thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
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contradiction ;
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contradiction ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
contradiction ;
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thesis ;
thesis ;
contradiction ;
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thesis ;
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 ;
thesis ;
thesis ;
contradiction ;
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contradiction ;
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thesis ;
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thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b
D c= S
let Y ;
S ` is Cauchy
q in P ;
V in F ;
y in N ;
x in T ;
m < n ;
m <= n ;
n > 1 ;
let r ;
t in I ;
n <= 4 ;
M is finite ;
let X ;
Y c= Z ;
A // M ;
let U ;
a in D ;
q in Y ;
let x ;
1 <= l ;
1 <= w ;
let G ;
y in N ;
f = {} ;
let x ;
x in Z ;
let x ;
F is one-to-one ;
e <> b ;
1 <= n ;
f is special ;
S misses C
t <= 1 ;
y divides m ;
P divides M ;
let Z ;
let x ;
y c= x ;
let X ;
let C ;
x _|_ p ;
o is monotone ;
let X ;
A = B ;
1 < i ;
let x ;
let u ;
k <> 0 ;
let p ;
0 < r ;
let n ;
let y ;
f is onto ;
x < 1 ;
G c= F ;
a is_>=_than X ;
T is continuous ;
d <= a ;
p <= r ;
t < s ;
p <= t ;
t < s ;
let r ;
D <= E ;
assume e > 0 ;
assume 0 < g ;
p in C ;
x in X ;
Y ` in Y ;
assume 0 < g ;
not c in Y ;
not v in L ;
2 in z ` ;
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 ` <= b ` ;
assume b in X ;
assume k <> 1 ;
f = product l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is from of squares ;
assume m > 0 ;
assume A c= B ;
X is lower
assume A <> {} ;
assume X <> {} ;
assume F <> {} ;
assume G is open ;
assume f is dilatation ;
assume y in W ;
y \not <= x ;
A ` in B ` ;
assume i = 1 ;
let x be element ;
x ` = 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 , v be Vertex of G ;
let G be _Graph , v be Vertex of G ;
let a be Complex ;
let x be element ;
let x be element ;
let C be FormalContext , a , b be Element of C ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
let y be Real ;
X c= f . a
let y be element ;
let x be element ;
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 ;
k1 be Integer ;
let X be set ;
let a be element ;
let x be element ;
let x be element ;
let q be element ;
let x be element ;
assume f is being_homeomorphism ;
let z be element ;
a , b // K ;
let n be Nat ;
let k be Nat ;
B ` c= B ` ;
set s = 1 ;
n >= 0 + 1 ;
k c= k + 1 ;
R1 c= R ;
k + 1 >= k ;
k c= k + 1 ;
let j be Nat ;
o , a // Y ;
R c= Cl G ;
Cl B = B ;
let j be Nat ;
1 <= j + 1 ;
arccot is_differentiable_on Z ;
exp 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 Element of REAL ;
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 Nat ;
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 REAL ;
cluster uparrow x -> being being being being being set ;
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 >= ks ;
G . y <> 0 ;
let X be RealNormSpace , x be Point of X ;
a in X ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , M be Subset of V ;
assume x in \cal M ;
k < s . a ;
not t in { p } ;
let Y be set , a be Real ;
M , L are_isomorphic ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded & L is lower-bounded ;
rng f = Y ;
GK 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 ` = a ` + 1 ;
x ` = a * y ` ;
rng D c= A ;
assume x in K1 ;
1 <= ii ;
1 <= ii ;
px c= PI ;
1 <= ii ;
1 <= ii ;
LMP C in L ;
1 in dom f ;
let seq ;
set C = a * B ;
x in rng f ;
assume f is_differentiable_on X ;
I = dom A ;
u in dom p ;
assume a < x + 1 ;
s-7 is bounded ;
assume I c= P1 ;
n in dom I ;
let Q ;
B c= dom f ;
b + p _|_ a ;
x in dom g ;
F-14 is continuous ;
dom g = X ;
len q = m ;
assume A2 : x in A2 ;
cluster R \ S -> real-valued ;
sup D in S ;
x << sup D ;
b1 >= Z1 & b2 >= b1 ;
assume w = 0. V ;
assume x in A . i ;
g in the carrier of X ;
y in dom t ;
i in dom g ;
assume P [ k ] ;
Let Let Let Let Let C be f1 ;
x9 is increasing ;
let e2 be element ;
- b divides b ;
F c= \tau ( F ) ;
Gseq is non-decreasing ;
Gseq is non-decreasing ;
assume v in H . m ;
assume b in [#] B ;
let S be non void ManySortedSign , f be Function of S , V ;
assume P [ n ] ;
assume union S is independent & A is finite ;
V is Subspace of V ;
assume P [ k ] ;
rng f c= NAT ;
assume ex_inf_of X , L ;
y in rng f ;
let s , I be set , a be Element of S ;
b ` c= b9 ` ;
assume not x in NAT + \ { {} } ;
A /\ B = { a } ;
assume len f > 0 ;
assume x in dom f ;
b , a // o , c ;
B in Bf1 ;
cluster product p -> non empty ;
z , x // x , p ;
assume x in rng N ;
cosec is_differentiable_in x & cosec is_differentiable_in x ;
assume y in rng S ;
let x , y ;
i2 < i1 & i1 < i2 ;
a * h in a * H ;
p , q in Y ;
Observe that sqrt I is left ideal ;
q1 in A1 & q2 in A2 ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A2 c= A1 ;
\hbox { \boldmath $ n $ } < n ;
assume A c= dom f ;
Re ( f ) is_integrable_on M ;
let k , m ;
a , a // b , b ;
j + 1 < k + 1 ;
m + 1 <= n1 ;
g is_differentiable_in x0 & g is_differentiable_in x0 ;
g is_differentiable_in x0 & g is_differentiable_in x0 ;
assume O is symmetric ;
let x , y ;
let j0 be Nat ;
[ y , x ] in R ;
let x , y ;
assume y in conv A ;
x in Int V ;
let v be VECTOR of V ;
P3 halts_on s , P2 = P +* I ;
d , c // a , b ;
let t , u ;
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 , x be Point of X ;
[ a , b ] in R ;
x + w < y + w ;
{ a , b } >= c ;
let B be Subset of A , a be Real ;
let S be non empty ManySortedSign ;
let x be variable of f , g ;
let b be Element of X , x be Element of X ;
R [ x , y ] ;
x ` = x ;
b \ x = 0. X ;
<* d *> in D |^ 1 ;
P [ k + 1 ] ;
m in dom ( mn ) ;
h2 . a = y ;
P [ n + 1 ] ;
Observe G * F is pre. ;
let R be non empty multMagma , a be Element of R ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p ` ;
assume f | X is lower ;
x in rng co & x in rng co ;
let x be Element of B ;
let t be Element of D ;
assume x in Q -Seg ( a ) ;
set q = s ^\ k ;
let t be VECTOR of X ;
let x be Element of A ;
assume y in rng p ` ;
let M be mamamaid ;
let N be non empty Subset of the carrier of M ;
let R be RelStr with finite R ;
let n , k be Nat ;
let P , Q be RelStr ;
P = Q /\ [#] S ;
F . r in { 0 } ;
let x be Element of X ;
let x be Element of X ;
let u be VECTOR of V ;
reconsider d = x as FinSeq-Location ;
assume I is not lim \HM { a } ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < vN ;
x <= c2 . x ;
x in F ` ;
Observe that S --> T is ManySortedSet ;
assume t1 <= t2 & t2 <= t2 ;
let i , j be even Integer ;
assume F1 <> F2 & F2 <> F1 ;
c in Intersect ( union R ) ;
dom p1 = c & dom p2 = c ;
a = 0 or a = 1 ;
assume A1 <> A6 & A2 <> A2 ;
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 ) / 2 ;
1 <= len g1 & 1 <= len g2 ;
len ( s2 ) > 1 ;
z in dom ( f1 - f2 ) ;
1 in dom D2 & 1 in dom D2 ;
( p `2 ) ^2 = 0 ;
j2 <= width G & 1 <= j2 ;
len PI > 1 + 1 ;
set n1 = n + 1 ;
|. q-35 .| = 1 ;
let s be SortSymbol of S ;
HT ( i , i ) = i ;
X1 c= dom f & X2 c= dom g ;
h . x in h . a ;
let G be mod of as \times of as Element of succ ;
cluster m * n -> square ;
let k9 be Nat ;
i - 1 > m - 1 ;
R is transitive implies R ~ = R ~
set F = <* u , w *> ;
pp c= P3 & card I c= card J ;
I is_closed_on t , Q ;
assume [ S , x ] is real ;
i <= len ( f2 ^ g2 ) ;
p is FinSequence of X ;
1 + 1 in dom g ;
Sum R2 = n * r ;
cluster f . x -> complex-valued ;
x in dom ( f1 - f2 ) ;
assume [ X , p ] in C ;
BX c= XX & BX c= BX ;
n2 <= ( 2 |^ ( n2 + 1 ) ) ;
A /\ [: P , Q :] c= A ` ;
cluster -> x -valued for Function ;
let Q be Subset-Family of S , a be Subset of T ;
assume n in dom ( ( g | n ) ^ ) ;
let a be Element of R ;
t `1 in dom e2 & t `2 in dom e2 ;
N . 1 in rng N ;
- z in A \/ B ;
let S be SigmaField of X , T be Subset-Family of S ;
i . y in rng i ;
REAL c= dom f & rng f c= dom g ;
f . x in rng f ;
mt <= ( r / 2 ) ;
s2 in r-5 & s1 in r-5 ;
let z , z be complex number ;
n <= N . m ;
LIN q , p , s ;
f . x = waybelow x /\ B ;
set L = |[ S \to T ]| ;
let x be non positive ExtReal ;
let m be Element of M ;
f in union rng F1 & f in rng F2 ;
let K be add-associative right_zeroed right_complementable non empty doubleLoopStr , p be Polynomial of K ;
let i be Element of NAT ;
rng ( F * g ) c= Y ;
dom f c= dom x & rng f c= dom y ;
n1 < n1 + 1 & n2 + 1 <= n1 ;
n1 < n1 + 1 & n2 + 1 <= n1 ;
cluster T . X -> non empty ;
[ y2 , 2 ] = z ;
let m be Element of NAT ;
let S be Subset of R ;
y in rng ( S . k ) ;
b = sup dom f & b in dom f ;
x in Seg ( len q ) ;
reconsider X = [: D , D :] as set ;
[ a , c ] in E1 ;
assume n in dom h2 ;
w + 1 = ma1 ;
j + 1 <= j + 1 + 1 ;
k2 + 1 <= k1 ;
let i be Element of NAT ;
Support u = Support p & Support u = Support p ;
assume X is complete complete ;
assume f = g & p = q ;
n1 <= n1 + 1 & n2 <= n1 + 1 ;
let x be Element of REAL ;
assume x in rng ( s2 | X ) ;
x0 < x0 + 1 & x0 in dom f ;
len ( L ) = W ;
P c= Seg ( len A ) ;
dom q = Seg n & dom q = Seg n ;
j <= width M *' ;
let r8 be real-valued finite sequence of NAT ;
let k be Element of NAT ;
Integral ( M , P ) < +infty ;
let n be Element of NAT ;
assume z in ) -) ( 0 , A ) ;
let i be set ;
n - 1 = n-1 - 1 ;
len ( n | m ) = n ;
Set N = cell ( Z , c ) ;
assume x in X or x = X ;
x is midpoint of b , c ;
let A , B be non empty set , f be Function of A , B ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q & i in dom q ;
let s be Element of E |^ omega ;
let B1 be Basis of x , y ;
L1 /\ L2 = {} ;
L1 /\ L2 = {} ;
assume downarrow x = downarrow y ;
assume b , c // b , c ` ;
LIN q , c , c ;
x in rng ( f | At ) ;
set n8 = n + j ;
let D7 be non empty set , f be Function of D7 , D8 ;
let K be add-associative right_zeroed right_complementable non empty addLoopStr , M be Matrix of K ;
assume f ` = f & h ` = h ;
R1 - R2 is total & R2 - R2 is total ;
k in NAT & 1 <= k ;
let a be Element of G ;
assume x0 in [. a , b .] ;
K1 ` is open & ( sn max ) . p = ( sn ) ` ;
assume a , b are_maximal in C ;
a , b be Element of S ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = Integral ( M , f | E ) ;
cluster as n-WFF for nes) ;
not u in { b9 } ;
the carrier of f c= B ;
reconsider z = x as VECTOR of V ;
cluster the RelStr of L -> \prod ;
r (#) H is being PartFunc of X , REAL ;
s . intloc 0 = 1 ;
assume x in C & y in C ;
let U0 be strict non-empty non-empty MSAlgebra over S , a be Element of H ;
[ x , Bottom T ] is compact ;
i + 1 + k in dom p ;
F . i is stable Subset of M ;
r-35 in ( iff y in { x } ) ;
let x , y be Element of X ;
A , I be seq of X ;
[ y , z ] in [: X , Y :] ;
( that that that card Macro i ) = 1 and card Macro i = 2 ;
rng Sgm A = A ;
q |- All ( y , q ) ;
for n holds X [ n ] ;
x in { a } & x in d ;
for n holds P [ n ] ;
set p = |[ x , y , z ]| ;
LIN o , a , b ;
p . 2 = Z |^ Y ;
( D - B ) `2 = {} ;
n + 1 + 1 <= len g ;
a in [: Al ( ) , NAT :] ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
let I be Ideal of L ;
set g = f1 + f2 , h = f1 + f2 ;
a <= max ( a , b ) ;
i-1 < len G + 1 ;
g . i1 = f . i1 ;
x ` , y ` in A2 ;
( f /* s ) . k < r ;
set v = VAL g ;
i - k + 1 <= S ;
cluster non empty multiplicative for multMagma ;
x in support ( ( support t ) | ( dom t ) ) ;
assume a in [: the carrier of G , the carrier of G :] ;
i `1 <= len y-5 & i `2 <= width y ;
assume p divides b1 + b2 ;
M1 <= sup M1 & M2 <= sup M1 ;
assume x in ( W \ X ) ;
j in dom ( z | X ) ;
let x be Element of D ( ) ;
IC s4 = l1 & IC s4 = 0 ;
a = {} or a = { x } ;
set uG = Vertices G , uH = Vertices H ;
seq " (#) ( h " ) is non-zero ;
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
hcarrier c= h-14 \/ { h } ;
]. a , b .[ c= Z ;
X1 , X2 , X3 , X2 , X1 , X2 , X2 , X1 , X2 , X2 , X1 , X2 , X2 , X1 , X2 , X2 , X1 , X2 , X2 , X1
a in Cl ( union F \ G ) ;
set x1 = [ 0 , 0 ] ;
k + 1 - 1 = k - 1 ;
cluster -> complex-valued for Relation of INT ;
ex v st C = v + W ;
let IT be non empty addLoopStr , a 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 ;
sup B is upper & sup B = sup B ;
let L be non empty reflexive RelStr , N be net of L ;
R is reflexive & X is transitive implies R is transitive
E , g |= H / ( H , E ) ;
dom G /. y = a ;
sqrt ( 1 - 4 ) >= - r ;
G . p0 in rng G & G . I in rng G ;
let x be Element of FK , a be Element of K ;
D [ P , 0 ] ;
z in dom ( id B ) & z in dom ( id B ) ;
y in the carrier of N & y in the carrier of N ;
g in the carrier of H & g in the carrier of H ;
rng ( f | X ) c= [: X , Y :] ;
j ` + 1 in dom s1 ;
let A , B be strict Subgroup of G ;
let C be non empty Subset of R^1 ;
f . z1 in dom h & g . z1 in dom h ;
P . k1 in rng P & P . k1 in rng P ;
M = A3 +* {} & M = A3 +* {} ;
let p be FinSequence of REAL , a be Element of REAL ;
f . n1 in rng f & g . n1 in rng g ;
M . ( F . 0 ) in REAL ;
h | [. a , b .] = b ;
assume the distance of V , Q ;
let a be Element of ^ ( V ) ;
let s be Element of PK ;
let PP be non empty RelStr ;
let n be Nat ;
the carrier of g c= B ;
I = halt SCM R & I = ( the InstructionsF of R ) . I ;
consider b being element such that b in B ;
set BK = BCS K , BK = BCS K ;
l <= ( -> net of F ) . j ;
assume x in downarrow [ s , t ] ;
( x - y ) / ( x - y ) in ]. t - x , t .[ ;
x in ( JumpParts T ) . ( T . i ) ;
let h be Morphism of c , a ;
Y c= ( 1_ K ) .: ( the_rank_of K ) ;
A2 \/ A3 c= L1 \/ L2 & A2 \/ A3 c= L1 \/ L2 ;
assume LIN o , a , b ;
b , c // d1 , e2 ;
x1 , x2 , x3 , x4 , x5 , x5 , Real ;
dom <* y *> = Seg 1 & dom <* y *> = Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar s .| ;
