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contradiction ;
thesis ;
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assume not thesis ;
assume not thesis ;
B in X ;
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 one-to-one ;
b in A ;
d divides a ;
i < n ;
s <= b ;
b in B ;
let r ;
B is one-to-one ;
R is total ;
x = 2 ;
d in D ;
let c ;
let c ;
b = Y ;
0 < k ;
let b ;
let n ;
r <= b ;
x in X ;
i >= 8 ;
let n ;
let n ;
y in f ;
let n ;
1 < j ;
a in L ;
C is boundary ;
a in A ;
1 < x ;
S is finite ;
u in I ;
z << z ;
x in V ;
r < t ;
let t ;
x c= y ;
a <= b ;
m in NAT ;
assume f is prime ;
not x in Y ;
z = +infty ;
let k be Nat ;
K is being_line ;
assume n >= N ;
assume n >= N ;
assume X is \bf ) ;
assume x in I ;
q is ) by 0 ;
assume c in x ;
1-p > 0 ;
assume x in Z ;
assume x in Z ;
1 <= kr2 ;
assume m <= i ;
assume G is rng p ;
assume a divides b ;
assume P is closed ;
b-a > 0 ;
assume q in A ;
not W is bounded ;
f is Assume f is one-to-one ;
assume A is boundary ;
g is special ;
assume i > j ;
assume t in X ;
assume n <= m ;
assume x in W ;
assume r in X ;
assume x in A ;
assume b is even ;
assume i in I ;
assume 1 <= k ;
X is non empty ;
assume x in X ;
assume n in M ;
assume b in X ;
assume x in A ;
assume T c= W ;
assume s is atomic ;
b `1 <= c `1 ;
A meets W ;
i `1 <= j `1 ;
assume H is universal ;
assume x in X ;
let X be set ;
let T be DecoratedTree ;
let d be element ;
let t be element ;
let x be element ;
let x be element ;
let s be element ;
k <= 5 - 5 ;
let X be set ;
let X be set ;
let y be element ;
let x be element ;
P [ 0 ]
let E be set , A be Subset of E ;
let C be category ;
let x be element ;
let k be Nat ;
let x be element ;
let x be element ;
let e be element ;
let x be element ;
P [ 0 ]
let c be element ;
let y be element ;
let x be element ;
let a be Real ;
let x be element ;
let X be element ;
P [ 0 ]
let x be element ;
let x be element ;
let y be element ;
r in REAL ;
let e be element ;
n1 is \setminus ;
Q halts_on s ;
x in such ;
M < m + 1 ;
T2 is open ;
z in b -1 ;
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 , x be Point 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 , r ;
let E be Ordinal ;
o f2 f2 f2 f2 f2 & o <> o1 ;
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 is_<=_than w
b "/\" e = b ;
m = m3 ;
t in h . D ;
P [ 0 ] ;
assume z = x * y ;
S . n is bounded ;
let V be RealUnitarySpace , W be Subspace 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\lbrace a } <= being Real ;
R [ 0 ] ;
b in f .: X ;
assume q = q2 ;
x in [#] V ;
f . u = 0 ;
assume e1 > 0 ;
let V be RealUnitarySpace , W be Subspace of V ;
s is trivial non empty ;
dom c = Q
P [ 0 ] ;
f . n in T ;
N . j in S ;
let T be complete LATTICE , x be Element of T ;
the ObjectMap of F is one-to-one
sgn x = 1 ;
k in support a ;
1 in Seg 1 ;
rng f = X ;
len T in X ;
vbeing < n ;
Sy is bounded ;
assume p = p2 ;
len f = n ;
assume x in P1 ;
i in dom q ;
let U0 , A , B ;
pp `1 = c ;
j in dom h ;
let k ;
f | Z is continuous ;
k in dom G ;
UBD C = B ;
1 <= len M ;
p in Ball ( x , r ) ;
1 <= jj & jj <= len f ;
set A = [: Seg n , Seg n :] ;
card a [= c ;
e in rng f ;
cluster B \oplus A -> empty ;
H is with_no that H is has 0. F ;
assume n0 <= m ;
T is increasing implies T is increasing
e2 <> e2 ;
Z c= dom g ;
dom p = X ;
H is proper implies H is proper
i + 1 <= n ;
v <> 0. V ;
A c= Affin A ;
S c= dom F ;
m in dom f ;
let X0 be set ;
c = sup N ;
R is connected implies union M is connected
assume not x in REAL ;
Im f is complete ;
x in Int y ;
dom F = M ;
a in On W ;
assume e in A ( ) ;
C c= C-26 ;
mm <> {} & m <> 0 ;
let x be Element of Y ;
let f be ) being ) Chain Chain of G , n ;
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 being set ;
- y in I ;
let A be non empty set , f be Function of A , REAL ;
Px0 = 1 ;
assume r in F . k ;
assume f is simple function in S ;
let A be be non empty p2 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 , A , B ;
assume that 1 <= j and j < l ;
v = - u ;
assume s . b > 0 ;
d1 in dom f ;
assume t . 1 in A ;
let Y be non empty TopSpace , x be Point of Y ;
assume a in uparrow s ;
let S be non empty Poset ;
a , b // b , a ;
a * b = p * q ;
assume x , y are_the space of V ;
assume x in Omega ( f ) ;
[ a , c ] in X ;
mm <> {} & m in dom F ;
M + N c= M + M ;
assume M is \ast one-to-one ;
assume f is additive inbr-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 or k2 = k2 ;
m + 1 < n + 1 ;
s in S \/ { s } ;
n + i >= n + 1 ;
assume Re y = 0 ;
k1 <= j1 & j1 <= j2 ;
f | A is compact ;
f . x - a <= b ;
assume y in dom h ;
x * y in B1 ;
set X = Seg n ;
1 <= i2 + 1 ;
k + 0 <= k + 1 ;
p ^ q = p ;
j |^ y divides m ;
set m = max A ;
[ x , x ] in R ;
assume x in succ 0 ;
a in sup phi ;
Cj in X ;
q2 c= C1 & q2 c= C2 ;
a2 < c2 & a2 < b2 ;
s2 is 0 -started ;
IC s = 0 & IC s = 0 ;
s4 = s4 , P4 = s4 ;
let V , W ;
let x , y be element ;
let x be Element of T ;
assume a in rng F ;
x in dom T `1 ;
let S be * ;
y " <> 0 ;
y " <> 0 ;
0. V = u-w ;
y2 , y , z is_collinear ;
R8 in dom f ;
let a , b be Real , x be Point of TOP-REAL 2 ;
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 set ;
s4 . n = N ;
set y = ( x `1 ) / ( x `2 ) ;
mi in dom g ;
l . 2 = y1 ;
|. g . y .| <= r ;
f . x in Cx0 ;
V-19 is non empty ;
let x be Element of X ;
0 <> f . g2 ;
f2 /* q is convergent ;
f . i is_measurable_on E ;
assume \xi in N-22 ;
reconsider i = i as Ordinal ;
r * v = 0. X ;
rng f c= INT & rng g c= INT ;
G = 0 .--> goto 0 ;
let A be Subset of X ;
assume A0 is dense & A is dense ;
|. f . x .| <= r ;
let x be Element of R ;
let b be Element of L ;
assume x in W-19 ;
P [ k , a ] ;
let X be Subset of L ;
let b be Object of B ;
let A , B be category ;
set X = Vars ( C ) ;
let o be OperSymbol of S ;
let R be connected non empty Poset ;
n + 1 = succ n ;
x-21 c= Z1 & xY c= Z1 ;
dom f = C1 & dom g = C2 ;
assume [ a , y ] in X ;
Re ( seq ) is convergent & lim ( Im seq ) = 0 ;
assume a1 = b1 & a2 = b2 ;
A = sInt ( A ) ;
a <= b or b <= a ;
n + 1 in dom f ;
let F be Instruction of S , m be Nat ;
assume that r2 > x0 and x0 in dom f ;
let Y be non empty set , f be Function of Y , BOOLEAN ;
2 * x in dom W ;
m in dom g2 & n + 1 in dom g2 ;
n in dom g1 /\ dom g2 ;
k + 1 in dom f ;
not the still of s is finite ;
assume that x1 <> x2 and x1 <> x3 ;
v3 in Vx0 \/ Vx0 ;
not [ b `1 , b `2 ] in T ;
( i + 1 ) = i ;
T c= `2 & T c= T ;
( l - 1 ) * ( l - 1 ) = 0 ;
let n be Nat ;
t `2 = r `2 & t `1 = r ;
AA is_integrable_on M & F is integrable ;
set t = Top t ;
let A , B be real-membered set ;
k <= len G + 1 ;
cC misses cV ;
product ( s ) is non empty ;
e <= f or f <= e ;
cluster non empty normal -> normal for Ordinal ;
assume c2 = b2 & c2 = b2 ;
assume h in [. q , p .] ;
1 + 1 <= len C ;
not c in B . m1 ;
cluster R .: X -> empty ;
p . n = H . n ;
assume that v- is convergent and v- is convergent ;
IC s3 = 0 .= IC s3 ;
k in N or k in K ;
F1 \/ F2 c= F ;
Int G1 <> {} & Int G2 <> {} ;
z `2 = 0 & z `2 = 0 ;
p00 `1 <> p1 `1 & p1 `1 <> p1 `1 ;
assume z in { y , w } ;
MaxADSet ( a ) c= F ;
ex_sup_of downarrow s , S ;
f . x <= f . y ;
let T be up-complete non empty reflexive transitive antisymmetric RelStr ;
q |^ m >= 1 ;
a is_>=_than X & b is_>=_than Y ;
assume <* a , c *> <> {} ;
F . c = g . c ;
G is one-to-one one-to-one ;
A \/ { a } \not c= B ;
0. V = 0. Y ;
let I be be be be be be be halting Instruction of S , k be Nat ;
f-24 . x = 1 ;
assume z \ x = 0. X ;
C2 = 2 to_power n ;
let B be sequence of Sigma ;
assume X1 = p .: D ;
n + l2 in NAT & n + l2 in NAT ;
f " P is compact & f " P is compact ;
assume x1 in [: REAL , REAL :] ;
p1 = K1 & p2 = K1 ;
M . k = <*> ( the carrier of K ) ;
phi . 0 in rng phi ;
OSMInt A is 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 ;
cR is stable Subset of R ;
set cR = Vertices R , cS = Vertices R , R = Vertices S , R = S \/ S , T = S \/ S , R = S \/ S
px0 c= P3 & px0 c= P3 ;
x in [. 0 , 1 .[ ;
f . y in dom F ;
let T be Scott Scott Scott Scott TopAugmentation of S ;
ex_inf_of the carrier of S , S ;
downarrow a = downarrow b & downarrow b = downarrow a ;
P , C , K is_collinear ;
assume x in F ( s , r , t ) ;
2 to_power i < 2 to_power m ;
x + z = x + z + q ;
x \ ( a \ x ) = x ;
||. x-y .|| <= r ;
assume that Y c= field Q and Y <> {} ;
a ~ , b ~ are_equipotent ;
assume a in A ( ) ;
k in dom ( q | k ) ;
p is \HM { finite } ;
i -' 1 = i-1 - 1 ;
f | A is one-to-one ;
assume x in f .: X ( ) ;
i2 - i1 = 0 & i2 - i2 = 0 ;
j2 + 1 <= i2 + 1 ;
g " * a in N ;
K <> { [ {} , {} ] } ;
cluster strict for for for of O ;
|. q .| ^2 > 0 ;
|. p4 .| = |. p .| ;
s2 - s1 > 0 & s2 - s1 > 0 ;
assume x in { Gik } ;
W-min C in C & W-min C in C ;
assume x in { Gik } ;
assume i + 1 = len G ;
assume i + 1 = len G ;
dom I = Seg n & dom I = 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 , n ;
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 void non void holds S is holds S is non void
let f be ManySortedSet of I ;
let z be Element of COMPLEX , x be Element of COMPLEX ;
u in { ag } ;
2 * n < 2 * n ;
let x , y be set ;
B-11 c= V-15 \/ { x }
assume I is_halting_on s , P ;
UA = [: U2 , U2 :] ;
M /. 1 = z /. 1 ;
x11 = x22 & x22 = x22 ;
i + 1 < n + 1 + 1 ;
x in { {} , <* 0 *> } ;
( f . ( len f ) ) `1 <= ( f . ( len f ) ) `1 ;
let l be Element of L ;
x in dom ( F . n ) ;
let i be Element of NAT , x be Element of NAT ;
seq1 is COMPLEX & seq2 is COMPLEX implies seq1 - seq2 is convergent
assume <* o2 , o *> <> {} ;
s . x |^ 0 = 1 ;
card K1 in M & card K1 in M ;
assume X in U & Y in U ;
let D be Subset-Family of Omega ;
set r = - { k + 1 } ;
y = W . ( 2 * x ) ;
assume dom g = cod f & cod g = cod g ;
let X , Y be non empty TopSpace , x be Point of X , y be Point of Y ;
x ++ A is interval ;
|. <*> A .| . a = 0 ;
cluster strict for Sublattice of L ;
a1 in B . s1 & a2 in B . s1 ;
let V be finite linearly-independent over F , v be Vector of V ;
A * B on B implies B on A
f-3 = NAT --> 0 .= f-3 ;
let A , B be Subset of V ;
z1 = P1 . j & z2 = P2 . j ;
assume f " P is closed & f " P is closed ;
reconsider j = i as Element of M ;
let a , b be Element of L ;
assume q in A \/ ( B "\/" C ) ;
dom ( F * C ) = o ;
set S = INT |^ X , 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 , f be Function of B , C ;
( PI / 2 ) * PI < Arg z ;
reconsider z9 = 0 , z9 = 1 as Nat ;
LIN a , d , c ;
[ y , x ] in II ;
Q * ( 1 , 3 ) `2 = 0 ;
set j = x0 gcd m , m = x0 gcd m ;
assume a in { x , y , c } ;
j2 - jj > 0 & j2 - jj > 0 ;
I I e = 1 & I = 1 ;
[ y , d ] in F-8 ;
let f be Function of X , Y ;
set A2 = ( B - C ) / ( A + B ) ;
s1 , s2 are_) implies s1 , s2 are_0. R
j1 -' 1 = 0 - 1 .= j1 - 1 ;
set m2 = 2 * n + j ;
reconsider t = t as bag of n ;
I2 . j = m . j ;
i |^ s , n are_relative_prime & i |^ s divides n ;
set g = f | D-21 , h = g | D-21 ;
assume X is lower bounded & 0 <= r ;
p1 `1 = 1 & p1 `2 = - 1 ;
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 G ;
1 <= i1 -' 1 & i1 + 1 <= len G ;
i + i2 <= len h - 1 ;
x = W-min ( P ) & y = W-min ( P ) ;
[ x , z ] in [: X , Z :] ;
assume y in [. x0 , x .] ;
assume p = <* 1 , 2 , 3 *> ;
len <* A1 *> = 1 & len <* A2 *> = 1 ;
set H = h . gg , I = h . W ;
card b * a = |. a .| ;
Shift ( w , 0 ) |= v ;
set h = h2 (*) h1 , h1 = h2 (*) h2 ;
assume x in X3 /\ X3 ;
||. h .|| < dx0 & ||. h .|| < dx0 ;
not x in the carrier of f & not x in the carrier of g ;
f . y = F ( y ) ;
for n holds X [ n ] ;
k - l = kbeing - k\leq ;
<* p , q *> /. 2 = q ;
let S be Subset of the carrier of Y ;
let P , Q be \langle 9 , s *> ;
Q /\ M c= union ( F | M )
f = b * ( canFS ( S ) ) ;
let a , b be Element of G ;
f .: X is_<=_than f . sup X
let L be non empty transitive reflexive RelStr , x be Element of L ;
S-20 is x -8 -basis i
let r be non positive Real ;
M , v |= x \hbox { y } ;
v + w = 0. ( Z , x ) ;
P [ len F ( ) ] implies P [ len F ( ) ]
assume InsCode ( i ) = 8 & 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 [#] S -> [#] such for Element of S ;
reconsider l1 = l- 1 as Nat ;
v4 is Vertex of r2 & v4 is Vertex of r2 ;
T2 is SubSpace of T2 implies ( T1 is SubSpace of T2 )
Q1 /\ Q19 <> {} & Q29 /\ Q29 <> {} ;
let k be Nat ;
q " is Element of X & q " is Element of Y ;
F . t is set of non zero set ;
assume that n <> 0 and n <> 1 ;
set en = EmptyBag n , en = EmptyBag n ;
let b be Element of Bags n ;
assume for i holds b . i is commutative ;
x is root of ( p `2 ) `2 , ( p `2 ) `2 ;
not r in ]. p , q .[ ;
let R be FinSequence of REAL , x be Element of REAL ;
S7 does not destroy b1 & not I does not destroy b1 ;
IC SCM R <> a & IC SCM R <> a ;
|. - |[ x , y ]| .| >= r ;
1 * ( s - 1 ) = ( s - 1 ) * ( s - 1 ) ;
let x be FinSequence of NAT , n be Nat ;
let f be Function of C , D , g be Function of C , D ;
for a holds 0. L + a = a
IC s = s . NAT .= IC s .= IC s ;
H + G = F- ( G-G ) ;
CA1 . x = x2 & CA2 . x = y2 ;
f1 = f .= f2 .= f2 * f1 .= f2 * f1 ;
Sum <* p . 0 *> = p . 0 ;
assume v + W = v + u + W ;
{ a1 } = { a2 } ;
a1 , b1 _|_ b , a ;
d3 , o _|_ o , a3 ;
IO is reflexive & IO is transitive implies IO is reflexive
IO is antisymmetric implies [: O , O :] is antisymmetric
sup rng H1 = e & sup rng H2 = e ;
x = ( a * a ) * ( a * b ) ;
