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 ;
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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 ;
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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 ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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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 ;
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thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b
D c= S
let Y ;
S `1 is convergent ;
q in X ;
V ;
y in N ;
x in T ;
m < n ;
m <= n ;
n > 1 ;
let r ;
t in I ;
n <= 4 ;
M is finite ;
let X ;
Y c= Z ;
A // M ;
let U ;
a in D ;
q in Y ;
let x ;
1 <= l ;
1 <= w ;
let G ;
y in N ;
f = {} ;
let x ;
x in Z ;
let x ;
F is one-to-one ;
e <> b ;
1 <= n ;
f is special ;
S misses C
t <= 1 ;
y divides m ;
P divides M ;
let Z ;
let x ;
y c= x ;
let X ;
let C ;
x _|_ p ;
o is monotone ;
let X ;
A = B ;
1 < i ;
let x ;
let u ;
k <> 0 ;
let p ;
0 < r ;
let n ;
let y ;
f is onto ;
x < 1 ;
G c= F ;
a is_>=_than X ;
T is continuous ;
d <= a ;
p <= r ;
t < s ;
p <= t ;
t < s ;
let r ;
D is_<=_than E ;
assume e > 0 ;
assume 0 < g ;
p in X ;
x in X ;
Y ` in Y ;
assume 0 < g ;
not c in Y ;
not v in L ;
2 in z `1 ;
assume f = g ;
N c= b ` ;
assume i < k ;
assume u = v ;
I = J ;
B ` = b ` ;
assume e in F ;
assume p > 0 ;
assume x in D ;
let i be element ;
assume F is onto ;
assume n <> 0 ;
let x be element ;
set k = z ;
assume o = x ;
assume b < a ;
assume x in A ;
a `1 <= b `1 ;
assume b in X ;
assume k <> 1 ;
f = Product l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is from of squares ;
assume m > 0 ;
assume A c= B ;
X is lower
assume A <> {} ;
assume X <> {} ;
assume F <> {} ;
assume G is open ;
assume f is dilatation ;
assume y in W ;
y \not <= x ;
A ` in B ` ;
assume i = 1 ;
let x be element ;
x `1 = x `1 ;
let X be BCK-algebra ;
assume S is non empty ;
a in REAL ;
let p be set ;
let A be set ;
let G be _Graph , 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 be Element of C ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
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 ;
k1 be Integer ;
let X be set ;
let a be element ;
let x be element ;
let x be element ;
let q be element ;
let x be element ;
assume f is being_homeomorphism ;
let z be element ;
a , b // K ;
let n be Nat ;
let k be Nat ;
B ` c= B ` ;
set s = / ( 1 - s ) ;
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 . x in dom f ;
j < i0 ;
let j be Nat ;
n <= n + 1 ;
k = i + m ;
assume C meets S ;
n <= n + 1 ;
let n be Nat ;
h1 = {} ;
0 + 1 = 1 ;
o <> b3 ;
f2 is one-to-one ;
support p = {}
assume x in Z ;
i <= i + 1 ;
r1 <= 1 ;
let n be Nat ;
a "/\" b <= a ;
let n be Nat ;
0 <= r0 ;
let e be Real , x be Point of TOP-REAL 2 ;
not r in G . l
c1 = 0 ;
a + a = a ;
<* 0 *> in e ;
t in { t } ;
assume F is non discrete ;
m1 divides m ;
B * A <> {} ;
a + b <> {} ;
p * p > p ;
let y be ExtReal ;
let a be Int-Location , f be FinSeq-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 ;
cluster waybelow x -> non empty ;
X <> { x } ;
x in { x } ;
q , b // M ;
A . i c= Y ;
P [ k ] ;
2 to_power x in W ;
X [ 0 ] ;
P [ 0 ] ;
A = A |^ i ;
2 >= \bf \cdot s ;
G . y <> 0 ;
let X be RealNormSpace , x be Point of X ;
a in X ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , M be Subset of V ;
assume x in - M ;
k < s . a ;
not t in { p } ;
let Y be set , f be Function ;
M , L are_equipotent ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded & L is lower-bounded ;
rng f = Y ;
( G \/ { x } ) c= L ;
assume x in Cl Q ;
m in dom P ;
i <= len Q ;
len F = 3 ;
still_not-bound_in p = {} ;
z in rng p ;
lim b = 0 ;
len W = 3 ;
k in dom p ;
k <= len p ;
i <= len p ;
1 in dom f ;
b `1 = a `1 + 1 ;
x `1 = a * y `1 ;
rng D c= A ;
assume x in K1 ;
1 <= i-32 ;
1 <= i-32 ;
pare c= PI ;
1 <= ii & ii <= len G ;
1 <= ii & ii <= len G ;
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 or b1 >= Z1 ;
assume w = 0. V ;
assume x in A . i ;
g in being being being _ of X ;
y in dom t ;
i in dom g ;
assume P [ k ] ;
if C c= f , f , g holds 0 <= f
x9 is increasing & y9 is increasing ;
let e2 be element ;
- b divides b ;
F c= \tau ( F ) ;
( G is non-decreasing ) implies G is non-decreasing
( G is non-decreasing ) implies G is non-decreasing
assume v in H . m ;
assume b in [#] B ;
let S be non void ManySortedSign , f be Function ;
assume P [ n ] ;
assume union S is finite & finite S is finite ;
V is Subspace of V ;
assume P [ k ] ;
rng f c= NAT \/ { x }
assume ex_inf_of X , L ;
y in rng f ;
let s , I be set , F be Function ;
b ` ` c= b9 ` ;
assume not x in REAL ;
A /\ B = { a } ;
assume len f > 0 ;
assume x in dom f ;
b , a // o , c ;
B in B-24 ;
cluster product p -> non empty ;
z , x // x , p ;
assume x in rng N ;
cosec is_differentiable_in x & cosec is_differentiable_in x ;
assume y in rng S ;
let x , y be element ;
i2 < i1 & i1 <= len f ;
a * h in a * H ;
p , q in Y ;
redefine func sqrt I -> Ideal of L ;
q1 in A1 & q2 in A2 ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A2 c= A1 ;
an < n & bn < n ;
assume A c= dom f ;
Re ( f ) is_integrable_on M ;
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 a non-empty ManySortedSet of X ;
assume k in dom s ;
let r be non negative Real ;
assume x in F | M ;
let Y be Subset of S ;
let X be non empty TopSpace , A be Subset of X ;
[ a , b ] in R ;
x + w < y + w ;
{ a , b } >= c ;
let B be Subset of A , C be Subset of B ;
let S be non empty ManySortedSign ;
let x be variable , f be Function ;
let b be Element of X , x be Element of X ;
R [ x , y ] ;
x ` ` = x ;
b \ x = 0. X ;
<* d *> in D * ;
P [ k + 1 ] ;
m in dom ( n |-> 0 ) ;
h2 . a = y ;
P [ n + 1 ] ;
redefine func G * F -> ] Functor ;
let R be non empty multMagma , x be Element of R ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p ;
assume f | X is lower ;
x in rng co /\ L~ co ;
let x be Element of B ;
let t be Element of D ;
assume x in Q .vertices() ;
set q = s ^\ k ;
let t be VECTOR of X ;
let x be Element of A ;
assume y in rng p `1 ;
let M be as as as as mamaid id ;
let N be non empty \HM { the is non empty \HM { of M } } ;
let R be RelStr with finite is finite ;
let n , k be Nat ;
let P , Q be be be be be be be let let RelStr ;
P = Q /\ [#] S ;
F . r in { 0 } ;
let x be Element of X ;
let x be Element of X ;
let u be VECTOR of V ;
reconsider d = x as Int-Location ;
assume I is not \leq a ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < ( v + u ) ;
x <= c2 . x ;
x in F ` & y in F ` ;
redefine func S --> T -> ManySortedSet of I ;
assume that t1 <= t2 and t2 <= t2 ;
let i , j be even Integer ;
assume that F1 <> F2 and F2 <> F1 ;
c in Intersect ( union R ) ;
dom p1 = c & dom p2 = c ;
a = 0 or a = 1 ;
assume A1 <> A2 & A2 <> A1 ;
set i1 = i + 1 ;
assume a1 = b1 & a2 = b2 ;
dom g1 = A & dom g2 = B ;
i < len M + 1 ;
assume not -infty in rng G ;
N c= dom f1 /\ dom f2 ;
x in dom ( sec | dom sec ) ;
assume [ x , y ] in R ;
set d = ( x - y ) / ( x - y ) ;
1 <= len g1 & 1 <= len g2 ;
len s2 > 1 & len s2 > 1 ;
z in dom ( f1 | X ) ;
1 in dom ( D2 | indx ( D2 , D1 , j1 ) ) ;
( p `2 ) ^2 = 0 ;
j2 <= width G & 1 <= j1 ;
len PI > 1 + 1 ;
set n1 = n + 1 ;
|. q-35 .| = 1 ;
let s be SortSymbol of S ;
( n gcd i ) = i ;
X1 c= dom f & X1 c= dom g ;
h . x in h . a ;
let G be V be thesis , F be Subset of on ;
cluster m * n -> square ;
let k9 be Nat , x be Element of X ;
i - 1 > m - 1 ;
R is transitive implies R is transitive
set F = <* u , w *> ;
p-2 c= P3 & p`2 c= P3 ;
I is_closed_on t , Q ;
assume [ S , x ] is thesis ;
i <= len ( f2 ^ <* p *> ) ;
p is FinSequence of X ;
1 + 1 in dom g ;
Sum R2 = n * r ;
cluster f . x -> complex-valued ;
x in dom f1 /\ dom f2 ;
assume [ X , p ] in C ;
BX c= XX & BX c= X ;
n2 <= ( 2 * n ) / 2 ;
A /\ cP c= A ` ;
cluster x -valued -> constant for Function ;
let Q be Subset-Family of S , P be Subset of S ;
assume n in dom ( g2 | n ) ;
let a be Element of R ;
t `1 in dom ( e2 `1 ) ;
N . 1 in rng N ;
- z in A \/ B ;
let S be SigmaField of X , M be Element of S ;
i . y in rng i ;
[: REAL , REAL :] c= dom f ;
f . x in rng f ;
mt <= ( r / 2 ) ;
s2 in r-5 & s2 in r-5 ;
let z , z be complex number ;
n <= NN . m ;
LIN q , p , s ;
f . x = waybelow x /\ B ;
set L = |[ S \to T ]| ;
let x be non positive ExtReal ;
let m be Element of M ;
f in union rng ( F1 | n ) ;
let K be add-associative right_zeroed right_complementable non empty doubleLoopStr , M be Matrix of K ;
let i be Element of NAT ;
rng ( F * g ) c= Y
dom f c= dom x & dom g c= dom x ;
n1 < n1 + 1 & n1 + 1 <= n1 ;
n1 < n1 + 1 & n1 + 1 <= n1 ;
cluster <% T %> -> \overline { T } ;
[ y2 , 2 ] `1 = z ;
let m be Element of NAT ;
let S be Subset of R ;
y in rng ( S29 | X ) ;
b = sup dom f & b = sup dom f ;
x in Seg ( len q ) ;
reconsider X = D ( ) as set ;
[ a , c ] in E1 ;
assume n in dom ( h2 | n ) ;
w + 1 = ( a - 1 ) * a ;
j + 1 <= j + 1 + 1 ;
k2 + 1 <= k1 & k1 <= k2 ;
let i be Element of NAT ;
Support u = Support p & Support u = Support p ;
assume X is complete \frac m ;
assume that f = g and p = q ;
n1 <= n1 + 1 & n1 <= n2 + 1 ;
let x be Element of REAL ;
assume x in rng ( s2 | X ) ;
x0 < x0 + 1 & x0 + 1 < x0 + 1 ;
len ( L5 ^ L6 ) = W ;
P c= Seg ( len A ) ;
dom q = Seg n & dom q = Seg n ;
j <= width M @ ;
let seq1 be real-valued sequence of X ;
let k be Element of NAT ;
Integral ( M , P ) < +infty ;
let n be Element of NAT ;
assume z in being \tt at-LSeg ( 0 , A ) ;
let i be set ;
n - 1 = n-1 - 1 ;
len ( n * ( - n ) ) = n ;
\mathop { \rm \cal Z } c= F
assume x in X or x = X ;
x is midpoint of b , c ;
let A , B be non empty set , f be Function of A , B ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q & i in dom q ;
let s be Element of E * ;
let B1 be Basis of x , B2 be Basis of y ;
L3 /\ L2 = {} & L3 /\ L2 = {} ;
L1 /\ LSeg ( L2 , p2 ) = {} ;
assume downarrow x = downarrow y ;
assume b , c , b is_collinear ;
LIN q , c , c ;
x in rng ( f . -129 ) ;
set n8 = n + j ;
let D7 be non empty set , f be Function of D , REAL ;
let K be right_zeroed non empty addLoopStr , M be Matrix of K ;
assume that f `1 = f and h `2 = h ;
R1 - R2 is total & R1 - R2 is total ;
k in NAT & 1 <= k ;
let a be Element of G ;
assume x0 in [. a , b .] ;
K1 ` is open & K1 ` is open ;
assume a , b ) is maximal distance ;
let a , b be Element of S ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = Integral ( M , f ) ;
cluster n[ ] -> nes] for ;
not u in { ag } ;
the carrier of f c= B \/ { v }
reconsider z = x as VECTOR of V ;
cluster the Str of L -> \rangle ;
r (#) H is C [ 0 ] ;
s . intloc 0 = 1 ;
assume that x in C and y in C ;
let U0 be strict universal MSAlgebra over S , a be Element of U0 ;
[ x , Bottom T ] is compact ;
i + 1 + k in dom p ;
F . i is stable Subset of M ;
r-35 in : ( x in { y } ) & r in { x } ;
let x , y be Element of X ;
let A , I be such such that A is { \rm \hbox { - } set } ;
[ y , z ] in [: O , O :] ;
( card Macro i ) = 1 & card Macro i = 1 ;
rng Sgm ( A ) = A ;
q |- \!
