thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
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thesis ;
contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
contradiction ;
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contradiction ;
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contradiction ;
contradiction ;
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thesis ;
contradiction ;
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contradiction ;
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thesis ;
thesis ;
contradiction ;
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contradiction ;
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thesis ;
thesis ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b
D c= S
let Y ;
S ` is convergent
q in X ;
V ;
y in N ;
x in T ;
m < n ;
m <= n ;
n > 1 ;
let r ;
t in I ;
n <= 4 ;
M is finite ;
let X ;
Y c= Z ;
A // M ;
let U ;
a in D ;
q in Y ;
let x ;
1 <= l ;
1 <= w ;
let G ;
y in N ;
f = {} ;
let x ;
x in Z ;
let x ;
F is one-to-one ;
e <> b ;
1 <= n ;
f is special ;
S misses C
t <= 1 ;
y divides m ;
P divides M ;
let Z ;
let x ;
y c= x ;
let X ;
let C ;
x _|_ p ;
o is monotone ;
let X ;
A = B ;
1 < i ;
let x ;
let u ;
k <> 0 ;
let p ;
0 < r ;
let n ;
let y ;
f is onto ;
x < 1 ;
G c= F ;
a is_>=_than X ;
T is continuous ;
d <= a ;
p <= r ;
t < s ;
p <= t ;
t < s ;
let r ;
D is_<=_than 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 `2 <= b `2 ;
assume b in X ;
assume k <> 1 ;
f = Product l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is from 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 ` = x ` ` ;
let X be BCK-algebra ;
assume S is non empty ;
a in REAL ;
let p be set ;
let A be set ;
let G be _Graph , W be Walk of G ;
let G be _Graph , W be Walk of G ;
let a be Complex ;
let x be element ;
let x be element ;
let C be FormalContext , a , b 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 ;
let i be Nat ;
let x be element ;
n in NAT ;
let a be element ;
m in NAT ;
let u be element ;
i in NAT ;
let g be Function ;
Z c= NAT ;
l <= ma ;
let y be element ;
r2 in dom f ;
let x be element ;
let k1 be Integer ;
let X be set ;
let a be element ;
let x be element ;
let x be element ;
let q be element ;
let x be element ;
assume f is being_homeomorphism ;
let z be element ;
a , b // K ;
let n be Nat ;
let k be Nat ;
B ` c= B ` ;
set s = / ( 1 + 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 TOP-REAL 2 ;
not r in G . l
c1 = 0 ;
a + a = a ;
<* 0 *> in e ;
t in { t } ;
assume F is non discrete ;
m1 divides m ;
B * A <> {} ;
a + b <> {} ;
p * p > p ;
let y be ExtReal ;
let a be Int-Location , I be Program of SCM+FSA ;
let l be Nat ;
let i be Nat ;
let r ;
1 <= i2 ;
a "\/" c = c ;
let r be Real ;
let i be Nat ;
let m be Nat ;
x = p2 ;
let i be Nat ;
y < r + 1 ;
rng c c= E
Cl R is boundary ;
let i be Nat ;
R2 in dom f ;
cluster uparrow x -> non empty ;
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 >= G2 + implies n <= len implies 2 * n >= 0
G . y <> 0 ;
let X be RealNormSpace , x be Point of X ;
a in X ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , W be Subspace of V ;
assume x in - - M ;
k < s . a ;
not t in { p } ;
let Y be set , f be set ;
M , L are_equipotent ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded & L is upper-bounded ;
rng f = Y ;
( G . n ) c= L ;
assume x in Cl Q ;
m in dom P ;
i <= len Q ;
len F = 3 ;
Free p = {} ;
z in rng p ;
lim b = 0 ;
len W = 3 ;
k in dom p ;
k <= len p ;
i <= len p ;
1 in dom f ;
b `1 = a `1 + 1 ;
x ` = a * y ` ;
rng D c= A ;
assume x in K1 ;
1 <= i-32 ;
1 <= i-32 ;
pp c= PI & pp c= PI ;
1 <= i-15 ;
1 <= i-15 ;
LMP C in L ;
1 in dom f ;
let seq , n ;
set C = a * B ;
x in rng f ;
assume f is_continuous_on X ;
I = dom A ;
u in dom p ;
assume a < x + 1 ;
s-7 is bounded ;
assume I c= P1 & I c= P2 ;
n in dom I ;
let Q ;
B c= dom f ;
b + p _|_ a ;
x in dom g ;
F-14 is continuous ;
dom g = X ;
len q = m ;
assume A2 : x in A2 ;
cluster R \ S -> real-valued ;
sup D in S ;
x << sup D ;
b1 >= Z1 or b1 >= Z1 ;
assume w = 0. V ;
assume x in A . i ;
g in being _ of X ;
y in dom t ;
i in dom g ;
assume P [ k ] ;
len Let C c= f ;
x4 is increasing ;
let e2 be element ;
- b divides b ;
F c= \tau ( F ) ;
IT is non-decreasing implies IT is non-decreasing
IT is non-decreasing implies IT is non-decreasing
assume v in H . m ;
assume b in [#] B ;
let S be non void ManySortedSign , X be set ;
assume P [ n ] ;
assume union S is finite independent & finite S is finite ;
V is Subspace of V ;
assume P [ k ] ;
rng f c= NAT ;
assume ex_inf_of X , L ;
y in rng f ;
let s , I be set , J be set ;
b ` c= b9 ` ;
assume not x in REAL ;
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 & i2 <= len f ;
a * h in a * H ;
p , q in Y ;
redefine func sqrt I -> Ideal of L ;
q1 in A1 & q2 in A2 ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A2 c= A1 ;
an < n & bn < n ;
assume A c= dom f ;
Re f is_integrable_on M & Im f is_integrable_on M ;
let k , m be element ;
a , a // b , b ;
j + 1 < k + 1 ;
m + 1 <= n1 ;
g is_differentiable_in x0 & g is_differentiable_in x0 ;
g is_continuous_in x0 & g is_continuous_in x0 ;
assume O is symmetric transitive ;
let x , y be element ;
let j0 be Nat ;
[ y , x ] in R ;
let x , y be element ;
assume y in conv A ;
x in Int V ;
let v be Vector of V ;
P3 halts_on s & P3 halts_on s ;
d , c // a , b ;
let t , u be set ;
let X be set ;
assume k in dom s ;
let r be non negative Real ;
assume x in F | M ;
let Y be Subset of S ;
let X be non empty TopSpace , A be Subset of X ;
[ a , b ] in R ;
x + w < y + w ;
{ a , b } >= c ;
let B be Subset of A , C be Subset of B ;
let S be non empty ManySortedSign ;
let x be variable , f be Function ;
let b be Element of X , c be Element of Y ;
R [ x , y ] ;
x ` = x & y ` = y ;
b \ x = 0. X ;
<* d *> in D * ;
P [ k + 1 ] ;
m in dom ( mn ) ;
h2 . a = y ;
P [ n + 1 ] ;
redefine func G * F -> ] ] ;
let R be non empty multMagma , x be Element of R ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p ;
assume f | X is lower ;
x in rng co /\ 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 `2 ;
let M be as m mamaid ;
let N be non empty K K is $ < N ;
let R be RelStr with finite finite finite finite finite finite is Nat ;
let n , k be Nat ;
let P , Q be be be be be be be be let let let let let let \HM of P ;
P = Q /\ [#] S ;
F . r in { 0 } ;
let x be Element of X ;
let x be Element of X ;
let u be Vector of V ;
reconsider d = x as Int-Location ;
assume I is not \leq _ o ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < vseq ;
x <= c2 . x ;
x in F ` & y in F ` ;
redefine func S --> T -> ManySortedSet of I ;
assume that t1 <= t2 and t2 <= t2 ;
let i , j be even Integer ;
assume that F1 <> F2 and F2 <> F1 ;
c in Intersect ( union R ) ;
dom p1 = c & dom p2 = c ;
a = 0 or a = 1 ;
assume A1 <> A2 & A2 <> A1 ;
set i1 = i + 1 ;
assume a1 = b1 & a2 = b2 ;
dom g1 = A & dom g2 = A ;
i < len M + 1 ;
assume not -infty in rng G ;
N c= dom ( f1 + f2 ) ;
x in dom ( sec * sec ) ;
assume [ x , y ] in R ;
set d = ( x - y ) / ( x - y ) ;
1 <= len g1 & 1 <= len g2 ;
len s2 > 1 & len s2 > 1 ;
z in dom ( f1 + f2 ) ;
1 in dom ( D2 | indx ( D2 , D1 , j ) ) ;
p `2 = 0 & p `2 = 0 ;
j2 <= width G & j2 <= width G ;
len PI > 1 + 1 ;
set n1 = n + 1 ;
|. q-35 .| = 1 ;
let s be SortSymbol of S ;
gcd ( i , i ) = i ;
X1 c= dom f & X1 c= dom f ;
h . x in h . a ;
let G be \cap of \cap \mathopen { w } ;
cluster m * n -> square ;
let k9 be Nat , k be Nat ;
i - 1 > m - 1 ;
R is transitive & R is transitive implies R is transitive
set F = <* u , w *> ;
p-2 c= P3 & p`2 c= P3 ;
I is_closed_on t , Q & I is_halting_on t , Q ;
assume [ S , x ] is thesis ;
i <= len ( f2 ^ g2 ) ;
p is FinSequence of X & q is FinSequence of X ;
1 + 1 in dom g ;
Sum R2 = n * r ;
cluster f . x -> complex-valued ;
x in dom f1 & y in dom f2 ;
assume [ X , p ] in C ;
BX c= XX & BX c= XX ;
n2 <= ( 2 to_power ( n2 + 1 ) ) ;
A /\ cP c= A ` ;
cluster x -valued -> constant for Function ;
let Q be Subset-Family of S , P be Subset of S ;
assume that n in dom g2 and m <= n ;
let a be Element of R ;
t `2 in dom ( e2 `2 ) ;
N . 1 in rng N ;
- z in A \/ B ;
let S be SigmaField of X , M be Element of S ;
i . y in rng i ;
REAL c= dom f & rng f c= dom f ;
f . x in rng f ;
mt <= ( r / 2 ) ;
s2 in r-5 & s2 in r-5 ;
let z , z be complex number ;
n <= ( N . m ) ;
LIN q , p , s ;
f . x = waybelow x /\ B ;
set L = [' S , T = S , S = T ;
let x be non positive ExtReal ;
let m be Element of M ;
f in union rng ( F1 | n ) ;
let K be add-associative right_zeroed right_complementable non empty doubleLoopStr , p be Polynomial of K ;
let i be Element of NAT , k be Nat ;
rng ( F * g ) c= Y
dom f c= dom x & rng f c= dom y ;
n1 < n1 + 1 & n1 <= len f ;
n1 < n1 + 1 & n1 <= len f ;
cluster 1. ( X , X ) -> \overline W ;
[ y2 , 2 ] `2 = z ;
let m be Element of NAT ;
let S be Subset of R ;
y in rng S[: S , T :] ;
b = sup ( dom f ) ;
x in Seg ( len q ) ;
reconsider X = D ( ) as set ;
[ a , c ] in E1 ;
assume that n in dom h2 and n + 1 in dom h2 ;
w + 1 = ( - a ) + 1 ;
j + 1 <= j + 1 + 1 ;
k2 + 1 <= k1 & k2 + 1 <= k2 ;
let i be Element of NAT ;
Support u = Support p & Support u = Support p ;
assume X is complete implies X is complete .| ;
assume f = g & p = q ;
n1 <= n1 + 1 & n1 <= n2 + 1 ;
let x be Element of REAL ;
assume x in rng ( s2 ^\ k ) ;
x0 < x0 + 1 & x0 < x0 + 1 ;
len ( L5 * L5 ) = W ;
P c= Seg ( len A ) ;
dom q = Seg n & dom q = Seg n ;
j <= width ( M @ ) ;
let seq1 be real-valued sequence of X ;
let k be Element of NAT ;
Integral ( M , P ) < +infty ;
let n be Element of NAT ;
assume z in being such := ) _ 0 ( 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 midpoint of b , c ;
let A , B be non empty set , f be Function of A , B ;
set d = dim ( p ) ;
let p be FinSequence of L , x be Element of L ;
Seg i = dom q & dom q = Seg i ;
let s be Element of E * ;
let B1 be Basis of x , B2 be Basis of y ;
L3 /\ L2 = {} & L3 /\ L2 = {} ;
L1 /\ L2 = {} & L1 /\ L2 = {} ;
assume downarrow x = downarrow y ;
assume b , c // b , c ;
LIN q , c , c & LIN q , c , b ;
x in rng f-129 & x in rng f-129 ;
set nn8 = n + j ;
let DD be non empty set , f be Function of DD , REAL ;
let K be add-associative right_zeroed right_complementable non empty addLoopStr , M be Matrix of K ;
assume f `2 = f & h `2 = h ;
R1 - R2 is total & R1 - R2 is total ;
k in NAT & 1 <= k implies k <= n
let a be Element of G ;
assume x0 in [. a , b .] ;
K1 ` is open & K1 ` is open ;
assume a , b are_maximal in C ;
let a , b be Element of S ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = Integral ( M , f ) ;
cluster -> ne] for dist of M ;
not u in { ag } ;
the carrier of f c= B \/ { x } ;
reconsider z = x as Vector of V ;
cluster the which is \rangle for non empty RelStr ;
r (#) H is C [ X ] ;
s . intloc 0 = 1 & s . intloc 0 = 2 ;
assume that x in C and y in C ;
let U0 be strict universal MSAlgebra over S , A be Subset of U0 ;
[ x , Bottom T ] is compact ;
i + 1 + k in dom p ;
F . i is stable Subset of M ;
r-35 in ( { y } ) & r-35 in ( { y } ) ;
let x , y be Element of X ;
let A , I be such such that A is such \bf \vert I .| ;
[ y , z ] in [: O , O :] ;
( card Macro i ) = 1 & card Macro i = 1 ;
rng Sgm A = A & rng Sgm A = A ;
q |- \! to All ( y , q ) ;
for n holds X [ n ] ;
x in { a } & x in d ;
for n holds P [ n ] ;
set p = |[ x , y , z ]| ;
LIN o , a , b & LIN o , a , b ;
p . 2 = Z / Y ;
D2 `2 = {} & D2 `2 = {} ;
n + 1 + 1 <= len g ;
a in [: CQC-WFF ( Al ) , D ( ) :] ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
let I be Ideal of L ;
set g = f1 + f2 , h = f2 + f1 , i = j + 1 ;
a <= max ( a , b ) ;
i-1 < len G + 1-1 ;
g . 1 = f . i1 & g . 1 = f . i2 ;
x ` , y ` in A2 ;
( f /* s ) . k < r ;
set v = VAL g ;
i -' k + 1 <= S ;
cluster associative associative non empty for multMagma ;
x in support ( support ( t ) ) ;
assume a in [: the carrier of G ( ) , the carrier of G ( ) :] ;
i `2 <= len ( y `2 ) ;
assume that p divides b1 + b2 and b1 divides b2 ;
M1 <= sup M1 & M1 <= sup M2 implies M1 <= M2
assume x in W-min ( X ) & y in W-min ( X ) ;
j in dom ( z | i ) ;
let x be Element of D ( ) ;
IC s4 = l1 & IC s4 = l1 ;
a = {} or a = { x } ;
set uG = Vertices G , uH = Vertices G ;
seq " is non-zero & seq " is non-zero implies seq " is non-zero
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
hcn c= h-14 & hcn c= h-14 ;
]. a , b .[ c= Z ;
X1 , X2 are_elements & X2 , X1 are_elements implies X1 , X2 are_elements
a in Cl ( union ( F \ G ) ) ;
set x1 = [ 0 , 0 ] ;
k + 1 -' 1 = k - 1 ;
cluster real-valued -> { INT -valued for Relation ;
ex v st C = v + W ;
let IT be non empty addLoopStr , x be Element of IT ;
assume V is Abelian add-associative right_zeroed right_complementable ;
XY \/ Y in \sigma ( L ) ;
reconsider x = x as Element of S ;
max ( a , b ) = a & max ( a , b ) = b ;
sup B is upper & sup B is upper Subset of T ;
let L be non empty reflexive RelStr , X be Subset of L ;
R is reflexive transitive & R is transitive implies R is transitive
E , g |= the_right_argument_of ( H , E ) ;
dom G `2 /. y = a ;
( 1 / 4 ) * 4 >= - r ;
G . p0 in rng G & G . p0 in rng G ;
let x be Element of FF , y be Element of FF ;
D [ P-6 , 0 ] & D [ Pd ] ;
z in dom ( id B ) & z in dom ( id B ) ;
y in the carrier of N & y in the carrier of N ;
g in the carrier of H & h in the carrier of G ;
rng fl c= [: NAT , NAT :] & rng fl c= [: NAT , NAT :] ;
j `2 + 1 in dom s1 & j `2 + 1 in dom s2 ;
let A , B be strict Subgroup of G ;
let C be non empty Subset of REAL ;
f . z1 in dom h & h . z2 in dom h ;
P . k1 in rng P & P . k1 in rng P ;
M = AM +* {} .= AM +* {} ;
let p be FinSequence of REAL , n be Nat ;
f . n1 in rng f & f . n1 in rng f ;
M . ( F . 0 ) in REAL ;
holds holds holds holds holds not |. a .| = b-a ;
assume the distance of V , Q is_<* v *> ;
let a be Element of ^ ( V ) ;
let s be Element of PL , x be Element of PL ;
let Px be non empty thesis , P be Subset of \rm be Subset of X ;
let n be Nat ;
the carrier of g c= B & the carrier of g c= the carrier of B ;
I = halt SCM R & I = halt SCM R ;
consider b being element such that b in B ;
set BK = BCS ( K , P , S ) ;
l <= ( -> id ( F . j ) ) ;
assume x in downarrow [ s , t ] ;
x `2 in uparrow t & x `2 in uparrow t ;
x in dom ( JumpParts T ) & x in dom ( JumpParts T ) ;
let h be Morphism of c , a ;
Y c= ( 1. ( K , len Y ) ) & Y c= ( the carrier of K ) ;
A2 \/ A3 c= L1 & A2 \/ A3 c= L2 ;
assume LIN o , a , b & LIN o , a , b ;
b , c // d1 , e2 & b , c // d , e2 ;
x1 , x2 in Y & x2 in Y ;
