sn ) <= 0 ;
assume for d7 being Element of NAT st d7 <= d7 holds ( for i being Element of NAT st i <= 8 holds d7 . i = s2 . ( d7 ) ) & ( for i being Element of NAT st i in 8 holds s1 . i = s2 . ( d7 ) ) & ( for i being Element of NAT st i in 8 holds s1 . i = s2 . i ) ;
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 Sphere ( x , r ) st e = Ball ( x , r ) /\ Ball ( y , r ) & e in Ball ( x , r ) /\ Ball ( y , r ) & e in Ball ( x , r ) /\ Ball ( y , r ) ;
given r such that 0 < r and for s st 0 < s holds 0 < s or ex x1 be Point of CNS st x1 in dom f & ||. x1 - x0 .|| < s & |. x1 - x0 .| < s & |. x1 - x0 .| < s & |. x1 - x0 .| < s & |. x1 - x0 .| < s & |. x1 - x0 .| < s ;
( p | x ) | ( ( x | x ) | ( x | x ) ) = ( ( ( x | x ) | ( x | x ) ) | ( x | x ) ) | ( ( x | x ) | ( x | x ) ) .= ( ( x | x ) | ( x | x ) ) | ( x | x ) ;
assume that x , x + h in dom sec and x in dom sec and ( Let x , h ) . x = ( 4 * sin . x + 2 * cos . x ) / ( sin . x + 2 * cos . x ) / ( sin . x + 2 * cos . x ) ^2 and sin . x = ( 4 * sin . x + 2 * cos . x ) / ( cos . x + 2 * cos . x ) ^2 ;
assume that i in dom A and len A > 1 and len B > 1 and B c= the set of ( the set of K ) and A = ( the { of A , the carrier of K ) --> ( the carrier of K ) and B = ( the -F of A ) * ( i , j ) and B = ( the -F of A ) * ( i , j ) ;
for i being non zero Element of NAT st i in Seg n holds i divides n or i = n or i = 1 or i <> n & i <> n & i <> n & i <> n & j <> n & j <> n & j <> n implies h . i = <* 1. F_Complex , 1. F_Complex *> or h . j = 1. F_Complex \ <* 1. F_Complex *> & h . i = 1. F_Complex \ <* 1. F_Complex *>
( ( b1 'imp' b2 ) '&' ( c1 'imp' c2 ) ) '&' ( ( a1 'or' b1 ) '&' ( a2 'or' b2 ) ) '&' ( ( a1 'or' b1 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) ) '&' ( ( a1 'or' b1 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) ) '&' ( ( a1 'or' b1 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a1 'or' b2 ) '&' ( a1 'or' b2 ) '&' ( a1 'or' b2 ) '&' ( a2 'or' b2 ) ) '&' ( a1 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a1 'or' b2 ) '&' ( a1 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a1 'or' b2 ) '&' ( a1 'or' b2 ) '&' ( a1 'or' b2 ) '&' ( a1 'or' b2 ) '&' ( a1 'or'
assume that for x holds f . x = ( ( ( - cot * ( sin + cos ) ) * ( sin + cos ) ) `| Z ) . x and x in dom ( ( - cot * ( sin + cos ) ) * ( sin + cos ) ) and for x st x in Z holds ( ( ( - cot * ( sin + cos ) ) * ( sin + cos ) ) `| Z ) . x = - cos . x ;
consider R8 , I8 be Real such that R8 = Integral ( M , Re ( F . n ) ) and I8 = Integral ( M , Re ( F . n ) ) and I8 = Integral ( M , Im 8 ) and I8 = Integral ( M , Im ( F . n ) ) and R8 = Integral ( M , Im ( F . n ) ) ;
