thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b
D c= S
let Y ;
S ` is convergent
q in X ;
let V ;
y in N ;
x in T ;
m < n ;
m <= n ;
n > 1 ;
let r ;
t in I ;
n <= 4 ;
M is finite ;
let X ;
Y c= Z ;
A // M ;
let U ;
a in D ;
q in Y ;
let x ;
1 <= l ;
1 <= w ;
let G ;
y in N ;
f = {} ;
let x ;
x in Z ;
let x ;
F is one-to-one ;
e <> b ;
1 <= n ;
f is special ;
S misses C
t <= 1 ;
y divides m ;
P divides M ;
let Z ;
let x ;
y c= x ;
let X ;
let C ;
x _|_ p ;
o is monotone ;
let X ;
A = B ;
1 < i ;
let x ;
let u ;
k <> 0 ;
let p ;
0 < r ;
let n ;
let y ;
f is onto ;
x < 1 ;
G c= F ;
a is_>=_than X ;
T is continuous ;
d <= a ;
p <= r ;
t < s ;
p <= t ;
t < s ;
let r ;
D is_<=_than E ;
assume e > 0 ;
assume 0 < g ;
p in X ;
x in X ;
Y ` in Y ;
assume 0 < g ;
not c in Y ;
not v in L ;
2 in z `2 ;
assume f = g ;
N c= b ` ;
assume i < k ;
assume u = v ;
I = J ;
B ` ` = b ` ;
assume e in F ;
assume p > 0 ;
assume x in D ;
let i be element ;
assume F is onto ;
assume n <> 0 ;
let x be element ;
set k = z ;
assume o = x ;
assume b < a ;
assume x in A ;
a `2 <= b `2 ;
assume b in X ;
assume k <> 1 ;
f = product l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is from squares ;
assume m > 0 ;
assume A c= B ;
X is lower ;
assume A <> {} ;
assume X <> {} ;
assume F <> {} ;
assume G is open ;
assume f is dilatation ;
assume y in W ;
y \not <= x ;
A ` in B ` ;
assume i = 1 ;
let x be element ;
x `2 = x `2 ;
let X be BCK-algebra ;
assume S is non empty ;
a in REAL ;
let p be set ;
let A be set ;
let G be _Graph , W be Walk of G ;
let G be _Graph , W be Walk of G ;
let a be Complex ;
let x be element ;
let x be element ;
let C be FormalContext , a , b be Real ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
let y be Real ;
X c= f . a
let y be element ;
let x be element ;
i be Nat ;
let x be element ;
n in NAT ;
let a be element ;
m in NAT ;
let u be element ;
i in NAT ;
let g be Function ;
Z c= NAT ;
l <= ma ;
let y be element ;
r2 in dom f ;
let x be element ;
let k1 be Integer ;
let X be set ;
let a be element ;
let x be element ;
let x be element ;
let q be element ;
let x be element ;
assume f is being_homeomorphism ;
let z be element ;
a , b // K ;
let n be Nat ;
let k be Nat ;
B ` c= B ` ;
set s = as Real ;
n >= 0 + 1 ;
k c= k + 1 ;
R1 c= R ;
k + 1 >= k ;
k c= k + 1 ;
let j be Nat ;
o , a // Y ;
R c= Cl G ;
Cl B = B ;
let j be Nat ;
1 <= j + 1 ;
arccot is_differentiable_on Z ;
exp_R is_differentiable_in x ;
j < i0 ;
let j be Nat ;
n <= n + 1 ;
k = i + m ;
assume C meets S ;
n <= n + 1 ;
let n be Nat ;
h1 = {} ;
0 + 1 = 1 ;
o <> b3 ;
f2 is one-to-one ;
support p = {}
assume x in Z ;
i <= i + 1 ;
r1 <= 1 ;
let n be Nat ;
a "/\" b <= a ;
let n be Nat ;
0 <= r0 ;
let e be Real , x be Point of TOP-REAL n ;
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 , b be Int-Location ;
let l be Nat ;
let i be Nat ;
let r ;
1 <= i2 ;
a "\/" c = c ;
let r be Real ;
let i be Nat ;
let m be Nat ;
x = p2 ;
let i be Nat ;
y < r + 1 ;
rng c c= E
Cl R is boundary ;
let i be Nat ;
R2 in X ;
cluster uparrow x -> in ;
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 * ;
\cap L~ holds \cap n >= \bf s ;
G . y <> 0 ;
let X be RealNormSpace , x be Point of X ;
a in X ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , W be Subspace of V ;
assume x in - - M ;
k < s . a ;
not t in { p } ;
let Y be set , f be Function of Y , X ;
M , L are_equipotent ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded lower-bounded ;
rng f = Y ;
G8 c= L & G8 c= L ;
assume x in Cl Q ;
m in dom P ;
i <= len Q ;
len F = 3 ;
and 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 ` = a ` + 1 ;
x `2 = a * y `2 ;
rng D c= A ;
assume x in K1 ;
1 <= i-32 ;
1 <= i-32 ;
pp c= cos .: { p } ;
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 is closed ;
cluster R \ S -> real-valued ;
sup D in S ;
x << sup D ;
b1 >= Z1 + 1 ;
assume w = 0. V ;
assume x in A . i ;
g in the carrier of Assume X ;
y in dom t ;
i in dom g ;
assume P [ k ] ;
.= C . f ;
x9 is increasing & x9 is increasing ;
let e2 be element ;
- - b divides b ;
F c= \tau ( F ) ;
Gseq is non-decreasing & Gseq is non-decreasing ;
Gseq is non-decreasing & Gseq is non-decreasing ;
assume v in H . m ;
assume b in [#] B ;
let S be non void ManySortedSign , A be non-empty MSAlgebra over S ;
assume P [ n ] ;
assume union S is finite independent ;
V is Subspace of V ;
assume P [ k ] ;
rng f c= NAT * ;
assume ex_inf_of X , L ;
y in rng f ;
let s , I be set , f be Function of s , I ;
b ` ` c= b9 ` ;
assume not x in INT ;
A /\ B = { a } ;
assume len f > 0 ;
assume x in dom f ;
b , a // o , c ;
B in B-24 ;
cluster product p -> non empty ;
z , x // x , p ;
assume x in rng N ;
cosec is_differentiable_in x & cosec is_differentiable_in x ;
assume y in rng S ;
let x , y be element ;
i2 < i1 & i1 < i2 ;
a * h in a * H ;
p , q in Y ;
redefine func sqrt I -> Ideal of L ;
q1 in A1 & q2 in A2 ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A2 c= A1 ;
an < n & n < k ;
assume A c= dom f ;
Re f is_integrable_on M ;
let k , m be element ;
a , a \equiv b , b ;
j + 1 < k + 1 ;
m + 1 <= n1 ;
g is_differentiable_in x0 & g is_differentiable_in x0 ;
g is_continuous_in x0 & g is_continuous_in x0 ;
assume 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 with non-empty ManySortedSet of I ;
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 , Y be Subset of X ;
[ a , b ] in R ;
x + w < y + w ;
{ a , b } is_>=_than c ;
let B be Subset of A , C be Subset of B ;
let S be non empty ManySortedSign ;
let x be variable , f be Function of X , REAL ;
let b be Element of X , x be Element of X ;
R [ x , y ] ;
x ` ` = x & x ` = x ;
b \ x = 0. X ;
<* d *> in D * ;
P [ k + 1 ] ;
m in dom mnn ;
h2 . a = y ;
P [ n + 1 ] ;
redefine func G * F -> ] | C ;
let R be non empty multMagma , a be Element of R ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p ;
assume f | X is lower ;
x in rng co /\ rng co ;
let x be Element of B ;
let t be Element of D ;
assume x in Q .vertices() ;
set q = s ^\ k ;
let t be VECTOR of X ;
let x be Element of A ;
assume y in rng p `2 ;
let M be | mamaid ;
let N be non empty \HM of \HM { the } \HM { carrier of M } ;
let R be RelStr with finite finite finite for RelStr ;
let n , k be Nat ;
let P , Q be RelStr ;
P = Q /\ [#] S ;
F . r in { 0 } ;
let x be Element of X ;
let x be Element of X ;
let u be VECTOR of V ;
reconsider d = x as Int-Location ;
assume I is not [ a , b ] ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < v8 ;
x <= c2 . x ;
x in F ` & y in F ` ;
redefine func S --> T -> * ;
assume that t1 <= t2 and t1 <= t2 ;
let i , j be even Integer ;
assume that F1 <> F2 and F1 <> F2 ;
c in Intersect ( union R ) ;
dom p1 = c & dom p2 = c ;
a = 0 or a = 1 ;
assume A1 <> A2 & A1 <> A2 ;
set i1 = i + 1 ;
assume a1 = b1 & a2 = b2 ;
dom g1 = A & dom g2 = B ;
i < len M + 1 ;
assume not -infty in rng G ;
N c= dom f1 /\ dom f2 ;
x in dom ( sec * f ) ;
assume [ x , y ] in R ;
set d = x / y ;
1 <= len g1 & 1 <= len g2 ;
len s2 > 1 & len s2 > 1 ;
z in dom f1 /\ dom f2 ;
1 in dom D2 & 1 <= i ;
p `2 / |. p .| = 0 ;
j2 <= width G & j2 <= width G ;
len cos > 1 + 1 ;
set n1 = n + 1 ;
|. q-35 .| = 1 ;
let s be SortSymbol of S ;
( i , i ) gcd i = i ;
X1 c= dom f & X1 c= dom g ;
h . x in h . a ;
let G be in thesis , F be Subset of on ;
cluster m * n -> square ;
let k9 be Nat , k be Nat ;
i - 1 > m - 1 ;
R is transitive implies field R is transitive
set F = <* u , w *> ;
p-2 c= P3 & p`2 c= P3 ;
I is_halting_on t , Q ;
assume [ S , x ] is thesis ;
i <= len f2 & i <= len f2 ;
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= X3 & BX c= XX ;
n2 <= ( 2 - 1 ) / 2 ;
A /\ cP c= A ` ;
cluster x -valued -> constant for Function ;
let Q be Subset-Family of S , P be Subset of Q ;
assume n in dom g2 & n + 1 in dom g2 ;
let a be Element of R ;
t `2 in dom e2 & t `2 = e2 ;
N . 1 in rng N ;
- z in A \/ B ;
let S be in of X , M be Element of S ;
i . y in rng i ;
REAL c= dom f & dom f c= dom g ;
f . x in rng f ;
mt <= r / 2 ;
s2 in r-5 & s2 in r-5 ;
let z , z be complex number ;
n <= Nk . 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 | X ) ;
let K be add-associative right_zeroed right_complementable non empty doubleLoopStr , V be Subset of V ;
let i be Element of NAT , k be Nat ;
rng ( F * g ) c= Y
dom f c= dom x & dom g c= dom y ;
n1 < n1 + 1 & n1 + 1 < n2 ;
n1 < n1 + 1 & n1 + 1 < n2 ;
cluster 1. T -> \overline W & T is \overline W ;
[ y2 , 2 ] `2 = z ;
let m be Element of NAT , n be Nat ;
let S be Subset of R ;
y in rng ( S\cdot ) ;
b = upper_bound dom f & b = upper_bound dom f ;
x in Seg ( len q ) ;
reconsider X = D ( ) as set ;
[ a , c ] in E1 ;
assume n in dom ( h2 | n ) ;
w + 1 = ma + 1 ;
j + 1 <= j + 1 + 1 ;
k2 + 1 <= k1 + 1 ;
let i be Element of NAT ;
Support u = Support p & Support u = Support p ;
assume X is complete thesis implies X is complete
assume that f = g and p = q ;
n1 <= n1 + 1 & n2 + 1 <= n1 + 1 ;
let x be Element of REAL , r be Real ;
assume x in rng ( s2 | X ) ;
x0 < x0 + 1 & x0 < x0 + 1 ;
len Carrier ( L ) = W ;
P c= Seg ( len A ) ;
dom q = Seg n .= dom p ;
j <= width ( M @ ) ;
let seq1 be real-valued FinSequence , seq2 be convergent Real_Sequence ;
let k be Element of NAT , n be Nat ;
Integral ( M , P ) < +infty ;
let n be Element of NAT , x be Element of X ;
assume z in being \tt _- ( 0 , A ) ;
let i be set ;
n - 1 = n-1 - 1 ;
len n-27 = n & len n-27 = n ;
\cal ( Z , c ) c= F
assume x in X or x = X ;
x is midpoint of b , c ;
let A , B be non empty set , f be Function of A , B ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q & dom q = Seg i ;
let s be Element of E * ;
let B1 be Basis of x , B2 be Basis of x ;
Carrier ( L3 /\ L2 ) = {} ;
L1 /\ LSeg ( p1 , p2 ) = {} ;
assume downarrow x = downarrow y ;
assume b , c // b , c ;
LIN q , c , c ;
x in rng f-129 & x in rng f-129 ;
set nbeing = n + j ;
let D7 be non empty set , f be FinSequence of D , i be Nat ;
let K be right_zeroed non empty addLoopStr , V be Subset of K ;
assume f opp = f & h opp = h ;
R1 - R2 is total & R2 - R1 is total ;
k in NAT & 1 <= k implies k <= n
let a be Element of G ;
assume x0 in [. a , b .] ;
K1 ` is open & K1 is open ;
assume a , b is maximal in C ;
let a , b be Element of S ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = Integral ( M , f ) ;
cluster n^ -> ness] for ;
not u in { ag } ;
the carrier of f c= B \/ { v } ;
reconsider z = x as VECTOR of V ;
cluster the carrier of L -> being x1 ;
r (#) H is C non-zero ;
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 : r in y & r in y ;
let x , y be Element of X ;
let A , I be } \rm the carrier of X ;
[ y , z ] in [: O , O :] ;
( \subseteq being Nat ) & ( len Macro i ) = 1 ;
rng Sgm ( A ) = A ;
q |- \! such that All ( y , q ) |- q ;
for n holds X [ n ] ;
x in { a } & x in d ;
for n holds P [ n ] ;
set p = |[ x , y , z ]| ;
LIN o , a , b ;
p . 2 = Z |^ Y ;
( ( DD2 ) `2 ) ^2 = {} ;
n + 1 + 1 <= len g ;
a in [: CQC-WFF ( Al ) , { x } :] ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
let I be Ideal of L ;
set g = f1 + f2 , h = f2 + f3 ;
a <= max ( a , b ) ;
i-1 < len G + 1-1 ;
g . 1 = f . i1 ;
x ` , y ` in A2 ;
( f /* s ) . k < r ;
set v = VAL g ;
i - k + 1 <= S ;
cluster associative for non empty multMagma ;
x in support ( ( support t ) | support ( k ) ) ;
assume a in [: the carrier of G , the carrier of G :] ;
i `2 <= len ( y | i ) ;
assume that p divides b1 + b2 and p divides b2 ;
M1 <= upper_bound M1 & M2 <= upper_bound M1 implies M1 + M2 <= M1 + M2
assume x in W-min ( X ) & y in W-min ( X ) ;
j in dom ( z | i ) ;
let x be Element of D ( ) ;
IC s4 = l1 .= ( 0 + 1 ) ;
a = {} or a = { x } ;
set us = Vertices G , us = Vertices G , us = Vertices G , us = Vertices G , us = Vertices G , us = Vertices G , us
seq " is non-zero & seq " is non-zero implies seq " is non-zero
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
hK1 c= h-14 & hK1 c= h-14 ;
]. a , b .[ c= Z ;
X1 , X2 are_separated implies X1 , X2 are_separated & X2 , X1 are_separated
a in Cl ( union F \ G ) ;
set x1 = [ 0 , 0 ] ;
k + 1 - 1 = k - 1 ;
cluster real-valued -> INT -valued for Relation ;
ex v st C = v + W ;
let IT be non empty addLoopStr , x be Element of IT ;
assume V is Abelian add-associative right_zeroed right_complementable ;
X-21 \/ Y in \sigma ( L ) ;
reconsider x `2 = x as Element of S ;
max ( a , b ) = a ;
sup B is upper Subset of B & sup B is upper Subset of B ;
let L be non empty reflexive RelStr , X be Subset of L ;
