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contradiction ;
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contradiction ;
thesis ;
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assume not thesis ;
assume not thesis ;
B in X ;
a <> c
T c= S
D c= B
c in X ;
b in X ;
X ;
b in D ;
x = e ;
let m ;
h is onto ;
N in K ;
let i ;
j = 1 ;
x = u ;
let n ;
let k ;
y in A ;
let x ;
let x ;
m c= y ;
F is one-to-one ;
let q ;
m = 1 ;
1 < k ;
G is finite ;
b in A ;
d divides a ;
i < n ;
s <= b ;
b in B ;
let r ;
B is one-to-one ;
R is total ;
x = 2 ;
d in D ;
let c ;
let c ;
b = Y ;
0 < k ;
let b ;
let n ;
r <= b ;
x in X ;
i >= 8 ;
let n ;
let n ;
y in f ;
let n ;
1 < j ;
a in L ;
C is boundary ;
a in A ;
1 < x ;
S is finite ;
u in I ;
z << z ;
x in V ;
r < t ;
let t ;
x c= y ;
a <= b ;
m in NAT ;
assume f is z -element ;
not x in Y ;
z = +infty ;
let k be Nat ;
K ` is being_line ;
assume n >= N ;
assume n >= N ;
assume X is let ;
assume x in I ;
q is ) by 0 ;
assume c in x ;
p > 0 ;
assume x in Z ;
assume x in Z ;
1 <= k} ;
assume m <= i ;
assume G is finite ;
assume a divides b ;
assume P is closed ;
b-a > 0 ;
assume q in A ;
W is not bounded ;
f is Assume f is Assume g is Assume f is one-to-one ;
assume A is boundary ;
g is special ;
assume i > j ;
assume t in X ;
assume n <= m ;
assume x in W ;
assume r in X ;
assume x in A ;
assume b is even ;
assume i in I ;
assume 1 <= k ;
X is non empty ;
assume x in X ;
assume n in M ;
assume b in X ;
assume x in A ;
assume T c= W ;
assume s is atomic ;
b `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 - -2 ;
let X be set ;
let X be set ;
let y be element ;
let x be element ;
P [ 0 ]
let E be set , A be Subset of E ;
let C be category ;
let x be element ;
let k be Nat ;
let x be element ;
let x be element ;
let e be element ;
let x be element ;
P [ 0 ]
let c be element ;
let y be element ;
let x be element ;
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 k1 & n2 is k1 ;
Q halts_on s ;
x in for \in holds x in for y st y in that y in that y in that x in that y in that y
M < m + 1 ;
T2 is open ;
z in b \mathclose a ;
R2 is well-ordering ;
1 <= k + 1 ;
i > n + 1 ;
q1 is one-to-one ;
let x be trivial set ;
PP is one-to-one ;
n <= n + 2 ;
1 <= k + 1 ;
1 <= k + 1 ;
let e be Real ;
i < i + 1 ;
p3 in P ;
p1 in K ;
y in C1 ;
k + 1 <= n ;
let a be Real , A be Subset of TOP-REAL n ;
X |- r => p ;
x in { A } ;
let n be Nat ;
let k be Nat ;
let k be Nat ;
let m be Nat ;
0 < 0 + k ;
f is_differentiable_in x ;
let x0 , r ;
let E be Ordinal ;
o on 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 , W be Subspace of V ;
not s in Y |^ 0 ;
rng f is_<=_than w
b "/\" e = b ;
m = m3 ;
t in h . D ;
P [ 0 ] ;
assume z = x * y ;
S . n is bounded ;
let V be RealUnitarySpace , A be Subset of V ;
P [ 1 ] ;
P [ {} ] ;
C1 is component ;
H = G . i ;
1 <= i `1 + 1 ;
F . m in A ;
f . o = o ;
P [ 0 ] ;
a\rbrace <= being Real ;
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 trivial 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 Arrows of F is one-to-one
sgn x = 1 ;
k in support a ;
1 in Seg 1 ;
rng f = X ;
len T in X ;
vbeing < n ;
Sy is bounded ;
assume p = p2 ;
len f = n ;
assume x in P1 ;
i in dom q ;
let UA , G , H ;
pp = c & pp = d ;
j in dom h ;
let k ;
f | Z is continuous ;
k in dom G ;
UBD C = B ;
1 <= len M ;
p in Ball ( x , r ) ;
1 <= j & j <= len f ;
set A = \mathclose { as set ;
card a [= c ;
e in rng f ;
cluster B ++ A -> empty ;
H is with_no for ;
assume n0 <= m ;
T is increasing ;
e2 <> e2 . e ;
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 in union M
assume not x in REAL ;
Im ( f , x ) is complete ;
x in Int y ;
dom F = M ;
a in On W ;
assume e in A ( ) ;
C c= C-26 ;
mm <> {} & m <> 0 ;
let x be Element of Y ;
let f be ) ] Chain Chain of P ;
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 in v `2 ;
- y in I ;
let A be non empty set , f be Function of A , REAL ;
Px0 = 1 ;
assume r in F . k ;
assume f is simple function in S ;
let A be [ \cdot A ] ;
rng f c= NAT * ;
assume P [ k ] ;
ff <> {} ;
o be Ordinal ;
assume x is sum of f ;
assume not v in { 1 } ;
let IX , I , J ;
assume that 1 <= j and j < l ;
v = - u ;
assume s . b > 0 ;
d1 in dom f ;
assume t . 1 in A ;
let Y be non empty TopSpace , A 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 <> 0 ;
M + N c= M + M ;
assume M is \mathclose { hh\overline ;
assume f is additive for bbrst } is closed ;
let x , y be element ;
let T be non empty TopSpace ;
b , a // b , c ;
k in dom Sum p ;
let v be Element of V ;
[ x , y ] in T ;
assume len p = 0 ;
assume C in rng f ;
k1 = k2 or k2 = k2 ;
m + 1 < n + 1 ;
s in S \/ { s } ;
n + i >= n + 1 ;
assume Re y = 0 ;
k1 <= j1 & j1 <= len f ;
f | A is non empty implies f | A is non empty
f . x - x <= 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 ;
godo in X ;
q2 c= C1 & q2 c= C2 ;
a2 < c2 & c2 < c2 implies c1 + c2 < c2
s2 is 0 -started & s2 is 0 -started ;
IC s = 0 & IC s = 0 ;
s4 = s4 & s4 = s4 ;
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 \ of L ;
y " <> 0 & y " <> 0 ;
y " <> 0 & y " <> 0 ;
0. V = u-w ;
y2 , y , y is_collinear ;
R8 in X ;
let a , b be Real , x be Point of TOP-REAL 2 ;
let a be Object of C ;
let x be Vertex of G ;
let o be object of C , a be Object of C ;
r '&' q = P \lbrack l .] ;
let i , j be Nat ;
let s be State of A , x be set ;
s4 . n = N ;
set y = x `1 , z = y `2 ;
mi in dom g & mi in dom g ;
l . 2 = y1 ;
|. g . y .| <= r ;
f . x in CX0 ;
V-19 is non empty ;
let x be Element of X ;
0 <> f . g2 ;
f2 /* q is convergent ;
f . i is_measurable_on E ;
assume \xi in N-22 ;
reconsider i = i as Ordinal ;
r * v = 0. X ;
rng f c= INT & rng g c= INT ;
G = 0 .--> goto 0 ;
let A be Subset of X ;
assume A0 is dense & A is open ;
|. f . x .| <= r ;
let x be Element of R ;
let b be Element of L ;
assume x in W-19 ;
P [ k , a ] ;
let X be Subset of L ;
let b be Object of B ;
let A , B be category ;
set X = Vars ( C ) ;
let o be OperSymbol of S ;
let R be connected non empty Poset ;
n + 1 = succ n ;
x-21 c= Z1 & xY c= Z1 ;
dom f = C1 & dom g = C2 ;
assume [ a , y ] in X ;
Re seq is convergent & Im seq is convergent ;
assume a1 = b1 & a2 = b2 ;
A = sInt ( A ) ;
a <= b or b <= a ;
n + 1 in dom f ;
let F be Instruction of S , i be Nat ;
assume 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 + 1 in dom g1 ;
k + 1 in dom f ;
the still of { s } is finite ;
assume that x1 <> x2 and y1 <> y2 ;
v3 in Vx0 & v2 in Vx0 ;
not [ b `1 , b ] in T ;
i-35 + 1 = i ;
T c= <> * ( T , X ) ;
l `1 = 0 & l `2 = 0 ;
let n be Nat ;
t `2 = r & t `2 = s ;
AA is_integrable_on M & F is integrable ;
set t = Top t ;
let A , B be real-membered set ;
k <= len G + 1 ;
C ( ) misses V ( ) ;
product ( s ) is non empty ;
e <= f or f <= e ;
cluster non empty normal for Ordinal ;
assume c2 = b2 & c2 = b1 ;
assume h in [. q , p .] ;
1 + 1 <= len C ;
not c in B . m1 ;
cluster R .: X -> empty ;
p . n = H . n ;
assume that vseq is convergent and vseq is convergent and lim vseq = 0 ;
IC s3 = 0 & IC s3 = 0 ;
k in N or k in K ;
F1 \/ F2 c= F ;
Int G1 <> {} & Int G2 <> {} ;
z `2 = 0 or z `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 is_>=_than X & b is_>=_than Y ;
assume <* a , c *> <> {} ;
F . c = g . c ;
G is one-to-one one-to-one ;
A \/ { a } \not c= B ;
0. V = 0. Y .= 0. V ;
let I be be be be be be the ) Instruction of S , k be Nat ;
f-24 . x = 1 ;
assume z \ x = 0. X ;
p4 = 2 to_power n ;
let B be SetSequence of Sigma ;
assume X1 = p .: D ;
n + l2 in NAT & n + l2 in NAT ;
f " P is compact & f " P is compact ;
assume that x1 in REAL and x2 in REAL ;
p1 = ( K + L ) . p2 ;
M . k = <*> REAL ;
phi . 0 in rng phi ;
OSMis closed ;
assume z0 <> 0. L & z0 <> 0. L ;
n < ( N . k ) ;
0 <= seq . 0 & seq . 0 <= seq . 0 ;
- q + p = v ;
{ v } is Subset of B ;
set g = f `| 1 ;
cR is stable Subset of R ;
set cR = Vertices R ;
px0 c= P3 & px0 c= P3 ;
x in [. 0 , 1 .[ ;
f . y in dom F ;
let T be Scott Scott TopAugmentation of S ;
ex_inf_of the carrier of S , S ;
\vert 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 - x .|| <= r ;
assume that Y c= field Q and Y <> {} ;
a ~ , b ~ are_isomorphic ;
assume a in A ( i ) ;
k in dom ( q | ( i + 1 ) ) ;
p is non empty \HM { finite } ;
i - 1 = i-1 - 1 ;
f | A is one-to-one ;
assume x in f .: X ( ) ;
i2 - i1 = 0 & i2 - i2 = 0 ;
j2 + 1 <= i2 & j2 + 1 <= len f ;
g " * a in N ;
K <> { [ {} , {} ] } ;
cluster strict for for for for for for } ;
|. q .| ^2 > 0 ;
|. p4 .| = |. p .| ;
s2 - s1 > 0 & s2 - s1 > 0 ;
assume x in { Gij } ;
W-min C in C & W-min C in C ;
assume x in { Gij } ;
assume i + 1 = len G ;
assume i + 1 = len G ;
dom I = Seg n & dom I = Seg n ;
assume that k in dom C and k <> i ;
1 + 1-1 <= i + j ;
dom S = dom F & dom F = dom G ;
let s be Element of NAT , n be Nat ;
let R be ManySortedSet of A ;
let n be Element of NAT ;
let S be non empty non void non void holds S is holds S is with_2 -\subseteq the carrier of S ;
let f be ManySortedSet of I ;
let z be Element of F_Complex , p be FinSequence of COMPLEX ;
u in { ag } ;
2 * n < 2 * n ;
let x , y be set ;
B-11 c= V-15 \/ { x } ;
assume I is_halting_on s , P & I is_halting_on s , P ;
UA = ( the carrier of ( TOP-REAL 2 ) ) ;
M /. 1 = z /. 1 ;
x11 = x22 & x22 = x22 ;
i + 1 < n + 1 + 1 ;
x in { {} , <* 0 *> } ;
( f | X ) . x <= ( f | X ) . x ;
let l be Element of L ;
x in dom ( F . 0 ) ;
let i be Element of NAT , k be Nat ;
seq1 is COMPLEX -valued & seq2 is COMPLEX -valued implies seq1 - seq2 is COMPLEX -valued
assume <* o2 , o *> <> {} ;
s . x |^ 0 = 1 ;
card K1 in M & card K1 in card M ;
assume that X in U and Y in U ;
let D be non empty set ;
set r = { q . ( k + 1 ) } ;
y = W . ( 2 * PI ) ;
assume dom g = cod f & cod g = cod f ;
let X , Y be non empty TopSpace , A be Subset of X ;
x ++ A is interval ;
|. <*> A .| . a = 0 ;
cluster strict for SubLattice of L ;
a1 in B . s1 & a2 in B . s2 ;
let V be finite < F , A be Subset of V ;
A * B on B , A ;
f-3 = NAT --> 0 .= fs1 ;
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 , T = X ;
z in dom ( A --> y ) ;
P [ y , h . y ] ;
{ x0 } c= dom f & { x0 } c= dom f ;
let B be non-empty ManySortedSet of I , A be ManySortedSet of I ;
PI / 2 < Arg z & Arg z < PI / 2 ;
reconsider z9 = 0 , z9 = 1 as Nat ;
LIN a , d , c & LIN a , d , c ;
[ y , x ] in II ;
Q * ( 3 , 3 ) = 0 & Q * ( 3 , 1 ) = 0 ;
set j = x0 gcd m , m = x0 gcd m ;
assume a in { x , y , c } ;
j2 - jj > 0 & j2 - jj > 0 ;
I \! \mathop { phi } = 1 ;
[ y , d ] in F-8 ;
let f be Function of X , Y ;
set A2 = ( B - C ) / 2 ;
s1 , s2 are_card ( X \ { x } ) ;
j1 -' 1 = 0 & j2 -' 1 = 1 ;
set m2 = 2 * n + j ;
reconsider t = t as bag of n ;
I2 . j = m . j ;
i |^ s , n are_relative_prime & i |^ s , n are_relative_prime ;
set g = f | D-21 ;
assume that X is lower bounded and 0 <= r ;
p1 `1 = 1 & p2 `2 = - 1 ;
a < p3 `1 & p3 `1 < b ;
L \ { m } c= UBD C ;
x in Ball ( x , 10 ) ;
not a in LSeg ( c , m ) ;
1 <= i1 -' 1 & i1 + 1 <= len G ;
1 <= i1 -' 1 & i1 + 1 <= len G ;
i + i2 <= len h & 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 & width <* A1 *> = 1 ;
set H = h . gg ;
card b * a = |. a .| ;
Shift ( w , 0 ) |= v ;
set h = h2 (*) h1 , h1 = h2 (*) h2 ;
assume x in ( X /\ 4 ) ;
||. h - c .|| < d1 & ||. h - c .|| < d ;
not x in the carrier of f . i ;
f . y = F ( y ) ;
for n holds X [ n ] ;
k - l = kbeing - k\leq ;
<* p , q *> /. 2 = q ;
let S be Subset of the carrier of Y ;
let P , Q be \vert s be + t ;
Q /\ M c= union ( F | M )
f = b * ( canFS S ) ;
let a , b be Element of G ;
f .: X is_<=_than f . sup X
let L be non empty transitive reflexive RelStr , X be Subset of L ;
S-20 is x -[ i , x ] ;
let r be non positive Real ;
M , v |= x \hbox \hbox { = } y ;
v + w = 0. ( Z ) ;
P [ len F ( ) ] ;
assume InsCode ( i ) = 8 & InsCode ( i ) = 8 ;
the zero of M = 0 & the carrier of M = { 0 } ;
cluster z * seq -> summable for Real_Sequence ;
let O be Subset of the carrier of C ;
||. f .|| | X is continuous ;
x2 = g . ( j + 1 ) ;
cluster -> [#] for Element of ( AllSymbolsOf S ) ;
reconsider l1 = l- 1 as Nat ;
v4 is Vertex of r2 & v4 is Vertex of G ;
T2 is SubSpace of T2 implies ( the carrier of T2 ) is SubSpace of T2
Q1 /\ Q19 <> {} & Q1 /\ Q19 <> {} ;
let k be Nat ;
q " is Element of X & q " is Element of X ;
F . t is set & F . t is set ;
assume that n <> 0 and n <> 1 ;
set en = EmptyBag n , en = EmptyBag n ;
let b be Element of Bags n ;
assume for i holds b . i is commutative ;
x is root of ( p `2 ) , ( p `2 ) ;
not r in ]. p , q .] ;
let R be FinSequence of REAL , x be Element of REAL ;
not ( S is not empty or not S is non empty ) ;
IC SCM R <> a & IC SCM R <> a ;
|. - |[ x , y ]| .| >= r ;
1 * seq = seq & seq is convergent & lim seq = 0 ;
let x be FinSequence of NAT , n be Nat ;
let f be Function of C , D , g be Function of C , D ;
for a holds 0. L + a = a
IC s = s . NAT .= IC s . NAT ;
H + G = F- ( G-G ) ;
CP1 . x = x2 & CP1 . x = y2 ;
f1 = f .= f2 .= f2 * f1 .= f2 ;
Sum <* p . 0 *> = p . 0 ;
assume v + W = v + u + W ;
{ a1 } = { a2 } ;
a1 , b1 _|_ b , a ;
d3 , o _|_ o , a3 implies a3 , a3 _|_ o
I is reflexive & I is transitive implies I is transitive
IL is antisymmetric implies C is antisymmetric & x in C
sup rng H1 = e & sup rng H2 = e ;
x = ( a * a9 ) * ( a * b9 ) ;
|. p1 .| ^2 >= 1 ^2 - 1 ^2 ;
assume j2 -' 1 < 1 & j2 + 1 < width G ;
rng s c= dom f1 /\ dom f2 & rng s c= dom f1 ;