ex k being Element of NAT st k = k & 0 < d & for q be Element of product G st q in X & ||. q-r - partdiff ( f , q , k ) .|| < r holds ||. partdiff ( f , x , k ) .|| < r ;
x in { x1 , x2 , x3 , x4 , x5 , x5 , x6 , x5 , x6 , 7 , 8 , 7 , 8 } iff x in { x1 , x2 , x3 , x4 , x5 , 6 , 7 } or x in { x1 , x2 , x3 , x4 , x5 } or x in { x1 , x2 , x3 , x4 } or x in { x1 , x2 , x3 , x4 } or x in { x4 , x5 , 6 }
G * ( j , ii ) `2 = G * ( 1 , ii ) `2 .= G * ( 1 , ii ) `2 .= G * ( 1 , ii ) `2 .= G * ( 1 , ii ) `2 .= G * ( 1 , ii ) `2 .= G * ( 1 , ii ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 + G * ( 1 , j2 ) `2 .= G * ( 1 , j2 + G * ( 1 , j2 ) `2 .= G
f1 * p = p .= ( ( the Arity of S1 ) +* ( the Arity of S2 ) ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o ;
func tree ( T , P , T1 , T2 ) -> DecoratedTree means : Def3 : q in it iff q in P & for p st p in P holds p in P or p = q or p = p or p in P & q in P & p in P & q in P & p <> q ;
F /. ( k + 1 ) = F . ( k + 1 -' 1 ) .= FA2 . ( p . ( k + 1 -' 1 ) , k + 1 -' 1 ) .= FA2 . ( p . ( k + 1 -' 1 ) , k -' 1 ) .= FA2 . ( p . ( k + 1 -' 1 ) , k -' 1 ) .= FA2 . ( p . k , p . ( k + 1 -' 1 ) ) .= FA2 . ( p . k ;
for A , B , C being Matrix of n , K st len B = len C & width B = width C & len B = width C & len B > 0 & len A > 0 & len B > 0 & len B > 0 & len A > 0 & len B > 0 & len B > 0 & len A > 0 & width B > 0 & len B > 0 & width B > 0 & len B > 0 & len A > 0 & width B = 0 & width B > 0 & width B > 0 & width B > 0 & width B > 0 & width B > 0 & width B > 0 & width B > 0 & width B > 0 & len B > 0 & len B > 0 & len B > 0 & len B > 0 & len B > 0 & len B > 0 & len A > 0 & len A > 0 & width B > 0 & width B > 0 & width B > 0
seq . ( k + 1 ) = 0. F_Complex + seq . ( k + 1 ) .= ( Partial_Sums seq ) . ( k + 1 ) .= Partial_Sums ( seq ) . k + ( Partial_Sums ( seq ) ) . ( k + 1 ) .= Partial_Sums ( seq ) . k + ( Partial_Sums ( seq ) ) . ( k + 1 ) .= Partial_Sums ( seq ) . k + ( Partial_Sums ( seq ) ) . ( k + 1 ) .= Partial_Sums ( seq ) . ( k + 1 ) ;
assume x in ( the carrier of Cy ) \/ ( the carrier of Cy ) & y in ( the carrier of Cy ) \/ ( the carrier of Cy ) & x <> y & y in the carrier of Cy ;
defpred P [ Element of NAT ] means for f st len f = $1 holds ( VAL g ) . ( f /. ( k + 1 ) ) = ( VAL g ) . ( f /. ( k + 1 ) ) '&' ( VAL g ) . ( f /. ( k + 1 ) ) & ( VAL g ) . ( k + 1 ) = ( VAL g ) . ( f /. k ) '&' ( VAL g ) . ( f /. k ) ;
assume that 1 <= k and k + 1 <= len f and f is FinSequence of TOP-REAL 2 and [ i + 1 , j ] in Indices G and f /. k = G * ( i , j ) and f /. k = G * ( i + 1 , j ) and f /. ( k + 1 ) = G * ( i , j ) ;
assume that sn < 1 and q `1 > 0 and ( q `2 / |. q .| - sn ) / ( 1 + sn ) >= sn and q `1 / |. q .| - sn and q `2 / |. q .| - sn < 0 and q `1 / |. q .| - sn < sn and q `1 / |. q .| - sn < 0 and q `2 / |. q .| < sn and q `2 / |. q .| < 0 and q `2 < 0 ;