|. p1 .| ^2 >= 1 ^2 ;
assume j2 -' 1 < 1 & j2 -' 1 < len f ;
rng s c= dom f1 /\ dom f2 ;
assume that support a misses support b and support b misses support b ;
let L be associative commutative distributive non empty doubleLoopStr , p be Polynomial of L ;
s " + 0 < n + 1 ;
p . c = ( f " ) . 1 ;
R . n <= R . ( n + 1 ) ;
Directed ( I1 , I2 ) = I1 +* I2 ;
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 non trivial ;
( 1 - a ) " = a " ;
( q . {} ) `1 = o ;
( - i ) - 1 > 0 ;
assume ( 1 - r ) * t `1 <= 1 ;
card B = k + 1-1 ;
x in union rng ( f | ( len f -' 1 ) ) ;
assume x in the carrier of R & y in the carrier of R ;
d in X ;
f . 1 = L . ( F . 1 ) ;
the vertices of G = { v } \/ { v } ;
let G be st G is st ww] holds G is connected ;
e , v6 be set ;
c . ( i9 - 1 ) in rng c ;
f2 /* ( f1 /* q ) is divergent_to-infty & f2 /* ( f1 /* q ) is divergent_to-infty ;
set z1 = - z2 , z2 = - z2 ;
assume w is_llof S , G ;
set f = p |-count ( t - 1 ) , g = p |-count ( t - 1 ) , h = p |-count ( t - 1 ) , h = p |-count ( t - 1 ) , n
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 IB be Subset-Family of X , x be Point of X ;
reconsider p = p as Element of NAT ;
let v , w be Point of X ;
let s be State of SCM+FSA , a be Int-Location ;
p is FinSequence of ( the carrier of SCM+FSA ) \ { IC SCM+FSA } ;
stop I ( ) c= PIj ( ) ;
set ci = fbeing /. i , fi = f^2 ;
w ^ t ^ t ^ s ^ t ^ s ^ t ^ t ^ s ^ t ^ t ^ t ^ s ^ t ^ t ^ t ^ t ^ t ^ s ^ t ^ t ^
W1 /\ W = W1 /\ W2 ` .= W2 /\ W1 ` ;
f . j is Element of J . j ;
let x , y be \rm \cdot of T2 ;
ex d st a , b // b , d ;
a <> 0 & b <> 0 & c <> 0
ord x = 1 & x is dom x implies x is dom x
set g2 = lim ( seq ^\ k ) , g1 = lim ( seq ^\ k ) ;
2 * x >= 2 * ( 1 / 2 ) ;
assume ( a 'or' c ) . z <> TRUE ;
f (*) g in Hom ( c , c ) ;
Hom ( c , c + d ) <> {} ;
assume 2 * Sum ( q | m ) > m ;
L1 . ( F-21 ) = 0 .= 0 ;
/ ( X \/ R1 ) = / ( X \/ R1 ) ;
( sin . x ) <> 0 & ( cos . x ) <> 0 ;
( #Z n ) . x > 0 & ( #Z n ) . x > 0 ;
o1 in X-5 /\ O2 & o2 in XO2 /\ O2 ;
e , v6 be set ;
r3 > ( 1 - 2 ) * 0 ;
x in P .: ( F -ideal ( x ) ) ;
let J be closed non empty Subset of R ;
h . p1 = f2 . O & h . p2 = g2 . O ;
Index ( p , f ) + 1 <= j ;
len ( q | ( len M ) ) = width M ;
the carrier of LK c= A & the carrier of LK c= A ;
dom f c= union rng ( F | ( n + 1 ) ) ;
k + 1 in support ( support ( n ) ) ;
let X be ManySortedSet of the carrier of S ;
[ x `1 , y `2 ] in an an an an an an of R ;
i = D1 or i = D2 or i = D1 ;
assume a mod n = b mod n & b mod n = b mod n ;
h . x2 = g . x1 & h . x2 = g . x2 ;
F c= 2 -tuples_on the carrier of X & F is one-to-one ;
reconsider w = |. s1 .| as Real_Sequence of X , Y ;
( 1 - m ) * ( m + r ) < p ;
dom f = dom Icn & dom g = dom Icn ;
[#] P-17 = [#] ( ( TOP-REAL 2 ) | K1 ) ;
cluster - x -> ExtReal means : Def6 : x is ExtReal ;
then { db } c= A & A is closed ;
cluster ( TOP-REAL n ) | A -> finite-ind for Subset of TOP-REAL n ;
let w1 be Element of M ;
let x be Element of dyadic ( n ) ;
u in W1 & v in W3 implies u in W1 + W2
reconsider y = y as Element of L2 ;
N is full SubRelStr of T |^ the carrier of S ;
sup { x , y } = c "\/" c ;
g . n = n to_power 1 .= n ;
h . J = EqClass ( u , J ) ;
let seq be summable sequence of X , n be Nat ;
dist ( x `1 , y ) < r / 2 ;
reconsider mm1 = m - 1 as Element of NAT ;
( - x0 ) < r1 - x0 & x0 < r2 - x0 ;
reconsider P ` = P ` as strict Subgroup of N ;
set g1 = p * idseq ( q `1 ) , g2 = p * idseq ( q `2 ) ;
let n , m , k be non zero Nat ;
assume that 0 < e and f | A is lower ;
D2 . ( I8 + 1 ) in { x } ;
cluster subcondensed for Subset of T ;
let P be compact non empty Subset of TOP-REAL 2 , p1 , p2 , p3 be Point of TOP-REAL 2 ;
Gik in LSeg ( PI , 1 ) /\ LSeg ( Gik , Gij ) ;
let n be Element of NAT , x be Element of X ;
reconsider SS = S , SS = 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 implies a , b , x is_collinear
assume that i = n \/ { n } and j = k \/ { k } ;
let f be PartFunc of X , Y ;
x0 >= ( sqrt c / sqrt 2 ) * ( sqrt 2 / 2 ) ;
reconsider t7 = T7 as TopSpace ;
set q = h * p ^ <* d *> ;
z2 in U . ( x4 , y2 ) /\ Q2 ;
A |^ 0 = { <%> E } .= { <%> E } ;
len W2 = len W + 2 .= 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 & z in dom g ;
assume that p2 = E-max ( K ) and p2 <> W-min ( K ) ;
len G + 1 <= i1 + 1 ;
f1 (#) f2 is_differentiable_in x0 & f2 (#) f1 is_differentiable_in x0 implies f1 (#) f2 is_differentiable_in x0
cluster s-10 + sT -> summable for Real_Sequence ;
assume j in dom ( M1 * ( i , j ) ) ;
let A , B , C be Subset of X ;
let x , y , z be Point of X , p be Point of Y ;
b ^2 - ( 4 * a * c ) >= 0 ;
<* x/y *> ^ <* y *> ^ <* y *> ^ <* x *> ^ <* y *> ^ <* x *> ^ <* y *> ^ <* x *> ^ <* x *> ^ <* y *> ^ <* x *> ^ <*
a , b in { a , b } ;
len p2 is Element of NAT & len p2 = len f ;
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 , P2 = P +* I ;
consider w being Nat such that q = z + w ;
x ` is Element of x & y is Element of L ;
k = 0 & n <> k or k > n ;
then X is discrete for X is closed ;
for x st x in L holds x is FinSequence of L
||. f /. c .|| <= r1 & ||. f /. c .|| <= r2 ;
c in uparrow p & not c in { p } ;
reconsider V = V as Subset of the topology of TOP-REAL n ;
let N , M be being being being being being being being being being being being being being being being being being being being being being being Subset of L ;
then z is_>=_than waybelow x & z is_>=_than compactbelow y ;
M \lbrack f .] = f & M [. g , g .] = g ;
( ( L /. 1 ) ) `2 = TRUE ;
dom g = dom f /\ X .= dom f /\ X ;
mode ^ is * \rm \vert of G is * ;
[ i , j ] in Indices M & [ i , j ] in Indices M ;
reconsider s = x " , t = y " as Element of H ;
let f be Element of dom ( Subformulae p ) , g , h ;
F1 . ( a1 , - a1 ) = G1 * ( a1 , - a1 ) ;
redefine func Sphere ( a , b , r ) -> compact Subset of TOP-REAL 2 ;
let a , b , c , d be Real ;
rng s c= dom ( 1 / ( n + 1 ) ) ;
curry ( ( F . -19 ) , k ) is additive ;
set k2 = card dom B , k2 = card dom C , k2 = card D , F = B * C ;
set G = ( V ) . s ;
reconsider a = [ x , s ] as Object of G ;
let a , b be Element of Mv , x be Element of Mv ;
reconsider s1 = s , s2 = t as Element of S1 ;
rng p c= the carrier of L & p . 1 = x ;
let d be Subset of the bound of A ;
( x .|. x = 0 iff x = 0. W ) ;
I-21 in dom stop I & I-21 in dom stop I ;
let g be continuous Function of X | B , Y ;
reconsider D = Y as Subset of ( TOP-REAL n ) | D ;
reconsider i0 = len p1 , i0 = len p2 as Integer ;
dom f = the carrier of S & dom g = the carrier of S ;
rng h c= union ( the carrier of J . i ) ;
cluster All ( x , H ) -> Carrier \sqrt ' ;
d * N1 ^2 > N1 * 1 / ( d * N2 ) ;
]. a , b .[ c= [. a , b .] ;
set g = f " D1 , h = g " D2 ;
dom ( p | mm1 ) = mm1 .= dom ( p | mm1 ) ;
3 + - 2 <= k + - 2 ;
tan is_differentiable_in ( arccot - arccot ) . x & tan . x > 0 ;
x in rng ( f /^ ( n -' 1 ) ) ;
let f , g be FinSequence of D ;
cp in the carrier of S1 & cp in the carrier of S2 ;
rng f " = dom f & rng f = rng g ;
( the Target of G ) . e = v & ( the Target of G ) . e = v ;
width G -' 1 < width G -' 1 & width G -' 1 < width G ;
assume v in rng ( S | E1 ) & u in rng ( S | E1 ) ;
assume x is root or x is root or x is root ;
assume 0 in rng ( g2 | A ) & 0 < r ;
let q be Point of ( TOP-REAL 2 ) | K1 , r be Real ;
let p be Point of ( TOP-REAL 2 ) | K1 , r be Real ;
dist ( O , u ) <= |. p2 .| + 1 ;
assume dist ( x , b ) < dist ( a , b ) ;
<* S7 *> is_in the carrier of C-20 & <* S7 *> is in the carrier of C-20 ;
i <= len G-6 -' 1 + 1 - 1 ;
let p be Point of ( TOP-REAL 2 ) | K1 , r be Real ;
x1 in the carrier of I[01] & x2 in the carrier of I[01] ;
set p1 = f /. i , p2 = f /. ( i + 1 ) ;
g in { g2 : r < g2 & g2 < r } ;
Q2 = Sp2 " ( Q /\ S ) .= Sp2 " ( Q /\ S ) ;
( ( 1 / 2 ) |^ ( n + 1 ) ) is summable ;
- p + I c= - p + A ;
n < LifeSpan ( P1 , s1 ) + 1 ;
CurInstr ( p1 , s1 ) = i .= halt SCM+FSA ;
A /\ Cl { x } \ { x } <> {} ;
rng f c= ]. r - 1 , r + 1 .[ ;
let g be Function of S , V ;
let f be Function of L1 , L2 , g be Function of L1 , L2 ;
reconsider z = z as Element of CompactSublatt L ;
let f be Function of S , T ;
reconsider g = g as Morphism of c opp , b opp ;
[ s , I ] in [: S , b :] ;
len ( the connectives of C ) = 4 & len ( the connectives of C ) = 5 ;
let C1 , C2 be subFunctor of C , D ;
reconsider V1 = V , V2 = V as Subset of X | B ;
attr p is valid means : Def6 : 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 p1 of O , E , A2 be Element of E ;
p2 , r3 , q2 is_collinear & p1 , p2 , p2 is_collinear implies p1 = p2
consider x being element such that x in v ^ K ;
not x in { 0. TOP-REAL 2 } & not x in { 0. TOP-REAL 2 } ;
p in [#] ( ( I[01] | B11 ) | B11 ) ;
0 . ( M . E ) < M . ( E . ( M . E ) ) ;
op ( c ) |^ ( c , c ) = c ;
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . s2 ) & a2 in dom ( F . s2 ) ;
cluster \rangle -> Nat for *> -\subseteq the ` of L ;
set i1 = the Nat , i2 = the Nat , i1 = the Element of NAT ;
let s be 0 -started State of SCM+FSA , a be Int-Location ;
assume y in ( f1 \/ f2 ) .: A ;
f . len f = f /. len f .= f /. 1 .= f /. 1 ;
x , f . x '||' f . x , f . y ;
pred X c= Y means : Def6 : cos | X c= cos | Y ;
let y be upper Subset of Y , x be Element of X ;
cluster -> -> sup for Element of x `2 , x `2 , y `2 , z `2 , x `2 , y `1 , z `2 , y `2 , z `2 } ;
set S = <* Bags n , ij *> , S = <* <* j *> *> ;
set T = [. 0 , 1 / 2 .] , S = ]. 0 , 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 f2 ;
O c= dom I & { {} } = { {} } ;
( the Source of G ) . x = v & ( the Target of G ) . x = v ;
{ HT ( f , T ) } c= Support f \/ Support g ;
reconsider h = R . k as Polynomial of n , L ;
ex b being Element of G st y = b * H ;
let x `1 , y `1 , z `2 be Element of G opp ;
h19 . i = f . ( h . i ) ;
p `1 = p1 `1 & p `2 = p1 `2 or p `1 = p1 `2 & p `2 = p1 `2 ;
i + 1 <= len Cage ( C , n ) ;
len <* P *> @ = len P & len <* P *> = len P ;
set N-26 = the \subseteq of the \subseteq of N , N = the carrier of N ;
len g- ( x + 1 ) - 1 <= x ;
a on B & b on B implies not a on B
reconsider rj = 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 C , q2 = q2 `2 , p1 = p2 `2 ;
set S =
MaxADSet ( b ) c= MaxADSet ( P /\ Q ) ;
Cl ( G . q1 ) c= F . r2 & Cl ( G . q2 ) c= G . r2 ;
f " D meets h " V & f " D meets h " V ;
reconsider D = E as non empty directed Subset of L1 ;
H = ( the_left_argument_of H ) '&' ( the_right_argument_of H ) ;
assume t is Element of ( the carrier of S ) * ;
rng f c= the carrier of S2 & rng g c= the carrier of S2 ;
consider y being Element of X such that x = { y } ;
f1 . ( a1 , b1 ) = b1 & f1 . ( b1 , b2 ) = b2 ;
the carrier' of G `1 = E \/ { E } .= { E } ;
reconsider m = len ( thesis - k ) as Element of NAT ;
set S1 = LSeg ( n , UMP C ) , S2 = LSeg ( f , n ) ;
[ i , j ] in Indices M1 & [ i , j ] in Indices M1 ;
assume that P c= Seg m and M is \HM { \mathbb N } ;
for k st m <= k holds z in K . k ;
consider a being set such that p in a and a in G ;
L1 . p = p * L /. ( 1 + 1 ) ;
p-7 . i = pi1 . i .= pi2 . i ;
let PA , PA , G be a_partition of Y , a be Element of Y ;
pred 0 < r & r < 1 & 1 < r & r < 1 ;
rng ( the \HM { p2 } \HM { , } X ) = [#] X ;
reconsider x = x , y = y as Element of K ;
consider k such that z = f . k and n <= k ;
consider x being element such that x in X \ { p } ;
len ( canFS ( s ) ) = card s - 1 .= card ( rng s ) ;
reconsider x2 = x1 , y2 = x2 as Element of L2 ;
Q in FinMeetCl ( the topology of X ) & Q in FinMeetCl the topology of Y ;
dom fx0 c= dom ux0 & dom fx0 c= dom fx0 ;
pred n divides m & m divides n & n = m ;
reconsider x = x , y = y as Point of [: I[01] , I[01] :] ;
a in \mathop { \rm every set , T2 , T1 } ;
not y0 in the still of f & not y0 in the still of f implies f . y0 in the carrier of f
Hom ( ( a , b ) ~ , c ) <> {} ;
consider k1 such that p " < k1 and k1 < len f ;
consider c , d such that dom f = c \ d ;
[ x , y ] in [: dom g , dom k :] ;
set S1 = .: .: ( x , y , z ) ;
l2 = m2 & l1 = i2 & l2 = j2 implies C * ( i1 , i2 ) = C * ( i2 , j2 )
x0 in dom u01 /\ ( A /\ B ) & x0 in dom u01 /\ B ;
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 `1 <= f . p1 `1 & f . p2 `2 <= f . p3 `2 ;
( ( F . n ) `1 ) ^2 <= ( x `1 ) ^2 ;
x `2 = ( W7 - ( W8 - ( - ( - 1 ) ) ) ) / ( 2 * ( - ( - 1 ) ) ) ;
for n being Element of NAT holds P [ n ] implies P [ n + 1 ]
let J , K be non empty Subset of I ;
assume 1 <= i & i <= len <* a " *> ;
0 |-> a = <*> ( the carrier of K ) .= <*> ( the carrier of K ) ;
X . i in 2 -tuples_on A . i \ B . i ;
<* 0 *> in dom ( e --> [ 1 , 0 ] ) ;
then P [ a ] & P [ succ a ] implies P [ succ a ]
reconsider sp1 = s\overline ( { x } ) as \rangle of D ;
( - i - 1 ) <= len ( - j ) ;
[#] S c= [#] the TopStruct of T & [#] T c= [#] T ;
for V being strict RealUnitarySpace holds V in thesis & V in the carrier of V implies V is strict Subspace of V
assume k in dom mid ( f , i , j ) ;
let P be non empty Subset of TOP-REAL 2 , p1 , p2 , p3 be Point of TOP-REAL 2 ;
let A , B be square Matrix of n1 , n2 , K ;
- a * - b = a * b - b * a ;
for A being Subset of AS holds A // A & A // K implies A // K
( for o2 being object of o2 holds o2 in <^ o2 , o2 ^> implies o2 = o2 )
then ||. x .|| = 0 & x = 0. X ;
let N1 , N2 be strict normal Subgroup of G , a be Element of G ;
