thesis ;
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thesis ;
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contradiction ;
contradiction ;
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thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
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contradiction ;
contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
thesis ;
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contradiction ;
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contradiction ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
B in B ;
a <> c
T c= S
D c= B
c in X ;
b in X ;
X ;
b in D ;
x = e ;
let m ;
h is onto ;
N in K ;
let i ;
j = 1 ;
x = u ;
let n ;
let k ;
y in A ;
let x ;
let x ;
m c= y ;
F is one-to-one ;
let q ;
m = 1 ;
1 < k ;
G is prime ;
b in A ;
d divides a ;
i < n ;
s <= b ;
b in B ;
let r ;
B is one-to-one ;
R is total ;
x = 2 ;
d in D ;
let c ;
let c ;
b = Y ;
0 < k ;
let b ;
let n ;
r <= b ;
x in X ;
i >= 8 ;
let n ;
let n ;
y in f ;
let n ;
1 < j ;
a in L ;
C is boundary ;
a in A ;
1 < x ;
S is finite ;
u in I ;
z << z ;
x in V ;
r < t ;
let t ;
x c= y ;
a <= b ;
m in NAT ;
assume f is prime ;
not x in Y ;
z = +infty ;
let k be Nat ;
K ` is being_line ;
assume n >= N ;
assume n >= N ;
assume X is 1 ;
assume x in I ;
q is measurable ;
assume c in x ;
let 1-r > 0 ;
assume x in Z ;
assume x in Z ;
1 <= k.| ;
assume m <= i ;
assume G is prime ;
assume a divides b ;
assume P is closed ;
\bf 2 > 0 ;
assume q in A ;
W is not bounded ;
f is IC ;
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 negative ;
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 <= \rm A-2 ;
let X be set ;
let X be set ;
let y be element ;
let x be element ;
P [ 0 ]
let E be set , A be Subset of E ;
let C be Category ;
let x be element ;
let k be Nat ;
let x be element ;
let x be element ;
let e be element ;
let x be element ;
P [ 0 ]
let c be element ;
let y be element ;
let x be element ;
let a be Real ;
let x be element ;
let X be element ;
P [ 0 ]
let x be element ;
let x be element ;
let y be element ;
r in REAL ;
let e be element ;
n1 is invertible ;
Q halts_on s ;
x in such ;
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 ;
P3 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 , f be PartFunc of REAL , REAL ;
X |- r => p ;
x in { A } ;
let n be Nat ;
let k be Nat ;
let k be Nat ;
let m be Nat ;
0 < 0 + k ;
f is_differentiable_in x ;
let x0 ;
let E be Ordinal ;
o <=' 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 , M be Subset of V ;
not s in Y |^ 0 ;
rng f <= w
b "/\" e = b ;
m = m3 ;
t in h . D ;
P [ 0 ] ;
assume z = x * y ;
S . n is bounded ;
let V be RealUnitarySpace , M be Subset of V ;
P [ 1 ] ;
P [ {} ] ;
C1 is as Subset of X ;
H = G . i ;
1 <= i `1 + 1 ;
F . m in A ;
f . o = o ;
P [ 0 ] ;
ab <= being Real ;
R [ 0 ] ;
b in f .: X ;
assume q = q2 ;
x in [#] V ;
f . u = 0 ;
assume e1 > 0 ;
let V be RealUnitarySpace , M be Subset of V ;
s is trivial & s is non empty ;
dom c = Q
P [ 0 ] ;
f . n in T ;
N . j in S ;
let T be complete LATTICE , a , b be Element 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 ;
vA2 < n ;
Smax is bounded ;
assume p = p2 ;
len f = n ;
assume x in P1 ;
i in dom q ;
let U0 ;
pp = c ;
j in dom h ;
let k ;
f | Z is continuous ;
k in dom G ;
UBD C = B ;
1 <= len M ;
p in being element ;
1 <= j9 ;
set A = Cl A ;
card a [= c ;
e in rng f ;
cluster B \oplus A -> empty ;
H |= the \rangle ;
assume x0 <= m ;
T is increasing ;
e2 <> e2 ;
Z c= dom g ;
dom p = X ;
H is proper ;
i + 1 <= n ;
v <> 0. V ;
A c= Affin A ;
S c= dom F ;
m in dom f ;
let X0 be set ;
c = sup N ;
R is connected implies union M is connected ;
assume not x in REAL ;
Im f is complete ;
x in Int y ;
dom F = M ;
a in On W ;
assume e in A ( ) ;
C c= C-26 ;
mm <> {} ;
let x be Element of Y ;
let f be let let yielding FinSequence , i be Nat ;
not n in Seg 3 ;
assume X in f .: A ;
assume that p <= n and p <= m ;
assume not u in { v } ;
d is Element of A ;
A |^ b misses B ;
e in v `2 ;
- y in I ;
let A be non empty set , a be element ;
P0 = 1 ;
assume r in F . k ;
assume f is simple ;
let A be Incountable set ;
rng f c= NAT ;
assume P [ k ] ;
f6 <> {} ;
let o be Ordinal ;
assume x is sum of squares ;
assume not v in { 1 } ;
let I1 ;
assume that 1 <= j and j < l ;
v = - u ;
assume s . b > 0 ;
d1 in NAT ;
assume t . 1 in A ;
let Y be non empty TopSpace , f be Function of Y , BOOLEAN ;
assume a in ]. s , t .[ ;
let S be non empty RelStr ;
a , b // b , a ;
a * b = p * q ;
assume x , y are_the space ;
assume x in [#] ( f ) ;
[ a , c ] in X ;
mm <> {} ;
M + N c= M + M ;
assume M is connected hhhhhz ;
assume f is (#) mab-v-r-n-directed ;
let x , y be element ;
let T be non empty TopSpace ;
b , a // b , c ;
k in dom Sum p ;
let v be Element of V ;
[ x , y ] in T ;
assume len p = 0 ;
assume C in rng f ;
k1 = k2 & k2 = k2 ;
m + 1 < n + 1 ;
s in S \/ { s } ;
n + i >= n + 1 ;
assume Re y = 0 ;
k1 <= j1 & k1 <= len G ;
f | A is Sum f ;
f . x be Real ;
assume y in dom h ;
x * y in B1 ;
set X = Seg n ;
1 <= i2 + 1 ;
k + 0 <= k + 1 ;
p ^ q = p ;
j |^ y divides m ;
set m = max A ;
[ x , x ] in R ;
assume x in succ 0 ;
a in sup phi ;
Cj in C ;
q2 c= C1 & not q2 in C1 ;
a2 < c2 & a2 < a1 ;
s2 is 0 -started ;
IC s = 0 ;
s4 = s4 . a & s4 . b = s3 . b ;
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 empty Subset of L ;
y " <> 0 ;
y " <> 0 ;
0. V = uw ;
y2 , y are_u1 ;
R8 is open ;
let a , b be Real , f be PartFunc of REAL , REAL ;
let a be object of C ;
let x be Vertex of G ;
let o be Object of C , a be element ;
r '&' q = P \lbrack l \rbrack ;
let i , j be Nat ;
let s be State of A , a be element ;
s4 . n = N ;
set y = ( x `1 ) ^2 ;
mi in dom g ;
l . 2 = y1 ;
|. g . y .| <= r ;
f . x in C0 ;
V1 is non empty ;
let x be Element of X ;
0 <> f . g2 ;
f2 /* q is convergent ;
f . i is_measurable_on E ;
assume \xi in N-22 ;
reconsider i = i as Ordinal ;
r * v = 0. X ;
rng f c= INT & f . 0 = 0 ;
G = 0 .--> goto 0 ;
let A be Subset of X ;
assume A9 is dense & A is open ;
|. 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 RelStr ;
n + 1 = succ n ;
xx c= Y1 & m c= m ;
dom f = C1 & dom f = C1 ;
assume [ a , y ] in X ;
Re ( seq ^\ k ) is convergent ;
assume a1 = b1 & b1 = b2 ;
A = sInt A ( ) ;
a <= b or b <= a ;
n + 1 in dom f ;
let F be sequence of S , a be element ;
assume r2 > x0 & x0 < r2 ;
let Y be non empty set , f be Function of Y , BOOLEAN ;
2 * x in dom W ;
m in dom g2 & m + 1 in dom g2 ;
n in dom g1 ;
k + 1 in dom f ;
not the still not bound in { s } ;
assume x1 <> x2 & x1 <> x3 ;
v2 in V1 & v2 in V1 ;
not [ b `1 , b `2 ] in T ;
ii + 1 = i ;
T c= ] ( T ) ;
( l . 1 ) `1 = 0 ;
let n be Nat ;
( t `2 ) ^2 = r ;
A=> is_integrable_on M ;
set t = Top t ;
let A , B be real-membered set ;
k <= len G + 1 ;
[: C , C :] misses [: C , C :] ;
Product seq is non empty ;
e <= f or f <= e ;
cluster -> non empty and normal for set ;
assume c2 = b2 & c1 = c2 ;
assume h in [. q , p .] ;
1 + 1 <= len C ;
not c in B . m1 ;
cluster R .: X -> empty ;
p . n = H . n ;
assume v-4 is Cauchy sequence of X ;
IC s3 = 0 & IC s3 = 1 ;
k in N or k in K ;
F1 \/ F2 c= F ;
Int G1 <> {} & G1 is open ;
( z `2 ) ^2 = 0 ;
p10 <> p1 & p1 <> 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 antisymmetric non empty reflexive RelStr ;
q |^ m >= 1 ;
a >= X & b >= Y ;
assume <* a , c *> <> {} ;
F . c = g . c ;
G is one-to-one ;
A \/ { a } \not c= B ;
0. V = 0. Y .= 0. V ;
let I be <= T , J be Instruction of S ;
f-24 . x = 1 ;
assume z \ x = 0. X ;
C4 = 2 to_power n ;
let B be SetSequence of \Sigma ;
assume X1 = p .: D ;
n + l2 in NAT & l2 in NAT ;
f " P is compact ;
assume x1 in REAL + ( REAL + 1 ) ;
p1 = K1 & p2 = K1 ;
M . k = <*> REAL ;
phi . 0 in rng phi ;
MMMMInt A is closed ;
assume z0 <> 0. L ;
n < N . k ;
0 <= seq . 0 & seq . 1 <= seq . 0 ;
- q + p = v ;
{ v } is Subset of B ;
set g = f /. 1 ;
[: R , R :] is stable ;
set R = Vertices R , s = Vertices R , t = the carrier of R ;
p0 c= P3 & p1 = p2 ;
x in [. 0 , 1 .] ;
f . y in dom F ;
let T be Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott of S ;
inf the carrier of S = {} ;
sup downarrow a = sup downarrow b ;
P , C , K is_collinear ;
assume x in F ( s , r ) ;
2 to_power i < 2 to_power m ;
x + z = x + z + q ;
x \ ( a \ x ) = x ;
||. \mathopen { \Vert } x-y .|| <= r ;
assume that Y c= field Q and Y <> {} ;
a ~ , b ~ are_isomorphic ;
assume a in A ( ) ;
k in dom ( q4 * q ) ;
p is \kern1pt empty FinSequence of S ;
i -' 1 = i-1 ;
f | A is one-to-one ;
assume x in f .: X ;
i2 - i1 = 0 ;
j2 + 1 <= i2 ;
g " * a in N ;
K <> { [ {} , {} ] } ;
cluster strict 2 -number -> strict for *> ;
|. q .| ^2 > 0 ;
|. p4 .| = |. p .| ;
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 <= i + j ;
dom S = dom F & for i be Nat st i in dom F holds F . i = G . i ;
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 holds holds S is non void holds the TopStruct of S = the TopStruct of S ;
let f be ManySortedSet of I ;
let z be Element of COMPLEX , a be Element of COMPLEX ;
u in { \hbox { \boldmath $ g $ } } ;
2 * n < 2 * n ;
let x , y be set ;
B-11 c= V1 & V is open ;
assume I is_halting_on s , P ;
U1 = U2 & U2 = U1 ;
M /. 1 = z /. 1 ;
xx = xx & xx = [ m , x ] ;
i + 1 < n + 1 + 1 ;
x in { {} , <* 0 *> } ;
f7 <= f6 & f7 <= f6 ;
let l be Element of L ;
x in dom ( F . 0 ) ;
let i be Element of NAT ;
r8 is ( len C ) -valued ;
assume <* o2 , o *> <> {} ;
s . x |^ 0 = 1 ;
card K1 in M & card K1 in card M ;
assume X in U & Y in U ;
let D be \rangle of Omega ;
set r = { q + 1 } ;
y = W . ( 2 * x ) ;
assume dom g = cod f & cod g = cod f ;
let X , Y be non empty TopSpace , f be Function of X , Y ;
x \oplus A is interval ;
|. <*> A .| . a = 0 ;
cluster strict for Sublattice of L ;
a1 in B . s1 & a1 in B . s1 ;
let V be finite .[ over F , a be Element of V ;
A * B on B , A ;
f-3 = ( NAT --> 0 ) +* ( NAT --> 1 ) ;
let A , B be Subset of V ;
z1 = P1 . j & z2 = P1 . j ;
assume f " P is closed ;
reconsider j = i as Element of M ;
let a , b be Element of L ;
assume q in A \/ ( B "\/" C ) ;
dom ( F * C ) = o ;
set S = INT |^ X , T = 2 |^ X ;
z in dom ( A --> y ) ;
P [ y , h . y ] ;
{ x0 } c= dom f ;
let B be non-empty ManySortedSet of I , f be Function of B , C ;
sqrt ( PI / 2 ) < Arg z ;
reconsider z9 = 0 as Nat ;
LIN a , d , c ;
[ y , x ] in [: I , I :] ;
( Q ) `1 = 0 & ( Q `1 ) `2 = 1 ;
set j = x0 div m , m = x0 / m ;
assume a in { x , y , c } ;
j2 - ( j - 1 ) > 0 ;
I not ( I , 1 ) string = 1 ;
[ y , d ] in F-8 ;
let f be Function of X , Y ;
set A2 = ( B |^ C ) * ( B |^ C ) ;
s1 , s2 are_in R ;
j1 -' 1 = 0 & j1 -' 1 = 1 ;
set m2 = 2 * n + j ;
reconsider t = t as bag of n ;
I2 . j = m . j ;
i |^ s , n are_relative_prime ;
set g = f | D-21 ;
assume that X is lower bounded and 0 <= r ;
( 1 - ( ( p1 `1 / p1 `2 ) ^2 ) ) ^2 = 1 ;
a < ( p3 `1 ) ^2 + ( p3 `2 ) ^2 ;
L \ { m } c= UBD C ;
x in Ball ( x , 10 ) ;
not a in LSeg ( c , m ) ;
1 <= i1 -' 1 & i1 -' 1 <= i2 ;
1 <= i1 -' 1 & i1 -' 1 <= i2 ;
i + i2 <= len h ;
x = W-min ( P ) ;
[ x , z ] in X ~ ;
assume y in [. x0 , x .] ;
assume p = <* 1 , 2 , 3 *> ;
len <* A1 *> = 1 ;
set H = h . -3 , I = h . -3 , J = h . ( len h + 1 ) , K = h . ( len h + 1 ) , N = len
card b * a = |. a .| ;
Shift ( w , 0 ) |= v ;
set h = h2 ** h1 , h1 = h1 ** h2 , h2 = h2 ** h1 , h2 = h1 ** h2 , h2 = h2 (#) h1 , h2 = h2 (#) h1 , h2
assume x in X0 /\ ( X1 union X2 ) ;
||. h .|| < dx0 & 0 < dx0 ;
not x in the carrier of f & not x in the carrier of f ;
f . y = F ( y ) ;
for n holds X [ n ] ;
k - l = kl2 ;
<* p , q *> /. 2 = q ;
let S be Subset of the lattice of Y ;
let P , Q be ^ of s ;
Q /\ M c= union ( F | M )
f = b * canFS ( S ) ;
let a , b be Element of G ;
f .: X <= f . sup X ;
let L be non empty transitive RelStr , f be Function of L , L ;
S-20 is x -let i -Basis of n ;
let r be non positive Real ;
M , v |= x \hbox { y } ;
v + w = 0. ( Z ) ;
P [ len F ] implies P [ F ] ;
assume InsCode ( i + 5 ) = 8 ;
the zero of M = 0 & the carrier of M = 0 ;
cluster z * seq -> summable for Real_Sequence ;
let O be Subset of the carrier of C ;
|. f .| " X is continuous ;
x2 = g . ( j + 1 ) ;
cluster -> ( S , S ) -valued for string .. S ;
reconsider l1 = l- 1 as Nat ;
v4 is Vertex of r2 & v4 is Vertex of G ;
TT2 is SubSpace of T2 & f is continuous ;
Q1 /\ Q1 <> {} ;
let k be Nat ;
q " is Element of X ;
F . t is set of S ;
assume that n <> 0 and n <> 1 ;
set e1 = EmptyBag n , e2 = EmptyBag n , e2 = EmptyBag n ;
let b be Element of Bags n ;
assume for i holds b . i is commutative ;
x is root of ( p ) . 2 ;
not r in ]. p , q .[ ;
let R be FinSequence of REAL , a be Element of REAL ;
not S7 does not destroy b1 & not LIN b1 , b2 , T ;
IC SCM ( R ) <> a ;
|. p - |[ x , y ]| .| >= r ;
1 * ( seq ^\ k ) = seq . ( seq ^\ k ) ;
let x be FinSequence of NAT , a be Element of NAT ;
let f be Function of C , D , a be element ;
