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contradiction ;
contradiction ;
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thesis ;
thesis ;
contradiction ;
contradiction ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
contradiction ;
contradiction ;
thesis ;
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contradiction ;
contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
thesis ;
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contradiction ;
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contradiction ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
B in A ;
a <> c
T c= S
D c= B
c in X ;
b in X ;
X in F ;
b in D ;
x = e ;
let m ;
h is onto ;
N in K ;
let i ;
j = 1 ;
x = u ;
let n ;
let k ;
y in A ;
let x ;
let x ;
m c= y ;
F is one-to-one ;
let q ;
m = 1 ;
1 < k ;
G is prime ;
b in A ;
d divides a ;
i < n ;
s <= b ;
b in B ;
let r ;
B is one-to-one ;
R is total ;
x = 2 ;
d in D ;
let c ;
let c ;
b = Y ;
0 < k ;
let b ;
let n ;
r <= b ;
x in X ;
i >= 8 ;
let n ;
let n ;
y in f ;
let n ;
1 < j ;
a in L ;
C is boundary ;
a in A ;
1 < x ;
S is finite ;
u in I ;
z << z ;
x in V ;
r < t ;
let t ;
x c= y ;
a <= b ;
m in NAT ;
assume f is prime ;
not x in Y ;
z = +infty ;
k be Nat ;
K ` is being_line ;
assume n >= N ;
assume n >= N ;
assume X is 1 -element ;
assume x in I ;
q is there 0 ;
assume c in x ;
p > 0 ;
assume x in Z ;
assume x in Z ;
1 <= k12 ;
assume m <= i ;
assume G is cyclic ;
assume a divides b ;
assume P is closed ;
\bf 2 > 0 ;
assume q in A ;
W is non bounded ;
f is Then f is simple ;
assume A is boundary ;
g is special ;
assume i > j ;
assume t in X ;
assume n <= m ;
assume x in W ;
assume r in X ;
assume x in A ;
assume b is even ;
assume i in I ;
assume 1 <= k ;
X is non empty ;
assume x in X ;
assume n in M ;
assume b in X ;
assume x in A ;
assume T c= W ;
assume s is atomic ;
b `1 <= c `1 ;
A meets W ;
i `1 <= j `1 ;
assume H is universal ;
assume x in X ;
let X be set ;
let T be Tree ;
let d be element ;
let t be element ;
let x be element ;
let x be element ;
let s be element ;
k <= A-2 ;
let X be set ;
let X be set ;
let y be element ;
let x be element ;
P [ 0 ]
let E be set , A , B be Subset of E ;
let C be Category ;
let x be element ;
k be Nat ;
let x be element ;
let x be element ;
let e be element ;
let x be element ;
P [ 0 ]
let c be element ;
let y be element ;
let x be element ;
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 subset of TOP-REAL 2 ;
Q halts_on s ;
x in such ;
M < m + 1 ;
T2 is open ;
z in b there a ;
R2 is well-ordering ;
1 <= k + 1 ;
i > n + 1 ;
q1 is one-to-one ;
let x be trivial set ;
P3 is one-to-one ;
n <= n + 2 ;
1 <= k + 1 ;
1 <= k + 1 ;
let e be Real ;
i < i + 1 ;
p4 in P ;
p1 in K ;
y in C1 ;
k + 1 <= n ;
let a be Real , A be Subset of TOP-REAL 2 ;
X |- r => p ;
x in { A } ;
let n be Nat ;
let k be Nat ;
let k be Nat ;
let m be Nat ;
0 < 0 + k ;
f is_differentiable_in x ;
let x0 ;
let E be Ordinal ;
o \mathord 4 ;
O <> O2 ;
let r be Real ;
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 , X be Subset of V ;
not s in Y |^ 0 ;
rng f <= w
b "/\" e = b ;
m = m2 ;
t in h . D ;
P [ 0 ] ;
assume z = x * y ;
S . n is bounded ;
let V be RealUnitarySpace , A , B be Subset of V ;
P [ 1 ] ;
P [ {} ] ;
C1 meets L~ f ;
H = G . i ;
1 <= i `1 + 1 ;
F . m in A ;
f . o = o ;
P [ 0 ] ;
aa <= b ;
R [ 0 ] ;
b in f .: X ;
assume q = q2 ;
x in [#] V ;
f . u = 0 ;
assume e1 > 0 ;
let V be RealUnitarySpace , A , B be Subset of V ;
s is non trivial & s is non trivial ;
dom c = Q ;
P [ 0 ] ;
f . n in T ;
N . j in S ;
let T be complete LATTICE , A , B be Subset of T ;
the object map of F is one-to-one
sgn x = 1 ;
k in support a ;
1 in Seg 1 ;
rng f = X ;
len T in X ;
vA2 < n ;
S\mathopen { - } S } is bounded ;
assume p = p2 ;
len f = n ;
assume x in P1 ;
i in dom q ;
let U2 ;
pp `2 = 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 has \cdot the \rangle ;
assume x0 <= m ;
T is increasing for sequence of REAL ;
e2 <> e1 & e2 <> e2 ;
Z c= dom g ;
dom p = X ;
H is proper implies H is proper
i + 1 <= n ;
v <> 0. V ;
A c= Affin A ;
S c= dom F ;
m in dom f ;
X0 be set ;
c = sup N ;
R is connected implies union M in M
assume not x in REAL ;
Im f is complete ;
x in Int y ;
dom F = M ;
a in On W ;
assume e in [: A , B :] ;
C c= C1. R ;
mm <> {} ;
x be Element of Y ;
let f be For oriented Chain , g be Function ;
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 | ( \cal G ) ;
- y in I ;
let A be non empty set , B be Subset of A ;
P0 = 1 ;
assume r in F . k ;
assume f is simple ;
let A be |^ |^ X ;
rng f c= NAT ;
assume P [ k ] ;
f9 <> {} ;
o be Ordinal ;
assume x is sum of squares ;
assume not v in { 1 } ;
let IF ;
assume that 1 <= j and j < l ;
v = - u ;
assume s . b > 0 ;
d1 in NAT ;
assume t . 1 in A ;
Y be non empty TopSpace , A be Subset 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 connected hhhhhhhh;
assume f is (#) marrrrr-r\lbrack ;
let x , y be element ;
let T be non empty TopSpace ;
b , a // b , c ;
k in dom Sum p ;
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 & j1 <= j2 ;
f | A is continuous ;
f . x real <= b ;
assume y in dom h ;
x * y in B1 ;
set X = Seg n ;
1 <= i2 + 1 ;
k + 0 <= k + 1 ;
p ^ q = p ;
j |^ y divides m ;
set m = max A ;
[ x , x ] in R ;
assume x in succ 0 ;
a in sup phi ;
CH in X ;
q2 c= C1 & q2 c= C1 ;
a2 < c2 & c2 < c2 ;
s2 is 0 -started ;
IC s = 0 ;
s4 = s4 & s4 = s4 ;
let V ;
let x , y be element ;
x be Element of T ;
assume a in rng F ;
x in dom T ` ;
let S be MSAlgebra over L ;
y " <> 0 ;
y " <> 0 ;
0. V = uw -] ;
y2 , y , z is_collinear ;
R8 ;
let a , b be Real , r be Real ;
let a be Object of C ;
let x be Vertex of G ;
let o be Object of C , a , b be Object of C ;
r '&' q = P \lbrack l \rbrack ;
let i , j be Nat ;
s be State of A , a , b be element ;
s4 . n = N ;
set y = ( x - 1 ) / 2 ;
NAT in dom g ;
l . 2 = y1 ;
|. g . y .| <= r ;
f . x in C0 ;
V1 is non empty ;
let x be Element of X ;
0 <> f . g2 ;
f2 /* q is convergent ;
f . i is_measurable_on E ;
assume \xi in NH ;
reconsider i = i as Ordinal ;
r * v = 0. X ;
rng f c= INT ;
G = 0 .--> goto 0 ;
let A be Subset of X ;
assume A9 is dense & open is dense ;
|. f . x .| <= r ;
x be Element of R ;
let b be Element of L ;
assume x in W-19 ;
P [ k , a ] ;
let X be Subset of L ;
let b be Object of B ;
let A , B be category ;
set X = Vars C ;
let o be OperSymbol of S ;
let R be connected non empty Poset ;
n + 1 = succ n ;
x9 c= Z1 & y9 c= Z1 ;
dom f = [: C1 , C2 :] ;
assume [ a , y ] in X ;
Re ( seq . n ) is convergent ;
assume a1 = b1 & a2 = b2 ;
A = ( sInt A ) ` ;
a <= b or b <= a ;
n + 1 in dom f ;
let F be instruction of S , T ;
assume r2 > x0 & x0 < r2 ;
Y be non empty set , M be Matrix of Y ;
2 * x in dom W ;
m in dom g2 & n + 1 in dom g2 ;
n in dom g1 ;
k + 1 in dom f ;
not the still not bound in { s } ;
assume x1 <> x2 & x2 <> x3 ;
v1 in V1 & v2 in V1 ;
not [ b `1 , b `2 ] in T ;
i9 + 1 = i ;
T c= \rangle ( T ) ;
( l - 1 ) * ( l - 1 ) = 0 ;
n be Nat ;
( t `2 ) ^2 = r ;
Aj is_integrable_on M & f is_integrable_on M ;
set t = Bottom t ;
let A , B be real-membered set ;
k <= len G + 1 ;
[: V , C :] misses [: V , C :] ;
Product seq is non empty ;
e <= f or f <= e ;
cluster -> non empty for normal sequence ;
assume c2 = b2 & c2 = b3 ;
assume h in [. q , p .] ;
1 + 1 <= len C ;
not c in B . m1 ;
cluster R .: X -> empty ;
p . n = H . n ;
assume v9 is Cauchy & vseq is convergent ;
IC Comput ( P3 , s3 , 0 ) = 0 ;
k in N or k in K ;
F1 \/ F2 c= F ;
Int G1 <> {} & Int G2 <> {} ;
( z `2 ) ^2 = 0 ;
p10 <> p1 & p2 <> p1 & p1 <> p2 ;
assume z in { y , w } ;
MaxADSet ( a ) c= F ;
ex_sup_of downarrow s , S ;
f . x <= f . y ;
let T be up-complete up-complete non empty 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 , full ;
A \/ { a } \not c= B ;
0. V = 0. V .= 0. V ;
let I be the \cal SCM , S be Instruction of S ;
f-24 . x = 1 ;
assume z \ x = 0. X ;
C4 = 2 |^ n ;
let B be sequence of Sigma ;
assume X1 = p .: D ;
n + l2 in NAT ;
f " P is compact ;
assume x1 in REAL & x2 in REAL ;
p1 = K0 & p2 = K1 ;
M . k = <*> REAL ;
phi . 0 in rng phi ;
MMMorder A is closed ;
assume z0 <> 0. L ;
n < N7 . k ;
0 <= seq . 0 ;
- q + p = v ;
{ v } is Subset of B ;
set g = f /. 1 ;
[: R , R :] is stable implies R is stable
set R = Vertices R , S = Vertices R ;
p0 c= P3 & p4 c= P2 ;
x in [. 0 , 1 .] ;
f . y in dom F ;
let T be Scott Scott Scott Scott Scott Scott of S ;
ex_inf_of the carrier of S , S ;
sup downarrow a = sup downarrow b ;
P , C , K is_collinear ;
assume x in F ( s , r , t ) ;
2 |^ i < 2 |^ m ;
x + z = x + z + q ;
x \ ( a \ x ) = x ;
||. \mathopen { \Vert } yielding .|| <= r ;
assume that Y c= field Q and Y <> {} ;
a ~ , b ~ are_isomorphic ;
assume a in A ( ) ;
k in dom ( q | 4 ) ;
p is card $ S ;
i -' 1 = i-1 ;
f | A is one-to-one ;
assume x in f .: [: X , Y :] ;
i2 - i1 = 0 & i1 - i1 = 0 ;
j2 + 1 <= i2 & j2 + 1 <= j2 ;
g " * a in N ;
K <> { [ {} , {} ] } ;
cluster strict \mathbb 1 -> strict for 2 -element ;
|. q .| ^2 > 0 ;
|. p4 .| = |. p .| ;
s2 - s1 > 0 ;
assume x in { Gij } ;
W-min C in C & W-min C in C ;
assume x in { Gij } ;
assume i + 1 = len G ;
assume i + 1 = len G ;
dom I = Seg n & rng I c= Seg n ;
assume that k in dom C and k <> i ;
1 + 1-1 <= i + 1-1 ;
dom S = dom F ;
let s be Element of NAT ;
let R be ManySortedSet of A ;
let n be Element of NAT ;
let S be non empty non void holds cluster non void holds S is with_b1 -\HM } -NAT ;
let f be ManySortedSet of I ;
let z be Element of COMPLEX , x , y be Element of COMPLEX ;
u in { \hbox { \boldmath $ g } } ;
2 * n < 2 * n ;
x , y in X ;
B-11 c= V1 & V c= V1 ;
assume I is_halting_on s , P ;
U2 = U2 | U2 .= U2 | U2 ;
M /. 1 = z /. 1 ;
x9 = x9 & y9 = y9 & x9 = y9 ;
i + 1 < n + 1 + 1 ;
x in { {} , <* 0 *> } ;
f9 <= ( f . x ) `1 & f . x <= 1 ;
l be Element of L ;
x in dom ( F . -17 ) ;
let i be Element of NAT ;
r8 is ( len r ) -element ;
assume <* o2 , o *> <> {} ;
s . x |^ 0 = 1 ;
card K1 in M & card K1 in M ;
assume that X in U and Y in U ;
let D be \rangle in \in \in \in Omega ;
set r = { q + 1 } ;
y = W . ( 2 * x9 ) ;
assume dom g = cod f & cod g = cod f ;
let X , Y be non empty TopSpace , A be Subset of X ;
x \oplus A is interval ;
|. <*> A .| . a = 0 ;
cluster strict for Sublattice of L ;
a1 in B . s1 & a2 in B . s1 ;
let V be finite VectSp of F , F be Function of V , W ;
A * B on B , A ;
f-3 = NAT --> 0 .= 0 ;
A , B be Subset of V ;
z1 = P1 . j & z2 = P2 . 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 ;
let B be non-empty ManySortedSet of I , A be ManySortedSet of I ;
sqrt 2 / 2 < Arg z ;
reconsider z9 = 0 as Nat ;
LIN a , d , c ;
[ y , x ] in II ;
( Q Q ) * ( 3 , 1 ) = 0 ;
set j = x0 div m , n = 0 ;
assume a in { x , y , c } ;
j2 - ( j - 1 ) > 0 ;
I \! phi = 1 ;
[ y , d ] in F-8 ;
let f be Function of X , Y ;
set A2 = ( sqrt B ) / 2 ;
s1 , s2 are_` , T & s1 , s2 are_` ;
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 & n mod 2 = 1 ;
set g = f | D-21 ;
assume that X is lower bounded and 0 <= r ;
( 1 - ( ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ) ^2 = 1 ;
a < ( p3 `1 ) ^2 + ( p4 `2 ) ^2 ;
L \ { m } c= UBD C ;
x in Ball ( x , 10 ) ;
not a in LSeg ( c , m ) ;
1 <= i1 -' 1 + 1 ;
1 <= i1 -' 1 + 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 . -3 ;
card b * a = |. a .| ;
Shift ( w , 0 ) |= v ;
set h = h2 ** h1 ;
assume x in ( 3 /\ 4 ) /\ ( X /\ 4 ) ;
||. h .|| < d1 & ||. h .|| < d ;
not x in the carrier of f & not x in { v } ;
f . y = F ( y ) ;
for n holds X [ n ] ;
k -' l = kl2 ;
<* p , q *> /. 2 = q ;
let S be Subset of the lattice of T ;
P , Q be EqRel 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 reflexive transitive RelStr , F be Function of L , L ;
S-20 is x -min min i , min K -min i
let r be non positive Real ;
M , v |= x \hbox { y } } ;
v + w = 0. ( Z ) ;
P [ len F ] ;
assume InsCode ( i ) = 8 & InsCode ( i ) = 8 ;
the zero of M = 0 & the carrier of M = 0 ;
cluster z (#) seq -> summable ;
let O be Subset of the carrier of C ;
||. f .|| | X is continuous ;
x2 = g . ( j + 1 ) ;
cluster -> non \vert for Element of S ;
reconsider l1 = l-1 as Nat ;
v9 is Vertex of r2 & v2 is Vertex of r2 ;
T3 is SubSpace of T2 | D ;
Q1 /\ Q1 <> {} ( T | A ) ;
k be Nat ;
q " is Element of X ;
F . t is set of S -by M ;
assume n <> 0 & n <> 1 ;
set e = EmptyBag n , f = EmptyBag n ;
let b be Element of Bags n ;
assume for i holds b . i is commutative ;
x is root implies ( p . x ) `2 = ( p . x ) `2
not r in ]. p , q .[ ;
let R be FinSequence of REAL , r be Real ;
S7 does not destroy b1 , b1 ;
IC SCM R <> a & IC SCM R <> a ;
|. - |[ x , y ]| .| >= r ;
1 * ( seq . n ) = seq . n * ( seq . n ) ;
x be FinSequence of NAT ;
f be Function of C , D , a , b be Element of C ;
for a holds 0. L + a = a
IC s = s . NAT .= s . NAT ;
H + G = F-GG ;
C1 . x = x2 & C1 . x = x2 ;
f1 = f . x .= f2 . x .= f2 . x ;
Sum <* p . 0 *> = p . 0 ;