for n holds X [ n ] ;
x in { a } & x in d ;
for n holds P [ n ] ;
set p = |[ x , y , z ]| ;
LIN o , a , b & LIN o , a , b ;
p . 2 = Z / Y ;
( DD ) `2 = {} ;
n + 1 + 1 <= len g ;
a in [: CQC-WFF ( Al ) , NAT :] ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
let I be Ideal of L ;
set g = f1 + f2 , h = g1 + g2 ;
a <= max ( a , b ) ;
i-1 < len G + 1-1 ;
g . 1 = f . i1 ;
x `1 , y `2 in A2 ;
( f /* s ) . k < r ;
set v = VAL g ;
i -' k + 1 <= S ;
cluster associative -> invertible for non empty multMagma ;
x in support ( ( support ( t ) ) * ( b * x ) ) ;
assume a in [: the carrier of G ( ) , the carrier of G ( ) :] ;
i `1 <= len ( y `1 ) ;
assume that p divides b1 + b2 and p divides b1 + b2 ;
M <= sup M1 & M <= sup M2 ;
assume x in W-min ( X ) & y in L~ f ;
j in dom ( z | k ) ;
let x be Element of D ( ) ;
IC s4 = l1 & IC s4 = l ;
a = {} or a = { x } ;
set uG = Vertices G , uG = Vertices G ;
seq " is non-zero & seq " is non-zero implies seq " is non-zero
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
hcn c= h-14 & hK1 c= dom h ;
]. a , b .[ c= Z ;
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 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 = x as Element of S ;
max ( a , b ) = a ;
sup B is upper & sup B is upper ;
let L be non empty reflexive antisymmetric RelStr , x be Element of L ;
R is reflexive & R is transitive ;
E , g |= the_right_argument_of ( H , E ) ;
dom G `2 /. y = a ;
( 1 / 4 ) * ( 1 / 4 ) >= - r ;
G . p0 in rng G & G . p0 in rng G ;
let x be Element of FF , y be Element of F ;
D [ P-6 , 0 ] & D [ D ] ;
z in dom ( id B ) & z in dom ( id B ) ;
y in the carrier of N & y in the carrier of N ;
g in the carrier of H & h in the carrier of H ;
rng ( f | X ) c= [: NAT , NAT :] ;
j `2 + 1 in dom s1 & j + 1 in dom s1 ;
let A , B be strict Subgroup of G ;
let C be non empty Subset of REAL ;
f . z1 in dom h & h . z2 in dom h ;
P . k1 in rng P & P . k1 in rng P ;
M = A9 +* ( {} , { {} } ) ;
let p be FinSequence of REAL , n be Nat ;
f . n1 in rng f & f . n1 in rng g ;
M . ( F . 0 ) in REAL ;
holds holds holds not ( |. a - b .| ) = - ( a - b ) ;
assume the distance of V , Q is_\mathbb v ;
let a be Element of ^ ( V ) ;
let s be Element of PL , x be Element of PL ;
let Pa be non empty thesis , P be Subset of X ;
n be Nat ;
the carrier of g c= B & the carrier of g c= B ;
I = halt SCM R & I = halt SCM R ;
consider b being element such that b in B ;
set BK = BCS ( K , n ) ;
l <= j & j <= j implies j <= sup ( F . j )
assume x in downarrow [ s , t ] ;
( x `2 ) `2 in uparrow t & ( x `2 ) in uparrow t ;
x in ( JumpParts T ) \/ { {} } ;
let h be Morphism of c , a ;
Y c= [: R , R :] & Y c= [: R , R :] ;
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 & dom <* y *> = Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar s .| ;
[ x , x `2 ] in [: X , X :] ;
for n be Nat holds 0 <= x . n
[' a , b '] = [. a , b .] ;
cluster -> non closed for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 in P & q2 , q1 is_collinear ;
dom M1 = Seg n & dom M2 = Seg n ;
x = [ x1 , x2 ] & y = [ y1 , y2 ] ;
R , Q be ManySortedSet of A ;
set d = ( 1 / ( n + 1 ) ) ;
rng ( g2 ) c= dom W & ( g2 ) . x = 0 ;
P . ( [#] Sigma \ B ) <> 0 ;
a in field R & a = b ;
let M be non empty Subset of V , v be Element of V ;
let I be Program of SCM+FSA , a be Int-Location ;
assume x in rng ( the InternalRel of R ) ;
let b be Element of the carrier of T ;
dist ( e , z ) - r-r > 0 ;
u1 + v1 in W2 & v1 in W2 + W3 ;
assume func support L -> Subset of rng G ;
let L be lower-bounded antisymmetric transitive antisymmetric transitive 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 Bool M , a be Element of M ;
0 <= Arg a & Arg a < 2 * PI ;
o , a9 // o , y & o , a // o , y ;
{ v } c= the carrier of l & { v } c= the carrier of l ;
let x be variable of A ;
assume x in dom ( uncurry f ) & y in dom ( uncurry f ) ;
rng F c= ( product f ) |^ X ;
assume that D2 . k in rng D and D2 . k in rng D ;
f " . p1 = 0 & f " . p2 = 0 ;
set x = the Element of X , y = the Element of Y ;
dom Ser ( G ) = NAT & rng Ser ( G ) = NAT ;
n be Element of NAT ;
assume LIN c , a , e1 & LIN c , a , e1 ;
cluster -> natural for FinSequence of NAT ;
reconsider d = c , e = d as Element of L1 ;
( v2 |-- I ) . X <= 1 ;
assume x in the carrier of f & y in the carrier of g ;
conv @ S c= conv ( A \/ B ) ;
reconsider B = b as Element of the topology of T ;
J , v |= P \lbrack UP . ( l + 1 ) , v .] ;
redefine func J . i -> non empty TopSpace ;
ex_sup_of Y1 \/ Y2 , T & ex_sup_of Y1 \/ Y2 , T ;
W1 , W2 are_well \rbrace & W1 , W2 are_not contradiction ;
assume x in the carrier of R & y in the carrier of R ;
dom ( n |-> 0 ) = Seg n & dom ( n --> 0 ) = Seg n ;
s4 misses s2 & s4 misses s4 ;
assume ( a 'imp' b ) . z = TRUE ;
assume that X is open and f = X --> d ;
assume [ a , y ] in 0. ( f ) ;
assume that } c= dom } I and { d } c= K ;
Im ( lim seq ) = 0 & Im ( lim seq ) = 0 ;
( sin . x ) ^2 <> 0 & ( sin . x ) ^2 <> 0 ;
sin is_differentiable_on Z & cos is_differentiable_on Z implies cos * sin is_differentiable_on Z & for x st x in Z holds ( ( cos * sin ) `| Z ) . x = sin . x / x + cos . x / x /
6 . n = t3 . n & 6 . n = 1 ;
dom ( ( dom ( F | A ) ) ) c= dom F ;
W1 . x = W2 . x & W1 . 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 .: P c= P ;
h . p4 = g2 . I & h . p2 = g2 . I ;
( G /. 1 ) `1 = U /. 1 .= G * ( 1 , k ) `1 ;
f . rr1 in rng f & f . rr1 in rng f ;
i + 1 + 1 - 1 <= len f - 1 ;
rng F = rng ( F | ( Seg k ) ) ;
mode then f2 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 m c= B & the carrier of m c= B ;
not [ y , x ] in id ( X ) ;
1 + p .. f <= i + len f ;
seq ^\ k1 is lower & seq ^\ k1 is lower ;
len ( F /. i ) = len I & len ( F /. i ) = len I ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be Complex , x be Element of X ;
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 of be \langle T *> ;
cluster directed-sups-preserving -> directed-sups-preserving for Function of L , L ;
f . j1 in K . j1 & f . j2 in K . j1 ;
redefine func J => y -> total for I -valued Function ;
K c= 2 -tuples_on the carrier of T ;
F . b1 = F . b2 & F . b2 = F . b1 ;
x1 = x or x1 = y or x1 = z ;
pred a <> {} means : Def1 : ( a - 1 ) / a = 1 ;
assume that a c= b and b in a ;
s1 . n in rng s1 & s1 . n in rng s1 ;
{ o , b2 } on C2 & { o , b2 } on C2 ;
LIN o , b , b9 & LIN o , b , b ;
reconsider m = x as Element of Funcs ( V , E ) ;
let f be non constant non trivial FinSequence of D ;
let FF2 be non empty seq of X ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in F-8 ;
reconsider pp2 = x , pp2 = y as Subset of m ;
A , B , C be Element of R ;
redefine func strict non empty for ^ is strict non empty be \rbrace ;
rng c `1 misses rng ( e `1 ) & rng c `1 c= rng ( e `2 ) ;
z is Element of gr { x } & z is Element of gr { x } ;
not b in dom ( a .--> p1 ) ;
assume that k >= 2 and P [ k ] ;
Z c= dom ( cot * ( f1 - f2 ) ) ;
the component of Q c= UBD A & the component of Q c= UBD A ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( 1 / ( 1 + x ^2 ) ) ;
pred f = u means : Def1 : a * f = a * u ;
for n holds P1 [ n ] implies P1 [ n + 1 ]
{ x . O : x in L } <> {} ;
let x be Element of V . s ;
a , b be Nat ;
assume that S = S2 and p = p2 and q = p1 ;
gcd ( n1 , n2 , k ) = 1 & gcd ( n1 , n2 , k ) = 1 ;
set ok = * ( i , k ) , ok = ( - 1 ) * ( i , k ) ;
seq . n < |. r1 .| & |. seq . n .| < r ;
assume that seq is increasing and r < 0 ;
f . ( y1 , x1 ) <= a & f . ( y1 , x1 ) <= b ;
ex c being Nat st P [ c ] & c <= n ;
set g = { n to_power 1 where n is Element of NAT : n in NAT } ;
k = a or k = b or k = c ;
a9 , b9 , c9 , a9 , b9 , c9 , b9 , c9 , a9 , b9 , c9 , c9 , a9 , b9 , c9 , a9 , b9 , c9 , a9 , b9 , c9 , c9 , a9 , b9
assume that Y = { 1 } and s = <* 1 *> ;
IS1 . x = f . x .= 0 .= 0. ( TOP-REAL 2 ) ;
W3 .last() = W3 . 1 & W3 .last() = W3 . 1 ;
cluster non trivial finite -> finite for Walk of G ;
reconsider u = u as Element of Bags X ;
A in B ^ implies A , B ^ ;
x in { [ 2 * n + 3 , k ] } ;
1 >= ( q `1 / |. q .| - cn ) / ( 1 - cn ) ;
f1 is_as as as as as as as as _ _ _\leq ( f1 ^ f2 ) ;
( f /. 1 ) `2 <= ( q `2 ) * ( 1 + ( q `2 / q `1 ) ^2 ) ;
h is_the carrier' of Cage ( C , n ) ;
b `2 <= p `2 & p `2 <= p `2 or b `2 >= p `2 & p `2 <= p `2 ;
let f , g be Function of X , Y ;
S * ( k , k ) <> 0. K ;
x in dom ( max ( - f , g ) ) ;
p2 in NO . p1 & p2 in NO . p2 ;
len ( the_right_argument_of H ) < len ( H ) & len ( H ) = len ( H ) ;
F [ A , FF . A ] & F . A = F . ( A , F . A ) ;
consider Z such that y in Z and Z in X ;
pred 1 in C means : Def1 : A c= C |^ A ;
assume that r1 <> 0 or r2 <> 0 and r1 < r2 ;
rng q1 c= rng C1 & rng q2 c= rng C2 ;
A1 , L , A3 , A3 is_collinear & A1 , L , A2 is_collinear ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in C & c in C implies a , b , c , d is_collinear
then S is atomic & P-2 [ S ] ;
Cl ( Int [#] T ) = [#] T & Cl ( Int ( T ) ) = [#] T ;
f12 | A2 = f2 | A1 & f12 | A2 = f2 | A2 ;
0. M in the carrier of W & 0. M in the carrier of W ;
v , v be Element of M ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ V & V c= Y \ Z ;
let X be Subset of [: S , T :] ;
consider H1 such that H = 'not' H1 and H1 in X ;
1_ 1 c= ( ( t * ( p - 1 ) ) / ( p - 1 ) ) ;
0 * a = 0. R .= a * 0 ;
A |^ ( 2 , 2 ) = A ^^ ( A * ) ;
set vbeing /. n = ( v /. n ) `1 , vn = ( v /. n ) `1 ;
r = 0. ( REAL-NS n ) & ||. 0. ( REAL-NS n ) .|| = 0 ;
( f . p4 ) `1 >= 0 & ( f . p2 ) `2 >= 0 ;
len W = len ( W | ( W . n ) ) .= len W ;
f /* ( s * G ) is divergent_to+infty ;
consider l be Nat such that m = F . l ;
t8 e e e e e e & not e in dom b1 & not e in dom b1 ;
reconsider Y1 = X1 , Y2 = X2 as SubSpace of X ;
consider w such that w in F and not x in w ;
let a , b , c , d be Real ;
reconsider i = i as non zero Element of NAT ;
c . x >= id ( L . x ) ;
\sigma ( T ) \/ omega ( T ) is Basis of T ;
for x being element st x in X holds x in Y
cluster [ x1 , x2 ] -> non pair for element ;
downarrow a /\ downarrow t is Ideal of T & a /\ downarrow t is Ideal of T
let X be set , NAT , f be non empty set ;
rng f = being \rm being \rm as \rm <* of S , X ;
let p be Element of B , x be Element of the carrier' 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 = q1 & p2 = q2 & p1 = p2 ;
assume that gRR1 in the right of g and gRR1 in the carrier of g ;
let A1 , A2 be Point of S , A be Subset of S ;
x in h " P /\ [#] T1 & x in h " P /\ [#] T1 ;
1 in Seg 2 & 1 in Seg 3 & 2 in Seg 3 ;
reconsider X-5 = X , X\rangle as non empty Subset of Tsuch that X = { y } ;
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the Source of G ) -valued ;
n1 <= i2 + len g2 & n1 <= len g2 implies ( f /. n1 ) `1 <= ( f /. ( n1 + 1 ) ) `1
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume that v in the carrier' of G2 and u in the carrier' of G2 ;
y = Re y + ( Im y ) * i ;
( - ( ( - 1 ) * p ) ) ^2 = 1 ;
x2 is differentiable & ( for a st a in ]. a , b .[ holds a - b <= x ) implies x2 is differentiable
rng M5 c= rng D2 & rng M5 c= rng D1 ;
for p being Real st p in Z holds p >= a
( ( X --> f ) * ( X --> f ) ) . x = proj1 . x ;
( seq ^\ m ) . k <> 0 & ( seq ^\ m ) . k <> 0 ;
s . ( G . ( k + 1 ) ) > x0 ;
( p \cap M ) . 2 = d & ( p /\ M ) . 3 = d ;
A \oplus ( B \ominus C ) = ( A \oplus B ) \ominus C
h \equiv gg . ( mod ( P , T ) -ideal ( a , b , T ) ) ;
reconsider i1 = i-1 , i2 = i - 1 as Element of NAT ;
let v1 , v2 be VECTOR of V , v be VECTOR of V ;
for V being strict Subspace of V holds V is Subspace of [#] V
reconsider i = i , j = j as Element of NAT ;
dom f c= [: C ( ) , D ( ) :] ;
x in ( the Element of B ) . n & x in ( the Element of B ) . n ;
len [ [ 2 , len f2 ] , [ 2 , len f2 ] ] in Indices f2 ;
pp1 c= the topology of T & pp1 c= the topology of T ;
]. r , s .] c= [. r , s .] ;
let B2 be Basis of T2 , a be Element of T1 ;
G * ( B * A ) = ( B * A ) * ( B * A ) ;
assume that p , u , v is_collinear and u , v , w be Element of V ;
[ z , z ] in union rng ( F | X ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S , y = ( $1 .. S ) + 1 ;
LIN a1 , a3 , b1 & LIN a1 , b1 , b1 & LIN b1 , b2 , b2 ;
f " ( f .: x ) = { x } ;
dom w2 = dom r12 & dom r12 = dom r12 ;
assume that 1 <= i and i <= n and j <= n ;
( ( ( g2 ) . O ) `2 ) ^2 <= 1 ;
p in LSeg ( E . i , F . i ) ;
IL * ( i , j ) = 0. K ( ) ;
|. f . ( s . m ) - g .| < g1 ;
( ex q st q . x in rng ( q | k ) ) & ( q | k ) . x = q . x ;
Carrier Lxy misses Carrier ( Lxy ) ` & Carrier ( Lxy ) misses Carrier ( LR2 ) ;
consider c being element such that [ a , c ] in G ;
assume that Nreal = obeing and o8 = o8 and N8 = N8 ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= ( F . Cmin ( C , Cmin ) ) ;
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [: [. f . j , f . j .] , { f . j } :] ;
pred 0 <= x & x <= 1 & x ^2 <= 1 ;
p `1 - q `1 <> 0. TOP-REAL 2 & p `2 - q `1 = 0 ;
