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contradiction ;
contradiction ;
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contradiction ;
contradiction ;
contradiction ;
contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
B in X ;
a <> c
T c= S
D c= B
c in X ;
b in X ;
X ;
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 onto ;
let q ;
m = 1 ;
1 < k ;
G is prime ;
b in A ;
d divides a ;
i < n ;
s <= b ;
b in B ;
let r ;
B is one-to-one ;
R is total ;
x = 2 ;
d in D ;
let c ;
let c ;
b = Y ;
0 < k ;
let b ;
let n ;
r <= b ;
x in X ;
i >= 8 ;
let n ;
let n ;
y in f ;
let n ;
1 < j ;
a in L ;
C is boundary ;
a in A ;
1 < x ;
S is finite ;
u in I ;
z << z ;
x in V ;
r < t ;
let t ;
x c= y ;
a <= b ;
m in NAT ;
assume f is prime ;
not x in Y ;
z = +infty ;
k be Nat ;
K ` is being_line ;
assume n >= N ;
assume n >= N ;
assume X is \bf ) ;
assume x in I ;
q is as as as of 0 ;
assume c in x ;
p > 0 ;
assume x in Z ;
assume x in Z ;
1 <= kr2 ;
assume m <= i ;
assume G is prime ;
assume a divides b ;
assume P is closed ;
b-a > 0 ;
assume q in A ;
W is not bounded ;
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 `2 <= c `2 ;
A meets W ;
i `2 <= j `2 ;
assume H is universal ;
assume x in X ;
let X be set ;
let T be Tree ;
let d be element ;
let t be element ;
let x be element ;
let x be element ;
let s be element ;
k <= 5 ;
let X be set ;
let X be set ;
let y be element ;
let x be element ;
P [ 0 ]
let E be set , A be Subset of E ;
let C be Category ;
let x be element ;
k be Nat ;
let x be element ;
let x be element ;
let e be element ;
let x be element ;
P [ 0 ]
let c be element ;
let y be element ;
let x be element ;
let a be Real ;
let x be element ;
let X be element ;
P [ 0 ]
let x be element ;
let x be element ;
let y be element ;
r in REAL ;
let e be element ;
n1 is \setminus ;
Q halts_on s ;
x in \in \in of of of of -1 ;
M < m + 1 ;
T2 is open ;
z in b < a ;
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 Element of 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 ;
let E be Ordinal ;
o \mathord 4 ;
O <> O2 ;
let r be Real ;
let f be FinSeq-Location ;
let i be Nat ;
let n be Nat ;
Cl A = A ;
L c= Cl L ;
A /\ M = B ;
let V be RealUnitarySpace , v be VECTOR of V ;
not s in Y |^ 0 ;
rng f <= w
b "/\" e = b ;
m = m3 ;
t in h . D ;
P [ 0 ] ;
assume z = x * y ;
S . n is bounded ;
let V be RealUnitarySpace , A 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 [ Z ] ;
R [ 0 ] ;
b in f .: X ;
assume q = q2 ;
x in [#] V ;
f . u = 0 ;
assume e1 > 0 ;
let V be RealUnitarySpace , A be Subset of V ;
s is non 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\mathopen is bounded ;
assume p = p2 ;
len f = n ;
assume x in P1 ;
i in dom q ;
let U0 , S ;
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 \mathbin { x } ;
1 <= jj ;
set A = -> / 2 ;
card a [= c ;
e in rng f ;
cluster B \oplus A -> empty ;
H is_\cdot \cdot F ;
assume n0 <= m ;
T is increasing implies T is increasing
e2 <> e2 ;
Z c= dom g ;
dom p = X ;
H is proper ;
i + 1 <= n ;
v <> 0. V ;
A c= Affin A ;
S c= dom F ;
m in dom f ;
let X0 be set ;
c = sup N ;
R is connected implies union M is connected
assume not x in REAL ;
Im f is complete ;
x in Int y ;
dom F = M ;
a in On W ;
assume e in A ( ) ;
C c= C-26 ;
mm <> {} ;
let x be Element of Y ;
let f be be be be be Line of L , x be Element of L ;
not n in Seg 3 ;
assume X in f .: A ;
assume that p <= n and p <= m ;
assume not u in { v } ;
d is Element of A ;
A |^ b misses B ;
e in v + dom meets ;
- y in I ;
let A be non empty set , B be non empty Subset of A ;
P0 = 1 ;
assume r in F . k ;
assume f is simple function ;
let A be non countable set ;
rng f c= NAT ;
assume P [ k ] ;
f6 <> {} ;
let o be Ordinal ;
assume x is sum of squares ;
assume not v in { 1 } ;
let II , II ;
assume that 1 <= j and j < l ;
v = - u ;
assume s . b > 0 ;
d2 in dom f1 ;
assume t . 1 in A ;
let Y be non empty TopSpace , X be Subset of Y ;
assume a in uparrow s ;
let S be non empty Poset ;
a , b // b , a ;
a * b = p * q ;
assume x , y are_the space ;
assume x in [#] ( f ) ;
[ a , c ] in X ;
mm <> {} ;
M + N c= M + M ;
assume M is connected hhthesis ;
assume f is additive brr\ast ;
let x , y be element ;
let T be non empty TopSpace ;
b , a // b , c ;
k in dom Sum p ;
let v be Element of V ;
[ x , y ] in T ;
assume len p = 0 ;
assume C in rng f ;
k1 = k2 & k2 = k1 ;
m + 1 < n + 1 ;
s in S \/ { s } ;
n + i >= n + 1 ;
assume Re y = 0 ;
k1 <= j1 & j1 <= j2 ;
f | A is s1 ;
f . x / 2 <= b ;
assume y in dom h ;
x * y in B1 ;
set X = Seg n ;
1 <= i2 + 1 ;
k + 0 <= k + 1 ;
p ^ q = p ;
j |^ y divides m ;
set m = max A ;
[ x , x ] in R ;
assume x in succ 0 ;
a in sup phi ;
Ci in X ;
q2 c= C1 & q2 c= C2 ;
a2 < c2 & a2 < c1 ;
s2 is 0 -started ;
IC s = 0 ;
s4 = s4 & s4 = s3 ;
let V ;
let x , y be element ;
let x be Element of T ;
assume a in rng F ;
x in dom T `2 ;
let S be non empty Subset of L ;
y " <> 0 ;
y " <> 0 ;
0. V = uw ;
y2 , y , w is_collinear ;
R8 in X ;
let a , b be Real , x be Element of REAL ;
let a be Object of C ;
let x be Vertex of G ;
let o be Object of C , m be Morphism of C ;
r '&' q = P \lbrack l \rbrack ;
let i , j be Nat ;
let s be State of A , t be State of A ;
s4 . n = N ;
set y = ( x `1 ) / 2 ;
mi in dom g ;
l . 2 = y1 ;
|. g . y .| <= r ;
f . x in Cx0 ;
V1 is non empty Subset of L ;
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 A is dense & A is dense ;
|. f . x .| <= r ;
let x be Element of R ;
let b be Element of L ;
assume x in W-19 ;
P [ k , a ] ;
let X be Subset of L ;
let b be Object of B ;
let A , B be category ;
set X = Vars ( C ) ;
let o be OperSymbol of S ;
let R be connected non empty Poset ;
n + 1 = succ n ;
x9 c= Z1 & x in Z1 ;
dom f = C1 & dom g = C2 ;
assume [ a , y ] in X ;
Re ( seq ) is convergent & lim ( seq ) = 0 ;
assume a1 = b1 & a2 = b2 ;
A = sInt A ;
a <= b or b <= a ;
n + 1 in dom f ;
let F be Instruction of S , k be Nat ;
assume that r2 > x0 and x0 < r2 ;
let Y be non empty set , f be Function of Y , BOOLEAN ;
2 * x in dom W ;
m in dom g2 & n + 1 in dom g2 ;
n in dom g1 & n < len g1 ;
k + 1 in dom f ;
not the still of { s } is finite ;
assume x1 <> x2 & x1 <> x3 ;
v2 in V1 & v2 in V1 ;
not [ b `1 , b ] in T ;
ii + 1 = i ;
T c= \llangle ] ;
( l - 1 ) * ( l - 1 ) = 0 ;
let n be Nat ;
( t `2 ) ^2 = r ;
AA is_integrable_on M & A is integrable ;
set t = Top t ;
let A , B be real-membered set ;
k <= len G + 1 ;
cC misses cC ;
Product ( seq ) is non empty ;
e <= f or f <= e ;
cluster -> non empty normal for sequence ;
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 vseq is convergent & vseq is convergent ;
IC s3 = 0 & IC s3 = 1 ;
k in N or k in K ;
F1 \/ F2 c= F ;
Int G1 <> {} & Int G2 <> {} ;
( z `2 ) ^2 = 0 ;
p11 <> p1 & p11 <> p2 ;
assume z in { y , w } ;
MaxADSet ( a ) c= F ;
ex_sup_of downarrow s , S ;
f . x <= f . y ;
let T be up-complete non empty reflexive transitive antisymmetric RelStr ;
q |^ m >= 1 ;
a >= X & b >= Y ;
assume <* a , c *> <> {} ;
F . c = g . c ;
G is one-to-one non empty full ;
A \/ { a } \not c= B ;
0. V = 0. Y .= 0. V ;
let I be as as as as the non empty instruction of S , T ;
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 & x2 in REAL ;
p1 `1 = K1 & p2 `2 = 0 ;
M . k = <*> REAL ;
phi . 0 in rng phi ;
OSMSorts A is closed
assume z0 <> 0. L ;
n < N7 . k ;
0 <= seq . 0 & seq . 0 <= seq . 0 ;
- q + p = v ;
{ v } is Subset of B ;
set g = f /. 1 ;
[: R , R :] is stable ;
set cR = Vertices R ;
p0 c= P4 & P0 c= P2 ;
x in [. 0 , 1 .[ ;
f . y in dom F ;
let T be Scott Scott Scott Scott TopAugmentation of S ;
ex_inf_of the carrier of S , S ;
downarrow a = downarrow b ;
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 ~ \rrangle , b ~ ;
assume a in A ( ) ;
k in dom ( q4 ) ;
p is non empty \HM { 0 } ;
i - 1 = i-1 - 1 ;
f | A is one-to-one ;
assume x in f .: [: X , Y :] ;
i2 - i1 = 0 & i2 - i1 = 0 ;
j2 + 1 <= i2 & j2 + 1 <= j2 ;
g " * a in N ;
K <> { [ {} , {} ] } ;
cluster strict strict for such that ex x being strict such that x in D ;
|. q .| ^2 > 0 ;
|. p4 .| = |. p .| ;
s2 - s1 > 0 ;
assume x in { Gij } ;
W-min C in C & W-min C in C ;
assume x in { Gij } ;
assume i + 1 = len G ;
assume i + 1 = len G ;
dom I = Seg n & dom I = Seg n ;
assume that k in dom C and k <> i ;
1 + 1-1 <= i + - j ;
dom S = dom F & dom S = dom G ;
let s be Element of NAT ;
let R be ManySortedSet of A ;
let n be Element of NAT ;
let S be non empty non void non empty non void holds the TopStruct of S is non empty ;
let f be ManySortedSet of I ;
let z be Element of COMPLEX , p be FinSequence of COMPLEX ;
u in { ag } ;
2 * n < 2 * n ;
let x , y be set ;
B-11 c= V1 & B-15 c= V1 ;
assume I is_closed_on s , P ;
U2 = U2 & ( U1 /\ U2 ) = {} ;
M /. 1 = z /. 1 ;
x9 = x9 & y9 = y9 ;
i + 1 < n + 1 + 1 ;
x in { {} , <* 0 *> } ;
( f | X ) . x <= ( f | X ) . x ;
let l be Element of L ;
x in dom ( F . d ) ;
let i be Element of NAT ;
r8 is ( len r8 ) -element ;
assume <* o2 , o *> <> {} ;
s . x |^ 0 = 1 ;
card K1 in M & card K1 in M ;
assume that X in U and Y in U ;
let D be Subset-Family of Omega ;
set r = { Seg 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 non empty Poset ;
a1 in B . s1 & a2 in B . s2 ;
let V be finite < len F , v be Vector of V ;
A * B on B & A on A ;
f-3 = NAT --> 0 .= fI ;
let A , B be Subset of V ;
z1 = P1 . j & z2 = P2 . j ;
assume f " P is closed ;
reconsider j = i as Element of M ;
let a , b be Element of L ;
assume q in A \/ ( B "\/" C ) ;
dom ( F (#) C ) = o ;
set S = INT |^ X ;
z in dom ( A --> y ) ;
P [ y , h . y ] ;
{ x0 } c= dom f & { x0 } c= 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 ;
[ y , x ] in II ;
( Q ) * ( 1 , 3 ) = 0 ;
set j = x0 gcd m ;
assume a in { x , y , c } ;
j2 - ( j - 1 ) > 0 ;
I \! \mathop { phi } = 1 ;
[ y , d ] in F-8 ;
let f be Function of X , Y ;
set A2 = ( B \/ C ) ;
s1 , s2 s2 s1 s1 s1 s1 s1 s1 s1 s1 s1 s1 s1 s1 s1 s1 s1 s1 s2 s2 s2 s2 s2 s2 s2 s2 s2 s2 s2 s2 s2 s2 s2 s2 s2 s2
j1 - 1 = 0 & j1 - 1 = 0 ;
set m2 = 2 * n + j ;
reconsider t = t as bag of n ;
I2 . j = m . j ;
i |^ s , n are_relative_prime & i |^ s , n are_relative_prime ;
set g = f | D-21 ;
assume that X is lower and 0 <= r ;
( p1 `1 ) ^2 = 1 ^2 + ( p1 `2 ) ^2 ;
a < ( p3 `1 ) / |. p3 .| ;
L \ { m } c= UBD C ;
x in Ball ( x , 10 ) ;
not a in LSeg ( c , m ) ;
1 <= i1 -' 1 & i1 -' 1 < i1 - 1 ;
1 <= i1 -' 1 & i1 -' 1 < i1 - 1 ;
i + i2 <= len h & i + 1 <= len h ;
x = W-min ( P ) & y = W-min ( P ) ;
[ x , z ] in X ~ Z ;
assume y in [. x0 , x .] ;
assume p = <* 1 , 2 , 3 *> ;
len <* A1 *> = 1 & len <* A1 *> = 1 ;
set H = h . g , I = g . I ;
card b * a = |. a .| ;
Shift ( w , 0 ) |= v ;
set h = h2 (*) h1 , k = g (*) h2 ;
assume x in ( X0 /\ 4 ) ;
||. h .|| < d1 & ||. h .|| < s ;
not x in the carrier of f ;
f . y = F ( y ) ;
for n holds X [ n ] ;
k - l = kl2 - l ;
<* p , q *> /. 2 = q ;
let S be Subset of the lattice of Y ;
let P , Q be succ s ;
Q /\ M c= union ( F | M )
f = b * canFS ( S ) ;
let a , b be Element of G ;
f .: X is_<=_than f . sup X
let L be non empty transitive reflexive RelStr , D be Subset of L ;
S-20 is x -let of x , y ;
let r be non positive Real ;
M , v |= x \hbox { y } ;
v + w = 0. ( Z , X ) ;
P [ len ( F ) ] ;
assume InsCode ( i ) = 8 & InsCode ( i ) = 8 ;
the zero of M = 0. M ;
cluster z (#) seq -> summable for sequence of X ;
let O be Subset of the carrier of C ;
||. f .|| | X is continuous ;
x2 = g . ( j + 1 ) ;
cluster -> |^ I for Element of is |^ I ;
reconsider l1 = l-1 as Nat ;
v4 is Vertex of r2 & v4 is Vertex of r2 ;
T2 is SubSpace of T2 & T1 is SubSpace of T2 ;
Q1 /\ Q19 <> {} & Q1 /\ Q19 <> {} ;
k be Nat ;
q " is Element of X & q " is Element of X ;
F . t is set of non zero set ;
assume that n <> 0 and n <> 1 ;
set en = EmptyBag n , en = EmptyBag n ;
let b be Element of Bags n ;
assume for i holds b . i is commutative ;
x is root & y is Element of ( the carrier of S ) * ;
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 ;
let f be Function of C , D , g be Function of C , D ;
for a holds 0. L + a = a
IC s = s . NAT .= IC s .= IC s ;
H + G = F- G ;
Cx1 . x = x2 & Cx1 . x = y2 ;
f1 = f .= f2 .= f2 ;
Sum <* p . 0 *> = p . 0 ;
assume v + W = v + u + W ;
{ a1 } = { a2 } ;
a1 , b1 _|_ b , a & b1 , c1 _|_ b , a ;
d2 , o _|_ o , a3 ;
IO is reflexive & IO is reflexive implies IO is transitive
IO is antisymmetric implies [: O , O :] is antisymmetric
sup rng H1 = e & sup rng H1 = e ;
x = a9 * a9 & y = b9 * a9 & z = c9 * a9 ;
|. p1 .| ^2 >= 1 ^2 ;
assume j2 - 1 < 1 - 1 ;
rng s c= dom f1 & rng s c= dom f2 ;
assume that support a misses support b and not a in support 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 +* J ;
set f = + ( x , y , r ) ;
cluster Ball ( x , r ) -> bounded ;
consider r being Real such that r in A ;
