thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
contradiction ;
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contradiction ;
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thesis ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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thesis ;
contradiction ;
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contradiction ;
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contradiction ;
contradiction ;
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thesis ;
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thesis ;
thesis ;
contradiction ;
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contradiction ;
contradiction ;
thesis ;
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thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b ;
D c= S
let Y ;
S `1 is convergent
q ;
V is open ;
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 >= X ;
T is continuous ;
d <= a ;
p <= r ;
t < s ;
p <= t ;
t < s ;
let r ;
D <= E ;
assume e > 0 ;
assume 0 < g ;
p ;
x in X ;
Y `2 in Y ;
assume 0 < g ;
c in Y ;
v in L ;
2 in z `2 ;
assume f = g ;
N c= b `1 ;
assume i < k ;
assume u = v ;
I = J ;
B `2 = b `2 ;
assume e in F ;
assume p > 0 ;
assume x in D ;
let i be element ;
assume F is onto ;
assume n <> 0 ;
let x be element ;
set k = z ;
assume o = x ;
assume b < a ;
assume x in A ;
a `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 generated from squares ;
assume m > 0 ;
assume A c= B ;
X is lower
assume A <> {} ;
assume X <> {} ;
assume F <> {} ;
assume G is open ;
assume f is dilatation ;
assume y in W ;
y \not <= x ;
A `2 in B `2 ;
assume i = 1 ;
let x be element ;
x `2 = x `2 ;
let X be LE X ;
assume S is non bounded ;
a in REAL ;
let p be set ;
let A be set ;
let G be graph , x be set ;
let G be graph , x be set ;
let a be Element of V ;
let x be element ;
let x be element ;
let C be FormalContext , x be set ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
let y be Real ;
X c= f . a
let y be element ;
let x be element ;
let i be Nat ;
let x be element ;
n in NAT ;
let a be element ;
m in NAT ;
let u be element ;
i in NAT ;
let g be Function ;
Z c= NAT ;
l <= a ;
let y be element ;
r2 < 1 ;
let x be element ;
k1 be Integer ;
let X be set ;
let a be element ;
let x be element ;
let x be element ;
let q be element ;
let x be element ;
assume f is being_homeomorphism ;
let z be element ;
a , b // K ;
let n be Nat ;
let k be Nat ;
B ` c= B ` ;
set s = 2 ;
n >= 0 + 1 ;
k c= k + 1 ;
R1 c= R ;
k + 1 >= k ;
k c= k + 1 ;
let j be Nat ;
o , a // Y , a ;
R c= Int G ;
Cl B = B ;
let j be Nat ;
1 <= j + 1 ;
arccot is_differentiable_on Z ;
the function exp is differentiable ;
j < i2 ;
let j be Nat ;
n <= n + 1 + 1 ;
k = i + m ;
assume C meets S ;
n <= n + 1 + 1 ;
let n be Nat ;
h1 = {} ;
0 + 1 = 1 ;
o <> b1 ;
f2 is one-to-one ;
support p = {} ;
assume x in Z ;
i <= i + 1 + 1 ;
r1 <= 1 ;
let n be Nat ;
a "/\" b <= a ;
let n be Nat ;
0 <= r1 ;
let e be Real , x be Real ;
r in G . l ;
c1 = 0 ;
a + a = a ;
<* 0 *> in e ;
t in { t } ;
assume F is discrete ;
m1 divides m ;
B * A <> {} ;
a + b <> {} ;
p * p > p ;
let y be ExtReal ;
let a be Int-Location , x be Int-Location ;
let l be Nat ;
let i be Nat ;
let r ;
1 <= i2 ;
a "\/" c = c ;
let r be Real ;
let i be Nat ;
let m be Nat ;
x = p2 ;
let i be Nat ;
y < r + 1 ;
rng c c= E ;
Int R is unital ;
let i be Nat ;
R ;
cluster uparrow x -> closed ;
X <> { x } ;
x in { x } ;
q , b // M ;
A . i c= Y ;
P [ k ] ;
2 |^ x in W ;
X [ 0 ] ;
P [ 0 ] ;
A = A |^ i ;
D1 >= s ;
G . y <> 0 ;
let X be complex normed space , x be Real ;
a in X ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , W be Subspace of V ;
assume x in indices M ;
k < s . a ;
t in { p } ;
let Y be real-membered set , X be Subset of Y ;
M , L are_isomorphic ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded ;
rng f = Y ;
{ G } c= L ;
assume x in Cl Q ;
m in dom P ;
i <= len Q ;
len F = 3 ;
'not' 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 `2 = a `2 + 1 ;
x `2 = a * y `2 ;
rng D c= A ;
assume x in K ;
1 <= i2 ;
1 <= i2 ;
p9 c= \pi ;
1 <= i2 ;
1 <= i2 ;
UMP C in L ;
1 in dom f ;
let seq ;
set C = a * B ;
x in rng f ;
assume f is Lipschitzian ;
I = dom A ;
u in dom p ;
assume a < x + 1 ;
s2 is bounded ;
assume I c= P1 ;
n in dom I ;
let Q ;
B c= dom f ;
b + p * p _|_ a ;
x in dom g ;
F1 is continuous ;
dom g = X ;
len q = m ;
assume A2 : : A is open ;
cluster R \ S -> real-valued ;
ex_sup_of D , S ;
x << sup D ;
b1 >= b2 ;
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 ] ;
W-min C c= f .: C ;
x1 is increasing ;
let e be element ;
- b divides b ;
F c= 5 ( F ) ;
G1 is continuous ;
G1 is continuous ;
assume v in H . m ;
assume b in [#] B ;
let S be non void Signature , x be set ;
assume P [ n ] ;
assume union S is independent & not contradiction ;
V is Subspace of V ;
assume P [ k ] ;
rng f c= NAT ;
assume inf X in L ;
y in rng f ;
let s , I be set , x be set ;
b `2 c= b1 `2 ;
assume x in REAL + 1 ;
A /\ B = { a } ;
assume len f > 0 ;
assume x in dom f ;
b , a // o , c ;
B in { Bnnnnnnnnnnnnnnnnnnnnnnnnnnnn
cluster product p -> non empty ;
z , x // x , p ;
assume x in rng N ;
cosec is differentiable ;
assume y in rng S ;
let x , y be element ;
i2 < i2 + 1 ;
a * h in H * H ;
p , q in Y ;
cluster sqrt I -> non empty for ideal of I ;
q1 in A1 ;
i + 1 <= 2 + 2 ;
A1 c= A2 & A2 c= A1 implies A1 c= A2
\hbox { \boldmath $ n $ } < n ;
assume A c= dom f ;
Re ( f + g ) is_integrable_on M ;
let k , m be Nat ;
a , b \equiv b , c ;
j + 1 < k + 1 + 1 ;
m + 1 <= n1 + 1 ;
g is_differentiable_on REAL ;
g is continuous ;
assume O is symmetric ;
let x , y be element ;
let j 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 ;
s3 halts_on s ;
d , c // a , b ;
let t , u be set ;
let X be set ;
assume k in dom s ;
let r be non negative Real ;
assume x in F | M ;
let Y be Subset of S ;
let X be non empty TopSpace , x be Point of X ;
[ a , b ] in R ;
x + w < y + w ;
not a , b >= c ;
let B be Subset of A , x be Element of A ;
let S be non empty many sorted signature ;
let x be variable of f & x in dom f ;
let b be Element of X , x be Element of X ;
R [ x , y ] ;
x " = x " ;
b \ x = 0. X ;
<* d *> in D |^ 1 ;
P [ k + 1 ] ;
m in dom ( n |-> 0 ) ;
h2 . a = y ;
P [ n + 1 ] ;
cluster G * F -> -> -> -> -> -> odd ;
let R be non empty doubleLoopStr , x be Element of R ;
let G be graph ;
let j be Element of I ;
a , p // x , p ;
assume f | X is bounded ;
x in rng ( o + 1 ) ;
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 ;
let M be maid id id REAL & Sum id REAL is maid ;
let N be non empty Point of Point of Point Nmode Point of M ;
let R be RelStr structure ;
let n , k be Nat ;
let P , Q be reflexive 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 FinSequence of INT ;
assume I is not LIN a , I ;
let n , k be Nat ;
let x be Point of T ;
f +* g c= f +* g ;
assume m < v1 ;
x <= c2 . x ;
x in F " { x } ;
cluster S --> T -> non empty ;
assume t1 <= t2 & t1 <= t2 ;
let i , j be Nat ;
assume F1 <> F2 ;
c in Intersect ( R ) ;
dom p1 = c ;
a = 0 or a = 1 ;
assume A1 : A2 <> {} & for A st A in A1 holds A in A2 ;
set i1 = i + 1 ;
assume a1 = b1 & a2 = b2 ;
dom ( g * f ) = A ;
i < len M + 1 ;
assume - \infty \notin rng G ;
N c= dom ( f1 + f2 ) ;
x in dom ( sec ) ;
assume [ x , y ] in R ;
set d = ( x + y ) * ( x + y ) ;
1 <= len g1 + len g2 ;
len s2 > 1 + 1 ;
z in dom ( f1 + f2 ) ;
1 in dom ( D2 | Seg len D2 ) ;
( p `2 ) ^2 = 0 ;
j1 <= width G ;
len \pi > 1 + 1 ;
set n1 = n + 1 ;
|. q1 .| = 1 ;
let s be SortSymbol of S ;
gcd ( i , j ) = i ;
X1 c= dom f /\ dom g ;
h . x in h . a ;
let G be O_GF of p ;
cluster m * n -> square ;
let k1 be Nat ;
i -' 1 + 1 > m ;
R is reflexive ;
set F = <* u , w *> ;
p1 c= p3 ;
I is_closed_on t , Q ;
assume [ S , x ] is V1 ;
i <= len ( f2 - f1 ) ;
p is FinSequence of X ;
1 + 1 in dom g ;
Sum R = n * r ;
cluster f . x -> real-valued ;
x in dom ( f1 + f2 ) ;
assume [ X , p ] in C ;
{ B } c= { X } ;
i2 <= len i2 + len j2 ;
A /\ { p1 } c= A ` ;
cluster x .--> x -> NAT for Function ;
let Q be Subset-Family of S , x be Element of S ;
assume n in dom ( - g ) ;
a be Element of R ;
t `2 in dom ( e `2 ) ;
N . 1 in rng N ;
- z in A \/ B \/ C \/ B ;
let S be SigmaField of X , A be Element of S ;
i . y in rng i ;
REAL c= dom ( f + g ) ;
f . x in rng f ;
reconsider t = ( r * t ) as Element of REAL ;
s2 in { r where r is Real : r <= r & r <= 1 } ;
let z , z be Element of number ;
n <= N . m ;
LIN q , p , s ;
f . x = \twoheaddownarrow x /\ B ;
set L = [ S \to T ] ;
let x be non negative Real , r be Real ;
m is Element of M ;
f in union rng F1 ;
let K be add-associative right_zeroed right_complementable complementable for L being add-associative right_zeroed right_complementable complementable right_complementable non empty doubleLoopStr ;
let i be Element of NAT ;
rng ( F * g ) c= Y ;
dom f c= dom ( x - y ) ;
n1 < n + 1 + 1 ;
n1 < n + 1 + 1 ;
cluster { \bf T } ( X ) -> universal for Lattice ;
[ y , 2 ] = z ;
let m be Element of NAT ;
let S be Subset of R ;
y in rng S29 ;
b = sup dom ( f + g ) ;
x in Seg len q ;
reconsider X = { D } as set ;
[ a , c ] in E ;
assume n in dom ( h + c ) ;
w + 1 = - ( a - 1 ) ;
j + 1 <= j + 1 + 1 ;
k2 + 1 <= k2 + 1 ;
i be Element of NAT ;
Support u = Support p ;
assume X is being_indexed by m ;
assume that f = g and p = g ;
n1 <= n + 1 + 1 ;
let x be Element of REAL ;
assume x in rng s1 ;
x0 < r1 + r ;
len L5 = len L5 + len <* 5 *> ;
P c= Seg len A ;
dom q = Seg n ;
j <= width M ;
let r1 be real-valued FinSequence of REAL ;
let k be Element of NAT ;
\int P + d < + \infty ;
let n be Element of NAT ;
assume z in atatatatatat*> ( A ) ;
let i be set ;
n -' 1 + 1 = n - 1 ;
len n1 = n + 1 ;
LowerCone ( Z , c ) c= F ;
assume x in X or x = X ;
x is Element of b , c , d , x , y be Element of L ;
let A , B be non empty set , x be set ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q ;
let s be Element of E |^ \omega ;
B be Basis of x , y ;
non empty doubleLoopStr /\ L2 = {} ;
L1 /\ L2 = {} ;
assume \mathopen { \downarrow } x = \mathopen { \downarrow } y ;
assume b , c // b , c ;
LIN q , c , c9 ;
x in rng ( f1 + f2 ) ;
set n1 = n + j + 1 ;
let D1 be non empty set , D2 be non empty set , D1 be set ;
let K be add-associative right_zeroed right_complementable complementable for p be Polynomial of K ;
assume that f ' = f and h = g ;
R1 - R2 - R1 + R1 - R2 - R1 + R1 - R2 ;
k in NAT & 1 <= k ;
a be Element of G ;
assume x0 in [. a , b .] ;
K is open ;
assume a < b & b is_a_maximal distance in C ;
a , b , c , d is_collinear ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = \int f , M ;
cluster -> -> binary binary binary binary binary binary binary binary binary binary binary binary binary binary also also also also also positive ;
u in { \hbox { \boldmath $ g } } : not contradiction } ;
the support of f c= B ;
reconsider z = x as VECTOR of V ;
let the "/\" of L is for x being Element of L holds x is Element of L ;
r (#) H is qua qua qua qua Function of X , X ;
s . intloc 0 = 1 ;
assume x in C & y in C ;
let U1 be strict strict strict strict strict Subgroup of U0 ;
[ x , \bot T ] is compact ;
i + 1 in dom p ;
F . i is stable Subset of M ;
rry in ( ry ) ;
let x , y be Element of X ;
A , I be LE I , A , P ;
[ y , z ] in { O where O is Element of O : O in O } ;
LE \rbrace , goto ( i + 1 ) , i + 1 , j ) = 1 ;
rng Sgm A = A ;
q \vdash p1 ;
for n holds X [ n ] ;
x in { a } & x in { d } ;
for n holds P [ n ] ;
set p = [ x , y ] ;
LIN o9 , a9 , b9 ;
p . 2 = Z |^ Y ;
( D1 ) `2 = ( D1 `2 ) `2 ;
n + 1 + 1 <= len g ;
a in Indices ( Al ) ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
I be ideal of L ;
set g = f1 + f2 + f3 ;
a <= max ( a , b ) ;
i-1 < len G + len G + 1-1 ;
g . 1 = f . ( i + 1 ) ;
x `2 , y `2 in A2 ;
( f /* s ) . k < r ;
set v = VAL g ;
i -' k + 1 <= len S ;
cluster -> multiplicative for Element of G ;
x in support ( t ) ;
assume a in { [ 0 , 1 ] } ;
i `2 <= len ( y `2 ) `2 ;
assume p divides b1 + b2 ;
M1 <= M1 & M1 <= M2 implies M1 * M2 = M1 * M1
assume x in W .: ( X ) ;
j in dom ( z * z ) ;
let x be Element of D ;
IC s2 = IC s2 + 1 ;
a = {} or a = { x } ;
set u = Vertices G , v = G , u = G , w = H , v = H , w = H , u = H , w = H , v = H , w
seq " (#) ( seq " ) is non-zero ;
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
h1 c= h1 ;
]. a , b .[ c= Z ;
X1 , X2 , x3 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 be Point of X ;
a in Cl ( union F \ ( union F \ G ) ) ;
set x1 = [ 0 , 0 , 1 ] ;
k + 1 -' 1 + 1 = k + 1 - 1 ;
cluster R -> -> real-valued for Relation of REAL ;
ex v st C = v + W ;
let G1 be non empty 1-sorted , x be Element of G1 ;
assume V is Abelian add-associative right_zeroed right_complementable complementable ;
X1 \/ Y in by by by the topology of L ;
reconsider x = x as Element of S ;
max ( a , b ) = a ;
ex_sup_of B , A , B ;
