thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
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contradiction ;
contradiction ;
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contradiction ;
thesis ;
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contradiction ;
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contradiction ;
thesis ;
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thesis ;
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thesis ;
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contradiction ;
thesis ;
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contradiction ;
thesis ;
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contradiction ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
B in X ;
a <> c
T c= S
D c= B
c in X ;
b in X ;
X in Y ;
b in D ;
x = e ;
let m ;
h is onto ;
N in K ;
let i ;
j = 1 ;
x = u ;
let n ;
let k ;
y in A ;
let x ;
let x ;
m c= y ;
F is one-to-one ;
let q ;
m = 1 ;
1 < k ;
G is prime ;
b in A ;
d divides a ;
i < n ;
s <= b ;
b in B ;
let r ;
B is one-to-one ;
R is total ;
x = 2 ;
d in D ;
let c ;
let c ;
b = Y ;
0 < k ;
let b ;
let n ;
r <= b ;
x in X ;
i >= 8 ;
let n ;
let n ;
y in f ;
let n ;
1 < j ;
a in L ;
C is boundary ;
a in A ;
1 < x ;
S is finite ;
u in I ;
z << z ;
x in V ;
r < t ;
let t ;
x c= y ;
a <= b ;
m in NAT ;
assume f is prime ;
not x in Y ;
z = -infty ;
let k be Nat ;
K ` is being_line ;
assume n >= N ;
assume n >= N ;
assume X is 1 ;
assume x in I ;
q is from 0 ;
assume c in x ;
'not' p > 0 ;
assume x in Z ;
assume x in Z ;
1 <= kLet ;
assume m <= i ;
assume G is prime ;
assume a divides b ;
assume P is closed ;
\bf 2 > 0 ;
assume q in A ;
W is non bounded ;
f is that g is that 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 negative ;
b ` <= c ` ;
A meets W ;
i ` <= j ` ;
assume H is universal ;
assume x in X ;
let X be set ;
let T be Tree ;
let d be element ;
let t be element ;
let x be element ;
let x be element ;
let s be element ;
k <= 5 ;
let X be set ;
let X be set ;
let y be element ;
let x be element ;
P [ 0 ]
let E be set , A be Subset of E ;
let C be Category ;
let x be element ;
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 iso ;
Q halts_on s ;
x in \in \rbrack ;
M < m + 1 ;
T2 is open ;
z in b set ;
R2 is well-ordering ;
1 <= k + 1 ;
i > n + 1 ;
q1 is one-to-one ;
let x be trivial set ;
PP is one-to-one ;
n <= n + 2 ;
1 <= k + 1 ;
1 <= k + 1 ;
let e be Real ;
i < i + 1 ;
p3 in P ;
p1 in K ;
y in C1 ;
k + 1 <= n ;
let a be Real , b be Real ;
X |- r => p ;
x in { A } ;
let n be Nat ;
let k be Nat ;
let k be Nat ;
let m be Nat ;
0 < k + k ;
f is_differentiable_in x ;
let x0 ;
let E be Ordinal ;
o \mathord 4 ;
O <> O2 ;
let r be Real ;
let f be FinSeq-Location ;
let i be Nat ;
let n be Nat ;
Cl A = A ;
L c= Cl L ;
A /\ M = B ;
let V be RealUnitarySpace , M be Subset of V ;
not s in Y |^ 0 ;
rng f <= w ;
b "/\" e = b ;
m = m3 ;
t in h . D ;
P [ 0 ] ;
assume z = x * y ;
S . n is bounded ;
let V be RealUnitarySpace , M be Subset of V ;
P [ 1 ] ;
P [ {} ] ;
C1 is L~ f ;
H = G . i ;
1 <= i ^2 + 1 ;
F . m in A ;
f . o = o ;
P [ 0 ] ;
aZ <= being Real ;
R [ 0 ] ;
b in f .: X ;
assume q = q2 ;
x in [#] V ;
f . u = 0 ;
assume e1 > 0 ;
let V be RealUnitarySpace , M be Subset of V ;
s is trivial & s is non empty ;
dom c = Q ;
P [ 0 ] ;
f . n in T ;
N . j in S ;
let T be complete LATTICE , a 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 ;
v|^ n < n ;
Smax is bounded ;
assume p = p2 ;
len f = n ;
assume x in P1 ;
i in dom q ;
let U , S ;
pp = c ;
j in dom h ;
let k ;
f | Z is continuous ;
k in dom G ;
UBD C = B ;
1 <= len M ;
p in \mathbin { x } ;
1 <= jj ;
set A = Cl A ;
card a [= c ;
e in rng f ;
cluster B \oplus A -> empty ;
H is_\cdot the \mathclose { or } ;
assume x0 <= m ;
T is increasing ;
e2 <> e2 ;
Z c= dom g ;
dom p = X ;
H is proper implies H = G
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= CN ;
mm <> {} ;
let x be Element of Y ;
let f be let ) ;
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 Let ( \HM { the } \HM { carrier } \HM { of G } ) ;
- y in I ;
let A be non empty set , a be Element of A ;
P0 = 1 ;
assume r in F . k ;
assume f is simple ;
let A be \lbrack set ;
rng f c= NAT ;
assume P [ k ] ;
f6 <> {} ;
let o be Ordinal ;
assume x is sum of squares ;
assume not v in { 1 } ;
let I1 ;
assume that 1 <= j and j < l ;
v = - u ;
assume s . b > 0 ;
d1 in dom g ;
assume t . 1 in A ;
let Y be non empty TopSpace , x be Point of Y ;
assume a in ]. s , t .[ ;
let S be non empty RelStr ;
a , b // b , a ;
a * b = p * q ;
assume x , y are_the space ;
assume x in [#] ( f ) ;
[ a , c ] in X ;
mm <> {} ;
M + N c= M + M ;
assume M is connected hh) ;
assume f is IExec bbbr-r) ;
let x , y ;
let T be non empty TopSpace ;
b , a // b , c ;
k in dom Sum p ;
let v be Element of V ;
[ x , y ] in T ;
assume len p = 0 ;
assume C in rng f ;
k1 = k2 & k2 = k2 ;
m + 1 < n + 1 ;
s in S \/ { s } ;
n + i >= n + 1 ;
assume Re y = 0 ;
k1 <= j1 & k1 <= len G ;
f | A is continuous ;
f . x ^2 <= b ;
assume y in dom h ;
x * y in B1 ;
set X = Seg n ;
1 <= i2 + 1 ;
k + 0 <= k + 1 ;
p ^ q = p ;
j |^ y divides m ;
set m = max A ;
[ x , x ] in R ;
assume x in succ 0 ;
a in sup phi ;
Cf in X ;
q2 c= C1 & q2 c= C1 ;
a2 < c2 & b2 < c ;
s2 is 0 -started ;
IC s = 0 ;
s4 = s4 & s4 = c ;
let V ;
let x , y ;
let x be Element of T ;
assume a in rng F ;
x in dom T ` ;
let S be PI of L ;
y " <> 0 ;
y " <> 0 ;
0. V = u-uw ;
y2 , y are_\! to w ;
R8 in X ;
let a , b be Real , x be Element of REAL ;
let a be object of C ;
let x be Vertex of G ;
let o be object of C , a be object of C ;
r '&' q = P \lbrack l , l .] ;
let i , j be Nat ;
let s be State of A , a be element ;
s4 . n = N ;
set y = ( x `1 ) ^2 ;
mi in dom g ;
l . 2 = y1 ;
|. g . y .| <= r ;
f . x in CH ;
V1 is non empty ;
let x be Element of X ;
0 <> f . g2 ;
f2 /* q is convergent ;
f . i is_measurable_on E ;
assume \xi in N-22 ;
reconsider i = i as Ordinal ;
r * v = 0. X ;
rng f c= INT & rng f c= INT ;
G = 0 .--> goto 0 ;
let A be Subset of X ;
assume A9 is dense & A2 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 , Y = Vars C ;
let o be OperSymbol of S ;
let R be connected RelStr ;
n + 1 = succ n ;
x9 c= Z1 & x9 c= Z1 implies x9 c= y9
dom f = C1 & dom g = C2 ;
assume [ a , y ] in X ;
Re ( seq ) is convergent ;
assume a1 = b1 & a2 = b2 ;
A = ( sInt A ) . i ;
a <= b or b <= a ;
n + 1 in dom f ;
let F be Instruction of S , s be State of S ;
assume r2 > x0 & x0 in dom f ;
let Y be non empty set , f be Function of Y , BOOLEAN ;
2 * x in dom W ;
m in dom ( ( g | X ) ^ ) ;
n in dom g1 & n in dom g2 ;
k + 1 in dom f ;
the still not bound in { s } ;
assume x1 <> x2 & y1 <> y2 ;
v2 in V1 & v2 in V1 ;
not [ b `1 , b `2 ] in T ;
i9 + 1 = i ;
T c= ] ( T ) ;
( l - 1 ) * ( l - 1 ) = 0 ;
let n be Nat ;
( t `2 ) ^2 = r ;
AA is_integrable_on M & AA c= M ;
set t = Top t ;
let A , B be real-membered set ;
k <= len G + 1 ;
[: C , C :] misses [: C , C :] ;
product ( F | n ) is non empty ;
e <= f or f <= e ;
cluster -> non empty for normal sequence ;
assume c2 = b2 & c1 = b2 ;
assume h in [. q , p .] ;
1 + 1 <= len C ;
not c in B . m1 ;
cluster R .: X -> empty ;
p . n = H . n ;
assume v-4 is Cauchy & lim vK = 0 ;
IC s3 = 0 & IC s3 = 0 ;
k in N or k in K ;
F1 \/ F2 c= F \/ G ;
Int G1 <> {} & Int G1 <> {} ;
( z `2 ) ^2 = 0 ;
p10 <> p1 & p10 <> p2 ;
assume z in { y , w } ;
MaxADSet ( a ) c= F ;
ex_sup_of downarrow s , S ;
f . x <= f . y ;
let T be up-complete reflexive transitive antisymmetric RelStr ;
q |^ m >= 1 ;
a >= X & b >= Y ;
assume <* a , c *> <> {} ;
F . c = g . c ;
G is one-to-one , one-to-one ;
A \/ { a } c= B ;
0. V = 0. Y ;
let I be Instruction of S , s be State of S ;
fp . x = 1 / ( x ^2 ) ;
assume z \ x = 0. X ;
C4 = 2 to_power n ;
let B be SetSequence of Sigma ;
assume X1 = p .: D ;
n + l2 in NAT & l2 in NAT ;
f " P is compact ;
assume x1 in REAL n + 1 ;
p1 = K1 & p2 = K1 ;
M . k = <*> REAL ;
phi . 0 in rng phi ;
MMML is closed ;
assume z0 <> 0. L ;
n < N . k ;
0 <= seq . 0 - lim seq ;
- q + p = v ;
{ v } is Subset of B ;
set g = f /. 1 ;
[: R , S :] is stable ;
set GR = Vertices R , S = R ;
pp c= P3 & px c= P3 ;
x in [. 0 , 1 .] ;
f . y in dom F ;
let T be Scott Scott TopAugmentation of S ;
ex_inf_of the carrier of S , T ;
downarrow a = downarrow b & downarrow b = { 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 ;
||. an - y .|| <= r ;
assume Y c= field Q & Y <> {} ;
a ~ , b ~ ;
assume a in A ( ) ;
k in dom ( q ^ <* x *> ) ;
p is \rbrack ;
i - 1 = i-1 ;
f | A is one-to-one ;
assume x in f .: [: X , Y :] ;
i2 - i1 = 0 ;
j2 + 1 <= i2 ;
g " * a in N ;
K <> { [ {} , {} ] } ;
cluster -> of \rm from in \rm ;
|. q .| ^2 > 0 ;
|. p4 .| = |. p .| ;
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 k in dom C & k <> i ;
1 + 1-1 <= i + 1-1 ;
dom S = dom F & rng F c= dom G ;
let s be Element of NAT ;
let R be ManySortedSet of A ;
let n be Element of NAT ;
let S be non empty non void non empty non void ManySortedSign ;
let f be ManySortedSet of I ;
let z be Element of F_Complex , a be Element of F_Complex ;
u in { b9 , g } ;
2 * n < 2 * n ;
x , y be set ;
BW c= V1 & V c= V ;
assume I is_closed_on s , P ;
U2 = U & U2 = U \/ { {} } ;
M /. 1 = z /. 1 ;
x9 = x9 & x9 = y9 ;
i + 1 < n + 1 + 1 ;
x in { {} , <* 0 *> } ;
f9 <= ( f | n ) . ( f . n ) ;
l be Element of L ;
x in dom ( F | k ) ;
let i be Element of NAT ;
r8 is ( COMPLEX , n ) -valued ;
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 * x9 ) ;
assume dom g = cod f & cod g = cod g ;
let X , Y be non empty TopSpace , f be Function of X , Y ;
x \oplus A is interval ;
|. <*> A .| . a = 0 ;
cluster strict SubSublattice of L -> strict ;
a1 in B . s1 & a1 in B . s1 ;
let V be finite VectSp of F , a be Element of V ;
A * B on B implies A on B
fg = NAT --> 0 ;
let A , B be Subset of V ;
z1 = P1 . j & z2 = P1 . j ;
assume f " P is closed ;
reconsider j = i as Element of M ;
a , b be Element of L ;
assume q in A \/ ( B "\/" C ) ;
dom ( F * C ) = o ;
set S = INT |^ X ;
z in dom ( A --> y ) ;
P [ y , h . y ] ;
{ x0 } c= dom f & f | X is continuous ;
let B be non-empty ManySortedSet of I , A be ManySortedSet of I ;
sqrt ( PI / 2 ) < Arg z ;
reconsider z9 = 0 as Nat ;
LIN a , d , c ;
[ y , x ] in IN ;
( Q ) * ( 3 , 3 ) = 0 ;
set j = x0 div m ;
assume a in { x , y , c } ;
j2 - ( jj - 1 ) > 0 ;
I -phi = 1 ;
[ y , d ] in F-8 ;
let f be Function of X , Y ;
set A2 = ( B |^ C ) * ( A |^ C ) ;
s1 , s2 are_/ 2 ;
j1 - 1 = 0 & j1 - 1 = 0 ;
set m2 = 2 * n + j ;
reconsider t = t as bag of n ;
I2 . j = m . j ;
i |^ s , n are_relative_prime ;
set g = f | D-21 ;
assume X is lower & 0 <= r ;
( p1 `1 ) ^2 = 1 ;
a < ( p3 `1 ) ^2 + ( p3 `2 ) ^2 ;
L \ { m } c= UBD C ;
x in Ball ( x , 10 ) ;
not a in LSeg ( c , m ) ;
1 <= i1 - 1 & i1 - 1 <= i1 - 1 ;
1 <= i1 - 1 & i1 - 1 <= i1 - 1 ;
i + i2 <= len h ;
x = W-min ( P ) & x in P ;
[ x , z ] in X ~ ;
assume y in [. x0 , x .] ;
assume p = <* 1 , 2 , 3 *> ;
len <* A1 *> = 1 ;
set H = h . ( g . I ) ;
card b * a = |. a .| ;
Shift ( w , 0 ) |= v ;
set h = h2 (*) h1 , h1 = h1 (*) h2 ;
assume x in ( X0 /\ X1 ) ;
||. h .|| < dx0 & 0 < dx0 ;
not x in the carrier of f ;
f . y = F ( y ) ;
for n holds X [ n ] ;
k - l = kl2 - l ;
<* p , q *> /. 2 = q ;
let S be Subset of the lattice of Y ;
P , Q be ] Subset of s ;
Q /\ M c= union ( F | M )
f = b * canFS ( S ) ;
let a , b be Element of G ;
f .: X <= f . sup X ;
let L be non empty transitive RelStr , N be net of L ;
S-20 is x -basis of i , K ;
let r be non positive Real ;
M , v |= x \hbox { y } ;
v + w = 0. ( Z ) ;
P [ len F ] implies F ( len F ) = F ( len F )
assume InsCode ( i ) = 8 & InsCode ( i ) = 8 ;
the zero of M = 0 implies M = 0
cluster z * seq -> summable ;
let O be Subset of the carrier of C ;
||. f .|| | X is continuous ;
x2 = g . ( j + 1 ) ;
cluster -> Element of AllTermsOf S ;
reconsider l1 = l-1 as Nat ;
v4 is Vertex of r2 & v4 is Vertex of G ;
TT2 is SubSpace of T2 & TT2 is SubSpace of T2 ;
Q1 /\ Q19 <> {} ;
let k be Nat ;
q " is Element of X & q " is Element of X ;
F . t is set of of zero M ;
assume n <> 0 & n <> 1 ;
set en = EmptyBag n , em = EmptyBag n ;
let b be Element of Bags n ;
assume for i holds b . i is commutative ;
x is root implies ( p `1 ) ^2 = ( p `2 ) ^2
not r in ]. p , q .] ;
let R be FinSequence of REAL , a be Element of REAL ;
S7 does not destroy b1 & S7 does not cluster b2 ;
IC SCM R <> a & IC R <> b ;
|. - |[ x , y ]| .| >= r ;
1 * ( seq . n ) = seq . ( seq . n ) ;
let x be FinSequence of NAT , a be Element of NAT ;
let f be Function of C , D , g be Function of C , D ;
for a holds 0. L + a = a
IC s = s . NAT & s . NAT = s . NAT ;
H + G = F- ( G-G ) ;
C1 . x = x2 & C1 . x = y2 ;
f1 = f .= f2 .= f1 ^ f2 ;
Sum <* p . 0 *> = p . 0 ;
assume v + W = v + u + W ;
{ a1 } = { a2 } & { a2 } = { a1 } ;
a1 , b1 _|_ b , a ;
d1 , o _|_ o , a3 ;
IN is reflexive & IN is reflexive implies IN is reflexive
IC is antisymmetric implies [: C , C :] is antisymmetric
sup rng ( H1 , H2 ) = e ;
x = a9 * a9 & y = c9 * a9 ;
|. p1 .| ^2 >= 1 ^2 ;
assume j2 -' 1 < 1 - 1 ;
rng s c= dom f1 & rng s c= dom f2 ;
assume support a misses support b & not a in support b ;
let L be associative non empty doubleLoopStr , a be Element of L ;
s " + 0 < n + 1 ;
p . c = ( f " ) . 1 ;
R . n <= R . ( n + 1 ) ;
Directed ( I1 , I2 ) = I1 " ;
set f = + ( x , y , r ) ;
cluster Ball ( x , r ) -> bounded ;
consider r being Real such that r in A ;
cluster -> non empty for Function ;
let X be non empty directed Subset of S ;
let S be non empty full SubRelStr of L ;
cluster <* *> . N , \subseteq <* 1 *> . N -> complete ;
( 1 - a " ) = a ;
( q . {} ) `1 = o ;
n - ( i -' 1 ) > 0 ;
assume ( 1 - 2 ) ^2 <= t ^2 ;
card B = k + 1 ;
x in union rng ( f | n ) ;
assume x in the carrier of R & y in the carrier of R ;
d in dom f ;
f . 1 = L . ( F . 1 ) ;
the vertices of G = { v } & { v } c= the vertices of G ;
let G be : ww] ;
e , v9 , e , f ;
c . ( i9 - 1 ) in rng c ;
f2 /* q is divergent_to+infty & f2 /* q is divergent_to+infty ;
set z1 = - z2 , z2 = - z2 , z2 = - z2 ;
assume w is_lllof S , G ;
set f = p \! \mathop { t } , g = p \! \mathop { t } , h = p ;
let c be Object of C ;
assume ex a st P [ a ] ;
let x be Element of REAL-NS m ;
let IK be Subset-Family of X ;
reconsider p = p as Element of NAT ;
let v , w be Point of X ;
let s be State of SCM+FSA , a be Int-Location ;
p is FinSequence of ( the InstructionsF of SCM+FSA ) * ;
stop I c= P & stop I c= P & card I = card J ;
set ci = fc /. i , cj = fc /. i , cj = c /. i , cj = c /. i , cj = c /. i , cj = c
w ^ t ^ s ^ t ^ s ^ t ^ s ^ t ^ s ^ t ^ t ^ s ^ t ^ s ^ t ^ t ^ s ^ t ^ t ^ s ^ t ^
W1 /\ W = W1 /\ W ` + W ;
f . j is Element of J . j ;
let x , y be Element of T2 ;
ex d st a , b // b , d ;
a <> 0 & b <> 0 ;
ord ( x ) = 1 & x is \sum ( x ) is \sum ( x ) ;
set g2 = lim ( seq ^\ k ) , g1 = lim seq ;
2 * x >= 2 * ( 1 / 2 ) ;
assume ( a 'or' c ) . z <> TRUE ;
f (*) g in Hom ( c , c ) ;
Hom ( c , c + d ) <> {} ;
assume 2 * Sum ( q | m ) > m ;
L1 . ( F1 . L1 ) = 0 ;
h \/ R1 = ( the carrier of X ) \/ R1 ;
( ( - sin ) (#) ( sin - cos ) ) . x <> 0 ;
( ( exp_R * exp_R ) `| Z ) . x > 0 ;
o1 in [: X , Y :] /\ [: X , Y :] ;
e , v9 , e , f ;
r3 > ( 1 - 2 ) * 0 ;
x in P .: ( F -ideal ) ;
let J be closed Ideal of R , I be Ideal of R ;
h . p1 = f2 . O & h . p2 = g2 . I ;
Index ( p , f ) + 1 <= j ;
len ( q | M ) = width M & width ( q | M ) = width M ;
the carrier of .|| c= A & the carrier of .|| c= A ;
dom f c= union rng ( F | -10 ) ;
k + 1 in support ( support ( n ) ) ;
let X be ManySortedSet of the carrier of S ;
[ x `1 , y `2 ] in ( InnerVertices R ) ;
i = D1 or i = D2 or i = D1 ;
assume a mod n = b mod n ;
h . x2 = g . x1 & h . x2 = g . x2 ;
F c= 2 |^ ( the carrier of X ) ;
reconsider w = |. s1 .| as Real_Sequence ;
sqrt ( 1 / m * m + r ) < p ;
dom f = dom ( I --> finite Function ) ;
[#] P-17 = [#] ( ( TOP-REAL 2 ) | K1 ) ;
cluster - x -> ExtReal equals - x + x ;
then { d1 } c= A & A is closed ;
cluster TOP-REAL n -> finite-ind for non empty TopSpace ;
let w1 be Element of M ;
let x be Element of dyadic ( n ) ;
u in W1 & v in W2 implies u in 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 \mathclose of X , n be Element of NAT ;
dist ( x , y ) < ( r / 2 ) ;
reconsider mm = m as Element of NAT ;
x- x0 < r1 - x0 & r1 - x0 < r1 - x0 ;
reconsider P = P ` as strict Subgroup of N ;
set g1 = p * idseq ( q `1 ) , g2 = idseq ( q `1 ) ;
let n , m , k be non zero Nat ;
assume that 0 < e and f | A is lower ;
D2 . ( ID2 . ( ID2 . ( ID2 . n ) ) ) in { x } ;
cluster -> subcondensed for Subset of T ;
let P be compact non empty Subset of TOP-REAL 2 , p1 , p2 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 SK = S as Subset of T ;
dom ( i .--> X ` ) = { i } ;
let X be non-empty ManySortedSet of S ;
let X be non-empty ManySortedSet of S ;
op ( 1 ) c= { [ {} , {} ] } ;
reconsider m = k-1 as Element of NAT ;
reconsider d = x as Element of C ( ) ;
let s be 0 -started State of SCMPDS , P be Subset of SCMPDS ;
let t be 0 -started State of SCMPDS , Q ;
b , b ` , x , y is_collinear ;
assume i = n \/ { n } & j = k \/ { k } ;
let f be PartFunc of X , Y ;
Nx0 >= sqrt ( c ^2 + sqrt ( c ^2 ) ) ;
reconsider t7 = T" as Point of TOP-REAL 2 ;
set q = h * p ^ <* d *> ;
z2 in U . ( y2 ) /\ Q2 . ( z2 . ( y2 ) ) /\ Q1 . ( z2 . ( y2 . ( z2 . ( z2 . ( z2 . ( z2 . ( z2 . ( z2 . ( z2 .
