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contradiction ;
contradiction ;
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thesis ;
thesis ;
contradiction ;
contradiction ;
contradiction ;
contradiction ;
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thesis ;
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contradiction ;
contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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thesis ;
contradiction ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
B ;
a <> c
T c= S
D c= B
c in X ;
b in X ;
X ;
b in D ;
x = e ;
let m ;
h is onto ;
N in K ;
let i ;
j = 1 ;
x = u ;
let n ;
let k ;
y in A ;
let x ;
let x ;
m c= y ;
F is one-to-one ;
let q ;
m = 1 ;
1 < k ;
G is prime ;
b in A ;
d divides a ;
i < n ;
s <= b ;
b in B ;
let r ;
B is one-to-one ;
R is total ;
x = 2 ;
d in D ;
let c ;
let c ;
b = Y ;
0 < k ;
let b ;
let n ;
r <= b ;
x in X ;
i >= 8 ;
let n ;
let n ;
y in f ;
let n ;
1 < j ;
a in L ;
C is boundary ;
a in A ;
1 < x ;
S is finite ;
u in I ;
z << z ;
x in V ;
r < t ;
let t ;
x c= y ;
a <= b ;
m in NAT ;
assume f is prime ;
not x in Y ;
z = +infty ;
k be Nat ;
K is being_line ;
assume n >= N ;
assume n >= N ;
assume X is \bf ;
assume x in I ;
q is as as Nat ;
assume c in x ;
z > 0 ;
assume x in Z ;
assume x in Z ;
1 <= kr2 ;
assume m <= i ;
assume G is prime ;
assume a divides b ;
assume P is closed ;
b-a > 0 ;
assume q in A ;
W is not bounded ;
f is means : Def1 : 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 + 5 ;
let X be set ;
let X be set ;
let y be element ;
let x be element ;
P [ 0 ]
let E be set , A be non empty Subset of E ;
let C be category ;
let x be element ;
k be Nat ;
let x be element ;
let x be element ;
let e be element ;
let x be element ;
P [ 0 ]
let c be element ;
let y be element ;
let x be element ;
let a be Real ;
let x be element ;
let X be element ;
P [ 0 ]
let x be element ;
let x be element ;
let y be element ;
r in REAL ;
let e be element ;
n1 is \setminus ;
Q halts_on s ;
x in that x in that x in that y in \in that y in \in that x ;
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 ;
PM is one-to-one ;
n <= n + 2 ;
1 <= k + 1 ;
1 <= k + 1 ;
let e be Real ;
i < i + 1 ;
p3 in P ;
p1 in K ;
y in C1 ;
k + 1 <= n ;
let a be Real , x be Point of TOP-REAL 2 ;
X |- r => p ;
x in { A } ;
let n be Nat ;
let k be Nat ;
let k be Nat ;
let m be Nat ;
0 < 0 + k ;
f is_differentiable_in x ;
let x0 , r ;
let E be Ordinal ;
o : o >= 4 ;
O <> O2 ;
let r be Real ;
let f be FinSeq-Location ;
let i be Nat ;
let n be Nat ;
Cl A = A ;
L c= Cl L ;
A /\ M = B ;
let V be RealUnitarySpace , W be Subspace of V ;
not s in Y to_power 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 , W be Subspace of V ;
P [ 1 ] ;
P [ {} ] ;
C1 is component ;
H = G . i ;
1 <= i `2 + 1 ;
F . m in A ;
f . o = o ;
P [ 0 ] ;
a\rrangle <= being Real ;
R [ 0 ] ;
b in f .: X ;
assume q = q2 ;
x in [#] V ;
f . u = 0 ;
assume e1 > 0 ;
let V be RealUnitarySpace , W be Subspace of V ;
s is trivial & s is non empty ;
dom c = Q
P [ 0 ] ;
f . n in T ;
N . j in S ;
let T be complete LATTICE , x be Element of T ;
the ObjectMap of F is one-to-one
sgn x = 1 ;
k in support a ;
1 in Seg 1 ;
rng f = X ;
len T in X ;
vbeing < n ;
S\HM is bounded ;
assume p = p2 ;
len f = n ;
assume x in P1 ;
i in dom q ;
let U2 , U1 , U2 ;
pp = 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 `1 ;
1 <= ( j + 1 ) ;
set A = \it number ;
card a [= c ;
e in rng f ;
cluster B \oplus A -> empty ;
H has \rrangle has { \HM { the } \HM { as } \HM { 0. } } ;
assume n0 <= m ;
T is increasing ;
e2 <> e1 & e2 <> e2 ;
Z c= dom g ;
dom p = X ;
H is proper ;
i + 1 <= n ;
v <> 0. V ;
A c= Affin A ;
S c= dom F ;
m in dom f ;
let X0 be set ;
c = sup N ;
R is connected implies union M is connected
assume not x in REAL ;
Im f is complete ;
x in Int y ;
dom F = M ;
a in On W ;
assume e in A ( ) ;
C c= C-26 ;
mm <> {} & mm <> {} ;
let x be Element of Y ;
let f be ) ;
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 `2 ;
- y in I ;
let A be non empty set , x be Element of A ;
Px0 = 1 ;
assume r in F . k ;
assume f is simple ;
let A be l countable set ;
rng f c= NAT * ;
assume P [ k ] ;
f6 <> {} ;
let o be Ordinal ;
assume x is sum of lim ] ;
assume not v in { 1 } ;
let II , J ;
assume that 1 <= j and j < l ;
v = - u ;
assume s . b > 0 ;
d2 in dom f ;
assume t . 1 in A ;
let Y be non empty TopSpace , x be Point 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 <> {} & mm <> {} ;
M + N c= M + M ;
assume M is \mathclose hh/. ;
assume f is additive for bb-nst ;
let x , y be element ;
let T be non empty TopSpace ;
b , a // b , c ;
k in dom Sum p ;
let v be Element of V ;
[ x , y ] in T ;
assume len p = 0 ;
assume C in rng f ;
k1 = k2 & k2 = k2 ;
m + 1 < n + 1 ;
s in S \/ { s } ;
n + i >= n + 1 ;
assume Re y = 0 ;
k1 <= j1 & k1 <= len f ;
f | A is non empty ;
f . x being Z <= b ;
assume y in dom h ;
x * y in B1 ;
set X = Seg n ;
1 <= i2 + 1 ;
k + 0 <= k + 1 ;
p ^ q = p ;
j |^ ( y divides m ) ;
set m = max A ;
[ x , x ] in R ;
assume x in succ 0 ;
a in sup phi ;
Cv in X ;
q2 c= C1 & q2 c= C2 ;
a2 < c2 & a2 < c1 ;
s2 is 0 -started State of SCMPDS ;
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 { F } ;
y " <> 0 & y " <> 0 ;
y " <> 0 & y " <> 0 ;
0. V = u-w ;
y2 , y are_not zero ;
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 , m be Morphism of C ;
r '&' q = P \lbrack l \rbrack ;
let i , j be Nat ;
let s be State of A , n be Nat ;
s4 . n = N ;
set y = ( x `1 ) ^2 ;
mi in dom g & mi in dom g ;
l . 2 = y1 ;
|. g . y .| <= r ;
f . x in Cx0 ;
V1 is non empty & V2 is non empty ;
let x be Element of X ;
0 <> f . g2 ;
f2 /* q is convergent & 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 that A0 is dense and A is dense ;
|. f . x .| <= r ;
let x be Element of R ;
let b be Element of L ;
assume x in W-19 ;
P [ k , a ] ;
let X be Subset of L ;
let b be Object of B ;
let A , B be category ;
set X = Vars ( C ) ;
let o be OperSymbol of S ;
let R be connected non empty Poset ;
n + 1 = succ n ;
x9 c= Z1 & x9 c= Z1 ;
dom f = C1 & dom g = C2 ;
assume [ a , y ] in X ;
Re ( seq ) is convergent & lim ( Im seq ) = 0 ;
assume a1 = b1 & a2 = b2 ;
A = sInt ( A ) ;
a <= b or b <= a ;
n + 1 in dom f ;
let F be Instruction of S , i be Nat ;
assume r2 > x0 & x0 < r2 ;
let Y be non empty set , x be Element of Y ;
2 * x in dom W ;
m in dom ( g2 | m ) ;
n in dom g1 & m in dom g2 ;
k + 1 in dom f ;
the still of not s in { s } ;
assume that x1 <> x2 and x2 <> x3 ;
v1 in [: V1 , V2 :] ;
not [ b `1 , b ] in T ;
i-35 + 1 = i ;
T c= and T c= G ;
( l `1 ) ^2 = 0 ;
let n be Nat ;
( t `2 ) ^2 = r ;
AA is_integrable_on M & AA is integrable ;
set t = Top t ;
let A , B be real-membered set ;
k <= len G + 1 ;
[: C ( ) , C ( ) :] misses [: C ( ) , C ( ) :] ;
product ( s ) is non empty ;
e <= f or f <= e ;
cluster non empty normal for sequence ;
assume c2 = b2 & c2 = b3 ;
assume h in [. q , p .] ;
1 + 1 <= len C ;
not c in B . m1 ;
cluster R .: X -> empty ;
p . n = H . n ;
assume that v-4 is convergent and lim vseq = 0 ;
IC s3 = 0 & IC s3 = 0 ;
k in N or k in K ;
F1 \/ F2 c= F \/ G ;
Int ( G1 \/ G2 ) <> {} ;
( z `2 ) ^2 = 0 ;
p01 `1 <> p1 `1 & p01 `2 <> p1 `2 ;
assume z in { y , w } ;
MaxADSet ( a ) c= F ;
ex_sup_of downarrow s , S ;
f . x <= f . y ;
let T be up-complete non empty reflexive transitive RelStr ;
q |^ m >= 1 ;
a is_>=_than X & b is_>=_than Y ;
assume <* a , c *> <> {} ;
F . c = g . c ;
G is one-to-one one-to-one full full ;
A \/ { a } \not c= B ;
0. V = 0. Y ;
let I be halting Instruction of S , S ;
f-24 . x = 1 ;
assume z \ x = 0. X ;
C4 = 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 x1 in [: the carrier of S , the carrier of T :] ;
p1 = K1 & p2 = K1 & p3 = K1 ;
M . k = <*> ( the carrier of V ) ;
phi . 0 in rng phi ;
OSMInt A is 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 ;
[: the carrier of R , the carrier of R :] is stable ;
set cR = Vertices R ;
p0 c= P3 & p2 c= P3 ;
x in [. 0 , 1 .[ ;
f . y in dom F ;
let T be Scott Scott Scott Scott Scott of S ;
ex_inf_of the carrier of S , S ;
\HM { a } = the carrier of b ;
P , C , K is_collinear ;
assume x in F ( s , r , t ) ;
2 to_power i < 2 to_power m ;
x + z = x + z + q ;
x \ ( a \ x ) = x ;
||. x-y .|| <= r ;
assume that Y c= field Q and Y <> {} ;
a ~ , b ~ \rrangle , b ~ are_isomorphic ;
assume a in A ( i ) ;
k in dom ( q | Seg n ) ;
p is non empty \HM { finite set } ;
i -' 1 = i-1 - 1 ;
f | A is one-to-one ;
assume x in f .: [: X ( ) , Y ( ) :] ;
i2 - i1 = 0 & i2 = 0 ;
j2 + 1 <= i2 & j2 + 1 <= j2 ;
g " * a in N ;
K <> { [ {} , {} ] } ;
cluster strict 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 Element of NAT ;
let R be ManySortedSet of A ;
let n be Element of NAT ;
let S be non empty non void non void non empty non void non empty non void ManySortedSign ;
let f be ManySortedSet of I ;
let z be Element of COMPLEX , x be Element of COMPLEX ;
u in { ag } ;
2 * n < ( 2 * n ) ;
let x , y be set ;
B-11 c= [: V1 , V1 :] ;
assume I is_closed_on s , P ;
U2 = U2 & U2 = U2 implies U2 is closed
M /. 1 = z /. 1 ;
x9 = x9 & y9 = y9 ;
i + 1 < n + 1 + 1 ;
x in { {} , <* 0 *> } ;
( f | n ) <= ( f | n ) ;
let l be Element of L ;
x in dom ( ( F . n ) | X ) ;
let i be Element of NAT , x be Element of NAT ;
r8 is COMPLEX -valued & r8 is ( len r ) -element ;
assume <* o2 , o *> <> {} ;
s . x |^ 0 = 1 ;
card K1 in M & card K1 in N ;
assume that X in U and Y in U ;
let D be \frac of Omega ;
set r = Seg ( k + 1 ) ;
y = W . ( 2 * x ) ;
assume dom g = cod f & cod g = cod f ;
let X , Y be non empty TopSpace , x be Point of X ;
x \oplus A is interval ;
|. <*> A .| . a = 0 ;
cluster strict for Sublattice of L ;
a1 in B . s1 & a2 in B . s2 ;
let V be finite field of F , W be Subspace of V ;
A * B on B & A on C ;
f-3 = NAT --> 0 ;
let A , B be Subset of V ;
z1 = P1 . j & z2 = P2 . j ;
assume f " P is closed & f " P is closed ;
reconsider j = i as Element of M ;
let a , b be Element of L ;
assume q in A \/ ( B "\/" C ) ;
dom ( F * C ) = o ;
set S = INT |^ X ;
z in dom ( A --> y ) ;
P [ y , h . y ] ;
{ x0 } c= dom f & { x0 } c= dom f ;
let B be non-empty ManySortedSet of I , A be ManySortedSet of I ;
( PI / 2 ) * PI < Arg z ;
reconsider z9 = 0 , z9 = 1 as Nat ;
LIN a , d , c & LIN a , d , c ;
[ y , x ] in [: I , I :] ;
( Q ) * ( 1 , 3 ) = 0 ;
set j = x0 div m , i = x0 mod m ;
assume a in { x , y , c } ;
j2 - ( j - 1 ) > 0 ;
I = I as 1 -element Element of S ;
[ y , d ] in [: F-8 , F-8 :] ;
let f be Function of X , Y ;
set A2 = ( B \/ C ) /\ ( C \/ D ) ;
s1 , s2 are_) & s2 , s1 are_) implies s1 , s2 are_T
j1 -' 1 = 0 & j2 -' 1 = j2 ;
set m2 = 2 * n + j ;
reconsider t = t as bag of n ;
I2 . j = m . j ;
i |^ s , n are_relative_prime & i |^ n , m are_relative_prime ;
set g = f | D-21 , h = g | D-21 ;
assume that X is lower and 0 <= r ;
( p1 `1 ) ^2 = 1 ^2 & ( p1 `2 ) ^2 = 1 ^2 ;
a < p3 `1 & p3 `1 < a ;
L \ { m } c= UBD C ;
x in Ball ( x , 10 ) ;
not a in LSeg ( c , m ) ;
1 <= i1 -' 1 & i1 + 1 <= i2 ;
1 <= i1 -' 1 & i1 + 1 <= i2 ;
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 *> = 2 ;
set H = h . g9 , I = h . x , J = h . y , M = h . z , N = h . z , N = h . x , S
card b * a = |. a .| ;
Shift ( w , 0 ) |= v ;
set h = h2 (*) h1 , h1 = h2 (*) h2 ;
assume x in ( X3 /\ X2 ) ;
||. h .|| < d1 & ||. h .|| < s ;
not x in the carrier of f & not x in the carrier of g ;
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 + q be \rm \rm \HM { of s } ;
Q /\ M c= union ( F | M )
f = b * ( canFS ( S ) ) ;
let a , b be Element of G ;
f .: X is_<=_than f . sup X
let L be non empty transitive reflexive RelStr , x be Element of L ;
S-20 is x -f1 -basis i
let r be non positive Real ;
M , v |= x \hbox { y } ;
v + w = 0. ( Z ) ;
P [ len ( F | n ) ] ;
assume InsCode ( i ) = 8 & InsCode ( i ) = 8 ;
the zero of M = 0 & the zero of M = 0 ;
cluster z * seq -> summable for Real_Sequence ;
let O be Subset of the carrier of C ;
||. f .|| | X is continuous ;
x2 = g . ( j + 1 ) ;
cluster -> [#] S -> [#] < ^2 ;
reconsider l1 = l-1 as Nat ;
v4 is Vertex of r2 & v4 is Vertex of G ;
T2 is SubSpace of T2 & T2 is SubSpace of T2 ;
Q1 /\ Q19 <> {} & Q1 /\ Q19 <> {} ;
k be Nat ;
q " is Element of X & q is Element of X ;
F . t is set of empty & F . t is J ;
assume that n <> 0 and n <> 1 ;
set en = EmptyBag n , en = EmptyBag n ;
let b be Element of Bags n ;
assume for i holds b . i is commutative ;
x is root of ( p `2 ) ^2 & y is root ;
not r in ]. p , q .[ ;
let R be FinSequence of REAL , x be Element of REAL ;
S7 does not destroy b1 & S7 does not destroy b1 ;
IC SCM R <> a & IC SCM R <> a ;
|. - |[ x , y ]| .| >= r ;
1 * ( seq . n ) = seq . n ;
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 ;
H + G = F- ( GG ) ;
Cx1 . x = x2 & Cy1 . x = y2 ;
f1 = f .= f2 .= ( f | X ) ;
Sum <* p . 0 *> = p . 0 ;
assume v + W = v + u + W ;
{ a1 } = { a2 } & { a2 } = { a2 } ;
a1 , b1 _|_ b , a & b1 , c1 _|_ a , b ;
d3 , o _|_ o , a3 & d3 , o _|_ a3 , o ;
I is reflexive & I is transitive implies I is transitive
I is antisymmetric & I is antisymmetric implies I is antisymmetric
upper_bound rng ( H1 | n ) = e & upper_bound rng ( H1 | n ) = e ;
x = ( a * ( - 1 ) ) * ( - 1 ) ;
|. p1 .| ^2 >= 1 ^2 ;
assume j2 -' 1 < 1 & j2 + 1 < j2 ;
rng s c= dom f1 & rng s c= dom f2 ;
assume support a misses support b & support b misses support a ;
let L be associative commutative non empty doubleLoopStr , x be Element of L ;
s " + 0 < n + 1 ;
p . c = ( f " ) . 1 ;
R . n <= R . ( n + 1 ) ;
Directed ( I1 , I2 ) = I1 & Directed ( I2 , I2 ) = I2 ;
set f = + ( x , y , r ) ;
cluster Ball ( x , r ) -> bounded ;
consider r being Real such that r in A ;
cluster non empty NAT -defined NAT -defined Function ;
let X be non empty directed Subset of S ;
let S be non empty full SubRelStr of L ;
cluster <* [ ] , ] *> -> complete non trivial ;