let L be non empty reflexive RelStr , X be Subset of L ;
R is reflexive & R is reflexive ;
E |= _ { g } ( H ) ;
dom G = a ;
sqrt ( 1 - 4 * r ) >= - r ;
G . ( p1 , p2 ) in rng G ;
let x be Element of F1 , y be Element of F2 ;
D [ P1 ] ;
z in dom ( id B ) ;
y in the carrier of N ;
g in the carrier of H ;
rng Sgm ( dom f1 ) c= NAT ;
j `2 + 1 in dom s1 ;
A , B , C be strict Subgroup of G ;
let C be non empty Subset of REAL ;
f . z1 in dom h ;
P . ( k + 1 ) in rng P ;
M = { A } +* ( {} , {} ) ;
let p be FinSequence of REAL ;
f . n1 in rng f ;
M . ( F . ( F . 0 ) ) in REAL ;
degree [. a , b .] = b-a ;
assume the InternalRel of V is symmetric & for x being Element of V holds x in Q iff x in Q ;
let a be Element of V ;
let s be Element of P1 ;
let p1 be non empty RelStr ;
let n be Nat ;
the support of g c= B ;
I = halt R .= ( halt R ) * ( I , l ) ;
consider b being element such that b in B ;
set BK = BCS K ;
l <= rng ( F . j ) ;
assume x in ]. s , t .[ ;
( x - t ) . x in ]. t , t .[ ;
x in Reconsider Reconsider Reconsider as Element of product ( T . i ) as Element of product ( T . i ) ;
let h be Morphism of c , b ;
Y c= { \bf R ( ) : not contradiction } ;
A2 \/ A1 \/ A2 c= A1 \/ A2 \/ A1 \/ A2 \/ A3 \/ A3 ;
assume LIN o9 , a9 , b9 ;
b , c // d , e ;
x1 , x2 , x3 , x5 , x5 , x5 , x5 , x5 , x5 , x5 be Element of Y ;
dom <* y *> = Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar .| ;
[ x , x ] in X \times X ;
for n be Nat holds 0 <= x . n ;
[ a , b ] = [ a , b ] ;
cluster -> non empty for Subset of T ;
x = h . ( f . ( z + 1 ) ) ;
q1 , q2 , q1 is_collinear ;
dom M1 = Seg n ;
x = [ x1 , x2 ] , x3 ] ;
R , Q be ManySortedSet of A ;
set d = ( 1 - 1 ) * ( n + 1 ) ;
rng ( g2 * g1 ) c= dom ( g2 * g1 ) ;
( P ` ) \ B ` <> 0 ;
a in field R & a in field R ;
let M be non empty Subset of V , V be non empty Subset of V ;
let I be Program of SCM+FSA ;
assume x in rng ( R * ( R * ( x , y ) ) ) ;
let b be Element of the lattice of T ;
dist ( e , z ) > r-r ;
u + v in W1 + W2 ;
assume support L misses rng ( L * F ) ;
let L be lower-bounded RelStr , X be Subset of L ;
assume [ x , y ] in { a } ;
dom ( A * e ) = NAT ;
let a , b be Vertex of G ;
let x be Element of bool ( M ) ;
0 <= Arg a * PI ;
o , a // o , y ;
not v in the support of l ;
let x be bound of A ;
assume x in dom ( uncurry f ) ;
rng F c= ( product f ) .: X ;
assume D2 . k in rng D2 ;
f " . p1 = 0 ;
set x = the Element of X ;
dom Ser ( G ) = NAT ;
n be Element of NAT ;
assume LIN c , d , e1 ;
cluster finite -> finite for FinSequence of NAT ;
reconsider d = c as Element of L1 ;
( v2 \rightarrow I ) . ( X --> 0 ) <= 1 ;
assume x in the carrier of Lin ( A ) ;
conv @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
reconsider B = b as Element of T ;
J |= P ! l l ( P ! l l ( P ! l l l ) ) ;
cluster the topology of J . i -> non empty ;
ex_sup_of Y \/ { Y } , T \/ { Y } , T ;
W1 is_field W1 & W2 is_field W1 c= field W1 ;
assume x in the carrier of R ;
dom ( n |-> 0. K ) = Seg n ;
s1 misses s2 or s1 misses s2 ;
assume ( a 'imp' b ) . z = TRUE ;
assume that X is open and f is continuous ;
assume [ a , y ] in Indices ( f * ( a , b ) ) ;
assume that that that that that I c= J and I c= J and J c= P and J c= P and J c= P and J c= P and J c= P and J c= P and J c= P and J c= P
Im ( ( lim seq ) - lim seq ) = 0 ;
( the function sin ) . x <> 0 ;
the function sin is differentiable & for x st x in Z holds ( ( \HM { the } \HM { function } \HM { cos } ) ) `| Z ) . x = - 1 / ( x + a )
6 . n = t . n .= t . n ;
dom ( even even ) c= dom F /\ dom ( F - - 1 ) ;
W1 . x = W1 . x + W2 . x ;
y in W \/ ( W . ( len W + 2 ) ) ;
k2 <= len ( - - 1 ) + 1 ;
x * a \equiv y * ( m * ( m * ( m * ( n , m ) ) ) ;
proj2 .: S c= ( TOP-REAL 2 ) .: P ;
h . p1 = g . p2 .= g . p2 ;
G1 = G1 * ( 1 , 1 ) ;
f . r1 in rng f ;
i + 1 + 1 <= len f + len g ;
rng F = rng ( F * ( len F ) ) ;
mode partial partial -> unital -> unital for PartFunc of REAL , REAL ;
[ x , y ] in A ~ ;
x1 . o in { L1 . o } ;
the support of \setminus m c= B ;
[ y , x ] in id X ;
1 + p .. f <= i + len f + len f + p .. f ;
seq ^\ k is bounded ;
len F = len F + len <* x *> ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be complex number ;
Comput ( P , s , n ) = s ;
k <= k + len p ;
reconsider c = {} as Element of L ;
let Y be with_Chain of T ;
cluster -> id L -> also cluster id L -> also \mathord ;
f . ( j + 1 ) in K . ( j + 1 ) ;
cluster J => J -> total for total ;
K c= 2 |^ ( n + 1 ) ;
F . ( b1 , b2 ) = F . ( b1 , b2 ) ;
x1 = x or x2 = y ;
attr a <> {} means a <> 0. G & a * ( a * b ) = 1 ;
assume cf a c= b & a c= b ;
s1 . n in rng s1 ;
not o , b , c is_collinear ;
LIN o9 , a9 , b9 ;
reconsider m = x as Element of Funcs ( V , REAL ) ;
let f be FinSequence of D ;
let F1 be non empty TopSpace ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , f . 2 ] in F ;
reconsider p1 = x as Point of m ;
A , B , C , D , E , F , J , J , M being Element of R ;
cluster -> non empty for TopSpace ;
rng c misses rng ( c1 - c2 ) ;
z is Element of gr { { x } ;
b in dom ( a .--> ( p +* ( p +* ( p +* ( p +* ( p +* ( p +* q ) ) ) ) ) ) ;
assume that k >= 2 and k >= 2 ;
Z c= dom ( ( tan + cot ) (#) cot ) ;
the Sorts of Q c= the Sorts of Q ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( ( f1 + f2 ) (#) ( f2 + g2 ) ) ;
attr f = u * v ;
for n , P being l , P being l , P being l , l being Nat st P [ l ] holds P [ l + 1 ]
{ x : x in L } <> {} ;
x is Element of V . s ;
a , b , c are_relative_prime ;
assume that S = p1 and p = p2 and p = p2 and p = p1 ;
gcd ( n1 , n ) = 1 ;
set o = ( - 1 ) * ( ( - 1 ) * ( - 1 ) ) ;
seq . n < |. ( ( seq . n ) - ( seq . n ) ) .| ;
assume that s1 is increasing and s1 < 1 ;
f . x1 , f . x2 ]| <= a ;
ex c being Nat st P [ c ] ;
set g = { n |^ ( n + 1 ) where n is Nat : n in NAT } ;
k = a or k = b ;
a1 in B & a2 in C ;
assume Y = { 1 } & s = <* 1 *> ;
I . x = f . x .= f . x ;
W1 . ( W1 + W2 ) = W1 . ( W2 + W1 . ( W1 + W2 ) ) ;
let G be subgraph of G , a be Point of G ;
reconsider u = u as Element of Bags n ;
A in B implies A |^ B = B |^ ( n + 1 )
x in { 2 * n + 3 * n + 3 * n + 3 * n + 3 * n ;
1 >= ( ( q `1 ) / |. q .| ) * ( |. q .| ) ;
f1 is_sub\smallfrown f2 & f2 is_subsub\smallfrown f1 implies for i being Nat st i in dom f1 holds f1 . i = f2 . i
( f /. ( q `2 ) ) `2 <= ( f /. ( q `2 ) ) `2 ;
h is_indexed indexed indexed by C ;
( b - p ) `2 <= ( p `2 ) `2 ;
let f , g be Function of X , REAL ;
S /. k <> 0. K ;
x in dom ( - ( - f ) ) ;
p2 in NAT . ( p1 + p2 ) ;
len ( H ) < len ( H ) + len ( H ) ;
F [ A , F . A ] ;
consider Z such that y in Z and Z in X ;
attr 1 in C implies A c= C ;
assume r1 <> 0 or r1 <> 0 ;
rng q1 c= rng q2 & rng q2 c= rng p1 ;
A1 , A2 , p3 , p1 is_collinear ;
y in rng f & y in rng f implies y in { x }
f /. i + f /. ( i + 1 ) in L~ f ;
b in \alpha=0 ( p , S , T ) ;
then S is negative & S is negative ;
Cl ( [#] T ) = ( Cl ( Cl ( Cl ( Cl ( A ) ) ) ) ;
f1 | ( A \/ B ) = f2 ;
0. M in the carrier of W ;
v , u ] in M ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ Z ;
let X be Subset of S \times T ;
consider H1 such that H = 'not' H1 and H is negative ;
{ \bf 1 } c= t * ( t * r ) ;
0 * a = 0. R * a .= a * ( a * b ) .= a * ( b * a ) ;
A |^ 2 = A |^ 2 ;
set v1 = { v where v is Element of V : not contradiction } , v2 = { v where v is VECTOR of V : not contradiction } ;
r = 0. TOP-REAL n , ( ( TOP-REAL n ) | K1 ) ;
( f . p3 ) `2 >= 0 ;
len W = len ( W + ( len W ) ) + len W ;
f /* ( s * ( s * ( n + k ) ) ) is convergent ;
consider l being Nat such that m = F . l ;
t1 < t1 & t1 <> t2 & t1 <> t2 & t1 <> t2 & t1 <> t2 & t1 <> t2 & t1 <> t2 & t1 <> t2 & t1 <> t2 & t1 <> t2 & t1 <> t2 & t1 <> t2 & t1 <> t2
reconsider X1 = X1 as SubSpace of X ;
consider w such that w in F and x in w ;
let a , b , c be Real ;
reconsider i = i as non zero Element of NAT ;
c . x >= id L . x ;
where T is Basis of T : \omega is Basis of T & \omega is Basis of T ;
for x being element st x in X holds x in Y
cluster [ x1 , x2 ] -> pair for Relation of A , B ;
downarrow a /\ \mathopen { \downarrow } t \mathclose { ^ { \rm c } } is ideal of T ;
let X be with_\hbox { - } \sum ( F , G ) -> with_\hbox { - } \sum ( F , G ) ;
rng f = elements ( the carrier of X ) ;
let p be Element of B ( ) , x be Element of B ( ) ;
max ( N , 2 ) >= N ;
0. X <= b * ( m * ( m * n ) ) ;
assume that i in I and R . i = R . i ;
i = j & j = ( i - j ) * ( i - j ) ;
assume that gggggggggggggggggggggggggassume assume gggggggggggggassume assume assume ggggg
let A1 , A2 be Point of S ;
x in h " ( ( ( P /\ ( ( P /\ ( ( P /\ Q ) ) ) /\ Q ) ) ) ;
1 in Seg 2 & 1 in Seg 2 ;
reconsider X1 = X as non empty Subset of T ;
x in ( the Arrows of B ) . i ;
cluster E . n -> ( the Sorts of G ) . n -> ( the Sorts of G ) . n ;
n1 <= i2 + len j2 + len j2 + 1 ;
( i + 1 ) + 1 = i + 1 ;
assume v in the carrier of G ;
y = \Re ( y ) + ( Im ( y ) ) ;
( gcd ( - 1 ) ) = 1 - ( p - 1 ) ;
x0 in ]. a , b .[ ;
rng ( M1 * M2 ) c= dom ( M1 * M2 ) ;
for p being Real st p in Z holds p >= a
|[ X , f ]| in [: X , Y :] ;
( seq ^\ k ) . m <> 0 ;
s . ( G . ( k + 1 ) ) > 0 ;
( p - M ) . 2 = d ;
A \ominus ( B \ominus C ) = ( A \ominus B ) + ( B \ominus C ) ;
h \equiv ( g mod ( m , n ) ) * ( g mod n ) ;
reconsider i1 = i-1 as Element of NAT ;
let v1 , v2 be VECTOR of V ;
mode (0). V is Element of V equals the carrier of V ;
reconsider i2 = i as Element of NAT ;
dom f c= [: C , D :] \/ { D } ;
x in ( the \alpha=0 of B ) . n ;
len L2 in Seg len L2 + len L2 ;
p1 c= the topology of T ;
]. r , s .[ c= [. r , s .] ;
B2 be Basis of T ;
G * ( B * A ) = ( the Sorts of A ) * ( B * A ) ;
assume that p , u , v is_collinear and u <> v ;
[ z , y ] in union rng F ;
( 'not' b ) . x 'or' b . x = TRUE ;
deffunc F ( set ) = $1 -total ( $1 , $2 ) ;
LIN a1 , a2 , a2 , a3 ;
f " . x = { x } ;
dom ( - w ) = dom ( - r ) /\ dom ( - r ) ;
assume 1 <= i & i <= n ;
( g2 . O ) `2 <= 1 ;
p in LSeg ( E . i , F . i ) ;
I /. i = 0. K ;
|. f . s . m - g . m .| < g ;
q1 . x in rng ( p1 + p2 ) ;
LRRRRnna misses ga ;
consider c being element such that [ a , c ] in G ;
assume N19 = N29 ;
q . ( j + 1 ) = q . ( j + 1 ) ;
rng F c= { F . ( C . ( C . ( C . ( C . C ) ) ) } ;
P . ( B \/ C ) c= 0 + ( B + C ) ;
f . j in { f . j where i is Nat : j in dom f } ;
attr 0 <= x & x <= 1 implies x <= 1 ;
p `2 <> 0. TOP-REAL 2 ;
cluster LaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaiT -> non empty ;
x is Element of S \times T ;
<^ a , b ^> is one-to-one ;
|. i - j .| <= - i ;
the carrier of ( TOP-REAL 2 ) | ( ( ( TOP-REAL 2 ) | ( ( ( ( TOP-REAL 2 ) | P ) ) | P ) ) ) = ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | P ) ) .= ( TOP-REAL 2
! * ( n + 1 ) > 0 * ( n + 1 ) ;
S c= ( A1 /\ A2 ) /\ ( A1 /\ ( A2 /\ A1 ) ) ;
a1 , a2 // a2 , b1 ;
then dom A <> {} ;
1 + ( 2 * k + 1 ) = 2 * k + 2 * k + 2 * k ;
x Joins v1 , v2 , G implies ex y being Vertex of G st x in X & y in Y & x = y ;
set v2 = v1 /. i , u2 = v2 /. i , v2 = u2 /. i , u2 = u2 /. i , u2 = u2 /. i , u2 = u2 /. i , u2 = u2 /. i , u2 = u2 /. i , u2
x = r . n .= ( r (#) ( F ) ) . n .= r * ( F . n ) ;
f . s in the carrier of S2 & f . s in the carrier of S2 ;
dom g = the carrier of I[01] ;
p in Lower_Arc ( P ) /\ Lower_Arc ( P ) ;
dom ( d * ( i + 1 ) ) = A * ( i + 1 ) ;
0 < ( p - z ) + ( p - z ) ;
e . ( m + 1 ) <= e . ( m + 1 ) ;
B \ominus X c= B \ominus X ;
- \infty < \int ( g | B ) + g | B ;
cluster O \tt : F F is \tt X ;
let U1 , U2 be non-empty MSAlgebra over S ;
Proj ( i , n ) * g is_differentiable_on X ;
x , y , z , u is_collinear ;
reconsider p1 = p . x as Subset of V ;
x in the carrier of Lin ( A ) ;
let I , J be Program of SCM+FSA ;
assume - a is Point of X ;
Int Cl ( A \/ B ) c= Cl ( A \/ B ) ;
assume for A being Subset of X holds A is open ;
assume q in Ball ( x , r ) ;
( - p2 ) `2 <= - ( p2 `2 ) `2 ;
Int ( Q ` ) ` = ( ( ( ( ( the carrier of T ) ` ) ` ) ` ) ` ) ` ;
set S = the carrier of T ;
set I = ( f , n ) * ( f , n ) , J = ( f , n ) * ( f , n ) ;
len p -' n = len p - n + 1 ;
A is permutation of Seg ( len A ) ;
reconsider nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn
1 <= j + 1 + len ( s - t ) ;
q1 , q2 , q1 , q2 , q1 is_collinear ;
a1 in the carrier of S1 & a2 in the carrier of S2 & a2 in the carrier of S1 & a2 in the carrier of S2 ;
c1 /. n = c1 . n .= c2 . n ;
let f be FinSequence of TOP-REAL 2 ;
y = ( ( f1 * f2 ) `| Z ) . x ;
consider x being element such that x in \mathclose \mathclose \mathclose { \rm c } ;
assume r in ( dist ( o , P ) ) .: P ;
set i2 = len h + len h ;
h . ( j + 1 ) in rng h ;
Line ( M1 , k ) = M1 * ( i , j ) ;
reconsider m = ( x - 2 * PI * PI * PI ) as Element of REAL ;
U1 , U2 , U1 , U2 , u2 , u2 , u1 , u2 , u2 , u2 , u2 , u1 , u2 , u2 , u2 , u2 , u1 , u2 , u2 , u2 , u1 , u2 , u2 , u1 , u2 ,
set P = Line ( a , d ) ;
len p1 < len ( - 1 ) + 1 + 1 ;
for T1 , T2 being Scott net of L , x being Element of L holds x in { x } iff x in { x }
then x <= y ;
set M = n -\hbox { - } ;