[ x , x ` ] in X ~ ;
for n be Nat holds 0 <= x . n
|[ a , b ]| = [. a , b .] ;
cluster -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> non closed ;
x = h . ( f . z1 ) ;
q1 , q2 , q1 is_collinear & q2 , q1 , q2 is_collinear ;
dom M1 = [: Seg n , Seg n :] ;
x = [ x1 , x2 ] & y = [ y1 , y2 ] ;
R , Q be ManySortedSet of A ;
set d = ( 1 / n + 1 ) ;
rng ( ( g " ) " ) c= dom W ;
P . ( [#] Sigma \ B ) <> 0 ;
a in field R & a = b ;
let M be non empty Subset of V , a be Real ;
let I be Program of SCM+FSA , a be Int-Location ;
assume x in rng ( ( R * S ) * ( R * S ) ) ;
let b be Element of the lattice of T ;
dist ( e , z ) > r-r ;
u1 + v1 in W2 & v1 + v2 in W1 ;
assume the carrier of L misses rng G ;
let L be lower-bounded antisymmetric transitive RelStr ;
assume [ x , y ] in a9 ;
dom ( A * e ) = NAT ;
a , b being Vertex of G ;
let x be Element of Bool ( M ) ;
0 <= 2 * PI ;
o , a9 // o , y & o <> c9 ;
{ v } c= the carrier of l ;
let x be variable of A ;
assume x in dom ( ( uncurry f ) . i ) ;
rng F c= ( product f ) |^ X ;
assume D2 . k in rng D ;
f " . p1 = 0 & f " . p2 = 1 ;
set x = the Element of X , y = the Element of Y ;
dom Ser ( G ) = NAT & rng Ser ( G ) c= NAT ;
let n be Element of NAT ;
assume LIN c , a , e1 ;
cluster -> finite for FinSequence of NAT ;
reconsider d = c as Element of L1 ;
( v2 |-- I ) . X <= 1 ;
assume x in the carrier of f & y in the carrier of f ;
conv @ A c= conv @ A & conv @ A c= conv @ A ;
reconsider B = b as Element of the carrier of T ;
J , v |= P \lbrack l , l2 .] ;
cluster J . i -> non empty TopSpace ;
ex_sup_of Y1 \/ Y2 , T & ex_sup_of Y1 \/ Y2 , T ;
W1 is_\! > ( W1 + W2 ) implies W1 is well-ordering
assume x in the carrier of R & y in the carrier of R ;
dom ( n --> 0 ) = Seg n & dom ( n --> 0 ) = Seg n ;
s4 misses s4 \/ { p2 } ;
assume ( a 'imp' b ) . z = TRUE ;
assume X is open & f = X --> d ;
assume [ a , y ] in Indices ( f | X ) ;
assume that that that that ' I c= J and dom } J c= K and card I c= K ;
Im ( lim seq ) = 0 & Im ( seq ) = 0 ;
( ( - sin ) (#) ( sin - cos ) ) . x <> 0 ;
sin is_differentiable_on Z & cos is_differentiable_on Z implies cos is_differentiable_on Z & cos is_differentiable_on Z
t6 . n = t6 . n ;
dom ( element ) c= dom F ;
W1 . x = W2 . x & W2 . x = W2 . x ;
y in W { x } \/ W { y } ;
k9 <= len ( v | k ) & k <= len ( v | k ) ;
x * a \equiv y * a . ( mod m ) ;
proj2 .: S c= proj2 .: P & proj2 .: P c= P ;
h . p4 = g2 . I & h . p2 = ( h . I ) `1 ;
G6 = U /. 1 & G6 = U . 1 ;
f . rr1 in rng f & rr2 in rng f ;
i + 1 + 1 <= len - 1 ;
rng F = rng ( F | A ) & F | A is one-to-one ;
mode then empty multMagma of G is well unital associative non empty multiplicative loop ;
[ x , y ] in A ~ ;
x1 . o in L2 . o & x2 . o in rng x2 ;
the carrier of m - m c= B ;
not [ y , x ] in id X ;
1 + p .. f <= i + len f ;
seq ^\ k1 is lower & seq ^\ k1 is lower ;
len ( F-12 ^ I ) = len I + len I ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be complex number , a be Real ;
Comput ( P , s , n ) = s ;
k <= k + 1 & k + 1 <= len p ;
reconsider c = {} T as Element of L ;
let Y be Element of of of of \mathbin { \mid [ } X ] } ;
cluster ex L being Function of L , L st L is directed-sups-preserving
f . j1 in K . j1 & f . j1 in K . j1 ;
cluster J => y -> total for ( J --> x ) -valued Function ;
K c= 2 |^ the carrier of T ;
F . b1 = F . b2 & F . b2 = G . b2 ;
x1 = x or x1 = y or x1 = z ;
pred a <> {} means : Def6 : ( a * a ) = 1 ;
assume that cf c= b and b in a ;
s1 . n in rng s1 & s1 . n in rng s1 ;
{ o , b2 } on C2 & { o , b2 } on C2 ;
LIN o , b , b9 & LIN o , b , c ;
reconsider m = x as Element of Funcs ( V , C ) ;
let f be non constant finite sequence of D ;
let F2 be non empty TopSpace , f be Function of I[01] , TOP-REAL 2 ;
assume h is being_homeomorphism & y = h . x ;
[ f . 1 , w ] in F-8 ;
reconsider p9 = x as Subset of m -tuples_on the carrier of K ;
A , B , C be Element of R ;
cluster non empty strict for non empty as strict an order ;
rng c `1 misses rng e2 & rng e c= rng e2 ;
z is Element of gr { x } & z in { x } ;
not b in dom ( a .--> p1 ) ;
assume k >= 2 & P [ k ] ;
Z c= dom ( ( - cot ) * cot ) ;
the component of Q c= UBD A & ( Q /\ P ) = ( Q /\ P ) /\ ( Q /\ P ) ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( 1 / 2 ) & g2 in dom ( 1 / 2 ) ;
pred f = u means : Def6 : a * f = a * u ;
for n holds P1 [ prop n ] ;
{ x . O : x in L } <> {} ;
let x be Element of V . s ;
a , b be Nat ;
assume S = S2 & p = p2 & q = p1 ;
gcd ( n1 , n2 ) = 1 & gcd ( n1 , n2 ) = 1 ;
set o9 = ( a * _ + b * _ ) , c9 = ( a * _ + b * _ + c * _ ) ;
seq . n < |. r1 .| & |. seq . n .| < r ;
assume that seq is increasing and r < 0 ;
f . ( y1 , x1 ) <= a ;
ex c being Nat st P [ c ] ;
set g = { n to_power 1 : n in dom f } ;
k = a or k = b or k = c ;
a9 , b9 , c9 , d9 , d9 , P ;
assume Y = { 1 } & s = <* 1 *> ;
I1 . x = f . x .= 0 ;
W3 . 1 = W3 . 1 & W . 2 = W . 2 ;
cluster -> trivial for subgraph of G , finite , being finite _Graph ;
reconsider u = u as Element of Bags X ;
A in B ^ implies A , B are_that A , B are_that B , A are_that B , A are_that A , B are_that B , A are_that B , A are_that A , B are_
x in { [ 2 * n + 3 , k ] } ;
1 >= ( q `1 ) ^2 / ( |. q .| ) ^2 ;
f1 is_] ] ] and f2 is_] ] or f1 is_\upharpoonright ( len f1 ) ;
( f . q ) `2 <= ( q `2 ) ^2 / ( |. q .| ) ^2 ;
h is_the carrier of Cage ( C , n ) ;
( b `2 ) ^2 / ( p `2 ) ^2 <= ( p `2 ) ^2 / ( p `2 ) ^2 ;
let f , g be Element of X , h be Function of X , Y ;
S * ( k , k ) <> 0. K ;
x in dom ( f - g ) ;
p2 in NN . p1 & p2 in NN . p2 ;
len ( the_right_argument_of H ) < len ( H ) ;
F [ A , F1 ( ) . A ] ;
consider Z such that y in Z and Z in X ;
pred 1 in C means : Def6 : A c= C |^ A ;
assume r1 <> 0 or r2 <> 0 ;
rng q1 c= rng C1 & rng C1 c= rng C1 ;
A1 , L , A3 , A3 , A2 , A3 , A3 , A2 , A3 ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in element ( p , SK ) & b in K ;
then S is negative implies P-2 [ S ] ;
Cl ( [#] T ) = [#] T & Cl ( [#] T ) = [#] T ;
f12 | A2 = f2 & f12 | A2 = f1 | A2 ;
0. M in the carrier of W & 0. M in the carrier of W ;
v , v ` be Element of M ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ V ;
let X be Subset of S ~ ;
consider H1 such that H = 'not' H1 and H1 in H ;
1_ K c= ( t * \HM { the } \HM { Real } ) * ( x - 1 ) ;
0 * a = 0. R .= a * 0 .= a * 0 ;
A |^ ( 2 , 2 ) = A ^^ A ;
set vY = v4 /. n , vY = v4 /. n ;
r = 0. ( \langle REAL-NS n , REAL-NS n *> * ||. f .|| ) ;
( f . p4 ) `1 / ( 1 + ( p4 `1 / |. p4 .| ) ) ^2 >= 0 ;
len W = len ( W ^ ( I ^ J ) ) ;
f /* ( s * G ) is divergent_to+infty & f /* ( s * G ) is divergent_to+infty ;
consider l being Nat such that m = F . l ;
t8 / ( W , 7 ) does not contradiction & not b1 on a1 & b2 on a1 ;
reconsider Y1 = X1 , Y2 = X2 as SubSpace of X ;
consider w such that w in F and not x in w ;
let a , b , c , d be Real ;
reconsider i = i as non zero Element of NAT ;
c . x >= ( id L ) . x ;
\sigma ( T ) \/ omega ( T ) is Basis of T ;
for x being element st x in X holds x in Y
cluster [ x1 , x2 , x3 ] -> pair ;
downarrow a /\ downarrow t is Ideal of T ;
let X be non empty set , N be non empty set ;
rng f = element ( S , X ) ;
let p be Element of B , a be Element of the carrier of S ;
max ( N1 , 2 ) >= N1 & max ( N1 , 2 ) >= N2 ;
0. X <= b |^ ( m * mmmmmmmmmA ) ;
assume i in I & R1 . i = R ;
i = j1 & p1 = q1 & p2 = q2 ;
assume gR in the right of g & gR in the carrier of g ;
let A1 , A2 be Subset of S , x be Point of S ;
x in h " P /\ [#] T1 & x in h " P ;
1 in Seg 2 & 1 in Seg 3 & 2 in Seg 3 ;
reconsider X-5 = X as non empty Subset of Tsuch that X = TN \/ TN and XN = XN ;
x in ( the Arrows of B ) . i & x in the Arrows of B ;
cluster E|^ n -> ( the Source of G ) -valued ;
n1 <= i2 + len g2 & n2 <= len g2 ;
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume v in the carrier' of G2 & v in the carrier' of G1 ;
y = Re y + ( Im y ) * i ;
( ( - 1 ) |^ p ) divides 1 ;
x2 is_differentiable_on ]. a , b .[ & ( for x st x in ]. a , b .[ holds ( f | ]. a , b .[ ) . x = ( f | ]. a , b .[ ) . x
rng ( M | D2 ) c= rng ( M | D2 ) ;
for p be Real st p in Z holds p >= a
( \bf X ) * ( f | X ) = proj1 * ( f | X ) ;
( seq ^\ m ) . k <> 0 ;
s . ( G . ( k + 1 ) ) > x0 ;
( p -Path M ) . 2 = d ;
A \oplus ( B \ominus C ) = ( A \oplus B ) \ominus C
h \equiv gg . ( ( mod P ) . ( g mod P ) ) ;
reconsider i1 = 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 i9 = i as Element of NAT , m be Element of NAT ;
dom f c= [: C , D :] & dom g = C ;
x in ( the sup of B ) . n & x in ( the sup of B ) . n ;
len func func func f1 , f2 , f3 , f4 *> in Seg ( len f1 + 2 ) ;
p9 c= the topology of T & p9 c= the topology of T ;
]. r , s .[ c= [. r , s .] ;
let B2 be Basis of T2 , a be Subset of T2 ;
G * ( B * A ) = ( the Arrows of o1 ) * ( A , B ) ;
assume that p , u are_\! to u and q , v are_not zero ;
[ z , z ] in union rng ( F | A ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S implies $1 in dom S ;
LIN a1 , a3 , b1 & LIN a1 , a2 , 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 ) ^2 / ( 1 + ( ( g2 ) . I ) `2 ) ^2 ) <= 1 ;
p in LSeg ( E . i , F . i ) ;
IK * ( i , j ) = 0. K ;
|. f . ( s . m ) - g .| < g1 ;
q9 . x in rng ( q | A ) & q9 . x in rng ( q | A ) ;
Carrier ( \lbrace H , F , G ) misses Carrier ( H , G ) ` ;
consider c being element such that [ a , c ] in G ;
assume that $ \mathop { N, N|[ a , b ]| } = o9 and not LIN o , Nbeing Element of NAT ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= ( F |^ C ) " { {} } ;
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [. f . j , g . j .] ;
pred 0 <= x & x ^2 <= x ^2 ;
p `1 - q `2 <> 0. TOP-REAL 2 & p `2 - q `2 <> 0. TOP-REAL 2 ;
cluster ) aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaL ;
let x be Element of S ~ ;
<^ F , F ^> is one-to-one of F , F ;
|. i .| <= - ( - 2 |^ n ) ;
the carrier of I[01] = dom P & the carrier of I[01] = dom P ;
exp_R * ( n + 1 ) ! > 0 * PI ;
S c= ( A1 /\ A2 ) /\ ( A1 /\ A2 ) ;
a3 , a4 // a3 , b3 & a3 , a4 // a3 , b3 ;
then dom A <> {} & dom A <> {} ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x joins X , Y & y in X & x in Y ;
set v2 = v4 /. ( i + 1 ) , v4 = v4 /. ( i + 1 ) , v5 = v5 ;
x = r . n .= ( r . n ) / 2 ;
f . s in the carrier of S2 & f . s in the carrier of S2 ;
dom g = the carrier of I[01] & dom g = the carrier of I[01] ;
p in Upper_Arc ( P ) /\ Upper_Arc ( P ) ;
dom d2 = A2 & dom d2 = A2 & dom d2 = A2 ;
0 < ( p / ( ||. z .|| + 1 ) ) / ( ||. z .|| + 1 ) ;
e . ( m1 + 1 ) <= e . ( m1 + 1 ) ;
B \ominus X \/ B \ominus Y c= B \ominus X /\ Y
-infty < Integral ( M , Im ( g | B ) ) ;
cluster O \HM { \tt and F is \HM { an where F is an operation of X : F is an operation of X } -> Element 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 & g `| X = g `| X ;
x , y , z be Point of X , p be Point of X ;
reconsider p9 = p . x as Subset of V ;
x in the carrier of Lin ( A ) & x in the carrier of Lin ( A ) ;
let I , J be parahalting Program of SCM+FSA , a be Int-Location ;
assume - a is lower & - b is lower implies - a is lower ;
Int Cl ( A ) c= Cl ( Cl A ) ;
assume for A being Subset of X holds Cl A = A ;
assume q in Ball ( |[ x , y ]| , r ) ;
( p2 `2 ) ^2 / ( p2 `2 ) ^2 <= ( p2 `2 ) ^2 / ( p2 `2 ) ^2 ;
Cl Q ` = [#] ( T | A ` ) .= A ;
set S = the carrier of T , N = the carrier of T ;
set IK = ' ( f |^ n ) , IK = ' ( f |^ n ) , IK = ' ( f | n ) , IK = ' ( f | n ) , IK = ' ( f | n ) , IK
len p - n = len ( p - n ) ;
A is Permutation of Swap ( A , x , y ) ;
reconsider nnan = n\rbrace - ( nan + 1 ) as Element of NAT ;
1 <= j + 1 & j + 1 <= len ( s | n ) ;
let q9 , q9 be Element of M , a , b be Element of M ;
a9 in the carrier of S1 & b9 in the carrier of S2 ;
c1 /. n1 = c1 . n1 & c2 /. n1 = c2 . n1 ;
let f be FinSequence of TOP-REAL 2 , a be Real ;
y = ( ( f * SN ) . x ) * ( ( f * SN ) . x ) ;
consider x being element such that x in an " A ;
assume r in ( ( dist o ) .: P ) ;
set i2 = ( n + 1 ) - h + 1 ;
h2 . ( j + 1 ) in rng h2 & h2 . ( j + 1 ) in rng h2 ;
Line ( M29 , k ) = M . i ;
reconsider m = ( x - 2 ) / 2 as Element of ( len x - 2 ) -tuples_on REAL ;
let U1 , U2 be Subspace of U0 , a be Element of U1 ;
set P = Line ( a , d ) ;
len p1 < len p2 + 1 & len p2 = len p1 + 1 ;
T1 , T2 be Scott Scott Scott TopAugmentation of L , a be Element of the topology of L ;
then x <= y implies ( x \ y ) c= ( x \ y ) \ ( x \ y ) ;
set M = n -{ m } , N = n -{ m } ;
reconsider i = x1 , j = x2 as Nat ;
rng ( ( the Arity of S ) * ( the Arity of S ) ) c= dom H ;
z1 " = z1 " & z1 " = z1 " * z1 " ;
x0 - r / 2 in L /\ dom f & f . x0 in L /\ dom f ;
then w is strict implies rng w /\ L <> {} ;
set x9 = ( x9 ^ <* Z *> ) ^ <* Z *> ;
len w1 in Seg ( len w1 + 1 ) & len w2 = len w1 + 1 ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of PFuncs ( V , { k } ) ;
x . n = ( |. a . n .| ) * ( |. b . n .| ) ;
( p `1 ) ^2 / ( |. p .| ) ^2 <= ( |. p .| ) ^2 / ( |. p .| ) ^2 ;
rng ( g " ) c= L~ ( g " ) \/ rng ( g " ) ;
reconsider k = i-1 * ( l + j ) as Nat ;
for n be Nat holds F . n is Seg ( n + 1 ) ;
reconsider x9 = x9 , y9 = y9 as Element of M ;
dom ( f | X ) = X /\ dom f ;
p , a // p , c & b , a // c , c ;
reconsider x1 = x as Element of REAL m , x2 be Element of REAL m ;
assume i in dom ( a * p ^ q ) ;
m . ( \hbox { \boldmath $ g $ } , g ) = p . ( ( \hbox { \boldmath $ g $ } , p ) , g ) ;
a / ( s . ( m + n ) ) * ( n + 1 ) <= 1 ;
S . ( n + k + 1 ) c= S . ( n + k ) ;
assume B1 \/ C1 = B2 \/ C2 & C2 \/ C1 = C2 \/ C1 ;
X . i = { x1 , x2 , x3 , x4 , x5 , M } . i ;