for a holds 0. L + a = a
IC s = s . NAT .= s . NAT ;
H + G = F- ( GG ) ;
C1 . x = x2 & C1 . x = y2 ;
f1 = f .= f2 .= f1 . ( len f1 ) .= f2 . ( len f1 ) ;
Sum <* p . 0 *> = p . 0 ;
assume v + W = v + u + W ;
{ a1 } = { a2 } ;
a1 , b1 _|_ b , a ;
a3 , o _|_ o , a3 ;
I1 is_reflexive implies C is reflexive & C is reflexive & C is transitive
I is_antisymmetric implies C is antisymmetric & C is antisymmetric & C is antisymmetric & C is antisymmetric
sup rng ( H1 . n ) = e . ( n + 1 ) ;
x = a9 * ( a * a9 ) .= a * ( a * b ) ;
|. p1 .| ^2 >= 1 ;
assume j2 -' 1 < j2 -' 1 ;
rng s c= dom f1 /\ dom f2 ;
assume support a misses support b & not a in support b ;
let L be associative non empty doubleLoopStr , a be Element of L ;
s " + 0 < n + 1 ;
p . c = f-1 . 1 ;
R . n <= R . ( n + 1 ) ;
Directed I1 = I1 ;
set f = + ( x , y , r ) ;
cluster Ball ( x , r ) -> bounded ;
consider r being Real such that r in A ;
cluster non empty NAT -valued for Function ;
let X be non empty directed Subset of S ;
let S be non empty full full SubRelStr of L ;
cluster <* *> . N , N *> -> complete complete complete for non trivial Real ;
sqrt ( 1 + a ) = a ;
( q . {} ) `1 = o ;
n - ( i -' 1 ) > 0 ;
assume sqrt ( 1 + 2 ) <= t `1 ;
card B = k + 1 - 1 ;
x in union rng ( f | n ) ;
assume x in the carrier of R & y in the carrier of R ;
d in D ;
f . 1 = L . ( F . 1 ) ;
the vertices of G = { v } ;
let G be : : Let G be : 1 <= G ;
e , v6 , v6 , v1 , v2 , V ;
c . i9 in rng c & c . i9 in rng c ;
f2 /* q is divergent_to+infty & f2 /* ( seq ^\ k ) is divergent_to+infty ;
set z1 = - z2 , z2 = - z2 , z2 = - z1 , z2 = - z2 , z2 = - z2 , z1 = - z2 , z2 = - z2 , z2 = - z2 , z2
assume w is Element of llas ( G , H ) ;
set f = p \! \mathop { t } , g = p \! \mathop { t } , h = p ;
let c be Object of C ;
assume ex a st P [ a ] ;
let x be Element of REAL-NS m , a be Element of REAL m ;
let I1 be Subset-Family of X ;
reconsider p = p as Element of NAT ;
let v , w be Point of X ;
let s be State of SCM+FSA , a be element ;
p is FinSequence of ( the InstructionsF of SCM+FSA ) ;
stop I c= ( card I + 1 ) ;
set ci = f^2 /. i , cj = 1 ;
w ^ t seq w ^ s ^ s ^ t ^ t ^ s ^ t ^ s ^ t ^ s ^ t ^ t ^ s ^ t ^ s ^ t ^ t ^ s ^ t ^
W1 /\ W = W1 /\ W .= ( W1 /\ W2 ) /\ ( W2 /\ W1 ) ;
f . j is Element of J . j ;
let x , y be Element of T2 ;
ex d st a , b // b , d ;
a <> 0 & b <> 0 ;
ord x = 1 & x is positive ;
set g2 = lim ( seq ^\ k ) , g1 = lim ( seq ^\ k ) ;
2 * x >= 2 * ( 1 / 2 ) ;
assume ( a 'or' c ) . z <> TRUE ;
f (*) g in Hom ( c , c ) ;
Hom ( c , c + d ) <> {} ;
assume 2 * Sum ( q | m ) > m ;
L1 . ( F1 . ( F . k ) ) = 0 ;
( id X ) \/ R1 = ( id X ) \/ R1 .= ( id X ) \/ R1 ;
( sin * sin ) . x <> 0 ;
( ( exp_R * exp_R ) `| Z ) . x > 0 ;
o1 in ( X /\ O2 ) /\ ( X /\ O2 ) ;
e , v6 , v6 , v1 , v2 , V ;
r3 > ( 1 / 2 ) * 0 ;
x in P .: ( F -ideal ) ;
let J be closed Ideal of R , I be non empty Subset of R ;
h . p1 = f2 . O ;
Index ( p , f ) + 1 <= j ;
len ( q * M ) = width M & width ( q * M ) = width M ;
the carrier of L c= A & the carrier of K c= A ;
dom f c= union rng ( F | X ) ;
k + 1 in support ( ( n ) | k ) ;
let X be ManySortedSet of the carrier of S ;
[ x `1 , y `2 ] in InnerVertices ( R ) ;
i = D1 or i = D2 ;
assume a mod n = b mod n ;
h . x2 = g . x1 & h . x2 = g . x2 ;
F c= 2 to_power the carrier of X implies F is closed
reconsider w = |. s1 .| as Real_Sequence ;
sqrt ( 1 / m * m + r ) < p ;
dom f = dom ( I +* ( i , I ) ) ;
[#] P-17 = [#] ( ( TOP-REAL 2 ) | K1 ) .= [#] ( ( TOP-REAL 2 ) | K1 ) ;
cluster - x -> ExtReal equals - x ;
then { d1 } c= A ;
cluster TOP-REAL n -> finite-ind for Subset of TOP-REAL n ;
let w1 be Element of M ;
let x be Element of dyadic ( n ) ;
u in W1 & v in W2 implies u = v + ( W1 + W2 )
reconsider y = y as Element of L2 ;
N is full full SubRelStr of ( T |^ n ) ;
sup { x , y } = c "\/" c ;
g . n = n to_power 1 .= n ;
h . J = EqClass ( u , J ) ;
let seq be summable sequence of X ;
dist ( x `1 , y ) < r / 2 ;
reconsider mm = m - 1 as Element of NAT ;
x- x0 < r1 - x0 & r1 < r2 - x0 ;
reconsider P = P `1 as strict Subgroup of N ;
set g1 = p * idseq ( q `1 ) , g2 = q `2 ;
let n , m , k be non zero Nat ;
assume that 0 < e and f | A is lower ;
D2 . I2 in { x } & D2 . I2 in { x } ;
cluster -> 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 ) ;
let n be Element of NAT , a be Element of REAL ;
reconsider ST = S as Subset of T ;
dom ( i .--> X ` ) = { i } ;
let X be non-empty ManySortedSet of S ;
let X be non-empty ManySortedSet of S ;
op 1 c= { [ {} , {} ] } ;
reconsider m = mm - 1 as Element of NAT ;
reconsider d = x as Element of C ( ) ;
let s be 0 -started State of SCMPDS , a be element ;
let t be 0 -started State of SCMPDS ;
b , b , x , y , z is_collinear ;
assume that i = n \/ { k } and j = k \/ { k } ;
let f be PartFunc of X , Y ;
N2 >= sqrt ( c ^2 + sqrt ( c ^2 + d ^2 ) ) ;
reconsider t7 = T-1 as TopSpace ;
set q = h * p ^ <* d *> ;
z2 in U . 4 /\ Q2 . 6 /\ Q2 . 6 ;
A |^ 0 = { <* E *> } ;
len W2 = len W + 2 & len W2 = len W + 1 ;
len h2 in dom h2 & len h2 = len h2 ;
i + 1 in Seg ( len s2 ) ;
z in dom g1 /\ dom f & g1 . z in dom f ;
assume p2 = E-max ( K ) & p1 = E-max ( K ) ;
len G + 1 <= i1 + 1 ;
f1 (#) f2 is convergent & lim ( f1 (#) f2 ) = x0 ;
cluster seq + seq ^\ n -> summable for Real_Sequence ;
assume j in dom M1 & i in dom M1 ;
let A , B , C be Subset of X ;
let x , y , z be Point of X , a be Real ;
b ^2 - ( 4 * a * c ) >= 0 ;
<* xy *> ^ <* y *> \subseteq x ;
a , b in { a , b } ;
len p2 is Element of NAT & len p1 = len p2 ;
ex x being element st x in dom R & y = R . x ;
len q = len ( K (#) G ) ;
s1 = Initialize ( Initialized s ) , P1 = P +* I , P2 = P +* I , P3 = P +* I , s3 = Comput ( P3 , s3 , 1 ) , P4 = P3 ;
consider w be Nat such that q = z + w ;
x ` is Element of L & x ` is ` ;
k = 0 & n <> k or k > n ;
then X is discrete ;
for x st x in L holds x is FinSequence ;
||. f /. c .|| <= r1 & ||. f /. c .|| <= r2 ;
c in ]. p , q .[ & not c in { p } ;
reconsider V = V as Subset of the topology of TOP-REAL n ;
let N , M be \hbox { N , X } ;
then z >= waybelow x & z >= compactbelow y ;
M | [. f , g .] = f & M | [. g , g .] = g ;
( ( ( n + 1 ) ) to_power 1 ) to_power 1 = TRUE ;
dom g = dom f /\ X .= X ;
mode Walk of G is : number : for n being Nat holds P [ n ] ;
[ 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 & f . x = {} ;
F1 . ( a1 , - a2 ) = G1 . ( a1 , - a2 ) ;
cluster term ( a , b , r ) -> compact ;
let a , b , c , d be Real ;
rng s c= dom ( 1 / 2 ) ;
curry ' ( F , k ) is additive ;
set k2 = card dom B - 1 , k1 = card dom B - 1 ;
set G = over X , S = X ;
reconsider a = [ x , s ] as \HM of G ;
let a , b be Element of Mr , n be Nat ;
reconsider s1 = s , s2 = s as Element of ( len s ) -tuples_on the carrier of S ;
rng p c= the carrier of L & p . 1 = 0. L ;
let d be Subset of the Sorts of A ;
( x .|. x = 0 iff x = 0. W ) ;
I1 . k in dom stop I & m . k in dom stop I ;
let g be continuous Function of X | B , Y ;
reconsider D = Y as Subset of TOP-REAL n ;
reconsider i0 = len p1 - 1 as Integer ;
dom f = the carrier of S & f . 0 = the carrier of S ;
rng h c= union ( the carrier of J ) ;
cluster All ( x , H ) -> One -reconsider ;
d * N1 ^2 > N1 * 1 ;
]. a , b .[ c= [. a , b .] ;
set g = f " D1 , f = f " D2 , A = f " D2 ;
dom ( p | ( ( m + 1 ) -tuples_on NAT ) ) = NAT ;
3 + - 2 <= k + - 2 ;
tan is_differentiable_in ( arccot * arccot ) . x ;
x in rng ( f /^ p ) & x in rng ( f /^ p ) ;
let f , g be FinSequence of D ;
cp in the carrier of S1 & cp in the carrier of S1 ;
rng f " { 0 } = dom f ;
( the Source of G ) . e = v ;
width G -' 1 < width G -' 1 & 1 <= j & j + 1 <= width G ;
assume v in rng ( S | E1 ) & v in rng ( S | E1 ) ;
assume x is root or x is root or x is root & x is root ;
assume 0 in rng ( g2 | A ) & 0 < r ;
let q be Point of TOP-REAL 2 , a , b be Real ;
let p be Point of TOP-REAL 2 , a , b be Real ;
dist ( O , u ) <= |. p2 .| + 1 ;
assume dist ( x , b ) < dist ( a , b ) ;
<* S *> is in the carrier' of C-20 & <* S *> is Element of the carrier' of C-20 ;
i <= len ( G | ( len G -' 1 ) ) ;
let p be Point of TOP-REAL 2 , a , b be Real ;
x1 in the carrier of I[01] & x2 in the carrier of I[01] ;
set p1 = f /. i , p2 = f /. ( i + 1 ) ;
g in { g2 : r < g2 & g2 < r } ;
Q2 = Sp2 " ( Q . ( Q . ( Q . i ) ) ) ;
( ( 1 / 2 ) to_power ( n + 1 ) ) to_power k is summable ;
- p + I c= - p + A ;
n < LifeSpan ( P1 , s1 ) + 1 ;
CurInstr ( p1 , s1 ) = i .= ( I . IC s1 ) .= ( I . IC s1 ) ;
A /\ Cl { x } \ { x } <> {} ;
rng f c= ]. r , r + 1 .[ ;
let g be Function of S , V ;
let f be Function of L1 , L2 ;
reconsider z = z as Element of InclPoset ( L ) ;
let f be Function of S , T ;
reconsider g = g as Morphism of c opp , b ;
[ s , I ] in S ~ ( A , B ) ;
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 : Def2 : All ( x , p ) is valid ;
assume that X c= dom f and f .: X c= dom g and f .: X c= dom g ;
H |^ a " is Subgroup of H ;
let A1 be |^ of O , B1 be Element of E ;
p2 , r3 , q3 is_collinear & q2 , q2 , q3 is_collinear ;
consider x being element such that x in v ^ K ;
not x in { 0. TOP-REAL 2 } ;
p in [#] ( ( TOP-REAL 2 ) | B ) & p in [#] ( ( TOP-REAL 2 ) | B ) ;
0 . E < M . ( E . ( E . m ) ) ;
^ ( c , d ) = c ;
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . s2 ) ;
cluster -> \uparrow -| for Scott E -from of L ;
set i1 = the Nat of S , i2 = the carrier of S , n = the carrier of S , n = len S ;
let s be 0 -started State of SCM+FSA , a be element ;
assume y in ( f1 \/ f2 ) .: A ;
f . ( len f ) = f /. len f ;
x , f . x '||' f . x , f . y ;
redefine pred X c= Y means : Def2 : cos . X c= cos . Y ;
let y be upper Subset of Y , x be element ;
cluster -> \mathbb i for Nat ;
set S = <* Bags n , i9 *> , i = the carrier of L ;
set T = [. 0 , 1 .] , G = [. 0 , 1 .] , H = [. 0 , 1 .] ;
1 in dom mid ( f , 1 , 1 ) ;
sqrt ( 4 * PI ) < sqrt ( 2 * PI ) ;
x2 in dom ( f1 (#) f2 ) /\ dom ( f1 (#) f2 ) ;
O c= dom I & { {} } = { {} } ;
( the Source of G ) . x = v ;
{ HT ( f , T ) } c= Support f ;
reconsider h = R . k as Polynomial of n , L ;
ex b being Element of G st y = b * H ;
let x , y , z be Element of G ` ;
h19 . i = f . ( h . i ) ;
( p `1 ) ^2 = ( p1 `1 ) ^2 + ( p1 `2 ) ^2 ;
i + 1 <= len Cage ( C , n ) ;
len <* P *> ^ ( P ^ Q ) = len P + 1 ;
set Nthen = the CastNode of N , the InternalRel of N = the InternalRel of N ;
len g-2 + ( x + 1 ) - 1 <= x ;
a on B & b on B implies a on B ;
reconsider r-12 = r * I . v as FinSequence ;
consider d such that x = d and a [= d ;
given u such that u in W and x = v + u ;
len f /. ( \downharpoonright n ) = len \mathbb n - n ;
set q2 = q2 , q3 = q2 , q3 = q2 , q3 = q2 ;
set S = 1 , S1 = 2 , S2 = 3 , S2 = 4 ;
MaxADSet ( b ) c= MaxADSet ( P /\ Q ) ;
Cl ( G . q1 ) c= F . r2 ;
f " D meets h " V & f " V meets h " V ;
reconsider D = E as non empty directed Subset of L1 ;
H = ( the_left_argument_of H ) '&' ( ( H ) '&' ( H ) ) ;
assume t is Element of ( F . X ) . s ;
rng f c= the carrier of S2 & f . 0 = f . 1 ;
consider y being Element of X such that x = { y } ;
f1 . ( a1 , b1 ) = b1 . ( a1 , b1 ) ;
the carrier' of G = E \/ { E } ;
reconsider m = len , k = len - 1 as Element of NAT ;
set S1 = LSeg ( n , UMP C ) ;
[ i , j ] in Indices M1 & M1 * ( i , j ) = M1 * ( i , j ) ;
assume that P c= Seg m and M is \HM { i } ;
for k st m <= k holds z in K . k ;
consider a being set such that p in a and a in G ;
L1 . p = p * ( L /. 1 ) ;
pp . i = ( p . i ) * ( p . i ) ;
let PA , G be a_partition of Y ;
redefine attr 0 < r & 1 < 1 & r < 1 ;
rng ( AffineMap ( a , X ) ) = [#] X ;
reconsider x = x , y = y as Element of K ;
consider k such that z = f . k and n <= k ;
consider x being element such that x in X \ { p } ;
len ( canFS ( s ) ) = card s - 1 .= card s - 1 ;
reconsider x2 = x1 , y2 = x2 as Element of L2 ;
Q in FinMeetCl ( ( the topology of X ) | A ) ;
dom f0 c= dom ( u | E ) & dom ( u | E ) = E ;
redefine pred n divides m & m divides n ;
reconsider x = x as Point of I[01] , I be Point of I[01] ;
a in ) implies a in the carrier of T2 & b in the carrier of T2
not y0 in the still of f & not ( not ex y st y in the bound not bound not y in rng f ) ;
Hom ( ( a \times b ) , c ) <> {} ;
consider k1 such that p " < k1 and k1 < len p and p . k1 = f . k1 ;
consider c , d such that dom f = c \ d ;
[ x , y ] in dom g & [ y , k ] in dom g ;
set S1 = Let is non empty ManySortedSign ;
l1 = m2 & l1 = l2 & l2 = l2 & l2 = m2 ;
x0 in dom ( ( u | A ) | A ) /\ A ;
reconsider p = x as Point of ( TOP-REAL 2 ) | K1 ;
I[01] = [: I[01] , I[01] :] | [: B , B :] .= [: B , B :] ;
f . p4 <= f . p1 & f . p2 <= f . p1 ;
( ( F . x ) `1 ) ^2 <= ( ( F . x ) `1 ) ^2 ;
( x `2 ) ^2 = ( W `1 ) ^2 + ( W `2 ) ^2 ;
for n being Element of NAT holds P [ n ] ;
let J , K be non empty Subset of I ;
assume 1 <= i & i <= len <* a " *> ;
0 |-> a = <*> the carrier of K & 0 -tuples_on the carrier of K ;
X . i in 2 -tuples_on A . i \ B . i ;
