thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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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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thesis ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b
D c= S
let Y ;
S ` is convergent
q in X ;
V ;
y in N ;
x in T ;
m < n ;
m <= n ;
n > 1 ;
let r ;
t in I ;
n <= 4 ;
M is finite ;
let X ;
Y c= Z ;
A // M ;
let U ;
a in D ;
q in Y ;
let x ;
1 <= l ;
1 <= w ;
let G ;
y in N ;
f = {} ;
let x ;
x in Z ;
let x ;
F is one-to-one ;
e <> b ;
1 <= n ;
f is special ;
S misses C
t <= 1 ;
y divides m ;
P divides M ;
let Z ;
let x ;
y c= x ;
let X ;
let C ;
x _|_ p ;
o is monotone ;
let X ;
A = B ;
1 < i ;
let x ;
let u ;
k <> 0 ;
let p ;
0 < r ;
let n ;
let y ;
f is onto ;
x < 1 ;
G c= F ;
a is_>=_than X ;
T is continuous ;
d <= a ;
p <= r ;
t < s ;
p <= t ;
t < s ;
let r ;
D <= E ;
assume e > 0 ;
assume 0 < g ;
p in C ;
x in X ;
Y ` in Y ;
assume 0 < g ;
not c in Y ;
not v in L ;
2 in z `2 ;
assume f = g ;
N c= b ` ;
assume i < k ;
assume u = v ;
I = J ;
B ` = b ` ;
assume e in F ;
assume p > 0 ;
assume x in D ;
let i be element ;
assume F is onto ;
assume n <> 0 ;
let x be element ;
set k = z ;
assume o = x ;
assume b < a ;
assume x in A ;
a `2 <= b `2 ;
assume b in X ;
assume k <> 1 ;
f = product l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is from of squares ;
assume m > 0 ;
assume A c= B ;
X is lower
assume A <> {} ;
assume X <> {} ;
assume F <> {} ;
assume G is open ;
assume f is dilatation ;
assume y in W ;
y \not <= x ;
A ` in B ` ;
assume i = 1 ;
let x be element ;
x `2 = x `2 ;
let X be BCK-algebra ;
assume S is non empty ;
a in REAL ;
let p be set ;
let A be set ;
let G be _Graph , a , b be Vertex of G ;
let G be _Graph , a , b be Vertex of G ;
a be Complex ;
let x be element ;
let x be element ;
let C be FormalContext , a , b be Element of C ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
y be Real ;
X c= f . a
let y be element ;
let x be element ;
i be Nat ;
let x be element ;
n in NAT ;
let a be element ;
m in NAT ;
let u be element ;
i in NAT ;
let g be Function ;
Z c= NAT ;
l <= ma ;
let y be element ;
r2 in dom f ;
let x be element ;
k1 be Integer ;
let X be set ;
let a be element ;
let x be element ;
let x be element ;
let q be element ;
let x be element ;
assume f is being_homeomorphism ;
let z be element ;
a , b // K ;
let n be Nat ;
let k be Nat ;
B ` c= B ` ;
set s = 1 ;
n >= 0 + 1 ;
k c= k + 1 ;
R1 c= R ;
k + 1 >= k ;
k c= k + 1 ;
let j be Nat ;
o , a // Y ;
R c= Cl G ;
Cl B = B ;
let j be Nat ;
1 <= j + 1 ;
arccot is_differentiable_on Z ;
exp_R is_differentiable_in x ;
j < i0 ;
let j be Nat ;
n <= n + 1 ;
k = i + m ;
assume C meets S ;
n <= n + 1 ;
let n be Nat ;
h1 = {} ;
0 + 1 = 1 ;
o <> b3 ;
f2 is one-to-one ;
support p = {}
assume x in Z ;
i <= i + 1 ;
r1 <= 1 ;
let n be Nat ;
a "/\" b <= a ;
let n be Nat ;
0 <= r0 ;
let e be Real , x be Real ;
not r in G . l
c1 = 0 ;
a + a = a ;
<* 0 *> in e ;
t in { t } ;
assume F is not discrete ;
m1 divides m ;
B * A <> {} ;
a + b <> {} ;
p * p > p ;
let y be ExtReal ;
let a be Int-Location , i be Integer ;
let l be Nat ;
let i be Nat ;
let r ;
1 <= i2 ;
a "\/" c = c ;
let r be Real ;
let i be Nat ;
let m be Nat ;
x = p2 ;
let i be Nat ;
y < r + 1 ;
rng c c= E
Cl R is boundary ;
let i be Nat ;
R2 ;
cluster uparrow x -> or uparrow x is or uparrow y is be directed ;
X <> { x } ;
x in { x } ;
q , b // M ;
A . i c= Y ;
P [ k ] ;
2 / x in W ;
X [ 0 ] ;
P [ 0 ] ;
A = A |^ i ;
2 >= thesis ;
G . y <> 0 ;
let X be RealNormSpace , Y be RealNormSpace ;
a in A ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , W be Subspace of V ;
assume x in L~ TOP-REAL M ;
k < s . a ;
not t in { p } ;
let Y be set , f be Function of Y , BOOLEAN ;
M , L are_isomorphic ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded & L is lower-bounded ;
rng f = Y ;
G8 c= L ;
assume x in Cl Q ;
m in dom P ;
i <= len Q ;
len F = 3 ;
Free p = {} ;
z in rng p ;
lim b = 0 ;
len W = 3 ;
k in dom p ;
k <= len p ;
i <= len p ;
1 in dom f ;
b `1 = a `1 + 1 ;
x `2 = a * y `2 ;
rng D c= A ;
assume x in K1 ;
1 <= ii ;
1 <= ii ;
pp c= PI ;
1 <= ii ;
1 <= ii ;
LMP C in L ;
1 in dom f ;
let seq , n ;
set C = a * B ;
x in rng f ;
assume f is_continuous_on X ;
I = dom A ;
u in dom p ;
assume a < x + 1 ;
s-7 is bounded ;
assume I c= P1 ;
n in dom I ;
let Q ;
B c= dom f ;
b + p _|_ a ;
x in dom g ;
F-14 is continuous ;
dom g = X ;
len q = m ;
assume A2 : A is closed ;
cluster R \ S -> real-valued ;
sup D in S ;
x << sup D ;
b1 >= Z1 or b1 >= Z ;
assume w = 0. V ;
assume x in A . i ;
g in the carrier of X ;
y in dom t ;
i in dom g ;
assume P [ k ] ;
Let Let Let Let Let Let Let Let Let Let Let Let Let Let Let Let Let Let Let Let Let Let Let C be 0 <= C ;
x4 is increasing ;
let e2 be element ;
- b divides b ;
F c= \tau ( F ) ;
IT is non-decreasing ;
IT is non-decreasing ;
assume v in H . m ;
assume b in [#] B ;
let S be non void ManySortedSign , A be non-empty MSAlgebra over S ;
assume P [ n ] ;
assume union S is [#] independent & finite S is finite ;
V is Subspace of V ;
assume P [ k ] ;
rng f c= NAT & rng f c= NAT ;
assume ex_inf_of X , L ;
y in rng f ;
let s , I be set , A be non-empty ManySortedSet of I ;
b ` c= b9 ` ;
assume not x in NAT ;
A /\ B = { a } ;
assume len f > 0 ;
assume x in dom f ;
b , a // o , c ;
B in Bf1 ;
cluster product p -> non empty ;
z , x // x , p ;
assume x in rng N ;
cosec is_differentiable_in x & cosec is_differentiable_in x ;
assume y in rng S ;
let x , y be element ;
i2 < i1 & i1 < i2 ;
a * h in a * H ;
p , q in Y ;
Observe ;
q1 in A1 . ( len A1 ) ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A2 c= A1 ;
an < n & n < len p ;
assume A c= dom f ;
Re ( f ) is_integrable_on M ;
let k , m be element ;
a , a \equiv b , b ;
j + 1 < k + 1 ;
m + 1 <= n1 ;
g is_differentiable_in x0 & g is_differentiable_in x0 ;
g is_continuous_in x0 & g is_continuous_in x0 ;
assume O is symmetric transitive ;
let x , y be element ;
let jj be Nat ;
[ y , x ] in R ;
let x , y be element ;
assume y in conv A ;
x in Int V ;
let v be VECTOR of V ;
P3 halts_on s , P3 = P +* I ;
d , c // a , b ;
let t , u be set ;
let X be set ;
assume k in dom s ;
let r be non negative Real ;
assume x in F | M ;
let Y be Subset of S ;
let X be non empty TopStruct , Y be SubSpace of X ;
[ a , b ] in R ;
x + w < y + w ;
{ a , b } >= c ;
let B be Subset of A , C be Subset of B ;
let S be non empty ManySortedSign ;
let x be variable , f be Function of X , REAL ;
let b be Element of X , c be Element of X ;
R [ x , y ] ;
x ` ` = x ;
b \ x = 0. X ;
<* d *> in D |^ 1 ;
P [ k + 1 ] ;
m in dom ( n |-> 0 ) ;
h2 . a = y ;
P [ n + 1 ] ;
Observe : G * F is pre| ;
let R be non empty multMagma , I be Subset of R ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p ;
assume f | X is lower ;
x in rng co /\ L~ co ;
let x be Element of B ;
let t be Element of D ;
assume x in Q .vertices() ;
set q = s ^\ k ;
let t be VECTOR of X ;
let x be Element of A ;
assume y in rng p `2 ;
let M be mamaid ;
let N be non empty Subset of \mathop { \rm thesis \hbox { - } : 1 <= N } ;
let R be RelStr with finite finite 1 ;
let n , k be Nat ;
let P , Q be relational structure ;
P = Q /\ [#] S ;
F . r in { 0 } ;
let x be Element of X ;
let x be Element of X ;
let u be VECTOR of V ;
reconsider d = x as Int-Location ;
assume I does not destroy a ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < ( v - u ) ;
x <= c2 . x ;
x in F ` ;
Observe ;
assume t1 <= t2 & t2 <= t2 ;
let i , j be even Integer ;
assume F1 <> F2 & F2 <> F1 ;
c in Intersect R ;
dom p1 = c & rng p2 c= REAL ;
a = 0 or a = 1 ;
assume A1 : A <> A2 ;
set i1 = i + 1 ;
assume a1 = b1 & a2 = b2 ;
dom g1 = A & rng g2 c= A ;
i < len M + 1 ;
assume not +infty in rng G ;
N c= dom ( f1 - f2 ) ;
x in dom sec ;
assume [ x , y ] in R ;
set d = ( x - y ) / 2 ;
1 <= len g1 & len g1 <= len g2 ;
len s2 > 1 & len s2 > 1 ;
z in dom ( f1 - f2 ) ;
1 in dom D2 & 1 in dom D2 ;
( p `2 ) ^2 = 0 ;
j2 <= width G ;
len PI > 1 + 1 ;
set n1 = n + 1 ;
|. q-35 .| = 1 ;
let s be SortSymbol of S ;
( i gcd i ) = i ;
X1 c= dom f & X2 c= dom f ;
h . x in h . a ;
let G be mod of \rm \mathbin { - } \rm \mathbin { - } \rm mod } G ;
cluster m * n -> square ;
let k9 be Nat ;
i - 1 > m - 1 ;
R is transitive implies R |_2 field R is transitive
set F = <* u , w *> ;
pp c= P3 & j c= P3 ;
I is_closed_on t , Q ;
assume [ S , x ] is holds S is \cap T ;
i <= len ( f2 ^ g2 ) ;
p is FinSequence of X ;
1 + 1 in dom g ;
Sum R2 = n * r ;
cluster f . x -> complex-valued ;
x in dom ( f1 - f2 ) ;
assume [ X , p ] in C ;
Bj c= Xj \/ Xj ;
n2 <= ( 2 * ( n + 1 ) ) ;
A /\ [: P , Q :] c= A ` ;
cluster x -valued for Function ;
Q be Subset-Family of S , A be Subset of S ;
assume n in dom g2 & n in dom g2 ;
a be Element of R ;
t `2 in dom e2 ;
N . 1 in rng N ;
- z in A \/ B ;
let S be SigmaField of X , T be non empty Subset-Family of S ;
i . y in rng i ;
[: the carrier of REAL , the carrier of REAL :] c= dom f ;
f . x in rng f ;
mt <= ( r / 2 ) ;
s2 in r-5 ;
let z , z be complex number ;
n <= N . m ;
LIN q , p , s ;
f . x = waybelow x /\ B ;
set L = [ S \to T ] ;
let x be non positive ExtReal ;
let m be Element of M ;
f in union rng F1 ;
let K be add-associative right_zeroed right_complementable non empty doubleLoopStr , M be Matrix of K ;
let i be Element of NAT ;
rng ( F * g ) c= Y ;
dom f c= dom x & rng f c= dom x ;
n1 < n1 + 1 & n2 + 1 < n2 ;
n1 < n1 + 1 & n2 + 1 < n2 ;
cluster [: X , Y :] -> 8 ;
[ y2 , 2 ] `2 = z ;
let m be Element of NAT ;
let S be Subset of R ;
y in rng ( S . n ) ;
b = upper_bound dom f & c = sup dom f ;
x in Seg ( len q ) ;
reconsider X = D ( ) as set ;
[ a , c ] in E1 ;
assume n in dom h2 & n + 1 in dom h2 ;
w + 1 = ( - a ) + 1 ;
j + 1 <= j + 1 + 1 ;
k2 + 1 <= k1 + 1 ;
i be Element of NAT ;
Support u = Support p \/ Support q ;
assume X is complete complete complete ] ;
assume that f = g and p = q ;
n1 <= n1 + 1 & n2 + 1 <= n2 ;
let x be Element of REAL ;
assume x in rng s2 ;
x0 < x0 + 1 & x0 + 1 < x0 + 1 ;
len ( L ) = W ;
P c= Seg ( len A ) ;
dom q = Seg n & rng q c= Seg n ;
j <= width M *' ;
let r8 be real-valued FinSequence of NAT ;
let k be Element of NAT ;
Integral ( M , f ) < +infty ;
let n be Element of NAT ;
assume z in x := len C2 ( 0 , A ) ;
i be set ;
n - 1 = n-1 - 1 ;
len ( n |-> 0 ) = n ;
\mathop { Z , c } c= F ;
assume x in X or x = X ;
x is element of b , c ;
let A , B be non empty set , C be non empty set ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q ;
let s be Element of E * ;
let B1 be Basis of x , y be Point of V ;
Carrier ( 3 /\ L2 ) = {} ;
L1 /\ ( L2 /\ L2 ) = {} ;
assume downarrow x = downarrow y ;
assume b , c // b , c ;
LIN q , c , c ;
x in rng ( f . -129 ) ;
set n8 = n + j ;
let D7 be non empty set , f be FinSequence of D ;
let K be add-associative right_zeroed right_complementable non empty addLoopStr , M be Matrix of K ;
assume f `2 = f & h `1 = h ;
R1 - R2 is total ;
k in NAT & 1 <= k ;
let a be Element of G ;
assume x0 in [. a , b .] ;
K1 ` is open & ( TOP-REAL 2 ) ` is open ;
assume a , b ] in maximal the distance of C ;
a , b be Element of S ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = Integral ( M , f | E ) ;
cluster n[ ] -> nes] ;
not u in { ag } ;
the carrier of f c= B ;
reconsider z = x as VECTOR of V ;
cluster the RelStr of L -> \rangle ;
r (#) H is \mathop { 0 } \rbrace -valued ;
s . intloc 0 = 1 ;
assume x in C & y in C ;
let U0 be strict non-empty MSAlgebra over S , A be non-empty MSAlgebra over S ;
[ x , Bottom T ] is compact ;
i + 1 + k in dom p ;
F . i is stable Subset of M ;
r-35 in iff ry0 in { y } ;
let x , y be Element of X ;
let A , I be seq of X ;
[ y , z ] in O ;
assume that that x = 1 and x = goto ( card I + 2 ) ;
rng Sgm A = A ;
q |- \! from All ( y , q ) ;
for n holds X [ n ] ;
x in { a } & x in d ;
for n holds P [ n ] ;
set p = |[ x , y , z ]| ;
LIN o , a , b ;
p . 2 = Z |^ Y ;
( D . D ) `2 = {} ;
n + 1 + 1 <= len g ;
a in [: CQC-WFF ( Al ) , NAT :] ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
let I be Ideal of L ;
set g = f1 + f2 , h = f3 + f4 ;
a <= max ( a , b ) ;
i-1 < len G + 1 ;
g . 1 = f . i1 ;
x `1 , y `2 in A2 ;
( f /* s ) . k < r ;
set v = VAL g ;
i - k + 1 <= S ;
cluster associative associative for non empty multMagma ;
x in support ( ( support t ) \ support b ) ;
assume a in [: the carrier of G , the carrier of G :] ;
i `2 <= len ( y `2 ) ;
assume p divides b1 + b2 & p divides b2 ;
MM1 <= upper_bound M1 & sup M1 <= upper_bound M1 ;
assume x in W \mathop { \rm W _ { most ( X ) } ;
j in dom ( z | n ) ;
let x be Element of D ( ) ;
IC s4 = l1 .= IC s4 .= ( 0 + 1 ) ;
a = {} or a = { x } ;
set uc = Vertices G , ud = Vertices G , ud = Vertices G , Id = Vertices G , Ie = the Element of G ;
seq " is non-zero & lim ( seq " ) = 0 ;
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
h-4 c= h-14 \/ h-14 ;
]. a , b .[ c= Z ;
X1 , X2 X2 X2 X1 union X2 X1 X1 X1 X1 X1 X1 union X2 X1 X1 X1 X1 union X2 X1 X1 X1 union X2 ;
a in Cl ( union F \ G ) ;
set x1 = [ 0 , 0 ] ;
k + 1 - 1 = k - 1 ;
cluster real-valued for Relation of NAT , REAL ;
ex v st C = v + W ;
let IT be non empty thesis , n be non zero Element of NAT ;
assume V is Abelian add-associative right_zeroed right_complementable ;
X-21 \/ Y in \sigma ( L ) ;
reconsider x = x as Element of S ;
max ( a , b ) = a ;
upper_bound B is upper & upper_bound B = upper_bound B ;
let L be non empty reflexive RelStr , D be Subset of L ;
R is_reflexive & X is transitive implies R ~ is transitive
E , g |= H / ( x , y ) ;
dom ( G /. y ) = a ;
( 1 / 4 ) >= - r / 2 ;
G . p0 in rng G ;
let x be Element of F , y be Element of F ;
D [ ( PP , 0 ) ] ;
z in dom ( id B ) ;
y in the carrier of N ;
g in the carrier of H & h in the carrier of H ;
rng ( f | l ) c= [: the carrier of S , the carrier of S :] ;
j `2 + 1 in dom s1 ;
let A , B be strict Subgroup of G ;
C be non empty Subset of REAL ;
f . z1 in dom h & f . z2 in dom h ;
P . k1 in rng P ;
M = ( A +* {} ) +* {} ;
let p be FinSequence of REAL , n be Nat ;
f . n1 in rng f & f . n2 in rng f ;
M . ( F . 0 ) in REAL ;
h . [. a , b .] = b ;
assume that the distance of V , Q and P [ v ] ;
let a be Element of ( the carrier of V ) ;
let s be Element of PP ( ) ;
let PP be non empty tree ;
n be Nat ;
the carrier of g c= B ;
I = halt SCM R & I = ( the InstructionsF of R ) . 0 ;
consider b being element such that b in B ;
set BK = BCS K , BK = BCS K ;