reconsider i = x1 , j = x2 as Element of NAT ;
rng ( the Arity of S ) c= dom ( the Arity of S ) ;
z1 " * z " = z1 " * z " ;
x0 - r in L /\ L /\ L ;
then w is which is which is non empty ;
set x1 = { x where x is Element of NAT : x in dom ( f1 + f2 ) & x in dom ( f1 + f2 ) ;
len ( w - y ) in Seg len ( w - y ) ;
( uncurry f ) . x = g . y ;
be Element of PFuncs ( V , { k } ) ;
x . n = ( |. a .| ) . n ;
( p `1 ) * ( ( p `1 ) * ( p `1 ) ) `1 + ( p `1 ) * ( p `1 ) ) `1 ;
rng ( g * f ) c= L~ ( g * f ) ;
reconsider k = i-1 * ( l + 1 ) as Nat ;
for n be Nat holds F . n is with_\hbox { - \infty ;
reconsider x1 = x as VECTOR of M ;
dom ( f | X ) = X /\ dom ( f | X ) ;
p , a // p , c & p , a // p , a ;
reconsider x1 = x as Element of REAL m ;
assume i in dom ( a * p ) ;
m . ( \hbox { \boldmath $ g $ } } ) = p . ( m + 1 ) ;
a |^ ( s |^ ( m + n ) ) - ( s |^ ( m + n ) ) <= 1 - 1 ;
S . ( n + k ) c= S . ( n + k ) ;
assume A1 \/ A2 \/ A1 \/ A2 = ( A1 \/ A2 ) \/ ( ( A2 \/ A1 ) \/ ( A1 \/ A2 ) ;
X . i = { x1 , x2 , x3 , x4 } . i ;
r2 in dom ( h + c ) ;
internal _ ( R , a ) = a & b-~ = b ;
F1 is closed & F1 is closed implies for n being Nat holds F1 . n = t . n + t . n
set T = \vert --> --> --> --> --> 0 , { 1 } , { 1 } , { 0 } , { 1 } , { 1 } , { 1 } , { 1 } , { 1 } , { 1 } , { 1 } , { 1
Int Cl ( Cl ( Cl ( Cl Int Cl ( Cl A ) ) ) ) c= Cl ( Cl ( Int A ) ) ;
consider y being Element of L such that c . y = x ;
rng F1 = { F1 . x where F1 is Element of F : F1 . x = F1 . x } ;
G1 " ( { c } \/ S ) c= B \/ S \/ S \/ S \/ S ;
f1 /* ( g1 ^ ) is convergent & f2 /* ( g1 ^ ) = f1 ^ f2 ;
set R1 = the Point of TOP-REAL 2 ;
assume that n + 1 >= 1 and n + 1 >= 1 ;
k2 be Element of NAT ;
reconsider p1 = u as Element of \mathclose { -1 } ;
g . x in dom f & g . x in dom f ;
assume that 1 <= n and n + 1 <= len f and n + 1 <= len f ;
reconsider T = b * N as Element of G |^ N ;
len ( - p1 ) <= len ( - p1 ) + len ( - p1 ) ;
x " in the carrier of ( A * ) ;
[ i , j ] in Indices the indices of M1 & [ i , j ] in Indices M1 ;
for m be Nat holds Re ( F . m ) = ( Im ( F . m ) ) * ( Im ( ( F . m ) ) )
f . x = a . i .= a . i ;
let f be PartFunc of REAL , REAL ;
rng f = the carrier of \prod ( A * ) ;
assume s1 = sqrt ( 2 * ( 2 * r ) ) ;
attr a > 1 & a > 1 implies a * b > 1 ;
let A , B , C be Subset of TOP-REAL 2 ;
reconsider X1 = X , X2 = Y as Subset of X ;
let f be PartFunc of REAL , REAL ;
r * ( v1 - v2 ) < r * ( v1 - v2 ) ;
assume that V is Subspace of X and for x being Element of X holds x in V implies x in V ;
t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 be Element of T ;
Q [ { e } \/ { f . e } ] ;
g .. Cage ( C , n ) = z .. Cage ( C , n ) ;
|. [ x , v ] - [ y , v ] .| = vvy ;
- f . w = - ( L . w ) ;
z -' y <= x - y + z ;
sqrt ( 1 - ( ( p `1 / p `1 ) ) ^2 ) > 0 ;
assume X is LE 0 , 1 , 0 , 1 ;
F . 1 = v1 & F . 2 = v2 & F . 3 = v2 ;
( f | X ) . x = f . x ;
( the function tan of tan ) . x in dom ( tan * tan ) ;
i2 = ( f /. len f + 1 ) ) `2 ;
X1 = X1 \/ X2 \/ X1 \/ X2 \/ X0 \/ X0 \/ X0 \/ { X0 } .= { X0 } \/ { X0 } \/ { X0 } \/ { X0 } \/ { X0 } \/ { X0 } \/ { X0 } \/ { X0 } \/ { X0 } \/ {
[. a , b .] = { a , b } ;
let V , W be VectSp of K ;
dom ( g * f ) = the carrier of TOP-REAL 2 ;
dom ( - f2 ) = the carrier of TOP-REAL 2 ;
( proj2 | X ) .: X = ( proj2 | X ) .: X ;
f . ( x , y ) = h . ( x , y ) ;
x0 < r1 & r1 < x0 + r1 ;
|. ( f /* s ) . k - ( f /* s ) . k .| < r ;
len ( Line ( A , i ) ) = len A + len B ;
S\mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak \mathfrak
reconsider f = v + u as Function of X , the carrier of Y ;
( intloc 0 ) in dom ( Initialized ( s ) ) ;
i1 <> i2 & i2 <> j2 & i2 <> j2 & i2 <> j2 & i2 <> j2 & i2 <> j2 & i2 <> j2 & i2 <> j2 & i2 <> j2 & i2 <> j2 & i2 <> j2 & i2 <> j2 & j2 <> j2 & i2 <> j2 & i2 <> j2 & i2 <> j2
sqrt ( r ^2 + r ^2 ) + r ^2 = r ^2 + r ^2 + r ^2 ;
for x st x in Z holds ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( - 1 ) (#) ( - 1 ) ) ) ) `| Z ) . x = - 1 / ( x - 1 ) * ( - 1 ) *
reconsider q2 = ( ( - x ) * ( ( - x ) * ( ( - x ) * ( - x ) ) ) ) as Element of REAL ;
( 0 qua Nat ) + 1 <= i + 1 + 1 ;
assume f in the carrier of X ;
F . a = H . ( x , y ) ;
{} in { {} , C ( ) : not contradiction } ;
dist ( a * ( s * ( ( s * ( n + 1 ) ) ) ) , h ) < r ;
1 in the carrier of \lbrack 0 , 1 .] ;
( - p2 ) * ( - p1 ) > - ( - p2 ) * ( - p2 ) ;
|. r1 .| = |. r1 .| * |. r1 .| ;
reconsider S-14 = 8 as Element of Seg ( len 8 ) ;
( A \/ B ) \/ ( A \/ B ) c= A \/ B \/ ( B \/ C )
DW . ( DW ) = DW . ( W . ( len W + 1 ) ) + W . ( W . ( len W + 1 ) ) ;
i1 = ( - n ) + ( - 1 ) & ( - 1 ) * ( - 1 ) = - 1 * ( - 1 ) * ( - 1 ) ;
f . a [= f . ( O , L ) "\/" ( f . ( O , L ) ) ;
attr f = v & g = u + v ;
I . n = \int F . n + M . n ;
\raise .4ex .4ex .4ex \hbox { \boldmath $ T , T , f , g , h be Function of T , S , s be Real ;
a = VERUM ( A ) or a = VERUM ( A ) ;
reconsider k2 = s . ( b , k2 ) as Element of NAT ;
( Comput ( P , s , 4 ) ) . a = ( Initialize s ) . a ;
L~ ( M1 M1 M1 ) meets L~ M2 ;
set h = the continuous Function of X , TOP-REAL n ;
set A = { L . ( k + 1 ) where k is Nat : k < n } ;
for H st H is negative holds H is negative
set with with with ' , a , b , c , d be Element of S ;
Hom ( a , b ) c= Hom ( a , b ) ;
( 1 - n ) * ( 1 - n ) < ( 1 - n ) * ( 1 - n ) ;
( l , 1 ) `2 = [ l , 1 ] `2 ;
y +* ( i , y ) in dom g ;
let p be Element of QC-WFF ( A , p ) ;
X /\ X1 c= dom ( - ( 1 - ( 2 * x ) ) ) ;
p2 in rng ( f ^ g ) ;
1 <= indx ( D2 , D1 , j ) + 1 ;
assume x in { p2 } /\ K \/ { p2 } /\ K ;
- 1 - ( ( ( ( ( ( ( ( ( O ) O ) ) * ( ( O ) ) * ( ( O ) ) * ( ( O ) ) * ( ( O ) ) * ( ( O ) ) * ( ( O ) ) * ( ( O
f , g be Function of I[01] , TOP-REAL 2 ;
k1 -' k2 = ( k - 1 ) - 1 ;
rng seq c= ]. x0 - r , x0 + r .[ ;
g2 in ]. x0 - r , x0 + r .[ ;
sgn ( p , K ) = - - p * ( - p * ( - p ) * ( - p * ( - p * ( - p * p ) ) ) ;
consider u being Nat such that b = p |^ u * u + u * v ;
given a being Ordinal such that a in Sum ( A ) and for n being Nat st n <= m holds Sum ( A ) = a * ( A . n ) ;
Cl ( Cl Cl Cl ( Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Int Cl Cl Int Cl Cl Cl Int Cl Cl Cl Int Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl
len t = len ( t + <* n *> ) + len t + ( <* n *> ) .= len t + len <* n *> + 1 ;
v1 = v + u + u + v .= v + u + u + u ;
deffunc v <> DataLoc ( t . a , k1 ) ;
g . s = sup ( { d } ^ s ) ;
( \dot { y } ) . s = s . y ;
not s < t implies s < t
s " * s \ ( s * t ) = s * t " * s * t " ;
defpred P [ Nat ] means B + $1 in A ;
( 339 + 1 ) * ( 339 + 1 ) = 339 * ( 339 + 1 ) ;
U /. ( succ A ) = ( U /. ( succ A ) ) `1 ;
reconsider y = y as Element of REAL ;
consider i2 being Integer such that y = p * ( i2 * i2 ) + ( i2 * j2 ) ;
reconsider p = Y | Seg ( k + 1 ) as FinSequence of NAT ;
set f = ( S , U ) \! \mathop { \rm \hbox { - } TruthEval } ( z , n ) ;
consider Z being set such that lim s in Z and Z in F ;
let f be Function of I[01] , TOP-REAL 2 , b being Point of TOP-REAL 2 ;
=> ( M . ( n + i ) ) <> 1 ;
ex r being Real st x = r & r <= 1 & r <= 1 ;
R1 , R2 , R1 , R2 be Element of REAL n -tuples_on REAL n ;
reconsider l = 0. V as Linear_Combination of A ;
set r = |. e .| + |. s .| + |. s .| + |. s .| + |. t .| ;
consider y being Element of S such that z <= y and y in X ;
a `2 = ( a `2 ) `2 ;
||. x0 - y0 .|| < r ;
b1 , a1 // b1 , a1 ;
1 <= ( k + 1 ) - 1 + 1 & k + 1 <= len ( k + 1 ) - 1 ;
( - p ) * ( - p ) * ( - p ) = - p * ( - p ) * ( - p * ( - p ) ) ;
( ( - q ) * ( - q ) ) * ( - q ) < ( - q ) * ( - q ) ;
W-min ( C ) in LSeg ( ( ( R /. 1 ) , 1 ) ) ;
consider e being Element of NAT such that a = 2 * e + 1 ;
Re ( ( lim F ) | D ) = Re ( ( lim F ) | D ) ;
LIN b , a , c or LIN b , c , d ;
p `1 , a `2 // a `1 , b `2 or p `2 = b `2 ;
g . n = a * ( f . n ) .= a * ( f . n ) .= a * ( f . n ) ;
consider f being Subset of X such that e = f and f is one-to-one ;
F | ( [: N , S :] , S :] ) = F * ( F * ( N , S ) ) ;
q in LSeg ( q , v ) \/ { q } ;
Ball ( m , r ) c= Ball ( m , r ) ;
the carrier of ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( ( TOP-REAL 2 ) | K ) ) ) ) = { 0. TOP-REAL 2 } ;
rng ( ( the function tan ) (#) ( tan + tan ) ) `| Z ) = [. - 1 , 1 .] ;
assume that Re ( seq ) is summable and Im ( seq ) = Im ( seq ) ;
||. ( - seq . n ) - ( seq . n ) .|| < e ;
set g = O --> 1 ;
reconsider t2 = { t where t is 0 .| * ( t + 1 ) * ( t + 1 ) * ( t + 1 ) * ( t + 1 ) * ( t + 1 ) * ( t + 1 ) ) * ( t + 1 ) * ( t + 1 ) * ( t +
reconsider x1 = s1 as sequence of REAL n , x2 = s2 as sequence of REAL n ;
assume that \mathop { \rm carrier' } ( C ) meets \widetilde { \cal L } ( \pi ) and \pi * ( 1 + 1 ) in \widetilde { \cal L } ( \pi ) ;
- ( 1 - ( card ( A ) ) ) < F . n - 1 ;
set d = \bf min ( ( |. x1 - x2 .| , |. x2 - x3 .| ) , |. x1 - x3 .| ) , |. x2 - x3 .| = |. x1 - x3 .| ;
2 |^ ( 2 * 1 ) -' 2 = 2 |^ ( 2 * 1 ) - 2 * ( 2 * 1 ) ;
dom ( - v ) = Seg len ( - v ) ;
set x1 = - ( x1 - x2 ) , x2 = - ( x1 - x2 ) , x3 = - ( x1 - x2 ) , x3 = - ( x1 - x2 ) , x3 = - ( x1 - x2 ) , x4 = - ( x1 - x2 ) , x4 = - ( x1 - x2
assume for n being Element of X holds ( F . n ) ` <= ( F . n ) ` ;
assume 0 <= |. means : for i st i in dom T holds T . i = T . i ;
for A being Subset of X holds c . ( c . A ) = c . A
the support of L1 + L2 + L2 + L1 c= the carrier of L1 + L2
'not' ( x => p ) => ( p => ( x => p ) ) is valid ;
( f | n ) /. k = f /. k ;
reconsider Z = { [ {} , {} ] } as Element of the carrier of K ;
Z c= dom ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( f f f ) ) ) ) ) ) ) ) ) ) ) ) ) ) `| Z ) ;
|. 0. TOP-REAL 2 - ( ( TOP-REAL 2 ) | K1 ) - sn ) .| < r ;
\mathop { \rm ConsecutiveSet2 D \ { A } , B , C ) c= \mathop { \rm ConsecutiveSet2 } ( A , B ) ;
E = dom ( L1 + L2 ) & E is nonnegative ;
C |^ ( A + B ) = C |^ ( A + B ) ;
the carrier of ( W1 + W2 ) c= the carrier of ( W1 + W2 ) ;
I . IC Comput ( P , s , 2 ) = P . IC Comput ( P , s , 2 ) , 2 ) ;
attr x > 0 implies x * ( x * ( 1 - x ) ) = x * ( 1 - x ) ;
LSeg ( f ^ g , i ) = LSeg ( f ^ g , j ) ;
consider p being Point of T such that C = { p where p is Point of T : p in C } ;
b , c , d is_collinear & c , d , d is_collinear implies b , c , d is_collinear
assume f = id the carrier of C & g is Function of the carrier of C , the carrier of C ;
consider v such that v <> 0. V and f . v = L . v ;
let l be Linear_Combination of {} ( ) ;
reconsider g = f " as Function of U , ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( ( TOP-REAL 2 ) | ( ( ( TOP-REAL 2 ) | ( ( ( TOP-REAL 2 ) ) | K1 ) ) ) ) ) ) , ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) |
A1 in the carrier of ( ( TOP-REAL 2 ) | ( ( ( ( ( ( ( ( ( ( ( TOP-REAL 2 ) | ( K ) ) | K ) ) ) ) ) ) ) | ( ( ( TOP-REAL 2 ) | K ) | K ) ) ) ;
|. x - y .| = - x .= y - y ;
set S = \mathop { \rm Segm ( x , y , c ) ;
Fib ( n ) * ( 5 * n ) >= 5 * n + 1 * n ;
v1 /. k = ( the Sorts of A ) . k + ( the Sorts of A ) . k ;
0 mod i mod ( i mod ( i mod j ) ) = i mod ( i mod j ) ;
the carrier of M1 = [: the carrier of M1 , the carrier of M1 :] ;
Line ( pnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn
h . x1 , y . x2 = [ x1 , y1 ] ;
|. f .| (#) ( |. f .| (#) ( |. b .| ) ) .| is nonnegative ;
assume x = ( a ^ b ) ^ <* x *> ;
M1 is closed on I , P ;
DataLoc ( t . a , 4 ) = t . a + 4 ;
x + y < x + y + x & x + y = y + x + y + x ;
LIN c , d , c & LIN d , c , d ;
f1 . ( 1 , t ) = f . ( 0 , t ) .= f . ( 1 , t ) ;
x + ( y + z ) = ( x + y ) + ( y + z ) ;
f1 . a = f1 . a .= f2 . a .= f2 . a ;
( p `1 ) * ( ( p `2 ) * ( p `2 ) ) `1 + ( p `2 ) * ( p `2 ) ) `1 ;
set Rnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn
( p `1 ) ^2 >= ( p `2 ) ^2 ;
consider p such that p = { p : p < p & p < s & s < p & p in X & p in X & p in X ;
|. ( f /* s ) . l - ( f /* s ) . l .| < r ;
Segm ( M , p ) = Segm ( M , p ) ;
len ( Line ( N , k ) ) = width ( N , k ) ;
f1 /* ( s ^\ k ) is convergent & f2 /* ( s ^\ k ) = f1 /* ( s ^\ k ) ;
f . x1 = x1 & f . x2 = x2 ;
len f <= len f + len g & len f + len g = len f + len g + len g ;
dom ( Proj ( i , n ) ) = REAL ( m , n ) ;
n = k * ( 2 * t ) + ( 2 * t ) * ( 2 * t ) ;
dom B = 2 ^ ( A * B ) ;
consider r such that r _|_ x and r in X and r in X ;
reconsider B1 = the carrier of Y as Subset of X ;
1 in the carrier of [. - 1 , 1 .] ;
let L be complete continuous continuous continuous continuous continuous LATTICE , x be Element of L ;
[ g1 , g2 ] in Indices ( I ) \ { [ g1 , g2 ] } ;
set S2 = 1GateCircStr ( x , y ) , S2 = 1GateCircuit ( y , c ) ;
assume that f1 is_differentiable_on REAL and f2 = #Z n ;
reconsider y = ( a " ) * ( a " ) as Element of L ;
dom s = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 7 , 7 , 8 , 7 , 7 , 7 , 7 , 7 , 7 , 8 , 7 , 7 , 8 , 7 , 7 , 7 , 8 , 7 } & 8 <> 7 & 8 <> 7 & 8 <>
( min ( g - f ) ) . c <= h . c ;
set G2 = the subgraph of G , the subgraph of G , the topology of G , the topology of G = the topology of G , the topology of G = the topology of G , the topology of G = the topology of G ;
