thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
B ;
a <> c
T c= S
D c= B
c in X ;
b in X ;
X ;
b in D ;
x = e ;
let m ;
h is onto ;
N in K ;
let i ;
j = 1 ;
x = u ;
let n ;
let k ;
y in A ;
let x ;
let x ;
m c= y ;
F is one-to-one ;
let q ;
m = 1 ;
1 < k ;
G is prime ;
b in A ;
d divides a ;
i < n ;
s <= b ;
b in B ;
let r ;
B is one-to-one ;
R is total ;
x = 2 ;
d in D ;
let c ;
let c ;
b = Y ;
0 < k ;
let b ;
let n ;
r <= b ;
x in X ;
i >= 8 ;
let n ;
let n ;
y in f ;
let n ;
1 < j ;
a in L ;
C is boundary ;
a in A ;
1 < x ;
S is finite ;
u in I ;
z << z ;
x in V ;
r < t ;
let t ;
x c= y ;
a <= b ;
m in NAT ;
assume f is prime ;
not x in Y ;
z = +infty ;
k be Nat ;
K ` is being_line ;
assume n >= N ;
assume n >= N ;
assume X is \bf ;
assume x in I ;
q is as as Nat ;
assume c in x ;
p > 0 ;
assume x in Z ;
assume x in Z ;
1 <= kr2 ;
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 non ] ;
assume A is boundary ;
g is special ;
assume i > j ;
assume t in X ;
assume n <= m ;
assume x in W ;
assume r in X ;
assume x in A ;
assume b is even ;
assume i in I ;
assume 1 <= k ;
X is non empty ;
assume x in X ;
assume n in M ;
assume b in X ;
assume x in A ;
assume T c= W ;
assume s is atomic ;
b `1 <= c `1 ;
A meets W ;
i `1 <= j `1 ;
assume H is universal ;
assume x in X ;
let X be set ;
let T be Tree ;
let d be element ;
let t be element ;
let x be element ;
let x be element ;
let s be element ;
k <= 5 - 1 ;
let X be set ;
let X be set ;
let y be element ;
let x be element ;
P [ 0 ]
let E be set , f be Function of E , F ;
let C be category ;
let x be element ;
k be Nat ;
let x be element ;
let x be element ;
let e be element ;
let x be element ;
P [ 0 ]
let c be element ;
let y be element ;
let x be element ;
let a be Real ;
let x be element ;
let X be element ;
P [ 0 ]
let x be element ;
let x be element ;
let y be element ;
r in REAL ;
let e be element ;
n1 is r2 ;
Q halts_on s ;
x in \in \in \in that ;
M < m + 1 ;
T2 is open ;
z in b 0 ;
R2 is well-ordering ;
1 <= k + 1 ;
i > n + 1 ;
q1 is one-to-one ;
let x be trivial set ;
PM is one-to-one
n <= n + 2 ;
1 <= k + 1 ;
1 <= k + 1 ;
let e be Real ;
i < i + 1 ;
p3 in P ;
p1 in K ;
y in C1 ;
k + 1 <= n ;
let a be Real , A be Subset of TOP-REAL 2 ;
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 : o ;
O <> O2 ;
let r be Real ;
let f be FinSeq-Location ;
let i be Nat ;
let n be Nat ;
Cl A = A ;
L c= Cl L ;
A /\ M = B ;
let V be 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 component ;
H = G . i ;
1 <= i `1 + 1 ;
F . m in A ;
f . o = o ;
P [ 0 ] ;
a-0 <= non < or aLet <= b ;
R [ 0 ] ;
b in f .: X ;
assume q = q2 ;
x in [#] V ;
f . u = 0 ;
assume e1 > 0 ;
let V be RealUnitarySpace , M be Subset of V ;
s is trivial & s is non empty ;
dom c = Q ;
P [ 0 ] ;
f . n in T ;
N . j in S ;
let T be complete LATTICE , f be Function of T , S ;
the Arrows of F is one-to-one
sgn x = 1 ;
k in support a ;
1 in Seg 1 ;
rng f = X ;
len T in X ;
vbeing < n ;
Sp2 is bounded ;
assume p = p2 ;
len f = n ;
assume x in P1 ;
i in dom q ;
let U1 , U2 , U2 , U1 , U2 ;
pp = c ;
j in dom h ;
let k ;
f | Z is continuous ;
k in dom G ;
UBD C = B ;
1 <= len M ;
p in \mathbin { x } ;
1 <= jj ;
set A = \it Boolean ;
card a [= c ;
e in rng f ;
cluster B \oplus A -> empty ;
H is non empty and H is non empty ;
assume n0 <= 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 c= union M
assume not x in REAL ;
Im f is complete ;
x in Int y ;
dom F = M ;
a in On W ;
assume e in A ( ) ;
C c= C-26 ;
mm <> {} ;
let x be Element of Y ;
let f be let is let g be let L , x be Element of L ;
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 + dom ( \HM { the carrier of G } ) ;
- y in I ;
let A be non empty set , F be Function of A , B ;
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 II , J ;
assume that 1 <= j and j < l ;
v = - u ;
assume s . b > 0 ;
d1 in dom f ;
assume t . 1 in A ;
let Y be non empty TopSpace , f be Function of Y , Z ;
assume a in uparrow s ;
let S be non empty Poset ;
a , b // b , a ;
a * b = p * q ;
assume x , y are_the space ;
assume x in Omega f ;
[ a , c ] in X ;
mm <> {} ;
M + N c= M + M ;
assume M is \mathclose hhhz ;
assume f is Let bbrr-rst ;
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 = k1 ;
m + 1 < n + 1 ;
s in S \/ { s } ;
n + i >= n + 1 ;
assume Re y = 0 ;
k1 <= j1 & k1 <= len f ;
f | A is non empty ;
f . x - b <= b ;
assume y in dom h ;
x * y in B1 ;
set X = Seg n ;
1 <= i2 + 1 ;
k + 0 <= k + 1 ;
p ^ q = p ;
j |^ y divides m ;
set m = max A ;
[ x , x ] in R ;
assume x in succ 0 ;
a in sup phi ;
Cc in X ;
q2 c= C1 & q2 c= C2 ;
a2 < c2 & a2 < b2 ;
s2 is 0 -started ;
IC s = 0 ;
s4 = s4 - 1 ;
let V ;
let x , y be element ;
let x be Element of T ;
assume a in rng F ;
x in dom T ` ;
let S be Sub\overline of L ;
y " <> 0 ;
y " <> 0 ;
0. V = u-w ;
y2 , y , w is_collinear ;
R8 in X ;
let a , b be Real , c be Real ;
let a be object of C ;
let x be Vertex of G ;
let o be object of C , m be Morphism of C ;
r '&' q = P \lbrack l \rbrack ;
let i , j be Nat ;
let s be State of A , x be Element of S ;
s4 . n = N ;
set y = ( x `1 ) ^2 ;
NAT in dom g ;
l . 2 = y1 ;
|. g . y .| <= r ;
f . x in Cx0 ;
not V 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 NF ;
reconsider i = i as Ordinal ;
r * v = 0. X ;
rng f c= INT & rng f c= INT ;
G = 0 .--> goto 0 ;
let A be Subset of X ;
assume that X0 is dense and X0 is dense ;
|. f . x .| <= r ;
x be Element of R ;
let b be Element of L ;
assume x in W-19 ;
P [ k , a ] ;
let X be Subset of L ;
let b be object of B ;
let A , B be category ;
set X = Vars ( C ) ;
let o be OperSymbol of S ;
let R be connected non empty Poset ;
n + 1 = succ n ;
xY c= Z1 & xY c= Z1 ;
dom f = C1 & dom g = C2 ;
assume [ a , y ] in X ;
Re ( seq ^\ k ) is convergent ;
assume a1 = b1 & b1 = b2 ;
A = ( sInt A ) ` ;
a <= b or b <= a ;
n + 1 in dom f ;
let F be Instruction of S , i be Nat ;
assume that r2 > x0 and x0 < r2 ;
let Y be non empty set , f be Function of Y , BOOLEAN ;
2 * x in dom W ;
m in dom g2 & n <= m ;
n in dom g1 /\ dom g2 ;
k + 1 in dom f ;
the still of S is finite ;
assume that x1 <> x2 and x2 <> x3 ;
v1 in V1 & v2 in V1 ;
not [ b `1 , b ] in T ;
i-35 + 1 = i ;
T c= and T c= ] ;
( l `1 ) ^2 = 0 ;
let n be Nat ;
( t `2 ) = r ;
AA is integrable & f | A is bounded ;
set t = Top t ;
let A , B be real-membered set ;
k <= len G + 1 ;
C ( ) misses V ( ) ;
Product ( s ) is non empty ;
e <= f or f <= e ;
cluster non empty normal for sequence ;
assume c2 = b2 & c1 = c2 ;
assume h in [. q , p .] ;
1 + 1 <= len C ;
not c in B . m1 ;
cluster R .: X -> empty ;
p . n = H . n ;
assume that vseq is convergent and vseq is convergent ;
IC s3 = 0 & IC s3 = 0 ;
k in N or k in K ;
F1 \/ F2 c= F ;
Int G1 <> {} & Int G2 <> {} ;
( z `2 ) = 0 ;
p11 <> p1 or p11 <> p2 ;
assume z in { y , w } ;
MaxADSet ( a ) c= F ;
ex_sup_of downarrow s , S ;
f . x <= f . y ;
let T be up-complete non empty reflexive transitive RelStr ;
q |^ m >= 1 ;
a is_>=_than X & b is_>=_than Y ;
assume <* a , c *> <> {} ;
F . c = g . c ;
G is one-to-one , full , full ;
A \/ { a } \not c= B ;
0. V = 0. Y .= 0. V ;
let I be non empty the InstructionsF of S , S ;
f-24 . x = 1 ;
assume z \ x = 0. X ;
C4 = 2 to_power n ;
let B be SetSequence of Sigma ;
assume X1 = p .: D ;
n + l2 in NAT ;
f " P is compact ;
assume x1 in REAL & x2 in REAL ;
p1 = ( K + 1 ) * ( K + 1 ) ;
M . k = <*> REAL ;
phi . 0 in rng phi ;
OSMis closed ;
assume z0 <> 0. L & z0 <> 0. L ;
n < N7 . k ;
0 <= ( seq . 0 ) ;
- q + p = v ;
{ v } is Subset of B ;
set g = f /. 1 ;
R ( ) is stable of R ;
set cR = Vertices R , cR = Vertices R ;
p0 c= P3 & p0 c= P3 ;
x in [. 0 , 1 .[ ;
f . y in dom F ;
let T be Scott Scott Scott Scott Scott Scott Scott of S ;
ex_inf_of the carrier of S , S ;
downarrow a = downarrow b ;
P , C , K is_collinear ;
assume x in F ( s , r , t ) ;
2 to_power i < 2 to_power m ;
x + z = x + z + q ;
x \ ( a \ x ) = x ;
||. x-y .|| <= r ;
assume that Y c= field Q and Y <> {} ;
a ~ , b ~ ] , c ~ ;
assume a in A ( ) ;
k in dom ( q | k ) ;
p is \HM { finite } -defined FinSequence of S ;
i -' 1 = i-1 ;
f | A is one-to-one ;
assume x in f .: X ( ) ;
i2 - i1 = 0 & i1 - i2 = 0 ;
j2 + 1 <= i2 ;
g " * a in N ;
K <> { [ {} , {} ] } ;
cluster strict for for for for . of A ;
|. q .| ^2 > 0 ;
|. p4 .| = |. p .| ;
s2 - s1 > 0 - s1 ;
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 & rng I c= Seg n ;
assume that k in dom C and k <> i ;
1 + 1-1 <= i + j ;
dom S = dom F /\ dom G ;
let s be Element of NAT , x be Element of NAT ;
let R be ManySortedSet of A ;
let n be Element of NAT , x be Element of NAT ;
let S be non empty non void non empty non void holds S is holds S is non empty ;
let f be ManySortedSet of I ;
let z be Element of F_Complex , v be Element of COMPLEX ;
u in { ag } ;
2 * n < 2 * n ;
let x , y be set ;
B-11 c= ( V ) . n ;
assume I is_halting_on s , P ;
U2 = U2 & U2 = U2 implies U2 = U2
M /. 1 = z /. 1 ;
x9 = x9 & y9 = y9 ;
i + 1 < n + 1 + 1 ;
x in { {} , <* 0 *> } ;
( f . i ) <= ( f . i ) ;
let l be Element of L ;
x in dom ( F . k ) ;
let i be Element of NAT , k be Nat ;
r8 is COMPLEX -valued ;
assume <* o2 , o *> <> {} ;
s . x |^ 0 = 1 ;
card K1 in M & card K1 in M ;
assume that X in U and Y in U ;
let D be Subset-Family of Omega ;
set r = q | { k + 1 } ;
y = W . ( 2 * PI ) ;
assume dom g = cod f & cod g = cod f ;
let X , Y be non empty TopSpace , f be Function of X , Y ;
x \oplus A is interval ;
|. <*> A .| . a = 0 ;
cluster strict for SubLattice of L ;
a1 in B . s1 & a2 in B . s2 ;
let V be finite finite < be VectSp of F , v be Vector of V ;
A * B on B , A ;
f-3 = NAT --> 0 .= fg ;
let A , B be Subset of V ;
z1 = P1 . j & z2 = P2 . j ;
assume f " P is closed ;
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 = INT |^ X ;
z in dom ( A --> y ) ;
P [ y , h . y ] ;
{ x0 } c= dom f /\ dom g ;
let B be non-empty ManySortedSet of I , A be ManySortedSet of I ;
PI / 2 < Arg z / 2 ;
reconsider z9 = 0 , z9 = 1 as Nat ;
LIN a , d , c ;
[ y , x ] in IF ;
( Q ) * ( 1 , 3 ) = 0 ;
set j = x0 div m , i = x0 mod m ;
assume a in { x , y , c } ;
j2 - ( j - 1 ) > 0 ;
I \! \mathop { phi } = 1 ;
[ y , d ] in F-8 ;
let f be Function of X , Y ;
set A2 = ( B - C ) / ( A - B ) ;
s1 , s2 are_/ 2 implies s1 , s2 are_/ 2
j1 -' 1 = 0 & j1 -' 1 = 0 ;
set m2 = 2 * n + j ;
reconsider t = t as bag of n ;
I2 . j = m . j ;
i |^ s , n are_relative_prime ;
set g = f | D-21 ;
assume that X is lower and 0 <= r ;
( p1 `1 ) ^2 = 1 ^2 .= 1 ^2 ;
a < p3 `1 & p3 `1 < b ;
L \ { m } c= UBD C ;
x in Ball ( x , 10 ) ;
not a in LSeg ( c , m ) ;
1 <= i1 -' 1 & i1 -' 1 <= len f ;
1 <= i1 -' 1 & i1 -' 1 <= len f ;
i + i2 <= len h - 1 ;
x = W-min ( P ) & y = W-min ( P ) ;
[ x , z ] in X ~ ;
assume y in [. x0 , x .] ;
assume p = <* 1 , 2 , 3 *> ;
len <* A1 *> = 1 & len <* A1 *> = 1 ;
set H = h . g , I = h . I ;
card b * a = |. a .| ;
Shift ( w , 0 ) |= v ;
set h = h2 (*) h1 , k = k (*) h2 ;
assume x in cluster cluster cluster cluster cluster X2 /\ 4 -> non empty ;
||. h .|| < d1 & ||. h .|| < d ;
not x in the carrier of f .: the carrier of L ;
f . y = F ( y ) ;
for n holds X [ n ] ;
k - l = k\leq <= k\leq ;
<* p , q *> /. 2 = q ;
let S be Subset of the lattice of Y ;
let P , Q be succ s ;
Q /\ M c= union ( F | M )
f = b * ( canFS ( S ) ) ;
let a , b be Element of G ;
f .: X is_<=_than f . sup X
let L be non empty transitive RelStr , X be Subset of L ;
S-20 is x -let \leq x -f1 ;
let r be non positive Real ;
M , v |= x \hbox { y } ;
v + w = 0. ( Z ) ;
P [ ( len F ) + 1 ] ;
assume InsCode ( i ) = 8 or InsCode ( i ) = 8 ;
the zero of M = 0 & the zero of M = 0 ;
cluster z * seq -> summable for Real_Sequence ;
let O be Subset of the carrier of C ;
||. f .|| | X is continuous ;
x2 = g . ( j + 1 ) ;
cluster -> non empty for Element of S ;
reconsider l1 = l-1 as Nat ;
v4 is Vertex of r2 & v4 is Vertex of G ;
T2 is SubSpace of T2 implies ( for x being Point of T2 holds x in the carrier of T2 )
Q1 /\ Q19 <> {} & Q1 /\ Q19 <> {} ;
k be Nat ;
q " is Element of X & q " is Element of X ;
F . t is set of non zero & F . t is non empty ;
assume that n <> 0 and n <> 1 ;
set en = EmptyBag n , en = EmptyBag n ;
let b be Element of Bags n ;
assume for i holds b . i is commutative ;
x is root of ( p `2 ) ^2 & y is root ;
not r in ]. p , q .[ ;
let R be FinSequence of REAL , a be Real ;
S7 does not destroy b1 , S8 = b1 " ;
IC SCM R <> a & IC S = IC S ;
|. - |[ x , y ]| .| >= r ;
1 * ( s . n ) = seq . n * ( s . n ) ;
let x be FinSequence of NAT , y be Element of NAT ;
let f be Function of C , D , g be Function of C , D ;
for a holds 0. L + a = a
IC s = s . NAT .= IC s . NAT ;
H + G = F\hbox { - } G } ;
Cx1 . x = x2 & Cx1 . x = y2 ;
f1 = f .= f2 .= ( f | X ) . x ;
Sum <* p . 0 *> = p . 0 ;
assume v + W = v + u + W ;
{ a1 } = { a2 } .= { a2 } ;
a1 , b1 _|_ b , a ;
d1 , o _|_ o , a3 ;
IF is reflexive & IF is reflexive implies F + F is reflexive
IO is antisymmetric implies ( for x st x in CO holds x in O )
sup rng H1 = e & sup rng H2 = e ;