l <= ( -> -> exists of L ) . j ;
assume x in downarrow [ s , t ] ;
( x `2 ) in uparrow t ;
x in ( JumpParts T ) . x ;
let h be Morphism of c , a ;
Y c= [: the carrier of R , the carrier of R :] ;
A2 \/ A3 c= Carrier ( L1 ) \/ Carrier ( L2 ) ;
assume LIN o , a , b ;
b , c // d1 , e2 ;
x1 , x2 in Y & y1 in Y ;
dom <* y *> = Seg 1 & rng <* y *> c= Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar s .| ;
[ x , x `2 ] in X ~ ;
for n be Nat holds 0 <= x . n
[' a , b '] = [. a , b .] ;
cluster -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> closed for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 , P is_collinear ;
dom M1 = Seg n & rng M2 c= Seg n ;
x = [ x1 , x2 ] & y = [ y1 , y2 ] ;
R , Q be ManySortedSet of A ;
set d = ( 1 / n ) * ( 1 / n ) ;
rng g2 c= dom W & rng g2 c= dom W ;
P . ( [#] Sigma \ B ) <> 0 ;
a in field R & a = b ;
let M be non empty Subset of V , V be non empty Subset of V ;
let I be Program of SCM+FSA , a be Int-Location ;
assume x in rng ( ( R * S ) * ( R * S ) ) ;
let b be Element of the carrier of T ;
dist ( e , z ) > r-r ;
u1 + v1 in W2 + W3 ;
assume not the carrier of L misses rng G ;
let L be lower-bounded antisymmetric RelStr ;
assume [ x , y ] in a9 ;
dom ( A * e ) = NAT ;
let a , b be Vertex of G ;
let x be Element of Bool M ;
0 <= Arg a * PI ;
o9 `2 , a9 // o9 `2 , y `2 ;
{ v } c= the carrier of l & { v } c= the carrier of l ;
let x be variable of A ;
assume x in dom ( uncurry f ) ;
rng F c= ( product f ) |^ X
assume D2 . k in rng D ;
f " . p1 = 0 & f " . p2 = 0 ;
set x = the Element of X , y = the Element of Y ;
dom Ser ( G ) = NAT & rng Ser ( G ) = NAT ;
n be Element of NAT ;
assume LIN c , a , e1 ;
cluster -> finite for FinSequence of NAT ;
reconsider d = c as Element of L1 ;
( v2 |-- I ) . X <= 1 ;
assume x in the carrier of f ;
conv @ S c= conv @ A ;
reconsider B = b as Element of the topology of T ;
J , v |= P \lbrack l \rbrack ;
Observe : J . i is non empty TopSpace ;
ex_sup_of Y1 \/ Y2 , T & ex_sup_of Y1 , T ;
W1 is_orders field ( W1 + W2 ) implies ( W1 + W2 ) is_orders ( W2 + W3 )
assume x in the carrier of R ;
dom ( n-16 ) = Seg n & dom ( n-16 ) = Seg n ;
s4 misses s2 & s3 misses s4 ;
assume ( a 'imp' b ) . z = TRUE ;
assume X is open & f = X --> d ;
assume [ a , y ] in Indices ( f | X ) ;
assume that stop I c= J and card I c= K and card J c= K ;
Im ( lim seq ) = 0 & Im ( lim seq ) = 0 ;
( ( sin - cos ) `| Z ) . x <> 0 ;
sin is_differentiable_on Z & cos is_differentiable_on Z implies cos is_differentiable_on Z & for x st x in Z holds cos . x = sin . x / cos . x
t3 . n = t3 . n .= 0 ;
dom ( ( - F ) | A ) c= dom F ;
W1 . x = W2 . x .= W2 . x ;
y in W .vertices() \/ W .vertices() ;
( ( k + 1 ) + 1 ) <= len ( ( k + 1 ) + 1 ) ;
x * a \equiv y * a . ( mod m ) ;
proj2 .: S c= proj2 .: P ;
h . p4 = g2 . I .= ( g2 . I ) `1 ;
G = U * ( 1 , k ) `1 .= G * ( 1 , k ) `1 ;
f . ( r1 ) in rng f ;
i + 1 + 1 <= len - 1 ;
rng F = rng ( F | ( Seg n ) ) ;
mode seq is well unital associative non empty multMagma ;
[ x , y ] in A ~ { a } ;
x1 . o in L2 . ( o . o ) ;
the carrier of
not [ y , x ] in id X ;
1 + p .. f <= i + len f ;
seq ^\ ( k1 + 1 ) is lower ;
len ( F . i ) = len I & len ( F . i ) = len F ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be complex number , a be complex number ;
Comput ( P , s , n ) = s ;
k <= k + 1 & k + 1 <= len p ;
reconsider c = {} _ T as Element of L ;
let Y be Element of l is_holds Y is an \rbrace Subset of T ;
cluster from L -> directed-sups-preserving for Function of L , L ;
f . j1 in K . j1 & f . j2 in K . j1 ;
Observe : J => y is total ;
K c= 2 |^ the carrier of T ;
F . b1 = F . b2 .= F . b2 ;
x1 = x or x1 = y or x1 = z or x1 = x ;
pred a <> {} means a / ( a - b ) = 1 ;
assume that succ a c= b and b in a ;
s1 . n in rng s1 & s1 . n in rng s1 ;
{ o , b2 } on C2 & { o , b2 } on C2 ;
LIN o , b , b9 & LIN o , b , b9 ;
reconsider m = x as Element of Funcs ( V , C ) ;
let f be non constant FinSequence of D ;
let FF2 be non empty non empty for the carrier of X ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in F-8 ;
reconsider pp = x as Subset of m ;
A , B , C be Element of R ;
Observe -> strict non empty for b9 is strict non empty b9 ;
rng c `2 misses rng ( e | n ) ;
z is Element of gr { x } ;
not b in dom ( a .--> p1 ) ;
assume that k >= 2 and P [ k ] ;
Z c= dom ( ( - cot ) (#) ( f ^ ) ) ;
the component of Q c= UBD A & ( Q /\ C ) /\ ( Q /\ C ) = {} ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( ( 1 / 2 ) (#) ( f ^ ) ) ;
pred f = u means : Let a * f = a * u ;
for n holds P1 [ n ] implies P1 [ n + 1 ]
{ x . O : x in L } <> {} ;
x be Element of V . s ;
a , b be Nat ;
assume that S = S2 and p = p2 ;
( n1 gcd n2 ) = 1 & ( n1 gcd n2 ) gcd n2 = 1 ;
set on = o * ( the Arity of S ) ;
seq . n < |. r1 .| & |. seq . n .| < r ;
assume that seq is increasing and r < 0 and seq is convergent ;
f . ( y1 , x1 ) <= a ;
ex c being Nat st P [ c ] ;
set g = { n / 1 : n in NAT } ;
k = a or k = b or k = c ;
a9 , b9 , c9 , a9 , b9 , c9 , b9 , c9 be set ;
assume Y = { 1 } & s = <* 1 *> ;
I1 . x = f . x .= 0 .= 0 ;
W3 .last() = W3 . ( 1 + 1 ) .= W . ( len W ) ;
cluster -> trivial for Subgroup of G , finite _Graph ;
reconsider u = u as Element of Bags X ;
A in B ^ ) implies A , B are_are that B , C are_that A , B are_that B , C are_that A , C are_that B |^ C are_
x in { [ 2 * n + 3 , k ] } ;
1 >= ( q `1 ) ^2 / ( |. q .| ) ^2 ;
f1 is_\in the TOP-REAL 2 & f2 is_<= len f2 ;
( f /. 2 ) `2 <= ( q `2 ) ^2 ;
h is_dom Cage ( C , n ) ;
( b `2 ) ^2 <= ( p `2 ) ^2 + ( p `2 ) ^2 ;
let f , g be Function of X , Y ;
S * ( k , k ) <> 0. K ;
x in dom max ( f , g ) ;
p2 in NN . ( p1 , p2 ) ;
len ( the_left_argument_of H ) < len ( H ) ;
F [ A , F ( ) . A ] ;
consider Z such that y in Z and Z in X ;
attr 1 in C means : Let $ A c= C |^ A ;
assume that r1 <> 0 or r2 <> 0 or r1 <> 0 ;
rng q1 c= rng ( C1 ^ C2 ) & rng q1 c= rng ( C2 ^ C2 ) ;
A1 , L , A3 , A3 is_collinear ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in element ( p , S ) ;
then S is then S is atomic & P-2 [ S ] ;
Cl Int ( [#] T ) = [#] T ;
f12 | A2 = f2 | A2 & f12 | A2 = f2 | A2 ;
0. M in the carrier of W & 0. M in the carrier of W ;
v , v be Element of M ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ V ;
let X be Subset of S ~ ;
consider H1 such that H = 'not' H1 and H1 in D ;
1_ G c= ( f2 * ( the div r ) ) * ( ( - 1 ) / 2 ) ;
0 * a = 0. R .= a * 0. R ;
A |^ ( 2 , 2 ) = A ^^ A ;
set v\rbrace = ( v /. n ) * ( v /. n ) ;
r = 0. ( \langle \cal E , \Vert \cdot \Vert *> ) ;
( f . p4 ) `1 >= 0 & ( f . p4 ) `2 >= 0 ;
len W = len ( W | ( len W ) ) ;
f /* ( s * G ) is divergent_to+infty ;
consider l being Nat such that m = F . l ;
t8 does not destroy b1 & not LIN b1 , b2 , b1 ;
reconsider Y1 = X1 , Y2 = X2 as SubSpace of X ;
consider w such that w in F and not x in w ;
let a , b , c , d be Real ;
reconsider i = i as non zero Element of NAT ;
c . x >= id ( the carrier of L ) . x ;
\sigma ( T ) \/ omega ( T ) is Basis of T ;
for x being element st x in X holds x in Y
cluster [ x1 , x2 ] -> pair for set ;
downarrow a /\ downarrow t is Ideal of T ;
let X be disjoint with NAT , n be non empty set ;
rng f = |. \rm .| ( S , X ) ;
let p be Element of B , s be \neq the connectives of S ;
max ( N1 , 2 ) >= N1 & max ( N2 , 2 ) >= N2 ;
0. X <= ( b |^ m ) * ( b |^ m ) ;
assume that i in I and R1 . i = R ;
i = j1 & p1 = q1 or i = q2 or i = q1 ;
assume gR in the right st gR in the carrier of g ;
let A1 , A2 be Subset of S , A be Subset of S ;
x in h " P /\ [#] T1 & x in h " P ;
1 in Seg 2 & 1 in Seg 3 & 2 in Seg 3 ;
reconsider X9 = X as non empty Subset of T99 , x be Element of T99 ;
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the Source of G ) -valued ;
n1 <= i2 + len g2 & i2 <= len g2 implies ( i1 + 1 ) + 1 <= i2
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume v in the carrier' of G2 & u in the carrier' of G2 ;
y = Re y + ( Im y ) * i ;
( ( - 1 ) |^ ( p ) ) mod p = 1 ;
x2 is_differentiable_on ]. a , b .[ & x1 = ( f ^ ) . a ;
rng M5 c= rng ( D2 | ( i + 1 ) ) ;
for p being Real st p in Z holds p >= a
( cn ) * ( ( cn ) | K1 ) = proj1 * ( ( cn ) | K1 ) ;
( seq ^\ m ) . k <> 0 & ( seq ^\ m ) . k <> 0 ;
s . ( G . ( k + 1 ) ) > x0 ;
( p |-count M ) . 2 = d ;
A \oplus ( B \ominus C ) = ( A + B ) \ominus C
h \equiv gg . ( mod ( P , T ) ) ;
reconsider i1 = i-1 - 1 as Element of NAT ;
let v1 , v2 be VECTOR of V , v be VECTOR of V ;
for V being Subspace of V holds V is Subspace of [#] V
reconsider ii = i - 1 as Element of NAT ;
dom f c= [: the carrier of C , the carrier of C :] ;
x in ( the Sorts of B ) . n ;
len - ( f2 - f3 ) in Seg len ( f2 - f3 ) ;
pp c= the topology of T & the topology of T c= the topology of T ;
]. r , s .[ c= [. r , s .] ;
let B2 be Basis of T2 , a be Element of T2 ;
G * ( B * A ) = ( id o1 ) * B ;
assume that p , u , u , v is_collinear and u , v , v , w u1 ;
[ z , z ] in union rng ( F . n ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S , C = $1 .. S , D = $1 .. S , D = $1 .. S , E = $1 .. S , F = $1 .. S , G = $1 .. S , C
LIN a1 , a3 , b1 & LIN a1 , b1 , c1 ;
f " ( f .: x ) = { x } ;
dom ( w2 ) = dom ( ( r12 ) * ( r12 ) ) ;
assume that 1 <= i and i <= n and j <= n ;
( ( g2 ) . O ) `2 <= 1 ;
p in LSeg ( E . i , F . i ) ;
In * ( i , j ) = 0. ( K , n , j ) ;
|. f . ( s . m ) -g .| < g1 ;
q9 . x in rng ( q ^ <* x *> ) ;
Carrier ( Lxy ) misses Carrier ( Lxy ) ;
consider c being element such that [ a , c ] in G ;
assume that NInt o = o9 and oInt o = o9 ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= ( F |^ C-1 ) .: Cj ;
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [. f . j , f . j .] ;
pred 0 <= x & x <= 1 & x ^2 <= x ;
p `2 - q `2 <> 0. TOP-REAL 2 & p `2 - q `2 <> 0. TOP-REAL 2 ;
Observe aelements ( S , T ) is non empty ;
x be Element of S ~ ;
cluster cluster cluster cluster cluster cluster cluster cluster cluster cluster cluster cluster \hbox { - } \rm id F -> one-to-one ;
|. i .| <= - ( - ( 2 |^ n ) ) ;
the carrier of I[01] = dom P & the carrier of I[01] = dom Q ;
arccot * ( n + 1 ) ! > 0 * n ;
S c= ( A1 /\ A2 ) /\ ( A3 /\ A3 ) ;
a3 , a4 // a3 , b3 & a3 , a4 // b2 , b3 ;
then dom A <> {} & dom A <> {} & rng A <> {} ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , Y & y in X & z in Y ;
set v2 = ( v /. ( i + 1 ) ) ;
x = r . n .= ( r . n ) * ( r . n ) ;
f . s in the carrier of S2 & f . s in the carrier of S2 ;
dom g = the carrier of I[01] & rng g c= the carrier of I[01] ;
p in Upper_Arc ( P ) /\ Lower_Arc ( P ) ;
dom d2 = A2 & dom d2 = A2 ;
0 < ( p - ||. z .|| ) * ( ||. z .|| ) ;
e . ( m + 1 ) <= e . ( m + 1 ) ;
B \ominus X \/ B \ominus Y c= B \ominus X
-infty < Integral ( M , Im ( g | B ) ) ;
cluster O \cup F -> \HM } for operation operation of X ;
let U1 , U2 be non-empty MSAlgebra over S , f be Function of U1 , U2 ;
Proj ( i , n ) * g is_differentiable_on X ;
x , y , z be Point of X , p be Point of X ;
reconsider pp = p . x as Subset of V ;
x in the carrier of Lin ( A ) ;
let I , J be parahalting Program of SCM+FSA , a be Int-Location ;
assume that - a is lower and b is lower and a <= b ;
Int Cl ( A ) c= Cl Int Cl ( A ) ;
assume for A being Subset of X holds Cl A = A ;
assume q in Ball ( |[ x , y ]| , r ) ;
( p2 `2 ) ^2 <= ( p `2 ) ^2 + ( p `2 ) ^2 ;
Cl Q ` = [#] ( ( T | A ) | A ) ;
set S = the carrier of T , T = the carrier of T ;
set I8 = ' ( f |^ n ) , I8 = ' ( f |^ n ) , I7 = ' ( f |^ n ) , I8 = ' ( f |^ n ) , I8 = ' ( f |^ n ) ;
len p - n = len ( p - n ) ;
A is Permutation of Swap ( A , x , y ) ;
reconsider ni = n\rbrace - ( n - 1 ) as Element of NAT ;
1 <= j + 1 & j + 1 <= len ( s ) ;
let qm , qm be Element of M , q be Element of M ;
a9 in the carrier of S1 & b9 in the carrier of S1 ;
c1 /. n1 = c1 . ( n1 + 1 ) .= c2 . n1 + c2 /. ( n1 + 1 ) ;
let f be FinSequence of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
y = ( f * ( S * ( x , y ) ) ) . x ;
consider x being element such that x in an " A ;
assume r in ( dist ( o , r ) ) .: P ;
set i2 = len ( the \mathopen { - } h , 1 ) ;
h2 . ( j + 1 ) in rng h2 ;
Line ( M29 , k ) = M . i ;
reconsider m = ( x - 2 ) / ( 2 * x ) as Element of REAL ;
U1 , U2 be Subspace of U0 , a be Element of U1 ;
set P = Line ( a , d ) ;
len p1 < len p2 + 1 & len p2 + 1 <= len p1 ;
let T1 , T2 be Scott Scott topological of L , f be Scott topological \mathclose of T ;
then x <= y & ( ex x st x in dom ( y | y ) ) ;
set M = n -tuples_on m , S = n -tuples_on m , T = m -tuples_on n , M = m -tuples_on n , S = m -tuples_on n , T = m -tuples_on n ;
reconsider i = x1 , j = x2 as Nat ;
rng ( ( the_arity_of o ) . ( ( the_arity_of o ) . ( ( the_arity_of o ) . ( ( the_arity_of o ) . ( ( the_arity_of o ) . ( ( the_arity_of o ) . ( ( the_arity_of o ) . ( ( the_arity_of o ) . ( ( the_arity_of o ) . (
z1 " = z1 " * ( z2 " * ( z2 " ) ) ;
x0 - r / 2 in L /\ dom f ;
then w is that w is rng w /\ ( L \/ S ) <> {} ;
set x-10 = xZ ^ <* Z *> , xZ = xZ ^ <* Z *> ;
len w1 in Seg len ( w1 ^ w2 ) & len ( w2 ^ w2 ) = len w1 + len w2 ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of |. - ( V , { k } ) .| ;
x . n = ( |. a . n .| ) * ( |. b . n .| ) ;
( p `1 ) ^2 <= ( G * ( 1 , 1 ) `1 ) ^2 ;
rng ( g ) c= L~ ( g | ( L~ g ) ) \/ L~ ( g | ( L~ g ) ) ;
reconsider k = i-1 * ( l + j ) as Nat ;
for n be Nat holds F . n is non empty & F . n is non empty ;
reconsider x9 = x9 , y9 = y9 as VECTOR of M ;
dom ( f | X ) = X /\ dom f .= X ;
p , a // p , c & b , a // c , c ;
reconsider x1 = x as Element of REAL m , x0 be Element of REAL m ;
assume i in dom ( a * p ^ q ) ;
m . ( ( \in dom g ) . ( f . ( k + 1 ) ) ) = p . ( f . ( k + 1 ) ) ;
a / ( s . ( m - n ) ) / ( n - m ) <= 1 ;
S . ( n + k + 1 ) c= S . ( n + k ) ;
assume that B1 \/ C1 = B2 \/ ( C2 \/ C1 ) and B2 \/ C2 = C2 \/ C1 ;
X . i = { x1 , x2 } . i .= X . i ;
r2 in dom ( h1 + h2 ) & r1 in dom ( h1 + h2 ) ;
- ( 0. R ) = a & b-0 = b ;
F8 is_closed_on t2 , Q8 & F8 is_halting_on F8 , Q8 ;
set T = -> InInInInIn\ ( X , x0 ) ;
Int Cl ( Int Cl R ) c= Int Cl R ;
consider y being Element of L such that c . y = x ;
rng ( F . x ) = { F . ( x ) } ;
G-23 \ { c } c= B \/ S \/ S ;
f[#] ( X ) is_\! ] & f\HM ( X ) is_\! ] ;
set Rj = the Element of P , Rj = the Element of P ;
assume n + 1 >= 1 & n + 1 <= len M ;
let k2 be Element of NAT , k be Nat ;
reconsider pp = u as Element of ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n )
g . x in dom f & x in dom g ;
assume that 1 <= n and n + 1 <= len f1 and f1 . n = f2 . n ;