R is reflexive transitive & X c= X implies R is transitive
E , g |= the_right_argument_of ( H , E ) ;
dom G /. y = a ;
( 1 / 4 ) * ( - r ) >= - r ;
G . p0 in rng G & G . p0 in rng G ;
let x be Element of FF , y be Element of FF ;
D [ P-6 , 0 ] ;
z in dom ( id B ) /\ dom ( id B ) ;
y in the carrier of N & y in the carrier of N ;
g in the carrier of H & h in the carrier of H ;
rng fset c= NAT & rng fD c= NAT ;
j `2 + 1 in dom s1 & j + 1 in dom s2 ;
let A , B be strict Subgroup of G ;
let C be non empty Subset of REAL ;
f . z1 in dom h & h . z2 in dom h ;
P . k1 in rng P & P . k2 in rng P ;
M = ( A +* {} ) +* {} .= A ;
let p be FinSequence of REAL , r be Real ;
f . n1 in rng f & f . n2 in rng f ;
M . ( F . 0 ) in REAL ;
( h , b ) `2 = b-a ;
assume the distance of V , Q is_<* v *> ;
let a be Element of op ( V ) ;
let s be Element of Py ( ) ;
let PA be non empty as non empty \rm \rm RelStr ;
let 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 BM = BCS ( K , n ) ;
l <= ( j ) implies j <= len ( F . j ) ;
assume x in downarrow [ s , t ] ;
x `2 in uparrow t & x `2 in uparrow t ;
x in ( JumpParts T ) \/ ( JumpParts T ) ;
let h be Morphism of c , a ;
Y c= 1. R & Y c= 1. R implies Y c= X
A2 \/ A3 c= Carrier ( L1 ) \/ Carrier ( L2 ) ;
assume LIN o , a , b ;
b , c // d1 , e2 ;
x1 , x2 in Y & x2 in Y ;
dom <* y *> = Seg 1 .= Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar s .| ;
[ x , x `2 ] in [: X , X :] ;
for n being Nat holds 0 <= x . n
[' a , b '] = [. a , b .] ;
cluster -> non empty for Subset of T ;
x = h . f . z1 ;
q1 , q2 in P & q2 in P ;
dom M1 = [: Seg n , Seg n :] ;
x = [ x1 , x2 ] ;
let R , Q be ManySortedSet of A ;
set d = 1 / ( n + 1 ) ;
rng ( g2 ) c= dom W & rng ( g2 ) c= dom W ;
P . ( [#] Sigma \ B ) <> 0 ;
a in field R & a = b ;
let M be non empty Subset of V , V be Subset of V ;
let I be Program of SCM+FSA , a be Int-Location ;
assume x in rng ( the InternalRel of R ) ;
let b be Element of the carrier of T ;
dist ( e , z ) - r-r > r-r ;
u1 + v1 in W2 & v1 in W1 & v2 in W2 ;
assume the carrier of L misses rng G ;
let L be lower-bounded antisymmetric RelStr ;
assume [ x , y ] in a9 ;
dom ( A * e ) = NAT & dom ( A * e ) = NAT ;
let a , b be Vertex of G ;
let x be Element of ( ( M . i ) ) * ;
0 <= Arg a & Arg a < 2 * PI ;
o9 , a9 // o9 , y & o9 , b9 // o9 , y ;
{ v } c= the carrier of l & { v } c= the carrier of l ;
let x be bound variable of A ;
assume x in dom ( uncurry f ) /\ dom ( uncurry f ) ;
rng F c= ( product f ) * ;
assume D2 . k in rng D & D2 . k in rng D ;
f " . p1 = 0 & f " . p2 = 1 ;
set x = the Element of X , y = the Element of Y ;
dom Ser ( G ) = NAT & rng Ser ( G ) = NAT ;
let n be Element of NAT ;
assume LIN c , a , e1 ;
cluster -> R yielding for FinSequence of NAT ;
reconsider d = c as Element of L1 , L2 ;
( v2 |-- I ) . X <= 1 ;
assume x in the carrier of f & x in the carrier of g ;
conv @ S c= conv @ A & conv @ S c= conv @ A ;
reconsider B = b as Element of the topology of T ;
J , v |= P ! ( l ) ;
redefine func J . i -> non empty TopSpace ;
ex_sup_of Y1 \/ Y2 , T & ex_sup_of Y1 , T ;
W1 is_field W1 & W2 is_field W1 implies W1 + W2 is connected
assume x in the carrier of R & y in the carrier of R ;
dom n-16 = Seg n & dom n-16 = Seg n ;
( ( the carrier of S ) \ { x } ) misses ( ( the carrier of S ) \ { x } ) ;
assume ( a 'imp' b ) . z = TRUE ;
assume X is open & f = X --> d ;
assume [ a , y ] in implies [ a , y ] in being Element of product f ;
assume that I c= J and / I c= J and I c= K ;
Im ( lim seq , g ) = 0 ;
( ( sin * ( f ^ ) ) `| Z ) . x <> 0 ;
sin is_differentiable_on Z & cos is_differentiable_on Z implies ( cos * ( f ^ ) ) `| Z = f
t3 . n = t3 . n .= s . n ;
dom ( ( - x ) (#) F ) c= dom F ;
W1 . x = W2 . x & W2 . x = W2 . x ;
y in W .vertices() \/ W .vertices() \/ W .vertices() ;
( k + 1 ) <= len ( v | k ) ;
x * a \equiv y * a . ( mod m ) ;
proj2 .: S c= proj2 .: P & proj2 .: S c= proj2 .: P ;
h . p4 = g2 . I & h . I = g2 . I ;
G6 = U2 /. 1 .= U2 /. ( 1 + 1 ) ;
f . rr1 in rng f & f . rr2 in rng f ;
i + 1 + 1-1 <= len - 1 ;
rng F = rng FF2 & rng FF2 = rng FF2 ;
mode T is well unital associative non empty multMagma ;
[ x , y ] in [: A , { a } :] ;
x1 . o in L2 . o & x2 . o in L2 . o ;
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 implies seq is lower
len ( F | I ) = len I .= len ( F | I ) ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be complex number , a be Real ;
Comput ( P , s , n ) = s ;
k <= k + 1 & k + 1 <= len p ;
reconsider c = {} T as Element of L ;
let Y be Element of be Element of be Element of be Element of be 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 NAT -defined Function ;
K c= 2 -tuples_on the carrier of T ;
F . b1 = F . b2 & F . b2 = F . b2 ;
x1 = x or x1 = y or x1 = z ;
pred a <> {} means : Def2 : ( a - 1 ) / a = 1 ;
assume that cf a c= b and b in a ;
s1 . n in rng s1 & s1 . n in rng s1 ;
{ o , b2 } on C2 & { o , b1 } on C2 ;
LIN o9 , b , b9 & LIN o9 , b9 , a9 ;
reconsider m = x as Element of Funcs ( V , C ) ;
let f be non trivial non trivial FinSequence of D ;
let FF2 be non empty element , F be non empty that F is non empty
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in F-8 ;
reconsider pp2 = x , pp2 = y as Subset of m -tuples_on the carrier of K ;
let A , B , C be Element of R ;
redefine func strict strict non empty as strict normal transitive ;
rng c `2 misses rng ( e `2 ) \/ rng ( e `2 ) ;
z is Element of gr { x } & z is Element of gr { y } ;
not b in dom ( a .--> p1 ) ;
assume that k >= 2 and P [ k ] ;
Z c= dom ( cot * ( f ^ ) ) & Z c= dom ( cot * ( f ^ ) ) ;
the component of Q c= UBD A & UBD A c= UBD A ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( 1 / ( n + 1 ) ) ;
pred f = u means : Def2 : a * f = a * u ;
for n holds P1 [ n ] implies P1 [ n + 1 ]
{ x . O : x in L } <> {} ;
let x be Element of V . s ;
let a , b be Nat ;
assume that S = S2 and p = p2 and p = p1 ;
( n1 gcd n2 ) = 1 & ( n1 gcd n2 ) = 1 ;
set oz = * ( ( INT * ) + ( INT * ) ) ;
seq . n < |. r1 .| & seq . n < |. r2 .| ;
assume that seq is increasing and r < 0 ;
f . ( y1 , x1 ) <= a ;
ex c being Nat st P [ c ] ;
set g = { n to_power 1 where n is Nat : n <= 1 } ;
k = a or k = b or k = c ;
aa , ag , bg , bh , bh , bh , bh , bh , g be set ;
assume Y = { 1 } & s = <* 1 *> ;
Is1 . x = f . x .= 0 .= Is2 . x ;
W3 .last() = W3 . 1 .= W3 . ( 1 + 1 ) ;
cluster trivial -> finite finite for A , B be A , V be set ;
reconsider u = u as Element of Bags X ;
A in B ^ -> implies A , B ^ is $ / A
x in { [ 2 * n + 3 , k ] } ;
1 >= ( q `1 / |. q .| - cn ) / ( 1 + cn ) ;
f1 is_as as as as as as as as as as as as as as as as as the the rng of f ;
( f `2 ) ^2 <= ( q `2 ) ^2 & ( f `2 ) ^2 <= ( q `2 ) ^2 ;
h is_the carrier of Cage ( C , n ) ;
b `2 <= p `2 & p `2 <= ( p `2 ) * ( 1 + 1 ) `2 ;
let f , g be /. of X , Y ;
S * ( k , k ) <> 0. K ( ) ;
x in dom max ( - f , - g ) ;
p2 in NV . p1 & p2 in NV . p2 ;
len ( the_right_argument_of H ) < len ( H ) + len ( H ) ;
F [ A , F-14 . A ] ;
consider Z such that y in Z and Z in X ;
pred 1 in C means : Def2 : A c= C * ;
assume r1 <> 0 or r2 <> 0 ;
rng q1 c= rng C1 & rng q2 c= rng C2 ;
A1 , L , A3 , A3 , v2 , w , z be set ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in C & not p in C implies b in C
then S is negative means P-2 [ S ] ;
Cl Int [#] T = [#] T & Cl Int [#] T = [#] T ;
f12 | A2 = f2 & f12 | A2 = f2 | A2 ;
0. M in the carrier of W & 0. M in the carrier of W ;
let v , v be Element of M ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ V & Y \ V c= Y \ Z ;
let X be Subset of [: S , T :] ;
consider H1 such that H = 'not' H1 and H1 is finite ;
1. ( 1 , 1 ) c= ( \mathop { \rm where } r is Element of 1 , 1 ) * ;
0 * a = 0. R .= a * 0 ;
A |^ ( 2 , 2 ) = A ^^ A ;
set v, v' = ( v /. n ) `1 , v' = ( v /. n ) `1 ;
r = 0. ( REAL-NS n ) & 0 * r = 0. ( REAL-NS n ) ;
( f . p4 ) `1 >= 0 & ( f . p4 ) `2 >= 0 ;
len W = len ( W | ( W L ) ) .= len W ;
f /* ( s * G ) is divergent_to-infty ;
consider l being Nat such that m = F . l ;
t8 does not ^ ( W7 ) does not ^ ( ( len s ) .--> 1 ) ;
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 ] -> pair for element ;
downarrow a /\ downarrow t is Ideal of T & downarrow a /\ downarrow t is Ideal of T ;
let X be N with \hbox { NAT , D , E , F be Function of X , D ;
rng f = being Element of being I \lbrace S , X } ;
let p be Element of B , the connectives of S ;
max ( N1 , 2 ) >= N1 & max ( N1 , 2 ) >= N2 ;
0. X <= b |^ ( m * mm1 ) ;
assume that i in I and R1 . i = R ;
i = j1 & p1 `1 = q1 `1 & i = q2 `1 ;
assume gR in the right of g & gR in the carrier of g ;
let A1 , A2 be Point of S , A be Subset of X ;
x in h " P /\ [#] T1 & x in h " P /\ [#] T1 ;
1 in Seg 2 & 1 in Seg 3 implies 1 in Seg 3
reconsider X-5 = X , X, Ximplies X] = X as non empty Subset of Telement ;
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the Target of G ) -valued ;
n1 <= i2 + len g2 & n2 + 1 <= len g2 ;
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume v in the carrier' of G2 & u in the carrier' of G1 ;
y = Re y + ( Im y ) * i ;
( ( - 1 ) |^ ( p -' 1 ) ) mod p = 1 ;
x2 is_differentiable_on ]. a , b .[ & ( for x st x in ]. a , b .[ holds x = a ) implies ( for x st x in ]. a , b .[ holds ( x = a ) & ( x = b )
rng M5 c= rng D2 & rng M5 c= rng ( D2 | Seg n ) ;
for p being Real st p in Z holds p >= a
( for x being Point of X holds f . x = proj1 * f . x ) implies f is continuous
( seq ^\ m ) . k <> 0 & ( seq ^\ m ) . k <> 0 ;
s . ( G . ( k + 1 ) ) > x0 ;
( p -Path ( M ) ) . 2 = d ;
A ++ ( B \ominus C ) = ( A ++ B ) \ominus C
h \equiv gg . ( mod P ) , g . ( 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 ;
mode Subspace of 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 ;
len } in Seg ( len f2 ) & len ( f1 | k ) = len f1 ;
pp1 c= the topology of T & pp2 c= the topology of T ;
]. r , s .] c= [. r , s .] ;
let B2 be Basis of T2 , f be Function of T2 , T2 ;
G * ( B * A ) = ( B * o1 ) * A ;
assume that p , u , u , v is_collinear and u , v , v , w be Element of V ;
[ z , z ] in union rng ( F | X ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S , y = $1 .. S , z = $1 .. S ;
LIN a1 , a3 , b1 & LIN a1 , a3 , b1 ;
f " ( f .: x ) = { x } ;
dom w2 = dom r12 & dom r12 = dom r12 ;
assume that 1 <= i and i <= n and j <= n ;
( ( ( g2 ) . O ) `2 ) ^2 <= 1 ;
p in LSeg ( E . i , F . i ) ;
Iu * ( i , j ) = 0. K ;
|. f . ( s . m ) - g .| < g1 ;
q9 . x in rng ( q | ( Seg n ) ) ;
Carrier ( LLet ) misses Carrier ( L7 ) & Carrier ( L7 ) misses Carrier ( L7 ) ;
consider c being element such that [ a , c ] in G ;
assume Nthesis = o/. ( o, o ) & Ny = oy ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= ( F |^ CZ ) " { 0 } ;
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [. f . j , f . ( j + 1 ) .] ;
pred 0 <= x & x <= 1 means : Def2 : x ^2 <= x ;
p `2 - q `2 <> 0. TOP-REAL 2 & p `2 - q `2 = 0. TOP-REAL 2 ;
redefine func \cal a] ( S , T ) -> non empty set ;
let x be Element of [: S , T :] ;
( the non empty \hbox { - } Map } ) is one-to-one ;
|. i .| <= - - ( 2 to_power n ) & - ( 2 to_power n ) <= - ( 2 to_power n ) ;
the carrier of I[01] = dom P & the carrier of I[01] = the carrier of I[01] ;
} * ( n + 1 ) ! > 0 * } ;
S c= ( A1 /\ A2 ) /\ A3 & S c= ( A2 /\ A3 ) /\ A3 ;
a3 , a4 // b3 , b3 or a3 , a4 // b3 , b3 ;
then dom A <> {} & dom A <> {} & dom A <> {} ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , Y , G2 & x = y implies x = y
set v2 = ( v /. ( i + 1 ) ) `1 ;
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] & rng g = the carrier of I[01] ;
p in Upper_Arc ( P ) /\ Lower_Arc ( P ) & p in Upper_Arc ( P ) ;
dom d2 = [: A2 , A2 :] & dom d2 = [: A2 , A2 :] ;
0 < p / ( ||. z .|| + 1 ) ;
e . ( m3 + 1 ) <= e . m3 ;
B \ominus X \/ B \ominus Y c= B \ominus X /\ Y
-infty < Integral ( M , Im ( g | B ) ) ;
cluster O := F -> \HM { \HM { is } \HM { o } } -valued for operation of X ;
let U1 , U2 be non-empty MSAlgebra over S , F be ManySortedFunction of U1 , U2 ;
Proj ( i , n ) * g is_differentiable_on X & Proj ( i , n ) * g is_differentiable_on X ;
let x , y , z be Point of X , p be Point of X ;
reconsider pp = p . x , pp = p . y as Subset of V ;
x in the carrier of Lin ( A ) & x in the carrier of Lin ( A ) ;
let I , J be parahalting Program of SCM+FSA , a be Int-Location ;
assume that - a is lower and - a is Element of - X ;
Int Cl A c= Cl Int Cl A & Int Cl A c= Cl Int Cl A ;
assume for A being Subset of X holds Cl A = A ;
assume q in Ball ( |[ x , y ]| , r ) ;
p2 `2 <= p `2 & p `2 <= p2 `2 or p2 `2 >= p `2 & p2 `2 <= p `2 ;
