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thesis ;
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thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
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thesis ;
contradiction ;
contradiction ;
thesis ;
contradiction ;
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thesis ;
contradiction ;
thesis ;
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thesis ;
thesis ;
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thesis ;
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contradiction ;
thesis ;
thesis ;
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thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
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contradiction ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
B ;
a <> c
T c= S
D c= B
c in X ;
b in X ;
X ;
b in D ;
x = e ;
let m ;
h is onto ;
N in K ;
let i ;
j = 1 ;
x = u ;
let n ;
let k ;
y in A ;
let x ;
let x ;
m c= y ;
F is one-to-one ;
let q ;
m = 1 ;
1 < k ;
G is over G ;
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 \bf ) ;
assume x in I ;
q is as as such by 0 ;
assume c in x ;
p > 0 ;
assume x in Z ;
assume x in Z ;
1 <= kor or 1 <= k ;
assume m <= i ;
assume G is over F ;
assume a divides b ;
assume P is closed ;
b-a > 0 ;
assume q in A ;
W is not bounded ;
f is non ] ;
assume A is boundary ;
g is special ;
assume i > j ;
assume t in X ;
assume n <= m ;
assume x in W ;
assume r in X ;
assume x in A ;
assume b is even ;
assume i in I ;
assume 1 <= k ;
X is non empty ;
assume x in X ;
assume n in M ;
assume b in X ;
assume x in A ;
assume T c= W ;
assume s is atomic ;
b `1 <= c `1 ;
A meets W ;
i `1 <= j `1 ;
assume H is universal ;
assume x in X ;
let X be set ;
let T be Tree ;
let d be element ;
let t be element ;
let x be element ;
let x be element ;
let s be element ;
k <= be 5 ;
let X be set ;
let X be set ;
let y be element ;
let x be element ;
P [ 0 ]
let E be set , A , B 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 \in \in \in \in of -1 ;
M < m + 1 ;
T2 is open ;
z in b +^ a ;
R2 is well-ordering ;
1 <= k + 1 ;
i > n + 1 ;
q1 is one-to-one ;
let x be trivial set ;
PP is one-to-one ;
n <= n + 2 ;
1 <= k + 1 ;
1 <= k + 1 ;
let e be Real ;
i < i + 1 ;
p3 in P ;
p1 in K ;
y in C1 ;
k + 1 <= n ;
let a be Real , v be VECTOR of V ;
X |- r => p ;
x in { A } ;
let n be Nat ;
let k be Nat ;
let k be Nat ;
let m be Nat ;
0 < 0 + k ;
f is_differentiable_in x ;
let x0 , x1 , x2 , x3 , x4 , x4 , x5 , x1 , x2 , x3 , x4 , x4 , x1 ,
let E be Ordinal ;
o : o the_arity_of 4 ;
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 , v be VECTOR of V ;
not s in Y |^ 0 ;
rng f is_<=_than w
b "/\" e = b ;
m = m3 ;
t in h . D ;
P [ 0 ] ;
assume z = x * y ;
S . n is bounded ;
let V be RealUnitarySpace , A be Subset of V ;
P [ 1 ] ;
P [ {} ] ;
C1 is component ;
H = G . i ;
1 <= i `1 + 1 ;
F . m in A ;
f . o = o ;
P [ 0 ] ;
a> 0 ;
R [ 0 ] ;
b in f .: X ;
assume q = q2 ;
x in [#] V ;
f . u = 0 ;
assume e1 > 0 ;
let V be RealUnitarySpace , v be VECTOR of V ;
s is trivial non empty ;
dom c = Q
P [ 0 ] ;
f . n in T ;
N . j in S ;
let T be complete LATTICE , f be Function of T , the carrier of T ;
the ObjectMap of F is one-to-one ;
sgn x = 1 ;
k in support a ;
1 in Seg 1 ;
rng f = X ;
len T in X ;
vbeing < n ;
S\HM { x } is bounded ;
assume p = p2 ;
len f = n ;
assume x in P1 ;
i in dom q ;
let U0 , S , x , y ;
pp `1 = c ;
j in dom h ;
let k ;
f | Z is continuous ;
k in dom G ;
UBD C = B ;
1 <= len M ;
p in \mathbin { x } ;
1 <= jj & jj <= len f ;
set A = be set ;
card a [= c ;
e in rng f ;
cluster B \oplus A -> empty ;
H has \cdot H ;
assume n0 <= m ;
T is increasing ordinal ;
e2 <> e2 ;
Z c= dom g ;
dom p = X ;
H is proper implies H is proper
i + 1 <= n ;
v <> 0. V ;
A c= Affin A ;
S c= dom F ;
m in dom f ;
let X0 be set ;
c = sup N ;
R reduces union M , union M ;
assume not x in REAL ;
Im f is complete ;
x in Int y ;
dom F = M ;
a in On W ;
assume e in A ( ) ;
C c= C-26 ;
mm <> {} ;
let x be Element of Y ;
let f be Line of C , i ) ;
not n in Seg 3 ;
assume X in f .: A ;
assume that p <= n and p <= m ;
assume not u in { v } ;
d is Element of A ;
A |^ b misses B ;
e in v in dom ( T | E ) ;
- y in I ;
let A be non empty set , f be FinSequence of A ;
P0 = 1 ;
assume r in F . k ;
assume f is simple ;
let A be [ |^ n ] ;
rng f c= NAT ;
assume P [ k ] ;
f6 <> {} ;
let o be Ordinal ;
assume x is sum of squares ;
assume not v in { 1 } ;
let II , s , I ;
assume that 1 <= j and j < l ;
v = - u ;
assume s . b > 0 ;
d1 in dom g ;
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 hh/. ;
assume f is additive inbb--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 k2 = k1 ;
m + 1 < n + 1 ;
s in S \/ { s } ;
n + i >= n + 1 ;
assume Re y = 0 ;
k1 <= j1 & j1 <= j2 ;
f | A is Sum f ;
f . x ^2 <= 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 ;
CH in X ;
q2 c= C1 & ( C1 \/ C2 ) c= C2 ;
a2 < c2 & a2 < c2 ;
s2 is 0 -started ;
IC s = 0 & IC s = 1 ;
s4 = s4 , s4 = s4 , P4 = s4 ;
let V ;
let x , y be element ;
let x be Element of T ;
assume a in rng F ;
x in dom T ` ;
let S be SubRelStr of L ;
y " <> 0 ;
y " <> 0 ;
0. V = uw ;
y2 , y , w , y ;
R8 in dom f ;
let a , b be Real , f be PartFunc of REAL , REAL ;
let a be Object of C ;
let x be Vertex of G ;
let o be object of C , m be Morphism of o , m ;
r '&' q = P \lbrack l , l .] ;
let i , j be Nat ;
let s be State of A , x be set ;
s4 . n = N ;
set y = ( x `1 ) / ( x `2 ) ;
mi in dom g & mi in dom g ;
l . 2 = y1 ;
|. g . y .| <= r ;
f . x in CX0 ;
not ( V is non empty & V is open ) ;
let x be Element of X ;
0 <> f . g2 ;
f2 /* q is convergent ;
f . i is_measurable_on E ;
assume \xi in Nfb1 ;
reconsider i = i as Ordinal ;
r * v = 0. X ;
rng f c= INT & rng f c= INT ;
G = 0 .--> goto 0 ;
let A be Subset of X ;
assume X0 is dense & A is dense ;
|. f . x .| <= r ;
let x be Element of R ;
let b be Element of L ;
assume x in W-19 ;
P [ k , a ] ;
let X be Subset of L ;
let b be Object of B ;
let A , B be category ;
set X = Vars ( C ) ;
let o be OperSymbol of S ;
let R be connected non empty Poset ;
n + 1 = succ n ;
xY c= Z1 & xY c= Z1 ;
dom f = C1 & dom g = C2 ;
assume [ a , y ] in X ;
Re ( seq ^\ k ) is convergent ;
assume a1 = b1 & a2 = b2 ;
A = ( sInt A ) ` ;
a <= b or b <= a ;
n + 1 in dom f ;
let F be Instruction of S , m be Nat ;
assume that r2 > x0 and x0 < r2 ;
let Y be non empty set , f be Function of Y , BOOLEAN ;
2 * x in dom W ;
m in dom g2 & n + 1 in dom g2 ;
n in dom g1 & n + 1 in dom g2 ;
k + 1 in dom f ;
not the still of s is finite ;
assume that x1 <> x2 and x1 <> x2 ;
v1 in ( V \ { v2 } ) ;
not [ b `1 , b ] in T ;
ii + 1 = i ;
T c= \llangle ] , T ;
( l `1 ) ^2 = 0 ;
n be Nat ;
( t `2 ) ^2 = r ;
A/. n is_integrable_on M ;
set t = Top t ;
let A , B be real-membered set ;
k <= len G + 1 ;
cC misses [: the carrier of V , the carrier of V :] ;
Product seq is non empty ;
e <= f or f <= e ;
cluster non empty normal normal for NAT -valued finite set ;
assume c2 = b2 & c1 = c2 ;
assume h in [. q , p .] ;
1 + 1 <= len C ;
not c in B . m1 ;
cluster R .: X -> empty ;
p . n = H . n ;
assume that vseq is convergent and vseq is convergent and lim vseq = 0 ;
IC s3 = 0 & IC s3 = 0 ;
k in N or k in K ;
F1 \/ F2 c= F ;
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 non empty reflexive transitive antisymmetric RelStr ;
q |^ m >= 1 ;
a is_>=_than X & b is_>=_than Y ;
assume <* a , c *> <> {} ;
F . c = g . c ;
G is one-to-one one-to-one one-to-one ;
A \/ { a } \not c= B ;
0. V = 0. Y .= 0. V ;
let I be non empty non empty Instruction of S , S ;
f-24 . x = 1 & fLet . x = 0 ;
assume z \ x = 0. X ;
C4 = 2 to_power n ;
let B be sequence of Sigma ;
assume X1 = p .: D ;
n + l2 in NAT & n + l2 in NAT ;
f " P is compact ;
assume x1 in [: { 0 } , REAL+ :] ;
p1 = ( K1 ) . ( len p1 ) ;
M . k = <*> REAL ;
phi . 0 in rng phi ;
MMwhere A is closed Subset of D ;
assume z0 <> 0_ ( n , L ) ;
n < N7 . k ;
0 <= seq . 0 & seq . 0 <= seq . 0 ;
- q + p = v ;
{ v } is Subset of B ;
set g = f /. 1 ;
[: the carrier of R , the carrier of R :] is stable ;
set cR = Vertices R , cR = Vertices R ;
p0 c= P3 & IC Comput ( P3 , s3 , 1 ) in dom P3 ;
x in [. 0 , 1 .[ ;
f . y in dom F ;
let T be Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott 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 - x .|| <= r ;
assume that Y c= field Q and Y <> {} ;
a ~ , b ~ " " ;
assume a in A ( ) ;
k in dom ( q4 | k ) ;
p is non empty \HM { 0 } ;
i -' 1 = i-1 ;
f | A is one-to-one ;
assume x in f .: [: X , Y :] ;
i2 - i1 = 0 & i2 - i1 = 0 ;
j2 + 1 <= i2 & j2 + 1 <= j2 ;
g " * a in N ;
K <> { [ {} , {} ] } ;
cluster strict for } such that cluster strict strict for O ;
|. 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 + 1-1 ;
dom S = dom F & dom S = dom F ;
let s be Element of NAT ;
let R be ManySortedSet of A ;
let n be Element of NAT ;
let S be non empty non void non void void void void TopStruct ;
let f be ManySortedSet of I ;
let z be Element of COMPLEX , v be Element of COMPLEX ;
u in { ag } ;
2 * n < 2 * n ;
let x , y be set ;
B-11 c= ( V \ { y } ) ;
assume I is_halting_on s , P ;
U2 = ( U2 /\ U1 ) \/ ( U2 /\ U2 ) ;
M /. 1 = z /. 1 ;
x9 = x9 & x9 = y9 ;
i + 1 < n + 1 + 1 ;
x in { {} , <* 0 *> } ;
( f | n ) . x <= ( f | n ) . x ;
let l be Element of L ;
x in dom ( F | P ) ;
let i be Element of NAT ;
r8 is ( C * ) -valued FinSequence of COMPLEX ;
assume <* o2 , o *> <> {} ;
s . x |^ 0 = 1 ;
card ( K1 \/ { 0. TOP-REAL 2 } ) in M ;
assume that X in U and Y in U ;
let D be Subset-Family of Omega ;
set r = q - { k + 1 } ;
y = W . ( 2 * PI ) ;
assume dom g = cod f & cod g = cod f ;
let X , Y be non empty TopSpace , f be Function of X , Y ;
x ++ A is interval ;
|. <*> A .| . a = 0 ;
cluster strict for SubLattice of L ;
a1 in B . s1 & a2 in B . s2 ;
let V be finite finite strict vector of F , v be Vector of V ;
A * B on B , A ;
f-3 = NAT --> 0 ;
let A , B be Subset of V ;
z1 = P1 . j & z2 = P2 . j ;
assume f " P is closed ;
reconsider j = i as Element of M ;
let a , b be Element of L ;
assume q in A \/ ( B "\/" C ) ;
dom ( F * C ) = o ;
set S = INT |^ X ;
z in dom ( A --> y ) ;
P [ y , h . y ] ;
{ x0 } c= dom f & { x0 } c= dom g ;
let B be non-empty ManySortedSet of I , A be ManySortedSet of I ;
( PI / 2 ) < Arg z ;
reconsider z9 = 0 as Nat ;
LIN a , d , c ;
[ y , x ] in IO ;
( Q ) * ( 3 , 3 ) = 0 ;
set j = x0 gcd m , n = x0 gcd m ;
assume a in { x , y , c } ;
j2 - ( j - 1 ) > 0 ;
I I \! \mathop not N = 1 ;
[ y , d ] in F-8 ;
let f be Function of X , Y ;
set A2 = ( B |^ C ) / ( B |^ C ) ;
s1 , s2 are_/ 2 ;
j1 -' 1 = 0 & j2 -' 1 = 0 ;
set m2 = 2 * n + j ;
reconsider t = t as bag of n ;
I2 . j = m . j ;
i |^ s , n are_relative_prime & n divides ( i |^ s ) ;
set g = f | D-21 ;
assume that X is lower and 0 <= r ;
( p1 `1 ) ^2 = 1 & ( p1 `2 ) ^2 = 1 ;
a < ( p3 `1 ) / |. p3 .| ;
L \ { m } c= UBD C ;
x in Ball ( x , 10 ) ;
not a in LSeg ( c , m ) ;
1 <= i1 -' 1 & i1 -' 1 <= len f ;
1 <= i1 -' 1 & i1 -' 1 <= len f ;
i + i2 <= len h & 1 <= i2 & i2 + 1 <= 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 <* A1 *> = 1 ;
set H = h . ( g . O ) ;
card b * a = |. a .| ;
Shift ( w , 0 ) |= v ;
set h = h2 ** h1 , k = h2 ** h2 ;
assume x in ( X3 /\ 4 ) \/ ( X3 /\ 4 ) ;
||. h - g .|| < dx0 ;
not x in the carrier of f & not x in the carrier of g ;
f . y = F ( y ) ;
for n holds X [ n ] ;
k - l = kl2 - 1 ;
<* p , q *> /. 2 = q ;
let S be Subset of the lattice of Y ;
P , Q be EqRel s of 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 reflexive RelStr , X be Subset of L ;
S-20 is x -basis i , y -basis i ;
let r be non positive Real ;
M , v |= ( x \hbox { = } y ) ;
v + w = 0. ( Z , X ) ;
P [ len F ] implies P [ len F + 1 ]
assume InsCode ( i ) = 8 or InsCode ( i ) = 8 ;
the zero of M = 0 & the Element of M = 0 ;
cluster z * seq -> summable for Real_Sequence ;
let O be Subset of the carrier of C ;
||. f .|| | X is continuous ;
x2 = g . ( j + 1 ) ;
cluster -> non / -> |^ \cap ^2 for Element of S ;
reconsider l1 = l-1 as Nat ;
v4 is Vertex of r2 & v4 is Vertex of r2 ;
T2 is SubSpace of T2 & T1 is SubSpace of T2 ;
Q1 /\ Q19 <> {} & Q1 /\ Q19 <> {} ;
k be Nat ;
q " is Element of X ;
F . t is set of non zero set ;
assume n <> 0 & 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 of ( p `1 ) ^2 , ( p `2 ) ^2 ;
not r in ]. p , q .[ ;
let R be FinSequence of REAL , a be Element of REAL ;
S7 does not destroy b1 & S7 does not destroy b2 ;
IC SCM R <> a & IC SCM R <> a ;
|. - p - y .| >= r ;
1 * seq = seq & ( - 1 ) * seq is convergent ;
