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 ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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 ;
y in N ;
x in T ;
m < n ;
m <= n ;
n > 1 ;
let r ;
t in I ;
n <= 4 ;
M is finite ;
let X ;
Y c= Z ;
A // M ;
let U ;
a in D ;
q in Y ;
let x ;
1 <= l ;
1 <= w ;
let G ;
y in N ;
f = {} ;
let x ;
x in Z ;
let x ;
F is one-to-one ;
e <> b ;
1 <= n ;
f is special ;
S misses C
t <= 1 ;
y divides m ;
P divides M ;
let Z ;
let x ;
y c= x ;
let X ;
let C ;
x _|_ p ;
o is monotone ;
let X ;
A = B ;
1 < i ;
let x ;
let u ;
k <> 0 ;
let p ;
0 < r ;
let n ;
let y ;
f is onto ;
x < 1 ;
G c= F ;
a is_>=_than X ;
T is continuous ;
d <= a ;
p <= r ;
t < s ;
p <= t ;
t < s ;
let r ;
D <= E ;
assume e > 0 ;
assume 0 < g ;
p in X ;
x in X ;
Y ` in Y ;
assume 0 < g ;
not c in Y ;
not v in L ;
2 in z `2 ;
assume f = g ;
N c= b ` ;
assume i < k ;
assume u = v ;
I = J ;
B ` = b ` ;
assume e in F ;
assume p > 0 ;
assume x in D ;
let i be element ;
assume F is onto ;
assume n <> 0 ;
let x be element ;
set k = z ;
assume o = x ;
assume b < a ;
assume x in A ;
a `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 BCK-algebra ;
assume S is non empty ;
a in REAL ;
let p be set ;
let A be set ;
let G be _Graph , v be Vertex of G ;
let G be _Graph , v be Vertex 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 ;
let y be Real ;
X c= f . a
let y be element ;
let x be element ;
i be Nat ;
let x be element ;
n in NAT ;
let a be element ;
m in NAT ;
let u be element ;
i in NAT ;
let g be Function ;
Z c= NAT ;
l <= ma ;
let y be element ;
r2 in dom h ;
let x be element ;
let k1 be Integer ;
let X be set ;
let a be element ;
let x be element ;
let x be element ;
let q be element ;
let x be element ;
assume f is being_homeomorphism ;
let z be element ;
a , b // K ;
let n be Nat ;
let k be Nat ;
B ` c= B ` ;
set s = Set 1 ;
n >= 0 + 1 ;
k c= k + 1 ;
R1 c= R ;
k + 1 >= k ;
k c= k + 1 ;
let j be Nat ;
o , a // Y ;
R c= Cl G ;
Cl B = B ;
let j be Nat ;
1 <= j + 1 ;
arccot is_differentiable_on Z ;
exp_R is_differentiable_in x ;
j < i0 ;
let j be Nat ;
n <= n + 1 ;
k = i + m ;
assume C meets S ;
n <= n + 1 ;
let n be Nat ;
h1 = {} ;
0 + 1 = 1 ;
o <> b3 ;
f2 is one-to-one ;
support p = {}
assume x in Z ;
i <= i + 1 ;
r1 <= 1 ;
let n be Nat ;
a "/\" b <= a ;
let n be Nat ;
0 <= r0 ;
let e be Real , x be Point of X ;
not r in G . l
c1 = 0 ;
a + a = a ;
<* 0 *> in e ;
t in { t } ;
assume F is not discrete ;
m1 divides m ;
B * A <> {} ;
a + b <> {} ;
p * p > p ;
let y be ExtReal ;
let a be Int-Location , I be Program of SCM+FSA ;
let l be Nat ;
let i be Nat ;
let r ;
1 <= i2 ;
a "\/" c = c ;
let r be Real ;
let i be Nat ;
let m be Nat ;
x = p2 ;
let i be Nat ;
y < r + 1 ;
rng c c= E
Cl R is boundary ;
let i be Nat ;
R2 ;
cluster uparrow x -> 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 ;
2 >= \mathclose { \rm \hbox { - } s } ;
G . y <> 0 ;
let X be RealNormSpace , x be Point of X ;
a in X ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , M be Subset of V ;
assume x in mid M ;
k < s . a ;
not t in { p } ;
let Y be set , f be Function of Y , BOOLEAN ;
M , L are_isomorphic ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded ;
rng f = Y ;
G8 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 `1 = a * y `1 ;
rng D c= A ;
assume x in K1 ;
1 <= i-32 ;
1 <= i-32 ;
pif c= PI , then pV c= PI ;
1 <= i-15 ;
1 <= i-15 ;
UMP C in L ;
1 in dom f ;
let seq , seq1 , seq2 ;
set C = a * B ;
x in rng f ;
assume f is_continuous_on X ;
I = dom A ;
u in dom p ;
assume a < x + 1 ;
s-7 is bounded ;
assume I c= P1 ;
n in dom I ;
let Q ;
B c= dom f ;
b + p _|_ a ;
x in dom g ;
F-14 is continuous ;
dom g = X ;
len q = m ;
assume A2 is closed ;
cluster R \ S -> real-valued ;
sup D in S ;
x << sup D ;
b1 >= Z1 ;
assume w = 0. V ;
assume x in A . i ;
g in the \rm being carrier of X ;
y in dom t ;
i in dom g ;
assume P [ k ] ;
if C c= f , g holds 0 <= f
x4 is increasing ;
let e2 be element ;
- b divides b ;
F c= \tau ( F ) ;
Gseq is non-decreasing ;
Gseq is non-decreasing ;
assume v in H . m ;
assume b in [#] B ;
let S be non void ManySortedSign , A be non-empty MSAlgebra over S ;
assume P [ n ] ;
assume union S is finite independent ;
V is Subspace of V ;
assume P [ k ] ;
rng f c= NAT ;
assume ex_inf_of X , L ;
y in rng f ;
let s , I be set , A be ManySortedSet of I ;
b ` c= b ` ;
assume not x in NAT + ;
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 is_differentiable_in x ;
assume y in rng S ;
let x , y be element ;
i2 < i1 & i1 < i2 ;
a * h in a * H ;
p , q in Y ;
redefine func sqrt I ;
q1 in A1 & q2 in A2 ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A2 c= A1 ;
an < n & n <= k ;
assume A c= dom f ;
Re f is_integrable_on M ;
let k , m be element ;
a , a \equiv b , b ;
j + 1 < k + 1 ;
m + 1 <= n1 ;
g is_differentiable_in x0 & g is_differentiable_in x0 ;
g is_continuous_in x0 & g is_continuous_in x0 ;
assume that O is symmetric and O is symmetric ;
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 +* 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 , A be Subset of X ;
[ a , b ] in R ;
x + w < y + w ;
{ a , b } >= c ;
let B be Subset of A , C be Subset of A ;
let S be non empty ManySortedSign ;
let x be variable of f , g be element ;
let b be Element of X , c be Element of X ;
R [ x , y ] ;
x ` = x ;
b \ x = 0. X ;
<* d *> in D |^ 1 ;
P [ k + 1 ] ;
m in dom ( mn1 + 1 ) ;
h2 . a = y ;
P [ n + 1 ] ;
redefine func G * F -> |^ ;
let R be non empty multMagma , a be Element of R ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p ;
assume f | X is lower ;
x in rng co /\ rng co ;
let x be Element of B ;
let t be Element of D ;
assume x in Q .vertices() ;
set q = s ^\ k ;
let t be VECTOR of X ;
let x be Element of A ;
assume y in rng p `1 ;
let M be as mamamaid ;
let N be non empty for \mathop { N } is non empty Subset of M ;
let R be RelStr with finite finite ;
let n , k be Nat ;
let P , Q 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 does not lim s ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < v8 ;
x <= c2 . x ;
x in F ` ;
redefine func S --> T -> * ;
assume that t1 <= t2 and t2 <= t2 ;
let i , j be even Integer ;
assume that F1 <> F2 and F2 <> G2 ;
c in Intersect ( union R ) ;
dom p1 = c & dom p2 = c ;
a = 0 or a = 1 ;
assume A1 <> A2 & A2 <> A1 ;
set i1 = i + 1 ;
assume a1 = b1 & b1 = b2 ;
dom g1 = A & dom g2 = A ;
i < len M + 1 ;
assume not -infty in rng G ;
N c= dom f1 /\ dom f2 ;
x in dom sec /\ dom sec ;
assume [ x , y ] in R ;
set d = ( x - y ) / ( x - y ) ;
1 <= len g1 & g1 /. 1 = g1 . 1 ;
len s2 > 1 & len s2 > 1 ;
z in dom f1 /\ dom f2 ;
1 in dom D2 & 1 in dom D2 ;
( p `2 ) ^2 = 0 ;
j2 <= width G & 1 <= j1 ;
len PI > 1 + 1 ;
set n1 = n + 1 ;
|. q-35 .| = 1 ;
let s be SortSymbol of S ;
if i = i holds i = 1 ;
X1 c= dom f & X2 c= dom f ;
h . x in h . a ;
let G be \setminus id of on ;
cluster m * n -> square ;
let kk be Nat , k be Nat ;
i - 1 > m - 1 ;
R is transitive implies field R c= field R
set F = <* u , w *> ;
p-2 c= P3 & p-2 c= P3 ;
I is_halting_on t , Q ;
assume [ S , x ] is .| ;
i <= len f2 implies ( f2 /. i ) `1 <= ( f2 /. i ) `1
p is FinSequence of X ;
1 + 1 in dom g ;
Sum R2 = n * r ;
cluster f . x -> complex-valued ;
x in dom f1 /\ dom f2 ;
assume [ X , p ] in C ;
BF 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 T ;
assume n in dom g2 ;
let a be Element of R ;
t `1 in dom e2 ;
N . 1 in rng N ;
- z in A \/ B ;
let S be SigmaField of X , T be non empty TopSpace ;
i . y in rng i ;
REAL c= dom f /\ dom g ;
f . x in rng f ;
mt <= ( r / 2 ) * 2 ;
s2 in r-5 & s2 in r-5 ;
let z , z be complex number ;
n <= NN . m ;
LIN q , p , s ;
f . x = waybelow x /\ B ;
set L = [' S , T '] ;
let x be non positive ExtReal ;
let m be Element of M ;
f in union rng F1 ;
let K be add-associative right_zeroed right_complementable associative commutative associative associative distributive non empty doubleLoopStr , A be Subset of K ;
let i be Element of NAT , k be Nat ;
rng ( F * g ) c= Y
dom f c= dom x & dom g c= dom x ;
n1 < n1 + 1 & n2 + 1 < n2 + 1 ;
n1 < n1 + 1 & n2 + 1 < n2 + 1 ;
cluster { T } -> \overline W ;
[ y2 , 2 ] = z ;
let m be Element of NAT , n be 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 /\ dom h2 ;
w + 1 = ( a + 1 ) ;
j + 1 <= j + 1 + 1 ;
k2 + 1 <= k1 + 1 ;
let i be Element of NAT ;
Support u = Support p & Support u = Support p ;
assume X is complete complete holds X is complete ;
assume that f = g and p = q ;
n1 <= n1 + 1 & n2 + 1 <= n2 + 1 ;
let x be Element of REAL , y be Real ;
assume x in rng s2 ;
x0 < x0 + 1 / 2 ;
len L5 = W & len L5 = W ;
P c= Seg ( len A ) ;
dom q = Seg n & dom q = Seg n ;
j <= width M *' ;
let seq1 be real-valued sequence of X ;
let k be Element of NAT ;
Integral ( M , P ) < +infty ;
let n be Element of NAT , x be Element of NAT ;
assume z in \cup being \cup being as as as as as of as such that z in A ;
let i be set ;
n -' 1 = n-1 - 1 ;
len ( n-27 ) = n & len ( nu ) = n ;
\mathop { Z } c= F ;
assume x in X or x = X ;
x is LIN of b , c ;
let A , B be non empty set , f be Function of A , B ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q .= Seg i ;
let s be Element of E ^ ;
let B1 be Basis of x , B2 be Basis of x ;
L3 /\ L2 = {} ;
L1 /\ LSeg ( L2 , p2 ) = {} ;
assume downarrow x = downarrow y ;
assume b , c _|_ b `1 , c `2 ;
LIN q , c , c ;
x in rng ( f | f-129 ) ;
set n8 = n + j ;
let D7 be non empty set , f be Function of D7 , D ;
let K be right_zeroed non empty addLoopStr , M be Matrix of K ;
assume that f opp = f and h opp = h ;
R1 - R2 is total & R1 - R2 is total ;
k in NAT & 1 <= k ;
let a be Element of G ;
assume x0 in [. a , b .] ;
K1 ` is open & K1 ` is open ;
assume that a , b ] is maximal distance ;
let a , b be Element of S ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = Integral ( M , f ) ;
cluster non ns[ for \geq nlen s\lbrack
not u in { ag } ;
the carrier of f c= B ;
reconsider z = x as VECTOR of V ;
cluster the RelStr of L -> empty ;
r (#) H is as as PartFunc of X , REAL ;
s . intloc 0 = 1 ;
assume that x in C and y in C ;
let U0 be strict universal algebra , A be non-empty MSAlgebra over S , B 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 : ( ex y being Element of L st y in : x <= y ) ;
let x , y be Element of X ;
let A , I be |^ of X ;
[ y , z ] in [: O , O :] ;
( not ( card Macro i ) = 1 ) implies card i = card i
rng Sgm A = A ;
q |- p \! \mathop { y } q ;
for n holds X [ n ] ;
x in { a } & x in d ;
for n holds P [ n ] ;
set p = |[ x , y , z ]| ;
LIN o , a , b ;
p . 2 = Z / Y ;
( D = {} ) & ( D = {} ) ;
n + 1 + 1 <= len g ;
a in [: NAT , NAT :] ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
let I be Ideal of L ;
set g = f1 + f2 , h = f2 + f3 ;
a <= max ( a , b ) ;
i-1 < len G + 1-1 ;
g . 1 = f . i1 ;
x `1 , y `2 ] in A2 ;
( f /* s ) . k < r ;
set v = VAL g ;
i - k + 1 <= S ;
cluster associative associative for non empty multMagma ;
x in support ( ( support t ) + ( support b ) ) ;
assume a in [: the carrier of G , the carrier of G :] ;
i `1 <= ( y `1 ) / ( y `1 ) ;
assume p divides b1 + b2 & b1 divides b2 ;
M <= sup M1 & M <= sup M2 ;
assume x in W-min ( X ) & y in W-min ( X ) ;
j in dom ( z | k ) ;
let x be Element of D ( ) ;
IC s4 = l1 & IC s4 = l1 ;
a = {} or a = { x } ;
set uG = Vertices G , uG = Vertices G ;
seq " is non-zero & ( seq " ) (#) ( seq " ) is non-zero ;
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
h-4 c= hF & hF c= hF ;
]. a , b .[ c= Z ;
X1 , X2 are_separated implies X1 , X2 are_separated
a in Cl ( union F \ G ) ;
set x1 = [ 0 , 0 ] , x2 = [ 0 , 1 ] , x3 = [ 0 , 0 ] , x4 = [ 0 , 1 ] , x4 = [ 0 , 1
k + 1 - 1 = k - 1 ;
cluster empty for Relation of NAT , NAT ;
ex v st C = v + W ;
let IT be non empty addLoopStr , p be Point of IT ;
assume V is Abelian add-associative right_zeroed right_complementable associative associative associative associative associative distributive ;
XY \/ Y in \sigma ( L ) ;
reconsider x = x as Element of S ;
max ( a , b ) = a ;
sup B is upper & sup B is upper ;
let L be non empty reflexive RelStr , X be Subset of L ;
R is reflexive & R is transitive implies R + R is transitive
E , g |= the_right_argument_of H implies E , g |= f
dom G `2 = a ;
( 1 / 4 ) >= - r ;
G . p0 in rng G & G . p0 in rng G ;