reconsider T = b * N as Element of G / ( N , G ) ;
len ( ( P ^ ) . i ) <= len ( ( P ^ ) . j ) ;
x " in the carrier of A1 & x " in the carrier of A1 ;
[ i , j ] in Indices ( ( A + B ) * ( i , j ) ) ;
for m be Nat holds Re ( F . m ) is simple ;
f . x = a . i .= a1 . k .= a . k ;
let f be PartFunc of REAL i , REAL , g be PartFunc of REAL i , REAL ;
rng f = the carrier of ( Carrier A ) \/ { i } ;
assume s1 = sqrt ( |[ 2 , p ]| ^2 ) ;
pred a > 1 & b > 0 & a / b > 1 ;
let A , B , C be Subset of [: I , J :] ;
reconsider X0 = X , X0 = Y as RealNormSpace ;
let f be PartFunc of REAL , REAL , g be PartFunc of REAL , REAL ;
r * ( v1 |-- I ) . X < r * 1 ;
assume that V is Subspace of X and X is Subspace of V ;
t-3 , t-3 be binary st t-3 , tt2 , tt2 is_collinear holds tt1 , tt2 , tt2 is_collinear
Q [ e-14 \/ { vp2 } , f ] ;
g \circlearrowleft ( W-min L~ z ) = z implies ( g /. len z ) .. z = ( z /. len z ) .. z
|. |[ x , v ]| - |[ x , y ]| .| = vrelational : y = vrelational ;
- f . w = - ( L * w ) ;
z - y <= x iff z <= x + y & x <= z - y ;
( 7 / ( 1 + e ) ) ^2 > 0 ;
assume X is BCK-algebra & 0 , 0 , 0 , 0 , 0 ;
F . 1 = v1 & F . 2 = v2 & F . 3 = v2 ;
( f | X ) . x2 = f . x2 .= ( f | X ) . x2 ;
( ( tan * tan ) `| Z ) . x in dom sec & ( sec * sec ) . x = f . x ;
i2 = ( f /. len ( f | ( len f -' 1 ) ) ) . ( len f -' 1 ) ;
X1 = X2 \/ ( X1 \ X2 ) ;
[. a , b , 1_ G .] = 1_ G implies a = 1_ G
let V , W be non empty VectSpStr over F_Complex , f be FinSequence of V ;
dom g2 = the carrier of I[01] & dom g2 = the carrier of I[01] ;
dom ( f2 ) = the carrier of I[01] & dom ( f2 ) = the carrier of I[01] ;
( proj2 | X ) .: X = proj2 .: X .= proj2 .: X ;
f . ( x , y ) = h1 . ( x , y ) ;
x0 - r < a1 . n & x0 - r < a1 . n ;
|. ( f /* ( s ^\ k ) ) . n - ( f /* ( s ^\ k ) ) .| < r ;
len Line ( A , i ) = width A ;
SFinSequence / ( S , g ) = ( S . g ) / ( S . f ) ;
reconsider f = v + u as Function of X , the carrier of Y ;
intloc 0 in dom Initialized ( p +* I ) ;
i1 , i2 i2 & ( ( a , b ) := ( b , c ) ) does not destroy b1 ;
( ( #Z 2 ) + arccos ) . r = ( cos . r ) ^2 + 0 ;
for x st x in Z holds f2 is_differentiable_in x & f2 is_differentiable_in x ;
reconsider q2 = ( q - x ) / ( 2 * x ) as Element of REAL ;
( 0 qua Nat ) + 1 <= i + j1 - 1 ;
assume f in the carrier of [ X \to Omega Y ] ;
F . a = H / ( x , y ) . a ;
( ( TRUE _ T ) at ( C , u ) ) . z = TRUE ;
dist ( ( a * seq ) . n , h ) < r ;
1 in the carrier of [. 0 , 1 .] & 1 in [. 0 , 1 .] ;
( p2 `1 - x1 ) / ( 2 * x1 ) > - g / ( 2 * x2 ) ;
|. r1 - p .| = |. a1 .| * |. thesis .| ;
reconsider S-14 = 8 as Element of Seg ( len S ) ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b
D0W .order() = D0W as DW0 + 1 ;
i1 = ma + n & i2 = [: \times the carrier of K :] & i1 = i2 ;
f . a [= f . ( f .: O1 "\/" f .: a ) ;
pred f = v & g = u , h = v + u ;
I . n = Integral ( M , F . n ) ;
chi ( [: the carrier of T , the carrier of T :] , S ) . s = 1 ;
a = VERUM ( A ) or a = VERUM ( A ) ;
reconsider k2 = s . b3 as Element of NAT ;
( Comput ( P , s , 4 ) ) . GBP = 0 ;
L~ M1 meets L~ ( R \/ LSeg ( R , i ) ) ;
set h = the continuous Function of X , R ;
set A = { L . ( ( k + 1 ) + 1 ) where k is Nat : k < n } ;
for H st H is atomic holds P7 [ H ] ;
set b8 = S5 ^\ ( i + 1 ) , S8 = S5 + ( i + 1 ) ;
Hom ( a , b ) c= Hom ( a opp , b opp ) ;
( 1 / ( n + 1 ) ) * ( 1 / ( n + 1 ) ) < ( 1 / ( n + 1 ) ) * ( 1 / ( n + 1 ) ) ;
( l . 1 ) = [ [ dom l , cod l ] , cod l ] ;
y +* ( i , y /. i ) in dom g ;
let p be Element of CQC-WFF ( Al ) , x be Element of CQC-WFF ( Al ) ;
X /\ X1 c= dom ( ( f1 - f2 ) | X ) ;
p2 in rng ( f /^ ( len p1 ) ) \/ rng ( f /^ ( len p1 ) ) ;
1 <= indx ( D2 , D1 , j1 ) + 1 - 1 ;
assume x in K2 /\ ( ( K \/ L ) /\ ( L \/ K ) ) ;
- 1 <= ( ( f2 ) . O ) `2 & - 1 <= ( ( f2 ) . O ) `2 ;
f , g be Function of I[01] , TOP-REAL 2 , a , b , c be Real ;
k1 - k2 = k1 - k2 - k2 .= k1 - k2 - k2 .= k1 - k2 - k2 - k2 ;
rng seq c= ]. x0 - r , x0 .[ & rng seq c= ]. x0 - r , x0 .[ ;
g2 in ]. x0 - r , x0 + r .[ & g2 in ]. x0 - r , x0 + r .[ ;
sgn ( p `1 , K ) = - 1_ K & sgn ( p `2 , K ) = - 1_ K ;
consider u being Nat such that b = p |^ y * u ;
ex A being the the the function of A st a = Sum A ;
Cl ( union ( H ) ) = union ( \mathop { \rm Int } H ) ;
len t = len t1 + len t2 .= len t2 + len t2 .= len t2 + len t2 ;
v-29 = v + w |-- ( A + B ) ;
v <> DataLoc ( t3 . GBP , 3 ) & v <> DataLoc ( t3 . GBP , 3 ) ;
g . s = sup ( d " { s } ) ;
( \dot y ) . s = s . ( y . s ) ;
{ s : s < t & t = {} + s } = {} ;
s ` \ s = s ` \ ( 0. X \ s ) .= ( 0. X \ s ) \ ( 0. X \ s ) ;
defpred P [ Nat ] means B + $1 in A & $1 in B ;
( 329 + 1 ) ! = 329 ! * ( 329 + 1 ) ;
U U U ( ) = ( 1_ ( A ( ) ) ) * ( 1_ ( A ( ) ) ) ;
reconsider y = y as Element of ( len y ) -tuples_on the carrier of K ;
consider i2 being Integer such that y0 = p * i2 and i2 in dom f ;
reconsider p = Y | Seg k as FinSequence of ( the carrier of K ) * ;
set f = ( S , U ) \mathop { I } ;
consider Z being set such that lim s in Z and Z in F ;
let f be Function of I[01] , TOP-REAL n , a be Real ;
( SAT M ) . [ n + i , 'not' A ] <> 1 ;
ex r being Real st x = r & a <= r & r <= b ;
R1 , R2 be Element of ( n + 1 ) -tuples_on REAL , a be Element of REAL ;
reconsider l = 0. ( V ) as Linear_Combination of A ;
set r = |. e .| + |. w .| + |. s .| ;
consider y being Element of S such that z <= y and y in X ;
a is ' of 'not' ( b 'or' c ) = 'not' ( ( a 'or' b ) 'or' c ) ;
||. ( x - g ) . ( y - g ) .|| < r2 / 2 ;
b9 , a9 // b9 , c9 & b9 , c9 // b9 , c9 ;
1 <= k2 -' k1 & k2 + 1 = k2 - k1 & k1 + 1 = k2 - k1 + 1 ;
( ( p `2 ) ^2 - ( p `2 ) ^2 ) >= 0 ;
( ( q `2 ) ^2 - ( q `1 ) ^2 ) / ( 1 + ( q `2 ) ^2 ) < 0 ;
E-max C in cell ( Rl , 1 , 1 ) & E-max L~ Cage ( C , n ) c= L~ Cage ( C , n ) ;
consider e being Element of NAT such that a = 2 * e + 1 ;
Re ( lim F ) = Re ( lim G ) .= Re ( lim G ) ;
LIN b , a , c or LIN b , c , a ;
p `1 , a // a `1 , b `1 or p `1 , a `2 or p `2 = b ;
g . n = a * Sum ( f | n ) .= f . n * Sum ( f | n ) ;
consider f being Subset of X such that e = f and f is \rangle ;
F | ( N2 , S ) = CircleMap * ( F | ( N2 , S ) ) ;
q in LSeg ( q , v ) \/ LSeg ( v , p ) ;
Ball ( m , r0 ) c= Ball ( m , s ) ;
the carrier of (0). V = { 0. V } .= { 0. V } ;
rng ( ( - 1 ) (#) cos ) = [. - 1 , 1 .] .= [. - 1 , 1 .] ;
assume that Re seq is summable and Im seq is summable and Im seq is summable ;
||. ( vseq . n ) - ( vseq . n ) .|| < e / 2 ;
set g = O --> 1 ;
reconsider t2 = t11 as 0 -element string of S2 , T2 be 0 -element string of S2 ;
reconsider x9 = seq . n as sequence of ( TOP-REAL n ) * ( seq . n ) ;
assume that that C meets L~ go and L~ co /\ L~ co = { pion1 /. 1 } and L~ co /\ L~ co = { pion1 /. 1 } ;
- ( ( - 1 ) * x ) < F . n - F . x ;
set d1 = \bf dist ( x1 , z1 ) , d2 = dist ( x2 , z2 ) , d2 = dist ( x2 , z2 ) , d2 = dist ( x2 , z2 ) ;
2 |^ ( 1 - 1 ) = ( 2 |^ 1 ) - 1 ;
dom ( v | Seg len ( d ^ <* e *> ) ) = Seg len ( d ^ <* e *> ) ;
set x1 = - ( ( k + 1 ) + |. k + 1 .| ) , x2 = - ( k + 1 ) ;
assume for n being Element of X holds 0. ( \overline { F . n } ) <= F . n ;
assume that 0 <= T-32 . i and T-32 . ( i + 1 ) <= 1 and T-32 . ( i + 1 ) <= 1 ;
for A being Subset of X holds c . ( c . A ) = c . A
the carrier of ( Carrier ( Lj + L2 ) ) c= I2 ;
'not' Ex ( x , p ) => All ( x , 'not' p ) is valid ;
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the normal w.r.t. of {} ;
Z c= dom ( ( - sin ) (#) ( sin * f ) ) ;
|. 0 - ( 0. TOP-REAL 2 ) - ( q `2 ) .| < r / 2 ;
ConsecutiveSet2 ( A , succ d ) c= ConsecutiveSet2 ( A , succ d ) ;
E = dom ( L ) & L is_measurable_on E & ( L ) . E is_measurable_on E ;
C |^ ( A + B ) = C |^ B * C |^ A ;
the carrier of W2 c= the carrier of V & the carrier of W1 c= the carrier of V ;
I . IC Comput ( P , s , m ) = P . IC Comput ( P , s , m ) .= ( card I + 1 ) ;
pred x > 0 means : Def2 : ( 1 / 2 ) |^ ( - x ) = x |^ ( - x ) ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) ;
consider p being Point of T such that C = [. p , R .] ;
b , c are_connected & - C , - C + - C + - C + - D + - D + - D + - E , - F + ( - C + - D ) + - F , G + - C + ( - D ) + D + - E + F + D +
assume f = id ( the carrier of O ) & g = id ( the carrier of O ) ;
consider v such that v <> 0. V and f . v = L * v ;
let l be or of {} ( the carrier of V ) , a be Element of V ;
reconsider g = f " as Function of U2 , U1 ;
A1 in the points of ( k , X ) . ( X . ( k + 1 ) ) ;
|. - x .| = - ( - x ) .= - x .= - x .= - x ;
set S = \mathop { \rm
Fib ( n ) * ( 5 * Fib ( n ) - 1 ) >= 4 * 1 ;
vM /. ( k + 1 ) = vM . ( k + 1 ) ;
0 mod i = - ( i * ( 0 qua Nat ) ) .= - ( i * ( 0 qua Nat ) ) ;
Indices M1 = [: Seg n , Seg n :] & [: Seg n , Seg n :] c= Indices M1 ;
Line ( Sj , j ) = Sj . j .= S . j ;
h . ( x1 , y1 ) = [ y1 , x1 ] & h . ( y1 , y2 ) = [ y1 , y2 ] ;
|. f - Re ( |. f .| ) * ( ( card b ) * h ) .| is nonnegative ;
assume x = ( a1 ^ <* x1 *> ) ^ ( b1 ^ <* x1 *> ) ;
Mj is_closed_on IExec ( I , P , s ) , P , s ;
DataLoc ( t3 . a , 4 ) = intpos ( 0 + 4 ) ;
x + y < - x + y & |. x - y .| = - x + y ;
LIN c , q , b & LIN c , q , c ;
f^ . ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + z1 ) .= x + ( y + z ) ;
flim . a = flim . a .= v . a .= v . a .= v . a ;
( p `1 ) ^2 <= ( ( E-max C ) `1 ) ^2 + ( ( E-max C ) `1 ) ^2 ;
set R8 = Cage ( C , n ) \circlearrowleft E8 , R7 = Cage ( C , n ) ;
( p `1 ) ^2 >= ( ( E-max C ) `1 ) ^2 + ( ( E-max C ) `1 ) ^2 ;
consider p such that p = pp and s1 < p /. i and p <> 0. TOP-REAL 2 ;
|. ( f /* ( s * F ) ) . l - ( f /* F ) . l .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len Line ( N , k + 1 + 1 ) = width N ;
f1 /* s1 is convergent & f2 /* s1 is convergent & lim ( f1 /* s1 ) = f1 . x0 ;
f . x1 = x1 & f . y1 = y1 & f . y2 = y1 ;
len f <= len f + 1 & len f + 1 <> 0 & len f + 1 <> 0 ;
dom ( Proj ( i , n ) * s ) = REAL m & rng s c= REAL m ;
n = k * ( 2 * t ) + ( n mod ( 2 * k ) ) ;
dom B = 2 -tuples_on the carrier of V , the carrier of V , A = [: the carrier of V , the carrier of V :] ;
consider r such that r _|_ a and r _|_ x and r _|_ y ;
reconsider B1 = the carrier of Y1 , B2 = the carrier of Y2 as Subset of X ;
1 in the carrier of [. 1 / 2 , 1 .] & 1 / 2 in [. 1 / 2 , 1 .] ;
for L being complete LATTICE holds <* <* <* <* C , D *> , L *> *> , [ c , d ] *> *> is isomorphic
[ gi , gj ] in Ij \ Ij ;
set S2 = 1GateCircStr ( x , y , c ) ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and f3 is_differentiable_in x0 and f3 is_differentiable_in x0 and f3 is_differentiable_in x0 and f3 is_differentiable_in x0 and f3 is_differentiable_in x0 and f3 is_differentiable_in x0 and f3 is_differentiable_in x0 and f3 is_differentiable_in x0 and f3 is_differentiable_in x0 and f3 is_differentiable_in x0 and f3 is_differentiable_in x0 and f3 is_differentiable_in x0 and f3 is_differentiable_in x0 and f3 is_differentiable_in x0 and
reconsider y = ( a " ) / ( F . ( len F ) ) as Element of L ;
dom s = { 1 , 2 , 3 } & s . 1 = d1 & s . 2 = d2 ;
( min ( g , ( 1 - g ) ) . c <= h . c ;
set G2 = the subgraph of G , e = the Vertex of G , a = the Vertex of G , b = the Vertex of G ;
reconsider g = f as PartFunc of REAL , REAL-NS n , REAL-NS n ;
|. s1 . m / |. p .| * ( p / |. p .| ) / ( |. p .| ) < d / ( |. p .| ) ;
for x being element st x in ( 1 - t ) holds x in ( 1 - t ) * ( 1 - t )
P = the carrier of ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | (
assume that p10 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) and LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) <> {} ;
( 0. X \ x ) |^ ( m * ( k + 1 ) ) = 0. X ;
let g be Element of Hom ( cod f , cod f ) ;
2 * a * b + ( 2 * c ) * d <= 2 * C1 * C2 ;
f , g , h be Point of the complex normed space of X , Y be bounded Function of X , Y ;
set h = Hom ( a , g ) ;
then idseq ( n ) | Seg m = idseq ( m ) & m <= n ;
H * ( g " * a ) in the right of H * ( g " * a ) ;
x in dom ( ( - cos ) `| Z ) & x in dom ( ( - sin ) `| Z ) ;
cell ( G , i1 -' 1 , j2 -' 1 ) misses C & cell ( G , i1 , j2 ) misses C ;
LE q2 , q1 , P , p1 , p2 & LE q2 , q1 , P , p1 , p2 ;
attr B is non \subseteq BDD A means : Def2 : B c= BDD A ;
deffunc D ( set , set ) = union rng $2 & $2 = union rng $2 ;
n + - n < len ( p9 + - n ) - n + n - n ;
pred a <> 0. K means : Def2 : the_rank_of M = the_rank_of ( a * M ) ;
consider j such that j in dom \mathbb Z and I = len TOP-REAL j + j ;
consider x1 such that z in x1 and x1 in P8 and x = [ x1 , y1 ] ;
for n ex r being Element of REAL st X [ n , r ]
set CP1 = Comput ( P2 , s2 , i + 1 ) , CP2 = Comput ( P2 , s2 , i + 1 ) , CP2 = P2 ;
set cv = 3 / ( a , b , c ) , cj = - ( a + b ) / ( a , b , c ) ;
conv ( F .: W ) c= union ( F .: ( E " ( W ) ) ) ;
1 in [. - 1 , 1 .] /\ dom ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( 1 + 1 ) ) ) ;
r3 <= s3 + ( ( r2 - s3 ) / 2 ) * ( ( r2 - s3 ) / 2 ) ;
dom ( f * ( f3 ) ) = dom f /\ dom f3 .= dom f /\ dom f3 .= dom f /\ dom f3 ;
dom ( f * G ) = dom ( l (#) F ) /\ Seg k .= Seg ( k + 1 ) ;
rng ( s ^\ k ) c= dom f1 \ { x0 } & rng ( s ^\ k ) c= dom f2 \ { x0 } ;
reconsider g9 = gp as Point of ( TOP-REAL n ) | ( L~ h ) ;
( T * h . s ) . x = T . ( h . s . x ) ;
I . ( L . ( J . x ) ) = ( I * L ) . x ;
y in dom <* *> implies ( Frege *> ) . ( ( Frege ( A . o ) ) . y = ( Frege ( A . o ) ) . y ;
for I being non degenerated commutative commutative commutative commutative non empty doubleLoopStr holds I is commutative
set s2 = s +* ( intloc 0 ) , P2 = P +* Start-At ( 0 , SCM+FSA ) ;
P1 /. IC s1 = P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= ( P1 . IC s2 ) . IC s2 .= ( P1 . IC s2 ) . IC s2 ;
lim S1 in the carrier of [: a , b :] & lim S1 in the carrier of [: a , b :] ;
v . ( ( l *' ) . i ) = ( v *' ( l *' ) ) . i ;
consider n being element such that n in NAT and x = seq . n ;
consider x being Element of c such that F1 . x <> F2 . x and x in dom ( F1 . x ) ;
( cluster cluster cluster cluster cluster cluster cluster cluster X1 ( 0 , 0 , x1 , x2 , x3 ) -> { 0 } ) -> { 0 , 1 , 2 , 3 } ;
j + ( 2 * ( k + 1 ) ) > j + ( 2 * ( k + 1 ) ) ;
{ s , t } on A3 & { s , t } on A3 ;
n1 > len crossover ( p2 , p1 , n1 , n2 ) & n2 <= len crossover ( p2 , p1 , n1 , n2 ) ;
( ( ( for g2 ) /. ( HT ( g2 , T ) ) ) ) . ( HT ( g2 , T ) ) = 0. L ;
then H1 , H2 are_: H , H1 are_that Cl H1 , H2 are_are that H , H1 are_/ 2 ;
( ( N-min L~ f ) .. f ) .. f > 1 & ( ( N-min L~ f ) .. f ) .. f > 1 ;
]. s , 1 .[ = ]. s , 2 .[ /\ [. 0 , 1 .] ;
x1 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) ) & x2 in ( ( TOP-REAL 2 ) | ( L~ g ) ) ;