A |^ 0 = { <* \rangle *> } & A |^ 1 = { <* E *> } ;
len W2 = len W + 2 & len W2 = len W ;
len ( h2 ) in dom h2 & len ( h2 ) = len h2 ;
i + 1 in Seg ( len s2 ) ;
z in dom g1 /\ dom f & z in dom g1 /\ dom g ;
assume p2 `1 = E-bound ( K ) & p2 `2 <= N-bound ( K ) ;
len G + 1 <= i1 + 1 ;
f1 (#) f2 is convergent & f2 (#) ( f1 (#) f2 ) is convergent ;
cluster s-10 + s\in + ( |. s .| + 1 ) ;
assume j in dom ( M1 ^ M2 ) ;
let A , B , C be Subset of X ;
x , y , z be Point of X , p be Point of X ;
b / ( 4 * a * c ) >= 0 ;
<* xxy *> ^ <* y *> \subseteq x ;
a , b in { a , b } ;
len p2 is Element of NAT & len p2 = len p1 + 1 ;
ex x being element st x in dom R & y = R . x ;
len q = len ( K (#) G ) ;
s1 = Initialize Initialized s , P1 = P +* I , P2 = P +* I , P3 = Comput ( P +* I , s2 , 1 ) , P4 = P +* I ;
consider w being Nat such that q = z + w ;
x ` is Element of L & x ` is ` ;
k = 0 & n <> k or k > n ;
then X is discrete for X being Subset of X ;
for x st x in L holds x is FinSequence of L
||. f /. c .|| <= r1 & ||. f /. c .|| <= r2 ;
c in ]. p , q .[ & not c in { p } ;
reconsider V = V as Subset of the topology of TOP-REAL n ;
N , M be being being being being being being being \hbox of L ;
then z is_>=_than compactbelow x & z is_>=_than compactbelow y ;
M [. f , g .] = f & M [. g , f .] = g ;
( ( L~ z ) /. 1 ) = TRUE ;
dom g = dom f & g = f |^ X ;
mode Pholds of G is \cal holds G is : 1 <= len G ;
[ i , j ] in Indices M & [ i , j ] in Indices M ;
reconsider s = x " as Element of H ;
let f be Element of ( Subformulae p ) -tuples_on the carrier of Subformulae q ;
F1 . ( a1 , - a2 ) = G1 & F2 . ( a1 , - a2 ) = G2 ;
Observe ( a , b , r ) is compact ;
let a , b , c , d be Real ;
rng s c= dom ( 1 / 2 ) & rng s c= dom ( 1 / 2 ) ;
curry ( ( F . -19 ) , k ) is additive ;
set k2 = card dom B - 1 , k1 = card dom B - 1 ;
set G = coprod ( X ) ;
reconsider a = [ x , s ] as Object of G ;
let a , b be Element of Mf , M be Subset of V ;
reconsider s1 = s as Element of ( the carrier of S1 ) * ;
rng p c= the carrier of L & rng p c= the carrier of L ;
let d be Subset of the Sorts of A ;
( x | x ) = 0 iff x = 0. W ;
I-21 in dom stop I & Ik in dom stop I ;
let g be continuous Function of X | B , Y ;
reconsider D = Y as Subset of TOP-REAL n ;
reconsider i0 = len p1 - 1 as Integer ;
dom f = the carrier of S & dom g = the carrier of S ;
rng h c= union ( the carrier of J ) & rng h c= the carrier of J ;
cluster All ( x , H ) -> reconsider reconsider All of x , H ;
d * N1 ^2 > N1 * 1 / 2 ;
]. a , b .[ c= [. a , b .] ;
set g = f " D1 , g1 = f " D2 ;
dom ( p | ( Seg m ) ) = REAL m ;
3 + - 2 <= k + - 2 ;
tan is_differentiable_in ( ( #Z 2 ) * ( exp_R + arccot ) ) . x ;
x in rng ( f /^ ( p .. f ) ) ;
let f , g be FinSequence of D ;
[: p , q :] in the carrier of S1 & [: p , q :] in the carrier of S2 ;
rng f " = dom f & rng f = dom f ;
( the Source of G ) . e = v ;
width G - 1 < width G - 1 ;
assume v in rng ( S | E1 ) & v in rng ( S | E1 ) ;
assume x is root or x is root of g or x is root ;
assume 0 in rng ( ( g2 ) | A ) ;
let q be Point of TOP-REAL 2 , a , b be Real ;
let p be Point of TOP-REAL 2 , a be Real ;
dist ( O , u ) <= |. p2 .| + 1 ;
assume dist ( x , b ) < dist ( a , b ) ;
<* S7 *> is_the ] of the Matrix of C-20 ;
i <= len ( G | ( i1 -' 1 ) ) ;
let p be Point of TOP-REAL 2 , a be Real ;
x1 in the carrier of I[01] & x2 in the carrier of I[01] ;
set p1 = f /. i , p2 = f /. j ;
g in { g2 : r < g2 & g2 < r } ;
Q2 = Sp2 " . ( Q . ( Q . ( Q . i ) ) ) ;
( ( 1 / 2 ) |^ ( n + 1 ) ) is summable ;
- p + I c= - p + A + - A ;
n < LifeSpan ( P1 , s1 ) & n < len s1 ;
CurInstr ( p1 , s1 ) = i & CurInstr ( p2 , s2 ) = i ;
A /\ Cl { x } <> {} ;
rng f c= ]. r , r + 1 .[ ;
let g be Function of S , V ;
let f be Function of L1 , L2 , g be Function of L1 , L2 ;
reconsider z = z 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 ~ ( A , B ) ;
len ( the connectives of C ) = 4 & len ( the connectives of C ) = 5 ;
let C1 , C2 be SubFunctor of C , D ;
reconsider V1 = V as Subset of X | B ;
attr p is valid means : Def3 : All ( x , p ) is valid ;
assume X c= dom f & f .: X c= dom g & g .: X c= dom g ;
H |^ a " * a is Subgroup of H & H is Subgroup of H ;
let A1 be Let B1 of O , A1 , A2 be Element of E ;
p2 , r3 , q3 is_collinear & q2 , q1 , q2 is_collinear ;
consider x being element such that x in v ^ K ;
not x in { 0. TOP-REAL 2 } & not x in { 0. TOP-REAL 2 } ;
p in [#] ( I[01] | B0 ) & p in the carrier of ( TOP-REAL 2 ) | K1 ;
0 ( ) < M . ( EN . n ) ;
^ ( c ^ ) = c ;
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . s2 ) & a2 in dom ( F . s2 ) ;
cluster -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> from ---`1 ;
set i1 = the Nat , i2 = the Element of NAT ;
let s be 0 -started State of SCM+FSA , k be Nat ;
assume y in ( f1 \/ f2 ) .: A ;
f . ( len f ) = f /. len f ;
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 , y be Element of X ;
cluster ( x `1 ) ^2 -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> cluster -> -> -> -> -> -> -> -> natural for FinSequence ;
set S = <* Bags n , i9 *> , T = <* `| n , Let S *> ;
set T = [. 0 , 1 / 2 .] , G = [. 0 , 1 .] ;
1 in dom mid ( f , 1 , 1 ) ;
sqrt ( 4 * PI ) < sqrt ( 2 * PI ) ;
x2 in dom ( f1 + f2 ) /\ dom ( f1 + f2 ) ;
O c= dom I & { {} } = { {} } ;
( the Source of G ) . x = v ;
{ HT ( f , T ) } c= Support f ;
reconsider h = R . k as Polynomial of n , L ;
ex b being Element of G st y = b * H ;
let x , y , z be Element of G ` ;
h19 . i = f . ( h . i ) ;
( p `1 ) ^2 / ( 1 + ( p `2 ) ^2 ) = ( p `1 ) ^2 / ( 1 + ( p `2 ) ^2 ) ;
i + 1 <= len Cage ( C , n ) ;
len <* P *> |^ 1 = len P & len <* P *> = 1 ;
set NN = the b9 of N , NN = the over N ;
len g: x + ( x + 1 ) - 1 <= x ;
a on B & b on B & a on C ;
reconsider r-12 = r * I . v as FinSequence of REAL ;
consider d such that x = d and a \in d ;
given u such that u in W and x = v + u ;
len f /. \downharpoonright n = len \mathbb n - n ;
set q2 = N-min L~ Cage ( C , n ) , q1 = Cage ( C , n ) ;
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 /\ W ) ;
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 ( F . X ) . ( ( id X ) . 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 . ( a1 , b1 ) ;
the carrier' of G ` = E \/ { E } ;
reconsider m = len ' - k as Element of NAT ;
set S1 = LSeg ( n , UMP C ) , S2 = LSeg ( n , UMP C ) ;
[ i , j ] in Indices ( M1 + M2 ) ;
assume P c= Seg m & M is Seg m ;
for k st m <= k holds z in K . k ;
consider a being set such that p in a and a in G ;
L1 . p = p * ( 1 / p ) ;
pp . i = p9 . i & pp . i = p9 . i ;
let PA , G be a_partition of Y , BOOLEAN , a be Function of Y , BOOLEAN ;
pred 0 < r & 1 < 1 implies 1 < r & r <= 1 ;
rng ( AffineMap ( a , X ) ) = [#] X & rng ( AffineMap ( a , X ) ) = [#] X ;
reconsider x = x , y = y as Element of K ;
consider k such that z = f . k and n <= k ;
consider x being element such that x in X \ { p } ;
len ( canFS ( s ) ) = card s & len ( canFS ( s ) ) = card s ;
reconsider x2 = x1 , y2 = x2 as Element of L2 ;
Q in FinMeetCl ( ( the topology of X ) \/ the topology of Y ) ;
dom ( f | u ) c= dom ( u | v ) ;
pred n divides m & m divides n ;
reconsider x = x as Point of I[01] , I = the carrier of I[01] ;
a in ;
not y0 in the still of f & not ( ex g st g in the bound of f & not ( g in the bound of f ) ) ;
Hom ( ( a , b ) --> c , c ) <> {} ;
consider k1 such that p " < k1 and k1 < len p ;
consider c , d such that dom f = c \ d ;
[ x , y ] in dom g & k in dom g ;
set S1 = Let x , y = y , z = x , y = y ;
l2 = m2 & l2 = i2 & l2 = j2 & E = i2 ;
x0 in dom ( u01 /\ A01 ) & x0 in dom ( u01 /\ A01 ) ;
reconsider p = x as Point of TOP-REAL 2 , p1 , p2 be Point of TOP-REAL 2 ;
I[01] = ( I[01] | B01 ) | B01 .= ( the carrier of I[01] ) /\ B01 ;
f . p4 <= _ P P . p1 & f . p2 <= _ P . p2 ;
( ( F . n ) `1 ) ^2 <= ( x ^2 ) ^2 + ( x ^2 ) ;
( x `2 ) ^2 = ( W . ( n + 1 ) ) ^2 ;
for n being Element of NAT holds P [ n ] ;
J , K be non empty Subset of I ;
assume 1 <= i & i <= len <* a " *> ;
0 |-> a = <*> ( the carrier of K ) & 0 -tuples_on the carrier of K = 0 ;
X . i in 2 |^ ( A . i ) \ B . i ;
<* 0 *> in dom ( e --> [ 1 , 0 ] ) ;
then P [ a ] implies P [ succ a ] ;
reconsider sp2 = sp2 as } of D ( ) ;
( Seg ( i -' 1 ) ) <= len r1 - j ;
[#] S c= [#] ( T | the carrier of S ) & the TopStruct of T = the TopStruct of T ;
for V being strict non empty addLoopStr holds V in iff V in W
assume k in dom mid ( f , i , j ) ;
let P be non empty Subset of TOP-REAL 2 , p1 , p2 be Point of TOP-REAL 2 ;
let A , B be Matrix of n1 , K ;
- a * - b = a * b - b * a ;
for A being Subset of AA holds A // A & A c= A ;
id o2 in <* o2 , o2 , o1 , 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 ) | divset ( D2 , D1 ) ) ;
b = Q . ( len QK - 1 ) + 1 ;
f2 * f1 /* ( s ^\ k ) is divergent_to+infty & f2 * f1 /* ( s ^\ k ) is divergent_to+infty ;
reconsider h = f * g as Function of N4 , G ;
assume that a <> 0 and Polynom ( a , b , c ) >= 0 ;
[ t , t ] in the InternalRel of A & [ t , t ] in the InternalRel of A ;
( v |-- E ) | n is Element of TT & v | n in TT ;
{} = the carrier of L1 + L2 & the carrier of L1 = the carrier of L2 ;
Directed I is_closed_on Initialized s , P & Directed I is_closed_on Initialized s , P ;
Initialized ( p ) = Initialize ( p +* q ) , p2 = p +* q , s2 = p +* q , s3 = p +* q , s3 = p +* q , s4 = p +* q , s4 = p +* q , s4 = p +* q ,
reconsider N2 = N1 as strict net of R2 , R2 ;
reconsider Y = Y as Element of <* Ids ( L ) , \subseteq \rangle ;
"/\" ( ( uparrow p ) \ { p } , L ) <> p ;
consider j being Nat such that i2 = i1 + j and j in dom f ;
not [ s , 0 ] in the carrier of S2 & not [ s , 0 ] in the InternalRel of S2 ;
mm in ( B '&' C ) '/\' D \ { {} } ;
n <= len ( P + Q ) + len ( P + Q ) ;
( x1 - x2 ) / ( y1 - y2 ) = ( x2 - y2 ) / ( y1 - y2 ) ;
InputVertices S = { x1 , x2 , x3 , x4 , x5 , x5 , x5 , 7 , 8 , 8 } ;
let x , y be Element of FTTTT1 ( n ) ;
p = |[ p `1 , p `2 ]| & p `2 = p `2 ;
g * 1_ G = h " * g * h .= g * h ;
let p , q be Element of U ( V , C ) ;
x0 in dom ( x1 - x2 ) /\ dom ( y1 - y2 ) ;
( R qua Function ) " = R " & ( R " ) " = R ;
n in Seg ( len ( f /^ n ) ^ p ) ;
for s be Real st s in R holds s <= s2 & s <= 1 ;
rng s c= dom ( ( f2 * f1 ) ^ ) /\ dom ( f2 * f1 ) ;
synonym for for for for for X being Subset of ex f being Function of X , the carrier of X st f = X & f is one-to-one ;
1_ K * 1_ K = 1_ K & 1_ K * ( 1_ K ) = 1_ K ;
set S = Segm ( A , P1 , Q1 ) , Q1 = Segm ( A , s1 , Q1 ) ;
ex w st e = ( w - f ) . w & w in F ;
curry ( P+* ( k , X ' ) ) # x is convergent ;
cluster -> open for Subset of TK st F is open & F is open holds F is open ;
len f1 = 1 .= len ( f3 ^ f2 ) .= len f3 + len f3 ;
sqrt ( i * p ) < ( 2 * p ) ^2 ;
let x , y be Element of \rm Sub ( U0 ) ;
b1 , c1 // b9 , c9 & b1 , c1 // c , c9 ;
consider p being element such that c1 . j = { p } ;
assume f " { 0 } = {} & f is total ;
assume IC Comput ( F , s , k ) = n ;
Reloc ( J , card I + card J + 3 ) not a in dom J ;
goto ( card I + 1 ) not c on c , P & not c in dom I ;
set m3 = LifeSpan ( p3 , s3 ) , P4 = LifeSpan ( p2 , s2 ) ;
IC SCMPDS in dom ( Initialize ( s ) +* ( a , k1 ) ) ;
dom t = the carrier of SCM R & dom t = the carrier of SCM R ;
( ( E-max L~ f ) .. f ) .. f = 1 ;
let a , b be Element of Let ( V , C ) ;
Cl ( union F ) c= Cl ( union F ) ;
the carrier of X1 union X2 misses ( A1 \/ A2 ) \/ ( A1 \/ A2 ) ;
assume not LIN a , f . a , g . a ;
consider i being Element of M such that i = d6 and i in dom f ;
then Y c= { x } or Y = {} or Y = { x } ;
M , v |= H1 / ( ( y , v ) / ( y , x ) ) ;
consider m being element such that m in Intersect ( FF . 0 ) and x = ( Intersect ( FF . 0 ) ) . m ;
reconsider A1 = support u1 , A2 = support v1 as Subset of X ;
card ( A \/ B ) = k-1 + ( 2 * 1 ) ;
assume a1 <> a3 & a2 <> a3 & a3 <> a4 & a3 <> a4 ;
cluster s -\mathop { V } -> .| for string of S ;
L2 /. n2 = L2 . n2 & L2 /. n2 = L2 . n2 ;
let P be compact non empty Subset of TOP-REAL 2 , p1 , p2 be Point of TOP-REAL 2 ;
assume that r-7 in LSeg ( p1 , p2 ) and r-7 in LSeg ( p1 , p2 ) ;
let A be non empty compact Subset of TOP-REAL n , a be Real ;
assume [ k , m ] in Indices ( D * ( k , m ) ) ;
0 <= ( ( 1 / 2 ) |^ p ) / p ;
( F . N ) | EN . x = +infty ;
pred X c= Y & Z c= V & X \ V c= Y \ Z ;
( y * z ) * ( z * z ) <> 0. I & ( y * z ) * ( z * y ) = 0. I ;
1 + card ( X-18 ) <= card ( u \/ { w } ) ;
set g = z \circlearrowleft ( ( L~ z ) | ( L~ z ) ) ;
then k = 1 implies p . k = <* x , y *> . k ;
cluster -> ( the Element of C ) -one-to-one for Function of X , the carrier of S ;
reconsider B = A as non empty Subset of TOP-REAL n , a be Real ;
let a , b , c be Function of Y , BOOLEAN ;
L1 . i = ( i .--> g ) . i .= g . i ;
Plane ( x1 , x2 , x3 , x4 ) c= P & Line ( x2 , x3 , x4 ) c= P ;
n <= indx ( D2 , D1 , j1 ) + 1 & n <= len D1 ;
( ( ( g2 ) . O ) `1 ) ^2 = - 1 & ( ( g2 ) . O ) ^2 = - 1 ;
j + p .. f - len f <= len f - len f ;
set W = W-bound C , S = N-bound C , E = E-bound C , N = E-bound C ;
S1 . ( a ` , e ) = a + e .= a ` ;
1 in Seg width ( M * ( ColVec2Mx p ) ) ;
dom ( i (#) Im ( f ) ) = dom ( Im f ) /\ dom Im ( f ) ;
being ' ( x ) ` = W ( a *' ( a , p ) ) ;
set Q = |= ( >= g , f , h ) , S = g ;
cluster -> many sorted for ManySortedSet of U1 means : Let : for ManySortedSet of U1 holds it = an Carrier F ;
attr ex A st F = { A } & F is discrete ;
reconsider z9 = reproj ( i , x ) as Element of product G ;
rng f c= rng f1 \/ rng f2 & rng f1 c= dom f1 \/ rng f2 ;
consider x such that x in f .: A and x in f .: C ;
f = <*> ( the carrier of F_Complex ) & ( the carrier of F_Complex ) c= ( the carrier of F_Complex ) ;
E , j |= All ( x1 , x2 ) & E , j |= H ;
