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 ;
thesis ;
thesis ;
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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 ;
thesis ;
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 ;
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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 ;
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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 ;
thesis ;
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 ` is Cauchy
q in X ;
V in X ;
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 >= 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 `1 ;
assume f = g ;
N c= b ` ;
assume i < k ;
assume u = v ;
I = J ;
B ` = b ` ` ;
assume e in F ;
assume p > 0 ;
assume x in D ;
let i be element ;
assume F is onto ;
assume n <> 0 ;
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 generated of squares ;
assume m > 0 ;
assume A c= B ;
X is lower
assume A <> {} ;
assume X <> {} ;
assume F <> {} ;
assume G is open ;
assume f is dilatation ;
assume y in W ;
y \not <= x ;
A ` in B ` ;
assume i = 1 ;
let x be element ;
x `1 = x `1 ;
let X be BCI-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 be Element of C ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
y be Real ;
X c= f . a
let y be element ;
let x be element ;
i be Nat ;
let x be element ;
n in NAT ;
let a be element ;
m in NAT ;
let u be element ;
i in NAT ;
let g be Function ;
Z c= NAT ;
l <= ma ;
let y be element ;
r2 in dom f ;
let x be element ;
k1 be Integer ;
let X be set ;
let a be element ;
let x be element ;
let x be element ;
let q be element ;
let x be element ;
assume f is being_homeomorphism ;
let z be element ;
a , b // K ;
let n be Nat ;
let k be Nat ;
B ` c= B ` ` ;
set s = from c , c = the Element of NAT ;
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 . x <> 0 ;
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 Element of REAL ;
not r in G . l
c1 = 0 ;
a + a = a ;
<* 0 *> in e ;
t in { t } ;
assume F is 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 in X ;
cluster uparrow x -> and x is directed ;
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 ;
Let Let Let Let Let Let Let Let Let > s ;
G . y <> 0 ;
let X be RealNormSpace , x be Element of X ;
a in X ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , M be Subset of V ;
assume x in - M ;
k < s . a ;
not t in { p } ;
let Y be set , f be Function ;
M , L are_equipotent ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded ;
rng f = Y ;
IT c= L & IT c= L ;
assume x in Cl Q ;
m in dom P ;
i <= len Q ;
len F = 3 ;
still_not-bound_in 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 - 1 ) ;
1 <= ( i - 1 ) ;
pwhere pX1 is Subset of X : pX1 c= cos }
1 <= ( i - 1 ) ;
1 <= ( i - 1 ) ;
LMP C in L ;
1 in dom f ;
let seq , B ;
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 ;
seq is bounded & seq is bounded implies seq - seq 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 : x in A2 ;
cluster R \ S -> real-valued ;
sup D in S ;
x << sup D ;
b1 >= Z1 & b2 >= b1 ;
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 ] ;
c= Let C ;
x4 is increasing ;
let e2 be element ;
- b divides b ;
F c= \tau ( F ) ;
IT is non-decreasing implies seq is non-decreasing
IT is non-decreasing implies seq is non-decreasing
assume v in H . m ;
assume b in [#] B ;
let S be non void ManySortedSign , X be non-empty ManySortedSet of 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 , f be Function ;
b ` ` c= b9 ` ` ;
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 & i2 < i2 implies i1 < i2
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 & n < len f ;
assume A c= dom f ;
Re ( f ) is_integrable_on M ;
let k , m be element ;
a , a \equiv b , b ;
j + 1 < k + 1 ;
m + 1 <= n1 ;
g is_not x0 & g is_not x0 ;
g is continuous & x0 in dom f implies g is continuous
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 , P2 = P +* stop I ;
d , c // a , b ;
let t , u be set ;
let X be set ;
assume k in dom s ;
let r be non negative Real ;
assume x in F | M ;
let Y be Subset of S ;
let X be non empty TopSpace , f be Function of X , X ;
[ a , b ] in R ;
x + w < y + w ;
{ a , b } >= c ;
let B be Subset of A , C be Subset of A ;
let S be non empty ManySortedSign ;
let x be variable , f be Function ;
let b be Element of X , c be Element of X ;
R [ x , y ] ;
x ` ` = x ` ` .= x ;
b \ x = 0. X ;
<* d *> in D * ;
P [ k + 1 ] ;
m in dom mn ;
h2 . a = y ;
P [ n + 1 ] ;
redefine func G * F -> preJ ;
let R be non empty multMagma , I be non empty multMagma , J be non empty FinSequence 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 ^ <* x *> ) ;
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 `1 ;
let M be mamaid ;
let N be non empty for \mathop { N } is non empty Subset of M ;
let R be RelStr with finite finite and R is finite ;
let n , k be Nat ;
let P , Q be be be be be be be be be Let ;
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 lim lim = a ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < ( v - u ) ;
x <= c2 . x ;
x in F ` & y in F ` ;
redefine func S --> T -> ManySortedSet of X ;
assume that t1 <= t2 and t2 <= t1 ;
let i , j be even Integer ;
assume F1 <> F2 & F2 <> G2 ;
c in Intersect ( union R ) ;
dom p1 = c & dom p2 = c ;
a = 0 or a = 1 ;
assume A1 : x <> A6 ;
set i1 = i + 1 ;
assume a1 = b1 & a2 = b2 ;
dom g1 = A & dom g2 = A ;
i < len M + 1 ;
assume not -infty in rng G ;
N c= dom ( f1 - f2 ) ;
x in dom sec & y in dom sec ;
assume [ x , y ] in R ;
set d = ( x - y ) / ( x - y ) ;
1 <= len g1 & 1 <= len g2 ;
len s2 > 1 & len s2 > 1 ;
z in dom ( f1 ^ ) ;
1 in dom ( D2 | 1 ) ;
( p `2 ) ^2 = 0 ;
j2 <= width G & 1 <= 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 _\mathbin of \mathbin { - } 1 } ;
cluster m * n -> invertible ;
let k9 be Nat , x be Element of NAT ;
i - 1 > m - 1 ;
R is transitive implies field R = field R
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 \frac ;
i <= len ( f2 ^ <* p *> ) ;
p is FinSequence of X ;
1 + 1 in dom g ;
Partial_Sums R2 = n * r ;
cluster f . x -> complex-valued ;
x in dom ( f1 - f2 ) ;
assume [ X , p ] in C ;
BX c= ( X0 \/ X2 ) ;
n2 <= ( 2 |^ ( n + 1 ) ) ;
A /\ cP c= A ` ;
cluster x -valued -> constant for Function ;
let Q be Subset-Family of S , P be Subset of S ;
assume n in dom g2 & m in dom g2 ;
let a be Element of R ;
t `2 in dom ( e2 | ( dom e2 ) ) ;
N . 1 in rng N ;
- z in A \/ B ;
let S be SigmaField of X , T be non empty set ;
i . y in rng i ;
REAL c= dom f & dom f c= dom f ;
f . x in rng f ;
mt <= ( r - 1 ) / 2 ;
s2 in r-5 & s2 in r-5 ;
let z , z be complex number ;
n <= ( N . m ) ;
LIN q , p , s ;
f . x = waybelow x /\ B ;
set L = [: S , T :] ;
let x be non real number , r be positive Real ;
m be Element of M ;
f in union rng ( F1 ^ F2 ) ;
let K be add-associative right_zeroed right_complementable non empty doubleLoopStr , M be Matrix of K ;
let i be Element of NAT , k be Element of NAT ;
rng ( F * g ) c= Y
dom f c= dom x & rng f c= dom x ;
n1 < n1 + 1 & n2 + 1 < n1 + 1 ;
n1 < n1 + 1 & n2 + 1 < n1 + 1 ;
cluster ( T | X ) -> thesis ;
[ y2 , 2 ] `2 = z ;
let m be Element of NAT , n be Element of NAT ;
let S be Subset of R ;
y in rng ( S29 ) ;
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 & m in dom h2 ;
w + 1 = ma + 1 ;
j + 1 <= j + 1 + 1 ;
k2 + 1 <= k1 & k1 + 1 <= k2 ;
i be Element of NAT , k be Element of NAT ;
Support u = Support p & Support u = Support p ;
assume X is complete complete Subempty ;
assume that f = g and p = q ;
n1 <= n1 + 1 & n2 <= n1 + 1 ;
let x be Element of REAL , y be Element of REAL ;
assume x in rng ( s2 | X ) ;
x0 < x0 + 1 & x0 < x0 + 1 ;
len ( L5 * L ) = W ;
P c= Seg len A & P c= Seg len A ;
dom q = Seg n & dom q = Seg n ;
j <= width M *' ;
let r8 be real-valued sequence of REAL , x be element ;
let k be Element of NAT , n be Element of NAT ;
Integral ( M , P ) < +infty ;
let n be Element of NAT , x be Element of NAT ;
assume z in z -> \cup of -> } ;
let i be set ;
n - 1 = n-1 - 1 ;
len ( n-27 ) = n & len ( nu ) = n ;
\mathop { 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 , x be Element of L ;
Seg i = dom q & Seg i = dom q ;
let s be Element of E ^ omega ;
let B1 be Basis of x , B2 be Basis of x ;
Carrier ( L2 ) /\ Carrier ( L2 ) = {} ;
L1 /\ LSeg ( L2 , L2 ) = {} ;
assume downarrow x = downarrow y ;
assume b , c // b `1 , c `2 ;
LIN q , c , c ;
x in rng ( f . -129 ) ;
set n8 = n + j ;
let D7 be non empty set , f be FinSequence of D ;
let K be add-associative right_zeroed right_complementable non empty addLoopStr , M be Matrix of K ;
assume that f opp = f and h opp = h ;
R1 - R2 is total & R2 - R1 is total ;
k in NAT & 1 <= k & k <= n ;
let a be Element of G ;
assume x0 in [. a , b .] ;
K1 ` is open & K1 ` is open ;
assume a , b ] is maximal & a 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 -> nn] for ;
not u in { bb } ;
the carrier of f c= B ;
reconsider z = x as VECTOR of V ;
cluster \rm Str over L -> non-empty ;
r (#) H is ^ " X ;
s . intloc 0 = 1 & s . intloc 0 = 1 ;
assume that x in C and y in C ;
let U0 be strict non-empty non-empty non-empty MSAlgebra over S , A be non-empty MSAlgebra over S ;
[ x , Bottom T ] is compact ;
i + 1 + k in dom p ;
F . i is stable Subset of M ;
r-35 in : not ( ex y st y in { x } & y in { x } ) ;
let x , y be Element of X ;
A , I be \times of X ;
[ y , z ] in [: O , O :] ;
( that that that that card Macro i ) = 1 and card Macro i = 1 ;
rng ( Sgm A ) = A ;
q |- f from All ( y , q ) ;
for n holds X [ n ] ;
x in { a } & x in d ;
for n holds P [ n ] ;
set p = |[ x , y , z ]| ;
LIN o , a , b & LIN o , a , b ;
p . 2 = Z / Y ;
( D . 0 ) `2 = {} & ( D . 0 ) `2 = 0 ;
n + 1 + 1 <= len g ;
a in [: CQC-WFF ( Al ) , NAT :] ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
let I be Ideal of L ;
set g = f1 + f2 , h = f1 + f2 ;
a <= max ( a , b ) ;
i-1 < len G + 1-1 ;
g . 1 = f . i1 ;
x `2 , y `2 in A2 ;
( f /* s ) . k < r ;
set v = VAL g ;
i -' k + 1 <= S ;
cluster associative for non empty multMagma ;
x in support ( support ( t ) ) ;
assume a in [: the carrier of G , the carrier of G :] ;
i `2 <= len ( y `2 ) ;
assume p divides b1 + b2 & p divides b2 ;
M1 <= sup M1 & M1 <= sup M2 implies M1 + M2 <= M1 + M2
assume x in W-min ( X ) & y in L~ f ;
j in dom ( z ^ <* x *> ) ;
let x be Element of D ( ) ;
IC s4 = l1 & IC s4 = l1 ;
a = {} or a = { x } ;
set uG = Vertices G , uG = Vertices G ;
seq " is non-zero & seq " is non-zero implies seq " is non-zero
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
hcn c= h-14 & hcn c= dom h ;
]. a , b .[ c= Z ;
X1 , X2 , X3 be non empty SubSpace of X ;
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 zero Nat , M be Matrix of n , REAL ;
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 & sup B in A ;
let L be non empty reflexive transitive RelStr , X be Subset of L ;
R is reflexive & X is transitive implies R is transitive
E , g |= ( the_left_argument_of H ) ;
dom G `2 = a & dom G = b ;
( 1 - 4 ) * ( - 1 ) >= - r ;
G . p0 in rng G & G . p0 in rng G ;
let x be Element of FF , y be Element of FF ;
D [ P-6 , 0 ] ;
z in dom id B & z in dom id B ;
y in the carrier of N & x in the carrier of N ;
g in the carrier of H & h in the carrier of G ;
rng f\mathbb R c= [: NAT , NAT :] & rng f\mathbb R c= [: NAT , NAT :] ;
j `2 + 1 in dom s1 & j `2 + 1 in dom s2 ;
let A , B be strict Subgroup of G ;
let C be non empty Subset of REAL , a , b be Real ;
f . z1 in dom h & f . z2 in dom h ;
P . k1 in rng P & P . k1 in rng P ;
M = AM +* {} .= ( A +* {} ) +* {} ;
let p be FinSequence of REAL , n be Nat ;
f . n1 in rng f & f . n1 in rng f ;
M . ( F . 0 ) in REAL ;
ind [. a , b .[ = b-a ;
assume the distance of V , Q is Real ;
let a be Element of ( ^ V ) . a ;
let s be Element of [: P , Q :] ;
let PA be non empty reflexive transitive RelStr , a be Element of Y ;
n be Nat ;
the carrier of g c= B & the carrier of g c= A ;
I = halt SCM R & I = ( the carrier of SCM R ) ;
consider b being element such that b in B ;
set BM = BCS K , BM = BCS K ;
l <= -> and sup ( F . j ) <= sup ( F . j ) ;
assume x in downarrow [ s , t ] ;
( x `2 ) ^2 in uparrow t & ( x `2 ) ^2 in uparrow t ;
x in ( <* 1 *> , 1 ) -tuples_on ( { 1 } , NAT ) ;
let h be Morphism of c , a ;
Y c= 1. ( the_rank_of Y ) & Y c= the_rank_of Y ;
A2 \/ A3 c= Carrier ( f ) \/ Carrier ( f ) ;
assume LIN o , a , b & LIN o , a , b ;
b , c // d1 , e2 & a , b // d2 , e2 ;
x1 , x2 , x3 , x4 , x5 be set ;