redefine func aa] ( S , T ) -> non empty set ;
let x be Element of [: S , T :] ;
the ObjectMap of F . ( a , b ) is one-to-one ;
|. i .| <= - ( - 2 to_power n ) & |. i .| <= - ( - 2 to_power n ) ;
the carrier of I[01] = dom P & the carrier of I[01] = dom Q ;
} * ( n + 1 ) ! > 0 * ! ;
S c= ( A1 /\ A2 ) /\ A3 & S /\ A2 c= A1 /\ A2 ;
a3 , a4 // b3 , b3 & a3 , a4 // b3 , b3 ;
then dom A <> {} & dom A <> {} & A <> {} ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , G2 & y in X implies x = y
set v2 = ( v /. ( i + 1 ) ) ;
x = r . n .= ( r . n ) * ( r . n ) ;
f . s in the carrier of S2 & f . s in the carrier of S2 ;
dom g = the carrier of I[01] & rng g = the carrier of I[01] ;
p in Upper_Arc ( P ) /\ Lower_Arc ( P ) ;
dom d2 = [: [: A2 , A1 :] , [: A2 , A2 :] :] ;
0 < ( p / ||. z .|| + 1 ) / ( ||. z .|| + 1 ) ;
e . ( m + 1 ) <= e . ( m + 1 ) ;
B \ominus X \/ B \ominus Y c= B \ominus X /\ Y
-infty < Integral ( M , Im ( g | B ) ) ;
cluster O := F -> ' for operation of X ;
let U1 , U2 be non-empty MSAlgebra over S , a be Element of U1 ;
Proj ( i , n ) * g is_differentiable_on X & g is_differentiable_on X ;
x , y , z be Point of X , p be Point of X ;
reconsider p0 = p . x , p0 = p . x as Subset of V ;
x in the carrier of Lin ( A ) & x in the carrier of Lin ( B ) ;
let I , J be parahalting Program of SCM+FSA , a be Int-Location ;
assume that - a is lower and a in { - a } ;
Int ( Cl A ) c= Cl ( Int ( 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 & p `2 <= p `2 ;
Cl Q ` = [#] ( T | P ` ) ;
set S = the carrier of T , T = the carrier of T ;
set I8 = ( -> Element of -> ( ) * ) * ( m , n ) ;
len p - n = len ( thesis - n ) .= len p - n ;
A is Permutation of Swap ( A , x , y ) ;
reconsider n6 = n7 - ( n + 1 ) as Element of NAT ;
1 <= j + 1 & j + 1 <= len ( s | j ) ;
q\subseteq { q where q is Element of M : q in A } ;
a9 in the carrier of S1 & b9 in the carrier of S2 ;
c1 /. n1 = c1 . n1 & c2 /. n1 = c1 . n1 ;
let f be FinSequence of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
y = ( f * ( S . x ) ) . x ;
consider x being element such that x in being element such that x in being } ;
assume r in ( dist ( o ) ) .: P ;
set i2 = ( n , h ) `1 , i1 = ( n , h ) `1 , i2 = ( n , h ) `1 , i2 = ( n , h ) `1 , i1 = ( 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 , n = ( x - 1 ) / 2 as Element of REAL ;
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 = len p2 + 1 ;
let T1 , T2 be Scott Scott let L , x be Element of T1 , y be Element of T2 ;
then x <= y & ( x in : x in { y } ) ;
set M = n -{ m } , N = n -\hbox { m } ;
reconsider i = x1 , j = x2 as Nat ;
rng ( the_arity_of a9 ) c= dom H & ( the_arity_of o ) . ( ( the_arity_of o ) /. i ) = s ;
z1 " = z1 " & z1 " = z1 " & z1 " = z1 " ;
x0 - r / 2 in L /\ dom f & x0 - r / 2 in dom f ;
then w is that rng w /\ ( S \/ AllSymbolsOf S ) <> {} ;
set x9 = x9 ^ <* Z *> , y9 = y9 ^ <* Z *> , z9 = x9 ^ <* Z *> ;
len w1 in Seg len w1 & len w1 = len w2 & len w1 = len w2 ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of PFuncs ( V , { k } ) ;
x . n = ( |. a . n .| ) * ( x . n ) ;
p `1 <= G * ( i1 , 1 ) `1 & p `1 <= G * ( i1 , 1 ) `1 ;
rng ( g ) c= L~ ( g | ( len g ) ) ;
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 & dom ( f | X ) = X ;
p , a // p , c & b , a // c , c ;
reconsider x1 = x , y1 = y , y2 = z as Element of REAL m ;
assume i in dom ( a * p ^ q ) ;
m . ag = p . ag & m . bg = p . bg ;
a / ( s . m - s . n ) / ( s . m - s . n ) <= 1 ;
S . ( n + k + 1 ) c= S . ( n + k ) ;
assume that B1 \/ C1 = B2 \/ C2 and B2 \/ C1 = C2 \/ C1 ;
X . i = { x1 , x2 } . i .= x1 . i ;
r2 in dom ( h1 + h2 ) & r1 < r2 + h2 ;
- - 0. R = a & b-0 = b ;
FF is_closed_on t2 , Q2 & FF is_halting_on t2 , Q2 & FF is_halting_on t1 , Q1 ;
set T = non in { the InInInof X , x0 } ;
Int ( Cl ( Int R ) ) c= Int ( R ) ;
consider y being Element of L such that c . y = x ;
rng F\mathop = { F\mathop . x } & rng F\mathop { x } = { x } ;
G-23 ` \ { c } c= B \/ S ;
fbeing Relation of [: X , Y :] , X & X c= Y ;
set RQ = the Element of P , RQ = the Element of Q ;
assume that n + 1 >= 1 and n + 1 <= len M ;
let k2 be Element of NAT , k be Element of NAT ;
reconsider pcSubset = u , pcSubset = v as Element of ( then n + 1 ) -tuples_on NAT ;
g . x in dom f & x in dom g implies x in dom f
assume that 1 <= n and n + 1 <= len f1 ;
reconsider T = b * N as Element of ( G / N ) ;
len Pt <= len P-35 & len Pt <= len P-35 ;
x " in the carrier of A1 & x " in the carrier of A1 ;
[ i , j ] in Indices ( A @ ) & [ i , j ] in Indices ( A @ ) ;
for m be Nat holds Re ( F . m ) is simple ;
f . x = a . i .= a1 . k .= a1 . k ;
let f be PartFunc of REAL i , REAL n , x be Point of REAL m ;
rng f = the carrier of ( Carrier A ) & f . x = 1 ;
assume that s1 = sqrt ( ( p - r ) / 2 ) ;
pred a > 1 & b > 0 implies a / b > 1 ;
let A , B , C be Subset of II , a , b , c be Element of II ;
reconsider X0 = X , Y0 = Y as RealNormSpace ;
let f be PartFunc of REAL , REAL , x be Element of REAL ;
r * ( v1 |-- I ) . X < r * 1 ;
assume that V is Subspace of X and X is Subspace of V ;
let t9 , t-3 be Relation of the carrier of S , s ;
Q [ e-14 \/ { vy } , f . vy } ] ;
g -: W-min L~ z implies ( g /. 1 ) .. z < ( g /. 1 ) .. z
|. |[ x , v ]| - |[ x , y ]| .| = v\rrangle ;
- f . w = - ( L * w ) .= - ( L * w ) ;
z - y <= x iff z <= x + y & y <= z ;
( 7 / ( 1 + e ) ) / ( 1 + e ) > 0 ;
assume X is BCK-algebra of 0 , 0 , 0 , 0 , 0 ;
F . 1 = v1 & F . 2 = v2 & F . 3 = v2 ;
( f | X ) . x2 = f . x2 & ( f | X ) . x2 = f . x2 ;
( tan | Z ) . x in dom sec & ( sec | Z ) . x = f . x ;
i2 = ( f /. len f ) & ( f /. 1 ) = ( f /. 1 ) ;
X1 = X2 \/ ( X1 \ X2 ) & X2 = X1 union X2 ;
[. a , b , 1_ G .] = 1_ G & a * b = 1_ G ;
let V , W be non empty VectSpStr over F_Complex , v be VECTOR of V ;
dom g2 = the carrier of I[01] & dom g2 = the carrier of I[01] & g2 . 1 = p2 ;
dom f2 = the carrier of I[01] & dom f2 = the carrier of I[01] & f2 . 1 = p2 ;
( proj2 | X ) .: X = proj2 .: X & ( proj2 | X ) .: X = proj2 .: X ;
f . ( x , y ) = h1 . ( x `1 , y `2 ) ;
x0 - r < a1 . n & a1 . n < x0 + r ;
|. ( f /* s ) . k - ( G /* s ) . k .| < r ;
len Line ( A , i ) = width A & width Line ( A , i ) = width A ;
Sbeing @ = ( S . g ) @ & S . g = ( S . g ) @ ;
reconsider f = v + u as Function of X , the carrier of Y ;
intloc 0 in dom Initialized p & intloc 0 in dom Initialized p ;
i1 , i2 , i3 , i3 , Nat & I does not destroy b1 , b , c ;
arccos r + arccos r = ( PI / 2 ) + 0 .= PI / 2 + 0 ;
for x st x in Z holds f2 is_differentiable_in x & f2 is_differentiable_in x & f2 . x > 0
reconsider q2 = ( q - x ) / ( 1 + x ) as Element of REAL ;
( 0 qua Nat ) + 1 <= i + j1 & i + 1 <= len G ;
assume f in the carrier of [: X , Omega Y :] ;
F . a = H / ( x , y ) . a ;
( ( {} T ) at ( C , u ) ) . ( ( {} T ) . u ) = TRUE ;
dist ( ( a * seq ) . n , h ) < r ;
1 in the carrier of [. 0 , 1 .] & 1 in dom f ;
p2 `1 - x1 > - g & p2 `1 - x1 < p2 `1 - g ;
|. r1 - thesis .| = |. a1 .| * |. thesis .| ;
reconsider S-14 = 8 , S-14 = 8 as Element of ( len S ) -tuples_on NAT ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b
D0W .( ) = D0W .( ) + 1 ;
i1 = ( a + n ) * n & i2 = ( - 1 ) * n ;
f . a [= f . ( f . O1 "\/" ( a "\/" b ) ) ;
pred f = v & g = u implies f + g = v + u ;
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 ^ R5 ) or L~ ( M1 ^ M2 ) meets L~ ( R4 ^ R5 ) ;
set h = the continuous Function of X , R , x be Point of X ;
set A = { L . ( k9 . n ) where k is Element of NAT : P [ k ] } ;
for H st H is atomic holds P [ H ] ;
set bA = S5 ^\ ( i + 1 ) , SA = S5 ^\ ( i + 1 ) ;
Hom ( a , b ) c= Hom ( a opp , b opp ) ;
( 1 / ( n + 1 ) ) * ( 1 / ( n + 1 ) ) < ( 1 / ( n + 1 ) ) * ( 1 / ( n + 1 ) ) ;
( l `1 ) `1 = [ dom l `1 , cod l `2 ] `1 .= [ dom l `1 , cod l `2 ] ;
y +* ( i , y /. i ) in dom g & x = g . i ;
let p be Element of CQC-WFF ( Al ) , x be Element of CQC-WFF ( Al ) ;
X /\ X1 c= dom ( f1 - f2 ) /\ dom ( f2 - f1 ) ;
p2 in rng ( f /^ ( len f -' 1 ) ) & p1 = f /. ( len f -' 1 ) ;
1 <= indx ( D2 , D1 , j1 ) & 1 <= indx ( D2 , D1 , j1 ) ;
assume x in ( K1 /\ K0 \/ ( K1 /\ K0 ) /\ K0 ) /\ K0 ;
- 1 <= ( ( f2 ) . O ) `2 & ( ( f2 ) . O ) `2 <= 1 ;
let f , g be Function of I[01] , TOP-REAL 2 , x be Point of TOP-REAL 2 ;
k1 -' k2 = k1 - k2 & k1 -' k2 = k1 - k2 ;
rng seq c= ]. x0 , x0 + r .[ & rng seq c= dom f /\ ]. 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 as as as as - normal sequence of W st a = Sum A ;
Cl ( union HH ) = union ( ( union H ) \/ ( union H ) ) ;
len t = len t1 + len t2 & len t = len t1 + len t2 ;
v-29 = v + w |-- v + AA .= v + A ;
cv <> DataLoc ( t0 . GBP , 3 ) & cv <> DataLoc ( t0 . GBP , 3 ) ;
g . s = upper_bound ( d " { s } ) & s <= t ;
( \dot , y ) . s = s . ( \dot , y ) ;
{ s : s < t } in [: the carrier of I[01] , the carrier of I[01] :] implies t = {}
s ` \ s = s ` \ 0. X & s ` \ s = 0. X ;
defpred P [ Nat ] means B + $1 in A & not ( $1 in B & not $1 in A ) ;
( 3be + 1 ) ! = 333 ! * ( 3be + 1 ) ;
<% U %> . ( succ A ) = <% U . ( A , B ) %> . ( A , B ) ;
reconsider y = y as Element of ( len y ) -tuples_on COMPLEX ;
consider i2 being Integer such that y0 = p * i2 and i2 in dom f ;
reconsider p = Y | ( Seg k ) , q = Y | ( Seg k ) as FinSequence of NAT ;
set f = ( S , U ) \mathop \mathop { \it Boolean } , g = ( S , U ) \mathop { \it Boolean } , f = ( S , U ) \mathop { \it true } , g = ( S , U ) \mathop { \it true } , d = (
consider Z being set such that lim s in Z and Z in F ;
let f be Function of I[01] , TOP-REAL n , x be Point of TOP-REAL 2 ;
( SAT M ) . [ n + i , 'not' A ] <> 1 ;
ex r being Real st x = r & a <= r & r <= b ;
R1 , R2 be Element of ( REAL n ) * , x be Element of REAL n ;
reconsider l = 0. ( Lin ( A ) ) , v = 0. ( Lin ( A ) ) ;
set r = |. e .| + |. n .| + |. w .| + a ;
consider y being Element of S such that z <= y and y in X ;
a be being being being being being being being being being being being being Element of Y holds a 'or' ( b 'or' c ) = 'not' ( a 'or' b )
||. x9 - g9 .|| < r2 & ||. x - g .|| < r ;
b9 , a9 // b9 , c9 & b9 , c9 // c9 , a9 & b9 , c9 // c9 , a9 ;
1 <= k2 -' k1 & k1 + 1 = k2 & k2 + 1 = k1 + 1 ;
( p `2 / |. p .| - sn ) / ( 1 + sn ) >= 0 ;
( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 < 0 ;
E-max C in right_cell ( Rmax , 1 ) & E-max C in rng Rmax ( Rmax , 1 ) ;
consider e being Element of NAT such that a = 2 * e + 1 ;
Re ( lim F ) | D = Re ( lim G ) .= Re ( lim F ) ;
LIN b , a , c or LIN b , c , a ;
p `1 , a // a `1 , b or p `1 , a // b `1 , a `2 ;
g . n = a * Sum ( f | 1 ) .= f . n ;
consider f being Subset of X such that e = f and f is strict ;
F | ( [: N2 , S :] ) = CircleMap * ( F | [: N2 , S :] ) ;
q in LSeg ( q , v ) \/ LSeg ( v , p ) ;
Ball ( m , r0 ) c= Ball ( m , s ) & Ball ( x , r ) c= Ball ( x , r ) ;
the carrier of (0). V = { 0. V } & the carrier of V = { 0. V } ;
rng ( cos | [. - 1 , 1 .] ) = [. - 1 , 1 .] ;
assume that Re seq is summable and Im seq is summable and Im seq is summable ;
||. ( vseq . n ) - ( vseq . n ) .|| < e / ( ||. ( vseq . n ) .|| + e / ( ||. ( vseq . n ) .|| + e / ( ||. ( vseq . n ) .|| + 1 ) ) ) ;
set g = O --> 1 ;
reconsider t2 = t11 as 0 -started string of S2 , t2 = t22 as Element of S2 ;
reconsider x9 = seq . n , y9 = seq . ( n + 1 ) as sequence of REAL n ;
assume that C meets L~ go and C meets L~ pion1 and not E-max L~ go meets L~ pion1 and not E-max L~ go c= L~ pion1 ;
- ( Partial_Sums ( 1 / 2 ) ) . n < F . n - ( Partial_Sums ( 1 / 2 ) ) . x ;
set d1 = \bf dist ( x1 , z1 ) , d2 = dist ( x2 , z2 ) , d2 = dist ( x1 , z2 ) ;
2 |^ ( |. 00 .| - 1 ) = 2 |^ ( |. \rm 00 .| - 1 ) ;
dom vb2 = Seg len d6 & dom vb2 = Seg len d6 ;
set x1 = - k2 + |. k2 .| + 4 , x2 = - k2 + 4 ;
assume for n being Element of X holds 0. <= F . n & 0. <= F . n ;
assume that 0 <= T-32 . i and T-32 . ( i + 1 ) <= 1 ;
for A being Subset of X holds c . ( c . A ) = c . A
the carrier of LT + L2 c= I2 & the carrier of LT + L2 c= the carrier of LT ;
'not' Ex ( x , p ) => All ( x , p ) is valid ;
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the normal normal \Rightarrow of f ;
Z c= dom ( ( sin * f1 ) `| Z ) & Z c= dom ( ( sin * f1 ) `| Z ) ;
|. 0. TOP-REAL 2 - ( q `1 / |. q .| - cn ) .| < r / 2 ;
o ( ) c= ConsecutiveSet2 ( A , succ d ( ) ) & o ( ) c= succ ( A ( ) , succ d ( ) ) ;
E = dom L8 & L . ( E . n ) is_measurable_on E & L . ( E . n ) = L . ( E . n ) ;
C / ( A + B ) = C / B * C / A ;
the carrier of W2 c= the carrier of V & the carrier of W1 c= the carrier of V ;
I . IC s2 = P . IC s2 .= ( I . IC s2 ) ;
pred x > 0 means : Def1 : ( 1 / x ) ^2 = x ^2 / ( 1 + x ^2 ) ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) ;
consider p being Point of T such that C = [: [. p , q .] , p :] ;
b , c are_connected & - C , - C + - C + D + D + E + F + J + M + N + E + F + M + N + D + E + F + J + M + N + E + F + M + N + D + E + F +
