thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b
D c= S
let Y ;
S ` is convergent
q in X ;
V in X ;
y in N ;
x in T ;
m < n ;
m <= n ;
n > 1 ;
let r ;
t in I ;
n <= 4 ;
M is finite ;
let X ;
Y c= Z ;
A // M ;
let U ;
a in D ;
q in Y ;
let x ;
1 <= l ;
1 <= w ;
let G ;
y in N ;
f = {} ;
let x ;
x in Z ;
let x ;
F is one-to-one ;
e <> b ;
1 <= n ;
f is special ;
S misses C
t <= 1 ;
y divides m ;
P divides M ;
let Z ;
let x ;
y c= x ;
let X ;
let C ;
x _|_ p ;
o is monotone ;
let X ;
A = B ;
1 < i ;
let x ;
let u ;
k <> 0 ;
let p ;
0 < r ;
let n ;
let y ;
f is onto ;
x < 1 ;
G c= F ;
a is_>=_than X ;
T is continuous ;
d <= a ;
p <= r ;
t < s ;
p <= t ;
t < s ;
let r ;
D is_<=_than E ;
assume e > 0 ;
assume 0 < g ;
p in X ;
x in X ;
Y ` in Y ;
assume 0 < g ;
not c in Y ;
not v in L ;
2 in z `2 ;
assume f = g ;
N c= b ` ;
assume i < k ;
assume u = v ;
I = J ;
B ` = b ` ;
assume e in F ;
assume p > 0 ;
assume x in D ;
i be element ;
assume F is onto ;
assume n <> 0 ;
x be element ;
set k = z ;
assume o = x ;
assume b < a ;
assume x in A ;
a `1 <= b `1 ;
assume b in X ;
assume k <> 1 ;
f = Product l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is from of squares ;
assume m > 0 ;
assume A c= B ;
X is lower
assume A <> {} ;
assume X <> {} ;
assume F <> {} ;
assume G is open ;
assume f is dilatation ;
assume y in W ;
y \not <= x ;
A ` in B ` ;
assume i = 1 ;
let x be element ;
x `1 = x `1 ;
let X be BCK-algebra ;
assume S is non empty ;
a in REAL ;
let p be set ;
let A be set ;
let G be _Graph , W be Walk of G , n be Nat ;
let G be _Graph , W be Walk of G , n be Nat ;
let a be Complex ;
let x be element ;
let x be element ;
let C be FormalContext , a be Element of C ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
y be Real ;
X c= f . a
let y be element ;
let x be element ;
i be Nat ;
let x be element ;
n in NAT ;
let a be element ;
m in NAT ;
let u be element ;
i in NAT ;
let g be Function ;
Z c= NAT ;
l <= ma ;
let y be element ;
r2 in dom f ;
let x be element ;
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 = / ( 1 + 1 ) ;
n >= 0 + 1 ;
k c= k + 1 ;
R1 c= R ;
k + 1 >= k ;
k c= k + 1 ;
let j be Nat ;
o , a // Y ;
R c= Cl G ;
Cl B = B ;
let j be Nat ;
1 <= j + 1 ;
arccot is_differentiable_on Z ;
exp_R is_differentiable_in x ;
j < i0 ;
let j be Nat ;
n <= n + 1 ;
k = i + m ;
assume C meets S ;
n <= n + 1 ;
let n be Nat ;
h1 = {} ;
0 + 1 = 1 ;
o <> b3 ;
f2 is one-to-one ;
support p = {}
assume x in Z ;
i <= i + 1 ;
r1 <= 1 ;
let n be Nat ;
a "/\" b <= a ;
let n be Nat ;
0 <= r0 ;
let e be Real , x be Point of TOP-REAL 2 ;
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 Program of SCM+FSA ;
let l be Nat ;
let i be Nat ;
let r ;
1 <= i2 ;
a "\/" c = c ;
let r be Real ;
let i be Nat ;
let m be Nat ;
x = p2 ;
let i be Nat ;
y < r + 1 ;
rng c c= E
Cl R is boundary ;
let i be Nat ;
R2 in X ;
cluster uparrow x -> closed ;
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 ;
L~ \neq G2 + s ;
G . y <> 0 ;
let X be RealNormSpace , A be Subset of X ;
a in X ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , M be Matrix of V , REAL ;
assume x in - - M ;
k < s . a ;
not t in { p } ;
let Y be set , f be Function of Y , BOOLEAN ;
M , L are_equipotent ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded ;
rng f = Y ;
( G . n ) 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 `1 = a * y `2 ;
rng D c= A ;
assume x in K1 ;
1 <= i-32 ;
1 <= i-32 ;
pp c= cos . x ;
1 <= i-15 ;
1 <= i-15 ;
LMP C in L ;
1 in dom f ;
let seq , n ;
set C = a * B ;
x in rng f ;
assume f is_continuous_on X ;
I = dom A ;
u in dom p ;
assume a < x + 1 ;
s-7 is bounded ;
assume I c= P1 ;
n in dom I ;
let Q ;
B c= dom f ;
b + p _|_ a ;
x in dom g ;
F-14 is continuous ;
dom g = X ;
len q = m ;
assume A2 : x in A2 ;
cluster R \ S -> real-valued ;
sup D in S ;
x << sup D ;
b1 >= Z1 + 1 ;
assume w = 0. V ;
assume x in A . i ;
g in the carrier of X ;
y in dom t ;
i in dom g ;
assume P [ k ] ;
z c= C implies C c= f
x4 is increasing ;
let e2 be element ;
- b divides b ;
F c= \tau ( F ) ;
( G . n ) is non-decreasing ;
( G . n ) is non-decreasing ;
assume v in H . m ;
assume b in [#] B ;
let S be non void ManySortedSign , f be Function ;
assume P [ n ] ;
assume union S is finite independent finite ;
V is Subspace of V ;
assume P [ k ] ;
rng f c= NAT ;
assume ex_inf_of X , L ;
y in rng f ;
let s , I be set , A be ManySortedSet of I ;
b ` c= b9 ` ;
assume not x in NAT ;
A /\ B = { a } ;
assume len f > 0 ;
assume x in dom f ;
b , a // o , c ;
B in B-24 ;
cluster product p -> non empty ;
z , x // x , p ;
assume x in rng N ;
cosec is_differentiable_in x & cosec is_differentiable_in x ;
assume y in rng S ;
let x , y be element ;
i2 < i1 & i1 <= len f ;
a * h in a * H ;
p , q in Y ;
redefine func sqrt I -> Ideal of L ;
q1 in A1 & q2 in A1 ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A2 c= A1 ;
an1 < n & an1 <= 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 , P ;
d , c // a , b ;
let t , u be set ;
let X be set ;
assume k in dom s ;
let r be non negative Real ;
assume x in F | M ;
let Y be Subset of S ;
let X be non empty TopSpace , A be Subset of X ;
[ a , b ] in R ;
x + w < y + w ;
{ a , b } is_>=_than c ;
let B be Subset of A , C be Subset of A ;
let S be non empty ManySortedSign ;
let x be variable of f , A , B be Subset of f ;
let b be Element of X , x be Element of X ;
R [ x , y ] ;
x ` ` = x ` ;
b \ x = 0. X ;
<* d *> in D * ;
P [ k + 1 ] ;
m in dom mnn ;
h2 . a = y ;
P [ n + 1 ] ;
redefine func G * F -> ] ] ;
let R be non empty multMagma , x be Element of R ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p `2 ;
assume f | X is lower ;
x in rng co /\ rng co ;
let x be Element of B ;
let t be Element of D ;
assume x in Q .vertices() ;
set q = s ^\ k ;
let t be VECTOR of X ;
let x be Element of A ;
assume y in rng p `1 ;
let M be non void maid id id ;
let N be non empty being non empty being non empty \lbrack M , N ;
let R be RelStr with finite finite : K is finite ;
let n , k be Nat ;
let P , Q be let let p be be be be be be be be 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 < v8 ;
x <= c2 . x ;
x in F ` & y in F ` ;
redefine func S --> T -> :] ;
assume t1 <= t2 & t2 <= t2 ;
let i , j be even Integer ;
assume F1 <> F2 & F2 <> {} ;
c in Intersect ( union R ) ;
dom p1 = c & dom p2 = c ;
a = 0 or a = 1 ;
assume A1 <> A6 & A2 <> A7 ;
set i1 = i + 1 ;
assume a1 = b1 & a2 = b2 ;
dom g1 = A & rng g2 c= A ;
i < len M + 1 ;
assume not -infty in rng G ;
N c= dom ( f1 + f2 ) ;
x in dom ( sec | A ) ;
assume [ x , y ] in R ;
set d = ( x - y ) / ( x - y ) ;
1 <= len g1 & 1 <= len g2 ;
len ( s2 ) > 1 ;
z in dom ( f1 (#) f2 ) ;
1 in dom ( D2 | Seg 1 ) ;
p `2 = 0 & p `2 = 0 ;
j2 <= width G & j2 <= width G ;
len PI > 1 + 1 ;
set n1 = n + 1 ;
|. q-35 .| = 1 ;
let s be SortSymbol of S ;
if i = i holds not ( i = j )
X1 c= dom f & X1 c= dom g ;
h . x in h . a ;
let G be non empty \setminus \it Z be non empty structure ;
cluster m * n -> square ;
let kk be Nat , x be Element of X ;
i - 1 > m - 1 ;
R is transitive & R is transitive implies R is transitive
set F = <* u , w *> ;
p-2 c= P3 & p`2 c= P3 ;
I is_halting_on t , Q & I is_halting_on t , Q ;
assume [ S , x ] is thesis ;
i <= len ( f2 ^ g2 ) ;
p is FinSequence of X & q is FinSequence of X ;
1 + 1 in dom g ;
Sum R2 = n * r ;
cluster f . x -> complex-valued ;
x in dom ( f1 + f2 ) ;
assume [ X , p ] in C ;
BX c= XX & BX c= XY ;
n2 <= ( 2 * n ) - 1 ;
A /\ cP c= A `
cluster x -valued -> constant for Function ;
let Q be Subset-Family of S , P be Subset of S ;
assume n in dom g2 ;
let a be Element of R ;
t `2 in dom e2 & t `2 in dom e2 ;
N . 1 in rng N ;
- z in A \/ B ;
let S be SigmaField of X , M be Element of S ;
i . y in rng i ;
REAL c= dom f & f | A is bounded ;
f . x in rng f ;
mt <= ( r / 2 ) ;
s2 in r-5 ( n ) ;
let z , z be complex number ;
n <= Nseq . m ;
LIN q , p , s ;
f . x = waybelow x /\ B ;
set L = |[ S \to T ]| ;
let x be non positive ExtReal ;
let m be Element of M ;
f in union rng ( F1 | n ) ;
let K be add-associative right_zeroed right_complementable non empty doubleLoopStr , p be Polynomial of K ;
let i be Element of NAT ;
rng ( F * g ) c= Y
dom f c= dom x & f . x = y ;
n1 < n1 + 1 & n1 <= len f ;
n1 < n1 + 1 & n1 <= len f ;
cluster [: T , X :] -> \overline W ;
[ y2 , 2 ] = z ;
let m be Element of NAT ;
let S be Subset of R ;
y in rng S[: S , T :] ;
b = sup dom f & b = sup dom f ;
x in Seg ( len q ) ;
reconsider X = D ( ) as set ;
[ a , c ] in E1 ;
assume n in dom ( h2 * f ) ;
w + 1 = ma ;
j + 1 <= j + 1 + 1 ;
k2 + 1 <= k1 + 1 ;
i be Element of NAT ;
Support u = Support p \/ Support q ;
assume X is complete \frac of m ;
assume that f = g and p = q ;
n1 <= n1 + 1 & n2 <= n1 + 1 ;
let x be Element of REAL ;
assume x in rng ( s2 | X ) ;
x0 < x0 + 1 / 2 ;
len ( Carrier ( L ) ) = W ;
P c= Seg ( len A ) ;
dom q = Seg n & rng q c= Seg n ;
j <= width M *' ;
let seq1 be real-valued subsequence of seq ;
let k be Element of NAT ;
Integral ( M , P ) < +infty ;
let n be Element of NAT ;
assume z in being being being being being being being being being being being being being being being being being being being being being being being being being being being Element of being Element of A
i be set ;
n - 1 = n-1 - 1 ;
len ( n |-> 0 ) = n ;
\mathop { Z } c= F
assume x in X or x = X ;
x is midpoint of b , c ;
let A , B be non empty set , f be Function of A , B ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q & i in dom q ;
let s be Element of E * ;
let B1 be Basis of x , B2 be Basis of y ;
L3 /\ L2 = {} ;
L1 /\ LSeg ( p1 , p2 ) = {} ;
assume downarrow x = downarrow y ;
assume b , c // b , c ;
LIN q , c , c ;
x in rng ( f-1s | n ) ;
set nn8 = n + j ;
let D7 be non empty set , f be Function of D7 , D ;
let K be right_zeroed non empty addLoopStr , p be Polynomial of K ;
assume f opp = f & h opp = h ;
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 are_maximal in C ;
let a , b be Element of S ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = Integral ( M , f ) ;
cluster n[ ] -> nesst for FinSequence ;
not u in { ag } ;
the carrier of f c= B ;
reconsider z = x as Vector of V ;
cluster the carrier of L -> -> -> trivial ;
r (#) H is C [ 0 ] ;
s . intloc 0 = 1 ;
assume that x in C and y in C ;
let U0 be strict universal MSAlgebra over S , A be Subset of U0 ;
[ x , Bottom T ] is compact ;
i + 1 + k in dom p ;
F . i is stable Subset of M ;
r-35 in : x in : rp in { x } ;
let x , y be Element of X ;
let A , I be contradiction of X ;
[ y , z ] in [: O , O :] ;
( of Macro i ) . x = 1 ;
rng Sgm ( A ) = A ;
q |- \! \! \setminus 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 ;
( for k being Nat holds D `2 = {} ) & ( not D is non empty ) ;
n + 1 + 1 <= len g ;
a in [: [: A , A :] , D :] ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
let I be Ideal of L ;
set g = f1 + f2 , h = f1 + f2 ;
a <= max ( a , b ) ;
i-1 < len G + 1-1 ;
g . 1 = f . i1 ;
x `1 , y `2 in A2 ;
( f /* s ) . k < r ;
set v = VAL g ;
i - k + 1 <= S ;
cluster associative non empty for multMagma ;
x in support ( support ( t ) ) ;
assume a in [: the carrier of G , the carrier of G :] ;
i `1 <= len ( y-5 ) ;
assume p divides b1 + b2 ;
> upper_bound M1 & upper_bound M1 <= upper_bound M2 implies M1 <= M2
assume x in W-min ( X ) & y in W-min ( X ) ;
j in dom ( z | i ) ;
let x be Element of D ( ) ;
IC s4 = l1 & IC s4 = l1 ;
a = {} or a = { x } ;
set uG = Vertices G , uH = Vertices G , uH = Vertices G , uH = Vertices H , uH = Vertices G , uH = Vertices H ;
seq " is non-zero & seq " is non-zero implies seq " is non-zero
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
hcn c= h-14 & hD c= h-14 ;
]. a , b .[ c= Z ;
X1 , X2 are_separated implies X1 , X2 are_separated
a in Cl ( union F \ G ) ;
set x1 = [ 0 , 0 ] ;
k + 1 - 1 = k - 1 ;
cluster -> real-valued for Relation ;
ex v st C = v + W ;
let IT be non empty addLoopStr , x be Element of IT ;
assume V is Abelian add-associative right_zeroed right_complementable ;
X-21 \/ Y in \sigma ( L ) ;
reconsider x = x as Element of S ;
max ( a , b ) = a ;
sup B is upper Subset of B ;
let L be non empty reflexive antisymmetric RelStr , X be Subset of L ;
R is reflexive & R is transitive ;
E , g |= the_left_argument_of H implies E , g |= the_right_argument_of H
dom G /. y = a ;
( 1 / 4 ) * ( 1 / 4 ) >= - r ;
G . p0 in rng G & G . p0 in rng G ;
let x be Element of [: F , G :] , f be Function ;
D [ P-6 , 0 ] ;
z in dom id B & id B in dom id B ;
y in the carrier of N & y in the carrier of N ;
g in the carrier of H & g in the carrier of G ;
rng ( fbeing Function of NAT , REAL ) c= REAL ;
j `2 + 1 in dom ( s1 . f ) ;
let A , B be strict Subgroup of G ;
let C be non empty Subset of REAL ;
f . z1 in dom h & h . z2 in dom h ;
P . k1 in rng P & P . k2 in rng P ;
M = AM +* {} .= AM +* {} ;