let x be Element of FF , y be Element of F ;
D [ P-6 , 0 ] ;
z in dom id ( B ) & z in dom id ( B ) ;
y in the carrier of N & y in the carrier of N ;
g in the carrier of H & h in the carrier of G ;
rng ( f | X ) c= NAT & rng ( f | X ) c= NAT ;
j `1 + 1 in dom s1 ;
let A , B be strict Subgroup of G ;
let C be non empty Subset of R^1 ;
f . z1 in dom h & h . z2 in dom h ;
P . k1 in rng P & P . k1 in rng P ;
M = ( A +* {} ) +* {} .= A +* {} ;
let p be FinSequence of REAL , r be Real ;
f . n1 in rng f & f . n2 in rng f ;
M . ( F . 0 ) in REAL ;
( - a ) / 2 = b-a ;
assume the distance of V , Q is v ;
let a be Element of ^ ( V ) ;
let s be Element of PH ( s ) ;
let Pc be non empty \rm RelStr , f be Function of P , Q ;
let n be Nat ;
the carrier of g c= B & the carrier of g c= B ;
I = halt SCM+FSA .= ( the InstructionsF of SCM+FSA ) . 0 ;
consider b being element such that b in B ;
set BM = BCS ( K , n ) ;
l <= ( -> monotone ) ;
assume x in downarrow [ s , t ] ;
( x `2 ) in uparrow t ;
x in ( <* T *> ) . 1 ;
let h be Morphism of c , a ;
Y c= ( the carrier of the_rank_of Y ) \ { A } ;
A2 \/ A3 c= L1 \/ L2 \/ L2 \/ L1 ;
assume LIN o , a , b ;
b , c // d1 , e2 ;
x1 , x2 , x3 , x4 , x5 , x5 , x5 , x6 , x6 , x5 , x6 , x6 , x6 , x6 , x5 , Real ;
dom <* y *> = Seg 1 .= Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar s .| ;
[ x , x `2 ] in X ~ ;
for n being Nat holds 0 <= x . n
[' a , b '] = [. a , b .] ;
cluster -> \hbox closed for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 in P & q1 <> q2 ;
dom M1 = Seg n & dom M2 = Seg n ;
x = [ x1 , x2 ] & y = [ y1 , y2 ] ;
let R , Q be ManySortedSet of A ;
set d = ( 1 / ( n + 1 ) ) ;
rng g2 c= dom W & ( W + L ) . n = W . n ;
P . ( [#] Sigma \ B ) <> 0 ;
a in field R & a = b ;
let M be non empty Subset of V , v be Element of V ;
let I be Program of SCM+FSA , a be Int-Location ;
assume x in rng ( the InternalRel of R ) ;
let b be Element of the carrier of T ;
dist ( e , z ) - re > re ;
u1 + v1 in W2 & v1 + v2 in W1 + W2 ;
assume that Carrier L misses rng G and not x in rng G ;
let L be lower-bounded antisymmetric transitive antisymmetric RelStr ;
assume [ x , y ] in a9 ;
dom ( A * e ) = NAT ;
let a , b be Vertex of G ;
let x be Element of Bool M , y be Element of Bool M ;
0 <= 2 * PI ;
o9 , a9 // o9 , y & o9 , c9 // o9 , y ;
{ v } c= the carrier of l & { v } c= the carrier of l ;
let x be bound of A ;
assume x in dom ( uncurry f ) & y in dom ( uncurry f ) ;
rng F c= ( product f ) |^ X
assume that D2 . k in rng D and D . k = 0 ;
f " . p1 = 0 & f . p2 = 0 ;
set x = the Element of X , y = the Element of X ;
dom Ser ( G ) = NAT & rng Ser ( G ) = NAT ;
let n be Element of NAT ;
assume LIN c , a , e1 ;
cluster finite for FinSequence of NAT ;
reconsider d = c , e = d as Element of L1 ;
( v2 |-- I ) . X <= 1 ;
assume x in the carrier of f & y in the carrier of g ;
conv @ S c= conv ( A ) & conv @ S c= conv @ S ;
reconsider B = b as Element of the topology of T ;
J , v |= P \lbrack l , r .] ;
redefine func J . i -> non empty TopSpace ;
ex_sup_of Y1 \/ Y2 , T & ex_sup_of Y1 , T ;
W1 is_2 implies R |_2 field W1 c= field ( W1 + W2 )
assume x in the carrier of R & y in the carrier of S ;
dom nn = Seg n & dom nn = 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 ;
assume that not Reloc I c= J and not D c= J and not D in dom Reloc ( J , k ) ;
Im ( lim seq ) = 0 & Im ( seq ) = 0 ;
( sin . x ) <> 0 & ( sin . x ) <> 0 ;
sin is_differentiable_on Z & cos is_differentiable_on Z ;
6 . n = t3 . n .= s . n ;
dom ( ( - 1 ) (#) F ) c= dom F ;
W1 . x = W2 . x .= W2 . x ;
y in W .vertices() \/ W .vertices() ;
( for k being Nat st k <= len vM holds vM . k = v ) implies k <= len vM
x * a \equiv y * a . ( mod m ) ;
proj2 .: S c= proj2 .: P & proj2 .: S c= proj2 .: P ;
h . p4 = g2 . I .= ( h . p2 ) . I ;
IT = ( U /. 1 ) `1 .= ( U /. 1 ) `1 ;
f . rx1 in rng f & f . rx2 in rng f ;
i + 1 + 1 <= len - 1 ;
rng F = rng ( F . 0 ) .= rng F ;
mode multiplicative non empty multMagma of G is well unital associative associative associative non empty multMagma ;
[ x , y ] in A ~ ;
x1 . o in L2 . o & x2 . o in L2 . o ;
the carrier of ( m + 1 ) c= B ;
not [ y , x ] in id ( X ) ;
1 + p .. f <= i + len f ;
seq ^\ k1 is lower & seq ^\ k1 is lower ;
len ( F . ( len F ) ) = len I ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be Complex , c be Complex ;
Comput ( P , s , n ) = s ;
k <= k + 1 & k + 1 <= len p ;
reconsider c = {} T as Element of L ;
let Y be *> non empty Chain of T ;
cluster directed-sups-preserving for Function of L , L ;
f . j1 in K . j1 & f . j2 in K . j1 ;
redefine func J => y -> total Function of 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 ;
redefine pred a <> {} means : Def4 : ( a / a ) = 1 ;
assume that not a c= b and b in a ;
s1 . n in rng s1 & s1 . n in rng s1 ;
{ o , b2 } on C2 & { o , b2 } on C2 ;
LIN o , b , b9 & LIN o , b , c ;
reconsider m = x , n = y as Element of Funcs ( V , C ) ;
let f be non constant FinSequence of D , p be Point of D ;
let FW2 be non empty element , F be Function of FW2 , TOP-REAL n ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in F-8 ;
reconsider pp2 = x , pp2 = y as Subset of m ;
let A , B , C be Element of R ;
redefine func strict non empty for sqrt 5 -> strict non empty b9 -\overline ;
rng c `2 misses rng ( e | n ) ;
z is Element of gr { x } & z is Element of gr { x } ;
not b in dom ( a .--> p1 ) ;
assume that k >= 2 and P [ k ] ;
Z c= dom ( cot * cot ) /\ dom cot ;
the component of Q c= UBD ( A ) & ( L~ f ) ` c= ( L~ f ) ` ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( 1 / ( f ^ ) ) ;
redefine pred f = u means : Def4 : a * f = a * u ;
for n holds P1 [ n ] implies P1 [ n + 1 ]
{ x . O : x in L } <> {} ;
let x be Element of V . s ;
a , b be Nat ;
assume that S = S2 and p = S2 and q = S1 . ( i + 1 ) ;
gcd ( n1 , n2 ) = 1 & gcd ( n1 , n2 ) = 1 ;
set os = 2 * PI , os = 2 * PI , os = 2 * PI , os = 2 * PI , os = 2 * PI , os = 2 * PI , os
seq . n < |. r1 .| & |. seq . n .| < r ;
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 ] ;
set g = { n / 1 } , h = g / 2 ;
k = a or k = b or k = c ;
( a , b ) , ( b , c ) [: a , b :] ;
assume that Y = { 1 } and s = <* 1 *> ;
Ix1 . x = f . x .= 0 .= 0 ;
W3 .last() = W3 . 1 & W2 .last() = W3 . 1 ;
cluster trivial for Walk of G , finite _Graph ;
reconsider u = u as Element of Bags X ;
A in B ^ \bullet implies A , B \lbrack
x in { [ 2 * n + 3 , k ] } ;
1 >= ( q `1 / |. q .| - cn ) ;
f1 is__ _ X & f2 is__ _ X implies f1 - f2 is non empty
( f /. i ) `2 <= ( q `2 ) ^2 + ( q `2 ) ^2 ;
h is_the carrier of Cage ( C , n ) ;
( b `2 ) ^2 <= ( p `2 ) ^2 + ( p `2 ) ^2 ;
let f , g be |. is Function of X , Y ;
S * ( k , k ) <> 0. K ;
x in dom ( ( - 1 ) (#) f ) ;
p2 in NO . p1 & p2 in NO . p1 ;
len ( the_left_argument_of H ) < len ( H ) & len ( the_right_argument_of H ) < len ( H ) ;
F [ A , FF . A ] ;
consider Z such that y in Z and Z in X ;
hence 1 in C implies A c= C |^ A ;
assume that r1 <> 0 or r2 <> 0 and r1 < r2 ;
rng q1 c= rng ( C1 ^ C2 ) & rng q1 c= dom C2 ;
A1 , L , A3 , A2 , A3 be non empty set ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in u ( p , Ss ) & c in u ( p , Ss ) ;
then S is atomic not atomic & P-2 [ S ] ;
Cl ( [#] T ) = [#] T .= [#] T ;
f12 | A2 = f2 | ( A1 \/ A2 ) ;
0. M in the carrier of W & 0. M in the carrier of W ;
let v , v be Element of M ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ V & V c= Y \ Z ;
let X be Subset of S ~ ;
consider H1 such that H = 'not' H1 and H1 in M ;
1_ 1 c= ( t * ( p - 1 ) ) * ( p - 1 ) ;
0 * a = 0 * 0 .= a * 0 .= 0 ;
A |^ ( 2 , 2 ) = A ^^ A ;
set vY = ( v /. n ) , vY = ( v /. n ) ;
r = 0. ( REAL-NS n ) .= ||. 0. ( REAL-NS n ) .|| ;
( f . p4 ) `1 >= 0 & ( f . p2 ) `1 >= 0 ;
len W = len ( W | ( len W ) ) ;
f /* ( s * G ) is divergent_to+infty ;
consider l being Nat such that m = F . l ;
t16 . ( a , b ) does not destroy b1 . ( b , a ) ;
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 - 1 as non zero Element of NAT ;
c . x >= id ( L . x ) ;
\sigma ( T ) \/ omega ( T ) is Basis of T ;
for x being element st x in X holds x in Y
cluster [ x1 , x2 ] -> non pair for set ;
downarrow a /\ downarrow t is Ideal of T ;
let X be \hbox { \mathbb N } , F be non empty set ;
rng f = \rm \rm *> ( S , X ) ;
let p be Element of B , x be SortSymbol of S ;
max ( N1 , 2 ) >= N1 & max ( N2 , 2 ) >= N2 ;
0. X <= b |^ ( m * mm1 ) ;
assume that i in I and R1 . i = R ;
i = j1 & p1 = q1 & p2 = q2 implies i + 1 = j + 1
assume that gR in the right of g and gR in the carrier of g ;
let A1 , A2 be Point of S , A be Subset of S ;
x in h " P /\ [#] T1 & x in h " P ;
1 in Seg 2 & 1 in Seg 3 implies 1 in Seg 3
reconsider X-5 = X , T] = Y as non empty Subset of Tsuch that X = Y ;
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the Source of G ) -valued ;
n1 <= i2 + len g2 & n2 <= len g2 + len g1 ;
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume that v in the carrier' of G2 and v <> 0. G2 ;
y = Re y + ( Im y ) * i ;
( - ( - 1 ) ) ^2 = 1 ;
x2 is_differentiable_on ]. a , b .[ & x2 is_differentiable_on ]. a , b .[ ;
rng M5 c= rng ( D2 | Seg ( k + 1 ) ) ;
for p being Real st p in Z holds p >= a
( for x being Point of X holds f . x = proj1 . x ) implies f is continuous
( 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 ) .|
h \equiv gg . ( mod P ) & g in g . ( mod P ) ;
reconsider i1 = i-1 - 1 as Element of NAT ;
let v1 , v2 be VECTOR of V , v be VECTOR of V ;
for V being Subspace of V holds V is Subspace of [#] V
reconsider i-7 = i , i29 = j as Element of NAT ;
dom f c= [: C ( ) , D ( ) :] ;
x in ( the Sorts of B ) . n ;
len that len that that len that that 2 in Seg len f2 and len f1 = len f2 + 1 ;
pm1 c= the topology of T & pm1 c= the topology of T ;
]. r , s .[ c= [. r , s .] ;
let B2 be Basis of T2 , A be Subset of T2 ;
G * ( B * A ) = ( id o1 ) * ( B * A ) ;
assume that p , u ] and u , v , q is_collinear and p , u , q is_collinear ;
[ z , z ] in union rng ( F | ( X /\ Y ) ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S , G = S . $1 ;
LIN a1 , a3 , b1 & LIN a1 , b1 , b2 ;
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 / 2 ;
p in LSeg ( E . i , F . i ) ;
IC * ( i , j ) = 0. K .= 0. K ;
|. f . ( s . m ) - g .| < g1 ;
q7 . x in rng ( q7 | X ) ;
Carrier ( L7 ) misses ( Carrier ( L7 ) ) ` ;
consider c being element such that [ a , c ] in G ;
assume that N|^ o, oconsider o8 = o8 and o8 = o8 ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= ( F |^ ( C + 1 ) ) " { F . C }
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [: [. f . j , f . j .] , Q :] ;
redefine 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 \neq aacontinuous ( S , T ) ;
let x be Element of S ~ ;
( the Arrows of F ) . ( a , b ) is one-to-one ;
|. i .| <= - ( - 2 |^ n ) / ( n + 1 ) ;
the carrier of I[01] = dom P & P . 0 = P . 1 ;
} * ( n + 1 ) ! > 0 * n ;
S c= ( A1 /\ A2 ) /\ ( A2 /\ A3 ) ;
a3 , a4 // b3 , b2 & a1 , b1 // b2 , b3 ;
then dom A <> {} & dom A <> {} & A <> {} ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , Y & y Joins 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 = the carrier of I[01] ;
p in Upper_Arc ( P ) /\ Upper_Arc ( P ) ;
dom d2 = [: A , A :] & dom d1 = [: A , A :] ;
0 < ( p / ( ||. z .|| + 1 ) ) / ( ||. z .|| + 1 ) ;
e . ( m + 1 ) <= e . ( m + 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 , f be Function of U1 , U2 ;
Proj ( i , n ) * g is_differentiable_on X ;
x , y , z be Point of X , p be Point of X ;
reconsider p0 = p . x , p0 = p . x as Subset of V ;
x in the carrier of Lin ( A ) & x in the carrier of Lin ( B ) ;
let I , J be parahalting Program of SCM+FSA , a be Int-Location ;
assume that - a is lower and - a is lower and - a is bounded ;
Int Cl ( A ) c= Cl ( Int Cl A ) ;
assume for A being Subset of X holds Cl A = A ;
assume q in Ball ( |[ x , y ]| , r ) ;
( p2 `2 ) ^2 <= ( p `2 ) ^2 + ( p `2 ) ^2 ;
Cl Q ` = [#] ( T | A ) .= [#] ( T | A ) ;
set S = the carrier of T , T = the carrier of S ;
set I8 = -> f |^ n , I8 = f |^ n ;
len p - n = len ( p - n ) - n ;
A is Permutation of Swap ( A , x , y ) ;
reconsider nn = nwhere nn - 1 <= nn - 1 ;
1 <= j + 1 & j + 1 <= len ( s | X ) ;
let qbeing Let of M , qbeing Let of M , q be State of M ;
( a in the carrier of S1 & b in the carrier of S2 ) ;
c1 /. n1 = c1 . n1 & c2 /. n1 = c2 . n1 ;