dom <* y *> = Seg 1 & dom <* y *> = Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar s .| ;
[ x , x `2 ] in [: X , X :] ;
for n be Nat holds 0 <= x . n
[' a , b '] = [. a , b .] ;
cluster -> non empty for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 in P & 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 & rng g2 c= dom W ;
P . ( [#] Sigma \ B ) <> 0 ;
a in field R & a = b implies a = b
let M be non empty Subset of V , v , u be Element of V ;
let I be Program of SCM+FSA , J be Program of SCM+FSA ;
assume x in rng ( ( id dom ( R * S ) ) * ( id dom S ) ) ;
let b be Element of the carrier of T ;
dist ( e , z ) - r-r > r-r ;
u1 + v1 in W2 & v1 in W2 & v2 in W1 ;
assume func support L misses rng G & L . x = L . x ;
let L be lower-bounded antisymmetric transitive non empty RelStr ;
assume [ x , y ] in a9 & [ y , x ] in C2 ;
dom ( A * e ) = NAT & rng ( A * e ) c= NAT ;
let a , b be Vertex of G ;
let x be Element of Bool M , y be Element of Bool M ;
0 <= Arg a & Arg a < 2 * PI ;
o9 , a9 // o9 , y & o9 , b9 // o9 , y ;
{ v } c= the carrier of l & Carrier ( l ) c= the carrier of A ;
let x be variable of A ;
assume x in dom ( uncurry f ) & y in dom ( uncurry f ) ;
rng F c= ( product f ) |^ X
assume D2 . k in rng D & D2 . k in rng D ;
f " . p1 = 0 & f " . p2 = 0 ;
set x = the Element of X , y = the Element of Y ;
dom Ser ( G ) = NAT & rng Ser ( G ) = NAT ;
let n be Element of NAT ;
assume LIN c , a , e1 & LIN c , a , e1 ;
cluster -> finite for FinSequence of NAT ;
reconsider d = c , e = d as Element of L1 ;
( v2 |-- I ) . X <= 1 ;
assume x in the carrier of support ( f ) ;
conv @ S c= conv A & conv @ S c= conv @ A ;
reconsider B = b as Element of the carrier of T ;
J , v |= P \lbrack UP \rbrack ;
redefine func J . i -> non empty TopSpace equals J . i ;
ex_sup_of Y1 \/ Y2 , T & ex_sup_of Y1 , T ;
W1 is_well field W1 & W2 is_carrier of W1 implies W1 + W2 is_carrier of W2
assume x in the carrier of R & y in the carrier of S ;
dom nM = Seg n & dom nM = Seg n ;
s4 misses s2 & s4 misses s4 ;
assume ( a 'imp' b ) . z = TRUE ;
assume that X is open and f = X --> d ;
assume [ a , y ] in non implies [ a , y ] in a ;
assume that card I c= J and Reloc ( J , card I ) c= K ;
Im ( lim seq ) = 0 & Im ( lim seq ) = 0 ;
( sin . x ) <> 0 & ( sin . x ) <> 0 ;
sin is_differentiable_on Z & cos is_differentiable_on Z implies cos is_differentiable_on Z & for x st x in Z holds cos . x <> 0
t3 . n = t3 . n & t3 . n = s . n ;
dom ( ( dom \rbrace ) \ { x } ) c= dom F ;
W1 . x = W2 . x & W2 . x = W1 . x ;
y in W .vertices() \/ W .vertices() \/ W .vertices() ;
( k + 1 ) <= len ( v | k ) ;
x * a \equiv y * a . ( mod m ) ;
proj2 .: S c= proj2 .: P & proj2 .: S c= proj2 .: P ;
h . p4 = g2 . I & h . p2 = g2 . I ;
( for G be Subset of ( TOP-REAL 2 ) | ( G * ( 1 , 1 ) ) ) holds G * ( 1 , 1 ) `1 = G * ( 1 , 1 ) `1
f . rr1 in rng f & f . rr2 in rng f ;
i + 1 + 1 - 1 <= len - 1 ;
rng F = rng ( F | ( rng F ) ) .= rng F ;
mode then of A is well unital associative non empty multMagma ;
[ x , y ] in [: A , { a } :] ;
x1 . o in L2 . o & x2 . o in { x } ;
the carrier of support ( m ) c= B & the carrier of support ( m ) c= A ;
not [ y , x ] in id X & [ y , x ] in id X ;
1 + p .. f <= i + len f ;
seq ^\ k1 is lower & seq ^\ k1 is lower implies seq is lower
len ( F | ( len F ) ) = len I ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be complex number , p be Point of TOP-REAL 2 ;
Comput ( P , s , n ) = s ;
k <= k + 1 & k + 1 <= len p ;
reconsider c = {} T , d = {} as Element of L ;
let Y be Element of be \mathclose { -1 } of
cluster directed-sups-preserving for Function of L , L ;
f . j1 in K . j1 & f . j2 in K . j1 ;
redefine func J => y -> total ( J , I ) -valued Function ;
K c= 2 -tuples_on the carrier of T implies K is finite
F . b1 = F . b2 & F . b2 = F . b1 ;
x1 = x or x1 = y or x1 = z ;
pred a <> {} means : : : a = 1 & a = 1 ;
assume that a c= b and b in a and c 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 , b ;
reconsider m = x as Element of Funcs ( V , C ) ;
let f be non trivial FinSequence of D ;
let FF2 be non empty \cal X ;
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 , a , b , c be Element of R ;
redefine func strict non empty <* a9 *> -> strict non empty be be be Subset of V ;
rng c `2 misses rng ( ee `2 ) \/ rng ( ee `2 ) ;
z is Element of gr { x } & z is Element of gr { x } ;
not b in dom ( a .--> p1 ) & not b in dom ( a .--> p1 ) ;
assume that k >= 2 and P [ k ] ;
Z c= dom ( cot * ( cot - cot ) ) & Z c= dom ( cot * ( cot - cot ) ) ;
the component of Q c= ( UBD A ) /\ ( ( L~ f ) ` ) ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( 1 / ( f . x ) ) ;
pred f = u means : : : a * f = a * u ;
for n holds P1 [ ( for i holds P1 [ i ] ;
{ x . O : x in L } <> {} ;
let x be Element of V . s ;
let a , b be Nat ;
assume that S = S2 and p = p2 and q = p1 ;
gcd ( n1 , n2 , k ) = 1 & gcd ( n1 , n2 , k ) = 1 ;
set oI = ( - 1 ) * ( - 1 ) , oI = ( - 1 ) * ( - 1 ) ;
seq . n < |. r1 .| & |. seq . n .| < r ;
assume that seq is increasing and r < 0 and seq is increasing ;
f . ( y1 , x1 ) <= a & f . ( y2 , x2 ) <= a ;
ex c being Nat st P [ c ] & c <= n ;
set g = { n to_power 1 : n in NAT } ;
k = a or k = b or k = c ;
a\leq aA , aB , aB , bB , bB ;
assume Y = { 1 } & s = <* 1 *> ;
IW1 . x = f . x .= 0 .= 0 ;
W3 .first() = W3 . 1 & W3 .last() = W3 . 1 ;
cluster trivial -> finite for \HM of G , finite and G is finite ;
reconsider u = u , v = v as Element of Bags X ;
A in B @ implies A , B are_that A , B are_that A , B are_that A , B are_that A , B are_that A , B are_that A , B are_that A , B are_that B ,
x in { [ 2 * n + 3 , k ] } ;
1 >= ( q `1 / |. q .| - cn ) / ( 1 - cn ) ;
f1 is__ _ X implies f2 is__ _ X & f1 is__ _ X
f `2 <= q `2 & q `2 <= ( f /. i ) `2 ;
h is_\HM { implies Cage ( C , n ) /. 1 in rng Cage ( C , n )
b `2 <= p `2 & p `2 <= p `2 or b `2 >= p `2 & p `1 <= b `1 ;
let f , g be Function of X , Y ;
S * ( k , k ) <> 0. K ( ) ;
x in dom ( max ( - f , g ) ) ;
p2 in NN & p2 in NN & p2 in NN ;
len ( the_left_argument_of H ) < len ( H ) & len ( H ) < len ( H ) ;
F [ A , F-14 ( A ) ] & F [ A , F-14 ( A ) ] ;
consider Z such that y in Z and Z in X ;
pred 1 in C means : : : A c= C & A c= C ;
assume that r1 <> 0 or r2 <> 0 and r1 < r2 ;
rng q1 c= rng C1 & rng q2 c= rng C2 ;
A1 , L , A3 , A3 , A2 , A3 , A3 , A2 be set ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in 4 & 4 in C implies b in C & a in C
then S is atomic implies P-2 [ S ] ;
Cl Int [#] T = [#] T & Cl Int [#] T = [#] T ;
f12 | A2 = f2 | A2 & f12 | A2 = f2 | A2 ;
0. M in the carrier of W & 0. M in the carrier of W ;
let v , v be Element of M ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ V & V c= Y \ Z ;
let X be Subset of [: S , T :] ;
consider H1 such that H = 'not' H1 and H1 in X ;
( 1_ G ) c= ( ( t * ( ( p - 1 ) / ( p - 1 ) ) ) * ( p - 1 ) ) ;
0 * a = 0. R .= a * 0 .= 0 ;
A |^ ( 2 , 2 ) = A ^^ A & A c= A ;
set vFinSequence = v4 /. n , v4 = v4 /. n ;
r = 0. ( REAL-NS n ) & ||. 0. ( REAL-NS n ) - 0. ( REAL-NS n ) .|| = 0. ( REAL-NS n ) ;
( f . p4 ) `1 >= 0 & ( f . p4 ) `2 >= 0 ;
len W = len ( W -as Subset of ( W -<= ( len W ) ) ) ;
f /* ( s * G ) is divergent_to-infty & f /* ( s * G ) is divergent_to-infty ;
consider l be Nat such that m = F . l ;
t16 / W8 does not destroy b1 & t8 does not destroy b1 ;
reconsider Y1 = X1 , Y2 = X2 as SubSpace of X ;
consider w such that w in F and not x in w ;
let a , b , c , d , e , f be Real ;
reconsider i = i , j = j as non zero Element of NAT ;
c . x >= id ( L . x ) & c . x >= id ( L . x ) ;
\sigma ( T ) \/ omega ( T ) is Basis of T ;
for x being element st x in X holds x in Y
cluster [ x1 , x2 ] -> pair for set ;
downarrow a /\ downarrow t is Ideal of T & downarrow a /\ downarrow t is Ideal of T ;
let X be set , N be non empty set , M be non empty set ;
rng f = \vert implies S = \ ( S , X )
let p be Element of B , x be Element of the connectives of S ;
max ( N1 , 2 ) >= N1 & max ( N1 , 2 ) >= N2 ;
0. X <= b |^ ( m * mm1 ) ;
assume that i in I and R1 . i = R and R2 . i = R ;
i = j1 & p1 = q1 & p2 = q2 & p1 = q2 ;
assume gRRRRRX in the carrier of g ;
let A1 , A2 be Point of S , A2 be Point of T ;
x in h " P /\ [#] T1 & x in h " P /\ [#] T2 ;
1 in Seg 2 & 1 in Seg 3 & 2 in Seg 3 ;
reconsider X-5 = X , Ximplies X' = Y as non empty Subset of T<* X *> ;
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the Source of G ) -valued ;
n1 <= i2 + len g2 & n2 <= len g2 implies ( f /. n1 ) = f /. ( n1 + 1 )
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume v in the carrier' of G2 & u in the carrier' of G2 ;
y = Re y + ( Im y ) * i ;
len ( ( - 1 ) |^ ( p -' 1 ) ) = 1 ;
x2 is differentiable on ]. a , b .[ & ( for x st x in ]. a , b .[ holds x - a <= x ) implies ( ex x st x in ]. a , b .[ )
rng M5 c= rng D2 & rng M5 c= rng D2 ;
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 -Path M ) . 2 = d & ( p -Path M ) . 3 = d ;
A \oplus ( B \ominus C ) = ( A \oplus B ) \ominus C
h \equiv gg . ( ( mod P ) * ( - 1 ) ) ;
reconsider i1 = i-1 , i2 = i-1 as Element of NAT ;
let v1 , v2 be VECTOR of V , v be VECTOR of V ;
for V being Subspace of V holds V is Subspace of [#] V
reconsider i-7 = i , im2 = j as Element of NAT ;
dom f c= [: C ( ) , D ( ) :] ;
x in ( the Element of B ) . n & ( the sequence of B ) . n in rng ( the sequence of B ) ;
len } in Seg ( len ( f2 | i ) ) & len ( f1 | i ) = len f1 ;
pp1 c= the topology of T & pp2 c= the topology of T ;
]. r , s .] c= [. r , s .] & ]. r , s .] c= [. r , s .] ;
let B2 be Basis of T2 , B be Basis of T2 ;
G * ( B * A ) = ( id o1 ) * A .= A ;
assume that p , u , u be <> 0. V and u , v , w be Element of V ;
[ z , z ] in union rng ( F | X ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S , C = S . $1 ;
LIN a1 , a3 , b1 & LIN a1 , b1 , b1 & LIN b1 , b2 , 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 & ( ( g2 ) . I ) `2 <= 1 ;
p in LSeg ( E . i , F . i ) ;
IB * ( i , j ) = 0. K ( ) ;
|. f . ( s . m ) - g .| < g1 ;
( ex q st q . x in rng ( q | k ) ) & ( q | k ) . x = q . x ;
Carrier Lxy misses Carrier Lxy & Carrier Lxy misses Carrier L\pi ;
consider c being element such that [ a , c ] in G ;
assume that Nreal = o\HM and ofor = ost holds ofor = ost ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= ( F . Cj ) /\ ( F . Cj ) ;
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [. f . j , f . j .] ;
pred 0 <= x & x <= 1 & x ^2 <= 1 ;
p `2 - q `2 <> 0. TOP-REAL 2 & p `2 - q `2 = 0. TOP-REAL 2 ;
redefine func aaS ( S , T ) -> Subset of S ;
let x be Element of [: S , T :] ;
j is one-to-one & F ( a , b ) is one-to-one ;
|. i .| <= - ( 2 to_power n ) & |. i .| <= - ( 2 to_power n ) ;
the carrier of I[01] = dom P & the carrier of I[01] = dom P ;
} * ( n + 1 ) ! > 0 * PI ;
S c= ( A1 /\ A2 ) /\ A3 & S c= ( A1 /\ A2 ) /\ A3 ;
a3 , a4 // b3 , b3 or a3 , a4 // b3 , b3 ;
then dom A <> {} & dom A <> {} & A is finite ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , Y & y is Vertex of G2 implies x = y
set v2 = ( v /. ( i + 1 ) ) ;
x = r . n .= ( r . n ) * ( r . n ) ;
f . s in the carrier of S2 & f . s in the carrier of S2 ;
dom g = the carrier of I[01] & rng g = the carrier of I[01] ;
p in Upper_Arc ( P ) /\ Lower_Arc ( P ) & q in Lower_Arc ( P ) /\ Lower_Arc ( P ) ;
dom d2 = [: A2 , A2 :] & dom d2 = [: A2 , A1 :] ;
0 < ( p / ||. z .|| + 1 ) / ( ||. z .|| + 1 ) ;
e . ( m3 + 1 ) <= e . ( m3 + 1 ) ;
B \ominus X \/ B \ominus Y c= B \ominus X /\ Y
-infty < Integral ( M , Im ( g | B ) ) ;
cluster O := F -> 1 -element 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 & Proj ( i , n ) * g is_differentiable_on X ;
let x , y , z be Point of X , p be Point of X ;
reconsider pp = p . x , pV = q . x as Subset of V ;
x in the carrier of Lin ( A ) & y in the carrier of Lin ( A ) ;
let I , J be parahalting Program of SCM+FSA , a be Int-Location ;
assume that - a is lower and b is lower and a is lower ;
Int Cl A c= Cl Int Cl A & Int Cl A c= Cl Int Cl A ;
assume for A being Subset of X holds Cl A = A & Cl A = Cl A ;
assume q in Ball ( |[ x , y ]| , r ) ;
p2 `2 <= p `2 & p `2 <= p2 `2 or p2 `2 >= p `2 & p `1 <= p2 `2 ;
Cl Q ` = [#] ( TL ) & Cl Q = [#] ( TL ) ;
set S = the carrier of T , T = the carrier of S ;
set I8 = for f |^ n , I8 = -> Element of NAT ;
len p -' n = len ( thesis - n ) .= len p - n ;
A is Permutation of Swap ( A , x , y ) ;
reconsider n6 = n8 - ( n + 1 ) as Element of NAT ;
1 <= j + 1 & j + 1 <= len ( s ) ;
let q\subseteq , qSet be Subset of M , q be Element of M ;
a9 in the carrier of S1 & b9 in the carrier of S2 & x = [ a9 , b9 ] ;
c1 /. n1 = c1 . n1 & c2 /. n1 = c2 . n1 ;
let f be FinSequence of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
y = ( ( f * SS ) . x ) & y = ( f * ( S . x ) ) . y ;
consider x being element such that x in be element such that x in \cal _ in A ;
assume r in ( ( dist ( o ) ) .: P ) ;
set i2 = ( n , h ) , i1 = ( n + 1 ) + 1 , i2 = ( n + 1 ) + 1 ;
h2 . ( j + 1 ) in rng h2 & h2 . ( j + 1 ) in rng h2 ;
Line ( M29 , k ) = M . i & Line ( M29 , k ) = M . i ;
reconsider m = ( x - 2 ) / ( x - 2 ) as Element of ExtREAL ;
let U1 , U2 be strict Subspace of U0 , A be Subset of U0 ;
set P = Line ( a , d ) ;
len p1 < len p2 + 1 & len p2 <= len p1 + 1 ;
let T1 , T2 be Scott Scott Scott Scott of L , x be Point of T1 ;
then x <= y & ( for x being Element of Y holds ( x in { y } ) ) ;
set M = n -\hbox { m } , N = n -\hbox { m } , S = n -\hbox { m } , T = n -tuples_on X ;
reconsider i = x1 , j = x2 , k = x3 as Nat ;
rng ( the_arity_of ( a9 ) ) c= dom H & rng ( the_arity_of ( a9 ) ) c= dom H ;
z1 " = z9 " & z2 " = z2 " & z1 = z2 " ;
x0 - r / 2 in L /\ dom f & f . x0 = r / 2 ;
then w is that rng w /\ ( rng w ) <> {} & S is non empty ;
set x-10 = x-9 ^ <* Z *> , xX2 = xZ ^ <* Z *> ;
len w1 in Seg ( len w1 + len w2 ) & len w1 in Seg ( len w1 + len w2 ) ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of PFuncs ( V , { k } ) ;
x . n = ( |. a . n .| ) * ( |. b . n .| ) ;
p `1 <= ( Gik `1 ) * ( 1 , 1 ) `1 & p `1 <= ( Gik `1 ) * ( 1 , 1 ) `1 ;
rng ( g | X ) c= L~ ( g | X ) \/ rng ( g | X ) ;
reconsider k = i-1 * ( l + j ) , j = i + j ;
for n be Nat holds F . n is \HM { -infty } ;
reconsider x-10 = x-7 , xM = xM , xM = xM as Vector of V ;
dom ( f | X ) = X /\ dom f & dom ( f | X ) = X ;
p , a // p , c & b , a // c , c ;
reconsider x1 = x , y1 = y , y2 = z as Element of REAL m ;
assume i in dom ( a * p ^ q ) & j in dom ( a * p ) ;
m . ag = p . ag & m . bg = q . bg ;
a / ( s . m - n ) / ( s . n - n ) <= 1 ;
S . ( n + k + 1 ) c= S . ( n + k ) ;
assume that B1 \/ C1 = B2 \/ C2 and B2 \/ C2 = B2 \/ C1 ;