assume v + W = v + u + W ;
{ a1 } = { a2 } ;
a1 , b1 _|_ b , a ;
b3 , o _|_ o , a3 ;
IX is reflexive & IX is reflexive implies X is reflexive
Iy is antisymmetric implies [: C , C :] is antisymmetric
sup ( rng H1 ) = e & sup ( rng H2 ) = e ;
x = ( a * a9 ) * ( a * b ) ;
|. p1 .| ^2 >= 1 ^2 ;
assume j2 -' 1 < j2 -' 1 ;
rng s c= dom f1 & rng s c= dom f2 ;
assume support a misses support b & not b in support b ;
let L be associative non empty doubleLoopStr , F be non empty doubleLoopStr ;
s " + 0 < n + 1 ;
p . c = f9 . 1 .= f . c ;
R . n <= R . ( n + 1 ) ;
Directed ( I1 ) = I1 . 0 .= 1 ;
set f = + ( x , y , r ) ;
cluster Ball ( x , r ) -> bounded ;
consider r being Real such that r in A ;
cluster non empty for NAT -defined Function ;
let X be non empty directed Subset of S ;
let S be non empty full full full SubRelStr of L ;
cluster <* I1 . N , \rbrack .] -> complete for non trivial Real ;
sqrt ( 1 - a " ) = a ;
( q . {} ) `1 = o ;
n - ( i -' 1 ) > 0 ;
assume sqrt ( 1 - 2 ) <= t `1 ;
card B = k + 1 - 1 ;
x in union rng ( f | -9 ) ;
assume x in the carrier of R & y in the carrier of R ;
d in X ;
f . 1 = L . F . 1 ;
the vertices of G = { v } ;
let G be : *> is : Let ;
e , v9 , v , w , y is_collinear ;
c . ( i9 - 1 ) in rng c ;
f2 /* q is divergent_to+infty & f2 /* ( f1 /* s ) is divergent_to+infty ;
set z1 = - z2 , z2 = - z2 , z2 = - z2 , z2 = - z1 , z2 = - z2 ;
assume w is llllas of S , G ;
set f = p \! \mathop { t } , g = p \! \mathop { t } ;
c be Object of C ;
assume ex a st P [ a ] ;
let x be Element of REAL-NS m ;
let IX be Subset-Family of X , Y be Subset of X ;
reconsider p = p as Element of NAT ;
v , w be Point of X ;
let s be State of SCM+FSA , I be Program of SCM+FSA ;
p is FinSequence of SCM & q is FinSequence of the carrier of SCM ;
stop I c= P3 +* stop I ;
set ci = fT2 /. i , cj = f /. i ;
w ^ t seq seq seq seq ^ s ;
W1 /\ W = W1 /\ W /\ W ` ;
f . j is Element of J . j ;
let x , y be Element of T2 , T be Element of T2 ;
ex d st a , b // b , d ;
a <> 0 & b <> 0 & c <> 0 ;
ord x = 1 & x is \sum x ;
set g2 = lim ( seq ^\ k ) , g1 = lim ( seq ^\ k ) ;
2 * x >= 2 * sqrt ( 1 + ( 2 * x ) ^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 = 0 & L1 . F1 = 1 ;
( the InternalRel of X ) \/ R1 = the InternalRel of X ;
( sin * sin ) . x <> 0 & ( sin * cos ) . x <> 0 ;
( ( - exp ) * ( exp_R ) ) . x > 0 ;
o1 in ( X /\ O2 ) /\ O2 ;
e , v9 , v , w , y is_collinear ;
r3 > ( 1 - 2 ) * 0 ;
x in P .: ( F -ideal ) ;
J be closed Ideal of R , f be left ideal of R ;
h . p1 = f2 . O & h . O = g2 . I ;
Index ( p , f ) + 1 <= j ;
len ( q - p ) = width M & width ( q - p ) = width M ;
the carrier of K c= A & K c= A ;
dom f c= union rng ( F | X ) ;
k + 1 in support ( ( support ( n ) ) ) ;
let X be ManySortedSet of the carrier of S ;
[ x `1 , y `2 ] in ( ( InnerVertices R ) * ( the InternalRel of R ) ) ;
i = D1 or i = D2 or i = D1 ;
assume a mod n = b mod n ;
h . x2 = g . x1 & h . x2 = f . x2 ;
F c= 2 -tuples_on the carrier of X & F is one-to-one ;
reconsider w = |. s1 .| as Real_Sequence ;
sqrt ( 1 - m * r + r ) < p ;
dom f = dom ( I --> -4 ) .= I ;
[#] ( P-17 | K1 ) = [#] ( ( TOP-REAL 2 ) | K1 ) ;
cluster - x -> ExtReal equals - x ;
then { d } c= A ;
cluster [: TOP-REAL n , TOP-REAL n :] -> finite-ind ;
let w1 be Element of M ;
x be Element of dyadic ( n ) ;
u in W1 & v in W2 & u in W2 ;
reconsider y = y as Element of L2 ;
N is full full full SubRelStr of T |^ the carrier of S ;
sup { x , y } = c "\/" c ;
g . n = n |^ 1 .= n ;
h . J = EqClass ( u , J ) ;
let seq be \Vert for x be Point of X ;
dist ( x `1 , y `2 ) < sqrt ( r ^2 ) ;
reconsider mm = m as Element of NAT ;
x- x0 < r1 - x0 & r1 < x0 + x0 ;
reconsider P ` = P ` as strict Subgroup of N ;
set g1 = p * ( idseq q " ) ;
let n , m , k be non zero Nat ;
assume that 0 < e and f | A is lower bounded ;
D2 . ( I . 8 ) in { x } ;
cluster -> subcondensed for Subset of T ;
P be compact non empty Subset of TOP-REAL 2 , a , b , c , d be Real ;
Gik in LSeg ( cos , 1 ) /\ LSeg ( cos , 1 ) ;
n be Element of NAT , x be Element of NAT ;
reconsider S8 = S as Subset of T | A ;
dom ( i .--> X ) = { i } ;
let X be non-empty ManySortedSet of S ;
let X be non-empty ManySortedSet of S ;
op ( { {} } , {} ) c= { [ {} , {} ] }
reconsider m = .[ as Element of NAT ;
reconsider d = x as Element of [: the carrier of C , the carrier of C :] ;
let s be 0 -started State of SCMPDS , P be Initialize s of SCMPDS ;
let t be 0 -started State of SCMPDS , Q ;
b , b , x , y , x , y is_collinear ;
assume that i = n \/ { n } and j = k \/ { k } ;
let f be PartFunc of X , Y ;
N2 >= sqrt ( c ^2 - sqrt ( c ^2 - d ^2 ) ) ;
reconsider t7 = T" as Point of TOP-REAL 2 ;
set q = h * p ^ <* d *> ;
z2 in U . 4 /\ Q2 . 6 /\ Q1 . 6 /\ Q2 . 6 ;
A |^ 0 = { <* \rangle *> } ;
len W2 = len W + 2 & len W1 = len W + 2 ;
len h2 in dom h2 & len h2 in dom h2 & len h2 = len h2 ;
i + 1 in Seg ( len s2 ) ;
z in dom g1 /\ dom f & z in dom f /\ dom g ;
assume p2 = E-max ( K ) & p4 = E-max ( K ) ;
len G + 1 <= i1 + 1 ;
f1 (#) f2 is convergent & f2 (#) f1 is convergent & lim ( f1 (#) f2 ) = x0 ;
cluster seq + seq - seq -> summable for Real_Sequence ;
assume j in dom M1 & i in dom M1 ;
let A , B , C be Subset of X ;
x , y , z be Point of X , a , b , c be Real ;
b ^2 - ( 4 * a * c ) >= 0 ;
<* xy *> ^ <* y *> \subseteq x ;
a , b in { a , b } ;
len p2 is Element of NAT & len p1 = len p2 & len p1 = len p2 ;
ex x being element st x in dom R ;
len q = len ( K (#) G ) ;
s1 = Initialize Initialized s , P1 = Initialize Initialized s , P2 = P +* stop I ;
consider w be Nat such that q = z + w ;
x ` ` is ` iff x ` is ` & x ` is ` ;
k = 0 & n <> k or k > n ;
then X is discrete for A being Subset of X ;
for x st x in L holds x is FinSequence of REAL
||. f /. c .|| <= r1 & ||. f /. c .|| <= r1 ;
c in ]. p , q .[ & not c in { p } ;
reconsider V = V as Subset of the carrier of _ { n } ;
N , M be 1 -element implies N , M are_isomorphic
then z >= compactbelow x & z >= compactbelow y ;
M [. f , f .] = f & M [. g , g .] = g ;
( ( ( to_power 1 ) 1 ) to_power 1 ) /. 1 = TRUE ;
dom g = dom f .: X ;
mode \cal If is Int st G is \cal holds W is \cal
[ i , j ] in Indices M & [ i , j ] in Indices M ;
reconsider s = x " as Element of H ;
let f be Element of dom Subformulae p & f . x = 0 ;
F1 . ( a1 , - a2 ) = G1 . ( a1 , a2 ) ;
cluster rectangle ( a , b , r ) -> 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 ) , s3 = card ( dom B ) ;
set G = coprod X ;
reconsider a = [ x , s ] as Object of G ;
let a , b be Element of Mx , M be Matrix of n , K ;
reconsider s1 = s as Element of S1 . s ;
rng p c= the carrier of L & p . 1 c= the carrier of L ;
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 ;
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 & rng f c= the carrier of S ;
rng h c= union ( the carrier of J ) & rng h c= the carrier of J ;
cluster All ( x , H ) -> One -reconsider ;
d * N1 ^2 > N1 * 1 / 1 ;
]. a , b .[ c= [. a , b .] ;
set g = f " D1 | D1 , f = f " D2 ;
dom ( p | ( m + 1 ) ) = NAT ;
3 + - 2 <= k + - 2 ;
tan is_differentiable_in ( arctan * arccot ) . x ;
x in rng ( f /^ ( p -' 1 ) ) ;
f , g be FinSequence of D ;
[: p , q :] in the carrier of S1 & [: p , q :] in the carrier of S2 ;
rng f " { 0 } = dom f /\ dom g " { 0 } ;
( the Target of G ) . e = v & ( the Target of G ) . e = v ;
width G -' 1 < width G - 1 & width G - 1 < width G ;
assume v in rng ( S | E1 ) ;
assume x is root or x is root or x is root ;
assume 0 in rng ( ( g2 | A ) | A ) ;
let q be Point of ( TOP-REAL 2 ) | K1 , r be Real ;
let p be Point of ( TOP-REAL 2 ) | K1 , r be Real ;
dist ( O , u ) <= |. p2 .| + 1 ;
assume dist ( x , b ) < dist ( a , b ) ;
<* S *> is in the carrier of C-20 & <* S *> is in the carrier of C-20 ;
i <= len ( G | -6 -' 1 ) + 1 ;
let p be Point of ( TOP-REAL 2 ) | K1 , r be Real ;
x1 in the carrier of I[01] & x2 in the carrier of I[01] & x3 in the carrier of I[01] ;
set p1 = f /. i , p2 = f /. ( i + 1 ) ;
g in { g2 : r < g2 & g2 < r } ;
Q2 = Sp2 " ( Q " ) .= Q " ( Q " ) ;
( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 )
- p + I c= - p + A + A ;
n < LifeSpan ( P1 , s1 ) + 1 & n < LifeSpan ( P2 , s2 ) ;
CurInstr ( p1 , s1 ) = i .= ( m + 1 ) ;
A /\ Cl { x } \ { x } <> {} ;
rng f c= ]. r , r + 1 .[ ;
g be Function of S , V ;
f be Function of L1 , L2 , g be Function of L1 , L2 ;
reconsider z = z as Element of ( Ids L ) . s ;
f be Function of S , T ;
reconsider g = g as Morphism of c opp , b opp ;
[ s , I ] in S ~ ;
len ( the connectives of C ) = 4 & len ( the connectives of C ) = 5 ;
let C1 , C2 be Sub\/ C2 , F be subfunctor of C ;
reconsider V1 = V as Subset of X | B ;
attr p is valid means : Def1 : All ( x , p ) is valid ;
assume that X c= dom f and f .: X c= dom g and f .: X c= dom g ;
H |^ a " is Subgroup of H |^ a ;
let A1 be Let A2 of O , E1 be Element of O ;
p2 , r3 , r3 is_collinear & q2 , q2 , q3 is_collinear ;
consider x being element such that x in v ^ K ;
not x in { 0. TOP-REAL 2 } & not x in { 0. TOP-REAL 2 } ;
p in [#] ( ( TOP-REAL 2 ) | B ) & p in [#] ( ( TOP-REAL 2 ) | B ) ;
0 . E < M . ( E8 ) ;
( c / c ) / c = c ;
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . s2 ) & a2 in dom ( F . s2 ) ;
cluster -> \uparrow -| distributive -> | for distributive LATTICE ;
set i1 = the natural number , i2 = the Element of NAT , n = the Element of NAT ;
let s be 0 -started State of SCM+FSA , I be Program of SCM+FSA ;
assume y in ( f1 \/ f2 ) .: A ;
f . len f = f /. len f .= f /. len f ;
x , f . x '||' f . x , f . y ;
attr X c= Y means : Def1 : cos .: X c= cos .: Y ;
y be upper bound of Y , T , x be element ;
cluster ( x `1 ) / 2 -> non number equals ( x `1 ) / 2 ;
set S = <* Bags n , i9 *> , R = <* i *> ;
set T = [. 0 , 1 / 2 .] , G = [. 1 / 2 , 1 .] ;
1 in dom mid ( f , 1 , 1 ) ;
sqrt ( 4 * PI ) < sqrt ( 2 * PI ) ;
x2 in dom f1 /\ dom f2 & x1 in dom f1 /\ dom f2 ;
O c= dom I & { {} } = { {} } ;
( the Target of G ) . x = v & ( the Target of G ) . x = v ;
{ HT ( f , T ) } c= Support f ;
reconsider h = R . k as Polynomial of n , L ;
ex b being Element of G st y = b * H ;
let x , y , z be Element of G opp ;
h19 . i = f . h . i ;
( p `1 ) ^2 = ( p1 `1 ) ^2 & ( p `2 ) ^2 = ( p2 `1 ) ^2 ;
i + 1 <= len Cage ( C , n ) ;
len <* P *> = len P & len <* P *> = len P ;
set NN = the verify that the subsets of N = the InternalRel of N and the InternalRel of N = the InternalRel of N ;
len g-f - ( x + 1 ) - 1 <= x ;
a on B & b on B & a on B implies a on B
reconsider r-12 = r * I . v as FinSequence of REAL ;
consider d such that x = d and a [= d and a [= c ;
given u such that u in W and x = v + u ;
len f /. len f = len f - n ;
set q2 = E-max L~ Cage ( C , n ) , q2 = E-max L~ Cage ( C , n ) ;
set S = { S1 , S2 } , T = { S2 } , F = { S2 } , G = { S2 } , H = { S2 } , S = { S2 } , T = { S2 } , T = { S2 } ,
MaxADSet ( b ) c= MaxADSet ( P /\ Q ) ;
Cl ( G . q1 ) c= F . r2 ;
f " D meets h " V & h " V meets h " V ;
reconsider D = E as non empty directed Subset of L1 ;
H = the_left_argument_of H '&' ( the_right_argument_of H ) ;
assume t is Element of ( \mathfrak F ) . X ;
rng f c= the carrier of S2 & rng f c= the carrier of S2 ;
consider y being Element of X such that x = { y } ;
f1 . ( a1 , b1 ) = b1 . ( a1 , b1 ) ;
the carrier' of G ` = E \/ { E } ;
reconsider m = len \mathbb k - 1 as Element of NAT ;
set S1 = LSeg ( n , UMP C ) , S2 = LSeg ( n , UMP C ) ;
[ i , j ] in Indices M1 & [ i , j ] in Indices M1 ;
assume P c= Seg m & M is \HM { \vert } is ' ;
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 ) ;
p0 . i = p1 . i & p2 . i = p2 . i ;
let PA , PA be a_partition of Y , G be a_partition of Y ;
attr 0 < r & 1 < 1 & r < 1 implies 1 / 2 < r ;
rng \lbrace \lbrace a , X , Y \rbrace = [#] ( X , Y ) ;
reconsider x = x , y = y as Element of K ;
consider k such that z = f . k and n <= k and k <= n ;
consider x being element such that x in X \ { p } ;
len ( canFS ( s ) ) = card s .= card s ;
reconsider x2 = x1 , y2 = x2 as Element of L2 ( ) ;
Q in FinMeetCl ( ( the topology of X ) | the topology of X ) ;
dom ( f | 0 ) c= dom u & rng ( u | 0 ) c= dom u ;
pred n divides m & m divides n & n divides m ;
reconsider x = x as Point of [: I[01] , I[01] :] | K1 ;
a in // // // // the carrier of T2 , the carrier of T2 ;
not y0 in the still of f & not ( not y in dom ( f . x ) ) ;
Hom ( ( a , b ) \times c , c ) <> {} ;
consider k1 such that p " < k1 and p " < k1 and p " < p ;
consider c , d such that dom f = c \ d ;
[ x , y ] in dom g & k in dom g ;
set S1 =
l1 = m2 & l1 = m2 & l1 = m2 & l2 = m2 & l2 = y2 ;
x0 in dom u /\ ( ( u + v ) (#) ( u + v ) ) ;
reconsider p = x as Point of ( TOP-REAL 2 ) | K1 ;
I[01] = ( I[01] | B ) | B01 .= ( I[01] | B ) | B01 ;
f . p4 <= f . p1 & f . p2 <= f . p2 ;
( F . x ) `1 <= ( x `1 ) / 2 & x `2 <= ( x `2 ) / 2 ;
( x `2 ) ^2 = ( ( W . n ) `1 ) ^2 .= ( W . n ) `1 ;
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 < a ;