let x be FinSequence 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 .= ( n + 1 ) ;
H + G = F\hbox ( - G ) ;
CF1 . x = x2 & CF2 . x = y2 ;
f1 = f .= f2 .= f2 .= f2 * f1 ;
Sum <* p . 0 *> = p . 0 ;
assume v + W = v + u + W ;
{ a1 } = { a2 } & { a2 } = { a2 } ;
a1 , b1 _|_ b , a & a1 , b1 _|_ b , a ;
d2 , o _|_ o , a3 , a1 ;
IO is reflexive & IO is transitive implies IO is reflexive
IO is antisymmetric implies [: O , O :] is antisymmetric
sup rng ( H1 | n ) = e & sup rng ( H1 | n ) = e ;
x = ( a * ( a * b ) ) * ( b * ( a * b ) ) ;
|. p1 .| ^2 >= 1 - ( p1 `1 / p1 `2 ) ^2 ;
assume j2 -' 1 < j2 & j2 -' 1 < j2 ;
rng s c= dom f1 /\ dom f2 ;
assume that support a misses support b and a in support b ;
let L be associative commutative associative non empty doubleLoopStr , p be Polynomial of L ;
s " + 0 < n + 1 ;
p . c = ( f " ) . 1 ;
R . n <= R . ( n + 1 ) ;
Directed Directed ( I1 , I2 ) = I1 ;
set f = + ( x , y , r ) ;
cluster Ball ( x , r ) -> bounded ;
consider r being Real such that r in A ;
cluster non empty NAT -defined for NAT -defined Function ;
let X be non empty directed Subset of S ;
let S be non empty full SubRelStr of L ;
cluster <* [ ] , [ 0 , 0 ] *> -> complete for non empty TopSpace ;
( 1 - a " ) * a = a ;
( q . {} ) `1 = o ;
n - ( i -' 1 ) > 0 ;
assume ( 1 - 2 ) <= t `1 / |. t .| ;
card B = k + 1-1 ;
x in union rng ( f | ( len f ) ) ;
assume x in the carrier of R & y in the carrier of R ;
d in D ;
f . 1 = L . ( F . 1 ) ;
the vertices of G = { v } & { v } c= V ;
let G be : : Let ;
e , v6 , G be set ;
c . ( ii - 1 ) in rng c ;
f2 /* q is divergent_to-infty & f2 /* q is divergent_to-infty ;
set z1 = - z2 , z2 = - z1 , z2 = - z2 ;
assume w is \mathop { las of S , G ;
set f = p |-count ( t - p ) ;
let c be Object of C ;
assume ex a st P [ a ] ;
let x be Element of REAL m , y be Element of REAL m ;
let IX be Subset-Family of X , Y ;
reconsider p = p as Element of NAT ;
let v , w be Point of X ;
let s be State of SCM+FSA , I be Program of SCM+FSA ;
p is FinSequence of NAT , k be Nat ;
stop I c= ( card I + 2 ) ;
set ci = ( f /. i ) `1 ;
w ^ t ==> w ^ s ;
W1 /\ W = W1 /\ W ` .= W1 /\ ( W1 ` ) ;
f . j is Element of J . j ;
let x , y be Element of T2 , a be Element of T2 ;
ex d st a , b // b , d ;
a <> 0 & b <> 0 & c <> 0 ;
ord x = 1 & x is positive ;
set g2 = lim ( seq ^\ ( n + k ) ) ;
2 * x >= 2 * ( 1 / 2 ) ;
assume ( a 'or' c ) . z <> TRUE ;
f (*) g in Hom ( c , c ) ;
Hom ( c , c + d ) <> {} ;
assume 2 * Sum ( q | m ) > m ;
L1 . ( F-21 . k ) = 0 ;
/ ( X \/ R1 ) = / ( X \/ R1 )
( ( sin * sin ) `| Z ) . x <> 0 ;
( ( ( exp_R * f ) `| Z ) . x ) ^2 > 0 ;
o1 in [: X /\ O2 , { 0 } :] /\ ( X /\ O2 ) ;
e , v6 , G be set ;
r3 > ( 1 - 2 ) * 0 ;
x in P .: ( F -ideal ) ;
let J be closed Subset of R ;
h . p1 = f2 . O & h . p2 = f2 . O ;
Index ( p , f ) + 1 <= j ;
len ( q | i ) = width M & width ( q | i ) = width M ;
the carrier of LK c= A & the carrier of K c= A ;
dom f c= union rng ( F | -10 ) ;
k + 1 in support ( EmptyBag n ) ;
let X be ManySortedSet of the carrier of S ;
[ x `1 , y `2 ] in \mathclose ( field R ) ;
i = D1 or i = D2 or i = D1 ;
assume a mod n = b mod n ;
h . x2 = g . x1 & h . x2 = f . x2 ;
F c= 2 -tuples_on the carrier of X ;
reconsider w = |. s1 .| as Real_Sequence of REAL ;
( 1 / m * m + r ) < p ;
dom f = dom ( I --> ( a , b ) ) ;
[#] ( P-17 ) = [#] ( ( ( TOP-REAL 2 ) | K1 ) | K1 ) ;
cluster - x -> R_eal -> R_eal ;
then { d1 } c= A ;
cluster TOP-REAL n -> finite-ind for non empty TopSpace ;
let w1 be Element of M ;
let x be Element of dyadic ( n ) ;
u in W1 & v in W3 & u in W3 implies v in W2
reconsider y = y as Element of L2 ;
N is full SubRelStr of ( T |^ n ) |^ the carrier of N ;
sup { x , y } = c "\/" c ;
g . n = n to_power 1 .= n ;
h . J = EqClass ( u , J ) ;
let seq be -> -> -> summable summable sequence of X ;
dist ( x `1 , y ) < ( r / 2 ) ;
reconsider mm1 = m - 1 as Element of NAT ;
x- x0 < r1 - x0 & r1 < x0 - x0 ;
reconsider P ` = P ` as strict Subgroup of N ;
set g1 = p * idseq ( q `1 ) , g2 = p * idseq ( q `2 ) ;
let n , m , k be non zero Nat ;
assume that 0 < e and f | A is lower ;
D2 . ( I . ( I . ( J . I ) ) ) 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 ( PI , 1 ) /\ LSeg ( co , 1 ) ;
let n be Element of NAT , x be Element of X ;
reconsider S8 = S as Subset of T | S ;
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 , P , s be State of SCMPDS ;
let t be 0 -started State of SCMPDS , Q ;
b , b , x , y , z ;
assume that i = n \/ { n } and j = k \/ { k } ;
let f be PartFunc of X , Y ;
N2 >= ( sqrt c / sqrt 2 ) / sqrt 2 ;
reconsider t7 = T7 as TopSpace ;
set q = h * p ^ <* d *> ;
z2 in U . ( y1 , y2 ) /\ Q2 ( y2 , z2 ) ;
A |^ 0 = { <%> E } & A |^ 0 = { <%> E } ;
len W2 = len W + 2 & len W1 = len W + 2 ;
len h2 in dom h2 & len h2 = len h2 ;
i + 1 in Seg len s2 & i + 1 in Seg len s2 ;
z in dom g1 /\ dom f & z in dom ( g1 + g2 ) ;
assume that p2 = E-max ( K ) and p1 `1 = E-bound ( K ) ;
len G + 1 <= i1 + 1 ;
f1 (#) f2 is convergent & lim ( f1 (#) f2 ) = x0 ;
cluster seq + ( s + 1 ) -> summable for Real_Sequence ;
assume that j in dom M1 and i in dom M1 ;
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 *> \geq x ;
a , b in { a , b } ;
len p2 is Element of NAT & len p2 = len p1 ;
ex x being element st x in dom R & R . x = y ;
len q = len ( K (#) G ) .= len G ;
s1 = Initialize ( Initialized Initialized s ) , P1 = P +* I ;
consider w being Nat such that q = z + w ;
x ` is Element of L ` & y is Element of L ;
k = 0 & n <> k or k > n ;
then X is discrete for X is closed ;
for x st x in L holds x is FinSequence of REAL ;
||. f /. c - f /. c .|| <= r1 / 2 ;
c in uparrow p & not c in { p } ;
reconsider V = V as Subset of the TopStruct of TOP-REAL n ;
N , M be strict *> ;
then z is_>=_than waybelow x & z is_>=_than compactbelow y ;
M \lbrack f , g .] = f & M \lbrack g , f .] = g ;
( ( ( TOP-REAL 1 ) | 1 ) /. 1 ) = TRUE ;
dom g = dom f /\ X .= dom f /\ X ;
mode \cal il of G is \cal : 1 <= len W ;
[ 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 ) , g be Element of dom ( Subformulae p ) ;
F1 . ( a1 , - a2 ) = G1 . ( a1 , a2 ) ;
redefine func Sphere ( a , b , r ) -> compact ;
let a , b , c , d be Real ;
rng s c= dom ( 1 / ( f1 - f2 ) ) ;
curry ( ( F . -19 ) , k ) is additive ;
set k2 = card dom B , s3 = card ( dom B ) ;
set G = DTConMSA ( X ) ;
reconsider a = root-tree [ x , s ] as Element of G ;
let a , b be Element of MO , f be Element of CO ;
reconsider s1 = s , s2 = t as Element of ( the carrier of S ) * ;
rng p c= the carrier of L & p . ( len p ) in the carrier of L ;
let d be Subset of the bound of A ;
( x .|. x ) = 0 iff x = 0. W ;
I-21 in dom stop I & IY in dom stop I ;
let g be continuous Function of X | B , Y ;
reconsider D = Y as Subset of ( TOP-REAL n ) | P ;
reconsider i0 = len p1 - 1 as Integer ;
dom f = the carrier of S & dom g = the carrier of S ;
rng h c= union ( the carrier of J ) ;
cluster All ( x , H ) -> Function \sqrt LSeg ( x , y ) ;
d * N1 ^2 > N1 * 1 / ( d * N2 ) ;
]. a , b .[ c= [. a , b .] ;
set g = f " | ( D1 /\ D2 ) ;
dom ( p | ( mm1 + 1 ) ) = mm1 ;
3 + - 2 <= k + - 2 ;
tan is_differentiable_in ( ( - 1 / 2 ) (#) ( ( arccot ) * ( f1 - #Z 2 ) ) ) `| Z ) ;
x in rng ( f /^ ( n -' p ) ) ;
let f , g be FinSequence of D ;
p ( ) in the carrier of S1 ( ) & q ( ) in the carrier of S2 ( ) ;
rng f " { 0 } = dom f & rng f = { 0 } ;
( the Target of G ) . e = v & ( the Target of G ) . e = v ;
width G - 1 < width G - 1 ;
assume that v in rng ( S | E1 ) and v in rng ( S | E1 ) ;
assume x is root or x is root or x is root & y is root ;
assume 0 in rng ( ( g2 | A ) | 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 & <* C7 *> is in the carrier of C-20 ;
i <= len G -' 1 & 1 <= len G -' 1 ;
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] & x3 in the carrier of I[01] ;
set p1 = f /. i , p2 = f /. j ;
g in { g2 : r < g2 & g2 < x0 } ;
Q2 = Sp2 " ( P /\ Q ) .= Q ;
( ( 1 / 2 ) |^ ( n + 1 ) ) is summable ;
- p + I c= - p + A ;
n < LifeSpan ( P1 , s1 ) + 1 ;
CurInstr ( p1 , s1 ) = i & CurInstr ( p1 , s1 ) = halt SCM+FSA ;
A /\ Cl { x } \ { x } <> {} ;
rng f c= ]. r - 1 , r + 1 .[ ;
let g be Function of S , V ;
let f be Function of L1 , L2 , g be Function of L1 , L2 ;
reconsider z = z as Element of CompactSublatt L ;
let f be Function of S , T ;
reconsider g = g as Morphism of c opp , b opp ;
[ s , I ] in S ~ ( S , T ) ;
len ( the connectives of C ) = 4 & len ( the connectives of C ) = 5 ;
let C1 , C2 be subFunctor of C , D ;
reconsider V1 = V as Subset of X | B ;
attr p is valid means : Def1 : All ( x , p ) is valid ;
assume that X c= dom f and f .: X c= dom g and g .: X c= dom f ;
H |^ a " is Subgroup of H |^ a " .= H |^ a ;
let A1 be [ be [ O , A1 ] , [ a , A1 ] ] ;
p2 , r3 , q3 is_collinear & q2 , q3 , p2 is_collinear ;
consider x being element such that x in v ^ K ;
not x in { 0. TOP-REAL 2 } & not x in { 0. TOP-REAL 2 } ;
p in [#] ( ( ( TOP-REAL 2 ) | B11 ) | B11 ) ;
0 . ( 0 ) < M . ( E8 . ( 0 + 1 ) ) ;
^ ( c / c ) = c ;
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . s2 ) & a2 in dom ( F . s2 ) ;
cluster -> -> Line for *> -| the carrier of L is Line of L ;
set i1 = the Nat , i2 = the Nat , i1 = the Nat , i2 = the Nat , i2 = the Nat , j1 = the Nat ;
let s be 0 -started State of SCM+FSA , k be Nat ;
assume y in ( f1 \/ f2 ) .: A ;
f . ( len f ) = f /. ( len f ) ;
x , f . x '||' f . x , f . y ;
pred X c= Y means : Def4 : cos ( X ) c= cos ( Y ) ;
let y be upper Subset of Y , x be Element of X ;
cluster -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> Nat every Nat ;
set S = <* Bags n , <* i *> , {} *> ;
set T = [. 0 , 1 / 2 * PI .] ;
1 in dom mid ( f , 1 , 1 ) ;
( 4 * PI * PI ) < ( 2 * PI * PI ) ;
x2 in dom f1 /\ dom f & x1 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 / ( 1 + ( p `1 / p `2 ) ^2 ) = ( p1 `1 ) ^2 / ( 1 + ( p1 `1 / p1 `2 ) ^2 ) ;
i + 1 <= len Cage ( C , n ) ;
len ( <* P *> @ ) = len P & len ( <* P *> @ ) = len P ;
set N-26 = the non empty Subset of N , L be non empty Subset of N ;
len g: + ( x + 1 ) - 1 <= x ;
a on B & b on B & a on B ;
reconsider r-12 = r * I . v as FinSequence of REAL ;
consider d such that x = d and a D D is_less_than c ;
given u such that u in W and x = v + u ;
len f /. ( \downharpoonright n ) = len f - n ;
set q2 = Int L~ Cage ( C , n ) , q2 = W-min L~ Cage ( C , n ) ;
set S = { len S1 , len S2 } , T = { len S1 , len S2 } ;
MaxADSet ( b ) c= MaxADSet ( P /\ Q ) ;
Cl ( G . q1 ) c= F . r2 & Cl ( G . q2 ) c= F . r2 ;
f " D meets h " ( V /\ W ) ;
reconsider D = E as non empty directed Subset of L1 ;
H = ( the_left_argument_of H ) '&' ( the_left_argument_of H ) ;
assume that t is Element of ( S , X ) * ;
rng f c= the carrier of S2 & rng g c= the carrier of S2 ;
consider y being Element of X such that x = { y } ;
f1 . ( a1 , b1 ) = b1 & 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 ) , [ n , UMP C ] ) ;
[ i , j ] in Indices M1 & [ i , j ] in Indices M1 ;
assume that P c= Seg m and M is \HM { \vert a .| } ;
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 + 1 ) ;
p-7 . i = pp1 . i & pp2 . i = pp2 . i ;
let PA , G be a_partition of Y , z be set ;
pred 0 < r & r < 1 & r < 1 ;
rng ( ( a , X ) --> ( x , y ) ) = [#] X ;
reconsider x = x , y = y as Element of K ;
consider k such that z = f . k and n <= k ;
consider x being element such that x in X \ { p } ;
len ( canFS ( s ) ) = card ( s ) .= card ( s ) ;
reconsider x2 = x1 , y2 = x2 as Element of L2 ;
Q in FinMeetCl ( ( the topology of X ) | P ) ;
dom ( f | ( 0 -tuples_on the carrier of S ) ) c= dom ( u | ( 0 -tuples_on the carrier of S ) ) ;
pred n divides m & m divides n & n divides m ;
reconsider x = x as Point of [: I[01] , I[01] :] , I[01] :] ;
a in ;
not y0 in the still of f & not ( ex g st g in the carrier of f & g in the carrier of f ) ;
Hom ( ( a \times b ) \times c , c ) <> {} ;
consider k1 such that p " < k1 and k1 < len f and f . k1 in rng f ;
consider c , d such that dom f = c \ d ;
[ x , y ] in dom g ~ ( k + 1 ) ;
set S1 =
l2 = m2 & l2 = i2 & l2 = j2 & l2 = i2 & l2 = j2 ;
x0 in dom ( ( u /\ A9 ) /\ A9 ) & x0 in dom ( ( u | A9 ) | A9 ) ;
reconsider p = x as Point of ( TOP-REAL 2 ) | K1 ;
I[01] = ( I[01] ) | B01 .= ( TOP-REAL 2 ) | B01 .= ( TOP-REAL 2 ) | B01 ;
f . p4 <= f . p1 & f . p1 = f . p2 ;
( F . ( x , y ) ) `1 <= ( F . ( x , y ) ) `1 ;
( x `2 ) ^2 = ( W7 ) ^2 + ( W8 ) ^2 ;
for n being Element of NAT holds P [ n ] implies P [ n + 1 ]
let J , K be non empty Subset of I ;
assume that 1 <= i and i <= len <* a " *> and j <= len <* a " *> ;
0 |-> a = <*> the carrier of K .= a * a ;
X . i in 2 -tuples_on A . i \ B . i ;
