thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b
D c= S
let Y ;
S ` is Cauchy
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 <= E ;
assume e > 0 ;
assume 0 < g ;
p in A ;
x in X ;
Y `2 in Y ;
assume 0 < g ;
not c in Y ;
not v in L ;
2 in z `2 ;
assume f = g ;
N c= b `2 ;
assume i < k ;
assume u = v ;
I = J ;
B `2 = b `2 ;
assume e in F ;
assume p > 0 ;
assume x in D ;
let i be element ;
assume F is onto ;
assume n <> 0 ;
let x be element ;
set k = z ;
assume o = x ;
assume b < a ;
assume x in A ;
a `2 <= b `2 ;
assume b in X ;
assume k <> 1 ;
f = product l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is generated 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 `2 = x `2 ;
let X be BCK-algebra ;
assume S is non empty ;
a in REAL ;
let p be set ;
let A be set ;
let G be _Graph , a , b be set ;
let G be _Graph , a , b be set ;
let a be Complex ;
let x be element ;
let x be element ;
let C be FormalContext , a , b be Element of C ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
let y be Real ;
X c= f . a
let y be element ;
let x be element ;
let i be Nat ;
let x be element ;
n in NAT ;
let a be element ;
m in NAT ;
let u be element ;
i in NAT ;
let g be Function ;
Z c= NAT ;
l <= ma ;
let y be element ;
r2 in dom f ;
let x be element ;
let k1 be Integer ;
let X be set ;
let a be element ;
let x be element ;
let x be element ;
let q be element ;
let x be element ;
assume f is being_homeomorphism ;
let z be element ;
a , b // K ;
let n be Nat ;
let k be Nat ;
B ` c= B ` ;
set s = from Z ;
n >= 0 + 1 ;
k c= k + 1 ;
R1 c= R ;
k + 1 >= k ;
k c= k + 1 ;
let j be Nat ;
o , a // Y ;
R c= Cl G ;
Cl B = B ;
let j be Nat ;
1 <= j + 1 ;
arccot is_differentiable_on Z ;
exp is_differentiable_in x ;
j < i0 ;
let j be Nat ;
n <= n + 1 ;
k = i + m ;
assume C meets S ;
n <= n + 1 ;
let n be Nat ;
h1 = {} ;
0 + 1 = 1 ;
o <> b3 ;
f2 is one-to-one ;
support p = {}
assume x in Z ;
i <= i + 1 ;
r1 <= 1 ;
let n be Nat ;
a "/\" b <= a ;
let n be Nat ;
0 <= r0 ;
let e be Real , x be Real ;
not r in G . l
c1 = 0 ;
a + a = a ;
<* 0 *> in e ;
t in { t } ;
assume F is not discrete ;
m1 divides m ;
B * A <> {} ;
a + b <> {} ;
p * p > p ;
let y be ExtReal ;
let a be Int-Location , i be Nat ;
let l be Nat ;
let i be Nat ;
let r ;
1 <= i2 ;
a "\/" c = c ;
let r be Real ;
let i be Nat ;
let m be Nat ;
x = p2 ;
let i be Nat ;
y < r + 1 ;
rng c c= E
Cl R is boundary ;
let i be Nat ;
R2 in dom f1 ;
cluster downarrow x -> or uparrow x is or uparrow x is or uparrow x is or x is or x is or x is or
X <> { x } ;
x in { x } ;
q , b // M ;
A . i c= Y ;
P [ k ] ;
2 to_power x in W ;
X [ 0 ] ;
P [ 0 ] ;
A = A |^ i ;
- Let s >= os ;
G . y <> 0 ;
let X be RealNormSpace , x be Point of X ;
a in A ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , W be Subspace of V ;
assume x in - M ;
k < s . a ;
not t in { p } ;
let Y be set , x be set ;
M , L are_isomorphic ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded lower-bounded ;
rng f = Y ;
G8 c= L ;
assume x in Cl Q ;
m in dom P ;
i <= len Q ;
len F = 3 ;
Free p = {} ;
z in rng p ;
lim b = 0 ;
len W = 3 ;
k in dom p ;
k <= len p ;
i <= len p ;
1 in dom f ;
b `1 = a `1 + 1 ;
x `2 = a * y `2 ;
rng D c= A ;
assume x in K1 ;
1 <= ii ;
1 <= ii ;
[: p\hbox , r :] c= PI ;
1 <= ii ;
1 <= ii ;
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 : A is closed ;
cluster R \ S -> real-valued ;
sup D in S ;
x << sup D ;
b1 >= Z1 & b2 >= Z1 ;
assume w = 0. V ;
assume x in A . i ;
g in the carrier of into X ;
y in dom t ;
i in dom g ;
assume P [ k ] ;
Let -1 c= f ;
x9 is increasing ;
let e2 be element ;
- b divides b ;
F c= \tau ( F ) ;
Gseq is non-decreasing ;
Gseq is non-decreasing ;
assume v in H . m ;
assume b in [#] B ;
let S be non void ManySortedSign , x be set ;
assume P [ n ] ;
assume union S is [#] independent ;
V is Subspace of V ;
assume P [ k ] ;
rng f c= NAT ;
assume ex_inf_of X , L ;
y in rng f ;
let s , I be set , x be set ;
b `2 c= ( b `2 ) * ( b `2 ) ;
assume not x in INT ;
A /\ B = { a } ;
assume len f > 0 ;
assume x in dom f ;
b , a // o , c ;
B in B-24 ;
cluster product p -> non empty ;
z , x // x , p ;
assume x in rng N ;
cosec is_differentiable_in x ;
assume y in rng S ;
let x , y be element ;
i2 < i1 & i2 < i2 ;
a * h in a * H ;
p , q in Y ;
cluster sqrt I -> left ideal ;
q1 in A1 & q2 in A2 ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A2 c= A1 ;
\hbox { \boldmath $ p $ } , r ) < n ;
assume A c= dom f ;
Re f is_integrable_on M ;
let k , m be element ;
a , a \equiv b , b ;
j + 1 < k + 1 ;
m + 1 <= n1 ;
g is_differentiable_in x0 & g is_differentiable_in x0 ;
g is_continuous_in x0 & g is_differentiable_in x0 ;
assume O is symmetric transitive ;
let x , y be element ;
let j0 be Nat ;
[ y , x ] in R ;
let x , y be element ;
assume y in conv A ;
x in Int V ;
let v be Vector of V ;
P3 halts_on s , P3 = P +* J ;
d , c // a , b ;
let t , u be set ;
let X be set ;
assume k in dom s ;
let r be non negative Real ;
assume x in F | M ;
let Y be Subset of S ;
let X be non empty TopSpace , x be Point of X ;
[ a , b ] in R ;
x + w < y + w ;
{ a , b } is_>=_than c ;
let B be Subset of A , x be Element of B ;
let S be non empty ManySortedSign ;
let x be variable , f be Function of x , f ;
let b be Element of X , x be Element of X ;
R [ x , y ] ;
x ` ` = x ;
b \ x = 0. X ;
<* d *> in D |^ 1 ;
P [ k + 1 ] ;
m in dom ( - n ) ;
h2 . a = y ;
P [ n + 1 ] ;
cluster G * F -> pre| ;
let R be non empty multMagma , a , b 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 & y in rng co ;
let x be Element of B ;
let t be Element of D ;
assume x in Q .vertices() ;
set q = s ^\ k ;
let t be VECTOR of X ;
let x be Element of A ;
assume y in rng p `1 ;
let M be mamaid ;
let N be non empty Subset of \mathop { \rm being \mathop { \rm that N } ;
let R be RelStr with finite finite Anumber ;
let n , k be Nat ;
let P , Q be RelStr ;
P = Q /\ [#] S ;
F . r in { 0 } ;
let x be Element of X ;
let x be Element of X ;
let u be VECTOR of V ;
reconsider d = x as Int-Location ;
assume I does not destroy 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 ` ;
cluster S --> T -> such that S --> T is such that S is such is such
assume t1 <= t2 & t2 <= t2 ;
let i , j be even Integer ;
assume F1 <> F2 & F2 <> F1 ;
c in Intersect ( union R ) ;
dom ( p1 ^ p2 ) = c ;
a = 0 or a = 1 ;
assume A1 : A <> A2 & A <> A1 ;
set i1 = i + 1 ;
assume a1 = b1 & a2 = b2 ;
dom ( g1 * g2 ) = A ;
i < len M + 1 ;
assume not - +infty in rng G ;
N c= dom ( f1 + f2 ) ;
x in dom ( sec * sec ) ;
assume [ x , y ] in R ;
set d = ( x - y ) / ( x - y ) ;
1 <= len ( g1 ^ g2 ) ;
len s2 > 1 & len s2 > 1 ;
z in dom ( f1 + f2 ) ;
1 in dom ( D2 | Seg len D2 ) ;
( p `2 ) ^2 = 0 ;
j2 <= width G & j2 <= width G ;
len PI > 1 + 1 ;
set n1 = n + 1 ;
|. ( q `1 ) ^2 .| = 1 ;
let s be SortSymbol of S ;
lcm ( i , i ) = i ;
X1 c= dom f & X2 c= dom g ;
h . x in h . a ;
let G be mod of \mathbin { - } \rm Ball ( u , r ) ;
cluster m * n -> square ;
let k9 be Nat , k be Nat ;
i -' 1 > m ;
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 .| ;
i <= len ( f2 | n ) ;
p is FinSequence of X ;
1 + 1 in dom g ;
Sum R2 = n * r ;
cluster f . x -> complex-valued ;
x in dom ( f1 + f2 ) ;
assume [ X , p ] in C ;
BX c= ( 3 + 1 ) \ { x } ;
n2 <= ( 2 |^ ( n2 + 1 ) ) ;
A /\ [: A , B :] c= A `
cluster x -valued -> constant for Function ;
let Q be Subset-Family of S , a be Element of Q ;
assume n in dom ( g2 * f1 ) ;
let a be Element of R ;
t `1 in dom ( e2 . e ) ;
N . 1 in rng N ;
- z in A \/ B ;
let S be Subset-Family of X , x be Element of S ;
i . y in rng i ;
REAL c= dom f & dom ( f ^ ) = REAL ;
f . x in rng f ;
mt <= ( r / 2 ) ;
s2 in r-5 & s1 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 , S T = T --> T ;
let x be non positive ExtReal ;
let m be Element of M ;
f in union ( rng F1 ) ;
let K be add-associative right_zeroed right_complementable non empty doubleLoopStr , A be Matrix of K ;
let i be Element of NAT ;
rng ( F * g ) c= Y
dom f c= dom x & rng f c= dom x ;
n1 < n1 + 1 + 1 ;
n1 < n1 + 1 + 1 ;
cluster [: T , T :] -> \overline { A } ;
[ y2 , 2 ] `1 = z ;
let m be Element of NAT ;
let S be Subset of R ;
y in rng ( S | k ) ;
b = sup dom f & a = sup dom f ;
x in Seg ( len q ) ;
reconsider X = D as set ;
[ a , c ] in E1 ;
assume n in dom ( h2 + c ) ;
w + 1 = ( - a ) + 1 ;
j + 1 <= j + 1 + 1 ;
k2 + 1 <= k1 + 1 ;
let i be Element of NAT ;
Support u = Support ( p ) \/ Support ( q ) ;
assume X is complete complete Sub] of m ;
assume that f = g and p = q ;
n1 <= n1 + 1 + 1 ;
let x be Element of REAL ;
assume x in rng s2 & y in rng s2 ;
x0 < x0 + 1 & x0 + r < x0 + 1 ;
len ( L (#) F ) = W ;
P c= Seg ( len A ) ;
dom q = Seg n & rng q c= Seg n ;
j <= width M *' ;
let r8 be real-valued sequence of REAL ;
let k be Element of NAT ;
Integral ( M , P ) < +infty ;
let n be Element of NAT ;
assume z in len of of of ) -C2 ( 0 , A ) ;
let i be set ;
n - 1 = n-1 ;
len ( n |-> u ) = n ;
\mathop { Z , c } c= F
assume x in X or x = X ;
x is midpoint of b , c ;
let A , B be non empty set , x be Element of A ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q & dom q = Seg i ;
let s be Element of E -tuples_on omega ;
let B1 be Basis of x , y ;
LSeg ( 3 , L2 ) /\ L2 = {} ;
L1 /\ LSeg ( f , p2 ) = {} ;
assume downarrow x = downarrow y ;
assume b , c // b , c ;
LIN q , c , c ;
x in rng ( f | -1-129 ) ;
set n8 = n + j ;
let D7 be non empty set , f be FinSequence of D ;
let K be add-associative right_zeroed right_complementable non empty addLoopStr , f be Function of K , K ;
assume that f `2 = f and h `1 = h ;
R1 - R2 is total implies 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 ] in maximal in the maximal of C ;
let a , b be Element of S ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = Integral ( M , f | E ) ;
cluster n[ A ] -> ne[
not u in { \hbox { \boldmath $ g } , f } ;
the carrier of f c= B \/ { x } ;
reconsider z = x as Vector of V ;
cluster the Str of L -> \rangle ;
r (#) H is qua 0 -defined PartFunc of X , REAL ;
s . intloc 0 = 1 ;
assume x in C & y in C ;
let U0 be strict non-empty MSAlgebra over S , A be non-empty MSAlgebra over S ;
[ x , Bottom T ] is compact ;
i + 1 + k in dom p ;
F . i is stable Subset of M ;
r-35 in iff ( x in iff r in { y } ) ;
let x , y be Element of X ;
let A , I be |. such that A is |. [ A , I ] .| ;
[ y , z ] in [: O , O :] ;
( that that that card Macro i = 1 and card Macro i = 1 ;
rng Sgm A = A ;
q |- p ^' All ( y , q ) ;
for n holds X [ n ] ;
x in { a } & x in d ;
for n holds P [ n ] ;
set p = |[ x , y , z ]| ;
LIN o , a , b & LIN o , a , b ;
p . 2 = Z |^ Y ;
( D . 0 ) `2 = {} ;
n + 1 + 1 <= len g ;
a in [: [. Al ( ) , D ( ) :] ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
let I be Ideal of L ;
set g = f1 + f2 , h = g1 + g2 ;
a <= max ( a , b ) ;
i-1 < len G + 1 ;
g . 1 = f . ( i1 + 1 ) ;
x `1 , y `2 in A2 ;
( f /* s ) . k < r ;
set v = VAL g ;
i -' k + 1 <= S ;
cluster associative for non empty multMagma ;
x in ( support ( t ) ) \ { x } ;
assume a in [: the carrier of G , the carrier of G :] ;
i `2 <= len ( y `2 ) ;
assume p divides b1 + b2 & p divides b2 + b3 ;
p0 <= upper_bound ( M1 + M2 ) & ( M1 + M2 ) <= sup ( M1 + M2 ) ;
assume x in ( W-min X ) `2 ;
j in dom ( z (#) ( y (#) z ) ) ;
let x be Element of D ( ) ;
IC s4 = l1 .= l1 .= l1 .= l1 ;
a = {} or a = { x } ;
set uG = Vertices G , uH = Vertices G ;
seq " is non-zero & ( seq " ) (#) ( seq " ) is non-zero ;
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
h-4 c= h-14 \/ { h } ;
]. a , b .[ c= Z ;
X1 , X2 X2 , X2 be SubSpace of X ;
a in Cl ( union F \ G ) ;
set x1 = [ 0 , 0 ] ;
k + 1 - 1 = k ;
cluster real-valued -> real-valued for Relation of INT ;
ex v st C = v + W ;
let IT be non empty doubleLoopStr , x be Element of IT ;
assume V is Abelian add-associative right_zeroed right_complementable non empty doubleLoopStr ;
XY \/ Y in \sigma ( L ) ;
reconsider x = x as Element of S ;
max ( a , b ) = a ;
sup B is upper & sup B in B ;
let L be non empty reflexive antisymmetric RelStr , x be Element of L ;
R is reflexive & X is transitive implies R + ( X + Y ) is transitive
E , g |= ( H , ( H , x ) ) ;
dom ( G /. y ) = a ;
( 1 - 4 ) >= - r ;
G . p0 in rng G & G . I in rng G ;
let x be Element of FF , a be Element of FF ;
D [ ( P , 0 ) `1 , 0 ] ;
z in dom ( id B ) & z in dom ( id B ) ;
y in the carrier of N & x in the carrier of N ;
g in the carrier of H & h in the carrier of G ;
rng ( f | l ) c= [: NAT , NAT :] ;
j `2 + 1 in dom s1 & j + 1 in dom s2 ;
let A , B be strict Subgroup of G ;
let C be non empty Subset of REAL ;
f . z1 in dom h & f . z2 in dom h ;
P . ( k1 + 1 ) in rng P ;
M = ( A +* {} ) +* ( A .--> {} ) ;
let p be FinSequence of REAL , r be Real ;
f . n1 in rng f & f . n2 in rng f ;
M . ( F . 0 ) in REAL ;
h = h & b = b ;
assume that the distance of V , Q and Q is open ;
let a be Element of ^ ( V ) ;
let s be Element of P ( ) , a be Element of P ( ) ;
let PH be non empty RelStr ;
let n be Nat ;
the carrier of g c= B & the carrier of g c= B ;
I = halt SCM R & I = ( the carrier of R ) \ { {} } ;
consider b being element such that b in B ;
set BK = BCS K , BK = BCS K ;
l <= ( -> i ) & ( j - 1 ) <= ( j - 1 ) ;
assume x in downarrow [ s , t ] ;
( x `2 ) ^2 in uparrow t ;
x in ( JumpParts T ) . ( T . x ) ;
let h be Morphism of c , a ;
Y c= [: the carrier of K , the carrier of K :] ;
A2 \/ A3 c= Carrier ( A2 \/ A3 ) ;
assume LIN o , a , b & LIN o , a , b ;
b , c // d1 , e2 ;
x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8
dom <* y *> = Seg 1 & dom <* y *> = Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar s .| ;
[ x , x `2 ] in X ~ ;
for n being Nat holds 0 <= x . n
[' a , b '] = [. a , b .] ;