assume that support a misses support b and support b misses support b ;
let L be associative commutative associative non empty doubleLoopStr , p be Polynomial of L ;
s " + 0 < n + 1 ;
p . c = ( f " ) . 1 ;
R . n <= R . ( n + 1 ) ;
Directed ( I1 , I2 ) = I1 +* I ;
set f = + ( x , y , r ) ;
cluster Ball ( x , r ) -> bounded ;
consider r being Real such that r in A ;
cluster non empty -> NAT -defined for NAT -defined Function ;
let X be non empty directed Subset of S ;
let S be non empty full SubRelStr of L ;
cluster <* [ ] , N . 1 *> -> complete for non trivial TopSpace ;
1 / ( a " ) = a " .= 1 ;
( q . {} ) `1 = o ;
( - i ) - 1 > 0 ;
assume 1 / 2 <= t `1 & t `2 <= 1 ;
card B = k + 1-1 ;
x in union rng ( f | ( len f ) ) ;
assume x in the carrier of R & y in the carrier of R ;
d in X ;
f . 1 = L . ( F . 1 ) ;
the vertices of G = { v } ;
let G be finite connected ww_Graph ;
e , v9 , v be set ;
c . ( i - 1 ) in rng c ;
f2 /* q is divergent_to-infty & f2 /* q is divergent_to-infty implies f1 /* q is divergent_to-infty
set z1 = - z2 , z2 = - z2 , z1 = - z2 , z2 = - z1 ;
assume w is_ll_of S , G ;
set f = p |-count ( t ) , g = 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 , x be Point of X ;
reconsider p = p as Element of NAT ;
let v , w be Point of X ;
let s be State of SCM+FSA , I be Program of SCM+FSA ;
p is FinSequence of the carrier of SCM+FSA & q is FinSequence of the carrier of SCM+FSA ;
stop I ( ) c= P-12 & stop I ( ) c= P-12 ;
set ci = fbeing /. i , fD = f3 ;
w ^ t ^ t ^ s ^ w ^ t ^ w ^ t ^ w ^ w ^ t ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^
W1 /\ W = W1 /\ W ` .= W ;
f . j is Element of J . j ;
let x , y be \rm \vert Subset of T2 ;
ex d st a , b // b , d ;
a <> 0 & b <> 0 & c <> 0
ord x = 1 & x is dom and y is dom and y is dom ;
set g2 = lim ( seq ^\ k ) , g1 = lim ( seq ^\ k ) ;
2 * x >= 2 * ( 1 / 2 ) ;
assume ( a 'or' c ) . z <> TRUE ;
f (*) g in Hom ( c , c ) ;
Hom ( c , c + d ) <> {} ;
assume 2 * Sum ( q | m ) > m ;
L1 . F-21 = 0 & L2 . F-21 = 1 ;
h \/ R1 = h \/ R1 & R1 = h \/ R2 ;
( ( sin - cos ) `| Z ) . x <> 0 ;
( ( #Z n ) * ( exp_R + f ) ) . x > 0 ;
o1 in X-5 /\ O2 & o2 in XO2 /\ O2 ;
e , v9 , v be set ;
r3 > ( 1 - 2 ) * 0 ;
x in P .: ( F -ideal of L ) ;
let J be closed non empty Subset of R ;
h . p1 = f2 . O & h . p2 = g2 . O ;
Index ( p , f ) + 1 <= j ;
len ( q | M ) = width M & width ( q | M ) = width M ;
the carrier of `1 c= A & the carrier of `1 c= A ;
dom f c= union rng ( F | ( n + 1 ) ) ;
k + 1 in ( support ( s ) ) ;
let X be ManySortedSet of the carrier of S ;
[ x `1 , y `2 ] in ( \HM { the carrier of R } ) ;
i = D1 or i = D2 or i = D1 ;
assume a mod n = b mod n & b mod n = b mod n ;
h . x2 = g . x1 & h . x2 = g . x2 ;
F c= 2 -tuples_on the carrier of X & F is one-to-one ;
reconsider w = |. s1 .| as Real_Sequence of X , REAL ;
1 / ( m * m + r ) < p ;
dom f = dom Icn & dom g = dom Icn ;
[#] P-17 = [#] ( K-2 ) .= [#] ( TOP-REAL 2 ) ;
cluster - x -> ExtReal equals x - x + x ;
then { db } c= A & A is closed ;
cluster TOP-REAL n -> finite-ind for Subset of TOP-REAL n ;
let w1 be Element of M , w2 be Element of S ;
let x be Element of dyadic ( n ) ;
u in W1 & v in W3 implies u in W1 & v in W2
reconsider y = y as Element of L2 ( ) ;
N is full SubRelStr of T |^ the carrier of S ;
sup { x , y } = c "\/" c ;
g . n = n / 1 .= n ;
h . J = EqClass ( u , J ) ;
let seq be summable sequence of X , n be Nat ;
dist ( x `1 , y ) < r / 2 ;
reconsider mm1 = m - 1 as Element of NAT ;
x- x0 < r1 - x0 & x0 < r2 - x0 ;
reconsider P ` = P ` as strict Subgroup of N ;
set g1 = p * idseq ( q `1 ) ;
let n , m , k be non zero Nat ;
assume that 0 < e and f | A is lower ;
D2 . ( I8 + 1 ) in { x } ;
cluster -> subcondensed for Subset of T ;
let P be compact non empty Subset of TOP-REAL 2 , p1 , p2 be Point of TOP-REAL 2 ;
Gik in LSeg ( cos , 1 ) /\ LSeg ( Gik , Gij ) ;
let n be Element of NAT , x be Element of X ;
reconsider S8 = S , S8 = T as Subset of T ;
dom ( i .--> X ` ) = { i } ;
let X be non-empty ManySortedSet of S ;
let X be non-empty ManySortedSet of S ;
op ( 1 ) c= { [ {} , {} ] } ;
reconsider m = mm - 1 as Element of NAT ;
reconsider d = x `2 as Element of C ( ) ;
let s be 0 -started State of SCMPDS , a be Int_position , k1 be Integer ;
let t be 0 -started State of SCMPDS , Q be ] means t = s ;
b , b , x , y , x , y be Element of P ;
assume that i = n \/ { n } and j = k \/ { k } ;
let f be PartFunc of X , Y ;
x0 >= ( sqrt c ) / ( sqrt 2 ) & ( sqrt c ) / ( 2 * n ) >= 0 ;
reconsider t9 = T" as TopSpace , T = T | A ;
set q = h * p ^ <* d *> ;
z2 in U . ( 4 + 1 ) /\ Q2 & z2 in Q ;
A |^ 0 = { <%> E } & A |^ 1 = A ;
len W2 = len W + 2 & len W = len W + 2 ;
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 & ( g1 + g2 ) . z in dom f ;
assume that p2 = E-max ( K ) and p3 = E-max ( K ) ;
len G + 1 <= i1 + 1 & i1 + 1 <= len G ;
f1 (#) f2 is convergent & lim ( f1 (#) f2 ) = lim ( f1 , x0 ) ;
cluster s-10 + sT -> summable for Real_Sequence ;
assume j in dom ( M1 /. i ) ;
let A , B , C be Subset of X ;
let x , y , z be Point of X , p be Point of TOP-REAL n ;
b ^2 - ( 4 * a * c ) >= 0 ;
<* x/y *> ^ <* y *> ^ <* y *> ^ <* x *> ^ <* y *> ;
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 L ;
k = 0 & n <> k or k > n ;
then X is discrete for A being Subset of X ;
for x st x in L holds x is FinSequence of REAL
||. f /. c .|| <= r1 & ||. f /. c .|| <= r2 ;
c in uparrow p & not c in { p } ;
reconsider V = V as Subset of the topology of TOP-REAL n ;
let N , M be being being being being being being being 0 Element of L ;
then z is_>=_than waybelow x & z is_>=_than compactbelow y ;
M \lbrack f , g .] = f & M \lbrack g , f .] = g ;
( ( ( >= 1 ) to_power 1 ) ) /. 1 = TRUE ;
dom g = dom f /\ X .= dom f /\ X ;
mode : il is \HM { of G , { v } } is \HM { w } ;
[ i , j ] in Indices M & [ i , j ] in Indices M ;
reconsider s = x " , t = y " as Element of H ;
let f be Element of ( the carrier of Subformulae p ) \ { p } ;
F1 . ( a1 , - a1 ) = G1 . ( a1 , - a1 ) ;
redefine func Sphere ( a , b , r ) -> compact Subset of TOP-REAL 2 ;
let a , b , c , d be Real , p be Point of TOP-REAL 2 ;
rng s c= dom ( 1 / ( n + 1 ) ) ;
curry ( F-19 . ( k , k ) ) is additive ;
set k2 = card ( dom B ) , k1 = card ( dom B ) ;
set G = DTConMSA ( X ) ;
reconsider a = [ x , s ] as Object of G ;
let a , b be Element of ML , M be Matrix of n , K ;
reconsider s1 = s , s2 = t as Element of S1 . s ;
rng p c= the carrier of L & rng q c= the carrier of L ;
let d be Subset of the Sorts of A ;
( x .|. x ) = 0 iff x = 0. W
I-21 in dom stop I & stop I in dom stop I ;
let g be continuous Function of X | B , Y ;
reconsider D = Y as Subset of ( TOP-REAL n ) | A ;
reconsider i0 = len p1 , i0 = len p2 as Integer ;
dom f = the carrier of S & dom g = the carrier of T ;
rng h c= union ( the carrier of J . i ) ;
cluster All ( x , H ) -> \cal thesis ;
d * N1 / 2 > N1 * 1 / 2 ;
]. a , b .] c= [. a , b .] ;
set g = f " | D1 , h = f " | D2 ;
dom ( p | mm1 ) = mm1 & dom ( p | mm2 ) = mm2 ;
3 + - 2 <= k + - 2 & k + - 2 <= k + - 2 ;
tan is_differentiable_in ( arccot . x ) & ( arccot ) . x > 0 ;
x in rng ( f /^ n ) implies x in rng ( f /^ n )
let f , g be FinSequence of D ;
cp in the carrier of S1 & cp in the carrier of S2 ;
rng f " = dom f & rng f = rng f ;
( the Target of G ) . e = v & ( the Target of G ) . e = v ;
width G - 1 < width G - 1 & width G - 1 < width G ;
assume v in rng ( S | E1 ) & w in rng ( S | E1 ) ;
assume x is root or x is root or x is root ;
assume 0 in rng ( g2 | A ) & 0 < r ;
let q be Point of TOP-REAL 2 , r be Real ;
let p be Point of TOP-REAL 2 , r be Real ;
dist ( O , u ) <= |. p2 .| + 1 ;
assume dist ( x , b ) < dist ( a , b ) ;
<* S *> is_the carrier of C-20 & <* S *> is_the carrier of C-20 ;
i <= len ( G * ( i -' 1 , j ) ) ;
let p be Point of TOP-REAL 2 , r be Real ;
x1 in the carrier of I[01] & x2 in the carrier of I[01] & x3 in the carrier of I[01] ;
set p1 = f /. i , p2 = f /. ( i + 1 ) ;
g in { g2 : r < g2 & g2 < r } ;
Q2 = Sthesis " { x } & Q = Sy " { y } ;
( ( 1 / 2 ) |^ ( n + 1 ) ) is summable ;
- p + I c= - p + A & - p + I c= - p + I ;
n < LifeSpan ( P1 , s1 ) + 1 & Comput ( P2 , s2 , n ) = Comput ( P2 , s2 , n ) ;
CurInstr ( p1 , s1 ) = i .= CurInstr ( p2 , s2 ) ;
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 :] ;
len ( the connectives of C ) = 4 & len ( the connectives of C ) = 3 ;
let C1 , C2 be subV of C , a be Object of C ;
reconsider V1 = V , V2 = V as Subset of X | B ;
attr p is valid means : Def8 : All ( x , p ) is valid ;
assume that X c= dom f and f .: X c= dom g and g .: X c= dom f ;
H |^ ( a " ) is Subgroup of H & H |^ ( a " ) is Subgroup of H ;
let A1 be p1 of O , E1 , A2 be Element of E ;
p2 , r3 , q3 is_collinear & q2 , r3 , q3 is_collinear implies p2 , r3 , r3 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 the carrier of I[01] ;
0 . 0 < M . E8 . n ;
op ( c , c ) |^ a = c ;
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . s2 ) & a2 in dom ( F . s2 ) ;
cluster -> Nat for Nat -\subseteq the carrier of L ;
set i1 = the Nat , i2 = the carrier of G ;
let s be 0 -started State of SCM+FSA , P be Subset of SCM+FSA ;
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 : Def8 : cos | X c= cos | Y ;
let y be upper Subset of Y , x be Element of X ;
cluster the carrier of x `1 -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> Nat for holds x in { x } ;
set S = <* Bags n , il *> , T = <* T *> ;
set T = [. 0 , 1 / 2 .] , G = [. 0 , 1 .] ;
1 in dom mid ( f , 1 , 1 ) ;
( 4 * PI ) / PI < ( 2 * PI ) / PI ;
x2 in dom f1 /\ dom f & x2 in dom f1 /\ dom f ;
O c= dom I & { {} } = { {} } ;
( the Target of G ) . x = v & ( the Target of G ) . x = v ;
{ HT ( f , T ) } c= Support f ;
reconsider h = R . k as Polynomial of n , L ;
ex b being Element of G st y = b * H ;
let x `1 , y , z `2 , t be Element of G opp ;
h19 . i = f . ( h . i ) ;
p `1 = p1 `1 & p `2 = p2 `2 or p `1 = p1 `2 ;
i + 1 <= len Cage ( C , n ) ;
len <* P *> @ = len P & width <* P @ *> = 1 ;
set N-26 = the non empty Subset of N , NN = the carrier of N ;
len g: g: + ( x + 1 ) - 1 <= x ;
a on B & b on B & not b on B implies not a 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 ^2 - n ;
set q2 = N-min L~ Cage ( C , n ) , q2 = q2 ;
set S =
MaxADSet ( b ) c= MaxADSet ( P /\ Q ) ;
Cl ( G . q1 ) c= F . r2 ;
f " D meets h " V & f " D meets h " V ;
reconsider D = E as non empty directed Subset of L1 ;
H = ( the_left_argument_of H ) '&' ( the_right_argument_of H ) ;
assume t is Element of ( the carrier of S ) * ;
rng f c= the carrier of S2 & rng g c= the carrier of S2 ;
consider y being Element of X such that x = { y } ;
f1 . ( a1 , b1 ) = b1 & f2 . ( a1 , b1 ) = b2 ;
the carrier' of G ` = E \/ { E } .= { E } ;
reconsider m = len ( thesis - k ) as Element of NAT ;
set S1 = LSeg ( n , UMP C ) , S2 = LSeg ( UMP C , UMP C ) ;
[ i , j ] in Indices M1 & [ i , j ] in Indices M1 ;
assume that P c= Seg m and M is \HM { i } is } is non empty Subset of TOP-REAL n ;
for k st m <= k holds z in K . k ;
consider a being set such that p in a and a in G ;
L1 . p = p * L /. ( 1 + 1 ) ;
p-7 . i = pp1 . i .= pp1 . i ;
let PA , PA be a_partition of Y , PA be Subset of Y ;
attr 0 < r & r < 1 means : Def8 : 1 < r & r < 1 ;
rng the \overline of \mathbin ( a , X ) = [#] 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 ( rng ( s ) ) ;
reconsider x2 = x1 , y2 = x2 as Element of L2 ;
Q in FinMeetCl ( the topology of X ) & the topology of X = { x } ;
dom fx0 c= dom ( ux0 ) & dom ( fx0 ) c= dom ( fx0 ) ;
pred n divides m & m divides n means : Def8 : n = m ;
reconsider x = x , y = y as Point of [: I[01] , I[01] :] ;
a in ) implies product the carrier of T2 = { x } \/ { y }
not y0 in the carrier of f & not y0 in the carrier of g implies f . y0 in ( the carrier of g ) ;
Hom ( ( a ~ ) , c ) <> {} ;
consider k1 such that p " < k1 and k1 < len f and f . k1 = f . k1 ;
consider c , d such that dom f = c \ d ;
[ x , y ] in [: dom g , dom k :] ;
set S1 =
3 = m2 & l1 = i2 & l2 = j2 implies ( 1 - 1 ) * ( l1 - 1 ) = l1
x0 in dom ( u01 /\ A01 ) & ( G . x0 ) in dom ( G . x0 ) ;
reconsider p = x , q = y as Point of ( TOP-REAL 2 ) | K1 ;
I[01] = R^1 | B01 & ( TOP-REAL 2 ) | B01 = ( TOP-REAL 2 ) | B01 ;
f . p4 <= f . p1 & f . p2 <= f . p3 ;
( F . x ) `1 <= ( x `1 ) `1 & ( F . x ) `2 <= ( x `2 ) `2 ;
x `2 = W7 & x `2 = W8 or x `2 = W7 & x `2 = W8 ;
for n being Element of NAT holds P [ n ] implies P [ n + 1 ]
let J , K be non empty Subset of I ;
assume 1 <= i & i <= len <* a " *> ;
0 |-> a = <*> ( the carrier of K ) & 0. K = 0. K ;
X . i in 2 -tuples_on ( A . i \ B . i ) ;
<* 0 *> in dom ( e --> [ 1 , 0 ] ) ;
then P [ a ] & P [ succ a ] implies P [ succ a ] ;
reconsider sT = sT , s/. i as /\ ( the carrier of D ) ;
( - i ) - 1 <= len ( thesis - j ) - 1 ;
[#] S c= [#] the TopStruct of T & the TopStruct of T = the TopStruct of T ;
for V being strict RealUnitarySpace for W being Subspace of V holds V in the carrier of W implies V is strict Subspace of 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 square Matrix of n1 , K ;
- a * - b * - b = a * b - b * a ;
for A being Subset of AS holds A // A implies A // ( A \/ B )
( for o2 being object of B st o2 in <^ o2 , o2 ^> holds <^ o2 , o1 ^> <> {} ) ;
then ||. x - y .|| = 0 & x = 0. X ;
let N1 , N2 be strict normal Subgroup of G , x be Element of G ;
j >= len ( upper_volume ( g , D1 ) | ( len D2 ) ) ;