reconsider g = f as PartFunc of REAL , REAL n , REAL ;
|. s1 . m - s1 . m .| < d / 2 ;
for x being element st x in u holds ( u + v ) . x = u + v
P = the carrier of ( TOP-REAL n ) | K1 ;
assume p1 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ;
( 0. X ) \ ( x \ ( m \ k ) ) = 0. X ;
g is Element of Hom ( cod ( f , g ) ) ;
2 * a + b * c + d * c + d * a * c + d * a * c * c + d * a * c * c + d * a * c + d * c * a * c + d * c + d * a * c + d * a * c + d * c + d
f , g be PartFunc of X , REAL ;
set h = Hom ( a , f ) , g = hom ( a , f ) , h = hom ( a , f ) ;
then idseq ( n + m ) = m * ( n + m ) ;
H * ( g " * ( a " ) ) in the carrier of H ;
x in dom ( ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 )
cell ( G , i2 , j1 ) misses C ;
LE q1 , q2 , P & p1 , q2 , q1 , q2 , p1 & q1 , q2 , p1 , p2 & p1 , q2 , p1 , q1 , p1 , p1 , p2 , p1 , p1 , p2 , p1 , p1 , p1 , p2 , p1 , p1 , p1 , p1 , p1 , p2 , p1
attr B is Subset of TOP-REAL 2 means : DefDef: B is component & B is ` & B ` = B ` ;
deffunc D ( set , set ) = $2 $2 $2 $2 $2 $2 $2 $2 $2 $2 $2 $2 $2 $2 $2 $2 $2 $2 $2 $2 $2 $2 $2 $2 $2 = ( $2 " ) * ( $2 " ) ;
n + 1 - n < len p1 + len p2 + n - 1 + ( n - 1 ) ;
attr a <> 0. K K ;
consider j such that j in dom b1 and I = b1 + J . j ;
consider x1 such that z in { x1 : x1 in P & x2 in P & x1 in P & x2 in P & x3 in P & x3 in P & x3 in P & x4 in P & x4 in P & x4 in P ;
for n being Element of NAT holds X [ n ] ;
set C1 = Comput ( P1 , s1 , i ) , C2 = P1 . ( i + 1 ) , C1 = P1 . ( i + 1 ) , C2 = P2 . ( i + 1 ) , C2 = P2 . ( i + 1 ) , C2 = P2 . ( i + 1 ) , C2 = P2
set v = 3 * ( a , b ) , w = 3 * ( a , b ) , u = 3 * ( a , b , c ) , v = 3 * ( a , b , d ) , w = 3 * ( a , b , d ) , w = 3 * ( a , b
Int Int @ ^ @ @ W c= ( F ^ @ W ) .: W ;
1 in [. - 1 , 1 .] /\ dom ( - 1 ) * ( - 1 ) , - 1 .] ;
s3 <= s1 + s2 + s3 * ( 1 - s1 ) ;
dom ( f * ( - 1 ) ) = dom ( f * ( - 1 ) ) /\ dom ( - 1 ) ;
dom ( f * F ) = dom ( l * F ) /\ dom ( l * F ) .= dom ( l * F ) /\ dom ( l * F ) ;
rng ( s ^\ k ) c= dom ( f1 + f2 ) /\ dom f2 /\ dom f2 \ dom f2 ;
reconsider g1 = { p where p is Point of TOP-REAL 2 : p `1 >= p `1 & p `1 >= p `1 & p `2 >= p `1 } as Subset of TOP-REAL 2 ;
( T * h ) . s = ( h * ( s * ( s * x ) ) ) . s ;
I . ( J . ( J . ( J . x ) ) ) = ( J . ( J . x ) ) * ( J . ( J . x ) ) ;
y in dom f\alpha=0 implies PPPPPPPPPPPPPPPPPPPPPPPPPPPe ( A , o )
for I being non trivial doubleLoopStr for A being Subset of I holds A is commutative
set s2 = s +* ( a , I ) , s1 = ( s +* I ) +* ( a , I ) , s2 = P +* I , s2 = P +* I , s2 = P +* I , p = P +* I , p = P +* I , p = P +* I ;
P1 /. IC Comput ( P1 , s1 , i + 1 ) ) = P1 . IC Comput ( P1 , s1 , i + 1 ) .= P1 . IC Comput ( P1 , s1 , i + 1 ) ;
lim ( S1 , S2 ) in the carrier of ( TOP-REAL 2 ) | ( ( ( TOP-REAL 2 ) | ( ( ( ( ( a ) ) | B ) ) ) ) ;
v . ( l . i ) = ( v . i ) * ( l . i ) ;
consider n be element such that n in NAT and x = seq . n ;
consider x being Element of c such that F1 . x <> 0. F and F1 . x <> 0. F ;
Function ( X , 0 , 1 ) , ( X , 0 ) --> ( x , 0 ) --> ( x , 0 ) ) = { x } ;
j + ( 2 * k + 1 ) > j + ( 2 * k + 1 ) ;
not s , r , s , t is_collinear & t on { t , s , t } ;
n1 > len ( p1 , p2 ) - len ( p1 , p2 ) ;
g1 . ( g2 . ( g2 . ( g2 . ( g2 . ( g2 . ( g2 . ( g2 . ( g2 . ( g2 . n ) ) ) ) ) ) = 0. L ;
then H is \kern1pt & H is \kern1pt ;
( E-max L~ Cage ( C , n ) ) .. f >= 1 ;
]. s , 1 .[ /\ ]. s , 1 .[ = ]. s , 1 .[ /\ ]. s , 1 .[ ;
x1 in [#] ( ( TOP-REAL 2 ) | ( ( ( ( ( TOP-REAL 2 ) | ( K ) ) | K ) ) ) ) ;
let f1 , f2 be PartFunc of REAL , REAL ;
DigA ( tNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN
I = [ d , <* d , e *> ] & I = [ d , e , f ] ;
u in { [ a , b ] : not contradiction } ;
( w | p ) | ( p | p ) = p ;
consider v2 such that v2 in W1 and x in W2 and v2 in W2 and v2 in W1 ;
for y st y in rng F holds y in rng F implies ex n st n = a |^ n
dom ( g * ( g * ( K , n ) ) ) = K ;
ex x being element st x in ( the Sorts of A ) \/ ( ( the Sorts of A ) . s & x = ( the Sorts of A ) . s ;
ex x be element st x in ( ( the Sorts of O ) \/ ( ( the Sorts of O ) . s ) & x = ( the Sorts of O ) . s ;
f . x in the carrier of ]. - r , s .[ ;
( the carrier of X ) \/ { 0. X } /\ ( the carrier of X ) <> {} ;
p1 /\ LSeg ( p1 , p2 ) c= { p1 } ;
( b + b--\hbox { - } ) < r ;
ex_sup_of { x , y } , L & x "\/" y in { x , y } ;
for x being element st x in X holds P [ x , u ]
consider z being Point of G1 such that z = y and z in the carrier of G1 and z in G1 ;
( the addF of cluster the carrier of One by the carrier of One by the carrier of One Funcs ( X , Y ) ) . v = e ;
len ( w ^ <* w *> ) + 1 = len ( w ^ <* w *> ) + 1 ;
assume q in the carrier of ( TOP-REAL 2 ) | K1 ;
f | E = g | E .= g | E ;
reconsider i1 = x , i2 = y as Element of NAT ;
( a * A ) * ( a * B ) = ( a * A ) * ( a * B ) ;
assume ex n1 being Element of NAT st f |^ n1 = f |^ n1 ;
Seg ( len ( ( ( f ^ g ) ) ) ) = dom ( ( f ^ g ) - g ) ;
( Complement A1 ) . m c= ( A1 . m ) . n ;
f1 . p = p1 & f2 . p = p2 & ( f1 . p ) . q = p1 & ( f2 . q ) . p = p2 implies for p being Element of TOP-REAL 2 st p in dom ( f2 * p ) holds ( f2 * p ) . p = p * ( ( f2 * p ) . p )
FinS ( F , Y ) = FinS ( F , X ) + ( F , Y ) ;
( x | y ) | z = z ;
( |. x .| |^ n ) * ( |. x .| |^ n ) <= ( |. x .| |^ n ) * ( |. x .| |^ n ) ;
Sum ( F ) = Sum ( f + g ) & Sum ( F + g ) = Sum ( F + g ) + Sum ( F + g ) ;
assume for x , y being set st x in Y holds x in Y ;
assume that W1 is Subspace of W2 and W2 is Subspace of W1 and W1 is Subspace of W2 and W2 is Subspace of W1 and W1 = W2 ;
||. ( t . x ) - ( t . x ) .|| = ||. ( t . x ) - ( t . x ) .|| ;
assume i in dom D & i in dom ( f | A ) ;
( ( - p ) * ( - p ) ) * ( - p ) ) * ( - p ) <= - - - p * ( - p ) * ( - p * ( - p ) ) ;
g | Sphere ( p , r ) = g | ( TOP-REAL n ) ;
set NN = ( the InternalRel of N ) . ( n + 1 ) ;
let T be non empty TopSpace , x be Point of T ;
width B = ( B * ( i , j ) ) .= n ;
attr a <> 0. X implies ( a (#) ( A \ B ) ) (#) ( a (#) ( B \ B ) ) = ( a (#) ( B \ B ) ) ;
then f is_continuous_on SVF1 ( 2 , f , u ) ;
assume that a > 0 and a <> 0 and a * b = 0 ;
v1 , v2 , w , v1 , w , w , w , y , w , y is_collinear ;
p2 /. ( IC Comput ( p2 , s2 , i ) ) = p2 . ( len p2 + 1 ) .= p2 . ( len p2 + 1 ) ;
ind ( ( B1 | b ) | b ) = ( B1 | b ) | b .= ( B1 | b ) | b ;
[ a , A ] in the Line of K ;
m in ( the carrier' of C ) . ( the carrier' of C ) ;
( CompF ( a , P , G ) ) . z = TRUE ;
reconsider \varphi = \varphi , \varphi = \varphi , \varphi = \varphi , \varphi = \varphi , \varphi = l , \varphi = l , \varphi = l , l = l , l = l , \varphi = l , l = l , \varphi = l , l = l , \varphi = l , \varphi = l , l = l , l = l , l = l , l =
len ( s1 - s1 ) * ( ( s1 - s1 ) * ( 1 - s1 ) ) > 0 ;
\delta _ { D ( ) ) * ( f . ( ( ( ( ( ( ( ( ( the the carrier of A ) ) ) * ( f . x ) ) ) ) ) ) * ( f . ( ( - x ) ) * ( f . x ) ) ) ) ) < r ;
[ f1 , f2 ] in the carrier of [: A , B :] ;
the carrier of ( TOP-REAL 2 ) | K1 = ( TOP-REAL 2 ) | K1 & ( TOP-REAL 2 ) | K1 = ( TOP-REAL 2 ) | K1 & ( TOP-REAL 2 ) | K1 ) = ( TOP-REAL 2 ) | K1 ;
consider z being element such that z in dom ( - g ) and p = g . z and g . z = - g . z ;
[#] V = { 0. V } .= { 0. V } .= { 0. V } ;
consider P2 being FinSequence of K such that rng P2 = M and for i being Nat st i in dom P2 holds P2 . i = F ( i ) ;
assume that x1 in dom ( f | X ) and x1 in dom ( f | X ) and x2 in dom ( f | X ) and x1 in X and x2 in X ;
h1 = f ^ <* p *> .= f ^ <* p *> .= h ^ <* p *> ^ <* p *> .= h ^ <* p *> ^ <* p *> .= h ^ <* p *> ^ <* p *> ^ <* p *> .= h ^ <* p *> ^ <* p *> ^ <* p *> ^ <* p *> .= h ^ <* p *> ^ <* p *> ^ <* p *> .=
c /. [ b , c ] = [ c , d ] .= [ c , d ] ;
reconsider t1 = p , t2 = p , s2 = p as Point of C ;
( - 1 ) * ( - 1 ) in the carrier of ( TOP-REAL 2 ) | ( ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( K ) ) ) ) ;
ex W being Subset of X st p in W & h .: W c= V ;
( h . p1 ) `2 = C * ( ( h . p1 ) `2 + ( h . p1 ) `2 ) `2 + ( h . p1 `2 ) `2 ;
R . b = 2 * b .= 2 * b .= 2 * b * b .= 2 * b * b ;
consider \lambda such that B = 1- C * ( 1 - r1 ) and 0 <= r1 and r1 <= 1 ;
dom g = dom ( ( the Sorts of A ) * ( ( the Sorts of A ) * ( ( the Sorts of A ) * ( the Sorts of A ) ) ) ) ;
[ P . ( l . ( l . ( l . ( l . ( l . ( l . ( l . ( l . ( l . . ( . . . l . . . . . . . . x ) ) ) , l . ( l . ( l . ( l . ( l . ( l . ( l . ( l . ( l . ( l .
set s2 = Initialize s , P1 = P ;
reconsider M = mid ( z , i , j ) as Matrix of n , K ;
y in product ( the support of J +* ( V , v ) ) ;
1 / ( |[ 0 , 1 ]| ) = 1 & 1 <= 1 & ( 1 - |[ 0 , 1 ]| ) / ( 1 + |[ 1 , 1 ]| ) = 1 / ( 1 + |[ 1 , 1 ]| ) ;
assume x in the carrier of g or x in the carrier of g & x in the carrier of g ;
consider M being strict Subgroup of the Sorts of A such that a = M and M is SubSpace of A and M is Subspace of A ;
for x st x in Z holds ( ( ( ( - 1 ) (#) ( ( tan ) ) * ( ( tan + tan ) ) ) `| Z ) . x <> 0
len W1 + ( W2 + W1 ) = len W1 + ( W2 + W1 + W2 ) .= len W1 + len W2 + ( W1 + W2 ) ;
reconsider h1 = v1 . n - ( v1 . n ) * v1 as VECTOR of X ;
( i -' len p ) mod ( len p + 1 ) in dom ( p - 1 ) ;
assume that s2 is U and s1 is {} and s2 in the carrier of s2 and s2 in the carrier of s2 and s2 in the carrier of s2 and s2 in the carrier of s2 and s2 in the carrier of s2 and s2 in the carrier of s2 and s2 in the carrier of s2 and s2 in the carrier of s2 and s2 in the carrier of s2 and s2 in
( AffineMap ( x2 , x3 , x4 ) ) * ( |[ 0 , x2 ]| ) = |[ 0 , x3 ]| ;
for u being element st u in Bags n holds ( p + u ) . u = p . u + ( p . u )
for B being Subset of E st B in E holds B is open iff B is open
ex a being Point of X st a in A & a in A & a in A & a in A ;
set W1 = [: p , q :] \/ { p } , W2 = { p } \/ { q } , { p } , { q } , { p } , { q } } , { q } = { p } , { q } , { q } , { q } = { p } , { q } , { q } } , { q } =
x in { X where X is Element of L : X is Subset of L } ;
the carrier of W1 /\ ( the carrier of W1 + W2 ) c= the carrier of W1 + ( the carrier of W1 + W2 ) ;
id ( ( a + b ) * ( id a + b ) ) = id a * ( a + b ) .= a * ( b + b ) ;
( dom ( X --> 0 ) ) . x = ( X --> 0 ) . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) ;
p => ( q => r ) in TAUT ( Al ;
set \pi = LSeg ( G * ( i2 , k ) , G * ( i2 , k ) ) , G = G * ( i2 , k ) ;
set \pi = LSeg ( G * ( i2 , k ) , G * ( i2 , k ) ) , G = G * ( i2 , k ) ;
- 1 + 1 <= - 1 + 1 ;
( reproj ( 1 , z ) ) . x in dom ( ( ( ( ( f - f ) * ( f1 - f2 ) ) ) ) . x ) ;
assume that b1 . r = b1 and b2 . r = b2 . r and b1 . r = b2 . r and b1 . r = b2 . r ;
ex P st a on P & b on P & a on P & b on P & a on P & b on P & a on P & b on P & a on P & b on P & b on P & a on P & b on P & b on P & a on P & b on P & a on P & b on P
reconsider g1 = g * f as Element of X * ( h * g ) as Element of X ;
consider v1 being Element of T such that Q = ( ( the \downarrow of T ) " ) . v1 and for x being Element of T st x in ( the carrier of T ) . v1 holds x in { ( the carrier of T ) . v1 } ;
n in { i where i is Nat : i < n + 1 & i < n + 1 } ;
( F /. i ) `2 >= ( F /. i ) `2 ;
assume that p1 = { p : p `1 >= - 1 & p `2 >= - 1 & p `2 >= - 1 & p `2 >= 1 and p `2 >= 1 and p `2 >= 0 and p `2 >= 0 and p `2 >= 0 and p `1 >= 0 and p `1 >= 0 and p `1 >= 0 and p `1 >= 0 and p `1 >= 0 and p `1 >= 0 ;
SVF1 ( A , succ ( A , succ ( A , succ ( A , succ ( A , succ ( A , succ ( A , succ ( A , succ ( succ ( A , succ ( succ ( A , succ ( succ ( A ) ) ) ) ) ) ) ) ) ) = ( ( A succ ( A , succ ( A , succ ( A , succ ( A , succ ( A , succ ( A , succ (
set I = ( a , I ) := ( a , I ) , J = ( a , I ) := ( a , I ) ;
for i be Nat st 1 < i & i < len z holds z . i = z . i + z . i
X c= ( the carrier of L1 ) \/ ( the carrier of L1 ) ;
consider x9 being Element of GF ( p ) such that x9 = a * x9 + x9 and x9 <> 0 & x9 <> 0. GF ( p ) ;
reconsider f3 = { f1 where f1 is Element of D : f1 is continuous & f2 is continuous & f2 is continuous } as non empty Subset of D ;
ex O being set st O in S & O c= O & O c= O & O c= O & O c= O & O c= O & O c= O & O c= O & O c= O & O c= O & O c= O & O c= O & O c= O & O c= O & O c= O & O c= O & O c= O & O c= O & O c= O & O c= O & O
consider n be Nat such that for m being Nat st n <= m holds S . m in U ;
f * g * ( reproj ( i , x ) ) = ( f * g ) * ( reproj ( i , x ) ) ;
defpred P [ Nat ] means A + ( 1 - $1 ) * ( $1 - 1 ) * ( $1 - 1 ) * ( $1 - 1 ) ;
the carrier of right left right right diff ( g , d ) ) = the carrier of right right right right diff ( g , d ) , d ) ;
reconsider p1 = x , p2 = y , p3 = z as Point of TOP-REAL 2 ;
consider g such that - g = y and g . x = y and g . y = 0 ;
for n being Element of NAT holds X [ n ] ;
len ( - ( x ^ y ) ) = len ( - y ) + len ( - y ) .= len ( - y ) + len ( - y ) ;
for x being element st x in X holds ( the carrier of X ) . x = ( the carrier of X ) . x
LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = {} ;
func -> Function of X , Y means : DefDef: for x being set st x in X holds it . x = x ;