x = ( a * ( 1 / a ) ) * ( 1 / a ) ;
|. p1 .| ^2 >= 1 ^2 ;
assume that j2 -' 1 < 1 and j2 -' 1 < len f ;
rng s c= dom f1 /\ dom f2 ;
assume that support a misses support b and not a in support b ;
let L be associative commutative associative non empty doubleLoopStr , p be Polynomial of L ;
s " + 0 < n + 1 ;
p . c = ( f " ) . 1 ;
R . n <= R . ( n + 1 ) ;
Directed I1 = I1 +* ( card I + card J + 3 ) ;
set f = + ( x , y , r ) ;
cluster Ball ( x , r ) -> bounded ;
consider r being Real such that r in A ;
cluster non empty NAT -defined for NAT -defined Function ;
let X be non empty directed Subset of S ;
let S be non empty full SubRelStr of L ;
cluster <* [ ] , N . N *> -> complete for non trivial TopSpace ;
( 1 / a ) " = a ;
( q . {} ) `1 = o ;
( - i ) - 1 > 0 ;
assume that 1 / 2 <= t `1 and t `1 <= 1 ;
card B = k + 1-1 - 1 ;
x in union rng ( f | X ) ;
assume x in the carrier of R & y in the carrier of S ;
d in D ;
f . 1 = L . ( F . 1 ) ;
the vertices of G = { v } & { v } c= G ;
let G be : holds G is : 2 ;
e , v6 be set ;
c . ( i + 1 ) in rng c ;
f2 /* ( q ^\ k ) is divergent_to-infty ;
set z1 = - z2 , z2 = - z2 , z2 = - z1 , z2 = - z2 , z1 = z2 , z2 = z2 , z2 = - z2 ;
assume w is llas of S , G ;
set f = p |-count t , g = p |-count t , h = p |-count t , t = p |-count t , n = p |-count t , m = p |-count t , n = p |-count t
let c be Object of C ;
assume ex a st P [ a ] ;
let x be Element of REAL m , y be Element of REAL m ;
let IF be Subset-Family of X , A be Subset of X ;
reconsider p = p as Element of NAT ;
let v , w be Point of X ;
let s be State of SCM+FSA , I be Program of SCM+FSA , a be Int-Location ;
p is FinSequence of ( the InstructionsF of SCM+FSA ) * ;
stop I ( ) c= P-12 ( ) ;
set ci = f^ ( f /. i ) ;
w ^ t Q Q implies w ^ t ^ t ^ w ^ t ^ w ^ t ^ w ^ w ^ t ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^
W1 /\ W = W1 /\ W ` .= W1 /\ W2 ;
f . j is Element of J . j ;
let x , y be \rm \rm \cdot of T2 ;
ex d st a , b // b , d ;
a <> 0 & b <> 0 & c <> 0 ;
ord x = 1 & x is positive implies x is positive
set g2 = lim ( seq ^\ k ) ;
2 * x >= 2 * ( 1 / 2 ) ;
assume ( a 'or' c ) . z <> TRUE ;
f (*) g in Hom ( c , c ) ;
Hom ( c , c + d ) <> {} ;
assume 2 * Sum ( q | m ) > m ;
L1 . ( F-21 ) = 0 & L1 . ( FH ) = 0 ;
( the carrier of X ) \/ R1 = the carrier of X ;
( sin . x ) <> 0 & ( sin . x ) <> 0 ;
( ( exp_R . x ) ^2 ) > 0 ;
o1 in [: X /\ O2 , X /\ O2 :] ;
e , v6 be set ;
r3 > ( 1 - 2 ) * 0 ;
x in P .: ( F -ideal ( L ) ) ;
let J be closed Subset of R , I be left ideal non empty Subset of R ;
h . p1 = f2 . O & h . O = g2 ;
Index ( p , f ) + 1 <= j ;
len ( q | k ) = width M .= width M ;
the carrier of CK c= A & the carrier of CK c= A ;
dom f c= union rng ( F | n ) ;
k + 1 in support ( ( support n ) + ( support b ) ) ;
let X be ManySortedSet of the carrier of S ;
[ x `1 , y ] in ( an \/ R2 ) ;
i = D1 or i = D2 or i = D1 ;
assume a mod n = b mod n & b mod n = 0 ;
h . x2 = g . x1 & h . x2 = h . x2 ;
F c= 2 -tuples_on the carrier of X ;
reconsider w = |. s1 .| as Real_Sequence of X ;
( 1 / m * m + r ) < p ;
dom f = dom Iq .= dom Iq ;
[#] P-17 = [#] ( ( TOP-REAL 2 ) | K1 ) .= K1 ;
cluster - x -> ExtReal for ExtReal ;
then { d1 } c= A & A is closed ;
cluster ( TOP-REAL n ) | ( [#] TOP-REAL n ) -> finite-ind ;
let w1 be Element of M ;
let x be Element of dyadic ( n ) ;
u in W1 & v in W3 implies u + v in W3
reconsider y = y , z = z as Element of L2 ;
N is full SubRelStr of ( T |^ the carrier of S ) ;
sup { x , y } = c "\/" c ;
g . n = n to_power 1 .= n ;
h . J = EqClass ( u , J ) ;
let seq be -> -> -> -> -> -> -> -> summable sequence of X ;
dist ( x `1 , y ) < r / 2 ;
reconsider mm = m , mn = n as Element of NAT ;
- x0 < r1 - x0 & r1 < x0 + r2 ;
reconsider P ` = P ` as strict Subgroup of N ;
set g1 = p * ( idseq q `1 ) , g2 = p * ( idseq q `1 ) ;
let n , m , k be non zero Nat ;
assume that 0 < e and f | A is lower ;
D2 . ( I8 . I ) in { x } ;
cluster subcondensed -> subopen for Subset of T ;
let P be compact non empty Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
Gik in LSeg ( cos , 1 ) /\ LSeg ( cos , 1 ) ;
let n be Element of NAT , x be Element of NAT ;
reconsider S8 = S , S8 = T as Subset of T ;
dom ( i .--> X `1 ) = { i } ;
let X be non-empty ManySortedSet of S ;
let X be non-empty ManySortedSet of S ;
op ( 1 ) c= { [ {} , {} ] }
reconsider m = mm - 1 as Element of NAT ;
reconsider d = x as Element of C ( ) ;
let s be 0 -started State of SCMPDS , k be Nat ;
let t be 0 -started State of SCMPDS , Q ;
b , b , x , y , x is_collinear ;
assume that i = n \/ { n } and j = k \/ { k } ;
let f be PartFunc of X , Y ;
x0 >= ( sqrt c ) / ( sqrt c ) ;
reconsider t7 = T7 " as Point of Euclid n ;
set q = h * p ^ <* d *> ;
z2 in U . ( y2 ) /\ Q . ( z2 + y2 ) ;
A |^ 0 = { <%> E } .= { <%> E } ;
len W2 = len W + 2 & len W2 = len W + 2 ;
len h2 in dom h2 & len h2 in dom h2 ;
i + 1 in Seg ( len s2 + 1 ) ;
z in dom g1 /\ dom f /\ dom g ;
assume p2 = E-max ( K ) & p3 = E-max ( K ) ;
len G + 1 <= i1 + 1 + 1 ;
f1 (#) f2 is convergent & lim ( f1 (#) f2 ) = x0 ;
cluster ( seq + s ) -> summable for Real_Sequence ;
assume that j in dom M1 and i in dom M1 ;
let A , B , C be Subset of X ;
x , y , z be Point of X , p be Point of X ;
b ^2 - ( 4 * a * c ) >= 0 ;
<* x/y *> ^ <* y *> \geq x ;
a , b in { a , b } ;
len p2 is Element of NAT & len p2 = len p1 + 1 ;
ex x being element st x in dom R & y = R . x ;
len q = len ( K (#) G ) .= len G ;
s1 = Initialize ( Initialized s ) , P1 = P +* I ;
consider w being Nat such that q = z + w ;
x ` ` is to of x ` ;
k = 0 & n <> k or k > n ;
then X is discrete for A is closed ;
for x st x in L holds x is FinSequence ;
||. f /. c .|| <= r1 & ||. f /. c .|| <= r1 ;
c in uparrow p & not c in { p } ;
reconsider V = V as Subset of the TopStruct of TOP-REAL n ;
let N , M be being being being being being being being being being being being { sts of L ;
then z is_>=_than waybelow x & z is_>=_than compactbelow y ;
M | [. f , g .] = f & M | [. g , g .] = g ;
( ( TOP-REAL 1 ) /. 1 ) = TRUE .= TRUE ;
dom g = dom f -tuples_on X & dom g = dom f -tuples_on X ;
mode : of G is \cal : for W being Walk of G holds W is : NAT
[ i , j ] in Indices M & [ i , j ] in Indices M ;
reconsider s = x " as Element of H . s ;
let f be Element of dom ( Subformulae p ) , g be Element of dom ( Subformulae p ) ;
F1 . ( a1 , - a2 ) = G1 . ( a1 , - a2 ) ;
redefine func E ( a , b , r ) -> compact ;
let a , b , c , d be Real ;
rng s c= dom ( 1 / ( f1 + f2 ) ) ;
curry ( F-19 , k ) is additive ;
set k2 = card dom B , k1 = card dom B , k2 = card dom C ;
set G = or the Sorts of A ;
reconsider a = [ x , s ] as Object of G ;
let a , b be Element of MO , c be Element of M ;
reconsider s1 = s , s2 = t as Element of ( the carrier of S1 ) ;
rng p c= the carrier of L & p . ( len p ) = 0. L ;
let d be Subset of the Sorts of A ;
( x .|. x = 0 iff x = 0. W ) ;
I-21 in dom stop I & Ik in dom stop I ;
let g be continuous Function of X | B , Y ;
reconsider D = Y as Subset of ( TOP-REAL n ) | P ;
reconsider i0 = len p1 - 1 as Integer ;
dom f = the carrier of S & rng f = the carrier of T ;
rng h c= union ( the carrier of J ) & h . 0 = 1 ;
cluster All ( x , H ) -> P\mathopen ;
d * N1 / ( 1 - d ) > N1 * 1 / ( 1 - d ) ;
]. a , b .[ c= [. a , b .] ;
set g = f " ( D1 /\ D2 ) ;
dom ( p | ( NAT \ NAT ) ) = NAT ;
3 + - 2 <= k + - 2 ;
tan is_differentiable_in ( arccot * arccot ) . x ;
x in rng ( f /^ ( n + 1 ) ) ;
let f , g be FinSequence of D ;
p ( ) in the carrier of S1 & p ( ) in the carrier of S1 ;
rng f " = dom f & rng f = dom f ;
( the Source of G ) . e = v ;
width G - 1 < width G - 1 & width G - 1 < width G - 1 ;
assume that v in rng ( S | E1 ) and u in E ;
assume x is root or x is root or x is root ;
assume that 0 in rng ( g2 | A ) and 0 < g2 . 0 ;
let q be Point of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
let p be Point of TOP-REAL 2 , x be Point of TOP-REAL 2 ;
dist ( O , u ) <= |. p2 .| + 1 ;
assume dist ( x , b ) < dist ( a , b ) ;
<* S7 *> is_in the carrier of C-20 ( C ) ;
i <= len ( G | ( i2 -' 1 ) ) ;
let p be Point of TOP-REAL 2 , x be Point of TOP-REAL 2 ;
x1 in the carrier of I[01] & x2 in the carrier of I[01] ;
set p1 = f /. i , p2 = f /. ( i + 1 ) ;
g in { g2 : r < g2 & g2 < r } ;
Q2 = Sp2 " ( Q /\ R /\ S /\ S /\ Q /\ S /\ R /\ S /\ Q /\ S /\ Q /\ S /\ Q /\ S /\ Q /\ S /\ Q /\ S /\ Q /\ S /\ S /\
( 1 / 2 ) (#) ( 1 / 2 ) is summable ;
- p + I c= - p + A ;
n < LifeSpan ( P1 , s1 ) + 1 & n + 1 <= card I ;
CurInstr ( p1 , s1 ) = i .= ( 0 + 1 ) ;
A /\ Cl { x } \ { x } <> {} ;
rng f c= ]. r , r + 1 .[ ;
let g be Function of S , V ;
let f be Function of L1 , L2 , g be Function of L1 , L2 ;
reconsider z = z , t = y as Element of CompactSublatt L ;
let f be Function of S , T ;
reconsider g = g as Morphism of c opp , b opp ;
[ s , I ] in S ~ & [ s , I ] in S ~ ;
len ( the connectives of C ) = 4 & len ( the connectives of C ) = 4 ;
let C1 , C2 be subcategory of C , C2 ;
reconsider V1 = V as Subset of X | B , V1 = V as Subset of X ;
attr p is valid means : Def4 : All ( x , p ) is valid ;
assume that X c= dom f and f .: X c= dom g and g .: X c= dom f ;
H |^ ( a " ) is Subgroup of H |^ a ;
let A1 be [ be [ of O , E ] , A1 ] ;
p2 , r3 , q3 is_collinear & q2 , q3 , q3 is_collinear ;
consider x being element such that x in v ^ K ;
not x in { 0. TOP-REAL 2 } or x in { 0. TOP-REAL 2 } ;
p in [#] ( I[01] | B11 ) & p in [#] ( I[01] | B11 ) ;
0 . 0 < M . ( E . n ) ;
^ ( c / a ) = c ;
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . s2 ) & a2 in dom ( F . s2 ) ;
cluster -> Line for Sublattice of L ;
set i1 = the Nat , i2 = the Element of NAT ;
let s be 0 -started State of SCM+FSA , k be Nat ;
assume y in ( f1 \/ f2 ) .: A ;
f . ( len f ) = f /. len f .= f /. 1 ;
x , f . x '||' f . x , f . y ;
attr X c= Y means : Def4 : cos .: X c= cos .: Y ;
let y be upper Subset of Y , x be Element of X ;
cluster -> -> -> -> of ( - 1 ) -element for Relation ;
set S = <* Bags n , <* i1 *> , <* i2 *> *> ;
set T = [. 0 , 1 / 2 .] , S = [. 1 , 1 .] ;
1 in dom mid ( f , 1 , 1 ) ;
( 4 * PI ) / PI < ( 2 * PI ) / PI ;
x2 in dom f1 /\ dom f & x2 in dom f1 /\ dom f ;
O c= dom I & { {} } = { {} } ;
( the Source of G ) . x = v & ( the Source of G ) . x = v ;
{ HT ( f , T ) } c= Support f ;
reconsider h = R . k as Polynomial of n , L ;
ex b being Element of G st y = b * H ;
let x , y , z be Element of G opp , a , b be Element of G ;
h19 . i = f . ( h . i ) ;
( p `1 ) ^2 = ( p1 `1 ) ^2 + ( p2 `1 ) ^2 ;
i + 1 <= len Cage ( C , n ) ;
len <* P *> ^2 = len P & len <* P *> = len P ;
set N-26 = the or the or of N = the or of N = the or of N ;
len g\rrangle + ( x + 1 ) - 1 <= x ;
a on B & b on B & c on B ;
reconsider rv = r * I . v as FinSequence of REAL ;
consider d such that x = d and a [= d ;
given u such that u in W and x = v + u ;
len f /. ( \downharpoonright n ) = len ( f /^ n ) ;
set q2 = N-min L~ Cage ( C , n ) , q2 = q2 /. 1 ;
set S = { S1 , S2 , 3 } , T = { S1 , S2 , 3 } ;
MaxADSet ( b ) c= MaxADSet ( P /\ Q ) ;
Cl ( G . q1 ) c= F . r2 & Cl ( G . q2 ) c= G . r2 ;
f " D meets h " ( V /\ W ) ;
reconsider D = E as non empty directed Subset of L1 ;
H = ( the_left_argument_of H ) '&' ( the_right_argument_of H ) ;
assume that t is Element of ( the Sorts of Free ( S , X ) ) . s ;
rng f c= the carrier of S2 & rng f c= the carrier of S2 ;
consider y being Element of X such that x = { y } ;
f1 . ( a1 , b1 ) = b1 . ( a1 , b1 ) ;
the carrier' of G ` = E \/ { E } .= { E } ;
reconsider m = len ( thesis - k ) as Element of NAT ;
set S1 = LSeg ( n , UMP C ) , S2 = LSeg ( n , UMP C ) ;
[ i , j ] in Indices M1 & [ i , j ] in Indices M1 ;
assume that P c= Seg m and M is \HM { i } is of Seg m ;
for k st m <= k holds z in K . k ;
consider a being set such that p in a and a in G ;
L1 . p = p * L /. 1 .= L /. 1 ;
p-7 . i = p1 . i .= p1 . i ;
let PA , PA , G be a_partition of Y , a be Element of Y ;
If 0 < r & r < 1 , then 1 < ( 1 - r ) * 1 ;
rng ( ( a , X ) --> ( x , y ) ) = [#] X ;
reconsider x = x , y = y as Element of K ;
consider k such that z = f . k and n <= k ;
consider x being element such that x in X \ { p } ;
len ( ( canFS ( s ) ) | ( ( len s ) ) ) = card ( ( s ) | ( len s ) ) ;
reconsider x2 = x1 , y2 = x2 as Element of L2 ;
Q in FinMeetCl ( ( the topology of X ) \/ the topology of Y ) ;
dom ( f . 0 ) c= dom ( u . 0 ) & dom ( f . 0 ) c= dom f ;
redefine pred n divides m & m divides n implies n = m ;
reconsider x = x as Point of [: I[01] , I[01] :] ;
a in ) implies ( for b being let c being Element of T2 holds c in X )
not y0 in the still of f & not ( ex y st y in the carrier of f & not ( y in the carrier of f ) ;
Hom ( ( a ~ ) , c ) <> {} ;
consider k1 such that p " < k1 and k1 < k and k1 < k ;
consider c , d such that dom f = c \ d ;
[ x , y ] in [: dom g , dom k :] ;
set S1 = \kern1pt \kern1pt ( x , y , z ) ;
l1 = m2 & l1 = i2 & l2 = j2 implies l1 + l2 = i2 + ( l + 1 )
x0 in dom ( u /\ A ) /\ dom ( v + u ) ;