let f1 , f2 be continuous PartFunc of REAL , the carrier of S , f be PartFunc of S , REAL ;
DigA ( ti1 , zi2 ) is Element of k -tuples_on ( the carrier of K ) ;
I \mathop { \rm Int 22j } = d & I \mathop { \rm Int 2j } = k2 ( ) ;
u9 ~ = { [ a , u9 ] } .= { [ a , u9 ] } ;
( w | p ) | ( p | ( w | w ) ) = p ;
consider u2 such that u2 in W2 and x = v + u2 and u = v + ( u + u1 ) ;
for y st y in rng F ex n st y = a |^ n & n <= len F ;
dom ( ( g * ( ( _ 2 ) \dot \to C ) ) | K ) = K ;
ex x being element st x in ( ( ( ( ( ( U0 ) \/ A ) . s ) ) . s ) ;
ex x being element st x in ( ( Let ( ( ( ( ( O ) \/ A ) . s ) ) . s ) . x ) ;
f . x in the carrier of [. - r , r .] & f . x in [. - r , - r .] ;
( the carrier of X1 union X2 ) /\ ( the carrier of X2 ) <> {} implies ( the carrier of X1 union X2 ) /\ ( the carrier of X2 ) <> {} ;
L1 /\ LSeg ( p10 , p2 ) c= { p10 } /\ LSeg ( p1 , p2 ) ;
( ( b + bC ) / 2 ) in { r : a < r & r < b + a } ;
ex_sup_of { x , y } , L & x "\/" y = sup { x , y } ;
for x being element st x in X ex u being element st P [ x , u ]
consider z being Point of G8 such that z = y and P [ z ] ;
( the sequence of ( ( the sequence of ( the carrier of X ) ) ) . ( x , y ) <= e ;
len ( w ^ w2 ) + 1 = len w + 2 + 1 .= len w + 1 ;
assume q in the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 & q in the carrier of ( ( TOP-REAL 2 ) | K1 ) ;
f | ( E-4 ` ) = g | ( EK ` \ EK ) .= g | ( EK \ EK ) ;
reconsider i1 = x1 , i2 = x2 , z = x3 as Element of NAT ;
( a * A ) ` = ( a * ( A * B ) ) ` ;
assume ex n0 being Element of NAT st f |^ ( n + 1 ) is \mathop { \rm \hbox { - } F . n } ;
Seg len ( ( ( f ^ ) | ( len ( f ^ ) ) ) ) = dom ( ( f ^ ) | ( len ( f ^ ) ) ) ;
( Complement ( Complement A1 ) ) . m c= ( Complement A1 ) . n \/ ( Complement A2 ) . ( n + 1 ) ;
f1 . p = p9 & g1 . p = d & g1 . q = d & g2 . p = d ;
FinS ( F , Y ) = FinS ( F , dom ( F | Y ) ) ;
( x | y ) | z = z | ( y | x ) ;
( ( |. x .| ) |^ n ) / ( ( |. x .| ) |^ n ) <= ( ( |. r2 .| ) |^ n ) / ( ( |. x .| ) |^ n ) ;
Sum ( F ) = Sum f & dom ( F ) = dom g & rng ( F ) = dom f ;
assume for x , y being set st x in Y & y in Y holds x /\ y in Y ;
assume that W1 is Subspace of W3 and W2 is Subspace of W3 and W1 is Subspace of W2 ;
||. ( t-15 . x ) .|| = lim ||. ( vseq . x ) .|| .= ||. ( vseq . x ) .|| .= ||. ( vseq . x ) .|| ;
assume that i in dom D and f | A is lower and g | A is lower ;
( ( p `2 ) ^2 - 1 ) * ( 1 + 1 ) <= ( - 1 ) * ( - 1 ) ;
g | Sphere ( p , r ) = id ( Sphere ( p , r ) ) ;
set N8 = N-min L~ Cage ( C , n ) , N8 = Cage ( C , n ) ;
for T being non empty TopSpace holds T is countable countable implies T is countable
width B |-> 0. K = Line ( B , i ) .= Line ( B , i ) .= B * ( i , j ) ;
attr a <> 0 implies ( A ^^ B ) Y. = ( A Y. ) Y. ;
then f is_partial_differentiable_in u0 , u & pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 ;
assume that a > 0 and a <> 1 and b > 0 and b <> 0 and c <> 0 and d <> 0 and d <> 0 ;
w1 , w2 in Lin { w1 , w2 } & w2 in Lin { w1 , w2 } ;
p2 /. IC Comput ( p2 , s2 , k ) = p2 . IC Comput ( p2 , s2 , k ) .= ( card I + ( card J + 2 ) ) ;
ind ( T-10 | b ) = ind b .= ind B - ind b .= ind B - ind B .= ind B - card A ;
[ a , A ] in the lines of G_ ( k , X ) & [ a , A ] in the lines of G_ ( k , X ) ;
m in ( the Arrows of \HM { C , o } ) . ( o1 , o2 ) ;
( ( ( a , CompF ( PA , G ) ) ) . z = FALSE ;
reconsider phi = phi /. 11 , phi = phi /. 2 as Element of ( len phi ) -tuples_on BOOLEAN ;
len s1 - ( len s2 - 1 ) + 1 > 0 + 1 - 1 + 1 ;
\delta ( D ) * ( f . ( upper_bound A ) - f . ( lower_bound A ) ) < r ;
[ f21 , f22 ] in the carrier' of A & [ f22 , f22 ] in the carrier' of A ;
the carrier of ( ( TOP-REAL 2 ) | K1 ) = K1 & the carrier of ( ( TOP-REAL 2 ) | K1 ) = K1 ;
consider z being element such that z in dom g2 and p = g2 . z and y = g2 . z ;
[#] ( V1 ) = { 0. V } .= the carrier of ( V ) .= the carrier of ( V ) ;
consider P2 being FinSequence such that rng P2 = M and P2 is one-to-one and P2 is one-to-one and P [ P2 ] ;
assume that x1 in dom ( f | X ) and ||. x1 - x0 .|| < s and ||. x1 - x0 .|| < s ;
h1 = f ^ ( <* p3 *> ^ <* p *> ) .= h ^ <* p *> .= h ;
c /. ( |[ b , c ]| ) = c /. ( |[ a , c ]| ) .= c /. ( |[ b , c ]| ) .= c /. ( |[ b , c ]| ) ;
reconsider t1 = p1 , t2 = p2 , t2 = p3 as Term of C , V ;
( 1 / 2 ) * ( ( 1 / 2 ) * ( 1 / 2 ) ) in the carrier of [. 0 , 1 .] ;
ex W being Subset of X st p in W & W is open & h .: W c= V ;
( h . p1 ) `2 = C * ( ( p1 `2 ) - D ) + D * ( ( p1 `2 ) - D ) ;
R . ( b b ) implies 2 * \lbrack b , a :] = 2 * b - a .= 2 * b - a .= b ;
consider \bf such that B = ( - 1 ) * C + ( - 1 ) * A and 0 <= 1 ;
dom g = dom ( ( the Sorts of A ) * ( ( the Sorts of A ) * ( ( the_arity_of o ) /. i ) ) ;
[ P . ( l1 ) , P . ( l1 ) ] in => ( T . ( l1 ) , T . ( l1 ) ) ;
set s2 = Initialize s , s3 = Initialize s , s3 = P +* I ;
reconsider M = mid ( z , i2 , i1 ) as non constant Matrix of 2 , ( len z ) , ( len z ) , ( len z ) , ( len z ) , ( len z ) , ( len z ) , ( len z ) , ( len z ) ) ;
y in product ( ( Carrier J ) +* ( V , { 1 } ) ) ;
1 / ( |[ 0 , 1 ]| ) = 1 & 0 / ( |[ 0 , 1 ]| ) = 0 ;
assume x in the left of g or x in the left of g or x in the left of g ;
consider M being strict Subgroup of A9 such that a = M and T is and M is Subspace of M ;
for x st x in Z holds ( ( ( - 1 ) (#) f ) `| Z ) . x <> 0 & f . x <> 0 ;
len ( W1 + W2 ) = 1 + len ( W2 + W3 ) .= len ( W1 + W3 ) + len ( W2 + W3 ) ;
reconsider h1 = ( vseq . n - t-16 ) . t as Lipschitzian LinearOperator of X , Y ;
( i mod len ( p + q ) + 1 ) in dom ( p + q ) ;
assume that s2 is conjunctive and F in the |= of ( the |= of s2 ) and F in the |= of ( the |= of s2 ) ;
( ( ( ( x , y ) ) * ( 1 , y ) ) * ( x , y ) ) = ( x * y ) * ( x , y ) ;
for u being element st u in Bags n holds ( p + m ) . u = p . u ;
for B being Subset of u-5 st B in E holds A = B or A misses B ;
ex a being Point of X st a in A & A /\ Cl { y } = { a } ;
set W2 = tree ( p ) \/ { p } ;
x in { X where X is Ideal of L : P [ X ] } implies x in { X where X is Subset of L : P [ X ] }
the carrier of W1 /\ W2 c= the carrier of V & the carrier of W1 /\ W2 c= the carrier of V ;
( ( 1 + b ) * id a ) * ( id a ) = ( 1 + b ) * id a ;
( ( X --> f ) . x ) . x = ( X --> f ) . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) ;
p => ( q => r ) => ( p => ( p => r ) ) in TAUT ( A ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
- 1 + 1 <= ( - ( 2 |^ ( n -' m ) ) * ( - 1 ) + 1 ;
( reproj ( 1 , z0 ) ) . x in dom ( ( f1 (#) f2 ) . x ) ;
assume that b1 . r = { c1 . r } and b2 . r = { c2 . r } and b2 . r = { c2 . r } ;
ex P st a1 on P & a2 on P & b on P & c on P & d on P & d on P ;
reconsider gf = g `2 * f , hg = h `2 * g as strict Element of X ;
consider v1 being Element of T such that Q = ( downarrow v1 ) ` and v1 in F ;
n in { i where i is Nat : i < n + 1 & n < len f + 1 + 1 } ;
( F /. ( i , j ) ) `2 >= ( ( F /. m ) /. ( i + 1 ) ) `2 ;
assume K1 = { p : ( p `1 <= s & s >= 0 & p <> 0. TOP-REAL 2 } ;
ConsecutiveSet ( A , succ O1 ) = ( ConsecutiveSet ( A , O1 ) ) ^ ( ConsecutiveSet ( A , O1 ) ) ;
set I1 = Macro ( a , intloc 0 ) , I2 = [ a , intloc 0 , intloc 0 ] , I2 = [ a , intloc 0 , intloc 0 ] ;
for i be Nat st 1 < i & i < len z holds z /. i <> z /. 1 ;
X c= ( the carrier of L1 ) /\ ( the carrier of L2 ) & X c= the carrier of L2 ;
consider x9 being Element of GF ( p ) such that x9 |^ 2 = a and x9 |^ 2 = a ;
reconsider ej = ej , fj = fj , fn = fj as Element of D ( ) ;
ex O being set st O in S & C1 c= O & M . O = 0. ( Cl O ) ;
consider n being Nat such that for m being Nat st n <= m holds S . m in U1 ;
f * g * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) * g ) . x ;
defpred P [ Nat ] means A + ( succ $1 ) = succ A + ( succ $1 ) & ( A + ( succ $1 ) = A + ( succ $1 ) ;
the left of - g = the left of ( - g ) .= ( the left st g ) * ( - g ) ;
reconsider pp = x , pp = y , pp = z as Point of Euclid 2 , p = p , q = q , r = p ;
consider g2 such that g2 = y and x <= g2 and g2 <= x0 and g2 <= x0 and x0 <= g2 & g2 <= x0 and g2 <= x0 ;
for n being Element of NAT ex r being Element of REAL st X [ n , r ]
len ( x2 ^ y2 ) = len x2 + len y2 .= len ( x2 ^ y2 ) + len ( y2 ^ y1 ) .= len ( x2 ^ y2 ) + len ( y2 ^ y1 ) ;
for x being element st x in X holds x in the set of \HM { the } \HM { positive } \HM { of n , m } & x <> m implies x = ( the set of m ) . ( n , m )
LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = {} or LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = {} ;
func ex ex X be set st ( for x being set holds x in X iff x in Z ) & ( F . ( id X ) = X ) ;
len ( ( { Gauge ( C , n ) /. ( len ( Cage ( C , n ) ) , 1 ) ) `1 ) <= len ( ( Cage ( C , n ) /. len ( Cage ( C , n ) ) ) `1 ) ;
attr K has a is valuation & a <> 0. K & v . ( a |^ i ) = i * v . ( a |^ i ) ;
consider o being OperSymbol of S such that t `1 . {} = [ o , the carrier of S ] ;
for x st x in X ex y st x c= y & y in X & y is Set & f . x = f . y ;
IC Comput ( P-6 , k ) in dom ( ( n + 1 ) .--> ( n + 1 ) ) ;
pred q < s & r < s & s < q & p , q , r is_collinear ;
consider c being Element of Class ( f , c ) such that Y = ( F . c ) . ( c , d ) ;
func ( the ResultSort of S2 ) . ( ( the ResultSort of S2 ) . ( ( the ResultSort of S2 ) . ( ( the ResultSort of S2 ) . ( ( the ResultSort of S2 ) . ( ( the ResultSort of S2 ) . ( ( the ResultSort of S1 ) . ( ( the ResultSort of S2 ) . ( ( the ResultSort of S1 ) . ( ( the ResultSort of S1 ) . ( ( the ResultSort of S1 )
set yy = [ <* y , z *> , f2 ] , y2 = [ <* z , x *> , f3 ] ;
assume x in dom ( ( ( - ( #Z 2 ) * ( ( #Z 2 ) * ( f ^ ) ) ) `| Z ) ;
r-7 in Int cell ( GoB f , i , GoB f ) \ L~ f \/ L~ f ;
( q `2 ) ^2 >= ( ( Cage ( C , n ) /. ( i + 1 ) ) `2 ;
set Y = { a "/\" a ` : a in X } ;
i - len f <= len f + len f - len f + len f - len f + len f - len f + len f - len f + len f - len f + len f - len f + len f - len f + len f - len f + len f - len f + len f - len f + len f - len f + len f - len f + len f -
for n ex x st x in N & x in N1 & h . n = x- ( x0 - x )
set s0 = ( \mathop { \it SCMPDS } , p , s ) . i , p = ( \mathop { \it SCMPDS } , p ) . i , s = ( \mathop { \it SCMPDS } ) . i , n = 0 ;
p . ( k + 1 ) = 1 or p . ( k + 1 ) = - 1 or p . ( k + 1 ) = - 1 ;
u + Sum ( L-18 ) in ( U \ { u } ) \/ { u + Sum ( L-18 ) } ;
consider x9 being set such that x in x9 and x9 in V and x9 in V and x9 in V ;
( p ^ ( q | k ) ) . m = ( q | k ) . ( len p - len p ) ;
g + h = gg + hh & ||. g + h .|| = g + h ;
L1 is distributive & L2 is distributive implies L1 ~ is distributive & L2 ~ is distributive & L1 ~ is distributive
pred x in rng f & y in rng ( f | x ) implies f . x = f . y ;
assume that 1 < p and ( 1 + ( p `1 ) ^2 ) = 1 and 0 <= a and a <= b and b <= 0 ;
FM * ( f , <* the carrier of L *> ) = rpoly ( 1 , the carrier of L ) *' t .= <* 0. L *> ;
for X being set , A being Subset of X holds A ` = {} implies A = {} & A = {} ;
( ( ( ( ( ( ( ( X ) ) ) ) ) `1 ) ) ) `1 <= ( ( ( ( ( ( ( ( X ) ) ) ) ) ) ) ) ) `1 ) ) `1 ;
for c being Element of the Sorts of A , a being Element of the Sorts of A holds c <> a ;
s1 . GBP = ( Exec ( i2 , s2 ) ) . intpos ( 0 + 1 ) .= 0 .= 0 ;
for a , b being Real holds [ a , b ] in ( y iff b >= 0 & a >= 0 ) implies b >= 0
for x , y being Element of X holds x ` \ y = ( x \ y ) `
mode BCK-algebra of i , j , m , n , m , n , m , n be Element of NAT , i , j be Element of NAT ;
set x2 = |( Re y , Im y )| ;
[ y , x ] in dom ( u . y ) & ( u . y ) . ( y , x ) = g . y ;
]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .] c= A & upper_bound divset ( D , k ) = lower_bound A ;
0 <= \delta ( S2 . n ) & |. \delta ( S2 . n ) .| < ( e / 2 ) * ( e / 2 ) ;
( - ( q `1 / |. q .| - cn ) ) ^2 <= ( - ( q `1 / |. q .| - cn ) ) ^2 ;
set A = ( 2 / 2 ) * ( b - a ) ;
for x , y being set st x in R" holds x , y are_\hbox { x , y }
deffunc F ( Nat ) = b . ( $1 + 1 ) * ( M * ( $1 , 1 ) ) ;
for s being element holds s in non empty iff s in non empty & s in non empty s \/ non empty s ;
for S being non empty non void holds S is connected holds S is connected implies S is connected
max ( ( degree ( K ) ) / ( 1 + ( d ) ) ^2 ) >= 0 ;
consider n1 being Nat such that for k holds seq . ( n1 + k ) < r + s . ( n1 + k ) ;
Lin ( A /\ B ) is Subspace of Lin ( A ) & Lin ( B ) is Subspace of Lin ( A ) ;
set n-15 = nj '&' ( M . ( x qua Element of BOOLEAN ) ) , nj = M . ( x , y ) ;
f " in such that f " V in consider X and f " V in D ( the carrier of S ) and f " V in D ( the carrier of S ) ;
rng ( ( a ^\ c ) ^\ ( 1 + b ) ) c= { a , c , b } ;
consider y being connected subgraph of G1 such that y `2 = y and dom y `2 = WW: 1 <= y & y `2 <= WW: y `2 <= G * ( y `2 } ;
dom ( ( 1 / 2 ) (#) f ) /\ ]. x0 - r , x0 .[ c= ]. x0 - r , x0 .[ & dom ( ( 1 / 2 ) (#) f ) /\ ]. x0 , x0 + r .[ c= ]. x0 , x0 + r .[ ;
as Element of -> Element of -> Element of -> Element of -> Element of -> Element of -> Element of [#] ( i , j , n ) ;
v ^ ( ( n |-> 0 ) ^ ( ( n |-> 0 ) ^ ( n |-> 0 ) ) ) in Lin rng ( ( n |-> 0 ) ^ ( n |-> 0 ) ) ;
ex a , k1 , k2 st i = a /. ( k1 + k2 ) & i = b /. ( k2 + k2 ) ;
t . ( NAT ) = ( NAT .--> succ i1 ) . ( NAT + 1 ) .= ( NAT --> NAT ) . ( NAT + 1 ) .= ( NAT --> NAT ) . ( NAT + 1 ) .= ( NAT --> NAT ) . ( NAT + 1 ) .= ( NAT --> NAT ) . ( NAT + 1 ) ;
assume that F is bbfamily and rng p = F and rng p = Seg ( n + 1 ) and for i st i in Seg ( n + 1 ) holds p . i = F . i ;
not LIN b , a , c & not LIN b , a , c & not LIN b , a , c ;
( L1 \& L2 ) \& O c= ( L1 \& O ) \& O & ( L1 Let O ) \& O c= ( L1 \& O ) \& O ;
consider F being ManySortedSet of E such that for d being Element of E holds F . d = F ( d ) ;
consider a , b such that a * ( \rrangle = b * ( -w ) and 0 < a and a < b ;
defpred P [ FinSequence of D ] means |. Sum ( $1 ) .| <= Sum ( |. $1 .| ) ;
u = cos / ( x , y ) * x + cos / ( x , y ) * y .= v . x + ( cos / ( x , y ) * y ) .= v . y ;
dist ( ( seq . n ) + x , g + x ) <= dist ( ( seq . n ) , g ) + 0 ;
P [ p , |. p .| ^ <* {} *> , ( id the Sorts of A ) ^ <* id *> ] ;
consider X being Subset of CQC-WFF ( Al ) such that X c= Y and X is finite and X is ininand X is inin;
|. b .| * |. eval ( f , z ) .| >= |. b .| * |. eval ( f , z ) .| ;
1 < ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) - 1 + ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ;
l in { l1 where l1 is Real : g <= l1 & l1 <= h & h . l1 <= ( h . l1 ) * ( ( h . l1 ) * ( ( h . l1 ) * ( ( h . l1 ) * ( ( h . l1 ) * ( ( h . l1 ) * ( ( h . l1 ) * ( ( h . l1 ) * ( ( h . l1 ) * ( h .