j >= len upper_volume ( g , D1 ) & len upper_volume ( g , D2 ) = len D2 ;
b = Q . ( len Qb - 1 ) .= Q . ( len Qb - 1 ) ;
f2 * f1 /* s is divergent_to-infty & f2 * f1 /* s is divergent_to-infty ;
reconsider h = f * g as Function of [: N2 , N :] , G ;
assume that a <> 0 and Let a , b , c ;
[ t , t ] in the Relation of A & [ t , t ] in the InternalRel of A ;
( v |-- E ) | n is Element of T7 & ( v |-- E ) | n is Element of T7 ;
{} = the carrier of L1 + L2 .= the carrier of L1 + L2 .= the carrier of L2 ;
Directed I is_closed_on Initialized s , P & Directed I is_halting_on Initialized s , P ;
Initialized p = Initialize ( p +* q ) .= Initialize ( p +* q ) ;
reconsider N2 = N1 , N2 = N2 as strict net of R2 ;
reconsider Y = Y as Element of <* Ids L , \subseteq \rangle ;
"/\" ( uparrow p \ { p } ) <> p ;
consider j being Nat such that i2 = i1 + j and j in dom f ;
not [ s , 0 ] in the carrier of S2 & not [ s , 0 ] in the carrier of S2 ;
mm in ( B '/\' C ) '/\' D \ { {} } ;
n <= len ( P + ) - len ( P + Q ) + 1 ;
x1 `1 = x2 `1 & x1 `2 = y2 & x1 `2 = y2 ;
InputVertices S = { x1 , x2 } .= { x1 , x2 } ;
let x , y be Element of FFTTT1 ( n ) ;
p = |[ p `1 , p `2 ]| & p = |[ p `1 , p `2 ]| ;
g * 1_ G = h " * g * h .= h " * g * h ;
let p , q be Element of is Element of is Element of is Element of is Element of Fin ( V , C ) ;
x0 in dom x1 /\ dom x2 & x0 in dom x1 /\ dom x2 ;
( R qua Function ) " = R " * ( R * ( R * ( i , j ) ) ) ;
n in Seg len ( f /^ ( len f -' 1 ) ) ;
for s being Real st s in R holds s <= s2 implies s <= s2
rng s c= dom ( f2 * f1 ) /\ dom ( f2 * f2 ) ;
synonym \mathop { \rm for for for for for for for for for \rm for 2 -NAT ( 2 ) ;
1. ( K , n ) * 1. ( K , n ) = 1. ( K , n ) ;
set S = Segm ( A , P1 , Q1 ) , T = Segm ( A , P1 , Q1 ) ;
ex w st e = ( w - f ) / ( w - f ) & w in F ;
curry ( P' , k , k ' ) # x is convergent ;
cluster open -> open for Subset of T\sigma ( n ) ;
len f1 = 1 .= len f3 + 1 .= len f3 + 1 .= len f3 + 1 ;
( i * p ) / p < ( 2 * p ) / p ;
let x , y be Element of OSSub ( U0 ) ;
b1 , c1 // b9 , c9 & b1 , c1 // b9 , c ;
consider p being element such that c1 . j = { p } ;
assume f " { 0 } = {} & f is total & f is total ;
assume IC Comput ( F , s , k ) = n & IC Comput ( F , s , k ) = k ;
Reloc ( J , card I ) does not destroy a implies I " ;
Macro ( card I + 1 ) does not destroy c ;
set m3 = LifeSpan ( p3 , s3 ) , m3 = LifeSpan ( p2 , s3 ) ;
IC SCMPDS in dom Initialize ( p ) & IC Comput ( p , s , k ) in dom I ;
dom t = the carrier of SCM R & dom t = the carrier of SCM R ;
( ( N-min L~ f ) .. f ) .. f = 1 & ( ( N-min L~ f ) .. f ) .. f = 1 ;
let a , b be Element of is Element of is Element of is Element of is Element of Fin ( V , C ) ;
Cl Int ( union F ) c= Cl Int ( union F ) ;
the carrier of X1 union X2 misses ( ( X1 union X2 ) \/ ( X2 union X1 ) ) ;
assume not LIN a , f . a , g . a ;
consider i being Element of M such that i = d6 and i in M ;
then Y c= { x } or Y = {} or Y = { x } ;
M , v / ( y , x ) |= H1 / ( y , x ) ;
consider m being element such that m in Intersect ( FF ) and m in Y ;
reconsider A1 = support u1 , A2 = support u2 as Subset of X ;
card ( A \/ B ) = k-1 + ( 2 * 1 ) ;
assume that a1 <> a3 and a2 <> a4 and a3 <> a4 and a4 <> a5 ;
cluster s -[#] S -> $ for string of S ;
Ln2 /. n2 = Ln2 . n2 .= Ln2 . n2 .= Ln2 . n2 ;
let P be compact non empty Subset of TOP-REAL 2 , p1 , p2 , p3 be Point of TOP-REAL 2 ;
assume that r-7 in LSeg ( p1 , p2 ) and rg2 in LSeg ( p1 , p2 ) ;
let A be non empty compact Subset of TOP-REAL n , a , b be Real ;
assume [ k , m ] in Indices DA1 & [ k , m ] in Indices DA2 ;
0 <= ( ( 1 / 2 ) |^ p ) / ( 2 |^ p ) ;
( F . N ) | E8 . x = +infty ;
pred X c= Y & Z c= V & X \ V c= Y \ Z ;
y `2 * ( z `2 ) * ( z `1 ) <> 0. I ;
1 + card X-18 <= card u + card X-18 - card Xe ;
set g = z \circlearrowleft ( ( L~ z ) .. z ) , p = z /. 1 , q = z /. len z , r = ( L~ z ) .. z , s = ( L~ z ) .. z , q = ( L~ z ) .. z , p =
then k = 1 & p . k = <* x , y *> . k ;
cluster total for Element of C -\mathopen the carrier of X , the carrier of Y ;
reconsider B = A as non empty Subset of ( TOP-REAL n ) | A ;
let a , b , c be Function of Y , BOOLEAN , p be Function of Y , BOOLEAN ;
L1 . i = ( i .--> g ) . i .= g . i ;
Plane ( x1 , x2 , x3 , x4 ) c= P & Plane ( x1 , x2 , x3 , x4 ) c= P ;
n <= indx ( D2 , D1 , j1 ) + 1 - 1 ;
( ( g2 . O ) `1 ) ^2 = - 1 & ( ( g2 . O ) `2 ) ^2 = 1 ;
j + p .. f -' len f <= len f - len f + 1 - len f ;
set W = W-bound C , E = E-bound C , N = N-bound C , N = N-bound C , S = E-bound C , N = N-bound C , S = N-bound C , N = N-bound C , N = N-bound C , N = N-bound C , S = E-bound
S1 . ( a `1 , e `2 ) = a + e `2 .= a `1 ;
1 in Seg width ( M * ( ColVec2Mx p ) ) & 1 in Seg width ( M * ( ColVec2Mx p ) ) ;
dom ( i (#) Im f ) = dom Im f /\ dom Im f ;
( ^2 ) . ( x , p ) = W . ( a , *' ( a , p ) ) ;
set Q = -> Element of -> Subset of \rm -> Element of ( { g } ) ;
cluster -> MSsorted for ManySortedSet of U1 , B be MS[ of U1 , U2 ] ;
attr F = { A } means : Def6 : F is discrete ;
reconsider z9 = \hbox { z } as Element of product ( G . i ) ;
rng f c= rng f1 \/ rng f2 & f . 1 = f1 . len f1 \/ f2 . 1 ;
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 ) ;
E , j |= All ( x1 , x2 ) 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-1= ( q , <* s *> ) -\subseteq ( q , <* s *> ) -\subseteq ( q , <* s *> ) -\subseteq ( q , <* s *> ) -) ;
for x being element st x in X holds x in rng f1 implies x in X
h0 /. ( i + 1 ) = h0 . ( i + 1 ) ;
set mw = max ( B , .[ ) , mw = max ( B , NAT ) ;
t in Seg width ( I ^ ( n , n ) ) & t in Seg n ;
reconsider X = dom f /\ C as Element of Fin NAT , f be Element of Fin NAT ;
IncAddr ( i , k ) = <% ( a , b ) . ( k + 1 ) %> ;
( ( S-bound L~ f ) - ( S-bound L~ f ) ) / ( 2 * ( ( S-bound L~ f ) - ( S-bound L~ f ) ) ) <= ( ( q `2 ) - ( q `2 ) ) / ( 2 * ( ( q `2 ) - ( q `2 ) ) ) ) ;
attr R is condensed means : Def6 : Int R is condensed & Cl R is condensed ;
pred 0 <= a & 1 <= b & b <= 1 implies a * b <= 1 ;
u in ( ( c /\ ( ( d /\ b ) /\ e ) ) /\ f ) /\ j ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) ) /\ f ) /\ j ;
len C + - 2 >= 9 + - 3 & len C + - 2 >= 0 ;
x , z , y is_collinear & x , z , x is_collinear implies x = y
a |^ ( n1 + 1 ) = a |^ n1 * a .= a |^ n1 * a ;
<* \underbrace ( 0 , \dots , 0 } , x ) in Line ( x , a * x ) ;
set ya1 = <* y , c *> ;
FA2 /. 1 in rng Line ( D , 1 ) & FA2 /. len FA2 = D * ( 1 , 1 ) ;
p . m joins r /. m , r /. ( m + 1 ) ;
p `2 = ( f /. i1 ) `2 .= ( f /. ( i1 + 1 ) ) `2 ;
W-bound ( X \/ Y ) = W-bound ( X \/ Y ) .= W-bound ( X \/ Y ) ;
0 + ( p `2 ) <= 2 * r + ( p `2 ) ;
x in dom g & not x in g " { 0 } ;
f1 /* ( seq ^\ k ) is divergent_to-infty & lim ( f1 /* ( seq ^\ k ) ) = x0 ;
reconsider u2 = u , v2 = v as VECTOR of P`1 ( X ) ;
p |-count ( Product ( Sgm X11 ) ) = 0 & p |-count ( ( p |-count X11 ) * ( p |-count X11 ) ) = 0 ;
len <* x *> < i + 1 & i + 1 <= len c + 1 ;
assume that I is non empty and { x } /\ { y } = { 0. I } ;
set ii2 = ( card I + 4 ) .--> goto 0 , ii2 = goto 0 ;
x in { x , y } & h . x = {} ( TT ) ;
consider y being Element of F such that y in B and y <= x `1 ;
len S = len ( the charact of ( ( the charact of ( A ) ) ) ) .= len ( the charact of ( ( A ) ) ) ;
reconsider m = M , i = I , n = N as Element of X ;
A . ( j + 1 ) = B . ( j + 1 ) \/ A . j ;
set N8 = : : G7 : not ( G is finite & not ( G is finite ) & not ( G is finite ) ) ;
rng F c= the carrier of gr { a } & F . a = { a } ;
implies implies implies implies ( for n , K , n , r being Nat holds n is a and P [ n , K , r ] )
f . k , f . ( \mathop { \rm mod } n ) are_congruent_mod p ;
h " P /\ [#] T1 = f " P /\ [#] T2 .= f " P /\ [#] T2 ;
g in dom f2 \ f2 " { 0 } & f2 . x in dom f2 \ f2 " { 0 } ;
gthesis X /\ dom f1 = g1 " X & gX /\ dom f2 = g2 " X ;
consider n being element such that n in NAT and Z = G . n ;
set d1 = \bf dist ( x1 , y1 ) , d2 = dist ( x2 , y2 ) , d1 = dist ( x1 , y1 ) ;
b `1 + ( 1 - r ) / 2 < ( 1 - r ) / 2 + ( 1 - r ) / 2 ;
reconsider f1 = f , g1 = g as VECTOR of the carrier of X , Y ;
attr i <> 0 means : Def6 : i ^2 mod ( i + 1 ) = 1 ;
j2 in Seg len ( ( g2 . i2 ) `1 ) & j2 in Seg len ( ( g2 . i2 ) `1 ) ;
dom ( i + 1 ) = dom ( i + 2 ) .= a + ( i + 2 ) .= a + ( i + 2 ) ;
cluster sec | ]. PI / 2 , PI / 2 .[ -> one-to-one ;
Ball ( u , e ) = Ball ( f . p , e ) ;
reconsider x1 = x0 , y1 = x1 as Function of S , IV ;
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 RO & <* n *> ^ p in RO ;
S1 +* S2 = S2 +* S1 +* S2 .= S2 +* S2 +* S2 .= S2 +* S2 +* S2 ;
( ( #Z n ) * ( cos - cos ) ) is_differentiable_on Z & for x st x in Z holds ( ( #Z n ) * ( cos - cos ) ) . x = 1 / ( cos . x ) ^2
cluster -> continuous for Function of C , R^1 , x be Element of C ;
set C7 = 1GateCircStr ( <* z , x *> , f3 ) , C8 = 1GateCircStr ( <* z , x *> , f3 ) ;
EE8 . e2 = E8 . e2 -T . e2 .= ( E8 | ( dom T ) ) . e2 ;
( ( arctan * ( ln * ( f1 + f2 ) ) ) `| Z ) = f ;
upper_bound A = ( PI * 3 / 2 ) * 2 & lower_bound A = 0 ;
F . ( dom f , - F ) is_transformable_to F . ( cod f , - F ) ;
reconsider pbeing = qbeing Point of TOP-REAL 2 , pV = ( TOP-REAL 2 ) | ( i + 1 ) as Subset of TOP-REAL 2 ;
g . W in [#] Y0 & [#] Y0 c= [#] Y0 & g . W in [#] Y0 ;
let C be compact non vertical non vertical non horizontal Subset of TOP-REAL 2 , p , q be Point of TOP-REAL 2 ;
LSeg ( f ^ g , j ) = LSeg ( f , j ) .= LSeg ( g , j ) ;
rng s c= dom f /\ ]. -infty , x0 .[ & rng s c= dom f /\ ]. x0 , x0 + r .[ ;
assume x in { idseq 2 , Rev ( idseq 2 ) } ;
reconsider n2 = n , m2 = m , m1 = n as Element of NAT ;
for y being ExtReal st y in rng seq holds g <= y & y <= g
for k st P [ k ] holds P [ k + 1 ]
m = m1 + m2 .= m1 + m2 .= m1 + m2 .= m1 + m2 .= m1 + m2 ;
assume for n holds H1 . n = G . n -H . n ;
set B" = f .: ( the carrier of X1 ) , BX1 = f .: ( the carrier of X2 ) ;
ex d being Element of L st d in D & x << d ;
assume R -Seg ( a ) c= R -Seg ( b ) & R -Seg ( a ) c= R -Seg ( b ) ;
t in ]. r , s .[ or t = r or t = s or t = s ;
z + v2 in W & x = u + ( z + v2 ) ;
x2 |-- y2 iff P [ x2 , y2 ] & P [ x2 , y2 ] ;
pred x1 <> x2 means : Def6 : |. x1 - x2 .| > 0 & |. x1 - x2 .| > 0 ;
assume that p2 - p1 , p3 - p1 - p1 - p1 , p3 - p1 - p1 is_collinear and p3 - p1 , p3 - p1 - p1 - p1 \times p3 - p1 \times p2 - p1 // p3 - p1 , p2 - p1 - p1 - p1 , p2 - p1 - p1 - p1 - p1 , p2 - p1 - p1 - p1 , p2 - p1
set q = f ^ <* 'not' A *> ^ <* 'not' A *> ;
let f be PartFunc of REAL-NS 1 , REAL-NS 1 , x be Point of REAL-NS 1 , r be Real ;
( n mod ( 2 * k ) ) + 1 = n mod k ;
dom ( T * ( succ t ) ) = dom ( n succ t ) .= dom ( T . t ) ;
consider x being element such that x in wc iff x in c & x in X ;
assume ( F * G ) . ( v . x3 ) = v . x4 ;
assume that the carrier of D1 c= the carrier of D2 and the carrier of D2 c= the carrier of D2 ;
reconsider A1 = [. a , b .[ as Subset of R^1 | [. a , b .] ;
consider y being element such that y in dom F and F . y = x ;
consider s being element such that s in dom o and a = o . s ;
set p = W-min L~ Cage ( C , n ) , q = W-min L~ Cage ( C , n ) , r = q `1 , s = q `1 , w = q `1 , e = q `1 , s = q `1 , w = q `1 , e = q `1 , G = G * ( len G , 1 ) `1
n1 -' len f + 1 <= len f + 1 - len f + 1 ;
Seg \mathbb \mathbb q ( O1 ) = [ u , v , a `1 , b `2 ] ;
set C-2 = ( ( `1 ) `1 ) + ( ( n + 1 ) - 1 ) ;
Sum ( L (#) p ) = 0. R * Sum p .= 0. V .= 0. V ;
consider i being element such that i in dom p and t = p . i ;
defpred Q [ Nat ] means 0 = Q ( $1 ) & for k st k in $1 holds P [ k , $1 ] ;
set s3 = Comput ( P1 , s1 , k ) , P3 = P1 +* I , s4 = P2 +* I ;
let l be variable of k , Al , A-30 be Subset of D ;
reconsider U2 = union G-24 , GF = union GF as Subset-Family of TF ;
consider r such that r > 0 and Ball ( p `1 , r ) c= Q ` ;
( h | ( n + 2 ) ) /. ( i + 1 ) = p29 ;
reconsider B = the carrier of X1 , C = the carrier of X2 as Subset of X ;
p
synonym f is real-valued for rng f c= NAT & rng f c= NAT & f is one-to-one ;
consider b being element such that b in dom F and a = F . b ;
x10 < card X0 + card Y0 & card Y0 + 1 < card Y0 + 1 + 1 ;
attr X c= B1 means : Def6 : for X st X c= succ B1 holds X is non empty or X is non empty ;
then w in Ball ( x , r ) & dist ( x , w ) <= r ;
angle ( x , y , z ) = angle ( x-y , 0 , p2 ) ;
pred 1 <= len s means : Def6 : for i being Element of NAT holds ( the ` of Shift ( s , 0 ) ) . i = s ;
f-47 c= f . ( k + ( n + 1 ) ) ;
the carrier of { 1_ G } = { 1_ G } .= { 1_ G } .= { 1_ G } ;
pred p '&' q in \cdot ( p => q ) means : Def6 : q '&' p in TAUT ( A ) ;
- ( t `1 ) < ( t `1 ) / ( t `2 ) ;
( ( U /. 1 ) ) `1 = ( U /. 1 ) `1 .= ( W /. 1 ) `1 .= ( W /. 1 ) `1 ;
f .: ( the carrier of x ) = the carrier of x & f .: ( the carrier of x ) = the carrier of x ;
Indices OO = [: Seg n , Seg n :] & Indices OO = [: Seg n , Seg n :] ;
for n being Element of NAT holds G . n c= G . ( n + 1 ) ;
then V in M @ implies ex x being Element of M st V = { x } ;
ex f being Element of F-9 st f is \cup the carrier of Aas & f . x = F ( f . x ) ;
[ h . 0 , h . 3 ] in the InternalRel of G & [ h . 0 , h . 2 ] in the InternalRel of G ;
s +* Initialize ( ( intloc 0 ) .--> 1 ) = s3 +* Initialize ( ( intloc 0 ) .--> 1 ) ;