X . i = { x1 , x2 } . i .= X . i ;
r2 in dom ( h1 + h2 ) & r1 in dom ( h1 + h2 ) ;
\mathclose { 0. R } = a & b-0 = b & b0 = b ;
FF is_closed_on t3 , Q8 & FF is_halting_on Q8 , Q7 ;
set T = non empty non empty such that for X holds X in k1 implies X is non empty ;
Int Cl ( Int Cl R ) c= Int Cl R & Int Cl ( Int Cl R ) c= Int Cl R ;
consider y being Element of L such that c . y = x ;
rng F<* x *> = { Fthere . x } .= { F ( x ) } ;
G-23 \ { c } c= B \/ S & not c in S ;
f\rm is Relation of [: X , X :] , X & f\rm is Relation of X , Y ;
set RQ = the Point of ( P + Q ) , RQ = the Point of ( P + Q ) ;
assume that n + 1 >= 1 and n + 1 <= len M ;
let k2 be Element of NAT , k1 , k2 be Element of NAT ;
reconsider pcSubset = u , pcSubset = v , ppSubset = w as Element of ( REAL n ) ;
g . x in dom f & x in dom g implies f . x = g . x
assume that 1 <= n and n + 1 <= len f1 and f1 /. n = f1 /. ( n + 1 ) ;
reconsider T = b * N as Element of ( G / ( N , 1 ) ) * ;
len Pt <= len Pt & len Pt <= len P-35 ;
x " in the carrier of A1 & y " in the carrier of A2 ;
[ i , j ] in Indices ( A @ ) & [ i , j ] in Indices ( A @ ) ;
for m be Nat holds Re ( F . m ) is simple function in S
f . x = a . i .= a1 . k .= a1 . k ;
let f be PartFunc of REAL i , REAL n , x be Element of REAL m ;
rng f = the carrier of ( Carrier A ) & f . x = f . x ;
assume s1 = sqrt ( 2 / ( p |^ 2 ) - 1 ) ;
pred a > 1 & b > 0 & a to_power b > 1 implies a to_power b > 1 ;
let A , B , C be Subset of Ik & A c= B & B c= C ;
reconsider X0 = X , Y0 = Y , Y1 = Z as RealNormSpace ;
let f be PartFunc of REAL , REAL , x be Element of REAL ;
r * ( v1 |-- I ) . X < r * 1 ;
assume that V is Subspace of X and X is Subspace of V and X is Subspace of V ;
let t-3 , t-4 be Relation of T , t-4 ;
Q [ e-14 \/ { v-14 } , f ( ) , f ( ) , f ( ) , f ( ) , f ( ) , f ( ) , f ( ) ) ] ;
g :- W-min L~ z = z implies ( W-min L~ z ) .. z < ( W-min L~ z ) .. z
|. |[ x , v ]| - |[ x , y ]| .| = v\rrangle - vx ;
- f . w = - ( L (#) w ) .= - ( L (#) w ) ;
z - y <= x iff z <= x + y & y <= z ;
( 7 / p1 ) to_power ( 1 / e ) > 0 ;
assume X is BCK-algebra of 0 , 0 , 0 , 0 ;
F . 1 = v1 & F . 2 = v2 & F . 3 = v2 ;
( f | X ) . x2 = f . x2 & ( f | X ) . x2 = f . x2 ;
( ( for x st x in dom tan holds sec . x = 1 / ( 1 + x ^2 ) ) & sec . x <> 0
i2 = ( f /. len f ) & ( f /. len f ) = ( f /. len f ) ;
X1 = X2 \/ ( X1 \ X2 ) & X2 = X1 \/ ( X2 \ X1 ) ;
[. a , b , 1_ G .] = 1_ G & a * b = 1_ G ;
let V , W be non empty VectSpStr over F_Complex , f be FinSequence of F_Complex ;
dom g2 = the carrier of I[01] & rng g2 = the carrier of I[01] & rng g2 = the carrier of I[01] ;
dom f2 = the carrier of I[01] & dom g2 = the carrier of I[01] & rng g2 = the carrier of I[01] ;
( proj2 | X ) .: X = proj2 .: X & proj2 .: X = proj2 .: X ;
f . ( x , y ) = h1 . ( x `1 , y `2 ) ;
x0 - r < a1 . n & a1 . n < x0 + r ;
|. ( f /* s ) . k - ( f /* s ) . k .| < r ;
len Line ( A , i ) = width A & len Line ( B , i ) = width B ;
SFinSequence / ( g , f ) = ( S . g ) / ( g , f ) ;
reconsider f = v + u as Function of X , the carrier of Y ;
( intloc 0 ) in dom Initialized ( p ) implies ( Initialized p ) . 0 = p . 0
i1 := i2 , i3 , i3 , I I does not destroy b1 , P & I does not destroy b2 , P ;
arccos r + arccos r = ( PI / 2 ) + 0 .= PI / 2 + 0 ;
for x st x in Z holds f2 is_differentiable_in x & f2 is_differentiable_in x & for x st x in Z holds f2 . x > 0
reconsider q2 = ( q - x ) / ( 1 + x ) , q1 = ( q - x ) / ( 1 + x ) as Element of REAL ;
( 0 qua Nat ) + 1 <= i + j1 & j + 1 <= j + 1 ;
assume f in the carrier of [' X , Omega Y '] & g in the carrier of X ;
F . a = H / ( x , y ) . a ;
( ( ex T being Element of T ) at ( C , u ) ) = TRUE implies ( T . u ) = TRUE
dist ( ( a * seq ) . n , h ) < r ;
1 in the carrier of [. 0 , 1 .] & 1 in dom ( f | [. 0 , 1 .] ) ;
p2 `1 - x1 > - g & p1 `2 - x1 > - g & p2 `2 - x1 < p2 `2 - x1 ;
|. r1 - TOP-REAL n .| = |. a1 .| * |. thesis .| ;
reconsider S-14 = 8 , S-14 = 8 as Element of ( Seg 8 ) -tuples_on the carrier of K ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b
D0W .= D0W .& D0W .= D0W .;
i1 = ma + n & i2 = K + n & i1 = K + n ;
f . a [= f . ( f . O1 "\/" a ) ;
pred f = v & g = u , h = v + u ;
I . n = Integral ( M , F . n ) + Integral ( M , F . n ) ;
chi ( T1 , T1 ) . s = 1 & chi ( T2 , T2 ) . s = 1 ;
a = VERUM ( A ) or a = VERUM ( A ) or a = VERUM ( A ) ;
reconsider k2 = s . b3 , k2 = s . b3 , k2 = s . b3 as Element of NAT ;
( Comput ( P , s , 4 ) ) . GBP = 0 ;
L~ M1 meets L~ ( R4 ^ R5 ) or L~ ( R4 ^ R5 ) meets L~ ( R4 ^ R5 ) ;
set h = the continuous Function of X , R , x be Point of X ;
set A = { L . ( k9 . n ) where k is Element of NAT : k <= n } ;
for H st H is atomic holds P7 [ H ] ;
set bA = S5 ^\ ( i + 1 ) , SA = S5 /^ ( i + 1 ) , bA = S5 ;
Hom ( a , b ) c= Hom ( a opp , b opp ) ;
( 1 / ( n + 1 ) ) < ( 1 / s ) " ;
l `1 = [ dom l , cod l ] & l `2 = [ dom l , cod l ] ;
y +* ( i , y /. i ) in dom g & y in dom g ;
let p be Element of CQC-WFF ( Al ) , P be Subset of CQC-WFF ( Al ) ;
X /\ X1 c= dom ( f1 - f2 ) & X1 c= dom ( f1 - f2 ) ;
p2 in rng ( f /^ p1 ) & p2 in rng ( f /^ p1 ) ;
1 <= indx ( D2 , D1 , j1 ) & indx ( D2 , D1 , j1 ) <= len D2 ;
assume x in ( ( K /\ K0 ) \/ ( K /\ K0 ) ) /\ K0 ;
- 1 <= ( ( f2 ) . O ) `2 & ( ( f2 ) . O ) `2 <= 1 ;
let f , g be Function of I[01] , TOP-REAL 2 , p be Point of TOP-REAL 2 ;
k1 -' k2 = k1 - k2 + 1 .= k1 - k2 + 1 ;
rng ( seq ^\ k ) c= ]. x0 , x0 + r .[ & ( seq ^\ k ) . n in dom ( f1 + f2 ) ;
g2 in ]. x0 - r , x0 + r .[ & g2 in ]. x0 - r , x0 + r .[ ;
sgn ( p `1 , K ) = - ( 1_ K ) .= - ( 1_ K ) ;
consider u being Nat such that b = p |^ y * u and u in A ;
ex A being Line of B st a = Sum A & A in W ;
Cl ( union Ha ) = union ( ( Z \/ { a } ) /\ ( Z \/ { b } ) ) ;
len t = len t1 + len t2 & len t1 = len t1 + len t2 ;
v-29 = v + w |-- v + AA & vd = v + AA ;
v ( ) <> DataLoc ( t0 . GBP , 3 ) & v ( ) . intpos ( 0 + 3 ) = s ( ) . intpos ( 0 + 3 ) ;
g . s = sup ( d " { s } ) & g . s = sup ( d " { s } ) ;
( \dot y ) . s = s . ( \dot y ) & ( \dot y ) . s = s ;
{ s : s < t } in REAL implies t = {} or t = {}
s ` \ s = s ` \ 0. X & s ` \ s = 0. X ;
defpred P [ Nat ] means B + $1 in A & not ( ex k st k in A & not ( k in B ) ) ;
( 339 + 1 ) ! = 3339 ! * ( 339 + 1 ) ;
( for A st succ A in T holds A c= ( T . A ) ) implies A = {}
reconsider y = y , z = z as Element of ( len y ) -tuples_on ( the carrier of K ) ;
consider i2 being Integer such that y0 = p * i2 and i2 in dom f ;
reconsider p = Y | Seg k , q = Y | Seg k as FinSequence of NAT ;
set f = ( S , U ) \mathop { \it Boolean } , g = ( S , U ) \mathop { \it Boolean } , h = ( S , U ) \mathop { \it true } , f = ( S , U ) \mathop { \it true } , f = ( S
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 , r be Real ;
( ( SAT M ) . [ n + i , 'not' A ] ) <> 1 ;
ex r being Real st x = r & a <= r & r <= b ;
let R1 , R2 be Element of REAL n , x be Element of REAL n , y be Element of REAL n ;
reconsider l = ( 0. ( GF ) ) , r = ( 0. ( V ) ) 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 be being being being being being being being being being being being being Element of Y holds ( a 'or' b ) 'or' c = 'not' ( a 'or' b )
||. xy0 - ( g - ( g - ( g - f ) ) ) .|| < r2 ;
b9 , a9 // b9 , c9 & b9 , a9 // c9 , a9 & b9 , a9 // c9 , a9 ;
1 <= k2 -' k1 & k1 + 1 = k2 & k2 + 1 = k2 + 1 ;
( p `2 / |. p .| - sn ) / ( 1 + sn ) >= 0 ;
( q `2 / |. q .| - sn ) / ( 1 + sn ) < 0 ;
E-max C in cell ( Rm , 1 , 1 ) & ( E-max C in L~ Rm ) implies ( E-max C in L~ Rm )
consider e being Element of NAT such that a = 2 * e + 1 and e in dom f ;
Re ( ( lim F ) | D ) = Re ( ( lim G ) | D ) ;
LIN b , a , c or LIN b , c , a & LIN b , a , c ;
p `1 , a `2 // a `1 , b `2 or p `2 , a `2 // b `1 , a `2 ;
g . n = a * Sum ( f | n ) .= f . n ;
consider f being Subset of X such that e = f and f is .= f ;
F | ( N2 , S ) = CircleMap * ( F | ( N2 , S ) ) ;
q in LSeg ( q , v ) \/ LSeg ( v , p ) \/ LSeg ( v , p ) ;
Ball ( m , r0 ) c= Ball ( m , s ) & Ball ( x , r0 ) c= Ball ( x , r ) ;
the carrier of (0). V = { 0. V } & the carrier of (0). V = { 0. V } ;
rng ( cos | [. - 1 , 1 .] ) = [. - 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 / ( 1 + e ) ;
set g = O --> 1 ;
reconsider t2 = t11 , t2 = t22 as 0 0 string of S2 , D ;
reconsider x-29 = seq , xA = ( seq ^\ n ) as sequence of REAL n ;
assume that C meets L~ Cage ( C , n ) and L~ Cage ( C , n ) meets L~ go ;
- ( ( - 1 ) (#) ( 1 / ( n + 1 ) ) ) < F . n - ( - 1 ) * ( 1 / ( n + 1 ) ) ;
set d1 = non empty \bf dist ( x1 , z1 ) , d2 = dist ( x2 , z2 ) , d2 = dist ( x1 , z2 ) , d2 = dist ( x2 , y2 ) , d2 = dist ( x2 , y2 ) , d1 = dist ( x2 , y2 ) , d2 = dist ( y2
2 |^ ( 100 -' 1 ) = ( 2 |^ 100 ) - 1 .= 1 ;
dom vb2 = Seg ( len db2 ) .= dom ( db1 ^ <* d *> ) ;
set x1 = - k2 + |. k2 .| + 4 , x2 = - k2 + 4 ;
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 ;
for A being Subset of X holds c . ( c . A ) = c . A
the carrier of LT + L2 c= I2 & the carrier of LT c= the carrier of LT ;
'not' Ex ( x , p ) => All ( x , p ) is valid ;
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the normal normal \hbox { {} } ;
Z c= dom ( ( - 1 / ( n + 1 ) ) (#) ( f1 - f2 ) ) ;
|. 0. TOP-REAL 2 - ( q `2 / |. q .| - sn ) .| < r ;
ConsecutiveSet2 ( B , succ B ) c= ConsecutiveSet2 ( A , succ ( d , B ) ) ;
E = dom L8 & L8 c= E & L8 c= dom L8 ;
C / ( A + B ) = C / B * C / A ;
the carrier of W2 c= the carrier of V & the carrier of W1 c= the carrier of V ;
I . IC s2 = P . IC s2 .= ( I . IC s2 ) .= ( I . IC s2 ) ;
pred x > 0 means : : : 1 / x = x / ( - 1 ) ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) /\ LSeg ( g , i ) ;
consider p being Point of T such that C = [. p , R .] and p in A ;
b , c are_connected & - C , - C + - C + D + D + E + F + G + D + E + F + G + D + E + F + J + M + D + E + F + J + M + D + F + J + M + D +
assume f = id ( the carrier of O1 ) & f in the carrier of O1 & g in the carrier of O ;
consider v such that v <> 0. V and f . v = L * v ;
let l be Linear_Combination of {} ( ( the carrier of V ) --> { v } ) ;
reconsider g = f " as Function of U2 , U1 , f " , f " ;
A1 in the Points of G_ ( k , X ) & A2 in the Points of G_ ( k , X ) ;
|. - x .| = - ( - x ) .= - x .= - x .= - x ;
set S = ) , T = ) , S = ) ;
Fib ( n ) * ( 5 * Fib ( n ) - 1 ) >= 4 * be Element of NAT ;
vM /. ( k + 1 ) = vM . ( k + 1 ) ;
0 mod i = - ( i * ( 0 qua Nat ) ) .= - ( i * 0 qua Nat ) ;
Indices M1 = [: Seg n , Seg n :] & dom M1 = Seg n & dom M2 = Seg n ;
Line ( S<> j , j ) = S<> Line ( Sj , i ) ;
h . ( x1 , y1 ) = [ y1 , x1 ] & h . ( y1 , y2 ) = [ y2 , y1 ] ;
|. f .| - Re ( |. f .| * h ) is nonnegative ;
assume x = ( a1 ^ <* x1 *> ) ^ b1 & y = ( a1 ^ <* x1 *> ) ^ b1 ;
ME is_closed_on IExec ( I , P , s ) , P & ME is_halting_on s , P ;
DataLoc ( t3 . a , 4 ) = intpos ( 0 + 4 ) .= intpos ( 0 + 4 ) ;
x + y < - x + y & |. x .| = - x & |. y .| = - y ;
LIN c , q , b & LIN c , q , c & LIN c , q , c & LIN c , q , b & LIN c , q , c & LIN a , b , c & LIN a , b , c & LIN a , b , c & a , b // c , d
f\rangle . ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + z1 ) .= x1 + ( y1 + z1 ) ;
flet a1 . a = f{ a } & v in InputVertices S & [ v , w ] in InputVertices S ;
p `1 <= ( E-max C ) `1 & ( E-max C ) `1 <= ( E-max C ) `1 & ( E-max C ) `2 <= ( E-max C ) `1 ;
set R8 = Cage ( C , n ) , E7 = Cage ( C , n ) , E8 = Cage ( C , n ) , R7 = Gauge ( C , n ) , R8 = Gauge ( C , n ) , R7 = Gauge ( C , n ) , R8 = Gauge (
p `1 >= ( E-max C ) `1 & p `1 >= ( E-max C ) `1 or p `1 >= ( E-max C ) `1 ;
consider p such that p = p-20 and s1 < p & p < s2 ;
|. ( f /* ( s * F ) ) . l - ( f /* ( s * F ) ) . l .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len Line ( N , k + 1 + 1 ) = width N & width ( N @ ) = width N ;
f1 /* s1 is convergent & f2 /* s1 is convergent & lim ( f1 /* s1 ) = f1 /* s1 ;
f . x1 = x1 & f . y1 = y1 & f . y2 = y2 & f . y2 = y2 ;
len f <= len f + 1 & len f + 1 <> 0 & len f + 1 <> 0 ;
dom ( Proj ( i , n ) * s ) = REAL m & rng ( Proj ( i , n ) * s ) c= REAL n ;
n = k * ( 2 * t ) + ( n mod ( 2 * k ) ) ;
dom B = 2 -tuples_on ( the carrier of V ) \ { {} } & rng B c= 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 in dom ( - 1 / 2 ) ;
for L being complete LATTICE holds <* <* \mathclose { \mathbb C } , L *> , L are_isomorphic implies L is isomorphic
[ gi , gj ] in Ii \ Ij implies [ gi , gj ] in Ii \ Ij
set S2 = 1GateCircStr ( x , y , c ) , S2 = 1GateCircStr ( y , c , d ) ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and for r st r < x0 ex g st r < g & g < x0 & g in dom ( f2 * f1 ) ;
reconsider y = ( a ` ) * ( ( F ` ) * ( G ` ) ) , z = ( F ` ) * ( G ` ) as Element of L ;
dom s = { 1 , 2 , 3 } & s . 1 = d1 & s . 2 = d2 ;
( min ( g , ( 1 - 1 ) ) (#) f ) . c <= h . c ;
set G3 = the \HM { of G , v = the Vertex of G , v = the Vertex of G , X = the set of G , Y = the set of X ;
reconsider g = f as PartFunc of REAL n , REAL-NS n , REAL-NS n , REAL-NS n ;
|. s1 . m / p .| < d / ( p |^ m ) & |. s1 . m .| < d ;
for x being element st x in ( for u being element st u in ( k * t ) holds u in k ) implies x in k
P = the carrier of ( TOP-REAL n ) | Px0 & P = the carrier of ( TOP-REAL n ) | Px0 ;
assume that p00 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) and p2 in LSeg ( p1 , p2 ) /\ LSeg ( p2 , p2 ) ;
( 0. X \ x ) to_power ( m * ( k + 1 ) ) = 0. X ;
let g be Element of Hom ( cod f , \square ) , f be Morphism of dom f ;
2 * a * b + ( 2 * c * d ) <= 2 * C1 * C2 + ( 2 * c * d ) ;
let f , g , h be Point of the complex normed space of X , Y , h be Function of X , Y ;
set h = Hom ( a , g (*) f ) ;
then idseq ( n ) | Seg m = idseq ( m ) & m <= n implies m <= n ;
H * ( g " * a ) in the right * ( g " * a ) ;
x in dom ( ( cos * sin ) `| Z ) & ( ( cos * sin ) `| Z ) . x = ( cos . x ) ^2 ;
cell ( G , i1 , j2 -' 1 ) misses C & cell ( G , i2 , j2 ) misses C ;
LE q2 , p4 , P , p1 , p2 & LE q2 , p , P , p1 , p2 implies LE q2 , p , P , p1 , p2
pred B is component of A means : : for B being Subset of T holds B c= BDD A ;
deffunc D ( set , set ) = union rng $2 & $2 = union rng $2 ;
n + - n < len ( pl + - n ) + - n ;
pred a <> 0. K means : : : for M st M in rng ( a * M ) holds the_rank_of M = the_rank_of M ;
consider j such that j in dom sthesis and I = len sPI + j and len B = len PI + j ;
consider x1 such that z in x1 and x1 in P8 and not x1 in P8 ;
for n ex r being Element of REAL st X [ n , r ] & r <= n ;
set Ck1 = Comput ( P2 , s2 , i + 1 ) , Ck1 = Comput ( P2 , s2 , i + 1 ) , Ck1 = Comput ( P2 , s2 , i + 1 ) , Ck1 = Comput ( P2 , s2 , i + 1 ) , Ck1 = Comput ( P2 , s2