r2 in dom ( h1 + h2 ) & r2 in dom ( h1 + h2 ) ;
\cal R = a & b-0 - bR = b - a ;
F8 is_closed_on t1 , Q1 & P8 is_halting_on t1 , Q1 & F8 is_halting_on t2 , Q1 ;
set T = ) for X = -> InInIn0 ( X , x0 ) ;
Int Cl ( Int R ) c= Int Cl R & Cl R c= Cl R ;
consider y being Element of L such that c . y = x ;
rng ( FX ) = { FX . x } & rng FX c= dom ( FX . x ) ;
G-23 ( { c } ) c= B \/ S \/ S ;
f[#] A is Relation of [: X , X :] , X & fP is Function of X , X ;
set Rf = the Point of P , Rf = the Point of Q ;
assume n + 1 >= 1 & n + 1 <= len M ;
let k2 be Element of NAT , k be Element of NAT ;
reconsider p9 = u as Element of ( TOP-REAL n ) | ( ( TOP-REAL n ) | K1 ) ;
g . x in dom f & x in dom g implies f . x = g . x
assume that 1 <= n and n + 1 <= len f1 ;
reconsider T = b * N as Element of G / ( N , G ) ;
len ( P\mathopen ( 2 , n ) ) <= len ( P\mathopen ( 2 , n ) ) ;
x " in the carrier of A1 & x " in the carrier of A1 ;
[ i , j ] in Indices ( A * N ) ;
for m be Nat holds Re ( F ) . m is simple ;
f . x = a . i .= a1 . k .= a1 . k ;
let f be PartFunc of REAL-NS i , REAL-NS n , D be Subset of REAL n ;
rng f = the carrier of product A & f . 1 = product ( A --> { 1 } ) ;
assume s1 = sqrt ( 2 * ( p / 2 ) ) ^2 ) ;
pred a > 1 & b > 0 implies a |^ b > 1 ;
let A , B , C be Subset of [: I , J :] ;
reconsider X0 = X , Y1 = Y as non empty TopSpace ;
let f be PartFunc of REAL , REAL n , a be Real ;
r * ( v1 |-- I ) . X < r * 1 ;
assume that V is Subspace of X and X is Subspace of V ;
let tK , tK be Relation of the carrier of K ;
Q [ e-14 \/ { v-5 } , f ] ;
g \circlearrowleft ( W-min L~ z ) = z & g /. ( len z ) = z ;
|. |[ x , v ]| - |[ x , y ]| .| = v: y in REAL ;
- f . w = - ( L * w ) ;
z - y <= x iff z <= x + y & y <= z + x
sqrt ( 7 / p1 ^2 + 1 ) > 0 ;
assume X is BCK-algebra implies X is BCK-algebra & 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 ;
F . 1 = v1 & F . 2 = v2 & F . 3 = v1 ;
( f | X ) . x2 = f . x2 & ( f | X ) . x2 = f . x2 ;
( ( tan * tan ) `| Z ) . x in dom ( sec * cot ) ;
i2 = ( f /. len f ) & i1 = len f - 1 ;
X1 = X2 \/ ( X1 \ X2 ) & X2 \ X1 = X2 \ X1 ;
[. a , b , 1_ G .] = 1_ G implies a = b
let V , W be non empty addLoopStr , f be Function of V , W ;
dom ( ( g2 ) | [. 0 , 1 .] ) = the carrier of I[01] ;
dom ( f2 ) = the carrier of I[01] & dom ( f2 ) = the carrier of I[01] ;
( proj2 | X ) .: X = proj2 .: X & ( proj2 | X ) .: X = proj2 .: X ;
f . ( x , y ) = h1 . ( x `1 , y `2 ) ;
x0 - r < a1 . n & a1 . n < x0 + r ;
|. ( f /* s ) . k - Gx0 .| < r ;
len Line ( A , i ) = width A & width ( A @ ) = width A ;
SY / ( g , f ) = ( S . g ) / ( g , f ) ;
reconsider f = v + u as Function of X , the carrier of Y ;
( intloc 0 ) in dom Initialized Initialized ( p +* I ) & ( Initialized ( p +* I ) ) . a = 1 ;
i1 , i2 & ( i3 , i2 ) := ( b2 , b3 ) & not ( b1 , b2 , c , d is_collinear & b1 , b2 , c is_collinear & b2 , c , d is_collinear ) ;
exp_R + ( 1 / 2 ) * ( 1 / 2 ) = ( cos . ( PI / 2 ) ) ^2 ;
for x st x in Z holds f2 is_differentiable_in x & f2 is_differentiable_in x ;
reconsider q2 = ( q - x ) / 2 as Element of REAL ;
( 0 qua Nat ) + 1 <= i + j1 & 1 <= j + 1 ;
assume f in the carrier of [ X \to [#] Y , [#] Y ] ;
F . a = H / ( ( x , y ) / ( x , y ) ) ;
( TRUE T , u ) -at ( C , u ) = TRUE & ( T , u ) -Let ( C , u ) = TRUE ;
dist ( ( a * seq ) . n , h ) < r ;
1 in the carrier of [. 0 , 1 .] & 1 <= len G ;
( p2 `1 ) ^2 - x1 > - g ^2 / 2 - g ^2 ;
|. r1 - ] = |. a1 .| * |. thesis .| ;
reconsider S-14 = 8 as Element of Seg 8 ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b
DkW = DW .on ( 2 |^ n ) + 1 ;
i1 = ma + n & i2 = K + n & i1 = K + n ;
f . a [= f . ( f .: O1 "\/" ( f . a ) ) ;
pred f = v & g = u & f + g = v + u ;
I . n = Integral ( M , F . n ) ;
( chi ( T , T1 ) ) . s = 1 / ( s . s ) ;
a = VERUM ( A ) or a = VERUM ( A ) or a = VERUM ( A ) ;
reconsider k2 = s . b3 as Element of NAT ;
( Comput ( P , s , 4 ) ) . GBP = 0 ;
L~ M1 meets L~ ( R1 + R2 ) & L~ ( R1 + R2 ) meets L~ ( R2 + R2 ) ;
set h = the continuous Function of X , ( the carrier of X ) | A ;
set A = { L . ( k9 . n ) : not contradiction } ;
for H st H is negative holds PK [ H ] ;
set b9 = S5 ^\ ( i + 1 ) , S = ( i + 1 ) + 1 ;
Hom ( a , b ) c= Hom ( a opp , b opp ) ;
( 1 / n + 1 ) < ( 1 / s ) " ;
( l ) `1 = [ dom l , cod l ] & ( l ) `2 = cod l ;
y +* ( i , y /. i ) in dom g ;
let p be Element of CQC-WFF ( Al ) , A be Subset of D ;
X /\ X1 c= dom ( f1 - f2 ) /\ dom ( f1 - f2 ) ;
p2 in rng ( f /^ p1 ) \/ rng ( f /^ p2 ) ;
1 <= indx ( D2 , D1 , j1 ) & j1 <= indx ( D2 , D1 , j1 ) ;
assume x in ( ( K /\ ( 3 + 1 ) ) \/ ( K /\ ( 3 + 1 ) ) ) ;
- 1 <= ( ( f2 . O ) `2 ) / ( 1 - ( f2 . O ) `2 ) ;
let f , g be Function of I[01] , TOP-REAL 2 , a , b , c be Real ;
k1 - k2 = k1 - k2 & k1 - k2 = k2 - k2 + k2 ;
rng seq c= ]. x0 - r , x0 + r .[ implies seq is convergent & lim seq = lim ( seq + r )
g2 in ]. x0 - r , x0 + r .[ & g2 in ]. x0 - r , x0 + r .[ ;
sgn ( p `1 , K ) = - 1_ K & sgn ( p `1 ) = - 1_ K ;
consider u being Nat such that b = ( p |^ y ) * u ;
ex A being \/ the carrier of T st a = Sum A & A c= B ;
Cl ( union ( H ) ) = union ( Cl ( H ) ) & Cl ( H ) c= union ( H ) ;
len t = len t1 + len t2 & len t1 = len t2 + len t2 ;
v-29 = v + w |-- A + AA ;
v <> DataLoc ( ( t . GBP ) , 3 ) & v . DataLoc ( t . GBP , 3 ) = s . DataLoc ( t . GBP , 3 ) ;
g . s = ( sup d ) " { s } ;
( \dot y ) . s = s . ( y | s ) ;
{ s : s < t & t <= s } c= NAT implies t = {} ;
s ` \ s = s ` \ ( 0. X ) \ ( 0. X ) ;
defpred P [ Nat ] means B + $1 in A & A c= B + $1 ;
( 339 + 1 ) ! = 33222222299 * ( 33+ 1 ) ;
U U /. ( succ A ) = ( 1_ U ) . ( A , A ) ;
reconsider y = y as Element of ( len y ) -tuples_on COMPLEX ;
consider i2 being Integer such that y0 = p * i2 and i2 in dom f ;
reconsider p = Y | Seg k as FinSequence of ( Seg k ) \ { i } ;
set f = ( S , U ) \mathop ( I , z ) ;
consider Z being set such that lim s in Z and Z in F ;
let f be Function of I[01] , TOP-REAL n , R^1 , a be Real ;
( SAT M ) . [ n + i , 'not' A ] <> 1 ;
ex r being Real st x = r & a <= r & r <= b ;
R1 , R2 be Element of REAL n , R2 be Element of REAL n ;
reconsider l = 0. ( V ) as Linear_Combination of A ;
set r = |. e .| + |. w .| + |. s .| ;
consider y being Element of S such that z <= y and y in X ;
a ' 'or' ( b 'or' c ) = 'not' ( ( a 'or' b ) 'or' c ) ;
||. ( x9 - g ) . 2 - g .|| < r2 - g / 2 ;
b9 , a9 // b9 , c9 & b9 , c9 // c9 , c9 ;
1 <= k2 -' k1 & k2 + 1 = k2 - k1 + 1 & k2 + 1 = k2 - k1 + 1 ;
sqrt ( ( p `2 / |. p .| - sn ) / ( 1 - sn ) ) ^2 >= 0 ;
sqrt ( ( q `2 / |. q .| - sn ) / ( 1 - sn ) ) ^2 ) < 0 ;
E-max C in right_cell ( R , 1 , 1 ) & E-max C in rng R ;
consider e being Element of NAT such that a = 2 * e + 1 ;
Re ( lim F ) | D = Re ( lim G ) .= Re ( lim G ) ;
LIN b , a , c or LIN b , c , a ;
p `1 , a // a `1 , b `1 or p `1 , a `2 = b ;
g . n = a * Sum ( f | n ) .= f . n ;
consider f being Subset of X such that e = f and f is being w ;
F | ( N2 ~ ) = CircleMap * ( F | N2 ) .= CircleMap * ( F | N2 ) ;
q in LSeg ( q , v ) \/ LSeg ( v , p ) ;
Ball ( m , r0 ) c= Ball ( m , s ) & Ball ( m , r ) c= Ball ( m , s ) ;
the carrier of (0). V = { 0. V } & the carrier of W = { 0. V } ;
rng ( ( - 1 ) (#) cos ) = [. - 1 , 1 .] & dom ( ( - 1 ) (#) cos ) = [. - 1 , 1 .] ;
assume Re ( seq ) is summable & Im ( seq ) is summable ;
||. ( vseq . n ) - ( tseq . n ) .|| < e ;
set g = O --> 1 ;
reconsider t2 = t11 as 0 -started string of S2 , S2 = ( the carrier of S2 ) | D ;
reconsider x9 = seq as sequence of REAL-NS n , s1 , s2 be Element of REAL n ;
assume that that that C meets L~ go and L~ pion1 meets L~ pion1 and p in L~ pion1 and q in L~ pion1 and q in L~ pion1 and q in L~ pion1 and q in L~ pion1 and p in L~ pion1 and q in L~ pion1 and p in L~ pion1 and q in L~ pion1 ;
- ( ( 1 - sqrt 5 ) / 2 ) < F . n - ( 1 / 2 ) ;
set d1 = being dist of ( |. x1 .| ) * ( z1 , z2 ) , d2 = |. z1 .| * ( z2 , z2 ) ;
2 |^ ( q -' 1 ) = 2 |^ ( q -' 1 ) - 1 ;
dom ( vK ) = Seg ( len dK ) & dom ( vK ) = Seg len ( K ) ;
set x1 = - k2 + |. k2 .| + 2 * ( - k1 + k2 ) ;
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 ( Carrier ( l ) + L2 ) c= I & the carrier of ( Carrier ( l ) ) \/ { v } ;
'not' All ( x , p ) => All ( x , p ) is valid ;
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the normal w.r.t. of A ;
Z c= dom ( ( - sin * f1 ) ^ ) /\ dom ( ( - cos * f1 ) ^ ) ;
|. 0. TOP-REAL 2 - ( q `2 / |. q .| - sn ) .| < r / 2 ;
ConsecutiveSet2 \ { B , ConsecutiveSet ( A , succ d ) } c= ConsecutiveSet2 ( A , \ { d } ) ;
E = dom ( L | E ) & L | E is_measurable_on E & L | E is_measurable_on E ;
C |^ ( A + B ) = C |^ B * C ;
the carrier of W2 c= the carrier of V & the carrier of W1 c= the carrier of V ;
I . IC Comput ( P , s , 2 ) = P . IC Comput ( P , s , 2 ) ;
pred x > 0 means : Def1 : x = x |^ ( - 1 ) ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) ;
consider p being Point of T such that C = [. p , R .] and p in A ;
b , c are_connected & - C , - C are_connected implies - C , - C are_connected
assume f = id ( the carrier of C ) & g is Function of ( the carrier of C ) , the carrier of C ;
consider v such that v <> 0. V and f . v = L * v ;
let l be Linear_Combination of {} ( the carrier of V ) , a be Element of V ;
reconsider g = f " as Function of U2 , U1 , U2 ;
A1 in the topology of ( G . k ) & A2 : A1 c= the topology of ( G . k ) ;
|. - x .| = - ( - x ) .= - x .= - x .= - x ;
set S = many sorted set , A = many sorted set ;
Fib ( n ) * ( 5 * Fib n ) >= 4 * ( 5 * Fib n ) ^2 ;
vk /. ( k + 1 ) = vk . ( k + 1 ) ;
0 mod i = ( i * ( 0 qua Nat ) ) mod ( i * ( 0 qua Nat ) ) ;
Indices M1 = [: Seg n , Seg n :] & len M1 = width M2 ;
Line ( S\mathopen , j ) = S\mathopen ( j , i ) ;
h . ( x1 , y1 ) = [ y1 , x1 ] & h . ( y1 , y2 ) = y1 ;
|. f .| - ( Re ( |. f .| ) (#) ( ( b (#) h ) ) ) ) is nonnegative ;
assume x = ( a1 ^ <* x1 *> ) ^ b1 & y = ( a1 ^ b1 ) ^ b2 ;
MW is_closed_on IExec ( I , P , s ) , P & MW is_halting_on s , P ;
DataLoc ( t3 . a , 4 ) = intpos ( 0 + 4 ) ;
x + y < - x + y & |. x - y .| = - x + y ;
LIN c , q , b & LIN c , q , c ;
f^ . ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + y1 + ( y1 + z1 ) .= x + y ;
f_ { let a } . a = f_ { a } & v in InputVertices S & v in InputVertices S ;
( p `1 ) ^2 / ( ( E-max C ) ^2 ) <= ( E-max C ) ^2 / ( ( E-max C ) ^2 ) ;
set R8 = Cage ( C , n ) \circlearrowleft E8 , E8 = Cage ( C , n ) \circlearrowleft E8 ;
( p `1 ) ^2 >= ( ( E-max C ) `1 ) ^2 / ( ( E-max C ) `1 ) ^2 ) ;
consider p such that p = p9 and s1 < p & p in P and s < p ;
|. ( f /* ( s * F ) ) . l - ( G . ( s * F ) ) . l .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len Line ( N , k + 1 ) = width N & width N = width N ;
f1 /* s1 is convergent & f2 /* s1 is convergent & lim ( f1 /* s1 ) = lim ( f1 /* s1 ) ;
f . x1 = x1 & f . y1 = y1 & f . y2 = y2 ;
len f <= len f + 1 & len f + 1 <> 0 ;
dom ( Proj ( i , n ) * s ) = REAL m & rng ( Proj ( i , n ) * s ) = REAL m ;
n = k * ( 2 * t ) + ( n mod 2 ) ;
dom B = 2 -tuples_on the carrier of V & rng B 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 / 2 in [. 1 / 2 , 1 .] ;
for L being complete LATTICE holds ex a , b being Element of ex c being Function st L , a are_isomorphic & a , b are_isomorphic
[ gi , gj ] in [: I , I :] \ I & gj = gj \ I ;
set S2 = 1GateCircStr ( x , y , c ) ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 ;
reconsider y = ( a ` ) / ( F . n ) as Element of L ;
dom s = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8
( min ( g , ( 1 - g ) ) ) . c <= h . c ;
set GG2 = the subgraph of G , GG2 = the subgraph of G , GG2 = the as Vertex of G ;
reconsider g = f as PartFunc of REAL n , REAL-NS n , REAL-NS n ;
|. s1 . m - p .| / |. ( p . m ) - p .| < d / ( p . m ) - p / ( p . m ) ;
for x being element st x in ( ~ u ) holds x in ( ~ t ) . x
P = the carrier of ( TOP-REAL n ) | K1 & Q = the carrier of ( TOP-REAL n ) | K1 ;
assume p10 in LSeg ( p1 , p2 ) /\ LSeg ( p10 , p2 ) /\ LSeg ( p10 , p2 ) ;
( 0. X \ x ) |^ ( m + k ) = 0. X ;
let g be Element of Hom ( cod f , cod g ) ;
2 * a * b + ( 2 * c ) * d <= 2 * C1 * C2 ;
f , g , h be PartFunc of the carrier of X , the carrier of Y , z be Element of X ;
set h = Hom ( a , g (*) f ) ;
then idseq ( n ) | Seg m = idseq ( m ) & m <= n ;
H * ( g " * a ) in the right of H & H * ( g " * a ) in the carrier of H ;
x in dom ( ( - 1 ) (#) ( ( #Z 2 ) * ( exp_R ^ ) ) ) ;
cell ( G , i1 , j2 -' 1 ) misses C & cell ( G , i1 , j1 -' 1 ) misses C ;
LE q2 , p2 , P & LE q2 , p2 , P & LE q2 , p2 , P & LE q2 , p2 , P ;
attr B is Let of A means : Def6 : B c= BDD A & B c= BDD B ;
deffunc D ( set , set , set ) = union rng $2 & $2 = $2 ;
n + - n < len ( p ^ <* n *> ) + ( n - 1 ) ;
pred a <> 0. K means : Def7 : the_rank_of ( M * ( a , M ) ) = the_rank_of ( a * M ) ;
consider j such that j in dom ' ' and I = len ' - j + 1 ;
consider x1 such that z in x1 and x1 in PK and x = [ x1 , y1 ] ;
for n ex r being Element of REAL st X [ n , r ]
set C1 = Comput ( P2 , s2 , i + 1 ) , C1 = Comput ( P2 , s2 , i + 1 ) , C1 = P2 ;
set \cal v = 3 / ( 2 |^ { a , b , c } ) , w = 3 / ( 2 |^ { a , b , c } ) , y = 2 |^ { a , b , c } , z = - 1 / ( 2 |^ { a , b , c } ) , w =
conv @ W c= union ( F .: ( E .: W ) ) & card ( F .: ( E .: W ) ) c= card ( F .: ( E .: W ) ) ) ;
1 in [. - 1 , 1 .] /\ dom ( ( #Z 2 ) * ( ( #Z 2 ) * ( #Z 2 ) ) ) ;
r3 <= s2 + ( r2 - r2 ) * ( ( 1 - r2 ) * ( 1 - r2 ) ) ;
dom ( f (#) ( f3 (#) ( f1 + f2 ) ) ) = dom f /\ dom ( f3 (#) ( f1 + f2 ) ) ;
dom ( f * G ) = dom ( l (#) F ) /\ Seg k .= Seg ( k + 1 ) ;
rng ( s ^\ k ) c= dom f1 \ { x0 } & rng ( s ^\ k ) c= dom f2 \ { x0 } ;