len ( the carrier of K ) + len ( the carrier of K ) + len ( the carrier of K ) + 1 ) <= len ( the carrier of K ) + 1 ;
attr K is implies for a being Element of K holds a * ( - b * ( - b * ( - b * ( - b * a ) ) ) = - a * ( - b * a ) ) ;
consider o being OperSymbol of S such that t . {} = [ o , the carrier of S ] and o = [ o , the carrier of S ] ;
for x st x in X holds x in X iff x in X
IC Comput ( P1 , s1 , k + 1 ) in dom I ;
attr q < s & r < s implies r < s & s < q ;
consider c being Element of Class ( F , c ) such that Y = ( F . c ) * ( F . c ) ;
the carrier' of S = the carrier' of S & the carrier' of S = the carrier' of S ;
set y = [ <* y , z *> , f1 ] , f2 = [ <* z , x *> , f2 ] ;
assume x in dom ( ( ( ( ( ( ( ( ( ( the the function ) ) ) (#) ( f ) ) `| Z ) ) ) `| Z ) ) ;
r2 in LSeg ( f , i ) /\ LSeg ( f , j ) ;
( q `2 ) ^2 >= ( ( q `2 ) ^2 ;
set Y = { a where a is Element of L : a in X } ;
i -' len f + len g + ( - 1 ) + 1 <= len f + len g - 1 + 1 ;
for n , x , y st x in N holds h . n = x + y
set s1 = ( \mathop { \it false } ( a , I ) ) . i , s2 = ( \mathop { \it true } ( a , I ) ) . i , s2 = ( \mathop { \it true } ( a , I ) ) . i ;
p . ( k + 1 ) = 1 or p . ( k + 1 ) = 1 & p . ( k + 1 ) = 1 ;
u + Sum ( L + u ) in ( U \ { u } ) \/ ( { u } \/ { v } ) ;
consider x1 being set such that x in { x1 } and x1 in { x2 } and x2 in { x1 } and x1 in { x2 } ;
( p ^ <* p *> ) . m = ( p ^ <* p *> ) . m .= ( p ^ <* p *> ) . m ;
g + h = g1 + g2 & g + h = g2 + ( g1 + g2 ) ;
L1 is distributive implies for L being distributive distributive distributive distributive distributive distributive distributive distributive distributive distributive distributive distributive distributive & for x being Element of L holds ( L . x ) in the carrier of L
attr x in rng f & y in rng f implies x in rng f & y in dom f & x in dom f & y = f . x ;
assume that 1 < p and p < 1 and p in A and p in A and q in A ;
Fuuuuuuuuuuuuuuuuuuuuufunc ( f , g ) ) = ( ( ( f , g ) ) \ast ( f , g ) ) + ( f , h ) .= ( f , h ) \ast ( f , h ) ;
let X be set , A be Subset of X ;
( E-max L~ Cage ( C , n ) ) `1 <= ( E-max L~ Cage ( C , n ) ) `1 ;
let c be Element of the bound of A ( ) , a , b be Element of A ( ) ( ) ;
s1 . GBP = ( Initialize s2 ) . intpos ( i + j ) .= s1 . intpos ( i + j ) ;
let a , b be Real , x , y be Real st x in ( y + 1 ) holds x + y in ( y + 1 ) * ( y + 1 )
for x , y being Element of X holds ( x \ y ) \ ( y \ x ) = ( x \ y ) \ ( y \ x )
mode mode commutative -> commutative means : DefDef: for i , j st i < j & j < n holds i * j = j * ( i , j ) ;
set x1 = ( Re ( y ) ) | ( ( Im y ) | ( y , z ) ) ;
[ y , x ] in dom ( u + v ) & u . ( y + x ) = g . ( u , x ) ;
]. ( inf divset ( D , k ) ) , ( ( - ( k + 1 ) ) * ( ( - 1 ) * ( ( k + 1 ) * ( ( - 1 ) * ( k + 1 ) ) ) ) ) ) ) c= A ;
0 <= delta ( 2 , n ) & |. ( delta ( 2 , n ) ) . m - ( 2 * n ) . m .| < e ;
( - ( q `1 ) ) ^2 + ( - q `1 ) ^2 + ( q `1 ) ^2 ) ^2 + ( q `1 ) ^2 + ( q `1 / q `2 ) ^2 ) ^2 ;
set A = sqrt ( 2 * PI ) ;
for x , y being set st x in { x } & y in { y } holds x in { y }
deffunc F ( Nat ) = b * ( $1 + 1 ) ;
for s being element holds s in dom f \/ dom g iff s in dom f \/ dom g & s in dom f \/ dom g
let L be non void transitive antisymmetric transitive RelStr ;
max ( ( degree ( ( ( ( ( z ) / |. z .| ) ) ) ) ) ) ) + ( ( |. z .| ) ) ^2 ) >= 0 ;
consider n1 being Nat such that for n being Nat holds seq . n < r + 1 and for n being Nat holds seq . n < r ;
Lin ( A /\ B ) is Subspace of Lin ( B ) & Lin ( B ) = Lin ( B ) + Lin ( B ) ;
set nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn
f .: ( X \/ Y ) in [: X , Y :] & f .: ( X \/ Y ) in [: X , Y :] ;
rng ( a ^\ c ) \/ ( c ^\ c ) c= { a } \/ { b + c } ;
consider y being Point of G1 , x being Point of G1 such that y = y and x in y and y in dom ( G ) and x in y and y in dom ( G ) and x in dom ( G ) and y in dom ( G ) ;
dom ( ( f1 + f2 ) /* c ) /\ dom ( f2 + f1 ) c= dom ( f1 + f2 ) /\ dom f2 /\ dom ( f2 + f1 ) ;
AffineMap ( i , j , n ) is Element of TOP-REAL n ;
v ^ ( ( n |-> 0. K ) ) in ( the carrier of K ) \/ ( the carrier of K ) ;
ex a , k1 , k2 being Nat st i = a & k1 = b & k2 = c & k2 = c & k2 = c & k2 = c & k2 = c ;
t . ( NAT + 1 ) = ( NAT --> ( ( the InstructionsF of K ) . ( n + 1 ) ) ) . ( t . ( n + 1 ) ) .= t . ( t . ( n + 1 ) ) ;
assume that F is bbbbbbbbbbfamily and p is b and p is b ;
not LIN b , a , c & not b , c // b , d ;
( L1 --> L2 ) . ( L1 L1 L1 L1 . ( L1 . ( L1 . ( L1 . x ) ) ) ) = ( L1 . ( L1 . x ) ) ;
consider F being ManySortedSet of E such that for n being Element of NAT holds F . n = F ( n ) ;
consider a , b such that a * ( ww ) = b * ( ww ) + ( - b * w ) and 0 <= b * ( - b * w ) ;
defpred P [ FinSequence of D ] means for i being Nat st i in dom ( ( the carrier of D ) * ( $1 , 1 ) ) holds ( ( the carrier of D ) * ( ( the carrier of D ) * ( ( the carrier of D ) * ( ( the carrier of D ) * ( ( the carrier of D ) * ( ( f , 1 ) ) ) ) ) . i = ( the carrier
u = ( ( - 1 ) * x ) * ( - 1 ) .= x * ( - 1 ) * ( - 1 ) .= x * ( - 1 ) * x .= x * ( - 1 * x ) .= x * ( - 1 * x ) .= x * ( - 1 * x ) .= x * ( - 1 * x ) .= x * ( - 1 * x ) .= x * ( -
dist ( seq . n + x ) + ( x , y ) <= ( ( ( seq . n ) + x ) ) + ( x , y ) ;
P [ p , |. p .| ] ;
consider X being Subset of CQC \hbox { - } WFF } such that X c= Y and X is finite and X is finite ;
|. b .| * |. b .| >= |. b .| * |. b .| ;
1 < ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ) ;
l in { { ( ( ( l . ( m + 1 ) ) ) where l is Real : l <= m & l <= 1 } ;
( Partial_Sums ( G ) ) . n <= ( ( G . n ) * ( ( G . n ) * ( ( G . n ) * ( ( G . n ) * ( ( G . n ) * ( ( G . n ) * ( ( ( ( G . n ) * ( ( G . n ) * ( ( ( G . n ) * ( ( ( G . n ) * ( ( G
f . y = x * ( ( - 1 ) * y ) .= x * ( - 1 ) * y .= x * ( - 1 * y ) .= x * ( - 1 * y ) .= x * ( - 1 * y ) .= x * ( - 1 * y ) .= x * ( - 1 * y ) .= x * ( - 1 * y ) .= x * ( - 1 * y ) ;
NIC ( goto ( i1 , i2 ) , ( i1 , i2 ) ) = { i1 , i2 , j2 } ;
LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = { p1 } ;
product ( the support of I ) +* ( i , j ) in { { 0 , j } } ;
Following ( s , n ) = Following ( s , n ) .= Following ( s , n ) ;
W is with_( - 1 ) * ( 1 - 1 ) * ( 1 - 1 ) ) * ( 1 - 1 ) * ( 1 - 1 ) * ( 1 - 1 ) * ( 1 - 1 ) * ( 1 - 1 ) ) * ( 1 - 1 ) * ( 1 - 1 ) * ( 1 - 1 ) * ( 1 - 1 ) ) * ( 1 - 1 ) * ( 1 - 1 ) * ( 1 - 1 ) * (
f /. i <> f /. ( i + 1 ) ;
M |= _ { x , y , z } ( ( { x , y , z ) ) . ( x , y ) } ( x , y ) ) ;
len ( P1 ^ P2 ) + len ( P2 ^ s2 ) in dom ( P1 ^ P2 ) ;
A |^ ( n + 1 ) c= A |^ ( n + 1 ) & A |^ ( n + 1 ) c= A |^ ( n + 1 ) ;
for R st R \ { q : |. q .| < a } \ { q .| } holds |. q .| >= a
consider n1 being element such that n1 in dom p1 and p1 = p1 . n1 and p2 = p1 . n1 and p1 . n1 = p2 . n1 ;
consider X being set such that X in Q and for x being set st x in X holds x in X and x in X ;
CurInstr ( P3 , s3 ) <> ( ( P3 , s3 ) . ( l + 1 ) ) `1 ;
for v be VECTOR of F1 holds ( - v ) * ( - v ) = - ( v * ( - v ) )
for \varphi st \varphi in X holds ( for n st n >= m holds \varphi . n = n + 1 ) holds \varphi . n = n + 1
rng ( Sgm ( dom ( dom ( ( ( ( the carrier of TOP-REAL 2 ) | K ) ) | K ) ) ) ) ) c= dom ( ( - ( TOP-REAL 2 ) | K ) ) ;
ex c being FinSequence of D st len c = k & len c = k & for n being Nat st n in dom c holds ( p . n ) * c = p . n * c . n ;
Arity ( a , b ) = <* <* a , b *> , <* b , c *> *> ;
consider f1 being Function of X , REAL such that f1 = ( |. f1 .| ) | X and f1 is continuous ;
a1 = b1 & a2 = b2 & a2 = b1 implies a1 = b2 * a2 * b1 + b2 * b2
D2 . indx ( D2 , D1 , j1 ) + 1 ) = D2 . ( indx ( D2 , D1 , j1 ) + 1 ) ;
f . ( |. r .| - r .| ) = |. r .| - r .= |. r .| - r ;
consider n being Nat such that for m being Nat st n <= m holds C . m = ( C . n ) * ( C . m ) ;
consider d being Real such that for a , b being Real st a in X holds ( a * b ) * ( b * a ) = a * ( b * a ) ;
||. L /. h - ( K + 1 ) * ( h - c ) .|| <= p + ( K + 1 ) * ( - c ) ;
attr F is commutative means : DefDef: for b being Element of X holds F . b = F . ( b , f . b ) ;
p = 1- ( - ( p `1 ) ) * ( - p `1 ) ) .= - p `1 * ( - p `1 ) .= - p `1 * ( - p `1 ) * ( - p `1 / p `1 ) .= - p `1 * ( - p `1 / p `1 ) .= - p `1 * ( - p `1 / p `1 ) .= - p `1 * ( - p `1 / p `1 ) .= - p `1 * ( - p
consider z1 such that b , z1 // b , z1 and b <> z1 and for x , y being Element of X st x in A & x <> y holds x = y ;
consider i being Nat such that ( Arg ( s ) ) . i = s . i + ( Im s ) . i ;
consider g such that g is one-to-one and g is one-to-one and g . x = f . x ;
assume that A = { p1 } and A c= { p2 } and A c= { p1 } and A c= { p2 } and A c= { p2 } and { p1 } c= { p2 } and { p1 } c= { p2 } and { p2 } in { p2 } and { p1 } in { p2 } and { p1 } in { p2 } and { p2 } in { p2 } and { p1 } in { p2 } and { p2 } in {
attr F is associative means : Def: for x , y being Element of F holds F . x = F . y ;
ex x being Element of NAT st m = x & m = i + 1 & m < i & i < j ;
consider k2 being Nat such that k2 in dom ( ( the Sorts of A ) . i ) and k2 in dom ( the Sorts of A ) and k2 in dom ( the Sorts of A ) and k2 in dom ( the Sorts of A ) and k2 in dom ( the Sorts of A ) and k2 in dom ( the Sorts of A ) and k2 in dom the Sorts of A ;
s = r * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) ) ;
F1 . ( id the carrier of A ) = [ f , g . ( id A ) ] ;
not p "\/" ( { p } "\/" { q } , { p } ) = { p } "\/" ( { p } "\/" { q } , { p } ) ;
consider z being element such that z in dom ( ( ( ( ( F - F ) ) `| Z ) ) `| Z ) and ( ( ( ( F - F ) `| Z ) ) `| Z ) . z = ( ( ( ( ( ( ( F - F ) (#) ( F - F ) ) `| Z ) . z ) ) ;
for x , y being element st x in dom f & y in dom f & x = f . y holds f . x = g . y
cell ( G , i , j ) = { |[ r , s ]| : r <= G & s <= G * ( i + 1 , j ) `1 } ;
consider e being element such that e in dom ( T | E ) and e in E and e in { x } ;
( F /. b1 ) * ( b1 /. b2 ) = ( ( the Mx2Tran of K ) * ( b1 , b2 ) ) * ( b1 , b2 ) ) * ( b1 , b2 ) .= ( b1 /. b2 ) * ( b1 , b2 ) ;
- 1 = - 1 * ( ( - 1 ) * ( - 1 ) ) .= - 1 * ( - 1 ) * ( - 1 ) .= - 1 * ( - 1 ) * ( - 1 ) .= - 1 * ( - 1 ) * ( - 1 ) .= - 1 * ( - 1 ) * ( - 1 ) .= - 1 * ( - 1 ) * ( - 1 ) * ( - 1 ) .= - 1 * ( -
attr x in dom f /\ dom g & for x st x in dom f holds g . x = f . x ;
len ( f1 + f2 ) = len ( f1 + f2 ) + len ( f2 + g2 ) .= len ( f1 + f2 ) + ( f2 + g2 ) .= len ( f1 + f2 ) + ( f2 + g2 ) .= len ( f1 + f2 ) + len ( f2 + g2 ) + len ( f2 + g2 ) .= len ( f1 + f2 ) + len ( f2 + g2 ) + ( f2 + g2 ) + ( f2 + g2 )
'not' ( a , 'not' 'not' b ) '&' ( 'not' ( a , 'not' b ) ) = 'not' ( a , 'not' b ) ;
LSeg ( E . ( E . ( k + 1 ) ) , F . ( k + 1 ) ) c= ( E . ( k + 1 ) ) /\ ( E . ( k + 1 ) ) ;
x \ ( a \ ( x \ ( a \ ( a \ x ) ) ) = x \ ( a \ x ) .= ( x \ ( a \ x ) ) .= ( x \ ( a \ x ) ) ;
k - th = ( the carrier of K ) . ( n + 1 ) .= ( the carrier of K ) . n - 1 .= ( the carrier of K ) . n - 1 ;
for s being State of A , n being Nat holds Following ( s , n ) . ( n + 1 ) = n + 1
for x st x in Z holds ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( x - 1 ) ) ) ) . x = - 1 / ( x - 1 ) * ( - 1 ) )
support ( n gcd ( n gcd m ) ) c= support ( n gcd ( n gcd m ) ) ;
reconsider t = u as Function of the carrier of A , the carrier of B ( ) , the carrier of B ( ) ( ) , the carrier of B ( ) ) ( ) as Function of the carrier of B ( ) ( ) , the carrier of B ( ) ( ) ( ) , the carrier of B ( ) ) ) ;
- ( a * sqrt ( 1 + a * sqrt ( 1 + a * sqrt ( 1 + a ) ) ) ) <= - ( a * sqrt ( 1 + a * sqrt ( 1 + a ) ) ) ;
\varphi . a = g . a & for a being Element of A holds ( g . a ) . a = g . a ;
assume i in dom ( F ^ <* p *> ) & j in dom ( F ^ <* p *> ) ;
not x1 in { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x6 } \/ { x1 , x2 , x3 , x4 , x5 , x5 , x5 } \/ { x4 , x5 , x5 } \/ { x2 , x3 , x5 , x5 } \/ { x1 , x5 , x5 , x5 , x5 } \/ { x1 , x2 , x3 , x5 } \/ { x1 , x5 , x5
the Sorts of U1 /\ ( the Sorts of U2 ) c= the Sorts of ( the Sorts of U1 ) ) . o ;
( - ( 2 * a ) ) * ( - ( 2 * a ) ) * ( - b * a ) ) + ( 2 * a ) * ( - b * a ) * ( - b * a ) ) * ( - b * a ) ;
consider W1 being element such that for z being element st z in W1 holds W1 . z in W1 . z ;
assume ( the carrier' of S ) . o = <* a *> & ( the carrier' of S ) . o = <* a *> ;
Z = dom ( ( ( ( ( ( ( ( ( arccot ) ) - arccot ) ) * ( ( - arccot ) ) * ( ( - arccot ) * ( ( - arccot ) * ( ( - arccot ) * ( ( - arccot ) * ( ( - arccot ) * ( ( - arccot ) * ( ( - arccot ) * ( ( - arccot ) ) ) ) ) ) ) ) ) ) ;
integral ( f , S , T ) = integral ( f , S , T ) & for i be Nat st i in dom ( f , S ) holds ( f , S ) . i = integral ( f , T , i ) ;
[ X , f . ( x1 , x2 ) ] in [: X , { x1 , x2 } :] ;
len ( - ( M1 * M2 ) ) = n & width M1 = n ;
attr X1 \/ X2 \/ Y2 is open means : only : ( X1 union X2 ) union ( X1 union X2 ) = ( X1 union X2 ) union ( X2 union X0 ) ;
let L be lower-bounded antisymmetric transitive RelStr , X be Subset of L ;
reconsider f1 = F . ( ( id the carrier of C ) ) . ( f1 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . f1 f1 .