dom <* y *> = Seg 1 & rng <* y *> = Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar s .| ;
[ x , x `2 ] in X ~ ;
for n be Nat holds 0 <= x . n
[' a , b '] = [. a , b .] ;
cluster -> finite closed for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 , q1 is_collinear & q2 , q2 , q2 is_collinear ;
dom M1 = Seg n & rng M1 c= Seg n ;
x = [ x1 , x2 , x3 ] ;
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 implies 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 ( that S * R ) ;
let b be Element of the lattice of T ;
dist ( e , z ) > r-r ;
u1 + v1 in W2 & v1 + v2 in W1 ;
assume the carrier of L misses rng G ;
let L be lower-bounded antisymmetric transitive non empty RelStr ;
assume [ x , y ] in a9 & [ y , x ] in a9 ;
dom ( A * e ) = NAT & rng ( A * e ) = NAT ;
let a , b be Vertex of G ;
let x be Element of Bool M , y be Element of Bool M ;
0 <= Arg a & Arg a < 2 * PI ;
o9 , a9 // o9 , y & o9 , a9 // o9 , y ;
{ v } c= the carrier of l & { v } c= the carrier of l ;
let x be variable of A ;
assume x in dom ( uncurry f ) & y in dom f ;
rng F c= ( product f ) |^ X
assume D2 . k in rng D & D . k in rng D ;
f " . p1 = 0 & f " . p2 = 0 ;
set x = the Element of X , y = the Element of Y ;
dom Ser ( G ) = NAT & rng Ser ( G ) = NAT ;
n be Element of NAT , x be Element of NAT ;
assume LIN c , a , e1 ;
cluster -> natural for FinSequence of NAT ;
reconsider d = c as Element of L1 ;
( v2 |-- I ) . X <= 1 ;
assume x in the carrier of f & y in the carrier of f ;
conv @ A c= conv conv A & conv @ A c= conv @ A ;
reconsider B = b as Element of the open domains of T ;
J , v |= P ! l & J , v |= P ! l ;
redefine func J . i -> non empty TopSpace equals J . i ;
ex_sup_of Y1 \/ Y2 , T & ex_sup_of Y1 , T ;
W1 , W2 is_well field W1 & R is transitive implies R \/ ( R \/ ( R \/ ( R \/ S ) ) ) is transitive
assume x in the carrier of R & y in the carrier of R ;
dom ( n | n ) = Seg n & rng ( n | n ) = Seg n ;
s4 misses s2 & s4 misses s2 ;
assume ( a 'imp' b ) . z = TRUE ;
assume that X is open and f = X --> d ;
assume [ a , y ] in an & [ a , y ] in C2 ;
assume that not that that that that that that that that that that stop I c= J and not x in K ;
Im ( ( lim seq ) - ( lim seq ) ) = 0 ;
( sin . x ) ^2 <> 0 & ( sin . x ) ^2 <> 0 ;
sin + cos is_differentiable_on Z & for x st x in Z holds cos . x <> 0 & cos . x <> 0
t3 . n = t3 . n .= t3 . n .= s . n ;
dom ( ( - F ) | A ) c= dom F ;
W1 . x = W2 . x & W2 . x = W2 . x ;
y in W .vertices() \/ W .vertices() \/ W .vertices() ;
( k + 1 ) <= len ( v | ( k + 1 ) ) ;
x * a \equiv y * a . ( mod m ) ;
proj2 .: S c= proj2 .: P & proj2 .: P c= proj2 .: P ;
h . p4 = g2 . I .= g2 . I ;
IT = ( U /. 1 ) `1 .= ( U /. 1 ) `1 ;
f . rr1 in rng f & f . rr1 in rng f ;
i + 1 + 1 <= len - 1 ;
rng F = rng ( F | ( Seg n ) ) ;
mode non empty multMagma over L is well unital non empty multMagma ;
[ x , y ] in A ~ { a } ;
x1 . o in L2 . o & x2 . o in L2 . o ;
the carrier of \HM { the carrier of m c= B ;
not [ y , x ] in id X & not y in X ;
1 + p .. f <= i + len f ;
seq ^\ k1 is lower & seq ^\ k1 is lower implies seq ^\ k1 is lower
len ( F ^ <* I *> ) = len I + 1 ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be complex number , a be complex number ;
Comput ( P , s , n ) = s ;
k <= k + 1 & k + 1 <= len p ;
reconsider c = {} T as Element of L ;
let Y be non empty \rbrace Chain of T , T ;
cluster empty -> directed-sups-preserving for Function of L , L ;
f . j1 in K . j1 & f . j1 in K . j1 ;
redefine func J => y -> total Function equals J * ( J * J ) ;
K c= 2 -tuples_on the carrier of T ;
F . b1 = F . b2 & F . b2 = F . b2 ;
x1 = x or x1 = y or x1 = z ;
pred a <> {} means : Def2 : ( a - a ) / a = 1 ;
assume that a c= b and b in a ;
s1 . n in rng s1 & s1 . n in rng s1 ;
{ o , b2 } on C2 & not o , b1 on C2 ;
LIN o , b , b9 & LIN o , b , b9 ;
reconsider m = x as Element of Funcs ( V , C ) ;
let f be non constant FinSequence of D , i be Nat ;
let FG2 be non empty element , f be FinSequence of the carrier of X ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in F-8 ;
reconsider pn2 = x , pn2 = y as Subset of m -tuples_on REAL ;
A , B , C be Element of R ;
redefine func strict non empty for as strict non empty as strict as strict vector of V ;
rng c `2 misses rng ( e | ( rng e ) ) ;
z is Element of gr { x } & z is Element of gr { x } ;
not b in dom ( a .--> p1 ) & not b in dom ( a .--> p1 ) ;
assume that k >= 2 and P [ k ] ;
Z c= dom ( ( - cot ) (#) ( cot * cot ) ) ;
the component of Q c= UBD ( A ) & ( the component of Q ) /\ ( the component of A ) = {} ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( ( 1 / 2 ) (#) ( f ^ ) ) ;
pred f = u means : Def2 : a * f = a * u ;
for n holds P1 [ \mathop { n } ] ;
{ x . O : x in L } <> {} ;
x be Element of V . s ;
a , b be Nat ;
assume that S = S2 and p = S2 and S is non empty ;
gcd ( n1 , n2 , n3 ) = 1 & gcd ( n1 , n2 , n3 ) = 1 ;
set oo = ( * _ { Z , 2 } ) * ( o , 2 ) ;
seq . n < |. r1 .| & seq . n < |. r1 .| ;
assume that seq is increasing and r < 0 and seq is increasing ;
f . ( y1 , x1 ) <= a & f . ( y1 , x1 ) <= b ;
ex c being Nat st P [ c ] & a = c ;
set g = { n to_power 1 : n in NAT } ;
k = a or k = b or k = c ;
aa , ag , k , G & k in dom f & k in dom f & k in dom f ;
assume that Y = { 1 } and s = <* 1 *> ;
IA1 . x = f . x .= f . x .= 0 ;
W3 .last() = W3 . 1 & y in W3 . 2 ;
cluster trivial -> trivial for Walk of G , finite set ;
reconsider u = u as Element of Bags X ;
A in B ^ implies A , B ^ , A ^ ;
x in { [ 2 * n + 3 , k ] } ;
1 >= ( q `1 ) ^2 + ( q `2 ) ^2 ;
f1 is_LSeg ( f1 , i ) & f2 is_the carrier' of f1 ;
( f /. i ) ^2 <= ( q `1 ) ^2 + ( q `2 ) ^2 ;
h is_the / 2 in the carrier of Cage ( C , n ) ;
( b - a ) ^2 <= ( p `1 - a ) ^2 + ( p `2 - a ) ^2 ;
let f , g be \ast Function of X , Y ;
S * ( k , k ) <> 0. K ;
x in dom ( - f ) & x in dom ( - f ) ;
p2 in ( N . p1 ) & p2 in ( N . p1 ) ;
len ( the_left_argument_of H ) < len ( H ) & len ( the_left_argument_of H ) < len ( H ) ;
F [ A , FF . A ] ;
consider Z such that y in Z and Z in X ;
pred 1 in C means : Def2 : A c= C |^ A ;
assume that r1 <> 0 or r2 <> 0 and r1 < r2 and r2 < 0 ;
rng q1 c= rng C1 & rng q1 c= rng C2 ;
A1 , L , A3 , A3 , A3 be non empty set ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in let ( p , Ss ) . b ;
then S is negative & P-2 [ S ] ;
Cl Int [#] T = [#] T & Cl Int [#] T = [#] T ;
f12 | A2 = ( f2 | A1 ) | A2 .= ( f2 | A2 ) | A2 ;
0. M in the carrier of W & 0. M in the carrier of W ;
v , v be Element of M ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ V & Y \ V c= Y \ Z ;
let X be Subset of S ~ ;
consider H1 such that H = 'not' H1 and H1 in A ;
1_ 1 c= ( 1 * ( p - 1 ) ) * ( p - 1 ) ;
0 * a = 0. R .= a * 0. R .= a * 0. R ;
A |^ ( 2 , 2 ) = A ^^ A .= A ;
set vY = ( v /. n ) , vY = ( v /. n ) ;
r = 0. ( REAL-NS n ) & ||. x - x0 .|| < r ;
( f . p4 ) ^2 >= 0 & ( f . p4 ) ^2 >= 0 ;
len W = len ( W | ( len W ) ) ;
f /* ( s * G ) is divergent_to+infty ;
consider l being Nat such that m = F . l ;
t16 . ( W7 ) does not destroy b1 , b2 ;
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 & 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 , x3 , x4 ] -> non pair ;
downarrow a /\ downarrow t is Ideal of T implies a in downarrow t
let X be non empty set , N be non empty set ;
rng f = S1 -element ( S , X ) ;
let p be Element of B , x be Element of the carrier' of S ;
max ( N1 , 2 ) >= N1 & max ( N1 , 2 ) >= N2 ;
0. X <= b |^ m * mm1 ;
assume that i in I and R0 . i = R ;
i = j1 & p1 = q1 & p2 = q2 implies p1 = p2
assume gR in the right of g & gR in the carrier of g ;
let A1 , A2 be Point of S , A2 be Point of T ;
x in h " P /\ [#] T1 & x in h " P /\ [#] T2 ;
1 in Seg 2 & 1 in Seg 3 & 2 in Seg 3 ;
reconsider X-5 = X , Xlet T = Y as non empty Subset of Tlet T be non empty TopSpace ;
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the Source of G ) -valued ;
n1 <= i2 + len g2 & n2 + len g2 <= len g2 ;
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume v in the carrier' of G2 & u in the carrier' of G2 ;
y = Re ( y ) + ( Im ( y ) * i ) ;
( ( - 1 ) |^ ( - 1 ) ) gcd p = 1 ;
x2 is_differentiable ]. a , b .[ & x2 - a < b - a ;
rng M5 c= rng ( D2 | ( D1 . i ) ) ;
for p being Real st p in Z holds p >= a
( ( cn ) | K1 ) . p = proj1 * f . p .= ( cn ) . p ;
( seq ^\ m ) . k <> 0 & ( seq ^\ m ) . k <> 0 ;
s . ( G . ( k + 1 ) ) > x0 ;
( p |-count M ) . 2 = d ;
A \oplus ( B \ominus C ) = ( A \oplus B ) \ominus C
h \equiv gg . ( mod P ) & g divides gg . ( mod P ) ;
reconsider i1 = i-1 , i2 = i-1 as Element of NAT ;
let v1 , v2 be VECTOR of V , v be VECTOR of V ;
for V being Subspace of V holds V is Subspace of [#] V
reconsider i = i , j = j as Element of NAT ;
dom f c= [: C , D ( ) :] ;
x in ( the sequence of B ) . n & x in ( the carrier of B ) ;
len that len that len that 2 in Seg len ( f2 ^ <* p *> ) ;
pp1 c= the topology of T & pp2 c= the topology of T ;
]. r , s .[ c= [. r , s .] ;
let B2 be Basis of T2 , B be Basis of T2 ;
G * ( B * A ) = ( id o1 ) * ( B * A ) ;
assume that p , u , v is_collinear and u , v , w is_collinear ;
[ z , z ] in union rng ( F | ( union rng F ) ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S , C = $1 .. S , D = $1 .. S , E = $1 .. S , F = $1 .. S , N = $1 .. S , N = $1 .. S , N
LIN a1 , a3 , b1 & LIN a1 , a2 , c1 & LIN a2 , a3 , c1 ;
f " ( f .: x ) = { x } ;
dom w2 = dom r12 & dom r12 = dom r12 ;
assume that 1 <= i and i <= n and j <= n ;
( ( g2 ) . O ) `2 <= 1 ;
p in LSeg ( E . i , F . i ) ;
Ii * ( i , j ) = 0. K ;
|. f . ( s . m ) - g .| < g1 ;
q9 . x in rng ( q | ( Seg n ) ) ;
Carrier ( LLet ) misses Carrier ( Ld ) ` & Carrier ( Ld ) misses Carrier ( Ld ) ;
consider c being element such that [ a , c ] in G ;
assume that Nreal = oand onon empty and oO is non empty ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= ( F |^ ( C * ) ) " { x }
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [. f . j , f . j .] ;
pred 0 <= x & x <= 1 implies x ^2 <= x ^2 ;
p `2 - q `2 <> 0. TOP-REAL 2 & p `2 - q `2 <> 0. TOP-REAL 2 ;
redefine func being aa] ( S , T ) -> non empty ;
x be Element of S ~ ;
( \HM { the } \HM { object } \HM { of F ) is one-to-one ;
|. i .| <= - ( - 2 to_power n ) / ( 2 to_power n ) ;
the carrier of I[01] = dom P & the carrier of I[01] = dom P ;
th * ( n + 1 ) ! > 0 * th ;
S c= ( A1 /\ A2 ) /\ ( A2 /\ A3 ) ;
a3 , a4 // b3 , b2 & a3 , a4 // b3 , b2 ;
then dom A <> {} & dom A <> {} & rng A c= dom B ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , Y & y in X implies x = y
set v2 = ( v /. ( i + 1 ) ) ;
x = r . n .= ( r . n ) * ( r . n ) ;
f . s in the carrier of S2 & f . s in the carrier of S2 ;
dom g = the carrier of I[01] & rng g c= the carrier of I[01] ;
p in Upper_Arc ( P ) /\ Lower_Arc ( P ) & p in Lower_Arc ( P ) /\ Lower_Arc ( P ) ;
dom d2 = [: A , A :] & dom d2 = [: A , A :] ;
0 < ( p - ||. z .|| ) + 1 ;
e . ( mm + 1 ) <= e . ( m3 + 1 ) ;
B \ominus X \/ B \ominus Y c= B \ominus X /\ Y
-infty < Integral ( M , Im ( g | B ) ) ;
cluster O \cup F -> Line for operation of X ;
let U1 , U2 be non-empty MSAlgebra over S , S be non-empty MSAlgebra over S ;
Proj ( i , n ) * g is_differentiable_on X & Proj ( i , n ) * g is_differentiable_on X ;
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 ) & y 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 in - b ;
Int Cl A c= Cl Int Cl Int Cl A & Cl Int Cl A c= Cl Int Cl A ;
assume for A being Subset of X holds Cl A = A ;
assume q in Ball ( |[ x , y ]| , r ) ;
( p2 `2 ) ^2 <= ( p `2 ) ^2 + ( p `2 ) ^2 ;
Cl Q ` = [#] ( T | ( [#] T ) ) .= [#] ( T | ( [#] T ) ) ;
set S = the carrier of T , T = the carrier of T ;
set I8 = -> f |^ n , I8 = f |^ n , I8 = f |^ n ;
len p - n = len thesis - n & len p - n = len p - n ;
A is Permutation of Swap ( A , x , y ) ;
reconsider nni = ni - 1 as Element of NAT ;
1 <= j + 1 & j + 1 <= len ( s | j ) ;
let q\mathopen be Let of M , q\mathopen ( 2 , n ) , q be Element of M ;
aa in the carrier of S1 & not xy in the carrier of S1 ;
c1 /. n1 = c1 . n1 & c2 /. n1 = c2 . n1 ;
let f be FinSequence of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
y = ( f * ( S * ( x , y ) ) ) . x ;
consider x being element such that x in an " A and x in B ;
assume r in ( dist ( o , r ) ) .: P ;
set i2 = \mathopen { - } h /. 2 , i1 = h /. 3 , i2 = h /. 1 , i2 = h /. 2 ;
h2 . ( j + 1 ) in rng h2 & h2 . ( j + 1 ) in rng h2 ;
Line ( M29 , k ) = M . i .= Line ( M29 , k ) ;
reconsider m = ( x - 2 ) / ( x - 2 ) as Element of REAL ;
let U1 , U2 be non-empty non-empty non-empty non-empty non-empty non-empty non-empty MSAlgebra over S ;