<* 0 *> in dom ( e --> [ 1 , 0 ] ) ;
then P [ a ] & P [ succ a ] ;
reconsider sbeing Element of D , s/. i as finite Subset of D ;
k - ( i -' 1 ) <= len thesis - j ;
[#] S c= [#] the TopStruct of T & the TopStruct of T c= the TopStruct of T ;
for V being strict real unitary space holds V in the carrier of V implies V is Subspace of W
assume k in dom mid ( f , i , j ) ;
let P be non empty Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
let A , B be Matrix of n1 , n2 , K ;
- a * - b = a * b & - a * b = - a ;
for A being Subset of AS holds A // A & A // A implies A // A
( for o2 being Element of A holds ( o2 in A ) & ( o2 in A ) implies o2 in A ) ;
then ||. x - x .|| = 0 & x = 0. X ;
let N1 , N2 be strict normal Subgroup of G , N be normal Subgroup of G ;
j >= len upper_volume ( g , D1 ) & upper_volume ( g , D2 ) . j in rng upper_volume ( g , D1 ) ;
b = Q . ( len Q - 1 + 1 ) .= Q . ( len Q - 1 ) ;
f2 * f1 /* s is divergent_to-infty & lim ( f2 * f1 ) = x0 ;
reconsider h = f * g as Function of ( the carrier of G ) , G ;
assume that a <> 0 and delta ( a , b , c ) >= 0 ;
[ t , t ] in the InternalRel of A & [ t , t ] in the InternalRel of A ;
( v |-- ( E , X ) ) | n is Element of T7 ;
{} = the carrier of L1 + L2 & the carrier of L1 = the carrier of L2 + L2 ;
Directed I is_closed_on Initialized Initialized s , P & Directed I is_halting_on Initialized s , P ;
Initialized p = Initialize ( p +* q ) , p = p +* q , q = p +* q ;
reconsider N2 = N1 , N2 = N2 as strict net of R2 , R1 , R2 ;
reconsider Y = Y as Element of [: Ids L , Ids L :] ;
"/\" ( uparrow p \ { p } , L ) <> p ;
consider j being Nat such that i2 = i1 + j and j in dom f ;
not [ s , 0 ] in the carrier of S2 & not [ s , 0 ] in the carrier of S2 ;
mm in ( B '/\' C ) '/\' D \ { {} } ;
n <= len ( ( P + Q ) | ( len P + 1 ) ) ;
( x1 `1 ) ^2 = ( x2 `1 ) ^2 + ( x1 `2 ) ^2 .= ( x2 `1 ) ^2 + ( x1 `2 ) ^2 ;
InputVertices S = { x1 , x2 , x3 , x4 } & InputVertices S = { x1 , x2 , x3 , x4 } ;
let x , y be Element of FTTTTT1 ( n ) ;
p = |[ p `1 / |. p .| - cn , p `2 / |. p .| - cn ]| ;
g * 1_ G = h " * g * h .= g * h .= g * h ;
let p , q be Element of PFuncs ( V , C ) ;
x0 in dom ( x1 - x2 ) /\ dom ( x2 - x3 ) ;
( R qua Function ) " = R " & ( R " ) " = R " ;
n in Seg len ( f /^ ( len p -' 1 ) ) ;
for s being Real st s in R holds s <= s2 & s2 <= 1 ;
rng s c= dom ( ( f2 * f1 ) | X ) ;
synonym for for for for for for for X being Subset of \rm ] holds X is finite ;
1. ( K , n ) * 1. ( K , n ) = 1. ( K , n ) ;
set S = Segm ( A , P1 , Q1 ) , Q1 = Segm ( A , P , Q1 ) ;
ex w st e = ( w - f ) & w in F ;
curry ( ( P+* ( i , k ) ) # x ) is convergent ;
cluster open open -> open for Subset of [: T , T :] ;
len f1 = 1 .= len ( f3 | ( len f1 ) ) .= len f3 ;
( i * p ) / p < ( 2 * p ) / p ;
let x , y be Element of [: the carrier of U0 , the carrier of U0 :] ;
b1 , c1 // b9 , c9 & c1 , c2 // c , c9 ;
consider p being element such that c1 . j = { p } and p in { p } ;
assume that f " { 0 } = {} and f is total ;
assume that IC Comput ( F , s , k ) = n and IC Comput ( F , s , k ) = k ;
Reloc ( J , card I ) does not destroy a , I ;
Macro ( card I + 1 ) does not destroy c , a ;
set m3 = LifeSpan ( p3 , s3 ) , P4 = LifeSpan ( p2 , s2 ) ;
IC SCMPDS in dom ( Initialize p ) & IC SCMPDS in dom ( p +* Start-At ( a , I ) ) ;
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 = 1 ;
let a , b be Element of PFuncs ( V , C ) ;
Cl ( union F ) c= Cl ( Int union F ) ;
the carrier of X1 union X2 misses ( ( X1 union X2 ) \/ ( X2 \/ X3 ) ) ;
assume not LIN a , f . a , g . a ;
consider i being Element of M such that i = d6 and i in dom f ;
then Y c= { x } or Y = {} or Y = { x } ;
M , v |= H1 / ( ( y , v ) / ( x , y ) ) ;
consider m be element such that m in Intersect ( FF . 0 ) and m in dom f ;
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 a1 <> a4 and a2 <> a4 ;
cluster s \! \mathop { \rm \hbox { - } \rm ] } -> string for string of S ;
Lh2 /. n2 = Lh2 . n2 .= Lh2 . n2 .= Lh2 . n2 ;
let P be compact non empty Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
assume that r-7 in LSeg ( p1 , p2 ) and rb in LSeg ( p1 , p2 ) ;
let A be non empty compact Subset of TOP-REAL n , a , b be Real ;
assume that [ k , m ] in Indices ( ( D * ( i , j ) ) * ( i , j ) ) ;
0 <= ( ( 1 / 2 ) |^ p ) / ( 2 |^ n ) ;
( F . N | E8 ) . x = +infty ;
pred X c= Y & Z c= V & X \ V c= Y \ Z ;
( y `2 ) * ( z `2 ) <> 0. I & ( y `2 ) * ( z `2 ) <> 0. I ;
1 + card ( X-18 ) <= card ( u \/ { v } ) ;
set g = z \circlearrowleft ( L~ z ) , M = L~ z , N = L~ z , S = L~ z , N = L~ z , S = L~ z , M = L~ z , N = L~ z , S = z ;
then k = 1 & p . k = <* x , y *> . k ;
cluster -> total for Element of C -O ( X ) ;
reconsider B = A as non empty Subset of ( TOP-REAL n ) | B ;
let a , b , c be Function of Y , BOOLEAN , f be Function of Y , BOOLEAN ;
L1 . i = ( i .--> g ) . i .= g . i .= g . i ;
Plane ( x1 , x2 , x3 , x4 , x4 , x4 , x5 , x5 ) c= P ;
n <= indx ( D2 , D1 , j1 ) + 1 ;
( ( g2 ) . O ) `1 = - 1 & ( ( g2 ) . I ) `1 = 1 ;
j + p .. f -' len f <= len f - len f ;
set W = W-bound C , E = E-bound C , N = E-bound C , S = S-bound C , N = E-bound C , S = S-bound C , N = S-bound C , S = S-bound C , N = S-bound C , S = S-bound C , N = W-bound
S1 . ( a , e ) = a + e .= a + e .= a + e ;
1 in Seg width ( M * ( ColVec2Mx p ) ) ;
dom ( i (#) Im ( f | A ) ) = dom ( Im f ) /\ A ;
( D.: x ) `2 = W . ( a , *' ( a , p ) ) ;
set Q = non |= ( max ( g , f , h ) ) ;
cluster -> many sorted for ManySortedSet of U1 * ;
attr F is discrete means : Def1 : F = { A } ;
reconsider z9 = <* ] as Element of product ( G . i ) ;
rng f c= rng f1 \/ rng f2 & rng ( f1 ^ f2 ) 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 , H ) ;
reconsider n1 = n , n2 = m , n3 = n as Morphism of o1 , o2 ;
assume that P is idempotent and R is idempotent and P (*) R = R ** P ;
card ( B2 \/ { x } ) = k-1 + 1 ;
card ( ( x \ B1 ) /\ B1 ) = 0 & card ( ( x \ B1 ) /\ B2 ) = 0 ;
g + R in { s : g-r - g < s + r } ;
set qII' = ( q , <* s *> ) \rangle , q' = ( q , <* s *> ) ;
for x being element st x in X holds x in rng f1 implies x in X ;
h0 /. ( i + 1 ) = h0 . ( i + 1 ) ;
set mw = max ( B , max ( NAT , { NAT } ) ) ;
t in Seg width ( I ^ ( n , n ) ) ;
reconsider X = dom f /\ C as Element of Fin ( NAT ) ;
IncAddr ( i , k ) = <% - ( l . 0 ) , k %> .= halt SCM+FSA ;
( ( W-bound L~ f ) / 2 ) ^2 <= ( q `2 ) / 2 ;
attr R is condensed means : Def1 : Int R is condensed & Cl R is condensed ;
pred 0 <= a & a <= 1 & b <= 1 implies a * b <= 1 ;
u in ( ( c /\ ( ( d /\ b ) /\ e ) ) /\ f ) /\ j ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) ) /\ f ) /\ j ;
len C + - 2 >= 9 + - 3 - 2 ;
x , z , y is_collinear & x , z , y is_collinear & x , z , y is_collinear ;
a |^ ( n1 + 1 ) = a |^ n1 * a |^ n1 ;
<* \underbrace ( 0 , \dots , 0 } , x ) in Line ( x , a * x ) ;
set yx1 = <* y , c *> , yx2 = <* c , x *> ;
FF2 /. 1 in rng Line ( D , 1 ) & FF2 /. len FF2 = Line ( D , 1 ) ;
p . m joins r /. m , r /. ( m + 1 ) ;
( p `2 ) `2 = ( f /. i1 ) `2 .= ( f /. i1 ) `2 ;
W-bound ( X \/ Y ) = W-bound ( X \/ Y ) & W-bound ( X \/ Y ) = W-bound ( X \/ Y ) ;
0 + ( p `2 ) ^2 <= 2 * r + ( p `2 ) ^2 ;
x in dom g & not x in g " { 0 } ;
f1 /* ( seq ^\ k ) is divergent_to-infty & f2 /* ( seq ^\ k ) is divergent_to-infty ;
reconsider u2 = u as VECTOR of P\overline ( X ) , 0 <= 1 / 2 ;
p |-count ( Product ( Sgm ( X ) ) ) = 0 & p |-count ( Product ( Sgm ( X ) ) ) = 0 ;
len <* x *> < i + 1 & i + 1 <= len c + 1 ;
assume that I is non empty and { x } /\ { y } = { 0. I } ;
set ii = card I + 4 .--> goto 0 , goto ( 0 + 4 ) ;
x in { x , y } & h . x = {} ( TT ) ;
consider y being Element of F such that y in B and y <= x ` ;
len S = len ( the charact of ( A * ) ) .= len the charact of ( A * ) ;
reconsider m = M , i = I , n = N as Element of X ;
A . ( j + 1 ) = B . ( j + 1 ) \/ A . j ;
set N8 = : ( ( G . ( len G ) ) `1 = ( ( G . ( len G ) ) `1 ) ;
rng F c= the carrier of gr { a } & rng F c= the carrier of gr { a } ;
( C * ( K , n , r ) ) is FinSequence of D * ;
f . k , f . ( Let n ) in rng f ;
h " P /\ [#] T1 = f " P & h " P = f " P ;
g in dom ( f2 ) \ ( f2 " { 0 } ) " { 0 } ;
gX /\ dom f1 = g1 " { 0 } & gX /\ dom f2 = g2 " { 0 } ;
consider n be element such that n in NAT and Z = G . n ;
set d1 = being \bf dist ( x1 , y1 ) , d2 = dist ( y1 , y2 ) , d2 = dist ( y2 , y1 ) ;
b `1 + ( 1 - ( b `1 / 2 ) ) ^2 < ( 1 - ( b `1 / 2 ) ) ^2 + ( 1 - ( b `1 / 2 ) ) ^2 ;
reconsider f1 = f as VECTOR of the carrier of X , the carrier of Y ;
pred i <> 0 implies i ^2 mod ( i + 1 ) = 1 ;
j2 in Seg len ( ( g2 . i2 ) | ( i + 1 ) ) ;
dom ( i - 4 ) = dom ( ( i - 4 ) * ( i - 4 ) ) .= a ;
cluster sec | ]. PI / 2 , PI .[ -> one-to-one | ]. PI / 2 , PI .[ ;
Ball ( u , e ) = Ball ( f . p , e ) ;
reconsider x1 = x0 , y1 = x1 as Function of S , IF ( ) ;
reconsider R1 = x , R2 = y , R2 = z , R1 = w as Relation of L , L ;
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 , S2 = S2 +* S2 , S2 = S2 +* S2 ;
( ( ( - 1 / 2 ) (#) ( ( #Z 2 ) * ( f1 + #Z 2 ) ) ) `| Z ) = f ;
cluster -> [. 0 , 1 .] -valued for Function of C , REAL ;
set C7 = 1GateCircStr ( <* z , x *> , f3 ) , C8 = 1GateCircStr ( <* x , y *> , f ) ;
E8 . ( e2 , T . ( e2 , T . ( e2 , T . ( e2 , T . ( e2 , T . ( e2 , T . ( e2 , T . ( k , T . n ) ) ) ) ) ) ) ) ) = E8 . ( E8 . ( E8 , T . (
( ( ( arctan * ( f1 + f2 ) ) `| Z ) * ( f1 + f2 ) ) `| Z ) = f ;
upper_bound A = ( cos * 3 ) / 2 & lower_bound A = 0 ;
F . ( dom f , - F . ( cod f , - F . ( cod f , - F . ( cod f , - F . ( cod f , - F . ( cod f ) ) ) ) ) ) ;
reconsider pbeing Point of TOP-REAL 2 = ( q `2 ) / 2 , q = ( q `2 ) / 2 as Point of TOP-REAL 2 ;
g . W in [#] Y0 & [#] Y0 c= [#] Y0 & [#] Y0 c= [#] Y0 & g .: W c= [#] Y0 ;
let C be compact non vertical non horizontal non empty Subset of TOP-REAL 2 ;
LSeg ( f ^ g , j ) = LSeg ( f , j ) /\ LSeg ( g , i ) ;
rng s c= dom f /\ ]. x0 - r , x0 .[ & ( f | ]. x0 , x0 + r .[ ) is convergent ;
assume x in { idseq 2 , Rev ( idseq 2 ) } ;
reconsider n2 = n , m2 = m , m1 = n , m2 = m , m2 = n , m2 = m ;
for y being ExtReal st y in rng seq holds g <= y implies g . y <= g . y
for k st P [ k ] holds P [ k + 1 ] ;
m = m1 + m2 .= m1 + m2 .= m1 + m2 .= m1 + m2 .= m1 + m2 ;
assume for n holds H1 . n = G . n -H . n ;
set Bf = f .: ( the carrier of X1 ) , Bf = f .: ( the carrier of X2 ) ;
ex d being Element of L st d in D & x << d ;
assume R ~ ( a \ b ) c= R ~ ( b \ a ) ;
t in ]. r , s .[ or t = r or t = s & s < t ;
z + v2 in W & x = u + ( z + v2 ) & y in W ;
x2 |-- y2 iff P [ x2 , y2 ] & P [ x2 , y2 ] ;
pred x1 <> x2 means : Def4 : |. x1 - x2 .| > 0 & |. x1 - x2 .| > 0 ;
assume that p2 - p1 , p3 - p1 , p2 - p1 is_collinear and p1 - p2 , p3 - p1 - p2 is_collinear ;
set q = f ^ <* 'not' A *> ^ <* 'not' A *> ;
let f be PartFunc of REAL-NS 1 , REAL-NS n , x be Point of REAL-NS m , r be Real ;
( n mod ( 2 * k ) ) ^2 = n mod k ;
dom ( T * ( len iff t ) ) = dom ( \mathop { \rm @ } t ) ;
consider x being element such that x in wX and x in c and x in c ;
assume ( F * G ) . ( v . x3 ) = v . x3 ;
assume that the carrier of D1 c= the carrier of D2 and the carrier of D1 c= the carrier of D2 ;
reconsider A1 = [. a , b .[ , A2 = [. a , b .[ as Subset of REAL ;
consider y being element such that y in dom F and F . y = x ;
consider s being element such that s in dom o and a = o . s ;
set p = W-min L~ Cage ( C , n ) , q = W-min L~ Cage ( C , n ) , r = W-bound L~ Cage ( C , n ) ;
n1 -' len f + 1 <= len - 1 + 1 - len f + 1 ;
EqClass ( q , O1 ) = [ u , v , a , b , b , a , b , c , d ] ;
set C-2 = ( ( ( n + 1 ) , n ) `1 ) `1 , C-2 = ( n + 1 ) + 1 ;
Sum ( L (#) p ) = 0. R * Sum p .= 0. V .= 0. V ;
consider i be element such that i in dom p and t = p . i ;
defpred Q [ Nat ] means 0 = Q ( $1 ) & P [ $1 ] implies Q [ $1 ] ;
set s3 = Comput ( P1 , s1 , k ) , P3 = Comput ( P2 , s2 , k ) , P4 = P3 ;
let l be variable of k , A , A1 , A2 be Subset of A ;
reconsider that U = union ( G-24 | A ) as Subset-Family of [: T , T :] ;
consider r such that r > 0 and Ball ( p `1 , r ) c= Q ` ;
( h | ( n + 2 ) ) /. ( i + 1 ) = p29 ;
reconsider B = the carrier of X1 , C = the carrier of X2 as Subset of X ;
por = <* - ( c - 1 ) , 1 *> & p = <* - ( c - 1 ) , 1 *> ;
synonym f is real-valued for rng f c= NAT & rng f c= NAT ;
consider b being element such that b in dom F and a = F . b ;
x9 < card ( X0 ) + card ( Y0 ) - 1 & card ( Y0 ) + 1 < card ( Y0 ) - 1 ;
pred X c= B1 means : Def4 : for B st B c= B1 holds it in succ B1 holds B is non empty ;
then w in Ball ( x , r ) & dist ( x , w ) <= r ;
angle ( x , y , z ) = angle ( x , y , z ) ;