consider w being FinSequence of I such that the carrier of w = { w where w is Element of I : w in P & w in P } ;
g . ( a |^ ( b |^ 0 ) ) = g . ( a |^ ( b |^ 0 ) ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) ;
assume for i being Nat st i in dom f holds f . i = ( z . i ) * z ;
ex L being Subset of X st L4 = L & for x being Point of X st x in L holds L . x = L . x ;
( the carrier of C ) /\ ( the carrier of C ) c= the carrier of C ;
reconsider o = o as Element of Args ( o , A ) . s , p = ( the Sorts of A ) . s as Element of Args ( o , A ) . s ;
1 * ( x1 + x2 ) + ( x2 * y2 ) * ( x2 + y2 ) = x1 * x2 + ( y2 * y2 ) * ( x2 * y2 ) .= x1 * x2 + ( y2 * y2 ) * ( x2 * y2 ) + ( y2 * y2 ) * ( x2 * y2 ) + ( y2 * y2 ) * ( x2 * y2 ) + ( y2 * y2 ) * ( x2 * y2 ) ;
E . ( E . ( E . ( E . i ) ) ) = E . ( E . i ) .= E . ( E . i ) ;
reconsider u1 = the carrier of U /\ ( the carrier of U ) as non empty Subset of U ;
( x "/\" z ) "\/" ( y "/\" z ) <= ( x "/\" z ) "\/" ( y "/\" z ) ;
|. f . ( s1 + s1 ) .| < |. f . ( s1 + s1 ) .| ;
LSeg ( Gauge ( C , n ) * ( i , j ) , ( Gauge ( C , n ) * ( i , j ) ) ) , ( ( Gauge ( C , n ) * ( i , j ) ) ) ) `2 ) is vertical ;
( f | Z ) /. x = ( f `| Z ) . x + ( f `| Z ) . x ;
g . ( c * f . c ) <= h . ( f . c ) * f . c + g . c ;
( f + g ) | divset ( D , i ) = f | divset ( D , i ) + g | divset ( D , i ) ;
attr f is FinSequence of the carrier of K means : Def: for x being Element of K holds x in the carrier of K holds x * ( f . x ) = f . ( x * a ) ;
len ( - ( - ( p ) ) ) = len ( - p ) & width ( - p ) * ( - p ) ) = len p - len p + len q ;
let n , i be Nat ;
SVF1 ( 1 , 2 , u0 ) * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( ( 2 * ( ( ( ( ( ( ( ( ( ( ( f1 f1 f ) - 3 ) ) ) * ( ( 2 * ( 2 * ( 2 * ( 2 * ( ( f1 f1 f1 - 3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) * ( 2 * ( 2 *
attr a <> 0 & a <> 0 & a * ( a * b ) = 0 implies a * ( b * a ) = 0 & a * ( b * a ) = 0 & a * ( b * a ) = 0 & b * a * a = 0 & b * a = 0 & b * a = 0 ;
for c being set st c in ]. a , b .[ holds c in ]. a , c .[
assume that v1 is linearly independent and for u , v st u in V holds u + v in W and u + v in W and u + v in W and u + v in W and u + v in W and u + v in W and u + v in W and u + v in W and u + v in W and u + v in W and u + v in W and u + v in W and u + v in W and u + v in W and u + v in W and u + v in W
z * ( x1 - x2 ) + ( - x2 * ( x1 - x2 ) ) * ( x1 - x2 ) * ( x1 - x2 ) in N & z * ( - x2 * ( x1 - x2 ) ) in N * ( - x2 * ( x1 - x2 ) ) ;
rng ( ( P1 +* P2 ) * ( i + 1 ) ) ) = Seg len ( P1 + P2 ) * ( i + 1 ) ;
consider s2 being Real such that s2 is convergent and for n being Nat holds s2 . n = s1 . n ;
h . n = ( h . n ) " * ( h . n ) & ( h . n ) * ( h . n ) = ( h . n ) * ( h . n ) ;
( Partial_Sums ( seq ) ) . m = ( seq . m ) * ( seq . n ) .= ( seq . n ) * ( seq . n ) .= ( seq . n ) * ( seq . n ) ;
( Comput ( P1 , s1 , i ) ) . b = ( Comput ( P1 , s1 , i ) ) . b .= ( Comput ( P1 , s1 , i ) ) . b ;
- v = - ( - v ) * ( - v ) & - v * ( - v ) * ( - v ) = - v * ( - v * v ) ;
ex_sup_of ( the carrier of [: { k } , D :] , D :] , D ) = ( the carrier of D ) .: D .= ( the carrier of D ) .: D .= ( the carrier of D ) .: D .= ( the carrier of D ) .: D .= ( the carrier of D ) .: D .= ( the carrier of D ) .: D .= the carrier of D ;
A |^ ( k + 1 ) = ( A |^ ( k + 1 ) ) |^ ( k + 1 ) .= A |^ ( k + 1 ) ;
let L be add-associative right_zeroed right_complementable complementable for p , q be Element of L holds p + q = p + q
( f . p ) `1 = ( f . p ) `1 + ( f . p ) `1 ;
let a be non zero Nat , n be Nat ;
consider A1 being countable countable countable countable countable countable countable countable countable countable countable countable countable countable Subset of A such that r is countable and for p being Element of A holds p . p in { p } and p . ( p . p ) } = { p } ;
for X being non empty TopSpace , x being Point of X , y being Point of X st x in X holds x + y in X
not [ x1 , y1 ] in { x1 , y1 , y2 ] } ;
h . ( f . O ) = [ A , f . O ] * ( f . O ) + ( f . O ) * ( f . O ) * ( f . O ) ;
( Gauge ( C , n ) * ( i , j ) , i ) in L~ Cage ( C , n ) * ( i , j ) ;
let m be Nat ;
( f * F ) . x = f . ( F . x ) & ( f * F ) . x = f . ( F . x ) ;
let L be lower-bounded LATTICE , a , b be Element of L ;
consider b being element such that b in dom ( H . ( x , y ) ) and z = H . ( x , y ) ;
assume that x in dom ( F * ( x , y ) ) and y in dom ( F * ( x , y ) ) and x in dom ( F * ( x , y ) ) ;
assume that not e in W . ( n + 1 ) and e in W . ( n + 1 ) and e in W . ( n + 1 ) ;
( \delta ( h , n ) ) . x = ( 2 * ( h . n ) ) * ( 2 * ( 2 * n ) ) ;
j + 1 = i + len ( h + c ) .= i + ( j + 1 ) + 1 .= i + ( j + 1 ) + 1 ;
^ ( S \ast f ) = ( S \ast f ) . ( id dom f ) .= ( S \ast f ) . ( id dom f ) .= ( S \ast f ) . ( id dom f ) .= ( S \ast f ) . ( id dom f ) .= ( S \ast f ) . ( id dom f ) .= ( S . f ) . ( id dom f ) .= ( S . f ) . ( id dom f ) .= ( S . f ) . ( id dom f ) .= (
consider H such that H is one-to-one and H is one-to-one and H is one-to-one ;
attr R is TOP-REAL 2 means : Def: for p , q being Point of TOP-REAL 2 st p in R & q in R holds p >= q ;
dom ( ( ( ( X --> f ) ) `| X ) ) = ( ( X --> f ) `| X ) . x .= ( X --> f ) . x ;
ex_sup_of ( ( TOP-REAL 2 ) | ( ( ( ( ( TOP-REAL 2 ) | C ) ) | ( C ) ) | ( C ) ) , ( ( TOP-REAL 2 ) | C ) ) ) , ( ( TOP-REAL 2 ) | C ) | C ) ) ) is compact Subset of ( ( ( ( TOP-REAL 2 ) | C ) | C ) | C ) ;
for r be Real st 0 < r holds |. S . m - r .| < r
i * ( ( ( f - g ) * ( i - g ) ) ) = i * ( ( f - g ) * ( i - g ) ) .= i * ( i - g * ( i - g * ( i - g * ( i - g * ( i - g * ( i - g * ( i - g * ( i - g * ( i - g * ( i - g * ( j - g ) * ( i - g * ( i - g * ( i - g * ( i -
consider f being Function of X , REAL such that dom f = 2 and for x being element st x in X holds f . x = x and for x being element st x in X holds f . x = a * x ;
consider g1 , g2 being element such that g1 in [#] Y and g2 in [#] ( ( | X ) | X ) and g1 in ( ( TOP-REAL 2 ) | X ) and g2 in ( ( TOP-REAL 2 ) | X ) and g2 in ( ( TOP-REAL 2 ) | X ) and g2 in ( ( TOP-REAL 2 ) | X ) /\ X ) ;
func d \! > ( n * m ) -> Nat means : Def1 : for n being Nat holds it . n = ( n * m ) * ( n * m ) ;
[: [: the carrier of [: [: [: the carrier of [: [: [: [: [: the carrier of [: [: [: [: [: [: the carrier of [: [: [: TOP-REAL :] , TOP-REAL :] , the carrier of :] , the carrier of :] :] :] , ( ( TOP-REAL 2 ) ) ) ) :] .= ( ( TOP-REAL 2 ) * ( ( ( TOP-REAL 2 ) ) * ( ( TOP-REAL 2 ) ) * ( ( TOP-REAL 2 ) ) * ( ( TOP-REAL 2 ) ) ) ) .= ( ( TOP-REAL 2 ) * ( ( TOP-REAL 2 ) * ( (
t = h . ( B . ( C . ( C . ( C . C ) ) ) or t . ( C . C ) = h . ( C . C ) or t . ( C . C ) = t . ( C . C ) ;
consider m1 being Nat such that for n being Nat st n >= m1 holds dist ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( n n - ( seq . seq ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . n ) ;
( ( - q ) * ( - q ) ) * ( - q ) ) * ( - q ) = - ( q * ( - q ) ) * ( - q * ( - q ) ) ;
h . ( i + 1 ) = h . ( i + 1 ) + ( h . ( i + 1 ) ) ;
consider o being Element of the carrier' of S , x being Element of S such that a = [ o , the carrier' of S ] and o = [ o , the carrier' of S ] ;
let L be RelStr , a , b be Element of L ;
||. h1 . n - h1 . n .|| = |. h1 . n - h1 . n .| .= |. h1 . n - h1 . n .| .= |. h1 . n - h1 . n .| .= |. h1 . n - h1 . n .| ;
( ( - ( ( ( ( the function ) ) * ( f + g ) ) ) `| Z ) ) . x = ( - ( 1 + ( x + h ) * ( f + g ) ) ) . x .= - ( 1 + ( x + h ) * ( f + h ) ) . x .= - ( 1 + ( x + h ) * ( f + h ) ) . x ;
attr r = F .: ( p , q ) & r in F .: ( p , q ) ;
sqrt ( r ^2 + r ^2 ) + r ^2 + r ^2 + r ^2 ) ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 ) ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2
let i be Nat , n be Nat ;
then a <> 0. R & a * ( a * b ) = 0. R * ( a * b ) ;
p . ( j + 1 ) * ( p . ( j + 1 ) ) = ( p . ( j + 1 ) ) * ( p . ( j + 1 ) ) ;
deffunc F ( Nat ) = L . ( ( ( h ^\ n ) * ( h ^\ n ) ) * ( h ^\ n ) ) ;
assume that the carrier of ( H . ( the carrier of H ) ) = f .: ( the carrier of H ) and the carrier of H = { f . ( the carrier of H ) } ;
Args ( o , Free ( S , X ) ) = ( ( the Sorts of Free ( S , X ) ) . o ) ;
H = n + 1 + ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( n + 1 ) ) ) ) ) .= n + 1 * ( 2 * ( 2 * ( n + 1 ) ) ) .= n + 1 * ( 2 * ( n + 1 ) ) ;
( O + ( - 1 ) * ( O + 1 ) ) * ( ( - 1 ) * ( O + 1 ) ) ) * ( O + 1 ) ) = ( O + ( - 1 ) * ( O + 1 ) ) * ( O + 1 ) ) * ( O + 1 ) * ( O + 1 ) ) * ( O + 1 ) ) * ( O + 1 ) * ( O + 1 ) ) * ( O + 1 ) ;
F1 .: ( dom F1 /\ ( dom F1 /\ dom F2 ) ) = ( F1 /\ ( dom F1 /\ ( dom F2 ) ) .= ( F1 /\ ( dom F1 /\ ( dom F2 ) ) ) .= ( F1 /\ ( dom F1 ) /\ ( dom F1 ) ) .= ( F1 /\ ( dom F1 ) /\ ( dom F1 ) ) .= ( F1 /\ ( dom F1 ) ) .= ( F1 /\ ( dom F1 ) /\ ( dom F1 ) ) .= ( F1 /\ ( dom F1 ) ;
attr b <> 0 & b <> 0 implies a * b = b * ( a * b ) + ( b * c ) ;
dom ( f +* g ) = dom ( f +* g ) \/ dom g .= dom ( f +* g ) \/ dom g .= dom f \/ dom g \/ dom g .= dom f \/ dom g \/ dom g \/ dom h .= dom f \/ dom g \/ dom h \/ dom h .= dom f \/ dom g \/ dom h \/ dom h .= dom f \/ dom h \/ dom h \/ { x } ;
for i be set st i in dom g holds g /. i = u * ( i , j ) ;
g * P = g * P .= g * P .= g * P .= ( g * P ) * P .= g * P * P * P .= g * P * P * P * P .= ( g * P ) * P .= ( g * P ) * P .= ( g * P ) * P .= ( g * P ) * P * P .= ( g * P ) * P * P * P ;
consider i , j such that f . i = ( s - 1 ) * ( i - j ) and i <> j and i - j * ( i - j ) = i * ( i - j ) ;
h1 | [. a , b .] = ( ( a , b ) (#) ( ( b , c ) (#) ( ( a , c ) (#) ( b , c ) ) ) .= ( a , b ) (#) ( b , c ) ) (#) ( b , c ) .= ( a , b ) (#) ( b , c ) .= ( a , b ) (#) ( b , c ) ) ;
[ s1 , s2 ] in the InternalRel of G & s1 in the carrier of G implies s1 = s2 & s2 in the carrier of G
then H is negative & H is negative ;
attr f1 is total means : only : for p , q being Element of REAL st p in dom f1 & q in dom f1 & p in dom ( f1 + f2 ) holds f1 + f2 = f2 + ( ( f1 + f2 ) /* q ) . p ;
z1 in W1 + W2 & z2 in W1 + W2 implies z1 + z2 in W1 + W2 + ( W2 + W1 + W1 + W2 )
p = 1 * ( p * ( 1 - p ) ) .= 1 * ( p * ( 1 - p ) ) .= 1 * ( 1 - p * ( 1 - p * p ) ) .= 1 * ( 1 - p * p ) ;
for r1 being Real st for n being Nat holds ( for n being Nat holds ( ( for n being Nat holds ( ( n + 1 ) * ( n + 1 ) ) ) * ( ( n + 1 ) * ( n + 1 ) ) ) . n = ( n + 1 ) * ( n + 1 ) )
Index ( p1 , p2 ) meets L~ Cage ( C , n ) or ( p1 , p2 ) in L~ Cage ( C , n ) ) ;
||. f . ( g . ( k + 1 ) ) - g . ( k + 1 ) .|| <= ||. g . ( k + 1 ) - g . ( k + 1 ) .|| ;
assume h = ( B +* ( A +* B ) ) +* ( C +* ( A +* B ) ) ;
|. ( ( the lower of T ) . n ) - ( the Sorts of T ) . n ) .| <= e * ( ( the Sorts of T ) . n ) - ( the Sorts of T ) . n ;
( the Sorts of is_Sorts of A ) . e = [ the Sorts of A , the Sorts of A ] , [ the Sorts of A ] ] ;
not x1 in { x , y } & y in { x , y } implies x1 in { y , x }
attr A = [. 0 , PI * PI * ( 2 * PI * ( 2 * PI * ( 2 * PI * ( 2 * PI * ( 2 * PI * ( 2 * PI * PI * ( 2 * PI * ( 2 * PI * * ( 2 * PI * ( 2 * PI * ( 2 * PI * PI * ( 2 * PI * * ( 2 * PI * ( 2 * * ( 2 * PI * ( 2 * * ( 2 * PI * ( 2 * ( 2 * * ( 2 * * ( 2 * * ( 2 * * ( 2 * *
p `2 is Permutation of dom ( ( f " ) * ( f " ) ) & p `2 = ( f " ) * ( p `2 ) ;
for x , y st x in A & y in A holds ( x + y ) * ( x + y ) = ( x + y ) * ( x + y )
( - ( p2 `1 ) ) * ( ( - p2 `1 ) * ( - p2 `1 ) ) ) * ( - ( - ( p2 `1 ) ) * ( - ( - ( p2 `1 ) ) * ( - ( p2 `1 ) * ( - ( p2 `1 ) ) * ( - ( p2 `1 ) * ( - ( p2 `1 ) * ( - ( p2 `1 ) ) * ( - ( p2 `1 / p2 `1 ) ) ) ) ) ) ) ) = - ( - ( - ( p2 `1 / ( - ( p2 `1 / ( - ( p2 `1 / ( - ( p2
let f be PartFunc of C , D ;
assume for x being Element of Y st x in EqClass ( z , A ) holds ( x in B ) implies ( x in B ) & ( x in B ) & ( x in B ) implies x in B ) ;
consider F1 being Function of n , REAL such that dom F1 = n and for k being Nat st k in n holds F1 . k = F ( k ) ;
ex u , v st u <> v & u <> v & u <> v & u <> v & u <> v & u <> v & u <> v & u <> v & u <> v & u <> v & v <> v & u <> v & u <> v & u <> v & u <> v & u <> v & u <> v implies u <> v & u <> v & u <> v & u <> v & u <> v & u <> v & u <> v & u <> v & u <> v & u <> v & u <> v & u <> v & u <> v & u
let G be Group , A , B be Subset of G ;
for s be Real st s in dom ( ( ( ( ( ( ( ( ( f ^ ) ) (#) ( f ) ) ) `| Z ) ) `| Z ) holds ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( f ^ ) (#) ( f ) ) ) ) `| Z ) ) `| Z ) ) `| Z ) ) . x = ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( f ^ ) ) (#) ( f ) ) `| Z ) ) ) ) `| Z ) ) ) `| Z ) ) ) . x ) ) ^2 )
width ( ( - ( ( ( - ( ( - ( b ) b ) ) ) ) ) ) = len ( - ( b * ( - a ) ) ) .= len ( - ( b * ( - a ) ) ) .= len ( - ( b * ( - a ) ) ) ;
