thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
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thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
contradiction ;
contradiction ;
thesis ;
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thesis ;
thesis ;
contradiction ;
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contradiction ;
contradiction ;
thesis ;
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thesis ;
thesis ;
thesis ;
contradiction ;
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contradiction ;
thesis ;
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thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b
D c= S
let Y ;
S ` is Cauchy
q in X ;
V in W ;
y in N ;
x in T ;
m < n ;
m <= n ;
n > 1 ;
let r ;
t in I ;
n <= 4 ;
M is finite ;
let X ;
Y c= Z ;
A // M ;
let U ;
a in D ;
q in Y ;
let x ;
1 <= l ;
1 <= w ;
let G ;
y in N ;
f = {} ;
let x ;
x in Z ;
let x ;
F is one-to-one ;
e <> b ;
1 <= n ;
f is special ;
S misses C
t <= 1 ;
y divides m ;
P divides M ;
let Z ;
let x ;
y c= x ;
let X ;
let C ;
x _|_ p ;
o is monotone ;
let X ;
A = B ;
1 < i ;
let x ;
let u ;
k <> 0 ;
let p ;
0 < r ;
let n ;
let y ;
f is onto ;
x < 1 ;
G c= F ;
a is_>=_than X ;
T is continuous ;
d <= a ;
p <= r ;
t < s ;
p <= t ;
t < s ;
let r ;
D <= E ;
assume e > 0 ;
assume 0 < g ;
p in C ;
x in X ;
Y ` in Y ;
assume 0 < g ;
not c in Y ;
not v in L ;
2 in z ` ;
assume f = g ;
N c= b ` ;
assume i < k ;
assume u = v ;
I = J ;
B ` = b ` ;
assume e in F ;
assume p > 0 ;
assume x in D ;
let i be element ;
assume F is onto ;
assume n <> 0 ;
let x be element ;
set k = z ;
assume o = x ;
assume b < a ;
assume x in A ;
a `1 <= b `1 ;
assume b in X ;
assume k <> 1 ;
f = product l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is from from squares ;
assume m > 0 ;
assume A c= B ;
X is lower
assume A <> {} ;
assume X <> {} ;
assume F <> {} ;
assume G is open ;
assume f is dilatation ;
assume y in W ;
y \not <= x ;
A ` in B ` ;
assume i = 1 ;
let x be element ;
x ` = x ` ` ;
let X be BCI-algebra ;
assume S is non empty ;
a in REAL ;
let p be set ;
let A be set ;
let G be _Graph , v be set ;
let G be _Graph , v be set ;
let a be VECTOR of X ;
let x be element ;
let x be element ;
let C be FormalContext , a , b be Real ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
let y be Real ;
X c= f . a
let y be element ;
let x be element ;
let i be Nat ;
let x be element ;
n in NAT ;
let a be element ;
m in NAT ;
let u be element ;
i in NAT ;
let g be Function ;
Z c= NAT ;
l <= ma ;
let y be element ;
r2 in dom f ;
let x be element ;
let k1 be Integer ;
let X be set ;
let a be element ;
let x be element ;
let x be element ;
let q be element ;
let x be element ;
assume f is being_homeomorphism ;
let z be element ;
a , b // K ;
let n be Nat ;
let k be Nat ;
B ` c= B ` ;
set s = which is LSeg ( f , 1 ) ;
n >= 0 + 1 ;
k c= k + 1 ;
R1 c= R ;
k + 1 >= k ;
k c= k + 1 ;
let j be Nat ;
o , a // Y ;
R c= Cl G ;
Cl B = B ;
let j be Nat ;
1 <= j + 1 ;
arccot is_differentiable_on Z ;
exp is_differentiable_in x ;
j < i0 ;
let j be Nat ;
n <= n + 1 ;
k = i + m ;
assume C meets S ;
n <= n + 1 ;
let n be Nat ;
h1 = {} ;
0 + 1 = 1 ;
o <> b3 ;
f2 is one-to-one ;
support p = {}
assume x in Z ;
i <= i + 1 ;
r1 <= 1 ;
let n be Nat ;
a "/\" b <= a ;
let n be Nat ;
0 <= r0 ;
let e be Real , x be Real ;
not r in G . l
c1 = 0 ;
a + a = a ;
<* 0 *> in e ;
t in { t } ;
assume F is not discrete ;
m1 divides m ;
B * A <> {} ;
a + b <> {} ;
p * p > p ;
let y be ExtReal ;
let a be Int-Location , I be Program of SCM+FSA ;
let l be Nat ;
let i be Nat ;
let r ;
1 <= i2 ;
a "\/" c = c ;
let r be Real ;
let i be Nat ;
let m be Nat ;
x = p2 ;
let i be Nat ;
y < r + 1 ;
rng c c= E
Cl R is boundary ;
let i be Nat ;
R2 in 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 ;
2 >= \lbrack x , s .] ;
G . y <> 0 ;
let X be RealNormSpace , x be Point of X , r be Real ;
a in X ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , A be Subset of V ;
assume x in \cal M ;
k < s . a ;
not t in { p } ;
let Y be set , f be Function of Y , BOOLEAN ;
M , L are_isomorphic ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded & L is lower-bounded ;
rng f = Y ;
GK c= L ;
assume x in Cl Q ;
m in dom P ;
i <= len Q ;
len F = 3 ;
Free p = {} ;
z in rng p ;
lim b = 0 ;
len W = 3 ;
k in dom p ;
k <= len p ;
i <= len p ;
1 in dom f ;
b `1 = a `1 + 1 ;
x ` = a * y ` ;
rng D c= A ;
assume x in K1 ;
1 <= i9 ;
1 <= i9 ;
px c= PI ;
1 <= ii ;
1 <= ii ;
LMP C in L ;
1 in dom f ;
let seq ;
set C = a * B ;
x in rng f ;
assume f is_continuous_on X ;
I = dom A ;
u in dom p ;
assume a < x + 1 ;
s-7 is bounded ;
assume I c= P1 ;
n in dom I ;
let Q ;
B c= dom f ;
b + p _|_ a ;
x in dom g ;
F-14 is continuous ;
dom g = X ;
len q = m ;
assume A2 is closed ;
cluster R \ S -> real-valued ;
sup D in S ;
x << sup D ;
b1 >= Z1 & b2 >= Z1 ;
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 ] ;
Let Let Let Let Let Let Let Let Let C be f ;
x9 is increasing ;
let e2 be element ;
- b divides b ;
F c= \tau ( F ) ;
Gseq is non-decreasing ;
Gseq is non-decreasing ;
assume v in H . m ;
assume b in [#] B ;
let S be non void ManySortedSign , x be set ;
assume P [ n ] ;
assume union S is linearly-independent & S is independent ;
V is Subspace of V ;
assume P [ k ] ;
rng f c= NAT * ;
assume ex_sup_of X , L ;
y in rng f ;
let s , I be set , A be Subset of S ;
b ` c= b9 ` \/ b ` ;
assume not x in NAT + \ NAT ;
A /\ B = { a } ;
assume len f > 0 ;
assume x in dom f ;
b , a // o , c ;
B in B-24 ;
cluster product p -> non empty ;
z , x // x , p ;
assume x in rng N ;
cosec is_differentiable_in x ;
assume y in rng S ;
let x , y be element ;
i2 < i1 & i1 < i2 ;
a * h in a * H ;
p , q in Y ;
Observe : sqrt I is left ideal ;
q1 in A1 & q2 in A1 ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A2 c= A1 ;
( n + 1 ) < n ;
assume A c= dom f ;
Re ( f ) is_integrable_on M ;
let k , m be element ;
a , a \equiv b , b ;
j + 1 < k + 1 ;
m + 1 <= n1 ;
g is_differentiable_in x0 & g is_differentiable_in x0 ;
g is_continuous_in x0 & g is_differentiable_in x0 ;
assume O is symmetric & O is symmetric ;
let x , y be element ;
let j0 be Nat ;
[ y , x ] in R ;
let x , y be element ;
assume y in conv A ;
x in Int V ;
let v be VECTOR of V ;
P3 halts_on s & P3 halts_on s ;
d , c // a , b ;
let t , u be set ;
let X be set ;
assume k in dom s ;
let r be non negative Real ;
assume x in F | M ;
let Y be Subset of S ;
let X be non empty TopSpace , A be Subset of X ;
[ a , b ] in R ;
x + w < y + w ;
{ a , b } >= c ;
let B be Subset of A , C be Subset of B ;
let S be non empty ManySortedSign ;
let x be variable of f , g ;
let b be Element of X , x be set ;
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 ] ;
Observe that G * F is pre-1 ;
let R be non empty multMagma , x be Element of R ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p `2 ;
assume f | X is lower ;
x in rng pion1 & y in rng pion1 ;
let x be Element of B ;
let t be Element of D ;
assume x in Q .vertices() ;
set q = s ^\ k ;
let t be VECTOR of X ;