let f be FinSequence of TOP-REAL 2 , p be Point of TOP-REAL 2 , r be Real ;
y = ( f * SS ) . x .= ( f * SS ) . x ;
consider x being element such that x in be Assume _ is IC ;
assume r in ( dist ( o ) ) .: P ;
set i2 = ( for h being \hbox { - } corner L~ h ) .. h = 2 ;
h2 . ( j + 1 ) in rng h2 /\ rng h2 ;
Line ( M29 , k ) = M . i .= Line ( M29 , k ) ;
reconsider m = ( x - 1 ) / 2 as Element of ( len x - 1 ) -tuples_on REAL ;
let U1 , U2 be Subspace of U0 , a be Element of U0 ;
set P = Line ( a , d ) ;
len p1 < len p2 + 1 & len p2 + 1 < len p2 ;
let T1 , T2 be complete Scott Subset of L , f be Function of T1 , T2 ;
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 a9 ) c= dom H & ( the_arity_of a9 ) . o = ( the_arity_of a9 ) . o ;
z1 " = ( z " ) * ( z " ) .= ( z " ) * ( z " ) ;
x0 - r / 2 in L /\ dom f & x0 - r / 2 in dom f ;
then w is that rng w /\ L <> {} & rng w /\ L <> {} ;
set x-10 = ( x ^ <* Z *> ) ^ <* Z *> ;
len w1 in Seg len w1 & len w2 in Seg len w1 & w1 = w1 + w2 ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of PFuncs ( V , { k } ) ;
x . n = ( |. a . n .| ) * ( |. b . n .| ) ;
( p `1 ) ^2 / ( 1 + ( p `1 ) ) ^2 <= ( G * ( 1 , j ) ) `1 ;
rng ( g ) c= L~ ( g | ( len g + 1 ) ) ;
reconsider k = i-1 * ( l + j ) as Nat ;
for n being Nat holds F . n is \HM { -infty } ;
reconsider x9 = x9 , y9 = y9 as VECTOR of M ;
dom ( f | X ) = X /\ dom f /\ X ;
p , a // p , c & b , a // c , c ;
reconsider x1 = x , y1 = y as Element of REAL m ;
assume i in dom ( a * p ^ q ) ;
m . ( ag . k ) = p . ( ag . k ) ;
a / ( s . m - n ) / Q <= 1 / 2 ;
S . ( n + k + 1 ) c= S . ( n + k ) ;
assume that B1 \/ C1 = B2 \/ C2 and B2 \/ C2 = C2 \/ C1 ;
X . i = { x1 , x2 } . i .= x2 ;
r2 in dom ( h1 + h2 ) & r1 + h2 in dom h1 /\ dom h2 ;
- - 0 = a & b-0 = b ;
F8 is_halting_on t , Q & F8 is_halting_on t , Q ;
set T = for X being InInInof X , x0 , x1 be Point of X holds x0 in X ;
Int Cl ( Int R ) c= Int ( Cl R ) ;
consider y being Element of L such that c . y = x ;
rng F' = { F' . x } .= { F' . x } ;
G-23 ( { c } ) c= B \/ S & not c in S ;
f[#] X is_X , X & f | X is Function of X , X ;
set RF = the Point of ( P | P ) , RF = the Point of ( ( TOP-REAL 2 ) | P ) ;
assume that n + 1 >= 1 and n + 1 <= len M ;
let k2 be Element of NAT , k be Element of NAT ;
reconsider p/. u = u , p/. v = v as Element of ( ( TOP-REAL n ) | K1 ) ;
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 , G ) ;
len P\bf 1 <= len P-35 & P\bf 1 <= len P-35 ;
x " in the carrier of A1 & x " in the carrier of A1 ;
[ i , j ] in Indices ( A * ( i , j ) ) ;
for m be Nat holds Re ( F . m ) is simple
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 ( A . i ) .= the carrier of ( A . i ) ;
assume that s1 = sqrt ( 2 / ( p `1 - r ) ^2 ) and s2 = sqrt ( 2 / ( p `1 - r ) ^2 ) ;
attr a > 1 & b > 0 , 1 , 1 , 2 ;
let A , B , C be lines of IW , a be k1 k1 k1 k1 k1 of NAT ;
reconsider X0 = X , Y0 = Y as RealNormSpace , x = X /\ Y ;
let f be PartFunc of REAL , REAL , g be PartFunc of REAL , REAL ;
r * ( v1 |-- I ) . X < r * 1 ;
assume that V is Subspace of X and X is Subspace of V and V is Subspace of X ;
let t-3 , t-3 be Relation of n -tuples_on BOOLEAN , f be Function of n -tuples_on BOOLEAN , BOOLEAN ;
Q [ eF \/ { vF } , f ] implies Q [ eF , f ]
g \circlearrowleft ( W-min L~ z ) = z implies ( W-min L~ z ) .. z < ( W-min L~ z ) .. z
|. |[ x , v ]| - |[ x , y ]| .| = v\rrangle ;
- f . w = - ( L * w ) .= - ( L * w ) ;
z - y <= x iff z <= x + y & y <= z + x ;
( 7 / p1 ) ^2 > 0 / ( 1 / e ) ;
assume X is BCK-algebra of 0 , 0 , 0 , 0 , 0 , 0 ;
F . 1 = v1 & F . 2 = v2 & F . 3 = v1 ;
( f | X ) . x2 = f . x2 .= ( f | X ) . x2 ;
( ( tan | Z ) . x ) in dom sec /\ dom tan ;
i2 = ( f /. len f ) `2 .= ( f /. len f ) `2 .= ( f /. len f ) `2 ;
X1 = X2 \/ ( X1 \ X2 ) .= X2 \/ ( X1 \ X2 ) ;
[. a , b , 1_ G .] = 1_ G & a = 1_ G ;
let V , W be non empty VectSpStr over F_Complex , V be Subspace of V ;
dom g2 = the carrier of I[01] & dom g2 = the carrier of I[01] & g2 is continuous ;
dom f2 = the carrier of I[01] & dom f2 = the carrier of I[01] & f2 . 0 = 1 ;
( proj2 | X ) .: X = proj2 .: X .= proj2 .: X ;
f . ( x , y ) = h1 . ( x `1 , y `2 ) ;
x0 - r < a1 . n & x0 < x0 + r / 2 ;
|. ( f /* s ) . k - G3 .| < r ;
len Line ( A , i ) = width A .= width A ;
SFinSequence / ( g , h ) = ( S . g ) / ( g , h ) ;
reconsider f = v + u as Function of X , the carrier of Y ;
intloc 0 in dom Initialized p & IC Comput ( p , s , k ) in dom p ;
i1 , i2 , i3 , i3 , i2 not contradiction & I does not destroy b , a , b , c ;
arccos r + arccos r = ( PI / 2 ) + 0 ;
for x st x in Z holds f2 is_differentiable_in x & for x st x in Z holds f2 is_differentiable_in x ;
reconsider q2 = ( q / x ) / ( x - y ) as Element of REAL ;
( 0 qua Nat ) + 1 <= i + j1 + 1 ;
assume that f in the carrier of [: X , Omega Y :] and g in the carrier of [: X , Omega Y :] ;
F . a = H / ( ( x , y ) / ( x , y ) ) ;
( not ( ex u being Element of T st C = TRUE & u in C ) ) implies C = {}
dist ( ( a * seq ) . n , h ) < r / 2 ;
1 in the carrier of [. 0 , 1 .] & 1 in dom f ;
( p2 `1 ) ^2 - x1 > - g / 2 - g / 2 ;
|. r1 - `2 .| = |. a1 .| * |. thesis - a .| ;
reconsider S-14 = 8 as Element of Seg 8 ( ) ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b
D0W = D0W .9 + 1 ;
i1 = ma + n & i2 = K + n & z = a + n ;
f . a [= f . ( f .: O1 "\/" f . a ) ;
attr f = v & g = u , h = v + u ;
I . n = Integral ( M , F . n ) ;
chi ( T1 , S ) . s = 1 / ( s . s ) ;
a = VERUM ( A ) or a = VERUM ( A ) ;
reconsider k2 = s . b3 , k1 = s . b3 as Element of NAT ;
( Comput ( P , s , 4 ) ) . GBP = 0 ;
L~ M1 meets L~ ( R4 * ( i , j ) ) or L~ M1 /\ L~ M2 = { i } ;
set h = the continuous Function of X , R , x be Point of X ;
set A = { L . ( ( k + 1 ) + 1 ) where k is Nat : k in dom L } ;
for H st H is atomic holds P7 [ H ] ;
set b = S5 ^\ ( i + 1 ) , c = S5 ^\ ( i + 1 ) ;
Hom ( a , b ) c= Hom ( a opp , b opp ) ;
( 1 / ( n + 1 ) ) < ( 1 / s ) " ;
( l `1 = [ dom l , cod l ] ) `1 .= [ l , cod l ] `1 ;
y +* ( i , y /. i ) in dom g & y . i in dom g ;
let p be Element of QC-WFF ( Al ( ) ) , x be Element of D ( ) ;
X /\ X1 c= dom ( f1 - f2 ) /\ dom ( f2 - f1 ) ;
p2 in rng ( f /^ ( k + 1 ) ) & p2 in rng ( f /^ k ) ;
1 <= indx ( D2 , D1 , j1 ) + 1 - 1 ;
assume x in ( L2 /\ K0 ) \/ ( ( L2 /\ K0 ) /\ K0 ) ;
- 1 <= ( ( f2 ) . O ) `2 & - 1 <= ( ( f2 ) . O ) `2 ;
let f , g be Function of I[01] , TOP-REAL 2 , a , b be Real ;
k1 -' k2 = k1 - k2 + k2 - k2 .= k1 - k2 + k2 - k2 ;
rng ( seq ^\ k ) c= ]. x0 , x0 + r .[ & rng ( seq ^\ k ) c= dom ( f2 * f1 ) ;
g2 in ]. x0 , x0 + r .[ & g2 in ]. x0 , x0 + r .[ ;
sgn ( p `1 , K ) = - ( 1_ K ) .= - ( 1_ K ) ;
consider u being Nat such that b = p |^ ( y * u ) ;
ex A being Line of B 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 t1 + len t2 ;
v = v + w |-- v + ( w |-- A ) . v ;
v <> DataLoc ( t . GBP , 3 ) & v . DataLoc ( t . GBP , 3 ) = 0 ;
g . s = sup ( d " { s } ) .= s . s ;
( \dot y ) . s = s . ( y . s ) ;
{ s : s < t } in NAT implies t = {} ( NAT , REAL )
s ` \ s = s ` \ ( 0. X \ s ) .= ( 0. X \ s ) \ ( 0. X \ s ) ;
defpred P [ Nat ] means B + $1 in A & B + $1 in A ;
( 339 + 1 ) ! = 3339 ! * ( 339 + 1 ) ;
( U . succ A ) = ( ( U . ( U . ( U . ( U . ( U . ( U . ( U . ( U . ( U . ( U . ( U . ( U . ( U . ( U . ( U . ( U . (
reconsider y = y as Element of ( len y ) -tuples_on COMPLEX ;
consider i2 being Integer such that y0 = p * i2 and 1 <= i2 and i2 <= n ;
reconsider p = Y | ( Seg k ) as FinSequence of ( Seg k ) ;
set f = ( S , U ) \mathop { F . z } ;
consider Z being set such that lim s in Z and Z in F ;
let f be Function of I[01] , TOP-REAL n , x be Point of TOP-REAL n , y be Point of TOP-REAL n ;
( ( M + i ) . [ n + i , 'not' A ] ) . [ n + i , 1 ] <> 1 ;
ex r being Real st x = r & a <= r & r <= b ;
R1 , R2 be Element of REAL n , x be Element of REAL n , y be Element of REAL n ;
reconsider l = 0. ( V ) , r = 0. ( A ) as Linear_Combination of A ;
set r = |. e .| + |. n .| + |. w .| + |. s .| + 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 ) ;
||. ( x9 - y9 ) * ( y9 - g2 ) .|| < r2 / 2 * ||. x9 - g2 .|| ;
( b , a9 // b9 , c9 ) & ( b , c // c9 , a9 ) ;
1 <= k2 -' k1 & k2 + k1 = k2 - k1 & k2 + k1 = k2 - k1 + k1 ;
( p `2 / |. p .| - sn ) >= 0 ;
( q `2 / |. q .| - sn ) < 0 & ( q `2 / |. q .| - sn ) < 0 ;
E-max C in cell ( RR , 1 , 1 ) & ( E-max C ) `2 = ( R /. 1 ) `2 ;
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 // a `1 , b `1 or p `1 , a // b `1 , b ;
g . n = a * Sum ( f | n ) .= f . n * a ;
consider f being Subset of X such that e = f and f is empty ;
F | ( N2 , S ) = CircleMap * ( F | [: N2 , S :] ) ;
q in LSeg ( q , v ) \/ LSeg ( v , p ) ;
Ball ( m , r0 ) c= Ball ( m , s ) & Ball ( m , r0 ) c= Ball ( m , s ) ;
the carrier of (0). V = { 0. V } .= { 0. V } .= { 0. V } ;
rng ( cos | [. - 1 , 1 .] ) = [. - 1 , 1 .] ;
assume that Re seq is summable and Im ( seq ) is summable and Im ( seq ) is summable ;
||. ( vseq . n ) - ( vseq . n ) .|| < e / 2 ;
set g = O --> 1 ;
reconsider t2 = t11 as 0 -element string of S2 , Q = { t } as Element of S ;
reconsider x9 = seq . n , y9 = seq . n as sequence of ( TOP-REAL n ) | X ;
assume that not E-max L~ Cage ( C , n ) meets L~ go and not E-max L~ Cage ( C , n ) meets L~ pion1 ;
- ( ( Cl 1 ) . n ) < F . n - ( - ( 1 / 2 ) . n ) ;
set d1 = \overline ( dist ( x1 , z1 ) ) , d2 = dist ( x2 , z2 ) , d2 = dist ( x1 , z2 ) ;
2 |^ ( 100 -' 1 ) = 2 |^ ( 100 - 1 ) ;
dom ( v | ( Seg len ( v | ( Seg len v ) ) ) ) = Seg len ( v | ( Seg len v ) ) ;
set x1 = - k2 + |. k2 + 1 .| + 4 * ( 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 . A = c . A
the carrier of ( L\lbrace x } + L2 ) c= I2 & the carrier of ( L2 + L1 ) c= I ;
'not' Ex ( x , p ) => All ( x , p ) is valid ;
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the normal normal \Rightarrow of {} ;
Z c= dom ( ( sin * f1 ) `| Z ) ;
|. 0. TOP-REAL 2 - q .| < r / 2 - q `1 / 2 ;
ConsecutiveSet2 ( A , B ) c= ConsecutiveSet2 ( A , *> , L ) implies A c= L
E = dom ( L (#) F ) & L (#) F is_measurable_on E & ( L (#) F ) is_measurable_on E ;
C / ( A + B ) = C / B * C ;
the carrier of W2 c= the carrier of V & the carrier of W1 c= the carrier of V implies W1 + W2 c= V
I . IC s2 = P . IC s2 .= ( 0 + 1 ) .= ( 0 + 1 ) ;
attr x > 0 means : Def4 : ( 1 / x ) = x / ( 1 - x ) ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) .= LSeg ( f , i ) ;
consider p being Point of T such that C = [. p , g .] and p in C ;
b , c are_connected & - C , - C + - C + - C + - C + - C + - C + - C + C + - C + - C + C + - C + - C + C + - C + C + - C + C + - C + C +
assume that f = id the carrier of O and g is Function of O , O and f is Function of O , O ;
consider v such that v <> 0. V and f . v = L * v ;
let l be Linear_Combination of {} ( {} ( the carrier of V ) ) ;
reconsider g = f " as Function of U2 , U1 , U2 , U2 , U2 be Function of U2 , U2 ;
A1 in the Points of G_ ( k , X ) & A2 in the Points of k ;
|. - x .| = - x .= - x .= - x .= - x ;
set S = ) ( x , y , c ) ;
Fib ( n ) * ( 5 * Fib ( n ) - 1 ) >= 4 * 8 ;
v /. ( k + 1 ) = 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 , j ) = S\mathopen ( j , i ) .= Sj ;
h . ( x1 , y1 ) = [ y1 , x1 ] & h . ( y1 , y2 ) = [ y2 , y1 ] ;
|. f .| - Re ( |. f .| ) * ( ( card b ) * h ) is nonnegative ;
assume that x = ( a1 ^ <* x1 *> ) ^ b1 and y = ( a1 ^ <* x1 *> ) ^ b1 ;
MI is_halting_on IExec ( I , P , s ) , P & M is_halting_on s , P ;
DataLoc ( t3 . a , 4 ) = intpos ( 0 + 4 ) .= 0 ;
x + y < - x + y & |. x .| = - x + y & |. x .| = - x + y ;
LIN c , q , b & LIN c , q , c & LIN c , q , b ;
f| ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + z1 ) .= x1 + ( y1 + z1 ) ;
flet f . a = f\in . a & v in InputVertices S & v in InputVertices S ;
( p `1 ) ^2 <= ( E-max C ) `1 & ( E-max C ) `1 <= ( E-max C ) `1 ;
set R8 = Cage ( C , n ) :- E8 , R8 = Cage ( C , n ) ;
( p `1 ) ^2 >= ( E-max C ) `1 & ( E-max C ) `1 >= ( E-max C ) `1 ;
consider p such that p = p-20 and s1 < p and p < s2 and s2 < p ;
|. ( f /* ( s * F ) ) . l - GM .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len Line ( N , k + 1 + 1 ) = width N ;
f1 /* s1 is convergent & f2 /* s1 is convergent & ( f2 /* s1 ) . n = f2 . ( lim s1 ) ;
f . x1 = x1 & f . y1 = y1 & f . y2 = y2 ;
len f <= len f + 1 & len f + 1 <> 0 & len f + 1 <> 0 ;
dom ( Proj ( i , n ) * s ) = REAL m .= dom ( ( Proj ( i , n ) * s ) ) ;