f | ]. - 1 , 1 .[ = f | ]. - 1 , 1 .[ & for x st x in ]. - 1 , 1 .[ holds f . x = - 1 / ( x - 1 ) / ( x - 1 )
attr X is closed means : DefDef: for n being Nat holds ( n >= 1 ) * ( n + 1 ) & n >= 1 & n >= 1 implies x in X & x in X & x in X & y in X & x in X & y in X & x in X & y in X & x in X & y in X & x in X & y in X ;
Z = dom ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( f f f ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) `| Z ) ;
func |[ l . ( k + 1 ) , l . ( k + 1 ) ]| -> Subset of V equals { l . ( k + 1 ) : l . ( k + 1 ) <= l . ( k + 1 ) & l . ( k + 1 ) in l . ( k + 1 ) & l . ( k + 1 ) in l . ( k + 1 ) ;
let L be non empty reflexive transitive RelStr , x be Element of L ;
for s being Element of NAT holds ( ( the Element of NAT ) * ( ( the carrier of K ) * ( the carrier of K ) ) ) . s = ( the carrier of K ) * ( the carrier of K ) ) . s
then z /. 1 = ( E-max L~ z ) .. z + ( E-max L~ z ) .. z ;
len ( p ^ <* 0 *> ) = len p + len <* 0 *> .= len p + len <* 1 *> + len <* 1 *> .= len p + len <* 1 *> + len <* 0 *> .= len p + len <* 1 *> + len <* 1 *> + len <* 1 *> + len <* 1 *> .= len p + len <* 1 *> + len <* 1 *> + len <* 0 *> + len <* 1 *> .= len p + len <* 1 *> + len <* 1 *> + len <* 1 *> + len <* 1 *> + len <* 1 *> + len <* 1 *> + len <* 1 *> + len <*
assume that Z c= dom ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) * ( ( - 1 ) ) ) ) ) ) ) ) and for x st x in Z holds ( ( - 1 ) (#) ( ( - 1 ) (#) ( - 1 ) ) ) . x ) ) and ( ( - 1 ) (#) ( - 1
let L be add-associative right_zeroed right_complementable complementable distributive for p , q , r being Element of L holds p * r = p * r + ( p * r )
consider f being Function of the carrier of A , the carrier of B such that for x being element st x in A holds f . x = F ( x ) ;
dom ( x + y ) = Seg ( len ( x + y ) ) .= Seg len ( x + y ) .= Seg len ( x + y ) .= len ( x + y ) ;
for S being category , f being Morphism of C , D holds ( the Morphism of C ) . ( f , g ) = ( the Morphism of C ) . ( f , g )
ex a st a = - ( a - b ) & ( - a ) * ( - b ) = - a * ( - b * ( - a ) ) ;
a in Free ( H , ( { x } , ( ( { x } , { y } ) ) ) ;
let C be \sum of ( the carrier of V ) , A , B , C be Subset of V ;
( - ( ( - ( ( p `1 ) / |. p .| - sn ) ) ) ) * ( ( - sn ) ) * ( ( - sn ) ) * ( - sn ) ) ) = - ( - ( ( - ( p `1 / p `1 ) ) ) * ( - ( p `1 / p `1 ) ) ) * ( - ( p `1 / p `1 ) ) ) * ( - ( p `1 / p `1 ) ) ) * ( - ( p `1 / p `1 ) ) * ( - ( p `1 / p `1 ) ) ) * ( - ( p `1 / p `1 ) ) * ( - ( p `1 / p `1 /
attr u = <* x1 , x2 , x3 *> & u = <* x1 , x2 , x3 *> & u = x3 & u = x4 ;
then ( t . {} ) `1 in ( t . {} ) `1 & t . {} in ( t . {} ) `1 ;
Valid ( p , J ) . v = v . v .= v . v ;
assume that for x , y being Element of S st x <= y holds x <= y and y <= x ;
func Class ( R , x ) -> Subset of R equals R .: ( ( R , x ) ) ;
defpred P [ Nat ] means ( for x being Nat st x in dom ( ( the InternalRel of G ) * ( ( the InternalRel of G ) * ( ( the InternalRel of G ) * ( x , $1 ) ) ) ) . x ) & ( the InternalRel of G ) * ( x , $1 ) ) . x = ( the InternalRel of G ) * ( x , $1 ) ) . x ;
then dim ( W1 + W2 ) = 0 & for n being Nat holds ( W1 + W2 ) . n = ( W1 + W2 ) . n ;
mamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamamama
d = { x where x is Element of F : x in { x } & x in { y } & y in { y } ;
consider g such that x = g and g . x = f . x and g . x = f . x ;
x + ( x + y ) = x + ( y + ( x + y ) ) .= x + ( y + ( x + y ) ) .= x + ( y + ( x + y ) ) .= x + ( y + ( x + y ) ) .= x + ( y + ( x + y ) ) .= x + ( y + ( x + y ) ) .= x + ( y + y ) ;
k1 + 1 in dom ( f | ( k1 + 1 ) ) ;
assume that p1 is Point of TOP-REAL 2 and p2 in P and p1 in P and p1 in P and p1 in P and p2 in P and p1 in P and p1 in P and p1 in P and p1 in P and p2 in P and p1 in P and p1 in P and p1 in P and p1 in P and p1 in P and p1 in P and p1 in P and p1 in P and p1 in P and p1 in P and p1 in P and p1 in P and p1 in P and p1 in P and p1 in P and p1 in P and p2 in P and p1 in P and p1 in P and p1 in P and p1 in P and p1 in P and p1 in P and p1 in P and p1 in P and p1 in P and p1 in P and p1 in
reconsider a1 = a , a2 = b , a2 = c , a3 = d , a2 = d , a2 = c , a2 = d , a3 = d , a2 = d , a2 = c , a2 = d , a3 = d , a2 = d , a2 = = d , a2 = = = d , a2 = = = d , a3 = = d , a2 = = = d , a2 = = d , a3 = d , a2 = d , a2 = d , a2 = = = = d , a2 = = = = d , a2 = = = = = = = d , a2 = = = d , a3 = d , a2 = = = = = d , a2 = = = d , a2 = d , a2 = = = = d , a2 =
reconsider FFFFFFFFFFFf = t * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F
LSeg ( f , i + 1 ) = LSeg ( f /. i + 1 ) ;
\int P . ( m + 1 ) - P . ( m + 1 ) .| <= P . ( m + 1 ) - P . ( m + 1 ) ;
assume that dom ( f1 + f2 ) = dom ( f1 + f2 ) and for x st x in dom ( f1 + f2 ) holds f1 . x = ( f1 + f2 ) . x ;
consider v such that v = y and dist ( u , v ) < r and dist ( u , v ) < r ;
let G be strict strict Group , a , b be Element of G ;
consider B being Function of Seg ( n + L ) , the carrier of V such that for x being element st x in Seg ( n + L ) holds P [ x , B . x ] ;
reconsider K = { p where p is Point of TOP-REAL 2 : p `1 >= p `1 & p `1 >= - p `1 } as Subset of TOP-REAL 2 ;
( - ( C - B ) ) * ( 1 - B * ( 1 - B * ( 1 - B * ( C * ( 1 - B * ( C * ( 1 - B * B ) ) ) ) ) * ( 1 - B * ( 1 - B * B ) ) ) * ( 1 - B * ( 1 - B * B * B ) ) ) * ( 1 - B * B * B ) ) * ( 1 - B * B * B - B * B ) ) * ( 1 - B * B ) ) * ( 1 - B * B * B ) * ( 1 - B * B * B * B * B ) ) * ( 1 - B *
for x being Element of X , y being Element of X st |. ( ( ( Re F ) . n ) ) .| <= ( |. ( Im F ) . n ) .| ) holds |. ( ( Im F ) . n ) .| <= ( |. ( Im F ) . n ) .|
len ( @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
v . ( x , y ) = ( x , y ) . ( x , y ) .= ( x , y ) . ( x , y ) ;
consider r being Element of M such that M |= _ { x } ( { x } , y ) and for m being Element of M st m in { x } holds ( x , y ) |= ( x , y ) ) ;
func ( 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 \ ( w \ ( w \ ( w \ ( w \ ( w \ ( w \ ( w \ ( w
s2 . b = ( s1 . b ) . b .= s1 . b .= s2 . b ;
for n being Nat holds ( for n being Nat holds ( ( ( n + 1 ) * ( n + 1 ) ) ) * ( ( n + 1 ) * ( n + 1 ) ) ) * ( ( n + 1 ) * ( n + 1 ) ) ) = ( n * ( n + 1 ) ) * ( n + 1 )
set F = S \! \mathop { \rm \hbox { - } count } ( X ) ;
( Partial_Sums ( s ) ) . ( n + 1 ) = ( Partial_Sums ( s ) ) . n + ( Partial_Sums ( s ) ) . n ;
consider L , R such that for x , y st x in N holds ( f | Z ) . x = L . ( f . y ) ;
func the carrier of TOP-REAL 2 -> Subset of ( ( TOP-REAL 2 ) | ( ( ( ( ( ( ( a , b ) , c ) ) , d ) ) ) | ( ( ( ( ( TOP-REAL 2 ) ) | ( ( ( ( ( a , b ) , c ) ) , d ) ) ) ) ` ) ` ) ` ) ` ) ` ) ` ;
a * b + c * ( - c * b ) + ( - c * a ) * ( - c * b ) * ( - c * b ) * ( - c * b ) + ( - c * a ) * ( - c * b ) ) * ( - c * b ) * ( - c * b ) * ( - c * c ) ) * ( - c * b - c * c ) * ( - c * b ) ;
v . ( x1 , x2 ) = v . ( x1 , x2 ) .= v . ( x1 , x2 ) ;
( Q ^ <* x *> ) . ( ( Q ^ <* y *> ) ) . ( x ^ y ) = ( Q ^ <* x *> ) . ( y ^ ( x ^ y ) ) .= ( Q ^ <* y *> ) . ( x ^ y ) ;
Sum ( F - G ) = r * ( F - G ) .= r * ( F - G ) .= r * ( F - G ) + r * ( F - G ) .= r * ( F - G ) + ( G - G ) * ( F - G ) ) .= r * ( F - G ) + ( G - ( G - F ) * ( ( - G ) * ( ( - G ) * ( ( - G ) * ( ( ( - G ) * ( ( - G ) * ( ( - G ) * ( ( - G ) * ( ( - G ) * ( ( - G ) * ( ( - G ) * ( ( - G ) * ( ( - G )
( the InternalRel of G ) . ( ( the carrier of G ) . ( len G ) ) = ( the carrier of G ) . ( 1 , j ) ;
defpred X [ Element of NAT ] means ( ( Partial_Sums ( seq ) ) . $1 ) * ( ( seq . $1 ) * ( seq . $1 ) ) * ( seq . $1 ) ) * ( seq . ( seq . $1 ) * ( seq . $1 ) ) * ( seq . ( seq . $1 ) * ( seq . $1 ) * ( seq . $1 ) * ( seq . $1 ) * ( seq . $1 ) ;
the Arity of S = ( the carrier' of S ) . ( the carrier' of S ) .= ( the carrier' of S ) . ( the carrier' of S ) .= ( the carrier' of S ) . ( the carrier' of S ) .= ( the carrier' of S ) . ( the carrier' of S ) ;
( X --> Y ) '&' ( X --> Y ) = X '&' ( Y --> Y )
for a , b being Element of S , s being Element of S st s = ( n + 1 ) * ( a , b ) holds s . ( n + 1 ) = ( n + 1 ) * ( a , b )
E |= { x , y , z } => ( ( x , y ) => ( y , z ) ) => ( x , y ) => ( y , z ) ) ;
ex R being 1-sorted , p being Element of R st R = ( p . i ) * ( p . i ) & p . i = p . i * ( p . i ) * ( p . i ) ;
[. a , b .[ + ( b + ( a + b ) ) .] is Element of REAL ;
Comput ( P , s , 2 ) . ( 2 + 1 ) = ( a , b ) := ( a , b ) .= ( a , b ) := ( b , d ) ;
card ( h1 + h2 ) = ( h1 + h2 ) * ( h1 + h2 * ( h1 + h2 * ( h1 + h2 * ( h1 + h2 * ( h1 + h2 * ( h1 + h2 * ( h1 + h2 * ( h1 * ( h1 + h2 * ( h1 + h2 * ( h1 + h2 * ( h1 + h2 * ( h1 + h2 * ( h1 + h2 * ( h1 + h2 * ( h1 + h2 * ( h1 + h2 * ( h1 + h2 * ( h1 + h2 * ( h1 + h2 * ( h1 + h2 * ( h1 + h2 * ( h1 + h2 * ( h1 + h2 * ( h1 * ( h1 + h2 * ( h1 * ( h1 + h2 * ( h1 * ( h1
( ( f + g ) /* c ) . n = ( f + g ) . n .= ( f + g ) . n ;
len C - ( len C ) + len ( - 1 ) * ( len C ) + 1 ) = len C - ( len C - 1 ) + 1 ;
dom ( r (#) ( r (#) f ) ) = dom ( r (#) f ) /\ dom ( r (#) f ) .= dom ( r (#) f ) /\ dom ( r (#) f ) .= dom ( r (#) f ) /\ dom ( r (#) f ) .= dom ( r (#) f ) /\ dom ( r (#) f ) .= dom ( r (#) f ) /\ dom ( r (#) f ) ) .= dom ( r (#) f ) /\ dom ( r (#) f ) .= dom ( r (#) f ) /\ dom ( r (#) f ) .= dom ( r (#) f ) /\ dom ( r (#) f ) /\ dom ( r (#) f ) /\ dom ( r (#) f ) /\ dom ( r (#) f ) /\ dom ( r (#) f ) /\ dom ( r (#) f ) /\ dom ( r (#) f ) /\ dom
defpred P [ Nat ] means for n being Nat st n >= $1 holds ( 2 * n ) * ( 2 * n ) + 1 ) * ( 2 * n ) * ( 2 * n + 1 ) * ( 2 * n ) ;
consider f being Function of REAL , REAL n , REAL such that f = ( n + 1 ) (#) f and for x being Element of REAL n holds f . x = ( n + 1 ) (#) f . x ;
consider c being Function of S , T such that c = Function ( the Sorts of A ) , ( the Sorts of A ) . c and c in { 0. S } and c in { 0. S } and c in { 0. S } and c in { 0. S } and c in { 0. S } and c in { 0. S } and c in { 0. S } ;
consider y being Element of Y such that a = "\/" ( { [ x , y ] , [ x , y ] ] : [ y , z ] in R & [ y , z ] in R & [ y , z ] in R } ;
assume that A c= Z and ( ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) * ( ( - 1 ) * ( - 1 ) ) ) ) ) ) ) ) ) and A = ( - 1 ) (#) ( ( - 1 ) * ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( ( - 1 ) ) ) and A ) and A =
( GoB f ) * ( i , j ) `1 = ( GoB f ) * ( i , j ) `1 .= ( GoB f ) * ( i , j ) `1 ;
dom ( ( ( ( ( ( ( the Sorts of A ) * ( ( ( ( ( A ) ) * ( ( ( A ) ) * ( ( ( A * ( ( ( ( A ) ) * ( ( ( A ) ) * ( ( ( ( A * * ( ( ( A ) ) * ( ( ( A * ( ( ( A ) ) * ( ( ( A ) ) * ( ( ( A ) ) * ( ( ( A ) ) * ( ( ( ( ( ) ) * ( ( ( A ) ) * ( ( ( A ) ) * ( ( ( ( A ) ) * ( ( ( A ) ) * ( ( ( ( ( ) ) * ( ( ( ( A ) ) * ( ( ( ( A ) ) * ( ( ( ( ( A )
consider G1 , G2 being Morphism of G1 , G2 being Morphism of G2 , G2 such that G1 <= G2 and G1 * G2 = G1 * G2 and G1 * G2 = G1 * G2 and G1 * G2 * G2 * G2 * G2 * G1 = G1 * G1 * G1 * G2 * G2 * G2 * G1 ;
func - f -> PartFunc of C , REAL means : Def: for c being Point of C holds it . c = - f . c & for c being Element of C st c in dom it holds it . c = - f . c & for c being Element of C st c in dom it holds it . c = - f . c & for c being Element of C st c in dom it holds it . c = - f . c & for c being Element of C st c in dom it holds it . c = - f . c & for c being Element of C holds it . c = - f . c & c in dom it . c = - f . c = - f . c & c . c = - f . c & c in dom ( - f . c = -
consider \varphi being increasing such that \varphi is continuous and for a st a in \varphi . a holds ( for n being Nat holds \varphi . n = ( for n being Nat holds \varphi . n = n ) and for n being Nat st n <= n holds \varphi . n = ( n + 1 ) * ( n + 1 ) ) * ( n + 1 ) ;
consider i1 , i2 being Nat such that [ i1 , j1 ] in Indices GoB f and ( f /. k ) * ( i1 , j1 ) `1 = ( GoB f ) * ( i1 , j1 ) `1 and ( GoB f ) * ( i1 , j1 ) `1 = ( GoB f ) * ( i1 , j1 ) `1 ;
consider i , n such that n <> 0 & n <> 0 & n < 0 & n < i & i < n & n < m holds ( n + 1 ) * ( i + 1 ) = n * ( i + 1 ) ;
assume that 0 in Z and for x st x in Z holds ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( x - 1 ) ) ) `| Z ) . x = - 1 / ( x - 1 ) * ( x - 1 ) ) and ( - 1 / ( x - 1 ) ) * ( x - 1 ) ) = - 1 / ( x - 1 ) * ( x - 1 ) ) and x <> 0 and x <> 0 and x <> 0 and 1 + 1 / ( x - 1 ) ;
cell ( GoB ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
ex C1 being Subset of X st s = { C1 where C1 is Subset of Y : C1 in F & C1 in F } & ( for p being Point of X st p in F holds p in F & p in F & p in F & p in F & p in F & p in F & p in F & p in F & p in F & p in F & p in F & p in F & p in F & p in F & p in F & p in F & p in F & p in F & p in F & p in F & p in F & p in F & p in F & p in F & p in F & p in F & p in G & p in G & p in G & p in G & p in G ) implies p in G
