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thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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thesis ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b
D c= S
let Y ;
S ` is convergent ;
q in X ;
V ;
y in N ;
x in T ;
m < n ;
m <= n ;
n > 1 ;
let r ;
t in I ;
n <= 4 ;
M is finite ;
let X ;
Y c= Z ;
A // M ;
let U ;
a in D ;
q in Y ;
let x ;
1 <= l ;
1 <= w ;
let G , H ;
y in N ;
f = {} ;
let x ;
x in Z ;
let x ;
F is one-to-one ;
e <> b ;
1 <= n ;
f is special ;
S misses C
t <= 1 ;
y divides m ;
P divides M ;
let Z ;
let x ;
y c= x ;
let X ;
let C ;
x _|_ p ;
o is monotone ;
let X ;
A = B ;
1 < i ;
let x ;
let u ;
k <> 0 ;
let p ;
0 < r ;
let n ;
let y ;
f is onto ;
x < 1 ;
G c= F ;
a is_>=_than X ;
T is continuous ;
d <= a ;
p <= r ;
t < s ;
p <= t ;
t < s ;
let r ;
D is_<=_than E ;
assume e > 0 ;
assume 0 < g ;
p in X ;
x in X ;
Y ` in Y ;
assume 0 < g ;
not c in Y ;
not v in L ;
2 in z `2 ;
assume f = g ;
N c= b ` ;
assume i < k ;
assume u = v ;
I = J ;
B ` = b ` ;
assume e in F ;
assume p > 0 ;
assume x in D ;
let i be element ;
assume F is onto ;
assume n <> 0 ;
let x be element ;
set k = z ;
assume o = x ;
assume b < a ;
assume x in A ;
a `1 <= b `1 ;
assume b in X ;
assume k <> 1 ;
f = product l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is from squares ;
assume m > 0 ;
assume A c= B ;
X is lower
assume A <> {} ;
assume X <> {} ;
assume F <> {} ;
assume G is open ;
assume f is dilatation ;
assume y in W ;
y \not <= x ;
A ` in B ` ;
assume i = 1 ;
let x be element ;
x `1 = x `1 ;
let X be BCI-algebra ;
assume S is non empty ;
a in REAL ;
let p be set ;
let A be set ;
let G be _Graph , e be set ;
let G be _Graph , e be set ;
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 = Set ;
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 * ( f1 - f2 ) is_differentiable_on Z ;
exp_R is_differentiable_in x ;
j < i0 ;
let j be Nat ;
n <= n + 1 ;
k = i + m ;
assume C meets S ;
n <= n + 1 ;
let n be Nat ;
h1 = {} ;
0 + 1 = 1 ;
o <> b3 ;
f2 is one-to-one ;
support p = {}
assume x in Z ;
i <= i + 1 ;
r1 <= 1 ;
let n be Nat ;
a "/\" b <= a ;
let n be Nat ;
0 <= r0 ;
let e be Real , x be 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 Program of SCM+FSA ;
let l be Nat ;
let i be Nat ;
let r ;
1 <= i2 ;
a "\/" c = c ;
let r be Real ;
let i be Nat ;
let m be Nat ;
x = p2 ;
let i be Nat ;
y < r + 1 ;
rng c c= E
Cl R is boundary ;
let i be Nat ;
R2 ;
cluster uparrow x -> \mathclose { \rm c } ;
X <> { x } ;
x in { x } ;
q , b // M ;
A . i c= Y ;
P [ k ] ;
2 to_power x in W ;
X [ 0 ] ;
P [ 0 ] ;
A = A |^ i ;
Let Let Let Let Let Let Let Let non \hbox \hbox - s ;
G . y <> 0 ;
let X be RealNormSpace , A be Subset of X ;
a in X ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , M be Subset of V ;
assume x in - M ;
k < s . a ;
not t in { p } ;
let Y be set , f be Function ;
M , L are_isomorphic ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded ;
rng f = Y ;
G8 c= L ;
assume x in Cl Q ;
m in dom P ;
i <= len Q ;
len F = 3 ;
still_not-bound_in p = {} ;
z in rng p ;
lim b = 0 ;
len W = 3 ;
k in dom p ;
k <= len p ;
i <= len p ;
1 in dom f ;
b `1 = a `1 + 1 ;
x `1 = a * y `2 ;
rng D c= A ;
assume x in K1 ;
1 <= i-32 ;
1 <= i-32 ;
p^ p^ c= PI ;
1 <= i-15 ;
1 <= i-15 ;
LMP C in L ;
1 in dom f ;
let seq , n ;
set C = a * B ;
x in rng f ;
assume f is_continuous_on X ;
I = dom A ;
u in dom p ;
assume a < x + 1 ;
s-7 is bounded ;
assume I c= P1 ;
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 >= Z1 ;
assume w = 0. V ;
assume x in A . i ;
g in being being being being being being _ of X ;
y in dom t ;
i in dom g ;
assume P [ k ] ;
non empty if C c= f ;
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 , X be non-empty ManySortedSet of S ;
assume P [ n ] ;
assume union S is finite independent ;
V is Subspace of V ;
assume P [ k ] ;
rng f c= NAT ;
assume ex_inf_of X , L ;
y in rng f ;
let s , I be set , A be ManySortedSet of I ;
b ` c= b9 ` \/ b ` ;
assume not x in mQ ;
A /\ B = { a } ;
assume len f > 0 ;
assume x in dom f ;
b , a // o , c ;
B in B-24 ;
cluster product p -> non empty ;
z , x // x , p ;
assume x in rng N ;
cosec is_differentiable_in x & cosec is_differentiable_in x ;
assume y in rng S ;
let x , y be element ;
i2 < i1 & i2 < len f ;
a * h in a * H ;
p , q in Y ;
redefine func sqrt I -> Ideal of L ;
q1 in A1 & q2 in A2 ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A2 c= A1 ;
accnn < n ;
assume A c= dom f ;
Re ( f ) is_integrable_on M ;
let k , m be element ;
a , a \equiv b , b ;
j + 1 < k + 1 ;
m + 1 <= n1 ;
g is_differentiable_in x0 & g is_differentiable_in x0 ;
g is_continuous_in x0 & g is_differentiable_in x0 ;
assume O is symmetric transitive ;
let x , y be element ;
let j0 be Nat ;
[ y , x ] in R ;
let x , y be element ;
assume y in conv A ;
x in Int V ;
let v be VECTOR of V ;
P3 halts_on s & P3 halts_on s ;
d , c // a , b ;
let t , u be set ;
let X be set ;
assume k in dom s ;
let r be non negative Real ;
assume x in F | M ;
let Y be Subset of S ;
let X be non empty TopSpace , A be Subset of X ;
[ a , b ] in R ;
x + w < y + w ;
{ a , b } is_>=_than c ;
let B be Subset of A , x be Element of B ;
let S be non empty ManySortedSign ;
let x be variable , f be Function ;
let b be Element of X , x be Element of X ;
R [ x , y ] ;
x ` ` = x ` ;
b \ x = 0. X ;
<* d *> in D * ;
P [ k + 1 ] ;
m in dom mn ;
h2 . a = y ;
P [ n + 1 ] ;
redefine func G * F -> | Functor ;
let R be non empty multMagma , A be Subset of R ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p `2 ;
assume f | X is lower ;
x in rng co /\ L~ co ;
let x be Element of B ;
let t be Element of D ;
assume x in Q .vertices() ;
set q = s ^\ k ;
let t be VECTOR of X ;
let x be Element of A ;
assume y in rng p `2 ;
let M be as as as as mamaid ;
let N be non empty \HM } is non empty Subset of M ;
let R be RelStr with finite finite for n be Nat ;
let n , k be Nat ;
let P , Q be be be be be be be be let let RelStr ;
P = Q /\ [#] S ;
F . r in { 0 } ;
let x be Element of X ;
let x be Element of X ;
let u be VECTOR of V ;
reconsider d = x as Int-Location ;
assume I is not carrier of A ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < v8 ;
x <= c2 . x ;
x in F ` /\ F ` ;
redefine func S --> T -> * -defined Function ;
assume that t1 <= t2 and t2 <= t2 ;
let i , j be even Integer ;
assume that F1 <> F2 and F2 <> G2 ;
c in Intersect ( union R ) ;
dom p1 = c & dom p2 = c ;
a = 0 or a = 1 ;
assume A1 : x <> A6 ;
set i1 = i + 1 ;
assume a1 = b1 & a2 = b2 ;
dom g1 = A & dom g2 = A ;
i < len M + 1 ;
assume not -infty in rng G ;
N c= dom f1 /\ dom f2 ;
x in dom sec /\ dom sec ;
assume [ x , y ] in R ;
set d = ( x - y ) / ( x - y ) ;
1 <= len g1 & g1 /. 1 = g1 /. 1 ;
len s2 > 1 & len s2 > 1 ;
z in dom f1 /\ dom f2 ;
1 in dom ( D2 | 1 ) ;
( p `2 / |. p .| - sn ) = 0 ;
j2 <= width G & j2 <= width G ;
len PI > 1 + 1 ;
set n1 = n + 1 ;
|. q-35 .| = 1 ;
let s be SortSymbol of S ;
gcd ( i , i ) = i ;
X1 c= dom f & X1 c= dom f ;
h . x in h . a ;
let G be \cap \it \it let G be non empty \cup { x } ;
cluster m * n -> invertible ;
let k9 be Nat , n be Element of NAT ;
i - 1 > m - 1 ;
R is transitive implies field R = field R
set F = <* u , w *> ;
p-2 c= P3 & p-2 c= P3 ;
I is_closed_on t , Q & I is_halting_on t , Q ;
assume [ S , x ] is is thesis ;
i <= len ( f2 | i ) ;
p is FinSequence of X ;
1 + 1 in dom g ;
Sum R2 = n * r ;
cluster f . x -> complex-valued ;
x in dom f1 /\ dom f2 ;
assume [ X , p ] in C ;
BX c= XX \/ XY ;
n2 <= ( 2 * n ) - 1 ;
A /\ [: P , Q :] c= A `
cluster x -valued -> constant for Function ;
let Q be Subset-Family of S , A be Subset of S ;
assume n in dom g2 & n + 1 in dom g2 ;
let a be Element of R ;
t `2 in dom e2 & t `2 in dom e2 ;
N . 1 in rng N ;
- z in A \/ B ;
let S be SigmaField of X , f be PartFunc of X , REAL ;
i . y in rng i ;
[: REAL , REAL :] c= dom f ;
f . x in rng f ;
mt <= ( r / 2 ) * ( 1 - r ) ;
s2 in r-5 & s2 in r-5 ;
let z , z be complex number ;
n <= Nseq . m ;
LIN q , p , s ;
f . x = waybelow x /\ B ;
set L = [: S , T :] ;
let x be non positive ExtReal ;
let m be Element of M ;
f in union rng ( F1 | n ) ;
let K be add-associative right_zeroed right_complementable non empty doubleLoopStr , p be Polynomial of K ;
let i be Element of NAT ;
rng ( F (#) g ) c= Y
dom f c= dom x & dom f c= dom x ;
n1 < n1 + 1 & n2 + 1 <= len f ;
n1 < n1 + 1 & n2 + 1 <= len f ;
cluster [: X , X :] -> 8 ;
[ y2 , 2 ] `2 = z ;
let m be Element of NAT ;
let S be Subset of R ;
y in rng ( S | A ) ;
b = sup dom f & b = sup dom f ;
x in Seg ( len q ) ;
reconsider X = D ( ) as set ;
[ a , c ] in E1 ;
assume n in dom h2 /\ dom h2 ;
w + 1 = ma + 1 ;
j + 1 <= j + 1 + 1 ;
k2 + 1 <= k1 + 1 ;
let i be Element of NAT ;
Support u = Support p \/ Support q ;
assume X is complete \frac of m ;
assume that f = g and p = q ;
n1 <= n1 + 1 & n2 <= len f ;
let x be Element of REAL , r be Real ;
assume x in rng ( s2 | X ) ;
x0 < x0 + 1 & x0 + 1 < x0 + 1 ;
len ( Carrier ( L ) ) = W ;
P c= Seg ( len A ) ;
dom q = Seg n .= Seg n ;
j <= width M *' ;
let seq1 be real-valued sequence of X ;
let k be Element of NAT ;
Integral ( M , P ) < +infty ;
let n be Element of NAT ;
assume z in z := ] ( 0 , A ) ;
let i be set ;
n -' 1 = n-1 - 1 ;
len ( n-27 ) = n ;
\mathop { Z } c= F ;
assume x in X or x = X ;
x is midpoint of b , c ;
let A , B be non empty set , f be Function of A , B ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q .= dom ( p | i ) ;
let s be Element of E * ;
let B1 be Basis of x , B be Basis of x ;
L3 /\ L2 = {} implies L1 /\ L2 = {} ;
L1 /\ LSeg ( p00 , p2 ) = {} ;
assume downarrow x = downarrow y ;
assume b , c // b , c ;
LIN q , c , c ;
x in rng ( f . -1-129 ) ;
set n8 = n + j ;
let D7 be non empty set , f be Function of D7 , D ;
let K be add-associative right_zeroed right_complementable non empty addLoopStr , M be Matrix of K ;
assume f `1 = f & h `2 = h ;
R1 - R2 is thesis implies R1 + R2 is thesis
k in NAT & 1 <= k ;
let a be Element of G ;
assume x0 in [. a , b .] ;
K1 ` is open & ( TOP-REAL 2 ) | K1 is open ;
assume a , b are_maximal distance in C ;
let a , b be Element of S ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = Integral ( M , f ) ;
cluster nas -> nes[ ;
not u in { ag } ;
the carrier of f c= B \/ { x }
reconsider z = x as VECTOR of V ;
cluster the Str of L -> \rangle ;
r (#) H is being being sequence of X ;
s . intloc 0 = 1 ;
assume that x in C and y in C ;
let U0 be strict universal MSAlgebra over S , A be Subset of U0 ;
[ x , Bottom T ] is compact ;
i + 1 + k in dom p ;
F . i is stable Subset of M ;
r-35 in : ( y in : not ( y in { x } ) & x in { y } ) ;
let x , y be Element of X ;
let A , I be seq of X ;
[ y , z ] in [: O , O :] ;
of dom Macro i , a be Int-Location ;
rng Sgm ( A ) = A ;
q |- p -" { All ( y , q ) } ;
for n holds X [ n ] ;
x in { a } & x in d ;
for n holds P [ n ] ;
set p = |[ x , y , z ]| ;
LIN o , a , b & LIN o , b , b ;
p . 2 = Z ^ Y ;
( D ) `2 = {} & ( D ) `2 = {} ;
n + 1 + 1 <= len g ;
a in [: CQC-WFF ( Al ) , NAT :] ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
I be Ideal of L ;
set g = f1 + f2 , h = f2 + f3 ;
a <= max ( a , b ) ;
i-1 < len G + 1-1 ;
g . 1 = f . i1 ;
x `1 , y `2 in A2 ;
( f /* s ) . k < r ;
set v = VAL g ;
i -' k + 1 <= S ;
cluster associative associative associative non empty for multMagma ;
x in support ( support ( support ( t ) ) ) ;
assume a in [: the carrier of G , the carrier of G :] ;
i `1 <= len ( y `1 ) ;
assume p divides b1 + b2 & p divides b2 ;
M1 <= upper_bound M1 & M1 <= M2 implies M1 + M2 <= M2 + M1
assume x in W-min ( X ) & y in W-min ( X ) ;
j in dom ( z | i ) ;
let x be Element of D ( ) ;
IC s4 = l1 & IC s4 = 0 ;
a = {} or a = { x } ;
set uG = Vertices G , uG = Vertices G , c = ( the carrier of G ) \ { c } ;
seq " is non-zero implies ( seq " ) is non-zero & lim ( seq " ) = 0
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
hcn c= h-14 & hcn c= dom h ;
]. a , b .[ c= Z ;
X1 , X2 are_separated implies X1 , X2 are_separated
a in Cl ( union ( F \ G ) ) ;
set x1 = [ 0 , 0 ] ;
k + 1 -' 1 = k ;
cluster real-valued -> INT -valued for Relation ;
ex v st C = v + W ;
let IT be non empty zero RelStr , f be Function of IT , TOP-REAL 2 ;
assume V is Abelian add-associative right_zeroed right_complementable ;
X-21 \/ Y in \sigma ( L ) ;
reconsider x = x as Element of S ;
max ( a , b ) = a ;
sup B is upper & sup B is upper ;
let L be non empty reflexive RelStr , X be Subset of L ;
R is reflexive transitive & R is transitive ;
E , g |= the_right_argument_of ( H ) ;
dom G `2 = a & dom G `2 = b ;
( 1 / 4 ) >= - r ;
G . p0 in rng G & G . p2 in rng G ;
let x be Element of [: F , G :] , y be Element of F ;
D [ P-6 , 0 ] & for n holds P-6 . n = 0 ;
z in dom id [: B , B :] ;
y in the carrier of N & y in the carrier of N ;
g in the carrier of H & h in the carrier of G ;
rng ( f | X ) c= NAT & rng ( f | X ) c= X ;
j `2 + 1 in dom s1 & j `2 + 1 in dom s1 ;
let A , B be strict Subgroup of G ;
let C be non empty Subset of REAL ;
f . z1 in dom h & h . z2 in dom h ;
P . k1 in rng P & P . k1 in rng P ;
M = ( A +* {} ) +* ( A , A ) ;
let p be FinSequence of REAL , n be Element of NAT ;
f . n1 in rng f & f . n2 in rng f ;
M . ( F . 0 ) in REAL ;
diameter [. a , b .[ = b-a ;
assume the distance of V , Q is v ;
let a be Element of ^ ( V ) ;
let s be Element of PL , x be Element of PL ;
let Py be non empty RelStr , x be Element of Py ;
n be Nat ;
the carrier of g c= B & the carrier of g c= B ;
I = halt SCM R & I = halt SCM R ;
consider b being element such that b in B ;