assume that f = id ( the carrier of O1 ) and f in the carrier of O1 and g in the carrier of O1 ;
consider v such that v <> 0. V and f . v = L * v ;
let l be Linear_Combination of {} ( ( the carrier of V ) --> { v } ) ;
reconsider g = f " as Function of U2 , U1 , ( U1 /\ U2 ) ;
A1 in the Points of G_ ( k , X ) & A2 in the carrier of G_ ( k , X ) ;
|. - x .| = - ( - x ) .= - x .= - x ;
set S = is non empty ;
Fib ( n ) * ( 5 * Fib ( n ) ) ^2 >= 4 * be Element of NAT ;
vM /. ( k + 1 ) = vM . ( k + 1 ) ;
0 mod i = - ( i * ( 0 qua Nat ) ) .= - ( i * ( 0 qua Nat ) ) ;
Indices M1 = [: Seg n , Seg n :] & len M1 = n & width M1 = n ;
Line ( S\mathopen { i , j } , j ) = S\mathopen { i , j } ;
h . ( x1 , y1 ) = [ y1 , x1 ] & h . ( x1 , y1 ) = [ y1 , x1 ] ;
|. f .| - Re ( |. f .| * ( card b * h ) ) is nonnegative ;
assume x = ( a1 ^ <* x1 *> ) ^ b1 & y = ( a1 ^ <* x1 *> ) ^ b1 ;
ME is_closed_on IExec ( I , P , s ) , P & ME is_halting_on s , P ;
DataLoc ( t3 . a , 4 ) = intpos ( 0 + 4 ) .= intpos ( 0 + 4 ) ;
x + y < - x + y & |. x .| = - x + y ;
LIN c , q , b & LIN c , q , c & LIN c , q , b ;
f| ( 1 , t ) . ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + z1 ) .= x1 + y1 ;
fp1 . a = fa1 . a & v in InputVertices S & [ v , w ] in InputVertices S ;
p `1 <= ( E-max C ) `1 & p `1 <= ( E-max C ) `1 ;
set R8 = Cage ( C , n ) -: E8 , E8 = Cage ( C , n ) -: E8 ;
p `1 >= ( E-max C ) `1 & p `1 >= ( E-max C ) `1 ;
consider p such that p = p-20 and s1 < p & p < s2 ;
|. ( f /* ( s * F ) ) . l - ( G * F ) . l .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len Line ( N , k + 1 + 1 ) = width N .= width N ;
f1 /* s1 is convergent & f2 /* s1 is convergent & lim ( f1 /* s1 ) = lim ( ( f1 /* s1 ) /* s1 ) ;
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 & rng ( Proj ( i , n ) * s ) c= REAL m ;
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 dom f ;
for L being complete LATTICE holds <* <* \mathclose { a } , L *> *> , L are_isomorphic
[ gi , gj ] in Ii \ Ij implies [ gi , gj ] in [: I , I :]
set S2 = 1GateCircStr ( x , y , c ) , S2 = 1GateCircStr ( y , c , d ) ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and for r st r < x0 ex g st r < g & g < x0 & g in dom ( f1 + f2 ) ;
reconsider y = ( a ` ) ` , z = ( a ` ) ` , t = ( 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 as Vertex of G , v , x be set ;
reconsider g = f as PartFunc of REAL n , REAL-NS n , REAL-NS n ;
|. s1 . m / p .| / |. p .| < d / p / p ;
for x being element st x in ( for t being element st t in ( ( t t ) holds x in s ) ) holds x in ( ( t ) \ s )
P = the carrier of ( TOP-REAL n ) | P & Q = the carrier of ( TOP-REAL n ) | P ;
assume that p00 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) and p01 in { p1 } ;
( 0. X \ x ) to_power ( m * ( k + 1 ) ) = 0. X ;
let g be Element of Hom ( cod f , \square ) ;
2 * a * b + ( 2 * c * d ) <= 2 * C1 * C2 ;
let f , g , h be Point of the carrier of X , x be Point of Y ;
set h = Hom ( a , g (*) f ) ;
then idseq ( n ) | ( Seg m ) = idseq ( m ) | ( Seg m ) ;
H * ( g " * a ) in the right of H & H * ( g " * a ) in the right of H ;
x in dom ( ( cos | ]. - 1 , 1 .[ ) | ]. - 1 , 1 .[ ) ;
cell ( G , i1 , j2 -' 1 ) misses C & cell ( G , i1 , j2 ) misses C ;
LE q2 , p , P , p1 , p2 or LE p2 , p , P , p1 , p2 ;
attr B is not bounded means : Def1 : B c= BDD A & not B c= BDD A ;
deffunc D ( set , set ) = union rng $2 & $2 = union rng $2 ;
n + - n < len ( p ^ <* n *> ) + - n & p . ( n + 1 ) = p . ( n + 1 ) ;
attr a <> 0. K means : Def1 : the_rank_of M = the_rank_of ( a * M ) ;
consider j such that j in dom /\ /\ dom thesis and I = len } + j ;
consider x1 such that z in x1 and x1 in P8 and x = [ x1 , x2 ] ;
for n ex r being Element of REAL st X [ n , r ]
set CS1 = Comput ( P2 , s2 , i + 1 ) , CS2 = Comput ( P2 , s2 , i + 1 ) , CS2 = Comput ( P2 , s2 , i + 1 ) , CS2 = Comput ( P2 , s2 , i + 1 ) , CS2 = Comput ( P2 , s2
set cv = 3 / 3 / ( a , b , c ) , cv = - ( a , b , c ) ;
conv @ W c= union ( F .: ( E " W ) ) & conv @ W c= union ( F .: ( E " W ) ) ;
1 in [. - 1 , 1 .] /\ dom ( arccot * ( f1 + f2 ) ) ;
r3 <= s0 + ( ( |. v2 - v1 .| ) / ( 2 * ( 1 + 1 ) ) ) ;
dom ( f (#) f4 ) = dom f /\ dom f4 & dom ( f (#) f4 ) = dom f /\ dom f4 ;
dom ( f (#) G ) = dom ( l (#) F ) /\ Seg k .= dom ( G (#) F ) ;
rng ( s ^\ k ) c= dom f1 \ { x0 } & rng ( s ^\ k ) c= dom f2 ;
reconsider g9 = gp , gq = gq , gr = gr as Point of TOP-REAL n1 ;
( T * h . s ) . x = T . ( h . s . x ) ;
I . ( L . ( J . x ) ) = ( I * L ) . ( J . x ) ;
y in dom <* `1 *> & ( Frege ( Frege ( A . o ) ) ) . y = ( ( Frege ( A . o ) ) . y ) ;
for I being non degenerated commutative Ring holds the carrier of I is commutative commutative non empty doubleLoopStr
set s2 = s +* Initialize ( ( intloc 0 ) .--> 1 ) , P2 = P +* I +* J ;
P1 /. IC s1 = P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 ;
lim S1 in the carrier of [. a , b .] & ( for x st x in the carrier of [. a , b .] holds x <= a ) implies S1 is convergent
v . ( U-13 . i ) = ( v *' lw ) . i .= ( v *' lw ) . i ;
consider n being element such that n in NAT and x = ( cn M ) . n ;
consider x being Element of c such that F1 . x <> F2 . x and F1 . x <> F2 . x ;
card ( X , 0 , x1 , x2 , x3 ) = { E } & card ( X , 0 , x1 , x2 , x3 ) = 1 ;
j + ( 2 * k9 ) + m1 > j + ( 2 * k9 ) + ( 2 * k9 ) ;
{ s , t } on A3 & { s , t } on B3 ;
n1 > len crossover ( p2 , p1 , n1 ) & n2 >= len crossover ( p2 , p1 , n1 ) ;
mg . HT ( mg , T ) = 0. L & mg . HT ( mg , T ) = 0. L ;
then H1 , H2 are_) & ( ( H , H1 ) / ( 2 , n ) ) / ( 2 , n ) is C / ( 2 , n ) ;
( N-min L~ ( f | ( L~ f ) ) ) .. ( f | ( L~ f ) ) > 1 & ( f /. ( len f ) ) .. ( f /^ ( len f ) ) ) .. ( f /^ ( len f ) ) = 1 ;
]. s , 1 .[ = ]. s , 2 .] /\ [. 0 , 1 .] ;
x1 in [#] ( ( TOP-REAL 2 ) | L~ g ) & x2 in ( ( TOP-REAL 2 ) | L~ g ) ;
let f1 , f2 be continuous PartFunc of REAL , the carrier of S , x be Point of S ;
DigA ( t-23 , z9 ) is Element of k -tuples_on ( k -tuples_on ( the carrier of K ) ) ;
I is \mathop 22) & I is k2 & I is k2 & I is not empty ;
[: u , { u } :] = { [ a , u9 ] } & [: u , { u } :] = [: { u } , { u } :] ;
( w | p ) | ( p | ( w | w ) ) = p ;
consider u2 such that u2 in W2 and x = v + u2 and u2 in W2 and u1 in W1 ;
for y st y in rng F ex n st y = a |^ n & P [ n ]
dom ( ( g * ( f . ( V \dot \to C ) ) ) | K ) = K ;
ex x being element st x in ( ( the Sorts of U0 ) \/ A ) . s ;
ex x being element st x in ( and ( for s being element st s in OO \/ A ) holds P [ s ] ) ;
f . x in the carrier of [. - r , r .] & f . x = r ;
( the carrier of X1 union X2 ) /\ ( ( the carrier of X1 ) \/ ( the carrier of X2 ) ) <> {} ;
L1 /\ LSeg ( p01 , p2 ) c= { p10 } /\ LSeg ( p1 , p2 ) ;
( b + ( b\cap be ) ) / 2 in { r : a < r & r < b } ;
ex_sup_of { x , y } , L & x "\/" y = sup { x , y } ;
for x being element st x in X ex u being element st P [ x , u ]
consider z being Point of GX such that z = y and P [ z ] and z in A ;
( the sequence of ( ( the carrier of ( the carrier of X ) ) ) ) . ( being Real ) <= e ;
len ( w ^ w2 ) + 1 = len w + 2 + 1 .= len w + 1 ;
assume that q in the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 and q `2 = 0 ;
f | E-4 ` = g | E-4 ` & g | E-4 = g | E-4 ` & g | E-4 = g | E-4 ;
reconsider i1 = x1 , i2 = x2 , z = x3 , i1 = x3 , i2 = x4 , j1 = x4 , j2 = x4 , j1 = 7 , j2 = 8 + 1 ;
( a * A * B ) @ = ( a * ( A * B ) ) @ ;
assume ex n0 being Element of NAT st f to_power n0 is such & f to_power n0 is < r ;
Seg len ( ( len ( f2 ) ) | ( dom ( f2 ) ) ) = dom ( ( len ( f2 ) ) | ( dom ( f2 ) ) ) ;
( Complement A1 ) . m c= ( Complement A1 ) . n & ( Complement A2 ) . m c= ( Complement A1 ) . n ;
f1 . p = ( p - 1 ) * ( p - 1 ) & ( p - 1 ) * ( p - 1 ) = d * ( p - 1 ) ;
FinS ( F , Y ) = FinS ( F , dom ( F | Y ) ) ;
( x | y ) | z = z | ( y | x ) ;
( |. x .| to_power n ) / ( ( |. x .| to_power n ) ) <= ( ( |. x .| to_power n ) ) / ( ( |. x .| to_power n ) ) ;
Sum F-12 = Sum f & dom F-12 = dom g & for x st x in dom F-12 holds F-12 . x = g . x ;
assume for x , y being set st x in Y & y in Y holds x /\ y in Y ;
assume that W1 is Subspace of W3 and W2 is Subspace of W3 and W1 /\ W2 is Subspace of W3 and W2 /\ W3 = { v } ;
||. ( t . x ) - ( t . x ) .|| = lim ||. ( ( t . x ) - ( t . x ) ) .|| ;
assume that i in dom D and f | A is lower and g | A is lower ;
( ( p `2 ) - the / 2 ) ^2 <= ( ( - 1 ) * ( - ( p `2 ) ) ) ^2 ;
g | Sphere ( p , r ) = id ( Sphere ( p , r ) ) & g . p = r ;
set N8 = N-min L~ Cage ( C , n ) , N8 = width Gauge ( C , n ) ;
for T being non empty TopSpace holds T is countable implies the TopStruct of T is countable
width B |-> 0. K = Line ( B , i ) .= B * ( i , i ) .= B * ( i , j ) ;
pred a <> 0 means : Def1 : ( A \+\ B ) -- a = ( A Y. a ) \+\ ( B f2 a ) ;
then f is_is_is_is_is_or pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 & pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 ;
assume that a > 0 and a <> 1 and b > 0 and c > 1 and d > 0 ;
w1 , w2 in Lin { w1 , w2 } & w2 in Lin { w1 , w2 } ;
p2 /. IC s2 = p2 . IC s2 .= ( p2 . IC s2 ) .= ( p2 . IC s2 ) ;
ind ( T-10 | b ) = ind b .= ind B .= ind ( T-10 | b ) ;
[ a , A ] in the \cdot of ( 1 - 2 ) * ( 1 - 2 ) & [ a , A ] in Indices ( 1 - 2 ) ;
m in ( the Arrows of C ) . ( o1 , o2 ) & m in ( the Arrows of C ) . ( o1 , o2 ) ;
( ( a 'imp' CompF ( PA , G ) ) . z ) 'or' ( ( a 'imp' CompF ( PA , G ) ) . z ) = FALSE ;
reconsider phi = phi /. 11 , phi = phi /. 22 , phi = phi . 11 as Element of ^2 ;
len s1 - 1 * ( len s2 - 1 ) + 1 > 0 + 1 * ( len s2 - 1 ) ;
delta ( D ) * ( f . ( upper_bound A ) - lower_bound A ) < r ;
[ f21 , f22 ] in [: the carrier' of A , the carrier' of B :] ;
the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 = K1 & ( ( TOP-REAL 2 ) | K1 ) | K1 = K1 ;
consider z being element such that z in dom g2 and p = g2 . z and x = g2 . z ;
[#] V1 = { 0. V1 } .= the carrier of ( V + W ) .= { 0. V1 } ;
consider P2 being FinSequence such that rng P2 = M and P2 is one-to-one and for k being Nat st k in dom P2 holds P2 . k = F ( k ) ;
assume that x1 in dom ( f | X ) and ||. x1 - x0 .|| < s ;
h1 = f ^ ( <* p3 *> ^ <* p *> ) .= h ^ <* p *> ^ <* p *> .= h ^ <* p *> ;
c /. ( |[ b , c ]| ) `1 = c /. ( |[ a , c ]| ) `1 .= a ;
reconsider t1 = p1 , t2 = p2 , t2 = p3 , t1 = p2 as Term of C , V ;
( 1 - 2 ) * ( 1 - 2 ) in the carrier of [. 1 / 2 , 1 .] ;
ex W being Subset of X st p in W & W is open & h .: W c= V ;
( h . p1 ) `2 = C * ( p1 `2 + D ) `2 .= C * ( p1 `2 + D ) `2 ;
R . b - b <= 2 * - b .= 2 * - b .= - b ;
consider \rrangle such that B = ( - 1 ) * C + ( - 1 ) * A and 0 <= 1 ;
dom g = dom ( ( the Sorts of A ) * ( a , b ) ) & rng ( ( the Sorts of A ) * ( a , b ) ) c= dom ( ( the Sorts of A ) * ( a , b ) ) ;
[ P . U6 , P . U6 ] in => ( ( P . l6 ) \/ ( Q . l6 ) ) ;
set s2 = Initialize s , P2 = P +* I ;
reconsider M = mid ( z , i2 , i1 ) , N = len z - 1 , M = z - 1 , N = len z - 1 as Element of REAL ;
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 & y in the right of g & x in the carrier of g ;
consider M being strict strict Subgroup of A9 such that a = M and T is strict Subgroup of M ;
for x st x in Z holds ( ( #Z 2 ) * f ) . x <> 0 & ( ( #Z 2 ) * f ) . x = 1
len W1 + len W2 + m = 1 + len W3 + m .= len W3 + len W3 + m .= len W3 + m + 1 ;
reconsider h1 = ( vseq . n ) - ( t-16 . n ) as Lipschitzian LinearOperator of X , Y ;
( i - 1 ) mod ( len ( p + q ) + 1 ) in dom ( p + q ) ;
assume that s2 is_for s1 , F , s2 such that s1 in the { H : not contradiction } and s2 in the carrier of s2 } ;
( ( ( for x , y st x in dom ( x , y ) ) holds x , y are_relative_prime ) implies ( x , y ) in Y
for u being element st u in Bags n holds ( p `1 + 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 = W \/ ( p \/ W1 ) , W3 = W \/ W2 ;
x in { X where X is Ideal of L : X in F & X is finite } ;
the carrier of W1 /\ W2 c= the carrier of W1 & the carrier of W1 /\ W2 c= the carrier of W1 & the carrier of W1 c= the carrier of W2
( 1 - a ) * ( id a ) = ( 1 - a ) * ( id a ) .= ( 1 - a ) * ( id a ) ;
( dom ( X --> f ) ) . x = ( X --> dom f ) . x .= ( X --> dom f ) . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) , y = LSeg ( g , m ) /\ LSeg ( g , n ) ;
p => ( q => r ) => ( p => ( q => 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 - 1 ) / ( n - m ) ) + 1 ) ) / ( n - m ) ) ;
( 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 . r ;
ex P st a1 on P & a2 on P & b on P & c on P & a , b , c is_collinear ;
reconsider gf = g `1 * f `1 , hg = h `2 * g `2 as strict Element of X ;
consider v1 being Element of T such that Q = ( downarrow v1 ) ` and v1 in ( downarrow v2 ) ` ;
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 `2 >= 0 & p <> 0. TOP-REAL 2 } ;
ConsecutiveSet ( A , succ O1 ) = ( ( ConsecutiveSet ( A , O1 ) ) * ( succ O1 ) ) * ( ( succ O1 ) * ( A , O1 ) ) ;