for M being non empty TopSpace , x being Point of M , f being Function of M , M st x = x `1 holds ex f being sequence of M st f is sequence of M & for n being Element of NAT holds f . n = Ball ( x `1 , ( 1 / ( n + 1 ) ) * ( 2 * n ) )
defpred P [ Element of omega ] means f1 is_differentiable_on Z & f2 is_differentiable_on Z & for x st x in Z holds ( ( f1 - f2 ) `| Z ) . x = ( ( f1 - f2 ) `| Z ) . x - ( ( f1 - f2 ) `| Z ) . x / ( ( ( f1 - f2 ) `| Z ) . x ) ^2 ) ;
defpred P1 [ Nat , Point of CNS ] means ( $1 in Y & $2 in Y & $2 = ( $1 + 1 ) * ( f /. $1 ) & ( $1 + 1 ) * ( f /. $1 ) < ( f /. ( $1 + 1 ) ) * ( f /. ( $1 + 1 ) ) & ( f /. $1 ) * ( f /. ( $1 + 1 ) ) < ( f /. ( $1 + 1 ) ) * ( f /. ( $1 + 1 ) ) ;
( f ^ mid ( g , 2 , len g ) ) . i = ( mid ( g , 2 , len g ) ) . i .= g . ( i -' len f + 1 ) .= g . ( i -' len f + 2 ) .= g . ( i -' len f + 1 ) .= g . ( i -' len f + 2 ) .= g . ( i -' len f + 2 ) .= g . ( i -' len f + 2 ) ;
( 1 / 2 * n0 + 2 * ( n + 2 ) ) * ( 2 * n0 + 2 * ( n + 2 ) ) = ( 1 / 2 * ( n + 2 ) ) * ( 2 * ( n + 2 ) ) .= 1 / 2 * ( n + 2 ) * ( n + 2 ) * ( n + 2 ) .= 1 / 2 * ( n + 2 ) * ( n + 2 ) .= 1 / 2 * ( n + 2 ) * ( n + 2 ) * ( n + 2 ) * ( n + 2 ) * ( n + 2 ) * ( n + 2 * ( n + 2 ) * ( n + 2 ) * ( n + 2 ) * ( n + 2 ) * ( n + 2 ) * ( n + 2 * ( n + 2 ) * ( n + 2 ) .= ( n + 2 * ( n + 2 ) .= ( 1 / 2 * ( n + 2 * ( n + 2 ) * ( n + 2 ) .= 1 / 2
defpred P [ Nat ] means for G being non empty finite strict finite symmetric RelStr st G is \rm A1 , , symmetric symmetric RelStr for x being set st x in the carrier of G & x in the carrier of G holds the InternalRel of G = ( the InternalRel of G ) \/ ( the InternalRel of G ) & the InternalRel of G = ( the InternalRel of G ) \/ ( the InternalRel of G ) ;
assume that f /. 1 in Ball ( u , r ) and 1 <= m and m <= len ( - 1 ) and for i st 1 <= i & i <= len ( - 1 ) & ( - 1 <= i & i <= len ( - 1 ) holds not ( f /. i ) in Ball ( u , r ) & not ( f /. i ) in Ball ( u , r ) & not ( f /. i ) in Ball ( u , r ) ) ;
defpred P [ Element of NAT ] means ( Partial_Sums ( cos ) . $1 ) * ( Partial_Sums ( cos ) . $1 ) . ( 2 * $1 ) = ( Partial_Sums ( cos ) . $1 * ( Partial_Sums ( cos ) . $1 ) ) * ( Partial_Sums ( cos ) . $1 ) * ( Partial_Sums ( cos ) . $1 ) . ( 2 * $1 ) ;
for x being Element of product F holds x is FinSequence of G & dom x = I & for i being set st i in dom x holds x . i in ( the carrier of product F ) & for i being set st i in dom x holds x . i in ( the carrier of product F ) & x . i = ( the carrier of product F ) . i
( x " ) |^ ( n + 1 ) = ( 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 ;