gcd ( ( the carrier of gcd gcd gcd gcd gcd gcd gcd gcd ( A , gcd gcd gcd gcd gcd gcd gcd gcd gcd gcd ( A , gcd gcd gcd gcd ( A , gcd gcd gcd gcd gcd gcd gcd ( A , gcd gcd gcd gcd gcd gcd gcd gcd gcd gcd A , gcd gcd A ) ) ) , ( A , gcd gcd gcd ( A , gcd gcd gcd A ) ) ) ) = 1 ;
RI = ( the Sorts of A ) . ( ( the Sorts of A ) . ( s , I ) ) .= ( the Sorts of A ) . s .= ( the Sorts of A ) . s ;
CurInstr ( P3 , s3 ) = ( P3 , s3 ) . ( a , b ) .= ( P3 . a ) + ( P3 . b ) .= ( P3 . b ) + ( P3 . b ) .= ( \it it . b ) + ( ( P1 . b ) + ( P2 . b ) ) .= ( P1 . b ) + ( P2 . b ) ;
p1 /\ ( ( TOP-REAL 2 ) | ( ( ( ( ( TOP-REAL 2 ) ) | ( K ) ) | ( K ) ) ) ) = ( TOP-REAL 2 ) | ( K \/ ( K \/ { 1 } ) /\ ( K \/ { 1 } ) /\ ( K \/ { 1 } ) /\ ( K \/ { 1 } ) /\ ( K } ) .= ( TOP-REAL 2 ) /\ ( K \/ { 1 } \/ { 1 } ) /\ ( K \/ { 1 } ) .= ( TOP-REAL 2 ) /\ ( K \/ { 1 } \/ { 1 } ) /\ ( K \/ { 1 } \/ { 1 } ) /\ ( K \/ { 1 } ) /\ ( K \/ { 1 } \/ { 1 } ) /\ ( K \/ { 1 } ) /\ ( { 1 } ) /\ ( { 1 } ) \/ ( K
func mode bound -> bound of A -> Subset of the bound of A means : Def1 : for p being Element of A holds p . p in { x : p in A } ;
let a , b be Element of REAL ;
defpred P [ Nat ] means for i , j st i < j & j < n holds ( i + j ) * ( i + j ) = ( i + j ) * ( i + j ) ;
attr A1 : the carrier of C is associative means : Def: for x being Element of C holds x in the carrier of C iff x in the carrier of C & x in the carrier of C & x in the carrier of C ;
( ||. f .|| | X ) . c = ( ||. f .|| | X ) . c .= ( ||. f .|| | X ) . c .= ( ||. f .|| | X ) . c ;
|. q .| ^2 + ( |. q .| ) ^2 + ( |. q .| ) ^2 + ( |. q .| ) ^2 + ( |. q .| ) ^2 + ( |. q .| ) ^2 + ( |. q .| ) ^2 + ( |. q .| ) ^2 + ( |. q .| ) ^2 + ( |. q .| ) ^2 + ( |. q .| ) ^2 + ( |. q .| ) ^2 ) ;
for F being Subset-Family of T st F is open & for A being Subset of T st A in F holds A in F
assume that len F >= 1 and for k st k in dom F holds F . k = G . k and for n st n in dom F holds F . n = G . ( n + 1 ) ;
i |^ ( n + 1 ) - i |^ ( n + 1 ) = i |^ ( n + 1 ) - i |^ ( n + 1 ) .= i |^ ( n + 1 ) - i |^ ( n + 1 ) ;
consider q being Chain of G , V such that r = q and for n being Nat st n <= len q holds p . n = q . n and q . n = p . n and q . n = p . n and q . n = p . n and q . n = p . n and q . n = p . n and q . n = p . n and q . n = p . n and q . n = p . n and q . n = p . n and p . n = p . n and p . n = p . n ;
defpred P [ Element of NAT ] means for n being Element of NAT holds ( ( ( g * f ) `| Z ) . n = ( g * f ) . n ;
let A be -> -> -> -> commutative for Matrix of REAL , REAL ;
consider s being FinSequence of the carrier of R such that s . i = a * s and for i being Nat st i in dom s holds s . i = a * s . i ;
func |. ( x , y ) .| -> Element of REAL equals ( x , y ) * ( x , y ) ;
consider g1 being FinSequence of REAL such that for n being Nat holds ( for n being Nat holds ( g . n ) = ( g . n ) * ( f . n ) and for n being Nat holds g . n = ( g . n ) * ( f . n ) and g . n = ( g . n ) * ( f . n ) ) and g . n = ( g . n ) * ( f . n ) * ( f . n ) and g . n = ( g . n ) * ( f . n ) * ( f . n ) * ( f . n ) and g . n = ( g . n ) * ( f . n ) * ( f . n ) and g . n = ( g . n ) * ( f . n ) and g . n = ( g . n ) * ( f . n ) * ( f . n ) *
then p1 >= len ( p1 + p2 ) & p1 = p2 + p1 ;
( q `1 ) * ( q `1 ) + ( q `1 ) * ( q `1 ) * ( q `1 ) * ( q `1 ) ) * ( q `1 ) * ( q `1 ) + ( q `1 ) * ( q `1 ) * ( q `1 ) + ( q `1 / ( q `1 ) * ( q `1 ) ) ) ) * ( q `1 / ( q `1 ) * ( q `1 / ( q `1 ) ) ) ) * ( q `1 ) + ( q `1 / ( q `1 / ( q `1 / ( q `1 / ( q `1 / ( q `1 / ( q `1 ) ) ) * ( q `1 / ( q `1 / ( q `1 / ( q `1 ) ) ) * ( q `1 ) ) * ( q `1 ) ) ) * ( q `1 ) * ( q `1 ) * ( q `1 ) * ( q `1 / ( q `1 /
F1 . ( len F1 + 1 ) = F1 . ( len F1 + 1 ) .= F1 . ( len F1 + 1 ) .= F1 . ( len F1 + 1 ) .= F1 . ( F1 . ( len F1 + 1 ) ) .= F1 . ( F1 . ( len F1 + 1 ) ) .= F1 . ( F1 . ( len F1 + 1 ) ) .= F1 . ( F1 . ( len F1 + 1 ) ) .= F1 . ( F1 . ( len F1 + 1 ) ;
consider k1 being Nat such that k1 + k = ( a |^ k1 ) * ( ( a |^ k1 ) |^ k1 ) + ( b |^ k1 ) * ( a |^ k1 ) ;
consider B being Subset of A , C being Subset of A such that B = { A where C is Subset of A : C in F & C c= F & C c= F & C c= F & C c= F & C c= F and C c= F and C is open and C c= F and C c= F and C c= F and C c= F and C c= F and C c= F and C c= F and C c= F and C c= F and F c= F and F c= F and F c= F and F is open and F is open and F is open and F is open and F is open and F is open and F is open and F is open and F is open and F is open and F is open and F is open and F is open and F is open and F is open and F is open and F is open and F is open and F is open and F is
v2 . ( F1 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . F2 F2 F2 F2 . F2 F2 F2 F2 F2 F2 ) ) ) ) ) ) ) .= ( F2 . ( F2 . F2 F2 F2 F2 F2 . F2 ) ) ) * ( F2 . F2 . F2 . F2 ) .= ( F2 . F2 ) . ( F2 . F2 F2 F2 . F2 ) ) * ( F2 . F2 ) .= ( F2 . F2 ) * ( F2 . F2 ) * ( F2 . F2 ) * F2 . F2 * F2 . F2 ) * F2 * F2 . F2 . F2 * F2 . F2 * F2 . F2 * F2 . F2 * F2 . F2 * F2 . F2 * F2 ) * F2 . F2 ) .= ( F2 . F2 . F2 . F2 . F2 . F2 ) * F2 .
dom IExec ( I , P , s ) = { IC s } \/ { IC s } .= { IC s } \/ { IC s } .= { IC s } \/ { IC s } .= IC s ;
ex dbeing Real st for h being Real st h . ( ( ( - h ) * ( ( - h ) * ( ( - h ) * ( - h ) * ( - h ) * ( - h ) ) ) ) ) & h . ( - h * ( - h * ( - h ) * ( - h * ( - h ) * ( - h ) * ( - h ) ) ) ) ) = - h * ( - h * ( - h * ( - h * ( - h * ( - h * ( - h * ( - h * ( - h * ( - h * ( - h * ( - h * ( - h * ( - h * ( - h * ( - h * ( - h * ( - h * ( - h * ( - h * ( - h * ( - h * ( - h * ( - h *
LSeg ( G * ( len G , 1 ) , G * ( len G , 1 ) ) \/ { G * ( len G , 1 ) + |[ 1 , 1 ]| } c= { |[ 1 , 1 ]| } ;
LSeg ( h , i ) = ( h /. ( i + 1 ) ) `1 + ( h /. ( i + 1 ) ) `1 .= ( h /. ( i + 1 ) ) `1 + ( h /. ( i + 1 ) ) `1 ;
A = { p where p is Point of TOP-REAL 2 : p `1 >= p `1 & p `1 >= - p `1 & p `2 >= - p `1 & p `2 >= - p `1 & p `2 >= - p `1 & p `1 <= - p `1 & p `1 <= - p `1 & p `1 <= - p `1 & p `1 <= - p `1 & p `1 <= - p `1 & p `1 <= - p `1 & p `1 <= - p `1 & - p `1 & - p `1 <= - p `1 & - p `1 <= - p `1 & - p `1 & - p `1 <= - p `1 & - p `1 & - p `1 & - p `1 <= - p `1 & - p `1 <= - p `1 & - p `1 & - p `1 & - p `1 & - p `1 & - p `1 <= - p `1 & - p `1 <= - p `1 & - p `1 <= - p `1 & - p `1 <= - p
( - x ) * ( ( - x ) * ( - x ) ) = ( - 1 ) * ( - x ) .= - 1 * ( - x ) .= - 1 * ( - x ) * ( - x ) .= - 1 * ( - x * x ) .= - 1 * ( - x * x ) .= - 1 * ( - x * x ) .= - 1 * ( - x * x ) .= - 1 * ( - x * x ) .= - 1 * ( - 1 * x ) .= - 1 * ( - 1 * x ) * ( - 1 * x ) * ( - 1 * x * x * x * x * x * x * x .= - 1 * x * x .= - 1 * x * x * x .= 1 * x * x * x * x .= 1 * x * x * x .= 1 * x * x * x * x .= 1 * x * x * x * x *
0 * ( ( 1 - p ) * ( ( p - p ) * ( 1 - p ) ) ) + ( 1 - p * ( 1 - p ) * ( 1 - p * ( 1 - p ) * ( 1 - p ) ) ) ) = 1 * ( 1 - p * ( 1 - p `1 ) ) + ( 1 - p * ( 1 - p `1 ) ) ;
( - ( ( - ( ( ( p `1 ) - ( p `1 ) - ( p `1 ) ) ) ) ) * ( ( - ( p `1 ) - ( p `1 ) ) ) * ( - ( p `1 ) - ( p `1 ) ) ) ) .= - ( - ( p `1 ) - ( p `1 ) * ( p `1 ) ) * ( - ( p `1 ) - ( p `1 ) * ( p `1 ) ) ) * ( - ( p `1 ) ) * ( - ( p `1 ) * ( p `1 ) ) * ( - ( p `1 ) * ( p `1 ) * ( p `1 ) ) ) * ( p `1 ) ) * ( p `1 ) * ( p `1 ) ) * ( - ( p `1 ) * ( p `1 ) ) * ( p `1 ) * ( p `1 ) ) ) * ( p `1 ) ) * ( p `1 ) * ( p `1 ) ) ) .= - ( p
func Shift ( f , h ) -> PartFunc of REAL means : Def: for x being Element of REAL holds it . x = ( f + h ) . x ;
assume that 1 <= k and k + 1 <= len G and k + 1 <= width G and k + 1 <= width G and k + 1 <= width G and G * ( 1 , j ) `1 = G * ( 1 , j ) `1 and G * ( 1 , k ) `1 and G * ( 1 , k ) `2 = G * ( 1 , k ) `1 and G * ( 1 , k ) `2 = G * ( 1 , k ) `1 and G * ( 1 , k ) `1 = G * ( 1 , k ) `1 and G * ( 1 , k ) `1 and G * ( 1 , k + 1 , k + 1 , j + 1 , k + 1 ) `1 and G * ( 1 , j + 1 ) `1 and G * ( 1 , j + 1 ) `1 and G * ( 1 , j + 1 ) `1 and G * ( 1 , j + 1 ) `1 and G * ( 1 , j
attr y in Free ( H ) implies ex x , y st x in ( H . ( x , y ) ) & y in ( H . ( x , y ) ) & x in ( H . ( x , y ) ) ;
defpred P [ Element of NAT ] means for p being Element of NAT holds p |^ ( p |^ ( p |^ $1 ) ) = p |^ ( p |^ ( p |^ $1 ) ) ;
func ^\ ( C ) -> non empty Subset of X equals { A where A is Subset of X : A in F } ;
[#] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( TOP-REAL n ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) = ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
rng ( F | ( ( S . 0 ) ) ) = { [ 0 , 1 ] } or for x being Element of X st x in { 0 , 1 } holds F . x = 1 ;
( f " ) . i = ( f " ) . i .= ( f " ) . i .= ( f " ) . i .= ( f " ) . i ;
consider P1 , P2 being Subset of TOP-REAL 2 such that P1 = { p1 : p1 in P1 & p2 in P1 & p1 in P1 & p1 in P1 & p1 in P2 and p1 in P2 and p1 <> p2 and p1 in P2 and p1 <> p2 and p1 in P2 and p1 in P2 and p1 in P2 and p1 <> p2 and p1 in P2 and p1 in P2 and p1 <> p2 and p1 in P2 and p1 <> p2 and p1 <> p2 and p1 in P2 and p1 in P2 and p1 in P2 and p1 in P2 and p1 in P2 and p1 <> p2 and p1 in P2 and p1 <> p2 and p1 in P2 and p1 <> p2 and p1 in P2 and p1 <> p2 and p1 in P2 and p1 <> p2 and p1 in P2 and p1 <> p2 and p1 <> p2 and p1 in P2 and p1 in P2 and p1 in P2 and p1 in P2 and p1 in P2 and p1 in P2 and p1 in P2 and p1 in P2 and p1 in P2 and p1 <> p2 and p1 in P2 and p1 in P2 and p1 in P2 and p1 in P2
f . ( p1 , p2 ) = |[ 1 , 0 ]| .= |[ - 1 , 0 ]| ;
( |[ a , b ]| ) * ( |[ a , b ]| ) = ( |[ a , b ]| ) * |[ a , b ]| .= |[ a , b ]| .= |[ |[ a , b ]| .= |[ a , b ]| ;
let T be non empty TopSpace , A be Subset of T ;
for i , j being Nat st i in dom F & j in dom F holds F . i = F . j
for x st x in Z holds ( ( ( ( ( ( ( arctan ) ) - ( ( arctan ) ) ) * ( ( arctan ) ) * ( ( arctan ) ) * ( ( arctan ) ) ) ) ) `| Z ) . x = ( ( ( arctan ) ) . ( ( arctan ) ) . x ) ^2 )
synonym f `| Z for f `| Z for f + g ;
then X1 meets X2 & X2 meets X1 & X1 union X2 = X2 union X1 union X2 implies X1 union X2 = X2 union X1 union X0 union X0 union X0 ;
ex N being Neighbourhood of x0 st N c= dom ( ( f + g ) * ( h + c ) ) & for N st N c= dom ( f + g ) holds ( f + g ) * ( h + c ) = ( f + g ) * ( h + c )
( sqrt ( 1 - ( ( ( ( ( p1 `1 / p2 ) ) ) ) ) ^2 ) ) ^2 ) + ( ( ( ( ( - p2 ) ) ^2 ) ^2 ) ) ^2 + ( ( ( - p2 ) ) ^2 ) ^2 ) ^2 ) ^2 ) ^2 + ( ( ( - p2 ) ^2 ) ^2 ) ^2 ) ^2 + ( ( - p2 ) ^2 ) ^2 ) ^2 ) ^2 + ( ( - p2 ) ^2 ) ^2 + ( ( - p2 `1 ) ^2 ) ^2 ) ^2 + ( ( - p2 `1 ) ^2 ) ^2 ) ^2 + ( ( - p2 `1 ) ^2 + ( ( - p2 `1 ) ^2 ) ^2 ) ^2 ) ^2 + ( ( - p2 `1 ) ^2 + ( ( - p2 `1 ) ^2 ) ^2 ) ^2 + ( ( - p2 `1 ) ^2 ) ^2 ) ^2 + ( ( - p2 `1 ) ^2 ) ^2 ) ^2 + ( ( - ( p2 `1 ) ^2 ) ^2 ) ^2 + ( ( - p2 `1 ) ^2 + ( - p2 `1 ) ^2 + ( ( - p2 `1 ) ^2 + ( - p2 `1 ) ^2 ) ^2 ) ^2 + ( -
( ( ( 1 - ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( - ( 1 1 1 / ( ( ( ( ( - 1 1 / ( ( ( ( ( - 1 1 ) ( ( ( - 1 / ( ( ( ( ( ( - 1 / ( ( ( ( - 1 ) ( ( - 1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) `| Z ) = ( ( ( 1 - 1 ) ) ) `| Z ) ) `| Z ) ) . x ) ) & ( ( ( ( 1 - 1 ) ) `| Z ) ) . x ) = ( ( ( ( ( 1 - 1 ) ) (#) ( ( ( ( - 1 ) ) ) . x ) ) . x ) ) `| Z ) . x ) * ( ( ( x - 1 ) ) `| Z ) . x ) * ( ( ( ( x - 1 ) ) (#) ( ( ( ( x - 1 ) ) ) . x ) ) * ( ( x ) ) ) . x ) ) & ( ( ( x - 1 )
assume that for x st x in Z holds ( ( ( - ( x + a ) * ( x + a ) ) * ( x + a ) ) * ( x - a ) ) and ( ( - a ) * ( x - a ) ) * ( x - a * ( x - a * ( x - a ) ) ) ) * ( x - a * ( x - a * ( x - a * a ) ) ) ) and ( x - a * ( x - a * x + a * x ) ) * ( x - a * x + a * x ) ) * ( x - a * x ) ) * ( x - a * x ) * ( x - a * x + a * x ) ) * ( x - a * x ) * ( x - a * x ) = ( - a * x + a * x ) * ( x - a * x ) * ( x - a * x ) ) and ( ( - a * x + a * x ) * ( x - a * x + a * x + a * x ) = ( - a * x + a
consider X1 , X2 being Subset of X such that t = X1 and ( X1 union X2 ) is open and X1 is open and X2 is open and X1 is open and X2 is open and X1 is open and X2 is open and X1 is open and X2 is open and X1 is open and X2 is open ;
card ( S . ( n + 1 ) ) = ( d . ( n + 1 ) ) * ( a |^ ( n + 1 ) ) .= ( d |^ ( n + 1 ) ) * ( a |^ ( n + 1 ) ) .= ( d |^ ( n + 1 ) ) * ( a |^ ( n + 1 ) ) .= ( d |^ ( n + 1 ) ) * ( a |^ ( n + 1 ) ) ;
( - ( E - D ) ) * ( ( E - D ) * ( ( - D ) * ( ( - D ) * ( ( - D ) * ( ( - D ) * ( ( - D ) * ( ( - D ) * ( ( - D ) * ( ( - D ) * ( ( - D ) * ( ( - D ) * ( ( - D ) * ( ( - D ) * ( - D ) * ( ( - D ) * ( ( - D ) * ( ( - D ) * ( - D ) * ( - ( D * ( - ( D ) * ( - D ) * ( - D ) * ( - ( D ) ) ) ) ) ) ) ) ) ) ) * ( - ( D ) * ( - D ) * ( - ( D ) ) ) * ( ( - D ) * ( - D ) ) ) * ( - ( D * ( - ( - D ) * ( - ( - D ) * ( - ( D ) ) ) ) = ( - D ) * ( - D ) ) ) ) * ( - D ) ) ) *