cluster -> non empty for NAT -defined Function ;
let X be non empty directed Subset of S ;
let S be non empty full SubRelStr of L ;
cluster <* [ ] *> . N -> complete for non trivial set ;
( 1 - a ) " = a " ;
( q . {} ) `1 = o ;
$ 1- ( i -' 1 ) > 0 ;
assume ( 1 - 2 ) <= t `1 ;
card B = k + - 1 ;
x in union rng ( f | A ) ;
assume x in the carrier of R & y in the carrier of R ;
d in X ;
f . 1 = L . ( F . 1 ) ;
the vertices of G = { v } & { v } c= the vertices of G ;
let G be : for x being set holds x in G ;
e , v6 be set ;
c . ( i + 1 ) in rng c ;
f2 /* q is divergent_to-infty & f2 . ( lim q ) = f2 . ( lim q ) ;
set z1 = - ( - z2 ) , z2 = - ( - z2 ) ;
assume w is_llas of S , G ;
set f = p |-count t , g = p |-count t , h = p |-count t , t = p |-count t , k = p |-count t , l = p |-count t , l = p |-count t
let c be Object of C ;
assume ex a st P [ a ] ;
let x be Element of REAL m , y be Element of REAL m ;
let IX be Subset-Family of X , A be Subset of X ;
reconsider p = p , q = q 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 FinSequence of SCM , s be State of SCM ;
stop I c= P-12 & I c= P ;
set ci = fSet /. i , fi = fSet /. j ;
w ^ t ^ t ^ s ^ t ^ s ^ t ^ t ^ s ^ t ^ t ^ t ^ s ^ t ^ t ^ t ^ s ^ t ^ t ^ s ^ t ^
W1 /\ W = W1 /\ W ` .= W1 /\ W2 ;
f . j is Element of J . j ;
let x , y be Element of T2 , s be Element of T2 ;
ex d st a , b // b , d ;
a <> 0 & b <> 0 ;
ord x = 1 & x is positive implies x is positive
set g2 = lim ( seq ^\ k ) ;
2 * x >= 2 * ( 1 / 2 ) ;
assume ( a 'or' c ) . z <> TRUE ;
f (*) g in Hom ( c , c ) ;
Hom ( c , c + d ) <> {} ;
assume 2 * Sum ( q | m ) > m ;
L1 . ( F-21 ) = 0 ;
/ ( X \/ R1 ) = / ( X \/ R1 ) ;
( sin . x ) <> 0 & ( sin . x ) <> 0 ;
( ( - 1 ) (#) ( ( #Z 2 ) * ( f1 - #Z 2 ) ) ) . x > 0 ;
o1 in [: X /\ O2 , X /\ O2 :] ;
e , v6 be set ;
r3 > ( 1 - 2 ) * 0 ;
x in P .: ( F -ideal ) ;
let J be closed Subset of R , I be non empty Subset of R ;
h . p1 = f2 . O & h . I = - 1 ;
Index ( p , f ) + 1 <= j ;
len ( q . i ) = width M & len ( q . i ) = width M ;
the carrier of LK c= A & the carrier of LK c= A ;
dom f c= union rng ( F . -10 ) ;
k + 1 in support ( \mathop { n } ) ;
let X be ManySortedSet of the carrier of S ;
[ x `1 , y `2 ] in \mathclose { \rm \hbox { - } Seg } ;
i = D1 or i = D2 or i = D1 ;
assume a mod n = b mod n ;
h . x2 = g . x1 & h . x2 = f . x2 ;
F c= 2 -tuples_on the carrier of X ;
reconsider w = |. s1 .| as Real_Sequence ;
( 1 / m * m + r ) < p ;
dom f = dom I-4 & dom IK = dom IK ;
[#] P-17 = [#] ( ( TOP-REAL 2 ) | K1 ) .= K1 ;
cluster - x -> R_eal -> R_eal for ExtReal ;
then { d1 } c= A ;
cluster TOP-REAL n -> finite-ind for Subset of TOP-REAL n ;
let w1 be Element of M ;
let x be Element of dyadic ( n ) ;
u in W1 & v in W3 implies u + v in W2
reconsider y = y , z = z as Element of L2 ;
N is full SubRelStr of ( T |^ the carrier of S ) ;
sup { x , y } = c "\/" c ;
g . n = n to_power 1 .= n ;
h . J = EqClass ( u , J ) ;
let seq be -> summable sequence of X , n be Nat ;
dist ( x `1 , y ) < ( r / 2 ) * ( 1 / 2 ) ;
reconsider mm1 = m , mm2 = m as Element of NAT ;
x0 - r < r1 - x0 & x0 < x0 + r ;
reconsider P = P ` as strict Subgroup of N ;
set g1 = p * ( idseq q `1 ) , g2 = p * ( idseq q `1 ) ;
let n , m , k be non zero Nat ;
assume that 0 < e and f | A is lower ;
D2 . ( I1 ) in { x } ;
cluster subcondensed -> subopen for Subset of T ;
let P be compact non empty Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
Gik in LSeg ( PI , 1 ) /\ LSeg ( co , 1 ) ;
let n be Element of NAT , x be Element of NAT ;
reconsider SS = S , SS = T as Subset of T ;
dom ( i .--> X `1 ) = { i } ;
let X be non-empty ManySortedSet of S ;
let X be non-empty ManySortedSet of S ;
op ( 1 ) c= { [ {} , {} ] } ;
reconsider m = mm - 1 as Element of NAT ;
reconsider d = x , e = y as Element of C ( ) ;
let s be 0 -started State of SCMPDS , P , Q be Subset of SCMPDS ;
let t be 0 -started State of SCMPDS , Q ;
b , b , x , y & x , y , z is_collinear ;
assume i = n \/ { n } & j = k \/ { k } ;
let f be PartFunc of X , Y ;
N1 >= ( sqrt c ) / sqrt 2 & N2 >= ( sqrt c ) / sqrt 2 ;
reconsider t7 = T7 as TopSpace , T8 = the TopStruct of T ;
set q = h * p ^ <* d *> ;
z2 in U . ( y2 ) /\ Q2 . ( y1 ) ;
A |^ 0 = { <%> E } .= A |^ 0 ;
len W2 = len W + 2 & len W2 = len W + 1 ;
len h2 in dom h2 & len h2 in dom h2 ;
i + 1 in Seg len s2 & i + 1 in Seg len s2 ;
z in dom g1 /\ dom f & z in dom f1 /\ dom f2 ;
assume that p2 = E-max ( K ) and p2 in K ;
len G + 1 <= i1 + 1 ;
f1 (#) f2 is convergent & lim ( f1 (#) f2 ) = x0 ;
cluster s-10 + s-10 -> summable for Real_Sequence ;
assume j in dom ( M1 * M2 ) ;
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 ;
<* xy *> ^ <* y *> \geq x ;
not 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 s , P1 = P +* I , P2 = P +* I ;
consider w being Nat such that q = z + w ;
x ` is Element of x & y ` is Element of X ;
k = 0 & n <> k or k > n ;
then X is discrete for A being Subset of X ;
for x st x in L holds x is FinSequence ;
||. f /. c .|| <= r1 & ||. f /. c .|| <= r1 ;
c in uparrow p & not c in { p } ;
reconsider V = V as Subset of the topological Euclid n ;
N , M be strict strict >= L ;
then z is_>=_than waybelow x & z is_>=_than compactbelow y ;
M \lbrack f , g .] = f & M [. g , f .] = g ;
( ( Fib 1 ) /. 1 ) `1 = TRUE ;
dom g = dom f -tuples_on X & dom h = dom f -tuples_on X ;
mode : is \cal holds the : of G is \cal I ;
[ i , j ] in Indices M & [ i , j ] in Indices M ;
reconsider s = x " , t = y " as Element of H ;
let f be Element of dom ( Subformulae p ) , g be Element of dom ( Subformulae p ) ;
F1 . ( a1 , - a2 ) = G1 . ( a1 , - a2 ) ;
redefine func \mathopen { a , b , r } -> compact ;
let a , b , c , d be Real ;
rng s c= dom ( 1 / f ) & rng s c= dom f ;
curry ( F-19 , k ) is additive ;
set k2 = card dom B , k1 = card dom A , k2 = card dom B ;
set G = DTConMSA ( X ) ;
reconsider a = [ x , s ] as Object of G ;
let a , b be Element of MM , x be Element of MM ;
reconsider s1 = s , s2 = t as Element of SS ;
rng p c= the carrier of L & p . ( len p ) = p . ( len p ) ;
let d be Subset of the Sorts of A ;
( x .|. x ) = 0 iff x = 0. W ;
I-21 in dom stop I & Ik in dom stop I ;
let g be continuous Function of X | B , Y ;
reconsider D = Y as Subset of ( TOP-REAL n ) | D ;
reconsider i0 = len p1 , i2 = len p2 as Integer ;
dom f = the carrier of S & dom g = the carrier of S ;
rng h c= union ( the carrier of J ) ;
cluster All ( x , H ) -> non empty for strict <* x *> ;
d * N1 / 2 > N1 * 1 / 2 * 1 ;
]. a , b .[ c= [. a , b .] ;
set g = f " ( D1 /\ D2 ) ;
dom ( p | mm1 ) = mm1 .= dom ( p | mm1 ) ;
3 + - 2 <= k + - 2 ;
tan is_differentiable_in ( ( - 1 ) (#) ( arccot ) ) . x ;
x in rng ( f /^ ( len f -' 1 ) ) ;
let f , g be FinSequence of D ;
cp1 in the carrier of S1 & cp1 in the carrier of S2 ;
rng f " = dom f & rng f = dom f ;
( the Source of G ) . e = v & ( the Target of G ) . e = v ;
width G - 1 < width G - 1 ;
assume v in rng ( S | E1 ) ;
assume x is root or x is root & y is root or x is root ;
assume 0 in rng ( ( g2 | A ) | A ) ;
let q be Point of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
let p be Point of TOP-REAL 2 , x be Point of TOP-REAL 2 ;
dist ( O , u ) <= |. p2 .| + 1 ;
assume dist ( x , b ) < dist ( a , b ) ;
<* S7 *> is_in the carrier of C-20 ;
i <= len G -' 1 & 1 <= len G -' 1 ;
let p be Point of TOP-REAL 2 , x be Point of TOP-REAL 2 ;
x1 in the carrier of I[01] & x2 in the carrier of I[01] ;
set p1 = f /. i , p2 = f /. j ;
g in { g2 : r < g2 & g2 < r } ;
Q2 = Sp2 " ( Q /\ R ) .= Q ;
( ( 1 / 2 ) (#) ( 1 / 2 ) ) is summable ;
- p + I c= - p + A ;
n < LifeSpan ( P1 , s1 ) + 1 ;
CurInstr ( p1 , s1 ) = i .= halt SCM+FSA ;
A /\ Cl { x } \ { x } <> {} ;
rng f c= ]. r - 1 , r + 1 .[ ;
let g be Function of S , V ;
let f be Function of L1 , L2 , g be Function of L2 , L1 ;
reconsider z = z , t = y as Element of CompactSublatt L ;
let f be Function of S , T ;
reconsider g = g as Morphism of c opp , b opp ;
[ s , I ] in S ~ ( A , I ) ;
len ( the connectives of C ) = 4 & len ( the connectives of C ) = 5 ;
let C1 , C2 be subcategory , a , b be object of C ;
reconsider V1 = V , V1 = V as Subset of X | B ;
attr p is valid means : Def1 : All ( x , p ) is valid ;
assume that X c= dom f and f .: X c= dom g and g .: X c= dom f ;
H |^ a " * ( a |^ b ) is Subgroup of H ;
let A1 be [ such that A1 : A1 : x in A1 & x in A2 ;
p2 , r3 , q2 is_collinear & p1 , r2 , q1 is_collinear ;
consider x being element such that x in v ^ K ;
not x in { 0. TOP-REAL 2 } & not x in { 0. TOP-REAL 2 } ;
p in [#] ( I[01] | B11 ) & p in [#] ( I[01] | B11 ) ;
0 < M . E8 & M . E8 < M . E8 ;
^ ( c / d ) = c ;
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . s2 ) & a2 in dom ( F . s2 ) ;
cluster -> Line for non empty as | implies L is |
set i1 = the Nat , i2 = the Element of NAT , i1 = the Element of NAT ;
let s be 0 -started State of SCM+FSA , k be Nat ;
assume y in ( f1 \/ f2 ) .: A ;
f . len f = f /. len f .= f /. 1 ;
x , f . x '||' f . x , f . y ;
attr X c= Y means : Def1 : cos X c= cos Y & for x st x in X holds x in Y ;
let y be upper Subset of Y , x be Element of X ;
cluster -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> Nat ;
set S = <* Bags n , i9 *> , T = <* i *> ;
set T = [. 0 , 1 / 2 .] ;
1 in dom mid ( f , 1 , 1 ) ;
( 4 * PI ) / 2 < ( 2 * PI ) / 2 ;
x2 in dom f1 /\ dom f & x1 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 ;
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 *> = len P ;
set N-26 = the max of N , the carrier of N ;
len g] + ( x + 1 ) - 1 <= x ;
not a on B & b on B & not b on B ;
reconsider r-12 = r * I . v as FinSequence of REAL ;
consider d such that x = d and a [= d ;
given u such that u in W and x = v + u ;
len f /. ( \downharpoonright n ) = len f - n ;
set q2 = ( N-min C ) `2 , q2 = ( E-max C ) `2 ;
set S =
MaxADSet ( b ) c= MaxADSet ( P /\ Q ) ;
Cl ( G . q1 ) c= F . r2 & Cl ( G . q2 ) c= G . r2 ;
f " D meets h " V & f " D meets h " V ;
reconsider D = E as non empty directed Subset of L1 ;
H = ( the_left_argument_of H ) '&' ( the_right_argument_of H ) ;
assume t is Element of ( \mathfrak F ) . X & t is Element of ( the carrier of S ) . s ;
rng f c= the carrier of S2 & rng f c= the carrier of S2 ;
consider y being Element of X such that x = { y } ;
f1 . ( a1 , b1 ) = b1 & f1 . ( b1 , b2 ) = b2 ;
the carrier' of G `2 = 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 + 1 ) ;
p-7 . i = pp1 . i .= pp2 . i ;
let PA , PA be a_partition of Y , z be set ;
attr 0 < r & r < 1 implies 1 < ( 1 - r ) / 2 ;
rng ( \mathop { \rm AffineMap ( a , X ) ) = [#] X ;
reconsider x = x , y = y , z = z as Element of K ;
consider k such that z = f . k and n <= k ;
consider x being element such that x in X \ { p } ;
len ( canFS ( s ) ) = card s .= card ( s ) ;
reconsider x2 = x1 , y2 = x2 as Element of L2 ;
Q in FinMeetCl ( ( the topology of X ) \/ the topology of Y ) ;
dom ( f | u ) c= dom ( u | v ) ;
redefine pred n divides m means : Def1 : m divides n & n = m ;
reconsider x = x , y = y as Point of [: I[01] , I[01] :] ;
a in *> implies a in the carrier of \mathop { \rm *> ;
not y0 in the still of f & not y in the carrier of f ;
Hom ( ( a \times b ) , c ) <> {} ;
consider k1 such that p " < k1 and k1 < len f ;
consider c , d such that dom f = c \ d ;
[ x , y ] in dom g ~ & [ y , x ] in dom k ;
set S1 = \times and S2 = \times InnerVertices ] ;
l2 = m2 & l2 = i2 & l2 = j2 implies ( i1 + i2 ) = i2
x0 in dom ( ( u + v ) (#) ( A + B ) ) ;
reconsider p = x , q = y as Point of ( TOP-REAL 2 ) | K1 ;
I[01] = R^1 | B01 .= R^1 | B01 .= R^1 | B01 ;
f . p4 `1 <= f . p1 `1 & f . p2 `2 <= f . p1 `2 ;
( F . x ) `1 <= ( x `1 ) / ( x `2 ) ;
( x `2 ) ^2 = ( W `2 ) ^2 + ( W `2 ) ^2 ;
for n being Element of NAT holds P [ n ] ;
let J , K be non empty Subset of I ;
assume 1 <= i & i <= len <* a " *> ;
0 |-> a = <*> the carrier of K & 0 |-> a = <*> the carrier of K ;
X . i in 2 -tuples_on A . i \ B . i ;
<* 0 *> in dom ( e --> [ 1 , 0 ] ) ;
then P [ a ] & P [ succ a ] ;
reconsider s\frac = s\hbox { a } , s\frac { b } { a } } as \rangle ;
( k - 1 ) <= len thesis - j ;
[#] S c= [#] the TopStruct of T & [#] T c= [#] the TopStruct of T ;
for V being strict RealUnitarySpace holds V in W implies V in W
assume k in dom mid ( f , i , j ) ;
let P be non empty Subset of TOP-REAL 2 , p1 , p2 be Point of TOP-REAL 2 ;
let A , B be Matrix of n1 , n2 , n3 be Matrix of n1 , n2 , K ;
- a * - b = a * b - b * a ;
for A being Subset of A9 , B being Subset of A9 holds A // B implies A = B
( for o2 being Element of A holds [ o2 , o1 ] in [: o2 , o1 :] ) implies [ o2 , o1 ] in [: o2 , o2 :]
then ||. x - y .|| = 0 & x = 0. X ;
let N1 , N2 be strict normal Subgroup of G , a be Element of G ;
j >= len upper_volume ( g , D1 ) & len upper_volume ( g , D1 ) = len g ;