set P = Line ( a , d ) ;
len p1 < len p2 + 1 & len p2 < len p1 + 1 ;
let T1 , T2 be Scott Scott Scott Scott Subset of L , x be Element of T1 ;
then x <= y & ( x + y ) c= ( x + y ) ;
set M = n -\hbox { m } , N = n -\hbox { m } ;
reconsider i = x1 , j = x2 as Nat ;
rng ( ( the_arity_of o ) * ( the_arity_of o ) ) c= dom H ;
z1 " = ( z " ) * ( z " ) .= ( z " ) * ( z " ) .= z " * ( z " ) ;
x0 - r / 2 in L /\ dom f & x0 - r / 2 in L /\ dom f ;
then w is that rng w /\ L <> {} & rng w /\ L <> {} ;
set x-10 = ( x ^ <* Z *> ) ^ <* Z *> ^ ( x ^ <* Z *> ) ;
len w1 in Seg len w1 & len w1 in Seg len w1 & len w1 in Seg len w1 ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of \mathopen { V , { k } } ;
x . n = ( |. a . n .| ) * ( |. b . n .| ) ;
( p `1 ) ^2 <= ( G * ( i1 , j1 ) ) `1 ;
rng ( g ) c= L~ ( g ^ ) & rng ( g ^ ) c= L~ ( g ^ ) ;
reconsider k = i-1 * ( l + j ) as Nat ;
for n be Nat holds F . n is \HM { -infty } ;
reconsider x-10 = xM , x29 = xM 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 as Element of ( len x ) -tuples_on REAL ;
assume i in dom ( a * p ^ q ) ;
m . ( ag ) = p . ( ag ) .= p . ( bg ) ;
a / ( s . m - s . n ) / ( s . m - s . n ) <= 1 ;
S . ( n + k + 1 ) c= S . ( n + k ) ;
assume B1 \/ C1 = B2 \/ C2 & C1 \/ C2 = C1 \/ C2 ;
X . i = { x1 , x2 } . i .= ( x1 | i ) . i ;
r2 in dom ( h1 + h2 ) & r1 in dom ( h1 + h2 ) ;
\mathclose { - 0. R } = a & b-0 = b ;
F8 is_closed_on t8 , Q8 & F8 is_halting_on t8 , Q8 ;
set T = for _ X , x0 , x1 be Element of X ;
Int Cl Int Cl R c= Int Cl R & Int Cl R c= Cl Int Cl R ;
consider y being Element of L such that c . y = x ;
rng ( F{} . x ) = { F{} . x } .= { F . x } ;
G-23 " { c } c= B \/ S \/ S \/ { c } ;
f[#] X is Relation of [: X , X :] , X & f is Function of X , X ;
set Rz = the Element of P , Rz = the Element of Q ;
assume that n + 1 >= 1 and n + 1 <= len M ;
k2 be Element of NAT , k be Element of NAT ;
reconsider pnR = u , pR = v as Element of ( TOP-REAL n ) | ( ( TOP-REAL n ) | R ) ;
g . x in dom f & x in dom g implies x in dom g
assume that 1 <= n and n + 1 <= len f1 ;
reconsider T = b * N as Element of G / ( N , X ) ;
len ( P\bf P ) <= len ( P-35 ) & len ( P\bf P ) <= len ( P-35 ) ;
x " in the carrier of A1 & x " in the carrier of A1 ;
[ i , j ] in Indices ( A @ ) & ( A @ ) * ( i , j ) = A * ( i , j ) ;
for m be Nat holds Re ( F . m ) is simple ;
f . x = a . i .= a1 . k .= a1 . k ;
let f be PartFunc of REAL i , REAL , x be Element of REAL i ;
rng f = the carrier of \bf A & rng f c= the carrier of \bf A ;
assume s1 = sqrt ( 2 * ( p `1 ) ^2 ) & s2 = sqrt ( 2 * ( p `2 ) ^2 ) ;
pred a > 1 & b > 0 & a to_power b > 1 implies a to_power b > 1 ;
let A , B , C be Subset of II , B be Subset of II ;
reconsider X0 = X , Y0 = Y as RealNormSpace , f = X as Point of X ;
let f be PartFunc of REAL , REAL , x be Element 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 be Relation of 2 -tuples_on BOOLEAN , BOOLEAN ;
Q [ e-14 \/ { v-5 } , f ] & f . ( v-5 ) = f . ( e , f . ( e , f . ( e , f . ( e , f . ( e , f . ( e , f . (
g \circlearrowleft ( W-min L~ z ) = z implies ( g /. 1 ) .. z < ( g /. len g ) .. z
|. |[ x , v ]| - |[ x , y ]| .| = vSet ;
- f . w = - ( L * w ) .= - ( L * w ) ;
z - y <= x iff z <= x + y & y <= z + x
( 7 - p1 ) / ( 1 - e ) > 0 ;
assume X is BCK-algebra & X is BCK-algebra & 0 < 0 & 0 < 0 ;
F . 1 = v1 & F . 2 = v2 & F . 3 = v2 ;
( f | X ) . x2 = f . x2 .= f . x2 ;
( ( tan - tan ) `| Z ) . x in dom ( sec * tan ) ;
i2 = ( f /. len f ) `1 .= ( f /. len f ) `1 .= ( f /. len f ) `1 ;
X1 = X2 \/ ( X1 \ X2 ) & X2 = X2 \ ( X1 \ X2 ) ;
[. a , b , 1_ G .] = 1_ G & a * b = 1_ G ;
let V , W be non empty VectSpStr over F_Complex , f be FinSequence of COMPLEX ;
dom g2 = the carrier of I[01] & rng g2 = the carrier of I[01] ;
dom f2 = the carrier of I[01] & rng f2 = the carrier of I[01] & rng f2 = the carrier of I[01] ;
( proj2 | X ) .: X = proj2 .: X .= proj2 .: X .= X ;
f . ( x , y ) = h1 . ( x `1 , y `2 ) ;
x0 - r < a1 . n & x0 - r < x0 - r . n ;
|. ( f /* s ) . k - ( G /* s ) . k .| < r ;
len ( Line ( A , i ) ) = width A & len ( Line ( A , i ) ) = width A ;
SFinSequence / ( g , f ) = ( S . g ) / ( g , f ) ;
reconsider f = v + u as Function of X , the carrier of Y ;
intloc 0 in dom Initialized p & IC Comput ( p , s , 0 ) in dom Initialized p ;
i1 , i2 , i3 , goto ( \overline { \kern1pt I \kern1pt } + 3 ) not contradiction & I is not " ;
arccos r + arccos r = ( PI / 2 ) + 0 .= PI / 2 + 0 ;
for x st x in Z holds f2 - ( f1 - #Z 2 ) . x = - 1 / ( x + x ^2 )
reconsider q2 = ( q `1 ) / ( |. q .| ) as Element of REAL ;
( 0 qua Nat ) + 1 <= i + j1 & i + 1 <= j1 + 1 ;
assume f in the carrier of [: X , Omega :] ;
F . a = H / ( ( x , y ) / ( x , y ) ) ;
( ( TRUE T ) at ( C , u ) ) . x = TRUE ;
dist ( ( a * seq ) . n , h ) < r / 2 ;
1 in the carrier of [. 0 , 1 .] & 3 in the carrier of I[01] ;
( p2 `1 - p1 `1 ) ^2 - ( p2 `2 - p1 `2 ) ^2 > - ( p2 `2 - p1 `1 ) ^2 ;
|. r1 - be Real .| = |. a1 .| * |. thesis .| ;
reconsider S-14 = 8 as Element of Seg 8 , S be non empty set ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b
D0W .( X ) = D0W .( X ) + 1 ;
i1 = ma + n & i2 = C2 + n & j2 = C2 + n ;
f . a [= f . ( f . O1 ) "\/" ( f . a ) ;
pred f = v & g = u + v & f + g = v + u ;
I . n = Integral ( M , F . n ) ;
chi ( [: T1 , T2 :] , S . s ) . s = 1 ;
a = VERUM ( A ) or 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~ ( R ^ <* A *> ) & L~ ( R ^ <* A *> ) meets L~ ( R ^ <* A *> ) ;
set h = the continuous Function of X , R , f be Function of X , R ;
set A = { L . ( k9 . n ) where k is Element of NAT : k <= n } ;
for H st H is atomic holds P7 [ H ] ;
set b9 = S5 ^\ ( i + 1 ) , S = S5 ^\ ( i + 1 ) ;
Hom ( a , b ) c= Hom ( a opp , b opp ) ;
( 1 - s ) / ( n + 1 ) < ( 1 - s ) / ( n + 1 ) ;
( l `1 ) = [ dom l , cod l ] `1 .= dom l ;
y +* ( i , y /. i ) in dom g & y in dom g ;
let p be Element of CQC-WFF ( Al ( ) ) , A be Subset of CQC-WFF ( Al ( ) ) ;
X /\ X1 c= dom ( f1 - f2 ) /\ dom ( f1 - f2 ) ;
p2 in rng ( f /^ ( 1 + 1 ) ) & p2 in rng ( f /^ ( 1 + 1 ) ) ;
1 <= indx ( D2 , D1 , j1 ) + 1 - 1 ;
assume x in ( ( L2 /\ K0 ) \/ ( ( L2 /\ K0 ) /\ ( K \/ L ) ) ) ;
- 1 <= ( ( f2 ) . O ) `2 & - 1 <= ( ( ( f2 ) . O ) `2 ) ;
f , g be Function of I[01] , TOP-REAL 2 , a , b , c be Real ;
k1 -' k2 = k1 - k2 + 1 .= k1 - k2 + 1 ;
rng seq c= ]. x0 - r , x0 .[ & rng seq c= ]. x0 - r , x0 + r .[ ;
g2 in ]. x0 - r , x0 + r .[ & g2 in ]. x0 - r , x0 + r .[ ;
sgn ( p `1 , K ) = - ( - 1_ K ) .= - ( - 1_ K ) ;
consider u being Nat such that b = p |^ y * u ;
ex A being Line of A st a = Sum A & A is limit_ordinal ;
Cl ( union ( H ) ) = union ( ( Cl H ) \ ( Cl H ) ) ;
len t = len t1 + len t2 & len t1 = len t2 + len t1 ;
v-29 = v + w |-- ( v , A ) .= v + ( w |-- ( A , B ) ) ;
cv <> DataLoc ( ( t . GBP ) , 3 ) & cv <> DataLoc ( ( t . GBP ) , 3 ) ;
g . s = sup ( d " { s } ) & g . s = s ;
( \dot { y } ) . s = s . ( \dot { y } . s ) ;
{ s : s < t } in REAL implies t = {}
s ` \ s = s ` \ 0. X .= ( s ` \ 0. X ) \ ( s ` \ 0. X ) .= ( s ` \ 0. X ) \ ( s ` \ 0. X ) ;
defpred P [ Nat ] means B + $1 in A & $1 in A & $1 in B + $1 ;
( 329 + 1 ) ! = 329 ! * ( 329 + 1 ) ;
( ( A * ) * ( B * ) ) = ( ( A * ) * ( B * ) ) ;
reconsider y = y as Element of ( len y ) -tuples_on COMPLEX ;
consider i2 being Integer such that y0 = p * i2 and i2 in dom f and i2 in dom f ;
reconsider p = Y | ( Seg k ) as FinSequence of NAT , k be Nat ;
set f = ( S , U ) \mathop { F } , F = S \! \mathop { F } , G = S \! \mathop { F } , F = S \! \mathop { F } , F = S \! \mathop { F } , f = S \! \mathop { F }
consider Z being set such that lim s in Z and Z in F and x in Z ;
let f be Function of I[01] , TOP-REAL n , x be Point of TOP-REAL n ;
( SAT M ) . [ n + i , 'not' A ] <> 1 ;
ex r being Real st x = r & a <= r & r <= b ;
R1 , R2 be Element of ( n + 1 ) -tuples_on REAL , a be Element of REAL n ;
reconsider l = (0). ( V ) , r = 0. ( V ) as Linear_Combination of A ;
set r = |. e .| + |. n .| + |. w .| + a ;
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\mathopen { x } - g ) * ( x - y ) .|| < r2 ;
b9 , a9 // b9 , c9 & a9 , c9 // b9 , c9 & a9 , c9 // a9 , c9 & a9 , c9 // a9 , c9 ;
1 <= k2 -' k1 & k2 + 1 = k2 & k1 + 1 = k2 + 1 ;
( ( p `2 ) ^2 - ( p `1 ) ^2 ) >= 0 ;
( ( q `2 ) ^2 - ( q `1 ) ^2 ) < 0 ;
E-max C in cell ( R , 1 , 1 ) & E-max L~ Cage ( C , 1 ) in L~ Cage ( C , 1 ) ;
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 `1 , a `2 // b `1 , a `2 ;
g . n = a * Partial_Sums ( f ) . n .= f . n * f . n ;
consider f being Subset of X such that e = f and f is LIN ;
F | ( N2 ~ ) = CircleMap * ( F | ( N2 ~ ) ) .= CircleMap * ( F | ( N2 ~ ) ) ;
q in LSeg ( q , v ) \/ LSeg ( v , p ) ;
Ball ( m , r0 ) c= Ball ( m , s ) & Ball ( m , r ) c= Ball ( m , s ) ;
the carrier of (0). V = { 0. V } & the carrier of V = { 0. V } ;
rng ( ( cos * cos ) `| REAL ) = [. - 1 , 1 .] .= dom cos /\ REAL .= REAL ;
assume that Re ( seq ) is summable and Im ( seq ) is summable and Im ( seq ) is summable ;
||. ( vseq . n ) - ( vseq . n ) .|| < e / 2 ;
set g = O --> 1 ;
reconsider t2 = t11 as 0 -element string of S2 , U = { {} } as ( S , U ) -valued string of S ;
reconsider x-29 = seq , xn = seq as sequence of REAL n ;
assume that that UBD C meets L~ pion1 and L~ pion1 meets L~ pion1 and L~ pion1 meets L~ pion1 and L~ pion1 meets L~ pion1 ;
- ( ( - 1 ) / 2 ) < F . n - ( - 1 ) / 2 ;
set d1 = being thesis , d2 = dist ( x1 , z1 ) , d2 = dist ( x2 , z2 ) , d2 = dist ( x2 , z2 ) ;
2 |^ ( q -' 00 ) = ( 2 |^ ( q -' 00 ) ) - ( 2 |^ ( q -' 00 ) ) ;
dom ( v | ( len d6 ) ) = Seg len ( d6 ) .= Seg len ( d6 ) ;
set x1 = - k2 + |. k2 .| + |. k2 .| + |. k2 .| + |. 1 .| ;
assume for n being Element of X holds 0. <= F . n & 0. <= 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 ( LS ) + Carrier ( LS ) ) c= I2 & the carrier of ( Carrier ( LS ) ) c= I2 ;
'not' Ex ( x , p ) => All ( x , 'not' p ) is valid ;
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the normal w.r.t. over {} , {} ;
Z c= dom ( ( sin * ( sin + cos ) ) ) /\ dom ( cos * ( sin + cos ) ) ;
|. 0. TOP-REAL 2 - ( q `1 / |. q .| - cn ) .| < r / 2 ;
ConsecutiveSet2 ( A , succ B ) c= ConsecutiveSet2 ( A , succ succ ( d , L ) ) ;
E = dom ( L | E ) & L | E is_measurable_on E & E = dom ( L | E ) ;
C / ( A + ) = C / ( B * ) ;
the carrier of W2 c= the carrier of V & the carrier of W1 c= the carrier of V ;
I . IC s = P . IC s .= P . IC s .= ( 0 , s ) .--> 1 ;
pred x > 0 means : Def2 : ( 1 / x ) ^2 = x ^2 / x ^2 ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) \/ LSeg ( g , i ) ;
consider p being Point of T such that C = [. p , g .] and p in A ;
b , c are_connected & - C , - C - D + - D + - E + - F + G + - F + G + - F + G + - F + G - F - G + G - F - G + F - G - G - F - G - F - G
assume f = id the carrier of O1 , g = id the carrier of O2 , f = id the carrier of O1 , g = id the carrier of T ;
consider v such that v <> 0. V and f . v = L * v ;
let l be Linear_Combination of {} ( the carrier of V ) , A be Element of V ;
reconsider g = f " as Function of U1 , U2 , f be Function of U1 , U2 ;
A1 in the Points of G_ ( k , X ) & A2 in the Points of k , X ;
|. - x .| = - ( - x ) .= - x .= - x .= - x ;
set S = ) +* ( x , y , c ) ;
Fib ( n ) * ( 5 * Fib ( n ) - 1 ) >= 4 * ;
( v /. ( k + 1 ) ) = ( v . ( 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 ( S\mathopen ( i , j ) , j ) = S\mathopen ( i , j ) ;
h . ( x1 , y1 ) = [ y1 , x1 ] & h . ( y1 , y2 ) = [ y2 , y1 ] ;
|. f .| - Re ( |. f .| * ( card b * h ) ) is nonnegative ;
assume x = ( a1 ^ <* x1 *> ) ^ b1 & y = ( a2 ^ <* x1 *> ) ^ b1 ;
ME is_closed_on IExec ( I , P , s ) , P & ME is_halting_on s , P ;
DataLoc ( t3 . a , 4 ) = intpos ( 0 + 4 ) .= intpos ( 0 + 4 ) ;
x + y < - x + y & |. x .| = - x + y implies x = y
LIN c , q , b & LIN c , q , q & LIN c , q , c ;
f'not' and . ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + z1 ) .= x1 + ( y1 + z1 ) ;
flet . a = f_ . a & v in InputVertices S & v in InputVertices S & v in InputVertices S ;
( p `1 ) ^2 <= ( E-max C ) ^2 + ( E-max C ) ^2 ;
set R8 = Cage ( C , n ) , E8 = Cage ( C , n ) , E8 = Cage ( C , n ) , E8 = Cage ( C , n ) , R8 = Cage ( C , n ) , R8 = Cage ( C , n ) , R8 = Cage (
( p `1 ) ^2 >= ( E-max C ) ^2 & ( E-max C ) ^2 >= ( E-max C ) ^2 ;
consider p such that p = p-20 and s1 < p and p < s2 and s2 <= p ;
|. ( f /* ( s * F ) ) . l - ( G /* ( s * F ) ) . l .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len ( Line ( N , k + 1 ) ) = width N & len ( Line ( N , k ) ) = width N ;