<* 0 *> in dom ( e --> [ 1 , 0 ] ) ;
then P [ a ] & P [ succ a ] ;
reconsider s\rbrace = s\rbrace as \rangle of D ( ) , D ( ) ;
- ( i -' 1 ) <= len - j ;
[#] S c= [#] T & [#] T c= [#] the TopStruct of T ;
for V being strict RealUnitarySpace holds V in consider W being Subspace of V ;
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 square Matrix of n1 , K ;
- a * - b = a * b ;
for A being Subset of A9 holds A // A implies A = B
id o2 in <* o2 , o2 , o3 *> & id o2 in <^ o2 , o2 ^> ;
then ||. x .|| = 0 & x = 0. X ;
let N1 , N2 be strict normal Subgroup of G ;
j >= len upper_volume ( g , D1 ) - len g + 1 ;
b = Q . ( len Q - 1 ) + 1 ;
f2 * f1 /* ( s ^\ k ) is divergent_to+infty ;
reconsider h = f * g as Function of [: N2 , G :] , G ;
assume that a <> 0 and Polynom ( a , b , c ) >= 0 ;
[ t , t ] in the InternalRel of A & [ t , t ] in the InternalRel of A ;
( v |-- E ) | n is Element of T7 ;
{} = the carrier of L1 + L2 & L1 + L2 = the carrier of L1 + L2 ;
Directed I is halting & Initialized s is halting & Initialized s = P +* I ;
Initialized ( p ) = Initialize ( p +* q ) .= p +* q ;
reconsider N2 = N1 as strict net of R2 , R2 ;
reconsider Y = Y as Element of <* Ids L , \subseteq \rangle ;
"/\" ( { p } , L ) \ { p } <> p ;
consider j be Nat such that i2 = i1 + j and len f = k + j ;
not [ s , 0 ] in the carrier of S2 & not [ s , 0 ] in the InternalRel of S2 ;
mm in ( B \wedge C ) '/\' D /\ { {} } ;
n <= len ( ( P + Q ) | ( len P + 1 ) ) ;
( x1 `1 ) ^2 = ( x2 `1 ) ^2 + ( x1 `2 ) ^2 ;
InputVertices S = { x1 , x2 } & InputVertices S = { x1 , x2 } ;
let x , y be Element of FTTTTTTTTTTTTTT1 ( n ) ;
p = |[ p `1 , p `2 ]| & p `2 = p `2 ;
g * 1_ G = h " * g * h .= g " * h ;
let p , q be Element of PFuncs ( V , C ) ;
x0 in dom ( f1 (#) f2 ) /\ dom ( f1 (#) f2 ) ;
( R qua Function ) " = R " & ( R " ) " = R " ;
n in Seg len ( f /^ p ) & n <= len ( f /^ p ) ;
for s be Real st s in R holds s <= s2 ;
rng s c= dom ( f2 * f1 ) /\ dom f2 ;
synonym for for for for for for for for X being Subset of \rm ex f being Function of X , f st f is one-to-one & X is finite ;
1_ K * 1_ K = 1_ K & 1_ K * 1_ K = 1_ K ;
set S = Segm ( A , P1 , Q1 ) , Q1 = Segm ( A , P1 , Q1 ) ;
ex w st e = ( w - f ) * w & w in F ;
curry ' ( P\rbrack , k ' ) # x is convergent ;
cluster -> open for Subset of TT | the topology of T ;
len f1 = 1 .= len f3 .= len f3 + 1 .= len f3 + 1 ;
sqrt ( i * p ) < sqrt ( 2 * p ) ;
let x , y be Element of \rm Sub Sub ;
b1 , c1 // b9 , c9 & b1 , c1 // c , c9 ;
consider p being element such that c1 . j = { p } ;
assume that f " { 0 } = {} and f is total ;
assume IC Comput ( F , s , k ) = n ;
Reloc ( J , card I ) does not destroy a ;
goto ( card I + 1 ) not contradiction ;
set m1 = LifeSpan ( p3 , s3 ) , m2 = LifeSpan ( p2 , s2 ) + 2 ;
IC SCMPDS in dom ( Initialize ( p +* I ) +* I ) ;
dom t = the carrier of SCM & dom t = the carrier of SCM R ;
( E-max L~ f ) .. f = 1 & ( E-max L~ f ) .. f = 1 ;
let a , b be Element of PFuncs ( V , C ) ;
Cl union Int ( union F ) c= Cl Int union union F ;
the carrier of X1 union X2 misses ( A1 \/ A2 ) ;
assume not LIN a , f . a , g . a ;
consider i being Element of M such that i = d6 and i in A ;
then Y c= { x } or Y = {} or Y = { x } ;
M , v |= H1 / ( ( y , x ) \leftarrow ( y , x ) ) ;
consider m be element such that m in Intersect ( F . 0 ) ;
reconsider A1 = support u1 as Subset of X ;
card ( A \/ B ) = k-1 + ( 2 * 1 ) ;
assume that a1 <> a3 and a2 <> a4 and a3 <> a4 and a4 <> a4 ;
cluster s , S \mathop { \rm \hbox { - } :] -> 1 -element for string of S ;
L2 /. n2 = L2 . n2 .= L2 . 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 r-7 in LSeg ( p1 , p2 ) ;
let A be non empty compact Subset of TOP-REAL n , a be Real ;
assume [ k , m ] in Indices ( D * ( k , m ) ) ;
0 <= ( ( 1 / 2 ) |^ p ) * ( ( 1 / 2 ) |^ p ) ;
( F . N | E ) . x = +infty ;
attr X c= Y & Z c= V implies X \ V c= Y \ Z ;
( y `2 ) * ( ( z `2 ) * ( y `2 ) ) <> 0. I ;
1 + card ( X \/ Y ) <= card ( u \/ v ) ;
set g = z \circlearrowleft ( E-max L~ z ) , p = E-max L~ z ;
then k = 1 & p . k = <* x , y *> . k ;
cluster -> total for Element of C such that for X being Subset of C -one-to-one ( X ) holds X is total ;
reconsider B = A as non empty Subset of TOP-REAL n , a be Real ;
let a , b , c be Function of Y , BOOLEAN ;
L1 . i = ( i .--> g ) . i .= g . i ;
Plane ( x1 , x2 , x3 , x4 ) c= P ;
n <= indx ( D2 , D1 , j1 ) + 1 - 1 ;
( ( g2 . O ) `1 ) ^2 = - 1 ;
j + p .. f -' len f <= len f -' 1 + len f -' 1 ;
set W = W-bound C , E = E-bound C , N = E-bound C , S = E-bound C , S = E-bound C , N = E-bound C , S = E-bound C , S = E-bound C , N = E-bound C , S = E-bound C , S = E-bound
S1 . ( a `1 , e `1 ) = a + e .= a ;
1 in Seg width ( M * ( ColVec2Mx p ) ) ;
dom ( i (#) Im ( f ) ) = dom ( Im ( f ) ) ;
| ( Dx `1 ) = W . ( a *' ( a , p ) ) ;
set Q = non empty Element of i2 ( g , f , h ) ;
cluster -> many sorted for ManySortedSet of U1 means ex F being ManySortedFunction of U1 , U2 st F = R ;
attr F = { A } ;
reconsider z9 = *> as Element of product \overline G ;
rng f c= rng f1 \/ rng f2 & rng f c= dom f1 \/ rng f2 ;
consider x such that x in f .: A and x in f .: C ;
f = <*> the carrier of F_Complex & f = <*> the carrier of F_Complex & f = id the carrier of F_Complex ;
E , j |= All ( x1 , x2 ) / H ;
reconsider n1 = n as Morphism of o1 , o2 ;
assume that P is idempotent and R is idempotent and P (*) R = R ** P ;
card B2 \/ { x } = k-1 + 1 ;
card ( ( x \ B1 ) /\ B1 ) = 0 ;
g + R in { s : g-r < s & s < g + r } ;
set q00 = ( q , <* s *> ) \rangle , q00 = ( q , <* s *> ) \rangle , q00 = s ;
for x being element st x in X holds x in rng f1 & x in X ;
h0 /. ( i + 1 ) = h0 . ( i + 1 ) ;
set mw = max ( B , min ( B , max ( NAT , NAT ) ) ) ;
t in Seg width ( I ^ ( n , n ) ) ;
reconsider X = dom f , C = [: NAT , { {} } :] as Element of Fin NAT ;
IncAddr ( i , k ) = ( halt SCM+FSA ) . ( l + k ) .= i ;
( E-max L~ f ) `2 <= ( q `2 ) ^2 ;
attr R is condensed means : Def2 : Int R is condensed & Cl R is condensed ;
redefine attr 0 <= a & b <= 1 implies a * b <= 1 ;
u in ( ( c /\ ( ( d /\ b ) /\ e ) /\ f ) /\ j ) /\ j ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) /\ f ) /\ j ) /\ j ;
len C + - 2 >= 9 + - 3 + 3 ;
x , z , y is_collinear & x , z , x is_collinear implies x = y
a |^ ( n1 + 1 ) = a |^ ( n1 + 1 ) * a ;
<* \underbrace ( 0 , \dots , 0 ) *> in Line ( x , a * x ) ;
set y1 = <* y , c *> ;
F2 /. 1 in rng Line ( D , 1 ) ;
p . m joins r /. m , r /. ( m + 1 ) ;
( p `2 ) ^2 = ( f /. i1 ) ^2 + ( f /. ( i1 + 1 ) ) ^2 ;
W-bound ( X \/ Y ) = W-bound ( X \/ Y ) & W-bound ( X \/ Y ) = E-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 Pfrom X , the carrier of \rm \rm \hbox { - } \rm Real } ;
p \! \mathop { \rm \hbox { - } count ( ( Product ( Sgm X ) ) ) = 0 ;
len <* x *> < i + 1 & i + 1 <= len c ;
assume that I is non empty and { x } /\ { y } = { 0. I } ;
set ii = ( card I + 4 ) .--> ( goto 0 ) , goto 4 = goto 0 ;
x in { x , y } & h . x = {} T & h . x = {} ;
consider y being Element of F such that y in B and y <= x `1 ;
len S = len ( the charact of A ) & 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 = : \HM { G : G c= dom ( G-15 . n ) } ;
rng F c= the carrier of gr { a } & F . a = the carrier of gr { a } ;
indx ( Q , F ) . ( K , n ) is a K -valued FinSequence ;
f . k , f . ( mod n ) in rng f ;
h " P /\ [#] T1 = f " P & h " P = f " P ;
g in dom f2 \ f2 " { 0 } & f2 " { 0 } = f2 " { 0 } ;
gX /\ dom g1 = g1 " { g1 . x } .= g1 " { 0 } ;
consider n being element such that n in NAT and Z = G . n ;
set d1 = be element of being Real , f be Function of REAL , REAL ;
b `1 + sqrt ( 1 + ( b `1 / b ) ^2 ) < ( 1 + ( b `1 / b ) ^2 ) ;
reconsider f1 = f as VECTOR of the carrier of X , Y ;
redefine attr i <> 0 implies i ^2 mod ( i + 1 ) = 1 ;
j2 in Seg len ( g2 . ( i2 + 1 ) ) & j <= len ( g2 . ( i2 + 1 ) ) ;
dom ( i - 2 ) = dom ( i - 2 ) .= a ;
cluster sec | ]. PI / 2 , PI / 2 .[ -> one-to-one ;
Ball ( u , e ) = Ball ( f . p , e ) ;
reconsider x1 = x0 as Function of S , T ;
reconsider R1 = x , R2 = y , R2 = z as Relation of L , L ;
consider a , b being Subset of A such that x = [ a , b ] ;
( <* 1 *> ^ p ) ^ <* n *> in RH ;
S1 +* S2 = S2 +* ( S1 +* S2 ) +* ( S2 +* S2 ) +* ( S2 +* S2 ) ) ;
( ( ( - 1 ) (#) ( ( #Z 2 ) * ( #Z 2 ) ) ) `| Z ) = f ;
cluster -> continuous for Function of C , R^1 ;
set C7 = 1GateCircStr ( <* z , x *> , f3 ) ;
E\langle e2 . e2 *> = ( E . e2 ) -T . ( e2 . e2 ) ;
( ( arctan * arccot ) `| Z ) . x = ( ( arctan * arccot ) `| Z ) . x ;
upper_bound A = ( PI * 3 / 2 ) * ( PI * 2 ) & lower_bound A = 0 ;
F . ( dom f , - F . ( cod f ) ) is Functor of F . ( cod f , - F . ( cod f ) ) , F . ( cod f , - F . ( cod f ) ) ;
reconsider pp = qelement as Point of ( TOP-REAL 2 ) | K1 , ( TOP-REAL 2 ) | K1 ;
g . W in [#] ( Y | 0 ) & [#] ( Y | 0 ) c= [#] ( Y | 0 ) ;
let C be compact non vertical non vertical non horizontal Subset of TOP-REAL 2 ;
LSeg ( f ^ g , j ) = LSeg ( f , j ) ;
rng s c= dom f /\ ]. x0 , x0 + r .[ & for x st x in ]. x0 , x0 + r .[ holds f . x < x0 ;
assume x in { idseq ( 2 ) , Rev ( idseq 2 ) } ;
reconsider n2 = n , m2 = m - n as Element of NAT ;
for y being ExtReal st y in rng seq holds g <= y
for k st P [ k ] holds P [ k + 1 ] ;
m = m1 + m2 .= m1 + m2 .= m1 + m2 .= m1 + m2 ;
assume for n holds H1 . n = G . n -H . n ;
set BB = f .: ( the carrier of X1 ) , BB = f .: the carrier of X2 , BB = f .: the carrier of X2 ;
ex d being Element of L st d in D & x << d ;
assume R " ( a \ b ) c= R " ( a \ b ) ;
t in ]. r , s .[ or t = r & t = s ;
z + v2 in W & x = u + ( z + v2 ) ;
x2 |-- y2 iff P [ x2 , y2 ] & P [ x2 , y2 ] ;
redefine attr x1 <> x2 means : Def2 : |. x1 - x2 .| > 0 & |. x1 - x2 .| > 0 ;
assume p2 - p1 , p3 - p1 , p4 - p1 , p1 - p1 , p2 - p1 is_collinear ;
set q = ( ex f st f = ( 'not' 'not' 'not' A ) ^ <* 'not' 'not' A *> ) ;
let f be PartFunc of \langle REAL-NS 1 , REAL-NS 1 *> , REAL-NS 1 , REAL-NS 1 ;
( n mod ( 2 * k ) ) ! = n mod k ;
dom ( T * ( succ t ) ) = dom ( succ t ) .= dom ( T . t ) ;
consider x being element such that x in w and x in c ;
assume ( F * G ) . ( v . x3 ) = v . x4 ;
assume that the Sorts of D1 c= the Sorts of D2 and the Sorts of D1 c= the Sorts of D2 and the Sorts of D1 c= the Sorts of D2 ;
reconsider A1 = [. a , b .] as Subset of R^1 | [. a , b .] ;
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-min L~ Cage ( C , n ) ;
n1 -' len f + 1 <= len - len f + 1 + 1 ;
ConsecutiveSet ( q , O1 ) = [ u , v , a ] ;
set C-2 = ( ( `1 ) `1 ) . ( k + 1 ) ;
Sum ( L * p ) = 0. R .= 0. V .= 0. V ;
consider i being element such that i in dom p and t = p . i ;
defpred Q [ Nat ] means 0 = Q ( $1 ) & for n st n >= $1 holds P [ n , s . n ] ;
set s3 = Comput ( P1 , s1 , k ) , P4 = P1 +* I , P4 = P1 +* I , P4 = P1 +* I , P4 = P1 +* I , P4 = Comput ( P1 , s1 , k ) , P4 = P1 +* I , P4 = Comput ( P1 , s1 , k )
let l be variable of k , A , B be Element of l ;
reconsider U1 = union ( G . k ) as Subset-Family of T-24 . k ;
consider r such that r > 0 and Ball ( p `1 , r ) c= Q ` ;
( h | ( n + 2 ) ) /. ( i + 1 ) = pp ;
reconsider B = the carrier of X1 , C = the carrier of X2 as Subset of X ;
p[#] L = <* - ( c + 1 ) , 0. L *> .= <* - ( c + 1 ) , 0. L *> ;
synonym f is real-valued for rng f & rng f c= NAT & f is one-to-one ;
consider b be element such that b in dom F and a = F . b ;
x0 < card ( X0 ) + card ( X0 ) - card ( X0 ) + card ( X0 ) - 1 ;
attr X c= B1 means : Def2 : for B1 st B1 c= B1 holds X c= B1 & B1 c= B1 ;
then w in Cl Ball ( x , r ) & dist ( x , w ) <= r ;
angle ( x , y , z ) = angle ( \HM { the } \HM { function } , 0 , 0 , 2 , 0 ) ;
redefine attr 1 <= len s means : Def2 : len ( s * the _ of A ) = len s ;
fY. c= f . ( k + ( n + ( n + 1 ) ) ) ;
the carrier of { 1_ G } = { 1_ G } & the carrier of G = { 1_ G } ;
redefine pred p '&' q in TAUT ( A ) means : Def2 : q '&' p in TAUT ( A ) ;
- ( t `1 / t `2 ) ^2 < ( ( t `1 ) ^2 + ( t `2 ) ^2 ) ;
U1 . 1 = U1 /. 1 .= ( U1 /. 1 ) `1 .= ( U1 . 1 ) `1 .= ( U1 . 1 ) `1 ;
f .: ( the carrier of x ) = the carrier of x & f . ( the carrier of x ) = the carrier of x ;
Indices O = [: Seg n , Seg n :] & len O = n & width O = n ;
for n being Element of NAT holds G . n c= G . ( n + 1 ) ;
then ex V st V in M |^ \square & ex x being Element of M st V = { x } ;
ex f being Element of F-9 st f is \setminus & f is \setminus & f is one-to-one ;
[ h . 0 , h . 3 ] in the InternalRel of G ;
s +* Initialize ( ( intloc 0 ) .--> 1 ) = s3 +* Initialize ( ( intloc 0 ) .--> 1 ) ;
|[ w1 , v1 ]| `1 <> 0. TOP-REAL 2 & |[ w1 , v1 ]| `2 <> 0. TOP-REAL 2 ;
reconsider t = t as Element of INT -tuples_on INT ;
C \/ P c= [#] ( ( ( [#] ( ( ( ( ( ( ( A ) ) | A ) ) | A ) ) \ A ) ) ) ) ) ;
f " V in ( the topology of X ) /\ D ( the topology of X ) & f " ( the topology of X ) /\ D ( the topology of X ) ;
x in [#] ( ( the carrier of F ) /\ A ) & x in [#] ( F ) /\ A ;