X . i in 2 -tuples_on A . i \ B . i ;
<* 0 *> in dom ( e --> [ 1 , 0 ] ) ;
then P [ a ] implies P [ succ a ] ;
reconsider s\rbrace = seq . t as terminal of D , X ;
( Seg i -' 1 ) <= len \mathbb j - 1 ;
[#] S c= [#] T & the TopStruct of T c= the TopStruct of T ;
for V being strict real number holds V in W1 iff V in W2
assume k in dom mid ( f , i , j ) ;
let P be non empty Subset of TOP-REAL 2 , p , q be Point of TOP-REAL 2 ;
let A , B be Matrix of n1 , K , n , m be Nat ;
- a * b - b * a = a * b ;
for A being being_line holds A // K implies A // K
( id o2 ) in <* o2 , o2 , o1 *> & ( id o2 ) in dom ( the InternalRel of o1 ) ;
then ||. x .|| = 0 & x = 0. X ;
let N1 , N2 be strict normal Subgroup of G , a , b be Element of G ;
j >= len ( upper_volume ( g , D1 ) | j1 ) ;
b = Q . ( len Q - 1 ) + 1 ;
f2 * f1 /* s is divergent_to+infty & f2 * f1 is divergent_to+infty ;
reconsider h = f * g as Function of [: N2 , G :] , G ;
assume that a <> 0 and integral ( a , b , c ) >= 0 ;
[ t , t ] in the InternalRel of A & [ t , t ] in the InternalRel of A ;
( v |-- E ) | n is Element of T7 & v | n is Element of T7 ;
{} = the carrier of L1 + L2 & L2 + L2 = the carrier of L2 ;
Directed I is_halting_on Initialized s , P & Directed I is_halting_on Initialized s , P ;
Initialized ( p ) = Initialize ( ( p +* q ) +* ( i , k ) ) ;
reconsider N2 = N1 as strict net of R2 , the carrier of S2 ;
reconsider Y = Y as Element of <* Ids L , \subseteq \rangle ;
"/\" ( { p } \ { p } , L ) <> p ;
consider j be Nat such that i2 = i1 + j and j < len f ;
not [ s , 0 ] in the carrier of S2 & [ s , 0 ] in the carrier of S2 ;
m in ( B '&' C ) '/\' D /\ D ;
n <= len ( P6 + len ( P6 ) ) + len ( P6 + 1 ) ;
( x1 - x2 ) / ( x2 - x1 ) = ( x2 - x1 ) / ( x2 - x1 ) ;
InputVertices S = { x1 , x2 } & InputVertices S = { x1 , x2 } ;
let x , y be Element of FTTTTTTTTTT1 ( n ) ;
p = |[ p `1 , p `2 ]| & p = |[ p `1 , p `2 ]| ;
g * 1_ G = h " * g * h " .= g " * h " ;
let p , q be Element of V , C be Element of V ;
x0 in dom ( x1 - x2 ) /\ dom ( x2 - x3 ) ;
( R qua Function ) " = R " " ( R " ) ;
n in Seg len ( f /^ ( len p -' 1 ) ) ;
for s be Real st s in R holds s <= s2 & s <= 1
rng s c= dom ( f2 * f1 ) & rng s c= dom ( f2 * f1 ) ;
synonym ex X being Subset of ex R being Subset of \rm st X = R & R is \hbox { - } \sum R } ;
1_ K * 1_ K = 1_ K * 1_ K .= 1_ K * 1_ K ;
set S = Segm ( A , P1 , Q1 ) , Q1 = Segm ( A , Q1 ) ;
ex w st e = ( w - f ) * w & w in F ;
( curry ( PZ , k ) # x ) # x is convergent ;
cluster -> open for Subset of T7 | P ;
len f1 = 1 .= len f3 .= len f3 + 1 .= len f3 + 1 .= len f3 + 1 ;
sqrt ( i * p ) < sqrt ( 2 * p ) ;
let x , y be Element of ( the Sorts of U0 ) . 0 ;
b1 , c1 // b9 , c9 & b1 , c1 // b9 , c9 ;
consider p being element such that c1 . j = { p } and p in F ;
assume that f " { 0 } = {} and f is total ;
assume IC Comput ( F , s , k ) = n ;
Reloc ( J , card I ) not a := s , a ;
Macro ( card I + 1 ) not c does not destroy c , b ;
set m1 = LifeSpan ( p3 , s3 ) , m2 = LifeSpan ( P3 , s3 ) , P4 = Comput ( P3 , s3 , 1 ) , P4 = P3 ;
IC SCMPDS in dom Initialize ( ( intloc 0 ) .--> 1 ) ;
dom t = the carrier of SCM R & dom t = the carrier of SCM R ;
( E-max L~ f ) .. f = 1 & ( E-max L~ f ) .. f = 1 ;
let a , b be Element of V , C be Element of V ;
Cl ( union Int F ) c= Cl Int union F ;
the carrier of X1 union X2 misses ( A1 \/ A2 ) ;
assume not LIN a , f . a , g . a ;
consider i being Element of M such that i = d6 and i in A ;
then Y c= { x } or Y = {} or Y = { x } ;
M , v / ( y , x ) / ( x , y ) |= H ;
consider m be element such that m in Intersect ( F ) and m in dom F ;
reconsider A1 = support u1 as Subset of X | ( support b1 ) ;
card ( A \/ B ) = k-1 + ( 2 * 1 ) ;
assume a1 <> a3 & a2 <> a4 & a3 <> a4 & a4 <> a4 ;
cluster s \! \mathop { V } -> 0 -element for string of S ;
L2 /. ( n2 + 1 ) = L2 . ( n2 + 1 ) ;
let P be compact non empty Subset of TOP-REAL 2 , a , b , c be Real ;
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 , b , c be Real ;
assume [ k , m ] in Indices ( D1 ^ D2 ) ;
0 <= ( ( 1 / 2 ) |^ p ) / 2 ;
( F . N | E8 ) . x = +infty ;
attr X c= Y & Z c= V & X \ V c= Y \ Z ;
( y * z ) * ( z * ( z * x ) ) <> 0. I ;
1 + card ( X \ u ) <= card u + card v - u ;
set g = z \circlearrowleft ( E-max L~ z ) , h = z .. z ;
then k = 1 implies p . k = <* x , y *> . k ;
cluster -> total for Element of C -\to ( X , Y ) ;
reconsider B = A as non empty Subset of TOP-REAL n , a , b 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 & P c= Q ;
n <= indx ( D2 , D1 , j1 ) + 1 & indx ( D2 , D1 , j1 ) + 1 <= len D2 ;
( ( g2 ) . O ) `1 = - 1 & ( ( g2 ) . I ) `1 = - 1 ;
j + p .. f -' len f <= len f - 1 ;
set W = W-min ( C ) , S = E-max ( C ) ;
S1 . ( a , e ) = a + e .= a + e ;
1 in Seg width ( M * ( ColVec2Mx p ) ) ;
dom ( i (#) Im ( f ) ) = dom ( Im f ) ;
being being being W . x `1 = W . ( a *' ( a , p ) ) ;
set Q = ( ( empty ( g , f , h ) ) . x ;
cluster -> binary for ManySortedSet of U1 means ex U1 being ManySortedSet of U1 st U1 is MS\rm ] ;
attr F = { A } means : Def1 : F is discrete ;
reconsider z9 = *> as Element of product G . ( i + 1 ) ;
rng f c= rng f1 \/ rng f2 & rng ( f1 - f2 ) c= dom f1 \/ rng f2 ;
consider x such that x in f .: A and x in f .: C ;
f = <*> ( the carrier of F_Complex ) & f is ( the carrier of F_Complex ) --> 0. F_Complex ;
E , j |= All ( x1 , x2 ) & E , j |= H ;
reconsider n1 = n as Morphism of o1 , o2 , o2 be Morphism of o1 , o2 ;
assume that P is idempotent and R is idempotent and P ** R = R .: P ;
card ( B2 \/ { x } ) = k-1 + 1 ;
card ( x \ B1 ) /\ B1 = 0 ;
g + R in { s : g-r - r < s & s < g + r } ;
set q\it \it \it ' = ( q , <* s *> ) := ( s , <* s *> ) ;
for x being element st x in X holds x in rng f1 ;
h1 /. ( i + 1 ) = h1 . ( i + 1 ) ;
set mw = max ( B , ( min B ) ) , mw = min ( B , NAT ) ;
t in Seg width ( I ^ ( n , n ) ) & t in dom ( I ^ ( n , n ) ) ;
reconsider X = dom f \ C , Y = rng f as Element of Fin NAT ;
IncAddr ( i , k ) . x = ( 0. SCM+FSA ) . x + k .= i ;
( E-max L~ f ) .. f <= ( q `2 ) .. f & ( q `2 <= ( q `2 ) .. f ) .. f ;
attr R is condensed means : Def1 : Int R is condensed & Cl R is condensed ;
attr 0 <= a & b <= 1 & a * b <= 1 implies a * b <= 1 ;
u in ( ( c /\ ( d /\ b ) ) /\ e ) /\ f /\ j ;
u in ( ( c /\ ( d /\ e ) ) /\ b ) /\ f /\ j ;
len C + - 2 >= 9 + - 3 + 3 ;
x , z , y is_collinear & x , z , x is_collinear implies x , z , y is_collinear
a |^ n1 + 1 = a |^ n1 * a |^ n1 ;
<* \underbrace ( 0 , 0 , 0 ) , x *> in Line ( x , a ) ;
set y1 = <* y , c *> ;
F2 /. 1 in rng Line ( D , 1 ) & F /. 1 in rng Line ( D , 1 ) ;
p . m joins r /. m , r /. ( m + 1 ) , r /. ( m + 1 ) ;
( p `2 ) ^2 = ( f /. i1 ) `2 .= ( f /. i1 ) `2 ;
( W-min ( X \/ Y ) ) `1 = W-bound ( X \/ Y ) & ( W-min ( X \/ Y ) ) `1 = E-bound ( X \/ Y ) ;
0 + ( p `2 ) ^2 <= 2 * r + ( p `2 ) ^2 ;
x in dom g & not x in g " { 0 } ;
f1 /* ( seq ^\ k ) is divergent_to+infty & f2 /* ( seq ^\ k ) is divergent_to+infty ;
reconsider u2 = u as VECTOR of \mathop { \rm \hbox { - } \rm Real } ;
p \! \mathop { Product ( Sgm X ) } = 0 & p \! \mathop { PI ( Sgm X ) = 0 ;
len <* x *> < i + 1 & i + 1 <= len c + 1 ;
assume that I is non empty and { x } /\ { y } = { 0. I } ;
set i2 = card I + 4 .--> goto 0 , goto 0 = goto 0 ;
x in { x , y } & h . x = {} T & h . y = {} T ;
consider y being Element of F such that y in B and y <= x `1 ;
len S = len ( the charact of ( A ) ) & len ( the charact of ( A ) ) = len the charact of ( A ) ;
reconsider m = M , i = I , n = N as Element of X ;
A . ( j + 1 ) = B . ( j + 1 ) \/ A . j ;
set N8 = : \HM { G2 : G2 is open } c= the carrier of G ;
rng F c= the carrier of gr ( { a } , the carrier of G ) ;
( ( the means Q \ F ) . K , n , r ) is a finite sequence ;
f . k , f . ( mod n ) in rng f ;
h " P /\ [#] T1 = f " P /\ [#] T2 .= f " P ;
g in dom ( f2 \ f2 " { 0 } ) \ f2 " { 0 } ;
g\cal X /\ dom f1 = g1 " { g } .= g1 " { g } ;
consider n be element such that n in NAT and Z = G . n ;
set d1 = ]. b , x1 .[ , d2 = ]. b , x1 .[ , d2 = ]. b , x2 .[ , d2 = ]. b , x1 .[ , d2 = ]. b , x1 .[ ;
b `1 + sqrt ( 1 + ( b `1 / b ) ^2 ) < ( 1 + ( b `1 / b ) ^2 ) ;
reconsider f1 = f as VECTOR of the carrier of X , Y be bounded bounded bounded Function of X , Y ;
attr i <> 0 implies i ^2 mod ( i + 1 ) mod ( i + 1 ) = 1 ;
j2 in Seg len ( ( g2 ) . i2 ) & ( ( g2 ) . i2 ) `2 = ( ( ( g2 ) . i2 ) `2 ;
dom ( i - j ) = dom ( i - j ) .= a .= a ;
cluster sec | ]. PI / 2 , PI / 2 .[ -> one-to-one for Function of ]. PI / 2 , PI .[ , R^1 ;
Ball ( u , e ) = Ball ( f . p , e ) ;
reconsider x1 = x0 as Function of S , IF ( ) | IF ( ) | IF ( ) | IF ( ) | IF ( ) | IF ( ) | o ;
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 RL ;
S1 +* S2 = S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2
( ( - 1 ) (#) ( ( #Z 2 ) * ( ( #Z 2 ) * ( #Z 2 ) ) ) is_differentiable_on Z ;
cluster -> total for Function of C , REAL ;
set C7 = 1GateCircStr ( <* z , x *> , f3 ) , C7 = 1GateCircStr ( <* z , x *> , f3 ) ;
Esuch that E8 . e2 = E8 . ( e2 + 1 ) -T . ( e2 + 1 ) ;
( ( arctan * arctan ) `| Z ) . x = ( arctan * arctan ) . x .= ( arctan * arccot ) . x ;
sup A = ( cos * 3 ) / 2 & inf A = 0 ;
F . ( dom f , - F ) is_transformable_to F . ( cod f , - F . ( cod f , - F ) ) ;
reconsider p9 = q9 as Point of ( TOP-REAL 2 ) | K1 , ( TOP-REAL 2 ) | K1 = ( TOP-REAL 2 ) | K1 ;
g . W in [#] ( Y | 0 ) & [#] ( Y | 0 ) c= [#] ( Y | 0 ) ;
let C be compact non vertical non vertical non horizontal Subset of TOP-REAL 2 , n be Nat ;
LSeg ( f ^ g , j ) = LSeg ( f , j ) ;
rng s c= dom f /\ ]. - r , x0 .[ & for x st x in ]. - r , x0 .[ holds f . x < 0
assume x in { ( idseq 2 ) . ( ( idseq 2 ) . ( i + 1 ) ) } ;
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 .= m1 + m2 .= m1 + m2 ;
assume for n holds H1 . n = G . n -H . n ;
set Bf = f .: ( the carrier of X1 ) , Bf = f .: ( the carrier of X2 ) ;
ex d being Element of L st d in D & x << d ;
assume R ~ . a c= R ~ . b & R ~ . b c= R ~ . 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 ] ;
attr x1 <> x2 means x1 - x2 > 0 & |. x1 - x2 .| > 0 ;
assume p2 - p1 , p3 - p1 , p4 - p1 , p2 - p1 is_collinear ;
set q = ( \mathbb f ) ^ <* 'not' A *> ;
f be PartFunc of \langle REAL-NS 1 , \Vert * \Vert \rangle , REAL-NS 1 , REAL-NS 1 ;
( n mod 2 * k ) + 1 - k = n mod 2 ;
dom ( T * succ t ) = dom ( succ t ) .= dom <* t *> ;
consider x be element such that x in w and x in c and x in c ;
assume ( F * G ) . v = v . ( x3 . x3 ) ;
assume the Sorts of D1 c= the Sorts of D2 & the Sorts of D2 c= the Sorts of D2 ;
reconsider A1 = [. a , b .] as Subset of R^1 | [. a , b .] ;
consider y being element such that y in dom F and F . y = x ;
consider s being element such that s in dom o and a = o . s ;
set p = W-min L~ Cage ( C , n ) , q = W-min L~ Cage ( C , n ) ;
n1 -' len f + 1 - 1 <= len g + 1 - 1 ;
ConsecutiveSet ( q , O1 ) = [ u , v , a , b , a , b , b , c ] ;
set C-2 = ( ( `1 ) `1 ) . ( k + 1 ) , CG = ( ( `1 ) `1 ) . ( k + 1 ) ;
Sum ( L * p ) = 0. R * Sum p .= 0. V .= 0. V ;
consider i being element such that i in dom p and t = p . i ;
defpred Q [ Nat ] means 0 = Q ( ) & $1 < 1 implies P [ $1 ] ;
set s3 = Comput ( P1 , s1 , k ) , s4 = Comput ( P2 , s2 , k ) , P4 = Comput ( P2 , s2 , k ) , P4 = Comput ( P2 , s2 , k ) , P4 = Comput ( P2 , s2 , k ) , P4 = Comput ( P2 , s2
l be variable of k , A , A1 be Subset of A ;
reconsider U2 = union ( G . k ) as Subset-Family of T | A , T | A ;
consider r such that r > 0 and Ball ( p `1 , r ) c= Q ` ;
( h | ( n + 2 ) ) /. ( i + 1 ) = p2 ;
reconsider B = the carrier of X1 , C = the carrier of X2 as Subset of X ;
p0 = <* - c , 1 , 1 *> .= <* - c , 1 , 1 *> ;
synonym f is real-valued means : Def1 : rng f c= NAT & rng f c= NAT ;
consider b be element such that b in dom F and a = F . b ;
x9 < card ( ( X \ Y ) \/ card ( Y \ { x } ) ) + 1 & card ( ( X \ Y ) \/ { x } ) = 1 ;
attr X c= B1 means : Def1 : for B holds \mathop { \rm _ X } c= succ B ;
then w in Ball ( x , r ) & dist ( x , w ) <= r ;
angle ( x , y , z ) = angle ( \HM { the } \HM { function } , 0 , 0 ) ;
attr 1 <= len s means : Def1 : len ( s , 0 ) = 0 & for i being Element of NAT holds s . i = s . i ;
fY. c= f . ( k + ( n + 1 ) ) ;
the carrier of { 1_ G } = { 1_ G } & the carrier of G = { 1_ G } ;
pred p '&' q in TAUT ( A ) means q '&' p in TAUT ( A ) & q '&' p in TAUT ( A ) ;
- ( t `1 ) < ( - t `1 ) / 2 ;
U1 . 1 = U2 /. 1 .= U2 /. 1 .= U2 . 1 .= U2 . 1 .= U2 /. 1 ;
f .: ( the carrier of x ) = the carrier of x & f .: ( the carrier of x ) = the carrier of x ;
Indices ( O * ( i , j ) ) = [: Seg n , Seg n :] ;
for n being Element of NAT holds G . n c= G . ( n + 1 )
then V in M @ ;
ex f being Element of F-9 st f is_if and f is w.r.t. A9 & f . x = f . x ;
[ h . 0 , h . 3 ] in the InternalRel of G & [ 1 , 0 ] in the InternalRel of G ;
s +* Initialize ( ( intloc 0 ) .--> 1 ) . intloc 0 = s3 . intloc 0 .= s . intloc 0 ;
|[ w1 , v1 ]| `1 - |[ w1 , w2 ]| `2 <> 0. TOP-REAL 2 ;