b = Q . ( len Qc - 1 + 1 ) ;
f2 * f1 /* s is divergent_to-infty & f2 * f1 /* s is divergent_to-infty & f2 * f1 /* s is divergent_to-infty ;
reconsider h = f * g as Function of [: N2 , G :] , G ;
assume that a <> 0 and Let ( a , b , c ) >= 0 ;
[ t , t ] in the Relation of A & [ t , t ] in the carrier of A ;
( v |-- E ) | n is Element of ( T . n ) ;
{} = the carrier of ( L1 + L2 ) & the carrier of ( L1 + L2 ) = {} ;
Directed I is
Initialized p = Initialize ( p +* q ) , p +* I ;
reconsider N2 = N1 , N2 = N2 as strict net of R2 ;
reconsider Y = Y as Element of ( Ids L ) , the carrier of 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 & [ s , 0 ] in the carrier of S2 ;
mm in ( B '&' C ) '/\' D \ { {} } ;
n <= len ( P + ( len P + 1 ) ) & n <= len ( P + ( len P + 1 ) ) ;
x1 `1 = x2 & y1 `1 = y2 & y1 `2 = y2 & y2 = z2 ;
InputVertices S = { x1 , x2 } & InputVertices S = { x1 , x2 } ;
let x , y be Element of FTTT1 ( n ) ;
p = |[ p `1 , p `2 ]| & p = |[ p `1 , p `2 ]| ;
g * 1_ G = h " * g * h .= h " * ( g * h ) ;
let p , q be Element of is Element of PFuncs ( V , C ) ;
x0 in dom ( x1 | X ) & ( x1 | X ) . x0 = x1 . x0 ;
( R qua Function ) " " = R " & ( R " ) " = R " ;
n in Seg len ( f /^ ( i -' 1 ) ) ;
for s being Real st s in R holds s <= s2 implies ( s - s1 ) / 2 <= s2
rng s c= dom ( f2 * f1 ) /\ dom ( f2 * f1 ) ;
synonym ex X being Subset of such that X is finite for X is finite Subset of is finite ;
1. ( K , n ) * 1. ( K , n ) = 1. ( K , n ) ;
set S = Segm ( A , P1 , Q1 ) , T = Segm ( A , P1 , Q1 ) ;
ex w st e = ( w - f ) & w in F ;
( curry ' ( P+* ( i , k ) ) ) # x is convergent ;
cluster open -> open for Subset of [: T , T :] ;
len f1 = 1 .= len f3 .= len f2 .= len f3 + 1 .= len f2 ;
( i * p ) / p < ( 2 * p ) / p ;
let x , y be Element of OSSub ( U0 ) ;
b1 , c1 // b9 , c9 & b1 , c1 // b9 , c9 implies b1 = c1
consider p being element such that c1 . j = { p } ;
assume f " { 0 } = {} & f is total ;
assume IC Comput ( F , s , k ) = n & IC Comput ( F , s , k ) = n ;
Reloc ( J , card I ) does not destroy a implies J is not " ;
( Macro ( card I + 1 ) ) does not contradiction & ( I is not contradiction ) ;
set m3 = LifeSpan ( p3 , s3 ) , P4 = P +* I , P4 = P +* I ;
IC SCMPDS in dom Initialize ( p +* I ) & IC SCMPDS in dom Initialize ( p +* I ) ;
dom t = the carrier of SCM R & dom t = the carrier of SCM R ;
( ( N-min L~ f ) .. f ) .. f = 1 & ( ( N-min L~ f ) .. f ) .. f = 1 ;
let a , b be Element of thesis , C be Element of is Element of is Element of is Element of Fin ( V , C ) ;
Cl union Int ( union F ) c= Cl Int ( union F ) ;
the carrier of ( X1 union X2 ) misses ( ( A \/ B ) \/ ( A \/ C ) ) ;
assume not LIN a , f . a , g . a ;
consider i being Element of M such that i = d6 and i in X ;
then Y c= { x } or Y = {} or Y = { x } ;
M , v / ( y , ( y , x ) ) |= H ;
consider m being element such that m in Intersect ( FF . m ) and x = ( Intersect FF ) . m ;
reconsider A1 = support u1 , A2 = support u2 as Subset of X ;
card ( A \/ B ) = k-1 + ( 2 * 1 ) ;
assume that a1 <> a3 and a2 <> a4 and a3 <> a4 and a4 <> a5 ;
cluster s , U ) -> -> -> -> -> -> -> -> -> -> -> -> -> -> k1 for string of S ;
LL2 /. n2 = LL2 . n2 .= LL2 . n2 .= LL2 . n2 ;
let P be compact non empty Subset of TOP-REAL 2 , p1 , p2 be Point of TOP-REAL 2 ;
assume that r-7 in LSeg ( p1 , p2 ) and rg2 in LSeg ( p1 , p2 ) ;
let A be non empty compact Subset of TOP-REAL n , a be Real ;
assume [ k , m ] in Indices DD1 & [ k , m ] in Indices DD2 ;
0 <= ( 1 / 2 ) |^ ( p p ) & ( 1 / 2 ) |^ ( p ) <= 1 ;
( F . N ) | E8 . x = +infty ;
attr X c= Y means : Def8 : Z c= V \ Y ;
y `2 * z `2 <> 0. I & y `2 * z `2 <> 0. I ;
1 + card X-18 <= card u & card X-18 <= card X-18 + card X-18 ;
set g = z \circlearrowleft ( ( L~ z ) .. z ) , 2 = ( ( L~ z ) .. z ) .. z ;
then k = 1 & p . k = <* x , y *> . k ;
cluster total for Element of C -\mathop { X } , D ;
reconsider B = A , C = B as non empty Subset of TOP-REAL n ;
let a , b , c be Function of Y , BOOLEAN , p be Function of Y , BOOLEAN ;
L1 . i = ( i .--> g ) . i .= g . i ;
Plane ( x1 , x2 , x3 , x4 ) c= P & Plane ( x1 , x2 , x3 , x4 ) c= P ;
n <= indx ( D2 , D1 , j1 ) & indx ( D2 , D1 , j1 ) <= len D2 ;
( ( ( g2 ) . O ) `1 ) ^2 = - 1 & ( ( ( g2 ) . O ) `1 ) ^2 = 1 ;
j + p .. f - len f <= len f - len f + 1 - len f ;
set W = W-bound C , S = S-bound C , E = E-bound C , N = N-bound C , N = N-bound C ;
S1 . ( a `1 , e `2 ) = a + e `2 .= a `2 ;
1 in Seg width ( M * ( ColVec2Mx p ) ) ;
dom ( i (#) Im ( f , x ) ) = dom ( Im ( f , x ) ) ;
( ^2 ) . x = W . ( a , *' ( a , p ) ) ;
set Q = ( -> Element of -> Element of ( x , f , h ) ;
cluster MSsorted -> non-empty for ManySortedSet of U1 , ( the Sorts of U1 ) ;
attr F = { A } means : ex A st F = { A } ;
reconsider z9 = \hbox { y where y is Element of product G : y in X } as Subset of product G ;
rng f c= rng f1 \/ rng f2 & rng f1 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 ) implies E , j |= H
reconsider n1 = n , n2 = m as Morphism of o1 , o2 ;
assume that P is idempotent and R is idempotent and P ** R = R ** P ;
card ( B2 \/ { x } ) = k-1 + 1 ;
card ( ( x \ B1 ) /\ B1 ) = 0 & card ( ( x \ B1 ) /\ B2 ) = 1 ;
g + R in { s : g-r - s < s & s < g + r } ;
set q-19 = ( q , <* s *> ) -) , q-18 = ( q , <* s *> ) -) ;
for x being element st x in X holds x in rng f1 implies x in rng f2
h0 /. ( i + 1 ) = h0 . ( i + 1 ) ;
set mw = max ( B , ( the carrier of R ) --> NAT ) ;
t in Seg width ( I ^ ( n , n ) ) & t in Seg width ( I ^ ( n , n ) ) ;
reconsider X = dom ( f | C ) as Element of Fin NAT ;
IncAddr ( i , k ) = <% l . k , l . k %> ;
S-bound L~ f <= q `2 & q `2 <= ( q `2 ) / ( |. q .| ) ;
attr R is condensed means : |. Int R is condensed & Cl R is condensed & Cl R is condensed ;
attr 0 <= a & a <= 1 & b <= 1 implies a * b <= 1 ;
u in ( ( c /\ ( ( d /\ b ) /\ e ) ) /\ f ) /\ j ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) ) /\ f ) /\ j ;
len C + - 2 >= 9 + - 3 & len C + - 2 >= 0 ;
x , z , y is_collinear & x , z , x is_collinear implies x = y
a |^ ( n1 + 1 ) = a |^ n1 * a .= a |^ n1 * a ;
<* \underbrace ( 0 , \dots , 0 *> , x ) *> in Line ( x , a * x ) ;
set yy1 = <* y , c *> ;
FF2 /. 1 in rng Line ( D , 1 ) & FF2 /. 1 in rng Line ( D , 1 ) ;
p . m joins r /. m , r /. ( m + 1 ) ;
p `2 = ( f /. i1 ) `2 .= ( f /. i1 ) `2 .= ( f /. i1 ) `2 ;
W-bound ( X \/ Y ) = W-bound ( X \/ Y ) & W-bound ( X \/ Y ) = W-bound ( X \/ Y ) ;
0 + ( p `2 ) ^2 <= 2 * r + ( p `2 ) ^2 ;
x in dom g & not x in g " { 0 } ;
f1 /* ( seq ^\ k ) is divergent_to-infty & f2 /* ( seq ^\ k ) is divergent_to-infty ;
reconsider u2 = u , v2 = v as VECTOR of P`1 , REAL ;
p |-count ( Product Sgm ( X11 ) ) = 0 & ( p |-count ( X11 ) ) = 0 ;
len <* x *> < i + 1 & i + 1 <= len c + 1 ;
assume that I is non empty and { x } /\ { y } = { 0. I } ;
set ii2 = ( card I + 4 ) .--> goto 0 , ii2 = goto 0 , ii2 = goto 0 ;
x in { x , y } & h . x = {} ( TT ) ;
consider y being Element of F such that y in B and y <= x `2 ;
len S = len ( the charact of A0 ) & len ( the charact of A0 ) = len the charact of A0 ;
reconsider m = M , i = I , n = N as Element of X ;
A . ( j + 1 ) = B . ( j + 1 ) \/ A . j ;
set N8 = G \ V , G7 = G \ { v } ;
rng F c= the carrier of gr { a } & the carrier of gr { a } = the carrier of gr { a } ;
Comput ( P , K , n , r ) is in rng the P of [: P , { r } :] ;
f . k , f . ( thesis ) . ( mod n ) in rng f ;
h " P /\ [#] T1 = f " P /\ [#] T2 .= [#] T2 /\ [#] T2 .= [#] T2 ;
g in dom f2 \ f2 " { 0 } & f2 . ( len f2 ) in dom f2 ;
gthesis /\ X /\ dom f1 = g1 " X & gX /\ dom f2 = { g1 } ;
consider n being element such that n in NAT and Z = G . n ;
set d1 = thesis , d2 = dist ( x1 , y1 ) , d2 = dist ( x2 , y2 ) ;
b `1 + 1 / 2 < 1 / 2 + ( 1 / 2 ) ;
reconsider f1 = f as VECTOR of the carrier of the carrier of X , Y ;
attr i <> 0 means : Def8 : i / ( i + 1 ) mod ( i + 1 ) = 1 ;
j2 in Seg len ( g2 . i2 ) & ( g2 . i2 ) . j2 = ( g2 . i2 ) . j2 ;
dom ( i4 * i4 ) = dom i2 .= a .= ( the carrier of G ) ;
cluster sec | ]. PI / 2 , PI / 2 .[ -> one-to-one for Function of ]. PI / 2 , PI / 2 .[ ;
Ball ( u , e ) = Ball ( f . p , e ) ;
reconsider x1 = x0 , y1 = x1 as Function of S , IV , IV ;
reconsider R1 = x , R2 = y , R1 = z as Relation of L ;
consider a , b being Subset of A such that x = [ a , b ] ;
( <* 1 *> ^ p ) ^ <* n *> in RL ;
S1 +* S2 = S2 +* S1 +* S2 .= S1 +* S2 +* S2 .= S1 +* S2 +* S2 ;
( ( #Z n ) * ( cos + sin ) ) is_differentiable_on Z & ( ( #Z n ) * ( cos + sin ) ) `| Z = f ;
cluster -> [. 0 , 1 .] -valued for Function of C , REAL ;
set C7 = 1GateCircStr ( <* z , x *> , f3 ) , C8 = 1GateCircStr ( <* z , x *> , f2 ) ;
E\in . e2 -carrier of ( ( e . e2 ) -T ) & EE . e2 -carrier of ( ( e . e2 ) -T ) ;
( ( ( 1 / 2 ) (#) ( ( #Z 2 ) * ( ln ) ) ) `| Z ) = f ;
upper_bound A = ( PI * 3 / 2 ) * 2 & lower_bound A = 0 & upper_bound A = 0 ;
F . ( dom f , - F ) is_transformable_to F . ( cod f , - F ) ;
reconsider pNAT = q`2 , pNAT = p`2 as Point of TOP-REAL 2 ;
g . W in [#] Y0 & [#] Y0 c= [#] Y0 & g . W in [#] Y0 ;
let C be compact non vertical non horizontal Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
LSeg ( f ^ g , j ) = LSeg ( f , j ) ;
rng s c= dom f /\ ]. -infty , x0 .[ & f | ]. x0 , x0 + r .[ is convergent ;
assume x in { idseq 2 , Rev ( idseq 2 ) } ;
reconsider n2 = n , m2 = m , m2 = n , m2 = m as Element of NAT ;
for y being ExtReal st y in rng seq holds g <= y & y <= g
for k st P [ k ] holds P [ k + 1 ] ;
m = m1 + m2 .= m1 + m2 .= m1 + m2 .= m1 + m2 .= m1 + m2 ;
assume for n holds H1 . n = G . n -H . n ;
set B" = f .: the carrier of X1 , BX2 = f .: the carrier of X2 ;
ex d being Element of L st d in D & x << d ;
assume R -Seg ( a ) c= R -Seg ( b ) & R -Seg ( a ) c= R -Seg ( b ) ;
t in ]. r , s .] or t = r or t = s ;
z + v2 in W & x = u + ( z + v2 ) ;
x2 |-- y2 iff P [ x2 , y2 ] & P [ y2 , y2 ] or P [ y2 , y2 ] ;
pred x1 <> x2 means : 2 <> 0 & for x being Element of REAL st x in dom x1 holds |. x1 - x2 .| > 0 ;
assume that p2 - p1 , p3 - p1 , p2 - p1 is_collinear and p2 - p1 , p3 - p1 , p3 - p1 is_collinear ;
set q = ( R ^ f ) ^ <* 'not' A *> ;
let f be PartFunc of REAL-NS 1 , REAL-NS 1 , g be PartFunc of REAL 1 , REAL-NS 1 , REAL n ;
( n mod ( 2 * k ) ) + 1 = n mod k ;
dom ( T * ( $1 , n ) ) = dom ( n , dom ( n , T . ( n + 1 ) ) ) ;
consider x being element such that x in wc iff x in c & x in X ;
assume ( F * G ) . v . x3 = v . x4 ;
assume that the carrier of D1 c= the carrier of D2 and the carrier of D1 c= the carrier of D2 and the carrier of D2 c= the carrier 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 ) , r = q , s = q , w = q , e = q , w = s , e = s , w = q , e = s , w = q , w = s , e = s , w =
n1 - len f + 1 <= len ( - 1 ) + 1 - len f + 1 ;
rng c= rng ( q , O1 ) & rng ( q , O1 ) = { a , v , b } ;
set C-2 = ( ( Seg ( k ) ) `1 ) . ( k + 1 ) ;
Sum ( L (#) p ) = 0. R * Sum p .= 0. V * 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 ) & for n holds P [ n , $1 ] ;
set s3 = Comput ( P1 , s1 , k ) , P3 = P2 +* I ;
let l be variable of k , A , A-30 be Subset of A ;
reconsider UA = union G-24 , Gd = union Gd as Subset-Family of Td ;
consider r such that r > 0 and Ball ( p `1 , r ) c= Q ` ;
( h | ( n + 2 ) ) /. ( i + 1 ) = p29 ;
reconsider B = the carrier of X1 , C = the carrier of X2 as Subset of X ;
p$ 9 = <* - vs , 1 , 1 *> .= <* - vs , 1 *> ;
synonym f is real-valued for rng f c= NAT & rng f c= NAT & f is one-to-one & f is one-to-one ;
consider b being element such that b in dom F and a = F . b ;
x10 < card ( X0 ) + card ( Y0 ) & ( x in Y0 ) & ( x in Y0 ) implies x in X
attr X c= B1 means : Defooo) : for A st A in X holds _ X c= succ ( B1 , A ) ;
then w in Ball ( x , r ) & dist ( x , w ) <= r ;
angle ( x , y , z ) = angle ( x-y , 0 , PI ) ;
attr 1 <= len s means : Def: for s being Element of NAT holds ( s . 0 ) . 1 = s ;
f-47 c= f . ( k + ( n + 1 ) ) ;
the carrier of { 1_ G } = { 1_ G } & the carrier of { 1_ G } = { 1_ G } ;
pred p '&' q in \cdot ( p '&' q ) means : Def8 : q '&' p in \cdot ( p '&' q ) ;
- ( t `1 ) ^2 < ( t `1 ) ^2 & ( t `2 ) ^2 < ( t `2 ) ^2 ;
( 9 . 1 ) = ( 9 /. 1 ) .= ( 9 . 1 ) .= ( 9 . 1 ) ;
f .: ( the carrier of x ) = the carrier of x & f .: ( the carrier of x ) = the carrier of x ;
Indices OX = [: Seg n , Seg n :] & Indices OX = [: Seg n , Seg n :] ;
for n being Element of NAT holds G . n c= G . ( n + 1 ) ;
then V in M @ ex x being Element of M st V = { x } ;
ex f being Element of F-9 st f is H & f is <> 0. ( K , n ) & f is H ;
[ h . 0 , h . 3 ] in the InternalRel of G & [ h . 0 , h . 3 ] in the InternalRel of G ;
s +* Initialize ( ( intloc 0 ) .--> 1 ) = s3 +* Initialize ( ( intloc 0 ) .--> 1 ) ;
|[ w1 , v1 ]| - ( v1 - v2 ) <> 0. TOP-REAL 2 & |[ w1 , v1 - ( v2 - v1 ) ]| in W2 ;
reconsider t = t as Element of INT * , ( the carrier of X ) * ;
C \/ P c= [#] ( GX | ( [#] GX \ A ) ) & A c= [#] GX ;
f " V in ( the carrier of X ) /\ D & D = the carrier of ( the carrier of X ) /\ D ;
x in [#] ( ( the carrier of A ) /\ A ) implies x in the carrier of ( ( the carrier of F ) /\ A )
g . x <= h1 . x & h . x <= h1 . x & h1 . x <= h2 . x ;
InputVertices S = { xy , y , z } \/ { xy , z } .= { xy , y } ;
for n being Nat st P [ n ] holds P [ n + 1 ] ;
set R = Line ( M , i , a * Line ( M , i ) ) ;
assume that M1 is being_line and M2 is being_line and M3 is being_line and M3 is being_line and M3 is being_line and M2 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 ^ F2 ) .= len ( ( Len F1 ) ^ ( Len F2 ) ) ;