g . x <= h1 . x & h . x <= h1 . x ;
InputVertices S = { xy , y , z } & InputVertices S = { xy , y , z } ;
for n be Nat st P [ n ] holds P [ n + 1 ] ;
set R = Line ( M , i ) * Line ( M , i ) ;
assume M1 is being_line & M2 is being_line & M1 is being_line & M2 is being_line ;
reconsider a = f4 . i0 - 1 , b = f1 . ( i0 -' 1 ) as Element of K ;
len B2 = Sum Len ( F1 ^ F2 ) + Sum Len ( F2 ^ F1 ) .= len F1 + len F2 ;
len ( ( the R of n ) * ( i , j ) ) = n & len ( ( R * i ) * ( i , j ) ) = n ;
dom max ( f + g , h ) = dom ( f + g ) .= dom f /\ dom h .= dom f /\ dom h ;
( for n holds ( for m holds seq . n = sup Y1 ) ) implies for n holds ( seq ^\ n ) . n = ( seq ^\ ( n + m ) ) . n
dom ( p1 ^ p2 ) = dom ( f ^ p2 ) .= dom ( f ^ p2 ) .= dom ( f ^ p2 ) ;
M . [ 1 , y ] = 1 * v1 .= ( 1 - 1 ) * v1 .= ( 1 - 1 ) * v1 .= ( 1 - 1 ) * v1 ;
assume that W is non trivial and W |. W .| c= the \frac of G2 , G ;
C6 /. i1 = G1 * ( i1 , i2 ) & C6 = G2 * ( i1 , i2 ) ;
C8 |- 'not' All ( x , p ) 'or' p . ( x , y ) ;
for b st b in rng g holds lower_bound rng f[ f[ fa , b ] , fa ] <= b
- ( ( q1 `1 ) ^2 + ( q1 `2 ) ^2 ) = 1 ;
( LSeg ( c , m ) \/ LSeg ( l , k ) ) \/ LSeg ( l , k ) c= R ;
consider p be element such that p in { x } and p in L~ f and x = f /. p ;
Indices X = [: Seg n , Seg n :] & len X = n & width X = n ;
cluster s => ( q => p ) -> valid ;
Im ( ( Partial_Sums ( F ) ) . m ) . x is measurable & ( Partial_Sums ( F ) . m ) . x is measurable ;
cluster f . ( x1 , x2 ) -> Element of D ( ) ;
consider g being Function such that g = F . t & Q [ t , g . t ] ;
p in LSeg ( ( N-min Z ) * ( ( NW-corner Z ) * ( 1 , 1 ) ) , ( NW-corner Z ) * ( 1 , 1 ) ) ;
set R8 = R |^ 1 , R8 = ]. b , +infty .[ , R8 = ]. a , b .[ , R8 = ]. a , b .[ , R8 = ]. a , b .[ , + \infty .[ ;
IncAddr ( I , k ) = AddTo ( d , d1 ) .= goto IC Comput ( P , s , k ) ;
seq . m <= ( the ^\ ( seq . m ) ) . k & ( the ^\ ( seq . m ) ) . k <= ( the ^\ ( seq . m ) ) . k ;
a + b = ( a ` *' b ` ` ) ` .= ( a ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` .= a ` ` ` ` ` ` ;
id ( X /\ Y ) = id ( X /\ Y ) .= id X /\ id Y .= id X ;
for x being element st x in dom h holds h . x = f . x ;
reconsider H = U1 \/ U2 as non empty Subset of U0 ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) /\ f ) /\ m ) /\ m ;
consider y being element such that y in Y and P [ y , inf B ] ;
consider A being finite Subset of R such that card A = ( \alpha R ) * A and for x being Element of R st x in A holds x in A ;
p2 in rng ( f |-- p1 ) \ rng <* p1 *> & not p2 in rng <* p1 *> ;
len s1 - 1 > 0 & len s2 - 1 > 0 ;
( N-min L~ f ) `2 = ( E-max L~ f ) `2 .= ( E-max L~ f ) `2 .= ( E-max L~ f ) `2 ;
Ball ( e , r ) c= LeftComp Cage ( C , k + 1 ) ;
f . a1 ` = f . a1 ` .= ( a1 ` ` ) ` .= ( a1 ` ) ` ` .= ( a1 ` ) ` ;
( seq ^\ k ) . n in ]. - r , x0 .[ & ( seq ^\ k ) . n in ]. - r , x0 .[ ;
gg . seq = g . ( s . ( seq . ( seq . m ) ) ) .= g . ( seq . ( seq . m ) ) ;
the InternalRel of S is transitive implies ( the InternalRel of S ) \/ ( the InternalRel of S ) \/ ( the InternalRel of S )
deffunc F ( Ordinal , Ordinal ) = phi . ( $2 , $2 ) ;
F . ( s1 . a1 ) = F . ( s2 . a1 ) .= F . ( s1 . a1 ) ;
x `1 = A . o .= Den ( o , A . a ) .= Den ( o , A . a ) ;
Cl ( f " P1 ) c= f " ( Cl P1 ) & f " P1 c= f " P1 & f " P1 = f " P1 ;
FinMeetCl ( ( the topology of S ) \ { {} } ) c= the topology of T ;
synonym o is \bf for o <> \ast means for o <> \ast & o <> {} & o <> {} & o <> {} ;
assume that X = Y + Z and card X <> card Y and card X <> card Y and card Y <> card Z ;
the such that s <= 1 + ( the such that s = 1 + ( the such that s = ( the such | s ) +* ( 1 + ( s | s ) ) ) . 1 ) ;
LIN a , a1 , d or b , c // b1 , c1 ;
e /. 1 = 0 & e . 2 = 1 & e . 3 = 0 ;
EE in S1 & not E in { N . m } ;
set J = ( l , u ) If not contradiction , then ( l , u ) If not contradiction ;
set A1 = Following ( a9 , b9 ) , A2 = Following ( s , 2 ) , A2 = Following ( s , 2 ) , A1 = Following ( s , 2 ) ;
set c9 = [ <* cin , d *> , '&' ] , A2 = [ <* cin , d *> , '&' ] , A1 = [ <* cin , c *> , '&' ] , A2 = [ <* cin , d *> , '&' ] ;
x * z `1 * x " in x * ( z * N ) " * ( x * N ) ;
for x being element st x in dom f holds f . x = g2 . x ;
Int cell ( f , 1 , G ) c= RightComp f \/ RightComp f \/ L~ f \/ L~ f ;
U is an arc of W-min C , E-max C & W-min L~ Cage ( C , n ) in rng g ;
set f-17 = f @ @ @ g , f@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
attr S1 is convergent & S2 is convergent & for n holds ( S1 - S2 ) . n is convergent & lim ( S1 - S2 ) = lim ( S1 - S2 ) ;
f . ( 0 qua Nat ) = ( 0 qua Nat ) + a .= a ;
cluster -> \cal be Let -Let strict for non empty RelStr ;
consider d being element such that R reduces b , d and R reduces c , d ;
not b in dom Start-At ( ( card I + 2 ) , SCMPDS ) ;
( z + a ) + x = z + ( a + y ) .= z + ( a + y ) ;
len ( l (#) ( a * ( A * 0 ) ) ) = len l ;
t4 is ( {} \/ rng ( t . {} ) ) -valued ;
t = <* F . t *> ^ ( C . p ^ q ) .= q . ( p ^ q ) ;
set pp = W-min L~ Cage ( C , n ) , p = W-min L~ Cage ( C , n ) , q = W-min L~ Cage ( C , n ) ;
( ( i -' 1 ) + 1 ) - ( i -' 1 ) = ( i -' 1 ) - ( i -' 1 ) ;
consider u being Element of L such that u = u ` and u in D and u in D ;
len ( ( width ( \hbox { \boldmath $ G $ } } ) |-> a ) ) = width ( ( the _ { G , a } ) --> a ) ;
F3 . x in dom ( ( G * the_arity_of o ) . x ) ;
set H2 = the carrier of H2 , H1 = the carrier of H1 , H2 = the carrier of H2 ;
set H1 = the carrier of H1 , H2 = the carrier of H2 , E = the carrier of H2 ;
( Comput ( P , s , 6 ) ) . intpos ( m + 6 ) = s . intpos ( m + 6 ) ;
IC Comput ( Q2 , t , k ) = ( l + 1 ) - 1 .= ( card I + 1 ) - 1 ;
dom ( ( cos * sin ) `| Z ) = REAL & for x st x in Z holds ( ( cos * sin ) `| Z ) . x = sin . x / ( cos . x ) ^2 )
cluster <* l *> ^ phi -> ( 1 + 1 ) -element for string of S ;
set b5 = [ <* A1 , cin *> , '&' ] , d5 = [ <* cin , y2 *> , '&' ] , d5 = [ <* A1 , cin *> , '&' ] , d5 = [ <* A1 , cin *> , '&' ] , d5 = [ <* A1 , cin *> , '&' ] ;
Line ( Segm ( M `1 , P , Q ) , x ) = L * Sgm Q ;
n in dom ( ( the Sorts of A ) * the_arity_of o ) ;
cluster f1 + f2 -> continuous for PartFunc of REAL , REAL ;
consider y be Point of X such that a = y and ||. y - x .|| <= r ;
set x3 = t2 . DataLoc ( s2 . SBP , 2 ) , x2 = s . DataLoc ( s . SBP , 2 ) , x3 = s . DataLoc ( s . SBP , 2 ) , x4 = s . DataLoc ( s . SBP , 2 ) , x4 = s . DataLoc ( s . SBP , 2 ) , x4 = s
set p0 = stop I , s2 = stop I , s3 = stop I , s3 = stop I , s4 = stop I , P4 = stop I , P4 = stop I , P4 = stop I , P4 = stop I , s4 = stop I , P4 = stop I , P4 = stop I , P4 = stop I , s4 = 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 } \/ { C , D } ;
let A , B , C , D , E , F , J , M , N , N , F , J , M , N , N , F , M , N , N , F , J , M , N , F , M , N , N , F , J , M , N , F , J , M ,
|. p2 .| ^2 - ( ( p2 `1 ) ^2 + ( p2 `2 ) ^2 ) >= 0 ;
l - 1 + 1 = n-1 * ( l1 - 1 ) + ( l1 - 1 ) ;
x = v + ( a * w1 + ( b * w2 ) + c * w2 ) + ( c * w2 + c * w2 ) ;
the TopStruct of L = TopSpaceMetr (# the TopStruct of L , the TopStruct of L #) ;
consider y being element such that y in dom H1 and x = H1 . y and x = H1 . y ;
fH \ { n } = Free ( All ( 'not' v1 , H ) ) \/ Free ( H ) ;
for Y being Subset of X st Y is summable & Y is not empty holds Y is non empty implies Y is non empty
2 * n in { N : 2 * Sum ( p | N ) = N & N > 0 } ;
for s being FinSequence holds len ( the { - } *> * ( the { - } Shift of s ) ) = len s
for x st x in Z holds exp_R . x = exp_R . x / ( 1 + x ^2 )
rng ( h2 * f2 ) c= the carrier of ( ( TOP-REAL 2 ) | K1 ) & rng ( ( - 1 ) | K1 ) c= the carrier of ( ( TOP-REAL 2 ) | K1 ) ;
j + ( len f - len f ) <= len f + ( len f - len f ) - len f ;
reconsider R1 = R * I as PartFunc of REAL , REAL-NS n , REAL-NS n , REAL-NS m ;
C8 . x = s1 . ( a + 1 ) .= C8 . x + s1 . x .= C8 . x + s1 . x .= C8 . x ;
power ( F_Complex , n ) . ( z , n ) = 1 .= ( x |^ n ) * ( x |^ n ) .= ( x |^ n ) * ( x |^ n ) ;
t at ( C , s ) = f . ( the connectives of S ) . t ;
support ( f + g ) c= support f \/ ( C /\ support g ) ;
ex N st N = j1 & 2 * Sum ( ( r | N ) | N ) > N ;
for y , p st P [ p ] holds P [ All ( y , p ) ]
{ [ x1 , x2 ] } is Subset of [: X1 , X2 :] ;
h = ( i , j = h , id B . i ) .= H . i .= H . i ;
ex x1 being Element of G st x1 = x & x1 * N c= A & N * N c= A ;
set X = ( ( |. ( |. q .| ) .| ) * ( ( |. q .| ) * ( ( |. q .| ) * ( ( q .| ) * ( ( q .| ) * ( ( q `2 ) * ( q `2 ) ) ) ) ) ) ) ) ;
b . n in { g1 : x0 < g1 & g1 < x0 & g1 < x0 } ;
f /* s1 is convergent & f /. x0 = lim ( f /* s1 ) ;
the lattice of lattice Y = the lattice of the lattice of Y & the carrier of Y = the carrier of X & the carrier of Y = the carrier of X ;
'not' ( a . x ) 'or' b . x 'or' a . x = TRUE ;
2 = len ( q2 ^ r1 ) + len r1 .= len ( q2 ^ r1 ) + len r1 .= len ( q2 ^ r1 ) + len r1 ;
( ( 1 / a ) (#) ( sec * f1 - id Z ) - id Z ) is_differentiable_on Z ;
set K1 = upper \ ( lim ( H , A ) `| H ) , D2 = ( lim ( H , A ) `| H ) , E = ( lim H ) `| H ;
assume e in { ( w1 - w2 ) : w1 in F & w2 in G & w1 in F } ;
reconsider da = dom a `1 , db = dom F `1 , db = dom F `1 , dG = dom F `1 , dG = dom F `1 , dG = dom F , dG = dom F , dG = dom G , dG = dom F , dG = dom F , dG = dom G , dG = dom F
LSeg ( f /^ j , q ) = LSeg ( f , j + q .. f -' 1 ) .= LSeg ( f , j + q .. f -' 1 ) ;
assume X in { T . N2 , K . N2 } ;
assume that Hom ( d , c ) <> {} and <* f , g *> * f1 = <* f , g *> * f1 ;
dom S[. S . n , L . n .] = dom S /\ Seg n .= dom ( L . n ) .= Seg n ;
x in H |^ a implies ex g st x = g |^ a & g in H |^ a ;
a * ( ( 0 qua Nat ) * n + 1 ) = a `1 - ( 0 qua Nat ) * n .= a ;
D2 . j in { r : lower_bound A <= r & r <= upper_bound A } ;
ex p being Point of TOP-REAL 2 st p = x & P [ p ] & p `2 <= 0 ;
for c holds f . c <= g . c implies f ^ @ c = @ @ c ;
dom ( f1 (#) f2 ) /\ X c= dom ( f1 (#) f2 ) /\ X .= dom ( f1 (#) f2 ) /\ X .= dom ( f1 (#) f2 ) /\ X .= dom f1 /\ X /\ X .= dom f1 /\ X .= X /\ dom f2 ;
1 = ( p * p ) / p .= p * ( 1 / p ) .= p * ( 1 / p ) .= p * ( 1 / p ) ;
len g = len f + len <* x + y *> .= len f + 1 + 1 .= len f + 1 ;
dom ( ( F | [: N1 , N2 :] ) | [: N1 , S :] ) = dom ( F | [: N1 , S :] ) /\ [: N1 , S :] ;
dom ( ( f . t ) * I ) = dom ( ( f . t ) * g . t ) ;
assume a in ( "\/" ( ( T |^ \alpha ) , F ) ) .: D , ( id the carrier of S ) .: D ) ;
assume that g is one-to-one and ( the carrier' of S ) /\ rng g c= dom g and for x being Element of S st x in dom g holds g . x = f . x ;
( ( x \ y ) \ z ) \ ( ( x \ z ) \ ( y \ z ) ) = 0. X ;
consider f such that f * f = id b and f * f = id a & f * f = id b ;
( ( cos * cos ) `| [. - 1 , 1 .] ) | [. - 1 , 1 .] is increasing ;
Index ( p , co ) <= len LS - LS .. LS + 1 & LS . len LS -' 1 = Gij .. LS -' 1 + 1 ;
t1 , t2 , 3 , 4 , 5 , 6 , 7 , 8 , 6 , 8 , 7 , 8 , 8 , 6 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 ,
( ( ( Frege ( F , G ) ) . h ) . h ) . ( h . ( h . ( h . ( h . ( h . k ) ) ) ) ) ) . ( h . ( h . ( h . k ) ) ) ) . ( h . ( h . ( h . k ) ) ) . ( h . ( h . k ) )
then P [ f . i0 ] & F ( f . i0 + 1 ) < j & F ( f . i0 + 1 ) < j ;
Q [ ( D . ( x , 1 ) ) `1 , F . ( x , 1 ) ] ;
consider x being element such that x in dom ( F . s ) and y = ( F . s ) . x ;
l . i < r . i & [ l . i , r . i ] is r of G . i ;
the Sorts of A2 = ( the Sorts of S2 ) * ( the Arity of S2 ) .= ( the Sorts of A2 ) * ( the Arity of S2 ) ;
consider s being Function such that s is one-to-one & dom s = NAT & rng s c= F & s . 0 = s . 1 ;
dist ( b1 , b2 ) <= dist ( b1 , a ) + dist ( b2 , a ) ;
( ( for n holds ( for C be Subset of TOP-REAL 2 ) | ( len C ) ) ) `1 = W-bound L~ Cage ( C , n ) ;
q `2 <= ( ( UMP L~ Cage ( C , n ) ) * ( i + 1 ) ) `2 ;
LSeg ( f | i2 , i ) /\ LSeg ( f | i2 , j ) = {} ;
given a being ExtReal such that a <= I and A = ]. a , I1 .[ and a = ]. a , I1 .[ ;
consider a , b be complex number such that z = a & y = b & z + y = a + b ;
set X = { b |^ n where b is Element of NAT : not contradiction } , Y = { b |^ n : not contradiction } ;
( ( x * y ) \ z ) \ ( x * y ) \ z = 0. X ;
set xy = [ <* xy , yz , zx *> , '&' ] , yz = [ <* xy , yz *> , '&' ] , zx = [ <* z , x *> , '&' ] , zx = [ <* z , x *> , '&' ] , zx = [ <* z , x *> , '&' ] , zx = [ <* z , x *> , '&' ] ;
lr /. len ( lr /. 1 ) = ( lr /. len ( lr /. 1 ) ) `1 ;
sqrt ( ( ( q `2 ) - sn ) / ( 1 + sn ) ) ^2 = 1 ;
sqrt ( ( p `1 ) ^2 - ( p `2 ) ^2 ) < 1 ;
( ( ( ( ( X \/ Y ) \ Y ) ) \/ Y ) \/ Y ) \/ Y ) = ( ( X \/ Y ) \/ Y ) \/ ( ( X \/ Y ) \ Y ) ;
( ( s1 - s1 ) / ( k + 1 ) ) . n = ( s1 . k - s1 . k ) / ( k + 1 ) ;
rng ( ( h + c ) ^\ n ) c= dom SVF1 ( 1 , f , u0 ) ;
the carrier of X = the carrier of X & the carrier of X = the carrier of X & the carrier of X = the carrier of X ;
ex p4 st p4 = p4 & |. p4 - |[ a , b ]| .| = r ;
set ch = chi ( X , A ) , Ah = chi ( X , A ) , Ah = chi ( X , A ) , Ah = chi ( X , A ) , Ah = chi ( X , A ) , Ah = chi ( X , A ) ;
R |^ ( 0 * n ) = \mathop { Il ( X , X ) } .= R |^ ( n + 1 ) .= R |^ ( n + 1 ) ;