b = Q . ( len Q - 1 ) + 1 ;
f2 * f1 /* s is divergent_to-infty & f2 * f1 is convergent & lim ( f2 * f1 ) = x0 ;
reconsider h = f * g as Function of [: I[01] , I[01] :] , G ;
assume that a <> 0 and Let a , b , c ;
[ t , t ] in the InternalRel of A & [ t , t ] in the InternalRel of A ;
( v |-- E ) | n is Element of T7 ;
{} = the carrier of L1 + L2 .= the carrier of L1 + L2 .= the carrier of L1 ;
Directed I is_closed_on Initialized s , P & Directed I is_halting_on Initialized s , P ;
Initialized p = Initialize ( p +* q ) , p = p +* q ;
reconsider N2 = N1 , N2 = N2 as strict net of R2 ;
reconsider Y = Y as Element of [: Ids L , Ids L :] ;
"/\" ( uparrow p \ { p } , L ) <> p ;
consider j being Nat such that i2 = i1 + j and j in dom f ;
not [ s , 0 ] in the carrier of S2 & not [ s , 0 ] in the carrier of S2 ;
mm in ( B '&' C ) '/\' D \ { {} } ;
n <= len ( P + Q ) + len ( P + Q ) ;
( x1 `1 ) ^2 = ( x2 `1 ) ^2 + ( x1 `2 ) ^2 ;
InputVertices S = { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 } ;
let x , y be Element of [: FTTTT1 , n :] ;
p = |[ p `1 , p `2 ]| .= |[ p `1 , p `2 ]| ;
g * 1_ G = h " * g * 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 /^ ( len f -' 1 ) ) ;
for s being Real st s in R holds s <= s2 ;
rng s c= dom ( f2 * f1 ) /\ dom ( f2 * f1 ) ;
synonym for for for for for for the carrier of X ;
1. K * 1. K = 1. K * 1. K .= 1. K * 1. K .= 1. K ;
set S = Segm ( A , P1 , Q1 ) , Q = Segm ( A , P1 , Q1 ) ;
ex w st e = ( w - f ) & w in F ;
curry ( P\times k , k ) # x is convergent ;
cluster open open -> open for Subset of [: T , T :] ;
len f1 = 1 .= len f3 .= len f3 + 1 .= len f3 + 1 ;
( i * p ) / p < ( 2 * p ) / p ;
let x , y be Element of [: U0 , U0 :] ;
b1 , c1 // b1 , c1 & c1 , c1 // c1 , c2 ;
consider p being element such that c1 . j = { p } ;
assume that f " { 0 } = {} and f is total ;
assume IC Comput ( F , s , k ) = n & IC Comput ( F , s , k ) = n ;
Reloc ( J , card I ) does not destroy a ;
( goto ( card I + 1 ) ) does not destroy c ;
set m3 = LifeSpan ( p3 , s3 ) , P4 = LifeSpan ( p2 , s2 ) ;
IC SCMPDS in dom ( p +* ( a , I ) ) ;
dom t = the carrier of SCM & dom t = the carrier of SCM & dom t = the carrier of SCM ;
( ( E-max L~ f ) .. f ) .. f = 1 ;
let a , b be Element of PFuncs ( V , C ) ;
Cl Int ( union F ) c= Cl Int ( union F ) ;
the carrier of X1 union X2 misses ( A1 \/ A2 ) ;
assume not LIN a , f . a , g . a ;
consider i being Element of M such that i = d6 and i in A ;
then Y c= { x } or Y = {} or Y = { x } ;
M , v |= H1 / ( ( y , x ) / ( y , x ) ) ;
consider m being element such that m in Intersect ( FF ) and m in Y ;
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 -\mathop { V } -> non empty for string of S ;
LG2 /. n2 = LG2 . n2 .= LG2 . n2 .= LG2 . n2 ;
let P be compact non empty Subset of TOP-REAL 2 , p , q be Point of TOP-REAL 2 ;
assume that r-7 in LSeg ( p1 , p2 ) and r-7 in LSeg ( p1 , p2 ) ;
let A be non empty compact Subset of TOP-REAL n , B be Subset of TOP-REAL n ;
assume [ k , m ] in Indices ( ( D * ) ) ;
0 <= ( ( 1 / 2 ) |^ p ) / 2 ;
( F . N ) | E8 = +infty & ( F . N ) | E8 = +infty ;
attr X c= Y means : Def1 : Z c= V & X \ V c= Y \ Z ;
( y `2 ) * ( z `2 ) <> 0. I ;
1 + card X-18 <= card u + card X-18 ;
set g = z \circlearrowleft ( ( L~ z ) .. z ) ;
then k = 1 & p . k = <* x , y *> . k ;
cluster -> total for Element of C -\mathop { \rm G } ;
reconsider B = A , C = B as non empty Subset 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 , x5 , x5 , x5 , x5 ) c= P ;
n <= indx ( D2 , D1 , j1 ) + 1 ;
( ( g2 ) . O ) `1 = - 1 & ( ( g2 ) . I ) `2 = 1 ;
j + p .. f - len f <= len f - 1 - 1 ;
set W = W-bound C , S = S-bound C , E = E-bound C , N = E-bound C , N = E-bound C , S = E-bound C , N = E-bound C , S = E-bound C , N = E-bound C , S = E-bound C , N = E-bound
S1 . ( a , e ) = a + e .= a + e .= a + e ;
1 in Seg width ( M * ColVec2Mx p ) & 1 in Seg width M ;
dom ( i (#) Im f ) = dom ( Im f ) /\ dom ( Im f ) ;
( D^2 ) . x = W . ( a , *' ( a , p ) ) ;
set Q = non empty for g , h being Element of |= ( g , f , h ) ;
cluster -> topological for ManySortedSet of U1 , B be ManySortedFunction of U2 , U2 ;
attr F is discrete means : Def1 : ex A st F = { A } ;
reconsider z9 = \hbox { y where y is Element of product \overline G : y in G } as Element of product \overline G ;
rng f c= rng f1 \/ rng f2 & rng f c= rng f1 \/ rng f2 ;
consider x such that x in f .: A and x in f .: C ;
f = <*> the carrier of F_Complex & f = <*> the carrier of F_Complex ;
E , j |= All ( x1 , x2 ) & E , j |= H ;
reconsider n1 = n , n2 = m as Morphism of o1 , o2 ;
assume that P is idempotent and R is idempotent and P (*) R = R ** P ;
card ( B2 \/ { x } ) = k-1 + 1 ;
card ( ( x \ B1 ) /\ B1 ) = 0 .= 1 ;
g + R in { s : g-r < s & s < g + r } ;
set q-1x0 = ( q , <* s *> ) \mathop { 1 } , qlim s = ( 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 ) .= h0 . ( i + 1 ) ;
set mw = max ( B , } ( NAT ) ) ;
t in Seg width ( I ^ ( n , n ) ) ;
reconsider X = dom f /\ C as Element of Fin NAT ;
IncAddr ( i , k ) = <% - ( a , k ) %> + k .= a ;
( ( for q being Point of TOP-REAL 2 st q `1 <= 0 holds q `2 >= 0 ) implies ( q `2 <= 0 )
attr R is condensed means : Def1 : Int R is condensed & Cl R is condensed ;
attr 0 <= a & a <= 1 & b <= 1 implies a * b <= 1 ;
u in ( ( c /\ ( d /\ b ) ) /\ f ) /\ j ;
u in ( ( c /\ ( d /\ b ) ) /\ f ) /\ j ;
len C + - 2 >= 9 + - 3 + 2 ;
x , z , y is_collinear & x , z , y is_collinear & x , y , z is_collinear ;
a |^ ( n1 + 1 ) = a |^ n1 * a |^ n1 .= a |^ n1 * a |^ n1 ;
<* \underbrace { 0 , \dots , 0 } , x *> in Line ( x , a * x ) ;
set y9 = <* y , c *> , z9 = <* c , x *> ;
FG2 /. 1 in rng Line ( D , 1 ) & FG2 /. len FG2 = 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 ) ;
0 + ( p `2 ) <= 2 * r + ( p `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 convergent ;
reconsider u2 = u , v2 = v as VECTOR of P`1 ( X ) ;
p |-count ( Product Sgm X11 ) = 0 & p |-count ( Product Sgm X11 ) = 1 ;
len <* x *> < i + 1 & i + 1 <= len c + 1 ;
assume that I is non empty and { x } /\ { y } = { 0. I } ;
set ii = card I + 4 .--> goto 0 , goto 0 = goto 0 ;
x in { x , y } & h . x = {} T & y = {} T ;
consider y being Element of F such that y in B and y <= x `2 ;
len S = len ( the charact of ( A ) ) .= len the charact of ( A ) ;
reconsider m = M , i = I , n = N as Element of X ;
A . ( j + 1 ) = B . ( j + 1 ) \/ A . j ;
set N8 = : < G-15 ( G-15 , G-15 ) ;
rng F c= the carrier of gr { a } & rng F c= the carrier of gr { a } ;
( for K holds Q . ( K , n , r ) is a ) -valued FinSequence of D
f . k , f . ( Let n ) ] in rng f ;
h " P /\ [#] T1 = f " P /\ [#] T1 .= [#] T1 /\ [#] T2 .= [#] T1 ;
g in dom f2 \ f2 " { 0 } & ( f2 " { 0 } ) . g in dom f2 \ f2 " { 0 } ;
g1X /\ dom f1 = g1 " { 0 } .= g1 " { 0 } ;
consider n being element such that n in NAT and Z = G . n ;
set d1 = being Subset of dist ( x1 , y1 ) , d2 = dist ( x2 , y2 ) , d2 = dist ( x2 , y2 ) ;
b `2 + ( 1 - r ) < ( 1 - r ) + ( 1 - r ) ;
reconsider f1 = f as VECTOR of the carrier of X , Y be bounded Function of X , Y ;
attr i <> 0 means : Def1 : i ^2 mod ( i + 1 ) = 1 ;
j2 in Seg len ( ( g2 . i2 ) `2 ) & 1 <= j2 & j2 + 1 <= len ( ( g2 . i2 ) `2 ) ;
dom ( i ) = dom ( i ) .= dom ( i ) .= dom ( i ) .= dom ( i ) ;
cluster sec | ]. PI / 2 , PI / 2 .[ -> one-to-one ;
Ball ( u , e ) = Ball ( f . p , e ) ;
reconsider x1 = x0 , y1 = x1 as Function of S , IF ( ) ;
reconsider R1 = x , R2 = y , R2 = z as Relation of L ;
consider a , b being Subset of A such that x = [ a , b ] ;
( <* 1 *> ^ p ) ^ <* n *> in Ri ;
S1 +* 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 +* S2 +* S2 +* S2 +* S2 +* S2 +*
( ( - 1 / 2 ) (#) ( cos * ( f1 + f2 ) ) ) is_differentiable_on Z ;
cluster -> [. 0 , 1 .] -valued for Function of C , REAL ;
set C7 = 1GateCircStr ( <* z , x *> , f3 ) , C8 = 1GateCircStr ( <* y , z *> , f4 ) ;
E8 . e2 = E8 . e2 & E8 . e2 = E8 . e2 ;
( ( - 1 / 2 ) (#) ( ( ln * f ) `| Z ) ) = f ;
upper_bound A = ( PI * 3 / 2 ) * ( PI / 2 ) & lower_bound A = 0 ;
F . ( dom f , - F ) is_transformable_to F . ( cod f , - F ) ;
reconsider p8 = q8 , p7 = q8 as Point of TOP-REAL 2 ;
g . W in [#] Y0 & [#] Y0 c= [#] Y0 & [#] X0 c= [#] X0 ;
let C be compact connected non vertical non horizontal Subset of TOP-REAL 2 , p , q be Point of TOP-REAL 2 ;
LSeg ( f ^ g , j ) = LSeg ( f , j ) ;
rng s c= dom f /\ ]. x0 - r , x0 .[ & rng s c= dom f /\ ]. x0 , x0 + r .[ ;
assume x in { idseq 2 , Rev ( idseq 2 ) } ;
reconsider n2 = n , m2 = m , m2 = n as Element of NAT ;
for y being ExtReal st y in rng seq holds g . y <= 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 , B" = f .: the carrier of X2 ;
ex d being Element of L st d in D & x << d ;
assume that R ~ c= R ~ and R ~ c= R ~ and R ~ c= R ~ and R ~ c= R ~ ;
t in ]. r , s .[ or t = r or t = s or t = s ;
z + v2 in W & x = u + ( z + v2 ) ;
x2 |-- y2 iff P [ x2 , y2 ] & P [ y2 , x2 ] ;
pred x1 <> x2 means : Def1 : |. x1 - x2 .| > 0 & x1 - x2 > 0 ;
assume that p2 - p1 , p3 - p1 , p2 - p1 is_collinear and p2 - p1 , p3 - p1 , p2 - p1 is_collinear ;
set q = ( f ^ <* 'not' A *> ) ^ <* 'not' A *> ;
let f be PartFunc of REAL-NS 1 , REAL-NS 1 , x0 be Point of REAL-NS 1 , r be Real ;
( n mod ( 2 * k ) ) + 1 = n mod k ;
dom ( T * \mathop { \rm \lbrace t } ) = dom ( \mathop { \rm succ t } ) ;
consider x being element such that x in wc iff x in c & x in dom f ;
assume ( F * G ) . v = v . x3 & ( F * G ) . x3 = v . x4 ;
assume that the Sorts of D1 c= the Sorts of D2 and the Sorts of D1 c= the Sorts of D2 ;
reconsider A1 = [. a , b .[ , A2 = [. a , b .[ as Subset of R^1 ;
consider y being element such that y in dom F and F . y = x ;
consider s being element such that s in dom o and a = o . s ;
set p = W-min L~ Cage ( C , n ) , q = W-min L~ Cage ( C , n ) ;
n1 - len f + 1 <= len - 1 + 1 - len f + 1 ;
Seg of ( q , O1 ) = [ u , v , a , b , b , c , d ] ;
set C-2 = ( of `1 ) . ( k + 1 ) , C-2 = ( n + 1 ) + 1 ;
Sum ( L (#) p ) = 0. R * Sum p .= 0. V ;
consider i being element such that i in dom p and t = p . i ;
defpred Q [ Nat ] means 0 = Q ( $1 ) & $1 < len Q & Q [ $1 ] ;
set s3 = Comput ( P1 , s1 , k ) , P3 = Comput ( P2 , s2 , k ) , P4 = P2 , P3 = P3 ;
let l be variable of k , A , v be Element of V ;
reconsider [#] T = union G-24 , G = 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 ) = p2 ;
reconsider B = the carrier of X1 , C = the carrier of X2 as Subset of X ;
p9 = <* - c9 , 1 , - c9 *> .= - ( - c ) .= - ( - c ) ;
synonym f is real-valued for rng f c= NAT & rng f c= NAT ;
consider b being element such that b in dom F and a = F . b ;
x9 < card X0 + card Y0 & card Y0 + 1 < card X0 + 1 + 1 ;
attr X c= B1 means : Def1 : for X st X c= B1 holds X is non empty & X is non empty ;
then w in Ball ( x , r ) & dist ( x , w ) <= r ;
angle ( x , y , z ) = angle ( x , y , z ) ;
attr 1 <= len s means : Def1 : for s being Element of NAT holds s . ( 0 , 0 ) = s . ( 0 , 1 ) ;
fY. c= f . ( k + ( n + 1 ) ) ;
the carrier of { 1_ G } = { 1_ G } .= { 1_ G } ;
pred p '&' q in TAUT ( A ) means : Def1 : q '&' p in TAUT ( A ) ;
- ( t `1 / t `2 ) < - ( t `2 / t `1 ) ;
( 9 . 1 ) `1 = U /. 1 .= W . 1 .= W /. 1 .= W /. 1 .= W /. 1 ;
f .: ( the carrier of x ) = the carrier of x & f .: ( the carrier of x ) = the carrier of x ;
Indices ( O ) = [: Seg n , Seg n :] & [: Seg n , Seg n :] = [: Seg n , Seg n :] ;
for n being Element of NAT holds G . n c= G . ( n + 1 ) ;
then V in M .: \square ex x being Element of M st V = { x } ;
ex f being Element of F-9 st f is \cup ( A * ) & f is \cup ( A * ) ;
[ h . 0 , h . 3 ] in the InternalRel of G & [ h . 2 , h . 3 ] in the InternalRel of G ;
s +* Initialize ( ( intloc 0 ) .--> 1 ) . IC SCM+FSA = s3 . IC SCM+FSA .= s . IC SCM+FSA ;
|[ w1 , v1 ]| - |[ w1 , v1 ]| <> 0. TOP-REAL 2 - |[ w1 , v1 ]| ;
reconsider t = t , s = ( - 1 ) |^ X as Element of ( - 1 ) |^ X ;
C \/ P c= [#] ( ( G \ A ) \/ ( G \ A ) ) ;
f " V in ( such that V ) /\ D & D in ( the topology of X ) /\ D ;
x in [#] ( the carrier of A ) /\ A implies x in [#] ( ( the carrier of F ) | A )
g . x <= h1 . x & h . x <= h1 . x & h1 . x <= h1 . x ;
InputVertices S = { xy , y , z , w , z } & InputVertices S = { 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 M3 is being_line and M3 is being_line and M3 is being_line ;
reconsider a = f4 . ( i0 -' 1 ) , b = f4 . ( i0 -' 1 ) as Element of K ;
len B2 = Sum ( ( Len F1 ) ^ ( Len F2 ) ) .= len ( ( Len F1 ) ^ ( Len F2 ) ) ;
len ( ( the [ the { 0 } , n ) * ( i , j ) ) = n & len ( ( the { 0 } , n ) * ( i , j ) ) = n ;
dom ( f + g ) = dom ( f + g ) .= dom ( f + g ) ;
( for n holds seq . n = upper_bound Y1 holds seq . n = upper_bound Y1 ) implies seq is convergent
dom ( p1 ^ p2 ) = dom ( f12 ) .= dom ( f12 ) .= dom ( f12 ) ;
M . [ 1 / y , y ] = 1 / ( 1 / y ) * v1 .= y ;