( 1 - a ) " = a ;
( q . {} ) `1 = o ;
- ( i -' 1 ) > 0 ;
assume ( 1 / 2 ) <= t `1 & t `2 <= 1 ;
card B = k + - 1 ;
x in union rng ( f | ( len f ) ) ;
assume x in the carrier of R & y in the carrier of R ;
d in D ;
f . 1 = L . ( F . 1 ) ;
the vertices of G = { v } & the vertices of G = { v } ;
let G be : wwgraph ;
let e , v9 be set , x be set ;
c . ( i9 + 1 ) in rng c & c . ( i9 + 1 ) in rng c ;
f2 /* q is divergent_to+infty & f2 /* q is divergent_to+infty ;
set z1 = - z2 , z2 = - z1 , z2 = - z2 ;
assume w is llas of S , G ;
set f = p |-count ( t - p ) , g = p |-count ( t - p ) , h = p |-count ( t - p ) , n = p |-count ( t - p ) , n
let c be Object of C ;
assume ex a st P [ a ] ;
let x be Element of REAL m , y be Element of REAL m ;
let IB be Subset-Family of X , I be set ;
reconsider p = p , q = q as Element of NAT ;
let v , w be Point of X ;
let s be State of SCM+FSA , I be Program of SCM+FSA ;
p is finite implies p is non empty & p is non empty
stop I ( ) c= P-12 & stop I ( ) c= P-12 ;
set ci = ( f /. i ) `1 , cj = ( f /. j ) `1 , cj = ( f /. j ) `1 , cj = ( f /. j ) `2 , c
w ^ t Q Q Q Q Q ^ t ;
W1 /\ W = W1 /\ W ` .= W1 /\ W ;
f . j is Element of J . j ;
let x , y be \rm \HM { of T2 } ;
ex d st a , b // b , d ;
a <> 0 & b <> 0 & c <> 0 ;
ord x = 1 & x is not positive ;
set g2 = lim ( seq , n ) ;
2 * x >= 2 * ( 1 / 2 ) ;
assume ( a 'or' c ) . z <> TRUE ;
f (*) g in Hom ( c , c ) ;
Hom ( c , c + d ) <> {} ;
assume 2 * Sum ( q | m ) > m ;
L1 . F-21 = 0 & L1 . F-21 = 1 ;
/ ( X \/ R1 ) = id X & id ( X \/ R1 ) = id X ;
( sin . x ) ^2 <> 0 & ( sin . x ) ^2 <> 0 ;
( exp_R . x ) ^2 > 0 & exp_R . x > 0 ;
o1 in [: X /\ O2 , X /\ O2 :] ;
let e , v9 be set , x be set ;
r3 > ( 1 - r2 ) * 0 ;
x in P .: ( F -ideal ) ;
let J be closed non empty Subset of R ;
h . p1 = f2 . O & h . O = g2 . I ;
Index ( p , f ) + 1 <= j ;
len ( q | ( len M ) ) = width M ;
the carrier of CK c= A ;
dom f c= union rng ( F-10 . n ) ;
k + 1 in support ( support ( n ) ) ;
let X be ManySortedSet of the carrier of S ;
[ x `1 , y `2 ] in ( \HM { the } \HM { carrier of R ) ;
i = D1 or i = D2 or i = D1 ;
assume a mod n = b mod n & b mod n = 0 ;
h . x2 = g . x1 & h . x2 = h . x2 ;
F c= 2 -tuples_on the carrier of X & F is one-to-one ;
reconsider w = |. s1 .| as Real_Sequence ;
( 1 / m * m + r ) < p ;
dom f = dom ( I . 0 ) & dom g = dom ( I . 0 ) ;
[#] ( P-17 ) = [#] ( ( TOP-REAL 2 ) | K1 ) ;
cluster - x -> ExtReal for ExtReal ;
then { d } c= A & A is closed ;
cluster ( TOP-REAL n ) | ( i + 1 ) -> finite-ind ;
let w1 be Element of M ;
let x be Element of dyadic ( n ) ;
u in W1 & v in W3 & W3 in W3 implies u + v in W3
reconsider y = y , z = z as Element of L2 ;
N is full SubRelStr of T |^ the carrier of S ;
sup { x , y } = c "\/" c ;
g . n = n to_power 1 .= n ;
h . J = EqClass ( u , J ) ;
let seq be summable sequence of X , n be Nat ;
dist ( x `2 , y ) < ( r / 2 ) ;
reconsider mm = m , mn = n as Element of NAT ;
x- x0 < r1 - x0 & x - x0 < r2 - x0 ;
reconsider P = P ` as strict Subgroup of N ;
set g1 = p * idseq ( q `2 ) , g2 = p * idseq ( q `2 ) ;
let n , m , k be non zero Nat ;
assume that 0 < e and f | A is lower ;
D2 . ( I8 ) in { x } ;
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 ;
Gij in LSeg ( cos , 1 ) /\ LSeg ( cos , 1 ) ;
let n be Element of NAT , x be Element of X ;
reconsider ST = S , ST = T as Subset of T ;
dom ( i .--> X `2 ) = { 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 as Element of C ( ) ;
let s be 0 -started State of SCMPDS , n be Nat ;
let t be 0 -started State of SCMPDS , Q be t be State of SCMPDS ;
b , b , x , y be element ;
assume that i = n \/ { n } and j = k \/ { k } ;
let f be PartFunc of X , Y ;
N2 >= ( sqrt ( c ^2 - 1 ) / ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 *
reconsider t7 = T-1 as TopSpace , T7 = T7 as Point of TOP-REAL 2 ;
set q = h * p ^ <* d *> ;
z2 in U . ( y2 ) /\ Q2 . ( y2 ) ;
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 & z in dom g ;
assume p2 = E-max ( K ) & p1 <> 0. TOP-REAL 2 ;
len G + 1 <= i1 + 1 & 1 <= i2 ;
f1 (#) f2 is convergent & f2 /* ( seq + k ) = f2 /* seq ;
cluster ( seq + s ) - ( x - s ) -> summable ;
assume that j in dom M1 and i in Seg n ;
let A , B , C be Subset of X ;
let x , y , z be Point of X , p be Point of X ;
b ^2 - ( 4 * a * c ) >= 0 ;
<* x/ y *> ^ <* y *> Q Q Q Q Q Q Q Q [ x ] ;
a , b in { a , b } ;
len p2 is Element of NAT & len p1 = len p2 ;
ex x being element st x in dom R & y = R . x ;
len q = len ( K (#) G ) & width ( K (#) G ) = width G ;
s1 = Initialize Initialized s , P1 = P +* I , P2 = P +* J , P2 = P +* J , s2 = Comput ( P2 , s2 , 1 ) , P2 = P2 +* J , P3 = P3 ;
consider w being Nat such that q = z + w ;
x ` ` is Element of x & y is Element of X ;
k = 0 & n <> k or k > n ;
then X is discrete for A being Subset of X ;
for x st x in L holds x is FinSequence ;
||. f /. c .|| <= r1 & ||. f /. c .|| <= r ;
c in uparrow p & not c in { p } ;
reconsider V = V as Subset of the is \hbox { $ T } ;
let N , M be \mathbin { H } ;
then z is_>=_than waybelow x & z is_>=_than compactbelow y ;
M | [. f , g .] = f & M | [. g , g .] = g ;
( ( ( ( L ) to_power 1 ) * ( L to_power 1 ) ) to_power 1 ) to_power 1 = TRUE ;
dom g = dom f -tuples_on X & dom g = dom f ;
mode *> is * \rm <= <= len 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 ( Subformulae p ) -tuples_on Subformulae ( f ) ;
F1 . ( a1 , - 1 ) = G1 & F1 . ( a2 , - 1 ) = G2 ;
redefine func being set , a , b , r be Real ;
let a , b , c , d be Real ;
rng s c= dom ( 1 / ( f1 + f2 ) ) ;
curry ( F-19 , k ) is additive ;
set k2 = card ( dom B ) , k1 = card ( dom B ) ;
set G = the Sorts of A ;
reconsider a = [ x , s ] as 0. of G ;
let a , b be Element of MF , x be Element of MF ;
reconsider s1 = s , s2 = t as Element of ( the carrier of S ) ;
rng p c= the carrier of L & rng p c= the carrier of L ;
let d be Subset of the Sorts of A ;
( x .|. x ) = 0 iff x = 0. W ;
I-21 in dom stop I & Ik in dom stop I ;
let g be continuous Function of X | B , Y ;
reconsider D = Y as Subset of ( TOP-REAL n ) | D ;
reconsider i0 = len p1 , i2 = len p2 as Integer ;
dom f = the carrier of S & rng g = the carrier of S ;
rng h c= union ( ( the carrier of J ) --> ( the carrier of L ) )
cluster All ( x , H ) -> non \widetilde LSeg ;
d * N1 / ( 1 - d ) > N1 * 1 / ( 1 - d ) ;
]. a , b .[ c= [. a , b .] ;
set g = f " | D1 , h = f " | D2 ;
dom ( p | ( mm ) ) = ( m + 1 ) -tuples_on NAT ;
3 + - 2 <= k + - 2 & k + - 2 <= k + - 2 ;
tan is_differentiable_in ( arccot * arccot ) . x & tan . x > 0 ;
x in rng ( f /^ ( n -' 1 ) ) ;
let f , g be FinSequence of D ;
[: p , q :] in the carrier of S1 & [: p , q :] in the carrier of S2 ;
rng f " = dom f & rng f = dom f ;
( the Target of G ) . e = v & ( the Target of G ) . e = v ;
width G - 1 < width G - 1 & width G - 1 < width G ;
assume v in rng ( S | E1 ) ;
assume x is root or x is root or x is root ;
assume 0 in rng ( g2 | A ) & 0 < g2 . A ;
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 ) ;
<* S7 *> is_in the carrier of C-20 & <* S7 *> is in the carrier of C-20 ;
i <= len ( G * ( len G -' 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 = Sp2 " ( Q ) .= ( S * R ) " ( Q ) ;
( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( 1 / 2 ) ) ) (#) ( ( 1 / 2 ) (#) ( 1 / 2 ) ) ) is summable ;
- p + I c= - p + A ;
n < LifeSpan ( P1 , s1 ) + 1 & n <= LifeSpan ( P2 , s2 ) ;
CurInstr ( p1 , s1 ) = i .= halt SCM+FSA ;
A /\ Cl { x } \ { x } <> {} ;
rng f c= ]. r , r + 1 .[ ;
let g be Function of S , V ;
let f be Function of L1 , L2 , g be Function of 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 , T :] ;
len ( the connectives of C ) = 4 & len ( the connectives of C ) = 4 ;
let C1 , C2 be subFunctor of C , D ;
reconsider V1 = V as Subset of X | B ;
attr p is valid means : Def1 : All ( x , p ) is valid ;
assume that X c= dom f and f .: X c= dom g and g .: X c= dom f ;
H |^ ( a " ) is Subgroup of H & H |^ a is Subgroup of H ;
let A1 be p1 of O , A2 be Element of E ;
p2 , r3 , q2 is_collinear & p1 , r2 , q1 is_collinear & p2 , q2 , q2 is_collinear ;
consider x being element such that x in v ^ K ;
not x in { 0. TOP-REAL 2 } & not x in { 0. TOP-REAL 2 } ;
p in [#] ( I[01] | B11 ) & p in [#] ( I[01] | B11 ) ;
0 < M . E8 & M . E8 < M . E8 ;
^ ( c , c ) ` = c ;
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . s2 ) & a2 in dom ( F . s2 ) ;
cluster -> 0. for > w -| the carrier of L ;
set i1 = the Nat , i2 = the Nat , i1 = i2 = the Nat , i2 = the Nat , j1 = the Nat , j2 = the Nat ;
let s be 0 -started State of SCM+FSA , I be Program of SCM+FSA ;
assume y in ( f1 \/ f2 ) .: A ;
f . ( len f ) = f /. len f .= f /. 1 ;
x , f . x '||' f . x , f . y ;
pred X c= Y means : Def1 : cos ( X ) c= cos ( Y ) ;
let y be upper Subset of Y , x be Element of Y ;
cluster -> -> -> -> -> -> -> -> -> -> of \rm \leq implies x is l \rm div ;
set S = <* Bags n , ih *> , T = <* i , j *> , S = <* j *> , T = <* i , j *> , S = <* j , k *> , T = <* i , j *> , T
set T = [. 0 , 1 / 2 .] , S = [. 1 / 2 , 1 .] ;
1 in dom mid ( f , 1 , 1 ) ;
( 4 * PI * PI ) / 2 < ( 2 * PI ) / 2 ;
x2 in dom f1 /\ dom f & x2 in dom ( f1 + f2 ) ;
O c= dom I & { {} } = { {} } ;
( the Target of G ) . x = v & ( the Target of G ) . x = v ;
{ HT ( f , T ) } c= Support f ;
reconsider h = R . k as Polynomial of n , L ;
ex b being Element of G st y = b * H ;
let x , y , z be Element of G opp ;
h19 . i = f . ( h . i ) ;
( p `1 ) ^2 = ( p1 `1 ) ^2 + ( p1 `2 ) ^2 ;
i + 1 <= len Cage ( C , n ) ;
len <* P *> = len P & width <* P *> = width P ;
set N-26 = the \subseteq of the seq of N , Ny = the seq of N ;
len g\vert + ( x + 1 ) - 1 <= x ;
a does not on B & b does not on B ;
reconsider rr = r * I . v as FinSequence of REAL ;
consider d such that x = d and a is_less_than d [= c ;
given u such that u in W and x = v + u ;
len f /. ( \downharpoonright n ) = len ^2 n .= n ;
set q2 = \hbox { q2 : q2 `2 <= q1 `2 & q1 `2 <= q2 `2 } ;
set S =
MaxADSet ( b ) c= MaxADSet ( P /\ Q ) ;
Cl ( G . q1 ) c= F . r2 & Cl ( G . q2 ) c= G . q2 ;
f " D meets h " V & h " 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 ) & ( the_right_argument_of H ) '&' ( the_right_argument_of H ) = the_right_argument_of H ;
assume t is Element of ( gF ) . X ;
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 . ( a2 , b2 ) = b2 ;
the carrier' of G ` = E \/ { E } .= { E } ;
reconsider m = len thesis - k as Element of NAT ;
set S1 = LSeg ( n , UMP C ) , S2 = LSeg ( n , UMP C ) , S1 = LSeg ( m , UMP C ) , S2 = LSeg ( n , UMP C ) , S2 = LSeg ( m , UMP C )
[ i , j ] in Indices M1 & [ i , j ] in Indices M1 ;
assume that P c= Seg m and M is \HM { i } is of K ;
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 = pi1 . i & pi2 . i = pi2 . i ;
let PA , PA , G be a_partition of Y , PA be set ;
pred 0 < r & r < 1 means : Def1 : 1 < r & r < 1 ;
rng ( ( the on of 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 ( s ) ;
reconsider x2 = x1 , y2 = x2 as Element of L2 ;
Q in FinMeetCl ( ( the topology of X ) \/ the topology of Y ) ;
dom ( f | u ) c= dom ( f | u ) & dom ( f | u ) c= dom ( f | u ) ;
pred n divides m & m divides n & n = m ;
reconsider x = x , y = y , z = z as Point of I[01] ;
a in dom the
not y0 in the still of f & not y0 in the carrier' of f ;
Hom ( ( a [: b , c :] ) , c ) <> {} ;
consider k1 such that p " < k1 and k1 < p " ;
consider c , d such that dom f = c \ d and dom g = d ;
[ x , y ] in [: dom g , dom k :] ;
set S1 =
l1 = m2 & l1 = l2 & l2 = j2 & l2 = j2 implies l1 = l2
x0 in dom ( u01 ) /\ ( dom ( f | A ) ) ;
reconsider p = x , q = y , r = z as Point of TOP-REAL 2 ;
I[01] = R^1 | B01 & I[01] = ( TOP-REAL 2 ) | B01 ;
f . p4 <= _ P P & f . p1 = f . p2 ;
( ( ( F . x ) `1 ) / ( 1 + x ) ) ^2 <= ( ( F . x ) `2 ) ^2 ;
( x `2 ) ^2 = ( W `2 ) ^2 + ( W `2 ) ^2 ;
for n being Element of NAT holds P [ n ] ;
let J , K be non empty Subset of I ;
assume 1 <= i & i <= len <* a " *> ;
0 |-> a = <*> ( the carrier of K ) & ( 0 -tuples_on the carrier of K ) = ( 0 -tuples_on the carrier of K ) ;
X . i in 2 -tuples_on A . i \ B . i ;
<* 0 *> in dom ( e --> [ 1 , 0 ] ) ;
then P [ a ] & P [ succ a ] & Q [ succ a ] ;
reconsider sp2 = s\frac ( 1 - s ) as \rangle , sp2 = sp2 ;
( - ( i -' 1 ) ) <= len thesis - j ;
[#] S c= [#] T & [#] T c= the TopStruct of T ;
for V being strict RealUnitarySpace holds V in and V is Subspace of W implies V is 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 Matrix of n1 , K , n be Nat ;
- a * - b * - b = a * b - a * b ;
for A being Subset of AS holds A // A & A // A
( for o2 being element st o2 in dom <* o2 , o2 *> holds o2 . ( o1 , o2 ) = F ( o2 , o2 ) ) ;
then ||. x .|| = 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 upper_volume ( g , D2 ) = len D2 ;
b = Q . ( len Q- 1 + 1 ) .= Q . ( len Q- 1 + 1 ) ;
f2 * f1 /* s is divergent_to+infty & f2 * f1 /* s is divergent_to+infty ;
reconsider h = f * g as Function of [: N1 , N2 :] , G ;
assume that a <> 0 and delta ( a , b , c ) >= 0 ;
[ t , t ] in the InternalRel of A & [ t , t ] in the InternalRel of A ;
( v |-- ( E , X ) ) | n is Element of T7 ;
{} = the carrier of L1 + L2 & the carrier of L1 = the carrier of L2 + L2 ;
Directed I is_closed_on Initialized s , P & Directed I is_halting_on Initialized s , P ;
Initialized p = Initialize ( p +* q ) , p2 = p +* I , p3 = p +* I ;
reconsider N2 = N1 , N2 = N2 as strict net of R2 ;
reconsider Y = Y as Element of <* Ids L , \subseteq *> ;
"/\" ( uparrow p , L ) \ { p } <> p ;
consider j being Nat such that i2 = i1 + j and j in dom f ;
not [ s , 0 ] in the carrier of S2 & not [ s , 0 ] in the carrier of S2 ;
mm in ( B '/\' C ) '/\' D \ { {} } ;
n <= len ( P + ( len P ) ) + len ( P + ( len P ) ) ;
( x1 `1 ) ^2 = ( x2 `1 ) ^2 + ( x2 `2 ) ^2 ;
InputVertices S = { x1 , x2 } & InputVertices S = { x1 , x2 } ;
let x , y be Element of FT1 ( n ) ;
p = |[ p `1 / |. p .| - cn ]| ;