reconsider n1 = n as Morphism of o1 , o2 , o2 , o2 be Morphism of o2 , o2 ;
assume P is idempotent & R is idempotent & P (*) R = R ** P ;
card ( B2 \/ { x } ) = k-1 + 1 ;
card ( ( x \ B1 ) /\ A1 ) = 0 ;
g + R in { s : g-r < s & s < g + r } ;
set qi2 = ( q , <* s *> ) \mathop { 1 } , qi2 = ( q , <* s *> ) \mathop { 1 } ;
for x being element st x in X holds x in rng f1 & x in X ;
h0 /. ( i + 1 ) = h0 . ( i + 1 ) ;
set mw = max ( B , min ( NAT , { 0 } ) ) ;
t in Seg width ( I ^ ( n , n ) ) ;
reconsider X = dom f , C = C as Element of Fin NAT ;
IncAddr ( i , k ) = ( ( l + k ) + k ) ;
( ( GoB f ) * ( 1 , 1 ) ) `2 <= ( q `2 ) * ( 1 , 1 ) `2 ;
attr R is condensed means : Def6 : ( Cl R ) is condensed & Cl R is condensed ;
pred 0 <= a & b <= 1 & a * b <= 1 ;
u in ( ( c /\ ( ( d /\ b ) /\ e ) ) /\ f ) /\ j ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) ) /\ f ) /\ j ;
len C + ( - 2 ) >= 9 + ( - 3 ) ;
x , z , y is_collinear & x , z , y is_collinear ;
a |^ ( n1 + 1 ) = a |^ n1 * a ;
<* \underbrace ( 0 , \dots , 0 , 0 ) *> in Line ( x , a * x ) ;
set yc = <* y , c *> ;
F2 /. 1 in rng Line ( D , 1 ) & F2 /. 1 in rng Line ( D , 1 ) ;
p . m joins r /. m , r /. ( m + 1 ) ;
( p `2 ) ^2 = ( f /. i1 ) ^2 & ( f /. i1 ) ^2 = ( f /. i1 ) ^2 ;
W-bound ( X \/ Y ) = W-bound ( X \/ Y ) & W-bound ( X \/ Y ) = E-bound X ;
0 + ( p `2 ) ^2 <= 2 * r + ( p `2 ) ^2 ;
x in dom g & not x in g " { 0 } ;
f1 /* ( seq ^\ k ) is divergent_to+infty & f2 /* ( seq ^\ k ) is divergent_to+infty ;
reconsider u2 = u as VECTOR of P(# X , Y #) , the carrier of X #) ;
p \! \mathop { Product ( Sgm X ) . ( Product ( Sgm X ) ) . ( len p ) ) = 0 ;
len <* x *> < i + 1 & i <= len c + 1 ;
assume that I is non empty and { x } /\ { y } = { 0. I } ;
set ii = card I + 4 , goto 0 = ( card I + 4 ) ;
x in { x , y } & h . x = {} T & y in T ;
consider y being Element of F such that y in B and y <= x ` ;
len S = len ( the charact of ( A ) ) & len ( the charact of ( A ) ) = 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 NN = : G-15 : G-15 ( G-15 , G-15 ) ;
rng F c= the carrier of gr { a } implies F is one-to-one
f in implies ( for K holds F . K , n , r ) is Function-yielding ;
f . k , f . ( Let n ) in rng f & f . ( n + 1 ) in rng f ;
h " ( P ) /\ [#] T1 = f " ( P /\ [#] T2 ) ;
g in dom ( f2 " ) \ f2 " { 0 } & ( f2 " ) . g in dom ( ( f2 " ) " { 0 } ) ;
gN /\ dom f1 = g1 " ( g " ) .= ( g " ) * ( g " ) ;
consider n be element such that n in NAT and Z = G . n ;
set d1 = being dist of ( dist ( x1 , y1 ) ) , d2 = dist ( x2 , y2 ) ;
b `1 + sqrt ( 1 + ( b `1 / b `1 ) ^2 ) < ( 1 + sqrt ( 1 + ( b `1 / b `2 ) ^2 ) ) ;
reconsider f1 = f as VECTOR of the carrier of X , Y be Subset of Y ;
pred i <> 0 implies i |^ ( i + 1 ) mod ( i + 1 ) = 1 ;
j2 in Seg ( len ( g2 . i2 ) ) & 1 <= j2 & j2 <= len ( g2 . i2 ) ;
dom ( i - 1 ) = dom ( i - 1 ) .= Seg ( len G - 1 ) .= dom G ;
cluster sec | ]. PI / 2 , PI / 2 .[ -> one-to-one ;
Ball ( u , e ) = Ball ( f . p , e ) ;
reconsider x1 = x0 as Function of S , IF ( ) , IF ( ) ;
reconsider R1 = x , R2 = y as Relation of L , L ;
consider a , b being Subset of A such that x = [ a , b ] ;
( <* 1 *> ^ p ) ^ <* n *> in RN ;
S1 +* S2 = S2 +* ( S1 +* S2 ) & S2 +* ( S1 +* S2 ) = S1 +* ( S2 +* S2 ) ;
( ( ( #Z n ) * ( exp_R + exp_R + exp_R ) ) `| Z ) = ( exp_R + exp_R + exp_R + exp_R + exp_R + exp_R - exp_R + exp_R - exp_R ) ;
cluster -> continuous for Function of C , REAL n , REAL n ;
set C7 = 1GateCircStr ( <* z , x *> , f3 ) , C7 = 1GateCircStr ( <* x , y *> , f3 ) ;
E8 . e2 = ( e2 . e2 ) -T & E8 . e2 = ( e2 . e2 ) -T ;
( ( arctan * arccot ) `| Z ) = ( ( arctan * arccot ) `| Z ) . ( ( exp_R . x ) ^2 ) .= ( ( #Z 2 ) * ( ( #Z 2 ) * ( exp_R . x ) ^2 ) ) .= ( ( #Z 2 ) * ( exp_R . x ) ) ^2 ;
sup A = ( PI * 3 ) / 2 & inf A = 0 ;
F . ( dom f , - F ) is_transformable_to F . ( cod f , - F ) ;
reconsider p9 = q9 as Point of TOP-REAL 2 , q = ( q `2 ) / |. q .| as Point of TOP-REAL 2 ;
g . W in [#] ( Y | the carrier of X ) & [#] ( Y | the carrier of X ) c= [#] ( Y | the carrier of X ) ;
let C be compact non vertical non vertical non horizontal non horizontal Subset of TOP-REAL 2 , a , b be Real ;
LSeg ( f ^ g , j ) = LSeg ( f , j ) ;
rng s c= dom f /\ ]. - r , x0 .[ & f | ]. - r , x0 .[ is bounded ;
assume x in { ( idseq 2 ) . ( ( idseq 2 ) . ( len p ) ) } ;
reconsider n2 = n , m2 = m as Element of NAT ;
for y being ExtReal st y in rng seq holds g <= y
for k st P [ k ] holds P [ k + 1 ]
m = m1 + m2 .= m1 + m2 + m2 .= m1 + m2 + m2 .= m1 + m2 ;
assume for n holds H1 . n = G . n -H . n ;
set Bf = f .: ( the carrier of X1 , X2 ) , Bg = f .: ( the carrier of X2 , X2 ) ;
ex d being Element of L st d in D & x << d ;
assume R ~ ( a , b ) c= R ~ ( b , a ) ;
t in ]. r , s .] or t = r or t = s ;
z + v2 in W & x = u + ( z + v2 ) ;
x2 |-- y2 iff P [ x2 , y2 ] & P [ x2 , y2 ] ;
pred x1 <> x2 means : Def3 : |. x1 - x2 .| > 0 & x1 <> x2 ;
assume p2 - p1 , p3 - p1 , p2 - p1 , p3 - p1 , p2 - p1 , p3 - p1 , p2 - p1 , p3 - p1 , p2 - p1 , p2 - p1 , p3 - p1 , p2 - p1 , p2 - p1 , p3 - p1 , p2 - p1 , p2 - p1 , p3 - p1 , p2 - p1 , p2
set q = ( } f ^ <* 'not' 'not' 'not' 'not' A *> ) ^ <* 'not' 'not' A *> ;
let f be PartFunc of REAL-NS 1 , REAL-NS n , REAL-NS n , x be Element of REAL n ;
( n mod ( 2 * k ) ) ! = n mod k ;
dom ( T * succ t ) = dom ( \rbrack t , dom succ t ) ;
consider x being element such that x in wf and x in c ;
assume ( F * G ) . ( v . x3 ) = v . x3 ;
assume that the carrier of D1 c= the carrier of D2 and the carrier of D1 c= the carrier of D2 and the carrier of D1 = 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 = W-bound L~ Cage ( C , n ) , s = S-bound L~ Cage ( C , n ) , s = S-bound L~ Cage ( C , n ) , s = S-bound L~ Cage ( C , n ) , w =
n1 - len f + 1 - len f + 1 <= len g - len f + 1 - len f ;
ConsecutiveSet ( q , O1 , a ) = [ u , v , a , b , c ] ;
set C-2 = ( ( `1 ) `1 ) . ( k + 1 ) , C-2 = ( ( n + 1 ) ) `1 ;
Sum ( L * p ) = 0. R .= 0. V .= 0. V ;
consider i being element such that i in dom p and t = p . i ;
defpred Q [ Nat ] means 0 = Q ( $1 ) & for n holds P [ n ] ;
set s3 = Comput ( P1 , s1 , k ) , s4 = Comput ( P2 , s2 , k ) , P4 = P2 ;
let l be variable of k , A , B be Element of l ;
reconsider U2 = union Gf1 as Subset-Family of [: T , T :] , the topology of T :] ;
consider r such that r > 0 and Ball ( p `1 , r ) c= Q ` ;
( h | ( n + 2 ) ) /. ( i + 1 ) = p9 /. ( i + 1 ) ;
reconsider B = the carrier of X1 , C = the carrier of X2 as Subset of X ;
p9 = <* - c9 , 1 , - c9 , - c9 , - T *> & p9 = <* - c9 , - c9 , - T *> ;
synonym f is complex-valued means : Def6 : rng f c= NAT & rng f c= NAT & f is one-to-one ;
consider b being element such that b in dom F and a = F . b ;
x9 < card ( X0 \/ Y ) & x9 in ( X1 \/ X2 ) \/ ( Y \/ X2 ) ;
pred X c= B1 means : Defost X c= succ B1 & X c= succ A1 & X c= B1 ;
then w in Ball ( x , r ) & dist ( x , w ) <= r ;
angle ( x , y , z ) = angle ( \HM { the } \HM { function } , 0 , PI ) ;
pred 1 <= len s means : Def8 : len ( s , 0 ) = len s & for i being Nat holds s . i = s . i ;
fm c= f . ( k + ( n + 1 ) ) ;
the carrier of { 1_ G } = { 1_ G } & the carrier of G = { 1_ G } ;
pred p '&' q in the carrier of W means : Def8 : q '&' p in the carrier of W & q '&' p in the carrier of W ;
- ( t `1 / t `2 ) ^2 < ( ( t `1 ) ^2 / t `2 ) ^2 ;
U2 . 1 = U /. 1 .= ( W /. 1 ) .= ( W . 1 ) ;
f .: ( the carrier of x ) = the carrier of x & f . ( the carrier of x ) = f . ( the carrier of x ) ;
Indices ( ( - A ) * ( i , j ) ) = [: Seg n , Seg n :] ;
for n being Element of NAT holds G . n c= G . ( n + 1 ) ;
then V in M .: \square & ex x being Element of M st V = { x } ;
ex f being Element of F-9 st f is_\upharpoonright Aand x = f & y = f . x ;
[ h . 0 , h . 3 ] in the InternalRel of G & [ h . 2 , h . 3 ] in the InternalRel of G ;
s +* Initialize ( ( intloc 0 ) .--> 1 ) = s3 +* ( ( intloc 0 ) .--> 1 ) ;
|[ w1 , v1 ]| - |[ w1 , v1 ]| <> 0. TOP-REAL 2 & |[ w1 , v1 ]| - |[ w1 , v1 ]| = |[ w1 , v1 ]| ;
reconsider t = t as Element of INT ( ) * ;
C \/ P c= [#] ( ( the carrier of ( ( the carrier of ( ( | | A ) \ A ) ) \ A ) ) ;
f " V in ( the topology of X ) /\ ( the topology of Y ) . ( the carrier of V ) ;
x in [#] ( ( the carrier of F ) /\ A ) /\ the carrier of ( F ) /\ the carrier of ( F ) ;
g . x <= h1 . x & h . x <= h1 . x & h . x <= h1 . x ;
InputVertices S = { xy , yz , zx , \mathop { xy , A1 , cin } , { xy , cin , cin } } ;
for n be Nat st P [ n ] holds P [ n + 1 ]
set R = Line ( M , i ) * ( Line ( M , i ) ) ;
assume M1 is being_line & M2 is being_line & M3 is being_line & M2 is being_line & M3 is being_line & M2 is being_line ;
reconsider a = f0 . ( i0 -' 1 ) as Element of K ;
len B2 = Sum Len ( F1 ^ F2 ) & len ( F1 ^ F2 ) = len ( F1 ^ F2 ) + len ( F2 ^ B2 ) ;
len ( ( the _ of K ) * ( i , j ) ) = n & ( ( the _ of K ) * ( i , j ) ) = n ;
dom ( f + g ) = dom ( f + g ) /\ dom ( g + h ) ;
( the Sorts of seq ) . n = ( the Sorts of Y1 ) . n & ( the Sorts of seq ) . n = ( the Sorts of X1 ) . n ;
dom ( p1 ^ p2 ) = dom ( f ^ g ) .= dom ( f ^ g ) .= dom ( f ^ g ) ;
M . [ 1 , y ] = 1 * v1 .= ( 1 - 1 ) * v1 .= ( 1 - 1 ) * v1 ;
assume that W is non trivial and W { x } c= the \frac of G2 and W . x in the carrier of G2 ;
C6 /. i1 = G1 * ( i1 , i2 ) & C6 = G2 * ( i1 , i2 ) ;
C8 |- 'not' All ( x , p ) 'or' 'not' p . ( x , y ) ;
for b st b in rng g holds inf rng fmb <= b - 1
- ( ( q1 `1 / |. q1 .| - cn ) / ( 1 - cn ) ) = 1 ;
( LSeg ( c , m ) \/ { ml } ) \/ LSeg ( l , k ) c= R ;
consider p being element such that p in { x } and p in L~ f and x = f . p ;
Indices ( X @ ) = [: Seg n , Seg n :] & len ( X @ ) = width ( X @ ) ;
cluster s => ( q => p ) => ( q => ( s => p ) ) -> valid ;
Im ( ( Partial_Sums F ) . m ) = ( Partial_Sums F ) . m & Im ( ( Partial_Sums F ) . m ) = ( Partial_Sums F ) . m ;
cluster f . ( x1 , x2 ) -> Element of D ( ) ;
consider g being Function such that g = F . t & Q [ t , g . t ] ;
p in LSeg ( ( N-min Z ) . ( 1 , 1 ) , ( NW-corner Z ) . ( 1 , 1 ) ) ;
set R8 = R |^ 1 , R8 = R |^ 1 , R8 = R |^ 2 , R8 = R |^ 2 , R8 = R |^ 2 , R8 = R |^ 2 , R8 = R |^ ( 1 + 1 ) , R8 = R |^ ( 1 + 1 ) , R8 =
IncAddr ( I , k ) = AddTo ( d , d1 ) .= goto ( ( card I + 1 ) + k ) ;
seq . m <= ( ( seq ^\ k ) ^\ n ) . ( ( seq ^\ k ) ^\ n ) ;
a + b = ( a ` *' ) + ( b ` ` ) .= ( a ` + b ` ) + ( b ` ) ;
id ( X /\ Y ) = id ( X /\ Y ) /\ id ( X /\ Y ) ;
for x being element st x in dom h holds h . x = f . x ;
reconsider H = U1 \/ U2 as non empty Subset of U0 ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) /\ f ) /\ m ) /\ m ;
consider y being element such that y in Y and P [ y , inf B ] ;
consider A being finite stable Subset of R such that card A = card ( R ~ ) and card A = card ( R ~ ) ;
p2 in rng ( f |-- p1 ) \ rng <* p1 *> & p2 in rng <* p1 *> ;
len s1 - 1 > 0 & len s2 - 1 > 0 & len s1 - 1 > 0 ;
( ( N-min L~ f ) | ( len f ) ) `2 = ( ( N-min L~ f ) | ( len f ) ) `2 ;
Ball ( e , r ) c= LeftComp Cage ( C , k + 1 ) & e in LeftComp Cage ( C , k + 1 ) ;
f . a1 ` = f . a1 ` & f . a1 = ( a1 ` ) ` ;
( seq ^\ k ) . n in ]. x0 - r , x0 .[ & ( seq ^\ k ) . n in ]. x0 , x0 + r .[ ;
gg . seq = g . ( ( seq | X ) . n ) .= ( g | X ) . ( seq . n ) ;
the InternalRel of S is Real & the InternalRel of S is connected implies the InternalRel of S is connected
deffunc F ( Ordinal , Ordinal ) = phi . ( $2 , $2 ) ;
F . ( s1 . a1 ) = F . ( s2 . a1 ) & F . ( s1 . a1 ) = F . ( s1 . a1 ) ;
x `1 = A . o .= Den ( o , A . a ) ;
Cl ( f " P1 ) c= f " P1 & ( f " P1 ) " P1 c= ( f " P1 ) " P1 ;
FinMeetCl ( ( the topology of S ) | the topology of T ) c= the topology of T ;
synonym o is " means : Def11 : o <> \ast & o <> * & o <> * ;
assume that X = Y + 1 and card X <> card Y and card Y <> card X and X c= Y ;
the finite empty empty Subset of S <= 1 + ( the card of s ) * ( the card s ) ) ;
LIN a , a1 , d or b , c // b1 , c1 or b , c // a1 , c1 ;
e2 . 1 = 0 & e2 . 2 = 1 & e2 . 3 = 0 ;
EK in SK & EK in [: { N } , K :] & EK in [: { N } , K :] ;
set J = ( l , u ) \mathop { {} } ;
set A1 = 1GateCircStr ( a9 , b9 , c ) , A2 = Following ( s , 2 ) , A2 = Following ( s , 2 ) ;
set c9 = [ <* c9 , d9 *> , '&' ] , A2 = [ <* A1 , cin *> , '&' ] , us = [ <* cin , non } , '&' ] , us = [ <* cin , non } , '&' ] ;
x * z ` * x " in x * ( z * N ) * x " ;
for x being element st x in dom f holds f . x = f3 . x + g . x
Int cell ( f , 1 , G ) c= RightComp f \/ RightComp f \/ L~ f \/ L~ f \/ L~ f ;
U2 is_an_arc_of W-min ( C , n ) , E-max ( C , n ) & ( E-max C ) `1 = E-bound C & ( E-max C ) `1 <= E-bound C ;
set f9 = f @ @ g "/\" @ @ f ;
attr S1 is convergent & S2 is convergent means : Def7 : S1 - S2 is convergent & for n holds S1 . n = ( S1 . n ) - ( S2 . n ) ;
f . ( 0 + 1 ) = ( 0 qua Ordinal ) + a .= a + b .= a + b ;
cluster -> \mathclose for RelStr -be reflexive non empty reflexive transitive RelStr -symmetric -symmetric for non empty non empty non empty RelStr ;
consider d being element such that R reduces b , d and R reduces c , d and R reduces c , d ;
not b in dom Start-At ( ( card I + 2 ) , SCMPDS ) & not b in dom Start-At ( ( card I + 2 ) , SCMPDS ) ;
( z + a ) + x = z + ( a + y ) .= z + a + y ;
len ( l |^ ( ( a |^ 0 ) .--> x ) ) = len l ;
t4 ^ {} ( {} \/ { {} } ) is ( {} \/ rng t4 ) -valued Function ;
t = <* F . t *> ^ ( C . p ^ q ) .= C . ( p ^ q ) ^ C . ( q ^ p ) ;
set pp = W-min L~ Cage ( C , n ) , p1 = W-min L~ Cage ( C , n ) , p2 = W-min L~ Cage ( C , n ) ;
( k - i + 1 ) = ( k - i ) + ( k - i ) ;
consider u being Element of L such that u = u ` and u in D and u in D and u in D ;
len ( ( width ( ( ( B - G ) |-> a ) ) ^ ( b - G ) ) ) ) = width ( ( ( B - G ) ^ ( b - G ) ) ^ ( b - G ) ) ;
F3 . x in dom ( ( G * the_arity_of o ) . x ) ;