Cl ( Q ` ) = [#] ( TQ ) .= [#] ( TQ ) ;
set S = the carrier of T , T = the carrier of T ;
set I8 = for f be FinSequence of TOP-REAL n st f is one-to-one holds f is one-to-one ;
len p - n = len ( p - n ) .= len p ;
A is Permutation of Swap ( A , x , y ) ;
reconsider n6 = ni - 1 as Element of NAT ;
1 <= j + 1 & j + 1 <= len ( s | k ) ;
let q\subseteq , q\lbrace q , r } be Subset of M ;
a9 in the carrier of S1 & b9 in the carrier of S1 ;
c1 /. n1 = c1 . n1 & c2 /. n1 = c1 . n1 ;
let f be FinSequence of TOP-REAL 2 , p be Point of TOP-REAL 2 , r be Real ;
y = ( ( f * S8 ) * S8 ) . x ;
consider x being element such that x in be be element such that x in be } ;
assume r in ( dist ( o ) ) .: P ;
set i2 = ( n , h ) `1 , i1 = ( n , h ) `1 , i2 = ( n , h ) `1 , i2 = ( n , h ) `1 , i2 = ( n , h ) `1 , i2 = ( n
h2 . ( j + 1 ) in rng h2 & h2 . ( j + 1 ) in rng h2 ;
Line ( M29 , k ) = M . i .= Line ( M29 , k ) ;
reconsider m = ( x - 1 ) / 2 as Element of ExtREAL ;
let U1 , U2 be strict Subspace of U0 , a be Element of U1 ;
set P = Line ( a , d ) ;
len p1 < len p2 + 1 & len p1 + 1 < len p2 ;
let T1 , T2 be Scott topological as Scott topological as of L , T be Scott Function of L , T ;
then x <= y & ( x <= y implies x in : y in { x } ) ;
set M = n -\in ( Seg m ) ;
reconsider i = x1 , j = x2 as Nat , k = k - 1 as Nat ;
rng ( ( the_arity_of o ) /. i ) c= dom ( H . i ) ;
z1 " = z9 " & z2 " = z2 " & z1 = z2 " ;
x0 - r / 2 in L /\ dom f & x0 - r / 2 in dom f ;
then w is that rng w /\ L <> {} & rng w /\ L <> {} ;
set x9 = xx ^ <* Z *> , y9 = xx ^ <* Z *> , z9 = xx ^ <* Z *> , x9 = x ^ y , y9 = y ^ z , y9 = z ^ x , z9 = z ^ y , y9 = z ^ y , z9
len w1 in Seg ( len w1 ) & len w1 = len w2 ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of PFuncs ( V , { k } ) ;
x . n = ( |. a . n .| ) * ( 1 / ( n + 1 ) ) ;
p `1 <= Gik `1 & p `1 <= G * ( 1 , 1 ) `1 ;
rng ( g | 1 ) c= L~ ( g | 1 ) & rng ( g | 1 ) c= rng ( g | 1 ) ;
reconsider k = i-1 * ( l + j ) as Nat ;
for n being Nat holds F . n is \HM { -infty } ;
reconsider x9 = x9 , y9 = y9 as VECTOR of M ;
dom ( f | X ) = X /\ dom f .= X ;
p , a // p , c & b , a // c , c ;
reconsider x1 = x , y1 = y as Element of REAL m ;
assume i in dom ( a * p ^ q ) ;
m . ag = p . ag .= p . cg ;
a to_power ( s . m - n ) / Q <= 1 ;
S . ( n + k + 1 ) c= S . ( n + k ) ;
assume that B1 \/ C1 = B2 \/ C2 and B2 \/ C2 = C2 \/ C1 ;
X . i = { x1 , x2 } . i .= { x1 , x2 } . i ;
r2 in dom ( h1 + h2 ) & r1 in dom ( h1 + h2 ) ;
- 0. R = a & b-0 = b ;
( F is closed & Q [ t ] ) & ( F is closed implies Q [ t ] ) ;
set T = the { X , \rm such that X = { x0 } ;
Int Cl Int R c= Int R & Int Cl R c= Cl Int R ;
consider y being Element of L such that c . y = x ;
rng ( F . x ) = { F . x } & rng ( F . x ) = { F . x } ;
G-23 ( { c } ) \/ G and B c= B \/ S ;
fN is_X ( ) , X ( ) * ;
set RQ = the Point of P , RQ = the carrier of Q ;
assume that n + 1 >= 1 and n + 1 <= len M ;
let k2 be Element of NAT , k be Nat ;
reconsider pc/. i = u , ppi = v as Element of ( TOP-REAL n ) | ( i + 1 ) ;
g . x in dom f & x in dom g implies g . x = h . x
assume that 1 <= n and n + 1 <= len f1 ;
reconsider T = b * N as Element of carr ( G ) ;
len PM <= len PM & len PM <= len PM ;
x " in the carrier of A1 & x " in the carrier of A2 ;
[ i , j ] in Indices ( A * ( i , j ) ) ;
for m being Nat holds Re ( F . m ) is simple ;
f . x = a . i .= a1 . k .= a1 . k ;
let f be PartFunc of REAL i , REAL n , x0 be Element of REAL m ;
rng f = the carrier of ( ( Carrier A ) * ( i , j ) ) ;
assume s1 = sqrt ( 2 - p ) / ( 2 |^ ( 2 |^ ( 2 |^ ( 2 |^ ( 2 |^ ( 2 |^ ( 2 |^ ( 2 |^ ( 2 |^ ( 2 |^ ( 2 |^ ( 2 |^ ( 2 |^ ( 2 |^ (
pred a > 1 & b > 0 means : Def2 : a / b > 1 ;
let A , B , C be Subset of IV ;
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 , tt2 be Relation of X , Y ;
Q [ e-14 \/ { vN } , f . v-5 ] , f . eN , f . eN ] ;
g \circlearrowleft W-min L~ z = z implies ( g /. 1 ) .. z < ( g /. len z ) .. z
|. |[ x , v ]| - |[ x , y ]| .| = vLet ;
- f . w = - ( L * w ) .= - ( L * w ) ;
z - y <= x iff z <= x + y & y <= z ;
( 7 / p1 ) to_power ( ( 1 / e ) * e ) > 0 ;
assume X is BCK-algebra of 0 , 0 , 0 , 0 , 0 , 1 ;
F . 1 = v1 & F . 2 = v2 & F . 3 = v2 ;
( f | X ) . x2 = f . x2 & ( f | X ) . x2 = f . x2 ;
( ( ( 1 / 2 ) (#) sec ) `| Z ) . x = f . x ;
i2 = ( f /. len f ) `1 .= ( f /. len f ) `1 ;
X1 = X2 \/ ( X1 \ X2 ) & X2 = X1 \/ X2 ;
[. a , b , 1_ G .] = 1_ G & 1_ G = 1_ G ;
let V , W be non empty VectSpStr over F_Complex , f be Function of V , COMPLEX ;
dom g2 = the carrier of I[01] & rng g2 c= the carrier of I[01] ;
dom f2 = the carrier of I[01] & dom g2 = the carrier of I[01] & rng g2 c= the carrier of I[01] ;
( proj2 | X ) .: X = proj2 .: X .= proj2 .: X .= X ;
f . ( x , y ) = h1 . ( x `2 , y `2 ) ;
x0 - r / ( n + 1 ) < a1 . n ;
|. ( f /* s ) . k - GM .| < r ;
len Line ( A , i ) = width A .= width A ;
SFinSequence .: S = ( S . g ) .: S .= S .: S ;
reconsider f = v + u as Function of X , the carrier of Y ;
intloc 0 in dom Initialized p & intloc 1 in dom Initialized p ;
i1 := i2 := i3 does not contradiction & i1 does not destroy b1 implies i1 does not contradiction
arccos r + arccos r = PI / 2 + 0 ;
for x st x in Z holds f2 is_differentiable_in x & for x st x in Z holds f2 . x = 1 / ( x + a )
reconsider 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 ) ) = TRUE ;
dist ( ( a * seq ) . n , h ) < r ;
1 in the carrier of [. 0 , 1 .] & 1 in the carrier of I[01] ;
p2 `1 - x1 > - g / 2 & p2 `1 - g / 2 > - g / 2 ;
|. r1 - thesis .| = |. a1 .| * |. thesis .| .= |. - 1 .| ;
reconsider S-14 k = 8 as Element of Seg 8 ( ) ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b
D0W .D] = D0W .2 + 1 ;
i1 = ma + n & i2 = K + n implies i1 = i2
f . a [= f . ( f . O1 "\/" a ) ;
pred f = v & g = u , f + g ;
I . n = Integral ( M , F . n ) ;
chi ( T1 , S ) . s = 1 & chi ( T1 , 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 or L~ M1 meets L~ R4 & L~ M1 meets L~ R4 ;
set h = the continuous Function of X , R , g = the carrier of X ;
set A = { L . ( k9 . n ) where k is Nat : k <= n } ;
for H st H is atomic holds P7 [ H ] ;
set bA = S5 ^\ ( iA + i ) , SA = S5 ^\ ( i + j ) ;
Hom ( a , b ) c= Hom ( a opp , b opp ) ;
1 / ( n + 1 ) < 1 / ( s " ) ;
l `1 = [ dom l , cod l ] `1 .= [ dom l , cod l ] `1 .= dom l ;
y +* ( i , y /. i ) in dom g & y +* ( i , y /. i ) in dom g ;
let p be Element of CQC-WFF ( Al ) , x be Element of D , y be Element of D ;
X /\ X1 c= dom ( f1 - f2 ) /\ dom ( f2 - f3 ) ;
p2 in rng ( f /^ 1 ) & p2 in rng ( f /^ 1 ) ;
1 <= indx ( D2 , D1 , j1 ) & 1 <= indx ( D2 , D1 , j1 ) ;
assume x in ( ( ( K /\ K0 ) \/ ( K /\ K0 ) ) /\ K0 ;
- 1 <= ( ( f2 ) . O ) `2 & - 1 <= ( ( f2 ) . I ) `2 ;
let f , g be Function of I[01] , ( TOP-REAL 2 ) | P , p1 , p2 be Point of TOP-REAL 2 ;
k1 -' k2 = k1 - k2 .= k1 - k2 .= k1 -' k2 ;
rng seq c= ]. x0 , x0 + r .[ & rng seq c= ]. x0 , x0 + r .[ ;
g2 in ]. x0 , x0 + r .[ & g2 in ]. x0 , x0 + r .[ ;
sgn ( p `1 , K ) = - 1_ K & sgn ( p `2 , K ) = - 1_ K ;
consider u being Nat such that b = p |^ y * u ;
ex A being subset of T st a = Sum A ;
Cl ( union HF ) = union ( ( Cl H ) \/ ( Cl H ) ) ;
len t = len t1 + len t2 .= len t1 + len t2 .= len t1 + 1 ;
v-29 = v + w |-- v + A8 & v-29 = v + A8 ;
v ( ) <> DataLoc ( t0 . GBP , 3 ) & v ( ) . a = 0 ;
g . s = sup ( d " { s } ) & g . s = sup ( d " { s } ) ;
( \dot y ) . s = s . ( \dot y ) ;
{ s : s < t } in INT implies t = {}
s ` \ s = s ` \ 0. X & s ` \ s = 0. X ;
defpred P [ Nat ] means B + $1 in A & B + $1 in A ;
( 339 + 1 ) ! = 335 ! * ( 339 + 1 ) ;
1. ( A , succ A ) = 1. ( A , A ) .= 1. ( A , A ) ;
reconsider y = y as Element of COMPLEX * ( len y ) ;
consider i2 being Integer such that y0 = p * i2 and i2 <= n ;
reconsider p = Y | ( Seg k ) as FinSequence of NAT , the carrier of K ;
set f = ( S , U ) \mathop { \rm \hbox { - } of \rm that f is ( S , U ) u ;
consider Z being set such that lim s in Z and Z in F ;
let f be Function of I[01] , ( TOP-REAL n ) | P , p1 , p2 be Point of TOP-REAL n ;
( ( 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 , a be Real , b be Real ;
reconsider l = 0. ( { 0. V } ) as Linear_Combination of A ;
set r = |. e .| + |. n .| + |. w .| + a ;
consider y being Element of S such that z <= y and y in X ;
a is being being being being being being being being being being set holds 'not' ( b 'or' c ) = 'not' ( ( a 'or' b ) 'or' c )
||. x9 - gg .|| < r2 & ||. x1 - g .|| < r2 ;
b9 , a9 // b9 , c9 & b9 , c9 // c9 , a9 ;
1 <= k2 -' k1 & k1 + 1 = k2 or 1 + 1 = k2 + 1 ;
( ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) ^2 >= 0 ;
( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 < 0 ;
E-max C in right_cell ( Rv , 1 ) & E-max C in L~ Rv ;
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 , a // a , b or p , a // b , a ;
g . n = a * Sum ff1 .= f . n ;
consider f being Subset of X such that e = f and f is strict ;
F | ( N2 , S ) = CircleMap * ( F | N2 ) .= CircleMap * ( F | N2 ) ;
q in LSeg ( q , v ) \/ LSeg ( v , p ) ;
Ball ( m , r0 ) c= Ball ( m , s ) & Ball ( m , r0 ) c= Ball ( m , r ) ;
the carrier of (0). V = { 0. V } & 0. V in { 0. V } ;
rng ( cos | [. - 1 , 1 .] ) = [. - 1 , 1 .] ;
assume that Re seq is summable and Im seq is summable and Im seq is summable and Im seq is summable ;
||. ( vseq . n ) - ( vseq . m ) .|| < e / ( ||. vseq . n .|| + e / ( ||. vseq . m .|| + e ) ) ;
set g = O --> 1 ;
reconsider t2 = t11 as 0 -started string of S2 , the carrier of S2 ;
reconsider x9 = seq . n , y9 = seq . n as sequence of REAL n ;
assume that C meets C and C meets L~ go and not E-max C in L~ go /\ L~ pion1 ;
- ( Partial_Sums ( 1 ) ) . n < F . n - x ;
set d1 = being non empty thesis , c = dist ( x1 , z1 ) , d2 = dist ( x2 , z2 ) ;
2 |^ ( ( q -' 1 ) div 2 ) = 2 |^ ( ( q -' 1 ) div 2 ) ;
dom vk2 = Seg ( len d6 ) .= Seg ( len d6 ) .= dom v6 ;
set x1 = - k2 + |. k2 .| , y1 = - k2 + 1 , y2 = - 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 ( LX + L2 ) ) c= I2 & the carrier of ( Carrier ( LX + 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 normal \hbox { {} } ;
Z c= dom ( ( - 1 ) (#) ( ( id Z ) ^ ) ) ;
|. 0. TOP-REAL 2 - ( q `1 / |. q .| - cn ) .| < r ;
A \ \bf \ ( B , L ) c= A & A c= A ;
E = dom Carrier ( L ) & Carrier ( L ) is_measurable_on E implies ( L (#) ( L (#) ( E | 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 ss2 = P . IC ss2 .= ( I +* J ) . IC ss2 ;
pred x > 0 means : Def2 : 1 / x = x to_power ( - 1 ) ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) ;
consider p being Point of T such that C = [. p , R .] and p in C ;
b , c are_connected & - C , - C + - C + ( - C , - C + - C + - C + ( - C , - C ) ) are_connected ;
assume that f = id the carrier of OL and g = id the carrier of OL ;
consider v such that v <> 0. V and f . v = L * v ;
let l be ] Linear_Combination of {} ( ( the carrier of V ) --> { 0. V } ) ;
reconsider g = f " as Function of U2 , U1 , ( the carrier of U1 ) --> { {} } ;
A1 in the carrier of G_ ( k , X ) & A2 in the carrier of G ;
|. - x .| = - - ( - x ) .= x - x .= - x ;
set S = is is is is is is is ;
( Fib n ) * ( 5 * ( 5 * ( 5 / ( n + 1 ) ) ) >= 4 * be Element of NAT ;
xseq /. ( k + 1 ) = ( v . ( k + 1 ) ) ;
0 mod i = - ( i * ( 0 qua Nat ) ) .= - ( i mod i ) ;
Indices M1 = [: Seg n , Seg n :] & Indices M1 = [: Seg n , Seg n :] ;
Line ( S, j ) = SFinSequence . j .= SLine ( Sm , j ) ;
h . ( x1 , y1 ) = [ y1 , x1 ] & h . ( x2 , y2 ) = [ y2 , x2 ] ;
|. f .| - Re ( |. f .| * ( card 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 IExec ( I , P , s ) , P ;
DataLoc ( t3 . a , 4 ) = intpos ( 0 + 4 ) .= 0 ;
x + y < - x + y & |. x .| = - x + y ;
LIN c , q , b & LIN c , q , c & LIN c , q , c ;
f, t . ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + z1 ) .= x1 + ( y1 + z1 ) ;
flet f1 . a = f{ a } & v in InputVertices S & [ v , f1 . a ] in InputVertices S ;
p `1 <= ( E-max C ) `1 & ( E-max C ) `1 <= ( E-max C ) `1 ;
set R8 = Cage ( C , n ) :- E7 , R8 = Cage ( C , n ) ;
p `1 >= ( E-max C ) `1 & p `2 >= ( E-max C ) `2 ;
consider p such that p = pp and s1 < p /. i and p in L~ f and f /. i = f /. ( i + 1 ) ;
|. ( f /* ( s * F ) ) . l - GM .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len Line ( N , k + 1 + 1 ) = width N ;
f1 /* s1 is convergent & f2 /* s1 is convergent & lim ( f1 /* s1 ) = lim ( f2 , x0 ) ;
f . x1 = x1 & f . y1 = y1 & f . y2 = y2 ;
len f <= len f + 1 & len f + 1 <> 0 implies f . ( len f + 1 ) = f . ( len f + 1 )
dom ( Proj ( i , n ) * s ) = REAL m .= dom ( ( Proj ( i , n ) * s ) ) ;