g * 1_ G = h " * g * h * h ;
let p , q be Element of thesis of being Element of being Element of being Element of being Element of V ;
x0 in dom x1 /\ dom x2 & x0 in dom x1 /\ dom x2 & x1 . x0 = x2 . x0 ;
( R qua Function ) " = R " * ( R * ( R * ( R * ( R * ( R * ( S , S ) ) ) ) ) ;
n in Seg len ( f /^ ( len f -' 1 ) ) ;
for s being Real st s in R holds s <= s2 implies s <= s2
rng s c= dom ( f2 * f1 ) & rng s c= dom ( f2 * f1 ) ;
synonym for for for for for for X being Subset of \mathop { \rm being Subset of X } ;
1. K * 1. K = 1_ K & 1. K * 1. K = 1_ K ;
set S = Segm ( A , P1 , Q1 ) , T = Segm ( A , P1 , Q1 ) , S = Segm ( P2 , P1 , Q1 ) , T = Segm ( P2 , P1 , Q1 ) , S = Segm ( P2 , P1 , Q1 ) ,
ex w st e = ( w - f ) / 2 & w in F ;
curry ' ( P+* ( P+* ( P+* ( P+* ( P+* ( P+* ( P+* ( P+* ( P+* ( P+* ( P+* ( P+* ( P+* ( P+* ( P+* ( P+* ( P+* ( P+*
cluster open -> open for Subset of T\sigma ( T ) ;
len f1 = 1 .= len f3 .= len f3 + len f3 .= len f3 + 1 ;
( i * p ) / p < ( 2 * p ) / p ;
let x , y be Element of OSSub ( U0 ) ;
b1 , c1 // b9 , c9 & b1 , c1 // b9 , c & b1 , c1 // c , d ;
consider p being element such that c1 . j = { p } ;
assume that f " { 0 } = {} and f is total ;
assume that IC Comput ( F , s , k ) = n and IC Comput ( F , s , k ) = n ;
Reloc ( J , card I + 2 ) does not destroy a ;
goto ( card I + 1 ) does not destroy c ;
set m3 = LifeSpan ( p3 , s3 ) , m3 = LifeSpan ( p3 , s3 ) , P4 = LifeSpan ( p3 , s3 ) , P4 = LifeSpan ( p3 , s3 ) , P4 = LifeSpan ( p3 , s3 ) , D = Comput ( p3 , s3 ,
IC SCMPDS in dom Initialize p & IC SCMPDS in dom Initialize p & IC SCMPDS in dom p ;
dom t = the carrier of SCM R & dom t = the carrier of SCM R ;
( ( N-min L~ f ) .. f ) .. f = 1 & ( ( E-max L~ f ) .. f ) .. f = ( ( E-max L~ f ) .. f ) .. f ;
let a , b be Element of thesis , x be Element of V ;
Cl Int ( union F ) c= Cl Int ( union F ) ;
the carrier of X1 union X2 misses ( ( X1 union X2 ) union ( X1 union X2 ) ) ;
assume not LIN a , f . a , g . a , g . b ;
consider i being Element of M such that i = d6 and i in dom f ;
then Y c= { x } or Y = {} or Y = { x } ;
M , v |= H1 / ( ( y , x ) / ( y , x ) ) ;
consider m being element such that m in Intersect ( FF . m ) and x = Intersect ( FF . m ) ;
reconsider A1 = support u1 , A2 = support v1 as Subset of X ;
card ( A \/ B ) = k-1 + ( 2 * 1 ) ;
assume that a1 <> a3 and a2 <> a4 and a3 <> a4 and a4 <> a4 and a4 <> a5 ;
cluster s | ( s , V ) -> non empty for string of S ;
Ln2 /. n2 = Ln2 . n2 .= Ln2 . n2 .= Ln2 . n2 ;
let P be compact non empty Subset of TOP-REAL 2 , p1 , p2 be Point of TOP-REAL 2 ;
assume that r-7 in LSeg ( p1 , p2 ) and r-7 in LSeg ( p1 , p2 ) ;
let A be non empty compact Subset of TOP-REAL n , x be Point of TOP-REAL n ;
assume [ k , m ] in Indices ( ( D | ( i + 1 ) ) ) ;
0 <= ( ( 1 / 2 ) |^ p ) / ( 2 |^ n ) ;
( F . N | E8 ) . x = +infty ;
pred X c= Y & Z c= V & X \ V c= Y \ Z ;
( y `2 ) * ( z `2 ) <> 0. I & ( y `2 ) * ( z `2 ) <> 0. I ;
1 + card ( X-18 ) <= card u & card ( X-18 ) <= card ( X \/ Y ) ;
set g = z \circlearrowleft ( ( E-max L~ z ) .. z ) ;
then k = 1 & p . k = <* x , y *> . k ;
cluster -> total for Element of C -\mathop { 0 } ;
reconsider B = A as non empty Subset of TOP-REAL n , C be Subset of TOP-REAL n ;
let a , b , c be Function of Y , BOOLEAN , x be Element of Y ;
L1 . i = ( i .--> g ) . i .= g . i ;
Plane ( x1 , x2 , x3 ) c= P & Plane ( x2 , x3 , x4 ) c= P ;
n <= indx ( D2 , D1 , j1 ) + 1 & n <= len D1 ;
( ( g2 ) . O ) `1 = - 1 & ( g2 ) . I = 1 ;
j + p .. f - len f <= len f - len f - len f ;
set W = W-bound C , S = S-bound C , E = E-bound C , N = N-bound C , G = Gauge ( C , n ) , G = Gauge ( C , n ) , G = Gauge ( C , n ) , G = Gauge ( C ,
S1 . ( a `2 , e `2 ) = a + e `2 .= a `2 ;
1 in Seg width ( M * ( ColVec2Mx p ) ) & 1 in Seg width ( M * ( ColVec2Mx p ) ) ;
dom ( i (#) Im ( f ) ) = dom Im ( f ) /\ dom Im ( g ) ;
( ^2 ) = W . ( a , *' ( a , p ) ) ;
set Q = ( -> ( _ of ( g , f , h ) ) . ( x , h ) ) ;
cluster -> MSsorted for ManySortedSet of U1 * -valued Relation of U1 , U2 ;
attr A is discrete means : Def1 : ex F being Subset-Family of A st F = { A } ;
reconsider z9 = \hbox { z } , z9 = z as Element of product \overline 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 ) & g = <*> ( the carrier of F_Complex ) ;
E , j |= All ( x1 , x2 , H ) & E , j |= All ( x2 , H ) ;
reconsider n1 = n , n2 = m , n3 = n as Morphism of o1 , o2 ;
assume that P is idempotent and R is idempotent and P (*) R = R ** P ;
card ( B2 \/ { x } ) = k-1 + 1 ;
card ( ( x \ B1 ) /\ B1 ) = 0 & card ( x \ B1 ) = 0 ;
g + R in { s : g-r < s & s < g + r } ;
set qlim = ( q , <* s *> ) : q in dom f & p in f .: X } ;
for x being element st x in X holds x in rng ( f1 + f2 )
h0 /. ( i + 1 ) = h0 . ( i + 1 ) ;
set mw = max ( B , ( B * A ) ) ;
t in Seg width ( ( I ^ ( n , n ) ) * ( I ^ ( n , n ) ) ) ;
reconsider X = dom f /\ C as Element of Fin NAT ;
IncAddr ( i , k ) = <% - l %> + k .= <% - l %> ;
S-bound L~ f <= q `2 & q `2 <= q `2 & q `2 <= N-bound L~ f ;
attr R is condensed means : Def1 : Int R is condensed & Cl R is condensed ;
pred 0 <= a & a <= 1 & b <= 1 implies a * b <= 1 ;
u in ( ( c /\ ( ( d /\ b ) /\ e ) ) /\ f ) /\ j ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) ) /\ f ) /\ j ;
len C + - 2 >= 9 + - 3 & len C + - 2 >= 0 ;
x , z , y is_collinear & x , z , y is_collinear & x , z , y is_collinear ;
a |^ ( n1 + 1 ) = a |^ n1 * a ;
<* \underbrace ( 0 , \dots , 0 ) *> in Line ( x , a * x ) ;
set y9 = <* 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 & p `2 = ( f /. i1 ) `2 ;
W-bound ( X \/ Y ) = W-bound ( X \/ Y ) & W-bound ( X \/ Y ) = E-bound ( X \/ Y ) ;
0 + ( p `2 ) ^2 <= 2 * r + ( p `2 ) ^2 ;
x in dom g & not x in g " { 0 } ;
f1 /* ( seq ^\ k ) is divergent_to+infty & f2 /* ( seq ^\ k ) is divergent_to+infty ;
reconsider u2 = u , v2 = v as VECTOR of ( X | Y ) ;
p |-count ( Product Sgm ( X11 ) ) = 0 & p |-count ( Product Sgm ( 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 , ii2 = goto 0 , ii2 = goto 0 , ii2 = goto 0 , ii2 = goto 0 , ii2 = goto 0 , ii2 = goto 0
x in { x , y } & h . x = {} ( TF , T ) ;
consider y being Element of F such that y in B and y <= x `2 ;
len S = len ( the charact of ( ( the charact of ( A ) ) * the charact of ( B ) ) ) ;
reconsider m = M , i = I , n = N as Element of X ;
A . ( j + 1 ) = B . ( j + 1 ) \/ A . j ;
set N8 = ( G ) \ ( G-15 ) , N8 = ( G ) \ ( G-15 ) ;
rng F c= the carrier of gr { a } & rng F c= the carrier of gr ( { a } ) ;
P is and P is and ( F , n , r ) -valued implies P is in P
f . k , f . ( mod n ) are_relative_prime in rng f & ( f . m ) mod n in rng f ;
h " P /\ [#] ( T1 | P ) = f " P /\ [#] ( T2 | P ) ;
g in dom f2 \ ( f2 " { 0 } ) & ( f2 " { 0 } ) \ ( f2 " { 0 } ) c= dom f2 ;
g+ X /\ dom f1 = g1 " X & X = dom g1 & Y = g1 " { 0 } ;
consider n being element such that n in NAT and Z = G . n ;
set d1 = being thesis , d2 = dist ( x1 , y1 ) , d2 = dist ( x2 , y2 ) , d2 = dist ( x2 , y2 ) , d2 = dist ( x2 , y2 ) , d2 = dist ( y2 , y2 ) , d2 = dist ( y2 , y2 ) ,
b `2 / ( 1 + ( 1 + 1 ) ) ^2 < ( 1 + ( 1 + 1 ) ) ^2 ;
reconsider f1 = f , g1 = g as VECTOR of the carrier of X ;
pred i <> 0 means : Def1 : i ^2 mod ( i + 1 ) = 1 ;
j2 in Seg ( len ( g2 . i2 ) ) & j2 in Seg ( len g2 ) ;
dom ( i4 * i4 ) = dom ( i4 * i4 ) .= a ;
cluster sec | ]. PI / 2 , PI / 2 .[ -> one-to-one for Function of ]. PI / 2 , PI .[ , REAL ;
Ball ( u , e ) = Ball ( f . p , e ) ;
reconsider x1 = x0 , y1 = y0 as Function of S , T ;
reconsider R1 = x , R2 = y , R1 = z as Relation of L , L ;
consider a , b being Subset of A such that x = [ a , b ] ;
( <* 1 *> ^ p ) ^ <* n *> in Rq ;
S1 +* S2 = S2 +* S1 +* S2 +* S2 & S2 +* S2 = S1 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 ;
( ( exp_R * ( cos + cos ) ) `| Z ) = f ;
cluster -> [. 0 , 1 .] -valued for Function of C , REAL ;
set C7 = 1GateCircStr ( <* z , x *> , f3 ) , C8 = 1GateCircStr ( <* x , y *> , f3 ) ;
E8 . e2 = E8 . e2 .= ( - 2 ) * T . e2 .= ( - 2 ) * T . e2 ;
( ( ( arctan * ( f1 + f2 ) ) `| Z ) = f ;
upper_bound A = ( PI * 3 / 2 ) * 2 & lower_bound A = 0 ;
F . ( dom f , - F . ( cod f ) ) is_transformable_to F . ( cod f , - F . ( cod f ) ) ;
reconsider pbeing Point of TOP-REAL 2 , p8 = q8 as Point of TOP-REAL 2 ;
g . W in [#] Y0 & [#] Y0 c= [#] Y0 & g . W = [#] Y0 ;
let C be compact non vertical non horizontal Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
LSeg ( f ^ g , j ) = LSeg ( f , j ) /\ LSeg ( g , i ) ;
rng s c= dom f /\ ]. -infty , x0 + r .[ & rng s c= dom f /\ ]. x0 , x0 + r .[ ;
assume x in { idseq ( 2 ) , Rev ( idseq 2 ) } ;
reconsider n2 = n , m2 = m , n1 = n , n2 = m as Element of NAT ;
for y being ExtReal st y in rng seq holds g <= y + r
for k st P [ k ] holds P [ k + 1 ] ;
m = m1 + m2 .= m1 + m2 .= m1 + m2 + m2 .= m1 + m2 + m2 ;
assume for n holds H1 . n = G . n -H . n ;
set B" = f .: ( the carrier of X1 ) , BX1 = f .: ( the carrier of X2 ) , BX2 = f .: ( the carrier of X1 ) , BX2 = f .: ( the carrier of X2 ) , BX2 = f .: ( the carrier of X2 ) , B
ex d being Element of L st d in D & x << d ;
assume that R -Seg ( a ) c= R -Seg ( b ) and R -Seg ( a ) c= R -Seg ( b ) ;
t in ]. r , s .[ or t = r or t = s or t = s ;
z + v2 in W & x = u + ( z + v2 ) & y = v + ( z + v1 ) ;
x2 |-- y2 iff P [ x2 , y2 ] & P [ x2 , y2 ] ;
pred x1 <> x2 means : Def1 : |. x1 - x2 .| > 0 & |. x1 - x2 .| > 0 ;
assume that p2 - p1 , p3 - p1 , p3 - p1 , p1 - p3 - p3 .| is linearly-independent and p1 - p2 - p3 , p3 - p1 + p3 - p1 , p1 - p3 be 0. V ;
set q = ( f | 'not' A ) ^ <* 'not' A *> ;
let f be PartFunc of REAL-NS 1 , REAL-NS n , x be Point of REAL-NS 1 , r be Real ;
( n mod ( 2 * k ) ) + 1 = n mod k ;
dom ( T * ( P , T . t ) ) = dom ( T * ( P , T . t ) ) ;
consider x being element such that x in wc iff x in c & x in d ;
assume ( F * G ) . v = v . x4 & ( F * G ) . x3 = v . x4 ;
assume that the Sorts of D1 c= the Sorts of D2 and the Sorts of D1 c= the Sorts of D2 and the Sorts of D2 c= the Sorts of D1 ;
reconsider A1 = [. a , b .[ , A2 = [. a , b .[ as Subset of REAL ;
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 ) , G = Gauge ( C , n ) , G = Gauge ( C , n ) , G = Gauge ( C , n ) , G = Gauge ( C , n ) , G = Gauge ( C , n ) , G
n1 -' len f + 1 <= len f + 1 - len f + 1 - 1 + 1 - 1 + 1 ;
\lbrace \lbrace \lbrace q , O1 , a , b , c , d } , b , c } = { u , v , a , b , c } ;
set C-2 = ( ( ( `1 ) `1 ) + ( G `2 ) `2 ) ;
Sum ( L (#) p ) = 0. R * Sum p .= 0. V * Sum p .= Sum ( L (#) p ) ;
consider i being element such that i in dom p and t = p . i ;
defpred Q [ Nat ] means 0 = Q ( $1 ) & $1 <= n implies $1 <= n ;
set s3 = Comput ( P1 , s1 , k ) , P3 = P1 +* I , s4 = P2 +* J , s4 = P2 +* J , P4 = P3 ;
let l be variable of k , A-30 , A be Subset of k -tuples_on A ;
reconsider U2 = union ( G-24 ) , a = union ( G-24 ) as Subset-Family of TS ;
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 ;
pccr = <* - c , 1 *> .= <* - c , 1 *> ;
synonym f is real-valued means : Def1 : rng f c= NAT & rng f c= NAT ;
consider b being element such that b in dom F and a = F . b ;
x0 < card ( X0 ) + card ( Y0 ) & x0 < ( 1 - 1 ) * ( 1 - 1 ) ;
attr X c= B1 means : Def1 : for B1 , B2 being Subset of X st B1 c= B2 & B2 c= B1 holds X = B2 ;
then w in Ball ( x , r ) & dist ( x , w ) <= r ;
angle ( x , y , z ) = angle ( x-y , 0 , z ) ;
pred 1 <= len s means : Def1 : ( for i being Element of NAT holds s . i = s . i ) & ( for i being Element of NAT st i in dom s holds s . i = s . i ) ;
fJ c= f . ( k + ( n + 1 ) ) ;
the carrier of { 1_ G } = { 1_ G } & the carrier of { 1_ G } = { 1_ G } ;
pred p '&' q in TAUT ( A ) means q in TAUT ( A ) & p in TAUT ( A ) & q in TAUT ( A ) ;
- ( ( t `1 ) / |. t .| - sn ) < ( t `1 ) / |. t .| - sn ;
U2 . 1 = U2 /. 1 .= ( U2 /. 1 ) `1 .= ( U2 /. 1 ) `1 .= ( U2 /. 1 ) `1 ;
f .: ( the carrier of x ) = the carrier of x & f .: ( the carrier of x ) = the carrier of x ;
Indices ( ( the carrier of O1 ) --> ( the carrier of O1 , the carrier of O1 ) ) = [: the carrier of O1 , the carrier of O1 :] ;
for n being Element of NAT holds G . n c= G . ( n + 1 ) ;
then V in M / \square & ex x being Element of M st V = { x } ;
ex f being Element of F-9 st f is \cup ( A * B ) & f is \setminus of A * B ;
[ h . 0 , h . 3 ] in the InternalRel of G & [ h . 2 , h . 3 ] in the InternalRel of G ;
s +* Initialize ( ( intloc 0 ) .--> 1 ) = s3 +* Initialize ( ( intloc 0 ) .--> 1 ) ;
|[ w1 , v1 ]| Let |[ w1 `1 , v1 `2 ]| <> 0. TOP-REAL 2 & |[ w1 , v1 ]| <> 0. TOP-REAL 2 ;
reconsider t = t as Element of INT * , s be Element of INT * ;
C \/ P c= [#] ( ( ( ( ( ( ( ( ( ( ( G ) \ A ) \/ A ) ) \/ B ) \/ C ) \/ D ) ) \/ A ) ) ;
f " V in ( the topology of X ) /\ D & f " ( the topology of X , the topology of Y ) = D ;
x in [#] ( ( the carrier of A ) /\ ( delta ( F ) ) ) ;
g . x <= h1 . x & h . x <= h1 . x & h . y <= h2 . x ;
InputVertices S = { xy , y , z } & InputVertices S = { xy , y , z } ;
for n being Nat st P [ n ] holds P [ n + 1 ] ;
set R = ( Line ( M , i ) * Line ( M , i ) ) ;
assume that M1 is being_line and M2 is being_line and M3 is being_line and M3 is being_line and M3 is being_line ;
reconsider a = f4 . ( i0 -' 1 ) , b = f4 . ( i0 -' 1 ) as Element of K ;
len B2 = Sum Len ( F1 ^ F2 ) + len ( F2 ^ F1 ) .= len B2 + 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 ( f | 12 ) & dom ( p1 ^ p2 ) = dom ( f | 12 ) ;
M . [ 1 / y , y ] = 1 / ( 1 / y ) * v1 .= y ;
assume that W is not trivial and W .vertices() c= the carrier' of G2 and W is not trivial and W is not trivial ;
C6 * ( i1 , i2 ) = G1 * ( i1 , i2 ) .= G1 * ( i1 , i2 ) ;
C8 |- 'not' Ex ( x , p ) 'or' p . ( x , y ) ;
for b st b in rng g holds lower_bound rng f\lbrace b \rbrace <= b * ( upper_bound rng fdom f )
- ( ( ( q1 `1 ) / |. q1 .| - cn ) / ( 1 + cn ) ) = 1 ;
( LSeg ( c , m ) \/ [: NAT , NAT :] ) \/ [: { l } , NAT :] c= R ;
consider p being element such that p in such that p in such and p in L~ f and x = f /. p ;