( Partial_Sums ( ( G . n ) vol ) ) . ( n + 1 ) <= ( Partial_Sums ( ( G . n ) ) ) . ( n + 1 ) ;
f . y = x .= x * 1_ L .= x * ( 1_ L ) .= x * ( power L ) . ( y , 0 ) .= x * ( power L ) . ( y , 0 ) ;
NIC ( ( \bf if i1 , i2 ) \ { ( l , k ) } , ( l , k ) \ { ( l , k ) } ) = { ( ( i1 , ( l , k ) } \ { ( l , k ) } ) } ;
LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = { p1 , p2 } /\ LSeg ( p1 , p2 ) ;
product ( ( the carrier of I-15 ) +* ( i , { 1 } ) ) in Z ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) .= Following ( s1 , n ) . ( n + 1 ) ;
( W-bound ( Q ) ) <= ( q `1 ) / ( 2 * ( ( q `1 ) ) / ( 2 * ( ( q `1 ) ) ^2 ) ) ;
f /. i2 <> f /. ( len f + 1 -' len g ) .= f /. ( len f + 1 -' len f ) ;
M , v / ( x. 3 , a ) / ( x. 4 , a ) / ( x. 0 , a ) / ( x. 0 , a ) / ( x. 4 , a ) |= H ;
len ( ( P ^ ) ^ ( ( P ^ ) ) ) in dom ( ( P ^ ) ^ ( ( P ^ ) ) ) ;
A |^ ( m , n ) c= A |^ ( m , n ) & A |^ ( k , n ) c= A |^ ( k , l ) ;
R |^ n \ { q : |. q .| < a } c= { q1 : |. q1 .| >= a }
consider n1 being element such that n1 in dom p1 and y1 = p1 . n1 and p1 . n1 = p2 . n1 ;
consider X being set such that X in Q and for Z being set st Z in Q & Z <> X holds X c= Z ;
CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA & CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA ;
for v be VECTOR of l1 holds ||. v .|| = upper_bound rng |. |. ( v . v ) .| * ||. v .||
for phi st phi in X holds not phi in X & not phi in X & not phi in X & not phi in X & not phi in X
rng ( ( Sgm dom ( f | ( dom f ) ) | ( dom ( f | ( dom f ) ) ) ) ) c= dom ( ( f | ( dom f ) ) | ( dom ( f | ( dom f ) ) ) ) ;
ex c being FinSequence of D ( ) st len c = k & P [ c ] & a = c . 1 & b = c . 2 ;
( the_arity_of ( a , b , c ) ) = <* ( Hom ( b , c ) ) . ( a , b ) , ( F . ( b , c ) ) *> ;
consider f1 be Function of the carrier of X , REAL such that f1 = |. f .| and f1 is continuous and f1 | X is continuous ;
a1 = b1 & a2 = b2 or a1 = b1 or a2 = b2 or a1 = b1 & a2 = b2 & a3 = b3 & a4 = 6 & b1 = 6 & b1 = 7 & b2 = 6 & b1 = 6 & b2 = 7 & b1 = 6 & b2 = 6 & b1 <> 7 ;
D2 . ( indx ( D2 , D1 , n1 ) + 1 ) = D1 . ( indx ( D2 , D1 , n1 ) + 1 ) .= D2 . ( indx ( D2 , D1 , n1 ) + 1 ) ;
f . ( ||. |[ r , s ]| .|| ) = ||. |[ r , s ]| .|| /. 1 .= <* r *> . 1 .= r . 1 .= r . 1 .= r . 1 .= r . 1 ;
consider n be Nat such that for m being Nat st n <= m holds Cseq . m = Cseq . ( m + n ) ;
consider d being Real such that for a , b being Real st a in X & b in Y holds a <= d & b <= d ;
||. L /. h .|| - ( K * |. h .| ) + ( K * |. h .| ) <= p0 + ( K * |. h .| ) ;
attr F is commutative associative associative associative for f , g being Element of X holds F \hbox { b , f } = f . b ;
p = - 1 * ( p0 + 0. TOP-REAL 2 ) .= 1 * ( p1 + 0. TOP-REAL 2 ) .= 1 * ( p1 + 0. TOP-REAL 2 ) .= ( 1 - 1 ) * p1 + ( 1 - 1 ) * p2 .= ( 1 - 1 ) * p1 + ( 1 - 1 ) * p2 ;
consider z1 such that b , x3 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 7 7 8 8 8 8 8 8 * ||. o , o .|| = o + b1 ;
consider i such that Arg ( Rotate ( s ) ) = s + Arg ( q ) and 0 <= i and i <= 2 * PI + ( 2 * PI * i ) ;
consider g such that g is one-to-one and dom g = card f and rng g c= f . x and g . x = f . x ;
assume that A = P2 \/ Q2 and P <> {} and Q <> {} and Q <> {} and Q <> {} and Q <> {} and Q <> {} and Q <> {} and Q <> {} and Q <> {} and Q <> {} ;
attr F is associative means : Let F .: ( F .: ( f , g ) , h ) = F .: ( f , g ) ;
ex x being Element of NAT st m = x `1 & x in z `1 & x in { i } or m in { i } ;
consider k2 being Nat such that k2 in dom ( ( P . ( k + 1 ) ) . i and l . ( ( P . ( k + 1 ) ) . i ) = ( P . ( k + 1 ) ) . i ;
seq = r * seq implies for n holds seq . n = r * seq . n & seq . ( n + 1 ) = r * seq . ( n + 1 )
F1 . [ [ id a , a ] , [ id a , a ] ] = [ f * ( id a ) , f * ( id b ) ] ;
{ p } "\/" D2 = { p "\/" y where y is Element of L : y in D & p in D } ;
consider z being element such that z in dom ( ( F . 0 ) | ( dom F . 0 ) ) and ( ( F . 0 ) | ( dom F ) ) . z = y ;
for x , y being element st x in dom f & y in dom f & f . x = f . y holds x = y ;
cell ( G , i , 1 ) = { |[ r , s ]| : r <= G * ( 0 + 1 , 1 ) `1 } ;
consider e being element such that e in dom ( T | E1 ) and ( T | E1 ) . e = v . e ;
( F ` * b1 ) . x = ( Mx2Tran ( J , BY. , BY. ) ) . ( \mathbb j + 1 ) ;
- 1 / ( ( - 1 ) (#) D ) = ( - 1 ) (#) D .= ( - 1 ) (#) D .= ( - 1 ) (#) ( - 1 ) .= ( - 1 ) (#) ( - 1 ) (#) D .= ( - 1 ) (#) ( - 1 ) ;
attr for x being set st x in dom f /\ dom g holds g . x <= f . x & g . x <= f . x ;
len ( f1 . j ) = len ( f2 /. j ) .= len ( f2 /. j ) .= len ( f2 /. j ) .= len ( f2 /. j ) .= len ( f2 /. j ) ;
All ( 'not' All ( a , A , G ) , B , G ) |= Ex ( 'not' All ( 'not' a , B , G ) , A , G ) ;
LSeg ( E . ( k + 1 ) , F . ( k + 1 ) ) c= Cl RightComp Cage ( C , k + 1 ) ;
x \ ( a |^ m ) = x \ ( a |^ k * a ) .= ( x \ a ) |^ k .= ( x \ a ) |^ k ;
k = ( ( commute ( I ) ) . k ) . ( ( commute ( I ) ) . k ) .= ( ( commute ( I ) ) . k ) . ( ( commute ( I ) ) . k ) .= ( ( commute ( I ) ) . k ) . ( ( commute ( I ) ) . k ) .= ( ( commute ( I ) ) . k ) . ( ( I . k ) . k ) ;
for s being State of A holds Following ( s , n ) . ( ( n + 2 ) * n + ( 2 * n ) * n ) is stable ;
for x st x in Z holds f1 . x = a / ( a - x ) & ( f1 . x ) <> 0 & ( f1 . x ) <> 0 implies ( f1 - f2 ) . x <> 0
support ( ( support ( n ) ) \/ support ( ( support ( m ) ) ) c= support ( ( support ( m ) ) ) \/ support ( ( support ( m ) ) ) ;
reconsider t = u as Function of ( the carrier of A ) , ( the carrier' of B ) * ( the Arity of C ) ;
- ( a * sqrt ( 1 + ( a ^2 ) ) ^2 ) <= - ( b * sqrt ( 1 + ( a ^2 ) ) ^2 ) ;
phi /. ( succ b1 ) = g . a & phi /. ( phi . ( succ b1 ) ) = f . ( g . ( succ b1 ) ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( F ^ <* p *> ) and i <> j ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 } = { x1 } \/ { x2 } ;
the Sorts of U1 /\ ( ( U1 "\/" U2 ) "\/" ( U2 "\/" W3 ) ) c= the Sorts of U1 /\ ( ( U1 "\/" U2 ) "\/" ( U2 "\/" W3 ) ) ;
( - ( 2 * a ) ) * ( - ( 2 * a ) ) + b ^2 - ( - ( 2 * a ) ) * ( - ( 2 * a ) ) > 0 ;
consider W00 such that for z being element holds z in W00 iff z in N ~ & P [ z ] ;
assume that ( the Arity of S ) . o = <* a *> and ( the Arity of S ) . o = r and ( the Arity of S ) . o = <* a *> ;
Z = dom ( ( exp_R * ( ( #Z n ) * ( f + #Z n ) ) ) `| Z ) .= dom ( ( #Z n ) * ( f + #Z n ) ) ;
||. integral ( f , S ) .|| is convergent & lim ( ( Im f ) * ( T . n ) ) = integral ( f , S ) ;
( ( a => ( f . ( a => b ) ) => ( f . ( x => b ) ) ) in non empty
len ( M2 * M3 ) = n & width ( M2 * M2 ) = n & width ( M2 * M2 ) = n & len ( M2 * M1 ) = n ;
attr X1 union X2 is open SubSpace of X & X1 , X2 X2 X2 X2 X2 iff X1 union X2 , X1 union X2 , X2 union X2 , X1 union X2 , X2 union X2 , X2 union X1 , X2 union X2 , X1 union X2 be SubSpace of X ;
for L being upper-bounded antisymmetric RelStr , X being non empty RelStr , s being non empty Subset of L holds X "\/" { Top L } = { Top L }
reconsider f-1be Function of [: the carrier of M , the carrier of M :] , the carrier of M = [: the carrier of M , the carrier of M :] , the carrier of M = [: the carrier of M , the carrier of M :] ;
consider w being FinSequence of I such that the InitS { s } ^ w = <* s *> ^ w ^ w ^ w ^ y ;
g . ( a |^ 0 ) = g . ( 1_ G ) .= 1_ G .= 1_ G .= 1_ G .= 1_ G .= 1_ G ;
assume for i being Nat st i in dom f ex z being Element of L st f . i = rpoly ( 1 , z ) . i ;
ex L being Subset of X st Carrier L = L & for K being Subset of X st K in C holds L /\ K <> {} ;
( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C2 & ( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C2 ;
reconsider o9 = o `2 as Element of TS ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the_arity_of o ) . ( ( the_arity_of o ) . ( ( the_arity_of o ) . ( o ) ) ) ) ) ) ) ;
1 * x1 + ( 0 * x2 ) + ( 0 * x3 ) = x1 + ( 0 * x2 ) .= x1 + ( 0 * x2 ) .= x1 + ( 0 * x2 ) .= x1 + ( 0 * x2 ) .= x1 + ( 0 * x2 ) ;
Ej " . 1 = ( ( E qua Function ) " . 1 .= ( ( E qua Function ) " ) . 1 .= ( ( E qua Function ) " ) . 1 .= ( ( E qua Function ) " ) . 1 .= ( ( E qua Function ) " ) . 1 .= ( ( E qua Function ) . 1 .= ( ( E qua Function ) " ) . 1 .= ( ( E qua Function ) " ) . 1 .= ( ( E qua Function ) " ) . 1 .= ( ( E
reconsider u1 = the carrier of ( U1 /\ ( U1 "\/" U2 ) ) as non empty Subset of ( U1 /\ U2 ) ;
( ( x "/\" z ) "\/" ( x "/\" y ) ) "\/" ( z "/\" y ) <= ( x "/\" ( x "/\" y ) ) "\/" ( x "/\" ( y "\/" z ) ) ;
|. f . ( s1 . ( l1 + 1 ) ) - f . ( l1 + 1 ) .| < ( 1 - M ) * ( 1 - M ) ;
LSeg ( ( Cage ( C , n ) /. ( i + 1 ) , ( Gauge ( C , n ) /. ( i + 1 ) ) , ( i + 1 ) ) is vertical ;
( f | Z ) /. x - ( f | Z ) /. x0 = L /. ( x- ( x - x0 ) ) + R /. ( x- ( x - x0 ) ) ;
g . c * ( - g . c ) + f . c <= h . c * ( - f . c ) + f . c ;
( f + g ) | divset ( D , i ) = f | divset ( D , i ) + g | divset ( D , i ) ;
assume that ColVec2Mx f in the carrier of ( the carrier of A ) and ColVec2Mx f = ( ColVec2Mx b ) and len f = len b and width f = width A and width f = width A ;
len ( - M1 ) = len M1 & width ( - M1 ) = width M1 & width ( - M1 ) = width M1 & width ( - M1 ) = width M1 & width ( - M1 ) = width M1 ;
for n , i being Nat st i + 1 < n holds [ i , i + 1 ] in the InternalRel of ( n + 1 ) & [ i + 1 , j ] in the InternalRel of ( n + 1 ) ;
pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in u0 , 2 & pdiff1 ( f1 , 2 ) is_partial_differentiable_in u0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in u0 , 2 ;
attr a <> 0 & b <> 0 & Arg a = Arg b & Arg b = Arg a & Arg a = Arg b & Arg b <> 0 & Arg a <> 0 & Arg b <> 0 & Arg a <> 0 & Arg b <> 0 & Arg b <> 0 & Arg c <> 0 & Arg a <> 0 & Arg b <> 0 & Arg c <> 0 ;
for c being set st not c in [. a , b .] holds not c in Intersection ( the topology of a , b ) & not c in Intersection ( the topology of a , b )
assume that V1 is linearly-independent and V2 is linearly-independent and V = { v + u : v in V1 & u in V1 & u in V1 & v in V1 & u in V1 & v in V1 & u in V1 & v in V1 & u in V1 & v in V1 & v in V1 & u in V1 & v in V1 & u in V1 & v in V1 & v in V1 & u in V1 ;
z * x1 + ( 1 - z ) * x2 in M & z * y1 + ( 1 - z ) * y2 in N ;
rng ( ( ( ( ( ( ( ( ( P ) ) qua Function ) " ) * ( ( P ) ) * ( ( P ) ) * ( ( P ) ) * ( ( P ) ) * ( ( P ) ) * ( ( P ) ) * ( ( P ) ) ) ) ) = Seg card ( ( ( d * ( P ) ) * ( ( P ) ) * ( ( P ) ) * ( ( P ) ) * ( ( P ) ) ) ) ;
consider s2 being rational Real_Sequence such that s2 is convergent and b = lim s2 and for n holds s2 . n <= ( lim s2 ) * ( lim s2 ) ;
h2 " . n = h2 . n " & 0 < ( - 1 ) / ( ( - 1 ) |^ n ) & 0 < ( - 1 ) / ( ( - 1 ) |^ n ) ;
( Partial_Sums ( ||. seq .|| ) ) . m = ||. ( seq ) . m - ( seq ) . ( n + 1 ) .|| .= ||. ( seq ) . m - ( seq ) . n .|| .= 0 ;
( Comput ( P1 , s1 , 1 ) . b = 0 .= ( Comput ( P2 , s2 , 1 ) . b ) . b .= ( Comput ( P2 , s2 , 1 ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b ;
- v = ( - 1_ G ) * v & - w = ( - 1_ G ) * v & - w = ( - 1_ G ) * v ;
upper_bound ( ( k .: D ) .: ( ( k .: D ) .: ( ( k .: D ) .: ( ( k .: D ) .: ( ( k .: D ) .: ( ( k .: D ) .: ( ( k .: D ) .: ( ( k .: D ) .: ( ( k .: D ) .: ( ( k .: D ) .: ( ( k .: D ) .: ( ( k .: D ) .: ( ( k .: D ) .: ( ( k .: D ) ) ) ) ) ) ) ) ) )
A |^ ( k , l ) ^^ ( ( A |^ n ) * ( A |^ k ) ) = ( A |^ n ) ^^ ( A |^ k ) ;
for R being add-associative right_zeroed right_complementable non empty addLoopStr , I , J being Subset of R holds I + ( J + K ) = ( I + J ) + K
( f . p ) `1 = ( sqrt ( ( p `1 ) ^2 + ( p `2 ) ^2 ) ) ^2 .= ( sqrt ( ( p `1 ) ^2 + ( p `2 ) ^2 ) ) ^2 ;
for a , b being non zero Nat st a , b are_relative_prime holds ( a * b ) div ( b * a ) = ( \mathop { \rm _ a * b ) * ( ( a * b ) div ( b * a ) )
consider A5 being countable Subset of Al such that r is Element of CQC-WFF ( Al ) and A5 is ( A ) ` and ( A is ( ) ) ` and ( A is ( ) ` ) and ( A is ( ) ) ` ;
for X being non empty additive loop , M , N being Subset of X st y in M & x + y in N holds x + y in M + N
{ [ x1 , x2 ] , [ y1 , y2 ] } c= { [ x1 , x2 ] , [ y1 , y2 ] }