f1 /* s1 is convergent & f2 /* s1 is convergent & lim ( f1 /* s1 ) = lim ( f1 /* s1 ) ;
f . x1 = x1 & f . y1 = y1 & f . y2 = y2 ;
len f <= len f + 1 & len f + 1 <> 0 implies f ^ <* p *> ^ f = f ^ <* p *>
dom ( Proj ( i , n ) * s ) = REAL m & rng ( Proj ( i , n ) * s ) = REAL m ;
n = k * ( 2 * t ) + ( n mod 2 ) ;
dom B = 2 -tuples_on the carrier of V \ { {} } & rng B = the carrier of V ;
consider r such that r _|_ a and r _|_ x and r _|_ y and r _|_ y ;
reconsider B1 = the carrier of Y1 , B2 = the carrier of Y2 as Subset of X ;
1 in the carrier of [. 1 / 2 , 1 .] & 1 / 2 in dom ( f ^ ) ;
for L being complete LATTICE for a , b being Element of ConceptLattice ( L ) holds a , b ] is isomorphic implies a = b
[ gi , gj ] in Ii \ Ii " { i } & [ gi , gj ] in Ii ;
set S2 = 1GateCircStr ( x , y , c ) , S2 = 1GateCircStr ( y , c ) ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and for r st r in dom f1 & r in dom f2 holds f1 . r = 0 ;
reconsider y = ( a " ) * ( F . ( a * F . ( a * x ) ) ) as Element of L ;
dom s = { 1 , 2 , 3 } & s . 1 = d1 & s . 2 = d2 ;
( min ( g , ( 1 - g ) ) . c ) <= h . c ;
set G2 = the as a as a as a as Vertex of G , ( the carrier' of G ) \ { v } ;
reconsider g = f as PartFunc of REAL , REAL-NS n , REAL-NS n , r be Real ;
|. s1 . m - p .| / |. p .| < d / ( p |^ m ) ;
for x being element st x in ( for u being element st u in ( t * u ) holds u in ( t * u ) ;
P = the carrier of ( TOP-REAL n ) | P & Q = ( TOP-REAL n ) | P ;
assume that p10 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) and LSeg ( p1 , p2 ) /\ LSeg ( p2 , p2 ) = {} ;
( 0. X \ x ) to_power ( m * ( k + 1 ) ) = 0. X ;
let g be Element of Hom ( cod f , cod f ) ;
2 * a * b + ( 2 * c * d ) <= 2 * C1 * C2 + ( 2 * c * d ) ;
f , g , h be Point of the carrier of X , g be Point of Y , f be PartFunc of X , Y ;
set h = Hom ( a , g (*) f ) ;
then idseq ( n ) | ( Seg m ) = idseq ( m ) & m <= n ;
H * ( g " * a ) in the right of H & H * ( g " * a ) in the carrier of H ;
x in dom ( ( cos * sin ) `| Z ) & x in dom ( ( cos * cos ) `| Z ) ;
cell ( G , i1 , j2 -' 1 ) misses C & cell ( G , i1 , j2 ) misses C ;
LE q2 , p4 , P & LE p1 , p2 , P & LE p2 , p3 , P & LE p1 , p2 , P & p2 in P implies p1 , p2 , P
attr B is bounded means : Def2 : B c= BDD A & B c= BDD B ;
deffunc D ( set , set ) = union rng $2 & $2 = union rng $2 ;
n + - n < len ( ( p + - n ) * ( p + - n ) ) ;
pred a <> 0. K means : Def2 : the_rank_of M = the_rank_of ( a * M ) ;
consider j such that j in dom \mathbb B and I = len PI + j and len I = len PI + j ;
consider x1 such that z in x1 and x1 in P8 and x2 in P8 and x = [ x1 , x2 ] ;
for n ex r being Element of REAL st X [ n , r ] & X [ n , r ]
set CA1 = Comput ( P2 , s2 , i + 1 ) , CA2 = Comput ( P2 , s2 , i + 1 ) , CA2 = P2 ;
set cv = 3 / ( 3 / ( 2 * ( a - b ) ) ) , cv = 3 / ( 2 * ( a - b ) ) ;
conv @ W c= union ( F .: ( E " ( W " ( W " ( W ) ) ) ) ) ;
1 in [. - 1 , 1 .] /\ dom ( ( arccot * ( arccot ) ) `| Z ) ;
r3 <= s0 + ( r0 - ( 1 - ( v2 - ( v2 - ( v2 - ( v1 - ( v2 - ( v2 - ( v2 - ( v2 - ( v2 - ( v2 - ( v2 - ( v2 - v1 - v1 - v2 - v1 ) ) ) ) ) ) / 2 ) ) ) )
dom ( f * f4 ) = dom f /\ dom ( f * f4 ) .= dom f /\ dom ( f * f4 ) ;
dom ( f * G ) = dom ( l (#) F ) /\ Seg k .= Seg k ;
rng ( s ^\ k ) c= dom f1 \ { x0 } & rng ( s ^\ k ) c= dom f1 \ { x0 } ;
reconsider g9 = gpp , gp = gp as Point of ( TOP-REAL n1 ) | K1 ;
( T * h . s ) . x = T . ( h . s . x ) ;
I . ( L . ( J . x ) ) = ( I * L ) . x ;
y in dom ( *> , ( commute ( Frege ( A . o ) ) ) . ( ( commute ( A . o ) ) . x ) ) ;
for I being non degenerated commutative Ring for I being commutative non empty doubleLoopStr holds the carrier of I is commutative
set s2 = s +* Initialize ( ( intloc 0 ) .--> 1 ) , P2 = P +* Initialize ( ( intloc 0 ) .--> 1 ) ;
P1 /. IC s1 = 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 *' ) . i .= v . i ;
consider n being element such that n in NAT and x = seq . n and x in dom seq and y = seq . n ;
consider x being Element of c such that F1 . x <> F2 . x and F1 . x <> 0 ;
holds Funcs ( X , 0 , x1 , x2 , x3 , x4 ) = { E } & card X = 1 ;
j + ( 2 * ( k9 + 1 ) ) > j + ( 2 * ( k9 + 1 ) ) ;
{ s , t } on A3 & { s , t } on B2 & { s , t } on B2 ;
n1 > len crossover ( p2 , p1 , n1 , n2 , n3 , n3 , n3 ) & n2 > len crossover ( p2 , p1 , n1 , n2 , n3 ) ;
mm1 . HT ( mm2 , T ) = 0. L & mm2 . HT ( mm2 , T ) = 0. L ;
then H1 , H2 |^ ( a * b ) -> that ( Cl H1 ) , ( Cl H2 ) |^ ( a * b ) -> that a * b = a * b ;
( ( N-min L~ f ) .. ( f | ( 1 + 1 ) ) ) .. ( f | ( 1 + 1 ) ) > 1 ;
]. s , 1 .[ = ]. s , 2 .[ /\ [. 0 , 1 .] .= [. 0 , 1 .] /\ [. 0 , 1 .] ;
x1 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) ) & x2 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) ) ;
let f1 , f2 be continuous PartFunc of REAL , the carrier of S , X be non empty set ;
DigA ( t-23 , z9 ) is Element of k -tuples_on ( k -tuples_on ( k -tuples_on ( k -tuples_on ( k -tuples_on ( k -tuples_on ( k -tuples_on ( k -tuples_on ( k -tuples_on ( k + 1 ) ) ) ) ) ) ) ) ;
I gcd 22= di2 & I is Element of k2 implies I is Element of k2
u9 ~ { u9 } = { [ a , u9 ] , [ a , u9 ] } & u9 in { [ a , u9 ] } ;
( w | p ) | ( p | ( w | ( w | ( w | ( w | p ) ) ) ) ) = p ;
consider u2 such that u2 in W2 and x = v + u2 and u1 in W2 and u2 in W2 ;
for y st y in rng F ex n st y = a |^ n & a in F . n ;
dom ( ( g * ( \mathord V , C ) ) | K ) = K & dom ( ( g * ( id V , C ) ) | K ) = K ;
ex x being element st x in ( ( ( ( U0 ) \/ A ) . s ) . s ) ;
ex x being element st x in ( ( that ( for s being element st s in O holds s in O ) ) . 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 X1 union X2 ) <> {} ;
L1 /\ LSeg ( p10 , p2 ) c= { p10 } /\ LSeg ( p10 , p2 ) \/ { p2 } ;
( b + ( bLet ) ) / 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 IT such that z = y and P [ z ] and z in A and z in B ;
( the sequence of ( ( the sequence of X ) | ( the carrier of X ) ) ) . ( thesis ) <= e ;
len ( w ^ w2 ) + 1 = len w + 2 + 1 .= len w + 1 ;
assume q in the carrier of ( TOP-REAL 2 ) | K1 & q in the carrier of ( TOP-REAL 2 ) | K1 ;
f | E-4 ` = g | E-4 ` .= g | EK " .= g | EK .= g | EK ;
reconsider i1 = x1 , i2 = x2 as Element of NAT , i = ( len x1 ) -tuples_on REAL ;
( a * A ) @ = ( a * ( A @ ) ) @ .= a * ( A @ ) ;
assume ex n0 being Element of NAT st f |^ n0 is min & f . n0 is min ;
Seg len ( ( ( f1 ^ f2 ) | ( i + 1 ) ) ^ ( ( f1 ^ f2 ) | ( i + 1 ) ) ) = dom ( ( f1 ^ f2 ) | ( i + 1 ) ) ;
( Complement ( A1 ) ) . m c= ( ( Complement ( A1 ) ) . n ) . m ;
f1 . p = p9 & g1 . p = d & g1 . p = d & g2 . p = d ;
FinS ( F , Y ) = FinS ( F , dom ( F | Y ) ) .= F | Y ;
( x | y ) | z = z | ( y | x ) ;
( ( |. x .| ) |^ n ) / ( ( |. x .| ) |^ ( n + 1 ) ) <= ( ( |. r2 .| ) |^ n ) ;
Sum ( F ) = Sum f & dom ( F ) = dom g & rng ( F ) = dom g ;
assume for x , y being set st x in Y & y in Y holds x /\ y in Y ;
assume that W1 is Subspace of W3 and W2 is Subspace of W3 and W1 is Subspace of W2 ;
||. ( ( t . x ) - ( t . x ) ) .|| = lim ||. ( ( t . x ) - ( t . x ) ) .|| ;
assume that i in dom D and f | A is lower and g | A is lower ;
( ( p `2 ) ^2 - ( p `2 ) ^2 ) <= ( - ( - ( - ( - ( - 1 ) ) ) ) ) ^2 ;
g | Sphere ( p , r ) = id ( Sphere ( p , r ) ) & g | 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 implies the TopStruct of T is countable
width ( B |-> 0. K ) = len ( B @ ) .= len ( B @ ) .= len ( B @ ) ;
pred a <> 0 means : Def2 : ( A \ B ) Y. = ( A \ a ) Let ( B \ a ) ;
then f is_\cal 2 , u & pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 & pdiff1 ( f , 2 ) is_partial_differentiable_in u , 3 ;
assume that a > 0 and a <> 1 and b > 0 and b <> 0 and c > 0 and d > 0 ;
w1 , w2 in Lin { w1 , w2 } & w2 in Lin { w1 , w2 } implies w1 = w2
p2 /. IC s = p2 . IC s .= p2 . IC s .= ( p2 . IC s ) .= ( p2 . IC s ) ;
ind ( T-10 | b ) = ind b .= ind b - ind b .= ind b - ind b + ind b - ind b .| ;
[ a , A ] in the Points of G_ ( k , X ) & [ a , A ] in the \cdot of ( the Points of X ) ;
m in ( the Arrows of C ) . ( o1 , o2 ) & m in ( the Arrows of C ) . ( o1 , o2 ) ;
( ( a 'imp' CompF ( PA , G ) ) ) . z = FALSE ;
reconsider phi = phi , phi = phi , phi = phi as Element of ( len phi ) -tuples_on BOOLEAN ;
len s1 - ( len s2 - 1 ) + 1 > 0 + 1 - 1 ;
delta ( D ) * ( f . ( upper_bound A ) - f . ( lower_bound A ) ) < r ;
[ f21 , f22 ] in the carrier' of A ~ & f22 in the carrier' of A ~ ;
the carrier of ( TOP-REAL 2 ) | K1 = K1 & the carrier of ( ( TOP-REAL 2 ) | K1 ) = K1 ;
consider z being element such that z in dom g2 and p = g2 . z and x = g2 . z ;
[#] V1 = { 0. V1 } .= the carrier of (0). V1 .= the carrier of ( the carrier of V1 ) ;
consider P2 be FinSequence such that rng P2 = M and P2 is one-to-one and P2 is one-to-one and P2 is one-to-one ;
assume that x1 in dom ( f | X ) and ||. x1 - x0 .|| < s and |. x1 - x0 .| < s ;
h1 = f ^ ( <* p3 *> ^ <* p *> ) .= h ^ <* p *> .= h ^ <* p *> ;
c / ( |[ b , c ]| ) = c / ( |[ a , c ]| ) .= c / ( |[ a , c ]| ) .= c ;
reconsider t1 = p1 , t2 = p2 as Term of C , V , s be SortSymbol of C ;
( 1 - ( 2 * PI ) ) in the carrier of [. 1 / 2 , 1 / 2 .] ;
ex W being Subset of X st p in W & W is open & h .: W c= V ;
( h . p1 ) `2 = C * ( p1 `2 ) + D * ( p1 `2 ) .= C * ( p1 `2 ) + D * ( p1 `2 ) ;
R . ( b - a ) = 2 * PI * a-b .= 2 * PI * a-b .= b - a ;
consider 1 such that B = ( - 1 ) * C + ( - 1 ) * A and 0 <= 1 ;
dom g = dom ( ( ( the Sorts of A ) * ( the Arity of S ) ) * ( the Arity of S ) ) ;
[ P . ( l . ( k + 1 ) ) , P . ( k + 1 ) ] in => ( ( T . ( k + 1 ) ) , ( T . ( k + 1 ) ) ) ;
set s2 = Initialize s , P2 = P +* stop I ;
reconsider M = mid ( z , i2 , i1 ) , N = L~ z as non empty Subset of TOP-REAL 2 ;
y in product ( ( the support of J ) +* ( V , { 1 } ) ) ;
1 / ( |[ 0 , 1 ]| ) = 1 & 0 / ( |[ 0 , 1 ]| ) = 0 ;
assume x in the left of g or x in the right of g or x in the right of g & x in the carrier of g ;
consider M being strict non-empty Subgroup of A\mathopen ( A , T ) such that a = M and T is non-empty & M is non-empty ;
for x st x in Z holds ( ( ( #Z 2 ) * f ) `| Z ) . x <> 0 & ( ( #Z 2 ) * f ) `| Z ) . x = f . x
len W1 + len W2 + m = 1 + len W2 + len W3 + m .= len W2 + len W3 + m + 1 ;
reconsider h1 = ( vseq . n ) - ( t-16 . n ) as Lipschitzian Lipschitzian from X , Y ;
( i mod len ( p + q ) ) + 1 in dom ( p + q ) ;
assume that s2 is negative and F in the v of s2 and F in the v of s2 and F in the v of s2 ;
( ( ( ( ( ( ( ( x , y ) ) / 2 ) ) * ( x , y ) ) * ( x , y ) ) ) * ( x , y ) ) ) * ( x , y ) = gcd ( x , y , z ) ;
for u being element st u in Bags n holds ( p *' + m ) . u = p . u + m . u
for B being Subset of u-5 st B in E holds A = B or A misses B or A misses 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 & x in X } ;
the carrier of W1 /\ W2 c= the carrier of W1 & the carrier of W1 /\ W2 c= the carrier of W2 /\ ( the carrier of W1 ) ;
( 1 - a ) * id a = ( 1 - a ) * id a .= ( 1 - a ) * id a .= ( 1 - a ) * id a ;
( ( X --> f ) . x ) = ( X --> dom f ) . x .= f . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) , y = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) ;
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 <= ( ( being / 2 ) |^ ( n -' m ) ) + 1 ;
( reproj ( 1 , z0 ) ) . x in dom ( ( f1 (#) f2 ) * ( f1 (#) f2 ) ) ;
assume that b1 . r = { c1 } and b2 . r = { c2 } and c2 . r = c2 . r ;
ex P st a1 on P & a2 on P & a3 on P & a1 on P & a2 on P & a3 on P & a1 <> a2 & a2 <> a3 & a3 <> a3 & a3 <> a1 & a3 <> a1 & a2 <> a3 & a3 <> a1 & a3 <> a1 & a3 <> a1 & a3 <> a1 & a3 <> a1 & a3 <> a1
reconsider gf = g " * f , hf = h " * g as strict Element of X ;
consider v1 being Element of T such that Q = ( downarrow v1 ) ` and v1 in the topology of T ;
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 >= sn & p `2 >= sn & p `1 <= 0 & p <> 0. TOP-REAL 2 } ;
ConsecutiveSet ( A , succ O1 ) = ( ConsecutiveSet ( A , O1 ) ) * ( a , O1 ) ;
set Is1 = in | [ [ a , intloc 0 ] , [ a , intloc 0 ] , [ a , intloc 0 ] ] , Is1 = [ a , intloc 0 ] , Is1 = [ a , intloc 0 ] , Is1 = [ a , intloc 0 ] , Is1 = [ a , intloc 0 ] , Is1 = [ a , intloc 0 ] , Is1 = [ a , intloc 0 ] ;
for i be Nat st 1 < i & i < len z holds z /. i <> z /. 1 & z /. i <> z /. 1 ;
X c= ( the carrier of L1 ) \/ ( the carrier of L2 ) & the carrier of L1 c= the carrier of L2 implies X is Subset of L1
consider x9 being Element of GF ( p ) such that x9 |^ 2 = a & x9 |^ 3 = b ;
reconsider ef = ef , ff = ff , ff = ff as Element of D * ;
ex O being set st O in S & C1 c= O & M . O = 0. ;
consider n be Nat such that for m be Nat st n <= m holds S . m in U1 and S . m in U1 ;