set cv = 3 / 4 , cv = 3 / 4 , cv = 3 / 4 , cv = 5 / 4 , cv = 5 ;
conv @ W c= union ( F .: ( E " W ) ) & conv @ W c= union ( F .: ( E " W ) ) ;
1 in [. - 1 , 1 .] /\ dom ( arccot * ( arccot ) ) & 1 in dom ( arccot * ( arccot ) ) ;
r3 <= s0 + ( r0 - ( |. v2 - v1 .| ) / ( 2 * ( 1 + ( 1 + 1 ) ) ) ) ;
dom ( f (#) f4 ) = dom f /\ dom f4 .= dom ( f (#) f4 ) .= dom f ;
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 , gq = gq , gp = gp , gq = gq as Point of TOP-REAL n ;
( T * h . s ) . x = T . ( h . s . x ) ;
I . ( L . ( J . x ) ) = ( I * L ) . ( J . x ) ;
y in dom ( |^ ( ( Frege A ) * ( commute ( A . o ) ) ) ) ;
for I being non degenerated commutative Ring holds the carrier of I is commutative commutative commutative commutative non empty doubleLoopStr ;
set s2 = s +* Initialize ( ( intloc 0 ) .--> 1 ) , P2 = P +* I , P3 = P +* I , s4 = Comput ( P3 , s3 , 1 ) , P4 = P3 ;
P1 /. IC s1 = P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 ;
lim S1 in the carrier of [. a , b .] & ( for n holds S1 . n = a ) implies S1 is convergent & lim ( S1 + S2 ) = a
v . ( lpp . i ) = ( v *' lU ) . i .= ( v *' lU ) . i ;
consider n being element such that n in NAT and x = ( sn succ n ) . n ;
consider x being Element of c such that F1 . x <> F2 ( x ) and F1 . x <> F2 ( x ) ;
card Funcs ( X , 0 , x1 , x2 ) = { E } & card ( X \ { x1 } ) = { E } ;
j + ( 2 * k9 ) + m1 > j + ( 2 * k9 ) + ( 2 * k9 ) ;
{ s , t } on A3 & { s , t } on B2 & { s , t } on B2 ;
n1 > len crossover ( p2 , p1 , n1 ) & n2 >= len crossover ( p2 , p1 , n1 ) ;
mg . ( HT ( mg , T ) ) = 0. L & mg . ( HT ( mg , T ) ) = 0. L ;
then for H1 , H2 be \emptyset st H , H1 / ( 2 , m ) / ( 2 , m ) / ( 2 , m ) are_relative_prime holds H1 , H2 / ( 2 , m ) / ( 2 , m ) / ( 2 , m ) are_relative_prime ;
( N-min L~ ( f | ( L~ f ) ) ) .. ( f /^ ( len f -' 1 ) ) > 1 ;
]. s , 1 .[ = ]. s , 2 .] /\ [. 0 , 1 .] .= ]. 0 , 1 .] ;
x1 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) ) & x2 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) ) ;
let f1 , f2 be continuous PartFunc of REAL , the carrier of S , x be Point of S ;
DigA ( t-23 , z9 ) is Element of k ( ) -tuples_on ( k ( ) ) ;
I \leq d2k1 & I is k2 & I is k2 & I is k2 ;
[: u , { u } :] = { [ a , uu ] } & [: u , { u } :] = [: { a } , { u } :] ;
( w | p ) | ( p | ( w | w ) ) = p ;
consider u2 such that u2 in W2 and x = v + u2 and u1 in W2 and u2 in W2 ;
for y st y in rng F ex n st y = a |^ n & a <= n ;
dom ( ( g * ( ( f } \dot \to C ) | K ) ) = K & dom ( ( f * ( f /\ K ) ) | K ) = K ;
ex x being element st x in ( ( ( ( ( U0 ) \/ A ) ) . s ) ) & x in ( ( ( ( ( ( ( ( ( ( ( ( ( ( U0 ) . s ) ) . s ) ) . s ) ) ) ) ) ) . s ) ;
ex x being element st x in ( ( for O being element st O in ( the Sorts of A ) ) . s holds x in ( the Sorts of 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 ) /\ ( the carrier of X2 ) <> {} ;
L1 /\ LSeg ( p00 , p2 ) c= { p10 } & L1 /\ LSeg ( p00 , p2 ) c= { p00 } ;
( b + b- 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 GX such that z = y and P [ z ] and z in A ;
( the sequence of ( ( the carrier of ( ( the carrier of X ) ) * ) ) ) . ( b ) <= e ;
len ( w ^ w2 ) + 1 = len w + 2 + 1 .= len w + 1 ;
assume q in the carrier of ( TOP-REAL 2 ) | K1 & q in the carrier of ( TOP-REAL 2 ) | K1 ;
f | E-4 ` = g | E-4 ` & f | Ed = g | Ed ` ;
reconsider i1 = x1 , i2 = x2 , j1 = x3 , j2 = x4 , i1 = x4 , i2 = x4 , j1 = i2 , j2 = x4 , j2 = 7 , i1 = 8 , i2 = 7 , j1 = 8 , j2 = 7 , i1 = 8 , i2 = 7 , i2 = 8 , j2 = 7 , 7 = 8 , 8 =
( a * A ) @ = ( a * ( A * B ) ) @ ;
assume ex n0 being Element of NAT st f to_power n0 is NAT & f to_power n0 is < r ;
Seg len ( ( ( the support of f2 ) * ( len ( f2 ) ) ) ) = dom ( ( the support of f2 ) * ( len ( f2 ) ) ) ;
( Complement ( A ^\ n ) ) . m c= ( ( Complement ( A ^\ n ) ) ) . m ;
f1 . p = p9 & g1 . p = d & g2 . p = b & g2 . p = d & g2 . p = c ;
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 & rng ( G ) c= dom ( F ) ;
assume for x , y being set st x in Y & y in Y holds x /\ y in Y ;
assume that W1 is Subspace of W3 and W2 is Subspace of W3 and W1 is Subspace of W3 and W2 is Subspace of W3 ;
||. t-15 . x .|| = lim ||. ( xT ) . x .|| .= ||. ( x - x0 ) * ( x - x0 ) .|| ;
assume that i in dom D and f | A is lower bounded and g | A is lower bounded and g | A is lower bounded ;
( p `2 ) ^2 - 1 <= ( - 1 ) ^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 ) , N7 = ( N-min L~ Cage ( C , n ) ) .. Cage ( C , n ) ;
for T being non empty TopSpace holds T is countable countable implies the TopStruct of T is countable countable
width B |-> 0. K = Line ( B , i ) .= B @ .= B @ .= B @ ;
pred a <> 0 , b means : : : ( A \+\ B ) c= ( A Y. a ) \+\ ( B f2 ) ;
then f is_is_is_\cal 2 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 c > 1 and a > 0 and c > 0 ;
w1 , w2 in Lin { w1 , w2 } & w2 in Lin { w1 , w2 } ;
p2 /. IC s2 = p2 . IC s2 .= ( p2 . IC s2 ) .= ( p2 . IC s2 ) .= ( p2 . IC s2 ) ;
ind ( T-10 | b ) = ind b .= ind B .= ind B .= ind B ;
[ a , A ] in the \cdot of ( \cap \cap \cap { a } ) & [ a , A ] in the \cdot of ( the \leq of A ) ;
m in ( the Arrows of C ) . ( o1 , o2 ) & ( the Arrows of C ) . ( o1 , o2 ) = ( the Arrows of C ) . ( o2 , o2 ) ;
( a 'imp' CompF ( PA , G ) ) . z = FALSE & ( a 'imp' CompF ( PA , G ) ) . z = FALSE ;
reconsider phi = phi , phi = phi , phi = phi , phi = phi , phi = phi , phi = I , phi = I , N = I , M = I , N = I , M = I , N = I , N = I , M = I , N = I , N = I , M = I , N = I , N =
len s1 - ( len s2 - 1 ) + 1 > 0 + 1 - 1 ;
delta ( D ) * ( f . ( upper_bound A ) - f . ( lower_bound A ) ) < r ;
[ f21 , f22 ] in the carrier' of [: A , B :] & [ f22 , f22 ] in the carrier' of [: A , B :] ;
the carrier of ( TOP-REAL 2 ) | K1 = K1 & the carrier of ( ( TOP-REAL 2 ) | K1 ) = K1 ;
consider z being element such that z in dom g2 and p = g2 . z and q = g2 . z ;
[#] V1 = { 0. V1 } .= the carrier of ( (0). V1 ) /\ ( [#] V1 ) .= { 0. V1 } ;
consider P2 be FinSequence such that rng P2 = M and P2 is one-to-one and for k be Nat st k in dom P2 holds P2 . k = P2 . k ;
assume that x1 in dom ( f | X ) and ||. x1 - x0 .|| < s and |. x1 - x0 .| < s ;
h1 = f ^ ( <* p3 *> ^ <* p *> ) .= h ^ <* p *> .= h ;
c / ( |[ b , c ]| ) = c .= c / ( |[ a , c ]| ) .= c / ( |[ a , c ]| ) ;
reconsider t1 = p1 , t2 = p2 , t2 = p3 , t1 = p2 as Term of C , V , s ;
( 1 / 2 ) in the carrier of [. 1 / 2 , 1 .] & ( 1 / 2 ) * ( 1 / 2 ) in the carrier of [. 0 , 1 .] ;
ex W being Subset of X st p in W & W is open & h .: W c= V ;
( h . p1 ) `2 = C * ( p1 `2 ) + D .= C * ( p1 `2 ) + D ;
R . b - b .| = 2 * ( - b ) .= 2 * PI * b .= b ;
consider 1 such that B = ( - 1 ) * ] + ( - 1 ) * A and 0 <= 1 ;
dom g = dom ( ( the Sorts of A ) * ( a , f ) ) .= dom ( a * f ) ;
[ P . ( U7 ) , P . ( l7 ) ] in => ( ( P => 'not' ( P => ( P => ( Q => ( P => ( Q => ( Q => ( P => q ) ) ) ) ) ) ) ) ;
set s2 = Initialize s , P2 = P +* I ;
reconsider M = mid ( z , i2 , i1 ) , N = len z as Element of ( TOP-REAL 2 ) ;
y in product ( ( Carrier J ) +* ( V , { 1 } ) ) & y in product ( ( Carrier J ) +* ( V , { 1 } ) ) ;
1 / ( |[ 0 , 1 ]| ) = 1 & 0 / ( |[ 0 , 1 ]| ) = 0 ;
assume x in the left of g or x in the left of g or x in the right of g ;
consider M be strict Subgroup of AF such that a = M and T is strict Subspace of M and M is strict Subgroup of A ;
for x st x in Z holds ( ( ( #Z n ) * f ) `| Z ) . x <> 0
len W1 + len W2 + m = 1 + len W3 + m .= len W3 + len W3 + m .= len W3 + len W3 ;
reconsider h1 = ( vseq . n ) - t-16 as Lipschitzian LinearOperator of X , Y ;
( i mod len ( p + q ) ) + 1 in dom ( p + q ) ;
assume that s2 is_or F in the { of s2 : F in the { H : not contradiction } and s2 in the { H } } ;
( ( ( ( ( ( ( x - y ) - 1 ) ) * ( x - y ) ) ) * ( x - y ) ) ) * ( x - y ) = gcd ( x , y , 3 ) ;
for u being element st u in Bags n holds ( p `2 + 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 , W1 = tree ( q ) , W2 = tree ( p ) ;
x in { X where X is Ideal of L : for X being Ideal of L st X in F holds X is Subset of L } ;
the carrier of W1 /\ W2 c= the carrier of W1 & the carrier of W1 c= the carrier of W2 & the carrier of W1 c= the carrier of W2 ;
( 1 / a ) * ( a + b ) = ( 1 / a ) * ( a + b ) ;
( ( X --> f ) ) . x = ( X --> dom f ) . x .= ( X --> f ) . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) , y = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) ;
p => ( q => r ) => ( p => q => ( p => r ) ) in TAUT ( A ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) , G * ( i1 , k ) = LSeg ( G * ( i1 , k ) , G * ( i1 , k ) ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) , G * ( i1 , k ) = LSeg ( G * ( i1 , k ) , G * ( i1 , k ) ) ;
- 1 + 1 <= ( ( ( ( 2 |^ n ) -' m ) + 1 ) + 1 ) + 1 ;
( reproj ( 1 , z0 ) ) . x in dom ( f1 (#) f2 ) & ( reproj ( 1 , z0 ) ) . x in dom ( f1 (#) f2 ) ;
assume that b1 . r = { c1 } and b2 . r = { c2 } and b1 . r = { c2 } ;
ex P st a1 on P & a2 on P & b on P & c on P & d on P & c on P & d on P & d on P & c on P & d on P & d on P & c on P & d on P & d on P & d on P & c on P & d on P & d on P
reconsider gf = g `2 * f `1 , hg = h `2 * g `1 , hh = h `2 * g `2 as strict Element of X ;
consider v1 being Element of T such that Q = ( downarrow v1 ) ` and v1 in A and v1 in B ;
n in { i where i is Nat : i < n0 + 1 & i < n + 1 } ;
( F * ( i , j ) ) `2 >= ( F * ( m , k ) ) `2 & ( F * ( m , k ) ) `2 >= ( F * ( m , k ) ) `2 ;
assume K1 = { p : p `1 / |. p .| >= cn & p <> 0. TOP-REAL 2 } ;
ConsecutiveSet ( A , succ O1 ) = ( ( A , O1 ) set ) set , f = ( A , O1 ) set , g = ( A , O2 ) --> ( A , O2 ) ;
set Ik1 = Macro ( a , intloc 0 ) , Ik1 = AddTo ( a , intloc 0 ) , Ik1 = SubFrom ( a , intloc 0 ) , Ik1 = SubFrom ( a , intloc 0 ) , Ik1 = ;
for i being Nat st 1 < i & i < len z holds z /. i <> z /. 1 ;
X c= ( the carrier of L1 ) /\ ( the carrier of L2 ) & the carrier of L1 c= the carrier of L2 implies X c= the carrier of L1
consider x9 being Element of GF ( p ) such that x9 |^ 2 = a and not x9 in { x } ;
reconsider ee = ee , fe = fe , fe = fe , fe = fe as Element of D ;
ex O being set st O in S & C1 c= O & M . O = 0. ;
consider n being Nat such that for m being Nat st n <= m holds S . m in U1 and S . n in U1 ;
f * g * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) . x ) ;
defpred P [ Nat ] means A + succ $1 = succ A & for k st k in dom A holds P [ k , A . k ] ;
the left dom of - g = the left of g & the left of - g = the left of g ;
reconsider p\mathopen = x , p\mathopen = y , p\mathopen = z , p\mathopen = w , p\mathopen = y as Point of TOP-REAL 2 ;
consider g3 such that g3 = y and x <= g3 and g3 <= x0 and x0 <= y and y <= x0 ;
for n being Element of NAT ex r being Element of REAL st X [ n , r ] & r <= n ;
len ( x2 ^ y2 ) = len x2 + len y2 .= len ( x2 ^ y2 ) + len y2 .= len ( x2 ^ y2 ) + len y2 ;
for x being element st x in X holds x in the set of the set of positive iff x in the set of ( n -tuples_on NAT )
LSeg ( p10 , p2 ) /\ LSeg ( p1 , p2 ) = {} & LSeg ( p1 , p2 ) /\ LSeg ( p2 , p2 ) = {} ;
func such such that ( for X being set holds X in implies X in implies X in [: dom [: h , f :] ) & X is finite ;
len ( -> ( non empty set ) , CC be Subset of ( len CC ) -tuples_on ( the carrier of G ) ) ;
pred K is with_a , a , b be Element of K , i , j be Nat ;
consider o being OperSymbol of S such that t `1 . {} = [ o , the carrier of S ] and o in rng t ;
for x st x in X ex y st x c= y & y in X & y is NAT & f . x = f . y
IC Comput ( P-6 , sd , k ) in dom ( Pan ) & IC Comput ( Pd , sd , k ) in dom ( Pan ) ;
pred q < s & r < s & ]. r , s .] c= ]. p , q .] implies p <= r & s <= q ;
consider c being Element of Class ( f , c ) such that Y = ( F . c ) `1 and c in X ;
func the ResultSort of S2 -> Function of the carrier' of S2 , the carrier' of S2 means : + : it = id the carrier' of S2 & for x being Element of 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 ( ( ( #Z n ) * ( arccot ) ) `| Z ) & x in dom ( ( #Z n ) * ( arccot ) ) ;
r-7 in Int cell ( GoB f , i , GoB f ) \ L~ f & ri2 in ( L~ f ) \ L~ f ;
q `2 >= ( Cage ( C , n ) * ( i + 1 , 1 ) ) `2 & ( Cage ( C , n ) * ( i + 1 , 1 ) ) `2 >= ( Cage ( C , n ) * ( i + 1 , 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 ;
for n ex x st x in N & x in N1 & h . n = - ( x0 - x ) / ( n + 1 )
set s0 = ( \mathop { a , I , p , s ) . i , s0 = ( \mathop { a , I , p , s ) . i , s0 = ( \mathop { a , I , p , s ) . i , s0 = ( \mathop { a , I , p , s ) . i , s0 = ( \mathop { a , I , p , s )
p ( ) . k = 1 or p ( ) . 0 = - 1 or p ( ) . 0 = 1 & p ( ) . 1 = - 1 ;
u + Sum L-18 in ( U \ { u } ) \/ { u + Sum L-18 } ;
consider xset being set such that x in xT and xT in V1 and xT c= V1 and xT c= V1 and xT c= V1 ;
( p ^ ( q | k ) ) . m = ( q | k ) . ( ( len p ) - len p ) ;
g + h = gg + hh & for X holds Nat holds Nat ( g + h , X ) = g + h
L1 is distributive & L2 is distributive implies L1 /\ L2 is distributive & L1 /\ L2 is distributive iff L1 /\ L2 is distributive
pred x in rng f & y in rng ( f | x ) & f . x = f . y implies x = f . y ;
assume that 1 < p and ( 1 - p ) * q + ( 1 - p ) * q = 1 and 0 <= a and a <= b ;
F* ( f , s* t ) = rpoly ( 1 , t ) *' t + ( 0. F_Complex ) .= ( 0. F_Complex ) *' t ;
for X being set , A being Subset of X holds A ` = {} implies A = {} & A = {} & A = {} ;
( N-min X ) `1 <= ( ( N-min X ) `1 ) & ( ( N-min X ) `2 ) <= ( ( N-min X ) `2 ) & ( ( N-min X ) `2 ) <= ( ( N-min X ) `2 ) ;
for c being Element of the \geq the \geq A , a being Element of the \subseteq of A holds c <> a
s1 . GBP = ( Exec ( i2 , s2 ) ) . GBP .= s2 . GBP .= Exec ( i2 , s2 ) . GBP .= s2 . GBP .= s . GBP ;
for a , b being Real holds |[ a , b ]| in ( y >= 0 implies a >= 0 ) & b >= 0 implies a = 0
for x , y being Element of X holds x ` \ y = ( x \ y ) ` & y \ x = ( x \ y ) `
mode BCK-algebra of i , j , m , n , m , n , m , m , n , m , m , n , m , n ;
set x2 = |( Re ( y - Im ( y - Im ( y - Im ( y - Im ( y - Im ( y - Im ( y - Im ( y - - Im ( y - Im ( y - - Im ( y - - - y ) ) ) ) ) ) ) ) , - ( y - Im ( y - Im ( y - Im ( y - Im ( y - -
[ y , x ] in dom u5 & u5 . ( y , x ) = g . y & u5 . ( y , x ) = g . y ;
]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .[ c= A & upper_bound divset ( D , k ) = upper_bound divset ( D , k ) ;
0 <= delta ( S2 ) . n & |. delta ( S2 ) . n .| < ( e / ( 2 |^ n ) ) * ( e / ( 2 |^ n ) ) ;
( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) <= ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ;
set A = ( 2 / b-a ) ;
for x , y being set st x in R" holds x , y are_\hbox { - } implies x , y are_\hbox { - }
deffunc FF2 ( Nat ) = b . $1 * ( M * G ) . ( $1 , j ) ;
