thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
contradiction ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
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thesis ;
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contradiction ;
thesis ;
thesis ;
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thesis ;
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thesis ;
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thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
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thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
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contradiction ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
B ;
a <> c
T c= S
D c= B
c in X ;
b in X ;
X ;
b in D ;
x = e ;
let m ;
h is onto ;
N in K ;
let i ;
j = 1 ;
x = u ;
let n ;
let k ;
y in A ;
let x ;
let x ;
m c= y ;
F is rng ;
let q ;
m = 1 ;
1 < k ;
G is finite ;
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 .= X ;
assume x in I ;
q is as as as Nat ;
assume c in x ;
of p ;
assume x in Z ;
assume x in Z ;
1 <= kr2 ;
assume m <= i ;
assume G is finite ;
assume a divides b ;
assume P is closed ;
b-a > 0 ;
assume q in A ;
W is not bounded ;
f is Assume f is Seg one-to-one ;
assume A is boundary ;
g is special ;
assume i > j ;
assume t in X ;
assume n <= m ;
assume x in W ;
assume r in X ;
assume x in A ;
assume b is even ;
assume i in I ;
assume 1 <= k ;
X is non empty ;
assume x in X ;
assume n in M ;
assume b in X ;
assume x in A ;
assume T c= W ;
assume s is atomic ;
b `1 <= c `1 ;
A meets W ;
i `1 <= j `1 ;
assume H is universal ;
assume x in X ;
let X be set ;
let T be Tree ;
let d be element ;
let t be element ;
let x be element ;
let x be element ;
let s be element ;
k <= 5 + 1 ;
let X be set ;
let X be set ;
let y be element ;
let x be element ;
P [ 0 ]
let E be set , A 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 ;
let a be Real ;
let x be element ;
let X be element ;
P [ 0 ]
let x be element ;
let x be element ;
let y be element ;
r in REAL ;
let e be element ;
n1 is , ;
Q halts_on s ;
x in for of of -1 holds x in \in \in that x
M < m + 1 ;
T2 is open ;
z in b \rm \hbox { - } ;
R2 is well-ordering ;
1 <= k + 1 ;
i > n + 1 ;
q1 is one-to-one ;
let x be trivial set ;
PM is one-to-one ;
n <= n + 2 ;
1 <= k + 1 ;
1 <= k + 1 ;
let e be Real ;
i < i + 1 ;
p3 in P ;
p1 in K ;
y in C1 ;
k + 1 <= n ;
let a be Real , x be Real ;
X |- r => p ;
x in { A } ;
let n be Nat ;
let k be Nat ;
let k be Nat ;
let m be Nat ;
0 < 0 + k ;
f is_differentiable_in x ;
let x0 , r ;
let E be Ordinal ;
o : o <= 4 ;
O <> O2 ;
let r be Real ;
let f be FinSeq-Location ;
let i be Nat ;
let n be Nat ;
Cl A = A ;
L c= Cl L ;
A /\ M = B ;
let V be RealUnitarySpace , M be Subset of V ;
not s in Y |^ 0 ;
rng f is_<=_than w
b "/\" e = b ;
m = m3 ;
t in h . D ;
P [ 0 ] ;
assume z = x * y ;
S . n is bounded ;
let V be RealUnitarySpace , M be Subset of V ;
P [ 1 ] ;
P [ {} ] ;
C1 is component ;
H = G . i ;
1 <= i `2 + 1 ;
F . m in A ;
f . o = o ;
P [ 0 ] ;
aZ <= non < \pi ;
R [ 0 ] ;
b in f .: X ;
assume q = q2 ;
x in [#] V ;
f . u = 0 ;
assume e1 > 0 ;
let V be RealUnitarySpace , M be Subset of V ;
s is trivial & s is non empty ;
dom c = Q
P [ 0 ] ;
f . n in T ;
N . j in S ;
let T be complete LATTICE , X be Subset of T ;
the ObjectMap of F is one-to-one
sgn x = 1 ;
k in support a ;
1 in Seg 1 ;
rng f = X ;
len T in X ;
vbeing < n ;
S\HM is bounded ;
assume p = p2 ;
len f = n ;
assume x in P1 ;
i in dom q ;
let U2 , U1 , U2 ;
pp `1 = c ;
j in dom h ;
let k ;
f | Z is continuous ;
k in dom G ;
UBD C = B ;
1 <= len M ;
p in `1 /\ L~ x ;
1 <= j9 & j9 <= len f ;
set A = st :] ;
card a [= c ;
e in rng f ;
cluster B \oplus A -> empty ;
H is has no ex F ;
assume n0 <= m ;
T is increasing ;
e2 <> e1 & e2 <> e2 ;
Z c= dom g ;
dom p = X ;
H is proper ;
i + 1 <= n ;
v <> 0. V ;
A c= Affin A ;
S c= dom F ;
m in dom f ;
let X0 be set ;
c = sup N ;
R is_connected in union M ;
assume not x in REAL ;
Im f is complete ;
x in Int y ;
dom F = M ;
a in On W ;
assume e in A ( ) ;
C c= C-26 ;
mm <> {} ;
let x be Element of Y ;
let f be ) Int of X , x be element ;
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 + \HM { v } ;
- y in I ;
let A be non empty set , f be Function of A , B ;
Px0 = 1 ;
assume r in F . k ;
assume f is simple ;
let A be \cdot ^ ;
rng f c= NAT ;
assume P [ k ] ;
ff <> {} ;
let o be Ordinal ;
assume x is sum of from squares ;
assume not v in { 1 } ;
let II , A , B ;
assume that 1 <= j and j < l ;
v = - u ;
assume s . b > 0 ;
d1 in d2 ;
assume t . 1 in A ;
let Y be non empty TopSpace , f be Function of Y , X ;
assume a in uparrow s ;
let S be non empty Poset ;
a , b // b , a ;
a * b = p * q ;
assume x , y are_the space ;
assume x in Omega ( f ) ;
[ a , c ] in X ;
mm <> {} ;
M + N c= M + M ;
assume M is \mathclose hhhz ;
assume f is cbrst rst } is crst } ;
let x , y be element ;
let T be non empty TopSpace ;
b , a // b , c ;
k in dom Sum p ;
let v be Element of V ;
[ x , y ] in T ;
assume len p = 0 ;
assume C in rng f ;
k1 = k2 & k2 = k2 ;
m + 1 < n + 1 ;
s in S \/ { s } ;
n + i >= n + 1 ;
assume Re y = 0 ;
k1 <= j1 & k1 <= len f ;
f | A is Sum ;
f . x being Z <= b ;
assume y in dom h ;
x * y in B1 ;
set X = Seg n ;
1 <= i2 + 1 ;
k + 0 <= k + 1 ;
p ^ q = p ;
j |^ y divides m ;
set m = max A ;
[ x , x ] in R ;
assume x in succ 0 ;
a in sup phi ;
Cj in X ;
q2 c= C1 & q2 in C2 ;
a2 < c2 & a2 < c1 ;
s2 is 0 -started ;
IC s = 0 ;
s4 `2 = s4 `2 ;
let V ;
let x , y be element ;
let x be Element of T ;
assume a in rng F ;
x in dom T ` ;
let S be <> of L ;
y " <> 0 ;
y " <> 0 ;
0. V = uw ;
y2 , y , w is_collinear ;
R8 ;
let a , b be Real , x be Point of X ;
let a be Object of C ;
let x be Vertex of G ;
let o be Object of C , a be Object of C ;
r '&' q = P \lbrack l \rbrack ;
let i , j be Nat ;
let s be State of A , n be Nat ;
s4 . n = N ;
set y = ( x `1 ) / ( 1 + x `1 ) ;
mi in dom g & mi in dom g ;
l . 2 = y1 ;
|. g . y .| <= r ;
f . x in C2 ;
V1 is non empty ;
let x be Element of X ;
0 <> f . g2 ;
f2 /* q is convergent ;
f . i is_measurable_on E ;
assume \xi in N-22 ;
reconsider i = i as Ordinal ;
r * v = 0. X ;
rng f c= INT & rng f c= INT ;
G = 0 .--> goto 0 ;
let A be Subset of X ;
assume that A0 is dense and A is dense ;
|. f . x .| <= r ;
let x be Element of R ;
let b be Element of L ;
assume x in W-19 ;
P [ k , a ] ;
let X be Subset of L ;
let b be Object of B ;
let A , B be category ;
set X = Vars ( C ) ;
let o be OperSymbol of S ;
let R be connected non empty Poset ;
n + 1 = succ n ;
x-21 c= Z1 & xY c= Z1 ;
dom f = C1 & dom g = C2 ;
assume [ a , y ] in X ;
Re ( seq ^\ k ) 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 , i be Nat ;
assume r2 > x0 & x0 < r2 ;
let Y be non empty set , f be Function of Y , Z ;
2 * x in dom W ;
m in dom g2 & n + 1 in dom g2 ;
n in dom g1 /\ dom g2 ;
k + 1 in dom f ;
the still of s is finite ;
assume that x1 <> x2 and y1 <> y2 ;
v1 in V1 & v2 in V1 ;
not [ b `1 , b `2 ] in T ;
i-35 + 1 = i ;
T c= and T c= and T is finite ;
( l `1 ) ^2 = 0 ;
n be Nat ;
( t `2 ) ^2 = r ^2 ;
AA is_integrable_on M & f | A is integrable ;
set t = Top t ;
let A , B be real-membered set ;
k <= len G + 1 ;
[: C , C :] misses [: V , C :] ;
product ( s ) is non empty ;
e <= f or f <= e ;
cluster non empty normal for Ordinal ;
assume c2 = b2 & c1 = b1 ;
assume h in [. q , p .] ;
1 + 1 <= len C ;
not c in B . m1 ;
cluster R .: X -> empty ;
p . n = H . n ;
assume that vseq is convergent and lim vseq = 0 ;
IC s3 = 0 & IC s3 = 0 ;
k in N or k in K ;
F1 \/ F2 c= F \/ G ;
Int G1 <> {} & Int G2 <> {} ;
( z `2 ) ^2 = 0 ;
p01 <> p1 & p2 <> p3 ;
assume z in { y , w } ;
MaxADSet ( a ) c= F ;
ex_sup_of downarrow s , S ;
f . x <= f . y ;
let T be up-complete non empty reflexive transitive RelStr ;
q |^ m >= 1 ;
a is_>=_than X & b is_>=_than Y ;
assume <* a , c *> <> {} ;
F . c = g . c ;
G is one-to-one , full ;
A \/ { a } \not c= B ;
0. V = 0. Y .= 0. V ;
I be non empty non empty Instruction of S ;
f-24 . x = 1 ;
assume z \ x = 0. X ;
C2 = 2 to_power n ;
let B be SetSequence of Sigma ;
assume X1 = p .: D ;
n + l2 in NAT ;
f " P is compact ;
assume x1 in [: REAL , REAL :] ;
p1 `1 = K0 & p2 `2 = p2 `2 ;
M . k = <*> REAL ;
phi . 0 in rng phi ;
OSMas is closed ;
assume z0 <> 0. L & z0 <> 0. L ;
n < ( N . k ) ;
0 <= ( seq . 0 ) . 0 ;
- q + p = v ;
{ v } is Subset of B ;
set g = f `| 1 ;
[: the carrier of R , the carrier of R :] is stable ;
set cR = Vertices R , R = Vertices R ;
px0 c= P3 & px0 c= P3 ;
x in [. 0 , 1 .[ ;
f . y in dom F ;
let T be Scott Scott Scott Scott Scott of S ;
ex_inf_of the carrier of S , S ;
A2 = downarrow b & a in the carrier of S ;
P , C , K is_collinear ;
assume x in F ( s , r , t ) ;
2 to_power i < 2 to_power m ;
x + z = x + z + q ;
x \ ( a \ x ) = x ;
||. x-y .|| <= r ;
assume that Y c= field Q and Y <> {} ;
a ~ , b ~ are_equipotent ;
assume a in A ( i ) ;
k in dom ( q | k ) ;
p is \HM of } is FinSequence of S ;
i -' 1 = i-1 ;
f | A is one-to-one ;
assume x in f .: X ( ) ;
i2 - i1 = 0 & i2 - i2 = 0 ;
j2 + 1 <= i2 & j2 + 1 <= i2 ;
g " * a in N ;
K <> { [ {} , {} ] } ;
cluster strict for REAL ;
|. q .| ^2 > 0 ;
|. p4 .| = |. p .| ;
s2 - s1 > 0 & s2 - s1 > 0 ;
assume x in { Gij } ;
W-min ( C ) in C & W-min ( C ) in C ;
assume x in { Gij } ;
assume i + 1 = len G ;
assume i + 1 = len G ;
dom I = Seg n .= dom F ;
assume that k in dom C and k <> i ;
1 + 1-1 <= i + j ;
dom S = dom F .= dom G ;
let s be Element of NAT ;
let R be ManySortedSet of A ;
let n be Element of NAT ;
let S be non empty non void non void non empty non void ManySortedSign ;
let f be ManySortedSet of I ;
let z be Element of COMPLEX , x be Element of COMPLEX ;
u in { ag } ;
2 * n < 2 * n ;
let x , y be set ;
B-11 c= [: V1 , V1 :] ;
assume I is_closed_on s , P & I is_halting_on s , P ;
U2 = U2 & U2 = U2 implies U2 is open
M /. 1 = z /. 1 ;
x11 = x22 & x22 = x22 ;
i + 1 < n + 1 + 1 ;
x in { {} , <* 0 *> } ;
( f | n ) . x <= ( f | n ) . x ;
let l be Element of L ;
x in dom ( F | k ) ;
let i be Element of NAT ;
r8 is COMPLEX -valued ;
assume <* o2 , o *> <> {} ;
s . x |^ 0 = 1 ;
card K1 in M & card K1 in card M ;
assume that X in U and Y in U ;
let D be Subset-Family of Omega ;
set r = - { k + 1 } ;
y = W . ( 2 * x ) ;
assume dom g = cod f & cod g = cod f ;
let X , Y be non empty TopSpace , f be Function of X , Y ;
x \oplus A is interval ;
|. <*> A .| . a = 0 ;
cluster strict for SubLattice of L ;
a1 in B . s1 & a2 in B . s2 ;
let V be finite over F , A be Subset of V ;
A * B on B , A ;
f-3 = NAT --> 0 ;
let A , B be Subset of V ;
z1 = P1 . j & z2 = P1 . j ;
assume f " P is closed ;
reconsider j = i as Element of M ;
let a , b be Element of L ;
assume q in A \/ ( B "\/" C ) ;
dom ( F (#) C ) = o ;
set S = INT , X = INT ;
z in dom ( A --> y ) ;
P [ y , h . y ] ;
{ x0 } c= dom f /\ dom g ;
let B be non-empty ManySortedSet of I , A be ManySortedSet of I ;
( PI / 2 ) < Arg z ;
reconsider z9 = 0 , z9 = 1 as Nat ;
LIN a , d , c & LIN a , d , c ;
[ y , x ] in II ;
( Q ) * ( 1 , 3 ) = 0 ;
set j = x0 gcd m , i = x0 gcd m ;
assume a in { x , y , c } ;
j2 - j2 > 0 & j2 - j2 > 0 ;
I = I as 1 -element & I = 1 ;
[ y , d ] in [: F-8 , F-8 :] ;
let f be Function of X , Y ;
set A2 = ( B - C ) / ( A , B ) ;
s1 , s2 are_\in the carrier of S ;
j1 -' 1 = 0 & j2 -' 1 = 0 ;
set m2 = 2 * n + j ;
reconsider t = t as bag of n ;
I2 . j = m . j ;
i |^ s , n are_relative_prime ;
set g = f | D-21 , h = f | D-21 ;
assume that X is lower and 0 <= r ;
( p1 `1 ) ^2 = 1 ^2 + ( p2 `2 ) ^2 ;
a < p3 `1 & p3 `1 < b ;
L \ { m } c= UBD C ;
x in Ball ( x , 10 ) ;
not a in LSeg ( c , m ) ;
1 <= i1 -' 1 & i1 + 1 <= len f ;
1 <= i1 -' 1 & i1 + 1 <= len f ;
i + i2 <= len h implies i + 1 <= len h
x = W-min ( P ) & x in L~ f ;
[ x , z ] in X ~ ;
assume y in [. x0 , x .] ;
assume p = <* 1 , 2 , 3 *> ;
len <* A1 *> = 1 ;
set H = h . g9 , I = h . x , J = h . y , K = h . z , N = h . x , M = h . y , N
card b * a = |. a .| ;
Shift ( w , 0 ) |= v ;
set h = h2 ** h1 , h1 = h2 ** h2 ;
assume x in X2 /\ ( X1 union X2 ) ;
||. h .|| < d1 & ||. h .|| < s ;
not x in the carrier of f . ( x + 1 ) ;
f . y = F ( y ) ;
for n holds X [ n ] ;
k - l = kl1 - l ;
<* p , q *> /. 2 = q ;
let S be Subset of the carrier of Y ;
let P , Q be + + s ;
Q /\ M c= union ( F | M )
f = b * ( CFS ( S ) ) ;
let a , b be Element of G ;
f .: X is_<=_than f . sup X
let L be non empty transitive reflexive RelStr , X be Subset of L ;
S-20 is x -thesis of x -f1 -basis i
let r be non positive Real ;
M , v |= x \hbox { y } ;
v + w = 0. Z .= 0. Z ;
P [ ( len F ) + 1 ] ;
assume InsCode ( i ) = 8 or InsCode ( i ) = 8 ;
the zero of M = 0 & the zero of M = 0 ;
cluster z * seq -> summable for Real_Sequence ;
let O be Subset of the carrier of C ;
||. f .|| | X is continuous ;
x2 = g . ( j + 1 ) ;
cluster non empty for Element of \ast S ;
reconsider l1 = l- 1 as Nat ;
v4 is Vertex of r2 & v4 is Vertex of G ;
3 is SubSpace of T2 & 3 is SubSpace of T2 ;
Q1 /\ Q19 <> {} & Q1 /\ Q1 <> {} ;
k be Nat ;
q " is Element of X & q is Element of X ;
F . t is set of empty \rm \hbox { - } \rm st M , t is_;
assume that n <> 0 and n <> 1 ;
set en = EmptyBag n , en = EmptyBag n ;
let b be Element of Bags n ;
assume for i holds b . i is commutative ;
x is root of ( p `2 ) ^2 , ( p `2 ) ^2 ;
not r in ]. p , q .[ ;
let R be FinSequence of REAL , x be Element of REAL ;
S7 does not destroy b1 & S7 does not destroy b1 ;
IC SCM R <> a & IC SCM R <> a ;
|. - - |[ x , y ]| .| >= r ;
1 * ( seq ^\ k ) = seq ^\ k ;
let x be FinSequence of NAT , n be Element of NAT ;
let f be Function of C , D , g be Function of C , D ;
for a holds 0. L + a = a
IC s = s . NAT .= succ IC s ;
H + G = F- ( GG ) ;
Cx1 . x = x2 & Cy1 . x = y1 ;
f1 = f , f2 = g , f3 = h ;
Sum <* p . 0 *> = p . 0 ;
assume v + W = v + u + W ;
{ a1 } = { a2 } & { a2 } = { a2 } ;