set BK = BCS K , BK = BCS K ;
l <= -> & j <= j implies ( j - 1 ) * ( j - 1 ) <= j
assume x in downarrow [ s , t ] ;
( x `2 ) ^2 in uparrow t & ( x `2 ) ^2 in uparrow t ;
x in O implies dom ( JumpParts T ) = { 1 , 2 }
let h be Morphism of c , a ;
Y c= [: the carrier of the_rank_of Y , the carrier of the_rank_of Y :] ;
A2 \/ A3 c= Carrier ( L1 ) \/ Carrier ( L2 ) ;
assume LIN o , a , b & LIN o , b , c ;
b , c // d1 , e2 or b , c // d2 , e2 ;
x1 , x2 , x3 , x4 , x5 be set ;
dom <* y *> = Seg 1 .= dom <* y *> ;
reconsider i = x as Element of NAT ;
set l = |. ar s .| ;
[ x , x `2 ] in X ~ ;
for n be Nat holds 0 <= x . n
[' a , b '] = [. a , b .] ;
cluster finite closed closed closed for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 , q1 is_collinear & q1 <> q2 ;
dom M1 = Seg n & dom M2 = Seg n ;
x = [ x1 , x2 ] & y = [ y1 , y2 ] ;
let R , Q be ManySortedSet of A ;
set d = ( 1 / n + 1 ) ;
rng g2 c= dom W & rng g2 c= dom ( W ^ <* x *> ) ;
P . ( [#] Sigma \ B ) <> 0 ;
a in field R & a = b ;
let M be non empty Subset of V , A be Subset of V ;
let I be Program of SCM+FSA , a be Int-Location ;
assume x in rng ( the InternalRel of R ) & y in rng ( the InternalRel of S ) ;
let b be Element of the carrier of T ;
dist ( e , z ) - r-r > r-r ;
u1 + v1 in W2 & v1 + v2 in W1 ;
assume that the carrier of L misses rng G and the carrier of L is finite ;
let L be lower-bounded antisymmetric transitive antisymmetric RelStr ;
assume [ x , y ] in a9 ;
dom ( A * e ) = NAT .= dom e ;
let a , b be Vertex of G ;
let x be Element of Bool ( M ) , i be Element of I ;
0 <= Arg a * PI ;
o9 , a9 // o9 , y & o9 , b9 // c9 , y ;
{ v } c= the carrier of l & { v } c= the carrier of l ;
let x be bound of A ;
assume x in dom ( uncurry uncurry f ) & y in dom uncurry f ;
rng F c= ( product f ) |^ X
assume D2 . k in rng D & D2 . k in rng D ;
f " . p1 = 0 & f " . p2 = 0 ;
set x = the Element of X , y = the Element of Y ;
dom Ser ( G ) = NAT & rng Ser ( G ) c= NAT ;
n be Element of NAT ;
assume LIN c , a , e1 & LIN c , a , e1 ;
cluster -> increasing for FinSequence of NAT ;
reconsider d = c , e = d as Element of L1 ;
( v2 |-- I ) . X <= 1 ;
assume x in the carrier of f & y in the carrier of f ;
conv @ S c= conv @ A & conv @ S c= conv @ A ;
reconsider B = b as Element of the carrier of T ;
J , v |= P ! ( P ! l ) ;
redefine func J . i -> non empty TopSpace ;
ex_sup_of Y1 \/ Y2 , T & ex_sup_of Y1 , T ;
W1 is well W & R is well field R implies R is transitive
assume x in the carrier of R & y in the carrier of S ;
dom ( n |-> 0 ) = Seg n .= Seg n ;
s4 misses s4 & s5 misses s4 ;
assume ( a 'imp' b ) . z = TRUE ;
assume that X is open and f = X --> d ;
assume [ a , y ] in an implies [ a , y ] in an ;
assume that function Sgm I c= J and not x in dom Reloc ( J , K ) ;
Im ( ( lim seq ) , ( lim seq ) ) = 0 ;
( ( sin (#) sin ) `| Z ) . x <> 0 ;
sin * ( cos - cos ) is_differentiable_on Z & cos * ( cos - cos ) is_differentiable_on Z ;
t3 . n = t3 . n & for n holds t1 . n = s . n ;
dom ( non empty set ) c= dom F & dom ( F | A ) c= dom F ;
W1 . x = W2 . x & W2 . x = W2 . x ;
y in W .vertices() \/ W .vertices() \/ W .vertices() ;
( ( k + 1 ) + 1 ) <= len ( ( k + 1 ) + 1 ) ;
x * a \equiv y * a . ( mod m ) ;
proj2 .: S c= proj2 .: P & proj2 .: ( S /\ T ) = T ;
h . p4 = g2 . I & h . I = 1 ;
G6 = ( U /. 1 ) `1 .= G * ( 1 , 1 ) `1 ;
f . rp1 in rng f & f . rp1 in rng f ;
i + 1 + 1-1 <= len f ;
rng F = rng ( F | ( n + 1 ) ) ;
mode non empty multMagma is well unital associative non empty multMagma ;
[ x , y ] in A ~ { a } ;
x1 . o in L2 . o & x2 . o in { x } ;
the carrier of support ( m ) c= B ;
not [ y , x ] in id X ;
1 + p .. f <= i + len f ;
seq ^\ k1 is lower & seq ^\ k1 is lower ;
len ( F | ( i + 1 ) ) = len F ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be complex number , p be Point of TOP-REAL 2 ;
Comput ( P , s , n ) = s ;
k <= k + 1 & k + 1 <= len p ;
reconsider c = {} T as Element of L ;
let Y be Element of be Element of \HM { the carrier of T } ;
cluster directed-sups-preserving -> directed-sups-preserving for Function of L , L ;
f . j1 in K . j1 & f . j2 in K . j2 ;
redefine func J => y -> total Function equals J => ( J --> x ) ;
K c= 2 -tuples_on the carrier of T ;
F . b1 = F . b2 & F . b2 = F . b2 ;
x1 = x or x1 = y or x1 = z ;
redefine pred a <> {} means a = 1 & a = 1 ;
assume that ca c= b and b in a ;
s1 . n in rng s1 & s1 . n in rng s1 ;
{ o , b2 } on C2 & { o , b2 } on C2 ;
LIN o , b , b9 & LIN o , b , b9 ;
reconsider m = x as Element of Funcs ( V , C ) ;
let f be non trivial FinSequence of D ;
let Fk2 be non empty element , F be Function of X , Y ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in [: F-8 , { w } :] ;
reconsider pp2 = x , pp2 = y as Subset of m ;
let A , B , C be Element of R ;
redefine func strict non empty for implies a is strict normal ;
rng c `2 misses rng ee `2 & rng c `2 misses rng ee `2 ;
z is Element of gr { x } & z is Element of gr { x } ;
not b in dom ( a .--> p1 ) & a in dom ( b .--> p1 ) ;
assume that k >= 2 and P [ k ] ;
Z c= dom ( ( - cot ) (#) ( ( id Z ) ^ ) ) ;
the component of Q c= UBD A & the component of Q c= UBD A ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( 1 / 2 ) & g2 in dom ( 1 / 2 ) ;
redefine pred f = u means a * f = a * u ;
for n holds P1 [ succ n ] implies P1 [ n + 1 ]
{ x . O : x in L } <> {} ;
let x be Element of V . s ;
let a , b be Nat ;
assume that S = S2 and p = p2 and q = p1 ;
gcd ( n1 , n2 , n3 ) = 1 & gcd ( n2 , n3 , n3 ) = 1 ;
set oI = a * ( - 1 / 2 ) , oI = a * ( - 1 / 2 ) ;
seq . n < |. r1 .| & |. seq . n .| < r ;
assume that seq is increasing and r < 0 ;
f . ( y1 , x1 ) <= a & f . ( y1 , x2 ) <= b ;
ex c being Nat st P [ c ] & c <= n ;
set g = { n |^ 1 } , n = 1 ;
k = a or k = b or k = c ;
a9 , b9 , c9 , b is_collinear & b9 , c9 , c is_collinear ;
assume that Y = { 1 } and s = <* 1 *> ;
If1 . x = f . x .= 0 .= 0 ;
W3 .last() = W3 . 1 & y in W3 . ( len W + 1 ) ;
cluster trivial finite for Walk of G , v , w ;
reconsider u = u as Element of Bags X ;
A in B ^ -> \in of A implies A , B are_that A , B are_that A , B are_
x in { [ 2 * n + 3 , k ] } ;
1 >= ( q `1 / |. q .| - cn ) / ( 1 + cn ) ;
f1 is_<= <= <= <= ( f2 ) - 1 ;
( f /. i ) `2 <= ( q `2 ) ^2 + ( q `2 ) ^2 ;
h is_\HM { the } \HM { Go-board ( C , n ) ) ;
( b `2 ) ^2 <= ( p `2 ) ^2 + ( p `2 ) ^2 ;
let f , g be V of X , Y ;
S * ( k , k ) <> 0. K ;
x in dom max ( ( - 1 ) (#) f , ( - 1 ) (#) f ) ;
p2 in [: N , N :] & p2 in [: N , N :] ;
len ( the_left_argument_of H ) < len ( H ) + len ( H ) ;
F [ A , F-14 ( A ) , F-14 ( A ) ] ;
consider Z such that y in Z and Z in X ;
redefine pred 1 in C means A c= C * ;
assume that r1 <> 0 or r2 <> 0 and r1 <> 0 ;
rng q1 c= rng ( C1 ^ C2 ) & rng ( C2 ^ C1 ) c= rng C2 ;
A1 , L , A3 , A3 , v2 , w be Element of L ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in C ( p , SQ ) & c in C ( p , SQ ) ;
then S is atomic & P-2 [ S ] ;
Cl Int ( [#] T ) = [#] T & Cl ( [#] T ) = [#] T ;
f12 | A2 = f2 | A2 & f12 | A2 = f2 | 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 & V c= Y \ Z ;
let X be Subset of S ~ ;
consider H1 such that H = 'not' H1 and H1 in X ;
1_ ( 1 , 1 ) c= ( t * p1 ) / ( 1 - t ) ;
0 * a = 0. R .= a * 0 .= a * 0 ;
A |^ ( 2 , 2 ) = A ^^ A ;
set vFinSequence = ( v /. n ) `1 , vn = ( v /. n ) `1 , vn = ( v /. n ) `1 , vn = ( v /. n ) `1 , vn = (
r = 0. ( REAL-NS n ) * ( ||. x .|| " ) ;
( f . p4 ) `1 >= 0 & ( f . p2 ) `2 >= 0 ;
len W = len ( W U ) + len ( W ) ;
f /* ( s * G ) is divergent_to+infty & f /. x0 in dom f ;
consider l being Nat such that m = F . l ;
t16 . a2 does not destroy b1 ;
reconsider Y1 = X1 , Y2 = X2 as SubSpace of X ;
consider w such that w in F and not x in w ;
let a , b , c , d be Real ;
reconsider i = i as non zero Element of NAT ;
c . x >= id ( L . x ) ;
\sigma ( T ) \/ omega ( T ) is Basis of T ;
for x being element st x in X holds x in Y
cluster [ x1 , x2 , x3 , x4 ] -> pair ;
downarrow a /\ downarrow t is Ideal of T implies a is Ideal of T
let X be with \hbox { NAT , D } , f be PartFunc of X , D ;
rng f = being Element of S2 \rm set ( S , X ) ;
let p be Element of B , x be SortSymbol of S ;
max ( N1 , 2 ) >= N1 & max ( N2 , 2 ) >= N2 ;
0. X <= b |^ ( m * mm1 ) ;
assume that i in I and R1 . i = R ;
i = j1 & p1 = q1 & p2 = q2 implies p1 = p2
assume gRRRRRRRRRRg in the carrier of g ;
let A1 , A2 be Point of S , A be Subset of S ;
x in h " P /\ [#] T1 implies x in h " P
1 in Seg 2 & 1 in Seg 3 implies ( 1 - 2 ) * ( 1 - 2 ) = 1
reconsider X-5 = X , Xelement = Y as non empty Subset of Tsuch that X is non empty and Y is non empty ;
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the Source of G ) -valued ;
n1 <= i2 + len g2 & n2 <= len g1 ;
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume v in the carrier' of G2 & v in the carrier' of G2 ;
y = Re y + ( Im y ) * i ;
( ( - 1 ) |^ p ) gcd ( p |^ n ) = 1 ;
x2 is_differentiable_on ]. a , b .[ & x2 in ]. a , b .[ ;
rng ( M | D2 ) c= rng ( D2 | D2 ) ;
for p be Real st p in Z holds p >= a
( ( X /\ Y ) \ f ) " { 0 } = proj1 * f " { 0 } ;
( seq ^\ m ) . k <> 0 implies ( seq ^\ m ) . k = ( seq ^\ m ) . k
s . ( G . ( k + 1 ) ) > x0 ;
( p |-count M ) . 2 = d & ( p |-count M ) . 3 = p ;
A \oplus ( B " C ) = ( A \oplus B ) " C
h \equiv gg . ( ( mod P ) + ( mod P ) ) ;
reconsider i1 = i-1 , i2 = i-1 as Element of NAT ;
let v1 , v2 be VECTOR of V , v be VECTOR of V ;
for V being Subspace of V holds V is Subspace of [#] V
reconsider i-7 = i , im2 = j as Element of NAT ;
dom f c= [: C ( ) , D ( ) :] ;
x in ( the Element of B ) . n /\ B . n ;
len [ [ f2 , f2 ] , [ f2 , f1 ] ] in Indices ( f2 ) ;
pp1 c= the topology of T & the topology of T c= the topology of T ;
]. r , s .[ c= [. r , s .] ;
let B2 be Basis of T2 , A be Subset of T2 ;
G * ( B * A ) = ( id o1 ) * A ;
assume that p , u , q is_collinear and p , q , q is_collinear ;
[ z , z ] in union rng ( F | X ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S , C = $1 .. S ;
LIN a1 , a3 , b1 & LIN a1 , b1 , c1 & LIN a2 , a3 , c1 ;
f " ( f .: x ) = { x } ;
dom w2 = dom r12 & dom r12 = dom r12 ;
assume that 1 <= i and i <= n and j <= n ;
( ( g2 ) . O ) `2 <= 1 ;
p in LSeg ( E . i , F . i ) ;
IK * ( i , j ) = 0. K ;
|. f . ( s . m ) -g .| < g1 ;
q9 . x in rng q9 & q9 . x in rng q9 ;
Carrier ( [: X , Y :] , [: X , Y :] :] misses [: X , Y :] , [: Y , X :] :]
consider c being element such that [ a , c ] in G ;
assume that Nreal = olet and olet i be Element of NAT ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= [: F-12 , Cc :] " { x }
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [: [. f . j , f . i .] :] ;
redefine pred 0 <= x & x ^2 <= 1 ;
p `2 - q `2 <> 0. TOP-REAL 2 & p `2 - q `2 <> 0. TOP-REAL 2 ;
redefine func being aa] ( S , T ) -> non empty ;
let x be Element of S ~ ;
the ObjectMap of F is one-to-one & the ObjectMap of F is one-to-one ;
|. i .| <= - - ( 2 to_power n ) & |. i .| <= - ( 2 to_power n ) ;
the carrier of I[01] = dom P & the carrier of I[01] = the carrier of I[01] ;
! * ( n + 1 ) ! > 0 * n ;
S c= ( A1 /\ A2 ) /\ A3 & S /\ S c= ( A1 /\ A2 ) /\ A3 ;
a3 , a4 // b3 , b3 & a3 , a4 // b3 , b3 ;
then dom A <> {} & dom A <> {} & dom ( A * ) = {} ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , Y & y in X implies x = y
set v2 = ( v /. ( i + 1 ) ) `1 , v1 = ( v /. ( i + 1 ) ) `2 ;
x = r . n .= r4 . n .= r4 . n ;
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 Lower_Arc ( P ) /\ Lower_Arc ( P ) & p in Lower_Arc ( P ) /\ Lower_Arc ( P ) ;
dom d2 = [: A , A :] & dom d2 = [: A , A :] ;
0 < ( p / ||. z .|| + 1 ) / ( ||. z .|| + 1 ) ;
e . ( m3 + 1 ) <= e . ( m3 + 1 ) ;
B \ominus X \/ B " Y c= B " X /\ Y
-infty < Integral ( M , Im ( g | B ) ) ;
cluster O := F -> \HM } -for operation of X ;
let U1 , U2 be non-empty MSAlgebra over S , f be Function of U1 , U2 ;
Proj ( i , n ) * g is_differentiable_on X & Proj ( i , n ) * g is_differentiable_on X ;
let x , y , z be Point of X , p be Point of X ;
reconsider px0 = p . x , px0 = p . x0 as Subset of V ;
x in the carrier of Lin ( A ) & y in the carrier of Lin ( B ) ;
let I , J be parahalting Program of SCM+FSA , a be Int-Location ;
assume that - a is lower and - b is lower and - a is lower ;
Int Cl A c= Cl Int Cl A & Cl 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 / |. p2 .| <= ( - 1 ) * ( - 1 ) ;
Cl ( Q ` ) = [#] ( TX ) .= [#] ( TX ) ;
set S = the carrier of T , T = the carrier of S ;
set I8 = TOP-REAL n , I8 = TOP-REAL n , I8 = TOP-REAL n ;
len p - n = len ( thesis - n ) + n ;
A is Permutation of Swap ( A , x , y ) ;
reconsider n6 = n7 - 1 as Element of NAT ;
1 <= j + 1 & j + 1 <= len ( s | Seg n ) ;
let q9 , q9 be Element of M , q be Element of M ;
a9 in the carrier of S1 & b9 in the carrier of S2 ;
c1 /. n1 = c1 . n1 & c2 /. n1 = c1 . n1 ;
let f be FinSequence of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
y = ( f * ( S . x ) ) . x .= ( f * ( S . x ) ) . x ;
consider x being element such that x in be Assume A in } ;
assume r in ( dist ( o ) ) .: P ;
set i2 = ( n + 1 ) - 1 , h = ( n + 1 ) - 1 ;
h2 . ( j + 1 ) in rng h2 /\ rng h2 ;
Line ( M29 , k ) . i = M . i .= Line ( M29 , k ) . i ;
reconsider m = ( x - 1 ) / 2 , n = ( x - 1 ) / 2 as Element of NAT ;
let U1 , U2 be strict Subspace of U0 , A be Subset of U1 ;
set P = Line ( a , d ) ;
len p1 < len p2 + 1 & p2 . ( len p1 + 1 ) = p2 . ( len p2 + 1 ) ;
let T1 , T2 be Scott topological let L be Scott thesis of L , x be Element of T1 ;
then x <= y & ( for x st x in dom f holds f . x c= y ) ;
set M = n -tuples_on the carrier of K , A = ( n -tuples_on the carrier of K ) ;