n = k * ( 2 * t ) + ( n mod ( 2 * t ) ) ;
dom B = 2 -tuples_on the carrier of V \ { {} } .= the carrier of V ;
consider r such that r _|_ a and r _|_ x and r _|_ y ;
reconsider B1 = the carrier of Y1 , B2 = the carrier of Y2 as Subset of X ;
1 in the carrier of [. 1 / 2 , 1 .] & 1 / 2 <= 1 ;
for L being complete LATTICE for A being Subset of lattice ( rng F ) , L st A , L are_isomorphic holds A is isomorphic
[ gi , gj ] in II \ II & [ gi , gj ] in II \ 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 ( f2 * f1 ) holds f2 * f1 is_differentiable_in x0 ;
reconsider y = ( a ` ) / ( F . ( k + 1 ) ) , z = ( a ` ) / ( F . k ) as Element of L ;
dom s = { 1 , 2 , 3 } & s . 1 = d1 & s . 2 = d2 ;
( min ( g , 1 ) ) . c <= h . c & ( min ( g , 1 ) ) . c <= h . c ;
set G3 = the \HM { of G , v = the Vertex of G , v = the Vertex of G , v = the Vertex of G ;
reconsider g = f as PartFunc of REAL n , REAL-NS n , REAL-NS n ;
|. s1 . m / p .| < d / p / p / p / p / p ;
for x being element st x in ( ( for u being element st u in ( t ) holds u in x ) holds x in B
P = the carrier of ( TOP-REAL n ) | Px0 .= ( TOP-REAL n ) | Px0 ;
assume that p10 in LSeg ( p1 , p2 ) /\ LSeg ( p11 , p2 ) /\ LSeg ( p11 , p2 ) <> {} ;
( 0. X \ x ) |^ ( m * ( k + 1 ) ) = 0. X ;
let g be Element of Hom ( cod f , cod f ) ;
2 * a * b + ( 2 * c * d ) <= 2 * C1 * C2 + ( 2 * c ) * ( a * b ) ;
let f , g , h be Point of the carrier of X , f be PartFunc of X , Y ;
set h = Hom ( a , g (*) f ) ;
then idseq ( n ) | Seg m = idseq ( m ) | Seg n .= m ;
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 * sin ) ;
cell ( G , i1 , j2 -' 1 ) misses C & cell ( G , i1 , j1 -' 1 ) misses C ;
LE q2 , p4 , P , p1 , p2 & LE q2 , p2 , P , p1 , p2 ;
attr B is component means : Def4 : B c= BDD A & B c= BDD B ;
deffunc D ( set , set ) = union rng $2 & $2 = union rng $2 ;
n + - n < len ( p + - n ) + ( - n ) ;
attr a <> 0. K means : Def4 : the_rank_of M = the_rank_of ( a * M ) & the_rank_of M = the_rank_of ( a * M ) ;
consider j such that j in dom \mathbb of of of dom s.[ and I = len of PI + j and 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 ]
set CC1 = Comput ( P2 , s2 , i + 1 ) , CC2 = P2 +* I ;
set cv = 3 / ( 2 * ( a , b , c ) ) , cv = 2 * ( a , b , c ) ;
conv @ W c= union ( F .: ( E " W ) ) & conv @ W c= union ( F .: ( E " W ) ) ;
1 in [. - 1 , 1 .] /\ dom ( arccot * ( arccot * ( arccot * arccot ) ) ) ;
r3 <= s0 + ( r0 - ( v2 - v1 ) ) / ( 2 * ( v2 - v1 ) ) ;
dom ( f * f4 ) = dom f /\ dom ( f3 (#) f4 ) .= dom f /\ dom f3 ;
dom ( f (#) G ) = dom ( l (#) F ) /\ Seg k .= Seg k ;
rng ( s ^\ k ) c= dom f1 \ { x0 } & rng ( s ^\ k ) c= dom f2 \ { x0 } ;
reconsider g9 = gp , gp = gp as Point of ( TOP-REAL n1 ) | K1 ;
( T * h . s ) . x = T . ( h . s . x ) ;
I . ( J . x ) = ( I * L ) . ( J . x ) ;
y in dom ( *> *> , ( ( Frege A ) . o ) & y in dom ( ( Frege A ) . o ) ;
for I being non degenerated integral of I holds the carrier of I is commutative commutative associative commutative non empty doubleLoopStr
set s2 = s +* Initialize ( ( intloc 0 ) .--> 1 ) , P2 = P +* I +* J +* J ;
P1 /. IC s1 = P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 ;
lim S1 in the carrier of [. a , b .] & ( for x st x in the carrier of [. a , b .] holds x <= x ) ;
v . ( l-13 . i ) = ( v *' lw ) . i .= ( v *' lw ) . i ;
consider n being element such that n in NAT and x = ( sn | P ) . n ;
consider x being Element of c such that F1 . x <> F2 . x and x in F1 . x ;
as Function of Funcs ( X , 0 , x1 , x2 , x3 ) , E ;
j + ( 2 * ( k + m1 ) ) + m1 > j + ( 2 * ( k + m1 ) ) ;
{ s , t } on A3 & { s , t } on B2 & { s , t } on B2 ;
n1 > len crossover ( p2 , p1 , n1 , n2 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n3 , n3 , n2 , n3 , n3 , n4 , n2 , n3 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n2 , n3 , n4 , n4 , n4 , n4
( mg1 ) . HT ( mg2 , T ) = 0. L & ( mg1 ) . HT ( mg2 , T ) = 0. L ;
then H1 , H2 are_that card H1 , H2 are_that card H2 , H1 |^ n are_that card H2 = card H2 ;
( ( N-min L~ f ) .. ( f | 1 ) ) .. ( f | 1 ) > 1 & ( ( N-min L~ f ) .. ( f | 1 ) ) .. ( f | 1 ) > 1 ;
]. s , 1 .[ = ]. s , 2 .[ /\ [. 0 , 1 .] ;
x1 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) ) & x2 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) ) ;
let f1 , f2 be continuous PartFunc of REAL , the carrier of S , x0 be Point of S ;
DigA ( tmax ( k , z ) , i ) is Element of k -tuples_on ( k -tuples_on ( k + 1 ) ) ;
I is Element of NAT & I is Element of NAT & I is Element of k2 implies I is k2 & I is Element of NAT
( u ~ ) = { [ a , u9 ] } .= { [ a , u9 ] } .= { [ a , u9 ] } ;
( w | p ) | ( w | w ) = p ;
consider u2 such that u2 in W2 and x = v + u2 and u1 in W2 and u2 in W1 and v = v + u2 ;
for y st y in rng F ex n st y = a |^ n & F . n = F . ( n + 1 )
dom ( ( g * ( ( f . x ) \dot \to C ) ) | K ) = K ;
ex x being element st x in ( ( the Sorts of U0 ) \/ A ) . s & x in ( the Sorts of U0 ) . s ;
ex x being element st x in ( ( that ) \/ A ) . s & x in ( ( the Sorts of O1 ) \/ A ) . s ;
f . x in the carrier of [. - r , r .] & f . x in [. - r , r .] ;
( the carrier of X1 union X2 ) /\ ( the carrier of X1 ) <> {} & ( the carrier of X1 union X2 ) /\ ( the carrier of X2 ) <> {} ;
L1 /\ LSeg ( p01 , p2 ) c= { p10 } /\ LSeg ( p11 , p2 ) ;
( b + ( be - a ) ) / 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 G8 such that z = y and P [ z ] and z in A and z in B ;
( the sequence of ( ( the carrier of X ) | D ) ) . ( x , y ) <= e ;
len ( w ^ w2 ) + 1 = len w + 2 + 1 .= len w + 1 ;
assume that q in the carrier of ( TOP-REAL 2 ) | K1 and q in the carrier of ( TOP-REAL 2 ) | K1 ;
f | E-4 ` = g | EK ` .= g | EK ` .= g | EK ;
reconsider i1 = x1 , i2 = x2 , z = x3 as Element of NAT ;
( a * A * B ) ` = ( a * ( A * B ) ) ` ;
assume ex n0 being Element of NAT st f |^ n0 is is is is is
Seg len ( ( ( f1 ^ f2 ) | ( i + 1 ) ) ) = dom ( ( ( f1 ^ f2 ) | ( i + 1 ) ) ) ;
( Complement ( A ) ) . m c= ( Complement ( A ) ) . n ;
f1 . p = p9 & g1 . ( p , q ) = d & f1 . ( q , p ) = d ;
FinS ( F , Y ) = FinS ( F , dom ( F | Y ) ) .= FinS ( F , Y ) ;
( x | y ) | z = z | ( y | x ) ;
( |. x .| to_power n ) / ( n + 1 ) <= ( r2 to_power n ) / ( n + 1 ) ;
Sum ( F ) = Sum f & dom ( F ) = dom g & for x st x in dom F holds F . x = f . x ;
assume for x , y being set st x in Y & y in Y holds x /\ y in Y ;
assume that W1 is Subspace of W3 and W2 is Subspace of W3 and W1 is Subspace of W2 and W2 is Subspace of W3 ;
||. t-15 . x .|| = lim ||. ( x - y ) .|| .= ||. ( x - y ) .|| .= ||. x - y .|| ;
assume that i in dom D and f | A is lower and g | A is lower and g | A is lower ;
( p `2 ) ^2 - ( q `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 Line ( B , i ) .= len B .= len B .= len B .= len B ;
attr a <> 0 means : Def4 : ( A \+\ B ) Let a = ( A Y. ) Let ( B be element ;
then f is_\cal 3 implies pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 & pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 ;
assume that a > 0 and a <> 1 and b > 0 and b <> 1 and c > 0 ;
w1 , w2 in Lin { w1 , w2 } & w2 in Lin { w2 , w1 } implies w1 = w2
p2 /. IC s = p2 . IC s .= ( ( IC s ) + 1 ) * ( ( IC s ) + 1 ) ;
ind ( T-10 | b ) = ind b .= ind b - 1 .= ind b - 1 .= ind b - 1 ;
[ a , A ] in the \cdot of G_ ( k , X ) & [ a , A ] in the \cdot of G ;
m in ( the Arrows of C ) . ( o1 , o2 ) & m in ( the Arrows of C ) . ( o2 , o2 ) ;
( for z being Element of Y holds ( a , CompF ( PA , G ) ) . z = FALSE ) implies a = TRUE
reconsider phi = phi /. 11 , phi = phi /. 11 , phi = phi /. ( 11 + 1 ) as Element of ( the carrier of z1 ) * ;
len s1 - ( len s2 - 1 ) + 1 > 0 + 1 - 1 ;
delta ( D ) * ( f . ( upper_bound A ) - 0 ) < r ;
[ f21 , f22 ] in [: the carrier' of A , the carrier' of B :] ;
the carrier of ( ( TOP-REAL 2 ) | K1 ) = K1 & the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 = K1 ;
consider z being element such that z in dom g2 and p = g2 . z and z in rng g2 ;
[#] V1 = { 0. V1 } .= the carrier of ( ( the carrier of V1 ) | V1 ) .= the carrier of ( ( the carrier of V1 ) | 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 s in dom ( f | X ) ;
h1 = f ^ ( <* p3 *> ^ <* p *> ) .= h ^ <* p3 *> .= h ^ <* p *> ;
c / ( |[ b , c ]| ) = c .= |[ a , c ]| / ( |[ b , c ]| ) .= c ;
reconsider t1 = p1 , t2 = p2 , t1 = p3 as Term of C , V ;
( 1 / 2 ) in the carrier of [. 1 / 2 , 1 .] & ( 1 / 2 ) * ( 1 / 2 ) in the carrier of [. 1 / 2 , 1 .] ;
ex W being Subset of X st p in W & W is open & h .: W c= V ;
( h . p1 ) `2 = C * ( p1 `2 + D * ( p1 `2 ) + D * ( p1 `2 ) + D * ( p1 `2 ) + D * ( p1 `2 ) + D * ( p2 `2 ) + D * ( p2 `2 ) ;
R . b - a = 2 * - b .= 2 * b - a .= b ;
consider - 1 such that B = ( - 1 ) * C + ( - 1 ) * A and 0 <= 1 ;
dom g = dom ( ( the Sorts of A ) * ( the_arity_of o ) ) .= dom ( ( the Sorts of A ) * ( the_arity_of o ) ) ;
[ P . ( l . ( k + 1 ) ) , P . ( l . ( k + 1 ) ) ] in => ( P . ( l . k ) ) ;
set s2 = Initialize s , P2 = P +* I ;
reconsider M = mid ( z , i2 , i1 ) , N = L~ z as non empty Subset of TOP-REAL 2 ;
y in product ( ( the Sorts of J ) +* ( V , { 1 } ) ) ;
1 / ( |[ 0 , 1 ]| ) = 1 & 0 / ( |[ 0 , 1 ]| ) = 0 ;
assume that x in the left of g or x in the left of g and y = ( the right of g ) . x ;
consider M being strict Subgroup of Aj such that a = M and T is Subgroup of M and M is Subgroup of A ;
for x st x in Z holds ( ( ( ( #Z 2 ) * f ) + f ) `| Z ) . x <> 0
len W1 + len W2 + m = 1 + len W2 + m + 1 .= len W1 + len W2 + m + 1 ;
reconsider h1 = ( vseq . n ) - ( vseq . n ) as Lipschitzian LinearOperator of X , Y ;
( i mod len ( p + q ) ) + 1 in dom ( p + q ) ;
assume that s2 is negative and F in the { of s2 : not contradiction } and F in the { of s2 : not contradiction } ;
( ( ( ( ( x , y ) ) * ( 1 , 3 ) ) * ( 1 , 2 ) ) ) * ( 1 , 2 ) = ( ( x , y ) * ( 1 , 2 ) ) * ( 1 , 2 ) ;
for u being element st u in Bags n holds ( p *' m ) . u = p . u
for B be 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 and W1 = ( the Sorts of A ) . ( len p + 1 ) ;
x in { X where X is Ideal of L : X is directed & x in X } ;
the carrier of W1 /\ W2 c= the carrier of W1 & the carrier of W1 /\ W2 c= the carrier of W1 /\ W2 ;
( for a , b being Real holds ( a + b ) * id a = ( 1 - a ) * ( b + a )
( ( X --> f ) . x ) = ( X --> dom f ) . x .= ( X --> f ) . x ;
set x = 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 <= ( ( 2 |^ ( n -' m ) ) + 1 ) + 1 ;
( reproj ( 1 , z0 ) ) . x in dom ( f1 (#) f2 ) /\ dom ( f2 (#) f3 ) ;
assume that b1 . r = { c1 } and b2 . r = { c2 } and b2 . r = c2 . r ;
ex P st a1 on P & a2 on P & b2 on P & a1 , a2 on P & b1 on P & b2 on P & b2 on P ;
reconsider gf = g `1 * f `2 , hf = h `1 * g `2 as strict Element of X ;
consider v1 being Element of T such that Q = ( downarrow v1 ) ` and v1 in F and v1 in G ;
n in { i where i is Nat : i < n0 + 1 & i < n + 1 } ;
( F * ( i , j ) ) `2 >= ( F * ( m , k ) ) `2 ;
assume that K1 = { p : p `1 >= sn & p `1 >= 0 & p <> 0. TOP-REAL 2 } ;
ConsecutiveSet ( A , succ O1 ) = ( ConsecutiveSet ( A , O1 ) ) ^ ( ConsecutiveSet ( A , O1 ) ) ;
set Ik1 = in dom [ SubFrom ( a , intloc 0 ) , SubFrom ( 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 ) & X c= ( the carrier of L1 ) /\ the carrier of L2 ;
consider x9 be Element of GF ( p ) such that x9 |^ 2 = a & x9 |^ 2 = b ;
reconsider en = en , fn = fn , fn = fn as Element of D ( ) ;
ex O being set st O in S & C1 c= O & M . O = 0. ( X , L ) & M . O = 0. ( X , L ) ;
consider n being Nat such that for m being Nat st n <= m holds S . m in U1 and n <= m ;
f * g * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) * g ) . x ;
defpred P [ Nat ] means A + succ $1 = succ A & A + succ $1 = succ A + ( succ $1 ) ;
the left left of - g = the left & the left of - g = the left of - g & the left of - g = ( - 1 ) * ( - g ) ;
reconsider pp = x , p\mathopen = y , p\mathopen = z , p\mathopen = x , p\mathopen = y as Point of Euclid 2 ;
consider g2 such that g2 = y and x <= g2 and g2 <= x0 and x0 <= g2 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 + len y2 + len y2 .= len x2 + len y2 + len y2 ;
for x being element st x in X holds x in the set of ( the set of TOP-REAL ( n + 1 ) ) | X & x in X ;
LSeg ( p11 , p2 ) /\ LSeg ( p1 , p11 ) = {} or LSeg ( p11 , p2 ) /\ LSeg ( p11 , p2 ) = {} ;
func that ) ( X ) -> set equals [: [: X , X :] and [: X , X :] c= [: X , X :] ;