reconsider t = t as Element of INT ( ) , X ( ) ;
C \/ P c= [#] ( ( G | ( [#] G \ A ) ) \ A ;
f " V in ( ( X \ V ) /\ D ) . ( \alpha , the carrier of X ) ;
x in [#] ( ( [#] \alpha ) /\ A ) /\ ( A ` ) ;
g . x <= h1 . x & h . x <= h1 . x ;
InputVertices S = { xy , yz , zx } & InputVertices ( y , z ) = { xy , yz , zx } ;
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 & M2 is being_line & M3 is being_line ;
reconsider a = f0 . ( i0 -' 1 ) as Element of K ;
len ( B ^ ( Len F1 ) ) = Sum ( ( Len F1 ) ^ ( Len F2 ) ) .= Sum ( ( Len F1 ) ^ ( Len F2 ) ) ;
len ( ( the R of n ) * ( i , j ) ) = n & len ( ( R * ( i , j ) ) = n ;
dom max ( f + g , f + g ) = dom ( f + g ) ;
( for n holds seq . n = sup Y1 ) implies for n holds seq . n = sup Y1
dom ( p1 ^ p2 ) = dom ( f ^ p2 ) .= dom ( f ^ p2 ) ;
M . [ 1 , y ] = 1 * v1 .= v1 * v1 .= v1 * v1 .= v1 ;
assume W is non trivial & W .first() c= the carrier of G2 & W .first() c= the carrier of G2 ;
C6 * ( i1 , i2 ) `1 = G1 * ( i1 , i2 ) `1 & C6 * ( i2 , j1 ) `2 = G2 * ( i1 , j1 ) `2 ;
C8 |- 'not' All ( x , p ) 'or' p . ( x , y ) ;
for b st b in rng g holds inf ( rng f-r ) <= b
- sqrt ( ( q `1 ) ^2 + ( q `2 ) ^2 ) = 1 ;
( LSeg ( c , m ) \/ LSeg ( l , k ) ) c= R ;
consider p be element such that p in element and p in L~ x and x in L~ f and x = f /. p ;
Indices ( X @ ) = [: Seg n , Seg n :] & [: Seg n , Seg n :] = [: Seg n , Seg n :] ;
cluster s => ( q => p ) -> valid ;
( Im ( ( Partial_Sums F ) . m ) ) . x is_measurable_on E ;
cluster f . x1 , x2 ( ) -> ( len x1 ) -element for Element of D ( ) ;
consider g being Function such that g = F . t and Q [ t , g ] ;
p in LSeg ( NW-corner Z , NW-corner Z ) \/ LSeg ( NW-corner Z , NW-corner Z ) ;
set R8 = R |^ 1 , R8 = ]. b , + \infty .[ ;
IncAddr ( I , k ) = AddTo ( d , I ) .= goto ( d + k ) ;
seq . m <= ( the Sorts of seq ) . k & ( the Sorts of seq ) . k <= ( the Sorts of seq ) . k ;
a + b = ( a ` *' ) *' ( b ` ` ` ` ` ` ` ) ;
id ( X /\ Y ) = id X /\ id Y .= id X ;
for x being element st x in dom h holds h . x = f . x ;
reconsider H = U1 \/ U2 as non empty Subset of U0 | U1 , U2 = U2 | U2 ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) /\ f ) /\ j /\ m ;
consider y being element such that y in Y and P [ y , inf B ] ;
consider A being finite stable set of R such that card A = ( card R ) * 1 and A is finite ;
p2 in rng ( f |-- p1 ) \ rng <* p1 *> & rng <* p1 *> c= rng <* p1 *> ;
len s1 - 1 > 0 & len s2 - 1 > 0 & len s2 - 1 > 0 ;
( E-max ( P ) ) `2 = ( E-max ( P ) ) `2 .= ( E-max ( P ) ) `2 ;
Ball ( e , r ) c= LeftComp Cage ( C , k + 1 ) \/ LeftComp Cage ( C , k + 1 ) ;
f . a1 ` ` = f . a1 ` ` .= f . a1 ` ` .= ( f . a1 ) ` ;
( seq ^\ k ) . n in ]. - r , x0 .[ & ( seq ^\ k ) . n in ]. - r , x0 .[ ;
gg . seq = g . ( seq . seq . k ) .= g . ( seq . k ) ;
the InternalRel of S is non empty & ( the InternalRel of S ) c= ( the InternalRel of R ) \/ ( the InternalRel of S ) ;
deffunc F ( Ordinal , set ) = phi . $2 & phi . $2 = phi . $2 ;
F . s1 . a1 = F . ( s2 . a1 ) .= ( s2 . a1 ) . a1 ;
x `1 = A . o .= Den ( o , A . a ) .= Den ( o , A . a ) ;
Cl ( f " P1 ) c= f " P1 & f " P1 c= f " P1 ;
FinMeetCl ( ( the topology of S ) | the topology of T ) c= the topology of T ;
synonym o is J means : Def1 : o <> \ast p & o <> * p ;
assume that X = Y and card Y = card X + 1 and card X <> 0 and card Y <> 0 ;
the card of s <= 1 + ( the card ( the card of s ) ) ;
LIN a , a1 , d or b , c // b1 , c1 or b , c // b1 , c1 ;
e /. 1 = 0 & e /. 2 = 1 & e . 3 = 0 & e /. 4 = 0 ;
EE in SE & not EE in SE & not EE in { N } ;
set J = ( l , u ) If I is ( l , u ) If ;
set A1 = ]| , A2 = ]| , C = and2 ( a9 , b9 , c ) , A2 = InputVertices S1 ;
set c9 = [ <* c9 , A1 *> , '&' ] , A2 = [ <* A1 , cin *> , '&' ] , f4 ] , f4 = [ <* cin , A1 *> , '&' ] , f4 ] , f4 = [ <* A1 , cin *> , '&' ] , f4 = [ <* A1 , cin *> , '&' ] , f4 ] ;
x * z * x " in x * ( z * N " ) ;
for x being element st x in dom f holds f . x = g2 . x ;
Int cell ( GoB f , 1 , width GoB f ) c= RightComp f \/ RightComp f \/ RightComp f ;
U is_an_arc_of W-min C , W-min C , W-min C , W-min C , E-max C , W-min C , W-min C , W-min C , W-min C , W-min C , E-max C , E-max C , E-max C , E-max C , E-max C ;
set f9 = f @ "/\" @ g , @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ ;
attr S1 is convergent means for S2 , S2 being convergent convergent & S2 is convergent & lim S2 = x0 & lim S2 = x0 ;
f . ( 0 + 1 ) = ( 0 qua Ordinal ) + a .= a ;
cluster -> \mathclose { \rm c } -be reflexive transitive transitive for 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 ) = len l - ( 0 qua Nat ) .= len l - ( 0 qua Nat ) ;
t4 *> \setminus {} is ( {} \/ rng ( t ^ <* t1 *> ) ) -valued FinSequence of rng t ;
t = <* F . t *> ^ ( C . p ) ^ q .= ( C . p ) ^ q ;
set pp = W-min L~ Cage ( C , n ) , q = W-min L~ Cage ( C , n ) , r = W-min L~ Cage ( C , n ) ;
( k -' i + 1 ) - ( i + 1 ) = ( k - i ) + 1 - ( i + 1 ) ;
consider u being Element of L such that u = u ` ` ` and u in D and u in D ;
len ( ( width ( b |-> a ) ) |-> a ) = width ( ( len ( b |-> a ) ) |-> a ) .= width ( b |-> a ) ;
F3 . x in dom ( ( G * the_arity_of o ) . x ) ;
set H2 = the carrier of H2 , H1 = the carrier of H1 , H2 = the carrier of H2 ;
set H1 = the carrier of H1 , H2 = the carrier of H2 , H = the carrier of H1 ;
( Comput ( P , s , 6 ) ) . intpos m = s . intpos m .= s . intpos m ;
IC Comput ( P3 , t , k ) = ( l + 1 ) .= ( l + 1 ) ;
dom ( ( ( - 1 ) (#) ( ( #Z 2 ) * ( #Z 2 ) ) ) `| Z ) = REAL ;
cluster <* l *> ^ phi -> ( 1 + 0 ) string of S ;
set b9 = [ <* A1 , cin *> , '&' ] , c9 = [ <* cin , A1 *> , '&' ] , real ] , real ] ;
Line ( Segm ( M , P , Q ) , x ) . x = L * Sgm Q . x ;
n in dom ( ( the Sorts of A ) * the_arity_of o ) ;
cluster f1 + f2 -> continuous for PartFunc of REAL , the carrier of S ;
consider y be Point of X such that a = y and ||. \mathopen { \Vert \mathclose { \Vert } } <= r ;
set x3 = t1 . DataLoc ( s . GBP , 2 ) , x4 = s . DataLoc ( s . GBP , 2 ) , P4 = s . DataLoc ( s . GBP , 2 ) , P4 = s . DataLoc ( s . GBP , 2 ) , P4 = s . DataLoc ( s . GBP , 2 ) , P4 = s
set p0 = stop I , s2 = stop I , s3 = stop I ;
consider a being Point of D2 such that a in W1 and b = g . a and a in W2 and b in W2 ;
{ A , B , C , D } = { A , B } \/ { C , D } ;
let A , B , C , D , E , F , J be Function of A , B , M , N be set ;
|. p2 .| ^2 - ( ( p2 `1 ) ^2 ) >= 0 ;
l -' 1 + 1 = n-1 * ( l1 + 1 ) + 1 - 1 ;
x = v + ( a * w1 + b * w2 ) + ( c * w2 ) ;
the TopStruct of L = TopSpaceMetr (# the topology of L , the topology of T #) .= the TopStruct of ( T | L ) ;
consider y being element such that y in dom H1 and x = H1 . y and x = H1 . y ;
f9 \ { n } = ( Free All ( v1 , H ) ) \ ( Free ( v1 , H ) ) ;
for Y being Subset of X st Y is summable holds Y is iff Y is iff Y is iff Y is iff Y is iff Y is iff X is iff Y is iff Y is iff Y is iff Y is iff Y is iff X is iff Y is iff Y is iff Y is iff Y is iff Y is iff Y is iff Y is iff Y is summable
2 * n in { N : 2 * Sum ( p | N ) = N & N > 0 } ;
for s being FinSequence holds len ( the { - } ) = len s & len ( the { - } ) = len s
for x st x in Z holds exp_R * ( exp_R * f ) is_differentiable_in x & for x st x in Z holds ( exp_R * f ) . x = 1
rng ( ( h2 * f2 ) | X ) c= the carrier of ( TOP-REAL 2 ) | X & rng ( ( g2 * f2 ) | X ) c= the carrier of ( TOP-REAL 2 ) | X ;
j + 1- len f <= len f + ( len f - len f ) - len f ;
reconsider R1 = R * I as PartFunc of REAL , REAL-NS n , REAL-NS n ;
C8 . x = s1 . ( a - 1 ) .= C8 . ( a - 1 ) .= C8 . ( a - 1 ) .= C8 . ( a - 1 ) ;
power ( F_Complex , n ) . ( z , n ) = 1 .= x |^ n .= x |^ n ;
t at ( C , s ) . t = f . ( the connectives of S ) . t ;
support ( f + g ) c= support f \/ ( support g ) ;
ex N st N = j1 & 2 * Sum ( ( r | N ) | N ) > N ;
for y , p st P [ p ] holds P [ All ( y , p ) ]
{ [ x1 , x2 ] } is Subset of [: X1 , X2 :] , [: X2 , X3 :] :] ;
h . i = j |-- h , id B = id B . i as Function of B , B . i ;
ex x1 being Element of G st x1 = x & x1 * x1 c= A & x1 * x1 c= A ;
set X = ( ( |. q .| ) . O1 ) `1 , Y = ( |. q .| ) . O , 4 = ( |. q .| ) . O , 5 = ( |. q .| ) . O , 6 = |. q .| ;
b . n in { g1 : x0 - r < g1 & g1 < x0 & g1 < x0 } ;
f /* s1 is convergent & f /. x0 = lim ( f /* s1 ) ;
the lattice of the lattice of lattice Y = the lattice of the lattice of the lattice of Y & the carrier of T = the carrier of the lattice of Y ;
'not' ( a . x ) '&' b . x 'or' a . x '&' 'not' ( b . x ) = FALSE ;
2 = len ( q2 ^ r1 ) + len r1 .= len ( q2 ^ r1 ) + len r1 .= len ( q2 ^ r1 ) + len r1 ;
( ( 1 / a ) (#) ( ( sec * f1 ) - id Z ) ) is_differentiable_on Z ;
set K1 = upper ( lim ( f , H ) ) , D2 = lim ( f , H ) , H = lim ( f , H ) , N1 = lim ( f , H ) ;
assume e in { \frac w1 , w2 : w1 in F & w2 in G } ;
reconsider d7 = dom a `1 , d6 = dom F `1 as finite set ;
LSeg ( f /^ q , j ) = LSeg ( f , j + q .. f -' 1 ) ;
assume X in { T . N2 , K : h . N2 = [: N , K :] } ;
assume that Hom ( d , c ) <> {} and <* f , g *> * f1 = <* f , g *> * f1 ;
dom S.] = dom S /\ Seg n .= dom L6 /\ Seg n .= Seg n /\ Seg n .= Seg n /\ Seg n .= Seg n ;
x in H |^ a implies ex g st x = g |^ a & g in H |^ a
( a (#) ( 0 , 1 ) ) . ( a , 1 ) = a ' - ( 0 * n ) .= a ;
D2 . j in { r : lower_bound A <= r & r <= upper_bound A } ;
ex p being Point of TOP-REAL 2 st p = x & P [ p ] & p `2 <= 0 & p <> 0. TOP-REAL 2 ;
for c holds f . c <= g . c implies f @ . c <= g @ @ @ c
dom ( f1 (#) f2 ) /\ X c= dom ( f1 (#) f2 ) /\ X .= dom ( f1 (#) f2 ) /\ X .= dom ( f1 (#) f2 ) /\ X .= dom ( f1 (#) f2 ) /\ X ;
1 = ( p * p ) / p .= p * ( p / p ) .= p * ( p / p ) .= p * ( p / p ) ;
len g = len f + len <* x + y *> .= len f + 1 + 1 .= len f + 1 ;
dom ( F | [: N1 , S1 :] ) = dom ( F | [: N1 , S1 :] ) .= [: N1 , S1 :] ;
dom ( f . t ) * I . t = dom ( f . t ) * g . t ;
assume a in ( "\/" ( T |^ \alpha ) ) .: D , F .: D ] ;
assume that g is one-to-one and ( the carrier' of S ) /\ rng g c= dom g and g . ( len g ) c= dom g ;
( ( x \ y ) \ z ) \ ( ( x \ z ) \ ( y \ z ) ) = 0. X ;
consider f such that f * f = id b and f * f = id b and f * f = id a ;
( ( cos * cos ) | [. 0 , PI / 2 .] is increasing ;
Index ( p , co ) <= len LS - Index ( Gij , LS ) + 1 - Index ( Gij , LS ) ;
t1 , t2 , t2 , t1 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t2 , t2 , t1 , t2 ,
( ( ( ( Frege ( ( Frege F ) . h ) ) . h ) . h <= ( ( Frege ( ( curry G ) . h ) ) . h ) . h ;
then P [ f . i0 ] & F ( f . i0 + 1 ) < j & F ( f . i0 + 1 ) < j ;
Q [ D . ( [ D , 1 ] , F . ( D , 1 ) ] , F . ( D . ( D , 1 ) ) ] ;
consider x being element such that x in dom ( F . s ) and y = F . s ;
l . i < r . i & [ l . i , r . i ] is \setminus of G . i ;
the Sorts of A2 = ( the Sorts of A2 ) --> ( the carrier of S2 ) .= ( the Sorts of A2 ) +* ( the Sorts of A1 ) ;
consider s being Function such that s is one-to-one and dom s = NAT and rng s = { s } and s is one-to-one and rng s c= { s } ;
dist ( b1 , b2 ) <= dist ( b1 , a ) + dist ( a , b ) ;
( ( for n holds Cage ( C , n ) /. len Cage ( C , n ) ) `1 = W . ( n + 1 ) `1 ;
q <= ( UMP Upper_Arc L~ Cage ( C , 1 ) ) .. ( ( UMP L~ Cage ( C , 1 ) ) + 1 ;
LSeg ( f | i2 , i ) /\ LSeg ( f | i2 , j ) = {} ;
given a being ExtReal such that a <= II and A = ]. a , I .] and a < I and for i being element st i in I holds a < I . i ;
consider a , b be Complex such that z = a and y = b and z + y = a + b ;
set X = { b |^ n where b is Element of NAT : not contradiction } ;
( ( x * y ) \ z ) \ ( x * y ) = 0. X ;
set xy = [ <* xy , yz , z1 *> , f2 ] , zx = [ <* yz , z1 *> , f3 ] , f4 ] , f4 = [ <* xy , z1 *> , f3 ] , f4 = [ <* z , x *> , f3 ] , f4 ] , f4 = [ <* z , x *> , f3 ] ;
lk /. len ( lk ) = ( lk ) . ( len ( lk ) ) .= ( lk ) . ( ( k + 1 ) ) ;
sqrt ( ( ( q `1 ) ^2 - ( q `2 ) ^2 ) = 1 - ( ( q `2 ) ^2 ) ;
sqrt ( ( ( p `1 ) - ( p `2 ) ) ^2 < 1 ^2 / ( ( p `2 ) - ( p `2 ) ) ^2 ;
( ( ( ( X \/ Y ) \ Y ) \/ X ) \/ Y ) = ( ( X \/ Y ) \ Y ) \/ X .= X \/ Y ;
( s1 - ( s1 - s2 ) ) . k = s1 . k - s2 . k .= s1 . k - s2 . k ;
rng ( ( h + c ) ^\ n ) c= dom SVF1 ( 1 , f , u0 ) ;
the carrier of ( X , the carrier of X ) = the carrier of ( X , the carrier of Y ) --> 0. X & the carrier of ( X , Y ) --> 0. X = the carrier of Y ;
ex p4 st p4 = p4 & |. p4 - |[ a , b ]| .| = r & |. p4 - |[ a , b ]| .| = r ;
set h = IExec ( X , A , J ) , A = IExec ( X , A , J ) , B = IExec ( I , A , J ) ;
R |^ ( 0 * n ) = Il ( X , X ) .= R |^ n |^ 0 .= R |^ n ;
( Partial_Sums ( ( ( curry F ) . 0 ) ) . n is nonnegative & ( Partial_Sums ( ( ( curry F ) . 0 ) ) . n is nonnegative ;
f2 = C7 . ( E7 , the carrier of V , the carrier of K ) .= C7 . ( C7 , the carrier of K ) ;
S1 . b = s1 . b .= S2 . b .= S2 . b .= S2 . b .= S2 . b ;
p2 in LSeg ( p2 , p1 ) /\ LSeg ( p1 , p2 ) \/ 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 & the connectives of S = ( the connectives of S ) . 11 ;
set phi = ( l1 , l2 ) \HM { l2 } , phi = ( X , l2 ) \HM { l2 } ;
synonym p is invertible means : Def1 : HT ( p , T ) = 1 & HT ( p , T ) = 1 ;