let x be Element of A ;
assume y in rng p `2 ;
let M be mamaid ;
let N be non empty m1 \looparrowleft \mathop { \rm \mathclose { \rm c } } ;
let R be RelStr with finite R ;
let n , k be Nat ;
let P , Q be relational structure ;
P = Q /\ [#] S ;
F . r in { 0 } ;
let x be Element of X ;
let x be Element of X ;
let u be VECTOR of V ;
reconsider d = x as Int-Location ;
assume I is not \rbrace ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < v9 ;
x <= c2 . x ;
x in F ` & y in F ` ;
Observe --> T is ^2 ;
assume that t1 <= t2 and t2 <= t2 ;
let i , j be even Integer ;
assume F1 <> F2 & F2 <> F1 ;
c in Intersect ( union R ) ;
dom p1 = c & dom p2 = c ;
a = 0 or a = 1 ;
assume A1 <> A2 & A2 <> A1 ;
set i1 = i + 1 ;
assume that a1 = b1 and b1 = b2 ;
dom g1 = A & dom g1 = A ;
i < len M + 1 ;
assume not -infty in rng G ;
N c= dom ( f1 (#) f2 ) ;
x in dom sec /\ dom sec ;
assume [ x , y ] in R ;
set d = ( x / y ) / 2 ;
1 <= len ( g1 | 1 ) ;
len ( s2 - s1 ) > 1 ;
z in dom ( f1 (#) f2 ) ;
1 in dom D2 & 1 in dom D2 ;
( p `2 ) ^2 = 0 ;
j2 <= width G & 1 <= j & j <= width G ;
len PI > 1 + 1 ;
set n1 = n + 1 ;
|. q-35 .| = 1 ;
let s be SortSymbol of S ;
( i *' ( i , i ) ) = i ;
X1 c= dom f & X2 c= dom f ;
h . x in h . a ;
let G be mod of as mod of as Element of succ k ;
cluster m * n -> square ;
let k9 be Nat , k be Nat ;
i - 1 > m - 1 ;
R is_transitive field R & R is_field R ;
set F = <* u , w *> ;
pp c= P3 & I c= P3 ;
I is_halting_on t , Q ;
assume [ S , x ] is \frac { x } , S ] is \frac { x } ;
i <= len ( f2 | 1 ) ;
p is FinSequence of X & q is FinSequence of X ;
1 + 1 in dom g ;
Sum R2 = n * r ;
cluster f . x -> complex-valued ;
x in dom ( f1 (#) f2 ) ;
assume [ X , p ] in C ;
BX c= ( 3 - 1 ) * ( 3 - 1 ) ;
n2 <= ( 2 * ( 2 * n ) ) / 2 ;
A /\ ( P ` ) c= A ` ;
cluster x -valued -> $ -valued for Function ;
let Q be Subset-Family of S , A be Subset of T ;
assume n in dom ( g2 | n ) ;
let a be Element of R ;
t `1 in dom e2 & t `2 in dom e2 ;
N . 1 in rng N ;
- z in A \/ B ;
let S be SigmaField of X , x be Element of X ;
i . y in rng i ;
REAL c= dom f & dom f c= dom g ;
f . x in rng f ;
mt <= ( r / 2 ) * ( 1 / 2 ) ;
s2 in r-5 & s1 in r-5 ;
let z , z , t be complex number ;
n <= N . m ;
LIN q , p , s ;
f . x = waybelow x /\ B ;
set L = [ S \to T ] ;
let x be non positive ExtReal ;
let m be Element of M ;
f in union rng F1 & f in rng F2 ;
let K be add-associative right_zeroed right_complementable associative non empty doubleLoopStr , p be Polynomial of K ;
let i be Element of NAT , a be Element of REAL ;
rng ( F * g ) c= Y ;
dom f c= dom x & rng f c= dom x ;
n1 < n1 + 1 & n1 < n2 + 1 ;
n1 < n1 + 1 & n1 < n2 + 1 ;
cluster [: X , Y :] -> \overline ;
[ y2 , 2 ] = z ;
let m be Element of NAT , n be Nat ;
let S be Subset of R ;
y in rng ( S | [. p , q .] ) ;
b = upper_bound dom f & b = upper_bound dom f ;
x in Seg ( len q ) ;
reconsider X = D as set ;
[ a , c ] in E1 ;
assume n in dom ( h2 | n ) ;
w + 1 = ( a * 1 ) ;
j + 1 <= j + 1 + 1 ;
k2 + 1 <= k1 & k1 <= k2 ;
let i be Element of NAT ;
Support u = Support p \/ Support q ;
assume X is complete complete \frac m , n ;
assume that f = g and p = q ;
n1 <= n1 + 1 & n1 <= n2 + 1 ;
let x be Element of REAL n ;
assume x in rng ( s2 | X ) ;
x0 < x0 + 1 & x0 < x0 + 1 ;
len ( L * F ) = W ;
P c= Seg ( len A ) ;
dom q = Seg n & dom q = Seg n ;
j <= width M *' ;
let r8 be real-valued sequence of REAL ;
let k be Element of NAT ;
Integral ( M , P ) < +infty ;
let n be Element of NAT ;
assume z in ) max ( 0 , A ) ;
let i be set ;
n - 1 = n-1 - 1 ;
len ( n |-> u ) = n ;
Set N = \rbrack Z , c ;
assume x in X or x = X ;
x is midpoint of b , c ;
let A , B be non empty set , f be FinSequence of B , g be FinSequence of B , n be Nat ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q ;
let s be Element of E ^ omega ;
let B1 be Basis of x , y ;
Carrier ( 3 /\ L2 ) = {} ;
L1 /\ LSeg ( p10 , p2 ) = {} ;
assume downarrow x = downarrow y ;
assume b , c // b , c ` ;
LIN q , c , c ;
x in rng ( f-1z | n ) ;
set n8 = n + j ;
let D7 be non empty set , f be FinSequence of D , i be Nat ;
let K be add-associative right_zeroed right_complementable non empty addLoopStr , f be FinSequence of K ;
assume that f `1 = f and h `1 = h ;
R1 - R2 is total & R2 is total ;
k in NAT & 1 <= k & k <= n ;
let a be Element of G ;
assume x0 in [. a , b .] ;
K1 ` is open & ( sn - sn ) | K1 is open ;
assume that a , b ] in maximal and C is maximal in C ;
let a , b be Element of S ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = Integral ( M , f | E ) ;
cluster n\lbrack for \subseteq nA1 ;
not u in { [: g , g :] } ;
the carrier of f c= B ;
reconsider z = x as VECTOR of V ;
cluster the bounded Str of L -> \rangle ;
r (#) H is partial ;
s . intloc 0 = 1 ;
assume that x in C and y in C ;
let U0 be strict non-empty MSAlgebra over S , A be non-empty MSAlgebra over S ;
[ x , Bottom T ] is compact ;
i + 1 + k in dom p ;
F . i is stable Subset of M ;
r-35 in LSeg ( y , x ) & riff x in { y } ;
let x , y be Element of X ;
let A , I be Assume that I is Assume of X ;
[ y , z ] in [: O , A :] ;
st that that that that x in Macro i and x in dom Macro i ;
rng Sgm A = A & rng Sgm A = A ;
q |- |- All ( y , q ) ;
for n holds X [ n ] ;
x in { a } & x in d ;
for n holds P [ n ] ;
set p = |[ x , y , z ]| ;
LIN o , a , b & LIN o , a , b ;
p . 2 = Z / Y ;
( D . 0 ) `2 = {} ;
n + 1 + 1 <= len g ;
a in CQC-WFF ( Al ) & b in CQC-WFF ( Al ) ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
let I be Ideal of L ;
set g = f1 + f2 , h = f1 + f2 ;
a <= max ( a , b ) ;
i-1 < len G + 1 - 1 ;
g . 1 = f . i1 ;
x ` , y ` in A2 ;
( f /* s ) . k < r ;
set v = VAL g ;
i - k + 1 <= S ;
cluster associative -> associative for multMagma ;
x in support ( support ( t ) ) ;
assume a in [: the carrier of G , the carrier of G :] ;
i `1 <= len ( y `1 ) & i `2 <= ( y `2 ) ;
assume p divides b1 + b2 & p divides b1 + b2 ;
M1 <= upper_bound ( M1 + M2 ) & M1 <= M2 + M1 ;
assume x in W-min ( X ) ;
j in dom ( z | n ) ;
let x be Element of D ( ) ;
IC s4 = l1 & IC s4 = l1 ;
a = {} or a = { x } ;
set uG = Vertices G , c = c ;
seq " (#) ( seq " ) is non-zero ;
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
h-4 c= h-14 \/ h-14 \/ h-14 ;
]. a , b .[ c= Z ;
X1 , X2 X2 , x3 , x4 , x5 , M ;
a in Cl ( union F \ G ) ;
set x1 = [ 0 , 0 ] ;
k + 1 - 1 = k - 1 ;
cluster p2 -valued -> complex-valued for Relation ;
ex v st C = v + W ;
let IT be non empty addLoopStr , x be Element of IT ;
assume V is Abelian add-associative right_zeroed right_complementable associative ;
XY \/ Y in \sigma ( L ) ;
reconsider x = x as Element of S ;
max ( a , b ) = a ;
sup B is upper & sup B in B ;
let L be non empty reflexive RelStr , f be Function of L , L ;
R is_reflexive implies X is transitive
E , g |= H ( x , y ) ;
dom G /. y = a ;
( 1 - 4 ) >= - r ;
G . p0 in rng G & G . I in rng G ;
let x be Element of Fs , y be Element of Fs ;
D [ P-6 , 0 , 0 ] ;
z in dom ( id B ) /\ dom ( id B ) ;
y in the carrier of N & x in the carrier of N ;
g in the carrier of H & g in the carrier of G ;