set IS1 = Macro ( a , intloc 0 ) , IS2 = AddTo ( a , intloc 0 ) , IS2 = AddTo ( a , intloc 0 ) , IS2 = let ( card I + 1 ) , IS2 = goto 0 , IS2 = goto 0 , IS2 = goto 0 , IS2 = goto 0 , IS2 = goto 0 , IS2 = goto 0 , IS2 = goto 0 , IS2
for i being Nat st 1 < i & i < len z holds z /. i <> z /. 1 & z /. i <> z /. 1
X c= ( the carrier of L1 ) /\ ( the carrier of L2 ) & X c= ( the carrier of L1 ) /\ ( the carrier of L2 ) ;
consider x9 being Element of GF ( p ) such that x9 |^ 2 = a & x |^ 2 = a ;
reconsider e3 = e4 , f-18 = f5 , f-18 = f5 as Element of D ;
ex O being set st O in S & C1 c= O & M . O = 0. ;
consider n being Nat such that for m being Nat st n <= m holds S . m in U1 and S . n in U2 ;
f * g * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) * ( x - x0 ) ) ;
defpred P [ Nat ] means A + succ $1 = succ A & ( for k st k in A holds P [ k , A . k ] ) implies P [ A ] ;
the left of - g = the left of g & the carrier of - g = the carrier of g & the carrier of - g = the carrier of g ;
reconsider p\mathopen = x , p\mathopen = y , p\mathopen = z , p\mathopen = z as Point of TOP-REAL 2 ;
consider g3 such that g3 = y and x <= ex g2 being Real st g2 <= x0 & g2 <= x0 & g2 <= x ;
for n being Element of NAT ex r being Element of REAL st X [ n , r ]
len ( x2 ^ y2 ) = len x2 + len y2 & len ( x2 ^ y2 ) = len x2 + len y2 ;
for x being element st x in X holds x in the set of ( the set of n0 ) & ( x in X implies x in Y )
LSeg ( p01 , p2 ) /\ LSeg ( p1 , p2 ) = {} & LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = {} ;
func such that ( union X ) -> set equals ( the carrier of [: h , X :] ) \/ [: dom ( id X , X :] ) ;
len ( { o } ^ { C /. 1 } ) <= len ( C ^ <* a *> ) + len ( C ^ <* a *> ) ;
attr K is has a , a & a <> 0. K implies v . ( a |^ i ) = a * v . a ;
consider o being OperSymbol of S such that t `1 . {} = [ o , the carrier of S ] and o in rng t ;
for x st x in X ex y st x c= y & y in X & y is \mathop of f . x
IC Comput ( P-6 , sseq , k ) in dom ( sseq ( a , I ) +* I ) & IC Comput ( Pseq , sseq ( a , I ) , k ) in dom I ;
pred q < s means : Def1 : r < s & ]. r , s .[ c= ]. p , q .[ & s <= q ;
consider c being Element of Class ( f , c ) such that Y = ( F . c ) `1 and c in X ;
func the ResultSort of S2 -> Function of the carrier' of S2 , the carrier' of S2 means : Def1 : for x being Element of the carrier' of S2 holds it . x = x ;
set y9 = [ <* y , z *> , f2 ] , z9 = [ <* z , x *> , f3 ] ;
assume x in dom ( ( ( #Z 2 ) * ( arccot ) ) `| Z ) & x in dom ( ( #Z 2 ) * ( arccot ) ) ;
r-7 in Int cell ( GoB f , i , ( GoB f ) * ( i , 1 ) + ( GoB f ) * ( i , 1 ) ) & r-7 in cell ( GoB f , i + 1 , ( GoB f ) * ( i , 1 ) ) ;
q `2 >= ( Cage ( C , n ) /. ( i + 1 ) ) `2 & q `2 >= ( Cage ( C , n ) /. ( i + 1 ) ) `2 ;
set Y = { a "/\" a ` : a in X } ;
i - len f <= len f + len f - len f & i + len f <= len f - len f + len f ;
for n ex x st x in N & x in N1 & h . n = x- ( x0 - x )
set s0 = ( \mathop { a , I , p , s ) . i , s1 = ( \mathop { a , I , p , s ) . i , s1 = ( \mathop { a , I , p , s ) . i , s1 = ( \mathop { a , I , p , s ) . i , s1 = ( \mathop { a , I , p , s )
p ( k ) . 0 = 1 or p ( k ) . 0 = - 1 or p ( k ) . 0 = 1 ;
u + Sum L-18 in ( U \ { u } ) \/ { u + Sum L-18 } ;
consider x9 being set such that x in x9 and x9 in V1 and x = [ x9 , the carrier of L ] ;
( p ^ ( q | k ) ) . m = ( q | k ) . ( ( len p ) - len p ) ;
g + h = gg + hh & f + g = g + h & f + h = 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 ) implies f / x = f / y & f . x = f / y ;
assume that 1 < p and ( 1 - p ) * q + ( 1 - p ) * q = 1 and 0 <= a and a <= 1 ;
F* ( f , of M ) = rpoly ( 1 , t ) *' t + 0. F_Complex .= ( 0. F_Complex ) *' ;
for X being set , A being Subset of X holds A ` = {} implies A = X & A = X
( N-min X ) `1 <= ( ( N-min X ) `1 or ( ex x st x in X & x in X & x in Y ) & x in X ) ;
for c being Element of the \geq the \geq the \langle *> of A , a , b being Element of the bound A holds c <> a
s1 . GBP = ( Exec ( i2 , s2 ) ) . GBP .= s2 . GBP .= s2 . GBP .= s2 . GBP ;
for a , b being Real holds |[ a , b ]| in ( y >= 0 ) & b >= 0 implies a >= 0
for x , y being Element of X holds x ` \ y = ( x \ y ) ` & y \ x = y \ x
mode BCK-algebra of i , j , m , n , m , n , m , m , n ;
set x2 = |( Re ( y - x ) , Im ( x - y ) )| ;
[ 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 ) * ( ( e / 2 ) |^ n ) ;
( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) <= ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ;
set A = ( 2 / b-a ) ;
for x , y being set st x in R" holds x , y are_\hbox { - } F . x , F . y }
deffunc FF2 ( Nat ) = b . $1 * ( M * G ) . $1 & ( M * G ) . $1 = ( M * G ) . $1 ;
for s being element holds s in -> Element of \rm \rm \rm -> set iff s in -> Element of \rm \rm \rm \rm \rm <= ( f \/ g )
for S being non empty non void non void non empty non void holds S is connected iff S is connected
max ( degree ( z `1 ) , degree ( z `2 ) ) >= 0 & degree ( z `1 ) >= 0 ;
consider n1 being Nat such that for k holds seq . ( n1 + k ) < r + s and r < ( seq ^\ k ) . n1 ;
Lin ( A /\ B ) is Subspace of Lin ( A ) & Lin ( B /\ A ) is Subspace of Lin ( B ) ;
set n-15 = ( n '&' ( M . x qua Element of BOOLEAN ) ) . ( ( n + 1 ) -tuples_on BOOLEAN ) ;
f " V in ' ( X , p ) & f " V in D ( the carrier of X , p ) & f " V in D ( the carrier of X , p ) ;
rng ( ( a *> ) +* ( 1 , b ) ) c= { a , c , b } \/ { c } ;
consider y being as as p1 of G1 such that y `1 = y and dom y `1 = WW
dom ( 1 / f ) /\ ]. -infty , x0 .[ c= ]. -infty , x0 + r .[ & dom ( ( 1 / f ) (#) f ) = ]. x0 , x0 + r .[ ;
as Element of f2 ( i , j , n , r ) & ( i , j , n ) is Morphism of i , j , n , - r ;
v ^ ( ( n |-> 0 ) ^ ( n |-> 0 ) ) in Lin ( rng ( ( B | c1 ) ^ ( B | c2 ) ) ) ;
ex a , k1 , k2 st i = a /. k1 & j = b /. k2 & k1 <> k2 & k2 <> k2 ;
t . NAT = ( NAT .--> succ i1 ) . NAT .= succ ( ( NAT .--> succ i1 ) . NAT ) .= succ ( ( NAT .--> succ i1 ) . NAT ) .= ( NAT .--> succ i1 ) . NAT ;
assume that F is bbfamily and rng p = F and dom p = Seg ( n + 1 ) and for i st i in Seg ( n + 1 ) holds p . i = F . i ;
not LIN b , b9 , a & not LIN a , a9 , c & LIN b , b9 , a & not LIN c , a , b
( L1 \HM { a } \HM { or L2 } ) \& O c= ( L1 => L2 ) => ( L1 => ( L2 => O ) )
consider F being ManySortedSet of E such that for d being Element of E holds F . d = F ( d ) ;
consider a , b such that a * ( u1 - w ) = b * ( not w - y ) and 0 < a & a < b ;
defpred P [ FinSequence of D ] means |. Sum $1 .| <= Sum ( |. $1 .| ) & ( for k st k in dom $1 holds $1 . k = ( |. $1 .| ) . k ) ;
u = cos / ( x , y ) . v * x + ( cos / ( x , y ) . v ) * y .= v ;
dist ( ( seq . n ) + x , g + x ) <= dist ( ( seq . n ) , g ) + 0 ;
P [ p , |. p .| (#) |. p .| , {} ( the Sorts of A ) . p ] ;
consider X being Subset of CQC-WFF ( Al ) such that X c= Y and X is finite and X is ininand inX in X ;
|. 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 & h <= g } ;
vol ( ( G . n ) vol ) <= vol ( ( ( G . n ) vol ) ) * vol ( ( G . n ) vol ) ) ;
f . y = x .= x * ( 1_ L ) .= x * ( power L ) .= x * ( power L ) ;
NIC ( <% a , b %> , ( succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1
LSeg ( p01 , p2 ) /\ LSeg ( p1 , p2 ) = { p1 } /\ LSeg ( p1 , p2 ) ;
Product ( ( the carrier of I-15 ) +* ( i , { 1 } ) ) in ( Z . i ) \/ ( ( the carrier of I-15 ) \/ { 1 } ) ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) .= Following ( s1 , n ) .= Following ( s2 , n ) ;
W-bound ( Qs2 ) <= ( q1 `1 ) & ( for i st i in dom ( q1 `1 ) holds ( ( q1 `1 ) / ( i + 1 ) ) ^2 <= ( ( q1 `1 ) / ( i + 1 ) ) ^2 ;
f /. i2 <> f /. ( ( len f + len g ) -' 1 ) & f /. ( len f + 1 ) = f /. ( len f + 1 ) ;
M , f / ( x. 3 , a ) / ( x. 4 , a ) / ( x. 0 , a ) / ( x. 4 , a ) / ( x. 0 , a ) / ( x. 0 , a ) ) |= H ;
len ( ( P ^ Q ) ^ ( P ^ Q ) ) in dom ( ( P ^ Q ) ^ ( P ^ Q ) ) ;
A |^ ( n , n ) c= A |^ ( m , n ) & A |^ ( k , n ) c= A |^ ( k , l ) ;
( R |^ n ) \ { q : |. q .| < a } c= { q1 : |. q1 .| >= a }
consider n1 being element such that n1 in dom p1 and y1 = p1 . n1 and p1 . n1 = p2 . n1 ;
consider X being set such that X in Q and for Z being set st Z in Q & Z <> X holds X \not c= Z ;
CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA & CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA ;
for v be VECTOR of l1 holds ||. v .|| = upper_bound rng |. ( seq_id ( v ) ) .| & ||. v .|| <= |. ( |. v .| ) .| ;
for phi holds phi in X implies ( phi in X & not phi in X & phi in X ) & phi in X
rng ( Sgm dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f |
ex c being FinSequence of D ( ) st len c = k & a = c & a = c & a = c ;
the_arity_of ( a , b , c ) = <* Hom ( b , c ) , Hom ( a , b ) *> .= <* Hom ( a , b ) , Hom ( b , c ) *> ;
consider f1 be Function of the carrier of X , REAL such that f1 = |. f .| and f1 is continuous and f1 . 0 = r ;
a1 = b1 & a2 = b2 or a1 = b2 & a2 = b1 & b1 = b2 & b2 = b1 & b1 = b2 & b2 = b1 ;
D2 . indx ( D2 , D1 , n1 + 1 ) = D1 . ( n1 + 1 ) & D2 . ( indx ( D2 , D1 , n1 + 1 ) + 1 ) = D2 . ( n1 + 1 ) ;
f . ( ||. |[ r .|| , 1 ]| ) = ||. |[ r .|| , 1 ]| .|| .= <* r *> .= <* r *> ;
consider n being Nat such that for m being Nat st n <= m holds C-25 . n = C-25 . m ;
consider d being Real such that for a , b being Real st a in X & b in Y holds a <= d & b <= b ;
||. L /. h .|| - ( K * |. h .| ) + ( K * |. h .| ) <= p0 + ( K * |. h .| ) ;
attr F is commutative means : Def1 : for b being Element of X holds F -Sum { b } = f . b ;
p = - ( - p0 + 0. TOP-REAL 2 ) .= 1 * ( p0 `1 ) + ( 0. TOP-REAL 2 ) .= ( 1 - p `1 ) * ( p0 `1 ) + ( - p `2 ) * ( p2 `1 ) ;
consider z1 such that b `1 , x3 , x1 , x1 , x2 , x3 is_collinear and o , x1 , x2 is_collinear and o , x1 , x2 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 g . x = f . x ;
assume that A = P2 \/ Q2 and P2 <> {} and Q2 <> {} and Q2 misses Q2 and P2 /\ Q2 = {} and Q2 /\ Q2 = {} ;
attr F is associative means : Def1 : F .: ( F .: ( f , g ) , h ) = F .: ( f , F .: ( g , h ) ) ;
ex x being Element of NAT st m = x `1 & x in z `1 & m in { i } or m in { i } ;
consider k2 being Nat such that k2 in dom P-2 and l in PW1 . k2 and P [ k2 , i , k ] ;
seq = r (#) seq implies for n holds seq . n = r * seq . n & seq . n = r * seq . ( n + 1 )
F1 . [ ( id a ) , [ a , a ] ] = [ f * ( id a ) , [ a , b ] ] ;
{ p } "\/" D2 = { p "\/" y where y is Element of L : y in D2 & p in D2 } & { p } "\/" D2 = { p "\/" q where q is Element of L : q in D1 & q in D2 } ;
consider z being element such that z in dom ( ( dom F ) | ( dom F ) ) and ( ( dom F ) | ( 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 `1 * b1 ) . x = ( Mx2Tran ( J , Bf2 , Bthesis ) ) . ( thesis /. j ) .= ( ( Mx2Tran J ) * ( b1 /. j ) ) /. j ;
- 1 / ( - 1 ) = ( m (#) D ) | n .= ( m (#) D ) | n .= ( ( m (#) D ) (#) ( - 1 ) ) | n .= ( ( ( - 1 ) (#) ( - 1 ) ) | n ) ;
attr for x being set st x in dom f /\ dom g holds g . x <= f . x & ( - g ) . x <= 0 ;
len ( f1 . j ) = len f2 /. j .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) ;
All ( 'not' All ( a , 'not' A , G ) , B , G ) . z '&' Ex ( 'not' All ( a , B , G ) , A , G ) . z = TRUE ;
LSeg ( E . k0 , F . k0 ) c= Cl ( RightComp Cage ( C , k + 1 ) \/ RightComp Cage ( C , k + 1 ) ) ;
x \ a |^ m = x \ ( a |^ k * a ) .= ( x \ a ) |^ k * a ;
k -ininininininininin-inin-inin-in-in-in-in-in-in-in-in-in-in-in-in-in-in-in-in-in--in--in--------------------------------------------
for s being State of Aex n being Nat st 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 ) implies f1 - f2 is continuous
support ( ( support n ) \/ support ( support ( m ) ) ) c= support ( ( support n ) \/ support ( m ) ) \/ support ( ( support n ) \/ support ( m ) ) ;
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 ) . a = g . a & phi /. ( g . a ) = f . ( g . a ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( ( F ^ <* p *> ) ^ <* p *> ) and i = j ;
{ x1 , x2 , x3 , x4 } = { x1 } \/ { x2 , x3 , x4 } .= { x1 , x2 , x3 , x4 } \/ { x4 , x5 } ;
the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U1 & the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U1 /\ ( U1 "\/" U2 ) ;
( - ( 2 * a * ( b - a ) ) / ( 2 * a ) ) ^2 - delta ( a , b , c ) / ( 2 * a ) ^2 > 0 ;
consider W00 such that for z being element holds z in W00 iff z in N & P [ z ] & P [ z ] ;
assume that ( the Arity of S ) . o = <* a *> and ( the Arity of S ) . o = <* r *> and ( the Arity of S ) . o = <* r *> ;
Z = dom ( ( exp_R * ( arccot ) - ( arccot * ( f1 + f2 ) ) ) / ( f1 + f2 ) ) ;
sum ( f , SS1 ) is convergent & lim ( f , SS1 ) = integral ( f , SS1 ) & lim ( f , SS1 ) = integral ( f , SS1 ) ;
( X . ( a9 => f ) => ( g => x ) ) => ( x9 => ( g => x ) ) in thesis
len ( M2 * M3 ) = n & width ( ( M2 * M3 ) * ( M2 * M3 ) ) = n & width ( ( M2 * M3 ) * ( M2 * M3 ) ) = n ;
attr X1 union X2 means : Def1 : X1 , X2 are_separated & X2 , X1 are_separated & X1 , X2 are_separated & X2 , X1 are_separated & X2 , X1 are_separated ;