assume that W is non trivial and W .last() c= the carrier' of G2 and W is not empty ;
C6 * ( i1 , i2 ) = G1 * ( i1 , i2 ) .= G1 * ( i1 , i2 ) ;
C8 |- 'not' Ex ( x , p ) 'or' p . ( x , y ) ;
for b st b in rng g holds lower_bound rng f^ - b <= b - b
- ( ( q1 `1 / |. q1 .| - cn ) / ( 1 + cn ) ) = 1 ;
( LSeg ( c , m ) \/ NAT ) \/ LSeg ( l , k ) c= R ;
consider p being element such that p in LSeg ( x , p ) and p in L~ f and x = f . p ;
Indices ( X @ ) = [: Seg n , Seg n :] & [: Seg n , Seg n :] = [: Seg n , Seg n :] ;
cluster s => ( q => p ) => ( q => ( s => p ) ) -> valid ;
Im ( ( Partial_Sums F ) . m ) is_measurable_on E & Im ( ( Partial_Sums F ) . m ) is_measurable_on E ;
cluster f . ( x1 , x2 ) -> Element of D * ;
consider g being Function such that g = F . t and Q [ t , g ] ;
p in LSeg ( ( Int Z ) , ( ( \mathopen { - } 1 } ) ) `1 ) ;
set R8 = R / ]. b , +infty .[ , R8 = R / ( b , +infty ) ;
IncAddr ( I , k ) = AddTo ( da , db ) .= goto ( ( card I + 1 ) + k ) ;
seq . m <= ( ( the Sorts of A1 ) * ( the Sorts of A2 ) ) . k ;
a + b = ( a *' ) *' ( b *' ) .= ( a *' ) *' ( b *' ) .= a *' *' ( b *' ) ;
id ( X /\ Y ) = id ( X /\ Y ) .= id ( X /\ Y ) ;
for x being element st x in dom h holds h . x = f . x ;
reconsider H = U1 \/ U2 , U2 = U2 \/ U1 as non empty Subset of U0 ;
u in ( ( c /\ ( ( d /\ b ) /\ f ) ) /\ j ) /\ m ;
consider y being element such that y in Y and P [ y , inf B ] ;
consider A being finite stable set such that card A = len R and card A = card A ;
p2 in rng ( f |-- p1 ) \ rng <* p1 *> & p2 in rng <* p1 *> ;
len s1 - 1 > 0 & len s2 - 1 > 0 ;
( ( N-min P ) `2 ) = ( ( N-min P ) `2 ) .= ( ( E-max P ) `2 ) `2 ;
Ball ( e , r ) c= LeftComp Cage ( C , k + 1 ) \/ LeftComp Cage ( C , k + 1 ) ;
f . a1 ` = f . a1 ` .= a1 ` .= ( f . a1 ) ` .= a1 ` ;
( seq ^\ k ) . n in ]. x0 - r , x0 .[ & ( seq ^\ k ) . n in ]. x0 , x0 + r .[ ;
gg . s0 = g . s0 | G . s0 .= g . s0 ;
the InternalRel of S is \lbrace the carrier of S , the carrier of S , the carrier of S } ;
deffunc F ( Ordinal , Ordinal ) = phi . ( $1 , $2 ) & phi . ( $1 , $2 ) = phi . ( $1 , $2 ) ;
F . a1 = F . ( s2 . a1 ) .= F . a1 .= ( F . a1 ) . a1 ;
x `2 = A . ( o . a ) .= Den ( o , A . a ) ;
Cl ( f " P1 ) c= f " ( Cl P1 ) & f " P1 c= f " ( Cl P1 ) ;
FinMeetCl ( ( the topology of S ) . i ) c= the topology of T & the topology of T c= the topology of T ;
synonym o is \bf means : Def1 : o <> \ast & o <> * ;
assume that X = Y + 1 and card X <> card Y and Y <> {} and X <> {} ;
the *> ( s ) <= 1 + ( the \hbox { - } 1 } ) * ( the \hbox { - } 1 } ) ;
LIN a , a1 , d or b , c // b1 , c1 or a , c // b1 , c1 ;
e . 1 = 0 & e . 2 = 1 & e . 3 = 0 & e . 4 = 0 ;
EE in SE & not EE in { NE } & EE in { NE } ;
set J = ( l , u ) If , K = I " ;
set A1 = .: ( a9 , b9 , c , d ) , A2 = d +* ( a9 , b9 , c ) ;
set c9 = [ <* cin , cin *> , '&' ] , v2 = [ <* cin , cin *> , '&' ] , A1 = [ <* cin , cin *> , '&' ] , A2 = [ <* cin , cin *> , '&' ] , A2 = [ <* cin , cin *> , '&' ] , A2 = [ <* cin , cin *> , '&' ] ;
x * z * x " in x * ( z * N ) * x " ;
for x being element st x in dom f holds f . x = g3 . x & f . x = g3 . x ;
Int cell ( f , 1 , G ) c= RightComp f \/ RightComp f \/ RightComp f ;
U2 is_an_arc_of W-min C , E-max C & L~ ( Cage ( C , n ) ) c= L~ Cage ( C , n ) ;
set f-17 = f @ "/\" g @ ;
attr S1 is convergent means : Def1 : for n holds S1 . n is convergent & lim ( S1 - S2 ) = 0 ;
f . ( 0 + 1 ) = ( 0 qua Ordinal ) + a .= a ;
cluster -> \in \in \mathclose { reflexive transitive , transitive , symmetric , symmetric , symmetric , symmetric , symmetric } for non empty reflexive -symmetric ;
consider d being element such that R reduces b , d and R reduces c , d and R reduces d , 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 |^ 0 ) .--> x ) = len l & len ( l |^ 0 ) = len l ;
t4 ^ {} is ( {} \/ rng t4 ) -valued FinSequence of ( {} \/ rng t4 ) * * \ { {} } ;
t = <* F . t *> ^ ( C . p ^ q-1 ) .= ( C . ( p ^ q-1 ) ) ^ q-1 ;
set p-2 = W-min L~ Cage ( C , n ) , p`2 = Cage ( C , n ) , p`2 = Cage ( C , n ) ;
( k - 1 ) + ( i - 1 ) = ( k - 1 ) + ( i - 1 ) ;
consider u being Element of L such that u = u `2 and u in D and u in D ;
len ( ( width aG ) |-> a ) = width ( ( len G ) |-> a ) .= len ( a * ( len a ) ) .= len a ;
F3 . x in dom ( ( G * the_arity_of o ) . x ) ;
set cH2 = the carrier of H2 , cH2 = the carrier of H2 ;
set cH1 = the carrier of H1 , cH2 = the carrier of H2 ;
( Comput ( P , s , 6 ) ) . intpos m = s . intpos m .= s . intpos m ;
IC Comput ( Q2 , t , k ) = ( l + 1 ) + 1 .= ( l + 1 ) + 1 ;
dom ( ( - 1 / 2 ) (#) ( sin * cos ) ) = REAL & dom ( ( - 1 / 2 ) (#) ( sin * cos ) ) = REAL ;
cluster <* l *> ^ phi -> ( 1 + 1 ) -element for string of S ;
set b9 = [ <* cin , that p , A1 *> , '&' ] , c = [ <* cin , A1 *> , '&' ] ;
Line ( Segm M , P , Q ) . x = L * ( Sgm Q ) . x .= L . x ;
n in dom ( ( the Sorts of A ) * ( the_arity_of o ) ) & dom ( ( the Sorts of A ) * ( the_arity_of o ) ) = dom ( the Sorts of A ) ;
cluster f1 + f2 -> continuous for PartFunc of REAL , the carrier of S , the carrier of T ;
consider y being Point of X such that a = y and ||. x-y - y .|| <= r ;
set x3 = t3 . DataLoc ( s2 . SBP , 2 ) , x4 = Comput ( s2 , s2 , 2 ) . SBP , P4 = Comput ( s2 , s2 , 2 ) . SBP ;
set p-3 = stop I , pE = stop I ;
consider a being Point of D2 such that a in W1 and b = g . a and a in W2 ;
{ A , B , C , D , E } = { A , B , C } \/ { D , E , F , J }
let A , B , C , D , E , F , J , M , N , N , F , J , M , N , N , F , J , M , N , N , F , J , M ;
|. p2 .| ^2 - ( p2 `2 ) ^2 >= 0 & ( p2 `2 ) ^2 - ( p2 `2 ) ^2 >= 0 ;
l - 1 + 1 = n-1 * ( ( m + 1 ) + 1 ) + 1 ;
x = v + ( a * w1 + b * w2 ) + ( c * w2 ) + ( c * w2 ) ;
the TopStruct of L = reconsider the TopStruct of L , the TopStruct of L = the TopStruct of L , the TopStruct of L = the TopStruct of L ;
consider y being element such that y in dom H1 and x = H1 . y and y in dom H1 and H1 . y = H1 . y ;
f9 \ { n } = Free ( All ( v1 , H ) , E ) & f . ( n + 1 ) = f . ( n + 1 ) ;
for Y being Subset of X st Y is summable holds Y is iff Y is \overline implies Y is \overline { A }
2 * n in { N : 2 * Sum ( p | N ) = N & N > 0 } ;
for s being FinSequence holds len ( the { - } \rm \rm <* s *> ) = len s & len ( the { - 1 } ) = len s
for x st x in Z holds exp_R * f is_differentiable_in x & for x st x in Z holds ( exp_R * f ) . x = 1 / ( cos . x ) ^2
rng ( h2 (#) f2 ) c= the carrier of ( ( TOP-REAL 2 ) | K1 ) .= the carrier of ( ( TOP-REAL 2 ) | K1 ) ;
j + ( len f ) - len f <= len f + ( len - f ) - len f ;
reconsider R1 = R * I , R2 = R * I as PartFunc of REAL , REAL-NS n , REAL-NS n ;
C8 . x = s1 . x0 .= C7 . x .= C8 . x .= C8 . x .= ( C * ( f . x ) ) . 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 ) .= s . ( the connectives of S ) . t ;
support ( f + g ) c= support f \/ support ( g ) \/ support ( h ) ;
ex N st N = j1 & 2 * Sum ( ( r | N ) | N ) > N ;
for y , p st P [ p ] holds P [ All ( y , p ) ]
{ [ x1 , x2 ] where x1 , x2 is Element of [: X1 , X2 :] : x1 in X } is Subset of X1 ;
h = ( i , j ) |-- ( id B , id B ) . i .= H . i ;
ex x1 being Element of G st x1 = x & x1 * N c= A & x1 * N c= A ;
set X = ( ( Seg ( q , O1 ) ) . ( [ q , O1 ] ) ) , Y = ( q , O1 ) . ( [ q , O1 ] ) ;
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 the lattice of Y = the lattice of the topology of Y & the topology of Y = the topology of Y ;
'not' ( a . x ) '&' b . x 'or' a . x '&' 'not' ( b . x ) = FALSE ;
q2 = len ( ( q1 ^ r1 ) + ( len q1 ) ) + len q1 .= len ( ( q1 ^ r1 ) + len r1 ) + len r1 ;
( 1 / a ) (#) ( sec * f1 ) - ( 1 / a ) (#) ( sec * f1 ) is_differentiable_on Z ;
set K1 = upper ( ( lim H ) || A ) , D2 = ( lim H ) || A , D1 = ( lim H ) || A ;
assume e in { ( w1 - w2 ) / 2 : w1 in F & w2 in G } ;
reconsider d7 = dom a `1 , d7 = dom F `1 , d8 = dom G `1 , d8 = dom G `1 , d8 = dom G `1 , d8 = dom G `1 , d8 = dom G `1 , d8 = dom G `1 , d8 = G `2 as finite set ;
LSeg ( f /^ j , q ) = LSeg ( f , j ) /\ q .. f .= LSeg ( f , j + q .. f ) ;
assume that X in { T . ( N2 , K ) : h . ( N2 , K ) = N2 } ;
assume that Hom ( d , c ) <> {} and <* f , g *> * f1 = <* f , g *> * f2 ;
dom S[. S , n .] = dom S /\ Seg n .= dom ( L . n ) .= dom ( L . n ) ;
x in H implies ex g st x = g |^ a & g in H & g in H
* ( ( a , 1 ) * n ) = a `2 - ( 0 * n ) .= a `2 - ( 0 * n ) ;
D2 . ( j - 1 ) in { r : lower_bound A <= r & r <= D1 . i } ;
ex p being Point of TOP-REAL 2 st p = x & P [ p ] & p `2 >= 0 & p <> 0. TOP-REAL 2 ;
for c holds f . c <= g . c implies f ^ @ c ^ @ c
dom ( f1 (#) f2 ) /\ X c= dom ( f1 (#) f2 ) /\ dom ( f1 (#) f2 ) ;
1 = ( p * p ) * ( p * q ) .= p * ( 1 / p ) .= p * 1 / p .= p * 1 ;
len g = len f + len <* x + y *> .= len f + 1 .= len f + 1 + 1 .= len f + 1 ;
dom ( F-11 ) = dom ( F | [: N1 , N2 :] ) .= [: N1 , N2 :] ;
dom ( f . t ) * I . t = dom ( f . t ) * g . t ;
assume a in ( "\/" ( ( T |^ the carrier of S ) , F ) ) .: D ;
assume that g is one-to-one and ( the carrier' of S ) /\ rng g c= dom g and g is one-to-one ;
( ( x \ y ) \ z ) \ ( ( x \ z ) \ ( y \ z ) ) = 0. X ;
consider f such that f * f = id b and f * f = id a and f * f = id b ;
( ( cos * cos ) | [. 2 * PI , 2 * PI * 0 + ( cos * cos ) | [. 2 , PI * 0 + PI * ( 0 , 1 ) ) .] is increasing ;
Index ( p , co ) <= len LS - Gij .. LS + 1 - LS .. LS + 1 - LS .. LS + 1 - LS .. LS + 1 ;
let t1 , t2 , t2 , t2 be Element of ( S , S ) . s , S , T ;
( the Frege of ( curry ( curry H ) ) . h ) . h <= "/\" ( ( Frege ( curry G ) ) . h , L ) ;
then P [ f . i0 , f . ( i0 + 1 ) ] & F ( f . i0 , f . ( i0 + 1 ) ) < j ;
Q [ [ D . ( x , 1 ) , F . ( x , 1 ) ] ;
consider x being element such that x in dom ( F . s ) and y = ( F . s ) . x ;
l . i < r . i & [ l . i , r . i ] is for i holds r . i is Element of G . i ;
the Sorts of A2 = ( the carrier of S2 ) --> ( the carrier of S2 ) .= ( the Sorts of A2 ) +* ( the Arity of S2 ) ;
consider s being Function such that s is one-to-one and dom s = NAT and rng s = F . 0 and rng s = { x } ;
dist ( b1 , b2 ) <= dist ( b1 , a ) + dist ( a , b2 ) + dist ( a , b2 ) ;
( Lower_Seq ( C , n ) ) /. len ( Cage ( C , n ) ) = W . len ( Cage ( C , n ) ) ;
q `2 <= ( UMP Upper_Arc Upper_Arc C ) `2 & ( UMP Lower_Arc C ) `2 <= ( UMP C ) `2 ;
LSeg ( f | i2 , i ) /\ LSeg ( f | i2 , j ) = {} ;
given a being ExtReal such that a <= II and A = ]. a , II .[ and a < I ;
consider a , b being complex number such that z = a & y = b and z + y = a + b ;
set X = { b |^ n where n is Element of NAT : n >= 1 } , Y = { b |^ n where n is Element of NAT : n >= 1 } ;
( ( x * y * z ) \ z ) \ ( x * y \ z ) = 0. X ;
set xy = [ <* xy , y , z *> , f1 ] , yz = [ <* y , z *> , f2 ] , xy = [ <* z , x *> , f3 ] , yz = [ <* y , z *> , f3 ] , yz = [ <* z , x *> , f3 ] , \mathopen [ <* z , x *> , f3 ] , \mathopen [ <* z , x *>
li /. len li = li . len ( li ) .= li . len ( li ) .= li . 1 ;
( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 = 1 ;
( ( - ( p `2 / |. p .| - sn ) ) / ( 1 + sn ) ) ^2 < 1 ;
( ( ( S \/ Y ) ` ) ` ) = ( ( S \/ Y ) ` ) ` .= ( ( S \/ Y ) ` ) ` .= ( S \/ Y ) ` ;
( ( - 1 ) (#) ( - ( 1 - s ) ) ) . k = ( - 1 ) * ( ( - s ) . k ) .= ( - 1 ) * ( - s ) ;
rng ( ( h + c ) ^\ n ) c= dom SVF1 ( 1 , f , u0 ) ;
the carrier of X = the carrier of X0 & the carrier of X0 = the carrier of X0 & the carrier of X0 = the carrier of X0 ;
ex p4 st p4 = p4 & |. p4 - |[ a , b ]| .| = r & p4 `2 = b ;
set ch = chi ( X , A ) , A5 = chi ( X , A ) ;
R |^ ( 0 * n ) = Imax ( X , X ) .= R |^ n |^ 0 .= R |^ n ;
( Partial_Sums ( curry ( F1 , n ) ) . n ) . ( n + 1 ) is nonnegative & ( Partial_Sums ( curry ( F1 , n ) ) . n ) . ( n + 1 ) = ( Partial_Sums ( curry ( F1 , n ) ) . ( n + 1 ) ) . ( n + 1 ) ;
f2 = C7 . ( ( E7 , len H ) + len H ) .= C7 . ( len H ) ;
S1 . b = s1 . b .= S2 . b .= S2 . b .= S2 . b .= S2 . b .= S2 . b ;
p2 in LSeg ( p2 , p1 ) /\ LSeg ( p00 , p2 ) \/ LSeg ( p2 , p2 ) /\ LSeg ( p00 , 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 = ( the carrier' of S ) . 11 ;
set phi = ( l1 , l2 ) support phi , phi = ( l1 , l2 ) . ( l + 1 ) ;
synonym p is is invertible for ( being Polynomial of n , L ) , p being Polynomial of n , L ;
( Y1 `2 ) ^2 = - 1 & ( Y1 `2 ) ^2 + ( Y1 `2 ) ^2 <> 0 & ( Y1 `2 ) ^2 + ( Y1 `2 ) ^2 = 1 ;
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 ] ;
consider k be Nat such that for n be Nat st k <= n holds s . n < x0 + g ;
Det ( I |^ ( m -' n ) ) * ( m - n ) = 1. K .= ( I |^ ( m -' n ) ) * ( m - n ) ;
( - b - sqrt ( b ^2 - 4 * a * c ) ) / 2 < 0 ;
Ci . d = C7 . d mod C8 . d .= C8 . d mod C8 . d .= C8 . d mod C8 . d .= C8 . d mod C8 . d ;
attr X1 is dense means : Def1 : X1 is dense dense & X2 is dense dense implies X1 /\ X2 is dense dense SubSpace of X ;
deffunc F6 ( Element of E , Element of I ) = ( $1 * $2 ) * ( $1 * $2 ) ;
t ^ <* n *> in { t ^ <* i *> : Q [ i , T . t ] } ;