reconsider g9 = gp as Point of TOP-REAL n1 , p1 , p2 be Point of TOP-REAL n1 ;
( T * h . s ) . x = T . ( h . s ) ;
I . ( L . ( J . x ) ) = ( I * L ) . ( J . x ) ;
y in dom an an an dom <* *> implies ( commute ( A ) ) . ( ( Frege ( A ) ) . o ) = ( ( A . o ) ) . ( ( A . o ) ) ;
for I being non degenerated commutative commutative commutative commutative commutative non empty doubleLoopStr holds I is commutative iff I is commutative
set s2 = s +* ( ( intloc 0 ) .--> 1 ) , P2 = P +* ( ( intloc 0 ) .--> 1 ) , P3 = P +* ( ( intloc 0 ) .--> 1 ) , s4 = Comput ( P +* ( a , I ) +* ( 1 , I ) +* ( 1 + 1 ) , P4 = Comput ( P
P1 /. IC s1 = P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 ;
lim S1 in the carrier of [. a , b .] & for x be Element of REAL st x in the carrier of I[01] holds x in dom ( ( a (#) b ) | [. a , b .] )
v . ( l-13 . i ) = ( v *' lw ) . i ;
consider n be element such that n in NAT and x = ( sn Sorts ) . n ;
consider x be Element of c such that F1 . x <> F2 ( x ) and F1 . x <> 0 ;
card X ( X , 0 , x1 , x2 , x3 , x4 , x5 , x5 , F , J ) = { E } ;
j + ( 2 * k9 ) + m1 > j + ( 2 * ( k + 1 ) ) ;
{ s , t } on A3 & { s , t } on A3 ;
n1 > len crossover ( p2 , p1 , n1 , n2 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n2 , n3 , n2 , n3 , n3 , n2 , n3 , n2 , n3 , n3 , n2 , n3 , n2 , n3 , n2
( mf1 ) . ( HT ( ( ( mf2 ) ) , T ) ) = 0. L ;
then H1 , H2 are_are that ( H1 , H2 are_are Indices H1 ) & ( H1 , H2 are_are are are are are are are are are are are are are are are are are are are holds ( H1 , H2 ) , ( H1 , H2 ) are__ { H1 , H2 }
( ( N-min L~ f ) | ( L~ f ) ) .. f > 1 & ( f | ( len f ) ) .. f > 1 ;
]. s , 1 .[ = ]. s , 2 .] /\ [. 0 , 1 .] ;
x1 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) ) & x2 in ( L~ g ) /\ ( L~ g ) ;
let f1 , f2 be continuous PartFunc of REAL , REAL , x0 be Element of REAL n ;
DigA ( ta1 , z9 ) is Element of k -tuples_on k -tuples_on k -tuples_on k * ;
I = d2222I & I = I & I = k2 or I = k2 & I = k2 & I = k2 ;
uK ~ = { [ a , uK ] } & uK in { [ a , uK ] } ;
( w | p ) | ( p | ( w | w ) ) = p ;
consider u2 such that u2 in W2 and x = v + u2 and u2 in W2 and x = v + u2 ;
for y st y in rng F ex n st y = a |^ n & P [ y ]
dom ( ( g * ( {} , C ) ) | K ) = K ;
ex x being element st x in ( ( the Sorts of U0 ) \/ A ) . s & x in ( the Sorts of U1 ) . s ;
ex x being element st x in ( ( ( ( ( $1 \/ A ) \/ B ) \/ C ) \/ A ) . s ) & x in B ;
f . x in the carrier of [. - r , r .] & f . x in [. - r , r .] ;
( the carrier of X1 union X2 ) /\ ( the carrier of X2 ) <> {} & the carrier of X1 = the carrier of X2 ;
L1 /\ LSeg ( p10 , p2 ) c= { p10 } /\ { p10 } ;
sqrt ( b + ( be ) ^2 ) in { r : a < r & r < b + ( be ) } ;
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 GK such that z = y and P [ z ] ;
( the sequence of ( \overline the carrier of M ) ) . ( x , y ) <= e ;
len ( w ^ w2 ) + 1 = len w + 2 + 1 .= len w + 2 ;
assume q in the carrier of ( TOP-REAL 2 ) | K1 & ( sn D ) | K1 = ( sn -FanMorphE ) | K1 ;
f | E-4 ` = g | EK ` .= ( f | EK ) | EK ` .= ( f | EK ) | EK ;
reconsider i1 = x1 , i2 = x2 , j1 = y2 as Element of NAT ;
( a * A ) ` = ( a * ( A * B ) ) ` ;
assume ex n2 being Element of NAT st f |^ n2 is \mathop { 0 , 1 } ;
Seg len ( ( for f being FinSequence st f in dom ( ( for i being Nat holds f . i = ( F ^ <* x *> ) ) . i ) holds ( ( F ^ <* x *> ) . i ) . i = ( F ^ <* x *> ) . i
( Complement ( Complement A1 ) ) . m c= ( Complement A1 ) . n \/ ( Complement A1 ) . m ;
f1 . p = p9 & g1 . ( p9 . p ) = d & g1 . ( p9 . p ) = d ;
FinS ( F , Y ) = FinS ( F , dom ( F | Y ) ) , C = FinS ( F , Y ) ;
( x | y ) | z = z | ( y | x ) ;
( ( |. x .| to_power n ) * ( cos . n ) ) ^2 <= ( ( r2 to_power n ) * ( cos . n ) ) ^2 ;
Sum ( F ) = Sum f & dom ( F ) = dom g & rng ( F ) c= dom ( F ) ;
assume for x , y being set st x in Y holds x /\ y in Y ;
assume W1 is Subspace of W2 & W2 is Subspace of W1 & W1 is Subspace of W2 & W2 is Subspace of W1 ;
||. ( t-15 . x ) .|| = lim ( ( x - y ) * ( x - y ) ) ;
assume that i in dom D and f | A is lower and g | A is lower ;
sqrt ( ( p `2 ) ^2 + ( p `2 ) ^2 ) <= sqrt ( ( p `2 ) ^2 ) ;
g | Sphere ( p , r ) = id ( the carrier of TOP-REAL 2 ) & g | Sphere ( p , r ) = id the carrier of TOP-REAL 2 ;
set NN = ( N-min L~ Cage ( C , n ) ) .. ( Cage ( C , n ) ) ;
for T being non empty TopSpace holds T is countable implies the TopStruct of T is countable ;
width B |-> 0. K = Line ( B , i ) .= width ( B @ ) .= width ( B @ ) ;
pred a <> 0 implies ( A /\ B ) Y. = ( A Y. ) with ( B Y. ) Y. ;
then f is_partial differentiable of u , 3 & pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 & pdiff1 ( f , 2 ) is_partial_differentiable_in u , 3 ;
assume that a > 0 and a <> 1 and b <> 0 and c <> 0 and c <> 0 ;
w1 , w2 in Lin { w1 , w2 , w2 , w1 , w2 , w2 , w2 , w1 , w2 } ;
p2 /. IC Comput ( p2 , s2 , k ) = p2 . IC Comput ( p2 , s2 , k ) .= ( card I + 1 ) ;
ind ( T-10 | b ) = ind b .= ind b .= ind b ;
[ a , A ] in the InternalRel of G_ ( k , X ) & [ a , A ] in the InternalRel of G_ ( k , X ) ;
m in ( the Arrows of C ) . ( o1 , o2 ) & m in ( the Arrows of C ) . ( o1 , o2 ) ;
( ( a 'imp' CompF ( PA , G ) ) ) . z = FALSE & ( a 'imp' CompF ( PA , G ) ) . z = FALSE ;
reconsider phi = phi , phi = phi as Element of phi ;
len s1 - ( len s2 - 1 ) + 1 > 0 + 1 - 1 ;
\delta ( D ) * ( f . ( sup A ) - lower_bound A ) < r ;
[ f21 , f22 ] in the InternalRel of A & f22 = f22 ;
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 g2 . z = x ;
[#] V1 = { 0. V1 } .= the carrier of V1 .= the carrier of V1 .= the carrier of V1 ;
consider P2 be FinSequence such that rng P2 = M and P2 is one-to-one and P2 is one-to-one ;
assume that x1 in dom ( f | X ) and ||. x1 - x0 .|| < s and ||. x1 - x0 .|| < s ;
h1 = f ^ <* p3 *> ^ ( <* p *> ^ <* p1 *> ) .= h ^ <* p1 *> ^ ( <* p2 *> ^ <* p2 *> ) .= h1 ^ <* p2 *> ;
c / ( b , c ) = c .= c / ( a , c ) .= c / ( a , c ) ;
reconsider t1 = p1 , t2 = p2 , t2 = p1 as Element of C ( ) ;
sqrt ( 1 - ( 2 * ( 1 - 2 ) ) ^2 ) in the carrier of ( TOP-REAL 2 ) | K1 ;
ex W being Subset of X st p in W & W is open & h .: W c= V ;
( h . p1 ) `2 = C * ( p1 `2 ) + D * ( p2 `2 ) + D * ( p1 `2 ) ;
R . ( b ~ ) = 2 * r2 .= r2 * r2 .= r2 * r2 ;
consider I1 such that B = ( - 1 ) * C + ( - 1 ) * A and 0 <= I1 & I1 <= 1 ;
dom g = dom ( ( the Sorts of A ) * ( a , b ) ) & dom ( ( the Sorts of A ) * ( a , b ) ) = dom ( ( the Sorts of A ) * ( a , b ) ) ;
[ P . ( l + 1 ) , P . ( l + 1 ) ] in => ( T . ( l + 1 ) , T . ( l + 1 ) ) ;
set s2 = Initialize s , P2 = P +* I ;
reconsider M = mid ( z , i2 , i1 ) , i1 = mid ( z , i1 , i2 ) as Element of REAL 2 ;
y in product ( ( the support of J ) +* ( V , { 1 } ) ) ;
1 / ( 0 , 1 ) = 1 & 0 / ( 0 , 1 ) = 0 ;
assume x in the left of g or x in the left of g or x in the right of g or x in the right of g ;
consider M being strict Subspace of A\mathopen { A } , T being strict Subgroup of M such that a = M and T is Subspace of T ;
for x st x in Z holds ( ( ( #Z n ) * f ) + ( #Z n ) * f ) . x <> 0 ;
len W1 + len W2 = 1 + len W2 + len W1 .= len ( W1 + W2 ) + len W1 .= len ( W1 + W2 ) + len W2 ;
reconsider h1 = ( vseq . n - t-16 ) . ( t-16 . n ) as Lipschitzian LinearOperator of X , Y ;
( i mod len ( p + q ) ) + 1 in dom ( p + q ) ;
assume that s2 is negative and F in the |= of the |= of ( the |= of ( s2 ) ) and F in the |= of the |= of ( s2 ) and F in the |= of ( s2 ) ;
( ( the _ of SCMPDS ) . ( x , y ) ) * ( x , y ) = gcd ( x , y , z ) ;
for u being element st u in Bags n holds ( p + m ) . u = p . u + m . u
for B being Subset of u-5 st B in E holds A = B or A misses B or A misses B ;
ex a being Point of X st a in A & Cl ( { y } ) = { a } ;
set W2 = ( p \/ W1 ) \/ ( p \/ W2 ) ;
x in { X where X is Ideal of L : for x being Element of L holds x in X iff x in X } ;
the carrier of W1 /\ W2 c= the carrier of W1 & the carrier of W1 /\ W2 c= the carrier of W2 ;
( 1 + a ) * ( id a ) = ( 1 + a ) * ( id a ) ;
( dom ( X --> f ) ) . x = ( X --> dom f ) . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) ;
p => ( q => r ) => ( p => q ) in TAUT ( A ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
- 1 + 1 <= ( ( $1 + 2 ) |^ ( n -' m ) ) + 1 ;
( reproj ( 1 , z0 ) ) . x in dom ( f1 (#) f2 ) & ( f1 (#) f2 ) . x in dom ( f1 (#) f2 ) ;
assume that b1 . r = { c1 } and b2 . r = { c2 . r } and b2 . r = { c2 . r } ;
ex P st a1 on P & a2 on P & a1 on P & a2 on P & a1 on P & a2 on P & a1 on P & a2 on P & a1 on P & a2 on P & a1 on P & a2 on P & a1 on P & a2 on P & a1 on P & a2 on P & a1 on P & a2 on P
reconsider gf = g * f as strict Subgroup of X * g , h = g * g as strict Subgroup of X ;
consider v1 being Element of T such that Q = ( downarrow v1 ) ` and v1 in the topology of T ;
n in { i where i is Nat : i < n2 + 1 & i < n + 1 + 1 & i < n + 1 + 1 + 1 ;
( F /. ( i , j ) ) `2 >= ( F /. ( m , k ) ) `2 ;
assume K1 = { p : ( p `1 / |. p .| - cn ) / ( 1 - cn ) <= cn & p <> 0. TOP-REAL 2 } ;
ConsecutiveSet ( A , succ O1 ) = ( ConsecutiveSet ( A , O1 ) ) . ( O1 , O1 ) ;
set I1 = Macro ( a , intloc 0 ) , I1 = AddTo ( a , intloc 0 ) , I1 = goto 0 , I1 = goto 0 , I1 = goto 0 , I1 = goto 0 , I1 = goto 0 , I1 = goto 0 , I1 = goto 0 , I2 = goto 0 , I2 = goto 0 , I1 = goto 0 , I1 = goto 0 ,
for i be Nat st 1 < i & i < len z holds z /. i <> z /. 1 ;
X c= ( the carrier of L1 ) \/ ( the carrier of L2 ) & the carrier of L1 c= the carrier of L2 ;
consider x9 being Element of GF ( p ) such that x9 |^ 2 = a and x9 |^ 2 = a |^ 2 ;
reconsider eN = eN , fN = fN as Element of D ( ) ;
ex O being set st O in S & C1 c= O & M . O = 0. ( Cl O ) ;
consider n be Nat such that for m be Nat st n <= m holds S . m in U1 and S . m in U2 ;
f * g * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) * ( x - x0 ) ) . x ;
defpred P [ Nat ] means A + ( $1 + 1 ) = succ A & A = ( $1 + 1 ) + 1 ;
the left of ( - g ) = the left of ( - g ) & the right of ( - g ) = ( - g ) * ( ( - g ) * ( - g ) ) ;
reconsider p9 = x , p9 = y , q9 = z as Point of TOP-REAL 2 , s = z as Real ;
consider g2 such that g2 = y and x <= g2 & g2 <= x and g2 <= y and g2 <= x ;
for n being Element of NAT ex r being Element of REAL st X [ n , r ]
len ( x2 ^ y2 ) = len x2 + len y2 & len ( x2 ^ y2 ) = len x2 + len y2 ;
for x being element st x in X holds x in the set of \HM { the } \HM { set } : not contradiction } & ( ex n st x in X & not x in X & x in X ) ;
LSeg ( p10 , p2 ) /\ LSeg ( p1 , p2 ) = {} or LSeg ( p10 , p2 ) /\ LSeg ( p1 , p2 ) = {} ;
func ) ( X ) -> set means : Defit : for x being set holds x in it iff x in it iff x in X & x in X & x in X ;
len ( ( C | ( len C | 1 ) ) ^ ( C /^ 1 ) ) ) <= len ( ( C /^ 1 ) ^ ( C /^ 1 ) ) ;
pred K is every doubleLoopStr means a <> 0. K & v . ( a |^ i ) = i * v . ( a |^ i ) ;
consider o being OperSymbol of S such that t `1 . {} = [ o , the carrier of S ] and o in the carrier' of S ;
for x st x in X ex y st x c= y & y in X & y is Set of f
IC Comput ( P1 , s1 , k ) in dom ( Comput ( P2 , s2 , k ) ) ;
pred q < s & r < s implies ]. r , s .] c= ]. p , q .] & s <= q & q <= s & s <= q ;
consider c being Element of Class ( F . c , F . c ) such that Y = ( F . c ) * ( F . c ) ;
func the ResultSort of S2 -> ResultSort of the carrier of S2 means : Def4 : the ResultSort of S2 = the ResultSort of S2 & the ResultSort of S2 = the ResultSort of S2 ;
set y9 = [ <* y , z *> , f2 ] , z9 = [ <* z , x *> , f3 ] ;
assume x in dom ( ( ( #Z 2 ) * ( exp_R + arccot ) ) ^ ) /\ dom ( ( #Z 2 ) * ( exp_R + arccot ) ) ) ;
r-7 in Int cell ( GoB f , i , width GoB f ) \ { ( GoB f ) * ( i , 1 ) + ( GoB f ) * ( i , 1 ) + ( GoB f ) * ( i + 1 , 1 ) } ;
( q `2 ) ^2 >= ( ( Cage ( C , n ) ) /. ( i + 1 ) ) `2 ;
set Y = { a "/\" a ` : a in X } ;
i - len f <= len f + len f1 - len f + len f - len f + len f - len f + len f - len f + len f - len f + len f - len f + len f -' len f + 1 - len f + 1 - len f + 1 - len f + 1 - len f - len f + 1 - len f + 1 - len f +
for n holds ex x st x in N & x in N1 & h . n = x- ( x0 - h . n )
set s0 = ( Let ( a , I , p , s ) ) . i , s1 = ( s , p ) . i , s2 = ( s , p ) . i , s2 = ( s , p ) . i , s2 = ( s , p ) . i , s3 = ( s , p ) . i , s3 = ( s , p ) . i , s3 = ( s
( p . k ) . 0 = 1 or ( p . k ) . 0 = - 1 or ( p . k ) . 1 = - 1 ;
u + Sum L-18 in ( U \ { u } ) \/ { u + Sum L-18 } ;
consider x9 being set such that x in x9 and x9 in V1 and x9 in V1 and x9 in V1 and y = [ x9 , y9 ] ;
( p ^ ( q | k ) ) . m = ( q | k ) . ( len p + k ) ;
g + h = gg + hh & A1 + A1 = h + g + h + ( h + g ) ;
L1 is distributive & L2 is distributive implies L1 ~ is distributive & L1 ~ is distributive & L2 ~ is distributive & L1 ~ is distributive & L1 ~ is distributive & L2 ~ is distributive
pred x in rng f & y in rng ( f | x ) implies f | y = f | x ;
assume that 1 < p and ( 1 + ( p `1 / p `2 ) ^2 ) = 1 and 0 <= a & a <= b & b <= 1 ;
F* ( f , the ^ 2 ) = rpoly ( 1 , the ^ the 0. of F_Complex ) *' .= ( 1 , the carrier of F_Complex ) *' ;
for X being set , A being Subset of X holds A ` = {} implies A = {} & A = {} & A = {} ;
( ( NW-corner X ) `1 ) ^2 / ( ( NW-corner X ) `2 ) ^2 <= ( ( NW-corner X ) `1 ) ^2 / ( ( NW-corner X ) `2 ) ^2 ) ;
for c being Element of the \rbrack of the \rbrack of A , a being Element of the Sorts of A holds c <> a implies c <> a
s1 . GBP = ( Exec ( i2 , s2 ) ) . DataLoc ( s2 . GBP , 2 ) .= s2 . DataLoc ( s2 . GBP , 2 ) .= s2 . DataLoc ( s2 . GBP , 2 ) .= s2 . DataLoc ( s2 . GBP , 2 ) .= s2 . DataLoc ( s2 . GBP , 2 ) ;
for a , b being Real holds [ a , b ] in ( y iff b >= 0 ) & ( y >= 0 implies y = x ) implies y = x