reconsider i = x1 , j = x2 , k = x3 as Nat ;
rng ( ( the_arity_of a9 ) | ( the carrier of S ) ) c= dom H ;
z1 " = ( z " * z ) * ( z " * z ) .= z " * ( z " * z ) ;
x0 - r / 2 in L /\ dom f & x0 - r / 2 in dom f ;
then w is that rng w /\ L <> {} & rng w /\ L <> {} ;
set x-10 = ( x ^ <* Z *> ) ^ <* Z *> ;
len w1 in Seg ( len w1 ) & len w2 = len w1 ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of PFuncs ( V , { k } ) ;
x . n = ( |. a . n .| ) * ( A . n ) ;
( p `1 ) ^2 / ( G * ( i , j ) ) ^2 <= ( G * ( i , j ) ) ^2 ;
rng ( g ) c= L~ ( g | ( len g ) ) \/ rng ( g | ( len g ) ) ;
reconsider k = i-1 * ( i + j ) as Nat ;
for n be Nat holds F . n is \HM { -infty } ;
reconsider x9 = x9 , y9 = y9 , z9 = y9 as VECTOR of M ;
dom ( f | X ) = X /\ dom f .= X /\ dom f ;
p , a // p , c & b , a // c , c ;
reconsider x1 = x , y1 = y , y2 = z as Element of REAL m ;
assume i in dom ( a * p ^ q ) ;
m . ag = p . ag & m . cg = p . cg ;
a / ( s . m - n ) / ( m + 1 ) <= 1 ;
S . ( n + k + 1 ) c= S . ( n + k ) ;
assume that B1 \/ C1 = B2 \/ C2 and C2 \/ C2 = C2 \/ C2 ;
X . i = { x1 , x2 } . i .= { x1 , x2 } . i ;
r2 in dom ( h1 + h2 ) & r1 < r2 & r2 < r2 ;
that \mathclose ( 0. R ) = a and b-0 = b ;
F8 is_closed_on t1 , Q1 & F8 is_halting_on t1 , Q1 & F8 is_halting_on t1 , Q ;
set T = l , X = l , x0 = l , Y = l , X = l , Y = l , Z = l \/ { x0 } ;
Int Cl ( Int Cl R ) c= Int Cl R & Cl Int Cl R c= Cl Int Cl R ;
consider y being Element of L such that c . y = x ;
rng ( F[: f , g :] ) = { F[: f , g :] } ;
G-23 " { c } c= B \/ S \/ S \/ { c } ;
fbeing Relation of [: X , X :] , X & X is Relation of [: X , X :] , Y ;
set RQ = the Point of P , RQ = the Point of Q , f = the Function of P , Q ;
assume that n + 1 >= 1 and n + 1 <= len M ;
k2 be Element of NAT , k be Element of NAT ;
reconsider pcSubset = u , pcSubset = v , pcSubset = w as Element of ( 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 ) ;
len Pt <= len Pt & len Pt <= len P-35 ;
x " in the carrier of A1 & x " in the carrier of A2 ;
[ i , j ] in Indices ( ( A + B ) * ( i , j ) ) ;
for m be Nat holds Re ( F ) . m is simple Function of S , REAL
f . x = a . i .= a1 . k .= a1 . k ;
let f be PartFunc of REAL i , REAL , x be Element of REAL i ;
rng f = the carrier of ( Carrier A ) \/ { i } .= { i } ;
assume s1 = sqrt ( 2 * p ) - ( 1 - s1 ) ;
pred a > 1 & b > 0 implies a / b > 1 ;
let A , B , C be Subset of lines I , a , b be Real ;
reconsider X0 = X , Y0 = Y as RealNormSpace , X = Y ;
let f be PartFunc of REAL , REAL , x0 be Element of REAL ;
r * ( v1 |-- I ) . X < r * 1 ;
assume that V is Subspace of X and X is Subspace of V ;
let t-3 , t-3 be binary st t-3 , t-3 , t-3 be binary st t-3 , t-3 , t-3 be binary st t-3 , tt1 , tt1 be Element of , tt1 ;
Q [ e-14 \/ { v-14 } , f ( ) . va1 , f ( ) . a1 , f ( ) . a2 , f ( ) . a2 , f ( ) . a2 , f ( ) . a3 , f ( ) . a4 ,
g \circlearrowleft ( W-min L~ z ) = z implies ( W-min L~ z ) .. z < ( W-min L~ z ) .. z
|. |[ x , v ]| - |[ x , y ]| .| = v\vert x - y .| ;
- f . w = - ( L (#) w ) .= - ( L (#) w ) ;
z - y <= x iff z <= x + y & y <= z ;
( 7 / p1 ) to_power ( 1 / e ) > 0 ;
assume X is BCK-algebra of 0 , 0 , 0 , 0 , 0 ;
F . 1 = v1 & F . 2 = v2 & F . 3 = v1 ;
( f | X ) . x2 = f . x2 & ( f | X ) . x2 = f . x2 ;
( ( tan | X ) . x ) in dom sec /\ dom sec ;
i2 = ( f /. len f ) `1 & ( f /. len f ) `2 = ( f /. len f ) `2 ;
X1 = X2 \/ ( X1 \ X2 ) & X2 = X2 \/ ( X1 \ X2 ) ;
[. a , b , 1_ G .] = 1_ G & a * b = 1_ G ;
let V , W be non empty VectSpStr over F_Complex , f be Function of V , W ;
dom g2 = the carrier of I[01] & dom g2 = the carrier of I[01] & rng g2 = the carrier of I[01] ;
dom f2 = the carrier of I[01] & dom f2 = the carrier of I[01] & rng f2 = the carrier of I[01] ;
( proj2 | X ) .: X = proj2 .: X .= proj2 .: X ;
f . ( x , y ) = h1 . ( x `1 , y `2 ) ;
x0 - r < a1 . n & x0 < a1 . n ;
|. ( f /* s ) . k - Gx0 .| < r ;
len Line ( A , i ) = width A .= len Line ( A , i ) ;
SFinSequence ^ S = ( S . g ) @ .= ( S . g ) @ ;
reconsider f = v + u as Function of X , the carrier of Y ;
intloc 0 in dom Initialized p & intloc 0 in dom Initialized p ;
i1 , i2 , i3 , i3 , Nat , I is not empty & I is not empty ;
arccos r + arccos r = ( PI / 2 + 0 ) / ( 2 * PI ) ;
for x st x in Z holds f2 is_differentiable_in x & f2 . x > 0 ;
reconsider q2 = ( q - x ) / ( 1 - x ) , q2 = ( q - x ) / ( 1 - x ) as Element of REAL ;
( 0 qua Nat ) + 1 <= i + j1 & i + 1 <= len f ;
assume f in the carrier of [: X , Omega Y :] ;
F . a = H / ( x , y ) . a ;
( ( TRUE T ) at ( C , u ) ) . z = TRUE ;
dist ( ( a * seq ) . n , h ) < r ;
1 in the carrier of [. 0 , 1 .] & 1 in dom ( f | [. 0 , 1 .] ) ;
( p2 `1 - x1 ) / ( 1 - g ) > - ( - g ) / ( 1 - g ) ;
|. r1 - `2 .| = |. a1 .| * |. ` .| ;
reconsider S-14 8 = 8 as Element of ( Seg 8 ) -tuples_on the carrier of K ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b
D0W .= D0W .C + 1 ;
i1 = [: a + n , n :] & i2 = [: b , n :] ;
f . a [= f . ( f .: O1 "\/" a ) ;
pred f = v & g = u , h = v + u ;
I . n = Integral ( M , F . n ) | E ;
chi ( [: T1 , S :] , S ) . s = 1 ;
a = VERUM ( A ) or a = VERUM ( A ) ;
reconsider k2 = s . b3 , k2 = s . b3 as Element of NAT ;
( Comput ( P , s , 4 ) ) . GBP = 0 ;
L~ M1 meets L~ ( R4 ^ ( p , q ) ) \/ LSeg ( p , q ) ;
set h = the continuous Function of X , R , x be Point of X ;
set A = { L . ( ( k + 1 ) + 1 ) where k is Nat : k in dom L } ;
for H st H is atomic holds P7 [ H ] ;
set b\HM = S5 ^\ ( i + 1 ) , Sseq = Sseq ^\ ( i + 1 ) ;
Hom ( a , b ) c= Hom ( a opp , b opp ) ;
( 1 / n + 1 ) < ( 1 / s ) " ;
( l `1 ) `1 = [ dom l , cod l ] `2 .= dom l `1 ;
y +* ( i , y /. i ) in dom g & y +* ( i , y ) in dom g ;
let p be Element of CQC-WFF ( Al ) , x be Element of CQC-WFF ( Al ) ;
X /\ X1 c= dom ( f1 - f2 ) /\ dom ( f2 - f1 ) ;
p2 in rng ( f /^ ( i + 1 ) ) & p2 in rng ( f /^ ( i + 1 ) ) ;
1 <= indx ( D2 , D1 , j1 ) + 1 - indx ( D2 , D1 , j1 ) + 1 ;
assume x in ( ( TOP-REAL 2 ) /\ K0 ) \/ ( ( ( TOP-REAL 2 ) | K0 ) /\ K0 ) ;
- 1 <= ( ( ( f2 ) . O ) `2 ) / ( 1 + 1 ) ;
let f , g be Function of I[01] , TOP-REAL 2 , a , b , c , d be Real ;
k1 -' k2 = k1 - k2 + 1 .= k1 - k2 + 1 ;
rng ( seq ^\ k ) c= ]. x0 , x0 + r .[ & ( for n holds seq . n < x0 ) implies seq is convergent & lim ( seq ) = 0
g2 in ]. x0 , x0 + r .[ /\ dom f & g2 in ]. x0 , x0 + r .[ /\ dom f ;
sgn ( p `1 , K ) = - ( 1_ K ) .= - ( 1_ K ) ;
consider u being Nat such that b = p |^ y * u ;
ex A being as as as Y. of omega st a = Sum A ;
Cl ( union H ) = union ( ( union H ) \/ ( union G ) ) ;
len t = len t1 + len t2 & len t1 = len t2 + len t1 ;
v-29 = v + w |-- ( A , B ) & v-29 = v |-- ( A , B ) ;
cv <> DataLoc ( t0 . GBP , 3 ) & card I + 3 = 2 ;
g . s = sup ( d " { s } ) & g . s = s ;
( \dot \dot y ) . s = s . ( \dot y ) ;
{ s : s < t } in [: NAT , NAT :] implies t = {}
s ` \ s = s ` \ ( 0. X \ s ) .= ( 0. X \ ( 0. X \ s ) ) ` ;
defpred P [ Nat ] means B + $1 in A implies A + $1 in B + A ;
( 329 + 1 ) ! = 3329 ! * ( 329 + 1 ) ;
[: U , { A } :] = [: T , { A } :] ;
reconsider y = y as Element of COMPLEX , x be Element of COMPLEX ;
consider i2 being Integer such that y0 = p * i2 and i2 in dom f ;
reconsider p = Y | ( Seg k ) as FinSequence of NAT ;
set f = ( S , U ) \mathop \mathop { {} } , g = S S S \mathop { {} } ;
consider Z being set such that lim s in Z and Z in F ;
let f be Function of I[01] , TOP-REAL n , a , b be Real ;
( SAT M ) . [ n + i , 'not' A ] <> 1 ;
ex r being Real st x = r & a <= r & r <= b ;
let R1 , R2 be Element of REAL n , x be Element of REAL n ;
reconsider l = 1_ ( Lin ( A ) ) , r = Sum ( l ) as Linear_Combination of A ;
set r = |. e .| + |. n .| + |. w .| + a ;
consider y being Element of S such that z <= y and y in X ;
a 'or' b 'or' c = 'not' ( ( a 'or' b ) 'or' c ) ;
||. x-> - ( g - x0 ) .|| < r2 / 2 ;
b9 , a9 // b9 , c9 & b9 , c9 // b9 , c9 & b9 , c9 // b9 , c9 ;
1 <= k2 -' k1 & k2 + 1 = k2 & k2 + 1 = k2 ;
( ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) ^2 >= 0 ;
( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 < 0 ;
E-max L~ Cage ( C , n ) in cell ( GoB f , 1 , 1 ) ;
consider e being Element of NAT such that a = 2 * e + 1 ;
Re ( ( lim F ) | D ) = Re ( ( lim G ) | D ) ;
LIN b , a , c or LIN b , c , a ;
p `1 , a // a `1 , b or p `2 , a // b `1 , a `2 ;
g . n = a * Sum ( f | n ) .= f . n ;
consider f being Subset of X such that e = f and f is \lbrace 0 , 1 } ;
F | ( [: N2 , S :] ) = CircleMap * ( | [: N2 , S :] ) ;
q in LSeg ( q , v ) \/ LSeg ( v , p ) \/ LSeg ( p , q ) ;
Ball ( m , r0 ) c= Ball ( m , s ) & Ball ( x , s ) c= Ball ( x , s ) ;
the carrier of (0). V = { 0. V } .= { 0. V } ;
rng ( ( cos | [. - 1 , 1 .] ) | [. - 1 , 1 .] ) = [. - 1 , 1 .] ;
assume that Re ( seq ) is summable and Im ( seq ) is summable and Im ( seq ) is summable ;
||. ( vseq . n ) - ( vseq . m ) .|| < e / 2 ;
set g = O --> 1 ;
reconsider t2 = t11 as $ 0 as string of S2 , X , Y ;
reconsider x-29 = seq . n , xM = seq . m as sequence of REAL n ;
assume that C meets C and L~ Cage ( C , n ) meets L~ go and LSeg ( pion1 , 1 ) meets L~ co ;
- ( ( 1 / 2 ) . n ) < F . n - ( x - x0 ) ;
set d1 = dist ( x1 , z1 ) , d2 = dist ( x2 , z1 ) , d2 = dist ( x2 , y2 ) ;
2 |^ ( x -' 1 ) = 2 |^ ( x -' 00 ) - 1 ;
dom ( v | ( Seg len ( v | ( i + 1 ) ) ) ) = Seg len ( v | ( i + 1 ) ) ;
set x1 = - k2 + |. k2 .| + 4 , x2 = - k2 + 1 ;
assume for n being Element of X holds 0. <= F . n & 0. <= F . n ;
assume that 0 <= T-32 . i and T-32 . ( i + 1 ) <= 1 ;
for A being Subset of X holds c . ( c . A ) = c . A
the carrier of Carrier ( LS + L2 ) c= I2 & the carrier of Carrier ( LS + L2 ) c= I2 ;
'not' Ex ( x , p ) => All ( x , 'not' p ) is valid ;
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the normal is normal normal of {} ;
Z c= dom ( ( ( - 1 ) (#) ( sin * f1 ) ) `| Z ) ;
|. 0. TOP-REAL 2 - q .| < r / 2 + |. q - p .| / 2 ;
not not not not ConsecutiveSet2 ( A , succ d ) c= ConsecutiveSet ( A , succ d ) ;
E = dom ( L (#) F ) & L (#) G is_measurable_on E & ( L (#) F ) | E is_measurable_on E ;
C / ( A + B ) = C / B * C / A ;
the carrier of W2 c= the carrier of V & the carrier of W1 c= the carrier of V ;
I . IC s2 = P . IC s2 .= succ IC s2 .= succ IC s2 .= succ IC s2 ;
pred x > 0 means : Def3 : ( 1 - x ) / ( 1 - x ) = x / ( 1 - x ) ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) \/ LSeg ( g , i ) ;
consider p being Point of T such that C = [: [. p , q .] , { p } :] ;
b , c are_connected & - C , - C - a + b + c are_connected ;
assume that f = id the carrier of O1 and f is Function of O1 , O2 and f is Function of O1 , O2 ;
consider v such that v <> 0. V and f . v = L * v ;
let l be Linear_Combination of {} ( ( the carrier of V ) \ { v } ) ;
reconsider g = f " as Function of U2 , U1 , U2 ;
A1 in the Points of G_ ( k , X ) & A2 in the Points of G ;
|. - x .| = - ( - x ) .= - ( - x ) .= - ( - x ) ;
set S = ) (# x , y , c #) ;
Fib ( n ) * ( 5 * Fib ( n ) - 1 ) >= 4 * be Nat ;
vseq /. ( k + 1 ) = ( vseq /. k ) `1 .= ( vseq /. k ) `1 ;
0 mod i = - ( i * ( 0 qua Nat ) ) .= - ( i * ( 0 qua Nat ) ) ;
Indices M1 = [: Seg n , Seg n :] & Indices ( M1 + M2 ) = [: Seg n , Seg n :] ;
Line ( S\mathopen , j ) . j = S\mathopen ( j , i ) . j ;
h . ( x1 , y1 ) = [ y1 , x1 ] & h . ( y1 , y2 ) = [ y1 , y2 ] ;
|. f .| - Re ( |. f .| (#) ( b (#) h ) ) is nonnegative ;
assume that x = ( a1 ^ <* x1 *> ) ^ b1 and y = ( a1 ^ <* x1 *> ) ^ b1 ;
ME is_closed_on IExec ( I , P , s ) , P & ME is_halting_on s , P ;
DataLoc ( t3 . a , 4 ) = intpos ( 0 + 4 ) .= intpos ( 0 + 4 ) ;
x + y < - x + y & |. x - y .| = - x + y ;
LIN c , q , b & LIN c , q , c & LIN c , q , c ;
f| ( 1 , t ) . ( 0 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + z1 ) .= y1 + ( y2 + z2 ) ;
flet let let let f be Function , a , b be set , v be Element of InputVertices S ;
( p `1 ) ^2 <= ( ( E-max C ) `1 ) ^2 + ( ( E-max C ) `1 ) ^2 ;
set R8 = Cage ( C , n ) \circlearrowleft E8 , R7 = Cage ( C , n ) ;
( p `1 ) ^2 >= ( E-max C ) ^2 + ( E-max C ) ^2 ;
consider p such that p = p-20 and s1 < p & p `2 <= s2 ;
|. ( f /* ( s * F ) ) . l - GM .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len Line ( N , k + 1 + 1 ) = width N .= len ( Line ( N , k + 1 ) ) ;
f1 /* s1 is convergent & f2 /* s1 is convergent & lim ( f1 /* s1 ) = lim ( f2 /* s1 ) ;
f . x1 = x1 & f . y1 = y1 & f . y2 = y1 ;
len f <= len f + 1 & len f + 1 <> 0 implies f . ( len f + 1 ) <> 0
dom ( Proj ( i , n ) * s ) = REAL m .= dom ( Proj ( i , n ) * s ) ;
n = k * ( 2 * t ) + ( n mod ( 2 * t ) ) ;
dom B = 2 -tuples_on the carrier of V & dom B = the carrier of V ;
consider r such that r _|_ a and r _|_ x and r _|_ y ;
reconsider B1 = the carrier of Y1 , B2 = the carrier of Y2 as Subset of X ;
1 in the carrier of [. 1 / 2 , 1 .] & 1 / 2 in dom f ;
for L being complete LATTICE holds <* <* \mathclose ( C ) , L *> *> , L are_isomorphic ;
[ gi , gj ] in Ii \ Ij ~ & [ gi , gj ] in Ij \ Ij ;
set S2 = 1GateCircStr ( x , y , c ) , S2 = 1GateCircStr ( y , c , d ) ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 ;
reconsider y = ( a ` ) / ( F . i ) , z = ( a ` ) / ( F . i ) as Element of L ;
dom s = { 1 , 2 , 3 } & s . 1 = d1 & s . 2 = d2 ;
( min ( g , ( 1 - g ) ) . c ) <= h . c ;
set G3 = the set of G , v = the Vertex of G , X = the set of G , Y = the set of G , X = the set of G , Y = the set of G , X = the set of G , Y = the set of G , X = the set of Y , Z
reconsider g = f as PartFunc of REAL , REAL-NS n , REAL-NS n , r be Real ;
|. s1 . m - p .| / |. p .| < d / ( p + 1 ) ;
for x being element st x in ( for u being element st u in ( for t being element st t in ( I ) holds u in I ) holds P [ x , u ]