reconsider p = x , q = y as Point of ( TOP-REAL 2 ) | K1 ;
I[01] = R^1 | B01 .= ( TOP-REAL 2 ) | B01 .= ( TOP-REAL 2 ) | B01 ;
f . p4 <= _ P P f . p1 , f . p2 , f . p3 , f . p4 , f . p4 , f . p1 , f . p2 ;
( F . ( x `1 ) ) ^2 <= ( x `1 ) ^2 + ( x `2 ) ^2 ;
( x `2 = ( W . ( len W ) ) ) `2 .= ( W . ( len W ) ) `2 ;
for n being Element of NAT holds P [ n ] implies P [ n + 1 ]
let J , K be non empty Subset of I ;
assume that 1 <= i and i <= len <* a " *> ;
0 |-> a = <*> the carrier of K & 0 |-> a = <*> the carrier of K ;
X . i in 2 -tuples_on ( A . i \ B . i ) ;
<* 0 *> in dom ( e --> [ 1 , 0 ] ) ;
then P [ a ] implies P [ succ a ] & P [ succ a ]
reconsider sbeing being finite \rangle , s/. k being ' of D ;
( k - 1 ) <= len ( an - 1 ) ;
[#] S c= [#] the TopStruct of T & [#] T c= [#] the TopStruct of T ;
for V being strict RealUnitarySpace holds V in the carrier of V implies V is Subspace of W
assume that k in dom mid ( f , i , j ) and k <= j ;
let P be non empty Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
let A , B be Matrix of n1 , n2 , K be Matrix of n1 , n2 , K ;
- a * - b = a * b - b * a ;
for A being being_line of AS holds A // A or A // A ;
( for o2 being object of o2 st o2 in <^ o2 , o2 ^> holds [ o2 , o2 ] in <^ o2 , o2 ^> ;
then ||. x .|| = 0 & x = 0. X ;
let N1 , N2 be strict normal Subgroup of G , N be normal Subgroup of G ;
j >= len ( upper_volume ( g , D1 ) | j1 ) ;
b = Q . ( len Q - 1 + 1 ) ;
f2 * f1 /* ( s ^\ k ) is divergent_to+infty ;
reconsider h = f * g as Function of [: N2 , N :] , 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 ( T | n ) | n ;
{} = the carrier of L1 + L2 .= the carrier of L1 + L2 .= the carrier of L1 + L2 ;
Directed I is_halting_on Initialized s , P & Directed I is_halting_on Initialized s , P ;
Initialized p = Initialize ( p +* q ) , p +* I = p +* I ;
reconsider N2 = N1 , N2 = N2 as strict net of R1 , R2 ;
reconsider Y = Y as Element of \langle Ids L , \subseteq \rangle ;
"/\" ( uparrow p , L ) <> p ;
consider j being Nat such that i2 = i1 + j and j in dom f ;
not [ s , 0 ] in the carrier of S2 or not [ s , 0 ] in the carrier of S2 ;
mm in ( B '/\' C ) '/\' D \ { {} } ;
n <= len ( P + Q ) & n <= len ( P + Q ) ;
( x1 `1 ) = x2 `1 & ( x2 `1 ) = x3 `1 ;
InputVertices S = { x1 , x2 } & InputVertices S = { x1 , x2 } ;
let x , y be Element of FTT1 ( n ) ;
p = |[ p `1 , p `2 ]| .= |[ p `1 , p `2 ]| ;
g * 1_ G = h " * g * h .= h * g .= h * g ;
let p , q be Element of PFuncs ( V , C ) ;
x0 in dom ( x1 /\ dom x2 ) /\ dom ( x2 /\ dom x3 ) ;
( R qua Function ) " = R " .= ( R " ) " .= R ;
n in Seg len ( f /^ ( len f -' 1 ) ) ;
for s being Real st s in R holds s <= s2 implies s <= 1
rng s c= dom ( f2 * f1 ) /\ dom ( f2 * f1 ) ;
synonym for for for for 2 -tuples_on X for 2 -tuples_on X ;
1_ K * 1_ K = 1_ K * ( 1_ K ) .= 1_ K * 1_ K ;
set S = Segm ( A , P1 , Q1 ) , T = Segm ( A , P2 , Q1 ) ;
ex w st e = ( w / f ) & w in F ;
curry ( P+* ( k , x ) ) # x is convergent ;
cluster open open -> open for Subset of [: T , T :] ;
len f1 = 1 .= len f3 .= len f3 + 1 .= len f3 + 1 ;
( i * p ) / p < ( 2 * p ) / p ;
let x , y be Element of OSSub ( U0 ) ;
b1 , c1 // b9 , c & b1 , c1 // b2 , c2 ;
consider p being element such that c1 . j = { p } ;
assume that f " { 0 } = {} and f is total and f is total ;
assume that IC Comput ( F , s , k ) = n and IC Comput ( F , s , k ) = n ;
Reloc ( J , card I ) does not destroy a , P ;
goto ( card I + 1 ) does not destroy c ;
set m3 = LifeSpan ( p3 , s3 ) , m3 = LifeSpan ( p3 , s3 ) ;
IC SCMPDS in dom ( p +* ( a , I ) ) ;
dom t = the carrier of SCM & dom t = the carrier of SCM & t . 0 = t . 1 ;
( ( E-max L~ f ) .. f ) .. f = 1 & ( E-max L~ f ) .. f = 1 ;
let a , b be Element of PFuncs ( V , C ) ;
Cl ( Int F ) c= Cl ( Int F ) ;
the carrier of X1 union X2 misses ( ( X1 union X2 ) \/ ( X2 union X3 ) ) ;
assume not LIN a , f . a , g . a , g . b ;
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 / ( y , x ) / ( y , x ) |= H ;
consider m be element such that m in Intersect ( FF . m ) and x = f . m ;
reconsider A1 = support u1 , A2 = 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 <> a5 and a5 <> a5 ;
cluster s -k2 -> $ for string of S ;
LG2 /. n2 = LG2 . n2 .= LG2 . n2 .= LG2 . n2 ;
let P be compact non empty Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
assume that r-7 in LSeg ( p1 , p2 ) and rp29 in LSeg ( p2 , p3 ) ;
let A be non empty compact Subset of TOP-REAL n , x be Point of TOP-REAL n ;
assume that [ k , m ] in Indices ( D * ( i , j ) ) ;
0 <= ( 1 / 2 ) |^ p / ( 1 / p ) ;
( F . N | E8 ) . x = +infty ;
attr X c= Y means : Def4 : Z c= V & X \ V c= Y \ Z ;
( y `2 ) * ( z `2 ) <> 0. I & ( y `2 ) * ( z `2 ) <> 0. I ;
1 + card ( X-18 ) <= card ( u + card ( X ) ) ;
set g = z :- ( ( L~ z ) /. 1 ) , M = z /. len z , N = L~ z , S = z /. len z , T = z /. 1 , N = z /. len z , S = z /. len z , T = z
then k = 1 implies p . k = <* x , y *> . k ;
cluster -> total for Element of C -\mathop { 0 } , D ;
reconsider B = A as non empty Subset of ( TOP-REAL n ) | B as Subset of TOP-REAL n ;
let a , b , c be Function of Y , BOOLEAN , p be Function of Y , BOOLEAN ;
L1 . i = ( i .--> g ) . i .= g . i .= g . i ;
Plane ( x1 , x2 , x3 ) c= P & Plane ( x1 , x2 , x3 ) c= P ;
n <= indx ( D2 , D1 , j1 ) + 1 - 1 ;
( ( g2 ) . O ) `1 = - 1 & ( g2 ) . O = 1 ;
j + p .. f - len f <= len f - len f + p .. f - 1 ;
set W = W-bound C , E = W-bound C , N = S-bound C ;
S1 . ( a `1 , e `1 ) = a + e `1 .= a ;
1 in Seg width ( M * ( ColVec2Mx p ) ) ;
dom ( i (#) Im f ) = dom Im f /\ dom Im f ;
( that ^2 ) . x = W . ( a , *' ( a , p ) ) ;
set Q = non |= ( max ( g , f , h ) ) ;
cluster -> many sorted for ManySortedSet of U1 , U2 ;
attr F = { A } means : Def4 : F is discrete ;
reconsider z9 = \hbox ( x , y ) as Element of product \overline G ;
rng f c= rng f1 \/ rng f2 & rng ( f1 + f2 ) 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 implies f = <*> ( the carrier of F_Complex )
E , j |= All ( x1 , x2 , H ) implies E , j |= H
reconsider n1 = n , n2 = m as Morphism of o1 , o2 ;
assume that P is idempotent and R is idempotent and P ** R = R ** P ;
card ( B2 \/ { x } ) = k-1 + 1 ;
card ( ( x \ B1 ) /\ B1 ) = 0 implies card ( x \ B1 ) = 0
g + R in { s : g-r < s & s < g + r } ;
set q-1x0 = ( q , <* s *> ) \mathop { 1 } ;
for x being element st x in X holds x in rng f1 implies x in X
h0 /. ( i + 1 ) = h0 . ( i + 1 ) ;
set mw = max ( B , } ( 2 ) ) , mw = max ( B , 2 ) ;
t in Seg width ( I ^ ( n , n ) ) ;
reconsider X = dom f /\ C as Element of Fin ( NAT * ) ;
IncAddr ( i , k ) = <% - l . ( k + k ) %> ;
( ( q `2 ) ) ^2 <= ( q `2 ) ^2 & ( q `2 ) ^2 <= ( q `2 ) ^2 ;
attr R is condensed means : Def4 : Int R is condensed & Cl R is condensed ;
redefine pred 0 <= a & b <= 1 & a * b <= 1 ;
u in ( ( c /\ ( ( d /\ b ) /\ e ) ) /\ f ) /\ j ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) ) /\ f ) /\ j ;
len C + - 2 >= 9 + - 3 + 2 - 2 ;
x , z , y is_collinear & x , z , y is_collinear implies x = y
a |^ ( n1 + 1 ) = a |^ n1 * a |^ n1 ;
<* \underbrace ( 0 , \dots , 0 ) *> in Line ( x , a * x ) ;
set y9 = <* y , c *> ;
FG2 /. 1 in rng Line ( D , 1 ) & FG2 /. len FG2 = F . 1 ;
p . m Joins r /. m , r /. ( m + 1 ) ;
( p `2 = ( f /. i1 ) `2 .= ( f /. i1 ) `2 .= ( f /. i1 ) `2 ;
W-bound ( X \/ Y ) = W-bound ( X \/ Y ) .= W-bound ( X \/ Y ) ;
0 + ( p `2 ) <= 2 * r + ( p `2 ) ^2 ;
x in dom g & not x in g " { 0 } implies x in dom g
f1 /* ( seq ^\ k ) is divergent_to+infty & f2 /* ( seq ^\ k ) is divergent_to+infty ;
reconsider u2 = u , v2 = v as VECTOR of P`1 , u = v as VECTOR of X ;
p |-count ( Product Sgm ( X ) ) = 0 & p |-count ( Product Sgm ( X ) ) = 0 ;
len <* x *> < i + 1 & i + 1 <= len c + 1 ;
assume that I is not empty and { x } /\ { y } = { 0. I } ;
set ii2 = card I + 4 .--> goto 0 , ii2 = goto 0 , ii2 = goto 0 ;
x in { x , y } & h . x = {} T implies x = y
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 S ;
reconsider m = M , i = I , n = N as Element of X ;
A . ( j + 1 ) = B . ( j + 1 ) \/ A . j ;
set N8 = : ( Ge ) `1 = ( Ge ) `1 ;
rng F c= the carrier of gr { a } & ( for x being Element of G st x in F holds x is strict Subgroup of G ) implies F is Subgroup of G
\langle Q *> is in rng ( Q * ( K , n , r ) ) & Q is in rng Q ;
f . k , f . ( Let n ) ] in rng f ;
h " P /\ [#] T1 = f " P /\ [#] T2 .= f " P /\ [#] T2 ;
g in dom f2 \ f2 " { 0 } & f2 . ( g . 0 ) = f2 . ( g . 0 ) ;
gfinite X /\ dom f1 = g1 " X /\ dom ( f1 | X ) ;
consider n being element such that n in NAT and Z = G . n ;
set d1 = \overline ( dist ( x1 , y1 ) ) , d2 = dist ( x2 , y2 ) ;
b `1 + 1 / 2 < ( 1 + 1 ) / 2 + 1 / 2 ;
reconsider f1 = f as VECTOR of the carrier of X , Y ;
attr i <> 0 means : Def4 : i ^2 mod ( i + 1 ) = 1 ;
j2 in Seg len ( g2 . i2 ) & j in Seg len ( g2 . i2 ) ;
dom ( i ) = dom ( i ) .= dom ( i ) .= dom ( i ) ;
cluster sec | ]. PI , PI / 2 .[ -> one-to-one ;
Ball ( u , e ) = Ball ( f . p , e ) ;
reconsider x1 = x0 , y1 = x0 as Function of S , I ;
reconsider R1 = x , R2 = y , R1 = z as Relation of L ;
consider a , b being Subset of A such that x = [ a , b ] ;
( <* 1 *> ^ p ) ^ <* n *> in RH & <* n *> ^ p in RH ;
S1 +* S2 = S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2
( ( - 1 ) (#) ( cos * sin ) ) is_differentiable_on Z ;
cluster -> [. 0 , 1 .] -valued for Function of C , REAL ;
set C7 = 1GateCircStr ( <* z , x *> , f3 ) , C8 = 1GateCircStr ( <* z , x *> , f3 ) ;
E8 . e2 = E8 . e2 .= ( - T ) . e2 .= ( T . e2 ) . e2 ;
( ( arctan * arccot ) `| Z ) = ( arctan * arccot ) `| Z ;
upper_bound A = PI * 2 / 2 & lower_bound A = 0 & lower_bound A = 0 ;
F . ( dom f , - F . ( cod f ) ) = F . ( cod f , - F . ( cod f ) ) ;
reconsider pbe Point of TOP-REAL 2 , p8 = ( q `2 ) as Point of TOP-REAL 2 ;
g . W in [#] Y0 & [#] Y0 c= [#] Y0 & g . W c= [#] Y0 ;
let C be compact non vertical non vertical non horizontal Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
LSeg ( f ^ g , j ) = LSeg ( f , j ) .= LSeg ( f , i ) ;
rng s c= dom f /\ ]. x0 , x0 + r .[ & rng ( f /* s ) c= dom f /\ ]. x0 , x0 + r .[ ;
assume x in { idseq ( 2 ) , Rev ( idseq 2 ) } ;
reconsider n2 = n , m2 = m , n1 = n , n2 = m 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 + m2 .= m1 + m2 + m2 ;
assume for n holds H1 . n = G . n -H . n ;
set B" = f .: ( the carrier of X1 ) , B" = f .: ( the carrier of X2 ) ;
ex d being Element of L st d in D & x << d ;
assume that R ~ c= R ~ and R ~ c= R ~ and R ~ c= R ~ and R c= R ~ ;
t in ]. r , s .[ or t = r or t = s ;
z + v2 in W & x = u + ( z + v2 ) ;
x2 |-- y2 iff P [ x2 , y2 ] or P [ x2 , y2 ] or P [ y2 , x2 ] ;
redefine pred x1 <> x2 means : Def4 : |. x1 - x2 .| > 0 & |. x1 - x2 .| > 0 ;
assume that p2 - p1 , p3 - p1 , p3 - p1 is_collinear and p2 - p3 , p3 - p1 - p3 - p1 is_collinear ;
set q = ( -1 f ) ^ <* 'not' A *> ;
let f be PartFunc of REAL-NS 1 , REAL-NS n , g be PartFunc of REAL-NS 1 , REAL-NS n , REAL n ;
( n mod ( 2 * k ) ) + 1 = n mod k ;
dom ( T * ( that ) ) = dom ( that dom ( T * ( ) ) ) ;
consider x being element such that x in wc iff x in c & x in c ;
assume ( F * G ) . v = v . x3 & ( F * G ) . x3 = v . x3 ;
assume that the Sorts of D1 c= the Sorts of D2 and the Sorts of D1 c= the Sorts of A and the Sorts of D1 c= the Sorts of A ;
reconsider A1 = [. a , b .[ , A2 = [. a , b .] as Subset of R^1 ;
consider y being element such that y in dom F and F . y = x ;
consider s being element such that s in dom o and a = o . s ;
set p = W-min L~ Cage ( C , n ) , q = W-min L~ Cage ( C , n ) , r = W-bound L~ Cage ( C , n ) ;
n1 - len f + 1 <= len - ( len f + 1 ) + 1 - len f ;
st ( q , O1 ) = [ u , v , a , b , c ] ;
set C-2 = ( ( `1 ) `1 ) + ( ( G ) `1 ) ;
Sum ( L * p ) = 0. R * Sum p .= 0. V * Sum p .= 0. V ;
consider i being element such that i in dom p and t = p . i ;
defpred Q [ Nat ] means 0 = Q ( $1 ) & $1 <= n implies P [ $1 ] ;
set s3 = Comput ( P1 , s1 , k ) , P3 = P1 +* I , s4 = P2 +* I , P4 = P3 ;
let l be [: of k , A , B :] , P be l of k , A ;
reconsider U1 = union G-24 as Subset-Family of ( T | A ) , B be Subset-Family of T ;
consider r such that r > 0 and Ball ( p `1 , r ) c= Q ` ;
( h | ( n + 2 ) ) /. ( i + 1 ) = p29 /. ( n + 2 ) ;
reconsider B = the carrier of X1 , C = the carrier of X2 as Subset of X ;
pthesis = <* - c , 1 , 1 *> .= <* - c , 1 , 1 *> ;
synonym f is complex-valued means : Def4 : rng f c= NAT & rng f c= NAT ;
consider b being element such that b in dom F and a = F . b ;
x20 < card X0 + card Y0 - card Y0 + card Y0 - card Y0 + card Y0 - 1 < card X0 + card Y0 - 1 ;
attr X c= B1 means : Def4 : for B holds X c= \mathop { B , X } ;
then w in Ball ( x , r ) & dist ( x , w ) <= r ;
angle ( x , y , z ) = angle ( x , y , z ) ;
redefine attr 1 <= len s means : Def4 : for x being Element of NAT holds ( the _ of s ) . x = s ;
fsqrt c= f . ( k + ( n + 1 ) ) ;
the carrier of { 1_ G } = { 1_ G } .= { 1_ G } .= { 1_ G } ;
redefine pred p '&' q in TAUT ( A ) & q in TAUT ( A ) implies p '&' q in TAUT ( A ) ;
- ( t `1 ) < ( t `1 ) ^2 / ( 1 - t `1 ) ^2 ;