set H2 = the carrier of H2 , y2 = the carrier of H ;
set H1 = the carrier of H1 , H2 = the carrier of H2 ;
( Comput ( P , s , 6 ) ) . intpos ( m + 6 ) = s . intpos ( m + 6 ) ;
IC Comput ( P3 , t , k ) = l + 1 & IC Comput ( P3 , s3 , k ) = 0 ;
dom ( ( ( id Z ) (#) ( cos * sin ) ) `| Z ) = dom ( ( id Z ) (#) ( cos * cos ) ) ;
cluster <* l *> ^ phi phi -> ( 1 + 1 ) -element for string of S ;
set b9 = [ <* A1 , cin *> , '&' ] , Z = [ <* A1 , cin *> , '&' ] , e = [ <* A2 , A1 *> , '&' ] , f = [ <* A1 , cin *> , '&' ] , e = [ <* A2 , A1 *> , '&' ] , f = [ <* A1 , A2 *> , '&' ] , e = [ <*
Line ( Segm ( M @ , P , Q ) , x ) = L * Sgm Q ;
n in dom ( ( the Sorts of A ) * ( the_arity_of o ) ) & ( the Sorts of A ) . ( ( the Sorts of A ) . ( o , n ) ) = ( the Sorts of A ) . ( ( the Sorts of A ) . ( o , n ) ) ;
cluster f1 + f2 -> continuous for PartFunc of REAL , REAL n , the carrier of S ;
consider y being Point of X such that a = y and ||. y - x .|| <= r ;
set x3 = ( t . DataLoc ( s2 . SBP , 2 ) , 2 ) , y1 = ( t . SBP ) , y2 = ( t . SBP ) , y1 = ( t . SBP ) , y2 = ( t . SBP ) , y2 = ( t . SBP ) , y1 = ( t . SBP ) , y2 = (
set pp = stop I , p1 = P +* I , p2 = Comput ( p1 , s1 , 1 ) , s2 = Comput ( p2 , s2 , 1 ) , p2 = Comput ( p2 , s2 , 1 ) , s2 = p2 ;
consider a being Point of D2 such that a in W1 and b = g . a and a in W2 ;
{ A , B , C , D } = { A , B , C , D } \/ { E , F , J , M , N , F , J , M , N , N , F , J , M , N , N , N , A , N , A , N , A , J , M , N ,
let A , B , C , D , E , F , J , M , N , N , F , J , M , N , N , F , J , M , N , N , F , J , M , N , N , M , N , N , A , N , A , N , A , N ,
|. p2 .| ^2 - ( ( p2 `2 / |. p2 .| - cn ) / ( 1 - cn ) ) ^2 >= 0 ;
l - 1 + 1 = n-1 * ( l + 1 ) + ( 1 + 1 ) ;
x = v + ( a * w1 + b * w2 ) + ( c * w2 ) + ( c * w2 ) ;
the TopStruct of L = TopSpaceMetr (# the TopStruct of N , the TopStruct of N #) & the TopStruct of N = the TopStruct of N ;
consider y being element such that y in dom H1 and x = H1 . y and y in H . x ;
f9 \ { n } = ( Free All ( v1 , H ) ) \ { n } & f . ( n + 1 ) = f . ( n + 1 ) ;
for Y being Subset of X st Y is summable & Y is number holds Y is non empty iff Y is non empty
2 * n in { N : 2 * Sum ( p | N ) = N & N > 0 } ;
for s being FinSequence holds len ( the _ of A ) = len s & len ( the _ of A ) = len s & width ( the _ of A ) = width s
for x st x in Z holds exp_R * ( exp_R * f ) is_differentiable_in x & exp_R * ( exp_R * f ) is_differentiable_in x ;
rng ( h2 * ( f2 - g2 ) ) c= the carrier of ( TOP-REAL 2 ) | K1 .= the carrier of ( TOP-REAL 2 ) | K1 ;
j + ( len f ) - len f <= len f + ( len f - len f ) - len f ;
reconsider R1 = R * I as PartFunc of REAL n , REAL-NS n , REAL-NS n ;
C8 . x = s1 . ( a1 . x ) .= C8 . ( a1 . x ) .= C8 . ( a1 . x ) ;
power F_Complex . ( z , n ) = 1 .= ( x |^ n ) * ( x |^ n ) .= ( x |^ n ) * ( x |^ n ) ;
t at ( C , s ) = f . ( the that s in the that s in the elements of S & t in the Sorts of C & s = f . ( the connectives of S ) . s ;
support ( f + g ) c= ( support f ) \/ ( { C } ) ;
ex N st N = j1 & 2 * Sum ( ( r | N ) | N ) > N ;
for y , p st P [ p ] holds P [ All ( y , p ) ]
{ [ x1 , x2 ] } is Subset of [: X1 , X2 :] & { [ x1 , x2 ] } in the InternalRel of X1 ;
h = ( i , j ) |-- ( id B , id B ) .= H . i .= H . i ;
ex x1 being Element of G st x1 = x & x1 * N c= A & N c= A ;
set X = ( ( ConsecutiveSet ( q , O1 ) ) . ( O1 , O2 ) ) `1 , Y = ( ( ConsecutiveSet ( q , O1 ) ) . ( O1 , O2 ) ) `1 , Z = ( a , b ) `1 ;
b . n in { g1 : x0 - r < g1 & g1 < x0 + r } ;
f /* s1 is convergent & f /. x0 = lim ( f /* s1 ) implies f /* s1 is convergent & lim s1 = lim ( f /* s1 )
the lattice of lattice ( T ) = the lattice of the lattice of T & the carrier of T = the carrier of T & the carrier of T = the carrier of T ;
'not' ( a . x ) '&' b . x 'or' a . x '&' 'not' ( b . x ) '&' 'not' ( b . x ) '&' 'not' ( b . x ) '&' 'not' ( b . x ) = TRUE ;
q2 = len ( ( q ^ r1 ) ^ r1 ) + len ( ( q ^ r1 ) ^ r2 ) .= len ( q ^ r2 ) + len ( ( q ^ r1 ) ^ r2 ) ;
( ( 1 / a ) (#) ( sec * f1 ) - id Z ) - ( id Z ) * ( ( id Z ) ^ ) is_differentiable_on Z ;
set K1 = upper ( lim ( H , A ) | H ) , D2 = ( lim H ) | A , D1 = ( lim H ) | A ;
assume e in { ( w1 - w2 ) / 2 : w1 in F & w2 in G & w1 in F } ;
reconsider d' = dom a as Function of dom F `1 , dom F , dom F , dom F be finite set ;
LSeg ( f /^ j , q ) = LSeg ( f , j + q .. f -' 1 ) ;
assume X in { T . ( N2 , K1 ) : h . ( N2 , K1 ) = ( N . ( N2 , K1 ) ) `1 } ;
assume ( for d , c holds <* f , g *> * f1 = <* f , g *> * f1 ) & ( for d holds f1 . d = <* f , g *> * f1 ) ;
dom S[. S , n .] = dom S /\ Seg n .= dom ( L | Seg n ) .= Seg n /\ dom ( L | Seg n ) .= Seg n ;
x in H |^ a implies ex g st x = g |^ a & g in H |^ a
( a (#) ( n , 1 ) ) . ( a , 1 ) = a ` - ( 0 * n ) .= a ` - ( 0 * n ) .= a ` - ( 0 * n ) .= a ` - ( 0 * n ) ;
D2 . j in { r : lower_bound A <= r & r <= D1 . i & D1 . i <= D2 . i } ;
ex p being Point of TOP-REAL 2 st p = x & P [ p ] & p <> 0. TOP-REAL 2 ;
for c holds f . c <= g . c implies f ^ @ c ^ @ g ^ @ c in @ @ c
dom ( f1 (#) f2 ) /\ X c= dom ( f1 (#) f2 ) /\ X .= dom ( f1 (#) f2 ) /\ X .= dom ( f1 (#) f2 ) /\ X .= dom ( f1 (#) f2 ) /\ X ;
1 = ( p * p ) * ( p * q ) .= p * ( q * p ) .= p * ( q * p ) ;
len g = len f + len <* x + y *> .= len f + 1 + 1 .= len f + 1 ;
dom ( F | [: N1 , S :] ) = dom ( F | [: N1 , S :] ) .= [: N , S :] ;
dom ( f . t ) * I . t = dom ( f . t ) * g . t ;
assume a in ( "\/" ( ( T |^ the carrier of S ) ) .: D ) .: D ) ;
assume that g is one-to-one and ( the carrier' of S ) /\ rng g c= dom g and g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . x ) ) ) ) ) ) ) ) = g . ( g . ( g . ( g . ( g
( ( x \ y ) \ z ) \ ( ( x \ y ) \ z ) = 0. X ;
consider f such that f * f = id b & f * f = id b & f * f = id b ;
( ( cos * cos ) | [. 0 , PI / 2 .] ) is increasing ;
Index ( p , co ) <= len LS - Index ( Gij , LS ) + 1 - Index ( Gij , LS ) ;
t1 , t2 , 3 , 4 , 5 , 6 , 6 , 7 , 8 , 8 , 8 , 6 , 8 , 7 , 8 , 8 , 8 , 9 , 8 , 8 , 8 , 8 , 9 , 8 , 8 , 8 , 7 , 8 , 8 , 9 , 8 , 8 , 8 , 9 , 8 , 8 , 8 , 7 , 8 ,
( ( ( Frege ( F ) ) . h ) . ( h . ( ( curry G ) . h ) ) ) <= ( ( Frege ( F ) ) . h ) . ( h . ( h . ( h . ( h . ( h . ( h . ( h . ( h . ( h . ( h . ( h . ( h . ( h . ( h
then P [ f . i0 , f . i0 ] & F ( f . i0 , i ) < j ;
Q [ ( D . ( x , 1 ) ) `1 , 1 ] , F ( ( D . ( x , 1 ) ) `1 ) ] ;
consider x being element such that x in dom ( F . s ) and y = ( F . s ) . x ;
l . i < r . i & [ l . i , r . i ] is r of G . i ;
the Sorts of A2 = ( the Sorts of A2 ) +* ( the Sorts of A2 ) & the Sorts of A2 = ( the Sorts of A2 ) +* ( the Sorts of A1 ) ;
consider s being Function such that s is one-to-one & dom s = NAT & rng s c= F & s is one-to-one ;
dist ( b1 , b2 ) <= dist ( b1 , a ) + dist ( b2 , a ) ;
( ( C , n ) /. len ( C , n ) ) `1 = W & ( C , n ) /. len ( C , n ) ) `1 = W ;
q `2 <= ( ( UMP C ) * ( ( UMP C ) * ( 1 , 1 ) ) + ( ( UMP C ) * ( 1 , 1 ) ) ) `2 ;
LSeg ( f | i2 , i ) /\ LSeg ( f | i2 , j ) = {} ;
given a being ExtReal such that a <= IK and A = ]. a , IK .[ and a in A and b in A and c in A ;
consider a , b being complex number such that z = a & y = b & z + y = a + b ;
set X = { b |^ n where b is Element of NAT : b in X & a <= b } , Y = { b } ;
( ( x * y * z ) \ x ) \ z = 0. X ;
set x9 = [ <* xy , \mathopen { z } , \mathopen { z } ] , y9 = [ <* y , z *> , f1 ] , z9 = [ <* z , x *> , f2 ] , x9 = [ <* y , z *> , f3 ] , y9 = [ <* z , x *> , f3 ] , y9 = [ <* z , y *> , f3 ] ;
Carrier ( l ) /. len l = ( l . ( len l ) ) * ( l . ( len l ) ) ;
sqrt ( ( q `2 / |. q .| - sn ) / ( 1 - sn ) ) ^2 ) = 1 ;
sqrt ( ( p `2 / |. p .| - sn ) / ( 1 - sn ) ) ^2 ) < 1 ;
( ( ( ( X \/ Y ) \ { {} } ) \/ X ) \/ Y ) \/ X = ( ( X \/ Y ) \ X ) \/ Y ) \/ X .= ( ( X \/ Y ) \ Y ) \/ X .= ( X \/ Y ) \ Y ;
( ( s1 - s1 ) . k ) . k = ( s1 . k - s1 . k ) / ( s1 . k - s1 . k ) ;
rng ( ( h + c ) ^\ n ) c= dom SVF1 ( 1 , f , u0 ) /\ dom SVF1 ( 1 , f , u0 ) ;
the carrier of X = the carrier of X & the carrier of X = the carrier of X & the carrier of X = the carrier of X ;
ex p4 st p3 = p4 & |. p4 - |[ a , b ]| .| = r & |. p4 - |[ a , b ]| .| = r ;
set h = ( chi ( X , A ) ) | ( X /\ Af ) ;
R |^ ( ( 0 * n ) * n ) = \mathop Il ( X , X ) .= R |^ n * ( 0 * n ) .= R |^ ( n + 1 ) ;
( Partial_Sums ( ( F . -19 ) ) ) . n is nonnegative & ( Partial_Sums ( ( F . -19 ) ) ) . n is nonnegative & ( Partial_Sums ( ( F . -19 ) ) ) . n <= ( Partial_Sums ( ( F . I ) ) ) . n ;
f2 = CK . ( EK . ( EK . ( len K ) ) , K . ( len K ) ) ;
S1 . b = s1 . b .= s2 . b .= s2 . b .= s2 . b .= s2 . b ;
p2 in LSeg ( p2 , p1 ) /\ LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ;
dom ( f . t ) = Seg n & dom ( I . t ) = Seg n & rng ( I . t ) = Seg n ;
assume o = ( the connectives of S ) . 11 & 11 in ( the carrier' of S ) . 11 ;
set phi = ( l1 , l2 ) \mathop ( l , l2 ) , phi = ( l , l2 ) \mathop ( l , l2 ) , phi = ( l , l2 ) . ( l , l2 ) , phi = ( l , l2 ) . ( l , l2 ) , phi = ( l , l2 ) . ( l , l2 ) , phi = ( l , l2 ) . ( l , l2 ) , phi =
synonym p is invertible means : Def6 : ( p = q ) & ( p = q ) & ( p = q implies p = q ) ;
( Y1 - Y2 ) / ( 1 - j1 ) = ( - 1 ) / ( 1 - j1 ) & ( - 1 ) / ( 1 - j1 ) <> 0 ;
defpred X [ Nat , set , set , set ] means P [ $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2
consider k be Nat such that for n be Nat st k <= n holds s . n < x0 + g ;
Det ( I |^ ( ( m -' n ) -' n ) , K ) = 1_ K & ( m - n ) * ( ( m - n ) * ( K |^ n ) ) = 1_ K ;
sqrt ( - b ^2 - sqrt ( b ^2 - 4 * a * c ) ) < 0 ;
CK . d = CK . ( dK . d ) mod CK . ( dK . d ) .= CK . ( dK . d ) mod K . ( dK . d ) ;
attr X1 is dense & X2 is dense & X1 /\ X2 is dense implies X1 /\ X2 is dense SubSpace dense SubSpace of X1 ;
deffunc F6 ( Element of E , Element of I , Element of I ) = ( $1 , $2 ) * $2 + ( $1 , $2 ) * $2 ;
t ^ <* n *> in { t ^ <* i *> : Q [ i , T . ( t ^ <* n *> ) ] } ;
( x \ y ) \ x = ( x \ x ) \ y .= y \ x .= y \ x .= y \ x .= y \ x ;
for X being non empty set holds X is Basis of <* X , product <* Y *> , product <* X *> *>
synonym A , B are_means : as : Cl ( A \/ B ) misses Cl ( A \/ B ) & Cl ( A \/ B ) misses Cl ( A \/ B ) ;
len ( M @ ) = len p & width ( M @ ) = width ( M @ ) & width ( M @ ) = width ( M @ ) ;
J = { x where x is Element of K : 0 < v & x < y & v = f . x } ;
( Sgm ( Seg m ) ) . d - ( Sgm ( Seg m ) ) . e <> 0 ;
lower_bound divset ( D2 , k + k2 ) = D2 . ( k + k2 ) - D2 . ( k + k2 ) ;
g . r1 = ( - 1 ) * r1 + 1 & dom h = [. 0 , 1 .] & rng h c= [. 0 , 1 .] ;
|. a .| * ||. f .|| = 0 * ||. 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 ;
IC Exec ( i , s1 ) + n = IC Comput ( P2 , s2 , n ) .= ( 0 + n ) + n ;
IC Comput ( P , s , 1 ) = IC Comput ( P , s , 9 ) .= 5 + 9 .= 5 + 9 .= 9 ;
( IExec ( W6 , Q , t ) ) . intpos ( 0 + 1 ) = t . intpos ( 0 + 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 or y <= x ;
integral ( f , C ) | X = f . ( upper_bound C ) - lower_bound C .= ( upper_bound C ) * ( lower_bound C ) - lower_bound C .= ( lower_bound C ) * ( lower_bound C ) - lower_bound C ;
for F , G being one-to-one st rng F misses rng G & rng G misses rng G holds F ^ G is one-to-one
||. R /. ( h + c ) .|| < e1 * ( K + 1 ) * ( K + 1 ) ;
assume a in { q where q is Element of M : dist ( z , q ) <= r & q <= r } ;
set p4 = [ 2 , 1 ] .--> [ 2 , 0 ] ;
consider x , y being Subset of X such that [ x , y ] in F and x c= d and y c= d and x c= y ;
for y , x being Element of REAL st y ` in Y & x in X & y ` <= x ` holds y ` <= x `
func |. p ^ q .| -> \rbrack of A means : Def10 : for p , q st p in it holds it . p = ( p ^ q ) . ( p ^ q ) ;
consider t being Element of S such that x ` , y ` '||' z , t & x , y '||' y , t ;
dom x1 = Seg ( len x1 ) & len y1 = len y1 & len x1 = len y1 & len y1 = len y2 ;
consider y2 be Real such that x2 = y2 and 0 <= y2 and y2 <= 1 and y2 <= 1 and y2 <= 1 and y2 <= 1 ;
||. f | X /* s1 .|| = ||. f .|| | X & ||. f .|| | X is convergent & lim ( f | X ) = lim ( f | X ) ;
( the InternalRel of A ) ~ = ( the InternalRel of A ) ~ /\ Y .= {} \/ ( the InternalRel of A ) ~ .= {} ;
assume that i in dom p and for j being Nat st j in dom q holds P [ i , j ] and P [ i + 1 ] and P [ i + 1 ] ;
reconsider h = f | [: X , Y :] as Function of X , Y , X , Y ;
u1 in the carrier of W1 & u2 in the carrier of W2 & u2 in the carrier of W1 & v2 in the carrier of W2 implies ( the carrier of W1 ) \/ the carrier of W2 = the carrier of W2
defpred P [ Element of L ] means M <= f . $1 & f . $1 <= g . $1 & f . $1 <= g . $1 ;
succ ( u , a , v ) = s * x + ( - s ) * y + ( - s ) * y .= b - s * y + ( - s ) * y .= b - s * y + ( - s ) * y .= b - s * y + ( - s ) * y ;
- ( an - y ) = - x + ( - y ) .= - x + y .= - y + ( - x ) .= - x + y .= - y + x .= - x + y ;
given a being Point of Gf such that for x being Point of Gf holds a , x are_Assume f and a , x are_not collinear ;
fY. = [ dom ( @ ( f2 , g2 ) ) , cod ( ( @ g2 ) . ( g , g2 ) ) ] ;
for k , n be Nat st k <> 0 & k < n & k < n holds k divides n & k divides n implies k divides n
for x being element holds x in A |^ d iff x in ( ( A ` ) |^ d ) ` & x in ( ( A ` ) |^ d ) `
consider u , v being Element of R , a being Element of A such that l /. i = u * a * v ;
- ( ( - ( p `1 / |. p .| - cn ) ) / ( 1 - cn ) ) ^2 ) > 0 ;
U-13 . k = LS . ( F . k ) & F . k in dom ( L . k ) & F . k in rng ( L . k ) ;
set i2 = AddTo ( a , i , - n ) , i1 = goto ( - n + 1 ) , i2 = goto ( - n ) , i2 = goto ( - n ) ;
attr B is >= ) means : Def5 : for S holds -\mathop ( B , S ) . S = ( B , S ) . ( S ) ;
a9 "/\" D = { a "/\" d where d is Element of N : d in D & d in D } ;
|( exp_R . ( q9 - q9 ) , q9 . ( q9 - q9 ) )| * |( exp_R . ( q9 - q9 ) , exp_R . ( q9 - q9 ) )| >= |( exp_R . ( q9 - q9 ) , exp_R . ( q9 - q9 ) )| ;