( Partial_Sums ( ( ( F . -19 ) ) . n ) ) . ( ( F . -19 ) . n ) is nonnegative & ( ( F . -19 ) . n ) . ( ( F . -19 ) . n ) is nonnegative ;
f2 = C7 . ( ( E , ( V , ( K , len ( K , len ( K , len ( K , len ( K , len ( K , len ( K , len ( K , len ( K , len ( K , len ( K , len K , len K ) ) ) ) ) ) ) ) ) ) ) ;
S1 . b = s1 . b .= s2 . b .= s2 . b .= ( s1 . b ) `1 .= s2 . b ;
p2 in LSeg ( p2 , p1 ) /\ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) ;
dom ( f . t ) = Seg n & dom ( I . t ) = Seg n & dom ( I . t ) = Seg n ;
assume o = ( the connectives of S ) . 11 & ( the connectives of S ) . 11 in ( the connectives of S ) . 11 ;
set phi = ( l1 , l2 ) \HM { l2 , l2 } , N = ( l2 , l2 ) \HM { {} } , D = { {} } , D = { {} } , E = { {} } , D = { {} } , E = { {} } , D = { {} } , D = { {} } , E = { {} } , E = { {} } , D = { {} } ,
synonym p is invertible for ( p , T ) *' ( q , T ) = 1. L ;
( Y1 `2 = - 1 & ( Y1 `2 = - 1 ) * ( Y1 `2 ) + 1 ) * ( Y1 `2 - 1 ) * ( Y1 `2 - 1 ) * ( Y1 `2 - 1 ) * ( Y1 `2 - 1 ) * ( Y1 `2 - 1 ) * ( Y1 `2 - 1 ) * ( Y1 - 1 ) * ( Y1 - 1 ) * ( Y1 - 1 ) * ( Y1 - 1
defpred X [ Nat , set ] means P [ $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $1 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2
consider k be Nat such that for n be Nat st k <= n holds s . n < x0 + g ;
Det I |^ ( ( m -' n ) mod ( m -' n ) ) = 1. K .= ( m - n ) mod ( m -' n ) ;
sqrt ( - b ^2 - ( - b * c ) ) * ( - a * c ) < 0 ;
Cr . d = Cr . ( dd . d ) mod Cr . ( dd . d ) .= Cr . ( dd . d ) mod Cr . ( dd . d ) ;
attr X1 is dense & X2 is dense & X1 is dense implies X1 /\ X2 is dense SubSpace of X ;
deffunc F ( Element of E , Element of I , Element of E ] means $1 = $2 * $2 & $2 = ( $1 , $2 ) * ( $2 , $2 ) ;
t ^ <* n *> in { t ^ <* i *> : Q [ i , T . t ] } ;
( x \ y ) \ x = ( x \ x ) \ y .= y \ 0. X .= 0. X ;
for X being non empty set holds X is Basis of <* X , product <* Y *> *> & X is Basis of <* X , product <* Y *> *>
synonym A , B means : Def2 : Cl ( A \/ B ) misses Cl ( B \/ C ) & A misses B & B misses A ;
len Mp = len p & width Mp = width M & width Mp = width M implies p = q
J = { x where x is Element of K : 0 < v . x & v . x < 1 } ;
( Sgm ( support m ) ) . d - ( Sgm ( support m ) ) . e <> 0 ;
lower_bound divset ( D2 , k + k2 ) = D2 . ( k + k2 ) - D2 . ( k + k2 ) ;
g . r1 = - 2 * r1 + 1 & dom h = [. 0 , 1 .] & rng h c= [. 0 , 1 .] ;
|. a .| * ||. f .|| = 0 * ||. f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| .= 0 ;
f . x = ( h . x ) `1 & g . x = ( h . x ) `2 ;
ex w st w in dom B1 & <* 1 *> ^ s = <* 1 *> ^ w & s = <* 1 *> ^ w ;
[ 1 , {} , <* d1 *> ] in ( { 0 , {} } \/ { {} , {} } ) \/ ( { {} , {} } \/ { {} , {} } ) ;
IC Exec ( i , s1 ) + n = IC Exec ( i , s2 ) + n .= IC Exec ( i , s2 ) + n .= IC Exec ( i , s2 ) , Comput ( i , s2 , n ) ) ;
IC Comput ( P , s , 1 ) = IC Comput ( P , s , 1 ) .= 5 + ( card I + 1 ) .= ( card I + 1 ) + 1 .= ( card I + 1 ) + 1 ;
( IExec ( W6 , Q , t ) ) . intpos ( m + 1 ) = t . intpos ( m + 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 holds x <= y or y <= x ;
|( integral ( f , C ) , f /. ( upper_bound X ) , f /. ( lower_bound X ) , f /. ( lower_bound X ) , f /. ( lower_bound X ) , f /. ( lower_bound X ) , f /. ( lower_bound X ) , f /. ( lower_bound X ) , f /. ( lower_bound X ) , f /. ( lower_bound X ) , f /. ( lower_bound X ) ) *] REAL ;
for F , G being one-to-one FinSequence st rng F misses rng G & G is one-to-one holds F ^ G is one-to-one
||. R /. ( L . h ) - R /. ( K . h ) .|| < e1 * ( K . h + R . h ) ;
assume a in { q where q is Element of M : dist ( z , q ) <= r } ;
set p4 = [ 2 , 1 ] .--> [ 2 , 0 ] ;
consider x , y being Subset of X such that [ x , y ] in F and x c= d and y c= d ;
for y , x being Element of REAL st y `1 in Y & x `1 in X holds y `1 <= x `1 ;
func |. p (#) : p .| -> variable of A means : Def2 : for i being Nat st i in dom it holds it . i = min ( NBI . i , p . i ) ;
consider t being Element of S such that x `1 , y `1 '||' z `1 , t `2 & x , y '||' z `1 , t `2 ;
dom x1 = Seg ( len x1 ) & len x1 = len l1 & for i st i in dom x1 holds x1 . i = ( l1 . i ) * ( l1 . i ) ) ;
consider y2 being Real such that x2 = y2 and 0 <= y2 and y2 <= 1 and 0 <= y2 and y2 <= 1 ;
||. f | X /* s1 .|| = ||. f .|| | X .= ||. f .|| /* s1 .= ||. f .|| /. ( lim s1 ) .|| .= ||. f /. ( lim s1 ) .|| ;
( the InternalRel of A ) ` = {} or ( the InternalRel of A ) ` = {} \/ {} .= {} .= {} ;
assume that i in dom p and for j be Nat st j in dom q holds P [ i , j ] and for i be Nat st i in dom q & i < j holds P [ i , j ] ;
reconsider h = f | [: X , Y :] as Function of [: X , Y :] , [: X , Y :] ;
u1 in the carrier of W1 & u2 in the carrier of W2 & u2 in the carrier of W1 & u2 in the carrier of W2 ;
defpred P [ Element of L ] means M <= f . $1 & f . $1 <= f . $1 & f . $1 <= f . $1 ;
T . ( u , a , v ) = s * x + ( - ( s * x + y ) ) .= b ;
- ( - ( - x ) ) = - x + ( - y ) .= - ( - x ) + ( - y ) .= - ( - x ) + ( - y ) .= - ( - x ) + ( - y ) .= - ( - x ) + ( - y ) ;
given a being Point of IT such that for x being Point of IT holds a , x are_not collinear ;
f\rbrace = [ dom @ ( f2 , cod ( f2 , cod ( f1 , cod ( f2 , cod ( f2 , cod ( f1 , cod ( f2 , cod ( f1 , cod ( f2 , cod ( f1 , cod ( f1 , cod ( f2 , cod ( f1 , cod ( f1 , cod ( f2 , x0 ) ) ) ) ) ) ) ) , cod ( f2 , cod ( f2 ,
for k , n be Nat st k <> 0 & k < n & n <= n holds k divides n implies ( k * n ) div ( k * n ) = ( k * n ) div ( k * n )
for x being element holds x in A |^ d iff x in ( ( A ` ) |^ d ) ` ) ` & not x in ( ( A ` ) |^ d ) ` ;
consider u , v being Element of R , a being Element of A such that l /. i = u * a * v ;
- ( ( p `1 / |. p .| - sn ) / ( 1 + sn ) ) ^2 > 0 ;
Carrier ( L-13 ) . k = ( Carrier ( L ) ) . k & F . k in dom ( Carrier ( L ) ) ;
set i2 = AddTo ( a , i , - n ) , i1 = goto ( - n + 1 ) , i2 = goto ( - n ) , i2 = goto ( - n ) , i1 = goto ( - n ) , i2 = goto ( - n ) , i2 = goto ( - n ) , i2 = goto ( - n ) ;
attr B is \overline means : Def2 : for S being non empty ManySortedSign holds ( for B being Element of \sum ( ( B ) | B ) ) . S = ( B . S ) . S ) ;
a9 "/\" D = { a "/\" d where d is Element of N : d in D & d in D } ;
|( \square , q9 . ( q9 - q19 ) , REAL . ( q9 - q19 ) )| >= |( \square , 0 )| ;
( - f ) . ( upper_bound A ) = ( ( - f ) . ( upper_bound A ) ) * ( upper_bound A ) .= ( - f ) . ( upper_bound A ) ;
( G * ( len G , k ) `1 ) `1 = ( G * ( len G , k ) `1 ) `1 .= G * ( 1 , k ) `1 ;
( Proj ( i , n ) * ( proj ( i , n ) * ( h ) ) ) . 3 = <* ( proj ( i , n ) * ( h ) ) . 3 *> ;
f1 + f2 * reproj ( i , x ) is_differentiable_in ( ( reproj ( i , x ) ) * reproj ( i , x ) ) . x0 ;
redefine attr ( tan * tan ) . x <> 0 & ( tan * tan ) . x <> 0 ;
ex t being SortSymbol of S st t = s & h1 . ( t . x ) = h2 . ( t . x ) ;
defpred C [ Nat ] means ( P8 . $1 is <= $1 & A8 . $1 is as n -A8 ;
consider y being element such that y in dom p9 and q8 . i = p9 . y and ( for i being element st i in dom p9 holds q8 . i = y . i ) ;
reconsider L = product ( { x1 } +* ( index ( B ) , l ) ) as product of Carrier A ;
for c being Element of C ex d being Element of D st T . ( id c ) = id d & c = d & d = c
Comput ( f , n , p ) = ( f | n ) ^ <* p . ( n + 1 ) *> .= f ;
( f * g ) . x = f . ( g . x ) & ( f * h ) . x = f . ( h . x ) ;
p in { ( 1 / 2 ) * ( G * ( i + 1 , j ) + G * ( i + 1 , j ) ) } ;
f `1 - cp = ( f | ( n , L ) ) *' ( - ( f | ( n , L ) ) ) .= ( f - ( - ( f | ( n , L ) ) ) ) *' ( - ( f | ( n , L ) ) ) ;
consider r be Real such that r in rng ( f | divset ( D , j ) ) and r < m + s ;
f1 . [ ( r2 - r1 ) / ( 2 * ( r2 - r1 ) ) , ( r1 - r1 ) / ( 2 * ( r2 - r1 ) ) , r1 - r1 ) ] in f1 .: [: W1 , W2 :] ;
eval ( a | ( n , L ) , x ) = ( a | ( n , L ) ) . x .= a . ( n , L ) .= a . ( n , L ) ;
z = DigA ( t9 , x9 ) .= DigA ( t9 , x ) .= DigA ( t9 , x ) .= DigA ( t9 , x ) .= ( DigA ( t9 , x ) ) . x ;
set H = { Intersect S where S is Subset-Family of X : S c= G & S c= F } , G = { {} , { {} } } , H = { {} , { {} , { {} } } , H = { {} , { {} , { {} } } , H = { {} , { {} , { {} } } , H = { {} , { {} , { {} , {} } } , H = { {} , {
consider S19 being Element of D ( ) , d being Element of D ( ) such that S `1 = S19 ^ <* d *> and d = ( S ^ <* d *> ) . ( len S + 1 ) ;
assume that x1 in dom f and x2 in dom f and f . x1 = f . x2 and f . x2 = f . x1 ;
- 1 <= ( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) / ( 1 + sn ) ;
0. K is Linear_Combination of A & Sum ( L ) = 0. V implies Sum ( L ) = 0. V & Sum ( L ) = 0. V
let k1 , k2 , x4 , k2 , x4 , x5 , 6 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 9 be Element of NAT ;
consider j being element such that j in dom a and j in g " { k } and x = a . j ;
H1 . x1 c= H1 . x2 or H1 . x2 c= H1 . ( x2 . x1 ) or H1 . ( x2 . x1 ) c= H1 . ( x2 . x1 ) ;
consider a being Real such that p = *> * p1 + ( a * p2 ) and 0 <= a and a <= 1 ;
assume that a <= c & c <= d and [ a , b ] c= dom f and [ a , b ] in 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 : not contradiction } ;
( T * b1 ) . y = L * b2 /. ( b1 /. y ) .= ( F * b1 ) . y .= ( F * b2 ) . y ;
g . ( s , I ) . x = s . y & g . ( s , I ) . y = |. s . x - g . y .| ;
( log ( 2 , k ) ) ^2 >= ( log ( 2 , k ) ) ^2 + ( log ( 2 , k ) ) ^2 ;
then p => q in S & not x in the still of A & not x in S & not x in S ;
dom ( the InitS of r-10 ) misses dom ( the InitS of r-10 ) & dom ( the InitS of r-10 ) = dom ( the \setminus of r-10 ) ;
synonym f is extended real means : Def2 : for x being set st x in rng f holds x is ExtReal ;
assume for a being Element of D holds f . { a } = a & for X being Subset-Family of D holds f . ( f .: X ) = f . union X ;
i = len p1 .= len p3 + len <* x *> .= len p1 + len <* x *> .= len p1 + len <* x *> .= len p1 + len <* x *> .= len p1 + len <* x *> .= len p1 + len <* x *> .= len p1 + 1 ;
( l . ( 1 , 3 ) ) `1 = ( g . ( 1 , 3 ) ) `1 + ( g . ( 1 , 3 ) ) `1 - ( g . ( 1 , 3 ) ) `1 + ( g . ( 1 , 3 ) ) `1 ) ;
CurInstr ( P2 , Comput ( P2 , s2 , l ) ) = halt SCM+FSA .= halt SCM+FSA .= ( halt SCM+FSA ) . IC SCM+FSA .= ( IC Comput ( P2 , s2 , l ) ) .= ( IC Comput ( P2 , s2 , l ) ) ;
assume for n be Nat holds ||. ( seq . n ) - ( seq . n ) .|| <= ( seq . n ) - ( seq . n ) ;
sin . r2 = sin . ( r * cos . ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . x ) ) ) ) ) ) ) ) .= 0 ;
set q = |[ g1 . ( t1 . t ) , g2 . ( t2 . t ) ]| , g1 . ( t1 . t ) ]| ;
consider G being sequence of S such that for n being Element of NAT holds G . n in \mathopen { \rm \mathopen { - } ( F . n ) } ;
consider G such that F = G and ex G1 , G2 st G1 in S3 & G2 in S3 & G = ( the carrier of G ) \/ ( the carrier of G ) ;
the root of \llangle x , s \rrangle in ( ( the Sorts of Free ( C , X ) ) * ( the Arity of S ) ) . s ;
Z c= dom ( ( exp_R * ( ( exp_R * f1 ) ) / ( exp_R * f1 ) ) ) ;
for k be Element of NAT holds ( ( Im f ) . k ) . k = ( ( Im f ) . k ) * ( ( Im f ) . k )
assume that - 1 < n and ( q `2 / |. q .| - sn ) / ( 1 + sn ) < 0 and ( q `2 / |. q .| - sn ) / ( 1 + sn ) < 0 ;
assume that f is continuous and a < b and c < d and f . a = g . c and f . b = d and f . c = d ;
consider r being Element of NAT such that s8 = Comput ( P1 , s1 , r ) and r <= q and r <= s ;
LE f /. ( i + 1 ) , f /. ( j + 1 ) , L~ f , f /. ( len f -' 1 ) , f /. ( len f -' 1 ) ;
assume that x in the carrier of K and y in the carrier of K and inf { x , y } in the carrier of K ;
assume f +* ( i1 , phi ) in ( proj ( F , i2 ) ) " ( proj ( F , i2 ) ) " ( Carrier ( F , i2 ) ) ) ;
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 : not contradiction } ;
consider l be Nat such that for m be Nat st l <= m holds ||. s1 . m - f /. x0 .|| < g ;
consider t be VECTOR of product G such that mt = ||. Dt . t .|| and ||. t . t .|| <= 1 ;
assume that the ] that the [#] of v = 2 and v ^ <* 0 *> in dom p and v ^ <* 1 *> in dom p and p ^ <* 1 *> in dom p ;
consider a being Element of the \rm lines of [ X1 , X2 , X3 ] such that a on A & b on A & c 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 . i holds p . i in D . i
defpred R [ element ] means ex x , y st [ x , y ] = $1 & P [ x , y ] & P [ y , x ] ;
L~ f2 = union { LSeg ( p0 , p1 ) , LSeg ( p1 , p2 ) , LSeg ( p1 , p2 ) } .= { LSeg ( p1 , p2 ) , LSeg ( p1 , p2 ) } ;
i -' len h11 + 2 - 1 < i -' len h11 + 2 - 1 + 1 + 1 - 1 + 1 ;
for n be Element of NAT st n in dom F holds F . n = |. ( n\cal F . n ) .| + |. ( F . n ) .|
for r , s1 , s2 , r , s , t , t , r , s , t , t , r , s , t , r , s , t , r , s , t , s , t , r , s , t , r , s , t , s , t , r , s , t , s , t , t , s , r , s , t , s , t , r , s , t , s , r , s , t
assume v in { G where G is Subset of T2 : G in B2 & G c= B1 & G c= B2 & G c= B1 & G c= B2 & G c= B2 & G c= B1 & G c= B2 & G c= B1 & G c= B2 & G c= B1 & G c= B2 ;
let g be Element of A , X be Element of INT , b be Element of INT , c be Element of ( A , B ) | ( A , B ) | ( A , B ) | ( A , B ) ;