rng f\mathbb R c= [: { 0 } , REAL+ :] ;
j `2 + 1 in dom s1 & j `2 < s1 `2 ;
let A , B be strict Subgroup of G ;
let C be non empty Subset of R^1 , A be Subset of REAL ;
f . z1 in dom h & f . z2 in dom h ;
P . k1 in rng P & P . k1 in rng P ;
M = ( A +* {} ) +* {} .= {} ;
let p be FinSequence of REAL , a be Real ;
f . n1 in rng f & f . n1 in rng f ;
M . ( F . 0 ) in REAL ;
ind [. a , b .] = b - a ;
assume that the distance of V , Q and P is .| ;
let a be Element of ^ ( V , C ) ;
let s be Element of PP ( ) ;
let Pf be non empty RelStr , x be Element of Rf ;
let n be Nat ;
the carrier of g c= B ;
I = halt SCM R & I = ( the carrier of R ) --> NAT ;
consider b being element such that b in B ;
set BK = BCS K , BK = BCS K ;
l <= ( -> & j <= len ( F . j ) ;
assume x in downarrow [ s , t ] ;
( x `2 ) ^2 in ]. t `1 , t `2 .] ;
x in ( JumpParts T ) . ( T . x ) ;
let h be Morphism of c , a ;
Y c= ( the carrier of K ) \/ ( the carrier of K ) ;
A2 \/ A3 c= Carrier ( L ) \/ Carrier ( L ) ;
assume LIN o , a , b ;
b , c // d1 , e2 ;
x1 , x2 , x3 , x4 , x5 , x5 , M ;
dom <* y *> = Seg 1 & dom <* y *> = Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar s .| ;
[ x , x ] `1 in X ~ ;
for n being Nat holds 0 <= x . n
|[ a , b ]| = [. a , b .] ;
cluster -> -> \alpha for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 , q1 is_collinear & q2 , q1 , q2 is_collinear ;
dom M1 = Seg n & dom M2 = Seg n ;
x = [ x1 , x2 ] & y = [ x1 , x2 ] ;
let R , Q be ManySortedSet of A ;
set d = ( 1 / ( n + 1 ) ) / ( n + 1 ) ;
rng ( ( g | X ) | X ) c= dom W ;
P . ( [#] Sigma \ B ) <> 0 ;
a in field R & a = b ;
let M be non empty Subset of V , A be Subset of V ;
let I be Program of SCM+FSA , a be Int-Location ;
assume x in rng ( that S * R ) and y in rng ( R * S ) ;
let b be Element of the carrier of T ;
dist ( e , z ) > r-r ;
u1 + v1 in W2 & v1 in W2 ;
assume that the carrier of L misses rng G and the carrier of L in rng G ;
let L be lower-bounded antisymmetric RelStr , R be Relation of L ;
assume [ x , y ] in a9 ;
dom ( A * e ) = NAT ;
let a , b be Vertex of G ;
let x be Element of Bool ( M ) ;
0 <= Arg a & Arg a < 2 * PI ;
o9 , a9 // o9 , y & o9 , c9 // o9 , y ;
{ v } c= the carrier of l & { v } c= the carrier of l ;
let x be variable of A ;
assume x in dom ( ( uncurry f ) . x ) ;
rng F c= ( product f ) |^ X
assume D2 . k in rng D & D . k in rng D ;
f " . p1 = 0 & f " . p2 = 1 ;
set x = the Element of X , y = the Element of X ;
dom Ser ( G ) = NAT & dom Ser ( G ) = NAT ;
let n be Element of NAT ;
assume LIN c , a , e1 ;
cluster -> finite for FinSequence of NAT ;
reconsider d = c as Element of L1 ( ) ;
( v2 |-- I ) . X <= 1 ;
assume x in the carrier of f & y in the carrier of f ;
conv @ S c= conv @ A & conv @ S c= conv @ A ;
reconsider B = b as Element of the carrier of T ;
J , v |= P \lbrack l , l1 \rbrack ;
Observe that the TopStruct of J . i is non empty TopStruct ;
ex_sup_of Y1 \/ Y2 , T & ex_sup_of Y1 , T ;
W1 is_well field ( W1 + W2 ) & W2 is_field ( W2 + W1 ) ;
assume x in the carrier of R & y in the carrier of R ;
dom ( n |-> 0 ) = Seg n & dom ( n --> 0 ) = Seg n ;
s4 misses s4 & s4 misses s4 ;
assume ( a 'imp' b ) . z = TRUE ;
assume X is open & f = X --> d ;
assume [ a , y ] in Indices ( f | n ) ;
assume that stop I c= J and that that x in K and x in K and x in J ;
Im ( ( lim seq ) , x ) = 0 ;
( ( - sin ) `| Z ) . x <> 0 ;
sin is_differentiable_on Z & cos is_differentiable_on Z implies cos is_differentiable_on Z & cos is_differentiable_on Z
t6 . n = t3 . n .= 0 ;
dom ( non empty set ) c= dom F ;
W1 . x = W2 . x .= ( W2 . x ) ;
y in W .vertices() \/ W .vertices() \/ W .vertices() ;
( k + 1 ) <= len ( v | k ) ;
x * a \equiv y * a . ( mod m ) ;
proj2 .: S c= proj2 .: P & proj2 .: P c= proj2 .: P ;
h . p4 = g2 . I .= ( g2 . I ) `1 ;
G6 = U /. 1 .= G * ( 1 , 1 ) `1 ;
f . r1 in rng f & f . r1 in rng f ;
i + 1 + 1 - 1 <= len - 1 ;
rng F = rng ( F | 2 ) .= rng ( F | 2 ) ;
mode reconsider then is well unital associative non empty multMagma ;
[ x , y ] in A ~ { a } ;
x1 . o in L2 . o & x2 . o in L2 . o ;
the carrier of m c= B & the carrier of m c= B ;
not [ y , x ] in id ( X ) ;
1 + p .. f <= i + len f ;
seq ^\ ( k1 + 1 ) is lower ;
len ( F . m ) = len I .= len ( F . m ) ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be complex number , p be Point of TOP-REAL 2 ;
Comput ( P , s , n ) = s ;
k <= k + 1 & k + 1 <= len p ;
reconsider c = {} T as Element of L ;
let Y be Element of of of of of \mathbin { {} } ;
cluster strict for Function of L , L ;
f . j1 in K . j1 & f . j2 in K . j1 ;
Observe that J => y is total for Function ;
K c= 2 |^ ( the carrier of T ) ;
F . b1 = F . b2 & F . b2 = F . b2 ;
x1 = x or x1 = y or x1 = z ;
pred a <> {} means a |^ a = 1 ;
assume that succ a c= b and b in a ;
s1 . n in rng s1 & s1 . n in rng s1 ;
{ o , b2 } on C2 & { o , b2 } on C2 ;
LIN o , b , b9 & LIN o , b , b9 ;
reconsider m = x as Element of Funcs ( V , C ) ;
let f be non constant FinSequence of D ;
let FG2 be non empty RelStr , f be Function of I[01] , TOP-REAL 2 ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in F-8 ;
reconsider pp = x as Subset of m -tuples_on REAL ;
let A , B , C be Element of R ;
Observe that ex a being non empty natural number st a is strict and a is strict compact ;
rng c `1 misses rng ( e | i ) & not c in rng ( e | i ) ;
z is Element of gr { x } & z in { x } ;
not b in dom ( a .--> p1 ) ;
assume that k >= 2 and P [ k ] ;
Z c= dom ( ( - cot ) `| Z ) & Z c= dom ( ( - cot ) `| Z ) ;
the component of Q c= UBD A & UBD Q c= UBD A ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( 1 / 2 ) & g1 in dom ( 1 / 2 ) ;
pred f = u means : Def6 : a * f = a * u ;
for n holds P1 [ n ] implies P1 [ n + 1 ]
{ x . O : x in L } <> {} ;
let x be Element of V . s ;
let a , b be Nat ;
assume that S = S2 and p = p2 and S is closed ;
gcd ( n1 , n2 ) = 1 & gcd ( n1 , n2 ) = 1 ;
set o9 = a * ( - b ) , z2 = a * ( - b ) , z2 = a * ( - b ) ;
seq . n < |. r1 .| & |. r1 - x0 .| < r ;
assume that seq is increasing and r < 0 and 0 < r ;
f . ( y1 , x1 ) <= a ;
ex c being Nat st P [ c ] ;
set g = { n / 1 : n in NAT } ;
k = a or k = b or k = c ;
a9 , b9 , c9 , a9 , b9 , c9 , c9 , a9 , b9 , c9 , b9 , c9 , a9 , b9 ;
assume that Y = { 1 } and s = <* 1 *> ;
I1 . x = f . x .= 0 .= 0 ;
W3 .last() = W3 . 1 & W2 . 1 = W . 1 ;
cluster -> trivial for Walk of G , V , E , F , G , G , H ;
reconsider u = u as Element of Bags X ;
A in B ^ ) implies A , B are_that A , B are_that B , A are_
x in { [ 2 * n + 3 , k ] } ;
1 >= ( q `1 ) ^2 / |. q .| ;
f1 is_the ] in the ] & f2 is_the ] to f1 implies f1 is in the carrier of TOP-REAL 2
( f . I ) `2 <= ( q `2 ) ^2 / ( |. q .| ) ^2 ;
h is_the carrier of Cage ( C , n ) ;
( b `2 ) ^2 / ( p `2 ) ^2 <= ( p `2 ) ^2 / ( p `1 ) ^2 ;
let f , g be Element of X , f , g be Function of X , Y ;
S * ( k , k ) <> 0. K ;
x in dom ( max ( - f , - f ) ) ;
p2 in NN . ( p1 , p2 ) & p2 in NN . ( p2 , p1 ) ;
len ( H ) < len ( H ) ;
F [ A , F-14 ( A ) , F ( A ) ] ;
consider Z such that y in Z and Z in X ;
pred 1 in C means : Def6 : A c= C |^ A ;
assume that r1 <> 0 or r2 <> 0 and r2 <> 0 ;