cluster -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> closed for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 , q1 is_collinear & q2 , q1 , q2 is_collinear ;
dom ( M1 + M2 ) = Seg n & dom ( M1 + M2 ) = Seg n ;
x = [ x1 , x2 ] & y = [ y1 , y2 ] ;
let R , Q be ManySortedSet of A ;
set d = ( 1 / n + 1 ) ;
rng ( g2 ) c= dom W & rng ( g2 ) c= dom W ;
P . ( [#] Sigma \ B ) <> 0 ;
a in field R & a = b ;
let M be non empty Subset of V , A be Subset of V ;
let I be Program of SCM+FSA , a be Int-Location ;
assume x in rng ( ( L * R ) * S ) ;
let b be Element of the carrier of T ;
dist ( e , z ) - r-r > r-r ;
u1 + v1 in W2 + W3 + W3 ;
assume that not L misses rng G and not L in rng F ;
let L be lower-bounded antisymmetric antisymmetric transitive antisymmetric RelStr ;
assume [ x , y ] in a9 ;
dom ( A * e ) = NAT ;
let a , b be Vertex of G ;
let x be Element of Bool M ;
0 <= Arg a * PI ;
o9 , a9 // o9 , y & o9 , a9 // o9 , 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 D2 . k in rng D & D2 . k in rng D ;
f " ( p1 ) = 0 & f " ( p2 ) = 0 ;
set x = the Element of X \ { x } ;
dom Ser ( G ) = NAT & rng Ser ( G ) = NAT ;
let n be Element of NAT ;
assume LIN c , a , e1 & LIN c , e1 , e1 ;
cluster -> natural for FinSequence of NAT ;
reconsider d = c as Element of L1 ;
( v2 |-- I ) . X <= 1 ;
assume x in the carrier of f & y in the carrier of f ;
conv @ S c= conv @ A & conv @ S c= conv @ A ;
reconsider B = b as Element of the carrier of T ;
J , v |= P \lbrack l , P ( ) .] ;
cluster J . i -> non empty for TopStruct ;
ex_sup_of Y1 \/ Y2 , T & ex_sup_of Y1 , T ;
W1 is_not field ( W1 + W2 ) & W2 is_not field ( W1 + W2 ) ;
assume x in the carrier of R & y in the carrier of R ;
dom ( n |-> ( n + 1 ) ) = Seg n ;
s4 misses s2 & s4 misses s2 ;
assume ( a 'imp' b ) . z = TRUE ;
assume that X is open and f = X --> d ;
assume [ a , y ] in Indices ( f | X ) ;
assume that that function I c= J and dom function function function ( function J ) c= K ;
Im ( seq ) . n = 0 ;
( sin . x ) ^2 <> 0 & ( sin . x ) ^2 <> 0 ;
sin is_differentiable_on Z & cos is_differentiable_on Z implies cos is_differentiable_on Z & for x st x in Z holds cos . x > 0
6 . n = t0 . n .= ( n + 1 ) ;
dom ( - ( - F ) ) c= dom F ;
W1 . x = W2 . x .= ( W1 . x ) . x ;
y in W .vertices() \/ W .vertices() \/ W .vertices() ;
( ( k + 1 ) + 1 ) + 1 <= len ( ( k + 1 ) + 1 ) ;
x * a \equiv y * a . ( mod m ) ;
proj2 .: S c= proj2 .: P & proj2 .: S c= proj2 .: P ;
h . p4 = g2 . I .= ( g . I ) `1 ;
G = U /. 1 .= G * ( 1 , k ) `1 ;
f . ( rr1 ) in rng f ;
i + 1 + 1 <= len - 1 ;
rng F = rng ( F | n ) .= dom F ;
mode Y. is well unital non empty multMagma ;
[ x , y ] in A ~ ;
x1 . o in L2 . o & x2 . o in L2 . o ;
the carrier of ( support } ) c= B ;
not [ y , x ] in id ( X ) ;
1 + p .. f <= i + len f ;
( seq ^\ k1 ) ^\ 1 is lower ;
len ( F ^ <* a *> ) = len I + len <* a *> ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be complex number , x be Element of REAL ;
Comput ( P , s , n ) = s ;
k <= k + 1 & k + 1 <= len p ;
reconsider c = {} T as Element of L ;
let Y be Element of of of of of of of of of of T ;
cluster -> implies ex L being Function of L , L st L is directed-sups-preserving
f . j1 in K . j1 & f . j1 in K . j1 ;
cluster J => y -> total for total Relation ;
K c= 2 -tuples_on the carrier of T ;
F . b1 = F . b2 .= F . b2 ;
x1 = x or x1 = y or x1 = z ;
attr a <> {} means : Def3 : ( a / a ) = 1 ;
assume that succ a c= b and b in a ;
s1 . n in rng s1 & s1 . n in rng s2 ;
{ o , b2 } on C2 & { o , b2 } on C2 ;
LIN o , b , b9 & LIN o , b , b9 ;
reconsider m = x as Element of Funcs ( V , C ) ;
let f be non trivial FinSequence of D ;
let Fs2 be non empty TopSpace , x be Point of TOP-REAL 2 ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in F-8 ;
reconsider ps2 = x as Subset of m ;
let A , B , C be Element of R ;
cluster strict non empty for for \mathopen { - } u is strict ;
rng c `1 misses rng ( e `1 ) & rng ( e `2 ) c= rng ( e `2 ) ;
z is Element of gr { x } & z in gr { x } ;
not b in dom ( a .--> p1 ) ;
assume that k >= 2 and P [ k ] ;
Z c= dom ( cot * cot ) ;
the component of Q c= UBD A & UBD ( Q ) = UBD ( Q ) ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( 1 / ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2
attr f = u means : Def3 : a * f = a * u ;
for n holds P1 [ n ] implies P1 [ n + 1 ]
{ x . O : x in L } <> {} ;
let x be Element of V . s ;
let a , b be Nat ;
assume that S = S2 and p = p2 and q = q2 ;
gcd ( n1 , n2 , n2 ) = 1 & gcd ( n1 , n2 , n2 , n3 ) = 1 ;
set o9 = a * ( - b ) , a9 = a * ( - b ) ;
( seq . n ) < |. r1 .| & ( seq . n ) < r1 ;
assume that seq is increasing and r < 0 ;
f . ( y1 , x1 ) <= a ;
ex c being Nat st P [ c ] ;
set g = { n |^ 1 : n in dom f } ;
k = a or k = b or k = c ;
a9 , b9 , c9 , a9 , b9 , c9 , c9 , a9 , b9 , c9 , b9 , c9 , a9 , b9 , c9 ;
assume that Y = { 1 } and s = <* 1 *> ;
IS1 . x = f . x .= 0 .= 0 ;
W3 .last() = W3 . 1 .= ( W . 1 ) `2 ;
cluster -> trivial for Subgroup of G , finite _Graph ;
reconsider u = u as Element of Bags X ;
A in B ^ implies A , B are_that A , B are_that A , B are_that B , A are_that A , B are_that A , B are_that B , A |^ B are_that A , B |^ B
x in { [ 2 * n + 3 , k ] } ;
1 >= ( q `1 / |. q .| - cn ) / ( 1 + cn ) ;
f1 is_tcluster _of f2 , 1 & f2 is_tcluster f1 ;
( f /. i ) `2 <= ( q `2 ) ^2 ;
h is_the carrier of Cage ( C , n ) ;
( b `2 ) ^2 <= ( p `2 ) ^2 & ( p `2 ) ^2 <= ( p `2 ) ^2 ;
let f , g be Function of X , Y ;
S * ( k , k ) <> 0. K ;
x in dom ( max ( - f , - f ) ) ;
p2 in NF . ( p1 , p2 ) & p2 in NF . ( p1 , p2 ) ;
len ( ( the_left_argument_of H ) . 1 ) < len ( H . 1 ) ;
F [ A , F-14 . A ] ;
consider Z such that y in Z and Z in X ;
attr 1 in C means : Def3 : A c= C & A c= C ;
assume that r1 <> 0 or r2 <> 0 ;
rng ( q1 ^ q2 ) c= rng ( q1 ^ q2 ) ;
A1 , L , A2 , A3 , A3 , A2 , A3 , A3 , A2 ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in element ( p , SF ) & c in dom ( p , SF ) ;
then S is negative & P-2 [ S ] ;
Cl Int [#] T = [#] T .= [#] T ;
( f | A2 ) | A2 = f2 | A2 .= ( f | A2 ) | A2 ;
0. M in the carrier of W & 0. M in the carrier of W ;
let v , v be Element of M ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ V
let X be Subset of S ~ ;
consider H1 such that H = 'not' H1 and H1 in M ;
( - 1 ) * ( - 1 ) in ( - 1 ) * ( - 1 ) * ( - 1 ) ;
0 * a = 0. R .= a * 0 ;
A |^ ( 2 , 2 ) = A ^^ A ;
set vbeing = ( v /. n ) * ( v /. n ) ;
r = 0. ( \langle \cal E , \Vert \cdot \Vert *> ) ;
( f . p4 ) `1 >= 0 & ( f . p4 ) `2 >= 0 ;
len W = len ( W .last() ) .= len ( W \Omega ) ;
f /* ( s * G ) is divergent_to-infty & f /* ( s * G ) is divergent_to-infty ;
consider l being Nat such that m = F . l ;
t8 / ( W + 1 ) does not destroy b1 & not ( not b1 in W & not b2 in W ) ;
reconsider Y1 = X1 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 set ;
downarrow a /\ downarrow t is Ideal of T ;
let X be non empty set , F be non empty set , f be Function of X , NAT ;
rng f = S1 ( ) \/ W ;
let p be Element of B , x be the \mathclose of S ;
max ( N1 , 2 ) >= N1 & max ( N1 , 2 ) >= N2 ;
0. X <= b |^ ( m * mmm1 ) ;
assume that i in I and R1 . i = R ;
i = j1 & ( p1 = q1 ) & ( p2 = q2 implies q1 = q2 ) implies i = j
assume gR in the right & FR in the carrier of R ;
let A1 , A2 be Point of S , A2 be Point of T ;
x in h " P /\ [#] ( T1 | P ) ;
1 in Seg 2 & 1 in Seg 3 & 1 in Seg 3 ;
reconsider X-5 = X as non empty Subset of Tsuch that X = { x } and Y is non empty ;
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the Source of G ) -valued ;
n1 <= i2 + len ( g2 ) & n2 + len ( g2 ) <= len ( g2 ) ;
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume that v in the carrier' of G2 and u in the carrier' of G1 ;
y = Re y + ( Im y ) * i ;
( - ( - 1 ) |^ ( p -' 1 ) ) = 1 ;
x2 is_differentiable_on ]. a , b .[ & x2 is_differentiable_on ]. a , b .[ ;
rng ( M5 ) c= rng ( D2 | ( Seg n ) ) ;
for p being Real st p in Z holds p >= a
( ( - 1 ) (#) f ) | ( dom f ) = ( - 1 ) (#) f ;
( seq ^\ m ) . k <> 0 ;
s . ( G . ( k + 1 ) ) > x0 ;
( p -Path ( M , p ) ) . 2 = d ;
A \oplus ( B \ominus C ) = ( A \oplus B ) \ominus C
h \equiv gg . ( mod P ) , gg . ( mod P ) ;
reconsider i1 = i-1 - 1 as Element of NAT ;
let v1 , v2 be VECTOR of V , v be VECTOR of V ;
for V being Subspace of [#] V holds V is Subspace of [#] V
reconsider ii = i - 1 as Element of NAT ;
dom f c= [: C , D :] ;
x in ( the sequence of B ) . n & x in ( the InternalRel of B ) . n ;
len \rbrace in Seg ( len f2 + len f3 ) & len ( f1 + f2 ) = len f1 + len f2 ;
pA1 c= the topology of T & the topology of T c= the topology of T ;
]. r , s .[ c= [. r , s .] ;
let B2 be Basis of T2 , x be Element of T2 ;
G * ( B * A ) = ( the Arrows of o1 ) * ( the Arity of o2 ) ;
assume that p , u , v , u and u , v , v , u , v , w ;
[ z , z ] in union rng ( F | X ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S & $1 in S ;
LIN a1 , a3 , b1 & LIN a2 , a3 , b3 ;
f " ( f .: x ) = { x } ;
dom ( w2 - r ) = dom ( r12 - r ) .= dom ( r12 - r ) ;
assume that 1 <= i and i <= n and j <= n ;
( ( g2 ) . O ) `2 <= 1 - ( ( g2 ) . I ) `2 ;
p in LSeg ( E . i , F . i ) ;
Ij * ( i , j ) = 0. K ;
|. f . ( s . m ) - g .| < g1 ;
( q7 . x ) in rng ( ( q | n ) | n ) ;
Carrier ( ( Carrier ( l ) ) \ { p } ) misses Carrier ( l ) ;
consider c being element such that [ a , c ] in G ;
assume that Nreal = o\rbrack and for o being Element of O holds o <> a ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= ( ( F |^ C ) * ( F |^ D ) ) ;
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [. f . j , f . j .] ;
attr 0 <= x & x <= 1 implies x ^2 <= x ^2 ;
p `2 - q `2 <> 0. TOP-REAL 2 & p `2 - q `2 <> 0. TOP-REAL 2 ;
cluster \rbrace aa\circ ( S , T ) -> non empty ;
let x be Element of S ~ ;
cluster cluster cluster cluster cluster cluster cluster cluster cluster -> \hbox { - } -> one-to-one ;
|. i .| <= - ( - 2 |^ n ) / ( - 2 |^ n ) ;
the carrier of I[01] = dom P & the carrier of I[01] = dom Q ;
} * ( n + 1 ) ! > 0 * n ;
S c= ( A1 /\ A2 ) /\ ( A2 /\ A3 ) ;
a3 , a4 // b3 , b3 & a3 , a4 // b3 , b3 ;
then dom A <> {} & dom A <> {} & rng A <> {} ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , Y & y in X & z in 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] .= the carrier of I[01] .= the carrier of I[01] ;
p in Upper_Arc ( P ) /\ Upper_Arc ( P ) ;
dom ( d2 * A ) = [: A2 , A2 :] & dom ( d2 * A ) = 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 -> \HM { \HM { an } : F is \HM { \bf \rbrace } -> an operation of X
let U1 , U2 be non-empty MSAlgebra over S , A be MSAlgebra over S ;
Proj ( i , n ) * g is_differentiable_on X ;
let x , y , z be Point of X , p be Point of X ;
reconsider p0 = p . x as Subset of V ;
x in the carrier of Lin ( A ) & x in the carrier of W ;
let I , J be parahalting Program of SCM+FSA , a be Int-Location ;
assume that - a is lower and - b is lower and a <= - b ;
Int Cl ( A \/ B ) c= Cl Int Cl ( A \/ B ) ;
assume for A being Subset of X holds Cl A = A ;
assume q in Ball ( |[ x , y ]| , r ) ;
( p2 `2 ) ^2 <= ( p2 `1 ) ^2 & ( p2 `2 ) ^2 <= ( p2 `2 ) ^2 ;
Cl Q ` = [#] ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) |
set S = the carrier of T , T = the carrier of T ;
set I8 = ' ( f |^ n ) , I8 = ' ( f |^ n ) ;
len ( p /^ n ) = len ( p /^ n ) .= len p ;
A is Permutation of dom ( Swap ( A , x , y ) ) ;
reconsider n6 = n6 - 1 as Element of NAT ;
1 <= j + 1 & j + 1 <= len ( s | [. p , q .] ) ;
let q\mathopen { - } ( M + N ) , q\mathopen { - } ( M + N ) } , q\mathopen { - } ( M + N + N ) } , q = M + ( N + N ) ;
a9 in the carrier of S1 & b9 in the carrier of S1 ;
c1 /. ( n1 + 1 ) = c1 . ( n1 + 1 ) .= c2 . ( n1 + 1 ) ;
let f be FinSequence of TOP-REAL 2 , p be Point of TOP-REAL 2 , r be Real ;
y = ( f * ( S * ( x , y ) ) ) . x ;
consider x being element such that x in an " A ;
assume r in ( ( dist ( o , r ) ) .: P ) ;
set i2 = len ( n |-> \hbox { - } ) ;
h2 . ( j + 1 ) in rng h2 & h2 . ( j + 1 ) in rng h2 ;
Line ( M29 , k ) = M . i .= Line ( M29 , k ) ;
reconsider m = ( x - 2 ) / ( x - 2 ) as Element of ( - 2 ) ;
let U1 , U2 be Subspace of U0 , a be Element of U1 ;
set P = Line ( a , d ) ;
len ( p1 ^ p2 ) < len p2 + 1 + len p2 ;
let T1 , T2 be Scott Scott Scott Scott X1 of L ;
then x <= y & ( ex x st x in dom ( y | x ) ) ;
set M = n -tuples_on ( m -tuples_on ( k , m ) ) ;
reconsider i = x1 , j = x2 as Nat ;
rng ( ( the_arity_of o ) . a ) c= dom H & dom ( ( the_arity_of o ) . a ) = dom H ;
z1 " = ( z1 " ) * ( z1 " ) .= ( z1 " ) * ( z1 * ( z1 * ( z1 * ( z1 * ( z1 * ( z1 * ( z1 * ( z1 * ( z1 * ( z1 * ( z1 * ( z1 * ( z1
x0 - r / 2 in L /\ dom f & x0 - r / 2 in dom f ;
then w is that w is is non empty and rng w <> {} & w is non empty ;
set x9 = x9 ^ <* Z *> ^ ( x ^ <* Z *> ^ ( x ^ <* Z *> ) ) ;
len ( w1 ^ w2 ) in Seg len ( w1 ^ w2 ) ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of PFuncs ( V , { k } ) ;
x . n = ( |. a . n .| ) * ( |. b . n .| ) ;
( p `1 ) ^2 <= ( G * ( 1 , 1 ) ) `1 ;
rng ( g | ( len g ) ) c= L~ ( g | ( len g ) ) ;
reconsider k = i-1 * ( l + j ) as Nat ;
for n being Nat holds F . n is \HM { -infty } & F is \HM { -infty }
reconsider x9 = x9 , y9 = y9 as VECTOR of M ;
dom ( f | X ) = X /\ dom f .= X ;
p , a // p , c & b , a // c , c ;
reconsider x1 = x as Element of REAL m , x2 be Element of REAL m ;
assume i in dom ( a * p ^ q ) ;
m . ( ( k + 1 ) + 1 ) = p . ( k + 1 ) ;
a / ( s . m - n ) / ( s . n - n ) <= 1 ;
S . ( n + k + 1 ) c= S . ( n + k ) ;
assume that B1 \/ C2 = B2 \/ C2 and B2 \/ C2 = B2 \/ C2 and B1 \/ B2 = B2 \/ C2 ;
X . i = { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 ,
r2 in dom ( h1 + h2 ) & r1 in dom ( h1 + h2 ) ;
- ( 0. R ) = a & b-0 ( R ) = b ;
F8 is_closed_on t1 , Q & F8 is_closed_on t2 , Q ;
set T = -> InInInInj of X ;
Int Cl ( Cl ( Cl R ) ) c= Int Cl R ;
consider y being Element of L such that c . y = x ;