len ( Cage ( C , n ) ) <= len ( Cage ( C , n ) ) & len ( Cage ( C , n ) ) <= len ( Cage ( C , n ) ) ;
attr K is has a , v & a <> 0. K implies v . ( a |^ i ) = i * v . ( a |^ i ) ;
consider o being OperSymbol of S such that t . {} = [ o , the carrier of S ] and t . {} = [ o , the carrier of S ] ;
for x st x in X ex y st x c= y & y in X & y is Z & f . x = f . y
IC Comput ( P-6 , k ) in dom ( san +* I ) & IC Comput ( Pd , k ) in dom ( san +* I ) ;
attr q < s means : Def4 : r < s & s < q implies ]. r , s .[ c= ]. p , q .[ ;
consider c being Element of Class ( f , c ) such that Y = ( F . c ) `1 and c in Class ( f , c ) ;
func the ResultSort of S2 -> Function of the carrier' of S2 , the carrier' of S2 means : Def4 : for x being set st x in the carrier' of S2 holds it . x = id ( the carrier' of S2 ) ;
set y9 = [ <* y , z *> , f2 ] , z9 = [ <* z , x *> , f3 ] ;
assume x in dom ( ( ( - arccot ) * ( arccot ) ) `| Z ) & x in dom ( ( - arccot ) * ( arccot ) ) ;
r-7 in Int cell ( GoB f , i , GoB f ) \ L~ f \/ L~ f /\ L~ f & rrf in ( L~ f ) ` ;
( q `2 ) >= ( ( Cage ( C , n ) ) /. ( i + 1 ) ) `2 ;
set Y = { a "/\" a ` : a in X } ;
i - len f <= len f + len f1 - len f + len f - len f + len f - len f + 1 - len f + 1 - len f + 1 - len f + 1 - len f + 1 - len f + 1 <= len f - len f + 1 - len f + 1 - len f + 1 - len f + 1 - len f + 1 - len f
for n ex x st x in N & x in N1 & h . n = x- x0 & h . n = x0 + R . n
set s0 = ( \mathop { a , I , p , s ) . i , s0 = ( s , p , s ) . i , s0 = ( s , p , s ) . i , s0 = ( s , p , s ) . i , s0 = ( s , p , s ) . i , s0 = ( s , p , s ) . i ;
p . k . 0 = 1 or p . k . 0 = - 1 or p . 0 = 1 or p . 0 = 1 ;
u + Sum L-18 in ( U \ { u } ) \/ { u + Sum L-18 } ;
consider x9 being set such that x in x9 and x9 in V1 and x9 in V1 and x = [ x9 , x9 ] ;
( p ^ ( q | k ) ) . m = ( q | k ) . ( ( len p ) - 1 ) ;
g + h = gg + hg & f + g = g + h + g ;
L1 is distributive & L2 is distributive implies L1 ~ is distributive & L2 ~ is distributive & L1 ~ is distributive & L2 ~ is distributive
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 + ( 1 - p ) * q = 1 and 0 <= a and a <= b ;
F* ( f , <* A1 *> ) = rpoly ( 1 , 1 ) *' t + 1. L .= 1. L + 0. L .= 0. L ;
for X being set , A being Subset of X holds A ` = {} implies A = {} & A = {} implies A = {}
( ( N-min X ) `1 ) ^2 <= ( ( E-max X ) `1 ) ^2 & ( ( E-max X ) `1 ) `1 <= ( ( E-max X ) `1 ) ^2 ;
for c being Element of the Sorts of A , a being Element of the Sorts of A holds c <> a implies c <> a
s1 . GBP = ( Exec ( i2 , s2 ) ) . GBP .= Exec ( i2 , s2 ) . GBP .= Exec ( i2 , s2 ) . GBP .= 0 ;
for a , b being Real holds |[ a , b ]| in ( y ) ` implies b >= 0 & a >= 0
for x , y being Element of X holds x ` \ y = ( x \ y ) ` & y \ x = 0. X implies x = y
mode BCK-algebra of i , j , m , n , m , n , m , m be BCK-algebra of i , j , m , n , m , m ;
set x2 = |( Re ( y ) , Im ( x ) )| ;
[ y , x ] in dom u5 & u5 . ( y , x ) = g . y ;
]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .[ c= A & A c= A /\ divset ( D , k ) ;
0 <= delta ( S2 . n ) & |. delta ( S2 . n ) .| < ( e / 2 ) * ( 2 |^ n ) ;
( - ( q `1 ) ) ^2 <= ( - ( q `1 ) ) ^2 + ( - ( q `1 ) ) ^2 ;
set A = 2 / ( b-a ) ;
for x , y being set st x in R" holds x , y are_holds x , y are_\hbox { $ \subseteq $ }
deffunc FF2 ( Nat ) = b . $1 * ( M * G ) . $1 * ( M * G ) . $1 ;
for s being element holds s in -> ( f 'or' g ) iff s in ( f \/ g ) . s
for S being non empty non void non empty non void holds S is connected iff S is connected
max ( ( degree ( z ) ) , ( degree ( z ) ) ) >= 0 & ( degree ( z ) ) ) . ( ( degree ( z ) ) ) >= 0 ;
consider n1 being Nat such that for k holds seq . ( n1 + k ) < r + s and seq . k < x0 + s ;
Lin ( A /\ B ) is Subspace of Lin ( A ) & Lin ( B ) is Subspace of Lin ( A ) & Lin ( A ) = Lin ( B ) ;
set n-15 = n-15 '&' ( M . x qua Element of BOOLEAN ) , n-15 = M . ( n + 1 ) , n-15 = M . ( n + 1 ) ;
f " V in the topology of X & f " V in D & f " V in D & f .: V c= D implies f .: V c= D
rng ( ( a ^\ c ) ^\ ( 1 , b ) ) c= { a , c } \/ { b } ;
consider y being as as , and y is WW: 1 <= y & y in WWE and x `1 = WWE ;
dom ( 1 / f ) /\ ]. x0 , x0 + r .[ c= ]. x0 , x0 + r .[ /\ ]. x0 , x0 + r .[ ;
as Morphism of as ( i , j , n , r ) & ( i , j , n ) = ( i + j ) * r ;
v ^ ( n-3 |-> 0 ) in Lin ( rng ( ( B | c1 ) | ( dom ( B | c2 ) ) ) ) & v ^ ( B | c1 ) = v ;
ex a , k1 , k2 st i = a := ( k1 , k2 ) & i = ( a , k1 ) := k2 ;
t . NAT = ( NAT .--> succ i1 ) . NAT .= succ ( NAT .--> succ i1 ) .= succ ( 5 .--> succ i1 ) .= ( NAT .--> succ i1 ) . NAT .= NAT ;
assume that F is bbfamily and rng p = F and rng p = Seg ( n + 1 ) and for k be Nat st k in Seg ( n + 1 ) holds P [ k , F . k ] ;
not LIN b , b9 , a & not LIN a , c , b & not LIN a , c , b & a , c // b , c
( L1 Let L2 ) \& O c= ( L1 \& O ) \& O & ( L2 Let O ) Let O = ( L1 \& O ) \& O ;
consider F be ManySortedSet of E such that for d being Element of E holds F . d = F ( d ) and for d be Element of E holds F . d = G ( d ) ;
consider a , b such that a * ( v + u ) = b * ( -w ) and 0 < a and 0 < b ;
defpred P [ FinSequence of D ] means |. Partial_Sums ( $1 ) .| <= Sum |. $1 .| & Partial_Sums ( $1 ) . $1 <= Sum |. $1 .| ;
u = cos . ( x , y ) * x + cos . ( y , v ) * y .= cos . ( x , y ) * y .= v ;
dist ( ( seq . n ) + x , g + x ) <= dist ( ( seq . n ) , g ) + 0 ;
P [ p , |. p .| (#) |. p .| , {} ] implies P [ p , ( p - id the Sorts of A ) . p ]
consider X being Subset of CQC-WFF ( Al ( ) ) such that X c= Y and X is finite and X is finite and X is inininininin;
|. b .| * |. eval ( f , z ) .| >= |. b .| * |. eval ( f , z ) .| ;
1 < ( ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ) .. Cage ( C , n ) + 1 ;
l in { l1 where l1 is Real : g <= l1 & l1 <= h & h . l1 <= g . l1 & l . l1 <= h . l1 } ;
( Partial_Sums ( G ) . n ) * vol <= ( Partial_Sums ( ( G ) . n ) ) * vol ( ( G . n ) ) ;
f . y = x .= x * 1_ L .= x * ( ( power L ) . ( y , 0 ) ) .= x * ( ( power L ) . ( y , 0 ) ) ;
NIC ( <% i1 , i2 %> , k ) = { i1 , succ ( i1 , k ) } .= { i1 , succ ( i1 , k ) } ;
LSeg ( p10 , p2 ) /\ LSeg ( p1 , p11 ) = { p1 } /\ LSeg ( p11 , p2 ) /\ LSeg ( p1 , p11 ) ;
Product ( ( the carrier of I-15 ) +* ( i , { 1 } ) ) in Z & Product ( ( the carrier of I-15 ) +* ( i , { 1 } ) ) in Z ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) .= Following ( s1 , n ) .= Following ( s2 , n ) ;
W-bound Qb <= ( q1 `1 ) ^2 & ( q1 `1 ) ^2 <= ( q1 `1 ) ^2 + ( q2 `1 ) ^2 ;
f /. i2 <> f /. ( ( i1 + len g -' 1 ) + len f -' 1 ) & f /. ( i1 + 1 ) <> f /. ( i1 + 1 ) ;
M , f / ( x. 3 , a ) / ( x. 4 , f / ( x. 0 , a ) ) / ( x. 4 , a ) / ( x. 0 , a ) |= H ;
len ( P7 ^ P6 ) in dom ( ( P ^ Q ) ^ <* ( P ^ Q ) . ( len P + 1 ) *> ) ;
A |^ ( n , n ) c= A |^ ( m , n ) & A |^ ( k , n ) c= A |^ ( k , l ) ;
R |^ n \ { q : |. q .| < a } c= { q1 : |. q1 .| >= a }
consider n1 being element such that n1 in dom p1 and y1 = p1 . n1 and p1 . n1 = p2 . n1 and p2 . n1 = p2 . n1 ;
consider X being set such that X in Q and for Z being set st Z in Q & Z <> X holds X c= Z ;
CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA & CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA ;
for v be VECTOR of l1 holds ||. v .|| = upper_bound rng |. ( id the carrier of X ) . v .| & ||. v .|| <= |. v .| ;
for phi st phi in X holds phi in X implies ( not phi in X & phi in X & phi in X )
rng ( Sgm ( dom ( f | ( dom f ) ) ) c= dom ( f | ( dom ( f | ( dom f ) ) ) ) ;
ex c being FinSequence of D ( ) st len c = k & P [ c ] & a = c & c = a ;
( the_arity_of ( a , b , c ) ) = <* ( h . b , c ) , ( h . c ) *> .= <* h . c , h . c *> ;
consider f1 be Function of the carrier of X , REAL such that f1 = |. f .| and f1 is continuous and f1 . 0 = r ;
a1 = b1 & a2 = b2 or a1 = b1 & a2 = b2 & b1 = b2 & b2 = b3 or a1 = b3 & b1 = b2 & b2 = b3 ;
D2 . ( indx ( D2 , D1 , n1 ) + 1 ) = D2 . ( n1 + 1 ) .= D2 . ( n1 + 1 ) .= D2 . ( n1 + 1 ) ;
f . ( ||. r .|| ) = ||. ( r ) . 1 .|| .= ||. r .|| . 1 .= r . 1 .= r . 1 .= r . 1 ;
consider n being Nat such that for m being Nat st n <= m holds C-25 . n = C-25 . m and C-25 . m = C-25 . m ;
consider d being Real such that for a , b being Real st a in X & b in Y holds a <= d and d <= b ;
||. L /. h .|| - ( K * |. h .| ) + ( K * |. h .| ) <= p0 + ( K * |. h .| ) ;
attr F is commutative means : Def4 : for b being Element of X holds F \hbox { b } = f . b & F is commutative ;
p = - 1 * p0 + 0 * p0 .= 1 * p0 + 0 * p0 .= 1 * p0 + 0 * p0 .= 1 * p0 + 0 * p2 .= 1 * p0 + 0 * p2 .= 1 * p2 + 0 * p3 ;
consider z1 such that b , x3 , x1 is_collinear and o , x1 , x2 is_collinear and o , x1 , x2 is_collinear and o <> x2 ;
consider i such that Arg ( Rotate ( f ) ) = s + Arg ( q ) and 2 * PI * i = PI + ( 2 * PI * i ) ;
consider g such that g is one-to-one and dom g = card f ( x ) and rng g = f . x and g is one-to-one and g is one-to-one ;
assume that A = P2 \/ Q2 and P2 <> {} and ( for x st x in P2 holds x in P2 ) and ( x in P2 implies x in P2 ) ;
attr F is associative means : Def4 : F .: ( F , f ) = F .: ( f , F ) ;
ex x being Element of NAT st m = x `1 & x in z `1 & x < i or m in { i } & m in { i } ;
consider k2 being Nat such that k2 in dom P-2 and l in P-2 . k2 and k = P-2 . k2 + 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 ) , f * ( id b ) ] ;
{ p } "\/" D2 = { p "\/" y where y is Element of L : y in D2 & p in D1 } "\/" { p "\/" q where q is Element of L : q in D2 } ;
consider z being element such that z in dom ( ( dom F ) . 0 ) and ( ( F . 0 ) . 0 ) . 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 , j ) = { |[ r , s ]| : r <= G * ( 0 + 1 , 1 ) `1 & G * ( 0 , 1 ) `2 <= s } ;
consider e being element such that e in dom ( T | E1 ) and ( T | E1 ) . e = v and e in ( T | E1 ) . e ;
( F `1 * b1 ) . x = ( Mx2Tran ( ( J , of n , B\rrangle ) ) . ( ( ( J , Bthesis ) . j ) ) ;
- 1 = ( m (#) D ) | n .= ( m (#) D ) | n .= ( m (#) D ) | n .= ( ( m (#) D ) | n ) .= ( ( m (#) D ) | n ) * ( ( m (#) D ) | n ) ;
attr x being set means : Def4 : x in dom f /\ dom g holds g . x <= f . x & g . x <= g . x ;
len ( f1 . j ) = len f2 /. j .= len ( f2 /. j ) .= len ( f2 /. j ) .= len ( f2 /. j ) ;
All ( 'not' All ( a , A , G ) , B , G ) '<' Ex ( 'not' All ( a , B , G ) , A , G ) ;
LSeg ( E . k , F . ( k + 1 ) ) c= Cl RightComp Cage ( C , k + 1 ) & LSeg ( E , k + 1 ) c= RightComp Cage ( C , k + 1 ) ;
x \ a |^ m = x \ ( a |^ k * a ) .= ( x \ a ) \ a |^ k .= ( x \ a ) \ a |^ k ;
k -th inininininin-inin-in-in-in-in-in= ( commute Isuch that k -inin-in-in-in-in-in) . k .= ( commute I|^ k ) . k .= ( commute I|^ k ) . k ;
for s being State of A holds Following ( s , n ) . 0 + ( n + 2 ) * n is stable & Following ( s , n ) . 0 is stable
for x st x in Z holds f1 . x = a ^2 & ( f1 - f2 ) . x <> 0 & ( f1 - f2 ) . x <> 0
support ( ( support ( m ) ) \/ support ( m ) ) c= support ( ( m ) ) \/ support ( m ) ;
reconsider t = u as Function of ( the carrier of A ) , ( the carrier of B ) * the carrier of C , ( the carrier of B ) * the carrier of C ;
- ( a * sqrt ( 1 + b ^2 ) ) <= - ( b * sqrt ( 1 + a ^2 ) ) ;
phi /. ( succ b1 ) = g . a & phi /. ( succ b1 ) = f . ( g . ( succ b1 ) ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( ( F ^ <* p *> ) | ( i + 1 ) ) and i <> j ;
{ x1 , x2 , x3 , x4 } = { x1 } \/ { x2 } .= { x1 } \/ { x2 } .= { x1 } \/ { x2 } .= { x1 } \/ { x2 } ;
the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U1 & the Sorts of U1 /\ U2 c= the Sorts of U1 /\ ( U2 "\/" W3 ) ;
( - ( 2 * a ) * ( b / ( 2 * a ) ) + b / ( 2 * a ) ) ^2 > 0 ;
consider W00 such that for z being element holds z in W00 iff z in N ~ N & P [ z ] and W [ 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 ( ( exp_R * ( arccot + arccot ) ) `| Z ) /\ dom ( ( arccot * ( arccot + arccot ) ) `| Z ) ;
sum ( f , SS1 ) is convergent & lim ( ( lim f ) (#) ( S . m ) ) = integral ( f , SS1 ) * ( lim S ) ;
( X . ( ( a . f ) => ( g . ( x9 => y9 ) ) ) ) in cluster ( X . ( a => f ) ) ;
len ( M2 * M3 ) = n & width ( M2 * M3 ) = n & width ( M2 * M3 ) = n & width ( M2 * M3 ) = n ;
attr X1 \/ X2 is open means : Def4 : X1 , X2 are_separated & X2 , Y2 are_separated & X1 , X2 are_separated & X1 , X2 are_separated & X2 , Y2 are_separated ;
for L being upper-bounded antisymmetric RelStr , X being non empty Subset of L for X being non empty Subset of L holds X "\/" { Top L } = { Top L }