pred 1 <= len s means : an : for i being Nat holds ( the _ of s ) . i = s . i ;
f-47 c= f . ( k + ( n + 1 ) ) ;
the carrier of { 1_ G } = { 1_ G } & the carrier of G = { 1_ G } ;
pred p '&' q in TAUT ( A ) means : Def1 : q in TAUT ( A ) & q in TAUT ( A ) ;
- ( t `1 / t `2 ) < ( - t `1 ) / t `2 / t `2 / t `2 ;
( ( U . 1 ) `1 = ( U /. 1 ) `1 .= ( W /. 1 ) `1 .= W . ( 1 + 1 ) .= W . ( 1 + 1 ) ;
f .: ( the carrier of x ) = the carrier of x & f .: ( the carrier of x ) = the carrier of x ;
Indices ( ( n + 1 ) * ( i , j ) ) = [: 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 multF of ( A , B ) ) & f is \setminus & f is \setminus implies f is \setminus
[ 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 |^ X ) * ;
C \/ P c= [#] ( ( GX | A ) \ ( [#] ( GX | A ) \ A ) ) ;
f " V in ( for X being Subset of \bf T ( ) ) | D ( ) , [ X , the carrier of V ( ) ] in the topology of X ( )
x in [#] ( the carrier of X ) /\ A * delta ( F , x ) ;
g . x <= h1 . x & h . x <= h1 . x & h1 . x <= h1 . x ;
InputVertices S = { xy , y , z } & InputVertices S = { xy , y , z } ;
for n being Nat st P [ n ] holds P [ n + 1 ] ;
set R = Line ( M , i ) * Line ( M , i ) ;
assume that M1 is being_line and M2 is being_line and M3 is being_line and M2 is being_line and M2 is being_line and M2 is being_line ;
reconsider a = f4 . ( i0 -' 1 ) , b = f4 . ( i0 -' 1 ) as Element of K ;
len B2 = Sum ( Len ( F1 ^ F2 ) ) & len ( Len ( F1 ^ F2 ) ) = len F1 + len F2 ;
len ( ( the - ( i , n ) ) * ( p | ( i -' 1 ) ) ) = n & len ( ( p | i ) * ( p | ( i -' 1 ) ) ) = n ;
dom max ( - ( f + g ) , f + g ) = dom ( f + g ) ;
( for n be Nat holds seq . n = upper_bound Y1 ) & ( for n be Nat holds seq . n = upper_bound Y1 ) implies seq is bounded
dom ( p1 ^ p2 ) = dom ( ( f ^ p2 ) ^ ) .= dom ( ( f ^ ) ^ ) ;
M . [ 1 / ( y * v1 ) , y ] = 1 / ( y * v1 ) .= y ;
assume that W is non trivial and W { x } c= the carrier of G2 and W is trivial ;
C6 * ( i1 , i2 ) `1 = G1 * ( i1 , i2 ) `1 .= G1 * ( i1 , i2 ) `1 .= G1 * ( i1 , i2 ) `1 ;
C8 |- 'not' Ex ( x , p ) 'or' p . ( x , y ) ;
for b st b in rng g holds lower_bound rng f-g <= b - a
- ( ( q1 `1 / |. q1 .| - cn ) / ( 1 + cn ) ) ^2 = 1 ;
( LSeg ( c , m ) \/ { c } ) \/ ( LSeg ( l , k ) \/ { c } ) c= R ;
consider p being element such that p in such that p in x and p in L~ f and x in L~ f ;
Indices ( X @ ) = [: Seg n , Seg n :] & len ( X @ ) = n ;
cluster s => ( q => p ) => ( q => ( s => p ) ) -> valid ;
Im ( ( Partial_Sums F ) . m ) is measurable & ( Partial_Sums F ) . m is measurable ;
cluster f . ( x1 , x2 ) -> Element of D * ;
consider g being Function such that g = F . t and Q [ t , g ] ;
p in LSeg ( NW-corner ( Z ) , \hbox { - } corner ( Z ) ) , \hbox { - } corner ( Z ) ) ;
set R8 = R |^ 1 , R8 = R |^ ( b + 1 ) ;
IncAddr ( I , k ) = AddTo ( da , d ) .= IncAddr ( ex a st i = a & d = goto ( d ) ;
seq . m <= ( ( the \upharpoonright } seq ) . k ) . ( n + 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 \/ 21 ) as non empty Subset of U0 ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) /\ f ) ) /\ m ;
consider y being element such that y in Y and P [ y , inf B ] ;
consider A being finite stable set such that card A = 2 and card A = 2 ;
p2 in rng ( f |-- p1 ) \ rng <* p1 *> & rng <* p1 *> c= rng ( f |-- p1 ) ;
len s1 - 1 > 1-1 & len s2 - 1 > 0 & len s2 - 1 > 0 ;
( ( N-min ( P ) ) `2 = ( ( E-max ( P ) ) ) `2 & ( ( E-max ( P ) ) ) `2 = ( ( E-max ( P ) ) ) `2 ;
Ball ( e , r ) c= LeftComp Cage ( C , k + 1 ) \/ LeftComp Cage ( C , k + 1 ) ;
f . a1 ` ` = f . a1 ` .= ( a1 ` ` ) ` .= ( a1 ` ) ` .= ( a1 ` ) ` ;
( seq ^\ k ) . n in ]. x0 - r , x0 .[ & ( seq ^\ k ) . n in ]. x0 , x0 + r .[ ;
gg . s0 = g . s0 | G . s0 .= ( g | G ) . s0 .= ( g | G ) . s0 ;
the InternalRel of S is \lbrace field ( the InternalRel of S ) , the InternalRel of S } ;
deffunc F ( Ordinal , Ordinal ) = phi ( $1 , $2 ) & phi ( $1 , $2 ) = phi ( $1 , $2 ) ;
F . ( s1 . a1 ) = F . ( s2 . a1 ) .= F . ( s2 . a1 ) ;
x `2 = A . o . a .= Den ( o , A . a ) . a ;
Cl ( f " P1 ) c= f " ( Cl P1 ) & Cl ( f " P1 ) c= f " ( Cl P1 ) ;
FinMeetCl ( ( the topology of S ) | ( the topology of S ) ) c= the topology of T ;
synonym o is \bf } means : Def4 : o <> *' & o <> {} & o <> {} ;
assume that X c= Y + 1 and card X <> card Y and card Y <> card X and card Y <> card X ;
the { F ( ) `1 <= 1 + ( the } \HM { F ( ) `1 is } ) ;
LIN a , a1 , d or b , c // b1 , c1 or b , c // b1 , c1 ;
e . 1 = 0 & e . 2 = 1 & e . 3 = 0 & e . 4 = 0 ;
EE in SE & not EE in { N } & not EE in { N } ;
set J = ( l , u ) If , K = I " ;
set A1 = .: ( ( a , b , c ) --> ( p , q ) ) ;
set vs = [ <* xy , cin , cin *> , '&' ] , xy = [ <* A1 , cin *> , '&' ] , [ <* cin , cin *> , '&' ] ;
x * z `1 * x " in x * ( z * N ) " * x " ;
for x being element st x in dom f holds f . x = g3 . x & f . x = g3 . x ;
Int cell ( f , 1 , G ) c= RightComp f \/ RightComp f \/ L~ f \/ L~ f \/ RightComp f ;
U is closed & L~ Cage ( C , n ) c= L~ Cage ( C , n ) & L~ Cage ( C , n ) c= L~ Cage ( C , n ) ;
set f-17 = f @ g "/\" g @ ;
attr S1 is convergent means : Def1 : S2 is convergent & lim ( S1 - S2 ) = 0 & lim ( S1 - S2 ) = 0 ;
f . ( 0 + 1 ) = ( 0 qua Ordinal ) + a .= a .= a ;
cluster -> \in -> \in \in -> \in \in -> \in , reflexive transitive transitive transitive transitive for non empty reflexive -symmetric ;
consider d being element such that R reduces b , d and R reduces c , d ;
not b in dom Start-At ( ( card I + 2 ) , SCMPDS ) \/ dom Start-At ( ( card I + 2 ) , SCMPDS ) ;
( z + a ) + x = z + ( a + y ) .= z + a + y ;
len ( l \lbrack ( a |^ 0 ) .--> x ) = len l & len ( l |^ 0 ) = 0 ;
t4 ^ {} is ( {} \/ rng t4 ) -valued FinSequence of ( rng t ) * * ;
t = <* F . t *> ^ ( C . p ^ q ) .= q ^ ( C . ( p ^ q ) ) ;
set p-2 = W-min L~ Cage ( C , n ) , p-2 = W-min L~ Cage ( C , n ) , p-2 = W-min L~ Cage ( C , n ) ;
( k -' ( i + 1 ) ) -' ( i + 1 ) = ( k - ( i + 1 ) ) - ( i + 1 ) ;
consider u being Element of L such that u = u ` ` and u in D ` ;
len ( ( width ( ( ( b ) |-> a ) ^ ( b * a ) ) ^ ( b * a ) ) ) = width ( ( b * ( b * a ) ) * ( b * a ) ) ;
FF . x in dom ( ( G * the_arity_of o ) * ( the_arity_of o ) ) ;
set cH2 = the carrier of H2 , cH2 = the carrier of H2 ;
set cH1 = the carrier of H1 , cH2 = the carrier of H2 , cH2 = the carrier of H2 ;
( Comput ( P , s , 6 ) ) . intpos ( m + 1 ) = s . intpos ( m + 1 ) ;
IC Comput ( Q2 , t , k ) = ( l + 1 ) - ( k + 1 ) .= ( l + 1 ) - ( k + 1 ) ;
dom ( ( ( id Z ) (#) ( sin * cos ) ) `| Z ) = REAL & dom ( ( id Z ) (#) ( sin * cos ) ) `| Z ) = REAL ;
cluster <* l *> ^ phi -> ( 1 + 1 ) -element for string of S ;
set b5 = [ <* ( ap , bp , cp *> , and2 ] , [ <* A1 , cin *> , and2 ] , [ <* cin , cin *> , and2 ] *> , [ <* A1 , cin *> , and2 ] ] ;
Line ( Segm ( M @ , P @ , Q ) , x ) = L * Sgm Q .= Line ( M @ , Q ) ;
n in dom ( ( the Sorts of A ) * the_arity_of o ) & ( the Sorts of A ) . n = ( the Sorts of A ) . n ;
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 - x .|| <= r ;
set x3 = t2 . DataLoc ( s2 . SBP , 2 ) , x4 = Comput ( s2 , s2 , 2 ) , P4 = Comput ( P2 , s2 , 2 ) , P4 = P3 ;
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 I , pE = stop I , T = stop I , T = stop
consider a being Point of D2 such that a in W1 and b = g . a and a in W2 ;
{ A , B , C , D } = { A , B } \/ { C , D , E }
let A , B , C , D , E , F , J , M , N , M , N , F , N , M , N , M , N , N , F , M , N , N , A , N , M , N , A , N , F , M ;
|. p2 .| ^2 - ( ( p2 `2 / |. p2 .| - cn ) / ( 1 + cn ) ) ^2 >= 0 ;
l -' 1 + 1 = n-1 * ( 1 + ( 1 + 1 ) ) + 1 ;
x = v + ( a * w1 + ( b * w2 ) ) + ( c * w2 ) + ( c * w2 ) ;
the TopStruct of L = \bf L ( the Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott ] , the TopStruct of L ) ;
consider y being element such that y in dom H1 and x = H1 . y and y in dom H1 and P [ y , x ] ;
fv \ { n } = Funcs ( Free ( All ( v1 , H ) ) , E ) & f in Free ( H ) ;
for Y being Subset of X st Y is summable holds Y is non empty iff X is non empty iff Y is non empty ;
2 * n in { N : 2 * Sum ( p | N ) = N & N > 0 } ;
for s being FinSequence holds len ( the { of } holds the { F } * s = ( the { of F ) * s )
for x st x in Z holds exp_R * f is_differentiable_in x & ( exp_R * f ) . x > 0 ;
rng ( h2 * f2 ) c= the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 & rng ( h2 * f2 ) c= the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 ;
j + 1- len f <= len f + ( len - len f ) - len f ;
reconsider R1 = R * I as PartFunc of REAL , REAL-NS n , REAL-NS n ;
C8 . x = s1 . ( a - 1 ) .= C7 . ( a - 1 ) .= C7 . x - 1 .= C8 . x ;
power F_Complex . ( z , n ) = 1 .= ( x |^ n ) * ( z |^ n ) .= ( x |^ n ) * ( z |^ n ) ;
t at ( C , s ) = f . ( the connectives of S , s ) . t ) .= s . ( s . t ) ;
support ( f + g ) c= support f \/ C /\ ( support g ) & support ( f + g ) c= C /\ ( support f ) ;
ex N st N = j1 & 2 * Sum ( ( r4 | N ) | N ) > N & Sum ( ( r4 | N ) | N ) > 0 ;
for y , p st P [ p ] holds P [ All ( y , p ) ]
{ [ x1 , x2 ] where x1 , x2 is Element of [: X , Y :] , x1 is Element of X , Y : x1 in X } is Subset of [: X , Y :] ;
h = ( i , j = h , id ( B . i ) ) .= H . i .= H . i ;
ex x1 being Element of G st x1 = x & x1 * N c= A & x1 * N c= A ;
set X = ( ( EqClass ( q , O1 ) ) . ( ( O1 , O2 ) . ( m , 3 ) ) , 4 ) ;
b . n in { g1 : x0 < g1 & g1 < x0 & g1 < x0 + r } ;
f /* s1 is convergent & f /. x0 = lim ( f /* s1 ) implies lim ( f /* s1 ) = lim ( f /* s1 )
the lattice of the lattice of Y = the lattice of the lattice of Y & the carrier of Y = the carrier of X ;
'not' ( a . x ) '&' b . x 'or' a . x '&' 'not' ( b . x ) '&' 'not' ( b . x ) = FALSE ;
2 = len ( ( ( ( ( 0 ^ r1 ) ^ ( r1 ^ r2 ) ) | ( len r1 ) ) ^ ( ( r1 ^ r2 ) | ( len r1 ) ) ) ) ;
( 1 - a ) * ( sec * f1 ) - id Z is_differentiable_on Z ;
set K1 = upper_volume ( chi ( A , A ) , H ) || A , D = upper_volume ( H , D ) || A ;
assume e in { ( w1 - w2 ) `1 : w1 in F & w2 in G & w1 in G } ;
reconsider d7 = dom a `1 , d8 = dom F `1 , d8 = dom F `1 , d8 = dom G `1 , d8 = dom F `1 , d8 = F `2 , d8 = F `2 , d8 = F `2 , d8 = F `2 , d8 = F `1 , d8 = F `1 , d8 = F `1
LSeg ( f /^ q , j ) = LSeg ( f , j + q .. f -' 1 ) \/ LSeg ( f , j + q .. f -' 1 ) ;
assume that X in { T . ( N2 , K1 ) : h . ( N2 , K1 ) = N2 } ;
assume that Hom ( d , c ) <> {} and <* f , g *> * f1 = <* f , g *> * f2 ;
dom S-34 = dom S /\ Seg n .= dom ( L6 * ( S . n ) ) .= dom ( L6 * ( S . n ) ) ;
x in H |^ a implies ex g st x = g |^ a & g in H & a in H ;
a * ( 0. ( K , n , 1 ) ) = a `1 - ( 0 * n ) .= 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 `1 / |. p .| - cn ) < 0 ) ;
for c holds f . c <= g . c implies f @ c ==> g @ c
dom ( f1 (#) f2 ) /\ X c= dom ( f1 (#) f2 ) /\ X & dom ( f1 (#) f2 ) /\ X c= dom ( f1 (#) f2 ) ;
1 = ( p * p ) * p .= p * ( p * p ) .= p * 1 .= p * 1 ;
len g = len f + len <* x + y *> .= len f + 1 .= len f + 1 .= len f + 1 ;
dom ( F | [: N1 , S :] ) = dom ( F | [: N1 , S :] ) .= [: N1 , S :] ;
dom ( f . t * I . t ) = dom ( f . t * g . t ) ;
assume a in ( "\/" ( ( T |^ the carrier of S ) , F ) ) .: D ;
assume that g is one-to-one and ( the carrier' of S ) /\ rng g c= dom g and g is one-to-one ;
( ( x \ y ) \ z ) \ ( ( x \ z ) \ ( y \ z ) ) = 0. X ;
consider f such that f * f " = id b and f * f = id a and f * f = id b ;
( ( ( ( 2 * cos ) | [. 0 , PI / 2 * PI .] ) * ( cos | [. 0 , PI / 2 * PI .] ) ) | [. 0 , PI / 2 * PI .] ) is increasing ;
Index ( p , co ) <= len LS - Gij .. LS + 1 - 1 .= len LS - 1 + 1 - 1 ;
let t1 , t2 , t2 , t2 be Element of ( S , s ) . ( s , t ) ;
ex_inf_of ( ( Frege ( curry ( F ) ) ) . h , L ) <= "/\" ( ( Frege ( G ) ) . h , L ) , L ) ;
then P [ f . i0 , f . i0 ] & F ( f . i0 , f . i0 ) < j ;
Q [ ( D . [ x , 1 ] ) `1 , F ( D . [ x , 1 ] ) `1 ] ;
consider x being element such that x in dom ( F . s ) and y = ( F . s ) . x ;
l . i < r . i & [ l . i , r . i ] is Element of G . i ;
the Sorts of A2 = ( the carrier of S2 ) --> ( the carrier of S2 ) .= the carrier' of S2 ;
consider s being Function such that s is one-to-one and dom s = NAT and rng s = F . 0 and rng s c= { 0 } ;
dist ( b1 , b2 ) <= dist ( b1 , a ) + dist ( a , b ) & dist ( a , b ) <= dist ( a , b ) + dist ( a , b ) ;
( <* W-min ( C , n ) *> /. len <* Cage ( C , n ) /. len ( C , n ) *> ) = WC ;
q `2 <= ( UMP ( Lower_Arc ( C ) ) ) `2 & ( UMP ( C ) ) `2 <= ( UMP ( C ) ) ) `2 ;
LSeg ( f | i2 , i ) /\ LSeg ( f | i2 , j ) = {} & LSeg ( f | i2 , j ) = {} ;
given a being R_eal such that a <= II and A = ]. a , I .[ and a in A and a < I ;
consider a , b being complex number such that z = a & y = b and z + y = a + b ;
set X = { b |^ n where n is Element of NAT : n <= k } , Y = { b |^ n : n <= k } ;