DataPart Comput ( P +* I , Initialize s , LifeSpan ( P +* I , Initialize s ) ) = DataPart Comput ( P +* I , Initialize s , LifeSpan ( P +* I , Initialize s ) ) .= DataPart Comput ( P +* I , Initialize s , LifeSpan ( P +* I , Initialize s ) ) .= DataPart Comput ( P +* I , Initialize s , LifeSpan ( P +* I , Initialize s ) ) ;
given r such that 0 < r and ]. x0 - r , x0 .[ c= dom f1 /\ dom f2 and for g st g in ]. x0 , x0 + r .[ holds f1 . g <= f1 . g & for g st g in ]. x0 , x0 + r .[ holds f1 . g <= f2 . g & f2 . 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 holds f1 . x = f2 . x ) and ( for x st x in X holds f1 . x = ( - 1 ) * ( - 1 ) ) and ( f1 | X ) . x = ( - 1 ) * ( - 1 ) and ( - 1 ) * ( - 1 ) * ( - 1 ) = ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( - 1 ) * ( ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1
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 finite & x is finite
Support ( e *' p ) in { m *' p where m is Polynomial of n , L : ex p being Polynomial of n , L st p in Support ( m ) & ex i being Element of NAT st i in dom ( m ) & p . i = ( m *' p ) . i & p . i = ( m *' p ) . i & p . i = ( m *' p ) . i & p . i = ( m *' p ) . i ;
( 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 ) ;
ex p1 being Element of CQC-WFF ( Al ) st p1 = g . p1 & for g being Function of [: [: the carrier of Al , the carrier of Al :] , the carrier of Al :] st P [ g , p1 , g ] holds P [ g , p1 , g ] & P [ g , p1 , g ] ;
( mid ( f , i , len f -' 1 ) ^ <* f /. ( len f -' 1 ) *> ) /. j = ( mid ( f , i , len f -' 1 ) ^ mid ( f , i , len f -' 1 ) ) /. j .= ( mid ( f , i , len f -' 1 ) ^ mid ( f , i , len f -' 1 ) ) /. j .= f /. ( j + 1 ) ;
( ( p ^ q ) ^ r ) . ( len p + k ) = ( ( p ^ q ) ^ r ) . ( len p + k ) .= ( ( p ^ q ) ^ r ) . ( len p + k ) .= ( ( p ^ q ) ^ r ) . ( len p + k ) .= ( ( p ^ q ) ^ r ) . ( len p + k ) .= ( p ^ r ) . ( len p + k ) .= ( p ^ r ) . ( len p + k ) .= ( p ^ r ) . ( len p + k ) .= ( p ^ r ) . ( len p + k ) .= ( p ^ r ) . ( len p + ( p ^ r ) .= ( p ^ r ) . ( len p + k ) .= ( p ^ r ) . ( len p + k ) .= ( p ^ r ) . ( len p + k ) .= ( p ^ r ) . ( len p + k ) . ( len p + k ) .= ( p ^ r ) . ( len p + k ) .= ( p ^ r ) . ( len
len mid ( upper_volume ( D2 , D1 ) , indx ( D2 , D1 , j1 ) + 1 ) = indx ( D2 , D1 , j1 ) - indx ( D2 , D1 , j1 ) + 1 .= indx ( D2 , D1 , j1 ) - 1 .= indx ( D2 , D1 , j1 ) + 1 .= indx ( D2 , D1 , j1 ) - 1 ;
x * y * z = Mz . ( x * y , z * z9 ) .= x * ( y * z ) .= ( x * ( y * z ) ) * ( y * z ) .= ( x * ( y * z ) ) * ( y * z ) .= ( x * ( y * z ) ) * ( y * z ) .= ( x * y ) * ( x * z ) .= x * y * z ;