for L being upper-bounded antisymmetric transitive RelStr , X being non empty Subset of L holds X "\/" { Top L } = { Top L }
reconsider f-129 = F2 . ( b , b ) , f-129 = F2 . ( b , a ) , f-129 = F2 . ( b , a ) , f-129 = F2 . ( b , a ) , f-129 = F2 . ( b , a ) , f-129 = F2 . ( b , a ) , f-129 = F2 . ( b , a ) , f-129 = F2 . ( b , a ) , f-1
consider w being FinSequence of I such that the InitS of M , w -seq ( <* s *> ^ w ) ^ <* s *> ^ 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 ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w
g . ( a |^ 0 ) = g . ( 1_ G ) .= 1_ H .= ( g . a ) |^ 0 .= ( g . a ) |^ 0 ;
assume for i being Nat st i in dom f ex z being Element of L st f . i = rpoly ( 1 , z ) & f . i = z ;
ex L being Subset of X st Carrier L = C & for K being Subset of X st K in C holds L /\ K <> {} ;
( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C1 & ( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C1 /\ ( the carrier' of C2 ) ;
reconsider oY = o `1 , oY = p `2 , oY = p `2 , o = p `2 , o = p `2 , o = p `2 , Y = p `2 , o = p `2 , Y = p `2 , Z = { p } , Y = { p } , Z = { p } , Y = { p } , Z = { p } , Y = { p } , Z = { p } , Z = { p } , Y = { p } , Z =
1 * x1 + ( 0 * x2 ) + ( 0 * x3 ) = x1 + <* \underbrace ( 0 , \dots , 0 ) , 0 , 0 , 1 *> .= x1 + <* \underbrace ( 0 , \dots , 0 ) *> .= x1 + x2 ;
Ek " . 1 = ( Ek qua Function ) " . 1 .= ( ( 1 - k ) |^ 1 ) " . 1 .= ( ( 1 - k ) |^ 1 ) " ;
reconsider u1 = the carrier of U1 /\ ( U1 "\/" U2 ) , u2 = the carrier of U2 /\ ( U1 "\/" U2 ) as non empty Subset of U0 ;
( ( x "/\" z ) "\/" ( x "/\" y ) ) "\/" ( z "/\" y ) <= ( x "/\" ( z "\/" y ) ) "\/" ( z "/\" ( x "\/" y ) ) ;
|. f . ( s1 . l1 + 1 ) - f . ( s1 . l1 + 1 ) .| < ( 1 / ( |. M .| + 1 ) ) * ( 1 / ( |. M .| + 1 ) ) ;
LSeg ( ( Lower_Seq ( C , n ) ) /. ( i + 1 ) , ( \hbox { \boldmath $ p $ } } ) is vertical ;
( f | Z ) /. x - ( f | Z ) /. x0 = L /. ( x- x ) + R /. ( x- x ) ;
g . c * ( - ( g . c ) * f . c ) + f . c <= h . c * ( ( - g . c ) * f . c ) + f . c ;
( f + g ) | divset ( D , i ) = f | divset ( D , i ) + g | divset ( D , i ) ;
assume that ColVec2Mx f in the set of ColVec2Mx A and ColVec2Mx b in the carrier of ColVec2Mx A and len ColVec2Mx f = width A and width ColVec2Mx f = width A and width ColVec2Mx f = width A and width ColVec2Mx f = width A ;
len ( - M3 ) = len M1 & width ( - M3 ) = width M1 & width ( - M2 ) = width M1 & width ( - M2 ) = width M1 ;
for n , i being Nat st i + 1 < n holds [ i , i + 1 ] in the InternalRel of ( ( the InternalRel of ( n + 1 ) ) \/ the InternalRel of ( ( n + 1 ) -tuples_on the carrier of ( n + 1 ) ) )
pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 2 ;
attr a <> 0 & b <> 0 & Arg a = Arg b implies Arg ( - a ) = Arg ( - b ) & Arg ( - a ) = Arg ( - b ) ;
for c being set st not c in [. a , b .] holds not c in Intersection ( the non empty set of the topology of T , a ) & not c in Intersection ( the topology of T , a )
assume that V1 is linearly-independent and V2 is linearly-independent and V2 = { v + u : v in V1 & u in V1 & u in V1 & v in V2 } and V1 = V1 /\ V2 and V1 = V2 /\ V2 ;
z * x1 + ( 1 - z ) * x2 in M & z * ( x1 + ( 1 - z ) * x2 ) in N ;
rng ( ( PS1 qua Function ) " * SS1 ) = Seg ( card dS1 ) .= dom ( ( PS1 ) " * SS1 ) ;
consider s2 being rational number such that s2 is convergent and b = lim s2 and for n holds s2 . n <= b . n and s2 . n <= b . n ;
h2 " . n = h2 . n " & 0 < - ( 1 / ( ( 1 - ( ( 1 - ( 2 / ( n + 1 ) ) ) * ( 1 / ( n + 1 ) ) ) ) ) ) ;
( Partial_Sums ( ||. seq1 .|| ) ) . m = ||. seq1 .|| . m .= ||. ( seq . m ) .|| .= ||. ( seq . m ) ) .|| .= ( ||. seq .|| ) . m .= ( ||. seq .|| ) . m ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b ;
- v = ( - 1_ ( GX ) ) * v & - w = ( - 1_ ( GX ) ) * v & - w = ( - 1_ ( GX ) ) * v ;
sup ( ( k .: D ) .: D ) = sup ( ( k .: D ) .: ( D .: D ) ) .= sup ( ( k .: D ) .: ( D .: D ) ) .= sup ( ( k .: D ) .: D ) ;
A |^ ( k , l ) ^^ ( A |^ ( n , .. A ) ) = ( A |^ ( k , .. A ) ) ^^ ( A |^ ( k , .. A ) ) ;
for R being add-associative right_zeroed right_complementable non empty addLoopStr , I , J being Subset of R holds I + ( J + K ) = ( I + J ) + K
( f . p ) `1 = ( p `1 ) ^2 + sqrt ( 1 + ( p `1 / p `2 ) ^2 ) .= ( p `1 ) ^2 + ( p `2 ) ^2 ;
for a , b being non zero Nat st a , b are_relative_prime holds ( for n being Nat holds support ( a * b ) = support ( a ) + support ( b ) ) & ( a * b ) = support ( a ) + support ( b )
consider A9 being countable Nat such that r is Element of CQC-WFF ( Al ) and A9 is ( ) \overline { A } and A is ( ) -connected and A is ( ) -connected ;
for X being non empty addLoopStr for M being Subset of X , x , y being Point of X st x in M & y in M holds x + y in M
{ [ x1 , x2 ] , [ y1 , y2 ] } c= [: { x1 , y1 } , { x2 } :] \/ [: { x2 , y2 } , { y2 } :] ;
h . ( f . O ) = |[ A * ( f . O ) `1 + B , C * ( f . O ) `2 + D ]| ;
( Gauge ( C , n ) * ( k , i ) ) `1 in L~ Upper_Seq ( C , n ) /\ L~ Lower_Seq ( C , n ) & ( Gauge ( C , n ) * ( k , i ) ) `1 = E-bound L~ Cage ( C , n ) ;
cluster m , n are_relative_prime means : such : the carrier of it is prime & for p being prime Nat holds p divides m & p divides n & p divides n ;
( f * F ) . x1 = f . ( F . x1 ) & ( f * F ) . x2 = f . ( F . x2 ) ;
for L being LATTICE , a , b , c being Element of L st a \ b <= c & b \ a <= c holds a \+\ b <= c
consider b being element such that b in dom ( H / ( x , y ) ) and z = ( H / ( x , y ) ) . b ;
assume that x in dom ( F * g ) and y in dom ( F * g ) and ( F * g ) . x = ( F * g ) . y ;
assume ex e being element st e Joins W . 1 , W . 5 , W . 3 , W . 7 , W . 7 , W . 8 , W . 7 , W . 8 , W . 8 , G ;
( $1 (#) ' ( f ) ) . ( 2 * n ) = ( h (#) ( f | A ) ) . ( 2 * n + ( n * h ) ) ;
j + 1 = ( - len h11 ) + 1 .= i + 1 - len h11 + 2 .= i + 1 - len h11 + 2 - 1 .= i + 1 - len h11 + 2 - 1 ;
( ^ ( S , f ) ) . f = S *' . ( ( S , f ) . ( f , g ) ) .= S . ( ( S , f ) . ( f , g ) ) .= S . ( f , g ) ;
consider H such that H is one-to-one and rng H = the carrier of L2 and Sum ( L2 * H ) = Sum ( L2 * H ) and Sum ( L2 * H ) = Sum ( L2 * H ) ;
attr R is \mathopen means : Def1 : p in R & p <> q implies ex P st P is special & p in P & q in P & p in R ;
dom product ( X --> f ) = meet ( dom ( X --> f ) ) .= meet ( ( X --> dom f ) . f ) .= meet ( ( X --> dom f ) . f ) .= dom f ;
upper_bound ( proj2 .: ( Upper_Arc C /\ Lower_Arc C /\ from ( w /\ E-bound C ) ) ) <= upper_bound ( ( proj2 .: ( C /\ from ( w /\ E-bound C ) ) /\ from ( w /\ 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 - p .| < r
i * f-28 - ( i * y\in * ( i * y ) ) = i * ( ( i * y ) - ( i * f ) ) .= i * ( ( i * y ) - ( i * f ) ) ;
consider f being Function such that dom f = 2 -tuples_on X and for Y being set st Y in 2 -tuples_on X holds f . Y = F ( Y ) ;
consider g1 , g2 being element such that g1 in [#] Y and g2 in union C and g = [ g1 , g2 ] and [ g1 , g2 ] in C and [ g2 , g2 ] in C ;
func d |-count n -> Nat means : such : d |^ n divides n & d |^ ( n + 1 ) divides ( d |^ n ) & ( not d divides n ) & not 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 = F . J or t = M ;
consider m1 be Nat such that for n st n >= m1 holds dist ( ( seq . n ) , ( seq . n ) ) < 1 / ( ( seq . n ) * ( seq . n ) ) ;
( ( q `1 ) / |. q .| ) ^2 <= ( ( q `1 ) / |. q .| ) ^2 & ( ( q `1 ) / |. q .| ) ^2 <= ( ( q `1 ) / |. q .| ) ^2 ;
h0 . ( i + 1 + 1 ) = h21 . ( i + 1 + 1 -' len h11 + 2 -' 1 ) .= h21 . ( i + 1 + 1 -' len h11 + 2 -' 1 ) ;
consider o being Element of the carrier' of S , x2 being Element of { the carrier of S } such that a = [ o , x2 ] and [ o , x2 ] in R ;
for L being RelStr , a , b being Element of L holds a <= { b } iff a >= b & b >= a
||. h1 .|| . n = ||. ( h1 . n ) .|| .= |. ( h . n ) .| .= ||. ( h . n ) .|| .= ||. ( h . n ) .|| .= ||. ( h . n ) .|| ;
( ( - 1 ) (#) ( ( #Z 2 ) * ( f1 - f2 ) ) ) . x = f . x - ( ( #Z 2 ) * ( f1 - f2 ) ) . x .= ( ( - 1 ) (#) ( f1 - f2 ) ) . x ;
pred r = F .: ( p , q ) means : Def1 : len r = min ( len p , len q ) & for i st i in dom p holds p . i = q . i ;
( r\mathop ^2 + ( r\mathop ^2 + ( rmax + 1 ) ^2 ) / 2 ) ^2 + ( r\mathop ^2 + ( rmax + 1 ) ^2 ) / 2 ) ^2 <= ( r ^2 + ( r ^2 + 1 ) ^2 ) ;
for i being Nat , M being Matrix of n , K st i in Seg n holds Det M = Sum ( ( Det M ) @ ) & Det M = ( Det ( M @ ) ) . i
then a <> 0. R & a " * ( a * v ) = 1 * v & a " * ( a * v ) = 1 * v & a * v = 1 * v ;
p . ( j - 1 ) * ( q *' r ) . ( i + 1 -' j ) = Sum ( p . ( j - 1 ) * r3 ) .= Sum ( p . ( j -' 1 ) * r3 ) ;
deffunc F ( Nat ) = L . 1 + ( ( R /* h ) ^\ n ) . $1 * ( ( R /* h ) ^\ n ) " ;
assume that the carrier of H1 = f .: ( the carrier of H1 ) and the carrier of H2 = f .: ( the carrier of H2 ) and the carrier of H1 = f .: ( the carrier of H2 ) and the carrier of H2 = 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 ) ) * ( the Arity of S ) ) . o ;
H1 = n + 1 / ( |. 2 to_power ( n + 1 ) .| + h ) .= n + 1 / ( |. 2 to_power ( n + 1 ) .| + h ) .= n + 1 / ( |. 2 to_power ( n + 1 ) .| + h ) ;
( O1 `1 = 0 & O `2 = 0 & O `2 = 1 & O `2 = 0 or O `1 = 1 & O `2 = 0 & O `2 = 1 ) & O <> 0 & O <> 0 & O <> 0 & O <> 0 & O <> 0 & O <> 0 & O <> 0 & O <> 0 & O <> 0 & O <> 0 & O <> 0 & O <> 0 & O <> 0 & O <> 0 implies O is non empty
F1 .: ( dom F1 /\ dom F2 ) = F1 .: ( dom F1 /\ dom F2 ) .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } ;
attr b <> 0 & d <> 0 & b <> d & ( a - b ) = ( e - d ) / ( b - b\rbrack ) & ( a - b ) / ( b - b\lbrace b \rbrace ) = ( ( - e ) / ( b - b\lbrace b \rbrace ) ) / ( b - b\lbrace b \rbrace ) ;
dom ( ( f +* g ) | D ) = dom ( f +* g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D .= dom ( f | D ) /\ D .= dom ( f | D ) ;
for i being set st i in dom g ex u , v being Element of B st g /. i = u * a & u in B & v in A & v in B
g `1 * P `2 * g `2 " = g `2 * ( g `1 * P `2 ) * g `2 .= g `2 * ( g `2 * P `2 ) " .= g `2 * ( g `2 * P `2 ) ;
consider i , s1 such that f . i = s1 and not ( ex s1 st f . ( i + 1 ) <> s1 & s1 . ( i + 1 ) <> s1 ) & not ( s1 is empty & s1 is empty ) ;
h5 | ]. a , b .[ = ( g | ]. a , b .[ ) | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ ;
[ s1 , t1 ] , [ t2 , t2 ] are_connected & [ s2 , t2 ] , [ t2 , t2 ] are_connected & [ s2 , t2 ] , [ t2 , t2 ] are_connected ;
then H is negative & H is non empty & H is non empty & H is non empty & H is non empty implies H is non empty iff H is non empty iff H is non empty iff H is non implies H is not implies H is not implies H is not implies H is not implies H is not implies H is not implies H is not implies H is not conjunctive
attr f1 is total means : Def1 : f1 is total & f2 is total & ( for c st c in dom f1 holds f1 . c = c * ( f2 . c ) " ) & ( f1 . c = c * ( f2 . c ) " ) & ( f1 . c = ( f2 * ( f2 * f1 ) ) " ) ;
z1 in W2 -Seg ( z2 ) or z1 = z2 & not z1 in W2 & not z1 in W2 & not z1 in W1 & not z2 in W2 & not z1 in W2 & not z1 in W1 & not z2 in W2 & not z1 in W1 & not z1 in W2 & not z1 in W2 & not z1 in W1 & not z1 in W2
p = 1 * p .= a " * a * p .= a " * ( b * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) ;
for seq1 be Real_Sequence for K be Real st for n be Nat holds seq1 . n <= K holds upper_bound rng ( seq1 ^\ n ) <= upper_bound rng ( seq1 ^\ n )
C meets ( L~ go \/ L~ pion1 or C meets ( L~ pion1 \/ L~ pion1 ) or C meets ( L~ pion1 \/ L~ pion1 ) or C meets ( L~ pion1 \/ L~ pion1 ) or C meets ( L~ pion1 \/ L~ pion1 ) or C meets ( L~ pion1 \/ L~ pion1 ) or C meets ( L~ pion1 \/ L~ pion1 ) or C meets ( L~ pion1 \/ L~ co ) ;
||. f . ( g . ( k + 1 ) ) - g . ( g . k ) .|| <= ||. g . 1 - g . 0 .|| * ( K * ( K to_power k ) ) ;
assume h = ( ( B .--> B ) +* ( C .--> C ) +* ( D .--> E ) +* ( E .--> F ) +* ( F .--> J ) +* ( M .--> N ) +* ( M .--> M ) +* ( N .--> N ) +* ( N .--> N ) +* ( M .--> N ) +* ( M .--> N ) +* ( M .--> N ) ) +* ( M .--> N ) +* ( M .--> N ) ) +* ( M .--> M ) +* ( N .--> N .--> N ) +* ( M .--> N ) +* ( M .--> N .--> N ) +* ( M .--> N
|. ( ( ex H be Real st ( ex n be Element of NAT st ( for k be Element of NAT st k <= n holds ( ( H . k ) (#) ( T . k ) ) ) ) . n ) .| <= e * ( ( b-a ) . n ) ;
( ( ( the Sorts of A ) . i ) . e ) = [ ( the s2 at ( v , the carrier of IC ) ) . e , ( the Sorts of IC ) . e ] .= [ ( the Sorts of IC ) . e , ( the Sorts of IC ) . e ] ;
{ x1 , x1 , x1 , x1 , x1 , x1 , x2 , x3 , x4 } = { x1 , x1 , x2 , x3 , x4 } .= { x1 , x2 , x3 , x4 } .= { x1 , x2 , x3 } ;
assume that A = [. 0 , 2 * PI .] and integral ( ( #Z n ) * ( cos - sin ) , A ) = 0 and integral ( ( #Z n ) * ( sin - cos ) , A ) = 0 ;
p `1 is Permutation of dom f1 & p `2 " = ( Sgm Y ) " * p & p `2 = ( Sgm Y ) " * p & p `2 = ( Sgm Y ) " * p ;
for x , y st x in A holds |. ( 1 - 1 ) (#) ( f . x ) - ( 1 - 1 ) (#) ( f . y ) ) <= 1 * |. ( f . x ) - ( f . y ) .|
p2 `2 = |. q2 .| * ( ( q2 `2 / |. q2 .| - sn ) / ( 1 + sn ) ) .= ( ( q2 `2 / |. q2 .| - sn ) / ( 1 + sn ) ) / ( 1 + sn ) ;