a1 , b1 _|_ b , a ;
d1 , o _|_ o , a3 & d1 , o _|_ a3 , o ;
II is reflexive & II is transitive ;
IO is antisymmetric implies [: the carrier of X , the carrier of Y :] is antisymmetric
upper_bound rng H1 = e & upper_bound rng H2 = e ;
x = ( a * ( - 1 ) ) * ( - 1 ) ;
|. p1 .| ^2 >= 1 ;
assume j2 -' 1 < j2 & j2 + 1 < j2 ;
rng s c= dom f1 /\ dom f2 & rng s c= dom f1 /\ dom f2 ;
assume that support a misses support b and a <> b ;
let L be associative commutative non empty doubleLoopStr , p be Polynomial of L ;
s " + 0 < n + 1 ;
p . c = ( f " ) . 1 ;
R . n <= R . ( n + 1 ) ;
Directed I1 = I1 , I2 = I2 , I2 = I2 ;
set f = + ( x , y , r ) ;
cluster Ball ( x , r ) -> bounded ;
consider r be Real such that r in A ;
cluster non empty -> NAT -defined for Function ;
let X be non empty directed Subset of S ;
let S be non empty full SubRelStr of L ;
cluster <* [ ] , 0 ] *> -> complete non trivial ;
( 1 - a " ) * ( 1 - a " ) = a ;
( q . {} ) `1 = o ;
n - ( i - 1 ) > 0 ;
assume ( 1 / 2 ) <= t `1 & t `2 <= 1 ;
card B = k + 1-1 ;
x in union rng ( f | X ) ;
assume x in the carrier of R & y in the carrier of S ;
d in X ;
f . 1 = L . ( F . 1 ) ;
the vertices of G = { v } & { v } c= the vertices of G ;
let G be let G be } wwfinite _Graph ;
e , v6 be set , x be set ;
c . ( i - 1 ) in rng c ;
f2 /* q is divergent_to+infty & f2 /* q is divergent_to+infty ;
set z1 = - z2 , z2 = - z1 , z2 = - z2 , z1 = - z2 , z2 = - z1 , z2 = - z2 ;
assume w is \mathop llas of S , G ;
set f = p |-count ( t - p ) , g = p |-count ( t - p ) , h = p |-count ( t - p ) , n = p |-count ( t - p ) , n
let c be Object of C ;
assume ex a st P [ a ] ;
let x be Element of REAL m , y be Element of REAL m ;
let IX be Subset-Family of X , f be Function of X , Y ;
reconsider p = p as Element of NAT ;
let v , w be Point of X ;
let s be State of SCM+FSA , I be Program of SCM+FSA ;
p is finite implies p is non empty & p is non empty
stop I ( ) c= PGij ( ) ;
set ci = f^ ( f /. i ) ;
w ^ t ^ s ^ ( p ^ q ) ^ q ^ s ^ p ^ q ^ q ^ ( p ^ q ) ^ ( p ^ q ) ^ ( p ^ q ) ^ ( p
W1 /\ W = W1 /\ W ` .= the carrier of W1 ;
f . j is Element of J . j ;
let x , y be \rm \rm \rm \hbox { - } of T2 ;
ex d st a , b // b , d ;
a <> 0 & b <> 0 implies c <> 0
ord x = 1 & x is not a ;
set g2 = lim ( seq ^\ k ) , g1 = lim ( seq ^\ k ) ;
2 * x >= 2 * ( 1 / 2 ) ;
assume ( a 'or' c ) . z <> TRUE ;
f (*) g in Hom ( c , c ) ;
Hom ( c , c + d ) <> {} ;
assume 2 * Sum ( q | m ) > m ;
L1 . F-21 = 0 & L1 . F-21 = 1 ;
x1 \/ R1 = x1 & y1 = x2 ;
( ( sin (#) sin ) `| Z ) . x <> 0 ;
( ( #Z n ) * ( f ^ ) ) . x > 0 ;
o1 in ( X /\ O2 ) /\ ( X /\ O2 ) ;
e , v6 be set , x be set ;
r3 > ( 1 - 2 ) * 0 ;
x in P .: ( F -ideal ( L ) ) ;
let J be closed non empty Subset of R ;
h . p1 = f2 . O & h . O = g2 . I ;
Index ( p , f ) + 1 <= j ;
len ( q | i ) = width M .= len ( p | i ) ;
the carrier of LK c= A & the carrier of LK c= A ;
dom f c= union rng ( F | X ) ;
k + 1 in support ( ( support ( n ) ) | k ) ;
let X be ManySortedSet of the carrier of S ;
[ x `1 , y `2 ] in ( an \/ { y } ) ;
i = D1 or i = D2 or i = D1 ;
assume a mod n = b mod n & b mod n = b mod n ;
h . x2 = g . x1 & h . x2 = f . x2 ;
F c= 2 -tuples_on the carrier of X
reconsider w = |. s1 .| as Real_Sequence ;
( 1 / m * m + r ) < p ;
dom f = dom ( I --> ( a , b ) ) ;
[#] P-17 = [#] ( ( TOP-REAL 2 ) | K1 ) .= K1 ;
cluster - x -> ExtReal for ExtReal ;
then { d1 } c= A & A is closed ;
cluster TOP-REAL n -> finite-ind for non empty TopSpace ;
let w1 be Element of M , a be Element of M ;
let x be Element of dyadic ( n ) ;
u in W1 & v in W3 implies u + v in W2
reconsider y = y , z = z as Element of L2 ;
N is full SubRelStr of [: T , T :] ;
sup { x , y } = c "\/" c ;
g . n = n to_power 1 .= n ;
h . J = EqClass ( u , J ) ;
let seq be summable sequence of X , n be Nat ;
dist ( x `1 , y ) < ( r / 2 ) ;
reconsider mm = m , mn = n as Element of NAT ;
x- x0 < r1 - x0 & r1 < x0 + r2 ;
reconsider P ` = P ` as strict Subgroup of N ;
set g1 = p * idseq ( q `2 ) , g2 = p * idseq ( q `2 ) ;
let n , m , k be non zero Nat ;
assume that 0 < e and f | A is lower ;
D2 . I8 in { x } & D2 . I8 in { x } ;
cluster subcondensed condensed -> subopen for Subset of T ;
let P be compact non empty Subset of TOP-REAL 2 , p , q be Point of TOP-REAL 2 ;
Gik in LSeg ( cos , 1 ) /\ LSeg ( cos , 1 ) ;
n be Element of NAT , p be Point of TOP-REAL n ;
reconsider S8 = S , S8 = T as Subset of T ;
dom ( i .--> X ` ) = { i } ;
let X be non-empty ManySortedSet of S ;
let X be non-empty ManySortedSet of S ;
op ( 1 ) c= { [ {} , {} ] } ;
reconsider m = mm as Element of NAT ;
reconsider d = x as Element of C ( ) ;
let s be 0 -started State of SCMPDS , a be Int-Location ;
let t be 0 -started State of SCMPDS , Q be State of SCMPDS ;
b , b , x , y , z is_collinear ;
assume that i = n \/ { n } and j = k \/ { k } ;
let f be PartFunc of X , Y ;
x0 >= ( sqrt ( c ^2 - 1 ) / sqrt ( 1 - c ^2 ) ) ;
reconsider t7 = T7 as TopSpace , T7 = T7 as TopSpace ;
set q = h * p ^ <* d *> ;
z2 in U . ( y2 , y1 ) /\ Q2 & z2 in Q /\ Q2 ;
A |^ 0 = { <%> E } & A |^ 0 = { <%> E } ;
len W2 = len W + 2 .= len W + 1 ;
len h2 in dom h2 & len h2 in dom h2 ;
i + 1 in Seg ( len s2 ) ;
z in dom g1 /\ dom f & z in dom f1 /\ dom f2 ;
assume p2 `1 = ( E-max ( K ) ) `1 & p2 `2 <= ( E-max ( K ) ) `1 ;
len G + 1 <= i1 + 1 ;
f1 (#) f2 /* x0 is convergent & lim ( f1 (#) f2 ) = x0 ;
cluster s-10 + s-10 - p -> summable ;
assume that j in dom M1 and i = len M1 ;
let A , B , C be Subset of X ;
let x , y , z be Point of X , p be Point of X ;
b ^2 - 4 * ( a * c ) >= 0 ;
<* xx : <* y *> ^ <* y *> Q Q Q Q Q Q Q Q Q [ x ] } P ;
a , b in { a , b } ;
len p2 is Element of NAT & len p2 = len p1 + 1 ;
ex x being element st x in dom R & y = R . x ;
len q = len ( K (#) G ) .= len G ;
s1 = Initialize Initialized s , P1 = P +* I , P2 = P +* J ;
consider w being Nat such that q = z + w ;
x ` is Element of x implies x ` is Element of X
k = 0 & n <> k or k > n ;
then X is discrete for A is closed ;
for x st x in L holds x is FinSequence ;
||. f /. c .|| <= r1 & ||. f /. c .|| <= 1 ;
c in uparrow p & not c in { p } ;
reconsider V = V as Subset of the TopStruct of TOP-REAL n ;
N , M be \mathbin >= >= >= L ;
then z is_>=_than waybelow x /\ compactbelow y ;
M | M = f & M | [. g , g .] = g ;
( ( ( TOP-REAL 1 ) | 1 ) /. 1 ) = TRUE ;
dom g = dom f /\ X .= dom f /\ X ;
mode : such that G is ^ of W , G ;
[ i , j ] in Indices ( M * ( i , j ) ) ;
reconsider s = x " , t = y " as Element of H ;
let f be Element of ( dom ( f | ( dom f ) ) ) , p be Element of ( dom f ) ;
F1 . ( a1 , - a2 ) = G1 . ( a1 , - a2 ) ;
redefine func Sphere ( a , b , r ) -> compact ;
let a , b , c , d be Real ;
rng s c= dom ( 1 / ( f1 - f2 ) ) ;
curry ' ( F-19 , k ) is additive ;
set k2 = card dom B , k2 = card dom C , k2 = card card D ;
set G = DTConMSA ( X ) ;
reconsider a = [ x , s ] as of G ;
let a , b be Element of ML , f be Function of ML , M ;
reconsider s1 = s , s2 = t as Element of ( S ) * ;
rng p c= the carrier of L & p . ( len p ) = p . ( len p ) ;
let d be Subset of the bound of A ;
( x .|. x = 0 iff x = 0. W ) ;
I-21 in dom stop I & card I-21 = card I + 1 ;
let g be continuous Function of X | B , Y ;
reconsider D = Y as Subset of ( TOP-REAL n ) | D ;
reconsider i0 = len p1 , j1 = len p2 as Integer ;
dom f = the carrier of S & dom g = the carrier of S ;
rng h c= union ( ( the carrier of J ) --> { x } )
cluster All ( x , H ) -> non empty for set ;
d * N1 / ( 2 * n ) > N1 * 1 ;
]. a , b .[ c= [. a , b .] ;
set g = f " D1 , h = f " D2 , f = f " D2 ;
dom ( p | mm1 ) = mm1 .= dom ( p | mm1 ) ;
3 + - 2 <= k + - 2 & - 2 <= k + - 2 ;
tan is_differentiable_in ( ( arccot * arccot ) `| Z ) . x ;
x in rng ( f /^ ( n + 1 ) ) ;
let f , g be FinSequence of D ;
[: p , q :] in the carrier of S1 & [: p , q :] in the carrier of S2 ;
rng f " = dom f & rng f = dom f ;
( the 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 ) & v in rng ( S | E1 ) ;
assume x is root or x is root & x is root ;
assume 0 in rng ( g2 | A ) & 0 < r ;
let q be Point of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
let p be Point of TOP-REAL 2 , a , b be Real ;
dist ( O , u ) <= |. p2 .| + 1 ;
assume dist ( x , b ) < dist ( a , b ) ;
<* S7 *> is_in the carrier of C-20 ( C ) ;
i <= len ( G-6 /^ 1 ) + 1 ;
let p be Point of TOP-REAL 2 , a , b be Real ;
x1 in the carrier of I[01] & x2 in the carrier of I[01] ;
set p1 = f /. i , p2 = f /. j ;
g in { g2 : r < g2 & g2 < x0 } ;
Q2 = Sthesis " ( Q ) .= ( the carrier of S ) \/ Q ;
( ( 1 / 2 ) (#) ( 1 / 2 ) ) (#) ( 1 / 2 ) is summable ;
- p + I c= - p + A implies - p + I c= - p + I
n < LifeSpan ( P1 , s1 ) + 1 & I ( ) . a = s . a ;
CurInstr ( p1 , s1 ) = i .= CurInstr ( p2 , s2 ) ;
A /\ ( Cl { x } \ { x } ) <> {} ;
rng f c= ]. r , r + 1 .[ /\ dom f ;
let g be Function of S , V ;
let f be Function of L1 , L2 , g be Function of L2 , L1 ;
reconsider z = z as Element of CompactSublatt L , x be Element of L ;
let f be Function of S , T ;
reconsider g = g as Morphism of c opp , b opp ;
[ s , I ] in [: S , A :] ;
len ( the connectives of C ) = 4 & len ( the connectives of C ) = 5 ;
let C1 , C2 be subcategory of C , a be object of C1 , b be object of C2 ;
reconsider V1 = V , V1 = V as Subset of X | B ;
attr p is valid means : Def3 : All ( x , p ) is valid ;
assume that X c= dom f and f ^ \circ X c= dom g ;
H |^ a " is Subgroup of H & H |^ a = H ;
let A1 be p1 of O , E be Element of E ;
p2 , r3 , q2 is_collinear & q1 , q2 , q2 is_collinear ;
consider x being element such that x in v ^ K ;
not x in { 0. TOP-REAL 2 } \/ { 0. TOP-REAL 2 } ;
p in [#] ( I[01] | B11 ) & p in [#] ( I[01] | B11 ) ;
0 in M . ( E . n ) & M . ( E . n ) < M . ( E . n ) ;
^ ( c , b ) / ( c , a ) = c ;
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . s2 ) & a2 in dom ( F . s2 ) ;
cluster -> on \mathbin ' -' being for Lattice of L ;
set i1 = the Nat , i2 = the Element of NAT , x = [ x , y ] ;
let s be 0 -started State of SCM+FSA , a be Int-Location ;
assume y in ( f1 \/ f2 ) .: A ;
f . ( len f ) = f /. ( len f ) ;
x , f . x '||' f . x , f . y ;
attr X c= Y means : Def3 : cos ( X ) c= cos ( Y ) ;
let y be upper Subset of Y , x be Element of Y ;
cluster -> non \rm \rm A1 -> non \rm \rm \rm \rm \hbox { - } \sum f -> non empty ;
set S = <* Bags n , i *> , T = <* i *> , S = <* i *> , T = <* i *> , S = <* i *> , T = <* i *> , T = <* i *> , S = i ,
set T = [. 0 , 1 / 2 .] , S = [. 1 / 2 , 1 .] ;
1 in dom mid ( f , 1 , 1 ) ;
( 4 * PI ) / PI < ( 2 * PI ) / PI ;
x2 in dom f1 /\ dom f & x2 in dom f1 /\ dom f2 ;
O c= dom I & { {} } = { {} } ;
( the Source of G ) . x = v & ( the Target of G ) . x = v ;
{ HT ( f , T ) } c= Support f \/ Support ( f , T ) ;
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 + ( p1 `2 ) ^2 ;
i + 1 <= len Cage ( C , n ) ;
len <* P *> @ = len P & len <* P *> = 1 ;
set N-26 = the Element of the seq of N , N-26 = the seq of N ;
len g\vert + ( x + 1 ) - 1 <= x ;
not a on B & b on B & c on B ;
reconsider r-12 = r * I . v as FinSequence of REAL ;
consider d such that x = d and a D is_less_than c ;
given u such that u in W and x = v + u ;
len f /. ( \downharpoonright n ) = len ( n |-> p ) ;
set q2 = Int L~ Cage ( C , n ) , q2 = ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ;
set S =
MaxADSet ( b ) c= MaxADSet ( P /\ Q ) ;
Cl ( G . q1 ) c= F . r2 & Cl ( G . q2 ) c= G . q2 ;
f " D meets h " ( V /\ W ) & f " D meets h " ( V /\ W ) ;
reconsider D = E as non empty directed Subset of L1 ;
H = ( the_left_argument_of H ) '&' ( the_right_argument_of H ) & ( the_right_argument_of H ) '&' ( the_right_argument_of H ) = ( the_right_argument_of H ) '&' ( the_right_argument_of H )
assume t is Element of ( S , X ) . s & t is Element of ( S , X ) . s ;
rng f c= the carrier of S2 & rng g c= the carrier of S2 ;
consider y being Element of X such that x = { y } ;
f1 . ( a1 , b1 ) = b1 & f1 . ( b1 , b2 ) = b2 ;
the carrier' of G ` = E \/ { E } .= { E } ;
reconsider m = len ( thesis - k ) as Element of NAT ;
set S1 = LSeg ( n , UMP C ) , S2 = LSeg ( n , UMP C ) ;
[ i , j ] in Indices ( M1 + M2 ) ;
assume that P c= Seg m and M is \HM { i } ;
for k st m <= k holds z in K . k ;
consider a being set such that p in a and a in G ;
L1 . p = p * L /. 1 .= p * L /. 1 ;
p-7 . i = p1 . i & p1 . i = p1 . i ;
let PA , G be a_partition of Y , A be Subset of Y ;
pred 0 < r & r < 1 implies 1 < ( 1 - r ) / 2 ;
rng ( ( a , X ) --> ( a , b ) ) = [#] X ;
reconsider x = x , y = y as Element of K ;
consider k such that z = f . k and n <= k ;
consider x being element such that x in X \ { p } ;
len ( ( canFS ( s ) ) | ( n + 1 ) ) = card ( n + 1 ) ;
reconsider x2 = x1 , y2 = x2 as Element of L2 ;
Q in FinMeetCl ( the topology of X ) & Q is finite ;
dom ( f | u ) c= dom ( f | u ) & dom ( f | v ) c= dom f ;
redefine pred n divides m & m divides n implies n = m ;
reconsider x = x , y = y as Point of [: I[01] , I[01] :] ;
a in ' ( the carrier of T2 , the carrier of T2 ) ;
not y0 in the still of f & not y0 in the carrier of f ;
Hom ( ( a ~ ) , c ) <> {} ;
consider k1 such that p " < k1 and k1 < len f ;
consider c , d such that dom f = c \ d and dom f = c \/ d ;
[ x , y ] in [: dom g , dom k :] ;
set S1 = Let x , S2 = y , S2 = z ;
l1 = m2 & l2 = i2 & l2 = j2 implies l1 = i2
x0 in dom ( u + v ) /\ ( dom ( v + u ) ) ;
reconsider p = x , q = y , r = z as Point of Euclid 2 ;
I[01] = R^1 | ( B , B ) & R^1 | ( B , B ) is continuous ;
f . p4 `2 <= f . p1 `2 & f . p2 `2 <= f . p1 `2 ;
( ( F . x ) `1 ) ^2 <= ( ( F . x ) `2 ) ^2 ;
( x `2 ) ^2 = ( W `2 ) ^2 + ( W `2 ) ^2 ;
for n being Element of NAT holds P [ n ] implies P [ n + 1 ]
let J , K be non empty Subset-Family of I ;
assume 1 <= i & i <= len <* a " *> ;
0 |-> a = <*> the carrier of K .= ( 0 -tuples_on the carrier of K ) --> a ;
X . i in 2 -tuples_on A . i \ B . i ;
<* 0 *> in dom ( e --> [ 1 , 0 ] ) ;
then P [ a ] & P [ succ a ] & Q [ succ a ] ;