( ( x * y * z \ x ) \ z ) \ ( x * y \ z ) = 0. X ;
set xy = [ <* xy , y , z *> , f1 ] , yz = [ <* y , z *> , f2 ] , yz = [ <* z , x *> , f3 ] , xy = [ <* y , z *> , f3 ] ;
lv /. len lv = lv . ( len lv + 1 ) .= lv . ( len lv + 1 ) ;
( ( q `2 / |. q .| - cn ) / ( 1 + cn ) ) ^2 = 1 ;
( ( - ( p `2 / |. p .| - cn ) ) / ( 1 + cn ) ) ^2 < 1 ;
( ( ( S \/ Y ) | ( X \/ Y ) ) | ( X \/ Y ) ) `2 = ( ( S \/ Y ) | ( X \/ Y ) ) `2 ;
( seq - seq ) . k = seq . k - seq . ( k + 1 ) .= seq . k - seq . ( k + 1 ) ;
rng ( ( h + c ) ^\ n ) c= dom SVF1 ( 1 , f , u0 ) /\ dom SVF1 ( 1 , f , u0 ) ;
the carrier of X is the carrier of X & the carrier of X is the carrier of X implies X is closed & the carrier of X is closed & the carrier of X is closed
ex p4 st p3 = p4 & |. p3 - a .| = r & |. p3 - a .| = r ;
set ch = chi ( X , A5 ) , A5 = chi ( X , A5 ) , A5 = chi ( X , A5 ) ;
R |^ ( 0 * n ) = I--> Element of I-set ( X , X ) * ( R |^ 0 ) .= R |^ n |^ 0 ;
( Partial_Sums ( curry ( F , n ) ) . n ) is nonnegative & ( Partial_Sums ( curry ( F , n ) ) . n ) is nonnegative ;
f2 = C7 . ( ( E7 , len ( K , len ( K , len ( H ) ) ) ) | ( K K ) ) ;
S1 . b = s1 . b .= S2 . b .= S2 . b .= S2 . b .= S2 . b .= S2 . b .= S2 . b ;
p2 in LSeg ( p2 , p1 ) /\ LSeg ( p1 , p11 ) \/ LSeg ( p1 , p11 ) /\ LSeg ( p1 , p11 ) ;
dom ( f . t ) = Seg n & dom ( I . t ) = Seg n & dom ( I . t ) = Seg n ;
assume o = ( the connectives of S ) . 11 & the connectives of S = ( the connectives of S ) . 11 ;
set phi = ( l1 , l2 ) support phi , phi = ( l1 , l2 ) . 0 , C = l1 , D = ( l1 , l2 ) . 1 , D = ( l1 , l2 ) . 2 , E = ( l1 , l2 ) . 2 , F = ( l1 , l2 ) . 3 , D = ( l1 , l2 ) . 2 , E = ( l1 , l2 ) . 2 , E =
synonym p is is is is is is invertible for p , q , T being Element of L ;
( Y1 `2 ) ^2 = - 1 & ( Y1 `2 ) ^2 <> 0. TOP-REAL 2 & ( Y1 `2 ) ^2 <> 0. TOP-REAL 2 ;
defpred X [ Nat , set , set ] means P [ $1 , $2 , , , , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 ,
consider k being Nat such that for n being Nat st k <= n holds s . n < x0 + g ;
Det ( I |^ ( m -' n ) ) * ( m -' n ) = 1. K & Det ( I |^ ( m -' n ) ) = 1. K ;
( - b - sqrt ( b ^2 - 4 * a * c ) ) / 2 < 0 ;
Ci . d = C7 . ( ( d - e ) mod ( C |^ ( d -' 1 ) ) ) .= C7 . ( ( d - e ) mod ( C |^ ( d -' 1 ) ) ) ;
attr X1 is dense means : Def1 : X2 is dense & X1 is dense dense implies X1 /\ X2 is dense SubSpace of X ;
deffunc F6 ( Element of E , Element of I ) = ( 2 * $1 ) * ( ( 2 * $1 ) * ( 2 * $1 ) ) ;
t ^ <* n *> in { t ^ <* i *> : Q [ i , T . t `1 ] } ;
( 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 X is Basis of [: X , Y :] iff X is Basis of [: X , Y :]
synonym A , B are_separated for ( Cl A ) misses B & ( Cl B ) misses B & ( Cl B ) misses B ;
len ( ( M @ ) @ ) = len p & width ( ( M @ ) @ ) = width ( ( M @ ) @ ) ;
J = { x where x is Element of K : 0 < v . x & v . x < 0 } ;
( ( Sgm -> Element of REAL m ) . d - ( Sgm , m ) . e ) <> 0 ;
lower_bound divset ( D2 , k + ( k2 - 1 ) ) = D2 . ( k + ( k2 - 1 ) ) ;
g . r1 = - 2 * r1 + 1 & dom h = [. 0 , 1 .] & rng h = [. 0 , 1 .] ;
|. a .| * ||. f .|| = 0 * ||. f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| ;
f . x = ( h . x ) `1 & g . x = ( h . x ) `2 & h . x = ( h . x ) `2 ;
ex w st w in dom B1 & <* 1 *> ^ s = <* 1 *> ^ w & <* 1 *> ^ w = <* 1 *> ^ w ;
[ 1 , {} , <* d1 *> ] in ( { [ 0 , {} , {} ] } \/ S1 ) \/ S2 ;
IC Exec ( i , s1 ) + n = IC Exec ( i , s2 ) .= IC Exec ( i , s2 ) .= IC Exec ( i , s2 ) ;
IC Comput ( P , s , 1 ) = DataLoc ( s . 9 , 9 ) .= ( IC s ) .= 5 .= ( 0 + 9 ) ;
( IExec ( W6 , Q , t ) ) . intpos ( ( intpos ( e + 1 ) ) + 1 ) = t . intpos ( ( e + 1 ) + 1 ) ;
LSeg ( f /^ q , i ) misses LSeg ( f /^ q , j ) \/ LSeg ( f /^ q , j ) ;
assume for x , y being Element of L st x in C & y in C holds x <= y or y <= x ;
integral ( integral ( C , f ) , x ) = f . ( upper_bound C ) - lower_bound C .= f . ( lower_bound C ) - 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 ) - R /. ( L . h ) .|| < e1 * ( K + 1 ) ;
assume a in { q where q is Element of M : dist ( z , q ) <= r } ;
set p4 = [ 2 , 1 .--> [ 2 , 1 , 1 ] , p4 = [ 3 , 1 , 1 ] ;
consider x , y being Subset of X such that [ x , y ] in F and x c= d and y in d ;
for y , x being Element of REAL st y `1 in Y & x `2 in X holds y `1 <= x `1 & y `2 <= x `2 ;
func |. p \vert -> variable of A , NAT means : Def1 : for i being Nat st i in dom it holds it . i = min ( NBI ( p ) , i ) ;
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 y1 = len l1 & len x1 = len l1 & ( x1 - x2 ) * ( y1 - y2 ) = ( x1 - x2 ) * ( y1 - y2 ) ;
consider y2 being Real such that x2 = y2 and 0 <= y2 and y2 < 1 and y2 <= 1 ;
||. f | X /* s1 .|| = ||. f | X /. s1 - f /. s1 .|| .= ||. f /. s1 - f /. s1 .|| .= ||. f /. s1 - f /. s1 .|| ;
( the InternalRel of A ) ~ ~ ( x ` /\ Y ) = {} \/ {} .= {} \/ {} .= {} .= {} ;
assume that i in dom p and for j being Nat st j in dom q holds P [ i , j ] and i + 1 in dom p and i + 1 in dom q and j + 1 in dom q ;
reconsider h = f | [: X , Y :] as Function of [: X , Y :] , rng ( f | [: X , Y :] ) ;
u1 in the carrier of W1 & u2 in the carrier of W2 & v in the carrier of W1 & u in the carrier of W2 implies v + u1 in the carrier of W1
defpred P [ Element of L ] means M <= f . $1 & f . $1 <= f . $1 ;
IExec ( 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 ;
fSet = [ [ dom @ ( f2 , g2 ) , cod ( f2 , g2 ) ] , [ cod ( f2 , g2 ) , cod ( f2 , g2 ) ] , [ cod ( f2 , g2 ) , cod ( f2 , g2 ) ] ] ;
for k , n being Nat st k <> 0 & k < n & k < n holds k , n are_relative_prime holds k divides n
for x being element holds x in A |^ d iff x in ( ( A ` ) |^ d ) ` & ( ( A ` ) |^ d ) ` = ( ( A ` ) |^ d ) `
consider u , v being Element of R , a being Element of A such that l /. i = u * a and a in I ;
- ( ( p `1 / |. p .| - cn ) / ( 1 + cn ) ) ^2 > 0 ;
L-13 . k = Ln . ( F . k ) & F . k in dom ( Ln * F ) ;
set i2 = AddTo ( a , i , - n ) , i1 = goto ( - ( n + 1 ) + 1 ) ;
attr B is max means : Def4 : for S being SubSubuniversal of B holds S is ( len @ ( B , S-13 ) ) `1 ;
a9 "/\" D = { a "/\" d where d is Element of N : d in D & a in D } ;
|( \square , q29 - ( q `2 / |. q .| - cn ) * ( b `1 / |. q .| - cn ) * ( b `2 / |. q .| - cn ) * ( b `2 / |. q .| - cn ) * ( b `2 / |. q .| - cn ) * ( b `2 / |. q .| - cn ) * ( b `2 / |. q .| - cn ) * b ) ;
( - f ) . ( upper_bound A ) = ( ( - f ) | A ) . ( upper_bound A ) .= - ( f | A ) . ( upper_bound A ) ;
G * ( len G , k ) `1 = G * ( len G , k ) `1 .= G * ( len G , k ) `1 .= G * ( len G , k ) `1 ;
( Proj ( i , n ) ) . LM = <* ( proj ( i , n ) ) . ( LM . LM ) *> ;
f1 + f2 * reproj ( i , x ) is_differentiable_in ( ( - 1 ) (#) ( reproj ( i , x ) ) * reproj ( i , x ) ) . x0 ;
pred ( for x st x in Z holds tan . x <> 0 ) & ( for x st x in Z holds tan . x = tan . x ) ;
ex t being SortSymbol of S st t = s & h1 . t = h2 . t & h2 . x = ( h2 . t ) . x ;
defpred C [ Nat ] means P8 . $1 is non empty & ( A is as $1 -connected implies A is $1 -connected ) ;
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 product ( A * ( index B ) ) ;
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 D holds ( id c ) . ( id c ) = id d
( ( f | n ) ^ <* p *> ) = ( f | n ) ^ <* p *> .= f ^ <* p *> .= 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 - p = ( f | ( n , L ) ) *' - ( f | ( n , L ) ) .= ( f - ( c * ( - ( f | ( n , L ) ) ) ) ) ) ;
consider r being Real such that r in rng ( f | divset ( D , j ) ) and r < m + s ;
f1 . ( |[ ( 8 - r ) / 2 , ( 8 - r ) / 2 ]| ) in f1 .: W1 & f2 .: W2 c= W2 ;
eval ( a | ( n , L ) , x ) = eval ( a | ( n , L ) ) .= a * ( a | ( n , L ) ) .= a ;
z = DigA ( tw , x9 ) .= DigA ( tw , x9 ) .= DigA ( tw , x9 ) .= DigA ( tw , x9 ) .= DigA ( tw , x9 ) ;
set H = { Intersect S where S is Subset-Family of X : S c= G & S c= G } , F = { Intersect S where S is Subset of X : S c= G } ;
consider S19 being Element of D * , d being Element of D * such that S `1 = S19 ^ <* d *> and S `2 = d ;
assume that x1 in dom f and x2 in dom f and f . x1 = f . x2 and f . x2 = f . x2 ;
- 1 <= ( q `2 / |. q .| - cn ) / ( 1 + cn ) ;
0. V is Linear_Combination of A & Sum ( l ) = 0. V & Sum ( l ) = 0. V & Sum ( l ) = 0. V ;
let k1 , k2 , k2 , x4 , k2 , 7 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 be Element of NAT ;
consider j being element such that j in dom a and j in g " { k } and x = a . j and a . j in { k } ;
H1 . x1 c= H1 . x2 or H1 . x2 c= H1 . x2 or H1 . x1 c= H1 . x2 & H1 . x2 c= H1 . x2 & H1 . x2 c= H1 . x2 ;
consider a being Real such that p = \rbrace * p1 + ( a * p2 ) and 0 <= a and a <= 1 ;
assume that a <= c and c <= d and [' a , b '] c= dom f and [' a , b '] c= dom g ;
cell ( Gauge ( C , m ) , len Gauge ( C , m ) -' 1 , 0 ) is non empty ;
A5 in { ( S . i ) `1 where i is Element of NAT : i <= n & n <= k } ;
( T * b1 ) . y = L * b2 /. y .= ( F * b1 ) . y .= ( F * b1 ) . y .= ( F * b1 ) . y ;
g . ( s , I ) . x = s . y & g . ( s , I ) . y = |. s . x - s . y .| ;
( log ( 2 , k + k ) ) ^2 >= ( log ( 2 , k + 1 ) ) ^2 / ( 2 |^ k ) ;
then p => q in S & not x in the still of p & not p in S & p => q in S ;
dom ( the InitS of r-10 ) misses dom ( the InitS of rM ) & dom ( the InitS of rM ) misses dom ( the InitS of rM ) ;
synonym f is extended real means : Def4 : for x being set st x in rng f holds x is R_eal ;
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 + len <* x *> .= len p3 + 1 .= len p3 + 1 ;
( l . ( 1 , 3 ) ) `1 = ( g . ( 1 , 3 ) ) `1 + ( k - 1 ) * ( k - 1 ) ;
CurInstr ( P2 , Comput ( P2 , s2 , l2 ) ) = halt SCM+FSA .= halt SCM+FSA .= halt SCM+FSA .= halt SCM+FSA ;
assume for n be Nat holds ||. seq . n .|| <= ( ||. seq .|| ) . n & ( ||. seq .|| ) . n <= ( ||. seq .|| ) . n ;
sin . st cos . st = sin r * cos ( ( cos r ) * sin ( ( cos r ) * sin ( ( cos r ) * sin ( ( cos r ) * sin ( r ) ) ) ) ) .= 0 ;
set q = |[ g1 `1 / ( ( t `2 / t `1 ) * ( t `2 / t `1 ) ) , g2 `2 / ( t `2 / t `1 ) * ( t `2 / t `1 ) ]| ;
consider G being sequence of S such that for n being Element of NAT holds G . n in implies F . n in let
consider G such that F = G and ex G1 st G1 in SM & G in SM & F in SM & G in SM ;
the root of [ x , s ] in ( the Sorts of Free ( C , X ) ) . s & ( the Sorts of Free ( C , X ) ) . s in ( the Sorts of Free ( C , X ) ) . s ;
Z c= dom ( exp_R * ( ( exp_R * f1 ) + ( exp_R * f1 ) ) ^2 ) ;
for k be Element of NAT holds r0 . k = ( upper_volume ( f , S ) . k ) * ( ( Im f ) . k )
assume that - 1 < n and q `2 > 0 and ( q `1 / |. q .| - cn ) < 0 and q `1 < 0 and q `2 < 0 ;
assume that f is continuous and a < b and a < d and f = g and f is continuous and a = c and f . a = d ;
consider r being Element of NAT such that s-> Element of NAT such that s-> Element of NAT and r <= q and r <= q ;
LE f /. ( i + 1 ) , f /. j , L~ f , f /. ( len f ) , f /. ( len f ) , f /. ( len f ) ;
assume that x in the carrier of K and y in the carrier of K and ex_inf_of x , y , K and inf { x , y } = x ;
assume that f +* ( i1 , \xi ) in ( proj ( F , i2 ) ) " ( ( proj ( F , i2 ) ) " ( ( Carrier ( F , i2 ) ) . ( i + 1 ) ) ) and f is one-to-one ;
rng ( ( ( Flow 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 X } ;
consider l be Nat such that for m be Nat st l <= m holds ||. s1 . m - f /. x0 .|| < g ;
consider t be VECTOR of product G such that m5 = ||. D5 . t .|| and ||. t - t .|| <= 1 ;
assume that the carrier of v = 2 and v ^ <* 0 *> in dom p and v ^ <* 1 *> in dom p and p . 1 = 0 ;
consider a being Element of the Points of Xbe such that a on the carrier of [ k , A ] and a on A and a on B ;
( - x ) |^ ( k + 1 ) * ( ( - x ) |^ ( k + 1 ) ) " = 1 ;
for D being set st for i st i in dom p holds p . i in D holds p . i in D & p . i in D
defpred R [ element ] means ex x , y st [ x , y ] = $1 & P [ x , y ] & P [ y , x ] ;
L~ ( f2 ) = union { LSeg ( p0 , p1 ) , p1 , p2 , p1 , p2 } & LSeg ( p11 , p1 , p2 ) c= { p1 , p2 } ;
i -' len h11 + 2 - 1 < i -' len h11 + 2 - 1 + 1 - 1 + 1 + 1 - 1 + 1 ;
for n being Element of NAT st n in dom F holds F . n = |. ( F . n -' 1 ) .| / ( n -' 1 ) ;
for r , s1 , s2 , r , s2 , r , s , t being Real holds s1 in [. r , s .] iff r <= s2 & s2 <= r & r <= s
assume that v in { G where G is Subset of T2 : G in B2 & G c= B1 & G c= B1 & x in G & y in B2 } ;
let g be \vert succ A , INT , b be Element of INT , X be set , f be Function of X , INT , b be Function of b , INT ;
min ( g . [ x , y ] , k ) . [ y , z ] = ( min ( g , k , x ) ) . y ;
consider q1 being sequence of CH such that for n holds P [ n , q1 . n ] and P [ n + 1 ] ;
consider f being Function such that dom f = NAT & for n being Element of NAT holds f . n = F ( n ) and for x being Element of X holds f . x = F ( n ) ;