U1 . 1 = U2 /. 1 .= U2 /. ( 1 + 1 ) .= U2 . ( 1 + 1 ) .= U2 /. 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 :] & [: Seg n , Seg n :] = [: Seg n , Seg n :] ;
for n being Element of NAT holds G . n c= G . ( n + 1 )
then V in M .: \square & ex x being Element of M st V = { x } ;
ex f being Element of F-9 st f is \cup ( A * ) & f is \setminus of A * ;
[ h . 0 , h . 3 ] in the InternalRel of G & [ h . 0 , h . 2 ] in the InternalRel of G ;
s +* Initialize ( ( intloc 0 ) .--> 1 ) = s3 +* ( 0 , 1 ) ;
|[ w1 , v1 ]| - |[ w1 , v1 ]| <> 0. TOP-REAL 2 & |[ w1 , v1 ]| - |[ w1 , v1 ]| = 0. TOP-REAL 2 ;
reconsider t = t as Element of ( INT , INT ) * ;
C \/ P c= [#] ( ( G | A ) \ A ) & C /\ P c= [#] ( G | A ) ;
f " V in ( the topology of X ) /\ D ( the carrier of X ) ;
x in [#] ( the carrier of A ) /\ A /\ delta ( F ) ;
g . x <= h1 . x & h . x <= h1 . x & h . x <= h . x ;
InputVertices S = { xy , y , z } .= { xy , y , z } \/ { z } ;
for n being Nat st P [ n ] holds P [ n + 1 ]
set R = Line ( M , i ) , a = Line ( M , i ) , b = Line ( M , i ) , c = Line ( M , i ) , d = Line ( M , i ) , e = Line ( M , i ) , e = Line ( M , i ) , e =
assume that M1 is being_line and M2 is being_line and M3 is being_line and M2 is being_line and M1 + M2 is being_line ;
reconsider a = f4 . ( i0 -' 1 ) , b = f4 . ( i0 -' 1 ) as Element of K ;
len B2 = Sum Len ( F1 ^ F2 ) .= len ( F1 ^ F2 ) + len ( F2 ^ F1 ) ;
len ( ( the
dom ( ( f + g ) | X ) = dom ( f + g ) /\ X ;
( the Sorts of seq ) . n = upper_bound Y1 & ( the Sorts of seq ) . n = upper_bound Y1 ;
dom ( p1 ^ p2 ) = dom ( f ^ p1 ) .= dom ( f ^ p2 ) .= dom ( f ^ p1 ) ;
M . [ 1 / y , y ] = 1 / ( 1 * v1 ) * v1 .= 1 / 1 * v1 .= 1 ;
assume that W is non trivial and W { v } c= the carrier' of G2 and W is not trivial ;
C6 /. i1 = G1 * ( i1 , i2 ) .= G1 * ( i1 , i2 ) .= G1 * ( i1 , i2 ) ;
C8 |- 'not' Ex ( x , p ) 'or' p . ( x , y ) ;
for b st b in rng g holds inf rng f-0 <= b * ( upper_bound rng f-0 )
- ( ( q1 `1 ) / |. q1 .| ) = 1 / ( |. q1 .| ) .= ( |. q1 .| ) / |. q1 .| ;
( LSeg ( c , m ) \/ { l } ) \/ LSeg ( l , k ) c= R ;
consider p be element such that p in LSeg ( x , p ) and p in L~ f and x = f . p ;
Indices ( X @ ) = [: Seg n , Seg n :] & [: Seg n , Seg n :] = [: Seg n , Seg n :] ;
cluster s => ( q => p ) -> valid ;
Im ( ( Partial_Sums F ) . m ) is_measurable_on E & Im ( ( Partial_Sums F ) . m ) is_measurable_on E ;
cluster f . ( x1 , x2 ) -> Element of D ( ) * ;
consider g being Function such that g = F . t and Q [ t , g ] ;
p in LSeg ( ( Int Z ) , ( implies ex g being Function of Z , TOP-REAL 2 ) | Z ) . p = ( Z + 1 ) * ( g + f )
set R8 = R / ( ]. b , +infty .[ ) , R8 = R / ( ]. b , +infty .[ ) ;
IncAddr ( I , k ) = SubFrom ( da , db ) .= SubFrom ( da , db ) .= ( n + k ) + k ;
seq . m <= ( ( the Sorts of A ) * ( seq ^\ k ) ) . n ;
a + b = ( a ` *' b ) ` .= ( a ` *' b ) ` .= ( a ` *' b ) ` ;
id ( X /\ Y ) = id ( X /\ Y ) /\ id ( X /\ Y ) ;
for x being element st x in dom h holds h . x = f . x
reconsider H = U1 \/ U2 , U1 = U2 as non empty Subset of ( the carrier of U0 ) * ;
u in ( ( c /\ ( ( d /\ e ) /\ f ) ) /\ j ;
consider y being element such that y in Y and P [ y , inf B ] ;
consider A being finite stable set of R such that card A = ( the carrier of R ) and A is finite ;
p2 in rng ( f |-- p1 ) \ rng <* p1 *> & p2 in rng ( f |-- p1 ) \ { p2 } ;
len s1 - 1 > 0 & len s2 - 1 > 0 & len s2 - 1 > 0 ;
( N-min ( P ) ) `2 = ( N-min ( P ) ) `2 .= ( E-max ( P ) ) `2 ;
Ball ( e , r ) c= LeftComp Cage ( C , k + 1 ) /\ L~ Cage ( C , k + 1 ) ;
f . a1 ` = f . a1 ` .= ( f . a1 ) ` .= ( f . a1 ) ` ;
( seq ^\ k ) . n in ]. x0 , x0 + r .[ & ( seq ^\ k ) . n in ]. x0 , x0 + r .[ ;
gg . s0 = g . s0 | G . ( s0 . s0 ) .= g . ( s0 . s0 ) ;
the InternalRel of S is \lbrace the carrier of S , the carrier of S , the carrier of S , the carrier of S #) ;
deffunc F ( Ordinal , Ordinal ) = phi . ( $1 , $2 ) ;
F . s1 . a1 = F . s2 . a1 .= F . s2 . a1 .= ( F . s2 ) . a1 ;
x `2 = A . o . a .= Den ( o , A . a ) . a ;
Cl ( f " P1 ) c= f " ( Cl P1 ) & Cl ( f " P1 ) c= f " P1 ;
FinMeetCl ( ( the topology of S ) . i ) c= the topology of T & FinMeetCl ( ( the topology of T ) . i ) c= the topology of T ;
synonym o is \bf means : Def4 : o <> \ast & o <> {} & o <> {} ;
assume that X |^ k = Y |^ k + card X and card X <> card Y and X <> Y ;
the { F } <= 1 + ( the { F } ) . ( len F + 1 ) & ( the { F } ) . ( len F + 1 ) = F . ( len F + 1 ) ;
LIN a , a1 , d or b , c // b1 , c1 or a , c // b1 , c1 ;
e . 1 = 0 & e . 2 = 1 & e . 3 = 0 & e . 4 = 0 ;
E in SX1 & not E in { NX2 } & not E in { NX2 } ;
set J = ( l , u ) If = l , u = u , v = v , w = u , u = v , v = u , w = v , u = v , v = u , w = v , u = v , v = u , w = v , u = v , v = u , w = v , u = v
set A1 = .| ( ( a , b , c ) , d ) , A2 = ( a , b , c ) , A2 = ( a , b , c ) ;
set c9 = [ <* c , 8 *> , '&' ] , d9 = [ <* d , c *> , '&' ] , A1 = [ <* c , 8 *> , '&' ] , A2 = [ <* d , c *> , '&' ] , p3 = [ <* c , 8 *> , '&' ] , A2 = [ <* c , 8 *> , '&' ] , R =
x * z `1 * x " in x * ( z * N ) * x " ;
for x being element st x in dom f holds f . x = g3 . x & f . x = f . x
Int cell ( f , 1 , G ) c= RightComp f \/ RightComp f \/ L~ f \/ L~ f \/ L~ f ;
U2 is_an_arc_of W-min ( C ) , E-max ( C ) , E-max ( C ) , E-max ( C ) ;
set f-17 = f .: @ g , f-17 = f .: @ g ;
attr S1 is convergent means : Def4 : S2 is convergent & ( for n holds ( n - 1 ) * ( S2 . n ) is convergent & lim ( S2 . n ) = x0 ;
f . ( 0 + 1 ) = ( 0 qua Ordinal ) + a .= a + ( 0 qua Ordinal ) .= a ;
cluster -> \in \mathclose \mathclose Let reflexive transitive transitive transitive transitive for reflexive transitive transitive RelStr , F be Function of F , F ;
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 ) & not b in dom Start-At ( ( card I + 2 ) , SCMPDS ) ;
( z + a ) + x = z + ( a + y ) .= z + a + y ;
len ( l \lbrack a , b .] ) = len l & len ( l (#) x ) = len l ;
t4 \+\ {} is ( {} \/ rng t4 ) -valued FinSequence of ( {} \/ rng t4 ) * * -valued Function ;
t = <* F . t *> ^ ( C . p ^ q ) .= ( C . t ^ q ) ^ ( C . t ) ;
set p-2 = W-min L~ Cage ( C , n ) , p-2 = W-min L~ Cage ( C , n ) , p`1 = W-bound L~ Cage ( C , n ) ;
( k -' ( i + 1 ) ) = ( k - ( i + 1 ) ) + ( i - ( i + 1 ) ) ;
consider u ` being Element of L such that u = u ` ` and u in D ` ;
len ( ( width ( ( a |-> b ) ) |-> a ) ) = width ( a |-> b ) .= len ( a |-> b ) ;
FF . x in dom ( ( G * the_arity_of o ) . x ) ;
set H2 = the carrier of H2 , H1 = the carrier of H2 , H2 = the carrier of H2 ;
set H1 = the carrier of H1 , H2 = the carrier of H2 , P = the carrier of H2 ;
( Comput ( P , s , 6 ) ) . intpos ( m + 6 ) = s . intpos ( m + 6 ) ;
IC Comput ( Q8 , t , k ) = ( l + 1 ) - ( k + 1 ) ;
dom ( ( cos * sin ) `| Z ) = REAL & dom ( cos * sin ) = dom ( cos * cos ) /\ dom cos ;
cluster <* l *> ^ phi -> ( 1 + 1 ) -element for string of S ;
set b5 = [ <* thesis , A1 *> , <* y , z *> , '&' ] , b5 = [ <* z , x *> , '&' ] ;
Line ( Segm ( M `1 , P , Q ) , x ) = L * ( Sgm Q ) ;
n in dom ( ( the Sorts of A ) * the_arity_of o ) & n in dom ( ( the Sorts of A ) * the_arity_of o ) ;
cluster f1 + f2 -> continuous for PartFunc of REAL , the carrier of S , the carrier of T ;
consider y be Point of X such that a = y and ||. x - y .|| <= r ;
set x3 = t . DataLoc ( s2 . SBP , 2 ) , x4 = t . DataLoc ( s2 . SBP , 2 ) , 6 = t . SBP , 7 = t . SBP , 8 = t . SBP , 8 = t . SBP , 8 = t . SBP , 8 = t . SBP , 8 = t . SBP ,
set p-3 = stop I ( ) , pE = stop I ( ) ;
consider a being Point of D2 such that a in W1 and b = g . a and a in W2 ;
{ A , B , C , D , E } = { A , B } \/ { C , D , E , F , J }
let A , B , C , D , E , F , J , M , N , F , J , M , N , N , F , J , M , N , N , A , J , M , N , A , N , A , J , M , N , A , N , A , J , M ;
|. p2 .| ^2 - ( p2 `2 ) ^2 - ( p2 `2 ) ^2 >= 0 ;
l -' 1 + 1 = n-1 * ( ( m + 1 ) + ( 1 + 1 ) ) ;
x = v + ( a * w1 + b * w2 ) + ( c * w2 ) + ( c * w2 ) + ( c * w2 ) + ( c * w2 ) + ( c * w2 ) = v + ( c * w2 ) ;
the TopStruct of L = reconsider the TopStruct of L , the TopStruct of L = the TopStruct of L , the TopStruct of L = the TopStruct of L ;
consider y being element such that y in dom H1 and x = H1 . y and y in dom H1 and x = H1 . y ;
fv \ { n } = ( Free All ( v1 , H ) ) \ { n } .= ( Free ( v1 , H ) ) \ { n } ;
for Y being Subset of X st Y is summable holds Y is iff Y is non empty & X is non empty
2 * n in { N : 2 * Sum ( p | N ) = N & N > 0 } ;
for s being FinSequence holds len ( the { of A } ) = len s & for i being Nat st i in dom s holds s . i = ( the { of A } ) . i
for x st x in Z holds exp_R * f is_differentiable_in x & exp_R * f is_differentiable_in x ;
rng ( h2 * f2 ) c= the carrier of ( ( TOP-REAL 2 ) | ( [#] ( TOP-REAL 2 ) ) ) | K1 ;
j + - len f <= len f + ( len g - len f ) - len f + ( len g - len f ) ;
reconsider R1 = R * I as PartFunc of REAL n , REAL-NS n , REAL-NS n ;
C8 . x = s1 . ( a - 1 ) .= C8 . x - 1 .= C8 . x - 1 ;
power ( F_Complex ) . ( z , n ) = 1 .= ( x |^ n ) |^ ( z , n ) .= x |^ n ;
t at ( C , s ) = f . ( the connectives of S ) . t & t . ( t , I ) = s . ( t , I ) ;
support ( f + g ) c= support f \/ ( support g ) /\ support ( f + g ) ;
ex N st N = j1 & 2 * Sum ( ( r | N ) | N ) > N & N > 0 ;
for y , p st P [ p ] holds P [ All ( y , p ) ]
{ [ x1 , x2 ] } is Subset of [: X1 , X2 :] & { [ x1 , x2 ] } is Subset of [: X2 , Y2 :] ;
h = ( i , j ) |-- id ( B , i ) .= H . i .= H . i ;
ex x1 being Element of G st x1 = x & x1 * N c= A & N c= A & N c= A ;
set X = ( ( st ( q , O1 ) ) `1 , ( q , O2 ) ) `1 , X = ( ( q , O1 ) `1 , 4 = ( q , O1 ) `1 , 5 = ( q , O1 ) `1 , 5 = ( q , O1 ) `1 , 6 = ( q , O1 ) `1 , 5 = ( q , O1 ) `1 , 6 = ( q , O1 ) `1 ,
b . n in { g1 : x0 < g1 & g1 < x0 + r } ;
f /* s1 is convergent & f /. x0 = lim ( f /* s1 ) implies f /* s1 is convergent & lim ( f /* s1 ) = x0
the lattice of the lattice of Y = the lattice of the lattice of the topology of Y & the carrier of X = the carrier of X & the carrier of X = the carrier of Y ;
'not' ( a . x ) '&' b . x 'or' a . x '&' 'not' ( b . x ) = FALSE ;
2 = len ( ( q ^ r1 ) ^ ( p ^ r1 ) ) + len ( ( q ^ r1 ) ^ ( p ^ r1 ) ) ;
( 1 / a ) (#) ( sec * f1 ) - id Z is_differentiable_on Z ;
set K1 = integral ( ( lim H ) || ( A , H ) ) , D2 = ( ( lim H ) || A ) , K1 = ( lim H ) || A ;
assume e in { ( w1 - w2 ) : w1 in F & w2 in G & w1 - w2 in G } ;
reconsider d7 = dom a `1 , d8 = dom F `1 , d8 = F `1 as finite set ;
LSeg ( f /^ q , j ) = LSeg ( f , j ) /\ LSeg ( f , q ) .= LSeg ( f , j + q .. f ) ;
assume that X in { T . ( N2 , K1 ) : h . N2 = N2 } and X in { N2 } ;
assume that Hom ( d , c ) <> {} and <* f , g *> * f1 = <* f , g *> * f2 ;
dom S29 = dom S /\ Seg n .= dom L6 /\ Seg n .= Seg n /\ Seg n .= Seg n /\ Seg n ;
x in H |^ a implies ex g st x = g |^ a & g in H & a in H
* ( ( 0 , 1 ) --> ( a , 1 ) ) = a `1 - ( 0 * n ) .= a `1 ;
D2 . ( j - 1 ) in { r : lower_bound A <= r & r <= upper_bound A } ;
ex p being Point of TOP-REAL 2 st p = x & P [ p ] & p `1 = x & p `2 <= 0 ;
for c holds f . c <= g . c implies f ^ @ c <= g @ c
dom ( f1 (#) f2 ) /\ X c= dom ( f1 (#) f2 ) /\ X /\ dom ( f2 (#) f1 ) ;
1 = ( p * p ) / p .= p * ( 1 / p ) .= 1 / p ;
len g = len f + len <* x + y *> .= len f + 1 + 1 .= len f + 1 + 1 ;
dom F-11 = dom ( F | [: N1 , S :] ) .= [: N1 , S :] ;
dom ( f . t * I . t ) = dom ( f . t * g . t ) ;
assume a in ( "\/" ( ( T |^ the carrier of S ) , F ) ) .: D ;
assume that g is one-to-one and ( the carrier' of S ) /\ rng g c= dom g and g is one-to-one and g is one-to-one ;
( ( x \ y ) \ z ) \ ( ( x \ z ) \ ( y \ z ) ) = 0. X ;
consider f such that f * f = id b and f * f = id b and f * g = id b and f = id b ;
( cos | [. 2 * PI , 0 + PI / 2 .] ) | [. PI , PI / 2 .] is increasing
Index ( p , co ) <= len LS - Index ( Gij , LS ) + 1 - Index ( Gij , LS ) + 1 - 1 ;
t1 , t2 , t be Element of ( the Sorts of A ) . s , t2 be Element of ( the Sorts of A ) . s ;
"/\" ( ( Frege curry H ) . h , L ) <= "/\" ( ( Frege ( curry G ) . h ) , L ) ;
then P [ f . i0 ] & F ( f . i0 ) < j & F ( f . i0 ) < j ;
Q [ ( D . x ) `1 , F . [ D . x , 1 ] ] ;
consider x being element such that x in dom ( F . s ) and y = ( F . s ) . x ;
l . i < r . i & [ l . i , r . i ] is for of G . i holds l . i is a < r . i
the Sorts of A2 = ( the carrier of S2 ) --> ( the carrier of S2 ) .= ( the Sorts of A1 ) +* ( the Sorts of A2 ) ;
consider s being Function such that s is one-to-one and dom s = NAT & rng s = F . 0 and rng s c= NAT ;
dist ( b1 , b2 ) <= dist ( b1 , a ) + dist ( a , b2 ) + dist ( b2 , a ) ;
( Lower_Seq ( C , n ) ) /. len ( Cage ( C , n ) ) = W /. len ( Cage ( C , n ) ) ;
q `2 <= ( UMP Upper_Arc L~ Cage ( C , 1 ) ) `2 & q `2 <= ( UMP L~ Cage ( C , 1 ) ) `2 ;
LSeg ( f | i2 , i ) /\ LSeg ( f | i2 , j ) = {} implies LSeg ( f | i2 , i ) = {}
given a being ExtReal such that a <= II and A = ]. a , I .[ and a < I ;
consider a , b be Complex such that z = a & y = b and z + y = a + b ;
set X = { b |^ n where n is Element of NAT : n <= k & k <= n } ;
( ( x * y * z ) \ x ) \ z = 0. X ;