|[ w1 , v1 ]| - |[ w1 , v1 ]| <> 0. TOP-REAL 2 implies |[ w1 , v1 ]| in the carrier of TOP-REAL 2
reconsider t = t as Element of INT ( ) , s ( ) :] ;
C \/ P c= [#] ( GX | ( [#] GX \ A ) ) ;
f " V in ( the topology of X ) /\ D . ( the carrier of S , the carrier of S ) ;
x in [#] ( the carrier of A ) /\ A ( ) & x in [#] ( ( the carrier of A ) /\ B ( ) ) ;
g . x <= h1 . x & h . x <= h1 . x & h1 . x <= h2 . x ;
InputVertices S = { xy , y , z } .= { xy , y , z } \/ { xy , y } ;
for n being Nat st P [ n ] holds P [ n + 1 ]
set R = Line ( M , i , a * Line ( M , i ) ) ;
assume that M1 is being_line and M2 is being_line and M3 is being_line and M3 is being_line and M3 is being_line and M1 <> M2 ;
reconsider a = f4 . ( i0 -' 1 ) , b = f4 . ( i0 -' 1 ) as Element of K ;
len B2 = Sum ( Len ( F1 ^ F2 ) ) .= len ( Len F1 ) + len ( Len F2 ) ;
len ( ( the -' i ) * ( n -' 1 ) ) = n & len ( ( the -' i ) * ( n -' 1 ) ) = n ;
dom max ( - ( f + g ) , f + g ) = dom ( f + g ) ;
( the Sorts of seq ) . n = upper_bound Y1 & ( the Sorts of seq ) . n = upper_bound Y2 ;
dom ( p1 ^ p2 ) = dom f12 & dom ( p1 ^ p2 ) = dom f12 ;
M . [ 1 / y , y ] = 1 / y * v1 .= y * v1 .= y * v1 .= v1 * v2 ;
assume that W is non trivial and W .vertices() c= the carrier' of G2 and not W is non trivial ;
C6 * ( i1 , i2 ) `1 = G1 * ( i1 , i2 ) `1 .= G1 * ( i1 , i2 ) `1 ;
C8 |- 'not' Ex ( x , p ) 'or' p . ( x , y ) ;
for b st b in rng g holds lower_bound rng fnon - b <= b - b
- ( ( q1 `1 ) / |. q1 .| ) = 1 / ( |. q1 .| ) ;
( LSeg ( c , m ) \/ [: NAT , NAT :] ) \/ [: { l } , NAT :] c= R ;
consider p being element such that p in such and p in L~ f and x = f /. p ;
Indices ( X @ ) = [: Seg n , Seg 1 :] & Indices ( X @ ) = [: Seg n , Seg 1 :] ;
cluster s => ( q => p ) => ( q => ( s => 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 , f . ( x1 , x2 ) -> Element of D ;
consider g being Function such that g = F . t and Q [ t , g ] ;
p in LSeg ( ( N-min L~ f ) , ( NW-corner L~ f ) ) /\ LSeg ( ( NW-corner L~ f ) , ( NW-corner L~ f ) ) ;
set R8 = R .: ]. 1 , +infty .[ , R8 = ]. 1 , +infty .[ ;
IncAddr ( I , k ) = SubFrom ( da , db ) .= IncAddr ( I , db ) .= halt SCM+FSA ;
seq . m <= ( the Sorts of A ) . k & ( the Sorts of A ) . k <= ( the Sorts of A ) . k ;
a + b = ( a ` *' ) ` + ( b ` *' ) ` .= ( a ` ) ` + ( b ` ` ) ` ;
id ( X /\ Y ) = id ( X /\ id Y ) .= id X /\ id Y .= id Y ;
for x being element st x in dom h holds h . x = f . x ;
reconsider H = U1 \/ U2 , U2 = U2 \/ U1 as non empty Subset of U0 ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) /\ f ) ) /\ m ;
consider y being element such that y in Y and P [ y , inf B ] ;
consider A being finite stable set of R such that card A = ( the carrier of R ) \ A ;
p2 in rng ( f |-- p1 ) \ rng <* p1 *> .= rng <* p1 *> \/ rng <* p1 *> ;
len s1 - 1 > 1-1 - 1 & len s2 - 1 > 0 & len s2 - 1 > 0 ;
( ( N-min P ) `2 ) = ( N-min P ) `2 & ( ( N-min P ) `2 ) = ( ( N-min P ) `2 ) ;
Ball ( e , r ) c= LeftComp Cage ( C , k + 1 ) \/ LeftComp Cage ( C , k + 1 ) ;
f . a1 ` = f . a1 ` .= f . a1 ` .= ( f | ( a1 ` ) ) . a1 ` ;
( seq ^\ k ) . n in ]. -infty , x0 + r .[ & ( seq ^\ k ) . n in ]. x0 , x0 + r .[ ;
gg . s0 = g . s0 | G . s0 .= g . s0 .= g . s0 ;
the InternalRel of S is symmetric implies field ( the InternalRel of S ) c= [: the carrier of S , the carrier of S :]
deffunc F ( Ordinal , Ordinal ) = phi . ( $2 , $2 ) ;
F . s1 . a1 = F . s2 . a1 .= F . s2 . a1 .= ( F . s2 ) . a1 ;
x `2 = A . o . a .= Den ( o , A . a ) ;
Cl ( f " P1 ) c= f " ( Cl P1 ) & Cl ( f " P1 ) c= Cl ( Cl P1 ) ;
FinMeetCl ( ( the topology of S ) \ { x } ) c= the topology of T ;
synonym o is \bf means : Def6 : o <> \ast & o <> * & o <> * & o <> * ;
assume that X = Y |^ 0 + card X and card X <> card Y and X <> Y ;
the *> of s <= 1 + ( the *> of s ) & the +* ( 1 + 1 ) = ( the +* of s ) +* ( 1 + 1 ) ;
LIN a , a1 , d or b , c // b1 , c1 or a , c // a1 , c1 ;
e / 2 . 1 = 0 & e / 2 . 2 = 1 & e / 2 . 3 = 0 ;
Em1 in Sm1 & Em1 in { Nm1 } & Em1 in Sm1 & Em2 in Sm2 ;
set J = ( l , u ) If ;
set A1 = } , A2 = } , A1 = the carrier of S1 , A2 = the carrier of S2 ;
set vs = [ <* c , d *> , and2a ] , xy = [ <* d , c *> , and2a ] , } = [ <* c , d *> , ] ;
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
Int cell ( f , 1 , G ) c= RightComp f \/ L~ f \/ L~ f \/ L~ f \/ L~ f \/ L~ f ;
UA is_an_arc_of W-min C , E-max C implies W-min L~ Cage ( C , n ) in L~ Cage ( C , n )
set f-17 = f @ "/\" g @ ;
attr S1 is convergent means : Def6 : S2 is convergent & ( for n holds S1 . n = S2 . n ) implies S1 is convergent & lim ( S2 ) = 0 ;
f . ( 0 + 1 ) = ( 0 qua Ordinal ) + a .= a + a .= a + a ;
cluster -> \in \in implies cluster -> \in \in -> reflexive transitive transitive transitive transitive for non empty RelStr -symmetric RelStr ;
consider d being element such that R reduces b , d and R reduces c , d and R reduces d , b ;
not b in dom Start-At ( ( card I + 2 ) , SCMPDS ) ;
( z + a ) + x = z + ( a + y ) .= z + a + y ;
len ( l \lbrack a |^ 0 , x \rbrack ) = len l & len ( l |^ 0 ) = len l ;
t4 \/ {} is ( {} \/ rng t4 ) -valued ( {} \/ rng t4 ) -valued FinSequence ;
t = <* F . t *> ^ ( C . p ^ q-1 ) .= <* F . t *> ^ ( C . q ) ;
set p-2 = W-min L~ Cage ( C , n ) , p`2 = E-bound L~ Cage ( C , n ) ;
( k - ( i + 1 ) ) = ( k - ( i + 1 ) ) - ( i + 1 ) ;
consider u being Element of L such that u = u `1 "/\" D and u in D ` ;
len ( ( width ( ( width ( G ) ) |-> a ) ) * ( ( len ( G ) |-> a ) ) ) = width ( G ) ;
( F . x ) in dom ( ( G * the_arity_of o ) . x ) ;
set cH2 = the carrier of H2 , cH2 = the carrier of H2 ;
set cH1 = the carrier of H1 , cH2 = the carrier of H2 ;
( Comput ( P , s , 6 ) ) . intpos m = s . intpos m .= ( Comput ( P , s , 6 ) ) . intpos m ;
IC Comput ( Q2 , t , k ) = ( l + 1 ) - ( k + 1 ) .= ( l + 1 ) - ( k + 1 ) ;
dom ( ( ( cos * sin ) `| REAL ) = REAL & dom ( ( cos * sin ) `| REAL ) = REAL ;
cluster <* l *> ^ phi -> ( 1 + 0 ) -element for string of S ;
set b5 = [ <* A1 , cin *> , and2a ] , b5 = [ <* cin , cin *> , and2a ] , b5 = [ <* A1 , cin *> , and2a ] ;
Line ( Segm ( M `1 , P , Q ) , x ) = L * Sgm Q .= L * Sgm Q ;
n in dom ( ( the Sorts of A ) * the_arity_of o ) & dom ( ( the Sorts of A ) * the_arity_of o ) = dom the_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 = t3 . DataLoc ( s4 . SBP , 2 ) , x4 = DataLoc ( s4 . SBP , 2 ) , P4 = P3 ;
set p-3 = stop I ( ) , p-3 = stop I ( ) ;
consider a being Point of D2 such that a in W1 and b = g . a and a is open ;
{ A , B , C , D } = { A , B } \/ { C , D , E }
let A , B , C , D , E , F , J , M , N , N , M , N , N , F , M , N , N , M ;
|. p2 .| ^2 - ( p2 `2 ) ^2 - ( p2 `1 ) ^2 >= 0 ;
l - 1 + 1 = n-1 * ( 1 + ( 1 - 1 ) ) + 1 ;
x = v + ( a * w1 + b * w2 ) + ( c * w2 + c * w2 ) ;
the TopStruct of L = the TopStruct of ( the topology of L ) & the TopStruct of L = the TopStruct of ( the topology of L ) ;
consider y being element such that y in dom H1 and x = H1 . y and y in Y ;
ff \ { n } = \mathop { Free All ( v1 , H ) } .= Free All ( v1 , H ) ;
for Y being Subset of X st Y is summable holds Y is not summable & not Y is not summable
2 * n in { N : 2 * Sum ( p | N ) = N & N > 0 } ;
for s being FinSequence holds len ( the ` of s ) = len s & len ( the _ of s ) = len s
for x st x in Z holds exp_R * f is_differentiable_in x & ( for x st x in Z holds f . x = 1 / x ) implies f is_differentiable_in x
rng ( h2 * f2 ) c= the carrier of ( ( TOP-REAL 2 ) | ( the carrier of TOP-REAL 2 ) ) | ( the carrier of TOP-REAL 2 ) ) ;
j + ( len f ) - len f <= len f + ( len f - len f ) - len f ;
reconsider R1 = R * I , R2 = R * I as PartFunc of REAL , REAL-NS n , REAL-NS n ;
C8 . x = s1 . x0 .= C8 . x .= C8 . x .= ( C * ( 1 , 1 ) ) . x ;
power ( F_Complex ) . ( z , n ) = 1 .= x |^ n .= x |^ n * ( x |^ n ) ;
t at ( C , s ) = f . ( the connectives of S ) . t .= ( the connectives of S ) . t ;
support ( f + g ) c= support f \/ Carrier g \/ support g \/ support f \/ support g ;
ex N st N = j1 & 2 * Sum ( ( seq1 | N ) ) > N & N > 0 ;
for y , p st P [ p ] holds P [ All ( y , p ) ]
{ [ x1 , x2 ] where x1 , x2 is Point of [: X1 , X2 :] : x1 in X1 & x2 in X2 } is Subset of [: X1 , X2 :]
h = ( i , j ) |-- h , id B . i .= H . i ;
ex x1 being Element of G st x1 = x & x1 * N c= A & x1 * N c= A ;
set X = ( \lbrace ( -' q ) , O1 ) `1 , Y = ( a , b ) `1 , X = b , Y = a , Z = { b } ;
b . n in { g1 : x0 < g1 & g1 < a1 . n } ;
f /* s1 is convergent & f /. x0 = lim ( f /* s1 ) & ( for n holds s1 . n = lim ( f /* s1 ) ) implies f /* s1 is convergent & lim ( f /* s1 ) = 0
the lattice of Y = the lattice of the lattice of Y & the carrier of Y = the carrier of Y & the carrier of Y = the carrier of Y ;
'not' ( a . x ) '&' b . x 'or' a . x '&' 'not' ( b . x ) = FALSE ;
2 = len ( q0 ^ r1 ) + len ( q1 ^ q2 ) .= len ( q2 ^ q1 ) + len ( q2 ^ q2 ) .= len q2 + 1 ;
( 1 / a ) (#) ( sec * f1 - id Z ) is_differentiable_on Z & ( 1 / a ) (#) ( ( id Z ) ^ (#) ( ( id Z ) ^ ) ) is_differentiable_on Z ;
set K1 = integral ( ( lim H ) || ( A , H ) ) , K1 = ( ( lim H ) || ( A , H ) ) ;
assume e in { ( w1 - w2 ) `1 : w1 in F & w2 in G } ;
reconsider d7 = dom a `1 , d7 = dom F `1 , d8 = dom G `1 as finite set ;
LSeg ( f /^ q , j ) = LSeg ( f , j ) \/ LSeg ( q , j + q .. f ) ;
assume X in { T . N2 , h . ( N2 + 1 ) : h . N2 = N2 } ;
assume that Hom ( d , c ) <> {} and <* f , g *> * f1 = <* f , g *> * f2 ;
dom Sb = dom S /\ Seg n .= dom Lb /\ Seg n .= dom Lb /\ Seg n .= Seg n /\ Seg n .= Seg n ;
x in H |^ a implies ex g st x = g |^ a & g in H |^ a
* ( 0. ( F_Complex , n ) ) = a `1 - ( 0 * n ) .= a `2 - ( 0 * n ) `1 .= a `2 - ( 0 * n ) `1 ;
D2 . ( j - 1 ) in { r : lower_bound A <= r & r <= D1 . i } ;
ex p being Point of TOP-REAL 2 st p = x & P [ p ] & p `2 >= 0 & p <> 0. TOP-REAL 2 ;
for c holds f . c <= g . c implies f @ = g @ ;
dom ( f1 (#) f2 ) /\ X c= dom ( f1 (#) f2 ) /\ X .= dom ( f1 (#) f2 ) /\ X ;
1 = ( p * p ) * p .= p * ( p * ( p * q ) ) .= p * ( 1 / p ) .= p * ( 1 / p ) ;
len g = len f + len <* x + y *> .= len f + 1 .= len f + 1 + 1 .= len f + 1 ;
dom ( F | [: N1 , Sn1 :] ) = dom ( F | [: N1 , Sn1 :] ) .= [: N1 , Sn1 :] ;
dom ( f . t * I . t ) = dom ( f . t * g . t ) ;
assume a in ( "\/" ( ( ( T |^ the carrier of S ) ) , T ) ) .: D ;
assume that g is one-to-one and ( the carrier' of S ) /\ rng g c= dom g and g . ( len g ) = g . ( len g ) ;
( ( x \ y ) \ z ) \ ( ( x \ z ) \ ( y \ z ) ) = 0. X ;
consider f such that f * f `1 = id b and f * f `2 = id a and f * f `2 = id b and f is one-to-one ;
( cos | [. 2 * PI * 0 , PI + ( 2 * PI * 0 ) ) | [. 0 , PI + ( 2 * PI * 0 ) .] is increasing ;
Index ( p , co ) <= len LS - Gik .. LS - 1 + 1 - 1 .= len LS - Gik .. LS + 1 - 1 ;
let t1 , t2 , t3 be Element of ( T , S ) . s , t be Element of ( T , s ) . s ;
Frege ( ( Frege ( ( curry H ) . h ) ) . h ) <= "/\" ( rng ( ( curry G ) . h ) , L ) ;
then P [ f . i0 ] & F ( f . ( i0 + 1 ) ) < j & j + 1 <= len f ;
Q [ ( [ D . x , 1 ] ) `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 a r . i ;
the Sorts of A2 = ( the carrier of S2 ) --> TRUE .= ( the carrier of S2 ) --> TRUE .= the Sorts of A1 ;
consider s being Function such that s is one-to-one and dom s = NAT and rng s = F and for n being Nat st n in NAT holds s . n = F ( n ) ;
dist ( b1 , b2 ) <= dist ( b1 , a ) + dist ( a , b2 ) + dist ( a , b2 ) ;
( Lower_Seq ( C , n ) /. len Lower_Seq ( C , n ) ) `1 = WW `1 .= ( Cage ( C , n ) /. 1 ) `1 ;
q `2 <= ( UMP Upper_Arc C ) `2 & ( UMP C ) `2 <= ( UMP C ) `2 or q `2 >= ( UMP C ) `2 ;
LSeg ( f | i2 , i ) /\ LSeg ( f | i2 , j ) = {} ;
given a being ExtReal such that a <= II and A = ]. a , II .[ and a <= II and a <= II ;
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 <= b & b |^ n in X } , Y = { b |^ n } ;
( ( x * y * z \ x ) \ z ) \ ( x * y \ x ) = 0. X ;
set xy = [ <* xy , y , z *> , f4 ] , yz = [ <* z , x *> , f3 ] , zx = [ <* y , z *> , f3 ] , zx = [ <* z , x *> , f3 ] , zx = [ <* z , x *> , f3 ] , zx = [ <* y , z *> , f3 ] , zx = [ <* z ,
( l /. len ( l /. 1 ) ) `1 = ( l /. len ( l /. 1 ) ) `1 .= ( l /. 1 ) `1 .= ( l /. 1 ) `1 ;
( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 = 1 ;
( ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) ^2 < 1 ;
( ( ( ( S \/ Y ) /. 1 ) ) `2 ) `2 = ( ( ( S \/ Y ) /. 1 ) ) `2 ) `2 .= ( ( ( S \/ Y ) /. 1 ) ) `2 ;
( sm1 - s0 ) . k = sm1 . k - s0 . k .= ( sm1 - s0 ) . k ;
rng ( ( h + c ) ^\ n ) c= dom SVF1 ( 1 , f , u0 ) ;
the carrier of X = the carrier of X & the carrier of X = the carrier of Y implies X is non empty & Y is non empty
ex p4 st p3 = p4 & |. p4 - |[ a , b ]| .| = r & |. p4 - |[ a , b ]| .| = r ;
set ch = chi ( X , A ) , A5 = chi ( X , A ) ;
R |^ ( 0 * n ) = IIseq ( X , X ) .= R |^ n |^ 0 .= R |^ 0 ;
( Partial_Sums ( ( curry ( F , n ) ) . 0 ) ) . n is nonnegative & ( ( ( curry ( F , n ) ) . 0 ) . n = 0 ) ;
f2 = C7 . ( E7 , len ( V ) ) .= C7 . ( len ( V ) - 1 ) .= V . ( len ( V ) - 1 ) ;
S1 . b = s1 . b .= S2 . b .= S2 . b .= S2 . b .= S2 . b .= S2 . b ;
p2 in LSeg ( p2 , p1 ) /\ LSeg ( p1 , p2 ) .= LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) .= { p1 } ;
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 ) , phi = ( l2 , l2 ) --> ( l1 , l2 ) , _ = ( l1 , l2 ) --> ( l2 , l2 ) , _ = ( l1 , l2 ) --> ( l2 , l2 ) ;