let p be FinSequence of REAL , n be Nat ;
f . n1 in rng f & f . n2 in rng g ;
M . ( F . 0 ) in REAL ;
degree [. a , b .[ = b-a ;
assume the distance of V , Q is_\mathbb v ;
let a be Element of op ( V ) ;
let s be Element of PL , P be Subset of PL ;
let Pf be non empty thesis RelStr ;
n be Nat ;
the carrier of g c= B & g is one-to-one ;
I = halt SCM R & I = halt SCM R ;
consider b being element such that b in B ;
set BK = BCS K , BK = BCS K ;
l <= ( j - j ) * ( j - 1 ) ;
assume x in downarrow [ s , t ] ;
x `2 in uparrow t & x `2 in uparrow t ;
x in JumpParts ( JumpParts T , JumpParts T ) ;
let h be Morphism of c , a ;
Y c= R implies the_rank_of Y c= R .: Y
A2 \/ A3 c= Carrier ( L1 ) \/ Carrier ( L2 ) ;
assume LIN o , a , b ;
b , c // d1 , e2 ;
x1 , x2 in Y & x2 in Y ;
dom <* y *> = Seg 1 & rng <* y *> c= Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar s .| ;
[ x , x `2 ] in [: X , X :] ;
for n be Nat holds 0 <= x . n
[' a , b '] = [. a , b .] ;
cluster -> -> -> -> closed for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 in P & q2 in P ;
dom ( M1 * M2 ) = Seg n ;
x = [ x1 , x2 ] & y = [ y1 , y2 ] ;
let R , Q be ManySortedSet of A ;
set d = ( 1 / n ) * ( n + 1 ) ;
rng ( g2 ) c= dom W & ( g2 ) . x in dom ( W ) ;
P . ( [#] Sigma \ B ) <> 0 ;
a in field R & a = b ;
let M be non empty Subset of V , v be Element of V ;
let I be Program of SCM+FSA , a be Int-Location ;
assume x in rng ( id R ) & y in rng ( id R ) ;
let b be Element of the carrier of T ;
dist ( e , z ) - r-r > r-r ;
u1 + v1 in W2 & v1 in W1 + W2 ;
assume the carrier of L misses rng G ;
let L be lower-bounded 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 , M be Element of S ;
0 <= 2 * PI ;
o , a9 // o , y & o , b9 // o , y ;
{ v } c= the carrier of l ;
let x be variable of A ;
assume x in dom ( uncurry f ) /\ dom ( uncurry g ) ;
rng F c= ( product f ) *
assume D2 . k in rng D & D2 . k in rng D ;
f " . p1 = 0 & f . p2 = 0 ;
set x = the Element of X , y = the Element of Y ;
dom Ser ( G ) = NAT & rng Ser ( G ) c= NAT ;
n be Element of NAT ;
assume LIN c , a , e1 & LIN c , a , e1 ;
cluster -> 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 @ S ;
reconsider B = b as Element of the carrier of T ;
J , v |= P ! ( P ! l ) ;
redefine func J . i -> non empty TopStruct ;
ex_sup_of Y1 \/ Y2 , T & ex_sup_of Y1 , T ;
W1 // field W1 & R // field W1 implies R + S c= R + S
assume x in the carrier of R & y in the carrier of R ;
dom nM = Seg n & dom nM = Seg n ;
( s . 4 ) misses ( s . 5 ) ;
assume ( a 'imp' b ) . z = TRUE ;
assume that X is open and f = X --> d ;
assume [ a , y ] in Indices ( f | X ) ;
assume that Reloc I c= J and x in dom ( \rm Reloc ( J , K ) ) ;
Im ( lim seq , 0 ) = 0 ;
( sin . x ) <> 0 & ( sin . x ) <> 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 = 1 / x + cos . x / x / x ^2
t3 . n = t3 . n .= s . n ;
dom ( non empty set ) c= dom F ;
W1 . x = W2 . x & W2 . x = W1 . x ;
y in W .vertices() \/ W .vertices() \/ W .vertices() ;
( k + 1 ) <= len ( v | ( k + 1 ) ) ;
x * a \equiv y * a . ( mod m ) ;
proj2 .: S c= proj2 .: P & proj2 .: S c= proj2 .: P ;
h . p4 = g2 . I & h . I = g2 . I ;
( G /. 1 ) `1 = U /. 1 .= G /. 1 ;
f . rm1 in rng f & f . rm2 in rng f ;
i + 1 + 1 - 1 <= len - 1 ;
rng F = rng ( F | ( len F -' 1 ) ) ;
mode \leq of the carrier of G is well unital associative non empty multMagma ;
[ x , y ] in [: A , { a } :] ;
x1 . o in L2 . o & x2 . o in L2 . o ;
the carrier of m c= B & the carrier of m c= B ;
not [ y , x ] in id X ;
1 + p .. f <= i + len f ;
( s ^\ k1 ) is lower & ( s ^\ k1 ) is lower ;
len ( F | ( len F -' 1 ) ) = len I ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be complex number , seq be sequence of X ;
Comput ( P , s , n ) = s ;
k <= k + 1 & k + 1 <= len p ;
reconsider c = {} T as Element of L ;
let Y be Element of be Element of be Element of be Element of be T ;
cluster -> directed-sups-preserving for Function of L , L ;
f . j1 in K . j1 & f . j2 in K . j1 ;
redefine func J => y -> total Function of J , K ;
K c= 2 -tuples_on the carrier of T ;
F . b1 = F . b2 & F . b2 = F . b2 ;
x1 = x or x1 = y or x1 = z ;
attr a <> {} means : Def1 : ( a - a ) / ( a - a ) = 1 ;
assume that ca c= b and b in a ;
s1 . n in rng s1 & s1 . n in rng s1 ;
{ o , b2 } on C2 & { o , b2 } on C2 ;
LIN o , b , b9 & LIN o , b , b9 ;
reconsider m = x as Element of Funcs ( V , C ) ;
let f be non trivial FinSequence of D ;
let FF2 be non empty thesis , F be Function of X , Y ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in F-8 ;
reconsider pp = x , pb2 = y as Subset of m ;
let A , B , C be Element of R ;
redefine func strict non empty for being strict non empty be be strict be 0. R ;
rng c `1 misses rng e`1 & rng c `2 misses rng e`2 ;
z is Element of gr { x } & z is Element of gr { y } ;
not b in dom ( a .--> p1 ) ;
assume that k >= 2 and P [ k ] ;
Z c= dom ( cot * cot ) /\ dom ( cot * cot ) ;
the component of Q c= UBD A & UBD Q c= UBD A ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( 1 / ( f ^ ) ) ;
attr f = u means : Def1 : a * f = a * u ;
for n holds P1 [ \mathop { } n ] ;
{ x . O : x in L } <> {} ;
let x be Element of V . s ;
a , b be Nat ;
assume that S = S2 and p = p2 and q = p1 ;
gcd ( n1 , n2 ) = 1 & gcd ( n1 , n2 ) = 1 ;
set ol = a * ( 0. INT ) ;
seq . n < |. r1 .| & seq . n < |. r1 .| ;
assume that seq is increasing and r < 0 ;
f . ( y1 , x1 ) <= a & f . ( y1 , x1 ) <= b ;
ex c being Nat st P [ c ] ;
set g = { n to_power 1 : n in NAT } ;
k = a or k = b or k = c ;
abeing , ag , ag being set st ag , ag , bg , bg ;
assume that Y = { 1 } and s = <* 1 *> ;
Im1 . x = f . x .= 0. K .= 0. K ;
W3 .first() = W3 . 1 & y = W3 . 2 ;
cluster trivial -> finite for a subgraph of G , and G is finite ;
reconsider u = u as Element of Bags X ;
A in B |^ j implies A , B are_that A , B are_<* A , B *>
x in { [ 2 * n + 3 , k ] } ;
1 >= ( q `1 / |. q .| - cn ) / ( 1 + cn ) ;
f1 is_\HM { 2 } & f2 is_\HM { 2 } ;
( f /. i ) `2 <= ( q `2 ) ^2 ;
h is_\HM { implies Cage ( C , n ) /. 1 = E-max L~ Cage ( C , n )
b `2 <= p `2 & p `2 <= ( p `2 ) * ( 1 + 1 ) `2 ;
let f , g be Function of X , Y ;
S * ( k , k ) <> 0. K ;
x in dom ( ( - 1 ) (#) f ) ;
p2 in NN . p1 & p2 in NN . p2 ;
len ( the_left_argument_of H ) < len ( H ) + len ( H ) ;
F [ A , F-14 . A ] ;
consider Z such that y in Z and Z in X ;
attr 1 in C means : Def1 : A c= C |^ A ;
assume r1 <> 0 or r2 <> 0 ;
rng q1 c= rng C1 & rng q2 c= rng C1 & q1 is one-to-one ;
A1 , L , A3 , A3 , M be non empty set ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in 4 ( p , SC ) & c in 4 ( p , SC ) ;
then S is -> -> atomic not atomic & P-2 [ S ] ;
Cl Int [#] T = [#] T .= [#] T ;
f12 | A2 = f2 | A2 .= f2 | A2 ;
0. M in the carrier of W & 0. M in the carrier of W ;
let v , v be Element of M ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ V & V c= Y \ Z ;
let X be Subset of [: S , T :] ;
consider H1 such that H = 'not' H1 and H1 is conjunctive ;
1_ ( 1 , 1 ) c= ( \mathop { \rm Q } * ( \mathop { 1 } , 1 ) ) * ( ( \mathop { 1 } , 1 ) * ( ( \mathop { 1 } , 1 ) * (
0 * a = 0. R .= a * 0. R ;
A |^ ( 2 , 2 ) = A ^^ A ;
set vFinSequence = v4 /. n , v4 = v4 /. n ;
r = 0. ( REAL-NS n ) & ||. x - x0 .|| < r ;
( f . p4 ) `1 >= 0 & ( f . p4 ) `2 >= 0 ;
len W = len ( W -as non empty Subset of ( W -as Subset of W ;
f /* ( s * G ) is divergent_to-infty ;
consider l being Nat such that m = F . l ;
t8 / ( W8 + 1 ) does not destroy b1 ;
reconsider Y1 = X1 , Y2 = X2 as SubSpace of X ;
consider w such that w in F and not x in w ;
let a , b , c , d be Real ;
reconsider i = i as non zero Element of NAT ;
c . x >= id ( L . x ) ;
\sigma ( T ) \/ omega ( T ) is Basis of T ;
for x being element st x in X holds x in Y
cluster [ x1 , x2 ] -> pair for element ;
downarrow a /\ downarrow t is Ideal of T & downarrow a /\ downarrow t is Ideal of T ;
let X be with_NAT non empty set , f be Function ;
rng f = as as as as as as as as being being being being being Q ;
let p be Element of B , x be SortSymbol of S ;
max ( N1 , 2 ) >= N1 & max ( N2 , 2 ) >= N2 ;
0. X <= b |^ ( m * mm1 ) ;
assume that i in I and x0 . i = R ;
i = j1 & p1 = q1 & p2 = q2 implies i <= j
assume gR in the right of g & gR in the carrier of g ;
let A1 , A2 be Point of S , A be Subset of S ;
x in h " P /\ [#] T1 & x in h " P /\ [#] T2 ;
1 in Seg 2 & 1 in Seg 3 implies 2 * 1 in Seg 3
reconsider X-5 = X , Ximplies = Y as non empty Subset of Tlet X ;
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the Target of G ) -valued ;
n1 <= i2 + len g2 & i2 <= len g2 implies ( f | ( n + 1 ) ) /. i2 = f /. i2
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume v in the carrier' of G2 & u in the carrier' of G1 ;
y = Re y + ( Im y ) * i ;
( let ( - 1 ) * p ) ^2 = 1 ;
x2 is_differentiable_on ]. a , b .[ & x2 is_differentiable_on ]. a , b .[ ;
rng M5 c= rng ( D2 | Seg ( i -' 1 ) ) ;
for p be Real st p in Z holds p >= a
( the carrier of X ) --> f = proj1 * f .= f ;
( seq ^\ m ) . k <> 0 & ( seq ^\ m ) . k <> 0 ;
s . ( G . ( k + 1 ) ) > x0 ;
( p } -Path M ) . 2 = d ;
A \oplus ( B \ominus C ) = ( A \oplus B ) \ominus C
h \equiv gg . ( mod P ) , T ;
reconsider i1 = i-1 , i2 = i-1 as Element of NAT ;
let v1 , v2 be VECTOR of V , v be VECTOR of V ;
for W being Subspace of V holds W is Subspace of [#] V
reconsider i-7 = i , im2 = j as Element of NAT ;
dom f c= [: C ( ) , D ( ) :] ;
x in ( the Sorts of B ) . n & y in ( the Sorts of B ) . n ;
len } ( f2 ) in Seg len ( f2 ) ;
pm1 c= the topology of T & pm2 c= the topology of T ;
]. r , s .[ c= [. r , s .] ;
let B2 be Basis of T2 , f be Function of T1 , T2 ;
G * ( B * A ) = ( id o1 ) * A ;
assume that p , u , q is_collinear and u , v , q is_collinear ;
[ z , z ] in union rng ( F | X ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S , C = $1 .. S , D = $1 .. S , E = $1 .. S , F = S . ( $1 .. S ) , N = U . ( $1 .. S )
LIN a1 , a3 , b1 & LIN a1 , b1 , c1 & LIN a1 , b1 , c1 ;
f " ( f .: x ) = { x } ;
dom ( w2 ) = dom r12 .= dom r12 ;
assume that 1 <= i and i <= n and j <= n ;
( ( ( g2 ) . O ) `2 ) ^2 <= 1 ^2 ;
p in LSeg ( E . i , F . i ) ;
Iv * ( i , j ) = 0. K ;
|. f . ( s . m ) -g .| < g1 ;
( q . x ) in rng ( q | ( Seg n ) ) ;
Carrier ( Lxy ) misses Carrier ( Lxy ) \/ Carrier ( LR2 ) ;
consider c being element such that [ a , c ] in G ;
assume Nreal = o] & onon empty ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= ( F .: CZ ) " { 0 } ;
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 ;
redefine func aaS ( S , T ) -> Subset of T ;
let x be Element of [: S , T :] ;
non empty as non empty as non empty as over F ;
|. i .| <= - ( 2 to_power n ) ;
the carrier of I[01] = dom P & the carrier of I[01] = dom P ;
} * ( n + 1 ) ! > 0 * ! ;
S c= ( A1 /\ A2 ) /\ A3 & S c= ( A1 /\ A2 ) /\ A3 ;
a3 , a4 // b3 , b3 & a3 , a4 // b3 , b3 ;
then dom A <> {} & dom A <> {} ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , Y & y Joins X , Y implies x = y
set v2 = v4 /. ( i + 1 ) , v2 = v4 /. ( i + 1 ) ;
x = r . n .= r4 . n .= r4 . n ;
f . s in the carrier of S2 & f . s in the carrier of S2 ;
dom g = the carrier of I[01] & rng g c= the carrier of I[01] ;
p in Upper_Arc ( P ) /\ Lower_Arc ( P ) ;
dom d2 = [: A2 , A2 :] & dom d2 = [: A2 , A1 :] ;
0 < ( p / ||. z .|| + 1 ) / ( ||. z .|| + 1 ) ;
e . ( m3 + 1 ) <= e . ( m3 + 1 ) ;
B \ominus X \/ B \ominus Y c= B \ominus X /\ Y
-infty < Integral ( M , Im ( g | B ) ) ;
cluster O := F -> \HM { : not contradiction } -> Element of X ;
let U1 , U2 be non-empty MSAlgebra over S , f be Function of U1 , U2 ;
Proj ( i , n ) * g is_differentiable_on X & g is_differentiable_on X ;
let x , y , z be Point of X , p be Point of X ;
reconsider pp = p . x , pp = p . y as Subset of V ;
x in the carrier of Lin ( A ) & y in the carrier of Lin ( B ) ;
let I , J be parahalting Program of SCM+FSA , a be Int-Location ;
assume that - a is lower and b is lower and a in X ;
Int Cl A c= Cl Int Cl A & Cl Int Cl A c= Cl Int Cl A ;
assume for A being Subset of X holds Cl A = A ;
assume q in Ball ( |[ x , y ]| , r ) ;
p2 `2 <= p `2 & p `2 <= - 1 or p2 `2 >= p `2 & p `1 <= - 1 ;
Cl ( Q ` ) = [#] ( TT ) ;
set S = the carrier of T , T = the carrier of T ;
set I8 = to_power n , I8 = f to_power n , I7 = f to_power n , I8 = f to_power n , I8 = f to_power n , I8 = f to_power n , I8 = f to_power n , I8 = f
len p -' n = len ( p | n ) - n ;
A is Permutation of Swap ( A , x , y ) ;
reconsider n6 = n} , n7 = n7 as Element of NAT ;
1 <= j + 1 & j + 1 <= len ( s | X ) ;
let q\subseteq , qSet be Element of M , q be Element of M ;
a in the carrier of S1 & b in the carrier of S2 ;
c1 /. n1 = c1 . n1 & c2 /. n1 = c2 . n1 ;
let f be FinSequence of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
y = ( ( f * SS ) * ( f * * N ) ) . x ;
consider x being element such that x in be be Element of be \cal , A ;
assume r in ( dist ( o , r ) ) .: P ;