for x , y being Element of X holds x ` \ y = ( x \ y ) ` & y \ x = y \ x ;
mode BCK-algebra of i , j , m , n , m , n be Element of NAT , i , j be Element of NAT ;
set x2 = |( ( Re y ) , ( Im x ) ^2 , ( Im y ) ^2 , ( Im y ) ^2 , ( Im y ) ^2 , ( Im y ) ^2 , ( Im y ) ^2 ) ;
[ y , x ] in dom u5 & u5 . ( y , x ) = g . y ;
]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .] c= A & A = [. lower_bound A , upper_bound A .] ;
0 <= ( \delta ( S ) . n ) & |. ( \delta ( S ) . n ) .| < ( e / 2 ) |^ n ;
( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) <= ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ;
set A = ( 2 / b ) / 2 ;
for x , y being set st x in R" holds x , y are_are_holds x , y are_are_to x
deffunc F2 ( Nat ) = b . ( $1 + 1 ) * ( M * G ) . ( $1 + 1 ) * ( M * G ) . ( $1 + 1 ) ) ;
for s being element holds s in -> Element of S iff s in -> Element of S \/ ( -> Element of S ) \/ ( -> Element of S \/ S ) ;
for S being non empty non void non empty non void non empty ManySortedSign st S is connected holds S is connected ;
max ( degree ( z ) , degree ( z ) ) >= 0 & degree ( z ) >= 1 ;
consider n1 be Nat such that for k holds seq . ( n1 + k ) < r + s ;
Lin ( A /\ B ) is Subspace of Lin ( A ) & Lin ( B ) is Subspace of Lin ( A ) & Lin ( B ) = Lin ( B ) ;
set n-15 = n-15 ( M . ( x , y ) qua Element of BOOLEAN ) , n-15 = ( M . ( x , y ) ) . ( y , z ) ;
f " V in ( the topology of X ) & f " V in D & f " V in D & f " V in D & f " V in D implies f " V in D
rng ( ( a (#) c ) \mathbin { 1 } ) c= { a , c , b , c } ;
consider y being Wsubgraph of G1 such that y ` = y and dom y ` = WWholds y ` in WWWf ;
dom ( ( 1 / 2 ) (#) ( f ^ ) ) /\ ]. x0 - r , x0 .[ c= ]. x0 - r , x0 .[ ;
as Element of -> Element of -> Element of -> Element of -> Element of -> Element of -> Element of -> Element of ( n , n , k ) -tuples_on the carrier of K ;
v ^ ( n-3 |-> 0 ) in Lin ( ( B \ { 0 } ) \/ { 1 } ) & v ^ ( ( B \ { 0 } ) \/ { 1 } ) = v ;
ex a , k1 , k2 st i = a := k1 & i = b := k2 & i = c := k2 & i = c := k2 ;
t . NAT = ( ( NAT .--> i1 ) +* ( i1 , i2 ) ) . NAT .= ( i1 + 1 ) .= ( i1 + 1 ) .= ( i1 + 1 ) .= ( i1 + 1 ) .= ( i1 + 1 ) .= ( i1 + 1 ) .= ( i1 + 1 ) .= ( i1 + 1 ) .= ( i1 + 1 ) .= ( i1 + 1 ) ;
assume F is bbbfamily & rng p = F & dom p = Seg ( n + 1 ) & rng p c= Seg ( n + 1 ) ;
not LIN b , b9 , a & not LIN b , a , c & not b , c // a , c ;
( L1 \HM { A1 } to L2 ) exists O st O c= ( L1 on O ) => ( L2 on O ) & ( L1 on O ) => ( L2 on O ) = ( L1 on O ) => ( L2 on O )
consider F be 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 * ( .] - w ) = b * ( -w ) and 0 < a & 0 < b ;
defpred P [ FinSequence of D ] means |. Sum ( $1 ) .| <= Sum ( $1 ) & for i st i in dom $1 holds ( Sum ( $1 ) ) . i = Sum ( $1 ) ;
u = cos . ( x , y ) * x + cos . ( x , y ) * y ) .= cos . ( x , y ) * y .= v . ( x , y ) * y ;
dist ( ( seq . n ) + x , g + x ) <= dist ( ( seq . n ) + g , g ) + 0 ;
P [ p , |. p .| : p in [: the carrier of A , the carrier of A :] & ( the Sorts of A ) . p = ( the Sorts of A ) . p
consider X being Subset of Al such that X c= Y and X is finite and X is ininand X is ininin;
|. b .| * |. eval ( f , z ) .| >= |. b .| * |. eval ( f , z ) .| ;
1 < ( ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ) .. Gauge ( C , n ) ;
l in { l1 where l1 is Real : g <= l1 & l1 <= h & g <= h & l1 <= g & l <= h } ;
( Partial_Sums ( ( G . n ) ) * vol ) . n <= ( Partial_Sums ( G . n ) ) . ( n + 1 ) ;
f . y = x .= x * 1_ L .= ( 1_ L ) * ( 1_ L ) .= x * ( 1_ L ) .= ( 1_ L ) * ( ( 1_ L ) * ( y , y ) ) ;
NIC ( halt SCM+FSA , i1 ) = { i1 , ( l + 1 ) , ( l + 1 ) .= { ( l + 1 ) , ( l + 1 ) } ;
LSeg ( p10 , p2 ) /\ LSeg ( p1 , p2 ) = { p1 } /\ LSeg ( p2 , p1 ) ;
product ( ( the carrier of I-15 ) +* ( i , { 1 } ) ) in Z & product ( ( i , { 1 } ) +* ( i , { 1 } ) ) in Z ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) .= Following ( s1 , n ) .= Exec ( s1 , s2 ) ;
( W -bound ( Q ) ) ^2 <= ( q1 `1 ) ^2 + ( q1 `2 ) ^2 & ( q1 `2 ) ^2 <= ( q1 `2 ) ^2 + ( q1 `2 ) ^2 ;
f /. i2 <> f /. ( ( i1 + 1 ) -' len f ) & f /. ( ( i1 + 1 ) -' 1 ) = f /. ( ( i1 + 1 ) -' 1 ) ;
M , v / ( x. 3 , x. 0 ) / ( x. 4 , x. 0 ) |= H / ( x. 0 , x. 4 , x. 0 ) ;
len ( ( P ^ ) ) in dom ( ( P ^ ) ) & len ( ( P ^ ) ) = len ( ( P ^ ) ) + len ( ( P ^ ) ) + len ( ( P ^ ) ) + len ( ( P ^ ) ) + len ( ( P ^ ) ) ) ;
A |^ ( m , n ) c= ( A |^ m , n ) & A |^ ( k , n ) c= A |^ ( k , n ) ;
TOP-REAL n \ { q : |. q .| < a & |. q .| >= a } c= { q1 : |. q1 .| >= a }
consider n1 being element such that n1 in dom p1 and y1 = p1 . n1 and p1 . n1 = p2 . n1 and p1 . n1 = p2 . ( p1 . n1 ) ;
consider X being set such that X in Q and for Z being set st Z in Q & Z <> X holds X c= Z ;
CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA & CurInstr ( P3 , s3 ) = halt SCM+FSA ;
for v be VECTOR of l1 holds ||. v .|| " = upper_bound ( |. ( v . v ) .| " ) & ||. v . v .|| = |. ( v . v ) .| " .|
for phi holds phi in X implies phi phi in X & phi phi in X & phi phi in X & phi phi in X & phi phi in X
rng ( ( Sgm dom ( f | A ) ) | ( dom ( f | A ) ) ) c= dom ( ( f | A ) | A ) ;
ex c being FinSequence of D ( ) st len c = k & P [ c ] & a = c & b = c ;
( \mathop { \rm Arity } ( a , b , c ) ) . ( g , f ) = <* F ( b , c ) , F ( g , f ) *> ;
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 = b1 & a2 = b2 & a3 = b3 & a1 = b3 & a3 = b3 & a1 = b3 & a3 = b3 ;
D2 . ( indx ( D2 , D1 , n1 ) + 1 ) = D1 . ( n1 + 1 ) .= D1 . ( n1 + 1 ) ;
f . ( ]. r , s .[ ) = ]. |. r .| .| /. 1 .= <* r . 1 *> .= <* r . 1 *> .= <* r . 1 *> .= r . 1 ;
consider n be Nat such that for m be Nat st n <= m holds Cc . m = Cc . m ;
consider d be Real such that for a , b be Real st a in X & b in Y holds a <= b & b <= d ;
||. L /. h .|| - ( K * |. h .| ) + ( K * |. h .| ) - ( K * |. h .| ) <= p1 + ( K * |. h .| ) ;
attr F is commutative associative means : Def4 : for b being Element of X holds F -{ b } = f . b ;
p = - ( p + 0. TOP-REAL 2 ) + 0. TOP-REAL 2 .= 1 * ( p + 0. TOP-REAL 2 ) + 0. TOP-REAL 2 .= 1 * ( p + 0. TOP-REAL 2 ) + 0. TOP-REAL 2 .= ( p + 0. TOP-REAL 2 ) + 0. TOP-REAL 2 .= ( p + 0. TOP-REAL 2 ) + 0. TOP-REAL 2 ;
consider z1 such that b , x3 , x1 is_collinear and o , x1 , y1 is_collinear and o <> 0 & o <> 0 ;
consider i such that Arg ( ( Rotate ( s ) ) . q ) = s + Arg ( q ) . i and 0 <= i and i <= 2 * PI * PI ;
consider g such that g is one-to-one and dom g = card ( f . x ) and rng g c= dom ( g . x ) and g . x = f . x ;
assume that A = P2 \/ Q2 and Q1 <> {} and Q1 <> {} and Q1 is open and Q1 is open & Q1 is open & Q1 is open & Q1 is open & Q1 is open & Q1 is open & Q is open & Q is open & Q is open & Q is open & Q is open & Q is open & Q is open & Q is open & Q is open & Q is open & Q is closed ;
attr F is associative means : Def4 : F .: ( f .: ( f , g ) , h ) = F .: ( f , g ) ;
ex x being Element of NAT st m = x & x in z & x in { i } or m in { i } & i in { i } ;
consider k2 being Nat such that k2 in dom P-2 and l = P-2 . ( k2 + 1 ) and len ( P-2 ^ <* k2 *> ) = k + 1 ;
seq = r (#) seq implies for n holds seq . n = r * seq . n & seq . n = r * seq . ( n + 1 )
F1 . [ ( id a ) , [ a , a ] ] = [ f * ( id a ) , f . ( id a ) ] ;
{ p } "\/" D2 = { p "\/" y where y is Element of L : y in D & p in D & y in D } ;
consider z being element such that z in dom ( ( dom F ) --> ( x , y ) ) and ( ( F . x ) | ( dom F ) ) . z = y ;
for x , y being element st x in dom f & y in dom f holds x = y & x = f . y
Int cell ( G , i , 1 ) = { |[ r , s ]| : r <= ( G * ( 0 , 1 ) `1 ) `1 & s <= G * ( 1 , 1 ) `1 } ;
consider e being element such that e in dom ( T | E1 ) and ( T | E1 ) . e = v ;
( F ` * b1 ) . x = ( Mx2Tran ( ( J , T ) , T ) ) . ( T . j ) .= ( Mx2Tran ( J , T ) ) . ( T . j ) ;
- 1 / ( - 1 ) = ( mD ) (#) ( D | n ) .= ( mD | n ) (#) ( D | n ) .= ( ( n |-> 0. K ) (#) ( D | n ) ) .= ( ( n |-> 0. K ) (#) ( D | n ) ) (#) ( ( n |-> 0. K ) (#) ( D | n ) ) .= ( ( n |-> 0. K ) (#) ( D | n ) ) ;
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 ) ) .= len ( ( f2 . j ) ) ;
All ( 'not' All ( a , A , G ) , B , G ) |= All ( All ( a , B , G ) , A , G ) ;
LSeg ( E . k , F . ( k + 1 ) ) c= Cl RightComp Cage ( C , n + 1 ) \/ { G * ( k , i ) + 1 } ;
x \ ( a |^ m ) = x \ ( ( a |^ k ) * a ) .= ( x \ a ) \ a ` .= ( x \ a ) \ a ` ;
k -{ \it it } -in{ I ( ) -> Element of S ( ) , F ( set ) -> Element of S ( ) , Element of NAT , Element of NAT } : ex i being Element of NAT st i = ( commute ( I ) ) . i & F . i = ( commute ( I ) ) . i ;
for s being State of A2 holds Following ( s , n ) . ( 0 + 1 ) + ( n + 2 ) * ( n + 1 ) is stable ;
for x st x in Z holds f1 . x = a / ( a - x ) & ( f1 - x ) / ( a - x ) <> 0 & ( - f1 ) / ( a - x ) <> 0 ;
support ( support ( n ) ) \/ support ( ( support ( m ) ) \ { x } ) c= support ( ( support ( n ) ) \ { x } ) ) ;
reconsider t = u as Function of ( the carrier of A ) , ( the carrier of B ) * the Arity of C ;
- ( a * sqrt ( 1 + b ^2 ) ) <= - ( b * sqrt ( 1 + b ^2 ) ) ;
phi /. ( succ b1 ) = g . a & phi /. ( g . ( g . ( g . b1 ) ) ) = f . ( g . ( g . b1 ) ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( F ^ <* p *> ) and i in dom ( F ^ <* p *> ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 ,
the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U2 & the Sorts of U2 c= the Sorts of U2 implies the Sorts of U2 c= the Sorts of U1
( - ( 2 * a * b + sqrt 2 ) ) / ( 2 * a + sqrt 2 ) - ( 2 * a + sqrt 2 ) / ( 2 * a + sqrt 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 = <* a *> ;
Z = dom ( ( exp_R * ( arccot * ( arccot * ( f1 + #Z 2 ) ) ) ) / ( f1 + #Z 2 ) ) ) ;
lim ( f , SS1 ) is convergent & lim ( f , SS1 ) = lim ( f , S ) - lim ( g , S ) ;
( X ( ) ) => ( g . ( x , y ) ) => ( ( x9 => y9 ) => ( x9 => y9 ) ) in being Element of being Element of l ( ) ;
len ( ( M2 * M3 ) * ( ( M1 ~ ) * ( i , j ) ) ) = n & width ( ( M2 * M1 ) * ( i , j ) ) = n ;
attr X1 union X2 is open means : Def1 : X1 , X2 , X1 , X2 , X2 , X1 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X2 , X1 , X2 , X2 , X1 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X1 , X2 ,
for L being lower-bounded antisymmetric antisymmetric RelStr , X being non empty Subset of L holds X "\/" { Top L } = { Top L }
reconsider f9 = ( F . b ) * ( ( F . b ) * ( F . c ) ) as Function of ( the Sorts of M ) . ( ( F . b ) * ( F . c ) ) , M ;
consider w being FinSequence of I such that the InitS of M , <* s *> ^ w ^ w ^ ( <* s *> ^ w ) ^ w ^ ( <* s *> ^ w ) ^ w ^ ( <* s *> ^ w ) ;
g . ( a |^ 0 ) = g . ( 1_ G ) .= 1_ H .= g . ( 1_ G ) .= g . ( 1_ G ) .= g . ( 1_ G ) .= g . ( 1_ G ) ;
assume for i be Nat st i in dom f ex z being Element of L st f . i = rpoly ( 1 , z ) & z in rng 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 <> {} & K /\ L <> {} ;
( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C1 & ( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C1 ;
reconsider o9 = o `1 , w = o `1 , y = o `2 as Element of TS ( ( the Sorts of A ) * ( the_arity_of o ) ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . o ) ) ;
1 * x1 + ( 0 * x2 + 0 * x3 ) = x1 + ( 0 * x3 + 0 * x4 ) .= x1 + ( 0 * x3 + 0 * x4 ) .= x1 + 0 * x2 + 0 * x3 .= x1 + 0 * x3 + 0 * x3 + 0 * x4 .= x1 + 0 * x2 + 0 * x3 .= 1 * x1 + 0 * x2 + 0 * x3 + 0 * x3 + 0 * x3 + 0 * x3 .= 1 * x1 + 0 * x2 + 0
EK " . 1 = ( ( E qua Function ) " . 1 ) " .= ( ( E qua Function ) " . 1 ) .= ( ( E qua Function ) " . 1 ) " .= ( ( E qua Function ) " . 1 ) " .= ( E " ) . 1 .= ( E " ) . 1 .= ( E " ) . 1 .= ( E " ) . 1 .= ( E " ) . 1 ;
reconsider u12 = the carrier of U1 /\ ( U1 "\/" U2 ) , u12 = the carrier of U2 /\ ( U1 "\/" U2 ) as non empty Subset of U2 ;
( ( x "/\" z ) "\/" ( x "/\" y ) ) "\/" ( z "/\" y ) <= ( x "/\" z ) "\/" ( y "\/" z ) ;
|. f . ( s1 . l1 + 1 ) - ( s1 . l1 + 1 ) - ( s1 . l1 + 1 ) .| < ( 1 - M ) * ( s1 . l1 + 1 ) ;
LSeg ( ( Lower_Seq ( C , n ) ) * ( i , j ) , ( Lower_Seq ( C , n ) ) * ( i + 1 , j ) ) , ( LSeg ( Cage ( C , n ) ) * ( i , j ) ) ) is vertical ;
( f | Z ) /. x - ( f | Z ) /. x0 = L /. ( x- ( x - x0 ) ) + R /. ( x- ( x - x0 ) ) ;
g . c * ( - g . c ) * ( g . c ) + f . c <= h . c * ( - g . c ) + f . c ;
( f + g ) | divset ( D , i ) = f | divset ( D , i ) + g | divset ( D , i ) .= f | divset ( D , i ) + g | divset ( D , i ) ;
assume that ColVec2Mx ( f ) in the carrier of Let ( A ) and ColVec2Mx ( b ) in the carrier of ( A ) and len ( b ) = width ( A ) and width ( b ) = width ( A ) ;
len ( - ( M1 + M2 ) ) = len M1 & width ( - ( M1 + M2 ) ) = width M1 & width ( - ( M1 + M2 ) ) = width M1 ;
for n , i being Nat st i + 1 < n holds [ i , i + 1 ] in the InternalRel of ( ( TOP-REAL n ) | ( the carrier of TOP-REAL n ) ) \/ ( the InternalRel of ( TOP-REAL n ) | ( the carrier of TOP-REAL n ) )
pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in 2 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in 2 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in 2 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in 2 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in 2 , 1 ;
pred a <> 0 & b <> 0 & Arg a = Arg b & Arg b = Arg ( - a ) & Arg ( - b ) = Arg ( - b ) ;
for c being set st not c in [. a , b .] holds not c in Intersection ( the topology of X , a ) & not c in ( the topology of X ) \/ ( the topology of X )
assume that V1 is linearly-independent and V2 is linearly-independent and V1 is linearly-independent and V1 is linearly-independent and V1 is the carrier of V1 and V1 is the carrier of V1 and V1 is the carrier of V1 ;