rng ( Flen F ) = { Flen ( F . x ) } ;
G-23 \ { c } c= B \/ S \/ S ;
f[#] A is_\! ] implies X , A is_\! ]
set RF = the Element of P , RF = the Element of P ;
assume that n + 1 >= 1 and n + 1 <= len M ;
let k2 be Element of NAT , k be Nat ;
reconsider ppj = u as Element of ( ( TOP-REAL n ) | ( Seg n ) ) ;
g . x in dom f & x in dom g implies x = g . x
assume that 1 <= n and n + 1 <= len f1 ;
reconsider T = b * N as Element of G / N ;
len ( ( P * ( i , j ) ) @ ) <= len ( ( P * ( i , j ) ) @ ) ;
x " in the carrier of [: A1 , A2 :] & x in the carrier of A1 ;
[ i , j ] in Indices ( ( A + B ) * ( i , j ) ) ;
for m being Nat holds Re ( F . m ) is simple function of S
f . x = a . i .= a1 . k .= a1 . k ;
let f be PartFunc of REAL i , REAL , x be Element of REAL i ;
rng f = the carrier of \bf \llangle A , { a } :] ;
assume s1 = sqrt ( 2 * ( p |^ 2 ) - ( p |^ 2 ) ) ;
attr a > 1 & b > 0 & a |^ b > 1 ;
let A , B , C be Subset of the lines of K ;
reconsider X0 = X , Y0 = Y as RealNormSpace of X ;
let f be PartFunc of REAL , REAL , g be PartFunc of REAL , REAL ;
r * ( v1 |-- I ) . X < r * 1 ;
assume that V is Subspace of X and X is Subspace of V ;
let t-3 , t-3 be Relation of the carrier of K ;
Q [ ( e \/ { v } ) `1 , ( e \/ { v } ) `2 ] ;
g \circlearrowleft ( W-min L~ z ) = z & g /. ( len g ) = z ;
|. |[ x , v ]| - |[ x , y ]| .| = vrelational ;
- f . w = - ( L * w ) .= - ( L * w ) ;
z -' y <= x iff z <= x + y & z <= y + x
( 7 / ( 1 + e ) ) / ( 1 + e ) > 0 ;
assume X is BCK-algebra & 0 , 0 , 0 , 0 , 0 ;
F . 1 = v1 & F . 2 = v2 & F . 3 = v2 ;
( f | X ) . x2 = f . x2 .= ( f | X ) . x2 ;
( ( tan (#) sec ) `| Z ) . x in dom ( sec (#) sec ) ;
i2 = ( f /. len f ) * ( len f + 1 ) ;
X1 = X2 \/ ( X1 \ X2 ) .= X2 \/ ( X1 \ X2 ) ;
[. a , b , 1_ G .] = 1_ G & [. a , b , 1_ G .] = 1_ G ;
let V , W be non empty VectSpStr over F_Complex , f be [: V , W :] ;
dom g2 = the carrier of I[01] .= the carrier of I[01] .= the carrier of I[01] ;
dom ( f2 | the carrier of I[01] ) = the carrier of I[01] .= the carrier of I[01] ;
( proj2 | X ) .: X = proj2 .: X .= proj2 .: X .= ( proj2 | X ) .: X ;
f . ( x , y ) = h1 . ( x `1 , y `2 ) ;
x0 - r < a1 . n & x0 - r < a1 . n ;
|. ( f /* s ) . k - ( f /* ( s + k ) ) . n .| < r ;
len ( Line ( A , i ) ) = width A ;
SFinSequence / ( g , f ) = ( S . g ) / ( g , f ) ;
reconsider f = v + u as Function of X , the carrier of Y ;
intloc 0 in dom ( Initialized p ) & IC ( p +* I ) in dom ( p +* I ) ;
( i1 , i2 ) := ( ( i2 , 3 ) , ( i2 , 4 ) ) & not ( i1 , i2 , a3 , 4 ) does not destroy b3 ;
arcsin + arccos r = ( cos r ) + 0 & 1 + cos r = 0 ;
for x st x in Z holds f2 is_differentiable_in x & f2 is_differentiable_in x ;
reconsider q2 = ( q - x ) / ( q - x ) as Element of REAL ;
( 0 qua Nat ) + 1 <= i + j1 + 1 ;
assume f in the carrier of [ X \to Omega Y , Omega Y ] ;
F . a = H / ( ( x , y ) / ( x , y ) ) ;
( TRUE , T ) at ( C , u ) = TRUE ;
dist ( ( a * seq ) . n , h ) < r ;
1 in the carrier of [. 0 , 1 .] & 1 in dom ( 1 / 2 ) ;
( p2 `1 ) ^2 - x1 > - g / 2 ;
|. r1 - p .| = |. a1 .| * |. thesis .| ;
reconsider S-14 = 8 as Element of Seg 8 , x be Element of Seg 8 ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b
DkW .order() = DW .1 + 1 ;
i1 = ma + n & i2 = K + n & j2 = K + n ;
f . a [= f . ( f . O1 ) "\/" f . ( a "\/" b ) ;
attr f = v & g = u & f + g = v + u ;
I . n = Integral ( M , F . n ) ;
chi ( [: T1 , T2 :] , S ) . s = 1 ;
a = VERUM ( A ) or a = VERUM ( A ) ;
reconsider k2 = s . b3 as Element of NAT ;
( Comput ( P , s , 4 ) ) . GBP = 0 ;
L~ ( M1 + M2 ) meets L~ ( M2 + M1 ) ;
set h = the continuous Function of X , R , x be Element of X ;
set A = { L . ( k + n ) : not contradiction } ;
for H st H is negative holds P7 [ H ] ;
set b8 = S5 \ ( i + 1 ) , S8 = ( i + 1 ) \ { i } ;
Hom ( a , b ) c= Hom ( a `1 , b `2 ) ;
( 1 / ( n + 1 ) ) / ( n + 1 ) < ( 1 / ( n + 1 ) ) ;
( l ) `1 = [ dom l , cod l ] `1 .= [ l , cod l ] `1 ;
y +* ( i , y /. i ) in dom g ;
let p be Element of [: Al ( ) , D ( ) :] ;
X /\ X1 c= dom ( ( f1 - f2 ) | X ) ;
p2 in rng ( f /^ ( len p1 + 1 ) ) & p2 in rng ( f /^ ( len p1 + 1 ) ) ;
1 <= indx ( D2 , D1 , j1 ) + 1 - 1 ;
assume x in ( ( K /\ / ( 1 + ( K + 1 ) ) ) /\ ( K + 1 ) ) ;
- 1 <= ( ( f2 . O ) `2 ) / ( 1 + ( f2 . O ) `2 ) ;
let f , g be Function of I[01] , TOP-REAL 2 , a , b be Real ;
k1 -' k2 = k1 - k2 + 1 .= k2 - 1 + 1 .= k2 - 1 + 1 ;
rng ( seq ^\ k ) c= ]. x0 - r , x0 + r .[ ;
g2 in ]. x0 - r , x0 + r .[ & g2 in ]. x0 - r , x0 + r .[ ;
sgn ( p `1 , K ) = - 1_ K & sgn ( p `2 , K ) = - - 1 ;
consider u being Nat such that b = ( p |^ y ) * u ;
ex A being .] st a = Sum A & ex A being Ordinal st a = Sum A ;
Cl ( union ( H ) ) = union ( ( Cl H ) \ ( Cl H ) ) ;
len t = len t1 + len t2 .= len t1 + len t2 .= len t2 + len t1 ;
v-29 = v + w |-- ( v + A ) ;
cv <> DataLoc ( t . GBP , 3 ) & cv <> DataLoc ( t . GBP , 3 ) ;
g . s = sup ( d " { s } ) .= sup ( d " { s } ) ;
( \dot { y } ) . s = s . ( \dot { y } . s ) ;
{ s : s < t } in INT implies t = {}
s ` \ s = s ` \ ( s \ t ) .= ( s \ t ) \ ( s \ t ) ;
defpred P [ Nat ] means B + $1 in A & C in A ;
( 339 + 1 ) ! = 32139 * ( 3be Element of NAT ) ;
U U = ( U , A ) --> ( ( U , A ) . ( i + 1 ) ) ;
reconsider y = y as Element of ( len y ) -tuples_on COMPLEX ;
consider i2 being Integer such that y0 = p * i2 and i2 in dom f ;
reconsider p = Y | Seg k as FinSequence of ( the carrier of K ) * ;
set f = ( S , U ) \mathop { I } , g = S , h = S , f = S \! \mathop { I } , F = S \! \mathop { I } , G = S \! \mathop { I } , F = S \! \mathop { I } ,
consider Z being set such that lim s in Z and Z in F ;
let f be Function of I[01] , TOP-REAL n , R^1 , a be Real ;
( SAT M ) . [ n + i , 'not' A ] <> 1 ;
ex r being Real st x = r & a <= r & r <= b ;
let R1 , R2 be Element of REAL n , a be Real ;
reconsider l = 0. ( V ) as Linear_Combination of A ;
set r = |. e .| + |. w .| + |. w .| + a ;
consider y being Element of S such that z <= y and y in X ;
a is \HM { \rm ' ( b 'or' c ) } = 'not' ( ( a 'or' b ) 'or' c )
||. ( x9 - g ) * ( y9 - g2 ) .|| < r2 ;
b9 , a9 // b9 , c9 & b9 , c9 // b9 , c9 & b9 , c9 // c9 , a9 ;
1 <= k2 -' k1 & k2 + 1 = k2 + 1 & k2 + 1 = k2 + 1 ;
( ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) ^2 >= 0 ;
( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 < 0 ;
E-max L~ Cage ( C , n ) in LSeg ( ( R /. 1 ) , ( R /. 1 ) ) ;
consider e being Element of NAT such that a = 2 * e + 1 ;
Re ( lim F ) = Re ( lim G ) .= Re ( lim G ) ;
LIN b , a , a or LIN b , c , a ;
p `1 , a // a `1 , b or p `1 , a // b `1 , a `2 ;
g . n = a * Sum ( f | n ) .= f . n * ( f | n ) . n ;
consider f being Subset of X such that e = f and f is odd ;
F | ( N2 , S ) = CircleMap * CircleMap .= CircleMap * ( F | ( N2 , S ) ) ;
q in LSeg ( q , v ) \/ LSeg ( v , p ) ;
Ball ( m , r0 ) c= Ball ( m , s ) ;
the carrier of (0). V = { 0. V } .= { 0. V } ;
rng ( ( - 1 ) (#) cos ) = [. - 1 , 1 .] & dom ( - 1 ) (#) cos = [. - 1 , 1 .] ;
assume that Re ( seq ) is summable and Im ( seq ) is summable and Im ( seq ) is summable ;
||. ( ( vseq . n ) - ( vseq . m ) ) .|| < e / 2 ;
set g = O --> 1 ;
reconsider t2 = t11 as 0 -element string of S2 , t2 = ( S , U ) {} ) string of S2 ;
reconsider x9 = seq . ( n + 1 ) as sequence of REAL n ;
assume that that that \mathop { \rm E _ { max } } meets L~ go and p in L~ co and q in L~ co and q in L~ co and p in L~ co and q in L~ co ;
- ( ( - 1 ) / ( n + 1 ) ) < F . n - ( - 1 ) / ( n + 1 ) ;
set d1 = being in being dist ( x1 , z1 ) , d2 = dist ( x2 , z2 ) , d2 = dist ( x2 , z2 ) ;
2 |^ ( |. 100 .| -' 1 ) = 2 |^ ( |. 100 .| - 1 ) ;
dom ( ( - ( 2 * ( len ( d * ( a , b ) ) ) ) ) * ( a , b ) ) = Seg len ( ( - ( 2 * ( a , b ) ) ) ) ;
set x1 = - ( k2 + 1 ) , x2 = - ( k2 + 1 ) , x3 = - ( k2 + 1 ) ;
assume for n being Element of X holds 0. ( X , x ) <= F . n ;
assume that 0 <= T-32 . i and ( T-32 . i ) `1 <= 1 ;
for A being Subset of X holds c . ( c . A ) = c . A
the carrier of ( Carrier ( L\mathopen { 0. } V , p ) ) c= I2 ;
'not' Ex ( x , p ) => All ( x , p ) is valid ;
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the normal w.r.t. over {} ;
Z c= dom ( ( - 1 / 2 ) (#) ( sin + cos ) ) ;
|. 0. TOP-REAL 2 - ( q `1 / |. q .| - cn ) .| < r ;
ConsecutiveSet ( A , succ B ) c= ConsecutiveSet2 ( A , succ succ d ) ;
E = dom ( L (#) F ) & L (#) F is_measurable_on E & L (#) F is_measurable_on E ;
C |^ ( A + B ) = C |^ B * C
the carrier of W2 c= the carrier of V & the carrier of W1 c= the carrier of V ;
I . IC Comput ( P , s , k ) = P . IC Comput ( P , s , k ) ;
attr x > 0 means : Def3 : ( 1 - x ) |^ ( - 1 ) = x |^ ( - 1 ) ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) \/ LSeg ( g , i ) ;
consider p being Point of T such that C = [. p , R .] and p in C ;
b , c are_connected connected & - C , - C + - C + D + D + - D + 1 .] where C , D being Path of C , D st C , D + C + D + - D + 1 .] is Path ;
assume that f = id the carrier of O and g = id the carrier of O and h = id the carrier of O ;
consider v such that v <> 0. V and f . v = L * v ;
let l be Linear_Combination of {} ( the carrier of V ) ;
reconsider g = f " as Function of U2 , U1 , U2 ;
A1 in the carrier of ( k + 1 ) & A2 : not ( A1 , A2 ) on A1 & ( A2 , A1 ) on A2 ;
|. - x .| = - ( - x ) .= - x .= - x .= - x ;
set S = ) (# { x , y , c , d , e , f #) ;
( ( 5 + 5 ) * ( 5 * ( 5 + 6 ) ) ) / ( 5 * ( 5 + 6 ) ) >= 4 * ( 5 * ( 5 + 6 ) ) ;
vM /. ( k + 1 ) = vM . ( k + 1 ) ;
0 mod i = - ( i * ( 0 qua Nat ) ) mod i ;
Indices ( M1 + M2 ) = [: Seg n , Seg n :] & Indices ( M1 + M2 ) = [: Seg n , Seg n :] ;
Line ( S\mathopen { i , j } , j ) = Sj . i .= j ;
h . ( x1 , y1 ) = [ y1 , x1 ] & h . ( y1 , y2 ) = [ y1 , y2 ] ;
|. f .| + Re ( |. f .| * ( ( b - a ) * h ) ) is nonnegative ;
assume x = ( a1 ^ <* x1 *> ) ^ ( b1 ^ <* x2 *> ) ^ ( b1 ^ <* x1 *> ) ;
Mj is_closed_on IExec ( I , P , s ) , P & Mj is_halting_on s , P ;
DataLoc ( t3 . a , 4 ) = intpos ( 0 + 4 ) .= intpos ( 0 + 4 ) ;
x + y < - x + y & |. x - y .| = - x + y & |. x - y .| = - x ;
LIN c , q , b & LIN c , q , b & LIN c , q , c ;
fsuch . ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + z1 ) .= x + ( y + z ) ;
flim . a = f . a & flim a in InputVertices S & flim a in InputVertices S ;
( p `1 ) ^2 <= ( ( E-max C ) `1 ) ^2 & ( p `2 ) ^2 <= ( ( E-max C ) `1 ) ^2 ;
set R8 = Cage ( C , n ) \circlearrowleft E8 , R7 = Cage ( C , n ) ;
( p `1 ) ^2 >= ( ( E-max C ) `1 ) ^2 + ( p `2 ) ^2 ;
consider p such that p = ( p-20 ) . i and s1 < p /. i ;
|. ( f /* ( s * F ) ) . l - ( f /* ( s * F ) ) . l .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len ( Line ( N , k + 1 ) ) = width N ;
f1 /* ( s ^\ k ) is convergent & f2 /* ( s ^\ k ) is convergent & lim ( f2 /* ( s ^\ k ) ) = 0 ;
f . ( x1 , y1 ) = x1 & f . ( y1 , y2 ) = y1 & f . ( y1 , y2 ) = y2 ;
len f <= len f + 1 & len f + 1 <> 0 implies f ^ <* f *> ^ g = f ^ g
dom ( Proj ( i , n ) * s ) = REAL m & dom ( Proj ( i , n ) * s ) = REAL m ;
n = k * ( 2 * t ) + ( n mod ( 2 * t ) ) ;
dom B = 2 -tuples_on the carrier of V .= [: the carrier of V , the carrier of V :] ;
consider r such that r _|_ a and r _|_ x and r _|_ y ;
reconsider B1 = the carrier of [: Y1 , Y2 :] as Subset of [: X1 , X2 :] ;
1 in the carrier of [. 1 / 2 , 1 .] & 1 / 2 in dom ( 1 / 2 ) ;
for L being complete LATTICE holds <* <* <* \mathbb L *> , <* A *> *> , [ <* A , A *> , [ A , B ] *> ] is isomorphic
[ gi , gj ] in Ij \ ( 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 f3 is_differentiable_in x0 and f3 is_differentiable_in x0 and f3 is_differentiable_in x0 and f2 is_differentiable_in x0 and f3 is_differentiable_in x0 and f3 is_differentiable_in x0 and f3 is_differentiable_in x0 and f3 is_differentiable_in x0 and f3 is_differentiable_in x0 and f3 is_differentiable_in x0 and f3 is_differentiable_in x0 and f3 is_differentiable_in x0 and f3 is_differentiable_in x0 and f3 is_differentiable_in x0 ;
reconsider y = ( a ` ) / ( a ` ) , z = ( a ` ) / ( a ` ) as Element of L ;
dom s = { 1 , 2 , 3 } & s . 1 = d1 & s . 2 = d2 ;
( min ( g , ( 1 , 1 ) ) ) . c <= h . c ;
set GG1 = the as Vertex of G , the carrier' of G , v = the Vertex of G ;
reconsider g = f as PartFunc of REAL , \langle REAL-NS n , \Vert \cdot \Vert \rangle ;
|. s1 . m - p .| < d / ( p + 1 ) ;
for x being element st x in ( ( 0 ) -tuples_on NAT ) holds x in ( 0 -tuples_on NAT )
P = the carrier of ( ( TOP-REAL n ) | P ) .= the carrier of ( ( TOP-REAL n ) | P ) ;
assume that p00 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) and LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = {} ;
( 0. X \ x ) |^ ( m * ( k + 1 ) ) = 0. X ;
let g be Element of Hom ( cod f , cod f ) ;
2 * a * b + ( 2 * c ) * d <= 2 * ( C1 * C2 ) ;
let f , g , h be Point of the complex normed space of X , Y ;
set h = Hom ( a , g ) ;
then Seg ( n + 1 ) = Seg ( m + 1 ) & m <= n + 1 ;
H * ( g " * a ) in the right of H * ( g " * a ) ;
x in dom ( ( - 1 / 2 ) (#) ( sin + cos ) ) ;
cell ( G , i1 , j2 -' 1 ) misses C & cell ( G , i1 , j2 -' 1 ) misses C ;
LE q2 , q2 , P & LE q2 , q1 , P & LE q2 , q2 , P & LE q2 , q2 , P & LE q2 , q2 , P & LE q2 , q2 , P & LE q2 , q2 , P & LE q2 , q2 , P & LE q2 , q2 , P & LE q2 ,
attr B is Let A component of A means : Def3 : B c= BDD A & B c= BDD A ;
deffunc D ( set , set ) = union rng $2 & $2 in union rng $2 & $2 in union rng $2 ;
n + - n < len ( ( p + q ) - n ) + ( - n ) ;
attr a <> 0. K means : Def3 : the_rank_of M = the_rank_of ( a * M ) ;
consider j such that j in dom sY. and I = len ( \/ { j } ) + j ;
consider x1 such that z in x1 and x1 in ( P \ { x } ) and x in P ;
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 ) ;
set cv = 3 -tuples_on { a , b , c , d } , cR = 2 -tuples_on { a , b , c , d } ;
conv @ W c= union ( F .: ( E " ( W ) ) ) ;
1 in [. - 1 , 1 .] /\ dom ( ( - 1 ) (#) ( arccot ) ) ;
r3 <= s0 + ( ( r2 - s2 ) / |. v2 - v2 .| ) ;