set xy = [ <* xy , y , z *> , f1 ] , yz = [ <* y , z *> , f2 ] , yz = [ <* z , x *> , f3 ] , yz = [ <* x , y *> , f3 ] , yz = [ <* z , x *> , f3 ] , yz = [ <* y , z *> , f3 ] , xy = [ <* z ,
lv /. len lv = ( l . ( len lv ) ) * ( l . ( len l ) ) .= ( l . ( len l ) ) * ( l . ( len l ) ) ;
( ( q `2 ) / |. q .| - sn ) / ( 1 + sn ) = 1 ;
( ( p `2 / |. p .| - sn ) / ( 1 - sn ) ) / ( 1 - sn ) < 1 ;
( ( ( ( S \/ Y ) \/ X ) \/ Y ) \/ X ) `2 = ( ( ( S \/ Y ) \/ X ) \/ Y ) `2 ;
( seq - seq ) . k = seq . k - ( seq . k - seq . k ) .= ( seq ^\ k ) . k - seq . k ;
rng ( ( h + c ) ^\ n ) c= dom SVF1 ( 1 , f , u0 ) /\ dom SVF1 ( 1 , f , u0 ) ;
the carrier of X is the carrier of X0 & the carrier of X0 = the carrier of X & the carrier of X = the carrier of X0 ;
ex p4 st p3 = p4 & |. p4 - |[ a , b ]| .| = r & |. p4 - |[ a , b ]| .| = r ;
set h = chi ( X , A ) , A = chi ( X , A ) , B = chi ( X , A ) ;
R |^ ( 0 * n ) = I\HM ( X , X ) .= R |^ n |^ 0 .= R |^ 0 ;
( Partial_Sums ( curry ( F1 , n ) ) ) . n is nonnegative & ( Partial_Sums ( F2 , n ) ) . n is nonnegative ;
f2 = C7 . ( ( the EESet of V , len H ) . ( len H ) ) .= H . ( len H ) ;
S1 . b = s1 . b .= S2 . b .= S2 . b .= S2 . b .= S2 . b .= S2 . b ;
p2 in LSeg ( p2 , p1 ) /\ LSeg ( p2 , p11 ) /\ LSeg ( p1 , p11 ) c= { p2 } /\ LSeg ( p11 , p2 ) ;
dom ( f . t ) = Seg n & dom ( I . t ) = Seg n & rng ( I . t ) = Seg n ;
assume o = ( the connectives of S ) . 11 & o in ( the carrier' of S ) . 11 ;
set phi = ( l1 , l2 ) <> ( l1 , l2 ) . ( l + 1 ) , phi = ( l1 , l2 ) . ( l + 1 ) , phi = ( l1 , l2 ) . ( l + 1 ) , phi = ( l1 , l2 ) . ( l + 1 ) , phi = ( l1 , l2 ) . ( l + 1 ) ;
synonym p is is invertible for p , q be Polynomial of n , L means : Def4 : p = 1 & q = 1 & p = q ;
( Y1 = - 1 & Y2 <> {} & Y1 <> {} & Y2 <> {} & Y1 <> {} & Y2 <> {} implies Y1 = {} ) implies ( Y1 = Y2 )
defpred X [ Nat , set , set ] means P [ $1 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 ] ;
consider k being Nat such that for n being Nat st k <= n holds s . n < x0 + g / 2 ;
Det ( I |^ ( m -' n ) ) * ( m - n ) = 1. K * ( - n ) .= ( - 1_ K ) * ( - n ) ;
( - b - sqrt ( b ^2 - ( 4 * a * c ) ) ) / 2 < 0 ;
Cs . d = Cs . ( ( d mod 2 ) + 1 ) mod Cs . ( d mod 2 ) .= Cs . d mod 2 ;
attr X1 is dense dense means : Def4 : X2 is dense dense & X1 /\ X2 is dense implies X1 /\ X2 is dense ;
deffunc FF ( Element of E , Element of I , Element of I ) = $1 * $2 & $2 = $1 * $2 ;
t ^ <* n *> in { t ^ <* i *> : Q [ i , T . t ] } ;
( x \ y ) \ x = ( x \ x ) \ y .= y ` \ 0. X .= 0. X ;
for X being non empty set for Y being Subset-Family of X holds X is Basis of <* X , \subseteq the topology of X , the topology of Y *>
synonym A , B are_separated means : Def4 : ( Cl A ) misses B & ( Cl B ) misses Cl ( Cl B ) ;
len M8 = len p & width M8 = width M & width M8 = width M & width M8 = width M & width M8 = width M ;
J . v = { x where x is Element of K : 0 < v . x & x < 1 } ;
( Sgm ( Seg m ) ) . d - ( Sgm ( Seg m ) ) . e <> 0 ;
inf divset ( D2 , k + k2 ) = D2 . ( k + k2 - 1 ) .= D2 . ( k + k2 - 1 ) ;
g . r1 = - 2 * r1 + 1 & dom h = [. 0 , 1 .] & rng h c= [. 0 , 1 .] ;
|. a .| * ||. f .|| = 0 * ||. f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| ;
f . x = ( h . x ) `1 & g . x = ( h . x ) `2 & h . x = ( h . x ) `2 ;
ex w st w in dom B1 & <* 1 *> ^ s = <* 1 *> ^ w & <* 1 *> ^ s = <* 1 *> ^ w ;
[ 1 , {} , <* d1 *> ] in ( { [ 0 , {} ] } \/ S1 ) \/ S2 ;
IC Exec ( i , s1 ) + n = IC Exec ( i , s2 ) + n .= ( 0 + n ) + n ;
IC Comput ( P , s , 1 ) = IC ( s , 9 ) .= 5 + 9 .= 5 + 9 .= ( card I + 9 ) ;
( IExec ( W6 , Q , t ) ) . intpos ( 8 + 1 ) = t . intpos ( 8 + 1 ) ;
LSeg ( f /^ q , i ) misses LSeg ( f /^ q , j ) & LSeg ( f /^ q , i ) misses LSeg ( f /^ q , j ) ;
assume for x , y being Element of L st x in C holds x <= y or y <= x or x <= y ;
integral ( f , C ) = f . ( upper_bound X ) - f . ( lower_bound X ) .= f . ( upper_bound X ) - f . ( lower_bound X ) ;
for F , G being one-to-one FinSequence st rng F misses rng G holds F ^ G is one-to-one & F ^ G is one-to-one & F ^ G is one-to-one
||. R /. ( L . h ) .|| < e1 * ( K + 1 * ||. h .|| ) ;
assume a in { q where q is Element of M : dist ( z , q ) <= r } ;
set p4 = [ 2 , 1 ] .--> [ 2 , 0 , 1 ] ;
consider x , y being Subset of X such that [ x , y ] in F and x c= d and y c= d ;
for y , x being Element of REAL st y ` in Y & x in X ` holds y ` <= x ` ` + y ` ;
func |. p .| -> variable of A , NAT equals min ( NBI , p ) . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( 1 - 1 ) ) ) ) ) ) ) ) ) ;
consider t `1 being Element of S such that x `1 , y `2 '||' z `1 , t `2 and x `1 , z `2 '||' y `1 , t `2 ;
dom x1 = Seg len x1 & len x1 = len x2 & len y1 = len y1 & len y1 = len y1 & len y1 = len y1 & len y1 = len y2 ;
consider y2 being Real such that x2 = y2 and 0 <= y2 and y2 < 1 / 2 and y2 <= 1 / 2 ;
||. f | X /* s1 .|| = ||. f /* s1 .|| & ||. f /* s1 .|| = ||. f /* s1 .|| ;
( the InternalRel of A ) ` ` /\ Y = {} \/ {} .= {} \/ {} .= {} \/ {} .= {} .= {} .= {} ;
assume that i in dom p and for j being Nat st j in dom q holds P [ i , j ] and i + 1 in dom p and i + 1 in dom p and p . i = p . j ;
reconsider h = f | X ( ) as Function of X ( ) , rng h ( ) , rng h ( ) ;
u1 in the carrier of W1 & u2 in the carrier of W2 & v = ( the carrier of W1 ) + ( the carrier of W2 ) + ( the carrier of W1 ) + ( the carrier of W2 ) = the carrier of W1 + W2 ;
defpred P [ Element of L ] means M <= f . $1 & f . $1 <= f . $1 & f . $1 <= f . $1 ;
l . ( u , a , v ) = s * x + ( - ( s * x ) + y ) .= b ;
- ( x-y ) = - x + - y .= - x + - y .= - x + - y .= x + - y ;
given a being Point of GX such that for x being Point of GX holds a , x + a are_<= x and a , x + b are_
fSet = [ [ [ dom ( f ^ f2 ) , cod ( f ^ g2 ) ] , cod ( f ^ g2 ) ] , cod ( f ^ g2 ) ] ;
for k , n being Nat st k <> 0 & k < n & n is prime & k , n are_relative_prime holds k , n are_relative_prime & k , n are_relative_prime
for x being element holds x in A |^ d iff x in ( ( A ` ) |^ d ) ` & x in ( A ` ) `
consider u , v being Element of R , a being Element of A such that l /. i = u * a and a in A ;
- ( ( p `1 / |. p .| - cn ) / ( 1 + cn ) ) > 0 ;
L-13 . k = LF . ( F . k ) & F . k in dom LF & F . k in dom LF ;
set i2 = SubFrom ( a , i , - n ) , i1 = goto - n ;
attr B is \frac means : Def4 : Subthat Subthat [ B , SZ ] = ( B ) `1 & S = ( B ) `1 ;
a9 "/\" D = { a "/\" d where d is Element of N : d in D & d in D } ;
|( \square , q29 - ( q - q1 ) * |( REAL , q29 )| >= |( \square , 1 )| * |( q , q1 )| ;
( - f ) . ( upper_bound A ) = ( ( - f ) | A ) . ( upper_bound A ) .= ( - f ) . ( upper_bound A ) ;
G * ( len G , k ) = G * ( len G , k ) .= G * ( len G , k ) .= G * ( len G , k ) ;
( Proj ( i , n ) ) . LM = <* ( proj ( i , n ) ) . ( LM ) . ( LM . x ) *> ;
f1 + f2 * reproj ( i , x ) is_differentiable_in ( ( - 1 ) (#) ( reproj ( i , x ) ) ) . ( x + x0 ) ;
redefine func ( cos . x ) <> 0 & ( tan . x ) <> 0 & ( tan . x ) <> 0 & ( tan . x ) <> 0 ;
ex t being SortSymbol of S st t = s & h1 . t = h2 . ( t . x ) & ( for x being set st x in dom h1 holds h1 . x = F ( x ) ) ;
defpred C [ Nat ] means P8 . $1 is non empty & ( A is non empty implies A is non empty or A is empty ) & ( A is non empty implies A is non empty or A is non empty ) ;
consider y being element such that y in dom ( p | i ) and ( q | i ) . y = ( p | i ) . y ;
reconsider L = product ( { x1 } +* ( index B , l ) ) as Basis of ( A . ( index B ) ) ;
for c being Element of C ex d being Element of D st T . ( id c ) = id d & for c being Element of C st c in dom T holds T . ( c , d ) = id d
( for f , n , p being FinSequence holds f = ( f | n ) ^ <* p *> .= f ^ <* p *> ^ <* p *> ;
( f * g ) . x = f . ( g . x ) & ( f * h ) . x = f . ( h . x ) ;
p in { 1 / 2 * ( G * ( i + 1 , j ) + G * ( i + 1 , j + 1 ) ) } ;
f `2 - p = ( f | ( n , L ) ) *' - ( f *' ) .= ( f *' ) - ( f *' ) ;
consider r being Real such that r in rng ( f | divset ( D , j ) ) and r < m + s ;
f1 . ( |[ ( r . 8 ) `1 , ( r . 8 ) `2 ]| ) in f1 .: W1 & f2 . ( |[ r . 8 , ( r . 8 ) `2 ]| in W2 .: W2 ;
eval ( a | ( n , L ) , x ) = eval ( a | ( n , L ) ) .= a * ( x | n ) .= a * x ;
z = DigA ( tx , x9 ) .= DigA ( ty , x ) .= DigA ( ty , x ) .= DigA ( ty , x ) ;
set H = { Intersect S where S is Subset-Family of X : S c= G & S c= G } , F = { Intersect S where S is Subset-Family of X : S c= G } ;
consider S19 being Element of D such that S ` = S19 ^ <* d *> and S = S19 ^ <* d *> ;
assume that x1 in dom f and x2 in dom f and f . x1 = f . x2 and f . x2 = f . x2 and f . x3 = f . x3 ;
- 1 <= ( q `2 / |. q .| - sn ) / ( 1 + sn ) ;
0. ( V ) is Linear_Combination of A & Sum ( L ) = 0. V implies Sum ( L ) = 0. V & Sum ( L ) = 0. V
let k1 , k2 , k2 , x4 , k2 , 6 , 7 , 8 , 8 , 6 , 8 , 7 , 8 be Element of NAT , p be FinSequence of NAT ;
consider j being element such that j in dom a and j in g " { k `2 } and x = a . j and j in dom a ;
H1 . x1 c= H1 . x2 or H1 . x2 c= H1 . x2 or H1 . x1 c= H1 . x2 or H1 . x2 c= H1 . x2 & H1 . x2 c= H1 . x3 ;
consider a being Real such that p = \hbox p1 * p1 + ( a * p2 ) and 0 <= a and a <= 1 ;
assume that a <= c & c <= d and [' a , b '] c= dom f and [' a , b '] c= dom g ;
cell ( Gauge ( C , m ) , 1 , width Gauge ( C , m ) -' 1 , 0 ) is non empty ;
A5 in { ( S . i ) `1 where i is Element of NAT : i <= n & n <= len S } ;
( T * b1 ) . y = L * b2 /. y .= ( F `1 * b1 ) . y .= ( F `1 * b2 ) . y ;
g . ( s , I ) . x = s . y & g . ( s , I ) . y = |. s . x - s . y .| ;
( log ( 2 , k + k ) ) ^2 >= ( log ( 2 , k + 1 ) ) ^2 / ( 2 * k ) ;
then p => q in S & not x in the carrier of S & not p => All ( x , q ) in S & not x in S ;
dom ( the InitS of rn ) misses dom ( the InitS of rn ) & dom ( the InitS of rn ) misses dom ( the InitS of rn ) ;
synonym f is extended real means : Def4 : 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 p3 + len <* x *> .= len p3 + 1 + 1 .= len p3 + 1 + 1 ;
( l ) `1 = ( g ) `1 ) `1 + ( k ) `1 .= ( g ) `1 + ( k ) - ( k ) .= ( g ) `1 + ( k - 1 ) ;
CurInstr ( P2 , Comput ( P2 , s2 , l ) ) = halt SCM+FSA .= CurInstr ( P2 , Comput ( P2 , s2 , l ) ) .= halt SCM+FSA ;
assume for n be Nat holds ||. seq . n .|| <= ( R . n ) & ( R . n ) is summable & ( R . n ) is summable & ( R . n ) is summable ;
sin . ( PI ) = sin . ( PI ) * cos . ( PI ) .= cos . ( PI ) * sin . ( PI ) .= 0 ;
set q = |[ g1 . t `1 , g2 . t `2 , g1 . t `2 ]| , g2 = |[ g2 . t `1 , g2 . t `2 ]| , g1 = |[ g2 , g2 ]| ;
consider G being sequence of S such that for n being Element of NAT holds G . n in implies G . n in implies G . n = ( G . n ) . n ;
consider G such that F = G and ex G1 st G1 in SM & G = [: G1 , G2 :] & ( for G being strict Subgroup of M st G in SM holds G is open ) ;
the root of [ x , s ] in ( the Sorts of Free ( C , X ) ) . s & t in ( the Sorts of Free ( C , X ) ) . s ;
Z c= dom ( exp_R * ( f + ( exp_R * f1 ) ) ) /\ dom ( exp_R * ( f + f1 ) ) ;
for k be Element of NAT holds seq1 . k = ( ( Im ( Im ( f ) ) ) . k ) * ( ( Im ( f ) ) . k )
assume that - 1 < n and n > 0 and ( q `2 ) > 0 and ( q `1 ) < 0 and ( q `1 ) < 0 and q `1 < 0 ;
assume that f is continuous and a < b and f is continuous and c < d and f . a = c and f . b = d and f . c = d ;
consider r being Element of NAT such that sd = Comput ( P1 , s1 , r ) and r <= q and r <= q ;
LE f /. ( i + 1 ) , f /. j , L~ f , f /. 1 , f /. ( len f + 1 ) , f /. len f ;
assume that x in the carrier of K and y in the carrier of K and inf { x , y } = inf { x , y } and x <= y ;
assume that f +* ( i1 , \xi ) in ( proj ( F , i2 ) ) " ( A ) and f . ( i1 + 1 ) in ( proj ( F , i2 ) ) .: A ;
rng ( ( ( ( Flow M ) ~ | ( the carrier of M ) ) | ( the carrier' of M ) ) ) c= the carrier' of M ;
assume z in { ( the carrier of G ) \ { t } where t is Element of T : t in the carrier of T } ;
consider l be Nat such that for m be Nat st l <= m holds ||. s1 . m - x0 .|| < g / 2 ;
consider t be VECTOR of product G such that mt = ||. D5 . t .|| and ||. t .|| <= 1 / 2 * ||. t .|| ;
assume that the ] that the ] degree degree v = 2 and v ^ <* 0 *> in dom p and v ^ <* 1 *> in dom p and p . 1 = v ;
consider a being Element of the Points of [ X , A ] , A being Element of the Points of [ X , A ] such that a on A and b on A ;
( - x ) |^ ( k + 1 ) * ( ( - x ) |^ k ) " = 1 / ( ( - x ) |^ k ) " ;
for D being set st for i st i in dom p holds p . i in D holds p . i is FinSequence of D & p . i = p . i
defpred R [ element ] means ex x , y st [ x , y ] = $1 & P [ x , y ] & P [ y ] & P [ x ] ;
L~ f2 = union { LSeg ( p0 , p2 ) , LSeg ( p10 , p2 ) } .= { p2 } \/ { p2 } .= { p2 } \/ { p2 } ;
i - len h11 + 2 - 1 < i - len h11 + 2 - 1 + 2 - 1 + 1 - 1 + 2 - 1 + 2 - 1 + 2 - 1 + 2 - 1 < i + 2 - 1 ;
for n being Element of NAT st n in dom F holds F . n = |. ( nX . ( n -' 1 ) ) .| ;
for r , s1 , s2 , s3 being Real holds r in [. s1 , s2 .] iff s1 <= r & r <= s2 & s1 <= s2 & s2 <= 1
assume v in { G where G is Subset of T2 : G in B2 & G c= z & G c= z & z in G & G c= z } ;
let g be \vert succ A , Z be Element of INT , b be Element of INT , c be Element of Z ;