( Y1 `2 ) ^2 = - 1 & ( ( Y1 `1 ) ) ^2 <> 0 & ( ( Y1 `2 ) ^2 = 1 ) & ( ( Y1 `2 ) ^2 = ( Y1 `2 ) ^2 ;
defpred X [ Nat , set , set ] means P [ $1 , $2 , $1 , $2 , $1 , $2 , $1 , $2 , $1 , $1 , $2 , $1 , $2 , $1 , $2 , $1 , $2 , $1 , $2 , $1 , $2 , $1 , $2 , $1 , $2 , $1 , $2 , $2 , $1 , $2 , $1 , $2 , $1 , $2 , $1 , $2 , $1 , $2
consider k be Nat such that for n be Nat st k <= n holds s . n < x0 + g ;
Det ( I |^ ( m -' n ) , ( m -' n ) * ( m -' n ) ) = 1_ K ;
sqrt ( b - sqrt ( b ^2 - 4 * a * c ) ) < 0 ;
Cy . d = Cy . d mod Cy . d mod Cy . d .= Cy . d mod Cy . d ;
attr X1 is dense means X1 is dense & X2 is dense & X1 meet X2 is dense implies X1 meet X2 is dense SubSpace dense SubSpace of X ;
deffunc F6 ( Element of E , Element of I ) = $1 * ( 2 |^ ( $1 + 1 ) ) ;
t ^ <* n *> in { t ^ <* i *> : Q [ i , T . i *> } ;
( x \ y ) \ x = ( x \ x ) \ y .= y ` \ x ` .= 0. X ;
for X being non empty set holds X is Basis of <* X , Y *> iff X is Basis of <* X , Y *>
synonym A , B are_separated means A misses ( Cl A ) ` & ( Cl B ) misses ( Cl B ) ` ;
len M8 = len p & width M8 = width p & width M8 = width p & width M8 = width p & width M8 = width p & width M8 = width p ;
J . v = { x where x is Element of K : 0 < v . x } ;
( Sgm ( mod m ) ) . d - ( Sgm ( mod m ) ) . e <> 0 ;
lower_bound divset ( D2 , k + k2 ) = D2 . ( k + k2 - 1 ) .= D2 . ( k + k2 - 1 ) ;
g . r1 = - 2 * r1 + 1 & dom h = [. 0 , 1 .] & rng h = [. 0 , 1 .] ;
|. a .| * ||. f .|| = 0 * ||. f .|| .= ||. a * f .|| .= ||. a * f .|| .= 0 * ||. f .|| .= 0 ;
f . x = ( h . x ) `1 & g . x = ( h . x ) `2 ;
ex w st w in dom B1 & <* 1 *> ^ s = <* 1 *> ^ w & <* 1 *> ^ s = w ^ w ;
[ 1 , {} , <* d1 *> ] in ( { [ 0 , {} , {} ] } \/ S1 ) \/ S2 ;
IC Exec ( i , s1 ) + n = IC Exec ( i , s2 ) + n .= IC Exec ( i , s2 ) + n .= IC Exec ( i , s2 ) ;
IC Comput ( P , s , 1 ) = DataLoc ( s . a , 9 ) .= 5 + 9 .= 5 + 9 .= 5 ;
( IExec ( W6 , Q , t ) ) . intpos ( i + 1 ) = t . intpos ( i + 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 x <= y or y <= x ;
integral ( f , C ) . x = f . ( upper_bound C ) - f . ( lower_bound C ) ;
for F , G being one-to-one FinSequence st rng F misses rng G holds F ^ G is one-to-one
||. R /. L - R /. h .|| < e1 * ( K + 1 ) * ( K + 1 ) ;
assume a in { q where q is Element of M : dist ( z , q ) <= r } ;
set p4 = [ 2 , 1 ] .--> [ 2 , 1 ] , p1 = [ 2 , 1 ] .--> [ 2 , 1 ] ;
consider x , y being Subset of X such that [ x , y ] in F and x c= d and y in d and x in d ;
for y , x being Element of REAL st y `1 in Y & x in X holds y `1 <= x `1 & y `1 <= x `1
func |. p \vert ^ <* p *> -> variable of A equals min ( NI , p ) ^ <* p *> ;
consider t being Element of S such that x `1 , y '||' z , t and x `1 , z '||' y , t and x , z '||' y , t ;
dom x1 = Seg len x1 & len x2 = len x1 & len x1 = len x2 & len x1 = len x2 & len x1 = width x2 & len x1 = width x2 ;
consider y2 be Real such that x2 = y2 and 0 <= y2 and y2 <= 1 and 0 <= y2 and y2 <= 1 and x2 <= 1 and y2 <= 1 and - 1 <= y2 and y2 <= 1 ;
||. f /* s1 .|| = ||. f /* s1 .|| .= ||. f /. s1 .|| .= ||. f /. s1 .|| .= ||. f /. s1 .|| ;
( the InternalRel of A ) ~ = {} ( the carrier of A ) /\ Y .= {} ( the carrier of A ) /\ Y .= {} ( the carrier of A ) ;
assume that i in dom p and for j be Nat st j in dom q holds P [ i , j ] and for i be Nat st i in dom p holds P [ i , j ] and i + 1 in dom p and i + 1 in dom p and i + 1 in dom q and i + 1 in dom q ;
reconsider h = f | [: X , Y :] as Function of [: X , Y :] , [: X , Y :] ;
u1 in the carrier of W1 & u2 in the carrier of W1 & u2 in the carrier of W1 & v2 in the carrier of W1 & u2 in the carrier of W1 & u2 in the carrier of W1 ;
defpred P [ Element of L ] means M <= f . $1 & f . $1 <= f . $1 ;
T . ( u , a , v ) = s * x + ( - s * x ) .= b - s ;
- ( - ( - y ) ) = - x + ( - y ) .= - x + ( - y ) .= - x + ( - y ) .= - x ;
given a being Point of Gx such that for x being Point of Gx holds a , x ] in \rm \hbox { - } ;
f9 = [ dom @ ( f2 @ ) , cod ( f2 @ ) ] & [ @ ( f2 @ ) , @ ( f2 @ ) ] in dom ( @ ( f2 @ ) ) ;
for k , n being Nat st k <> 0 & k < n & n <> 0 holds k mod n = 1
for x being element holds x in A |^ d iff x in ( A ` ) |^ d `
consider u , v being Element of R , a being Element of A such that l /. i = u * a and u . i = a * v ;
( - sqrt ( ( p `1 / |. p .| - cn ) / ( 1 + cn ) ) ^2 > 0 ;
LS . k = LS . ( F . k ) & F . k in dom ( L . k ) ;
set i2 = AddTo ( a , i , - n ) , n = AddTo ( a , i , - n ) , i = - n ;
attr B is \cap S is \overline means : Def1 : for x holds S ( x , y ) = ( B ( ) . x ) `1 ;
a9 " D = { a "/\" d where d is Element of N : d in D } ;
|( \square , q9 )| * |( q , q9 )| - |( q , q )| * |( q , q )| >= |( q , q )| * |( q , q )| ;
( - f ) . sup A = ( - f ) . sup A .= ( - f ) . sup A .= ( - f ) . sup A ;
( G * ( len G , k ) `1 ) `1 = ( G * ( len G , k ) `1 ) `1 .= ( G * ( len G , k ) ) `1 .= ( G * ( 1 , k ) ) `1 ;
( Proj ( i , n ) * ( g - h ) ) . 3 = <* ( proj ( i , n ) ) . ( g - h ) *> . 3 .= ( proj ( i , n ) ) . 3 ;
f1 + f2 * reproj ( i , x ) is_differentiable_in ( reproj ( i , x ) . i ) ;
attr ( tan * tan ) . x <> 0 & ( tan * tan ) . x <> 0 ;
ex t being SortSymbol of S st t = s & h1 . t = h2 . t & t . a = h2 . a ;
defpred C [ Nat ] means P8 . $1 is \HM { A } & A8 is } -then A2 is the carrier of A ;
consider y being element such that y in dom ( p | i ) and q8 . i = ( p | i ) . y and q8 . i = ( p | i ) . y ;
reconsider L = product ( { x1 } ) \mathbin ( index ( B ) , l ) as Basis of A ;
for c being Element of C ex d being Element of D st T . ( id c ) = id d & ( id c ) . ( id c ) = id d
LE f , n , p , T , i , T , i , b be Element of ( f | n ) * ;
( f (#) g ) . x = f . ( g . x ) & ( f (#) h ) . x = f . ( h . x ) ;
p in { 1 } * ( G * ( i + 1 , j ) + G * ( i + 1 , j ) } ;
f `1 - p = ( f | ( n , L ) ) *' - ( f | ( n , L ) ) .= f . ( c - p ) ;
consider r be Real such that r in rng ( f | divset ( D , j ) ) and r < m + s ;
f1 . [ ( r2 - r2 ) / 2 , ( r1 - r2 ) / 2 ] in f1 .: [: W1 , W2 :] ;
eval ( a | ( n , L ) , x ) = ( a | ( n , L ) ) . x .= a * ( b | ( n , L ) ) . x .= a * ( b | ( n , L ) ) . x ;
z = DigA ( tk , x ) .= DigA ( tk , x ) .= DigA ( tk , x ) .= DigA ( tk , x ) .= DigA ( tk , x ) ;
set H = { Intersect S where S is Subset-Family of X : S c= G & S c= G } ;
consider S19 being Element of D such that S `1 = S19 ^ <* d *> and S `2 = d ^ <* d *> and S `2 = d ^ <* d *> ;
assume x1 in dom f & x2 in dom f & x3 in dom f & x2 in dom f & f . x1 = f . x2 ;
- 1 <= ( sqrt ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 / ( 1 + cn ) ^2 ;
0. V is Linear_Combination of A & Sum ( 0. V ) = 0. V & Sum ( 0. V ) = 0. V & Sum ( 0. V ) = 0. V ;
let k1 , k2 , k2 , x4 , k2 , x4 , x5 , x5 , x5 be Int-Location , a , b , c , d be Int-Location , b , c , d be Int-Location ;
consider j be element such that j in dom a and j in g " { k } and x = a . j and x = a . j ;
H1 . x1 c= H1 . x2 or H1 . x2 c= H1 . x2 or H1 . x2 c= H1 . x2 or H1 . x2 c= H1 . x2 ;
consider a being Real such that p = a * p1 + ( a * p2 ) and 0 <= a and a <= 1 and a <= 1 and a <= 1 ;
assume that a <= c and d <= b and [ a , b ] c= dom f and [ a , b ] in dom g and g . a 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 * b1 ) . y .= ( F * b1 ) . y ;
g . ( s , I ) . x = s . y & g . ( s , I ) . y = |. s . x - g . y .| ;
( log ( 2 , k ) ) ^2 >= ( log ( 2 , k ) ) ^2 / ( 2 * ( k + 1 ) ) ^2 ;
then p => q in S & not x in the still of p & not x in S & not p in S ;
dom ( the initial of r-10 ) misses dom ( the initial of r-10 ) & dom ( the initial of r-10 ) misses dom ( the initial of r-10 ) ;
synonym f is ExtReal means : as : 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 p1 + len <* x *> .= len p1 + len <* x *> .= len p1 + 1 + 1 .= len p1 + 1 + 1 .= len p1 + 1 + 1 .= len p1 + 1 + 1 ;
( l , 3 ) `1 = ( g . ( k + 3 ) ) `1 + ( k , 3 ) `1 .= ( g . ( k + 3 ) ) `1 + ( k , 3 ) `1 ;
CurInstr ( P2 , Comput ( P2 , s2 , l ) ) = halt SCM+FSA .= ( halt SCM+FSA ) . IC SCM+FSA .= ( halt SCM+FSA ) . IC SCM+FSA .= ( halt SCM+FSA ) . IC SCM+FSA .= ( halt SCM+FSA ) . IC SCM+FSA .= ( IC SCM+FSA ) ;
assume for n be Nat holds ||. ( seq . n ) - ( seq . n ) .|| <= R . n & R . n < r ;
sin . ( \HM { the } \HM { function } ) = sin . r * cos . ( - ( cos . s ) ) .= 0 ;
set q = |[ g1 `1 , g2 `2 ]| , g2 = |[ g2 `1 , g2 `2 ]| , f3 `2 = |[ g2 `1 , g2 `2 ]| , t = |[ g2 `1 , t `2 ]| ;
consider G being sequence of S such that for n being Element of NAT holds G . n in holds G . n in holds ( ( the Sorts of F ) . n in ) ;
consider G such that F = G and ex G1 , G2 being Subset of X st G1 in SX & G2 in SX & G = [: G1 , G2 :] & G = [: G1 , G2 :] ;
the root of [ x , s ] in ( the Sorts of Free ( C , X ) ) . s & ( the Sorts of Free ( C , X ) ) . s in ( the Sorts of Free ( C , X ) ) . s ;
Z c= dom ( ( exp_R * ( exp_R + ( exp_R * f1 ) ) ) ;
for k be Element of NAT holds ( r . k ) = ( ( Im f ) . k ) * ( ( Im g ) . k )
assume that - 1 < n and ( q `2 / |. q .| - cn ) < 0 and ( q `2 / |. q .| - cn ) < 0 and ( q `2 / |. q .| - cn ) < 0 ;
assume that f is continuous and a < b and c < d and f . a = g and f . b = g . c and f . c = g . d ;
consider r being Element of NAT such that seq = Comput ( P1 , s1 , r ) and r <= q and r <= q and r <= p ;
LE f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) ;
assume that x in the carrier of K and y in the carrier of K and inf { x , y } in the carrier of K and inf { x , y } in K ;
assume f /^ ( i1 , \xi ) . ( i , \xi ) in ( proj ( F , i2 ) ) " ( proj ( F , i2 ) ) " ( ( proj ( F , i2 ) ) " ) ;
rng ( ( ( Flow M ) | ( the carrier of M ) ) c= the carrier' of M & ( ( the carrier of M ) | ( the carrier of M ) ) c= the carrier' of M ;
assume z in { ( the carrier of G ) \ { t } where t is Element of T : not contradiction } ;
consider l be Nat such that for m be Nat st l <= m holds ||. s1 . m - g .|| < g ;
consider t be VECTOR of product G such that NAT = ||. D5 .|| and ||. t .|| <= 1 and ||. t .|| <= 1 ;
assume that the carrier of v = 2 and v ^ <* 0 *> in dom p and v ^ <* 1 *> in dom p and v ^ <* 1 *> in dom p and v ^ <* 1 *> in dom p ;
consider a being Element of the points of [ X1 , X2 ] , A being Element of the \rm element such that a on A and not a on A and a on A and b on A ;
( - x ) |^ ( k + 1 ) * ( ( - x ) |^ ( k + 1 ) " ) = 1 ;
for D being set st i in dom p holds p . i in D & p . i in D holds p . i in D
defpred R [ element ] means ex x , y st [ x , y ] = $1 & P [ x , y ] ;
L~ f2 = union { LSeg ( p1 , p2 ) , LSeg ( p1 , p2 ) } .= LSeg ( p1 , p2 ) \/ LSeg ( p2 , p1 ) ;
i -' len h11 + 2 - 1 < i + 2 - 1 + 1 & i + 1 - 1 + 1 < i + 1 - 1 + 1 ;
for n be Element of NAT st n in dom F holds F . n = |. ( n -' 1 ) * ( n -' 1 ) .|
for r , s1 , s2 being Real holds r in [. s1 , s2 .] iff r <= s1 & s1 <= s2 & s2 <= 1 & s1 <= 1 & s2 <= 1 & s1 <= 1 & s2 <= 1
assume v in { G where G is Subset of T2 : G in B & G c= B & G c= B & G c= B } ;
let g be \subseteq G -:] , X be Element of INT , b be Element of INT -tuples_on ( X , b ) , c , d be Element of INT ;
min ( g . [ x , y ] , k ) . [ y , z ] = ( min ( g . k , x ) ) . y ;
consider q1 be sequence of CH such that for n holds P [ n , q1 . n ] and for n holds q1 . n = F ( n ) ;
consider f being Function such that dom f = NAT and for n being Element of NAT holds f . n = F ( n ) and f . n = F ( n ) ;
reconsider B-6 = B /\ O , O = O /\ Z , Z = O as Subset of B | Z ;
consider j be Element of NAT such that x = the holds 1 <= j and j <= n and 1 <= j and j <= n and 1 <= n and n <= len f and f . j = f . n ;
consider x such that z = x and card ( x . O2 ) in card ( x . O2 ) and x in L1 and x in L2 and x in L2 ;
( C * _ T4 ( k , n2 ) ) . 0 = C . ( ( C * _ ( k , n2 ) ) . 0 ) .= C . ( ( k + 1 ) - 1 ) ;
dom ( X --> rng f ) = X & dom ( X --> f ) = X & rng ( X --> f ) = dom ( X --> f ) ;
( E-max L~ Cage ( C , n ) ) `2 <= ( ( E-max L~ Cage ( C , n ) ) .. ( Cage ( C , n ) ) & ( E-max L~ Cage ( C , n ) ) `2 <= ( E-max L~ Cage ( C , n ) ) `2 ;
synonym x , y are_not collinear means : as : x = y or ex l being there y being there r being there l being there 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 x << y iff x << y ;
( ( 1 / 2 ) (#) ( ( #Z 2 ) * ( #Z 2 ) ) is_differentiable_on REAL ;
defpred P [ Element of omega ] means ( the partial of A1 ) . $1 = A1 . $1 & ( the partial of A1 ) . $1 = A2 . $1 ;
IC Comput ( P , s , 2 ) = succ IC Comput ( P , s , 1 ) .= ( 0 + 1 ) .= 6 .= 6 + 1 .= 6 + 1 .= 6 + 1 .= 6 + 1 ;
f . x = f . g1 * f . g2 .= f . g2 * ( g . g2 ) .= f . g2 * ( g . g2 ) .= f . g2 * ( g . g2 ) .= f . g2 * ( g . g2 ) .= f . g2 * ( g . g2 ) ;
( M * ( F . n ) ) . n = M . ( ( ( ( canFS ( Omega ) ) . n ) ) .= M . ( ( canFS ( Omega ) ) . n ) .= M . ( ( canFS ( Omega ) ) . n ) ;
the carrier of L1 + L2 c= ( the carrier of L1 ) \/ ( the carrier of L2 ) \/ ( the carrier of L2 ) ;
pred a , b , c , x , y , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z be set ;