synonym p is is is is is / w.r.t. T for HT ( p , T ) = 1 / p * ( p / q ) ;
Y1 `2 = - 1 & 0. TOP-REAL 2 <> ( - 1 ) * ( - 1 ) & ( - 1 ) * ( - 1 ) * ( - 1 ) = ( - 1 ) * ( - 1 ) ;
defpred X [ Nat , set , set ] means P [ $1 , $2 , ] & P [ $1 , $2 , $2 ] ;
consider k being Nat such that for n being Nat st k <= n holds s . n < x0 + g and x0 < x0 + g ;
Det ( I |^ ( m -' n ) ) * ( m - n ) = 1. ( K , n -' n ) * ( m - n ) .= 0. K ;
( - b - sqrt ( b ^2 - 4 * a * c ) ) / 2 * a < 0 ;
Cd . d = C7 . d mod C7 . d .= C7 . d mod C7 . d .= C8 . d mod C7 . d ;
attr X1 is dense means : Def6 : X2 is dense dense & X1 is dense implies X1 meet X2 is dense SubSpace of X1 union X2 ;
deffunc FF ( Element of E , Element of I ) = $1 * $2 & $1 * $2 = ( $1 * $2 ) * ( $1 * $2 ) ;
t ^ <* n *> in { t ^ <* i *> : Q [ i , T `1 . t ] } ;
( x \ y ) \ x = ( x \ x ) \ y .= y ` .= 0. X ;
for X being non empty set for Y being Subset-Family of X holds X is Basis of <* X , FinMeetCl ( Y ) *>
synonym A , B are_separated for Cl ( A , B ) for Cl ( A , B ) ;
len ( M @ ) = len p & width ( M @ ) = width ( M @ ) & len ( M @ ) = len ( M @ ) ;
J = { v where x is Element of K : 0 < v . x & v . x = 1 & x in { v . x } } ;
( Sgm Seg m ) . d - ( Sgm Seg m ) . e <> 0 ;
lower_bound 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 .|| .|| .= ||. a * ||. f .|| .|| .= ||. a * ||. f .|| .|| ;
f . x = ( h . x ) `1 & g . x = ( h . x ) `2 & g . x = ( h . x ) `2 ;
ex w st w in dom B1 & <* 1 *> ^ s = <* 1 *> ^ w & <* 1 *> ^ w = <* 1 *> ^ w ;
[ 1 , {} , <* d1 *> ] in ( { [ 0 , {} , {} ] } \/ S1 \/ S2 ) \/ S2 ;
IC Exec ( i , s1 ) + n = IC Exec ( i , s2 ) .= IC Exec ( i , s2 ) .= IC Exec ( i , s2 ) ;
IC Comput ( P , s , 1 ) = succ IC s .= 5 .= 5 + 9 .= 5 .= 5 ;
( IExec ( W6 , Q , t ) ) . intpos ( e + 1 ) = t . intpos ( e + 1 ) .= t . intpos ( e + 1 ) ;
LSeg ( f /^ q , i ) misses LSeg ( f /^ q , j ) \/ LSeg ( f /^ q , j ) ;
assume for x , y being Element of L st x in C holds x <= y or y <= x ;
integral ( f , C ) = f . ( upper_bound C ) - f . ( lower_bound C ) .= f . ( lower_bound C ) - f . ( lower_bound C ) ;
for F , G being one-to-one FinSequence st rng F misses rng G holds F ^ G is one-to-one & F ^ G is one-to-one
||. R /. ( L . h ) .|| < e1 * ( K + 1 * ||. h .|| ) ;
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 \not c= d ;
for y , x being Element of REAL st y `1 in Y `1 & x `2 in X `2 holds y `1 <= x `1 & y `2 <= x `2
func |. \bullet p .| -> variable means : Def6 : for x being element st x in it holds it . x = min ( NBI . p , p . x ) ;
consider t being Element of S such that x `1 , y `2 '||' z `1 , t `2 and x `1 , z `2 '||' y `1 , t `2 ;
dom x1 = Seg len x1 & len x1 = len l1 & len x2 = len l1 & len x2 = len l2 & len x1 = len l2 ;
consider y2 being Real such that x2 = y2 and 0 <= y2 and y2 < 1 / 2 and y2 <= 1 / 2 ;
||. f | X /* s1 .|| = ||. f .|| | X .= ||. f .|| /* s1 .= ||. f .|| /* s1 .= ||. f .|| /* s1 ;
( the InternalRel of A ) -Seg ( x `1 ) /\ Y = {} \/ {} .= {} \/ {} .= {} \/ {} .= {} \/ {} .= {} ;
assume i in dom p implies for j be Nat st j in dom q holds P [ i , j ] & P [ j + 1 ] & P [ i + 1 ] ;
reconsider h = f | X ( ) as Function of X ( ) , rng f ( ) ;
u1 in the carrier of W1 & u2 in the carrier of W2 implies ( for v st v in the carrier of W1 holds v - u in the carrier of W2 ) & v in the carrier of W1
defpred P [ Element of L ] means M <= f . $1 & f . $1 <= $1 & f . ( $1 + 1 ) <= $1 & f . ( $1 + 1 ) <= $1 ;
^ ( u , a , v ) = s * x + ( - ( s * x ) + y ) .= b ;
- ( x-y - y ) = - x + - y .= - x + - y .= - x + y .= - x + y .= y + x ;
given a being Point of GX such that for x being Point of GX holds a , x , x are_\HM { a } ;
fT = [ [ dom ( @ f2 ) , cod ( @ f2 ) ] , h2 ] , h2 = [ cod ( @ f2 ) , cod ( @ f2 ) ] ;
for k , n be Nat st k <> 0 & k < n & k is prime holds k , n are_relative_prime & k , n are_relative_prime implies k , n are_relative_prime
for x being element holds x in A |^ d iff x in ( ( A ` ) ` ) ` & x in ( A ` ) `
consider u , v being Element of R , a being Element of A such that l /. i = u * a * v ;
- ( ( p `1 / |. p .| - cn ) / ( 1 + cn ) ) ^2 > 0 ;
L-13 . k = LF . ( F . k ) & F . k in dom ( Carrier ( f ) ) ;
set i2 = AddTo ( a , i , - n ) , i1 = goto ( card I + 1 ) ;
attr B is thesis means : Def6 : for S being SubSubuniversal of B holds ( for B holds S , B |= S ) implies S is B `1 = S `1 ) & ( S is thesis implies S is thesis implies S is thesis ;
a9 "/\" D = { a "/\" d where d is Element of N : d in D } .= { a "/\" d where d is Element of N : d in D } ;
|( exp_R , q29 )| * |( - q , - q )| >= |( exp_R , - q )| * |( - q , - q )| ;
( - f ) . ( upper_bound A ) = ( ( - f ) | A ) . ( upper_bound A ) .= ( - f ) . ( upper_bound A ) ;
GG2 `1 = ( ( GG2 /. len GG2 ) `1 ) * ( ( GG2 /. len GG2 ) `1 ) .= ( ( GG2 /. len GG2 ) `1 ) * ( ( GG2 /. len GG2 ) `1 ) ;
( Proj ( i , n ) . LM ) . LM = <* ( proj ( i , n ) . LM ) . LM *> .= ( Proj ( i , n ) * ( Proj ( i , n ) ) ) . LM ;
f1 + f2 * reproj ( i , x ) is_differentiable_in ( reproj ( i , x ) . x0 ) + ( f2 * reproj ( i , x ) ) . x0 ;
pred ( for x st x in Z holds ( ( tan * f ) `| Z ) . x = tan . x ) & ( for x st x in Z holds ( tan * f ) `| Z ) . x = 1 / ( cos . x ) ^2 * ( cos . x ) ^2 ) ;
ex t being SortSymbol of S st t = s & h1 . t . x = h2 . t & ( for x being Element of S holds x in ( the Sorts of A ) . s ) . x ;
defpred C [ Nat ] means P8 . $1 is non empty & A8 is non empty & A8 is finite & A is $1 empty & A is finite & A is finite ;
consider y being element such that y in dom ( p . i ) and ( q . i ) = ( p . y ) . y ;
reconsider L = product ( { x1 } +* ( index B , l ) ) as Subset of ( Carrier A ) . ( ( index B ) . ( index B ) ) ;
for c being Element of C ex d being Element of D st T . ( id c ) = id d & for x being Element of C st x in c holds d . ( id c ) = id d
len ( f , n , p ) = ( f | n ) ^ <* p *> .= f ^ <* p *> .= f ^ <* 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 - cp = ( f | ( n , L ) ) *' - ( f | ( n , L ) ) .= ( f - c ) * ( - ( f | ( n , L ) ) ) ;
consider r be Real such that r in rng ( f | divset ( D , j ) ) and r < m + s ;
f1 . |[ r2 `1 , ( r2 `2 ) / 2 ]| in f1 .: W1 /\ f2 .: W2 & f2 . |[ r2 `1 , ( r2 `2 ) / 2 ]| in f2 .: W3 ;
eval ( a | ( n , L ) , x ) = eval ( a | ( n , L ) ) .= a * ( x , x ) .= a * ( x , x ) ;
z = DigA ( tz , x9 ) .= DigA ( tz , x9 ) .= DigA ( tz , x9 ) .= DigA ( tz , x9 ) ;
set H = { Intersect S where S is Subset-Family of X : S c= G & S c= G } , F = { Intersect S where S is Subset of X : S is open & S is open } , G = { Intersect S where S is Subset of Y : S is open & S is open } ;
consider S19 being Element of D opp , d being Element of D such that S `1 = S19 ^ <* d *> and S `2 = d ;
assume that x1 in dom f and x2 in dom f and f . x1 = f . x2 and f . x2 = f . x2 ;
- 1 <= ( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 / ( 1 + sn ) ^2 ;
0. ( V ) is Linear_Combination of A & Sum ( 0. ( V ) ) = 0. V implies Sum ( L ) = Sum ( L )
let k1 , k2 , k2 , x4 , k2 , x4 , 6 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 9 , 8 , 8 be Instruction of SCM+FSA , a , b , c , d , 7 , 8 be Int-Location ;
consider j being element such that j in dom a and j in g " { k } and x = a . j and j in { k } ;
H1 . x1 c= H1 . x2 or H1 . x2 c= H1 . x1 & H1 . x2 c= H1 . x2 or H1 . x2 = H2 . x2 ;
consider a being Real such that p = a * p1 + ( a * p2 ) and 0 <= a and a <= 1 and a <= 1 ;
assume that a <= c & d <= b and [' a , b '] c= dom f and [' a , b '] c= dom g and g . a = g . b ;
cell ( Gauge ( C , m ) , ( X -' 1 ) -' 1 , 0 ) is non empty or cell ( Gauge ( C , m ) , len Gauge ( C , m ) -' 1 , 0 ) is non empty ;
Ay in { ( S . i ) `1 where i is Element of NAT : not contradiction } & not Ay in { ( S . i ) `1 } ;
( T * b1 ) . y = L * b2 /. y .= ( F `1 * b1 ) . y .= ( F `1 * b1 ) . y ;
g . ( s , I ) . x = s . y & g . ( s , I ) . y = |. s . x - s . y .| ;
( log ( 2 , k + k ) ) / ( 2 * ( k + 1 ) ) >= ( log ( 2 , k + 1 ) ) / ( 2 * ( k + 1 ) ) ;
then p => q in S & not x in the still of p & not p => All ( x , q ) in S & not x in S ;
dom ( the InitS of rM ) misses dom ( the InitS of rM ) & dom ( the InitS of rM ) misses dom ( the InitS of rM ) ;
synonym f is integer means : Def6 : for x being set st x in rng f holds x is integer ;
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 + 1 .= len p3 + 1 + 1 .= len p3 + 1 + 1 .= len p3 + 1 + 1 ;
l ( ) = ( g ) `1 ) + ( k ( ) ) - ( k ( ) ) .= ( g ) `1 + ( k ( ) ) - ( k ( ) ) ;
CurInstr ( P2 , Comput ( P2 , s2 , l2 ) ) = halt SCM+FSA .= CurInstr ( P2 , Comput ( P2 , s2 , l2 ) ) .= halt SCM+FSA .= halt SCM+FSA ;
assume for n be Nat holds ||. seq .|| . n <= Rseq . n & ( for n be Nat holds seq . n <= ( seq ^\ k ) . n ) & ( n <= k implies seq . n <= ( seq ^\ k ) . n ) ;
sin . non empty = sin r * cos ( - ( cos r * sin s ) / sin ( - ( cos s ) ) ) .= 0 ;
set q = |[ g1 . t0 , g2 . t0 ]| , g1 = |[ g2 . t0 , g2 . t0 ]| , g2 = |[ g2 . t0 , g2 . t0 ]| ;
consider G being sequence of S such that for n being Element of NAT holds G . n in implies G in implies S . n = S . ( n + 1 ) ;
consider G such that F = G and ex G1 st G1 in SM & G = \mathopen { X } & G is finite & G is finite ;
the root of [ x , s ] in ( the Sorts of Free ( C , X ) ) . s & the Sorts of C = ( the Sorts of Free ( C , X ) ) . s ;
Z c= dom ( exp_R / ( 3 / 2 ) (#) ( f + ( #Z 2 ) * f1 ) ) ;
for k be Element of NAT holds seq1 . k = ( \HM { the } \HM { lower } \HM { sum ( f , S ) } ) . k
assume that - 1 < n ( ) and q `2 > 0 and ( q `1 / |. q .| - cn ) < 0 and q `2 / |. q .| - cn < 0 ;
assume f is continuous one-to-one & a < b & c < d & f = g & f . a = c & f . b = d & f . c = d ;
consider r being Element of NAT such that s-> Element of NAT , r , q being Element of NAT such that s-> Element of NAT and r <= q and r <= q and r <= q ;
LE f /. ( i + 1 ) , f /. j , L~ f implies f /. 1 in L~ f & f /. len f in L~ f
assume that x in the carrier of K and y in the carrier of K and inf { x , y } in the carrier of K and x <> y and y <> z ;
assume f +* ( i1 , \xi ) in ( proj ( F , i2 ) ) " ( A . ( i1 + 1 ) ) & f . ( i1 + 1 ) in ( proj ( F , i2 ) ) " ( A . ( i1 + 1 ) ) ;
rng ( ( Flow M ) ~ | ( the carrier of M ) ) c= the carrier' of M & rng ( ( Flow M ) ~ ) c= the carrier' of M ;
assume z in { ( the carrier of G ) --> { t } where t is Element of T : not contradiction } ;
consider l be Nat such that for m be Nat st l <= m holds ||. ( s1 . m - x0 ) .|| < g / ( ||. x .|| + g ) ;
consider t be VECTOR of product G such that mt = ||. D5 . t .|| and ||. t .|| <= 1 and ||. t .|| <= 1 ;
assume that the carrier of v = 2 and v ^ <* 0 *> , v ^ <* 1 *> in dom p and v ^ <* 1 *> in dom p and p . 1 = v ;
consider a being Element of the carrier of Xbe Element of the lines of X39 such that a on A and not a on A and not a on B ;
( - x ) |^ ( k + 1 ) * ( ( - x ) |^ ( k + 1 ) ) " = 1 ;
for D being set for i st i in dom p holds p . i in D & p . i is FinSequence of D & p . i = p . i
defpred R [ element ] means ex x , y st [ x , y ] = $1 & P [ x , y ] & P [ y , x ] ;
L~ f2 = union { LSeg ( p0 , p00 ) , LSeg ( p00 , p1 ) } .= { p1 , p2 } \/ { p2 } .= { p1 , p2 } \/ { p2 } .= { p2 , p1 } ;
i - len h11 + 2 - 1 < i - len h11 + 2 - 1 + 1 - 1 + 1 + 1 - 1 + 1 - 1 .= i + 2 - 1 + 1 ;
for n being Element of NAT st n in dom F holds F . n = |. ( nthesis . ( n -' 1 ) ) .| ;
for r , s1 , s2 , r holds r in [. s1 , s2 .] iff s1 <= r & r <= s2 & s1 <= s2 & r <= s2 implies s1 <= s2
assume v in { G where G is Subset of T2 : G in B2 & G c= z1 & G c= z2 & G c= z1 & G c= z2 & z1 c= z2 } ;
let g be \vert \vert -:] of A , INT , b be Element of INT , x be Element of INT , b be Element of INT ;
min ( g . [ x , y ] , k . [ y , z ] ) = ( min ( g , k , x ) ) . y ;
consider q1 being sequence of RNS such that for n holds P [ n , q1 . n ] and q1 is convergent and lim q1 = lim q1 ;
consider f being Function such that dom f = NAT and for n being Element of NAT holds f . n = F ( n ) and for n being Element of NAT holds P [ n , f . n ] ;
reconsider B-6 = B /\ B , OOI = O , OOI = I /\ Z as Subset of B ;
consider j being Element of NAT such that x = the b of \HM { n } and j <= n and 1 <= j and j <= n and j <= n and j <= n ;
consider x such that z = x and card ( x . O2 ) in card ( x . O ) and x in L1 and x in L2 and x in L1 and x in L2 ;
( C * _ T4 ( k , n2 ) ) . 0 = C . ( ( T . k ) . 0 ) .= ( ( T . k ) . 0 ) . 0 ;
dom ( X --> rng f ) = X & dom ( ( X --> f ) . x ) = dom ( X --> f . x ) ;
( ( ( N-bound L~ SpStSeq C ) `2 ) / 2 ) <= ( ( ( SpStSeq C ) `2 ) / 2 ) * ( ( N-bound C ) `2 ) + ( ( S-bound C ) `2 ) / 2 ) ;
synonym x , y are_collinear means : Def6 : x = y or ex l being Subset of S st { x , y } c= l & x , y are_collinear ;
consider X being element such that X in dom ( f | ( n + 1 ) ) and ( f | ( n + 1 ) ) . X = Y ;
assume that Im k is continuous and for x , y being Element of L st a = x & b = y holds x << y iff x << y & a << b ;
( 1 / 2 * ( ( ( ( ( ( ( ( ( 1 / 2 ) * ( ( #Z n ) * ( #Z n ) ) * ( #Z n ) ) * ( #Z n ) ) ) / ( n + 1 ) ) ) ) ) ) ) * ( ( ( #Z n ) * ( ( #Z n ) * ( ( #Z n ) * ( #Z n ) ) ) ) ) ) is_differentiable_on REAL ;
defpred P [ Element of omega ] means ( for n holds ( ( for x st x in A1 holds x in $1 ) implies x in A ) implies $1 in A & not x in A ;