n = k * ( 2 * t ) + ( n mod ( 2 * k ) ) ;
dom B = 2 -tuples_on the carrier of V & rng B c= the carrier of V ;
consider r such that r _|_ a and r _|_ x and r _|_ y ;
reconsider B1 = the carrier of Y1 , B2 = the carrier of Y2 as Subset of X ;
1 in the carrier of [. 1 / 2 , 1 .] & 1 / 2 in the carrier of I[01] ;
for L being complete LATTICE for A being Subset of L holds <* \HM { A } , L *> , L are_isomorphic
[ gi , gj ] in [: I , I :] \ [: I , I :] ;
set S2 = 1GateCircStr ( x , y , c ) ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and for r st r in dom ( f2 * f1 ) holds ( f2 * f1 ) . r = 0 ;
reconsider y = ( a ` ) * ( ( a ` ) * ( a ` ) ) 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 non empty set with G -\mathbin { v } , { v } = the carrier of G , { v } , { v } , { v } , { v } } ;
reconsider g = f as PartFunc of REAL , REAL-NS n , REAL-NS n , REAL-NS n ;
|. s1 . m / p .| < d / ( p |^ ( p -' 1 ) ) ;
for x being element st x in ( for u being element st u in ( ( the carrier of L ) \ { t } ) holds x in ( ( the carrier of L ) \ { t } )
P = the carrier of ( ( TOP-REAL n ) | Px0 ) & Q = the carrier of ( ( TOP-REAL n ) | Px0 ) ;
assume that p00 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) and p2 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ;
( 0. X \ x ) |^ ( m * ( k + 1 ) ) = 0. X ;
let g be Element of Hom ( cod f , \square ) , f be Morphism of C ;
2 * a * b + ( 2 * c * d ) <= 2 * C1 * C2 ;
let f , g , h be Point of the complex normed space of bounded Function of X , Y , h be Function of X , Y ;
set h = Hom ( a , g (*) f ) ;
then idseq n | ( Seg m ) = idseq m & m <= n implies m <= n & n <= m ;
H * ( g " * a ) in the right of H & g * ( g " * a ) in the carrier of H ;
x in dom ( ( - cos * sin ) `| Z ) & x - x0 in dom ( ( - cos * sin ) `| Z ) ;
cell ( G , i1 , j2 -' 1 ) misses C & cell ( G , i1 , j2 -' 1 ) misses C ;
LE q2 , p4 , P , p1 , p2 & LE q2 , p2 , P , p1 , p2 ;
attr B is an component of A means : Def2 : B c= BDD A ;
deffunc D ( set , set ) = union rng $1 & $2 = union rng $2 & $2 = union rng $1 ;
n + - n < len ( p + - n ) + ( - n ) ;
attr a <> 0. K means : Def2 : the_rank_of M = the_rank_of ( a * M ) ;
consider j such that j in dom /\ /\ dom /\ I and I = len ( l + j ) ;
consider x1 such that z in x1 and x1 in PA and not [ x1 , x2 ] in R ;
for n ex r being Element of REAL st X [ n , r ] & r <= n ;
set Cs1 = Comput ( P2 , s2 , i + 1 ) , Cs2 = Comput ( P2 , s2 , i + 1 ) , Cs2 = P2 ;
set cv = 3 / ( 2 * ( a , b , c ) ) , cv = 3 / ( 2 * ( a , b , c ) ) ;
conv @ W c= union ( F .: ( E " W ) ) & conv @ W c= union ( F .: ( E " W ) ) ;
1 in [. - 1 , 1 .] /\ dom ( arccot * ( arccot ) ) & 1 in [. - 1 , 1 .] ;
r3 <= s0 + ( r0 / ( |. v2 - v1 .| ) ) / ( |. v2 - v1 .| ) ;
dom ( f (#) f4 ) = dom f /\ dom f4 .= dom ( f (#) f4 ) .= dom f /\ dom f4 ;
dom ( f (#) G ) = dom ( l (#) F ) /\ Seg k .= dom ( l (#) F ) ;
rng ( s ^\ k ) c= dom f1 \ { x0 } & rng ( s ^\ k ) c= dom f2 \ { x0 } ;
reconsider gg = gp , gp = gp as Point of ( TOP-REAL n1 ) | ( L~ g ) ;
( T * h . s ) . x = T . ( h . s . x ) ;
I . ( L . J . x ) = ( I * L ) . ( J . x ) ;
y in dom ( the mapping of ( F . o ) ) & ( ( Frege ( A . o ) ) . y = ( ( the mapping of A ) . y ) ;
for I being non degenerated commutative 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 . ( lpp . i ) = ( v *' lpp ) . i .= v . i ;
consider n be element such that n in NAT and x = ( sn " ) . n and f . n = ( sn " ) . n ;
consider x being Element of c such that F1 . x <> F2 . x and x <> {} ;
card Funcs ( X , 0 , x1 , x2 ) = { Ed } & card X = k ;
j + ( 2 * k9 ) + m1 > j + ( 2 * k9 ) + m1 ;
{ s , t } on A3 & { s , t } on B2 & { s , t } on A3 ;
n1 > len crossover ( p2 , p1 , n1 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n4 , a9 ) ;
mg . HT ( mg , T ) = 0. L .= 0. L ;
then H1 , H2 are_isomorphic implies ( H , H1 ) / ( 2 , n ) / ( 2 , n ) is \emptyset ;
( N-min L~ f ) .. ( f | ( L~ f ) ) > 1 & ( N-min L~ f ) .. ( f | ( len f ) ) > 1 ;
]. s , 1 .] = ]. s , 2 .] /\ [. 0 , 1 .] .= ]. s , 1 .] ;
x1 in [#] ( ( ( TOP-REAL 2 ) | L~ g ) | L~ g ) & x2 in [#] ( ( TOP-REAL 2 ) | L~ g ) ;
let f1 , f2 be continuous PartFunc of REAL , the carrier of S , x be Point of S ;
DigA ( t-23 , z9 ) is Element of k -tuples_on ( the carrier of K ) ;
I is d 2\mathop = ddom ( d\lbrace k2 } ) & I is k2 -\mathbin I " ;
[: { u } , { u9 } :] = { [ a , u9 ] } & [: { u } , { u9 } :] = { [ a , u9 ] } ;
( w | p ) | ( p | ( w | w ) ) = p ;
consider u2 such that u2 in W2 and x = v + u2 and u2 in W2 and u1 = v + u2 ;
for y st y in rng F ex n st y = a |^ n & a |^ n = y
dom ( ( g * ( ( g } \dot \to C ) | K ) ) = K ;
ex x being element st x in ( ( ( U0 ) \/ A ) . s ) & x in ( ( ( U0 ) \/ A ) . s ) ;
ex x being element st x in ( ( ( the carrier of OS ) \/ A ) . s ;
f . x in the carrier of [. - r , r .] & f . x in [. - r , r .] ;
( the carrier of X1 union X2 ) /\ ( ( the carrier of X1 ) /\ ( the carrier of X2 ) ) <> {} ;
L1 /\ LSeg ( p00 , p2 ) c= { p01 } & LSeg ( p1 , p2 ) /\ LSeg ( p01 , p2 ) c= { p01 } ;
( b + ( bs ) ) / 2 in { r : a < r & r < b } ;
ex_sup_of { x , y } , L & x "\/" y = sup { x , y } ;
for x being element st x in X ex u being element st P [ x , u ]
consider z being Point of GX such that z = y and P [ z ] and z in A ;
( the 0. of ( ( the carrier of ( ( the carrier of X ) ) ) ) . ( x - y ) <= e ;
len ( w ^ w2 ) + 1 = len w + 2 + 1 .= len w + 1 ;
assume q in the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 & q in the carrier of ( ( TOP-REAL 2 ) | K1 ) ;
f | EK1 ` = g | EK1 ` .= g | EK1 ` .= g | EK1 ` .= g | EK1 ;
reconsider i1 = x1 , i2 = x2 , j1 = y2 as Element of NAT , i be Element of NAT ;
( a * A * B ) ` = ( a * ( A * B ) ) ` ;
assume ex n0 being Element of NAT st f to_power n0 is + 1 & f . n0 = n0 ;
Seg len ( ( ( len f2 ) | ( len f1 ) ) ) = dom ( ( len f2 ) | ( len f1 ) ) ;
( Complement ( A ) ) . m c= ( ( Complement ( A ) ) . n ) . ( ( Complement ( A ) ) . m ) ;
f1 . p = p9 & g1 . ( p9 . p9 ) = d & g1 . ( p9 . p9 ) = d ;
FinS ( F , Y ) = FinS ( F , dom ( F | Y ) ) .= F | Y ;
( x | y ) | z = z | ( y | x ) ;
( |. x .| to_power n ) / ( ( r |^ n ) * ( ( r |^ n ) / ( ( r |^ n ) * ( r |^ n ) ) ) <= r ;
Sum ( F-12 ) = Sum f & dom F-12 = dom g & for i be Nat st i in dom Fc holds Fc . i = F ( i ) ;
assume for x , y being set st x in Y & y in Y holds x /\ y in Y ;
assume that W1 is Subspace of W3 and W2 is Subspace of W3 and W1 is Subspace of W3 and W2 is Subspace of W3 ;
||. ( t-15 . x ) .|| = lim ||. ( x9 - y9 ) .|| .= ||. ( x9 - y9 ) .|| .= ||. ( x9 - y9 ) .|| ;
assume that i in dom D and f | A is lower and g | A is lower and g | A is lower ;
( ( p `2 ) ^2 - 1 ) <= ( ( - ( - ( p `2 / |. p .| - sn ) ) / ( 1 + sn ) ) ^2 ;
g | Sphere ( p , r ) = id ( Sphere ( p , r ) ) .= id ( the carrier of TOP-REAL n ) ;
set N8 = N-min L~ Cage ( C , n ) , N7 = N-min L~ Cage ( C , n ) , N8 = N-min L~ Cage ( C , n ) ;
for T being non empty TopSpace holds T is countable countable implies the TopStruct of T is countable countable
width B |-> 0. K = Line ( B , i ) .= B * ( i , i ) .= B * ( i , j ) ;
attr a <> 0 means for A , B being Real holds ( A \+\ B ) \+\ a = ( A ** a ) \+\ ( B ** a ) ;
then f is_is_\mathbin is_is_or pdiff1 ( f , 1 ) & pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 ;
assume that a > 0 and a <> 1 and b > 0 and b <> 0 and c > 0 ;
w1 , w2 in Lin { w1 , w2 } & w2 in Lin { w2 , w1 } ;
p2 /. IC s-7 = p2 . IC s-7 .= p2 . IC s-7 .= p2 . IC s-7 ;
ind ( T-10 | b ) = ind b .= ind B .= ind B .= ind ( T-10 | b ) ;
[ a , A ] in the carrier of Line ( A9 , 1 ) & [ a , A ] in the carrier of Line ( A9 , 1 ) ;
m in ( the Arrows of C ) . ( o1 , o2 ) & m in ( the Arrows of C ) . ( o2 , o1 ) ;
( ( a , CompF ( PA , G ) ) . z ) = FALSE & ( a , CompF ( PA , G ) ) . z = FALSE ;
reconsider phi = phi /. 11 , phi = phi /. 11 , phi = phi /. 12 as Element of ( S , X ) * ;
len s1 - 1 * ( len s2 - 1 ) + 1 > 0 + 1 ;
delta ( D ) * ( f . ( upper_bound A ) - f . ( lower_bound A ) ) < r ;
[ f21 , f21 ] in [: the carrier' of A , the carrier' of A :] ;
the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 = K1 & the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 = K1 ;
consider z being element such that z in dom g2 and p = g2 . z and g2 . z = y ;
[#] V1 = { 0. V1 } .= the carrier of (0). V1 .= { 0. V1 } .= { 0. V1 } ;
consider P2 be FinSequence such that rng P2 = M and P2 is one-to-one and P2 is one-to-one and P2 is one-to-one ;
assume that x1 in dom ( f | X ) and ||. x1 - x0 .|| < s and ||. x1 - x0 .|| < s ;
h1 = f ^ ( <* p3 *> ^ <* p *> ) .= h ^ <* p *> .= h ^ <* p *> ;
c /. |[ b , c ]| = c .= c /. |[ a , c ]| .= c /. |[ a , c ]| ;
reconsider t1 = p1 , t2 = p2 , t2 = p3 as term of C , V , f be Function of C , V ;
1 / 2 in the carrier of [. 1 / 2 , 1 .] & 1 / 2 in the carrier of [. 1 / 2 , 1 .] ;
ex W being Subset of X st p in W & W is open & h .: W c= V ;
( h . p1 ) `2 = C * p1 `2 + D * p1 `2 .= C * p1 `2 + D * p1 `2 ;
R . b ` = 2 * - a-b .= 2 * - a-b .= a-b ;
consider x1 such that B = ( - 1 ) * x1 + ( - 1 ) * A and 0 <= x1 & x1 <= 1 ;
dom g = dom ( ( the Sorts of A ) * a9 ) & dom ( ( the Sorts of A ) * b9 ) = dom ( ( the Sorts of A ) * b9 ) ;
[ P . U6 , P . l6 ] in ( the carrier of Tk ) \ { {} } ;
set s2 = Initialize s , P2 = P +* I ;
reconsider M = mid ( z , i2 , i1 ) as non empty compact Subset of ( TOP-REAL 2 ) | ( L~ z ) ;
y in product ( ( Carrier J ) +* ( V , { 1 } ) ) ;
1 / ( |[ 0 , 1 ]| ) = 1 & 0 / ( |[ 0 , 1 ]| ) = 0 ;
assume x in the left of g or x in the left of g & x in the carrier of g ;
consider M be strict Subspace of Aex T be strict Subspace of M st a = M & T is strict & the carrier of T = M ;
for x st x in Z holds ( ( ( 1 / 2 ) (#) f ) `| Z ) . x <> 0
len W1 + len W2 + m = 1 + len W3 + m .= len W3 + len W3 + m .= len W3 + 1 ;
reconsider h1 = ( vseq . n ) - t-16 as Lipschitzian LinearOperator of X , Y ;
( - ( i mod len ( p + q ) ) + 1 ) in dom ( p + q ) ;
assume that s2 is_or s1 in the carrier of s2 and F in the carrier of s2 and F c= the carrier of s2 ;
( ( ( ( ( ( ( ( ( m , y ) , 3 ) ) , 1 ) ) , 1 ) ) , 1 ) ) * ( ( ( ( ( ( ( ( m , y ) , 3 ) ) , 1 ) ) * ( ( ( ( m , y ) , 3 ) ) * ( ( ( m , y ) , 3 ) )
for u being element st u in Bags n holds ( p `2 + m ) . u = p . u
for B be Subset of u-5 st B in E holds A = B or A misses B ;
ex a being Point of X st a in A & A /\ Cl { y } = { a } ;
set W2 = tree ( p ) \/ W1 , W3 = p ^ q , W2 = p ^ q , W3 = p ^ q , W2 = q ^ r , W3 = p ^ r , W2 = q ^ r , W3 = p ^ r , W3 = q ^ r , W3 = p ^ r , U = q ^ r , E = p ^ r , F
x in { X where X is Ideal of L : X is Ideal of L } ;
the carrier of W1 /\ W2 c= the carrier of W1 & the carrier of W1 /\ W2 c= the carrier of W2 ;
( ( a + b ) (#) id a ) * ( id a ) = ( a + b ) (#) ( id a ) ;
( ( X --> f ) . x ) = ( X --> dom f ) . x .= f . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) ;
p => ( q => r ) => ( p => q => ( p => r ) ) in TAUT ( A ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
- 1 + 1 <= ( ( i - 2 ) / ( n - m ) ) + 1 ;
( reproj ( 1 , z0 ) ) . x in dom ( f1 (#) f2 ) & ( reproj ( 1 , z0 ) ) . x in dom ( f1 (#) f2 ) ;
assume that b1 . r = { c1 } and b2 . r = { c2 } and b1 . r = c1 . c2 ;
ex P st a1 on P & a2 on P & a3 on P & a1 , a2 on P & a3 , a2 on P ;
reconsider gf = g * f opp , hg = h * g opp as strict Element of X ;
consider v1 being Element of T such that Q = ( downarrow v1 ) ` and v1 in V & v1 in V ;
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 >= cn * |. p .| & p <> 0. TOP-REAL 2 } ;
( A , succ O1 ) = ( ( A , O1 ) --> ( A , O2 ) ) * ( A , O1 ) ;
set Is1 = in in Macro SubFrom ( a , intloc 0 ) , Is2 = SubFrom ( a , intloc 0 ) , Is2 = SubFrom ( a , intloc 0 ) , Is2 = goto 2 , Is2 = goto 3 , Is2 = goto 3 , Is2 = goto 4 , Is2 = goto 4 ;
for i be Nat st 1 < i & i < len z holds z /. i <> z /. 1 ;
X c= ( [: the carrier of L1 , the carrier of L2 :] \/ [: the carrier of L1 , the carrier of L2 :] ) ;
consider x9 be Element of GF ( p ) such that x9 |^ 2 = a and ex x be Element of GF ( p ) st x = a |^ 2 ;
reconsider eM = e4 , fN = fN , fN = fN as Element of D * ;
ex O being set st O in S & C1 c= O & M . O = 0. <= M & M . ( O + 1 ) = 0. ;
consider n being Nat such that for m being Nat st n <= m holds S . m in U1 and S . n in U2 ;
f (#) g * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) . x ) ;
defpred P [ Nat ] means A + succ $1 = succ A + $1 & ( A + ) = A + ( A + $1 ) ;
the left of - g = the left of g & the carrier of - g = the carrier of g ;