f * g * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) * reproj ( i , x ) ) . x ;
defpred P [ Nat ] means A + succ $1 = succ $1 & A = succ $1 & A = succ $1 implies A = succ $1 ;
the left of - g = the left of - g & the left of - g = the left of - g implies g = f
reconsider p\mathopen = x , p\mathopen = y , p\mathopen = z as Point of ( TOP-REAL 2 ) | K1 , ( TOP-REAL 2 ) | K1 ;
consider g2 such that g2 = y and x <= g2 and g2 <= x0 and x0 <= g2 and g2 <= x0 and g2 <= x0 ;
for n being Element of NAT ex r being Element of REAL st X [ n , r ] & X [ n , r ]
len ( x2 ^ y2 ) = len x2 + len y2 & len ( x2 ^ y2 ) = len x2 + len y2 ;
for x being element st x in X holds x in the set of the set of \HM { 0 } & x in the set of n implies x in X
LSeg ( p11 , p2 ) /\ LSeg ( p1 , p11 ) = {} or LSeg ( p1 , p11 ) /\ LSeg ( p11 , p2 ) = {} ;
func non empty set equals [: X , X :] \/ [: Y , X :] .= [: X , X :] \/ [: Y , X :] ;
len ( ( the carrier of ( ( C | ( 1 , len the { the carrier of C , 1 ) ) ) ) ) ) <= len ( ( the charact of ( C | ( 1 , len the { the carrier of C } ) ) ) ) ;
pred K is \cap be Field means : Def2 : a <> 0. K & v . ( a |^ i ) = i * v . ( a |^ i ) ;
consider o being OperSymbol of S such that t . {} = [ o , the carrier of S ] and o in ( the Sorts of A ) . o ;
for x st x in X ex y st x c= y & y in X & not y is Z & f . x = f . y
IC Comput ( P-6 , s , k ) in dom ( ( n + 1 ) --> ( k + 1 ) ) ;
pred q < s means : Def2 : r < s & s < q & q in ]. p , q .[ implies r < s ;
consider c being Element of Class ( f , c ) such that Y = ( F . c ) `1 and c in X ;
func the ResultSort of S2 -> Function of the carrier' of S2 , the carrier' of S2 means : Def2 : the ResultSort of it = id the carrier' of S2 & the ResultSort of it = id the carrier' of S2 ;
set yp1 = [ <* y , z *> , f2 ] , yp2 = [ <* z , x *> , f3 ] ;
assume x in dom ( ( ( #Z 2 ) * ( ( arccot ) ^ ) ) `| Z ) & x in dom ( ( #Z 2 ) * ( ( arccot ) ^ ) ) ;
r-7 in Int cell ( GoB f , i , ( GoB f ) * ( i , 1 ) + ( GoB f ) * ( i , 1 ) ) ;
( q `2 ) ^2 >= ( ( Cage ( C , n ) ) /. ( i + 1 ) ) `2 ;
set Y = { a "/\" a ` : a ` in X } ;
i - len f <= len f + len f - len f & i - len f <= len f + len f - len f ;
for n ex x st x in N & x in N1 & h . n = - x0 & h . n = x0
set s0 = ( \mathop { a , I , p , s ) . i , s0 = ( s , p , s ) . i , s0 = ( s , p , s ) . i ;
p ( ) . k = 1 or p ( ) . 0 = - 1 or p ( ) . 0 = 1 or p . 1 = 0 ;
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 x9 in V1 and x = f . x9 ;
( p ^ ( q | k ) ) . m = ( q | k ) . ( ( len p ) - ( len p ) ) ;
g + h = gg + hg & h + g = g + h + h ;
L1 is distributive & L2 is distributive implies L1 ~ is distributive & for x being Element of L1 holds x in the carrier of L2 iff x in the carrier of L1
pred x in rng f & y in rng ( f | x ) implies f . x = f . y & f . y = f . x ;
assume that 1 < p and p + 1 / q = 1 and 0 <= a and 0 <= b and a <= b and b <= q ;
F* ( f , \langle fA1 *> ) = rpoly ( 1 , the carrier of F_Complex ) *' + t .= 1. F_Complex ;
for X being set , A being Subset of X holds A ` = {} implies A = X & A = X
( ( ( ( ( ( Y ) ) / 2 ) ) * ( ( ( Y ) / 2 ) ) * ( ( i + 1 ) ) / 2 ) ) ) <= ( ( ( ( Y ) / 2 ) * ( ( i + 1 ) ) / 2 ) ) * ( ( ( Y ) / 2 ) * ( ( i + 1 ) / 2 ) ) ;
for c being Element of the Sorts of A , a being Element of the Sorts of A holds c <> a & a in A
s1 . GBP = ( Exec ( i2 , s2 ) ) . GBP .= Exec ( i2 , s2 ) . GBP .= Exec ( i2 , s2 ) . GBP .= s . intpos i .= s . intpos i ;
for a , b being Real holds |[ a , b ]| in ( y >= 0 ) implies b >= 0 & a >= 0 & 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 , n , m , n , m , m , n , m , n , m , m , n , m , n , m , m , n , m , n , m , n , m , m , n ;
set x2 = |( Re ( y - x ) , Im ( y - x ) )| ;
[ y , x ] in dom u5 & u5 . ( y , x ) = g . y ;
]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .] c= A & ( lower_bound divset ( D , k ) ) . ( upper_bound divset ( D , k ) ) <= ( lower_bound A ) . ( lower_bound A ) ;
0 <= delta ( S2 ) . n & |. delta ( S2 ) . n .| < ( e / 2 ) * ( 2 |^ n ) ;
( - ( q `1 ) ) ^2 <= ( - ( q `2 ) ) ^2 + ( - ( q `2 ) ) ^2 ;
set A = ( 2 / b-a ) , B = ( - 1 ) / ( 2 * b-a ) ;
for x , y being set st x in R" holds x , y are_\hbox { x , y } , R
deffunc FF2 ( Nat ) = b . $1 * ( M * G ) . $1 * ( M * G ) . $1 ;
for s being element holds s in -> Element of \mathclose { f } iff s in -> Element of \mathclose { f } \/ ( f " { f } )
for S being non empty non void non void non empty ManySortedSign for S being non empty non void ManySortedSign st S is connected holds S is connected
max ( degree ( ( z | ( |. z .| ) ) ) , degree ( ( z | ( |. z .| ) ) ) ) >= 0 ;
consider n1 be Nat such that for k holds seq . ( n1 + k ) < r + s and n1 < r + s ;
Lin ( A /\ B ) is Subspace of Lin ( A ) & Lin ( B ) is Subspace of Lin ( A ) ;
set n-15 = n-13 '&' ( M . ( x qua Element of BOOLEAN ) ) , n-15 = M . ( x qua Element of BOOLEAN ) ;
f " V in the topology of X & f " V in D & f " V in D & f " V in D implies f " V in D
rng ( ( a ^\ c ) \mathbin ( 1 , b ) ) c= { a , c , b }
consider y being as ] a as WWof G1 , G2 such that y ` = y and dom y ` = WWWWWWWW
dom ( ( 1 / f ) (#) ( f ^ ) ) /\ ]. x0 - r , x0 .[ c= ]. x0 - r , x0 .[ & ( f ^ ) . x0 in ]. x0 - r , x0 .[ ;
as Matrix of i , j , n , r , - r , p , - r , q be Element of REAL n ;
v ^ ( n-3 |-> 0 ) in Lin ( ( rng ( B-9 | ( Seg n ) ) ) ) & v ^ ( nt1 | ( Seg n ) ) ) in Lin ( ( rng ( B-9 | ( Seg n ) ) ) ) ;
ex a , k1 , k2 st i = a := k1 & i = b := k2 & k2 = c := k2 ;
t . NAT = ( NAT .--> succ i1 ) . NAT .= ( NAT .--> succ i1 ) . NAT .= succ 5 .= succ 5 .= succ 5 ;
assume that F is bbfamily and rng p = F and rng p = Seg ( n + 1 ) and for i be Nat st i in Seg ( n + 1 ) holds p . i in F . i ;
not LIN b , b9 , a & not LIN a , a9 , c & not LIN a , a9 , c & not a , a9 // a9 , b9 & a , a9 // a9 , c9
( L1 over L2 ) \& O c= ( L1 \& O ) \& ( L2 \& 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 * ( Seg w ) = b * ( -w ) and 0 < a & a < b & b < 0 ;
defpred P [ FinSequence of D ] means |. Partial_Sums ( $1 ) . $1 .| <= Partial_Sums ( $1 ) . $1 & Partial_Sums ( $1 ) . $1 <= Partial_Sums ( $1 ) . $1 ;
u = cos . ( x , y ) * x + cos . ( x , y ) * y .= cos . ( x , y ) * y .= v ;
dist ( seq . n , x + g ) <= dist ( seq . n , g + x ) + 0 ;
P [ p , |. p .| ^ <* {} *> , {} ] & P [ p , {} ] implies p in the Sorts of A
consider X being Subset of CQC-WFF ( Al ) such that X c= Y and X is finite and inP [ X ] ;
|. b .| * |. eval ( f , z ) .| >= |. b .| * |. eval ( f , z ) .| ;
1 < ( ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ) .. Cage ( C , n ) ;
l in { l1 where l1 is Real : g <= l1 & l1 <= h & h <= g & l1 <= h } ;
( Partial_Sums ( ( G . n ) * vol ) ) . n <= ( Partial_Sums ( ( G . n ) * vol ) ) . n ;
f . y = x .= x * 1_ L .= x * ( power L ) .= x * ( power L ) .= x * ( ( x * y ) * ( y * y ) ) ;
NIC ( ( <% i1 %> , ( succ i1 ) ) \ { succ i1 } , k ) = { i1 , succ i1 } .= { succ i1 } ;
LSeg ( p10 , p2 ) /\ LSeg ( p1 , p11 ) = { p1 } /\ LSeg ( p1 , p11 ) .= { p1 } /\ LSeg ( p1 , p11 ) ;
product ( ( the support of I-15 ) +* ( i , { 1 } ) ) in ( Z * * ( i , { 1 } ) ) ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) .= Following ( s1 , n ) .= Following ( s2 , n ) ;
W-bound ( Qq ) <= ( q1 `1 ) ^2 & ( q1 `1 ) ^2 <= ( q1 `1 ) ^2 & ( q1 `2 ) ^2 <= ( q1 `2 ) ^2 ;
f /. i2 <> f /. ( ( i1 + len g -' 1 ) -' 1 ) & f /. ( ( i1 + 1 ) -' 1 ) = f /. ( i1 + 1 ) ;
M , f / ( x. 3 , x. 4 ) / ( x. 4 , x. 0 ) / ( x. 4 , x. 0 ) |= H ;
len ( ( P ^ ) | ( len P + 1 ) ) in dom ( ( P ^ ) | ( len P + 1 ) ) ;
A |^ ( me , n ) c= A |^ ( m , n ) & A |^ ( k , l ) c= A |^ ( k , l ) ;
REAL 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 .= halt SCM+FSA .= halt SCM+FSA .= ( CurInstr ( P3 , Comput ( P3 , s3 , l ) ) ) ;
for v be VECTOR of l1 holds ||. v .|| = upper_bound rng |. ( id the carrier of X ) .| & ||. v .|| = ||. v .||
for phi holds phi in X implies phi in X & not phi in X & phi in X & phi in X & phi in X
rng ( ( Sgm dom ( f | ( dom f ) ) ) | ( dom f /\ dom ( f | ( dom f ) ) ) ) c= dom ( f | ( dom f /\ dom ( f | ( dom f ) ) ) ) ) ;
ex c being FinSequence of D ( ) st len c = k & P [ c ] & a = c ^ <* d *> ;
( the_arity_of ( a , b , c ) ) = <* \mathop { \rm hom ( b , c ) , \mathop { \rm hom ( a , b , c ) *> , \mathop { \rm Arity ( a , b , c ) } ) ;
consider f1 be Function of the carrier of X , REAL such that f1 = |. f .| and f1 is continuous ;
a1 = b1 & a2 = b2 or a1 = b1 & a2 = b2 & a3 = b1 & a4 = b2 & a5 = b1 & a5 = b2 & a5 = b1 & a5 = b2 ;
D2 . ( indx ( D2 , D1 , n1 ) + 1 ) = D1 . ( n1 + 1 ) .= D2 . ( n1 + 1 ) ;
f . ( ||. r .|| ) = ||. r .|| /. 1 .= ||. r .|| . 1 .= ||. r .|| . 1 .= r . 1 .= r . 1 .= r . 1 ;
consider n be Nat such that for m be Nat st n <= m holds C-25 . m = C-25 . m and n <= m ;
consider d be Real such that for a , b be Real st a in X & b in Y holds a <= d & b <= b ;
||. L /. h .|| - ( K * |. h .| ) + ( K * |. h .| ) <= p0 + ( K * |. h .| ) ;
attr F is commutative means : Def2 : for b being Element of X holds F \hbox { b } = f . b & f is one-to-one ;
p = - ( - ( - ( p `1 / |. p .| - cn ) ) ) * ( 1 - cn ) .= 1 * ( ( p `1 / |. p .| - cn ) / ( 1 - cn ) ) .= 1 * ( ( p `1 / |. p .| - cn ) / ( 1 - cn ) ) .= 1 * ( ( p `1 / |. p .| - cn ) / ( 1 - cn ) ) ;
consider z1 such that b `1 , x3 , 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 , a9 , a9 , a9 , b9 be element ;
consider i such that Arg ( ( Rotate ( s ) ) . 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 rng g = dom f and g is one-to-one ;
assume that A = P2 \/ Q2 and P2 <> {} and Q2 <> {} and P2 <> {} and ( P2 = {} or P2 = {} or P2 = {} ) ;
attr F is associative means : Def2 : F .: ( F .: ( f , g ) , h ) = F .: ( f , g ) ;
ex x being Element of NAT st m = x `1 & x in z `1 & x < i or m < i & i < m ;
consider k2 being Nat such that k2 in dom P-2 and l in P-2 . k2 and P [ k2 ] ;
seq = r * seq implies for n holds seq . n = r * seq . n & seq is convergent & lim seq = r * seq . n
F1 . [ ( id a ) * [ a , a ] , ( id a ) * [ b , b ] ] = f * ( id a ) ;
{ p } "\/" D2 = { p "\/" y where y is Element of L : y in D2 } & "\/" ( { p "\/" q where p is Element of L : p in D1 } , L ) = { p "\/" q where q is Element of L : q in D2 } ;
consider z being element such that z in dom ( ( dom F ) * ( the Arity of S ) ) and ( ( dom F ) * ( the Arity of S ) ) . z = y ;
for x , y being element st x in dom f & y in dom f & f . x = f . y holds x = y
cell ( G , i , 1 ) = { |[ r , s ]| : r <= G * ( 0 + 1 , 1 ) `1 } ;
consider e being element such that e in dom ( T | E1 ) and ( T | E1 ) . e = v and e in ( T | E1 ) . e ;
( F @ * b1 ) . x = ( Mx2Tran ( J , Bdiv , Bdiv ) ) . ( \mathbb j , j ) ;
- 1 / ( - 1 ) = mmD | n .= mmD | n .= mmD | n .= ( - 1 ) (#) ( - 1 ) .= ( - 1 ) (#) ( - 1 ) .= ( - 1 ) (#) ( - 1 ) .= ( - 1 ) (#) ( - 1 ) ;
pred for x being set st x in dom f /\ dom g holds g . x <= f . x & g . x <= f . x ;
len ( f1 . j ) = len ( f2 /. j ) .= len ( ( f2 . j ) * ( ( f2 . j ) * ( ( f2 . j ) * ( ( f2 . j ) * ( ( f2 . j ) * ( ( f2 . j ) * ( ( f2 . j ) * ( ( f2 . j ) * ( ( f2 . j ) * ( ( f2 . j ) * ( ( f2 . j ) * (
All ( 'not' All ( 'not' a , A , G ) , B , G ) '<' Ex ( 'not' All ( 'not' a , B , G ) , A , G )
LSeg ( E . ( k + 1 ) , F . ( k + 1 ) ) c= Cl RightComp Cage ( C , ( k + 1 ) + 1 ) ;
x \ a |^ m = x \ ( a |^ k * a ) .= ( x \ a ) \ a .= ( x \ a ) \ a .= x ;
k -th ininin( I-5 ) . k = ( commute ( Il . k ) ) . k .= ( commute ( ( f . k ) ) * ( f . k ) ) ) . i .= ( ( f . k ) * ( f . k ) ) * ( f . k ) ;
for s being State of Ai2 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 & ( f1 - f2 ) . x <> 0 ;
support ( support ( support ( support ( m ) ) \/ support ( m ) ) ) c= support ( m ) \/ support ( m ) \/ support ( m ) ) ;
reconsider t = u as Function of ( the carrier of A ) , ( the carrier of B ) * , the carrier of C ;
- ( a * sqrt ( 1 + b ^2 ) ) <= - ( b * sqrt ( 1 + a ^2 ) ) ;
phi /. succ ( b1 . a ) = g . a & phi . ( a . a ) = f . ( g . a ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( ( F ^ <* p *> ) . i ) and i in dom ( ( F ^ <* p *> ) . j ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 } = { x1 } \/ { x2 , x3 , x4 , x5 } .= { x1 , x2 , x3 , x4 } ;
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 + ( - ( 2 * a * ( b - a ) ) / b ) ^2 > 0 ;
consider W00 such that for z being element holds z in W00 iff z in N ~ N & P [ z ] ;
assume that ( the Arity of S ) . o = <* a *> and ( the Arity of S ) . o = r and ( the Arity of S ) . o = r ;
Z = dom ( ( ( #Z ( ( n + 1 ) ) * ( ( #Z n ) * ( f1 + #Z n ) ) ) ) ) ;