( for n be Nat holds s . n <= ( ( - 1 ) (#) s ) . n & ( - 1 ) (#) s . n <= ( - 1 ) (#) s . n ;
attr - 1 <= r & r <= 1 implies ( ( - 1 ) (#) ( arccot ) ) . r = - 1 / ( 1 + r ^2 ) ;
seq in { p ^ <* n *> where n is Nat : p ^ <* n *> in T1 } ;
|[ x1 , x2 , x3 ]| . 2 - |[ y1 , y2 ]| . 2 - |[ y1 , y2 ]| . 2 = x2 - y2 ;
attr for m be Nat holds F . m is nonnegative means for n be Nat holds ( Partial_Sums F ) . n is nonnegative ;
len ( ( G . z ) (#) ( G . y ) ) = len ( ( G . x ) (#) ( G . y ) ) .= len ( G . y ) .= len ( G . y ) ;
consider u , v being VECTOR of V such that x = u + v and u in W1 /\ W2 and v in W2 /\ W3 and u in W2 /\ W3 ;
given F being FinSequence of NAT such that F = x and dom F = n and rng F c= { 0 , 1 } and for k being Nat st k in n holds Sum F = k and Sum ( F | k ) = k ;
0 = 1-r * u , u2 = 1 * u , u1 = 1 * u - ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) ;
consider n be Nat such that for m be Nat st n <= m holds |. ( f # x ) . m - ( lim ( f # x ) ) . n .| < e ;
cluster non empty for \mathclose { \rm c } } is distributive for \hbox { ( { \rm c } _ 2 ) , ( { \rm c } _ 2 ) } , ( { \rm c } _ 2 ) } is Boolean
"/\" ( B , L ) = Bottom ( B , L ) .= "/\" ( B , L ) .= "/\" ( { {} } , L ) .= "/\" ( { {} } , L ) .= "/\" ( { {} } , L ) .= "/\" ( { {} } , L ) .= "/\" ( { {} } , L ) ;
sqrt ( r ^2 + ( r ^2 + ( r ^2 + ( r ^2 ) ) ^2 ) <= sqrt ( r ^2 + ( r ^2 ) ) ^2 + ( r ^2 + ( r ^2 + ( r ^2 ) ) ^2 ) ;
for x being element st x in A /\ dom ( f `| X ) holds ( f `| X ) . x >= r2
2 * r1 - c * ( |[ a , c ]| - ( 2 * r1 - 1 ) ) = 0. TOP-REAL 2 + ( 2 * r1 - 1 ) * ( |[ b , c ]| - ( 2 * r1 - 1 ) ) ;
reconsider p = P * ( \square , 1 ) , q = a " * ( ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( K / n ) ) ) ) ) ) ) ) ) ) as FinSequence of K ;
consider x1 , x2 being element such that x1 in downarrow s and x2 in downarrow t and x = [ x1 , x2 ] and [ x2 , x3 ] in t and x = [ x1 , x2 ] ;
for n be Nat st 1 <= n & n <= len q1 holds q1 . n = ( upper_bound ( g , ( len g ) ) ) . n
consider y , z being element such that y in the carrier of A and z in the carrier of A and i = [ y , z ] and i = [ y , z ] and i = [ y , z ] ;
given H1 , H2 being strict Subgroup of G such that x = H1 and y = H2 and H1 , H2 are_isomorphic and H1 , H2 are_isomorphic and H1 , H2 are_isomorphic and H1 , H2 are_isomorphic ;
for S , T being non empty RelStr , d being Function of T , S st T is complete holds d is monotone & d is monotone
[ a + 0 , b ] in ( the carrier of cLin ( A ) ) \ ( the carrier of cW2 ) & [ b , a ] in the carrier of [: cW2 , the carrier of V :] ;
reconsider mm = max ( len F1 , len ( p . n ) * <* x *> ) as Element of NAT ;
I <= width GoB ( GoB ( GoB h , len GoB h ) , GoB ( GoB h , 1 ) ) & width GoB ( GoB h , width GoB h ) = width GoB ( GoB h , 1 ) ;
f2 /* q = ( f2 /* ( f1 /* s ) ) ^\ k .= ( f2 * f1 ) /* s .= ( f2 * f1 ) ^\ k .= ( f2 * f1 ) /* s ;
attr A1 \/ A2 is linearly-independent means : Def1 : A1 : A2 misses A2 & A2 /\ ( A1 \/ A2 ) = { 0. V } & ( for x holds ( A1 /\ A2 ) /\ ( A2 \/ A1 ) = { 0. V } ) ;
func A -carrier C -> set equals union { A . s where s is Element of R : s in A } ;
dom ( Line ( v , i + 1 ) ) ^ ( ( Line ( v , m ) ) /. ( \square , 1 ) ) = dom ( F ^ <* G . ( i , 1 ) *> ) ;
cluster [ x , 4 , 4 , 5 , 6 ] -> [ x , 4 , 5 , 7 ] , [ x , 4 , 7 ] , [ x , 4 , 7 ] ] -> to [ x , 4 , 7 ] , [ x , 5 ] , [ x , 6 ] ] -> to x , 4 , 5 , 6 , 8 , 7 , 8 , 8 , 8 ] ;
E , ( All ( x2 , x1 ) ) . ( x2 , x1 ) = All ( x2 , x2 ) . ( x2 , x1 ) '&' All ( x2 , x1 ) '&' ( x2 , x1 ) '&' ( x2 , x1 ) '&' ( x2 , x1 ) '&' ( x2 , x1 ) '&' ( x2 , x1 ) '&' ( x2 , x1 ) '&' ( x2 , x1 ) '&' ( x2 , x1 ) '&' ( x2 , x1 ) |= ( x2 , x1 ) ;
F .: ( id X , g ) . x = F . ( id X , g . x ) .= F . ( id X , g . x ) .= F . ( id X , g . x ) .= F . ( id X , g . x ) .= F . ( id X , g . x ) ;
R . ( h . m ) = F . x0 + h . ( m + 1 ) - h . ( h . m ) .= F . x0 - F . x0 ;
cell ( G , ( X -' 1 , Y ) , ( Y + 1 ) \ L~ f ) meets ( L~ f ) \/ ( L~ f ) ;
IC Comput ( P2 , s2 , k ) = IC Comput ( P2 , s2 , k ) .= IC Comput ( P2 , s2 , k ) .= ( IC Comput ( P2 , s2 , k ) ) .= ( IC Comput ( P2 , s2 , k ) ) .= ( IC Comput ( P2 , s2 , k ) .= ( IC Comput ( P2 , s2 , k ) ) .= ( IC Comput ( P2 , s2 , k ) ) ;
sqrt ( ( - ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 ) > 0 ;
consider x0 be element such that x0 in dom a and x0 in dom g and y = g " . x0 and x0 in dom g and y = a . x0 and x0 in dom g and g . x0 = a . x0 ;
dom ( r1 (#) ( ( r1 (#) ( b (#) ( A . m ) ) ) ) = dom ( ( a (#) ( b (#) ( A . m ) ) ) ) .= dom ( ( a (#) ( b (#) ( A . m ) ) ) .= dom ( ( a (#) ( b (#) ( A . m ) ) ) .= dom ( ( a (#) ( b (#) ( A . m ) ) ) ;
d-7 . [ y , z ] = ( ( ( y , z ) `2 ) `1 - ( y , z ) `2 ) `2 .= ( ( y , z ) `2 ) - ( y , z ) `2 ;
attr for i being Nat holds C . i = A . i /\ B . i & C . i c= C . i /\ C . i ;
assume that x0 in dom f and f is continuous and f . x0 = ( f . x0 ) * ( f . x0 ) and for x st x in dom f holds f . x <> 0 and f . x0 <> 0 ;
p in Cl A implies for K being Basis of p , Q being Subset of T st Q in K & Q meets K holds A meets Q
for x be Element of REAL n st x in Line ( x1 , x2 ) holds |. y1 - y2 .| <= |. y1 - y2 .|
func => <*> a -> set means : Def1 : a in it iff a in it & for b being Ordinal st a in it holds it . b c= b ;
[ a1 , a2 , a3 , a4 ] in ( the carrier of A ) ~ & [ a1 , a2 , a3 ] in ( the carrier of A ) ~ ;
ex a , b being element st a in the carrier of S1 & b in the carrier of S2 & x = [ a , b ] & [ b , a ] in the carrier of S2 & [ a , b ] in the carrier of S2 & [ b , a ] in the carrier of S2 ;
||. ( vseq . n ) - ( vseq . m ) .|| * ||. x - y .|| < ( e * ||. x .|| ) * ||. x - y .|| ;
then for Z being set st Z in { Y where Y is Element of I7 : F c= Y } holds z in Z & z in Z ;
sup compactbelow ( s , t ) = [ sup ( { s } ) , sup ( { s } ) ] .= sup { s . ( s . ( s . ( s . ( s . ( s . ( s . ( s . ( s . ( s . ( s , t ) ) ) ) ) where s is Element of S : s . ( s . ( s . ( s , t ) ) ) } , T ] } ;
consider i , j being Element of NAT such that i < j and [ y , f . j ] in Ij and [ f . i , f . j ] in Ij and [ f . i , f . j ] in Ij ;
for D being non empty set , p , q being FinSequence of D st p c= q holds ex p being FinSequence of D st p = q & p ^ q = q & p ^ q = q
consider e1 being Element of the affine of X such that c9 , a9 // a9 , b9 and a , a9 // a9 , c9 and a , a9 // a9 , c9 and a , a9 // c9 , c9 and a , a9 // c9 , c9 and a , b // a9 , c9 and a , b // c9 , c9 and a , c // a9 , c9 and a , b // a9 , c9 and a , b // c9 , c9 and a , b // a9 , c9 and a , b // c9 , c9 and a , b // c9 , c9 and a , b // a9 , c9 and
set U2 = I \! \mathop { {} } , U2 = I \! \mathop { {} } , F = I \! \mathop { {} } ;
|. q2 .| ^2 = ( ( |. q2 .| ) ^2 + ( |. q2 .| ) ^2 .= ( |. q2 .| ) ^2 + ( |. q2 .| ) ^2 .= |. q2 .| ^2 ;
for T being non empty TopSpace , x , y being Element of [: T , T :] , T being Element of [: T , T :] holds x "\/" y = x "\/" y & x "/\" y = x "/\" y
dom ( signature U1 ) = dom ( ( the charact of U1 ) | ( the carrier of U1 ) ) & dom ( ( the charact of U1 ) | ( the carrier of U1 ) ) = dom ( ( the charact of U1 ) | ( the carrier of U1 ) ) ;
dom ( h | X ) = dom h /\ X .= dom h /\ X .= dom ( h | X ) /\ X .= dom ( h | X ) /\ X .= dom ( h | X ) .= dom ( h | X ) /\ X .= dom ( h | X ) .= dom ( h | X ) ;
for N1 , N2 being Element of [: G , H :] holds dom ( h . K ) = N & rng ( h . K ) = [: N , I :]
( mod ( u , m ) + mod ( v , m ) ) . i = ( mod ( u , m ) ) . i + ( mod ( v , m ) ) . i ;
- ( q `1 ) ^2 < - 1 or ( q `2 / |. q .| - cn ) / ( 1 + cn ) <= - ( q `1 / |. q .| - cn ) / ( 1 + cn ) ;
attr r1 = f9 & r2 = f9 & r1 = f2 & r2 = f9 & r1 = f2 & r2 = f1 & r1 = f2 & r2 = f3 & r1 = f2 & r2 = f3 ;
vseq . m is bounded Function of X , the carrier of Y & x9 = ( vseq . m ) . x & x9 = ( vseq . m ) . x ;
attr a <> b & b <> c & angle ( a , b , c ) = PI & angle ( b , c , a ) = 0 & angle ( b , c , a ) = 0 ;
consider i , j being Nat , r being Real such that p1 = [ i , r ] and p2 = [ i , s ] and r < s and s < t and t < t and t < s and t < s ;
|. p .| ^2 - ( 2 * |( p , q )| ) ^2 + |. q .| ^2 = |. p .| ^2 ^2 + |. q .| ^2 ;
consider p1 , q1 be Element of [: X , Y :] such that y = p1 ^ q1 and q1 ^ q2 = p1 ^ q1 and p1 ^ q1 = p2 ^ q1 and q1 ^ q2 = p1 ^ q1 and p1 ^ q1 = p2 ^ q2 ;
( the carrier of ( A ) ) . ( r1 , r2 , s1 , s2 , s2 ) = ( ( A ) . ( r2 , s2 ) ) . ( s2 , s1 ) .= ( ( A ) . ( r2 , s2 ) ) . ( s2 , s1 ) .= ( ( A ) . ( r2 , s2 ) ) . ( s2 , s1 ) ;
( for w holds ( for w holds w in proj2 .: ( A /\ Vertical_Line w ) ) & ( proj2 .: ( A /\ Vertical_Line w ) ) /\ ( proj2 .: ( A /\ Vertical_Line w ) is non empty ) implies proj2 .: ( A /\ Vertical_Line w ) is non empty
s , ( k |= H1 H1 ) implies s |= H1 iff s |= H2 iff s |= H1 |= H2 & s |= H1 '&' H2 iff s |= H1 '&' H2
len ( s + 5 ) = card ( support b1 ) + 1 .= card ( support b1 ) + 1 .= card ( support b2 ) + 1 .= ( len b1 ) + 1 .= ( len b1 ) + 1 .= ( len b1 ) + 1 .= ( len b1 ) + 1 .= len b2 + 1 .= len b1 ;
consider z being Element of L1 such that z >= x and z >= y and for z being Element of L1 st z >= x holds z >= y and z >= x ;
LSeg ( UMP D , |[ ( ( E-bound D ) + E-bound D ) / 2 , ( ( E-bound D ) / 2 ]| ) / 2 , ( ( E-bound D ) / 2 ) / 2 ]| /\ D = { UMP D } ;
lim ( ( ( ( f `| N ) / g ) /* b ) = ( ( f `| N ) / g ) /* b .= ( ( f `| N ) / g ) . 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 ) - ( R . k ) .|| < r holds ||. ( seq . k ) - ( R . k ) .|| < r
for X being set , P , Q being a_partition of X st x in a & P in P & Q in P & x in Q & P in Q & P /\ Q <> {} holds a = b
Z c= dom ( ( ( - exp ) (#) ( exp_R * f ) ) \ ( ( exp_R * f ) " { 0 } ) " { 0 } implies f " { 0 } c= dom ( ( - 1 ) (#) ( exp_R * f ) " { 0 } )
ex j be Nat st j in dom ( l ^ <* x *> ) & j < i & i < len ( l ^ <* x *> ) & y = i + 1 & z = i + 1 & z = i + 1 ;
for u , v being VECTOR of V , r being Real st 0 < r & u in N & v in N holds r * u + ( r * v ) in N
A , Int ( A , B ) / A , Int ( A , B ) / A / B / C / A / ( A , B ) / A / ( A , B ) / A / ( A , B ) / ( A , B ) / A / ( A , B ) / ( A , B ) / ( A , B ) / ( A , B ) / ( A , B ) / ( A , B ) / A / B / A / B ) , A / ( A , B ) / ( A , B ) / ( A , B ) / ( A , B ) / ( A , B
- Sum <* v , u , w *> = - ( v + u + u ) .= - ( v + u ) + u .= - ( v + u ) + u .= - ( v + u ) .= - ( v + u ) ;
( Exec ( a := b , s ) ) . IC SCM = ( Exec ( a := b , s ) ) . IC SCM .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s ;
consider h being Function such that f . a = h and dom h = I and for x being element st x in I holds h . x in ( the carrier of J ) . x ;
for S1 , S2 being non empty reflexive RelStr for D being non empty Subset of S1 , S1 being non empty Subset of S2 holds 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 ex z st z in X & z = x & x = y or z = x & y = z or z = x & x = y or z = y ;
E-max L~ Cage ( C , n ) in rng ( Cage ( C , n ) \circlearrowleft W-min L~ Cage ( C , n ) ) & E-max L~ Cage ( C , n ) in rng ( Cage ( C , n ) \circlearrowleft W-min L~ Cage ( C , n ) ) ;
for T , T being decorated tree , p , q being Element of dom T st p element T holds ( T -tree ( p , T ) ) . q = T . q & ( T -tree ( p , T ) ) . q = T . q
[ i2 + 1 , j2 ] in Indices G & [ i2 , j2 ] in Indices G & f /. k = G * ( i2 + 1 , j2 ) & f /. k = G * ( i2 + 1 , j2 ) ;
cluster the _ of k , n ) divides k & k divides n & n divides m & n divides k & m divides n & n divides k ;
dom F " = the carrier of X2 & rng F " { 0 } = the carrier of X1 & rng F " { 0 } = the carrier of X2 & rng F = the carrier of X2 & rng F c= the carrier of X1 & F " { 0 } = the carrier of X2 ;
consider C be finite Subset of V such that C c= A and card C = n and the carrier of V = A and the carrier of W = A and C is linearly-independent and A is linearly-independent and C is linearly-independent ;
V is prime implies for X , Y being Element of \langle the topology of T , \subseteq \rangle st X /\ Y c= V holds X c= Y or X c= Y or Y c= V or X c= V or X c= Y or X c= Y or X c= Y
set X = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } , Y = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } ;
angle ( p1 , p3 , p4 , p4 ) = 0 .= 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 `1 / |. q .| - cn ) / ( 1 + cn ) ;
ex f being Function of I[01] , TOP-REAL 2 st f is continuous one-to-one one-to-one & rng f = P & f . 0 = p1 & f . 1 = p2 & f . 0 = p4 & f . 1 = p4 & f . 0 = p4 & f . 1 = p4 & f . 1 = p4 ;
attr f is partial differentiable on on u0 means : Def1 : SVF1 ( 2 , 3 ) is_differentiable_in u0 & SVF1 ( 2 , 3 ) is_differentiable_in z0 & SVF1 ( 2 , 3 ) is_differentiable_in z0 ;
ex r , s st x = |[ r , s ]| & G * ( len G , 1 ) `1 < r & r < G * ( 1 , 1 ) `1 & s < G * ( 1 , 1 ) `2 & s < G * ( 1 , 1 ) `2 ;
assume that f is_sequence_on G and 1 <= t and t <= len G and 1 <= width G and G * ( t , width G ) `2 >= G * ( t , width G ) `2 and G * ( t , width G ) `2 >= G * ( t , width G ) `2 ;