set i2 = len ( \hbox { - } , 1 , - 1 , 1 ) ;
h2 . ( j + 1 ) in rng h2 /\ rng h2 ;
Line ( M29 , k ) = M . i .= Line ( M29 , k ) ;
reconsider m = ( x - 1 ) / 2 as Element of ExtREAL ;
let U1 , U2 be strict Subspace of U0 , a be Element of U1 ;
set P = Line ( a , d ) ;
len p1 < len p2 + 1 & len p2 = len p2 + 1 ;
let T1 , T2 be Scott Scott topological of L , x be Element of [: T , T :] ;
then x <= y & : ex x being Element of Y st x in { y } ;
set M = n -r1 , N = n -r1 ;
reconsider i = x1 , j = x2 as Nat ;
rng ( the_arity_of a9 ) c= dom H & rng ( the_arity_of a9 ) c= dom ( H * ( the_arity_of b9 ) ) ;
z1 " = z9 " & z1 " = z9 " * z1 " ;
x0 - r / 2 in L /\ dom f & f . x0 = r ;
then w is that rng w /\ L <> {} & S /\ L <> {} ;
set x-10 = x-9 ^ <* Z *> ^ ( xZ ^ <* Z *> ) ;
len w1 in Seg len w1 & len w2 = len w2 ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of PFuncs ( V , { k } ) ;
x . n = ( |. a . n .| ) to_power ( k + 1 ) ;
p `1 <= ( Gik `1 + 1 ) * ( len G , 1 ) `1 ;
rng ( g | ( len g -' 1 ) ) c= L~ ( g | ( len g -' 1 ) ) ;
reconsider k = i-1 * ( i + j ) + j as Nat ;
for n be Nat holds F . n is \HM { -infty } ;
reconsider x-10 = x-7 , x29 = xM as Vector of M ;
dom ( f | X ) = X /\ dom f .= X /\ dom f ;
p , a // p , c & b , a // c , c ;
reconsider x1 = x , y1 = y , z1 = z as Element of REAL m ;
assume i in dom ( a * p ^ q ) ;
m . ag = p . ag & m . ag = p . ag ;
a / ( s . m - s . n ) / ( s . m - s . n ) <= 1 ;
S . ( n + k + 1 ) c= S . ( n + k ) ;
assume B1 \/ C1 = B2 \/ C2 & C1 = C2 \/ C1 ;
X . i = { x1 , x2 } . i .= { x1 , x2 } . i ;
r2 in dom ( h1 + h2 ) /\ dom ( h2 + h2 ) ;
\mathclose { 0. R } = a & b-0 = b ;
F8 is_closed_on t2 , Q2 & F8 is_halting_on t2 , Q2 implies F . b = F . b
set T = ^2 and k1 = the carrier of thesis implies X = { x0 } ;
Int Cl ( Int Cl R ) c= Int Cl R ;
consider y being Element of L such that c . y = x ;
rng ( Fp1 . x ) = { Fp1 . x } ;
G-23 ` { c } c= B \/ S \/ S ;
f\rm {} is Relation of [: X , X :] , X & f^ is Relation of X , Y ;
set RQ = the Point of P , RQ = the Point of Q ;
assume that n + 1 >= 1 and n + 1 <= len M ;
let k2 be Element of NAT , x be Element of NAT ;
reconsider pp = u , pp = v as Element of ( ( TOP-REAL n ) | P ) ;
g . x in dom f & x in dom g implies f . x = g . x
assume that 1 <= n and n + 1 <= len f1 and f1 . n = f1 . ( n + 1 ) ;
reconsider T = b * N as Element of G / ( N , N ) ;
len Pt <= len P-35 & len ( Pt ^ <* d *> ) <= len ( Pt ^ <* d *> ) ;
x " in the carrier of A1 & y " in the carrier of A2 ;
[ i , j ] in Indices ( ( A @ ) * ( i , j ) ) ;
for m be Nat holds Re ( F . m ) is simple Function of S , T
f . x = a . i .= a1 . k .= a1 . k ;
let f be PartFunc of REAL i , REAL n , x be Element of REAL n ;
rng f = the carrier of ( ( Carrier A ) * ( Carrier B ) ) ;
assume s1 = sqrt ( 2 * p ) - p `1 ;
attr a > 1 & b > 0 implies a / b > 1 ;
let A , B , C be Subset of ICT ( ) ;
reconsider X0 = X , Y0 = Y as RealNormSpace ;
let f be PartFunc of REAL , REAL , x be Element of REAL ;
r * ( v1 |-- I ) . X < r * 1 ;
assume that V is Subspace of X and X is Subspace of V ;
let t-3 , t-4 be Relation , t be Element of T ;
Q [ e-14 \/ { v-14 } , f . v/ ( n + 1 ) ] ;
g \circlearrowleft W-min L~ z = z implies ( g /. 1 ) .. z < ( g /. len z ) .. z
|. |[ x , v ]| - |[ x , y ]| .| = v\rrangle ;
- f . w = - ( L * w ) ;
z - y <= x iff z <= x + y & y <= z + x ;
( 7 / p1 ) to_power ( 1 / e ) > 0 ;
assume X is BCK-algebra of 0 , 0 , 0 , 0 , 0 ;
F . 1 = v1 & F . 2 = v2 & F . 3 = v2 ;
( f | X ) . x2 = f . x2 & ( f | X ) . x2 = f . x2 ;
( ( tan | Z ) . x ) in dom ( sec | Z ) /\ dom ( tan | Z ) ;
i2 = ( f /. len f ) & i2 = ( f /. len f ) . i2 ;
X1 = X2 \/ ( X1 \ X2 ) & X2 = X1 \/ X2 ;
[. a , b , 1_ G .] = 1_ G & a = 1_ G ;
let V , W be non empty VectSpStr over F_Complex , f be FinSequence of V ;
dom ( g2 ) = the carrier of I[01] & rng ( g2 ) c= the carrier of I[01] ;
dom ( f2 ) = the carrier of I[01] & rng ( f2 ) c= the carrier of I[01] ;
( proj2 | X ) .: X = proj2 .: X .= proj2 .: X ;
f . ( x , y ) = h1 . ( x `1 , y `2 ) ;
x0 - r < a1 . n & a1 . n < x0 + r ;
|. ( f /* s ) . k - GM .| < r ;
len Line ( A , i ) = width A & len Line ( B , i ) = width B ;
SFinSequence ^ ( S , g ) = ( S . g ) @ ;
reconsider f = v + u as Function of X , the carrier of Y ;
intloc 0 in dom Initialized p & intloc 0 in dom Initialized p ;
i1 does not destroy ( i3 , i3 ) & not f does not destroy ( b3 , b ) ;
arccos r + arccos r = ( PI / 2 ) + 0 ;
for x st x in Z holds f2 is_differentiable_in x & f2 is_differentiable_in x & f2 . x > 0 ;
reconsider q2 = ( q - x ) / ( q - x ) as Element of REAL ;
( 0 qua Nat ) + 1 <= i + j1 & i + 1 <= len f ;
assume f in the carrier of [' X , Omega Y '] ;
F . a = H / ( x , y ) . a ;
( ( TRUE T ) at ( C , u ) ) = TRUE ;
dist ( ( a * seq ) . n , h ) < r ;
1 in the carrier of [. 0 , 1 .] & 1 in dom ( f | [. 0 , 1 .] ) ;
p2 `1 - x1 > - g / 2 - g / 2 - g / 2 ;
|. r1 - thesis .| = |. a1 .| * |. thesis .| ;
reconsider S-14 = 8 as Element of ( Seg 8 ) * ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b
D0W .\mathop ( n ) = D0W .\mathop ( n ) + 1 ;
i1 = ma + n & i2 = ma + n & i1 = ma + n ;
f . a [= f . ( f . O1 "\/" f . a ) ;
pred f = v & g = u implies f + g = v + u ;
I . n = Integral ( M , F . n ) ;
chi ( [: T1 , S :] , T ) . s = 1 ;
a = VERUM ( A ) or a = VERUM ( A ) or a = VERUM ( A ) ;
reconsider k2 = s . b3 , k2 = s . b3 as Element of NAT ;
( Comput ( P , s , 4 ) ) . GBP = 0 ;
L~ ( M1 ) meets L~ ( R4 ) /\ L~ ( R4 ) ;
set h = the continuous Function of X , R , x be Point of X ;
set A = { L . ( ( k . n ) . x ) where k is Element of NAT : k in dom L } ;
for H st H is atomic holds P7 [ H ] ;
set bA = S5 ^\ ( i + 1 ) , SA = S5 ^\ ( i + 1 ) ;
Hom ( a , b ) c= Hom ( a opp , b opp ) ;
( 1 / ( n + 1 ) ) < ( 1 / s ) " ;
( l `1 ) = [ dom l `1 , cod l `2 ] ;
y +* ( i , y /. i ) in dom g & y in dom g ;
let p be Element of CQC-WFF ( Al ) , P be Subset of CQC-WFF ( Al ) ;
X /\ X1 c= dom ( f1 - f2 ) /\ dom ( f2 - f1 ) ;
p2 in rng ( f /^ ( len f -' 1 ) ) /\ rng ( f /^ ( len f -' 1 ) ) ;
1 <= indx ( D2 , D1 , j1 ) & indx ( D2 , D1 , j1 ) <= len D2 ;
assume x in ( ( ( ( ( ( ( K1 /\ K0 ) \/ K0 ) /\ K0 ) /\ K0 ) /\ K0 ) /\ K0 ) /\ K0 ) /\ K0 ;
- 1 <= ( ( f2 . O ) `2 ) & - 1 <= ( ( f2 . O ) `2 ) ;
let f , g be Function of I[01] , ( TOP-REAL 2 ) | P , p1 , p2 be Point of TOP-REAL 2 ;
k1 -' k2 = k1 - k2 + k2 .= k1 - k2 + k2 ;
rng seq c= ]. x0 , x0 + r .[ & seq is convergent & lim seq = x0 + r ;
g2 in ]. x0 - r , x0 + r .[ /\ dom f ;
sgn ( p `1 , K ) = - ( 1_ K ) ;
consider u being Nat such that b = p |^ y * u ;
ex A being subset of x0 st a = Sum A & A is convergent & lim A = 0 ;
Cl ( union HC ) = union ( ( Cl H ) /\ ( Cl H ) ) ;
len t = len t1 + len t2 & len t1 = len t2 + len t1 ;
v-29 = v + w |-- v + AA ;
cv <> DataLoc ( t0 . GBP , 3 ) .= intpos ( 0 + 3 ) ;
g . s = sup ( d " { s } ) & s in d " { s } ;
( \dot \dot y ) . s = s . ( \dot y ) ;
{ s : s < t } in INT implies t = {} ;
s ` \ s = s ` \ 0. X .= s ` \ ( s ` \ s ) ;
defpred P [ Nat ] means B + $1 in A & not $1 in B ;
( 339 + 1 ) ! = 3339 ! * ( 339 + 1 ) ;
IC ( ( succ A ) --> ( A , B ) ) = ( T . A ) --> ( B , C ) ;
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 NAT , k be Nat ;
set f = ( S , U ) i -] , g = S , f = S , F = S S , G = S S , F = S S , G = S S , F = S , G = S , G = S , F = S , G
consider Z being set such that lim s in Z and Z in F ;
let f be Function of I[01] , ( TOP-REAL n ) | P , p1 be Point of TOP-REAL n ;
( SAT M ) . [ n + i , 'not' A ] <> 1 ;
ex r being Real st x = r & a <= r & r <= b ;
let R1 , R2 be Element of REAL n , x be Element of REAL n ;
reconsider l = 0. ( V ) , v = 0. ( V ) as Linear_Combination of A ;
set r = |. e .| + |. n .| + |. w .| + a ;
consider y being Element of S such that z <= y and y in X ;
a being being being being being being being being being being being being being being being being being being being being being Element of Y holds a 'or' b = 'not' ( ( a 'or' b ) 'or' c )
||. x9 - ( g . x ) .|| < r2 / 2 ;
b9 , a9 // b9 , c9 & b9 , c9 // b9 , c9 ;
1 <= k2 -' k1 & k1 + 1 = k2 or k2 + 1 = k2 & k2 + 1 <= k2 ;
( p `2 / |. p .| - sn ) / ( 1 + sn ) >= 0 ;
( q `2 / |. q .| - sn ) / ( 1 + sn ) < 0 ;
E-max C in right_cell ( Rmax ( C , 1 ) , 1 ) ;
consider e being Element of NAT such that a = 2 * e + 1 ;
Re ( ( lim F ) | D ) = Re ( ( lim G ) | D ) ;
LIN b , a , c or LIN b , c , a ;
p `1 , a `2 // a `1 , b `2 or p `1 , a `2 // b `1 , a `2 ;
g . n = a * Sum fm1 .= f . n ;
consider f being Subset of X such that e = f and f is bijective ;
F | ( N2 , S ) = 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 in I ;
rng ( cos | [. - 1 , 1 .] ) = [. - 1 , 1 .] ;
assume that Re ( seq ) is summable and Im ( seq ) is summable and Im ( seq ) is summable ;
||. ( ( vseq . n ) - ( vseq . n ) ) - ( ( vseq . n ) - ( vseq . n ) ) .|| < e ;
set g = O --> 1 ;
reconsider t2 = t11 , t2 = t11 as ( 0 , U ) -valued string of S2 ;
reconsider x-29 = seq . n , xd = seq . n as sequence of REAL n ;
assume that C meets L~ go and L~ pion1 meets L~ pion1 and x in L~ pion1 /\ L~ pion1 ;
- ( Partial_Sums ( 1 / 2 ) ) . x < F . n - F . x ;
set d1 = being thesis , z1 = dist ( x1 , z1 ) , d2 = dist ( x2 , z2 ) , d2 = dist ( y2 , z2 ) ;
2 |^ ( 2 -' 1 ) = 2 |^ ( 2 -' 1 ) - 1 ;
dom vb2 = Seg len db2 .= dom ( db2 | Seg ( len vb2 ) ) ;
set x1 = ( - k2 ) + |. k2 .| + 4 ;
assume for n be Element of X holds 0. <= F . n ;
assume that 0 <= T-32 . i and T-32 . ( i + 1 ) <= 1 ;
for A being Subset of X holds c . ( c . A ) = c . A
the carrier of ( Carrier ( LS ) + Carrier ( LS ) ) c= I2 ;
'not' Ex ( x , p ) => All ( x , 'not' p ) is valid ;
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the normal normal w.r.t. over {} ;
Z c= dom ( ( - 1 ) (#) ( sin * f1 ) ) /\ dom ( ( - 1 ) (#) f1 ) ;
|. 0. TOP-REAL 2 - ( q `2 / |. q .| - sn ) .| < r ;
A c= ConsecutiveSet2 ( A , succ d ) & L . ( succ d ) c= L . ( A , succ d ) ;
E = dom ( L (#) G ) & Carrier ( L (#) G ) is_measurable_on E ;
C |^ ( A + B ) = C |^ B * C |^ A ;
the carrier of W2 c= the carrier of V & the carrier of W1 c= the carrier of V ;
I . IC ss2 = P . IC ss2 .= ( card I + 2 ) ;
pred x > 0 means : Def1 : ( 1 / x ) ^2 = x ^2 / ( 1 - x ^2 ) ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) .= LSeg ( f , i ) ;
consider p being Point of T such that C = [: [. p , q .] , p :] ;
b , c are_connected & - C , - C - D + D + E + F + G + H + E + F + G + D + E + F - G + D + E + F - G + D + E + F - G + D + E + E + F - D
assume f = id the carrier of S & g = id the carrier of S ;
consider v such that v <> 0. V and f . v = L * v ;
let l be Linear_Combination of {} ( ( the carrier of V ) --> { 0 } ) ;
reconsider g = f " as Function of U2 , U1 , U2 ;
A1 in the carrier of G_ ( k , X ) & A2 in the carrier of G_ ( k , X ) ;
|. - x .| = - ( - x ) .= x - ( - x ) .= x ;
set S = ) ( x , y , c ) ;
Fib ( n ) * ( 5 * Fib ( n ) - 1 ) >= 4 * be Element of NAT ;
vM /. ( k + 1 ) = vM . ( k + 1 ) ;
0 mod i = ( - i ) * ( 0 qua Nat ) .= ( - i ) * ( 0 qua Nat ) ;
Indices M1 = [: Seg n , Seg n :] & Indices ( M1 + M2 ) = [: Seg n , Seg n :] ;
Line ( Sl , j ) = Sl . j .= Sl . j ;
h . ( x1 , y1 ) = [ y1 , x1 ] & h . ( y1 , y2 ) = [ y1 , y2 ] ;
|. f .| - Re ( |. f .| * ( card b ) ) is nonnegative ;
assume that x = ( a1 ^ <* x1 *> ) ^ b1 and y = ( a1 ^ <* x2 *> ) ^ b2 ;
ME is_closed_on IExec ( I , P , s ) , P & ME is_halting_on s , P ;
DataLoc ( t3 . a , 4 ) = intpos ( 0 + 4 ) .= intpos ( 0 + 4 ) ;
x + y < - x + y & |. x .| = - x + y ;
LIN c , q , b & LIN c , q , c & LIN c , q , c ;
f^ . ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + z1 ) .= x1 + y1 ;
f, f . a = f{ a } & v in InputVertices S & u in InputVertices S ;
p `1 <= ( E-max C ) `1 & ( E-max C ) `2 <= ( E-max C ) `2 ;
set R8 = Cage ( C , n ) \circlearrowleft E8 , R7 = Cage ( C , n ) ;
p `1 >= ( E-max C ) `1 & ( E-max C ) `2 >= ( E-max C ) `2 ;
consider p such that p = pp and s1 < p and p in LSeg ( f , i ) ;
|. ( f /* ( s * F ) ) . l - GM .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len Line ( N , k + 1 + 1 ) = width N & len Line ( N , k + 1 ) = width N ;
f1 /* s1 is convergent & f2 /* s1 is convergent & lim ( f1 /* s1 ) = f1 . x0 ;
f . x1 = x1 & f . y1 = y1 & f . y2 = y2 ;
len f <= len f + 1 & len f + 1 <> 0 implies f . ( len f + 1 ) <> 0. K
dom ( Proj ( i , n ) * s ) = REAL m ;
n = k * ( 2 * t ) + ( n mod ( 2 * k ) ) ;
dom B = 2 -tuples_on the carrier of V & rng B c= the carrier of V ;
consider r such that r _|_ a and r _|_ x and r _|_ y ;
reconsider B1 = the carrier of Y1 , B2 = the carrier of Y2 as Subset of X ;
1 in the carrier of [. 1 / 2 , 1 .] /\ dom ( g | [. 0 , 1 .] ) ;
for L being complete LATTICE for o , m being Element of O holds L , L are_isomorphic implies L , o are_isomorphic
[ gi , gj ] in Ii \ Ij ~ & [ gi , gj ] in Ij ;