reconsider s\mathclose = s\hbox { a } as w of D , a be Element of D ;
( - i ) <= len ( - j ) ;
[#] S c= [#] the TopStruct of T & [#] S is SubSpace of the TopStruct of T ;
for V being strict RealUnitarySpace holds V in being Subspace of V implies V in the carrier of W
assume k in dom mid ( f , i , j ) & k + 1 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 = a * b & - a * - b = - a * b ;
for A being Subset of AS holds A // A implies A is being_line
( for o2 being object of o2 st o2 in <^ o2 , o2 ^> holds [ o2 , o2 ] in f . ( o1 , o2 ) ) ;
then ||. x .|| = 0 implies x = 0. X & x = 0. X ;
let N1 , N2 be strict normal Subgroup of G , a be Element of G ;
j >= len upper_volume ( g , D1 ) & indx ( D2 , D1 , j ) <= len upper_volume ( g , D1 , j ) ;
b = Q . ( len Q - 1 + 1 ) .= Q . ( len Q - 1 ) ;
f2 * f1 /* s is divergent_to+infty & f2 * f1 is divergent_to+infty ;
reconsider h = f * g as Function of [: N2 , I[01] :] , G ;
assume that a <> 0 and Let a , b , c ;
[ t , t ] in the Relation of A & [ t , t ] in the carrier of B ;
( v |-- E ) | n is Element of ( T . n ) -tuples_on the carrier of T ;
{} = the carrier of L1 + L2 & { v } is Subset of L1 + L2 ;
Directed I is_closed_on Initialized s , P & Directed I is_halting_on Initialized s , P ;
Initialized p = Initialize ( p +* q ) , p1 = p +* I ;
reconsider N2 = N1 , N2 = N2 as strict net of R2 ;
reconsider Y = Y as Element of \langle Ids L , \subseteq \rangle ;
"/\" ( uparrow p , L ) \ { p } <> p ;
consider j being Nat such that i2 = i1 + j and j in dom f ;
not [ s , 0 ] in the carrier of S2 & [ s , 1 ] in the carrier of S2 ;
mm in ( B '&' C ) '/\' D \ { {} } ;
n <= len ( P + Q ) + len ( P + Q ) ;
( x1 `1 ) ^2 = ( x2 `1 ) ^2 + ( x3 `2 ) ^2 .= ( x2 `1 ) ^2 + ( x3 `2 ) ^2 ;
InputVertices S = { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 } ;
let x , y be Element of FT1 ( n ) , n be Element of NAT ;
p = |[ p `1 , p `2 ]| & p `2 = |[ p `1 , p `2 ]| ;
g * 1_ G = h " * g * h * h .= h " * g * h ;
let p , q be Element of PFuncs ( V , C ) ;
x0 in dom x1 /\ dom x2 & x0 in dom x1 /\ dom x2 ;
( R qua Function ) " = R " * ( R " ) .= R " * ( R " ) ;
n in Seg ( len ( f /^ n ) ) & ( f /^ n ) . n = p ;
for s be Real st s in R holds s <= s2 implies s1 <= s2
rng s c= dom ( f2 * f1 ) /\ dom ( f2 * f1 ) ;
synonym for for for for for for for for X being Subset of 2 , A being Subset of X holds A is finite ;
1_ K * 1_ K = 1_ K & 1_ K * 1_ K = 1_ K ;
set S = Segm ( A , P1 , Q1 ) , T = Segm ( A , P1 , Q1 ) ;
ex w st e = ( w / f ) . w & w in F ;
curry ' ( P+* ( i , k ) ) # x is convergent ;
cluster open -> open for Subset of [: T , T :] ;
len f1 = 1 .= len f3 + 1 .= len f3 + 1 ;
( i * p ) / p < ( 2 * p ) / p ;
let x , y be Element of OSSub ( U0 ) , s be Element of S ;
b1 , c1 // b9 , c & b1 , c1 // b9 , c ;
consider p being element such that c1 . j = { p } ;
assume that f " { 0 } = {} and f is total ;
assume that IC Comput ( F , s , k ) = n and IC Comput ( F , s , k ) = 0 ;
Reloc ( J , card I + 1 ) does not destroy a ;
goto ( card I + 1 ) does not destroy c ;
set m3 = LifeSpan ( p3 , s3 ) , s3 = Comput ( p3 , s3 , 1 ) , P3 = P3 ;
IC SCMPDS in dom Initialize p & IC SCMPDS in dom Initialize p & IC SCMPDS in dom Initialize p ;
dom t = the carrier of SCM & dom t = the carrier of SCM & t . {} = s . {} ;
( ( E-max L~ f ) .. f ) .. f = 1 ;
let a , b be Element of PFuncs ( V , C ) ;
Cl ( union Int F ) c= Cl ( Int union F ) ;
the carrier of X1 union X2 misses ( ( X1 union X2 ) union ( X2 union X1 ) ) ;
assume not LIN a , f . a , g . a , g . a ;
consider i being Element of M such that i = dI and i in A ;
then Y c= { x } or Y = { x } ;
M , v / ( y , x ) / ( y , x ) |= H ;
consider m be element such that m in Intersect ( FF . m ) and x = Intersect ( FF . m ) ;
reconsider A1 = support u1 , A2 = support v1 as Subset of X ;
card ( A \/ B ) = k-1 + ( 2 * 1 ) ;
assume that a1 <> a3 and a2 <> a4 and a3 <> a4 and a4 <> a5 ;
cluster s -\bf -> string for string of S ;
Ln2 /. n2 = Ln2 . n2 .= Ln2 . n2 ;
let P be compact non empty Subset of TOP-REAL 2 , p , q be Point of TOP-REAL 2 ;
assume r-7 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ;
let A be non empty compact Subset of TOP-REAL n , p be Point of TOP-REAL n ;
assume that [ k , m ] in Indices ( ( - 1 ) * ( i , j ) ) ;
0 <= ( ( 1 / 2 ) |^ p ) / p ;
( F . N | E8 ) . x = +infty ;
pred X c= Y means : Def3 : Z c= V & X \ V c= Y \ Z ;
( y `2 ) * ( z `2 ) <> 0. I & ( y `2 ) * ( z `2 ) <> 0. I ;
1 + card ( X-18 ) <= card ( u \/ { v } ) ;
set g = z \circlearrowleft ( ( E-max L~ z ) .. z ) ;
then k = 1 & p . k = <* x , y *> . k ;
cluster -> total for Element of C -function of X , the carrier of S ;
reconsider B = A as non empty Subset of TOP-REAL n , p be Point of TOP-REAL n ;
let a , b , c be Function of Y , BOOLEAN , f be Function of Y , BOOLEAN ;
L1 . i = ( i .--> g ) . i .= g . i .= g . i ;
Plane ( x1 , x2 , x3 , x4 ) c= P & Plane ( x2 , x3 , x4 ) c= P ;
n <= indx ( D2 , D1 , j1 ) + 1 - indx ( D2 , D1 , j1 ) + 1 ;
( ( g2 ) . O ) `1 = - 1 & ( ( g2 ) . I ) `1 = - 1 ;
j + p .. f -' len f <= len f -' len f + 1 - len f ;
set W = W-bound C , E = E-bound C , G = Gauge ( C , n ) ;
S1 . ( a `1 , e `2 ) = a + e .= a `1 ;
1 in Seg width ( ( M * ( ColVec2Mx p ) ) * ( ColVec2Mx p ) ) ;
dom ( i (#) Im ( f , x ) ) = dom Im ( f , x ) ;
that ^2 . ( x `2 ) = W . ( a , *' ( a , p ) ) ;
set Q = ( -> Element of \rm \rm \rm : _ 0 ( g , f , h ) ) ;
cluster -> topological for ManySortedSet of U1 , B be ManySortedSet of U2 ;
attr F = { A } means : Def3 : F is discrete ;
reconsider z9 = \hbox { y } as Element of product ( G . i ) ;
rng f c= rng f1 \/ rng f2 & f1 . 0 = f1 . 1 \/ f2 . 2 ;
consider x such that x in f .: A and x in f .: C ;
f = <*> the carrier of F_Complex & f = <*> the carrier of F_Complex implies f = <*> the carrier of F_Complex
E , j |= All ( x1 , x2 , x3 , x4 ) implies E , j |= H
reconsider n1 = n , n2 = m , n3 = n as Morphism of o1 , o2 ;
assume that P is idempotent and R is idempotent and P ** R = R ** P ;
card ( B2 \/ { x } ) = k-1 + 1 ;
card ( ( x \ B1 ) /\ B1 ) = 0 implies card ( x \ B1 ) = 0
g + R in { s : g-r < s & s < g + r } ;
set q0x0 = ( q , <* s *> ) \mathop { 1 } ;
for x being element st x in X holds x in rng f1 implies x in X
h0 /. ( i + 1 ) = h0 . ( i + 1 ) ;
set mw = max ( B , dom ( R | NAT ) ) ;
t in Seg width ( I ^ ( n , n ) ) & t in Seg n ;
reconsider X = dom f /\ C , Y = dom f /\ C as Element of Fin NAT ;
IncAddr ( i , k ) = <% - l , - l , - l , - l %> + k ;
( S-bound L~ f ) `2 <= ( q `2 ) ^2 & ( q `2 ) ^2 <= ( q `2 ) ^2 ;
attr R is condensed means : Def3 : Int R is condensed & Cl R is condensed ;
pred 0 <= a & a <= 1 & b <= 1 implies a * b <= 1 ;
u in ( ( c /\ ( ( d /\ b ) /\ e ) ) /\ f ) /\ j ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) /\ f ) /\ j ) /\ j ;
len C + - 2 >= 9 + - 3 + - 3 ;
x , z , y is_collinear & x , z , z is_collinear implies x , y , z , x is_collinear
a |^ ( n1 + 1 ) = a |^ n1 * a ;
<* \underbrace ( 0 , \dots , 0 ) *> in Line ( x , a * x ) ;
set y9 = <* y , c *> ;
Fk2 /. 1 in rng Line ( D , 1 ) & Fk2 /. 1 in rng Line ( D , 1 ) ;
p . m joins r /. m , r /. ( m + 1 ) ;
( p `2 ) ^2 = ( f /. i1 ) ^2 + ( f /. i2 ) ^2 ;
W-bound ( X \/ Y ) = W-bound ( X \/ Y ) & W-bound ( X \/ Y ) = W-bound ( X \/ Y ) ;
0 + ( p `2 ) ^2 <= 2 * r + ( p `2 ) ^2 ;
x in dom g & not x in g " { 0 } implies x in dom g
f1 /* ( seq ^\ k ) is divergent_to+infty & f2 /* ( seq ^\ k ) is divergent_to+infty ;
reconsider u2 = u , v2 = v as VECTOR of P`1 ( X , Y ) ;
p |-count ( Product Sgm ( X11 ) ) = 0 & p |-count ( Product Sgm ( X11 ) ) = 0 ;
len <* x *> < i + 1 & i + 1 <= len c ;
assume that I is non empty and { x } /\ { y } = { 0. I } ;
set ik2 = card I + 4 .--> goto 0 , ik2 = goto ( 0 + 4 ) ;
x in { x , y } & h . x = {} ( Ty , Ty ) ;
consider y being Element of F such that y in B and y <= x ` ;
len S = len ( the charact of ( ( the charact of A ) . 0 ) ) .= len the charact of ( A . 0 ) ;
reconsider m = M , i = I , n = N as Element of X ;
A . ( j + 1 ) = B . ( j + 1 ) \/ A . j ;
set N8 = : ( G is : ( G is finite ) & ( G is finite ) ) ;
rng F c= the carrier of gr { a } & rng F c= the carrier of gr { a } ;
P is and P is and for K , n , r being FinSequence st P is every FinSequence of K holds P is FinSequence of K
f . k , f . ( Let ( Let n ) + 1 ) in rng f ;
h " P /\ [#] ( T1 | P ) = f " P /\ [#] ( T1 | P ) ;
g in dom f2 \ ( f2 " { 0 } ) & ( f2 " { 0 } ) " { 0 } c= dom f2 ;
g+ gX /\ dom f1 = g1 " X & gX /\ dom f1 = dom g1 ;
consider n being element such that n in NAT and Z = G . n ;
set d1 = dist ( x1 , y1 ) , d2 = dist ( x2 , y2 ) , d2 = dist ( x2 , y2 ) ;
b `1 + ( 1 / 2 ) < ( 1 / 2 ) + ( 1 / 2 ) ;
reconsider f1 = f , f2 = g as VECTOR of the carrier of X , Y ;
pred i <> 0 means i ^2 mod ( i + 1 ) = 1 ;
j2 in Seg ( len ( g2 . i2 ) ) & j2 = ( g2 . i2 ) . j2 ;
dom ( i ) = dom ( i ) .= a .= dom ( i ) .= a ;
cluster sec | ]. PI / 2 , PI .[ -> one-to-one ;
Ball ( u , e ) = Ball ( f . p , e ) .= Ball ( u , e ) ;
reconsider x1 = x0 , y1 = x1 , y2 = x2 as Point of S ;
reconsider R1 = x , R2 = y , R1 = z as Relation of L , L ;
consider a , b being Subset of A such that x = [ a , b ] ;
( <* 1 *> ^ p ) ^ <* n *> in RH ;
S1 +* S2 = S2 +* S2 +* S2 +* S2 & S2 +* S2 +* S2 = S1 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2
( ( #Z n ) * ( ( #Z n ) * ( ( #Z n ) * ( ( #Z n ) * ( ( #Z n ) * ( ( #Z n ) * ( ( #Z n ) * ( ( #Z n ) * ( ( #Z n ) * ( ( #Z n ) * ( ( #Z n ) * ( (
cluster -> continuous for Function of C , REAL , x be Real ;
set C7 = 1GateCircStr ( <* z , x *> , f3 ) , C8 = 1GateCircStr ( <* z , x *> , f3 ) ;
Ea1 . e2 = E8 . e2 & E8 . e2 = E8 . e2 ;
( ( ( arctan * ( f1 + f2 ) ) `| Z ) = f ;
upper_bound A = ( PI * 3 / 2 ) * ( 2 * PI ) & lower_bound A = 0 ;
F . ( dom f , - F ) is_transformable_to F . ( cod f , - F ) & F . ( dom f , - F ) = F . ( dom f , - F ) ;
reconsider pbeing being Point of TOP-REAL 2 , q = ( q `2 / |. q .| - sn ) / ( 1 + sn ) as Point of TOP-REAL 2 ;
g . W in [#] Y0 & [#] Y0 c= [#] Y0 & [#] Y0 c= [#] Y0 ;
let C be compact non vertical non horizontal Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
LSeg ( f ^ g , j ) = LSeg ( f , j ) \/ LSeg ( g , i ) ;
rng s c= dom f /\ ]. x0 - r , x0 .[ & f . x0 < 0 ;
assume x in { idseq ( 2 ) , Rev ( idseq ( 2 ) ) } ;
reconsider n2 = n , n2 = m , n3 = n + 1 as Element of NAT ;
for y being ExtReal st y in rng seq holds g <= y implies 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 B" = f .: ( the carrier of X1 ) , BX1 = f .: the carrier of X2 ;
ex d being Element of L st d in D & x << d ;
assume that R " ( a ) c= R " ( b ) and R " ( a ) c= R " ( b ) ;
t in ]. r , s .[ or t = r or t = s & t < s ;
z + v2 in W & x = u + ( z + v2 ) ;
x2 |-- y2 iff P [ x2 , y2 ] & P [ x2 , y2 ] & P [ y2 , x2 ] ;
pred x1 <> x2 means : Def3 : |. x1 - x2 .| > 0 & |. x2 - y2 .| > 0 ;
assume p2 - p1 , p3 - p1 - p2 , p4 - p1 - p1 , p1 - p2 - p1 , p3 - p1 - p1 is_collinear ;
set q = ( be non empty v ^ <* 'not' A *> ) ^ <* 'not' A *> ;
let f be PartFunc of REAL-NS 1 , REAL-NS n , x be Point of REAL-NS 1 , r be Real ;
( n mod ( 2 * k ) ) + 1 = n mod k ;
dom ( T * ( succ t ) ) = dom ( succ t ) .= dom ( T . ( t + 1 ) ) ;
consider x being element such that x in wc and x in c ;
assume ( F (#) G ) . v . x3 = v . x3 & ( F (#) G ) . x3 = v . x4 ;
assume that the carrier of D1 c= the carrier of D2 and the carrier of D2 c= the carrier of D1 and the carrier of D2 c= the carrier of D2 ;
reconsider A1 = [. a , b .[ , B1 = [. a , b .] as Subset of R^1 ;
consider y being element such that y in dom F and F . y = x ;
consider s being element such that s in dom o and a = o . s ;
set p = W-min L~ Cage ( C , n ) , G = Gauge ( C , n ) * ( i , 1 ) , G = Gauge ( C , n ) * ( i , 1 ) , G = Gauge ( C , n ) * ( i , 1 ) , G = Gauge ( C , n ) * ( i ,
n1 -' len f + 1 <= len ( g | n1 ) + 1 - len f ;
\lbrace EqClass ( q , O1 ) , [ a , v , a , b ] } = { [ u , v , a , b ] } ;
set C-2 = ( ( n + 1 ) .--> ( G . ( k + 1 ) ) ) . ( k + 1 ) ;
Sum ( L (#) p ) = 0. R * Sum p .= 0. V * Sum p .= Sum ( L (#) p ) ;
consider i being element such that i in dom p and t = p . i ;
defpred Q [ Nat ] means 0 = Q ( $1 ) & $1 + 1 <= len f ;
set s3 = Comput ( P1 , s1 , k ) , P3 = Comput ( P2 , s2 , k ) , s4 = Comput ( P2 , s2 , k + 1 ) , P4 = Comput ( P2 , s2 , k + 1 ) , P4 = Comput ( P2 , s2 , k + 1 ) , P4
let l be variable of k , A , AA be Subset of k , x be element ;
reconsider U2 = union G-24 , G-24 = union G-24 as Subset-Family of [: T , T :] ;
consider r such that r > 0 and Ball ( p `1 , r ) c= Q ` ;
( h | ( n + 2 ) ) /. ( i + 1 ) = p29 ;
reconsider B = the carrier of X1 , C = the carrier of X2 as Subset of X ;
pccc: ( p , c ) = <* - 1 , 1 *> .= ( - 1 ) * ( - 1 ) ;
synonym f is real-valued means : Def3 : rng f c= NAT & rng f c= NAT ;
consider b being element such that b in dom F and a = F . b ;
x10 < card X0 + card Y0 & card ( Y0 \/ Y0 ) <= card X0 + 1 ;
attr X c= B1 means : Def3 : for B st B c= succ B1 holds X c= succ B & X c= B ;
then w in Ball ( x , r ) & dist ( x , w ) <= r ;
angle ( x , y , z ) = angle ( x , y , z ) ;
pred 1 <= len s means : Def3 : for i being Element of NAT holds ( for j being Element of NAT st j <= i holds s . j = F ( j ) ) ;
f[: f , k :] c= f . ( k + ( n + 1 ) ) ;
the carrier of { 1_ G } = { 1_ G } & the carrier of { 1_ G } = { 1_ G } ;
pred p '&' q in TAUT ( A ) means : Def3 : q '&' p in TAUT ( A ) & q '&' p in TAUT ( A ) ;
- ( t `1 / t `2 ) < ( t `2 / t `1 ) / t `1 ;
( ( U1 ) . 1 ) `1 = ( ( U1 /. 1 ) `1 ) `1 .= ( ( U1 /. 1 ) `1 ) `1 ;
f .: ( the carrier of x ) = the carrier of x & f .: ( the carrier of x ) = the carrier of x ;
Indices ( ( n + 1 ) -tuples_on the carrier of K ) = [: Seg n , Seg n :] ;
for n being Element of NAT holds G . n c= G . ( n + 1 )
then V in M .: \square & ex x being Element of M st V = { x } ;
ex f being Element of F-9 st f is is \cup & f is is \setminus & f is \setminus closed ;