( - f ) . ( sup A ) = ( - f ) . ( sup A ) .= - ( f . ( sup A ) ) .= - ( f . ( sup A ) ) ;
( G * ( len G , k ) ) `1 = ( G * ( len G , k ) ) `1 .= G * ( 1 , k ) `1 ;
( Proj ( i , n ) * ( h ) ) . x = <* ( proj ( i , n ) * ( h ) ) . x *> ;
f1 + f2 * reproj ( i , x ) = ( ( f1 + f2 ) * reproj ( i , x ) ) . x0 .= ( f1 + f2 ) . x ;
pred ( ( - cos * cos ) `| Z ) . x <> 0 & ( ( - cos * cos ) `| Z ) . x = ( ( - cos * cos ) `| Z ) . x ;
ex t being SortSymbol of S st t = s & h1 . t = h2 . ( t . {} ) & t . {} = ( t | ( the Sorts of A ) ) . ( t | ( the Sorts of A ) ) ;
defpred C [ Nat ] means P8 . $1 is n -AN & A8 : A8 : A8 : A8 : A8 : A8 : A8 : A8 : A8 : A8 : A8 : A8 c= A8 & A8 c= A8 & A8 c= A8 ;
consider y being element such that y in dom p9 and q9 . i = p9 . y and y in dom p9 and x = p9 . i ;
reconsider L = product ( { x1 } +* ( indx ( B ) , l ) ) as product of Carrier A ;
for c being Element of C ex d being Element of D st T . ( id c ) = id d & ( id d ) . ( id d ) = id d
( f | n , p ) = ( f | n ) ^ <* p *> .= f /^ n ^ <* p *> .= f /^ n ;
( f * g ) . x = f . ( g . x ) & ( f * h ) . x = f . ( h . x ) ;
p in { ( 1 - 2 ) * ( G * ( i + 1 , j ) + G * ( i + 1 , j + 1 ) ) } ;
f ` - { p } = ( ( c | n ) *' ( - ( f | n ) ) ) *' ( - ( f | n ) ) .= ( - ( f | n ) ) *' ( - ( f | n ) ) ;
consider r be Real such that r in rng ( f | divset ( D , j ) ) and r < m + s ;
f1 . ( |[ ( r2 - r2 ) / 2 , ( r2 - r2 ) / 2 ]| ) in f1 .: ( ( r2 - r2 ) / 2 , ( r2 - r2 ) / 2 ) ;
eval ( a | ( n , L ) , x ) = eval ( a | ( n , L ) ) , x ) .= a . ( ( n , L ) * x ) .= a . ( ( n , L ) * x ) ;
z = DigA ( t9 , x9 , y9 ) .= DigA ( t9 , x9 , y9 ) .= DigA ( t9 , x9 , y9 ) .= DigA ( t9 , x9 , y9 ) .= F9 . ( x , y , z ) ;
set H = { Intersect S where S is Subset-Family of X : S c= G & S c= G & S c= G } ;
consider S19 being Element of D * , d being Element of D such that S `1 = S19 ^ <* d *> and d in S and S /. ( len S ) = d ^ <* 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. K is Linear_Combination of A & Sum ( l ) = 0. K implies Sum ( l ) = Sum ( l ) & Sum ( l ) = Sum ( l )
let k1 , k2 , k2 , k2 , x4 , x5 , 6 , 7 , 8 , 8 , 7 , 8 , 8 , 9 , 8 , 7 , 8 , 8 , 9 , 8 , 8 , 7 , 8 , 9 be Element of NAT ;
consider j being element such that j in dom a and j in g " { k } and x = a . j ;
H1 . x1 c= H1 . x2 or H1 . x2 c= H1 . ( x1 . x2 ) or H1 . ( x1 . x2 ) c= H1 . ( x2 . x1 ) or H1 . ( x2 . x1 ) c= H2 . ( x2 . x1 ) ;
consider a being Real such that p = ( 1 - a ) * p1 + ( a * p2 ) * p2 and 0 <= a and a <= 1 and a <= 1 ;
assume that a <= c & c <= d & [ a , b ] c= dom f and [ a , b ] in dom g ;
cell ( Gauge ( C , m ) , ( len Gauge ( C , m ) ) -' 1 , 0 ) is non empty ;
A5 in { ( S . i ) `1 where i is Element of NAT : not contradiction } ;
( T * b1 ) . y = L * b2 /. y .= ( F * b2 ) . y .= ( F * b2 ) . y .= ( F * b2 ) . y ;
g . ( s , I ) . x = s . y & g . ( s , I ) . y = |. s . x - g . y .| ;
( log ( 2 , k ) ) to_power ( k + 1 ) >= ( log ( 2 , k ) ) to_power ( k + 1 ) ;
then p => q in S & not x in the still of S & not x in the carrier of S & not x in the carrier of S ;
dom ( the State of r-10 ) misses dom ( the State of r-10 ) & dom ( the --> ( the InstructionsF of r-10 ) ) = dom ( the --> ( the \mathclose of r-10 ) ) ;
synonym f is extended real means : Def3 : for x being set st x in rng f holds x is ExtReal ;
assume for a being Element of D holds f . { a } = a & for X being Subset-Family of D holds f .: ( f .: X ) = f . union X ;
i = len p1 .= len p3 + len <* x *> .= len p1 + len <* x *> .= len p1 + len <* x *> .= len p1 + len <* x *> .= len p1 + len <* x *> ;
( l , 3 ) `1 = ( g /. ( k + 1 ) ) `1 + ( k + 1 ) - ( k + 1 ) ) .= ( g /. ( k + 1 ) ) `1 + ( g /. ( k + 1 ) ) ;
CurInstr ( P2 , Comput ( P2 , s2 , l ) ) = halt SCM+FSA .= ( halt SCM+FSA ) . IC SCM+FSA .= ( ( 0 + l ) + 1 ) .= ( 0 + l ) ;
assume for n be Nat holds ||. seq .|| . n <= ( seq . n ) * ( seq . n ) & ( seq . n ) * ( seq . n ) <= ( seq . n ) * ( seq . n ) ;
sin . st ( for r2 holds sin . r2 = sin . ( cos . r2 - cos . ( 2 * PI * ( 2 * PI * ( 2 * PI * ( 2 * PI * ( 2 * PI * ( 2 * PI / 2 ) ) ) ) ) ) / 2 ) ) .= 0 ;
set q = |[ g1 . ( t1 , t2 ) `1 , g2 . ( t2 , t1 ) `2 ]| , g1 = |[ g1 . ( t2 , t1 ) `1 , g2 . ( t2 , t1 ) `2 ]| ;
consider G being sequence of S such that for n being Element of NAT holds G . n in holds G . ( F . n ) in ) & G . ( F . n ) in GAAAAS ;
consider G such that F = G and ex G1 st G1 in SX & G = ( the carrier of G ) \/ { x } ;
the root of ( x , s ) . [ x , s ] in ( the Sorts of Free ( C , s ) ) . ( ( the Sorts of C ) . s ) ;
Z c= dom ( exp_R * ( exp_R * ( exp_R + exp_R * ( exp_R + exp_R * ( exp_R + exp_R * f1 ) ) ) ) ) ;
for k be Element of NAT holds rF . k = ( ( Im f ) | ( S . k ) ) . k + ( ( Im f ) | ( S . k ) ) . k
assume that - 1 < n and ( q `2 / |. q .| - sn ) / ( 1 - sn ) < 0 and ( q `2 / |. q .| - sn ) / ( 1 - sn ) < 0 ;
assume that f is continuous and a < b and a < b and c < d and f . a = c and f . b = d ;
consider r being Element of NAT such that s8 = Comput ( P1 , s1 , i ) and r <= q and r <= q ;
LE f /. ( i + 1 ) , f /. j , L~ f & LSeg ( f , i ) = LSeg ( f /. ( i + 1 ) , f /. ( i + 1 ) ) ;
assume x in the carrier of K & y in the carrier of K & inf { x , y } in the carrier of K ;
assume f +* ( i1 , \xi ) in ( proj ( F , i1 ) ) " ( ( proj ( F , i2 ) ) " ) " ( ( proj ( F , i2 ) ) " ) ) ;
rng ( ( ( the InternalRel of M ) ~ ) | ( the carrier of M ) ) c= the carrier of M & the InternalRel of M = the InternalRel of M ;
assume z in { ( the carrier of G ) * : ex t being Element of G st t in { ( the carrier of G ) * : ex s being Element of G st s = ( the carrier of G ) * ( s , t ) } ;
consider l be Nat such that for m be Nat st l <= m holds ||. s1 . m - g .|| < s / 2 ;
consider t be VECTOR of product G such that mt = ||. Dt . t .|| & ||. t .|| <= 1 ;
Suppose the carrier of v = 2 implies v ^ <* 0 *> in dom p & v ^ <* 1 *> in dom p & v ^ <* 1 *> in dom p & p ^ <* 1 *> in dom p & p ^ <* 1 *> in dom p ;
consider a being Element of the points of Xas , A being Element of the lines of X\llangle A , B :] such that a on A and not a on A & b on A ;
( - x ) |^ ( k + 1 ) * ( ( - x ) |^ k ) " = 1 * ( ( - x ) |^ k ) " ;
for D being set st for i st i in dom p holds p . i in D holds p . i in D & p . i in D
defpred R [ element ] means ex x , y st [ x , y ] = $1 & P [ x , y ] ;
L~ f2 = union { LSeg ( p10 , p2 ) , LSeg ( p10 , p2 ) } \/ LSeg ( p1 , p2 ) } \/ LSeg ( p2 , p1 ) \/ LSeg ( p1 , p2 ) ;
i - len ( h11 + 2 ) + 1 - 1 < i - len ( h11 + 2 ) - 1 + 2 - 1 ;
for n be Element of NAT st n in dom F holds F . n = |. ( n -' 1 ) * ( n -' 1 ) .| ;
for r , s1 , s2 holds r in [. s1 , s2 .] iff s1 <= r & s1 <= s2 & s2 <= s2 & s1 <= s2 & s2 <= s2 & s2 <= s1 & s1 <= s2 & s2 <= s2 & s2 <= s2 & t1 <= s2 & t1 <= t2 & t1 <= t2
assume v in { G where G is Subset of T2 : G in B2 & G c= B1 & G c= B1 & G c= B2 & G c= B1 & G c= B2 & G c= B2 & G c= B1 & G c= B2 & G c= B1 & G c= B2 & G c= B2 & G c= B1 & G c= B2 & G c= B1 & G c= B2 & G c= B2 & G c= B1 & G c= B2 & G c= B2 & G c= B1 &
let g be Element of A , X be Element of INT , b be Element of INT , c be Element of ( 0 -tuples_on the carrier of K ) | ( b , c ) | ( b , c ) <> 0 ;
min ( g . [ x , y ] , k ) . [ y , z ] = ( min ( g . k , k ) ) . y ;
consider q1 being sequence of Cf such that for n holds P [ n , q1 . n ] and P [ q1 , q2 . n ] ;
consider f being Function such that dom f = NAT & for n being Element of NAT holds f . n = F ( n ) & f . ( n + 1 ) = F ( n ) ;
reconsider B-6 = B /\ [: B , C :] , \llangle Z , Z :] as Subset of [: B , C :] ;
consider j being Element of NAT such that x = the \ of ( n + 1 ) and 1 <= j and j <= n and 1 <= n and j <= n ;
consider x such that z = x and card ( x . O2 ) in card ( x . O1 ) & x in ( x . O1 ) & x in ( x . O1 ) & x in ( x . O1 ) . ( x . O1 ) ;
( C * _ T4 ( k , n2 ) ) . 0 = C . ( ( the ] of T4 ( k , n2 ) ) . 0 ) .= C . ( ( the sequence of T4 ( k , n2 ) ) . 0 ) ;
dom ( X --> rng f ) = X & dom ( X --> f ) = dom ( X --> f ) & rng ( X --> f ) = dom ( X --> f ) ;
( ( L~ SpStSeq C ) * ( i , j ) ) `2 <= ( ( L~ SpStSeq C ) * ( i , j ) ) `2 & ( ( L~ Cage ( C , n ) ) * ( i , j ) ) `2 <= ( ( L~ Cage ( C , n ) ) * ( i , j ) ) `2 ;
synonym x , y are_collinear means : Def1 : x = y or ex l being card of S st { x , y } c= l ;
consider X be element such that X in dom ( f | ( n + 1 ) ) and ( f | ( n + 1 ) ) . X = Y ;
assume that Im k is continuous and for x , y being Element of L st a = x & b = y & x << y holds a << b iff a << b & a << b ;
( 1 / 2 * ( ( #Z n ) * ( ( #Z n ) * ( #Z n ) ) ) ) is_differentiable_on REAL & ( #Z n ) * ( ( #Z n ) * ( #Z n ) ) ) is_differentiable_on REAL ;
defpred P [ Element of omega ] means ( for n holds ( ( $1 + 1 ) |^ n ) |^ ( n + 1 ) ) = A1 . ( ( $1 + 1 ) |^ n ) ;
IC Comput ( P , s , 2 ) = succ IC Comput ( P , s , 1 ) .= 6 + 1 .= 6 + 1 .= 6 + 1 .= 6 + 1 ;
f . x = f . g1 * f . g2 .= ( g1 * g2 ) . g2 .= ( g1 * g2 ) . g2 .= ( g1 * g2 ) . g2 .= ( g1 * g2 ) . g2 .= ( g1 * g2 ) . g2 .= ( g1 * g2 ) . g2 ;
( M * ( F-4 ) ) . n = M . ( ( ( ( \Omega ( [#] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( n | | n | n ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .= M . ( ( ( ( ( ( ( ( ( ( (
the carrier of L1 + L2 c= ( the carrier of L1 ) \/ ( the carrier of L2 ) \/ ( the carrier of L1 ) \/ ( the carrier of L2 ) ;
pred a , b , c , x , y , c , x , y , z , y , x , z , 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 , y , z , x , y , z , x , y , z , x , y , z , x , y
( the PartFunc of S , s ) . n <= ( the PartFunc of S , s ) . ( n + 1 ) * ( ( the PartFunc of S , s ) . n ) ;
pred - 1 <= r & r <= 1 implies ( ( - 1 ) (#) ( ( #Z 2 ) * ( r / 2 ) ) ) `| Z ) = - ( ( - 1 ) (#) ( ( #Z 2 ) * ( r / 2 ) ) ) ;
s8 in { p ^ <* n *> where p is FinSequence of T : p ^ <* n *> in T1 & p ^ <* n *> in T2 } ;
|[ x1 , x2 , x3 ]| . 2 - |[ y1 , y2 ]| . 2 - |[ y1 , y2 ]| . 2 = x2 - y2 ;
attr for m be Nat holds F . m is nonnegative means : every Nat holds ( Partial_Sums F ) . m is nonnegative & ( Partial_Sums F ) . m is nonnegative & ( Partial_Sums F ) . m is nonnegative ;
len ( ( \overline ( G , z ) ) * ( x , y ) ) = len ( ( \overline ( G , y ) ) * ( x , y ) ) + ( ( \overline ( G , y ) ) * ( x , y ) ) ) ;
consider u , v being VECTOR of V such that x = u + v and u in W1 /\ W2 and v in W2 and u in W2 /\ W3 ;
given F being FinSequence of NAT such that F = x & dom F = n & rng F c= { 0 , 1 } & for k st k in dom F holds F . k = k ;
0 = ( 1 * ] ) * u] iff 1 = ( ( - ( 1 - ( ] ) ) * ( ( - ( ] ) ) * ( ( - ( ( @ ) ) * ( @ @ ) ) ) ) ) / ( ( - ( ( - ( ( @ ) ) * ( @ @ ) ) ) ) ) ;
consider n be Nat such that for m be Nat st n <= m holds |. ( f # x ) . m - lim ( f # x ) .| < e ;
cluster -> Boolean -> Boolean for \hbox { ( 1 / 2 ) , ( 1 / 2 ) , ( 1 / 2 ) , ( 1 / 2 ) , ( 1 / 2 ) , ( 1 / 2 ) is Boolean ;
"/\" ( BB , L ) = Top ( ( the carrier of S ) --> ( {} , L ) ) .= "/\" ( [#] S , L ) .= "/\" ( ( the carrier of S ) --> ( {} , L ) ) .= "/\" ( ( the carrier of S ) --> ( {} , L ) ) .= "/\" ( ( the carrier of S ) --> ( {} , L ) ) .= "/\" ( {} , L ) ;
sqrt ( ( r ^2 + ( r ^2 ) ) ^2 ) + ( r ^2 + ( r ^2 ) ) ^2 ) <= sqrt ( ( r ^2 + ( r ^2 ) ) ^2 ) + sqrt ( ( r ^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 - c * |[ a , c ]| - ( 2 * r1 - c ) * |[ b , c ]| = 0. TOP-REAL 2 - ( 2 * r1 - c ) * |[ b , c ]| ;
reconsider p = P * ( \square , 1 ) , q = a " * ( ( - ( - ( - ( - ( K , n ) ) ) ) * ( ( 0. K , n ) ) ) 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 y = [ y1 , y2 ] ;
for n be Nat st 1 <= n & n <= len q1 holds q1 . n = ( ( g | ( len g ) ) | ( len g ) ) . n
consider y , z being element such that y in the carrier of A & z in the carrier of A & i = [ y , z ] ;
given H1 , H2 being strict Subgroup of G such that x = H1 & y = H2 & H = H1 & H = H2 & H1 , H2 // H , H1 & H = H2 ;
for S , T , T being non empty RelStr , d being Function of T , S st d is complete holds d is monotone & d is monotone & d is monotone
[ a + i , b ] in ( the carrier of F_Complex ) \/ ( the carrier of F_Complex ) & [ a , b ] in the carrier of K ;
reconsider mm = max ( len F1 , len ( p . n ) * ( x | ( n -' 1 ) ) ) as Element of NAT ;
I <= width GoB ( ( GoB ( h ) ) * ( 1 , width GoB ( h ) ) ) , ( GoB ( ( h ) ) * ( 1 , width GoB ( h ) ) ) ) , ( GoB ( ( h ) ) * ( 1 , width ( h ) ) ) ) ;
f2 /* q = ( f2 /* ( f1 /* s ) ) ^\ k .= ( f2 * f1 ) /* ( f1 /* s ) ) ^\ k .= ( f2 * f1 ) /* ( f1 /* s ) .= ( f2 * f1 ) . ( ( f1 /* s ) ^\ k ) .= ( f2 * f1 ) . ( ( f1 /* s ) ^\ k ) ;
attr A1 \/ A2 is linearly-independent means : Def8 : A1 : A1 misses A2 & A2 misses A1 & A2 /\ A1 = A2 & A1 /\ A2 = A2 /\ A1 & A2 /\ A1 = A2 /\ A1 ;
func A -the carrier of C -> set means : Def7 : union it = union { A ( s ) where s is Element of C : s in A & s in A } ;
dom ( ( Line ( v , i + 1 ) ) (#) ( ( ( ( v ^ ) ) (#) ( ( ( p ^ <* m *> ) (#) ( ( v ^ <* m *> ) (#) ( v ^ <* m *> ) ) ) ) ) ) ) = dom ( F ^ <* m *> ) ;
cluster [ x , ( x , 4 ) , ( x , x ) , ( x , y ) , x ) -> Function of x , x , y , z ;
E , ( All ( x2 , x3 ) ) |= All ( x2 , x3 ) '&' ( x2 '&' x3 ) '&' ( x3 '&' x4 ) ) & E , ( x2 '&' x3 ) |= x3 '&' ( x2 '&' x3 ) ;
F .: ( ( id X ) , g ) . x = F . ( id X , g . x ) .= F . ( g . x , g . x ) .= F . ( g . x , g . x ) .= F . ( g . x , g . x ) ;
R . ( h . m ) = F . ( x0 + h . m ) - ( h . m ) + ( h . m ) + ( h . m ) ;
cell ( G , ( X -' 1 , Y + 1 ) , ( Y -' 1 ) + ( Y -' 1 ) ) \ ( ( X -' 1 ) + ( Y -' 1 ) ) meets ( ( X -' 1 ) + ( Y -' 1 ) ) \ ( X -' 1 ) ;
IC Comput ( P2 , s2 , LifeSpan ( P2 , s2 ) ) = IC Comput ( P2 , s2 , LifeSpan ( P2 , s2 ) ) .= ( card I + 1 ) .= card I + ( card J + 2 ) .= card I + ( card J + 2 ) .= card I + card J + 2 .= card I + card I + 2 .= card I + card I + 2 ;