z * x1 + ( 1 - z ) * x2 + ( 1 - z ) * y2 in M & z * y1 + ( 1 - z ) * y2 in N ;
rng ( ( ( P qua Function ) " ) " ) = Seg ( card ( P " ) ) .= Seg ( card ( P " ) ) ) .= Seg ( card ( P " ) ) .= Seg ( card ( P " ) ) ;
consider s2 being integer number such that s2 is convergent and b = lim s2 and s2 . b <= ( lim s2 ) / 2 and s2 . b <= ( lim s2 ) / 2 ;
h2 " . n = h2 . n " & 0 < ( 1 - ( 1 / 2 ) ) |^ n & 0 < ( 1 / 2 ) |^ n ;
( Partial_Sums ( ||. seq .|| ) ) . m = ||. ( seq ) . m - ( seq . n ) .|| .= ||. ( seq . m ) - ( seq . n ) .|| .= 0 ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b ;
- v = ( - 1_ G ) * v & - w = ( - 1_ G ) * v + ( - 1_ G ) * v ;
sup ( ( k .: D ) .: D ) = sup ( ( k .: D ) .: D ) .= ( k .: D ) .: D .= ( k .: D ) .: D .= ( k .: D ) .: D .= ( k .: D ) .: D .= ( k .: D ) .: D ;
( A |^ k , l ) ^^ ( A |^ ( n , l ) , .. A ) = ( A |^ ( n , l ) , l ) ^^ ( A |^ ( k , l ) , .. A ) ;
for R being add-associative right_zeroed right_complementable non empty addLoopStr , I , J being Subset of R holds I + ( J + K ) = ( I + J ) + K
( f . p ) ^2 = ( p `1 ) ^2 + sqrt ( 1 + ( p `2 ) ^2 ) ) ^2 .= ( p `1 ) ^2 + sqrt ( 1 + ( p `1 ) ^2 ) ;
for a , b being non zero Nat st a , b are_congruent_mod p holds ( a * b ) mod p = ( a * b ) mod p & ( a * b ) mod p = ( a * b ) mod p
consider A5 being countable set such that r is Element of CQC-WFF ( Al ) & A5 : A5 : A5 : A is D & A c= A5 & A c= A5 & A c= A5 & A c= A5 ;
for X being non empty addLoopStr for M being Subset of X , x , y being Point of X st x in M & y in M holds x + y in M + M
{ [ x1 , x2 ] , [ y1 , y2 ] } c= { x1 , y1 , y2 } & { y1 , y2 } in [: { x1 , x2 } , { y1 , y2 } :] ;
h . ( f . O ) = |[ A * ( f . O ) + B , C * ( f . O ) + D ]| .= |[ A * ( f . O ) + D , C * ( f . O ) + D ]| ;
( Gauge ( C , n ) * ( k , i ) ) in L~ Lower_Seq ( C , n ) /\ L~ Lower_Seq ( C , n ) & ( Cage ( C , n ) ) * ( k , i ) in L~ Cage ( C , n ) /\ L~ Cage ( C , n ) ;
cluster m , n ) -> prime implies for Element of NAT st ( for p being prime Nat holds p divides m iff p divides n ) & ( not p divides n implies p divides n ) & ( not p divides n implies p divides n ) & ( not p divides n implies p divides n ) & ( not p divides n implies p divides n ) & ( p divides n implies p divides n )
( f * F ) . x1 = f . ( F . x1 ) & ( f * F ) . x2 = f . ( F . x2 ) & ( f * F ) . x2 = f . ( F . x2 ) ;
for L being Lattice , a , b , c being Element of L st a \ b <= c & b \ c <= c holds a "/\" b <= c
consider b being element such that b in dom ( H / ( x. 0 ) ) and z = H / ( x. 0 ) and y = ( H / ( x. 0 ) ) . b ;
assume x in dom ( F * g ) & y in dom ( F * g ) & ( F * g ) . x = ( F * g ) . y ;
assume ex e being element st e Joins W . 1 , W . 2 & e in G . 3 & e in G . 3 & e in G . 3 & e in G . 3 ;
( ( h h h ) ] [ f ] ) . ( 2 * n ) = ( h h h ) . ( 2 * n ) * ( h . n ) + ( h . n ) * ( h . n + h . n ) ;
j + 1 = j - len h11 + 1 + 1 .= i + 1 - len h11 + 2 - 1 + 1 .= i + 1 - len h11 + 2 - 1 ;
( S *' ) . ( f , g ) = S *' . ( ( S *' ) . ( f , g ) ) .= S . ( ( S *' ) . ( f , g ) ) .= S . ( f , g ) ;
consider H such that H is one-to-one and rng H = the carrier of L2 and Sum ( L2 ) = Sum ( L ) and Sum ( L ) = Sum ( L ) ;
attr R is <= means : Def6 : p in R & p <> q & p <> q & q in R & p in R & q in R & p <> q & q in R ;
dom ( product ( X --> f ) ) = meet ( ( X --> f ) ) .= meet ( ( X --> f ) . ( X --> f ) ) .= meet ( X --> f ) .= meet ( X --> f ) .= meet ( X --> f ) .= dom f ;
sup ( proj2 .: ( Upper_Arc C ) /\ Upper_Arc ( C ) ) <= sup ( proj2 .: ( Upper_Arc C ) /\ Vertical_Line ( w ) ) & sup ( proj2 .: ( Upper_Arc C ) /\ Vertical_Line ( w ) ) <= sup ( proj2 .: ( Upper_Arc C ) /\ Vertical_Line ( w ) ) ;
for r be Real st 0 < r ex n be Nat st for m be Nat st n <= m holds |. S . m - p . n .| < r
i * ( f - y ) = i * ( f - y ) .= i * ( f - y ) - ( i * y ) .= i * ( f - y ) - ( i * y ) .= i * ( f - y ) - ( i * y ) ;
consider f being Function such that dom f = 2 -tuples_on X and for Y being set st Y in 2 -tuples_on X holds f . Y = F ( Y ) ;
consider g1 , g2 being element such that g1 in [#] Y & g2 in union C and g = [ g1 , g2 ] and g1 in C and g2 in C and g2 in C and g2 in C and g2 in C ;
func d -Z -> Nat means : Def7 : d |^ n divides n & d |^ ( n + 1 ) divides d |^ ( n + 1 ) & d |^ ( n + 1 ) divides d |^ ( n + 1 ) ;
f9 . [ 0 , t ] = f . [ 0 , t ] .= ( - P ) . ( 2 * x ) .= ( - P ) . ( 2 * x ) .= a * ( - P . x ) .= - P . ( - P . x ) .= - P . ( - P . x ) .= - Q . ( - P . x ) .= - Q . x ;
t = h . D or t = h . B or t = h . C or t = h . D or t = h . E or t = h . F or t = J . E or t = F . J or t = M . N or t = N . N or t = N . N ;
consider m1 be Nat such that for n st n >= m1 holds dist ( ( seq . n ) , ( seq . n ) ) < 1 / ( ( seq . n ) to_power k ) ;
sqrt ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 <= sqrt ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 ) ;
h0 . ( i + 1 + 1 ) = h21 . ( i + 1 + 1 + 1 ) .= h21 . ( i + 1 + 1 ) .= h21 . ( i + 1 + 1 ) ;
consider o being Element of the carrier' of S , x2 being Element of { [ o , x2 ] } such that a = [ o , x2 ] and o <> [ o , x2 ] ;
for L being RelStr , a , b being Element of L holds a <= b iff a <= b & b <= a & a <= b implies a <= b
||. h1 .|| . n = ||. h1 . n .|| .= ||. h1 . n .|| .= ||. h1 . n .|| .= ||. h1 . n .|| .= ||. h1 . n .|| .= ||. h1 . n .|| ;
( ( - ( 1 / 2 ) (#) ( exp_R * exp_R ) ) ) . x = f . x - ( exp_R . x ) / ( exp_R . x ) .= ( - exp_R . x ) / ( exp_R . x ) ^2 .= - ( exp_R . x ) / ( exp_R . x ) ^2 .= - ( exp_R . x ) ^2 / ( exp_R . x ) ^2 ;
pred r = F .: ( p , q ) means : Def8 : len r = len ( p ^ q ) + len q ;
( ( r / 2 ) |^ 2 ) + ( ( r / 2 ) |^ 2 ) <= ( r / 2 ) |^ ( 2 + 1 ) + ( r / 2 ) |^ 2 ;
for i being Nat , M being Matrix of n , K st i in Seg n holds Det ( M , i ) = Sum ( Det ( M , i ) ) & Det ( M , i ) = ( width ( M @ ) ) * ( i , i )
then a <> 0. R implies a " * ( a * v ) = 1 * v & a " * v = 1 * v .= a * v ;
( p - 1 ) * ( q *' r ) . ( i + 1 ) = Sum ( p ) - ( q *' r ) . ( i + 1 ) * ( q *' r ) . ( i + 1 ) ;
deffunc F ( Nat ) = L . 1 + ( ( R /* h ) ^\ n ) * ( ( h ^\ n ) ^\ n ) " ) . ( ( h ^\ n ) . ( h ^\ n ) ) . ( h + n ) ) ;
assume that the carrier of H2 = f .: ( the carrier of H1 ) and the carrier of H2 = f .: ( the carrier of H2 ) and the carrier of H1 = the carrier of H2 and the carrier of H1 = 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 -] .= ( 2 to_power ( n + 1 ) ) * ( <* h . ( n + 1 ) *> ^ ( 2 to_power ( n + 1 ) ) *> ) .= n + 1 * ( 2 to_power ( n + 1 ) ) ;
( ( O - 1 ) / 3 ) * ( ( O - 1 ) / 3 ) = 0 & ( O - 1 ) / 3 = 0 & ( O - 1 ) / 3 = 0 & ( O - 1 ) / 3 = 0 ;
F1 .: ( dom F1 /\ dom F2 ) = F1 .: ( dom F1 /\ dom F2 ) .= F1 .: ( dom F1 /\ dom F2 ) .= ( f | ( Seg n ) ) /\ ( f | ( Seg n ) ) .= f . ( n + 2 ) ;
pred b <> 0 & d <> 0 & b <> d & ( a = b implies a = ( b - d ) / ( b - d ) ) / ( b - d ) ;
dom ( ( f +* g ) | D ) = dom ( ( f +* g ) | D ) /\ D .= ( dom f \/ g ) /\ D .= dom ( ( f +* g ) | D ) /\ D .= dom ( ( f +* g ) | D ) /\ D .= dom ( ( f +* g ) | D ) /\ D .= dom ( ( f +* g ) | D ) /\ D ;
for i being set st i in dom g ex u , v being Element of L st g /. i = u * a & v in B & u in C & v in C & v in C & u in C & v in C
g ` * P ` * g " = g ` * ( g " * P ` ) .= g ` * ( g " * P ` ) .= g ` * ( g " * P ` ) .= g ` * ( g " * P ` ) .= g ` * ( g " * P ` ) .= g ` * ( g " * P ` ) ;
consider i , s1 such that f . i = s1 and not ( s1 in dom s1 & not s2 in dom s1 & not s1 in dom s1 ) & not s1 . ( s1 . i ) <> s1 . ( s1 . i ) ;
h5 | ]. a , b .[ = ( g | ]. a , b .[ ) | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ ;
[ s1 , t1 ] , [ s2 , t2 ] ] in R & [ s2 , t2 ] in R & [ s1 , t2 ] in R & [ s2 , t2 ] in R ;
then H is negative & H is non negative & H is non negative & H is non -universal implies H is non -universal ;
attr f1 is total means : Def8 : ( 1 - f1 ) (#) ( f2 - f3 ) ) is total & ( f1 (#) ( f2 - f3 ) ) (#) ( f2 - f3 ) = ( f1 (#) ( f2 - f3 ) ) (#) ( f2 - f3 ) ;
z1 in W2 ` & z1 = z2 or z1 = z2 & z1 in W2 & z2 in W1 & z1 in W2 ` & z2 in W2 ` & z1 in W2 ` & z2 in W2 ` & z1 in W2 ` & z2 in W2 ` implies z1 = z2 + ( z2 + z1 )
p = 1 * p .= a " * a * p .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= ( a " * ( b " * q ) ) * ( b " * q ) ;
for rseq be Real_Sequence for K be Real st for n be Nat holds rseq . n <= K . ( n + 1 ) holds upper_bound rng ( seq ^\ k ) <= K . ( n + 1 )
( E-max C ) meets ( L~ go \/ L~ pion1 ) or ( E-max C ) meets ( L~ go \/ L~ pion1 ) or ( E-max C ) meets ( L~ co \/ L~ pion1 ) or ( E-max C ) meets ( L~ pion1 \/ L~ pion1 ) or ( E-max C ) meets ( L~ pion1 \/ L~ pion1 ) ;
||. ( f . ( g . k + 1 ) ) - ( g . k ) .|| <= ||. g . ( 1 - K ) .|| * ( K |^ k ) ;
assume h = ( ( B .--> ( C .--> B ) +* ( D .--> C ) ) +* ( E .--> F ) +* ( J .--> M ) +* ( N .--> N ) +* ( M .--> N ) +* ( M .--> N ) ) +* ( M .--> N ) +* ( M .--> N ) +* ( M .--> N ) ) +* ( M .--> N ) +* ( M .--> N ) ) ;
|. ( ( ( H . n ) || A8 ) . k ) - ( ( H . n ) . k ) .| <= e * ( ( H . n ) . k ) ;
(
{ x1 , x1 , y1 , x1 , y1 , x1 , y1 , x1 , y1 , x1 , y1 , x1 , y1 , x1 , y1 , x1 , y1 , x1 , y1 , x1 , y1 , x1 , y1 , x1 , y1 , x1 , y1 , x1 , y1 , x1 , y1 , x1 , y1 , x1 , y1 , x1 , y1 , x1 , y1 , y1 , x1 , y1 , x1 , y1 , x1 , y1 , x1 , y1 , x1 , y1 , y1 , x1 , y1 , x1 , y1 , y1 , x1 , y1 , y1 , y1 , x1
Suppose A = [. 0 , 2 * PI .] and integral ( A , A , B ) = ( ( #Z n ) * ( cos | A ) ) * ( cos | A ) + ( ( #Z n ) * ( cos | A ) ) * ( cos | A ) ;
p `1 is Permutation of dom f1 & p `1 " = ( Sgm Y ) " . ( p " ) & p " " . ( p " ) = ( Sgm Y ) " . ( p " ) * ( Sgm Y ) . ( p " ) ;
for x , y st x in A holds |. ( 1 - f ) . x - ( 1 - f . y ) .| <= 1 * |. ( f . x - f . y ) .|
( p2 `2 ) ^2 = |. q2 .| * ( ( ( q2 `2 ) ^2 - ( q2 `2 ) ^2 ) / ( 1 - ( q2 `2 ) ^2 ) ) ^2 ) .= ( ( q2 `2 ) ^2 - ( q2 `2 ) ^2 ) ^2 / ( 1 - ( q2 `2 ) ^2 ) ;
for f be PartFunc of the carrier of CC st dom f is compact & f | X is continuous & f | X is continuous holds f | X is compact & f | X is compact
assume for x being Element of Y st x in EqClass ( z , CompF ( B , CompF ( B , G ) ) ) holds ( All ( a , PA , G ) ) . x = TRUE ;
consider FK such that dom FK = n1 & for k be Nat st k in n1 holds Q [ k , FK . k ] ;
ex u , u1 st u <> u1 & u , u1 // v , v1 & u , u1 // v , v1 & u , v // v , v1 & u , v // v , u1 & u , v // v , v1 & v , u1 // v , v1 & u , v // v , v1 ;
for G being Group , A , B being non empty Subgroup of G , N being normal Subgroup of G holds ( N ` A ) * ( N ` A ) = N ` A * B ` B
for s be Real st s in dom F holds F . s = integral ( R to_power ( 0 + 1 ) , ( f to_power ( 0 + 1 ) ) * e ) ) & ( F to_power ( 0 + 1 ) ) . x = ( ( R to_power ( 0 + 1 ) ) * e ) . x
width AutMt ( f1 , b1 , b2 , b3 , b2 ) = len b2 & width AutMt ( f2 , b1 , b2 , b3 ) = width b2 & width AutMt ( f1 , b2 , b3 , b2 , b3 ) = width b2 & width AutMt ( f1 , b1 , b2 , b3 ) = width b2 ;
f | ]. - PI / 2 , PI / 2 .[ = f & f | ]. PI / 2 , PI / 2 .[ = f | ]. PI / 2 , PI / 2 .[ implies f | ]. PI / 2 , PI / 2 .[ = g | ]. PI / 2 , PI / 2 .[
assume that X is closed and a in X and a c= X and y in a .: [: f , { x } :] and x in a .: [: f , { y } :] ;
Z = dom ( ( ( #Z 2 ) * ( exp_R + arccot ) ) / ( exp_R + arccot ) ) ) /\ dom ( ( exp_R + exp_R + exp_R + exp_R ) / ( exp_R + exp_R + exp_R ) ) ;
func VERUM ( V ) -> Subset of V means : Def1 : 1 <= k & k <= len it & it . k = l . k & it . k = 1 ;
for L being non empty TopSpace , N being net of L , M being net of N st c is net of N & for c being Point of N st c is cluster holds c is cluster of N & c is continuous & c is continuous & c is continuous
for s being Element of NAT holds ( ( ( id the carrier of X ) + ( id the carrier of X ) ) ) . s = ( ( id the carrier of X ) + ( id the carrier of X ) ) . s
then z /. 1 = ( N-min L~ z ) .. z & ( N-min L~ z ) .. z < ( ( N-min L~ z ) .. z ) .. z ;
len ( p ^ <* ( 0 qua Real ) *> ^ <* ( 0 qua Real ) *> ^ ( <* 0 *> ^ ( <* 1 *> ^ ( p ^ q ) ) *> ) ) = len p + 1 .= len p + 1 + 1 .= len p + 1 ;
assume that Z c= dom ( ( - ( ln * f ) ) ^ ) and for x st x in Z holds f . x = exp_R . x / ( cos . x ) ^2 ) and f . x = exp_R . x / ( cos . x ) ^2 and f . x = exp_R . x / ( cos . x ) ^2 and f . x = 0 ;
for R being add-associative right_zeroed right_complementable Abelian associative well-unital distributive non empty doubleLoopStr , I , J being Subset of R , I being Ideal of R , J being Subset of R , I being Subset of R , J being Subset of R holds ( I + J ) *' ( I /\ J ) c= I /\ J
consider f being Function of [: B1 , B2 :] , B2 such that for x being Element of B1 holds f . x = F ( x ) & f . x = F ( x ) ;
dom ( x2 + y2 ) = Seg len x .= dom ( ( x ^ y ) ^ z ) .= Seg len ( x ^ y ) .= dom ( x ^ y ) .= dom ( x ^ y ) .= dom ( x ^ y ) .= dom ( x ^ y ) .= dom ( x ^ y ) ;
for S being preFunctor of C , B for c being Object of C holds ( S . ( id c ) ) . ( id c ) = id ( ( the Arrows of C ) . ( id c ) ) . ( id c )
ex a st a = a2 & a in f1 /\ f2 & ( for x st x in dom f1 /\ f2 holds f1 . x = f2 . x ) & ( for x st x in dom f1 holds f1 . x = f1 . x ) implies ( f1 - f2 ) . x = f2 . x - ( f1 . x ) / ( f1 . x )
a in Free ( H2 / ( x. 4 , x. k ) ) \/ { x. k , x. k , x. k } & a in Free ( H2 / ( x. k , x. k ) ) \/ { x. k , x. k } ;
for C1 , C2 being set , f , g being stable Function of C1 , C2 st f is stable & g is stable holds f = g & f = g & g = g
( W-min L~ go \/ L~ pion1 ) `1 = ( W-min L~ pion1 ) \/ ( L~ pion1 ) \/ ( L~ pion1 ) \/ ( L~ pion1 \/ L~ pion1 ) .= ( W-bound L~ pion1 ) \/ ( W-bound L~ pion1 ) \/ ( W-bound L~ pion1 ) \/ ( W-bound L~ pion1 ) \/ ( W-bound L~ pion1 ) \/ ( W-bound L~ pion1 ) \/ ( W-bound L~ pion1 ) ) ;
