thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b
D c= S
let Y ;
S `2 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 X ;
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 `1 = b `1 ;
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 `1 <= b `1 ;
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 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 `1 = x `1 ;
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 , W be Walk of G ;
let G be _Graph , W be Walk of G ;
let a be Complex ;
let x be element ;
let x be element ;
let C be FormalContext , a , b , c ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
let 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 ;
let 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 = b ;
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 Point of TOP-REAL 2 ;
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 Program of SCM+FSA ;
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 -> \mathclose { \rm c } ;
X <> { x } ;
x in { x } ;
q , b // M ;
A . i c= Y ;
P [ k ] ;
2 to_power x in W ;
X [ 0 ] ;
P [ 0 ] ;
A = A |^ i ;
L~ L~ o >= x ;
G . y <> 0 ;
let X be RealNormSpace , x be Point of X ;
a in X ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , W be Subspace of V ;
assume x in - M ;
k < s . a ;
not t in { p } ;
let Y be set , f be Function ;
M , L are_isomorphic ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded & L is upper-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 <= i-32 ;
1 <= i-32 ;
p\pi c= PI ;
1 <= ii & ii <= len G ;
1 <= ii & ii <= len G ;
LMP C in L ;
1 in dom f ;
let seq , seq1 , seq2 ;
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 is closed ;
cluster R \ S -> real-valued ;
sup D in S ;
x << sup D ;
b1 >= Z1 or b2 >= Z1 ;
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 ] ;
thesis ;
x4 is increasing ;
let e2 be element ;
- b divides b ;
F c= \tau ( F ) ;
Gseq is non-decreasing ;
Gseq 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 ;
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 ` & b ` c= y ;
assume not x in INT + ;
A /\ B = { a } ;
assume len f > 0 ;
assume x in dom f ;
b , a // o , c ;
B in B-24 ;
cluster product p -> non empty ;
z , x // x , p ;
assume x in rng N ;
cosec is_differentiable_in x & cosec . x > 0 ;
assume y in rng S ;
let x , y be element ;
i2 < i1 & i1 < len f ;
a * h in a * H ;
p , q in Y ;
redefine func sqrt I ;
q1 in A1 & q2 in A2 ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A2 c= A1 ;
an < n & bn < n ;
assume A c= dom f ;
Re ( f ) is_integrable_on M ;
let k , m be element ;
a , a // 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 j0 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 halts_on s ;
d , c // a , b ;
let t , u be set ;
let X be set with non-empty ManySortedSet of I ;
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 TopSpace , A be Subset 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 of f , A ;
let b be Element of X , x be Element of X ;
R [ x , y ] ;
x ` = x & y = y ;
b \ x = 0. X ;
<* d *> in D |^ 1 ;
P [ k + 1 ] ;
m in dom ( mn ) ;
h2 . a = y ;
P [ n + 1 ] ;
redefine func G * F -> | J ;
let R be non empty multMagma , x be Element 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 id mamaid id id id ;
let N be non empty seq of M ;
let R be RelStr with finite finite finite 1 ;
let n , k be Nat ;
let P , Q be be be be be be be be let RelStr ;
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 is not ` ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < v8 ;
x <= c2 . x ;
x in F ` & y in F ` ;
redefine func S --> T -> ManySortedSet of I ;
assume that t1 <= t2 and t2 <= t2 ;
let i , j be even Integer ;
assume that F1 <> F2 and F2 <> F1 ;
c in Intersect ( union R ) ;
dom p1 = c & dom p2 = c ;
a = 0 or a = 1 ;
assume A1 <> A2 & A2 <> A1 ;
set i1 = i + 1 ;
assume that a1 = b1 and a2 = b2 ;
dom g1 = A & dom g2 = B ;
i < len M + 1 ;
assume not -infty in rng G ;
N c= dom ( f1 + f2 ) ;
x in dom sec /\ dom sec ;
assume [ x , y ] in R ;
set d = x / y ;
1 <= len g1 & 1 <= len g2 ;
len s2 > 1 & len s1 > 1 ;
z in dom ( f1 + f2 ) ;
1 in dom ( D2 | indx ( D2 , D1 , j1 ) ) ;
( p `2 ) ^2 = 0 ;
j2 <= width G & j2 <= width G ;
len PI > 1 + 1 ;
set n1 = n + 1 ;
|. q-35 .| = 1 ;
let s be SortSymbol of S ;
gcd ( i , i ) = i ;
X1 c= dom f & X2 c= dom f ;
h . x in h . a ;
let G be not thesis ;
cluster m * n -> square ;
let kk be Nat , p be FinSequence ;
i - 1 > m - 1 ;
R is transitive & R is transitive ;
set F = <* u , w *> ;
p-2 c= P3 & p-2 c= P3 ;
I is_closed_on t , Q & I is_halting_on t , Q ;
assume [ S , x ] is thesis ;
i <= len ( f2 | k ) ;
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 ;
BX c= [ X , Y ] ;
n2 <= ( 2 to_power 4 ) ;
A /\ cP c= A `
cluster x -valued for Function ;
let Q be Subset-Family of S , P be Subset of T ;
assume n in dom g2 & m in dom g2 ;
let a be Element of R ;
t `2 in dom ( e2 `2 ) ;
N . 1 in rng N ;
- z in A \/ B ;
let S be SigmaField of X , M be Element of S ;
i . y in rng i ;
REAL c= dom f & dom f = REAL ;
f . x in rng f ;
mt <= ( r / 2 ) ;
s2 in r-5 & s1 in r-5 ;
let z , z be complex number ;
n <= Nseq . m ;
LIN q , p , s ;
f . x = waybelow x /\ B ;
set L = [' S , T '] ;
let x be non positive Real ;
let m be Element of M ;
f in union rng ( F1 | n ) ;
let K be add-associative right_zeroed right_complementable non empty doubleLoopStr , p be Polynomial of K ;
let i be Element of NAT , k be Nat ;
rng ( F * g ) c= Y
dom f c= dom x & rng f c= dom y ;
n1 < n1 + 1 & n2 + 1 <= len f ;
n1 < n1 + 1 & n2 + 1 <= len f ;
cluster 1. T -> \overline W ;
[ y2 , 2 ] = z ;
let m be Element of NAT , n ;
let S be Subset of R ;
y in rng ( S29 | X ) ;
b = sup dom f & b = sup dom f ;
x in Seg ( len q ) ;
reconsider X = D ( ) as set ;
[ a , c ] in E1 ;
assume n in dom ( h2 . n ) ;
w + 1 = a + 1 ;
j + 1 <= j + 1 + 1 ;
k2 + 1 <= k1 & k2 + 1 <= k2 ;
i be Element of NAT ;
Support u = Support p & Support u = Support p ;
assume X is complete Subthesis of m ;
assume f = g & p = q ;
n1 <= n1 + 1 & n2 <= n1 + 1 ;
let x be Element of REAL ;
assume x in rng ( s2 ^\ k ) ;
x0 < x0 + 1 & x0 + 1 < x0 + 1 ;
len ( L5 * L5 ) = W ;
P c= Seg ( len A ) ;
dom q = Seg n & dom q = Seg n ;
j <= width M *' ;
let seq1 be real-valued sequence of X ;
let k be Element of NAT ;
Integral ( M , P ) < +infty ;
let n be Element of NAT ;
assume z in len ( 0 := len A ) ;
i be set ;
n -' 1 = n-1 - 1 ;
len ( n * ( - n ) ) = n ;
\mathop { \rm Set } ( Z , c ) c= F
assume x in X or x = X ;
x is midpoint of b , c ;
let A , B be non empty set , F be Function of A , B ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q & dom q = Seg n ;
let s be Element of E ^ ;
let B1 be Basis of x , B ;
L3 /\ L2 = {} ;
L1 /\ L2 = {} implies L1 + L2 = L2 + L1
assume downarrow x = downarrow y ;
assume b , c // b , c ;
LIN q , c , c ;
x in rng ( f-129 | n ) ;
set nn8 = n + j ;
let D7 be non empty set , f be Function ;
let K be add-associative right_zeroed right_complementable non empty addLoopStr , M be Matrix of K ;
assume f `1 = f & h `2 = h ;
R1 - R2 is total & R2 - R1 is total ;
k in NAT & 1 <= k ;
let a be Element of G ;
assume x0 in [. a , b .] ;
K1 ` is open & K1 ` is open ;
assume that a , b ) is maximal distance in C ;
let a , b be Element of S ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = Integral ( M , f ) ;
cluster non ns] for \in nest ;
not u in { ag } ;
the carrier of f c= B \/ { v }
reconsider z = x as VECTOR of V ;
cluster the
r (#) H is as as as as as as as as as as as as as ;
s . intloc 0 = 1 ;
assume that x in C and y in C ;
let U0 be strict universal MSAlgebra over S , A be Element of U0 ;
[ x , Bottom T ] is compact ;
i + 1 + k in dom p ;
F . i is stable Subset of M ;
r-35 in : ( y in : y in { x } ) ;
let x , y be Element of X ;
let A , I be contradiction / 2 ;
[ y , z ] in [: O , O :] ;
( } Macro i ) = 1 & ( card Macro i ) = 2 ;
rng Sgm ( A ) = A ;
q |- \! such that q in L~ 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 & LIN o , a , b ;
p . 2 = Z / Y ;
( DD ) `2 = {} ;
n + 1 + 1 <= len g ;
a in [: CQC-WFF ( Al ) , { 0 } :] ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
let I be Ideal of L ;
set g = f1 + f2 , h = f1 + f2 ;
a <= max ( a , b ) ;
i-1 < len G + 1-1 ;
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 non empty multiplicative magma ;
x in support ( support ( support ( t ) ) ) ;
assume a in [: the carrier of G , the carrier of G :] ;
i `2 <= len ( y `2 ) ;
assume that p divides b1 + b2 and b1 divides b2 ;
M <= sup M1 & M <= sup M2 ;
assume x in W-min ( X ) & y in L~ f ;
j in dom ( z | k ) ;
let x be Element of D ( ) ;
IC s4 = l1 & IC s4 = l1 ;
a = {} or a = { x } ;
set uG = Vertices G , uH = Vertices G ;
seq " is non-zero & ( seq " ) (#) seq is non-zero ;
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
h-4 c= h-14 & hh2 c= h-14 ;
]. a , b .[ c= Z ;
X1 , X2 are_separated & X2 , X1 are_separated ;
a in Cl ( union F \ G ) ;
set x1 = [ 0 , 0 ] ;
k + 1 -' 1 = k - 1 ;
cluster -> real-valued for Relation ;
ex v st C = v + W ;
let IT be non empty addLoopStr , p be Point of IT ;
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 ;
sup B is upper Subset of B & sup B is upper ;
let L be non empty reflexive antisymmetric RelStr , x be Element of L ;
R is reflexive & X is transitive ;
E , g |= the_right_argument_of ( H ) ;
dom G `2 /. y = a ;
( 1 / 4 ) * ( 1 / 4 ) >= - r ;
G . p0 in rng G & G . p0 in rng G ;
let x be Element of FH , y be Element of FH ;
D [ P-6 , 0 ] ;
z in dom id ( B ) & z in dom id ( B ) ;
y in the carrier of N & y in the carrier of N ;
g in the carrier of H & h in the carrier of G ;
rng fl c= NAT & rng fl c= NAT ;
j `2 + 1 in dom s1 & j + 1 in dom s1 ;
let A , B be strict Subgroup of G ;
let C be non empty Subset of REAL ;
f . z1 in dom h & h . z2 in dom h ;
P . k1 in rng P & P . k2 in rng P ;
M = AM +* {} .= AM +* {} ;
let p be FinSequence of REAL , k be Nat ;
f . n1 in rng f & f . n2 in rng f ;
M . ( F . 0 ) in REAL ;
holds holds not |. a - b .| = b-a ;
assume the distance of V , Q is_V ;
let a be Element of ( the carrier of V ) ;
let s be Element of [: P , Q :] ;
let PA be non empty \cal RelStr , a be Element of Y ;
let n be Nat ;
the carrier of g c= B & the carrier of g c= the carrier of L ;
I = halt SCM R & I = halt SCM R ;
consider b being element such that b in B ;
set BM = BCS K , BM = BCS K ;
l <= j & j <= j implies ( j - 1 ) <= j
assume x in downarrow [ s , t ] ;
( x `2 ) ^2 in uparrow t & ( x `2 ) ^2 in uparrow t ;
x in O implies dom ( 1 -tuples_on the carrier of T ) = { 0 }
let h be Morphism of c , a ;
Y c= 1. R & Y c= dom ( R | Y ) ;
A2 \/ A3 c= L2 \/ L3 \/ L1 \/ L2 ;
assume LIN o , a , b & LIN o , a , b ;
b , c // d1 , e2 ;
x1 , x2 in Y & y1 in Y ;
dom <* y *> = Seg 1 & dom <* y *> = Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar s .| ;
[ x , x `2 ] in [: X , X :] ;
for n be Nat holds 0 <= x . n
[' a , b '] = [. a , b .] ;
cluster -> non empty -> closed for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 in P & q2 in P ;
dom M1 = Seg n & dom M2 = Seg n ;
x = [ x1 , x2 ] ;
let R , Q be ManySortedSet of A ;
set d = 1 / ( n + 1 ) ;
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 Subset of V ;
let I be Program of SCM+FSA , a be Int-Location ;
assume x in rng ( the InternalRel of R ) ;
let b be Element of the carrier of T ;
dist ( e , z ) - r-r > r-r ;
u1 + v1 in W2 & v1 in W1 + W2 ;
assume func support L -> Subset of rng G ;
let L be lower-bounded antisymmetric transitive antisymmetric RelStr ;
assume [ x , y ] in [: a , b :] ;
dom ( A * e ) = NAT & dom ( A * e ) = NAT ;
let a , b be Vertex of G ;
let x be Element of Bool ( M . i ) ;
0 <= Arg a & Arg a < 2 * PI ;
o , a9 // o , y & o , a9 // o , y ;
{ v } c= the carrier of l & { v } c= the carrier of l ;
let x be bound of A ;
assume x in dom ( uncurry uncurry f ) & y in dom ( uncurry uncurry f ) ;
rng F c= ( Product f ) |^ X
assume D2 . k in rng D & 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 ;
let n be Element of NAT ;
assume LIN c , a , e1 & LIN c , a , e1 ;
cluster -> \/ [ f ] -> ) -valued for FinSequence ;
reconsider d = c as Element of L1 ;
( v2 |-- I ) . X <= 1 ;
assume x in the carrier of f & y in the carrier of g ;
conv @ S c= conv @ A & conv @ S c= conv @ A ;
reconsider B = b as Element of the carrier of T ;
J , v |= P \lbrack l \rbrack ;
redefine func J . i -> non empty TopSpace equals J . i ;
ex_sup_of Y1 \/ Y2 , T & ex_sup_of Y1 , T ;
W1 is_field ( W1 + W2 ) & R is_field ( W1 + W2 ) ;
assume x in the carrier of R & y in the carrier of R ;
dom ( n --> ( n + 1 ) ) = Seg n .= Seg n ;