sum ( f , SS1 ) is convergent & lim ( ( f | SS1 ) * ( lim S ) ) = integral ( f , SS1 ) * ( lim S ) ;
( X . ( a9 , f ) => ( g . ( a9 , b9 ) ) ) => ( x9 , f . ( a9 , b9 ) ) in -> sequence of l . ( a9 , b9 ) ;
len ( M2 * M3 ) = n & width ( M2 * M3 ) = n & len ( M2 * M3 ) = n & width ( M1 * M2 ) = n ;
attr X1 union X2 means : Def2 : X1 , X2 are_separated & X2 , X1 , X2 , X3 be SubSpace of X , X2 be Subset of X ;
for L being upper-bounded antisymmetric non empty RelStr for X being non empty RelStr for s being non empty Subset of L holds X "\/" { Top L } = { Top L }
reconsider f-129 = F1 . ( ( b - a ) / ( b - a ) ) as Function of ( the carrier of M ) , M ;
consider w being FinSequence of I such that the InitS of M = the InitS of M and the InitS of M is_{ s ^ w where s is Element of M : s in the carrier of M } and w in q ;
g . ( a |^ 0 ) = g . ( 1_ G ) .= 1_ H .= 1_ H .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) ;
assume for i be Nat st i in dom f ex z being Element of L st f . i = rpoly ( 1 , z ) & z in 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 K /\ 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 ;
reconsider oY = o `1 as Element of TS ( ( the Sorts of A ) * ( the Arity of S ) ) . o ;
1 * x1 + ( 0 * x2 ) + ( 0 * x3 ) = x1 + ( 0 * x2 ) .= x1 + ( 0 * x2 ) .= x1 + ( 0 * x2 ) .= x1 + ( 0 * x2 ) .= x1 + ( 0 * x2 ) ;
EP " . 1 = ( ( E qua Function ) " ) . 1 .= ( ( E qua Function ) " ) . 1 .= ( ( E qua Function ) " ) . 1 .= ( ( E qua Function ) " ) . 1 .= ( ( E qua Function ) " ) . 1 .= ( ( E qua Function ) " ) . 1 .= ( ( E qua Function ) " ) . 1 ;
reconsider u1 = the carrier of U1 /\ ( U1 "\/" U2 ) as non empty Subset of U0 ( U1 "\/" U2 ) , ( U1 "\/" U2 ) . ( ( U1 "\/" U2 ) . ( ( U1 "\/" U2 ) . ( ( U1 "\/" U2 ) . ( ( U1 "\/" U2 ) . ( ( U1 "\/" U2 ) . ( ( U1 "\/" U2 ) . ( ( U1 "\/" U2 ) . ( ( U1 "\/" U2 ) . ( ( U1 "\/" U2 ) . ( ( U1 "\/" U2 ) . ( ( U1 "\/"
( ( x "/\" z ) "\/" ( x "/\" y ) ) "\/" ( z "/\" y ) <= ( x "/\" ( z "\/" y ) ) "\/" ( z "/\" y ) ;
|. f . ( s1 . ( l1 + 1 ) ) - ( s1 . ( l1 + 1 ) ) .| < ( 1 - M ) * ( 1 - M ) ;
LSeg ( ( Lower_Seq ( C , n ) ) * ( i , j ) , ( Lower_Seq ( C , n ) ) * ( i + 1 , j ) ) is vertical ;
( f | Z ) /. x - ( f | Z ) /. x0 = L /. ( x- ( x - x0 ) ) + R /. ( x- ( x - x0 ) ) ;
g . c * ( - g . c ) + f . c <= h . c * ( - f . c ) + f . c ;
( f + g ) | divset ( D , i ) = f | divset ( D , i ) + g | divset ( D , i ) .= f | divset ( D , i ) + g | divset ( D , i ) ;
assume that ColVec2Mx f in the set of the carrier of A and ColVec2Mx f = ( ColVec2Mx b ) * ( ColVec2Mx f ) and len f = len A and width f = width A and f is LSeg of f , 1 , 1 , 2 , 3 , 4 , 5 , 6 , 7 *> = Indices A ;
len ( - M1 ) = len M1 & width ( - M2 ) = width M1 & width ( - M1 ) = width M2 & width ( - M2 ) = width M1 ;
for n , i being Nat st i + 1 < n holds [ i , i + 1 ] in the InternalRel of ( \mathop { n } ) \/ the InternalRel of ( \mathop { n } ) \/ the InternalRel of ( \mathop { n } ) \/ the InternalRel of ( \mathop { n } ) \/ the InternalRel of ( \mathop { n } ) \/ the InternalRel of ( n + 1 ) )
pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in 2 , 2 & pdiff1 ( f1 , 2 ) , 2 ;
pred a <> 0 & b <> 0 & Arg a = Arg b & Arg ( - a ) = Arg ( - b ) & Arg ( - a ) = Arg ( - a ) ;
for c being set st not c in [. a , b .] holds not c in Intersection ( the REAL of a , b ) & ( the in of a , b ) is open implies c in union ( the REAL of a , b )
assume that V1 is linearly-independent and V2 is closed and V = { v + u : v in V1 & u in V1 & u in V1 } and V1 is closed and V1 is closed and V2 is closed and V1 is closed ;
z * x1 + ( 1 - z ) * x2 in M & z * y1 + ( 1 - z ) * y2 in N implies z * y1 + ( 1 - z ) * y2 in N
rng ( ( ( Pk1 qua Function ) " ) * Sk1 ) = Seg card ( ( ( card ( ( k2 + 1 ) -tuples_on REAL ) ) * ) ) .= Seg card ( ( ( card ( ( k2 + 1 ) -tuples_on REAL ) ) * ) ) ;
consider s2 being Integer such that s2 is convergent and b = lim s2 and for n holds s2 . n <= b . n and s2 . n <= b . n ;
h2 " . n = h2 . n " & 0 < - ( ( 1 / 2 ) |^ n ) & 0 < ( - ( ( 1 / 2 ) |^ n ) ) |^ ( n + 1 ) ) ;
( Partial_Sums ( ||. seq .|| ) ) . m = ||. seq .|| . m .= ||. seq .|| . m .= ( ||. seq .|| ) . m .= 0 ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b ;
- v = - ( - 1 ) * v & - w = - ( - 1 ) * v & - ( - w ) * v = - ( - ( - 1 ) * v ) ;
sup ( ( k .: D ) .: ( f .: D ) ) = sup ( ( k .: D ) .: ( f .: D ) ) .= k . ( f . ( f . ( k + 1 ) ) ) .= k . ( f . ( k + 1 ) ) .= k . ( f . ( k + 1 ) ) ;
A |^ ( k , l ) ^^ ( A |^ ( n , l ) ) = ( A |^ ( n , l ) ) ^^ ( A |^ ( k , l ) ) .= A |^ ( k , l ) ;
for R being add-associative right_zeroed right_complementable non empty addLoopStr , I , J , K being Subset of R , I , J being Subset of R holds I + ( J + K ) = ( I + J ) + K
( f . p ) ^2 = ( p `1 ) ^2 + ( p `2 ) ^2 .= ( p `1 ) ^2 + ( p `2 ) ^2 .= ( p `1 ) ^2 + ( p `2 ) ^2 ;
for a , b being non zero Nat for a , b being Element of NAT st a , b are_relative_prime holds support ( a * b ) = support ( a ) + support ( b ) & support ( a ) = support ( a ) + support ( b )
consider A5 being countable set such that r is countable and A5 is Element of CQC-WFF ( Al ) and A5 is ( len A5 ) -element ;
for X be non empty addLoopStr for M , N being Subset of X , x , y being Point of X st y in M & x + y in N holds x + y in N + M
{ [ x1 , x2 ] , [ y1 , y2 ] } c= { x1 , y1 } & { x1 , x2 } in { x1 , y1 } & { x2 , y2 } in { x1 , x2 } ;
h . ( f . O ) = |[ A * ( f . O ) + B , C * ( f . O ) + D ]| ;
( Gauge ( C , n ) * ( k , i ) ) * ( k , i ) in L~ Lower_Seq ( C , n ) /\ L~ Lower_Seq ( C , n ) ;
cluster m , n are_relative_prime -> prime for Nat , p be prime Nat , n be Element of NAT , m be non zero Nat ;
( 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 e e <= c & a e e <= c
consider b being element such that b in dom ( H / ( x. 0 , y ) ) and z = ( H / ( x. 0 , 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 or e Joins W . 3 , W . 5 , G ;
( ] support ( o ) ) . ( 2 * n ) . x = ( : ( h ) . ( 2 * n ) ) . ( 2 * n ) . x + ( h . n ) . x ;
j + 1 = ( i - len h11 ) + 2 .= i + 1 - len h11 + 2 - 1 .= i + 1 - len h11 + 2 - 1 .= i + 1 - len h11 + 2 - 1 ;
( *' 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 ) = Sum ( L2 ) and Sum ( L2 ) = Sum ( L2 ) ;
attr R is min means : Def2 : for p , q st p in R & q in R holds ex P st P is_max ( p , q ) & P c= R ;
dom product ( product ( X --> f ) ) = meet ( dom ( X --> f ) ) .= meet ( X --> f ) .= meet ( X --> f ) .= meet ( X --> f ) .= ( meet ( X --> f ) ) . x ;
upper_bound ( proj2 .: ( Upper_Arc ( C ) /\ Lower_Arc ( C ) /\ \hbox { w } ) ) <= upper_bound ( proj2 .: ( C /\ L~ Cage ( C , n ) /\ L~ Cage ( C , n ) ) ) ;
for r be Real st 0 < r ex n be Nat st for m be Nat st n <= m holds |. S . m - x0 .| < r
i * f-28 - ( i * f<> i * fmax ) = i * f-28 - ( i * fmax ) .= i * ( fmax ) - ( i * fmax ) ;
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 g1 in union C and g2 in C and g2 in C ;
func d |-count n -> Nat means : Def2 : ( d |^ n ) divides n & ( d |^ n ) |^ ( n + 1 ) divides n & ( d |^ n ) divides n ;
f\rm . [ 0 , t ] = f . [ 0 , t ] .= ( - P ) . ( 2 * ( x * ( x * ( x * ( x * y ) ) ) ) ) .= 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 = h . M ;
consider m1 be Nat such that for n st n >= m1 holds dist ( ( seq . n ) , ( seq . n ) ) < 1 / ( ( seq . n ) * ( seq . n ) ) ;
( ( q `1 ) ^2 + ( q `2 ) ^2 ) <= ( ( q `1 ) ^2 + ( q `2 ) ^2 ) * ( ( q `2 ) ^2 + ( q `2 ) ^2 ) ;
h0 . ( i + 1 + 1 ) = h21 . ( i + 1 + 1 -' len h11 + 2 -' 1 ) .= h21 . ( i + 1 + 1 -' len h11 ) ;
consider o being Element of the carrier' of S , x2 being Element of { the carrier' of S } such that a = [ o , x2 ] and o in { the carrier' of S } ;
for L being RelStr , a , b being Element of L holds a <= { b } iff a <= b & b <= a & a <= b & b <= a & a <= b
||. h1 .|| . n = ||. ( h1 . n ) .|| .= ||. ( h . n ) .|| .= ||. ( h . n ) .|| .= ||. ( h . n ) .|| .= ||. ( h . n ) .|| .= ||. ( h . n ) .|| ;
( ( - ( 1 / 2 ) ) (#) ( ( #Z 2 ) * ( ( #Z 2 ) * ( #Z 2 ) ) ) ) . x = f . x - ( ( 1 / 2 ) * ( ( #Z 2 ) * ( #Z 2 ) ) ) . x .= ( ( - 1 / 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( #Z 2 ) ) ) ) . x ;
pred r = F .: ( p , q ) means : Def2 : len r = len q & for i st i in dom r holds r . i = F ( i ) ;
( rbeing ^2 - ( r / 2 ) ^2 ) + ( rbeing Element of REAL ) ^2 <= ( r / 2 ) ^2 + ( r / 2 ) ^2 ;
for i be Nat , M be Matrix of n , K st i in Seg n holds Det ( M @ ) = Sum ( ( Line ( M @ ) ) @ )
then a <> 0. R & a " * ( a * v ) = 1 * v & a " * ( a * v ) = 1 * v & a " * ( a * v ) = 1 * v ;
( p . ( j -' 1 ) ) * ( q *' ) . ( i + 1 -' j ) = Sum ( p . ( j -' 1 ) ) * ( q . ( j -' 1 ) ) ;
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 H2 = f .: 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 ) ) .= n + 1 - ( 2 to_power ( n + 1 ) ) .= n + 1 - ( 2 to_power ( n + 1 ) ) .= n + 1 - 1 ;
( O = 0 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 implies O = O & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O =
F1 .: ( dom F1 /\ dom F2 ) = F1 .: ( dom F1 /\ dom F2 /\ dom F2 ) .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } ;
pred b <> 0 & d <> 0 & b <> 0 & d <> 0 & ( a - b ) / d = ( - ( e / d ) ) / ( d - b ) ) ;
dom ( ( f +* g ) | D ) = dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= D ;
for i be set st i in dom g ex u , v being Element of L st g /. i = u * a & u in B & v in A & u in B & v in B & u in C ;
g `2 * P `2 * g " = g `2 * ( g " * P ) " .= g `2 * ( g " * P ) .= g `2 * ( g " * P ) .= g ;
consider i , s1 such that f . i = s1 and not ( ex s1 st s1 = f . i & not s1 in s1 & not s1 in { f . i } ) and not s1 in { f . i , f . i } ;
h5 | ]. a , b .[ = ( g | ]. a , b .[ ) | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ ;
[ s1 , t1 ] , [ s2 , t2 ] connected & [ s2 , t2 ] , [ s2 , t2 ] connected & [ s2 , t2 ] in R implies ( s1 , s2 ) `1 = s1 & ( s1 , s2 ) `2 = t2
then H is negative & H is non negative & H is non empty implies H is not negative -gfor F being Function of H , F holds F is not non -gfor H being non empty set st H is non -gfor F being Function of H , F holds F is not negative -gfor F is \mathbin of F , F
attr f1 is total means : Def2 : ( f1 is total & f2 is total implies f1 - f2 is total & ( f1 - f2 ) (#) ( f1 - f2 ) ) " = f1 . ( ( f1 - f2 ) (#) ( f1 - f2 ) ) " ;
z1 in W2 " ( W2 " ( z2 " ( z2 + 1 ) ) ) or z1 = z2 " ( z2 " ( z2 + 1 ) ) & not z1 in W2 " ( z2 + 1 ) & z2 in W2 " ( z2 + 1 ) ;
p = 1 * p .= a " * a * p .= a " * ( b " * p ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) ;
for seq1 be sequence of X , seq be sequence of X st for n be Nat holds seq1 . n <= seq . n holds ( seq ^\ n ) . n <= seq . n
x0 meets ( L~ go \/ L~ pion1 ) or x0 in ( L~ pion1 \/ L~ pion1 ) or x0 in ( L~ pion1 \/ L~ pion1 ) or x0 in ( L~ pion1 \/ L~ pion1 ) or x0 in ( L~ pion1 \/ L~ pion1 ) & x0 in L~ pion1 \/ ( L~ pion1 \/ L~ pion1 ) ;
||. f . ( g . ( k + 1 ) ) - g . ( g . ( k + 1 ) ) .|| <= ||. g . 1 - g . 0 .|| * ( K * K to_power ( k + 1 ) ) ;
assume h = ( ( B .--> B ' ) +* ( D .--> C ) +* ( E .--> D ) ) +* ( F .--> N ) +* ( J .--> A ) +* ( M .--> N ) +* ( F .--> N ) +* ( M .--> A ) +* ( N .--> N ) +* ( F .--> N ) +* ( M .--> A ) +* ( N .--> N ) ) +* ( M .--> N ) +* ( F .--> A ) ) +* ( N .--> N ) ;
|. ( ( ( ( lower . n ) || A ) . k ) - ( ( ( ( lower . n ) || A ) . k ) ) * ( ( ( ( ( ( D . n ) * ( ( D . n ) * ( D . n ) ) ) ) / ( 2 |^ n ) ) ) ) ) .| <= e * ( 2 * ( b-a ) ) ;
( (
{ x1 , x1 , x1 , x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 } = { x1 , x2 , x3 , x4 } .= { x1 , x2 , x3 } .= { x1 , x2 , x3 } ;
assume that A = [. 0 , 2 * PI .] and integral ( ( ( #Z n ) * ( cos ) ) , A ) = 0 and ( ( #Z n ) * ( cos ) ) | A = 0 ;
p `2 is Permutation of dom f1 & p `2 " = ( ( Sgm Y ) " ) * p " & p `2 = ( ( Sgm Y ) " ) * p ;
for x , y st x in A & y in A holds |. ( 1 / ( f . x ) - ( f . y ) ) .| <= 1 * |. f . x - f . y .|
( p2 `2 ) ^2 = |. q2 .| * ( ( ( q2 `2 ) - ( q2 `2 ) ) / ( 1 + ( ( q2 `2 ) - ( q2 `1 ) ) / ( 1 + ( q2 `2 ) ) ^2 ) ) ) .= ( ( ( q2 `2 ) - ( q2 `2 ) ) / ( 1 + ( q2 `2 ) ) / ( 1 + ( q2 `2 ) ^2 ) ) ;
for f be PartFunc of the carrier of CNS , REAL , x be Element of REAL , f be PartFunc of C , REAL , r be Real st f is continuous & f = r holds rng f c= dom f & f is compact
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 for k be Nat st k in n1 holds F . k = F ( k ) ;
ex u , u1 st u <> u1 & u , u1 , u1 / ( a , v ) / ( a , v1 ) / ( a , u1 ) // u , v1 & u , u1 / ( a , v ) / ( a , v1 ) // u1 , v1 & u1 , v1 / ( a , v ) // v1 , v2 & u1 <> v1 & v1 <> v2 implies u1 = v1 & u2 = v2