dom ( f (#) ( f4 (#) ( f4 (#) ( f ^ ) ) ) ) = dom f /\ dom ( f ^ ) ;
dom ( f (#) G ) = dom ( l (#) F ) /\ Seg k .= Seg ( l + k ) ;
rng ( s ^\ k ) c= dom f1 \ { x0 } & rng ( s ^\ k ) c= dom f2 \ { x0 } ;
reconsider gg = gp as Point of TOP-REAL n , p be Point of TOP-REAL n ;
( T * h . s ) . x = T . ( h . s ) . x ;
I . ( L . ( J . x ) ) = ( I * L ) . ( J . x ) ;
y in dom <* *> & ( Frege *> ) . ( ( Frege ( A , o ) ) . y ) = <* x *> ;
for I being non degenerated commutative commutative Ring holds the carrier of I is commutative non empty doubleLoopStr
set s2 = s +* Initialize ( ( intloc 0 ) .--> 1 ) , s3 = Initialize s , s3 = Initialize s , P3 = P +* Initialize s , s4 = P +* Initialize s , s4 = P +* Initialize s , s4 = P +* Initialize s , s4 = Comput ( P3 , s3 , 1 ) , s4 = P3 ;
P1 /. IC Comput ( P1 , s1 , k ) = P1 . IC Comput ( P2 , s2 , k ) .= P1 . IC Comput ( P2 , s2 , k ) ;
lim ( S1 , a ) in the carrier of [: a , b :] & lim ( S1 , a ) = a ;
v . ( l-13 . i ) = ( v *' ( lpp . i ) ) . i ;
consider n being element such that n in NAT and x = ( sn Sorts ) . n ;
consider x being Element of c such that F1 . x <> F2 ( x ) and F2 . x <> 0 ;
Funcs ( X , 0 , x1 , x2 , x3 ) = { E } & { E , F , G } = { G , F , G } ;
j + ( 2 * ( k + 1 ) ) + m1 > j + ( 2 * ( k + 1 ) ) ;
{ s , t } on A3 & { s , t } on A3 & { s , t } on B2 ;
n1 > len crossover ( p2 , p1 , n1 , n2 ) & n2 >= len crossover ( p2 , p1 , n1 , n2 , n3 ) ;
( mg ) . HT ( mg , T ) = 0. L & ( mg ) . ( HT ( mg , T ) ) = 0. L ;
then H1 , H2 , H , H1 , H2 , H1 , H2 , H2 , H1 , H2 , H2 , H1 , H2 , H2 , H2 ;
( N-min L~ f ) .. f > 1 & ( N-min L~ f ) .. f > 1 implies ( N-min L~ f ) .. f > 1
]. s , 1 .[ = ]. s , 2 .[ /\ [. 0 , 1 .] ;
x1 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) ) .= { x1 , x2 } ;
let f1 , f2 be continuous PartFunc of REAL , REAL , x be Element of REAL , r be Real ;
DigA ( ti1 , ( k + 1 ) ) is Element of k -tuples_on ( k + 1 ) -tuples_on ( k + 1 ) -tuples_on ( k + 1 ) -tuples_on ( k + 1 ) -tuples_on ( k + 1 ) -tuples_on ( k + 1 ) -tuples_on ( k + 1 ) -tuples_on ( k + 1 ) -tuples_on ( k + 1 ) -tuples_on (
I \mathop { \rm 22\mathop { - } = d & I \mathop { - } = k2 ;
u9 ~ = { [ a , u9 ] , [ u , u9 ] , [ v , u9 ] } ;
( w | p ) | ( w | ( w | ( w | ( w | ( w | ( w | w ) ) ) ) ) ) = p ;
consider u2 such that u2 in W2 and x = v + u2 and u = v + u2 and u in W2 ;
for y st y in rng F ex n st y = a |^ n & F . n = a
dom ( ( g * ( /. i ) ) | K ) = K & dom ( ( g * ( ( id V ) --> a ) ) | K ) = K ;
ex x being element st x in ( ( the Sorts of U0 ) . s ) . s & ( the Sorts of U0 ) . s = {} ;
ex x being element st x in ( ( ( the Sorts of O1 ) \/ A ) . s ) . s ;
f . x in the carrier of [. - r , r .] & f . x in [. - r , - r .] ;
( the carrier of X1 union X2 ) /\ ( the carrier of X2 ) <> {} implies ( the carrier of X1 union X2 ) /\ ( the carrier of X2 ) <> {} ;
L1 /\ LSeg ( p1 , p2 ) c= { p11 } & LSeg ( p1 , p2 ) /\ LSeg ( p11 , p2 ) c= { p1 } ;
( b + ( bd - a ) ) / 2 in { r : a < r } ;
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 G8 such that z = y and ( for x being Point of G8 st x in z holds P [ x , z ] ) ;
( the sequence of ( the sequence of ( the carrier of X ) ) ) . ( thesis ;
len ( w ^ w2 ) + 1 = len w + 2 + 1 .= len w + 2 + 1 ;
assume q in the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 & q in the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 ;
f | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | D ) = g | ( ( TOP-REAL 2 ) | D ) ;
reconsider i1 = x1 , i2 = x2 , z = x3 as Element of ( the carrier of X ) * ;
( a * A ) @ = ( a * ( A @ ) ) @ ;
assume ex x0 being Element of NAT st f |^ ( n + k ) is \llangle x0 , x0 ] ;
Seg len ( ( the multF of f2 ) * ( i , j ) ) = dom ( ( the multF of f2 ) * ( i , j ) ) ;
( Complement ( Complement A ) ) . m c= ( Complement ( Complement A ) ) . n ;
f1 . p = p9 & g1 . p = d & g2 . p = d & g1 . p = c ;
FinS ( F , Y ) = FinS ( F , dom ( F | Y ) ) .= FinS ( F , Y ) ;
( x | y ) | z = z | ( y | x ) ;
( ( |. x .| ) |^ n ) / ( ( |. x .| ) |^ ( n + 1 ) ) <= ( ( |. r2 .| ) |^ n ) ;
Sum ( F ) = Sum f & dom ( F ) = dom g & rng ( F ) = dom g ;
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 ) + ( W2 ) = ( W1 + W2 ) + ( W2 + W3 ) ;
||. ( ( t . x ) .|| ) . x = lim ( ( ( t . x ) ) * ( ( t . x ) ) ) ;
assume that i in dom D and f | A is lower and g | A is lower ;
( ( p `2 ) / |. p .| ) ^2 <= ( |. p .| ) ^2 ;
g | Sphere ( p , r ) = id Sphere ( p , r ) .= id Sphere ( p , r ) ;
set N8 = ( N-min L~ Cage ( C , n ) ) .. ( Cage ( C , n ) ) ;
for T being non empty TopSpace holds T is countable implies the TopStruct of T is countable countable
width ( B |-> 0. K ) = len ( B |-> 0. K ) .= width ( B |-> 0. K ) .= width ( B |-> 0. K ) ;
attr a <> 0 means : Def3 : ( A \ B ) -- a = ( A Y. ) \ ( B Y. \ a ) ;
then f is_partial_differentiable_in u0 , 3 & pdiff1 ( f , 1 ) is_partial_differentiable_in 3 , 1 & pdiff1 ( f , 1 ) is_partial_differentiable_in 3 , 3 ;
assume that a > 0 and a <> 1 and b > 0 and b <> 0 and c > 0 and d > 0 ;
w1 , w2 in Lin { w1 , w2 } & w2 in Lin { w1 , w2 } ;
p2 /. IC Comput ( p2 , s , k ) = p2 . IC Comput ( p2 , s , k ) .= ( IC Comput ( p2 , s , k ) ) + 1 ;
ind ( T-10 | b ) = ind b .= ind b .= ind b .= ind b .= ind b ;
[ a , A ] in the InternalRel of thesis & [ a , A ] in the InternalRel of thesis implies [ a , A ] in the InternalRel of K
m in ( the Arrows of C ) . ( o1 , o2 ) & ( the Arrows of C ) . ( o1 , o2 ) = ( the Arrows of C ) . ( o1 , o2 ) ;
( ( a , CompF ( PA , G ) ) . z ) . z = TRUE ;
reconsider phi = phi /. 11 , phi = phi /. 2 , phi = phi /. 3 as Element of ( len phi ) -tuples_on BOOLEAN ;
len s1 - ( len s2 - 1 ) + 1 > 0 + 1 - ( len s2 - 1 ) ;
\delta ( D ) * ( f . ( upper_bound A ) - f . ( lower_bound A ) ) < r ;
[ f21 , f22 ] in the carrier' of A & [ f22 , f22 ] in the carrier' of A ;
the carrier of ( ( TOP-REAL 2 ) | K1 ) = ( ( TOP-REAL 2 ) | K1 ) | K1 .= K1 ;
consider z being element such that z in dom g2 and p = g2 . z and q = g2 . z ;
[#] ( V1 ) = { 0. V1 } .= the carrier of ( V1 ) \ { 0. V1 } .= the carrier of ( V1 ) \ { 0. V1 } ;
consider P2 being FinSequence such that rng P2 = M and P2 is one-to-one and P2 is one-to-one and P2 is one-to-one ;
assume that x1 in dom ( f | X ) and ||. x1 - x0 .|| < s and ||. x1 - x0 .|| < s ;
h1 = f ^ ( <* p3 *> ^ <* p *> ) .= h ^ <* p *> ^ <* p *> .= h ;
c / ( |[ b , c ]| ) = c / ( |[ a , c ]| ) .= c / ( |[ a , c ]| ) .= c / ( |[ a , c ]| ) ;
reconsider t1 = p1 , t2 = p2 , t2 = p3 as Term of C , V , s be Element of C ;
( 1 - 2 ) * ( 1 - 2 ) in the carrier of [. 1 , 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 + D * ( ( p1 `2 ) - D ) ;
R . b implies R . b \hbox { - } = 2 * - b .= 2 * b .= - b ;
consider ]| such that B = - 1 * C + ( - 1 ) * A and 0 <= 1 ;
dom g = dom ( ( the Sorts of A ) * ( the_arity_of o ) ) .= dom ( ( the Sorts of A ) * ( the_arity_of o ) ) ;
[ P . ( l + 1 ) , P . ( l + 1 ) ] in => ( T . ( l + 1 ) , T . ( l + 1 ) ) ;
set s2 = Initialize s , s3 = Initialize s , P2 = P +* stop I ;
reconsider M = mid ( z , i2 , i1 ) as Matrix of 2 , len z , width z ;
y in product ( ( the support of J ) +* ( V , { 1 } ) ) ;
1 / ( |[ 0 , 1 ]| ) = 1 & 0 / ( |[ 0 , 1 ]| ) = 0 ;
assume x in the left of g or x in the left of g or x in the left of g ;
consider M being strict Subgroup of A9 such that a = M and T is Subgroup of M and M is Subgroup of M ;
for x st x in Z holds ( ( ( - 1 / 2 ) (#) f ) `| Z ) . x <> 0
len ( W1 + W2 ) = 1 + len ( W2 + W3 ) .= 1 + len ( W3 + W3 ) .= len ( W1 + W3 ) + len ( W2 + W3 ) ;
reconsider h1 = ( vseq . n ) - ( t-16 . n ) as Lipschitzian Lipschitzian LinearOperator of X , Y ;
( i mod len ( p + q ) ) + 1 in dom ( p + q ) ;
assume that s2 is for F , s1 st F in the { s } and F in the { s } and F in the { s } and F in the not empty ;
( ( ( ( ( the non empty set ) , x ) , 1 ) , 1 ) div ( ( ( x - y ) div ( x - y ) ) ) ) mod ( x - y ) = ( ( x - y ) div ( x - y ) ) mod ( x - y ) ;
for u being element st u in Bags n holds ( p + m ) . u = p . u + m
for B being Subset of u-5 st B in E holds A = B or A misses B or A misses B ;
ex a being Point of X st a in A & A /\ Cl { y } = { a } ;
set W2 = tree ( p ) \/ ( rng <* p *> ) ;
x in { X where X is Ideal of L : for x being Element of L st x in X holds x is Ideal of L } ;
the carrier of ( W1 /\ W2 ) c= the carrier of ( W1 + W2 ) /\ the carrier of ( W1 + W2 ) ;
( for a , b holds ( 1 + a ) * id a = ( 1 + a ) * id a
( ( X --> f ) . x ) . x = ( X --> f ) . x .= ( X --> f ) . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) ;
p => ( q => r ) => ( p => ( p => r ) ) in TAUT ( A ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
- 1 + 1 <= ( - ( 2 |^ ( n -' m ) ) + 1 ) + 1 ;
( reproj ( 1 , z0 ) ) . x in dom ( ( f1 (#) f2 ) * ( z (#) ( z (#) f2 ) ) ) ;
assume that b1 . r = { c1 } and b2 . r = { c2 . r } and b2 . r = { c2 . r } ;
ex P st a1 on P & a2 on P & a3 on P & a2 on P & a3 on P & a4 on P & f . a2 = f . a2 ;
reconsider gf = g `1 * f `2 , hg = h `2 * g `2 as strict Element of X ;
consider v1 being Element of T such that Q = ( downarrow v1 ) ` and ( downarrow v1 ) ` = ( downarrow v1 ) ` ;
n in { i where i is Nat : i < ( n + 1 ) + 1 & i < n + 1 } ;
( F /. ( i , j ) ) `2 >= ( F /. ( m , k ) ) `2 ;
assume K1 = { p : ( p `1 / |. p .| - cn ) / ( 1 + cn ) >= 0 } ;
ConsecutiveSet ( A , succ O1 ) = ( ConsecutiveSet ( A , O1 ) ) * ( ConsecutiveSet ( A , O1 ) ) ;
set I1 = Macro ( a , intloc 0 , intloc 0 ) , I1 = [ a , intloc 0 , intloc 0 ] , I1 = [ a , intloc 0 , intloc 0 ] ;
for i be Nat st 1 < i & i < len z holds z /. i <> z /. 1
X c= ( the carrier of L1 ) \/ ( the carrier of L2 ) & the carrier of X c= the carrier of L2 ;
consider x9 being Element of GF ( p ) such that x9 |^ 2 = a and not x9 |^ 2 = a ;
reconsider ej = ( e - f ) / ( i + 1 ) , f-18 = ( e - f ) / ( i + 1 ) , ff-18 = ( e - f ) / ( i + 1 ) , fff\mathopen = ( e - f ) / ( i + 1 ) ;
ex O being set st O in S & C1 c= O & M . O = 0. ( Cl O ) & M . ( O , O ) = 0. ( Cl O ) ;
consider n being Nat such that for m being Nat st n <= m holds S . m in U1 and n in U1 ;
f (#) g * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) * g ) . x ;
defpred P [ Nat ] means A + ( $1 + 1 ) = succ A + $1 & ( $1 + 1 ) in dom A implies A = B ;
the left sqrt of - g = the left sqrt of - ( g - f ) .= ( - g ) * ( - g ) .= ( - g ) * ( - g ) ;
reconsider p\mathopen { - } , p\mathopen { - } , p\mathopen { - } , - ( - ( - ( - ( p - q ) ) ) / ( 1 + ( p - q ) ) ) ) / ( 1 + ( p - q ) ) ) = ( - ( p - q ) ) / ( 1 + ( p - q ) ) / ( 1 + ( p - q ) ) )
consider g2 such that g2 = y and x <= g2 and g2 <= x0 and x0 <= g2 and g2 <= x0 and g2 <= x0 and g2 <= x0 and g2 <= x0 ;
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 ( y2 ^ y1 ) .= len ( x2 ^ y2 ) + len ( y2 ^ y1 ) ;
for x being element st x in X holds x in the set of the set of m & x in the set of m implies x = ( the } of m ) . ( n + 1 )
LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = {} or LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = {} ;
func for X being set holds of of that of of that ( the carrier of X ) \ { {} } , { {} } :] c= [: X , X :] ;
len ( ( { w } /. 1 , ( w /. 1 ) ) | ( len w + 1 ) ) <= len ( ( w /. 1 ) | ( len w + 1 ) ) ;
attr K is a valuation means : Def3 : a <> 0. K & v . ( a |^ i ) = i * v . a ;
consider o being OperSymbol of S such that t `1 . {} = [ o , the carrier of S ] -tree p and o in dom ( t `1 ) and p in dom ( t `1 ) and p in dom ( t `2 ) ;
for x st x in X ex y st x c= y & y in X & y is a + x & y is a + x
IC Comput ( P-6 , k ) in dom ( ( n + 1 ) -tuples_on the carrier of K ) ;
attr q < s means : Def3 : r < s & s < q & ]. p , q .[ c= ]. p , q .[ ;
consider c being Element of Class ( f , c ) such that Y = ( F . c ) `1 and c in ( F . c ) `2 ;
func the ResultSort of S2 -> ResultSort of the carrier of S2 means : Def3 : the ResultSort of it = id the carrier' of S2 & the ResultSort of it = id the carrier' of S2 ;
set y9 = [ <* y , z *> , f2 ] , z9 = [ <* z , x *> , f3 ] ;
assume x in dom ( ( - ( exp_R * ( f1 + f2 ) ) ) `| Z ) ;
r-7 in Int cell ( GoB f , i , GoB f ) \ L~ f & r-7 in L~ f implies rj1 in L~ f
( q `2 ) ^2 >= ( ( ( Cage ( C , n ) ) * ( i , j ) ) `2 ) ^2 ;
set Y = { a "/\" a : a in X } ;
i -' len f <= len f + len g - len f + len g - 1 .= len f + len g - 1 .= len f + len g - 1 .= len f + len g - 1 + 1 .= len f + len g - 1 ;
for n holds ex x st x in N & x in N1 & h . n = x- ( x0 - h . n )
set s0 = ( ( a , I , p ) --> ( a , I ) ) . i , s0 = ( a , I , s ) . i ;
( for k holds ( p . k = 1 ) or ( p . k = - 1 ) implies p . k = 1 ) & ( for k st k in dom p holds p . k = 1 ) implies p . k = - 1 )
u + Sum ( L-18 ) in ( U \ { u } ) \/ { u + Sum ( L-18 ) } ;
consider x9 being set such that x in x9 and x9 in V1 and x9 in V1 and x9 in V1 and x9 in V1 ;
( p ^ ( q | k ) ) . m = ( q | k ) . ( ( m + len p ) + ( m + 1 ) ) ;
g + h = gg + h1 & ||. g + h .|| = g + h + h ;
L1 is distributive & L2 is distributive implies L1 , L2 are_isomorphic & L1 , L2 are_isomorphic & L2 , L1 be distributive non empty <* L1 , L2 *> is distributive
attr x in rng f & y in rng ( f | x ) & f | x = f | ( x , y ) ;
assume that 1 < p and ( 1 + p ) + q = 1 and 0 <= a and 0 <= b and a <= 1 and b <= 1 ;
F\circ ( f , <* <* M *> *> ) = rpoly ( 1 , 1 , M ) *' t .= 0. L ;
for X being set , A being Subset of X holds A ` = {} implies A = {} implies A = {} or A = {} ;
( ( N-min X ) `1 ) ^2 <= ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( X X ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) `1 ) ) `1 ) ^2 & ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( X
for c being Element of the \rbrack of the Sorts of A , a being Element of the Sorts of A holds c <> a
s1 . GBP = ( Exec ( i2 , s2 ) ) . GBP .= Exec ( i2 , s2 ) . GBP .= s . GBP .= s . GBP .= s . GBP .= s . GBP .= s . GBP .= s . GBP ;