rng q1 c= rng ( C1 ^ C2 ) & rng s1 c= rng C1 ;
A1 , L , A2 , A3 , A3 is_collinear ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in element & R c= R & not b in R ;
then S is negative ;
Cl ( [#] T ) = [#] ( T ) .= [#] ( T ) ;
f12 | A2 = f2 | A2 & f12 | A2 = f2 | A2 ;
0. M in the carrier of W & 0. M in the carrier of V ;
let v , v be Element of M ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ V & V c= Y \ Z ;
let X be Subset of S ~ ;
consider H1 such that H = 'not' H1 and H1 in A ;
1_ 1 c= ( t * t ) * ( ( t - 1 ) / t ) ;
0 * a = 0. R .= a * 0 .= a * 0 ;
A |^ ( 2 , 2 ) = A ^^ A ;
set vY = v4 /. n , vY = v4 /. n , vY = v4 /. n ;
r = 0. ( \langle \cal E *> , \Vert \cdot \Vert *> ) ;
( f . p4 ) `1 >= 0 & ( f . p2 ) `2 >= 0 ;
len W = len ( W | ( len W ) ) .= len ( W | ( len W ) ) ;
f /* ( s * G ) is divergent_to+infty ;
consider l being Nat such that m = F . l ;
t8 . W7 in { b1 , b2 } & not ( ex b1 st b1 in dom b1 & b2 in dom b2 & b1 in dom b2 & b2 in dom b1 & b2 in dom b2
reconsider Y1 = X1 , Y2 = X2 as SubSpace of X ;
consider w such that w in F and not x in w ;
let a , b , c , d , e , f , g , h be Real ;
reconsider i = i as non zero Element of NAT ;
c . x >= id L . x .= ( id L ) . x ;
\sigma ( T ) \/ omega ( T ) is Basis of T ;
for x being element st x in X holds x in Y
cluster [ x1 , x2 , x3 , x4 ] -> pair for element ;
downarrow a /\ downarrow t is Ideal of T ;
let X be with_\hbox { \mathbb N } , A be non empty set ;
rng f = \HM { \rm such } ( S , X ) ;
let p be Element of B , x be the the st x in the carrier' of S holds x is One ;
max ( N1 , 2 ) >= N1 & max ( N1 , 2 ) >= N2 ;
0. X <= ( b |^ m ) * ( ( m * mm1 ) * ( n + 1 ) ) ;
assume that i in I and R2 . i = R ;
i = j1 & p1 = q1 & p2 = q2 & q1 = q2 ;
assume that greal in the right of g and x in the carrier of g ;
let A1 , A2 be Point of S , x be Point of S ;
x in h " P /\ [#] T1 & x in h " P /\ [#] T1 ;
1 in Seg 2 & 1 in Seg 3 & 2 in Seg 3 ;
reconsider X-5 = X as non empty Subset of [: T , T :] | A ;
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the Source of G ) -valued ;
n1 <= i2 + len ( g | 1 ) & n2 <= len ( g | 1 ) ;
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume v in the carrier' of G2 & v in the carrier' of G1 ;
y = Re y + ( Im y ) * i ;
( - 1 ) , ( - 1 ) as Real ;
x2 is_differentiable_on ]. a , b .[ & x2 is_differentiable_on ]. a , b .[ ;
rng ( M * ( i , j ) ) c= rng D2 & len ( M * ( i , j ) ) = len D ;
for p being Real st p in Z holds p >= a
( cn max ( f ) ) . x = proj1 . x & ( cn max ( f ) ) . x = proj1 . x ;
( seq ^\ m ) . k <> 0 & ( seq ^\ m ) . k <> 0 ;
s . ( G . ( k + 1 ) ) > x0 ;
( p |-count M ) . 2 = d ;
A \oplus ( B \ominus C ) = ( A \oplus B ) Let C ;
h \equiv gg . ( mod P ) , gg . ( mod P ) ;
reconsider i1 = i-1 - 1 as Element of NAT ;
let v1 , v2 be VECTOR of V , v be VECTOR of V ;
for V being Subspace of V holds V is Subspace of [#] V
reconsider i9 = i - 1 as Element of NAT ;
dom f c= [: C , D :] & dom f = [: C , D :] ;
x in ( the inferior of B ) . n ;
len \rbrace in Seg ( len f2 + len g2 ) ;
pp1 c= the topology of T & P c= the topology of T ;
]. r , s .[ c= [. r , s .] ;
let B2 be Basis of T2 , x be Point of T2 ;
G * ( B * A ) = ( id o1 ) * ( id o2 ) ;
assume that p , u , v , u is_collinear and u , v , v , u1 u1 u1 u1 u2 ;
[ z , z ] in union rng ( F | D ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S , S = $1 .. S ;
LIN a1 , a3 , b1 & LIN a1 , b1 , b2 ;
f " ( f .: ( f .: x ) ) = { x } ;
dom ( w2 . i ) = dom ( r12 . i ) .= dom r12 ;
assume that 1 <= i and i <= n and j <= n ;
( ( ( g2 ) . O ) `2 ) ^2 <= 1 ;
p in LSeg ( E . i , F . i ) ;
Iu * ( i , j ) = 0. K ;
|. f . ( s . m ) - g .| < g1 ;
q9 . x in rng ( q | i ) & q . x in rng ( q | i ) ;
Carrier ( Lj ) misses Carrier ( Lj ) \/ Carrier ( Lj ) ;
consider c being element such that [ a , c ] in G ;
assume that for o9 being Element of NAT holds o9 in o9 & o9 in o ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= ( F-12 * ( Cpion1 * C ) ) " { 0 } ;
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [. f . j , f . j .] ;
pred 0 <= x & x <= 1 & x ^2 <= x ;
p `1 - q `2 <> 0. TOP-REAL 2 & p `2 - q `2 <> 0. TOP-REAL 2 ;
Observe that such that such that for S holds S , T is non empty ;
let x be Element of S ~ ;
cluster cluster cluster cluster cluster cluster cluster cluster cluster cluster \hbox \hbox { - } \rm id ( F ) -> one-to-one ;
|. i .| <= - ( - 2 |^ n ) / ( n + 1 ) ;
the carrier of I[01] = dom P & the carrier of I[01] = dom P ;
n * ( n + 1 ) ! > 0 * n ! ;
S c= ( A1 /\ A2 ) /\ ( A1 /\ A2 ) ;
a3 , a4 // b2 , b3 & a3 , a4 // b2 , b3 ;
then dom A <> {} & dom A <> {} & rng A c= {} ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x joins ( the Vertex of X ) . ( x , y ) , ( the Source of G2 ) . ( x , y ) ;
set v2 = ( v /. ( i + 1 ) ) * ( i + 1 ) ;
x = r . n .= ( r4 . n ) * ( r4 . n ) ;
f . s in the carrier of S2 & f . s in the carrier of S2 ;
dom g = the carrier of I[01] & dom g = the carrier of I[01] ;
p in Lower_Arc ( P ) /\ Lower_Arc ( P ) ;
dom d2 = [: A2 , A1 :] & dom d1 = [: A2 , A1 :] ;
0 < ( p / ||. z .|| + 1 ) / ( ||. z .|| + 1 ) ;
e . ( m + 1 ) <= e . ( m + 1 ) ;
B \ominus X \/ B \ominus Y c= B \ominus X /\ Y
-infty < Integral ( M , ( Im g ) | B ) ;
cluster O \cup F -> \HM { \HM { , where } is operation of X : not contradiction } -> Element of X
let U1 , U2 be non-empty MSAlgebra over S , f be Function of U1 , U2 ;
Proj ( i , n ) * g is_differentiable_on X ;
let x , y , z be Point of X , p be Point of X ;
reconsider pp = p . x as Subset of V ;
x in the carrier of Lin ( A ) & y in the carrier of Lin ( B ) ;
let I , J be parahalting Program of SCM+FSA , a be Int-Location ;
assume that - a is lower and a in - ( - X ) and b in - ( - X ) ;
Int Cl ( Cl A ) c= Cl ( Cl A ) ;
assume for A being Subset of X holds Cl A = A ;
assume q in Ball ( |[ x , y ]| , r ) ;
( ( p2 `2 ) / |. p2 .| ) ^2 <= ( ( p2 `2 ) ) ^2 / ( |. p2 .| ) ^2 ;
Cl Q ` = [#] ( ( TOP-REAL 2 ) | P ) .= [#] ( ( TOP-REAL 2 ) | P ) ;
set S = the carrier of T , S = the carrier of S ;
set I8 = ' ( f |^ n ) , I8 = ' ( f |^ n ) ;
len p -' n = len \langle - n *> .= len <* - n *> ;
A is Permutation of Funcs ( A , x , y ) ;
reconsider n6 = n8 - 1 as Element of NAT ;
1 <= j + 1 & j + 1 <= len ( s | [. 1 , 1 .] ) ;
let qc= [: M , M :] , qI = [: M , M :] ;
a9 in the carrier of S1 & b9 in the carrier of S2 ;
c1 /. n1 = c1 . n1 & c2 /. n1 = c1 . n1 ;
let f be FinSequence of TOP-REAL 2 , p be Point of TOP-REAL 2 , a be Real ;
y = ( f * SS ) . x .= ( f * SS ) . x ;
consider x being element such that x in ) and x in _ in A ;
assume r in ( ( dist ( o ) ) .: P ) .: P ;
set i2 = ( n + 1 ) div 2 , h = ( n + 1 ) div 2 ;
h2 . ( j + 1 ) in rng h2 & h2 . ( j + 1 ) in rng h2 ;
Line ( M29 , k ) = M . i .= Line ( M29 , k ) ;
reconsider m = ( x - 2 ) / 2 as Element of ( - 1 ) * ;
let U1 , U2 be Subspace of U0 , U2 be Subspace of U2 ;
set P = Line ( a , d ) ;
len p1 < len p2 + 1 & len p2 < len p1 + 1 ;
let T1 , T2 be Scott Scott topological or for x being Point of L holds x in the topology of L ;