Indices ( [: X , Y :] ) = [: Seg n , Seg 1 :] & [: X , Y :] = [: Seg n , Seg 1 :] ;
cluster s => ( q => p ) => ( q => ( s => p ) ) -> valid ;
Im ( ( Partial_Sums F ) . m ) is_measurable_on E & Im ( ( Partial_Sums F ) . m ) is_measurable_on E ;
cluster f . ( x1 , x2 ) -> Element of D * ;
consider g being Function such that g = F . t and Q [ t , g ] ;
p in LSeg ( ( Int Z ) , ( ( N-min Z ) / 2 ) ) implies p in LSeg ( ( Int Z ) , ( ( \mathopen Z ) / 2 ) )
set R8 = R / ( 1 / 2 ) , R8 = ( 1 / 2 ) (#) ( R / 2 ) ;
IncAddr ( I , k ) = SubFrom ( da , db ) .= goto ( da + 1 ) .= goto ( da + 1 ) ;
seq . m <= ( ( the Sorts of seq ) . k ) . n & ( ( seq ^\ k ) . n ) . n <= ( ( seq ^\ k ) . n ) . n ;
a + b = ( a ` *' b ) ` + ( a ` *' b ) ` .= ( a ` + a ` ) ` + ( 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 & h . x = f . x ;
reconsider H = U1 \/ U2 , U2 = U2 as non empty Subset of U0 , G = U1 /\ U2 , H = U2 /\ U1 as non empty Subset of U0 ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) ) /\ m ) /\ j ;
consider y being element such that y in Y and P [ y , inf B ] ;
consider A being finite stable set such that card A = ( the carrier of R ) \/ ( the carrier of S ) ;
p2 in rng ( f |-- p1 ) \ rng <* p1 *> & p2 in rng ( f |-- p1 ) \ rng <* p1 *> ;
len s1 - 1 > 1-1 & len s2 - 1 > 0 & len s2 - 1 > 0 ;
( ( N-min P ) `2 ) = ( ( N-min P ) `2 ) & ( ( N-min P ) `2 ) = ( ( N-min P ) `2 ) ;
Ball ( e , r ) c= LeftComp Cage ( C , k + 1 ) & Ball ( e , r ) c= LeftComp Cage ( C , k + 1 ) ;
f . a1 ` = f . a1 ` & f . a2 = f . a2 ` & f . a2 = f . a2 ` ;
( seq ^\ k ) . n in ]. -infty , x0 + r .[ & ( seq ^\ k ) . n in ]. x0 , x0 + r .[ ;
gg . s0 = g . s0 | G . s0 .= g . s0 .= g . s0 ;
the InternalRel of S is \lbrace the carrier of S , the carrier of S , the carrier of S \rbrace ;
deffunc F ( Ordinal , Ordinal ) = phi . ( $1 + 1 ) & phi . ( $2 ) = phi . ( $2 ) ;
F . a1 = F . ( s2 . a1 ) & F . a2 = F . ( s2 . a1 ) ;
x `2 = A . a .= Den ( o , A . a ) ;
Cl ( f " P1 ) c= f " ( Cl P1 ) & Cl ( f " P1 ) c= f " P1 ;
FinMeetCl ( ( the topology of S ) \/ the topology of T ) c= the topology of T & the topology of S = the topology of T ;
synonym o is \bf means : Def1 : o <> *' & o <> {} & o <> {} ;
assume that X = Y + Z and card X <> card Y and Y <> {} and X <> {} ;
the finite the { s } <= 1 + ( the *> of s ) & the finite Subset of ( the carrier of S ) * ;
LIN a , a1 , d or b , c // b1 , c1 & a , c // a1 , c1 or a , c // a1 , c1 ;
e / 2 . 1 = 0 & e / 2 . 2 = 1 & e / 2 . 3 = 0 ;
ES1 in SS1 & not ES2 in { NS1 } & not ES2 in SS2 ;
set J = ( l , u ) If ;
set A1 = Let ( ( a , b , c ) --> ( x , y , c ) ) ;
set vs = [ <* c , d *> , '&' ] , f3 = [ <* d , c *> , '&' ] , f4 = [ <* c , d *> , '&' ] , f4 = [ <* d , c *> , '&' ] , f4 = [ <* c , d *> , '&' ] , [ <* d , c *> , '&' ] , [ <* c ,
x * z `2 * x " in x * ( z * N ) * x " * x * ( z * N ) ;
for x being element st x in dom f holds f . x = g2 . x & g . x = g2 . x ;
Int cell ( f , 1 , G ) c= RightComp f \/ RightComp f & Int cell ( G , 1 , j ) c= RightComp f ;
U2 is_an_arc_of W-min C , E-max C & ( W-min C ) `1 <= ( E-max C ) `1 & ( E-max C ) `1 <= ( E-max C ) `1 ;
set f-17 = f @ g "/\" @ @ @ f ;
attr S1 is convergent means : Def1 : S2 is convergent & S1 is convergent & S2 is convergent & lim ( S1 - S2 ) = 0 ;
f . ( 0 + 1 ) = ( 0 qua Ordinal ) + a .= a ;
cluster -> 0. for non empty transitive RelStr -reflexive non empty non empty non void RelStr -reflexive RelStr ;
consider d being element such that R reduces b , d and R reduces c , d and R reduces d , c ;
not b in dom Start-At ( ( card I + 2 ) , SCMPDS ) & not a in dom Start-At ( ( card I + 2 ) , SCMPDS ) ;
( z + a ) + x = z + ( a + y ) .= z + a + y ;
len ( l \lbrack ( a |^ 0 ) .--> x ) = len l & len ( l |^ 0 ) = len l ;
t4 \+\ {} is ( {} \/ rng t4 ) -valued ( {} , {} ) -valued finite Function ;
t = <* F . t *> ^ ( C . p ^ ( C . q ) ) .= ( C . p ^ ( C . q ) ) ^ ( C . q ) ;
set 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 ) , pi = W-min L~ Cage ( C , n )
( k -' ( i + 1 ) ) = ( k - ( i + 1 ) ) + ( i + 1 ) ;
consider u being Element of L such that u = u ` ` and u in D ` ;
len ( ( width ( ( a |-> b ) ) * ( A @ ) ) ) = width ( ( a * A ) @ ) ;
FF . x in dom ( ( G * the_arity_of o ) . x ) & ( ( G * the_arity_of o ) . x ) = ( ( G * the_arity_of o ) . x ) . x ;
set H = the carrier of H2 , I = the carrier of H , J = the carrier of H , I = the carrier of H , J = the carrier of I ;
set H = the carrier of H1 , I = the carrier of H2 , J = the carrier of H2 , I = the carrier of H1 , J = the carrier of H2 , I = the carrier of H2 , J = the carrier of J , M = the carrier of G , I = the carrier of J , J = the carrier of G , M
( Comput ( P , s , 6 ) ) . intpos ( m + 6 ) = s . intpos ( m + 6 ) ;
IC Comput ( Q2 , t , k ) = ( l + 1 ) + 1 .= ( l + 1 ) + 1 ;
dom ( ( cos * sin ) `| REAL ) = REAL & dom ( ( cos * sin ) `| REAL ) = REAL ;
cluster <* l *> ^ phi -> ( 1 + 1 ) -element for string of S ;
set b5 = [ <* ( ( z + 1 ) , ( z + 1 ) *> , <* ( z + 1 ) , ( z + 1 ) *> ] ;
Line ( Segm ( M @ , P , Q ) , x ) = L * ( Sgm Q ) . x .= L * ( x , y ) ;
n in dom ( ( ( ( the Sorts of A ) * the_arity_of o ) * the_arity_of o ) . n ) ;
cluster f1 + f2 -> continuous for PartFunc of REAL , the carrier of S ;
consider y being Point of X such that a = y and ||. x-y .|| <= r ;
set x3 = t2 . DataLoc ( s4 . SBP , 2 ) , x4 = t2 . DataLoc ( s3 . SBP , 2 ) , P4 = s3 . DataLoc ( s3 . SBP , 3 ) , P4 = s3 . DataLoc ( s3 . SBP , 3 ) , P4 = s3 . SBP , P4 = s3 . SBP , P4 = s3 .
set p-3 = stop I ( ) , p-3 = Initialize s2 , p-3 = Initialize s2 , p-3 = Initialize s2 , pE = Initialize s2 , pE = Initialize s2 , pE = Initialize s2 , pE = Initialize s2 , pE = Initialize s2 , pE = Initialize s2 , pE = Initialize s2 ,
consider a being Point of D2 such that a in W1 and b = g . a and a in W2 ;
{ A , B , C , D } = { A , B , C } \/ { D , E , F , J }
let A , B , C , D , E , F , J , M , N , M , N , N , M be set ;
|. p2 .| ^2 - ( p2 `2 ) ^2 >= 0 & ( p2 `2 ) ^2 >= 0 ;
l -' 1 + 1 = n-1 * ( l + 1 ) + ( \setminus l ) + 1 ;
x = v + ( a * w1 + ( b * w2 ) ) + ( c * w2 ) + ( c * w2 ) ;
the TopStruct of L = , the TopStruct of L = \langle the Scott of L , the Scott Function of L , the topology of L *> ;
consider y being element such that y in dom H1 and x = H1 . y and y in H ;
( f \ { n } ) \ { n } = ( Free ( All ( v1 , H ) ) \ { n } ) ;
for Y being Subset of X st Y is summable & Y is not summable holds X is not summable
2 * n in { N : 2 * Sum ( p | N ) = N & N > 0 } ;
for s being FinSequence holds len ( the { - } Shift ( s ) ) = len s & len ( the { - } Shift ( s ) ) = len s
for x st x in Z holds exp_R * f is_differentiable_in x & exp_R * f is_differentiable_in x & f . x > 0 ;
rng ( h2 * f2 ) c= the carrier of ( ( TOP-REAL 2 ) | ( dom g2 ) ) | K1 ;
j + ( len f ) - len f <= len f + ( len g - len f ) - len f ;
reconsider R1 = R * I as PartFunc of REAL , REAL-NS n , I ;
C8 . x = s1 . x0 .= ( C . x0 ) * ( C . x0 ) .= C . x0 * ( C . x0 ) ;
power F_Complex = 1 .= ( x |^ n ) * ( z |^ n ) .= ( x |^ n ) * ( z |^ n ) ;
t at ( C , s ) = f . ( the connectives of S ) . t & t at ( C , s ) = f . ( t , I ) ;
support ( f + g ) c= support f \/ ( support g ) & support ( f + g ) c= support f \/ support g ;
ex N st N = j1 & 2 * Sum ( ( r4 | N ) | N ) > N & N > 0 ;
for y , p st P [ p ] holds P [ All ( y , p ) ] ;
{ [ x1 , x2 ] , [ x2 , y2 ] } is Subset of [: X1 , X2 :] ;
h = ( j |-- h , id B ) . i .= H . i .= H . i ;
ex x1 being Element of G st x1 = x & x1 * N c= A & N c= A ;
set X = ( ( \lbrace q , O1 } ) . ( q , 4 ) ) `1 , Y = ( ( { q , O1 } ) . ( q , 4 ) ) `1 , Z = ( { q , O1 } ) `1 , X = ( { q , O1 } ) . ( q , 4 ) , Y = ( { q , O1 } ) `1 , Z = { q , s } ;
b . n in { g1 : x0 < g1 & g1 < x0 & g1 < x0 + r } ;
f /* s1 is convergent & f /. x0 = lim ( f /* s1 ) & f /. x0 = lim ( f /* s1 ) ;
the lattice of Y = the lattice of the lattice of Y & the topology of Y = the topology of Y & the topology of Y = the topology of X ;
'not' ( a . x ) '&' b . x 'or' a . x '&' 'not' ( b . x ) '&' 'not' ( b . x ) = FALSE ;
q2 = len ( q0 ^ r1 ) + len q1 & q2 = len ( q2 ^ r1 ) + len q1 & q1 = q2 ^ r1 + len q1 ;
( ( 1 / a ) (#) ( sec * f1 ) - id Z ) is_differentiable_on Z & ( ( 1 / a ) (#) ( sec * f1 ) ) `| Z ) = f ;
set K1 = integral ( ( lim ( H || A ) ) || ( ( lim ( H || A ) ) ) , ( lim ( H || A ) ) || ( ( lim ( H || A ) ) ) ) ;
assume e in { ( w1 - w2 ) / ( w1 - w2 ) : w1 in F & w2 in G } ;
reconsider d7 = dom a `2 , d6 = dom F `2 , d8 = dom F `2 , d8 = dom F `2 , d7 = dom F `2 , d8 = dom F ;
LSeg ( f /^ q , j ) = LSeg ( f , j ) + q .. f .= 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 S29 = dom S /\ Seg n .= dom ( L | Seg n ) .= dom ( L | Seg n ) .= dom ( L | Seg n ) .= dom ( L | Seg n ) ;
x in H |^ a implies ex g st x = g |^ a & g in H & a in H
a * ( 0. ( INT , n ) ) = 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 < @ @ c ;
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 * 1 ;
len g = len f + len <* x + y *> .= len f + 1 + 1 .= len f + 1 + 1 .= len f + 1 ;
dom ( F-11 ) = dom ( F | [: N1 , S :] ) .= [: N1 , S :] ;
dom ( f . t * I . t ) = dom ( f . t * g . t ) ;
assume a in ( "\/" ( ( ( T |^ the carrier of S ) , F ) ) .: D ) ;
assume that g is one-to-one and ( the carrier' of S ) /\ rng g c= dom g and g is one-to-one ;
( ( x \ y ) \ z ) \ ( ( x \ z ) \ ( y \ z ) ) = 0. X ;
consider f such that f * f `2 = id b and f * f `2 = id a and f * g = id b ;
( cos | [. 2 * PI , 0 + 2 * PI .] ) | [. 2 * PI , 0 + 2 * PI .] is increasing ;
Index ( p , co ) <= len LS - Gij .. LS - 1 & Index ( p , LS ) + 1 <= len LS - 1 ;
let t1 , t2 , t2 , t1 be Element of ( T . s ) * , t2 be Element of ( T . s ) * ;
( -> "/\" ( ( Frege ( ( Frege H ) . h ) ) , L ) <= "/\" ( rng ( ( Frege G ) . h ) , L ) ;
then P [ f . ( i0 + 1 ) , F ( f . ( i0 + 1 ) ) ] & F ( f . ( i0 + 1 ) , F ( f . ( i0 + 1 ) ) ) < j ;
Q [ ( [ D . x , 1 ] ) `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 the Element of G . i ;
the Sorts of A2 = ( the carrier of S2 ) --> ( the carrier' of S1 ) .= ( the carrier' of S1 ) --> ( the carrier' of S2 ) ;
consider s being Function such that s is one-to-one and dom s = NAT and rng s = F and rng s c= F ;
dist ( b1 , b2 ) <= dist ( b1 , a ) + dist ( a , b2 ) + dist ( a , b2 ) ;
( Upper_Seq ( C , n ) /. len Upper_Seq ( C , n ) ) `1 = W & ( Cage ( C , n ) /. len Upper_Seq ( C , n ) ) `1 = W ;
q `2 <= ( UMP C ) `2 & ( UMP C ) `2 <= ( UMP C ) `2 ;
LSeg ( f | i2 , i ) /\ LSeg ( f | i2 , j ) = {} & LSeg ( f | i2 , j ) = {} ;
given a being ExtReal such that a <= II and A = ]. a , I .[ and a < I ;
consider a , b being complex number such that z = a & y = b & z + y = a + b ;
set X = { b |^ n where n is Element of NAT : n <= k } , Y = { b |^ n where n is Element of NAT : n <= k } , Z = { b |^ n : n <= k } ;
( ( x * y * z \ x ) \ z ) \ ( x * y \ x ) = 0. X ;
set xy = [ <* xy , y *> , f1 ] , yz = [ <* y , z *> , f2 ] , yz = [ <* z , x *> , f3 ] , yz = [ <* x , y *> , f3 ] , zx = [ <* y , z *> , f3 ] , zx = [ <* z , x *> , f3 ] , ] ;
lq /. len lq = lq . len lq .= ( lq | len lq ) /. 1 .= ( lq | len lq ) /. 1 ;
( ( ( ( q `2 / |. q .| - sn ) ) / ( 1 + sn ) ) / ( 1 + sn ) ) ^2 = 1 ;
( ( ( p `2 / |. p .| - sn ) / ( 1 - sn ) ) / ( 1 - sn ) ) ^2 < 1 ;
( ( ( ( S \/ Y ) `2 ) / 2 ) * ( ( S \/ Y ) `2 ) / 2 ) = ( ( S \/ Y ) / 2 ) * ( ( S \/ Y ) / 2 ) ;
( ( seq - seq ) . k ) = ( seq . k - seq . ( k + 1 ) ) / ( seq . k - seq . ( k + 1 ) ) ;
rng ( ( h + c ) ^\ n ) c= dom SVF1 ( 1 , f , u0 ) /\ dom SVF1 ( 1 , f , u0 ) ;
the carrier of X = the carrier of X & the carrier of X1 = the carrier of X1 & the carrier of X2 = the carrier of X2 implies X1 = X2
ex p4 st p3 = p4 & |. p4 - |[ a , b ]| .| = r & |. p4 - |[ a , b ]| .| = r ;
set h = chi ( X , A5 ) , A = ( X --> A ) | A ;
R to_power ( ( 0 * n ) ) = I\HM ( X , X ) .= R to_power ( n ) ;
( Partial_Sums ( curry ( F1 , n ) ) . n ) . x is nonnegative & ( ( curry ( F1 , n ) ) . x ) . x = ( ( Partial_Sums ( F1 , n ) ) . x ) . x ;
f2 = C7 . ( ( V , K , len H ) + 1 ) & f2 = C8 . ( ( V , K ) + 1 ) ;
S1 . b = s1 . b .= s2 . b .= s2 . b .= ( s2 * s2 ) . b .= ( s2 * s2 ) . b ;
p2 in LSeg ( p2 , p1 ) /\ LSeg ( p1 , p01 ) & p2 in LSeg ( p1 , p01 ) /\ LSeg ( p1 , p01 ) ;
dom ( f . t ) = Seg n & dom ( I . t ) = Seg n & dom ( I . t ) = Seg n ;
assume o = ( the connectives of S ) . 11 & o in ( the carrier' of S ) . 12 ;
set phi = ( l1 , l2 ) support phi , phi = ( l1 , l2 ) If , phi = ( l1 , l2 ) If , phi = ( l1 , l2 ) If , Q = ( l1 , l2 ) If , E = ( l1 , l2 ) If , E = ( l2 , l2 ) If , F = ( l1 , l2 ) $ , F = ( l1 , l2 ) $ , N = ( l2
synonym p is invertible for p is invertible & p = 1 implies p * ( p , T ) = ( p * ( p , T ) ) * ( p , T ) ;
( Y1 ) `2 = - 1 & 0. ( TOP-REAL 2 ) | Y1 <> ( TOP-REAL 2 ) | Y1 & ( Y1 is 0. TOP-REAL 2 ) | Y2 <> 0. ( TOP-REAL 2 ) | Y1 ;
defpred X [ Nat , set , set ] means P [ $2 , $2 ] & $2 = F ( $1 , $2 ) ;
consider k being Nat such that for n being Nat st k <= n holds s . n < x0 + g and s . n < x0 + g ;
Det ( I |^ ( ( m -' n ) * ( m -' n ) ) ) = 1_ K & Det ( I |^ ( m -' n ) ) = 1_ K ;
( - b - sqrt ( b ^2 - 4 * a * c ) ) / ( 2 * a * c ) < 0 ;
Cd . d = Cd . d mod Cd . d .= Cd . d mod Cd . e .= Cd . d mod Cd . e ;
attr X1 is dense means : Def1 : X2 is dense dense & X1 is dense dense & X2 is dense dense implies X1 union X2 is dense SubSpace of X & X1 is dense SubSpace of X ;
deffunc F6 ( Element of E , Element of I ) = $1 * ( ( $1 + 1 ) * ( ( $2 ) * ( $2 ) ) ;
t ^ <* n *> in { t ^ <* i *> : Q [ i , T ( t ) ] } ;
( x \ y ) \ x = ( x \ x ) \ y .= y ` \ 0. X .= 0. X ;
for X being non empty set for X being Subset-Family of X holds X is Basis of <* X , UniCl ( Y , X ) *>
synonym A , B are_means : Def1 : Cl ( A , B ) misses Cl ( B , C ) & A misses Cl ( A , B ) ;
len ( M @ ) = len p & width ( M @ ) = width ( M @ ) & width ( M @ ) = width ( M @ ) ;