for s being element holds s in -> element iff s in -> Element of \rm \rm \rm \rm : ( f 'or' g ) = ( f \/ g ) ;
for S being non empty non void non empty non void holds S is connected iff S is connected
max ( degree ( z `1 ) , degree ( z `2 ) ) >= 0 & degree ( z `2 ) >= 0 ;
consider n1 being Nat such that for k holds seq . ( n1 + k ) < r + s and r < x0 + s ;
Lin ( A /\ B ) is Subspace of Lin ( A ) & Lin ( B /\ A ) is Subspace of Lin ( B ) ;
set n-15 = nnS '&' ( M . x qua Element of BOOLEAN ) , nnS = ( M . x ) --> TRUE , nnS = ( M . x ) --> TRUE ;
f " V in ' ( X ) & f " V in D & f " V in D & f " V in D & f " V in D & f " V in D ;
rng ( ( a sequence c ) +* ( 1 , b ) ) c= { a , c , b } & rng ( ( a + c ) +* ( 1 , b ) ) c= { a , c } ;
consider y being p1 , -> \cdot as \HM of G1 such that y = y and dom y `1 = WWthesis & y `2 = WW
dom ( 1 / f ) /\ ]. -infty , x0 .[ c= ]. -infty , x0 .[ & ]. x0 , x0 + r .[ c= ]. x0 , x0 + r .[ ;
/\ j is {} , n , r , n be Element of REAL , i , j , - r , n be Element of NAT ;
v ^ ( n-3 |-> 0 ) in Lin ( rng ( B-9 | ( dom ( B-9 | ( dom ( B-9 ) ) ) ) ) ) ;
ex a , k1 , k2 st i = a /. k1 & j = b /. k2 & k2 = b /. k2 & k2 = b /. k2 ;
t . NAT = ( NAT .--> succ i1 ) . NAT .= ( NAT .--> succ i1 ) . NAT .= ( NAT --> succ i1 ) . NAT .= ( NAT --> succ i1 ) . NAT .= ( NAT --> succ i1 ) . NAT ;
assume that F is bbfamily and rng p = F and dom p = Seg ( n + 1 ) and rng p c= Seg ( n + 1 ) ;
not LIN b , b9 , a & not LIN a , a9 , c & LIN a , a9 , b & not LIN b , b9 , c
( L1 or L2 ) => O c= ( L1 => O ) => ( L2 => O )
consider F being ManySortedSet of E such that for d being Element of E holds F . d = F ( d ) and for d being Element of E holds F . d = G ( d ) ;
consider a , b such that a * ( u1 - w ) = b * ( cw - y ) and 0 < a and a < b ;
defpred P [ FinSequence of D ] means |. Sum $1 .| <= Sum |. $1 .| & for k be Nat st k in dom $1 holds |. $1 . k .| <= Sum |. $1 .| ;
u = cos . ( x , y ) * v + ( cos . ( x , y ) * y ) * v .= cos . ( x , y ) * v + ( cos . ( x , y ) * v ) .= v ;
dist ( ( seq . n ) + x , g + x ) <= dist ( ( seq . n ) , g ) + 0 ;
P [ p , |. p .| (#) |. p .| , {} ( the Sorts of A ) , id ( the Sorts of A ) ] ;
consider X being Subset of CQC-WFF ( Al ) such that X c= Y and X is finite and X is ininimplies X is ininand X is finite ;
|. 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 <= g } ;
vol ( ( G . n ) vol ) <= vol ( ( ( G . n ) vol ) ) + vol ( ( G . n ) vol ) ) ;
f . y = x .= x * ( ( 1_ L ) * ( y , 0 ) ) .= x * ( power L ) . ( y , 0 ) ;
NIC ( <% i1 , i2 %> , k ) = { i1 , succ ( i1 + k ) } .= { i1 , succ ( i1 + k ) } .= { succ ( i1 , i2 ) } ;
LSeg ( p00 , p2 ) /\ LSeg ( p1 , p2 ) = { p1 } /\ LSeg ( p1 , p2 ) .= { p1 } ;
Product ( ( the carrier of I-15 ) +* ( i , { 1 } ) ) in ( Z . i ) \/ ( Z . i ) ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) .= Following ( s1 , n ) .= Following ( s2 , n ) ;
W-bound Qs2 <= ( q1 `1 ) / ( |. q1 .| ) & ( q1 `1 ) / ( |. q1 .| ) <= ( q1 `1 ) / ( |. q1 .| ) ;
f /. i2 <> f /. ( ( i1 + len g ) -' 1 ) & f /. ( ( i1 + len g ) -' 1 ) <> f /. ( ( i1 + len g ) -' 1 ) ;
M , f / ( x. 3 , a ) / ( x. 4 , a ) / ( x. 0 , a ) / ( x. 4 , a ) / ( x. 0 , a ) |= H ;
len ( ( P ^ ) ) in dom ( ( P ^ ) ) & len ( ( P ^ ) ) = len ( ( P ^ ) ) + len ( ( P ^ ) ) ;
A |^ ( mn ) c= A |^ ( m , n ) & A |^ ( k , l ) c= A |^ ( k , l ) implies A = B |^ ( k , l )
REAL n \ { q : |. q .| < a } c= { q1 : |. q1 .| >= a }
consider n1 being element such that n1 in dom p1 and y1 = p1 . n1 and p1 . n1 = p1 . n1 ;
consider X being set such that X in Q and for Z being set st Z in Q & Z <> X holds X \not c= Z ;
CurInstr ( P3 , Comput ( P3 , s2 , l ) ) <> halt SCM+FSA & CurInstr ( P3 , Comput ( P3 , s2 , l ) ) <> halt SCM+FSA ;
for v be VECTOR of l1 holds ||. v .|| = upper_bound rng |. ( id the carrier of V ) .| & |. v .| = |. v .| & |. v .| = |. v .|
for phi holds phi in X implies ( phi in X & not phi in X & phi in X ) & ( phi in X implies phi in X )
rng ( Sgm dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f |
ex c being FinSequence of D ( ) st len c = k & P [ c ] & a = c & c = d ;
the_arity_of ( a , b , c ) = <* Hom ( b , c ) , Hom ( a , b ) *> & Hom ( a , b ) = { Hom ( a , b ) , Hom ( b , c ) } ;
consider f1 be Function of the carrier of X , REAL such that f1 = |. f .| and f1 is continuous and f1 is continuous ;
a1 = b1 & a2 = b2 or a1 = b2 & a2 = b1 & a3 = b2 & a4 = b3 or a1 = b1 & a2 = b2 & a3 = b3 ;
D2 . indx ( D2 , D1 , n1 + 1 ) = D1 . ( n1 + 1 ) & D2 . indx ( D2 , D1 , n1 + 1 ) = D2 . indx ( D2 , D1 , n1 ) ;
f . ( ||. r .|| ) = ||. ( r ) /. 1 ) .= ||. r .|| .= ||. r .|| .= ||. r .|| .= ||. r .|| .= ||. r .|| ;
consider n being Nat such that for m being Nat st n <= m holds C-25 . m = C-25 . m and C-25 . n = C-25 . m ;
consider d being Real such that for a , b being Real st a in X & b in Y holds a <= d & b <= c ;
||. L /. h - ( K * |. h .| ) + ( K * |. h .| ) + ( K * |. h .| ) <= p0 + ( K * |. h .| ) ;
attr F is commutative associative & for b being Element of X holds F -] { b } = f . b & F -\rm J = f . b ;
p = - ( - ( p0 + 0. TOP-REAL 2 ) ) * ( p0 - 0. TOP-REAL 2 ) .= 1 * ( p0 - 0. TOP-REAL 2 ) .= ( - ( p0 + 0. TOP-REAL 2 ) ) * ( p0 - 0. TOP-REAL 2 ) .= ( - ( p0 + 0. TOP-REAL 2 ) ) * ( p0 - 0. TOP-REAL 2 ) .= ( - ( p0 + 0. TOP-REAL 2 ) ) * ( p0 - 0. TOP-REAL 2 ) ;
consider z1 such that b , x3 , x3 , x1 is_collinear and o , x1 , x1 , x2 is_collinear and o , x1 , x2 is_collinear and o <> x1 and o <> x2 ;
consider i such that Arg ( ( Rotate ( s ) ) . q ) = s + Arg q + ( 2 * PI * i ) and i <= len ( 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 ;
assume that A = P2 \/ Q2 and P2 <> {} and Q2 misses P2 and P2 misses Q2 and P2 /\ Q2 = {} and P2 /\ Q2 = {} ;
attr F is associative means : : : F .: ( F .: ( f , g ) , h ) = F .: ( f , g ) ;
ex x being Element of NAT st m = x `2 & x in z `1 & x in { i } or m in { i } & x in { i } ;
consider k2 being Nat such that k2 in dom P-2 and l in PW1 . k2 and ( PW1 ) . k2 = ( PW2 ) . k2 ;
seq = r (#) seq implies for n holds seq . n = r * seq . n & ( for n holds seq . n = r * seq . n ) & ( for n holds seq . n = r )
F1 . [ ( ( id a ) * [ a , a ] ) , a ] = [ f * ( id a ) , f * ( id a ) ] ;
{ p } "\/" D2 = { p "\/" y where y is Element of L : y in D2 } & x in D1 & y in D2 } ;
consider z being element such that z in dom ( ( dom F ) * ( F . z ) ) and ( ( dom F ) * ( F . z ) ) . 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 , 1 ) `2 <= s } ;
consider e being element such that e in dom ( T | E1 ) and ( T | E1 ) . e = v and ( T | E1 ) . e = v ;
( F `1 * b1 ) . x = ( Mx2Tran ( J , sthesis , Bthesis ) ) . ( thesis , thesis /. j ) .= ( Mx2Tran J ) /. j ;
- 1 / ( mD ) = mm (#) D | n .= mm (#) D .= mm (#) ( - m ) .= ( Det M ) (#) ( - m ) .= ( Det M ) (#) ( - m ) ;
pred for x being set st x in dom f /\ dom g holds g . x <= f . x & ( - g ) . x <= f . x ;
len ( f1 . j ) = len f2 /. j .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) ;
All ( All ( a , 'not' A , G ) , B , G ) '<' Ex ( All ( a , B , G ) , A , G ) ;
LSeg ( E . k0 , F . k0 ) c= Cl RightComp Cage ( C , k + 1 ) & LSeg ( E , k + 1 ) c= RightComp Cage ( C , k + 1 ) ;
x \ a |^ m = x \ ( a |^ k * a ) .= ( x \ a ) |^ k \ a .= ( x \ a ) |^ k ;
k -inininininin-inin-inin-in-in-in-in-in-in-] = ( commute ( k ) ) . ( ( k + 1 ) - 1 ) .= ( ( k + 1 ) - 1 ) . ( k + 1 ) ) ;
for s being State of Aan holds Following ( s , n ) . ( 0 + ( 2 * n ) ) is stable & Following ( s , n ) . ( 2 * n + 1 ) is stable ;
for x st x in Z holds f1 . x = a ^2 & ( f1 - f2 ) . x <> 0 & ( f1 - f2 ) . x <> 0 ) implies f1 - f2 is continuous
support ( support ( n ) ) \/ support ( support ( m ) ) c= support ( max ( support ( n ) ) \/ support ( m ) ) ;
reconsider t = u as Function of ( the carrier of A ) /\ ( the carrier of B ) , the carrier' of C , f be Function of the carrier' of B , the carrier' of C ;
- ( a * sqrt ( 1 + b ^2 ) ) <= - ( b * sqrt ( 1 + a ^2 ) ) ;
phi / ( succ b1 ) . a = g . a & phi / ( b . a ) = f . ( g . a ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( ( F ^ <* p *> ) . i ) and i in dom ( F ^ <* p *> ) ;
{ x1 , x2 , x3 , x4 , x5 } = { x1 } \/ { x2 , x3 , x4 , x5 , x5 , x5 , x5 } \/ { x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 } ;
the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U1 & the Sorts of U2 c= the Sorts of U2 implies ( the Sorts of U1 ) . ( ( U1 "\/" U2 ) . ( ( U1 "\/" U2 ) . ( ( U1 "\/" U2 ) . ( ( U1 "\/" U2 ) . ( ( U1 "\/" U2 ) . ( ( U1 "\/" U2 ) . ( ( U1 "\/" U2 ) . ( ( U1 "\/" U2 ) . ( U1 "\/" U2 ) )
( - ( 2 * a * ( b - a ) ) ) ^2 + b ^2 - delta ( a , b , c ) ^2 > 0 ;
consider W00 such that for z being element holds z in W00 iff z in N & P [ z ] & P [ z ] ;
assume that ( the Arity of S ) . o = <* a *> and ( the Arity of S ) . o = r and ( the Arity of S ) . o = <* r *> and ( the Arity of S ) . o = <* r *> ;
Z = dom ( ( exp_R * ( arccot ) ) / ( f1 + #Z 2 ) ) & Z c= dom ( ( exp_R * ( arccot ) ) / ( f1 + #Z 2 ) ) ;
sum ( f , SS1 ) is convergent & lim ( \HM { the carrier of S , the carrier of T ) = integral ( f , SS1 ) & lim ( f , SS1 ) = integral ( f , SS1 ) ;
( X ( ) ) => ( ( a => b ) => ( ( x => y ) => ( x => y ) ) ) in
len ( M2 * M3 ) = n & width ( M3 * M2 ) = n & width ( M3 * M2 ) = n & len ( M2 * M3 ) = n & width ( M1 * M2 ) = n ;
attr X1 union X2 is open SubSpace of X means : : : X1 , X2 are_separated & X1 , X2 are_separated & X2 , X1 are_separated & X1 , X2 are_separated & X1 , X2 are_separated & X2 , X1 are_separated & X1 , X2 are_separated & X1 , X2 are_separated & X2 , X1 are_separated & X1 , X2 are_separated implies X1 , X2 \HM { , X2 } is open ;
for L being upper-bounded antisymmetric RelStr , X being non empty Subset of L holds X "\/" { Top L } = { Top L } & X "\/" { Top L } = { Top L }
reconsider f-122 = F1 . ( b `2 ) , f-1y = F2 . ( b `2 ) , f-1y = F2 . ( b `2 ) , f-1y = F2 . ( b `2 ) , f-1y = F2 . ( b `2 ) , f-1y = F2 . ( b `2 ) , f-1y = F2 . ( b `2 ) , f-1y = F2 . ( b `2 ) , f-1y = F2 . ( b `2 )
consider w being FinSequence of I such that the InitS of M is_\HM { w } ^ w and the InitS of M is_{ s } ^ w ;
g . ( a |^ 0 ) = g . ( 1_ G ) .= 1_ H .= ( g . a ) |^ 0 .= ( g . a ) |^ 0 .= ( g . a ) |^ 0 ;
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 ) = L & for K being Subset of X st K in C holds L /\ K <> {} ;
( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C1 & ( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C2 ;
reconsider oY = o `2 , oY = ( the Sorts of A ) . ( ( the_arity_of o ) /. i ) , oY = ( the Sorts of A ) . ( ( the_arity_of o ) /. i ) as Element of TS ( ( the Sorts of A ) . i ) ;
1 * x1 + ( 0 * x2 ) + ( 0 * x3 ) + ( 0 * x3 ) = x1 + ( \underbrace ( 0 , 0 , 0 ) ) .= x1 + ( 0 * x2 ) .= x1 + ( 0 * x3 ) .= x1 + ( 0 * x2 ) ;
Ea " . 1 = ( Ea qua Function ) " . 1 .= ( ( ( 1 - 2 ) |^ 1 ) ) " . 1 .= ( ( 1 - 2 ) |^ 1 ) " . 1 .= ( ( 1 - 2 ) |^ 1 ) " . 1 .= ( ( 1 - 2 ) |^ 1 ) " ) . 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 "/\" ( x "\/" y ) ) ;
|. f . ( s1 . ( l1 + 1 ) ) - f . ( s1 . ( l1 + 1 ) ) .| < ( 1 / ( |. M .| + 1 ) ) * ( 1 / ( |. M .| + 1 ) ) ;
LSeg ( ( Cage ( C , n ) /. ( i + 1 ) ) , ( ( Cage ( C , n ) /. ( i + 1 ) ) ) is vertical ;
( f | Z ) /. x - ( f | Z ) /. x0 = L /. ( x- ( x - x0 ) ) + R /. ( x- ( x - x0 ) ) ;
g . c * ( - g . c ) + f . c * ( - f . c ) + f . 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 ) ;
assume that ColVec2Mx f in the set of ColVec2Mx A and ColVec2Mx b in the carrier of ColVec2Mx A and len ColVec2Mx f = width A and width ColVec2Mx f = width A and width ColVec2Mx f = width A and len ColVec2Mx f = width A and width ColVec2Mx f = width A ;
len ( - M3 ) = len M1 & width ( - M3 ) = width M1 & width ( - M3 ) = width M1 & width ( - M3 ) = width M1 ;
for n , i being Nat st i + 1 < n holds [ i , i + 1 ] in the InternalRel of ( ( ( the InternalRel of ( n + 1 ) ) * ( i + 1 ) ) )
pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 ;
pred a <> 0 & b <> 0 & Arg a = Arg b & Arg ( - a ) = Arg ( - b ) & Arg ( - a ) = Arg ( - b ) ;
for c being set st not c in [. a , b .] holds not c in Intersection ( the } \HM { the } \HM { be set , a , b } , c )
assume that V1 is linearly-independent and V2 is linearly-independent and V1 = { v + u : v in V1 & u in V1 & u in V1 } and V1 = V1 /\ V2 ;
z * x1 + ( 1 / ( 1 - r ) ) * x2 in M & z * y1 + ( 1 / ( 1 - r ) ) * y2 in N implies z in N
rng ( ( Pk1 qua Function ) " * Sk1 ) = Seg ( card ( Pk1 ) ) .= Seg ( card ( Pk1 ) ) .= dom ( Pk1 ) ;
consider s2 being rational number such that s2 is convergent and b = lim s2 and for n holds s2 . n <= b . n and s2 . n <= c ;
h2 " . n = h2 . n " & 0 < - ( 1 / ( h2 . n ) ) & 0 < - ( 1 / ( h2 . n ) ) & ( - 1 / ( h2 . n ) ) < - ( 1 / ( h2 . n ) ) ;
( Partial_Sums ( ||. seq1 .|| ) ) . m = ||. ( seq1 . m ) - ( seq2 . m ) .|| .= ||. ( seq1 . m ) - ( seq2 . m ) .|| .= ||. ( seq1 . m ) - ( seq2 . m ) .|| .= ||. ( seq1 . m ) - ( seq2 . m ) .|| .= ||. ( seq1 . m ) - ( seq2 . m ) .|| .= ||. ( seq1 . m ) - ( seq2 . m ) .|| ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b ;
- v = ( - 1_ ( G ) ) * v & - w = ( - 1_ G ) * v & - w = ( - 1_ G ) * v & - w = - ( - 1_ G ) * v ;
sup ( ( k -tuples_on D ) .: D ) = sup ( ( k -tuples_on D ) .: ( D ) ) .= k -tuples_on D .= k -tuples_on D .= k -tuples_on D .= k -tuples_on D ;
A |^ ( k , l ) ^^ ( A |^ ( n , l ) ) = ( A |^ ( k , l ) ) ^^ ( A |^ ( k , l ) ) .= A |^ ( k , l ) ^^ ( A |^ ( k , l ) ) ;
for R being add-associative right_zeroed right_complementable non empty addLoopStr , I , J , K being Subset of R , I being Subset of R , J being Subset of R st I + J = ( I + J ) + K holds J = K
( f . p ) `1 = ( p `1 ) ^2 + sqrt ( 1 - ( p `2 / p `1 ) ^2 ) .= ( p `1 ) ^2 + ( p `2 / p `1 ) ^2 ;
for a , b being non zero Nat st a , b are_relative_prime holds support ( a * b ) = support ( a ) + support ( b ) & support ( a * b ) = support ( a ) + support ( b )
consider A5 being countable set such that r is countable & A5 is Element of CQC-WFF ( Al ) & A5 is ( Al ) -\overline { A } & A c= A & A c= A & A c= B & A c= B & A c= B & A c= B & A c= B & A c= B & A c= B & A c= B & A c= B & A c= B & A c= B & A c= B & A is D is countable ;
for X being non empty addLoopStr for M being Subset of X , x , y being Point of X st y in M holds x + y in M + M
{ [ x1 , x2 ] , [ y1 , y2 ] } c= { x1 , y1 } & { [ x1 , y2 ] , [ y1 , y2 ] } c= { x1 , y1 } & { x1 , x2 } c= { x1 , x2 } ;
h . ( f . O ) = |[ A * ( f . O ) `1 + B , C * ( f . O ) `2 + D ]| ;
( Gauge ( C , n ) * ( k , i ) ) in L~ Upper_Seq ( C , n ) /\ L~ Upper_Seq ( C , n ) & ( Cage ( C , n ) * ( k , i ) ) in L~ Upper_Seq ( C , n ) /\ L~ Upper_Seq ( C , n ) ;