P = the carrier of ( TOP-REAL n ) | K1 & Q = the carrier of ( TOP-REAL n ) | K1 ;
assume p00 in LSeg ( p1 , p2 ) /\ LSeg ( p01 , p2 ) \/ LSeg ( p2 , p1 ) /\ LSeg ( p01 , p2 ) ;
( 0. X \ x ) |^ ( m * ( k + 1 ) ) = 0. X ;
let g be Element of Hom ( cod f , dom f ) ;
2 * a * b + ( 2 * c * d ) <= 2 * C1 * C2 ;
let f , g , h be Point of the complex normed space of X , Y be non empty TopSpace ;
set h = Hom ( a , g (*) f ) ;
then idseq ( n ) | ( Seg m ) = idseq ( m ) | ( Seg m ) & m <= n ;
H * ( g " * a ) in the right of H & ( g " * a ) * ( g " * a ) in the right of H ;
x in dom ( ( ( - 1 / 2 ) (#) ( ( id Z ) ^ ) ) `| Z ) holds ( ( ( - 1 / 2 ) (#) ( ( id Z ) ^ ) ) `| Z ) . x = f . x
cell ( G , i1 , j2 -' 1 ) misses C & cell ( G , i1 , j2 -' 1 ) misses C ;
LE q2 , q1 , P , p1 , p2 & LE q1 , q2 , P , p1 , p2 implies LE q1 , q2 , P , p1 , p2
attr B is BDD closed means : Def3 : B c= BDD A & B is bounded ;
deffunc D ( set , Ordinal ) = union rng $2 & $2 = union rng $2 ;
n + - n < len ( p + - n ) + - n & n + - n < len p + n ;
attr a <> 0. K means : Def3 : the_rank_of M = the_rank_of ( a * M ) & the_rank_of M = the_rank_of ( a * M ) ;
consider j such that j in dom /\ /\ dom /\ I and I = len /\ j ;
consider x1 such that z in x1 and x1 in P8 and x = [ x1 , x2 ] ;
for n ex r being Element of REAL st X [ n , r ] & X [ n , r ]
set CP1 = Comput ( P2 , s2 , i + 1 ) , CP2 = Comput ( P2 , s2 , i + 1 ) , CP2 = Comput ( P2 , s2 , i + 1 ) , CP2 = Comput ( P2 , s2 , i + 1 ) , CP2 = Comput ( P2 , s2
set cv = 3 / ( 3 * ( a , b , c ) ) , cv = 4 * ( a , b , c ) ;
conv @ W c= union ( F .: ( E " ( W " ( E " ( W ) ) ) ) ) ;
1 in [. - 1 , 1 .] /\ dom ( arccot * ( 1 , 1 ) ) ;
r3 <= s0 + ( r0 - ( v2 - v1 ) ) / ( 2 * ( v2 - v1 ) ) ;
dom ( f (#) f4 ) = dom f /\ dom f4 .= dom ( f1 (#) f4 ) .= dom f1 /\ dom f2 ;
dom ( f (#) G ) = dom ( l (#) F ) /\ Seg k .= dom ( l (#) F ) ;
rng ( s ^\ k ) c= dom f1 \ { x0 } & ( f1 /* s ) . n in dom f2 ;
reconsider g9 = gp , gp = gp , gp = gp , gp = gp , gp = p , gp = p , gp = p , g = q , h = q , h = p , h = q , f = g ;
( T * h . s ) . x = T . ( h . s . x ) ;
I . ( L . J . x ) = ( I * L ) . ( J . x ) ;
y in dom *> & ( the mapping of commute ( A . o ) ) . y = ( ( Frege ( A . o ) ) . y ) ;
for I being non degenerated commutative Ring holds the carrier of I is commutative commutative non empty doubleLoopStr ;
set s2 = s +* Initialize ( ( intloc 0 ) .--> 1 ) , P2 = P +* Initialize ( ( intloc 0 ) .--> 1 ) ;
P1 /. IC s1 = P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 ;
lim S1 in the carrier of [. a , b .] & lim S1 in the carrier of [. a , b .] ;
v . ( l' . i ) = ( v *' l' ) . i .= ( v *' l' ) . i ;
consider n being element such that n in NAT and x = ( sn ) . n ;
consider x being Element of c such that F1 ( x ) <> F2 ( x ) and F2 ( x ) <> F2 ( x ) ;
Funcs ( X , 0 , x1 , x2 , x3 ) = { E } & card Funcs ( X , 0 , x2 , x3 ) = k ;
j + ( 2 * ( k + 1 ) ) + m1 > j + ( 2 * ( k + 1 ) ) ;
{ s , t } on A3 & { s , t } on A3 & { s , t } on B1 ;
n1 > len crossover ( p2 , p1 , n1 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , F , J ) ;
mg . ( HT ( mg , T ) ) = 0. L & ( HT ( mg , T ) ) . ( HT ( g , T ) ) = 0. L ;
then that H1 , H2 are_M and card H1 , H2 are_M and card H2 = 1 ;
( ( N-min L~ f ) .. f ) .. f > 1 & ( ( N-min L~ f ) .. f ) .. f > 1 ;
]. s , 1 .[ = ]. s , 2 .] /\ [. 0 , 1 .] .= ]. s , 1 .] /\ [. 0 , 1 .] ;
x1 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) ) & x2 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) ) ;
let f1 , f2 be continuous PartFunc of REAL , the carrier of S , x0 be Point of S ;
DigA ( t-23 , z9 ) is Element of k -tuples_on ( the carrier of K ) ;
I : I = d\lbrace k2 , k2 , k2 , k2 , k2 } = k2 & I = k2 ;
[: u , { u } :] = { [ a , u9 ] } & [: u , { v } :] = [: { a } , { v } :] ;
( w | p ) | ( p | ( w | p ) ) = p ;
consider u2 such that u2 in W2 and x = v + u2 and u2 in W2 and u1 in W3 ;
for y st y in rng F ex n st y = a |^ n & F . y = a |^ n
dom ( ( g * ( ( f , C ) | K ) ) | K ) = K ;
ex x being element st x in ( ( ( the Sorts of U0 ) \/ A ) . s ) . s & x in ( ( the Sorts of U0 ) . s ) . s ;
ex x being element st x in ( ( for s being element st s in ( ( for s being element st s in O1 holds s in O ) . s ) ) & x in A
f . x in the carrier of [. - r , r .] & f . x in [. - r , r .] ;
( the carrier of X1 union X2 ) /\ ( ( the carrier of X1 union X2 ) /\ ( the carrier of X2 ) ) <> {} ;
L1 /\ LSeg ( p00 , p2 ) c= { p01 } \/ LSeg ( p00 , p2 ) \/ LSeg ( p00 , p2 ) ;
( b + be - a ) / 2 in { r : a < r & r < b } ;
ex_sup_of { x , y } , L & x "\/" y = sup { x , y } ;
for x being element st x in X ex u being element st P [ x , u ]
consider z being Point of GX such that z = y and P [ z ] and z in G and z <> y ;
( the sequence of ( ( the carrier of X ) | ( the carrier of X ) ) ) . ( Y. ) <= e ;
len ( w ^ w2 ) + 1 = len w + 2 + 1 .= len w + 1 ;
assume that q in the carrier of ( TOP-REAL 2 ) | K1 and q in the carrier of ( TOP-REAL 2 ) | K1 ;
f | E-4 ` = g | E-4 ` & g | E-4 = g | E-4 ` ;
reconsider i1 = x1 , i2 = x2 , z = x3 , F = x4 , G = x5 ( x1 , x2 , x3 , x4 ) as Element of NAT ;
( a * A * B ) @ = ( a * ( A * B ) ) @ ;
assume ex n0 being Element of NAT st f to_power n0 is \mathop { 0 } & f to_power n0 is \mathop { 0 } ;
Seg ( len ( ( the FinSequence of f2 ) * ( f1 | i ) ) ) = dom ( ( the support of f2 ) * ( f1 | i ) ) ;
( Complement ( A ) ) . m c= ( Complement ( A ) ) . n /\ ( Complement ( A ) ) . m ;
f1 . p = p9 & g1 . ( p9 , q9 ) = d & f1 . ( p9 , q9 ) = e ;
FinS ( F , Y ) = FinS ( F , dom ( F | Y ) ) .= FinS ( F , Y ) ;
( x | y ) | z = z | ( y | x ) ;
( |. x .| to_power n ) / ( n + 1 ) <= ( r2 / ( n + 1 ) ) ;
Sum ( F ) = Sum f & dom ( F ) = dom g & rng ( F ) c= dom f ;
assume for x , y being set st x in Y & y in Y holds x /\ y in Y ;
assume that W1 is Subspace of W3 and W2 is Subspace of W3 and W3 is Subspace of W3 and W3 is Subspace of W3 ;
||. ( t . x ) - ( t . x ) .|| = lim ||. ( t . x ) - ( t . x ) .|| .= ||. ( t . x ) - ( t . x ) .|| ;
assume that i in dom D and f | A is lower and g | A is lower ;
( ( p `2 ) ^2 - 1 ) / ( - 1 ) <= ( - seq ) / ( - 1 ) ;
g | Sphere ( p , r ) = id Sphere ( p , r ) & g | Sphere ( p , r ) = id Sphere ( p , r ) ;
set N8 = ( N-min L~ Cage ( C , n ) ) .. Cage ( C , n ) ;
for T being non empty TopSpace holds T is countable countable implies the TopStruct of T is countable countable countable
width B |-> 0. K = Line ( B , i ) .= B * ( i , i ) .= B * ( i , j ) ;
attr a <> 0 means : Def3 : ( A \+\ B ) carrier a = ( A carrier a ) \+\ ( B carrier a ) ;
then f is_is_is_differentiable on 3 , u & pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 & pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 ;
assume that a > 0 and a <> 1 and b > 0 and c > 1 and b > 0 and c > 0 ;
w1 , w2 in Lin { w1 , w2 } & ( the carrier of X ) \ { w1 , w2 } c= Lin { w1 , w2 } ;
p2 /. IC s-7 = p2 . IC s-7 .= p2 . IC s-7 .= p2 . IC s-7 .= ( p2 /. IC sU ) . IC sU ;
ind ( T-10 | b ) = ind b .= ind b - ind b .= ind b - ind b ;
[ a , A ] in the carrier of Line ( AS , b ) & [ a , A ] in the carrier of Line ( AS , b ) ;
m in ( the Arrows of C ) . ( o1 , o2 ) & ( the Arrows of C ) . ( o1 , o2 ) = ( the Arrows of C ) . ( o2 , o2 ) ;
( ( a , CompF ( PA , G ) ) . z ) . z = FALSE ;
reconsider phi = phi , phi = phi , phi = phi , N = S , N = S , F = S S , F = S S , G = S S , G = S S , G = S S , N = S , F = S , G = S , G = S , F = S , G = S , G = S , G
len s1 - ( len s2 - 1 ) + 1 > 0 + 1 ;
delta ( D ) * ( f . ( upper_bound A ) - lower_bound A ) < r ;
[ f21 , f22 ] in [: the carrier' of A , the carrier' of B :] ;
the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 = K1 & the carrier of ( ( ( TOP-REAL 2 ) | K1 ) | K1 ) | K1 = K1 ;
consider z being element such that z in dom g2 and p = g2 . z and q = g2 . z ;
[#] V1 = { 0. V1 } .= the carrier of ( the carrier of V1 ) /\ the carrier of V1 .= the carrier of V1 ;
consider P2 being FinSequence such that rng P2 = M and P2 is one-to-one and P2 is one-to-one and P2 is one-to-one ;
assume that x1 in dom ( f | X ) and ||. x1 - x0 .|| < s and ||. x1 - x0 .|| < s ;
h1 = f ^ ( <* p3 *> ^ <* p *> ) .= h ^ <* p *> .= h ^ <* p *> ;
c / ( |[ b , c ]| ) = c .= |[ |[ a , c ]| , |[ b , c ]| ]| ;
reconsider t1 = p1 , t2 = p2 , t1 = p2 as Term of C , V , s be t be Element of C ;
( 1 / 2 ) in the carrier of [. 1 / 2 , 1 .] & ( 1 / 2 ) * ( 1 / 2 ) in the carrier of I[01] ;
ex W being Subset of X st p in W & W is open & h .: W c= V ;
( h . p1 ) `2 = C * ( p1 `2 ) + D .= ( h . p1 ) `2 + D ;
R . b - b = 2 * - b .= 2 * - b .= - b ;
consider \hbox such that B = - 1 * ] + ( - 1 ) * A and 0 <= 1 ;
dom g = dom ( ( the Sorts of A ) * ( ( the Sorts of A ) * ( the_arity_of o ) ) ) & dom ( ( the Sorts of A ) * ( ( the Sorts of A ) * ( the_arity_of o ) ) ) = dom ( ( the Sorts of A ) * ( the_arity_of o ) ) ;
[ P . U7 , P . ( U7 ) ] in => ( ( P . l ) . ( 'not' ( P . l ) ) , ( P . l ) ) ;
set s2 = Initialize s , P2 = P +* stop I ;
reconsider M = mid ( z , i2 , i1 ) , N = len z - 1 , M = z + ( 1 - 1 ) as non empty Subset of TOP-REAL 2 ;
y in product ( ( Carrier J ) +* ( V , { 1 } ) ) & y in product ( ( Carrier J ) +* ( V , { 1 } ) ) ;
1 / ( |[ 0 , 1 ]| ) = 1 & 0 / ( |[ 0 , 1 ]| ) = 0 ;
assume that x in the left of g or x in the left of g and y = g . x and x in the right of g ;
consider M being strict Subgroup of AS such that a = M and T is Subgroup of M and M is Subgroup of M ;
for x st x in Z holds ( ( ( #Z n ) + f ) `| Z ) . x <> 0 & ( ( ( #Z n ) + f ) `| Z ) . x = 0
len W1 + len W2 + m = 1 + len W3 + m .= len W1 + len W2 + m + 1 ;
reconsider h1 = ( vseq . n ) - tV as Lipschitzian Lipschitzian LinearOperator of X , Y ;
( - ( i + 1 ) mod ( len ( p + q ) ) + 1 ) in dom ( p + q ) ;
assume that s2 is U and F in the Element of s2 and s1 in the Element of s1 and s2 in the carrier of s1 and s2 in the carrier of s1 ;
( ( for x , y being Element of NAT holds x + y ) * ( x + y ) = gcd ( x , y ) * ( x + y ) ;
for u being element st u in Bags n holds ( p + m ) . u = p . u
for B be Subset of u-5 st B in E holds A = B or A misses B or A misses B
ex a being Point of X st a in A & A /\ Cl { y } = { a } ;
set W2 = tree ( p ) \/ W1 , W3 = tree ( q ) \/ W2 ;
x in { X where X is Ideal of L : X is finite & X is finite } ;
the carrier of W1 /\ W2 c= the carrier of W1 & the carrier of W1 /\ W2 = the carrier of W2 ;
( a + b ) * id a = ( a + b ) * id a .= ( a + b ) * id a .= ( a + b ) * id a ;
( ( X --> f ) . x ) . x = ( X --> dom f ) . x .= ( X --> dom f ) . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) , y = the Element of LSeg ( g , m ) /\ LSeg ( g , n ) ;
p => ( q => r ) => ( p => q => ( p => r ) ) in TAUT ( A )
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
- 1 + 1 <= ( ( being C - 2 ) |^ ( n -' m ) + 1 ) ;
( reproj ( 1 , z0 ) ) . x in dom ( f1 (#) f2 ) /\ dom ( f2 (#) f3 ) ;
assume that b1 . r = { c1 } and b2 . r = { c2 } and b2 . r = c1 ;
ex P st a1 on P & a2 on P & a3 on P & a4 on P & a4 on P & a5 on P & a5 on P & a4 on P & \lbrack a2 , a3 .] on P ;
reconsider gf = g `1 * f `2 , hg = h `2 * g `2 , hg = h `2 * g `2 as strict Element of X ;
consider v1 being Element of T such that Q = ( downarrow v1 ) ` and v1 in { x } and v1 in { y } ;
n in { i where i is Nat : i < n0 + 1 & i <= n + 1 } ;
( F * ( i , j ) ) `2 >= ( F * ( m , k ) ) `2 ;
assume K1 = { p : p `1 / |. p .| >= sn & p `2 <= 0 & p <> 0. TOP-REAL 2 } ;
ConsecutiveSet ( A , succ O1 ) = ( ConsecutiveSet ( A , O1 ) ) . ( succ O1 ) .= A . ( succ O1 ) ;
set If1 = Macro SubFrom ( a , intloc 0 ) , If2 = SubFrom ( a , intloc 0 ) , If2 = SubFrom ( a , intloc 0 ) , If2 = SubFrom ( a , intloc 0 ) ;
for i be Nat st 1 < i & i < len z holds z /. i <> z /. 1 ;
X c= ( the carrier of L1 ) /\ ( the carrier of L2 ) & X c= the carrier of L1 /\ the carrier of L2 ;
consider x9 be Element of GF ( p ) such that x9 |^ 2 = a & x9 |^ 3 = b ;
reconsider ee = ee , fe = fe , fe = fe , fe = f , fe = f , e = g , f = h , g = h , h = f , h = g , f = h , g = h , h = h , h = h , f = g , h = h , g = h , h = h , f = h , g = h ,
ex O being set st O in S & C1 c= O & M . O = 0. ( X , Y ) & M . ( O , O ) = 0. ( X , Y ) ;
consider n being Nat such that for m being Nat st n <= m holds S . m in U1 and n <= m ;
f (#) g * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) . x ) . x0 ;
defpred P [ Nat ] means A + succ $1 = succ A + $1 & A in succ $1 implies A = succ $1 + A ;
the left of - g = the left of - g & the left of - g = the left of - g implies g = f
reconsider p9 = x , q9 = y , p9 = z , q9 = x , q9 = y , p9 = z as Point of Euclid 2 ;
consider ex g2 such that g2 = y & x <= g2 & g2 <= x0 and x0 < g2 & g2 in dom f ;
for n being Element of NAT ex r being Element of REAL st X [ n , r ] & X [ n , r ]
len ( x2 ^ y2 ) = len x2 + len y2 .= len ( x2 ^ y2 ) + len ( y2 ^ y1 ) .= len ( x2 ^ y2 ) + len y1 ;
for x being element st x in X holds x in the set of ( the set of n0 ) | ( X \ { 0 } ) & ( x in X implies x in X )
LSeg ( p01 , p2 ) /\ LSeg ( p1 , p01 ) = {} & LSeg ( p1 , p01 ) /\ LSeg ( p01 , p2 ) = {} ;
func non empty set equals [: [: [: X , X :] , { x } :] , [: X , { x } :] :] \/ [: [: X , { x } :] , [: X , { x } :] :] ;
len ( -> [: { 0 } , { 1 } :] , [: { 1 } , { 1 } :] :] is len ( ( C | ( Seg 1 ) ) ) , ( C | ( Seg 1 ) ) ) ;
attr K is has a , a , b , c , a , b , c be Element of K ;
consider o being OperSymbol of S such that t `1 . {} = [ o , the carrier of S ] and t `2 = [ o , the carrier of S ] ;
for x st x in X ex y st x c= y & y in X & y is \mathop of f . x & f . y is x
IC Comput ( P-6 , scccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