min ( g . [ x , y ] , k ) = ( min ( g , k , x , z ) ) . y ;
consider q1 being sequence of CH such that for n holds P [ n , q1 . n ] and q1 . n = f . ( n + 1 ) ;
consider f being Function such that dom f = NAT & for n being Element of NAT holds f . n = F ( n ) and for x being Element of NAT holds f . x = F ( n ) ;
reconsider B-6 = B /\ O , OO = O , $ Z = O , Z = O , Z = O as Subset of B ;
consider j being Element of NAT such that x = the j j j and 1 <= j and j <= n and f . j = f . j ;
consider x such that z = x and card ( x . O2 ) in card ( x . O2 ) and x in L1 . O2 and x in L2 . O2 ;
( C * _ T4 ( k , n2 ) ) . 0 = C . ( ( ( of T4 ( k , n2 ) ) . 0 ) ) .= C . ( ( k + 1 ) + 1 ) ;
dom ( X --> rng f ) = X & dom ( X --> f ) = dom ( X --> f ) & dom ( X --> f ) = X --> f ;
( ( SpStSeq L~ SpStSeq C ) `2 <= ( ( SpStSeq L~ SpStSeq C ) `2 ) / ( 2 |^ ( ( ( L~ SpStSeq C ) `2 ) + ( S-bound L~ SpStSeq C ) `2 ) ) ;
synonym x , y are_collinear means : Def4 : x = y or ex l being Nat st { x , y } c= l & x in l ;
consider X be element such that X in dom ( f | ( n + 1 ) ) and ( f | ( n + 1 ) ) . X = Y ;
assume that Im k is continuous and for x , y being Element of L , a , b being Element of Im k st a = x & b = y holds x << b and a << b ;
( 1 / 2 * ( ( ( ( - 1 / 2 ) * ( ( ( ( #Z n ) * ( f1 + #Z n ) ) / ( 1 + 1 ) ) ) ) ) ) ) is_differentiable_on REAL ;
defpred P [ Element of omega ] means ( the partial of A1 ) . $1 = A1 . $1 & ( the partial of A2 ) . $1 = A2 . $1 ;
IC Comput ( P , s , 2 ) = succ IC Comput ( P , s , 1 ) .= ( 0 + 1 ) .= 6 + 1 .= 6 ;
f . x = f . g1 * f . g2 .= f . g1 * 1_ H .= f . g1 * 1_ H .= ( f . g1 ) * ( f . g2 ) .= ( f . g1 ) * ( f . g2 ) .= ( f . g1 ) * ( f . g2 ) ;
( M * F-4 ) . n = M . ( F( F . n ) ) .= M . ( ( ( canFS Omega ) . n ) ) .= M . ( ( canFS Omega ) . n ) ;
the carrier of L1 + L2 c= ( the carrier of L1 ) \/ ( the carrier of L2 ) & the carrier of L1 + L2 c= the carrier of L1 + ( the carrier of L2 ) ;
pred a , b , c , x , y , c , x , y , c , x , y , y , z , x , y , z ;
( the PartFunc of s ) . n <= ( the PartFunc of s ) . n * s . n & ( the Sorts of s ) . n <= ( the Sorts of s ) . n ;
attr - 1 <= r & r <= 1 implies ( arccot * ( arccot * ( arccot * ( arccot * ( arccot * ( arccot * f ) ) ) ) ) `| Z ) = - 1 ;
s in { p ^ <* n *> where n is Nat : p ^ <* n *> in T1 & n < len p + 1 } ;
|[ x1 , x2 , x3 ]| . 2 - |[ y1 , y2 , x3 ]| . 2 = x2 - |[ y2 , y2 ]| . 2 - |[ y1 , y2 , x3 ]| . 2 - |[ y1 , y2 , x3 ]| . 2 = x2 - y2 ;
attr m is nonnegative means : Def4 : F . m is nonnegative & ( Partial_Sums F ) . m is nonnegative & ( Partial_Sums F ) . m is nonnegative ;
len ( ( G . z ) ) = len ( ( ( G . ( x , y ) ) + ( G . ( y , z ) ) ) .= len ( G . ( x , y ) ) ;
consider u , v being VECTOR of V such that x = u + v and u in W1 /\ W2 and v in W2 /\ W3 and u in W3 /\ W3 and v in W2 /\ W3 ;
given F being finite FinSequence of NAT such that F = x and dom F = n and rng F c= { 0 , 1 } and Sum F = k and Sum ( F ) = k ;
0 = 1 * 'not' 1- ( 1 * uon ) iff 1 = ( ( 1 - ( - 1 ) ) * ( - ( 1 - ( - 1 ) ) * ( - ( 1 - ( 1 - ( 1 - ( 1 - 0 ) ) * ( - ( 1 - 0 ) ) * ( - ( 1 - 0 ) ) * ( - 1 ) ) ) ;
consider n be Nat such that for m be Nat st n <= m holds |. ( f # x ) . m - lim ( f # x ) .| < e ;
cluster -> Boolean for non empty _ of L , ( ( let L ) | ( D ) ) \hbox \hbox { $ ( ( ( \rm _ 2 ) | D ) ) | ( D ) is Boolean & ( ( _ 2 ) | D ) | D is Boolean ;
"/\" ( BF , L ) = Top B .= Top S .= "/\" ( I , L ) .= "/\" ( I , L ) .= "/\" ( I , L ) ;
( r / 2 ) ^2 + ( r / 2 ) ^2 <= ( r / 2 ) ^2 + ( r / 2 ) ^2 + ( r / 2 ) ^2 ;
for x being element st x in A /\ dom ( f `| X ) holds ( f `| X ) . x >= r2 & ( f `| X ) . x >= r2
2 * r1 - 2 * |[ a , c ]| - ( 2 * r1 - 2 * |[ b , c ]| ) = 0. TOP-REAL 2 - 2 * |[ b , c ]| ;
reconsider p = P * ( \square , 1 ) , q = a " * ( ( - ( - 1 ) ) * ( ( - ( K , n , 1 ) ) * ( 1 , 1 ) ) as FinSequence of K ;
consider x1 , x2 being element such that x1 in uparrow s and x2 in < t and x = [ x1 , x2 ] and [ x1 , x2 ] in < s and [ x2 , x3 ] in dom f ;
for n be Nat st 1 <= n & n <= len q1 holds q1 . n = ( ( upper_volume ( g , M7 ) ) | n ) . n
consider y , z being element such that y in the carrier of A and z in the carrier of A and i = [ y , z ] and i = [ y , z ] ;
given H1 , H2 being strict Subgroup of G such that x = H1 and y = H2 and H1 is Subgroup of H2 and H2 is Subgroup of H1 and H2 is Subgroup of H2 ;
for S , T being non empty RelStr , d being Function of T , S st T is complete holds d is directed-sups-preserving iff d is monotone & d is monotone
[ a + 0 , b + i ] in ( the carrier of F_Complex ) /\ ( the carrier of V ) & [ a + 0 , b + i ] in [: the carrier of F_Complex , the carrier of V :] ;
reconsider mm = max ( len F1 , len ( p . n ) * ( <* x *> |^ n ) ) as Element of NAT ;
I <= width GoB ( ( GoB h ) * ( 1 , 1 ) , ( GoB h ) * ( 1 , 1 ) ) & I <= width GoB ( ( GoB h ) * ( 1 , 1 ) ) `2 ;
f2 /* q = ( f2 /* ( f1 /* s ) ) ^\ k .= ( f2 * ( f1 /* s ) ) ^\ k .= ( f2 * ( f1 /* s ) ) ^\ k ;
attr A1 \/ A2 is linearly-independent means : Def4 : A1 is linearly-independent & A2 misses ( the carrier of A1 ) & ( for x being Element of A1 holds x in ( the carrier of A1 ) /\ ( the carrier of A2 ) ) implies ( x in ( the carrier of A1 ) /\ ( the carrier of A2 ) ) ;
func A -carrier of C -> set equals union { A . s where s is Element of R : s in C & s in C } ;
dom ( Line ( v , i + 1 ) (#) ( ( Line ( p , m ) ) * ( Line ( p , 1 ) ) ) ) = dom ( F ^ <* p . m *> ) ;
cluster [ x , 4 ] -> [ x , 4 ] , [ x , 4 ] , [ x , 4 ] , [ x , 5 ] ] -> reduces x , 4 & [ x , 5 ] in dom R ;
E , { All ( x2 , x1 ( ) ) } |= All ( x2 , x2 ( ) ) => ( x2 ( ) ) '&' ( x3 ( ) ) '&' ( x3 ( ) ) '&' ( x3 ( ) ) '&' ( x3 ( ) ) '&' ( x3 ( ) ) ) ;
F .: ( id ( X , g ) ) . x = F . ( id ( X , g ) . x ) .= F . ( x , g . x ) .= F . ( x , g . x ) ;
R . ( h . m ) = F . ( x0 + h . m ) - ( h . m ) + ( h . m ) - ( h . m ) ;
cell ( G , ( X -' 1 , Y ) , ( t + 1 ) ) \ ( ( t + 1 ) - ( t + 1 ) ) meets ( ( L~ f ) \ ( L~ f ) ) ;
IC Result ( P2 , s2 ) = IC IExec ( I , P , Initialize s ) .= card I .= card I + card J + card J + card J + card J + 3 .= card I + card J + card J + 3 .= card I + card J + card J + 3 ;
sqrt ( - ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 ) > 0 ;
consider x0 being element such that x0 in dom a and x0 in g " { k } and y = a . x0 and x0 in g .: { x0 } and x0 in { x0 } ;
dom ( r1 (#) chi ( A , C ) ) = dom chi ( A , C ) /\ dom ( chi ( A , C ) ) .= dom ( ( r1 (#) chi ( A , C ) ) | A ) .= dom ( ( r1 (#) chi ( A , C ) ) | A ) .= dom ( ( r1 (#) chi ( A , C ) ) | A ) .= A ;
d-7 . [ y , z ] = ( ( y - z ) * ( z - w ) ) * ( y - w ) .= ( ( y - z ) * z ) * ( z - w ) ;
attr i being Nat means : Def4 : C . i = A . i /\ B . i holds L~ C c= ( L~ C ) /\ ( L~ C ) ;
Suppose x0 in dom f and f is continuous and f is_differentiable_in x0 and ( for x st x in dom f holds f . x <> 0 ) implies f is_differentiable_in x0 & for x st x in dom f holds f . x = - f . x ) & for x st x in dom f holds f . x = - f . x ;
p in Cl A implies for K being Basis of p , Q being Subset of T st Q in K holds A meets Q implies A meets Q
for x being Element of REAL n st x in Line ( x1 , x2 ) holds |. y1 - y2 .| <= |. y1 - y2 .| + |. y2 - x2 .|
func {} -> /. Nat means : Def4 : a in it & for b being Ordinal st a in it holds it . b c= b ;
[ a1 , a2 , a3 ] in ( the carrier of A ) /\ ( the carrier of A ) & [ a1 , a2 ] in [: the carrier of A , the carrier of A :] ;
ex a , b being element st a in the carrier of S1 & b in the carrier of S2 & x = [ a , b ] & x = [ a , b ] ;
||. ( vseq . n ) - ( vseq . m ) .|| * ||. x - x0 .|| < ( e / ( ||. x .|| + ||. x .|| ) ) * ||. x - x0 .|| ;
then for Z being set st Z in { Y where Y is Element of I7 : F c= Z & z in Z } holds z in x & z in Z ;
sup compactbelow [ s , t ] = [ sup ( { x } ) , sup ( compactbelow [ s , t ] ) ] .= [ sup ( compactbelow s ) , sup ( compactbelow t ) ] ;
consider i , j being Element of NAT such that i < j and [ y , f . j ] in II and [ f . i , f . j ] in II and [ i , j ] in II ;
for D being non empty set , p , q being FinSequence of D st p c= q holds ex p being FinSequence of D st p ^ q = q & p ^ q = q ^ p
consider e39 being Element of the affine of X such that c9 , a9 // a9 , e and a9 <> c9 & c9 <> e & e <> e & e <> e & e <> x & e <> x & e <> x & e <> x & e <> x & e <> x & e <> x & e <> x & e <> x & e <> x & e <> x ;
set U2 = I \! \mathop { \vert I . m .| , U2 = I \! \mathop { \vert I . m .| } ;
|. q2 .| ^2 = ( |. q2 .| ) ^2 + ( |. 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 iff x "/\" y = x /\ y
dom signature U1 = dom ( the charact of U1 ) & Args ( o , MSAlg U1 ) = dom ( the charact of U1 ) & Args ( o , MSAlg U1 ) = dom ( ( the charact of U1 ) * ( the charact of U1 ) ) ;
dom ( h | X ) = dom h /\ X .= dom ( ( h | X ) | X ) .= dom ( ( h | X ) | X ) .= dom ( ( h | X ) | X ) .= dom ( h | X ) ;
for N1 , N2 being Element of G holds dom ( h . K1 ) = N & rng ( h . K1 ) = N & rng ( h . K1 ) c= N & rng ( h . K1 ) c= N
( mod ( u , m ) + mod ( v , m ) ) . i = ( mod ( u , m ) ) . i + ( mod ( v , m ) ) . i ;
- ( q `1 ) < - 1 or q `2 >= - 1 & - ( q `1 ) >= - 1 & - ( q `1 ) >= - 1 & - ( q `1 ) >= - 1 & - ( q `1 ) >= - 1 ;
attr r1 = f & r2 = f & for x st x in dom f holds r1 * f . x = ( f . x ) * ( f . x ) & ( f . x ) * ( f . x ) = ( f . x ) * ( f . x ) ;
vseq . m is bounded Function of X , the carrier of Y & x9 . m = ( ( vseq . m ) (#) ( vseq . m ) ) . x ;
attr a <> b & b <> c & angle ( a , b , c ) = PI & angle ( b , c , a ) = 0 & angle ( c , a , b ) = 0 ;
consider i , j being Nat , r , s being Real such that p1 = [ i , r ] and p2 = [ j , s ] and i < j and r < s and s < p2 ;
|. p .| ^2 - ( 2 * |( p , q )| ) ^2 + |. q .| ^2 = |. p .| ^2 + |. q .| ^2 - ( 2 * |. p .| ) ^2 ;
consider p1 , q1 being Element of [: X , Y :] such that y = p1 ^ q1 and q1 ^ q1 = p1 ^ q1 and len q1 = len q1 and len q1 = len q1 and len q1 = len q1 and len q1 = len q1 and len q1 = len q1 + 1 and len q1 = len q1 + 1 ;
( the _ of A ) . ( r1 , r2 , s1 , s2 , s1 , s2 , s2 , s2 , t2 ) = ( s2 * s1 ) * ( s2 * s2 ) .= ( s2 * s1 ) * s2 .= ( s2 * s1 ) * s2 ;
( ex w being Real st ( for A being Subset of TOP-REAL 2 st A = proj2 .: ( A /\ holds w is non empty ) ) & ( for B being Subset of TOP-REAL 2 st B = proj2 .: A holds B is non empty ) & ( for B being Subset of TOP-REAL 2 st B = B holds B is non empty or B is non empty ) implies B is non empty
s , ( ( k + 1 ) |= H1 , H2 ) iff s |= for k being Nat holds s |= ( H1 , k ) iff s |= ( H2 , k ) ) & s |= ( H2 , k )
len ( s + 1 ) = card ( support b1 ) + 1 .= card ( support b2 ) + 1 .= card ( support b1 ) + 1 .= card ( support b1 ) + 1 .= card ( support b1 ) + 1 .= card ( support b1 ) + 1 .= card ( support b1 ) + 1 ;
consider z being Element of L1 such that z >= x and z >= y and for z being Element of L1 st z >= x & z `1 >= y holds z `1 >= x `1 & z `2 >= y `1 ;
LSeg ( UMP D , |[ ( W-bound D ) + ( W-bound D ) / 2 , ( S-bound D ) + ( S-bound D ) / 2 ) ) /\ D = { UMP D } /\ D .= { UMP D } ;
lim ( ( ( f `| N ) / g ) /* b ) = lim ( ( f `| N ) / g ) .= ( ( f `| N ) / g ) . x0 .= ( ( f `| N ) / g ) . x0 ;
P [ i , pr1 ( f ) . i , pr1 ( f ) . ( i + 1 ) ] & pr1 ( f ) . ( i + 1 ) = pr1 ( f ) . ( i + 1 ) ;
for r be Real st 0 < r ex m be Nat st for k be Nat st m <= k holds ||. ( seq . k ) - ( R /* seq ) . k .|| < r
for X being set , P being a_partition of X , x , a , b being set st x in a & a in P & x in P & b in P & a in P holds a = b
Z c= dom ( ( ( - 1 ) (#) f ) `| Z ) /\ ( dom ( ( - 1 ) (#) f ) `| Z ) & Z = dom ( ( - 1 ) (#) f ) /\ dom f ;
ex j being Nat st j in dom ( l ^ <* x *> ) & j < i & y = ( l ^ <* x *> ) . j & i = 1 + j & j = len l + 1 & z = l . i & y = ( l ^ <* x *> ) . j ;
for u , v being VECTOR of V , r being Real st 0 < r & u in mid N & v in c= c= len ( - r ) holds r * u + ( - r ) * v in c= c= c= c= c= c= - N
A , Int ( 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 ) ) ;
- Sum <* v , u , w *> = - ( v + u + w ) .= - ( v + u ) + u .= - ( v + u ) + u .= - v + u ;
Exec ( a := b , s ) . IC SCM = ( Exec ( a := b , s ) ) . IC SCM .= Exec ( ( a := b , s ) , s ) . IC SCM .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= ( ( a := b ) , s ) . IC SCM ;
consider h being Function such that f . a = h and dom h = I and for x being element st x in I holds h . x in ( the carrier of J ) . x ;
for S1 , S2 being non empty reflexive RelStr , D being non empty directed Subset of S1 , D being non empty directed Subset of S2 , x being Element of S1 , y being Element of S2 holds cos ( x ) is directed iff x is directed & cos ( y ) is directed
card X = 2 implies ex x , y st x in X & y in X & x <> y & for z st z in X & z <> x holds z = x or z = y or z = x or z = y
E-max L~ Cage ( C , n ) in rng ( Cage ( C , n ) :- Cage ( C , n ) ) implies ( W-min L~ Cage ( C , n ) ) .. Cage ( C , n ) <= ( W-min L~ Cage ( C , n ) ) .. Cage ( C , n )
for T , T being DecoratedTree , p , q being Element of dom T st p >= q & ( T is with with dom p implies p in T ) & q in T implies p in T