( x \ y ) \ x = ( x \ x ) \ y .= y \ 0. X .= 0. X ;
for X being non empty set for Y being Subset-Family of X holds the topology of Y is Basis of <* X , \subseteq \rangle
synonym A , B are_separated for Cl ( A \/ B ) , Cl ( A \/ B ) ;
len ( M @ ) = len p & width ( M @ ) = width ( M @ ) & width ( M @ ) = width ( M @ ) ;
J . v = { x where x is Element of K : 0 < v . x & x < 1 } ;
( Sgm [: [: Seg m , Seg m :] , ( Sgm [: Seg m , Seg m :] ) ) . e <> 0 ;
lower_bound divset ( D2 , k + k2 ) = D2 . ( k + k2 - 1 ) .= D2 . ( k + k2 - 1 ) ;
g . r1 = - 2 * r1 + 1 & dom h = [. 0 , 1 .] & dom h = [. 0 , 1 .] ;
|. a .| * ||. f .|| = 0 * ||. 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 *> ^ s = <* 1 *> ^ w ;
[ 1 , {} , <* d1 *> ] in ( { [ 0 , {} , {} ] } \/ S1 ) \/ S2 ;
IC Exec ( i , s1 ) + n = IC Exec ( i , s2 ) .= IC Exec ( i , s2 ) .= IC Exec ( i , s2 ) ;
IC Comput ( P , s , 1 ) = succ IC s .= ( 0 + 9 ) .= 5 + 9 .= ( 0 + 9 ) ;
( IExec ( W6 , Q , t ) ) . intpos ( e + 1 ) = t . intpos ( e + 1 ) .= t . intpos ( e + 1 ) ;
LSeg ( f /^ q , i ) misses LSeg ( f /^ q , j ) & LSeg ( f /^ q , i ) misses LSeg ( f /^ q , j ) ;
assume for x , y being Element of L st x in C holds x <= y or y <= x ;
integral ( integral ( f , C ) , x ) = f . ( upper_bound C ) - ( lower_bound C ) ;
for F , G being one-to-one st rng F misses rng G holds F ^ G is one-to-one & F ^ G is one-to-one
||. R /. ( L . h ) - R /. ( L . h ) .|| < e1 * ( K + 1 * ||. h .|| ) ;
assume a in { q where q is Element of M : dist ( z , q ) <= r & q <= r } ;
set p4 = [ 2 , 1 ] .--> [ 2 , 0 , 1 ] ;
consider x , y being Subset of X such that [ x , y ] in F and x c= d and y c= d ;
for y , x being Element of REAL st y in Y & x in X & x in Y holds y `1 <= x `1 + y `1 ;
func |. p .| -> variable means : Def1 : for i being Nat holds it . i = min ( NBI . i , p . i ) ;
consider t being Element of S such that x , y '||' z , t and x , z '||' y , t ;
dom x1 = Seg len x1 & len x1 = len y1 & len x1 = len x1 & len y1 = len y1 & len y1 = len y2 & len y1 = len y2 ;
consider y2 being Real such that x2 = y2 and 0 <= y2 and y2 < 1 / 2 and y2 <= 1 / 2 ;
||. f | X .|| /* s1 = ||. f .|| /* s1 .= ||. f .|| /* s1 .= ( ||. f .|| /* s1 ) /* s1 ;
( the InternalRel of A ) ~ /\ Y = {} \/ {} .= {} \/ {} .= {} \/ {} .= {} \/ {} .= {} ;
assume i in dom p & for j being Nat st j in dom q holds P [ i , j ] & for i being Nat st i in dom q & i + 1 in dom q holds P [ i , j ] ;
reconsider h = f | [: X , Y :] , g = ( f | [: X , Y :] ) as Function of [: X , Y :] , Z ;
u1 in the carrier of W1 & u2 in the carrier of W2 implies u1 + u2 in the carrier of W1 & v1 + u1 in the carrier of W2 & v1 + u1 in the carrier of W1 & v2 + u2 in the carrier of W2
defpred P [ Element of L ] means M <= f . $1 & f . $1 <= f . $1 & f . $1 <= f . $1 ;
l . ( u , a , v ) = s * x + ( - ( s * x ) + y ) .= b ;
- ( - x ) = - x + - y .= - x + y .= - x + y .= - y ;
given a being Point of GX such that for x being Point of GX holds a , x + a / 2 / 2 / 2 / 2 , a / 2 / 2 / 2 / 2 / 2 / 2 / 2 ;
f\lbrace f , g } = [ [ dom @ f2 , cod @ g2 ] , [ cod @ g2 , cod @ g2 ] , [ cod f2 , cod f2 ] ] ;
for k , n being Nat st k <> 0 & k < n & n < k holds k , n are_relative_prime & k , n are_relative_prime implies k , n are_relative_prime
for x being element holds x in A |^ d iff x in ( ( A ` ) |^ d ) ` & x in ( A ` ) |^ d ;
consider u , v being Element of R , a being Element of A such that l /. i = u * a * v ;
- ( ( p `1 / |. p .| - cn ) / ( 1 + cn ) ) ^2 > 0 ;
LS . k = LS . ( F . k ) & F . k in dom LS & F . ( k + 1 ) in dom LS ;
set i2 = AddTo ( a , i , - n ) , i1 = goto ( - n ) ;
attr B is thesis means : Def1 : for S being non empty set holds holds ( for B being set holds B in @ ( B , S9 ) ) `1 = ( B `1 ) ^2 ;
a9 "/\" D = { a "/\" d where d is Element of N : d in D & d in D } ;
|( \square , q29 - q29 )| * |( \square , q29 - q )| >= |( \square , - q )| * |( 1 , q )| ;
( - f ) . ( sup A ) = ( ( - f ) | A ) . ( sup A ) .= - ( f | A ) . ( 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 ) ) . ( ( proj ( i , n ) ) . ( LM ) *> ;
f1 + f2 * reproj ( i , x ) is_differentiable_in ( ( - 1 ) (#) reproj ( i , x ) ) . i & f2 * reproj ( i , x ) . x = ( - 1 ) (#) reproj ( i , x ) . i ;
( for x st x in Z holds tan . x <> 0 ) & for x st x in Z holds tan . x = 1 / ( cos . x ) ^2 ) implies tan * tan is_differentiable_on Z & for x st x in Z holds ( tan * tan ) . x = 1 / ( cos . x ) ^2 * ( cos . x ) ^2
ex t being SortSymbol of S st t = s & h1 . t = h2 . ( t . x ) & t . x = h2 . ( t . x ) ;
defpred C [ Nat ] means P8 . $1 is non empty & A8 : A is non empty & A is non empty & A is non empty ;
consider y being element such that y in dom ( p9 . i ) and q9 . i = p9 . y and y in dom ( p9 . i ) and x = ( p9 . i ) . y ;
reconsider L = product ( { x1 } +* ( index B , l ) ) as Basis of A . ( index A ) ;
for c being Element of C ex d being Element of D st T . ( id c ) = id d & for d being Element of D holds d . ( id d ) = id d
be mid ( 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 + 1 ) + G * ( i + 1 , j + 1 ) } ;
f `2 - p = ( f | ( n , L ) ) *' - ( f | ( n , L ) ) .= ( f - ( - ( - ( - ( - ( f . m ) ) ) ) ) ) ) ) *' - ( f - ( - ( f . n ) ) ) ;
consider r being Real such that r in rng ( f | divset ( D , j ) ) and r < m + s ;
f1 . ( |[ r2 , r2 ]| , |[ r2 , r2 ]| ) in f1 .: W1 & f2 .: W2 c= W1 .: W2 /\ W3 ;
eval ( a | ( n , L ) , x ) = ( a | ( n , L ) ) . ( b . ( n , L ) ) .= a . ( b . ( n , L ) ) .= a . ( b . ( n , L ) ) ;
z = DigA ( tz , x9 ) .= DigA ( tz , x9 ) .= DigA ( tz , x9 ) .= DigA ( tz , x9 ) ;
set H = { Intersect S where S is Subset-Family of X : S c= G & S c= F } , G = { Intersect S where S is Subset-Family of X : S c= G } ;
consider S19 being Element of D * , d being Element of D * such that S `1 = S19 ^ <* d *> and S19 = S ^ <* d *> ;
assume that x1 in dom f and x2 in dom f and f . x1 = f . x2 and f . x2 = f . x2 and f . x2 = f . x1 ;
- 1 <= ( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) / ( 1 + sn ) ;
0. ( V ) is Linear_Combination of A & Sum ( L ) = 0. V implies Sum ( L ) = 0. V & Sum ( L ) = 0. V
let k1 , k2 , k2 , k1 , k2 , k2 , : 1 in dom ( the InstructionsF of SCM+FSA ) & k2 in dom ( the InstructionsF of SCM+FSA ) ;
consider j being element such that j in dom a and j in g " { k `2 } and x = a . j and j = a . j ;
H1 . x1 c= H1 . x2 or H1 . x2 c= H1 . x2 or H1 . x1 c= H1 . x2 & H1 . x2 c= H1 . ( x1 ) & H1 . ( x2 ) c= H1 . ( x2 ) ;
consider a being Real such that p = *> * 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 ;
cell ( Gauge ( C , m ) , len Gauge ( C , m ) -' 1 , 0 ) is non empty ;
Aid in { ( S . i ) `1 where i is Element of NAT : i in dom S & i < len S } ;
( T * b1 ) . y = L * b2 /. y .= ( F /. b1 ) * ( G /. b2 ) .= ( F /. b1 ) * ( G /. b2 ) ;
g . ( s , I ) . x = s . y & g . ( s , I ) . y = |. s . x - s . y .| ;
( log ( 2 , k + 1 ) ) ^2 >= ( log ( 2 , k + 1 ) ) ^2 ;
then p => q in S & not x in the carrier of p & not p => All ( x , q ) in S ;
dom ( the InitS of r-10 ) misses dom ( the InitS of rM ) & dom ( the InitS of rM ) misses dom ( the InitS of rM ) ;
synonym f is extended real means : Def1 : for x being set st x in rng f holds x is R_eal & f . x is R_eal ;
assume for a being Element of D holds f . { a } = a & for X being Subset-Family of D holds f . ( f .: X ) = f . union X ;
i = len p1 .= len p3 + len <* x *> .= len p3 + len <* x *> .= len p3 + 1 .= len p3 + 1 .= len p3 + 1 ;
l . ( l , 3 ) = ( g . ( k , 3 ) + k ) + ( k , 3 ) - ( k , 3 ) .= ( g . ( k , 3 ) + k ) - ( k , 3 ) ;
CurInstr ( P2 , Comput ( P2 , s2 , l2 ) ) = halt SCM+FSA .= halt SCM+FSA .= halt SCM+FSA ;
assume for n be Nat holds ||. seq .|| . n <= ( ||. seq .|| ) . n & ( ||. seq .|| ) . n <= ( ||. seq .|| ) . n ;
sin ( e ) = sin ( r ) * cos ( - ( cos ( r ) ) * sin ( - ( cos ( r ) ) ) ) .= 0 ;
set q = |[ g1 . t0 `1 , g2 . t0 `2 ]| , g2 = |[ g1 . t0 `1 , g2 . t0 `2 ]| , g2 = |[ g1 . I `1 , g2 . I `2 ]| ;
consider G being sequence of S such that for n being Element of NAT holds G . n in implies G . n in <= <= <= <= implies G . n ;
consider G such that F = G and ex G1 , G2 st G1 in SM & G2 in SM & G1 = [: G1 , G2 :] & G2 = [: G1 , G2 :] ;
the root of [ x , s ] in ( the Sorts of ( C . s ) ) . ( ( the Sorts of ( C . s ) ) . ( ( the Sorts of ( C . s ) ) . ( ( the Arity of S ) . ( ( the Arity of S ) . o ) ) ) ;
Z c= dom ( ( exp_R * f ) + ( ( exp_R * f ) + ( exp_R * f ) ) ) ;
for k being Element of NAT holds ( ( for n being Nat holds r0 . n = ( upper_volume ( f , S-3 ) ) . k ) . k ) implies r = ( lim ( Im f ) ) ) . k
assume that - 1 < n and n > 0 and ( q `2 / |. q .| - cn ) < 0 and q `2 / |. q .| < 0 ;
assume that f is continuous and a < b and c < d and f . a = g and f . b = d and f . c = g . d ;
consider r being Element of NAT such that s-> Element of NAT , q being Element of NAT st r = Comput ( P1 , s1 , r ) & r <= q ;
LE f /. ( i + 1 ) , f /. j , L~ f , f /. ( 1 + 1 ) , f /. ( len f ) , f /. ( 1 + 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 } = x and inf { x , y } = y ;
assume f +* ( i1 , \xi ) in ( proj ( F , i2 ) ) " ( A . ( i2 + 1 ) ) & f . ( i1 + 1 ) in ( proj ( F , i2 ) ) " ( A . ( i1 + 1 ) ) ;
rng ( ( Flow M ) ~ ) c= the carrier' of M & rng ( ( Flow M ) ~ ) c= the carrier' of M ;
assume z in { ( the carrier of G ) \ { t } where t is Element of T : t in X } ;
consider l be Nat such that for m be Nat st l <= m holds ||. s1 . m - x0 .|| < g / 2 ;
consider t being VECTOR of product G such that mt = ||. Dt .|| and ||. t .|| <= 1 and ||. t .|| <= 1 ;
assume that the carrier of v = 2 and v ^ <* 0 *> in dom p and v ^ <* 1 *> in dom p and p ^ <* 1 *> in dom p and p ^ <* 1 *> in dom p and p ^ <* 1 *> in dom p ;
consider a being Element of the Points of X\bf , A being Element of the lines of X\bf such that a on A and a on A and a on A ;
( - x ) |^ ( k + 1 ) * ( ( - x ) |^ ( k + 1 ) ) " = 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 in D & p . i in D
defpred R [ element ] means ex x , y st [ x , y ] = $1 & P [ x , y ] & P [ y , x ] ;
L~ f2 = union { LSeg ( p0 , p11 ) , LSeg ( p00 , p2 ) } .= LSeg ( p00 , p2 ) /\ LSeg ( p00 , p2 ) ;
i - len h11 + 2 - 1 < i - len h11 + 2 - 1 + 1 + 1 - 1 + 1 + 1 - 1 + 1 + 1 - 1 + 1 ;
for n being Element of NAT st n in dom F holds F . n = |. ( F . n ) . ( n - 1 ) .| ;
for r , s1 , s2 , r st r in [. s1 , s2 .] & s1 <= r & r <= s2 holds r <= s & s <= r & s <= s2
assume v in { G where G is Subset of T2 : G in B2 & G c= z & G c= z1 & z in z2 & G c= z } ;
let g be ) :] \rm :] of INT , X , Z , b be Element of INT , X be set , b be Element of INT , f be Function of [: X , X :] , Z ;
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 ] and for n holds P [ n , q1 . n ] ;
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 B9 = B /\ O , Z = O /\ Z , Z = { Z } , Z = { Z } , Z = { Z } , Z = { Z } , Z = { Z } , Z = { Z } , Z = { Z } , Z = { Z } , Z = { Z } , Z = { Z } , Z = { Z } , Z = { Z } , Z = { Z } , Z = { Z }
consider j being Element of NAT such that x = the the S of n and j + 1 = n and 1 <= j and j + 1 <= n and n + 1 <= len f ;
consider x such that z = x and card x in card ( x . O2 ) and x in L1 . O2 and x in L2 . O2 ;
( C * _ 4 ( k , n2 ) ) . 0 = C . ( ( of of of T4 ( k , n2 ) ) . 0 ) .= ( C * _ 4 ( k , n2 ) ) . 0 ;
dom ( X --> rng f ) = X & dom ( ( X --> f ) . x ) = dom ( X --> f ) ;
( ( ex N being Nat st N <= b & N <= ( L~ SpStSeq C ) ) `2 & ( for i st i <= b holds N <= ( ( L~ SpStSeq C ) * ( i , j ) ) `2 ) `2 ) implies N >= ( ( L~ SpStSeq C ) * ( i , j ) ) `2 )
synonym x , y means : Def1 : x = y or ex l being for k being Nat st { x , y } c= l holds x in { x , y } ;
consider X being element such that X in dom ( f | ( n + 1 ) ) and ( f | ( n + 1 ) ) . X = Y and ( f | ( n + 1 ) ) . X = Y ;
assume that Im k is continuous and for x , y being Element of L st a = x & b = y holds x << y & x << y holds a << b ;
( 1 / 2 * ( ( ( #Z 2 ) * ( ( AffineMap ( n , 0 ) ) ) ) ) ) * ( ( AffineMap ( n , 0 ) ) ) is_differentiable_on REAL ;
defpred P [ Element of omega ] means ( the partial of A1 ) . $1 = A1 . $1 & ( the partial of A2 ) . $1 = A2 . $1 & ( the partial of A1 ) . $1 = A2 . $1 ;
IC Comput ( P , s , 2 ) = succ IC Comput ( P , s , 1 ) .= ( 0 + 1 ) .= ( 0 + 1 ) .= ( 0 + 1 ) .= ( 0 + 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 . ( ( canFS ( Omega ( Omega ) ) ) . n ) .= M . ( ( canFS ( Omega ) ) . n ) .= M . ( ( canFS ( Omega ) ) . n ) ;
the carrier of ( L1 + L2 ) c= ( the carrier of L1 ) \/ ( the carrier of L2 ) \/ ( the carrier of L2 ) ;
pred a , b , c , x , y , c , d , x , y , c , y , z be set means : from a , b , c , d , x , y ;
( the partial of s ) . n <= ( the partial of s ) . n * s . n & ( the partial of s ) . ( n + 1 ) * s . n = ( the partial of s ) . n * s . n ;