[ h . 0 , h . 3 ] in the InternalRel of G & [ h . 2 , h . 3 ] in the InternalRel of G ;
s +* Initialize ( ( intloc 0 ) .--> 1 ) = s3 +* Initialize ( ( intloc 0 ) .--> 1 ) ;
|[ w1 , v1 ]| - |[ w1 , v1 ]| <> 0. TOP-REAL 2 & |[ w1 , v1 ]| - |[ w1 , v1 ]| = |[ w1 , v1 ]| ;
reconsider t = t as Element of INT * , n , m be Element of INT ;
C \/ P c= [#] ( GX | ( [#] GX \ A ) ) \/ [#] GX ;
f " V in ( the topology of X ) /\ D & f " ( the topology of X ) /\ D = D ;
x in [#] ( the carrier of A ) /\ delta ( A ) implies x in [#] ( the carrier of A ) /\ A
g . x <= h1 . x & h . x <= h1 . x implies h . x <= h . x
InputVertices S = { xy , y , z } \/ { xy , y , z } \/ { xy , y , z } ;
for n being Nat st P [ n ] holds P [ n + 1 ]
set R = ( Line ( M , i ) * Line ( M , i ) ) ;
assume that M1 is being_line and M2 is being_line and M3 is being_line and M1 is being_line and M2 is being_line ;
reconsider a = f4 . ( i0 -' 1 ) , b = f4 . ( i0 -' 1 ) as Element of K ;
len B2 = Sum ( Len F1 ^ F2 ) .= len ( Len F1 ) + len ( Len F2 ) ;
len ( ( the ` of n ) * ( i , j ) ) = n & len ( ( i , j ) * ( i , j ) ) = n ;
dom max ( f , g ) = dom ( f + g ) .= dom f /\ dom g ;
( for n be Nat holds seq . n = upper_bound Y1 ) implies for n be Nat holds seq . n = upper_bound Y1
dom ( p1 ^ p2 ) = dom ( f ^ <* p *> ) .= dom ( f ^ <* p *> ) .= dom f ;
M . [ 1 / y , y ] = 1 / ( 1 - y ) * v1 .= 1 / ( 1 - y ) * v1 ;
assume that W is non trivial and W .vertices() c= the carrier' of G2 and W .vertices() c= the carrier' of G2 ;
C6 * ( i1 , i2 ) `1 = G1 * ( i1 , i2 ) `1 .= G1 * ( i1 , i2 ) `1 ;
C8 |- 'not' Ex ( x , p ) 'or' p . ( x , y ) ;
for b st b in rng g holds lower_bound rng f\lbrace b - a , b + a .[ <= b
- ( ( q1 `1 / |. q1 .| - cn ) / ( 1 + cn ) ) = 1 ;
( LSeg ( c , m ) \/ [: l , k :] ) \/ LSeg ( l , k ) c= R ;
consider p being element such that p in such that p in LSeg ( x , p ) and p in L~ f ;
Indices ( ( X @ ) * ( i , j ) ) = [: Seg n , Seg n :] ;
cluster s => ( q => p ) => ( q => ( s => p ) ) -> valid ;
Im ( ( Partial_Sums F ) . m ) is measurable of E , REAL & ( Partial_Sums F ) . m is measurable ;
cluster f . ( x1 , x2 ) -> Element of D * & f . ( y1 , y2 ) -> Element of D * ;
consider g being Function such that g = F . t and Q [ t , g ] ;
p in LSeg ( ( Int Z ) , ( ( Int Z ) ` ) / 2 ) implies LSeg ( p , q ) c= LSeg ( p , q )
set R8 = R / 1 , R8 = ]. b , +infty .[ , R8 = ]. b , +infty .[ ;
IncAddr ( I , k ) = SubFrom ( da , db ) .= SubFrom ( da , db ) .= goto - ( card I + 1 ) ;
seq . m <= ( ( the Sorts of A ) * ( seq ^\ k ) ) . n & ( ( seq ^\ k ) * ( seq ^\ k ) ) . n <= ( ( seq ^\ k ) * ( seq ^\ k ) ) . n ;
a + b = ( a ` *' b ) ` .= ( a ` *' b ) ` .= ( a ` *' b ) ` ;
id ( X /\ Y ) = id X /\ id Y .= id ( X /\ Y ) ;
for x being element st x in dom h holds h . x = f . x ;
reconsider H = U1 \/ U2 , U2 = U1 \/ U2 as non empty Subset of U0 ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) /\ f ) /\ j ) /\ m ;
consider y being element such that y in Y and P [ y , lower_bound B ] ;
consider A being finite stable set of R such that card A = len R and card A = 1 ;
p2 in rng ( f |-- p1 ) \ rng <* p1 *> & p2 in rng ( f |-- p1 ) \ rng <* p1 *> ;
len s1 - 1 > 1-1 & len s2 - 1 > 0 & len s2 - 1 > 0 ;
( ( N-min P ) `2 ) = ( ( N-min P ) `2 ) & ( ( N-min 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 . a2 = f . a2 ` & f . a2 = f . a2 ` ;
( seq ^\ k ) . n in ]. -infty , x0 + r .[ & ( seq ^\ k ) . n in ]. x0 , x0 + r .[ ;
gg . s0 = g . s0 | G . s0 .= g . s0 | G . s0 ;
the InternalRel of S is \lbrace x , y } & the InternalRel of S is transitive ;
deffunc F ( Ordinal , Ordinal ) = phi . ( $2 , $1 ) & phi . ( $2 , $2 ) = phi . ( $2 , $2 ) ;
F . a1 = F . s2 & F . a1 = F . a1 & F . a2 = F . a2 ;
x `2 = A . o . a .= Den ( o , A . a ) ;
( Cl f ) " P1 c= f " ( Cl P1 ) & ( Cl f ) " P1 c= f " ( Cl P1 ) ;
FinMeetCl ( ( the topology of S ) \/ { i } ) c= the topology of T ;
synonym o is \bf means : Def3 : o <> \ast & o <> * & o <> * ;
assume that X c= Y + X and card X <> card Y and card Y <> card X and card X = card Y ;
the { F of s <= 1 + ( the *> of s ) & ( the { F } , s ) `2 = ( the Element of s ) `2 ;
LIN a , a1 , d or b , c // b1 , c1 & a , b // b1 , c1 ;
e / 2 . 1 = 0 & e / 2 . 2 = 1 & e / 2 . 3 = 0 ;
ES1 in SS1 & ES2 in SS2 implies ES1 in { NS1 }
set J = ( l , u ) If , K = ( l , u ) If , L = ( l , u ) If , M = ( l , u ) If , N = ( l , u ) If , N = ( l , u ) If ) ;
set A1 = } , A2 = [ <* a9 , b9 , c *> , A2 = [ <* b9 , c *> , d ] , A2 = [ <* c , d *> , e ] ;
set vs = [ <* xy , } , <* ] , '&' , sin ] , xy = [ <* A1 , cin *> , '&' ] , yz = [ <* A1 , cin *> , '&' ] , A1 = [ <* 8 , A1 *> , '&' ] , A2 = [ <* A1 , 8 *> , '&' ] , A2 = [ <* 8 , 8 *> , '&'
x * z ` * x " in x * ( z * N ) * x " * x " ;
for x being element st x in dom f holds f . x = g9 . x & f . x = g2 . x ;
Int cell ( f , 1 , G ) c= RightComp f \/ RightComp f \/ RightComp f \/ RightComp f \/ RightComp f \/ RightComp f
U2 is closed & ( W-min C ) `1 <= ( E-max C ) `1 & ( E-max C ) `1 <= ( E-max C ) `1 ;
set f-17 = f @ "/\" g @ ;
attr S1 is convergent means : Def3 : S2 is convergent & lim ( S1 - S2 ) = 0 ;
f . ( 0 + 1 ) = ( 0 qua Ordinal ) + a .= a ;
cluster -> \in be be be be be be be be reflexive transitive RelStr , F , G be non empty reflexive transitive non empty reflexive 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 \lbrack a , ( A * ) .--> x ) = len l & len ( l * a ) = len l ;
t4 \+\ {} is ( {} \/ rng t4 ) -valued FinSequence of ( {} \/ rng ( {} \/ rng ( {} \/ rng ( {} \/ rng ( {} \/ rng ( {} \/ rng ( {} \/ rng ( {} \/ rng ( {} \/ rng p ) ) ) ) ) ) ) * p ;
t = <* F . t *> ^ ( C . p ^ q ) .= ( C . p ^ q ) ^ ( C . q ^ r ) ;
set p-2 = W-min L~ Cage ( C , n ) , p-2 = W-bound L~ Cage ( C , n ) , p-2 = W-bound 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 ( ( a - b ) |-> ( a - b ) ) + ( b - a ) ) ) = width ( ( a - b ) |-> ( a - b ) ) ;
FM . x in dom ( ( G * the_arity_of o ) . x ) & ( ( G * the_arity_of o ) . x ) . x = ( G * the_arity_of o ) . x ;
set cH2 = the carrier of H2 , cH2 = the carrier of H2 , cH2 = the carrier of H2 ;
set cH1 = the carrier of H1 , cH2 = the carrier of H2 ;
( Comput ( P , s , 6 ) ) . intpos m = s . intpos m .= Comput ( P , s , 6 ) . intpos m ;
IC Comput ( Q2 , t , k ) = ( l + 1 ) + 1 .= ( l + 1 ) + 1 ;
dom ( ( ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) * ( ( 1 / 2 ) * ( 1 / 2 ) ) ) ) ) `| REAL ) ) = REAL ;
cluster <* l *> ^ phi -> ( 1 + not ( n + 1 ) ) -element for string of S ;
set b5 = [ <* a, A1 *> , <* A1 , A2 *> , <* A1 , A2 *> ] , b5 = [ <* A1 , A2 *> , <* A2 , A1 *> ] ;
Line ( Segm ( M @ , P , Q ) , x ) = L * ( Sgm Q ) . x .= Line ( M , i ) ;
n in dom ( ( the Sorts of A ) * ( the_arity_of o ) ) & ( ( the Sorts of A ) * ( the_arity_of o ) ) . n = ( the Sorts of A ) . n ;
cluster f1 + f2 -> continuous for PartFunc of REAL , the carrier of S ;
consider y be Point of X such that a = y and ||. y - x .|| <= r ;
set x3 = t3 . DataLoc ( s2 . SBP , 2 ) , x4 = Comput ( P1 , s1 , 2 ) . SBP , s4 = Comput ( P2 , s2 , 2 ) . SBP , P4 = Comput ( P2 , s2 , 2 ) . SBP , P4 = Comput ( P2 , s2 , 2 ) . SBP , P4 = Comput
set p-3 = stop I ( ) , p-3 = Initialize s ( ) ;
consider a being Point of D2 such that a in W1 and b = g . a and a in W1 ;
{ A , B , C , D , E } = { A , B } \/ { C , D }
let A , B , C , D , E , F , J , M , N , N , M , N , N , M , N , N , M , N , N , M , N , N , N , M , N , N , N , M , N , N , M , N , N , M ,
|. p2 .| ^2 - ( ( p2 `2 ) ^2 + ( p2 `2 ) ^2 ) >= 0 ;
l -' 1 + 1 = n-1 * ( x + ( - 1 ) ) + ( - 1 ) ;
x = v + ( a * w1 + b * w2 ) + ( c * w2 ) + ( c * w2 ) ;
the TopStruct of L = TopSpaceMetr ( the TopStruct of L ) & the TopStruct of L = TopSpaceMetr ( the TopStruct of L ) ;
consider y being element such that y in dom H1 and x = H1 . y and y in dom H2 ;
ff \ { n } = ( \mathop { v1 , v2 } \/ { v1 } ) \/ ( { v2 } \/ { v1 } ) ;
for Y being Subset of X st Y is summable holds Y is iff Y is iff X is element
2 * n in { N : 2 * Sum ( p | N ) = N & N > 0 } ;
for s being FinSequence holds len ( the { of s } ) = len s & len ( the { s } ) = len s
for x st x in Z holds exp_R * f is_differentiable_in x & ( exp_R * f ) . x > 0
rng ( h2 * f2 ) c= the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 & rng ( h2 * f2 ) c= the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 ;
j + ( len f ) - len f <= len f + ( len g ) - len f ;
reconsider R1 = R * I , R2 = R * I as PartFunc of REAL , REAL-NS n , REAL-NS n ;
C8 . x = s1 . x0 .= ( s1 . x0 ) * ( s2 . x0 ) .= ( s2 . x0 ) * ( s2 . x0 ) ;
power F_Complex . ( z , n ) = 1 .= ( x |^ n ) |^ ( z , n ) .= ( x |^ n ) |^ ( z , n ) ;
t at ( C , s ) = f . ( the connectives of S ) . t & t at ( C , s ) = f . t ;
support ( f + g ) c= support f \/ support g & support ( f + g ) c= support f \/ support g ;
ex N st N = j1 & 2 * Sum ( seq1 | N ) > N & N > 0 ;
for y , p st P [ p ] holds P [ All ( y , p ) ]
{ [ x1 , x2 ] , [ y1 , y2 ] } is Subset of [: X1 , X2 :] ;
h . i = ( j |-- h , id B ) . i .= H . i ;
ex x1 being Element of G st x1 = x & x1 * N c= A & x1 in A ;
set X = ( ( \lbrace q , O1 , O1 , O2 , O2 , O1 } ) . [ q , p , q ] ) ;
b . n in { g1 : x0 < g1 & g1 < x0 & g1 < x0 + r } ;
f /* s1 is convergent & f /. x0 = lim ( f /* s1 ) implies f /* s1 is convergent & lim ( f /* s1 ) = lim ( f /* s1 )
the lattice of Y = the lattice of the lattice of Y & the carrier of Y = the carrier of X & the carrier of Y = the carrier of Y ;
( 'not' a . x ) '&' b . x 'or' a . x '&' 'not' ( b . x ) '&' 'not' ( b . x ) = FALSE ;
q2 = len ( q0 ^ r1 ) + len q1 .= len q1 + len q2 + len q2 .= len q1 + len q2 + len q2 ;
( 1 / a ) (#) ( sec * f1 ) - id Z * ( ( 1 / a ) (#) ( sec * f1 ) ) is_differentiable_on Z ;
set K1 = integral ( ( lim ( lim ( H , A ) ) || ( A , B ) ) , D2 ) , A = ( ( lim ( H , A ) ) || ( A , B ) ) || ( A , B ) ;
assume e in { ( w1 - w2 ) / ( w1 - w2 ) : w1 in F & w2 in G } ;
reconsider d7 = dom a `1 , d7 = dom F `1 , d7 = dom F `1 , d7 = dom F `1 , d8 = dom F `1 , d7 = dom F `1 , d8 = dom F `1 , d8 = dom F `1 , d8 = dom F `1 , d8 = dom F `1 , d8 = dom F `1
LSeg ( f /^ j , j ) = LSeg ( f , j + q .. f ) .= LSeg ( f , j + q .. f ) ;
assume X in { T . N2 , K : h . N2 = N2 & h . I = I } ;
assume that Hom ( d , c ) <> {} and <* f , g *> * f1 = <* f , g *> * f2 ;
dom Sspace = dom S /\ Seg n .= dom ( L | n ) .= dom ( L | n ) .= dom ( L | n ) .= dom ( L | n ) ;
x in H |^ a implies ex g st x = g |^ a & g in H & a in H
a * ( 0. INT ) . ( a , 1 ) = a `1 - ( 0 * n ) .= a `2 - ( 0 * n ) ;
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 ^ @ g ^ @ f = g ^ @ f
dom ( f1 (#) f2 ) /\ X c= dom ( f1 (#) f2 ) /\ X & dom ( f1 (#) f2 ) /\ X c= dom f1 /\ X ;
1 = ( p * p ) / p .= p * ( p / p ) .= p * 1 ;
len g = len f + len <* x + y *> .= len f + 1 .= len f + 1 + 1 .= len f + 1 ;
dom ( F-11 | [: N1 , S :] ) = dom ( F | [: N1 , S :] ) .= [: N1 , S :] ;
dom ( f . t ) * I . t = dom ( f . t ) * g . t .= dom ( f . t ) * I . t ;
assume a in ( "\/" ( ( T |^ the carrier of S ) , F ) ) .: D & b in ( the carrier of S ) ;
assume that g is one-to-one and ( the carrier' of S ) /\ rng g c= dom g and g is one-to-one and g is one-to-one ;
( ( x \ y ) \ z ) \ ( ( x \ z ) \ ( y \ z ) ) = 0. X ;
consider f such that f * f `1 = id b and f * f `2 = id a and f is one-to-one ;
( ( cos | [. 2 * PI , 0 + 1 .] ) | [. 0 , 1 .] is increasing ;
Index ( p , co ) <= len LS - Index ( Gij , LS ) + 1 & Index ( Gij , LS ) + 1 <= len LS ;
let t1 , t2 , 3 be Element of ( the carrier of S ) * , s be Element of ( the carrier of S ) * ;
\langle -> Element of ( Frege ( curry ( curry H ) ) ) . h , ( Frege ( curry G ) ) . h *> <= J . j ;
then P [ f . i0 , f . ( i0 + 1 ) , f . ( i0 + 1 ) , f . ( i0 + 1 ) , f . ( i0 + 1 ) , f . ( i0 + 1 ) , f . ( i0 + 1 ) , f . ( i0 + 1 ) , f . ( i0 + 1 ) , f . ( i0 + 1 )
Q [ ( D . ( x , 1 ) ) `1 , F ( x , 1 ) ) `1 , F ( x , 2 ) ] ;
consider x being element such that x in dom ( F . s ) and y = ( F . s ) . x ;
l . i < r . i & [ l . i , r . i ] is for i holds r . i is for of G . i , G . i
the Sorts of A2 = ( the carrier of S2 ) --> { the carrier of S } .= the carrier of S ;
consider s being Function such that s is one-to-one and dom s = NAT and rng s = F and rng s c= F ;
dist ( b1 , b2 ) <= dist ( b1 , a ) + dist ( a , b2 ) & dist ( a , b2 ) <= dist ( a , b2 ) + dist ( b , b1 ) ;
( for n holds ( C /. ( len C ) ) /. n = W-min L~ Cage ( C , n ) ) implies ( C /. ( len C ) ) /. n = W-min L~ Cage ( C , n )
q `2 <= ( UMP Upper_Arc L~ Cage ( C , 1 ) ) `2 & ( UMP L~ Cage ( C , 1 ) ) `2 <= ( UMP L~ Cage ( C , 1 ) ) `2 ;
LSeg ( f | i2 , i ) /\ LSeg ( f | i2 , j ) = {} implies LSeg ( f | i2 , j ) = {}
given a be ExtReal such that a <= II and A = ]. a , II .] and a < II and a <= b ;
consider a , b be complex number such that z = a & y = b and z + y = a + b ;
set X = { b |^ n where n is Element of NAT : n <= m } , Y = { b |^ n where m is Element of NAT : m <= n } ;
( ( x * y * z ) \ x ) \ z = 0. X ;
set xy = [ <* xy , y *> , f1 ] , yz = [ <* y , z *> , f2 ] , zx = [ <* z , x *> , f3 ] , f4 = [ <* x , y *> , f3 ] , f4 = [ <* z , x *> , f3 ] , f4 = [ <* x , y *> , f3 ] , f4 = [ <* z , x *>
Uq /. ( len ( l ) ) = ( l . ( len ( l ) ) ) `1 .= ( l . ( len l ) ) `1 ;
( ( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) / ( 1 + sn ) ) ^2 = 1 ;
( ( ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) / ( 1 + sn ) ) ^2 < 1 ;
( ( ( ( S \/ Y ) \/ X ) \/ Y ) \/ X ) `2 = ( ( ( S \/ Y ) \/ X ) \/ Y ) `2 .= ( ( ( S \/ Y ) \/ X ) \/ X ) `2 ;
( ( s1 - s2 ) . k ) . k = ( s1 . k - s2 . k ) . k .= ( s1 . k - s2 . k ) . k ;
rng ( ( h + c ) ^\ n ) c= dom SVF1 ( 1 , f , u0 ) /\ dom SVF1 ( 1 , f , u0 ) ;
the carrier of X0 = the carrier of X0 & the carrier of X0 = the carrier of X0 implies X = the carrier of X0
ex p4 st p3 = p4 & |. p4 - |[ a , b ]| .| = r & |. p4 - |[ a , b ]| .| = 1 ;
set ch = chi ( X , A ) , Ah = chi ( X , A ) , Ah = chi ( X , A ) ;