for a , b being Real holds |[ a , b ]| in ( y ) implies b >= 0 & a >= 0 & b >= 0
for x , y being Element of X holds x ` \ y = ( x \ y ) ` & y \ x = ( x \ y ) `
mode BCK-algebra of i , j , m , n , m , n , m , k being Element of NAT , i , j , m , n , m , k be Element of n -tuples_on X ;
set x2 = |( ( Re y ) , ( Im y ) , y2 = ( Im y ) * ( Im y ) ;
[ y , x ] in dom ( u . y ) & ( u . y ) `1 = g . y & ( u . y ) `2 = g . y ;
]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .] c= A & A = [. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .] ;
0 <= \delta ( S2 . n ) & |. \delta ( S2 . n ) .| < e / 2 ;
( - ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ) <= ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ;
set A = ( 2 / b ) * ( 2 / a ) ;
for x , y being set st x in R" holds x , y , x , y , x , y , x , y ;
deffunc FF2 ( Nat ) = b . ( $1 * ( M * G ) . $1 ) * ( M * ( G * F ) . $1 ) ;
for s being element holds s in contradiction iff s in contradiction ( f 'or' g ) & s in contradiction ( f ) \/ contradiction ( g )
for S being non empty non void holds S is connected iff S is connected iff S is connected
max ( ( degree z ) / ( |. z .| ) , degree z ) >= 0 ;
consider n1 being Nat such that for k holds seq . ( n1 + k ) < r + s ;
Lin ( A /\ B ) is Subspace of Lin ( A ) & Lin ( B ) is Subspace of Lin ( A ) ;
set n-15 = n-13 '&' ( M . x qua Element of BOOLEAN ) , n-15 = ( M . x ) \bf 0. K ;
f " V in [ X , p ] & f " V in D & f " ( the carrier of X ) in D & f " ( the carrier of X ) in D ;
rng ( ( a , c ) := ( 1 , b ) ) c= { a , c , b }
consider y being as \vert as as many of G1 such that y `1 = y and dom y `2 = WWG1 and y `2 = WG1 ;
dom ( 1 / ( f . x ) ) /\ ]. x0 - r , x0 .[ c= ]. x0 - r , x0 + r .[ ;
as Element of v1 , j , r , s be Element of REAL n , i , j be Element of j , n , - r ;
v ^ ( ( n |-> 0 ) ^ <* 0 *> ) in Lin ( ( n |-> 0 ) ^ <* 1 *> ) ;
ex a , k1 , k2 st i = a := ( k1 , k2 ) & i = k2 & k2 = ( a , k1 ) := k2 ;
t . NAT = ( NAT .--> ( i1 , k ) ) . NAT .= ( NAT .--> ( i1 , k ) ) . NAT .= ( NAT --> ( i1 , k ) ) . NAT .= ( NAT --> ( i1 , k ) ) . ( k + 1 ) .= ( NAT --> ( i1 , k ) ) . ( k + 1 ) ;
assume that F is bbfamily and rng p = F and rng p = Seg ( n + 1 ) and rng p = Seg ( n + 1 ) ;
not LIN b , b9 , a & not LIN a , b , c & not LIN a , b , c & not LIN a , b , c
( L1 \HM { L1 \HM { or 0 } in O ) & ( L2 \HM { 0 } in O ) implies ( L1 \HM { 0 } \HM { 0 } ) is ( L2 \HM { 0 } ) \HM { {} } ) ;
consider F be ManySortedSet of E such that for d being Element of E holds F . d = F ( d ) and for d being Element of E holds F . d = G ( d ) ;
consider a , b such that a * ( \rrangle = b * ( -w ) and 0 < a and 0 < b ;
defpred P [ FinSequence of D ] means |. Sum ( $1 ) .| <= Sum ( $1 ) & for k be Nat st k in dom $1 holds ( $1 + 1 ) * ( $1 + 1 ) <= ( $1 + 1 ) * ( $1 + 1 ) ;
u = cos ( x , y ) . v * x + ( cos ( x , y ) . v ) .= cos ( x , y ) . v .= - cos ( x , y ) . v .= - cos ( x , y ) . v ;
dist ( ( seq . n ) + x , ( x + g ) + x ) <= dist ( ( seq . n ) + ( x + g ) ) + 0 ;
P [ p , |. p .| ^ <* {} *> , {} ] implies P [ p ]
consider X being Subset of Al such that X c= Y and X is finite and X is inininand X is ininininand X is inininand X is ininin;
|. b .| * |. eval ( f , z ) .| >= |. b .| * |. eval ( f , z ) .| ;
1 < ( ( N-min L~ Cage ( C , n ) ) .. Cage ( C , n ) ) .. Cage ( C , n ) ;
l in { l1 where l1 is Real : g <= l1 & l1 <= h & l1 <= h & l1 <= b } ;
( ( Partial_Sums ( G ) . n ) vol ( { x } ) ) * vol ( { x } , REAL ) <= ( Partial_Sums ( G ) . n ) * vol ( { x } , REAL ) ;
f . y = x .= x * 1_ L .= x * ( 1. L ) .= x * ( 1. L ) .= x * ( 1. L ) ;
NIC ( <% i1 , i2 %> , n ) = { ( i1 , ( n + 1 ) ) } .= { ( n + 1 ) , ( n + 1 ) } ;
LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = { p1 , p2 } .= { p1 , p2 } ;
product ( ( the Sorts of I-15 ) +* ( i , { 1 } ) ) in Z1 . i & product ( ( the Sorts of I-15 ) +* ( i , { 1 } ) ) in Z1 ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) | ( the carrier of S1 ) .= Following ( s1 , n ) ;
( W `1 ) ^2 <= ( q1 `1 ) ^2 & ( W `2 ) ^2 <= ( q1 `1 ) ^2 & ( W `2 ) ^2 <= ( ( q1 `2 ) ) ^2 ;
f /. i2 <> f /. ( len f + ( len g -' 1 ) + len f -' 1 ) ;
M , f / ( x. 3 , a ) / ( x. 4 , a ) / ( x. 0 , a ) / ( x. 4 , a ) / ( x. 0 , a ) / ( x. 4 , a ) / ( x. 0 , a ) / ( x. 0 , a ) ) |= H ;
len ( ( P ^ ) + ( P ^ ) ) in dom ( ( P ^ ) + ( P ^ ) ) ;
A |^ ( m , n ) c= A |^ ( m , n ) & A |^ ( k , l ) c= A |^ ( k , l ) ;
REAL n \ { q : |. q .| < a } c= { q1 : |. q1 .| >= a }
consider n1 being element such that n1 in dom p1 and y1 = p1 . n1 and ( for n being Nat st n in dom p1 holds p1 . n = F ( n ) ) ;
consider X being set such that X in Q and for Z being set st Z in Q & Z <> X holds X c= Z ;
CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA & CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA ;
for v be VECTOR of l1 holds ||. v .|| = upper_bound ( |. ( v - u ) .| ) & ||. v - u .|| = upper_bound ( |. ( v - u ) .| )
for phi st phi in X holds not phi in X & not phi in X & not phi in X implies not phi in X & not phi in X
rng ( ( Sgm dom ( f | ( dom f \ ( f | ( dom f \ ( dom f \ ( dom f ) \ ( dom f \ ( dom f \ ( dom f \ ( dom f \ ( dom f \ 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 ( ) & ( for a st a in dom c holds a . a = c . a ) & ( for a st a in dom c holds c . a = F ( a ) ) ;
the_result_sort_of ( a , b , c ) = <* Den ( b , c , a ) , Den ( b , c , a ) *> ;
consider f1 be Function of the carrier of X , REAL such that f1 = |. f .| and f1 is continuous and f1 is continuous ;
a1 = b1 & a2 = b2 or a1 = b2 & a2 = b3 & a3 = b3 or a2 = b2 & a3 = b3 & a4 = b3 & a4 = 6 & 8 = 6 & 8 = 6 & 8 = 7 & 8 = 8 & 8 = 8 & 7 = 8 & 8 = 8 & 8 = 8 & 8 = 8 & 8 = 8 & 8 = 8 & 8 = 8 ;
D2 . ( indx ( D2 , D1 , n1 ) + 1 ) = D2 . ( indx ( D2 , D1 , n1 ) + 1 ) .= D2 . ( indx ( D2 , D1 , n1 ) + 1 ) ;
f . ( ||. r .|| ) = ||. ||. ( r " ) * ( r * ( r * ( r * ( r * ( r * ( r * ( 1 / r ) ) ) ) ) ) ) .|| .= ||. r .|| * ( r * ( r * ( r * ( r * ( 1 / r ) ) ) ) .|| .= r * ( r * ( 1 / r ) ) .= r * ( r * ( 1 / r ) )
consider n being Nat such that for m being Nat st n <= m holds C-25 . m = C-25 . m ;
consider d being Real such that for a , b being Real st a in X & b in Y holds a <= b & b <= d ;
||. L /. h - ( K * |. h .| ) + ( K * |. h .| ) + ( K * |. h .| ) * ||. ( K * |. h .| ) + ( K * |. h .| ) * ||. ( K * |. h .| ) + K * |. h .| ) ) <= p0 ;
attr F is commutative means : Def3 : for b being Element of X holds F \hbox { b } = f . b ;
p = - 1 * p0 + 0. TOP-REAL 2 .= 1 * p0 + 0. TOP-REAL 2 .= 1 * p0 + 0. TOP-REAL 2 .= 1 * p0 + 0 * p0 .= 1 * p0 + 0 * p0 + 0 * p1 .= 1 * p0 + 0 * p2 + 0 * p2 .= 1 * p0 + 0 * p2 + 0 * p2 .= 1 * p0 + 0 * p2 + 0 * p3 ;
consider z1 such that b , x3 , x1 , z1 is_collinear and o , x1 , z1 is_collinear and o <> z1 and o <> z2 and o <> z1 and o <> z2 ;
consider i such that Arg ( ( Rotate ( s ) ) . q ) = s + Arg ( ( 2 * PI * i ) ) . i ;
consider g such that g is one-to-one and dom g = card ( f . x ) and rng g = { x , y } and rng g c= f . x and g is one-to-one ;
assume that A = P2 \/ Q2 and P2 <> {} and for x st x in P2 holds P2 . x in P2 and P2 . x in P2 and P2 . x in P2 and P2 . x in P2 and P2 . x in P2 and P2 . x in P2 and P2 . x in P2 and P2 . x in P2 and P2 . x in P2 and P2 . x in P2 and P2 . x in P2 and P2 . x in P2 and P2 .
attr F is associative means : Def3 : F .: ( F .: ( f , g ) , h ) = F .: ( f , g ) ;
ex x being Element of NAT st m = x `1 & x in z & x in { i } or m in { i } & x in { i } ;
consider k2 being Nat such that k2 in dom ( P . ( k + 1 ) ) and l in dom ( P . ( k + 1 ) ) and l = ( P . ( k + 1 ) ) ;
seq = r * seq implies for n holds seq . n = r * seq . n & seq is convergent & lim seq = r * seq . n & lim seq = r * seq . n
F1 . [ ( id a ) , [ a , a ] ] = [ f * ( id a , a ] , ( id a ) . [ a , a ] ] ;
{ p } "\/" D2 = { p "\/" y where y is Element of L : y in D2 & p in D2 } ;
consider z being element such that z in dom ( ( the _ of F ) . ( x , y ) ) and ( the _ of F ) . ( z , y ) = y ;
for x , y being element st x in dom f & y in dom f & f . x = f . y holds x = y
cell ( G , i , s ) = { |[ r , s ]| : r <= G * ( 0 , 1 ) `1 & s <= G * ( 0 , 1 ) `2 } ;
consider e being element such that e in dom ( T | E1 ) and ( T | E1 ) . e = v and ( T | E1 ) . e = v ;
( F `1 * ( b1 , b2 ) ) . x = ( ( Mx2Tran ( J , b1 , b2 ) ) * ( BY. , b3 ) ) . ( \mathbb j , j ) ;
- 1 / ( - 1_ K ) = ( - 1 ) (#) ( - 1 ) .= ( - 1 ) (#) ( - 1 ) .= ( - 1 ) (#) ( - 1 ) .= ( - 1 ) (#) ( - 1 ) .= ( - 1 ) (#) ( - 1 ) .= ( - 1 ) (#) ( - 1 ) .= ( - 1 ) (#) ( - 1 ) .= ( - 1 ) (#) ( - 1 ) ;
attr for x being set st x in dom f /\ dom g holds g . x <= f . x & g . x <= f . x ;
len ( f1 . j ) = len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) ;
All ( All ( 'not' a , A , G ) , B , G ) . z = Ex ( All ( 'not' a , B , G ) , A , G ) . z ;
LSeg ( E . ( k + 1 ) , F . ( k + 1 ) ) c= Cl RightComp Cage ( C , n + 1 ) ;
x \ a |^ m = x \ ( a |^ k * a ) .= ( x \ a ) |^ k .= ( x \ a ) |^ k ;
k -ininininthen ( k -inininin) . ( k + 1 ) = ( commute ( k , n ) ) . ( k + 1 ) .= ( ( k + 1 ) --> ( k + 1 ) ) . ( k + 1 ) .= ( ( k + 1 ) --> ( k + 1 ) ) . ( k + 1 ) .= ( ( k + 1 ) --> ( k + 1 ) ) . i ;
for s being State of A2 holds Following ( s , n ) . 0 + Following ( s , n ) . 1 is stable ;
for x st x in Z holds f1 . x = a |^ ( \bf 2 ) & ( f1 - f2 ) . x <> 0 implies f1 - f2 . x <> 0 & ( f1 - f2 ) . x <> 0 ) & ( f1 - f2 ) . x <> 0 implies f2 . x = 0 )
( support ( \mathop { \rm contradiction } ) \/ support ( m ) ) c= support ( m ) \/ support ( m ) \/ support ( m ) ;
reconsider t = u as Function of ( the carrier of A ) , ( the carrier' of B ) * the Arity of C ;
- ( a * sqrt ( 1 + b ^2 ) ) <= - ( b * sqrt ( 1 + b ^2 ) ) ;
phi /. ( succ b1 ) = g . a & phi . ( b , a ) = f . ( g . ( b , a ) ) & phi . ( b , a ) = f . ( g . ( b , a ) ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( F ^ <* p *> ) and i <> j ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 ,
the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U1 & the Sorts of U2 c= the Sorts of U2 ;
( - ( 2 * a ) ) / ( 2 * a ) + b / ( 2 * a ) - c > 0 ;
consider W00 such that for z being element holds z in W00 iff z in N ~ N & ( for z being element st z in N ~ N holds not z in N ) & ( not z in N ) ;
assume that ( the Arity of S ) . o = <* a *> and ( the Arity of S ) . o = r and ( the Arity of S ) . o = <* a *> and ( the Arity of S ) . o = r ;
Z = dom ( ( exp_R * ( arccot + arccot ) ) `| Z ) .= dom ( ( exp_R * ( arccot + arccot ) ) `| Z ) ;
sum ( f , SS1 ) is convergent & lim ( lim ( f , SS1 ) ) = integral ( f , S ) - lim ( lim ( f , S ) ) ;
( X ( ) => ( a => ( b => ( 'not' ( a => b ) ) ) ) ) => ( ( a => ( 'not' ( a => b ) ) ) => ( ( a => ( 'not' ( a => b ) ) => ( 'not' ( a => 'not' ( a => b ) ) ) ) ) ) in TAUT ( Al ( a ) ) ;
len ( M2 * ( M3 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * M1 ) ) ) ) ) ) ) ) = n & width ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * M1 ) ) ) ) ) ) ) = n ;
attr X1 \/ X2 is open & X1 , X2 X2 , X2 be SubSpace of X , X1 , X2 be SubSpace of X ;
for L being lower-bounded antisymmetric antisymmetric non empty RelStr for X being non empty Subset of L holds X "\/" { Top L } = { Top L }
reconsider f-1be = ( F . ( b , c ) ) `2 , f-16 = ( F . ( b , c ) ) `2 , f-16 = ( F . ( b , c ) ) `2 , f-16 = ( F . ( b , c ) ) `2 as Function of M , M ;
consider w being FinSequence of I such that the { of M = <* s *> ^ <* w *> ^ w ^ ( the { s } ^ w ) ^ q ) ;
g . ( a |^ 0 ) = g . ( 1_ G ) .= 1_ G .= 1_ G .= 1_ G .= 1_ G .= 1_ G ;
assume for i be Nat st i in dom f ex z being Element of L st f . i = rpoly ( 1 , z ) & z <> 0 ;
ex L being Subset of X st Carrier ( L ) = L & for K being Subset of X st K in C holds L /\ K <> {} & L /\ K <> {} ;
( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C2 & the carrier' of C2 = the carrier' of C2 implies the carrier' of C2 = the carrier' of C2
reconsider o9 = o `1 , p = o `2 , q = o `2 , r = o `1 , s = o `2 , p = o `2 , r = o `2 , s = o `1 , o = o `2 , p = o `1 , q = o `2 , r = o `2 , s = o `2 , s = o `1 , s = o `2 , p = o `1 , s = o `2 , r = o `2 , s = o `2 ;
1 * x1 + ( 0 * x2 + ( 0 * x3 ) ) + ( 0 * x2 + ( 0 * x3 ) ) = x1 + ( 0 * x2 + 0 * x3 ) .= x1 + ( 0 * x2 + 0 * x3 ) .= x1 + ( 0 * x2 + 0 * x3 ) .= x1 + ( 0 * x2 + 0 * x3 ) .= x1 + ( 0 * x2 + 0 * x3 ) .= 1 * x1 + ( 0 * x3 + 0 * x3 ) ;
( ( E " ) \ { 1 } ) \ ( E " { 1 } ) = ( E \ { 1 } ) \ ( E \ { 1 } ) .= ( E \ { 1 } ) \ { 1 } .= ( E \ { 1 } ) \ { 1 } .= ( E \ { 1 } ) \ { 1 } .= ( E \ { 1 } ) \ { 1 } .= ( E \ { 1 } ) \ { 1 } .= ( E \ { 1 } ) \ { 1
reconsider u1 = the carrier of U1 /\ ( ( U1 "\/" U2 ) "\/" ( U1 "\/" U2 ) ) as non empty Subset of ( U1 "\/" U2 ) ;
( ( x "/\" z ) "\/" ( x "/\" y ) ) "\/" ( z "/\" y ) <= ( x "/\" ( z "\/" y ) ) "\/" ( z "/\" ( x "\/" z ) ) ;
|. f . ( s1 . ( l1 + 1 ) - f . ( l1 + 1 ) ) .| < ( 1 / ( M + 1 ) ) * ( 1 / ( M + 1 ) ) ;
LSeg ( ( <* ( \mathop { \rm Cage ( C , n ) ) * ( i , j ) *> , ( \mathop { \rm Gauge ( C , n ) * ( i , j + 1 ) ) ) `1 ) is vertical ;
( f | Z ) /. x - ( f | Z ) /. x0 = L /. ( x- ( x - x0 ) ) + R /. ( x- ( x - x0 ) ) ;
g . c * ( - g . c ) + f . c * f . c <= h . c * ( - g . c ) + f . c ;
( f + g ) | divset ( D , i ) = f | divset ( D , i ) + g | divset ( D , i ) .= f | divset ( D , i ) + g | divset ( D , i ) ;
assume that ColVec2Mx f in the carrier of A and ColVec2Mx f in the carrier of A and ColVec2Mx f = ( ColVec2Mx f ) * ( ColVec2Mx f ) and len ( ColVec2Mx f ) = width A and width ( ColVec2Mx f ) = width A ;
len ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - M - M ) ) ) ) ) ) ) ) ) ) = len ( - ( - ( - ( - M - M ) ) ) ) ) & width ( - ( - ( - ( - M ) ) ) ) = width ( - ( - ( - M ) ) ) ;