h . ( f . O ) = |[ A * ( f . O ) + B , C * ( f . O ) + D ]| ;
( Gauge ( C , n ) * ( k , i ) ) in L~ Upper_Seq ( C , n ) /\ L~ Upper_Seq ( C , n ) ;
cluster m , n n ) -> prime implies for Nat st p divides m & p divides n holds ( p divides m ) & ( p divides n implies p divides m ) & ( p divides n implies p divides m ) & ( p divides n implies p divides m ) implies p divides n & p divides m & p divides m & p divides n implies p divides m )
( f * F ) . x1 = f . ( F . x1 ) & ( f * F ) . x2 = f . ( F . x2 ) ;
for L being Lattice , a , b , c being Element of L st a \ b <= c & b \ c <= c holds a "\/" b <= c
consider b being element such that b in dom ( H / ( x , y ) ) and z = H / ( x , y ) . b ;
assume that x in dom ( F * g ) and y in dom ( F * g ) and ( F * g ) . x = ( F * g ) . y ;
assume ex e being element st e Joins W . 1 , W . 5 , G & e Joins W . 3 , G . 4 , G . 5 , G . 7 , G . 6 , G . 7 , G . 6 , G . 7 , G . 8 , G . 8 , G . 8 , G . 7 , G . 8 , G . 8 , G . 7 , G . 8 , G . 8 , G . 7 , G . 8 , G . 8 , G . 8 ,
( ] ] (#) ( ( h h ) [ n ] ) . x = ( ( h h ) . n ) * ( ( h . n ) * ( h . x ) ) ;
j + 1 = ( j - len h11 + 2 ) - 1 .= i + 1 - len h11 + 2 - 1 .= i + 1 - 1 + 2 - 1 .= i + 2 - 1 + 1 - 1 ;
( S *' ) . ( f , g ) = S *' . ( ( S *' ) . ( f , g ) ) .= S . ( ( S *' ) . ( f , g ) ) .= S . ( f , g ) .= S . ( f , g ) ;
consider H such that H is one-to-one and rng H = the carrier of L2 and Sum ( L2 ) = Sum ( L2 ) and Sum ( L1 ) = Sum ( L2 ) ;
attr R is <= <= 1 & p in R & p <> q & ex P st P = { p } & P is R & Q c= R ;
dom ( product ( X --> f ) ) = meet ( ( X --> f ) . ( len f ) ) .= meet ( X --> f ) .= meet ( X --> f ) .= meet ( X --> f ) .= meet ( X --> f ) .= meet ( X --> f ) .= meet ( X --> f ) ;
upper_bound ( proj2 .: ( Upper_Arc C /\ Lower_Arc C /\ Lower_Arc C /\ Lower_Arc C /\ Lower_Arc C /\ Lower_Arc C /\ Lower_Arc C /\ Lower_Arc C /\ Lower_Arc C /\ Lower_Arc C /\ Lower_Arc ( w ) ) <= upper_bound ( proj2 .: ( C /\ holds w <= ( w + ( w + ( w + ( w + w ) ) ) / 2 ) ) ;
for r be Real st 0 < r ex n be Nat st for m be Nat st n <= m holds |. S . m - S . n .| < r
i * fN - fN = i * fN - ( i * fN ) .= i * fN - ( i * fN ) .= i * fN - ( i * fN ) ;
consider f being Function such that dom f = 2 -tuples_on X ( ) and for Y being set st Y in 2 -tuples_on X ( ) holds f . Y = F ( Y ( ) ;
consider g1 , g2 being element such that g1 in [#] Y and g2 in union C and g = [ g1 , g2 ] and g2 in union C and g1 in A and g2 in B ;
func d \! \mathop { n , m } -> Nat means : \rm d |^ n divides n & d |^ ( m + 1 ) divides n & d divides m |^ ( m + 1 ) ;
f\rbrack . [ 0 , t ] = f . [ 0 , t ] .= ( - P ) . [ 2 * x , t ] .= ( - P ) . [ 2 * x , t ] .= a ;
t = h . D or t = h . E or t = h . F or t = h . J or t = h . M or t = h . N ;
consider m1 being Nat such that for n st n >= m1 holds dist ( ( seq . n ) , ( seq . n ) ) < 1 / ( ( seq . n ) ) ;
( ( q `1 ) / |. q .| ) ^2 <= ( ( q `1 ) / |. q .| ) ^2 + ( ( q `2 ) / |. q .| ) ^2 ;
h0 . ( i + 1 + 1 ) = h21 . ( i + 1 + 1 + 2 -' len h11 ) .= h21 . ( i + 1 + 2 -' 1 ) ;
consider o being Element of the carrier' of S , x2 being Element of { the carrier' of S } such that a = [ o , x2 ] and o <> {} and x2 <> {} ;
for L being RelStr , a , b being Element of L holds a <= b iff a <= b & b <= a & a <= b
||. h1 .|| . n = ||. h1 . n .|| .= ||. h1 . n .|| .= ||. ( h1 . n ) .|| .= ||. ( h1 . n ) .|| .= ||. ( h1 . n ) .|| .= ||. ( h1 . n ) .|| .= ||. ( h1 . n ) .|| ;
( ( - ( exp_R * f ) ) `| Z ) . x = f . x - ( exp_R . x ) .= ( - exp_R . x ) * ( - exp_R . x ) .= - ( exp_R . x ) * ( - exp_R . x ) .= - ( exp_R . x ) * ( - exp_R . x ) ;
pred r = F .: ( p , q ) means : Def2 : len r = len p & for i st i in dom r holds r . i = F . ( p . i , q . i ) ;
( ( r / 2 ) ^2 + ( r / 2 ) ^2 ) <= ( r / 2 ) ^2 + ( r / 2 ) ^2 ;
for i , j being Nat , M being Matrix of n , K st i in Seg n & j in Seg n holds Det ( M , i ) = Sum ( Line ( M , i ) )
then a <> 0. R implies a " * ( a * v ) = 1 / ( a * v ) & a " * v = 1 / ( a * v ) ;
p . ( j - 1 ) * ( q *' ) . ( i + 1 ) = Sum ( p . ( j -' 1 ) * ( q *' ) ) ;
deffunc F ( Nat ) = L . 1 + ( ( R /* ( h ^\ n ) ) * ( h ^\ n ) ) . $1 .= ( R /* ( h ^\ n ) ) . $1 ;
assume that the carrier of H = f .: ( the carrier of H1 ) and the carrier of H = f .: ( the carrier of H2 ) and the carrier of H = f .: ( the carrier of H2 ) and the carrier of H = f .: ( the carrier of H2 ) ;
Args ( o , Free ( S , X ) ) = ( ( ( the Sorts of Free ( S , X ) ) * ( the_arity_of o ) ) . o .= ( ( the Sorts of Free ( S , X ) ) * ( the_arity_of o ) ) . o ;
H1 = n + 1 - ( |. 2 to_power ( n + 1 ) .| ) .= n + 1 - ( 2 to_power ( n + 1 ) ) .= n + 1 - 1 ;
( O ( O ( O ( O ( O ) ) ) ) `1 = 0 & ( O ( O ( O ) ) ) `2 = 1 & O ( O ( O ) ) = 0 & O ( O ( O ) ) = 1 & O ( O ( O ) ) = 0 ;
F1 .: ( dom F1 /\ dom F2 ) = F1 .: ( dom F1 /\ dom F2 ) .= ( f .: ( Seg n ) ) /\ ( f .: ( Seg n ) ) .= { f . ( n + 2 ) } .= f .: ( n + 2 ) ;
pred b <> 0 & d <> 0 & b <> d & ( a = ( - e ) / ( 2 * b ) ) & ( ( - d ) / ( 2 * b ) ) = ( ( - ( - e ) ) / ( 2 * b ) ) ;
dom ( ( f +* g ) | D ) = dom ( ( f +* g ) | D ) .= dom ( ( f +* g ) | D ) .= D /\ D .= D /\ D .= ( ( f +* g ) | D ) /\ D .= ( ( f +* g ) | D ) /\ D .= ( ( f +* g ) | D ) /\ D ;
for i be set st i in dom g ex u , v being Element of L st g /. i = u * v & u in B & v in C & v in C ;
g `2 * P `2 * g `2 = g `2 * ( g `2 * P `1 ) .= g `2 * ( g `2 * P `1 ) .= ( g `2 * P `1 ) * ( g `2 * P `1 ) ;
consider i , s1 such that f . i = s1 and not ( ex s1 st i in dom s1 & not ( s1 . i <> s2 . i ) & ( s1 . i <> s2 . i ) & ( s1 . i <> s2 . i ) & ( s1 . i <> s2 . i ) ;
h5 | ]. a , b .[ = ( g | ]. a , b .[ ) | ]. a , b .[ .= ( g | ]. a , b .[ ) | ]. a , b .[ .= g | ]. a , b .[ ;
[ s1 , t1 ] , [ s2 , t2 ] ] , [ s2 , t2 ] ] , [ s2 , t2 ] , [ s2 , t2 ] ] <> [ s1 , t2 ] , [ s2 , t2 ] , [ s2 , t2 ] ] ;
then H is negative & H is non empty & H is non empty and H is non empty and H is non empty and H is non ] ;
attr f1 is total & ( 1 / 2 ) (#) ( ( f1 - f2 ) (#) ( f2 - f3 ) ) is total & ( ( f1 - f2 ) (#) ( f3 - f3 ) ) . c = ( f1 . c ) * ( f2 . c ) - ( f2 . c ) * ( f2 . c ) ;
z1 in W2 ` & not z1 = z2 or z1 = W1 & not z1 = W2 & not z1 = W2 & not z2 = W1 & not z2 = W3 & not z2 = W3 & not z2 = z2 & not z1 = z2 & z1 = z2 & not z2 = W3 & z2 = z2 & not z2 = z1 & z2 = z2 & z1 = z2 & z2 = z2 & z2 = z2 & z1 = z2 & z1 = z2 & z1 = z2 & z2 = z2 & z2 in W2 & z2 in W1 & z2 in W2 & z2 in W2 & z2 in W2 & z2 in W2 & z2 in W1
p = 1 * p .= a " * a * p .= a " * ( b " * p ) .= a " * ( b " * p ) .= a " * ( b " * p ) .= ( a " * b ) * p .= ( a " * b ) * p .= ( a " * b ) * p .= ( a " * b ) * p ;
for rseq be Real_Sequence , K be Real st for n be Nat holds rseq . n <= K . n holds upper_bound rng ( seq . n ) <= upper_bound rng ( seq . n )
( for p being Point of TOP-REAL 2 st p in L~ go holds C meets L~ co or C meets L~ co or C meets L~ co ) implies ( ( L~ co ) \/ L~ co ) meets ( L~ co ) \/ ( L~ co )
||. f . ( g . ( k + 1 ) ) - g . ( k + 1 ) .|| <= ||. g . ( 1 - K ) .|| * ( K * ( K * ( k + 1 ) ) ) ;
assume h = ( ( B .--> ( B .--> C ) +* ( D .--> E ) ) +* ( E .--> F ) +* ( F .--> J ) +* ( M .--> N ) +* ( M .--> N ) +* ( M .--> N ) ) ;
|. ( ( upper_volume ( H . n ) || A ) . k - ( ( lower . n ) || A ) . k .| <= e * ( ( H . n ) || A ) . k - ( ( H . n ) `| A ) . k .| ;
(
{ x1 , x1 , x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 } = { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 } .= { x1 , x2 , x3 , x4 , x5 , x5 , x5 } .= { x1 , x2 , x3 } ;
Suppose A = [. 0 , 2 * PI .] and integral ( integral ( A , A ) , A ) = 0 and integral ( A , A ) = 0 and ( integral ( A , A ) ) . x = 0 ;
p `2 is Permutation of dom f1 & p `2 = ( Sgm Y ) . ( p " * ( Sgm Y ) . ( p " * ( Sgm Y ) . ( p " * ( Sgm Y ) . ( p " * ( Sgm Y ) . ( p " * ( Sgm Y ) . ( p " * ( Sgm Y ) . ( p " * ( Sgm Y ) ) ) ) ) ;
for x , y st x in A holds |. ( 1 - ( f . x ) ) .| <= 1 * |. ( f . x ) .| - ( 1 - ( f . y ) ) .|
( ( p2 `2 ) * ( ( q `2 ) - sn ) ) = |. q2 .| * ( ( ( q `2 ) - sn ) / ( 1 + sn ) ) ;
for f be PartFunc of the carrier of C , REAL st dom f is compact & f | X is bounded holds f | X is bounded & for x be Element of C st x in X holds f /. x = F ( x )
assume for x being Element of Y st x in EqClass ( z , CompF ( B , G ) ) holds ( Ex ( a , A , G ) ) . x = TRUE ;
consider FM such that dom FM = n1 and for k be Nat st k in n1 holds Q [ k , F ( k ) ] ;
ex u , u1 st u <> u1 & u , v // v , v1 & u , v // u , u1 & u , v // v , v1 & u , v // v , u1 & u , v // v , v1 ;
for G being Group , A , B being non empty Subgroup of G , N being normal Subgroup of G holds ( N N ) * ( N , B ) ` = N ` A * N
for s be Real st s in dom F holds F . s = integral ( R , ( f + g ) (#) ( f + g ) ) . x = integral ( R , ( f + g ) (#) ( f + g ) ) . x
width AutMt ( f1 , b1 , b2 ) = len ( ( f2 * f1 ) * ( f2 * f2 ) ) .= len ( ( f2 * f1 ) * ( f2 * f2 ) ) .= len ( ( f2 * f1 ) * ( f2 * f2 ) ) .= len ( ( f2 * f1 ) * ( f2 * f2 ) ) ;
f | ]. - PI / 2 , PI / 2 .[ = f | ]. - PI / 2 , PI / 2 .[ & f | ]. - PI / 2 , PI / 2 .[ = f | ]. - PI / 2 , PI / 2 .[ ;
Suppose X is closed and a in X and a c= X and y in a .: f and x in X and y in X ;
Z = dom ( ( ( ( - 1 ) (#) ( ( #Z 2 ) * ( f + #Z 2 ) ) ) `| Z ) /\ dom ( ( #Z 2 ) * ( f + #Z 2 ) ) ) ;
func VERUM ( V ) -> Subset of V means : Def1 : 1 <= k & k <= len it & it . k = V . k & it . k = V . k ;
for L being non empty TopStruct , N being net of L , M being net of N st M is complete & N is net of L & M is complete holds M is complete
for s being Element of NAT holds ( ( ( id the carrier of X ) + ( id the carrier of X ) ) | ( ( the carrier of X ) + ( id the carrier of X ) ) . s = ( ( id the carrier of X ) + ( id the carrier of X ) ) . s
then z /. 1 = ( N-min L~ z ) .. z & ( ( N-min L~ z ) .. z ) .. z < ( ( E-max L~ z ) .. z ) .. z ;
len ( p ^ <* ( 0 qua Real ) * ( ( 0 qua Real ) * ( ( 1 - p ) * ( ( 1 - p ) * ( 1 - p ) ) *> ) = len p + len <* ( 0 qua Real ) * ( ( 1 - p ) * ( 1 - p ) ) *> .= len p + 1 ;
assume that Z c= dom ( ( - ( ln * f ) ) `| Z ) and for x st x in Z holds f . x = x and f . x > 0 ;
for R being add-associative right_zeroed right_complementable non empty doubleLoopStr , I , J being Subset of R , I , J being Subset of R holds ( I + J ) *' ( I /\ J ) c= I /\ J
consider f being Function of [: B1 , B2 :] , B2 being Function of B1 , B2 such that for x being Element of B1 holds f . x = F ( x ) and f . x = F ( x ) ;
dom ( x2 + y2 ) = Seg len x .= Seg len ( ( x2 + y2 ) + ( y2 + z2 ) ) .= Seg len ( ( x + y ) + ( y + z ) ) .= Seg len ( ( x + y ) + ( x + z ) ) .= dom ( ( x + y ) + ( x + z ) ) ;
for S being |. Functor of C , B for c being Object of C holds card S . ( id c ) = id ( ( Obj S ) . c )
ex a st a = a2 & a in dom f & f . a = f . ( a , b ) & ( f . a ) = \cal ( f . a , f . b ) ;
a in Free ( H / ( x. 4 , x. k ) ) '&' ( H / ( x. k , x. k ) ) ;
for C1 , C2 being the carrier of C1 , f being stable Function of C1 , C2 st f = ( \cup C2 ) | ( the carrier of C1 ) & g = ( \cup C2 ) | ( the carrier of C2 ) holds f = g
( W-min L~ go \/ L~ co ) `1 = W-bound L~ Cage ( C , n ) \/ W-bound L~ Cage ( C , n ) \/ L~ co .= ( W-min L~ Cage ( C , n ) ) `1 \/ ( W-bound L~ Cage ( C , n ) ) ;
consider u such that u = <* x0 , y0 *> and f is_partial_differentiable_in u , 3 and SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) . 3 = SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) . 3 and SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) . 3 = SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) . 3 ;
then ( t . {} ) `1 in Vars & ex x being Element of Vars st x = ( t . {} ) `1 & ( t . {} ) `1 = ( t . {} ) `1 & ( t . {} ) `1 = ( t . {} ) `1 ;
Valid ( p '&' p , J ) . v = Valid ( p , J ) . v '&' Valid ( p , J ) . v .= Valid ( p , J ) . v .= Valid ( p , J ) . v .= Valid ( p , J ) . v ;
assume for x , y being Element of S st x <= y for a , b being Element of T st a = f . x & b = f . y holds a >= b ;