[ i2 + 1 , j2 ] in Indices G & [ i2 , j2 ] in Indices G & f /. k = G * ( i2 + 1 , j2 ) & f /. k = G * ( i2 + 1 , j2 ) ;
cluster ( k gcd n ) divides ( k gcd n ) & n divides ( k gcd n ) & ( k divides ( k gcd n ) ) & ( k divides ( k gcd n ) ) implies ( k divides ( k gcd n ) ) & ( k divides ( k gcd n ) ) & ( k divides ( k gcd n ) ) & ( k divides ( k gcd n ) ) ;
dom F " = the carrier of X2 & rng F = the carrier of X1 & F " = F " * F " & F " * F = F " * F " & F " * F = F " * F " * F "
consider C being finite Subset of V such that C c= A and card C = n and the \mathbb of V = Lin ( BM \/ C ) and C is linearly-independent and C is linearly-independent and C is linearly-independent and C is linearly-independent ;
V is prime implies for X , Y being Element of [: the topology of T , the topology of T :] st X /\ Y c= V holds X c= V or Y c= V
set X = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } , Y = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } ;
angle ( p1 , p3 , p4 ) = 0 .= angle ( p2 , p3 , p4 ) .= angle ( p3 , p4 ) .= angle ( p3 , p4 ) .= angle ( p3 , p4 ) .= angle ( p3 , p4 ) ;
- sqrt ( - ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 ) = - ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) .= - 1 ;
ex f being Function of I[01] , TOP-REAL 2 st f is continuous one-to-one & rng f = P & f . 0 = p1 & f . 1 = p2 & f . 0 = p3 & f . 1 = p4 & f . 1 = p4 & f . 0 = p2 & f . 1 = p4 ;
attr f is partial x0 means : Def4 : SVF1 ( 2 , pdiff1 ( f , 1 ) , u0 , u0 ) is_differentiable_in ( proj ( 2 , 3 ) ) . ( u0 + 1 ) & SVF1 ( 2 , f , u0 ) . u = ( proj ( 2 , 3 ) ) . u ;
ex r , s st x = |[ r , s ]| & G * ( len G , 1 ) `1 < r & r < G * ( len G , 1 ) `2 & s < G * ( 1 , 1 ) `2 & G * ( len G , 1 ) `2 < s & s < G * ( 1 , 1 ) `2 ;
assume that f is FinSequence of TOP-REAL 2 and 1 <= t and t <= len G and G * ( t , width G ) `2 >= ( GoB f ) * ( t , 1 ) `2 and G * ( t , 1 ) `2 <= ( GoB f ) * ( t , 1 ) `2 ;
attr i in dom G means : Def4 : r * ( f * reproj ( G , i ) ) = r * ( f * reproj ( G , i ) ) ;
consider c1 , c2 being bag of o1 + o2 such that ( decomp c ) /. k = <* c1 , c2 *> and c = c1 + c2 and c = c2 + c2 and c1 = c2 + 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 .= ( C ^ Y ) . ( k2 + 1 ) .= C . ( k2 + 1 ) .= C . ( k2 + 1 ) .= C . ( k2 + 1 ) .= C . ( k2 + 1 ) ;
attr M1 = len M2 means : Def4 : width M1 = width M2 & width M1 = width M2 & width M2 = width M2 & width M1 = width M2 & width M2 = width M2 & width M1 = width M2 & width M1 = width M2 ;
consider g2 be Real such that 0 < g2 and { y where y is Point of S : ||. y - x0 .|| < g2 & y in dom f & f . ( y - x0 ) < g2 & g2 in dom f } c= dom f /\ dom g ;
assume x < ( - b + sqrt ( let a , b , c ) ) / 2 or x > ( - b - sqrt ( a , b , c ) ) / 2 ;
( G1 '&' G2 ) . i = ( <* 3 *> ^ G1 ) . i & ( H1 '&' H2 ) . i = ( <* 3 *> ^ H1 ) . i & ( H1 '&' H2 ) . i = ( <* 3 *> ^ H1 ) . i ;
for i , j st [ i , j ] in Indices M3 + M1 holds ( M3 + M1 ) * ( i , j ) < M2 * ( i , j ) + M2 * ( i , j )
for f being FinSequence of NAT , i being Element of NAT st i in dom f & i divides len f holds i divides f /. i & i divides len f & i divides len f implies i divides f /. i & i divides f /. ( i + 1 )
assume F = { [ a , b ] where a , b is Subset of X : for c being set st c in B\bf X & a c= c holds b c= c } ;
b2 * q2 + ( b3 * q3 ) + ( - ( b2 * q2 ) + ( b1 * q2 ) ) = 0. TOP-REAL n + ( - ( b2 * q2 ) ) .= ( - ( b2 * q2 ) ) + ( - ( b2 * q2 ) ) .= ( - ( b2 * q2 ) ) + ( - ( b2 * q2 ) ) ;
Cl F = { D where D is Subset of T : ex B being Subset of T st D = Cl B & B in F & A c= D & B c= D & B c= D } ;
attr seq is summable means : Def4 : seq is summable & seq is summable & ( for n holds seq . n = Sum ( seq ^\ n ) ) & ( for n holds seq . n = Sum ( seq ^\ n ) ) & ( for n holds seq . n = Sum ( seq ^\ n ) ) ;
dom ( ( ( cn max ( 1 - cn ) ) | D ) = ( the carrier of ( TOP-REAL 2 ) ) | D ) /\ D .= the carrier of ( ( TOP-REAL 2 ) | D ) .= D ;
|[ X , Z ]| is full full non empty SubRelStr of ( [#] Z ) |^ the carrier of Z & [ X , Y ] is full full SubRelStr of ( [#] Z ) |^ the carrier of Z ;
G * ( 1 , j ) `2 = G * ( i , j ) `2 & G * ( i , j ) `2 <= G * ( i , j ) `2 & G * ( i , j ) `2 <= G * ( i , j ) `2 ;
synonym m1 c= m2 means : Def4 : for p being set st p in P holds the non empty Element of ( m + 1 ) -tuples_on NAT , x being set st x in P & the InternalRel of ( m + 1 ) \rm \hbox \hbox \hbox { - } Seg ( m + 1 ) = ( m + 1 ) -tuples_on NAT ;
consider a being Element of B ( ) such that x = F ( a ) and a in { G ( b ) where b is Element of A ( ) : P [ b ] } and a in A ( ) ;
attr IT is in the carrier of K means : Def4 : the multMagma of IT , the multF of IT #) & the multF of IT , the multF of IT #) = [ the carrier of IT , the carrier of IT ] & the multF of IT , the multF of IT #) = the multF of IT ;
sequence ( a , b , 1 ) + sequence ( c , d ) = b + the carrier of L .= b + the carrier of L .= b + d .= b + d .= b + d ;
cluster ( + _ ) -> $ -valued for Element of INT , i , j be Element of INT holds ( i , j ) = ( i , j ) --> ( i , j ) + ( j , i ) = ( i , j ) --> ( i , j ) + ( j , i ) --> ( j , i ) ;
- s2 * p1 + ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - s2 * p2 ) ) ) = ( - r2 ) * p1 + ( s2 * p2 - ( s2 * p2 - ( s2 * p2 ) ) ) * p2 ;
eval ( ( a | ( n , L ) ) *' p , x ) = eval ( a | ( n , L ) ) * eval ( p , x ) .= a * eval ( p , x ) .= a * eval ( p , x ) ;
assume that the TopStruct of S = the TopStruct of T and for D being non empty directed Subset of S , V being Subset of S st V in the topology of S holds V meets V and for V being open Subset of S st V in V holds V meets V ;
assume that 1 <= k & k <= len w + 1 and T-7 . ( ( q11 , w ) . k ) = ( T. ( q11 , w ) ) . k and T-7 . ( ( q11 , w ) . k ) = ( T-7 . ( q11 , w ) ) . k ;
2 * a |^ ( n + 1 ) + ( 2 * b |^ ( n + 1 ) ) >= ( a |^ n + ( b |^ ( n + 1 ) ) + ( b |^ ( n + 1 ) ) + ( a |^ ( n + 1 ) ) ;
M , v2 |= All ( x. 3 , All ( x. 0 , All ( x. 4 , H ) ) ) & M , v1 |= All ( x. 0 , ( x. 0 ) '&' ( ( x. 4 , H ) / ( x. 0 , H ) ) ) ;
assume that f is_differentiable_on l and for x0 st x0 in l holds 0 < f . x0 and for x1 st x1 in l holds f . x1 - f . x0 < f . x0 and for x1 st x1 in l holds f . x1 - f . x0 < f . x1 ;
for G1 being _Graph , W being Walk of G1 , e being set , G being Walk of G1 , v being Vertex of G2 st not e in W & v in W holds W is_Walk_from ( v ) , ( W ) . e , ( W ) . ( e ) , ( W ) . ( e ) , ( W ) . ( e ) , ( W ) . ( e ) , ( W ) . ( e ) ;
c9 is not empty iff ( for y st y is not empty & y is not empty or not ( ex x st x is not empty & y is not empty & not ( x is not empty or y is not empty ) & not ( x is not empty or y is not empty ) & not ( x is not empty or y is not empty ) ) ;
Indices GoB f = [: dom GoB f , Seg width GoB f :] & [: dom GoB f , Seg width GoB f :] = [: dom GoB f , Seg width GoB f :] & [: dom GoB f , Seg width GoB f :] c= [: dom GoB f , Seg width GoB f :] & [: dom GoB f , Seg width GoB f :] c= [: Seg width GoB f :] ;
for G1 , G2 , G3 being finite Subgroup of O , O being stable Subgroup of O st G1 is stable & G2 is stable & G1 is stable & G2 is stable holds G1 * G2 is stable & G1 * ( G2 * G1 ) = G1 * ( G2 * G2 )
UsedIntLoc ( int ) = { intloc 0 , intloc 1 , intloc 2 , intloc 3 , 1 , - 1 , 1 , - 1 , 1 , - 1 , 1 , - 1 , 1 , - 1 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 0 , 1 , 0 , 1 , 0 , 0 , 1 , 0 , 1 , 0 , 0 , 1 , 0 , 0 , 1 , 0 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 0 , 1 , 1 , 1 , 1 , 1 , 1 , 0 , 1 , 0 , 1 , 1 , 1 , 1 , 1 , 1 , 0 , 0 , 0 , 0 , 1 ,
for f1 , f2 be FinSequence of F st f1 ^ f2 is p -element & Q [ f1 ^ f2 ] & Q [ f2 ] holds Q [ f1 ^ f2 ] & Q [ f1 ^ f2 ] & Q [ f2 ^ f1 ] implies Q [ f1 ^ f2 ]
( p `1 ) ^2 / ( 1 + ( p `1 ) ^2 ) = ( q `1 ) ^2 / ( 1 + ( q `1 ) ^2 ) .= ( q `1 ) ^2 / ( 1 + ( q `1 ) ^2 ) ;
for x1 , x2 , x3 be Element of REAL n holds |( ( x1 - x2 ) , x1 - x2 )| = |( x1 , x1 - x2 )| + |( x3 , x2 - x3 )| + |( x3 , x2 - x3 )| + |. x2 - x3 .| + |. x3 - x3 .| + |. x3 - x3 .| + |. x2 - x3 .| + |. x3 - x3 .| + |. x3 - x3 .| + |. x3 - x3 .| + |. x3 - x3 .| < + |. x1 - x2 .| + |. x3 - x3 .| + |. x3 - x3 .| + |. x3 - x3 .| + |. x3 - x3 .| + |. x3 - x3 .| + |. x3 - x3 .| + |. x3 - x3 .| + |. x3 - x3 .| + |. x3 - x3 .|
for x st x in dom ( ( F | A ) | A ) holds ( ( ( F | A ) ) | A ) . ( - x ) = - ( ( ( F | A ) | A ) . x )
for T being non empty TopSpace , P being Subset-Family of T , B being Basis of T st P c= the topology of T for x being Point of T st B c= P & x in P holds P is Basis of x
( a 'or' b 'imp' c ) . x = 'not' ( ( a 'or' b ) . x ) 'or' c . x .= 'not' ( ( a . x ) 'or' b . x ) 'or' c . x .= TRUE '&' TRUE .= TRUE ;
for e being set st e in A ex X1 being Subset of X st e = [: X1 , Y1 :] & ex Y1 being Subset of X st e = [: Y1 , Y1 :] & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open
for i be set st i in the carrier of S for f being Function of [: S , S1 :] , S1 . i st f = H . i & F . i = f | ( i + 1 ) holds F . i = f | ( i + 1 )
for v , w st for y st x <> y holds w . y = v . y holds Valid ( VERUM ( Al ( ) ) , J ( ) ) . v = Valid ( VERUM ( Al ( ) ) , J ( ) ) . w
card D = card D1 + card D2 - card { i , j } .= ( c1 + c2 ) - ( c1 + c2 ) + ( c2 + c1 ) - ( c1 + c2 ) .= 2 * c1 + ( c2 + c2 ) - ( c1 + c2 ) .= 2 * c1 + ( c2 + c1 ) - ( c1 + c2 ) .= 2 * c2 + 1 - 1 ;
IC Exec ( i , s ) = ( s +* ( 0 .--> succ ( 0 + 1 ) ) ) . 0 .= ( 0 .--> ( 0 .--> 1 ) ) . 0 .= ( 0 .--> 1 ) . 0 .= ( 0 .--> 1 ) . 0 .= 0 ;
len f /. ( len f -' 1 ) + 1 = len f -' 1 + 1 .= len f - 1 + 1 .= len f - 1 + 1 .= len f - 1 + 1 .= len f - 1 + 1 .= len f - 1 + 1 ;
for a , b , c being Element of NAT st 1 <= a & 2 <= b & k <= a holds a <= b + b-2 or a = b + b-2 or a = b + b-2 or a = b + b-2 or a = b + b-2 ;
for f being FinSequence of TOP-REAL 2 , p being Point of TOP-REAL 2 , i being Nat st i in LSeg ( f , i ) & p in LSeg ( f , i ) holds Index ( p , f ) <= i & Index ( p , f ) <= len f
lim ( ( curry ( P+* ( k , n + 1 ) ) # x ) = lim ( ( curry ( P+* ( k , n ) ) # x ) ) + lim ( ( curry ( F+* ( k , n ) ) # x ) ) ;
z2 = g /. ( \downharpoonright n1 -' n2 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) ;
[ f . 0 , f . 3 ] in id ( the carrier of G ) \/ ( the InternalRel of G ) or [ f . 0 , f . 3 ] in the InternalRel of C6 & [ f . 0 , f . 2 ] in the InternalRel of C6 ;
for G being Subset-Family of B st G = { R [ X ] where R is Subset of A ( ) , Y is Subset of B ( ) st R in FF & Y in FF holds ( Intersect ( R , X ) ) = Intersect ( G , X ) holds ( Intersect ( R , X ) ) = Intersect ( G , X )
CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 ) ) = CurInstr ( P1 , Comput ( P1 , s1 , m1 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m1 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m1 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m1 ) ) .= halt SCMPDS ;
assume that a on M and b on M and c on N and d on N and p on N and a on M and c on N and p on N and p on M and a on N and p on N and p on M and c on N and p on N and p on M and p on N and p on N and p on M and p on N and p on N and p on N and p on N and p on N and p on M and p on N and p on N and p on N and p on N and p on N and p on N and p on N and p on N and p on M and p on N and p on N and p on M and p on N and p on N and p on N and p on N and p on N and p on M and p on N and p on M and p on N and p on M and p on M
Suppose T is \hbox { T _ 4 } and ex F being Subset-Family of T st F is closed & for n being Nat holds F . n is finite-ind & ( for m being Nat st m <= n holds F . m <= 0 ) and ( for n being Nat st n <= m holds F . n <= 0 ) implies T is finite-ind ) ;
for g1 , g2 st g1 in ]. r - g2 , r .[ & g2 in ]. r - g2 , r + g2 .[ holds |. f . g1 - f . g2 .| <= ( g1 - f ) . g2 - f . g2 .|
( ( - ( - ( x + y ) ) ) * ( z1 + z2 ) ) = ( ( - ( x + y ) ) * ( z2 + z2 ) ) * ( z2 + z2 ) .= ( ( - ( x + y ) ) * ( z2 + z2 ) ) * ( z2 + z2 ) ;
F . i = F /. i .= 0. R + r2 .= ( b + a ) to_power ( n + 1 ) .= <* ( ( n + 1 ) to_power ( n + 1 ) ) * ( b |^ ( n + 1 ) ) .= ( ( n + 1 ) to_power ( n + 1 ) ) to_power ( n + 1 ) .= ( ( n + 1 ) to_power ( n + 1 ) ) to_power ( n + 1 ) ;
ex y being set , f being Function st y = f . n & dom f = NAT & f . 0 = A ( ) & for n holds f . ( n + 1 ) = R ( n , f . n ) & for n holds f . n = R ( n , f . n ) ;
func f * F -> FinSequence of V means : Def4 : len it = len F & for i be Nat st i in dom it holds it . i = F /. i * ( F /. i ) * ( F /. i ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 } = { x1 , x2 , x3 } \/ { x4 , x5 , x5 } .= { x1 , x2 , x3 } \/ { x4 , x5 , x5 } .= { x1 , x2 } \/ { x3 } .= { x1 , x2 } \/ { x3 , x4 , x5 } ;
for n being Nat for x being set st x = h . n holds h . ( n + 1 ) = o ( x , n ) & x in InputVertices S ( x , n ) & o . ( n + 1 ) in InnerVertices S ( x , n ) & o . ( n + 1 ) in InnerVertices S ( x , n )
ex S1 being Element of QC-WFF ( Al ( ) ) st SubP ( P , l , e ) = S1 & ( for x being Element of CQC-WFF ( Al ( ) ) holds ( x is Element of D ( ) ) & ( x is Element of D ( ) ) implies S [ x ] ) & ( x is Element of D ( ) implies S [ x ] )