then x <= y & ( { x } c= { y } ) ;
set M = n -$ m , n = n -\hbox { m } , m = n -\hbox { m } , n = m -\hbox { m } ;
reconsider i = x1 , j = x2 as Nat ;
rng ( ( the_arity_of o ) | ( dom the_arity_of o ) ) c= dom H ;
z1 " = ( z1 " ) * ( z1 " ) .= ( z1 " ) * ( z1 " ) ;
x0 - r / 2 in L /\ dom f ;
then w is being being rng w string of S means : Def6 : rng w /\ S <> {} ;
set x9 = x9 ^ <* Z *> ^ <* Z *> ^ <* Z *> ^ <* Z *> ^ <* Z *> ^ <* Z *> ^ <* Z *> ^ <* Z *> ^ <* Z *> ^ <* Z *> ^ <* Z *> ^ <* Z *> ^ <* Z *>
len w1 in Seg ( len w1 + 1 ) & len w2 = len w1 + 1 ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of PFuncs ( V , { k } ) ;
x . n = ( |. a . n .| ) * ( |. a . n .| ) ;
( p `1 ) ^2 / ( G * ( 1 , 1 ) `1 ) ^2 <= ( G * ( 1 , 1 ) `1 ) ^2 ;
rng ( g | 1 ) c= L~ ( g | 1 ) \/ L~ ( g | 1 ) ;
reconsider k = i-1 * i-1 + j as Nat ;
for n being Nat holds F . n is \HM { -infty } ;
reconsider x9 = x9 , y9 = y9 as VECTOR of M ;
dom ( f | X ) = X /\ dom f .= X /\ dom f ;
p , a // p , c & b , a // c , c ;
reconsider x1 = x as Element of REAL m -tuples_on REAL m , r be Real ;
assume i in dom ( a * p ^ q ) ;
m . ( ( k + 1 ) + 1 ) = p . ( ( k + 1 ) + 1 ) ;
a / ( s . m ) - ( n / m ) <= 1 ;
S . ( n + k + 1 ) c= S . ( n + k ) ;
assume that B1 \/ C1 = B2 \/ C2 and B2 \/ C2 = A1 \/ A2 and B1 \/ B2 = A2 \/ A1 ;
X . i = { x1 , x2 , x3 , x4 , x5 } . i ;
r2 in dom ( h1 + h2 ) /\ dom ( h2 + h2 ) ;
\cal R = a & b-0 = b & b0 = b ;
F8 is_closed_on t1 , Q1 & P8 is_halting_on t1 , Q1 ;
set T = ^2 (# X , x0 , x1 #) ;
Int ( Cl ( Cl R ) ) c= ( Cl R ) \/ ( Cl R ) ;
consider y being Element of L such that c . y = x ;
rng ( FN . x ) = { FN . x } ;
Ga1 " { c } c= B \/ S \/ S \/ S ;
f[#] is Relation of [: X , Y :] , X , Y ;
set RF = the Element of P , RF = the Element of Q ;
assume that n + 1 >= 1 and n + 1 <= len M ;
let k2 be Element of NAT , k be Element of NAT ;
reconsider pp = u as Element of ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( K -' 1 ) ) ) ) ;
g . x in dom f & x in dom g implies x in dom g ;
assume that 1 <= n and n + 1 <= len f1 and f1 . n = f2 /. ( n + 1 ) ;
reconsider T = b * N as Element of ( G , N ) | ( N , S ) ;
len P5 <= len P-35 + len P-35 - 1 ;
x " in the carrier of A1 & x " in the carrier of A1 ;
[ i , j ] in Indices ( ( A * B ) * ( i , j ) ) ;
for m be Nat holds Re ( F . m ) is simple function
f . x = a . i .= a1 . k .= a1 . k ;
let f be PartFunc of REAL , REAL , x be Point of REAL-NS m ;
rng f = the carrier of Lin A & f . 1 = f . ( the carrier of A ) ;
assume s1 = sqrt ( 2 * p ) + sqrt ( 2 * p ) ;
pred a > 1 & b > 0 & a / b > 1 ;
let A , B , C be Subset of [: I , J :] ;
reconsider X0 = X , Y = Y as RealNormSpace of X , Y be Subset of Y ;
let f be PartFunc of REAL , REAL , x be Point of REAL ;
r * ( v1 |-- I ) . X < r * 1 ;
assume that V is Subspace of X and X is Subspace of V ;
let t-3 , t-3 be binary st t-3 in dom tt1 & tt2 in dom tt2 holds tt1 = tt2
Q [ e1 \/ { v9 } , f ] & f . ( v , v9 ) = f . ( v , v9 ) ;
g \circlearrowleft W-min L~ z = z & ( W-min L~ z ) .. z = z ;
|. |[ x , v ]| - |[ x , y ]| .| = v;
- ( f . w ) = - ( L * w ) ;
z - y <= x & z <= x + y implies z <= x + y
( 7 - 1 ) / ( ( 1 - e ) / ( 1 - e ) ) > 0 ;
assume X is BCI-algebra & 0 > 0 & 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ;
F . 1 = v1 & F . 2 = v2 & F . 3 = v2 ;
( f | X ) . x2 = f . x2 .= ( f | X ) . x2 ;
( ( tan * tan ) `| Z ) . x in dom sec & ( tan * tan ) . x > 0 ;
i2 = ( f /. len f ) `2 & ( f /. len f ) `2 = ( f /. 1 ) `2 ;
X1 = X2 \/ ( X1 \ X2 ) & X2 = ( X1 \ X2 ) \/ ( X2 \ X1 ) ;
[. a , b , 1_ G .] = 1_ G & [. a , b .] = 1_ G ;
let V , W be non empty VectSpStr over F_Complex , f be FinSequence of V , g be FinSequence of V ;
dom ( g2 ) = the carrier of I[01] & dom ( g2 ) = the carrier of I[01] ;
dom ( f2 | [. 0 , 1 .] ) = the carrier of I[01] .= [. 0 , 1 .] ;
( proj2 | X ) .: X = proj2 .: X .= proj2 .: X .= proj2 .: X ;
f . ( x , y ) = h1 . ( x , y ) .= f . ( x , y ) ;
x0 - r < a1 . n & x0 - r < a1 . n ;
|. ( f /* s ) . k - G . ( k + 1 ) .| < r ;
len Line ( A , i ) = width A & len Line ( A , i ) = width A ;
SFinSequence / 2 = ( S . ( g . ( g . x ) ) ) / 2 ;
reconsider f = v + u as Function of X , the carrier of Y ;
( intloc 0 ) in dom ( Initialized p ) \/ dom ( Initialized p ) ;
i1 , i2 , i3 , u1 , v1 , v2 , v2 , v1 , v2 , v2 , v1 , v2 , v2 , v1 , v2 , v2 , v1 , v2 , v2 , u1 , v1 , v2 , v2 , v1 , v2 ;
arccos + r / sqrt ( 1 + r ^2 ) = ( cos . x ) / sqrt ( 1 + r ^2 ) ;
for x st x in Z holds f2 is_differentiable_in x & f2 is_differentiable_in x ;
reconsider q2 = ( q - x ) / 2 as Element of REAL n ;
( 0 qua Nat ) + 1 <= i + j1 & ( 0 + 1 ) <= i + j1 ;
assume that f in the carrier of [ X \to Omega Y , Omega Y ] ;
F . a = H / ( ( x , y ) / ( x , y ) ) ;
( TRUE T ) at ( C , u ) = TRUE & ( T . ( C , u ) ) = TRUE ;
dist ( ( a * seq ) . n , h ) < r ;
1 in the carrier of [. 0 , 1 .] & 1 in dom ( [. 0 , 1 .] ) ;
( ( p2 `1 ) - x1 ) - x1 > - g & ( p2 `1 ) - x1 < g ;
|. r1 - p .| = |. a1 - p .| * |. thesis - p .| ;
reconsider S-14 = 8 as Element of Seg 8 , a be Element of Seg 8 ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b
DWW .last() = DWW .1 + 1 ;
i1 = [: a + n , K :] & i2 = [: K , n :] & n > 0 implies i1 = i2
f . a [= f . ( f . O1 , a "\/" ( f . a ) ) ;
pred f = v & g = u & f + g = v + u ;
I . n = Integral ( M , F . n ) ;
chi ( [: T1 , T2 :] , S ) . s = 1 ;
a = VERUM ( A ) or a = VERUM ( A ) or a = VERUM ( A ) ;
reconsider k2 = s . b3 as Element of NAT , k be Element of NAT ;
( Comput ( P , s , 4 ) ) . ( GBP + 1 ) = 0 ;
L~ ( M1 + M2 ) meets L~ ( R1 + R2 ) \/ LSeg ( R2 , 1 ) ;
set h = the continuous Function of X , ( the carrier of X ) | R ;
set A = { L . ( k . n ) : not contradiction } ;
for H st H is negative holds P7 [ H ] ;
set b8 = S5 \ ( i + 1 ) , S8 = S \ ( i + 1 ) ;
Hom ( a , b ) c= Hom ( a `1 , b `2 ) ;
( 1 / ( n + 1 ) ) * ( 1 / ( n + 1 ) ) < ( 1 / ( n + 1 ) ) * ( 1 / ( n + 1 ) ) ;
( l `1 ) = [ dom l , cod l ] & ( l `2 ) = cod l ;
y +* ( i , y /. i ) in dom g ;
let p be Element of CQC-WFF ( Al ( ) ) , x be Element of D ( ) ;
X /\ X1 c= dom ( ( f1 - f2 ) | X ) /\ dom ( ( f1 - f2 ) | X ) ;
p2 in rng ( f /^ ( len f -' 1 ) ) \/ rng ( f /^ 1 ) ;
1 <= indx ( D2 , D1 , j1 ) & 1 <= j1 & j1 <= len D1 ;
assume x in ( ( ( TOP-REAL 2 ) | K0 ) \/ ( ( TOP-REAL 2 ) | K0 ) ) \/ ( ( TOP-REAL 2 ) | K0 ) ;
- 1 <= ( ( f2 ) . O ) `2 & ( ( f2 ) . I ) `2 <= 1 ;
let f , g be Function of I[01] , TOP-REAL 2 , a , b , c , d be Real ;
k1 - k2 = k1 - k2 - k2 + 1 .= k1 - k2 + 1 - k2 ;
rng seq c= ]. x0 - r , x0 + r .[ & rng seq c= ]. x0 - r , x0 + r .[ ;
g2 in ]. x0 - r , x0 + r .[ & g2 in ]. x0 - r , x0 + r .[ ;
sgn ( p `1 , K ) = - 1_ K & sgn ( p `1 , K ) = - 1_ K ;
consider u being Nat such that b = p |^ y * u ;
pred a is the the the the the the normal normal sequence of A means : Def7 : a = Sum A ;
Cl ( union H ) = union ( ( Cl H ) \/ ( Cl H ) ) .= ( Cl H ) \/ ( Cl H ) ;
len t = len t1 + len t2 & len t1 = len t2 + len t1 ;
v-29 = v + w |-- v + AA .= v + AA ;
v <> DataLoc ( t1 . GBP , 3 ) & v <> DataLoc ( t1 . GBP , 3 ) ;
g . s = sup ( d " { s } ) ;