sqrt ( ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ) ^2 ) > 0 ;
consider x0 being element such that x0 in dom a and x0 in dom a and y = ( a " ) . x0 and x0 in { a . x0 } and a . x0 = ( a " ) . x0 ;
dom ( r1 (#) ( chi ( A , A ) ) , C ) = dom ( ( A * ( B * C ) ) , D ) .= dom ( ( A * ( B * C ) ) , D ) .= dom ( ( A * ( B * C ) ) , D ) .= dom ( ( A * ( B * C ) ) , D ) ;
d-7 . [ y , z ] = ( ( y , z ) `2 ) * ( y , z ) .= ( y , z ) `2 * ( y , z ) ;
pred for i being Nat holds C . i = A . i /\ B . i & C . i c= C . ( i + 1 ) /\ C . ( i + 1 ) ;
assume that x0 in dom f and f is continuous and f is continuous & for x st x in dom f holds f . x = - ( f . x ) / ( a - x ) ^2 ) and ( for x st x in dom f holds f . x = - ( a * x ) / ( a - x ) ^2 ) ;
p in Cl A implies for K being Basis of p st K in K & for Q being Subset of T st Q in K holds A meets Q & A meets Q & A meets Q & A meets Q & A meets Q & A meets Q & A meets Q & A meets Q implies A meets Q
for x be Element of REAL n st x in Line ( x1 , x2 ) holds |. y1 - y2 .| <= |. y1 - y2 .| & |. y1 - y2 .| <= |. y1 - y2 .|
func the \emptyset of <*> a -> Ordinal means : Def1 : a in it & for b being Ordinal st a in it holds it c= b & for b being Ordinal st b in a holds it . b = a ;
[ a1 , a2 , a3 ] in ( the carrier of A ) \/ ( the carrier of B ) & ( the InternalRel of A ) \/ ( the InternalRel of B ) = ( the InternalRel of A ) \/ ( the InternalRel of B ) ;
ex a , b being element st a in the carrier of S1 & b in the carrier of S2 & x = [ a , b ] & x = [ a , b ] ;
||. ( vseq . n ) - ( vseq . m ) .|| * ||. ( vseq . n ) - ( vseq . m ) .|| < e / ( ||. x .|| * ||. x .|| ) ;
then for Z being set st Z in { Y where Y is Element of IK : F c= Y & Z c= Y } holds z in Z ;
sup compactbelow ( s , t ) = [ sup compactbelow ( s , t ) , ( compactbelow ( s , t ) ) . ( s , t ) ] .= [ s , ( compactbelow ( s , t ) ) . ( s , t ) ] ;
consider i , j being Element of NAT such that i < j and [ y , f . i ] in [: I , I :] and [ f . i , f . j ] in [: I , I :] and [ f . i , f . j ] in [: I , I :] ;
for D being non empty set , p , q being FinSequence of D st p c= q & q in D holds ex x being FinSequence of D st p ^ q = q ^ x & p ^ ( q ^ x ) = p ^ x
consider e1 being Element of the carrier of X such that c9 , a9 // a9 , c9 and a9 <> c9 and a9 <> c9 and c9 <> b9 and a9 <> c9 and c9 <> c9 and a9 <> c9 and c9 <> b9 and c9 <> b9 ;
set U2 = I \! \mathop { {} } , N = I \! \mathop { {} } , F = S \! \mathop { {} } , N = S \! \mathop { {} } , F = S \! \mathop { {} } , N = S \! \mathop { {} } , N = S \! \mathop { {} } , N = S \! \mathop { {} } , N = S \! \mathop { {} } , N = S \! \mathop { {} } , N = S \! \mathop { {} } , N = S \! \mathop { {} } , N = S \! \mathop { {} } , F = S \! \mathop { {} } , F = S \! \mathop
|. q2 .| ^2 = ( ( |. q2 .| ) ^2 + ( |. q2 .| ) ^2 ) + ( |. q2 .| ) ^2 + ( |. q2 .| ) ^2 ) .= |. q2 .| ^2 + ( |. q2 .| ) ^2 ;
for T being non empty TopSpace , x , y being Element of [: the topology of T , the topology of T :] holds x "\/" y = x \/ y & x "/\" y = x /\ y implies x "/\" y = x /\ y
dom signature ( U1 ) = dom ( the charact of U1 ) & rng ( the charact of U1 ) = dom ( the charact of U2 ) & rng ( the charact of U2 ) = dom ( the charact of U2 ) ;
dom ( h | X ) = dom h /\ X .= dom ( h | X ) /\ X .= dom ( h | X ) /\ X .= dom ( h | X ) /\ X .= dom ( h | X ) /\ X .= dom ( h | X ) /\ X .= dom ( h | X ) /\ X .= dom ( h | X ) /\ X .= dom ( h | X ) /\ X .= dom ( h | X ) /\ X .= dom ( h | X ) ;
for N1 , N1 for K1 , N2 being Element of [: the carrier of G , the carrier of G :] holds ( h . ( h . K1 ) ) = ( h . K1 ) & ( h . ( h . K1 ) ) = ( h . K1 ) & ( h . ( h . K1 ) ) = ( h . K1 ) ;
( mod ( u , m ) ) . i = ( mod ( v , m ) ) . i + ( mod ( v , m ) ) . i .= ( mod ( v , m ) ) . i + ( mod ( v , m ) ) . i ;
- ( q `1 / |. q .| - cn ) / ( 1 + cn ) < - ( q `1 / |. q .| - cn ) / ( 1 + cn ) ;
pred r1 = f9 & r2 = f9 & r1 = f9 & r2 = f9 * ( f | i ) & s1 = ( f | i ) . ( ( f | i ) . ( g | i ) ) ;
v-4 . m is bounded Function of X , the carrier of Y & x9 . m = ( vseq . m ) * ( vseq . m ) & x9 . m = ( vseq . m ) * ( vseq . m ) ;
pred a <> b & b <> c & angle ( a , b , c ) = PI & angle ( b , c , a ) = PI & angle ( b , c , a ) = PI ;
consider i , j , r being Nat such that p1 = [ i , r ] and p2 = [ j , r ] and r < j and i < j and r < j and j < n ;
|. p .| ^2 - ( 2 * |( p , q )| ) ^2 + |. q .| ^2 = |. p .| ^2 + |. q .| ^2 ;
consider p1 , q1 being Element of ( X * ) such that y = p1 ^ q1 and q1 ^ q2 = p1 ^ q1 and p1 ^ q1 = q1 ^ q2 and q1 ^ q2 = q2 ^ q2 and p1 ^ q1 = q2 ^ q2 ;
, , r2 = ( 1 - r1 ) * ( s1 , s2 ) , s1 = ( s1 - s2 ) * ( s1 , s2 ) , s2 = ( s1 - s2 ) * ( s1 , s2 ) , s2 = ( s1 - s2 ) * ( s1 , s2 ) , s2 = ( s1 - s2 ) * ( s1 , s2 ) ;
( ( LMP A ) . ( w , z ) ) `2 = inf ( proj2 .: ( A /\ holds w in ( proj2 .: ( A /\ holds w in A ) ) ) & ( proj2 .: ( A /\ B ) ) . ( w , z ) is non empty ;
s , ( H , H1 ) |= H2 iff s |= H1 & s , ( H , H2 ) |= H2 iff s , ( H , H1 ) |= H2 & s , ( H , H1 ) |= H2
len ( s + t ) = card ( support b1 ) + 1 .= card ( support b1 ) + 1 .= card ( support b2 ) + 1 .= card ( support b1 ) + 1 .= card ( support b2 ) + 1 .= card ( support b1 ) + 1 .= card ( support b2 ) + 1 .= card ( support b2 ) + 1 .= card ( support b2 ) + 1 .= card ( support b2 ) + 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 >= y holds z >= y ;
LSeg ( ( UMP D ) . ( ( ( ( ( ( D ) . ( ( ( ( D ) ) . ( n + 1 ) ) ) ) + ( ( ( D ) . ( n + 1 ) ) ) + ( ( D ) . ( n + 1 ) ) ) ) ) , ( ( D ) . ( ( n + 1 ) ) + ( ( n + 1 ) ) + 1 ) ) ) /\ D = { ( ( n + 1 ) + 1 ) } ;
lim ( ( ( f `| N ) / g ) /* b ) = ( ( f `| N ) / g ) . b .= ( ( f `| N ) . b ) / ( ( g `| N ) . b ) ;
P [ i , pr1 ( f ) . i , pr1 ( f ) . ( i + 1 ) ] , pr1 ( f ) . ( i + 1 ) ] ;
for r be Real st 0 < r ex m be Nat st for k be Nat st m <= k holds ||. ( seq . k ) - ( seq . k ) .|| < r
for X being set , P being a_partition of X , x , y being set st x in a & y in P & x in P & y in P & x = a & y in P holds a = b
Z c= dom ( ( ( id Z ) ^ ) (#) ( ( #Z 2 ) ^ ) ) \ ( ( #Z 2 ) ^ ) " ) " { 0 } ;
ex j being Nat st j in dom ( l ^ <* x *> ) & j < i & y = ( l ^ <* x *> ) . j & z = ( l ^ <* x *> ) . j & z = ( l ^ <* x *> ) . j & z = ( l ^ <* x *> ) . j ;
for u , v being VECTOR of V for r being Real st 0 < r & u in N holds r * u + ( r * v ) in N holds r * u + ( r * v ) in N
A , Cl ( A , B ) , Cl ( A , B ) , Cl ( A , C ) , Cl ( A , B ) , Cl ( A , C ) , Cl ( A , B ) , Cl ( A , C ) , Cl ( A , B ) , Cl ( A , B ) , Cl ( A , B ) , Cl ( A , B ) , Cl ( A , B ) , Cl ( A , B ) , Cl ( A , B ) ) are_equipotent ;
- Sum <* v , u , w *> = - ( v + u ) + ( v + u ) .= - ( v + u ) + u .= - ( v + u ) + u .= - ( v + u ) + u .= - ( v + u ) + u .= - ( v + u ) + u .= - ( v + u ) ;
( Exec ( a := b , s ) ) . IC SCM+FSA = ( Exec ( a := b , s ) ) . IC SCM R .= succ IC s .= succ IC s .= ( ( 0 + 1 ) + 1 ) .= succ IC s .= succ IC s .= succ IC s .= ( ( 0 + 1 ) + 1 ) .= ( ( 0 + 1 ) + 1 ) .= ( ( 0 + 1 ) + 1 ) ;
consider h being Function such that f . a = h and dom h = I & for x being element st x in I holds h . x = ( the support of J ) . x ;
for S1 , S2 being non empty reflexive RelStr , D being non empty Subset of S1 , f being Function of S1 , S2 holds cos ( D ) is directed & cos ( D ) is directed & cos ( D ) is directed & cos ( D ) 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 & y = x or x = y & x = y or x = y or x = z & y = z or x = z & y = z & x = z or x = z or y = z & x = z or x = z & y = z or x = z & y = z & z = z & x = z or x = z & x = z & x = z & x = z & x = z & x = z & x =
( E-max L~ Cage ( C , n ) ) in rng ( Cage ( C , n ) ) & ( E-max L~ Cage ( C , n ) ) .. ( Cage ( C , n ) ) = ( E-max L~ Cage ( C , n ) ) .. ( Cage ( C , n ) ) ;
for T , T , p , q being decorated tree , T being Element of dom T st p ^ q ^ T = q ^ p holds ( T -tree ( p , T ) ) . q = T . ( q ^ p ) & ( T -tree ( p , T ) ) . q = T . ( q ^ p )
[ i2 + 1 , j2 ] in Indices G & [ i2 , j2 ] in Indices G & f /. k = G * ( i2 + 1 , j2 ) & f /. k = G * ( i2 + 1 , j2 ) ;
cluster ( k , n ) divides ( k , n ) & k divides ( k , n ) implies n divides k & ( k divides ( k + 1 ) ) & ( k divides ( k + 1 ) ) & ( k divides ( k + 1 ) ) & ( k divides ( k + 1 ) ) & ( k divides ( k + 1 ) implies ( k divides ( k + 1 ) ) implies ( k divides ( k + 1 ) ) & ( k divides ( k + 1 ) implies ( k divides ( k + 1 ) ) & ( k divides ( k + 1 ) implies ( k divides ( k + 1 ) ) &
dom F " = the carrier of X2 & rng F " = the carrier of X1 & F " is one-to-one & rng F = the carrier of X2 & F " is one-to-one & rng F = the carrier of X2 & rng F = the carrier of X1 & rng F = the carrier of X2 & rng F = the carrier of X2 & rng F = the carrier of X2 & rng F = the carrier of X1 & rng F = the carrier of X2 & rng F = the carrier of X2 & rng F = the carrier of X2 & rng F = the carrier of X2 & rng F = the carrier of X2 & rng F = the carrier of X2 &
consider C being finite Subset of V such that C c= A and card C = thesis and the carrier of W = A and the carrier of W = the carrier of W and the carrier of W = the carrier of W ;
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 or X c= V or Y c= V & X 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 , p4 ) .= angle ( p2 , p3 , p4 ) .= angle ( p2 , p3 , p4 ) .= angle ( p2 , p3 , p4 ) .= angle ( p2 , p3 , p4 ) ;
- sqrt ( ( - ( q `1 / |. q .| - cn ) ) / ( 1 - cn ) ) ^2 ) = - ( ( q `1 / |. q .| - cn ) / ( 1 - cn ) ) ^2 ) .= - ( ( q `2 / |. q .| - cn ) / ( 1 - cn ) ) ^2 .= - ( - ( q `2 / |. q .| - cn ) ) ^2 ;
ex f being Function of I[01] , TOP-REAL 2 st f is continuous one-to-one & rng f = P & f . 0 = p1 & f . 1 = p2 & f . 0 = p2 & f . 1 = p4 & f . 1 = p4 & f . 0 = p4 & f . 1 = p4 & f . 1 = p4 & f . 1 = p4 & f . 1 = p4 & f . 0 = p4 ;
pred f is_PartFunc u0 coordinate .] means : Def8 : SVF1 ( 2 , pdiff1 ( f , 1 ) , u0 ) . ( 2 * u ) - SVF1 ( 2 , pdiff1 ( f , 2 ) , u0 ) . ( 2 * u ) = ( proj ( 2 , 3 ) ) . ( u - y0 ) ;
ex r , s st x = |[ r , s ]| & ( G * ( len G , 1 ) , 1 ) `1 < r & r < G * ( 1 , 1 ) `1 & s < G * ( 1 , 1 ) `1 & r < G * ( 1 , 1 ) `2 ;
assume that f is_sequence_on G and 1 <= t and t <= len G and G * ( t , width G ) `2 >= ( ( G * ( t , width G ) ) `2 ) `2 and ( G * ( t , width G ) `2 ) `2 >= ( ( G * ( t , width G ) `2 ) `2 ) `2 ;
pred i in dom G means : Def8 : r * ( f (#) reproj ( i , x ) ) . i = r * ( reproj ( i , x ) . i ) ;
consider c1 , c2 being bag of o1 + o2 such that ( decomp c ) /. k = <* c1 , c2 *> and ( decomp c ) /. k = <* c1 , c2 *> & ( decomp c ) /. k = c1 + 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 & s1 < G * ( 1 , 1 ) `2 & s1 < G * ( 1 , 1 ) `2 } ;
Cl ( X ^ Y ) . k = the carrier of X . ( k2 + k ) .= ( C ^ ( i + k ) ) . ( ( k + 1 ) + 1 ) .= ( C ^ ( i + k ) ) . ( ( k + 1 ) + 1 ) .= ( C ^ ( i + k ) ) . ( ( k + 1 ) + 1 ) .= ( C ^ ( i + k ) ) . ( ( k + 1 ) + 1 ) .= ( C ^ ( i + 1 ) + 1 ) .= ( C ^ ( i + 1 ) + 1 ) . ( ( i + 1 ) + 1 ) .=
pred len M1 = len M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 implies M1 = M2 - M1 + M2 - M2 - M1 - M1 - M1 - M1 - M1 - M1 = M1 - M1 + M1 - M1 - M1 - M1 - M1 - M1 - M1 - M1 + M1 - M1 - M1 - M1 - M1 + M1 - M1 + M1 - M1 + M1 - M1 - M1 + M1 - M1 - M1 - M1
consider g2 be Real such that 0 < g2 and { y where y is Point of S : ||. y - x0 .|| < g2 & g2 . ( y - x0 ) - g2 . ( y - x0 ) .|| < g2 . ( y - x0 ) + g2 . ( y - x0 ) ;
assume x < ( - b + sqrt ( a , b , c ) ) * ( - sqrt ( a , b , c ) ) or x > - sqrt ( a , b , c ) * ( - sqrt ( a , b , c ) ) ;
( G1 '&' G2 ) . i = ( <* 3 *> ^ G1 ) . i & ( G1 ^ G2 ) . i = ( <* 3 *> ^ G1 ) . i & ( G1 ^ G2 ) . i = ( G1 ^ G2 ) . i & ( G1 ^ G2 ) . i = ( G1 ^ G2 ) . i ;
for i , j st [ i , j ] in Indices ( ( M + ( i + 1 ) ) * ( i , j ) ) ) holds ( ( M + ( i + 1 ) ) * ( i , j ) ) < ( M + ( i + 1 ) ) * ( i , j )
for f being FinSequence of NAT , i being Element of NAT st i in dom f & i in dom f holds i divides f /. ( i + 1 ) & i divides f /. ( i + 1 ) & i divides f /. ( i + 1 ) & i divides f /. ( i + 1 )
assume F = { [ a , b ] where a , b is set : for c st c in B & c in C holds a c= c & b c= c } ;
b2 * ( ( b2 - b3 ) * q3 ) + ( b2 - b3 ) * ( ( a1 - b2 ) * ( b1 - b2 ) ) + ( b2 - b3 ) * ( b2 - b3 ) = 0. TOP-REAL n + ( b2 - b3 ) * ( b1 - b2 ) .= ( ( a1 - a2 ) * ( b2 - b3 ) ) + ( ( b2 - b2 ) * ( b2 - b3 ) ) ;
Cl ( Cl F ) = { D where D is Subset of T : ex B being Subset of T st B = Cl ( D ) & D in Cl ( B ) & B c= Cl ( D ) & D c= Cl ( B ) & B c= Cl ( D ) & D c= Cl ( B ) & B c= Cl ( D ) & B c= Cl ( D ) & B c= Cl ( D ) ;
attr seq is summable means : Def8 : seq is summable & seq is summable & seq is summable & lim seq = Sum seq & seq is summable & lim seq = Sum seq & seq is summable & lim seq = Sum ( seq + seq ) ;
dom ( ( cn " ) | D ) = ( the carrier of ( TOP-REAL 2 ) | D ) /\ D .= ( the carrier of ( TOP-REAL 2 ) | D ) /\ D .= the carrier of ( ( TOP-REAL 2 ) | D ) /\ D .= the carrier of ( ( TOP-REAL 2 ) | D ) /\ D .= the carrier of ( ( TOP-REAL 2 ) | D ) /\ D .= the carrier of ( ( TOP-REAL 2 ) | D ) ;
[ X \to Z ] is full full SubRelStr of ( [#] Z ) |^ the carrier of Z & [ X \to Y ] is full SubRelStr of ( the carrier of Z ) |^ the carrier of Z ;
( G * ( 1 , j ) ) `2 = ( G * ( i , j ) ) `2 & ( G * ( i , j ) ) `2 <= ( G * ( i , j ) ) `2 ;
synonym m1 c= m2 & ( for p be set st p in P holds ( m1 in P iff p in P & not p in P & not p in P & not p in P ) & not p in P & not p in P & q in P & not q in P & not p in P & q in P & q in P & q in P & q in P & q in P & q in P & q in P & q in P & q in P implies q in P & q in P & q in P & q in P & q in P & q in P & q in P & q in P & q in P & q in P & q in P & q in P & q in P & q in P & q in