R |^ ( 0 * n ) = I\HM ( X , X ) .= R |^ n |^ 0 .= R |^ n ;
( Partial_Sums ( curry ( F-19 , n ) ) ) . n is nonnegative implies ( ( curry ( F-19 , n ) ) . n ) . x = ( ( Partial_Sums ( curry ( F-19 , n ) ) ) . x ) . x
f2 = C7 . ( E7 , K ) .= C7 . ( E7 , K ) .= ( the carrier of K ) --> ( f , g ) ;
S1 . b = s1 . b .= s2 . b .= s2 . b .= s2 . b .= s2 . b ;
p2 in LSeg ( p2 , p1 ) /\ LSeg ( p1 , p10 ) \/ LSeg ( p10 , p2 ) /\ LSeg ( p1 , p10 ) \/ LSeg ( p10 , p2 ) ;
dom ( f . t ) = Seg n & dom ( I . t ) = Seg n & dom ( I . t ) = Seg n ;
assume o = ( the connectives of S ) . 11 & ( the connectives of S ) . 11 in ( the carrier' of S ) . 11 ;
set phi = ( l1 , l2 ) \mathop { {} } , phi = ( l1 , l2 ) \mathop { {} } , N = S S , M = S S S S S S , N = S S , M = S S , N = S S , M = S S , N = S S , M = S S , N = S , N = S , M = S , N = S , M =
synonym p is is is invertible for p = 1 implies ( p - q ) / ( p - q ) = 1 / p ;
( Y1 `2 = - 1 ) & ( 0. TOP-REAL 2 ) <> ( Y1 `2 or ( Y1 `1 = - 1 ) & ( - 1 ) * ( Y1 `2 ) <> 0. TOP-REAL 2 ) implies Y1 is not empty
defpred X [ Nat , set , set ] means P [ $1 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2
consider k being Nat such that for n be Nat st k <= n holds s . n < x0 + g ;
Det ( I |^ ( m -' n ) ) * ( m -' n ) = 1. K & Det I * ( m -' n ) = 1. K ;
( - b - sqrt ( b ^2 - 4 * a * c ) ) / ( 2 * a * c ) < 0 ;
Cd . d = Cd . d mod Cd . e & Cd . e mod Cd . e = Cd . e mod Cd . e - Cd . e mod Cd . e mod Cd . e mod Cd . e mod Cd . e - 1 mod Cd . e mod Cd . e mod Cd . e mod Cd . e mod Cd . e )
attr X1 is dense means : Def3 : X1 is dense dense & X2 is dense implies X1 /\ X2 is dense SubSpace dense SubSpace SubSpace of X ;
deffunc F6 ( Element of E , Element of I ) = ( $1 * $2 ) * ( ( $1 * $2 ) * ( ( $1 * $2 ) * ( ( $1 * $2 ) * ( $2 ) ) ) ;
t ^ <* n *> in { t ^ <* i *> : Q [ i , T ( ) ] } ;
( x \ y ) \ x = ( x \ x ) \ y .= y ` .= 0. X ;
for X being non empty set for X being Subset-Family of X holds for Y being Subset-Family of X holds Y is Basis of [: X , FinMeetCl Y :]
synonym A , B are_separated means : Def3 : Cl ( A , B ) misses Cl ( B , C ) & A misses Cl ( B , C ) ;
len ( M @ ) = len p & width ( M @ ) = width ( M @ ) & width ( M @ ) = width ( M @ ) ;
J . v = { x where x is Element of K : 0 < v . x & v . x = 1 } ;
( Sgm ( Seg m ) ) . d - ( Sgm ( support ( m ) ) . e ) . d <> 0 ;
lower_bound divset ( D2 , k + k2 ) = D2 . ( k + k2 - 1 ) .= D2 . ( k + k2 - 1 ) ;
g . r1 = - 2 * r1 + 1 & dom h = [. 0 , 1 .] & rng h c= [. 0 , 1 .] ;
|. a .| * ||. f .|| = 0 * ||. f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| ;
f . x = ( h . x ) `1 & g . x = ( h . x ) `2 & h . y = ( h . y ) `2 ;
ex w st w in dom B1 & <* 1 *> ^ s = <* 1 *> ^ w & ( <* 1 *> ^ w ) ^ s = <* 1 *> ^ w ;
[ 1 , {} , <* d1 *> ] in ( { [ 0 , {} , {} ] } \/ S1 ) \/ S2 ;
IC Exec ( i , s1 ) + n = IC Exec ( i , s2 ) + n .= IC Exec ( i , s2 ) + n .= 0 ;
IC Comput ( P , s , 1 ) = succ IC s .= ( 5 + 9 ) .= 5 + 9 .= 5 ;
( IExec ( W6 , Q , t ) ) . intpos ( 8 + 1 ) = t . intpos ( 8 + 1 ) .= t . intpos ( 8 + 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 & y in C holds x <= y or y <= x ;
integral ( f , C ) . x = f . ( upper_bound C ) - f . ( lower_bound C ) .= f . ( lower_bound C ) - f . ( lower_bound C ) ;
for F , G being one-to-one FinSequence st rng F misses rng G holds F ^ G is one-to-one & ( F ^ G ) ^ <* G *> is one-to-one
||. R /. ( L . h ) .|| < e1 * ( K + 1 * ||. h .|| ) ;
assume a in { q where q is Element of M : dist ( z , q ) <= r } ;
set p4 = [ 2 , 1 ] .--> [ 2 , 0 , 1 ] ;
consider x , y being Subset of X such that [ x , y ] in F and x c= d and y in d and x \not c= d ;
for y , x being Element of REAL st y in Y & x in X holds y `1 <= x `1 + x `2
func |. p \bullet |. p .| -> variable of A means : Def3 : for x being element st x in it holds it . x = min ( NBI ( p ) , p ) ;
consider t being Element of S such that x `1 , y `2 '||' z `1 , t `2 and x `2 , z `2 '||' y `2 , t `2 ;
dom x1 = Seg ( len x1 ) & len x1 = len y1 & for i st i in Seg ( len x1 ) holds x1 . i = y1 . i ;
consider y2 being Real such that x2 = y2 and 0 <= y2 and y2 < 1 and not ( ex y1 , y2 st y1 in X & y2 in X ) ;
||. f | X /* s1 .|| = ||. f .|| | X & ||. f .|| | X = ||. f .|| | X & ||. f .|| | X = ||. f .|| | X ;
( the InternalRel of A ) ` /\ Y ` = {} \/ {} .= {} \/ {} .= {} \/ {} .= {} \/ {} .= {} ;
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 & i + 1 in dom p holds P [ i , j ] ;
reconsider h = f | X ( ) , g = ( f | X ( ) ) | X ( ) as Function of X ( ) , Y ( ) ;
u1 in the carrier of W1 & u2 in the carrier of W2 implies ( u1 + u2 ) + ( v1 + v2 ) = u1 + u2
defpred P [ Element of L ] means M <= f . $1 & f . $1 <= f . $1 & f . $1 <= f . $1 ;
l . ( u , a , v ) = s * x + ( - ( s * x ) + y ) .= b ;
- ( x-y ) = - x + - ( - y ) .= - x + - y .= - x + - y .= x + y .= x + y ;
given a being Point of GX such that for x being Point of GX holds a , x are_ed ed and a , x are_ed ed ;
fSet = [ [ dom ( f1 , f2 ) , cod ( f2 , g2 ) ] , h2 ] , h2 = [ cod ( f1 , f2 ) , cod ( f2 , g2 ) ] ;
for k , n being Nat st k <> 0 & k < n & n is prime holds k , n are_relative_prime & k , n are_relative_prime & k , n are_relative_prime holds k , n are_relative_prime
for x being element holds x in A |^ d implies x in ( ( A ` ) |^ d ) ` & ( ( A ` ) |^ d ) ` = ( A ` ) |^ d
consider u , v being Element of R , a being Element of A such that l /. i = u * a * v and a * v = v ;
- ( ( - ( p `1 / |. p .| - cn ) ) / ( 1 + cn ) ) ^2 > 0 ;
L-13 . k = Ln . ( F . k ) & F . k in dom ( L * F ) & F . k in dom ( L * F ) ;
set i2 = SubFrom ( a , i , - n ) , i1 = goto - ( n + 1 ) ;
attr B is thesis means : Def3 : for S being Subuniversal Subset of D holds ( S is ( ) `1 ) & ( S is ( ) `1 ) & ( S is ( ) `1 ) ;
a9 "/\" D = { a "/\" d where d is Element of N : d in D & a "/\" b = b } ;
|( \square , q29 )| * |( q , q29 )| + |( q , q29 )| * |( q , q1 )| >= |( \square , b )| * |( q , q1 )| + |( q , q1 )| * |( q , q1 )| ;
( - f ) . sup A = ( ( - f ) | A ) . sup A .= ( ( - f ) | A ) . sup A .= ( - f ) . sup A ;
G * ( len G , k ) `1 = G * ( len G , k ) `1 .= G * ( len G , k ) `1 .= G * ( len G , k ) `1 ;
( Proj ( i , n ) ) . LM = <* ( proj ( i , n ) ) . ( Carrier ( proj ( i , n ) ) . LM ) *> ;
f1 + f2 * reproj ( i , x ) is_differentiable_in ( reproj ( i , x ) + f2 * reproj ( i , x ) ) . x ;
redefine pred ( ( for x st x in Z holds ( ( tan (#) f ) `| Z ) . x = 1 ) & ( ( tan (#) f ) `| Z ) . x = 1 ) ;
ex t being SortSymbol of S st t = s & h1 . t = h2 . t & ( for x being set st x in dom h1 holds h1 . x = F ( x ) ) ;
defpred C [ Nat ] means P8 ( ) is as n -seq of ( n + 1 ) , A ( ) ;
consider y being element such that y in dom ( p | i ) and ( q | i ) . y = ( p | i ) . y ;
reconsider L = product ( { x1 } +* ( index B , l ) ) as Subset of ( Carrier A ) . ( index B ) ;
for c being Element of C ex d being Element of D st T . ( id c ) = id d & for c being Element of C st T . ( id c ) = id d holds c = d
be Element of ( f , n ) * p = ( f | n ) ^ <* p *> .= f ^ <* p *> ;
( f (#) g ) . x = f . g . x & ( f (#) h ) . x = f . ( h . x ) ;
p in { ( 1 - 2 ) * ( G * ( i + 1 , j ) + G * ( i + 1 , j + 1 ) ) } ;
f `2 - p = ( f | ( n , L ) ) *' ( - ( f | ( n , L ) ) ) .= ( - ( f | ( n , L ) ) ) *' ( - ( f | ( n , L ) ) ) ;
consider r be Real such that r in rng ( f | divset ( D , j ) ) and r < m + s ;
f1 . ( |[ r2 , r2 ]| ) in f1 .: [: W1 , W2 :] & f2 . ( |[ r2 , s2 ]| ) in f1 .: [: W1 , W2 :] ;
eval ( a | ( n , L ) , x ) = eval ( a | ( n , L ) , x ) .= a * eval ( x , x ) .= a * eval ( x , x ) ;
z = DigA ( tk , x9 ) .= DigA ( tk , ( k + 1 ) + 1 ) .= DigA ( tk , ( k + 1 ) + 1 ) .= DigA ( tk , ( k + 1 ) + 1 ) .= DigA ( tk , ( k + 1 ) + 1 ) ;
set H = { Intersect S where S is Subset-Family of X : S c= G & S c= G } , F = { Intersect S where S is Subset-Family of X : S c= G } , G = { Intersect S where S is Subset of X : S c= G } , G = { Intersect S where S is Subset of X : S is finite } ;
consider S19 being Element of D * , d being Element of D * such that S `1 = S19 ^ <* d *> and S `2 = d ;
assume that x1 in dom f and x2 in dom f and f . x1 = f . x2 and f . x2 = f . x2 ;
- 1 <= ( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) / ( 1 + sn ) ;
(0). ( V ) is Linear_Combination of A & Sum ( ( - L ) (#) ( - L ) ) = 0. V implies Sum ( - L ) = Sum ( - L ) * Sum ( - L )
let k1 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 be Nat , k2 be Integer , k2 be Integer , k2 be Integer , k2 be Integer ;
consider j being element such that j in dom a and j in g " { k `2 } and x = a . j and y = a . j ;
H1 . x1 c= H1 . x2 or H1 . x2 c= H1 . x3 or H1 . x3 c= H1 . x3 & H1 . x4 = H2 . x3 ;
consider a being Real such that p = L~ ( means * p1 + a * p2 ) and 0 <= a and a <= 1 and a <= 1 ;
assume that a <= c & c <= d and [' a , b '] c= dom f and [' a , b '] c= dom g and f . a = g . b ;
cell ( Gauge ( C , m ) , len Gauge ( C , m ) -' 1 , 0 ) is non empty ;
A5 in { ( S . i ) `1 where i is Element of NAT : i <= n } ;
( T * b1 ) . y = L * b2 /. y .= ( F `1 * b1 ) . y .= ( F `1 * b1 ) . y .= ( F `1 * b1 ) . y ;
g . ( s , I ) . x = s . y & g . ( s , I ) . y = |. s . x - s . y .| ;
( log ( 2 , k + k ) ) / ( 2 * k + 1 ) >= ( log ( 2 , k + 1 ) ) / ( 2 * k + 1 ) ;
then p => q in S & not x in the still of p & not p => All ( x , q ) in S & not x in S & p => All ( x , q ) in S ;
dom ( the InitS of r-10 ) misses dom ( the InitS of r-10 ) & ( the InitS of r-10 ) . ( ( the InitS of r-10 ) . ( ( the InitS of r-10 ) . ( ( the InitS of r-10 ) . ( ( the InitS of r-10 ) . ( n + 1 ) ) ) ) = the carrier of r-10 ;
synonym f is extended integer means : Def3 : for x being set st x in rng f holds x is integer ;
assume that for a being Element of D holds f . { a } = a and for X being Subset-Family of D holds f . ( f .: X ) = f . union X ;
i = len p1 .= len p3 + len <* x *> .= len p1 + len <* x *> .= len p1 + 1 .= len p1 + 1 + 1 .= len p1 + 1 ;
l ( ) = g /. ( 1 , 3 ) + ( k ( ) + 1 ) - ( k ( ) + 1 ) * ( e ( ) + 1 ) * ( e ( ) + 1 ) * ( e ( ) + 1 ) * ( e ( ) + 1 ) * ( e ( ) + 1 ) * ( e ( ) + 1 ) * ( e ( ) + 1 ) * ( e ( ) + 1 ) * ( e ( ) + 1 ) * ( e ( )
CurInstr ( P2 , Comput ( P2 , s2 , l ) ) = halt SCM+FSA .= CurInstr ( P2 , Comput ( P2 , s2 , l ) ) .= halt SCM+FSA ;
assume for n be Nat holds ||. seq . n .|| <= ( ||. seq .|| ) . n & ( ||. seq .|| ) . n <= 1 ;
sin . \mathclose { 0 } = sin r * cos ( ( - cos r ) * sin ( s ) ) .= 0 ;
set q = |[ g1 `1 / t `2 , g2 `2 / t `1 ]| , g1 = |[ g1 `1 / t `2 , g2 ]| , g2 = |[ g2 `1 / t `2 / t `2 , g2 `2 / t `2 ]| ;
consider G being sequence of S such that for n being Element of NAT holds G . n in implies G in implies for x being Element of S holds G . x in implies G . x in implies G . x in being Element of S
consider G such that F = G and ex G1 , G2 st G1 in SX & G2 in SX & G1 in SX & G2 in SX & G2 in X ;
the root of [ x , s ] in ( ( the Sorts of Free ( C , X ) ) * ( the Arity of S ) ) . s & ( the Sorts of Free ( C , X ) ) . s = ( the Sorts of Free ( C , X ) ) . s ;
Z c= dom ( ( exp_R * ( f1 + ( #Z 3 ) * f ) + ( ( #Z 3 ) * f1 ) ) `| Z ) ;
for k be Element of NAT holds seq1 . k = ( sum ( Im ( f ) , Sseq ) ) . k & ( Sum ( Im ( f ) , Sseq ) ) . k = Sum ( ( Im ( f ) ) , Sseq ) )
assume that - 1 < n and ( q `2 / |. q .| - sn ) < 0 and ( q `2 / |. q .| - sn ) < 0 and ( q `2 / |. q .| - sn ) < 0 ;
assume that f is continuous and a < b and c < d and f = g and f = h and f = h and f = k and g = k ;
consider r being Element of NAT such that s-> Element of NAT , s1 , s2 , r being Element of NAT such that r = Comput ( P1 , s1 , i ) and s1 <= s2 and s2 <= s2 ;
LE f /. ( i + 1 ) , f /. j , L~ f , f /. 1 , f /. ( len f ) , f /. ( len f ) ;
assume that x in the carrier of K and y in the carrier of K and inf { x , y } = inf { x , y } and x <= y ;
assume that f +* ( i1 , \xi ) in ( proj ( F , i2 ) ) " ( A . i2 ) and ( proj ( F , i2 ) ) . ( A . i2 ) = ( proj ( F , i2 ) ) . ( A . i2 ) ;
rng ( ( Flow M ) ~ | ( the carrier of M ) ) c= the carrier' of M & rng ( ( Flow M ) ~ | ( the carrier of M ) ) c= the carrier' of M ;
assume z in { ( the carrier of G ) \/ { t } where t is Element of T : t in X } ;
consider l be Nat such that for m be Nat st l <= m holds ||. s1 . m - x0 .|| < g / 2 * ||. x0 .|| + g / 2 * ||. x0 .|| ;
consider t be VECTOR of product G such that mt = ||. D . t .|| and ||. t .|| <= 1 and ||. t .|| <= 1 ;
assume that the ] degree of v = 2 and v ^ <* 0 *> in dom p and v ^ <* 1 *> in dom p and p . ( len p + 1 ) in dom p and p . ( len p + 1 ) in dom p ;
consider a being Element of the Points of X39 , A being Element of the Points of X39 such that a on A and a on A and a on A ;
( - x ) |^ ( k + 1 ) * ( ( - x ) |^ ( k + 1 ) ) " = 1 ;
for D being set st for i st i in dom p holds p . i in D holds p . i is FinSequence of D & p . i is FinSequence of D
defpred R [ element ] means ex x , y st [ x , y ] = $1 & P [ x , y ] & P [ y , x ] ;
L~ f2 = union { LSeg ( p0 , p10 ) , LSeg ( p10 , p1 ) } .= { p1 , p2 } \/ { p2 , p1 } \/ { p2 , p1 } ;
i -' len h11 + 2 - 1 < i - len h11 + 1 - len h11 + 2 - 1 + 1 - len h11 + 2 - 1 + 1 - len h11 + 2 - len h11 + 1 - len h11 + 1 - len h11 + 1 - len h11 + 1 - len h11 + 1 - len h11 + 1 - len h11 + 1 - len h11 + 1 - len h11 + 1 - len h11 + 1 - len h11 + 1 - len h11 + 1
for n being Element of NAT st n in dom F holds F . n = |. ( F . n ) . ( n -' 1 ) .| + |. ( F . n ) . ( n -' 1 ) .|
for r , s1 , s2 , s2 , s3 holds r in [. s1 , s2 .] iff s1 <= s2 & s2 <= 1 & s1 <= s2 & s2 <= 1
assume v in { G where G is Subset of T2 : G in B1 & G c= z1 & G c= z2 & G c= z1 } ;
let g be in Z -:] , X be INT -valued Function of [: the carrier of A , the carrier of A :] , the carrier of A ;
min ( g . [ x , y ] , k . [ y , z ] ) = ( min ( g , k , x ) ) . y ;
consider q1 being sequence of CNS such that for n holds P [ n , q1 . n ] & q1 . n = U ( n + 1 ) ;