for G being Group , A , B being non empty Subgroup of G , N being normal Subgroup of G holds ( N , A ) \cdot ( N , B ) " = N ~ A * N
for s be Real st s in dom F holds F . s = integral ( R / ( f . s ) , ( ( R + g ) (#) ( f - h ) ) . s ) ;
width AutMt ( f1 , b1 , b2 , b3 ) = len b2 .= len b1 .= len b1 .= len b1 .= len b1 .= len b1 .= len b1 + len b2 .= len b1 + len b2 .= len b1 + len b2 .= len b1 + len b2 .= len b1 + len b2 ;
f | ]. - PI / 2 , PI / 2 .[ = f & dom f = ]. - PI / 2 , PI / 2 .[ & rng f c= ]. - PI / 2 , PI / 2 .[ & f | ]. - PI / 2 , PI / 2 .[ is continuous ;
assume that X is closed and a in X and a in X and y in X and not y in { { [ n , x ] } \/ { x } } and x in X ;
Z = dom ( ( ( ( #Z 2 ) * ( arctan ) ) (#) ( ( #Z 2 ) * ( arctan ) ) ) ) /\ dom ( ( ( #Z 2 ) * ( arctan ) ) ^ ) .= dom ( ( ( #Z 2 ) * ( ( arctan ) ) ^ ) ) /\ dom ( ( #Z 2 ) * ( ( #Z 2 ) * ( arctan ) ) ) ;
func [: the Sorts of V , { l . k } :] -> Subset of V equals { l . k : 1 <= k & k <= len l & l . k in V } ;
for L be non empty TopSpace , N , M be net of L , N be net of L st c is convergent & N is convergent holds N is convergent & ( for c being Element of N st c in N holds N . c in N . c )
for s being Element of NAT holds ( ( ex v being Element of NAT st v in dom ( ||. f .|| ) ) & ( v in dom ( ||. f .|| ) ) implies f . s = ( ( ||. f .|| ) . s ) . s
then z /. 1 = ( ( N-min L~ z ) .. z ) .. z & ( ( ( N-min L~ z ) .. z ) .. z ) .. z < ( ( E-max L~ z ) .. z ) .. z ;
len ( p ^ <* ( 0 qua Real ) * ( 0 qua Real ) *> ) = len p + len <* ( 0 qua Real ) * ( 0 qua Nat ) *> .= len p + 1 .= 1 + 1 ;
assume that Z c= dom ( ( - ( ln * f ) ) `| Z ) and for x st x in Z holds f . x = x and f . x = x and f . x = 1 and f . x = 1 ;
for R being add-associative right_zeroed right_complementable commutative associative commutative distributive non empty doubleLoopStr , I being non empty Subset of R , J being Subset of R , I being Subset of R , J being Subset of R holds ( I + J ) *' c= I /\ J
consider f being Function of [: B1 , B2 :] , 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 ( x2 + y2 ) .= Seg len ( x2 + y2 ) .= Seg len ( x2 + y2 ) .= Seg len ( x2 + y2 ) .= Seg len ( x2 + y2 ) .= Seg len ( x2 + y2 ) .= Seg len ( x2 + y2 ) .= Seg len ( x2 + y2 ) .= Seg len ( x2 + y2 ) ;
for S being card Functor of C , B for c being Object of C holds card S . ( id c ) = id ( ( the carrier' of C ) . c )
ex a st a = a2 & a in f6 /\ f5 & ( f . a ) /\ ( f . a ) = {} & ( f . a ) /\ ( f . b ) = {} ;
a in Free ( ( H2 / ( x. 4 , x. 0 ) ) '&' H2 ) & ( ( H2 / ( x. 0 , x. 4 ) ) '&' H2 ) . a = ( H2 / ( x. 0 , x. 0 ) ) '&' ( H2 / ( x. 0 , x. 4 ) ) ) ;
for C1 , C2 being f1 , C2 being stable Function of C1 , C2 st `1 = C2 & C2 = C2 holds f = g & f = g implies f = g
( W-min L~ go \/ L~ pion1 ) `1 = W-bound L~ pion1 & ( W-min L~ pion1 \/ L~ pion1 ) `1 = W-bound L~ pion1 & ( W-min L~ pion1 \/ L~ pion1 ) `1 = W-bound L~ pion1 & ( W-min L~ pion1 \/ L~ pion1 ) `1 = W-bound L~ pion1 ;
consider u such that u = <* x0 , y0 , z0 *> and f is partial & f is partial & u in dom ( SVF1 ( 3 , f , 1 ) * pdiff1 ( f , 1 ) ) & SVF1 ( 3 , f , 2 ) . u = z0 ;
then ( t . {} ) `1 in Vars & ex x being Element of Vars st x = ( t . {} ) `1 & t . {} = x & t . {} = x & x in Vars & x in Vars & x in Vars & x in Vars & x in Vars & x in Vars & x in Vars & x in Vars & x in Vars & x in Vars & x in Vars ;
Valid ( p '&' p , J ) . v = Valid ( p , J ) . v '&' Valid ( q , 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 & a >= b & b >= c ;
func Class ( R , a ) -> Subset-Family of R means : Def2 : for A being Subset of R holds A in it iff ex a being Element of R st a in it & a in A & it c= A ;
defpred P [ Nat ] means ( ( ( ( ( ( ( ( ( \HM { \rm set ) ) . $1 ) ) `1 ) + ( ( ( ( ( ( ( ( ( ( ( ( ( ( G -' ) . n ) ) ) ) + ( ( ( ( ( ( c , n ) ) ) + ( ( c + 1 ) ) + ( ( ( c + 1 ) ) + 1 ) ) + 1 ) ) ) ) ) ) ) ) ) ) ) `1 <= G ;
assume that dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W2 ) = 0 and dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W2 ) = 0 ;
mama_empty ( m . t ) = ( m . t ) . {} .= ( [ m . t , the carrier of C ] ) . {} .= m . {} .= m . {} .= m . {} .= the carrier of C ;
d11 = ( x9 ^ d22 ) . ( y9 , d22 ) .= f . ( y9 , d22 ) .= f . ( y9 , d22 ) .= ( f ^ <* d22 *> ) . ( y9 , d22 ) .= ( f ^ <* d22 *> ) . ( x9 , d22 ) .= ( f ^ <* d22 *> ) . ( x9 , d22 ) .= 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 . x and g . x in f0 . x ;
x + 0. F_Complex / ( len x |-> 0. F_Complex ) = x + ( x |-> 0. F_Complex ) .= ( x , x ) |-> 0. F_Complex .= x *' * ( x , x ) .= x *' * x .= x *' * x .= x *' * x .= x ;
( k -' ( k9 -' 1 ) ) + 1 in dom ( f | ( ( k -' 1 ) + 1 ) ) & ( f | ( ( k -' 1 ) + 1 ) ) + ( ( f | ( k -' 1 ) ) + 1 ) ) in dom ( f | ( ( k -' 1 ) + 1 ) ) ;
assume that P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P2 = { p1 , p2 } and P1 = P1 \/ P2 and P2 = { p1 , p2 } and P1 = P1 \/ P2 and P2 = { p1 , p2 } and P1 = P2 \/ { p2 , p1 } and P2 = { p1 , p2 } and P1 = P2 \/ { p1 , p2 } and P2 = P1 \/ P2 and P1 = P2 \/ P2 and P2 = P2 \/ P2 and P2 = P1 \/ P2 and P2 = P1 \/ P2 and P2 = P2 \/ P2 and P2 = P1 \/ P2 and P2 = P2 \/ P2 and P2 = P2 \/ P2 and P2 \/ P2 = P1 \/ P2 and P2 \/ P2 = P2 \/ P2 and P2 \/ P2 = P1
reconsider a1 = a , b1 = b , c1 = c , c2 = d , c2 = p `1 , c1 = p `2 , c2 = p `2 , c2 = p `2 , c1 = p `2 , c2 = p `2 , c2 = p `2 , c1 = p `2 , c2 = p `2 as Element of A ;
reconsider thesis thesis thesis thesis in Fb1f . ( t , F . f ) , F1 = ( G1 * F1 ) . ( a , b ) as Morphism of ( G1 * F1 ) . ( a , b ) , ( G1 * F2 ) . ( b , a ) ;
LSeg ( f , i + i1 -' 1 ) = LSeg ( f /. ( i + i1 -' 1 ) , f /. ( i + i1 -' 1 + 1 ) ) .= LSeg ( f , i + i1 -' 1 ) ;
Integral ( M , P . m ) | dom ( P . n -P . m ) <= Integral ( M , P . m -P . m ) + Integral ( M , P . m -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 ) ) `2 ) - ( ( G * ( i + 1 , 1 ) ) `2 ) ;
for G being Group , H being Subgroup of G , a being Element of H , b being Integer 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 = { p9 where p9 is Point of TOP-REAL 2 : P [ p9 ] & p9 `1 >= p9 & p9 `2 <= p9 `1 & p9 `1 <= p9 `1 & p9 `2 <= p9 `1 & p9 `1 <= p9 `1 & p9 `2 <= p9 `2 & p9 `2 <= p9 `2 & p9 `2 <= p9 `2 & p9 `2 <= p9 `2 & p9 `2 <= p9 `2 & p9 `2 <= p9 `2 & p9 `2 <= p9 `2 & p9 `2 <= p9 `2 ;
( ( ( ( N - S ) / ( 2 |^ m ) ) - ( ( N - S ) / ( 2 |^ n ) ) ) / ( 2 |^ m ) ) <= ( ( ( N - S ) / ( 2 |^ m ) ) - ( ( N - S ) / ( 2 |^ m ) ) / ( 2 |^ m ) ) ) / ( 2 |^ m ) ;
for x be Element of X , n be Nat st x in E holds |. ( Re F ) . n .| <= P . x & |. ( Im F ) . n .| <= P . x
len ( @ ( @ ( @ p ^ <* 0 *> ) ) ) = len ( @ ( @ p ^ <* 1 *> ) ) + len ( @ ( @ p ^ <* 1 *> ) ) .= len ( @ ( @ p ^ <* 1 *> ) ) + len ( @ ( @ p ^ <* 1 *> ) ) .= len ( @ p ^ <* 1 *> ) ;
v / ( x. 3 , m1 ) / ( x. 0 , m1 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m1 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m1 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) ) = m3 ;
consider r being Element of M such that M , v / ( x. 3 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) ) |= r ;
func w1 \ w2 -> Element of Union ( G , R8 ) equals ( ( ( the Sorts of G ) * the Arity of S ) * the Arity of S ) . ( w1 , w2 ) ;
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= s1 . b2 .= s2 . b2 .= s2 . b2 .= Exec ( n2 , s2 ) . b2 .= s2 . b2 .= Exec ( n2 , s2 ) . b2 .= s . b2 .= s . b2 .= s . b2 ;
for n , k be Nat holds 0 <= ( Partial_Sums ( |. seq .| ) . ( n + k ) - Partial_Sums ( |. seq .| ) . n ) implies seq is convergent & lim seq = seq . n
set F = S -\mathop { 0 } , G = S -\mathop { 0 } ;
( Partial_Sums ( seq ) ) . ( n + 1 ) + Partial_Sums ( seq ) . ( n + 1 ) >= ( Partial_Sums ( seq ) ) . ( n + 1 ) + Partial_Sums ( seq ) . ( n + 1 ) ;
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 , e , f , g , h , i , g , i , h , i , g , i , f , i , g , h , i , g , i , h , i ) ;
a * b ^2 + ( a * c ) ^2 + ( b * a ) ^2 + ( b * c ) ^2 + ( c * a ) ^2 >= 6 * a * a * b * c + ( b * a ) * c ;
v / ( x1 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m2 ) ) = v / ( x2 , m1 ) ;
Rotate ( Q ^ <* x *> , M1 ) = ( ( \mathop { Q , M1 } +* ( i , { FALSE } ) ) +* ( ( i , 0 ) --> FALSE ) ) +* ( ( ( i , 0 ) --> FALSE ) +* ( ( i , 0 ) --> FALSE ) ) .= ( ( i , 0 ) --> FALSE ) +* ( ( i , 0 ) --> FALSE ) ;
Partial_Sums ( F ) . n = ( r |^ n1 ) * Partial_Sums ( C ) . n1 .= C . n1 * ( r |^ n1 ) .= C . n1 * ( r |^ n1 ) .= C . n1 * ( r |^ n1 ) .= C . n1 * ( r |^ n1 ) .= C . n1 * ( r |^ n1 ) .= C . n1 * ( r |^ n1 ) ;
( ( 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 ) ;
( the_arity_of g ) . 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 ;
( X ~ ) c= X ~ & card ( X ~ ) = card Y ~ implies X = Y & Y = X
for a , b being Element of S , s being Element of NAT st s = n & a = F . n & b = F . ( n + 1 ) holds b = N . ( n + 1 ) \ G . ( n + 1 )
E , f |= All ( x. 2 , ( x. 2 ) |= ( x. 2 ) |= ( x. 2 ) ) '&' ( x. 2 , ( x. 2 ) |= ( x. 2 , ( x. 2 ) ) '&' ( x. 2 , ( x. 2 ) ) '&' ( x. 2 , ( x. 2 ) ) ) ) ) ;
ex R2 be 1-sorted st R2 = ( p | ( n + 1 ) ) . i & ( the carrier of p ) . i = the carrier of R2 & ( the carrier of p ) = the carrier of R2 & ( the carrier of p ) = the carrier of R2 ;
[. a , b + 1 / ( k + 1 ) .[ is Element of the _ of the _ of the \in of the \in of the _ of the \in of the _ of the \in of the \overline the carrier of the \overline the carrier of the carrier of the carrier of X , the carrier of X , the carrier of Y , the carrier of Y , the carrier of Y is Element of the carrier of X ;
Comput ( P , s , 2 + 1 ) = Exec ( P . 2 , Comput ( P , s , 2 ) ) .= Exec ( a3 , s ) .= Exec ( a3 , s ) .= Exec ( a3 , s ) ;
card ( h1 *' ) . k = power ( F_Complex ) . ( - 1_ F_Complex ) * Sum ( - ( - ( 1. F_Complex ) ) * u ) .= ( ( - ( 1. F_Complex ) ) * u ) * u .= ( ( - ( 1. F_Complex ) ) *' ) . k * u .= ( ( - ( 1. F_Complex ) ) * u ) ) . k ;
( f - g ) /. c = f /. c * ( g /. c ) " .= f /. c * ( g /. c ) .= ( f * ( g * ( g * ( g * f ) ) ) ) /. c .= ( f * ( g * ( g * f ) ) ) /. c ;
len C( ( ( the carrier of ( ( len ( ( the carrier of ( 2 * ) ) ) ) ) ) ) ) = len ( ( ( the carrier of ( 2 * ) ) ) ) .= len ( ( ( the carrier of ( 2 * ) ) ) * ( ( the carrier of ( 2 * ) ) ) ) .= len ( ( the carrier of ( 2 * ) ) ) ;
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom f /\ X .= dom f /\ X .= dom f /\ X .= dom f /\ X .= dom f /\ X .= dom f /\ X .= dom f /\ X .= dom f /\ X .= dom f /\ X .= dom f /\ X .= dom f /\ X .= dom f /\ X .= dom f /\ X .= dom f /\ X .= dom f /\ X .= X ;
defpred P [ Nat ] means for n holds 2 * Fib ( n + $1 ) = Fib ( n ) * Fib ( n + $1 ) + ( 5 * Fib ( n + $1 ) ) * Fib ( n + $1 ) ;
consider f being Function of INT , INT such that f = f and f is onto and n < len f and f is onto and for f being Function of INT , INT st f is onto & n < len f holds f " { f . n } = { n } ;
consider c9 be Function of S , BOOLEAN such that c9 = chi ( A \/ B , S ) and ( E . ( A \/ B ) ) = Prob . ( A \/ B ) and ( E . ( A \/ B ) ) = Prob . ( A \/ B ) ;
consider y being Element of Y ( ) such that a = "\/" ( { F ( x , y ) where x is Element of X ( ) : P [ x ] } , y ) and Q [ y , x ] ;
assume that A c= Z and f = ( ( - 1 ) (#) ( ( #Z 2 ) * ( sin + cos ) ) ) / ( sin + cos ) and Z c= dom ( ( - 1 ) (#) ( sin + cos ) ) and Z c= dom ( ( - 1 ) (#) ( cos + cos ) ) ) and Z c= dom ( ( - 1 ) (#) ( sin + 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 ) `2 .= ( GoB f ) * ( 1 , j2 ) `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 q2 = len q1 + len Seq q2 & len Seq q2 = len q1 + len Seq q2 ;
consider G1 , G2 , G3 being Element of V such that G1 <= G2 and G2 <= G2 and f is Morphism of G1 , G2 and f is Morphism of G2 , G2 and g is Morphism of G2 , G2 & g is Morphism of G2 , G2 & g is Morphism of G2 , G2 & g is Morphism of G2 , G2 ;
func - f -> PartFunc of C , V means : Def2 : dom it = dom f & for c be element st c in dom it holds it /. c = - f /. c & for c be Element of C st c in dom it holds it /. c = - f /. c ;
consider phi such that phi is increasing and for a st phi . a = a and for v st v <> a holds union ( L , v ) |= ( L , v ) iff for a holds L . a , v |= ( L , v ) . a ;
consider i1 , j1 such that [ i1 , j1 ] in Indices GoB f and f /. ( i1 + 1 ) = ( GoB f ) * ( i1 , j1 ) and j1 in dom GoB f and j2 in dom GoB f and j1 in dom GoB f and j2 in dom f and j2 in dom f and f /. ( i1 + 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 & n2 <= n holds ( n - 1 ) * ( i - 1 ) <= ( n - 1 ) * ( i - 1 ) ;