pred q < s means : Def3 : r < s & ]. r , s .[ \not c= ]. p , q .[ & s <= q ;
consider c being Element of Class ( f , c ) such that Y = ( F . c ) `1 and c in X and c in X ;
func the ResultSort of S2 -> Function means : Def3 : the ResultSort of it = id the carrier of S2 & the ResultSort of it = id the carrier of S2 ;
set y9 = [ <* y , z *> , f2 ] , z9 = [ <* z , x *> , f3 ] ;
assume x in dom ( ( ( #Z n ) * ( ( #Z n ) * ( ( #Z n ) * ( ( #Z n ) * ( f1 + #Z n ) ) ) ) ) `| Z ) ;
r-7 in Int cell ( GoB f , i , GoB f ) \ { ( GoB f ) * ( i , j + 1 ) } & r-7 in { ( GoB f ) * ( i + 1 , j ) } ;
q `2 >= ( Cage ( C , n ) /. ( i + 1 ) ) `2 & ( Cage ( C , n ) /. ( i + 1 ) ) `2 >= ( Cage ( C , n ) /. ( i + 1 ) ) `2 ;
set Y = { a "/\" a ` : a in X } ;
i -' len f <= len f + len f1 - len f + len f1 - len f + len f1 - len f + len f1 - len f + len f1 - len f + len f1 - len f + len f1 - len f + len f1 - len f + len f1 - len f1 + len f1 - len f1 + len f1 - len f1 + len f1 - len f1 + len f1 -
for n ex x st x in N & x in N1 & h . n = x- ( x0 - r )
set s0 = ( \mathop { a , I , p , s ) . i , sI = p +* I +* J ;
p ( k ) . 0 = 1 or p ( k ) . 0 = - 1 or p ( k ) . 0 = - 1 & 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 f . x9 = f . x9 ;
( p ^ ( q | k ) ) . m = ( q | k ) . ( Seg len p ) .= ( p | k ) . ( len p ) ;
g + h = gg + h1 & for X holds f + g = g + h & f = h + h ;
L1 is distributive & L2 is distributive implies [: L1 , L2 :] is distributive & [: L1 , L2 :] is distributive
pred x in rng f & y in rng ( f | x ) implies f | x = f | y & f | x = f | y ;
assume that 1 < p and ( 1 - p ) / q + ( 1 - q ) / p = 1 and 0 <= a and a <= b ;
F* ( f , A1 ) = rpoly ( 1 , A1 ) *' t + rpoly ( 1 , A1 ) *' t .= rpoly ( 1 , A1 ) *' t ;
for X being set , A being Subset of X holds A ` = {} implies A = {} & A = {} implies A = {}
( ( ( ( ( ( ( X ) ) | X ) ) | X ) ) `1 <= ( ( ( ( ( ( ( X ) | X ) | X ) ) | X ) ) | X ) `1 ;
for c being Element of the function of the free of A , a being Element of the free of A holds c <> a implies c <> a
s1 . GBP = ( Exec ( i2 , s2 ) ) . GBP .= s2 . GBP .= Exec ( i2 , s2 ) . GBP .= s2 . GBP ;
for a , b being Real holds |[ a , b ]| in ( y >= 0 ) implies b >= 0 & a >= 0 & b >= 0
for x , y being Element of X holds x ` \ y = ( x \ y ) ` & y \ x = ( x \ y ) `
mode BCK-algebra of i , j , m , n , m , n , m , n , m , m , n , m , n , m , n , m , n , m , m , n , m , n , m , n , m , m , n , m , n , m , n , m , m , n ;
set x2 = |( Re ( y - x ) , Im ( y - x ) )| ;
[ y , x ] in dom u5 & ( y , x ) `1 = g . y & ( y , x ) `2 = g . y ;
]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .[ c= A & [. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .] c= A ;
0 <= delta ( S2 ) . n & |. delta ( S2 ) . n - 0 .| < ( e / 2 ) / ( n + 1 ) ;
( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) <= ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ;
set A = ( 2 / b-a ) ;
for x , y being set st x in R" holds x , y are_let x , y implies x , y are_$ let x , y
deffunc FF2 ( Nat ) = b . ( $1 + 1 ) * ( M * G ) . ( $1 + 1 ) * ( M * G ) . ( $1 + 1 ) ;
for s being element holds s in ( \rm \rm \rm \rm means : Let $ s being element holds s in ( \rm \rm \rm \rm : /. ( s , S ) ) ) implies s in ( { s } \/ { t } )
for S being non empty non void non void non empty non void non empty ManySortedSign st S is connected holds S is connected
max ( degree ( ( z `1 ) ^2 + ( z `2 ) ^2 ) , degree ( ( z `2 ) ^2 + ( z `2 ) ^2 ) ) >= 0 ;
consider n1 being Nat such that for k holds seq . ( n1 + k ) < r + s ;
Lin ( A /\ B ) is Subspace of Lin ( A /\ B ) & Lin ( A /\ B ) is Subspace of Lin ( B /\ A )
set n-15 = n' '&' ( M . x qua Element of BOOLEAN ) , n' = ( M . x qua Element of BOOLEAN ) * ( M . x qua Element of BOOLEAN ) ;
f " V in ' ( X ) & f " V in D ( the carrier of X ) & f " V in D ( the carrier of X ) implies f " V in D ( )
rng ( ( a , c ) sequence of ( c , 1 ) +* ( 1 , b ) ) c= { a , c , b } ;
consider y being being being \cdot of G1 such that y `1 = y and dom y `1 = WW\cdot y `1 and y `2 = WW\cdot y `2 ;
dom ( 1 / f ) /\ ]. - r , x0 .[ c= ]. - r , x0 .[ /\ dom f /\ ]. - r , x0 .[ ;
as Matrix of i , j , n , r , - r , ( i + j ) , ( i + j ) , ( i + j ) , ( i + j ) , ( i + j ) , ( i + j ) ) = ( i + j ) , ( i + j ) , ( i + j ) ) ;
v ^ ( n-3 |-> 0 ) in Lin ( ( rng ( B | ( n + 1 ) ) ) ) & ( n + 1 ) in Lin ( ( B | ( n + 1 ) ) ) ;
ex a , k1 , k2 st i = a /. k1 & j = b /. k2 & k2 = b . k2 & k2 = k2 ;
t . NAT = ( NAT .--> succ i1 ) . NAT .= succ i1 .= succ ( NAT .--> succ i1 ) .= succ ( 5 + 1 ) .= succ ( 5 + 1 ) ;
assume that F is bbfamily of X and rng p = F and dom p = Seg ( n + 1 ) and rng p = Seg ( n + 1 ) ;
( not LIN b , b9 , a ) & not LIN a , a9 , b & LIN a , a9 , c & LIN b , b9 , a ) & not LIN b , b9 , c
( L1 on L2 ) \& O c= ( L1 Let O ) Let ( L2 Let O ) Let ( L1 Let O ) ;
consider F be ManySortedSet of E such that for d be Element of E holds F . d = F ( d ) and for d be Element of E holds F . d = F ( d ) ;
consider a , b such that a * ( 0. V ) = b * ( -w ) and 0 < a and 0 < b ;
defpred P [ FinSequence of D ] means |. Sum ( $1 ) .| <= Sum ( |. $1 .| ) & Sum ( |. $1 .| ) <= Sum ( |. $1 .| ) ;
u = cos . ( x , y ) * x + ( cos . ( x , y ) * y ) .= cos . ( x , y ) * y + ( cos . ( x , y ) * y ) .= v ;
dist ( ( seq . n ) + x , g + x ) <= dist ( ( seq . n ) , g ) + 0 ;
P [ p , |. p .| ^ |. p .| , {} ] , id ( the Sorts of A ) . p , id ( the Sorts of A ) . p ] ;
consider X being Subset of CQC-WFF ( Al ) such that X c= Y and X is finite and X is ininininand X is ininininand X is inininand X is ininand X is ininininand X is inininand X is inininand X is finite ;
|. b .| * |. eval ( f , z ) .| >= |. b .| * |. eval ( f , z ) .| .= |. eval ( f , z ) .| ;
1 < ( ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ) .. Cage ( C , n ) + 1 ;
l in { l1 where l1 is Real : g <= l1 & l1 <= h & h <= g } ;
vol ( ( G . n ) vol ( { x } ) ) <= Sum ( ( G . n ) vol ( { x } ) ) ;
f . y = x .= x * 1_ L .= x * power L .= x * power L .= x * power L .= x * ( y * ( y * x ) ) ;
NIC ( <% i1 , i2 , j2 %> , n ) = { i1 , succ i2 , succ i2 , succ i2 , succ i2 , succ i2 , j2 } .= { i1 , i2 , j2 } ;
LSeg ( p00 , p2 ) /\ LSeg ( p1 , p01 ) = { p1 } \/ LSeg ( p01 , p2 ) .= { p1 } ;
product ( ( the carrier of I-15 ) +* ( i , { 1 } ) ) in [: Z , Z :] ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) .= Following ( s2 , n ) .= Following ( s2 , n ) ;
W-bound ( Qs2 ) <= ( q1 `1 ) / 2 & ( q1 `1 ) / 2 <= ( q1 `1 ) / 2 & ( q1 `2 ) / 2 <= ( q1 `2 ) / 2 ;
f /. i2 <> f /. ( ( i1 + len g -' 1 ) + 1 ) & f /. ( ( i1 + len g -' 1 ) + 1 ) = f /. ( i1 + 1 ) ;
M , v / ( x. 3 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) ) / ( x. 4 , m ) / ( x. 4 , m ) ;
len ( ( P ^ ) ^ ( Q ^ ) ) in dom ( ( P ^ ) ^ ( Q ^ ) ) & len ( ( P ^ ) ) + len ( Q ^ ) ) = len ( ( P ^ ) ) + len ( Q ^ ) ;
A |^ ( m , n ) c= A |^ ( m , n ) & A |^ ( k , n ) c= A |^ ( k , n ) implies A |^ ( k , n ) c= A |^ ( k , n )
R |^ n \ { q : |. q .| < a } c= { q1 : |. q1 .| >= a }
consider n1 being element such that n1 in dom p1 and y1 = p1 . n1 and p1 . n1 = p2 . n1 and p1 . n1 = p2 . n1 ;
consider X being set such that X in Q and for Z being set st Z in Q & Z <> X holds X \not c= Z ;
CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA & CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA ;
for v be VECTOR of l1 holds ||. v .|| = upper_bound rng |. ( id the carrier of V ) . v .| & ||. v .|| = upper_bound rng |. ( id the carrier of V ) . v .|
for phi st phi in X holds phi in X & ( phi in X & phi is not empty ) implies phi in X & phi is not empty
rng ( Sgm dom ( ( f | ( dom f ) ) | ( dom ( f | ( dom f ) ) ) ) ) c= dom ( ( f | ( dom f ) ) | ( dom ( f | ( dom f ) ) ) ) ) ;
ex c being FinSequence of D ( ) st len c = k & P [ c ] & a = c & P [ c ] ;
the_arity_of ( a , b , c ) = <* \mathop { \rm cod } ( b , c ) , \mathop { \rm cod ( a , b , c ) } , \mathop { \rm cod ( b , c ) } *> ;
consider f1 be Function of the carrier of X , REAL such that f1 = |. f .| and f1 is continuous and f1 . 0 = p ;
a1 = b1 & a2 = b2 or a1 = b2 & a2 = b2 & a3 = b3 or a1 = b3 & a2 = b2 & a3 = b3 & a4 = 6 ;
D2 . indx ( D2 , D1 , n1 + 1 ) = D1 . ( n1 + 1 ) & D2 . indx ( D2 , D1 , n1 + 1 ) = D2 . ( n1 + 1 ) ;
f . ( ||. r .|| ) = ||. ||. r .|| /. 1 .= ||. r .|| . 1 .= ||. r .|| . 1 .= x ;
consider n being Nat such that for m being Nat st n <= m holds C-25 . m = C-25 . m and C-25 . n = C-25 . n ;
consider d be Real such that for a , b be Real st a in X & b in Y holds a <= d & b <= b ;
||. L /. h - ( K * |. h .| ) + ( K * |. h .| ) * ( K * |. h .| ) <= p0 + K * ( K * |. h .| ) ;
attr F is commutative means : Def3 : for b being Element of X holds F \hbox { b } = f . b & f . b = f . b ;
p = - - ( - ( p0 `1 / 2 ) + 0. TOP-REAL 2 ) .= 1 * p0 `1 + 0. TOP-REAL 2 .= 1 * p0 `1 + 0. TOP-REAL 2 .= 1 * p0 `1 + 0. TOP-REAL 2 ;
consider z1 such that b , x3 , z1 is_collinear and o , x1 , z1 is_collinear and o <> x1 and o <> x2 and o <> z1 and o <> z1 and o <> z1 ;
consider i such that Arg ( Rotate ( s ) ) . q = s + Arg ( Rotate ( s ) ) . q and 0 <= ( 2 * PI * PI ) . i ;
consider g such that g is one-to-one and dom g = card f and rng g = { x } and g . x = f . x ;
assume that A = P2 \/ Q2 and Q1 <> {} and Q2 <> {} and Q2 is closed and for x being Element of NAT st x in Q1 holds x in Q1 and x in Q1 ;
attr F is associative means : Def3 : F .: ( f , g ) = F .: ( f , g ) & F .: ( g , h ) = F .: ( f , g ) ;
ex x being Element of NAT st m = x `1 & x in z `1 & x `2 < i or m in { i } ;
consider k2 being Nat such that k2 in dom P-2 and l = P-2 . k2 and ( for k2 being Nat st k2 in dom P-2 holds Pk1 . k2 = Pk1 . k2 ) ;
seq = r (#) seq implies for n holds seq . n = r * seq . n & ( for n holds seq . n = r * seq . n ) & ( for n holds seq . n = r * seq . n )
F1 . [ ( id a ) , [ a , a ] ] = [ f * ( id a ) , f * ( id a ) ] .= [ f , f ] ;
{ p } "\/" D2 = { p "\/" y where y is Element of L : y in D2 & p in D1 } \/ { p "\/" D2 where p is Element of L : p in D1 & p in D2 } ;
consider z being element such that z in dom ( dom ( ( dom F ) . 0 ) | ( dom ( F . 0 ) ) ) and ( ( F . 0 ) | ( dom ( F . 0 ) ) ) . z = y ;
for x , y being element st x in dom f & y in dom f & f . x = f . y holds x = y
Int cell ( G , i , j ) = { |[ r , s ]| : r <= G * ( 0 + 1 , 1 ) `1 } ;
consider e being element such that e in dom ( T | E1 ) and ( T | E1 ) . e = v ;
( F `1 * b1 ) . x = ( Mx2Tran ( J , BT , BT , BT ) ) . ( \mathbb j + 1 ) .= ( Mx2Tran J ) /. j ;
- 1 / ( n , m ) = mm (#) D | n .= mm (#) ( - 1 / ( n , m ) ) .= mm (#) ( - 1 / ( n , m ) ) .= Det M ;
attr x be set means : Def3 : for g be set st x in dom f /\ dom g holds g . x <= f . x & g is nonnegative ;
len ( f1 . j ) = len f2 .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) ;
All ( 'not' All ( a , A , G ) , B , G ) '<' Ex ( 'not' All ( a , B , G ) , A , G )
LSeg ( E . k0 , F . k0 ) c= Cl ( RightComp Cage ( C , k + 1 ) ) \/ { F . ( k + 1 ) } ;
x \ a |^ m = x \ ( a |^ k * a ) .= ( x \ ( a |^ k ) ) \ a .= ( x \ a ) |^ k ;
k -ininininininininn ( I ) . k = ( commute Isucc k ) . k .= ( ( commute Isucc k ) . k ) . i .= ( ( commute Isucc k ) . i ) . i ;
for s being State of Ai2 holds Following ( s , n ) . ( 0 + 2 * n + 1 ) is stable ;
for x st x in Z holds f1 . x = a ^2 & ( f1 - f2 ) . x <> 0 & f1 . x = 0 & f2 . x = 1 implies f1 - f2 is continuous
support ( support ( support ( n ) ) \/ support ( support ( m ) ) c= support ( ( support ( n ) ) \/ support ( m ) ) ;
reconsider t = u as Function of ( the carrier of A ) , ( the carrier of B ) * the carrier of F , ( the carrier of B ) * the carrier of F ;
- ( a * sqrt ( 1 + b ^2 ) ) <= - ( b * sqrt ( 1 + a ^2 ) ) ;
phi / ( succ b1 ) . a = g . a & phi / ( b , a ) . a = f . ( g . a ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( ( F ^ <* p *> ) . i ) and i <> j ;
{ x1 , x2 , x3 , x4 } = { x1 } \/ { x2 , x3 , x4 } \/ { x4 , x5 , x5 } .= { x1 , x2 , x3 , x4 } \/ { x4 , x5 , x5 } ;
the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U1 & the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U2 ;
( - ( 2 * a * ( b - a ) ) / b + b ^2 - delta ( a , b , c ) ) / delta ( a , b , c ) > 0 ;
consider W00 such that for z being element holds z in W00 iff z in N ~ & P [ z ] & ( for z being element st z in N ~ holds P [ z ] ) ;
assume that ( the Arity of S ) . o = <* a *> and ( the ResultSort of S ) . o = r and ( the ResultSort of S ) . o = <* r *> ;
Z = dom ( ( exp_R (#) ( ( #Z ( n + 1 ) ) * ( f1 + #Z ( n + 1 ) ) ) ) / ( f1 + #Z ( n + 1 ) ) ) ;
sum ( f , SS1 ) is convergent & lim sum ( f , SS1 ) = integral ( f , SS1 ) & lim ( sum ( f , SS1 ) ) = integral ( f , SS1 ) ;
( X ( a9 ) ) => ( g => ( xcnon empty set ) ) in -> Subset of iff ( X ( a9 ) => ( xx, xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx, xxxxxxx, xxxxxxxxxxxxxxx
len ( M2 * M3 ) = n & width ( M2 * M3 ) = n & width ( M1 * M2 ) = n & len ( M2 * M3 ) = n ;
attr X1 union X2 is an open SubSpace of X means : Def3 : X1 , X2 are_separated & X1 is open & X2 is open & X1 is open ;
for L being upper-bounded antisymmetric RelStr , X being non empty Subset of L holds X "\/" { Top L } = { Top L }
reconsider f-129 = F1 . ( b , b ) , f29 = F2 . ( b , b ) , f29 = F2 . ( b , a ) , f29 = F2 . ( b , a ) , f29 = F2 . ( b , a ) , f29 = F2 . ( b , a ) , f29 = F2 . ( b , a ) , f29 = F2 . ( b , a ) ;
consider w being FinSequence of I such that the InitS of M , the InitS of M ^ <* s *> ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ ( the InitS of M ) ^ w ^ w ^ w ^ w ^ ( the InitS of M ) ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ ( the InitS of M ) ^ w ) ^ w ^ (