for f being PartFunc of the carrier of CNS , REAL st dom f is compact & f is continuous holds rng f is compact & f . 0 = f . 1 & f . 1 = f . 1
assume for x being Element of Y st x in EqClass ( z , CompF ( B , G ) ) holds ( Ex ( a , CompF ( B , G ) ) . x ) . x = TRUE ;
consider FM such that dom FM = n1 and for k be Nat st k in n1 holds Q [ k , FM . k ] ;
ex u , u1 st u <> u1 & u , u1 // v , v1 & u , u1 // v , u1 & u , u1 // v , v1 & u , u1 // v , u1 & u , u1 // v , v1 & u , u1 // v , u1 & u , u1 // v , v1 & u1 , u1 // v , u1 & u , u1 // v , v1 & u1 , u1 // v , v1 & u1 , v1 // v , u1 & u , u1 // v , u1 implies u , u1 , v // v , v1 & u , u1 , v , u1 , v implies u
for G being Group , A , B being non empty Subgroup of G , N being normal Subgroup of G holds ( N ` A ) * ( N ` B ) = N ` A * ( N ` B )
for s be Real st s in dom F holds F . s = integral ( ( R to_power 0 ) (#) integral ( f , ( f + g ) / ( d + 1 ) ) ) ;
width AutMt ( f1 , b1 , b2 ) = len b2 .= len ( b1 * b1 ) .= len b1 .= len ( b1 * b2 ) .= len ( b1 * b2 ) .= len b1 .= len b1 .= len b1 ;
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 ;
assume that X is closed and a in X and a in X and y in a ^ { f . [ n , x ] } and x in a ;
Z = dom ( ( #Z 2 ) * ( arctan + arccot ) ) /\ dom ( ( #Z 2 ) * ( f1 + #Z 2 ) ) & Z = dom ( ( #Z 2 ) * ( f1 + #Z 2 ) ) /\ dom ( ( #Z 2 ) * ( f1 + #Z 2 ) ) ;
func [: l , l :] -> Subset of V means : Def1 : 1 <= l & l <= len l & 1 <= k & k <= len l & l . k = l . k ;
for L being non empty TopSpace , N being net of L , M being net of L st c is_be cluster M holds c is_is cluster cluster of N
for s being Element of NAT holds ( ( for v being Element of NAT holds ( for x being Element of NAT holds x in v iff x in C\rm id ( C\rm seq ) ) ) implies ( for x being Element of NAT holds x in C\rm seq ( v ) ) )
then z /. 1 = N-min L~ z & ( N-min L~ z ) .. z < ( ( N-min L~ z ) .. z ) .. z & ( ( N-min L~ z ) .. z ) .. z < ( ( N-min L~ z ) .. z ) .. z ;
len ( p ^ <* ( 0 qua Real ) *> ) = len p + len <* ( 0 qua Real ) * ( ( 0 qua Real ) * ( 0 qua Real ) ) .= len p + 1 .= len p + 1 ;
assume that Z c= dom ( ( - ln * f ) `| Z ) and for x st x in Z holds f . x = x & f . x > 0 & f . x > 0 ;
for R being add-associative right_zeroed right_complementable associative commutative associative well-unital distributive non empty doubleLoopStr , I 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 + y2 ) .= dom ( x (#) ( y (#) z ) ) .= dom ( x (#) ( y (#) z ) ) .= dom ( x (#) ( y (#) 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 ) is one-to-one
ex a st a = a2 & a in f6 /\ f5 & for x st x in f6 holds reconsider { f . x , f . x } = S ( x , a ) & { f . x , f . x } = S ( x , a ) ;
a in Free ( H / ( x. 4 , x. k ) ) '&' ( H2 / ( x. k , x. k ) ) & a in Free ( H / ( x. k , x. k ) ) & a in Free ( H / ( x. k , x. k ) ) ;
for C1 , C2 being v1 , C2 being strict non-empty Function of C1 , C2 st ( for f being Element of C1 holds f in C2 iff f = g ) & ( for x being Element of C1 holds f . x = g . x ) holds f = g
( W-min L~ go \/ L~ pion1 ) `1 = W-bound L~ go \/ ( L~ pion1 \/ L~ co ) or ( W-min L~ go \/ L~ co ) `1 = W-bound L~ go \/ W-bound L~ co or ( W-min L~ go \/ L~ co ) `1 = W-bound L~ go \/ W-bound L~ co ;
assume that u = <* x0 , y0 , z0 *> and f is_assume x0 and SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) = SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) and SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) = SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) ;
then ( t . {} ) `1 in Vars & ex x being Element of Vars st x = ( ( t . {} ) `1 ) & ( t . {} ) `2 = ( ( t . {} ) `1 ) & ( t . {} ) `2 = ( ( t . {} ) `1 ) ;
Valid ( p '&' p , J ) . v = Valid ( p , J ) . v '&' Valid ( p , J ) . v .= Valid ( p , J ) . v '&' Valid ( p , J ) . v .= Valid ( p , J ) . v ;
assume for x , y being Element of S st x <= y for a , b being Element of T st a = f . x & b = f . y holds a >= b ;
func Class R -> Subset-Family of R means : Def1 : for A being Subset of R holds A in it iff ex a being Element of R st a = Class ( R , a ) & it c= A ;
defpred P [ Nat ] means ( ( ( ( ( ( ( ( ) \cup ( { x } ) ) ) \/ { y } ) ) \/ { x } ) ) & ( ( ( ( ( G \ { x } ) \/ { y } ) ) \/ { x } ) ) ;
assume that dim ( W1 ) = 0 and dim ( W1 ) = 0 and V = { 0. ( W1 ) where W1 is Subspace of W1 : dim ( W1 ) = 0 & dim ( W1 ) = 0 & dim ( W1 ) = 0 & dim ( W1 ) = 0 & dim ( W1 ) = 0 & dim ( W1 ) = 0 ;
mas . ( m . t ) = ( m . t ) `1 .= ( [ m . t , the carrier of C ] `1 ) `1 .= ( [ m . t , the carrier of C ] `1 ) `1 .= m . t ;
d11 = x9 ^ d22 .= f ^ ( ( y , d22 ) --> ( x , y ) ) .= f ^ ( ( y , d22 ) --> ( x , y ) ) .= ( x ^ ( y , d22 ) ) ^ ( y ^ ( x , y ) ) .= ( x ^ ( y ^ ( x , y ) ) ) .= ( x ^ ( y ^ ( x , y ) ) ) ) ^ ( y ^ ( x , y ) ) .= ( x ^ ( x , y ) ) ;
consider g such that x = g and dom g = dom ( f . 0 ) and for x being element st x in dom ( f . 0 ) holds g . x in ( f . 0 ) . x ;
x + 0. F_Complex = x + len x |-> 0. F_Complex .= ( x , len x ) |-> 0. F_Complex .= ( x , len x ) |-> 0. F_Complex .= x *' ;
( k -' ( k + 1 ) ) + 1 in dom ( f /. ( k -' ( k + 1 ) ) ) & ( f /. ( k + 1 ) ) = ( f /. ( k + 1 ) ) + ( f /. ( k + 1 ) ) ;
assume that P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 = { p1 , p2 } and P1 = { p1 , p2 } and P1 /\ P2 = { p1 , p2 } and P1 /\ P2 = { p1 , p2 } and P1 /\ P2 = { p1 , p2 } and P1 /\ P2 = { p1 , p2 } and P1 /\ P2 = { p1 , p2 } and P1 /\ P2 = { p2 , p1 } ;
reconsider a1 = a , b1 = b , b1 = c , c1 = p , c2 = p , c2 = p , c1 = q , c2 = p , c2 = q , c1 = p , c2 = q , c2 = p , c2 = q , c1 = q , c2 = p , c2 = q , c2 = p , c1 = q , c2 = q , c2 = p , c2 = q , c2 = p , c2 = q , c1 = q , c2 = p , c2 = q , c2 = p , c2 = q , c2 = q , c2 = q , c2 = q , c2 = q , c2 = q , c2 = q , c2 = p , c2 = p , c1 = p , c2 = p , c2 = q , c2 =
reconsider set set set set set set set set set set = G1 . ( t , b ) * F1 . f , F2 = ( G1 * F1 ) . b , F1 = ( G1 * F2 ) . b , F2 = ( G1 * F2 ) . b , F2 = ( G2 * F2 ) . b , F2 = ( G2 * F2 ) . b ;
LSeg ( f , i + i1 -' 1 ) = LSeg ( f /. ( i + i1 -' 1 ) , f /. ( i + i1 -' 1 + 1 ) ) ;
Integral ( M ` . m , P . n ) <= Integral ( M ` . n , P . m ) & Integral ( M ` . m , P . n ) <= Integral ( M ` . m , P . n ) ;
assume that dom f1 = dom f2 and for x , y being element st [ x , y ] in dom f1 & [ y , x ] in dom f2 holds f1 . ( x , y ) = f2 . ( x , y ) ;
consider v such that v = y and dist ( u , v ) < min ( ( G * ( i , 1 ) `1 - G * ( i + 1 , 1 ) `1 ) , ( G * ( i + 1 , 1 ) `1 - G * ( i + 1 , 1 ) `1 ) ) ;
for G being Group , H being Subgroup of G , a being Integer st a = b holds for i being Integer holds a |^ i = b |^ i & a |^ i = b |^ i
consider B being Function of Seg ( S + L ) , the carrier of V1 such that for x being element st x in Seg ( S + L ) holds P [ x , B . x ] ;
reconsider K1 = { p where p is Point of TOP-REAL 2 : P [ p ] & p <> 0. TOP-REAL 2 } , K1 = { p : p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 } as Subset of ( TOP-REAL 2 ) | K1 ;
( ( N-bound C ) - ( S-bound C ) / 2 ) * ( ( S-bound C ) - ( S-bound C ) / 2 ) <= ( ( S-bound C ) - ( S-bound C ) / 2 ) * ( ( S-bound C ) - ( S-bound C ) / 2 ) ;
for x be Element of X , n be Nat st x in E holds |. Re ( F . n ) .| . x <= P . x & |. Im ( F . n ) .| <= P . x
len ( @ ( @ p ^ @ q ) ) = len ( @ p ^ <* 0 *> ) + len <* [ 2 , 0 ] *> .= len ( @ p ^ <* 0 *> ) + 1 .= len ( @ p ^ <* 1 *> ) + 1 ;
v / ( x. 3 , m1 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m3 ) / ( x. 0 , m3 ) ) = m3 / ( x. 4 , m3 ) ;
consider r being Element of M such that M , v / ( x. 3 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 0 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 0 , m ) ) |= H2 ;
func w1 \ w2 -> Element of Union ( G , R^ ) means : R : for w1 being Element of Union ( G , R^ ) holds it . ( ( the Sorts of G ) . ( w1 , w2 ) ) = ( the Sorts of G ) . ( ( the Sorts of G ) . ( w1 , w2 ) ) ;
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= s1 . b2 .= s2 . b2 .= ( s2 . b2 ) . 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 the carrier of S , G = S -\hbox { {} } ;
( Partial_Sums ( seq ) ) . K + Sum ( seq ) . ( K + 1 ) >= ( Partial_Sums ( seq ) ) . ( K + 1 ) + ( Partial_Sums ( seq ) ) . ( K + 1 ) ) & ( Partial_Sums ( seq ) ) . ( K + 1 ) >= 0 ;
consider L , R such that for x st x in N holds ( f | Z ) . x - ( f | Z ) . x0 = L . ( x- x ) + R . ( x - x0 ) ;
func the closed of \HM { a , b , c , d , e , f , g , h , i , x , y , z ) -> Subset of ( the \rm empty of rectangle ( a , b , c , d , e , f , i , x ) ) ` ;
a * b ^2 + ( a * c ) ^2 + ( b * a ) ^2 + ( b * c ) ^2 + ( c * a ) ^2 >= 6 * a * b * c ;
v / ( x1 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) = v / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) ;
+ ( Q ^ <* x *> , M1 ) = ( ( \mathop { Q } , M ) +* ( ( f ^ <* FALSE *> , FALSE ) --> ( f ^ <* FALSE *> , FALSE ) ) ) +* ( ( f ^ <* FALSE *> , FALSE ) --> ( f ^ <* FALSE *> , FALSE ) ) .= ( f ^ <* FALSE *> , M ) --> FALSE ;
Sum ( F ) = r |^ n1 * Sum ( Cv ) .= C . ( n1 + n1 ) .= C . ( n1 + n1 ) .= C . ( n1 + n1 ) .= C . ( n1 + n1 ) .= C . ( n1 + n1 ) ;
( GoB f ) * ( len GoB f , 1 ) `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 + 1 ) ) / ( 2 * $1 + 1 ) * ( $1 + 1 ) * ( $1 + 1 ) + b * ( $1 + 1 ) * ( $1 + 1 ) ;
the_arity_of g = ( the Arity of S ) . g .= ( ( the Arity of S ) * g ) . g .= ( ( the Arity of S ) * g ) . g .= ( ( the Arity of S ) * g ) . g .= ( ( the Arity of S ) * g ) . g .= ( ( the Arity of S ) * g ) . g ;
( X ~ ) ^ ( Z Z Z Z ) tolerates ( X ~ ) & card ( ( X ~ ) ^ ( Z Z ) ) = card ( X ~ ) & card ( ( X ~ ) ^ ( Z Z ) ) = card ( X ~ ) ;
for a , b being Element of S , s being Element of NAT st s = n & a = F . n & b = F . n holds b = N . ( s . n ) \ G . s ;
E , f |= All ( All ( x , All ( x , H ) ) '&' ( ( x. 0 ) .--> ( x. 1 ) ) '&' ( ( x. 2 ) .--> ( x. 2 ) ) '&' ( ( x. 0 ) .--> ( x. 1 ) ) '&' ( ( x. 2 ) .--> ( x. 1 ) ) '&' ( x. 2 ) ) ) ;
ex R2 being 1-sorted st R2 = ( p | ( n + 1 ) ) . i & ( for i being Element of NAT st i in dom p holds p . i = 0. K ) & ( for i being Element of NAT st i in dom p holds p . i = 0. K ) ;
[. a , b + 1 / ( k + 1 ) .[ is Element of the \in the \in of the carrier of X & ( the partial of f ) . ( k + 1 ) is Element of the carrier of X & ( the partial of f ) . ( k + 1 ) is Element of the carrier of X ) ;
Comput ( P , s , 2 + 1 ) = Exec ( P . 2 , Comput ( P , s , 2 ) ) .= Exec ( a3 , Comput ( P +* I , s , 2 ) ) .= Exec ( a3 , s ) . c .= s . c ;
card ( h1 ) . k = power ( F_Complex ) . ( ( - 1_ F_Complex ) * u , k ) * Sum u .= ( ( f *' ) . ( ( - 1_ F_Complex ) * u ) ) * u .= ( ( f *' ) . ( ( - 1_ F_Complex ) * u ) ) * u .= ( ( f *' ) . ( ( - 1_ F_Complex ) * u ) ) * u ;
( f / g ) /. c = f /. c * ( g /. c ) " .= f /. c * ( ( 1 / 2 ) (#) ( g / ( g / ( g / ( g / ( g / ( g / ( g / ( g / ( g / ( g - g ) ) ) ) ) ) ) ) ) .= ( f / ( g / ( g / ( g / ( g / ( g / ( g / ( g - g ) ) ) ) ) ) ) ) ) ) ;
len Cs - len ( ( C /. ( len C ) ) * ( len C ) ) = len C - len ( ( C /. ( len C ) ) * ( len C ) ) .= len C - len ( ( C /. ( len C ) ) * ( len C ) ) .= len C - len ( ( C /. ( len C ) ) * ( len C ) ) ;
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom f /\ X .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) ;
defpred P [ Nat ] means for n holds 2 * Fib ( n + $1 ) = Fib ( n ) * Fib ( n ) + ( 5 * Fib ( n ) * Fib ( n ) ) ;
consider f being Function of INT , INT such that f = f `1 and f is onto and for n st n < k holds f " { f . n } = { n } and f " { f . n } = { n } ;
consider vs being Function of S , BOOLEAN such that vs = chi ( A \/ B , S \/ B ) and E7 . ( A \/ B ) = Prob ( Q , D ) and E7 . ( c \/ ( A \/ B ) ) = Prob ( Q , D ) ;
consider y being Element of Y ( ) such that a = "\/" ( { F ( x , y ) where x is Element of X ( ) : P [ x ] } , L ( ) ) and P [ y ] ;
assume that A c= Z and f = ( #Z 2 ) * ( ( #Z 2 ) * ( f1 + f2 ) ) and for x st x in Z holds ( ( f1 + f2 ) `| Z ) . x = x * x + ( a - b ) / ( a - b ) and for x st x in Z holds ( ( f1 + f2 ) `| Z ) . x = a * x + b * x + b / ( a - b ) ;
( f /. i ) `2 = ( GoB f ) * ( 1 , j2 ) `2 .= ( GoB f ) * ( 1 , j2 ) `2 .= ( GoB f ) * ( 1 , j2 + 1 ) `2 .= ( GoB f ) * ( 1 , j2 + 1 ) `2 ;
dom Shift ( Seq q2 , len Seq q1 ) = { j + len Seq q1 where j is Nat : j in dom Seq q1 & i <= len Seq q2 } & dom Seq q1 = dom Seq q1 /\ dom Seq q2 ;
consider G1 , G2 , G3 being Element of V such that G1 <= G2 and G2 <= G2 and f in G1 and g in G2 and f in G2 and g in G2 and f = g * f ;
func - f -> PartFunc of C , V means : Def1 : dom it = dom f & for c st c in dom it holds it /. c = - f /. c & it /. c = - f /. c ;
consider phi such that phi is increasing and for a st phi . a = a & {} <> a for v holds union ( L , v ) in a & union ( L , v ) in H iff L . a c= H & L . a c= H ;