consider f being Function such that dom f = NAT & for n being Element of NAT holds f . n = F ( n ) and for n being Element of NAT holds f . n = F ( n ) ;
reconsider B-6 = B /\ O , OO = O , OO = I , I = I , I = J as Subset of B ;
consider j being Element of NAT such that x = ( the ` of n ) | j and j <= n and 1 <= j and j <= n ;
consider x such that z = x and card ( x . O2 ) in card ( x . O2 ) and x in L1 and x in L2 ;
( C * ) . ( k , n2 ) = C . ( ( _ of T4 ( k , n2 ) ) . 0 ) .= C . ( ( _ of T4 ( k , n2 ) ) . 0 ) ;
dom ( X --> rng f ) = X & dom ( X --> f ) = dom ( X --> f ) & dom ( X --> f ) = X ;
( S-bound L~ SpStSeq C ) `2 <= ( N-bound L~ SpStSeq C ) `2 & ( S-bound L~ SpStSeq C ) `2 <= ( S-bound L~ SpStSeq C ) `2 & ( S-bound L~ SpStSeq C ) `2 <= ( S-bound L~ SpStSeq C ) `2 ;
synonym x , y are_collinear means : Def3 : x = y or ex l being Subset of S st { x , y } c= l & ex l being Subset of S st { x , y } c= l ;
consider X being element such that X in dom ( f | ( n + 1 ) ) and ( f | ( n + 1 ) ) . X = Y ;
assume that Im k is continuous and for x , y being Element of L , a , b being Element of Im k st a = x & b = y holds x << y iff a << b ;
( 1 / 2 * ( ( ( ( ( ( ( ( ( ( ( ( ( ( r ) ) * ( ( ( r ) ) * ( 1 / 2 ) ) * ( 1 / 2 ) ) ) ) * ( ( ( 1 / 2 ) * ( 1 / 2 ) ) * ( 1 / 2 ) ) ) ) ) ) ) ) `| REAL ) ) = f ;
defpred P [ Element of omega ] means ( for n holds ( ( n + 1 ) to_power $1 ) to_power ( n + 1 ) = A1 . ( n + 1 ) ) implies ( ( n + 1 ) to_power ( n + 1 ) ) to_power ( n + 1 ) = A1 . ( n + 1 ) ;
IC Comput ( P , s , 2 ) = succ IC Comput ( P , s , 1 ) .= 6 + 1 .= 6 + 1 .= 6 + 1 ;
f . x = f . g1 * f . g2 .= f . g1 * 1_ H .= f . g1 * 1_ H .= f . g1 * 1_ H .= f . g1 * f . g2 ;
( M * F-4 ) . n = M . ( F-4 . n ) .= M . ( { ( canFS ( Omega ) ) . n } ) .= M . ( { ( canFS ( Omega ) ) . n } ) ;
the carrier of L1 + L2 c= ( the carrier of L1 ) \/ ( the carrier of L2 ) & the carrier of L1 + L2 c= the carrier of L1 & the carrier of L1 + L2 = the carrier of L2 ;
pred a , b , c , x , y , c , x , y , z , y , z , x , y , z , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , x , y , z , y , z , x , y , z , x
( the PartFunc of s ) . n <= ( ( the PartFunc of s ) . n ) * ( ( the Sorts of s ) . ( n + 1 ) ) & ( ( the Sorts of s ) . n ) * ( ( the Sorts of s ) . ( n + 1 ) ) <= ( ( the Sorts of s ) . n ) * ( ( the Sorts of s ) . ( n + 1 ) ) ;
pred - 1 <= r & r <= 1 implies ( ( - 1 ) (#) ( arccot * f ) ) . r = - ( 1 / r ) * ( 1 / r ) ;
seq in { p ^ <* n *> where n is Nat : p ^ <* n *> in T1 & p ^ <* n *> in T1 } implies ex n be Nat st p in T1 & p ^ <* n *> in T1
|[ x1 , x2 , x3 ]| . 2 - |[ y1 , y2 , x3 ]| . 2 = |[ x2 , y2 , x3 ]| . 2 - |[ y1 , y2 , x3 ]| . 2 - |[ y1 , y2 , x3 ]| . 2 - |[ y1 , y2 , x3 ]| . 2 - |[ y1 , y2 , x3 ]| . 2 - |[ y1 , y2 , x4 ]| . 2 ]| ;
attr m be Nat means : Def3 : F . m is nonnegative & ( Partial_Sums F ) . m is nonnegative implies ( Partial_Sums F ) . m is nonnegative ;
len ( w ) = len ( ( len ( G ) ) + len ( ( G ) ) + len ( ( G ) ) ) .= len ( ( G ) ) + len ( ( G ) ) + len ( ( y ) ) ;
consider u , v being VECTOR of V such that x = u + v and u in W1 /\ W2 and v in W1 /\ W3 and v in W2 /\ W3 ;
given F be finite Subset of NAT such that F = x and dom F = n and rng F c= { 0 , 1 } and Sum F = k + 1 and Sum F = k ;
0 = 0 * 0 * 0 ^2 + 1 * ( - ( - ( - 1 ) ) * ( - ( - ( 0 ) ) * ( - ( 0 ) ) * ( - ( 0 ) ) * ( - ( 0 ) ) * ( - ( 0 ) ) * ( - ( 0 ) ) ) ;
consider n be Nat such that for m be Nat st n <= m holds |. ( f # x ) . m - ( lim ( f # x ) ) . n .| < e ;
cluster } -> being being being being } Boolean non empty iff ( ( let L , M ) --> ( ( \in L ) --> ( x , y ) ) ) is Boolean & ( ( w + x ) --> ( y , z ) ) is Boolean
"/\" ( BB , {} ) = Top B .= Top S .= the carrier of S .= "/\" ( I , {} ) .= "/\" ( I , {} ) .= "/\" ( I , {} ) ;
( r / 2 ) ^2 + ( rbeing / 2 ) ^2 <= ( r / 2 ) ^2 + ( r / 2 ) ^2 + ( r / 2 ) ^2 ;
for x being element st x in A /\ dom ( f `| X ) holds ( f `| X ) . x >= r2 & ( f `| X ) . x >= r2
2 * r1 - c * |[ a , c ]| - ( 2 * r1 - c * |[ b , c ]| ) = 0. TOP-REAL 2 & ( 2 * r1 - c ) * |[ b , c ]| = 0. TOP-REAL 2 ;
reconsider p = P * ( ( - 1 ) * ( - ( - ( - ( K , n , 1 ) ) ) ) ) as FinSequence of K ;
consider x1 , x2 being element such that x1 in uparrow s and x2 in < t and x = [ x1 , x2 ] and [ x1 , x2 ] in F . ( x1 , x2 ) ;
for n be Nat st 1 <= n & n <= len q1 holds q1 . n = ( upper_volume ( g , M7 ) ) . ( n + 1 ) & q1 . n = ( upper_volume ( g , M7 ) ) . ( n + 1 )
consider y , z being element such that y in the carrier of A and z in the carrier of A and i = [ y , z ] and i = [ y , z ] ;
given H1 , H2 being strict Subgroup of G such that x = H1 & y = H2 and H1 is Subgroup of H2 and H2 is Subgroup of H2 and H2 is Subgroup of H2 ;
for S , T being non empty RelStr , d being Function of T , S , g being Function of T , S st d is complete holds d is monotone & g is monotone
[ a + 0 , b + ( i + 1 ) ] in ( the carrier of F_Complex ) /\ ( the carrier of V ) & [ a , b ] in [: the carrier of F_Complex , the carrier of V :] /\ [: the carrier of V , the carrier of V :] ;
reconsider mm = max ( len F1 , len ( p . n ) * ( x |^ n ) ) , mm = max ( len F1 , len ( p . n ) * ( x |^ n ) ) as Element of NAT ;
I <= width GoB ( ( GoB ( h ) ) * ( 1 , j ) ) & ( ( GoB ( h ) ) * ( 1 , j ) ) `2 <= ( GoB ( h ) ) * ( 1 , j + 1 ) `2 ;
f2 /* q = ( f2 /* ( f1 /* s ) ) ^\ k .= ( ( f2 * f1 ) /* s ) ^\ k .= ( ( f2 * f1 ) /* s ) ^\ k .= ( ( f2 * f1 ) /* s ) ^\ k ;
attr A1 \/ A2 is linearly-independent means : Def3 : A1 misses A2 & for x , y st x in A1 & y in A2 holds Lin ( A1 \/ A2 ) = Lin ( A1 \/ Lin ( A2 ) ) & Lin ( A1 \/ B ) = Lin ( A2 \/ Lin ( B ) ) ;
func A -carrier of C -> set means : Def3 : for s being Element of R holds s in it iff s in C & for x being Element of R st x in C holds it . x = A ( x ) ;
dom ( Line ( v , i + 1 ) ) ^ ( ( Line ( p , m ) ) * ( \square , 1 ) ) = dom ( F ^ <* 1 *> ) .= dom ( F ^ <* 1 *> ) .= dom ( F ^ <* 1 *> ) ;
cluster [ ( x `1 ) , ( x `2 ) ] -> non empty & [ ( x `2 ) , ( x `2 ) ] `1 = [ x `1 , ( x `2 ) ] `1 , ( x `2 ) ] `2 = [ x `1 , ( x `2 ) ] `2 ;
E , { All ( x2 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 ) } |= All ( x2 , x3 , x4 , x5 ) ;
F .: ( id X , g ) . x = F . ( id X , g . x ) .= F . ( id X , g . x ) .= F . ( x , g ) ;
R . ( h . m ) = F . x0 + h . ( m + 1 ) - ( h . m ) + ( h . ( m + 1 ) - ( h . m ) ) ;
cell ( G , X9 -' 1 , ( Y + 1 ) + ( t + 1 ) ) \ L~ f meets ( UBD L~ f ) \/ ( ( L~ f ) ` ) ;
IC Comput ( P2 , s2 , LifeSpan ( P2 , s ) ) = IC Comput ( P2 , s2 , LifeSpan ( P2 , s ) ) .= ( card I + 1 ) .= ( card I + 1 ) + 1 .= ( card I + 1 ) + 1 ;
sqrt ( ( - ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ) ^2 + ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ) ^2 ) > 0 ;
consider x0 being element such that x0 in dom a and x0 in g " { k } and y0 = a . x0 and x0 in dom ( g | X ) and x0 in X and x0 in X ;
dom ( r1 (#) chi ( A , A ) ) = dom chi ( A , A ) /\ dom chi ( A , A ) .= dom ( r1 (#) chi ( A , A ) ) /\ dom ( r2 (#) chi ( A , A ) ) .= dom ( r1 (#) chi ( A , A ) ) /\ dom ( r2 (#) chi ( A , A ) ) .= dom ( r2 (#) chi ( A , A ) ) ;
d-7 . [ y , z ] = ( ( y - z ) `2 ) * ( ( y - z ) `2 ) .= ( ( y - z ) `2 ) * ( ( y - z ) `2 ) ;
attr i be Nat means : Def3 : C . i = A . i /\ B . i & C . i c= A /\ B . i ;
assume that x0 in dom f and f is_continuous_in x0 and f is_continuous_in x0 and for r st x0 in dom f holds ||. f /. x0 - f /. x0 .|| < r & f /. x0 - f /. x0 .|| < r ;
p in Cl A implies for K being Basis of p , Q being Basis of p , Q being Subset of T st Q in K & p in Q holds A meets Q
for x being Element of REAL n st x in Line ( x1 , x2 ) holds |. y1 - y2 .| <= |. y1 - y2 .| & |. y1 - y2 .| <= |. y1 - y2 .|
func Sum <*> -> w \rm w -> Ordinal means : Def3 : a in it & for b being Ordinal st a in it holds it . b c= b & for b being Ordinal st b in it holds it . b c= a ;
[ a1 , a2 , a3 ] in ( the carrier of A ) /\ ( the carrier of A ) & [ a1 , a2 , a3 ] in [: the carrier of A , the carrier of A :] /\ [: the carrier of A , the carrier of A :] ;
ex a , b being element st a in the carrier of S1 & b in the carrier of S2 & x = [ a , b ] & [ a , b ] in [: the carrier of S1 , the carrier of S2 :] ;
||. ( vseq . n ) - ( vseq . m ) .|| * ||. x - x0 .|| < ( e / ( ||. x .|| + ||. x .|| ) * ||. x - x0 .|| ) * ||. x - x0 .|| ;
then for Z being set st Z in { Y where Y is Element of I7 : F c= Y & z in Z } holds z in x & z in Z ;
sup compactbelow [ s , t ] = [ sup ( { s } /\ compactbelow [ s , t ] ) , sup ( { t } /\ compactbelow [ s , t ] ) ] ;
consider i , j being Element of NAT such that i < j and [ y , f . j ] in II and [ f . i , z ] in II and [ f . i , z ] in II ;
for D being non empty set , p , q being FinSequence of D st p c= q holds ex p being FinSequence of D st p ^ q = q & p ^ q = q ^ p
consider e19 being Element of the carrier of X such that c9 , a9 // a9 , b9 and a9 <> b9 and c9 <> b9 and a9 <> b9 and a , b // c9 , b9 and a , c // a9 , b9 and a , b // a9 , b9 and a , b // a9 , b9 ;
set U2 = I \! \mathop { \vert : not contradiction } , U2 = I \! \mathop { \vert : |. I .| c= 1 } , U1 = { |. I .| : |. I .| c= 1 } , U2 = { |. I .| : |. I .| c= 1 } , U2 = { |. I .| : |. I .| c= 1 } , U2 = { |. I .| : |. I .| c= 1 } , U2 = { |. I .| : |. I .| c= 1 } , U2 = { |. I .| } , U2 = { |. I .| ^ |. I .| } , U2 = { |. I .| ^ |. I .| ^ |. I .|
|. q3 .| ^2 = ( |. q3 .| ) ^2 + ( |. q2 .| ) ^2 .= |. q .| ^2 + ( |. q2 .| ) ^2 .= |. q .| ^2 + ( |. q .| ) ^2 .= |. q .| ^2 + ( |. q .| ) ^2 ;
for T being non empty TopSpace , x , y being Element of [: the topology of T , the topology of T :] holds x "\/" y = x "\/" y implies x "/\" y = x /\ y
dom signature U1 = dom ( the charact of U1 ) & Args ( o , MSAlg U1 ) = dom ( the charact of U1 ) & Args ( o , MSAlg U1 ) = dom ( the charact of U1 ) & Args ( o , MSAlg U1 ) = dom ( the charact of U1 ) & dom ( the charact of U1 ) = dom ( the charact of U1 ) ;
dom ( h | X ) = dom h /\ X .= dom ( ||. h .|| | X ) .= dom ( ||. h .|| | X ) .= dom ( ||. h .|| | X ) .= dom ( ||. h .|| | X ) .= dom ( ||. h .|| | X ) .= dom ( ||. h .|| | X ) .= dom ( ||. h .|| | X ) ;
for N1 , N2 being Element of GX , h being Element of G st dom h = N & rng ( h . K1 ) = N & rng ( h . K1 ) c= N & h . ( h . K1 ) = N holds h . N1 = h . N2
( mod ( u , m ) + mod ( v , m ) ) . i = ( mod ( u , m ) + mod ( v , m ) ) . i .= ( mod ( v , m ) + mod ( v , m ) ) . i ;
- ( q `1 / |. q .| - cn ) < - ( q `1 / |. q .| - cn ) or - ( q `1 / |. q .| - cn ) >= - ( q `1 / |. q .| - cn ) & - ( q `1 / |. q .| - cn ) < - ( q `1 / |. q .| - cn ) ;
attr r1 = f9 & r2 = g9 & ex f , g st r1 * f = f9 & g * g = g9 & f * g = g9 & g * f = g9 & g * g = g9 & f * g = g9 ;
vseq . m is bounded Function of X , the carrier of Y & vseq . m = ( ( vseq . m ) + ( vseq . n ) ) . x & ( vseq . m ) + ( vseq . n ) . x = ( ( vseq . m ) + ( vseq . n ) ) . x ;
pred a <> b & b <> c & angle ( a , b , c ) = PI & angle ( b , c , a ) = 0 & angle ( b , c , a ) = 0 ;
consider i , j , r being Real such that p1 = [ i , r ] and p2 = [ j , s ] and i < j and j < len f and f . i = f . j ;
|. p .| ^2 - ( 2 * |( p , q )| ) ^2 + |. q .| ^2 = |. p .| ^2 + |. q .| ^2 + |. q .| ^2 ;
consider p1 , q1 being Element of X ( ) such that y = p1 ^ q1 and q1 in X ( ) and p1 ^ q1 = p1 ^ q1 and q1 in X ( ) and q1 in X ( ) and q1 in X ( ) and q2 in X ( ) and q1 = q1 ^ q2 ;
, , + ( r1 , s1 ) , s2 = ( s2 + s1 ) / 2 , s1 = ( s2 + s1 ) / 2 , s2 = ( s2 + s2 ) / 2 , s1 = ( s2 + s2 ) / 2 , s2 = ( s2 + s2 ) / 2 ;
( ( LMP A ) `2 ) = lower_bound ( proj2 .: ( A /\ Vertical_Line w ) ) & ( proj2 .: ( A /\ Vertical_Line w ) ) is non empty implies proj2 .: ( A /\ Vertical_Line w ) is non empty
s , ( k + 1 ) |= H1 => H2 iff s , ( k + 1 ) |= ( H ) & ( s , ( k + 1 ) ) . ( k + 1 ) |= ( H ) . ( k + 1 ) ;
len ( s ) + 1 = card ( support b1 ) + 1 .= card ( support b2 ) + 1 .= card ( support b2 ) + 1 .= card ( support b2 ) + 1 .= card ( support b2 ) + 1 .= card ( support b2 ) + 1 ;
consider z being Element of L1 such that z >= x and z >= y and for z being Element of L1 st z >= x & z `1 >= y holds z `2 >= z `2 ;
LSeg ( UMP D , |[ ( W-bound D + E-bound D ) / 2 , ( E-bound D + E-bound D ) / 2 ]| ) /\ D = { UMP D } ;
lim ( ( ( f `| N ) / g ) /* b ) = ( ( f `| N ) / g ) /* b .= lim ( ( f `| N ) / g ) .= lim ( ( f `| N ) / g ) ;
P [ i , pr1 ( f ) . i , pr1 ( f ) . ( i + 1 ) , pr1 ( f ) . ( i + 1 ) ] & pr1 ( f ) . ( i + 1 ) = pr1 ( f ) . i ;
for r be Real st 0 < r ex m be Nat st for k be Nat st m <= k holds ||. ( ( seq . k ) - ( seq . m ) ) .|| < r
for X being set , P being a_partition of X , x , a , b being set st x in a & a in P & b in P & x in P & b in P holds a = b
Z c= dom ( ( ( #Z n ) * f ) / ( ( #Z n ) * f ) ) \ ( ( ( #Z n ) * f ) " { 0 } ) & f | Z is continuous ;
ex j being Nat st j in dom ( l ^ <* x *> ) & j < i & y = ( l ^ <* x *> ) . j & i = 1 + j & j = len ( l ^ <* x *> ) & z = ( l ^ <* x *> ) . j ;
for u , v being VECTOR of V , r being Real st 0 < r & u in N holds r * u + ( 1-r * v ) in N +
A , Int A , Cl ( A , Cl ( A , B ) ) , Cl ( A , Cl ( A , B ) ) / ( Cl ( A , B ) , Cl ( A , B ) ) / ( Cl ( A , B ) , Cl ( A , B ) ) / ( Cl ( A , B ) , Cl ( A , B ) ) / ( Cl ( A , B ) , Cl ( A , B ) ) ) / ( Cl ( Cl ( A , B ) ) , Cl ( Cl ( A , B ) ) / ( Cl ( A , B ) ) / ( Cl ( A , B )
- Sum <* v , u , w *> = - ( v + u + w ) .= - ( v + u ) + ( w + w ) .= - ( v + u ) + ( w + w ) .= - ( v + u ) + ( w + w ) ;
Exec ( a := b , s ) . IC SCM R = ( Exec ( a := b , s ) ) . NAT .= Exec ( ( a := b ) , s ) . NAT .= 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 and h . x in ( the carrier of J ) . x ;
for S1 , S2 , D , E , T be non empty RelStr , f , g be Function of S1 , S2 , g be Function of S2 , T , h be Function of S2 , T , k be Element of S , g be Function of S2 , T st f = g & g = k & k <= n holds g . k 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 in X & x = z or x = y & y = z or x = z & y = z