attr i in dom G means r (#) ( f (#) reproj ( i , x ) ) = r (#) reproj ( i , x ) & r (#) reproj ( i , x ) = r (#) reproj ( i , x ) ;
consider c1 , c2 being bag of o1 + o2 such that ( ( decomp c ) /. k = <* c1 , c2 *> and ( ( decomp c ) /. k = <* c1 , c2 *> and ( ( decomp c ) /. k = c1 + c2 and ( decomp c ) /. k = c2 + c2 and ( decomp c ) /. k = c2 + c2 ;
u in { |[ r1 , s1 ]| : r1 < s1 & s1 < 1 & s1 < 1 } ;
Cl ( X ^ Y ) . k = the carrier of X . ( k2 + 1 ) .= C4 . ( k2 + 1 ) .= C4 . ( k2 + 1 ) .= C4 . ( k2 + 1 ) .= C4 . ( k2 + 1 ) .= C4 . ( k2 + 1 ) .= C4 . ( k2 + 1 ) ;
attr 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 ;
consider g2 be Real such that 0 < g2 and { y where y is Point of S : ||. y - x0 .|| < g2 & g2 < x0 } c= N2 and for g be Real st g in N2 holds ||. ( y - x0 ) . g - ( g . x0 ) .|| < g2 . g ;
assume x < ( - b + sqrt ( 2 * a , c ) ) / 2 or x > - b + sqrt ( 2 * a , c ) / 2 ;
( G1 '&' G2 ) . i = ( <* 3 *> ^ H1 ) . i & ( G1 '&' G2 ) . i = ( <* 3 *> ^ H1 ) . i & ( G1 '&' G2 ) . i = ( <* 3 *> ^ H1 ) . i ;
for i , j st [ i , j ] in Indices M2 holds ( M2 + M1 ) * ( i , j ) < M2 * ( i , j )
for f being FinSequence of NAT , i being Element of NAT st i in dom f holds i divides f /. i & i <= len f & i divides f /. i & i divides f /. i & i divides len f holds i divides f /. i
assume F = { [ a , b ] where a , b is Subset of X : for c being set st c in B holds a c= c & b c= c & a c= c & b c= c & a c= b & b c= c & c c= c ;
b2 * ( ( - b3 ) * ( - ( 3 * q2 ) + ( - ( 3 * q2 ) ) ) + ( - ( 3 * q2 ) ) * ( - ( 3 * q2 ) ) = 0. TOP-REAL n + ( 3 * q2 ) * ( - ( 3 * q2 ) ) ;
Cl ( Cl F ) = { D where D is Subset of T : ex B being Subset of T st D = Cl B & B in F & A /\ B c= Cl ( F | D ) } ;
attr seq is summable means : Def1 : seq is summable & seq is summable & seq is summable & lim seq = 0 implies seq + seq is summable & lim ( seq + seq ) = ( lim seq ) * ( lim seq ) + ( lim seq ) * ( lim seq ) ;
dom ( ( cn max D ) | D ) = ( the carrier of ( TOP-REAL 2 ) | D ) /\ D .= the carrier of ( TOP-REAL 2 ) | D .= the carrier of ( TOP-REAL 2 ) | D .= the carrier of ( TOP-REAL 2 ) | D .= the carrier of ( TOP-REAL 2 ) | D .= the carrier of ( TOP-REAL 2 ) | D ;
[ X \to Z ] is full full full full full full full full full full full SubRelStr SubRelStr full SubRelStr non empty full SubRelStr of ( [#] Z ) |^ the carrier of Z , the carrier of Z ] means [ X \to Y ] is full holds X is full
( G * ( 1 , j ) `2 = ( G * ( 1 , j ) `2 ) `2 & ( G * ( 1 , j ) `2 <= ( G * ( 1 , j ) `2 ) `2 ;
synonym m1 c= m2 means for for for for p , q being set st p in P & q in P holds the InternalRel of ( m1 + 1 ) . p <= ( the Subset of ( m1 + 1 ) ) . q & the Subset of ( m1 + 1 ) | ( m1 + 1 ) = ( the carrier of ( m1 + 1 ) ) | ( m1 + 1 ) ;
consider a being Element of [: B , C :] such that x = F ( a ) and a in { G ( b ) where b is Element of A ( ) : P [ b ] } and P [ b ] ;
func multiplicative empty multiplicative loop structure means : Def1 : the multiplicative of mas of it = \llangle the carrier of it , the carrier of it #) , the carrier of it = [: the carrier of it , the carrier of it :] ;
the carrier of \mathop { a , b , d } + \mathop ( c , d ) = b + \mathop ( c , d ) .= b + \mathop { \rm \HM { c + d } : c in b } + d .= b + \mathop { \rm \HM { c + d } : c in d } ;
cluster strict for non empty RelStr means : Def1 : for Element of INT holds ( for i , j being Element of INT st i in INT holds ( i , j ) = i + j & ( i , j ) = j + 1 ) & ( i = j implies ( i = j implies implies ( i = j ) = j ) ;
( - 2 * p1 ) + ( 2 * p2 ) - ( 2 * p2 ) = ( - 2 * p2 ) + ( 2 * p2 ) - ( 2 * 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 ) ;
assume that the TopStruct of S = the TopStruct of T and for D being non empty Subset of S st D in [#] T holds sup D meets ( [#] T ) and sup D meets ( [#] T ) and sup D meets ( [#] T ) and sup D meets ( [#] T ) and sup D meets ( [#] T ) /\ ( [#] T ) = [#] T ;
assume 1 <= k & k <= len w + 1 implies T-7 . ( ( ( q , w ) -succ k ) . ( q , w ) ) = ( T11 . ( q , w ) ) . ( q , w ) & ( T . ( q , w ) ) . ( q , w ) = ( T . ( q , w ) ) . ( q , w ) ;
2 * ( a |^ ( n + 1 ) ) + ( 2 * ( b |^ ( n + 1 ) ) >= ( a |^ n ) + ( b |^ ( n + 1 ) ) + ( b |^ n ) ;
M , v / ( x. 3 , 0 ) / ( x. 0 , 4 ) / ( x. 0 , x. 0 ) / ( x. 4 , x. 0 ) / ( x. 0 , x. 0 ) / ( x. 4 , x. 0 ) / ( x. 0 , x. 0 ) / ( x. 0 , x. 0 ) / ( x. 0 , x. 0 ) / ( x. 0 , x. 0 ) / ( x. 0 , x. 0 ) / ( x. 0 , x. 0 ) |= ( x. 0 , x. 0 ) ;
assume that f is_differentiable_on l and for x0 st x0 in l holds 0 < f . x0 and for x1 st x1 in l holds f . x1 < f . x0 and f . x0 < 0 ;
for G1 being _Graph , W being Walk of G1 , e being Vertex of G2 , e being Vertex of G2 st e in W holds not e in W iff not e in W & not e in W
c9 is non empty iff ( ( ex x1 , x2 st x1 is non empty & x2 is non empty or ( ex y1 st y1 is non empty & y2 is non empty ) & ( ( ex y2 st y1 is non empty & y2 is non empty ) & ( ( ex y1 st y1 is non empty or y2 is non empty ) ) & ( ( ex y2 st y1 is non empty or y2 is non empty ) ) & ( not y1 is non empty ) ;
Indices GoB f = [: dom GoB f , Seg width GoB f :] & [: dom GoB f , Seg width GoB f :] = [: dom GoB f , Seg width GoB f :] & [: dom GoB f , Seg width GoB f :] c= Indices GoB f & [: dom GoB f , Seg width GoB f :] c= Indices GoB f & [: dom GoB f , Seg width GoB f :] c= Indices GoB f & [: L~ f , Seg width GoB f :] c= Indices GoB f & [: L~ f , Seg width GoB f , Seg width GoB f :] c= Indices GoB f , Seg width GoB f :] c= Indices GoB f :] holds f /. 1 , Seg len GoB f :] c= Indices GoB f :]
for G1 , G2 being Group , O being strict Subgroup of O st G1 is stable & G2 is stable holds G1 is stable iff G1 is stable & G2 is stable & G2 is stable & G1 is stable & G2 is stable
UsedIntLoc ( ( ( intloc 0 ) .--> 1 ) ) = { intloc 0 , ( ( intloc 0 ) .--> 1 ) , ( ( intloc 0 ) .--> 1 ) , ( ( intloc 0 ) .--> 1 ) , ( ( intloc 0 ) .--> 1 ) , ( ( intloc 0 ) .--> 1 ) , ( ( intloc 0 ) .--> 1 ) .--> 1 , ( ( intloc 0 ) .--> 1 ) .--> 1 ) , ( ( intloc 0 ) .--> 1 ) ;
for f1 , f2 being FinSequence of F st f1 ^ f2 is p -element & for i being Nat st i in dom f1 holds Q [ i , f2 . i ] holds Q [ i , f1 . i ]
sqrt ( ( p `1 ) ^2 + ( p `2 ) ^2 ) = sqrt ( ( p `1 ) ^2 + ( p `2 ) ^2 ) .= sqrt ( ( p `1 ) ^2 ) + ( p `2 ) ^2 ;
for x1 , x2 , x3 , x4 being Element of REAL n holds |( x1 - x2 , x3 )| = |( x1 , x2 )| & |( x2 , x3 )| = |( x1 , x2 )| - |( x2 , x3 )|
for x st x in dom ( ( - ( - x ) ) | A ) holds ( ( - x ) | A ) . x = - ( x ) | A
for T being non empty TopSpace , P being Subset-Family of T st P c= the topology of T & for B being Basis of T ex x being Point of T st B c= P & P c= B holds P is Basis of T
( a 'or' b ) . x = 'not' ( ( a 'or' b ) . x ) 'or' c . x .= 'not' ( a . x ) 'or' b . x .= TRUE 'or' b . x .= TRUE ;
for e being set st e in A1 ex X1 being Subset of X st e = X1 & X1 is open & [: X1 , Y1 :] c= [: X1 , Y1 :] & [: X1 , Y1 :] c= [: Y1 , Y2 :] & [: X1 , Y1 :] c= [: Y1 , Y2 :]
for i be set st i in the carrier of S for f being Function of [: S1 , S2 :] , S1 . i st f = H . i & f . i = f | [: S1 , S2 :] holds F . i = f | [: S2 , S2 :] . i
for v , w st for y st x <> y holds w . y = v . y holds J . ( v . y ) = Valid ( VERUM ( Al , J ) , J ) . ( v . y ) = Valid ( VERUM ( Al , J ) , J ) . ( v . y )
card D = card D1 + card D2 - 1 .= card D1 + card D2 - 1 .= card D1 + 1 - 1 .= len D1 + 1 - 1 .= len D1 + 1 - 1 .= len D1 - 1 + 1 - 1 .= len D1 - 1 + 1 - 1 .= len D1 - 1 + 1 - 1 .= len D1 - 1 .= len D1 + 1 - 1 .= len D1 - 1 + 1 .= len D1 - 1 ;
IC Exec ( i , s ) = ( s +* ( 0 .--> s ) ) . 0 .= s . 0 .= s . 0 .= s . 0 .= s . 0 .= s . 0 .= s . 0 .= s . 0 .= s . 0 .= s . 0 ;
len f /. ( \downharpoonright i1 -' 1 ) + 1 = len f -' 1 + 1 .= len f -' 1 + 1 .= len f -' 1 + 1 .= len f -' 1 + 1 .= len f -' 1 + 1 .= len f -' 1 + 1 .= len f -' 1 + 1 .= len f -' 1 + 1 ;
for a , b , c being Element of NAT st 1 <= a & 2 <= b & k < n holds a <= b or a <= b + c or a = b + c or a = b + c
for f being FinSequence of TOP-REAL 2 , p being Point of TOP-REAL 2 st p in LSeg ( f , i ) & p in LSeg ( f , i ) holds Index ( p , f ) <= i & Index ( p , f ) + 1 <= Index ( p , f )
( ( curry ( P , yielding ) ) # x + ( ( curry ( P , -19 ) ) # x ) . x = ( lim ( ( curry ( P , -19 ) ) # x ) + ( lim ( ( curry ( P , -19 ) ) # x ) ) . x ;
z2 = g /. ( i -' n1 + 1 ) .= g /. ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) .= g . ( i -' n1 + 1 ) ;
[ f . 0 , f . 3 ] in id ( the carrier of G ) \/ ( the InternalRel of G ) or [ f . 0 , f . 3 ] in id ( the carrier of G ) ;
for G being Subset-Family of B st G = { R [ X ] where R is Subset of A , R is Subset of A ( ) : R in F ( ) & R in F ( ) & R in F ( ) holds ( Intersect ( R , R ) ) . X = Intersect ( R , R ) . X
CurInstr ( P1 , Comput ( P1 , s1 , m ) ) = CurInstr ( P1 , Comput ( P1 , s1 , m ) ) .= CurInstr ( P1 , Comput ( P2 , s2 , m ) ) .= CurInstr ( P2 , Comput ( P2 , s2 , m ) ) .= CurInstr ( P2 , Comput ( P2 , s2 , m ) ) .= CurInstr ( P2 , Comput ( P2 , s2 , m ) ) .= halt SCMPDS ;
assume that a on M and b on M and c on N and d on N and p on M and d on N and p on M and a on M and p on M and a on N and p on M and a on M and b on N and c on N and d on N and a on M and b on N and a on N and b on N and c on N and a on N and a on M and b on N and c 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 a on N and a on N and a on N and a on N and c on N and a on N and a on M and a on M and c on N and d on N and d on N and d on N and c on N and d on N and d on N
assume that T is \hbox { T _ 4 } and F is closed and ex F being Subset-Family of T st F is closed & for n being Nat holds F . n is finite-ind & for n being Nat holds F . n <= 0 and ind T <= n and ind T <= n and ind T <= n and ind T <= n and ind T <= n ;
for g1 , g2 st g1 in ]. r - r , r .[ & g2 in ]. r - r , r .[ & g1 < g2 holds |. f . g1 - f . g2 .| <= ( r1 - r ) / 2 * ( r1 - r )
( ( - 1 ) * ( ( - 1 ) * ( - 1 ) ) + ( - 1 ) * ( - 1 ) = ( - 1 ) * ( - 1 ) * ( - 1 ) + ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) ;
F . i = F /. i .= 0. R + r2 .= b |^ ( n + 1 ) .= b |^ ( n + 1 ) .= b |^ ( n + 1 ) .= b |^ ( n + 1 ) .= b |^ ( n + 1 ) .= b |^ ( n + 1 ) ;
ex y being set , f being Function st y = f . n & dom f = NAT & f . 0 = A ( ) & for n being Nat holds f . ( n + 1 ) = R ( n , f . n ) & f . ( n + 1 ) = R ( n , f . n ) ;
func f (#) F -> FinSequence of V means : Def1 : len it = len F & for i be Nat st i in dom F holds it . i = F . i * f . ( i - 1 ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 } = { x1 , x2 , x3 , x4 , x5 , x5 } \/ { x4 , x5 , x5 } \/ { x5 , x5 , x5 } \/ { x5 , x5 , x5 } \/ { x1 , x2 , x3 } \/ { x2 , x3 , x4 } \/ { x4 , x5 , x5 } \/ { x1 , x2 , x4 } \/ { x2 , x4 } ;
for n being Nat , x being set st x = h . n holds h . ( n + 1 ) = o . ( x , n ) & x in InputVertices ( S1 +* S2 ) & o . ( n + 1 ) in InnerVertices S1 & o . ( n + 1 ) in InnerVertices S2 ;
ex S1 being Element of QC-WFF ( Al ) st SubP ( P , l , e ) = S1 & ( for i holds S1 . i = S2 . i ) & ( for i holds S1 . i = S2 . i ) & ( for i holds S1 . i = S2 . i ) & ( for i holds S1 . i = S2 . i ) ;
consider P being FinSequence of GT2 such that p9 = product P and for i being Element of dom P ex t being Element of the carrier of K st P . i = t & ex i being Element of dom t st t = ( the multF of K ) . i & t . i = i and t . i = i and t . i = i and t . i = j ;
for T1 , T2 being strict non empty TopSpace , P being Basis of T1 , Q being Basis of T2 st the topology of T1 = the topology of T2 & the topology of T2 = the topology of T2 & the topology of T2 = the topology of T2 holds P is Basis of T2
assume that f is_partial_differentiable_in u0 , 3 and r (#) pdiff1 ( 3 , 3 ) is_differentiable_in u0 and r (#) pdiff1 ( 3 , 3 ) is_differentiable_in z0 and SVF1 ( 3 , 1 , 3 ) . z0 = r * SVF1 ( 3 , 1 , 3 ) . z0 ;
defpred P [ Nat , FinSequence of bool REAL ] means for F , G being Permutation of bool $1 st len F = $1 & G = F & G = F & F = G holds Sum ( F ^ G ) = Sum ( F ) * Sum ( G ) & Sum ( F ) = Sum ( G ) * Sum ( G ) ;
ex j st 1 <= j & j < width GoB f & ( GoB f ) * ( 1 , j ) `2 <= s & s <= ( GoB f ) * ( 1 , j + 1 ) `2 & ( GoB f ) * ( 1 , j + 1 ) `2 <= s & s < ( GoB f ) * ( 1 , j + 1 ) `2 ;
defpred U [ set , set ] means ex Fa1 being Subset-Family of T st $1 = Fa1 & for F being Subset-Family of T st F = Fa1 & F is discrete holds union F is discrete & union F is discrete & union F c= union ( Fa1 /\ the carrier of T ) ;
for p4 being Point of TOP-REAL 2 st LE p4 , p4 , P , p1 , p2 & LE p4 , p2 , P , p1 , p2 holds LE p4 , p2 , P , p1 , p2 & LE p4 , p1 , P , p1 , p2 & LE p2 , p1 , P , p1 , p2 & LE p1 , p2 , P , P , p1 , p2 & LE p1 , p2 , P , p1 , p2
f in St ( E , H ) & for g st g in f . y holds x = g . y iff for y st y in f . y holds f . y = f . ( y , x )
ex 8 being Point of TOP-REAL 2 st x = p2 & ( sqrt ( 1 - ( ( q `2 / |. q .| - cn ) / ( 1 + cn ) ) ^2 ) >= 8 & ( sqrt ( 1 - cn ) ) ^2 >= 8 & ( sqrt ( ( q `2 / |. q .| - cn ) / ( 1 + cn ) ) ^2 >= 8 ;