min ( g . [ x , y ] , k . [ y , z ] , h . ( y , z ] ) = ( min ( g . [ y , z ] , h . ( y , z ) ) ) . y ;
consider q1 being sequence of Cj such that for n holds P [ n , q1 . n , q1 . ( n + 1 ) ] ;
consider f being Function such that dom f = NAT and for n being Element of NAT holds f . n = F ( n ) ;
reconsider B9 = B /\ B , Z = O , Z = Z /\ O as Subset of B ;
consider j being Element of NAT such that x = the holds j = ( the thesis of n ) * ( i , j ) and 1 <= j & j <= n ;
consider x such that z = x and card ( x . O2 ) in card ( x . O2 ) and x in L1 . ( O . O2 ) and x in L1 . ( O . O2 ) ;
( C * _ ( k , n2 ) ) . 0 = C . ( ( ( C * _ 2 , n2 ) ) . 0 ) ) .= C . ( ( ( C * _ 2 , n2 ) ) . 0 ) ;
dom ( X --> rng f ) = X & dom ( X --> f ) = X & rng ( X --> f ) = dom ( X --> f ) ;
( ( NW-corner L~ SpStSeq L~ Cage ( C , n ) ) .. ( SpStSeq L~ Cage ( C , n ) ) <= ( ( NW-corner L~ Cage ( C , n ) ) .. ( NW-corner L~ Cage ( C , n ) ) ) .. ( NW-corner L~ Cage ( C , n ) ) ;
synonym x , y means : Def2 : x = y or ex l being Nat st { x , y } c= 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 & x << y holds a << y iff a << y ;
( ( 1 / 2 ) * ( ( ( cos * ( #Z n ) ) * ( #Z n ) ) ) `| REAL ) = ( REAL ) * ( ( #Z n ) * ( ( #Z n ) * ( #Z n ) ) ) ;
defpred P [ Element of omega ] means ( ( the partial of A1 ) * ( $1 + 1 ) ) . $1 = A1 . ( $1 + 1 ) & ( for n holds ( ( A1 ^\ ( $1 + 1 ) ) * ( $1 + 1 ) ) . n = A1 . ( $1 + 1 ) ) ;
IC Comput ( P , s , 2 ) = succ IC Comput ( P , s , 1 ) .= 6 + ( card I + 1 ) .= 6 + ( card I + 1 ) .= 6 + ( card I + 1 ) .= 6 + ( card I + 1 ) .= 6 + ( card I + 1 ) ;
f . x = f . g1 * f . g2 .= ( f . g1 ) * ( f . g2 ) .= ( f . g1 ) * ( f . g2 ) .= ( f . g2 ) * ( f . g2 ) .= ( f . g2 ) * ( f . g2 ) .= ( f . g2 ) * ( f . g2 ) ;
( M * ( F * ( ( A . n ) ) ) ) . x = M . ( ( ( F . n ) ) . x ) .= M . ( ( ( F . n ) ) . x ) ;
the carrier of L1 + L2 c= ( the carrier of L1 ) \/ ( the carrier of L2 ) \/ ( the carrier of L1 ) \/ ( the carrier of L2 ) ;
pred a , b , c , x , y , c , d , x , y , z , x , y , z , x , z , y , x , z , y , x , z , x , y , z , x , y , z , x , z , x , y , z , x , z ;
( the partial of s ) . n <= ( the partial of s ) . ( n + 1 ) * ( ( the Sorts of s ) . n ) ;
redefine attr - 1 <= r & ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( r - 1 ) ) ) ;
seq in { p ^ <* n *> where n is Nat : p ^ <* n *> in T1 & p ^ <* n *> in T1 } ;
|[ x1 , x2 , x3 , x4 ]| . 2 - |[ y1 , y2 ]| . 2 = x2 - x3 - x3 ;
attr for m be Nat holds F . m is nonnegative means : Def2 : ( Partial_Sums F ) . m is nonnegative & ( Partial_Sums F ) . m is nonnegative & ( Partial_Sums F ) . m is nonnegative ;
len ( ( \overline G ) * ( y , z ) ) = len ( ( \overline G ) * ( y , z ) ) + len ( \overline G ) * ( y , z ) ) ;
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 being FinSequence of NAT such that F = x & dom F = n & rng F c= { 0 , 1 } & Sum F = k ;
0 = ( 1 * a2 ) * u] , 1 = ( 1 - ( ( - 1 ) * a2 ) * ( ( - 1 ) * a2 ) ) ;
consider n be Nat such that for m be Nat st n <= m holds |. ( f # x ) . m - lim ( f # x ) .| < e ;
cluster non empty Boolean for RelStr means : Def2 : for \hbox { $ ( where is Element of L ) , c is Element of L st c in D & ( ( ( Y ) ) ` ) ` ) ` ) ` = { c } ;
"/\" ( B , {} ) = Top ( B ) .= "/\" ( [#] ( S ) , [#] ( S ) ) .= "/\" ( [#] ( S ) , [#] ( S ) ) .= "/\" ( [#] ( S ) , [#] ( S ) ) .= "/\" ( [#] ( S ) , [#] ( S ) ) .= "/\" ( [#] ( S ) , [#] ( S ) ) ;
sqrt ( ( r / 2 ) ^2 + ( r / 2 ) ^2 ) <= sqrt ( r ^2 + ( r / 2 ) ^2 ) ;
for x being element st x in A /\ dom ( ( f `| X ) `| A ) holds ( ( f `| X ) `| A ) . x >= r2
2 * r1 - ( 2 * r1 - c ) * ( 2 * r1 - c ) = 0. TOP-REAL 2 + ( 2 * r1 - c ) * ( 2 * r1 - c ) ;
reconsider p = P * ( \square , 1 ) , q = a " * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( 1 , 1 ) ) ) ) ) as FinSequence of K ;
consider x1 , x2 being element such that x1 in uparrow s and x2 in downarrow t & x = [ x1 , x2 ] ;
for n be Nat st 1 <= n & q1 . n = ( upper_bound rng ( g | divset ( D , n ) ) ) . i holds ( ( g | divset ( D , n ) ) . i ) . n = ( ( g | divset ( D , n ) ) . i ) * vol ( divset ( D , n ) )
consider y , z being element such that y in the carrier of A & z in the carrier of A and i = [ y , z ] ;
given H1 , H2 being strict Subgroup of G such that x = H1 & y = H2 & H1 , H2 // H1 , H2 & H1 , H2 // H1 , H2 ;
for S , T , d being Function of T , S , e being Function of S , T st d is directed-sups-preserving holds d is monotone & d is monotone & e is monotone
[ a + i , b ] in ( the carrier of F_Complex ) /\ ( the carrier of F_Complex ) & [ a , b ] in the carrier of K ;
reconsider mm = max ( len F1 . ( len F1 ) * ( p . ( n + 1 ) ) * ( p . ( n + 1 ) ) ) / ( n + 1 ) ) as Element of NAT ;
I <= width GoB ( ( GoB f ) * ( 1 , 1 ) ) & ( GoB f ) * ( 1 , 1 ) `2 <= ( GoB f ) * ( 1 , 1 ) `2 ;
f2 /* q = ( f2 /* ( f1 /* s ) ) ^\ k .= ( f2 * f1 ) /* s ) ^\ k .= ( f2 * f1 ) /* s ;
attr A1 \/ A2 is linearly-independent means : Def1 : A1 misses A2 & A2 misses A1 & ( for x being Element of V st x in A1 holds ( x in A2 ) /\ ( A1 \/ A2 ) ) = {} ;
func A -carrier of C -> set means : Def2 : union { A . s where s is Element of R : s in A } ;
dom ( Line ( v , i + 1 ) (#) ( ( ( ( F . m ) (#) ( ( ( F . m ) (#) ( ( F . m ) (#) ( F . m ) ) ) ) ) ) ) ) ) ) ) = dom ( F . m ) ;
cluster [ ( x `1 ) , ( x `2 ) , ( x `2 ) ] -> to [ x `1 , ( x `2 ) , ( x `2 ) ] , ( x `2 ) ) `1 , ( x `2 ) `2 ] -> to ;
E , E |= All ( x2 , ( x2 . ( x2 . ( x2 . ( x2 . ( x2 . ( x2 . ( x2 . ( x2 . ( x2 . x ) ) ) ) ) ) ) ) ) ) ) ) ) => ( ( x2 . ( x2 . ( x2 . ( x2 . ( x2 . ( x2 . x ) ) ) ) ) ) ;
F .: ( id X , g ) . x = F . ( id X , g . x ) .= F . ( id X , g . x ) .= F . ( id X , g . x ) .= F . ( id X , g . x ) ;
R . ( h . m ) = F . x0 + h . ( m + 1 ) - h . x0 + h . ( m + 1 ) - h . x0 .= ( h . m - h . m ) - h . x0 ;
cell ( G , ( X -' 1 , Y ) , ( Y -' 1 ) + ( Y -' 1 ) ) \ L~ f meets UBD L~ f ;
IC Comput ( P2 , s2 , LifeSpan ( P2 , s2 ) ) = IC Comput ( P2 , s2 , LifeSpan ( P2 , s2 ) ) .= card I + ( card I + 2 ) .= card I + ( card I + 2 ) ) .= card I + ( card I + 2 ) .= card I + ( card I + 2 ) + 2 ) ;
sqrt ( 1 - ( ( q `1 / |. q .| - sn ) / ( 1 + sn ) ) ^2 ) > 0 ;
consider x0 being element such that x0 in dom a and x0 in g " { k } and y = a . x0 and x0 in { a . k } ;
dom ( ( r1 (#) ( chi ( A , A ) ) ) `| REAL ) = dom ( chi ( A , A ) ) .= dom ( ( chi ( A , A ) ) | A ) .= dom ( ( r1 (#) ( chi ( A , A ) ) ) | A ) .= dom ( ( r1 (#) ( chi ( A , A ) ) ) | A ) .= dom ( r1 (#) ( chi ( A , A ) ) ) ;
d-7 . [ y , z ] = ( ( y , z ) `2 ) * ( ( y , z ) `2 ) * ( ( y , z ) `2 ) * ( ( y , z ) `2 ) * ( ( y , z ) `2 ) * ( ( y , z ) `2 ) * ( ( y , z ) `2 ) * ( ( y , z ) `2 ) * ( ( y , z ) `2 ) * ( ( y , z ) `2 ) * ( ( y , z ) `2 ) * ( y , z ) `2 ) ) ;
attr for i being Nat holds C . i = A . i /\ B . i & C . i c= C . i /\ C . i ;
assume that x0 in dom f and f is_continuous_in x0 and for x st x in dom f holds ||. f /. x - f /. x0 .|| < r & |. f /. x - f /. x0 .| < r ;
p in Cl A implies for K being Basis of p , Q being Basis of T st K in K & A meets Q & P meets Q & P meets Q & P meets Q & P meets Q & P meets Q & P meets Q & P meets Q & P meets Q implies P meets Q ;
for x be Element of REAL n st x in Line ( x1 , x2 ) holds |. y1 - y2 .| <= |. y1 - y2 .|
func => <*> a -> Ordinal means : Def2 : a in it & for b being Ordinal st a in it holds it . b = b ;
[ a1 , a2 , a3 ] in ( the carrier of A ) /\ ( the carrier of B ) & [ a1 , a3 ] in the InternalRel of A ;
ex a , b being element st a in the carrier of S1 & b in the carrier of S2 & x = [ a , b ] & x = [ a , b ] ;
||. ( vseq . n ) - ( vseq . m ) .|| * ||. ( vseq . m ) - ( vseq . m ) .|| < e * ( ||. x .|| ) + ||. x .|| ) ;
then for Z being set st Z in { Y where Y is Element of I : F . Z = x } & z in Z & z in Z ;
sup compactbelow ( s , t ) = [ sup Y , sup Y ] .= [ sup Y , sup Y ] .= [ sup Y , sup Y ] .= [ s , sup Y ] ;
consider i , j being Element of NAT such that i < j & [ y , f . i ] in [: I , I :] and [ f . i , f . j ] in [: I , I :] ;
for D being non empty set , p , q being FinSequence of D st p c= q & q = p ^ q holds ex p being FinSequence of D st p = q & p = q ^ p
consider e1 being Element of the affine of X such that c9 , a9 // a9 , c9 and not a , c // a9 , c9 and not a , c // a9 , c9 and a , c // a9 , c9 and a , c // a9 , c9 ;
set U2 = I \! \mathop { \rm \hbox { - } u } , C = I \! \mathop { \rm \hbox { - } u } , D = { u } , E = { u } , F = { u } , G = { u } , C = { u } , N = { u } , N = { u } , N = { u } , N = { u } , E = { u } , E = { u } , E = { u } , N = { u } , N = { u } , N = { u } , N = { u } , N = { u } , N
|. q2 .| ^2 = ( ( |. q2 .| ) ^2 + ( |. q2 .| ) ^2 + ( |. q2 .| ) ^2 ) .= |. q2 .| ^2 + ( |. q2 .| ) ^2 ;
for T being non empty TopSpace , x , y being Element of [: the topology of T , the topology of T :] holds x "\/" y = x "\/" y & x "/\" y = x "\/" y & x "/\" y = x "\/" y implies x "/\" y = x "\/" y
dom signature U1 = dom ( ( the charact of U1 ) | the carrier' of S1 ) & rng ( ( the charact of U2 ) | the carrier' of S1 ) = dom ( the charact of U1 ) /\ the carrier' of S1 ;
dom ( h | X ) = dom h /\ X .= dom h /\ X .= dom ( |. h .| | X ) .= dom h /\ X .= dom ( |. h .| | X ) .= dom h /\ X .= dom ( |. h .| | X ) .= dom h /\ X .= X .= dom ( |. h .| | X ) .= X ;
for N1 , N2 , N1 , N2 , N1 , N2 , N1 , N2 , N1 , N2 , N1 , N2 , N2 , N1 , N2 , N1 , N2 , N1 , N2 , N2 , N1 , N2 , N1 , N2 , N1 , N2 , N2 , N1 , N2 , N2 , N1 , N2 , N2 , N1 , N2 , N2 , N2 , N1 , N2 , N2 , N2 , N1 , N2 , N2 , N1 , N2 , N1 , N2 , N2 , N1 , N2 , N2 , N1 , N2 , N2 , N1 , N2 , N2 , N2 , N1 , N2 , N2 , N2 , N1 , N2 , N2
( mod ( u , m ) + mod ( v , m ) ) . i = ( mod ( u , m ) + mod ( v , m ) ) . i + ( mod ( v , m ) ) . i ) ;
- ( q `1 / |. q .| - sn ) < - 1 or ( q `1 / |. q .| - sn ) >= - ( q `1 / |. q .| - sn ) & ( q `2 / |. q .| - sn ) >= - sn & ( q `2 / |. q .| - sn ) >= - sn ;
attr r1 = f9 & r2 = f9 & r1 = f9 implies r1 * r2 = f9 * f & r1 * f = f * ( r1 * f ) ;
v-4 . m is bounded Function of X , the carrier of Y & ( for m be Nat st m in X holds xx . m = ( ( seq_id ( vseq . m , X ) ) ) . x ) * ( ( seq_id ( vseq . m , X ) ) . x ) ;
attr a <> b & b <> c & angle ( a , b , c ) = PI & angle ( b , c , d ) = PI ;
consider i , j being Nat such that p1 = [ i , r ] and p2 = [ i , s ] and r < j and i < len s and s . i = r . ( j + 1 ) ;
|. p .| ^2 - ( 2 * |( p , q )| ) ^2 + |. q .| ^2 = |. p .| ^2 + ( 2 * |. q .| ) ^2 ;
consider p1 , q1 being Element of ( X * ) such that y = p1 ^ q1 and p1 ^ q1 = p1 ^ q1 and p1 ^ q1 = q1 ^ q1 and p1 ^ q1 = q1 ^ q1 ^ q1 ;
L1 . ( 1 / 2 * r1 ) = ( s2 . ( 1 / 2 * r1 ) ) / 2 * r1 .= ( s2 . ( 1 / 2 * r1 ) ) / 2 ;
( ( for w being Element of A ) holds ( proj2 .: ( A /\ being ) ) /\ ( proj2 .: ( A /\ being Subset of T ) ) ) is non empty & proj2 .: ( ( Cl ( A /\ ( A /\ B ) ) ) /\ ( Cl ( A /\ B ) ) ) is non empty ;
s , ( H , H1 ) |= ( H1 , H2 ) |= ( H , H1 ) . ( k , ( H , H1 ) . ( k , m ) ) ;
len ( s + t ) = card ( support b1 ) + ( support b2 ) .= card ( support b1 ) + ( support b2 ) .= ( len b1 ) + ( len b2 ) .= ( len b1 ) + ( len b2 ) .= ( len b1 ) + ( len b2 ) ;
consider z being Element of L1 such that z >= x and z >= y & for z being Element of L1 st z >= x & z >= y holds z >= x ;
LSeg ( ( UMP D ) . ( |[ ( W-bound D ) / 2 , ( S-bound D ) / 2 ]| ) , ( ( W-bound D ) / 2 ) * ( ( W-bound D ) / 2 ) + ( S-bound D ) / 2 ) ) = { UMP D } ;
lim ( ( ( f `| N ) / ( g `| N ) ) /* b ) = lim ( ( f `| N ) / ( g `| N ) ) .= lim ( ( f `| N ) / ( g `| N ) ) ;
P [ i , pr1 ( f ) . i , pr1 ( f ) . ( i + 1 ) ] ;
for r be Real st 0 < r ex m be Nat st for k be Nat st m <= k holds ||. ( seq . k ) - ( R . k ) .|| < r
for X being set , P being a_partition of X , x , y being set st x in X & a in P & b in P & x = a & y in P & x = a & P [ b , x , y ] holds a = b
Z c= dom ( ( ( ( - 1 ) (#) ( ( #Z 2 ) * f ) ) / ( ( #Z 2 ) * f ) ) ) \ ( ( #Z 2 ) * f ) " { 0 } ) ;
ex j be Nat st j in dom ( l ^ <* x *> ) & j < i & y = ( l ^ <* x *> ) . j & i = 1 + j & j = i + j ;
for u , v being VECTOR of V , r being Real st 0 < r & u in N holds r * u + ( r * v ) in N
A , Int Cl A , Cl Int Cl A , Cl Int Cl A , Cl Int Cl A , Cl Int Cl A , Cl Int Cl A , Cl Int Cl A , Cl Int Cl A , Cl Int Cl A , Cl Int Cl A , Cl Int Cl A , Cl Int Cl A , Cl A , Cl Int A , Cl A , Cl A , Cl A , Cl A , Cl A , Cl A , Cl A , Cl A , Cl A , Cl A , Cl A , Cl A , Cl A , Cl A , Cl A , Cl A , Cl A , Cl A , Cl A , Cl A , Cl A ,
- Sum <* v , u , w *> = - ( v + u + u ) .= - ( v + u ) + u .= - ( v + u ) + u .= - ( v + u ) + u .= - ( v + u ) + u .= - ( v + u ) ;
( Exec ( a := b , s ) ) . IC SCM = ( Exec ( a := b , s ) ) . IC SCM .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= 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 & dom h = I & for x being element st x in I holds h . x in ( the support of J ) . x ;