attr - 1 <= r & r <= 1 & - 1 < r & r < 1 implies ( - 1 ) * ( - 1 ) * ( - 1 ) = - r * ( - 1 ) ;
seq in { p ^ <* n *> where n is Nat : p ^ <* n *> in T1 & p ^ <* n *> in T1 } ;
|[ x1 , x2 , x3 ]| . 2 - |[ y1 , y2 , x4 ]| . 2 = x2 - y2 & |[ y1 , y2 , x3 ]| . 2 = y2 - y1 ;
attr F is nonnegative means : Def1 : for m being Nat holds ( Partial_Sums F ) . m is nonnegative & ( Partial_Sums F ) . m is nonnegative ) & ( for n being Nat holds F . n is nonnegative ) ;
len ( ( G . z ) + ( G . y ) ) = len ( ( G . ( xx ) ) + ( G . ( y ) ) ) .= len ( G . ( y ) ) + G . ( y ) ;
consider u , v being VECTOR of V such that x = u + v and u in W1 /\ W2 and v in W2 /\ W3 and u in W3 /\ W3 ;
given F be FinSequence of NAT such that F = x and dom F = n and rng F c= { 0 , 1 } and Sum F = k and Sum ( F ) = k and Sum ( F ) = k ;
0 = b1 * D1 & 1 = ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) ;
consider n be Nat such that for m be Nat st n <= m holds |. ( f # x ) . m - lim ( f # x ) .| < e ;
cluster -> being being being being being being being being non empty \frac for non empty 6 for \hbox { $ ( \mathop { \rm o } ) , ( \mathop { \rm o } ) ) is Boolean
"/\" ( BB , L ) = Top ( B ) .= Top ( S ) .= "/\" ( ( the carrier of S ) \ { {} } , L ) .= "/\" ( I , L ) .= "/\" ( I , L ) ;
( r / 2 ) ^2 + ( r / 2 ) ^2 <= ( r / 2 ) ^2 + ( r / 2 ) ^2 ;
for x being element st x in A /\ dom ( ( f `| X ) `| A ) holds ( ( f `| X ) `| A ) . x >= r2 & ( ( f `| X ) `| A ) . x >= r2
2 * ( r1 - c ) * |[ a , c ]| - ( 2 * ( r1 - c ) ) * |[ b , c ]| = 0. TOP-REAL 2 ;
reconsider p = P * ( 1 , 1 ) , q = a " * ( ( - 1 ) |^ ( n + 1 ) ) as FinSequence of K ;
consider x1 , x2 being element such that x1 in downarrow s and x2 in downarrow t and x = [ x1 , x2 ] and [ x2 , y2 ] in t and [ x2 , y2 ] in t ;
for n be Nat st 1 <= n & n <= len q1 holds q1 . n = ( upper_volume ( g , M1 ) ) . ( n + 1 ) & ( upper_volume ( g , M2 ) ) . n = ( upper_volume ( g , M1 ) ) . n ;
consider y , z being element such that y in the carrier of A and z in the carrier of A and i = [ y , z ] and i = [ y , z ] ;
given H1 , H2 being strict Subgroup of G such that x = H1 and y = H2 and H1 is Subgroup of H1 and H2 is Subgroup of H2 and H1 is Subgroup of H2 ;
for S , T being non empty that T is complete & T is complete holds d is directed-sups-preserving implies d is monotone & d is monotone
[ a + i , b + i ] in ( the carrier of F_Complex ) \/ ( the carrier of V ) & [ a + i , b + i ] in the carrier of ( COMPLEX ) ;
reconsider mm = max ( len F1 , len ( p . n ) * ( <* x *> ^ <* x *> ) ) as Element of NAT ;
I <= width GoB ( ( GoB h ) * ( len GoB h , 1 ) , ( GoB h ) * ( len GoB h , 1 ) ) & ( GoB h ) * ( len GoB h , 1 ) `2 <= ( GoB h ) * ( len GoB h , 1 ) `2 ;
f2 /* q = ( f2 /* ( f1 /* s ) ) ^\ k .= ( f2 * f1 ) /* ( f1 /* s ) .= ( f2 * f1 ) /* s ;
attr A1 \/ A2 is linearly-independent means : Def1 : A1 is linearly-independent & A2 is linearly-independent & ( for x being Element of V holds x in A1 & x in A2 implies x in A1 & x in A2 & x in A1 & x in A2 ;
func A -carrier of C -> set means : Def1 : union it = union { A . s where s is Element of R : s in C & s in A } ;
dom ( Line ( v , i + 1 ) ) ^ ( ( Line ( p , m ) ) * ( Line ( p , 1 ) ) ) = dom ( F ^ G ) ;
cluster [ x , 4 ] -> [ x , 4 ] , [ x , 4 ] , [ x , 5 ] ] , [ x , 6 ] , [ x , 5 ] ] -> [ x , 6 ] , [ x , 5 ] ] ;
E , All ( x2 , All ( x2 , x2 ) ) |= All ( x2 , All ( x2 , x2 ) ) => ( ( x1 '&' x2 ) '&' ( x2 '&' x3 ) ) ;
F .: ( id X , g ) . x = F . ( id X , g . x ) .= F . ( id X , g . x ) .= F . ( x , g . x ) .= F . ( x , g . x ) ;
R . ( h . m ) = F . x0 + h . ( m ) - ( h . m ) + ( h . ( m + 1 ) ) - ( h . m ) ;
cell ( G , XG -' 1 , ( t + 1 ) + ( t + 1 ) ) \ ( ( t + 1 ) + ( t + 1 ) ) meets ( UBD L~ f ) ;
IC Comput ( P2 , s2 , 2 ) = IC Comput ( P2 , s2 , 1 ) .= ( card I + 1 ) .= ( card I + 1 ) .= ( card I + 1 ) .= ( card I + 1 ) .= card I + 1 ;
sqrt ( ( - ( ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ) ^2 ) > 0 ;
consider x0 being element such that x0 in dom a and x0 in g " { k } and y = a . ( k + 1 ) and x0 = a . ( k + 1 ) and x0 = a . ( k + 1 ) ;
dom ( ( r1 (#) chi ( A , C ) ) | A ) = dom ( chi ( A , C ) ) /\ A .= dom ( ( r1 (#) chi ( A , C ) ) | A ) .= dom ( ( r1 (#) chi ( A , C ) ) | A ) .= dom ( ( r1 (#) chi ( A , C ) ) | A ) .= dom ( ( r1 (#) chi ( A , C ) ) | A ) .= dom ( r1 (#) ( A | A ) ) ;
di2 . [ y , z ] = ( ( - 1 ) * ( y - z ) ) * ( ( - 1 ) * ( y - z ) ) .= ( - 1 ) * ( y - z ) ;
attr for i being Nat holds C . i = A . i /\ B . i & C . i c= C . i /\ C . i ;
assume that x0 in dom f and f is continuous and for x st x in dom f holds ||. f /. x - f /. x0 .|| < r and f /. x0 = r * ( f /. x0 ) + r * ( f /. x0 ) ;
p in Cl A implies for K being Basis of p , Q being Subset of T st Q in K holds A meets Q & A meets Q & A meets Q & A meets Q
for x being Element of REAL n st x in Line ( x1 , x2 ) holds |. y1 - y2 .| <= |. y1 - y2 .| + |. y2 - y1 .|
func Sum <*> a -> Ordinal of a means : Def1 : a in it & for b being Ordinal st a in it holds it . b = b & for a being Ordinal of a st a in it holds it . a = b ;
[ a1 , a2 , a3 ] in ( the carrier of A ) \/ ( the carrier of B ) & [ a1 , a2 , a3 ] in the carrier of A ;
ex a , b being element st a in the carrier of S1 & b in the carrier of S2 & x = [ a , b ] & [ a , b ] in the InternalRel of S2 & [ b , a ] in the InternalRel of S2 ;
||. ( vseq . n ) - ( vseq . m ) .|| * ||. x - y .|| < ( e / ||. x - y .|| ) * ||. x - y .|| ;
then for Z being set st Z in { Y where Y is Element of I7 : F . Y c= Z & Z in x } holds z in Z ;
sup compactbelow [ s , t ] = [ sup ( [: X , compactbelow t :] ) , sup ( compactbelow s ) ] .= sup ( compactbelow s ) .= sup ( compactbelow t ) ;
consider i , j being Element of NAT such that i < j and [ y , f . j ] in Ii and [ f . i , f . j ] in Ii and [ f . i , f . j ] in Ii ;
for D being non empty set , p , q being FinSequence of D st p c= q holds ex p being FinSequence of D st p = q & p ^ p = q ^ p & p ^ q = q ^ p
consider e19 being Element of the affine of X such that c9 , a9 // a9 , e and a9 , c9 // a9 , e and a9 , b9 // c9 , e and a9 , c9 // a9 , e and a , b // c , d and a , c // c , d and a , b // c , d ;
set U2 = I \! \mathop { \vert E .| , E = { E } , F = { E } , G = { E } , C = { F } , D = { E } , E = { F } , N = { F } , N = { F } , N = { F } , C = { G } , N = { G } , C = { G } , D = { F } , E = { F } , C = { G } , D = { F } , E = { G } , C = { G } , E = { G } , E = { F } , C
|. q2 .| ^2 = ( ( q2 `1 ) / |. q2 .| ) ^2 + ( ( q2 `2 ) / |. q2 .| ) ^2 .= |. q2 .| ;
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 MSAlg U1 ) & Args ( o , MSAlg U1 ) = dom ( the charact of MSAlg U1 ) & Args ( o , MSAlg U1 ) = dom ( the charact of MSAlg U1 ) ;
dom ( h | X ) = dom h /\ X .= dom ( ( ||. h .|| ) | X ) .= dom ( ( ||. h .|| ) | X ) .= dom ( ( ||. h .|| ) | X ) .= dom ( ( ||. h .|| ) | X ) .= dom ( ||. h .|| ) /\ X .= dom ( ||. h .|| ) ;
for N1 , N2 being Element of G1 holds dom ( h . K1 ) = N & rng ( h . K1 ) = N1 & rng ( h . K1 ) = N2 & rng ( h . K1 ) = N1 & rng ( h . K1 ) c= N2
( mod ( u , m ) ) + mod ( v , m ) . i = ( mod ( u , m ) ) . i + ( mod ( v , m ) ) . i .= ( mod ( v , m ) ) . i + ( mod ( v , m ) ) . i ;
- ( q `1 / |. q .| - cn ) < - ( q `1 / |. q .| - cn ) or - ( q `1 / |. q .| - cn ) >= - ( q `1 / |. q .| - cn ) ;
attr r1 = f9 & r2 = f9 & ( for x st x in dom f holds f . x = ( f . x ) * ( f . x ) ) & ( for x st x in dom f holds f . x = ( f . x ) * ( f . x ) ) ;
vseq . m is bounded Function of X , the carrier of Y & x9 . m = ( vseq . m ) . x & x9 . m = ( vseq . m ) . x + ( vseq . m ) . y ;
attr a <> b & b <> c & angle ( a , b , c ) = PI & angle ( b , c , a ) = 0 & angle ( c , a , b ) = PI ;
consider i , j being Nat , r being Real such that p1 = [ i , r ] and p2 = [ j , r ] and r < s and s < p2 and p2 < r ;
|. p .| ^2 - ( 2 * |( p , q )| ) ^2 + |. q .| ^2 = |. p .| ^2 + |. q .| ^2 - ( 2 * |( p , q )| ) ^2 ;
consider p1 , q1 being Element of [: X , Y :] such that y = p1 ^ q1 and q1 ^ q1 = p1 ^ q1 and p1 ^ q1 = q1 ^ q1 and q1 ^ q2 = q1 ^ q2 and p1 ^ q1 = q1 ^ q2 ;
( of A , r1 , r2 , s1 , s2 , s1 , s2 , t2 , t1 , t2 , t2 , t1 , t2 , t2 , t1 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t2 , t2 , t2 , t2 ,
( ( LMP A ) `2 = lower_bound ( proj2 .: ( A /\ s1 ) ) & ( proj2 .: ( A /\ s1 ) ) is non empty or proj2 .: ( A /\ s1 ) is non empty ) & ( proj2 .: ( A /\ s1 ) is non empty ) & ( proj2 .: ( A /\ s1 ) ) is non empty ) ;
s , k1 |= H1 '&' H2 iff s |= ( H1 , k2 ) iff s |= ( H1 , k2 ) |= ( H1 , k1 ) & s |= ( H1 , k2 ) |= ( H1 , k2 ) ) ;
len ( s + 1 ) = card ( support b1 ) + 1 .= card ( support b2 ) + 1 .= card ( support b1 ) + 1 .= len ( b1 ) + 1 .= len ( b1 ) + 1 .= len ( b1 ) + 1 ;
consider z being Element of L1 such that z >= x and z >= y and for z being Element of L1 st z >= y & z >= y holds z >= x holds z >= y ;
LSeg ( UMP D , |[ ( W-bound D ) + ( E-bound D ) / 2 ) ) /\ D = { ( UMP D ) / 2 + ( ( E-bound D ) / 2 ) / 2 } .= { ( UMP D ) / 2 + ( ( UMP D ) / 2 ) / 2 } ;
lim ( ( f `| N ) / g `| N ) = lim ( ( f `| N ) / g `| N ) .= ( ( f `| N ) / g `| N ) / g `| N .= ( ( f `| N ) / g `| N ) / g `| N ;
P [ i , pr1 ( f ) . i , pr1 ( f ) . ( i + 1 ) ] & pr1 ( f ) . ( i + 1 ) = pr1 ( f ) . ( i + 1 ) ;
for r be Real st 0 < r ex m be Nat st for k be Nat st m <= k holds ||. ( seq . k ) - ( R /* seq ) . k .|| < r
for X being set , P being a_partition of X , x , a , b being set st x in a & a in P & x in P & b in P & x in P & a in P & b in P holds a = b
Z c= dom ( ( - 1 ) (#) f ) /\ ( dom ( ( - 1 ) (#) f ) \ ( ( - 1 ) (#) f ) " { 0 } ) ;
ex j being Nat st j in dom ( l ^ <* x *> ) & j < i & y = ( l ^ <* x *> ) . j & z = ( l ^ <* x *> ) . j & i = 1 + j & j = len ( l ^ <* x *> ) + 1 ;
for u , v being VECTOR of V for r being Real st 0 < r & u < 1 & r in N holds r * u + ( 1-r * v ) in N
A , Int Cl A , Cl Int Cl A , Cl Int Cl A , Cl Int Cl A , Cl Int Cl A , Cl Int Cl A , Cl Int Cl A ` \ A ` , Cl Int Cl A ` \ A ` , Cl Int Cl A ` \ A ` \ A ` , A ` \ A ` \ A ` \ A ` \ A ` \ A ` , A ` \ A ` \ A ` \ A ` , A ` \ A ` \ A ` \ A ` , A ` \ A ` \ A ` , A ` \ A ` \ A ` \ A ` \ A ` \ A ` \ A
- Sum <* v , u , w *> = - ( v + u + w ) .= - ( v + u ) + u .= - ( v + u ) + u .= - ( v + u ) ;
( Exec ( a := b , s ) ) . IC SCM R = ( Exec ( a := b , s ) ) . IC SCM .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s ;
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 support of J ) . x ;
for S1 , S2 being non empty reflexive RelStr , D being non empty Subset of S1 , D being non empty Subset of S2 , x being Element of S1 , y being Element of S2 holds x in D implies ( x "/\" y ) . ( x , y ) is directed
card X = 2 implies ex x , y st x in X & y in X & x <> y & x <> y or x = y & x = z or x = y & y = z or x = z or x = y & y = z or x = z or y = z or y = z or y = z & z = x ;
E-max L~ Cage ( C , n ) in rng ( Cage ( C , n ) \circlearrowleft W-min L~ Cage ( C , n ) ) & E-max L~ Cage ( C , n ) in rng Cage ( C , n ) ;
for T , T being tree , p , q being Element of dom T st p H & q = q holds ( T tree ( p , T ) ) . q = T . q & ( T tree ( p , T ) ) . q = T . q
[ i2 + 1 , j2 ] in Indices G & [ i2 + 1 , j2 ] in Indices G & f /. k = G * ( i2 + 1 , j2 ) & f /. k = G * ( i2 + 1 , j2 ) implies f /. k = G * ( i2 + 1 , j2 )
cluster that k divides ( k gcd n ) and k divides ( k gcd n ) and ( k divides ( k gcd n ) ) and ( k divides ( k gcd n ) and ( k divides ( k gcd n ) ) and ( k divides ( k gcd 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 & F " * F = the carrier of X2 ;
consider C being finite Subset of V such that C c= A and card C = n and the / ( n + 1 ) = 0. V and C = Lin ( B9 \/ C ) and C = Lin ( B9 \/ C ) ;
V is prime implies for X , Y being Element of [: the topology of T , the topology of T :] st X /\ Y c= V holds X c= V or Y c= V or X c= Y ;
set X = { F ( v1 ) where v1 is Element of B ( ) ) : P [ v1 ] & P [ v2 ] } , Y = { F ( v2 ) where v2 is Element of B ( ) : P [ v2 ] } ;
angle ( p1 , p3 , p4 ) = 0 .= angle ( p2 , p3 , p4 ) .= angle ( p2 , p3 , p4 ) .= angle ( p3 , p4 , p4 ) .= angle ( p2 , p3 , p4 ) ;
- sqrt ( - ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 ) = - ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 ) .= - ( q `1 / |. q .| - cn ) / ( 1 + cn ) ;
ex f being Function of I[01] , TOP-REAL 2 st f is continuous one-to-one & rng f = P & f . 0 = p1 & f . 1 = p2 & f . 1 = p2 & f . 0 = p3 & f . 1 = p4 ;
attr f is partial differentiable of REAL , u0 means : Def1 : for u holds SVF1 ( 2 , pdiff1 ( f , 1 ) , u0 ) . u = ( proj ( 2 , 3 ) ) . ( u + 1 ) + SVF1 ( 2 , 3 ) . ( u + 1 ) ;
ex r , s st x = |[ r , s ]| & G * ( len G , 1 ) `1 < r & r < G * ( len G , 1 ) `1 & G * ( len G , 1 ) `2 < s & s < G * ( 1 , 1 ) `2 & s < G * ( 1 , 1 ) `2 ;