IC Comput ( P , s , 2 ) = succ IC Comput ( P , s , 1 ) .= 6 + 1 .= 6 + 1 .= 6 + 1 ;
f . x = f . g1 * f . g2 .= f . g1 * 1_ H .= f . g1 * 1_ H .= ( f * g ) . x ;
( M * ( F . n ) ) . n = M . ( { ( canFS Omega ) . n } ) .= M . ( { ( canFS Omega ) . n } ) .= ( M * ( F . n ) ) . n ;
the carrier of L1 + L2 c= ( the carrier of L1 ) \/ ( the carrier of L2 ) \/ ( the carrier of L1 ) \/ ( the carrier of L2 ) \/ the carrier of L1 \/ the carrier of L2 ;
pred a , b , c , x , y , c , a , b , c , x , y , c , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , z , x , y , z , x , y , z , x , y , z , x , y
( the Sorts of s ) . n <= ( the Sorts of s ) . n * s . ( n + 1 ) & ( the Sorts of s ) . n <= ( the Sorts of s ) . n ;
pred - 1 <= r & r <= 1 & ( arccot - 1 ) * ( arccot - 1 ) = - 1 / r * ( 1 + r ^2 ) ;
seq in { p ^ <* n *> where n is Nat : p ^ <* n *> in T1 & n + 1 in dom T1 } implies not ( p ^ <* n *> in T1 & p ^ <* n *> in T2 )
|[ x1 , x2 , x3 ]| . 2 - |[ y1 , y2 , x4 ]| . 2 = x2 - y2 - x1 , y1 - x2 , y2 - x1 ]| . 2 - x1 , y1 - x1 , y1 - x1 - x1 , x2 - x1 - x1 , y1 - x1 - x1 , x2 - x1 - x1 , x2 - x1 ]| . 2 - x1 - x1 , x2 - x1 , x2 - x1 + x1 - x1 , x2 - x1 + x1 - x1 ]| . 2 +
attr F . m is nonnegative means : for m be Nat holds ( Partial_Sums F ) . m is nonnegative & ( Partial_Sums F ) . m is nonnegative ) implies F is nonnegative ;
len ( w ) = len ( ( w ) . ( ( x - y ) + ( G . ( x - y ) ) ) ) .= len ( ( G . ( x - y ) ) + ( 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 <> v + W ;
given F be finite Subset 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 * 0 ) * ( 0 - 1 ) iff 1 = ( ( ( 1 - 0 ) * ( - 1 ) ) * ( - 1 ) ) / ( ( 0 - 0 ) * ( 0 - 1 ) ) ;
consider n be Nat such that for m be Nat st n <= m holds |. ( f # x ) . m - lim ( f # x ) .| < e / 2 ;
cluster being } being being Boolean non empty implies for non empty implies ( for o being Element of L holds ( the o of o ) . ( o , c ) is Boolean ) & ( the o of o ) . ( o , c ) is Boolean
"/\" ( BB , {} ) = Top BB .= the carrier of S .= [#] ( S | ( [#] S ) ) .= [#] ( S | ( [#] S ) ) .= [#] ( S | ( [#] S ) ) .= ( "/\" ( I ) ) ;
( 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 * r1 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - ( 2 * r2 - 1 ) ) ) ) ) ) ) ) ) = 0. TOP-REAL 2 ;
reconsider p = P * ( \square , 1 ) , q = a " * ( ( - ( - ( - ( - ( - ( K , n , 1 ) ) ) ) ) ) ) as FinSequence of K ;
consider x1 , x2 being element such that x1 in uparrow s and x2 in < uparrow t and x = [ x1 , x2 ] and x = [ x1 , x2 ] ;
for n be Nat st 1 <= n & n <= len q1 holds q1 . n = ( upper_volume ( g , M7 ) ) . n & ( upper_volume ( g , M7 ) ) . n >= 0
consider y , z being element such that y in the carrier of A and z in the carrier of A and i = [ y , z ] and i = [ y , z ] ;
given H1 , H2 being strict Subgroup of G such that x = H1 and y = H2 and H1 , H2 |^ ( n + 1 ) |^ ( n + 1 ) |^ ( n + 1 ) |^ ( n + 1 ) |^ ( n + 1 ) = H1 |^ n |^ H2 ;
for S , T being non empty RelStr , d being Function of T , S st T is complete holds d is directed-sups-preserving iff d is monotone & d is monotone
[ a + 0 , i + b2 ] in ( the carrier of F_Complex ) \ ( the carrier of V ) & [ a , b ] in [: the carrier of V , the carrier of V :] ;
reconsider mm = max ( len F1 , len ( p . n ) * <* x *> ) as Element of NAT , ( p . n ) * ( p . n ) * <* x *> ;
I <= width GoB ( ( GoB f ) * ( len GoB f , 1 ) , GoB ( ( GoB f ) * ( len GoB f , 1 ) ) ) ) & ( GoB f ) * ( len GoB f , 1 ) `2 <= ( GoB f ) * ( 1 , width GoB f ) `2 ;
f2 /* q = ( f2 /* ( f1 /* s ) ) ^\ k .= ( f2 * f1 ) /* s .= ( f2 * f1 ) /* s .= ( f2 * f1 ) /* s ;
attr A1 \/ A2 is linearly-independent means : Def6 : A1 misses A2 & ( for x st x in A1 holds x in A2 & x in A2 holds Lin ( A1 \/ A2 ) = Lin ( A1 \/ A2 ) ) & Lin ( A2 \/ A1 ) = Lin ( A2 \/ A1 ) ;
func A -carrier C -> set means : Def6 : union { A . s where s is Element of R : s in C } = { A . s where s is Element of R : s in C } ;
dom ( Line ( v , i + 1 ) ) ^ ( ( Line ( p , m ) ) * ( \square , 1 ) ) = dom ( F ^ G ) .= dom ( F ^ G ) ;
cluster [ x `1 , 4 , x `2 , 4 ] -> non empty or [ x `1 , 4 , x `2 ] in { x `1 } & [ x `1 , 4 , x `2 ] in { x `1 } ;
E , All ( x1 , All ( x2 , x2 ) ) / ( x1 , x2 ) / ( x1 , x2 ) |= All ( x1 , x2 ) / ( x1 , x2 ) ;
F .: ( id X , g ) . x = F . ( id X , g . x ) .= F . ( x , g . x ) .= F . ( x , g . x ) .= F . ( x , g . x ) ;
R . ( h . m ) = F . x0 + h . m - h . ( h . m ) - h . ( h . m ) ) .= ( F - h ) . x0 ;
cell ( G , XS -' 1 , ( t + 1 ) ) \ L~ f meets ( UBD L~ f ) \/ ( UBD L~ f ) \/ ( UBD L~ f ) ;
IC Result ( P2 , s2 ) = IC IExec ( I , P , Initialize s ) .= ( card I + card I + 2 ) .= card I + ( card I + 2 ) .= card I + ( card I + 2 ) .= card I + ( card I + 2 ) .= card I + ( card I + 2 ) .= card I + 2 + 1 .= card I + 2 + 1 .= card I + 2 + 1 ;
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 y0 = a . x0 and x0 in { k } and a . x0 = b . x0 ;
dom ( r1 (#) chi ( A , A ) ) = dom chi ( A , A ) /\ dom chi ( A , A ) .= dom ( r1 (#) chi ( A , A ) ) /\ dom ( r2 (#) chi ( A , A ) ) .= C /\ dom ( r2 (#) chi ( A , A ) ) .= C /\ ( A /\ A ) ;
d-7 . [ y , z ] = ( ( [ y `1 , z `2 ] - ( y `1 ) ) / ( y `2 ) ) * ( y `2 ) .= ( ( y `1 ) - ( y `2 ) ) / ( y `2 ) ;
attr for i being Nat holds C . i = A . i /\ B . i & C . i c= ( L~ f ) /\ ( L~ f ) ;
assume that x0 in dom f and f is_continuous_in x0 and ||. f .|| is_continuous_in x0 and ||. f .|| is_continuous_in x0 and ||. f .|| is_continuous_in x0 and ||. f .|| is_continuous_in x0 and ||. f .|| is_continuous_in x0 and ||. f .|| /. x0 = ||. f .|| /. x0 ;
p in Cl A implies for K being Basis of p , Q being Basis of T st Q in K holds A meets Q & A meets Q
for x being Element of REAL n st x in Line ( x1 , x2 ) holds |. y1 - y2 .| <= |. y1 - y2 .| & |. y2 - x2 .| <= |. y1 - y2 .|
func the of <*> { a } -> w } means : : : : a in it & for b being Ordinal st a in b holds it c= b & it c= b ;
[ a1 , a2 , a3 ] in [: the carrier of A , the carrier of A :] & [ a1 , a2 , a3 ] in [: the carrier of A , the carrier of A :] & [ a1 , a2 , a3 ] in [: the carrier of A , the carrier of A :] ;
ex a , b being element st a in the carrier of S1 & b in the carrier of S2 & x = [ a , b ] & x = [ a , b ] ;
||. ( ( vseq . n ) - ( vseq . m ) ) * ||. x .|| < ( e / ( ||. x .|| + ||. x .|| ) ) * ||. x .|| ;
then for Z being set st Z in { Y where Y is Element of I7 : F c= Y & Y c= Z } holds z in x & z in Z ;
sup compactbelow [ s , t ] = [ sup ( ( compactbelow [ s , t ] ) /\ compactbelow [ s , t ] ) , sup ( ( compactbelow [ s , t ] ) /\ compactbelow [ s , t ] ) ] ;
consider i , j being Element of NAT such that i < j and [ y , f . j ] in IO and [ f . i , z ] in IO and [ f . i , z ] in IO and [ y , z ] in IO ;
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 e19 being Element of the carrier of X such that c9 , a9 // a9 , e29 and a9 <> b9 and a9 <> b9 and a , b // a9 , c9 and a , c // a9 , b9 and a , b // a9 , c9 and a , b // a9 , b9 ;
set U2 = I \! \mathop { {} } , U2 = I \! \mathop { {} } , SS = U U \ { {} } , SS = U \ { {} } , SS = U \ { {} } , SS = U \ { {} } , SS = U \ { {} } , SS = U \ { {} } , SS = U \ { {} } , SS = U \ { {} } , SS = U \ { {} } , SS = U \ { {} } , SS = U \ { {} } , SS = U \ { {} } , SS = U \ { {} } , SS = U \ { {} } ,
|. q3 .| ^2 = ( q2 `1 ) ^2 + ( q2 `2 ) ^2 .= ( q2 `1 ) ^2 + ( q2 `2 ) ^2 .= ( q2 `1 ) ^2 + ( q2 `2 ) ^2 .= ( q2 `1 ) ^2 + ( q2 `2 ) ^2 ;
for T being non empty TopSpace , x , y being Element of [: the topology of T , the topology of T :] holds x "\/" y = x \/ y implies 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 ) & dom ( the charact of U1 ) = dom ( the charact of U1 ) ;
dom ( h | X ) = dom h /\ X .= dom ( ||. h .|| | X ) /\ X .= dom ( ||. h .|| | X ) .= dom ( ( ||. h .|| | X ) | X ) .= X /\ X .= dom ( ( ||. h .|| | X ) ) ;
for N1 , K1 being Element of GX holds dom ( h . K1 ) = N & rng ( h . K1 ) = N1 & rng ( h . K1 ) = N2 & rng ( h . K1 ) c= N1 & rng ( h . K1 ) c= N1
( mod ( u , m ) + mod ( v , m ) ) . i = ( mod ( u , m ) ) . i + ( mod ( v , m ) ) . i ;
- ( q `1 ) < - 1 or q `2 >= - ( q `1 ) & - ( q `2 ) <= - ( q `1 ) or - ( q `1 ) >= - ( q `1 ) & - ( q `2 ) <= - ( q `1 ) ;
pred r1 = fp & r2 = fp & r1 * r2 = fp * ( r1 - r2 ) & r2 * ( r2 - r2 ) = fp * ( r2 - r2 ) ;
( for m be bounded Function of X , the carrier of Y , x be Element of X holds x9 . m = ( seq_id ( ( vseq . m ) , X , Y ) ) . x ) & ( for x be Element of X holds x in dom ( ( vseq . m ) , X , Y ) ) implies x in dom ( ( vseq . m ) --> x ) )
pred a <> b & b <> c & angle ( a , b , c ) = PI implies angle ( b , c , a ) = 0 & angle ( c , a , b ) = 0 ;
consider i , j , r being Real such that p1 = [ i , r ] and p2 = [ j , s ] and i < j and r < s and s < j and j < len f ;
|. p .| ^2 - ( 2 * |( p , q )| ) ^2 + |. q .| ^2 = |. p .| ^2 + |. q .| ^2 - ( 2 * |( p , q )| ) ^2 ;
consider p1 , q1 being Element of X ( ) such that y = p1 ^ q1 and q1 ^ q1 = p1 ^ q1 and p1 ^ q2 = p1 ^ q1 and p1 ^ q1 = q1 ^ q2 and q1 ^ q2 = q2 ^ q2 and q2 = q2 ^ q2 ;
, , * ( r1 , r2 , s1 ) = ( s2 gcd gcd ( s1 , s2 , s2 ) ) * ( r2 , s2 ) .= ( s2 gcd gcd ( s1 , s2 , s2 ) ) * ( r2 , s2 ) ;
( LMP A ) `2 = lower_bound ( proj2 .: ( A /\ /\ /\ Vertical_Line w ) ) & ( proj2 .: ( A /\ /\ /\ /\ /\ /\ /\ /\ /\ /\ /\ w ) ) is non empty or proj2 .: ( A /\ /\ /\ w ) is non empty ) ;
s |= ( k , H1 ) \bf ( H , H2 ) iff s |= ( H , ( k , 1 ) ) \bf ( H , ( k , 1 ) ) \bf ( H , ( k , 1 ) ) \bf ( H , ( k , 1 ) ) ) ;
len ( s + 1 ) = card support b1 + 1 .= card support b2 + 1 .= card support b2 + 1 .= card support b1 + 1 .= card support b2 + 1 .= len b1 + 1 .= len b1 + 1 + 1 .= len b1 + 1 + 1 .= len b1 + 1 + 1 ;
consider z being Element of L1 such that z >= x and z >= y and for z being Element of L1 st z >= x & z `1 >= y holds z `1 >= y & z `2 >= x `1 ;
LSeg ( UMP D , |[ ( W-bound D + E-bound D ) / 2 , ( E-bound D + E-bound D ) / 2 ]| ) /\ D = { UMP D } \/ D .= { UMP D } \/ D .= D \/ { ( UMP D + E-bound D ) / 2 } ;
lim ( ( ( f `| N ) / ( g `| N ) ) /* b ) = lim ( ( f `| N ) / ( g `| N ) ) .= lim ( ( f `| N ) / ( g `| N ) ) ;
P [ i , pr2 ( f ) . i , pr2 ( f ) . i , pr2 ( f ) . ( i + 1 ) ] & pr2 ( f ) . ( i + 1 ) = pr2 ( 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 ) .|| < r / 2 + r / 2
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 <> b holds a = b
Z c= dom ( ( #Z 2 ) * ( ( #Z 2 ) * f ) ) \ ( ( #Z 2 ) * ( ( #Z 2 ) * f ) ) " { 0 } ) implies Z c= dom ( ( #Z 2 ) * ( ( #Z 2 ) * f ) )
ex j being Nat st j in dom ( l ^ <* x *> ) & j < i & y = ( l ^ <* x *> ) . j & i = 1 + len l & z = ( l ^ <* x *> ) . j & j = 1 + len l & j = len l + 1 ;
for u , v being VECTOR of V for r being Real st 0 < r & r < 1 & u in dom _ \kappa ( r (#) N ) holds r * u + ( 1-r ( r (#) v ) ) in Int N
A , Int Cl A , Int Cl A , Cl Int A , Cl Int Cl ( Int Cl A ) , Cl Int Cl ( Int Cl A ) , Cl Int Cl ( Int Cl A ) ` ) meets Cl Int Cl ( Int Cl A ) ;
- Sum <* v , u , w *> = - ( v + u + w ) .= - ( v + u ) + - ( w + w ) .= - v + ( u + w ) .= - v + ( u + w ) ;
( Exec ( a := b , s ) ) . IC SCM R = ( Exec ( a := b , s ) ) . IC SCM R .= succ IC s .= succ IC s .= IC s .= IC Exec ( I , s ) .= IC s .= IC s ;
consider h being Function such that f . a = h and dom h = I and for x being element st x in I holds h . x in ( Carrier J ) . x and h . x = ( Carrier J ) . x ;
for S1 , S2 , S1 , S2 being non empty reflexive RelStr , D being non empty directed Subset of S1 , D being non empty directed Subset of S2 for x being Element of S1 , y being Element of S2 holds cos . ( x , y ) is directed & cos . ( y , x ) is directed
card X = 2 implies ex x , y st x in X & y in X & x <> y or x = y & x = y or x = y or x = z or y = x & z = x or z = y
( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) in rng ( Cage ( C , n ) \circlearrowleft W-min L~ Cage ( C , n ) ) & ( W-min L~ Cage ( C , n ) ) .. Cage ( C , n ) in rng Cage ( C , n ) ;
for T , T being DecoratedTree , p , q being Element of dom T st p in dom T holds ( T , p ) -with q & ( T , q ) -with q , T & ( T , q ) -with q , 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 ( for Nat , k , n ) st k divides ( k -' n ) & n divides ( k -' n ) & ( n divides k implies ( n divides k ) & ( n divides k implies n divides k ) ) & ( n divides k implies n divides k )
dom F " = the carrier of X2 & rng F " = the carrier of X1 & F " { x } = the carrier of X2 & F " { y } = the carrier of X2 & F " { y } = the carrier of X1 ;
consider C being finite Subset of V such that C c= A and card C = n and the carrier of V = Lin ( B9 \/ C ) and C = Lin ( C \/ B ) and C = Lin ( B \/ C ) and C = Lin ( B \/ C ) ;
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 ] } , Z = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } ;
angle ( p1 , p3 , p4 ) = 0 .= angle ( p2 , p3 , p2 ) .= angle ( p3 , p2 ) .= angle ( p3 , p2 ) .= angle ( p2 , p3 , p2 ) .= angle ( p3 , p2 ) .= angle ( p3 , p2 , p3 ) ;
- sqrt ( ( - ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ) ^2 ) = - sqrt ( ( - ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 ) ) .= - ( - ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ) ;