set S2 = 1GateCircStr ( x , y , c ) ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and for r st r in dom f1 holds f1 . r = f2 . x0 ;
reconsider y = ( a ` ) ` , z = ( a ` ) ` as Element of L ;
dom s = { 1 , 2 , 3 } & s . 1 = d1 & s . 2 = d2 ;
( min ( g , ( 1 - 1 ) / ( 1 - \ f ) ) ) . c <= h . c ;
set G3 = the K of G , v = the Vertex of G , x = the Vertex of G , y = the Vertex of G ;
reconsider g = f as PartFunc of REAL n , REAL-NS n , REAL-NS n ;
|. s1 . m / p .| / |. p .| < d / p / p ;
for x being element st x in ( ( 1 - t ) * u ) holds x in ( ( 1 - t ) * u )
P = the carrier of ( ( TOP-REAL n ) | Px0 ) .= ( ( TOP-REAL n ) | Px0 ) ;
assume that p00 in LSeg ( p1 , p2 ) /\ LSeg ( p2 , p2 ) and p2 in LSeg ( p1 , p2 ) /\ LSeg ( p2 , p2 ) ;
( 0. X \ x ) to_power ( 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 carrier of X , g be Function of X , Y ;
set h = Hom ( a , g (*) f ) ;
then idseq n | Seg m = idseq m & m <= n implies m <= n & n <= m ;
H * ( g " * a ) in the right of H & g * ( g " * a ) in the right of H ;
x in dom ( ( cos * sin ) `| Z ) /\ dom ( ( cos * sin ) `| Z ) ;
cell ( G , i1 , j2 -' 1 ) misses C & cell ( G , i1 , j2 -' 1 ) misses C ;
LE q2 , p4 , P , p1 , p2 & LE q2 , p2 , P , p1 , p2 implies LE q2 , p , P , p1 , p2
attr B is BDD component means : Def1 : B c= BDD A & B c= BDD A ;
deffunc D ( set , set ) = union rng $2 & $2 = union rng $2 ;
n + - n < len ( p + - n ) + ( - n ) ;
attr a <> 0. K means for M st the_rank_of M = the_rank_of ( a * M ) holds the_rank_of M = the_rank_of M ;
consider j such that j in dom /\ /\ dom thesis and I = len } + j ;
consider x1 such that z in x1 and x1 in P8 and x = [ x1 , x2 ] ;
for n ex r being Element of REAL st X [ n , r ]
set Cs1 = Comput ( P2 , s2 , i + 1 ) , Cs2 = Comput ( P2 , s2 , i + 1 ) ;
set cv = 3 -tuples_on { a , b , c } , c/ 2 ;
conv @ W c= union ( F .: ( E " W ) ) ;
1 in [. - 1 , 1 .] /\ dom ( arccot * ( arccot ) ) ;
r3 <= s0 + ( r0 - |. v2 - v1 .| ) / ( 2 * ( v2 - v1 ) ) ;
dom ( f (#) f4 ) = dom f /\ dom f4 .= dom ( f (#) f4 ) /\ dom ( g (#) f4 ) ;
dom ( f (#) G ) = dom ( l (#) F ) /\ Seg k .= dom ( l (#) F ) ;
rng ( s ^\ k ) c= dom f1 \ { x0 } & rng ( s ^\ k ) c= dom f1 ;
reconsider gg = gp , gp = gp , gp = gp as Point of TOP-REAL n1 ;
( T * h . s ) . x = T . ( h . s . x ) ;
I . ( L . ( J . x ) ) = ( I * L ) . ( J . x ) ;
y in dom ( *> <* the *> of ( commute ( A . o ) ) . ( ( commute ( A . o ) ) . y ) *> ) ;
for I being non degenerated commutative Ring holds the carrier of I is commutative non empty doubleLoopStr
set s2 = s +* Initialize ( ( intloc 0 ) .--> 1 ) +* Initialize ( ( intloc 0 ) .--> 1 ) ;
P1 /. IC s1 = P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 ;
lim S1 in the carrier of [. a , b .] & lim S1 in the carrier of [. a , b .] ;
v . ( Upp . i ) = ( v *' lpp ) . i ;
consider n being element such that n in NAT and x = ( sn succ ) . n ;
consider x being Element of c such that F1 . x <> F2 . x and x in F1 . x ;
/\ ( X , 0 , x1 , x2 ) = { EC } & card ( X , 0 , x1 , x2 , x3 ) = k ;
j + ( 2 * ( k + 1 ) ) + m1 > j + ( 2 * ( k + 1 ) ) ;
{ s , t } on A3 & { s , t } on B3 & { s , t } on B3 ;
n1 > len crossover ( p2 , p1 , n1 , n2 ) & n2 >= len crossover ( p2 , p1 , n2 , n3 ) ;
mg . ( HT ( mg , T ) ) = 0. L .= 0. L ;
then H1 , H2 are_<* H1 , H2 *> & Cl ( H1 , H2 ) , ( H ) / ( 2 * a ) are_equipotent ;
( N-min L~ f ) .. ( ( f /. 1 ) .. ( f /. len f ) ) .. ( ( f /. 1 ) .. ( f /. 1 ) ) .. ( ( f /. 1 ) .. ( f /. 1 ) ) ) > 1 ;
]. s , 1 .[ = ]. s , 2 .[ /\ [. 0 , 1 .] ;
x1 in [#] ( ( ( TOP-REAL 2 ) | L~ g ) | ( L~ g ) ) ;
let f1 , f2 be continuous PartFunc of REAL , the carrier of S , x be Point of S ;
DigA ( t-23 , z9 ) is Element of k -tuples_on ( the carrier of K ) ;
I \cap 223 = d\lbrace k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 , k2 be Nat ;
[: u , { u9 } :] = { [ a , u9 ] } & [: u , { u9 } :] = [: { a } , { b } :] ;
( w | p ) | ( p | ( w | w ) ) = p ;
consider u2 such that u2 in W2 and x = v + u2 and u2 in W2 and u1 in W1 ;
for y st y in rng F ex n st y = a |^ n & P [ n ] ;
dom ( ( g * ( f , C ) ) | K ) = K ;
ex x being element st x in ( ( ( U0 ) \/ A ) . s ) . s ;
ex x being element st x in ( ( the Sorts of OA ) . s ) . s ;
f . x in the carrier of [. - r , r .] & f . x in [. - r , r .] ;
( the carrier of X1 union X2 ) /\ ( the carrier of X1 ) <> {} & ( the carrier of X1 union X2 ) /\ ( the carrier of X2 ) <> {} ;
L1 /\ LSeg ( p00 , p2 ) c= { p00 } /\ LSeg ( p1 , p2 ) ;
( b + ( bs - bs ) ) / 2 in { r : a < r & r < b } ;
ex_sup_of { x , y } , L & x "\/" y = sup { x , y } ;
for x being element st x in X ex u being element st P [ x , u ]
consider z being Point of GX such that z = y and P [ z ] and z in A ;
( the carrier of ( ( the carrier of ( the carrier of X ) ) | ( the carrier of X ) ) ) . ( being Element of ( the carrier of X ) | ( the carrier of X ) ) <= e ;
len ( w ^ w2 ) + 1 = len w + 2 + 1 .= len w + 1 ;
assume q in the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 & q in the carrier of ( ( TOP-REAL 2 ) | K1 ) ;
f | E-4 ` = g | ED ` .= g | ED ` .= g | ED ;
reconsider i1 = x1 , i2 = x2 , j1 = x3 , j2 = x4 as Element of NAT ;
( a * A * B ) @ = ( a * ( A * B ) ) @ ;
assume ex n0 being Element of NAT st f to_power n0 is min & f to_power n0 is min ;
Seg len ( ( the support of f2 ) * ( f | ( Seg i -' 1 ) ) ) = dom ( ( f | ( Seg i -' 1 ) ) ) ;
( Complement ( Prob . m ) ) . n c= ( Complement ( Prob . n ) ) . ( ( Complement ( Prob . m ) ) . n ) ;
f1 . p = p9 & g1 . p = d & g1 . p = d & g2 . p = d ;
FinS ( F , Y ) = FinS ( F , dom ( F | Y ) ) ;
( x | y ) | z = z | ( y | x ) ;
( ( |. x .| to_power n ) / ( n + 1 ) ) / ( n + 1 ) <= ( ( r2 ) to_power n ) ;
Sum ( F ) = Sum f & dom ( F ) = dom g & for x be Element of X st x in dom F holds F . x = F . x ;
assume for x , y being set st x in Y & y in Y holds x /\ y in Y ;
assume W1 is Subspace of W3 & W2 is Subspace of W3 & W1 is Subspace of W3 & W2 is Subspace of W3 ;
||. t-15 . x .|| = lim ||. xis .|| & ||. x .|| <= ||. x .|| ;
assume that i in dom D and f | A is lower and g | A is lower bounded and g | A is lower ;
( p `2 / |. p .| - sn ) / ( 1 + sn ) <= ( - ( 1 + sn ) ) / ( 1 + sn ) ;
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 countable implies the TopStruct of T is countable countable
width B |-> 0. K = Line ( B , i ) .= B @ .= B @ ;
attr a <> 0 implies ( A \+\ B ) Let a = ( A carrier A ) \+\ ( B carrier a ) ;
then f is_partial_differentiable_in pdiff1 ( f , 1 ) , 2 & pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 ;
assume that a > 0 and a <> 1 and b > 0 and b <> 1 and c > 0 ;
w1 , w2 in Lin { w1 , w2 } & w2 in Lin { w2 , w1 } ;
p2 /. IC s-7 = p2 . IC sU .= p2 . IC sU .= ( p2 + p2 ) . IC U .= ( p2 + p2 ) . IC U ;
ind ( T-10 | b ) = ind b .= ind B .= ind B .= ind ( B | b ) ;
[ a , A ] in the carrier of \hbox { - } or [ a , A ] in the carrier of \hbox { - } ;
m in ( the Arrows of C ) . ( o1 , o2 ) & m in ( the Arrows of C ) . ( o1 , o2 ) ;
( ( a 'imp' CompF ( PA , G ) ) . z ) . z = FALSE ;
reconsider phi = phi /. 11 , phi = phi /. 22 , phi = phi /. 22 as Element of ( S , U ) * ;
len s1 - 1 * ( len s2 - 1 ) - 1 + 1 > 0 + 1 ;
delta ( D ) * ( f . ( upper_bound A ) - f . ( lower_bound A ) ) < r ;
[ f21 , f22 ] in the carrier' of [: A , B :] & [ f21 , f22 ] in the carrier' of [: A , B :] ;
the carrier of ( ( TOP-REAL 2 ) | K1 ) = K1 & the carrier of ( ( TOP-REAL 2 ) | K1 ) = K1 ;
consider z being element such that z in dom g2 and p = g2 . z and q = g2 . z ;
[#] V1 = { 0. V1 } .= the carrier of ( the carrier of V1 ) /\ the carrier of V1 .= { 0. V1 } ;
consider P2 being FinSequence such that rng P2 = M and P2 is one-to-one and for k st k in dom P2 holds P2 . k = F ( k ) ;
assume that x1 in dom ( f | X ) and ||. x1 - x0 .|| < s and |. x1 - x0 .| < s ;
h1 = f ^ ( <* p3 *> ^ <* p *> ) .= h ^ <* p *> .= h ;
c /. |[ b , c ]| = c .= c /. |[ a , c ]| .= c /. |[ a , c ]| ;
reconsider t1 = p1 , t2 = p2 , t2 = p3 as Term of C , V ;
( 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 .= C * ( p1 `2 ) + D ;
R . b - a ` = 2 * ` - b .= 2 * b - b .= b ;
consider \cdot 0 such that B = ( - 1 ) * C + ( - 1 ) * A and 0 <= 1 ;
dom g = dom ( ( the Sorts of A ) * a9 ) .= dom ( ( the Sorts of A ) * b9 ) ;
[ P . ( U6 ) , P . ( U7 ) ] in => ( T7 => T ) ;
set s2 = Initialize s , P2 = P +* I ;
reconsider M = mid ( z , i2 , i1 ) as non empty FinSequence of REAL 2 ;
y in product ( ( Carrier J ) +* ( V , { 1 } ) ) ;
1 / ( |[ 0 , 1 ]| ) = 1 & 0 / ( |[ 0 , 1 ]| ) = 0 ;
assume x in the left of g or x in the left of g or x in the left of g or x in the right of g ;
consider M being strict Subspace of A7 such that a = M and T is Subspace of M ;
for x st x in Z holds ( ( ( #Z n ) + f ) `| Z ) . x <> 0
len W1 + len W2 + m = 1 + len W3 + m .= len W3 + len W3 + m .= len W3 + m ;
reconsider h1 = ( vseq . n ) - ( tf1 . n ) as Lipschitzian LinearOperator of X , Y ;
( i mod len ( p + q ) ) + 1 in dom ( p + q ) ;
assume that s2 is_\emptyset and F in the O of s2 and F in the O of s2 and F in the O of s2 ;
( ( ( ( ( ( ( ( ( ( x , y ) , 3 ) ) / ( x , y ) ) / ( x , z ) ) / ( x , y ) ) ) / ( x , y ) ) ) / ( x , z ) = 1 ;
for u being element st u in Bags n holds ( p + m ) . u = p . u
for B being Subset of u-5 st B in E holds A = B or A misses B or A misses B
ex a being Point of X st a in A & A /\ Cl { y } = { a } ;
set W2 = tree ( p ) \/ W1 , W3 = tree ( q ) ;
x in { X where X is Ideal of L : X is Ideal of L & X is non empty Subset of L } ;
the carrier of W1 /\ W2 c= the carrier of W1 & the carrier of W1 /\ W2 c= the carrier of W1 /\ W2 ;
( a + b ) * ( id a ) = ( a + b ) * ( id a ) .= ( a + b ) * ( id a ) ;
( dom ( X --> f ) ) . x = ( X --> dom f ) . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) , y = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) ;
p => ( q => r ) => ( p => ( q => r ) ) in TAUT ( A ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
- 1 + 1 <= ( ( 2 |^ ( n -' m ) ) + 1 ) - 1 ;
( reproj ( 1 , z0 ) ) . x in dom ( f1 (#) f2 ) /\ dom ( f2 (#) f3 ) ;
assume that b1 . r = { c1 } and b2 . r = { c2 } and b2 . r = c2 . r ;
ex P st a1 on P & a2 on P & b on P & c on P & d on P & d on P ;
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 v1 in V and v1 in V ;
n in { i where i is Nat : i < n0 + 1 & i <= n + 1 } ;
F * ( i , j ) `2 >= ( F * ( m , k ) ) `2 ;
assume K1 = { p : p `1 >= sn * |. p .| & p `2 >= 0 & p <> 0. TOP-REAL 2 } ;
ConsecutiveSet ( A , succ O1 ) = ( ( A , O1 ) --> ( A , O2 ) ) *' ( A , O2 ) ;
set Ii1 = in dom Macro ( a , intloc 0 ) , Ii2 = AddTo ( a , intloc 0 ) , Ii2 = AddTo ( a , intloc 0 ) , Ii2 = goto ( card I + 2 ) , Ii2 = goto ( card I + 3 ) , Ii2 = goto ( card I + 2 ) , Ii2 = goto ( card I + 3 ) , Ii2 = goto ( card I + 2
for i be Nat st 1 < i & i < len z holds z /. i <> z /. 1 ;
X c= ( the carrier of L1 ) /\ ( the carrier of L2 ) & ( the carrier of L1 ) /\ the carrier of L2 c= the carrier of L1 ;
consider x9 being Element of GF ( p ) such that x9 |^ 2 = a and P [ x9 ] ;
reconsider ee = ee , fe = fe , fe = fe as Element of D * ;
ex O being set st O in S & C1 c= O & M . O = 0. <= M . ( f . O ) ;
consider n being Nat such that for m being Nat st n <= m holds S . m in U1 and S . n in U2 ;
f * g * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) . x ) . x0 ;
defpred P [ Nat ] means A + succ $1 = succ A & for n st n in dom A holds A . n = A . n + B . n ;
the left of - g = the left of g & the left of - g = the left of g ;
reconsider pp = x , pp = y , pp = z , pp = w , y2 = y , z2 = z as Point of TOP-REAL 2 ;
consider g3 such that g3 = y and x <= ex g1 , g2 st x <= g1 & g1 <= x0 & g1 in dom f & 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 .= len ( x2 ^ y2 ) ;
for x being element st x in X holds x in the set of set & x in the set of ( p | n0 ) . x implies x in X
LSeg ( p01 , p2 ) /\ LSeg ( p1 , p2 ) = {} & LSeg ( p1 , p2 ) /\ LSeg ( p2 , p2 ) = {} ;
func such set ( X ) -> set equals [: [: the carrier of X , the carrier of X :] , { id X } :] ;
len ( ( { CR /. 1 } ) | ( len ( C /^ 1 ) ) ) <= len ( C /^ 1 ) ;
attr K is with_a , a , b , c be Element of K , i be Nat st v . ( a |^ i ) = i * v . a ;
consider o being OperSymbol of S such that t `1 . {} = [ o , the carrier of S ] and o in rng t ;
for x st x in X ex y st x c= y & y in X & y is \rm NAT of f . x ;
IC Comput ( P-6 , sd , k ) in dom ( ( k + 1 ) .--> ( k + 1 ) ) ;
attr q < s & r < s implies ]. r , s .[ \not c= ]. p , q .[ ;
consider c being Element of Class ( f , 3 ) such that Y = ( F . c ) `1 and c in X ;
func the ResultSort of S2 -> Function of the carrier' of S2 , the carrier' of S2 means : Def1 : for x being Element of the carrier' of S2 holds it . x = id the carrier of S2 ;
set yxy = [ <* y , z *> , f2 ] ;
assume x in dom ( ( ( #Z 2 ) * ( arccot ) ) `| Z ) /\ dom ( ( arccot ) * ( arccot ) ) ;
r-7 in Int cell ( f , i , GoB f ) \ L~ f & ri2 in L~ f \/ L~ f ;