s4 misses s2 & s4 in s4 ;
assume ( a 'imp' b ) . z = TRUE ;
assume that X is open and f = X --> d ;
assume [ a , y ] in an & [ a , y ] in the InternalRel of R ;
assume that not that that that that that not I c= J and not I c= J ;
Im ( lim seq ) = 0 & Im ( lim seq ) = 0 ;
( sin * cos ) . x <> 0 & ( sin * cos ) . x <> 0 ;
sin is_differentiable_on Z & cos is_differentiable_on Z implies cos * cos is_differentiable_on Z & for x st x in Z holds cos . x <> 0
t3 . n = t0 . n & t1 . n = t2 . n ;
dom ( ( dom ( F | A ) ) ) c= dom F ;
W1 . x = W2 . x & W2 . x = W1 . x ;
y in W .vertices() \/ W .vertices() \/ W .vertices() ;
( k <= len vM ) & ( k + 1 <= len vM ) ;
x * a \equiv y * a . ( mod m ) ;
proj2 .: S c= proj2 .: P & proj2 .: S c= proj2 .: P ;
h . p4 = g2 . I & h . I = g2 . I ;
G = ( U /. 1 ) `1 .= G * ( 1 , k ) `1 ;
f . rp1 in rng f & f . rp1 in rng f ;
i + 1 + 1 <= len - 1 ;
rng F = rng ( F | ( dom F ) ) .= dom F ;
mode then empty multMagma is well unital associative associative non empty multMagma ;
[ x , y ] in [: A , { a } :] ;
x1 . o in L2 . o & x2 . o in L2 . o ;
the carrier of m c= B & the carrier of m c= B ;
not [ y , x ] in id X ;
1 + p .. f <= i + len f ;
seq ^\ k1 is lower & seq ^\ k1 is lower ;
len F-12 = len I & len F-12 = len I ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be complex number , x be Point of X ;
Comput ( P , s , n ) = s ;
k <= k + 1 & k + 1 <= len p ;
reconsider c = {} T as Element of L ;
let Y be Element of of of of of P ;
cluster empty -> directed-sups-preserving for Function of L , L ;
f . j1 in K . j1 & f . j2 in K . j1 ;
redefine func J => y -> total ( J , I ) -defined Function ;
K c= 2 -tuples_on the carrier of T & K is finite ;
F . b1 = F . b2 & F . b2 = F . b2 ;
x1 = x or x1 = y or x1 = z ;
pred a <> {} means : Def1 : a - 1 = 1 ;
assume that 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 trivial non trivial FinSequence of D ;
let FF2 be non empty element ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in F-8 ;
reconsider pp2 = x , pp2 = y as Subset of m ;
let A , B , C be Element of R ;
redefine func strict non empty s3 _ l -> strict non empty be be \times over K ;
rng c `1 misses rng ( e[: , { 0 } :] ) ;
z is Element of gr { x } & z is Element of gr { x } ;
not b in dom ( a .--> p1 ) ;
assume that k >= 2 and P [ k ] ;
Z c= dom ( cot * cot ) /\ dom ( cot * cot ) ;
the component of Q c= UBD A & the component of Q c= UBD A ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( 1 / ( f . x ) ) ;
pred f = u means : Def1 : a * f = a * u ;
for n holds P1 [ \mathop { \rm VERUM } ]
{ x . O : x in L } <> {} ;
let x be Element of V . s ;
let a , b be Nat ;
assume that S = S2 and p = S2 ;
gcd ( n1 , n2 , k ) = 1 & gcd ( n2 , n2 , k ) = 1 ;
set ok = a * ( 0. INT ) , ok = a * ( 0. INT ) ;
seq . n < |. r1 .| & |. seq . n .| < r ;
assume that seq is increasing and r < 0 and seq is increasing ;
f . ( y1 , x1 ) <= a ;
ex c being Nat st P [ c ] & c <= n ;
set g = { n to_power 1 : n in NAT } ;
k = a or k = b or k = c ;
not ( ex a , b st a , b // a , b & a <> b ) ;
assume that Y = { 1 } and s = <* 1 *> ;
Ip1 . x = f . x .= 0 .= 0 ;
W3 .last() = W3 . 1 & W1 .last() = W3 . 2 ;
cluster trivial -> trivial finite for Walk of G ;
reconsider u = u as Element of Bags X ;
A in B ^ -> FinSequence of REAL implies A , B N N
x in { [ 2 * n + 3 , k ] } ;
1 >= ( q `1 / |. q .| - cn ) / ( 1 - cn ) ;
f1 is_<= _ { f , g } , T ;
( f /. i ) `2 <= ( q `2 ) ^2 ;
h is_the carrier of Cage ( C , n ) ;
( b `2 ) ^2 / ( |. b .| ) ^2 <= ( |. b .| ) ^2 ;
let f , g be Function of X , Y ;
S * ( k , k ) <> 0. K ;
x in dom max ( - f , - f ) ;
p2 in NH . p1 & p2 in NH . p1 ;
len ( the_left_argument_of H ) < len ( H ) & len ( H ) < len ( H ) ;
F [ A , F-14 . A ] ;
consider Z such that y in Z and Z in X ;
hence 1 in C & A c= C ^ B ;
assume that r1 <> 0 or r2 <> 0 and r1 <> 0 ;
rng q1 c= rng C1 & rng q2 c= rng C2 ;
let A1 , L , A2 , A3 , A3 be non empty set ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in C & c in C & d in C ;
then S is atomic & P-2 [ S ] ;
Cl Int [#] T = [#] T & 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 ;
let 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 , T :] ;
consider H1 such that H = 'not' H1 and H1 in X ;
1. ( K , 1 ) c= ( \mathop { 0 } * ( 1 / 2 ) ) ;
0 * a = 0. R .= a * 0. R .= a ;
A |^ ( 2 , 2 ) = A ^^ A ;
set vFinSequence = v4 /. n , v5 = v5 /. n ;
r = 0. ( REAL-NS n ) & ||. 0. ( REAL-NS n ) .|| = 0. ( REAL-NS n ) ;
( f . p4 ) `1 >= 0 & ( f . p4 ) `2 >= 0 ;
len W = len ( W Line ( W , i ) ) .= len W + 1 ;
f /* ( s * G ) is divergent_to+infty ;
consider l being Nat such that m = F . l ;
t8 does not destroy b1 & not t8 does not destroy 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 ( 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 ] -> non pair for element ;
downarrow a /\ downarrow t is Ideal of T & a /\ downarrow t is Ideal of T ;
let X be with \hbox { NAT , D } , f be Function ;
rng f = being Element of \rm \rm set ( S , X ) ;
let p be Element of B , the carrier' of S ;
max ( N1 , 2 ) >= N1 & max ( N2 , 2 ) >= N2 ;
0. X <= b |^ ( m * mm1 ) ;
assume that i in I and R1 . i = R ;
i = j1 & p1 = q1 & p2 = q2 & p1 = q2 ;
assume gRRRRRRRRg in the carrier of g ;
let A1 , A2 be Point of S , A be Subset of X ;
x in h " P /\ [#] T1 & x in h " P /\ [#] T1 ;
1 in Seg 2 & 1 in Seg 3 & 2 in Seg 3 ;
reconsider X-5 = X , Xseq = Y as non empty Subset of T<* T , S *> ;
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the Source of G ) -defined for Function ;
n1 <= i2 + len g2 & n2 <= len g2 + len g2 ;
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume that v in the carrier' of G2 and v in the carrier' of G1 ;
y = Re y + ( Im y ) * i ;
( ( - 1 ) * p ) gcd p = 1 ;
x2 is_differentiable_on ]. a , b .[ & x2 is_differentiable_on ]. a , b .[ ;
rng M5 c= rng D2 & len M5 = len D2 ;
for p be Real st p in Z holds p >= a
( 1. X ) * ( f /. x ) = proj1 . x .= ( 1. X ) * ( f /. x ) ;
( seq ^\ m ) . k <> 0 & ( seq ^\ m ) . k <> 0 ;
s . ( G . ( k + 1 ) ) > x0 ;
( p |-count M ) . 2 = d & ( p |-- M ) . 3 = d ;
A \oplus ( B \ominus C ) = ( A \oplus B ) \ominus C
h \equiv gg . ( mod P ) & g . ( mod P ) = gg . ( mod P ) ;
reconsider i1 = i-1 , i2 = i2 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 i-7 = i , im2 = j as Element of NAT ;
dom f c= [: C ( ) , D ( ) :] ;
x in ( the Sorts of B ) . n & ( the Sorts of B ) . n in rng the Sorts of A ;
len } in Seg ( len f2 ) & len ( f2 | n ) = n ;
pp1 c= the topology of T & pp1 c= the topology of T ;
]. r , s .] c= [. r , s .] ;
let B2 be Basis of T2 , f be Function of T1 , T2 ;
G * ( B * A ) = ( id o1 ) * A ;
assume that p , u , v , q is_collinear and p , q , v , q is_collinear ;
[ z , z ] in union rng ( F | X ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S , R = $1 .. S ;
LIN a1 , a3 , b1 & LIN a1 , a3 , b1 & LIN a1 , a3 , b2 ;
f " ( f .: x ) = { x } ;
dom w2 = dom r12 & dom ( r12 . x ) = dom r12 ;
assume that 1 <= i and i <= n and j <= n ;
( ( ( g2 ) . O ) `2 ) ^2 <= 1 ;
p in LSeg ( E . i , F . i ) ;
IK * ( i , j ) = 0. K ;
|. f . ( s . m ) - g .| < g1 ;
qq . x in rng ( q | Seg k ) & qq . x in rng ( q | Seg k ) ;
Carrier ( LLet ) misses Carrier ( LR1 ) ` & Carrier ( LR2 ) c= Carrier ( LR2 ) ;
consider c being element such that [ a , c ] in G ;
assume that Nreal = N\HM and for o being Element of NAT holds o <= o ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= ( F |^ C-1 ) \/ ( F |^ C-1 ) ;
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [: [. f . j , f . j .] , { f . j } :] ;
pred 0 <= x & x <= 1 & x ^2 <= 1 ;
p `2 - q `2 <> 0. TOP-REAL 2 & p `2 - q `2 <> 0. TOP-REAL 2 ;
redefine func being aa] ( S , T ) -> non empty RelStr ;
let x be Element of [: S , T :] ;
( the Arrows of F ) . ( a , b ) is one-to-one ;
|. i .| <= - ( - 2 to_power n ) ;
the carrier of I[01] = dom P & the carrier of I[01] = the carrier of I[01] ;
n * ( n + 1 ) ! > 0 * th ;
S c= ( A1 /\ A2 ) /\ A3 & S /\ A2 c= ( A1 /\ A2 ) /\ A3 ;
a3 , a4 // b3 , b2 & a3 , a4 // b3 , b3 ;
then dom A <> {} & dom A <> {} & rng A <> {} ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , Y & y in X implies x = y
set v2 = ( v /. ( i + 1 ) ) * v /. ( i + 1 ) ;
x = r . n .= ( r . n ) . x .= ( r . n ) . x ;
f . s in the carrier of S2 & f . s in the carrier of S2 ;
dom g = the carrier of I[01] & rng g = the carrier of I[01] ;
p in Upper_Arc ( P ) /\ Lower_Arc ( P ) /\ Lower_Arc ( P ) ;
dom d2 = [: [: A2 , A1 :] , [: A2 , A1 :] :] ;
0 < ( p / ||. z .|| + 1 ) / ( ||. z .|| + 1 ) ;
e . ( mm + 1 ) <= e . ( mm + 1 ) ;
B \ominus X \/ B \ominus Y c= B \ominus X /\ Y
-infty < Integral ( M , Im ( g | B ) ) ;
cluster O := F -> \HM { + } -valued for 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 , i ;
let x , y , z be Point of X , p be Point of X ;
reconsider p0 = p . x , p0 = p . x as Subset of V ;
x in the carrier of Lin ( A ) & 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 & 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 / |. p2 .| <= ( |. p2 .| ) / ( |. p2 .| ) ;
Cl Q ` = [#] ( T | P ) .= [#] ( T | P ) ;
set S = the carrier of T , T = the carrier of T ;
set I8 = for f |^ n , I8 = f |^ n ;
len p - n = len ( thesis - n ) .= len p - n ;
A is Permutation of Swap ( A , x , y ) ;
reconsider nnseq = n8 - 1 as Element of NAT ;
1 <= j + 1 & j + 1 <= len ( s | k ) ;
let q\subseteq , qSet , qJ be Element of M ;
( a in the carrier of S1 ) & ( a in the carrier of S1 ) ;
c1 /. n1 = c1 . n1 & c2 /. n1 = c2 . n1 & c1 /. n1 = c2 . n1 ;
let f be FinSequence of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
y = ( f * S8 ) . x .= ( f * S8 ) . x ;
consider x being element such that x in Assume that x in Assume that A = f . x ;
assume r in ( ( dist ( o ) ) .: P ) ;
set i2 = width ( the / 2 ) , i1 = len ( the / 2 ) ;
h2 . ( j + 1 ) in rng h2 & h2 . ( j + 1 ) in rng h2 ;
Line ( M29 , k ) = M . i .= Line ( M29 , k ) ;
reconsider m = ( x - 1 ) / 2 as Element of ( the carrier of X ) ;
let U1 , U2 be strict Subspace of U0 , A be Element of U1 ;
set P = Line ( a , d ) ;
len p1 < len p2 + 1 & len p1 + 1 <= len p2 ;
let T1 , T2 be Scott `1 of L , x be Point of T1 , y be Point of T2 ;
then x <= y & : -> : x in : x in { y } ;
set M = n -tuples_on ( the carrier of K ) ;
reconsider i = x1 , j = x2 as Nat ;
rng the_arity_of ( a9 , b9 ) c= dom H & rng the_arity_of ( a9 , b9 ) c= dom H ;
z1 " = ( z " ) * ( z " ) .= ( z " ) * ( z " ) ;
x0 - r / 2 in L /\ dom f & x0 - r / 2 in dom f ;
then w is that rng w /\ L <> {} & rng w /\ L <> {} ;
set x-10 = ( x ^ <* Z *> ) . ( x + 1 ) , x-10 = ( x ^ <* Z *> ) . ( x + 1 ) ;
len w1 in Seg ( len w1 + len w2 ) & len w2 in Seg ( len w1 + len w2 ) ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of PFuncs ( V , { k } ) ;
x . n = ( |. a . n .| ) * ( |. b . n .| ) ;
p `1 <= G * ( i1 , 1 ) `1 & p `1 <= G * ( i1 , 1 ) `1 ;
rng ( g | 1 ) c= L~ ( g | 1 ) \/ L~ ( g | 1 ) ;
reconsider k = i-1 * ( i + j ) as Nat ;
for n being Nat holds F . n is \HM { -infty } ;
reconsider x-10 = x-7 , x29 = xm2 , x29 = xm2 as VECTOR of M ;
dom ( f | X ) = X /\ dom f .= X /\ dom f ;
p , a // p , c & b , a // c , c ;
reconsider x1 = x , y1 = y , z1 = z as Element of REAL m ;
assume i in dom ( a * p ^ q ) ;
m . ( ag . ( bg . ( bg . ( bg . ( bg . ( bg . ( bg . ( len g ) ) ) ) ) ) ) = p . ( a . ( g . ( len g )
a / ( s . m - 1 ) / ( s . n - 1 ) <= 1 ;
S . ( n + k + 1 ) c= S . ( n + k ) ;
assume that B1 \/ C1 = B2 \/ C2 and B2 = C1 \/ C2 ;
X . i = { x1 , x2 } . i .= ( X * f ) . i ;
r2 in dom ( h1 + h2 ) & r2 in dom ( h1 + h2 ) ;
- - 0. R = a & b-0 = b ;
F8 is_closed_on t2 , Q & F8 is_halting_on t , Q & F8 is_halting_on t , Q ;
set T = -> InInInInInInof X , x0 , x1 ;
Int Cl Int R c= Int Cl R & Int Cl R c= Cl Int R ;
consider y being Element of L such that c . y = x ;
rng Fp1 = { Fp1 . x } .= { Fp1 . x } ;
G-23 " { c } c= B \/ S & G . c = G . c ;
f[#] is Relation of [: X , Y :] , X & f is Function of X , Y ;
set RQ = the Point of P , RQ = the carrier of Q ;
assume that n + 1 >= 1 and n + 1 <= len M ;
let k2 be Element of NAT , k be Nat ;
reconsider pSubset = u , pSubset = v as Element of ( TOP-REAL n ) | ( the carrier of TOP-REAL n ) ;
g . x in dom f & x in dom g implies f . x = g . x
assume that 1 <= n and n + 1 <= len f1 ;
reconsider T = b * N as Element of ( G / ( N , 1 ) ) ;
len Pt <= len P-35 & len Pt <= len P-35 ;
x " in the carrier of A1 & x " in the carrier of A1 ;
[ i , j ] in Indices ( A * ( i , j ) ) ;
for m be Nat holds Re ( F . m ) is simple Function of S , REAL
f . x = a . i .= a1 . k .= a1 . k ;