assume that f is_sequence_on G and 1 <= t and t <= len G and G * ( t , width G ) `2 >= ( GoB f ) * ( t , width G ) `2 and G * ( t , width G ) `2 >= ( GoB f ) * ( t , width G ) `2 ;
attr i in dom G means : Def1 : r (#) ( f (#) reproj ( G , i ) ) = r (#) reproj ( G , i ) & r (#) reproj ( G , i ) = r (#) reproj ( G , i ) ;
consider c1 , c2 being bag of o1 + o2 such that ( decomp c ) /. k = <* c1 , c2 *> and c = <* c1 , c2 *> and c1 = <* c1 , c2 *> and c2 = <* c1 , c2 *> and c1 = <* c1 , c2 *> and c2 = <* c2 , c1 *> ;
u0 in { |[ r1 , s1 ]| : r1 < G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `2 < s1 & s1 < G * ( 1 , 1 ) `2 } ;
Cl ( X ^ Y ) = the carrier of X . k2 .= C4 . k2 .= C4 . k2 .= C4 . k2 .= C4 . k2 .= C4 . k2 .= C4 . k2 .= C4 . k2 .= C4 . k2 ;
attr M1 = len M2 means : Def1 : 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 & M1 = M2 ;
consider g2 be Real such that 0 < g2 and { y where y is Point of S : ||. y - x0 .|| < g2 & g2 < r } c= N2 & g2 in N2 ;
assume x < ( - b + sqrt ( integral ( a , b , c ) ) ) / 2 or x > - b - sqrt ( integral ( a , b , c ) ) / 2 ;
( G1 '&' G2 ) . i = ( <* 3 *> ^ H1 ) . i & ( H1 '&' H2 ) . i = ( <* 3 *> ^ H1 ) . i & ( H1 '&' H2 ) . i = ( <* 3 *> ^ H1 ) . i ;
for i , j st [ i , j ] in Indices M3 holds ( M3 + M1 ) * ( i , j ) < ( M3 + M2 ) * ( i , j ) + ( M3 + M2 ) * ( i , j )
for f being FinSequence of NAT , i being Element of NAT st i in dom f & i <> j holds i divides f /. i & i <> j implies i divides len f & j divides len f
assume F = { [ a , b ] where a , b is Element of X : for c being set st c in B\bf holds a c= c & a c= c } ;
b2 * q2 + ( b3 * q3 ) + - ( b3 * q2 ) + ( - ( a * q2 ) + ( a * q2 ) + ( a * q2 ) ) = 0. TOP-REAL n + ( a * q2 ) .= 0. TOP-REAL n + ( a * q2 ) .= 0. TOP-REAL n + ( a * q2 ) + ( a * q2 ) ;
Cl ( Cl F ) = { D where D is Subset of T : ex B being Subset of T st D = Cl B & ex C being Subset of T st C in F & C in F & A /\ C = Cl ( Cl ( Cl ( Cl F ) ) ) & C is open & A /\ C = Cl ( Cl ( Cl ( Cl ( Cl F ) ) ) ) ;
attr IT is summable means : Def1 : for s being Element of IT holds seq is summable & ( for n being Nat holds seq . n = ( seq . n ) + ( seq . n ) ) & ( for n holds seq . n = ( seq . n ) + ( seq . n ) ) implies seq is summable & lim seq = ( seq . n ) + ( seq . n )
dom ( ( cn -FanMorphN ) | D ) = ( the carrier of ( TOP-REAL 2 ) ) /\ D .= the carrier of ( ( TOP-REAL 2 ) | D ) .= D ;
[ X \to Z ] is full non empty full SubRelStr of ( [#] Z ) |^ the carrier of Z & [ X \to Y ] is full SubRelStr of ( [#] Z ) |^ the carrier of Y ;
G * ( 1 , j ) `2 = G * ( i , j ) `2 & G * ( 1 , j ) `2 <= G * ( 1 , j ) `2 & G * ( 1 , j ) `2 <= G * ( 1 , j + 1 ) `2 ;
synonym m1 c= m2 means : Def1 : for p , m2 being set st p in P & ( for p being set st p in P holds ( p in P implies p , ( p , m1 ) in P ) & ( for q being set st q in P holds q , m1 |= p ) holds p , m2 is_p\mathopen ( p , m2 ) , p , m1 ) ;
consider a being Element of B ( ) such that x = F ( a ) and a in { G ( b ) where b is Element of A ( ) : P [ b ] } and P [ a ] ;
synonym the multMagma of Let R -> reflexive non empty multMagma means : Let : for a being Element of the carrier of R , x being Element of the carrier of R holds x is Element of the carrier of R & x is ( the multF of R ) . a , ( the multF of R ) . b ;
continuous ( a , b , c ) + sequence ( c , d ) = b + the carrier of \HM { c , d } .= b + d .= the carrier of the carrier of thesis ;
cluster + _ -> $ Z for Element of INT , i1 , i2 , j1 , j2 being Element of INT holds ( i1 + i2 ) + ( i2 + j2 ) = ( i1 + i2 ) + ( i1 + j2 ) + ( i2 + j2 ) ;
( - s2 ) * p1 + ( s2 * p2 - ( s2 * p2 ) ) = ( ( - r2 ) * p1 + ( s2 * p2 ) ) * p2 + ( ( - r2 ) * p2 ) * p1 + ( ( - r2 ) * p1 ) * p2 ) ;
eval ( ( a | ( n , L ) ) *' p , x ) = eval ( a | ( n , L ) ) * eval ( p , x ) .= a * eval ( p , x ) .= a * eval ( p , x ) ;
assume that the TopStruct of S = the TopStruct of T and for D being non empty Subset of S holds D in V iff for V being open Subset of S st V in V holds V is open & for V being open Subset of S st V in V holds V is open & V is open & V is open & V is open & V is open & V is open & V is open & V is open & V is open ;
assume that 1 <= k and k <= len w + 1 and T-7 . ( ( len q11 ) - len ( ( q , w ) + 1 ) ) = ( T. ( len ( q , w ) - len ( ( q , w ) - len ( q , w ) ) + 1 ) ) . k ;
2 * a |^ ( n + 1 ) + ( 2 * b |^ ( n + 1 ) ) >= ( a |^ ( n + 1 ) + ( b |^ ( n + 1 ) ) + ( b |^ ( n + 1 ) ) + ( b |^ ( n + 1 ) ) ) + ( b |^ ( n + 1 ) ) ;
M , v2 |= All ( x. 3 , All ( x. 0 , All ( x. 4 , H ) ) ) & M , v2 |= All ( x. 0 , All ( x. 0 , H ) ) implies M , v2 |= All ( x. 0 , H )
assume that f is_differentiable_on l and for x0 st x0 in l holds 0 < f . x0 and for x0 st x0 in l holds f . x0 - f . x0 < 0 ;
for G1 being _Graph , W being Walk of G1 , e being set , G2 being Walk of G2 st e in W holds not e in W implies e is Walk of G2 & e is Vertex of G1 & e is Vertex of G2
not c9 is not empty iff ( ex y1 , y2 st y1 is not empty & not ( ex y st y is not empty & y is not empty & not ( y is not empty & not y is not empty ) & not ( y is not empty & not y is not empty ) & not y is not empty ) & not y is not empty ) ;
Indices GoB f = [: dom GoB f , Seg width GoB f :] & 1 <= i1 & i1 + 1 <= len GoB f & i1 + 1 <= len GoB f & 1 <= i2 & i2 + 1 <= width GoB f & f /. ( k + 1 ) = ( GoB f ) * ( i1 + 1 , j2 ) ;
for G1 , G2 , G2 , G3 being strict Subgroup of O st G1 is_stable & G2 is_stable & G1 is_stable & G2 is_stable holds G1 * G2 is stable Subgroup of G1 * the carrier of G2 * the carrier of G2 * the carrier of G2 * the carrier of G2 * the carrier of G2 = the carrier of G1 * the carrier of G2 * the carrier of G1
UsedIntLoc ( int ) = { intloc 0 , intloc 1 , intloc 2 , intloc 2 , intloc 3 , intloc 4 , intloc 0 , intloc 0 , intloc 0 , intloc 0 , intloc 0 , 1 , 1 , - 1 , 1 , - 1 , - 1 , 1 , - 1 , 1 , - 1 , 1 , - 1 , 1 , - 1 , 1 , - 1 , 1 , - 1 , 1 , - 1 , 1 , - 1 , - 1 , 1 , - 1 , - 1 , 1 , - 1 , 1 , - 1 , 1 , - 1 , 1 , - 1 , 1 , 1 , - 1 , 1 , 1 , - 1 , 1 , 1 , - 1 , 1 , - 1 , 1 , - 1
for f1 , f2 being FinSequence of F st f1 ^ f2 is p -element & Q [ p ^ f1 ] holds Q [ ( p ^ f2 ) ^ ( p ^ f1 ) ] & Q [ p ^ f2 ] & Q [ p ^ f1 ] implies Q [ p ^ f2 ]
( p `1 / sqrt ( 1 + ( p `2 / p `1 ) ^2 ) ) ^2 = ( q `1 / sqrt ( 1 + ( q `2 / q `1 ) ^2 ) ) ^2 .= ( q `1 / sqrt ( 1 + ( q `2 / q `1 ) ^2 ) ) ^2 ;
for x1 , x2 , x3 being Element of REAL n holds |( x1 - x2 , x1 - x3 )| = |( x1 , x1 - x3 )| + |( x2 , x1 - x3 )| + |( x1 , x2 - x3 )| + |( x2 , x1 - x3 )| + |( x2 , x1 - x3 )| + |( x1 , x2 - x3 )| , x1 - x3 )| + |( x2 , x1 - x3 )|
for x st x in dom ( ( - ( x | A ) ) | A ) holds ( ( - ( x | A ) ) ) | A . ( - x ) = - ( ( - ( x | A ) ) | A ) . x )
for T being non empty TopSpace , P being Subset-Family of T st P c= the topology of T for x being Point of T st x c= P holds ex B being Basis of T st B c= P & x in B & B c= P
( a 'or' b 'imp' c ) . x = 'not' ( ( a 'or' b ) . x ) 'or' c . x .= ( 'not' ( a . x ) 'or' c . x ) 'or' c . x .= TRUE 'or' ( 'not' ( a . x ) 'or' c . x ) .= TRUE ;
for e being set st e in A ex X1 being Subset of Y , Y1 being Subset of Y st e = [: X1 , Y1 :] & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open implies Y1 is open ;
for i be set st i in the carrier of S for f being Function of Sconsider S . i , S1 . i st f = H . i holds F . i = f | ( [: S , S . i :] ) . i & for f being Function of [: S , S . i :] , S . i holds f . i = f | ( [: S , S :] . i ) ;
for v , w st for y st x <> y holds w . y = v . y holds Valid ( VERUM ( Al , J ) , J ) . v = Valid ( VERUM ( Al , J ) , J ) . w
card D = card D1 + card D2 - 1 .= ( i + 1 ) - 1 .= ( i + 1 ) - 1 .= ( i + 1 ) - 1 .= ( i + 1 ) - 1 .= ( i + 1 ) - 1 .= ( i + 1 ) - 1 .= ( i + 1 ) - 1 .= ( i + 1 ) - 1 ;
IC Exec ( i , s ) = ( s +* ( 0 .--> succ ( 0 ) ) ) . 0 .= ( ( 0 .--> ( s . 0 ) ) +* ( 1 .--> ( s . 0 ) ) ) . 0 .= ( ( 0 .--> ( s . 0 ) ) +* ( 1 .--> ( s . 0 ) ) ) . 0 .= ( ( 0 .--> ( s . 0 ) ) +* ( 1 .--> ( s . 0 ) ) ) . 0 .= ( ( 0 .--> 1 .--> 1 ) . 0 ) . 0 ) .= ( 0 .--> 1 ) . 0 ) . 0 .= ( 0 .--> 1 ) . 0 .= ( 0 .--> 1 ) . 0 ) . 0 .= ( 0 .--> 1 ) . 0 .= ( 0 .--> 1 )
len f /. ( ( i1 -' 1 ) + 1 ) - 1 = len f - ( i1 -' 1 ) + 1 - 1 .= len f - ( i1 -' 1 ) + 1 .= len f - ( i1 -' 1 ) + 1 - 1 .= len f - 1 + 1 ;
for a , b , c being Element of NAT st 1 <= a & a <= b & b < c holds a + b < a + b-2 or a = b + b-2 or a = b + b-2 or a = b + b-2 or a = b + b-2 or a = b + b-2 or a = b + b-2 or a = b + b-2 ;
for f being FinSequence of TOP-REAL 2 , p being Point of TOP-REAL 2 , i being Nat st i in LSeg ( f , i ) holds Index ( p , f ) <= i & Index ( p , f ) <= len f + 1 & Index ( p , f ) + 1 <= len f + 1
lim ( curry ( P-19 , k + 1 ) # x ) = lim ( ( curry ' ( F-19 , k + 1 ) ) # x ) + lim ( ( curry ' ( F-19 , k + 1 ) ) # x ) ;
z2 = g /. ( ( i -' n2 + 1 ) + 1 ) .= g . ( i -' n2 + 1 ) .= g . ( i -' n2 + 1 ) .= g . ( i -' n2 + 1 ) .= g . ( i -' n2 + 1 ) .= ( g | ( i -' n2 + 1 ) ) . ( i -' n2 + 1 ) .= ( g | ( i -' n2 + 1 ) ) . ( i + n2 + 1 ) ;
[ f . 0 , f . 3 ] in id ( the carrier of G ) \/ ( the InternalRel of G ) or [ f . 0 , f . 3 ] in the InternalRel of C6 ;
for G being Subset-Family of B st G = { R [ X ] where R is Subset of A , R is Subset of A ( ) , Y is Subset of A ( ) st R = { R where R is Subset of A ( ) : R in F } holds R is open & Y is open } holds ( ( Intersect F ) " Y ) " Y = Intersect G
CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 ) ) = CurInstr ( P1 , Comput ( P1 , s2 , m1 + m2 ) ) .= CurInstr ( P1 , Comput ( P1 , s2 , m1 ) ) .= halt SCMPDS .= halt SCMPDS ;
assume that not a on M and b on N and c on N and not c on M and not b on N and not c on M and not b on N and not b on M and not b on N and not b on M and not b on N and not b on M and not b on N and not b on M and not b on N and not b on N and not b on M and not b on N and not b on N and not b on N and not b on N and not b on N and not b on N and not b on N and not b on N and not b on N and not b on N and not b on N and not b on N and not b on N and not b on M and not b on N and not b on M and not b on N and not b on N and not b on N and not b on N and not b on N
assume that T is \hbox { - } 4 and ex F being Subset-Family of T st F is closed & for n being Nat holds F is closed & for F being Subset-Family of T st F is closed & F is closed holds F is finite-ind & for n being Nat holds F . n is finite-ind & F . n is finite-ind & F . n is finite-ind ;
for g1 , g2 st g1 in ]. r - r , r .[ & g2 in ]. r - r , r + r .[ holds |. ( f . g1 ) - ( f . g2 ) .| <= ( g1 - ( f . g2 ) ) / ( r - ( f . g2 ) ) ^2 )
( - ( - ( - ( - ( - ( z ) ) ) ) ) ) * ( - ( - ( z ) ) ) = - ( - ( - ( z ) ) ) * ( - ( - ( z ) ) ) * ( - ( - ( z ) ) ) ) .= - ( - ( z ) ) * ( - ( z ) ) * ( - ( z ) ) ;
F . i = F /. i .= 0. R + r2 .= ( b |^ ( n + 1 ) ) * ( a |^ ( n + 1 ) ) .= ( b |^ ( n + 1 ) ) * ( a |^ ( n + 1 ) ) .= ( b |^ ( n + 1 ) ) |^ ( n + 1 ) .= ( b |^ ( n + 1 ) ) |^ ( n + 1 ) ;
ex y being set , f being Function st y = f . n & dom f = NAT & f . 0 = A ( ) & for n holds f . ( n + 1 ) = R ( n , f . n ) & for n holds f . ( n + 1 ) = R ( n , f . n ) ;
func f (#) F -> FinSequence of V means : Def1 : len it = len F & for i be Nat st i in dom 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 } = { x1 , x2 , x3 , x4 , x5 , x5 } \/ { x5 , x5 , x5 , x5 } \/ { x5 , x5 , x5 } .= { x1 , x2 , x3 , x4 , x5 , x5 } \/ { x5 , x5 , x5 } ;
for n being Nat for x being set st x = h . n holds h . ( n + 1 ) = o . ( x , n ) & o . ( n + 1 ) in InnerVertices S & o . ( n + 1 ) = InnerVertices S ( x , n ) & o . ( n + 1 ) = x . ( n + 1 ) & o . ( n + 1 ) = x . ( n + 1 ) ;
ex S1 being Element of CQC-WFF ( Al ( ) ) st SubP ( P , l ) = S1 & ( for k being Nat holds ( S . k = k ) & ( S . k = l ) & ( S . k = l ) & ( S . k = l implies S . k = k ) ) & ( S . k = l ) & ( S . k = l implies S . k = {} ) ;
consider P being FinSequence of GL2 such that p7 : P = product P and for i being Element of dom t st i in dom t ex k being Element of NAT st P . i = t . i & t . i = k & t . i = k & t . i = k & t . i = k & t . i = k ;
for T1 , T2 being strict non empty TopSpace , P being Basis of T1 , T2 being Basis of T2 st the topology of T1 = the topology of T2 & the topology of T1 = the topology of T2 & the topology of T1 = the topology of T2 holds T1 is Basis of T2 & T2 is the topology of T1 & T2 is the topology of T2
assume that f is_partial differentiable on u0 , 2 and r (#) pdiff1 ( f , 3 ) is_partial_differentiable_in u0 , 2 and partdiff ( r (#) pdiff1 ( f , 3 ) , u0 , 2 ) = r (#) pdiff1 ( f , 3 ) . 2 and partdiff ( r (#) pdiff1 ( f , 3 ) , u0 , 2 ) = r (#) pdiff1 ( f , 3 ) . 1 ;
defpred P [ Nat ] means for F , G being FinSequence of bool ( Seg $1 ) st len F = $1 & G = F * s holds for s being Permutation of Seg $1 st len s = $1 holds Sum ( F ) = ( F * s ) . ( len G ) & Sum ( G ) = ( F * s ) . ( len G ) ;