( \dot \dot y ) . s = s . ( \dot y ) ;
{ s : s < t & t in Q + + 1 } = {} implies t = {}
s ` \ s = s ` \ ( 0. X \ s ) .= ( 0. X \ s ) \ ( 0. X \ s ) ;
defpred P [ Nat ] means B + $1 in A & not $1 in B ;
( 3be ! + 1 ) ! = 3333399 ! * ( 3be + 1 ) ;
U /. ( succ A ) = T . ( ( succ A ) . ( succ A ) ) .= T . ( succ A ) ;
reconsider y = y as Element of ( len y ) -tuples_on REAL ;
consider i2 being Integer such that y0 = p * i2 and i2 in dom f and 1 <= i2 and i2 <= len f ;
reconsider p = Y | Seg k as FinSequence of ( the carrier of K ) * ;
set f = ( S , U ) \mathop { N } , g = S ( ) ;
consider Z being set such that ( lim s ) in Z & Z in F ;
let f be Function of I[01] , TOP-REAL n , a be Real , b be Real ;
( SAT M ) . [ n + i , 'not' A ] <> 1 ;
ex r being Real st x = r & a <= r & r <= b ;
let R1 , R2 be Element of REAL n , x be Element of REAL n ;
reconsider l = 0. ( V ) as Linear_Combination of A ;
set r = |. e .| + |. w + n .| + |. w + n .| ;
consider y being Element of S such that z <= y and y in X ;
a 'or' ( b 'or' c ) = 'not' ( ( a 'or' b ) 'or' c ) ;
||. ( x9 - g2 ) . x - g2 .|| < r2 / 2 * ||. x - g2 .|| ;
b9 , a9 // b9 , c9 & c9 , a9 // b9 , c9 ;
1 <= k2 -' k1 & k1 + 1 <= k2 & k2 + 1 = k2 + 1 & k2 <= k1 ;
( ( p `2 ) - sn ) / ( 1 + sn ) >= 0 ;
( ( q `2 ) - sn ) / ( 1 + sn ) < 0 ;
( E-max C ) in LSeg ( ( R /. 1 ) , ( R /. 1 ) ) ;
consider e being Element of NAT such that a = 2 * e + 1 and e in A ;
Re ( ( lim F ) | D ) = Re ( ( lim G ) | D ) ;
LIN b , a , c or LIN b , c , a or LIN b , c , a ;
p `1 , a // a `1 , b or p `1 , a // b `1 , a `2 ;
g . n = a * Sum ( f | 1 ) .= f . n * ( f | 1 ) . n ;
consider f being Subset of X such that e = f and f is Cage ;
F | ( N2 ~ S ) = CircleMap * ( F | ( N2 ~ S ) ) .= CircleMap * ( F | ( N2 ~ ) ) ;
q in LSeg ( q , v ) \/ LSeg ( v , p ) \/ LSeg ( v , p ) ;
Ball ( m , r0 ) c= Ball ( m , s ) \/ Ball ( u , s ) ;
the carrier of (0). V = { 0. V } & the carrier of V = { 0. V } ;
rng ( ( - 1 ) (#) cos ) = [. - 1 , 1 .] .= [. - 1 , 1 .] ;
assume that Re ( seq ) is summable and Im ( seq ) is summable and Im ( seq ) is summable and Im ( seq ) is summable ;
||. ( vseq . n ) - ( t . n ) .|| < e / 2 ;
set g = O --> 1 ;
reconsider t2 = t11 as 0 string of S2 , t2 = ( 0 , 1 ) (#) t2 as 0 -element string of S2 ;
reconsider x9 = seq . n as sequence of REAL-NS n , x be Point of REAL-NS n ;
assume that that W-min L~ Cage ( C , n ) meets L~ pion1 and not W-min L~ Cage ( C , n ) meets L~ pion1 ;
- ( 1 - ( 1 - r ) ) < F . n - ( 1 - r ) . x ;
set d1 = c9 ( x1 , z1 ) , d2 = c9 ( y1 , z1 ) , d1 = h . ( y1 , z1 ) , d2 = h . ( y2 , z2 ) , d2 = h . ( y1 , z1 ) , d1 = h . ( y2 , z2 ) ;
2 |^ ( 1 -' 1 ) = 2 |^ ( 1 -' 1 ) - 1 ;
dom ( v | Seg len ( v | 1 ) ) = Seg len ( v | 1 ) .= Seg 1 ;
set x1 = - ( k2 + 1 ) + |. k2 + 1 .| , x2 = - ( k2 + 1 ) , x3 = - ( k2 + 1 ) ;
assume for n being Element of X holds 0. ( X , REAL ) <= F . n ;
assume that 0 <= T-32 . i and T-32 . ( i + 1 ) <= 1 and Tc . ( i + 1 ) <= 1 ;
for A being Subset of X holds c . ( c . A ) = c . A
the carrier of ( Carrier ( L^2 ) + L2 ) c= [: I , { 1 } :] ;
'not' All ( x , p ) => All ( x , p ) is valid ;
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the normal w.r.t. of A , {} ;
Z c= dom ( ( - 1 ) (#) ( ( sin * f1 ) `| Z ) ) ;
|. 0. TOP-REAL 2 - ( q `2 / |. q .| - sn ) .| < r / 2 ;
ConsecutiveSet2 \ B c= ConsecutiveSet2 ( A , ConsecutiveSet ( A , ConsecutiveSet ( d , L ) ) ) ;
E = dom ( L * F ) & L is_measurable_on E & L is measurable & L is measurable implies L * F is measurable
C / ( A + B ) = C / ( B * C ) ;
the carrier of W2 c= the carrier of V & the carrier of V c= the carrier of V ;
I . IC Comput ( P , s , m ) = P . IC Comput ( P , s , m ) .= I . IC Comput ( P , s , m ) ;
pred x > 0 means : Def6 : x |^ ( - 1 ) = x |^ ( - 1 ) ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) \/ LSeg ( g , i ) ;
consider p being Point of T such that C = [. p , g .] and p in C ;
b , c are_connected & - C , - C are_connected & - C , - C are_connected ;
assume that f = id ( the carrier of O ) and f is monotone and f is continuous ;
consider v such that v <> 0. V and f . v = L * v ;
let l be Linear_Combination of {} ( the carrier of V ) , a be Element of V ;
reconsider g = f " as Function of U2 , U1 , U2 , U2 be Function of U2 , U2 ;
A1 in the carrier of ( k + 1 ) & A2 in the carrier of ( k + 1 ) ;
|. - x .| = - ( x - x ) .= x - x .= - x ;
set S = ) +* 1GateCircStr ( x , y , c ) ;
( 5 * Fib n ) * ( 5 * Fib n ) >= 4 * log ( n , / sqrt 5 ) ;
v9 /. ( k + 1 ) = v9 . ( k + 1 ) .= v . ( k + 1 ) ;
0 mod i = - ( i * ( 0 qua Nat ) ) .= - ( i * ( 0 qua Nat ) ) ;
Indices ( M1 * M2 ) = [: Seg n , Seg n :] & len ( M1 * M2 ) = n ;
Line ( S\mathopen , j ) = S\mathopen ( j , i ) .= Sj . i ;
h . ( x1 , y1 ) = [ y1 , x1 ] & h . ( y1 , y2 ) = [ y1 , y2 ] ;
|. f - Re ( |. f .| ) * ( ( card b ) * h ) ) is nonnegative ;
assume x = ( a1 ^ <* x1 *> ) ^ <* x2 *> ^ <* x1 *> ^ <* x2 *> ^ <* x1 *> ;
MW is_closed_on IExec ( I , P , s ) , P & M is_halting_on s , P ;
DataLoc ( t3 . a , 4 ) = intpos ( 0 + 4 ) .= intpos ( 0 + 4 ) ;
x + y < - x + y & |. x - y .| = - x + y ;
LIN c , q , b & LIN c , q , c & LIN c , q , c ;
f^ . ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + z1 ) .= x1 + ( y1 + z1 ) ;
f8 . a = f8 . a & f8 . a in InputVertices S & f8 . b in InputVertices S ;
( p `1 ) ^2 / ( ( E-max C ) `1 ) ^2 <= ( ( E-max C ) `1 ) ^2 / ( ( E-max C ) `1 ) ^2 ;
set R8 = Cage ( C , n ) \circlearrowleft E8 , R8 = Cage ( C , n ) ;
( p `1 ) ^2 >= ( ( E-max C ) `1 ) ^2 / ( ( E-max C ) `1 ) ^2 ;
consider p such that p = p9 and s1 < p and p in P and p in P and p in P ;
|. ( f /* ( s * F ) ) . l - G . l .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len Line ( N , k + 1 ) = width N & len N = width N ;
f1 /* s1 is convergent & f2 /* s1 is convergent & lim ( f1 /* s1 ) = x0 ;
f . x1 = x1 & f . y1 = y1 & f . y2 = y2 ;
len f <= len f + 1 & len f + 1 <> 0 & len f + 1 <> 0 ;
dom ( Proj ( i , n ) * s ) = REAL m .= REAL m ;
n = k * ( 2 * t ) + ( n mod 2 ) ;
dom B = 2 -tuples_on the carrier of V , { {} } :] ;
consider r such that r _|_ a and r _|_ x and r _|_ y ;
reconsider B1 = the carrier of Y1 , B2 = the carrier of Y2 as Subset of X ;
1 in the carrier of [. 1 / 2 , 1 .] & 1 in [. 1 / 2 , 1 .] ;
for L being complete LATTICE holds \mathclose { \rm \hbox { - } \rm \hbox { - } ] is isomorphic
[ gi , gj ] in Ij \ Ij \ Ij \ Ij ;
set S2 = 1GateCircStr ( x , y , c ) , S2 = 1GateCircStr ( y , c , d ) ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and f1 is_differentiable_in x0 and f2 is_differentiable_in x0 ;
reconsider y = ( a ` ) / ( F ` ) as Element of L ;
dom s = { 1 , 2 , 3 } & s . 1 = d1 & s . 2 = d2 ;
( min ( g , ( 1 - g ) ) . c <= h . c ;
set GG2 = the subgraph of G , v = the Vertex of G , a = the Vertex of G , b = the Vertex of G , c = the Vertex of G , d = the Vertex of G , e = the Vertex of G ;
reconsider g = f as PartFunc of REAL , \langle REAL-NS n *> , REAL-NS n , REAL-NS n ;
|. s1 . m - ( p - p ) .| < d / ( p - q ) ;
for x being element st x in ~ ( u ) holds x in ~ ( t )