for n , i being Nat st i + 1 < n holds [ i , i + 1 ] in the InternalRel of ( TOP-REAL n ) & [ i , i + 1 ] in the InternalRel of ( TOP-REAL n ) ;
pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1 ;
attr a <> 0 & b <> 0 & Arg a = Arg b & Arg b = Arg a & Arg a = Arg b & Arg b = Arg b & Arg a = Arg b & Arg b = Arg c ;
for c being set st not c in [. a , b .] holds not c in Intersection ( the non empty set , the set , the set , the set of set , the set , the set of set , the set of set , the set is Element of the set of set , the set of set , the set of set )
assume that V1 is linearly-independent and V2 is closed and V2 = { v + u : v in V1 & u in V1 & u in V1 & v in V1 & u in V1 & v in V1 & u in V1 & v in V1 & u in V1 & v in V1 & u in V1 & v in V1 & u in V1 & u in V1 & v in V1 & u in V1 & v in V1 & u in V1 & v in V1 & u in V1 ;
z * ( x1 + ( 1 - z ) * x2 ) in M & z * ( y1 + ( 1 - z ) * y2 ) in N & z * ( y1 + ( 1 - z ) * y2 ) in N ;
rng ( ( ( ( P qua Function ) " ) * ( S qua Function ) ) * ( S * ( S * ( S * ( S * ( S * ( S , R ) ) ) ) ) ) ) ) = Seg card ( S * ( S * ( S * ( S * ( S * ( S * ( S * ( S * ( S * ( S , R ) ) ) ) ) ) ) ) ;
consider s2 being Integer such that s2 is convergent and b = lim s2 and for n holds s2 . n = lim s2 and for n holds s2 . n = lim s2 ;
h2 " . n = h2 . n " & 0 < ( - 1 ) / ( n + 1 ) & 0 < ( - 1 ) / ( n + 1 ) ;
( Partial_Sums ( ||. ( r (#) ( seq ^\ k ) ) .|| ) . m = ||. ( r (#) ( seq ^\ k ) ) . m ) .|| .= ||. ( r * ( seq ^\ k ) ) . m .|| .= ||. ( r * ( seq ^\ k ) ) . m .|| .= ||. ( r * ( seq ^\ k ) ) . m .|| .= ||. ( r * ( seq ^\ k ) ) . m .|| ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b ;
- v = ( - 1_ G ) * v & - w = ( - 1_ G ) * v & - w = ( - 1_ G ) * v & - w = ( - 1_ G ) * v ;
sup ( ( k \circ D ) .: D ) = sup ( ( k \circ D ) .: D ) .= ( k .: D ) . ( k , j ) .= ( k * D ) . ( k , j ) .= ( k * D ) . ( k , j ) .= ( k * D ) . ( k , j ) ;
A |^ ( k , l ) = ( A |^ ( n , l ) ) ^^ ( A |^ ( k , l ) ) .= ( A |^ ( k , l ) ) \ ( A |^ ( k , l ) ) ;
for R being add-associative right_zeroed right_complementable non empty addLoopStr for I , J , K being Subset of R holds I + ( J + K ) = ( I + J ) + K
( f . p ) `1 = ( p `1 ) / sqrt ( 1 + ( p `2 / p `1 ) ^2 ) .= ( p `1 ) / sqrt ( 1 + ( p `2 / p `1 ) ^2 ) ;
for a , b being non zero Nat st a , b are_relative_prime holds ( a * b ) div p = ( a * b ) div p & ( a * b ) div p = ( a * b ) div p
consider A9 being countable set such that r is countable & ( for A being Element of Al holds A is Element of D ) & ( for A being Element of D holds A is Element of D ) & ( for B being Element of D holds B in A ) implies A is D -sqrt ( B ) is D -sqrt ( A ) ;
for X being non empty addLoopStr for M , N being Subset of X st for x , y being Point of X st x in M & y in N holds x + y in M + N
{ [ x1 , x2 ] , [ y1 , y2 ] , [ y2 , z2 ] } c= { x1 , y1 , y2 } ;
h . ( f . O ) = |[ A * ( f . O ) `1 + B * ( f . I ) `1 , A * ( f . I ) `2 + D * ( f . I ) `2 + D * ( f . I ) `2 + D * ( f . I ) `2 ]| ;
( Gauge ( C , n ) * ( k , i ) ) `1 in L~ Upper_Seq ( C , n ) /\ L~ Upper_Seq ( C , n ) ;
cluster m , n ) are_relative_prime -> prime implies for Nat st ( for Element of NAT holds ( m divides n ) & ( for p being Nat st p divides m ) & p divides n ) & ( not p divides n ) & ( not p divides n implies ( m divides n ) ) & ( not p divides n ) & not ( m divides n implies ( m divides n ) ) & ( m divides n ) ) implies ( m divides n ) ) implies ( m divides n )
( f * F ) . x1 = f . ( F . x1 ) & ( f * F ) . x2 = f . ( F . x2 ) & ( f * F ) . x1 = f . ( F . x2 ) ;
for L being Lattice , a , b , c being Element of L st a \ b <= c & b \ c <= c holds a \ b <= c
consider b being element such that b in dom ( H / ( ( x , y ) \leftarrow ( 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 . 1 , G & W . 5 in G & e in W . 3 & W . ( 7 + 1 ) in G . ( 7 + 1 ) ;
( ' ' ( h ) ) . ( 2 * n ) = ( ' ' ( h ) ) . ( 2 * n ) .= ( h ' ( h ) ) . ( 2 * n ) ;
j + 1 = Seg ( len h11 + len h11 + 1 ) .= i + 1 + len h11 + 2 - 1 .= i + 1 + 1 + 1 - 1 .= i + 1 + 1 - 1 ;
( S *' ) . ( f , g ) = S *' . ( f , g ) .= S . ( ( S *' ) . ( f , g ) ) .= S . ( f , g ) .= S . ( f , g ) .= S . ( f , g ) ;
consider H such that H is one-to-one and rng H = the carrier of L2 and Sum ( L2 ) = Sum ( L2 ) and Sum ( L1 ) = Sum ( L2 ) and Sum ( L2 ) = Sum ( L2 ) ;
attr R is <= .| means : Def3 : for p , q st p in R & q in R holds ex P st P is special & p in P & q in P & p in P & q in R ;
dom ( product ( X --> f ) ) = meet ( ( X --> f ) . i ) .= meet ( ( X --> f ) . i ) .= meet ( X --> f ) .= meet ( X --> f ) .= meet ( X --> f ) .= meet ( X --> f ) .= meet ( X --> f ) .= meet ( X --> f ) .= meet ( X --> f ) ;
upper_bound ( proj2 .: ( Upper_Arc C /\ Upper_Arc C /\ Vertical_Line w ) ) <= upper_bound ( proj2 .: ( Upper_Arc C /\ Vertical_Line w ) ) & upper_bound ( proj2 .: ( Upper_Arc w /\ Vertical_Line w ) ) <= upper_bound ( proj2 .: ( Upper_Arc w /\ Vertical_Line w ) ) ;
for r be Real st 0 < r ex n be Nat st for m be Nat st n <= m holds |. S . m - 0 .| < r
i * ( f - f ) = i * ( f - ( i * y ) ) .= i * ( f - ( i * y ) ) .= i * ( f - ( i * y ) ) .= i * ( f - ( i * y ) ) ;
consider f being Function such that dom f = 2 -tuples_on X ( ) and for Y being set st Y in 2 -tuples_on X ( ) holds f . Y = F ( Y ) ;
consider g1 , g2 being element such that g1 in [#] Y and g2 in C and g2 in C and g = [ g1 , g2 ] and [ g1 , g2 ] in the InternalRel of Y ;
func d \! \mathop { n } -> Nat means : Def3 : d |^ ( n + 1 ) divides n & ( d |^ ( n + 1 ) ) divides ( n + 1 ) ;
f{ [ 0 , t ] , t ] } = f . [ 0 , t ] .= ( - P ) . [ 2 * x , t ] .= ( - P ) . [ 2 * x , t ] .= ( - P ) . [ 2 * x , t ] .= a ;
t = h . D or t = h . B or t = h . C or t = h . D or t = h . E or t = h . F or t = h . J or t = h . M ;
consider m1 be Nat such that for n st n >= m1 holds dist ( ( seq . n ) , ( seq . n ) ) < 1 / ( n + 1 ) ;
( ( q `1 ) / |. q .| ) ^2 <= ( ( q `1 ) / |. q .| ) ^2 ;
h0 . ( i + 1 + 1 + 1 ) = h21 . ( i + 1 + 1 + 1 + 1 ) .= h21 . ( i + 1 + 1 ) ;
consider o being Element of the carrier' of S , x2 being Element of { the carrier' of S : not contradiction } such that a = [ o , x2 ] and a = [ o , x2 ] ;
for L being RelStr , a , b being Element of L holds a <= { b } iff a <= b & b <= a & a <= b implies a <= b
||. h1 .|| . n = ||. h1 . n .|| .= ||. h1 . n .|| .= ||. ( h1 . n ) .|| .= ||. ( h1 . n ) .|| .= ||. ( h1 . n ) .|| .= ||. ( h1 . n ) .|| ;
( - ( ( - 1 ) (#) ( ( id Z ) ^ ) ) `| Z ) . x = f . x - ( ( - 1 ) (#) ( ( id Z ) ^ ) ) . x .= - ( x - a ) * ( x - a ) .= - ( x - a ) * ( x - a ) .= - ( x - a ) * ( x - a ) ;
attr r = F .: ( p , q ) means : Def3 : len r = len ( p ^ q ) & for i st i in dom r holds r . i = F ( i , len p ) ;
( ( r / 2 ) |^ 2 ) / ( r + ( r / 2 ) |^ 2 ) <= ( r / 2 ) |^ ( 2 + 1 ) ;
for i being Nat , M being Matrix of n , K st i in Seg n holds ( Det ( M @ ) ) . i = Sum ( ( Line ( M @ ) ) )
then a <> 0. R & a " * ( a * v ) = 1 / a * v & a " * ( a * v ) = 1 / a * v ;
( p . ( j -' 1 ) ) * ( q *' r ) . ( i + 1 -' j ) = Sum ( p . ( j -' 1 ) ) * ( q . ( j -' 1 ) ) ;
deffunc F ( Nat ) = L . 1 + ( ( R /* ( h ^\ n ) ) * ( h ^\ n ) ) " .= ( R /* ( h ^\ n ) ) . ( h ^\ n ) ;
assume that the carrier of H2 = f .: the carrier of H1 and the carrier of H2 = f .: the carrier of H2 and the carrier of H2 = the carrier of H2 and the carrier of H2 = the carrier of H1 and the carrier of H2 = the carrier of H2 ;
Args ( o , Free ( X , s ) ) = ( ( the Sorts of Free ( o , s ) ) * ( the Arity of S ) ) . o .= ( the Arity of S ) . o ;
H1 = n + 1 -tuples_on ( |. 2 to_power ( n + 1 ) .| ) .= n + 1 -tuples_on ( |. 2 to_power ( n + 1 ) .| ) .= n + 1 -tuples_on ( |. 2 to_power ( n + 1 ) .| ) ;
( O = 0 implies O = 1 ) & ( O = 1 implies O = 1 ) & ( O = 1 implies O = 1 ) & ( O = 1 implies O = 1 ) & ( O = 1 implies O = 1 ) & O = 1 implies O = 1 ) & O = 1 implies O = 1 ) & O = 1 implies O = O & O = 1 implies O = O & O = O & O = O & O = O & O = O & O = O & O = O & O = O & O = O & O = O & O = O
F1 .: ( dom ( F1 /\ dom F2 ) ) = F1 .: ( dom F1 /\ dom F2 /\ dom F2 .= { f /. ( n + 1 ) } .= { f /. ( n + 1 ) } .= ( F1 /. ( n + 1 ) ) * ( F1 /. ( n + 1 ) ) .= ( F1 /. ( n + 1 ) ) * ( F1 /. ( n + 1 ) ) ;
attr b <> 0 & d <> 0 & b <> d & ( a = d ) & ( b = d implies a = ( - b ) / ( d + c ) ) ;
dom ( ( f +* g ) | D ) = dom ( ( f +* g ) | D ) .= ( ( f +* g ) | D ) /\ D .= ( ( f +* g ) | D ) /\ D .= ( ( f +* g ) | D ) /\ D .= ( ( f +* g ) | D ) /\ D .= ( f +* g ) | D .= ( f +* g ) | D ;
for i being set st i in dom g ex u , v being Element of L st g /. i = u * a & for a being Element of L st a in dom g ex u being Element of B st u in A & v = g . a * v
g `2 * P `2 * g " = g `2 * ( g " * P ) .= g `2 * ( g " * P ) .= g `2 * ( g " * P ) .= g `2 * ( g " * P ) .= g ;
consider i , s1 such that f . i = s1 and not ( ex s st s in dom s1 & not ( s in { p } ) & not ( s in { p } ) & not ( not s in { p } ) ;
h5 | ]. a , b .[ = ( g | ]. a , b .[ ) | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ ;
[ s1 , t1 ] , [ s2 , t2 ] ] , [ s2 , t2 ] connected & [ s2 , t2 ] , [ s2 , t2 ] ] in connected & [ s2 , t2 ] , [ s2 , t2 ] ] in connected ;
then H is negative & H is non negative & H is non negative & H is non empty implies H is not an -\mathopen <* H *> ^ <* H *> ^ <* H *> ^ <* H *> ^ <* H *> ^ <* H *> ^ <* H *> ^ <* H *> ^ <* H *> ;
attr f1 is total means : Def3 : ( 1 - f1 ) (#) f2 is total & ( 1 - f1 ) (#) f2 is total & for c st c in dom f1 holds f1 . c = ( f1 . c ) (#) f2 . c ;
z1 in W2 " ( { z2 } ) or z1 = z2 & not z1 in W2 & not z2 in W2 & not ( z1 in W1 & not z2 in W2 & not ( z1 in W2 & not z2 in W2 & not ( z1 in W1 & not z2 in W2 ) & not ( z1 in W2 & not z2 in W2 ) & not ( z1 in W2 & not z2 in W2 & not ( z1 in W2 ) & not ( z1 in W2 ) & ( z1 in W2 ) & ( z1 in W2 & not ( z1 in W2 ) & ( z1 in W2 & not ( z1 in W2 ) & (
p = 1 * p .= a " * a * p .= a " * ( b " * p ) .= a " * ( b " * p ) .= a " * ( b " * p ) .= a " * ( b " * p ) .= ( a " * b ) * ( b " * p ) .= ( a " * b ) * ( b " * p ) .= ( a " * b ) * ( b " * p ) .= ( a " * b ) * ( b " * p ) ;
for rseq be Real_Sequence for K be Real st for n be Nat holds ( for k be Nat st n <= k holds ( seq . n ) <= K ) holds upper_bound rng ( seq ) <= upper_bound rng ( seq )
( for x being Element of TOP-REAL 2 st x in L~ go holds x in L~ go \/ L~ pion1 ) or x in L~ co or x in L~ co or x in L~ co & x in L~ co or x in L~ co or x in L~ co or x in L~ co or x in L~ co ;
||. f . ( g . ( k + 1 ) - g . ( k + 1 ) ) .|| <= ||. g . 1 - g . ( k + 1 ) .|| * ( K * ( k + 1 ) - K * ( k + 1 ) ) ;
assume h = ( ( B .--> ( C .--> D ) ) +* ( E .--> F ) ) +* ( J .--> M ) +* ( N .--> N ) +* ( M .--> N ) +* ( M .--> N ) ) ;
|. ( ( ( ( ( H . n ) || A ) `| A ) . k ) - ( ( ( H . n ) `| A ) . k ) .| <= e * ( ( ( H . n ) `| A ) . k ) ;
( (
{ x1 , x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8
assume that A = [. 0 , 2 * PI .] and integral ( ( exp_R (#) cos ) , A ) = 0 and integral ( ( exp_R (#) cos ) , A ) = 0 ;
p `1 is Permutation of dom f1 & p `2 " = ( Sgm Y ) " & p `2 " = ( Sgm Y ) " & p `2 = ( Sgm Y ) " & p `2 = ( Sgm Y ) " ;
for x , y st x in A holds |. ( 1 / ( x - y ) ) * ( 1 - ( x - y ) ) .| <= 1 * |. ( f . x - ( x - y ) ) * ( 1 - ( x - y ) ) .|
( p2 `2 ) ^2 = |. q2 .| * ( ( q2 `2 ) ^2 + ( q2 `1 ) ^2 ) .= |. q2 .| * ( ( q2 `2 ) ^2 + ( q2 `1 ) ^2 ) ;
for f be PartFunc of the carrier of CNS , REAL st dom f is compact & f is continuous & for x be Element of REAL st x in dom f holds f /. x = F ( x ) holds f /. x = G ( x )
assume for x being Element of Y st x in EqClass ( z , CompF ( B , G ) ) holds ( Ex ( a , PA , G ) ) . x = TRUE ;
consider FM such that dom FM = n1 and for k be Nat st k in n1 holds ( for k be Nat st k in n1 holds ( for x be Nat st x in n1 holds ( x , F . k ) = f . x ) ;
ex u , u1 st u <> u1 & u , v / ( u , v ) / ( u , v1 ) / ( u , u1 ) / ( u , v2 ) / ( u , v1 ) > 0 & u , v / ( u , v2 ) / ( u , v2 ) / ( u , v1 ) / ( u , v2 ) / ( u , v2 ) > 0 ;
for G being Group , A , B being non empty Subgroup of G , N being normal Subgroup of G holds ( N , A ) ` = N ` A & ( N , B ) ` = N ` A
for s be Real st s in dom F holds F . s = integral ( R , ( R + e ) * ( f + g ) ) . x .= integral ( R , ( R + e ) * ( f + g ) ) . x )
width AutMt ( f1 , b1 , b2 , b3 , b3 , b3 , b2 ) = len b2 .= width ( b1 + b2 + b3 ) .= width ( b1 + b2 + b3 ) .= width ( b2 + b3 ) .= width ( b2 + b3 ) .= width ( b2 + b3 ) .= width ( b2 + b3 ) .= width ( b2 + b3 ) .= width ( b2 + b3 ) ;
f | ]. - sqrt ( 1 - ( 2 * PI ) , - 1 ) = f | ]. - sqrt ( 1 - ( 2 * PI ) , - 1 ) , - sqrt ( 2 * PI ) + sqrt ( 2 * PI ) + - sqrt ( 2 * PI ) , - 1 ) ;
assume that X is closed and a in X and a in X and a in X and y in X and x in { [ n , x ] } and y in X and x in X ;
Z = dom ( ( ( - 1 / 2 ) (#) ( ( #Z 2 ) * ( f1 + #Z 2 ) ) ) `| Z ) /\ dom ( ( - 1 / 2 ) (#) ( ( #Z 2 ) * ( f1 + #Z 2 ) ) ) ;
func h1 . l -> Subset of V means : Def3 : for k st 1 <= k & k <= len l holds it . k in V & ( l . k ) `1 in V ;
for L being non empty TopSpace , N being net of L , M being net of L , x being Point of N st x is Point of N & for c being Point of N st c in N holds x is Point of N holds x is Point of N
for s being Element of NAT holds ( ( for v be Element of NAT holds ( ( for x be Element of NAT holds x in v + ( v + u ) ) ) ) & ( for x be Element of NAT holds x in v + ( u + ( v + u ) ) ) implies x = ( v + u ) + ( v + u )
then z /. 1 = ( N-min L~ z ) .. z & ( N-min L~ z ) .. z < ( N-min L~ z ) .. z ;
len ( p ^ <* ( 0 qua Real ) * ( 0 qua Real ) *> ) = len p + len <* ( 0 qua Real ) * ( 0 qua Real ) *> .= len p + 1 .= len p + 1 .= len p + 1 + 1 .= len p + 1 + 1 .= len p + 1 ;
assume that Z c= dom ( - ( ( - 1 / 2 ) (#) f ) and for x st x in Z holds f . x = x and f . x > 0 ;