len ( ( the ` of n ) * ( i , j ) ) = n & len ( ( i , j ) * ( i , j ) ) = n ;
dom max ( - ( f + g ) , f + g ) = dom ( f + g ) ;
( the Sorts of seq ) . n = upper_bound Y1 & ( the Sorts of seq ) . n = upper_bound Y2 ;
dom ( p1 ^ p2 ) = dom f12 & dom ( p1 ^ p2 ) = dom f12 ;
M . [ 1 / y , y ] = 1 / C * v1 .= y * v1 .= y * v1 ;
assume that W is not trivial and W .vertices() c= the carrier' of G2 and W is non empty and not v in W ;
godo /. i1 = G1 * ( i1 , i2 ) & card L~ godo = 1 & card L~ godo = 1 ;
C8 |- 'not' Ex ( x , p ) 'or' p . ( x , y ) ;
for b st b in rng g holds lower_bound rng f\rbrace <= b & upper_bound rng f\rbrace <= upper_bound rng f\rbrace
- ( ( q1 `1 / |. q1 .| - cn ) / ( 1 + cn ) ) = 1 ;
( LSeg ( c , m ) \/ ml ) \/ LSeg ( l , k ) c= R ;
consider p being element such that p in Ball ( x , r ) and p in L~ f and x = f . p ;
Indices ( X @ ) = [: Seg n , Seg 1 :] & Indices ( X @ ) = [: Seg n , Seg 1 :] ;
cluster s => ( q => p ) => ( q => ( s => p ) ) -> valid ;
Im ( ( Partial_Sums F ) . m , ( Partial_Sums F ) . n ) is_measurable_on E ;
cluster f . ( x1 , x2 ) -> Element of D * * \ { x1 } ;
consider g being Function such that g = F . t and Q [ t , g ] ;
p in LSeg ( N-min Z , p2 ) /\ LSeg ( o , \mathop { \rm S \hbox { - } bound } ( Z ) ) ;
set R8 = R |^ 1 \/ ]. b , +infty .[ ;
IncAddr ( I , k ) = SubFrom ( da , db ) .= SubFrom ( da , db ) .= IncAddr ( da , db ) ;
seq . m <= ( the Sorts of seq ) . k & ( the Sorts of seq ) . m <= ( the Sorts of seq ) . 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 , U1 = U1 \/ U2 as non empty Subset of U0 ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) /\ f ) /\ j /\ m ;
consider y being element such that y in Y and P [ y , lower_bound B ] ;
consider A being finite stable set of R such that card A = ( the carrier of R ) \ A ;
p2 in rng ( f |-- p1 ) \ rng <* p1 *> & p2 in rng <* p1 *> ;
len s1 - 1 > 1-1 & len s2 - 1 > 1-1 & len s2 - 1 > 0 ;
( ( N-min P ) `2 ) ^2 = ( ( N-min P ) `2 ) ^2 & ( ( N-min P ) `2 ) ^2 = ( ( N-min P ) `2 ) ^2 ;
Ball ( e , r ) c= LeftComp Cage ( C , k + 1 ) \/ LeftComp Cage ( C , k + 1 ) ;
f . a1 ` ` = f . a1 ` .= ( f | ( a1 ` ) ) . a1 ;
( seq ^\ k ) . n in ]. -infty , x0 + r .[ & ( seq ^\ k ) . n in dom f ;
gg . s0 = g . s0 | G . s0 .= g . s0 ;
the InternalRel of S is symmetric & the carrier of S is non empty & the carrier of T is non empty ;
deffunc F ( Ordinal , Ordinal ) = phi . ( $1 + 1 ) & phi . ( $1 + 1 ) = phi . ( $1 + 1 ) ;
F . s1 . a1 = F . s2 . a1 .= F . s2 . a1 .= s . a1 ;
x `2 = A . ( o . a ) .= Den ( o , A . a ) ;
Cl ( f " P1 ) c= f " ( Cl P1 ) & Cl ( f " P1 ) c= Cl ( f " ( Cl P1 ) ) ;
FinMeetCl ( ( the topology of S ) | ( the topology of T ) ) c= the topology of T ;
synonym o is \bf for o <> *' o & o <> * & o <> * & o <> * ;
assume that X = Y + = Y + 1 and card X <> card Y and card Y <> card X and card X <> card Y ;
the { the carrier of s <= 1 + ( the carrier of s ) & the carrier of s = { the carrier of s , the carrier of s } ;
LIN a , a1 , d or b , c // b1 , c1 or a , c // a1 , c1 ;
e / 2 . 1 = 0 & e / 2 . 2 = 1 & e / 2 . 3 = 0 ;
Ex in SS1 & not Ex in { Nx } implies Ex in { Nx }
set J = ( l , u ) If , K = I ;
set A1 = .| ( ( a , { c } , { d } ) , A2 = { d } , A1 = { c } ;
set vs = [ <* c , d *> , '&' ] , f3 = [ <* d , c *> , '&' ] , f4 = [ <* c , d *> , '&' ] ;
x * z `1 * x " in x * ( z * N ) * x " ;
for x being element st x in dom f holds f . x = ( g . x ) * ( g . x )
Int cell ( f , 1 , G ) c= RightComp f \/ RightComp f \/ L~ f \/ L~ f \/ L~ f ;
( for n holds n in dom ( W-min C ) implies ( W-min C ) /. n = E-max C ) implies ( W-min C ) /. n = W-min C
set f-17 = f @ "/\" ( g @ ) ;
attr S1 is convergent means : Def8 : S2 is convergent & lim ( S1 - S2 ) = 0 & ( lim S1 ) - ( lim S2 ) = 0 ;
f . ( 0 + 1 ) = ( 0 qua Ordinal ) + a .= a ;
cluster -> -> x9 -' for reflexive transitive non empty reflexive transitive RelStr , the carrier of G -symmetric reflexive non empty ;
consider d being element such that R reduces b , d and R reduces c , d and R reduces d , b ;
not b in dom Start-At ( ( card I + 2 ) , SCMPDS ) & not b in dom Start-At ( ( card I + 2 ) , SCMPDS ) ;
( z + a ) + x = z + ( a + y ) .= z + a + y ;
len ( l | [. a , b .] ) = len l & len ( l | [. a , b .] ) = len l ;
( t4 ^ {} ) is ( {} \/ rng t4 ) -valued FinSequence of D ;
t = <* F . t *> ^ ( C . p ^ q ) .= <* F . t *> ^ q ;
set p-2 = W-min L~ Cage ( C , n ) , p`2 = W-min L~ Cage ( C , n ) , pi = W-min L~ Cage ( C , n ) , pi = W-min L~ Cage ( C , n ) , pi = W-min L~ Cage ( C , n ) , pi = W-min L~ Cage ( C , n )
( k - ( i + 1 ) ) = ( k - ( i + 1 ) ) - ( i + ( i + 1 ) ) ;
consider u being Element of L such that u = u ` and u in D ` ;
len ( ( width E ) |-> a ) = width E & width ( ( width E ) |-> a ) = width E ;
( F . x ) . ( ( G * the_arity_of o ) . x ) in dom ( G * the_arity_of o ) ;
set cH2 = the carrier of H2 , cH1 = the carrier of H1 , cH2 = the carrier of H2 ;
set cH1 = the carrier of H1 , cH2 = the carrier of H2 ;
( Comput ( P , s , 6 ) ) . intpos m = s . intpos m .= Comput ( P , s , 6 ) . intpos m ;
IC Comput ( Q2 , t , k ) = ( l + 1 ) + 1 .= ( card I + 1 ) ;
dom ( ( ( - 1 / 2 ) (#) ( sin * sin ) ) `| REAL ) = REAL & dom ( ( - 1 / 2 ) (#) ( sin * sin ) ) = REAL ;
cluster <* l *> ^ phi -> ( 1 + 1 ) , ( D , D ) ) Sum phi -> string of S ;
set b5 = [ <* that ( ap , { 1 } ) , {} , {} ] , [ <* 1 , 2 *> , {} ] ] ;
Line ( Segm ( M @ , P , Q ) , x ) = L * Sgm Q .= Line ( M , i ) ;
n in dom ( ( the Sorts of A ) * ( the_arity_of o ) ) & ( the Sorts of A ) . n in 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 .|| <= r ;
set x3 = t2 . DataLoc ( ( s . SBP ) , 2 ) , x2 = s . SBP , x3 = s . SBP , x4 = s . SBP , x4 = s . SBP , x4 = s . SBP , x4 = s . SBP , 7 = s . SBP , 8 = s . SBP , 8 = s . SBP
set p-3 = stop I ( ) , ps2 = 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 }
let A , B , C , D , E , F , J , M , N , M , N , N , M , N , F , M , N , N , M , N , N , M ;
|. p2 .| ^2 - ( p2 `2 ) ^2 - ( p2 `1 ) ^2 + ( p2 `2 ) ^2 >= 0 ;
l - 1 + 1 = n-1 * ( l + 1 ) + ( mm - 1 ) ;
x = v + ( a * w1 + ( b * w2 ) ) + ( c * w2 ) ;
the TopStruct of L = , the TopStruct of L , the topology of T = the topology of L , the topology of T ;
consider y being element such that y in dom H1 and x = H1 . y and y in A ;
f9 \ { n } = \mathop { Free All ( v1 , H ) } & f . n = Free All ( v1 , H ) ;
for Y being Subset of X st Y is summable & Y is summable holds Y is * summable
2 * n in { N : 2 * Sum ( p | N ) = N & N > 0 } ;
for s being FinSequence holds len ( the { of \rm \rm <* s *> } ) = len s & len ( the { s } ) = len s
for x st x in Z holds exp_R * f is_differentiable_in x & ( exp_R * f ) . x > 0
rng ( h2 * f2 ) c= the carrier of ( ( TOP-REAL 2 ) | ( the carrier of TOP-REAL 2 ) ) | K1 ;
j + - len f <= len f + ( len - len f ) - len f + ( len f - len f ) ;
reconsider R1 = R * I , R2 = R * I as PartFunc of REAL , REAL-NS n , REAL-NS n ;
C8 . x = s1 . x0 .= C8 . x .= C8 . x .= C8 . x .= ( C * ( x , x0 ) ) . x ;
power ( F_Complex ) . ( z , n ) = 1 .= x |^ n .= x |^ n .= x |^ n ;
t at ( C , s ) = f . ( the connectives of S ) . t .= ( the connectives of S ) . t ;
support ( f + g ) c= support f \/ ( C /\ support g ) & support ( f + g ) c= support f \/ support g ;
ex N st N = j1 & 2 * Sum ( seq1 | N ) > N & N < 2 * Sum ( seq2 | N ) ;
for y , p st P [ p ] holds P [ All ( y , p ) ]
{ [ x1 , x2 ] where x1 is Point of [: X1 , X2 :] : x1 in X } is Subset of [: X1 , X2 :]
h = ( i , j ) |-- h .= ( i , j ) --> id B .= H . i ;
ex x1 being Element of G st x1 = x & x1 * N c= A & N c= A & x1 in N ;
set X = ( ( rng ( q , O1 ) ) `1 ) , Y = ( { ( d , O1 ) `1 , 4 } ) ;
b . n in { g1 : x0 < g1 & g1 < a1 . n } ;
f /* s1 is convergent & f /. x0 = lim ( f /* s1 ) & f /. x0 = lim ( f /* s1 ) ;
the carrier of the lattice of Y = the carrier of the open Subset of Y & the carrier of Y = the topology of Y & the topology of Y = the topology of Y ;
'not' ( a . x ) '&' b . x 'or' a . x '&' 'not' ( b . x ) '&' 'not' ( b . x ) = FALSE ;
( ( len q0 ) + len r1 ) = len ( q0 ^ r1 ) + len r1 .= len ( q ^ r1 ) + len r1 .= len q + 1 ;
( 1 / a ) (#) ( sec * f1 ) - id Z is_differentiable_on Z & ( ( 1 / a ) (#) ( sec * f1 ) ) is_differentiable_on Z ;
set K1 = upper_volume ( ( lim ( H , H ) || A ) , ( lim ( H , H ) || A ) ) ;
assume e in { ( w1 - w2 ) / ( w1 - w2 ) : w1 in F & w2 in G } ;
reconsider d7 = dom a `1 , d6 = dom F `1 , d6 = dom F `1 , d6 = dom G `1 as finite set ;
LSeg ( f /^ j , j ) = LSeg ( f , j ) /\ q .= LSeg ( f , j + q .. f ) ;
assume X in { T . ( N2 , K1 ) : h . ( N2 , K1 ) = N2 } ;
assume that Hom ( d , c ) <> {} and <* f , g *> * f1 = <* f , g *> * f2 ;
dom SM = dom S /\ Seg n .= dom LM .= Seg n /\ Seg n .= Seg n /\ Seg n .= Seg n ;
x in H |^ a implies ex g st x = g |^ a & g in H & g in H
* ( a , 1 ) = a `2 - ( 0 * n ) .= a `2 - ( 0 * n ) .= a `2 ;
D2 . j in { r : lower_bound A <= r & r <= D1 . i } ;
ex p being Point of TOP-REAL 2 st p = x & P [ p ] & p `2 <= 0 & p <> 0. TOP-REAL 2 ;
for c holds f . c <= g . c implies f @ @ g ^ @ g ;
dom ( f1 (#) f2 ) /\ X c= dom ( f1 (#) f2 ) /\ X & dom ( f1 (#) f2 ) /\ X c= dom ( f1 (#) f2 ) ;
1 = ( p * p ) / p .= p * ( p / p ) .= p * 1 .= p ;
len g = len f + len <* x + y *> .= len f + 1 .= len f + 1 .= len f + 1 ;
dom ( F-11 | [: N1 , S-23 :] ) = [: dom ( F | [: N1 , S-23 :] ) , dom ( F | [: N1 , S-23 :] ) ;
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 for x being Element of X holds g . x = f . x ;
( ( x \ y ) \ z ) \ ( ( x \ z ) \ ( y \ z ) ) = 0. X ;
consider f such that f * f `1 = id b and f `2 * f `2 = id a and f `2 = id b ;
( ( cos | [. 2 * PI * 0 , PI * 0 + 2 * PI * 0 + 2 * PI * 0 + 2 * PI * 0 + 2 * PI * 0 + 2 * PI * 0 ) ) is increasing ;
Index ( p , co ) <= len LS - Gij .. LS & Index ( Gij , LS ) <= len LS - Gij .. LS ;
let t1 , t2 , t3 be Element of ( the carrier of S ) * , s be Element of ( the carrier of T ) * ;
j . ( j . ( ( Frege ( curry H ) . h ) ) ) <= j . ( j . ( ( Frege G ) . h ) ) ;
then P [ f . i0 ] & F ( f . i0 + 1 ) < j & j < len f ;
Q [ [ D . x , 1 ] , F . [ D . x , 1 ] ] ;
consider x being element such that x in dom ( F . s ) and y = F . s . x ;
l . i < r . i & [ l . i , r . i ] is for of G . i ;
the Sorts of A2 = ( the carrier of S2 ) --> ( the carrier of S2 ) .= ( the carrier of S1 ) --> ( the carrier of S2 ) ;
consider s being Function such that s is one-to-one and dom s = NAT & rng s = F . 0 and for n being Nat holds s . n = F ( n ) ;
dist ( b1 , b2 ) <= dist ( b1 , a ) + dist ( a , b2 ) & dist ( a , b2 ) <= dist ( b2 , a ) + dist ( b2 , a ) ;
( Lower_Seq ( C , n ) /. len Lower_Seq ( C , n ) ) /. 1 = W-min L~ Cage ( C , n ) ;
q `2 <= ( UMP Upper_Arc C ) `2 & ( UMP C ) `2 <= ( UMP C ) `2 ;
LSeg ( f | i2 , i ) /\ LSeg ( f | i2 , j ) = {} & LSeg ( f | i2 , j ) /\ LSeg ( f , j ) = {} ;
given a being ExtReal such that a <= IA and A = ]. a , IA .] and a <= IA and a <= IA ;
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 <= m & b |^ n in n } ;
( ( x * y * z \ x ) \ z ) \ ( x * y \ x ) = 0. X ;
set xy = [ <* xy , y *> , f1 ] , yz = [ <* y , z *> , f2 ] , zx = [ <* z , x *> , f3 ] ;
l /. len l = ( l . ( len l ) ) & ( l . ( len l ) ) = ( l . ( len l ) ) ;
( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 = 1 - sn ^2 / ( 1 + sn ) ^2 ;
( ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) * ( 1 + sn ) < 1 ;
( ( ( S \/ Y ) \/ X ) `2 ) `2 = ( ( S \/ Y ) \/ X ) `2 .= ( ( S \/ Y ) \/ X ) `2 ;
( seq - seq ) . k = seq . k - seq . k .= ( seq ^\ k ) . k - seq . k .= ( seq ^\ k ) . k ;
rng ( ( h + c ) ^\ n ) c= dom SVF1 ( 1 , f , u0 ) /\ dom SVF1 ( 1 , f , u0 ) ;
the carrier of the carrier of X = the carrier of X & the carrier of X = the carrier of Y implies the carrier of X = the carrier of Y
ex p4 st p3 = p4 & |. p4 - |[ a , b ]| .| = r & |. p4 - |[ a , b ]| .| = r ;
set ch = chi ( X , A5 ) , A5 = chi ( X , A5 ) ;
R |^ ( 0 * n ) = I\HM ( X , X ) .= R |^ n |^ 0 .= R |^ 0 ;
( ( Partial_Sums ( curry ( F-19 , n ) ) ) . n ) is nonnegative & ( ( ( curry ( F-19 , n ) ) . n ) . x is nonnegative ;
f2 = C7 . ( E7 , K ) .= C8 . ( E7 , len H ) .= H . ( len H ) ;
S1 . b = s1 . b .= s2 . b .= ( S2 . b ) . b .= ( S2 . b ) . b ;
p2 in LSeg ( p2 , p1 ) /\ LSeg ( p2 , p2 ) or p2 in LSeg ( p2 , p1 ) /\ LSeg ( p1 , p2 ) ;
dom ( f . t ) = Seg n & dom ( I . t ) = Seg n & dom ( I . t ) = Seg n ;
assume o = ( the connectives of S ) . 11 & ( the connectives of S ) . 11 in ( the carrier' of S ) . 11 ;
set phi = ( l1 , l2 ) implies ( X , l2 ) is ( X , l2 ) contradiction ;
synonym p is is is is is is is is is is or p = 1 implies p is invertible ;
Y1 `2 = - 1 & 0. TOP-REAL 2 <> 0. TOP-REAL 2 & Y1 `2 <> 0. TOP-REAL 2 implies ( Y1 `2 <= - 1 ) & ( Y2 `1 <= ( Y1 `2 <= - 1 ) / ( Y1 `2 ) )
defpred X [ Nat , set , set ] means P [ $1 , $2 , $2 ] & P [ $1 , $2 , $2 ] ;
consider k being Nat such that for n being Nat st k <= n holds s . n < x0 + g and g in dom f ;
Det ( I |^ ( m -' n ) ) = 1. ( K , m -' n ) & Det ( I |^ ( m -' n ) ) = 1. ( K , m -' n ) ;
( - b - sqrt ( b ^2 - 4 * a * c ) ) / 2 * a < 0 ;
godo . d = ( C . da ) mod ( C . db ) .= ( C . da ) mod ( C . db ) .= ( C . da ) mod ( C . db ) ;
attr X1 is dense dense means : Def8 : X2 is dense dense & X1 /\ X2 is dense dense & X2 /\ X1 is dense dense implies X1 /\ X2 is dense dense ;