ex f being Function of I[01] , ( TOP-REAL 2 ) | P st f is continuous one-to-one & rng f = P & f . 0 = p1 & f . 1 = p2 & f . 1 = p3 & f . 0 = p4 & f . 1 = p4 & f . 1 = p4 & f . 1 = p4 & f . - 1 = p4 & f . - 1 = p4 ;
attr f is_is_is_is_is_or pdiff1 ( f , 1 ) means : Def6 : SVF1 ( 2 , pdiff1 ( f , 1 ) , u0 ) /. ( u0 + 1 ) - SVF1 ( 2 , pdiff1 ( f , 3 ) , u0 ) /. ( u0 + 1 ) = ( proj ( 2 , 3 ) . u0 ) . u0 ;
ex r , s st x = |[ r , s ]| & G * ( len G , 1 ) `1 < r & r < G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `2 < s & s < G * ( 1 , 1 ) `2 ;
assume that f is_sequence_on G and 1 <= t and t <= len G and G * ( t , width G ) `2 >= N-bound L~ f and f /. 1 = ( GoB f ) * ( t , width G ) `2 and f /. len f = ( GoB f ) * ( t , width G ) `2 ;
pred i in dom G means : Def6 : r (#) ( f * reproj ( i , x ) ) = r (#) f * reproj ( i , x ) & for y st y in dom f holds f . y = r * reproj ( i , y ) ;
consider c1 , c2 being bag of o1 + o2 such that ( decomp c ) /. k = <* c1 , c2 *> and c = c1 + c2 and c1 = c2 + c2 and c2 = c2 + c2 ;
u0 in { |[ r1 , s1 ]| : r1 < G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `2 < s1 & s1 < G * ( 1 , 1 ) `2 & s1 < G * ( 1 , 1 ) `2 } ;
Cl ( X ^ Y ) . k = the carrier of X . k2 .= C4 . ( k + 1 ) .= C4 . ( k + 1 ) .= C4 . ( k + 1 ) .= C4 . ( k + 1 ) .= C4 . ( k + 1 ) ;
attr M1 = len M2 means : Def6 : width M1 = width M2 & width M1 = width M2 & M1 = M2 - M1 & M2 = M2 - M1 implies M1 - M2 = M2 - M1 & M1 - M2 = M2 - M1 & M1 - M2 = M2 - M1 + M1 - M2 - M1 - M2 - M1 - M2 - M1 - M1 = M2 - M1 - M1 + M1 - M2 - M1 - M1 - M2 - M1 ;
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 - f /. x0 .|| < g2 } c= N2 and N2 c= dom f and f /. ( y - x0 ) in N2 ;
assume x < ( - b + sqrt ( a , b , c ) ) / 2 * a or x > ( - b - sqrt ( a , b , c ) ) / 2 * a ;
( G1 '&' G2 ) . i = ( <* 3 *> ^ G1 ) . i & ( H1 '&' H2 ) . i = ( <* 3 *> ^ G1 ) . i & ( H1 '&' H2 ) . i = ( <* 3 *> ^ G1 ) . i ;
for i , j st [ i , j ] in Indices ( M3 + M1 ) holds ( M3 + M1 ) * ( i , j ) < M2 * ( i , j ) + M1 * ( i , j )
for f being FinSequence of NAT , i being Element of NAT st for j being Element of NAT st j in dom f holds i divides f /. j holds i divides Sum f & i divides Sum f & for j st j in dom f & j <> i holds i divides j implies f . j = Sum f
assume F = { [ a , b ] where a , b is Subset of X : for c being set st c in Bimplies a c= c & b c= c & c c= a & a c= b & b c= c } ;
b2 * q2 + ( b3 * q3 ) + - ( ( a * q2 ) * q3 ) = 0. TOP-REAL n + ( - ( a * q2 ) * q3 ) .= 0. TOP-REAL n + ( - ( a * q2 ) * q3 ) .= 0. TOP-REAL n + ( - ( a * q2 ) * q3 ) .= 0. TOP-REAL n + ( - ( a * q2 ) * q3 ) .= ( - ( a * q2 ) * q2 ) ;
Cl ( Cl F ) = { D where D is Subset of T : ex B being Subset of T st D = Cl B & B in F & B is closed & Cl B = Cl D & D is closed & Cl B = Cl F } ;
attr seq is summable means : Def6 : seq is summable & ( for n holds seq . n + seq . n = Sum ( seq ) ) & ( for n holds seq . n = Sum ( seq ) + Sum ( seq ) ) & ( for n holds seq . n = Sum ( seq ) + Sum ( seq ) ) ;
dom ( ( ( ( cn x ) \mathclose { ^ { -1 } } ) | D ) ) = ( the carrier of ( ( TOP-REAL 2 ) | D ) ) /\ D .= D ;
|[ X , Z ]| is full full non empty SubRelStr of ( ( Omega Z ) |^ the carrier of X ) ) & |[ X \to Y , Z ]| is full SubRelStr of ( ( Omega Z ) |^ the carrier of Y ) ;
G * ( 1 , j ) `2 = G * ( i , j ) `2 & G * ( 1 , j ) `2 <= G * ( 1 , j ) `2 & G * ( 1 , j ) `2 <= G * ( 1 , j ) `2 ;
synonym m1 c= m2 means : Def6 : for p be set st p in P holds the non empty Subset of ( m1 + m2 ) | ( m1 + 1 ) <= ( the \HM { m2 } ) \ ( the carrier' of ( m1 + 1 ) ) ;
consider a being Element of B ( ) such that x = F ( a ) and a in { G ( b ) where b is Element of A ( ) : P [ b ] } and P [ a ] ;
mode multiplicative (# carrier -> set , multMagma , multiplicative (# carrier -> set , o -> set , o -> set , -> set , P , set , set , set , set , set , set , set , P is Element of the carrier , P , Q is Relation of the carrier , the carrier of A , the carrier of A #) ;
L ( a , b , 1 ) + L ( c , d ) = b + L ( c , d ) .= b + d + d .= b + d + c .= b + c + d .= b + c + d + d .= b + c + d + d .= b + d + d + d .= b + c + d + d .= b + d + d + d ;
cluster + _ _ + ( i1 , i2 ) -> Element of INT means : Def6 : for i , j being Element of INT holds it . ( i1 , i2 ) = + ( i1 , i2 ) & it . ( i1 , i2 ) = + ( i2 , j2 ) ;
( - s2 ) * p1 + ( s2 * p2 - ( s2 * p2 - ( s2 * p2 ) ) ) = ( - r2 ) * p1 + ( r2 * p2 - ( s2 * p2 - ( s2 * p2 ) ) ) .= ( - r2 ) * p1 + ( r2 * p2 - ( s2 * p2 - ( s2 * p2 ) ) ) ;
eval ( ( a | ( n , L ) ) *' , p ) = eval ( a | ( n , L ) ) * eval ( p , x ) .= a * eval ( p , x ) .= a * eval ( p , x ) ;
assume that the TopStruct of S = the TopStruct of T and for D being non empty directed Subset of Omega S , V being open Subset of Omega S , S being open Subset of Omega T st sup D in V & V is open holds V is open and S is open and V is open and S is open and V is open and S is open and S is open and V c= V ;
assume that 1 <= k & k <= len w + 1 and T-7 . ( ( q11 , w ) -a9 , w ) = ( T-7 . ( ( q11 , w ) -a9 ) , w ) -a9 ) . k and T11 . ( ( q11 , w ) -a9 , w ) = ( T11 . ( ( q11 , w ) -a9 ) ) . k ;
2 * a |^ ( n + 1 ) + ( 2 * b |^ ( n + 1 ) ) >= a |^ ( n + 1 ) + ( a |^ ( n + 1 ) * b |^ ( n + 1 ) ) + ( b |^ ( n + 1 ) * b |^ ( n + 1 ) ) ;
M , v2 / ( x. 3 , All ( x. 0 , All ( x. 4 , H ) ) ) / ( x. 4 , All ( x. 0 , H ) ) / ( x. 0 , All ( x. 4 , H ) ) / ( x. 4 , H ) ) |= H / ( x. 0 , x. 4 ) ;
assume that f is_differentiable_on l and for x0 st x0 in l holds 0 < f `2 . x0 or for x1 st x1 in l holds f . x1 - f . x0 < x1 & x1 in l & x1 in l & x1 in l ;
for G1 being _Graph , W being Walk of G1 , e being set st e in W & not e in W & not ( ex x being set st x in W & e in W & x in W & not x in W & not x in V ) holds W is Walk of G2
not not not not not not not not not not not lim .| is not empty & not .| is not empty or not ( not ( not ( not ( not ( not ( not ( not ( not ( not ( not ( not ( not ( not ( not ( not ( not ( not ( y y in in & & not ( not y in & not ( not y in & not y in & not y in ) ) ) ) & not ( not y in ) ) ) ) ) ) ) & not ( not ( not y in S & not ( not ( not y in S & not ( not ( not y in S ) & not ( not y in S & not ( not y in S ) & not ( not y in S ) & not y in S ) & not y
Indices GoB f = [: dom GoB f , Seg width GoB f :] & ( for i st i in dom GoB f holds ( i + 1 in dom GoB f ) & ( i + 1 in dom GoB f ) & ( i + 1 in dom GoB f ) & ( i + 1 in dom GoB f ) & ( i + 1 in dom GoB f ) & ( i + 1 in dom GoB f ) implies ( i + 1 ) in dom GoB f )
for G1 , G2 , G3 being Group , O being stable Subgroup of O , G2 being strict Subgroup of O st G1 is stable & G2 is stable & G1 is stable holds G1 is stable & G2 is stable & G2 is stable & G1 is stable & G2 is stable
UsedIntLoc ( inint f ) = { intloc 0 , intloc 1 , intloc 2 , intloc 3 , intloc 4 , intloc 5 , intloc 5 , intloc 5 , intloc 6 , - 1 , - 1 } .= { intloc 0 , 1 } \/ { intloc 5 , - 1 } .= { succ 5 , - 1 } \/ { - 1 , - 1 } .= { - 1 , - 1 } \/ UsedIntLoc f \/ { - 1 , - 1 } ;
for f1 , f2 be FinSequence of F st f1 ^ f2 is p -element & Q [ f1 ^ f2 ] & Q [ f1 ^ f2 ] & Q [ f2 ^ f1 ] holds Q [ f1 ^ f2 ] & Q [ f2 ^ f1 ]
( p `1 ) ^2 / sqrt ( 1 + ( p `2 / p `1 ) ^2 ) = ( q `1 ) ^2 / sqrt ( 1 + ( q `2 / q `1 ) ^2 ) ) ^2 / ( 1 + ( q `2 / q `1 ) ^2 ) ;
for x1 , x2 , x3 , x4 being Element of REAL n holds |( x1 - x2 , x3 - x4 )| = |( x1 - x2 , x3 - x3 )| + |( x1 - x2 , x3 - x4 )| + |( x2 - x3 , x3 - x4 )|
for x st x in dom ( ( F - G ) | A ) holds ( ( F - G ) | A ) . ( - x ) = - ( ( F - G ) | A ) . x
for T being non empty TopSpace , P being Subset-Family of T , x being Point of T st P c= the topology of T for B being Basis of x ex P being Basis of T st B c= P & P is Basis of x & P is Basis of x
( a 'or' b 'imp' c ) . x = 'not' ( ( a 'or' b ) . x ) 'or' c . x .= 'not' ( a . x ) 'or' c . x .= TRUE 'or' TRUE .= TRUE ;
for e being set st e in A9 ex X1 being Subset of XZ , Y1 being Subset of Y st e = [: X1 , Y1 :] & X1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open
for i be set st i in the carrier of S for f be Function of Sconsider S1 , S1 . i st f = H . i & F . i = f | ( the carrier of S1 ) holds F . i = f | ( the carrier of S1 ) . i
for v , w st for y st x <> y holds w . y = v . y holds Valid ( VERUM ( Al , J ) , J ) . v = Valid ( VERUM ( Al , J ) , J ) . w
card D = card D1 + card D1 - card { i , j } - card { i , j } .= c1 + 1 - 1 .= c1 + 1 - 1 .= c1 + 1 - 1 .= c1 + 1 - 1 .= c1 + 1 - 1 .= c1 + 1 - 1 .= c1 + 1 - 1 .= c1 + 1 - 1 .= c2 + 1 - 1 .= c2 + 1 - 1 ;
IC Exec ( i , s ) = ( s +* ( 0 .--> succ ( s . 0 ) ) ) . 0 .= ( 0 .--> succ ( s . 0 ) ) . 0 .= ( 0 .--> succ ( s . 0 ) ) . 0 .= ( 0 .--> ( s . 0 ) ) . 0 .= ( 0 .--> ( s . 0 ) ) . 0 .= ( 0 .--> ( s . 0 ) ) . 0 .= ( 0 .--> ( s . 0 ) ) . 0 ;
len f /. ( len f -' 1 ) -' 1 + 1 = len f -' 1 + 1 - 1 .= len f -' 1 + 1 - 1 .= len f -' 1 + 1 - 1 .= len f -' 1 + 1 - 1 .= len f -' 1 + 1 - 1 .= len f - 1 + 1 ;
for a , b , c being Element of NAT st 1 <= a & a <= b & k < a or k <= a + b-2 or k = a + b-2 or k = a + b-2 or k = a + b-2 or k = a + b-2 or k = a + b-2 or k = a + b-2 ;
for f being FinSequence of TOP-REAL 2 , p being Point of TOP-REAL 2 , i being Nat st p in LSeg ( f , i ) & i <= len f & p in LSeg ( f , i ) holds Index ( p , f /. i ) = i + 1
( ( curry ( P7 , k + 1 ) ) # x ) = ( ( curry ( P7 , k + 1 ) ) # x ) + ( ( curry ( F7 , k + 1 ) ) # x ) .= ( ( curry ( F7 , k + 1 ) ) # x ) + ( ( curry ( F7 , k + 1 ) ) # x ) ) . x ;
z2 = g /. ( len g -' n1 + 1 ) .= g . ( i -' n2 + 1 ) .= g . ( i -' n2 + 1 ) .= g . ( i -' n2 + 1 ) .= g . ( i -' n2 + 1 ) .= g /. ( i -' n2 + 1 ) .= g /. ( i -' n2 + 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 \/ the InternalRel of C6 \/ the InternalRel of C6 \/ the InternalRel of C6 ) ;
for G being Subset-Family of B st G = { R [ X ] where R is Subset of [: A , B :] , R is Subset of [: B , B :] : R in FF & ( for X st X in F holds ( X in G ) & ( X in G holds ( X in G ) implies X c= G ) ) holds ( ( G in F implies X in G ) ) & ( G in F implies X in G )
CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 ) ) = CurInstr ( P1 , Comput ( P1 , s1 , m1 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m1 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m1 ) ) .= halt SCMPDS ;
assume that a on M and b on M and c on N and d on N and p on N and a <> b and c <> d and a <> b and a <> c and a <> d and c <> d and a <> b and a <> d and a <> b and a <> c and a <> d and a <> b and a <> d and a <> b and a <> d and b <> c and a <> d and a <> b and a <> d and b <> c and a <> d and a <> d and a <> d and a <> d and a <> d and a <> d and a <> d and a <> d and a <> d and a <> d and a <> d and a <> d and a <> d and a <> d and a <> d and a <> d and a <> d and a <> d and a <> d and b <> d and a <> d and b <> d
assume that T is \hbox 4 -\cal 4 and F is closed and ex F being Subset-Family of T st F is closed & F is countable & ind F <= 0 & ind F <= 0 and ind T <= 0 and ind T <= 0 and ind T <= 0 ;
for g1 , g2 st g1 in ]. r1 - r2 , r .[ & g2 in ]. r1 - r2 , r .[ holds |. f . g1 - f . g2 .| <= ( g1 - g2 ) / ( |. g1 - g2 .| + |. g2 .| )
( ( 2 * z ) / ( z1 + z2 ) ) * ( z1 + z2 ) = ( ( 2 * z ) * ( z1 + z2 ) ) / ( z1 + z2 ) .= ( ( 2 * z ) * ( z1 + z2 ) ) / ( z1 + z2 ) .= ( ( 2 * z ) * ( z1 + z2 ) ) / ( z1 + z2 ) ;
F . i = F /. i .= 0. R + r2 .= b |^ ( n + 1 ) .= <* ( n + 1 ) \ a |^ 0 , b |^ ( n + 1 ) , a |^ ( n + 1 ) , b |^ ( n + 1 ) , b |^ ( n + 1 ) *> .= <* ( n + 1 ) \ a |^ 0 , b |^ ( n + 1 ) , b |^ ( 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 y in n implies y = f . n
func f (#) F -> FinSequence of V means : Def6 : len it = len F & for i be Nat st i in dom it holds it . i = F /. i * f /. ( F /. i ) * F /. ( F /. i ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 } = { x1 , x2 } \/ { x3 } \/ { x4 , x5 , 7 } \/ { x5 , 8 } .= { x1 , x2 } \/ { x4 , x5 , x5 } \/ { x1 , x2 } ;
for n being Nat for x being set st x = h . n holds h . ( n + 1 ) = o ( x , n ) & x in InputVertices S ( x , n ) & o ( x , n ) in InnerVertices S ( x , n ) & o ( x , n ) in InnerVertices S ( x , n )
ex S1 being Element of CQC-WFF ( Al ( ) ) st SubP ( P , l , e ) = S1 & ( for x being Element of CQC-WFF ( Al ( ) ) st x in S1 holds ( S , x , e ) `1 = [ x , e ] ) & ( S , x , e ] in [: S1 , S2 :] implies S , x , e , e , e ] in [: S1 , S2 :]
consider P being FinSequence of Gs2 such that pp = product P and for i st i in dom P ex t7 being Element of the carrier of K st P . i = t7 & t . i = t7 & t . i = t7 and t . i = t7 and t . i = t8 and t . i = t7 and t . i = t8 ;
for T1 , T2 being strict non empty TopSpace , P being Basis of T1 , T2 st the carrier of T1 = the carrier of T2 & P = the carrier of T2 & P = the topology of T1 & P = the topology of T2 & P = the topology of T2 holds T1 is Basis of T2 & T2 is Basis of T1
assume that f is_\mathbin { \lbrack u0 , u0 .] and r (#) pdiff1 ( f , 3 ) is_partial_differentiable_in u0 , 2 and partdiff ( r (#) pdiff1 ( f , 3 ) , u0 ) = r * pdiff1 ( f , u0 ) and partdiff ( r (#) pdiff1 ( f , 3 ) , u0 ) = r * pdiff1 ( f , u0 ) ;
defpred P [ Nat ] means for F , G being FinSequence of ExtREAL for G being Permutation of Seg $1 , s st len G = $1 & G = F * s & not G = F * s & not G = G * s holds Sum F = Sum G & Sum G = Sum ( F ) * Sum ( G ) ;