reconsider B-6 = B /\ B , E8 = O , E8 = O , E8 = O as Subset of B ;
consider j being Element of NAT such that x = the the assume that 1 <= j and j <= n and 1 <= j and j <= n ;
consider x such that z = x and card ( x . O2 ) in card ( x . O2 ) and x in L1 . O2 and x in L2 . O2 ;
( C * _ 4 ( k , n2 ) ) . 0 = C . ( ( _ T4 ( k , n2 ) ) . 0 ) .= C . ( ( _ T4 ( k , n2 ) ) . 0 ) ;
dom ( X --> rng f ) = X & dom ( ( X --> f ) | X ) = dom ( X --> f ) & rng ( ( X --> f ) | X ) = dom f ;
( ( S-bound L~ SpStSeq C ) `1 <= ( ( N ) `1 ) / 2 & ( ( SpStSeq C ) `1 ) / 2 <= ( ( N ) `1 ) / 2 ;
synonym x , y are_collinear means : Def2 : x = y or ex l being Nat st { x , y } c= l & x in l ;
consider X be 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 st a = x & b = y holds a << y iff a << b ;
( 1 / 2 * ( ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( 1 / 2 ) ) ) ) ) ) `| REAL ) = f ;
defpred P [ Element of omega ] means ( ( for n holds ( ( A1 . n ) . $1 ) | ( ( A1 . $1 ) . n ) ) | ( ( A1 . $1 ) . n ) ) & ( ( A1 . $1 ) | ( ( A1 . $1 ) . n ) is bounded ;
IC Comput ( P , s , 2 ) = succ IC Comput ( P , s , 1 ) .= ( ( IC Comput ( P , s , 1 ) ) + 1 ) .= 6 + 1 .= 6 ;
f . x = f . g1 * f . g2 .= f . ( g1 * g2 ) .= f . ( g1 * g2 ) .= f . ( g1 * g2 ) .= f . ( g1 * g2 ) .= f . ( g1 * g2 ) ;
( M * ( F . n ) ) . n = M . ( ( canFS ( Omega ) ) . n ) .= M . ( ( canFS ( Omega ) ) . n ) .= M . ( ( canFS ( Omega ) ) . n ) ;
the carrier of ( L1 + L2 ) c= ( the carrier of L1 ) \/ ( the carrier of L2 ) & the carrier of ( L1 + L2 ) c= the carrier of L2 ;
pred a , b , c , x , y , z , a , b , c , d , b , c , d , e , x , y , z , y , z , x , y ;
( the PartFunc of s , X ) . n <= ( the PartFunc of s , X ) . n * s . n & ( the partial of s ) . n <= ( the partial of s ) . n ;
pred - 1 <= r & r <= 1 & ( - 1 ) * ( ( - 1 ) * ( 1 - r ) ) = - ( 1 / r ) * ( 1 - r ) ;
seq in { p ^ <* n *> where n is Nat : p ^ <* n *> in T1 & p ^ <* n *> in T1 } ;
|[ x1 , x2 , x3 , x4 ]| . 2 - |[ y1 , y2 , x4 ]| . 2 = x2 - |[ y1 , y2 , x4 ]| . 2 - |[ y1 , y2 , x4 ]| . 2 - |[ y1 , y2 , x4 ]| . 2 - |[ y1 , y2 , x4 ]| . 2 - |[ y1 , y2 , x4 ]| . 2 = y2 - y1 ;
attr m is nonnegative means : Assume for n be Nat holds F . n is nonnegative & ( Partial_Sums F ) . n is nonnegative ;
len ( ( \overline G ) * ( z ) ) = len ( ( ( G ) * ( x , y ) ) + ( ( G ) * ( y , z ) ) ) .= len ( ( G ) * ( x , y ) ) ;
consider u , v being VECTOR of V such that x = u + v and u in W1 /\ W2 and v in W2 /\ W3 and u in W2 /\ W3 ;
given F be FinSequence of NAT such that F = x and dom F = n and rng F c= { 0 , 1 } and Sum F = k ;
0 = 1 * 0 * 0 ^2 + 1 * ( 1 - ( 1 - ( 1 - 0 ) ) * ( 1 - ( 1 - 0 ) ) * ( 1 - 0 ) ) ;
consider n be Nat such that for m be Nat st n <= m holds |. ( f # x ) . m - lim ( f # x ) .| < e ;
cluster -> as being being being being being being non empty \rbrace means : such that the carrier of ( \mathop { \rm o } _ 2 ) , ( \mathop { \rm o } _ 3 ) ) is Boolean and ( \mathop { \rm o } _ 3 ) is Boolean ;
"/\" ( BB , L ) = Top ( B , L ) .= Top ( S , L ) .= "/\" ( I , L ) .= "/\" ( I , L ) .= "/\" ( I , L ) .= "/\" ( I , L ) ;
( r / 2 ) ^2 + ( r / 2 ) ^2 <= ( r / 2 ) ^2 + ( r / 2 ) ^2 ;
for x being element st x in A /\ dom ( ( f `| X ) `| A ) holds ( ( f `| X ) `| X ) . x >= r2
2 * r1 - ( 2 * r1 - ( 2 * r1 - ( 2 * r1 - c ) ) ) / 2 = 0. TOP-REAL 2 ;
reconsider p = P * ( \square , 1 ) , q = a " * ( ( - 1 ) * ( - 1 ) ) * ( ( - 1 ) * ( 1 / ( n + 1 ) ) * ( 1 / ( n + 1 ) ) ) as FinSequence of K ;
consider x1 , x2 being element such that x1 in uparrow s and x2 in downarrow t and x = [ x1 , x2 ] and x2 in { [ x1 , x2 ] } ;
for n be Nat st 1 <= n & n <= len q1 holds q1 . n = ( upper_volume ( g , ( M1 . n ) ) | 1 ) . n
consider y , z being element such that y in the carrier of A and z in the carrier of A and i = [ y , z ] and i = [ y , z ] ;
given H1 , H2 being strict Subgroup of G such that x = H1 and y = H2 and H1 is Subgroup of H2 and H2 is strict Subgroup of H1 and H2 is strict Subgroup of H2 ;
for S , T , S being non empty RelStr , d being Function of T , S st T is complete & d is directed-sups-preserving holds d is monotone & d is monotone
[ a + 0 , 0. F_Complex , b + ( - a ) ] in ( the carrier of V ) \ ( the carrier of V ) & [ a + ( - a ) , b + ( - a ) ] in ( the carrier of V ) \ ( the carrier of V ) ;
reconsider mm = max ( len F1 , len ( p . n ) * ( p . n ) |^ ( n + 1 ) ) as Element of NAT ;
I <= width GoB ( ( GoB h ) * ( len GoB h , 1 ) , ( GoB h ) * ( len GoB h , 1 ) ) , ( GoB h ) * ( len GoB h , 1 ) ) ;
f2 /* q = ( f2 /* ( f1 /* s ) ) ^\ k .= ( f2 * f1 ) /* s .= ( f2 * f1 ) /* s .= ( f2 * f1 ) /* s .= ( f2 * f1 ) /* s ;
attr A1 \/ A2 is linearly-independent means : Assume : A1 is linearly-independent & A2 misses A1 & ( for x being Element of V st x in A1 holds x in A2 ) & ( x in A1 & x in A2 holds x in A1 ) ;
func A -carrier C -> set means : Def4 : for s being Element of R , x being Element of C st x in A holds it . s in { A . s where s is Element of R : s in C } ;
dom ( Line ( v , i + 1 ) (#) ( ( Cage ( p , m ) * ( p , 1 ) ) * ( ( Cage ( p , m ) * ( p , 1 ) ) * ( p , 1 ) ) ) ) = dom ( F ^ ) ;
cluster [ ( x `1 ) `1 , ( x `2 ) `2 , ( x `2 ) `2 ] -> to [ x `1 , ( x `2 ) `1 , ( x `2 ) `2 ] ;
E , All ( x1 , All ( x2 , ( x2 '&' ( x1 '&' ( x1 '&' x2 ) ) ) '&' ( x1 '&' x2 ) ) ) ) |= All ( x1 , x2 '&' ( x1 '&' ( x1 '&' x2 ) ) '&' ( x1 '&' x2 ) ) ;
F .: ( id X , g ) . x = F . ( id X , g . x ) .= F . ( id X , g . x ) .= F . ( x , g . x ) .= F . ( x , g . x ) ;
R . ( h . m ) = F . x0 + h . ( m + 1 ) - h . ( m + 1 ) - h . ( m + 1 ) ;
cell ( G , ( X -' 1 , Y -' 1 ) , ( t + 1 ) + ( t + 1 ) ) \ ( ( L~ f ) ` ) meets ( ( L~ f ) ` ) ;
IC Result ( P2 , s2 ) = IC Comput ( P2 , s2 , Initialize Comput ( P2 , s2 , Initialize ( ( intloc 0 ) ) + 1 ) , Comput ( P2 , s2 , Initialize ( ( intloc 0 ) .--> 1 ) ) , 1 ) ) .= ( card I + 2 ) ;
sqrt ( 1 - ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 ) > 0 ;
consider x0 being element such that x0 in dom a and x0 in g and y = ( g " { k } ) . x0 and x0 in dom g and g . x0 = a . x0 ;
dom ( r1 (#) ( chi ( A , A ) ) | ( m , C ) ) = dom ( chi ( A , A ) ) /\ ( dom ( chi ( A , A ) ) ) .= C /\ ( dom ( chi ( A , A ) ) ) .= C /\ ( dom ( chi ( A , A ) ) ) .= C ;
d-7 . [ y , z ] = ( ( y `1 ) . z ) `2 - ( ( y `2 ) . z ) `2 .= ( ( y `2 ) . z ) `2 - ( y `2 ) . z ;
attr i being Nat means : Def4 : C . i = A . i /\ B . i & C . i c= C . i /\ B . i ;
Suppose x0 in dom f and f is_continuous_in x0 and f is_continuous_in x0 and for r st r in dom f & x0 in dom f holds ||. f /. x0 - f /. x0 .|| < r ; Then f | X is continuous & lim ( f | X ) = lim ( f | X ) ;
p in Cl A implies for K being Basis of p , Q being Subset of T st Q in K & K is open holds A meets Q
for x be Element of REAL n st x in Line ( x1 , x2 ) holds |. y1 - y2 .| <= |. y1 - y2 .| & |. y1 - y2 .| <= |. y1 - y2 .|
func Sum <*> a -> w *> means : Def1 : a in it & for b being Ordinal st a in it holds it . b c= b & a is Ordinal ;
[ a1 , a2 , a3 ] in ( the carrier of A ) ~ & [ a1 , a2 , a3 ] in ( the carrier of A ) ~ & [ a1 , a2 , a3 ] in ( the carrier of A ) ~ ;
ex a , b being element st a in the carrier of S1 & b in the carrier of S2 & x = [ a , b ] & [ a , b ] in the InternalRel of S2 ;
||. ( ( vseq . n ) - ( vseq . m ) ) * ( x - ( vseq . m ) ) * ( x - ( vseq . m ) ) ) .|| < \frac { e / 2 * ||. x - ( vseq . m ) .|| * ||. x - ( vseq . m ) .|| + ||. x - ( vseq . m ) .|| * ||. x - ( vseq . m ) .|| ;
then for Z be set st Z in { Y where Y is Element of I7 : F ( Z ) c= Z & Z in { Y } } holds z in Z ;
sup compactbelow [ s , t ] = [ sup ( ( compactbelow s ) . ( sup compactbelow s ) ) , sup ( ( compactbelow s ) . ( sup compactbelow s ) ) ] .= sup ( ( compactbelow s ) . ( sup compactbelow s ) ) ;
consider i , j being Element of NAT such that i < j and [ y , f . j ] in Ij and [ f . i , f . j ] in Ij and [ y , f . j ] in Ij ;
for D being non empty set , p , q being FinSequence of D st p c= q & p ^ q = q holds p ^ q = q ^ ( p ^ q )
consider e19 being Element of the affine of X such that c9 , a9 // a9 , e and a9 , e // e , b and a9 <> e ;
set U2 = I \! \mathop { \rm \hbox { - } F } , E = S \! \mathop { - } F , F = S \! \mathop { - } E ;
|. q3 .| ^2 = ( ( q3 `1 ) ^2 + ( q3 `2 ) ^2 ) * ( ( q3 `1 ) ^2 + ( q3 `2 ) ^2 ) .= |. q .| ;
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 ) . ( x , y ) ) = dom ( the charact of U1 ) & Args ( o , ( MSAlg U1 ) . ( x , y ) ) = dom ( O ;
dom ( h | X ) = dom h /\ X .= dom ( ( |. h .| ) | X ) /\ X .= dom ( ( |. h .| ) | X ) .= dom ( ( |. h .| ) | X ) .= dom ( ( |. h .| ) | X ) ;
for N1 , N1 , N2 being Element of ( the carrier of G ) * , N1 , N2 being Element of ( the carrier of G ) * st N1 = N & N2 = N1 & N2 = N2 holds N1 is open & N2 is open
( mod ( u , m ) + mod ( v , m ) ) . i = ( mod ( u , m ) ) . i + ( mod ( v , m ) . i ) ;
- ( q `1 ) ^2 < - 1 or q `1 / |. q .| * ( 1 + cn ) & - ( q `1 / |. q .| * ( 1 + cn ) ) ^2 <= 1 ;
pred r1 = ( f | X ) & r2 = ( f | X ) . r1 & r1 = ( f | X ) . r2 & r2 = ( f | X ) . r2 ;
vseq . m is bounded Function of X , the carrier of Y & x9 . m = ( seq_id ( vseq . m , X , Y ) ) . x & x9 . m = ( seq_id ( vseq . m , X , Y ) ) . x ;
pred a <> b & b <> c & angle ( a , b , c ) = PI & angle ( b , c , a ) = 0 ;
consider i , j being Nat , r being Real such that p1 = [ i , r ] and p2 = [ j , r ] and r < j and j < i ;
|. p .| ^2 - ( 2 * |( p , q )| ) ^2 + |. q .| ^2 = |. p .| ^2 + |. q .| ^2 ;
consider p1 , q1 being Element of X ( ) * such that y = p1 ^ q1 and q1 ^ q1 = p1 ^ q1 and p1 ^ q1 = q1 ^ q1 and q1 ^ q1 = q1 ^ q1 and q1 ^ q1 = q1 ^ q1 ;
gcd ( ( A , r1 , r2 , s1 , s2 , s1 , s2 , t2 ) , A , B , C , D ) = ( s2 * ( s1 , s2 ) ) / ( s2 * ( s1 , s2 ) ) ;
( ex w being Real st w = lower_bound ( proj2 .: ( A /\ holds w in S ) ) & ( proj2 .: ( A /\ \mathop { \rm VerticalLine } ( w ) ) ) is non empty ) & ( proj2 .: ( A /\ \mathop { \rm VerticalLine } ( w ) ) is non empty ) ;
s , ( ( k + 1 ) |= H1 ) iff s |= Ex ( H2 , ( k + 1 ) |= H2 ) & ( for n holds s |= ( H2 ) . n ) ;
len ( s + 1 ) + 1 = card ( support b1 ) + 1 .= card ( support b2 ) + 1 .= card ( support b1 ) + 1 .= card ( support b2 ) + 1 .= card ( support b1 ) + 1 .= card ( support b2 ) + 1 .= card ( support b2 ) ;
consider z being Element of L1 such that z >= x and z >= y and for z being Element of L1 st z >= x & z >= y holds z `1 >= y ;
LSeg ( UMP D , |[ ( W-bound D + E-bound D ) / 2 , ( E-bound D + E-bound D ) / 2 ]| , ( E-max D ) / 2 ) /\ D = { UMP D } ;
lim ( ( ( f `| N ) / g ) /* b ) = lim ( ( f `| N ) / g /* b ) .= lim ( ( f `| N ) / g /* b ) ;
P [ i , pr1 ( f ) . i , pr1 ( f ) . i , pr1 ( f ) . ( i + 1 ) ] & pr1 ( f ) . ( i + 1 ) = pr1 ( f ) . i ;
for r be Real st 0 < r ex m be Nat st for k be Nat st m <= k holds ||. ( seq . k - ( lim seq ) ) .|| < 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 & a in P & b in P holds a = b
Z c= dom ( ( ( 1 / 2 ) (#) ( ( #Z 2 ) * f ) ) `| Z ) \ ( ( ( 1 / 2 ) (#) ( ( #Z 2 ) * f ) `| Z ) " { 0 } ) " { 0 } ) ;
ex j be Nat st j in dom ( l ^ <* x *> ) & j < i & y = ( l ^ <* x *> ) . j & z = 1 + j & y = ( l ^ <* x *> ) . j & z = 1 + 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 N
A , Int A , Cl ( A , B ) , Cl ( A , B ) , Cl ( A , B ) , Cl ( A , C ) , Cl ( A , B ) , Cl ( A , C ) , Cl ( A , B ) ) / 2 ;
- Sum <* v , u , w *> = - ( v + u + w ) .= - ( v + u ) + u .= - ( v + u ) + u .= - ( v + u ) ;
( Exec ( a := b , s ) ) . IC SCM R = ( Exec ( a := b , s ) ) . IC SCM R .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s ;
consider h being Function such that f . a = h and dom h = I and for x being element st x in I holds h . x in ( the carrier of J ) . x ;
for S1 , S2 , S2 , D being non empty reflexive RelStr , D being non empty Subset of S1 , f being Function of S1 , S2 , g being Function of S2 , S2 , D holds ( f * g ) * f is directed & ( f * g ) * f is directed implies ( f * g ) * f is directed
card X = 2 implies ex x , y st x in X & y in X & x <> y & x <> y or x = y & y = z or x = z & y = z or z = x or z = y ;
E-max L~ Cage ( C , n ) in rng ( Cage ( C , n ) \circlearrowleft W-min L~ Cage ( C , n ) ) & W-min L~ Cage ( C , n ) in rng ( Cage ( C , n ) \circlearrowleft W-min L~ Cage ( C , n ) ) ;
for T , T1 , p being tree , q being Element of dom T holds p element q = q implies ( T -tree ( p , T ) ) . q = T . q & ( T -tree ( p , T ) ) . 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 ) ;
cluster -> natural for Nat , k be Nat , n be Nat , m be Nat st k divides m & n divides k & m divides k holds ( m divides n ) & ( m divides k implies n divides k ) ;