deffunc FF ( Element of E , Element of I ) = $1 * $2 & $2 = ( $1 * $2 ) * ( $2 ) ;
t ^ <* n *> in { t ^ <* i *> : Q [ i , T . t ] } ;
( x \ y ) \ x = ( x \ x ) \ y .= y ` .= 0. X .= y ;
for X being non empty set for Y being Subset-Family of X holds X is Basis of <* X , FinMeetCl Y *> & Y is Basis of X
synonym A , B are_separated means : Def1 : Cl A misses B & A misses Cl B & B misses Cl A & A misses Cl B ;
len ( M . p ) = len p & width ( M . p ) = width M & len ( M . p ) = width M ;
J . v = { x where x is Element of K : 0 < v . x & v . x < 1 } ;
( Sgm ( Seg m ) ) . d - ( Sgm ( Seg m ) ) . e <> 0 ;
lower_bound divset ( D2 , k + k2 ) = D2 . ( k + k2 - 1 ) .= D2 . ( k + k2 - 1 ) ;
g . r1 = - 2 * r1 + 1 & dom h = [. 0 , 1 .] & rng h c= [. 0 , 1 .] ;
|. a .| * ||. f .|| = 0 * ||. f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| ;
f . x = ( h . x ) `1 & g . x = ( h . x ) `2 & ( h . x ) `2 = ( h . x ) `2 ;
ex w st w in dom B1 & <* 1 *> ^ s = <* 1 *> ^ w & <* 1 *> ^ w = <* 1 *> ^ w ;
[ 1 , {} , <* d1 *> ] in ( { [ 0 , {} , {} ] } \/ S1 ) \/ S2 \/ S2 ;
IC Exec ( i , s1 ) + n = IC Exec ( i , s2 ) .= IC Exec ( i , s2 ) .= IC Exec ( i , s2 ) ;
IC Comput ( P , s , 1 ) = IC Comput ( P , s , 9 ) .= 5 + 9 .= 9 + 9 .= 9 ;
( IExec ( W6 , Q , t ) ) . intpos ( e + 2 ) = t . intpos ( e + 2 ) .= t . intpos ( e + 2 ) ;
LSeg ( f /^ q , i ) misses LSeg ( f /^ q , j ) \/ LSeg ( f /^ q , j ) ;
assume for x , y being Element of L st x in C & y in C holds x <= y or y <= x ;
integral ( f , C ) . x = f . ( upper_bound C ) - f . ( lower_bound C ) .= f . ( lower_bound C ) - f . ( lower_bound C ) ;
for F , G being one-to-one FinSequence st rng F misses rng G holds F ^ G is one-to-one & F ^ G is one-to-one & F ^ G is one-to-one
||. R /. ( L . h ) .|| < e1 * ( K + 1 * ||. h .|| ) ;
assume a in { q where q is Element of M : dist ( z , q ) <= r } ;
set p4 = [ 2 , 1 ] .--> [ 2 , 0 , 1 ] ;
consider x , y being Subset of X such that [ x , y ] in F and x c= d and y \not c= d ;
for y , x being Element of REAL st y ` in Y ` & x in X ` holds y <= x ` & x <= y `
func |. o .| -> variable of A means : - p in it & for x being element st x in it holds it . x = min ( NBI ( p ) , x ) ;
consider t being Element of S such that x `1 , y `2 '||' z `1 , t `2 and x `1 , z `2 '||' y `1 , t `2 ;
dom x1 = Seg len x1 & len x1 = len l1 & for i st i in Seg len x1 holds x1 . i = ( l . i ) * ( x1 . i ) ;
consider y2 being Real such that x2 = y2 and 0 <= y2 and y2 <= 1 / 2 and y2 <= 1 / 2 ;
||. f | X /* s1 .|| = ||. f .|| | X & ||. f /. s1 .|| = ||. f /. s1 .|| & ||. f /. s1 .|| = ||. f /. s1 .|| ;
( the InternalRel of A ) -Seg ( x ` ) /\ Y = {} \/ {} .= {} \/ {} .= {} \/ {} .= {} \/ {} .= {} ;
assume i in dom p implies for j be Nat st j in dom q holds P [ i , j ] & i + 1 in dom p & for j st j in dom q holds P [ j , i ] ;
reconsider h = f | X ( ) , g = f | X ( ) , h = g | X ( ) as Function of X ( ) , Y ( ) ;
u1 in the carrier of W1 & u2 in the carrier of W2 & v in the carrier of W1 implies ( u + v ) + ( v + w ) in the carrier of W2
defpred P [ Element of L ] means M <= f . $1 & f . $1 <= $1 & f . $1 <= f . $1 & f . $1 <= f . $1 ;
^ ( u , a , v ) = s * x + ( - ( s * x ) + y ) .= b ;
- ( x-y - y ) = - x + - y .= - x + y .= - x + y .= - x + y .= y + x ;
given a being Point of GX such that for x being Point of GX holds a , x , x , a , x , y r r ;
fT = [ [ dom @ f2 , cod @ g2 ] , [ cod @ g2 , cod h2 ] ] .= [ cod f2 , cod g2 ] ;
for k , n being Nat st k <> 0 & k < n & n is prime holds 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 ;
US . k = US . ( F . k ) & F . k in dom ( Carrier ( A , k ) ) ;
set i2 = AddTo ( a , i , - n ) , i1 = goto ( card I + 2 ) ;
attr B is thesis means : Def8 : for S being non empty Subset of L holds S is B `1 & S is finite & S is finite ;
a9 "/\" D = { a "/\" d where d is Element of N : d in D } & a "/\" D = { a "/\" d where d is Element of N : d in D } ;
|( exp_R - q9 , q - q9 )| * |( exp_R - q9 , q - q9 )| * |( exp_R - q9 , q - q )| * |( REAL , q - q )| * |( REAL , q - q )| * |( REAL , q )|
( - f ) . sup A = ( ( - f ) | A ) . sup A .= ( ( - f ) | A ) . sup A ;
GG2 `1 = ( ( G * ( len G , k ) ) `1 ) `1 .= ( ( G * ( len G , k ) ) `1 ) `1 .= G * ( 1 , k ) `1 ;
( Proj ( i , n ) ) . LM = <* ( proj ( i , n ) ) . LM *> .= ( Proj ( i , n ) ) . LM ;
f1 + f2 * reproj ( i , x ) is_differentiable_in ( the - 1 ) * reproj ( i , x ) . x0 + ( f2 * reproj ( i , x ) ) . x0 ;
attr ( for x st x in Z holds ( ( tan * tan ) `| Z ) . x = tan . x ) & ( ( tan * tan ) `| Z ) . x = tan . x ;
ex t being SortSymbol of S st t = s & h1 . t . x = h2 . t . x & ( h . s ) . x = ( h . t ) . x ;
defpred C [ Nat ] means ( P . $1 ) is - $1 & ( P . $1 ) is D -seq implies ( P . $1 ) is not empty ) & ( P . $1 is not empty implies P . $1 is not empty ) ;
consider y being element such that y in dom ( ( p | i ) | ( i + 1 ) ) and ( ( p | i ) | ( i + 1 ) ) . y = ( p | i ) . y ;
reconsider L = product ( { x1 } +* ( index ( B ) , l ) ) as Subset of ( the carrier of ( 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 non empty set , f , p be FinSequence of n -tuples_on REAL , p be Element of ( n -tuples_on REAL ) ;
( f * g ) . x = f . ( g . x ) & ( f * h ) . x = f . ( h . x ) ;
p in { 1 / 2 * ( G * ( i + 1 , j ) + G * ( i + 1 , j + 1 ) ) } ;
f `2 - cp = ( - c ) | ( n , L ) *' - ( - ( - ( - ( - ( p + - L ) ) ) ) ) ;
consider r being Real such that r in rng ( f | divset ( D , j ) ) and r < m + s ;
f1 . ( |[ r2 `1 , r2 `2 ]| ) in f1 .: ( W1 /\ W2 ) & f2 . ( |[ r2 , r2 ]| ) in ( the carrier of TOP-REAL 2 ) ;
eval ( a | ( n , L ) , x ) = eval ( a | ( n , L ) , x ) .= a * ( x , x ) ;
z = DigA ( tv , x9 ) .= DigA ( tv , x9 ) .= DigA ( tv , x9 ) .= DigA ( tv , x9 ) .= DigA ( tv , x9 ) ;
set H = { Intersect S where S is Subset-Family of X : S c= G } , G = { Intersect S where S is Subset of X : S is open & S is open } , F = { Intersect S where S is Subset of X : S is open } ;
consider S19 being Element of D * , d being Element of D * such that S `1 = S19 ^ <* d *> and S19 = <* d *> ;
assume that x1 in dom f and x2 in dom f and f . x1 = f . x2 and f . x2 = f . x2 ;
- 1 <= ( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 & ( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 <= 1 ;
(0). V is Linear_Combination of A & Sum ( L ) is Linear_Combination of A & Sum ( L ) = 0. V implies Sum ( L ) = Sum ( L )
let k1 , k2 , k2 , k , k2 , k , 5 , 6 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 be Element of SCM+FSA ;
consider j being element such that j in dom a and j in g " { k `2 } and x = a . j and a . j = x ;
H1 . x1 c= H1 . x2 or H1 . x2 c= H1 . x1 & H1 . x2 c= H1 . x2 or H1 . x2 c= H1 . x2 & 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 & d <= b & [' a , b '] c= dom f and [' a , b '] c= dom g and [' a , b '] c= dom g ;
cell ( Gauge ( C , m ) , ( X X -' 1 ) -' 1 , 0 ) is non empty ;
Ay in { ( S . i ) `1 where i is Element of NAT : not contradiction } & { ( S . i ) `1 where i is Element of NAT : not contradiction } is Subset of NAT ;
( T * b1 ) . y = L * ( b2 /. y ) .= ( F `1 * b1 ) . y .= ( F `1 * b1 ) . y ;
g . ( s , I ) . x = s . y & g . ( s , I ) . y = |. s . x - s . y .| ;
( log ( 2 , k + k ) ) ^2 / ( ( log ( 2 , k + 1 ) ) ^2 ) >= ( log ( 2 , k + 1 ) ) ^2 / ( ( log ( 2 , k + 1 ) ) ^2 ) ;
then p => q in S & not x in the carrier of p & p => All ( x , q ) in S & 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 ) & dom ( the InitS of rM ) misses dom ( the InitS of rM ) ;
synonym f is integer ;
assume for a being Element of D holds f . { a } = a & for X being Subset-Family of D holds f . ( f .: X ) = f . union X ;
i = len p1 .= len p3 + len <* x *> .= len p3 + len <* x *> .= len p3 + 1 .= len p3 + 1 .= len p3 + 1 ;
l . ( l , 3 ) = g . ( ( g . ( 1 , 3 ) + ( k , 1 ) ) - ( e - ( g . ( 1 , 3 ) + e ) ) ) .= g . ( ( 1 , 3 ) + e - ( e - ( g . ( 1 , 3 ) + e ) ) ;
CurInstr ( P2 , Comput ( P2 , s2 , l2 ) ) = halt SCM+FSA .= CurInstr ( P2 , Comput ( P2 , s2 , l2 ) ) .= halt SCM+FSA ;
assume for n be Nat holds ||. seq .|| . n <= Rseq . n & Rseq is summable & Rseq is summable & lim seq = 0 & Rseq is summable & lim seq = 0 ;
sin . ( \vert non .| ) = sin . ( r ) * cos . ( - ( cos . s ) * sin . s ) .= 0 ;
set q = |[ g1 `1 . t0 , g2 `2 . t0 , f3 `2 . t0 ]| , f2 = |[ g2 . t0 , f3 . t0 ]| , f3 = |[ g2 . t0 , f3 . t0 ]| ;
consider G being sequence of S such that for n being Element of NAT holds G . n in implies G in implies for n being Element of NAT holds G . n in implies G . n in
consider G such that F = G and ex G1 st G1 in SM & G = ( the carrier of G1 ) \/ { H } ;
the root of [ x , s ] in ( the Sorts of Free ( C , X ) ) . s & ( the Sorts of Free ( C , X ) ) . s = s ;
Z c= dom ( exp_R * ( f + ( ( #Z 3 ) * f1 ) ) ) /\ dom ( ( #Z 3 ) * f1 ) ;
for k be Element of NAT holds seq1 . k = ( ( \HM { Im ( f , S ) ) . k ) . i & ( ( Im ( f , S ) ) . k ) . i = ( ( Im ( f , S ) ) . i ) . i
assume - 1 < n ( ) & q `2 > 0 & ( q `1 / |. q .| - cn ) < 0 & ( q `1 / |. q .| - cn ) < 0 ) & ( q `1 / |. q .| - cn ) < 0 ;
assume that f is continuous one-to-one and a < b and c < d and f . a = c and f . b = d and f . c = d ;
consider r being Element of NAT such that s-> Nat such that s-> Nat , r , q being Element of NAT st r = Comput ( P1 , s1 , r ) & q <= r ;
LE f /. ( i + 1 ) , f /. j , L~ f , f /. 1 , f /. ( len f ) , f /. ( len f ) ;
assume that x in the carrier of K and y in the carrier of K and ex_inf_of { x , y } , L and ex_inf_of x , y , L and x <= y ;
assume f +* ( i1 , \xi /. 1 ) in ( proj ( F , i2 ) " ( A . ( i + 1 ) ) ) " ( ( A . ( i + 1 ) ) \ { i } ) ;
rng ( ( ( ( ( ( ( the carrier of M ) ) ~ | ( the carrier of M ) ) ) | ( the carrier of M ) ) ) c= the carrier' of M ;
assume z in { ( the carrier of G ) \/ { t } where t is Element of T : t in A } ;
consider l be Nat such that for m be Nat st l <= m holds ||. s1 . m - x0 .|| < g / ( 2 |^ m ) * ||. x0 - x0 .|| ;
consider t being VECTOR of product G such that mt = ||. D5 . t .|| and ||. t .|| <= 1 and ||. t .|| <= 1 ;
assume that the carrier of v = 2 and v ^ <* 0 *> , v ^ <* 1 *> in dom p and p . ( len p + 1 ) in dom p and p . ( len p + 1 ) in dom q ;
consider a being Element of the carrier of X39 , A being Element of the carrier of X39 such that a on A and not a on A ;
( - x ) |^ ( k + 1 ) * ( ( - x ) |^ ( k + 1 ) ) " = 1 / ( ( - x ) |^ ( k + 1 ) ) ;
for D being set for i st i in dom p holds p . i in D & p . i is FinSequence of D & p . i is FinSequence of D
defpred R [ element ] means ex x , y st [ x , y ] = $1 & P [ x , y ] & P [ y , x ] ;
L~ f2 = union { LSeg ( p0 , p10 ) , LSeg ( p10 , p2 ) } .= { p2 , p1 } \/ { p2 } .= { p2 , p1 } \/ { p2 } ;
i - len h11 + 2 - 1 < i - len h11 + 2 - 1 + 1 + 1 - 1 + 1 + 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 = |. nthesis . ( n - 1 ) .| & F . ( n - 1 ) = |. n\frac .|
for r , s1 , s2 being Real holds r in [. s1 , s2 .] iff s1 <= r & r <= s2 & s2 <= s1 & s1 <= s2 & r <= s2
assume v in { G where G is Subset of T2 : G in B2 & G c= z1 & G c= z2 & G c= z1 } ;
let g be non-empty element of A , INT , h be Function of X , INT , b be Element of INT , b be Element of INT ;
min ( g . [ x , y ] , k ) . [ y , z ] = ( min ( g , k , x ) ) . y ;
consider q1 being sequence of CNS such that for n holds P [ n , q1 . n ] and q1 is convergent and lim q1 = lim q1 ;
consider f being Function such that dom f = NAT & for n being Element of NAT holds f . n = F ( n ) and for n being Element of NAT holds f . n = F ( n ) ;
reconsider B-6 = B /\ O , Od = O , Z = Z , Z = { A where A is Subset of B : A is Subset of X : A is open } as Subset of X ;
consider j being Element of NAT such that x = the ` of n and j <= n and 1 <= j & j <= n and j <= n ;
consider x such that z = x and card ( x . O2 ) in card ( x . O ) and x in L1 and x in L2 ;
( C * _ T4 ( k , n2 ) ) . 0 = C . ( ( _ T4 ( k , n2 ) ) . 0 ) .= C . ( ( _ T4 ( k , n2 ) ) . 0 ) ;
dom ( X --> rng f ) = X & dom ( ( X --> f ) . x ) = dom ( ( X --> f ) . x ) ;
S-bound L~ SpStSeq C <= ( b `2 ) * ( ( SpStSeq L~ SpStSeq C ) `2 ) & ( for i being Nat st i in dom SpStSeq C holds ( ( SpStSeq L~ SpStSeq C ) /. i ) `2 <= ( ( SpStSeq L~ SpStSeq C ) /. i ) `2
synonym x , y are_collinear means : l = x or ex l being for l being for k being Nat st { x , y } c= l holds l is card ( A \ { x , y } ) ;
consider X being element such that X in dom ( f | ( n + 1 ) ) and ( f | ( n + 1 ) ) . X = Y ;
assume that Im k is continuous and for x , y being Element of L for a , b being Element of Im k st a = x & b = y holds x << y iff a << b ;
1 / 2 * ( ( - ( PI / 2 ) ) * ( AffineMap ( n , 0 ) ) ) is_differentiable_on REAL & ( ( - ( PI / 2 ) ) * ( AffineMap ( n , 0 ) ) ) is_differentiable_on REAL ;
defpred P [ Element of omega ] means ( the Sorts of A1 ) . $1 = A1 . $1 & ( the Sorts of A2 ) . $1 = A2 . $1 & ( the Sorts of A1 ) . $1 = A2 . $1 ;
IC Comput ( P , s , 2 ) = succ IC Comput ( P , s , 1 ) .= 6 + 1 .= 6 + 1 .= 6 + 1 .= 6 ;
f . x = f . g1 * f . g2 .= f . g1 * 1_ H .= f . g1 * 1_ H .= ( f | ( the carrier of H ) ) . x ;
( M * F-4 ) . n = M . ( F-4 . n ) .= M . ( { ( canFS ( Omega ) ) . n } ) .= M . ( { ( canFS ( Omega ) ) . n } ) ;
the carrier of L1 + L2 c= ( the carrier of L1 ) \/ ( the carrier of L2 ) & the carrier of L1 + L2 c= the carrier of L2 & the carrier of L1 + L2 c= the carrier of L2 ;
pred a , b , c , x , y , c , x , y , c , y be Element of S , a , b , c , x , y be Element of S ;