consider a being Element of B ( ) such that x = F ( a ) and a in { G ( b ) where b is Element of B ( ) : P [ b ] } and P [ a ] ;
cluster multiplicative multiplicative for multiplicative over N for multiplicative over N , s being State of the carrier of N , the carrier of N , the carrier of N , the carrier of N , the carrier of N , the carrier of N , the carrier of N , the carrier of N , the carrier of N , the carrier of N , the carrier of N #) is Element of the carrier of N ;
( n , b ) + ( d , 1 ) = b + ( c , d ) + ( d + 1 ) .= n + ( b + d ) .= n + ( d + 1 ) .= n + ( b + d ) .= n + ( b + d ) .= n + ( b + d ) .= n + ( d + 1 ) ;
cluster strict non empty for Element of INT means : Let : for i1 , i2 being Element of INT holds it . ( i1 , i2 ) = ( i1 + i2 ) * ( i1 , i2 ) + ( i1 + i2 ) * ( i1 , i2 ) ;
- ( ( s2 * p1 + ( s1 * p2 ) - ( s1 * p2 ) ) ) = ( ( 1 - s1 ) * ( p2 + s2 ) ) * ( p2 + s2 * p2 ) ;
eval ( ( a | ( n , L ) ) *' p , x ) = eval ( a | ( n , L ) ) * eval ( p , x ) .= a * eval ( p , x ) .= a * eval ( p , x ) .= a * eval ( p , x ) .= a * eval ( p , x ) ;
assume that the TopStruct of S = the TopStruct of T and for D being non empty Subset of S st D in the topology of T holds D meets the topology of T and for V being non empty Subset of T st V in the topology of T holds V meets the topology of T and V is open ;
assume 1 <= k + 1 & k <= len w + 1 implies T-7 ( ( ( ( q , w ) -succ k ) ^ <* ( q , w ) . k *> ^ w ) ^ w ) . k = ( ( ( q , w ) -succ ( ( q , w ) -w ) ^ w ) . k ;
2 * ( a |^ ( n + 1 ) ) + ( 2 * ( b |^ ( n + 1 ) ) ) >= ( a |^ n ) + ( b |^ ( n + 1 ) ) + ( b |^ ( n + 1 ) ) ;
M , v / ( x. 3 , x. 0 ) / ( x. 0 , x. 4 , x. 0 ) / ( x. 0 , x. 0 ) |= All ( x. 4 , x. 0 , x. 0 ) / ( x. 4 , x. 0 ) ) ;
assume that f is_differentiable_on l and for x0 st x0 in l holds 0 < f . ( x - x0 ) or 0 < f . ( x - x0 ) & f . ( x - x0 ) < 0 ;
for G1 being _Graph , W being Walk of G1 , e being Vertex of G2 st e in W & e in W holds not ( W is Walk & e in W iff e in W & e in W & e in W implies e in W & e in W & e in W
c9 is non empty iff ( not ( ( ex y1 , y2 st y1 is not empty & y2 is not empty & not y1 is not empty & not y2 is not empty ) & not y1 is not empty ) & not ( not ( ( not y1 in not y1 in not y1 in not y1 & y2 in not y1 in not y1 & not y2 in y2 ) & not y1 in y2 ) & not y2 in y1 & not y2 in y1 & not y2 in y2 ) & not y2 in y1 & not y2 in y2 & not y2 in y2 & not y2 in y2 & not y1 in y2 & y1 in y2 & y1 in y2 & not y2 in y2 & y1 in y2 & not y2 in y2 & y1 in y2 & not y2 in y2 & y1 in y2 & y1 in y2 &
Indices GoB ( f ) = [: dom GoB ( f ) , dom GoB ( f ) :] & ( GoB ( f ) ) * ( i1 , j1 ) + ( GoB ( f ) ) * ( i1 , j1 + 1 ) ) = ( GoB ( f ) ) * ( i1 , j1 + 1 ) ;
for G1 , G2 , G2 , G1 , G2 being strict Subgroup of O st G1 is_stable & G2 is_stable & G1 is_stable & G2 is_stable & G1 is stable holds G1 * G2 is stable Subgroup of G & G2 * ( G1 * G2 ) is stable Subgroup of G & G1 * ( G1 * G2 ) is stable
UsedIntLoc ( ( the int of f ) +* ( 1 , ( the InstructionsF of S ) +* ( 1 , ( the InstructionsF of S ) +* ( 1 , ( the InstructionsF of S ) +* ( the InstructionsF of S ) +* ( the InstructionsF of S ) +* ( the InstructionsF of S ) +* ( the InstructionsF of S ) +* ( the InstructionsF of S ) +* ( the InstructionsF of S ) +* ( the carrier' of S ) ) ) ) ) = { f , g , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , ( the Sorts of S ) , h , h , ( the Sorts of S
for f1 , f2 being FinSequence of F st f1 ^ f2 is p -element & ( for i being Element of NAT holds Q [ i ] implies Q [ f1 ^ f2 ] ) & Q [ f1 ^ f2 ] holds Q [ f1 ^ f2 ^ f2 ]
sqrt ( ( p `1 ) ^2 + ( p `2 ) ^2 ) = sqrt ( ( q `1 ) ^2 + ( q `2 ) ^2 ) .= sqrt ( ( q `1 ) ^2 + ( q `2 ) ^2 ) .= sqrt ( q `1 ) ^2 + ( q `2 ) ^2 ) .= sqrt ( q `1 ) ^2 + ( q `2 ) ^2 ;
for x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , Real st ( x1 - x2 ) <> 0 & y1 <> 0 & y2 <> 0 & y2 <> 0 holds |( y1 , y2 )| = |( x1 , y1 - y2 )| + |( y1 , y2 )|
for x st x in dom ( ( the Sorts of A ) | A ) holds ( ( the Sorts of A ) | A ) . ( ( the Sorts of A ) | A ) = - ( ( the Sorts of A ) | A ) . ( ( the Sorts of A ) | A )
for T being non empty TopSpace , P being Subset of T st P c= the topology of T & P is Basis of T holds ex B being Basis of T st B c= P & P is Basis of T & P is Basis of T
( a 'or' b ) 'imp' c . x = 'not' ( ( a 'or' b ) . x ) 'or' c . x 'or' c . x .= 'not' ( a . x 'or' b . x ) 'or' 'not' ( b . x ) 'or' 'not' ( b . x 'or' c . x ) .= 'not' ( 'not' ( b . x ) 'or' 'not' ( b . x ) 'or' 'not' ( b . x ) 'or' 'not' ( b . x ) 'or' 'not' ( b . x ) 'or' 'not' ( b . x ) 'or' 'not' ( b . x ) 'or' 'not' ( b . x ) 'or' 'not' ( b . x ) 'or' 'not' ( b . x ) 'or' 'not' ( b . x ) 'or' 'not' ( b . x ) 'or' 'not' ( b . x ) 'or' 'not' ( b . x
for e being set st e in A8 ex X1 , Y1 being Subset of Y st e = X1 & Y1 = Y1 & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & 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
for i being set st i in the carrier of S for f being Function of [: S1 , S2 :] , S1 . i st f = H . i & f . i = F . i holds F . i = f | [: S1 , S2 :] . i
for v , w st for y st x <> y & w . y = v . y holds J . ( ( VERUM ( Al ) ) . ( y , w ) ) = Valid ( Al ( Al ) ) . ( y , w ) )
card D = card D1 + card D2 - card D1 + card D2 - 1 .= ( i + 1 ) - 1 + 1 - 1 .= ( i + 1 ) - 1 + 1 .= ( i + 1 ) - 1 + 1 .= ( i + 1 ) - 1 + 1 .= ( i + 1 ) - 1 + 1 .= ( i + 1 ) - 1 + 1 .= ( i + 1 ) - 1 + 1 .= ( i + 1 ) - 1 + 1 ;
IC Exec ( i , s ) = ( i .--> ( 0 qua Nat ) ) . ( 0 + 1 ) .= ( ( 0 .--> s ) . 0 ) . 0 .= ( ( 0 .--> s ) . 0 ) . 0 .= ( ( 0 .--> s ) . 0 ) . 0 .= ( ( 0 .--> s ) . 0 ) . 0 .= ( ( 0 .--> s ) . 0 .= ( ( 0 .--> s ) . 0 ) .= ( ( 0 .--> s ) . 0 ) . 0 .= ( ( 0 .--> s ) . 0 .= ( ( 0 .--> s ) . 0 .= ( ( 0 .--> s ) . 0 .= ( ( 0 .--> s ) . 0 .= ( ( 0 .--> s ) . 0 .= ( ( 0 .--> s
len f /. ( \downharpoonright i1 -' 1 ) + 1 - 1 + 1 = len f -' ( i1 -' 1 ) + 1 - 1 + 1 .= len f -' ( i1 -' 1 ) + 1 - 1 .= len f - ( i1 -' 1 ) + 1 - 1 .= len f - ( i1 -' 1 ) + 1 ;
for a , b , c being Element of NAT st 1 <= a & 2 <= b & a < b holds a <= b or a = b or a = b & b = c or a = b + b-2 or a = b + b-2 or a = b-2 or a = b-2 or a = b-2 or b = b-2 or a = b-2 or a = b-2 or a = b-2 or a = b-2 ;
for f being FinSequence of TOP-REAL 2 , p being Point of TOP-REAL 2 st p in LSeg ( f , i ) & f . ( Index ( p , f ) + 1 ) = f . ( Index ( p , f ) + 1 ) holds Index ( p , f ) <= i & Index ( p , f ) + 1 <= len f
lim ( ( curry ' ( k , X , 0 ) ) # x ) = lim ( ( curry ' ( k , X , 0 ) ) # x ) + lim ( ( ( curry ' ( k , X , 0 ) ) # x ) ) ) ;
z2 = g /. ( \downharpoonright n1 -' n2 + 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 ) .= 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 G & [ f . 2 , f . 3 ] in the InternalRel of G ;
for G being Subset-Family of B st G = { [ X , Y ] where X is Subset of A ( ) : X in F & Y in G & X in G } holds ( Intersect ( F ) ) . ( X , Y ) = ( Intersect ( F ) ) . ( X , Y )
CurInstr ( P1 , Comput ( P1 , s1 , m1 + 1 ) ) = CurInstr ( P1 , Comput ( P1 , s1 , m1 + 1 ) ) .= CurInstr ( P1 , Comput ( P2 , s2 , m1 ) ) .= ( CurInstr ( P2 , Comput ( P2 , s2 , m1 ) ) ) .= ( CurInstr ( P2 , s2 ) ) .= ( CurInstr ( P2 , s2 ) ) ;
assume that a on M and b on M and c on N and d on N and p on N and d on M and p on N and c on N and d on M and p on N and p on M and d on N and p on M and c on N and d on N and p on M and c on N and p on N and p on M and c on N and p on N and p on M and c on N and p on N and c on N and d on N and d on M and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on M and p on N and d on M and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N
attr T is \hbox { T _ 4 4 4 } means : Def4 : ex F being Subset-Family of T st F is closed & for n being Element of NAT holds F . n is finite-ind & ind F <= 0 & ind F <= 0 & ind F <= 0 & ind F <= 0 & ind F <= 0 ;
for g1 , g2 st g1 in ]. r1 - r2 , r .[ & g1 in ]. r1 - r2 , r .[ & |. g1 - g2 .| < g1 & g1 in ]. g1 - g2 , r .[ holds |. ( f - g ) . g1 - ( g - g2 ) . g2 .| <= ( ( f - g ) . g2 ) / ( |. g2 .| - ( g - g2 ) )
( ( exp_R * z ) + ( z1 * z ) ) + ( ( exp_R * z ) + ( - z2 ) ) = ( ( exp_R * z ) + ( ( - z2 ) * z ) + ( ( - z2 ) * z ) ;
F . i = F /. i .= 0. R + r2 .= <* ( n + 1 ) -tuples_on ( n + 1 ) -tuples_on ( n + 1 ) -tuples_on ( n + 1 ) -tuples_on ( n + 1 ) -tuples_on ( n + 1 ) -tuples_on ( n + 1 ) -tuples_on ( n + 1 ) -tuples_on ( n + 1 ) -tuples_on ( n + 1 ) -tuples_on ( n + 1 ) -tuples_on ( n + 1 ) -tuples_on ( n + 1 ) -tuples_on ( n + 1 ) -tuples_on ( n + 1 ) -tuples_on ( n + 1 ) -tuples_on ( n + 1 ) -tuples_on ( n + 1 ) -tuples_on ( n + 1 ) -tuples_on ( n + 1 ) -tuples_on ( n + 1 ) -tuples_on ( n + 1 ) -tuples_on ( n + 1 ) -tuples_on ( n + 1 ) -tuples_on ( n + 1 ) -tuples_on ( ( n + 1 ) ) .= <* (
ex y being set , f being Function st y = f . n & dom f = NAT & f . 0 = [: A , B :] & for n holds f . ( n + 1 ) = [: A , B :] . ( n + 1 ) ;
func f (#) F -> FinSequence of V means : Def6 : len it = len F & for i be Nat st i in dom F holds it . i = F . ( f . i ) * F . ( F . i ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , 8 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , 8 , x5 , 8 , x5 , x5 , x5 , x5 , x5
for n being Nat for x being set st x = h . n holds h . ( n + 1 ) = o ( x , n ) & x in InputVertices S ( x , n ) & h . ( n + 1 ) in InnerVertices S ( x , n ) & h . ( n + 1 ) in InnerVertices S ( x , n ) & h . ( n + 1 ) in InnerVertices S ( x , n ) ;
ex S1 being Element of CQC-WFF ( Al ) st SubP ( P , l ) = S1 & ( P , l ) . ( l + 1 ) = ( P , l ) . ( l + 1 ) & ( P , l ) . ( l + 1 ) = ( P , l ) . ( l + 1 ) & ( P , l ) . ( l + 1 ) = P . ( l + 1 ) ;
consider P being FinSequence of GK such that p9 = product P and for i being Element of dom P st i in dom P ex tK being Element of the carrier of K st P . i = tK & tK . i = t & tK . i = t . i & tK . i = t . i ;
for T1 , T2 being strict non empty TopSpace , P being Basis of T1 , T2 st the topology of T1 = the topology of T2 & the topology of T2 = the topology of T2 & the topology of T2 = the topology of T2 & the topology of T1 = the topology of T2 holds P is Basis of T2
Suppose f is_partial u0 coordinate u0 coordinate coordinate u & r (#) pdiff1 ( f , 3 ) is_partial_differentiable_in u0 , 2 & pdiff1 ( r (#) pdiff1 ( f , 3 ) , 3 ) is_partial_differentiable_in u0 , 2 & pdiff1 ( r (#) pdiff1 ( f , 3 ) , 3 ) . ( 3 + 1 ) = r * pdiff1 ( f , 3 ) . ( 3 + 1 ) ;
defpred P [ Nat ] means for F , G being FinSequence of ( Seg $1 ) st len F = $1 & G = F & G = G holds not ( for s being Permutation of Seg $1 st len s = $1 & s = G . ( len s ) holds Sum ( F ^ G ) = Sum ( F ^ G ) ;
ex j st 1 <= j & j < width GoB f & ( GoB f ) * ( 1 , j ) `2 <= ( GoB f ) * ( 1 , j + 1 ) `2 & ( GoB f ) * ( 1 , j + 1 ) `2 <= ( GoB f ) * ( 1 , j + 1 ) `2 ;
defpred U [ set , set ] means ex Fa1 being Subset-Family of T st $1 = Fa1 & $2 = ( the topology of T ) . $1 & ( for n holds union ( Fa1 . $1 ) is open & ( union ( Fa1 . $1 ) is open implies $2 is open ) & ( union ( Fa1 . $1 ) is open implies $2 is open ) & ( union ( F . $1 ) is open implies $2 is open ) ;
for p4 being Point of TOP-REAL 2 st LE p4 , p2 , P & LE p1 , p2 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p1 , p2 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2
f in the carrier of E ( H ( H ( H ( ) ) ) ) & for g st g . y <> f . y holds g in the carrier of E ( ) & g in the carrier of E ( ) & f in the carrier of E ( ) & g in the carrier of E ( ) & g in the carrier of E ( ) & g in the carrier of E ( ) & g in the carrier of E ( ) & g in the carrier of E ( ) & g in the carrier of E ( ) & g in the carrier of E ( ) & g in the carrier of E ( ) & g in the carrier of E ( ) & g in the carrier of E ( ) & g in the carrier of E ( ) & g in the carrier of E ( ) implies f in the carrier of E ( ) & g in the carrier of E ( ) & g in the carrier of E ( ) & g
ex p9 being Point of TOP-REAL 2 st x = p9 & ( ( for p st p in the carrier of TOP-REAL 2 holds ( p `1 / |. p .| - sn ) / ( 1 + sn ) ) ^2 ) & ( ( p `1 / |. p .| - sn ) / ( 1 + sn ) ) ^2 >= 0 & ( p <> 0. TOP-REAL 2 ) & ( p <> 0. TOP-REAL 2 ) & ( p <> 0. TOP-REAL 2 ) & ( p <> 0. TOP-REAL 2 ) & ( p <> 0. TOP-REAL 2 ) = 0. TOP-REAL 2 ) & ( p <> 0. TOP-REAL 2 ) = 0. TOP-REAL 2 ) & ( p <> 0. TOP-REAL 2 ) & ( p <> 0. TOP-REAL 2 ) <> 0. TOP-REAL 2 ) & ( p <> 0. TOP-REAL 2 ) . p <> 0. TOP-REAL 2 ) & ( p <> 0. TOP-REAL 2 ) & ( p <> 0. TOP-REAL 2 ) . p <> 0. TOP-REAL 2 ) . p <> 0. TOP-REAL 2 ) & (
assume for d7 being Element of NAT st d7 <= ( ( n + 1 ) -tuples_on the carrier of K ) & d7 <= ( n + 1 ) -tuples_on the carrier of K holds s1 . ( ( n + 1 ) -tuples_on the carrier of K ) = s1 . ( ( n + 1 ) -tuples_on the carrier of K ) & s1 . ( ( n + 1 ) -tuples_on the carrier of K ) . ( ( n + 1 ) -tuples_on the carrier of K ) ) = s1 . ( ( n + 1 ) -tuples_on the carrier of K ) . ( ( n + 1 ) . ( ( n + 1 ) -tuples_on the carrier of K ) . ( ( n + 1 ) . ( ( n + 1 ) -tuples_on the carrier of K ) . ( ( n + 1 ) -tuples_on the carrier of K ) . ( ( ( ( K ) . ( ( n + 1 ) . ( ( n + 1 ) . ( ( n + 1
assume that s <> t and s is Point of Sphere ( x , r ) and s is Point of Sphere ( x , r ) and s is not zero and ex e being Point of TOP-REAL n st e = Ball ( x , r ) & e = Ball ( x , r ) & e = Ball ( x , r ) ;
given r such that 0 < r and for s holds 0 < s or ex x1 , x2 being Point of CNS st x1 in dom f & x2 in dom f & ||. x1 - x2 .|| < s & ||. f /. x1 - f /. x2 .|| < r & ||. f /. x2 - f /. x2 .|| < r ;
( p | x ) | ( ( x | x ) | ( ( x | x ) | ( x | x ) ) ) = ( ( x | x ) | ( x | x ) ) | ( ( x | x ) | ( x | x ) | ( x | x ) ) ) | ( ( x | x ) | ( x | x ) | ( x | x ) ) ;