v . <* x , y *> - ( <* x0 , y0 *> ) * i = partdiff ( v , ( x - y ) ) * ( ( x - x0 ) * ( ( x - x0 ) * ( y - x0 ) ) + ( ( y - x0 ) * ( y - x0 ) ) * ( y - x0 ) ) + ( ( proj ( 1 , 1 ) * ( y - x0 ) ) * ( y - x0 ) ) + ( ( proj ( 1 , 1 ) * ( y - x0 ) * ( y - x0 ) * ( y - x0 ) ) + ( ( proj ( 1 , 1 ) * ( y - x0 ) * ( y - x0 ) ) * ( y - x0 ) * ( y - x0 ) ) * ( y - x0 ) * ( y - x0 ) ) + ( ( x - x0 ) ) + ( ( x - x0 ) ) + ( ( x - x0 ) * ( x - x0 ) * ( y - x0 ) * ( y - x0 ) * ( x - x0 ) ) + ( ( proj ( 1 , 1 ) * ( y - x0 ) *
i * i = <* 0 * ( - 1 ) - 0 * ( 0 * 0 ) + 0 * 0 , 0 * 0 + 0 * 0 , 0 * 0 + 0 * 0 , 0 * 0 + 0 * 0 + 0 * 0 , 0 * 0 + 0 * 0 + 0 * 0 , 0 * 0 + 0 * 0 + 0 * 0 , 0 * 0 + 0 * 0 + 0 * 0 , 0 * 0 + 0 * 0 + 0 * 0 , 0 * 0 + 0 * 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 * 0 * 0 + 0 * 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0
Sum ( L (#) F ) = Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) F1 ) + Sum ( L (#) F2 ) .= Sum ( L (#) F1 ) + Sum ( L (#) F2 ) .= Sum ( L (#) F1 ) + Sum ( L (#) F2 ) .= Sum ( L (#) F1 ) + Sum ( L (#) F2 ) .= Sum ( L (#) F1 ) + Sum ( L (#) F2 ) .= Sum ( L (#) F1 ) .= Sum ( L (#) F1 ) + Sum ( L (#) F2 ) .= Sum ( L (#) F1 ) + Sum ( L (#) F2 ) + Sum ( L (#) F2 ) + Sum ( L (#) F2 ) + Sum ( L (#) F2 ) + Sum ( L (#) F2 ) .= Sum ( L (#) F2 ) .= Sum ( L (#) F2 ) .= Sum ( L (#) F2 ) .= Sum ( L (#) F2 ) .= Sum ( L (#) F1 ) + Sum ( L (#) F2 ) .= Sum ( L (#) F2 ) + Sum ( L (#) F2 ) .= Sum ( L (#) F2 ) + Sum ( L (#) F2 ) .= Sum ( L (#) F2 ) + Sum ( L (#) F2
ex r be Real st for e be Real st 0 < e ex Y1 be finite Subset of X st Y1 is non empty & for Y1 be finite Subset of X st Y1 c= Y & Y1 c= Y & for Y1 be finite Subset of X st Y1 c= Y & Y1 c= Y holds |. ( - 1 ) * ( Y1 ) - ( - 1 ) * ( Y1 ) .| < r * ( Y1 )
( GoB f ) * ( i , j + 1 ) = f /. ( k + 2 ) & ( GoB f ) * ( i , j + 1 ) = f /. ( k + 1 ) or ( GoB f ) * ( i , j ) = f /. ( k + 1 ) or ( GoB f ) * ( i , j ) = f /. ( k + 1 ) & ( GoB f ) * ( i , j ) = f /. ( k + 1 ) ;
( ( cos - cos ) ^2 ) ^2 = ( - ( sin ^2 ) ^2 + ( cos ^2 ) ^2 ) .= ( - ( 1 / 2 ) ^2 ) * ( ( cos ^2 ) ^2 + ( cos ^2 ) ^2 ) .= ( - ( 1 / 2 ) ^2 ) * ( ( cos ^2 ) ^2 + ( cos ^2 ) ^2 ) .= ( - ( 1 / 2 ) ^2 ) * ( ( cos ^2 ) ^2 ) .= ( - ( 1 / 2 ) ^2 ;
( - ( - b ) + sqrt ( integral ( a , b , c ) ) ^2 ) < 0 & ( - ( - b ) + sqrt ( integral ( a , b , c ) ) ^2 ) < 0 or ( - ( - b ) + sqrt ( integral ( a , b , c ) ) ^2 ) < 0 ;
assume that ex_inf_of uparrow "\/" ( X , C ) , L and ex_sup_of X , L and "\/" ( X , C ) , L and "\/" ( X , C ) = "/\" ( uparrow "\/" ( X , C ) , L ) and "\/" ( X , C ) = "/\" ( ( uparrow "\/" ( X , C ) ) , L ) and "\/" ( X , C ) = "/\" ( X , L ) ;
( ( for i being Element of NAT st i in the Sorts of B ) holds ( i = j implies ( i = j implies j = i ) ) & ( i = j implies j = i ) & ( j = j implies j = i ) ) & ( j = i implies j = i implies j = i ) )