let f be PartFunc of REAL i , REAL , x be Point of REAL m ;
rng f = the carrier of ( Carrier A ) & f . x = ( Carrier A ) . x ;
assume that s1 = sqrt ( 2 / p ) - 1 and s2 = sqrt ( 2 / p ) - 1 ;
pred a > 1 & b > 0 & a / b > 1 ;
let A , B , C be Subset of [: I , J :] ;
reconsider X0 = X , Y0 = Y as RealNormSpace ;
let f be PartFunc of REAL , REAL , x be Point of REAL ;
r * ( v1 |-- I ) . X < r * 1 ;
assume that V is Subspace of X and X is Subspace of V ;
let t-3 , t-3 , tt2 be Relation of T , the carrier of S ;
Q [ e-14 \/ { v-14 } , f . vW1 , f . vW1 , f . vW2 ] ;
g \circlearrowleft W-min L~ z = z implies ( g /. 1 ) .. z < ( g /. len z ) .. z
|. |[ x , v ]| - |[ x , y ]| .| = v\rrangle ;
- f . w = - ( L (#) w ) .= - ( L (#) w ) ;
z - y <= x iff z <= x + y & y <= z ;
( 7 / ( 1 / e ) ) to_power ( 1 / e ) > 0 ;
assume X is BCK-algebra of 0 , 0 , 0 , 0 , 0 ;
F . 1 = v1 & F . 2 = v2 & F . 3 = v2 ;
( f | X ) . x2 = f . x2 & ( f | X ) . x2 = f . x2 ;
( tan * sec ) . x in dom ( sec * sec ) /\ dom ( sec * sec ) ;
i2 = ( f /. len f ) & ( f /. len f ) = ( f /. len f ) ;
X1 = X2 \/ ( X1 \ X2 ) & X2 = ( X1 \ X2 ) \/ ( X2 \ X1 ) ;
[. a , b , 1. G .] = 1_ G & a * b = 1_ G ;
let V , W be non empty VectSpStr over F_Complex , f be FinSequence of V ;
dom g2 = the carrier of I[01] & rng g2 = the carrier of I[01] & g2 . 0 = p2 ;
dom f2 = the carrier of I[01] & rng f2 = the carrier of I[01] & f2 . 0 = p2 ;
( proj2 | X ) .: X = proj2 .: X .= proj2 .: X .= X ;
f . ( x , y ) = h1 . ( x `1 , y `2 ) ;
x0 - r < a1 . n & x0 < x0 + r ;
|. ( f /* s ) . k - G . ( k + 1 ) .| < r ;
len Line ( A , i ) = width A & len Line ( A , i ) = width A ;
SFinSequence @ = ( S . g ) @ .= ( S . g ) @ ;
reconsider f = v + u as Function of X , the carrier of Y ;
intloc 0 in dom Initialized p & ( Initialized p ) . intloc 0 = 1 ;
i1 does not destroy ( i2 , 3 ) & not ( i1 does not destroy ( i2 , 3 ) ) does not destroy ( i2 , 3 ) ;
arccos r + arccos r = ( PI / 2 + 0 ) * ( PI / 2 + 0 ) ;
for x st x in Z holds f2 * ( f1 + f2 ) is_differentiable_in x ;
reconsider q2 = ( q - x ) / ( q - x ) as Element of REAL ;
( 0 qua Nat ) + 1 <= i + j1 & i + 1 <= j ;
assume f in the carrier of [: X , Omega Y :] ;
F . a = H / ( x , y ) . a ;
( ( {} T ) at ( C , u ) ) . x = TRUE ;
dist ( ( a * seq ) . n , h ) < r / 2 ;
1 in the carrier of [. 0 , 1 .] & 2 in the carrier of I[01] ;
( p2 `1 ) - x1 > - g & ( p2 `2 ) - g < ( p2 `2 ) - g ;
|. r1 - p .| = |. a1 .| * |. thesis .| .= |. - a1 .| ;
reconsider S-14 = 8 as Element of ( the carrier of K ) * ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b
D0W .\rbrack = D0W ..\rbrack + 1 ;
i1 = ma + n & i2 = K + n & i1 = K + n ;
f . a [= f . ( f . O1 "\/" a ) ;
pred f = v & g = u , h = v + u ;
I . n = Integral ( M , F . n ) ;
chi ( [: T1 , T2 :] , S ) . s = 1 .= ( the Sorts of T1 ) . s ;
a = VERUM ( A ) or a = VERUM ( A ) ;
reconsider k2 = s . b3 , k1 = s . b3 as Element of NAT ;
( Comput ( P , s , 4 ) ) . GBP = 0 ;
L~ M1 meets L~ ( R4 * R4 ) & L~ ( R4 * R4 * R4 ) misses L~ ( R4 * R4 * R5 ) ;
set h = the continuous Function of X , R , x be Point of X ;
set A = { L . ( k . n ) : n <= k } ;
for H st H is atomic holds P7 [ H ] ;
set b\HM = S5 ^\ ( i + 1 ) , Sseq = Sseq ^\ ( i + 1 ) ;
Hom ( a , b ) c= Hom ( a opp , b ) ;
( 1 / ( n + 1 ) ) < ( 1 / ( n + 1 ) ) ;
( l ) `1 = [ dom l , cod l ] `2 .= [ dom l , cod l ] `2 .= dom l ;
y +* ( i , y /. i ) in dom g & y +* ( i , y ) in dom g ;
let p be Element of CQC-WFF ( Al ) , P , l be Element of CQC-WFF ( Al ) ;
X /\ X1 c= dom ( f1 - f2 ) & f1 - f2 c= dom ( f1 - f2 ) ;
p2 in rng ( f /^ ( len f -' 1 ) ) & p2 in rng ( f /^ ( len f -' 1 ) ) ;
1 <= indx ( D2 , D1 , j1 ) & indx ( D2 , D1 , j1 ) <= len D2 ;
assume x in ( ( K /\ / 3 ) \/ ( K /\ 4 ) ) /\ ( K /\ 5 ) ;
- 1 <= ( ( f2 ) . O ) `2 & ( ( f2 ) . O ) `2 <= 1 ;
let f , g be Function of I[01] , ( TOP-REAL 2 ) | K1 , R^1 ;
k1 -' k2 = k1 - k2 + 1 .= k1 - k2 + 1 .= k1 - k2 ;
rng seq c= ]. x0 , x0 + r .[ & ( for n holds seq . n < x0 ) implies seq is convergent & lim seq = x0
g2 in ]. x0 , x0 + r .[ & g2 in ]. x0 , x0 + r .[ ;
sgn ( p `1 , K ) = - ( - 1_ K ) .= - ( 1_ K ) ;
consider u being Nat such that b = p |^ y * u ;
ex A being as as as as as - normal sequence of W st a = Sum A ;
Cl ( union H ) = union ( ( Cl H ) \/ ( Cl H ) ) ;
len t = len t1 + len t2 & len t1 = len t1 + len t2 + len t2 ;
v-29 = v + w |-- v + AA .= v + AA ;
cv <> DataLoc ( t0 . GBP , 3 ) & cv <> DataLoc ( t0 . GBP , 3 ) ;
g . s = sup ( d " { s } ) .= sup ( d " { s } ) ;
( \dot y ) . s = s . ( y . s ) ;
{ s : s < t } in INT & t = {} or t = {} & s = {} ;
s ` \ s = s ` \ 0. X .= ( s ` \ s ) ` .= ( s ` \ s ) ` ;
defpred P [ Nat ] means B + $1 in A & A . $1 = B . $1 ;
( 339 + 1 ) ! = 3339 ! * ( 339 + 1 ) ;
1. ( A , succ A ) = 1. ( A , succ A ) .= ( A , A ) --> 0. ( A , A ) ;
reconsider y = y as Element of COMPLEX , x be Element of COMPLEX ;
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 \mathop { \it Boolean } , g = ( S , U ) \mathop { \it Boolean } ;
consider Z being set such that lim s in Z and Z in F ;
let f be Function of I[01] , ( TOP-REAL n ) | P , p1 , p2 be Point of TOP-REAL n ;
( SAT M ) . [ n + i , 'not' A ] <> 1 ;
ex r being Real st x = r & a <= r & r <= b ;
let R1 , R2 be Element of REAL n , a , b be Real ;
reconsider l = 0. ( { v } ) , r = 0. ( A ) as Linear_Combination of A ;
set r = |. e .| + |. n .| + |. w .| + |. s .| ;
consider y being Element of S such that z <= y and y in X ;
a 'or' ( b 'or' c ) = 'not' ( ( a 'or' b ) 'or' c ) ;
||. ( x - g ) . x - g /. x0 .|| < r2 / 2 ;
b9 , a9 // b9 , c9 & b9 , c9 // b9 , c9 & b9 , c9 // a9 , c9 ;
1 <= k2 -' k1 & k2 + 1 = k2 & k2 + 1 = k2 + 1 ;
( ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) ^2 >= 0 ;
( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 < 0 ;
E-max C in cell ( Rg , 1 , 1 ) & ( E-max L~ g ) .. ( Rg ) = ( L~ g ) .. ( R ;
consider e being Element of NAT such that a = 2 * e + 1 ;
Re ( ( lim F ) | D ) = Re ( ( lim G ) | D ) ;
LIN b , a , c or LIN b , c , a ;
p `1 , a `2 // a `1 , b or p `2 , a `2 // b `2 , a `2 ;
g . n = a * Sum ( f | 1 ) .= f . n ;
consider f being Subset of X such that e = f and f is being being being set ;
F | ( N2 ~ ) = CircleMap * ( F | ( N2 ~ ) ) .= ( F | ( N2 ~ ) ) ;
q in LSeg ( q , v ) \/ LSeg ( v , p ) ;
Ball ( m , r0 ) c= Ball ( m , s ) & Ball ( m , r ) c= Ball ( m , r ) ;
the carrier of (0). V = { 0. V } .= { 0. V } ;
rng ( cos | [. - 1 , 1 .] ) = [. - 1 , 1 .] .= [. - 1 , 1 .] ;
assume that Re ( seq ) is summable and Im ( seq ) is summable ;
||. ( vseq . n ) - ( vseq . n ) .|| < e / ( ||. x .|| + e / ( ||. y .|| + e ) ) ;
set g = O --> 1 ;
reconsider t2 = t11 , t2 = 0 as string of S2 , I = the carrier of S2 ;
reconsider x-29 = seq , xseq = seq as sequence of REAL n ;
assume that C meets L~ Cage ( C , n ) and L~ Cage ( C , n ) meets L~ Cage ( C , n ) ;
- ( ( - 1 ) / ( 1 - r ) ) < F . n - ( - 1 ) / ( 1 - r ) ;
set d1 = dist ( x1 , z1 ) , d2 = dist ( x2 , z2 ) , d2 = dist ( x2 , z2 ) ;
2 |^ ( 100 -' 1 ) = 2 |^ ( 100 - 1 ) ;
dom vF2 = Seg ( len d6 ) .= dom d6 .= dom ( d7 ^ d8 ) ;
set x1 = - k2 + |. k2 .| + 4 , x2 = - k2 + 4 ;
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 ( LX + L2 ) ) c= I2 & the carrier of ( Carrier ( LX + 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 normal \hbox { - } \rm \mathbb N } ;
Z c= dom ( ( sin * ( f1 + f2 ) ) `| Z ) ;
|. 0. TOP-REAL 2 - q .| < r / 2 + |. q - p .| ;
o c= ConsecutiveSet2 ( A , succ ( d , B ) ) & ( for B st B in A holds d . B = F ( B ) ) ;
E = dom ( L (#) G ) & L (#) G is_measurable_on E & ( L (#) G ) | E is_measurable_on E ;
C / ( A + B ) = C / B * C ;
the carrier of W2 c= the carrier of V & the carrier of W1 c= the carrier of V implies W1 + W2 = W2
I . IC Comput ( P , s , k ) = P . IC Comput ( P , s , k ) .= ( card I + 1 ) ;
pred x > 0 means : Def1 : 1 / x = x / ( - 1 ) ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) .= LSeg ( f , k ) ;
consider p being Point of T such that C = [: [. p , q .] , p :] ;
b , c are_connected & - C , - C + - C + D + D + D + E + F + G + D + E + F + D + E + F + J + M + E + F + J + M + D + E + F + F + D + D +
assume f = id ( the carrier of O1 ) & g = id ( the carrier of O1 ) ;
consider v such that v <> 0. V and f . v = L * v ;
let l be Linear_Combination of {} ( ( the carrier of V ) --> 0. V ) ;
reconsider g = f " as Function of U2 , U1 , ( the carrier of U1 ) --> ( the carrier of U2 ) ;
A1 in the points of G_ ( k , X ) & A2 in the carrier of G_ ( k , X ) ;
|. - x .| = - ( - x ) .= - ( - x ) .= - x ;
set S = ) ( x , y , c ) ;
Fib n * ( 5 * Fib ( n ) - 1 ) >= 4 * \/ / 5 ;
vM /. ( k + 1 ) = vM . ( k + 1 ) .= vM . ( k + 1 ) ;
0 mod i = - ( i * ( 0 qua Nat ) ) .= - ( i * ( 0 qua Nat ) ) ;
Indices M1 = [: Seg n , Seg n :] & Indices M1 = [: Seg n , Seg n :] ;
Line ( St , j ) = St . j .= ( S . j ) * ( Line ( St , j ) ) ;
h . ( x1 , y1 ) = [ y1 , x1 ] & h . ( y1 , y2 ) = [ y1 , x2 ] ;
|. f - Re ( |. f .| * ( card b ) ) .| is nonnegative ;
assume that x = ( a1 ^ <* x1 *> ) ^ b1 and y = ( a1 ^ <* x2 *> ) ^ b2 ;
ME is_closed_on IExec ( I , P , s ) , P & ME is_halting_on IExec ( I , P , s ) , P ;
DataLoc ( t3 . a , 4 ) = intpos ( 0 + 4 ) .= intpos ( 0 + 4 ) ;
x + y < - x + y & |. x .| = - x + y & |. y .| = - y ;
LIN c , q , b & LIN c , q , c & LIN c , q , b ;
f| ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + z1 ) .= y1 + ( z1 + z2 ) ;
f, , f . a = f\in . a & v in InputVertices S & [ v , f . a ] in InputVertices S ;
p `1 <= ( E-max C ) `1 & ( E-max C ) `2 <= ( E-max C ) `2 ;
set R8 = Cage ( C , n ) \circlearrowleft E8 , R8 = Cage ( C , n ) ;
p `1 >= ( E-max C ) `1 & ( E-max C ) `2 <= ( E-max C ) `2 ;
consider p such that p = p-20 and s1 < p & p <= s2 ;
|. ( f /* ( s * F ) ) . l - G . l .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len Line ( N , k + 1 + 1 ) = width N .= width N ;
f1 /* s1 is convergent & f2 /* s1 is convergent & lim ( f1 /* s1 ) = x0 ;
f . x1 = x1 & f . y1 = y1 & f . y2 = y2 & f . y2 = y2 ;
len f <= len f + 1 & len f + 1 <> 0 & len f + 1 <> 0 ;
dom ( Proj ( i , n ) * s ) = REAL m & rng ( Proj ( i , n ) * s ) c= REAL n ;
n = k * ( 2 * t ) + ( n mod ( 2 * k ) ) ;
dom B = 2 -tuples_on the carrier of V & rng B = the carrier of V & rng B = 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 <= 1 ;
for L being complete LATTICE holds <* <* \mathclose { L } , L *> *> , L are_isomorphic
[ gi , gj ] in [: I , I :] \ ( I \ { j } ) ;
set S2 = 1GateCircStr ( x , y , c ) , S2 = 1GateCircStr ( y , c , d ) ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 ;
reconsider y = ( a ` ) ` , z = ( a ` ) ` , t = ( a ` ) ` as Element of L ;
dom s = { 1 , 2 , 3 } & s . 1 = d1 & s . 2 = d2 ;
( min ( g , ( 1 - g ) (#) f ) . c ) <= h . c ;
set G3 = the u of G , v = the Vertex of G , w = the Vertex of G , e = the Element of G ;
reconsider g = f as PartFunc of REAL , REAL-NS n , REAL-NS n ;
|. s1 . m / p .| / |. p .| < d / p / p ;
for x being element st x in consider ( for u being element st u in ( ( the carrier of R ) \ { 0 } ) holds x in ( the carrier of R )
P = the carrier of ( TOP-REAL n ) | P & Q = the carrier of ( TOP-REAL n ) | P ;
assume that p01 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) and LSeg ( p1 , p2 ) /\ LSeg ( p01 , p2 ) = { p1 } ;
( 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 ;
let f , g , h be Point of the carrier of X , g be Point of Y ;
set h = Hom ( a , g (*) f ) ;
then idseq ( n ) | ( Seg m ) = idseq ( m ) | ( Seg m ) & m <= n ;
H * ( g " * a ) in the right of H & I * ( g " * a ) in the carrier of H ;
x in dom ( cos / sin ^2 ) & ( cos / sin ^2 ) . x = ( cos . x ) ^2 / ( cos . x ) ^2 ;
cell ( G , i1 , j2 -' 1 ) misses C & cell ( G , i1 , j2 -' 1 ) misses C ;
LE q2 , p4 , P , p1 , p2 & LE q1 , q2 , P , p1 , p2 ;
pred B is an component of A means : Def1 : B c= BDD A & B c= A ;
deffunc D ( set , set ) = union rng $2 & $2 = union rng $2 ;
n + - n < len ( p + - n ) + - n & n + - n < len p + ( - n ) ;
pred a <> 0. K means : Def1 : the_rank_of M = the_rank_of ( a * M ) ;
consider j such that j in dom /\ /\ dom _ $ I and I = len _ { w + j : j in dom } ;
consider x1 such that z in x1 and x1 in P8 and x = [ x1 , x2 ] ;
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 / 4 * PI , cv = 2 * PI , cv = 3 / 4 * PI , cv = 4 * PI ;
conv @ W c= union ( F .: ( E " W ) ) & conv @ W c= union ( F .: ( E " W ) ) ;
1 in [. - 1 , 1 .] /\ dom ( arccot * arccot ) & 1 in dom ( arccot * arccot ) ;
r3 <= s0 + ( r0 - ( |. v2 - v1 .| ) / ( |. v2 - v1 .| ) ) ;
dom ( f (#) f4 ) = dom f /\ dom f4 .= dom ( f (#) f4 ) /\ dom ( g (#) f4 ) ;
dom ( f (#) G ) = dom ( l (#) F ) /\ Seg k .= Seg k ;