E-max L~ Cage ( C , n ) in rng ( Cage ( C , n ) \circlearrowleft Cage ( C , n ) ) & E-max L~ Cage ( C , n ) in rng ( Cage ( C , n ) \circlearrowleft Cage ( C , n ) ) ;
for T , T being Tree , p , q being Element of dom T st p in dom T holds ( T , q ) `1 = T . q & ( T , p ) `2 = T . q & ( T , q ) `2 = 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 that the gcd of k , n and k divides ( m * n ) and n divides ( m * n ) and ( m * n ) divides ( m * n ) and ( m * n ) divides ( m * n ) ;
dom F " = the carrier of X2 & rng F " = the carrier of X1 & F " = the carrier of X2 & F " = the carrier of X2 & F " = the carrier of X2 & F " = the carrier of X1 ;
consider C being finite Subset of V such that C c= A and card C = n and the carrier of V = n and the carrier of W = Lin ( B9 \/ C ) and Lin ( B \/ C ) = Lin ( C \/ B ) ;
V is prime implies for X , Y being Element of [: the topology of T , the topology of T :] st X /\ Y c= V holds X c= V or Y c= V or Y c= V
set X = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] & for v being Element of B ( ) , v1 ] holds P [ v1 , v ] } ;
angle ( p1 , p3 , p4 ) = 0 .= angle ( p2 , p3 , p4 ) .= angle ( p2 , p3 , p4 ) .= angle ( p2 , p3 , p4 ) .= angle ( p2 , p3 , p4 ) .= angle ( p2 , p3 , p4 ) ;
- sqrt ( ( - ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 ) = - sqrt ( ( - ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 ) .= - 1 ;
ex f being Function of I[01] , TOP-REAL 2 st f is continuous one-to-one & rng f = P & f . 0 = p1 & f . 1 = p2 & f . 0 = p3 & f . 1 = p4 & f . 1 = p4 & f . 0 = p4 & f . 1 = p4 & f . 1 = p4 & f . 1 = p4 ;
attr f is partial differentiable on on on u0 , u0 means : Def3 : SVF1 ( 2 , pdiff1 ( f , 1 ) , u0 ) . u0 = ( proj ( 2 , 3 ) , u0 ) . u0 + SVF1 ( 2 , 3 ) . u0 ;
ex r , s st x = |[ r , s ]| & G * ( len G , 1 ) `1 < r & r < G * ( 1 , 1 ) `1 & G * ( 1 , j ) `2 < s & s < G * ( 1 , j + 1 ) `2 ;
assume that f is_sequence_on G and 1 <= t and t <= len G and G * ( t , width G ) `2 >= ( GoB f ) * ( t , width G ) `2 and G * ( t , width G ) `2 >= ( GoB f ) * ( t , width G ) `2 ;
pred i in dom G means : Def3 : r (#) ( f (#) reproj ( i , x ) ) = r (#) reproj ( i , x ) & f . i = r (#) reproj ( i , x ) ;
consider c1 , c2 being bag of o1 + o2 such that ( decomp c ) /. k = <* c1 , c2 *> and c = ( decomp c1 ) /. k and c1 = ( decomp c1 ) /. ( k + 1 ) and c2 = ( decomp c1 ) /. ( k + 1 ) ;
u0 in { |[ r1 , s1 ]| : r1 < G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `2 < s1 & s1 < G * ( 1 , 1 ) `2 } ;
( ( X ^ Y ) . k ) = the carrier of X . k2 .= ( ( X ^ Y ) . k2 ) . k2 .= ( ( X ^ Y ) . k2 ) . k2 .= ( ( X ^ Y ) . k2 ) . k2 .= ( ( X ^ Y ) . k2 ) . k2 ;
pred len M1 = len M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 ;
consider g2 be Real such that 0 < g2 and { y where y is Point of S : ||. y - x0 .|| < g2 & g2 . y - x0 < g2 & g2 . y < x0 } c= N2 & N c= dom f & f . x0 - f . x0 < r & g2 . y < 0 } c= N2 ;
assume that x < ( - b + sqrt ( delta ( a , b , c ) ) / ( 2 * a ) ) or x > ( - b - sqrt ( delta ( a , b , c ) ) / ( 2 * a ) ) ;
( G1 '&' G2 ) . i = ( <* 3 *> ^ G1 ) . i & ( H1 '&' G1 ) . i = ( <* 3 *> ^ G1 ) . i & ( H1 '&' G1 ) . i = ( <* 3 *> ^ G1 ) . i ;
for i , j st [ i , j ] in Indices ( M3 + M1 ) holds ( M3 + M1 ) * ( i , j ) < M2 * ( i , j ) + M1 * ( i , j )
for f being FinSequence of NAT , i being Element of NAT st i in dom f & for j being Element of NAT st j in dom f holds i divides f /. j holds i divides ( f /. j )
assume F = { [ a , b ] where a , b is set , c is set : for a , b , c being set st a in Bimplies a c= c & b c= c } & ( for a , b st a in Bimplies b c= c ) & c c= a & a c= b ) ;
b2 * q2 + ( b3 * q3 ) + - ( ( b3 * q2 ) * q1 ) + - ( ( a * q2 ) * q2 ) = 0. TOP-REAL n + ( ( a * q2 ) * q1 ) .= 0. TOP-REAL n + ( a * q2 ) * q2 ;
Cl ( Cl ( Cl F ) where D is Subset of T : ex B being Subset of T st D = Cl B & ex B being Subset of T st B in F & A = Cl B & B in F } c= Cl ( Cl ( Cl F ) )
attr seq is summable means : Def3 : seq is summable & ( for n holds seq . n = Sum ( seq ) ) & ( for n holds seq . n = Sum ( seq ) ) implies seq is summable & ( for n holds seq . n = Sum ( seq ) ) & ( for n holds seq . n = Sum ( seq ) ) & ( for n holds seq . n = Sum ( seq ) ) ;
dom ( ( ( ( ( TOP-REAL 2 ) | D ) | D ) | D ) ) = ( the carrier of ( ( TOP-REAL 2 ) | D ) | D ) .= the carrier of ( ( TOP-REAL 2 ) | D ) | D .= D ;
[: X , Z :] is full SubRelStr of [: ( Omega Z ) |^ the carrier of X , ( Omega Z ) |^ the carrier of Y :] & [: X , Y :] is full SubRelStr of [: ( Omega Z ) |^ the carrier of Y , ( Omega Z ) |^ the carrier of Y :] ;
G * ( 1 , j ) `2 = G * ( i , j ) `2 & G * ( 1 , j ) `2 <= G * ( i , j ) `2 & G * ( 1 , j ) `2 <= G * ( i , j ) `2 ;
synonym m1 c= m2 means : Def3 : for p be set st p in P holds the } is \HM { p where p is Element of NAT : the } & the set of p <= ( m2 + 1 ) & the set of p <= ( m2 + 1 ) + 1 ;
consider a being Element of B ( ) such that x = F ( a ) and a in { G ( b ) where b is Element of A ( ) : P [ b ] } and a in A ( ) ;
synonym IT is multiplicative non empty multMagma means : Def3 : the carrier of it = [ the carrier of IT , the carrier of IT , the carrier of it , the carrier of it , the carrier of it , the carrier of it , the carrier of it , the carrier of it , the carrier of it , the carrier of it , the carrier of it , the carrier of it , the carrier of it , 0. it -> Relation of the carrier , the carrier , 0. it #) ;
sequence ( a , b ) + + ( c , d ) = b + ( the carrier of T , 1 ) .= b + ( a + c ) .= b + ( a + c ) .= ( a + c ) + b + c ;
cluster + _ -> in INT means : Def3 : for i1 , i2 , j1 , j2 being Element of INT holds it . ( i1 , i2 ) = + ( i1 , i2 ) + ( j1 , j2 ) + ( j2 , j2 ) ;
( - s2 ) * p1 + ( s2 * p2 ) - ( s2 * p2 ) = ( - r2 ) * p1 + ( s2 * p2 ) - ( s2 * p2 ) .= ( ( - s2 ) * p1 ) + ( s2 * p2 ) .= ( ( - s2 ) * p2 ) + ( s2 * p2 ) ;
eval ( ( a | ( n , L ) ) *' p , x ) = eval ( a | ( n , L ) ) * eval ( p , x ) .= eval ( a | ( n , L ) ) * eval ( p , x ) .= eval ( a | ( n , L ) ) * eval ( p , x ) ;
assume that the TopStruct of S = the TopStruct of T and for D being non empty directed Subset of S , V being open Subset of Omega S , D being open Subset of Omega S st D in V & V is open holds V meets V and V is open and for V being open Subset of S st V in V holds V meets V ;
assume that 1 <= k & k <= len w + 1 and T-7 . ( ( q , w ) -) . k = ( T-7 . ( q , w ) ) . ( ( q , w ) -) . k and TU . ( ( q , w ) -) . ( ( q , w ) -) . ( ( q , w ) -) ;
2 * a |^ ( n + 1 ) + ( 2 * b |^ ( n + 1 ) ) >= ( a |^ n + 1 ) + ( b |^ n + 1 ) + ( b |^ n + 1 ) ;
M , v / ( x. 3 , m ) / ( x. 0 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 0 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 0 , m ) / ( x. 4 , m ) / ( x. 4 , m ) ) / ( x. 0 , m ) / ( x. 4 , m ) ;
assume that f is_differentiable_on l and for x0 st x0 in l holds 0 < f . x0 & for x1 st x1 in l holds f . x1 - f . x0 < f . x0 and f . x1 - f . x0 < 0 ;
for G1 being _Graph , W being Walk of G1 , e being set , W being Walk of G2 , x being set st e in W & x in W holds not ex W being Walk of G1 st e in W & W is Walk of G2
not c9 is not empty iff not ( iff not ( ex y1 , y2 st y1 is not empty & not y2 is not empty & not q1 is not empty & not q2 is not empty ) & not not ( not q1 is not empty & not q2 is not empty & not q2 is not empty ) & not q1 is not empty & not q2 is not empty ) ;
Indices GoB f = [: dom GoB f , Seg width GoB f :] & ( GoB f ) * ( i1 + 1 , j ) in [: dom GoB f , Seg width GoB f :] & ( GoB f ) * ( i1 + 1 , j ) = ( GoB f ) * ( i1 + 1 , j ) ;
for G1 , G2 , G3 being strict Subgroup of O , a , b , c being Element of G1 , x being Element of G2 , y being Element of G2 , a being Element of G1 , b being Element of G2 st a in G1 & b in G2 holds x * y = a * x & a * y = b * x
UsedIntLoc ( int -insort non empty or I = { intloc 0 , intloc 1 , intloc 2 , intloc 0 , intloc 0 , 1 , 1 , 1 , 0 , 1 , 1 , 0 , 1 , 0 , 1 , 0 ) , 1 = [ 0 , 1 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 1 , 0 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 0 , 1 , 1 , 1 , 1 , 0 , 1 , 0 , 1 , 1 , 1 , 1 , 1 , 0 , 1 , 0 , 1 , 1 ,
for f1 , f2 being FinSequence of F st f1 ^ f2 is p -element & Q [ f1 ^ f2 ] & Q [ f2 ] holds Q [ f1 ^ f2 , f2 ^ f1 ] & Q [ f2 ^ f1 , f2 ^ f1 ]
( ( p `1 ) ^2 + ( sqrt ( 1 - ( p `1 / p `2 ) ^2 ) ) ^2 ) = ( ( q `1 / q `2 ) ^2 + ( q `1 / q `2 ) ^2 ) * ( 1 - ( q `1 / q `2 ) ^2 ) ) / ( 1 - ( q `1 / q `2 ) ^2 ) ;
for x1 , x2 , x3 , x4 being Element of REAL n holds |( x1 - x2 , x3 - x3 )| = |( x1 , x3 - x3 )| + |( x2 , x3 - x3 )| + |( x3 , x3 - x4 )| + |( x2 , x3 - x3 )| + |( x3 , x3 - x4 )| + |( x3 , x3 )| + |( x3 , x3 )| + |( x3 , x3 )| + |( x3 , x3 )| + |( x3 , x3 )| = |( x1 , x3 )| + 0 )| + |( x2 , x3 )| + |( x3 , x3 )| + |( x3 , x3 )| + |( x3 , x3 )| + |( x3 , x3 )| + |( x3 , x3 )| + |( x3 , x3 )| + |( x3 , x3 )| + |( x3 , x3 )|
for x st x in dom ( ( element | A ) | A ) holds ( ( ( ( ( the carrier of X ) | A ) | A ) | A ) . ( - x ) ) = - ( ( ( ( the carrier of X ) | A ) | A ) | A ) . ( - x )
for T being non empty TopSpace , P being Subset-Family of T , B being Subset-Family of T , x being Point of T , B being Point of T st B c= P & x in B holds P is Basis of x
( a 'or' b 'imp' c ) . x = 'not' ( ( a 'or' b ) . x ) 'or' c .= ( 'not' ( a 'or' b ) . x 'or' c ) 'or' c .= ( 'not' ( a 'or' b ) . x 'or' b ) 'or' c .= ( 'not' ( a 'or' b ) . x 'or' b ) 'or' c .= TRUE ;
for e being set st e in A ex X1 being Subset of X , Y1 being Subset of Y , Y2 being Subset of X st e = [: X1 , Y1 :] & Y1 is open & [: Y1 , Y2 :] c= [: Y1 , Y2 :] & [: Y1 , Y2 :] c= [: Y1 , Y2 :]
for i be set st i in the carrier of S for f be Function of [: S , T :] , S1 , S2 be Function of [: S , T :] , S1 , S2 be Function of [: S , T :] , S2 , S2 be Function of S , T st f = H . i & f = F | [: S1 , S2 :] holds f | [: S1 , S2 :] = f | [: S1 , S2 :]
for v , w st for y st x <> y holds w . y = v . y holds Valid ( VERUM ( Al ) , J ) . v = Valid ( VERUM ( Al ) , J ) . w
card D = card D1 + card D1 - card D2 .= ( i + 1 ) + ( i + 1 ) - 1 .= ( i + 1 ) + ( i + 1 ) - 1 .= ( i + 1 ) + 1 - 1 .= ( i + 1 ) - 1 .= ( i + 1 ) - 1 ;
IC Exec ( i , s ) = ( s +* ( 0 .--> succ ( s . 0 ) ) ) . 0 .= ( 0 .--> succ ( s . 0 ) ) . 0 .= succ ( s . 0 ) .= succ 0 ;
len f /. ( \downharpoonright i1 -' 1 ) -' 1 + 1 = len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) + 1 ;
for a , b , c being Element of NAT st 1 <= a & 2 <= b & k < a holds a <= b + b-2 or a = a + b-2 or b = a + b-2 or a = b + b-2 or a = a + b-2 or a = b + b-2 or a = a + b-2 or a = a + b-2
for f being FinSequence of TOP-REAL 2 , p being Point of TOP-REAL 2 , f being FinSequence of TOP-REAL 2 , i being Nat st i in LSeg ( f , i ) & p in LSeg ( f , i ) holds Index ( p , f ) <= Index ( p , f ) + Index ( p , f )
( curry ' ( P+* ( P+* ( i , k + 1 ) ) ) # x ) . x = ( ( curry ' ( P+* ( i , k + 1 ) ) ) # x ) . x + ( ( curry ' ( P+* ( i , k + 1 ) ) # x ) . x ) . x ;
z2 = g /. ( \downharpoonright n1 -' n2 + 1 ) .= g . ( i -' n2 + 1 ) .= g . ( i -' n2 + 1 ) .= g . ( i -' n2 + 1 ) .= g . ( i -' n2 + 1 ) .= g . ( i -' n2 + 1 ) .= g . ( i -' n2 + 1 ) ;
[ f . 0 , f . 3 ] in id ( the carrier of G ) \/ ( the InternalRel of G ) or [ f . 0 , f . 3 ] in the InternalRel of G & [ f . 0 , f . 3 ] in the InternalRel of G & [ f . 0 , f . 2 ] in the InternalRel of G ;
for G being Subset-Family of B st G = { R [ X ] where X is Subset of A , Y is Subset of B st X in F6 & Y in F6 holds ( for X being Subset of A st X in F6 holds X is finite ) holds ( ( for X being Subset of A holds X in F ) implies X is finite )
CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 ) ) = CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m1 ) ) .= halt SCMPDS .= halt SCMPDS ;
assume that a on M and b on M and c on N and d on N and p on M and a <> b and c <> d and p on M and a <> b and p <> d and p <> c and p <> d and a <> b and p <> d and p <> b and a <> b and p <> c and a <> b and p <> d and a <> b and p <> d and a <> b ;
assume that T is \hbox { T _ 4 } and F is closed and ex F be Subset-Family of T , A be finite-ind of T st F is closed & A is finite-ind & ind F <= 0 and ind ( T , A ) <= 0 and ind ( T , A ) <= 0 ;
for g1 , g2 st g1 in ]. r - s , r .[ & g2 in ]. r , r .[ holds |. f . g1 - f . g2 .| <= ( g1 - f ) / ( r - s ) & |. f . g2 - f . g2 .| <= ( g1 - f ) / ( r - s )
( ( - ( x + y ) ) * ( z1 + z2 ) ) = ( ( - ( x + y ) ) * ( z1 + z2 ) ) * ( z2 + z2 ) .= ( ( - ( x + y ) ) * ( z1 + z2 ) ) * ( z2 + z2 ) .= ( ( - ( x + y ) ) * ( z2 + y2 ) ) * ( z2 + z2 ) ;
F . i = F /. i .= 0. R + r2 .= ( b |^ n + 1 ) * ( a |^ n ) .= ( ( ( n + 1 ) + a |^ n ) + b |^ n ) * ( a |^ n ) .= ( ( n + 1 ) + a |^ n ) * ( b |^ n ) .= ( ( n + 1 ) + a |^ n ) * ( b |^ n ) ;
ex y being set , f being Function st y = f . n & dom f = A ( ) & f . 0 = A ( ) & for n holds f . ( n + 1 ) = R ( n , f . n ) & f . ( n + 1 ) = y ;
func f (#) F -> FinSequence of V means : Def3 : len it = len F & for i be Nat st i in dom it holds it . i = F /. i * F /. i & for i be Nat st i in dom it holds it . i = f /. i * F /. i ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 } = { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5
for n being Nat for x being set st x = h . n holds h . ( n + 1 ) = o ( x , n ) & x in InputVertices S ( x , n ) & o ( x , n ) in InnerVertices S ( x , n ) & o ( x , n ) in InnerVertices S ( x , n )
ex S1 being Element of CQC-WFF ( Al ) st SubP ( P , l , e ) = S1 & ( for S , x , y being Element of Al ( ) holds ( S , x , y ) `1 = x ) & ( S , x , y ) `1 = x & ( S , y , z ) `2 = y & ( S , x , y ) `1 = y ) ;
consider P being FinSequence of G8 such that p9 = Product P and for i st i in dom P ex t7 being Element of the carrier of G st P . i = t & ex t7 being Element of the carrier of G st P . i = t7 & t . i = t . i & t . i = ( the in of G ) . i ;