consider P being FinSequence of GL2 such that p7 = Product P and for i being Element of dom t ex t9 being Element of the carrier of G st P . i = t9 & t . i = t . i & t . i = t . i & t . i = t . i & t . i = t . i & t . i = t . i ;
for T1 , T2 being strict non empty TopSpace , P being Basis of T1 , T2 being Basis of T2 st the carrier of T1 = the carrier of T2 & P = the topology of T2 & P = the topology of T2 holds P = P & P = the topology of T1 & P = the topology of T2 implies P is Basis of T1
Suppose f is PartFunc of REAL , REAL . u . 3 Then r (#) pdiff1 ( f , 3 ) is_partial_differentiable_in u0 , 2 & partdiff ( r (#) pdiff1 ( f , 3 ) , u , 3 ) = r (#) pdiff1 ( f , u ) . 3 & partdiff ( r (#) pdiff1 ( f , 3 ) , u , 3 ) = r (#) pdiff1 ( f , u , 3 ) . 3 ;
defpred P [ Nat ] means for F , G being FinSequence of bool ( Seg $1 ) , s be Permutation of ( Seg $1 ) st len F = $1 & rng s = rng F & for s being Permutation of ( Seg $1 ) st s = F * s holds Sum ( F ) = Sum ( G ) * s ;
ex j st 1 <= j & j < width GoB f & ( ( GoB f ) * ( 1 , j ) ) `2 <= s & s * ( 1 , j ) `2 <= ( ( GoB f ) * ( 1 , j ) ) `2 & s * ( 1 , j ) `2 <= ( ( GoB f ) * ( 1 , j ) ) `2 ;
defpred U [ set , set ] means ex F-23 being Subset-Family of T st $2 = F-23 & F is open & union F-23 is open & for n being Nat holds F . n is open & F . n is discrete & F . n is discrete & F . n is discrete & F . n is discrete ;
for p4 being Point of TOP-REAL 2 st LE p4 , p1 , P , p1 , p2 & LE p4 , p1 , P , p1 , p2 holds LE p4 , p1 , P , p1 , p2 & LE p4 , p1 , P , p1 , p2 & LE p2 , p1 , P , p1 , p2 & LE p1 , p2 , P , p1 , p2
f in the carrier of D ( ) & for g st g <> f . y holds x = g implies for y st y in D ( ) holds g . y = y iff f . y = f . y ) & for x st x in D ( ) holds f . x = f . ( All ( x , H ) . x )
ex 8 being Point of TOP-REAL 2 st x = 8 & ( ( ( - 1 ) * ( |. 8 .| ) ) / ( |. 8 .| ) ) >= 8 & ( ( - 1 ) * ( |. 8 .| ) ) / ( |. 8 .| ) >= 0 & ( ( - 1 ) * ( |. 8 .| ) ) / ( |. 8 .| ) >= 0 ;
assume for d7 being Element of NAT st d7 <= ( n -`1 ) holds s1 . ( ( n -\hbox { t } ) . ( n + 1 ) ) = s2 . ( ( n -\hbox { t } ) . ( n + 1 ) ) & s1 . ( ( n + 1 ) + 1 ) = s2 . ( ( n + 1 ) + 1 ) ;
Suppose s <> t and s is Point of Sphere ( x , r ) and s is not Point of Sphere ( x , r ) and ex e being Point of E st { e } = Ball ( s , r ) /\ Sphere ( x , r ) and not e in Sphere ( s , t ) /\ Sphere ( x , r ) ;
given r such that 0 < r and for s st 0 < s ex x1 be Point of CNS st x1 in dom f & ||. x1 - x0 .|| < s & |. f /. x1 - f /. x0 .| < r and |. f /. x1 - f /. x0 .| < r ;
( p | x ) | ( p | ( x | x ) ) = ( ( ( x | x ) | x ) | ( x | x ) ) | ( p | ( x | x ) ) .= ( ( x | x ) | ( x | x ) ) | ( x | x ) ;
assume that x , x + h in dom sec and ( for x st x in dom sec holds ( cos . x ) = ( 4 * sin . x + cos . x ) * sin . x + cos . x * cos . x ) and cos . x = ( 4 * sin . x + cos . x ) ^2 + cos . x * cos . x ;
assume that i in dom A and len A > 1 and B c= 1 and for i st i in dom A & i < len B holds A * ( i , j ) = ( A * ( i , j ) ) * ( B * ( i , j ) ) and A * ( i , j ) = ( A * ( i , j ) ) * ( B * ( i , j ) ) ;
for i be non zero Element of NAT st i in Seg n holds i divides n or i = <* n *> or i = <* n *> & i <> n & i <> n & i <> n implies h . i = ( 1. F_Complex ) . ( n + i ) & h . i = ( 1. F_Complex F_Complex ) . ( n + i )
( ( ( b1 'imp' b2 ) '&' ( c1 'imp' c2 ) '&' ( c2 'imp' c2 ) ) '&' ( ( a1 'or' b1 ) '&' ( a2 'or' b2 ) '&' 'not' ( a2 '&' b2 ) '&' 'not' ( b2 '&' c2 ) ) '&' 'not' ( a2 '&' c2 ) '&' 'not' ( b2 '&' c2 ) '&' 'not' ( a2 '&' c2 ) '&' 'not' ( b2 '&' c2 ) '&' 'not' ( a2 '&' c2 ) '&' 'not' ( b2 '&' c2 ) '&' 'not' ( b2 '&' c2 ) '&' 'not' ( a2 '&' c2 ) '&' 'not' ( b2 '&' c2 ) '&' 'not' ( a2 '&' c2 ) '&' 'not' ( a2 '&' c2 ) '&' 'not' ( b2 '&' c2 ) '&' 'not' ( a2 '&' c2 ) '&' 'not' ( a2 '&' c2 ) '&' 'not' ( b2 '&' c2 ) '&' 'not' ( b2 '&' c2 ) '&' 'not' ( b2 '&' c2 ) '&' 'not' ( a2 '&' c2 ) '&' 'not' ( a2 '&' c2 ) '&' 'not' ( a2 '&' c2 ) '&' 'not' ( a2 '&' c2
assume that for x holds f . x = ( ( ( - cot * cot ) `| Z ) . x ) and for x st x in dom cot holds ( ( - cot * cot ) `| Z ) . x = cos . ( x- x ) and ( - cot * cot ) . x = cos . ( x- x ) and ( - cot * cot ) . x = cos . ( x- x ) ;
consider R8 , I8 be Real such that R8 = Integral ( M , Re F ) and I8 = Integral ( M , Im F ) and Integral ( M , Im F ) = Integral ( M , Im F ) and Integral ( M , Im F ) = Integral ( M , Im F ) + Integral ( M , Im F ) ;
ex k be Element of NAT st k = k & 0 < d & for q be Element of product G st q in X & ||. q-x - f /. x .|| < r holds ||. partdiff ( f , q , k ) - partdiff ( f , x , k ) .|| < r ;
x in { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 } or x in { x1 , x2 , x3 } or x in { x1 , x2 , x3 } or x in { x3 } or x in { x4 , x5 , x5 } ;
G * ( j , i ) `2 = G * ( 1 , i ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j
f1 * p = p .= ( ( the Arity of S1 ) * the Arity of S1 ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o ;
func tree ( T , P , T1 ) -> DecoratedTree means : Def4 : q in it iff ex p , q st p in P & q in T & p = q ^ r & p in T1 & q in T1 & p = r ^ q ;
F /. ( k + 1 ) = F . ( k + 1 -' 1 ) .= FV . ( p . ( k + 1 -' 1 ) , k + 1 -' 1 ) .= FV . ( k + 1 -' 1 ) .= F|^ ( p . k , k + 1 -' 1 ) .= FV . ( k + 1 -' 1 ) .= FV . k ;
for A , B , C being Matrix of len C st len B = len C & width B = width C & len B = width C & len B = width C & len B > 0 & len C > 0 & len A > 0 & len B > 0 & len C > 0 & len A > 0 & len B > 0 & len A = len B & width B = width C holds A * B = C * ( B * C )
seq . ( k + 1 ) = 0. F_Complex + seq . ( k + 1 ) .= ( Partial_Sums ( seq ) ) . k + ( Partial_Sums ( seq ) ) . ( k + 1 ) .= ( Partial_Sums ( seq ) ) . k + ( Partial_Sums ( seq ) ) . ( k + 1 ) .= ( Partial_Sums ( seq ) ) . k + ( Partial_Sums ( seq ) ) . ( k + 1 ) ;
assume that x in ( the carrier of CP ) ~ and y in ( the carrier of CP ) and x in ( the carrier of CP ) and y in ( the carrier of CP ) and z = [ x , y ] ;
defpred P [ Element of NAT ] means for f st len f = $1 holds ( for k st k in $1 holds f . k = ( ( VAL g ) . k ) '&' ( ( VAL g ) . k ) '&' ( ( VAL g ) . k ) ;
assume that 1 <= k and k + 1 <= len f and f is_sequence_on G and [ i , j ] in Indices G and [ i + 1 , j ] in Indices G and f /. k = G * ( i , j ) and f /. k = G * ( i , j ) and f /. ( k + 1 ) = G * ( i , j ) ;
assume that cn < 1 and q `1 > 0 and ( q `2 ) > 0 and ( q `2 <= 1 & q `2 <= 1 or q `1 >= 0 & q `1 = 1 & q `1 = 1 or q `1 = 1 & q `1 = 1 or q `1 = 1 & q `2 = 1 or q `1 = 1 & q `1 = 1 & q `2 = 1 ;
for M being non empty TopSpace , x being Point of M , f being Point of M st x = x `1 holds ex x being Point of M st for n being Element of NAT holds f . n = Ball ( x `1 , 1 / ( n + 1 ) ) & ex f being sequence of M st f is sequence of M & f is one-to-one
defpred P [ Element of omega ] means f1 is_differentiable_on Z & f2 is_differentiable_on Z & for x st x in Z holds ( ( f1 - f2 ) `| Z ) . x = ( ( f1 - f2 ) `| Z ) . x - ( ( f1 - f2 ) `| Z ) . x = ( ( f1 - f2 ) `| Z ) . x - ( f2 - g2 ) `| Z ;
defpred P1 [ Nat , Point of CNS ] means ( $1 in Y & $2 in Y & ||. $2 - x0 .|| < r & $2 - x0 .|| < r & ||. $2 - x0 .|| < r ) & ||. f /. $2 - f /. x0 .|| < r ;
( f ^ mid ( g , 2 , len g ) ) . i = ( mid ( g , 2 , len g ) ) . ( i -' 1 ) .= g . ( i -' 1 ) .= g . ( i -' 1 ) .= g . ( i -' 1 ) .= g . ( i -' 1 ) .= g . ( i -' 1 ) .= ( g . i ) . i ;
( 1 / 2 * n0 + 2 * ( n0 + 2 * ( n0 + 2 ) ) ) = ( 1 / 2 * ( ( n + 2 ) * ( n + 2 ) ) ) * ( ( n + 2 ) * ( n + 2 ) ) .= 1 / 2 * ( n + 2 ) * ( n + 2 ) .= 1 / 2 * ( n + 2 ) * ( n + 2 ) .= 1 / 2 * ( n + 2 ) ;
defpred P [ Nat ] means for G being non empty finite strict strict finite RelStr , F being Function of G , F st G is Let G being non empty finite non empty finite RelStr st G is LSeg & F is the carrier of G holds the RelStr of F = ( the RelStr of G ) \/ ( the RelStr of F ) & the RelStr of F = ( the RelStr of G ) \/ ( the RelStr of F ) ;
assume that not f /. 1 in Ball ( u , r ) and 1 <= m and m <= len f and for i st 1 <= i & i <= len f holds not f . i in Ball ( f , r ) and not f . ( m + 1 ) in Ball ( u , r ) and not f . m in Ball ( u , r ) and not f . m in Ball ( u , r ) ;
defpred P [ Element of NAT ] means ( Partial_Sums ( cos * cos ) ) . $1 = ( Partial_Sums ( cos * cos ) ) . $1 - ( cos * cos ) . $1 & ( cos * cos ) . $1 = ( Partial_Sums ( cos * cos ) ) . $1 - ( cos * cos ) . $1 ;
for x being Element of product F holds x is FinSequence of G & ( for i being set st i in dom F holds x . i = I . i ) & for i being set st i in dom F holds x . i in ( the carrier of G ) . i ) & for i being set st i in dom F holds F . i = ( the carrier of G ) . i
( x " ) |^ ( n + 1 ) = ( x " ) |^ n * x " .= ( x * x ) |^ n * x " .= ( x * x ) |^ n * x " .= ( x * x ) |^ n * x " .= ( x * x ) |^ n * x " .= ( x * x ) |^ n * x " .= ( x * x ) |^ n ;
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 ) .= DataPart Comput ( P +* I , s , LifeSpan ( P +* I , s ) + 3 ) ;
given r such that 0 < r and ]. x0 , x0 + r .[ c= dom ( f1 + f2 ) /\ dom ( f2 + f3 ) and for g st g in ]. x0 , x0 + r .[ /\ dom ( f2 + f3 ) holds ( f1 + f2 ) . g <= ( f1 + f2 ) . g ;
assume that X c= dom f1 /\ dom f2 and f1 | X is continuous and ( for x st x in X /\ dom f2 holds f2 . x = ( f1 + f2 ) | X ) and ( f1 - f2 ) | X is continuous and ( f1 - f2 ) | X is continuous and ( f2 - g2 ) | X is continuous and ( f1 + f2 ) | X is continuous ;
for L being continuous complete LATTICE for l being Element of L ex X being Subset of L st l = sup X & for x being Element of L st x in X holds x is directed & for x being Element of L st x in X holds x is directed & x is directed & x is directed
Support ( e *' A ) = { m *' p where m is Polynomial of n , L : ex p being Polynomial of n , L st p in Support ( m *' A ) & ex q being Polynomial of n , L st q = p & q = ( m *' A ) . ( m *' q ) & p . ( m + 1 ) = p . ( m + 1 ) ;
( f1 - f2 ) /* s1 = lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) .= lim ( ( f2 /* s1 ) - ( f2 /* s1 ) ) .= lim ( ( f2 /* s1 ) - ( f2 /* s1 ) ) .= lim ( ( f2 /* s1 ) - ( f2 /* s1 ) ) .= lim ( f2 /* s1 ) ;
ex p1 being Element of QC-WFF ( A ) st F . p1 = g . p1 & for g being Function of [: [: D ( ) , D ( ) :] , D ( ) ) st P [ g , p1 , g ] holds P [ g , p1 , g ] ;
( mid ( f , i , len f -' 1 ) ^ <* f /. ( len f -' 1 ) *> ) /. j = ( mid ( f , i , len f -' 1 ) ) /. j .= ( mid ( f , i , len f -' 1 ) ) /. j .= ( mid ( f , i , len f -' 1 ) ) /. j .= f /. ( j + 1 ) ;
( ( p ^ q ) ^ r ) . ( len p + k ) = ( ( p ^ q ) . ( len p + k ) ) .= ( ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( ( p ^ q ) . ( len p + k ) .= ( ( ( p ^ q ) . ( len p + k ) .= ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( ( p ^ q ) . ( len p + k ) .= ( ( p ^ r ) . ( len p + k ) .= ( ( p ^ r ) . ( len p + k ) ) . ( len p + k ) . ( len p + k ) .= ( ( p ^ r ) . ( len p + k ) .= ( ( p ^ r ) . ( len p + k
len mid ( ( ( D2 , D1 ) + 1 , indx ( D2 , D1 , j1 ) ) + 1 , indx ( D2 , D1 , j1 ) ) = indx ( D2 , D1 , j1 ) + 1 - 1 .= indx ( D2 , D1 , j1 ) + 1 - 1 .= indx ( D2 , D1 , j1 ) + 1 - 1 ;
x * y * z = MF . ( ( y * z ) * ( z * z ) ) .= ( x * ( y * z ) ) * ( y * z ) .= ( x * ( y * z ) ) * ( y * z ) .= ( x * ( y * z ) ) * ( y * z ) .= ( x * y ) * ( z * z ) ;
v . ( <* x , y *> ) - ( <* x0 , y *> ) . i = partdiff ( v , ( x - y ) ) * ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( x - x0 ) ) + ( ( proj ( 1 , 1 ) ) . ( x - x0 ) ) * ( ( proj ( 1 , 1 ) ) . ( x - x0 ) ) ;
i * i = <* 0 * ( - 1 ) - 0 * 0 , 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0
Sum ( L * F ) = Sum ( L * ( F1 ^ F2 ) ) .= Sum ( L * F1 ) + Sum ( L * F2 ) .= Sum ( L * F1 ) + Sum ( L * F2 ) .= Sum ( L * F1 ) + Sum ( L * F2 ) .= Sum ( L * F1 ) + Sum ( L * F2 ) .= Sum ( L * F1 ) + Sum ( L * F2 ) .= Sum ( L * F1 ) ;
ex r be Real st for e be Real st 0 < e ex Y1 be finite Subset of X st Y1 is non empty & ( for Y1 be finite Subset of X st Y1 is non empty & Y1 c= Y holds |. ( the carrier of X ) . Y1 - ( the carrier of X ) . Y2 .| < r )
( GoB f ) * ( i , j + 1 ) = f /. ( k + 2 ) & ( GoB f ) * ( i , j + 1 ) = f /. ( k + 1 ) or ( GoB f ) * ( i , j ) = f /. ( k + 1 ) or ( GoB f ) * ( i , j ) = f /. ( k + 1 ) ;
( ( cos . x ) ^2 ) ^2 = ( ( cos . x ) ^2 ) ^2 .= ( ( 1 / 2 ) ^2 ) ^2 .= ( ( 1 / 2 ) ^2 ) ^2 .= ( ( 1 / 2 ) ^2 ) ^2 .= ( ( 1 / 2 ) ^2 ) ^2 .= ( ( 1 / 2 ) ^2 ) ^2 .= ( 1 / 2 ) ^2 .= ( 1 / 2 ) ^2 ;
( - b + sqrt ( Let a , b , c ) ) / 2 * a < 0 & ( - b + sqrt ( a , b , c ) ) / 2 * a < 0 & ( - b + sqrt ( a , b , c ) ) / 2 * a < 0 or - b < - b & - b < c ;
ex_inf_of uparrow "\/" ( X , L ) /\ C & ex_sup_of ( X , L ) /\ C , L & "\/" ( ( subrelstr X ) /\ C ) = "/\" ( ( subrelstr X ) /\ C ) & "\/" ( ( subrelstr X ) /\ C ) = "/\" ( ( subrelstr X ) /\ C ) & "\/" ( ( subrelstr X ) /\ C ) = "/\" ( ( uparrow X ) /\ C ) ;
( for j being Element of NAT st j = i holds ( j = i implies j = i ) implies ( j = i implies j = i ) ) & ( for j being Element of NAT st j in i holds j = i implies j = i ) implies ( j = i implies j = i ) )