rng ( s ^\ k ) c= dom f1 \ { x0 } & f1 . x0 in dom f2 \ f2 . x0 ;
reconsider gg = gp , gq = gq , gq = gq as Point of TOP-REAL n ;
( T * h . s ) . x = T . ( h . s . x ) ;
I . ( L . ( J . x ) ) = ( I * L ) . ( J . x ) ;
y in dom ( *> <* *> ) implies commute ( Frege ( ( Frege ( A . o ) ) . o ) ) = ( ( Frege ( A . o ) ) . y ) . y
for I being non degenerated commutative commutative Ring holds the carrier of I is commutative commutative commutative commutative non empty doubleLoopStr
set s2 = s +* Initialize ( ( intloc 0 ) .--> 1 ) , P2 = P +* I +* J ;
P1 /. IC s1 = P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 ;
lim S1 in the carrier of [. a , b .] & lim S1 in the carrier of [. a , b .] ;
v . ( l-13 . i ) = ( v *' l.| ) . i .= ( v *' l.| ) . i ;
consider n being element such that n in NAT and x = ( sn " ) . n ;
consider x being Element of c such that F1 . x <> F2 . x and x in F2 . x ;
Funcs ( X , 0 , x1 , x2 , x3 ) = { E } & card ( X \ { x1 , x2 , x3 } ) = 1 ;
j + ( 2 * kk ) + m1 > j + ( 2 * kk ) + ( 2 * kk ) ;
{ s , t } on A3 & { s , t } on B3 & { s , t } on B3 ;
n1 > len crossover ( p2 , p1 , p2 , n1 , n2 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n3
mg . HT ( mg , T ) = 0. L & g . HT ( mg , T ) = 0. L ;
then H1 , H2 are_) & card ( H1 , H2 ) , ( H2 ) / ( 2 |^ n ) are_relative_prime ;
( N-min L~ f ) .. ( ( f /. len f ) .. ( f /. 1 ) ) + ( ( f /. len f ) .. ( f /. 1 ) .. ( f /. 1 ) ) .. ( f /. 1 ) .. ( f /. 1 ) .. ( f /. 1 ) .. ( f /. 1 ) .. ( f /. 1 )
]. s , 1 .[ = ]. s , 2 .] /\ [. 0 , 1 .] ;
x1 in [#] ( ( TOP-REAL 2 ) | L~ g ) & x2 in ( ( TOP-REAL 2 ) | L~ g ) . x2 ;
let f1 , f2 be continuous PartFunc of REAL , the carrier of S , x be Point of S ;
DigA ( t-23 , z9 ) is Element of k -tuples_on ( k -tuples_on ( k + 1 ) ) ;
I is Element of NAT & I is Element of NAT & I is Element of NAT & I is Element of NAT & I is Element of NAT ;
[: u , { u9 } :] = { [ a , u9 ] } .= [: { a } , { b } :] ;
( w | p ) | ( p | ( w | w ) ) = p ;
consider u2 such that u2 in W2 and x = v + u2 and u2 in W2 and v = v + u2 ;
for y st y in rng F ex n st y = a |^ n & n <= len F ;
dom ( ( g * f ) . ( f . K ) | K ) = K ;
ex x being element st x in ( ( the Sorts of U0 ) \/ A ) . s & x in ( the Sorts of U0 ) . s ;
ex x being element st x in ( the Sorts of O1 ) . s \/ A . s ;
f . x in the carrier of [. - r , r .] & f . x in [. - r , r .] ;
( the carrier of X1 union X2 ) /\ ( the carrier of X1 union X2 ) <> {} & ( the carrier of X1 union X2 ) /\ ( the carrier of X2 ) <> {} ;
L1 /\ LSeg ( p01 , p2 ) c= { p01 } /\ LSeg ( p01 , p2 ) ;
( b + bs0 ) / 2 in { r : a < r & r < b } ;
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 GX such that z = y and P [ z ] and z in A ;
( the sequence of ( ( the carrier of X ) --> ( the carrier of X ) ) ) . ( TOP-REAL n ) <= e ;
len ( w ^ w2 ) + 1 = len w + 2 + 1 .= len w + 1 + 1 .= len w + 1 ;
assume q in the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 & q in the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 ;
f | E-4 ` = g | E-4 ` & g | E11 = g | E11 ` & g | E11 = g | E11 ` ;
reconsider i1 = x1 , i2 = x2 , j1 = x3 , j2 = x4 as Element of NAT ;
( a * A * B ) @ = ( a * ( A * B ) ) @ ;
assume ex n0 being Element of NAT st f to_power n0 is
Seg len ( ( the thesis ) --> ( ( len f2 ) --> ( len f1 ) ) ) = dom ( ( the support of K ) --> ( len f1 ) ) ;
( Complement ( A . m ) ) . n c= ( Complement ( A . n ) ) . ( ( Complement ( A . m ) ) . n ) ;
f1 . p = p9 & g1 . p = q9 & g2 . p = d & g1 . p = q9 & g2 . p = g2 . p ;
FinS ( F , Y ) = FinS ( F , dom ( F | Y ) ) .= FinS ( F , Y ) ;
( x | y ) | z = z | ( y | x ) ;
( |. x .| |^ n ) / ( n + 1 ) <= ( ( r2 |^ n ) / ( n + 1 ) ) ;
Sum F-12 = Sum f & dom F-12 = dom g & for x st x in dom F-12 holds F-12 . x = g . x ;
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 /\ W2 is Subspace of W3 and W1 /\ W2 = W2 /\ W3 ;
||. ( t-15 . x ) .|| = lim ||. ( x - y ) .|| .= ||. ( x - y ) .|| .= ||. x - y .|| ;
assume that i in dom D and f | A is lower and g | A is lower and g | A is lower ;
( ( p `2 ) ^2 - 1 ) / ( 1 + ( - 1 ) ) <= ( ( - 1 ) / ( 1 + ( p `2 / p `1 ) ^2 ) ) / ( 1 + ( p `2 / p `1 ) ^2 ) ;
g | Sphere ( p , r ) = id ( Sphere ( p , r ) ) .= id ( Sphere ( p , r ) ) ;
set N8 = ( N-min L~ Cage ( C , n ) ) .. Cage ( C , n ) ;
for T being non empty TopSpace holds T is countable countable implies the TopStruct of T is countable
width B |-> 0. K = Line ( B , i ) .= B * ( i , j ) .= B * ( i , j ) ;
pred a <> 0 means : Def1 : ( A \+\ B ) Y. = ( A Y. ) \diffsym ( B f2 ) ;
then f is_\cal 2 , u & pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 & 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 a > 0 ;
w1 , w2 in Lin { w1 , w2 } & w2 in Lin { w1 , w2 } ;
p2 /. IC Comput ( p2 , s , k ) = p2 . IC Comput ( p2 , s , k ) .= ( I . IC Comput ( p2 , s , k ) ) ;
ind ( T-10 | b ) = ind b .= ind B .= ind ( T-10 | b ) .= ind b ;
[ a , A ] in the carrier of G_ ( Al , A ) & [ a , A ] in the carrier of G_ ( Al , A ) ;
m in ( the Arrows of C ) . ( o1 , o2 ) & ( the Arrows of C ) . ( o1 , o2 ) = ( the Arrows of C ) . ( o2 , o1 ) ;
( a , CompF ( PA , G ) ) . z = FALSE & ( a , CompF ( PA , G ) ) . z = FALSE ;
reconsider phi = phi , phi = phi , phi = phi , phi = phi as Element of ( S , D ) * ;
len s1 - 1 * ( len s2 - 1 ) + 1 > 0 + 1 * ( len s2 - 1 ) ;
delta ( D ) * ( f . sup A ) - lower_bound ( rng f ) < r ;
[ f21 , f22 ] in the carrier' of A & [ f21 , f22 ] in the carrier' of B ;
the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 = K1 & the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 = K1 ;
consider z being element such that z in dom g2 and p = g2 . z and q = g2 . z ;
[#] ( V1 + V2 ) = { 0. V } .= the carrier of ( (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 for k being Nat st k in dom P2 holds P2 . k = P2 . k ;
assume that x1 in dom ( f | X ) and ||. x1 - x0 .|| < s and |. f /. x1 - f /. x0 .| < s ;
h1 = f ^ ( <* p3 *> ^ <* p *> ) .= h ^ <* p *> .= h ^ <* p *> ;
c /. ( |[ b , c ]| ) = c /. c .= |[ a , c ]| .= |[ a , c ]| ;
reconsider t1 = p1 , t2 = p2 , t2 = p3 as Term of C , V ;
( 1 / 2 ) * ( 1 / 2 ) in the carrier of [. 1 / 2 , 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 .= ( h . p1 ) `2 + D ;
R . b - R . b = 2 * - b .= 2 * - b .= - b ;
consider \hbox such that B = 1- 1 * ] + ( - 1 ) * A and 0 <= 1 ;
dom g = dom ( ( the Sorts of A ) * ( a , b ) ) .= the carrier of S ;
[ P . ( l ) , P . ( l + 1 ) ] in => ( ( P . ( l + 1 ) ) . ( k + 1 ) ) ;
set s2 = Initialize s , P2 = P +* I ;
reconsider M = mid ( z , i2 , i1 ) , N = L~ z as non empty Subset of ( TOP-REAL 2 ) | ( L~ z ) ;
y in product ( ( Carrier J ) +* ( V , { 1 } ) ) & 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 & y in the carrier of g & x = g . x ;
consider M being strict Subgroup of AJ such that a = M and T is Subgroup of M and the carrier of M = the carrier of J ;
for x st x in Z holds ( ( ( #Z n ) * f ) + ( #Z n ) * f ) `| Z ) . x <> 0
len W1 + len W2 + m = 1 + len W3 + len W3 + m .= len W1 + len W2 + m .= len W1 + len W2 + m ;
reconsider h1 = ( vseq . n ) - t-16 as Lipschitzian LinearOperator of X , Y ;
( - ( len p + q ) mod ( len p + q ) ) in dom ( p + q ) ;
assume that s2 is_\emptyset and F in the Element of s2 and not F in the Element of s2 and not F in the Element of s2 ;
( ( ( ( ( ex x , y st x in Y ) ) * ( x , 3 ) ) + ( ( - 1 ) * ( x , 2 ) ) ) / ( 2 * ( x , 3 ) ) ) = gcd ( x , y , 3 ) ;
for u being element st u in Bags n holds ( p `2 + m ) . u = p . u + m . u
for B be Subset of u-5 st B in E holds A = B or A misses B or A = B ;
ex a being Point of X st a in A & A /\ Cl { y } = { a } ;
set W2 = tree p \/ W1 , W1 = tree p , W2 = tree q ;
x in { X where X is Ideal of L : X is non empty Subset of L & X is non empty Subset of L } ;
the carrier of W1 /\ W2 c= the carrier of W1 & the carrier of W1 /\ W2 c= the carrier of W1 /\ W2 implies W1 + W2 = W2
( 1 / a + b ) * id a = ( 1 / a + b ) * id a .= ( 1 / a + b ) * id a ;
( ( X --> f ) . x ) . x = ( X --> dom f ) . x .= f . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) , y = the Element of LSeg ( g , m ) /\ LSeg ( g , n ) ;
p => ( q => r ) => ( p => q ) in TAUT ( A ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
- 1 + 1 <= ( ( ( i - 2 ) |^ ( n -' m ) ) + 1 ;
( reproj ( 1 , z0 ) ) . x in dom ( f1 (#) f2 ) /\ dom ( f2 (#) f3 ) ;
assume that b1 . r = { c1 } and b2 . r = { c2 } and b1 . r = { c2 . r } ;
ex P st a1 on P & a2 on P & b on P & c on P & c on P ;
reconsider gf = g `1 * f `2 , hg = h `2 * g `2 as strict Element of X ;
consider v1 being Element of T such that Q = ( downarrow v1 ) ` and ( downarrow v1 ) ` = ( downarrow v1 ) ` ;
n in { i where i is Nat : i < n0 + 1 & i <= n + 1 } ;
( F * ( i , j ) ) `2 >= ( F * ( m , k ) ) `2 ;
assume K1 = { p : p `1 / |. p .| >= cn & p `2 >= 0 & p <> 0. TOP-REAL 2 } ;
ConsecutiveSet ( A , succ O1 ) = ( ConsecutiveSet ( A , O1 ) ) . ( ( succ O1 ) . ( ( succ O1 ) . ( ( succ O1 ) . ( ( succ O1 ) . ( ( succ O1 ) . ( ( succ O1 ) . ( ( succ O1 ) . ( succ O1 ) ) ) ) ) ) ;
set IW1 = in dom SubFrom ( a , intloc 0 ) , IW2 = SubFrom ( a , intloc 0 ) , IW2 = SubFrom ( a , intloc 0 ) , IW2 = SubFrom ( a , 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 ) & ( the carrier of L1 ) \/ ( the carrier of L2 ) c= the carrier of L1 ;
consider x9 being Element of GF ( p ) such that x9 |^ 2 = a & x9 |^ 2 = b ;
reconsider eX = eX , fX = fX , fY = fY as Element of D ;
ex O being set st O in S & C1 c= O & M . O = 0. ( \overline { A } ) ;
consider n being Nat such that for m being Nat st n <= m holds S . m in U1 and S . n in U2 ;
f * g * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) . x ) . x0 ;
defpred P [ Nat ] means A + succ $1 = succ A & ( A + ) = ( A + ) + ( B + $1 ) ;
the left of - g = the left of g & the left of - g = the left of g & the carrier of - g = the carrier of g ;
reconsider p\mathopen = x , p\mathopen = y , p\mathopen = z as Point of TOP-REAL 2 , a , b , c , d ;
consider g3 such that g3 = y and x <= ex g2 st g2 <= x & g2 <= x0 & x0 <= g2 & 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 ( y1 ^ y2 ) ;
for x being element st x in X holds x in the set of Real & x in the set of ( the set of 0 ) | X & x in the carrier of ( 0 -tuples_on REAL )
LSeg ( p01 , p2 ) /\ LSeg ( p1 , p2 ) = {} & LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = {} ;
func that ) of ) ( X ) -> set equals [: [: X , X :] , [: X , X :] :] ;
len ( ( { ( C /. len C ) *> ^ ( C /^ 1 ) ) ) <= len ( C | 1 ) + len ( C /^ 1 ) ;
pred K is has a , a & a <> 0. K & v . ( a |^ i ) = i * v . a ;
consider o being OperSymbol of S such that t `1 . {} = [ o , the carrier of S ] and t . {} = [ o , the carrier of S ] ;
for x st x in X ex y st x c= y & y in X & y is NAT & f . x is a + 1 / ( f . x )
IC Comput ( P-6 , sseq , k ) in dom ( Pseq ( ) ) & IC Comput ( Pseq , sseq , k ) in dom ( Pseq ( ) ) ;
pred q < s means : Def1 : r < s & s <= q & s <= q ;
consider c being Element of Class ( f , c ) such that Y = ( F . c ) `1 and ( F . c ) `2 = 3 ;
func the ResultSort of S2 -> Function of the carrier of S2 , the carrier of S2 means : Def1 : the ResultSort of it = the carrier of S2 & the ResultSort of it = the ResultSort of S2 ;
set y9 = [ <* y , z *> , f2 ] , z9 = [ <* z , x *> , f3 ] ;
assume x in dom ( ( ( #Z n ) * ( arccot ) ) `| Z ) & x in dom ( ( #Z n ) * ( arccot ) ) ;
r-7 in Int cell ( GoB f , i , GoB f ) \ L~ f & r-7 in cell ( GoB f , i , width GoB f ) \ L~ f ;
q `2 >= ( Cage ( C , n ) /. ( i + 1 ) ) `2 & ( Cage ( C , n ) /. ( i + 1 ) ) `2 >= ( Cage ( C , n ) ) `2 ;
set Y = { a "/\" a ` : a in X } ;
i -' len f <= len f + len f -' len f & i + 1 <= len f + len f - len f ;
for n ex x st x in N & x in N1 & h . n = x- ( x0 - h . n )
set s0 = ( \mathop { a , I , p , s ) . i , s1 = ( \mathop { a , I , p , s ) . i , s1 = ( \mathop { a , I , p , s ) . i , s1 = ( \mathop { a , I , p , s ) . i , s1 = ( \mathop { a , I , p , s )
p ( k ) . 0 = 1 or p ( k ) . 0 = - 1 or p ( k ) . 0 = - 1 & p ( k ) . 0 = - 1 ;
u + Sum ( L-18 ) in ( U \ { u } ) \/ { u + Sum ( L-18 ) } ;
consider x9 being set such that x in x9 and x9 in V1 and f . x = [ x9 , the carrier of K ] ;
( p ^ ( q | k ) ) . m = ( q | k ) . ( len p ) .= p . ( len p ) ;
g + h = gg + h1 & A1 + ( g + h ) = g + h & A2 + h = g + h ;
L1 is distributive & L2 is distributive implies [: L1 , L2 :] is distributive & the carrier of L1 = the carrier of L2 & the carrier of L1 = the carrier of L2 & the carrier of L1 = the carrier of L2
pred x in rng f & y in rng ( f | x ) means : Def1 : y in rng ( f | x ) & f . x = f . y ;