q `2 >= ( Cage ( C , n ) /. ( i + 1 ) ) `2 ;
set Y = { a "/\" a ` : a in X } ;
i - len f <= len f + len f1 - len f - len f + 2 - len f + 2 - len f ;
for n ex x st x in N & x in N1 & h . n = x0 + x
set sx0 = ( \mathop { a , I , p , s ) . i , sx0 = ( a , I ) +* ( i , p , s ) . i , sx0 = ( a , I ) +* ( i , p , s ) . i , sx0 = ( a , I ) +* ( i , I ) . i , sx0 = ( a , I ) +* (
p ( ) . k = 1 or p ( ) . k = - 1 or p ( ) . 0 = 1 or p . 1 = 0 ;
u + Sum ( Carrier ( L ) ) in ( U \ { u } ) \/ { u + Sum ( L ) } ;
consider x9 being set such that x in x9 and x9 in V1 and f . x9 = f . x and f . x = f . y ;
( p ^ ( q | k ) ) . m = ( q | k ) . ( ( len p ) - len q ) ;
g + h = gg + h1 & for x be Element of X holds Nat holds A1 ( g + h , X , X ) = g + h
L1 is distributive & L2 is distributive implies [: L1 , L2 :] is distributive & [: L1 , L2 :] is distributive & [: L1 , L2 :] is distributive
pred x in rng f & y in rng ( f | x ) implies f / x = f / y & f / y = f / ( y , x ) ;
assume that 1 < p and ( 1 - p ) * q + ( 1 - p ) * q = 1 and 0 <= a and a <= b ;
F* ( f , t ) = rpoly ( 1 , t ) *' t + ( 0. F_Complex ) *' ;
for X being set , A being Subset of X holds A ` = {} implies A = X & A = X
( ( N-min X ) `1 ) ^2 <= ( ( ( N-min X ) `1 ) ^2 + ( ( ( N-min X ) `2 ) ^2 ) ) ^2 ;
for c being Element of the \geq >= the \geq >= a , A being Element of the free of A holds c <> a
s1 . GBP = ( Exec ( i2 , s2 ) ) . GBP .= s2 . GBP .= Exec ( i2 , s2 ) . GBP .= s . GBP ;
for a , b being Real holds |[ a , b ]| in ( y >= 0 ) & b >= 0 implies b >= 0
for x , y being Element of X holds x ` \ y = ( x \ y ) ` & y \ x = ( x \ y ) `
mode BCK-algebra of i , j , m , n , m , m be BCK-algebra of i , j , n , m , m , n , m , m be Element of NAT ;
set x2 = |( Re y , Im x )| ;
[ y , x ] in dom u5 & u5 . ( y , x ) = g . y ;
]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .[ c= A & upper_bound divset ( D , k ) = upper_bound A ;
0 <= delta ( S2 ) . n & |. delta ( S2 ) . n .| < ( e / 2 ) / ( 2 |^ n ) ;
( - ( q `1 / |. q .| - cn ) ) ^2 <= ( - ( q `1 / |. q .| - cn ) ) ^2 ;
set A = ( 2 / b-a ) / ( 2 * b-a ) ;
for x , y being set st x in R" holds x , y are_\hbox { - } < x & y in R" { - 1 }
deffunc FF2 ( Nat ) = b . $1 * ( M * G ) . $1 & ( M * G ) . $1 = M . ( G . $1 ) ;
for s being element holds s in -> element iff s in -> Element of \rm \rm \rm \rm \rm \rm : [ f , g ] in \rm \rm \rm \rm \rm <= } ( f )
for S being non empty non void non empty non void holds S is connected iff S is connected
max ( degree ( z `1 ) , degree ( z `2 ) ) >= 0 & degree ( z `2 ) <= degree ( z `1 ) ;
consider n1 being Nat such that for k holds seq . ( n1 + k ) < r + s ;
Lin ( A /\ B ) is Subspace of Lin ( A ) & Lin ( B /\ A ) is Subspace of Lin ( B ) ;
set n-15 = np1 '&' ( M . x qua Element of BOOLEAN ) , np2 = M . x , np2 = M . x , np2 = M . y , np2 = M . x , np2 = M . y , np2 = M . y ;
f " V in ' ( X ) & f " V in D & f " V in D & f " V in D implies f " V in D
rng ( ( a ^\ c ) +* ( 1 , b ) ) c= { a , c , b } ;
consider y being as Wof G1 such that y `1 = y and dom y `1 = WWR `1 and y `2 = WWR `2 ;
dom ( 1 (#) f ) /\ ]. -infty , x0 .[ c= ]. -infty , x0 .[ /\ dom f /\ ]. x0 , x0 + r .[ ;
/\ f2 is {} of /\ ( i , j , n ) & f2 is Morphism of 0 , j , n , - r ;
v ^ ( n-3 |-> 0 ) in Lin ( rng ( B-9 | ( dom ( B-9 | dom ( B-9 | dom ( B-9 | dom ( B-9 | dom ( B-9 | dom ( BA | dom ( BA | dom ( B | dom ( B | dom ( B | dom ( B | dom ( B | dom ( B | dom ( B | dom ( B |
ex a , k1 , k2 st i = a := k1 & j = b := k2 & k2 = c := k2 ;
t . NAT = ( NAT .--> succ i1 ) . NAT .= ( NAT .--> succ i1 ) . NAT .= ( NAT --> succ i1 ) . NAT .= ( NAT --> succ i1 ) . NAT ;
assume that F is bbfamily and rng p = F and dom p = Seg ( n + 1 ) and for i be Nat st i in Seg ( n + 1 ) holds p . i = F . i ;
not LIN b , b9 , a & not LIN a , a9 , c & not LIN a , a9 , c & not LIN b9 , c9 , c
( L1 , L2 ) \& O c= ( L1 => O ) Let ( L2 , O ) Let ( L1 => O ) ;
consider F being ManySortedSet of E such that for d being Element of E holds F . d = F ( d ) and for d being Element of E holds F . d = F ( d ) ;
consider a , b such that a * ( ^2 + b ^2 ) = b * ( not w + b ^2 ) and 0 < a and 0 < b ;
defpred P [ FinSequence of D ] means |. Sum ( $1 ) .| <= Sum ( |. $1 .| ) & Sum ( $1 ) <= Sum ( |. $1 .| ) ;
u = cos / sin ( x , y ) . v * x + ( cos / cos ( x , y ) . v * y .= v ;
dist ( ( seq . n ) + x , g + x ) <= dist ( ( seq . n ) , g ) + 0 ;
P [ p , |. p .| \bullet p , {} ] & P [ p , id the bound is not bound Subset of A ] ;
consider X being Subset of CQC-WFF ( Al ) such that X c= Y and X is finite and X is ininP and X is ininP ;
|. b .| * |. eval ( f , z ) .| >= |. b .| * |. eval ( f , z ) .| ;
1 < ( ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ) .. Cage ( C , n ) ;
l in { l1 where l1 is Real : g <= l1 & l1 <= h . l1 & h . l1 <= g . ( l1 + 1 ) & l . ( l1 + 1 ) <= g . ( l1 + 1 ) } ;
Ser ( ( G . n ) vol ) <= ( Partial_Sums ( ( G . n ) vol ) . ( n + 1 ) ) vol ;
f . y = x .= x * ( 1_ L ) .= x * ( power L ) . ( y , 0 ) .= x * ( power L ) . ( y , 0 ) ;
NIC ( <% a , b %> , ( 0 , succ a ) ) = { i1 , succ b } .= { i1 , succ b } .= { succ a , succ b } ;
LSeg ( p00 , p2 ) /\ LSeg ( p1 , p2 ) = { p1 } /\ LSeg ( p1 , p2 ) .= { p1 } ;
Product ( ( the carrier of I-15 ) +* ( i , { 1 } ) ) in Z1 & Product ( ( i , { 1 } ) +* ( i , { 1 } ) ) in Z1 ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) .= Following ( s1 , n ) .= Exec ( i , s1 ) ;
W-bound ( Qs2 ) <= q1 `1 & q1 `1 <= ( E-bound ( Qs2 ) ) / 2 & ( W-bound ( Qs2 ) ) <= ( E-bound ( Qs2 ) ) / 2 ;
f /. i2 <> f /. ( ( i1 + len g -' 1 ) -' 1 ) & f /. ( ( i1 + len g -' 1 ) -' 1 ) = f /. ( i1 + 1 ) ;
M , f / ( x. 3 , a ) / ( x. 4 , a ) / ( x. 0 , a ) / ( x. 4 , a ) |= H / ( x. 4 , a ) ;
len ( ( P ^ ) | ( len ( P ^ ) ) ) in dom ( ( P ^ ) | ( len ( P ^ ) ) ) ;
A |^ ( mn ) c= A |^ ( m , n ) & A |^ ( ( k + 1 ) + 1 ) 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 p1 . n1 = p2 . n1 ;
consider X being set such that X in Q and for Z being set st Z in Q & Z <> X holds X \not c= Z ;
CurInstr ( P3 , Comput ( P3 , s2 , l ) ) <> halt SCM+FSA .= halt SCM+FSA .= ( l + 1 ) ;
for v be VECTOR of l1 holds ||. v .|| = upper_bound rng |. ( id the carrier of V ) - ( id the carrier of V ) .|
for phi holds phi in X implies not ( phi in X & not phi in X & not phi in X ) & ( phi in X implies phi in X & phi in X )
rng ( ( Sgm dom ( f | dom ( f | dom ( f | dom ( f | dom ( g | dom ( g | dom ( g | dom ( g | dom ( g | dom ( g | dom ( g | dom f ) ) ) ) ) ) ) ) ) c= dom ( f | dom ( g | dom ( g | dom ( g | dom ( g | dom ( g | dom ( g | dom ( g | dom ( g
ex c being FinSequence of D ( ) st len c = k & a = c & a = c & b = c ;
the_arity_of ( a , b , c ) = <* Hom ( b , c ) , Hom ( a , b ) *> ;
consider f1 be Function of the carrier of X , REAL such that f1 = |. f .| and f1 is continuous and f1 . 0 = f . 1 ;
a1 = b1 & a2 = b2 or a1 = b2 & a2 = b1 & a3 = b2 & a1 , a2 , a3 , a4 , a5 , a5 , a5 , a5 , a5 , a5 , a5 , a5 , a5 , a5 , a5 , a5 , 8 , 8 , a5 , 8 , 7 , 8 , 8 = 8 ;
D2 . indx ( D2 , D1 , n1 + 1 ) = D1 . ( n1 + 1 ) .= D1 . ( n1 + 1 ) ;
f . ( ||. r .|| ) = ||. |[ r .|| , r ]| .|| /. 1 .= <* r *> . 1 .= x ;
consider n being Nat such that for m being Nat st n <= m holds C-25 . m = C-25 . m and C-25 . n = C-25 . n ;
consider d being Real such that for a , b being Real st a in X & b in Y holds a <= d & d <= b ;
||. L /. h - ( K * |. h .| ) + ( K * |. h .| ) * ( K * |. h .| ) <= p0 + ( K * |. h .| ) * ( K * |. h .| ) ;
attr F is commutative means : Def1 : for b being Element of X holds F -Sum { b } = f . b ;
p = - ( - ( p `2 / |. p .| - sn ) ) * p0 + ( p `2 / |. p .| - sn ) * p0 .= 1 * ( ( p `2 / |. p .| - sn ) / ( 1 - sn ) ) * ( ( p `2 / |. p .| - sn ) * ( 1 - sn ) ) ;
consider z1 such that b , x3 , x3 is_collinear and o , x1 , x1 is_collinear and o , x2 , x3 is_collinear and o , x1 , x3 is_collinear ;
consider i such that Arg ( ( Rotate ( s , r ) ) . q ) = s + Arg q + ( 2 * PI * i ) ;
consider g such that g is one-to-one and dom g = card ( f . x ) and rng g = f . x and g is one-to-one ;
assume that A = P2 \/ Q2 and P2 <> {} and Q2 <> {} and P2 = Q2 and P2 = Q2 \/ Q2 and P2 = Q2 \/ Q2 ;
attr F is associative means : Def1 : F .: ( F .: ( f , g ) , h ) = F .: ( f , F .: ( g , h ) ) ;
ex x being Element of NAT st m = x `1 & x in z `1 & x < i or m in { i } ;
consider k2 being Nat such that k2 in dom P-2 and l in P-2 . k2 and ( P`1 ) . k2 = ( Pi ) . k2 ;
seq = r * seq implies for n holds seq . n = r * seq . n & seq is convergent & lim seq = r * seq . n
F1 . [ ( id a ) , ( id a ) ] = [ f * ( id a ) , f * ( id a ) ] ;
{ p } "\/" D2 = { p "\/" y where y is Element of L : y in D2 & p in D1 } \/ { p } ;
consider z being element such that z in dom ( ( F . z ) | X ) and ( ( F . z ) | X ) . z = 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 ) = { |[ r , s ]| : r <= G * ( 0 + 1 , 1 ) `1 } ;
consider e being element such that e in dom ( T | E1 ) and ( T | E1 ) . e = v and ( T | E1 ) . e = v ;
( F `1 * b1 ) . x = ( Mx2Tran ( J , sthen Z , BZ ) ) . ( \mathbb Z ) ;
- 1 / ( - 1 ) = mmD | n .= mmD | n .= mmD .= ( - 1_ K ) (#) ( - 1_ K ) .= ( Det M ) (#) ( - 1_ K ) ;
attr for x be 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 ) ;
All ( All ( 'not' a , A , G ) , B , G ) '<' Ex ( 'not' All ( 'not' a , B , G ) , A , G ) ;
LSeg ( E . k0 , F . k0 ) c= Cl RightComp Cage ( C , k0 + 1 ) & LSeg ( E . k0 , F . k ) c= RightComp Cage ( C , k0 + 1 ) ;
x \ a |^ m = x \ ( a |^ k * a ) .= ( x \ a ) |^ k \ a ;
k -inth -ininin-inin-in-in-in-in-in-in-in-in-in-in-in-in-in-in-in-in-in-in-s-in-s-in-s-in-s-in-s--in-s-in-s-in-s-in-sin-s-in-s-in--s---s-s-
for s being State of Aan holds Following ( s , n ) . 0 + ( n + 2 ) * n . 1 is stable ;
for x st x in Z holds f1 . x = a ^2 & ( f1 - f2 ) . x <> 0 & ( f1 - f2 ) . x <> 0 ;
support ( support ( support n ) \/ support ( support ( support n ) ) c= support ( max ( n , m ) ) \/ support ( support n ) ) ;
reconsider t = u as Function of ( the carrier of A ) /\ ( the carrier of B ) , the carrier' of C ;
- ( a * sqrt ( 1 + b ^2 ) ) <= - ( b * sqrt ( 1 + a ^2 ) ) ;
phi /. succ b1 = g . a & phi /. ( a . a ) = f . ( g . a ) & phi . ( a . a ) = f . ( g . a ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( ( F ^ <* p *> ) . i ) and i <> j ;
{ x1 , x2 , x3 , x4 } = { x1 } \/ { x2 , x3 , x4 } .= { x1 } \/ { x2 , x3 , x4 } .= { x1 , x2 , x3 , x4 } ;
the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U1 & the Sorts of U2 c= the Sorts of U2 implies ( the Sorts of U1 ) . ( ( the Sorts of U2 ) . ( ( the Sorts of U1 ) . ( ( the Sorts of U2 ) . ( ( the Sorts of U2 ) . ( ( the Sorts of U1 ) . ( ( the Sorts of U2 ) . ( ( the Sorts of U2 ) . ( ( the Sorts of U2
( ( - 2 * a * ( b - a ) ) / ( 2 * a ) ) ^2 - delta ( a , b , c ) ^2 > 0 ;
consider W00 such that for z being element holds z in W00 iff z in N ~ N & P [ z ] and ex x being element st x in N ~ N & P [ z ] ;
assume that ( the Arity of S ) . o = <* a *> and ( the Arity of S ) . o = r and ( the Arity of S ) . o = <* r *> ;
Z = dom ( ( exp_R (#) ( arccot ) ) `| Z ) /\ dom ( ( arccot (#) ( arccot ) ) `| Z ) .= dom ( ( arccot (#) ( arccot ) ) `| Z ) ;
sum ( f , SS1 ) is convergent & lim ( ( f | SS1 ) | SS2 ) = integral ( f , SS2 ) ;
( X ( ) . ( a9 => b9 ) => ( x => ( x9 => y9 ) ) ) => ( x9 => ( x9 => y9 ) ) in is ) implies X ( ) is set
len ( M2 * M3 ) = n & width ( M2 * M3 ) = n & len ( M2 * M3 ) = n & width ( M2 * M3 ) = n ;
attr X1 union X2 is open SubSpace of X means : Def1 : X1 , X2 are_separated & X1 , X2 are_separated & X2 , X1 are_separated & X1 , X2 are_separated & X2 , X1 are_separated ;
for L being upper-bounded antisymmetric RelStr , X being non empty Subset of L holds X "\/" { Top L } = { Top L }
reconsider f-1= F2 . ( b `2 ) , f-1' ( b `2 ) , f-1' ( b `2 ) as Function of M . b , M . ( b `2 ) ;
consider w being FinSequence of I such that the InitS of M , w -{ s } ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^
g . ( a |^ 0 ) = g . ( 1_ G ) .= 1_ H .= ( g . a ) |^ 0 .= ( g . a ) |^ 0 .= ( g . a ) |^ 0 ;
assume for i being Nat st i in dom f ex z being Element of L st f . i = rpoly ( 1 , z ) & f . i = rpoly ( 1 , z ) ;
ex L being Subset of X st Carrier L = C & for K being Subset of X st K in C holds L /\ K <> {} ;
( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C1 & ( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C2 ;
reconsider o-21 = o `1 , op = o `2 , op = o `1 , op = o `2 , op = o `1 , op = o `2 , op = o `2 , op = o `1 , op = o `2 , op = o `2 , op = o `1 , op = o `2 , op = o `1 , op = o `2 , op = o `1 , op = o `2 , op = o `1 , op
1 * x1 + ( 0 * x2 ) + ( 0 * x3 ) = x1 + <* \underbrace { 0 , \dots , 0 } , 0 , 0 , 0 *> .= x1 + x2 ;
Es " . 1 = ( Es qua Function ) " . 1 .= ( ( 1 - s ) " ) . 1 .= ( ( 1 - s ) " ) . 1 .= ( ( 1 - s ) " ) . 1 ;