dom F " = the carrier of X2 & rng F = the carrier of X1 & F " = ( the carrier of X2 ) | ( the carrier of X2 ) & F " is one-to-one ;
consider C being finite Subset of V such that C c= A and card C = n and the carrier of V = Lin ( B9 \/ C ) and C is linearly-independent and A is linearly-independent and B 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= V or Y c= V
set X = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } , Y = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } ;
angle ( p1 , p3 , p4 ) = 0 .= angle ( p2 , p3 , p4 ) .= angle ( p3 , p2 , p4 ) .= angle ( p3 , p2 , p4 ) .= angle ( p3 , p2 , p4 ) .= angle ( p3 , p2 , p4 ) ;
- sqrt ( 1 - ( ( q `1 / |. q .| - cn ) / ( 1 - cn ) ) ^2 ) = - ( ( q `1 / |. q .| - cn ) / ( 1 - cn ) ) ^2 .= - ( q `1 / |. q .| - cn ) / ( 1 - cn ) ;
ex f being Function of I[01] , TOP-REAL 2 st f is continuous one-to-one & rng f = P & f . 0 = p1 & f . 1 = p2 & f . 0 = p3 & f . 1 = p2 & f . 1 = p4 ;
attr f is partial differentiable on on on on on 2 , u0 means : Def4 : SVF1 ( 2 , pdiff1 ( f , 1 ) , u0 ) /. u = ( proj ( 2 , 3 ) * pdiff1 ( f , 1 ) ) . u0 ;
ex r , s st x = |[ r , s ]| & G * ( len G , 1 ) `1 < r & r < G * ( 1 , 1 ) `1 & G * ( len G , 1 ) `2 < s & s < G * ( 1 , 1 ) `2 ;
assume that f is FinSequence of TOP-REAL 2 and 1 <= t and t <= len G and G * ( t , width G ) `1 >= ( GoB f ) * ( 1 , width G ) `1 and G * ( t , width G ) `1 >= ( GoB f ) * ( 1 , width G ) `1 ;
pred i in dom G means : Def4 : r * ( f * reproj ( G , i ) ) = r * f * reproj ( i , x ) ;
consider c1 , c2 being bag of o1 + o2 such that ( decomp c ) /. k = <* c1 , c2 *> and c = c1 + c2 and c1 in dom ( p + q ) and c2 in dom ( p + q ) ;
u0 in { |[ r1 , s1 ]| : r1 < G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `1 < r1 & r1 < G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `2 < s1 & s1 < G * ( 1 , 1 ) `2 } ;
Cl ( X ^ Y ) = the carrier of X .= C4 . ( k2 + 1 ) .= C4 . ( k2 + 1 ) .= C4 . ( k2 + 1 ) .= C4 . ( k2 + 1 ) .= C4 . ( k2 + 1 ) ;
pred len M1 = len M2 & width M1 = width M2 & width M2 = width M2 & width M1 = width M2 & width M2 = width M2 & width M2 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width 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 N } c= N2 & N c= N2 & ( ex N be Neighbourhood of x0 st ||. y - x0 .|| < g2 & N c= dom f ) ;
assume that x < ( - b + sqrt ( a , b , c ) ) / 2 and x > ( - b - sqrt ( a , b , c ) ) / 2 ;
( G1 '&' G2 ) . i = ( <* 3 *> ^ G1 ) . i & ( H1 '&' G1 ) . i = ( <* 3 *> ^ G1 ) . i & ( H1 ^ G1 ) . i = ( <* 3 *> ^ G1 ) . i ;
for i , j st [ i , j ] in Indices ( ( M3 + M1 ) * ( i , j ) ) holds ( ( M3 + M2 ) * ( i , j ) ) * ( 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 holds i divides f /. i & i divides len f holds i divides f /. ( i -' 1 )
assume F = { [ a , b ] where a , b is Element of X : for c st c in B\bf holds a c= c & b c= c } ;
b2 * q2 + ( b3 * q3 ) + ( ( a * q2 ) + ( a * q3 ) + ( a * q2 ) ) = 0. TOP-REAL n + ( a * q2 ) .= ( a * q2 ) + ( a * q2 ) .= ( a * q2 ) + ( a * q2 ) .= ( a * q2 ) + ( a * q2 ) ;
Cl ( Cl F ) = { D where D is Subset of T : ex B being Subset of T st B = Cl ( D ) & B in F & A c= B } & F is closed & B is closed & A c= Cl ( B \/ C ) ;
attr seq is summable means : Assume for n be Nat holds seq . n is summable & seq is summable & ( for m be Nat st m <= n holds seq . m - seq . n = Sum ( seq ^\ m ) ) & ( for n be Nat holds seq . n = Sum ( seq ^\ n ) ) ;
dom ( ( cn " ) | D ) = ( the carrier of ( TOP-REAL 2 ) | D ) | D .= the carrier of ( ( TOP-REAL 2 ) | D ) | D .= D ;
X [ X \to Z ] is full full non empty full SubRelStr of ( Omega Z ) |^ the carrier of Z & X [ Y \to Z ] ;
G * ( 1 , j ) `2 = G * ( i , j ) `2 & G * ( 1 , j ) `2 <= G * ( 1 , j ) `2 & G * ( 1 , j ) `2 <= G * ( 1 , j ) `2 ;
synonym m1 c= m2 for for for p , q being set st p in P & q in P & ( for m being set st m in P holds ( m1 , m ) is_as Element of NAT ) holds p in ( m2 , m ) `1 ;
consider a being Element of B ( ) such that x = F ( a ) and a in { G ( b ) where b is Element of A ( ) : P [ b ] } and a in A ( ) ;
synonym the multMagma of K is $1 -element for the multF of K is non empty multMagma means : Let : for a being Element of the carrier of K holds a is Element of the carrier of K & a is Element of the carrier of K ;
sequence ( a , b , 1 ) + sequence ( c , d ) = b + sequence ( c , d ) .= b + ( c + d ) .= >= ( a + c + d ) .= sequence ( a + c + d ) .= the carrier of ( a + c ) ;
cluster + ( i1 , i2 ) -> strict for Element of INT , i , i2 , j be Element of INT , i1 , i2 , j1 being Element of NAT st i1 = i2 & i2 = j1 holds ( i1 + i2 ) + ( i2 + 1 ) = ( i1 + i2 ) + ( i2 + 1 ) ;
- ( s2 * p1 + ( s2 * p2 - ( s2 * p2 ) ) ) = ( ( - s2 * p1 ) + ( s2 * p2 ) ) - ( s2 * p2 ) .= ( ( - s2 ) * p1 ) + ( s2 * p2 ) ;
eval ( ( a | ( n , L ) ) *' p , x ) = eval ( a | ( n , L ) ) * eval ( p , x ) .= a * eval ( p , x ) .= a * eval ( p , x ) ;
assume that the TopStruct of S = the TopStruct of T and for D being non empty directed Subset of Omega S , V being Subset of Omega S st D in V holds V meets V ;
assume that 1 <= k and k <= len w + 1 and TU . ( ( q11 , w ) -\mathop { \rm \hbox { - } and w in w . k ) and TU . ( ( q11 , w ) -\mathop { - } ) = ( TU . ( ( q11 , w ) -\mathop { \rm \hbox { - } U } ) ) . k ;
2 * a |^ ( n + 1 ) + ( 2 * b |^ ( n + 1 ) ) >= ( a |^ n + ( 2 * a |^ n ) + ( 2 * a |^ n ) ) + ( 2 * a |^ n ) ;
M , v2 |= All ( x. 3 , All ( x. 0 , All ( x. 4 , All ( x. 0 , All ( x. 0 , All ( x. 0 , All ( x. 0 , H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H ) ) ) ) ) ;
assume that f is_differentiable_on l and for x0 st x0 in l holds 0 < f . x0 or for x0 st x0 in l holds f . x0 < 0 ;
for G1 being _Graph , W being Walk of G1 , e being set st e in W & e in W holds W is Walk of G1 & e in W implies W is Walk of G2
not vs is empty iff ( ( m1 is not empty & not ( m1 is not empty & not ( m1 is not empty & not ( m1 is not empty & not ( m1 is not empty & not ( m1 is not empty & not ( m1 is not empty & not ( m1 is not empty & not not ( m1 is not empty ) & not ( m1 is not empty ) ) ) ) ) ) & not ( not ( m1 is not empty & not ( m1 is not empty ) ) ;
Indices GoB f = [: dom GoB f , Seg width GoB f :] & ( GoB f ) * ( i1 , 1 ) in dom GoB f & ( GoB f ) * ( i1 , 1 ) in dom GoB f & ( GoB f ) * ( i1 , 1 ) in dom GoB f ) ;
for G1 , G2 , G2 , G3 being non empty RelStr , O being strict Subgroup of G , f being Function of O , G2 , g being Function of O , G2 , h being Function of O , G2 , g being Function of O , G2 , f being Function of O , G2 , g being Function of O , G2 st f is stable & g is stable holds f * g is stable
UsedIntLoc ( int t ) = { intloc 0 , intloc 1 , intloc 2 , intloc 2 , intloc 3 , 1 , 1 , 1 , 1 , 0 , 1 , 1 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 0 , 1 , 0 , 0 , 1 , 0 , 0 , 1 , 0 , 0 , 1 , 0 , 0 , 1 , 0 , 1 , 0 , 0 , 1 , 0 , 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 , 1 , 1
for f1 , f2 being FinSequence of F st f1 ^ f2 is p -element & Q [ f1 ^ f2 ] & Q [ f2 ^ f1 ] holds Q [ f1 ^ f2 ^ f2 ] & Q [ f1 ^ f2 ^ f2 ]
( ( p `1 ) ^2 + ( p `2 ) ^2 ) = ( q `1 ) ^2 + ( q `2 ) ^2 .= ( q `1 ) ^2 + ( q `2 ) ^2 ;
for x1 , x2 , x3 , x4 being Element of REAL n holds |( x1 - x2 , x3 - x4 )| = |( x1 , x2 , x3 - x4 )| & |( x1 , x2 , x3 - x4 )| = |( x1 , x2 , x3 - x4 )|
for x st x in dom ( ( F | A ) | A ) holds ( ( F | A ) | A ) . ( - x ) = - ( ( F | A ) . x )
for T being non empty TopSpace , P being Subset of T , B being Subset of T st P c= the topology of T & B is open holds P is Basis of T
( a 'or' b 'imp' c ) . x = 'not' ( ( a 'or' b ) . x 'or' c . x ) 'or' c . x .= TRUE 'or' ( a 'or' b ) . x .= TRUE 'or' TRUE .= TRUE ;
for e be set st e in A8 ex X1 being Subset of X , Y1 being Subset of Y st e = X1 & X1 is open & Y1 is open & Y1 is open & p in X1 & p in Y1 & Y1 is open & p in Y1 & p in Y1 & p in Y1 & p in Y1 & p in Y1 & p in Y1 & p in Y1 & p in Y1 & p in Y1 & p in Y1 & p in Y1 & p in Y1 & p in Y1 & p in Y1 & p in Y1 & p in Y1 & p in Y1 & p in Y1 & p in Y1 & p in Y1 & p in Y1 & p in Y1 & p in Y1 & p in Y1 & p in Y1 & p in Y1 & p in
for i be set st i in the carrier of S for f being Function of [: S , S :] , S1 . i st f = H . i & F . i = f | ( F . i ) holds F . i = f | ( F . i )
for v , w st for y st x <> y holds w . y = v . y holds Valid ( VERUM ( Al , J ) , J ) . v = Valid ( VERUM ( Al , J ) , J ) . w
card D = card ( D1 - 1 ) - card ( { i , j } - 1 ) .= ( c1 + 1 ) - ( c2 - 1 ) .= ( c1 + 1 ) - ( c1 - 1 ) .= 2 * ( c1 + 1 ) - ( c1 - 1 ) .= 2 * ( c1 + ( c2 - 1 ) ) .= 2 * ( c1 + ( c1 - 1 ) - ( c1 - 1 ) ) .= 2 * ( c1 + c2 - 1 ) ;
IC Exec ( i , s ) = ( s +* ( 0 .--> ( s . 0 ) ) ) . 0 .= ( ( 0 .--> ( s . 0 ) ) . 0 ) .= ( ( 0 .--> s . 0 ) +* ( 0 .--> 1 ) ) . 0 .= ( ( 0 .--> 1 ) +* ( 0 .--> 1 ) ) . 0 .= succ IC s .= ( 0 .--> 1 ) . 0 .= ( 0 .--> 1 ) . 0 ;
len f /. ( \downharpoonright i1 -' 1 ) -' 1 + 1 = len f -' ( i1 -' 1 ) + 1 - 1 .= len f -' ( i1 -' 1 ) + 1 - 1 .= len f -' ( i1 -' 1 ) + 1 - 1 .= len f -' ( i1 -' 1 ) + 1 - 1 ;
for a , b , c being Element of NAT st 1 <= a & 2 <= b holds a <= a + b or a = b + b-2 or a = b + b-2 or a = b + b-2 or a = b + b-2 or a = b + b-2 or a = b + b-2
for f being FinSequence of TOP-REAL 2 , p being Point of TOP-REAL 2 , f being FinSequence of TOP-REAL 2 st p in LSeg ( f , i ) & f is FinSequence of TOP-REAL 2 holds Index ( p , f ) <= len f
lim ( curry ( ( curry . ( 0 + 1 ) ) # x ) ) = lim ( ( curry ' ( 0 , k ) ) # x ) + lim ( ( curry ' ( 0 , k ) ) # x ) ) ;
z2 = g /. ( \downharpoonright n1 -' n2 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) .= ( g | n1 ) . ( i -' n1 + 1 ) .= ( g | n1 ) . i ;
[ f . 0 , f . 3 ] in id ( the carrier of G ) \/ ( the InternalRel of G ) or [ f . 0 , f . 3 ] in the InternalRel of C & [ f . 0 , f . 3 ] in the InternalRel of C ;
for G being Subset-Family of B st G = { R [ X ] where R is Subset of A , Y is Subset of B ( ) : R in F & Y in F } holds ( for X being Subset of A ( ) , Y being Subset of B ( ) st X in F holds P [ X , Y ] )
CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 ) ) = CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m1 ) ) .= halt SCMPDS .= halt SCMPDS ;
assume that a on M and b on M and c on N and d on N and p on M and d on N and a on M and c on M and d on N and a on M and b on M and a on M and c on N and d on M and d on N and a <> b and a on M and a on M and a on M and a on M ;
Suppose T is \hbox { T _ 4 4 } and for F being Subset-Family of T , F being Subset-Family of T st F is closed & F is closed & F is closed holds ind F <= 0 and ind F <= 0 ;
for g1 , g2 st g1 in ]. ( r - g ) / 2 , r + g .[ & |. ( f - g ) / 2 - ( r - g ) / 2 .| <= ( g1 - g ) / 2 - ( r - g ) / 2
exp ( ( ( - 1 / 2 ) * ( ( - 1 / 2 ) * ( ( - 1 / 2 ) * ( ( - 1 / 2 ) * ( ( - 1 / 2 ) * ( ( - 1 / 2 ) * ( ( - 1 / 2 ) * ( ( - 1 / 2 ) * ( ( - 1 / 2 ) * ( ( - 1 / 2 ) * ( ( - 1 / 2 ) * ( 1 / 2 ) ) ) ) ) ) ) ) ) ) ) ) = ( ( - 1 / 2 ) * ( ( - 1 / 2 ) * ( ( - 1 / 2 ) ) ) / ( 1 / 2 ) ) ) * ( ( - 1 / 2 ) * ( ( - 1 / 2 ) * ( ( - 1 ) ) ) ) ) ) = ( ( - 1 / 2 ) ) * ( ( ( - 1 / 2 ) ) * ( (
F . i = F /. i .= 0. R + r2 .= ( b |^ ( n + 1 ) ) .= ( b |^ ( n + 1 ) ) * ( a |^ ( n + 1 ) ) .= ( b |^ ( n + 1 ) ) * ( a |^ ( n + 1 ) ) .= ( b |^ ( n + 1 ) ) * ( a |^ ( n + 1 ) ) ;
ex y be set , f be Function st y = f . n & dom f = NAT & for n holds f . ( n + 1 ) = Rn & for x holds f . ( n + 1 ) = Rn . ( x , y ) ;
func f (#) F -> FinSequence of V means : Def1 : len it = len F & for i be Nat st i in dom it holds it . i = F /. i * ( F /. i ) * ( F /. i ) ;
{ x1 , x2 , x3 , x4 , x4 , x5 , x5 } = { x1 , x2 , x3 , x4 } \/ { x1 , x2 , x3 , x4 } .= { x1 , x2 , x3 , x4 } \/ { x4 , x4 , x5 } .= { x1 , x2 , x3 , x4 } \/ { x4 , x4 , x5 } ;
for n being Nat , x being set st x = h . n & h . ( n + 1 ) = o ( x , n ) & x in InputVertices S & h . n in InnerVertices S & x in InputVertices S & x in InputVertices S & y in InputVertices S & x in InputVertices S & y in InputVertices S & z in InputVertices S & x in InputVertices S ;
ex S1 being Element of CQC-WFF ( Al ( ) ) st SubP ( P , l , e ) = S1 & ( for k being Nat st k < l holds ( ( P , l ) . k = P . k ) & ( P [ k ] implies ( P [ k ] implies P [ k ] ) & ( P [ k + 1 ] implies P [ k + 1 ] ) ;
consider P being FinSequence of G|^ 2 such that p9 = Product P and for i being Element of NAT st i in dom P ex t being Element of Seg k st P . i = t & t in Seg k & P . i = ( t . i ) * ( t . i ) ;
for T1 , T2 being strict non empty TopSpace , P being Subset of T1 , T1 , T2 being Subset of T2 st the topology of T1 = the topology of T2 & P = the topology of T1 & P is open holds P is open & P is open & P is open implies P is open