assume that 1 < p and ( 1 - p ) * q + ( 1 - p ) * q = 1 and 0 <= a and a <= b ;
F* ( f , sA1 ) = rpoly ( 1 , M ) *' t + 1. F_Complex .= 0. F_Complex + 0. F_Complex .= 0. F_Complex ;
for X being set , A being Subset of X holds A ` = {} implies A = {} & A = {} & A = {} & A = {}
( N-min X ) `1 <= ( ( N-min X ) `1 ) / ( 1 + ( ( ( ( ( ( the carrier of X ) ) / ( 1 + ( the carrier of X ) ) / ( 1 + ( the carrier of X ) ) ) ) / ( 1 + ( the carrier of X ) ) ) ) ;
for c being Element of the *> of the function of A , a being Element of the \subseteq the free of A holds c <> a
s1 . GBP = ( Exec ( i2 , s2 ) ) . GBP .= s2 . GBP .= Exec ( i2 , s2 ) . GBP .= s2 . GBP .= Exec ( i2 , s2 ) . GBP .= s2 . GBP ;
for a , b being Real holds |[ a , b ]| in ( y >= 0 ) & b >= 0 implies b >= 0
for x , y being Element of X holds x ` \ y = ( x \ y ) ` & y \ x = ( x \ y ) `
mode BCK-algebra of i , j , m , n , m , m , n , m , m , n ;
set x2 = |( Re ( y - Im ( x - Im y ) ) , Im ( y - Im ( x - Im y ) ) )| ;
[ y , x ] in dom u5 & u5 . ( y , x ) = g . y & [ u , x ] in dom g ;
]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .] c= A & upper_bound divset ( D , k ) = upper_bound divset ( D , k ) ;
0 <= delta ( S2 ) . n & |. delta ( S2 ) . n - 0 .| < ( e / 2 ) / 2 ;
( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) <= ( - ( q `2 / |. q .| - cn ) ) / ( 1 + cn ) ;
set A = 2 / b-a ;
for x , y being set st x in R" holds x , y are_\hbox { - } x , the carrier of S
deffunc FF2 ( Nat ) = b . ( $1 + 1 ) * ( M * G ) . ( $1 + 1 ) ;
for s being element holds s in -> Element of S iff s in -> Element of \rm : _ 0 ( f , g ) \/ _ _ 0 ( f , g )
for S being non empty non void holds S is connected holds S is connected iff S is connected
max ( degree ( ( z `1 ) ^2 + ( z `2 ) ^2 ) , degree ( ( z `2 ) ^2 + ( z `2 ) ^2 ) ) >= 0 ;
consider n1 being Nat such that for k holds seq . ( n1 + k ) < r + s and seq . ( n1 + k ) < r + s ;
Lin ( A /\ B ) is Subspace of Lin ( A /\ B ) & Lin ( A /\ B ) is Subspace of Lin ( B )
set n-15 = n-13 '&' ( M . x qua Element of ( the carrier of X ) --> TRUE ) , n-15 = ( M . x ) --> TRUE ;
f " V in ' ( X ) & f " V in D & f " V in D & f " V in D & f " V in D ;
rng ( ( a ^\ c ) +* ( 1 , b ) ) c= { a , c , b } \/ { b } ;
consider y being set such that y `1 = y and dom y `1 = WWG and rng y `2 = WWG and y `2 = WWG ;
dom ( 1 / f ) /\ ]. - 1 , 0 .[ c= ]. - 1 , 1 .[ & dom ( 1 / f ) /\ ]. - 1 , 1 .[ c= dom ( 1 / f ) ;
as Morphism of j , j , n , r be Element of j , ( - r ) * ( - r ) ;
v ^ ( ( n |-> 0 ) ^ ( ( n |-> 0 ) ^ ( ( n |-> 0 ) ^ ( ( n |-> 0 ) ^ ( ( n |-> 0 ) ^ ( ( n |-> 0 ) ^ ( n |-> 0 ) ) ) ) ) ) in Lin ( rng ( ( n |-> 0 ) ^ ( n |-> 0 ) ) ) ;
ex a , k1 , k2 st i = a := k1 & j = b := k2 & k2 = b := k2 & k2 = b := k2 ;
t . NAT = ( NAT .--> succ i1 ) . NAT .= succ i1 .= succ i1 .= succ i1 .= ( NAT --> succ i1 ) . NAT .= ( NAT --> succ i1 ) . NAT .= ( NAT --> succ i1 ) . NAT ;
assume that F is bbfamily and rng p = F and dom p = Seg ( n + 1 ) and for i being Nat st i in Seg ( n + 1 ) holds p . i = F . i ;
not LIN b , b9 , a & not LIN a , a9 , c & LIN b9 , a9 , c & LIN b9 , a9 , a & LIN b9 , a9 , c
( L1 \rangle ) \& O c= ( L1 => L2 ) \& O & ( L2 = L1 => L2 ) Let O ;
consider F being ManySortedSet of E such that for d being Element of E holds F . d = F ( d ) and for d being Element of E holds F . d = G ( d ) ;
consider a , b such that a * ( 0. V ) = b * ( -w ) and 0 < a and 0 < b ;
defpred P [ FinSequence of D ] means |. Sum ( $1 ) .| <= Sum ( |. $1 .| ) ;
u = cos / ( x , y ) . v * x + ( cos / ( x , y ) . v ) * y .= v ;
dist ( ( seq . n ) + x , g + x ) <= dist ( ( seq . n ) , g ) + 0 ;
P [ p , |. p .| \bullet |. p .| , {} ( the Sorts of A ) . p , id ( the Sorts of A ) . p ] ;
consider X being Subset of CQC-WFF ( Al ) such that X c= Y and X is finite and X is non empty and innot X in X ;
|. b .| * |. eval ( f , z ) .| >= |. b .| * |. eval ( f , z ) .| ;
1 < ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) + 1 & ( W-min L~ Cage ( C , n ) ) .. Cage ( C , n ) <= ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ;
l in { l1 where l1 is Real : g <= l1 & l1 <= h & h <= g } ;
vol ( ( G . n ) vol ) <= vol ( ( G . n ) vol ) + vol ( ( G . n ) vol ) ;
f . y = x .= x * 1. L .= x * power L .= x * ( power L ) .= x * ( 1. L ) ;
NIC ( <% i1 , i2 %> , ( the carrier of SCM+FSA ) --> ( the carrier of SCM+FSA ) ) = { i1 , succ i2 } .= { i1 , succ i2 } .= { i1 , i2 } ;
LSeg ( p01 , p2 ) /\ LSeg ( p1 , p01 ) = { p1 } /\ LSeg ( p1 , p2 ) .= { p1 } ;
Product ( ( the carrier of I-15 ) +* ( i , { 1 } ) ) in [: Z , Z :] ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) .= Following ( s1 , n ) .= Exec ( i , s1 ) ;
W-bound ( Q `1 ) <= ( q `1 ) / ( |. q .| ) & ( |. q .| ) <= ( |. q .| ) / ( |. q .| ) ;
f /. i2 <> f /. ( ( len f + len g -' 1 ) + len g ) & f /. ( len f + 1 ) = f /. ( len f + 1 ) ;
M , f / ( x. 3 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) |= H ;
len ( ( P ^ ) ^ ( Q ^ ) ) in dom ( ( P ^ ) ^ ( Q ^ ) ) & len ( ( P ^ ) ) + ( Q ^ ) = len ( Q ^ ) + len ( Q ^ ) ;
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 \not 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 |. ( id the carrier of V ) . v .| & ||. v .|| = |. ( id the carrier of V ) . v .|
for phi holds phi in X implies phi in X & ( phi in X & phi in X ) & ( phi in X & phi in X )
rng ( Sgm dom ( f | dom ( f | dom ( f | dom ( f | dom q ) ) ) ) ) c= dom ( f | dom ( f | dom q ) ) ;
ex c being FinSequence of D ( ) st len c = k & P [ c ] & a = c & c = d ;
the_arity_of ( a , b , c ) = <* Hom ( b , c ) , Hom ( a , b ) *> .= <* Hom ( a , b ) , Hom ( b , c ) *> ;
consider f1 being Function of the carrier of X , REAL such that f1 = |. f .| and f1 is continuous and for x being Point of X holds f1 . x = f . x ;
a1 = b1 & a2 = b2 or a1 = b2 & a2 = b2 & b1 = b2 & b2 = b3 or b1 = b3 & b2 = b3 or b1 = b2 & b2 = b3 ;
D2 . indx ( D2 , D1 , n1 + 1 ) = D1 . ( n1 + 1 ) .= D2 . ( n1 + 1 ) .= D2 . ( n1 + 1 ) ;
f . ( ||. r .|| ) = ||. ( r (#) f ) /. 1 .|| .= <* r * f /. 1 *> .= r * f /. 1 .= r * f /. 1 .= r * f /. 1 ;
consider n being Nat such that for m being Nat st n <= m holds C-25 . m = C-25 . m and C-25 . n = C-25 . m ;
consider d being Real such that for a , b being Real st a in X & b in Y holds a <= d and d <= b ;
||. L /. h - ( K * |. h .| ) + ( K * |. h .| ) .|| <= p0 + ( K * |. h .| ) ;
attr F is commutative associative means : Def1 : for b being Element of X holds F -\hbox { b } f = f . b ;
p = - ( - p0 + 0. TOP-REAL 2 ) + 0. TOP-REAL 2 .= 1 * ( p0 `1 ) + 0. TOP-REAL 2 .= ( - 1 ) * ( p0 `1 ) + 0. TOP-REAL 2 .= ( - 1 ) * ( p0 `1 ) + 0. TOP-REAL 2 .= ( - 1 ) * ( p0 `1 ) .= ( - 1 ) * ( p0 `1 ) .= ( - 1 ) * ( p0 `1 ) ;
consider z1 such that b `1 , x3 , z1 is_collinear and o , x1 , z1 is_collinear and o <> z1 and o <> z1 and o <> z1 and o <> z1 ;
consider i such that Arg ( ( Rotate ( s , r ) ) . q ) = s + Arg q + ( 2 * PI * i ) ;
consider g such that g is one-to-one and dom g = card f . x and rng g = f . x and g . x = f . x ;
assume that A = P2 \/ Q2 and P2 <> {} and Q2 <> {} and P2 <> {} and P2 <> {} and Q2 <> {} and P2 <> {} and Q2 <> {} and P2 <> {} ;
attr F is associative means : Def1 : F .: ( f , g ) = F .: ( f , g ) & F .: ( g , h ) = F .: ( f , g ) ;
ex x being Element of NAT st m = x `1 & x in z `2 & x in { i } or m in { i } & m in { i } ;
consider k2 being Nat such that k2 in dom P-2 and l in P-2 . k2 and PW1 . k2 = PW1 . k2 ;
seq = r (#) seq implies for n holds seq . n = r * seq . n & seq . n = r * seq . n
F1 . [ ( id a ) , [ a , a ] ] = [ f * ( id a ) , [ a , b ] ] .= [ f , f ] ;
{ p } "\/" D2 = { p "\/" y where y is Element of L : y in D2 & p in D2 } & x in D1 "\/" D2 ;
consider z being element such that z in dom ( ( dom F ) . 0 ) and ( ( dom F ) . 0 ) . z = y ;
for x , y being element st x in dom f & y in dom f & f . x = f . y holds x = y
Int 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 and ( T | E1 ) . e = v ;
( F `1 * b1 ) . x = ( Mx2Tran ( J , BY. , Z , b1 ) ) . ( Y. /. j ) .= ( Mx2Tran J ) . j ;
- 1 / ( - 1 ) = mmD (#) D | n .= mmD (#) ( - 1 ) .= ( - 1_ K ) (#) ( - 1 ) .= ( - 1_ K ) (#) ( - 1 ) .= ( - 1_ K ) * ( - 1 ) .= ( - 1_ K ) ;
pred for x being set st x in dom f /\ dom g holds g . x <= f . x & g . x <= g . x ;
len ( f1 . j ) = len f2 /. j .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) ;
All ( 'not' a , A , G ) |= Ex ( 'not' All ( a , B , G ) , A , G ) ;
LSeg ( E . k0 , F . k0 ) c= Cl RightComp Cage ( C , k + 1 ) & LSeg ( E , k + 1 ) c= RightComp Cage ( C , k + 1 ) ;
x \ a |^ m = x \ ( a |^ k * a ) .= ( x \ a ) |^ k .= ( x \ a ) |^ k ;
k -inininininininininininin1 = ( commute ( I . k ) ) . ( i + 1 ) .= ( commute ( I . k ) ) . ( i + 1 ) .= ( ( commute I ) . ( i + 1 ) ) . ( i + 1 ) .= ( ( commute I ) . ( i + 1 ) ) . ( i + 1 ) ;
for s being State of A2 holds Following ( s , n ) . ( 0 + ( n + 2 ) * n + 1 ) is stable ;
for x st x in Z holds f1 . x = a ^2 & ( f1 - f2 ) . x <> 0 & ( f1 - f2 ) . x <> 0 ) implies f1 - f2 is_differentiable_on Z & for x st x in Z holds f1 . x = 1 / x - ( x ^2 )
support ( Sgm ( n ) \/ support ( Sgm ( n ) ) ) c= support max ( n , support ( m ) ) \/ support ( m ) ;
reconsider t = u as Function of ( the carrier of A ) \/ ( the carrier of B ) , the carrier' of C , f = ( the carrier of B ) \/ ( the carrier of C ) ;
- ( a * sqrt ( 1 + b ^2 ) ) <= - ( b * sqrt ( 1 + a ^2 ) ) ;
phi /. ( succ b1 ) = g . a & phi /. ( a . a ) = f . ( g . a ) & phi /. ( a . a ) = f . ( g . a ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( ( F ^ <* p *> ) . i ) and i = len ( F ^ <* p *> ) ;
{ x1 , x2 , x3 , x4 , x5 } = { x1 } \/ { x2 , x3 , x4 , x5 , x5 , x5 , x5 } \/ { x5 , x5 , x5 , x5 } .= { x1 } \/ { x2 , x3 , x4 , x5 } ;
the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U1 & the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U1 /\ ( U1 "\/" U2 ) ;
( - ( 2 * a * ( b - a ) ) + b ) / ( 2 * a * ( b - a ) ) > 0 ;
consider W00 such that for z being element holds z in W00 iff z in N & P [ z ] & P [ z ] ;
assume that ( the Arity of S ) . o = <* a *> and ( the Arity of S ) . o = r and ( the Arity of S ) . o = <* a *> and ( the Arity of S ) . o = a ;
Z = dom ( exp_R * ( arccot - arccot ) ) /\ dom ( ( arccot - arccot ) (#) ( arccot - arccot ) ) ;
sum ( f , SS1 ) is convergent & lim ( \HM { the carrier of S } ) = integral ( f , SS1 ) & lim ( f , SS1 ) = integral ( f , SS1 ) ;
( X . ( ( a . f ) => ( g . x ) ) => ( ( g . x ) => ( g . x ) ) ) in [: the carrier of l , the carrier of l :] ;
len ( M2 * M3 ) = n & width ( M2 * M3 ) = n & width ( M2 * M3 ) = n & width ( M2 * M1 ) = n ;
attr X1 union X2 is open SubSpace of X means : Def1 : X1 , X2 are_separated & ( X1 , X2 are_separated ) & ( X1 , X2 are_separated ) & ( X2 , X1 are_separated ) & ( X1 , X2 are_separated implies X2 , X1 are_separated ) ;
for L being upper-bounded antisymmetric RelStr , X being non empty Subset of L holds X "\/" { Top L } = { Top L }
reconsider f-129 = F1 . ( b `2 ) , f-129 = F2 . ( b `2 ) , f-129 = F2 . ( b `2 ) , f-129 = F2 . ( b `2 ) as Function of M , M ;
consider w being FinSequence of I such that the InitS of M = ( the InitS of M ) Carrier ( <* s *> ^ w ) and len w = len q ;
g . ( a |^ 0 ) = g . ( 1_ G ) .= 1_ H .= ( g . ( a |^ 0 ) ) |^ 0 .= ( g . ( a |^ 0 ) ) |^ 0 .= ( g . ( a |^ 0 ) ) |^ 0 .= ( g . ( a |^ 0 ) ) |^ 0 ;
assume for i be Nat st i in dom f ex z be Element of L st f . i = rpoly ( 1 , z ) & f . i = rpoly ( 1 , z ) ;
ex L being Subset of X st Carrier ( L ) = C & 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 C1 & ( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C1 & ( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C1 ;
reconsider o-21 = o `1 , oY = p `2 , oY = p `2 , oY = p `2 , oY = p `2 , oY = p `2 , oY = p `2 , oY = p `2 , oY = p `2 , oY = p `2 ;
1 * x1 + ( 0 * x2 ) + ( 0 * x3 ) = x1 + <* \underbrace { 0 , \dots , 0 } , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ) .= x1 + x2 + 0 ;
Ek " . 1 = ( Ek qua Function ) " . 1 .= ( ( 1 - k ) |^ 1 ) " .= ( ( 1 - k ) |^ 1 ) " .= ( ( 1 - k ) |^ 1 ) " .= ( ( 1 - k ) |^ 1 ) " ;
reconsider u1 = the carrier of U1 /\ ( U1 "\/" U2 ) , v1 = the carrier of U1 /\ ( U1 "\/" U2 ) as non empty Subset of U0 ;
( ( x "/\" z ) "\/" ( x "/\" y ) ) "\/" ( z "/\" y ) <= ( x "/\" ( z "\/" y ) ) "\/" ( z "/\" ( x "\/" y ) ) ;
|. f . ( s1 . ( l1 + 1 ) ) - f . ( s1 . l1 ) .| < ( 1 / |. M .| + 1 / ( |. M .| + 1 ) ) ;
LSeg ( ( Upper_Seq ( C , n ) ) /. ( i + 1 ) , ( Upper_Seq ( C , n ) ) /. ( i + 1 ) ) is vertical ;
( f | Z ) /. x - ( f | Z ) /. x0 = L /. ( x- x ) + R /. ( x- x0 ) ;