assume that u = <* x0 , y0 , z0 *> and f is_partial_differentiable_in z0 , 3 and SVF1 ( 3 , f , 1 ) u u , 3 and SVF1 ( 3 , f , u ) = SVF1 ( 3 , f , u ) ;
then ( t . {} ) `1 in Vars or ex x being Element of Vars st x = ( t . {} ) `1 & ( t . {} ) `1 = ( t . {} ) `1 & ( t . {} ) `1 = ( t . {} ) `1 & ( t . {} ) `1 = ( t . {} ) `1 ;
Valid ( p '&' p , J ) . v = Valid ( p , J ) . v '&' Valid ( p , J ) . v .= Valid ( p , J ) . v .= Valid ( p , J ) . v .= Valid ( p , J ) . v ;
assume for x , y being Element of S st x <= y for a , b being Element of T st a = f . x & b = f . y holds a >= b ;
func Class ( R , x ) -> Subset-Family of R means : Def6 : for A being Subset of R holds A in it iff ex a being Element of R st a in A & a in A & a c= A ;
defpred P [ Nat ] means ( ( ( ( ( ( G ) . $1 ) `1 ) `1 ) + 1 ) + 1 ) + 1 ) `1 c= G . ( ( ( G . $1 ) `1 ) + 1 ) + 1 ) & ( ( ( G . $1 ) `1 ) + 1 ) + 1 ) + 1 <= ( ( G . $1 ) `1 + 1 ) ;
assume that dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W2 ) = 0 and dim ( W1 ) = 0 and dim ( W2 ) = 0 and dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W1 ) = 0 ;
mama_empty ( m . t ) = ( m . t ) `1 .= ( m . t ) `1 .= ( m . t ) `1 .= ( m . t ) `1 .= ( m . t ) `1 .= ( m . t ) `1 .= ( m . t ) `1 .= ( m . t ) `1 ;
d11 = x9 ^ d22 .= ( y9 ^ d22 ) . ( y9 ^ d22 ) .= ( y9 ^ d22 ) . ( y9 ^ d22 ) .= ( y9 ^ d22 ) . ( y9 ^ d22 ) .= ( y9 ^ d22 ) . ( y9 ^ d22 ) .= ( x9 ^ d22 ) . ( y9 ^ d22 ) .= ( x9 ^ d22 ) . ( y9 ^ d22 ) .= x9 ^ d22 ^ d22 .= d22 ^ d22 ^ d22 ^ d22 ^ d22 ^ d22 .= d22 ^ d22 ^ d22 .= d22 ^ d22 ^ d22 ^ d22 .= d22 ^ d22 ^ d22 ^ d22 ^ d22 ^ d22 ^ d22
consider g such that x = g and dom g = dom ( f | X ) & for x being element st x in dom ( f | X ) holds g . x in ( f | X ) . x ;
x + 0. F_Complex = x + ( len x |-> 0. F_Complex ) .= ( x + ( len x |-> 0. F_Complex ) ) .= ( x + ( len x |-> 0. F_Complex ) ) .= ( x + ( len x |-> 0. F_Complex ) ) .= x ` + ( x * ( len x |-> 0. F_Complex ) ) .= x ` + ( x * ( len x ) ) .= x ` + x * ( x * ( len x ) ) .= x ` + x * ( x * ( len x ) ) .= x ` + x * ( x * ( len x ) ) .= x ` + x * ( x * ( x * ( x * ( x * ( x * ( x * ( len x ) ) .= x ` + x * ( x * ( len
( k -' ( k -' 1 ) ) + 1 in dom ( f | ( Seg ( ( k -' 1 ) + 1 ) ) ^ <* ( k -' 1 ) + 1 ) *> ) ;
assume that P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 = p1 \/ p2 and P1 = p2 \/ p1 and P1 = p1 \/ p2 and P1 = p2 \/ p1 and P1 = p2 \/ p1 and P2 = p1 \/ p2 and P1 = p2 \/ p1 and P1 = p2 \/ p2 and P1 = p1 \/ p2 and P1 = p2 \/ p1 \/ p2 and P1 = p2 \/ p1 and P1 = p2 \/ p2 and P1 = p2 \/ p2 and P1 = p2 \/ p2 and P1 = p2 and P1 = p1 \/ p2 and P1 = p1 \/ p2 and P1 = p2 \/ p2 and P1 = p1 \/ p2 and P1 = p2 \/ p2 and P1 = p2 \/ p2 and P1 = p1 \/ p2 and P1 = p1 \/ p2 and P1 = p1 \/ p2 and P1
reconsider a1 = a , b1 = b , c1 = c , c1 = d , c2 = p , c1 = q , c2 = p , c1 = r , c2 = s , c1 = s , c2 = s , c1 = s , c2 = r , c1 = s , c2 = s , c1 = s , c2 = s , c1 = s , c2 = s , c2 = s , c2 = s , c1 = s , c1 = s , c2 = s , c1 = s , c1 = s , c1 = s , c1 = s , c1 = s , c2 = s , c1 = s , c1 = s , c1 = s , c1 = s , c1 = s , c1 = s , c1 = s , c2 = s , c2 = s , c2 =
reconsider GtbFFFFf = G1 . ( t , F1 . b ) * F1 . ( F2 . b ) as Morphism of ( G1 * F1 ) . ( F2 . b ) , ( G1 * F2 ) . ( F2 . b ) * F2 . ( F2 . b ) * F2 . ( F2 . b ) * F2 . ( F2 . b ) * F2 . ( F2 . b ) * F2 . b ;
LSeg ( f , i + i1 -' 1 ) = LSeg ( f /. ( i + i1 -' 1 ) , f /. ( i + i1 -' 1 + 1 -' 1 ) ) , f /. ( i + i1 -' 1 + 1 -' 1 ) ) ;
Integral ( M , ( P . m ) | dom ( P . n ) -P . m ) <= Integral ( M , ( P . n ) -P . m ) ;
assume that dom f1 = dom f2 and for x , y being element st x in dom f1 & y in dom f2 holds f1 . x = f2 . y and f1 . y = f2 . x ;
consider v such that v = y and dist ( u , v ) < min ( ( G * ( i , 1 ) `1 ) - ( G * ( i , 1 ) `1 ) ) / 2 , ( G * ( i + 1 , 1 ) `1 ) - ( G * ( i + 1 , 1 ) `1 ) ) ;
for G being Group , H being Subgroup of G , a being Element of G st a = b holds for i being Integer st i in H holds a |^ i = b |^ i * H
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 = { p2 where p2 is Point of TOP-REAL 2 : P [ p2 `1 , p2 `2 ] & p2 `2 <= - p2 `1 & p2 `2 <= - p2 `1 & p2 `2 <= - p2 `1 & p2 `1 <= - p2 `2 & p2 `2 <= - p2 `1 & p2 `2 <= - p2 `1 & p2 `2 <= - p2 `1 & p2 `1 <= - p2 `1 & p2 `1 <= - p2 `1 & p2 `1 <= - p2 `1 & p2 `1 <= p2 `1 & p2 `1 <= - p2 `1 & p2 `1 or p2 `1 <= - p2 `1 & p2 `1 <= - p2 `1 & p2 `2 <= - p2 `1 & p2 `1 or p2 `1 <= p2 `1 & p2 `1 <= - p2 `1
sqrt ( ( ( ( N - S ) / 2 ) |^ m ) - ( ( N - S ) |^ m ) / 2 ) <= sqrt ( ( N - S ) |^ m ) - ( ( N - S ) |^ m ) / 2 ) ;
for x be Element of X , n be Nat st x in E holds |. ( Re F ) . n .| . x <= P . x & |. ( Im F ) . n .| <= P . x & |. ( Im F ) . n .| <= P . x
len @ H = len ( @ @ H ^ <* 0 *> ) + len <* 2 *> .= len ( @ H ^ <* 0 *> ) + len <* 2 *> .= len ( @ H ^ <* 1 *> ) + len <* 2 *> .= len ( @ H ^ <* 1 *> ) + len <* 2 *> .= len ( @ H ^ <* 1 *> ) + len ( @ H ^ <* 0 *> ) ;
v / ( x. 3 , m1 ) / ( x. 0 , m1 ) / ( x. 4 , m1 ) ) / ( x. 0 , m1 ) = ( x. 4 , m1 ) / ( x. 0 , m1 ) ;
consider r being Element of M such that M , v / ( x. 3 , m ) / ( x. 0 , m ) / ( x. 4 , m ) / ( x. 0 , m ) / ( x. 0 , m ) ) |= H2 and m1 = m ;
func w1 \ w2 -> Element of Union ( G , RA2 ) means : Def14 : for i being Element of NAT holds it . ( ( the Element of G ) . ( i , w ) ) = ( the Element of ( G , RA2 ) . ( i , w ) ) . ( ( the Element of G ) . ( i , w ) ) ;
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= Exec ( n2 , s2 ) . b2 .= Exec ( n2 , s2 ) . b2 .= Exec ( n2 , s2 ) . b2 .= s2 . b2 .= s2 . b2 .= ( s2 . b2 ) .= ( s2 . b2 ) .= ( s2 . b2 ) + ( s2 . b2 ) .= ( s2 . b2 ) + ( s2 . b2 ) .= ( s2 . b2 ) + ( s2 . b2 ) .= ( s2 . b2 ) + ( s2 . b2 ) + ( s2 . b2 ) + ( s2 . b2 ) .= ( s2 . b2 ) + ( s2 . b2 ) .= ( s2 . b2 ) + ( s2 . b2
for n , k be Nat holds 0 <= Partial_Sums ( |. seq .| ) . ( n + k ) - Partial_Sums ( |. seq .| ) . ( n + k ) + Partial_Sums ( |. seq .| ) . ( n + k ) - Partial_Sums ( |. seq .| ) . ( n + k )
set F = S -\mathop { {} } , G = S -\mathop { {} } ;
( Partial_Sums ( seq ) ) . ( n + 1 ) + Partial_Sums ( seq ) . ( n + 1 ) >= ( Partial_Sums ( seq ) ) . ( n + 1 ) + Partial_Sums ( seq ) . ( n + 1 ) ;
consider L , R such that for x st x in N holds ( f | Z ) . x - ( f | Z ) . x = L . ( x- Z ) + R . ( x- Z ) ;
func the closed of \HM { a , b , c , d } -> Subset of TOP-REAL 2 equals ( the Element of \HM { a , b , c , d } ) \/ ( the Element of TOP-REAL 2 ) ;
a * b ^2 + ( a * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c
v / ( x1 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) = v / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) ;
product ( Q ^ <* x *> ^ <* y *> ) = ( Q +* ( ( Q ^ <* x *> ) ^ ( Q ^ <* y *> ) ) ) +* ( ( Q ^ <* x *> ^ ( Q ^ <* y *> ) ) ) .= ( Q +* ( ( Q ^ <* x *> ^ ( Q ^ <* y *> ) ) ) +* ( ( Q ^ <* x *> ^ ( Q ^ <* y *> ) ) ) .= ( Q ^ <* x *> ^ ( Q ^ <* y *> ^ ( Q ^ <* x *> ^ ( Q ^ <* y *> ^ ( Q ^ <* x *> ^ ( Q ^ <* y *> ^ ( Q ^ <* x ^ <* x *> ^ ( Q ^ <* x
Sum ( FM ) = ( r |^ n1 ) * Sum ( CM ) .= ( C |^ n1 ) * ( ( n + 1 ) + 1 ) .= C . ( ( n + 1 ) + 1 ) .= C . ( ( n + 1 ) + 1 ) .= C . ( ( n + 1 ) + 1 ) .= C . ( ( n + 1 ) + 1 ) .= C . ( ( n + 1 ) + 1 ) .= C . ( ( n + 1 ) + 1 ) .= C . ( ( n + 1 ) + 1 ) .= C . ( ( n + 1 ) + 1 ) .= C . ( ( n + 1 ) + 1 ) .= C . ( ( n + 1
( ( GoB f ) * ( len GoB f , 2 ) ) `1 = ( ( GoB f ) * ( len GoB f , 1 ) ) `1 .= ( GoB f ) * ( 1 , 1 ) `1 .= ( GoB f ) * ( 1 , 1 ) `1 ;
defpred X [ Element of NAT ] means ( Partial_Sums ( s ) ) . ( $1 + 1 ) = ( Partial_Sums ( s ) ) . ( $1 + 1 ) * ( $1 + 1 ) + 1 ) * ( $1 + 1 ) + 1 * ( $1 + 1 ) * ( $1 + 1 ) + 1 * ( $1 + 1 ) * ( $1 + 1 ) + 1 * ( $1 + 1 ) * ( $1 + 1 ) + 1 * ( $1 + 1 ) * ( $1 + 1 ) + 1 * ( $1 + 1 ) * ( $1 + 1 ) * ( $1 + 1 ) + 1 * ( $1 + 1 ) * ( $1 + 1 ) * ( $1 + 1 ) * ( $1 + 1
( the Arity of S ) . g = ( the Arity of S ) . ( g . ( the Arity of S ) . g ) .= ( ( the Arity of S ) . g ) . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . x ) ) ) ) ) ) ) ) ) ) ) ) ) ) . ( f . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g .
( X ~ Y ) ^ Z tolerates X ~ Y & card ( X ~ ) c= card ( X ~ ) & card ( X ~ ) = card ( X ~ ) + card ( Y ~ ) ;
for a , b being Element of S for s being Element of NAT st s = n & a = F . n & b = F . n & s = G . ( n + 1 ) \ G . ( n + 1 ) holds b = G . ( n + 1 ) \ G . ( n + 1 )
E , f |= All ( All ( x , H ) , ( ( x. 2 ) --> ( x. 0 ) ) ) '&' ( ( x. 2 ) --> ( x. 1 ) ) '&' ( ( x. 2 ) --> ( x. 1 ) ) ) '&' ( ( x. 1 ) --> ( x. 1 ) ) ;
ex R2 being 1-sorted st R2 = ( p | ( n + 1 ) ) . i & ( the carrier of R1 ) . i = the carrier of R2 & ( the carrier of R2 ) . i = the carrier of R2 & ( the carrier of R2 ) . i = the carrier of R2 & ( the carrier of R2 ) . i = the carrier of R2 ;
[. a , b + sqrt ( k + 1 ) / ( k + 1 ) ) , ( the partial of f ) . ( k + 1 ) .] is Element of the carrier of ( the partial of f ) . ( k + 1 ) & ( the partial of f ) . ( k + 1 ) is Element of the carrier of ( the partial of f ) . ( k + 1 ) )
Comput ( P , s , 2 + 1 ) = Exec ( P . 2 , Comput ( P , s , 2 ) ) .= Exec ( ( a , k1 ) := ( card I + 2 ) ) .= Exec ( ( a , k1 ) := ( card I + 2 ) , s ) ;
card ( h1 ) . k = power ( F_Complex ) . ( ( - 1_ F_Complex ) . ( ( - 1_ F_Complex ) . k ) * u ) .= ( ( - 1_ F_Complex ) . ( - 1_ F_Complex ) ) * u .= ( ( - 1_ F_Complex ) . ( - 1_ F_Complex ) ) * u .= ( ( - 1_ F_Complex ) . ( - 1_ F_Complex ) ) * u .= ( ( - ( 1_ F_Complex ) . k ) * u ) * u .= ( ( - ( 1_ F_Complex ) . k ) * u ) * u .= ( ( - ( ( - ( 1_ F_Complex ) . k ) * u ) * u .= ( ( - ( 1_ F_Complex ) . k ) * u ) * u ) * u .= ( (
( f (#) g ) /. c = f /. c * ( g /. c ) " .= ( f (#) g ) /. c * ( g /. c ) .= ( ( f (#) g ) /. c ) " * ( ( g (#) f ) /. c ) .= ( ( f (#) g ) /. c ) " * ( ( g (#) f ) /. c ) .= ( ( f (#) g ) /. c ) " * ( ( g (#) g ) /. c ) .= ( ( ( f (#) g ) /. c ) " * ( ( g (#) g ) /. c ) " * ( ( g (#) g ) /. c ) " * ( ( g (#) ( g (#) f ) /. c ) " * ( ( g (#) f ) /. c ) " * ( ( g (#) f ) /. c ) " * (
len Cf - len ( ( C | ( len C | ( len C | ( len C | ( Seg ( len C | ( Seg ( len C | ( C | ( C | ( C | ( C | ( C | ( C | ( C | ( C | ( C | ( C | ( C | ( C | C ) ) ) ) ) ) ) ) ) ) ) ) ) ) = len ( ( C | ( D | ( D | ( D | ( D | ( D | ( D | ( D | ( D | ( D | ( D | ( D | ( D | ( D | ( D | ( D | ( D | ( D | ( D | ( D | ( D | ( D | ( D | ( D | ( D | ( D | ( D
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom ( ( r (#) f ) | X ) /\ X .= dom ( ( r (#) f ) | X ) /\ X .= dom ( ( r (#) f ) | X ) /\ X .= dom ( ( r (#) f ) | X ) /\ X .= dom ( ( r (#) f ) | X ) /\ X .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) /\ X .= dom ( ( r (#) f ) | X ) /\ X .= dom ( ( r (#) f ) | X ) /\ X ) /\ X .= dom ( ( r (#) f ) | X ) /\ X .= dom ( ( r (#) f ) /\ X ) /\ X .= dom ( (
defpred P [ Nat ] means for n holds 2 * Fib ( n + $1 ) = Fib ( n + $1 ) * Fib ( n + $1 ) + ( 5 * Fib ( n + $1 ) ) * Fib ( n + $1 ) + ( 5 * Fib ( n + $1 ) ) * Fib ( n + $1 ) ;
consider f being Function of INT , INT such that f = f & f is onto and f is onto and n < len f and f is onto and f " { 0 } = f " { 0 } ;
consider c9 being Function of S , BOOLEAN such that c9 = ( chi ( A \/ B , S ) ) . ( A \/ B ) and c9 = ( Prob * ( A \/ B ) ) . ( A \/ B ) and c9 = ( Prob * ( A \/ B ) ) . ( A \/ B ) ;
consider y being Element of [: X ( ) , Y ( ) :] such that a = "\/" ( { F ( x , y ) where x is Element of X ( ) : P [ x , y ] } , T ( x , y ) ) & P [ y , x ] ;
assume A c= Z & f = f & f = ( ( #Z 2 ) * ( exp_R + exp_R ) ) * ( exp_R + exp_R + exp_R ) + exp_R * ( exp_R + exp_R ) ) & f = exp_R * ( exp_R + exp_R + exp_R ) + exp_R * ( exp_R + exp_R + exp_R ) * ( exp_R + exp_R ) ;
( f /. i ) `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 q1 ) = { j + len Seq q2 where j is Element of NAT : j in dom Seq q2 & len Seq q2 = len q2 & len Seq q2 = len q2 & len Seq q2 = len q2 & len Seq q2 = len q2 & len Seq q2 = len q2 & len q1 = len q2 & len q1 = len q2 & len q2 = len q2 & len q1 = len q2 & len q2 = len q2 ;
consider G1 , G2 , G2 , H being Element of V such that G1 <= G2 & G2 <= G1 & g = G2 & f = G1 & g = G2 & f = G2 & g = G2 & g = G1 & g = G2 & f = G2 & g = G1 & g = G2 & g = G2 ;
func - f -> PartFunc of C , V means : Def5 : dom it = dom f & for c st c in dom it holds it /. c = - f /. c + f /. c ;
consider phi such that phi is increasing and for a st phi . a = a & {} <> a & for v st v in a holds L . ( v , a ) |= H & L . ( v , a ) |= H ;
consider i1 , j1 such that [ i1 , j1 ] in Indices GoB f and f /. ( i1 + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i1 + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i1 + 1 ) = ( GoB f ) * ( i1 , j1 ) ;
consider i , n such that n <> 0 and sqrt p = sqrt n and for i1 being Integer st i1 <> 0 & i1 <> 0 & i1 <> 0 holds sqrt ( n + 1 ) = sqrt ( n + 1 ) * ( n + 1 ) and sqrt ( n + 1 ) = sqrt ( n + 1 ) * ( n + 1 ) ;
assume that not 0 in Z and Z c= dom ( ( #Z 2 ) * ( ( #Z 2 ) * ( exp_R + arccot ) ) ) and for x st x in Z holds ( ( #Z 2 ) * ( exp_R + arccot ) ) . x > - 1 & ( ( #Z 2 ) * ( exp_R + arccot ) ) . x < 1 ;
cell ( G1 , i1 -' 1 , j1 -' 1 ) \ ( ( m + 1 ) -' 1 ) \ ( ( m + 1 ) -' 1 ) \ ( ( m + 1 ) -' 1 ) \ ( ( m + 1 ) -' 1 ) \ ( ( m + 1 ) -' 1 ) \ ( ( m + 1 ) -' 1 ) ) c= ( ( m + 1 ) \ ( ( m + 1 ) \ ( m + 1 ) ) \ ( m + 1 ) ) \ ( m + 1 ) \ ( m + 1 ) \ ( ( m + 1 ) \ ( ( m + 1 ) \ ( m + 1 ) \ ( m + 1 ) \ ( m + 1 ) \ ( ( m + 1 ) \ ( m + 1 ) \ ( m + 1 ) \ ( m + 1 ) \
ex Q1 being open Subset of X st s = Q1 & ex F1 being Subset of Y st F1 c= F & ( for a being Subset of Y st a in F1 & a in F1 holds F1 . a = ( F . a ) \/ ( F . b ) & ( F . b ) \/ ( F . b ) c= ( F . b ) \/ ( F . b ) ;