cluster m , n are_relative_prime means : : : : for p being prime Nat holds p divides m & p divides n & p divides n & p divides n ;
( f * F ) . x1 = f . ( F . x1 ) & ( f * F ) . x2 = f . ( F . x2 ) ;
for L being LATTICE , a , b , c being Element of L st a \ b <= c & b \ a <= c holds a \+\ b <= c
consider b being element such that b in dom ( H / ( x , y ) ) and z = ( H / ( x , y ) ) . b ;
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 , W . 3 , W . 4 , W . 5 , G & e Joins W . 1 , W . 5 , G ;
( ( r (#) o ) . ( 2 * n ) ) . x = ( r (#) ( delta ( f ) ) ) . ( 2 * n ) . x + ( r (#) ( n + 1 ) ) . x ;
j + 1 = ( - len h11 ) + 2 - 1 .= i + 1 - len h11 + 2 - 1 .= i + 1 - 1 .= i + 1 - 1 .= i + 1 - 1 ;
( ^ ( S , T ) ) . f = S *' . ( ( ^ ( S , T ) ) . f ) .= S *' . ( ( S , T ) . f ) .= S *' . ( f , f ) ;
consider H such that H is one-to-one and rng H = the carrier of L2 and Sum ( L2 ) = Sum ( L2 ) and for k st k in dom L2 holds Sum ( L1 ) = Sum ( L1 ) ;
attr R is be be } means : : : for p , q st p in R & q <> q holds ex P st P is special & p in P & q in P & p in P ;
dom product ( X --> f ) = meet ( dom ( X --> f ) ) .= meet ( X --> dom f ) .= meet ( X --> dom f ) .= meet ( X --> dom f ) .= dom f .= dom f ;
upper_bound ( proj2 .: ( Upper_Arc C /\ Upper_Arc C ) ) <= upper_bound ( proj2 .: ( Upper_Arc C /\ Vertical_Line C ) ) & upper_bound ( proj2 .: ( C /\ Vertical_Line C ) ) <= upper_bound ( proj2 .: ( C /\ Vertical_Line C ) ) ;
for r be Real st 0 < r ex n be Nat st for m be Nat st n <= m holds |. S . m - pp .| < r
i * f-28 - f\in = i * f-28 - ( i * f\overline ( f ) - j ) .= i * ( fN - j ) .= i * ( fN - j ) ;
consider f being Function such that dom f = 2 -tuples_on X and for Y being set st Y in 2 -tuples_on X holds f . Y = F ( Y ) and for x being set st x in 2 -tuples_on X holds f . x = F ( x ) ;
consider g1 , g2 being element such that g1 in [#] Y and g2 in union C and g = [ g1 , g2 ] and g1 in C and g2 in C and g2 in C ;
func d |^ n -> Nat means : : : : d |^ n divides d & d |^ n divides ( d |^ n ) & ( not d divides n & not d divides n ) & not d divides n ;
fj . [ 0 , t ] = f . [ 0 , t ] .= ( - P ) . ( 2 * x ) .= ( - P ) . ( x `1 ) .= a ;
t = h . D or t = h . B or t = h . C or t = h . E or t = h . F or t = h . J or t = M or t = N or t = M ;
consider m1 be Nat such that for n st n >= m1 holds dist ( ( seq . n ) , ( seq . n ) ) < 1 / ( n + 1 ) ;
( q `1 ) ^2 / ( |. q .| ) ^2 <= ( q `2 ) ^2 / ( |. q .| ) ^2 & ( |. q .| ) ^2 <= ( |. q .| ) ^2 / ( |. q .| ) ^2 ;
h0 . ( i + 1 + 1 ) = h21 . ( i + 1 + 1 -' len h11 + 2 -' 1 ) .= h21 . ( i + 1 + 1 -' len h11 + 2 -' 1 ) ;
consider o being Element of the carrier' of S , x2 being Element of { the carrier of S } such that a = [ o , x2 ] and [ o , x2 ] in dom f ;
for L being RelStr , a , b being Element of L holds a <= { b } iff a >= b & b >= a & a >= b
||. h1 .|| . n = ||. h1 . n .|| .= ||. h . n .|| .= ||. h . n .|| .= ||. h . n .|| .= ||. h . n .|| .= ||. h . n .|| .= ||. h . n .|| .= ||. h . n .|| ;
( ( - 1 ) (#) ( ( #Z n ) * ( f1 - f2 ) ) ) . x = f . x - ( ( #Z n ) * ( f1 - f2 ) ) . x .= ( ( - 1 ) (#) ( f1 - f2 ) ) . x ;
pred r = F .: ( p , q ) means : : : for r st len r = len p holds r = min ( p , q ) ;
( rbeing / 2 ) ^2 + ( rbeing / 2 ) ^2 / 2 <= ( r ^2 ) ^2 + ( rbeing / 2 ) ^2 / 2 ;
for i being Nat , M being Matrix of n , K st i in Seg n holds Det M = Sum ( (
then a <> 0. R & a " * ( a * v ) = 1 / R * v & a " * ( a * v ) = 1 / R * v & a " * ( a * v ) = 1 / R ;
p . ( j -' 1 ) * ( q *' r ) . ( i + 1 -' j ) = Sum ( p . ( j -' 1 ) * r3 ) .= Sum ( p ) * r3 .= Sum ( q ) * r3 ;
deffunc F ( Nat ) = L . 1 + ( ( R /* ( h ^\ n ) ) * ( h ^\ n ) ) " . $1 & ( ( R /* ( h ^\ n ) ) ) " . $1 = ( ( R /* ( h ^\ n ) ) ) " . ( h ^\ n ) ;
assume that the carrier of H1 = f .: ( the carrier of H2 ) and the carrier of H2 = f .: ( the carrier of H1 ) and the carrier of H1 = the carrier of H2 and the carrier of H1 = the carrier of H2 and the carrier of H2 = the carrier of H1 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 Arity of S ) . o ;
H1 = n + 1 -] iff |. ( 2 to_power ( n + 1 ) + h ) .| = n + 1 -H .= n + 1 -H .= n + 1 -H .= n + 1 -H ;
( O1 = 0 & O = 0 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O =
F1 .: ( dom F1 /\ dom F2 ) = F1 .: ( ( dom F1 ) /\ ( dom F2 ) ) .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } ;
pred b <> 0 & d <> 0 & b <> d & ( a - b ) / ( d - c ) = ( ( - e ) / ( d - c ) ) / ( d - c ) ;
dom ( ( f +* g ) | D ) = dom ( f +* g ) /\ D .= ( dom f \/ dom g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) ;
for i being set st i in dom g ex u , v being Element of L , a being Element of B st g /. i = u * a & a * v = a * v
g `2 * P `2 * g `2 " = g `2 * ( g `2 * P `2 ) * g `2 .= g `2 * ( g `2 * P `2 ) " .= g `2 * ( g `2 * P `2 ) " ;
consider i , s1 such that f . i = s1 and if ( not i in dom s1 & s1 . ( i + 1 ) <> s1 . ( i + 1 ) ) & s1 . ( i + 1 ) <> s1 . ( i + 1 ) ;
h5 | ]. a , b .[ = ( g | ]. a , b .[ ) | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ ;
[ s1 , t1 ] , [ s2 , t2 ] connected & [ s2 , t2 ] , [ s3 , t2 ] connected & [ s3 , t2 ] in connected implies ( ( s1 , t2 ) `1 ) `1 = ( ( s1 ) `1 ) / ( ( s2 ) `1 ) ^2 )
then H is negative & H is not atomic & H is not conjunctive implies H is not implies H is not implies H is not implies H is not implies H is not implies H is not implies H is not implies H is not implies H is not implies H is not implies H is not One
attr f1 is total means : : : for c be element st c is total holds ( f1 - f2 ) . c = f1 . c - f2 . c ) & ( f1 - f2 ) . c = f1 . c - f2 . c ;
z1 in W2 -Seg ( z2 ) or z1 = z2 & not z1 in W2 & not z1 in W2 & not z2 in W2 & not z1 in W1 & not z2 in W2 & not z1 in W2 & not z1 in W1 & not z2 in W2 & not z1 in W2 & not z1 in W1 & not z2 in W2 & not z1 in W1 & not z2 in W2 & not z1 in W2 & not z1 in W2
p = 1 * p .= a " * a * p .= a " * ( b * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) ;
for seq1 be Real_Sequence for K be Real st for n be Nat holds seq1 . n <= K holds upper_bound rng ( seq1 ^\ K ) <= upper_bound rng ( seq1 ^\ K ) & upper_bound rng ( seq1 ^\ K ) <= upper_bound rng ( seq1 ^\ K )
x0 meets ( L~ go \/ L~ pion1 ) or ( E-max C ) meets ( L~ pion1 \/ L~ pion1 ) or ( E-max C ) meets ( L~ pion1 \/ L~ co ) or ( E-max C ) meets ( L~ pion1 \/ L~ co ) or ( E-max C ) meets ( L~ pion1 \/ L~ co ) or ( E-max C \/ L~ pion1 ) meets L~ co or ( E-max C \/ L~ pion1 ) c= L~ co ;
||. f . ( g . ( k + 1 ) ) - g . ( g . k ) .|| <= ||. g . 1 - g . 0 .|| * ( K to_power k ) ;
assume h = ( ( B .--> B ' +* ( C .--> D ) +* ( D .--> E ) +* ( E .--> F ) +* ( F .--> J ) +* ( M .--> N ) +* ( M .--> M ) +* ( N .--> N ) +* ( M .--> N ) +* ( M .--> N ) +* ( M .--> M ) +* ( N .--> N ) ) +* ( M .--> N ) +* ( M .--> N ) ) +* ( M .--> N ) +* ( M .--> M .--> N ) +* ( N .--> M ) +* ( N .--> N ) +* ( N .--> M ) +*
|. ( ( delta ( H . n ) || A ) ) . k - ( ( lower ( H . n ) || A ) ) . k .| <= e * ( b-a ( H . n ) || A ) ;
( ( ( ( the Sorts of A ) * ( i , j ) ) ) . e ) = [ ( ( the Sorts of A ) * ( i , j ) ) . e , ( the Sorts of A ) * ( i , j ) ] ;
{ x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x2 , x1 , x1 , x2 , x1 , x2 , x1 , x2 , x3 } = { x1 , x2 , x3 } ;
assume that A = [. 0 , 2 * PI .] and integral ( ( #Z n ) * sin , A ) = 0 and integral ( ( #Z n ) * sin , A ) = 0 and integral ( ( #Z n ) * sin , A ) = 0 ;
p `2 is Permutation of dom f1 & p `2 " = ( Sgm Y ) " X * p " & p `2 " X = ( Sgm Y ) " X * Sgm Y " X * Sgm Y " X ;
for x , y st x in A holds |. 1 / ( f . x ) - 1 / ( f . y ) .| <= 1 * |. f . x - f . y .|
p2 `2 = |. q2 .| * ( ( q2 `2 / |. q2 .| - sn ) / ( 1 + sn ) ) - sn * ( ( q2 `2 / |. q2 .| - sn ) / ( 1 + sn ) ) ^2 ) .= ( q2 `2 / |. q2 .| - sn ) / ( 1 + sn ) ;
for f be PartFunc of the carrier of CNS , REAL st dom f is compact & f is continuous holds rng f is compact & rng f c= dom ( f | X ) & rng f c= dom ( f | X )
assume for x being Element of Y st x in EqClass ( z , CompF ( B , G ) ) holds ( Ex ( a , A , G ) ) . x = TRUE ) & ( 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 , u1 / v , u1 / v , v1 / ( u , u1 ) // u1 , v1 & u , u1 / ( u , v1 ) // u1 , v1 & u1 <> v1 / ( u , u1 ) & u1 <> v1 / ( u , u1 ) = u1 / ( u , u1 ) ;
for G being Group , A , B being non empty Subgroup of G , N being normal Subgroup of G holds ( N -normal A ) * ( N -normal A ) = N -normal A * N
for s be Real st s in dom F holds F . s = integral ( ( R to_power 0 ) (#) ( f - g ) ) & ( F to_power 0 ) . s = integral ( ( R to_power 0 ) (#) ( f - g ) ) ;
width AutMt ( f1 , b1 , b2 ) = len b2 .= len ( b1 * b1 ) .= width AutMt ( f2 , b1 , b2 ) .= width AutMt ( f1 , b1 , b2 ) .= width AutMt ( f2 , b1 , b2 ) .= len AutMt ( f1 , b1 , b2 ) ;
f | ]. - PI / 2 , PI / 2 .[ = f & dom f = ]. - PI / 2 , PI / 2 .[ & rng f c= ]. - PI / 2 , PI / 2 .[ & f | ]. - PI / 2 , PI / 2 .[ = f | ]. - PI / 2 , PI / 2 .[ ;
assume that X is closed and a in X and a c= X and y in a and y in { { [ n , x ] } \/ { y } and x in a and y in a ;
Z = dom ( ( ( #Z 2 ) * ( arctan + arccot ) ) `| Z ) /\ dom ( ( ( #Z 2 ) * ( arctan + arccot ) ) `| Z ) .= dom ( ( #Z 2 ) * ( arctan + arccot ) ) /\ Z .= dom ( ( #Z 2 ) * ( arctan + arccot ) ) ;
func V ( l ) -> Subset of V means : : : for k st 1 <= k & k <= len l holds it . k in V ( ) & ( not k in V ( ) & not l . k in V ( ) ) ;
for L being non empty TopSpace , N being net of L , M being net of N , N being net of L st c is net of N for c being Point of L st c is cluster cluster cluster cluster -> convergent for net of N holds c is continuous
for s being Element of NAT holds ( for v being Element of NAT holds ( for u being Element of Cq holds u in Cq iff u in Cq ) & v in Cq implies u in Cq
then z /. 1 = ( N-min L~ z ) .. z & ( N-min L~ z ) .. z < ( N-min L~ z ) .. z & ( N-min L~ z ) .. z < ( N-min L~ z ) .. z ;
len ( p ^ <* ( 0 qua Real ) *> ) = len p + len <* ( 0 qua Real ) *> .= len p + 1 .= len p + 1 .= len p + 1 .= len p + 1 .= len p + 1 ;
assume that Z c= dom ( - ( ln * f ) ) and for x st x in Z holds f . x = x and f . x > 0 & f . x < 1 ;
for R being add-associative right_zeroed right_complementable commutative associative well-unital distributive non empty doubleLoopStr , I being Ideal of R , J being Subset of R , I being Ideal of R , J being Ideal of I holds ( I + J ) *' ( I /\ J ) c= I /\ J
consider f being Function of [: B1 , B2 :] , B12 such that for x being Element of B1 , y being Element of B2 holds f . x = F ( x , y ) and f . y = F ( x , y ) ;
dom ( x2 + y2 ) = Seg ( len x ) .= Seg ( len ( x2 ) ) .= Seg ( len ( x2 ) ) .= dom ( x2 ) .= dom ( x (#) ( y (#) y ) ) .= dom ( x (#) y ) .= dom ( x (#) y ) ;
for S being Functor of C , B for c being object of C holds card S . ( id c ) = id ( ( Obj S ) . ( id c ) ) & id ( ( Obj S ) . ( id c ) ) = id ( ( Obj S ) . ( id c ) )
ex a st a = a2 & a in f6 /\ f5 & for x st x in f6 holds or ( f , a ) = or f , a in or f , a in or f , a in or f , a in or f , a in { f } & f , a in { f } & f , a in { f , g } ;
a in Free ( ( H / ( x. 4 , x. k ) ) '&' ( H2 / ( x. k , x. k ) ) ) & ( ( H / ( x. k , x. k ) ) '&' ( H2 / ( x. k , x. k ) ) ) '&' ( H / ( x. k , x. k ) ) = Free ( H2 / ( x. k , x. k ) ) ;
for C1 , C2 being v1 , C2 being stable Function of C1 , C2 st `1 = ( i + 1 ) * ( i + 1 ) holds f = g & f = g
( W-min ( L~ go \/ L~ co ) ) `1 = W-bound ( L~ go \/ L~ co ) & ( W-min ( L~ go \/ L~ co ) ) `1 = W-bound ( L~ go \/ L~ co ) & ( W-min ( L~ go \/ L~ co ) ) `2 = S-bound ( L~ go \/ L~ co ) ;
assume that u = <* x0 , y0 , z0 *> and f is_PartFunc of REAL 3 , u and SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) is_differentiable_in z0 and SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) = SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) ;
then ( t . {} ) `1 in Vars implies ex x being Element of Vars st x = ( t . {} ) `1 & t = x & ( t . {} ) `2 = ( x , s ) `2 & ( x = s ) `2 & ( x = s ) `2 & ( x = s ) & ( x = s ) & ( x = s ) & ( x = s ) & ( x = s ) implies x = s ) ;
Valid ( p '&' p , J ) . v = Valid ( p , J ) . v '&' Valid ( p , J ) . v .= Valid ( p , J ) . v '&' Valid ( p , J ) . v .= Valid ( p , J ) . v ;
assume for x , y being Element of S st x <= y for a , b being Element of T ~ st a = f . x & b = f . y holds a >= b ;
func Class R -> Subset-Family of R means : : : for A being Subset of R holds A in it iff ex a being Element of R st a in Class ( R , a ) & it c= A ;
defpred P [ Nat ] means ( ( ( ( ( ( ( ( ( ) ) . $1 ) ) `1 ) ) . ( n + 1 ) ) `1 ) ) c= G \cdot ( ( ( ( ( ( ( ( ( ( ( ( G . $1 ) ) `1 ) . ( n + 1 ) ) `1 ) . ( n + 1 ) ) `1 ) . ( n + 1 ) ) ) . ( n + 1 ) ) ;
assume that dim ( W1 ) = 0 and dim ( W2 ) = 0 and ( dim ( W1 ) = 0 implies ( the carrier of ( W1 ) ) c= the carrier of ( W2 ) & ( dim ( W2 ) ) c= the carrier of ( W1 ) ) & ( dim ( W1 ) ) c= the carrier of ( W2 ) ) ;
mamas ( m ) . t = ( m . t ) `1 .= ( [ m . {} , the carrier of C ] ) `1 .= ( [ m , the carrier of C ] ) `1 .= ( [ m , the carrier of C ] ) `1 .= m . {} .= m . {} ;
d11 = x22 ^ d22 .= f . ( y22 , d22 ) .= f . ( y22 , d22 ) .= ( f ^ d22 ) . ( d22 , d22 ) .= ( f ^ d22 ) . ( d22 , d22 ) .= ( f ^ d22 ) . ( d22 , d22 ) .= ( f ^ d22 ) . ( d22 , d22 ) .= ( f ^ d22 ) . ( d22 ) ;
consider g such that x = g and dom g = dom fx0 and for x being element st x in dom fx0 holds g . x in fx0 and f . x in fx0 ;
x + 0. F_Complex / ( len x ) = x + len x |-> 0. F_Complex .= ( x , len x ) |-> 0. F_Complex .= ( x , len x ) |-> 0. F_Complex .= x ` .= x ` .= x ` ;
( k -' kk + 1 ) in dom ( f /. ( ( k -' 1 ) + 1 ) ) & ( f /. ( ( k -' 1 ) + 1 ) ) = ( f /. ( k -' 1 ) ) + ( f /. ( k + 1 ) ) ;
assume that P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 \/ P1 \/ P2 \/ P1 \/ P2 and P1 = P1 \/ P2 \/ P2 \/ P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 and P1 = P1 \/ P2 and P1 = P1
reconsider a1 = a , b1 = b , b1 = c , c1 = p , c1 = p , c2 = p , c2 = q , c2 = r , c1 = s , c2 = r , c2 = s , c1 = s , c2 = r , c2 = s , c2 = s , c1 = r , c2 = s , c2 = s , c2 = s , c2 = r , c2 = s , c1 = s , c2 = s , c2 = r , c1 = s , c2 = s , c2 = s , c2 = s , c2 = s , c2 = s , c2 = s , c2 = s , c2 = s , c2 = s , c2 = r , c2 = s , c2 = s , c2 = s , c2 = s , c2 =
reconsider : _ tf = G1 . ( t . b ) * F1 . f , F1 . ( t . b ) * F2 . ( t . b ) , F2 . ( t . b ) * F2 . ( t . b ) * F2 . ( t . b ) * F2 . ( t . b ) * F2 . ( t . b ) * F2 . ( t . b ) * F2 . ( t . b ) * F2 . ( t . b ) * F2 . ( t . b ) , F2 . ( t . b ) * F2 . ( t . b ) , F2 . ( t . b ) * F2 . ( t . b ) * F2 . ( t . b ) * F2 . ( t . b ) * F2 . ( t .