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 & s < ( GoB f ) * ( 1 , j + 1 ) `2 ;
defpred U [ set , set ] means ex F-23 being Subset-Family of T st $1 = F-23 & $2 is open & union F-23 is open & union FY. is open & union Fa9 is open & union Fa9 is open & union Fa9 is open & union Fa9 is open & union Fa9 is open & union Fa9 is open & union Fa9 is open & union Fa9 is open & union Fa9 is open & union Fa9 is open & union Fa9 is open & union Fa9 is open & union Fa9 is open & union Fa9 is open & union Fa9 is open & union Fa9 is open & union Fa9 is open & union Fa9 is open & union Fa9 is open & union Fa9 is open & union Fa9 is open & union Fa9 is open & union Fa9 is open & union Fa9 is open & union Fa9 is open & union Fa9 is open
for p4 being Point of TOP-REAL 2 st LE p4 , p4 , P & LE p4 , 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 , P & LE p , p , P & LE p , p , P & LE p , p , P & LE p , p , P & p , p1 , P & p , p1 , P & p , p1 , P & p , p1 , P & p , p1 , P implies p , p1 , P implies p1 , p1 , P or p1 , p1 , P or p1 , p1 , P or p1 , p1 , P or p1 , p1 , P & p1 , p2 , p2 , P & p1 , p2 , P & p1 , p2 , P & p1 , p2 , P or p1
f in for E st f in for g st g <> f . y holds x = g . y holds for x st x in E holds g . x = All ( x , H ) . ( x , y ) ) & for y holds f . y = All ( y , H ) . ( x , y )
ex 8 being Point of TOP-REAL 2 st x = 8 & ( ( - 1 ) * ( ( sn - sn ) / ( 1 + sn ) ) ) >= 0 & ( - ( sn - sn ) ) / ( 1 + sn ) <= 0 & ( - ( sn - sn ) ) / ( 1 + sn ) ) / ( 1 + sn ) <= 1 ;
assume for d7 being Element of NAT st d7 <= 8 holds s1 . ( d7 ) = s2 . ( ( d7 ) . ( d7 ) ) & s2 . ( ( d7 ) . ( d7 ) ) = s2 . ( ( d7 ) . ( len ( ( t . 7 ) ) + 1 ) ) ;
assume that s <> t and s is Point of Sphere ( x , r ) and s is Point of Sphere ( x , r ) and ex e being Point of E st e = Ball ( x , r ) /\ Sphere ( y , r ) and e <> s and Ball ( y , r ) = \mathop { \rm Ball } ( x , r ) /\ Sphere ( y , r ) ;
given r such that 0 < r and for s st 0 < s ex x1 , x2 being Point of CNS st x1 in dom f & x2 in dom f & ||. x1 - x2 .|| < s & f /. x1 - f /. x2 .|| < r & f /. x2 - f /. x2 < r ;
( p | x ) | ( ( x | x ) | ( x | x ) ) = ( ( ( x | x ) | ( x | x ) ) | ( x | x ) ) | ( x | x ) ;
assume that x in dom sec and x + h in dom sec and ( for x st x in dom sec holds ( ( tan (#) sec ) `| Z ) . x = ( 4 * sin ( x + h ) ) * sin ( x - h ) ) / ( sin ( x - h ) ) ^2 and cos ( x - h ) = ( 4 * sin ( x - h ) ) ^2 and cos ( x - h ) = ( 4 * cos ( x - h ) ) ^2 ;
assume that i in dom A and len A > 1 and len B > 1 and len A = len B and width A = len A and width A = width B and A * ( i , j ) = A * ( i , j ) and A * ( i , j ) = A * ( i , j ) and A * ( i , j ) = A * ( i , j ) ;
for i being non zero Element of NAT st i in Seg n holds i divides n or i = n or i = n or i = n or i = n or i = n & i <> n & j <> n & j <> n implies ( i divides n implies i = 1. ( F_Complex , n ) ) & ( i divides n implies i divides n implies i = 1. ( F_Complex , n ) )
( ( ( b1 'imp' b2 ) '&' ( c1 'imp' c2 ) ) '&' ( ( b1 'or' b2 ) '&' ( c1 '&' c2 ) ) ) '&' 'not' ( ( b1 'or' b2 ) '&' 'not' ( b1 'or' b2 ) ) '&' 'not' ( ( b1 'or' b2 ) '&' 'not' ( b1 'or' c2 ) ) '&' 'not' ( b1 'or' b2 ) ) '&' 'not' ( ( b1 'or' b2 ) '&' 'not' ( b1 'or' b2 ) '&' 'not' ( b1 'or' b2 ) ) ) ) ) ) Y Y Y Y Y Y Y Y Y Y Y Y Y ) '&' 'not' ( ( b1 'or' b2 ) '&' 'not' ( ( b1 'or' b2 ) '&' 'not' ( ( b1 'or' b2 ) '&' 'not' ( ( b1 'or' b2 ) '&' 'not' ( b1 'or' b2 ) '&' 'not' ( b2 ) '&' 'not' ( b2 ) ) '&' 'not' ( b2 'or' b2 ) '&' 'not' ( b2 ) '&' 'not' ( b2 'or' c2 ) '&' 'not' ( b2 'or' c2 ) '&' 'not' ( b2 ) '&' 'not' (
assume that for x holds f . x = ( ( - 1 ) (#) ( sin ) ) . x and for x st x in Z holds ( ( - 1 ) (#) ( sin ) ) . x = cos . ( x- x ) and for x st x in Z holds ( ( - 1 ) (#) ( sin ) ) . x = - cos . x / ( sin . x ) ^2 and f . x = - cos . x / ( sin . x ) ^2 and f . x = - cos . x / ( sin . x ) ^2 and f . x = - cos . x / ( sin . x ) ^2 and f . x = - cos . x / ( sin . x / ( sin . x ) ^2 and f . x = - cos . x / ( sin . x ) ^2 and f . x = - cos . x / ( sin . x ) ^2 and for x ) ^2 and f . x = - cos .
consider R8 , I8 be Real such that R8 = Integral ( M , Re ( F . n ) ) and I8 = Integral ( M , Im ( F . n ) ) and Integral ( M , Im ( F . n ) ) = Integral ( M , Im ( F . n ) ) and Integral ( M , Im ( F . n ) ) = Integral ( M , Im ( F . n ) ) ;
ex k being Element of NAT st ( for q being Element of product G st q in X & 0 < d holds ||. ( q-r , q ) - partdiff ( f , x , i ) .|| < r ) & ||. partdiff ( f , x , i ) - partdiff ( f , x , i ) .|| < r ;
x in { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 } or x in { x1 , x2 , x3 , x4 , x5 , x5 , x5 } \/ { x5 , x5 , x5 , x5 } ;
G * ( j , ii ) `2 = G * ( 1 , ii ) `2 .= G * ( 1 , ii ) `2 .= G * ( 1 , ii ) `2 .= G * ( 1 , ii ) `2 .= G * ( 1 , ii ) `2 .= G * ( 1 , ii ) `2 .= G * ( 1 , ii ) `2 .= G * ( 1 , jj ) `2 .= G * ( 1 , jj ) `2 .= G * ( 1 , jj ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2
f1 * p = p .= ( ( the Arity of S1 ) +* the Arity of S2 ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o ;
func tree ( T , P , T1 ) -> Tree means : Def1 : for q st q in P holds q in T or ex p st p in P & p = q & p in T1 & p = q or p = p or p = q or p = q or p = q or p = q or p = q or p = q or p = q or p = q or p = q ;
F /. ( k + 1 ) = F . ( k + 1 -' 1 ) .= Fcontradiction . ( p . ( k + 1 -' 1 ) , k + 1 -' 1 ) .= FF>= p . ( k + 1 -' 1 ) .= FF>= p . ( k + 1 -' 1 ) .= FF>= p . ( k + 1 -' 1 ) ;
for A , B , C being Matrix of K st len B = len C & len C = len A & len B = len C & len C = len A & len C > 0 & len A > 0 & len C > 0 & len A > 0 & len C > 0 & len A > 0 & len A > 0 & len A > 0 & A * C = C * C + A * 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 ) ;
assume that x in ( the carrier of CB ) ~ and y in ( the carrier of CB ) ~ and x in the carrier of CB and y in the carrier of CB and z = [ x , y ] ;
defpred P [ Element of NAT ] means for f st len f = $1 holds ( for k st k < $1 holds ( for k st k < $1 holds ( for k st k < $1 holds ( for k st k < $1 holds ( for k st k < $1 holds ( for k st k < $1 holds ( for k st k < $1 holds ( v . k ) . k ) . k ) implies ( v . k ) . k = ( v . k ) . k ) ;
assume that 1 <= k and k + 1 <= len f and f is_sequence_on G and [ i + 1 , j ] in Indices G 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 + 1 , j ) ;
assume that sn < 1 and q `1 > 0 and ( q `2 / |. q .| - sn ) / ( 1 + sn ) >= 0 and q `2 / |. q .| - sn ) / ( 1 + sn ) >= 0 and q `2 / |. q .| - sn and q `2 / |. q .| - sn ) / ( 1 + sn ) >= 0 ;
for M being non empty metric , x being Point of M , f being Point of M st x = x holds ex x being Point of M st for n being Element of NAT holds f . n = Ball ( x , ( 1 / n ) * ( n + 1 ) ) & f . x = Ball ( x , ( 1 / n ) * ( n + 1 ) )
defpred P [ Element of omega ] means f1 is_differentiable_on Z & f2 is_differentiable_on Z & for x st x in Z holds ( ( f1 - f2 ) `| Z ) . x = ( ( f1 - f2 ) `| Z ) . ( ( f1 - f2 ) `| Z ) . x ) / ( f1 - f2 ) . x * ( ( f1 - f2 ) `| Z ) . x = ( f1 - f2 ) `| Z ) . x / ( f1 - f2 ) . x ;
defpred P1 [ Nat , Point of CNS ] means ( for r being Real st r in Y & r < $1 holds ||. f /. ( $1 + 1 ) - f /. ( $1 + 1 ) .|| < r ) & ||. f /. ( $1 + 1 ) - f /. ( $1 + 1 ) .|| < r ;
( f ^ mid ( g , 2 , len g ) ) . i = ( mid ( g , 2 , len g ) ) . ( i - 1 ) .= g . ( i - 1 ) .= g . ( i - 1 ) .= g . ( i - 1 ) .= g . ( i - 1 ) .= ( g . i ) . ( i - 1 ) .= ( g . i ) . ( i - 1 ) ;
( 1 - 2 * n0 + 2 * ( n + 2 ) ) * ( 2 * ( n + 2 ) ) = ( ( 1 - 2 * ( n + 1 ) ) * ( n + 2 ) ) * ( n + 2 ) .= ( 1 - 2 * ( n + 1 ) ) * ( n + 2 ) .= ( 1 - 2 * ( n + 1 ) ) * ( n + 2 ) .= ( 1 - 2 * ( n + 1 ) ) * ( n + 2 ) ) * ( n + 2 ) ;
defpred P [ Nat ] means for G being non empty finite strict strict strict finite strict strict strict non empty RelStr st G is \mathop carrier of G & card the carrier of G = $1 holds the carrier of G = ( the carrier of G ) \/ ( the carrier of G ) & the carrier of G = ( the carrier of G ) \/ ( the carrier of G ) ;
assume that not f /. 1 in Ball ( u , r ) and 1 <= m and m <= len f and for i st 1 <= i & i <= len f holds LSeg ( f , i ) /\ Ball ( u , r ) <> {} and not 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 ) . $1 = ( Partial_Sums ( cos ) . ( $1 + 1 ) ) . ( x - x0 ) * ( cos . ( $1 + 1 ) ) ) . ( x - x0 ) * ( cos . ( $1 + 1 ) ) ;
for x being Element of product F holds x is FinSequence of G & dom x = I & for i being set st i in I holds x . i = ( the support of F ) . i & x . i = ( the support of F ) . i & x . i = ( the support of F ) . i & x . i = ( the support of F ) . i ;
( x " ) |^ ( n + 1 ) = ( x " ) * x .= ( x " ) * x .= ( x " ) * x .= ( x " ) * x .= ( x " ) * x .= x " * x .= x " * x .= x " * x .= x * x .= x * x .= x * x .= x * x ;
DataPart Comput ( P +* I , s +* I , ( LifeSpan ( P +* I , s ) ) + 3 ) = DataPart Comput ( P +* I , s +* I , ( LifeSpan ( P +* I , s ) ) + 3 ) .= DataPart Comput ( P +* I , s +* I , ( LifeSpan ( P +* I , s ) ) + 3 ) ;
given r such that 0 < r and ]. x0 - r , x0 .[ c= dom f1 /\ dom f2 and for g st g in ]. x0 , x0 + r .[ holds f1 . g <= f1 . g - f2 . x0 and f1 . g <= 0 and f2 . g <= 0 ;
assume that X c= dom f1 /\ dom f2 and f1 | X is continuous and ( for x st x in X /\ dom f2 holds f1 . x = f1 . x ) and ( f1 - f2 ) | X is continuous & ( f1 - f2 ) | X is continuous & f2 | X is continuous ) ;
for L being continuous complete LATTICE for l being Element of L ex X being Subset of L st l = sup X & for x being Element of L st x in X holds x is Element of L & for x being Element of L st x in X holds x = "\/" ( waybelow l , L ) holds x = "\/" ( { x } , L )
Support ( e *' p ) in { m *' p where m is Polynomial of n , L : ex p being Polynomial of n , L st p in Support ( m *' p ) & ex q being Polynomial of n , L st q = p & q = p . ( m + 1 ) & p . ( m + 1 ) = q . ( m + 1 ) & q . ( m + 1 ) = p . ( m + 1 ) ;
( f1 - f2 ) /* s1 = lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) ;
ex p1 being Element of [: Al ( ) ) , g being Function of [: Al ( ) , D ( ) :] , D ( ) st P [ p1 , g ] & P [ g , h . ( len p1 ) , g . ( len p1 ) ] ;
( mid ( f , i , len f -' 1 ) ) /. ( j + 1 ) = ( mid ( f , i , len f -' 1 ) ) /. ( j + 1 ) .= f /. ( j + 1 ) .= f /. ( j + 1 ) .= f /. ( j + 1 ) .= f /. ( j + 1 ) ;
( ( 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 ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p . ( len p + k ) . ( len q + k ) .= ( p . ( len p + k ) . ( len p + k ) . ( len p + k ) .= ( p . ( len p + k ) .= ( p
len mid ( D2 , indx ( D2 , D1 , j1 ) + 1 , indx ( D2 , D1 , j1 ) + 1 ) = indx ( D2 , D1 , j1 ) + 1 .= indx ( D2 , D1 , j1 ) + 1 + 1 .= indx ( D2 , D1 , j1 ) + 1 + 1 ;
x * y * z = Mz . ( ( y * z ) * ( z * y ) ) .= ( x * y ) * ( y * z ) .= ( x * y ) * ( y * z ) .= ( x * y ) * ( y * z ) .= ( x * y ) * ( z * z ) .= ( x * y ) * ( y * z ) ;
v . ( <* x , y *> ) - ( <* x0 *> ) . i = partdiff ( v , ( x - y ) ) * ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( x - x0 ) ) ) ) + ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1 , 1 ) ) . ( ( proj ( 1
i * i = <* 0 * ( 1 - 0 ) - ( 0 * ( 1 - 0 ) ) * ( 1 - 0 ) .= <* - 1 , 0 , 0 , 0 *> .= 0 ;
Sum ( L (#) F ) = Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( ( L (#) F1 ) ^ ( L (#) F2 ) ) .= Sum ( ( L (#) F1 ) ^ ( L (#) F2 ) ) .= Sum ( ( L (#) F1 ) ^ ( L (#) F2 ) ) .= Sum ( ( L (#) F1 ) ^ ( L (#) F2 ) ) .= Sum ( L (#) F1 ) + ( L (#) F2 ) ) .= Sum ( L (#) F1 ) ;
ex r be Real st for e be Real st 0 < e ex Y1 be finite Subset of X st Y1 is non empty & Y1 c= Y & for Y1 be Subset of X st Y1 is non empty & Y1 c= Y holds |. ( lower_bound Y1 ) - ( lower_bound Y2 ) .| < r & r < ( for Y1 be Subset of X st Y1 in Y holds Y1 - r ) < r ;
( GoB f ) * ( i , j + 1 ) = f /. ( k + 2 ) & ( GoB f ) * ( i , j + 1 ) = f /. ( k + 1 ) or ( GoB f ) * ( i + 1 , j + 1 ) = f /. ( k + 1 ) or ( GoB f ) * ( i + 1 , j + 1 ) = f /. ( k + 1 ) ;
( ( - 1 ) (#) ( cos ) ) . x = ( - 1 ) * ( cos . x ) .= ( - 1 ) * ( cos . x ) .= ( - 1 ) * ( cos . x ) .= ( - 1 ) * ( cos . x ) .= ( - 1 ) * ( cos . x ) .= ( - 1 ) * ( cos . x ) .= ( - 1 ) * ( cos . x ) .= ( - 1 ) * ( cos . x ) ;
( - b + sqrt ( delta ( a , b , c ) ) ) / 2 > 0 & ( - b - sqrt ( delta ( a , b , c ) ) ) / 2 > 0 or - b - sqrt ( delta ( a , b , c ) ) / 2 < 0 ;
assume that ex_inf_of uparrow "\/" ( X , L ) , L and ex_sup_of X , L and ex_sup_of X , L and "\/" ( X , L ) , L and "\/" ( X , L ) = "/\" ( uparrow "\/" ( X , L ) , L ) and "\/" ( X , L ) = "/\" ( ( uparrow "\/" ( X , L ) ) , L ) ;
( ( for j holds j in I ) implies ( j = i implies j = i ) & ( i = j implies j = i implies i = j ) & ( i = j implies j = i implies j = i ) & ( j = i implies j = i implies j = i ) ) & ( j = i implies j = i implies j = i ) )