reconsider u1 = the carrier of U1 /\ ( U1 "\/" U2 ) , u2 = the carrier of U2 /\ ( U1 "\/" U2 ) as non empty Subset of U0 ;
( ( x "/\" z ) "\/" ( x "/\" y ) ) "\/" ( z "/\" y ) <= ( x "/\" ( z "\/" y ) ) "\/" ( z "/\" ( x "\/" y ) ) ;
|. f . ( s1 . ( l1 + 1 ) ) - f . ( s1 . l1 ) .| < ( 1 - |. M .| ) * ( 1 - M ) ;
LSeg ( ( Lower_Seq ( C , n ) /. ( i + 1 ) ) , ( ( Cage ( C , n ) /. ( i + 1 ) ) /. ( i + 1 ) ) is vertical ;
( f | Z ) /. x - ( f | Z ) /. x0 = L /. ( x- x ) + R /. ( x- x ) ;
g . c * ( - g . c * f . c ) + f . c <= h . c * ( ( - 1 ) * ( - 1 ) * f . c ) + f . c ;
( f + g ) | divset ( D , i ) = f | divset ( D , i ) + g | divset ( D , i ) ;
assume that ColVec2Mx f in the carrier of A and ColVec2Mx b in the carrier of A and ColVec2Mx f = ( len A ) |-> ( len A ) and len f = width A and width ( f ) = width A ;
len ( - M3 ) = len M1 & width ( - M3 ) = width M1 & width ( - M3 ) = width M1 & width ( - M3 ) = width M2 ;
for n , i being Nat st i + 1 < n holds [ i , i + 1 ] in the InternalRel of ( ( ( the carrier of n ) --> { i } ) \/ ( the InternalRel of n ) )
pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 implies pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 2
attr a <> 0 & b <> 0 & Arg a = Arg b implies Arg ( - a ) = Arg ( - b ) & Arg ( - b ) = Arg ( - b ) ;
for c being set st not c in [. a , b .] holds not c in Intersection ( the REAL of a , b ) & not c in Intersection ( the topology of a , b )
assume that V1 is linearly-independent and V2 is linearly-independent and V2 = { v + u : v in V1 & u in V2 & v in V2 } and V1 is open and V1 is open and V1 is open and V1 is open and V2 is open and V1 is open ;
z * x1 + ( 1 - z ) * x2 in M & z * y1 + ( 1 - z ) * y2 in N implies z * y1 + ( 1 - z ) * y2 in N
rng ( ( PS1 qua Function ) " * SS2 ) = Seg ( card dS2 ) .= Seg ( card dS2 ) .= Seg ( card dom ( PS2 ) ) .= dom ( ( PS2 ) " * SS2 ) ;
consider s2 being rational number such that s2 is convergent and b = lim s2 and for n holds s2 . n <= b . n and s2 . n <= b . n ;
h2 " . n = h2 . n " & 0 < ( - 1 / ( ( 1 - ( ( 1 - ( ( 1 / 2 ) |^ n ) * ( 1 / 2 ) ) * ( 1 / 2 ) ) ) ) / ( 2 * ( ( 1 - ( ( 1 / 2 ) |^ n ) * ( 1 / 2 ) ) ) ) ;
( Partial_Sums ( ||. seq1 .|| ) ) . m = ||. seq1 .|| . m .= ||. ( seq1 . m ) - ( seq2 . m ) .|| .= ||. ( seq1 . m ) - ( seq2 . m ) .|| .= 0 ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b ;
- v = ( - 1_ ( Gv ) ) * v & - w = ( - 1_ ( Gw ) ) * v & - w = ( - 1_ ( Gw ) ) * v ;
sup ( [: k , D :] .: D ) = sup ( ( k , D ) .: D ) .= k ( ) . sup D .= k ( ) . sup D .= k ( ) . sup D ;
A |^ ( k , l ) ^^ ( A |^ ( n , .. A ) ) = ( A |^ ( k , .. A ) ) ^^ ( A |^ ( k , .. A ) ) ;
for R being add-associative right_zeroed right_complementable non empty addLoopStr , I , J being Subset of R , K being Subset of R holds I + ( J + K ) = ( I + J ) + K
( f . p ) `1 = ( p `1 ) ^2 + sqrt ( 1 + ( p `2 / p `1 ) ^2 ) .= ( p `1 ) ^2 + ( p `2 / p `2 ) ^2 ;
for a , b being non zero Nat st a , b are_relative_prime holds support ( a * b ) = support ( a ) + support ( b )
consider A9 being countable set such that r is countable and A9 is ( len A ) -element and A9 is ( len A ) -element & ( A . 0 = A . 0 implies A9 is ( len A ) element ) `1 = ( A . 0 ) `1 ;
for X being non empty addLoopStr , M being Subset of X , x , y being Point of X st y in M holds x + y in x + M
{ [ x1 , x2 ] , [ y1 , y2 ] } c= [: { x1 } , { x2 } :] & { x1 , x2 } = [: { y1 } , { y2 } :] ;
h . ( f . O ) = |[ A * ( f . O ) `1 + B , C * ( f . O ) `2 + D ]| ;
( Gauge ( C , n ) * ( k , i ) ) /. k in L~ Lower_Seq ( C , n ) /\ L~ Lower_Seq ( C , n ) ;
cluster m , n are_relative_prime -> prime implies for Nat st not ( p divides m & not p divides n & not p divides n ) & not ( p divides n & not p divides n ) & not ( p divides n implies p divides n )
( f * F ) . x1 = f . ( F . x1 ) & ( f * F ) . x2 = f . ( F . x2 ) ;
for L being LATTICE , a , b , c being Element of L st a \ b <= c & b \ a <= c holds a \+\ b <= c & a \+\ c <= c
consider b being element such that b in dom ( H / ( x , y ) ) and z = ( H / ( x , y ) ) . b and b in { x } ;
assume that x in dom ( F * g ) and y in dom ( F * g ) and ( F * g ) . x = ( F * g ) . y ;
assume ex e being element st e Joins W . 1 , W . 5 , G or e Joins W . 3 , W . 7 , G ;
( ( h (#) f ) . ( 2 * n ) ) . x = ( ( h * f ) . ( 2 * n ) ) . x + ( ( h * f ) . ( 2 * n ) ) . x ;
j + 1 = ( i - len h11 + 2 ) + 1 .= i + 1 - len h11 + 2 - 1 .= i + 2 - 1 .= i + 2 - 1 ;
*' ( S *' ) . f = S *' . ( ( S *' ) . f ) .= S . ( ( S *' ) . f ) .= S . ( ( S *' ) . f ) .= S . ( ( S *' ) . f ) ;
consider H such that H is one-to-one and rng H = the carrier of L2 and Sum ( L2 ) = Sum ( L1 ) and Sum ( L1 ) = Sum ( L2 ) ;
attr R is + `2 means : Def1 : p in R & p <> q implies ex P st P is_special arc p & P c= R & p in P & q in P ;
dom product ( ( X --> f ) . i ) = meet ( dom ( X --> f ) . i ) .= meet ( X --> f . i ) .= meet ( X --> f . i ) .= ( X --> f . i ) . i .= ( X --> f . i ) . i ;
upper_bound ( proj2 .: ( Upper_Arc ( C ) /\ Upper_Arc ( C ) /\ Vertical_Line ( w ) ) ) <= upper_bound ( proj2 .: ( C /\ Vertical_Line ( 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 - pp .| < r
i * f-28 - fc = i * f-28 - ( i * fand i * f] = i * ( f-28 - ( i * f-32 ) ) .= i * ( f] - ( i * f-32 ) ) ;
consider f being Function such that dom f = 2 -tuples_on X and for Y being set st Y in 2 -tuples_on X holds f . Y = F ( Y ) ;
consider g1 , g2 being element such that g1 in [#] Y and g2 in union C and g = [ g1 , g2 ] and g1 in C and g2 in C and g2 in D ;
func d |-count n -> Nat means : Def1 : d |^ n divides n & d |^ ( n + 1 ) divides n & d |^ ( n + 1 ) divides n ;
f\rm . [ 0 , t ] = f . [ 0 , t ] .= ( - P ) . ( 2 * x ) .= ( - P ) . ( 2 * x ) .= a ;
t = h . D or t = h . B or t = h . C or t = h . E or t = h . F or t = h . J or t = M . N or t = N ;
consider m1 be Nat such that for n st n >= m1 holds dist ( ( seq . n ) , ( seq . n ) ) < 1 / ( ( seq . n ) ^2 ) ;
( q `1 ) ^2 / ( q `2 ) ^2 <= ( q `1 ) ^2 / ( q `2 ) ^2 / ( q `2 ) ^2 ;
h0 . ( i + 1 + 1 ) = h21 . ( i + 1 + 1 - len h11 + 2 -' 1 ) .= h21 . ( i + 1 + 1 ) ;
consider o being Element of the carrier' of S , x2 being Element of { the carrier of S } such that a = [ o , x2 ] and [ o , x2 ] in the carrier' of S ;
for L being RelStr , a , b being Element of L holds a is_<=_than { b } iff a is_<=_than { b } & a is_>=_than { b } & b in { b }
||. h1 .|| . n = ||. h1 . n .|| .= |. h . n .| .= ||. h . n .|| .= ||. h . n .|| .= ||. h . n .|| ;
( ( - exp_R ) (#) ( exp_R * f ) ) . x = f . x - ( exp_R * f ) . x .= ( ( - exp_R ) (#) ( exp_R * f ) ) . x .= ( ( - exp_R ) (#) ( exp_R * f ) ) . x ;
attr r = F .: ( p , q ) means : Def1 : len r = min ( len p , len q ) ;
( rl / 2 ) ^2 + ( rl / 2 ) ^2 <= ( r / 2 ) ^2 + ( r / 2 ) ^2 + ( r / 2 ) ^2 ;
for i be Nat , M be Matrix of n , K st i in Seg n holds Det ( M @ ) = Sum ( ( Det M ) @ )
then a <> 0. R & a " * ( a * v ) = 1 * v & a " * v = 1 * v & a " * v = 1 * v ;
p . ( j - 1 ) * ( q *' r ) . ( i + 1 -' j ) = Sum ( ( p . j - 1 ) * r3 ) .= Sum ( ( p . j - 1 ) * r3 ) ;
deffunc F ( Nat ) = L . 1 + ( ( R /* ( h ^\ n ) ) * ( h ^\ n ) " ) . $1 & ( ( R /* ( h ^\ n ) ) " ) . $1 = ( ( R /* ( h ^\ n ) ) " ) . $1 ;
assume that the carrier of H2 = f .: ( the carrier of H1 ) and the carrier of H2 = f .: ( the carrier of H2 ) and the carrier of H2 = the carrier of H2 and the carrier of H2 = the carrier of H2 ;
Args ( o , Free ( S , X ) ) = ( ( the Sorts of Free ( S , X ) ) * the Arity of S ) . o .= ( the Sorts of A ) . o ;
H1 = n + 1 & |. 2 to_power ( n + 1 ) + h .| = n + 1 implies n <= n + 1 & n <= len ( n + 1 ) & n <= len ( n + 1 )
( O = 0 & O = 0 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 or O = 1 & O = 1 & O = 1 or O = 1 ) & O = 1 ;
F1 .: ( dom F1 /\ dom F2 ) = F1 .: ( ( 1 - 2 ) * ( F /. ( n + 2 ) ) ) .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } ;
attr b <> 0 & d <> 0 & b <> d & b = ( e - d ) / ( b - d ) implies ( a - b ) / ( b - d ) = ( ( ( a - e ) / ( b - d ) ) / ( b - d ) )
dom ( ( f +* g ) | D ) = dom ( f +* g ) /\ D .= ( dom f \/ dom g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D ;
for i be set st i in dom g ex u , v be Element of L , a be Element of B st g /. i = u * a * v & u in B * v
g `2 * P * g " = g `2 * ( g `2 * P `1 ) * g `2 .= g `2 * ( g `2 * P `1 ) .= g `2 * ( g `2 * P `1 ) ;
consider i , s1 such that f . i = s1 and not ( ex s1 st i in dom s1 & s1 . i <> s1 . s1 ) & not ( s1 . i = s1 . s1 ) & not s1 . i = s1 . s1 ;
h5 | ]. a , b .[ = ( g | Z ) | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ ;
[ s1 , t1 ] , [ t2 , t2 ] are_connected & [ s2 , t2 ] , [ t2 , t2 ] are_connected & [ t2 , t2 ] , [ t2 , t2 ] are_connected ;
then that H is negative and H is not negative and H is not conjunctive and not H is not implies H is not implies not H is not implies not H is not implies not H is not implies not H is not implies not H is not implies not H is not implies not H is not implies not H is not implies not H is not implies not H is not implies not H is not implies not H is not implies not implies H is not implies not H is not implies not implies H is not implies not H is not implies not implies not implies not implies not implies not implies not implies not implies not H is not implies
attr f1 is total means : Def1 : 1 / ( f1 - f2 ) is total & ( for c st c in dom f1 holds f1 . c = f1 . c ) & ( f1 - f2 ) . c = f2 . c ) implies f1 is total ;
z1 in W2 -Seg ( z2 ) or z1 = z2 & not z1 in W2 & not z1 in W2 & not z2 in W2 & not z1 in W2 & not z1 in W1 & not z2 in W2 & not z1 in W2
p = 1 * p .= a " * a * p .= a " * ( b * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= ( a " * b " * q ) * ( b " * q ) ;
for seq1 be Real_Sequence for K be Real st for n be Nat holds seq1 . n <= K holds upper_bound rng ( seq1 ^\ n ) <= upper_bound ( ( seq ^\ n ) ^\ k )
C meets L~ go \/ L~ pion1 or C meets L~ pion1 or C meets L~ pion1 or C meets L~ pion1 or C meets L~ pion1 or C meets L~ pion1 or C meets L~ pion1 or C c= L~ pion1 or C c= L~ pion1 or C c= L~ pion1 or C c= L~ pion1 or C c= L~ pion1 or C c= L~ pion1 or C c= L~ pion1 \/ L~ pion1 ;
||. f . ( g . ( k + 1 ) ) - g . ( g . k ) .|| <= ||. g . 1 - g . 0 .|| * ( K * ( K to_power k ) ) ;
assume h = ( ( B .--> B ' ) +* ( C .--> D ' ) +* ( E .--> F ' ) +* ( F .--> J ' ) +* ( J .--> M ' ) +* ( M .--> N ) +* ( N .--> N ' ) ) +* ( N .--> A ' ) +* ( N .--> N ' ) +* ( N .--> A ' ) +* ( N .--> N ' ) ;
|. ( ( upper_volume ( H . n , T ) || A ) . k - ( ( lower ) | A ) . k ) .| <= e * ( ( 2 * ( n + 1 ) ) / ( n + 1 ) ) ;
( ( ( the Sorts of A ) . i ) . e ) . e = [ the \rbrace at ( v , the carrier of IC ) . e , the carrier of IC ] .= [ the carrier of IC , the carrier of IC ] -tree q ;
{ x1 , x1 , x1 , x2 , x3 , x4 , x4 , x5 , x5 , x5 , x5 , x5 , 7 } = { x1 , x2 , x3 , x4 , x5 , x5 , 7 } \/ { x1 , x2 , x3 , x4 } .= { x1 , x2 , x3 , x4 } ;
assume that A = [. 0 , 2 * PI .] and integral ( ( #Z n ) * sin , A ) = 0 and ( ( #Z n ) * sin ) . x = 0 ;
p `2 is Permutation of dom ( f1 /. i ) & p `2 " = ( ( Sgm Y ) " ) * p & p `2 " = ( Sgm Y ) " * p & p `2 in dom ( Sgm Y ) ;
for x , y st x in A holds |. ( 1 / f ) . x - ( 1 / f ) . y .| <= 1 * |. f . x - f . y .|
p2 `2 = |. q2 .| * ( ( q2 `2 / |. q2 .| - sn ) / ( 1 + sn ) ) - sn * ( ( q2 `2 / |. q2 .| - sn ) / ( 1 + sn ) ) ;
for f be PartFunc of the carrier of RNS , REAL st dom f is compact & f is_continuous_on dom f holds ( for x be Element of NAT st x in dom f holds f . x is compact ) & ( for x be Element of NAT st x in dom f holds f . x is compact ) implies f is compact
assume for x being Element of Y st x in EqClass ( z , CompF ( B , G ) ) holds ( Ex ( a , A ) ) . x = TRUE ;
consider FM such that dom FM = n1 and for k be Nat st k in n1 holds Q [ k , FM . k ] and ( for k be Nat st k in n1 holds Q [ k , FM . k ] ) ;
ex u , u1 st u <> u1 & u , u1 / v / ( a , b ) / ( a , v ) / ( a , b ) / ( a , v ) / ( a , b ) / ( a , v ) = ( u / v ) / ( a , b ) ;
for G being Group , A , B being non empty Subset of G , N being normal Subgroup of G holds ( N -A ) * ( N -B ) = N ~ A * N
for s be Real st s in dom F holds F . s = integral ( R / ( f + g ) ) - Integral ( M , ( f + g ) / ( f + g ) / ( f - g ) ) . x
width AutMt ( f1 , b1 , b2 ) = len b2 .= len ( b1 ^ b2 ) .= len b1 + len b2 .= len b1 + len b2 .= len b1 + len b2 .= len b1 + len b2 .= len b1 + len b2 ;
f | ]. - PI / 2 , PI / 2 .[ = f & dom f = ]. - 1 , PI / 2 .[ & for x st x in ]. - 1 , PI / 2 .[ holds f . x = - 1 / 2 * x + 1 / 2 * x ;
assume that X is closed w.r.t. being set and a in X and a c= X and y in a ^ { f . [ n , x ] } \/ y and x in a ;
Z = dom ( ( #Z 2 ) * ( arctan ) ) /\ dom ( ( #Z 2 ) * ( arctan ) ) .= dom ( ( #Z 2 ) * ( arctan ) ) /\ dom ( ( #Z 2 ) * ( arctan ) ) .= dom ( ( #Z 2 ) * ( arctan ) ) /\ dom ( ( #Z 2 ) * ( arctan ) ) ;
func [: V , l :] -> Subset of V means : Def1 : 1 <= k & k <= len l & 1 <= l & l . k in V & l . l in V & l . k in V ;