LSeg ( f , i + i1 -' 1 ) = LSeg ( f /. ( i + i1 -' 1 ) , f /. ( i + i1 -' 1 + 1 ) ) .= LSeg ( f /. ( i + i1 -' 1 ) , f /. ( i + i1 -' 1 ) ) ;
Integral ( M ` . m , P ) | dom ( P . n -to_power m ) <= Integral ( M ` . n , P . m ) & Integral ( M , P . m ) <= Integral ( M , P . n ) ;
assume that dom f1 = dom f2 and for x , y being element st [ x , y ] in dom f1 & [ x , y ] in dom f2 holds f1 . ( x , y ) = f2 . ( x , y ) ;
consider v such that v = y and dist ( u , v ) < min ( ( - ( G * ( i , 1 ) ) `1 ) , ( G * ( i + 1 , 1 ) `1 ) ) / ( 2 * ( ( G * ( i + 1 , 1 ) `1 ) `1 ) ) ;
for G being Group , H being Subgroup of G , a being Integer , b being Integer st a = b holds for i being Integer holds a |^ i = b |^ i & 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 = { p9 where p9 is Point of TOP-REAL 2 : P [ p9 ] & p9 <> 0. TOP-REAL 2 } , K1 = { p where p is Point of TOP-REAL 2 : P [ p ] & p <> 0. TOP-REAL 2 } as Subset of TOP-REAL 2 ;
( ( N-bound C ) - ( S-bound C ) / 2 ) / ( 2 |^ m ) <= ( ( N-bound C ) - ( S-bound C ) / ( 2 |^ m ) ) / ( 2 |^ m ) & ( S-bound C ) - ( S-bound C ) / ( 2 |^ m ) <= ( S-bound C ) - ( S-bound C ) / ( 2 |^ m ) ;
for x be Element of X , n be Nat st x in E holds |. Re ( F . n ) .| . x <= P . x & |. Im ( F . n ) .| . x <= P . x
len @ ( @ p ^ @ q ) = len ( @ p ^ @ q ) + len <* [ 2 , 0 ] *> .= len ( @ p ^ @ q ) + len <* [ 2 , 0 ] *> .= len ( @ p ^ @ q ) + len <* 2 , 0 *> .= len ( @ p ^ @ q ) + len ( @ q ^ @ p ) ;
v / ( x. 3 , m1 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) = m3 / ( x. 0 , m2 ) / ( x. 0 , m2 ) / ( x. 0 , m2 ) / ( x. 0 , m2 ) /
consider r being Element of M such that M , v / ( x. 3 , m3 ) / ( x. 4 , m3 ) / ( x. 3 , m3 ) / ( x. 4 , m3 ) / ( x. 3 , m3 ) / ( x. 4 , m3 ) / ( x. 3 , m3 ) / ( x. 4 , m3 ) / ( x. 3 , m3 ) / ( x. 4 , m3 ) / ( x. 3 , m3 ) / ( x. 4 , m3 ) / ( x. 3 , m3 ) / ( x. 4 , m3 ) / ( x. 0 , m3 ) / ( x. 0 , m3 ) / ( x. 0 , m3 ) / ( x. 0 , m3 ) / ( x. 0 , m3 )
func w1 \ w2 -> Element of Union ( G , R13 ) means : : : for w1 being Element of Union ( G , R\in ) holds it . ( ( ( G , R\in ) * ( the Arity of G ) ) . ( w1 , w2 ) ) = ( ( ( G , R\in ) * ( the Arity of G ) ) * ( the Arity of G ) ) . ( w1 , w2 ) ;
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= s1 . b2 .= ( Exec ( n2 , s2 ) ) . b2 .= s2 . b2 .= Exec ( n2 , s2 ) . b2 .= s2 . b2 .= s . b1 .= s . b1 .= s . b1 .= s . b2 ;
for n , k be Nat holds 0 <= ( Partial_Sums ( |. seq .| ) ) . ( n + k ) - ( Partial_Sums ( |. seq .| ) ) . ( n + k ) + ( Partial_Sums ( |. seq .| ) ) . ( n + k ) )
set F = S -implies F = S -B ;
( Partial_Sums ( seq ) ) . K + Partial_Sums ( seq ) . ( K + 1 ) >= ( Partial_Sums ( seq ) ) . ( K + 1 ) + ( Partial_Sums ( seq ) ) . ( K + 1 ) ) + ( Partial_Sums ( seq ) ) . ( K + 1 ) ;
consider L , R such that for x st x in N holds ( f | Z ) . x - ( f | Z ) . x0 = L . ( x- x0 ) + R . ( x - x0 ) ;
func \HM { closed \HM { a , b , c , d : a = the Element of \HM { rectangle ( a , b , c , d ) } , P , Q } -> Subset of TOP-REAL 2 equals { a , b , c , d } \/ P \/ Q ;
a * b ^2 + ( a * c ^2 + b * a ^2 ) + ( b * c ^2 + c * a ^2 ) + ( c * a ^2 + c * a ^2 ) >= 6 * a * a * b * c + c * a * b * c + c * a * b * c + c * a * b + c * a * b + c * a * b * c + c * a * b * c + c * a * b * c + c * a + c * a + c * a + c * a + c * a * b * a + c * a + c * a + c * a * b * c + c * a + c *
v / ( x1 , m1 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m1 ) = v / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) = v / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 )
+ ( Q ^ <* x *> , M1 ) = ( + ( Q , M1 ) +* ( ( L ^ <* x *> --> ( L ^ <* FALSE *> ) ) ) ) +* ( ( L ^ <* x *> --> ( L ^ <* FALSE *> ) ) ) .= ( ( L ^ <* x *> --> ( L ^ <* FALSE *> ) ) +* ( L ^ <* FALSE *> --> ( L ^ <* TRUE *> ) ) ) ;
Sum ( F ) = r |^ n1 * Sum ( Cnz ) .= C ( n1 ) * ( Cnz ) .= C ( n1 ) * ( Cnz ) .= C ( n1 ) * ( Cnz ) .= C ( n1 ) * ( Cnz ) .= C ( n1 ) * ( Cnz ) .= C ( n1 ) * ( Cnz ) ;
( GoB f ) * ( len GoB f , 1 ) `1 = ( ( GoB f ) * ( len GoB f , 1 ) ) `1 & ( GoB f ) * ( len GoB f , 1 ) `2 = ( ( GoB f ) * ( len GoB f , 1 ) ) `2 ;
defpred X [ Element of NAT ] means ( Partial_Sums s ) . $1 = ( a * ( $1 + 1 ) ) * ( $1 + 1 ) + b * ( $1 + 1 ) * ( $1 + 1 ) + b * ( $1 + 1 ) * ( $1 + 1 ) + b * ( $1 + 1 ) * ( $1 + 1 ) + b * ( $1 + 1 ) * ( $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 .= ( the Arity of S ) . g .= ( the Arity of S ) . g .= ( the Arity of S ) . g ;
( X ~ ) c= X ~ & card ( ( X ~ ) \ Y ) = card ( X ~ ) & card ( ( X ~ ) \ Y ) = card ( X ~ ) ;
for a , b being Element of S , s being Element of NAT st s = F . n & a = F . ( n + 1 ) & b = N . ( s + 1 ) holds b = N . ( s + 1 ) \ G . s
E , f |= All ( x. 2 , ( x. 0 ) .--> ( x. 2 ) ) => ( ( x. 0 ) .--> ( x. 1 ) ) => ( ( x. 2 ) .--> ( x. 0 ) ) '&' ( x. 1 ) ) => ( ( x. 2 ) '&' ( x. 1 ) ) '&' ( x. 2 ) '&' ( x. 0 ) ) = ( x. 1 ) '&' ( x. 1 ) '&' ( x. 2 ) ;
ex R2 being 1-sorted st R2 = ( p | n-11 ) . i & ( the carrier of p ) . i = the carrier of R2 & ( the carrier of p ) c= the carrier of R2 & ( the carrier of p ) c= the carrier of R2 ;
[. a , b + 1 / ( k + 1 ) .[ is Element of the \in the \rbrace of holds ( the partial F of f ) . ( k + 1 ) is Element of the carrier of S & ( the partial F of f ) . ( k + 1 ) is Element of the carrier of S & ( the partial F of f ) . ( k + 1 ) is Element of the carrier of S
Comput ( P , s , 2 + 1 ) = Exec ( P . 2 , Comput ( P , s , 2 ) ) .= Exec ( a3 := ( s . a ) , Comput ( P , s , 2 ) ) .= Exec ( a3 := ( s . a ) , s . b ) ;
card ( h1 ) = power ( F_Complex ) . ( ( - 1_ F_Complex ) * k , k ) * Sum u .= ( ( - 1_ F_Complex ) * ( - 1_ F_Complex ) ) * Sum u .= ( ( - 1_ F_Complex ) * ( - 1_ F_Complex ) ) * Sum u .= ( ( - 1_ F_Complex ) * ( - 1_ F_Complex ) ) * ( - 1_ F_Complex ) .= ( ( - 1_ F_Complex ) * ( - 1_ F_Complex ) ) * ( - 1_ F_Complex ) ;
( f / g ) /. c = f /. c * ( g /. c ) " .= f /. c * ( ( 1 / g ) * ( g /. c ) ) .= ( f (#) ( 1 / g ) ) /. c .= ( f (#) ( 1 / g ) ) /. c ;
len CC - len ( -> Element of NAT , CC ) = len CC - len ( C /^ 1 ) .= len ( C /^ 1 ) - len ( C /^ 1 ) .= len ( C /^ 1 ) - len ( C /^ 1 ) .= len ( C /^ 1 ) ;
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom f /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) ( f | X ) ) .= dom ( r (#) ( f | X ) ) .= dom ( r (#) ( f | X ) ) .= X /\ X .= dom ( r (#) ( f | X ) ) .= X ;
defpred P [ Nat ] means for n holds 2 * Fib ( n + $1 ) = Fib ( n ) * Fib ( n ) + ( 5 * Fib ( n ) * Fib ( n ) ) + ( 5 * Fib ( n ) * Fib ( n ) ) ;
consider f being Function of INT , INT such that f = f `1 and f is onto and for n st n < k holds f " { f . n } = { n } and f " { f . n } = { n } ;
consider c9 be Function of S , BOOLEAN such that c9 = chi ( A \/ B , S ) and ( for A , B st A in E holds ( A \/ B ) /\ E = Prob ( c , A ) ) and ( A /\ B ) /\ E = Prob ( c , B ) /\ E ;
consider y being Element of Y ( ) such that a = "\/" ( { F ( x , y ) where x is Element of X ( ) : P [ x ] } , L ( ) ) and for y being Element of X ( ) st P [ y ] holds P [ y ] ;
assume that A c= Z and f = ( - 1 ) (#) ( ( #Z 2 ) * ( ( #Z 2 ) * ( f1 + f2 ) ) ) and for x st x in Z holds ( ( #Z 2 ) * ( f1 + f2 ) ) . x = ( - 1 ) * ( f1 + f2 ) . x + ( #Z 2 ) * ( f2 + g2 ) . x ;
( f /. i ) `2 = ( GoB f ) * ( 1 , j2 ) `2 .= ( GoB f ) * ( 1 , j2 ) `2 .= ( GoB f ) * ( 1 , j2 + 1 ) `2 .= ( GoB f ) * ( 1 , j2 + 1 ) `2 .= ( GoB f ) * ( 1 , j2 + 1 ) `2 ;
dom Shift ( Seq q2 , len Seq q1 ) = { j + len Seq q1 where j is Nat : j in dom Seq q1 & len Seq q1 = len Seq q1 + len Seq q2 } .= dom Seq q1 \/ dom Seq q2 ;
consider G1 , G2 , G3 being Element of V such that G1 <= G2 and g \leq h and f in G1 and h in G2 and g in G2 and f in G2 and g in G2 and f in G2 and h in G2 and h in G2 and h = g * f ;
func - f -> PartFunc of C , V means : : : : dom it = dom f & for c st c in dom it holds it /. c = - f /. c & it /. c = - f /. c ;
consider phi such that phi is increasing and for a st phi . a = a and {} <> a for v holds union L |= ( union ( L , [. v , u .] ) ) iff L , L |= H & L , L |= H ;
consider i1 , j1 such that [ i1 , j1 ] in Indices GoB f and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) ;
consider i , n such that n <> 0 and sqrt p = ( i - n ) * ( i - 1 ) and for n1 being Nat st n1 <> 0 & n1 <= n holds sqrt p = ( i - n1 ) * ( i - 1 ) and sqrt p = ( i - n1 ) * ( i - 1 ) ;
assume that not 0 in Z and Z c= dom ( ( arccot * ( f1 + f2 ) ) `| Z ) and for x st x in Z holds ( ( 1 / 2 ) (#) ( f1 + f2 ) ) `| Z ) . x > - 1 & for x st x in Z holds ( ( 1 / 2 ) (#) ( f1 + f2 ) ) . x = - 1 / ( x ^2 ) and ( x - 1 ) * ( 1 / ( x ^2 ) ) ^2 = 1 / ( x ^2 ) and ( x ^2 ) = 1 / ( x ^2 ) and ( x ^2 ) * ( x ^2 ) = 1 / ( x ^2 ) = 1 / ( x ^2 ) and ( x ^2 ) * ( x ^2 ) = 1 / ( x ^2 ) * ( x ^2 ) *
cell ( G1 , i1 -' 1 , ( 2 |^ ( m -' 1 ) ) * ( Y1 -' 1 ) + 2 ) \ L~ ( ( f | ( m -' 1 ) ) * ( Y1 -' 1 ) ) c= BDD L~ f & ( ( f | ( m -' 1 ) ) * ( Y1 -' 1 ) ) \ L~ ( ( f | ( m -' 1 ) ) * ( Y1 -' 1 ) ) c= BDD L~ f ;
ex Q1 being open Subset of X st s = Q1 & ex Q1 being Subset-Family of Y st F1 c= F & Q1 is open & [#] ( Y | Q1 ) c= the topology of Y & [#] ( Y | Q1 ) c= the topology of Y & [#] ( Y | Q1 ) c= the topology of X ;
gcd ( A , ( gcd ( A1 , s1 , Amp ) , gcd ( A2 , s2 , Amp ) , gcd ( A1 , s1 , Amp ) , gcd ( A2 , s2 , Amp ) , gcd ( A1 , s1 , Amp ) , gcd ( A2 , s2 , Amp ) , gcd ( A1 , s1 , Amp ) , gcd ( A2 , s2 , Amp ) , gcd ( A2 , s2 , Amp ) , gcd ( A2 , s2 , Amp ) , 0. R ) = 1 ;
R8 = ( ( Result ( s2 ) ) . ( m2 + 1 ) ) . ( m2 + 1 ) .= ( ( Result ( s2 ) ) . ( m2 + 1 ) ) . ( m2 + 1 ) .= [ 3 , ( ( Result ( s2 ) ) . ( m2 + 1 ) ) . ( m2 + 1 ) ] .= [ 3 , ( ( Result ( s2 ) ) . ( m2 + 1 ) ) . ( m2 + 1 ) ] ;
CurInstr ( P-6 , Comput ( Pmeans , m1 + m2 ) ) = CurInstr ( P3 , Comput ( P3 , s3 , m3 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m3 ) ) .= CurInstr ( P3 , s3 ) .= CurInstr ( P3 , s3 ) .= CurInstr ( P3 , s3 ) .= CurInstr ( P3 , s3 ) .= CurInstr ( P3 , s3 ) .= halt ( P3 , s3 ) ;
P1 /\ P2 = ( { p1 } \/ LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) \/ LSeg ( p2 , p2 ) /\ LSeg ( p1 , p2 ) \/ LSeg ( p2 , p2 ) /\ LSeg ( p1 , p2 ) \/ { p2 } /\ LSeg ( p1 , p2 ) \/ { p2 } /\ LSeg ( p2 , p2 ) \/ { p2 } /\ LSeg ( p1 , p2 ) \/ { p2 } /\ LSeg ( p2 , p2 ) /\ LSeg ( p2 , p2 ) \/ { p2 , p2 ) \/ { p2 } /\ LSeg ( p2 , p2 ) /\ LSeg ( p2 , p2 ) /\ LSeg ( p2 , p2 ) /\ LSeg ( p2 , p2 ) \/ { p2 , p2 ) \/ { p2 , p2 ) \/ { p2 , p2 } ;
func not ( ex f being Subset of the Sorts of A ( ) st a in the carrier of A ( ) iff ex i , j st i in dom f & j in the carrier of A ( ) & a = f . i & a = f . j ) & a = f . j ;
for a , b being Element of F_Complex st |. a .| > |. b .| for f being Polynomial of F_Complex st f is \cup { b } holds f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or f is or
defpred P [ Nat ] means 1 <= $1 & $1 <= len g implies for i , j st [ i , j ] in Indices G & [ i , j ] in Indices G & [ i , j ] in Indices G & f /. i = G * ( $1 , j ) & f /. ( i + 1 ) = G * ( i , j ) ;
assume that C1 , C2 are_`2 and for f being State of C1 , g being State of C2 , s1 being State of C1 , s2 being State of C2 st s1 = s2 holds s1 is stable & s2 is stable & s1 is stable & s2 is stable & s2 is stable & s1 is stable & s2 is stable & s2 is stable holds s1 is stable & s2 is stable ;
( ||. f .|| | X ) . c = ||. f /. c .|| .= ||. f /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| ;
|. q .| ^2 = ( q `1 ) ^2 + ( q `2 ) ^2 & 0 + ( q `2 ) ^2 < ( q `1 ) ^2 + ( q `2 ) ^2 & 0 + ( q `1 ) ^2 < ( q `1 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 ;
for F being Subset-Family of TT st F is open & not {} in F for A , B being Subset of TT st A in F & A <> B & A <> B & A c= B holds card F c= card B & card A c= card B & card A c= card B & card A c= card B & card A c= card B & card A c= card B