J = { v where x is Element of K : 0 < v . x & v . x < 0 } ;
( Sgm ( Seg m ) . d - ( Sgm ( Seg m ) . e ) ) * ( m - 1 ) <> 0 ;
lower_bound divset ( D2 , k + k2 ) = D2 . ( k + k2 - 1 ) .= D2 . ( k + k2 - 1 ) .= D2 . ( k + 1 - 1 ) ;
g . r1 = - 2 * r1 + 1 & dom h = [. 0 , 1 .] & rng h = [. 0 , 1 .] & h is one-to-one ;
|. a .| * ||. f .|| = 0 * ||. f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| ;
f . x = ( h . x ) `1 & g . x = ( h . x ) `2 & h . y = ( h . y ) `2 ;
ex w st w in dom B1 & <* 1 *> ^ s = <* 1 *> ^ w & len w = len w + 1 ;
[ 1 , {} , <* d1 *> ] in ( { [ 0 , {} , {} ] } \/ S1 ) \/ S2 ;
IC Exec ( i , s1 ) + n = IC Exec ( i , s2 ) + n .= IC Exec ( i , s2 ) + n ;
IC Comput ( P , s , 1 ) = IC Comput ( P , s , 9 ) .= 5 + 9 .= 5 + 9 .= ( card I + 9 ) + 9 ;
( IExec ( W6 , Q , t ) ) . intpos ( 8 + 1 ) = t . intpos ( 8 + 1 ) .= t . intpos ( 8 + 1 ) ;
LSeg ( f /^ q , i ) misses LSeg ( f /^ q , j ) & LSeg ( f /^ q , j ) misses LSeg ( f /^ q , j ) ;
assume for x , y being Element of L st x in C holds x <= y or y <= x or y <= y ;
Integral ( M , ( f `| X ) . x ) = f . ( upper_bound C ) - ( f `| X ) . x .= ( r (#) f ) . x ;
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 c= d ;
for y , x being Element of REAL st y in Y & x in X ` holds y <= x ` & y <= x ` ;
func |. ( p \bullet <* p *> ) /. ( i + 1 ) -> variable of A equals min ( NBI . ( p , <* p *> ) , p ) ;
consider t being Element of S such that x `1 , y `2 '||' z `1 , t `2 & x `2 '||' y `2 & y `2 '||' t `2 ;
dom x1 = Seg ( len x1 ) & len x1 = len l1 & len x2 = len l1 & len x1 = len l1 & len x2 = len l1 & len x1 = len l1 ;
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 .|| | X ) /* s1 .= ( ||. f .|| | X ) /* s1 ;
( the InternalRel of A ) ` ` ` /\ Y = {} \/ {} .= {} \/ {} .= {} .= {} .= {} ;
assume i in dom p implies for j being Nat st j in dom q holds P [ i , j ] & i + 1 in dom p & j in dom p & p . i = p . j ;
reconsider h = f | X ( ) as Function of X ( ) , rng ( f | X ( ) ) ;
u1 in the carrier of W1 & u2 in the carrier of W2 & u1 in the carrier of W1 & v1 = ( the carrier of W1 ) /\ the carrier of W2 implies u1 = v1
defpred P [ Element of L ] means M <= f . $1 & f . $1 <= g . $1 & g . $1 <= h . $1 ;
not ( u , a , v , y ) = s * x + ( - ( s * x ) + y ) .= b ;
- ( - ( - ( - x ) ) ) = - ( x + - ( - y ) ) .= - ( x + y ) .= - ( x + y ) ;
given a being Point of GX such that for x being Point of GX holds a , x are_\sum ( G , a ) and x , a are_not ed ;
fT = [ [ dom ( @ f2 ) , cod ( @ g2 ) ] , [ cod ( @ g2 ) , cod ( @ g2 ) ] ] ;
for k , n being Nat st k <> 0 & k < n & n is prime holds k , n are_relative_prime & k , n are_relative_prime & k , n are_relative_prime
for x being element holds x in A |^ d iff x in ( ( A ` ) |^ d ) ` & ( ( A ` ) |^ d ) ` = ( ( A ` ) |^ d ) `
consider u , v being Element of R , a being Element of A such that l /. i = u * a * v ;
- ( ( - ( p `1 / |. p .| - cn ) ) / ( 1 + cn ) ) ^2 > 0 ;
L-13 . k = Ln . ( F . k ) & F . k in dom ( L * F ) & F . k in dom ( L * F ) ;
set i2 = SubFrom ( a , i , - n ) , i1 = goto ( - n + 1 ) , i2 = goto ( - n + 1 ) , i2 = goto ( - n + 1 ) , i2 = goto ( - n + 1 ) ;
attr B is thesis means : Def1 : for S being Subuniversal ( B , S-13 ) = ( B `1 ) \/ ( B `2 ) ;
a9 "/\" D = { a "/\" d where d is Element of N : d in D } & a "/\" b = ( a "/\" d ) "/\" ( a "/\" b ) ;
|( \square , q29 )| * |( q29 , q29 )| * |( q29 )| >= |( \square , q29 )| * |( q29 , q29 )| ;
( - f ) . sup A = ( ( - f ) | A ) . sup A .= ( ( - f ) | A ) . sup A .= ( ( - f ) | A ) . sup A ;
G * ( len G , k ) `1 = G * ( len G , k ) `1 .= G * ( len G , k ) `1 .= G * ( len G , k ) `1 ;
( Proj ( i , n ) ) . LM = <* ( proj ( i , n ) ) . LM *> .= ( proj ( i , n ) ) . LM ;
f1 + f2 * reproj ( i , x ) is_differentiable_in ( ( reproj ( i , x ) + f2 ) * reproj ( i , x ) ) . x0 ;
pred ( cos . x ) <> 0 & ( tan . x ) <> 0 & ( tan . x ) <> 0 & ( tan . x ) <> 0 & ( tan . x ) <> 0 ) ;
ex t being SortSymbol of S st t = s & h1 . t = h2 . t & ( for x being set st x in dom h1 holds h1 . x = F ( x ) ) ;
defpred C [ Nat ] means ( P . $1 is non empty & A is $1 empty & A is non empty & A is non empty & A is non empty & A is non empty & A is non empty ;
consider y being element such that y in dom p9 and q9 . i = p9 . y and p9 . i = q9 . i ;
reconsider L = product ( { x1 } +* ( index B , l ) ) as non empty Subset of ( A . index B ) ;
for c being Element of C ex d being Element of D st T . ( id c ) = id d & for c being Element of C st c in dom T holds T . ( id c ) = id d
( f , n , p ) = ( f | n ) ^ <* p *> .= f ^ <* p *> .= f ^ <* p *> ;
( f * g ) . x = f . ( g . x ) & ( f * h ) . x = f . ( h . x ) ;
p in { ( 1 - 2 ) * ( G * ( i + 1 , j ) + G * ( i + 1 , j + 1 ) ) } ;
f `2 - p = ( f | ( n , L ) ) *' ( - ( f | ( n , L ) ) ) .= ( f - ( f | ( n , L ) ) ) *' ( - ( f | ( n , L ) ) ) ;
consider r being Real such that r in rng ( f | divset ( D , j ) ) and r < m + s ;
f1 . ( |[ r2 , r2 ]| ) in f1 .: ( W1 /\ W2 ) & f2 .: ( W1 /\ W3 ) in f1 .: ( W2 /\ W3 ) ;
eval ( a | ( n , L ) , x ) = eval ( a | ( n , L ) , x ) .= a * ( x | ( n , L ) ) .= a * ( x | ( n , L ) ) ;
z = DigA ( ty , x9 ) .= DigA ( ty , x9 ) .= DigA ( ty , x9 ) .= DigA ( ty , x9 ) ;
set H = { Intersect S where S is Subset-Family of X : S c= G } , G = { Intersect S where S is Subset-Family of X : S c= G } , F = G , G = H \ { Intersect S where S is Subset-Family of X : S c= G } , H = { Intersect S } , G = { Intersect S where S is Subset of X : S c= G } , F = H \ { Intersect S } , G
consider S19 being Element of D * , d being Element of D * such that S `1 = S19 ^ <* d *> and S `2 = d ;
assume that x1 in dom f and x2 in dom f and f . x1 = f . x2 and f . x2 = f . x2 and f . x2 = f . x2 ;
- 1 <= ( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 / ( 1 + sn ) ^2 ;
0. V is Linear_Combination of A & Sum ( 1. V ) = 0. V & Sum ( L (#) F ) = 0. V & Sum ( L (#) F ) = 0. V ;
let k1 , k2 , k2 , x4 , k2 , x4 be Instruction of SCM+FSA , a be Int-Location , b be Int-Location ;
consider j being element such that j in dom a and j in g " { k `2 } and x = a . j and y = a . j ;
H1 . x1 c= H1 . x2 or H1 . x2 c= H1 . x1 or H1 . x2 c= H1 . x2 or H1 . x1 c= H1 . x2 & H1 . x2 c= H1 . x2 ;
consider a being Real such that p = len implies p * p1 + ( a * p2 ) * p2 and 0 <= a and a <= 1 ;
assume that a <= c & c <= d & [' a , b '] c= dom f and [' a , b '] c= dom g and [' a , b '] c= dom g ;
cell ( Gauge ( C , m ) , len Gauge ( C , m ) -' 1 , 0 ) is non empty ;
Ax in { ( S . i ) `1 where i is Element of NAT : not contradiction } ;
( T * b1 ) . y = L * b2 /. y .= ( F `1 * b1 ) . y .= ( F `2 * b1 ) . y ;
g . ( s , I ) . x = s . y & g . ( s , I ) . y = |. s . x - s . y .| ;
( log ( 2 , k + 1 ) ) to_power ( 2 * k + 1 ) >= ( log ( 2 , k + 1 ) ) to_power ( 2 * k + 1 ) ;
then that p => q in S and not x in the still of p & not x in S & not x in S & not x in S & not x in S ;
dom ( the InitS of r-10 ) misses dom ( the InitS of r-11 ) & dom ( the InitS of r-11 ) misses dom ( the InitS of r-11 ) ;
synonym f is extended integer means : Def1 : for x being set st x in rng f holds x is ExtReal ;
assume for a being Element of D holds f . { a } = a & for X being Subset-Family of D holds f . ( f .: X ) = f . union X ;
i = len p1 .= len p3 + len <* x *> .= len p3 + len <* x *> .= len p3 + len <* x *> .= len p3 + 1 .= len p3 + 1 ;
l ( ) = ( g /. ( 1 , 3 ) + ( k ( ) ) * ( k , 3 ) - ( e ( ) * ( k , 3 ) ) ) ;
CurInstr ( P2 , Comput ( P2 , s2 , l ) ) = halt SCM+FSA .= CurInstr ( P2 , Comput ( P2 , s2 , l ) ) ;
assume for n be Nat holds ||. seq .|| . n <= ( R * ||. seq .|| ) . n & ( R * ||. seq .|| ) . n <= ( R * ||. seq .|| ) . n ;
sin . ( M * ( K - 1 ) ) = sin . ( r * ( cos . ( K - 1 ) ) ) .= 0 ;
set q = |[ g1 . t0 , g2 . t0 ]| , r = |[ s . t0 , t . t0 ]| , s = |[ s . t0 , t . t0 ]| , t = |[ s . t0 , t . t0 ]| , s = |[ t . t0 , t . t0 ]| , t = |[ t . t0 , t . t0 ]| , s = |[ t . t0 , t . t0 ]| , t = |[ t . t0 ]| , t = |[ t . a , t . a ]|
consider G being sequence of S such that for n being Element of NAT holds G . n in G and G . n in G ;
consider G such that F = G and ex G1 st G1 in SM & G = \mathopen { X : X in G & X in G } ;
the root of [ x , s ] in ( the Sorts of Free ( C , X ) ) . s & ( the Sorts of Free ( C , X ) ) . s = [ x , s ] ;
Z c= dom ( exp_R * ( f + ( ( #Z 3 ) * ( f1 + #Z 3 ) ) ) ) & Z c= dom ( exp_R * ( f + ( #Z 3 ) * ( f1 + #Z 3 ) ) ) ;
for k being Element of NAT holds rx0 . k = ( sum ( Im ( f , S-3 ) , S-3 ) ) . k + ( Im ( f , S-3 ) ) . k ;
assume that - 1 < n ( ) and q `2 > 0 and ( q `2 <= 1 ) and ( q `2 <= 1 ) and ( q `2 <= 1 ) ;
assume that f is continuous one-to-one and a < b and f < c and f = g and f . a = c and g . b = d and f . c = d ;
consider r being Element of NAT such that s-> Element of NAT such that s-> Element of NAT and r <= q and q <= r ;
LE f /. ( i + 1 ) , f /. j , L~ f , f /. ( len f ) , 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 inf ( { x , y } , L ) = inf ( { x , y } , L ) ;
assume f +* ( i1 , \xi ) in ( proj ( F , i2 ) ) " ( ( proj ( F , i2 ) ) " ( A ) ) & ( proj ( F , i2 ) ) . ( ( proj ( F , i2 ) ) . ( A ) ) = ( proj ( F , i2 ) ) . ( A . ( A . ( i + 1 ) ) ;
rng ( ( ( ( ( ( ( ( ( ( ( ( ( ( the M ) ) ) | ( 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 the carrier of T } ;
consider l be Nat such that for m be Nat st l <= m holds ||. s1 . m - x0 .|| < g & g . x0 < g ;
consider t be VECTOR of product G such that mt = ||. D5 . t .|| and ||. t .|| <= 1 ;
assume that the degree degree of v = 2 and v ^ <* 0 *> in dom p and v ^ <* 1 *> in dom p and p . v = p . v ;
consider a being Element of the Points of X39 , A being Element of the lines of X39 such that a does not contradiction and A is not contradiction ;
( - x ) |^ ( k + 1 ) * ( ( - x ) |^ ( k + 1 ) ) " = 1 ;
for D being set st for i st i in dom p holds p . i in D holds p is FinSequence of D & for i being Nat st i in dom p holds p . i = F ( i )
defpred R [ element ] means ex x , y st [ x , y ] = $1 & P [ x , y ] & Q [ y , x ] ;
L~ f2 = union { LSeg ( p0 , p01 ) , LSeg ( p1 , p01 ) , LSeg ( p1 , p01 ) } .= { p1 , p2 } \/ { 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 - 1 + 1 - 1 + 1 - 1 + 1 - 1 < i + 2 - 1 + 1 ;
for n being Element of NAT st n in dom F holds F . n = |. ( ( F . n ) - ( F . n ) ) .| ;
for r , s1 , s2 , r st r in [. s1 , s2 .] & s1 <= s2 holds r <= ( s1 - s2 ) * ( s1 - s2 )
assume v in { G where G is Subset of T2 : G in B2 & G c= z1 & G c= z2 } ;
let g be \vert element of A , INT , ( b , a ) --> ( b , b ) <> 0 & ( a , b ) --> ( b , c ) <> 0 ;
min ( g . [ x , y ] , k ) = ( min ( g , k , x ) ) . y .= min ( g , k , x ) ;
consider q1 being sequence of CNS such that for n holds P [ n , q1 . n ] & q1 is convergent & q1 is convergent & q2 is convergent & lim q1 = lim q2 ;
consider f being Function such that dom f = NAT & for n being Element of NAT holds f . n = F ( n ) and f is one-to-one ;
reconsider B-6 = B /\ B , Od = O , Bd = I , Bd = I as Subset of B ;
consider j being Element of NAT such that x = the the ` of n and j <= n and 1 <= j & j <= n ;
consider x such that z = x and card ( x . O2 ) in card ( x . O2 ) and x . O2 in L1 & x . O2 in L2 ;
( C * _ T4 ( k , n2 ) ) . 0 = C . ( ( ( of T4 ( k , n2 ) ) * ( C * ( k , n2 ) ) ) . 0 ) ;
dom ( X --> rng f ) = X & dom ( ( X --> f ) | X ) = dom ( X --> f ) & dom ( ( X --> f ) | X ) = X ;
S-bound L~ SpStSeq C <= ( ( SpStSeq C ) `2 ) / 2 & ( ( SpStSeq C ) `2 <= ( ( SpStSeq C ) `2 ) / 2 & ( ( SpStSeq C ) `2 ) / 2 <= ( ( SpStSeq C ) `2 ) / 2 ) / 2 ;
synonym x , y are_collinear means : Def1 : x = y or ex l being \HM { x , y } c= l & x is Subset of S ;
consider X being element such that X in dom ( f | ( n + 1 ) ) and ( f | ( n + 1 ) ) . X = Y ;
assume that Im k is continuous and for x , y being Element of L , a , b being Element of Im k st a = x & b = y & x << y holds a << b ;
( 1 / 2 * ( ( ( - ( ( ( - ( ( - ( 0 ) ) ) * ( ( 0 ) ) / 2 ) ) * ( ( 0 - ( 0 ) ) / 2 ) ) ) ) ) ) ) ) is_differentiable_on REAL ;
defpred P [ Element of omega ] means ( ( the Sorts of A1 ) * ( $1 + 1 ) ) . x = A1 . $1 & ( ( the Sorts of A2 ) * ( $1 + 1 ) ) . x = A2 . $1 ;
IC Comput ( P , s , 2 ) = succ IC Comput ( P , s , 1 ) .= 6 + 1 .= 6 + 1 .= 6 + 1 ;
f . x = f . g1 * f . g2 .= f . g1 * 1_ H .= ( f . g1 ) * 1_ H .= ( f . g1 ) * ( g . g2 ) .= ( f . g1 ) * ( g . g2 ) .= ( f . g1 ) * ( g . g2 ) ;
( M * F-4 ) . n = M . ( ( canFS ( Omega ) ) . n ) .= M . ( { ( canFS ( Omega ) ) . n } ) .= M . ( ( canFS ( Omega ) ) . n ) ;
the carrier of L1 + L2 c= ( the carrier of L1 ) \/ ( the carrier of L2 ) & the carrier of L1 = the carrier of L1 \/ the carrier of L2 & the carrier of L1 = the carrier of L2 & L1 is the carrier of L2 ;
pred a , b , c , x , y be element means : Def1 : a , b // o , c & b , c // o , y & x , y // o , y & x , y // o , y ;
( the PartFunc of s ) . n <= ( ( the Sorts of s ) . n ) * ( ( the Sorts of s ) . n + ( the Sorts of s ) . n ) ;
pred - 1 <= r & r <= 1 & ( arccot ) . r = - 1 & ( arccot ) . r = - 1 & ( arccot ) . r = 1 ;
seq in { p ^ <* n *> where n is Nat : p ^ <* n *> in T1 & n in T1 } implies for x being element st x in T1 & x in T2 holds x in T1 & x in T2
|[ x1 , x2 , x3 ]| . 2 - |[ y1 , y2 , x3 ]| . 2 = x2 - |[ y2 , y2 ]| . 2 - |[ y1 , y2 , x3 ]| . 2 - |[ y1 , y2 , x3 ]| . 2 = y2 - |[ y1 , y2 ]| . 2 - |[ y1 , y2 , x3 ]| . 2 - |[ y1 , y2 , x4 ]| . 2 - |[ y1 , y2 ]| . 2 ]| . 2 = y2 - |[ y1 , y2 ]| . 2 -
attr m be Nat means : Def1 : F . m is nonnegative & ( Partial_Sums F ) . m is nonnegative & ( Partial_Sums F ) . m is nonnegative ;
len ( ( 2 * G ) . z ) = len ( ( ( 2 * G ) . ( x9 + 1 ) ) + ( ( 2 * G ) . ( y9 + 1 ) ) ) .= len ( ( 2 * G ) . ( x9 + 1 ) ) ;
consider u , v being VECTOR of V such that x = u + v and u in W1 /\ W2 and v in W3 /\ W3 and v in W3 /\ W3 ;
given F be finite Subset of NAT such that F = x and dom F = n & rng F c= { 0 , 1 } and Sum F = k and Sum F = k ;