( the carrier of s ) . n <= ( the carrier of s ) . n * s . ( n + 1 ) & ( the carrier of s ) . n <= ( the carrier of s ) . n ;
attr - 1 <= r & r <= 1 & ( arccot ) . r = - 1 / ( 1 + r ^2 ) & ( arccot ) . r = - 1 / ( 1 + r ^2 ) ;
seq in { p ^ <* n *> where n is Nat : p ^ <* n *> in T1 } implies ex n being Nat st p ^ <* n *> in T1 & p ^ <* n *> in T2
|[ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 ) ]| = x2 - 4 ;
attr F . m is nonnegative means : R : for m being Nat holds ( Partial_Sums F ) . m is nonnegative & ( Partial_Sums F ) . m is nonnegative ;
len ( ( G . z ) * ( ( G . x9 ) * ( y , z ) ) ) = len ( ( ( G . x9 ) * ( y , z ) ) * ( y , z ) ) .= len ( ( G . x9 ) * ( y , z ) ) ;
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 ;
given F being finite Subset of NAT such that F = x and dom F = n and rng F c= { 0 , 1 } and Sum F = k and Sum F = k ;
0 = 1 * ( - Im ) * ue iff 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 -> as being for for non empty Poset , ( ( let ( let L ) | D ) ) , ( ( ( \rm _ D ) | D ) ) , ( ( ( and ( ( and _ D ) | D ) ) ) ) is Boolean ;
"/\" ( B9 , {} ) = Top ( B9 ) .= the carrier of S .= the carrier of S .= the carrier of ( S | ( the carrier of T ) ) .= the carrier of ( S | ( the carrier of T ) ) ;
( r / 2 ) ^2 + ( r / 2 ) ^2 + ( r / 2 ) ^2 <= ( r / 2 ) ^2 + ( r / 2 ) ^2 + ( r / 2 ) ^2 ;
for x being element st x in A /\ dom ( f `| X ) holds ( f `| X ) . x >= r2 & ( f `| X ) . x >= r2
2 * r1 - c * |[ a , c ]| - ( 2 * r1 - ( 2 * r1 - ( 2 * r1 - ( 2 * r1 - c ) ) ) ) = 0. TOP-REAL 2 ;
reconsider p = P * ( \square , 1 ) , q = a " * ( ( - ( - ( K , n , 1 ) ) * ( ( - ( K , n , 1 ) ) * ( p , 1 ) ) ) as FinSequence of K ;
consider x1 , x2 being element such that x1 in uparrow s and x2 in < t and x = [ x1 , x2 ] and [ x2 , y2 ] in Indices G and [ x1 , x2 ] in Indices G ;
for n be Nat st 1 <= n & n <= len q1 holds q1 . n = ( ( upper_volume ( g , M ) ) | ( n + 1 ) ) . ( len ( ( upper_volume ( g , M ) | ( n + 1 ) ) ) )
consider y , z being element such that y in the carrier of A and z in the carrier of A and i = [ y , z ] and y in the carrier of A ;
given H1 , H2 being strict Subgroup of G such that x = H1 and y = H2 and H1 is Subgroup of H2 and H2 is Subgroup of H1 and H2 is Subgroup of H2 and H1 /\ H2 is Subgroup of H2 ;
for S , T being non empty Poset , d being Function of T , S st T is complete holds d is monotone implies d is monotone & d is monotone
[ a + 0. i , b2 ] in ( the carrier of F_Complex ) & [ a , 0. F_Complex ] in [: the carrier of F_Complex , the carrier of V :] & [ a , 0. F_Complex ] in [: the carrier of V , the carrier of V :] ;
reconsider mm = max ( len F1 , len ( p . n ) * ( <* x *> |^ n ) ) as Element of NAT ;
I <= width GoB ( ( GoB h ) * ( len GoB h , width GoB h ) , ( GoB h ) * ( len GoB h , width GoB h ) ) & ( GoB h ) * ( len GoB h , width GoB h ) `2 <= ( GoB h ) * ( 1 , width GoB h ) `2 ;
f2 /* q = ( f2 /* ( f1 /* s ) ) ^\ k .= ( f2 * ( f1 /* s ) ) ^\ k .= ( ( f2 * f1 ) /* s ) ^\ k .= ( ( f2 * f1 ) /* s ) ^\ k ;
attr A1 \/ A2 is linearly-independent means : Def8 : A1 is linearly-independent & A2 misses A2 & for v being Element of V holds Lin ( A1 /\ A2 ) = Lin ( A2 ) & Lin ( A2 ) = Lin ( A1 ) /\ Lin ( A2 ) ;
func A -carrier C -> set equals union { A . s where s is Element of R : s in C } where s is Element of R : s in C & s in C } ;
dom ( Line ( v , i + 1 ) (#) ( ( Line ( p , m ) ) * ( \square , 1 ) ) ) = dom ( F ^ <* ( L . m ) * ( i , 1 ) *> ) ;
cluster [ x `1 , 4 , x `2 ] , [ x `1 , 4 , x `2 ] , [ x `1 , 4 , x `2 ] , [ x `1 , 4 ] , [ x , 4 ] ] , [ x , 4 ] , [ x , 4 ] ] , [ x , 4 ] ] ;
E , { All ( x2 , ( x2 , x1 ) / ( x0 , x1 ) ) } => ( ( x2 , x1 ) / ( x0 , x1 ) ) '&' ( x1 , x2 ) / ( x0 , x1 ) ) |= ( ( x1 , x2 ) / ( x0 , x1 ) ) ;
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 ) - ( h . m ) + ( h . m ) ;
cell ( G , XX -' 1 , ( Y + 1 ) \ ( t + 1 ) ) \ L~ f meets ( UBD L~ f ) \/ ( UBD L~ f ) \/ ( UBD L~ f ) ;
IC Result ( P2 , s2 ) = IC IExec ( I , P , Initialize s ) .= card I .= card I + card J .= card I + card J + card J .= card I + card J + card J + 3 .= card I + card J + 3 .= card I + card J + 3 ;
sqrt ( ( - ( ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ) ^2 ) > 0 & ( - ( ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ) ^2 > 0 ;
consider x0 being element such that x0 in dom a and x0 in g " { k `2 } and y0 = a . ( k + 1 ) and y = a . ( k + 1 ) and a . x0 = b . ( k + 1 ) ;
dom ( r1 (#) chi ( A , C ) ) = dom chi ( A , C ) /\ ( A /\ A ) .= C /\ ( A /\ ( A /\ B ) ) .= C /\ ( A /\ ( A /\ B ) ) .= C /\ ( A /\ ( A /\ B ) ) ;
d-7 . [ y , z ] = ( ( [ y , z ] `1 - ( y - z ) ) / ( 1 - ( y - z ) ) ) * ( y - z ) .= ( ( y - z ) + ( y - z ) ) / ( 1 - ( y - z ) ) * ( y - z ) ;
attr i be Nat means : Def8 : C . i = A . i /\ B . i & L~ C c= ( L~ f ) /\ ( L~ f ) ;
assume that x0 in dom f and f is_continuous_in x0 and f is_continuous_in x0 and for r st r in dom f & 0 < r ex g st g < x0 & g in dom f & f /. g <> 0 ;
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
for x being Element of REAL n st x in Line ( x1 , x2 ) holds |. y1 - y2 .| <= |. y1 - y2 .| & |. y2 - x .| <= |. y1 - y2 .|
func /. <*> { a } -> w w } means : : : a in it & for b being w Ordinal Ordinal st a in b holds it . b c= b & it . a c= b ;
[ a1 , a2 , a3 ] in ( the carrier of A ) \/ ( the carrier of B ) & [ a1 , a2 , a3 ] in [: the carrier of A , the carrier of B :] ;
ex a , b being element st a in the carrier of S1 & b in the carrier of S2 & x = [ a , b ] & [ b , a ] in the carrier of S2 & [ a , b ] in the carrier of S2 ;
||. ( vseq . n ) - ( vseq . m ) .|| * ||. x - ( vseq . n ) .|| < e / ( ||. x - y .|| + e / ( ||. x - y .|| + e / ( ||. x - y .|| + e / ( ||. x - y .|| + e / ( ||. x - y .|| + e / ( ||. x - y .|| + e / ( ||. x - y .|| + e / ( ||. x - y .|| + e / ( ||. x - y .|| ) ) ) ) ) ) * ( x - y .|| ) ) ) ;
then for Z being set st Z in { Y where Y is Element of I7 : F c= Y & Y in Z } holds z in x & z in Z ;
sup compactbelow [ s , t ] = [ sup ( ( compactbelow [ s , t ] ) /\ ( compactbelow [ s , t ] ) ) , sup ( ( compactbelow [ s , t ] ) /\ ( compactbelow [ s , t ] ) ) ] ;
consider i , j being Element of NAT such that i < j and [ y , f . j ] in IT and [ f . i , z ] in IT and [ f . i , z ] in IT ;
for D being non empty set , p , q being FinSequence of D st p c= q ex p being FinSequence of D st p ^ q = q & p ^ q = q ^ p
consider e19 being Element of the carrier of X such that c9 , a9 // a9 , e29 and a9 <> b9 and b9 <> c9 and [ e , f ] in [: { e } , { f } :] ;
set U2 = I \! \mathop { x } , d = I \! \mathop { x } , e = I \! \mathop { x } , f = I \! \mathop { x } , g = I \! \mathop { x } , d = I \! \mathop { x } , d = I \! \mathop { x } , d = I \! \mathop { x } , d = I \! \mathop { x } , d = I \! \mathop { x } , d = I . ( I . ( I . ( I . ( I . ( I . ( I . ( I . ( I . ( I . ( I . ( I . ( I . (
|. q3 .| ^2 = ( ( q3 `1 ) ^2 + ( q2 `2 ) ^2 ) + ( ( q3 `1 ) ^2 + ( q2 `2 ) ^2 ) .= ( q `1 ) ^2 + ( q `2 ) ^2 .= ( q `1 ) ^2 + ( q `2 ) ^2 ;
for T being non empty TopSpace , x , y being Element of <* the topology of T , \subseteq \rangle holds x "\/" y = x \/ y & x "/\" y = x /\ y implies x "/\" y = x /\ y
dom signature U1 = dom ( the charact of U1 ) & Args ( o , MSAlg U1 ) = dom ( the charact of U1 ) & Args ( o , MSAlg U1 ) = dom ( the charact of U1 ) & Args ( o , MSAlg U1 ) = dom ( the charact of U1 ) ;
dom ( h | X ) = dom h /\ X .= dom ( ||. h .|| | X ) /\ X .= dom ( ||. h .|| | X ) .= dom ( ( ||. h .|| | X ) | X ) .= dom ( ( ||. h .|| | X ) | X ) ;
for N1 , K1 being Element of G8 holds dom ( h . K1 ) = N & rng ( h . K1 ) = N1 & rng ( h . K1 ) = K1 & rng ( h . K1 ) c= K1 & rng ( h . K1 ) c= K1
( mod ( u , m ) + mod ( v , m ) ) . i = ( mod ( u , m ) ) . i + ( mod ( v , m ) ) . i ;
- ( q `1 ) ^2 < - 1 or ( q `2 ) ^2 / ( |. q .| ) ^2 & ( q `1 ) ^2 / ( |. q .| ) ^2 <= ( - ( q `1 ) ) ^2 / ( |. q .| ) ^2 ) or ( q `1 ) ^2 / ( |. q .| ) ^2 <= ( - ( q `1 ) ) ^2 / ( |. q .| ) ^2 ;
attr r1 = f9 & r2 = f9 & for r being Real st r in dom f9 holds r1 * ( f . r ) = r2 * ( f . r ) & r2 * ( f . r ) = ( f . r ) * ( f . r ) ;
vseq . m is bounded Function of X , the carrier of Y & x9 . m = ( for x be Point of X , X be Subset of Y holds ( ( vseq . m ) - ( vseq . n ) ) ) . x = ( ( vseq . m ) - ( vseq . n ) ) . x
attr a <> b & b <> c & angle ( a , b , c ) = PI & angle ( b , c , a ) = 0 implies angle ( c , a , b ) = PI & angle ( b , a , c ) = PI
consider i , j being Nat , r , s being Real such that p1 = [ i , r ] and p2 = [ j , s ] and i < j and r < s and s < p2 ;
|. p .| ^2 - ( 2 * |( p , q )| ) ^2 + |. q .| ^2 = |. p .| ^2 ^2 + |. q .| ^2 - ( 2 * |( p , q )| ) ^2 ;
consider p1 , q1 being Element of X ( ) such that y = p1 ^ q1 and p1 ^ q1 = p2 ^ q2 and p1 ^ q1 = p2 ^ q2 and p1 ^ q1 = p2 ^ q2 and p2 ^ q2 = q2 ^ q1 ;
, ( the carrier of A ) . ( r1 , r2 , s1 , s2 , s2 ) = ( s2 - s1 ) * ( s1 - s2 ) .= ( s2 - s1 ) * ( s2 - s2 ) .= ( s2 - s1 ) * ( s2 - s2 ) ;
( ( LMP A ) `2 ) `2 = lower_bound ( proj2 .: ( A /\ /\ /\ /\ /\ /\ Ball ( w , r ) ) ) & ( proj2 .: ( A /\ /\ /\ Ball ( w , r ) ) ) is non empty ;
s , k |= H1 , ( k , j ) |= H2 iff s , k |= ( H1 , k ) '&' ( H2 , k ) = ( H1 , k ) '&' ( H2 , k )
len ( s + 1 ) = card ( support b1 ) + 1 .= card ( support b2 ) + 1 .= card ( support b2 ) + 1 .= card ( support b1 ) + 1 .= card ( support b1 ) + 1 .= card ( support b1 ) + 1 .= card ( support b1 ) + 1 .= card ( support b1 ) + 1 ;
consider z being Element of L1 such that z >= x and z >= y and for z being Element of L1 st z >= x & z >= y holds z `1 >= y & z `2 >= x `2 ;
LSeg ( UMP D , |[ ( W-bound D + E-bound D ) / 2 , ( E-bound D + E-bound D ) / 2 ]| ) /\ D = { UMP D } \/ { ( ( /\ D ) /\ D ) / 2 } ;
lim ( ( ( f `| N ) / g ) /* ( h ^\ N ) - ( f `| N ) /* b ) = lim ( ( f `| N ) / ( g `| N ) - ( f `| N ) /* b ) ;
P [ i , pr1 ( f ) . i , pr1 ( f ) . i , pr1 ( f ) . ( i + 1 ) , pr2 ( f ) . ( i + 1 ) ] ;
for r be Real st 0 < r ex m be Nat st for k be Nat st m <= k holds ||. ( seq . k ) - ( R-2 . 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 & a in P & b in P holds a = b
Z c= dom ( ( #Z n ) ) /\ ( dom ( ( #Z n ) * f ) \ ( ( #Z n ) * f ) " { 0 } ) & ( ( #Z n ) * f ) " { 0 } c= dom ( ( #Z n ) * f ) " { 0 } ) ;
ex j being Nat st j in dom ( l ^ <* x *> ) & j < i & y = ( l ^ <* x *> ) . j & i = 1 + len l & z = ( l ^ <* x *> ) . j & i = 1 + len l & j = len l + 1 ;
for u , v being VECTOR of V , r being Real st 0 < r & u < 1 & v in Seg N holds r * u + ( 1-r * v ) in N
A , Int A , Cl ( A , Cl ( A , B ) ) , Cl ( Int ( A , B ) , Cl ( A , C ) ) , Cl ( Int ( A , B ) , Cl ( A , C ) ) , Cl ( Int ( A , B ) , Cl ( A , C ) ) , Cl ( A , B ) , Cl ( A , C ) ) in / 2 ;
- Sum <* v , u , w *> = - ( v + u + w ) .= - ( v + u ) + w .= - ( v + u ) + w .= - ( v + u ) + w .= - ( v + u ) + w ;
( Exec ( a := b , s ) ) . IC SCM R = ( Exec ( a := b , s ) ) . IC SCM R .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= IC s .= IC s .= IC s .= IC s .= IC s ;
consider h being Function such that f . a = h and dom h = I and for x being element st x in I holds h . x in ( the carrier of J ) & h . x in ( the carrier of J ) ;
for S1 , S2 being non empty reflexive RelStr , D being non empty directed Subset of [: S1 , S2 :] , x being Element of [: S1 , S2 :] holds x is directed & x is directed & x is directed & x is directed
card X = 2 implies ex x , y st x in X & y in X & x <> y & for z st z in X holds z = x or z = y or z = y
E-max L~ Cage ( C , n ) in rng ( Cage ( C , n ) \circlearrowleft W-min L~ Cage ( C , n ) ) & W-min L~ Cage ( C , n ) in rng ( Cage ( C , n ) \circlearrowleft W-min L~ Cage ( C , n ) ) ;
for T , T being DecoratedTree , p , q being Element of dom T st p in dom T & q in dom T holds ( T -with p , T -with q ) . q = T . q
[ i2 + 1 , j2 ] , [ i2 , j2 ] ] in Indices G & f /. k = G * ( i2 + 1 , j2 ) & f /. k = G * ( i2 + 1 , j2 ) implies ( f /. k ) `1 = G * ( i2 + 1 , j2 ) `1
cluster ( k , n ) gcd ( k , n ) -> natural & n divides it & ( n divides m implies ( n divides m ) & ( n divides m implies ( n divides m ) & ( n divides m ) & ( n divides m ) implies ( n divides m ) & ( n divides m ) & ( n divides m implies n divides m ) ) implies n divides m
dom F " = the carrier of X2 & rng F " = the carrier of X1 & F " { x } = the carrier of X2 & F " { x } is one-to-one & F " { x } is one-to-one & F " { x } is one-to-one ;
consider C being finite Subset of V such that C c= A and card C = n and the carrier of V = Lin ( B9 \/ C ) and C = Lin ( B9 \/ C ) and Lin ( C \/ B ) = Lin ( B9 \/ C ) ;
V is prime implies for X , Y being Element of <* the topology of T , \subseteq the topology of T , the topology of T *> st X /\ Y c= V holds X c= V or Y c= V
set X = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } , Y = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } ;
angle ( p1 , p3 , p4 ) = 0 .= angle ( p2 , p3 , p4 ) .= angle ( p3 , p4 ) .= angle ( p2 , p3 , p4 ) .= angle ( p3 , p4 , p2 ) .= angle ( p3 , p4 ) .= angle ( p2 , p4 , p3 ) ;
- sqrt ( - ( ( q `1 / |. q .| - cn ) / ( 1 - cn ) ) ^2 ) = - sqrt ( ( - ( ( q `1 / |. q .| - cn ) ) / ( 1 - cn ) ) ^2 ) .= - sqrt ( ( - ( ( q `1 / |. q .| - cn ) / ( 1 - cn ) ) ^2 ) ) .= - 1 ;
ex f being Function of I[01] , TOP-REAL 2 st f is continuous one-to-one & rng f = P & f . 0 = p1 & f . 1 = p2 & f . 0 = p3 & f . 1 = p4 & f . 1 = p4 & f . 1 = p4 ;