assume that len F >= 1 and len F = k + 1 and len F = len G and for k st k in dom F holds H . k = g . ( F . k , G . k ) and for k st k in dom F holds H . k = g . ( F . k , G . k ) ;
i |^ ( ( ( mod ( n ) ) |^ ( ( p + k ) ) |^ s ) ) = i |^ s |^ s + i |^ k - ( i |^ k ) * ( i |^ k ) ) .= i |^ s * ( i |^ k ) - ( i |^ k ) * ( i |^ k ) - ( i |^ k ) * ( i |^ k ) .= i |^ ( ( k + 1 ) - 1 ) ;
consider q being oriented oriented Chain of G such that r = q and q <> {} and ( F . ( q . 1 ) ) = v1 & ( F . ( q . len q ) ) = v2 & ( F . ( q . 1 ) ) c= rng ( p ) & ( p . len p ) c= ( p . len p ) & ( p . len p ) c= ( p . len p ) ;
defpred P [ Element of NAT ] means $1 <= len ( s ) implies ( g = ( g , Z , I ) ^ ( g , I ) ) . ( len ( g , Z , I ) + $1 ) = ( ( g , Z , I ) ) . ( len ( g , Z , I ) + $1 ) ;
for A , B being Matrix of n , REAL , B being Matrix of n , REAL , C being Matrix of n , K st len ( A * B ) = len A & width ( A * B ) = width B & width ( A * B ) = width C & width ( A * B ) = width C & width ( A * B ) = width C & len ( B * C ) = width C holds A = C
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 = a * b ;
func |( x , y )| -> Element of COMPLEX means : : : for i be Nat holds it . ( Re x , Re y ) = |( Re ( x , Re y ) , ( Im ( x , Im y ) ) , ( Im ( x , Im y ) ) )| + ( Im ( x , Im y ) ) ^2 + ( Im ( x , Im y ) ) ^2 ;
consider g be FinSequence of FH such that g is continuous & rng ( g | A ) c= A & g . 1 = x1 & g . len g = x2 & for i st i in dom g holds g . i = f . i & g . ( len g ) = g . ( len g ) ;
then n1 >= len p1 & crossover ( p1 , p2 , n1 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , W , W , W , S , p1 , p2 , n1 , n2 , n3 , n3 , n3 , W , p1 , p2 , n1 , n2 , n3 , n3 , W , W , W , S , p1 , p2 , n1 , n2 , n3 , n3 , W , W , W , W , 7 , 7 , 7 , 7 , 7 , 7 , W , 7 , 8 , W , 7 , 7 , 7 , 8 , 8 , 7 , 8 , 7 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 7 , 8 , 8 , 7 , 7 , 7 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 7 , 7 , 7 , 7 , 7 , 7
( q `1 ) * a <= q `1 & - q `1 <= q `1 & - q `1 <= q `2 or q `1 >= q `1 & - q `2 <= - q `1 & q `1 <= - q `2 & - q `2 <= - q `1 or q `1 >= q `1 & - q `1 <= q `2 & q `2 <= - q `1 & - q `1 <= q `1 & q `1 <= - q `1 or q `1 >= q `2 & q `1 <= - q `1 & q `2 <= - q `1 & q `1 <= - q `1 & q `1 <= - q `1 & q `1 <= - q `1 & q `1 <= - q `1 & q `1 <= - q `1 or q `1 <= q `2 & q `2 <= - q `1 & q `2 <= - q `1 & q `1 <= - q `1 or q `1 <= q `2 & q `2 <= q `2 & q `2 <= -
Fv . ( len pv ) = Fv . ( p . ( len pv ) ) .= Fv . ( len ( pv ) + 1 ) .= vv . ( len ( pv ) + 1 ) .= vv . ( len ( pv ) + 1 ) .= vv . ( len ( pv ) + 1 ) .= ( len ( F ) ) + 1 .= len ( F ) ;
consider k1 being Nat such that k1 + k = 1 and a := k = ( <* a := intloc 0 *> ) ^ ( ( k --> intloc 0 ) ^ <* halt SCM+FSA *> ) ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* IC SCM+FSA *> ^ <* IC SCM+FSA *> ^ <* IC SCM+FSA *> ^ <* IC SCM+FSA *> ^ <* IC SCM+FSA *> ^ <* IC SCM+FSA *> ^ <* IC SCM+FSA *> ^ <* IC SCM+FSA *> ^ <* IC SCM+FSA *> ^ <* IC SCM+FSA *> ^ <* IC SCM+FSA *> ^ <* IC SCM+FSA *> ^ <* IC SCM+FSA *> ^ <* IC SCM+FSA *> ^ <* IC SCM+FSA *> ^ <* IC SCM+FSA *> ^ <* IC SCM+FSA *> ^ <* IC SCM+FSA *> ^ <* IC SCM+FSA *> ^ <* IC SCM+FSA *>
consider B8 being Subset of B1 , y8 being Function of B1 , A1 such that B8 is finite and D8 = \frac 0 and D8 = \frac 0 and D8 = \frac { A1 : A1 is Subset of B1 & A1 is Subset of A1 & A2 is Subset of A2 & A2 is Subset of A2 } and B8 c= the carrier of B1 and B8 c= the carrier of B2 ;
v2 . b2 = ( curry ( F2 , g ) * ( ( curry F ) . b2 ) ) . b1 .= ( curry ( F2 , g ) ) . b1 .= ( ( curry F ) * ( ( ( curry F ) . b1 ) ) ) . b1 .= ( ( curry F ) * ( ( ( curry F ) . b1 ) ) ) . b1 .= ( ( curry F ) * ( ( ( curry F ) . b1 ) ) ) . b1 .= ( ( ( ( ( ( ( ( ( ( ( ( b1 ) ) . b2 ) ) ) ) . b1 ) ) . b1 ) ) .= ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( b1 ) ) . b2 ) ) . b2 ) ) . b2 ) ) * ( ( ( ) ) . b2 )
dom IExec ( I-35 , P , Initialize s ) = the carrier of SCMPDS .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) ;
ex dbeing Real st d-32 > 0 & for h being Real st h > 0 & |. h .| < ( |. h .| ) holds |. h .| " * ||. ( R2 + R1 ) /. h .|| < e / ( ( |. h .| * ||. h .|| ) * ||. h .|| ) )
LSeg ( G * ( len G + 1 , - 1 ) , G * ( len G , 1 ) + |[ 1 , 0 ]| ) c= Int cell ( G , len G + 1 , 0 ) \/ { G * ( len G , 1 ) } ;
LSeg ( mid ( h , i1 , i2 ) , i ) = LSeg ( h /. ( i + i1 -' 1 ) , h /. ( i + i1 -' 1 + 1 ) ) .= LSeg ( h /. ( i + i1 -' 1 ) , h /. ( i + i1 -' 1 ) ) .= LSeg ( h /. ( i + i1 -' 1 ) , h /. ( i + i1 -' 1 ) ) ;
A = { q where q is Point of TOP-REAL 2 : LE p1 , q , P , p1 , p2 & LE p2 , p1 , P , p1 , p2 & LE p1 , p2 , P , p1 , p2 & LE p2 , p1 , P , p1 , p2 & LE p1 , p2 , P , p1 , p2 & LE p2 , p1 , P , p1 , p2 & LE p2 , p1 , P , p1 , p2 & LE p2 , p1 , P , p1 & LE p2 , p1 , P , p2 & LE p2 , p1 , P , p2 & LE p2 , p1 , P , p1 , p2 , p3 , p2 , p3 , p2 , p3 , p2 & LE p1 , p1 , p2 , p3 , p3 , p3 , p3 , p4 & LE p3 , p3 , P , p1 , p3 , p3 , p3 , p3 , p3 , p3 , p3 , p4 , p4 , p4 , p3 , p4 , p4 , p4 , p4 , p4 ,
( ( - x ) .|. y ) = ( - 1 ) * ( ( - 1 ) * y ) .= ( - 1 ) * ( ( - 1 ) * y ) .= ( - 1 ) * ( ( - 1 ) * y ) .= ( - 1 ) * ( - 1 ) .= ( - 1 ) * ( - 1 ) .= ( - 1 ) * ( - 1 ) .= ( - 1 ) * ( - 1 ) .= ( - 1 ) * ( - 1 ) ;
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 ) ^2 ) .= ( p `1 ) ^2 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) ;
( ( U * ( Wq ) ) * ( Wq ) ) = ( ( ( U * ( Wq ) ) * ( Wq ) ) ) * ( Wq ) .= ( ( ( U * ( Wq ) ) * ( Wq ) ) * ( Wq ) ) * ( ( U * ( q ) ) * ( ( U * ( q ) ) ) ) .= ( ( U * ( q ) ) * ( q ) ) ;
func Shift ( f , h ) -> PartFunc of REAL , REAL means : : : for x be Element of REAL , f be PartFunc of REAL , REAL st x in dom it holds it . x = - h . x + h . x & for x be Element of REAL st x in dom it holds it . x = - h . x + h . x ;
assume that 1 <= k and k + 1 <= len f and [ i , j ] in Indices G and [ i + 1 , j ] in Indices G and f /. k = G * ( i + 1 , j ) and f /. ( k + 1 ) = G * ( i , j ) and f /. ( k + 1 ) = G * ( i , j ) ;
assume that not y in Free H and not x in Free H and ( not x in Free H implies x in Free H ) & not x in Free ( H ) & not x in Free ( H ) & not x in Free ( H ) & not x in Free ( H ) & not x in Free ( H ) & not x in Free ( H ) & not x in Free ( H ) & not x in Free ( H ) ;
defpred P11 [ Element of NAT , Element of NAT ] means ( $1 = p |^ $1 implies $2 = p |^ $1 ) & ( $1 = p |^ $1 implies $2 = p |^ $1 ) & ( $1 = p |^ $1 implies $2 = p |^ $1 ) & ( $1 = p |^ $1 implies $2 = p |^ $1 ) & ( $1 = p |^ $1 implies $2 = p |^ $1 ) & ( not $1 = p |^ $1 implies $2 = p |^ $1 ) ;
func \sigma ( C ) -> non empty Subset-Family of X means : : : for A being Subset of X holds A in it iff for W being Subset of X st W c= A \ A holds C c= W & for W being Subset of X st W c= A \ W holds C c= W & C c= W implies C c= W ;
[#] ( ( dist ( P ) ) .: Q ) = ( dist ( P ) ) .: Q & lower_bound [#] ( ( dist ( P ) ) .: Q ) = ( lower_bound ( ( dist ( P ) ) .: Q ) ) & lower_bound ( ( dist ( P ) ) .: Q ) = lower_bound ( ( dist ( P ) ) .: Q ) & lower_bound ( ( dist ( P ) ) .: Q ) = lower_bound ( ( dist ( P ) ) .: Q ) ;
rng ( F | ( [: S , T :] ) ) = {} or rng ( F | ( [: S , T :] ) ) = { 1 } or rng ( F | ( [: S , T :] ) ) = { 2 } or rng ( F | ( [: S , T :] ) ) = { 1 } or rng ( F | ( [: S , T :] ) ) = { 2 } ;
( f " ( rng ( f | ( rng f ) ) ) ) . i = f . i " .= ( f . i ) " ( ( f . i ) " ( f . i ) ) .= ( f . i ) " ( f . i ) .= ( f . i ) " ( f . i ) .= ( f . i ) " ( f . i ) ;
consider P1 , P2 being non empty Subset of TOP-REAL 2 such that P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p2 , p1 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p2 , p1 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p2 , p2 & P1 is_an_arc_of p1 , p2 & P1 is_an_arc_of p1 , p2 , p2 holds P1 c= P2 ;
f . p2 = |[ ( p2 `1 ) ^2 + sqrt ( 1 - ( p2 `2 / p2 `1 ) ^2 ) , ( p2 `2 ) ^2 + ( p2 `2 ) ^2 ) ]| .= |[ ( p2 `1 ) ^2 + ( p2 `2 ) ^2 , ( p2 `2 ) ^2 + ( p2 `2 ) ^2 ]| ;
( ( ( a , X ) --> x ) " ) . x = ( ( ( ( a , X ) qua Function ) qua Function ) " ) . x .= ( ( ( a , X ) --> x ) " ) . x .= ( ( ( a , X ) --> x ) " ) . x .= ( ( a , X ) --> x ) . x .= ( ( a , X ) --> x ) . x ;
for T being non empty normal TopSpace , A , B being closed Subset of T , A being Subset of T st A <> {} & A misses B for p being Point of T , r being Point of S , p being Point of ( Element of dom G ) , r being Point of ( Element of dom G ) st p in A & r in A & p in B holds ( Element ( dom G ) ) . p <= r
for i , j st i + 1 in dom F for G1 , G2 being strict normal Subgroup of G st G1 = F . i & G2 = F . ( i + 1 ) & G2 = F . ( i + 1 ) holds G1 is strict Subgroup of G & G2 is strict Subgroup of G
for x st x in Z holds ( ( ( #Z 2 ) * ( arctan - arccot ) ) `| Z ) . x = ( ( #Z 2 ) * ( arctan - arccot ) ) `| Z ) . x - ( ( #Z 2 ) * ( arctan - arccot ) ) . x
synonym f is_right x0 means : : for x0 st x0 in dom f & for a st a in Z holds f . x0 = lim ( f , x0 ) & for x st x in Z holds f . x = ( f /* a ) . x - ( f /* a ) . x0 ;
then X1 , X2 are_separated & X1 misses Y1 & Y1 misses Y2 implies ex Y1 , Y2 being SubSpace of X st Y1 misses Y2 & Y1 misses X1 & Y2 misses X2 & Y1 misses X1 & Y1 misses X1 & Y2 misses X2 & Y1 misses Y2 & Y1 misses Y2 & Y1 misses Y2 & Y2 misses X1 & Y1 misses X2 & Y1 misses X1 & Y1 misses X1 & Y1 misses X2 implies X1 misses X2 & Y1 misses X1 & Y1 misses X1 & Y1 misses X2 & Y2 misses X1 & Y1 misses Y2 implies X1 misses X2 & Y1 misses Y2 & Y2 misses X2 & Y1 misses Y2 & Y2 misses X2 & Y2 misses X1 & Y1 misses X1 & Y1 misses X2 & Y1 misses Y2 & Y2 misses X2 & Y2 misses X2 & Y2 misses X2 & Y2 misses X2 & Y1 misses X2 & Y2 misses X2 implies X1 misses X2 implies X1 misses X2 & Y2 misses X2 & X1 misses X2 & X1 misses X2 implies X1 misses X2 & Y1 misses X2 & Y1 misses X2 & X1 misses X2 & Y1 misses X2 & Y1 misses X2 & Y1 misses X2 & X1 misses X2 & Y1 & Y1 misses X2 & Y1 misses X2 & Y1 misses X2 & Y1 misses 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 - SVF1 ( 1 , f , u ) . x0 = L . ( x - x0 ) + R . ( x - x0 ) + R . ( x - x0 )
( p2 `1 ) ^2 * sqrt ( 1 + ( p3 `1 ) ^2 ) >= ( p3 `1 ) ^2 * sqrt ( 1 + ( p3 `2 ) ^2 ) & ( p3 `1 ) ^2 + ( p3 `2 ) ^2 * sqrt ( 1 + ( p3 `1 ) ^2 ) >= ( p3 `1 ) ^2 * sqrt ( 1 + ( p3 `1 ) ^2 ) ;
( ( 1 / t1 ) (#) ||. f1 .|| ) to_power ( n + 1 ) = ( ( 1 / t1 ) (#) ||. f1 .|| ) to_power ( m + 1 ) & ( ( 1 / t2 ) (#) ||. f1 .|| ) to_power ( m + 1 ) = ( ( 1 / t ) (#) ||. f1 .|| ) to_power ( m + 1 ) & ( ( 1 / t ) (#) ||. f1 .|| ) to_power ( m + 1 ) = ( 1 / t ) to_power ( m + 1 ) ;
assume for x holds f . x = ( ( - 1 / ( sin . x ) ) (#) ( sin . x ) ) & x + h in dom ( ( - 1 / ( sin . x ) ) (#) ( sin * ( ( sin . x ) ) (#) ( sin * ( ( sin . x ) ) (#) ( sin * ( ( sin . x ) ) (#) ( sin * ( ( sin * ( ( sin * ( ( 1 / ( sin . x ) ) ) / ( sin . x ) ) ) ) ) ) & ( for x ) ) ) `| Z ) . x ) = ( - 1 / ( sin . x ) ) ) & ( ( sin * ( sin * ( x ) ) ) `| Z ) . x = ( - 1 / ( sin . x ) ) & ( ( sin . x ) (#) ( sin . x ) (#) ( sin * ( sin * ( sin * ( ( sin . x ) ) (#) ( sin * ( sin * ( sin * ( 1 / ( sin . x ) ) ) & ( ( ( ( sin . x ) ) ) `| Z ) . x ) = (
consider Xf1 being Subset of [: Y , X :] , Y1 being Subset of X , Y1 being Subset of Y such that Y1 = [: Xf1 , Y1 :] and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open ;
card ( S . n ) = card { [: d , Y :] + ( a * d ) where d is Element of GF ( p ) : [ d , Y ] in R } .= [: { d , Y :] \/ { d } , { d } :] .= [: { d , Y } , { d } :] \/ { d } , { d } :] ;
( W-bound D - W-bound D ) * ( ( W-bound D ) / 2 ) * ( ( W-bound D ) / 2 ) = ( W-bound D ) * ( ( W-bound D ) / 2 ) * ( ( W-bound D ) / 2 ) .= ( W-bound D ) * ( ( W-bound D ) / 2 ) * ( ( W-bound D ) / 2 ) .= ( W-bound D ) * ( ( W-bound D ) / 2 ) ;