0 = 0 * 0 * 0 ^2 + 1 * 0 = ( ( - 1 ) * ( - 1 ) ) * ( ( - 1 ) * ( - 1 ) ) ;
consider n being Nat such that for m being Nat st n <= m holds |. ( f # x ) . m - ( lim ( f # x ) ) .| < e ;
cluster -> as } for non empty implies ( ( let L ) | ( ( let c ) | ( c , d ) ) ) | ( ( c | ( c , d ) ) ) is Boolean non empty ;
"/\" ( BB , L ) = Top ( BB , L ) .= Top ( I , L ) .= "/\" ( I , L ) .= "/\" ( I , L ) .= "/\" ( I , L ) ;
( 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 - 2 * |[ a , c ]| - ( 2 * r1 - 2 * |[ b , c ]| ) = 0. TOP-REAL 2 - ( 2 * r1 - 2 * r2 ) ;
reconsider p = P * ( \square , 1 ) , q = a " * ( ( - ( - ( - ( K , n , 1 ) ) ) * ( ( - ( K , n , 1 ) ) * ( ( - ( K , n , 1 ) ) ) ) ) ) as FinSequence of K ;
consider x1 , x2 being element such that x1 in uparrow s and x2 in downarrow t and x = [ x1 , x2 ] and y = [ x1 , x2 ] ;
for n be Nat st 1 <= n & n <= len q1 holds q1 . n = ( ( upper_volume ( g , M7 ) ) | ( n + 1 ) ) . n
consider y , z being element such that y in the carrier of A and z in the carrier of A and i = [ y , z ] and x = [ y , z ] ;
given H1 , H2 being strict Subgroup of G such that x = H1 & y = H2 and H1 is Subgroup of H2 and H2 is Subgroup of H1 and H2 is Subgroup of H2 and H1 is Subgroup of H2 ;
for S , T being non empty that T , d being Function of T , S st T is complete holds d is directed-sups-preserving implies d is monotone & d is monotone & d is monotone
[ a + ( 0. F_Complex ) , b2 ] in ( the carrier of F_Complex ) /\ ( the carrier of V ) & [ a + ( 0. F_Complex ) , b2 ] in [: the carrier of V , the carrier of V :] ;
reconsider mm = max ( len F1 , len ( p . n ) * ( x |^ n ) ) , mm = max ( len F1 , len F2 ) as Element of NAT ;
I <= width GoB ( ( ( ( ( ( ( ( ( ( ( ( Y + Y ) ) ) * ( h + c ) ) / 2 ) ) * ( 2 * ( ( Y + 1 ) / 2 ) ) * ( 2 * ( 2 * ( 2 * ( Y + 1 ) ) ) ) ) + 2 ) ) ) & I <= len ( ( ( ( ( ( ( Y + 1 ) / 2 ) * ( 2 * ( Y + 1 ) ) / 2 ) ) * ( 2 * ( 2 * ( Y + 1 ) ) + 2 )
f2 /* q = ( f2 /* ( f1 /* s ) ) ^\ k .= ( f2 * ( f1 /* s ) ) ^\ k .= ( f2 * ( f1 /* s ) ) ^\ k .= ( f2 * ( f1 /* s ) ) ^\ k ;
attr A1 \/ A2 is linearly-independent means : Def1 : A1 is linearly-independent & A2 misses A2 & ( Lin ( A1 ) /\ Lin ( A2 ) ) /\ Lin ( A1 ) = (0). V & Lin ( A2 ) /\ Lin ( A1 ) = (0). V & Lin ( A1 ) /\ Lin ( A2 ) = (0). V ;
func A -carrier C -> set equals union { A . s where s is Element of R : s in C } & for x being Element of R st x in C holds s <= x ;
dom ( Line ( v , i + 1 ) (#) ( ( Line ( p , m ) ) * ( \square , 1 ) ) ) = dom ( F ^ G ) .= dom ( F ^ G ) ;
cluster [ ( x `1 ) , ( x `2 ) ] -> non empty & [ x `2 , ( x `2 ) ] `1 = x & [ x `2 , ( x `2 ) ) `2 = x `2 ;
E , f |= All ( All ( x2 , ( x2 'in' x1 ) '&' ( x2 'imp' x1 ) ) => All ( x2 , ( x2 '&' x1 ) '&' ( x2 '&' x1 ) ) ) ;
F .: ( id X , g ) . x = F . ( id X , g . x ) .= F . ( x , g . x ) .= F . ( x , g . x ) .= F . ( x , g . x ) ;
R . ( h . m ) = F . x0 + h . m - h . x0 / ( h . m ) (#) ( h . m - h . x0 ) / ( h . m ) (#) ( h . m - h . x0 ) / ( h . m - h . x0 ) ;
cell ( G , ( ( X -' 1 ) , ( Y + 1 ) ) \ L~ f ) meets ( UBD L~ f ) \/ ( ( L~ f ) \ ( L~ f ) ) ;
IC Comput ( P2 , s2 , i ) = IC Comput ( P2 , s2 , i ) .= card I .= card I + card J + 2 .= card I + card J + 2 .= card J + card J + 2 .= card J + card J + 2 ;
sqrt ( ( - ( ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ) ) ^2 ) > 0 ;
consider x0 being element such that x0 in dom a and x0 in dom a & y0 in g " { k } and y0 = a . x0 & x0 in dom g & g . x0 = b . x0 ;
dom ( r1 (#) chi ( A , A ) ) = dom ( chi ( A , A ) ) /\ dom ( chi ( A , A ) ) .= dom ( ( r1 (#) chi ( A , A ) ) | A ) .= dom ( ( r1 (#) chi ( A , A ) ) | A ) .= dom ( ( r1 (#) chi ( A , A ) ) | A ) .= A ;
d-7 . [ y , z ] = ( ( ( y , z ) `2 ) - ( ( y , z ) `2 ) ) * ( ( y , z ) `2 ) .= ( ( y , z ) `2 ) * ( ( y , z ) `2 ) ;
attr i being Nat means : Def1 : C . i = A . i /\ B . i & C . i c= C . i /\ C . i ;
assume that x0 in dom f and f is_continuous_in x0 and ||. f .|| is_continuous_in x0 and ||. f .|| is_continuous_in x0 and ||. f .|| is_continuous_in x0 & ||. f .|| is_continuous_in x0 & ||. f .|| is_continuous_in x0 & ||. f .|| is_continuous_in x0 ;
p in Cl A implies for K being Basis of p , Q being Basis of T st Q in K & A meets Q holds A meets Q
for x being Element of REAL n st x in Line ( x1 , x2 ) holds |. y1 - y2 .| <= |. y1 - y2 .| + |. y2 - x .|
func the NAT of <*> a -> w w Ordinal means : Def1 : a in it & for b being 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 ) ) \/ ( ( the carrier of C ) \/ ( the carrier of D ) ) & [ a1 , a2 , a3 ] in ( the carrier of A ) \/ ( the carrier of C ) ;
ex a , b being element st a in the carrier of S1 & b in the carrier of S2 & x = [ a , b ] & y = [ a , b ] ;
||. ( vseq . n ) - ( vseq . m ) .|| * ||. x .|| < ( e / ( ||. x .|| + 1 ) ) * ||. x .|| ;
then for Z being set st Z in { Y where Y is Element of [: I , I :] : F c= Y & Z in x } holds z in { x where x is Element of [: I , I :] : F . x = x } ;
upper_bound compactbelow [ s , t ] = [ sup ( ( sup ( { s } ) ) , sup ( ( compactbelow t ) ) ] , sup ( ( sup ( ( compactbelow t ) ) ) ) ] ;
consider i , j being Element of NAT such that i < j and [ y , f . j ] in II and [ f . i , z ] in II and [ y , z ] in II and [ z , y ] in II ;
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 <> c9 & b9 <> c9 & a , c9 // a9 , e & a , e // a9 , c9 & a , c // a9 , c9 & a , c // a9 , c9 & a , b // a9 , c9 & c , d // a9 , c9 & a , b // a9 , c9 ;
set U2 = I \! \mathop { \vert I .| } , U1 = I \! \mathop { |. I .| } , U2 = I \! \mathop { |. I .| } , U2 = I \! \mathop { |. I .| } , E = { |. I .| } , SS = { |. I .| } , SS = { |. I .| } , SS = { |. I .| } , SS = { |. I .| , E = { |. I .| } , SS = { E } , SS = { E } , SS = { E } , SS = { E } , SS = { E } , SS = { E } , SS = { E
|. q2 .| ^2 = ( |. q2 .| ) ^2 + ( |. q2 .| ) ^2 + ( |. q2 .| ) ^2 .= |. q .| ^2 + ( |. q2 .| ) ^2 .= |. q .| ^2 + ( |. q2 .| ) ^2 .= |. q .| ^2 + ( |. q2 .| ) ^2 ;
for T being non empty TopSpace , x , y being Element of [: the topology of T , the topology of T :] holds x "\/" y = x \/ y implies x "/\" y = x /\ y
dom signature U1 = dom ( the charact of U1 ) & Args ( o , MSAlg U1 ) = dom ( the charact of U1 ) & Args ( o , MSAlg U1 ) = dom ( the charact of U1 ) & Args ( o , MSAlg U1 ) = dom ( the charact of U1 ) & dom ( the charact of U1 ) = dom ( the charact of U1 ) ;
dom ( h | X ) = dom h /\ X .= dom ( ||. h .|| | X ) /\ X .= dom ( ( ||. h .|| | X ) | X ) .= dom ( ( ||. h .|| | X ) | X ) .= dom ( ( ||. h .|| | X ) | X ) .= dom ( ( ||. h .|| | X ) | X ) ;
for N1 , N1 being Element of ( the carrier of G ) * , ( h . K1 ) = N & rng ( h . K1 ) = N1 & rng ( h . K1 ) c= N1 & rng ( h . K1 ) c= N1 & rng ( h . K1 ) c= N1 & rng ( h . K1 ) c= N1 & rng ( h . K1 ) c= N1 & rng ( h . K1 ) c= N2
( ( mod ( u , m ) + mod ( v , m ) ) ) . i = ( mod ( u , m ) + mod ( v , m ) ) . i + ( mod ( v , m ) ) . i ;
- ( ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ) ^2 < - ( ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ) ^2 & - ( ( q `2 / |. q .| - cn ) / ( 1 + cn ) ) ^2 < - ( ( q `2 / |. q .| - cn ) / ( 1 + cn ) ) ^2 ;
pred r1 = fp & r2 = fp & r1 = fp & r2 = fp & for x st x in dom f holds f . x = ( f . x ) ^2 + ( f . r2 ) ^2 ;
( for m be bounded Function of X , the carrier of Y , Y st x9 . m = ( ( vseq . m ) (#) ( vseq . m ) ) . x holds ( vseq . m ) (#) ( vseq . m ) ) . x = ( ( vseq . m ) (#) ( vseq . m ) ) . x
pred a <> b & b <> c & b <> c & angle ( a , b , c ) = PI & angle ( b , c , a ) = PI & angle ( c , a , b ) = PI & angle ( c , a , b ) = PI ;
consider i , j being Nat , r , s being Real such that p1 = [ i , r ] and p2 = [ j , s ] and r < s and s < p2 and r < s ;
|. p .| ^2 - ( 2 * |( p , q )| ) ^2 + |. q .| ^2 = |. p .| ^2 + |. q .| ^2 + |. q .| ^2 ;
consider p1 , q1 being Element of [: X , Y :] such that y = p1 ^ q1 and q1 in X and p1 ^ q1 = p2 and q1 = q2 and q1 = q2 and q2 = q2 and p1 = q1 and q1 = q2 ;
( ( ( ( ( - 1 ) * ( r1 , r2 ) ) * ( r2 , s2 ) ) * ( r2 , s2 ) ) * ( r2 , s2 ) ) = ( ( ( ( - 1 ) * ( r2 , s2 ) ) * ( r2 , s2 ) ) * ( r2 , s2 ) ) ;
( ( LMP A ) `2 ) = lower_bound ( proj2 .: ( A /\ /\ /\ and ( proj2 .: ( A /\ \mathop { w } ) ) ) & proj2 .: ( ( proj2 .: ( A /\ \mathop { w } ) ) /\ ( proj2 .: ( A /\ \mathop { w } ) ) ) is non empty ;
s |= ( ( k , ( ( k , 1 ) element ) |= H1 ) iff s |= ( ( H , ( k , 1 ) ) element ) & ( ( H , ( k , 1 ) ) |= H2 ) & ( s |= ( ( H , ( k , 1 ) ) element ) ) ;
len ( s + 1 ) = card support b1 + 1 .= card support b2 + 1 .= card support b2 + 1 .= card support b2 + 1 .= card support b1 + 1 .= card support b2 + 1 .= len b1 + 1 + 1 .= len b1 + 1 + 1 .= len b1 + 1 + 1 .= len b1 + 1 + 1 ;
consider z being Element of L1 such that z >= x and z >= y and for z being Element of L1 st z >= x & z >= y holds z >= y & z >= x ;
LSeg ( UMP D , |[ ( W-bound D + E-bound D ) / 2 , ( N-bound D + ( E-bound D ) / 2 ) / 2 ]| , ( ( UMP D + ( S-bound D ) / 2 ) / 2 ) / 2 ) = { UMP D } ;
lim ( ( ( ( f `| N ) / g ) /* b ) - ( ( f `| N ) / g ) /* b ) = ( ( ( f `| N ) / g ) /* b ) / ( g `| N ) ) . 0 .= ( ( ( f `| N ) / g ) /* b ) . 0 ;
P [ i , pr1 ( f ) . i , pr1 ( f ) . ( i + 1 ) , pr2 ( f ) . ( i + 1 ) ] & pr1 ( f ) . ( i + 1 ) = pr1 ( f ) . ( i + 1 ) ;
for r be Real st 0 < r ex m be Nat st for k be Nat st m <= k holds ||. ( ( seq . k ) - ( R /* seq ) ) . k .|| < r
for X being set , P being a_partition of X , x , a , b being set st x in a & a in P & b in P & x in P & y in P & x <> a & y in P holds a = b
Z c= dom ( ( ( #Z 2 ) * ( ( #Z 2 ) * f ) - ( #Z 2 ) * f ) ) \ ( ( ( #Z 2 ) * ( ( #Z 2 ) * f ) ) " { 0 } ) ;
ex j being Nat st j in dom ( l ^ <* x *> ) & j < i & y = ( l ^ <* x *> ) . j & z = 1 + ( len l + 1 ) & z = ( l + 1 ) * ( len l + 1 ) & z = ( l + 1 ) * ( len l + 1 ) ;
for u , v being VECTOR of V , r being Real st 0 < r & r < 1 & u in dom ( holds r * u + ( 1-r * v ) in N ) holds r * u in N
A , Int Cl A , Cl Int Cl A , Cl Int Cl A , Cl Int Cl A , Cl Int Cl A , Cl Int Cl A , Cl Int Cl A , Cl Cl A , Cl Cl A , Cl Cl A , Cl Cl A , Cl Cl A , Cl Cl A , Cl Cl A , Cl Cl A , Cl Cl A , Cl Cl A , Cl Cl A , Cl Cl A , Cl Cl A , Cl Cl A , Cl Cl A , Cl Cl A , Cl Cl A , Cl ( A , Cl ( A , Cl ( A , Cl A , Cl ( A , Cl ( A , Cl ( A ,
- Sum <* v , u , w *> = - ( v + u + w ) .= - ( v + u ) + ( u + w ) .= - ( v + u ) + ( w + w ) .= - ( v + u ) + ( w + w ) .= - ( v + u ) ;
( 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 .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s ;
consider h being Function such that f . a = h and dom h = I and for x being element st x in I holds h . x in ( the carrier of J ) /\ ( the carrier of L ) and h . x = ( the carrier of J ) /\ ( the carrier of L ) ;
for S1 , S2 being non empty reflexive RelStr , D being non empty Subset of S1 , D being non empty directed Subset of S2 , x being Element of S1 , y being Element of S2 , z being Element of S2 st x in D & y in D & x <= y holds ( cos * ( x , y ) ) is directed
card X = 2 implies ex x , y st x in X & y in X & x <> y & z = y or z = x & z = y or z = y & z = x or z = y & z = x or z = y ;
E-max L~ Cage ( C , n ) in rng ( Cage ( C , n ) \circlearrowleft W-min L~ Cage ( C , n ) ) & E-max L~ Cage ( C , n ) in rng ( Cage ( C , n ) \circlearrowleft W-min L~ Cage ( C , n ) ) ;
for T , T being as as as as Tree , p , q being Element of dom T st p in dom T & q in dom T holds ( T -\hbox { p } ) . q = T . q & ( T -\hbox { q } ) . q = T . q
[ i2 + 1 , j2 ] in Indices G & [ i2 , j2 ] in Indices G & f /. k = G * ( i2 + 1 , j2 ) & f /. k = G * ( i2 + 1 , j2 ) & f /. k = G * ( i2 + 1 , j2 ) ;
cluster ( k gcd n ) divides ( k * n ) & n divides ( k * n ) & ( k divides ( k * n ) & ( k divides ( k * n ) ) & ( k divides ( k * n ) ) & ( k divides ( k * n ) ) implies ( k divides n ) & ( k divides n implies ( k divides n ) & ( k divides n ) & ( k divides n ) & ( k divides n ) & ( k divides n ) & ( k divides n ) & ( k divides n implies ( k divides n ) & ( k divides n implies ( k divides n ) & ( k divides
dom F " = the carrier of X2 & rng F " = the carrier of X1 & F " = the carrier of X2 & F " = F " * F & F " = F " * F " & F " * F = F " * F " * F " & F " * F = F " * F " * F " ;
consider C being finite Subset of V such that C c= A and card C = n and the carrier of V = Lin ( BM \/ C ) and C is linearly-independent & C is linearly-independent & C is linearly-independent & C is linearly-independent ;
V is prime implies for X , Y being Element of [: the topology of T , the topology of T :] st X /\ Y c= V & Y c= V holds X = Y or Y = V
set X = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } , Y = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } , Z = { F ( v2 ) : P [ v1 ] } , Z = { F ( v2 ) : P [ v2 ] } , Z = { Z ( ) : Q [ v1 ] } , Z = { Z ( ) where v2 is Element of B ( ) : P [ v2 ] } ;
angle ( p1 , p3 , p4 ) = 0 .= angle ( p2 , p3 , p4 ) .= angle ( p2 , p3 , p4 ) .= angle ( p3 , p4 ) + angle ( p4 , p1 , p2 ) .= angle ( p3 , p4 ) + angle ( p4 , p1 , p2 ) .= angle ( p3 , p4 ) + angle ( p4 , p1 , p2 ) ;
- sqrt ( ( - ( ( - ( q `1 / |. q .| - cn ) ) / ( 1 - cn ) ) ^2 ) ) = - sqrt ( ( - ( ( q `1 / |. q .| - cn ) ) / ( 1 - cn ) ) ^2 ) .= - ( ( - ( q `2 / |. q .| - cn ) ) / ( 1 - cn ) ) ^2 ) .= - ( - ( ( q `2 / |. q .| - cn ) ) ;
ex f being Function of I[01] , TOP-REAL 2 st f is continuous one-to-one & rng f = P & f . 0 = p1 & f . 1 = p2 & f . 1 = p2 & f . 0 = p3 & f . 1 = p4 & f . 1 = p4 & f . 1 = p4 & f . 2 = p4 & f . 3 = p4 & f . 1 = p4 & f . 3 = p4 ;
attr f is_is_is_is_differentiable on u0 means : Def1 : SVF1 ( 2 , pdiff1 ( f , 1 ) , u0 ) is continuous & SVF1 ( 2 , pdiff1 ( f , 3 ) , u0 ) . u = ( proj ( 2 , 3 ) * pdiff1 ( f , 3 ) ) . u ;
ex r , s st x = |[ r , s ]| & G * ( len G , 1 ) `1 < r & r < G * ( 1 , 1 ) `2 & s < G * ( 1 , 1 ) `2 & s < G * ( 1 , 1 ) `2 & s < G * ( 1 , 1 ) `2 ;