P = the carrier of ( TOP-REAL n ) | P & Q = ( TOP-REAL n ) | Q ;
assume that p10 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p11 ) and LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = { p1 } ;
( 0. X \ x ) |^ ( m * ( k + 1 ) ) = 0. X ;
let g be Element of Hom ( cod f , dom f ) ;
2 * a * b + ( 2 * c ) * d <= 2 * C1 * C2 ;
let f , g , h be Point of the complex normed space of X , Y be set ;
set h = Hom ( a , g (*) f ) ;
then Seg ( n ) | Seg m = idseq ( m ) & m <= n ;
H * ( g " * a ) in the right of H * ( g " * a ) ;
x in dom ( ( - 1 ) (#) ( cos * sin ) `| Z ) & x in dom ( ( - 1 ) (#) cos ) ;
cell ( G , i1 , j2 -' 1 ) misses C & not ( G * ( i1 , j1 -' 1 ) ) misses C ;
LE q2 , q1 , P & LE q2 , q2 , P & LE q2 , q2 , P ;
attr B is an component of A means : Def6 : B c= BDD A & B c= BDD B ;
deffunc D ( set , set ) = union rng $2 & $2 = union rng $2 ;
n + - n < len ( p + - n ) + ( - n ) ;
attr a <> 0. K means : Def6 : the_rank_of ( M * a ) = the_rank_of ( a * M ) ;
consider j such that j in dom \mathbb Z and I = len \mathbb Z + j and len I = len k + j ;
consider x1 such that z in x1 and x1 in PK and x = [ x1 , x2 ] ;
for n ex r being Element of REAL st X [ n , r ]
set C1 = Comput ( P2 , s2 , i + 1 ) , C1 = Comput ( P2 , s2 , i + 1 ) , C1 = P2 ;
set cp1 = 3 / ( b , c ) , cp2 = 2 / ( b , c ) , cp1 = 3 / ( b , c ) , cp2 = - 1 / ( b , c ) , cp1 = - 1 / ( b - c ) , cp2 = - 1 / (
conv @ W c= union ( F .: ( E " ( W " ( W " ( W " ( W " ( W " ( W " ( W ) ) ) ) ) ) ) ) ;
1 in [. - 1 , 1 .] /\ dom ( ( arccot ) * ( arccot ) ) ;
r3 <= s1 + ( s2 - s1 ) / ( s2 - s1 ) & s1 <= s2 + ( s2 - s1 ) / ( s2 - s1 ) ;
dom ( f (#) f3 ) = dom f /\ dom f3 .= dom ( f (#) f3 ) /\ dom f3 .= dom ( f (#) f3 ) ;
dom ( f (#) G ) = dom ( l (#) F ) /\ Seg k .= Seg k ;
rng ( s ^\ k ) c= dom f1 \ { x0 } & rng ( s ^\ k ) c= dom f2 \ { x0 } ;
reconsider g9 = gp as Point of TOP-REAL n , p be Point of TOP-REAL n , x be Point of TOP-REAL n ;
( T * h . s ) . x = T . ( h . s . x ) ;
I . ( L . ( J . x ) ) = ( I * L ) . ( J . x ) ;
y in dom \vert *> & ( Frege ( A ) ) . ( ( Frege ( A ) ) . o ) = ( ( Frege ( A ) ) . o ) ;
for I being non degenerated commutative commutative commutative commutative commutative commutative associative non empty doubleLoopStr holds I is commutative
set s2 = s +* Initialize ( ( intloc 0 ) .--> 1 ) , P1 = P +* Start-At ( 0 , SCM+FSA ) ;
P1 /. IC Comput ( P1 , s1 , k ) = P1 . IC Comput ( P2 , s2 , k ) .= P1 . IC Comput ( P2 , s2 , k ) ;
lim S1 in the carrier of [. a , b .] & lim s1 in [. a , b .] ;
v . ( l-13 . i ) = ( v *' ( lw ) ) . i .= ( v *' ( lw ) ) . i ;
consider n being element such that n in NAT and x = ( sn -FanMorphE ) . n ;
consider x being Element of c such that F1 . x <> F2 ( x ) and x in F1 ( x ) ;
Funcs ( X , 0 , x1 , x2 , x3 , x4 , x1 , x2 , x3 , x4 , x1 , x2 , x3 , x4 , x1 , x2 , x3 , x4 , x1 , x2 , x3 , x4 , x1 , x2 , x3 , x4 , x1 , x2 , x3 , x4 , x1 , x2 , x4 ,
j + ( 2 * ( k + 1 ) ) + m1 > j + ( 2 * ( k + 1 ) ) ;
{ s , t } on A3 & { s , t } on A3 & { s , t } on B1 ;
n1 > len crossover ( p2 , p1 , n1 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n4 , n4 , n4 , p2 , n3 , n4 , n4 , p2 , n3 , n3 , n4 , p2 , n3 , n3 , n4 , n3 , n4 , p2 , n3 , n3 , n4
( ( ( ( ( g2 ) ) * ( HT ( g2 , T ) ) ) , T ) ) . ( HT ( g2 , T ) ) = 0. L ;
then H , H1 , H2 are_isomorphic & ( H , H1 ) / ( x , H2 ) / ( x , H2 ) is that H1 , H2 / ( x , H2 ) / ( x , H2 ) are_isomorphic ;
( ( N-min L~ f ) .. ( f /^ 1 ) ) .. ( f /^ 1 ) > 1 & ( f /^ 1 ) .. ( f /^ 1 ) > 1 ;
]. s , 1 .[ = ]. s , 2 .] /\ [. 0 , 1 .] ;
x1 in [#] ( ( TOP-REAL 2 ) | L~ g ) & x2 in [#] ( ( TOP-REAL 2 ) | L~ g ) ;
let f1 , f2 be continuous PartFunc of REAL , REAL , x be Point of S , y be Point of S , r be Real ;
DigA ( ta1 , za2 ) is Element of k -tuples_on k -tuples_on k , k -tuples_on k , k ) ;
I \mathop { \rm 22j } = d3j & I \mathop { \rm \hbox { - } tree } = k2 ( k ) ;
u9 ~ = { [ a , u9 ] , u9 = { [ a , u9 ] } ;
( w | p ) | ( p | ( w | w ) ) = p ;
consider u2 such that u2 in W2 and x = v + u2 and u = v + u2 and u1 in W2 ;
for y st y in rng F ex n st y = a |^ n & a <= n
dom ( ( g * ( ( f * ( f \mathbb V ) ) | K ) ) | K ) = K ;
ex x being element st x in ( ( [#] U0 ) \/ A ) . s & x in ( ( [#] U0 ) \/ A ) . s ;
ex x being element st x in ( ( Let ( O ) \/ A ) \/ B ) . s & x in ( ( the Sorts of O ) \/ B ) . s ;
f . x in the carrier of [. - r , r .] & f . x in [. - r , 1 .] ;
( the carrier of X1 union X2 ) /\ ( the carrier of X2 ) <> {} implies ( the carrier of X1 union X2 ) /\ ( the carrier of X2 ) <> {}
L1 /\ LSeg ( p10 , p2 ) c= { p10 } /\ { p10 } ;
( b + ( be - be ) ) / 2 in { r : a < r & r < b + ( be - be ) } ;
ex_sup_of { x , y } , L & x "\/" y = sup { x , y } ;
for x being element st x in X ex u being element st P [ x , u ]
consider z being Point of G8 such that z = y and P [ z ] and z in A and z in B ;
( the sequence of \overline ( seq ) ) . ( n + 1 ) <= e ;
len ( w ^ w2 ) + 1 = len w + 2 + 1 .= len w + 2 ;
assume q in the carrier of ( TOP-REAL 2 ) | K1 & q in the carrier of ( TOP-REAL 2 ) | K1 ;
f | E-4 ` = g | E-4 ` .= ( g | E-4 ) | E-4 ` .= ( g | E-4 ) | E-4 ;
reconsider i1 = x1 , i2 = x2 , j2 = x3 as Element of NAT ;
( a * A * B ) / ( a * A ) = ( a * ( A * B ) ) / ( a * B ) ;
assume ex x0 being Element of NAT st f / ( n + 1 ) is \mathop of x0 & f . x0 = g . x0 ;
Seg len ( ( for f2 being FinSequence of D ) * st len ( ( f | 1 ) ^ <* x *> ) = dom ( ( f | 1 ) ^ <* x *> ) holds ( ( f | 1 ) ^ <* x *> ) ^ <* x *> = <* x *>
( Complement A1 ) . m c= ( ( Complement A1 ) . n ) . ( ( Complement A1 ) . m ) ;
f1 . p = p9 & g1 . p = d & g1 . p = d & g1 . p = d ;
FinS ( F , Y ) = FinS ( F , dom ( F | Y ) ) .= FinS ( F , Y ) ;
( x | y ) | z = z | ( y | x ) ;
( |. ( |. x .| ) |^ n ) / ( n + 1 ) .| <= ( ( |. ( r .| ) |^ n ) / ( n + 1 ) ) ;
Sum ( F ) = Sum f & dom ( F ) = dom g & rng ( F ) c= dom g ;
assume for x , y being set st x in Y & y in Y holds x /\ y in Y ;
assume that W1 is Subspace of W2 and W2 is Subspace of W1 and ( W1 is Subspace of W2 ) and ( W2 is Subspace of W1 ) ;
||. t-15 . x - lim ( ] = ||. x9 - x ) * ( ||. x9 - x .|| ) ;
assume that i in dom D and f | A is lower and g | A is lower and g | A is lower ;
( ( p `2 ) - cn ) / ( 1 + cn ) <= ( - 1 ) / ( 1 + cn ) ;
g | Sphere ( p , r ) = id ( the carrier of TOP-REAL 2 ) .= id ( the carrier of TOP-REAL 2 ) ;
set N8 = ( N-min L~ Cage ( C , n ) ) .. Cage ( C , n ) ;
for T being non empty TopSpace holds T is countable iff T is countable & the TopStruct of T is countable
width B |-> 0. K = Line ( B , i ) .= Line ( B , i ) .= B * ( i , j ) ;
pred a <> 0 means : Def: ( A \ B ) Y. = ( A Y. a ) \ ( B Y. a ) ;
then f is_differentiable on u0 & pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 & pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 ;
assume that a > 0 and a <> 1 and b > 0 and b <> 0 and c > 0 and d > 0 ;
w1 , w2 in Lin { w1 , w2 } & w2 in Lin { w1 , w2 } ;
p2 /. IC Comput ( p2 , s2 , k ) = p2 . IC Comput ( p2 , s2 , k ) .= ( IC Comput ( p2 , s2 , k ) ) .= ( IC Comput ( p2 , s2 , k ) ) ;