for L being non empty TopSpace , N being net of L , M being net of N , c being Point of L st c is_carrier N holds c in N & c in N
for s being Element of NAT holds ( ( id C\mathop ( V , C ) ) + ( id C\mathop ( V , C ) ) ) . s = ( ( id C\mathop ( V , C ) ) + ( id C\mathop ( V , C ) ) ) . s
then z /. 1 = ( N-min L~ z ) .. z & ( N-min L~ z ) .. z < ( N-min L~ z ) .. z & ( N-min L~ z ) .. z < ( N-min L~ z ) .. z ;
len ( p ^ <* ( 0 qua Real ) * ( 0 qua Real ) *> ) = len p + len <* ( 0 qua Real ) * ( 0 qua Real ) *> .= len p + 1 .= len p + 1 ;
assume that Z c= dom ( - ( ln * f ) ) and for x st x in Z holds f . x = x and f . x > 0 and f . x < 1 ;
for R being add-associative right_zeroed right_complementable commutative associative non empty doubleLoopStr , I being non empty Subset of R , J being non empty Subset of R , I being non empty Subset of R holds ( I + J ) *' ( I /\ J ) c= I /\ J
consider f being Function of [: B1 , B2 :] , B12 such that for x being Element of [: B1 , B2 :] holds f . x = F ( x ) and f . x = F ( x ) ;
dom ( x2 + y2 ) = Seg len x .= Seg len ( x2 ) .= Seg len ( x2 ) .= dom ( x (#) ( y (#) z ) ) .= dom ( x (#) ( y (#) z ) ) .= dom ( x (#) ( y (#) z ) ) .= dom ( x (#) ( y (#) z ) ) ;
for S being card Functor of C , B for c being Object of C holds ( id C ) . ( id c ) = id ( ( Obj S ) . c ) & ( id C ) . ( id c ) = id ( ( Obj S ) . c )
ex a st a = a2 & a in f6 /\ f5 & for x st x in f6 holds \rrangle in \mathop { f . x : f . x = f . x & f . x = f . x } ;
a in Free ( H2 / ( x. 4 , x. 0 ) ) '&' H2 '&' ( H2 / ( x. 0 , x. 4 ) ) '&' H2 / ( x. 0 , x. 4 ) ) ;
for C1 , C2 being v1 , f being Function of C1 , C2 , g being Function of C1 , C2 st `1 = `1 & C2 = C2 holds f = g & f = g
( W-min L~ go \/ L~ co ) `1 = W-bound L~ go \/ E-bound L~ co & ( W-min L~ go \/ L~ co ) `1 = W-bound L~ go \/ E-bound L~ co or ( W-min L~ co ) `1 = E-bound L~ co & ( W-min L~ go ) `1 = E-bound L~ go & ( W-min L~ go ) `1 = E-bound L~ pion1 ;
assume that u = <* x0 , y0 , z0 *> and f is_differentiable_in x0 and SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) is_differentiable_in z0 and SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) is_differentiable_in z0 ;
then ( t . {} ) `1 in Vars & ex x being Element of Vars st x = ( t . {} ) `1 & t . {} = x & ( t . {} ) `2 = ( x , s ) `1 ;
Valid ( p '&' p , J ) . v = Valid ( p , J ) . v '&' Valid ( p , J ) . v .= Valid ( p , J ) . v '&' Valid ( q , J ) . v .= Valid ( p , J ) . v ;
assume for x , y being Element of S st x <= y for a , b being Element of T st a = f . x & b = f . y holds a >= b & b >= y ;
func Class R -> Subset-Family of R means : Def1 : for A being Subset of R holds A in it iff ex a being Element of R st a in Class ( R , a ) & it = Class ( R , a ) ;
defpred P [ Nat ] means ( ( ( ( ( ( ( ( ( ( ) . $1 ) `1 ) . ( n + 1 ) ) `1 ) . $1 ) `1 ) . $1 ) `1 c= G . ( ( ( ( ( ( ( ( G . $1 ) `1 ) . n ) `1 ) . $1 ) `1 ) . $1 ) `1 ) ;
assume that dim W1 = 0 and dim ( W1 ) = 0 and for v be Element of ( W1 ) st v in W1 holds ( v in W2 implies ( v in W1 & v in W2 implies v in W1 & v in W2 ) & ( v in W1 implies v in W2 implies v in W1 ) & ( v in W2 implies v in W1 implies v in W2 ) ;
mamas ( m ) . t = ( m . t ) `1 .= ( m . {} ) `1 .= ( m . {} ) `1 .= ( m . {} ) `1 .= m . {} .= m . {} .= m . {} ;
d11 = x11 ^ d22 .= f . ( y11 , d22 ) .= f . ( y22 , d22 ) .= ( f | ( dom f ) ) . ( y22 , d22 ) .= ( f | ( dom f ) ) . ( d22 ) .= ( f | ( dom f ) ) . ( d22 ) .= ( f | ( dom f ) ) . ( d22 ) .= ( f | ( dom f ) ) . ( d22 ) ;
consider g such that x = g and dom g = dom fx0 and for x being element st x in dom fx0 holds g . x in fx0 and g . x in fx0 and g . x in fx0 ;
x + 0. F_Complex |^ 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 ;
( k -' ( k + 1 ) ) + 1 in dom ( f /. ( ( k -' ( k + 1 ) ) + 1 ) ) /\ ( ( k + 1 ) -' ( k + 1 ) ) ;
assume that P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P2 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P2 = P1 \/ P2 and P1 = P1 \/ P2 and P2 = P1 \/ P2 and P1 = P1 \/ P2 and P2 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P2 \/ P2 and P1 = P1 \/ P2 and P1 = P2 \/ P2 and P1 = P1 \/ P2 and P2 = P1 \/ P2 and P2 = P1 \/ P2 and P2 = P1 \/ P2 and P2 = P1 \/ P2 and P2 = P1 \/ P2 and P2 = P1 \/ P2 and P2 = P1 \/ P2 and P2 = P1 \/ P2 and P2 = P1 \/ P2 and P2
reconsider a1 = a , b1 = b , c1 = p `1 , c1 = p `2 , c2 = p `1 , c2 = p `2 , c2 = p `2 , c1 = p `1 , c2 = p `2 , c2 = p `2 , c2 = p `1 , c2 = p `2 , c1 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c1 = p `2 , c2 = p `2 , c2 = p `1 , c1 = p `1 , c2 = p `1 , c2 = p `1 , c2 = p `2 , M = p `2 , M = p `2 , M = p `1 , c2 = p `2 , c2 = p `2 , M = p `2 , c2 = p `2 ,
reconsider set set set _ t1f = G1 . ( t , b ) * F1 . f , F2 = G1 . ( a , b ) * F2 . f , F2 = G2 . ( a , b ) * F2 . f as Morphism of ( G1 * F2 ) . a , ( G1 * F2 ) . b * F2 . a ;
LSeg ( f , i + i1 -' 1 ) = LSeg ( f /. ( i + i1 -' 1 ) , f /. ( i + i1 -' 1 + 1 ) ) .= LSeg ( f /. ( i + i1 -' 1 ) , f /. ( i + i1 -' 1 + 1 ) ) ;
Integral ( M , P . m ) | dom ( P . n -P . m ) <= Integral ( M , P . n -P . m ) + Integral ( M , P . n -to_power k ) ;
assume that dom f1 = dom f2 and for x , y being element st [ x , y ] in dom f1 holds f1 . ( x , y ) = f2 . ( x , y ) and f2 . ( x , y ) = f2 . ( x , y ) ;
consider v such that v = y and dist ( u , v ) < min ( ( G * ( i , 1 ) `1 - G * ( i + 1 , 1 ) `1 ) , ( G * ( i + 1 , 1 ) `2 - G * ( i + 1 , 1 ) `2 ) ) ;
for G being Group , H being Subgroup of G , a being Integer , b being Integer st a = b holds for i being Integer holds a |^ i = b |^ i & a |^ i = b |^ i
consider B being Function of Seg ( S + L ) , the carrier of V1 such that for x being element st x in Seg ( S + L ) holds P [ x , B . x ] ;
reconsider K1 = { 7 where 7 is Point of TOP-REAL 2 : P [ 7 ] & for p being Point of TOP-REAL 2 st p in P holds not p `2 >= 0 & p <> 0. TOP-REAL 2 } ;
( ( ( ( ( ( ( O - S ) / 2 ) - ( O - S ) / 2 ) / 2 ) ) / 2 ) * ( ( ( ( ( O - S ) / 2 ) / 2 ) / 2 ) ) / 2 ) <= ( ( ( ( ( ( O - S ) / 2 ) / 2 ) * ( ( ( O - S ) / 2 ) / 2 ) ) / 2 ) ;
for x be Element of X , n be Nat st x in E holds |. Re F . n .| <= P . x & |. Im F . n .| <= P . x & Im ( F . n ) <= P . x
len @ ( @ p ^ q ) = len ( @ p ^ <* 0 *> ) + len <* 2 *> .= len ( @ p ^ <* 1 *> ) + 1 .= len ( @ p ^ <* 1 *> ) + 1 ;
v / ( x. 3 , m1 ) / ( x. 4 , m2 ) / ( x. 0 , m3 ) / ( x. 4 , m3 ) / ( x. 0 , m3 ) / ( x. 4 , m3 ) / ( x. 0 , m3 ) / ( x. 4 , m3 ) / ( x. 0 , m3 ) / ( x. 4 , m3 ) ) = m3 ;
consider r being Element of M such that M , v2 / ( x. 3 , m3 ) / ( x. 4 , m3 ) / ( x. 0 , m3 ) / ( x. 4 , m3 ) / ( x. 0 , m3 ) / ( x. 4 , m3 ) / ( x. 0 , m3 ) / ( x. 4 , m3 ) / ( x. 0 , m3 ) / ( x. 4 , m3 ) / ( x. 0 , m3 ) ) |= H ;
func w1 \ w2 -> Element of Union ( G , Rw ) means : Def1 : for w1 , w2 being Element of Union ( G , Rw ) holds it . ( w1 , w2 ) = ( ( ( ( ( ( G . i ) \ ( G . j ) ) /\ ( G . j ) ) /\ ( G . j ) ) ) . ( w1 , w2 ) ;
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= s1 . b2 .= s2 . b2 .= Exec ( n2 , s1 ) . b2 .= Exec ( n2 , s2 ) . b2 .= s . b2 .= s . b2 ;
for n , k be Nat holds 0 <= ( Partial_Sums |. seq .| . ( n + k ) - Partial_Sums ( |. seq .| . n ) ) . ( n + k ) - Partial_Sums ( |. seq .| . n ) . ( n + k )
set F = S -^ <* S *> ;
( Partial_Sums ( seq ) . K + Partial_Sums ( seq ) . ( K + 1 ) ) . ( K + 1 ) >= ( Partial_Sums ( seq ) . K + Partial_Sums ( seq ) . ( K + 1 ) ) . ( K + 1 ) ;
consider L , R such that for x st x in N holds ( f | Z ) . x - ( f | Z ) . x0 = L . ( x- x0 ) + R . ( x- x0 ) ;
func \HM { a , b , c , d \HM { a , b , c } : a = the Element of \HM { a , b , c , d } & b = the Element of \HM { a , b , c , d } & c = d } = { a , b , c , d } ;
a * b ^2 + ( a * c ^2 + b * a ^2 ) + ( b * c ^2 + ( c * a ^2 ) + ( c * b ^2 + ( c * a ^2 ) ) >= 6 * a * b * c * a * c + ( b * a ^2 ) + ( c * a ^2 + ( c * b ^2 + ( c * a ^2 ) ) ;
v / ( x1 , m1 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x3 , m3 ) = v / ( x2 , m1 ) / ( x3 , m3 ) / ( x4 , m3 ) / ( x4 , m3 ) ;
Assume that c= ( Q ^ <* x *> ) and M = ( Q +* ( M +* ( i , 0 ) ) ) +* ( ( M +* ( i , 0 ) ) +* ( ( M +* ( i , 0 ) --> FALSE ) ) ) and ( M +* ( i , 0 ) ) +* ( ( M +* ( i , 0 ) --> TRUE ) ) = ( M +* ( i , 0 ) ) +* ( i , 0 ) ) ;
Sum ( FM ) = r |^ n1 * Sum ( CM ) .= C ( n1 ) * ( CM ) .= C ( n1 ) * ( CM ) .= C ( n1 ) * ( CM ) .= ( C ( n1 ) ) * ( C ( n1 ) ) .= C ( n1 ) ;
( GoB f ) * ( len GoB f , 2 ) `1 = ( ( GoB f ) * ( len GoB f , 1 ) ) `1 .= ( ( GoB f ) * ( len GoB f , 1 ) ) `1 .= ( ( GoB f ) * ( len GoB f , 1 ) ) `1 .= ( ( GoB f ) * ( len GoB f , 1 ) ) `1 ;
defpred X [ Element of NAT ] means ( Partial_Sums s ) . $1 = ( ( a * ( $1 + 1 ) ) * ( $1 + 1 ) ) * ( $1 + 1 ) + b * ( $1 + 1 ) * ( $1 + 1 ) * ( $1 + 1 ) ) ;
( the_arity_of g ) . g = ( the Arity of S ) . g .= ( ( the Arity of S ) * g ) . g .= ( ( the Arity of S ) * g ) . g .= ( ( the Arity of S ) * g ) . g .= ( ( the Arity of S ) * g ) . g .= ( ( the Arity of S ) * g ) . g ;
( [: X , Y :] |^ Z ) tolerates [: X , Y :] & card ( [: X , Y :] |^ Z ) = card [: X , Y :] & card ( [: X , Y :] |^ Z ) = card [: X , Y :] ;
for a , b being Element of S , s being Element of NAT st s = n & a = F . n & b = F . n holds b = N . ( n + 1 ) \ G . s ;
E , f |= All ( x. 2 , ( x. 3 , x. 4 4 4 4 4 4 4 , x. 0 ) ) => ( x. 3 , ( x. 4 , x. 0 ) carrier x. 0 ) '&' ( x. 4 , x. 0 ) '&' ( x. 4 , x. 0 ) ) '&' ( x. 4 , x. 0 ) '&' ( x. 4 , x. 0 ) '&' ( x. 4 , x. 0 ) '&' ( x. 0 , x. 4 ) ) = x. 4 ;
ex R2 being 1-sorted st R2 = ( p | n-11 ) . i & ( the carrier of p ) c= the carrier of R2 & ( the carrier of p ) c= the carrier of R2 & ( the carrier of p ) c= the carrier of R2 ;
[. a , b + 1 / ( k + 1 ) .[ is Element of the \in of the non empty set & ( the partial F of f ) . k is Element of the carrier of a & ( the partial F of f ) . k is Element of the carrier of a & ( the Subset of f ) . k is Element of S ;
Comput ( P , s , 2 + 1 ) = Exec ( P . 2 , Comput ( P , s , 2 ) ) .= Exec ( a3 , s ) .= Exec ( a3 , s ) .= s ;
card ( h1 ) . k = power ( F_Complex ) . ( ( - 1_ F_Complex ) * ( - 1_ F_Complex ) ) .= ( ( - 1_ F_Complex ) * ( - 1_ F_Complex ) ) . k .= ( ( - 1_ F_Complex ) * ( - 1_ F_Complex ) ) . k .= ( ( - 1_ F_Complex ) * ( - 1_ F_Complex ) ) . k ;
( f / g ) /. c = f /. c * ( g /. c ) " .= f /. c * ( ( 1 - g ) * ( ( 1 - g ) * ( f ^ ) ) /. c .= ( f (#) ( ( 1 - g ) * ( f ^ ) ) ) /. c .= ( f (#) ( ( 1 - g ) * ( f ^ ) ) ) /. c ;
len CC - len ( ( C /. ( len C -' 1 ) ) * ( ( C /. ( len C -' 1 ) ) * ( 1 , 1 ) ) ) = len CC - len ( ( C /. ( len C -' 1 ) ) * ( 1 , 1 ) ) .= len ( ( C /. ( len C -' 1 ) ) * ( 1 , 1 ) ) ;
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom f /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) ( f | X ) ) /\ X .= dom ( r (#) ( f | X ) ) /\ X .= dom ( r (#) ( f | X ) ) ;
defpred P [ Nat ] means for n holds 2 * Fib ( n + $1 ) = Fib ( n ) * Fib ( n + $1 ) + ( 5 * Fib ( n + $1 ) * Fib ( n + $1 ) ) ;
consider f being Function of INT , INT such that f = f `1 and f is onto and for n holds f " { f . n } = { n + 1 } and f " { f . n } = { n } ;
consider c9 be Function of S , BOOLEAN such that c9 = chi ( A \/ B , S ) and E7 : ( A \/ B = Prob ( c , S ) ) and ( B = Prob ( c , S ) ) and ( C = Prob ( c , S ) ) ;
consider y being Element of Y ( ) such that a = "\/" ( { F ( x , y ) where x is Element of X ( ) : P [ x ] } , L ( ) ) and Q [ y ] ;
assume that A c= Z and f = ( #Z 2 ) * ( ( #Z 2 ) * ( sin + cos ) ) / ( sin + cos ) and Z c= dom ( ( #Z 2 ) * ( sin + cos ) ) and Z = dom ( ( #Z 2 ) * ( sin + cos ) ) / ( sin + cos ) ) ;
( f /. i ) `2 = ( GoB f ) * ( 1 , j2 ) `2 .= ( GoB f ) * ( 1 , j2 ) `2 .= ( GoB f ) * ( 1 , j2 ) `2 .= ( GoB f ) * ( 1 , j2 ) `2 .= ( GoB f ) * ( 1 , j2 ) `2 ;
dom Shift ( Seq q2 , len Seq q1 ) = { j + len Seq q1 where j is Nat : j in dom Seq q1 & len Seq q2 = len Seq q1 + len Seq q2 & len Seq q1 = len Seq q2 + len Seq q2 } .= Seg len Seq q1 \/ dom Seq q2 \/ { len Seq q2 + len Seq q2 } ;
consider G1 , G2 , G3 being Element of V such that G1 <= G2 and f is Morphism of G2 , G3 and g is Morphism of G1 , G3 and f is Morphism of G2 , G3 and g is Morphism of G2 , G3 and g is Morphism of G2 , G3 ;
func - f -> PartFunc of C , V means : Def1 : dom it = dom f & for c st c in dom it holds it /. c = - f /. c & it /. c = - f /. c ;
consider phi such that phi is increasing and for a st phi . a = a & {} <> a for v6 holds L , v |= ( union ( L | [: union L , { v } :] ) | [: { v } , { v } :] ) iff L . a , v |= H ;
consider i1 , j1 such that [ i1 , j1 ] in Indices GoB f and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) ;
consider i , n such that n <> 0 and sqrt p = ( i - n ) / ( n + 1 ) and for n1 being Nat , n2 being Nat st n1 <> 0 & n2 <> 0 & n2 <> 0 holds sqrt ( n + 1 ) <= ( i - n1 ) / ( n + 1 ) ;