g . c * ( - g . c ) + f . c <= h . c * ( - g . c ) + f . c ;
( f + g ) | divset ( D , i ) = f | divset ( D , i ) + g | divset ( D , i ) .= f | divset ( D , i ) + g | divset ( D , i ) ;
assume that ColVec2Mx f in the set of K and ColVec2Mx f in the carrier of K and len ColVec2Mx b = width A and width ColVec2Mx f = width A and width ColVec2Mx f = width A and width ColVec2Mx f = width A and width ColVec2Mx f = width A ;
len ( - M1 ) = len M1 & width ( - M1 ) = width M1 & width ( - M1 ) = width M2 & width ( - M1 ) = width M1 ;
for n , i being Nat st i + 1 < n holds [ i , i + 1 ] in the InternalRel of ( the InternalRel of thesis ) \/ the InternalRel of ( the InternalRel of G )
pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1 ;
pred a <> 0 & b <> 0 & Arg a = Arg b & Arg ( - a ) = Arg ( - b ) & Arg ( - a ) = Arg ( - b ) ;
for c being set st not c in [. a , b .] holds not c in Intersection ( the open of a , b , c ) & not c in Intersection ( the InternalRel of a , b , c )
assume that V1 is linearly-independent and V2 is linearly-independent and V2 is closed and v in V1 and u in V1 and v in V1 and v in V2 and v in V2 and v + u in V2 ;
z * x1 + ( 1 - z ) * x2 in M & z * y1 + ( 1 - z ) * y2 in N ;
rng ( ( PS1 qua Function ) " * SS2 ) = Seg ( card dS2 ) .= Seg ( card dS2 ) .= dom ( ( PS2 ) " * ( PS2 ) ) .= dom ( ( PS2 ) " * ( PS2 ) ) ;
consider s2 being rational Real_Sequence such that s2 is convergent and b = lim s2 and for n holds s2 . n <= b . n and ( for n holds s2 . n <= b . n ) & ( for n holds s2 . n <= b . n ) ;
h2 " . n = h2 . n " & 0 < h2 . n & 0 < - ( 1 / ( ( 1 - ( ( 1 / ( n + 1 ) ) * ( 1 / ( n + 1 ) ) ) ) ) ;
( Partial_Sums ( ||. seq1 .|| ) ) . m = ||. ( seq1 . m ) .|| .= ||. ( seq . m ) .|| .= ||. ( seq . m ) .|| .= ( ||. seq .|| ) . m .= ( ||. seq .|| ) . m .= ( ||. seq .|| ) . m ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b ;
- v = ( - 1_ G ) * v & - w = ( - 1_ G ) * v & ( - w ) * v = ( - 1_ G ) * v & ( - w ) * v = 0. G ;
sup ( ( k .: D ) .: D ) = sup ( ( k .: D ) .: D ) .= k . sup D .= sup ( k .: D ) .= sup ( k .: D ) .= sup ( k .: D ) .= sup ( k .: D ) .= sup ( k .: D ) .= sup ( k .: D ) ;
A |^ ( k , l ) ^^ ( A |^ ( n , .. A ) ) = ( A |^ ( k , .. A ) ) ^^ ( A |^ ( k , .. A ) ) ;
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 = ( p `1 ) ^2 + sqrt ( 1 + ( p `2 / p `1 ) ^2 ) .= ( p `2 ) ^2 + ( p `2 / p `1 ) ^2 ;
for a , b being non zero Nat st a , b are_relative_prime holds ( for n being Nat holds support ( a * b ) = support ( a ) + support ( b ) ) & ( a * b ) = support ( a ) + support ( b )
consider A5 being countable set such that r is Element of CQC-WFF ( Al ) & A5 is ( len A5 ) -element & ( not ( ex n being Nat st n in dom A5 & n < len A5 ) & not ( n < len A5 ) ) ;
for X being non empty addLoopStr for M being Subset of X , x , y being Point of X st y in M holds x + y in M + M
{ [ x1 , x2 ] , [ y1 , y2 ] } c= [: { x1 } , { x2 } :] \/ [: { y1 } , { y2 } :] ;
h . ( f . O ) = |[ A * ( f . O ) + B , C * ( f . O ) + D ]| ;
( Upper_Seq ( C , n ) ) * ( k , i ) in L~ Upper_Seq ( C , n ) /\ L~ Upper_Seq ( C , n ) & ( Cage ( C , n ) ) * ( k , i ) in L~ Upper_Seq ( C , n ) ;
cluster m , n are_relative_prime means : Def1 : the carrier of it is prime & for p being prime Nat holds p divides m & p divides n & p divides n ;
( 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 \ a <= c holds a "\/" b <= c
consider b being element such that b in dom ( H / ( x , y ) ) and z = ( H / ( x , y ) ) . b and ( H / ( x , y ) ) . b = ( 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 , W . 5 , G & W . 7 in G ;
( ( h (#) o ) . ( 2 * n ) ) . x = ( h * ( f . n ) ) . ( 2 * n ) .= ( ( h * f ) . ( x + h . n ) ) . ( x + h . n ) ;
j + 1 = ( len h ) + ( len h ) + 1 .= i + 1 - len h + 2 .= i + 1 - len h + 2 .= i + 1 - len h + 2 .= i + 1 - len h + 2 ;
( S *' ) . f = S *' . ( ( S *' ) . f ) .= S . ( ( S *' ) . f ) .= S . ( ( S *' ) . f ) .= ( S *' ) . f ;
consider H such that H is one-to-one and rng H = the carrier of L2 and Sum ( L2 * H ) = Sum ( L2 * H ) and Sum ( L2 * H ) = Sum ( L2 * H ) ;
attr R is + } means : Def1 : for p , q st p in R & q in R holds ex P st P is special & p in P & q in P & p in P ;
dom Product ( X --> f ) = meet ( dom ( X --> f ) ) .= meet ( ( X --> f ) . ( dom f ) ) .= meet ( dom ( X --> f ) ) .= dom f .= dom f .= dom ( X --> f ) .= dom ( X --> f ) .= dom ( X --> f ) .= dom ( X --> f ) .= dom ( X --> f ) .= dom ( X --> f ) .= dom ( X --> f ) .= dom ( X --> f ) .=
upper_bound ( proj2 .: ( Upper_Arc C /\ Lower_Arc C ) /\ from ( ( TOP-REAL 2 ) | ( Upper_Arc C ) ) .: ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( K ) ) ) ) ) ) ) ) <= upper_bound ( proj2 .: ( ( the carrier of TOP-REAL 2 ) /\ ( the carrier of TOP-REAL 2 ) ) ;
for r be Real st 0 < r ex n be Nat st for m be Nat st n <= m holds |. S . m - pp2 .| < r / 2
i * f-28 - fand f = i * fN - ( i * fN ) .= i * ( fN - ( i * fN ) ) .= i * ( fN - ( i * fN ) ) ;
consider f being Function such that dom f = 2 -tuples_on X ( ) & 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 f = [ g1 , g2 ] and g = [ g1 , g2 ] ;
func d |-count n -> Nat means : Def1 : d |^ n divides d & it |^ ( n + 1 ) divides d |^ n & it |^ ( n + 1 ) divides d |^ n ;
f\in . [ 0 , t ] = f . [ 0 , t ] .= ( - P ) . ( 2 * x ) .= ( - P ) . ( x , t ) .= a ;
t = h . D or t = h . B or t = h . C or t = h . E or t = h . F or t = h . J or t = M ;
consider m1 being Nat such that for n st n >= m1 holds dist ( ( seq . n ) , ( seq . n ) ) < 1 / ( ( seq . n ) + ( seq . n ) ) ;
( ( q `1 ) / |. q .| ) ^2 <= ( ( q `2 ) / |. q .| ) ^2 + ( ( q `2 / |. q .| ) ^2 ) ;
h0 . ( i + 1 + 1 ) = h0 . ( i + 1 + 1 -' len h11 + 2 -' 1 ) .= h0 . ( i + 1 + 1 -' len h11 + 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 , x2 ] in the carrier' of S ;
for L being RelStr , a , b being Element of L holds a <= { b } iff a <= b & b >= a & b >= a & b >= a
||. h1 .|| . n = ||. ( h1 . n ) .|| .= |. ( h . n ) .| .= ||. ( h . n ) .|| .= ||. ( h . n ) .|| .= ||. ( h . n ) .|| .= ||. ( h . n ) .|| .= ||. ( h . n ) .|| .= ||. ( h . n ) .|| ;
( ( - ( exp_R * f ) ) `| Z ) . x = f . x - ( exp_R * f ) . x .= ( ( - exp_R * f ) `| Z ) . x .= ( ( - exp_R * f ) `| Z ) . x .= ( ( - exp_R * f ) `| Z ) . x ;
pred r = F .: ( p , q ) means : Def1 : len r = min ( len p , len q ) & for i st i in dom r holds r . i = F . ( p . i ) ;
( r8 / 2 ) ^2 + ( r8 / 2 ) ^2 <= ( r ^2 + ( r ^2 + 1 ) / 2 ) ^2 + ( r ^2 + 1 ) ^2 ;
for i being Nat , M being Matrix of n , K st i in Seg n holds Det M = Sum ( ( Det M ) | ( i , j ) )
then a <> 0. R & a " * ( a * v ) = 1 * v & a " * v = 1 * v & a " * v = 1 * v ;
p . ( j -' 1 ) * ( q *' r ) . ( i + 1 -' j ) = Sum ( p . ( j -' 1 ) * ( q *' r ) ) ;
deffunc F ( Nat ) = L . 1 + ( ( R /* ( h ^\ n ) ) * ( h ^\ n ) " ) . $1 .= ( ( R /* ( h ^\ n ) ) " ) . $1 ;
assume that the carrier of H2 = f .: ( the carrier of H1 ) and the carrier of H2 = f .: ( the carrier of H2 ) and the carrier of H1 = the carrier of H2 and the carrier of H2 = the carrier of H2 and the carrier of H1 = the carrier of H2 and the carrier of H2 = the carrier of H2 ;
Args ( o , Free ( S , X ) ) = ( ( the Sorts of Free ( S , X ) ) * ( the Arity of S ) ) . o .= ( the Sorts of Free ( S , X ) ) . o ;
H1 = n + 1 & |. 2 to_power ( n + 1 ) + h .| = n + 1 & |. 2 to_power ( n + 1 ) + h .| = n + 1 & |. 2 to_power ( n + 1 ) + h .| = n + 1 ;
( O = 0 & ( O = 1 & O = 2 & O = 3 ) & ( O = 1 & O = 2 or O = 3 ) & ( O = 1 & O = 2 or O = 3 ) & ( O = 1 & O = 3 ) & ( O = 1 implies O = 3 ) ) ;
F1 .: ( dom F1 /\ dom F2 ) = F1 .: ( dom F1 /\ dom F2 ) .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } ;
pred b <> 0 & d <> 0 & b <> d & ( a - b ) / ( d - c ) = ( - e ) / ( d - b ) ;
dom ( ( f +* g ) | D ) = dom ( f +* g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ;
for i being set st i in dom g ex u , v being Element of L st g /. i = u * a & v = a * v & u in B & v in C
g `2 * P `2 * g `2 = g `2 * ( g `2 * P `2 ) * g `2 .= g `2 * ( g `2 * P `2 ) .= g `2 * ( g `2 * P `2 ) .= g `2 ;
consider i , s1 such that f . i = s1 and if ( not i in dom s1 & s1 . ( i + 1 ) <> s1 . ( i + 1 ) ) & s1 . ( i + 1 ) <> s1 . ( i + 1 ) & s1 . ( i + 1 ) <> s1 . ( i + 1 ) ;
h5 | ]. a , b .[ = ( g | ]. a , b .[ ) | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ ;
[ s1 , t1 ] , [ s2 , t2 ] are_connected & [ s2 , t2 ] , [ s3 , t2 ] are_connected & [ s3 , t2 ] , [ s3 , t2 ] are_connected & [ s3 , t2 ] `2 = [ s1 , t2 ] ;
then H is negative & H is non empty and H is non empty and H is non empty and H is non empty and H is non -ggex F being Function of H , D st F is not negative -g+* F & F is not an -gof H ;
attr f1 is total means : Def1 : f1 is total & f2 is total & ( for c st c in dom f1 holds f1 . c = f2 . c ) & ( f1 . c = f2 . c ) & ( f1 . c = f2 . c ) ;
z1 in W2 \mathclose ( W2 ) or z1 = z2 & not ( ex z2 st z2 in W2 & not ( z1 in W2 & not z2 in W1 & not z2 in W2 & not z2 in W1 & not z2 in W2 ) ) ;
p = 1 * p .= a " * a * p .= a " * ( b * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= a ;
for seq1 be Real_Sequence for K be Real_Sequence for n be Nat st for n be Nat holds seq1 . n <= K holds upper_bound rng ( seq1 ^\ n ) <= upper_bound rng ( seq1 ^\ n )
C meets L~ go \/ L~ pion1 or C /\ L~ pion1 = { go /. ( len go + 1 ) } or C /\ L~ co = { pion1 /. ( len pion1 + 1 ) } \/ L~ co or C /\ L~ co = { pion1 /. ( len pion1 + 1 ) } \/ L~ co ;
||. f . ( g . ( k + 1 ) ) - g . ( g . k ) .|| <= ||. g . ( 1 - g . k ) .|| * ( K to_power k ) ;
assume h = ( ( B .--> B ' ) +* ( C .--> D ' ) +* ( E .--> F ' ) +* ( F .--> J ) +* ( J .--> M ' ) +* ( M .--> N ) +* ( N .--> N ' ) +* ( F .--> N ) +* ( M .--> N ) +* ( N .--> M ) +* ( N .--> N ) +* ( N .--> M ) +* ( N .--> M ) +* ( N .--> N ) +* ( N .--> M ) +* ( N .--> N ) +* ( N .--> M ) +* ( N .--> M ) +* ( N .--> N )
|. ( ( lower_bound ( H . n ) || A ) . k - ( lower_bound ( H . n ) || A ) . k .| <= e * ( ( upper_bound ( H . n ) || A ) . k ) ;
( (
{ x1 , x1 , x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 } = { x1 , x2 , x3 , x4 } ;
assume that A = [. 0 , 2 * PI .] and integral ( ( exp_R * cos ) , A ) = - 1 & integral ( ( exp_R * cos ) , A ) = - 1 and integral ( ( exp_R * cos ) , A ) = - 1 ;
p `2 is Permutation of dom f1 & p `2 " = ( Sgm Y ) " * p & p `2 " * Sgm X = ( Sgm Y ) " * p & p `2 = Sgm Y ;
for x , y st x in A & y in A holds |. ( 1 / ( f . x ) - 1 ) / ( f . y ) .| <= 1 * |. f . x - f . y .|
p2 `2 = |. q2 .| * ( ( q2 `2 / |. q2 .| - sn ) / ( 1 + sn ) ) .= ( ( q2 `2 / |. q2 .| - sn ) / ( 1 + sn ) ) / ( 1 + sn ) ;
for f be PartFunc of the carrier of CNS , REAL , x be Point of CNS , r be Real st dom f is compact & f is continuous & x in dom f holds f | X is continuous
assume for x being Element of Y st x in EqClass ( z , CompF ( B , G ) ) holds ( Ex ( a , CompF ( B , G ) ) ) . x = TRUE ;
consider FM such that dom FM = n1 and for k be Nat st k in n1 holds Q [ k , FM . k ] and Q [ k , FM . k ] ;
ex u , u1 st u <> u1 & u , u1 // v , v1 & u , u1 // v , u1 & u , v1 // v , v1 & u1 , v1 // v , u1 & v , u1 // v , v1 & v , u1 // v , u1 & v , v1 // v , v1 & v , u1 // v , u1 & v , u1 // v , v1 & v , v1 // v , u1 & v , u1 // v , v1 & v , u1 // v , v1 & v , u1 // v , u1 implies v , u1 // v , v1 & v , v1 // v
for G being Group , A , B being non empty Subset of G , N being normal Subgroup of G holds ( N N ` A ) * ( N ` B ) = N ` A * N
for s be Real st s in dom F holds F . s = integral ( R , ( R + e ) (#) integral ( f , ( f + g ) (#) e ) ) . s ) ;
width AutMt ( f1 , b1 , b2 ) = len b2 .= len b1 .= len ( ( len b1 ) |-> ( len b2 ) ) .= len ( ( len b1 ) |-> ( len b1 ) ) .= len ( b1 * b1 ) .= len b1 ;
f | ]. - PI / 2 , PI / 2 .[ = f & dom f = ]. - 1 , 1 .[ & f | ]. - 1 , 1 .[ = f | ]. - 1 , 1 .[ ;
assume that X is closed w.r.t. as and a in X and a in X and y in a ^ f and x in { { [ n , x ] } \/ y : x in X } \/ a in X ;
Z = dom ( ( ( #Z n ) * ( arctan + arccot ) ) `| Z ) /\ dom ( ( ( #Z n ) * ( arctan + arccot ) ) `| Z ) .= dom ( ( #Z n ) * ( ( #Z n ) * ( f1 + #Z n ) ) `| Z ) ;
func [: V , V :] -> Subset of V means : Def1 : for k st 1 <= k & k <= len it holds it . k in [: V , V :] & it . k in [: V , V :] ;
for L being non empty TopSpace , N being net of L , M being net of N , c being Point of L st c is net of M & c is continuous holds c is continuous & c is continuous
for s being Element of NAT holds ( ( for v being Element of C\mathop ( C\mathop ( v , C\mathop ( v , C\mathop ( v , C\mathop ( v , C\mathop ( v , C\mathop ( v , C\mathop ( v , C\mathop ( v , C\mathop ( v , C\mathop ( v , C\mathop ( v , C\mathop ( v , C\mathop ( v , C\mathop ( v , C\mathop ( v , C\mathop ( v , Cq ) ) ) ) ) ) ) ) ) ) ) ) ) . s ) ) . s = ( ( ( v + C\mathop ( v ,