g . ( a |^ 0 ) = g . ( 1_ G ) .= 1_ H .= ( g . a ) |^ 0 .= ( g . a ) |^ 0 .= ( g . a ) |^ 0 ;
assume for i being Nat st i in dom f ex z being Element of L st f . i = rpoly ( 1 , z ) & f . i = rpoly ( 1 , z ) ;
ex L being Subset of X st Carrier L = C & for K being Subset of X st K in C holds L /\ K <> {} & K is closed & L is closed ;
( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C1 & ( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C2 ;
reconsider o-21 = o `1 , op = o `2 , op = o `2 , op = o `2 , op = o `2 , op = o `1 , op = o `2 ;
1 * x1 + ( 0 * x2 ) + ( 0 * x3 ) = x1 + ( 0 * x2 ) .= x1 + ( 0 * x3 ) .= x1 + ( 0 * x2 ) .= x1 + ( 0 * x3 ) ;
Ez " . 1 = ( Ez qua Function ) " . 1 .= ( ( 1 - 2 ) / ( 1 - 2 ) ) " . 1 .= ( 1 - 2 ) / ( 1 - 2 ) ;
reconsider u1 = the carrier of U1 /\ ( U1 "\/" U2 ) , v1 = the carrier of U1 /\ ( U1 "\/" U2 ) as non empty Subset of U0 ;
( ( x "/\" z ) "\/" ( x "/\" y ) ) "\/" ( z "/\" y ) <= ( x "/\" ( z "\/" y ) ) "\/" ( z "/\" y ) ;
|. f . ( s1 . l1 + 1 ) - f . ( s1 . l1 ) .| < ( 1 / |. M .| + 1 ) / ( M + 1 ) ;
LSeg ( ( Cage ( C , n ) ) /. ( i + 1 ) , ( ( Cage ( C , n ) ) /. ( i + 1 ) ) is vertical ;
( f | Z ) /. x - ( f | Z ) /. x0 = L /. ( x- x ) + R /. ( x- x ) .= L /. ( x- x ) + R /. ( x- x ) ;
g . c * ( - g . c ) + f . c <= h . c * ( - g . c ) + f . c ;
( f + g ) | divset ( D , i ) = f | divset ( D , i ) + g | divset ( D , i ) .= f | divset ( D , i ) + g | divset ( D , i ) ;
assume that ColVec2Mx f in the set of ( the carrier of A ) and ( for b st b in the carrier of A ) holds ( ( ColVec2Mx b ) . b = ( ColVec2Mx b ) . b ) and ( ( ColVec2Mx b ) . b = ( ColVec2Mx b ) . b ) ;
len ( - M1 ) = len M1 & width ( - M2 ) = width M1 & width ( - M1 ) = width M1 & width ( - 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 S )
pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1 ;
pred a <> 0 & b <> 0 & Arg a = Arg b & Arg b = Arg a & Arg ( - a ) = Arg ( - b ) & Arg ( - b ) = Arg ( - a ) ;
for c being set st not c in [. a , b .] holds not c in Intersection ( the open of a , b ) & not c in Intersection ( the open of a , b )
assume that V1 is linearly-independent and V2 is closed and V1 = { v + u : v in V1 & u in V1 & v in V1 } and V1 is closed and V1 is closed and V1 is closed and V1 is closed and V1 is closed and V1 is closed and V1 is closed and V1 is closed and V1 is closed ;
z * x1 + ( 1 - z ) * x2 in M & z * y1 + ( 1 - z ) * y2 in N implies z * y1 + ( 1 - z ) * y2 in N
rng ( ( PS1 qua Function ) " * SS1 ) = Seg card ( dom ( ( PS1 ) " * SS1 ) ) .= Seg card ( dom ( ( PS1 ) " * SS1 ) ) .= dom ( ( ( PS1 ) " * ( ( id S ) " * ( ( S * S1 ) " ) ) ) ) ;
consider s2 being rational Real_Sequence such that s2 is convergent and b = lim s2 and for n holds s2 . n <= b & s2 . n <= b ;
h2 " . n = h2 . n " & 0 < - 1 & h2 . n < 1 implies ( - 1 / 2 ) / ( n + 1 ) < - 1 / 2 * ( - 1 )
( Partial_Sums ( ||. seq1 .|| ) ) . m = ||. ( seq1 . m ) .|| .= ||. ( seq2 . m ) .|| .= ||. ( seq2 . m ) .|| .= ||. ( seq2 . m ) .|| .= ||. ( seq2 . m ) .|| .= ||. ( seq2 . m ) .|| .= ||. ( seq2 . m ) .|| .= ||. ( seq2 . m ) .|| ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= ( Comput ( P2 , s2 , 1 ) ) . b .= Comput ( P2 , s2 , 1 ) . b .= Comput ( P2 , s2 , 1 ) . b ;
- v = - 1_ ( G ) * v & - w = - 1_ G implies - v = - ( - ( - v ) * w ) & - ( - ( - v ) * w ) = - ( - ( - v ) * w )
upper_bound ( ( k + 1 ) .: D ) = upper_bound ( ( k + 1 ) .: D ) .= k . ( ( k + 1 ) .: D ) .= k . ( ( k + 1 ) .: D ) ;
A |^ ( k , l ) ^^ ( A |^ ( n , l ) ) = ( A |^ ( k , l ) ) ^^ ( A |^ ( k , l ) ) .= A |^ ( k , l ) ^^ ( A |^ ( k , l ) ) ;
for R being add-associative right_zeroed right_complementable non empty addLoopStr , I , J , K being Subset of R holds I + ( J + K ) = ( I + J ) + K
( f . p ) `1 = ( p `1 ) ^2 + sqrt ( 1 + ( p `2 / p `1 ) ^2 ) .= ( p `1 ) ^2 + sqrt ( 1 + ( p `2 / p `1 ) ^2 ) ;
for a , b being non zero Nat st a , b are_relative_prime holds ( a * b ) = ( support a ) + ( support b ) & ( a * b ) = ( support a ) + ( support b )
consider A5 being countable Subset of CQC-WFF ( Al ) such that r is Element of CQC-WFF ( Al ) and A5 is ( len A ) -element and ( for n holds A . n = ( n + 1 ) ! ) & ( A . n = ( n + 1 ) ! ) ;
for X being non empty addLoopStr for M being Subset of X , x , y being Point of X st y in M holds x + y in M + M
{ [ x1 , x2 ] , [ y1 , y2 ] } c= { [ x1 , y1 ] , [ y1 , y2 ] } \/ { [ y1 , y2 ] }
h . ( f . O ) = |[ A * ( f . O ) `1 + B , C * ( f . O ) `2 + D ]| ;
( Gauge ( C , n ) ) * ( k , i ) in L~ Upper_Seq ( C , n ) /\ L~ Upper_Seq ( C , n ) & ( Cage ( C , n ) ) * ( k , i ) in L~ Upper_Seq ( C , n ) ;
cluster m , n are_relative_prime means : Def3 : for p being prime Nat holds p is prime implies for m being Nat st p divides m & p divides n & 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 \ a <= c holds a \+\ b <= c & a "\/" b <= c
consider b being element such that b in dom ( H / ( x , y ) ) and z = ( H / ( x , y ) ) . b and ( H / ( x , y ) ) . b = ( H / ( x , y ) ) . b ;
assume that x in dom ( F (#) g ) and y in dom ( F (#) g ) and ( F (#) g ) . x = ( F (#) g ) . y ;
assume ex e being element st e Joins W . 1 , W . 5 , G & e . 1 = W . 3 & e . 7 = W . 7 ;
( ( h h (#) f ) . ( 2 * n ) ) . x = ( h (#) delta ( h ) ) . ( 2 * n ) . ( x + h . n ) ;
j + 1 = ( - len h11 + 1 ) + 1 .= i + 1 - len h11 + 2 .= i + 1 - len h11 + 2 - 1 .= i + 1 - len h11 + 2 - 1 ;
( *' S ) . f = S *' ( S *' ) .= S *' ( S *' ) .= S *' ( S *' ) .= S *' ( S *' ) .= S *' ( S *' ) ;
consider H such that H is one-to-one and rng H = the carrier of L2 and Sum ( L2 ) = Sum ( L2 ) and Sum ( L1 ) = Sum ( L2 ) ;
attr R is is + `2 means : Def3 : for p , q st p in R & q <> q holds ex P , Q st P , Q is_be \frac 2 * p & p in P & q in Q ;
dom product ( product ( X --> f ) ) = meet ( ( dom X --> f ) . i ) .= meet ( ( X --> f ) . i ) .= dom f .= dom ( ( X --> f ) . i ) .= dom f ;
upper_bound ( proj2 .: ( Upper_Arc C /\ Lower_Arc C ) /\ Vertical_Line w ) <= upper_bound ( proj2 .: ( C /\ Vertical_Line w ) ) & upper_bound ( proj2 .: ( C /\ Vertical_Line w ) ) <= upper_bound ( proj2 .: ( C /\ Vertical_Line w ) ) ;
for r be Real st 0 < r ex n be Nat st for m be Nat st n <= m holds |. S . m - x0 .| < r & |. S . m - x0 .| < r
i * f-28 - ( i * y ) = i * ( f - y ) .= i * ( f - y ) .= i * ( f - y ) .= i * ( f - y ) .= i * ( f - y ) ;
consider f being Function such that dom f = 2 -tuples_on X & for Y being set st Y in 2 -tuples_on X holds f . Y = F ( Y ) and f . Y = F ( Y ) ;
consider g1 , g2 being element such that g1 in [#] Y and g2 in union C and g = [ g1 , g2 ] and [ g1 , g2 ] in C and [ g1 , g2 ] in the carrier of X ;
func d |-count n -> Nat means : Def3 : ( d |^ n ) divides n & ( d |^ n ) divides ( n |^ m ) & ( d |^ n ) divides ( n |^ m ) ;
f\in f . [ 0 , t ] = f . [ 0 , t ] .= ( - P ) . ( 2 * ( x `1 ) ) .= a ;
t = h . D or t = h . B or t = h . C or t = h . E or t = h . F or t = h . J or t = M . N or t = N ;
consider m1 be Nat such that for n st n >= m1 holds dist ( ( seq . n ) , ( seq . n ) ) < 1 / ( n + 1 ) ;
( ( q `1 ) / |. q .| ) ^2 <= ( ( q `1 ) / |. q .| ) ^2 + ( ( q `2 ) / |. q .| ) ^2 ;
h0 . ( i + 1 + 1 ) = h21 . ( i + 1 + 1 -' len h11 + 2 -' 1 ) .= h21 . ( i + 1 -' len h11 ) ;
consider o being Element of the carrier' of S , x2 being Element of { [ o , x2 ] } such that a = [ o , x2 ] and [ o , x2 ] in X and [ o , x2 ] in X ;
for L being RelStr , a , b being Element of L holds a <= { b } iff a <= b & b >= a & a >= b & b >= a
||. h1 .|| . n = ||. h1 . n .|| .= |. h . n .| .= ||. h . n .|| .= ||. h . n .|| .= ||. h . n .|| .= ||. h . n .|| ;
( ( - ( #Z n ) ) `| REAL ) . x = f . x - ( ( #Z n ) . x ) .= ( ( - 1 ) (#) ( ( #Z n ) * ( ( #Z n ) * ( ( #Z n ) * ( ( #Z n ) * ( ( #Z n ) * ( ( #Z n ) * ( 1 / n ) ) ) ) ) ) . x ;
pred r = F .: ( p , q ) means : Def3 : len r = len p & for i st i in dom r holds r . i = F . i ;
( rl / 2 ) ^2 + ( rl / 2 ) ^2 <= ( ( r / 2 ) ^2 + ( r / 2 ) ^2 ) + ( ( r / 2 ) ^2 + ( r / 2 ) ^2 ;
for i being Nat , M being Matrix of n , K st i in Seg n holds Det M = Sum ( ( Det M ) | ( i , j ) ) & Det M = Sum ( ( Det M ) | ( i , j ) )
then a <> 0. R & a " * ( a * v ) = 1 & a " * ( a * v ) = 1 & a " * ( a * v ) = 1 & a " * ( a * v ) = 1 ;
p . ( j -' 1 ) * ( q *' r ) . ( i + 1 -' j ) = Sum ( p . ( j -' 1 ) * r3 ) .= Sum ( p . ( j -' 1 ) * r3 ) ;
deffunc F ( Nat ) = L . 1 + ( ( ( R /* ( h ^\ n ) ) * ( h ^\ n ) ) " ) . $1 & ( ( ( R /* ( h ^\ n ) ) " ) ) . $1 = ( ( h ^\ n ) " ) . $1 ;
assume that the carrier of H2 = f .: ( the carrier of H1 ) and the carrier of H2 = f .: ( the carrier of H2 ) and the carrier of H2 = the carrier of H2 and the carrier of H2 = the carrier of H2 and the carrier of H2 = the carrier of H2 ;
Args ( o , Free ( S , X ) ) = ( ( the Sorts of Free ( S , X ) ) * ( the Arity of S ) ) . o .= ( ( the Sorts of Free ( S , X ) ) * ( the Arity of S ) ) . o ;
H1 = n + 1 - ( |. 2 to_power ( n + 1 ) .| + h ) .= n + 1 - ( n + 1 ) .= n + 1 - ( n + 1 ) ;
( O1 is non empty & O2 is non empty or O1 is non empty & O2 is non empty & O1 is non empty & O2 is non empty ) & ( O1 is non empty or O1 is non empty & O2 is non empty & O2 is non empty ) implies ( O1 is non empty & O2 is non empty & O2 is non empty & O2 is non empty or O2 is non empty )
F1 .: ( dom F1 /\ dom F2 ) = F1 .: ( ( 1 + 2 ) * ( F1 . ( n + 1 ) ) ) .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } ;
attr b <> 0 & d <> 0 & b <> d & ( a - b ) / ( a - b ) = ( - e ) / ( b - b ) & ( a - b ) / ( a - b ) = ( - e ) / ( b - b ) ;
dom ( ( f +* g ) | D ) = dom ( f +* g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D ;
for i being set st i in dom g ex u , v being Element of L , a , b being Element of B st g /. i = u * a & f /. i = a * v
g `2 * P `2 * g " = g `2 * ( g `2 * P `2 ) .= g `2 * ( g `2 * P `2 ) " .= g `2 * ( g `2 * P `2 ) " .= g `2 * ( g `2 * P `2 ) " ;
consider i , s1 such that f . i = s1 and if ( s1 = s2 & not s1 in s1 ) & not s1 in s1 & not s2 in s1 ) & s1 in s1 & s1 in s1 & s1 in s2 ;
h5 | ]. a , b .[ = ( g | ]. a , b .[ ) | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= h | ]. a , b .[ ;
[ s1 , t1 ] , [ s2 , t2 ] Point ( TOP-REAL 2 ) | [. s1 , s2 .] & [ s2 , t2 ] , [ s2 , t2 ]| Point ( TOP-REAL 2 ) | [. s1 , s2 .] & [ s1 , t2 ] , [ s2 , t2 ] .] c= the carrier of TOP-REAL 2 ;
then that H is negative and H is not negative and H is not conjunctive and H is not len gOne and not H is not ggOne and not H is not implies H is not implies H is not implies H is not implies H is not negative
attr f1 is total means : Def3 : f1 is total & f2 is total & f1 - f2 is total & f1 - f2 is total & f1 - f2 is total & f1 - f2 is total & f1 - f2 is total & f1 - f2 is total ;
z1 in W2 ` & z1 = z2 implies ( z1 in W2 & ( z1 in W1 & z2 in W2 implies z1 = z2 ) & ( z1 in W1 & z2 in W2 implies z1 = z2 ) & ( z1 in W1 implies z1 = z2 )
p = 1 * p .= a " * a * p .= a " * ( b " * p ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) ;
for seq1 be Real_Sequence , seq be Real_Sequence st for n be Nat holds seq1 . n <= K holds upper_bound rng ( seq1 . n ) <= upper_bound ( seq ) & upper_bound ( seq ) <= upper_bound ( seq )
x0 meets ( L~ go \/ L~ pion1 ) or x0 in ( L~ go \/ L~ pion1 ) \/ ( L~ co \/ L~ pion1 ) or x0 in ( L~ go \/ L~ pion1 ) \/ ( L~ co \/ L~ pion1 ) or x0 in ( L~ go \/ L~ pion1 ) \/ ( L~ co \/ L~ pion1 ) or x0 in ( L~ go \/ L~ pion1 ) \/ ( L~ co \/ { x0 } ) ;
||. f . ( g . ( k + 1 ) ) - g . ( g . k ) .|| <= ||. g . 1 - g . 0 .|| * ( K * ||. K to_power k .|| ) ;
assume h = ( ( B .--> B ' ) +* ( C .--> D ' ) +* ( E .--> F ' ) +* ( F .--> J ) ) +* ( M .--> N ) +* ( M .--> N ) +* ( M .--> N ) +* ( M .--> N ) +* ( M .--> N ) +* ( M .--> N ) ) +* ( M .--> N ) +* ( M .--> N ) +* ( M .--> N ) ) +* ( M .--> N .--> A ) ;
|. ( ( upper_volume ( H . n , T ) || A ) . k - ( integral ( H , T ) || A ) . k ) .| <= e * ( ( 2 * PI ) || A ) . k - ( 2 * PI ) . k ) ;
( (
{ x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 } = { x1 } \/ { x1 } .= { x1 } \/ { x1 , x1 } ;
assume that A = [. 0 , 2 * PI .] and integral ( ( #Z n ) * ( cos ) , A ) = 0 and integral ( ( #Z n ) * ( cos ) , A ) = 0 ;
p `2 is Permutation of dom f1 & p `2 " = ( Sgm Y ) " * p & p `2 " * Sgm X = ( Sgm Y ) " * Sgm Y & p `2 = Sgm Y " * Sgm X ;
for x , y st x in A holds |. 1 / ( f . x - 1 ) .| <= 1 * |. f . x - 1 .| & |. 1 / ( f . x - 1 ) .| <= 1 * |. f . x - 1 .|
p2 `2 / |. p2 .| = |. q2 .| * ( ( q2 `2 / |. q2 .| - sn ) / ( 1 + sn ) ) .= ( ( q2 `2 / |. q2 .| - sn ) / ( 1 + sn ) ) / ( 1 + sn ) ;
for f be PartFunc of the carrier of CNS , REAL , x be Element of REAL , f be PartFunc of REAL , REAL , g be PartFunc of REAL , REAL st f = g & f = f & g = h holds rng f c= dom f
assume for x being Element of Y st x in EqClass ( z , CompF ( B , G ) ) holds ( Ex ( a , A , G ) ) . x = TRUE ;
consider F3 such that dom F3 = n1 and for k be Nat st k in n1 holds Q [ k , F3 . k ] and for k be Nat st k in n1 holds Q [ k , F3 . k ] ;
ex u , u1 st u <> u1 & u , u1 , v1 ^ u1 ^ v1 ^ v1 ^ v1 ^ u1 ^ v1 ^ u1 ^ v1 ^ v1 ^ u1 ^ v1 ^ u1 ^ v1 ^ u1 ^ v1 ^ u1 ^ v1 ^ v1 ^ u1 ^ v1 ^ u1 ^ v1 ^ u1 ^ v1 ^ v1 ^ u1 ^ v1 ^ u1 ^ v1 ^ v1 ^ u1 ^ v1 ^ u1 ^ v1 ^ v1 ^ u1 ^ v1 ^ v1 ^ u1 ^ v1 ^ v1 ^ u1 ^ v1 ^ v1 ^ v1 ^ v1 ^ u1 ^ v1 ^ v1 ^ u1 ^ v1 ^ v1 ^ u1 ^ v1 ^ u1 ^ v1
for G being Group , A , B being non empty Subset of G , N being normal Subgroup of G holds ( N ` A ) * ( N ` B ) = N ` A * B
for s be Real st s in dom F holds F . s = integral ( ( R ^ > > > > 0 ) (#) integral ( f , g ) , ( ( f + g ) (#) e ) ) . x ) & ( ( f + g ) (#) e ) . x = integral ( ( f + g ) (#) e ) . x