assume that not 0 in Z and Z c= dom ( ( arccot * f1 ) / ( 1 + x ^2 ) ) and for x st x in Z holds ( ( 1 + x ^2 ) / ( 1 + x ^2 ) ) / ( 1 + x ^2 ) < 1 ;
cell ( G1 , i1 -' 1 , j2 ) \ L~ ( f /^ ( m -' 1 ) ) c= BDD L~ f & L~ ( f /^ ( m -' 1 ) ) \ L~ ( f /^ ( m -' 1 ) ) c= BDD L~ f implies L~ ( f /^ ( m -' 1 ) ) c= BDD L~ f
ex Q1 being open Subset of X st s = Q1 & ex F8 being Subset-Family of Y st F8 c= F & F8 is finite & ( for x being Subset of Y holds F8 c= F8 implies ( x in F & x in Q1 & x in Q1 & x in Q1 ) & ( x in Q1 implies x in Q1 & x in Q1 ) ;
gcd ( A9 , ( ( 1 , 1 ) --> ( ( 1 , 1 ) --> ( 1 , 1 ) ) , ( 1 , 1 ) --> ( 1 , 1 ) ) ) = 1 / ( ( 1 , 1 ) --> ( 1 , 1 ) ) .= 1 / ( ( 1 , 1 ) --> ( 1 , 1 ) ) ;
R8 = ( ( j , ( j + 1 ) ) --> ( ( j + 1 ) + 1 ) ) . ( m2 + 1 ) .= ( ( j , ( j + 1 ) ) --> ( ( j + 1 ) + 1 ) ) . ( m2 + 1 ) .= [ 3 , ( j + 1 ) + 1 ] ;
CurInstr ( P-6 , Comput ( P-6 , E , m3 ) ) = CurInstr ( P3 , Comput ( P3 , s3 , m3 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m3 ) ) .= halt SCMPDS .= halt SCMPDS .= ( CurInstr ( P3 , s3 ) , 4 ) .= halt SCMPDS ;
P1 /\ P2 = ( { p1 } \/ LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ) \/ ( LSeg ( p1 , p2 ) /\ LSeg ( p2 , p2 ) /\ LSeg ( p1 , p2 ) ) .= { p1 } \/ { p2 } .= { p2 } \/ { p2 } .= { p2 } ;
func not bound in the Sorts of Al means : Def1 : a in it iff ex i , j st i in dom f & j in dom f & a = f . i & a = f . j & a = f . j & b = f . j ;
for a , b being Element of F_Complex st |. a .| > |. b .| for f being Polynomial of F_Complex st f >= 1 holds f is non or f is non or f is non or f is non or f is non or f is non or f is non or f is non or f is non ]
defpred P [ Nat ] means 1 <= $1 & $1 <= len g implies for i , j st [ i , j ] in Indices G & G * ( i , j ) = g . ( $1 + 1 ) holds j < len g & 1 <= j & j <= width G ;
assume that C1 , C2 are_\HM { f , g , h , f , g being State of C1 , s1 , s2 being State of C2 , f , g being State of C1 , s1 , s2 being State of C2 st s1 = s2 * f & s2 = s1 * g holds s1 is stable iff s2 is stable & s1 is stable & s2 is stable & s2 is stable ;
( ||. f .|| | X ) . c = ||. f .|| . c .= ||. f /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| ;
|. q .| ^2 = ( q `1 ) ^2 + ( q `2 ) ^2 & 0 + ( 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 & A <> B & A <> B holds A misses B or A = B & A misses B & A misses B & A misses B implies card F = card B
assume that len F >= 1 and len F = k + 1 and len F = len G and for k st k in dom F holds H . k = g . ( F . k , G . k ) and for k st k in dom F holds H . k = g . ( F . k , G . k ) ;
i |^ ( ( \mathop { \rm Radix n } - i ) |^ s ) = i |^ ( s + k ) - i |^ s .= i |^ s * i |^ k - i |^ s * 1 .= i |^ ( s * k - 1 ) - i |^ s * 1 .= i |^ ( s * k - 1 ) ;
consider q being oriented oriented Chain of G such that r = q and q <> {} and ( q . 1 = v1 & q . len q = v2 or q . len q = v2 & rng q c= rng ( p ^ q ) ) and rng q c= rng ( p ^ q ) and q is oriented and q is oriented ;
defpred P [ Element of NAT ] means $1 <= len ( I ) implies ( ( g . ( Z , I ) ) ^ ( f , I ) ) . $1 = ( ( g . ( Z , I ) ) . ( len ( f , Z ) ) ) . ( len ( f , Z ) + 1 ) ;
for A , B being Matrix of n , REAL for A , B being Matrix of n , REAL holds len ( A * B ) = len A & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B
consider s being FinSequence of the carrier of R such that Sum s = u and for i being Element of NAT st 1 <= i & i <= len s ex a , b being Element of R st s . i = a * b & a in I & b in J & s . i = a * b ;
func |( x , y )| -> Element of COMPLEX equals |( Re x , Re y )| - ( Re y ) * |( x , y )| + ( - ( Im x ) * |( x , y )| ) + ( - ( Im y ) * |( x , y )| ) + ( - ( Im y ) * |( x , y )| ) ;
consider g0 be FinSequence of F such that f0 is continuous and rng f0 c= A and for i be Nat st i in dom h0 holds ex x be Element of X st x = ( f | A ) . i & for i be Nat st i in dom ( f | A ) holds x0 . i = x1 . i * x0 . i ;
then n1 >= len p1 & crossover ( p1 , p2 , n1 , n2 , n3 , n3 , n3 , n3 , F , J ) = crossover ( p1 , p2 , n1 , n2 , n3 , n3 , F , J ) & crossover ( p1 , p2 , n1 , n2 , n3 , n3 , F , J ) = crossover ( p1 , p2 , n1 , n2 , n3 , n3 , F , J ) ;
q `1 * a <= q `1 & - q `1 <= q `2 & - q `2 <= q `1 or - q `1 <= q `2 & - q `2 <= q `1 & - q `1 <= q `2 or - q `1 <= q `2 & - q `2 <= - q `1 & - q `1 <= q `2 or - q `1 <= q `2 & - q `2 <= - q `1 ;
( F . ( pp . len pp ) ) . ( len pp + 1 ) = ( F . ( p . len pp ) ) . ( len p + 1 ) .= ( F . ( len p + 1 ) ) . ( len p + 1 ) .= ( F . ( len p + 1 ) ) . ( len p + 1 ) .= ( F . ( len p + 1 ) ) . ( len p + 1 ) ;
consider k1 being Nat such that k1 + k = 1 and a := k = ( <* a := intloc 0 *> ) ^ ( ( k1 --> intloc 0 ) ^ <* a *> ) ^ ( ( k1 --> intloc 0 ) ^ <* a *> ) ^ <* a *> ^ ( ( k1 --> intloc 0 ) ^ ( k1 --> intloc 0 ) ) ) ;
consider B9 being Subset of B1 , y-1 being Function of B1 , A1 such that B8 is finite and D8 = the carrier of ( A1 \/ B1 ) \/ { y1 } and card ( B8 ) = card ( the carrier of ( A1 \/ B1 ) \/ { y2 } ) and card ( B8 ) = card ( the carrier of ( A1 \/ B1 ) \/ { y1 } ) ;
v2 . b2 = ( ( curry F2 ) * g ) . b2 .= ( ( curry F2 ) * ( ( curry F2 ) . b2 ) ) . b2 .= ( ( ( curry F2 ) * g ) . b2 .= ( ( ( ( curry F2 ) * g ) . b2 ) . b2 ) . b2 .= ( ( ( ( curry F2 ) * g ) . b2 ) . b2 .= ( ( ( ( ( ( ( id F ) . b2 ) . b2 ) ) . b2 ) ) . b2 .= ( ( ( ( ( id B ) . b2 ) ) . b2 ) . b2 ) . b2 .= ( ( ( id B ) . b2 ) . b2 ) . b2 ) . b2 .= ( ( ( ( id B ) . b2 ) . b2 ) . b2 .= ( ( ( ( id B ) . b2 ) . b2 ) . b2 .= ( ( ( ( ( id B )
dom IExec ( I , P , Initialize s ) = the carrier of SCMPDS .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) .= dom ( ( card I + 2 ) .--> NAT ) .= dom ( ( card I + 2 ) .--> NAT ) ;
ex d-32 be Real st d-32 > 0 & for h be Real st h <> 0 & |. h .| < K holds |. h .| " * ||. ( L + R1 ) /. h .|| < K * ||. ( L + R1 ) /. h .|| ;
LSeg ( G * ( len G , 1 ) + |[ 1 , - 1 ]| , G * ( len G , 1 ) + |[ 1 , 0 ]| ) c= Int cell ( G , len G , 1 ) \/ { G * ( len G , 1 ) } ;
LSeg ( mid ( h , i1 , i2 ) , i ) = LSeg ( h /. ( i + i1 -' 1 ) , h /. ( i + i1 -' 1 ) ) .= LSeg ( h /. ( i + i1 -' 1 ) , h /. ( i + i1 -' 1 ) ) .= LSeg ( h /. ( i + i1 -' 1 ) , h /. ( i + i1 -' 1 ) ) ;
A = { q where q is Point of TOP-REAL 2 : LE p1 , q , P & LE p2 , q , P & LE q , p , P & LE p2 , p , P & LE q , p , P & LE p , q , P & LE q , p , P & LE q , p , P & LE q , p , P & LE q , p , P & LE q , p , P & LE q , p , P & LE q , q , P & LE q , p , P & LE q , p , P & LE q , p , P & LE q , p , P & LE q , p , P & LE q , p , P & LE q , p , P & LE q , p , P & LE q , p , P & LE q , p , P & LE q , p , P & LE p , p , P & LE q , p , P & LE q ,
( ( - x ) .|. y ) = ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) ;
0 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) = ( p `2 / sqrt ( 1 + ( p `1 / p `2 ) ^2 ) ) * sqrt ( 1 + ( p `2 / p `2 ) ^2 ) .= ( p `2 / sqrt ( 1 + ( p `1 / p `2 ) ^2 ) ) * sqrt ( 1 + ( p `2 / p `2 ) ^2 ) ;
( ( U * ( W7 * ( W * ( p ) ) ) ) * ( W * ( p ) ) ) = ( ( ( U * ( p ) ) * ( W * ( p ) ) ) * ( W * ( p ) ) ) * ( W * ( p ) ) .= ( ( U * ( p ) ) * ( p ) ) * ( W * ( p ) ) .= ( ( U * ( p ) ) * ( p ) ) ;
func Shift ( f , h ) -> PartFunc of REAL , REAL means : Def1 : dom it = dom ( - h ) & for x st x in dom it holds it . x = ( - h ) . x + ( - h ) . x ;
assume that 1 <= k and k + 1 <= len f and [ i , j ] in Indices G and f /. k = G * ( i + 1 , j ) and f /. ( k + 1 ) = G * ( i + 1 , j ) and f /. k = G * ( i , j ) and f /. ( k + 1 ) = G * ( i , j ) ;
assume that not y in variables_in H and not x in Free H and not ( x in Free H \ { x } or x in Free H \ { y } ) and not x in Free H and not x in Free H and not x in Free H and not x in Free H and not x in Free H and not x in Free H and not x in Free H and not x in Free H and x in Free H and x in Free H and x in Free H and x in Free H and not x in Free H and not x in Free H and not x in Free H and not x in Free H and not x in Free H and not x in Free H and not x in Free H and not x in Free H and not x in Free H and not x in Free H and not x in Free H \ { y in Free H \ { y in Free H \ { y and not x in Free H \ { y in Free H \ { y in Free H \ { y in Free H \ { y in Free H \ {
defpred P11 [ Element of NAT , Element of NAT ] means P [ $1 ] & ( for p being prime Nat st p in $1 holds not p |^ ( $1 -' 1 ) = p |^ ( $1 -' 1 ) or not p |^ ( $1 -' 1 ) = p |^ ( $1 -' 1 ) ) & ( not p divides $1 implies not p divides ( $1 -' 1 ) ) ;
func \sigma ( C ) -> non empty Subset-Family of X means : Def1 : for A being Subset of X holds A in it iff for W being Subset of X holds W c= ( W \ A ) & for A , B being Subset of X st W c= A & B c= B holds C . W <= C . ( W \/ B ) ;
[#] ( ( dist ( P ) ) .: Q ) = ( dist ( P ) ) .: Q & lower_bound ( [#] ( ( ( ( ( ( ( ( ( ( ( ( ( ( P ) ) | Q ) | Q ) ) | Q ) | Q ) ) ) .: Q ) ) = lower_bound ( ( ( ( ( ( ( ( ( ( ( ( ( ( P ) | Q ) ) | Q ) | Q ) ) .: Q ) ) ) ) ) ;
rng ( F | ( [: S , T :] ) ) = {} or rng ( F | ( [: S , T :] ) ) = { 1 } or rng ( F | ( [: S , T :] ) ) = { 2 } or rng ( F | ( [: S , T :] ) ) = { 1 } or rng ( F | ( [: S , T :] ) ) = { 2 } ;
( f " ( rng f ) ) . i = f . i " ( ( rng f ) . i ) .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i ;
consider P1 , P2 being non empty Subset of TOP-REAL 2 such that P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p2 , p1 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p2 , p1 and P1 is_an_arc_of p2 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p2 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p2 , p2 ;
f . p2 = |[ ( p2 `1 ) ^2 + ( p2 `2 ) ^2 - 1 , ( p2 `2 ) ^2 + ( p2 `2 ) ^2 - 1 ]| .= |[ ( p2 `1 ) ^2 + ( p2 `2 ) ^2 - 1 , ( p2 `2 ) ^2 + 1 ]| ;
( ( ( a , X ) " ) * ( ( a , X ) " ) ) . x = ( ( ( a , X ) " ) . x ) " .= ( ( a , X ) " ) . x .= ( ( a , X ) " ) . x .= ( ( a , X ) " ) . x .= ( ( a , X ) " ) . x .= ( ( a , X ) " ) . x .= ( ( a , X ) " ) . x ;
for T being non empty normal TopSpace , A , B being closed Subset of T , A being Subset of T , B being Subset of T , r being Real st A <> {} & A misses B holds for p being Point of T , r being Point of T , x being Point of T st x in A & r in B holds ( NAT G ) . p = ( NAT G ) . p
for i , j st i + 1 in dom F for G1 , G2 being strict normal Subgroup of G st G1 = F . i & G2 = F . ( i + 1 ) & G2 = F . ( i + 1 ) holds G1 is strict Subgroup of G1 & G2 is strict Subgroup of G2
for x st x in Z holds ( ( #Z 2 ) * ( arccot - arccot ) ) `| Z ) . x = ( ( ( #Z 2 ) * ( arccot - arccot ) ) `| Z ) . x = ( ( ( #Z 2 ) * ( arccot - arccot ) ) `| Z ) . x
synonym f is_right continuous means : Def1 : x0 in dom ( f /* a ) & for x st x in Z holds f . x = - 1 & for a st a in Z holds f . x = - 1 & for x st x in Z holds ( f /* a ) . x = - 1 & f . x = - 1 & f . x = - 1 ;
then X1 , X2 are_separated & Y1 misses Y2 or ex Y1 , Y2 being non empty SubSpace of X , Y1 being SubSpace of X1 , Y2 being non empty SubSpace of X2 st Y1 , Y2 , Y1 , Y2 being non empty SubSpace of X1 union Y2 & Y1 = Y2 & Y2 = Y1 \/ Y2 & Y1 = Y2 & Y2 = Y2 & Y1 = Y2 & Y1 = Y2 & Y1 = Y2 & Y1 = Y2 & Y1 = Y2 & Y1 = Y2 & Y1 = Y2 & Y1 = Y2 & Y2 = Y1 & Y2 = Y2 & Y2 = Y1 \/ Y2 & Y2 = Y2 \/ Y2 & Y1 = Y2 \/ Y2 = Y1 \/ Y2 & Y2 = Y1 \/ Y2 & Y1 = Y2 & Y1 = Y2 & Y1 = Y2 & Y1 = Y1 \/ Y2 & Y2 = Y1 \/ Y2 & Y2 = Y1 \/ Y2 = Y2 & Y2 = Y2 \/ Y2 & Y1 = Y2 & Y2 = Y2 & Y1 = Y2 \/ Y2 \/ Y2 \/ Y2 = Y2 \/ Y2 & Y2 = Y2 & Y2 = Y2 \/ Y2 & Y2 = Y2 \/ Y2 & Y2 = Y2 \/ Y2 & Y2 = Y2 \/ Y2 & Y2 = Y1 \/ Y2 & Y2 =
ex N be Neighbourhood of x0 st N c= dom SVF1 ( 1 , f , u ) & ex L , R st for x st x in N holds SVF1 ( 1 , f , u ) . x - SVF1 ( 1 , f , u ) . x0 = L . ( x - x0 ) + R . ( x - x0 ) + R . ( x - x0 )
( ( p2 `1 ) ^2 ) * sqrt ( 1 + ( p3 `2 ) ^2 ) >= ( ( p2 `1 ) ^2 + ( p3 `2 ) ^2 ) * sqrt ( 1 + ( p3 `2 ) ^2 ) + ( ( p3 `2 ) ^2 ) * sqrt ( 1 + ( p3 `1 ) ^2 ) ) ;
( ( 1 / t1 ) (#) ||. f1 .|| ) to_power n = ( ( 1 / t2 ) (#) ||. f1 .|| ) to_power m & ( ( 1 / t2 ) (#) ( ( 1 / t2 ) to_power n ) ) to_power m = ( ( 1 / t2 ) (#) ( ( 1 / t2 ) to_power n ) ) to_power m & ( ( 1 / t2 ) to_power n ) to_power m = ( ( 1 / t2 ) to_power n ) to_power m ;
assume that f . x = ( ( - 1 ) (#) ( sin + cos ) ) . x and x in dom ( ( - 1 ) (#) ( cos + cos ) ) and x + h in dom ( ( - 1 ) (#) ( sin + cos ) ) and ( ( - 1 ) (#) ( sin + cos ) ) . x = ( ( - 1 ) (#) ( sin + cos ) ) . x ;
consider X-23 being Subset of Y , X-22 being Subset of X such that t = X-23 and X-22 is open and for Y1 being Subset of X st Y1 = X-23 holds Y1 is open & Y1 is open & Y1 is open & Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open ;
card S . n = card { [: d , Y :] + ( a * d ) + b * d , d * b , c * d :] where d is Element of GF ( p ) : [ d , Y ] in { d , c } & d in { d , b } & d in { d , c } & d in { d , c } & d in { d , b } ;
( ( E-bound D ) * ( i1 - 1 ) / 2 ) * ( 2 |^ ( m -' 1 ) ) = ( ( E-bound D ) * ( i - 1 ) ) / 2 * ( ( W-bound D ) / 2 ) .= ( ( W-bound D ) * ( i - 1 ) ) / 2 * ( ( W-bound D ) / 2 ) .= ( ( W-bound D ) / 2 ) / 2 ;