assume for d being Element of NAT st d <= ( n - 1 ) holds s1 . ( n + 1 ) = s2 . ( n + 1 ) & s2 . ( n + 1 ) = s2 . ( n + 1 ) & s1 . ( n + 1 ) = s2 . ( n + 1 ) ;
assume that s <> t and s is Point of Closed-Interval-TSpace ( x , r ) and s is Point of Closed-Interval-TSpace ( x , r ) and not ex e being Point of Closed-Interval-TSpace ( x , r ) st e = |[ s , r ]| and not e in Ball ( x , r ) /\ Ball ( y , r ) and e = |[ s , r ]| and e <> s and e <> s and e <> s and e <> s and s <> t and s <> t and t <> t and t <> t and t <> t and t <> t and t <> t and t <> t and t <> t and t <> t and t <> t and t <> t and t <> t and t <> t and t <> t and t <> t and t <> s in Ball ( x in Ball ( x , r and t in Ball ( x , r and t in Ball ( x , r and t in Ball ( x , r ) and t
given r such that 0 < r and for s being Point of C1 holds 0 < s or ex x1 , x2 be Point of C1 st x1 in dom f & x2 in dom f & ||. x1 - x2 .|| < s & ||. x1 - x2 .|| < s & ||. f /. x1 - f /. x2 .|| < r ;
( p | x ) | ( ( x | x ) | ( x | x ) ) = ( ( x | x ) | ( x | x ) ) | ( x | x ) ;
assume that x , x + h in dom ( sec * sec ) and ( there exists $ h st ( \HM { the } \HM { function } ) = ( 2 * sec ) * ( sin * cos ) + ( 2 * cos ) * sin ) and for x st x in dom ( ( 2 * cos ) * sin ) holds ( ( 2 * cos ) `| Z ) . x = ( 4 * sin ) . x + ( 2 * cos . x ) ^2 ;
assume that i in dom A and len A > 1 and len A > 1 and width B > 1 and width B = 1 and width B = 1 and width B = 1 and width B = 1 and width B = 1 and width B = 1 and width B = 1 and width B = 1 and width B = 1 and width B = 1 and width B = 1 and width B = 1 and width B = 1 and width B = 1 and width B = 1 and len A = 1 and width B = 1 and width B = 1 and width B = 1 and width B = 1 and width B = 1 and width B = 1 and width B = 1 and width B = 1 and width B = 1 and width B = 1 and len A = 1 and width B = 1 and len A = 1 and len A = 1 and len A = 1 and len A = 1 and len A = 1 and len A = 1 and width B
for i being non zero Element of NAT st i in Seg n holds i divides n or i divides n or h = n implies h = ( 1. F_Complex ( n , L ) ) * ( i |^ n ) & ( i divides n implies h = ( 1. F_Complex ( n , L ) ) * ( i |^ n )
( ( b1 'imp' b2 ) '&' ( c1 '&' c2 ) ) '&' ( ( b1 'imp' b2 ) '&' ( c1 '&' c2 ) '&' ( c2 '&' c2 ) '&' ( c2 '&' c2 ) '&' ( c1 '&' c2 ) '&' ( c2 '&' c2 ) '&' ( c2 '&' c2 ) '&' ( c1 '&' c2 ) '&' ( c2 '&' c2 ) '&' ( c2 '&' c2 ) '&' ( c2 '&' c2 ) '&' ( c2 '&' c2 ) '&' ( c2 '&' c2 ) '&' ( c2 '&' c2 ) '&' ( c2 '&' c2 ) '&' ( c2 '&' c2 ) '&' ( c2 '&' c2 ) '&' ( c2 '&' c2 ) '&' ( c2 '&' c2 ) '&' ( c2 '&' c2 ) '&' ( c2 '&' c2 ) '&' ( c2 '&' c2 ) '&' ( c2 '&' c2 ) '&' ( c2 '&' c2 ) '&' ( c2 '&' c2 ) '&' ( c2 '&' c2 ) '&' ( c1 '&' c2 ) '&' ( c2 '&' c2 ) '&' ( c1 '&' c2 ) '&' ( c1 '&' c2 )
assume that for x holds f . x = ( ( - 1 ) (#) ( sin * cos ) ) . x and for x st x in dom ( ( - 1 ) (#) ( sin * cos ) ) holds ( ( - 1 ) (#) ( sin * cos ) ) . x = ( - 1 ) * ( sin . x ) and ( ( - 1 ) (#) ( sin * cos ) ) . x <> 0 ;
consider R8 , I-8 be Real such that R8 = Integral ( M , F . n ) and I8 = Integral ( M , F . n ) and I = Integral ( M , F . n ) and Integral ( M , F . n ) = Integral ( M , F . n ) and Integral ( M , F . n ) = Integral ( M , F . n ) ;
ex k be Element of NAT st ' = k & 0 < d & for q be Element of product G st q in X & 0 < q & ||. q] - f /. x0 .|| < r holds ||. partdiff ( f , q , x ) - partdiff ( f , x , y ) .|| < r
x in { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 } or x in { x1 , x2 , x3 , x4 } or x in { x2 , x3 , x4 } or x in { x1 , x2 , x4 } or x in { x2 , x3 , x4 } or x in { x2 , x4 , x4 } ;
( G * ( j , i ) `2 ) `2 = ( G * ( 1 , i ) `2 ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 .= ( G *
f1 * p = p .= ( the arity of S1 ) . ( ( the arity of S2 ) . o ) .= ( the arity of S2 ) . ( ( the arity of S2 ) . o ) .= ( the arity of S2 ) . ( ( the arity of S2 ) . o ) .= ( the arity of S2 ) . ( ( the arity of S2 ) . o ) .= ( the arity of S2 ) . o ;
func card T , P , T1 , T2 -> Tree means : Def1 : q in T iff q in T & not p in T or ex r st r in T & q = p ^ r or p in T & q in T & r in T & r in T ;
F /. ( k + 1 ) = F . ( k + 1 -' 1 ) .= FH . ( p . ( k + 1 -' 1 ) , k + 1 -' 1 ) .= FH . ( k + 1 -' 1 ) .= FH . ( k + 1 -' 1 ) .= FH . ( k + 1 -' 1 ) .= FH . ( k + 1 -' 1 ) ;
for A , B , C being Matrix of K st len B = len C & width C = width C & width B = width C & width B = width C & width B = width C & width B = width C & width B = width C & width B = width C & width B = width C & width B = width C & width A = width C & width B = width C & width A = width C & width B = width C & width B = width C & width B = width C & width A = width C & width B = width C & width B = width C & width B = width C & width B = width C & width B = width C & width B = width C & width B = width C = width C & width B = width C & width B = width C = width C = width C & width B = width C = width C & width B = width C = width C = width
seq . ( k + 1 ) = 0. F_Complex + seq . ( k + 1 ) .= ( Partial_Sums seq ) . ( k + 1 ) .= ( Partial_Sums seq ) . ( k + 1 ) .= ( Partial_Sums seq ) . ( k + 1 ) + Partial_Sums ( seq ) . ( k + 1 ) .= ( Partial_Sums seq ) . ( k + 1 ) + Partial_Sums ( seq ) . ( k + 1 ) .= ( Partial_Sums seq ) . ( k + 1 ) + Partial_Sums ( seq ) . ( k + 1 ) .= ( Partial_Sums seq ) . ( k + 1 ) . ( k + 1 ) . ( k + 1 ) . ( k + 1 ) . ( k + 1 ) . ( k + 1 ) . ( k + 1 ) . ( k + 1 ) . ( k + 1 ) . ( k + 1 ) . ( k + 1 ) . ( k + 1 ) . ( k + 1 ) . ( k + 1 ) . ( k
assume that x in ( the carrier of C1 ) & y in ( the carrier of C2 ) and z in ( the carrier of C2 ) and x = ( the carrier of C2 ) and y = ( the carrier of C2 ) and z = ( the carrier of C2 ) and x = ( the carrier of C2 ) . z ;
defpred P [ Element of NAT ] means for f st len f = $1 & ( for k st k in $1 holds ( ( VAL g ) . k = ( ( VAL g ) . k ) ) . ( f /. ( $1 + 1 ) ) = ( ( VAL g ) . ( k + 1 ) ) . ( f /. ( $1 + 1 ) ) ;
assume that 1 <= k and k + 1 <= len f and f /. k = G * ( i + 1 , j ) and [ i + 1 , j ] in Indices G and f /. k = G * ( i + 1 , j ) and f /. k = G * ( i + 1 , j ) ;
assume that s < 1 and ( q `1 / |. q .| - cn ) / ( 1 + cn ) >= 0 and ( q `2 / |. q .| - cn ) / ( 1 + cn ) >= 0 and ( p `2 / |. q .| - cn ) / ( 1 + cn ) >= 0 and ( p `2 / |. q .| - cn ) / ( 1 + cn ) >= 0 and ( p `2 / |. q .| - cn ) / ( 1 + cn ) >= 0 and ( p `2 / |. q .| - cn ;
for M being non empty metric , x being Point of M , f being Point of M st x = f holds ex x being Point of M st x = f . ( n + 1 ) & ex f being sequence of TopSpaceMetr \overline ( M ) st for n being Element of M holds f . n = Ball ( x , f . n )
defpred P [ Element of omega ] means f1 . $1 - f2 . ( $1 + 1 ) / ( 2 * $1 ) . x = f1 . ( $1 + 1 ) / ( 2 * $1 - 1 ) . x & ( f1 - f2 ) . x = f1 . ( $1 + 1 ) / ( 2 * $1 - 1 ) . x ;
defpred P1 [ Nat , Real ] means ( for r be Point of C st $1 in Y & ||. $2 - $2 .|| < r & ||. f /. $1 - f /. ( $1 + 1 ) .|| < r holds ||. f /. ( $1 + 1 ) - f /. /. ( $1 + 1 ) .|| < r ;
( f ^ mid ( g , 2 , len g ) ) . i = ( mid ( g , 2 , len g ) ) . i .= g . ( i -' 1 ) .= g . ( i -' 1 ) .= g . ( i -' 1 ) .= g . ( i -' 1 ) .= g . ( i -' 1 ) ;
sqrt ( 1 - 2 * ( n + 2 ) + 2 * ( n + 1 ) ) * ( 2 * ( n + 1 ) ) = ( sqrt ( 1 - 2 * ( n + 1 ) ) ) * ( 2 * ( n + 1 ) ) .= 1 * ( 2 * ( n + 1 ) ) .= 1 * ( 2 * ( n + 1 ) ) .= 1 * ( 2 * ( n + 1 ) ;
defpred P [ Nat ] means for G being non empty finite RelStr , F being ( the carrier of G ) , R being non empty RelStr st G is the carrier of G & the carrier of G = the carrier of G holds the carrier of G = the carrier of R & the carrier of G = the carrier of R & the carrier of F = the carrier of R ;
assume that not f /. 1 in Ball ( u , r ) and 1 <= m and m <= len f and for i st 1 <= i & i <= len f & 1 <= i & i <= len f holds not LSeg ( f , i ) /\ LSeg ( f , i ) <> {} and LSeg ( f , i ) /\ LSeg ( f , i ) <> {} ;
defpred P [ Element of NAT ] means ( Partial_Sums ( cos ) ) . $1 = ( Partial_Sums ( ( cos ) ) . $1 - ( Partial_Sums ( ( cos ) ) . $1 ) * ( Partial_Sums ( ( cos ) ) . $1 ) ;
for x being Element of product F holds x is FinSequence of G & for i being Element of dom x st x in I & x . i in dom ( the _ of F ) & x . i in dom ( the _ of F ) holds x . i in ( the carrier of F ) . i
( x " ) |^ ( n + 1 ) = ( x " ) |^ n * x " .= ( x " ) |^ n * x " .= ( x " ) |^ n * x " .= ( x " ) |^ n * x " .= ( x " ) |^ n .= x " |^ n * x " .= x " |^ n ;
DataPart ( Comput ( P +* I , s , LifeSpan ( P +* I , s ) ) ) = DataPart Comput ( P +* Directed I , Comput ( P +* Directed I , s , LifeSpan ( P +* Directed I , s ) ) ) .= DataPart Comput ( P +* Directed I , s , LifeSpan ( P +* Directed I , Initialize s ) ) .= DataPart Comput ( P +* Directed I , s , LifeSpan ( P +* Directed I , Initialize s ) ;
given r such that 0 < r and ]. x0 - r , x0 .[ c= dom ( f1 (#) f2 ) and for g st g in ]. x0 - r , x0 .[ holds f1 . g <= ( f1 (#) f2 ) . g and for g st g in ]. x0 , x0 + r .[ holds f2 . g <= ( f1 (#) f2 ) . g ;
assume that X c= dom f1 /\ X and f2 | X is continuous and for x st x in X holds ( f1 - f2 ) | X is continuous and for x st x in X holds ( f1 - f2 ) | X is continuous and for x st x in X holds ( f1 - f2 ) | X . x = ( f1 - f2 ) | X . x ;
for L being continuous complete LATTICE for l being Element of L ex X being Subset of L st l = sup X & for x being Element of L st x in X holds x is ` & for x being Element of L st x in X holds x is ` & x is prime
Support ( e *' A ) . i = { m *' ( m *' p ) where m is Element of NAT : m in dom ( m *' p ) & p . i = ( m *' ( m *' q ) ) . i } & ex m being Element of NAT st m in dom ( p *' q ) & p . i = ( m *' q ) . i } ;
( f1 - f2 ) /* s1 is convergent & ( f1 - f2 ) /. ( lim s1 ) = lim ( f1 /* s1 ) - ( f2 /* s1 ) .= lim ( f1 /* s1 ) - ( f2 /* s1 ) .= lim ( f2 /* s1 ) - ( f2 /* s1 ) .= lim ( f2 /* s1 ) .= lim ( f2 /* s1 ) ;
ex p1 being Element of QC-WFF ( Al ) st F . p = g . p1 & F . p = g . ( p `1 ) & for g being Function of QC-WFF ( Al ) , D st g = f . ( p `1 ) holds P [ g , g . ( p `1 ) ] ;
( mid ( f , i , len f -' 1 ) ) /. j = ( mid ( f , i , len f -' 1 ) ) /. j .= ( mid ( f , i , len f -' 1 ) ) /. j .= ( mid ( f , i , len f -' 1 ) ) /. j .= ( mid ( f , i , len f -' 1 ) ) /. j .= ( mid ( f , i , len f -' 1 ) /. j ;
( ( p ^ q ) . ( len p + k ) ) . n = ( ( p ^ q ) . ( len p + k ) ) . n .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . n .= ( p ^ q ) . n ;
len mid ( D2 , D1 , j1 ) + 1 = indx ( D2 , D1 , j1 ) + 1 .= indx ( D2 , D1 , j1 ) + 1 .= indx ( D2 , D1 , j1 ) + 1 .= indx ( D2 , D1 , j1 ) + 1 ;
x * y * z = M5 * ( x * ( y * z ) ) .= ( x * ( y * z ) ) * ( y * z ) .= ( x * ( y * z ) ) * ( y * z ) .= x * ( y * z ) .= x * ( y * z ) ;
v . ( <* x , y *> ) - v . ( <* y *> ) = v . ( <* x , y *> ) * ( <* y *> ) + ( ( proj ( 1 , 1 ) * ( proj ( 1 , 1 ) ) ) . ( <* y *> ) + ( proj ( 1 , 1 ) * ( proj ( 1 , 1 ) ) ) . ( y - x0 ) ;
i * i = <* 0 * ( 1 - i ) - ( 1 - i ) * ( 1 - i ) *> .= <* 0 * ( 1 - i ) - ( 1 - i ) * ( 1 - i ) *> .= <* 0 * ( 1 - i ) - ( 1 - i ) * ( 1 - i ) .= 0 ;
Sum ( L (#) F ) = Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) .= Sum ( L (#) F2 ) .= Sum ( L (#) ( F1 ^ F2 ) .= Sum ( L (#) ( F1 ^ F2 ) .= Sum ( L (#) ( F1 ^ F2 ) .= Sum ( L (#) ( F1 ^ F2 ) .= Sum ( L (#) F2 ) .= Sum ( L (#) F2 ) + Sum ( L (#) ( F1 ^ F2 ) + Sum ( L (#) ( F1 ^ F2 ) + Sum ( F1 ^ F2 ) + Sum ( L (#) F2 ) + Sum ( F1 ^ F2 ) + Sum ( F1 ^ F2 ) .= Sum ( L (#) ( F1 ^ F2 ) + Sum ( L (#) ( F1 ^ F2 )
ex r be Real st for e be Real st 0 < e ex Y be finite Subset of X st Y is non empty & for x be Real st Y in Y & x in Y holds |. ( f | Y ) . x - f . x .| < r
( GoB f ) * ( i + 1 , j ) `2 = f /. ( k + 2 ) & ( GoB f ) * ( i + 1 , j ) `2 = f /. ( k + 2 ) or ( GoB f ) * ( i + 1 , j + 1 ) `2 = f /. ( k + 2 ) or ( GoB f ) * ( i + 1 , j + 1 ) `2 = f /. ( k + 2 ) ;
( ( - 1 ) (#) cos ) . x = ( - 1 ) * ( cos . x ) .= ( - 1 ) * ( cos . x ) .= ( - 1 ) * ( cos . x ) .= ( - 1 ) * ( cos . x ) .= ( - 1 ) * ( cos . x ) .= ( - 1 ) * ( cos . x ) .= ( - 1 ) * ( cos . x ) .= ( - 1 ) * ( cos . x ) .= ( - 1 ) * ( cos . x ) .= ( - 1 ) * ( cos . x ) * ( cos . x ) * ( cos . x ) * ( cos . x ) * ( sin . x ) .= ( - 1 ) * ( sin . x ) * ( sin . x ) * ( sin . x ) * ( sin . x ) * ( sin . x ) * ( sin . x ) * ( sin . x ) * ( sin . x ) * ( sin . x ) * ( sin . x ) * ( sin . x ) * ( sin . x ) * ( sin . x ) .= ( - 1 ) *
( - sqrt ( b - a , c ) ) / 2 + ( - sqrt ( b - a , c ) ) / 2 > 0 & ( - sqrt ( b - a , c ) / 2 < 0 ) implies ( - sqrt ( b - a , c ) / 2 + ( - sqrt ( b - a , c ) / 2 ) / 2 < 0
assume that inf ( \mathopen { \uparrow } X ) = "\/" ( X , L ) and for C being Subset of L st C in X holds C is maximal & for X being Subset of L st X in X holds "\/" ( X , L ) = "/\" ( X , L ) and "\/" ( X , L ) = "\/" ( X , L ) ;
( ( ( for B , i , j ) --> ( id the Sorts of B ) ) . ( i , j ) = ( i , j ) --> ( id the Sorts of B ) ) . ( i , j ) .= ( i |-> ( j , i ) ) . ( i , j ) .= ( i |-> ( j , i ) ) . ( i , j ) .= ( i |-> ( j , i ) ) . ( i , j ) ;