reconsider f-129 = F1 . ( b . ( b . ( X , b ) ) ) as Function of [: M , M :] , M ;
consider w being FinSequence of I such that the InitS of M , the InitS of M ^ <* s *> ^ w ^ w ^ w ^ w ^ w ^ v ^ v ^ w ^ v ^ w ^ v ^ w ^ w ^ v ^ w ^ w ^ v ^ w ^ w ^ v ^ w ^ w ^ w ^ v ^ w ^ w ^ w ^ v ^ w ^ w ^ w ^ v ^ w ^ w ^ w ^ w ^ v ^ w ^ w
g . ( a |^ 0 ) = g . ( 1_ G ) .= 1_ H .= 1_ H .= 1_ H .= 1_ H .= 1_ H .= 1_ H ;
assume for i being Nat st i in dom f ex z being Element of L st f . i = rpoly ( 1 , z ) & f . i = rpoly ( 1 , z ) ;
ex L being Subset of X st Carrier L = C & for K being Subset of X st K in C holds L /\ K <> {} & K is open ;
( 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 , oY = o `2 , oY = o `2 , oY = o `2 as Element of TS ( ( the Sorts of A ) * ( the_arity_of o ) ) ;
1 * x1 + ( 0 * x2 ) + ( 0 * x3 ) = x1 + ( 0 * x2 + ( 0 * x3 ) ) .= x1 + ( 0 * x3 ) .= x1 + ( 0 * x2 ) .= x1 + ( 0 * x3 ) .= x1 + ( 0 * x3 ) ;
E " = ( ( 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 ) , u2 = the carrier of U2 /\ ( U1 "\/" U2 ) as non empty Subset of U0 ;
( ( x "/\" z ) "\/" ( x "/\" y ) ) "\/" ( z "/\" y ) <= ( x "/\" ( z "\/" y ) ) "\/" ( z "/\" y ) ;
|. f . ( s1 . ( l1 + 1 ) ) - f . ( s1 . ( l1 + 1 ) ) .| < ( 1 / |. M .| + 1 / ( M ) ) ;
LSeg ( ( Lower_Seq ( C , n ) ) * ( i , j ) , ( Cage ( C , n ) ) * ( i + 1 , j ) ) is vertical ;
( f | Z ) /. x - ( f | Z ) /. x0 = L /. ( x- x ) + R /. ( x- x ) .= L /. ( x- x ) + R /. ( x- x ) ;
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 \HM { the } \HM { set } and len f = width A and len f = width A and len f = width A and len f = width A and width f = width A ;
len ( - M1 ) = len M1 & width ( - M2 ) = width M1 & width ( - M2 ) = width M1 & width ( - M2 ) = width M1 ;
for n , i being Nat st i + 1 < n holds [ i , i + 1 ] in the InternalRel of ( ( the InternalRel of ( TOP-REAL n ) | n ) )
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 ( f2 , 2 ) is_partial_differentiable_in u0 , 2 & pdiff1 ( f1 , 2 ) is_partial_differentiable_in u0 , 2 ;
attr a <> 0 & b <> 0 & Arg a = Arg b & Arg b = Arg a implies Arg ( - a ) = Arg ( - b ) & Arg ( - a ) = Arg ( - b ) ;
for c being set st not c in [. a , b .] holds not c in Intersection ( the topology of a , b ) & not c in Intersection ( the topology of a , b )
assume that V1 is linearly-independent and V2 is closed and V = { v + u : v in V1 & u in V1 & v in V2 } ;
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 ( ( ( k2 + 1 ) " ) * Sk1 ) ) .= Seg card ( ( k2 + 1 ) " ) .= Seg card ( ( k2 + 1 ) " ) ;
consider s2 being Integer such that s2 is convergent and b = lim s2 and for n holds s2 . n <= b . n and ( lim s2 ) = ( lim s2 ) * ( lim s2 ) ;
h2 " . n = h2 . n " & 0 < h2 . n & 0 < - ( ( 1 / 2 ) |^ n ) / ( 2 |^ ( n + 1 ) ) ;
( Partial_Sums ( ||. seq .|| ) ) . m = ||. ( seq .|| ) . m .= ( ||. seq .|| ) . m .= 0 .= 0 ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= ( Comput ( P2 , s2 , 1 ) ) . b .= Comput ( P2 , s2 , 1 ) . b .= Comput ( P2 , s2 , 1 ) . b ;
- v = ( - 1_ G ) * v & - w = ( - 1_ G ) * v & - w = ( - 1_ G ) * v & - w = - ( - w ) * v ;
sup ( ( k .: D ) .: D ) = sup ( ( k .: D ) .: D ) .= k . ( sup D ) .= k . ( sup D ) .= k . ( sup D ) ;
A |^ ( k , l ) = ( A |^ ( n , l ) ) ^^ ( A |^ ( k , l ) ) .= A |^ ( k , l ) ;
for R being add-associative right_zeroed right_complementable associative commutative associative associative well-unital distributive non empty doubleLoopStr , I , J being Subset of R , K being Subset of R holds I + ( J + K ) = ( I + J ) + K
( f . p ) `1 = ( p `1 ) ^2 + ( p `2 ) ^2 .= ( p `1 ) ^2 + ( p `2 ) ^2 ;
for a , b being non zero Nat st a , b are_relative_prime holds ( for x being Nat st x , b are_relative_prime holds ( x = a ) + ( b - a ) * ( x + b ) ) implies a = ( n + 1 ) * ( x + b )
consider A5 being countable set such that r is Element of CQC-WFF ( Al ( ) ) & A5 is ( ( len A ( ) ) , ( len A ( ) ) ) + 1 ) = ( ( len A ( ) ) + 1 ) & ( A ( ) ) is ( len A ( ) ) -NAT ;
for X being non empty addLoopStr , M being Subset of X , x , y being Point of X st y in M holds x + y in x + M
{ [ x1 , x2 ] , [ y1 , y2 ] } c= { [ x1 , y1 ] } \/ { [ x2 , y2 ] } ;
h . ( f . O ) = |[ A * ( f . O ) + B , C * ( f . O ) + D * ( f . O ) + D * ( f . O ) + D * ( f . O ) + D * ( f . O ) + D * ( f . O ) + D * ( f . O ) + D * ( f . O ) ;
( Gauge ( C , n ) ) * ( k , i ) in L~ Lower_Seq ( C , n ) /\ L~ Lower_Seq ( C , n ) ;
cluster m , n are_relative_prime for Nat , p be prime Nat , n be Nat , m be Nat , p be prime 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 \ c <= c holds a \+\ b <= c
consider b being element such that b in dom ( H / ( x. 0 ) ) and z = ( H / ( x. 0 ) ) . b and ( H / ( x. 0 ) ) . b = ( H / ( x. 0 ) ) . b ;
assume that x in dom ( F * g ) and y in dom ( F * g ) and ( F * g ) . x = ( F * g ) . y ;
assume ex e being element st e Joins W . 1 , W . 5 , G & e Joins W . 3 , G ;
( h (#) o ) . ( 2 * n ) = ( ( h (#) f ) . n ) . ( 2 * n ) .= ( ( h (#) f ) . n ) . ( 2 * n ) ;
j + 1 = ( - len h11 + 2 ) + 1 .= i + 1 - 2 + 2 - 1 .= i + 1 - 2 - 1 .= i + 2 - 1 ;
*' ( S *' ) = S *' ( S *' ) .= S *' ( S *' ) .= S *' ( S *' ) .= S *' ( S *' ) .= S *' ( S *' ) ;
consider H such that H is one-to-one and rng H = the carrier of L2 and Sum ( L2 * H ) = Sum ( L2 * H ) and Sum ( L2 * H ) = Sum ( L2 * H ) ;
attr R is *> means : Def4 : for p , q st p in R & q <> p holds ex P st P is_.| p & P c= R & p in R ;
dom ( product ^ ( X --> f ) ) = meet ( ( dom X --> f ) . 0 ) .= meet ( ( X --> f ) . 0 ) .= meet ( ( X --> f ) . 0 ) .= ( ( X --> f ) . 0 ) /\ ( X --> f ) . 0 .= ( ( X --> f ) . 0 ) /\ ( X --> f ) . 0 ;
upper_bound ( proj2 .: ( Upper_Arc ( C ) /\ Vertical_Line w ) ) <= upper_bound ( proj2 .: ( Upper_Arc ( C ) /\ Vertical_Line w ) ) & upper_bound ( proj2 .: ( Upper_Arc ( C ) /\ Vertical_Line w ) ) <= upper_bound ( proj2 .: ( Upper_Arc ( C ) /\ Vertical_Line w ) ;
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 - f<> i * f-28 - ( i * fnc ) .= i * ( fN - ( i * fc ) ) .= i * ( fN - ( i * fc ) ) ;
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 g = [ g1 , g2 ] ;
func d |-count n -> Nat means : Def4 : ( d |^ n ) divides ( d |^ n ) & ( d |^ n ) divides ( d |^ n ) & ( d |^ n ) divides ( d |^ n ) ;
f\in f . [ 0 , t ] = f . [ 0 , t ] .= ( - P ) . ( 2 * x ) .= ( - P ) . ( 2 * x ) .= a ;
t = h . D or t = h . B or t = h . C or t = h . D or t = h . E or t = F . J ;
consider m1 being Nat such that for n st n >= m1 holds dist ( ( seq . n ) , ( seq . n ) ) < 1 / ( n + 1 ) ;
( q `1 ) ^2 / ( |. q .| ) ^2 <= ( ( q `1 ) ) ^2 / ( |. q .| ) ^2 ;
h0 . ( i + 1 + 1 ) = h21 . ( i + 1 + 1 + 1 ) .= h21 . ( i + 1 + 1 ) .= h21 . ( i + 1 + 1 ) ;
consider o being Element of the carrier' of S , x2 being Element of { [ o , x2 ] } such that a = [ o , x2 ] and o <= g ;
for L being RelStr , a , b being Element of L holds a <= { b } iff a <= b & b <= a & a <= b implies b <= a
||. h1 .|| . n = ||. ( h1 . n ) .|| .= ||. ( h . n ) .|| .= ||. ( h . n ) .|| .= ||. ( h . n ) .|| .= ||. ( h . n ) .|| ;
( ( - ( exp_R * f ) ) `| Z ) . x = f . x - exp_R . x .= - ( exp_R . x ) / ( exp_R . x ) .= - ( exp_R . x ) / ( exp_R . x ) ;
attr r = F .: ( p , q ) means : Def4 : len r = min ( len p , len q ) & for i st i in dom r holds r . i = min ( p . i , q . i ) ;
( r / 2 ) ^2 + ( r / 2 ) ^2 <= ( r / 2 ) ^2 + ( r / 2 ) ^2 + ( r / 2 ) ^2 ;
for i being Nat , M being Matrix of n , K st i in Seg n holds Det M = Sum ( ( Line ( M , i ) ) * ( Det M ) )
then a <> 0. R & a " * ( a * v ) = 1 * v & a " * v = 1 * v & a " * v = 1 * v ;
p . ( j -' 1 ) * ( q *' r ) . ( i + 1 -' j ) = Sum ( p . ( j -' 1 ) * ( q *' r ) ) ;
deffunc F ( Nat ) = L . 1 + ( ( R /* ( h ^\ n ) ) * ( h ^\ n ) " ) . $1 ;
assume that the carrier of H2 = f .: ( the carrier of H1 ) and the carrier of H2 = f .: ( the carrier of H2 ) and the carrier of H1 = the carrier of H2 and the carrier of H2 = the carrier of H2 ;
Args ( o , Free ( S , X ) ) = ( ( the Sorts of Free ( S , X ) ) * ( the Arity of S ) ) . o .= ( ( the Sorts of Free ( S , X ) ) * ( the Arity of S ) ) . o ;
H1 = n + 1 and |. 2 to_power ( n + 1 + h ) .| = n + 1 to_power ( n + 1 + h ) .= n + 1 to_power ( n + 1 + h ) ;
( O = 0 & 3 = 0 & 1 = O & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 implies O = O & O = O & O = O & O = O
F1 .: ( dom F1 /\ dom F2 ) = F1 .: ( dom F1 /\ dom F2 ) .= F1 .: ( dom F1 /\ dom F2 ) .= { f /. ( n + 2 ) } .= { f /. n } ;
attr b <> 0 & d <> 0 & b <> d & b <> d & ( a = b ) implies ( a = ( b - b) / ( b - b) ) / ( b - bc )
dom ( ( f +* g ) | D ) = dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) ;
for i be set st i in dom g ex u , v be Element of L st g /. i = u * a & u in B & v in B & u in C & v in C
g `2 * P `2 = g `2 * ( g `2 * P `1 ) * g `2 .= g `2 * ( g `1 * P `1 ) .= g `2 * ( g `1 * P `1 ) ;
consider i , s1 such that f . i = s1 and not ( ex i st i in dom f & f . ( i + 1 ) <> s1 ) & not ( ex s st s <> s1 & f . ( i + 1 ) <> s1 ) ;
h5 | ]. a , b .[ = ( g | ]. a , b .[ ) | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ ;
[ s1 , t1 ] , [ s2 , t2 ] are_connected & [ s2 , t2 ] , [ t2 , t2 ] are_connected & [ s2 , t2 ] , [ t2 , t2 ] are_connected ;
then H is negative & H is not negative & H is not conjunctive & H is not conjunctive -gfor F being Function of H , D holds F is not non empty ;
attr f1 is total means : Def4 : f1 is total & f2 is total & ( for c be Element of dom f1 holds f1 . c = ( f2 . c ) * ( f2 . c ) " ;
z1 in W2 ` & z1 = z2 ` or z1 = z2 & not z1 in W2 & not z2 in W2 & not z2 in W1 & not z2 in W2 & not z2 in W1 & not z2 in W2 & not z2 in W2
p = 1 * p .= a " * a * p .= a " * ( b * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) ;
for r be Real_Sequence , K be Real st for n be Nat holds seq1 . n <= K holds upper_bound rng ( seq ^\ k ) <= upper_bound rng ( seq ^\ k )
( for x being Point of TOP-REAL 2 st x in L~ go holds ( go /. x ) meets L~ pion1 or ( ex x being Point of TOP-REAL 2 st x in L~ pion1 & x in L~ pion1 ) holds not ( ex y being Point of TOP-REAL 2 st y in L~ pion1 & not y in L~ pion1 )
||. f . ( g . ( k + 1 ) ) - g . ( g . k ) .|| <= ||. g . 1 - g . 0 .|| * ( K to_power k ) ;
assume h = ( ( B .--> B ' ) +* ( D .--> C ' ) +* ( E .--> D ' ) +* ( F .--> J ' ) +* ( J .--> N ' ) +* ( M .--> N ' ) +* ( N .--> N ' ) +* ( M .--> N ' ) +* ( N .--> N ' ) +* ( M .--> N ' ) +* ( N .--> N ' ) +* ( N .--> N ' ) ) +* ( M .--> N ) ;
|. ( ( ( ( ( H . n ) || A ) || A ) . k ) - ( ( ( ( H . n ) || A ) . k ) || A ) . k .| <= e * ( ( ( H . n ) || A ) . k ) ;
( (
{ x1 , x1 , x1 , x1 , x1 , x1 , x2 , x3 , x4 } = { x1 , x1 , x2 , x3 } .= { x1 , x2 , x3 } .= { x1 } ;
Suppose A = [. 0 , 2 * PI .] and integral ( ( exp_R * sin ) , A ) = 0 and ( ( exp_R * cos ) | A ) . x = 0 ; Then ( ( exp_R * cos ) | A ) . x = 0 ;
p `1 is Permutation of dom f1 & p `1 " = ( Sgm Y ) " * p " .= ( Sgm Y ) " * p " .= ( Sgm Y ) " * p " .= ( Sgm X ) " * p ;
for x , y st x in A holds |. 1 / ( f . x ) - 1 / ( f . y ) .| <= 1 * |. f . x - 1 / ( f . y ) .|
( p2 `2 = |. q2 .| * ( ( q2 `2 / |. q2 .| - sn ) / ( 1 + sn ) ) ) / ( 1 + sn ) - cn * ( ( q2 `2 / |. q2 .| - sn ) / ( 1 + sn ) ) ;
for f be PartFunc of the carrier of CNS , REAL st dom f is compact & f is continuous & f is continuous holds rng f c= dom f & f | X is continuous
assume for x being Element of Y st x in EqClass ( z , CompF ( B , G ) ) holds ( Ex ( a , A , G ) ) . x = TRUE ;
consider FM such that dom FM = n1 and for k be Nat st k in n1 holds Q [ k , FM . k ] and for k be Nat st k in n1 holds FM [ k , FM . k ] ;
ex u , u1 st u <> u1 & u , u1 / ( 2 |^ ( n + 1 ) ) / ( 2 |^ ( n + 1 ) ) / ( 2 |^ ( n + 1 ) ) > 0 & u , u1 / ( 2 |^ ( n + 1 ) ) / ( 2 |^ ( n + 1 ) ) / ( 2 |^ ( n + 1 ) ) > 0 ;
for G being Group , A , B being non empty Subgroup of G , N being normal Subgroup of G holds ( N ` A ) * ( N ` B ) = N ` A * N ` B
for s be Real st s in dom F holds F . s = integral ( R ^ > 0 ) (#) ( ( f + g ) (#) e - ( f + g ) (#) e ) . x
width AutMt ( f1 , b1 , b2 ) = len b2 .= len ( b1 * b2 ) .= len ( b1 * b2 ) .= len b1 .= len b1 .= len b1 .= len b1 .= len b1 .= len b1 .= len b1 .= len b1 .= len b2 .= len b2 .= len b2 .= len b2 .= len b2 .= len b2 ;
f | ]. - PI / 2 , PI / 2 .[ = f & f | ]. - PI / 2 , PI / 2 .[ = f & f | ]. - PI / 2 , PI / 2 .[ is continuous ;