assume that f is_sequence_on G and 1 <= t and t <= len G and G * ( t , width G ) `2 >= N-bound L~ f and G * ( t , width G ) `2 >= N-bound L~ f and G * ( t , width G ) `2 >= N-bound L~ f and G * ( t , width G ) `2 <= N-bound L~ f ;
pred i in dom G means : Def1 : r * ( f * reproj ( i , x ) ) = r * f * reproj ( i , x ) + r * ( f * reproj ( i , x ) ) ;
consider c1 , c2 being bag of o1 + o2 such that ( decomp c ) /. k = <* c1 , c2 *> and c = c1 + c2 and ( decomp c ) /. k = c1 + c2 and ( decomp c ) /. k = c2 + c2 ;
u0 in { |[ r1 , s1 ]| : r1 < G * ( 1 , 1 ) `1 & s1 < G * ( 1 , 1 ) `2 & G * ( 1 , j ) `2 < s1 & s1 < G * ( 1 , j + 1 ) `2 } ;
( ( X ^ Y ) . k ) = the carrier of X . k2 .= ( ( C ^ Y ) . k2 ) . k .= ( C . k2 ) . k .= ( C . k2 ) . k .= ( C . k2 ) . k .= ( C . k2 ) . k ;
attr M1 = len M2 & width M1 = width M2 & M1 = M2 & M2 = M1 & M1 = M2 implies M1 - M2 = M2 - M1 & M1 - M2 = M2 - M1 & M1 - M2 = M2 - M1 & M1 - M2 = M2 - M2 & M1 - M2 = M2 - M1 & M1 - M2 = M2 - M1 ;
consider g2 being Real such that 0 < g2 and { y where y is Point of S : ||. y - x0 .|| < g2 & y in dom f & f . y = f . ( y - x0 ) + g2 & g2 in dom f } c= N2 ;
assume x < ( - b + sqrt ( delta ( a , b , c ) ) / ( 2 * a ) ) or x > ( - b - sqrt ( delta ( a , b , c ) ) / ( 2 * a ) ) ;
( G1 '&' G2 ) . i = ( <* 3 *> ^ G1 ) . i & ( H1 '&' H2 ) . 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 ) + M1 * ( i , j )
for f being FinSequence of NAT , i being Element of NAT st i in dom f & i divides f /. i holds i divides Sum ( f | ( i + 1 ) ) & i divides Sum ( f | ( i + 1 ) )
assume F = { [ a , b ] where a , b is set , c is set : for a , b being set st a in B\mathopen { \rbrack a , b .[ & b in B\mathopen { a , b .[ holds a c= c } ;
b2 * q2 + ( b3 * q3 ) + - ( ( a1 * q2 ) + ( a2 * q3 ) ) + ( ( a2 * q2 ) + ( a2 * q3 ) ) = 0. TOP-REAL n + ( ( a2 * q2 ) + ( a2 * q3 ) ) .= ( ( a2 * q2 ) + ( a2 * q3 ) ) + ( ( a2 * q2 ) + ( a2 * q3 ) ) ;
Cl ( Cl ( F ) ) = { D where D is Subset of T : ex B being Subset of T st D = Cl B & B in F & D in F } & F is closed & Cl ( Cl ( F ) ) = Cl ( Cl ( B ) ) ;
attr seq is summable means : Def1 : seq is summable & seq is summable & ( for n holds seq . n = ( seq . n ) * ( seq . n ) ) & ( seq is summable implies seq is summable & seq is summable ) implies seq is summable & seq is summable & lim seq = Sum ( seq ) + Sum ( seq )
dom ( ( ( ( ( ( ( ( TOP-REAL 2 ) | D ) ) | D ) ) | D ) ) = ( the carrier of ( ( TOP-REAL 2 ) | D ) ) /\ D ) .= the carrier of ( ( ( TOP-REAL 2 ) | D ) | D ) .= D ;
[: X , Z :] is full full non empty SubRelStr of [: Y , Z :] |^ the carrier of Z & [ X \to Y ] is full SubRelStr of [: Y , Z :] |^ the carrier of Z ;
G * ( 1 , j ) `2 = G * ( i , j ) `2 & G * ( 1 , j ) `2 <= G * ( 1 , j ) `2 & G * ( 1 , j ) `2 <= G * ( 1 , j ) `2 & G * ( 1 , j ) `2 <= G * ( 1 , j + 1 ) `2 ;
synonym m1 c= m2 means : Def1 : for p being set st p in P holds the } is the carrier of ( m , n ) \rm the carrier of ( m , n ) | the carrier of ( m , n ) = ( the $ m2 ) | the carrier of ( m , n ) | the carrier' of ( m , n ) ;
consider a being Element of B ( ) such that x = F ( a ) and a in { G ( b ) where b is Element of A ( ) : P [ b ] } and a in A ( ) ;
synonym mode multiplicative loop Let R -> non empty multMagma means : Def1 : the multMagma of it = [ ( the carrier of R ) , ( the multF of R ) ] means the multF of it = [ the carrier of R , ( the multF of R ) ] ;
sequence ( a , b ) + 1 + and and and sequence ( c , d ) = b + and sequence ( c , d ) = b + d + 1 .= b + ( c + d ) .= ( sequence ( a + c ) + ( d + d ) ) + ( ( d + c ) + ( d + c ) ) ;
cluster + ( i1 , i2 ) -> INT -valued for Element of INT , i , j be Element of INT , i1 , i2 be Element of INT , i2 be Element of INT st i1 = i2 & i2 = i2 & i1 = i2 & i2 = i2 holds ( i1 + i2 ) + ( i2 + 1 ) = i2 + ( i1 + i2 ) + ( i2 + 1 ) ;
( - s2 ) * p1 + ( s2 * p2 - ( s2 * p2 ) ) = ( ( - s2 ) * p1 + ( s2 * p2 ) ) * ( p1 - p2 ) + ( s2 * p2 ) * ( p1 - p2 ) .= ( ( ( - s2 ) * p1 ) + ( s2 * p2 ) ) * ( p1 - p2 ) ;
eval ( ( a | ( n , L ) ) *' p , x ) = eval ( a | ( n , L ) , x ) * eval ( p , x ) .= a * eval ( p , x ) * eval ( p , x ) .= a * eval ( p , x ) * eval ( q , x ) .= a * eval ( p , x ) ;
assume that the TopStruct of S = the TopStruct of T and for D being non empty directed Subset of S , V being open Subset of S st V in the topology of S & V = V holds V meets V and for W being open Subset of S st W in V & W is open & V is open holds W is open & V is open & W is open & V is open ;
assume that 1 <= k & k <= len w + 1 and T-7 . ( ( ( ( T , w ) | ( k + 1 ) ) | ( k + 1 ) ) ) = ( ( T , w ) | ( k + 1 ) ) | ( k + 1 ) ) | ( k + 1 ) ;
2 * a |^ ( n + 1 ) + ( 2 * b |^ ( n + 1 ) ) >= ( a |^ n + ( b |^ n ) * b |^ n ) + ( ( a |^ n ) * b |^ n ) + ( ( a |^ n ) * b |^ n ) + ( ( a |^ n ) * b |^ n ) ;
M , v2 |= All ( x. 3 , All ( x. 0 , All ( x. 4 , All ( x. 0 , All ( x. 4 , All ( x. 0 , All ( x. 0 , All ( x. 0 , All ( x. 0 , All ( x. 0 , All ( x. 0 , x. 0 , x. 0 ) ) ) ) ) ) ) ) ) ) ) ;
assume that f is_differentiable_on l and for x0 st x0 in l holds 0 < f . x0 & f . x0 < 0 or for x0 st x0 in l holds f . x0 < 0 & f . x0 < 0 & f . x0 < 0 ;
for G1 being _Graph , W being Walk of G1 , e being set , W being Walk of G2 , e being set st e in W .vertices() & e in W .vertices() & W is Walk of G holds e in W implies W is Walk of G
not ( not c is not empty iff not iff iff not ( not f1 is not empty & not f2 is not empty & not f1 is not empty & not f2 is not empty & not f1 is not empty & not f2 is not empty & not f1 is not empty & not f2 is not empty ) & not not not f1 is not empty & not f2 is not empty & not f2 is not empty ;
Indices GoB f = [: dom GoB f , Seg width GoB f :] & ( GoB f ) * ( i1 + 1 , j1 ) in [: dom GoB f , Seg width GoB f :] & ( GoB f ) * ( i1 + 1 , j1 ) in [: dom GoB f , Seg width GoB f :] & ( GoB f ) * ( i1 + 1 , j1 ) in [: dom GoB f , Seg width GoB f :] ;
for G1 , G2 , G3 being Group , O being stable Subgroup of G1 , G2 being stable Subgroup of G2 , A being set st G1 is stable & G2 is stable & A is stable & A is stable & B is stable holds ( G1 * G2 ) * A = ( G1 * A ) * ( G2 * A )
UsedIntLoc ( in4 ( f ) ) = { intloc 0 ( 1 ) , intloc 1 ( 2 ) , intloc 0 ( 2 ) , intloc 0 ( 2 ) , intloc 0 ( 2 ) , intloc 0 ( 2 ) , intloc 0 ( 2 ) , intloc 0 ( 2 ) , intloc 0 ( 2 ) , intloc 0 ( 2 ) , intloc 0 ( 2 ) , intloc 0 ( 2 ) , intloc 0 ( 2 ) , intloc 0 ( 2 ) , intloc 0 ) ;
for f1 , f2 be FinSequence of F st f1 ^ f2 is p -element & Q [ p ^ <* p *> ] & Q [ p ^ <* p *> ] & P [ p ^ <* p *> ] holds P [ f1 ^ f2 ] & Q [ p ^ <* p *> ] & Q [ p ^ <* p *> ] ;
( ( p `1 ) / sqrt ( 1 + ( p `2 / p `1 ) ^2 ) ) ^2 = ( ( q `1 ) / sqrt ( 1 + ( q `2 / q `1 ) ^2 ) ) ^2 + ( ( q `2 ) / sqrt ( 1 + ( q `2 / q `1 ) ^2 ) ) ^2 ;
for x1 , x2 , x3 being Element of REAL n holds |( x1 - x2 , x3 )| = |( x1 , x3 )| + |( x2 , x3 )| + |( x1 , x3 )| + |( x2 , x3 )| + |( x2 , x3 )| + |( x3 , x4 )| + |( x2 , x3 )| + |( x3 , x4 )| + |( x3 , x3 )| + |( x3 , x3 )| + |( x3 , x3 )|
for x st x in dom ( ( ( ( ( ( ( ( ( ( ( x - x ) ) ) | A ) ) | A ) ) | A ) ) ) ) holds ( ( ( ( ( ( ( ( x - x ) | A ) ) | A ) ) | A ) ) | A ) . x = - ( ( ( ( ( ( x - x ) | A ) | A ) | A ) ) . x )
for T being non empty TopSpace , P being Subset-Family of T , B being Basis of T st P c= the topology of T for x being Point of T st P c= B ex B being Basis of x st B c= P & x in B & P is Basis of B
( a 'or' b 'imp' c ) . x = 'not' ( ( a 'or' b ) . x ) 'or' c . x .= 'not' ( ( a . x ) 'or' c . x ) 'or' c . x .= TRUE 'or' ( ( a . x ) 'or' c . x ) 'or' c . x .= TRUE 'or' ( ( a . x ) 'or' c . x ) 'or' c . x .= 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 & Y2 is open & Y1 is open & Y1 is open & Y1 is open & Y2 is open
for i being set st i in the carrier of S for f being Function of [: S , T :] , 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 D1 - card { i , j } - 1 .= c1 + 1 - 1 + 1 - 1 .= c1 + 1 - 1 + 1 - 1 .= c1 + 1 - 1 + 1 - 1 .= c1 + 1 - 1 + 1 - 1 .= c1 + 1 - 1 + 1 - 1 .= c1 + 1 - 1 + 1 .= c1 + 1 - 1 + 1 ;
IC Exec ( i , s ) = ( s +* ( 0 .--> succ ( s . 0 ) ) ) . 0 .= ( 0 .--> ( s . 0 ) ) . 0 .= ( ( 0 .--> s . 0 ) ) . 0 .= ( ( 0 .--> s . 0 ) +* ( ( 0 .--> s . 0 ) ) ) . 0 .= ( ( 0 .--> s . 0 ) .--> ( ( 0 .--> s . 0 ) ) . 0 ) .= ( ( 0 .--> s ) . 0 ) . 0 .= ( ( 0 .--> s . 0 ) . 0 ) .= ( ( 0 .--> ( s . 0 ) . 0 .= ( ( 0 .--> s . 0 ) . 0 .= ( ( 0 .--> ( s . 0 ) ) . 0 .=
len f /. ( \downharpoonright i1 -' 1 + 1 ) = len f -' i1 + 1 - 1 + 1 .= len f -' i1 + 1 - 1 + 1 .= len f -' i1 + 1 - 1 + 1 .= len f -' 1 + 1 - 1 + 1 .= len f -' 1 + 1 - 1 + 1 .= len f - 1 + 1 - 1 + 1 ;
for a , b , c being Element of NAT st 1 <= a & 2 <= b & k < a or a <= b & k = a + b-2 or k = a + b-2 or k = a + b-2 or k = a + b + a or k = a + b + a ;
for f being FinSequence of TOP-REAL 2 , p being Point of TOP-REAL 2 , i being Element of NAT st p in LSeg ( f , i ) & i <= len f & i <= len f & p in LSeg ( f , i ) holds Index ( p , f ) <= Index ( p , f )
( curry ( ( curry ( ( P , k + 1 ) ) # x ) ) # x ) = ( ( curry ( ( curry ( P , k ) ) # x ) ) . x ) + ( ( curry ( ( curry ( P , k ) ) # x ) # x ) ) . x ;
z2 = g /. ( \downharpoonright n1 -' n2 + 1 ) .= g . ( i -' n2 + 1 + 1 ) .= g . ( i -' n2 + 1 + 1 ) .= g . ( i -' n2 + 1 + 1 ) .= g . ( i -' n2 + 1 + 1 ) .= g . ( i -' n2 + 1 + 1 ) .= g . ( i + 1 + 1 ) ;
[ f . 0 , f . 3 ] in id ( the carrier of G ) \/ ( the InternalRel of G ) or [ f . 0 , f . 3 ] in the InternalRel of G & [ f . 0 , f . 3 ] in the InternalRel of G & [ f . 2 , f . 3 ] in the InternalRel of G & [ f . 1 , f . 3 ] in the InternalRel of G ;
for G being Subset-Family of B st G = { R [ X ] where X is Subset of A ( ) , Y is Subset of B ( ) st X in F & Y in F & X c= Y holds ( Intersect G ) . X = Intersect G & ( Intersect G ) . Y = Intersect G & ( Intersect G ) . Y = Intersect G )
CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 ) ) = CurInstr ( P1 , Comput ( P1 , s1 , m1 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m2 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m2 ) ) .= halt SCMPDS .= halt SCMPDS ;
assume that a on M and b on M and c on N and d on M and p on N and c on M and d on N and p on P and c on P and d on P and p on Q and c on Q and d on P and p on Q and d on P and c on Q and d on P and d on Q and p on Q ;
assume that T is \hbox { T _ 4 } and F is closed and ex F being Subset-Family of T st F is closed & F is finite-ind & ind F <= 0 & ind T <= 0 and ind T <= 0 and ind T <= 0 ;
for g1 , g2 st g1 in ]. r - s , r .[ & g2 in ]. r - s .[ holds |. ( f . g1 ) - ( f . g2 ) .| <= ( ( f - g ) / ( r - s ) ) / ( r - s )
( ( - ( ( - 1 ) / ( n + 2 ) ) * ( ( - 1 ) / ( n + 1 ) ) ) * ( ( - 1 ) / ( n + 2 ) ) ) = ( ( - 1 ) / ( n + 1 ) ) * ( ( - 1 ) / ( n + 2 ) ) * ( ( - 1 ) / ( n + 2 ) ) ;
F . i = F /. i .= 0. R + r2 .= ( b |^ n ) + ( a |^ n ) .= ( b |^ n ) + ( a |^ n ) .= ( ( ( n + 1 ) ) / ( n + 1 ) ) * ( ( a |^ n ) / ( n + 1 ) ) .= ( ( b |^ n ) |^ n ) / ( n + 1 ) ;
ex y being set , f being Function st y = f . n & dom f = A ( ) & f . 0 = R ( ) & for n holds f . ( n + 1 ) = R ( n , f . n ) & f . ( n + 1 ) = R ( n , f . n ) ;
func f * F -> FinSequence of V means : Def1 : len it = len F & for i be Nat st i in dom it holds it . i = F /. i * ( F /. i ) & for i be Nat st i in dom it holds it . i = F /. ( F /. i ) * ( F /. i ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 } = { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 } \/ { x5 , x5 } \/ { x5 , 6 } ;
for n being Nat , 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 ) & o ( x , n ) in InnerVertices S ( x , n )
ex S1 being Element of CQC-WFF ( Al ) st SubP ( P , l , e ) = S1 & ( for S1 being Element of CQC-WFF ( Al ) st S1 = S2 & S1 is non empty & S2 is non empty & S1 is non empty & S2 is non empty & S1 is non empty & S2 is non empty & S1 is non empty & S2 is non empty & S2 is non empty & S2 is non empty & S2 is non empty & S2 is non empty & S2 is non empty & S2 is non empty & S2 is non empty & S2 is non empty & S2 is non empty & S2 is non empty & S2 is non empty & S2 is non empty & S2 is non empty & S2 is non empty & S2 is non empty & S2 is non empty & S2 is non empty & S2 is non empty & S2 is non empty & S2 is
consider P being FinSequence of Gq2 such that p9 = Product P and for i being Element of NAT st i in dom P ex t being Element of the carrier of G st P . i = t & t is i -element & P . i = ( t . i ) * ( t . i ) ;
for T1 , T2 being non empty TopSpace , P being Basis of T1 , T2 being Basis of T2 st the carrier of T1 = the carrier of T2 & P = the topology of T2 & P = the topology of T2 & P = the topology of T2 & P = the topology of T2 & P = the topology of T2 & P = the topology of T2 holds P is Basis of T1 & P is Basis of T2
assume that f is_is_finite , u0 and r (#) pdiff1 ( f , 3 ) is_partial_differentiable_in u0 , 2 and partdiff ( r (#) pdiff1 ( f , 3 ) , u0 , 3 ) = r * pdiff1 ( f , 3 ) . u0 & partdiff ( r (#) pdiff1 ( f , 3 ) , u0 , 3 ) = r * pdiff1 ( f , 3 ) . u0 ;
defpred P [ Nat ] means for F , G be FinSequence of REAL , G be Permutation of Seg $1 , s be Permutation of rng F st len s = $1 & G = F * s & not ( ex F be Permutation of Seg $1 st F = G * s & F is one-to-one & s = F * s ) & ( for i being Nat st i in rng F holds F . i = G . i ) & ( F is one-to-one implies F is one-to-one ) ;
ex j st 1 <= j & j < width GoB f & ( GoB f ) * ( 1 , j ) `2 <= s & s * ( 1 , j ) `2 < ( GoB f ) * ( 1 , j + 1 ) `2 & s * ( 1 , j + 1 ) `2 < ( GoB f ) * ( 1 , j + 1 ) `2 ;
defpred U [ set , set ] means ex Fa1 be Subset-Family of T st $2 = Fa1 & ( union Fa1 is open & union Fa1 is open & union Fa1 is open & union Fa1 is open & union Fa1 is open & union Fa1 is closed ) & ( union Fa1 is closed & union Fa1 is closed & union Fa1 is closed ) & ( union F is discrete discrete & union F is discrete discrete ) ;
for p4 being Point of TOP-REAL 2 st LE p4 , p1 , P , p1 , p2 & LE p4 , p1 , P , p1 , p2 & LE p4 , p1 , P , p1 , p2 & LE p4 , p1 , P , p1 , p2 & LE p4 , p1 , P , p1 , p2 holds LE p4 , p1 , P , p1 , p2 & LE p4 , p1 , P , p1 , p2