for R being add-associative right_zeroed right_complementable non empty doubleLoopStr , I , J being Subset of R , I being Ideal of R , J being Subset of R holds ( I + J ) *' ( I + J ) c= I /\ J
consider f being Function of [: B1 , B2 :] , B2 such that for x being Element of B1 , y being Element of B2 holds f . x = F ( x , y ) and f . y = F ( x , y ) ;
dom ( x2 + y2 ) = Seg len x .= Seg len ( x + ( - y ) ) .= Seg len ( ( x + ( - y ) ) + ( - y ) ) .= Seg len ( ( x + ( - y ) ) + ( - y ) ) .= Seg len ( ( x + y ) - ( - y ) ) .= dom ( x + y ) ;
for S being |. Functor of C , B for c being Object of C holds card S = ( id C ) . c & card S = ( id C ) . c & card S = ( id C ) . c
ex a st a = a2 & a in dom f /\ dom ( f | 5 ) & ( for x st x in 5 holds f . x = F ( x ) ) & ( for x st x in 5 holds f . x = F ( x ) ) implies for x st x in 5 holds f . x = F ( x ) ) ;
a in Free ( ( H / ( x. 4 , x. 0 ) ) '&' ( ( H / ( x. 4 , x. 0 ) ) '&' ( ( H / ( x. 4 , x. 0 ) ) ) / ( x. 4 , x. 0 ) ) ) ;
for C1 , C2 being non empty set , f , g being Function of C1 , C2 st for x being set , f being Function of C1 , C2 st x = f holds x = g iff x = y
( W-min L~ Cage ( C , n ) ) `1 = W-bound L~ Cage ( C , n ) & ( W-min L~ Cage ( C , n ) ) `1 = W-bound L~ Cage ( C , n ) implies ( W-min L~ Cage ( C , n ) ) `1 = W-bound L~ Cage ( C , n )
consider u , y0 , z0 being Point of REAL-NS 2 such that u = <* x0 , y0 , z0 *> and f is_partial_differentiable_in z0 , z0 and u in dom ( SVF1 ( 3 , pdiff1 ( f , 1 ) , 1 ) (#) pdiff1 ( f , 1 ) ) ;
then ( t . {} ) `1 in Vars & ex x being Element of Vars st x = ( t . {} ) `1 & ( t . {} ) `2 = ( x , y ) `2 & ( t . {} ) `1 = ( x , y ) `2 ;
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 .= 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 , a ) -> Subset-Family of R means : Def3 : for A being Subset of R holds A in it iff ex a being Element of R st a in A & A c= a & it = Class ( R , a ) ;
defpred P [ Nat ] means ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . $1 ) `1 ) `1 ) `1 ) `1 ) & ( ( ( ( ( ( ( ( ( ( G ) . $1 ) ) ) ) ) ) `1 ) `2 ) `2 <= G * ( 1 , 1 ) `2 ) ;
assume that dim ( W1 ) = 0 and dim ( U1 ) = 0 and dim ( U1 ) = 0 and dim ( U1 ) = 0 and dim ( U1 ) = 0 and dim ( U1 ) = 0 and dim ( U1 ) = 0 and dim ( U1 ) = 0 and dim ( U1 ) = 0 and dim ( U1 ) = 0 ;
mathat mathat ( m . t ) . {} = ( m . t ) . {} and ( m . t ) . {} = ( m . t ) . {} .= ( m . t ) . {} .= ( m . t ) . {} .= ( m . t ) . {} .= ( m . t ) . {} ;
d11 = ( x9 ^ <* d *> ) . ( ( y9 ^ <* d *> ) .= f . ( y9 ^ <* d *> ) .= ( x9 ^ <* d *> ) . ( x9 ^ <* d *> ) .= ( x9 ^ <* d *> ) . ( y9 ^ <* d *> ) .= ( x9 ^ <* d *> ) . ( y9 ^ <* d *> ) .= ( x9 ^ <* d *> ) . ( y9 ^ <* d *> ) .= ( x9 ^ <* d *> ) . ( y9 ^ <* d ^ <* d *> ) .= ( x9 ^ <* d *> ) . ( y9 ^ <* d ^ <* d *> ) .= ( x9 ^ <* d *> ) . ( y9 ^ <* d *> ) .= ( x9 ^ <* d *> ) . ( y9 ^
consider g such that x = g and dom g = dom f and for x being element st x in dom f holds g . x in f . x and g . x in f . x ;
x + 0. F_Complex = x + len x .= ( x + len x ) |-> 0. F_Complex .= ( x + ( len x |-> 0. F_Complex ) ) ^ ( x |-> 0. F_Complex ) .= ( x + ( len x |-> 0. F_Complex ) ) ^ ( x |-> 0. F_Complex ) .= x `1 + ( x - 0. F_Complex ) .= x `1 + ( x - 0. F_Complex ) .= x `2 + ( x - 0. F_Complex ) ;
( k -' ( k + 1 ) ) + 1 in dom ( f | ( k + 1 ) ) & ( k + 1 ) + 1 in dom ( 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 , p3 } and P2 = { p1 , p2 , p3 } and P1 = { p1 , p2 , p3 } and P2 = { p1 , p2 , p3 } and P2 = { p1 , p2 , p3 } and P2 = { p1 , p2 , p3 } and P2 = { p1 , p3 , p4 } and P2 = { p2 , p3 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p1 , p3 , p4 , p2 , p4 , p2 , p4 , p1 , p2 , p3 , p4 , p4 , p2 , p3 , p4 , p4 ,
reconsider a1 = a , b1 = b , c1 = c , c1 = d , c1 = p , c2 = p , c1 = q , c1 = p , c2 = r , c1 = s , c2 = s , c1 = s , c1 = p , c1 = q , c2 = s , c1 = r , c2 = s , c1 = s , c1 = s , c2 = s , c2 = s , c1 = s , c1 = s , c1 = s , c1 = s , c1 = s , c1 = p , c1 = p , c1 = q , c1 = p , c1 = s , c1 = s , c1 = s , c1 = s , c1 = p , c1 = s , c1 = s , c1 = p , c1 = s , c2 =
reconsider Gt1f = G1 . ( t , [. b , a .] ) * F1 . ( t , F2 . a ) as Morphism of ( G1 * F1 ) . ( t , F2 . b ) , ( G1 * F2 ) . ( t , a ) ;
LSeg ( f , i + i1 -' 1 ) = LSeg ( f /. ( i + 1 -' 1 + 1 ) , f /. ( i + 1 -' 1 + 1 ) ) ;
Integral ( P . m , P . n ) | dom ( P . m ) <= Integral ( M , ( P . m ) | dom ( P . m ) ) ;
assume that dom f1 = dom f2 and for x , y being element st [ x , y ] in dom f2 & [ x , y ] in dom f2 holds f1 . ( x , y ) = f2 . ( x , y ) ;
consider v such that v = y and dist ( u , v ) < min ( ( G * ( i , 1 ) `1 - ( G * ( i , 1 ) `1 ) - ( G * ( i + 1 , 1 ) `2 - ( G * ( i + 1 , 1 ) `2 - ( G * ( i + 1 , 1 ) `2 - ( G * ( i , 1 ) `2 - ( G * ( i + 1 , 1 ) `2 ) ) ) / 2 ) ) ;
for G being Group , H being Subgroup of G , a , b being Element of G st a = b & b = a & for i being Integer st i in dom a holds a |^ i = b |^ i & b |^ i = a |^ i
consider B being Function of Seg ( S + L ) , the carrier of V1 such that for x being element st x in Seg ( S + L ) holds P [ x , B . x ] ;
reconsider K1 = { p0 where 7 is Point of TOP-REAL 2 : P [ 7 ] & ( for p being Point of TOP-REAL 2 st p in P holds P [ p ] ) & ( for p being Point of TOP-REAL 2 st p in P holds P [ p ] ) implies P is Subset of TOP-REAL 2 ) ;
( ( ( ( ( N - S ) / |. N .| - ( N + 1 ) ) / ( 2 * ( N + 1 ) ) ) / ( 2 * ( N + 1 ) ) ) ) / ( 2 * ( N + 1 ) ) ) <= ( ( ( N - S ) / ( 2 * ( N + 1 ) ) ) / ( 2 * ( N + 1 ) ) ) ;
for x be Element of X , n be Nat st x in E holds |. Re ( F . n ) .| <= P . x & |. Im ( F . n ) .| <= P . x
len ( @ ( @ H ^ <* 0 *> ) ) = len ( @ H ^ <* 1 *> ) + len <* 1 *> .= len ( @ H ^ <* 1 *> ) + len <* 1 *> .= len ( @ H ^ <* 1 *> ) + len <* 1 *> .= len ( @ H ^ <* 1 *> ) + len <* 1 *> .= len ( @ H ^ <* 1 *> ) + len <* 1 *> .= len ( @ H ^ <* 1 *> ) + 1 ;
v / ( x. 3 , m1 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) ) = m3 ;
consider r being Element of M such that M , v2 / ( x. 3 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m
func ( ( the Nf2 of G , Rw ) --> ( the Element of G , ( the Element of G ) --> ( the carrier of G ) ) ) -> Element of Union ( ( the NRw of G , ( the carrier of G ) --> ( the carrier of G ) ) ) ;
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= ( Exec ( n2 , s2 ) ) . b2 .= ( Exec ( n2 , s2 ) ) . b2 .= ( Exec ( n2 , s2 ) ) . b2 .= ( Exec ( n2 , s2 ) ) . b2 .= ( Exec ( n2 , s2 ) ) . b2 .= ( Exec ( n2 , s2 ) ) . b2 .= ( Exec ( n2 , s2 ) ) . b2 .= ( Exec ( n2 , s2 ) ) . b2 .= ( Exec ( n2 , s2 ) ) . b2 .= ( Exec ( n2 , s2 ) ) . b2 .= ( Exec ( n2 , s2 ) ) . b2 .= ( Exec ( n2 , s2 ) )
for n , k being Nat holds 0 <= ( Partial_Sums ( |. seq .| ) . n ) . ( n + k ) - ( Partial_Sums ( |. seq .| ) . n ) . ( n + k ) - ( Partial_Sums ( |. seq .| ) . n ) . ( n + k )
set F = S \! \mathop { \vert S .| , G = S \! \mathop { \vert S .| } ;
( Partial_Sums ( seq ) ) . ( n + 1 ) + Partial_Sums ( seq ) . ( n + 1 ) >= ( Partial_Sums ( seq ) ) . ( n + 1 ) + Partial_Sums ( seq ) . ( n + 1 ) ;
consider L , R such that for x st x in N holds ( f | Z ) . x - ( f | Z ) . ( x- ( 1 / 2 ) ) = L . ( x- ( 1 / 2 ) ) + R . ( x- ( 1 / 2 ) ) ;
func the closed of \HM { a , b , c , d , e , f , g , h , i , i , f , g , i , h , i , g , i , h , i , g , i , h , i , g , i , h , i , g , i , i , h , i , g , i , h , i , g , i , i , h , i , g , i , h , i , g , i , j , g , i , h , i , g , i , j ) ;
a * b ^2 + ( a * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 >= 6 * a * a * b * c + ( b * c ) + ( b * c ) + ( b * c ) + ( b * c ) + ( b * c ) + ( b * c ) + ( b * c ) + ( b * c ) + ( b * c ) + ( b * c ) + ( b * c ) + ( b * c ) + ( b * c ) + ( b * c ) + ( b * c ) + ( b * c ) + ( b *
v / ( ( x1 , m1 ) / ( x2 , m2 ) ) / ( x2 , m1 ) / ( x2 , m2 ) = v / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m2 ) ;
Assume that + ( Q ^ <* x *> , M ) = ( ( Q ^ <* x *> ) ^ ( M ^ <* TRUE *> ) ) ^ ( M ^ <* TRUE *> ) and ( ( Q ^ <* x *> ) ^ ( M ^ <* TRUE *> ) ) ^ ( M ^ <* TRUE *> ) = ( Q ^ <* TRUE *> ) ^ ( M ^ <* TRUE *> ) ;
Sum ( ( F |^ ( n1 + 1 ) ) * Sum ( C |^ n1 ) ) = ( C |^ n1 ) * Sum ( C |^ n1 ) .= C . ( n1 + 1 ) * Sum ( C |^ n1 ) .= C . ( n1 + 1 ) * Sum ( C |^ n1 ) .= C . ( n1 + 1 ) * Sum ( C ^\ n1 ) .= C . ( n1 + 1 ) * Sum ( C ^\ n1 ) .= C . ( n1 + 1 ) .= C . ( n1 + 1 ) .= C . ( n1 + 1 ) .= C . ( n1 + 1 ) .= C . ( n1 + 1 ) .= C . ( n1 + 1 ) .= C . ( n1 + 1 ) .=
( ( GoB f ) * ( len GoB f , 1 ) ) `1 = ( ( GoB f ) * ( len GoB f , 1 ) ) `1 .= ( ( GoB f ) * ( len GoB f , 1 ) ) `1 .= ( ( GoB f ) * ( len GoB f , 1 ) ) `1 ;
defpred X [ Element of NAT ] means ( Partial_Sums ( s ) ) . $1 = ( a * ( $1 + 1 ) ) * ( $1 + 1 ) + b * ( $1 + 1 ) ;
( the_arity_of g ) . ( g . ( the Arity of S ) . ( g . ( the Arity of S ) ) ) = ( ( the Arity of S ) . ( g . ( the Arity of S ) . ( g . ( the Arity of S ) ) ) ) ) . ( g . ( g . ( the Arity of S ) ) ) .= ( g . ( the Arity of S ) . ( g . ( g . ( the Arity of S ) ) ) ) ;
( X ~ ) ^ ( Y ~ ) tolerates X ~ & card ( X ~ ) = card ( X ~ ) + card ( Y ~ ) & card ( X ~ ) = card ( X ~ ) + card ( Y ~ ) ;
for a , b being Element of S , s being Element of NAT st s = n . s & a = F . n & b = F . n & ( a = n implies s = b ) holds a = b
E , f |= All ( All ( x , All ( x , All ( x , All ( x , p ) ) ) , All ( x , All ( x , p ) ) ) => ( All ( x , All ( x , p ) ) ) ) ;
ex R2 being 1-sorted st R2 = ( p | ( n + 1 ) ) . i & ( the carrier of ( p | ( n + 1 ) ) ) . i = the carrier of ( p | ( n + 1 ) ) . i & ( the carrier of ( p | ( n + 1 ) ) ) . i = the carrier of ( p | ( n + 1 ) ) . i ;
[. a , b + ( 1 + k ) .[ is Element of the , the carrier of G & ( the partial F of G ) . ( k + 1 ) is Element of the carrier of G & ( the partial F of G ) . ( k + 1 ) is Element of the carrier of G & ( the partial F ) . ( k + 1 ) is Element of the carrier of G ;
Comput ( P , s , 2 + 1 ) = Exec ( P . 2 , Comput ( P , s , 2 ) ) .= Exec ( ( a , i ) := ( a , I ) ) .= Exec ( ( a , i ) := ( a , I ) ) ;
card ( h1 ) . k = power ( K , len ( - 1_ K , len ( - 1_ K , len ( - 1_ K , len ( - 1_ K , len ( - 1_ K , len ( - 1_ K , len ( - 1_ K , len ( - 1_ K , len - 1_ K , len - 1_ K , len - 1_ K , len - 1_ K , len - 1_ K , - 1_ K ) ) ) ) ) .= ( card ( f *' ( - 1_ K , len ( - 1_ K , len ( - 1_ K , len ( - 1_ K , len ( - 1_ K , len ( - 1_ K , len ( - 1_ K , len ( - 1_ K , len ( - 1_ K , len
( f / g ) / ( g / f ) = f / ( g / f ) .= ( f / g ) / ( g / f ) .= ( f / g ) / ( g / f ) .= ( ( f / g ) / ( g / f ) ) / ( g / f ) .= ( ( f / g ) / ( g / f ) ) / ( g / f ) .= ( ( f / g ) / ( g / f ) ) / ( g / f ) ;
len ( ( C /. len ( C /. 1 ) ) - ( len ( Gauge ( C , 1 ) ) ) + 1 ) = len ( ( C /. 1 ) - ( len ( C /. 1 ) ) + 1 ) .= len ( ( C /. 1 ) - ( C /. 1 ) ) + 1 .= len ( ( C /. 1 ) - ( C /. 1 ) ) ;
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) ( r (#) f ) .= dom ( r (#) ( r (#) f ) .= dom ( r (#) ( f | X ) .= dom ( r (#) f ) /\ X .= dom ( r (#)
defpred P [ Nat ] means for n holds 2 * Fib ( n + $1 ) = Fib ( n + $1 ) * Fib ( n + $1 ) + ( 5 * Fib ( n + $1 ) ) * Fib ( n + $1 ) + ( 5 * Fib ( n + $1 ) ) * Fib ( n + $1 ) ;
consider f being Function of INT , INT such that f = f and f is onto and ( for n being Nat st n < k holds f . n = F ( n ) ) & for x being Element of INT st x in Z holds f " { x } = { n } ) & ( for x being Element of INT st x in Z holds f " { x } = { n } ) ;
consider c9 being Function of S , BOOLEAN such that c9 = chi ( S , A ) and ( for A being Element of S holds E . A = chi ( S , A ) . A ) and ( for A being Element of S holds E . A = Prob . A ) ;
consider y being Element of Y ( ) such that a = "\/" ( { F ( x , y ) where x is Element of X ( ) : P [ x ] } , L ( ) & Q [ y , x ] } , L ( ) ;
assume that A c= Z and Z = dom f and f = ( ( - 1 ) (#) ( ( id Z ) ^ ) ) (#) ( ( id Z ) ^ ) and for x st x in Z holds f . x = 1 / x and f . x = - x and f . x > 0 ;
( f /. i ) `2 = ( ( GoB f ) * ( 1 , j2 + 1 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 + 1 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 + 1 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 + 1 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 + 1 ) ) `2 ;
dom ( Shift ( q2 , len q1 ) ) = { j + len ( Seq q1 ) + len ( Seq q2 ) + len ( Seq q2 ) + len ( Seq q2 ) + len ( Seq q2 ) + len ( Seq q2 ) + len ( q1 ) + len ( q1 ) + len ( q1 ) ;
consider G1 , G2 , H being Element of V such that G1 <= G2 and G2 <= H and f = G1 * ( G1 , G2 ) and g = G2 * ( G1 , G2 ) and f = G1 * ( G1 , G2 ) and g = G2 * ( G1 , G2 ) ;
func - f -> PartFunc of C , V means : Def3 : dom it = dom f & for c st c in dom it holds it /. c = - f /. c & for c st c in dom it holds it /. c = - f /. c ;
consider phi such that phi is increasing and for a st phi . a = a and for v st v <> {} holds not ( for a st a in dom L holds L . ( a , v ) = a ) & ( for a st a in dom L holds L . ( a , v ) = a ) ;
consider i1 , j1 such that [ i1 , j1 ] in Indices GoB f and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 + 1 ) and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 + 1 ) ;
consider i , n such that n <> 0 and sqrt ( p ) = sqrt ( i ^2 ) and for n1 being Integer st n1 <> 0 & n2 <> 0 & n < 0 holds sqrt ( ( i + 1 ) ^2 ) = sqrt ( ( i + 1 ) ^2 ) ;
assume that not 0 in Z and Z c= dom ( ( - 1 / 2 ) (#) ( ( - 1 / 2 ) (#) ( ( - 1 ) (#) ( ( - 1 ) / 2 ) (#) ( ( - 1 ) / 2 ) ) ) and for x st x in Z holds ( ( - 1 / 2 ) (#) ( ( - 1 ) / 2 ) (#) ( ( - 1 ) / 2 ) (#) ( ( - 1 ) / 2 ) ) ) ;
cell ( G1 , i1 -' 1 , i2 -' 1 ) \ LSeg ( G1 * ( i1 -' 1 , i2 -' 1 ) , G * ( i1 -' 1 , j1 -' 1 ) ) c= BDD L~ f \/ L~ Cage ( C , n + 1 ) \ L~ f ;
ex Q1 being open Subset of X st s = Q1 & ex Q1 being Subset of Y st Q1 c= F & ( for a being Real st a in Q1 holds ( a in Q1 ) & ( a in Q1 ) implies a in Q1 ) & ( a in Q1 implies a in Q1 ) & ( a in Q1 ) implies a in Q1 ) & b in Q1 ) ;