width ( AutMt ( f1 , b1 , b2 ) ) = len b2 .= len ( ( f2 , b1 ) | ( i + 1 ) ) .= len ( ( f2 , b1 ) | ( i + 1 ) ) .= len ( ( f2 , b1 ) | ( i + 1 ) ) ;
f | ]. - PI / 2 , PI / 2 .[ = f & dom f = ]. - PI / 2 , PI / 2 .[ & rng f c= ]. - PI / 2 , PI .[ & f | ]. - PI / 2 , PI / 2 .[ is continuous ;
assume that X is closed and a in X and a c= X and y in a ^ { f . n } and x in a \/ { f . n } and y in a \/ { f . n } ;
Z = dom ( ( ( ( ( 1 / 2 ) (#) ( ( #Z 2 ) * ( f1 + #Z 2 ) ) ) ) `| Z ) /\ dom ( ( ( 1 / 2 ) * ( f1 + #Z 2 ) ) `| Z ) ) ;
func [: V , { l } :] -> Subset of V means : Def3 : for k st 1 <= k & k <= len it holds it . k in V & for k st 1 <= k & k <= len it holds it . k in V ;
for L being non empty TopSpace , N being net of L , M being net of N , c being Point of N , N being net of L st c is net of M holds c is net of N
for s being Element of NAT holds ( ( for v be Element of NAT holds ( ( for u be Element of NAT holds u in ( v + u ) + ( u + u ) ) ) holds f . s = ( ( ( v + u ) + u ) + v ) ) . s ) implies f . s = ( ( v + u ) + u ) . s
then z /. 1 = ( N-min L~ z ) .. z & ( N-min L~ z ) .. z < ( ( N-min L~ z ) .. z ) .. z & ( ( N-min L~ z ) .. z ) .. z < ( ( N-min L~ z ) .. z ) .. z ;
len ( p ^ <* ( 0 qua Real ) *> ) = len p + len <* ( 0 qua Real ) * ( ( 0 qua Real ) + ( 0 qua Real ) * ( ( 0 qua Real ) + ( 0 qua Real ) ) *> ) .= len p + 1 .= len p + 1 ;
assume that Z c= dom ( - ( ln * f ) ) and for x st x in Z holds f . x = x and f . x > 0 and f . x > 0 ;
for R being add-associative right_zeroed right_complementable associative commutative distributive non empty doubleLoopStr , I being non empty Subset of R , J being Subset of R holds ( I + J ) *' ( I /\ J ) c= I /\ J implies I + J = I /\ J
consider f being Function of [: B1 , B2 :] , B1 such that for x being Element of B1 holds f . x = F ( x ) and f . x = F ( x ) and f . x = F ( x ) ;
dom ( x2 + y2 ) = Seg len x .= Seg len ( x2 + y2 ) .= Seg len ( x2 + y2 ) .= dom ( x (#) ( y | ( i + 1 ) ) ) .= dom ( x (#) ( y | ( i + 1 ) ) ) .= dom ( x | ( i + 1 ) ) ;
for S being Functor of C , B for c being Object of C holds card S = id ( ( Obj S ) . c ) & card ( id ( ( Obj S ) . c ) ) = id ( ( Obj S ) . c )
ex a st a = a2 & a in f6 /\ f5 & for a , b st a in f6 & b in f6 holds reconsider f . a = f2 . a , f . b = f . b , f . a = f . a , f . b = f . b , f . a = f . a , f . b = f . a , f . b ;
a in Free ( H2 / ( x. 4 , x. 0 ) ) '&' ( H2 / ( x. 0 , x. 4 ) ) & ( ( ( x. 0 , x. 4 ) / ( x. 0 , x. 4 ) ) / ( x. 0 , x. 4 ) ) / ( x. 0 , x. 4 ) ) / ( x. 0 , x. 4 ) = ( ( x. 4 , x. 0 ) / ( x. 4 , x. 0 ) ) / ( x. 0 ) ;
for C1 , C2 being v1 , f , g being Function of C1 , C2 st for a , b being set st a in C2 holds f = g holds f = g iff f = g
( W-min L~ go \/ L~ pion1 ) `1 = W-bound L~ go \/ E-bound L~ pion1 & ( W-min L~ go \/ W-bound L~ pion1 ) `1 = W-bound L~ go \/ E-bound L~ pion1 & ( W-min L~ go \/ W-bound L~ co ) `1 = E-bound L~ go \/ E-bound L~ co or W-bound L~ co = W-bound L~ go & W-bound L~ go = W-bound L~ go ;
Suppose u = <* x0 , y0 , z0 *> and f is_assume x0 & SVF1 ( 3 , f , u ) = SVF1 ( 3 , f , u ) . z0 and SVF1 ( 3 , f , u ) . z0 = SVF1 ( 3 , f , u ) . z0 ; Then SVF1 ( 3 , f , u ) . z0 = SVF1 ( 3 , f , u ) . z0 + SVF1 ( 3 , f , u ) . z0 ;
then ( t . {} ) `1 in Vars & ex x being Element of Vars st x = ( t . {} ) `1 & t . {} = x & ( t . {} ) `2 = ( t . {} ) `2 ;
Valid ( p '&' p , J ) . v = Valid ( p , J ) . v '&' Valid ( q , J ) . v .= Valid ( p , J ) . v '&' Valid ( q , J ) . v .= Valid ( p , J ) . v ;
assume for x , y being Element of S st x <= y for a , b being Element of T st a = f . x & b = f . y holds a >= b ;
func Class R -> Subset-Family of R means : Def3 : for A being Subset of R holds A in it iff ex a being Element of R st a in it & A c= Class ( R , a ) ;
defpred P [ Nat ] means ( ( ( \HM { the } \HM { vertices } } G ) . $1 ) `1 ) `1 c= G . ( ( the carrier' of G ) . $1 ) `1 & ( ( ( the Source of G ) . $1 ) `2 ) `2 = ( ( the Source of G ) . $1 ) `2 ;
assume that dim W1 = 0 and dim W1 = 0 and dim W2 = 0 and n = 0 and n = 0 and m = 0 and n = 0 and n = 0 and m = 0 and n = 0 and n = 0 and n = 0 and m = 0 and n = 0 or n = 0 or n = 0 or n = 0 or n = 0 or n = 0 ;
mama_/. ( m . t ) = ( m . t ) `1 .= ( [ m . t , the carrier of C ] `1 ) `1 .= ( [ m , the carrier of C ] `1 ) `1 .= m ;
d11 = x11 ^ d22 .= f . ( ( y , d22 ) `1 ) .= f . ( ( y , d22 ) `2 ) .= ( y , d22 ) `2 .= ( y , d22 ) `2 .= ( y , d22 ) `2 .= ( y , d22 ) `2 ;
consider g such that x = g and dom g = dom f and for x being element st x in dom f holds g . x in f . x and f . x = f . x ;
x + 0. F_Complex = x + len x |-> 0. F_Complex .= ( x + 0. F_Complex ) ^ ( x |-> 0. F_Complex ) .= ( x + 0. F_Complex ) ^ ( x |-> 0. F_Complex ) .= x ` ^ ( x |-> 0. F_Complex ) .= x ` ^ ( x |-> 0. F_Complex ) ;
( ( k -' ( k + 1 ) ) + 1 ) in dom ( f | ( ( k -' 1 ) + 1 ) ) & ( ( k + 1 ) + ( k + 1 ) ) + ( ( k + 1 ) + 1 ) in dom ( f | ( k + 1 ) ) ;
assume that P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 = { p1 , p2 } and P1 = { p1 , p2 } and P1 = { p1 , p2 } and P1 = { p1 , p2 } and P1 = { p2 , p1 } and P2 = { p2 , p1 } and P1 = { p2 , p1 } and P1 = { p2 , p1 } ;
reconsider a1 = a , b1 = b , c1 = c , c1 = p , c2 = p , c2 = p , c1 = q , c2 = p , c2 = p , c1 = q , c2 = p , c1 = r , c2 = p , c2 = q , c2 = r , c1 = r , c2 = p , c2 = r , c2 = p , c2 = q , c1 = r , c2 = p , c2 = r , c1 = s , c2 = s , c2 = s , c2 = s , c2 = s , c2 = s , c2 = s , c1 = s , c2 = s , c2 = s , c2 = s , c2 = s , c1 = s , c2 = s , c1 = s , c2 = s , c2 =
reconsider Gttt1f = G1 . ( t , b ) * F1 . f , Fsf = G1 . ( a , b ) , Fsf = G1 . ( a , b ) * F2 . ( b , a ) , Fsf = G1 . ( a , b ) * F2 . ( b , a ) ;
LSeg ( f , i + i1 -' 1 ) = LSeg ( f /. ( i + i1 -' 1 ) , f /. ( i + i1 -' 1 ) ) .= LSeg ( f , i + i1 -' 1 ) ;
Integral ( M , P . m ) | dom ( P . n ) <= Integral ( M , P . n ) * Integral ( M , P . m ) + Integral ( M , P . n ) * Integral ( M , P . m ) ;
assume that dom f1 = dom f2 and for x , y being element st [ x , y ] in dom f1 & f1 . x = f2 . y holds f1 . ( x , y ) = f2 . ( x , y ) ;
consider v such that v = y and dist ( u , v ) < min ( ( G * ( i , 1 ) ) `1 , ( G * ( i + 1 , 1 ) ) `2 ) ;
for G being Group , H being Subgroup of G , a being Integer st a = b holds for i being Integer , b being Integer st a = b holds for i being Integer holds a |^ i = b |^ i
consider B being Function of Seg ( S + L ) , the carrier of V1 such that for x being element st x in Seg ( S + L ) holds P [ x , B . x ] ;
reconsider K1 = { p0 where 7 is Point of TOP-REAL 2 : P [ 7 ] & for p being Point of TOP-REAL 2 , q being Point of TOP-REAL 2 st q in P & p in P & q `2 <= 0 } as Subset of TOP-REAL 2 ;
( ( ( ( N - S ) / ( m + 1 ) ) * ( 2 |^ m ) ) / ( 2 |^ n ) ) * ( 2 |^ n ) <= ( ( ( ( N - S ) / ( m + 1 ) ) * ( 2 |^ m ) ) / ( 2 |^ n ) ) * ( 2 |^ m ) ;
for x be Element of X , n be Nat st x in E holds |. Re ( F . n ) .| . x <= P . x & |. Im ( F . n ) .| . x <= P . x
len ( @ ( @ p ^ q ) ) = len ( @ ( @ p ^ <* 0 *> ) ) + len <* [ 2 , 0 ] *> .= len ( @ p ^ <* 1 *> ) + len ( @ q ) .= len ( @ p ^ <* 1 *> ) + len ( @ q ) ;
v / ( x. 3 , m1 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 0 , m2 ) / ( x. 0 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / (
consider r be Element of M such that M , v / ( x. 3 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m
func w1 \ w2 -> Element of Union ( G , R6 , R7 , R8 , R8 , R8 , R8 , R8 , R8 , R8 , R7 , R8 , R8 , R8 , R8 , R8 , R8 , R7 , R8 , R8 , R8 , R8 , R8 , R8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 ,
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= s1 . b2 .= s2 . b2 .= Exec ( n2 , s2 ) . b2 .= s2 . b2 .= Exec ( n2 , s2 ) . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 ;
for n , k be Nat holds 0 <= ( Partial_Sums |. seq .| ) . ( n + k ) - ( Partial_Sums ( |. seq .| ) ) . ( n + k ) + ( Partial_Sums ( |. seq .| ) ) . n ) + ( Partial_Sums ( |. seq .| ) ) . ( n + k ) ;
set F = S -\rm \hbox { - } ^ } ;
( Partial_Sums ( seq ) ) . ( K + 1 ) + Sum ( seq ) . ( K + 1 ) >= ( Partial_Sums ( seq ) ) . ( K + 1 ) + ( Partial_Sums ( seq ) ) . ( K + 1 ) ;
consider L , R such that for x st x in N holds ( f | Z ) . x - ( f | Z ) . x0 = L . ( x- x0 ) + R . ( x - x0 ) + R . ( x - x0 ) ;
func the closed of \HM { a , b , c , d } -> Subset of rectangle ( a , b , c , d ) equals ( the \rbrace of \HM { a , b , c , d } ) ` .= ( the \rbrace of TOP-REAL 2 ) | ( the carrier of K ) .= ( the be of TOP-REAL 2 ) | ( the carrier of K ) ;
a * b ^2 + ( a * c ^2 + b * a ^2 + c * b ^2 + c * a * b + c * b ^2 + c * b * c + a * b * c + b * c + c * a * b + c * b * c + a * b * c + c * b * c + a * b * c + a * c + b * c * b * c + a * c ) >= 6 * a * a * b * b * b * c + a * c + a * c + a * b * c + a * b * c + a * b * c + a * b * c + a * b * c + a *
v / ( x1 , m1 ) / ( x2 , m2 ) / ( x3 , m1 ) / ( x3 , m2 ) / ( x4 , m2 ) / ( x4 , m2 ) / ( x4 , m2 ) / ( x4 , m2 ) / ( x4 , m2 ) / ( x4 , m1 ) / ( x4 , m2 ) / ( x4 , m2 ) / ( x4 , m2 ) / ( x4 , m2 ) ;
+ ( Q ^ <* x *> , M ) = ( ( + Q ) +* ( M , { FALSE } ) +* ( M , { FALSE } ) ) +* ( ( M ^ <* x *> ) +* ( M , { FALSE } ) ) .= ( ( M ^ <* x *> ) +* ( M , { FALSE } ) ) +* ( M , { FALSE } ) ;
Sum ( FM | n1 ) = r |^ n1 * Sum ( CM | n1 ) .= C . n1 * ( ( C | n1 ) | n1 ) .= C . n1 * ( ( C | n1 ) | n1 ) .= C . n1 * ( ( C | n1 ) | n1 ) .= C . n1 * ( ( C | n1 ) | n1 ) ;
( GoB f ) * ( len GoB f , 2 ) `1 = ( GoB f ) * ( len GoB f , 1 ) `1 .= ( GoB f ) * ( len GoB f , 1 ) `1 .= ( GoB f ) * ( len GoB f , 1 ) `1 .= ( GoB f ) * ( len GoB f , 1 ) `1 ;
defpred X [ Element of NAT ] means ( Partial_Sums s ) . $1 = ( a * ( $1 + 1 ) + b * ( $1 + 1 ) ) / ( 2 * $1 + 1 ) * ( b * ( $1 + 1 ) ) / ( 2 * $1 + 1 ) * ( b * ( $1 + 1 ) ) ;
( the_arity_of g ) . g = ( the Arity of S ) . g .= ( [ ( the Arity of S ) . g , ( the ResultSort of S ) . g ] ) `1 .= [ g , ( the ResultSort of S ) . g ] `2 .= [ g , ( the ResultSort of S ) . g ) `1 .= [ g , ( the ResultSort of S ) . g ] `2 .= [ g , ( the ResultSort of S ) . g ) `2 ] ;
( X ~ ) c= X ~ & card ( X ~ ) = card ( X ~ ) implies card ( X , X ) = card ( X , X )
for a , b being Element of S , s being Element of NAT st s = n & a = F . n & b = F . n holds b = N . ( s + 1 ) \ G . ( s + 1 ) & a = N . ( s + 1 ) \ G . ( s + 1 )
E , f |= All ( All ( x , H ) , ( ( x. 0 ) .--> ( x. 1 ) ) ) => ( ( x. 1 ) .--> ( x. 2 ) ) => ( ( x. 1 ) .--> ( x. 2 ) ) ) => ( ( x. 1 ) .--> ( x. 2 ) ) ;
ex R2 being 1-sorted st R2 = ( p | ( n + 1 ) ) . i & ( the carrier of ( p | ( n + 1 ) ) ) . i = the carrier of ( ( p | n ) . i ) & ( the carrier of ( p | n ) ) . i = the carrier of ( p | n ) . i ) ;
[. a , b + 1 .[ is Element of the carrier of REAL & ( the partial of f ) . k is Element of the carrier of REAL & ( the partial of f ) . k is Element of the carrier of X & ( the \overline of f ) . k is Element of the carrier of X & ( the \overline of f ) . k is Element of the carrier of X
Comput ( P , s , 2 + 1 ) = Exec ( P . 2 , Comput ( P , s , 2 ) ) .= Exec ( a3 , s ) . a2 .= Exec ( a3 , s ) . a2 .= s . a2 .= s . a2 - s . a3 ;
card ( h1 ) . k = power F_Complex * ( - 1_ F_Complex ) .= ( - 1_ F_Complex ) . ( - 1_ F_Complex ) * Sum u .= ( - 1_ F_Complex ) . k * Sum u .= ( ( - 1_ F_Complex ) *' ) . k * Sum ( - 1_ F_Complex ) .= ( ( - 1_ F_Complex ) *' ) . k * Sum ( - h ) .= ( ( - 1_ F_Complex ) *' ) . k ;
( f / g ) /. c = f /. c * ( g /. c ) " .= f /. c * ( g /. c ) " .= ( f (#) g ) /. c * ( g /. c ) .= ( f (#) g ) /. c * ( g /. c ) .= ( f (#) g ) /. c * ( g (#) f ) /. c ;
len Cv - len ( -> Element of ( the carrier of ( ( the carrier of ( ( the carrier of ( C ) ) ) ) ) ) , ( the carrier of ( ( the carrier of ( C ) ) ) ) ) = len ( ( the carrier of ( the carrier of ( C ) ) ) ) .= len ( ( the carrier of ( C ) ) ) ) ;
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom f /\ X .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) ;
defpred P [ Nat ] means for n , m holds 2 * Fib ( n + $1 ) = Fib ( n ) * Fib ( n ) + ( 5 * Fib ( n ) * Fib ( n ) * Fib ( n ) ) + ( 5 * Fib ( n ) * Fib ( n ) * Fib ( n ) * Fib ( n ) * Fib ( n ) ) ;
consider f being Function of INT , INT such that f = f and f is onto and f is onto and for n st n < 1 holds f " { n } = { n } and f " { n } = { n } and f " { n } = { n } ;
consider vs be Function of S , BOOLEAN such that vs = chi ( A \/ B , S ) and E7 : ( for A , B holds c . A = Prob ( A \/ B ) ) & ( for A , B holds c . A = Prob ( B ) ) & ( for A holds c . A = Prob ( A ) ) & ( for A holds c . A = Prob ( B ) ) ;
consider y being Element of Y ( ) such that a = "\/" ( { F ( x , y ) where x is Element of X ( ) : P [ x ] } , y ) & Q [ y ] and P [ y ] and P [ y ] ;
assume that A c= Z and f = ( ( #Z n ) * ( ( #Z n ) * ( ( #Z n ) * ( f1 + #Z n ) ) / ( f1 + #Z n ) ) ) and Z = dom ( ( #Z n ) * ( f1 + #Z n ) ) and f = ( #Z n ) (#) ( f1 + #Z n ) ) / ( f1 + #Z n ) ;
G * ( i , j2 ) `2 = G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 ;
dom Shift ( Seq q2 , len Seq q1 ) = { j + len Seq q1 where j is Nat : j in dom Seq q1 & len Seq q1 = len Seq q1 & len Seq q1 = len q1 + len Seq q2 } ;
consider G1 , G2 , G3 being Morphism of V such that G1 <= G2 and g is Morphism of G1 and f = G1 & g = G2 and f = G2 and g = G2 and f = G1 and g = G2 and f = G2 and g = G2 and f = G1 and g = G2 ;
func - f -> PartFunc of C , V means : Def3 : dom it = dom f & for c st c in dom it holds it /. c = - f /. c & for c st c in dom it holds it /. c = - f /. c ;
consider phi such that phi is increasing and for a st phi . a = a & for v st v <> a for v1 st v1 in union rng L holds L . ( v | a ) = a * v & L . ( v | a ) = a * v ;