f in ) & for H st for y st g . y <> f . y holds x in Free ( E , H ) & g in ( ( the Sorts of H ) . ( g . y ) implies f in ( the Sorts of H ) . ( g . x ) & f in ( the Sorts of H ) . ( g . y ) & g in ( the Sorts of H ) . ( g . y )
ex 8 being Point of TOP-REAL 2 st x = 8 & ( ( ( ( ( ( ( ( ( ( ( p `2 ) / |. p .| - sn ) ) / |. p .| - sn ) ) / ( 1 - sn ) ) / ( 1 - sn ) ) ) * ( 1 - sn ) ) & ( ( ( ( ( ( ( p `2 / |. p .| - sn ) / ( 1 - sn ) ) / ( 1 - sn ) ) / ( 1 - sn ) ) ^2 ) >= 0 ) & ( ( ( ( p `2 / ( 1 - sn ) ) * ( 1 - sn ) ) / ( 1 - sn ) ) / ( 1 - sn ) ) * ( 1 - sn ) ) / ( 1 - sn ) ) ^2 >= 0 & ( ( 1 - sn ) ) * ( 1 - sn ) ) ^2 >= 0 & ( ( ( p `2 / ( 1 - sn ) ) * (
assume for d7 being Element of NAT st d7 <= d7 holds ( s1 . d7 = s2 . ( d7 ) & s2 . ( d7 ) = s2 . ( d7 ) & ( s2 . ( d7 ) = s2 . ( d7 ) ) & ( s2 . ( d7 ) = s2 . ( d7 ) ) ;
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 ) & t = Sphere ( s , t ) /\ Sphere ( x , r ) ;
given r such that 0 < r and for s st 0 < s holds not 0 < s or ex x1 be Point of CNS st x1 in dom f & ||. x1 - x0 .|| < s & |. f /. x1 - f /. x0 .|| < r & |. f /. x0 - f /. x0 .| < r ;
( p | x ) | ( p | ( x | x ) ) = ( ( ( x | x ) | x ) | ( x | x ) ) | p .= ( ( ( x | x ) | x ) | p ) | p .= ( ( ( x | x ) | x ) | p ) | p ;
assume that x , x + h / 2 in dom sec and ( for x st x in dom sec holds ( ( sec * sec ) `| Z ) . x = ( ( 4 * sin . x + h / 2 ) * sin . x ) ^2 + ( ( cos * sin ) `| Z ) . x ) ^2 and ( ( cos * sin ) `| Z ) . x = ( ( 4 * sin . x + h / 2 ) / ( cos . x ) ^2 ) ;
assume that i in dom A and len A > 1 and B c= the set of ( the set of A ) and A c= the set of ( the m of A ) \/ the m of ( the m of B ) and B = ( the 0. of A ) \/ ( the 0. K ) and B = ( the 0. K ) \/ ( the 0. K ) ;
for i being non zero Element of NAT st i in Seg n holds i divides n or i = <* 1. F_Complex , n *> & ( i divides n & n divides n & i <> 0 implies h . i = ( ( 1. F_Complex ) * ( i , n ) ) * ( h . n ) ) & ( i divides n implies h . i = ( ( 1. F_Complex ) * ( i , n ) ) * ( h . n ) )
( ( ( b1 'imp' b2 ) '&' ( c1 'imp' c2 ) '&' ( b1 'or' c2 ) '&' ( c1 '&' c2 ) ) '&' ( ( b1 'or' b2 ) '&' ( b1 'or' c2 ) '&' ( c1 '&' c2 ) '&' ( a1 'or' a2 ) '&' ( b1 'or' a2 ) '&' ( b1 'or' b2 ) '&' ( b1 'or' b2 ) '&' ( b1 'or' b2 ) '&' ( b1 'or' b2 ) '&' ( b1 'or' b2 ) '&' ( b1 'or' b2 ) '&' ( b1 'or' b2 ) '&' ( b1 'or' b2 ) '&' ( b1 'or' b2 ) '&' ( a1 'or' b2 ) '&' ( b1 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a2 'or' b2 ) '&' ( a1 'or' b2 ) '&' ( a1 'or' b2 ) '&' ( a1 'or' b2 ) '&' ( a2 'or' b2
assume that for x holds f . x = ( ( cot * sin ) `| Z ) . x and x in dom ( ( cot * sin ) `| Z ) & for x st x in Z holds ( ( cot * sin ) `| Z ) . x = - 1 / ( sin . x ) ^2 and for x st x in Z holds ( ( cot * sin ) `| Z ) . x = - 1 / ( sin . x ) ^2 and f . x = 1 / ( sin . x ) ^2 and f . x = 1 / ( sin . x ) ^2 and f . x = 1 / ( sin . x ) ^2 and f . x = 1 / ( sin . x ) ^2 and f . x = - 1 / ( sin . x ) ^2 and f . x = 1 / ( sin . x - cos . x - 1 / ( sin . x - 1 / ( sin . x ) ^2 and f . x = 1
consider R8 , I-8 be Real such that R8 = Integral ( M , Re ( F . n ) ) and I8 = Integral ( M , Im ( F . n ) ) and Integral ( M , Im ( F . n ) ) = Integral ( M , Im ( F . n ) ) + Integral ( M , Im ( F . n ) ) ;
ex k being Element of NAT st ( ex q being Element of product G st q = k & 0 < d & for q being Element of product G st q in X & ||. q-r .|| < d holds ||. partdiff ( f , q , k ) - partdiff ( f , x , k ) .|| < r ) & ||. partdiff ( f , q , k ) .|| < r ;
x in { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 } iff x in { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 } \/ { x5 , x5 }
G * ( j , ( G * ( j , i ) ) `2 ) = G * ( 1 , ( G * ( j , i ) ) `2 ) .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j ) `2
f1 * p = p .= ( ( the Arity of S1 ) * g ) . o .= ( ( the Arity of S2 ) * g ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S2 ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S2 ) . o ;
func \vert \vert T , P , T1 , T2 ] -> DecoratedTree means : Def1 : q in it iff q in P & for p st p in P holds p in T or ex q st q in P & q in T & p = q or p = q or q = p & q = p ;
F /. ( k + 1 ) = F . ( k + 1 -' 1 ) .= Fx0 ( p . ( k + 1 -' 1 ) , k + 1 -' 1 ) .= Fx0 ( p . ( k + 1 -' 1 ) , k + 1 ) .= Fx0 ( p . k , k + 1 -' 1 ) .= Fx0 ( p . k , k + 1 -' 1 ) .= Fx0 ( p . k , k + 1 ) ;
for A , B , C being Matrix of K st len B = len C & width B = width C & len B = width C & len C > 0 & len A > 0 & len B > 0 & len C > 0 & len A > 0 & len B > 0 & width A > 0 & width B = 0 & A = C * B & C * B = C * B + C * C * A
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 ) \/ ( the carrier of Cq ) and y in ( the carrier of Cq ) \/ ( the carrier of Cq ) and z = [ x , y ] and [ y , z ] in the InternalRel of Cq and [ x , z ] in the InternalRel of Cq ;
defpred P [ Element of NAT ] means for f st len f = $1 & ( for k st k = $1 holds ( ( VAL g ) . k = ( ( VAL g ) | 1 ) . k ) holds ( ( ( VAL g ) | ( k -' 1 ) ) ) . f = ( ( VAL g ) | ( k -' 1 ) ) . f ) '&' ( ( 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 > 0 and ( q `2 / |. q .| - sn ) / ( 1 + sn ) >= 0 and ( q `2 / |. q .| - sn ) / ( 1 + sn ) >= 0 and ( q `2 / |. q .| - sn ) / ( 1 + sn ) >= 0 and ( q `2 / |. q .| - sn ) / ( 1 + sn ) >= 0 ;
for M being non empty dist , x being Point of M , f being Point of M st x = x `2 holds ex x being Point of M st f . x = Ball ( x `1 , ( 1 / 2 ) * ( x `2 ) ) & f . x = Ball ( x `2 , ( 1 / 2 ) * ( x `2 ) )
defpred P [ Element of omega ] means f1 is_differentiable_on Z & f2 is_differentiable_on Z & ( for x st x in Z holds f1 . x = - x ) & ( for x st x in Z holds f1 . x = - x ) & ( for x st x in Z holds f1 . x = - x ) & ( f1 - f2 ) is_differentiable_on Z & ( f1 - f2 ) is_differentiable_on Z implies f1 is_differentiable_on Z & f2 is_differentiable_on Z & ( f1 - f2 ) is_differentiable_on Z & f2 is_differentiable_on Z & ( f1 - f2 ) | Z & ( f1 is_differentiable_on Z & f2 is_differentiable_on Z & f2 is_differentiable_on Z & ( f1 - f2 ) | Z & ( f1 - f2 ) | Z implies ( f1 - f2 ) | Z & ( f1 - f2 ) | Z & ( f1 - f2 ) . x = ( f1 - f2 ) . x = ( f1 - f2 ) . x = ( f1 - f2 ) . x = ( f1 - f2 ) . x = ( ( f1 - f2
defpred P1 [ Nat , Point of CNS ] means $2 in Y & ||. ( f /. $1 ) - ( f /. ( $1 + 1 ) ) .|| < r & ||. ( f /. ( $1 + 1 ) ) - ( f /. ( $1 + 1 ) ) .|| < r ;
( f ^ mid ( g , 2 , len g ) ) . i = ( mid ( g , 2 , len g ) ) . i .= ( g . ( i -' len g ) ) + ( g . ( i + 1 ) ) .= g . ( i -' len g ) + ( g . ( i + 1 ) ) .= g . ( i + 1 ) + ( g . ( i + 1 ) ) .= g . ( i + 1 ) ;
( 1 - 2 * ( 2 * n0 + 2 * ( 2 * n0 + 2 * n0 ) ) ) * ( 2 * ( 2 * n0 + 2 * n0 ) ) = ( ( 1 - 2 * ( 2 * n0 + 2 * n0 ) ) * ( 2 * n0 + 2 * ( 2 * n0 + 1 ) ) ) * ( 2 * ( 2 * n0 + 1 ) ) * ( 2 * n0 + 1 ) ) * ( 2 * ( 2 * n0 ) ) * ( 2 * ( 2 * n0 ) ) .= ( 1 / 2 * ( 2 * n0 ) * ( 2 * ( 2 * n0 ) ) * ( 2 * ( 2 * n0 ) ) * ( 2 * ( 2 * n0 ) * ( 2 * n0 ) .= ( 1 / 2 * ( 2 * n0 ) * ( 2 * n0 ) + 2 * ( 2 * n0 ) + 2 * ( 2 * n0 ) ) * ( 2 * ( 2 * n0 ) * ( 2 * n0 ) + 2 * ( 2
defpred P [ Nat ] means for G being non empty finite strict non empty finite strict strict non empty RelStr st G is space & card the carrier of G = $1 & the carrier of G in the carrier of G & the carrier of G in the carrier of G & the InternalRel of G in the carrier of G & the InternalRel of G in the carrier of G & the InternalRel of G in the carrier of G & the InternalRel of G in the carrier of G ;
assume that f /. 1 in Ball ( u , r ) and 1 <= m and m <= len ( - 1 ) and for i st 1 <= i & i <= len ( f | ( i + 1 ) ) /\ Ball ( u , r ) & ( not i in Ball ( u , r ) & not i in Ball ( u , r ) holds m . i <> 0 ) ;
defpred P [ Element of NAT ] means ( Partial_Sums ( ( cos * ( ( cos - r ) / ( 2 * $1 ) ) ) ) . ( 2 * $1 ) = ( Partial_Sums ( ( cos * ( ( cos - r ) / ( 2 * $1 ) ) ) ) . ( 2 * $1 ) ) * ( ( cos * ( ( cos - r ) / ( 2 * $1 ) ) / ( 2 * $1 ) ) ;
for x being Element of product F holds x is FinSequence of G & dom x = I & x . i = ( the carrier of F ) . i & for i being set st i in dom F holds x . i = ( the carrier of F ) . i & for i being set st i in dom F holds x . i = ( the carrier of F ) . i
( x " ) |^ ( n + 1 ) = ( ( x " ) |^ n ) * x .= ( ( x |^ n ) |^ n ) * x .= ( ( x |^ n ) |^ n ) * x .= ( ( x |^ n ) |^ n ) * x .= ( ( x |^ n ) |^ n ) * x .= ( ( x |^ n ) |^ n ) * x .= ( ( x |^ n ) |^ n ) |^ n ;
DataPart Comput ( P +* ( I , P +* I , Initialize s ) , LifeSpan ( P +* I , Initialize s ) + 3 ) = DataPart Comput ( P +* I , Initialize s , LifeSpan ( P +* I , Initialize s ) + 3 ) .= DataPart Comput ( P +* I , Initialize s , LifeSpan ( P +* I , Initialize s ) + 3 ) ;
given r such that 0 < r and ]. x0 , x0 + r .[ c= ( dom f1 /\ dom f2 ) /\ ( ]. x0 , x0 + r .[ ) and for g st g in ]. x0 , x0 + r .[ /\ dom f2 holds f1 . g <= ( f2 * f1 ) . g & f2 . g <= ( f2 * 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 f2 holds f1 | X is continuous ) and ( for r st r in X /\ dom f2 holds f2 | X is continuous ) & ( for r st r in X /\ dom f2 holds f2 | X is continuous ) implies f2 | X is continuous & ( f2 | X ) is continuous & ( f2 | X is continuous & ( f2 | X is continuous & ( f1 | X is continuous & f2 | X is continuous & f2 | X is continuous & f2 | X is continuous & ( f2 | X is continuous & ( f2 | X is continuous & ( f1 | X is continuous & ( f2 | X is continuous & ( f1 | X is continuous & f2 | X is continuous & ( f1 | X is continuous & ( f1 | X is continuous & f2 | X is continuous & f2 | X is continuous & f2 | X is continuous & ( f2 | X is continuous & ( f2 | X is continuous & ( f1 | X is
for L being continuous complete LATTICE for l being Element of L ex X being Subset of L st l = sup X & for x being Element of L st x in X holds x is Element of L & x is an & x is an & x is an & x is an L implies x is directed
Support ( e *' p ) in { Support ( m *' p ) where m is Polynomial of n , L : ex i being Element of NAT st i in dom ( m *' p ) & ( m *' p ) . i = ( m *' p ) . i & ( m *' p ) . i = ( m *' p ) . i ;
( f1 - f2 ) /* s1 = ( lim ( f1 /* s1 ) - ( f2 /* s1 ) ) - ( lim ( f2 /* s1 ) ) .= lim ( ( ( f1 /* s1 ) - ( f2 /* s1 ) ) ) - ( ( f2 /* s1 ) - ( ( f2 /* s1 ) - ( f2 /* s1 ) ) ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) ;
ex p1 being Element of CQC-WFF ( Al ) st F . p1 = g `2 & for g being Function of [: Al ( ) , D ( ) :] , D ( ) st P [ g ] & Q [ g , h ] holds P [ g , h . ( len F ) ] ;
( mid ( f , i , len f -' 1 ) ^ <* f /. ( len f -' 1 ) *> ) /. j = ( mid ( f , i , len f -' 1 ) ^ mid ( f , i , len f -' 1 ) ) /. j .= ( mid ( f , i , len f -' 1 ) ) /. j .= ( mid ( f , i , len f -' 1 ) ) /. j .= ( mid ( f , i , len f -' 1 ) /. j ;
( ( p ^ q ) ^ r ) . ( len p + k ) = ( ( p ^ q ) . ( len p + k ) ) . ( len q + k ) .= ( ( p ^ q ) . ( len p + k ) ) . ( len q + k ) .= ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( p . ( len p + k ) .= ( p . ( len p + k ) + ( p . ( len p + k ) .= ( p . ( len p + k ) + ( p . ( len p + k ) + ( p . ( len p + k ) .= ( p . ( len p + k ) .= ( p . ( len p + k ) + ( len q + k ) + ( p . ( len p + k ) + ( p . ( len p + k ) + ( p . ( len p + k ) .= ( p . ( len p + k ) + (
len mid ( upper_volume ( f , D2 ) , indx ( D2 , D1 , j1 ) + 1 , indx ( D2 , D1 , j ) ) = indx ( D2 , D1 , j1 ) - ( indx ( D2 , D1 , j1 ) + 1 ) + 1 .= indx ( D2 , D1 , j1 ) + 1 - 1 .= indx ( D2 , D1 , j1 ) + 1 - 1 ;
x * y * z = Mz . ( x * y , z ) .= x * ( y * z ) .= ( x * y ) * ( z * z ) .= ( x * y ) * ( z * z ) .= ( x * y ) * ( z * z ) .= ( x * y ) * ( z * z ) .= ( x * y ) * ( z * z ) ;
v . <* x , y *> + ( <* x0 , y0 *> ) * i = partdiff ( v , ( x - x0 ) * ( x - x0 ) + ( y - 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 - 0 ) * ( 0 * 0 ) + 0 * ( 0 * 0 ) + 0 * ( 0 * 0 ) + 0 * ( 0 * 0 ) + 0 * ( 0 * 0 ) .= <* - 1 * 0 , 0 * 0 , 0 * 0 + 0 * 0 , 0 * 0 + 0 * 0 + 0 * 0 , 0 * 0 + 0 * 0 + 0 * 0 , 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 , 0 * 0 + 0 * 0 , 0 * 0 + 0 * 0 , 0 * 0 + 0 * 0 * 0 , 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0
Sum ( L (#) F ) = Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( ( L (#) F1 ) ^ ( L (#) F2 ) ) .= Sum ( ( L (#) F1 ) ^ ( L (#) F2 ) ) + Sum ( ( L (#) F2 ) ) .= Sum ( L (#) F1 ) + Sum ( L (#) F2 ) .= Sum ( L (#) F1 ) + 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 (#) F2 ) + Sum ( L (#) F2 ) + Sum ( L (#) F1 ) .= Sum ( L (#) F1 ) + Sum ( L (#) F1 ) + Sum ( L (#) F1 ) + Sum ( L (#) F1 ) + Sum ( L (#) F1 ) + Sum ( L (#) F2 ) .= Sum ( L (#) F1 ) + Sum ( L (#) F1 ) + Sum ( L (#) F2 ) .= Sum
ex r be Real st for e be Real st 0 < e ex Y be finite Subset of X st Y is non empty & Y c= Y & for Y1 be finite Subset of X st Y1 c= Y & Y1 c= Y holds |. ( - 1 ) * ( Y - 1 ) .| < r * ( Y - 1 ) * ( Y - 1 ) ;
( GoB f ) * ( i , j + 1 ) = f /. ( k + 2 ) & ( GoB f ) * ( i , j + 1 ) = f /. ( k + 1 ) or ( GoB f ) * ( i , j ) = f /. ( k + 1 ) & ( GoB f ) * ( i , j + 2 ) = f /. ( k + 2 ) ;
( ( cos . x ) ^2 ) = ( ( r - 1 ) * ( cos . x ) ^2 + ( cos . x ) ^2 .= ( ( r - 1 ) * ( cos . x ) ) ^2 + ( ( cos . x ) ^2 ) .= ( ( r - 1 ) * ( cos . x ) ) ^2 + ( ( r - 1 ) * ( cos . x ) ^2 ) .= ( r * ( cos . x ) ^2 ) ;
- ( - b + sqrt ( delta ( a , b , c ) ) / ( 2 * a ) ) < 0 & - ( - b + sqrt ( delta ( a , b , c ) ) / ( 2 * a ) ) < 0 or - ( - ( - b + sqrt ( 2 * a , c ) ) / ( 2 * a ) ) / ( 2 * a ) < 0 ;
assume that ex_inf_of uparrow "\/" ( X , L ) , L and ex_sup_of X , L and ex_sup_of X , L and "\/" ( Y , L ) , L & "\/" ( X , L ) = "/\" ( uparrow "\/" ( X , L ) , L ) and for x being Element of L holds x < sup ( uparrow x /\ L ) & x <= sup ( uparrow x /\ L ) ;
( ( for j being Element of NAT st j in the Sorts of B ) . i = ( j = i ) |-- id ( ( B , i ) , ( B , j ) ) ** id ( ( B , i ) .: ( ( B , j ) .: ( ( B , i ) .: ( B , j ) ) ) ) = ( ( i , j ) ** id ( ( B , i ) .: ( B , j ) ) ) ** ( ( B , i ) , ( A , j ) ) ;