gcd ( A , ( the carrier of gcd ( A , B ) ) , 1 ) = 1 / ( ( the carrier of A ) * ( 1 , 1 ) ) .= 1 / ( ( the carrier of A ) * ( 1 , 1 ) ) ;
R8 = ( the for of ( the Sorts of A2 ) . ( ( the Sorts of A2 ) . ( ( the Sorts of A1 ) . ( ( the Sorts of A2 ) . ( ( the Sorts of A2 ) . ( ( the Sorts of A2 ) . ( ( the Sorts of A2 ) . ( ( the Sorts of A2 ) . ( ( the Sorts of A2 ) . ( m + 1 ) ) + 1 ) ) ) ) ) .= [ 3 , 4 ] ;
CurInstr ( P-6 , Comput ( P-6 , m1 + m2 ) ) = CurInstr ( P-6 , Comput ( P-6 , m1 + m2 ) ) .= CurInstr ( P-6 , Comput ( P-6 , m1 + m2 ) ) .= CurInstr ( P-6 , Comput ( P-6 , m1 + m2 ) ) .= halt SCMPDS .= halt SCMPDS ;
P1 /\ P2 = ( { p1 } \/ LSeg ( p1 , p2 ) ) /\ ( LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) ) .= ( LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) ) \/ ( LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) ) ;
func not ( ex f being Subset of the Sorts of A2 st a in the Sorts of A2 & not a in dom f & ex p st p in dom f & a = f . p ) & ( for i st i in dom f holds a . i = F ( i ) ) ;
for a , b being Element of F_Complex st |. a .| > |. b .| & for f being Polynomial of F_Complex st f >= 1 & f >= 0 holds f * ( - b ) is >= >= 0 implies f * ( - b ) is >= 0
defpred P [ Nat ] means ( for i , j st [ i , j ] in Indices G & [ i , j ] in Indices G & [ i , j ] in Indices G & [ j , i ] in Indices G & [ i , j ] in Indices G & [ j , j ] in Indices G ) & [ i , j ] in Indices G & [ j , j ] in Indices G & [ j , i ] in Indices G & [ j , i ] in Indices G & [ j , i ] in Indices G & [ j , i ] in Indices G & [ j , i ] in Indices G & [ j , i ] in Indices G & [ j , i ] in Indices G & [ j , i ] in Indices G & [ j , i ] in Indices G & [ j
assume that C1 , C2 , f , g , h , i , n , g being Element of C1 , f being Function of C2 , C2 , g being Function of C2 , C2 such that for x being Element of C1 , y being Element of C2 holds x = y iff f . x = g . y and g is stable and f is stable and g is stable ;
( ||. f .|| | X ) . c = ||. f /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .=
|. q .| ^2 = ( ( q `1 ) ^2 + ( q `2 ) ^2 ) ^2 + ( q `2 ) ^2 + ( q `1 ) ^2 + ( q `2 ) ^2 + ( q `1 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 ;
for F being Subset-Family of T7 st F is open & not {} in F & not {} in F & for A , B being Subset of T7 st A in F & B in F & A <> B & A is connected & B is connected & A is connected & B is connected holds card ( A \/ B ) c= card ( A \/ 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 F . k = G . ( k , i ) and for i st i in dom F holds F . i = G . ( k , i ) ;
i |^ ( ( \mathop { \rm mod } n ) - i ) |^ s = i |^ ( s + k ) - i .= i |^ ( ( k + 1 ) - i ) * ( ( k + 1 ) - i ) .= i |^ ( ( k + 1 ) - i ) * ( ( k + 1 ) - i ) ;
consider q being oriented Chain , v being oriented Chain of G such that r = q and q <> {} and ( for j st j in dom ( F . j ) holds ( F . j ) . ( len F + 1 ) = v1 ) and ( F . ( len F + 1 ) = ( F . ( j + 1 ) ) . ( len F + 1 ) ) ;
defpred P [ Element of NAT ] means $1 <= len ( g , Z ) implies ( g , Z ) . $1 = ( g , Z ) . ( len g + $1 ) & ( g , Z ) . ( len g + $1 ) = ( g , Z ) . ( len g + $1 ) ;
for A , B being Matrix of n , REAL for x , y being Element of n -tuples_on REAL holds len ( A * B ) = len x & width ( A * B ) = width ( A * B ) & width ( A * B ) = width ( A * B )
consider s being FinSequence of the carrier of R such that Sum s = u 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 in J & b in K & s . i = b * a ;
func |( ( x , y ) , ( x , y ) )| -> Element of COMPLEX equals |( ( ( Re x ) , ( Re y ) , ( Im y ) , ( Im y ) ) , ( Im y ) + ( Im y ) + ( Im y ) + ( Im y ) , ( Im y ) + ( Im y ) + ( Im y ) + ( Im y ) + ( Im y ) ) ;
consider g2 being FinSequence of FF such that g2 is continuous and rng g2 c= A and for x st x in A & x in A & g2 . 1 = F . x holds g2 . ( len g2 ) = F . x and g2 . ( len g2 ) = F . x and g2 . ( len g2 ) = F . x ;
then n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n2 >= len p1 & n3 >= len p1 & n3 >= len p1 & n2 >= len p1 & n3 >= len p1 & n3 >= len p1 & n2 >= len p1 & n3 >= len p1 & n3 >= len p1 & n3 >= len p1 & n3 >= len p1 & n3 >= len p1 & n2 >= len p1 + n2 >= len p1 + n2 + 1 ;
( q `1 ) * a <= ( q `1 ) * a & ( - q `1 ) * a <= ( q `2 ) * a & ( - q `1 ) * a <= ( q `1 ) * a & ( - q `1 ) * a <= ( q `2 ) * a & ( - q `1 ) * a <= ( q `1 ) * a & ( - q `1 ) * a <= ( q `1 ) * a & ( q `1 ) * a <= ( q `1 ) * a & ( - q `1 ) * a & ( - q `1 ) * a & ( - q `1 ) * a & ( - q `1 ) * a & ( - q `1 ) * a & ( - q `1 ) * a <= ( - q `1 ) * a & ( - q `1 <= ( - q `1 ) * a & ( - q `1 <= ( - q `2 ) * a &
( ( F . ( p . ( len p ) ) ) `1 ) `1 = ( ( F . ( len p ) ) `1 ) `1 .= ( ( F . ( len p ) ) `1 ) `1 .= ( ( F . ( len p ) ) `2 ) `1 .= ( ( F . ( len p ) ) `2 ) `1 .= ( ( F . ( len p ) ) `2 ) `2 .= ( ( F . ( len p ) ) `2 ) `2 ;
consider k1 being Nat such that k1 + 1 = 1 and a := k1 = ( ( a := intloc 0 ) .--> ( ( a := intloc 0 ) .--> 1 ) ) ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ ( a := intloc 0 ) ;
consider B8 being Subset of B1 , y8 being Function of B1 , B2 such that B8 : for x being Element of B1 , y being Element of B2 , a being Element of B1 , b being Element of B2 st a in B1 & b in B2 & a = b holds [ x , y ] in \mathop { a , b , c , d } ;
v2 . b2 = ( ( curry F2 ) * ( ( ( -> Function of F2 , F2 ) . b2 ) ) * ( ( ( the Function of F2 , F2 ) . b2 ) ) . b2 ) .= ( ( ( the Function of F2 , F2 ) . b2 ) * ( ( the Function of F2 , F2 ) . b2 ) ) . b2 .= ( ( the Y of F2 ) . ( b2 , b2 ) ) . b2 .= ( ( the Y of F2 ) . ( ( the Function of F2 ) . ( b2 , b2 ) ) . b2 ) . b2 ) . b2 .= ( ( ( the Function of F2 ) . ( ( F2 ) . b2 ) . b2 ) . b2 .= ( ( the Function of F2 ) . b2 ) . b2 .= ( ( ( ( the Function of F2 ) . b2 ) . b2 ) . b2 ) . b2 ) . b2 ) . b2 )
dom IExec ( I , P , Initialize s ) = the carrier of SCMPDS .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( 0 , SCMPDS ) ) .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( 0 , SCMPDS ) ) .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( 0 , SCMPDS ) ) .= dom ( IExec ( I , P , s ) +* Start-At ( 0 , SCMPDS ) ) .= dom ( ( card I ) ;
ex d-32 being Real st d-32 > 0 & for h be Real st h > 0 & |. h .| < d holds |. ( R * ( h + c ) ) - ( R * ( h + c ) ) .| < e ;
LSeg ( G * ( len G , 1 ) + |[ - 1 , 1 ]| , G * ( len G , 1 ) + |[ - 1 , 1 ]| ) c= Int cell ( G , len G , width G ) \/ { |[ - 1 , 1 ]| } ;
LSeg ( mid ( h , i1 , i2 ) , i ) = LSeg ( h /. ( i + 1 + 1 ) , h /. ( i + 1 + 1 + 1 ) ) .= LSeg ( h /. ( i + 1 + 1 + 1 + 1 + 1 ) , h /. ( i + 1 + 1 + 1 + 1 ) ) .= LSeg ( h /. ( i + 1 + 1 + 1 + 1 + 1 ) , h /. ( i + 1 + 1 + 1 + 1 ) ) .= LSeg ( h /. ( i + 1 + 1 + 1 + 1 + 1 ) ;
A = { q where q is Point of TOP-REAL 2 : LE q , p , P & LE q , p , P & LE q , p , P & LE p , q , P & LE q , p , P & LE p , q , P & LE p , q , P & LE q , p , P & LE p , q , P & LE q , p , P & LE q , p , P & LE p , q , P & LE q , p , P & LE p , q , P & LE p , q , P & LE p , q , P & LE p , q , P & LE q , q , P & LE p , q , P & LE p , q , P & LE q , q , P & LE q , q , P & LE p , q , P & LE p , q , P & LE p , q , P & LE p , q , P & LE p ,
( ( - x ) .|. y ) = - ( ( - 1 ) .|. y ) * ( ( - 1 ) .|. y ) .= ( - 1 ) * ( ( - 1 ) .|. y ) .= ( - 1 ) * ( - 1 ) * ( - 1 ) .= ( - 1 ) * ( - 1 ) * ( - 1 ) .= ( - 1 ) * ( - 1 ) .= ( - 1 ) * ( - 1 ) * ( - 1 ) .= ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) .= ( - 1 ) * ( - 1 ) .= ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) *
0 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) = ( ( - 1 ) * sqrt ( 1 + ( p `2 / p `1 ) ^2 ) ) * sqrt ( 1 + ( p `2 / p `1 ) ^2 ) ;
( U U - ( - 1 ) * ( ( W - 1 ) * ( W - 1 ) ) ) = ( ( U - 1 ) * ( W - 1 ) ) * ( W - 1 ) .= ( ( U - 1 ) * ( W - 1 ) ) * ( W - 1 ) .= ( ( U - 1 ) * ( W - 1 ) ) * ( W - 1 ) .= ( ( U - 1 ) * ( W - 1 ) ) * ( W - 1 ) ;
func Shift ( f , h ) -> PartFunc of REAL , REAL means : Def3 : dom it = REAL & for x be Element of REAL , y be Element of REAL st x in dom it holds it . x = - ( f . x ) & for x be Element of REAL st x in dom it holds it . x = - ( f . x ) * ( - h . x ) ;
assume that 1 <= k and k + 1 <= len f and [ i , j ] in Indices G and [ i + 1 , j ] in Indices G and f /. k = G * ( i , j ) and f /. k = G * ( i + 1 , j ) and f /. k = G * ( i , j ) and f /. k = G * ( i , j ) ;
( not y in Free H implies x in Free H & not x in Free H & not ( x in Free H implies x in Free H & not x in Free H ) & not x in Free ( H , ( x , y ) ) & not x in Free ( H , ( x , y ) ) ) ;
defpred P11 [ Element of NAT , Element of NAT ] means ( for p being prime Element of NAT st p = $1 & p is prime & p is prime & $1 <= $1 & $2 <= $1 & $2 <= k implies $2 = ( $1 + 1 ) |^ ( $1 + 1 ) ) |^ ( $1 + 1 ) ;
func \sigma ( C ) -> non empty Subset-Family of X means : Def3 : for A , B being non empty Subset-Family of X st A in it & B in it holds for A being Subset of X st A in it holds A in it iff for W being Subset of X st W in it holds A in W & A is open & B c= W ;
[#] ( ( dist ( ( ( ( ( ( ( O ) ) | Q ) | Q ) ) | Q ) ) ) .: Q ) = ( ( ( ( the carrier of ( ( ( O ) | R ) | Q ) | Q ) ) | Q ) ) .: Q ) .= ( ( the carrier of ( ( ( O ) | Q ) | Q ) | Q ) ) .: Q ;
rng ( F | [: S , T :] ) = {} or rng ( F | [: S , T :] ) = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 7 , 7 , 8 , 8 , 7 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 ,
( f " ( rng f ) ) . i = f . i .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) . i .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) . i .= ( f " ( rng f ) . i .= ( f " ( rng f ) . i ) . i .= ( f " ( rng f ) . i .= ( f " ( rng f ) . i .= ( f " ( rng f ) . i .= ( f " ( rng f ) . i .= ( f " ( rng f ) . i .= ( f " ( rng f ) . i ) .= ( f " ( rng f ) .
consider P1 , P2 being non empty Subset of TOP-REAL 2 such that P1 /\ P2 = { p1 , p2 } and P1 is closed and C = { p1 , p2 , p3 } and C = { p1 , p2 , p3 } and C = { p1 , p2 , p3 } and C = { p2 , p3 , p4 } and D = { p1 , p2 , p3 , p4 } and D = C \/ { p1 , p2 , p3 , p4 } and D = { p1 , p2 , p3 , p4 } and D = { p2 , p3 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p1 , p2 , p4 , p4 , p4 , p4 , p1 , p3 , p3 , p4 , p1 , p2 , p1 , p2 , p3 , p3 , p2 , p3 , p3 , p3 , p3 , p3 , p3 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p2 , p3
f . p2 = |[ ( p2 `1 ) ^2 + ( p2 `2 ) ^2 , ( p2 `2 ) ^2 + ( p2 `1 ) ^2 + ( p2 `2 ) ^2 ]| .= |[ ( p2 `1 ) ^2 + ( p2 `2 ) ^2 , ( p2 `2 ) ^2 + ( p2 `2 ) ^2 ]| ;
( ( AffineMap ( a , X ) ) " . x = ( ( AffineMap ( a , X ) ) . x ) " .= ( ( AffineMap ( a , X ) ) . x ) " .= ( ( AffineMap ( a , X ) ) . x ) " .= ( ( AffineMap ( a , X ) ) . x ) " .= ( ( AffineMap ( a , X ) ) . x ) " .= ( ( AffineMap ( a , X ) ) . x ) " .= ( ( - a ) . x ) " .= ( - a ) . x ) " .= ( - a ) . x ) * ( - a ) . x .= ( - a ) * ( - a ) * ( - a ) . x .= ( - a ) * ( - a ) * ( - a ) * ( - a ) * ( ( - a ) * ( - a ) * ( ( - a ) * ( - a ) * ( - a ) * ( - a ) * ( - a ) * ( - a ) * ( - a ) * ( - a ) * ( - a ) * ( - a ) * ( - a
for T being non empty normal TopSpace , A , B being closed Subset of T , A being Subset of T st A <> {} & A misses B & A misses B holds ( in Cl A ) \ { p } = ( Cl ( A \/ B ) ) \ { p } & ( for r being Real st r in Cl ( B \/ C ) holds p in Cl ( B \/ C ) ) implies p in Cl ( ( Cl ( B \/ C ) )
for i , j being strict Subgroup of G1 for G1 , G2 being strict Subgroup of G2 st G1 = F . i & G2 = F . i & G2 = G . j & G1 = G . j holds G1 = G2
for x st x in Z holds ( ( ( - 1 / 2 ) (#) ( ( #Z 2 ) * ( arctan + arccot ) ) ) `| Z ) . x = ( ( - 1 / 2 ) (#) ( ( #Z 2 ) * ( arctan + arccot ) ) `| Z ) . x
synonym f is right continuous & for for for x0 in dom f & for a st a in dom f & x0 in dom f & for x st x in dom f & x in Z holds f . x = a & for x st x in Z holds f . x = b & f . x > - b & f . x < - b ;
then ( X1 , X2 X2 X2 X2 ) meets ( X1 union X2 ) & ( X1 , X2 X2 X2 X2 ) meet ( X2 , X1 meet X2 ) = X1 union X2 & ( X1 , X2 meet X2 ) meet X2 = X2 meet X2 & ( X1 , X2 meet X2 ) meet X2 = X2 meet X2 implies X1 , X2 meet X2 , X1 union X2 , X2 meet X2 , X2 meet X2 , X2 meet X2 , X2 meet X2 , X2 meet X1 , X2 union X2 union X2 , X2 union X2 union X2 union X2 union X2 union X2 , X2 union X2 union X2 , X2 union X2 union X2 union X2 , X1 union X2 union X2 union X2 union X2 , X2 union X2 union X2 , X2 union X2 union X2 union X2 union X2 , X2 union X2 , X2 union X2 , X2 union X2 , X2 union X2 , X2 union X2 , X2 union X2 , X2 union X2 , X2 union X2 , X2 union X2 , X2 union X2 , X2 union X2 , X2 union X2 , X2 union X2 , X2 union X2 union X2 , X2 union X2 , X2 union X2 union X2 , X2 union X2 union X2 union X2 , X2 union X2 union
ex N being Neighbourhood of x0 st N c= dom SVF1 ( 1 , f , u ) & ex L st for x st x in N holds ( SVF1 ( 1 , f , u ) ) . x - ( SVF1 ( 1 , f , u ) ) . x0 = L . ( x - x0 ) + R . ( x - x0 ) ;
( ( - ( p2 `1 / p2 `2 ) ) * sqrt ( 1 + ( p2 `1 / p2 `2 ) ^2 ) ) / sqrt ( 1 + ( p2 `1 / p2 `2 ) ^2 ) >= ( - ( p2 `1 / p2 `2 ) * sqrt ( 1 + ( p2 `1 / p2 `2 ) ^2 ) ;
( ( 1 - t ) * ||. f1 .|| ) |^ ( m + 1 ) = ( 1 - t ) |^ ( m + 1 ) * ( ( 1 - t ) |^ m ) & ( ( 1 - t ) |^ ( m + 1 ) ) * ( ( 1 - t ) |^ m ) = ( 1 - t ) |^ ( m + 1 ) * ( ( 1 - t ) |^ m ) ;
assume that for x holds f . x = ( ( - 1 / ( 2 * x ) ) * ( sin . x ) ) and for x st x in Z holds f . x = - 1 / ( 2 * x ) and for x st x in Z holds f . x = - 1 / ( 2 * x ) and f . x = - 1 / ( 2 * x ) ;
consider Xi1 being Subset of Y , Y1 being open Subset of X such that Y1 is open and Y1 is open and ex Y1 being Subset of X st Y1 = Y1 & Y1 is open & ex Y1 being Subset of Y st Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open ;
card ( S . n ) = card { [ d , ( a |^ 3 ) * b ] , [ d , ( a |^ 3 ) * b ] , [ d , ( a |^ 3 ) * b |^ 3 ] } .= 1 + ( d + b ) * b .= ( d + b ) * c .= d + b ;
( ( W-bound D ) - ( W-bound D ) ) / ( 2 |^ m ) = ( W-bound D ) - ( W-bound D ) / ( 2 |^ m ) .= ( W-bound D ) / ( 2 |^ m ) .= ( W-bound D ) / ( 2 |^ m ) .= ( W-bound D ) / ( 2 |^ m ) ;