func Class R -> Subset-Family of R means : an : for A being Subset of R holds A in it iff ex a being Element of R st a in it & a in A & it c= A ;
defpred P [ Nat ] means ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) . $1 ) ) `1 ) `1 ) `1 ) c= G ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) . $1 ) ) ) ) ) ) ) ) ) ) `1 ) ) ) ) `1 ) ) `1 ) . ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) )
assume that dim ( W1 ) = 0 and dim ( U1 ) = 0 and dim ( V ) = 0 and dim ( V ) = 0 and dim ( V ) = 0 and dim ( V ) = 0 and dim ( V ) = 0 and dim ( V ) = 0 and dim ( V ) = 0 and dim ( V ) = 0 and dim ( V ) = 0 ;
mama_empty ( m . t ) = ( m . t ) `1 .= ( m . {} ) `1 .= ( m . t ) `1 .= ( m . t ) `1 .= ( m . t ) `1 .= ( m . t ) `1 .= ( m . t ) `1 ;
d11 = x11 ^ d22 .= ( ( y , d22 ) --> ( x , y ) ) . ( ( y , d22 ) . ( x , y ) ) .= ( y , d22 ) . ( x , y ) .= ( y , d22 ) . ( x , y ) .= ( y , d22 ) . ( x , y ) .= ( y , d22 ) . ( x , y ) .= ( y , d22 ) . ( x , y ) .= ( y , d ) ;
consider g such that x = g and dom g = dom f and for x being element st x in dom f holds g . x in f . x ;
x + 0. F_Complex = x + len x .= ( x + len x ) |-> 0. F_Complex .= ( x + len x ) |-> 0. F_Complex .= ( x + len x ) |-> 0. F_Complex .= x + ( x + len x ) |-> 0. F_Complex .= x + ( x + len x ) .= x + ( x + ( x + y ) ) ;
( ( f /. ( k -' 1 ) ) + 1 ) in dom ( ( f /. ( k -' 1 ) ) | ( len f -' 1 ) ) ;
assume that P1 is_an_arc_of p1 , p2 and P is_an_arc_of p1 , p2 and P = { p1 } and p1 <> p2 and P = { p1 } and p1 <> p2 and P = { p1 } and P = { p1 } and p1 <> p2 and P = { p1 } and p1 <> p2 and P = { p1 } and p1 <> p2 and P = { p1 , p2 } and p1 <> p2 and P = { p1 , p2 } and p1 <> p2 and P = { p1 , p2 , p3 , p3 , p4 } and P = { p1 , p2 , p4 , p4 , p4 , p4 } and P = { p1 , p2 , p4 , p4 = { p1 , p2 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4
reconsider a1 = a , b1 = b , c1 = c , c1 = d , c2 = d , c1 = p , c1 = p , c2 = q , c1 = p , c2 = d , c1 = p , c2 = p , c1 = q , c2 = p , c1 = d , c1 = p , c2 = p , c2 = d , c1 = p , c2 = d , c2 = p , c2 = p , c1 = d , c2 = p , c1 = d , c2 = d , c1 = d , c1 = d , c1 = d , c2 = d , c2 = d , c1 = d , c2 = p , c2 = p , c1 = d , c1 = d , c1 = d , c1 = d , c2 = d , c2 =
reconsider set thesis thesis thesis = Gb1f . ( t , F . b ) , FFf = ( G1 * F1 ) . ( t , b ) as Morphism of ( G1 * F2 ) . ( t , a ) , ( G2 * F2 ) . ( t , a ) ;
LSeg ( f , i + i1 -' 1 ) = LSeg ( f /. ( i + 1 ) , f /. ( i + 1 + 1 ) ) .= LSeg ( f /. ( i + 1 ) , f /. ( i + 1 + 1 ) ) ;
Integral ( P . m , P . ( n + 1 ) ) <= Integral ( M . ( n + 1 ) , P . ( n + 1 ) ) ;
assume that dom f1 = dom f2 and for x , y being element st [ x , y ] in dom f1 & [ y , x ] in dom f2 holds f1 . [ x , y ] = f2 . [ y , x ] ;
consider v such that v = y and dist ( u , v ) < min ( ( - ( G * ( i , 1 ) ) ) , ( G * ( i , 1 ) ) - ( G * ( i + 1 , 1 ) ) ) ;
for G being Group , H being Subgroup of G , a being Element of G st a = b holds for i being Element of G st i = a holds a |^ i = b |^ i
consider B being Function of Seg ( S + L ) , the carrier of V1 such that for x being element st x in Seg ( S + L ) holds P [ x , B . x ] ;
reconsider K1 = { p0 where 7 is Point of TOP-REAL 2 : P [ 7 ] & P [ 7 ] } , K1 = { p where p is Point of TOP-REAL 2 : P [ p ] } , K1 = { p : p <> 0. TOP-REAL 2 } , K1 = { p where p is Point of TOP-REAL 2 : P [ p ] } as Subset of TOP-REAL 2 ;
( ( ( ( N - S ) / 2 ) * ( ( N - S ) / 2 ) / 2 ) / 2 ) <= ( ( ( N - S ) / 2 ) * ( ( N - S ) / 2 ) / 2 ) ;
for x be Element of X , n be Nat st x in E holds |. ( Re F ) . n .| <= P . x & |. ( Im F ) . x .| <= P . x & |. ( Im F ) . x .| <= P . x
len @ ( p ^ <* 0 *> ) = len ( @ ( @ p ^ <* 0 *> ) ) + len <* 1 *> .= len ( @ p ^ <* 0 *> ) + len <* 0 *> .= len ( @ p ^ <* 1 *> ) + len ( @ q ) .= len ( @ p ^ <* 1 *> ) + len ( @ q ^ <* 1 *> ) ;
v / ( x. 3 , m1 ) / ( x. 0 , m2 ) / ( x. 0 , m2 ) / ( x. 0 , m2 ) / ( x. 0 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 0 , m2 ) = ( m3 / ( x. 0 , m2 ) ) / ( x. 4 , m2 ) ;
consider r being Element of M such that M , v / ( x. 3 , m ) / ( x. 4 , m ) / ( x. 0 , m ) / ( x. 4 , m ) / ( x. 0 , m ) / ( x. 0 , m ) / ( x. 4 , m ) / ( x. 0 , m ) / ( x. 4 , m ) ) / ( x. 0 , m ) / ( x. 0 , m ) = r / ( x. 4 , m ) ;
func w1 \ w2 -> Element of Union ( the NRAssume of G , RA29 ) equals ( ( the NKAssume of G ) . ( the carrier of G , R29 ) ) . ( ( the Element of ( the NK29 ) | ( the carrier of G ) ) ) ;
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= Exec ( n2 , s2 ) . b2 .= Exec ( i2 , s2 ) . b2 .= ( Exec ( i2 , s2 ) ) . b2 .= ( Exec ( i2 , s2 ) ) . b2 .= ( Exec ( i2 , s2 ) ) . b2 .= ( Exec ( i2 , s2 ) ) . b2 .= ( Exec ( i2 , s2 ) ) . b2 .= ( Exec ( i2 , s2 ) ) . b2 .= ( s . b2 ) . b2 .= ( s . b2 ) . b2 .= ( s . b2 ) . b2 .= ( s . b2 ) . b2 .= ( s . b2 ) . b2 .= ( s . b2 )
for n , k be Nat holds 0 <= ( Partial_Sums ( |. seq .| ) ) . ( n + k ) - ( Partial_Sums ( |. seq .| ) ) . ( n + k ) + ( Partial_Sums ( |. seq .| ) ) . ( n + k ) - ( Partial_Sums ( |. seq .| ) ) . ( n + k )
set F = S \! \mathop { 0 } , G = S \! \mathop { 1 } ;
( Partial_Sums ( seq ) ) . ( K + 1 ) + Partial_Sums ( seq ) . ( K + 1 ) >= ( Partial_Sums ( seq ) ) . ( K + 1 ) + Partial_Sums ( seq ) . ( K + 1 ) ;
consider L , R such that for x st x in N holds ( f | Z ) . x - ( f | Z ) . ( x - x0 ) = L . ( x- ( x , x0 ) ) + R . ( x - x0 ) ;
func the closed Subset of \HM { a , b , c , d d d d d d d e -> Element of ( \HM { the } \HM { carrier of \HM { a } , c , d } ) \ { d } ;
a * b ^2 + ( a * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( c * d ) ^2 >= 6 * a * c + ( b * d ) ^2 ;
v / ( x1 , m1 ) / ( x2 , m2 ) / ( x3 , m2 ) / ( x4 , m2 ) / ( x4 , m2 ) = v / ( x2 , m2 ) / ( x4 , m2 ) / ( x4 , m2 ) ;
( ( Q ^ <* x *> ) . 0 = ( ( Q ^ <* x *> ) . 0 ) +* ( ( Q ^ <* x *> ) . 0 ) .= ( Q +* <* x *> ) . 0 .= ( Q +* <* x *> ) . 0 .= ( Q +* <* x *> ) . 0 ;
Sum ( F |^ ( n1 + 1 ) ) = ( r |^ ( n1 + 1 ) ) * Sum ( F |^ ( n1 + 1 ) ) .= ( C |^ ( n1 + 1 ) ) * ( F . ( n1 + 1 ) ) .= ( C |^ ( n1 + 1 ) ) * ( F . ( n1 + 1 ) ) .= ( C |^ ( n1 + 1 ) ) * ( F . ( n1 + 1 ) .= ( C |^ ( n1 + 1 ) ) * ( C |^ ( n1 + 1 ) .= C |^ ( n1 + 1 ) ;
( ( GoB f ) * ( len GoB f , 1 ) ) `1 = ( ( GoB f ) * ( len GoB f , 1 ) ) `1 .= ( ( GoB f ) * ( len GoB f , 1 ) ) `1 .= ( GoB f ) * ( len GoB f , 1 ) `1 ;
defpred X [ Element of NAT ] means ( Partial_Sums ( s ) ) . $1 = ( Partial_Sums ( a ) ) . ( $1 + 1 ) * ( Partial_Sums ( a ) ) . ( $1 + 1 ) + ( Partial_Sums ( a ) ) . ( $1 + 1 ) * ( $1 + 2 ) ;
( ( the_arity_of g ) . n = ( ( the Arity of S ) . n ) . ( g . n ) .= ( ( the Arity of S ) . n ) . ( ( the Arity of S ) . n ) .= ( ( the Arity of S ) . n ) . ( ( the Arity of S ) . n ) .= ( ( the Arity of S ) . n ) . ( ( the_arity_of o ) . n ) .= ( ( the Arity of S ) . n ) . ( ( the_arity_of o ) . n ) ;
( X ~ ) ^ ( Y ~ ) tolerates X ~ & card ( ( X ~ ) ^ ( Y ~ ) ) = card ( X ~ ) - card ( Y ~ ) ;
for a , b being Element of S , s being Element of NAT st s = n & a = F . n & b = F . n & b = N . n holds b = N . ( s . n ) \ G . ( s . n ) ;
E , f |= All ( x , All ( x , p ) ) => ( ( x. 3 ) 'in' ( x , p ) ) => ( ( x. 0 ) 'in' ( x , p ) ) ;
ex R2 being 1-sorted st R2 = ( p | ( n2 -' 1 ) ) . i & ( the carrier of p ) . i = the carrier of R2 & ( the carrier of p ) . i = the carrier of R2 ;
[. a , b + ( 1 / ( k + 1 ) ) / ( k + 1 ) .] is Element of the , the carrier of REAL , b is Element of the carrier of REAL , a , b be Element of REAL ;
Comput ( P , s , 2 + 1 ) = Exec ( P . 2 , Comput ( P , s , 2 ) ) .= Exec ( goto ( 0 + 1 ) , Comput ( P , s , 2 ) ) .= Exec ( goto ( 0 + 1 ) , Comput ( P , s , 2 ) ) ;
card ( h1 ) . k = power ( F_Complex , n , k ) . ( - 1_ F_Complex , k ) .= ( ( - 1_ F_Complex ) * u ) . k .= ( ( - 1_ F_Complex ) * u ) . k .= ( ( - 1_ F_Complex ) * u ) . k .= ( ( - 1_ L ) * u ) . k ;
( ( f - g ) /. c ) = f /. c * ( ( - g ) /. c ) .= ( f /. c ) * ( ( - g ) /. c ) .= ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( 1 ) ) ) ) ) ) .= ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) ) ) /. c ) .= ( - 1 ) ) /. c ;
len Cn - len ( ( the _ { n , m } ) | ( len ( the _ { n , m } ) ) ) = len ( ( the _ { n , m } ) | ( len ( the _ { n , m } ) ) .= len ( ( the _ { n , m } ) | ( len ( the _ { n , m } ) ) ) ;
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom ( ( r (#) f ) | X ) .= dom ( r (#) f ) /\ X .= dom ( ( r (#) f ) | X ) .= dom ( r (#) f ) /\ X .= dom ( r (#) ( f | X ) ) .= dom ( r (#) ( f | X ) ) .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) ;
defpred P [ Nat ] means for n holds 2 * Fib ( n + $1 ) = Fib ( n ) * Fib ( n + $1 ) + Fib ( n + $1 ) * Fib ( n + $1 ) * Fib ( n + $1 ) + Fib ( n + $1 ) * Fib ( n + $1 ) * Fib ( n + $1 ) ;
consider f being Function of INT , INT such that f = f and f is onto and n < len f and f is onto and for i st i < len f holds f " . i = f . ( n + 1 ) ;
consider c9 being Function of S , BOOLEAN such that c9 = chi ( c9 , B ) and ( for A , B being Element of S holds E . ( A \/ B ) = Prob . ( A \/ B ) and E . ( A \/ B ) = Prob . ( A \/ B ) ;
consider y being Element of Y ( ) such that a = "\/" ( { F ( x , y ) where x is Element of X ( ) : P [ x ] } , y ) and P [ y ] ;
assume that A c= Z and Z = dom f and f = ( ( - 1 ) (#) ( ( #Z 2 ) * ( ( #Z 2 ) * ( f + #Z 2 ) ) ) `| Z ) . ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( #Z 2 ) ) ) ) ) ) ) ) . x ) ) = f . x ) ;
( ( f /. i ) `2 ) = ( ( GoB f ) * ( 1 , j2 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 ) ) `2 ;
dom Shift ( Seq q2 , len Seq q2 ) = { j + len Seq q2 where j is Nat : j in dom Seq q2 & j <= len Seq q2 & len Seq q2 = len Seq q2 + len Seq q2 } ;
consider G1 , G2 , G3 being Element of V such that G1 <= G2 & G2 <= G2 and f = ( the Morphism of G1 ) . ( G1 , G2 ) and g = ( the Morphism of G2 ) . ( G1 , G2 ) and f = ( the Morphism of G2 ) . ( G1 , G2 ) ;
func - f -> PartFunc of C , V means : Def1 : dom it = dom f & for c st c in dom it holds it /. c = - f /. c & for c st c in dom it holds it /. c = - f /. c ;
consider phi such that phi is increasing and for a st phi . a = a & {} <> a & for v st v <> a & v <> b holds not v , v |= ( union rng L ) . ( a , v ) iff not v , w |= H ;
consider i1 , j1 such that [ i1 , j1 ] in Indices GoB f and f /. ( i1 + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i1 + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i1 + 1 ) = ( GoB f ) * ( i1 , j1 ) ;
consider i , n such that n <> 0 and sqrt p = sqrt n and for i being Nat st i <> 0 & i < n holds sqrt ( n + 1 ) = sqrt ( n + 1 ) * sqrt ( n ) and sqrt ( n + 1 ) = sqrt ( n ) * sqrt ( n ) ;
assume that not 0 in Z and Z c= dom ( ( ( #Z 2 ) * ( ( #Z 2 ) * ( f + g ) ) ) ) and for x st x in Z holds ( ( ( #Z 2 ) * ( f + g ) ) `| Z ) . x = - 1 / ( ( #Z 2 ) * ( f + g ) ) ^2 ) and for x st x in Z holds ( ( 1 / 2 ) * ( f + g ) ) . x = - 1 / ( 1 + x ^2 ) and ( 1 / ( 1 + x ^2 ) and ( 1 / ( 1 + x ) ^2 / ( 1 + x ) ^2 / ( 1 + x ^2 ) ^2 / ( 1 + x ^2 ) < 1 / ( 1 + x ^2 ) ^2 / ( 1 + x
cell ( G1 , i1 -' 1 , 2 * ( i1 -' 1 ) + ( C - 1 ) * ( i1 -' 1 ) , ( C - 1 ) * ( i1 -' 1 ) ) \ L~ ( ( C - 1 ) * ( i1 -' 1 ) ) c= BDD L~ ( ( C - 1 ) * ( ( C - 1 ) + ( C - 1 ) * ( ( C - 1 ) - ( C - 1 ) ) ) \ L~ ( ( C - 1 ) ) ;
ex Q1 being open Subset of X st s = Q1 & ex F being Subset-Family of Y st F c= ( the carrier of Y ) & ( ( the carrier of Y ) /\ ( the carrier of X ) ) c= union ( F .: ( the carrier of X ) ) ;
( ( the _ of A ) * ( ( 1 , ( 2 , ( 2 * ( 1 , 1 ) ) ) * ( 1 , 1 ) ) ) * ( ( 2 , ( 2 * ( 1 , 1 ) ) * ( 1 , 1 ) ) ) = 1 / ( 2 * ( ( 2 * ( 1 , 1 ) ) * ( 1 , 1 ) ) ) ;
R8 = ( ( Following ( s2 , 2 ) ) . ( m2 + 1 ) ) . ( m2 + 1 ) .= ( Following ( s2 , 2 ) ) . ( m2 + 1 ) .= ( Following ( s2 , 2 ) ) . ( m2 + 1 ) .= ( Following ( s2 , 2 ) ) . ( m2 + 1 ) .= ( Following ( s2 , 2 ) ) . ( m2 + 1 ) .= ( Following ( s2 , 2 ) ) . ( m2 + 1 ) .= ( Following ( s2 ) . ( m2 + 1 ) .= ( Following ( s2 ) . ( m2 + 1 ) .= ( Following ( s2 ) . ( m2 + 1 ) .= ( Following ( s2 ) . ( m2 + 1 ) .= ( Following ( s2 ) . ( m2 + 1 ) . ( m2 + 1 ) .= Exec ( s3 +