then z /. 1 = ( N-min L~ z ) .. z & ( N-min L~ z ) .. z < ( N-min L~ z ) .. z & ( N-min L~ z ) .. z < ( E-max L~ z ) .. z ;
len ( p ^ <* ( 0 qua Real ) * ( 0 qua Real ) *> ) = len p + len <* ( 0 qua Real ) * ( 0 qua Real ) .= len p + 1 .= len p + 1 .= len p + 1 ;
assume that Z c= dom ( - ( ln * f ) ) and for x st x in Z holds f . x = x & f . x > 0 & f . x > 0 ;
for R being add-associative right_zeroed right_complementable left distributive non empty doubleLoopStr , I being Subset of R , J being Subset of R holds ( I + J ) *' ( I /\ J ) c= I /\ J
consider f being Function of [: B1 , B2 :] , B1 , B2 being Element of [: B1 , B2 :] such that for x being Element of [: B1 , B2 :] holds f . x = F ( x ) and f . x = F ( x ) ;
dom ( x2 + y2 ) = Seg len x .= Seg len x .= len ( x2 + y2 ) .= len ( x + y ) .= len ( x + y ) .= len ( x + y ) .= len ( x + y ) .= len ( x + y ) ;
for S being Functor of C , B for c being object of C holds card S . id c = id ( ( Obj S ) . c ) & ( Obj S ) . c = id ( ( Obj S ) . c )
ex a st a = a2 & a in f6 /\ f5 & or F ( a , f ) = or F ( a , f ) = Im ( f , a ) & F ( a , f ) = F ( a , f ) ;
a in Free ( H / ( x. 4 , x. k ) ) '&' ( H2 / ( x. k , x. k ) ) '&' ( H2 / ( x. k , x. k ) ) '&' ( H2 / ( x. k , x. k ) ) ;
for C1 , C2 being v1 , f being Function of C1 , C2 , g being Function of C1 , C2 st ( for x being Element of C1 holds f . x = g . x ) holds f = g
( W-min L~ go \/ L~ co ) `1 = W-bound L~ go \/ ( W-bound L~ co ) .= W-bound L~ go \/ W-bound L~ co .= W-bound L~ go \/ W-bound L~ co .= W-bound L~ go \/ W-bound L~ co .= W-bound L~ go \/ W-bound L~ co .= W-bound L~ go \/ W-bound L~ co .= W-bound L~ go ;
assume that u = <* x0 , y0 *> and f is_or f is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or f is_or g is_or g is_or f is_or f is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_or g is_
then ( t . {} ) `1 in Vars & ex x being Element of Vars st x = ( t . {} ) `1 & ( t . {} ) `2 = ( x . {} ) `2 & ( x = {} ) `2 & ( x = {} implies x = {} ) & ( x = {} ) & x = {} ) ;
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 & b >= a ;
func Class R -> Subset-Family of R means : Def1 : for A being Subset of R holds A in it iff ex a being Element of R st a in it & R . a = Class ( R , a ) ;
defpred P [ Nat ] means ( ( \HM { the } \HM { vertices } \HM { of G , v } ) `1 ) `1 c= G * ( $1 + 1 ) `1 & ( the Source of G ) `2 c= G * ( $1 + 1 ) `2 ;
assume that dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W2 ) = 0 and V = the carrier of ( W1 ) /\ ( W2 ) and f = f & f = f | ( the carrier of W1 ) & f = f | ( the carrier of W1 ) & f = f | ( the carrier of W1 ) ;
mamain ( m . t ) `1 = ( m . t ) `1 .= ( [ m . t , the carrier of C ] `1 ) `1 .= ( [ m . t , the carrier of C ] `2 ) `1 .= m . t ;
d11 = x11 ^ d22 .= f . ( ( y , d22 ) /. ( y , d22 ) ) .= f . ( ( y , d22 ) /. ( y , d22 ) ) .= ( f | d22 ) . ( y , d22 ) .= ( f | d22 ) . ( y , d22 ) .= ( f | d22 ) . ( y , d22 ) .= ( f | d22 ) . ( y , d22 ) ;
consider g such that x = g and dom g = dom f0 and for x being element st x in dom f0 holds g . x in f0 and g . x in f0 and g . x in f0 and g . x in f0 ;
x + 0. F_Complex / ( len x ) = x + len x |-> 0. F_Complex .= ( x + 0. F_Complex ) * x .= ( x + 0. F_Complex ) * x .= x + 0. F_Complex .= x ;
( k -' ( k + 1 ) ) + 1 in dom ( f | ( len ( k -' ( k + 1 ) ) ) | ( k + 1 ) ) & ( f | ( k + 1 ) ) . ( k + 1 ) = ( f | ( k + 1 ) ) . ( k + 1 ) ;
assume that P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 = { p1 , p2 } and P1 = { p1 , p2 } and P1 /\ P2 = { p1 , p2 } and P1 /\ P2 = { p1 , p2 } and P1 /\ P2 = { p1 , p2 } and P1 /\ P2 = { p1 , p2 } and P1 /\ P2 = { p1 , p2 } ;
reconsider a1 = a , b1 = b , b1 = c , c1 = p `1 , c1 = p `2 , c2 = p `2 , c2 = p `2 , c1 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c1 = p `2 , c2 = p `2 , c2 = p `2 , c1 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 ;
reconsider Gtt1f = G1 . ( t , b ) * F1 . f , FFf = G1 . ( t , a ) * F2 . f , FFf = G1 . ( t , a ) * F2 . f , FFf . ( t , b ) * F2 . f ;
LSeg ( f , i + i1 -' 1 ) = LSeg ( f /. ( i + i1 -' 1 ) , f /. ( i + i1 -' 1 + 1 ) ) .= LSeg ( f , i + i1 -' 1 ) ;
Integral ( M M , P . m ) | dom ( P . n ) <= Integral ( M , P . n ) | dom ( P . m ) ;
assume that dom f1 = dom f2 and for x , y being element st [ x , y ] in dom f1 & [ x , y ] in dom f2 holds f1 . ( x , y ) = f2 . ( x , y ) ;
consider v such that v = y and dist ( u , v ) < min ( ( G * ( i , 1 ) `1 - G * ( i + 1 , 1 ) `1 ) , ( G * ( i + 1 , 1 ) `2 - G * ( i + 1 , 1 ) `2 ) ) ;
for G being Group , H being Subgroup of G , a being Element of G , b being Element of H st a = b holds for i being Integer holds a |^ i = b |^ i & b |^ 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 = { 7 where 7 is Point of TOP-REAL 2 : P [ 7 ] & P [ 7 ] } , K1 = { p : P [ 7 ] } as Subset of ( TOP-REAL 2 ) | K1 ;
( ( N-bound C ) - ( S-bound C ) ) / 2 <= ( ( S-bound C ) - ( S-bound C ) ) / 2 + ( ( S-bound C ) - ( S-bound C ) ) / 2 + ( ( S-bound C ) - ( S-bound C ) ) / 2 ;
for x be Element of X , n be Nat st x in E holds |. Re ( F . n ) .| . x <= P . x & Im ( F . n ) <= P . x & Im ( F . n ) <= P . x
len ( @ @ ( @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
v / ( x. 3 , m1 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 4 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / (
consider r being Element of M such that M , v2 / ( x. 3 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m
func w1 \ w2 -> Element of Union ( G , R^ R^ R^ R^ R^ R^ R^ ( G , R^ R^ R^ R^ R^ w ) ) equals ( ( ( the Sorts of G ) * ( the Arity of G ) ) . ( ( the Arity of G ) . ( the Arity of G ) ) . ( ( the Arity of G ) . ( the Arity of G ) ) ;
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= s1 . b2 .= s2 . b2 .= Exec ( n2 , s2 ) . b2 .= s2 . b2 .= Exec ( n2 , s2 ) . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 ;
for n , k be Nat holds 0 <= ( Partial_Sums |. seq .| ) . ( n + k ) - ( Partial_Sums |. seq .| ) . ( n + k ) + ( Partial_Sums |. seq .| ) . ( n + k )
set F = S -\mathop { 0 } ;
( Partial_Sums ( seq ) ) . K + Partial_Sums ( seq ) . ( K + 1 ) >= ( Partial_Sums ( seq ) ) . K + ( Partial_Sums ( seq ) ) . ( K + 1 ) + ( Partial_Sums ( seq ) ) . K ;
consider L , R such that for x st x in N holds ( f | Z ) . x - ( f | Z ) . x0 = L . ( x- x ) + R . ( x- x ) ;
func the closed of \HM { a , b , c , d : a = the Element of \HM { \HM { a , b , c : b in P & c in P } , Q = the closed Subset of \HM { a , b , c , d } } ;
a * b ^2 + ( a * c ) ^2 + ( b * a ^2 + c * a ^2 ) + ( b * c ) ^2 + ( b * a ^2 + c * a ^2 + b * a ^2 + c * a ^2 ) >= 6 * a * b * c ;
v / ( x1 , m1 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m1 ) = v / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m2 ) ;
Rotate ( Q ^ <* x *> , M ) = ( ( \mathop { Q } ) +* ( { x } --> FALSE ) ) +* ( ( ( { x } --> FALSE ) +* ( { x } --> FALSE ) ) +* ( ( { x } --> FALSE ) +* ( { x } --> FALSE ) ) .= ( ( { x } --> FALSE ) +* ( { x } --> FALSE ) ) +* ( { x } --> FALSE ) ;
Partial_Sums ( F ) . n = r |^ ( n1 + 1 ) * Partial_Sums ( C ) . n1 .= C . ( n1 + 1 ) .= C . ( n1 + 1 ) .= C . ( n1 + 1 ) .= C . ( n1 + 1 ) .= C . ( n1 + 1 ) ;
( GoB f ) * ( len GoB f , 2 ) `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 = ( a * ( $1 + 1 ) ) * ( $1 + 1 ) + b * ( $1 + 1 ) * ( $1 + 1 ) + b * ( $1 + 1 ) ;
the_arity_of g = ( the Arity of S ) . g .= ( the Arity of S ) . g .= ( ( the Arity of S ) . g ) . g .= ( the Arity of S ) . g .= ( the Arity of S ) . g .= ( the Arity of S ) . g .= ( the Arity of S ) . g .= ( the Arity of S ) . g .= ( the Arity of S ) . g ;
( X ~ ) c= X ~ & card ( ( X ~ ) \/ Y ) = card ( X ~ ) & card ( ( X ~ ) \/ Y ) = card ( X ~ ) ;
for a , b being Element of S , s being Element of NAT st s = n & a = F . n & b = F . n & b = N . ( n + 1 ) \ G . s holds b = N . ( n + 1 ) \ G . s
E , f |= All ( x , ( ( x. 2 ) . ( x. 0 ) ) ) '&' ( ( x. 2 ) . ( x. 1 ) ) '&' ( ( x. 2 ) . ( x. 1 ) ) '&' ( ( x. 2 ) . ( x. 1 ) ) '&' ( ( x. 2 ) . ( x. 1 ) ) '&' ( ( x. 2 ) . ( x. 1 ) ) '&' ( x. 2 ) ) ;
ex R2 being 1-sorted st R2 = ( p | ( n + ( n + 1 ) ) ) . i & ( the carrier of p ) = the carrier of R2 & ( the carrier of p ) c= the carrier of R2 & ( the carrier of p ) c= the carrier of R2 ) ;
[. a , b + 1 / ( k + 1 ) .[ is Element of the \in of the set of set & ( the partial of f ) . ( k + 1 ) is Element of the carrier of a & ( the partial of f ) . ( k + 1 ) is Element of the carrier of a ) ;
Comput ( P , s , 2 + 1 ) = Exec ( P . 2 , Comput ( P , s , 2 ) ) .= Exec ( a3 , Comput ( P , s , 2 ) ) .= Exec ( a3 , Comput ( P , s , 2 ) ) .= Exec ( a3 , s ) ;
card ( h1 ) . k = power ( F_Complex ) . ( ( - 1. F_Complex ) * k , k ) .= ( ( - 1_ F_Complex ) * ( ( - 1_ F_Complex ) * k ) ) * u .= ( ( - 1_ F_Complex ) * ( - 1_ F_Complex ) ) * u .= ( ( - 1_ F_Complex ) * ( - 1_ F_Complex ) ) * ( - 1_ F_Complex ) .= ( ( - 1_ F_Complex ) * ( - 1_ F_Complex ) ) * u ;
( f / g ) /. c = f /. c * ( g /. c ) " .= f /. c * ( ( 1 / g ) /. c ) .= ( f (#) ( g (#) ( g / ( g / ( g / ( g - f ) ) ) ) ) ) /. c .= ( f (#) ( g / ( g / ( g - f ) ) ) ) /. c ;
len Cs - len ( ( C | ( len ( C | ( len C -' 1 ) ) ) ) = len ( C | ( len C -' 1 ) ) .= len ( ( C | ( len C -' 1 ) ) ) .= len ( ( C | ( len C -' 1 ) ) ) .= len ( ( C | ( len C -' 1 ) ) ) .= len ( ( C | ( len C -' 1 ) ) ) ;
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom 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 ) ) ;
defpred P [ Nat ] means for n holds 2 * Fib ( n + $1 ) = Fib ( n ) * Fib ( n ) + ( 5 * Fib ( n ) * Fib ( n ) + 5 * Fib ( n ) * Fib ( n ) ) ;
consider f being Function of INT , INT such that f = f `1 and f is onto and ( n + 1 ) < n & f " { f . n } = { n + 1 } and f " { f . n } = { n } ;
consider vs being Function of S , BOOLEAN such that c9 = chi ( A \/ B , S ) and E7 . ( A \/ B ) = Prob ( Q , S \/ B ) and E7 . ( A \/ B ) = Prob ( Q , S \/ B ) ;
consider y being Element of Y ( ) such that a = "\/" ( { F ( x , y ) where x is Element of X ( ) : P [ x ] } , L ( ) ) and Q [ y ] ;
assume that A c= Z and f = ( f `| Z ) (#) ( ( #Z 2 ) * ( sin + cos ) ) + ( ( id Z ) (#) ( cos + cos ) ) (#) ( ( id Z ) + sin + cos ) ) and Z = dom f and f = ( ( id Z ) (#) ( cos + cos ) ) (#) ( ( id Z ) (#) ( cos + cos ) ) ;
( 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 + 1 ) `2 .= ( GoB f ) * ( 1 , j2 + 1 ) `2 ;
dom Shift ( Seq q2 , len Seq q1 ) = { j + len Seq q1 where j is Nat : j in dom Seq q1 & j in dom Seq q2 } & len Seq q1 = len Seq q1 + len Seq q2 } ;
consider G1 , G2 , G3 being Element of V such that G1 <= G2 & G2 <= G1 & f is Morphism of G1 , G2 and g is Morphism of G2 , G3 and f is Morphism of G1 , G2 and g is Morphism of G2 , G3 and g is Morphism of G2 , G3 and f is Morphism of 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 + f /. c ;
consider phi such that phi is increasing and for a st phi . a = a & {} <> a and for v holds union rng L c= a & L . a c= union ( union ( L | [. v , v .] ) ) iff L . a in rng L & L . a c= H ;
consider i1 , j1 such that [ i1 , j1 ] in Indices GoB f and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) ;
consider i , n such that n <> 0 and sqrt p = ( i - n ) * ( i - 1 ) and for n1 being Nat st n1 <> 0 & n1 <= n & n <= len p holds sqrt p = ( i - n ) * ( i - 1 ) and ( i <= n implies p . n1 = 0 ) ;
assume that not 0 in Z and Z c= dom ( ( arccot * ( f1 + f2 ) ) `| Z ) and for x st x in Z holds ( ( 1 / 2 ) (#) ( ( arccot * ( f1 + f2 ) ) `| Z ) . x = - 1 / ( 1 + x ^2 ) ) and for x st x in Z holds 1 / ( 1 + x ^2 ) = 1 / ( 1 + x ^2 ) ;
cell ( G1 , i1 -' 1 , ( 2 |^ ( m -' 1 ) ) * ( ( Y -' 1 ) + ( 2 |^ ( m -' 1 ) ) * ( ( Y -' 1 ) + ( 2 |^ ( m -' 1 ) ) * ( ( Y -' 1 ) + ( Y -' 1 ) ) * ( ( Y -' 1 ) + ( Y -' 1 ) + ( Y -' 1 ) ) * ( ( Y -' 1 ) + ( Y -' 1 ) ) ) ) c= BDD L~ f1 ;
ex Q1 being open Subset of X st s = Q1 & ex F8 being Subset-Family of Y st F8 c= F & F8 is finite & ( for x being Point of Y , Q being Subset of X st Q in F8 & Q c= Q holds Q = Q ) & ( for x being Point of Y holds Q [ x , Q . x ] ) & ( for x being Point of Y holds Q [ x , Q . x ] ) ;
gcd ( ( 1. ( A , B ) ) , ( 1. ( A , B ) ) , ( 1. ( A , B ) ) , ( 1. ( A , B ) ) ) = 1 / ( ( 1. ( A , B ) ) * ( 1. ( A , B ) ) ) .= 1 / ( ( 1. ( A , B ) ) * ( 1. ( A , B ) ) ) ;
R8 = ( ( ( the InternalRel of s2 ) . ( 1 + 1 ) ) . ( m2 + 1 ) ) . ( m2 + 1 ) .= ( ( ( the Let of s2 ) . ( m2 + 1 ) ) . ( m2 + 1 ) ) . ( m2 + 1 ) .= [ 3 , ( the InternalRel of s2 ) . ( m2 + 1 ) ] .= [ 3 , ( the InternalRel of s2 ) . ( m2 + 1 ) ] ;