for S1 , S2 being non empty RelStr for D being non empty Subset of S1 , x being Element of S2 , y being Element of S2 , x being Element of S1 , y being Element of S2 holds cos . ( x , y ) is directed & cos . ( x , y ) is directed
card X = 2 implies ex x , y st x in X & y in X & x <> y & x = y or x = y & x = y & x = y & x = y ;
E-max L~ Cage ( C , n ) in rng ( Cage ( C , n ) \circlearrowleft W-min L~ Cage ( C , n ) ) ;
for T , T , p , q being FinSequence of dom T st p in dom T holds ( T -tree ( p , T ) ) . q = T . q & ( T -tree ( p , T ) ) . q = T . q
[ i2 + 1 , j2 ] in Indices G & [ i2 + 1 , j2 ] in Indices G & f /. k = G * ( i2 + 1 , j2 ) ;
cluster -> natural means : Def2 : k divides ( m + n ) & n divides m & ( k divides m ) & ( k divides n implies m divides k ) ;
dom F " = the carrier of X2 & rng F = the carrier of X1 & F " is one-to-one & F " is one-to-one & rng F = the carrier of X2 & F " { 0 } = the carrier of X1 & F " { 1 } = the carrier of X2 ;
consider C being finite Subset of V such that C c= A & card C = omega & the carrier of V = C & C is linearly-independent & C is linearly-independent & C is linearly-independent ;
V is prime implies for X , Y being Element of [: the topology of T , the topology of T :] st X /\ Y c= V holds X c= Y or X c= Y
set X = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } , Y = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } , Z = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } ;
angle ( p1 , p3 , p4 ) = 0 .= angle ( p2 , p3 , p4 ) .= angle ( p2 , p3 , p4 ) .= angle ( p2 , p3 , p4 ) .= angle ( p2 , p3 , p4 ) ;
- sqrt ( ( - ( q `1 / |. q .| - sn ) / ( 1 + sn ) ) ^2 ) = - ( - ( q `1 / |. q .| - sn ) / ( 1 + sn ) ) ^2 ) .= - ( - ( q `1 / |. q .| - sn ) ) ;
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 = p4 & f . 1 = p4 ;
attr f is partial differentiable on on on u0 means : Def2 : SVF1 ( 2 , f , 3 ) . u , SVF1 ( 2 , f , 3 ) . u ) & SVF1 ( 2 , f , 3 ) . u = ( proj ( 2 , 3 ) ) . u ;
ex r , s st x = |[ r , s ]| & G * ( len G , 1 ) `2 < s & s < G * ( 1 , 1 ) `2 & G * ( 1 , 1 ) `2 < s & s < G * ( 1 , 1 ) `2 } ;
assume that f is_sequence_on G and 1 <= t and t <= len G and G * ( t , width G ) `2 >= ( GoB f ) * ( t , width G ) `2 ;
redefine attr i in dom G means : Def2 : r (#) ( f (#) reproj ( i , x ) ) = r * f . ( reproj ( i , x ) ) ;
consider c1 , c2 being bag of o1 + o2 such that ( ( decomp c ) *' ( O *' ) ) /. k = <* c1 , c2 *> and c = c1 + c2 and ( ( decomp c ) *' ( O *' ) ) /. k = c1 + c2 ;
u0 in { |[ r1 , s1 ]| : r1 < G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `2 < s1 & s1 < G * ( 1 , 1 ) `2 } ;
Cl ( X ^ Y ) = the carrier of X . ( k2 + 1 ) .= ( C . ( k2 + 1 ) ) .= C . ( ( C . ( k2 + 1 ) ) ) .= C . ( ( C . ( k2 + 1 ) ) ) .= C . ( ( C . ( k2 + 1 ) ) ) .= C . ( ( C . ( k2 + 1 ) ) ) ;
attr M1 = len M2 means : Def2 : 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 & g2 . y - g2 . x0 < g2 . y } c= N2 ;
assume x < ( - b + sqrt ( a , b ) ) / ( 2 * a ) or x > - b + sqrt ( a , b ) ;
( G1 '&' G2 ) . i = ( <* 3 *> ^ G1 ) . i & ( G1 '&' G2 ) . i = ( <* 3 *> ^ G1 ) . i & ( G1 '&' G2 ) . i = ( <* 3 *> ^ G1 ) . i ;
for i , j st [ i , j ] in Indices M1 holds ( M1 + M2 ) * ( i , j ) < M2 * ( i , j )
for f being FinSequence of NAT , i being Element of NAT st i in dom f & i in dom f holds i divides f /. i & i divides f /. ( i + 1 )
assume F = { [ a , b ] where a is Subset of X : for c being set st c in B & c in B holds a c= c } ;
b2 * q2 + ( b3 * q3 + ( b3 * q3 ) + ( - ( q3 * q3 ) + ( - ( q3 * q3 ) + ( p1 * q3 ) + ( p1 * q3 ) ) ) ) = 0. TOP-REAL n ;
Cl Cl ( F ) = { D where D is Subset of T : ex B being Subset of T st B = Cl D & B is closed & D is closed & B is closed & B is closed & D is closed & B is closed & B = Cl D & B is closed & B is closed & B is closed & B is closed & B is closed ;
attr seq is summable means : Def2 : seq is summable & seq is summable & seq is summable & lim seq = Sum ( seq ) & seq is summable & lim seq = Sum ( seq ) ;
dom ( ( cn -FanMorphN ) | D ) = ( the carrier of ( TOP-REAL 2 ) | D ) /\ D .= the carrier of ( ( TOP-REAL 2 ) | D ) .= the carrier of ( ( TOP-REAL 2 ) | D ) .= the carrier of ( ( TOP-REAL 2 ) | D ) .= the carrier of ( ( TOP-REAL 2 ) | D ) | D ;
[ X \to Z ] is full full full full SubRelStr of ( [#] Z ) |^ the carrier of Z & [ X \to Y ] is full full SubRelStr of ( [#] Z ) |^ the carrier of Z ;
( G * ( 1 , j ) `2 ) `2 = ( G * ( 1 , j ) `2 ) `2 & ( G * ( 1 , j ) `2 ) `2 <= ( G * ( 1 , j ) `2 ) `2 ;
synonym m1 c= m2 means : Def2 : for for for p being set st p in P holds the non empty set of p = the empty empty set & the InternalRel of p = the InternalRel of p & the InternalRel of p = the ^ of q ;
consider a being Element of B ( ) such that x = F ( a ) and a in { G ( b ) where b is Element of B ( ) : P [ b ] } and P [ a ] ;
cluster multiplicative $1 -> multiplicative for multMagma over $1 means : Def2 : for s being State of $1 , the carrier of S holds [ s , the carrier of $1 ] in the carrier of $1 & the multF of s = the multF of $1 ;
( ( a , b ) + ( c , d ) ) + ( b , c ) = b + ( c + d ) .= ( a + c ) + ( b + d ) .= ( a + c ) + ( b + d ) .= ( a + c ) + ( b + d ) .= ( a + c ) + ( b + d ) ;
cluster strict for non empty addLoopStr means : Def1 : for Element of INT , i , j being Element of INT st i = j & j = i holds it . ( i , j ) = ( i + j ) * ( i , j ) ;
( - ( s2 * p1 + ( s1 * p2 ) - ( s1 * p2 + ( s2 * p2 + - s1 * p2 ) ) ) / ( 1 + ( s1 * p2 + - s1 * p2 ) ) ) = ( ( - 1 ) * ( ( s1 * p2 + - s1 * p2 ) - ( s1 * p2 + - s1 * p2 ) ) / ( 1 + ( s1 * p2 ) - s1 * p2 ) ) ;
eval ( ( a | ( n , L ) ) *' p , x ) = eval ( a | ( n , L ) ) * eval ( p , x ) .= a * eval ( p , x ) .= a * eval ( p , x ) ;
assume that the TopStruct of S = the TopStruct of T and for D being non empty Subset of S st D in [#] S holds D meets [#] S and D is open & D is open & D is open & D is open & D is open & D is open & D is open & D is open & D is open ;
assume that 1 <= k and k <= len w + 1 and T-7 . ( ( ( q , w ) \mathop { + } ) , w ) = ( T . ( ( q , w ) \mathop { + } ) ) . k ) ;
2 * a |^ ( n + 1 ) + ( 2 * b |^ ( n + 1 ) ) >= ( a |^ n ) + ( b |^ ( n + 1 ) ) + ( a |^ ( n + 1 ) ) + ( b |^ ( n + 1 ) ) + ( a |^ ( n + 1 ) ) + b |^ ( n + 1 ) ) ;
M , v / ( x. 3 , 0 ) / ( x. 4 , 0 ) |= ( All ( x. 3 , 4 ) ) / ( x. 4 , 0 ) ) / ( x. 4 , 0 ) / ( x. 0 , 0 ) ;
assume that f is_differentiable_on l and for x0 st x0 in l holds 0 < f . x0 & for x0 st x0 < l holds f . x0 < f . x0 & f . x0 < 0 ;
for G1 being _Graph , W being Walk of G1 , e being Vertex of G2 , x being Vertex of G1 , y being Vertex of G2 st e in W & x in W holds W . ( e ) = W . ( e )
not ] is not empty iff ( ( ex x1 , x2 , x3 , x4 , x4 , x5 , x1 , x2 , x3 , x4 , x4 , x1 , x2 , x3 , x4 ) & ( ( ex x2 , x3 , x4 , x4 , x2 , x4 ) ) & ( ( iff ( ex x2 , x3 , x4 , x4 , x4 ) ) ) & ( ( ex x2 , x3 , x4 , x4 , x4 , x4 ) ) ) & ( ( ex x2 , x3 , x4 , x4 , x4 , x2 , x4 , x4 , x4 , x4 , x4 , x4 , x4 , x4 , x4 , x4 , x4 , x4 , x4 , x4 , x4 , x2 , x4 , x4 , x4 , x2 , x4 , x4 , x4 , x4 , x4 , x4 , x4
Indices GoB f = [: dom GoB f , Seg width GoB f :] & ( GoB f ) * ( i , j ) = [: dom GoB f , Seg width GoB f :] & ( GoB f ) * ( i , j ) = ( GoB f ) * ( i , j ) ;
for G1 , G2 , G1 , G2 being Group holds G1 is stable & G2 is stable & G1 is stable & G2 is stable & G1 is stable & G2 is stable & G1 is stable implies G1 is stable
UsedIntLoc ( ( in1 ( a , I ) ) +* ( 1 , intloc 0 ) ) = { intloc 0 ( a , intloc 0 ) , intloc 0 ( a , intloc 0 ) , intloc 1 ( a , intloc 0 ) ) } ;
for f1 , f2 being FinSequence of F st f1 ^ f2 is p -element & f1 is p -element & f2 is ( len f1 ) -element & ( for i st i in dom f1 holds Q [ i , f1 . i ] ) holds Q [ f1 ^ f2 ]
sqrt ( ( p `1 ) ^2 + ( p `2 ) ^2 ) = sqrt ( ( p `1 ) ^2 + ( p `2 ) ^2 ) + ( p `2 ) ^2 + ( p `2 ) ^2 ) ;
for x1 , x2 , x3 , x4 , x5 , x2 , x4 , x4 , x5 , x5 , x5 , x2 , x4 , x4 , x5 , x5 , x2 , x4 , x5 , x5 , Real , x5 , x5 , Real , x5 , x5 , x5 , Real , x5 , x5 , x5 , Real , x5 , x5 , x5 , Real , x5 , Real , x5 , x5 , Real , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 7 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 7 , 7 , 8 , 8
for x st x in dom ( ( - ( ( - ( x - 1 ) ) (#) ( cos * cos ) ) `| Z ) holds ( ( ( - 1 ) (#) cos ) `| Z ) `| Z ) . x = - ( cos . x ) / ( cos . x ) ^2
for T being non empty TopSpace , P being Subset of T st P c= the topology of T & P is Basis of T holds P is Basis of T
( a 'or' b ) . x = 'not' ( ( a 'or' b ) . x ) 'or' c . x .= 'not' ( a . x ) 'or' b . x 'or' c . x .= TRUE ;
for e being set st e in A8 ex X1 , X2 being Subset of X st e = X1 & X1 is open & X2 is open & X1 is open & X2 is open & X1 is open & X2 is open & X1 is open & X2 is open & X1 is open & X2 is open & X1 is open & X2 is open & X1 is open & X2 is open ;
for i being set for f being Function of [: S ( ) , S ( ) :] , S ( ) :] , [: S ( ) , S ( ) :] st f = H ( i ) & f . i = f | [: F ( ) , S ( ) :] holds F . i = f | [: F ( ) , S ( ) :]
for v , w st for y st x <> y holds w . y = v . y holds J . ( v . y ) = J . ( v . y )
card D = card D1 + card D2 - card D1 .= ( i + 1 ) - card D1 + card D2 - 1 .= ( i + 1 ) - 1 + 1 .= ( i + 1 ) - 1 + 1 .= ( i + 1 ) - 1 + 1 .= ( i + 1 ) - 1 + 1 .= ( i + 1 ) - 1 ;
IC Exec ( i , s ) = ( s +* ( 0 .--> ( s . 0 ) ) ) . 0 .= s . 0 .= s . 0 .= s . 0 .= s . 0 .= s . 0 ;
len f /. ( \downharpoonright i1 -' 1 ) = len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) + 1 ;
for a , b , c being Element of NAT st 1 <= a & 2 <= b & k < a holds a + b = a + b or a = b + c + d
for f being FinSequence of TOP-REAL 2 , p being Point of TOP-REAL 2 , i being Nat st p in LSeg ( f , i ) & f /. 1 = f /. ( i + 1 ) holds Index ( p , f ) <= Index ( p , f ) + 1 )
lim ( ( curry ' ( P , k + 1 ) ) # x ) = lim ( ( curry ' ( P , k + 1 ) ) # x ) + lim ( ( curry ' ( P , k + 1 ) ) # x ) .= lim ( ( curry ' ( P , k + 1 ) ) # x ) ;
z2 = g /. ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) ;
[ f . 0 , f . 3 ] in id ( the carrier of G ) \/ ( the InternalRel of G ) \/ ( the InternalRel of G ) \/ ( the InternalRel of G ) \/ ( the InternalRel of G ) \/ ( the InternalRel of G ) ;
for G being Subset-Family of B st G = { R [ X ] where R is Subset of A ( ) , Y ( ) : R in F ( ) & P [ X ] } holds ( Intersect ( F ( ) ) ) . x = Intersect ( F ( ) ) . x )
CurInstr ( P1 , Comput ( P1 , s1 , m1 ) ) = CurInstr ( P1 , Comput ( P1 , s1 , m1 ) ) .= halt SCMPDS .= halt SCMPDS .= halt SCMPDS ;
assume that a on M and b on M and c on N and d on N and p <> d and a <> c and p <> d and a <> d and p <> d and a <> d and a <> d and p <> d and a <> d and a <> b and a <> d and a <> b and a <> d and a <> b ;
assume that T is \hbox { T _ 4 4 } and F is closed and ex F be Subset-Family of T st F is closed & for x being Element of T st x in F & x is finite-ind holds F is finite-ind & for x being Element of T st x in F holds x is finite-ind & x is finite-ind & F is finite-ind & x is finite-ind ;
for g1 , g2 st g1 in ]. r1 - r , r2 .[ & g2 in ]. r1 - r , r2 + r .[ & |. f . g1 - f . g2 .| <= ( r1 - r ) / ( |. g1 .| - r ) ) holds |. f . g1 - f . g2 .| <= ( r1 - r ) / ( |. g1 .| - r )
( ( exp_R * ( z + z2 ) ) + ( exp_R * ( z + z2 ) ) ) = ( ( exp_R * ( z + z2 ) ) + ( exp_R * ( z + z2 ) ) ) + ( - ( exp_R * ( z + z2 ) ) ) ) + ( - ( exp_R * ( z + z2 ) ) ) ;
F . i = F /. i .= 0 .= ( b |^ ( n + 2 ) ) * ( a |^ ( n + 1 ) ) .= <* b |^ ( n + 1 ) * a |^ ( n + 1 ) ) .= <* ( n + 1 ) * a |^ ( n + 1 ) ) * a |^ ( n + 1 ) *> ;
ex y being set , f being Function st y = f . n & dom f = NAT & for n being Nat holds f . ( n + 1 ) = A ( n ) & for n being Nat holds f . ( n + 1 ) = R ( n ) & f . ( n + 1 ) = R ( n ) ;
func f (#) F -> FinSequence of V means : Def1 : len it = len F & for i be Nat st i in dom F holds it . i = F . i * ( f . i ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x6 , x6 , x5 , x6 , x5 , x6 } = { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x6 , x5 , x6 , x5 , x5 , x6 } \/ { x1 , x2 , x3 , x4 , x5 } \/ { x2 , x3 , x4 , x5 , x5 } ;
for n being Nat , x being set st x = h . n & h . ( n + 1 ) = o . ( x , n ) & h . ( n + 1 ) in InnerVertices S & h . ( n + 1 ) in InnerVertices S & h . ( n + 1 ) in InnerVertices S ;
ex S1 being Element of CQC-WFF ( Al ( ) ) st ( for P being Element of CQC-WFF ( Al ( ) ) ) , e being Element of CQC-WFF ( Al ( ) ) , x being Element of CQC-WFF ( Al ( ) ) st P [ P ( ) , x ] holds P [ x ] ) ;
consider P being FinSequence of G2 such that p9 = product P and for i being Element of dom P st i in dom P ex t7 being Element of the carrier of K st P . i = t7 & for t being Element of K st t in dom P & t . i = t . i holds P . t = Q . t ;
for T1 , T2 being strict non empty TopSpace , P being Basis of T1 , T2 being Basis of T2 st the topology of T1 = the topology of T2 & the topology of T2 = the topology of T2 & the topology of T2 = the topology of T2 & the topology of T2 = the topology of T2 & the topology of T2 = the topology of T2 holds P is Basis of T2