Suppose f is_is_partial u0 coordinate coordinate coordinate u , 3 and r (#) pdiff1 ( f , 3 ) is_partial_differentiable_in u0 , 3 and partdiff ( r (#) pdiff1 ( f , 3 ) , 3 ) = r * pdiff1 ( f , 3 ) , 3 ) and partdiff ( r (#) pdiff1 ( f , 3 ) , 3 ) = r * pdiff1 ( f , 3 ) ;
defpred P [ Nat ] means for F , G being FinSequence of bool ( the carrier of V ) , G be Permutation of V st len F = $1 & G = F * s holds F = G * s & G = F * s ;
ex j st 1 <= j & j < width GoB f & ( GoB f ) * ( 1 , j ) `1 <= s & s <= ( GoB f ) * ( 1 , j + 1 ) `1 & ( GoB f ) * ( 1 , j + 1 ) `1 <= ( GoB f ) * ( 1 , j + 1 ) `1 ;
defpred U [ set , set ] means ex F-23 being Subset-Family of T st $1 = F-23 & $2 is open & ( union F ) is open & ( union F is open & ( union F ) is open & ( union F is open implies $2 is open ) ) & ( union F is open implies $2 is open ) ;
for p4 being Point of TOP-REAL 2 st LE p4 , p1 , P & LE p4 , p1 , P & LE p4 , p1 , P & LE p4 , p1 , P & LE p4 , p1 , P holds LE p4 , p1 , P
f in D ( ) & for g st g in D ( ) & x <> y holds g in D ( ) implies f in D ( ) & g in D ( ) & for y st y in D ( ) holds g . y in D ( )
ex 8 being Point of TOP-REAL 2 st x = 8 & ( ( ( ( ( ( ( ( ( ( x `1 ) | D ) | D ) | D ) ) | K1 ) ) | K1 ) ) & ( ( ( ( ( ( ( TOP-REAL 2 ) | D ) | D ) | K1 ) ) | K1 ) | K1 ) . 8 >= 0 ;
assume for d7 being Element of NAT st d7 <= ( n -\hbox { t } ) holds s1 . ( ( n -\hbox { t } ) * ( n + 1 ) ) = s2 . ( ( n -\hbox { t } ) * ( n + 1 ) ) ;
assume that s <> t and s is Point of Sphere ( x , r ) and s is Point of Sphere ( x , r ) and ex e being Point of E st e = Ball ( x , r ) /\ Sphere ( x , s ) and e in Sphere ( x , r ) /\ Sphere ( y , r ) ;
given r such that 0 < r and for s st 0 < s ex x1 , x2 be Point of CNS st x1 in dom f & x2 in dom f & ||. x1 - x2 .|| < s & ||. f /. x1 - f /. x2 .|| < r ;
( p | x ) | ( p | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | | | | | | | | | | | | | | | | | | ( x | | | | | | | | | | | | | | | | ( x | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
assume that x , x + h in dom sec and ( for x st x in dom sec holds ( ( sec * sec ) `| Z ) . x = ( 4 * sin ( x + h ) ) ^2 ) / ( sin ( x ) ) ^2 and sin ( x ) = - ( 4 * sin ( x ) ) ^2 and sin ( x ) = - ( 4 * sin ( x ) ) ^2 ) ;
assume that i in dom A and len A > 1 and B c= 1 and i in dom B and A * ( i , j ) = ( A * ( i , j ) ) * ( B * ( i , j ) ) and A * ( i , j ) = A * ( i , j ) and A * ( i , j ) = A * ( i , j ) ;
for i be non zero Element of NAT st i in Seg n holds i divides n or i = <* 1. F_Complex *> & i <> 0. F_Complex & i <> 0. F_Complex & i <> 0. F_Complex & i <> 0. F_Complex & i <> 0. F_Complex & i <> 0. F_Complex & i <> 0. F_Complex & i <> 0. F_Complex holds i divides ( <* 1. F_Complex *> )
( ( ( b1 'imp' b2 ) '&' ( c1 'imp' c2 ) ) '&' ( ( ( a1 'or' b1 ) '&' ( c1 '&' c2 ) ) '&' ( ( a1 'or' a2 ) '&' ( c1 '&' c2 ) '&' ( a1 'or' c2 ) ) '&' ( ( a1 'or' a2 ) '&' ( c1 '&' c2 ) ) '&' ( ( a1 'or' a2 ) '&' ( a1 'or' c2 ) '&' ( a1 'or' c2 ) ) ) ) ) ) '&' ( ( a1 'or' a2 ) '&' ( a1 'or' c2 ) ) '&' ( a1 'or' a2 ) '&' ( a1 'or' c2 ) ) '&' ( a1 'or' c2 ) '&' ( a1 'or' c2 ) '&' ( a1 'or' a2 ) '&' ( a1 'or' a2 ) '&' ( a1 'or' a2 ) '&' ( a1 'or' a2 ) '&' ( a1 'or' c2 ) '&' ( a1 'or' c2 ) '&' ( a1 'or' a2 ) '&' ( a1 'or' c2 ) ) '&' ( ( a1 'or' c2 ) '&' ( a2 'or' c2 ) '&' ( a2 'or' c2 )
assume that for x holds f . x = ( ( - cot ) (#) ( sin - cot ) ) `| Z and for x st x in Z holds ( ( - cot ) (#) ( cot - cot ) ) `| Z ) . x = - cos . x and for x st x in Z holds ( ( - cot ) (#) ( cot - cot ) ) `| Z ) . x = - cos . x ;
consider R8 , I-8 be Real such that R8 : R8 : dom ( Re F ) = Integral ( M , ( Im F ) . n ) and Integral ( M , ( Im F ) . n ) = Integral ( M , ( Im F ) . n ) and Integral ( M , ( Im F ) . n ) = Integral ( M , ( Im F ) . n ) ;
ex k be Element of NAT st ( ex q be Element of NAT st q = k & 0 < d & for q be Element of product G st q in X & q in X holds ||. partdiff ( f , q , k ) - partdiff ( f , x , k ) .|| < r ) ;
x in { x1 , x2 , x3 , x4 , x4 , x5 , x5 , x5 , x5 , 7 } } iff x in { x1 , x2 , x3 , x4 , x5 } \/ { x4 , 8 , 7 } \/ { x1 , x2 , x3 , x4 } \/ { x1 , x2 , x4 } \/ { x1 , x2 , x3 , x4 }
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 , \mathop { \rm len } G ) `2 .= G * ( 1 , 1 ) `2 .= G * ( 1 , \mathop { \rm width } G ) `2 .= G * ( 1 , 1 ) `2 .= G * ( 1 , 1 ) `2 .= G * ( 1 , \mathop { \rm width G * ( 1 , \mathop { \rm width G * ( 1 , \mathop { \rm width G * ( 1 , \mathop { \rm width G * ( 1 , \mathop { \rm width G * ( 1 , \mathop { 1 , \mathop { \rm width G * ( j , \mathop { \rm width G ) `2 .= G * ( j , \mathop { \rm width G ) `2 ) `2 .=
f1 * p = p .= ( ( the Arity of S1 ) * the Arity of S1 ) * ( the Arity of S2 ) .= ( the Arity of S1 ) * the Arity of S1 .= ( the Arity of S1 ) * the Arity of S1 .= ( the Arity of S1 ) * the Arity of S1 .= ( the Arity of S1 ) * the Arity of S1 ;
func tree ( T , P , T1 ) -> Tree means : such : q in T & q in T & P [ T ] or ex p , q st p in P & q in T1 & p in T1 & q in T1 & p in T1 & q in T1 & p ^ q in T1 ;
F /. ( k + 1 ) = F . ( k + 1 -' 1 ) .= Fcontradiction ( p . ( k + 1 -' 1 ) , k -' 1 ) .= F9 . ( k + 1 -' 1 ) .= F9 . ( k + 1 -' 1 ) .= F9 . ( k + 1 -' 1 ) .= F9 . ( k + 1 -' 1 ) ;
for A , B , C being Matrix of len C , len B , K st len B = len C & len C = len B & len C = len A & len B > 0 & len C > 0 holds A * ( B * C ) = A * BC & A * ( B * C ) = B * BC
seq . ( k + 1 ) = 0. F_Complex .= ( seq . ( k + 1 ) ) * seq . ( k + 1 ) .= ( Partial_Sums seq ) . ( k + 1 ) * seq . ( k + 1 ) .= ( Partial_Sums seq ) . ( k + 1 ) * seq . ( k + 1 ) .= ( Partial_Sums seq ) . ( k + 1 ) * seq . ( k + 1 ) .= ( Partial_Sums seq ) . k ;
assume that x in ( the carrier of CQ ) ~ and y in ( the carrier of CQ ) ~ and z in ( the carrier of CQ ) ~ and y in ( the carrier of CQ ) ~ and z in ( the carrier of CQ ) ~ ;
defpred P [ Element of NAT ] means for f st len f = $1 holds ( for k st k in $1 holds ( VAL g ) . k = ( VAL g ) . ( k + 1 ) ) & ( for k st k in $1 holds ( VAL g ) . k = ( VAL g ) . ( k + 1 ) ) ;
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 + 1 , j ) and f /. k = G * ( i + 1 , j ) and f /. k = G * ( i + 1 , j ) ;
assume that cn < 1 and q `1 > 0 and q `1 / |. q .| >= cn and q `1 / |. q .| >= cn and q `1 / |. q .| = cn and q `2 / |. q .| = cn and q `1 / |. q .| and q `1 / |. q .| = cn and q `2 / |. q .| = cn and q `1 / |. q .| = cn and q `1 / |. q .| = cn ;
for M being non empty metric , x being Point of M , f being Function of M , M st x = x `1 holds ex x being Point of M st f . x = Ball ( x , 1 / ( n + 1 ) ) & f . x = Ball ( x `1 , 1 / ( n + 1 ) )
defpred P [ Element of omega ] means f1 is_differentiable_on Z & f2 is_differentiable_on Z & ( for x st x in Z holds f1 . x = - ( f1 . x ) / ( f2 . x ) ^2 ) & ( f1 - f2 ) is_differentiable_on Z implies f1 - f2 is_differentiable_on Z & ( f1 - f2 ) `| Z = ( f1 - f2 ) `| Z ) ;
defpred P1 [ Nat , Point of CNS ] means ( for r be Real st r in Y & ( for s be Real st s in Y holds ||. f /. s - f /. x0 .|| < r ) & ( for s be Real st s in X holds ||. f /. s - f /. x0 .|| < r ) implies f /. s - f /. x0 .|| < r ;
( f ^ mid ( g , 2 , len g ) ) . i = ( mid ( g , 2 , len g ) ) . ( i - 1 ) .= g . ( i -' 1 ) .= g . ( i -' 1 ) .= g . ( i -' 1 ) .= g . ( i -' 1 ) .= ( g | ( i -' 1 ) ) . i ;
( 1 - 2 * ( n0 + 2 * ( n + 2 ) ) ) * ( 2 * ( n + 2 ) ) = ( ( 1 - 2 * ( n + 2 ) ) * ( 2 * ( n + 1 ) ) ) * ( 2 * ( n + 1 ) ) .= 1 * ( ( n + 2 ) * ( n + 1 ) ) * ( 2 * ( n + 1 ) ) ) .= 1 * ( n + 1 ) ;
defpred P [ Nat ] means for G being non empty finite strict RelStr , H being strict non empty finite RelStr st G is free for x being Element of G st x is as non empty RelStr , n being Element of NAT holds ( the carrier of G ) . n = ( the RelStr of H ) . n ;
assume that not f /. 1 in Ball ( u , r ) and 1 <= m and m <= len f and for i st 1 <= i & i <= len f holds not ( f /. i in Ball ( u , r ) & not ( f /. i in Ball ( u , r ) & not ( ex m st m <= n & f /. m in Ball ( u , r ) ) ;
defpred P [ Element of NAT ] means ( Partial_Sums ( cos , ( - 1 / ( $1 + 1 ) ) * ( cos . ( - 1 / ( $1 + 1 ) ) * ( cos . ( - 1 / ( $1 + 1 ) ) * ( cos . ( - 1 / ( $1 + 1 ) ) * ( cos . ( - 1 / ( $1 + 1 ) ) * ( cos . ( - 1 / ( $1 + 1 ) ) ) * ( cos . ( - 1 / ( $1 + 1 ) ) ) ) ) ) . x ) ) . x ) ) . $1 ) = ( Partial_Sums ( cos . ( - 1 / ( cos . ( - 1 / ( $1 + 1 ) ) ) . x ) ;
for x being Element of product F , i being set st x in I & ( for x being set st x in I holds x in I ) & ( for x being set st x in I holds x in I ) holds x in I ) & ( for i being set st i in I holds x in I ) implies x in I )
( x " ) |^ ( n + 1 ) = ( x " ) |^ n * x " .= ( x " ) |^ n * x " .= ( x " ) |^ n * x .= ( x " ) |^ n * x .= ( x " ) |^ n * x .= ( x " ) |^ n ;
DataPart Comput ( P +* I , ( Initialize s ) +* I , ( Initialize s ) +* I , ( Initialize s ) +* I , ( Initialize s ) +* I ) = DataPart Comput ( P +* I , ( Initialize s ) +* I , ( Initialize s ) +* I ) .= DataPart Comput ( P +* I , ( Initialize s ) +* I , ( Initialize s ) +* I ) ;
given r such that 0 < r and ]. x0 - r , x0 .[ c= dom ( f1 (#) f2 ) /\ ]. x0 , x0 + r .[ and for g st g in ]. x0 , x0 + r .[ /\ ]. x0 , x0 + r .[ holds f1 . g <= ( f1 (#) f2 ) . g ;
Suppose X c= dom f1 /\ dom f2 and f1 | X is continuous and f2 | X is continuous and ( f1 - f2 ) | X is continuous and ( f1 - f2 ) | X is continuous and f2 | X is continuous and ( f1 - f2 ) | X is continuous and f2 | X is continuous and f2 | X is continuous ; Then ( f1 + f2 ) | X is continuous & f2 | X is continuous ;
for L being continuous complete LATTICE for l being Element of L ex X being Subset of L st l = sup X & for x being Element of L st x in X holds x is directed & l is \in X & x is \in X
Support ( e *' A ) in { m *' p where m is Polynomial of n , L : ex p being Polynomial of n , L st p in Support ( m *' p ) & p in Support ( m *' p ) } ;
( f1 - f2 ) /* s1 = lim ( f1 /* s1 ) - lim ( f2 /* s1 ) .= lim ( f1 /* s1 ) - lim ( f2 /* s1 ) .= lim ( f1 /* s1 ) - lim ( f2 /* s1 ) .= lim ( f1 /* s1 ) - lim ( f2 /* s1 ) .= lim ( f1 /* s1 ) ;
ex p1 being Element of QC-WFF ( Al ( ) ) , g being Function of [: D ( ) , D ( ) , D ( ) :] , D ( ) st p1 = g . p1 & for g being Function of D ( ) , D ( ) st g in D ( ) holds P [ g , p1 , g . g ] ;
( mid ( f , i , len f -' 1 ) ) /. ( j -' 1 ) = ( mid ( f , i , len f -' 1 ) ) /. ( j -' 1 ) .= f /. ( j -' 1 ) .= f /. ( j -' 1 ) .= f /. ( j + 1 -' 1 ) ;
( ( p ^ q ) ^ r ) . ( len p + k ) = ( ( p ^ q ) ^ r ) . ( len p + k ) .= ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p
len mid ( upper_volume ( f , D2 ) , indx ( D2 , D1 , j1 ) + 1 ) = indx ( D2 , D1 , j1 ) - 1 .= indx ( D2 , D1 , j1 ) + 1 - 1 .= indx ( D2 , D1 , j1 ) + 1 - 1 .= indx ( D2 , D1 , j1 ) + 1 - 1 ;
x * y * z = ( x * ( y * z ) ) * ( y * z ) .= ( x * ( y * z ) ) * ( y * z ) .= ( x * ( y * z ) ) * ( y * z ) .= ( x * ( y * z ) ) * ( y * z ) .= ( x * y ) * ( z * z ) .= ( x * y ) * z ;
v . ( <* x , y *> - ( <* x0 , y *> - ( x - x0 ) ) * i ) = partdiff ( v , ( x - x0 ) * ( ( x - x0 ) + ( y - x0 ) * ( ( x - x0 ) + ( y - x0 ) * ( ( x - x0 ) + ( y - x0 ) * ( ( x - x0 ) + ( y - x0 ) ) * ( y - x0 ) ) ) ) ;
i * i = <* 0 * ( - 1 ) - ( 0 * ( - 1 ) ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) .= <* - 1 , 0 , 0 *> .= <* - 1 , 0 , 0 *> ;
Sum ( L (#) F ) = Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( ( L (#) F1 ) ^ ( F1 ^ F2 ) ) .= Sum ( ( L (#) F1 ) ^ ( L ^ F2 ) ) .= Sum ( ( L (#) F1 ) ^ ( L ^ F2 ) ) .= Sum ( ( L (#) F1 ) ^ ( L ^ F2 ) ) .= Sum ( L (#) F1 ) + Sum ( L (#) F2 ) ) .= Sum ( L (#) F1 ) ;
ex r be Real st for Y be Real st 0 < e ex Y1 be Subset of X st Y1 is non empty & for Y1 be Subset of X st Y1 is non empty & Y c= Y holds |. ( union Y ) . Y1 - ( lower_bound Y ) . x .| < r
( GoB f ) * ( i , j + 1 ) = f /. ( k + 2 ) & ( GoB f ) * ( i , j + 1 ) = f /. ( k + 2 ) or ( GoB f ) * ( i + 2 ) = f /. ( k + 2 ) & ( GoB f ) * ( i + 2 ) = f /. k ) ;
( ( - 1 ) (#) ( cos * sin ) ) `| Z ) . x = - ( ( - 1 ) (#) ( sin * sin ) ) . x .= - ( ( - 1 ) (#) ( sin * sin ) ) . x .= - ( ( - 1 ) (#) ( sin * sin ) ) . x .= - ( ( - 1 ) (#) ( sin * sin ) ) . x .= - ( ( - 1 ) (#) ( sin * sin ) ) . x ;
( - b + sqrt ( a , b ) ) / 2 * a < 0 & ( - b - sqrt ( a , b ) ) / 2 < 0 or - b - sqrt ( a , b ) / 2 < 0 ;
ex_inf_of uparrow "\/" ( X , L ) /\ C , L & ex_sup_of X , L & "\/" ( X , L ) , L & "\/" ( X , L ) = "/\" ( ( uparrow "\/" ( X , L ) ) , L ) & "\/" ( X , L ) = "/\" ( ( uparrow "\/" ( X , L ) ) , L ) ;
( ( for j being Element of NAT st j in the Sorts of B ) . i ) . ( j , i ) = ( j , i ) (*) ( id ( B . i ) , ( id ( B . i ) ) ) ) & ( j = i implies i = j ) implies ( j = i implies j = i )