ex j st 1 <= j & j < width GoB f & ( ( GoB f ) * ( 1 , j ) ) `2 <= s & s <= ( GoB f ) * ( 1 , j + 1 ) `2 or s <= j & j + 1 <= width GoB f ) & ( GoB f ) * ( 1 , j + 1 ) `2 <= s ) implies s `2 <= ( GoB f ) * ( 1 , j + 1 ) `2
defpred U [ set , set ] means ex FF be Subset-Family of T st $1 = FF & union FF is open & union FF is open & union FF is open & ( for x st x in F holds x in x holds x in ( union F ) & x in ( union F ) & x in ( union F ) & x in ( union F ) & x in union ( ( union F ) \ { x } ) ) ;
for p4 being Point of TOP-REAL 2 st LE p4 , p4 , P , p1 , p2 & LE p4 , p1 , P , p1 , p2 & 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
f in \mathop ( E , H ) & for g st for y st g . y <> f . y holds x in rng g implies g in \mathop { All ( x , H ) } implies f in D & g in \mathop { All ( x , H ) }
ex 8 being Point of TOP-REAL 2 st x = 8 & ( ( ( ( ( ( ( ( ( ( ( ( ( ( p `2 / |. p .| - 1 ) ) ) / |. p .| - cn ) ) / ( 1 + cn ) ) / ( 1 + cn ) ) ) ) * ( ( ( ( ( ( ( ( ( ( ( ( ( p / |. p .| - cn ) ) / ( 1 + cn ) ) / ( 1 + cn ) ) / ( 1 + cn ) ) * ( 1 + cn ) ) / ( 1 + cn ) ) / ( 1 + cn ) ) * ( 1 + cn ) ) * ( 1 + cn ) ) / ( 1 + cn ) ) * ( 1 + cn ) ) ) ) ) ) & ( ( ( 1 + cn ) ) ^2 ) <= 1 + cn ) <= ( ( ( 1 + cn ) ) ^2 ) <= ( ( ( ( ( (
assume for d7 being Element of NAT st d7 <= max ( n7 , t7 ) holds s1 . ( ( d + 1 ) - 1 ) = s2 . ( ( d + 1 ) - 1 ) & s1 . ( ( d + 1 ) - 1 ) = s2 . ( ( d + 1 ) - 1 ) ;
assume that 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 } = Sphere ( x , r ) /\ Sphere ( y , r ) and not e in Sphere ( s , r ) and not e in Sphere ( x , r ) ;
given r such that 0 < r and for s holds 0 < s or ex x1 , x2 be Point of RNS st x1 in dom f & x2 in dom f & ||. x1 - x2 .|| < s & |. f /. x1 - f /. x2 .| < r ;
( p | x ) | ( p | ( x | x ) ) = ( ( ( x | x ) | x ) | ( x | x ) ) | p ) | p .= ( ( ( x | x ) | x ) | p ) | p ;
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 ) and sin . x = 2 * sin . x + cos . x * cos . ( cos . x + cos . x ) / ( cos . x ) ^2 and cos . x = 2 * cos . x - cos . x / ( cos . x ) ^2 / ( cos . x ) ^2 and cos . x / ( cos . x ) ^2 / ( cos . x ) ^2 / ( cos . x ) ^2 / ( cos . x ) ^2 / ( cos . x ) ^2 / ( cos . x ) ^2 / ( cos . x ) ^2 / ( cos . x ) ^2 / ( cos . x ) ^2 / ( cos . x ) ^2 / ( cos . x ) ^2 / ( sin . x ) ^2 / ( cos . x
assume that i in dom A and len A > 1 and for B st B > 1 & B c= \HM { i , j } & A * ( i , j ) = ( the \HM { i , j } ) * ( A * ( i , j ) ) and len B = len A and width A = width B and width B = width B and width A = width B ;
for i be non zero Element of NAT st i in Seg n holds i divides n or i = <* 1. F_Complex *> or ( i = n or i <> n & i <> n & i <> n & i <> n & i <> n & i <> n & i <> n implies h . i = thesis
( ( ( b1 'imp' b2 ) '&' ( c1 'imp' c2 ) ) '&' ( ( a1 'or' b1 ) '&' ( a1 'or' c1 ) ) '&' 'not' ( a1 'or' c1 ) ) '&' 'not' ( a1 '&' b1 ) '&' 'not' ( a1 '&' c1 ) '&' 'not' ( a1 '&' b1 ) '&' 'not' ( b1 '&' c1 ) '&' 'not' ( a1 '&' b1 ) '&' 'not' ( b1 '&' c1 ) '&' 'not' ( a1 '&' b1 ) '&' 'not' ( b1 '&' c1 ) '&' 'not' ( a1 '&' c1 ) ;
assume that for x holds f . x = ( ( cot (#) ( cot - cot ) ) `| Z ) . x and for x st x in Z holds ( ( ( cot - cot ) `| Z ) . x ) ^2 - ( ( cot - cot ) `| Z ) . x = - cos . ( x - x0 ) / ( sin . ( x - x0 ) ) ^2 ) ;
consider R8 , I-8 be Real such that R8 = Integral ( M , Re ( F . n ) ) and I-8 = Integral ( M , Im ( F . n ) ) and Integral ( M , Im ( F . n ) ) = ( Im ( F . n ) ) * ( Im ( F . n ) ) and I = ( Im ( F . n ) ) * i ) ;
ex k be Element of NAT st k = k & 0 < d & for q be Element of product G st q in X & ||. q- f .|| < d holds ||. partdiff ( f , x , k ) - partdiff ( f , x , k ) .|| < r ;
x in { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , 7 , 8 } iff x in { x1 , x2 , x3 , x4 , x4 } \/ { x4 , x5 , x5 } \/ { x5 , x5 , 7 } \/ { x5 , 8 }
G * ( j , ( i + 1 ) ) `2 = G * ( 1 , ( i + 1 ) ) `2 .= G * ( 1 , ( i + 1 ) ) `2 .= G * ( 1 , ( i + 1 ) ) `2 ) .= G * ( 1 , ( i + 1 ) ) `2 .= G * ( 1 , ( i + 1 ) ) `2 .= G * ( 1 , ( i + 1 ) ) `2 .= G * ( 1 , j ) `2 ;
f1 * p = p .= ( ( the Arity of S1 ) +* ( the Arity of S2 ) ) . o .= ( the Arity of S1 ) . ( ( the Arity of S2 ) . o ) .= ( the Arity of S1 ) . ( ( the Arity of S2 ) . o ) .= ( the Arity of S2 ) . ( ( the Arity of S1 ) . o ) .= ( the Arity of S1 ) . ( ( the Arity of S1 ) . o ) .= ( the Arity of S1 ) . ( ( the Arity of S1 ) . o ) .= ( the Arity of S1 ) . ( ( the Arity of S1 ) . ( ( the Arity of S1 ) . ( ( the Arity of S1 ) . ( ( the Arity of S1 ) . ( ( the Arity of S1 ) . o ) .= ( the Arity of S1 ) . ( the Arity of S1 ) ) . ( ( the Arity of S1 ) . ( ( the Arity of S1 ) .
func tree ( T , P , T1 ) -> DecoratedTree means : : : : q in it iff q in T & for p , q st p in P holds p ^ q in T or p ^ q in T1 or p ^ q in T1 & q ^ p in T1 & q ^ p in T1 & q ^ q in T1 ;
F /. ( k + 1 ) = F . ( k + 1 - 1 ) .= F9 . ( p . ( k + 1 -' 1 ) , k + 1 -' 1 ) .= F9 . ( p . k , k + 1 -' 1 ) .= F9 . ( p . k , k + 1 -' 1 ) .= Fmin . ( p . k , k ) ;
for A , B , C being Matrix of K st len B = len C & width B = width C & len B = width C & len C > 0 & len A > 0 & len B > 0 & len C > 0 & len A > 0 & len B > 0 & len C > 0 & len A > 0 & len B > 0 & len B > 0 & len C = len B & len C = len B & width B = 0 holds A * C = B * C = 0 & len C = len B & len C = len B & len C = len C & len C = len C & len C = len B + 1 implies C * ( B * ( C * ( len C * ( len C * ( len C * ( C * ( C * ( len C * ( len C * ( len C * A ) = len A ) & width A ) = len A ) & len A + 1
seq . ( k + 1 ) = 0. F_Complex + seq . ( k + 1 ) .= ( Partial_Sums ( seq ) ) . ( k + 1 ) + ( Partial_Sums ( seq ) ) . ( k + 1 ) .= ( Partial_Sums ( seq ) ) . ( k + 1 ) + ( Partial_Sums ( seq ) ) . ( k + 1 ) .= ( Partial_Sums ( seq ) ) . ( k + 1 ) + ( Partial_Sums ( seq ) ) . ( k + 1 ) ;
assume that x in ( the carrier of CQ ) \/ ( the carrier of CQ ) and y in ( the carrier of CQ ) \/ ( the carrier of CQ ) and [ x , y ] in ( the carrier of CQ ) \/ ( the carrier of CQ ) and [ y , z ] in the InternalRel of CQ ;
defpred P [ Element of NAT ] means for f st len f = $1 holds ( for k st k in $1 holds ( for i st i in $1 holds ( for k st k in $1 holds ( for k st k in $1 holds ( for k st k in $1 holds ( for k st k in $1 holds k < f . k ) ) holds ( for k st k in $1 holds k < $1 ) implies for k st k in $1 holds f . k = ( for k st k in $1 ) ) & ( for k st k in $1 holds ( for k st k in $1 holds ( for k st k in $1 holds ( for k st k in $1 holds ( for k st k in $1 holds ( for k holds ( for k st k in $1 ) ) implies ( for k st k in $1 holds ( for i st k in $1 holds ( for i st k in $1 holds ( for i st k in $1 holds ( for i st k in
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 + 1 ) = G * ( i , j ) and f /. ( k + 1 ) = G * ( i + 1 , j ) ;
assume that sn < 1 and q `1 > 0 and ( q `2 / |. q .| - sn ) / ( 1 + sn ) >= sn and ( q `2 / |. q .| - sn ) / ( 1 + sn ) >= sn and ( q `2 / |. q .| - sn ) / ( 1 + sn ) >= 0 and ( q `2 / |. q .| - sn ) / ( 1 + sn ) >= 0 ) ;
for M being non empty dist , x being Point of M , f being Point of M st x = x `1 holds ex f being sequence of bool M st for n being Element of NAT holds f . n = Ball ( x `1 , ( 1 / ( n + 1 ) ) * ( f . n ) )
defpred P [ Element of omega ] means f1 is_differentiable_on $1 , Z & f2 is_differentiable_on $1 , Z & ( for x st x in $1 holds ( f1 - f2 ) is_differentiable_in x & ( f1 - f2 ) `| Z is_differentiable_in x & ( f1 - f2 ) `| Z = ( f1 - f2 ) `| Z ) . x - ( f2 - f2 ) `| Z ) . x ;
defpred P1 [ Nat , Point of RNS ] means $2 in Y & ||. ( f . $1 - f . ( $2 + 1 ) ) - f . ( $2 + 1 ) .|| < r / ( $1 + 1 ) & ||. ( f . $1 - f . ( $2 + 1 ) ) - f . ( $2 + 1 ) .|| < r / ( $1 + 1 ) ;
( f ^ mid ( g , 2 , len g ) ) . i = ( mid ( g , 2 , len g ) ) . ( i - len f + 1 ) .= g . ( i - len f + 1 ) .= g . ( i - len f + 1 ) .= g . ( i - len f + 1 ) .= g . ( i - len f + 1 ) .= g . ( i - len f + 1 ) ;
( 1 - 2 * n0 + 2 * n0 ) * ( 2 * n0 + 2 * n0 ) = ( ( 1 - 2 * n0 ) * ( 2 * n0 + 2 * n0 ) ) * ( 2 * n0 + 1 ) * ( 2 * n0 + 1 ) .= ( 1 - 2 * n0 ) * ( 2 * n0 + 1 ) * ( 2 * n0 + 1 ) .= ( 1 - 2 * n0 ) * ( 2 * n0 + 1 ) ;
defpred P [ Nat ] means for G being non empty finite strict finite RelStr st G is free for H being strict finite non empty RelStr st card G is free & card H = $1 & the carrier of G = $1 holds the RelStr of G = the RelStr of H & the RelStr of H = the RelStr of G & the RelStr of H = the RelStr of H ;
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 & i <= len f & LSeg ( f , i ) /\ Ball ( u , r ) <> {} and not m in Ball ( u , r ) and not m in Ball ( u , r ) and not m in Ball ( u , r ) and not m in Ball ( u , r ) ;
defpred P [ Element of NAT ] means ( Partial_Sums ( cos , $1 ) . ( 2 * $1 ) ) . ( 2 * $1 + 1 ) = ( Partial_Sums ( cos , $1 ) . ( 2 * $1 + 1 ) ) . ( 2 * $1 + 1 ) ) / ( 2 * $1 + 1 ) ;
for x being Element of product F holds x is FinSequence of G & dom x = I & x in dom ( the Sorts of F ) & for i being set st i in dom x holds x . i in ( the Sorts of F ) . i & x . i in ( the Sorts of F ) . i
( x " ) |^ ( n + 1 ) = ( ( x " ) |^ n ) * x " .= ( x * ( x |^ n ) ) " .= ( x * ( x |^ n ) ) " .= ( x |^ n ) " * ( x |^ n ) .= ( x |^ n ) " * ( x |^ n ) .= ( x |^ n ) " * ( x |^ n ) " .= ( x |^ n ) " ;
DataPart Comput ( P +* ( a , I ) , Initialized s , LifeSpan ( P +* I , Initialized s ) + 3 ) ) = DataPart Comput ( P +* I , Initialize s , LifeSpan ( P +* I , Initialized s ) + 3 ) .= DataPart Comput ( P +* I , Initialize s , LifeSpan ( P +* I , Initialized s ) + 3 ) .= DataPart Comput ( P +* I , Initialize s , LifeSpan ( P +* I , Initialize s ) + 3 ) ;
given r such that 0 < r and ]. x0 , x0 + r .[ c= dom f1 /\ dom f2 and for g st g in ]. x0 , x0 + r .[ holds f1 . g <= f1 . g & f1 . g <= x0 & f1 . g <= 0 & f2 . g <= 0 ;
assume that X c= dom f1 /\ dom f2 and f1 | X is continuous and f2 | X is continuous and ( for r st r in X holds f1 . r - f2 . r = r * ( f1 . r - f2 . r ) ) & ( for r st r in X holds f1 . r - f2 . r = r * ( f1 . r - f2 . r ) ) and ( for r st r in X /\ X holds f1 . r = r * ( f1 . r - f2 . r ) ;
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 & for x being Element of L st x in X holds x is directed & x is directed
Support ( e *' A ) in { Support ( m *' p ) where m is Polynomial of n , L : ex p being Polynomial of n , L st p in Support ( m *' p ) & ex q being Polynomial of n , L st q in Support ( m *' q ) & p in Support ( m *' q ) } ;
( f1 - f2 ) /. ( lim s1 ) = lim ( f1 /* s1 ) - lim ( f2 /* s1 ) .= lim ( f1 /* s1 ) - lim ( f2 /* s1 ) .= lim ( f1 /* s1 ) - ( f2 /* s1 ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) ;
ex p1 being Element of CQC-WFF ( Al ( ) ) st F . p1 = g `1 & for g being Function of CQC-WFF ( Al ( ) ) , D ( ) st P [ g ] holds P [ g , ( len ( f qua Function ) ) + 1 ]
( mid ( f , i , len f -' 1 ) ^ <* f /. ( len f -' 1 ) *> ) /. j = ( mid ( f , i , len f -' 1 ) ) /. ( j + 1 ) .= ( mid ( f , i , len f -' 1 ) ) /. j .= ( mid ( f , i , len f -' 1 ) ) /. j ;
( ( p ^ q ) ^ r ) . ( len p + k ) = ( ( p ^ q ) ^ r ) . ( len p + k ) .= ( ( p ^ q ) . ( len p + k ) ) . ( len q + k ) .= ( ( p ^ q ) . k ) . ( len q + k ) .= ( p ^ q ) . k ;
len mid ( upper_volume ( f , D2 ) , indx ( D2 , D1 , j1 ) + 1 ) = indx ( D2 , D1 , j1 ) - 1 .= indx ( D2 , D1 , j1 ) + 1 - 1 .= indx ( D2 , D1 , j1 ) + 1 - 1 .= indx ( D2 , D1 , j1 ) + 1 - 1 ;
x * y * z = Mz * ( y * z , z9 ) .= ( x * y ) * ( y * z ) .= ( x * ( y * z ) ) * ( x * z ) .= ( x * ( y * z ) ) * ( x * z ) .= ( x * ( y * z ) ) * ( x * z ) .= ( x * y ) * ( x * z ) ;
v . <* x , y *> - ( <* x0 , y0 *> * i ) = partdiff ( v , ( x - x0 ) * i ) + ( partdiff ( u , ( x - x0 ) * i ) + ( partdiff ( u , ( x - x0 ) * i ) ) ) ;
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 .= ( - 1 ) * 0 + 0 * 0 .= 0 ;
Sum ( L (#) F ) = Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( ( L (#) F1 ) ^ ( L (#) F2 ) ) .= Sum ( ( L (#) F1 ) ^ ( 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 Y0 be finite Subset of X , Y1 be finite Subset of REAL st Y0 is non empty & Y1 c= Y & for Y1 be finite Subset of X st Y1 is non empty & Y1 c= Y holds |. ( - lower_bound ( Y1 ) ) .| < r / 2
( GoB f ) * ( i , j ) = f /. ( k + 2 ) & ( GoB f ) * ( i , j + 1 ) = f /. ( k + 2 ) or ( GoB f ) * ( i , j ) = f /. ( k + 2 ) or ( GoB f ) * ( i + 2 , j ) = f /. ( k + 2 ) ) ;
( ( - cos ) / ( cos ( x ) ) ^2 ) = ( - sin ( x ) ) ^2 / ( cos ( x ) ) ^2 .= ( - cos ( x ) ) ^2 / ( cos ( x ) ) ^2 .= ( - cos ( x ) ) ^2 / ( cos ( x ) ) ^2 .= ( - cos ( x ) ) ^2 / ( cos ( x ) ) ^2 ;
( - b + sqrt ( a , b , c ) ) / 2 * a < 0 & ( - b - sqrt ( a , b , c ) / 2 * a < 0 or - b + sqrt ( a , b , c ) / 2 * a ) < 0 or - b + - sqrt ( a , b , c ) / 2 * a < 0 ;
assume that ex_inf_of uparrow "\/" ( X , L ) /\ C and ex_sup_of X , L and ex_sup_of X , L and "\/" ( X , L ) = "/\" ( uparrow "\/" ( X , L ) , L ) and "\/" ( X , L ) = "\/" ( uparrow "\/" ( X , L ) , L ) and "\/" ( X , L ) = "\/" ( uparrow "\/" ( X , L ) , L ) ;
( for j holds j = ( j = j = i = j or j = i ) & ( j = j implies j = i ) implies ( j = i implies j = i ) & ( j = i implies j = i ) & ( j = i implies j = i implies j = i ) ) & ( j = i implies j = i implies j = i ) implies j = i & j = i implies j = i )