consider i1 , j1 such that [ i1 , j1 ] in Indices GoB f and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) ;
consider i , n such that n <> 0 and sqrt p = ( i - n ) / ( n + 1 ) and for n1 being Nat st n1 <> 0 & n <= len p holds sqrt ( p . n1 ) <= ( i - n ) / ( n + 1 ) and sqrt ( p . n1 ) <= ( i - n ) / ( n + 1 ) ;
assume that not 0 in Z and Z c= dom ( ( arccot * f1 ) `| Z ) and for x st x in Z holds ( ( ( arccot * f1 ) `| Z ) . x ) > - 1 & ( ( ( #Z ( 1 / 2 ) ) * f1 ) `| Z ) . x = - 1 & ( ( #Z ( 1 / 2 ) ) * f1 ) `| Z ) . x = 1 ;
cell ( G1 , i1 -' 1 , ( 2 |^ ( m -' 1 ) ) * ( ( Y -' 1 ) + 2 ) ) \ L~ ( f1 | ( L~ f1 ) ) c= ( L~ f1 ) \/ ( L~ f2 ) ) & ( ( L~ f1 ) \/ ( L~ f1 ) ) /\ ( L~ f1 ) ;
ex Q1 being open Subset of X st s = Q1 & ex Q1 being Subset-Family of Y st Q1 c= F & ( for x being Point of Y holds x in Q1 & x in Q1 & x in Q1 ) & ( x in Q1 implies x in Q1 & x in Q1 ) ;
gcd ( A , ( ( 1 , 1 ) --> ( 1 , 1 ) ) , ( ( 1 , 1 ) --> ( 1 , 1 ) ) , ( ( 1 , 1 ) --> ( 1 , 1 ) ) ) = 1 / ( ( 1 , 1 ) --> ( 1 , 1 ) ) ;
R8 = ( ( the let 2 , m2 ) . ( 1 + 1 ) ) . ( m2 + 1 ) .= ( ( the - 3 ) * ( 1 + 1 ) ) . ( m2 + 1 ) .= [ 3 , ( the - 2 ) * ( 1 + 1 ) ) . ( m2 + 1 ) .= [ 3 , ( the - 2 ) * ( 1 + 1 ) ) . ( m2 + 1 ) ] ;
CurInstr ( P-6 , Comput ( Pmeans , m1 + m2 ) ) = CurInstr ( P3 , Comput ( P3 , s3 , m3 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m3 ) ) .= CurInstr ( P3 , s3 ) .= CurInstr ( P3 , s3 ) .= CurInstr ( P3 , s3 ) .= ( CurInstr ( P3 , s3 ) ) .= ( CurInstr ( P3 , s3 ) ) ;
P1 /\ P2 = ( { p1 } \/ LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) \/ { p2 } ) \/ ( { p2 } \/ { p2 } /\ LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ) \/ { p2 } \/ { p2 } /\ { p2 } \/ { p2 } /\ { p2 } \/ { p2 } /\ { p2 } ;
func 'not' f -> Subset of the carrier of Al means : Def1 : a in it iff ex i st i in dom f & a = f . i & for p st p in dom f holds p in it & p in the carrier of A & p in the carrier of A ;
for a , b being Element of F_Complex st |. a .| > |. b .| for f being Polynomial of F_Complex st f >= 1 & f is ] holds f is \cup ( f - ( b * card f ) )
defpred P [ Nat ] means 1 <= $1 & $1 <= len g implies for i , j st [ i , j ] in Indices G & 1 <= j & j <= len G & 1 <= i & i <= len G & G * ( i , j ) = G * ( i , j ) ;
assume that C1 , C2 are_\HM } and f , g are_\HM { s1 where s1 is State of C1 , s2 is State of C2 : s1 = s2 & s2 = f . s1 & s1 = s2 & s2 = f . s2 & s1 = s2 ;
( ||. f .|| | X ) . c = ||. f /. 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 & 0 + ( q `2 ) ^2 < ( q `1 ) ^2 + ( q `2 ) ^2 ;
for F being Subset-Family of TT st F is open & {} in F & A <> {} for A , B being Subset of TT st A in F & B in F & A misses B holds card A c= card B
assume that len F >= 1 and len F = k + 1 and len F = len G and for k st k in dom F holds H . k = g . ( F . k , G . k ) and for k st k in dom F holds H . k = g . ( F . k , G . k ) ;
i |^ ( ( ( ( ( m mod n ) |^ ( ( m - k ) ) |^ s ) ) ) ) = i |^ ( s + k ) - i |^ ( ( ( m - k ) |^ s ) * 1 ) ) .= i |^ ( ( ( ( i - k ) |^ s ) * 1 ) - i |^ ( ( ( ( k - k ) * 1 ) |^ s ) - 1 ) ) .= i |^ ( ( ( ( ( ( k - k ) * 1 ) * 1 ) - 1 ) ) ;
consider q being oriented oriented oriented oriented Chain of G such that r = q and q <> {} and ( F . ( q . 1 ) ) `1 = v1 & rng ( F . ( len F ) ) `1 = v2 & rng ( F . ( len F ) ) `1 = v2 & rng ( F . ( len F ) ) `1 = v2 ;
defpred P [ Element of NAT ] means $1 <= len ( I . ( Z ^ I ) ) implies ( ( g . ( Z ^ I ) ) . ( len ( g . ( Z ^ I ) ) ) ) . ( len ( g . ( Z ^ I ) ) ) = ( ( g . ( Z ^ I ) ) . ( len ( g . ( Z ^ I ) ) ) ) . ( len ( g . ( Z ^ I ) ) ) ;
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
consider s being FinSequence of the carrier of R such that Sum s = u and for i being Element of NAT st 1 <= i & i <= len s ex a , b being Element of R st s . i = a * b & a in I & b in J & s . i = b * a ;
func |( x , y )| -> Element of COMPLEX equals |( Re x , Re y )| - ( Re y ) * |( x , Im y )| + ( ( Im x ) * |( x , Im y )| ) + ( ( Im y ) * |( x , Im y )| ) + ( ( Im x ) * |( x , Im y )| ) ;
consider g be FinSequence of FF such that g is continuous and rng ( g ) c= A & g . 1 = x1 & g . len g = x2 & for i st i in dom g holds g . i = x1 & g . ( len g ) = y1 & g . ( len g ) = x2 ;
then n1 >= len p1 & crossover ( p1 , p2 , n1 , n2 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n2 , n3 , n2 , n3 , n2 , n3 , n2 , n3 , n2 , n3 , n2 , n3 , n2 , n3 , n3 , n2 , n3 , n2 , n3 , n2 , n3 , n2 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n2
( q `1 ) * a <= ( q `1 ) * a & - ( q `1 ) * a <= ( q `1 ) * a or ( q `1 ) * a >= ( q `1 ) * a & ( q `1 ) * a >= 0 or ( q `1 ) * a >= 0 ) & ( q `1 ) * a >= 0 & ( q `1 ) * a >= 0 ) implies a = 0 & b = 0
FF . ( p . ( len pp ) ) = FF . ( p . ( len p ) ) .= ( v /. ( len p ) ) .= v /. ( len p ) .= v /. ( len v ) .= v /. ( len v ) .= v /. ( len v ) .= v /. ( len v ) .= v /. ( len v ) .= v /. ( len v ) ;
consider k1 being Nat such that k1 + k = 1 and a := k = ( <* a *> ^ ( ( intloc 0 ) --> ( intloc 0 ) ) ) ^ <* ( ( intloc 0 ) .--> ( intloc 0 ) ) . ( a , intloc 0 ) *> ^ <* ( ( intloc 0 ) .--> ( intloc 0 ) ) . ( a , intloc 0 ) ) *> ;
consider B9 being Subset of B1 , y" being Function of B1 , A1 such that B8 is finite and D8 = the carrier of A1 and D8 = the carrier of B1 and D8 = the carrier of B1 and B8 = the carrier of B2 and B8 = the carrier of B1 and B8 = the carrier of B2 and B8 = the carrier of B1 ;
v2 . b2 = ( curry F2 ) * ( ( curry F2 ) . b2 ) .= ( ( curry F2 ) * ( ( ( ( ( the > F ) | B ) ) . b2 ) ) ) . ( ( ( ( ( the L of F ) | B ) . b2 ) ) ) .= ( ( ( ( ( the L of F ) | B ) ) | B ) ) . ( ( ( ( the ) of F ) | B ) ) . ( ( id B ) ) ) .= ( ( ( the ) of B ) . b2 ) ) . ( ( id B ) . b2 ) .= ( ( ( ( the carrier of B ) . b2 ) ) . ( ( id B ) . b2 ) .= ( ( ( ( the ) of B ) . b2 ) . b2 ) .= ( ( ( ( ) . b2 ) . ( ( ( ( ) . b2 ) . b2
dom IExec ( I-35 , P , Initialize s ) = the carrier of SCMPDS .= dom IExec ( I , P , Initialize s ) .= dom IExec ( I , P , Initialize s ) .= dom IExec ( I , P , Initialize s ) .= dom IExec ( I , P , Initialize s ) .= dom IExec ( I , P , Initialize s ) ;
ex d-32 be Real st d-32 > 0 & |. h .| < d & |. h .| " * ||. ( R2 + R1 ) /. h .|| < e / ( ( ||. R2 .|| + 1 ) * ( ||. ( R2 + R1 ) /. h .|| ) ) ;
LSeg ( G * ( len G , 1 ) + |[ - 1 , 0 ]| , G * ( len G , 1 ) + |[ 1 , 0 ]| ) c= Int cell ( G , len G , 0 ) \/ { |[ 1 , 0 ]| } ;
LSeg ( mid ( h , i1 , i2 ) , i ) = LSeg ( h /. ( i + i1 -' 1 ) , h /. ( i + 1 + 1 ) ) .= LSeg ( h /. ( i + i1 -' 1 ) , h /. ( i + 1 ) ) .= LSeg ( h /. ( i + i1 -' 1 ) , h /. ( i + 1 ) ) ;
A = { q where q is Point of TOP-REAL 2 : LE p1 , p2 , P & LE p2 , q , P & LE q , p , P & LE p2 , p , P & LE q , p , P & LE p , q , P & LE q , p , P & LE q , p , P & LE q , p , P & LE q , p , P & LE q , p , P & LE q , p , P & LE q , q , P & LE q , p , P & LE q , p , P & LE q , p , P & LE q , p , P & LE q , q , P & LE q , p , P & LE p , q , P & LE p , q , P & LE q , p , P & LE p , p , P & LE p , p , P & LE p , p1 , P & LE p , p , P & LE p ,
( ( - x ) .|. y ) = - ( ( - 1 ) * ( x .|. y ) ) .= ( - ( - 1 ) * ( x .|. y ) ) .= ( - ( - 1 ) * ( x .|. y ) ) .= ( ( - 1 ) * ( x .|. y ) ) .= ( ( - 1 ) * ( x .|. y ) ) ;
0 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) = ( p `1 ) ^2 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) .= ( p `1 ) ^2 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) ;
( U * ( W7 * ( W * ( 1 / 2 ) ) ) ) * ( ( W * ( 1 / 2 ) ) ) = ( ( U * ( W * ( 1 / 2 ) ) ) ) * ( ( W * ( 1 / 2 ) ) ) .= ( U * ( W * ( 1 / 2 ) ) ) ) * ( ( W * ( 1 / 2 ) ) ) ;
func Shift ( f , h ) -> PartFunc of REAL , REAL means : Def1 : dom it = dom ( - h ) & for x st x in dom it holds it . x = ( - h ) . x + ( - h ) . x ;
assume that 1 <= k and k + 1 <= len f and [ i , j ] in Indices G and 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 y in Free H and not x in Free H and not x in Free H and not y in Free H and not x in Free H and not x in Free H and not y in Free H and x in Free H ;
defpred P11 [ Element of NAT , Element of NAT ] means ( P [ $1 , p ] & ( P [ $1 , p ] implies P [ $1 , p ] ) & ( P [ $1 , p ] implies Q [ $1 , p ] ) & ( P [ $1 , p ] implies Q [ $1 , p ] ) ;
func \sigma ( C ) -> non empty Subset-Family of X means : Def1 : for A being Subset of X holds A in it iff for W being Subset of X st W c= X & W c= X \ A holds it . W = C . ( W \/ A ) ;
[#] ( ( dist ( P ) ) .: Q ) = ( dist ( P ) ) .: Q & lower_bound ( [#] ( ( dist ( P ) ) .: Q ) ) = lower_bound ( ( dist ( P ) ) .: Q ) & lower_bound ( [#] ( ( dist ( P ) ) .: Q ) ) = lower_bound ( ( dist ( P ) ) .: Q ) ;
rng ( F | ( [ S ] ^ <* 2 *> ) ) = {} or rng ( F | ( [ S ] ^ <* 2 *> ) ) = { 1 } or rng ( F | ( [ S ] ^ <* 2 *> ) ) = { 2 } or rng ( F | ( [ S ] ^ <* 2 *> ) ) = { 1 } ;
( f " ( rng f ) ) . i = f . i " * ( ( f . i ) " ) . ( ( f . i ) " ) .= ( f . i ) " * ( ( f . i ) " ) . ( ( f . i ) " ) .= ( f . i ) " ;
consider P1 , P2 being non empty Subset of TOP-REAL 2 such that P1 \/ P2 = { p1 , p2 } and P1 \/ P2 = { p1 , p2 } and P1 /\ P2 = { p1 , p2 } and P1 /\ P2 = { p1 , p2 } and P1 /\ P2 = { p1 , p2 } ;
f . p2 = |[ ( p2 `1 ) ^2 / sqrt ( 1 + ( p2 `1 / p2 `2 ) ^2 ) , ( p2 `1 ) ^2 / sqrt ( 1 + ( p2 `1 / p2 `2 ) ^2 ) ]| & ( f . p2 ) `1 = ( p2 `1 ) ^2 / sqrt ( 1 + ( p2 `1 / p2 `2 ) ^2 ) ;
( ( ( a , X ) --> x ) " ) . x = ( ( ( a , X ) --> x ) " ) . x .= ( ( a , X ) --> x ) . x .= ( ( a , X ) --> x ) . x .= ( ( a , X ) --> x ) . x .= ( ( a , X ) --> x ) . x .= ( ( a , x ) --> x ) . x ;
for T being non empty normal TopSpace , A , B being closed Subset of T st A <> {} & A misses B for p being Point of T , r being Real st p in A & r in B & p in A holds ( <* in ( Element G ) ) . p , r ) = ( ( Element G ) . p ) * ( ( <* r *> ^ p ) . r )
for i st i in dom F for G1 , G2 being strict normal Subgroup of G st G1 = F . i & G2 = F . i & G2 = F . ( i + 1 ) & G2 = F . ( i + 1 ) holds G1 , G2 K K
for x st x in Z holds ( ( ( #Z 2 ) * ( arctan - arccot ) ) `| Z ) . x = ( ( ( #Z 2 ) * ( arctan - arccot ) ) `| Z ) . x / ( 1 + x ^2 )
synonym f is_continuous means : for x0 st x0 in dom f & x0 in dom f & for a st a in ]. x0 , x0 + a .[ holds f . a = lim ( f , x0 ) & ( for x st x in ]. x0 , x0 + a .[ holds f . x = lim ( f , x0 ) ) & ( for x st x in ]. x0 , x0 + a .[ holds f . x = lim ( f , x0 ) ) ;
then X1 , X2 are_separated & X2 , X2 are_separated & ( X1 union X2 ) is SubSpace of X & ( X1 union X2 ) is SubSpace of X2 & ( X1 union X2 ) misses X2 & ( X1 union X2 ) misses X2 & X2 misses X1 & X1 misses X2 implies X1 union X2 is SubSpace of X2 & X2 misses X1 & X2 misses X2 & X2 misses X2 & X2 misses X2 & X2 misses X1 & X2 misses X2 & X2 misses X2 & X2 misses X1 & X2 misses X2 & X2 misses X2 & X2 misses X2 & X2 misses X2 & X2 misses X2 & X2 misses X2 & X2 misses X2 & X2 misses X2 & X2 misses X2 & X2 misses X2 & X2 misses X2 & X2 misses X2 & X2 misses X2 & X2 misses X2 & X2 misses X2 & X2 misses X2 & X2 misses X2 & X2 misses X2 & X1 misses X2 & X1 union X2 implies X1 union X2 implies X1 union X2 implies X1 union X2 implies X1 union X2 & X2 union X2 & X2 misses X2 & X2 misses X2 & X2 misses X2 & X2 misses X2 & X2 misses X2 & X2 union X2 & X2 misses X2 & X2 misses X2 & X2 misses X2 & X2 misses
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 ) . x = L . ( x - u ) + R . ( x - u ) ;
( p2 `1 ) ^2 * sqrt ( 1 + ( p3 `1 ) ^2 ) >= ( ( p3 `1 ) ^2 * sqrt ( 1 + ( p3 `1 ) ^2 ) ) ^2 + ( ( p3 `1 ) ^2 * sqrt ( 1 + ( p3 `1 ) ^2 ) ) ;
( ( 1 / t ) (#) ||. ( f1 - f2 ) .|| ) to_power n = ( ( 1 / t ) (#) ||. ( f1 - f2 ) .|| ) to_power m & ( ( 1 / t ) (#) ||. ( f1 - f2 ) .|| ) to_power m = ( ( 1 / t ) (#) ||. ( f1 - f2 ) .|| ) to_power ( m + 1 ) ;
assume that for x holds f . x = ( sin . x + cos . x ) / ( sin . x ) ^2 and x in dom ( sin * ( sin . x + cos . x ) ) and for x st x in dom ( sin * ( sin * ( sin * ( sin * ( cos * ( f1 + f2 ) ) ) ) ) holds ( sin * ( sin * ( f1 + f2 ) ) ) ) . x = ( sin . x + ( sin . x ) / ( sin . x ) ) / ( sin . x ) ^2 / ( sin . x ) ^2 and ( sin . x ) ^2 / ( sin . x ) ^2 / ( sin . x ) ^2 / ( sin . x ) ^2 and ( sin . x ) ^2 / ( sin . x ) ^2 / ( sin . x ) ^2 = ( sin . x ) ^2 and for x st x = 1 + ( sin . x ) ^2 and ( sin . x ) ^2 / ( sin . x ) ^2 and ( sin . x ) ^2 = ( sin . x ) ^2 / ( sin . x ) ^2 and ( sin . x )
consider Xf1 being Subset of Y , Y1 being Subset of X such that t = [: Xf1 , Y1 :] and Y1 is open and ex Y1 being Subset of X st Y1 = Y1 & Y1 is open & Y1 is open & Y1 c= Y1 & Y1 is open ;
card ( S . n ) = card { [: d , Y :] + ( a * d ) / b where d is Element of GF ( p ) : [ d , Y ] in R & [ d , Y ] in R } .= ( ( GF ( p ) ) * R ) \/ { d } .= ( ( GF ( p ) ) * R ) \/ { d } ;
( W-bound D - E-bound D ) / 2 * ( ( W-bound D ) / 2 ) * ( ( W-bound D ) / 2 ) = ( W-bound D ) - ( W-bound D ) / 2 * ( ( W-bound D ) / 2 ) .= ( W-bound D ) / 2 * ( ( W-bound D ) / 2 ) ;