Suppose x , h + x in dom sec and ( ( h + sec ) (#) ( sec * sec ) ) . x = ( 4 * ( sin . x ) + cos . x ) * ( sin . x ) ^2 + ( sin . x ) ^2 * ( cos . x ) ^2 + ( sin . x ) ^2 * ( sin . x ) ^2 + ( cos . x ) ^2 ) ;
assume that i in dom A and len A > 1 and len A = width B and B = A * ( i , j ) and B = A * ( i , j ) and B = A * ( i , j ) and B = A * ( i , j ) and C = B * ( i , j ) ;
for i being non zero Element of NAT st i in Seg n holds ( i divides n or i divides n or i divides n & i divides n & i divides n & h . i = ( n + 1 ) * ( i |^ n ) & h . ( i -' 1 ) = ( n + 1 ) * ( i |^ n )
( ( b1 => b2 ) '&' ( c1 '&' c2 ) ) '&' ( ( b1 '&' b2 ) '&' ( c1 '&' c2 ) ) '&' ( ( b1 '&' b2 ) '&' ( c1 '&' c2 ) '&' ( a1 '&' c2 ) '&' ( a2 '&' c2 ) '&' ( a1 '&' c2 ) '&' ( a2 '&' b2 ) '&' ( a1 '&' a2 '&' a3 ) '&' ( a1 '&' a2 '&' a3 ) '&' ( a1 '&' a2 '&' a3 ) '&' ( a1 '&' a2 '&' a3 ) '&' ( a2 '&' a3 ) '&' ( a2 '&' a3 ) '&' ( a1 '&' a3 ) '&' ( a1 '&' a3 ) '&' ( a2 '&' a3 ) '&' ( a1 '&' a3 ) '&' ( a1 '&' a3 '&' a3 '&' a3 '&' a3 '&' a3 '&' a3 '&' a3 '&' a3 '&' a3 '&' a3 '&' a3 '&' a3 '&' a3 '&' a3 '&' a3 '&' a3 '&' a3 '&' a3 '&' a4 '&' a4 '&' a3 '&' a3 '&' a3 '&' a3 '&' a3 '&' a3 '&' a4 '&' a3 '&' a3 '&'
assume that for x holds f . x = ( ( - cot * ( sin * cos ) ) ) . x and for x st x in dom ( ( cot * cos ) * ( sin * cos ) ) holds ( ( - cot * cos ) `| Z ) . x = ( ( - cot * cos ) `| Z ) . x ;
consider RK , IK be Real such that RK = Integral ( M , F . n ) & IK = Integral ( M , F . n ) & I = Integral ( M , F . n ) & I = Integral ( M , F . n ) & I = Integral ( M , F . n ) ;
ex k being Element of NAT st ' = k & 0 < d & for q be Element of product G st q in X & ||. q - x .|| < r holds ||. partdiff ( f , q , k ) - partdiff ( f , x , k ) .|| . ( k + 1 ) - partdiff ( f , x , k ) . ( k + 1 ) .|| < r ;
x in { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , 8 , x5 , x5 , x5 , 8 , x9 } ;
( G * ( j , ii ) ) `2 = ( G * ( 1 , jj ) ) `2 .= ( G * ( 1 , jj ) ) `2 .= ( G * ( 1 , jj ) ) `2 .= ( G * ( 1 , jj ) ) `2 .= ( G * ( 1 , jj ) ) `2 .= ( G * ( 1 , jj ) ) `2 .= ( G * ( 1 , jj ) ) `2 ;
f1 * p = p .= ( the Arity of S1 ) . ( ( the Arity of S1 ) . ( ( the Arity of S1 ) . o ) ) .= ( the Arity of S1 ) . ( ( the Arity of S1 ) . o ) .= ( the Arity of S1 ) . ( ( the Arity of S1 ) . o ) .= ( the Arity of S1 ) . ( ( the Arity of S1 ) . o ) ;
func tree ( T , P , T1 ) -> Function of T , T1 means : Def1 : q in T iff q in T & q = T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . 1 ) ) ) ) ) ) ) ) ) ) ) or ex p st p in T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T . ( T
F /. ( k + 1 ) = F . ( ( p . ( k + 1 ) ) -' 1 ) .= Fx0 . ( ( p . ( k + 1 ) ) -' 1 ) .= Fx0 . ( ( p . k ) -' 1 ) .= Fx0 . ( ( p . k ) -' 1 ) .= Fx0 . ( ( p . k ) -' 1 ) .= ( p . k ) - 1 .= ( p . k ) - 1 .= ( p . k ) - 1 .= ( p . k ) - 1 .= ( p . k ) - 1 .= ( p . k ) - 1 .= ( p . k ) - 1 .= ( p . k ) - 1 .= ( p . k ) - 1 .= ( p . k ) - 1 .= ( p . k + ( p . k + ( p . k ) - 1 .= ( p . k + ( p . k + ( p . k + 1 )
for A , B , C , D being Matrix of len C st len B = len C & C = len C & len A = width C & C * ( len A , C ) = C * ( A , B ) & len ( A * B ) = width C & width ( A * B ) = width C & width ( A * B ) = width C & width ( A * B ) = width C implies A * B ) = C * ( A * B )
seq . ( k + 1 ) = 0. ( ( seq ^\ k ) + seq ^\ k ) .= ( seq ^\ k ) + seq ^\ k .= ( seq ^\ k ) + seq ^\ k .= ( seq ^\ k ) + seq ^\ k .= ( seq ^\ k ) + seq ^\ k .= ( seq ^\ k ) + seq ^\ k .= seq ^\ k + seq ^\ k .= seq ^\ k + seq ^\ k .= seq ^\ k + seq ^\ k .= seq ^\ k + seq ^\ k .= seq ^\ k + seq ^\ k + seq ^\ k + seq ^\ k + seq ^\ k + seq ^\ k + seq ^\ k + seq ^\ k + seq ^\ k + seq ^\ k + seq ^\ k + seq ^\ k + seq ^\ k + seq ^\ k + seq ^\ k .= seq ^\ k + seq ^\ k + seq ^\ k + seq ^\ k + seq ^\ k + seq ^\ k + seq ^\ k + seq ^\ k + seq ^\ k
assume x in ( the carrier of C1 ) & y in ( the carrier of C1 ) & z in the carrier of C1 & x = ( the carrier of C1 ) \/ ( the carrier of C2 ) & y = ( the carrier of C1 ) \/ ( the carrier of C2 ) ;
defpred P [ Element of NAT ] means for f st len f = $1 & ( for k st k in dom f holds ( ( f | k ) . k ) . f = ( ( ( f | k ) . ( k + 1 ) ) . f ) . ( ( ( f | k ) . k ) . f ) * ( ( ( ( f | k ) . k ) . f ) . ( ( f | k ) . f ) . f ) ) ;
assume that 1 <= k and k + 1 <= len f and f /. k = G * ( i , j ) and [ i , j ] in Indices G and f /. k = G * ( i , j ) and f /. ( k + 1 ) = G * ( i , j ) and f /. ( k + 1 ) = G * ( i , j ) ;
assume that sn < 1 and ( q `2 / |. q .| - sn ) ^2 > 0 and ( p `2 / |. q .| - sn ) ^2 >= 0 and ( p <> 0. TOP-REAL 2 ) and ( p <> 0. TOP-REAL 2 ) ^2 >= 0 ;
for M being non empty TopSpace , x being Point of M st x = x holds ex f being Function of M , M st for n being Element of NAT holds f . n = Ball ( x , r ) & f . n = Ball ( x , r )
defpred P [ Element of omega ] means f1 is_differentiable_on Z & ( for x st x in Z holds f1 . x = 1 / ( x - a ) * ( x - a ) ) & ( for x st x in Z holds f1 . x = 1 / ( x - a ) * ( x - a ) ) ;
defpred P1 [ Nat , Point of C , Point of C ] means ( $1 in Y & $2 in Y & $2 = ( $1 + 1 ) * ( $2 - 1 ) ) & ( $1 - 1 ) * ( $2 - 1 ) - ( $2 - 1 ) * ( $2 - 1 ) + ( $1 - 1 ) * ( $2 - 1 ) ) < ( $1 - 1 ) * ( $2 - 1 ) + ( $1 - 1 ) * ( $2 - 1 ) ;
( f ^ mid ( g , 2 , len g ) ) . i = ( mid ( g , 2 , len g ) ) . ( i + 1 ) .= ( g | ( i + 1 ) ) . ( i + 1 ) .= g . ( ( i + 1 ) + 1 ) .= g . ( ( i + 1 ) + 1 ) .= g . ( ( i + 1 ) + 1 ) .= g . ( ( i + 1 ) + 1 ) ;
( 1 - 2 * n2 + 2 * n2 + 2 * n2 + 1 * n2 + 2 * n2 + 1 * n2 + 2 * n2 + 1 * n2 + 2 * n2 + 1 * n2 + 2 * n2 + 1 * n2 + 2 * n2 + 2 * n2 + 2 * n2 + 2 * n2 + 2 * ( n2 + 1 ) * n2 + 2 * ( n2 + 1 ) * n2 + 2 * ( n2 + 1 ) * n2 + 2 * ( n2 + 1 * n2 + 1 * n2 + 1 * n2 + 1 * n2 + 1 * ( n2 + 1 * n2 + 1 * n2 + 1 * n2 + 1 * n2 + 1 * n2 + 1 * n2 + 1 * n2 + 1 * n2 + 1 * n2 + 1 * ( n2 + 1 * n2 + 1 * n2 + 1 * n2 + 1 * n2 + 1 * n2 + 1 * n2 + 1 * n2 + 1 * n2 + 1 * n2 + 1 * ( n2 + 1 * n2 + 1 * (
defpred P [ Nat ] means for G being non empty finite strict finite RelStr st G is $1 , finite , symmetric , symmetric holds the carrier of G = the carrier of G & the carrier of G = the carrier of G & the carrier of G = the carrier of G & the carrier of G = the carrier of G & the carrier of G = the carrier of G & the carrier of G = the carrier of G & the carrier of G = the carrier of G & the carrier of G & the carrier of G = the carrier of G & the carrier of G & the carrier of G & the carrier of G = the carrier of G & the carrier of G & the carrier of G = the carrier of G & the carrier of G = the carrier of G & the carrier of G = the carrier of G & the carrier of G & the carrier of G & the carrier of G = the carrier of G & the carrier of G = the carrier of G & the carrier of G = the carrier of G & the carrier of G = the
assume that not f /. 1 in Ball ( u , r ) and 1 <= m and m <= len f and not ( f . m in Ball ( u , r ) & not ( f . m in Ball ( u , r ) ) & not ( ex u st u in Ball ( u , r ) & not u in Ball ( u , r ) ) ;
defpred P [ Element of NAT ] means ( Partial_Sums ( cos ) ) . ( $1 + 1 ) = Partial_Sums ( cos ) . ( $1 + 1 ) + ( Partial_Sums ( cos ) . ( $1 + 1 ) ) * ( ( Partial_Sums ( cos ) ) . ( $1 + 1 ) + 1 ) * ( ( Partial_Sums ( cos ) ) . ( $1 + 1 ) + 1 ) * ( ( Partial_Sums ( cos ) ) . ( $1 + 1 ) + 1 ) ) ;
for x being Element of product F holds x is FinSequence of product F & dom x = I & x in dom ( the _ of F ) & for i being set st i in dom x holds x . i in ( the _ of F ) . i & x . i in ( the _ of F ) . i
( x " ) |^ ( n + 1 ) = ( x " ) |^ n * x .= ( x |^ n ) |^ ( n + 1 ) .= ( x |^ n ) |^ ( n + 1 ) .= ( x |^ n ) |^ ( n + 1 ) .= ( x |^ n ) |^ ( n + 1 ) .= ( x |^ n ) |^ ( n + 1 ) .= ( x |^ n ) |^ ( n + 1 ) .= ( x |^ n ) |^ ( n + 1 ) * x .= ( x |^ n ) |^ ( n + 1 ) * x |^ ( n + 1 ) * x |^ ( n + 1 ) * x |^ ( n + 1 ) * x |^ ( n + 1 ) * x |^ ( n + 1 ) * x |^ ( n + 1 ) * x |^ ( n + 1 ) * x |^ ( n + 1 ) * x |^ ( n + 1 ) * x |^ ( n + 1 ) * x |^ ( n + 1 ) * x |^ ( n + 1 ) * x |^ ( n + 1 )
DataPart Comput ( P +* I , s1 , LifeSpan ( P +* I , s1 ) ) = DataPart Comput ( P +* I , s2 , LifeSpan ( P +* I , s1 ) + 3 ) .= DataPart Comput ( P +* I , s2 , LifeSpan ( P +* I , s1 ) + 3 ) .= DataPart Comput ( P +* I , s2 , LifeSpan ( P +* I , s2 ) + 3 ) ;
given r such that 0 < r and ]. x0 - r , x0 + r .[ c= dom ( f1 + f2 ) /\ ]. x0 - r , x0 + r .[ and for g st g in ]. x0 - r , x0 + r .[ holds ( f1 + f2 ) . g <= ( f1 + f2 ) . g ;
assume that X c= dom f1 /\ dom f2 and f1 | X is continuous and f2 | X is continuous and ( f1 - f2 ) | X is continuous and ( f1 - f2 ) | X is continuous & ( f1 - f2 ) | X is continuous & ( f1 - f2 ) | X is continuous ;
for L being continuous complete LATTICE for l being Element of L ex X being Subset of L st l = sup X & for x being Element of L st x in X holds x is prime & x is prime & l is prime & l is prime & l is prime & l is prime
Support ( e ) in { ( m *' p ) where m is Element of NAT : i in dom ( m *' p ) & ( m = len p ) & ( m = len p ) & ( m = len p implies ( m = len p ) & ( m = len p ) implies ( m = len p ) & ( m = len p ) implies ( m = len p ) + ( m -' 1 ) ) ;
( f1 - f2 ) /* ( f1 /* s ) = lim ( f1 /* s ) - f2 /* ( f1 /* s ) .= ( f1 /* s ) - f2 /* ( f1 /* s ) .= ( f1 /* s ) - f2 /* s .= ( f1 /* s ) - f2 /* s .= ( f1 /* s ) - f2 /* s ;
ex p1 being Element of CQC-WFF ( Al ) st F . ( p1 `1 ) = g . ( p1 `1 ) & for g being Function of [: Al , Al :] , D st g . ( p1 `1 ) = g . ( p1 `1 ) & g . ( p1 `1 ) = g . ( p1 `1 ) & g . ( p1 `1 ) = g . ( p1 `1 ) ;
( mid ( f , i , len f -' 1 ) ) /. ( j + 1 ) = ( mid ( f , i , len f -' 1 ) ) /. ( j + 1 ) .= ( mid ( f , i , len f -' 1 ) ) /. ( j + 1 ) .= ( mid ( f , i , len f -' 1 ) ) /. ( j + 1 ) .= ( mid ( f , i , len f -' 1 ) ) /. ( j + 1 ) ;
( ( p ^ q ) /^ k ) . ( len p + k ) = ( ( p ^ q ) /^ k ) . ( len p + k ) .= ( ( p ^ q ) /^ k ) . ( len p + k ) .= ( ( p ^ q ) /^ k ) . ( len p + k ) .= ( p ^ q ) /^ k .= ( p ^ q ) /^ k .= ( p ^ q ) /^ k .= ( p ^ q ) /^ k ;
len ( mid ( f , D1 , j1 ) + 1 ) = len ( ( D2 | indx ( D2 , D1 , j1 ) + 1 ) ) + 1 .= len ( D2 | indx ( D2 , D1 , j1 ) + 1 ) .= len ( D2 | indx ( D2 , D1 , j1 ) + 1 ) ;
x * y = ( x * y ) * ( y * z ) .= ( x * y ) * ( y * z ) .= ( x * y ) * ( y * z ) .= ( x * y ) * ( y * z ) .= ( x * y ) * ( y * z ) .= ( x * y ) * ( y * z ) .= ( x * y ) * ( y * z ) .= x * y * ( y * z ) .= x * y * ( y * z ) .= x * y * ( y * z ) * ( y * z ) .= x * ( y * z ) ;
v . <* x , y *> = <* x0 , y0 *> * ( <* x0 , y0 *> - ( y - x0 ) * ( y - x0 ) ) + ( ( ( y - x0 ) * ( y - x0 ) ) * ( y - x0 ) ) + ( ( ( y - x0 ) * ( y - x0 ) ) * ( y - x0 ) ) ) ;
i * i = <* 0 * ( 1 - i ) * ( 0 - i ) + ( 0 * ( i - i ) ) * ( i - i ) + ( 0 * ( i - i ) ) * ( i - i ) ) .= <* 0 * ( i - i ) + ( 0 * ( i - i ) ) * ( i - i ) + ( 0 * ( i - i ) ) * ( i - i ) ) .= ( 0 * ( i - i ) + ( 0 * ( i - i ) ) .= ( 0 * ( i - i ) + ( 0 * ( i - i ) + ( 0 * ( i - i ) ) + ( 0 * ( i - i ) ) .= ( 0 * ( i - i ) + ( 0 * ( i - i ) ) + ( 0 * ( i - i ) ) + ( 0 * ( i - i ) ) + ( 0 * ( i - i ) ) + ( 0 * ( i - i ) ) .= ( 0 * ( i - i ) + ( 0 * ( i - i ) + ( 0 * ( i
Sum ( L (#) F ) = Sum ( L (#) ( F1 ^ F2 ) ) + Sum ( ( L (#) F ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) ) + Sum ( ( L (#) F ) ^ ( L (#) ( F1 ^ F2 ) ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( ( L (#) ( F1 ^ F2 ) ) ) + Sum ( ( L (#) ( F1 ^ F2 ) ) ) .= Sum ( ( L (#) ( ( L (#) ( F1 ^ F2 ) ) ) + Sum ( ( L (#) ( F1 ^ F2 ) ) ) .= Sum ( L (#) ( ( L (#) F2 ) ) + Sum ( ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) + Sum ( ( L (#) ( F1 ^ F2 ) ) .= Sum ( ( L (#) ( F1 ^ F2 ) ) .= Sum ( ( L (#) ( F1 ^ F2 ) ) + Sum
ex r be Real st for e be Real st 0 < e ex Y1 be finite Subset of X st Y1 is non empty & Y1 c= Y & for Y1 be finite Subset of X st Y1 c= Y & Y1 c= Y holds |. ( Y1 + Y2 ) . Y1 - ( Y1 + Y2 ) . Y1 .| < r
( GoB f ) * ( i , j + 1 ) = f /. ( k + 2 ) & ( GoB f ) * ( i , j + 1 ) = f /. ( k + 1 ) or ( GoB f ) * ( i , j + 1 ) = f /. ( k + 1 ) or ( GoB f ) * ( i + 1 , j + 1 ) = f /. ( k + 1 ) & ( GoB f ) * ( i + 1 , j + 1 ) = f /. ( k + 1 ) & ( GoB f ) * ( i + 1 ) & ( GoB f ) * ( i + 1 , j + 1 ) ;
( ( - cos ) ^2 ) / ( ( cos * cos ) ^2 ) = ( ( r / ( cos * cos ) ^2 ) ) / ( ( cos * cos ^2 ) ) .= ( ( r / ( cos * cos ^2 ) ) ^2 ) / ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . x ) ) ) ) ) ^2 ) ) .= ( r / ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . x ) ) ) ) ) ) ^2 ) ) ) ) ^2 ) ) ^2 ) ) .= ( 1 / ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . ( cos . x ) ) ) ) ) ) ^2 ) )
( - b + sqrt ( a , b , c ) ) * ( 2 * a + sqrt ( b , c ) ) > 0 & ( - b + sqrt ( a , b , c ) ) * ( 2 * a + sqrt ( b , c ) ) < 0 implies ( - b + sqrt ( a , b , c ) ) * ( 2 * a + sqrt ( b , c ) ) < 0
Suppose inf ( \mathopen { \uparrow X , C ) /\ C ) = "\/" ( ( the InternalRel of L ) /\ C , L ) and for X holds X is maximal implies "\/" ( ( the InternalRel of L ) /\ C , L ) = "\/" ( ( the InternalRel of L ) /\ C , L ) & "\/" ( ( the InternalRel of L ) /\ C , L ) = "\/" ( ( the InternalRel of L ) /\ ( the InternalRel of L ) /\ ( the InternalRel of L ) ) ;
( ( the Sorts of B ) . ( j , i ) ) . ( j , i ) = ( j , i ) --> ( ( j , i ) --> ( ( j , i ) --> ( ( j , i ) --> ( j , i ) ) ) ) . ( j , i ) ) & ( j = i implies ( j = i implies ( j = i ) & ( j = i implies ( j = i ) ) & ( j = i implies ( j = i implies ( j = i implies ( j = i implies ( j = i implies ( j = i ) implies ( j = i ) ) & ( j = i implies ( j = i implies ( j = i implies ( j = i ) implies ( j = i ) = ( j = i ) & ( j = i implies ( j = i implies ( j = i implies ( j = i implies i = i implies ( j = i implies i = i implies ( j = i implies i = i ) implies ( j = i implies ( j = i implies ( j = i implies ( j = i implies ( j = i implies ( j