Suppose f is_partial_differentiable_in u0 , 3 . u . 3 & r (#) pdiff1 ( f , 3 ) is_partial_differentiable_in u0 , 3 . 1 & SVF1 ( 3 , 3 , u ) . 3 = r * SVF1 ( 3 , 3 , u ) . 1 , u . 3 * SVF1 ( 3 , 1 , u ) . 3 ; Then ( r (#) SVF1 ( 3 , 1 , u ) ) . 3 = r * SVF1 ( 3 , 1 , u ) . 3 ) ;
defpred P [ Nat ] means for F , G being FinSequence of REAL st len F = $1 & G = F & G = G * s holds Sum ( F ^ G ) = Sum ( F ) * G & Sum ( G ) = Sum ( F ) * G ) ;
ex j st 1 <= j & j < width GoB f & ( GoB f ) * ( 1 , j ) `2 <= s & s < ( GoB f ) * ( 1 , j ) `2 & ( GoB f ) * ( 1 , j ) `2 <= s `2 ;
defpred U [ set , set ] means ex Fa1 being Subset-Family of T st $1 = ( F . $1 ) & ( F . $1 = ( F . $1 ) ` ) & ( F . $1 = ( F . $1 ) ` ) & ( F . $1 = ( F . $1 ) ` ) & ( F . $1 = ( F . $1 ) ` ) & ( F . $1 = ( F . $1 ) ` ) ;
for p4 being Point of TOP-REAL 2 st LE p4 , p1 , P & LE p4 , p1 , P & LE p4 , p1 , P , p1 , p2 holds LE p4 , p1 , P , p1 , p2
f in St ( E , H ) & for g st g . y <> f . y holds g in St ( E , H ) implies f in St ( E , H ) & g in St ( E , H )
ex p0 being Point of TOP-REAL 2 st x = p2 & ( |. p2 .| ) * ( |. p2 .| ) >= 0 & ( |. p2 .| ) * ( |. p2 .| ) >= 0 & ( |. p2 .| ) * ( |. p2 .| ) >= 0 & ( |. p2 .| ) * ( |. p2 .| ) >= 0 ;
assume for d7 being Element of NAT st d7 <= d8 holds s1 . ( d7 . ( d7 . ( d7 . ( |. 7 . ( |. 7 . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . x ) ) ) ) ) ) ) ) ) ) ) ) ) = s1 . ( ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d . ( d
assume that s <> t and s is Point of Closed-Interval-TSpace ( x , r ) and s is not zero and ex e being Point of Closed-Interval-TSpace ( x , r ) st e = Ball ( x , r ) /\ Ball ( x , r ) & e in Ball ( x , r ) ;
given r such that 0 < r and for s be Point of CNS holds 0 < s or ex x1 , x2 be Point of CNS st x1 in dom f & x2 in dom f & |. x1 - x2 .| < s & |. x1 - x2 .| < r & |. x1 - x2 .| < s ;
( p | x ) | ( ( x | x ) | ( x | x ) ) = ( ( x | x ) | ( x | x ) ) | ( x | x ) .= ( ( x | x ) | ( x | x ) ) | ( x | x ) ;
assume that x , h + h in dom sec and ( exists h being Function of dom sec , dom sec ) | [. - h , h .] . x = ( 4 * PI * ( cos . x ) ) * sin . x + h . x * sin . x + h . x * sin . x * sin . x + h . x * sin . x + h . x * sin . x + h . x * sin . x ) ;
assume that i in dom A and len A > 1 and B > 1 and B = A * ( i , j ) and B = A * ( i , j ) and C = B * ( i , j ) ;
for i being non zero Element of NAT st i in Seg n holds not i divides n or h . i = <* 1. F_Complex *> & h . i = 1. F_Complex & h . i = 1. F_Complex & h . i = 1. F_Complex ;
( ( ( b1 => b2 ) '&' ( c1 '&' c2 ) ) '&' ( a1 'or' c1 ) ) '&' ( a1 'or' c1 ) '&' ( b1 'or' c1 ) ) '&' ( a1 'or' c1 ) '&' ( a1 'or' c1 ) '&' ( a1 'or' c1 ) ) '&' ( a1 'or' c1 ) '&' ( a1 'or' c1 ) '&' ( a1 'or' c1 ) '&' ( a1 'or' c1 ) '&' ( a1 'or' c1 ) '&' ( a1 'or' c1 ) '&' ( a1 'or' c1 ) '&' ( a1 'or' c1 ) '&' ( a1 'or' c1 ) '&' ( a1 'or' c1 ) '&' ( a1 'or' c1 ) '&' ( a1 'or' c1 ) '&' ( a1 'or' c1 ) '&' ( a1 'or' c1 ) '&' ( a1 'or' c1 ) '&' ( a1 'or' c1 ) '&' ( a1 'or' c1 ) '&' ( a1 'or' c1 ) '&' ( a1 'or' c1 ) '&' ( a1 'or' c1 ) '&' ( a1 'or' c1 ) '&' ( a1 'or' c1 ) '&' ( a1 'or' c1 ) '&' ( a1
assume that for x holds f . x = ( ( - 1 / 2 ) (#) ( sin - cos ) ) . x and for x st x in Z holds ( ( - 1 / 2 ) (#) ( sin - cos ) ) `| Z ) . x = - sin . x / ( sin . x ) ^2 and ( ( - 1 / 2 ) (#) ( sin - cos ) ) `| Z ) . x = - sin . x ;
consider R8 , IE be Real such that R8 = Integral ( M , F . n ) & I = Integral ( M , F . n ) & I = Integral ( M , F . n ) & I = Integral ( M , F . n ) ;
ex k be Element of NAT st ' = k & 0 < d & for q be Element of product G st q in X & ||. qa1 - qx0 .|| < d holds ||. partdiff ( f , q ) - partdiff ( f , x ) .|| < r ;
x in { x1 , x2 , x3 , x4 , x5 , x6 , x6 , x6 , x5 , x6 , x6 , x5 , x6 , x5 , x6 , x6 , x5 , x6 , x6 , x6 , x6 , x5 , x5 , x5 , x6 , x5 , x5 , x5 } ;
( G * ( j , i ) `2 ) `2 = ( G * ( 1 , i ) `2 ) `2 .= ( G * ( 1 , i ) `2 ) `2 .= ( G * ( 1 , i ) `2 ) `2 .= ( G * ( 1 , i ) `2 ) `2 ;
f1 * p = p .= ( ( the Arity of S1 ) * the Arity of S1 ) . ( ( the Arity of S1 ) . o ) .= ( the Arity of S1 ) . ( ( the Arity of S1 ) . o ) .= ( the Arity of S1 ) . ( ( the Arity of S1 ) . o ) .= ( the Arity of S1 ) . ( ( the Arity of S1 ) . o ) .= ( the Arity of S1 ) . ( ( the Arity of S1 ) . o ) ) .= ( the Arity of S1 ) . ( ( the Arity of S1 ) . o ) ;
func tree ( T , P , T1 ) -> tree means : Def2 : q in T & q in P & not p = q or ex p , q st p = p ^ q & p in P & q in T & p = q ;
F /. ( k + 1 ) = F . ( p . ( k + 1 ) ) .= F . ( p . ( k + 1 ) ) .= F . ( p . ( k + 1 ) ) .= F . ( p . ( k + 1 ) ) .= F . ( p . ( k + 1 ) ) .= F . ( p . ( k + 1 ) ) ;
for A , B , C being Matrix of K st len B = len C & width C = len C & width B = width C & width C = width C & width C = width C & width C = width C & width C = width C & width C = width C & width C = width C & width C = width C & width C = width C & C * C = C * C + C * C
seq . ( k + 1 ) = 0. F_Complex .= ( Partial_Sums ( seq ) ) . ( k + 1 ) .= ( Partial_Sums ( seq ) ) . ( k + 1 ) ) * ( Partial_Sums ( seq ) ) . ( k + 1 ) .= ( Partial_Sums ( seq ) ) . ( k + 1 ) * ( Partial_Sums ( seq ) ) . ( k + 1 ) ) .= ( Partial_Sums ( seq ) ) . ( k + 1 ) ) * ( Partial_Sums ( seq ) ) . ( k + 1 ) * ( Partial_Sums ( seq ) ) . ( k + 1 ) ) * ( Partial_Sums ( seq ) ) . ( k + 1 ) * ( Partial_Sums ( seq ) ) . ( k + 1 ) * ( Partial_Sums ( seq ) ) . ( k + 1 ) * ( Partial_Sums ( seq ) . ( k + 1 ) + Partial_Sums ( seq ) . ( k + 1 ) * Partial_Sums ( seq ) . ( k + 1 ) + Partial_Sums ( seq ) . (
assume that x in ( the carrier of C1 ) and y in ( the carrier of C1 ) and z = ( the carrier of C1 ) /\ ( the carrier of C1 ) and x = ( the carrier of C1 ) /\ ( the carrier of C1 ) and y = ( the carrier of C1 ) /\ ( the carrier of C1 ) ;
defpred P [ Element of NAT ] means for f being FinSequence of ( k + 1 ) -tuples_on BOOLEAN st len f = $1 & ( for k st k in dom f holds ( ( ( ( ( f | k ) ^ <* f . k *> ) ) ^ <* f . k *> ) ^ ( ( ( f | k ) ^ <* f . k *> ) ) ) holds ( ( ( ( f | k ) ^ <* f . k *> ) ) ^ ( ( ( f | k ) ^ <* f . k ) ) ) ) . ( ( ( f | k ) . ( k + 1 ) ) . ( k + 1 ) ) . ( k + 1 ) ) . ( k + 1 ) ) = ( ( ( ( f | k ) . ( k + 1 ) ) . ( ( ( ( ( k + 1 ) ) . ( ( ( ( k + 1 ) ) . ( ( ( ( f | k ) ) . ( ( f |
assume that 1 <= k and k + 1 <= len f and f /. k = G * ( i , j ) and [ i , j ] in Indices G and f /. k = G * ( i , j ) and f /. k = G * ( i , j ) ;
assume that cn < 1 and ( q `1 / |. q .| - sn ) / ( 1 + sn ) >= 0 and ( |. q .| - sn ) / ( 1 + sn ) >= 0 & ( |. q .| - sn ) / ( 1 + sn ) >= 0 ;
for M being non empty metric , x being Point of M , f being Function of M , ( M ` ) st x = f & ex r being Element of M ` st r = f & for n being Element of M st n >= r holds f . n = F ( n ) ) holds f . n = r . ( 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 - f2 ) `| Z ) . x = f1 . x - ( f2 . x ) / ( f1 . x ) ^2 ) & for x st x in Z holds ( ( f1 - f2 ) `| Z ) . x = - ( f1 . x ) / ( f1 . x ) ^2 ) ;
defpred P1 [ Nat , Point of C ] means ( for r be Point of C st r in Y & $2 = ( f . ( $1 + 1 ) ) - f . ( $1 + 1 ) ) & ||. ( f . ( $1 + 1 ) - f . ( $1 + 1 ) .|| < r ) & ||. f . ( $1 + 1 ) - f . ( $1 + 1 ) .|| < r ;
( f ^ mid ( g , 2 , len g ) ) . i = ( mid ( g , 2 , len g ) ) . ( i + 1 ) .= g . ( i + 1 ) .= g . ( i + 1 ) .= g . ( i + 1 ) ;
sqrt ( 1 + ( 2 * n + 2 * n ) ) * ( 2 * n + 1 ) = ( 1 + ( 2 * n ) ) * ( 2 * n + 1 ) ) * ( 2 * n + 1 ) .= 1 * ( 2 * n + 1 ) * ( 2 * n + 1 ) .= ( 1 * ( 2 * n + 1 ) ) * ( 2 * n + 1 ) ;
defpred P [ Nat ] means for G being finite non empty RelStr st G is the carrier of G & card the carrier of G = $1 + 1 & the carrier of G = $1 + 1 & the carrier of G = the carrier of G & the carrier of G = the carrier of G ;
assume that not f /. 1 in Ball ( u , r ) and not 1 <= m & m <= len f & not ex i st 1 <= i & i <= len f & LSeg ( f , i ) /\ LSeg ( f , i ) <> {} ;
defpred P [ Element of NAT ] means ( Partial_Sums ( cos ) ) . $1 = Partial_Sums ( cos ) . ( $1 + 1 ) * ( cos . ( $1 + 1 ) ) * ( cos . ( $1 + 1 ) ) * ( cos . ( $1 + 1 ) ) * ( cos . ( $1 + 1 ) ) * ( cos . ( $1 + 1 ) ) * ( cos . ( $1 + 1 ) ) * ( cos . ( $1 + 1 ) ) ) ;
for x being Element of product F holds x is FinSequence of G & dom x = I & for i being set st i in I holds x . i = ( i - 1 ) * ( i - 1 ) ) & for i be set st i in I & i <> I & i <> I holds x . i = ( i - 1 ) * ( i - 1 ) * ( i - 1 ) )
( ( x " ) |^ ( n + 1 ) ) * x = ( ( x " ) |^ ( n + 1 ) ) * x .= ( ( x " ) |^ ( n + 1 ) ) * x .= ( ( x |^ ( n + 1 ) ) * x ) * x .= ( ( x |^ ( n + 1 ) ) * x ) * x .= ( ( x |^ n ) * x ) * x .= ( x |^ ( n + 1 ) ) * x .= ( x |^ ( n + 1 ) ) * x .= ( x |^ ( n + 1 ) ) * x .= ( x |^ ( n + 1 ) ) * x ) * x .= ( x |^ ( n + 1 ) * x ) * x .= ( x |^ ( n + 1 ) * x ) * x .= ( x |^ ( n + 1 ) * x .= ( x |^ ( n + 1 ) * x ) * x |^ ( n + 1 ) * x ) * x .= ( x |^ ( n + 1 ) * x |^ ( n + 1 ) *
DataPart Comput ( P +* I , s ) = DataPart Comput ( P +* I , s , LifeSpan ( P +* I , s ) ) .= DataPart Comput ( P +* I , s , LifeSpan ( P +* I , s ) + 3 ) ) .= DataPart Comput ( P +* I , s , LifeSpan ( P +* I , s ) + 3 ) ;
given r such that 0 < r and ]. x0 - r , x0 + r .[ c= dom ( f1 + f2 ) /\ ]. x0 , x0 + r .[ & for g st g in ]. x0 , x0 + r .[ /\ ]. x0 , x0 + r .[ holds f1 . g <= f1 . g . g . g ;
assume that X c= dom f1 /\ X and f1 | X is continuous and for x st x in X /\ dom f2 holds ( f1 + f2 ) | X . x = f1 . x + f2 . x ;
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 prime & x is prime & x is prime
Support ( e *' A ) = { m *' ( p *' A ) where m is Element of NAT : m in dom ( p *' A ) & p . ( m + 1 ) = ( m *' A ) . ( m + 1 ) } ;
( f1 - f2 ) /* ( ( f1 /* s1 ) - f2 /* s1 ) = lim ( f1 /* s1 ) - f2 /* s1 ) .= lim ( f1 /* s1 ) - f2 /* s1 .= lim ( f1 /* s1 ) - f2 /* s1 ) .= lim ( f1 /* s1 ) - f2 /* s1 ) .= lim ( f1 /* s1 ) ;
ex p1 being Element of CQC-WFF ( Al ) st F . p1 = g . p1 & for g being Function of CQC-WFF ( Al ) , D st g . p1 = f . ( g . ( len g ) ) & for g being Function of CQC-WFF ( Al ) , D st g . ( len g ) = g . ( len g ) holds P [ g . ( len g ) ] ;
( mid ( f , i , len f -' 1 ) ) /. j = ( mid ( f , i , len f -' 1 ) ) /. ( j + 1 ) .= ( mid ( f , i , len f -' 1 ) ) . j .= ( mid ( f , i , len f -' 1 ) ) . j ;
( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) = ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( k + k ) .= ( p . ( k + k ) . ( k + k ) . ( k + k ) . ( k + k ) . ( k + k ) . ( k + k ) .= ( p . ( k + k ) .= ( p . ( k + k ) .= ( p . ( k + k ) ) . ( k + k ) .= ( p . ( k + k ) . ( k + k ) .= ( p . ( k + k ) .= ( p . ( k +
len ( upper_volume ( D2 , D1 ) + indx ( D2 , D1 , j1 ) ) - 1 = len ( D2 , D1 ) - 1 + ( indx ( D2 , D1 , j1 ) + 1 ) - 1 .= len ( D2 , D1 ) - 1 + 1 .= len ( D2 , D1 ) - 1 + 1 ;
x * y = ( M . ( ( y * z ) * ( y * z ) ) ) * ( M . ( y * z ) ) .= ( M . ( y * z ) ) * ( M . ( y * z ) ) .= ( M . ( y * z ) ) * ( M . ( y * z ) ) .= ( M . ( y * z ) ) * ( M . ( y * z ) ) .= x * ( M . ( y * z ) ) ;
v . ( <* x , y *> ) + i * ( <* x0 , y0 *> ) . i = proj ( 1 , 1 ) . ( y - x0 ) + ( proj ( 1 , 1 ) ) . ( y - x0 ) + ( proj ( 1 , 1 ) ) . ( y - x0 ) + ( proj ( 1 , 1 ) ) . ( y - x0 ) ) ;
i * i = <* 0 * ( 1 - i ) - i * ( 1 - i ) + i * ( 1 - i ) + i * ( 1 - i ) + i * ( 1 - i ) + i * ( 1 - i ) ) .= 0 ;
Sum ( L (#) F ) = Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) ) .= Sum ( ( L (#) ( F1 ^ F2 ) ) .= Sum ( ( L (#) ( F1 ^ F2 ) ) .= Sum ( ( ( F1 ^ F2 ) ) ) .= Sum ( ( L (#) ( F1 ^ F2 ) ) ) .= Sum ( ( L (#) ( F1 ^ F2 ) ) .= Sum ( ( F1 ^ F2 ) ) + Sum ( ( ( F1 ^ F2 ) ) + Sum ( ( ( ( F1 ^ F2 ) ) + Sum ( F1 ^ F2 ) ) + Sum ( ( F1 ^ F2 ) ) + Sum ( ( F1 ^ F2 ) ) + Sum ( ( F1 ^ F2 ) ) .= Sum ( ( ( ( F1 ^ F2 ) ) + Sum
ex r be Real st for e be Real st 0 < e ex Y1 be finite Subset of X st Y1 is non empty & for Y1 be Subset of X st Y1 is non empty & Y1 is non empty & Y1 is non empty & Y is non empty & Y is non empty & Y is non empty & Y is non empty & Y is non empty & Y is non empty implies Y is non empty ;
( GoB f ) * ( i , j ) = f /. ( k + 2 ) & ( GoB f ) * ( i , j ) = f /. ( k + 2 ) or ( GoB f ) * ( i , j ) = f /. ( k + 2 ) & ( GoB f ) * ( i , j ) = f /. ( k + 2 ) ;
( ( - 1 ) * ( cos ( x ) ) ) / ( cos ( x ) ) = ( - 1 ) * ( cos ( x ) ) / ( cos ( x ) ) .= ( - 1 ) * ( cos ( x ) ) / ( cos ( x ) ) .= ( - 1 ) * ( cos ( x ) ) / ( cos ( x ) ) ;
x- ( - b + sqrt ( a , b ) ) * 2 > 0 & - ( - b + sqrt ( a , b ) ) * 2 > 0 & - ( - b + sqrt ( a , b ) ) * 2 < 0 ;
assume that inf ( \mathopen { \uparrow } X ) /\ C is maximal and for X st X in C holds not X in C & not X in C & not X in C & not X in C & not X in C & not X in C & not X in C & not X in C & X in C & not X in C & X in C & X in C & X in C & X in C & X in C & X in C ) ) ;
( ( for B , i , j , k ) , i , j , k being Element of NAT holds ( j = i implies i = j ) & j = j implies ( j = i ) & ( j = j implies i = j ) ) & ( i = j implies i = j ) & ( i = j implies i = j ) & j = j ) implies i = j