consider i1 , j1 such that [ i1 , j1 ] in Indices GoB f and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) ;
consider i , n such that n <> 0 and sqrt ( p ) = ( i - n ) * ( i - 1 ) and for n1 being Integer , n2 being Integer st n1 <> 0 & n2 <> 0 & n2 <> 0 holds sqrt ( p . n1 , n2 ) <= sqrt ( i - n2 ) * sqrt ( i - n2 ) ;
assume that not 0 in Z and Z c= dom ( ( ( ( - 1 ) (#) ( ( #Z 2 ) * ( f1 - f2 ) ) / ( 1 + 1 ) ) ) and for x st x in Z holds ( ( ( - 1 ) (#) ( f1 - f2 ) ) / ( 1 + 1 ) ) `| Z ) . x = - 1 / x and ( ( ( - 1 ) (#) ( f1 - f2 ) / ( 1 + 1 ) ) `| Z ) . x = - 1 / x ^2 ) ;
cell ( G1 , i1 -' 1 , ( 2 |^ ( m -' 1 ) ) * ( Y1 -' 1 ) + ( ( 2 |^ ( m -' 1 ) ) * ( Y1 -' 1 ) ) * ( Y1 -' 1 ) ) c= BDD ( ( ( m -' 1 ) + 1 ) * ( Y1 -' 1 ) ) * ( Y1 -' 1 ) + ( ( ( m -' 1 ) + 1 ) * ( Y1 -' 1 ) ) * ( Y1 -' 1 ) ) ;
ex Q1 being open Subset of X st s = Q1 & ex F8 being Subset-Family of Y st F8 c= F & ( for a being Subset of Y , b being Subset of X st b in F8 & b in F8 holds b in ( the topology of Y ) \/ { a } ) & ( a in b implies b in b ) ;
gcd ( ( 1 , A ) * ( 1 , s1 ) , ( 1 , s1 ) * ( 1 , s1 ) ) = 1 / ( ( 1 - s1 ) * ( 1 , s1 ) ) .= 1 / ( ( 1 - s1 ) * ( 1 - s1 ) ) .= 1 / ( ( 1 - s1 ) * ( 1 - s1 ) ) .= 1 / ( 1 - s1 ) ;
R8 = ( ( the for of s2 ) . ( m2 + 1 ) ) . ( m2 + 1 ) .= ( ( the Let of s2 ) . ( m2 + 1 ) ) . ( m2 + 1 ) .= [ 3 , ( the let s1 ) . ( m2 + 1 ) ] .= [ 3 , ( the Element of s2 ) . ( m2 + 1 ) ] ;
CurInstr ( P-6 , Comput ( P-6 , smeans , m1 + 1 ) ) = CurInstr ( P3 , Comput ( P3 , s3 , m3 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m3 ) ) .= halt SCMPDS .= halt SCMPDS ;
P1 /\ P2 = ( { p1 } \/ LSeg ( p1 , p2 ) ) \/ ( LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) ) \/ ( LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) ) .= { p1 } \/ ( LSeg ( p1 , p2 ) \/ LSeg ( p2 , p2 ) ) \/ { p2 } ;
func not ( ex f being Subset of the Sorts of A ) st a in dom f & ( ex a , b , c st a in dom f & b in dom f & f . a = f . b ) & ( for i st i in dom f holds f . i = f . i ) & ( for i holds f . i = f . i ) ;
for a , b being Element of F_Complex st |. a .| > |. b .| for f being Polynomial of F_Complex st f >= 1 & f is non zero holds f is \cup ( f | ( b * f ) ) is and f is is is is \cup of ( b * f ) | ( b * f ) )
defpred P [ Nat ] means 1 <= $1 & $1 <= len g implies for i , j st [ i , j ] in Indices G & G * ( i , j ) = G * ( i , j ) & 1 <= j & j <= len G holds LSeg ( G * ( i , j ) , G * ( i , j ) ) = LSeg ( G * ( i , j ) , G * ( i , j ) ) ;
assume that C1 , C2 are_\HM { \lbrack f , g .] and for s1 , s2 being State of C1 , f being Function of C1 , C2 st s1 = s2 holds s1 = s2 & f = s2 & f = s1 & g = s2 & f = s2 ;
( ||. f .|| | X ) . c = ||. f .|| . c .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| ;
|. q .| ^2 = ( ( q `1 ) ^2 + ( q `2 ) ^2 ) + ( ( q `2 ) ^2 + ( q `2 ) ^2 ) & 0 + ( ( q `2 ) ^2 + ( q `2 ) ^2 ) < ( ( q `2 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 ;
for F being Subset-Family of T7 st F is open & {} in F & for A , B being Subset of T7 st A in F & B in F & A <> B holds card A = card B & card B = card A & card B = card B implies card A = card B
assume that len F >= 1 and len F = k + 1 and len F = len G and for k st k in dom F holds H . k = g . k and for k st k in dom F holds H . k = g . k * f . k ;
i |^ ( ( mod ( n , m ) - i ) |^ s ) = i |^ ( s + k ) - i |^ ( ( n + k ) - i ) .= i |^ ( s + ( k + 1 ) - i ) .= i |^ ( ( n + k ) - i ) .= i |^ ( ( n + k ) - i ) ;
consider q being oriented Chain of G such that r = q and q <> {} and ( for p st p in rng q holds q . p = v1 ) and ( for q st q in rng p holds q . p = v1 ) & ( for p st p in rng p holds p . q = v1 ) & ( for q st q in rng p holds q . q = v1 ) ;
defpred P [ Element of NAT ] means $1 <= len ( s . ( Z , I ) ) implies ( ( g . ( Z , I ) ) . $1 = ( ( f , Z ) ^ ( g , I ) ) . ( len ( g , Z ) ) + 1 ) & ( ( f , Z ) ^ ( g , I ) ) . ( len ( g , Z ) + 1 ) = ( ( f , Z ) ^ ( g , I ) ) . ( len ( g , Z ) ) . ( len ( f , Z ) ) ;
for A , B being Matrix of n , REAL holds len ( A * B ) = len A & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B
consider s being FinSequence of the carrier of R such that Sum s = u and for i being Element of NAT st 1 <= i & i <= len s ex a , b being Element of R st s . i = a * b & a in I & b in J & s . i = b * a ;
func |( x , y )| -> Element of COMPLEX means : Def3 : for i , j be Element of NAT holds |( Re ( x , y ) , ( Re ( x , y ) ) )| = |( Re ( x , y ) , ( Re ( x , y ) ) )| + ( |( ( x , y ) , ( Re ( x , y ) ) ) , ( Re ( x , y ) ) ) )| ;
consider g9 being FinSequence of F such that for g being FinSequence of F st g is continuous & rng g9 c= A & for x being Element of F st x in A & g . 1 = x1 & g . ( len g1 ) = x2 holds g1 . ( len g1 ) = y1 & g1 . ( len g1 ) = y2 ;
then n1 >= len p1 & n2 >= len p1 & n3 ( p1 , p2 , n1 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , p1 , p2 , p1 , p2 , n2 , n3 , n3 , p1 , p2 , p1 , p2 , p1 , p2 , p1 , p2 , p1 , p1 , p2 , p1 , p2 , p2 , p1 , p2 , p2 , p1 , p2 , p1 , p2 , p1 , p2 , p1 , p1 , p2 , p2 , p1 , p2 , p2 , p1 , p2 , p2 , n3 , p2 , p2 ,
( q `1 ) ^2 * a <= ( q `1 ) ^2 & - ( q `2 ) ^2 * a <= ( q `1 ) ^2 * a & - ( q `2 ) ^2 * a <= ( q `1 ) ^2 * a & - ( q `1 ) ^2 * a <= ( q `2 ) ^2 * a ;
Fv . ( p . ( len p ) ) = Fv . ( p . ( len p ) ) .= ( ( p . ( len p ) ) . ( len p ) ) . ( p . ( len p ) ) .= ( ( p . ( len p ) ) . ( len p ) ) . ( len p ) .= ( ( p . ( len p ) ) . ( len p ) ) . ( len p ) .= ( p . ( len p ) . ( len p ) . ( len p ) .= ( p . ( len p ) ) . ( len p ) .= ( p . ( len p ) ) . ( len p ) .= ( p . ( len p ) ) . ( len p ) .= ( p . ( len p ) .= ( p . ( len p ) .= ( ( p . ( len p ) .= ( ( p . ( len p
consider k1 being Nat such that k1 + k = 1 and a := k = ( <* a := intloc 0 *> ^ ( k1 --> 1 ) ) ^ ( ( a := intloc 0 ) .--> 1 ) ^ ( ( a := intloc 0 ) .--> 1 ) ^ ( ( a := intloc 0 ) .--> 1 ) ) ^ ( ( a := intloc 0 ) .--> 1 ) ;
consider B9 being Subset of B1 , y9 being Function of B1 , B2 such that B1 is finite and [: B1 , B2 :] = the carrier of B1 and for x being set st x in B1 holds [ x , y ] in the carrier of B1 and [ y , x ] in the carrier of B1 and [ y , x ] in the carrier of B1 and [ y , x ] in the carrier of B1 ;
v2 . b2 = ( ( curry F2 ) * g ) . b2 .= ( ( curry F2 ) . b2 ) . b2 .= ( ( ( curry F2 ) . b2 ) . b2 ) . b2 .= ( ( ( curry F2 ) . b1 ) . b2 ) . b2 .= ( ( ( curry F2 ) . b1 ) . b2 ) . b2 .= ( ( ( curry F2 ) . b1 ) . b2 ) . b2 .= ( ( ( curry F2 ) . b1 ) . b2 ) . b2 .= ( ( ( ( ( id B ) . b2 ) . b2 ) . b2 ) . b2 ) . b2 .= ( ( ( id B ) . b2 ) . b2 ) . b2 ) . b2 .= ( ( ( id B ) . b2 ) . b2 .= ( ( ( ( id B ) . b2 ) . b2 ) . b2 ) . b2 ) . b2 .= ( ( ( ( ( ( id
dom IExec ( I-35 , P , Initialize s ) = the carrier of SCMPDS .= dom IExec ( I , P , Initialize s ) .= dom IExec ( I , P , Initialize s ) .= dom IExec ( I , P , Initialize s ) .= dom IExec ( I , P , Initialize s ) .= dom IExec ( I , P , Initialize s ) .= dom IExec ( I , P , Initialize s ) ;
ex d-32 be Real st d-32 > 0 & for h be Real st h <> 0 & |. h .| < e holds |. h .| " * ||. ( R /* h ) . h .|| < e / ( 1 + e ) * ||. ( R /* h ) . h .|| ;
LSeg ( G * ( len G , 1 ) + |[ 1 , - 1 ]| , G * ( len G , 1 ) + |[ 1 , - 1 ]| ) c= Int cell ( G , len G , 1 ) \/ { |[ 1 , - 1 ]| }
LSeg ( mid ( h , i1 , i2 ) , i ) = LSeg ( h /. ( i + 1 -' 1 ) , h /. ( i + 1 -' 1 ) ) .= LSeg ( h /. ( i + 1 -' 1 ) , h /. ( i + 1 -' 1 ) ) .= LSeg ( h , i ) ;
A = { q where q is Point of TOP-REAL 2 : LE p1 , q , P , p1 , p2 & LE q1 , q2 , P , p1 , p2 & LE q1 , q2 , P , p1 , p2 & LE q1 , q2 , P , p1 , p2 & LE q1 , q2 , P , p1 , p2 & LE q1 , q2 , P , p1 , p2 & LE q1 , q2 , P , p1 , p2 } ;
( ( - x ) .|. y ) = - ( ( 1 - x ) .|. y ) * ( ( - x ) .|. y ) .= - ( ( 1 - x ) .|. y ) * ( ( - x ) .|. y ) .= ( ( - x ) .|. y ) * ( ( - x ) .|. y ) .= ( ( - x ) .|. y ) * ( ( - x ) .|. y ) .= ( ( - x ) .|. y ) .|. y ;
0 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) = ( ( p `1 / p `2 ) ^2 + ( p `2 / p `1 ) ^2 ) * sqrt ( 1 + ( p `2 / p `1 ) ^2 ) .= ( ( p `1 / p `2 ) ^2 + ( p `2 / p `1 ) ^2 ) * sqrt ( 1 + ( p `2 / p `1 ) ^2 ) ;
( ( U + W ) * ( W + U ) ) * ( thesis ) = ( ( ( U + W ) * ( W + U ) ) * ( W + U ) ) * ( W + U ) ) .= ( ( U + W ) * ( W + U ) ) * ( W + U ) .= ( U + W ) * ( W + U ) ) * ( W + U ) .= ( U + W ) * ( W + U ) * ( W + U ) ;
func Shift ( f , h ) -> PartFunc of REAL , REAL means : Def3 : dom it = dom f & for x st x in dom it holds it . x = ( x + h ) . x + ( x - h ) . x & for x st x in dom it holds it . x = ( x + h ) . x + ( x - h ) . x ;
assume that 1 <= k and k + 1 <= len f and [ i , j ] in Indices G and [ i + 1 , j ] in Indices G and f /. k = G * ( i + 1 , j ) and f /. k = G * ( i + 1 , j ) and f /. k = G * ( i , j ) ;
assume that not y in Free H and not x in Free H and not ( x in Free ( H ) and not x in Free ( H ) and not x in Free ( H ) and not x in Free ( H ) and not x in Free ( H ) and not x in Free ( H ) and not x in Free ( H ) and not x in Free ( H ) and y in Free ( H ) ;
defpred P11 [ Element of NAT , \HM Element of NAT , \HM \HM { \HM { p is prime } , \HM { is Element of NAT : P [ $1 ] } , $2 = ( p |^ $1 ) |^ ( 2 * $1 + 1 ) ) * ( ( p |^ $1 ) |^ ( 2 * $1 + 1 ) ) ;
func \sigma ( C ) -> non empty Subset-Family of X means : Def3 : for A being Subset of X holds A in it iff for W being Subset of X holds W in it iff for A , B being Subset of X st A c= B & A c= B holds C in it iff A c= B & C in it & B is finite ;
[#] ( ( dist ( P ) ) .: Q ) = ( dist ( P ) ) .: Q & lower_bound ( ( dist ( P ) ) .: Q ) = lower_bound ( ( dist ( P ) ) .: Q ) & lower_bound ( ( dist ( P ) ) .: Q ) = lower_bound ( ( dist ( P ) ) .: Q ) ;
rng ( F | ( [: S , T :] ) ) = {} or rng ( F | ( [: S , T :] ) ) = { 1 } or rng ( F | ( [: S , T :] ) ) = { 2 } or rng ( F | ( [: S , T :] :] ) = { 1 , 2 } ;
( f " ( rng f ) ) . i = f . i " . ( ( rng f ) . i ) .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i ;
consider P1 , P2 being non empty Subset of TOP-REAL 2 such that P1 is_an_arc_of p1 , p2 and P1 = { p1 , p2 } and P1 = { p1 , p2 } and P2 = { p2 , p1 } and P1 = { p2 , p1 } and P2 = { p2 , p1 } and P1 = { p2 , p1 } and P2 = { p2 , p1 } ;
f . p2 = |[ ( p2 `1 / sqrt ( 1 + ( p2 `2 / p2 `1 ) ^2 ) ) / sqrt ( 1 + ( p2 `2 / p2 `1 ) ^2 ) , ( p2 `2 / sqrt ( 1 + ( p2 `2 / p2 `1 ) ^2 ) ) ]| ;
( ( for a , X be set holds ( ( a , X ) " ) . x = ( ( ( a , X ) qua Function ) " ) . x ) " .= ( ( a , X ) " ) . x ) " .= ( ( a , X ) " ) . x + ( ( a , X ) " ) . x .= ( ( a , X ) " ) . x + ( a , X ) " ) .= ( a + b ) " * x ;
for T being non empty normal TopSpace , A , B being closed Subset of T , p being Point of T , r being Real st A <> {} & A misses B for p being Point of T , r being Real st p in A & A = ( in \mathbb R ) . r holds p in O implies p in O
for i , j st i + 1 in dom F for G1 , G2 being strict normal Subgroup of G st G1 = F . i & G2 = G . ( i + 1 ) & G2 = F . ( i + 1 ) holds G1 = G2 & G1 = G2
for x st x in Z holds ( ( ( ( #Z 2 ) * ( arctan ) - ( arccot ) ) / ( 1 + x ^2 ) ) `| Z ) . x = ( ( ( 1 / 2 ) * ( arctan ) - ( arctan ) / ( 1 + x ^2 ) ) `| Z ) . x
synonym f is_right x0 means : Def3 : for a st x0 in dom f & f . x0 = lim ( f /* a ) & for x st x in dom f & x in dom f holds f /. x = lim ( f /* a ) & f /. x0 = lim ( f /* a ) ;
then X1 , X2 are_separated & X1 misses X2 or ex Y1 , Y2 being non empty SubSpace of X st Y1 misses Y2 & Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & Y2 is SubSpace of X2 & Y2 is SubSpace of X2 & Y2 is SubSpace of X1 & Y1 is SubSpace of X2 & Y2 is SubSpace of X2 & Y2 is SubSpace of X2 & Y2 is SubSpace of X2 & Y2 is SubSpace of X1 & Y1 is SubSpace of X2 ;
ex N be Neighbourhood of x0 st N c= dom SVF1 ( 1 , f , u ) & ex L , R st for x st x in N holds SVF1 ( 1 , f , u ) . x - SVF1 ( 1 , f , u ) . x0 = L . ( x- ( 1 , f , u ) . x - SVF1 ( 1 , f , u ) . x0 )
( ( p2 `1 ) ^2 * sqrt ( 1 + ( p2 `2 / p2 `1 ) ^2 ) ) * ( ( p2 `2 ) ^2 + ( p2 `2 / p2 `1 ) ^2 ) >= ( ( ( p2 `1 ) ^2 + ( p2 `2 / p2 `1 ) ^2 ) * sqrt ( 1 + ( p2 `2 / p2 `1 ) ^2 ) ) * sqrt ( 1 + ( p2 `1 / p2 `1 ) ^2 ) ;
( ( 1 / t ) (#) ||. f1 .|| ) . x = ( ( 1 / t ) (#) ||. f1 .|| ) . x & ( ( 1 / t ) (#) ||. f1 .|| ) . x = ( ( 1 / t ) (#) ||. f1 .|| ) . x & ( ( 1 / t ) (#) ||. f1 .|| ) . x = ( ( 1 / t ) (#) ||. f1 .|| ) . x ;
assume that for x holds f . x = ( ( sin (#) ( sin + cos ) ) `| Z ) . x and x + h / 2 in dom ( sin (#) ( cos + cos ) ) and for x st x in Z holds ( ( sin (#) ( sin + cos ) ) `| Z ) . x = ( sin . x + sin . x ) / ( cos . x ) ^2 and ( cos (#) ( sin + cos ) ) `| Z ) . x = ( sin . x ) ^2 - ( sin . x ) ^2 - ( sin . x ) ^2 - ( sin . x ) ^2 - ( sin . x ) ^2 - ( sin . x ) ^2 - ( sin . x ) ^2 - ( sin . x ) ^2 - ( sin . x ) ^2 - ( sin . x ) ^2 - ( sin . x ) ^2 - ( sin . x ) ^2 - ( sin . x ) ^2 - ( sin . x ) ^2 - ( sin . x ) ^2 - ( sin . x ) ^2 - ( sin . x ) ^2 - ( sin . x ) ^2 - ( sin . x ) ^2 - ( sin .
consider X-23 being Subset of Y , Y1 being Subset of X such that t = [: Y1 , Y1 :] and Y1 is open and ex Y1 being Subset of X st Y1 = [: Y1 , Y1 :] & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open ;
card S . n = card { [: d , Y :] + ( a * b ) + 1 where d , Y is Element of GF ( p ) , Y is Element of GF ( p ) : [ d , Y ] in R & [ d , Y ] in R } .= [: { d , Y } , { b } :] \/ [: { b } , { b } :] \/ [: { b } :] , { b } :] ;
( W-bound D - W-bound D ) * ( ( W-bound D - W-bound D ) / ( m + 1 ) ) = ( W-bound D - W-bound D ) / ( m + 1 ) * ( ( W-bound D - W-bound D ) / ( m + 1 ) ) .= ( W-bound D - W-bound D ) / ( m + 1 ) * ( ( W-bound D - W-bound D ) / ( m + 1 ) ) .= ( W-bound D - W-bound D ) / ( m + 1 ) * ( m + 1 ) ;