for T1 , T2 being strict non empty TopSpace , P being Basis of T1 , T1 , T2 being Basis of T2 st the carrier of T1 = the carrier of T2 & the topology of T2 = the topology of T2 & the topology of T1 = the topology of T2 & the topology of T2 = the topology of T2 & the topology of T1 = the topology of T2 holds T1 is Basis of T1
assume that f is_\/ pdiff1 ( f , 3 ) and r (#) pdiff1 ( f , 3 ) is_partial_differentiable_in u0 , 2 and partdiff ( r (#) pdiff1 ( f , 3 ) , u0 , 3 ) = r * partdiff ( pdiff1 ( f , 3 ) , u0 , 3 ) and partdiff ( r (#) pdiff1 ( f , 3 ) , u0 , 3 ) = r * partdiff ( f , u0 , 3 ) ;
defpred P [ Nat ] means for F , G be FinSequence of REAL , G be Permutation of Seg $1 , s be Permutation of Seg $1 , f be Permutation of rng F st len G = $1 & for i be Nat st i in dom F holds Sum ( F , s ) = Sum ( F , s ) . i & Sum ( G , s ) = Sum ( F , s ) ;
ex j st 1 <= j & j < width GoB f & ( GoB f ) * ( 1 , j ) `2 <= s & s <= ( GoB f ) * ( 1 , j + 1 ) `2 & s <= ( GoB f ) * ( 1 , j + 1 ) `2 & ( GoB f ) * ( 1 , j + 1 ) `2 <= s ;
defpred U [ set , set ] means ex F-23 be Subset-Family of T st $1 = F-23 & union F-23 is open & union F-23 is open & for f be Function of T , the carrier of T st f is open & f is open holds f is discrete discrete of ( T . $1 ) , ( T . $1 ) ` ;
for p4 being Point of TOP-REAL 2 st LE p4 , p4 , P & LE p4 , p , P & LE p4 , p , P & LE p4 , p , P & LE p , p , P holds LE p4 , p , P & LE p , p , P & LE p , p , P & LE p , p , P & LE p , p , P & LE p , p , P & LE p , p , C & LE p , p , C & LE p , p4 , C & LE p , q , C & LE p , q , C & LE p , q , C & LE p , q , C & LE q , q , C & LE q , q , C & LE q , q , C & LE q , q , C & LE q , q , C & LE q , q , C & LE q , q , C & LE q , q , C & LE q
f in \rbrace ( E , H ) & ( for y st y <> x holds g . y = f . y ) implies for y st y in \rbrace ( E , H ) & ( for y st y in E holds f . y = f . y ) & ( for y st y in E holds f . y = f . y ) & ( y = x implies f . y = f . ( y , x ) )
ex 8 being Point of TOP-REAL 2 st x = 8 & ( ( for q being Point of TOP-REAL 2 st q in 8 holds |. q .| <= 8 ) & ( |. q .| ) ^2 >= 8 & ( |. q .| ) ^2 >= 8 implies |. q .| ^2 >= 8 & ( |. q .| ) ^2 + ( |. q .| ) ^2 ) >= 8 * ( |. q .| ) ^2 ;
assume for d7 being Element of NAT st d7 <= d7 holds ( for t being Element of NAT st t <= ( n + 1 ) holds s1 . t = s2 . t ) & ( for n being Nat st n in dom s1 holds s1 . n = s2 . n ) & s1 . t = s2 . t ) & s1 . t = s2 . t ;
assume that s <> t and s is Point of Sphere ( x , r ) and not s is Point of Sphere ( x , r ) and ex e being Point of E st e = Ball ( x , t ) & ex e being Point of E st e = Ball ( s , t ) & e in Ball ( x , r ) & Ball ( e , s ) c= Sphere ( x , r ) ;
given r such that 0 < r and for s st 0 < s ex x1 be Point of CNS st x1 in dom f & ||. x1 - x0 .|| < s & |. f /. x1 - f /. x0 .|| < r / 2 ;
( p | x ) | ( p | ( x | x ) ) = ( ( ( x | x ) | ( x | x ) ) | ( p | ( x | x ) ) ) | ( p | ( x | x ) ) .= ( ( x | x ) | ( x | x ) ) | p ;
assume that x , x + h in dom sec and ( for x st x in dom sec holds ( x + h ) (#) sec . x = ( 4 * sin ( x + h ) + sin ( x ) * sin ( x ) ) / sin ( x ) ^2 and sin . x = sin ( x ) * sin ( x ) + sin ( x ) * sin ( x ) * sin ( x ) ) ^2 ;
assume that i in dom A and len A > 1 and for i , j st i in dom A & j in dom A & i <> j holds A * ( i , j ) = ( ( i , j ) |-> ( i , j ) ) . ( i , j ) and ( i + j ) = ( i + j ) |-> ( i , j ) ;
for i be non zero Element of NAT st i in Seg n holds i divides n or i = n or i = n or i = n & i <> n & j <> n & j <> n implies h . i = <* 1. F_Complex , 1. F_Complex , 0. F_Complex *> & i <> n implies h . i = <* 1. F_Complex , 0. F_Complex , 0. F_Complex *>
( ( b1 => b2 ) '&' ( c1 => c2 ) ) '&' ( ( a1 'or' b1 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 )
assume that for x holds f . x = ( ( ( - 1 ) (#) ( sin * ( x - h ) ) (#) ( sin * ( x - h ) ) ) `| REAL ) . x and for x st x in dom ( ( - 1 ) (#) ( sin * ( x - h ) ) (#) ( sin * ( x - h ) ) ) holds ( ( - 1 ) (#) ( sin * ( x - h ) ) ) `| REAL ) . x = ( - 1 ) * ( sin * ( x - h ) ) . x ) / ( sin * ( x - h ) ) . x - ( sin . x - h ) . x - ( sin . x ) / ( sin . x ) / ( sin . x ) / ( sin . x ) / ( sin . x ) ^2 and ( sin . x ) ^2 and ( sin . x ) ^2 and ( sin . x ) ^2 and ( sin . x )
consider R8 , I8 be Real such that R8 = Integral ( M , Re ( F . n ) ) and I8 = Integral ( M , Re ( F . n ) ) and Integral ( M , Im ( F . n ) ) = Integral ( M , Im ( F . n ) ) + Integral ( M , Im ( F . n ) ) ;
ex k be Element of NAT st k = k & 0 < d & for q be Element of product G st q in X & ||. qthesis - f /. x .|| < r holds ||. partdiff ( f , q , x ) - partdiff ( f , x ) .|| < r ;
x in { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 } \/ 7 } \/ { A } \/ { A , N } \/ { A , 7 } \/ { A , 7 } \/ { A , 7 } \/ { A } \/ { A , 7 } \/ { A , 7 } \/ { A } \/ { A , 7 } \/ { A , 7 } \/ { A , 8
G * ( j , i ) `2 = G * ( 1 , i ) `2 .= G * ( j , i ) `2 .= G * ( j , i ) `2 .= G * ( j , i ) `2 .= G * ( j , i ) `2 .= G * ( j , i ) `2 .= G * ( j , i ) `2 .= G * ( j , i ) `2 .= G * ( j , i ) `2 ;
f1 * p = p .= ( ( the Arity of S1 ) * ( the Arity of S2 ) ) . o .= ( ( the Arity of S1 ) * ( the Arity of S2 ) ) . o .= ( ( the Arity of S1 ) * ( the Arity of S2 ) ) . o .= ( ( the Arity of S1 ) * ( the Arity of S1 ) ) . o .= ( ( the Arity of S1 ) * ( the Arity of S1 ) ) . o ) ;
func \vert tree ( T , P , T1 ) -> Tree means : Def3 : q in it iff q in P & for p st p in P holds p in P or ex p , q st p in P & q in P & p ^ q in P & p ^ q in P & p ^ q in P & p ^ q in P & p ^ q in P ;
F /. ( k + 1 ) = F . ( k + 1 -' 1 ) .= F\cdot ( p . ( k + 1 -' 1 ) , k + 1 -' 1 ) .= F\cdot ( p . k , k + 1 -' 1 ) .= F\cdot ( p . k , k + 1 -' 1 ) .= F\cdot ( p . k , k + 1 -' 1 ) .= FD * ( p . k , k ) ;
for A , B , C being Matrix of K st len B = len C & width B = width C & len A = width C & len B = len C & width A > 0 & len B > 0 & width A > 0 & len A > 0 & width A > 0 & width A > 0 & len B > 0 & width A > 0 holds A * ( B * C ) = ( A * B ) * C ) + ( B * C )
seq . ( k + 1 ) = 0. F_Complex + seq . ( k + 1 ) .= ( Partial_Sums ( seq ) ) . ( k + 1 ) + ( Partial_Sums ( seq ) ) . ( k + 1 ) .= ( Partial_Sums ( seq ) ) . ( k + 1 ) + ( Partial_Sums ( seq ) ) . ( k + 1 ) .= ( Partial_Sums ( seq ) ) . ( k + 1 ) + ( Partial_Sums ( seq ) ) . ( k + 1 ) ;
assume that x in ( the carrier of Cy ) \/ ( the carrier of Cy ) and y in ( the carrier of Cy ) \/ ( the carrier of Cy ) and [ x , y ] in the InternalRel of Cy and [ x , y ] in the InternalRel of Cy and [ x , y ] in the InternalRel of Cy ;
defpred P [ Element of NAT ] means for f st len f = $1 holds ( for k st k in dom f holds ( VAL g ) . k = ( VAL g ) . ( k + 1 ) ) & ( for i st i in dom g holds ( VAL g ) . ( k + 1 ) = ( VAL g ) . ( k + 1 ) ) ;
assume that 1 <= k and k + 1 <= len f and f /. k = G * ( i , j ) and [ i + 1 , j ] in Indices G and f /. k = G * ( i + 1 , j ) and f /. k = G * ( i + 1 , j ) and f /. k = G * ( i , j ) ;
assume that sn < 1 and ( q `1 > 0 & ( q `2 / |. q .| - sn ) / ( 1 + sn ) <= 0 ) and ( for p st p in sn holds p `2 / |. q .| - sn ) / ( 1 + sn ) <= 0 and ( p `2 / |. q .| - sn ) / ( 1 + sn ) <= 0 ) and ( p `2 / |. p .| - sn ) / ( 1 + sn ) and q `2 / |. q .| = 0 ;
for M being non empty dist , x being Point of M , f being Point of M , x being Point of M st x = x `1 holds ex f being sequence of M st f is sequence of rng ( x `2 ) & for n being Element of NAT holds f . n = Ball ( x `1 , ( 1 / n ) * ( x `2 ) )
defpred P [ Element of omega ] means f1 is_differentiable_on Z & f2 is_differentiable_on Z & for x st x in Z holds ( f1 - f2 ) . x = f1 . x - f2 . x / ( f1 . x - f2 . x ) / ( f1 . x ) ^2 & ( f1 - f2 ) . x = f1 . x - f2 . x / ( f1 . x - f2 . x ) / ( f1 . x - f2 . x ) ^2 ;
defpred P1 [ Nat , Point of CNS ] means ( $1 in Y & ||. s1 . $1 - $2 .|| < r & ||. f /. $2 - f /. x0 .|| < r ) implies ||. f /. $2 - f /. x0 .|| < r / 2 * ||. ( $1 - $2 ) - f /. x0 .|| < r / 2 * ||. ( $1 - $2 ) - f /. x0 .|| ;
( f ^ mid ( g , 2 , len g ) ) . i = ( mid ( g , 2 , len g ) ) . i .= ( mid ( g , 2 , len g ) ) . ( i + 1 ) .= g . ( i + 1 ) .= g . ( i + 1 ) .= g . ( i + 1 ) .= g . ( i + 1 ) ;
( 1 - 2 * n0 + 2 * ( n + 2 * ( n + 1 ) ) ) * ( 2 * ( n + 1 ) ) = ( ( 1 - 2 * ( n + 1 ) ) * ( n + 1 ) ) * ( 2 * ( n + 1 ) ) .= ( 1 - 2 * ( n + 1 ) ) * ( n + 1 ) .= ( 1 - 2 * ( n + 1 ) ) * ( n + 1 ) ;
defpred P [ Nat ] means for G being non empty strict finite RelStr , A , B being non empty finite RelStr st G is space & card the carrier of A = $1 & the carrier of B = the carrier of B & the carrier of A = the carrier of B & the InternalRel of B = the RelStr of A holds A is finite ;
assume that f /. 1 in Ball ( u , r ) and 1 <= m and m <= len ( - 1 ) and LSeg ( f , i ) /\ Ball ( u , r ) <> {} and LSeg ( f , i ) /\ Ball ( u , r ) <> {} and LSeg ( f , i ) <> {} and LSeg ( f , i ) /\ Ball ( u , r ) <> {} ;
defpred P [ Element of NAT ] means ( Partial_Sums ( ( cos (#) ( cos * ( ( cos * ( ( cos * ( ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos , ( cos * ( cos , \frac ) / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / /
for x be Element of product F , i be set st x in I & ( for x be set st x in I holds x in dom ( the support of F ) ) & ( for x be set st x in dom ( the Sorts of F ) holds ( x in I ) implies ( x in I ) ) & ( x in I implies x = ( the carrier of F ) . i )
( x " ) |^ ( n + 1 ) = ( ( x " ) * x ) |^ n * x .= ( ( x " ) |^ n ) * ( x |^ n ) .= ( ( x " ) |^ n ) |^ n * x .= ( ( x " ) |^ n ) |^ n * x .= ( ( x " ) |^ n ) |^ n * x .= ( ( x " ) |^ n ) * x ;
DataPart Comput ( P +* I , s , LifeSpan ( P +* I , Initialize s ) + 3 ) = DataPart Comput ( P +* I , s , LifeSpan ( P +* I , Initialize s ) + 3 ) .= DataPart Comput ( P +* I , s , LifeSpan ( P +* I , Initialize s ) + 3 ) .= DataPart Comput ( P +* I , s , LifeSpan ( P +* I , Initialize s ) + 3 ) ;
given r such that 0 < r and ]. x0 , x0 + r .[ c= ( dom f1 /\ dom f2 ) /\ dom ( f1 | ]. x0 , x0 + r .[ ) and for g st g in ]. x0 , x0 + r .[ /\ dom ( f2 | ]. x0 , x0 + r .[ ) holds f1 . g <= f1 . g ;
assume that X c= dom f1 /\ dom f2 and f1 | X is continuous and f2 | X is continuous and ( f1 | X ) is continuous and ( f1 | X ) is continuous and f2 | X is continuous and ( f1 | X ) is continuous and f2 | X is continuous and f2 | X is continuous and f2 | X is continuous and f2 | X is continuous ;
for L being continuous complete LATTICE for l being Element of L , X being Subset of L , x being Element of L st l = sup X & for X being Subset of L st X in X holds x is compact & for x being Element of L st x in X holds x is compact holds x is compact & x is compact
Support ( e /. i ) in { m *' p where m is Polynomial of n , L : ex p being Polynomial of n , L st p in Support ( m ) & ( p /. i ) `1 = p & ( p /. i ) `2 = q `2 & ( p /. i ) `2 = q `2 & p `2 <= q `2 ) ;
( f1 - f2 ) /. ( lim s1 ) = lim ( f1 /* s1 ) - lim ( f2 /* s1 ) .= lim ( f1 /* s1 ) - lim ( f2 /* s1 ) .= lim ( f1 /* s1 ) - lim ( f2 /* s1 ) .= lim ( f1 /* s1 ) - lim ( f2 /* s1 ) .= lim ( f1 /* s1 ) - lim ( f2 /* s1 ) ;
ex p1 being Element of CQC-WFF ( Al ) st p1 = g . p1 & for g being Function of [: [: D , D :] , D ( ) :] , [: D ( ) , D ( ) :] st P [ g , p1 , p2 , p1 , p2 , p3 , q2 ] holds P [ g , p1 , p2 , q2 , q2 , q3 , q3 , p4 , Real , Real )
( 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 ) ) . ( len p + k ) = ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) . ( len p + k ) . ( len p + k ) .= ( p . ( len p + k ) .= ( p . ( len p + k ) . ( len p + k ) . ( len p + k ) . ( len p + k ) .= ( p . ( len p + k ) .= ( ( p + k ) + ( p . ( len p + k ) .= ( ( p . ( len p + k ) + ( p . ( len p + k ) .= ( p . ( len p + k ) + ( p . ( len p + k ) + ( k + k ) .= ( p .
len mid ( upper_volume ( f , D2 ) , indx ( D2 , D1 , j1 ) + 1 , indx ( D2 , D1 , j1 ) + 1 ) = indx ( D2 , D1 , j1 ) - indx ( D2 , D1 , j1 ) + 1 .= indx ( D2 , D1 , j1 ) + 1 - indx ( D2 , D1 , j1 ) + 1 ;
x * y * z = Mz * ( y * z ) .= ( x * y ) * ( y * z ) .= ( x * y ) * ( y * z ) .= ( x * y ) * ( y * z ) .= ( x * y ) * ( y * z ) .= ( x * y ) * ( y * z ) .= ( x * y ) * ( y * z ) ;
v . ( <* x , y *> ) + ( <* x0 , y0 *> ) * i = partdiff ( v , ( x - x0 ) ) + partdiff ( u , ( x - x0 ) ) * ( ( x - x0 ) * ( ( x - x0 ) * ( y - x0 ) ) + partdiff ( u , ( x - x0 ) * ( y - x0 ) ) ) ;
i * i = <* 0 * ( - 1 ) - ( 0 * 0 ) + ( 0 * ( - 1 ) ) + ( - 1 ) * ( - 1 ) + ( - 1 ) * ( - 1 ) .= <* - 1 , 0 , 0 , 0 *> + ( - 1 ) * ( - 1 ) .= <* - 1 , 0 , 0 , 0 *> + ( - 1 ) * ( - 1 ) * ( - 1 ) .= <* - 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0
Sum ( L (#) F ) = Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) + Sum ( ( L (#) F2 ) ^ ( F1 ^ F2 ) ) .= Sum ( L (#) F1 ) + Sum ( ( L (#) F2 ) ^ ( F1 ^ F2 ) ) .= Sum ( L (#) F1 ) + Sum ( ( L (#) F2 ) ^ ( F1 ^ F2 ) ) .= Sum ( L (#) F1 ) + Sum ( F1 (#) F2 ) ;
ex r be Real st for e be Real st 0 < e ex Y1 be finite Subset of X st Y1 is non empty & for Y be finite Subset of X st Y is non empty & Y is finite & for Y1 be Subset of X st Y1 c= Y holds |. Sum ( Y1 ) - Sum ( Y2 ) .| < r / 2
( GoB f ) * ( i , j ) = f /. ( k + 2 ) & ( GoB f ) * ( i , j + 1 ) = f /. ( k + 1 ) or ( GoB f ) * ( i , j + 1 ) = f /. ( k + 1 ) or ( GoB f ) * ( i , j + 1 ) = f /. ( k + 1 ) ;
( ( ( r / 2 ) (#) ( cos * f ) ) `| REAL ) . x = ( ( r / 2 ) (#) ( cos * f ) ) . x .= ( ( 1 / 2 ) (#) ( cos * f ) ) . x .= ( ( 1 / 2 ) (#) ( cos * f ) ) . x .= ( ( 1 / 2 ) (#) ( cos * f ) ) . x ;
( - b + sqrt ( delta ( a , b , c ) ) / ( 2 * a ) ) < 0 & ( - b - sqrt ( delta ( a , b , c ) ) / ( 2 * a ) ) / ( 2 * a ) < 0 or - ( - b - sqrt ( delta ( a , b , c ) ) / ( 2 * a ) ) / ( 2 * a ) ) > 0 ;
assume that ex_inf_of uparrow X /\ C , L and ex_sup_of X , L and ex_sup_of X , L and for x st x in X holds "\/" ( X , L ) = "\/" ( uparrow "\/" ( X , L ) , L ) and "\/" ( X , L ) = "\/" ( uparrow "\/" ( X , L ) , L ) and "\/" ( X , L ) = "\/" ( uparrow "\/" ( X , L ) , L ) ;
( for j being Element of NAT holds ( j = j ) implies ( j = j ) implies ( j = i implies ( j = i implies j = i ) ) & ( j = i implies j = i ) & ( j = j implies j = i ) implies j = i )