assume that X is closed and a in X and a c= X and y in X and x in X and y in X and x in X and y in X ;
Z = dom ( ( ( ( - 1 / 2 ) (#) ( arctan ) ) `| Z ) /\ dom ( ( ( - 1 / 2 ) (#) ( arctan ) ) `| Z ) .= dom ( ( ( - 1 / 2 ) (#) ( arctan ) ) `| Z ) ;
func TAUT ( V ) -> Subset of V means : Def4 : for k st 1 <= k & k <= len l holds it . k in V & l . k in V ;
for L being non empty TopSpace , N being net of L , M being net of L , N being net of L st c is convergent holds N is net of N & for c being Point of L st c in N holds N . c is convergent
for s being Element of NAT holds ( for v being Element of NAT holds ( for x being Element of NAT holds x in dom ( ||. v .|| ) ) implies ||. ( v .|| ) . x = ( ( ||. v .|| ) . x ) * ( ||. v .|| )
then z /. 1 = ( N-min L~ z ) .. z & ( N-min L~ z ) .. z < ( ( N-min L~ z ) .. z ) .. z ;
len ( p ^ <* ( 0 qua Real ) + 1 ) = len p + len <* ( 0 qua Real ) + 1 *> .= len p + 1 .= len p + 1 ;
assume that Z c= dom ( - ( ln * f ) ) and for x st x in Z holds f . x = x and f . x > 0 and for x st x in Z holds f . x = x - 1 / ( sin . x ) and f . x > 0 ;
for R being add-associative right_zeroed right_complementable associative well-unital distributive non empty doubleLoopStr , I being 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 x .= len ( x2 + y2 ) .= len ( x2 + y2 ) .= len ( x + y2 ) .= len ( x + y ) .= len ( x + y ) ;
for S being card Functor of C , B for c being Object of C holds card ( id c ) = id ( ( ( the Arrows of C ) . c ) ) & ( id c ) . c = id ( ( ( the Arrows of C ) . c ) )
ex a st a = a2 & a in f6 /\ f5 & \mathop { f . a , f . b } = \mathop { f . a , f . b } & \mathop { f . a , f . b } = { f . a , f . b } ;
a in Free ( H2 / ( x. 4 , x. k ) ) '&' ( H2 / ( x. k , x. k ) ) / ( x. k , x. k ) / ( x. k , x. k ) ;
for C1 , C2 being \llangle C1 , C2 :] , f being stable Function of C1 , C2 st f = g & f = g holds f = g & f = g implies f = g
( W-min L~ go \/ L~ pion1 ) `1 = W-bound L~ go \/ W-bound L~ pion1 .= W-bound L~ pion1 \/ W-bound L~ pion1 .= W-bound L~ pion1 \/ W-bound L~ pion1 .= W-bound L~ pion1 \/ W-bound L~ pion1 .= W-bound L~ pion1 ;
assume that u = <* x0 , y0 , z0 *> and f is_differentiable on u and SVF1 ( 3 , f , u ) is_differentiable_in x0 and SVF1 ( 3 , f , u ) is_differentiable_in y0 ;
then ( t . {} ) `1 in Vars & ex x being Element of Vars st x = ( t . {} ) `1 & t . {} = x & t . {} = [ x , s ] ) or ex x being Element of Vars st x = [ x , s ] & t . {} = [ x , s ] ;
Valid ( p '&' p , J ) . v = Valid ( p , J ) . v '&' Valid ( q , J ) . v .= Valid ( p , J ) . v '&' Valid ( q , J ) . v .= Valid ( p , J ) . v ;
assume for x , y being Element of S st x <= y for a , b being Element of T st a = f . x & b = f . y holds a >= b ;
func Class ( R , x ) -> Subset-Family of R means : Def4 : for A being Subset of R holds A in it iff ex a being Element of R st a in it & A c= a ;
defpred P [ Nat ] means ( ( ( ( \HM { the carrier' of G ) \ { $1 } ) \/ { x } ) \/ { y } ) c= G ) & ( ( ( G ) \ { x } ) \/ { y } c= G ) ;
assume that dim W1 = 0 implies dim U2 = 0 & ( dim U2 = 0 implies ( S = 0 implies S = 0 ) & ( S = { 0 } implies S = { 0 } ) & ( S = { 0 } implies S = { 0 } ) & ( S = { 0 } implies S = { 0 } ) ;
mama_/. ( m . t ) = ( m . t ) `1 .= ( [ m . t , the carrier of C ] ) `1 .= [ m . t , the carrier of C ] `1 .= m . t ;
d11 = x9 ^ d22 .= f . ( ( y9 , d ) --> ( x9 , d ) ) .= f . ( y9 , d ) .= ( f | ( y9 , d ) ) . ( x9 , d ) .= ( f | ( y9 , d ) ) . ( y9 , d ) .= ( f | ( y9 , d ) ) . ( x9 , d ) .= ( f | ( y9 , d ) ) . ( y9 , d ) ;
consider g such that x = g and dom g = dom f and for x being element st x in dom f holds g . x in f . x and g . x in f . x ;
x + 0. F_Complex |^ ( len x ) = x + len x |-> 0. F_Complex .= ( x + ( len x |-> 0. F_Complex ) ) .: ( ( len x |-> 0. F_Complex ) ) .= x ` ;
( k -' ( k + 1 ) ) in dom ( f | ( k -' 1 ) ) /\ ( ( k -' ( k + 1 ) ) /\ ( k -' 1 ) ) ;
assume that P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 = P \/ { p2 } and P2 = P \/ { p1 , p2 } and P1 = P \/ { p2 } and P2 = P \/ { p1 , p2 } and P1 /\ P2 = { p2 } and P2 = P \/ { p1 , p2 } and P1 /\ P2 = { p2 } and P2 /\ P2 = { p1 , p2 } ;
reconsider a1 = a , b1 = b , c1 = c , c2 = d , c1 = c , c2 = d , c2 = c , c1 = d , c2 = c , c2 = d , c1 = c , c2 = d , c2 = c , c1 = d , c2 = d , c2 = c , c1 = d , c2 = c , c2 = d , c2 = d , c1 = c , c2 = d , c2 = d , c1 = c , c2 = d , c2 = d , c2 = d , c2 = d , c2 = c , c2 = d , c2 = d , c1 = d , c2 = d , c2 = d , c2 = d , c2 = d , c2 = c , c2 = c , c2 = d , c2 =
reconsider set set set FbFFF1f = G1 . ( t , b ) * F1 . ( t , b ) as Morphism of ( G1 * F1 ) . ( t , b ) * F2 . ( t , b ) ;
LSeg ( f , i + i1 -' 1 ) = LSeg ( f /. ( i + i1 -' 1 ) , f /. ( i + i1 -' 1 + 1 ) ) .= LSeg ( f , i + 1 ) ;
Integral ( P . m , P . n ) | dom ( P . n ) <= Integral ( M . n , P . m ) | dom ( P . n ) ;
assume that dom f1 = dom f2 and for x , y being element st [ x , y ] in dom f2 holds f1 . ( x , y ) = f2 . ( x , y ) and f2 . ( x , y ) = f2 . ( x , y ) ;
consider v such that v = y and dist ( u , v ) < min ( ( - G * ( i , 1 ) ) , ( - G * ( i , 1 ) ) ) ;
for G being Group , H being Subgroup of G , a being Integer , b being Integer st a = b holds for i being Integer st i in dom H holds a |^ i = b |^ i
consider B being Function of Seg ( S + L ) , the carrier of V1 such that for x being element st x in Seg ( S + L ) holds P [ x , B . x ] ;
reconsider K1 = { 7 where 7 is Point of TOP-REAL 2 : P [ 7 ] } , K1 = { 7 where 7 is Point of TOP-REAL 2 : P [ 7 ] } as Subset of TOP-REAL 2 ;
( ( ( ( N - S ) / 2 ) - ( ( N - S ) / 2 ) ) / 2 ) * ( ( ( N - S ) / 2 ) ) <= ( ( ( N - S ) / 2 ) - ( ( N - S ) / 2 ) / 2 ) * ( ( N - S ) / 2 ) ;
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 ^ <* 0 *> ) ) + len <* 0 *> .= len ( @ ( @ p ^ <* 0 *> ) ) + len ( @ p ^ <* 1 *> ) .= len ( @ p ^ <* 0 *> ) + len ( @ p ^ <* 1 *> ) ;
v / ( x. 3 , m1 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m1 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) = m3 ;
consider r being Element of M such that M , v / ( x. 3 , m ) / ( x. 4 , m ) / ( x. 0 , m ) / ( x. 4 , m ) / ( x. 0 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 0 , m ) / ( x. 4 , m ) / ( x. 0 , m ) / ( x. 4 , m ) / ( x. 0 , m ) / ( x. 4 , m ) / ( x. 0 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 0 , m ) / ( x. 0 ) / ( x. 0 ) / ( x. 0
func w1 \ w2 -> Element of Union ( G , R8 ) equals ( ( ( the Sorts of G ) * ( the Sorts of R8 ) ) . ( w1 , w2 ) ) . ( w1 , w2 ) ;
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= s1 . b2 .= Exec ( n2 , s2 ) . b2 .= Exec ( n2 , s2 ) . b2 .= Exec ( n2 , s2 ) . b2 .= Exec ( n2 , s2 ) . b2 .= ( Exec ( n2 , s2 ) ) . b2 .= ( Exec ( n2 , s2 ) ) . b2 .= ( s ) . b2 ;
for n , k be Nat holds 0 <= ( Partial_Sums |. seq .| ) . ( n + k ) - Partial_Sums ( |. seq .| ) . n + Partial_Sums ( |. seq .| ) . ( n + k ) - Partial_Sums ( |. seq .| ) . n ;
set F = S -_ _ _ F , G = S -_ _ _ F ;
Partial_Sums ( seq ) . ( n + 1 ) + Partial_Sums ( seq ) . ( n + 1 ) + Partial_Sums ( seq ) . n >= Partial_Sums ( seq ) . ( n + 1 ) + Partial_Sums ( seq ) . n + Partial_Sums ( seq ) . ( n + 1 ) ;
consider L , R such that for x st x in N holds ( f | Z ) . x = L . ( x- x ) + R . ( x- x ) and R . ( x- x ) = L . ( x- x ) + R . ( x- x ) ;
func the closed of \HM { a , b , c , d , e , f , g , h , i , f , i , g , h , i , f , i , g , h h h h h h h i i i i , g , i , h , i ) ;
a * b ^2 + ( a * c ^2 + b * c ^2 + c * a ^2 + b * c ^2 + c * a ^2 + d * b ^2 + c * a ^2 + d * b ^2 + c * a ^2 + d * b ^2 + c * b ^2 + d * c ^2 + c * a ^2 + d * b ^2 + c * c + d * b ^2 + c * a ^2 + d * b ^2 + c ^2 + d * b ^2 + d * b ^2 + d * c ^2 + d * c ^2 + d * c ^2 + c * c ^2 + d * c ^2 + c ^2 + d * c ^2 + c * c ^2 + d * c
v / ( x1 , m1 ) / ( x2 , m2 ) / ( x3 , m1 ) / ( x3 , m2 ) / ( x3 , m2 ) / ( x3 , m1 ) = v / ( x2 , m1 ) / ( x3 , m2 ) / ( x3 , m1 ) ;
Rotate ( Q ^ <* x *> , M ) = ( Rotate ( Q , M ) +* ( M , { FALSE } ) ) +* ( ( M ^ <* x *> --> FALSE ) ) +* ( ( M ^ <* x *> --> FALSE ) ) .= ( M ^ <* x *> --> TRUE ) +* ( M ^ <* x *> --> FALSE ) .= ( M ^ <* x *> --> TRUE ) ;
Sum ( F ) = r |^ n1 * Sum ( C ) .= C . n1 * ( C . n1 ) .= C . n1 * ( C . n1 ) .= C . n1 * ( C . n1 ) .= C . n1 * ( C . n1 ) .= C . n1 * ( C . n1 ) .= C . n1 * ( C . n1 ) .= C . 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 ) ) * ( $1 + 1 ) + b * ( $1 + 1 ) ;
( the_arity_of g ) = ( the Arity of S ) . g .= ( ( the Arity of S ) . g ) . g .= ( ( the Arity of S ) . g ) . g .= ( ( the Arity of S ) * g ) . g .= ( ( the Arity of S ) * g ) . g .= ( ( the Arity of S ) * g ) . g ;
( X ~ ) c= X ^2 & Z c= X ^2 implies card ( ( X ~ ) \/ ( Y ~ ) ) = card ( X ~ )
for a , b being Element of S , s being Element of NAT st s = n & a = F . n & b = F . n & b = F . n holds b = G . n \ G . n
E , f |= All ( All ( x , ( ( x. 2 ) . ( x. 0 ) ) ) , ( ( x. 2 ) . ( x. 0 ) ) ) => ( ( x. 2 ) . ( x. 0 ) ) '&' ( ( x. 2 ) . ( x. 0 ) ) '&' ( ( x. 2 ) . ( x. 0 ) ) ;
ex R2 being 1-sorted st R2 = ( p | ( n + 1 ) ) . i & ( the carrier of R1 ) = the carrier of R2 & ( the carrier of R2 ) = the carrier of R2 & ( the carrier of R2 ) c= the carrier of R2 ;
[. a , b + 1 / ( k + 1 ) .[ is Element of the _ of the _ of the carrier of X & ( the partial of X ) . k is Element of the carrier of X & ( the partial of X ) . k is Element of the \overline of X & ( the \overline of X ) . k is Element of the \overline of X ;
Comput ( P , s , 2 + 1 ) = Exec ( P . 2 , Comput ( P , s , 2 ) ) .= Exec ( a3 , Comput ( P , s , 2 ) ) .= Exec ( a3 , s ) . IC SCM+FSA .= ( ( a , b ) := ( card I + 2 ) ) . IC SCM+FSA ;
card ( h1 ) . k = power F_Complex . ( ( - 1_ F_Complex ) . k , k ) * Sum u .= ( ( - 1_ F_Complex ) *' ) . k * u .= ( ( - 1_ F_Complex ) *' ) . k * u .= ( ( - ( - 1_ F_Complex ) ) *' ) . k * u .= ( ( ( - ( - 1_ F_Complex ) *' ) *' ) . k ;
( f / g ) /. c = f /. c * ( g /. c ) " .= f /. c * ( g /. c ) .= ( f (#) g ) /. c .= ( f (#) g ) /. c .= ( f (#) g ) /. c ;
len CH - len ( the _ { ( C ) * ( len CH ) , ( C ) * ( len CH ) ) = len CH - len ( the _ { ( C ) * ( len CH ) , ( C ) * ( len CH ) ) .= len ( C - ( C - ( C - ( C - ( C - ( C - ( C - ( C - D ) ) ) ) ) ) ;
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom f /\ X .= dom ( f | X ) /\ X .= dom ( f | X ) .= dom ( f | X ) .= dom ( f | X ) /\ X .= dom ( f | X ) .= X /\ X .= dom ( f | X ) ;
defpred P [ Nat ] means for n holds 2 * Fib ( n + $1 ) = Fib ( n ) * Fib ( n ) + ( 5 * Fib ( n ) ) + ( 5 * Fib ( n ) ) ;
consider f being Function of INT , INT such that f = f ` and f is onto and for n st n < k holds f . n = n + 1 & f . n = n + 1 and f . n = n + 1 ;
consider c9 be Function of S , BOOLEAN such that c9 = chi ( A \/ B , S ) and ( for A being Element of S holds E . A = Prob . A ) & ( for A being Element of S holds E . A = Prob . A ) & ( for A being Element of S holds E . A = Prob . A ) ;
consider y being Element of Y ( ) such that a = "\/" ( { F ( x , y ) where x is Element of X ( ) ) : P [ x ] } and for y being Element of X ( ) st P [ y ] holds y in { F ( x ) where x is Element of X ( ) : P [ x ] } ;
assume that A c= Z and Z = dom f and f = ( ( - 1 / 2 ) (#) ( sin + cos ) ) / ( cos + sin ) and Z = dom f and f = ( - 1 / 2 ) (#) ( cos + cos ) ;
( f /. i ) `2 = ( ( GoB f ) * ( 1 , j2 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 ) ) `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 & q1 . j = ( Seq q1 ) . ( len q1 + 1 ) & q1 . ( len q1 + 1 ) = ( Seq q1 ) . ( len q1 + 1 ) ;
consider G1 , G2 , G3 being Element of V such that G1 <= G2 & G2 <= G1 & f = G1 & g = G2 & f = G2 & g = G1 & f = G2 and g = G2 and f = G1 & g = G2 & f = G2 & g = G2 ;
func - f -> PartFunc of C , V means : Def4 : dom it = dom f & for c st c in dom it holds it /. c = - f /. c & for c st c in dom it holds it /. c = - f /. c ;
consider phi such that phi is increasing and for a st phi . a = a & {} <> a for v st v in a holds union ( L , v ) |= ( L , v ) iff for v st v in a holds L , v |= v ;
consider i1 , j1 such that [ i1 , j1 ] in Indices GoB f and f /. i = ( GoB f ) * ( i1 , j1 ) and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) ;
consider i , n such that n <> 0 and sqrt p = ( i - n ) / ( n + 1 ) and for n1 being Nat st n1 <> 0 & n1 <= n holds sqrt p = ( i - n ) / ( n + 1 ) and n <= k and k <= n ;