gcd ( Agcd ( A1 , B1 , b2 , b3 , b3 , b2 , b3 , b2 , b3 , b3 ) , ( for r1 , r2 , s1 , s2 , s1 ) holds gcd ( Agcd ( A1 , B1 , s1 , s2 , s1 ) , ( for r1 , r2 , s2 , s1 , s2 , s1 ) holds s2 = s1 + s2 + s2
R8 = ( ( ( the State of s2 ) . ( 1 + 1 ) ) . ( m2 + 1 ) ) .= ( ( the Following of s2 ) . ( m2 + 1 ) ) . ( m2 + 1 ) .= ( the Following of s2 ) . ( m2 + 1 ) .= ( the Following of s2 ) . ( m2 + 1 ) .= ( the Following of s2 ) . ( m2 + 1 ) .= ( the Following of s2 ) . ( m2 + 1 ) .= ( the Following of s2 ) . ( m2 + 1 ) .= ( the +* s2 ) . ( m2 + 1 ) .= ( the +* s2 ) . ( m2 + 1 ) .= ( the +* s2 ) . ( m2 + 1 ) .= ( the +* s2 ) . ( m2 + 1 ) . ( m2 + 1 ) .= ( the +* s2 )
CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) = CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= halt SCMPDS .= ( CurInstr ( P3 , s3 ) ) . IC Comput ( P3 , s3 , m1 ) .= ( CurInstr ( P3 , s3 ) ) . IC Comput ( P3 , s3 , m1 ) .= ( CurInstr ( P3 , s3 ) ) . IC Comput ( P3 , s3 , m1 ) .= ( CurInstr ( P3 , s3 ) .= ( CurInstr ( P3 , s3 ) ) .= ( CurInstr ( P3 , s3 ) ) . IC Comput ( P3 , s3 ) .= ( CurInstr ( P3 , s3 ) .= ( CurInstr ( P3 , s3 ) ) . IC Comput ( P3 , s3 ) ) .= ( CurInstr ( P3 , s3 ) .= (
P1 /\ P2 = ( { p1 } \/ LSeg ( p1 , p2 ) \/ LSeg ( p2 , p1 ) ) \/ ( LSeg ( p1 , p2 ) \/ LSeg ( p2 , p1 ) ) \/ ( LSeg ( p1 , p2 ) \/ LSeg ( p2 , p2 ) ) \/ ( LSeg ( p1 , p2 ) \/ LSeg ( p2 , p2 ) ) \/ ( LSeg ( p1 , p2 ) \/ LSeg ( p2 , p2 ) ) ;
func -> Subset of the Sorts of Al means : Def5 : a in it iff ex p st p in it & a = f . p & p . ( a , b ) = f . ( a , b ) ;
for a , b being Element of F_Complex st |. a .| > |. b .| & |. b .| >= 1 holds a * ( - b ) is >= 1 & a * ( - b ) is >= 1 implies a * ( - b ) is >= >= 1 & a * ( - b ) is >= 0
defpred P [ Nat ] means 1 <= $1 & $1 <= len g implies G * ( i , j ) `1 = g . ( $1 + 1 ) & G * ( i , j ) `1 = g . ( $1 + 1 ) `1 & G * ( i , j ) `2 = g . ( $1 + 1 ) `2 ;
assume that C1 , C2 , C2 , C2 , g being Element of C1 , f being Function of C1 , C2 , g being Function of C2 , C2 such that s1 = g & g = f & g = g & f = g & g = h & g = h & g = h ;
( ||. f .|| | X ) . c = ||. f .|| . c .= ||. f .|| . c .= ||. f .|| . c .= ||. f .|| . c .= ||. f .|| . c .= ||. f .|| . c .= ||. f .|| . c .= ||. f .|| . c ;
|. q .| ^2 = ( q `1 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 * ( q `1 ) ^2 + ( q `2 ) ^2 * ( q `2 ) ^2 + ( q `2 ) ^2 ) .= ( q `1 ) ^2 + ( q `2 ) ^2 ;
for F being Subset-Family of T7 st F is open & {} in F & for A , B st A in F & B in F & A misses B holds A misses B & A c= B & A c= B & A c= B & A c= B & A c= B & A c= B implies A c= B & A c= B & A c= B & A c= B & A c= B & A c= B & A c= B & A c= B & B c= B & A c= B & B c= B & A c= B & B c= B implies A c= B & A c= B & A c= B & A c= B & A c= B & A c= B & A c= B & B c= B & B c= B & B c= B & B c= B & B c= B & B c= B & A c= 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 ) and for k st k in dom F holds H . k = f . ( F . k ) ;
i |^ ( ( Let ( n + 1 ) - i ) |^ s ) = ( i |^ ( s + k ) ) - ( i |^ k ) .= ( i |^ ( s + k ) ) - ( i |^ k ) .= ( i |^ ( s + k ) ) - ( i |^ k ) .= ( i |^ k ) - ( i |^ k ) ;
consider q being oriented oriented oriented Chain of G such that r = q and q <> {} and ( F . ( q . 1 ) ) = v1 & ( F . ( q . 1 ) ) = v2 & ( F . ( q . 1 ) ) . ( q . 1 ) = v2 & ( F . ( q . 1 ) ) . ( q . 1 ) = v1 ;
defpred P [ Element of NAT ] means $1 <= len ( g , Z , I ) implies ( ( ( g , Z ) ^ I ) . ( len g + $1 ) ) . ( ( g , Z ) . ( len g + $1 ) ) = ( ( ( g , Z ) ^ I ) . ( len g + $1 ) ) . ( len g + $1 ) ;
for A , B being Matrix of n , K holds len ( A * B ) = len A & width ( A * B ) = width B & width ( A * B ) = width A & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B implies width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width ( A * B ) = width ( A * B ) = width ( A * B ) =
consider s being FinSequence of the carrier of R such that Sum s = u & for i being Element of NAT st 1 <= i & i <= len s holds s . i = a * s . i & s . i = b * s . i & s . i = b * s . i ;
func |( x , y )| -> Element of COMPLEX equals |( ( Re x ) * ( Re y ) , ( Im y ) * ( Im y ) , ( Im y ) * ( Im y ) , ( Im y ) * ( Im y ) ) , ( Im y ) * ( Im y ) , ( Im y ) * ( Im y ) , ( Im y ) * ( Im y ) ) *] ;
consider g2 being FinSequence of Fg such that g2 is continuous and ( for k be Element of NAT holds g2 . k = ( g . k ) & ( g . k ) . ( g . k ) = ( g . k ) . ( g . k ) & g2 . ( g . k ) = ( g . k ) . ( g . k ) ;
then n1 >= len p1 & n2 >= len ( p1 , p2 ) & crossover ( p1 , p2 , n1 , n2 , n3 , n2 , n3 , n2 , n3 , n3 , n1 , n2 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n2 , n3 , n3 , n2 , n3 , n2 , n3 , n2 , n3 , n2 , n3 , n3 , n2 , n3 , n2 , n3 , n2 , n3 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n2 , n3 , n2 , n3 , n3 , n3 , n3 , n2 , n3 , n2 , n3 ,
( q `1 ) * a <= ( q `1 ) * a & ( q `2 ) * a <= ( q `1 ) * a & ( q `2 ) * a <= ( q `2 ) * a & ( q `2 ) * a <= ( q `2 ) * a & ( q `2 ) * a <= ( q `2 ) * a ;
FK . ( p9 . ( len p9 + 1 ) ) = ( FK . ( p9 . ( len p9 + 1 ) ) ) * ( ( p9 . ( len p9 + 1 ) ) ) .= ( ( the Sorts of A ) * ( ( p9 . ( len p9 ) ) ) ) * ( ( ( p9 . ( len p9 ) ) ) ) * ( ( ( p9 . ( len p9 ) ) ) ) * ( ( ( p9 . ( len p9 ) ) ) ) ) .= ( ( ( ( p9 . ( len p9 ) ) ) ) * ( ( ( p9 ) ) ) ) * ( ( ( ( p9 ) ) ) ) ) * ( ( ( ( p9 . ( len p9 ) ) ) ) ) * ( ( ( ( p9 . ( len p9 ) ) ) ) ) .= ( ( ( p9 ) ) ) ) * ( ( ( ( ( p9 )
consider k1 being Nat such that k1 + k = 1 and a := ( ( intloc 0 ) .--> 1 ) . ( ( intloc 0 ) .--> 1 ) = ( ( intloc 0 ) .--> 1 ) +* ( ( intloc 0 ) .--> 1 ) . ( ( intloc 0 ) .--> 1 ) ) ;
consider B8 being Subset of A1 , y1 being Function of B1 , A2 such that B1 is finite and y1 is finite and D1 = the carrier of A1 and B1 = the carrier of A2 and D1 is finite and B1 is finite and B1 is finite and B1 is finite ;
v2 . b2 = ( ( curry F2 ) * ( ( ( F2 . b2 ) * ( ( ( F2 . b2 ) * ( F2 . b2 ) ) ) ) ) . b2 ) .= ( ( ( F2 . b2 ) * ( F2 . b2 ) ) * ( ( F2 . b2 ) * ( ( F2 . b2 ) * ( F2 . b2 ) ) ) .= ( ( F2 . b2 ) * ( ( F2 . b2 ) * ( F2 . b2 ) ) ) .= ( ( F2 . b2 ) * ( ( F2 . b2 ) ) * ( ( F2 . b2 ) * ( ( F2 . b2 ) ) ) * ( ( F2 . b2 ) ) * ( ( ( F2 . b2 ) ) * ( ( F2 . b2 ) * ( ( F2 . b2 ) ) ) * ( ( F2 . b2 ) ) ) . b2 ) .= ( ( ( ( ( ( F2 . b2
dom IExec ( I , P , s ) = the carrier of SCMPDS .= dom ( IExec ( I , P , s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) .= dom ( IExec ( I , P , s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) .= dom ( IExec ( I , P , s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) ;
ex d-32 be Real st d-32 > 0 & for h be Real st h <> 0 & |. h .| < e & |. h .| " * ||. ( L . h ) - ( R . h ) + ( R . h ) ) .| < e holds |. h . h - ( R . h ) .| < e
LSeg ( G * ( len G , 1 ) + |[ 1 , 1 ]| , G * ( len G , 1 ) + |[ 1 , 0 ]| ) c= Int cell ( G , len G , 1 ) \/ { G * ( len G , 1 ) + |[ 1 , 0 ]| } ;
LSeg ( mid ( h , i1 , i2 ) , i ) = LSeg ( h /. ( i1 + 1 ) , h /. ( i1 + 1 ) ) , h /. ( i1 + 1 ) ) .= LSeg ( h /. ( i1 + 1 ) , h /. ( i1 + 1 ) ) .= LSeg ( h /. ( i1 + 1 ) , h /. ( i1 + 1 ) ) .= LSeg ( h /. ( i1 + 1 ) , h /. ( i1 + 1 ) ) ;
A = { q where q is Point of TOP-REAL 2 : LE q , p , P & LE q , p , P & LE p , q , P & LE q , p , P & LE p , q , P & LE p , q , P & LE q , p , P & LE p , q , P & LE q , p , P & LE p , q , P & LE q , p , P & LE p , q , P & q , q , P & LE p , q , P & LE p , q , P & LE p , q , P & LE p , q , P & LE p , q , P & LE p , q , P & LE p , q , P & LE p , q , P & LE p , q , P & LE p , q , P & LE p , q , P & LE p , q , P & LE p , q , P & LE p , q
( ( - x ) .|. y ) = ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) * ( x .|. y ) * ( x .|. y ) * ( x .|. y ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) * ( x .|. y ) .= ( - 1 )
0 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) = ( p `1 / p `2 ) ^2 + ( p `2 / p `2 ) ^2 ) * sqrt ( 1 + ( p `2 / p `2 ) ^2 ) .= ( p `1 / p `2 ) ^2 + ( p `2 / p `2 ) ^2 ;
( ( U * ( W - p ) ) * ( W - p ) ) * ( W - p ) = ( ( U * ( W - p ) ) * ( W - p ) ) * ( W - p ) .= ( U * ( W - p ) ) * ( W - p ) .= ( W * ( W - p ) ) * ( W - p ) .= ( W * ( W - p ) ) * ( W - p ) .= ( W * ( W - p ) ) * ( W - p ) .= ( W * ( W - p ) + ( W * ( W - p ) ;
func Shift ( f , h ) -> PartFunc of REAL n , REAL means : Def4 : dom it = dom h & for x be Element of REAL n holds it . x = ( x + h ) . x + h . x ;
assume that 1 <= k and k + 1 <= len f and [ i , j ] in Indices G and f /. ( k + 1 ) = G * ( i , j ) and f /. ( k + 1 ) = G * ( i , j ) and f /. ( k + 1 ) = G * ( i , j ) and f /. ( k + 1 ) = G * ( i , j ) ;
Suppose not y in Free H and x in Free H implies ( ( H \ { x } ) \/ { y } ) \/ { x } = ( ( H \ { y } ) \/ { y } ) \/ { x } ) \/ { y } & ( ( H \ { x } ) \/ { y } ) \/ { x } = ( ( H \ { y } ) \/ { y } ) \/ { y } ) ;
defpred P11 [ Element of NAT , Element of NAT ] means ( p |^ $1 ) * ( 2 |^ $1 ) = p |^ ( $1 + 1 ) & ( $1 |^ $2 ) * ( 2 |^ $1 ) = ( p |^ $1 ) * ( 2 |^ $1 ) & ( $1 |^ $2 ) * ( 2 |^ $1 ) = ( p |^ $1 ) * ( 2 |^ $1 ) ;
func \sigma ( C ) -> non empty Subset-Family of X means : Def8 : for A , B being Subset of X st A c= X & B c= X & A c= Y holds it . ( A \/ B ) <= C . ( A \/ B ) & for A , B being Subset of X st A c= B & B c= X holds it . ( A \/ B ) <= C . ( A \/ B ) ;
[#] ( ( dist ( P ) ) .: Q ) = ( ( dist ( P ) ) .: Q ) .: Q & ( ( dist ( P ) ) .: Q ) .: Q = ( ( dist ( P ) ) .: Q ) .: Q & ( ( dist ( P ) ) .: Q ) .: Q = ( ( dist ( P ) ) .: Q ) .: Q ;
rng ( F | ( [: S , T :] , { 2 } :] ) = {} or rng ( F | ( [: S , T :] , { 2 } :] ) ) = { 1 } or rng ( F | ( [: S , T :] , { 2 } :] ) = { 2 } or rng ( F | ( [: S , T :] , { 2 } :] ) ) = { 2 } or rng ( F | ( [: S , T :] , { 2 } ) ) = { 2 } ) ;
( f " ) . ( ( rng f ) \ ( f " ) . i ) = f . i \ ( f " ) . i .= ( f " ) . i \ ( f " ) . i .= ( f " ) . i \ ( f " ) . i .= ( f " ) . i \ ( f " ) . i .= ( f " ) . i \ ( f " ) . i .= ( f " ) . i \ ( f " ) . i .= ( f " ) . i \ ( f " ) . i \ ( f " ) . i \ ( f " ) . i \ ( f " ) . i \ ( f " ) . i \ ( f " ) . i \ ( f " ) . i \ ( f " ) . i \ ( f " ) . i ) \ ( f " ) . i \ ( f " ) . i \ ( f " ) . i \ ( f " ) . i \ ( f " ) . i \ ( f " ) . i \ ( f
consider P1 , P2 being non empty Subset of TOP-REAL 2 such that P1 /\ P2 = { p1 , p2 , p3 } and P1 \/ P2 = { p1 , p2 , p3 } and C = ( p1 \/ p2 ) \/ ( p1 \/ p2 ) and C = ( p1 \/ p2 ) \/ ( p1 \/ p2 ) and C = ( p1 \/ p2 ) \/ ( p1 \/ p2 ) ;
f . p2 = |[ ( p2 `1 ) ^2 + ( p2 `2 ) ^2 * p2 `1 ) ^2 + ( p2 `2 ) ^2 * p2 `1 .= ( p2 `1 ) ^2 + ( p2 `2 ) ^2 * p2 `1 ;
( \lbrace ( a , X ) " ) . x ) = ( ( \lbrace a , X ) " . x , ( a , X ) " . x ) .= ( ( a , X ) " . x ) " .= ( ( a , X ) " . x ) " .= ( ( a , X ) " . x ) " .= ( ( a , X ) " . x ) " .= ( a " ) . x .= ( ( a " ) " . x ) " .= ( ( a " ) . x ) " . x .= ( ( a " ) . x .= ( ( a " ) . x ) " . x .= ( ( x " ) . x .= ( ( a " ) . x .= ( ( a " ) . x ) " . x .= ( ( a " ) . x ) * ( ( ( x " ) * ( ( ( x " ) * ( ( x " ) * ( x " ) * ( x " ) * ( x " ) * ( x " ) * ( x " ) * ( x " ) * ( x " ) * ( x " ) * ( x " )
for T being non empty normal TopSpace , A , B being Subset of T st A <> {} & A misses B & A misses B & B misses ( A \/ B ) holds ( for p being Point of T st p in A & p in B holds p in A & p in B & p in A implies p in Cl ( A \/ B ) )
for i , j st i + 1 in dom F for G1 , G2 being strict Subgroup of G st G1 = F . i & G2 = G . ( i + 1 ) & G1 = G . ( i + 1 ) & G2 = G . ( i + 1 ) holds G1 = G2 & G2 = G . ( i + 1 ) & G1 = G . ( i + 1 ) & G2 = G . ( i + 1 )
for x st x in Z holds ( ( ( #Z 2 ) * ( exp_R + arccot ) ) / ( exp_R + arccot ) ) . x = ( ( exp_R + arccot ) / ( exp_R + arccot ) ) . x / ( exp_R . x + exp_R . x ) ^2
synonym f /* ( x0 - h ) -> right convergent means : as : for x st x in dom f & x in dom f holds f . x = ( f /* ( x0 - h ) ) . ( x - h ) ;
then X1 , X2 , X1 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X1 , X2 ,
ex N be Neighbourhood of x0 st N c= dom SVF1 ( 1 , f , u ) & ex L be Neighbourhood of 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 )
sqrt ( ( ( p2 `1 ) ^2 + ( p2 `2 ) ^2 ) ) ^2 + ( ( p2 `1 ) ^2 + ( p2 `2 ) ^2 ) ^2 ) >= sqrt ( ( p2 `1 ) ^2 + ( p2 `2 ) ^2 ) ;
( ( 1 - t1 ) * ( |. f1 .| " ) ) . ( x - x0 ) = ( ( 1 - t1 ) * ( |. f1 .| " .| ) ) . ( x - x0 ) & ( ( 1 - t1 ) * ( |. f1 .| " ) ) . ( x - x0 ) = ( ( 1 - s1 ) * ( |. f1 .| " ) ) . ( x - x0 ) ;
assume that for x holds f . x = ( ( 1 - sin . x ) * ( sin . x ) ) / ( sin . x ) ^2 - ( sin . x ) ^2 ) and for x holds f . x = ( sin . x ) ^2 / ( sin . x ) ^2 - ( sin . x ) ^2 / ( sin . x ) ^2 - ( sin . x ) ^2 / ( sin . x ) ^2 ;
consider X1 being Subset of Y , Y1 being Subset of X such that t = X1 and Y1 in A and Y1 = X1 /\ [#] Y 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 , ( d |^ 3 ) * ( a |^ 3 ) + b |^ 3 ] where d is Element of GF : d in { 0 , 1 } & d <> 0 } \/ { 1 } = { 0 } \/ { 1 , 2 } \/ { 2 , 3 } \/ { 3 } \/ { 3 , 1 } \/ { 2 , 3 } } ;
sqrt ( ( W-bound D ) ^2 - ( W-bound D ) ^2 + ( W-bound D ) ^2 ) * ( ( W-bound D ) ^2 - ( W-bound D ) ^2 ) ) = ( W-bound D ) ^2 - ( ( W-bound D ) ^2 - ( W-bound D ) ^2 ) .= ( W-bound D ) ^2 - ( ( W-bound D ) ^2 - ( ( W-bound D ) ^2 ) ) ^2 - ( ( W-bound D ) ^2 - ( ( W-bound D ) ^2 ) ) ^2 ) .= ( ( W-bound D ) ^2 - ( ( W-bound D ) ^2 + ( ( W-bound D ) ^2 ) ) ^2 - ( ( W-bound D ) ^2 ) .= ( ( W-bound D ) ^2 ) ^2 - ( ( W-bound D ) ^2 ) ^2 - ( ( W-bound D ) ^2 ) + ( ( W-bound D ) ^2 ) * ( ( W-bound D ) ^2 ) + ( ( W-bound D ) ^2 + ( ( W-bound D ) ^2 ) * ( ( W-bound D ) ^2 ) .= ( ( W-bound D ) ^2 + ( ( W-bound D ) ^2 ) + ( ( W-bound D ) ^2 + ( ( W-bound D ) ^2 ) + ( ( W-bound D ) ^2 + ( (