CurInstr ( P-6 , Comput ( P-6 , sseq , m3 + 1 ) ) = CurInstr ( P3 , Comput ( P3 , s3 , m3 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m3 ) ) .= CurInstr ( P3 , s3 ) .= CurInstr ( P3 , s3 ) .= CurInstr ( P3 , s3 ) .= CurInstr ( P3 , s3 ) .= CurInstr ( P3 , s3 ) .= CurInstr ( P3 , s3 ) .= 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 ) ) \/ { p2 } ) \/ { p1 } ;
func not bound in the Sorts of Al means : Def1 : a in it iff ex i st i in dom f & ex p st p in dom f & p = f . i & a in the Sorts of A & f . p = ( f . i ) . p & a in the Sorts of A & f . p = ( f . i ) . p ;
for a , b being Element of F_Complex st |. a .| > |. b .| & for f being Polynomial of F_Complex st f >= 1 holds f is \cap |. b .| is non empty & f is is \cap & f is *> implies f * ( - b * card f ) is \cup |. b * card f .|
defpred P [ Nat ] means 1 <= $1 & $1 <= len g & for i , j st [ i , j ] in Indices G & G * ( i , j ) = g . ( $1 + 1 ) & G * ( i , j ) = g . ( j + 1 ) & G * ( i , j ) = g . ( $1 + 1 ) ;
assume that C1 , C2 are_{} and for f , g being State of C1 , s1 , s2 being State of C2 , f being Function of C1 , C2 st f = s2 & g = s1 holds s1 * f is stable & s2 * f is stable & s1 * g is stable ;
( ||. f .|| | X ) . c = ||. f .|| . c .= ||. f /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( 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
|. q .| ^2 = ( q `1 ) ^2 + ( q `2 ) ^2 & 0 + ( q `2 ) ^2 < ( q `2 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^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 in F & A <> B & A <> B holds card F = card A & card F = card B & card F = card B & card A = 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 and for k st k in dom F & k <> k & k <> n holds H . k = g . ( F . k , G . k ) ;
i |^ ( \mathop { \rm mod n - i |^ s ) = i |^ ( s + k ) - i |^ s .= i |^ s * i |^ k - i |^ s * 1 .= i |^ ( s + k ) - i |^ s * 1 .= i |^ ( s + k ) - i |^ s * 1 .= i |^ ( s + k - i ) ;
consider q being oriented oriented Chain of G such that r = q and q <> {} and F8 . ( q . 1 ) = v1 and F7 . ( q . len q ) = v2 and rng q c= rng p-2 and q . ( len q ) = v2 and rng q c= rng p and p . ( len q ) = v1 . ( len p ) ;
defpred P [ Element of NAT ] means $1 <= len ( I . ( Z , I ) ) & ( g . ( Z , I ) ) . $1 = ( ( g . ( Z , I ) ) . ( len ( g . ( Z , I ) ) ) ) . ( len ( g . ( Z , I ) ) + $1 ) ) ;
for A , B being square Matrix of n , REAL holds len ( A * B ) = len A & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B
consider s being FinSequence of the carrier of R such that Sum s = u and for i being Element of NAT st 1 <= i & i <= len s ex a , b being Element of R st s . i = a * b & a in I & b in J & s . i = b * a ;
func |( x , y )| -> Element of COMPLEX equals |( ( Re x ) , ( Re y ) )| - ( ( Re x ) ^2 + ( Re y ) ^2 + ( Im y ) ^2 + ( Im y ) ^2 + ( Im y ) ^2 + ( Im y ) ^2 + ( Im y ) ^2 ) , ( Im x ) ^2 + ( Im y ) ^2 + ( ( Im y ) ^2 + ( Im y ) ^2 + ( Im y ) ^2 ) ;
consider f0 be FinSequence of FH such that f0 is continuous & rng f0 c= A & g2 . 1 = x1 & g2 . len g2 = x2 & g2 . len g2 = y1 & rng g2 = A & g2 . len g2 = y1 & rng g2 = B & g2 . len g2 = y2 & rng g2 = A & g2 . len g2 = y1 ;
then n1 >= len p1 & n2 >= len p1 & n1 >= len p2 & n2 >= len p1 & n1 >= len p1 & n2 >= len p1 & n1 >= len p2 & n2 >= len p1 & n2 >= len p1 & n1 >= len p2 & n2 >= len p1 & n2 >= len p1 & n2 >= len p1 & n2 >= len p1 & n2 >= len p1 & n2 >= len p1 & n2 >= len p1 & n2 >= len p1 + n2 + n2 + n2 + n3 + n3 ;
( q `1 ) * a <= ( q `1 ) * a & ( q `2 ) * a <= ( q `2 ) * a or q `1 >= ( q `2 ) * a & ( q `2 ) * a <= ( q `2 ) * a or q `1 >= ( q `2 ) * a ;
F6 . ( len p6 ) = F6 . ( p . ( len p ) ) .= F6 . ( len p ) .= v6 . ( len p ) .= v6 . ( len p ) .= v6 . ( len p ) .= v6 . ( len p ) .= ( v . ( len p ) ) .= v . ( len p ) ;
consider k1 being Nat such that k1 + k = 1 and a := k = ( <* a := intloc 0 *> ) ^ ( ( intloc 0 ) --> 1 ) ^ ( ( intloc 0 ) --> 1 ) ^ ( ( intloc 0 ) --> 1 ) ^ ( ( intloc 0 ) --> 1 ) ) ^ ( ( intloc 0 ) --> 1 ) ;
consider B8 being Subset of B1 , y8 being Function of B1 , A1 such that B8 is finite and D8 = the carrier of A1 and the carrier of 8 = the carrier of B1 and the carrier of 8 = the carrier of A1 and the carrier of 8 = the carrier of B1 and the carrier of 8 = the carrier of A1 and the carrier of 8 = the carrier of A2 ;
v2 . b2 = ( curry ( F2 , g ) * ( curry Map ( B , g ) ) ) . b2 .= ( ( curry F ) . ( ( curry Map ( B , g ) ) . b2 ) ) . b2 .= ( ( curry F ) . ( ( curry Map ( B , g ) ) . b2 ) . b2 .= ( ( curry F ) . ( ( curry Map ( B , g ) ) . b2 ) . b2 .= ( ( curry F ) . b1 ) . b2 .= ( ( ( curry F ) . b2 ) . b2 ) . b2 .= ( ( ( curry F ) . b2 ) . b2 .= ( ( ( F . b2 ) . b2 ) . b2 .= ( ( curry F ) . b2 ) . b2 .= ( ( ( ( ( ( curry F ) . b2 ) . b2 ) . b2 .= ( ( ( curry F ) . b2 ) . b2
dom IExec ( I-35 , P , Initialize s ) = the carrier of SCMPDS .= dom IExec ( I , P , Initialize s ) .= dom IExec ( I , P , Initialize s ) .= dom IExec ( I , P , Initialize s ) .= dom IExec ( I , P , Initialize s ) .= dom IExec ( I , P , Initialize s ) .= dom IExec ( I , P , Initialize s ) .= dom IExec ( I , P , Initialize s ) ;
ex d-32 be Real st d-32 > 0 & for h be Real st h <> 0 & |. h .| < d-32 holds |. h .| " * ||. ( R2 + R1 ) /. h .|| < e / ( ||. ( R2 + R1 ) /. h .|| + e / ( ||. ( R2 + R1 ) /. h .|| ) ) ) ;
LSeg ( G * ( len G , 1 ) + |[ 1 , - 1 ]| , G * ( len G , 1 ) + |[ 1 , 0 ]| ) c= Int cell ( G , len G , 1 ) \/ { G * ( len G , 1 ) + |[ 1 , 0 ]| } ;
LSeg ( mid ( h , i1 , i2 ) , i ) = LSeg ( h /. ( i + i1 -' 1 ) , h /. ( i + i1 -' 1 + 1 ) ) .= LSeg ( h , i ) .= LSeg ( h , i ) ;
A = { q where q is Point of TOP-REAL 2 : LE p1 , p2 , P , p1 , p2 & LE p2 , p1 , P , p1 , p2 & LE p1 , p2 , P , p1 , p2 & LE p2 , p1 , P , p1 , p2 & LE p1 , p2 , P , p1 , p2 & LE p2 , p1 , P , p1 , p2 & LE p2 , p1 , P , p2 & LE p2 , p1 , P , p1 , p2 } , P , p1 , p2 & LE p1 , p2 , P , p2 & LE p2 , p1 , p2 , P , p1 , p2 , P , p1 , p2 , P , p1 , p2 & LE p1 , p2 , P , p1 , p2 & LE p1 , p2 , P , p1 , p2 & LE p2 , p2 , P , p1 , p2 , p2 & LE p1 , p2 , P , p1 , p2 & LE p1 , p2 , P , p1 , p2 & LE
( ( - x ) .|. y ) = ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 )
0 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) = ( p `2 ) ^2 + ( p `2 / p `2 ) ^2 .= ( p `2 ) ^2 + ( p `2 / p `2 ) ^2 .= ( p `2 ) ^2 + ( p `2 / p `1 ) ^2 ;
( ( U . W ) * ( W7 * ( W7 ) ) ) * ( W . W ) = ( ( U . W ) * ( W7 ) ) * ( W . W ) .= ( ( U . W ) * ( W . W ) ) * ( W . W ) .= ( ( U . W ) * ( W . W ) ) * ( W . W ) .= ( ( U . W ) * ( W . W ) ) * ( W . W ) .= ( ( U . W ) ;
func Shift ( f , h ) -> PartFunc of REAL , REAL means : Def1 : dom it = dom ( - h ) & for x st x in dom it holds it . x = - h . x & for x st x in dom it holds it . x = ( - h ) . x + ( - h ) . x ;
assume that 1 <= k and k + 1 <= len f and [ i , j ] in Indices G and [ i + 1 , j ] in Indices G and f /. k = G * ( i + 1 , j ) and f /. ( k + 1 ) = G * ( i , j ) and f /. ( k + 1 ) = G * ( i , j ) ;
assume that not y in Free H and x in Free H and not x in Free H and not y in Free H and not x in Free H and not x in Free H and not x in Free H and not x in Free H and x in Free H and not x in Free H and x = H . ( x , y ) ;
defpred P11 [ Element of NAT , Element of NAT , Element of NAT , Element of NAT , Element of NAT , Element of NAT , Element of NAT , Element of NAT , Element of NAT , Element of REAL n , Element of REAL n st ( $1 = p |^ $2 & $2 = $1 |^ $2 & $2 = p |^ $2 & $2 = $1 |^ $2 ) & $2 = p |^ $2 & $2 = p |^ $1 & $2 = p |^ $1 & $2 = p |^ $1 & $2 = p |^ $1 & $2 = p |^ $1 & $2 = p |^ $1 & $2 = p |^ $1 & $2 = p |^ $1 & $2 = p |^ $1 & $2 = p |^ $1 & $2 = p |^ $1 & $2 = p |^ $1 & $2 = p |^ $1 & $2 = p |^ $1 & $2 = p |^ $1 & $2 = p |^ $1 & $2 = p |^ $1 & $2 = p |^ $1 & $2 = p |^ $1 & $2 = p |^ $1 & $2 = p |^ $1 & $2 = p |^ $1 & $2
func \sigma ( C ) -> non empty Subset-Family of X means : Def1 : for A being Subset of X holds A in it iff for W being Subset of X holds W in it iff for A being Subset of X holds A in it iff for W being Subset of X st W in it & W in it & A c= W holds C . W = C . W \/ C . A ;
[#] ( ( dist ( ( dist ( P ) ) ) .: Q ) ) = ( dist ( ( dist ( P ) ) .: Q ) ) .: Q & lower_bound [#] ( ( dist ( P ) ) .: Q ) = lower_bound ( ( dist ( P ) ) .: Q ) ) ;
rng ( F | ( [ S ] |^ 2 ) ) = {} or rng ( F | ( [ S ] |^ 2 ) ) = { 1 } or rng ( F | ( [ S ] |^ 2 ) ) = { 2 } or rng ( F | ( [ S ] |^ 2 ) ) = { 1 } or rng ( F | ( [ S ] |^ 2 ) ) = { 2 } ;
( f " ( rng f ) ) . i = f . i " . ( ( rng f ) . i ) .= ( f . i ) " . ( ( f . i ) " . ( f . i ) ) .= ( f . i ) " . ( f . i ) .= ( f . i ) " . ( f . i ) .= ( f . i ) " . ( f . i ) .= ( f . i ) " . i .= ( f . i ;
consider P1 , P2 being non empty Subset of TOP-REAL 2 such that P1 is_an_arc_of p1 , p2 and P1 /\ P2 = { p1 , p2 } and P1 = { p1 , p2 } and P1 = { p1 , p2 } and P2 = { p1 , p2 } and P1 = { p1 , p2 } and P2 = { p1 , p2 } ;
f . p2 = |[ ( p2 `1 ) ^2 + ( p2 `2 ) ^2 - ( p2 `2 ) ^2 + ( p2 `2 ) ^2 .= ( ( p2 `1 ) ^2 + ( p2 `2 ) ^2 - ( p2 `2 ) ^2 + ( p2 `2 ) ^2 ;
( ( ( AffineMap ( a , X ) ) " ) . x ) = ( ( ( AffineMap ( a , X ) ) qua Function ) . x ) " .= ( ( ( the carrier of X ) --> a ) . x ) " .= ( ( the carrier of X ) --> a ) . x .= ( ( the carrier of X ) --> a ) . x .= ( ( the carrier of X ) --> a ) . x .= ( ( the carrier of X ) --> a ) . x .= ( ( ( the carrier of X ) . x .= ( ( the carrier of X ) . x .= ( ( the carrier of X ) . x .= ( ( the carrier of X ) . x ) + ( ( the carrier of X ) . x .= ( ( the carrier of X ) . x .= ( ( the carrier of X ) . x .= ( ( the carrier of X ) . x ) + ( ( x ) ) . x .= ( ( ( the carrier of X ) . x ) + ( ( the carrier of X ) . x .= ( ( the carrier of X ) . x .= ( ( the carrier of X ) .
for T being non empty normal TopSpace , A , B being closed Subset of T st A <> {} & A misses B & B misses B for p being Point of T , r being Real st p in A & r in B & p in A & p in B holds ( <* in A , B *> ) . p = r * ( ( in > 0 ) * ( for p being Point of T holds p in A ) )
for i , j st i + 1 in dom F for G1 , G2 being strict normal Subgroup of G st G1 = F . i & G2 = F . i & G2 = F . ( i + 1 ) holds G1 * F is strict Subgroup of G1 & G1 * G = the Subgroup of G2
for x st x in Z holds ( ( ( #Z n ) * ( arccot - arccot ) ) `| Z ) . x = ( ( #Z n ) . ( x + 2 ) - 1 / ( 1 + x ^2 ) ) / ( 1 + x ^2 )
synonym f is right continuous means : Def1 : x0 in dom ( f /* a ) & ( for a st rng a c= dom f & a in dom f holds f /. a is convergent & ( for b st b in dom f holds f /. b is convergent ) & ( ex a st a in dom f & b in dom f holds f /. b = ( f /* a ) /. x0 ) ;
then X1 , X2 are_separated & ( X1 union X2 ) misses X2 & ( ex Y1 , Y2 being non empty SubSpace of X st Y1 misses Y2 & Y1 is open & Y2 is open & Y1 is open & Y2 is open & Y2 is open & Y1 is open & Y2 is open & Y2 is open & Y1 is open & Y2 is open & Y2 is open & Y1 is open & Y2 is open ) ;
ex N being Neighbourhood of x0 st N c= dom SVF1 ( 1 , f , u ) & ex L , R st for x st x in N holds SVF1 ( 1 , f , u ) . x - SVF1 ( 1 , f , u ) . x0 = L . ( x - x0 ) + R . ( x - x0 ) + R . ( x - x0 )
( p2 `1 ) * sqrt ( 1 + ( p3 `2 / p2 `1 ) ^2 ) >= ( ( 1 + ( p3 `2 / p3 `1 ) ^2 ) * sqrt ( 1 + ( p3 `2 / p3 `1 ) ^2 ) ) * sqrt ( 1 + ( p3 `2 / p3 `1 ) ^2 ) ;
( ( 1 / t ) (#) ||. ( f1 (#) ( g `| Z ) , ( g `| Z ) , ( g `| Z ) , ( g `| Z ) , ( g `| Z ) , ( g `| Z ) , ( g `| Z ) , ( g `| Z ) , ( g `| Z ) , ( g `| Z ) , ( g `| Z ) , ( g `| Z ) , ( g `| Z ) , ( g `| Z ) , ( g `| Z ) , ( g `| Z ) , ( g `| Z ) , ( g `| Z ) , ( g `| Z ) , ( g `| Z ) , ( g `| Z ) , ( g `| Z ) , ( g `| Z ) , ( g `| Z ) , ( g `| Z ) . ( g `| Z ) . ( x `| Z ) . ( x `| Z ) . ( x `| Z ) . ( x `| Z ) . ( x `| Z ) . ( x `| Z ) . ( x `| Z ) . ( x `| Z ) . ( x `| Z ) ) = ( ( g `| Z ) . ( x `| Z
assume that for x holds f . x = ( ( sin (#) cot ) `| Z ) . x and x + h / 2 in dom ( ( sin (#) cot ) `| Z ) and x + h / 2 in dom ( ( sin (#) cot ) `| Z ) and ( ( sin (#) cot ) `| Z ) . x = ( sin (#) cot ) `| Z ) . x ;
consider X-23 being Subset of Y , Y1 being Subset of X such that t = [: X1 , Y1 :] and Y1 is open and ex Y1 being Subset of X st Y1 = [: Y1 , Y1 :] & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open ;
card ( S . n ) = card { [: d , Y :] + ( a * b ) where d is Element of GF ( p ) : [ d , Y ] in R } .= { d , b } \/ { b , c } .= { d , b } \/ { c , d } .= { d , b } \/ { c , d } .= { d , b } \/ { c , d } ;
( W-bound D - W-bound D ) * ( ( i1 - W-bound D ) / 2 ) * ( ( i - W-bound D ) / 2 ) = ( W-bound D - W-bound D ) * ( ( i - W-bound D ) / 2 ) .= ( W-bound D - W-bound D ) * ( ( i - W-bound D ) / 2 ) * ( ( i - W-bound D ) / 2 ) .= ( ( W-bound D - W-bound D ) / 2 ) * ( ( i - W-bound D ) / 2 ) ;