assume that not 0 in Z and Z c= dom ( ( arccot * f1 ) `| Z ) and for x st x in Z holds ( ( arccot * f1 ) `| Z ) . x > - 1 & f1 . x < 1 & f1 . x < 1 & f1 . x < 1 ;
cell ( G1 , i1 -' 1 , ( 2 |^ ( m -' 1 ) ) * ( Y1 -' 1 ) ) \ ( Y1 ) c= ( ( L~ f ) ` ) \ ( ( L~ f ) ` ) ` & ( ( L~ f ) ` ) \ ( ( L~ f ) ` ) c= ( ( L~ f ) ` ) ;
ex Q1 being open Subset of X st s = Q1 & ex F8 being Subset-Family of Y st F8 c= F & F8 is finite & ( for n being Nat holds F8 [ n , F8 . n ] ) & ( for n being Nat st n in dom F8 holds F8 [ n , F8 . n ] ) & ( for n being Nat st n in dom F8 holds P8 [ n , n ] )
gcd ( ( the carrier of A ) , ( the carrier of A ) , ( the carrier of A ) , ( the carrier of A ) , ( the carrier of A ) , ( the InternalRel of A ) , ( the InternalRel of A ) , ( the InternalRel of A ) , ( the InternalRel of A ) , ( the InternalRel of A ) , ( the InternalRel of A ) , ( the InternalRel of A ) , ( the InternalRel of A ) , ( the InternalRel of A ) = 1
R8 = ( ( the _ of ( s2 ) ) . ( m1 + 1 ) ) . ( m2 + 1 ) .= ( ( the _ of ( s2 ) ) . ( m2 + 1 ) ) . ( m2 + 1 ) .= [ 3 , 1 ] ;
CurInstr ( P-6 , Comput ( P-6 , s , m1 + 1 ) ) = CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= halt SCMPDS .= ( CurInstr ( P3 , s3 ) ) . IC SCMPDS .= ( halt SCMPDS ) . IC SCMPDS .= ( halt SCMPDS ) . IC SCMPDS .= ( halt SCMPDS ) . IC SCMPDS .= ( halt SCMPDS ) . IC SCMPDS ;
P1 /\ P2 = ( { p1 } \/ LSeg ( p1 , p2 ) ) /\ LSeg ( p11 , p2 ) \/ ( LSeg ( p11 , p2 ) \/ LSeg ( p11 , p2 ) ) /\ LSeg ( p11 , p2 ) \/ { p2 } /\ LSeg ( p11 , p2 ) \/ { p2 } ;
func the Sorts of A -> Subset of the Sorts of A means : Def4 : a in it iff ex p st p in dom f & a = f . p & p in the Sorts of A & a = f . p ;
for a , b being Element of F_Complex st |. a .| > b for f being Polynomial of F_Complex st f >= 1 & f is ) for a being Element of F_Complex st a >= 1 & f is >= 0 holds f * ( - b ) is ] & f is ] implies ( - a ) * ( - b ) is ]
defpred P [ Nat ] means 1 <= $1 & $1 <= len g implies for i , j st [ i , j ] in Indices G & G * ( i , j ) = g . ( i , j ) holds G * ( i , j ) = G * ( i , j ) ;
Suppose C1 , C2 are_\vert & f is \vert and g is \vert and for s1 , s2 being State of C1 , f being State of C2 st f = g * f holds f is stable iff for x being State of C1 , y being State of C2 st x = f * g holds x is stable & y is stable & f . x = y * f . y ;
( ||. 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 .|| ;
|. q .| ^2 = ( q `1 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 & 0 + ( q `1 ) ^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 <> B for A , B being Subset of T7 st A in F & B <> {} & A <> B & A misses B holds card F c= card A & card ( A \/ B ) c= card B
assume that len F >= 1 and len F = k + 1 and len F = k + 1 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 |^ ( ( ( Let n ) |^ ( ( p |^ k ) - i ) ) = i |^ ( ( s |^ k ) - i ) .= i |^ ( ( s |^ k ) - i ) .= i |^ ( ( s |^ k ) - i ) .= i |^ ( ( s |^ k ) - i ) .= i |^ ( ( s |^ k ) - i ) ;
consider q being oriented oriented Chain of G such that r = q and q <> {} and F7 . ( q . 1 ) = v1 and ( for q being Element of G st q in rng holds q . ( q . 1 ) = v1 & ( for q being Element of G st q in rng q holds q . ( q . 1 ) = v1 ) and rng q c= rng ( p ^ <* q *> ) ;
defpred P [ Element of NAT ] means $1 <= len ( g , Z ) implies ( ( ( g , Z ) ^ I ) . $1 = ( ( g , Z ) ^ I ) . ( len g + $1 ) ) & ( ( g , Z ) ^ I ) . $1 = ( ( g , Z ) ^ I ) . ( len g + $1 ) ;
for A , B being square Matrix of n , REAL holds len ( A * B ) = len A & width ( A * B ) = width B & width ( A * B ) = width A & width ( A * B ) = width B & width ( A * B ) = width A & width ( B * B ) = width B
consider s being FinSequence of the carrier of R such that Sum s = u and for i being Element of NAT st 1 <= i & i <= len s ex a , b being Element of R st s . i = a * b & a in I & b in J & s . i = b * a ;
func |( x , y )| -> Element of COMPLEX equals |( Re x , Re y )| - ( Re y ) * |( x , y )| + ( - ( Re x ) * |( x , y )| ) + ( - ( Im y ) * |( x , y )| ) + ( - ( Im y ) * |( x , y )| ) ;
consider g9 being FinSequence of F such that g9 is continuous and rng g9 c= A and for k st k in A holds g1 . k = x1 & g1 . k = x2 and g1 . k = y1 and g1 . k = x0 and g1 . k = x0 and g1 . k = x0 and g1 . k = x0 and g1 . k = x0 ;
then n1 >= len p1 & n2 >= len p1 implies crossover ( p1 , p2 , n1 , n2 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , p1 , p2 , n1 , n2 , n3 , n2 , n3 , n3 , n4 , n4 , n4 , n4 , p1 , p2 , n2 , n3 , n2 , n3 , n2 , n3 , n2 , n3 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , p1 , n2 , n3 , n4 , n4 , n2 , n3 , n2 , n3 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n3 , n4 , n4 , n4 , n4 , n2 , n3 , n2 ,
( q `1 ) * a <= ( q `1 ) * a & - ( q `1 ) * a <= ( q `1 ) * a or q `1 >= - ( q `1 ) * a & - ( q `1 ) * a <= - ( q `1 ) * a ;
Fv . ( ( len pv ) + 1 ) = Fv . ( p . ( len p ) ) .= ( ( v . ( len p ) ) + 1 ) * ( v . ( len p ) ) .= ( v . ( len p ) ) * v . ( len p ) .= ( v . ( len p ) ) * v . ( len p ) .= v . ( len p ) ;
consider k1 being Nat such that k1 + k = 1 and a := k = ( <* a := intloc 0 *> ^ ( ( k + 1 ) --> 1 ) ) ^ ( ( k + 1 ) --> 1 ) ^ ( ( k + 1 ) --> 1 ) .= ( ( k + 1 ) --> 0 ) ^ ( ( k + 1 ) --> 1 ) ;
consider B8 being Subset of B1 , y8 being Function of B1 , B2 such that B8 is finite and D8 = the carrier of A1 and for x being set st x in B1 holds ( x in B1 or x in B2 or x in B1 or x in B2 ) and ( x in B2 or x in B1 or x in B2 or x in B2 or x in B2 or x in B2 ) ;
v2 . b2 = ( curry F2 , g ) * ( ( curry F2 ) . b2 ) .= ( ( curry F2 , g ) . b2 ) * ( ( ( curry F2 ) . b2 ) ) .= ( ( curry F2 ) . b1 ) * ( ( ( curry F2 ) . b2 ) .= ( ( curry F2 ) . b1 ) * ( ( ( curry F2 ) . b2 ) ) .= ( ( ( curry F2 ) . b1 ) * ( ( id F2 ) . b2 ) ;
dom IExec ( I-35 , P , Initialize s ) = the carrier of SCMPDS .= dom ( IExec ( I , P , Initialize s ) ) .= dom ( IExec ( I , P , Initialize s ) ) .= dom ( IExec ( I , P , Initialize s ) ) .= dom ( IExec ( I , P , Initialize s ) ) ;
ex d-32 be Real st d-32 > 0 & for h be Real st h <> 0 & |. h .| < e holds |. h .| " * ||. ( R + R1 ) /. h .|| < e / ( 1 + R ) * ||. ( R2 + R1 ) /. h .|| < e / ( 1 + R ) * ||. ( R2 + R1 ) /. h .|| ;
LSeg ( G * ( len G , 1 ) + |[ 1 , - 1 ]| , G * ( len G , 1 ) + |[ 1 , 0 ]| ) c= Int cell ( G , 0 , 0 ) \/ { G * ( len G , 1 ) + |[ 1 , 0 ]| } ;
LSeg ( mid ( h , i1 , i2 ) , i ) = LSeg ( h /. ( i + 1 ) , h /. ( i + 1 + 1 ) ) .= LSeg ( h /. ( i + 1 + 1 ) , h /. ( i + 1 + 1 ) ) .= LSeg ( h /. i , h /. ( i + 1 + 1 ) ) .= LSeg ( h /. i , h /. ( i + 1 + 1 ) ) ;
A = { q where q is Point of TOP-REAL 2 : LE p1 , p2 , P , p1 , p2 & LE p2 , p3 , P , p1 , p2 & LE p1 , p2 , P , p1 , p2 & LE p2 , p3 , P , p1 , p2 & LE p2 , p3 , P , p1 , p2 & LE p1 , p2 , P , p1 , p2 & LE p2 , p3 , P , p1 , p2 & LE p2 , p3 , P , p1 , p2 , p2 & LE p1 , p3 , P , p2 , p3 , p3 , p2 , p3 , p3 , p3 , P , p1 , p3 , p3 , P , p1 , p2 , p3 , p3 , p4 , p4 , p4 , p4 , p4 , p4 , P , p1 , p2 , p4 , p4 , p4 , p4 , P , p4 , p4 , p4 , p4 , p4 , P , p1 , p3 , p4 , p4 , P , p1 , p3 , p4 ,
( ( - x ) .|. y ) = - ( 1 / ( 1 - x ) * ( x .|. y ) ) .= ( - 1 / ( 1 - x ) ) * ( x .|. y ) .= ( - 1 / ( 1 - x ) ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * x .|. y .= ( - 1 ) * x .|. y ;
0 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) = ( p `1 ) ^2 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) .= ( p `1 ) ^2 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) .= ( p `1 ) ^2 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) ;
( ( U * ( W * ( W ) ) ) ) * ( W * ( W * ( W ) ) ) = ( ( ( U * ( W * ( W * ( W * ( W * ( W ) ) ) ) ) ) ) * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * L ) ) ) ) ) ) ) ) .= ( ( ( U * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W ) ) ) ) ) ) ) ) ) ) ) ) ) ) * ( W * ( W * (
func Shift ( f , h ) -> PartFunc of REAL , REAL means : Def4 : dom it = dom h & for x be Element of REAL st x in dom it holds it . x = ( h . x ) * f . x & for x be Element of REAL st x in dom it holds it . x = ( h . x ) * f . x ;
assume that 1 <= k and k + 1 <= len f and [ i , j ] in Indices G and [ i + 1 , j ] in Indices G and f /. k = G * ( i , j ) and f /. k = G * ( i , j ) and f /. k = 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 ) ;
defpred P11 [ Element of NAT , Element of NAT , Element of NAT ] means ( $1 = p |^ $1 implies $2 = ( $1 |^ $1 ) * ( p |^ $1 ) ) & ( $1 |^ $1 ) * ( p |^ $1 ) < ( $1 |^ $1 ) * ( p |^ $1 ) & ( $1 |^ $1 ) * ( p |^ $1 ) < ( $1 |^ $1 ) * ( p |^ $1 ) ;
func \sigma ( C ) -> non empty Subset-Family of X means : Def4 : for A , B being Subset of X holds A in it iff for W being Subset of X st W c= A & W c= B holds C c= W \ A & C c= B \ A ;
[#] ( ( dist ( ( dist ( P ) ) ) .: Q ) = ( ( dist ( P ) ) ) .: Q ) .: Q & inf [#] ( ( dist ( P ) ) .: Q ) = lower_bound ( ( dist ( P ) ) .: Q ) & inf ( ( dist ( P ) ) .: Q ) = lower_bound ( ( dist ( P ) ) .: Q ) ;
rng ( F | ( [: S , T :] ) ) = {} or rng ( F | ( [: S , T :] ) ) = { 1 } or rng ( F | ( [: S , T :] ) ) = { 1 } or rng ( F | ( [: S , T :] ) ) = { 2 } or rng ( F | ( [: S , T :] ) ) = { 1 } ;
( f " ( rng f ) ) . i = f . i " .= ( ( f . i ) " ) . i .= ( ( f . i ) " ) . i .= ( ( f . i ) " ) . i .= ( ( f . i ) " ) . i .= ( ( f . i ) " ) . i .= ( ( f . i ) " ) . i .= ( ( f . i ) " ) . i ;
consider P1 , P2 being non empty Subset of TOP-REAL 2 such that P1 is_an_arc_of p1 , p2 and P1 = { p1 , p2 } and P2 = { p1 , p2 } and P1 = { p2 } and P2 = { p1 , p2 } and P1 = { p2 , p1 } and P2 = { p2 , p1 } and P2 = { p1 , p2 } ;
f . p2 = |[ ( p2 `1 ) ^2 + ( p2 `2 ) ^2 + ( p2 `2 ) ^2 + ( p2 `2 ) ^2 * ( p2 `1 ) ^2 + ( p2 `2 ) ^2 + ( p2 `2 ) ^2 * ( p2 `1 ) ^2 + ( p2 `2 ) ^2 + ( p2 `2 ) ^2 ;
( ( for a being Real , X being non empty TopSpace , x being Point of X holds x = ( ( the carrier of X ) --> a ) " ) . x .= ( ( the carrier of X ) --> a ) . x .= ( ( the carrier of X ) --> a ) . x .= ( ( the carrier of X ) --> a ) . x .= ( ( the carrier of X ) --> a ) . x .= ( ( a + 1 ) * x ) . x ;
for T being non empty normal TopSpace , A , B being closed Subset of T , r being Real st A <> {} & A misses B & B misses B for p being Point of T st p in A & p in B holds p in ( ( in the topology of T ) | A ) & for r being Real st r in A holds p in ( ( ( in the topology of T ) | B ) . r )
for i , j st i in dom F for G1 , G2 being strict normal Subgroup of G st G1 = F . i & G2 = F . i & G2 = F . j holds G1 * G2 is strict Subgroup of G1 & G2 = F . i & G1 * G2 = G2 * G1
for x st x in Z holds ( ( arctan - arccot ) `| Z ) . x = ( ( arctan - arccot ) `| Z ) . x / ( 1 + x ^2 ) - ( arccot - arccot ) . x / ( 1 + x ^2 )
synonym f is right continuous means : Def4 : x0 = lim ( f /* a ) & for x st x in dom f holds f . x = lim ( f /* a ) & for x st x in dom f holds f . x = ( f /* a ) . x ;
then X1 , X2 are_separated & ( X1 misses X2 or Y1 misses Y2 or X1 misses X2 & ( X1 misses X1 or X2 misses X2 or X1 misses X2 or X1 misses X2 & X1 misses X2 or X1 misses X2 or X1 misses X2 or X2 misses X1 or X1 misses X2 or X1 misses X2 or X1 union X2 = X2 or X1 union X2 = X2 union X2 ;
ex N being Neighbourhood of x0 st N c= dom SVF1 ( 1 , f , u ) & ex L , R st for x st x in N holds ( SVF1 ( 1 , f , u ) ) . x = L . ( x- ( 1 , f , u ) ) + R . ( x - x0 ) + R . ( x - x0 )
( p2 `1 ) ^2 * sqrt ( 1 + ( p3 `1 ) ^2 ) >= ( ( p2 `1 ) ) ^2 * sqrt ( 1 + ( p3 `1 ) ^2 ) * sqrt ( 1 + ( p3 `1 ) ^2 ) ;
( ( 1 / t ) (#) ( ||. f1 .|| ) ) to_power ( n + 1 ) = ( 1 / t ) * ( ||. g1 .|| ) to_power ( n + 1 ) & ( ( 1 / t ) (#) ( ||. g1 .|| ) to_power n ) . x = ( 1 / t ) * ( ||. g1 .|| ) to_power ( n + 1 ) ;
assume that for x holds f . x = ( ( - 1 / 2 ) (#) ( sin * cos ) ) . x and x in dom ( ( - 1 / 2 ) (#) ( sin * cos ) ) and x in dom ( ( - 1 / 2 ) (#) ( cos * cos ) ) and x in dom ( ( - 1 / 2 ) (#) ( sin * cos ) ) ;
consider Xf1 being Subset of Y , Y1 being Subset of X such that t = [: Y1 , Y1 :] and Y1 is open and ex Y1 being Subset of X st Y1 = [: Y1 , Y1 :] & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open ;
card ( S . n ) = card { [: d , Y :] + ( a * d ) + b * d where d , b is Element of GF ( p ) : [ d , b ] in R & [ d , b ] in \mathop { d } } .= { d } + { b } .= { d } + { b } .= { d } ;
( W-bound D - W-bound D ) * ( ( i1 - 1 ) / 2 ) = ( W-bound D - ( W-bound D ) / 2 ) * ( ( i - 1 ) / 2 ) .= ( W-bound D - ( W-bound D ) / 2 ) * ( ( i - 1 ) / 2 ) .= ( W-bound D - ( W-bound D ) / 2 ) * ( i - 1 ) ;