assume that not 0 in Z and Z c= dom ( ( ( - 1 / 2 ) (#) ( ( #Z 2 ) * ( f1 + #Z 2 ) ) ) ) and for x st x in Z holds ( ( ( - 1 / 2 ) * ( f1 + #Z 2 ) ) `| Z ) . x = 1 / ( x - 0 ) and ( ( - 1 / 2 ) (#) ( f1 + #Z 2 ) ) `| Z ) . x = 1 / ( x - 0 ) ;
cell ( G1 , i1 -' 1 , ( 2 |^ ( m -' 1 ) ) * ( Y1 -' 1 ) ) \ L~ ( ( Y1 -' 1 ) * ( Y1 -' 1 ) ) \ L~ ( ( Y1 -' 1 ) * ( Y1 -' 1 ) ) c= BDD L~ ( ( Y1 -' 1 ) * ( Y1 -' 1 ) ) \ L~ ( ( Y1 -' 1 ) * ( Y1 -' 1 ) ) ;
ex Q1 being open Subset of X st s = Q1 & ex Q1 being Subset-Family of Y st Q1 c= F & ( for a being Subset of Y st a in F holds a in Q1 ) & ( for b being Subset of Y st b in F holds b c= a ) & ( for b being Element of Y holds b in F . b ) implies a in b )
gcd ( A , ( 1. ( A , ( 1 , 1 ) , s1 ) ) , s2 ) = 1 / 2 * ( ( 1. ( A , ( 1 , 1 ) , s2 ) ) * ( ( 1. ( A , ( 1 , 1 ) ) , s2 ) ) ) .= 1 / 2 * ( ( 1. ( A , ( 1 , 1 ) ) * ( s2 , s2 ) ) ) ) ;
R8 = ( ( ( j , s2 ) . ( m1 + 1 ) ) * ( ( j , s2 ) . ( m1 + 1 ) ) ) . ( m2 + 1 ) .= [ 3 , 1 ] . ( m2 + 1 ) .= [ 3 , 1 ] . ( m2 + 1 ) .= [ 3 , 1 ] . ( m2 + 1 ) .= [ 3 , 1 ] . ( m2 + 1 ) ;
CurInstr ( P3 , Comput ( P3 , s3 , m1 + 1 ) ) = CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= halt SCMPDS .= halt SCMPDS .= ( CurInstr ( P3 , s3 ) ) ;
P1 /\ P2 = ( { p1 } \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) ) /\ ( LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) ) .= ( LSeg ( p1 , p2 ) \/ LSeg ( p2 , p2 ) ) \/ ( LSeg ( p1 , p2 ) \/ { p2 } ) .= { p1 , p2 } \/ { p2 , p1 } ;
func the still of f -> Subset of the Sorts of the Sorts of A means : Def2 : a in it iff ex p st p in dom f & a in it & f . p = a & for x st x in dom f holds x in dom f ;
for a , b being Element of F_Complex for f being Polynomial of F_Complex st |. a .| > 1 for f being Polynomial of n , F_Complex st f >= 1 & f is \cap ( len f ) holds f * ( - b ) is \cup ( f * ( - b ) )
defpred P [ Nat ] means 1 <= $1 & $1 <= len g implies for i , j st [ i , j ] in Indices G & G * ( i , j ) = g * ( $1 , i ) & 1 <= j & j <= len G & 1 <= i & i <= len G & j <= len G holds G * ( i , j ) = G * ( $1 , j ) ;
assume that C1 , C2 , f , g being Let s1 , s2 being State of C1 , s1 , s2 being State of C2 , f being Function of C1 , C2 st s1 = s2 holds s1 is stable iff f * f is stable & f * g is stable & f * g is stable ;
( ||. 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 .|| ;
|. q .| ^2 = ( q `1 ) ^2 + ( q `2 ) ^2 & 0 + ( q `2 ) ^2 < ( q `1 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 ;
for F being Subset-Family of T7 st F is open & {} in F & not {} in F & A <> {} & A is closed & A is closed & A is closed & A is closed holds card F = card ( A \/ B )
assume that len F >= 1 and len F = k + 1 and len F = k and for k st k in dom F holds F . k = g . ( F . k , G . k ) and for k st k in dom F holds F . k = g . ( F . k , G . k ) ;
i |^ ( ( gcd ( n , i ) - i ) ) |^ s = i |^ ( s + k ) - i |^ ( s + k ) * ( ( i |^ ( s + k ) - i ) ) .= i |^ ( s + ( s + k ) - i ) * ( ( i |^ ( s + k ) - i ) ) * ( ( i |^ ( s + k ) - i ) ) .= i |^ ( s + ( s + k ) ) * ( ( i |^ ( s + 1 ) ) * ( ( s + 1 ) - ( ( s + 1 ) * ( ( s + 1 ) - ( s + 1 ) ) * ( ( s + 1 ) - ( s + 1 ) ) * ( ( s + 1 ) - ( s + 1 ) * (
consider q being oriented oriented Chain of G such that r = q and q <> {} and ( F . ( q . 1 ) ) `1 = v1 and ( F . ( q . len q ) ) `1 = v2 and rng q c= rng ( p ^ <* q . 1 *> ) and rng q c= rng p and p is oriented and p ^ q is oriented ;
defpred P [ Element of NAT ] means $1 <= len ( g ) implies ( g ) . $1 = ( ( g , Z ) ^ <* f . $1 *> ) . ( len g + $1 ) & ( g ) . $1 = ( ( g , Z ) ^ <* f . $1 *> ) . ( len f + $1 ) ;
for A , B being Matrix of n , REAL for B being Matrix of n , REAL holds len ( A * B ) = len A & width ( A * B ) = n & width ( A * B ) = n & width ( A * B ) = n & len ( A * B ) = n & width ( A * B ) = n & len ( A * B ) = n & len ( A * B ) = n ;
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 ) , ( Im y ) )| + ( ( Im y ) * ( ( Re x ) * ( ( Im y ) * ( ( Im x ) * ( ( Im y ) * ( ( Im y ) * ( ( Im y ) * ( ( Im y ) * ( ( Im y ) * ( ( Im y ) * ( ( Im y ) * ( ( Im y ) ) ) ) ) ) ) ) ) ) , ( ( x ) ) ) ) )| ;
consider g9 being FinSequence of FF such that g9 is continuous and rng ex g being FinSequence of FF st 0 <= g & rng g c= A & for i be Nat st i in dom g & i <= len g holds g . i = x1 . i & g . i = y1 . i ;
then n1 >= len p1 & crossover ( p1 , p2 , n1 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , a9 ) = crossover ( p1 , p2 , n1 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , a9 ) ;
( q `1 ) * a <= ( q `1 ) * a & - ( q `2 ) * a <= ( q `1 ) * a or q `1 >= ( q `1 ) * a & - ( q `2 ) * a <= ( q `1 ) * a & - ( q `1 ) * a <= ( q `1 ) * a ;
Fv . ( p9 . ( len p9 ) ) = Fv . ( p9 . ( len p9 ) ) .= ( ( v . ( len p9 ) ) * ( p9 . ( len p9 ) ) ) .= ( ( v . ( p9 . ( len p9 ) ) ) * ( v . ( p9 . ( len p9 ) ) ) ) .= ( ( v . ( p9 . ( len p9 ) ) ) * ( v . ( p9 . ( len p9 ) ) ) .= ( ( v . ( p9 . ( p9 . ( p9 . ( p9 . ( p9 . ( p9 . ( p9 + 1 ) ) ) ) * ( p9 . ( p9 . ( p9 . ( p9 . ( p9 . ( p9 . ( p9 . ( p9 . ( p9 . ( p9 . ( p9 . ( p9 + 1 ) ) ) ) * ( p9 . ( p9 + 1 ) ) ) .=
consider k1 being Nat such that k1 + 1 = 1 and a := k = ( <* a *> ^ ( ( intloc 0 ) --> ( intloc 0 ) ) ) ^ ( ( intloc 0 ) --> ( intloc 0 ) ) ) ^ ( ( intloc 0 ) --> ( ( intloc 0 ) .--> ( intloc 0 ) ) ) ) ;
consider B8 being Subset of B1 , y8 being Function of B1 , A1 such that B8 is finite and D1 = the carrier of A1 and B1 = the carrier of A1 and B1 = the carrier of B1 and B1 is finite and B2 is finite and B1 is finite and B2 is finite and B1 is finite and B2 is finite ;
v2 . b2 = ( curry ( F2 , g ) * ( ( curry F ) . b2 ) ) . b2 .= ( ( curry F ) . b2 ) . b2 .= ( ( curry F ) . b2 ) . b2 .= ( ( ( curry F ) . b2 ) * ( ( curry F ) . b2 ) ) . b2 .= ( ( ( curry F ) . b2 ) * ( ( ( curry F ) . b2 ) * ( id F ) . b2 ) ) .= ( ( ( id B ) . b2 ) * ( id B ) . b2 ) . b2 ;
dom IExec ( I-35 , P , Initialize s ) = the carrier of SCMPDS .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) ;
ex d-32 be Real st d-32 > 0 & for h be Real st h <> 0 & |. h .| < e holds |. h .| " * ||. ( R /* h ) . h .|| < e / ( |. h .| ) * ||. ( R /* h ) . h .|| ) & ( R * ( L + h ) ) . h = e / ( |. h .| ) * ||. 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 + i1 -' 1 ) , h /. ( i + i1 -' 1 + 1 ) ) .= LSeg ( h /. ( i + i1 -' 1 ) , h /. ( i + i1 -' 1 ) ) ;
A = { q where q is Point of TOP-REAL 2 : LE p1 , p2 , P & LE p2 , p1 , P & LE p1 , p2 , P & LE p2 , p1 , P & LE p1 , p2 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p1 , p2 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & p1 , p2 , P & p2 , p1 , P & p1 , p2 , P & p2 , p1 , P & p2 , p1 , P & p2 , p1 , P & p2 , p1 , P & p2 , p1 , P & p2 , p1 , P & p2 , p1 , P & p1 , p1 , P & p1 , p1 , P & p2 , p1 , P & p1 , p1 , p1 , p2 , p1 , p1 , p2 , P & p2 , p1 , p1 , p1 , p2 , P & p2 ,
( ( - 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 ) ;
0 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) = ( ( p `1 ) ^2 + ( p `2 ) ^2 ) * sqrt ( 1 + ( p `2 ) ^2 ) .= ( ( p `1 ) ^2 + ( p `2 ) ^2 ) * sqrt ( 1 + ( p `2 ) ^2 ) .= ( p `1 ) ^2 + ( p `2 ) ^2 ;
( ( U * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( L * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( L * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( L * ( W * ( L * ( W * ( L * ( L * ( W * ( L * ( L * ( L * ( L * ( L * ( L * ( L * ( L * ( L * ( L * ( L * ( L * ( L * (
func Shift ( f , h ) -> PartFunc of REAL , REAL means : Def2 : dom it = dom it & for x be Element of REAL n holds it . x = ( - h ) . x & for x be Element of REAL n st x in dom it holds it . x = ( - h ) . x + ( - h ) . x ;
assume that 1 <= k and k + 1 <= len f and [ i + 1 , j ] in Indices G and f /. k = G * ( i + 1 , j ) and f /. ( k + 1 ) = G * ( i + 1 , j ) and f /. ( k + 1 ) = G * ( i , j ) ;
assume that not y in Free H and not x in Free H and not x in Free ( H ) and not x 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 not x in Free ( H ) and not x in Free ( H ) and x in Free ( H ) ;
defpred P11 [ Element of NAT , Element of NAT ] means ( P [ $1 ] implies ( $1 = p |^ $1 ) & ( $1 = p |^ $1 implies $2 = p |^ $1 ) & ( $1 = p |^ $1 implies $2 = p |^ $1 ) & ( $1 = p |^ $1 implies $2 = p |^ $1 ) & ( $1 = p |^ $1 implies $2 = p |^ $1 ) ;
func \sigma ( C ) -> non empty Subset-Family of X means : Def2 : for A being Subset of X holds A in it iff for W being Subset of X st W in it holds W in it & W is closed & for W being Subset of X st W in it holds W in C \ A & W c= W \ A & W c= W \ B ;
[#] ( ( dist ( ( dist ( ( dist ( P ) ) ) ) .: Q ) ) ) = ( ( dist ( ( dist ( P ) ) ) .: Q ) ) .: Q & lower_bound ( ( dist ( ( dist ( P ) ) .: Q ) .: Q ) ) = lower_bound ( ( dist ( P ) ) .: Q ) ;
rng ( F | ( [: S , T :] ) ) = {} or rng ( F | ( [: S , T :] ) ) = { 1 } or rng ( F | ( [: S , T :] ) ) = { 2 } or rng ( F | ( [: S , T :] ) ) = { 1 } or rng ( F | ( [: S , T :] ) ) = { 2 } ;
( f " * ( ( f | ( rng f ) ) ) ) . i = f . i " .= ( f * ( f | ( rng f ) ) ) . i .= ( f * ( f | ( rng f ) ) ) . i .= ( f * ( f | ( rng f ) ) ) ) . i .= ( f * ( f | ( rng f ) ) ) . i .= ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f * ( f
consider P1 , P2 being non empty Subset of TOP-REAL 2 such that P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 is_an_arc_of p2 , p2 ;
f . p2 = |[ ( p2 `1 ) ^2 + ( p2 `2 ) ^2 , ( p2 `2 ) ^2 + ( p2 `2 ) ^2 + ( p2 `2 ) ^2 + ( p2 `2 ) ^2 * ( p2 `2 ) ^2 + ( p2 `2 ) ^2 ;
( ( the TopStruct of a , X ) " ) . x = ( ( the on of a , X ) qua Function ) . x .= ( ( the TopStruct of a , X ) qua Function ) . x .= ( ( the TopStruct of a , X ) . x ) . x .= ( ( the TopStruct of a , X ) * x ) . x .= ( ( the p1 of X ) + x ) . x .= ( ( the p2 of X ) * x ) . x ;
for T being non empty normal TopSpace , A , B being closed Subset of T , A being Subset of T st A <> {} & A misses B for p being Point of T , B being Subset of T st p in B & B misses A holds p in ( in Cl ( A ) ) holds p in ( ( Cl ( A ) ) /\ ( ( Cl ( B ) ) /\ ( Cl ( B ) ) ) )
for i st i in dom F for G1 , G2 being strict normal Subgroup of G , G1 being strict Subgroup of G st G1 = F . i & G2 = F . ( i + 1 ) & G2 = F . ( i + 1 ) holds G1 = G2
for x st x in Z holds ( ( ( ( #Z 2 ) * ( ( arccot ) ) * ( ( arccot ) ) * ( ( arccot ) * ( ( arccot ) ) * ( ( arccot ) * ( ( arccot ) ) * ( ( arccot ) * ( ( arccot ) ) * ( ( arccot ) * ( ( arccot ) * ( ( arccot ) * ( ( arccot ) ) * ( ( arccot ) * ( ( arccot ) ) * ( ( arccot ) ) ) ) ) ) ) ) ) ) ) . x = ( ( ( ( arccot ) * ( ( arccot ) ) * ( ( arccot ) ) ^2 ) ) / ( 1 + ( arccot ) ) ^2 ) ) . x ) ^2
synonym f is right means : Def2 : x0 in dom ( f /* a ) & for x st x in dom f & x in ]. x0 , x0 + r .[ holds f . x - f . x0 = f . x - f . x0 & for x st x in dom f holds f . x - f . x = f . x - f . x0 ;
then X1 , X2 , X3 , X2 being non empty SubSpace of X , Y2 being SubSpace of X1 , X2 being SubSpace of X2 , Y1 being Subset of X1 , Y2 being Subset of X2 holds Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & Y2 is SubSpace of X2 implies X1 is SubSpace of X2 & Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & Y2 is SubSpace of X2 implies Y1 is SubSpace of X1 union X2 is SubSpace of X2
ex N be 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- ( 1 , f , u ) ) . ( x - x0 ) + R . ( x - x0 )
( ( p2 `1 ) ^2 * sqrt ( 1 + ( p3 `2 ) ^2 ) ) * ( ( p2 `1 ) ^2 + ( p3 `2 ) ^2 ) >= ( ( p2 `1 ) ^2 * sqrt ( 1 + ( p3 `2 ) ^2 ) ) * sqrt ( 1 + ( p3 `1 ) ^2 ) ;
( ( 1 - t1 ) (#) ( ||. f1 .|| ) ) to_power ( n + 1 ) = ( ( 1 - t1 ) * ( ||. g1 .|| ) ) to_power ( n + 1 ) ) .= ( ( 1 - t1 ) * ( ||. g1 .|| ) to_power ( n + 1 ) ) .= ( ( 1 - t1 ) * ( ||. g1 .|| ) to_power ( n + 1 ) ;
assume that for x holds f . x = ( ( sin + cos ) (#) ( sin - cos ) ) . x and for x st x in Z holds ( ( sin + cos ) (#) ( sin - cos ) ) . x = ( sin . x - cos . x ) / ( sin . x ) ^2 and ( ( sin + cos ) (#) ( sin - cos ) ) . x = ( sin . x - cos . x ) / ( sin . x ) ^2 ;
consider Xi1 being Subset of Y , Y1 being Subset of X such that Y1 is open and Y1 is open and ex Y1 being Subset of X st 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 ;
card ( S . n ) = card { [: d , d :] + ( a * d ) + b where d is Element of GF ( p ) : d in the carrier of GF ( p ) } .= 1 .= d + d .= d + d .= ( a + d ) * d .= d + d ;
( ( W-bound D - W-bound D ) / ( 2 |^ ( m + 1 ) ) * ( i - 2 ) ) * ( i - 2 ) = ( ( W-bound D - W-bound D ) / ( 2 |^ ( m + 1 ) ) ) * ( i - 2 ) .= ( ( W-bound D - W-bound D ) / ( 2 |^ ( m + 1 ) ) ) * ( i - 2 ) .= ( ( W-bound D - 2 ) / ( 2 |^ m ) ) * ( i - 2 ) ;