attr f is partial differentiable on 3 , REAL means : R : SVF1 ( 2 , pdiff1 ( f , 1 ) , u0 ) is continuous & SVF1 ( 2 , pdiff1 ( f , 1 ) , u0 ) is continuous & SVF1 ( 2 , pdiff1 ( f , 3 ) , u0 ) . u0 = ( proj ( 2 , 3 ) ) . u0 ;
ex r , s st x = |[ r , s ]| & G * ( len G , 1 ) `1 < r & r < G * ( 1 , 1 ) `2 & G * ( 1 , 1 ) `2 < s & s < G * ( 1 , 1 ) `2 ;
assume that f is_sequence_on G and 1 <= t & t <= len G and G * ( t , width G ) `2 >= N-bound L~ f and LSeg ( G * ( t , width G ) , width G ) c= L~ f and LSeg ( G * ( t , width G ) , width G ) c= L~ f ;
attr i in dom G means : R : r * ( f * reproj ( i , x ) ) = r * ( reproj ( i , x ) . i ) & ( f * reproj ( i , x ) ) . x = r * ( reproj ( i , x ) . x ) ;
consider c1 , c2 being bag of o1 + o2 such that ( decomp c ) /. k = <* c1 , c2 *> and c /. k = c1 + c2 and ( decomp c ) /. k = c1 + c2 and ( decomp c ) /. k = c2 + c2 ;
u0 in { |[ r1 , s1 ]| : r1 < G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `2 < s1 & s1 < G * ( 1 , 1 ) `2 }
( ( X ^ Y ) . k ) = the carrier of X . k2 .= ( ( C ^ Y ) . k ) . ( ( C ^ Y ) . k ) .= ( ( C ^ Y ) . k ) . ( ( C ^ Y ) . k ) .= ( C ^ Y ) . ( ( C ^ Y ) . k ) ;
attr M1 = len M2 means : Def8 : len M1 = width M2 & for i st i in dom M1 holds M1 . i = M2 . i - M2 . i & M1 . i = M2 . i ;
consider g2 be Real such that 0 < g2 and { y where y is Point of S : ||. y - x0 .|| < g2 & y in dom f & f /. y = ( f | N ) /. ( y - x0 ) ) & ( f | N ) /. ( y - x0 ) = ( f | N ) /. ( y - x0 ) ;
assume x < ( - b + sqrt ( x0 , b , c ) ) / 2 * a or x > ( - b - sqrt ( x0 , b , c ) ) / 2 * a ;
( G1 '&' G2 ) . i = ( <* 3 *> ^ G1 ) . i & ( H1 '&' H2 ) . i = ( <* 3 *> ^ G1 ) . i & ( H1 '&' H2 ) . i = ( <* 3 *> ^ G1 ) . i ;
for i , j st [ i , j ] in Indices ( M3 + M1 ) holds ( M3 + M1 ) * ( i , j ) < M2 * ( i , j ) & M2 * ( i , j ) < M2 * ( i , j )
for f being FinSequence of NAT , i being Element of NAT , j being Element of NAT st j in dom f & i <= j holds i divides f /. j & ( i divides j implies f /. j = f /. j ) & ( i divides j implies f /. i = f /. j )
assume F = { [ a , b ] where a , b is Subset of X : for c being set st c in B39 & a c= c & b c= c } holds b in { c where c is set : c in B & a c= c } ;
b2 * q2 + ( b3 * q3 ) + - ( ( a * q2 ) + ( - ( a * q2 ) ) * ( q2 + q3 ) ) = 0. TOP-REAL n & - ( ( a * q2 ) + ( - ( a * q2 ) ) * ( q2 + q3 ) ) = 0. TOP-REAL n ;
Cl Cl F = { D where D is Subset of T : ex B being Subset of T st D = Cl B & B in F & B in Cl F } & Cl F = Cl D ;
attr seq is summable means : _ : seq is summable & seq is summable & ( for n holds seq . n = Sum ( seq ) ) & ( for n holds seq . n = Sum ( seq ) ) & ( for n holds seq . n = Sum ( seq ) ) & ( for n holds seq . n = Sum ( seq ) ) implies seq is summable & Sum ( seq ) = Sum ( seq ) ) ;
dom ( ( ( cn " ) | D ) | D ) = ( the carrier of ( TOP-REAL 2 ) ) /\ D .= the carrier of ( ( TOP-REAL 2 ) | D ) .= the carrier of ( ( TOP-REAL 2 ) | D ) | D .= ( ( TOP-REAL 2 ) | D ) | D .= ( ( TOP-REAL 2 ) | D ) ;
|[ X , Z ]| is full full non empty SubRelStr of ( Omega Z ) |^ the carrier of Z & |[ X , Y ] is full full SubRelStr of ( Omega Z ) |^ the carrier of Z implies X is full non empty
G * ( 1 , j ) `2 = G * ( i , j ) `2 & G * ( 1 , j ) `2 <= G * ( i , j ) `2 & G * ( 1 , j ) `2 <= G * ( i , j ) `2 ;
synonym m1 c= m2 means : Def3 : for p being set st p in P holds the carrier of m1 <= p & the carrier of m2 <= the carrier of m2 & the carrier of m1 <= the carrier of m2 & the carrier of m1 = the carrier of m2 & the carrier of m1 = the carrier of m2 & the carrier of m1 = the carrier of m2 ;
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 for b being Element of A ( ) holds P [ b ] ;
synonym the multMagma of R is reflexive & multiplicative seq is associative means : : : for a being Element of R holds [ a , the multF of R ] in { [ a , the multF of R ] where a is Element of R , a is Element of R : a in the carrier of R } ;
sequence ( a , b , 1 ) + sequence ( c , d , 1 ) = b + sequence ( c , d , 1 ) .= b + d .= b + d + c .= \mathop { + ( a , c , d ) } ;
cluster + ( i , j ) -> in INT means : R : for i1 , i2 being Element of INT holds it . ( i1 , i2 ) = + ( i , j ) & ( i , j ) in dom ( i , j ) implies ( i , j ) = i ) ;
( - s2 ) * p1 + ( s2 * p2 - ( s2 * p2 - ( s2 * p2 ) ) ) * p2 = ( - r2 ) * p1 + ( ( s2 - ( s2 * p2 - ( s2 * p2 ) ) ) * p2 ) .= ( ( - s2 ) * p1 ) + ( ( s2 - ( s2 * p2 ) ) ) * p2 ) ;
eval ( ( a | ( n , L ) ) *' , 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 directed Subset of Omega S , V being open Subset of Omega S , V being open Subset of Omega T st V in V & V is open & V is open & V is open & V is open holds V is open & V is open ;
assume that 1 <= k & k <= len w + 1 and T-7 . ( ( q , w ) -{ w } ) = ( T11 . ( q , w ) ) -) . k and T11 . ( ( q , w ) -) = ( T11 . ( q , w ) ) -) . k ;
2 * a |^ ( n + 1 ) + ( 2 * b |^ ( n + 1 ) ) >= a |^ ( n + 1 ) + ( ( a |^ n ) * b + ( b |^ n ) * a ) + ( ( a |^ n ) * b + ( b |^ n ) * a ) ;
M , v2 |= All ( x. 3 , All ( x. 0 , All ( x. 4 , H ) ) ) implies M , v / ( x. 4 , All ( x. 0 , H ) ) / ( x. 4 , ( x. 0 ) / ( x. 4 , ( x. 0 ) ) / ( x. 0 , H ) ) |= H
assume that f is_differentiable_on l and for x0 st x0 in l holds 0 < f ' ( x0 ) or for x0 st x0 in l holds f ' ( x0 ) - f ' ( x0 ) < 0 and for x1 st x1 in l holds f ' ( x1 ) - f ' ( x0 ) < 0 ;
for G1 being _Graph , W being Walk of G1 , e being set , G2 being Walk of G1 , e being set st not e in W & not e in W & ( W is Walk of G2 & e in W & v in W & W is Walk of G2 ) holds W is Walk of G1
not vs is not empty iff ( for y st y is not empty & y is not empty holds not y is not empty ) & not ( y is not empty or not y is not empty ) & not ( y is not empty or not y is not empty ) & not ( y is not empty or not y is not empty ) & not y is not empty ) ;
Indices GoB f = [: dom GoB f , Seg width GoB f :] & ( GoB f ) * ( i1 + 1 , j + 1 ) in Indices GoB f & ( GoB f ) * ( i1 + 1 , j + 1 ) in Indices GoB f ) implies ( GoB f ) * ( i1 + 1 , j + 1 ) in cell ( GoB f , i1 + 1 , j + 1 )
for G1 , G2 , G3 , G3 being strict Subgroup of O , O being stable Subgroup of G2 , G2 being stable Subgroup of G1 , G2 being stable Subgroup of G2 , G2 being stable Subgroup of G2 , G2 being stable Subgroup of G2 st G1 is stable & G2 is stable & G2 is stable holds G1 is stable Subgroup of G2
UsedIntLoc ( in4 ( f ) ) = { intloc 0 , intloc 1 , intloc 2 , intloc 3 , intloc 4 , intloc 4 , intloc 4 , intloc 5 , intloc 5 , intloc 6 , intloc 5 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 }
for f1 , f2 being FinSequence of F st f1 ^ f2 is p -element & Q [ f1 ^ f2 ] & Q [ f2 ^ f1 ] & Q [ f2 ^ f1 ] holds Q [ f1 ^ f2 ^ f2 ]
( p `1 ) ^2 / sqrt ( 1 + ( p `1 / p `2 ) ^2 ) = ( q `1 ) ^2 / sqrt ( 1 + ( q `1 / q `2 ) ^2 ) .= ( q `1 ) ^2 / sqrt ( 1 + ( q `1 / q `2 ) ^2 ) .= ( q `1 ) ^2 / sqrt ( 1 + ( q `1 / q `2 ) ^2 ) ;
for x1 , x2 , x3 , x4 being Element of REAL n holds |( x1 - x2 , x3 - x4 )| = |( x1 - x2 , x3 - x4 )| & |( x1 - x2 , x3 - x4 )| = |( x1 - x2 , x3 - x4 )| & |( x1 - x2 , x3 - x4 )| = |( x1 - x2 , x3 - x4 )|
for x st x in dom ( ( - x ) | A ) holds ( ( - x ) | A ) . ( - x ) = - ( ( - x ) | A ) . ( - x )
for T being non empty TopSpace , P being Subset-Family of T , x being Point of T st P c= the topology of T for B being Basis of x ex P being Basis of T st B c= P & P is Basis of x & P is Basis of x
( a 'or' b 'imp' c ) . x = 'not' ( ( a 'or' b ) . x ) 'or' c . x .= 'not' ( a . x 'or' b . x ) 'or' c . x .= TRUE 'or' TRUE .= TRUE 'or' TRUE .= TRUE .= TRUE ;
for e being set st e in [: A , Y1 :] 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
for i be set st i in the carrier of S for f be Function of Su . i , S1 . i st f = H . i & F . i = f | ( F . i ) holds F . i = f | ( F . 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 - card { i , j } .= ( c1 + 1 ) - ( i + 1 ) + ( 1 - 1 ) .= ( c1 + 1 ) - ( i + 1 ) + ( 1 - 1 ) .= 2 * c1 + ( i + 1 ) - ( i + 1 ) .= 2 * c1 + ( i + 1 ) - ( i + 1 ) ;
IC Exec ( i , s ) = ( s +* ( 0 .--> succ ( s . 0 ) ) ) . 0 .= ( 0 .--> succ ( s . 0 ) ) . 0 .= ( 0 .--> succ ( s . 0 ) ) . 0 .= succ IC s .= succ IC s .= succ IC s .= IC s ;
len f /. ( \downharpoonright i1 -' 1 ) -' 1 + 1 = len f /. ( \downharpoonright i1 -' 1 + 1 ) - 1 + 1 .= len f -' 1 + 1 - 1 .= len f -' 1 + 1 - 1 + 1 .= len f -' 1 + 1 - 1 .= len f - 1 + 1 - 1 + 1 .= len f - 1 + 1 ;
for a , b , c being Element of NAT st 1 <= a & a <= b & k <= a holds k < ( a + b-2 ) or k = a + b-2 or k = a + b-2 or k = a + b-2 or k = b + b-2 or k = a + b-2 or k = a + b-2
for f being FinSequence of TOP-REAL 2 , p being Point of TOP-REAL 2 , i being Nat st p in LSeg ( f , i ) & i <= len f & p in LSeg ( f , i ) holds Index ( p , f ) <= i
lim ( ( curry ( PT , k + 1 ) ) # x ) = lim ( ( curry ( PT , k + 1 ) ) # x ) + lim ( ( curry ( PT , k + 1 ) ) # x ) .= lim ( ( curry ( FT , k + 1 ) ) # x ) ;
z2 = g /. ( \downharpoonright n1 -' n2 + 1 ) .= g . ( i - n2 + 1 ) .= g . ( i - n2 + 1 ) .= g . ( i - n2 + 1 ) .= g . ( i - n2 + 1 ) .= g . ( i - n2 + 1 ) .= g . ( i - n2 + 1 ) ;
[ f . 0 , f . 3 ] in id ( the carrier of G ) \/ ( the InternalRel of G ) or [ f . 0 , f . 3 ] in the InternalRel of C6 & [ f . 0 , f . 3 ] in the InternalRel of C6 ;
for G being Subset-Family of B for R being Subset of [: A , B :] st G = { R [ X ] where R is Subset of A , Y is Subset of B : R in FF & Y in F } holds ( Intersect ( G ) ) . [ X , Y ] = Intersect ( G ) . [ X , Y ]
CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 ) ) = CurInstr ( P1 , Comput ( P1 , s1 , m2 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m2 ) ) .= CurInstr ( P1 , Comput ( P2 , s2 , m2 ) ) .= CurInstr ( P2 , Comput ( P2 , s2 , m2 ) ) .= halt SCMPDS ;
assume that a on M and b on M and c on N and d on N and p on M and a on M and c on N and p on M and a on M and c on N and p on M and a on M and p on M and a on M and c on N and p on N and a on M and c on N and p on N and a on M and a on M and b on N and a on M and c on N and a on N and a on M and b on N and b on N and b on N and a on N and b on N and b on N and b on N and b on N and c on N and c on N and b on N and c on N and a on M and b on N and a on N and c on N and c on N and c on M and c on M and c on N and c on M
assume that T is \hbox { T _ 4 } and T is as as as as non empty set and ex F be Subset-Family of T st F is closed & F is finite-ind & ind F <= 0 & ind T <= n & ind T <= n & ind T <= n & ind T <= n ;
for g1 , g2 st g1 in ]. r - g2 , r .[ & g2 in ]. r - g2 , r .[ holds |. f . g1 - f . g2 .| <= ( g1 - g ) / ( |. r - g2 .| + ( |. r - g2 .| + |. g - g2 .| ) / ( |. r - g2 .| )
( ( - ( z - z1 ) ) / ( z - z2 ) ) = ( ( - ( z - z1 ) ) / ( z - z2 ) ) * ( ( - ( z - z1 ) ) / ( z - z2 ) ) .= ( ( - ( z - z1 ) ) / ( z - z2 ) ) * ( z - z2 ) ;
F . i = F /. i .= 0. R + r2 .= ( b |^ n ) * ( a |^ ( n + 1 ) ) .= <* ( n + 1 ) |^ ( n + 1 ) , \dots , ( n + 1 ) |^ ( n + 1 ) , \dots , ( n + 1 ) |^ ( n + 1 ) , \dots , ( n + 1 ) |^ ( n + 1 ) , \dots , ( 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 ) = Rf . ( n , f . n ) & for n holds f . ( n + 1 ) = Rf . ( n + 1 ) & y = Rf . n ;
func f (#) F -> FinSequence of V means : Def6 : len it = len F & for i be Nat st i in dom it holds it . i = F /. i * ( F /. 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 , 7 , 8 , 8 , 7 , 8 , 8 , 7 , 8 } = { x1 , x2 , x3 } \/ { x4 , 8 , 7 , 8 } \/ { x5 , 8 , 7 , 8 } \/ { x5 , 8 , 7 , 8 } \/ { 6 }
for n being Nat for x being set st x = h . n holds h . ( n + 1 ) = o . ( x , n ) & x in InputVertices S ( x , n ) & o ( x , n ) in InnerVertices S ( x , n ) & o ( x , n ) in InnerVertices S ( x , n )
ex S1 being Element of CQC-WFF ( Al ( ) ) st SubP ( P , l , e ) = S1 & ( for e being Element of CQC-WFF ( Al ( ) ) holds ( S . e ) `1 is Element of CQC-WFF ( Al ( ) ) iff ( S . e ) `1 = e ) & ( S . e ) `1 = e ) & ( S . e ) `1 = e & ( S . e ) `1 = e ) & ( S . e = e ) `1 ;
consider P being FinSequence of G_ 2 such that pbeing = product P and for i st i in dom P ex t9 being Element of the carrier of K st P . i = t9 & ( ex t9 being Element of the carrier of K st P . i = t9 ) & ( ex t9 being Element of K st P . i = t9 ) & ( ex t9 being Element of K st t . i = t9 ) & ( t . i = t9 ) ;
for T1 , T2 being strict non empty TopSpace , P being Basis of T1 , T2 being Basis of T2 st the carrier of T1 = the carrier of T2 & P is Basis of T2 & P is Basis of T1 & P is Basis of T2 & P is Basis of T2 & P is Basis of T2 holds P is Basis of T1 & P is Basis of T2
assume that f is_\/ pdiff1 ( f , 3 ) and r (#) pdiff1 ( f , 3 ) is_partial_differentiable_in u0 , 2 and partdiff ( r (#) pdiff1 ( f , 3 ) , u0 , 2 ) = r * pdiff1 ( f , 3 ) and partdiff ( r (#) pdiff1 ( f , 3 ) , u0 , 2 ) = r * pdiff1 ( f , 3 ) ;
defpred P [ Nat ] means for F , G being FinSequence of ExtREAL for s being Permutation of REAL for G being Permutation of Seg $1 st len F = $1 & G = F * s & not G = F * s holds Sum ( F ) = Sum ( G ) & Sum ( G ) = Sum ( F ) * Sum ( G ) ;
ex j st 1 <= j & j < width GoB f & ( GoB f ) * ( 1 , j ) `2 <= s & s < ( GoB f ) * ( 1 , j + 1 ) `2 & ( GoB f ) * ( 1 , j + 1 ) `2 <= s & s < ( GoB f ) * ( 1 , j + 1 ) `2
defpred U [ set , set ] means ex FF be Subset-Family of T st $1 = FF & F is open & union FF is open & union FF is open & union FF is open & union FF is open & union FF is open & union FF is open & union FF is open & union FF is open & union FF is open & union FF is open & union FF is open & union FF is open & union FF is open & union FF is open ) ;