reconsider pp = x , pp = y , pp = z , pp = w as Point of Euclid 2 , r be Real ;
consider g3 such that g3 = y and x <= g3 and g3 <= x0 and for x st x in X holds g3 . x = F ( x ) ;
for n being Element of NAT ex r being Element of REAL st X [ n , r ] & r <= n ;
len ( x2 ^ y2 ) = len x2 + len y2 .= len ( x2 ^ y2 ) + len ( y2 ^ y2 ) ;
for x being element st x in X holds x in the set of positive ( n0 , n ) & x in the carrier of n ;
LSeg ( p11 , p2 ) /\ LSeg ( p1 , p11 ) = {} & LSeg ( p1 , p11 ) /\ LSeg ( p11 , p2 ) = {} ;
func of of ) -> set equals [: dom h , rng h :] by : : : : : for x being element st x in dom h holds h . x = ( id X ) . x ;
len ( ( { C /. len C , CC } ) | ( len C -' 1 ) ) <= len CC ;
attr K is with_a , a , 0. K , i be Nat , v be Element of K ;
consider o being OperSymbol of S such that t `2 . {} = [ o , the carrier of S ] and o in ( the carrier of S ) ;
for x st x in X ex y st x c= y & y in X & y is - f /. x
IC Comput ( P-6 , smeans : Def2 : IC Comput ( P5 , s5 , k ) in dom ( P5 ) ;
pred q < s means : Def2 : r < s & ]. r , s .] \not c= ]. p , q .] ;
consider c being Element of Class ( f , c ) such that Y = ( F . c ) `1 and c in X ;
func the ResultSort of S2 -> ( the carrier of S2 ) --> { the carrier of S1 } equals id the carrier of S2 .= the carrier of S1 ;
set y9 = [ <* y , z *> , f2 ] , z9 = [ <* z , x *> , f3 ] ;
assume x in dom ( ( ( 1 / 2 ) (#) ( arccot ) ) `| Z ) & x in dom ( ( 1 / 2 ) (#) ( arccot ) ) ;
r-7 in Int cell ( GoB f , i , GoB f ) \ L~ f & ri2 in Int cell ( GoB f , i , width GoB f ) ;
q `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 & i - len f <= len f + len g - len f ;
for n ex x st x in N & x in N1 & h . n = x- ( x0 - x )
set sx0 = ( ( a , I , p , s ) +* I ) . i , sx0 = ( a , I , p , s ) +* I ;
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 Vd and x9 in V and x9 in V and x9 in V ;
( p ^ ( q | k ) ) . m = ( q | k ) . ( - len p ) .= p . ( - len p ) ;
g + h = gg + hg1 & for X holds holds A1 + ( g , X ) = g + h
L1 is distributive & L2 is distributive implies [: L1 , L2 :] is distributive & [: L1 , L2 :] is distributive & [: L1 , L2 :] is distributive
pred x in rng f & y in rng ( f | x ) means : Def2 : f . x = f . y ;
assume that 1 < p and 1 / p + 1 / q = 1 and 0 <= a and 0 <= b ;
F* ( f , 6 ) = rpoly ( 1 , M ) *' t + 1. L .= 1. L + 1. L .= 0. L ;
for X being set , A being Subset of X holds A ` = {} implies A = X & A = {} or A = {}
( ( N-min X ) `1 <= ( ( \hbox { \boldmath $ p $ } ) `1 ) & ( ( ( the carrier of X ) `1 ) /\ ( ( the carrier of X ) `2 ) ) = {} ;
for c being Element of the - of A , a being Element of the free of A holds c <> a ;
s1 . GBP = ( Exec ( i2 , s2 ) ) . GBP .= s2 . GBP .= 0 ;
for a , b being Real holds |[ a , b ]| in ( y >= 0 ) & b >= 0 implies b >= 0
for x , y being Element of X holds x ` \ y = ( x \ y ) ` & y ` \ x = y ` ;
mode BCK-algebra of i , j , m , n , m , n be BCK-algebra of i , j , m , n , m , n ;
set x2 = |( Re ( y ) , Im ( x ) )| ;
[ y , x ] in dom u5 & u5 . ( y , x ) = g . y ;
]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .[ c= A & upper_bound divset ( D , k ) = upper_bound A ;
0 <= delta ( S2 . n ) & |. delta ( S2 . n ) .| < e / ( 2 |^ ( n + 1 ) ) ;
( - ( q `1 / |. q .| - cn ) ) ^2 <= ( - ( q `2 / |. q .| - cn ) ) ^2 ;
set A = 2 / b-a ;
for x , y being set st x in R" holds x , y are_\hbox { $ ( x , y ) }
deffunc FF2 ( Nat ) = b . $1 * ( M * G ) . $1 & ( M * G ) . $1 = b * ( M * G ) . $1 ;
for s being element holds s in Q ( f 'or' g ) iff s in Q ( f \/ g )
for S being non empty non void non void non empty non void TopSpace st S is H holds S is connected
max ( degree ( ( z `1 ) ^2 + 1 ) , degree ( ( z `2 ) ^2 ) ) >= 0 ;
consider n1 being Nat such that for k holds seq . ( n1 + k ) < r + s and for n being Nat holds seq . ( n + k ) < r + s ;
Lin ( A /\ B ) is Subspace of Lin ( A ) & Lin ( B ) is Subspace of Lin ( B ) ;
set n-15 = n-13 '&' ( M . x qua Element of BOOLEAN ) , n-15 = M . ( n + 1 ) , n-15 = M . ( n + 1 ) , n-15 = M . ( n + 1 ) , n-15 = M . ( n + 1 ) , n-15 = M . ( n + 1 ) , n-15 = M . ( n + 1 ) ;
f " V in ' ( X ) & f " V in D . ( the carrier of X , p ) & f " V in D . ( the carrier of X , p ) ;
rng ( ( a is :] ) +* ( 1 , b ) c= { a , c , b } ;
consider y being & as such that y ` = y and dom y ` = WWW
dom ( ( 1 / f ) (#) ( f `| ]. x0 - r , x0 .[ ) ) c= ]. - r , x0 .[ & ( ( 1 / f ) (#) ( f `| ]. x0 - r , x0 .[ ) ) . x0 in dom ( f `| ]. x0 - r , x0 .[ ) ;
as Morphism of i , j , n , r be Element of non empty Subset of non 1 , a , b , r be Real ;
v ^ ( ( n-3 |-> 0 ) in Lin ( rng ( B-9 | c1 ) ) & v ^ ( ( B-9 | c1 ) ^ ( B-9 | c1 ) ) in Lin ( rng ( B-9 | c1 ) ) ;
ex a , k1 , k2 st i = a /. k1 & k2 = b /. k2 & i <> k2 & i <> k2 ;
t . NAT = ( NAT .--> succ i1 ) . NAT .= succ ( i1 + 1 ) .= succ ( i1 + 1 ) .= ( NAT .--> succ i1 ) . NAT ;
assume that F is bbfamily and rng p = F and dom p = Seg ( n + 1 ) and for i be Nat st i in Seg ( n + 1 ) holds p . i = F . ( i + 1 ) ;
not LIN b , b9 , a & LIN a , a9 , c & LIN a , a9 , c & LIN a , a9 , c ;
( L1 -or L2 ) \& O c= ( L1 -\HM { x } ) \HM { x } & ( L2 -N ) Let x in ( L1 ;
consider F being ManySortedSet of E such that for d being Element of E holds F . d = F ( d ) and for d being Element of E holds F . d = G ( d ) ;
consider a , b such that a * ( 0. V ) = b * ( - w ) and 0 < a & a < b ;
defpred P [ FinSequence of D ] means |. Sum ( $1 ) .| <= Sum |. $1 .| & Sum ( |. $1 .| ) <= Sum |. $1 .| ;
u = cos . ( x , y ) * x + ( cos 2 ) * y .= v ;
dist ( ( seq . n ) + x , g + x ) <= dist ( ( seq . n ) , g ) + 0 ;
P [ p , |. p .| (#) ( p (#) {} ) , id ( the Sorts of A ) ] ;
consider X being Subset of CQC-WFF ( Al ) such that X c= Y and X is finite and X is ininininand X c= Y ;
|. b .| * |. eval ( f , z ) .| >= |. b .| * |. eval ( f , z ) .| ;
1 < ( ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ) .. Cage ( C , n ) ;
l in { l1 where l1 is Real : g <= l1 & l1 <= h } & l in { l1 : h <= g } ;
Ser ( ( ( G . n ) vol ) ) . i <= ( Partial_Sums ( ( G . n ) vol ) ) . i ;
f . y = x .= x * 1. L .= x * ( power L ) . ( y , 0 ) .= x * ( ( power L ) . ( y , 0 ) ) ;
NIC ( <% i1 , i2 %> , x ) = { i1 , succ i1 } & NIC ( i , x ) = { succ i1 , succ i2 } ;
LSeg ( p00 , p2 ) /\ LSeg ( p1 , p11 ) = { p1 } & LSeg ( p1 , p01 ) /\ LSeg ( p01 , p2 ) = { p1 } ;
product ( ( the carrier of I-15 ) +* ( i , { 1 } ) ) in ( Z . i ) ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) .= Following ( s1 , n ) ;
W-bound Qs2 <= q1 `1 & ( for i st i in Qs2 holds ( ex i st i in Qs2 & i <= len f ) & ( f /. i = f /. i ) implies ( f /. i = f /. ( i + 1 ) )
f /. i2 <> f /. ( ( len g + len g -' 1 ) -' 1 , f /. ( len f + 1 ) ) ;
M , f / ( x. 3 , a ) / ( x. 4 , a ) / ( x. 4 , a ) |= H / ( x. 4 , a ) ;
len ( ( P ^ ) ^ ( ( Q ^ ) ) ) in dom ( ( P ^ ) ^ ( Q ^ ) ) ;
A |^ ( mn ) c= A |^ ( m , n ) & A |^ ( k , l ) c= A |^ ( k , l ) ;
( REAL n ) \ { q : |. q .| < a } c= { q1 : |. q1 .| >= a }
consider n1 be element such that n1 in dom p1 and y1 = p1 . n1 and p1 . ( n1 + 1 ) = p1 . n1 ;
consider X being set such that X in Q and for Z being set st Z in Q & Z <> X holds X \not c= Z ;
CurInstr ( P3 , Comput ( P3 , s2 , l ) ) <> halt SCM+FSA & CurInstr ( P3 , Comput ( P3 , s2 , l ) ) <> halt SCM+FSA ;
for v be VECTOR of l1 holds ||. v .|| = upper_bound rng |. ( ( id the carrier of l1 ) ) .| & ||. v .|| = ||. v .||
for phi holds phi in X implies not ( phi in X & not phi in X & phi in X ) & ( phi in X implies phi in X )
rng ( ( Sgm dom f-6 ) | dom f-9 ) c= dom ( ( Sgm dom f-9 ) | dom f-9 ) ;
ex c being FinSequence of D ( ) st len c = k & a = c & for i being Nat st i in dom c holds c . i = F ( i ) ;
( the_arity_of ( o , b , c ) ) = <* ( o , b , c ) , ( o , c ) *> ;
consider f1 be Function of the carrier of X , REAL such that f1 = |. f .| and f1 is continuous and f1 . 0 = f . 1 ;
a1 = b1 & a2 = b2 or a1 = b1 & a2 = b2 & a3 = b3 or a1 = b1 & a3 = b3 & a4 = G1 ;
D2 . indx ( D2 , D1 , n1 + 1 ) = D1 . ( n1 + 1 ) .= D1 . ( n1 + 1 ) ;
f . ( ||. r .|| ) = ||. |[ r , r ]| .|| /. 1 .= <* r *> . 1 .= <* r *> . 1 .= x ;
consider n being Nat such that for m being Nat st n <= m holds C-25 . n = C-25 . m and for m being Nat st m <= n holds C-25 . m = C-25 . m ;
consider d be Real such that for a , b be Real st a in X & b in Y holds a <= d & d <= b ;
||. L /. h .|| - ( K * |. h .| ) + ( K * |. h .| ) <= p0 + ( K * |. h .| ) ;
attr F is commutative associative means : Def2 : for b being Element of X holds F -Sum ( { b } --> f ) = f . b ;
p = - ( - p0 + 0. TOP-REAL 2 ) .= 1 * p0 + 0. TOP-REAL 2 .= 1 * p0 + 0. TOP-REAL 2 .= 1 * p0 + 0 * p0 .= p0 ;
consider z1 such that b , x3 , x3 is_collinear and o , x1 , z1 is_collinear and o , x1 , z1 is_collinear and o , x1 , z1 is_collinear ;
consider i such that Arg ( Rotate ( s ) ) . q = s + Arg q + ( 2 * PI * i ) ;
consider g such that g is one-to-one and dom g = card ( f . x ) and rng g = f . x and for x st x in dom f holds g . x = F ( x ) ;
assume that A = P2 \/ Q2 and P2 <> {} and P2 <> {} and P2 misses Q2 and P2 misses Q2 and P2 misses Q2 and P2 = {} and P2 = {} ;
attr F is associative means : Def2 : F .: ( F .: ( f , g ) , h ) = F .: ( f , F .: ( g , h ) ) ;
ex x being Element of NAT st m = x `2 & x in z `1 & x < i or m in { i } ;
consider k2 being Nat such that k2 in dom P-2 and l in P-2 . k2 and ( P-2 . k2 ) = P-2 . k2 ;
seq = r (#) seq implies for n holds seq . n = r * seq . n & seq . n = r * seq . n & seq . n = r * seq . n
F1 . [ ( ( id a ) * [ a , a ] ) , [ a , b ] ] = [ f * ( id a ) , f * ( id a ) ] ;
{ p } "\/" D2 = { p "\/" y where y is Element of L : y in D2 } & p "\/" q in { p "\/" q where q is Element of L : q in D2 } ;
consider z being element such that z in dom ( ( dom F ) . z ) and ( ( dom F ) . 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 , 1 ) = { |[ r , s ]| : r <= G * ( 0 + 1 , 1 ) `1 } ;
consider e being element such that e in dom ( T | E1 ) and ( T | E1 ) . e = v ;
( F ` * b1 ) . x = ( Mx2Tran ( J , BZ , BZ ) ) . ( ( is ( i , j ) --> ( BZ ) ) /. x ) ;
- 1 / ( mM | n ) = mm (#) D | n .= mm (#) ( M | n ) .= mm (#) ( M | n ) .= Det M ;
attr x in dom f /\ dom g means : Def2 : for x being set st x in dom f /\ dom g holds ( - f ) . x <= f . x ;
len ( f1 . j ) = len f2 /. j .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) ;
All ( All ( 'not' a , A , G ) , B , G ) '<' Ex ( Ex ( 'not' a , B , G ) , A , G ) ;
LSeg ( E . k0 , F . k0 ) c= Cl RightComp Cage ( C , k0 + 1 ) & ( E . k0 ) `1 c= ( L~ Cage ( C , k0 + 1 ) ) `1 ;
x \ a |^ m = x \ ( ( a |^ k ) * a ) .= ( x \ ( a |^ k ) ) \ a ;
k -inininininininininin-in= ( commute Ip1 ) . k .= ( ( commute p1 ) . k ) . ( ( ( the carrier of S ) --> NAT ) ) .= ( ( the carrier of S ) --> NAT ) . k ;
for s being State of Al holds Following ( s , n ) . 0 + ( n + 2 ) * ( n + 1 ) is stable ;
for x st x in Z holds f1 . x = a ^2 & ( f1 - f2 ) . x <> 0 & ( f1 - f2 ) . x <> 0 ;
support ( ( support ( n ) ) \/ support ( m ) ) c= support ( ( support ( n ) ) \/ support ( m ) ) \/ support ( ( support ( n ) ) ) ;
reconsider t = u as Function of ( the carrier of A ) , ( the carrier of B ) * , the carrier of C ;
- ( a * sqrt ( 1 + b ^2 ) ) <= - ( b * sqrt ( 1 + a ^2 ) ) ;
phi /. ( succ b1 ) = g . a & phi /. ( g . ( g . 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 } .= { x1 } \/ { x2 , x3 , x4 } ;
the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U1 & the Sorts of U1 c= the Sorts of U2 & the Sorts of U1 c= the Sorts of U2 ;
( - ( 2 * a * ( b / 2 ) ) + b ) ^2 - delta ( a , b , c ) > 0 ;
consider W00 such that for z being element holds z in W00 iff z in [: N , N :] & P [ z , W00 ] ;
assume ( the Arity of S ) . o = <* a *> & ( the ResultSort of S ) . o = r & ( the ResultSort of S ) . o = <* r *> ;
Z = dom ( ( exp_R * ( arccot ) ) `| Z ) & f = ( ( exp_R * ( arccot ) ) `| Z ) . ( f . x ) ;
sum ( f , SD1 ) is convergent & lim ( f , SD1 ) = integral ( f , SD1 ) & lim ( f , SD1 ) = integral ( f , SD1 ) ;
( X ( a9 . f ) => ( g => ( x9 ) ) ) => ( x9 => ( x9 ) ) in is that ( X ( ) => ( x9 ) ) ;
len ( M2 * M3 ) = n & width ( M3 * M3 ) = n & width ( M3 * M3 ) = n & width ( M3 * M3 ) = n ;
attr X1 union X2 is open SubSpace of X means : Def2 : X1 , X2 are_separated & X1 , X2 are_separated & X2 , X1 are_separated & X1 , X2 are_separated implies X1 union X2 misses X2 ;
for L being upper-bounded antisymmetric RelStr , X being non empty Subset of L holds X "\/" { Top L } = { Top L } & X "\/" { Top L } = { Top L }