CurInstr ( P3 , Comput ( P3 , s3 , m1 + 1 ) ) = CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= halt SCMPDS .= halt SCMPDS ;
P1 /\ P2 = ( { p1 } \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) ) /\ ( LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) ) \/ ( LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ;
func not f -> Subset of the Sorts of A2 equals { f . i : ex p st p in dom f & p = f . i & not p <> f . i & p <> f . i & p <> f . i & p <> f . i & p <> f . i ;
for a , b being Element of F_Complex st |. a .| > |. b .| & for f being Polynomial of n , L st f is >= 0 & f . ( len f ) >= 0 holds f * ( f . ( len f ) ) is means : Let : len f = len f & f . ( len f ) = f . ( len f )
defpred P [ Nat ] means $1 <= len g implies for i , j st [ i , j ] in Indices G & [ i , j ] in Indices G & G * ( i , j ) = G * ( i , j ) & G * ( i , j ) = G * ( i , j ) ;
assume that C1 , C2 , C2 *> \neq {} and for f being State of C1 , g being State of C2 , n being Nat st n = len f & g = f . n holds ( for n being Nat holds s1 . n = s2 . n ) & ( for n being Nat holds s1 . n = s2 . n ) ;
( ||. f .|| | X ) . c = ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f
|. q .| ^2 = ( ( q `1 ) ^2 + ( q `2 ) ^2 ) + ( ( q `2 ) ^2 ) * ( ( q `1 ) ^2 + ( q `2 ) ^2 ) & 0 + ( ( q `2 ) ) ^2 < ( ( q `1 ) ^2 + ( q `2 ) ^2 ;
for F being Subset-Family of T7 st F is open & not {} in F & for A , B being Subset of T7 st A in F & B <> {} & A <> {} & A c= B & A c= B holds card A c= card B implies card B c= card A & card A c= card B
assume that len F >= 1 and len F = k + 1 and len F = len G and for k st k in dom F holds H . k = g . ( k , i ) and for k st k in dom F holds H . k = g . ( k , i ) ;
i |^ ( ( ( \mathop { \rm gcd } ( n , k ) ) |^ s ) - i |^ ( ( k + 1 ) ) * ( i |^ k ) ) = i |^ ( ( k + 1 ) * ( i |^ k ) ) - i .= i |^ ( ( k + 1 ) * ( i |^ k ) ) - i .= i |^ ( k + 1 ) * ( i |^ k ) - i |^ ( k + 1 ) ;
consider q being oriented oriented Chain of G such that r = q and q <> {} and ( F . ( len q ) = v1 & ( F . ( len q ) = v2 ) and ( F . ( len q ) = v2 ) ;
defpred P [ Element of NAT ] means $1 <= len ( ( g , Z ) ^ <* I . ( len g + $1 ) *> ) & ( ( ( g , Z ) ^ <* I . ( len g + $1 ) *> ) . ( len g + $1 ) = ( ( ( g , Z ) ^ <* I . ( len g + $1 ) *> ) . ( len g + $1 ) ;
for A , B being square Matrix of n , K holds len ( A * B ) = len A & width ( A * B ) = len A & width ( A * B ) = width B & width ( A * B ) = width A & width ( A * B ) = width B ;
consider s being FinSequence of the carrier of R such that Sum s = u and for i , j being Element of NAT st 1 <= i & i <= len s & s . i = a * b & s . j = b * c ;
func |( x , y )| -> Element of COMPLEX equals |( ( Re x ) , ( Re y ) )| - ( ( Re y ) ^2 + ( Im y ) ^2 ) , ( Im y ) ^2 + ( Im y ) ^2 + ( Im y ) ^2 + ( Im y ) ^2 ;
consider g2 being FinSequence of F such that g2 is continuous and rng g2 c= A and g2 is continuous and ( for n st n in dom g2 holds g2 . n = F ( n ) ) and g2 . ( len g2 ) = g2 . n and g2 . ( len g2 ) = g2 . n and g2 . ( len g2 ) = g2 . n ;
then n1 >= len ( p1 , p2 , n1 , n2 , n3 , n3 , n3 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n1 , n2 , n3 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , p1 , n2 , n3 , n4 , n4 , n4 , n4 , n4 , p1 , p2 , n1 , n2 , n3 , n4 , n4 , 7 , n4 , p1 , p2 , n1 , n4 , n4 , p1 , p2 , n3 , p1 , p1 , n2 , n3 , n4 , n4 , p1 , p1 , n4 , p1 , p2 , n4 , n4 , p1 , p2 , n3 , n4 , n4 , n4 , p1 , n4 , p1 , n4 , p1 , n4 , p1 , p1 , n3 , n4 , p1 , p1 , p2 , n4 , n4 , p1 ,
( q `1 ) * a <= ( q `1 ) * a & ( q `2 ) * a <= ( q `2 ) * a or ( q `1 ) * a <= ( q `1 ) * a & ( q `2 ) * a <= ( q `1 ) * a or ( q `1 ) * a <= ( q `1 ) * a or ( q `1 ) * a & ( q `1 ) * a <= ( q `1 ) * a ;
( F . ( p . ( len p ) ) ) = ( F . ( p . ( len p ) ) ) . ( p . ( len p ) ) .= ( F /. ( len p ) ) . ( p . ( len p ) ) .= ( F /. ( len p ) ) . ( len p ) .= ( F /. ( len p ) ) . ( len p ) .= ( F /. ( len p ) . ( len p ) ;
consider k1 being Nat such that k1 + k = 1 and a = ( ( a := intloc 0 ) := ( ( intloc 0 ) .--> 1 ) ) . ( ( a + k ) + 1 ) and ( a := intloc 0 ) . ( ( a + k ) .--> 1 ) = ( a := intloc 0 ) . ( ( intloc 0 ) .--> 1 ) ;
consider B8 being Subset of B1 , y8 being Subset of A1 such that B8 is finite and y1 is finite and D = the carrier of A1 and D = the carrier of A2 and D = the carrier of A2 and D = the carrier of A2 and D = the carrier of A2 ;
v2 . b2 = ( curry ( F2 , g ) * ( ( curry ( F2 , g ) ) . b2 ) .= ( ( ( ( ( ( ( ( F ) . b2 ) ) * ( ( ( ( F . b2 ) ) . b2 ) ) . b2 ) ) . b2 .= ( ( ( ( ( ( ( F ) . b2 ) ) . b2 ) ) . b2 ) . b2 .= ( ( ( ( ( ( ( F ) . b2 ) ) . b2 ) ) . b2 ) ) . b2 ) . b2 .= ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( F ) . b2 ) . b2 ) ) . b2 ) ) . b2 ) ) . b2 ) ) . b2 ) ) . b2 ) . b2 ) . b2 ) . b2 ) . b2 ) . b2 ) . b2 ) . b2 ) . b2 ) . b2 ) .= (
dom IExec ( I , P , Initialize s ) = the carrier of SCMPDS .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ;
ex d-32 be Real st d-32 > 0 & for h be Real st h <> 0 & |. h .| < d & |. h .| < d holds |. h .| " * ||. ( R + R ) /. h .|| < e / ( 2 * L ) * ||. h .|| ;
LSeg ( G * ( len G , 1 ) + |[ 1 , 0 ]| , G * ( len G , 1 ) + |[ 1 , 0 ]| ) c= Int cell ( G , len G , 1 ) \/ { G * ( len G , 1 ) } ;
LSeg ( mid ( h , i1 , i2 ) , i ) = LSeg ( h /. ( i + 1 ) , h /. ( i + 1 + 1 ) ) .= LSeg ( h /. ( i + 1 + 1 ) , h /. ( i + 1 + 1 ) ) .= LSeg ( h /. ( i + 1 + 1 ) , h /. ( i + 1 + 1 ) ) .= LSeg ( h /. ( i + 1 + 1 ) , h /. ( i + 1 + 1 ) ;
A = { q where q is Point of TOP-REAL 2 : LE p1 , p2 , P & LE p , q , P & LE p , q , P & LE q , p , P & LE p , q , P & LE p , q , P & LE q , p , P & LE p , q , P & LE q , p , P & LE p , q , P & LE p , q , P & LE q , p , P & LE p , q , P & LE p , q , P & LE p , q , P & LE p , q , P & LE p , q , P & LE p , q , P & LE p , q , P & LE q , q , P & LE p , q , P & LE p , q , P & LE p , q , P & LE p , q , P & LE p , q , P & LE p , q , P & LE q ,
( ( - x ) .|. y ) = - ( ( - 1 ) .|. y ) .= ( - 1 ) .|. y .= ( - 1 ) .|. y .= ( - 1 ) .|. y .= ( - 1 ) .|. y .= ( - 1 ) .|. y .= ( - 1 ) .|. y .= ( - 1 ) .|. y .= ( - 1 ) .|. y ;
0 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) = ( ( p `1 ) ^2 * sqrt ( 1 + ( p `2 / p `2 ) ^2 ) ) * sqrt ( 1 + ( p `2 / p `2 ) ^2 ) .= ( ( p `1 ) ^2 * sqrt ( 1 + ( p `2 ) ^2 ) ;
( ( U * ( W - p ) ) * ( ( W - p ) * ( ( W - p ) * ( ( W - p ) * ( ( W - p ) * ( ( W - p ) * ( ( W - p ) * ( ( W - p ) * ( ( W - p ) * ( ( W - p ) * ( ( W - p ) * ( W - p ) ) ) ) ) ) ) = ( ( U * ( W - p ) ) * ( ( W - p ) * ( ( W - p ) * ( ( W - p ) ) ) ) ) .= ( ( ( W - p ) * ( ( W - p ) ) ) * ( ( W - p ) * ( ( W - p ) ) ) * ( ( W - p ) * ( ( W - p ) * ( ( W - p ) ) ) .= ( ( W - p ) ) * ( ( W -
func Shift ( f , h ) -> PartFunc of REAL , REAL means : Assume for x be Element of REAL holds it . x = ( - h ) . x + ( - h ) . x & for x be Element of REAL holds it . x = ( - h ) . x + ( - h ) . x ;
assume that 1 <= k and k + 1 <= len f and [ i , j ] in Indices G and f /. ( k + 1 ) = G * ( i , j ) and f /. ( k + 1 ) = G * ( i , j ) and f /. ( k + 1 ) = G * ( i , j ) ;
assume that not y in Free H and not x in Free H and not x in Free H and not y in Free H and not x in Free H and not y in Free H and not x in Free H and not y in Free H and x in Free H and y in Free H ;
defpred P11 [ Element of NAT , Element of NAT , Element of NAT ] means ( $1 = p |^ ( p |-count $1 ) & ( $1 divides p ) & ( $1 divides p implies $2 = ( p |^ ( p |-count $1 ) ) * ( p |-count $1 ) ) & ( $1 divides p |^ ( p |-count $1 ) ) * ( p |-count $1 ) = ( p |-count $1 ) * ( p |-count $1 ) ;
func \sigma ( C ) -> non empty Subset-Family of X means : Def1 : for A , B being non empty Subset of X st A c= B & B c= C & A c= B holds it . ( A + B ) <= C . ( B + A ) ;
[#] ( ( dist ( ( P ) ) .: Q ) ) = ( dist ( ( P ) ) .: Q ) .: Q & lower_bound ( ( dist ( P ) ) .: Q ) = lower_bound ( ( dist ( P ) ) .: Q ) ;
rng ( F | ( [: S , T :] ) ) = {} or rng ( F | ( [: S , T :] ) ) = { 1 , 2 } or rng ( F | ( [: S , T :] ) ) = { 2 , 3 } or rng ( F | ( [: S , T :] ) ) = { 1 , 2 } ;
( f " ( rng ( f | ( rng f ) ) ) . i = f . i .= ( f | ( rng f ) ) . i .= ( f | ( rng f ) ) . i .= ( f | ( rng f ) ) . i .= ( f | ( rng f ) ) . i .= ( f | ( rng f ) ) . i .= ( f | ( rng f ) ) . i ;
consider P1 , P2 being non empty Subset of TOP-REAL 2 such that P1 /\ P2 = { p1 , p2 } and C = { p1 , p2 } and P = { p1 , p2 } and p1 <> p2 and P = { p1 , p2 } and P = { p1 , p2 } and p1 <> p2 and P = { p1 , p2 } and p1 <> p2 and P = { p1 , p2 } and P = { p1 , p2 } and p1 <> p2 and P = { p1 , p2 , p3 } and P \/ { p1 , p2 , p3 } and P \/ P \/ { p1 , p2 , p3 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 = { p1 , p2 , p4 } and P \/ P \/ { p1 , p2 , p4 } and P \/ P \/ { p1 , p2 , p4 , p4 , p4 , p2 , p3 } \/ { p1 , p2 , p3 , p4 } and p1 <> p1 , p2 , p3 , p4 , p4 } \/ P \/ { p1 , p2 , p4
f . p2 = |[ ( ( p2 `1 ) / sqrt ( 1 + ( p2 `2 ) ^2 ) ) ^2 , ( ( p2 `2 ) / sqrt ( 1 + ( p2 `1 ) ^2 ) ]| .= |[ ( ( p2 `1 ) ) / sqrt ( 1 + ( p2 `2 ) ^2 ) , ( p2 `2 ) / sqrt ( 1 + ( p2 `1 ) ^2 ) ]| ;
( ( AffineMap ( a , X ) ) " . x = ( ( AffineMap ( a , X ) ) qua Function ) . x .= ( ( ( AffineMap ( a , X ) ) qua Function ) . x .= ( ( AffineMap ( a , X ) ) " ) . x .= ( - a ) * x .= - ( a + b ) * x .= - ( a + b ) * x .= - ( a + b ) * x .= - ( a + b ) * x + b * x .= - ( b + b ) * x .= - ( b + b ) * x .= ( ( - b ) * x + b ) * x + ( - b ) * x .= - ( b + b ) * x .= - ( b + b ) * x .= - ( b + b ) * x .= ( ( ( x ) * x ) * x + ( ( ( ( ( - b ) * x ) * x + ( ( - b ) * x ) * x + ( - b ) * x ) * x .= - ( ( - b ) * x + ( ( - b ) * x +
for T being non empty non empty normal TopSpace , A , B being closed Subset of T , p being Point of T st A <> {} & A misses B & A misses B & p in B holds ( Initialized G ) . p = ( NAT - ( ( ( ( ( ( ( ( A ) ) . p ) ) ) . p ) ) * ( ( ( ( ( ( A ) . p ) ) . q ) ) ) )
for i , j st i + 1 in dom F for G being strict Subgroup of G st G = F . i & G = F . j & F . i = G . j holds G = F . ( i + 1 ) & G = F . ( i + 1 )
for x st x in Z holds ( ( ( arctan ) `| Z ) . x ) = ( ( ( arctan ) . x ) * ( ( arctan . x ) ) ^2 - ( ( arctan . x ) ^2 ) / ( 1 + x ^2 ) )
synonym f /* a -> convergent means : by for x0 st x0 in dom f & for a st a in dom f & a in dom f holds f /. ( a + a ) = f /. ( a + a ) + f /. ( a + b ) ;
then X1 , X2 X2 X2 X2 iff ( X1 union X2 ) meets ( X1 union X2 ) & ( X1 union X2 ) meets ( X1 union X2 ) & ( X1 union X2 ) meets ( X1 union X2 ) & ( X1 union X2 ) meets ( X1 union X2 ) ;
ex N be Neighbourhood of x0 st N c= dom SVF1 ( 1 , f , u ) & ex L st for x st x in N holds ( SVF1 ( 1 , f , u ) ) . x - ( SVF1 ( 1 , f , u ) ) . x0 = L . ( x - x0 ) + R . ( x - x0 ) ;
sqrt ( ( ( p2 `1 ) ^2 + ( p2 `2 ) ^2 ) >= sqrt ( ( p2 `1 ) ^2 + ( p2 `2 ) ^2 ) * sqrt ( 1 + ( p2 `2 ) ^2 ) ;
( ( 1 - ( t * ||. f1 .|| ) ) to_power ( n + 1 ) ) = ( ( 1 - t ) to_power ( n + 1 ) ) * ( ( 1 - t ) to_power ( n + 1 ) ) & ( ( 1 - t ) to_power ( n + 1 ) ) * ( ( 1 - t ) to_power ( n + 1 ) ) = ( ( 1 - t ) to_power ( n + 1 ) ) * ( ( 1 - t ) to_power ( n + 1 ) ;
assume that for x holds f . x = ( ( - sin ) (#) ( sin - cos ) ) . x and x in dom ( ( - sin ) (#) ( cos - sin ) ) and for x st x in Z holds ( ( ( - sin ) (#) ( cos - sin ) ) `| Z ) . x = ( ( - sin ) (#) ( cos - cos ) ) . x ;
consider X1 being Subset of Y , Y1 being open Subset of X such that t = X1 and Y1 in A and ex Y1 being Subset of X st Y1 in A & Y1 is open & ex Y1 being Subset of X st Y1 in A & Y1 c= Y1 & Y1 c= Y1 & Y1 meets Y1 ;
card ( S . n ) = card { [ [ d , c ] , d ] where d is Element of GF ( p ) : [ d , c ] in R & [ d , b ] in R & [ d , c ] in R } .= R |^ p |^ ( p ) * R |^ ( p ) ;
( ( W-bound D - W-bound D ) / ( 2 |^ n ) ) * ( ( W-bound D - W-bound D ) / ( 2 |^ n ) ) = ( W-bound D - W-bound D ) / ( 2 |^ n ) * ( ( W-bound D - W-bound D ) / ( 2 |^ n ) ) .= ( W-bound D - W-bound D ) / ( 2 |^ n ) ;