for pp being Point of TOP-REAL 2 st LE p4 , p , P , p1 , p2 & LE p , p , P , p1 , p2 & LE p , p , P , p1 , p2 & LE p , p , P , p1 , p2 & LE p , p , P , p1 , p2 & LE p , p , P , p1 , p2 holds LE p , p1 , P , p2
f in \mathop { E } ( H ) & for g st g . y <> f . y holds x in E implies g in \mathop { E } implies f in \mathop { All ( x , H ) } & g in \mathop { All ( x , H ) }
ex 8 being Point of TOP-REAL 2 st x = 8 & ( ( ( ( ( ( ( ( ( ( ( p `2 ) ) ) | D ) ) | D ) ) ) & ( ( ( ( ( ( p `2 ) | D ) ) | D ) ) | D ) ) & ( ( ( ( ( ( p `2 ) | D ) ) | D ) ) | D ) ) is continuous ;
assume for d7 being Element of NAT st d7 <= max ( d7 , t ) holds s1 . ( ( d - t7 ) / ( ( d - t7 ) + ( d - t7 ) ) / ( ( d - d7 ) + ( d - d7 ) ) / ( ( d - d7 ) + ( d - d7 ) ) ) <= s2 ;
assume that s <> t and s is Point of Sphere ( x , r ) and s is not Point of Sphere ( x , r ) and ex e being Point of E st { e } = Sphere ( s , t ) /\ Sphere ( x , r ) and e in Sphere ( s , t ) and e in Sphere ( x , r ) /\ Sphere ( x , r ) ;
given r such that 0 < r and for s holds 0 < s and for x1 , x2 being Point of CNS st x1 in dom f & x2 in dom f & ||. x1 - x2 .|| < s & |. f /. x1 - f /. x2 .| < r holds |. f /. x1 - f /. x2 .| < r / 2 ;
( p | x ) | ( p | ( ( x | x ) | ( x | x ) ) ) = ( ( ( x | x ) | ( x | x ) ) | ( ( x | x ) | p ) ) | ( ( ( x | x ) | p ) ) ) ;
assume that x , x + h / 2 in dom sec and ( for x st x in dom sec holds sin . x = ( 4 * ( sin . x + h / 2 ) ) * sin . x + cos . ( x + h / 2 ) * sin . x ) ^2 and sin | A is continuous ;
assume that i in dom A and len A > 1 and for B being non empty Subset of the carrier of K st B c= A & B c= the carrier of ( len A ) & A = ( len A ) \ { i } holds A is non empty or A is non empty or A is non empty & A is non empty or A is non empty or A is non empty or A is non empty & A is non empty or A is non empty & A is non empty & B is non empty & A is non empty & A is non empty or A is non empty & A is non empty & A is non empty or A is non empty & A is non empty or A is non empty & A is non empty & A is non empty & A is non empty & A is non empty or A is non empty or A is non empty or A is non empty or A is non empty or B is non empty or A is non empty or
for i be non zero Element of NAT st i in Seg n holds i divides n or i = <* 1. F_Complex *> or i <> 0. F_Complex & for i be Element of NAT st i in Seg n holds h . i = <* 1. F_Complex , 1. F_Complex *> & h . i = 1. F_Complex & h . i = 1. F_Complex \ { h . i }
( ( b1 'imp' b2 ) '&' ( c1 'imp' c2 ) ) '&' ( ( a1 'or' b1 ) '&' ( b1 'or' c1 ) ) '&' ( ( a1 'or' b1 ) '&' ( b1 'or' c1 ) ) '&' ( ( b1 'or' c1 ) '&' ( b1 'or' c1 ) '&' 'not' ( b2 '&' c2 ) ) '&' 'not' ( a1 '&' b1 ) '&' 'not' ( b1 '&' c1 ) '&' 'not' ( b1 '&' c1 ) '&' 'not' ( b2 '&' c2 ) ) '&' 'not' ( b1 '&' b2 )
assume that for x holds f . x = ( ( cot * ( ( cot * ( f - h ) ) ) ) `| Z ) . x and for x st x in Z holds ( ( ( cot * ( ( f - h ) / ( 2 * ( f - h ) ) ) ) `| Z ) . x = sin . ( x- / ( 2 * ( f - h ) ) ) ^2 ) ) ;
consider R8 , I-8 being Real such that R8 = Integral ( M , Re ( F . n ) ) and Integral ( M , Im ( F . n ) ) = Integral ( M , Im ( F . n ) ) and Integral ( M , Im ( F . n ) ) = Integral ( M , Im ( F . n ) ) + Integral ( M , Im ( F . n ) ) ) ;
ex k being Element of NAT st k0 = k & 0 < d & for q being Element of product G st q in X & ||. q- 1 .|| < d holds ||. partdiff ( f , q , k ) . q - partdiff ( f , x , k ) . p .|| < r
x in { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 } ;
G * ( j , i ) `2 = G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 ;
f1 * p = p .= ( ( the Arity of S1 ) +* ( the Arity of S2 ) ) . o .= ( the Arity of S1 ) . ( ( the Arity of S2 ) . o ) .= ( the Arity of S1 ) . ( ( the Arity of S2 ) . o ) .= ( the Arity of S1 ) . ( ( the Arity of S2 ) . o ) .= ( the Arity of S1 ) . ( ( the Arity of S1 ) . o ) .= ( the Arity of S1 ) . ( ( the Arity of S1 ) . ( ( the Arity of S1 ) . ( ( the Arity of S1 ) . ( ( the Arity of S1 ) . ( ( the Arity of S1 ) . ( ( the Arity of S1 ) . ( ( the Arity of S1 ) . ( ( the Arity of S1 ) . ( ( the Arity of S1 ) . ( ( the Arity of S1 ) . ( ( ( the Arity of S1 ) . ( ( the
func tree ( T , P , T1 ) -> DecoratedTree means : : : q in it iff q in T & for p st p in P holds p in T & q in T1 or ex r st p in P & r in T1 & p = r ^ q ;
F /. ( k + 1 ) = F . ( k + 1 - 1 ) .= Fq . ( k + 1 - 1 ) .= Fq . ( k + 1 - 1 ) .= Fq . ( k + 1 - 1 ) .= Fq . ( k + 1 - 1 ) .= Fq . ( k + 1 - 1 ) .= Fq . ( k + 1 - 1 ) .= Fq . ( k + 1 ) ;
for A , B , C being Matrix of K st len B = len C & width B = width C & len B = width C & len A > 0 & len B > 0 & len C > 0 & len A > 0 & len B > 0 & len A > 0 & width B > 0 & width A > 0 & width B = 0 holds A * ( B * C ) = B * ( B * C )
seq . ( k + 1 ) = 0. F_Complex + seq . ( k + 1 ) .= ( Partial_Sums seq ) . ( k + 1 ) + seq . ( k + 1 ) .= ( Partial_Sums seq ) . ( k + 1 ) + seq . ( k + 1 ) .= ( Partial_Sums seq ) . ( k + 1 ) + seq . ( k + 1 ) .= ( Partial_Sums seq ) . ( k + 1 ) + seq . ( k + 1 ) ;
assume that x in ( the carrier of CQ ) and y in ( the carrier of CQ ) and [ x , y ] in ( the carrier of CQ ) and [ y , x ] in ( the carrier of CQ ) and [ y , x ] in the carrier of CQ and [ y , x ] in the carrier of CQ ;
defpred P [ Element of NAT ] means for f st len f = $1 holds ( VAL g ) . ( k + 1 ) = ( VAL g ) . ( k + 1 ) & ( VAL g ) . ( k + 1 ) = ( VAL g ) . ( k + 1 ) & ( VAL g ) . ( k + 1 ) = ( VAL g ) . ( k + 1 ) ;
assume that 1 <= k and k + 1 <= len f and f is_sequence_on G and [ i , j ] in Indices G and [ i + 1 , j ] in Indices G and f /. k = G * ( i , j ) and f /. k = G * ( i , j ) and f /. ( k + 1 ) = G * ( i , j ) ;
assume that cn < 1 and ( q `1 / |. q .| - cn ) > 0 and ( q `2 / |. q .| - cn ) >= 0 or ( p `1 / |. q .| - cn ) >= 0 & ( p `1 / |. q .| - cn ) <= ( q `1 / |. q .| - cn ) and ( p `2 / |. p .| - cn ) <= ( q `1 / |. q .| - cn ) ;
for M being non empty TopSpace , x being Point of M , f being Point of M st x = x `1 holds ex f being sequence of M st for n being Element of NAT holds f . n = Ball ( x `1 , ( 1 - r ) * ( n + 1 ) ) & f . n = Ball ( x `1 , ( 1 - r ) * ( 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 . x ) - ( f2 . x ) ) / ( ( f1 . x ) ^2 ) & ( ( f1 - f2 ) `| Z ) . x = ( f1 . x ) - ( f2 . x ) ) / ( ( f1 . x ) ^2 ) ;
defpred P1 [ Nat , Point of CNS ] means $1 in Y & ||. $2 - $1 .|| < r & ||. ( f /. $1 ) - ( f /. ( $1 + 1 ) ) .|| < r & ||. ( f /. $1 ) - ( f /. ( $1 + 1 ) ) .|| < r & ||. ( f /. $1 ) - ( f /. ( $1 + 1 ) ) .|| < r ;
( f ^ mid ( g , 2 , len g ) ) . i = ( mid ( g , 2 , len g ) ) . i .= ( mid ( g , 2 , len g ) ) . ( i - len f + 1 ) .= g . ( i - len f + 1 ) .= g . ( i - len f + 1 ) .= g . ( i - len f + 1 ) ;
( 1 / 2 * n0 + 2 * n0 ) * ( 2 * n0 + 2 * n0 ) = ( 1 / 2 * n0 + 2 * n0 ) * ( 2 * n0 + 1 ) * ( 2 * n0 + 1 ) .= ( 1 / 2 * n0 + 1 ) * ( 2 * n0 + 1 ) * ( 2 * n0 + 1 ) .= ( 1 / 2 ) * ( 2 * n0 + 1 ) * ( 2 * n0 + 1 ) ;
defpred P [ Nat ] means for G being non empty strict finite symmetric RelStr for H being strict symmetric RelStr st G is space for x being Element of G st x is space & the carrier of G in H & the carrier of G in H holds the carrier of G = { the carrier of H } & the carrier of G in H & the carrier of G in H holds x in the carrier of H ;
assume that not f /. 1 in Ball ( u , r ) and 1 <= m & m <= len - ( f /. i ) and for i st 1 <= i & i <= len f & LSeg ( f , i ) /\ Ball ( u , r ) <> {} holds not f /. i in Ball ( u , r ) and not f /. i in Ball ( u , r ) ;
defpred P [ Element of NAT ] means ( Partial_Sums ( ( cos * ( ( cos - r ) / ( $1 + 1 ) ) ) ) ) . ( 2 * ( ( cos - r ) / ( $1 + 1 ) ) ) = ( Partial_Sums ( ( cos * ( ( cos - r ) / ( $1 + 1 ) ) ) ) ) . ( 2 * ( ( cos - r ) / ( $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 dom ( the Sorts of F ) holds x . i = ( ( the Sorts of F ) * ( the Arity of S ) ) . i ) & for i being set st i in dom ( the Sorts of F ) holds x . i = ( ( the Sorts of F ) * ( the Arity of S ) ) . i
( x " ) |^ ( n + 1 ) = ( ( x " ) |^ n ) * x " .= ( x |^ n ) " * x .= ( x |^ n ) " * x .= ( x |^ n ) " * x .= ( x |^ n ) " * x .= ( x |^ n ) " * x .= ( x |^ n ) " * x .= ( x |^ n ) " * x ;
DataPart Comput ( P +* ( I , P +* I , Initialized s ) , LifeSpan ( P +* I , Initialized s ) + 3 ) = DataPart Comput ( P +* I , Initialize s , LifeSpan ( P +* I , Initialized s ) + 3 ) .= DataPart Comput ( P +* I , Initialize s , LifeSpan ( P +* I , Initialized s ) ) ;
given r such that 0 < r and ]. x0 - r , x0 .[ c= ( dom f1 /\ dom f2 ) and for g st g in ]. x0 - r , x0 .[ holds f1 . g <= f1 . g & for g st g in ]. x0 - r , x0 .[ holds f1 . g <= f2 . g & f2 . g <= f1 . g ;
assume that X c= dom f1 /\ dom f2 and f1 | X is continuous and f2 | X is continuous and for r st r in X /\ dom ( f1 + f2 ) & r in X /\ dom ( f1 + f2 ) holds ( f1 + f2 ) | X is continuous & ( f1 + f2 ) | X is continuous & ( f1 + f2 ) | X is continuous & ( f1 + f2 ) | X is continuous & ( f1 + f2 ) | X is continuous ) ;
for L being continuous complete LATTICE for l being Element of L ex X being Subset of L st l = sup X & for x being Element of L st x in X holds x is compact & for x being Element of L st x in X holds x is compact & x is compact & for l being Element of L st l in ( waybelow l ) holds l . l is compact
Support e8 in { Support ( m *' p ) where m is Polynomial of n , L : ex i being Element of NAT , p being Polynomial of n , L st i in Support ( m *' p ) & p . i = ( m *' p ) . i & p . i = ( m *' q ) . i } ;
( f1 - f2 ) /. ( lim s1 ) = lim ( f1 /* s1 ) - lim ( 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 CQC-WFF ( Al ( ) ) st p1 = g `1 & for p being Function of [: CQC-WFF ( Al ( ) ) , D ( ) :] , D ( ) st P [ p , ( len p ) qua Nat ] holds P [ p , p1 , p . ( len p ) ] ;
( mid ( f , i , len f -' 1 ) ^ <* f /. j *> ) /. j = ( mid ( f , i , len f -' 1 ) ) /. j .= ( mid ( f , i , len f -' 1 ) ) /. j .= ( mid ( f , i , len f -' 1 ) ) /. j .= ( mid ( f , i , len f -' 1 ) ) . j .= ( mid ( f , i , len f -' 1 ) ) /. j ;
( ( p ^ q ) ^ r ) . ( len p + k ) = ( ( p ^ q ) ^ r ) . ( len p + k ) .= ( ( p ^ q ) ^ r ) . ( len p + k ) .= ( ( p ^ q ) ^ r ) . ( len p + k ) .= ( ( p ^ q ) ^ r ) . ( len p + k ) .= ( ( p ^ q ) ^ r ) . ( len p + k ) .= ( ( p ^ r ) . ( len p + k ) .= ( ( p ^ r ) . ( len p + k ) . ( len p + k ) .= ( ( p ^ r ) . ( len p + k ) .= ( ( p ^ r ) . ( k + k ) . ( k + k ) . ( k + k ) .= ( ( p ^ r ) . ( k + k ) . ( k + k ) .= ( ( p ^ r ) . ( k + k ) .= ( ( p ^ r ) . ( k + k ) .= ( ( p ^ r ) . ( k + k )
len mid ( upper_volume ( f , D2 ) , indx ( D2 , D1 , j1 ) + 1 ) = indx ( D2 , D1 , j1 ) + ( indx ( D2 , D1 , j ) + 1 ) - ( indx ( D2 , D1 , j ) + 1 ) .= indx ( D2 , D1 , j ) - ( indx ( D2 , D1 , j ) + 1 ) + 1 ;
x * y * z = MQ . ( ( y * z ) , z9 ) .= ( x * ( y * z ) ) * ( y * z ) .= ( x * ( y * z ) ) * ( y * z ) .= ( x * ( y * z ) ) * ( y * z ) .= ( x * ( y * z ) ) * ( y * z ) ;
v . <* x , y *> + ( <* x0 , y0 *> ) . i * x = partdiff ( v , ( x - x0 ) ) . ( x - x0 ) + ( proj ( 1 , 1 ) . ( x - x0 ) ) + ( proj ( 1 , 1 ) . ( x - x0 ) ) . ( x - x0 ) + ( proj ( 1 , 1 ) . ( x - x0 ) ) ;
i * i = <* 0 * ( - 1 ) - ( 0 * 1 ) - ( 0 * 0 ) , 0 * ( - 1 ) , 0 * ( - 1 ) , 0 * ( - 1 ) , 0 * ( - 1 ) , 0 * ( - 1 ) , 0 * ( - 1 ) + 0 * ( - 1 ) , 0 * ( - 1 ) + 0 * ( - 1 ) , 0 * ( - 1 ) , 0 * ( - 1 ) + 0 * ( - 1 ) + 0 * ( - 1 ) + 0 * ( - 1 ) + 0 * ( - 1 ) + 0 * ( - 1 ) + 0 * ( - 1 ) + 0 * ( - 1 ) + 0 * ( - 1 ) + 0 * ( - 1 ) + 0 * ( - 1 ) + 0 * ( - 1 ) + 0 * ( - 1 * ( - 1 ) + 0 * ( - 1 ) + ( - 1 * ( - 1 ) + 0 * ( - 1 ) + ( - 1 ) + 0 * ( - 1 ) + 0 * ( - 1 ) + 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 ) + Sum ( L (#) F2 ) ) .= Sum ( L (#) F1 ) + Sum ( L (#) F2 ) + Sum ( L (#) F1 ) + Sum ( L (#) F1 ) + Sum ( L (#) F2 ) + Sum ( L (#) F2 ) + Sum ( L (#) F2 ) .= Sum ( L (#) F1 ) + Sum ( ( L (#) F2 ) .= Sum ( L (#) F2 ) + Sum ( ( L (#) F2 ) + Sum ( L (#) F2 ) + Sum ( L (#) F2 ) + Sum ( L (#) F2 ) + Sum ( L (#) F2 ) + Sum ( L (#) F2 ) .= Sum ( L (#) F1 ) + Sum ( L
ex r be Real st for e be Real st 0 < e ex Y0 be finite Subset of X st Y0 is non empty & for Y1 be finite Subset of X st Y0 is non empty & Y1 c= Y & card ( Y1 ) = card Y1 & card ( Y1 ) = card Y1 & card ( Y1 ) = card Y1 & card ( Y1 ) = card Y1
( GoB f ) * ( i , j ) = f /. ( k + 2 ) & ( GoB f ) * ( i , j + 1 ) = f /. ( k + 2 ) or ( GoB f ) * ( i , j + 1 ) = f /. ( k + 1 ) ) & ( GoB f ) * ( i , j + 1 ) = f /. ( k + 2 ) or ( GoB f ) * ( i , j + 1 ) = f /. ( k + 2 ) ;
( ( - cos ) / ( sin . x ) ^2 ) = ( - sin . x ) ^2 / ( cos . x ) ^2 .= ( - sin . x ) ^2 / ( cos . x ) ^2 .= ( - 1 ) * ( sin . x ) ^2 / ( cos . x ) ^2 .= ( - 1 ) * ( sin . x ) ^2 / ( cos . x ) ^2 .= ( - 1 ) * ( sin . x ) ^2 / ( cos . x ) ^2 ;
( - b + sqrt ( x0 - b , c ) / ( 2 * a ) ) / ( 2 * a ) < 0 & ( - b + sqrt ( x0 , b ) / ( 2 * a ) ) / ( 2 * a ) < 0 or ( - b + sqrt ( x0 , c ) / ( 2 * a ) ) / ( 2 * a ) < 0
assume that ex_inf_of uparrow "\/" ( X , C ) , L and ex_sup_of X , L and ex_sup_of X , L and "\/" ( X , L ) = "/\" ( ( uparrow "\/" ( X , L ) ) , L ) and "\/" ( X , L ) = "/\" ( ( uparrow "\/" ( X , L ) ) , L ) and for x being Element of L st x in ( uparrow ( X , L ) ) holds x < "\/" ( X , L ) ;
( ( the Sorts of B ) . j ) . ( j , i ) = ( j = j ) |-- ( id the Sorts of B , ( j , j ) ) & ( j = j implies ( j = i implies j = i ) ) & ( j = i implies j = i ) implies j = i )