reconsider f-129 = F1 . ( b `2 ) as Function of ( the carrier of M ) , M . ( b `2 ) , M . ( b `2 ) ;
consider w being FinSequence of I such that the InitS of M is_ststst\langle ( s ) ^ w ^ w , q ^ w ^ w ^ w ^ w ^ w ^ w ^ w , q ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^
g . ( a |^ 0 ) = g . 1_ H .= 1_ H .= ( g . a ) |^ 0 .= ( g . a ) |^ 0 .= ( g . a ) |^ 0 ;
assume for i be Nat st i in dom f ex z be Element of L st f . i = rpoly ( 1 , z ) & f . i = z ;
ex L being Subset of X st Carrier L = L & for K being Subset of X st K in C holds L /\ K <> {} ;
( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= ( the carrier' of C1 ) /\ ( the carrier' of C2 ) & ( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= ( the carrier' of C1 ) /\ ( the carrier' of C2 ) ;
reconsider oY = o `2 as Element of ( ( the Sorts of A ) * ( ( the Arity of S ) . o ) ) . ( ( the Arity of S ) . o ) ;
1 * x1 + ( 0 * x2 ) + ( 0 * x3 ) = x1 + <* \underbrace ( 0 , \dots , 0 ) *> .= x1 + <* \underbrace ( 0 , \dots , 0 ) *> .= x1 + <* \underbrace ( 0 , 0 ) *> .= x1 ;
Es " . 1 = ( ( ( E qua Function ) qua Function ) " ) . 1 .= ( ( E qua Function ) " ) . 1 .= ( ( E qua Function ) " ) . 1 .= ( ( E qua Function ) " ) . 1 ;
reconsider u1 = the carrier of U1 /\ ( U1 "\/" U2 ) as non empty Subset of U0 & ( the carrier of U1 ) /\ ( U1 "\/" U2 ) = the carrier of U0 ;
( ( x "/\" z ) "\/" ( x "/\" y ) ) "\/" ( z "/\" y ) <= ( x "/\" ( z "\/" y ) ) "\/" ( z "/\" ( x "\/" y ) ) ;
|. f . ( s1 . ( l1 + 1 ) ) - f . ( s1 . l1 ) .| < 1 / ( |. M .| + 1 ) ;
LSeg ( ( Lower_Seq ( C , n ) ) * ( i , ( k + 1 ) ) , ( ( Lower_Seq ( C , n ) ) * ( i + 1 , k ) ) ) is vertical ;
( f | Z ) /. x - ( f | Z ) /. x0 = L /. ( x- x0 ) + R /. ( x- x0 ) ;
g . c * ( - ( g . c ) * f . c ) + f . c <= h . c * ( - ( g . c ) ) + f . c ;
( f + 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 f . b = width b ) and ( for i st i in dom f holds f . i = ( i ) * ( i , j ) ) ;
len ( - M3 ) = len M1 & width ( - M3 ) = width M1 & width ( - M3 ) = width M1 & width ( - M3 ) = width M1 ;
for n , i being Nat st i + 1 < n holds [ i , i + 1 ] in the InternalRel of ( ( the carrier of TOP-REAL n ) \ ( the carrier of ( TOP-REAL n ) \ ( the carrier of ( TOP-REAL n ) \ ( the carrier of ( TOP-REAL n ) \ ( the carrier of ( TOP-REAL n ) \ ( the carrier of ( TOP-REAL n ) \ ( the carrier of ( TOP-REAL n ) \ ( the carrier of ( TOP-REAL n ) ) ) ) ) ) ) ;
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 ( f2 , 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 ( f2 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) . 1 = f2 . 1 ;
pred a <> 0 & b <> 0 & Arg a = Arg b means : Def2 : Arg ( - a ) = Arg ( - b ) & Arg ( - a ) = Arg ( - a ) ;
for c being set st not c in [. a , b .] holds not c in Intersection ( the topology of a , b ) & not c in Intersection ( the topology of a , b )
assume that V1 is linearly-independent and V2 is linearly-independent and V2 = { v + u : v in V1 & u in V2 } and V1 is open & V2 is open & V2 is open ;
z * x1 + ( 1 / COMPLEX ) * x2 in M & z * ( y1 + ( 1 / COMPLEX ) * y2 ) in N implies z * x1 + ( 1 / COMPLEX ) * x2 in N
rng ( ( Pk1 qua Function ) " * Sk1 ) = Seg ( card dk1 ) .= Seg ( card dk1 ) .= Seg ( card Sk1 ) .= Seg ( card Sk1 ) ;
consider s2 being rational Real_Sequence such that s2 is convergent and b = lim s2 and for n holds s2 . n <= b and s2 . n <= b ;
h2 " . n = h2 . n " & 0 < - 1 / ( ( 1 / ( ( ( ( 1 / ( n + 1 ) ) * ( i + 1 ) ) ) ) ) ;
( Partial_Sums ( ||. seq1 .|| ) ) . m = ||. seq1 .|| . m .= ||. seq1 .|| . m .= ||. seq1 .|| . m .= ( ||. seq1 .|| ) . m .= ( ||. seq1 .|| ) . m .= ( ||. seq1 .|| ) . m ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= ( Comput ( P2 , s2 , 1 ) ) . b .= Comput ( P2 , s2 , 1 ) . b ;
- v = - 1_ GV * v & - w = - 1_ GV * w & - w = - 1_ GV * w & - w = - 1_ GV * w ;
upper_bound ( ( k .: D ) .: D ) = upper_bound ( ( k .: D ) ) .= k . ( sup D ) .= sup ( ( k .: D ) .: D ) .= sup ( ( k .: D ) .: D ) ;
A |^ ( k , l ) ^^ ( A |^ ( n , l ) ) = ( A |^ ( k , l ) ) ^^ ( A |^ ( k , l ) ) ;
for R being add-associative right_zeroed right_complementable non empty addLoopStr , I , J being Subset of R holds I + ( J + K ) = ( I + J ) + K
( f . p ) `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 Al being countable Nat such that r is countable & Al is Al and Al is Al and Al = ( A ) ! and Al = ( A ) ! ;
for X being non empty addLoopStr , M being Subset of X , x , y being Point of X st y in M holds x + y in x + M
{ [ x1 , x2 ] , [ y1 , y2 ] } c= [: { x1 , y1 } , { y2 } :] & { [ x1 , y2 ] , [ y1 , y2 ] } c= [: { x1 , 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 ) & ( Gauge ( C , n ) * ( k , i ) ) `1 = ( Gauge ( C , n ) * ( k , i ) ) `1 ;
cluster m , n are_relative_prime means : Def2 : for p being prime Nat st p divides m & p divides n holds p divides n & p divides n ;
( f * F ) . x1 = f . ( F . x1 ) & ( f * F ) . x2 = f . ( F . x2 ) ;
for L being LATTICE , a , b , c being Element of L st a \ b <= c & b \ a <= c holds a \+\ b <= c
consider b being element such that b in dom ( H / ( x , y ) ) and z = ( H / ( x , y ) ) . b ;
assume that x in dom ( F * g ) and y in dom ( F * g ) and ( F * g ) . x = ( F * g ) . y ;
assume ex e being element st e Joins W . 1 , W . 5 , G or e Joins W . 3 , W . 7 , G ;
( r (#) delta ( h ) ) . ( 2 * n ) . x = ( r (#) ( delta ( h ) ) ) . ( 2 * n + ( n * h ) ) . x ;
j + 1 = ( len h11 + 2 ) + 1 .= i + 1 - len h11 + 2 .= i + 1 - len h11 + 2 - 1 .= i + 1 ;
( *' ( S *' ) ) . f = S *' . ( ( opp f ) . f ) .= S . ( ( *' f ) . f ) .= S . f ;
consider H such that H is one-to-one and rng H = the carrier of L2 and Sum ( L2 * H ) = Sum ( L2 ) and Sum ( L1 ) = Sum ( L1 ) ;
attr R is + \mathbb means : Def2 : for p , q st p in R & q <> q holds ex P st P is special & P c= R & P c= R ;
dom product ( X --> f ) = meet ( ( X --> f ) . ( ( X --> f ) . ( X --> f ) ) ) .= meet ( X --> f ) .= dom f ;
upper_bound ( proj2 .: ( Upper_Arc C /\ E-bound C ) ) <= upper_bound ( proj2 .: ( C /\ E-bound C ) ) & upper_bound ( proj2 .: ( C /\ E-bound C ) ) <= upper_bound ( proj2 .: ( C /\ E-bound C ) ) ;
for r be Real st 0 < r ex n be Nat st for m be Nat st n <= m holds |. S . m - pp .| < r
i * fN - f\overline ( i * yN ) = i * fN - ( i * fN ) .= i * ( fN - fN ) .= i * ( fN - fN ) ;
consider f being Function such that dom f = 2 -tuples_on X & for Y being set st Y in 2 -tuples_on X holds f . Y = F ( Y ) ;
consider g1 , g2 being element such that g1 in [#] Y and g2 in union C and g = [ g1 , g2 ] and g = [ g1 , g2 ] ;
func d |-count n -> Nat means : Def2 : d |^ it divides n & d |^ ( it + 1 ) divides n & d divides n & d divides n ;
f\in . [ 0 , t ] = f . [ 0 , t ] .= ( - P ) . ( 2 * x ) .= a ;
t = h . D or t = h . B or t = h . C or t = h . E or t = h . F or t = h . J ;
consider m1 be Nat such that for n st n >= m1 holds dist ( ( seq . n ) , ( seq . n ) ) < 1 / ( n + 1 ) ;
( ( q `1 ) ^2 + ( q `2 ) ^2 ) <= ( ( q `2 ) ^2 / ( 1 + ( q `2 / q `1 ) ^2 ) ) ^2 / ( 1 + ( q `2 / q `1 ) ^2 ) ;
h0 . ( i + 1 + 1 ) = h21 . ( i + 1 + 1 -' len h11 + 2 -' 1 ) .= h21 . ( i + 1 + 1 -' len h11 + 2 -' 1 ) ;
consider o being Element of the carrier of S , x2 being Element of { the carrier of S } such that a = [ o , x2 ] and [ o , x2 ] in R ;
for L being RelStr , a , b being Element of L holds a is_<=_than { b } iff a is_<=_than { b } & b is_<=_than { a }
||. h1 .|| . n = ||. h1 . n .|| .= ||. h .|| . n .= ||. h .|| . n .= ||. h .|| . n .= ||. h .|| . n .= ||. h .|| . n .= ||. h .|| . n ;
( ( - ( #Z n ) ) * ( ( #Z n ) * ( f1 - f2 ) ) ) . x = f . x - ( ( #Z n ) * ( f1 - f2 ) ) . x .= ( ( - 1 ) * ( f1 - f2 ) ) . x ;
pred r = F .: ( p , q ) means : Def2 : len r = min ( len p , len q ) & for i being Nat st i in dom r holds r . i = F . ( p . i ) ;
( r\mathop / 2 ) ^2 + ( r\mathop / 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 ( ( B * M ) @ ) & Det M = ( ( B * M ) @ ) * ( i , j )
then a <> 0. R & a " * ( a * v ) = 1 * v & a " * ( a * v ) = 1 * v & a * ( a * v ) = 1 * v ;
p . ( j - 1 ) * ( q *' r ) . ( i + 1 -' j ) = Sum ( p . ( j -' 1 ) * r3 ) .= Sum ( p *' r ) ;
deffunc F ( Nat ) = L . 1 + ( ( R /* ( h ^\ n ) ) * ( h ^\ n ) " ) . $1 - ( ( R /* ( h ^\ n ) ) * ( h ^\ n ) ) . $1 ;
assume that the carrier of H1 = f .: ( the carrier of H2 ) and the carrier of H2 = f .: ( the carrier of H1 ) and the carrier of H1 = f .: ( the carrier of H2 ) and the carrier of H1 = the carrier of H2 ;
Args ( o , Free ( S , X ) ) = ( ( the Sorts of Free ( S , X ) ) * ( the Arity of S ) ) . o .= ( the Sorts of Free ( S , X ) ) . o ;
H1 = n + 1 -H .= n + 1 -H .= n + 1 -H .= n + 1 -H .= n + 1 -H .= n + 1 -H .= n + 1 -H ;
( ( O = 0 ) & ( O = 1 ) & ( O = 1 implies O = 1 ) & ( O = 1 implies O = 1 ) & ( O = 1 implies O = 1 ) & ( O = 1 implies O = 1 ) & ( O = 1 implies O = 1 ) & ( O = 1 implies O = 1 ) & ( O = 1 implies O = 1 ) ;
F1 .: ( dom F1 /\ dom F2 ) = F1 .: ( 1 / 2 ) .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } ;
attr b <> 0 & d <> 0 & b <> d & ( a / b ) = ( - e ) / ( b - b- d ) ;
dom ( ( f +* g ) | D ) = dom ( f +* g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D .= dom f /\ D ;
for i be set st i in dom g ex u , v be Element of L , a be Element of B st g /. i = u * a * v & u in A & v in B
g `2 * P * g " = g `2 * ( g * P " ) * g " .= g `2 * ( g * P " ) * ( g * P " ) .= g * ( g * P " ) ;
consider i , s1 such that f . i = s1 and if i = 1 & not f . ( i + 1 ) <> s1 & not f . ( i + 1 ) in { s1 } ;
h5 | ]. a , b .[ = ( g | Z ) | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .] ;
[ s1 , t1 ] , [ s2 , t2 ] are_connected & [ s2 , t2 ] , [ t2 , t2 ] are_connected & [ s2 , t2 ] , [ t2 , t2 ] are_connected & [ t2 , t2 ] , [ t2 , t2 ] are_connected ;
then H is negative & H is non negative & H is non empty & H is non empty & H is non -g\mathopen implies H is non empty ;
attr f1 is total means : Def2 : 1 / f2 is total & ( for c st c in dom f1 holds f1 . c = f1 . c * f2 . c ) & ( for c st c in dom f1 holds f1 . c = f2 . c ) ;
z1 in W2 -Seg ( z2 ) or z1 = z2 & not z1 in W2 & not z2 in W2 & ex z2 st z2 in W2 & z = z2 & z = z2 or z1 = z2 & z = z2 & z1 = z2 ;
p = 1 * p .= a " * a * p .= a " * ( b * q ) .= a " * ( b * q ) .= a " * ( b * q ) .= a " * ( b * q ) ;
for seq1 be Real_Sequence , K be Real st for n be Nat holds seq1 . n <= K holds upper_bound rng seq1 <= ( upper_bound rng seq1 ) * ( upper_bound rng seq1 ) + ( upper_bound rng seq1 ) * ( upper_bound rng seq1 )
C meets C or C meets L~ go \/ L~ pion1 or C meets L~ pion1 or C meets L~ pion1 or C meets L~ pion1 or C meets L~ pion1 or C meets L~ pion1 or C meets L~ pion1 or C meets L~ pion1 or C meets L~ pion1 or C meets L~ pion1 or C meets L~ pion1 ;
||. f . ( g . ( k + 1 ) ) - g . ( g . k ) .|| <= ||. g . 1 - 0 .|| * ( K * K to_power k ) ;
assume h = ( ( B .--> B ' ) +* ( C .--> D ' ) +* ( D .--> E ' ) +* ( E .--> F ' ) +* ( F .--> J ' ) +* ( J .--> M ' ) +* ( N .--> N ' ) +* ( F .--> J ' ) +* ( M .--> N ' ) ) ;
|. ( ( ( ( ( ( H . n ) || A ) , T ) ) . k ) - ( ( ( ( H . n ) || A ) ) . k ) .| <= e * ( ( ( ( ( H . n ) || A ) . k ) - ( ( ( H . n ) || A ) . k ) ) ) ;
( ( ( the Sorts of A ) * ( i , j ) ) . e = [ ( the ; of v ) . ( ( the carrier of I ) --> ( ( the carrier of I ) --> { 0 } ) ) . e , ( ( the carrier of I ) --> { 0 } ) . e ] ;
{ 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 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x2 , x2 , x2 , x2 , x2 , x2 , x2 , x2 , x2 , x2 , x2
assume that A = [. 0 , 2 * PI .] and integral ( ( exp_R (#) ( ( #Z n ) * ( sin ) ) , A ) = 0 and ( ( exp_R (#) ( ( #Z n ) * ( sin ) ) ) `| A ) = 0 ;
p `2 is Permutation of dom f1 & p `2 " = ( ( Sgm Y ) * p ) " * ( Sgm X ) " * ( Sgm Y ) " * ( Sgm X ) " * ( Sgm X ) " * ( Sgm X ) " ;
for x , y st x in A & y in A holds |. ( 1 / f ) . x - ( 1 / f ) . y .| <= 1 * |. f . x - f . y .|
p2 `2 = |. q2 .| * ( ( ( q2 `2 / |. q2 .| - sn ) / ( 1 + sn ) ) / ( 1 + sn ) ) .= ( ( q2 `2 / |. q2 .| - sn ) ) / ( 1 + sn ) ;
for f be PartFunc of the carrier of CNS , REAL st dom f is compact & f is_continuous_on dom f & f is_continuous_on dom f holds f is_continuous_on dom f & f | X is continuous & f | X is continuous
assume for x being Element of Y st x in EqClass ( z , CompF ( B , G ) ) holds ( Ex ( a , A , G ) ) . x = TRUE ;
consider FM such that dom FM = n1 and for k be Nat st k in n1 holds Q [ k , FM . k ] and ( for k be Nat st k in Seg n1 holds FM [ k , FM . k ] ) & ( for k be Nat st k in Seg n1 holds FM [ k , FM . k ] ) ;
ex u , u1 st u <> u1 & u , u1 / ( 2 > 0 ) & u , u1 / ( 2 > 0 ) & u , u1 / ( 2 > 0 ) & u1 , v1 / ( 2 |^ 0 ) // u , u1 & u1 / ( 2 |^ 0 ) // u1 , v1 ;
for G being Group , A , B being non empty normal Subgroup 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 to_power k ) (#) ( e to_power k ) , ( ( f + g ) (#) e to_power k ) ) ;
width AutMt ( f1 , b1 , b2 ) = len b2 .= width ( ( f2 * f1 ) * ( i , b2 ) ) .= width ( ( f2 * f1 ) * ( i , b2 ) ) .= width ( ( f2 * f1 ) * ( i , b2 ) ) ;
f | ]. - PI / 2 , PI / 2 .[ = f & dom f = ]. - 1 , 1 .[ & for x st x in ]. - 1 , 1 .[ holds f . x = - 1 / 2 * x + 1 / 2 * x & f . x = - 1 / 2 * x + 1 / 2 * x
assume that X is closed & a in X and a c= X and y in a and { { [ n , x ] } \/ y : x in a } in X ;
Z = dom ( ( ( 1 / 2 ) (#) ( arctan + arccot ) ) `| Z ) /\ dom ( ( ( 1 / 2 ) (#) ( arctan + arccot ) ) `| Z ) .= dom f /\ dom ( ( 1 / 2 ) (#) ( arctan + arccot ) ) ;
func ( V ) -> Subset of V means : Def2 : for l being Nat st 1 <= l & l <= len l & l . k in V & l . l in V ;
for L being non empty TopSpace , N being net of L , M being net of L , c being Point of N st c is_9 ( N ) holds c is continuous & c in N
for s being Element of NAT holds ( ( ( ( id C\mathop ) + ( id C\mathop ) ) (#) ( C\mathop ) ) `| ( the carrier of C\mathop ) ) . s = ( ( ( ( id C\mathop ) + ( C\mathop ) ) (#) ( C\mathop ) ) | ( the carrier of CV ) ) . s
then z /. 1 = N-min L~ z & ( N-min L~ z ) .. z < ( N-min L~ z ) .. z & ( N-min L~ z ) .. z < ( N-min L~ z ) .. z ;
len ( p ^ <* ( 0 qua Real ) *> ) = len p + len <* ( 0 qua Real ) *> .= len p + 1 .= len p + 1 .= len p + 1 ;
assume that Z c= dom ( - ( ln * f ) ) and for x st x in Z holds f . x = x and f . x > 0 ;
for R being right_zeroed right complementable non empty doubleLoopStr , I being Ideal of R , J being Subset of R holds ( I + J ) *' ( I /\ J ) c= I /\ J
consider f being Function of [: B1 , B2 :] , B12 such that for x being Element of [: B1 , B2 :] holds f . x = F ( x ) and f . x = F ( x ) ;
dom ( x2 + y2 ) = Seg len x .= Seg len ( x2 (#) z ) .= Seg len ( x2 (#) z ) .= dom ( x2 (#) z ) .= dom ( x2 (#) z ) .= dom ( x2 (#) z ) .= dom ( x2 (#) z ) ;
for S being Functor of C , B for c being object of C holds card S . ( id c ) = id ( ( Obj S ) . c ) & S . ( id c ) = id ( ( Obj S ) . c )
ex a st a = a2 & a in f6 /\ f5 & \cal or ( f , a ) = \cal ( f , a ) & for x st x in f6 holds x in Im ( f , a ) & x in Im ( f , a ) ;
a in Free ( H2 / ( x. 4 , x. 0 ) ) '&' H2 / ( x. 4 , x. 0 ) & a in Free ( H2 / ( x. 4 , x. 0 ) ) & a in Free ( H2 / ( x. 4 , x. 0 ) ) ;
for C1 , C2 being 1 , f , g being stable Function of C1 , C2 st `1 = ( C2 ) * ( f , g ) holds f = g & f = g
( W-min ( L~ go \/ L~ co ) ) `1 = W-bound ( L~ go \/ L~ co ) & ( W-min ( L~ go \/ L~ co ) ) `1 = W-bound ( L~ go \/ L~ co ) & ( W-min ( L~ go \/ L~ co ) ) `1 = W-bound ( L~ go \/ E-bound ( L~ go \/ L~ co ) ) ;
assume that u = <* x0 , y0 *> and f is_is_is_is_or SVF1 ( 3 , pdiff1 ( f , 1 ) , u and SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) . z0 = 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 ( p , J ) . v .= Valid ( p , J ) . v ;
assume for x , y being Element of S st x <= y for a , b being Element of T st a = f . x & b = f . y holds a >= b ;
func Class R -> Subset-Family of R means : Def2 : for A being Subset of R holds A in it iff ex a being Element of R st A = Class ( a , a ) & it = Class ( R , a ) ;
defpred P [ Nat ] means ( ( ( ( ( ( ( G ) | ( ( G ) . $1 ) ) `1 ) ) ) ) `1 c= G } & ( ( ( G is Function of the carrier of G , { ( G ) } ) `1 ) `1 ) c= G ;
assume that dim W1 = 0 and dim W1 = 0 and ( dim W2 = 0 implies dim W1 = 0 ) & ( for i be Nat st i in dom W1 holds 0. W1 = 0. W2 ) & ( for i be Nat st i in dom W1 holds W1 . i = 0. W2 ) & ( i in dom W1 implies W1 . i = 0. W2 ) ;
mama-ma[ ( m . t ) `1 , ( m . t ) `2 ] = ( m . [ m . t , the carrier of C ] ) `1 .= ( m . [ m . t , the carrier of C ] ) `1 .= m . t ;
d11 = x9 ^ d22 .= f . ( y9 , d22 ) .= f . ( y9 , d22 ) .= f . ( y9 , d22 ) .= ( f | ( y9 , d22 ) ) . ( y9 , d22 ) .= ( f | ( y9 , d22 ) ) . ( y9 , d22 ) .= ( f | ( y9 , d22 ) ) . ( y9 , d11 ) ;
consider g such that x = g and dom g = dom fx0 and for x being element st x in dom fx0 holds g . x in fx0 & g . x in fx0 ;
x + 0. F_Complex |^ ( len x ) = x + len x |-> 0. F_Complex .= ( x , len x ) |-> 0. F_Complex .= ( x , len x ) |-> 0. F_Complex .= x ` .= x ` ;
( k -' kk + 1 ) in dom ( f | ( len f -' ( k -' 1 ) ) ) & ( f | ( k -' ( k -' 1 ) + 1 ) ) . ( k -' ( k -' 1 ) + 1 ) = ( f | ( k -' ( k -' 1 ) + 1 ) ) . ( k -' ( k -' ( k -' 1 ) + 1 ) ) ;
assume that P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P2 = P1 \/ P2 and P1 = P1 \/ P2 and P2 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P2 = P1 \/ P2 and P1 = P2 \/ P2 ;
reconsider a1 = a , b1 = b , b1 = c , c1 = p `2 , c1 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c1 = p `2 , c2 = p `2 , c1 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c1 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c1 = p `2 , c2 = p `2 , c1 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2
reconsider Gtf = G1 . ( t * b ) as Morphism of ( G1 * F1 ) . a , ( G1 * F2 ) . b * ( G1 * F2 ) . b * G1 . a as Morphism of G1 . a , ( G1 * F2 ) . b ;
LSeg ( f , i + i1 -' 1 ) = LSeg ( f /. ( i + i1 -' 1 ) , f /. ( i + i1 -' 1 + 1 ) ) .= LSeg ( f , i + i1 -' 1 ) ;
Integral ( M , P . m ) | dom ( P . n -P . m ) <= Integral ( M , P . n ) | dom ( P . m -P . m ) ;
assume that dom f1 = dom f2 and for x , y being element st [ x , y ] in dom f1 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 ) / 2 , ( G * ( i + 1 , 1 ) `2 - G * ( i + 1 , 1 ) `2 ) ;
for G being Group , H being Subgroup of G , a being Element of G st a = b holds for i being Integer st i = b holds a |^ i = b |^ i & a |^ i = b |^ i
consider B being Function of Seg ( S + L ) , the carrier of V1 such that for x being element st x in Seg ( S + L ) holds P [ x , B . x ] ;
reconsider K1 = { pp where pp is Point of TOP-REAL 2 : P [ 7 ] } as Subset of TOP-REAL 2 & { p where p is Point of TOP-REAL 2 : P [ p ] } & { p where p is Point of TOP-REAL 2 : p `1 >= 0 } c= the carrier of ( TOP-REAL 2 ) | K1 & { p } c= the carrier of ( TOP-REAL 2 ) | K1 ;
( ( ( ( ( the carrier of C ) - ( S-bound C ) ) / ( 2 |^ m ) ) ) / ( 2 |^ n ) ) * ( ( ( ( ( the carrier of C ) - ( S-bound C ) ) / ( 2 |^ m ) ) / ( 2 |^ n ) ) ) * ( ( ( ( the carrier of C ) - ( S-bound C ) ) / ( 2 |^ m ) ) ) ) ) * ( ( ( ( ( ( the carrier of C ) - ( ( the carrier of C ) - ( 2 |^ m ) / ( 2 |^ m ) ) / ( 2 |^ n ) ) / ( 2 |^ n ) ) ) * ( 2 |^ n ) )
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 ) = len ( @ p ^ @ q ) + len <* [ 2 , 0 ] *> .= len ( @ p ^ @ q ) + len ( @ q ) .= len ( @ p ^ @ q ) + len ( @ q ) ;
v / ( x. 3 , m1 ) / ( x. 4 , m2 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) = m3 / ( x. 4 , m3 ) / ( x. 4 , m3 ) ;
consider r be Element of M such that M , v2 / ( x. 3 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) |= H2 ;
func w1 \ w2 -> Element of Union ( G , R^ R^ ) means : Def2 : for w1 , w2 being Element of Union ( G , R ) holds it . ( w1 , w2 ) = ( ( ( ( ( ( G , R ) * the Arity of S ) ) * the Arity of S ) . w1 ) . w2 ;
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= s1 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= ( s2 . b2 ) . b2 .= ( s2 . b2 ) . b2 .= ( s2 . b2 ) . b2 .= ( s2 . b2 ) . b2 ;
for n , k be Nat holds 0 <= ( Partial_Sums ( |. seq .| ) ) . ( n + k ) - ( Partial_Sums ( |. seq .| ) ) . ( n + k ) + ( Partial_Sums ( |. seq .| ) ) . ( n + k )
set F = S -\rm \hbox { - } \rm implies F is S -\rm \hbox { - } \rm implies F is S -\rm \hbox { - } \rm implies F is S -\rm \hbox { - }
( Partial_Sums seq ) . K + Partial_Sums ( seq ) . ( K + 1 ) >= ( Partial_Sums ( seq ) ) . ( K + 1 ) + ( Partial_Sums ( seq ) ) . ( K + 1 ) ;
consider L , R such that for x st x in N holds ( f | Z ) . x - ( f | Z ) . x0 = L . ( x- x0 ) + R . ( x - x0 ) ;
func the closed of rectangle ( a , b , c , d ) -> closed Subset of ( \HM { a , b , c , d } ) | ( the carrier of rectangle ( a , b , c , d ) ) ` , the carrier of rectangle ( a , b , c , d ) ;
a * b ^2 + ( a * c ) ^2 + ( b * a ) ^2 + ( b * c ) ^2 + ( c * a ) ^2 >= 6 * a * b * c ;
v / ( ( x1 , m1 ) / m2 ) / ( ( x2 , m1 ) / ( ( x2 , m1 ) / ( ( x3 , m2 ) / ( x3 , m ) ) ) = v / ( ( x2 , m1 ) / ( x3 , m ) ) / ( x3 , m ) ) ;
} ( Q ^ <* x *> , M ) = ( > ( ( Q , M ) +* ( ( N , M ) --> FALSE ) +* ( ( N , M ) --> FALSE ) ) +* ( ( ( N , M ) --> TRUE ) ) .= ( Q , M ) +* ( ( N , M ) --> TRUE ) ;
Sum ( FM ) = r |^ n1 * Sum CM .= C . n1 * ( r |^ n1 ) .= C . n1 * ( r |^ n1 ) .= ( C . n1 ) * ( r |^ n1 ) .= ( C . n1 ) * ( r |^ n1 ) .= ( C . n1 ) * ( r |^ n1 ) .= ( C . n1 ) * ( r |^ n1 ) ;
( GoB f ) * ( len GoB f , 2 ) `1 = ( ( GoB f ) * ( len GoB f , 1 ) ) `1 .= ( ( GoB f ) * ( 1 , 1 ) ) `1 .= ( ( GoB f ) * ( 1 , 1 ) ) `1 .= ( ( GoB f ) * ( 1 , 1 ) ) `1 ;
defpred X [ Element of NAT ] means ( Partial_Sums ( s ) ) . $1 = ( a * ( ( $1 + 1 ) * $1 ) + b * ( ( $1 + 1 ) * ( ( $1 + 1 ) * ( ( $1 + 1 ) * ( ( $1 + 1 ) * ( ( $1 + 1 ) * ( ( $1 + 1 ) * ( ( $1 + 1 ) * ( $1 + 1 ) ) + b ) ) ) ;
( ( the Arity of S ) * g ) . g = ( the Arity of S ) . [ g , g ] .= [ g . g , g . g ] .= [ g . g , g . g ] .= [ g . g , g . g ] .= [ g . g , g . g ] .= [ g . g , g . g ] ;
( [: X , Y :] |^ Z ) tolerates [: X , Y :] & card ( [: X , Y :] |^ Z ) = card [: X , Y :] ;
for a , b being Element of S , s being Element of NAT st s = n & a = F . ( n + 1 ) & b = F . ( n + 1 ) holds b = N . ( s + 1 ) \ G . s ;
E , f |= All ( All ( x. 2 , ( x. 2 ) \ ( x. 3 ) ) , ( x. 4 ) \ ( x. 4 ) ) => ( ( x. 4 ) \ ( x. 4 ) ) '&' ( ( x. 4 ) \ ( x. 4 ) ) ;
ex R2 being 1-sorted st R2 = ( p | nM ) . i & ( ( p | nM ) . i = the carrier of R2 & ( p | nM ) . i = the carrier of R2 ) & ( p | nM ) . i = the carrier of R2 ;
[. a , b + 1 / ( k + 1 ) .[ is Element of the _ of the carrier of f & ( the partial of f ) . k is Element of the carrier of a & ( the partial of f ) . k is Element of the carrier of a & ( the , f ) . k is Element of the carrier of a implies ( the Subset of f ) . k is Element of the carrier of a
Comput ( P , s , 2 + 1 ) = Exec ( P . 2 , Comput ( P , s , 2 ) ) .= Exec ( a3 := a2 , Comput ( P , s , 2 ) ) .= Exec ( a3 := a2 , Comput ( P , s , 2 ) ) ;
card ( h1 ) . k = power F_Complex . ( - 1_ F_Complex , k ) * Sum u .= ( ( - 1_ F_Complex ) *' ) . k * ( ( - 1_ F_Complex ) *' ) . k .= ( ( ( - 1_ F_Complex ) *' ) . k * ( ( - 1_ F_Complex ) *' ) ) * ( ( - 1_ F_Complex ) *' ) . k ;
( f / g ) /. c = f /. c * ( g /. c ) " .= f /. c * ( ( 1 / g ) * ( g /. c ) ) .= ( f * ( 1 / g ) ) /. c ;
len C( C ) - len ( ( C ) /. ( len ( ( C ) ) -' 1 ) ) = len C( ( C ) - len ( ( ( C ) ) -' 1 ) ) .= len ( ( ( C ) - len ( ( C ) ) -' 1 ) ) ;
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom f /\ X .= dom ( ( r (#) f ) | X ) .= dom ( r (#) ( f | X ) ) .= dom ( r (#) ( f | X ) ) ;
defpred P [ Nat ] means for n holds 2 * Fib ( n + $1 ) = Fib ( n ) * Fib ( n + $1 ) + ( 5 * Fib ( n + $1 ) ) * Fib ( n + $1 ) ;
consider f being Function of INT , INT such that f = f `1 and f is onto and f is increasing and for n being Nat st n < k + 1 holds f " { f . n } = { n } ;
consider vs be Function of S , BOOLEAN such that vs = chi ( A \/ B , S ) and ( for A , B being Element of S holds E . ( A \/ B ) = Prob . A + Prob . B ) & ( for A , B being Element of S holds c . ( A \/ B ) = Prob . A + Prob . B ) ;
consider y being Element of Y ( ) such that a = "\/" ( { F ( x , y ) where x is Element of X ( ) : P [ x ] } , Q [ y ] } and Q [ y ] ;
assume that A c= Z and Z = dom f and f = ( ( - 1 ) (#) ( ( #Z 2 ) * ( ( #Z 2 ) * ( f1 + f2 ) ) ) `| Z ) . ( ( f1 + f2 ) . ( f1 + f2 ) ) = f1 . ( ( f1 . ( x + 1 ) ) / ( ( x + 1 ) ^2 ) ) ;
( f /. i ) `2 = ( GoB f ) * ( 1 , j2 ) `2 .= ( GoB f ) * ( 1 , j2 + 1 ) `2 .= ( GoB f ) * ( 1 , j2 + 1 ) `2 .= ( GoB f ) * ( 1 , j2 + 1 ) `2 .= ( GoB f ) * ( 1 , j2 + 1 ) `2 ;
dom Shift ( Seq q2 , len Seq q1 ) = { j + len Seq q1 where j is Nat : j in dom Seq q1 } & dom Seq q2 = dom Seq q2 & len Seq q2 = len Seq q1 & len Seq q2 = len Seq q2 } ;
consider G1 , G2 , G3 being Element of V such that G1 <= G2 & G2 <= G3 and f is Morphism of G1 , G2 & g is Morphism of G2 , G3 & f is Morphism of G1 , G3 & g is Morphism of G2 , G3 & g is Morphism of G2 , G3 & g is Morphism of G2 , G3 & g is Morphism of G2 , G3 ;
func - f -> PartFunc of C , V means : Def2 : dom it = dom f & for c st c in dom it holds it /. c = - f /. c & for c be Element of C st c in dom it holds it /. c = - f /. c ;
consider phi such that phi is increasing and for a st phi . a = a & {} <> a for v holds union L |= ( ( union L ) | [. v , a .] ) iff L . a , v |= ( ( union L ) | [. v , a .] ) ;
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 1 <= i1 and i1 + 1 <= len GoB f ;
consider i , n such that n <> 0 and sqrt p = ( i / n ) * n and for i1 being Nat , n being Nat st n <> 0 & n <= len p & n <= len p holds n <= i & p . n <= n ;
assume that not 0 in Z and Z c= dom ( ( arccot * ( 1 / 2 ) ) (#) ( ( arccot * ( 1 / 2 ) ) (#) ( ( arccot * ( 1 / 2 ) ) (#) ( ( arccot * ( 1 / 2 ) ) (#) ( ( arccot * ( 1 / 2 ) ) (#) ( ( arccot * ( 1 / 2 ) ) (#) ( 1 / 2 ) ) ) ) and Z = dom f ;
cell ( G1 , i1 -' 1 , ( 2 |^ ( m -' 1 ) ) * ( Y -' 1 ) + 2 ) \ ( ( Y -' 1 ) * ( Y -' 1 ) ) c= BDD L~ f1 & ( ( Y -' 1 ) * ( Y -' 1 ) ) * ( Y -' 1 ) c= BDD L~ f1 ;
ex Q1 being open Subset of [: X , Y :] st s = Q1 & ex F1 being Subset-Family of [: Y , X :] st F1 c= F & F1 is open & [#] ( Y , X ) c= F1 & [#] ( Y , X ) c= F1 & ( for i being Element of X st i in Y holds Q1 . i = F2 . i ) & Q1 is open ) ;
( ( A gcd ( r1 , r2 , s1 , s2 , s2 , Amp ) ) , ( ( A gcd ( A , B , s1 , s2 , s1 ) ) , ( ( A gcd ( A , B , s1 , s2 , s2 ) ) , s2 , t2 ) ) = 1 / ( ( A * ( ( A , B , s1 , s2 ) , s2 ) ) ;
R8 = ( ( ( the Indices of s2 ) * ( 1 + 1 ) ) * ( ( the Indices of s2 ) * ( 1 + 1 ) ) ) . ( m2 + 1 ) .= ( ( ( ( the Indices of s2 ) * ( 1 + 1 ) ) * ( 1 + 1 ) ) * ( ( ( the Indices of s2 ) * ( 1 + 1 ) ) ) . m2 .= [ 3 , 4 ] ;
CurInstr ( P-6 , Comput ( P-6 , s2 , m1 + m3 ) ) = CurInstr ( P3 , Comput ( P3 , s3 , m3 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m3 ) ) .= halt SCMPDS .= halt SCMPDS .= halt SCMPDS .= halt SCMPDS .= halt SCMPDS ;
P1 /\ P2 = ( { p1 } \/ LSeg ( p1 , p11 ) ) /\ ( LSeg ( p11 , p2 ) \/ LSeg ( p11 , p2 ) ) .= ( { p1 } \/ LSeg ( p11 , p2 ) ) /\ ( LSeg ( p11 , p2 ) ) .= { p1 } \/ ( LSeg ( p11 , p2 ) /\ LSeg ( p11 , p2 ) ) ;
func the still of f -> Subset of the carrier of Al means : : : : a in it iff ex i , p st i in dom f & p = f . i & a in the carrier of f & p in the carrier of f & for i st i in dom f holds f . i = p ;
for a , b being Element of F_Complex st |. a .| > |. b .| for f being Polynomial of F_Complex st f >= 1 & f is \cap L~ f holds a * ( - b ) is \/
defpred P [ Nat ] means 1 <= $1 & $1 <= len g implies for i , j st [ i , j ] in Indices G & G * ( i , j ) = g . ( j + 1 ) holds j < i & i < j & j <= len g ;
assume that C1 , C2 are_`2 and for f , g being State of C1 , s1 , s2 being State of C2 st s1 = s2 * f & s2 = s2 * f holds s1 is stable iff s2 is stable & s1 is stable & s2 is stable & for n being Nat st n in dom s1 holds s1 is stable iff s2 is stable & s1 is stable ;
( ||. f .|| | X ) . c = ||. f .|| . c .= ||. f /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| ;
|. q .| ^2 = ( q `1 ) ^2 + ( q `2 ) ^2 & 0 + ( q `1 ) ^2 < ( q `1 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 ;
for F being Subset-Family of T7 st F is open & not {} in F & for A , B being Subset of T7 st A in F & B in F & A <> B holds A misses B & B = F & A misses B & B misses F
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 ;
i |^ ( ( ( ( ( ( ( ( ( p |^ n ) |^ ( i -' 1 ) ) ) ) |^ n ) ) ) ) ) = i |^ ( ( ( ( i |^ ( ( i -' 1 ) ) div ( i -' 1 ) ) ) ) ) .= i |^ ( ( ( ( i - 1 ) div ( i - 1 ) ) ) ) ;
consider q being oriented oriented Chain of G such that r = q and q <> {} and ( F . ( q . 1 ) = v1 & ( F . ( q . len q ) ) = v2 & ( F . ( q . len q ) ) `1 = v2 & ( F . ( q . len q ) ) `1 = v2 ;
defpred P [ Element of NAT ] means $1 <= len ( g , Z ) implies ( ( g , Z ) ^ I ) . $1 = ( ( ( g , Z ) ^ I ) . ( len ( g , Z ) + $1 ) ) . ( len ( g , Z ) + $1 ) ;
for A , B being square Matrix of n , REAL holds len ( A * B ) = len A & width ( A * B ) = width B & width ( A * B ) = width A & width ( A * B ) = width B & width ( A * B ) = width A & width ( 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 & s . i = a & s . b = b & s . i = b ;
func |( x , y )| -> Element of COMPLEX equals |( Re ( x , y ) , ( Re ( x , y ) ) )| - ( i * ( Re ( x , y ) ) ) + ( i * ( ( Im ( x , y ) ) ) ) ;
consider g1 be FinSequence of Fk such that g1 is continuous & rng g1 c= A & g1 . 1 = x1 & g1 . len g1 = x2 and for i be Nat st i in dom g1 & i <= len g1 holds g1 . i = y1 . i and g1 . ( len g1 ) = y2 . i ;
then n1 >= len p1 implies crossover ( p1 , p2 , n1 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n3 p1 , n2 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n2 , n3 , n2 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n3 , n2 , n3 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n2 , n3 , n3 ,
( q `1 ) * a <= q `1 & - ( q `1 ) * a <= q `1 or q `1 >= q `1 & - ( q `1 ) * a <= q `2 & - ( q `2 ) * a <= q `1 or q `1 >= q `1 & - ( q `1 ) * a <= q `2 & - ( q `1 ) * a <= q `2 ) ;
( F . ( p9 . ( len p9 ) ) ) = ( F . ( p . ( len p9 ) ) ) * ( ( F . ( len p9 ) ) .= ( F . ( len p9 ) ) * ( ( F . ( len p9 ) ) ) .= ( ( F . ( len p9 ) ) * ( ( F . ( len p9 ) ) ) .= ( ( F . ( len p9 ) ) * ( ( F . ( len p9 ) ) ) ;
consider k1 being Nat such that k1 + k = 1 and a := k = ( <* a := intloc 0 *> ^ ( ( k --> 1 ) --> ( k --> intloc 0 ) ) ) ^ ( ( k --> 1 ) .--> ( k --> intloc 0 ) ) ^ <* halt SCM+FSA *> ;
consider B8 being Subset of B1 , y8 being Function of B1 , A1 such that B8 is finite and D8 = the carrier of ( A1 , B1 ) and D8 = the carrier of ( A1 , B1 ) and D8 = the carrier of ( A1 , B1 ) and B8 = the carrier of ( A2 , B2 ) ;
v2 . b2 = ( curry ( F2 , g ) * ( ( curry F ) * ( ( the L of B ) . b2 ) ) ) . b2 .= ( ( curry F ) . ( ( ( ( ( ( the carrier of B ) --> g ) . b2 ) ) . b2 ) ) . b2 .= ( ( ( ( ( the carrier of B ) --> id C ) ) . b2 ) ) . b2 .= ( ( ( ( the thesis of C ) --> id C ) ) . b2 ) . b2 ) .= ( ( ( the carrier of C ) . b2 ) . b2 ) . b2 .= ( ( ( the carrier of C ) . b2 ) . b2 ) . b2 .= ( ( ( the carrier of C ) . b2 ) . b2 .= ( ( ( ( the carrier of C ) . b2 ) . b2 ) . b2 ) . b2 ) . b2 ) . b2 ) .
dom IExec ( I , P , Initialize s ) = the carrier of SCMPDS .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) ;
ex d-32 be Real st d-32 > 0 & for h be Real st h <> 0 & |. h .| < d-32 holds |. h .| " * ||. ( R2 * ( L + R1 ) ) /. h .|| < ( e / ( 1 + h ) ) * ||. ( R2 * ( L + R1 ) ) /. h .|| ;
LSeg ( G * ( len G , 1 ) + |[ 1 , 0 ]| , G * ( len G , 1 + 1 ) ) c= Int cell ( G , len G , 1 ) \/ { G * ( len G , 1 ) } ;
LSeg ( mid ( h , i1 , i2 ) , i ) = LSeg ( h /. ( i + i1 -' 1 ) , h /. ( i + i1 -' 1 ) ) .= LSeg ( h /. ( i + i1 -' 1 ) , h /. ( i + i1 -' 1 ) ) .= LSeg ( h /. ( i + i1 -' 1 ) , h /. ( i + i1 -' 1 ) ) ;
A = { q where q is Point of TOP-REAL 2 : LE p1 , q , P , p1 , p2 & LE q , p1 , P , p1 , p2 & LE q , p1 , P , p1 , p2 & LE q , p1 , P , p1 , p2 } ;
( ( - x ) .|. y ) = - ( ( - 1 ) * ( x .|. y ) ) * ( x .|. y ) .= ( - ( - 1 ) * ( x .|. y ) ) * ( x .|. y ) .= ( ( - 1 ) * ( x .|. y ) ) * ( x .|. y ) .= ( ( - 1 ) * ( x .|. y ) ) * ( x .|. y ) .= ( ( - 1 ) * ( x .|. y ) ) * ( x .|. y ) ;
0 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) = ( p `2 ) ^2 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) .= ( p `2 ) ^2 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) .= ( p `2 ) ^2 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) ;
( ( U * W ) * ( W7 ) ) * ( W * ( *> ) ) = ( ( ( U * W ) * ( W * ( W * ( W * n ) ) ) ) * ( W * ( W * n ) ) ) .= ( ( W * ( W * n ) ) * ( W * n ) ) * ( W * n ) .= ( W * ( W * n ) ) * ( W * n ) .= W * ( W * n ) ;
func Shift ( f , h ) -> PartFunc of REAL , REAL means : Def2 : dom it = - h & for x st x in dom h holds it . x = - h . x & for x st x in dom h holds it . x = ( - h . x ) * ( h . x ) ;
assume that 1 <= k and k + 1 <= len f and [ i , j ] in Indices G and f /. k = G * ( i + 1 , j ) and f /. ( k + 1 ) = G * ( i , j ) and f /. ( k + 1 ) = G * ( i , j ) ;
assume that not y in Free H and not x in Free H and ( not x in Free H ) and not x in Free H and not x in Free H and not x in Free H and not x in Free H and not x in Free H and not x in Free H and not x in Free H and not x in Free H ;
defpred P11 [ Element of NAT , Element of NAT , Element of NAT ] means ( ( ( p |^ $1 ) * ( ( p |^ $1 ) * ( ( p |^ ( $1 + 1 ) ) ) ) ) * ( ( ( p |^ ( $1 + 1 ) ) * ( ( p |^ ( $1 + 1 ) ) ) ) ) = ( p |^ ( $1 + 1 ) ) * ( ( p |^ ( $1 + 1 ) ) ) ;
func \sigma ( C ) -> non empty Subset-Family of X means : Def2 : for A being Subset of X holds A in it iff for W being Subset of X st W c= A \ A & W c= C \ A holds C . W <= C . ( W \/ A ) ;
[#] ( ( dist ( ( ( ( ( ( ( ( Q ) ) ) ) ) .: Q ) ) ) .: Q ) ) = ( ( dist ( ( ( ( ( Q ) ) ) .: Q ) ) ) .: Q ) ) ) & lower_bound ( ( ( ( ( ( ( ( ( Q ) ) .: Q ) ) .: Q ) ) .: Q ) ) = lower_bound ( ( ( ( ( ( ( ( Q ) ) .: Q ) ) .: Q ) ) ) ) ) ;
rng ( F | ( S |^ 2 ) ) = {} or rng ( F | ( S |^ 2 ) ) = { 1 } or rng ( F | ( S |^ 2 ) ) = { 2 } or rng ( F | ( S |^ 2 ) ) = { 1 } or rng ( F | ( S |^ 2 ) ) = { 2 } ;
( f " ( rng f ) ) . i = f . i " ( ( ( rng f ) . i ) ) .= f . i " ( ( rng f ) . i ) .= ( f " ( ( rng f ) . i ) ) " ( ( ( 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 P2 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 = P1 \/ P2 and P2 = P1 \/ P2 and P1 = P2 \/ P2 and P2 = P1 \/ P2 and P1 = P2 \/ P2 and P1 = P1 \/ P2 and P2 = P1 \/ P2 and P1 = P2 \/ P2 and P1 = P1 \/ P2 and P1 = P2 \/ P2 ;
f . p2 = |[ ( p2 `1 ) ^2 / sqrt ( 1 + ( p2 `2 / p2 `1 ) ^2 ) , ( p2 `2 ) ^2 / sqrt ( 1 + ( p2 `2 / p2 `1 ) ^2 ) ]| .= |[ ( p2 `1 ) ^2 / sqrt ( 1 + ( p2 `2 / p2 `1 ) ^2 ) , ( p2 `2 ) ^2 / sqrt ( 1 + ( p2 `2 / p2 `1 ) ^2 ) ]| ;
( ( \mathbin ( a , X ) ) " ) . x = ( ( \mathbin ( a , X ) qua Function ) " . x .= ( ( ( the carrier of X ) qua Function ) qua Function ) . x .= ( ( - a ) qua Function ) . x .= - a + u .= - a + u .= - a + u .= - a + u .= - a + u .= - a + u .= - a + u .= - a ;
for T being non empty normal TopSpace , A , B being closed Subset of T st A <> {} & A misses B for p being Point of T , r being Real st r in B & p in B & r in B holds ( in > G ) . p = ( ( in > G ) . p ) * ( ( in B ) . p )
for i , i st i + 1 in dom F for G1 , G2 being strict normal Subgroup of G st G1 = F . i & G2 = F . ( i + 1 ) & G2 = F . ( i + 1 ) & for i being Element of NAT st i in dom G1 holds G1 . i = G1 . ( i + 1 ) & G1 . i = G2 . i & G1 . ( i + 1 ) = G2 . ( i + 1 ) holds G1 is strict Subgroup of G1
for x st x in Z holds ( ( ( 1 / 2 ) (#) ( ( arccot ) * ( arccot ) ) `| Z ) . x = ( ( ( 2 / 2 ) * ( ( arccot ) * ( arccot ) ) `| Z ) . x / ( 1 + x ^2 ) )
synonym f is_continuous x0 means : Def2 : x0 in dom f & for a st rng a c= ]. x0 , x0 + r .[ & a is convergent & lim a = x0 & for x0 st x0 in dom f & x0 <> 0 holds f /. x0 = lim ( f /* a ) ;
then X1 , X2 are_separated implies ( X1 union X2 ) is SubSpace of X & ( X1 , X2 are_separated implies X1 , X2 are_separated ) & ( X1 , X2 are_separated implies X1 , X2 are_separated ) & ( X1 , X2 are_separated implies X1 , X2 are_separated ) & ( X1 , X2 are_separated implies X1 , X2 are_separated ) ;
ex N being Neighbourhood of x0 st N c= dom SVF1 ( 1 , f , u ) & ex L , R st for x st x in N holds SVF1 ( 1 , f , u ) . x - SVF1 ( 1 , f , u ) . x0 = L . ( x - x0 ) + R . ( x - x0 )
( p2 `1 ) * sqrt ( 1 + ( p3 `1 ) ^2 ) >= ( ( p3 `1 ) ^2 * sqrt ( 1 + ( p3 `2 ) ^2 ) ) * sqrt ( 1 + ( p3 `2 ) ^2 ) .= ( ( p3 `1 ) ^2 * sqrt ( 1 + ( p3 `2 ) ^2 ) ) * sqrt ( 1 + ( p3 `1 ) ^2 ) ;
( ( 1 / t1 ) (#) ( ||. f1 .|| ) ) . x = ( ( 1 / t2 ) (#) ( ||. f1 .|| ) ) . x & ( ( 1 / t2 ) (#) ( ||. g1 .|| ) ) . x = ( ( 1 / t2 ) (#) ( ( ( 1 / t2 ) (#) ( ||. f1 .|| ) ) ) . x ;
assume that for x holds f . x = ( ( - 1 ) (#) ( ( cot * f ) `| Z ) . x ) and x in dom ( ( 1 / 2 ) (#) ( ( cot * f ) `| Z ) ) and for x st x in Z holds ( ( ( 1 / 2 ) (#) ( ( cot * f ) `| Z ) ) `| Z ) . x = ( 1 / ( sin . x ) ^2 ) / ( sin . x ) ^2 ;
consider Xj1 being Subset of [: Y , Y :] , Y1 being Subset of X such that t = [: Xj1 , Y :] and Y1 is open and ex Y1 being Subset of Y , Y2 being Subset of Y st Y1 = [: Y1 , Y2 :] & Y1 is open & Y2 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open ;
card ( S . n ) = card { [: d , Y :] + b where d is Element of GF ( p ) : [ d , 1 ] in R & [ d , 1 ] in R } .= { d + 1 where d is Element of GF ( p ) : [ d , 1 ] in R } .= { d + 1 where d is Element of GF ( p ) : d in R } ;
( ( ( W-bound D ) - ( W-bound D ) ) / ( 2 |^ m ) ) * ( ( W-bound D ) - ( W-bound D ) ) = ( ( W-bound D ) - ( W-bound D ) ) * ( ( W-bound D ) - ( W-bound D ) ) .= ( ( W-bound D ) - ( W-bound D ) ) * ( ( W-bound D ) - ( W-bound D ) ) .= ( ( W-bound D ) - ( W-bound D ) ) * ( ( W-bound D ) ) ;