ind ( T-10 | b ) = ind b .= ind b .= ind b - ind b .= ind b - ind b ;
[ a , A ] in the Partial_Sums of G_ ( k , X ) & [ a , A ] in the InternalRel of X ;
m in ( the Arrows of C ) . ( o1 , o2 ) & ( the Arrows of C ) . ( o1 , o2 ) = ( the Arrows of C ) . ( o1 , o2 ) ;
( iff Ex ( a , CompF ( PA , G ) ) ) . z = TRUE & ( a 'imp' PA ) . z = TRUE ;
reconsider phi = phi /. 11 , phi = phi /. ( 11 + 1 ) as Element of from W , D ;
len s1 - ( len s2 - 1 ) * ( len s1 - 1 ) > 0 + 1 ;
\delta ( D ) * ( f . ( upper_bound A ) - f . ( lower_bound A ) ) < r ;
[ f21 , f22 ] in the carrier of [: A , B :] & [ f22 , f22 ] in the carrier of [: A , B :] ;
the carrier of ( TOP-REAL 2 ) | K1 = K1 & the carrier of ( TOP-REAL 2 ) | K1 = K1 ;
consider z being element such that z in dom g2 and p = g2 . z and x = g2 . z ;
[#] ( V1 ) = { 0. ( V1 , V1 ) } .= the carrier of ( V1 , V1 ) .= the carrier of ( V1 | V1 ) ;
consider P2 being FinSequence such that rng P2 = M and P2 is one-to-one and len P2 = len M and P . len P2 = M * ( i , j ) ;
assume that x1 in dom ( f | X ) and ||. x1 - x0 .|| < s and ||. x1 - x0 .|| < s ;
h1 = f ^ ( <* p3 *> ^ <* p *> ) .= h ^ <* p *> .= h ^ <* p *> ;
c /. ( |[ b , c ]| ) = c /. ( |[ a , c ]| ) .= c /. ( |[ a , c ]| ) .= c /. ( |[ a , c ]| ) ;
reconsider t1 = p1 , t2 = p2 , t2 = p as Term of C , V ;
( 1 - 2 ) * ( 1 - 2 ) in the carrier of [. 1 - 2 , 1 .] ;
ex W being Subset of X st p in W & W is open & h .: W c= V ;
( h . p1 ) `2 = C * ( ( p1 `2 ) ) + D * ( ( p1 `2 ) ) `2 ;
R . b b / 2 = 2 * \cal b - b .= 2 * b - b .= 2 * b ;
consider ] such that B = ( 1 - 0 ) * C + ( 0 - 0 ) * C and 0 <= 1 ;
dom g = dom ( ( the Sorts of A ) * ( the_arity_of o ) ) .= dom ( ( the Sorts of A ) * ( the_arity_of o ) ) ;
[ P . ( l1 , P . ( l1 , k ) ) , P . ( l1 , k ) ] in => ( T . ( l1 , k ) ) ;
set s2 = Initialize s , P2 = P +* I , s2 = P +* stop I , P2 = P +* stop I , s2 = LifeSpan ( P2 , s2 ) , P2 = P +* stop I , s2 = Comput ( P2 , s2 , 1 ) , P2 = Comput ( P2 , s2 , 1 ) , P2 = P2 +* stop I ;
reconsider M = mid ( z , i2 , i1 ) , i1 = len ( f | i2 ) as Element of TOP-REAL 2 ;
y in product ( ( the carrier of J ) +* ( V , { 1 } ) ) ;
1 / ( |[ 0 , 1 ]| ) = 1 & 0 / ( |[ 0 , 1 ]| ) = 0 & 1 / ( |[ 0 , 1 ]| ) = 0 ;
assume x in the left & x in the left of g or x in the left of g & y in the right of g ;
consider M being strict Subgroup of A such that a = M and T is SubSpace of M and M is Subgroup of M ;
for x st x in Z holds ( ( ( id Z ) (#) f ) `| Z ) . x <> 0 & f . x = 1 / ( x + a )
len ( W1 + W2 ) + len ( W2 + W3 ) = 1 + len ( W2 + W3 ) .= len ( W1 + W2 ) + len ( W2 + W3 ) ;
reconsider h1 = ( vseq . n - tf1 ) . t - tf1 . t as Lipschitzian LinearOperator of X , Y ;
( - ( len p + q ) ) + 1 in dom ( p + q ) ;
assume that s2 is conjunctive and F in the carrier of ( the carrier of ( TOP-REAL 2 ) | A ) and F is the carrier of ( TOP-REAL 2 ) | A ;
( ( ( for x , y , z ) ) * ( x , y , z ) ) * ( x , y , z ) = gcd ( x , y , z ) ;
for u being element st u in Bags n holds ( p + m ) . u = p . u + m . u
for B being Subset of u-5 st B in E holds A = B or A misses B or A misses B ;
ex a being Point of X st a in A & A /\ Cl { y } = { a } ;
set W2 = tree ( p ) \/ W1 \/ W2 ;
x in { X where X is Ideal of L : for x being Element of L holds x in X iff x in X } ;
the carrier of W1 /\ W2 c= the carrier of ( W1 + W2 ) & the carrier of ( W1 + W2 ) c= the carrier of ( W1 + W2 ) ;
( 1 + a ) * ( a + b ) * id ( a + b ) = ( 1 + a ) * ( b + a ) ;
( dom ( X --> f ) ) . x = ( X --> dom f ) . x .= ( X --> f ) . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) ;
p => ( q => r ) => ( p => q ) in TAUT ( A ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
- 1 + 1 <= ( - ( 2 |^ ( n -' m ) ) + 1 ) / 2 + 1 ;
( reproj ( 1 , z0 ) ) . x in dom ( ( f1 (#) f2 ) `| Z ) /\ dom ( f1 (#) f2 ) ;
assume that b1 . r = { c1 } and b2 . r = { c2 . r } and b2 . r = c2 . r ;
ex P st a1 on P & a2 on P & a3 on P & b1 on P & b2 on P & b2 on P & b1 on P & b2 on P ;
reconsider gf = g * f * f * g * h " as strict Subgroup of X , Y ;
consider v1 being Element of T such that Q = ( downarrow v1 ) ` and ( downarrow v2 ) ` = ( downarrow v2 ) ` ;
n in { i where i is Nat : i < n1 + 1 & n1 < n + 1 & n < m + 1 & m in dom f } ;
( F * ( i , j ) ) `2 >= ( ( F * ( m , k ) ) `2 ) / ( ( m , k ) `2 ) ;
assume K1 = { p : ( p `1 / |. p .| - sn ) / ( 1 + sn ) >= sn & p <> 0. TOP-REAL 2 } ;
ConsecutiveSet ( A , succ O1 ) = ( ConsecutiveSet ( A , O1 ) ) * ( ConsecutiveSet ( A , O1 ) ) ;
set I1 = Macro SubFrom ( a , intloc 0 , intloc 0 , intloc 0 , 1 ) , I2 = SubFrom ( a , intloc 0 , intloc 0 , intloc 0 ) , I2 = Macro ( a , intloc 0 , intloc 0 ) , I2 = intloc 0 ;
for i being Nat st 1 < i & i < len z holds z /. i <> z /. 1 & z /. i <> z /. 1
X c= ( the carrier of L1 ) \/ ( the carrier of L2 ) & ( the carrier of L1 ) \/ ( the carrier of L2 ) c= the carrier of L2 ;
consider x9 being Element of GF ( p ) such that x9 |^ 2 = a and x9 |^ 2 = a |^ 3 ;
reconsider ek = ek , fk = fk , fk = f-5 as Element of D * ;
ex O being set st O in S & C1 c= O & M . O = 0. ( X , Y ) ;
consider n being Nat such that for m being Nat st n <= m holds S . m in U1 and S . m in U2 ;
f * g * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) * g ) . x ;
defpred P [ Nat ] means A + succ $1 = succ A + 1 & A = succ $1 + 1 & A = succ $1 ;
the left } - ( - g ) = the left } - ( ( - g ) * ( - g ) ) .= the left of g ;
reconsider pp = x , pp = y , pp = z as Point of Euclid 2 , p = y , q = z , r = z ;
consider g2 such that g2 = y and x <= g2 & g2 <= x0 and g2 <= x0 and x0 <= g2 & g2 <= x0 and g2 <= x0 ;
for n being Element of NAT ex r being Element of REAL st X [ n , r , n ]
len ( x2 ^ y2 ) = len x2 + len y2 .= len x2 + len y2 .= len ( x2 ^ y2 ) + len y2 .= len ( x2 ^ y2 ) + len ( y2 ^ z2 ) ;
for x being element st x in X holds x in the set of \HM { the } \HM { set } , where n is Element of NAT : n <= len the _ of m } , n )
LSeg ( p10 , p2 ) /\ LSeg ( p1 , p11 ) = {} & LSeg ( p1 , p2 ) /\ LSeg ( p1 , p11 ) = {} ;
func ex ex X being set st ( for x being set holds x in X iff x in ( the carrier of X ) \/ ( id X ) ) ;
len ( mid ( Cg , len Cg , 1 ) ) <= len ( Cg | ( len Cg -' 1 ) ) ;
attr K is L means : Def21 : a <> 0. K & v . ( a |^ i ) = i * v . a ;
consider o being OperSymbol of S such that t `1 . {} = [ o , the carrier of S ] and o `1 = [ o , the carrier of S ] ;
for x st x in X ex y st x c= y & y in X & y is Ordinal of f . x
IC Comput ( P-6 , s1 , k ) in dom ( ( n + 1 ) --> ( k + 1 ) ) ;
pred q < s & r < s & s < q implies ]. p , q .] c= ]. p , q .] ;
consider c being Element of Class ( f , c ) such that Y = ( F . c ) * ( F . c ) and c in X ;
the ResultSort of S2 = id ( the carrier of S1 ) & the ResultSort of S2 = id ( the carrier of S2 ) ;
set yy = [ <* y , z *> , f2 ] , f3 = [ <* z , x *> , f3 ] ;
assume x in dom ( ( ( ( id Z ) (#) ( ( arccot ) * ( arccot ) ) `| Z ) ) /\ dom ( ( id Z ) (#) ( ( id Z ) * ( arccot ) ) `| Z ) ;
r-7 in Int cell ( GoB f , i , GoB f ) \ { ( GoB f ) * ( i , 1 ) } \/ L~ f \/ L~ f ;
( q `2 ) ^2 >= ( ( Cage ( C , n ) ) * ( i + 1 , 1 ) ) `2 ;
set Y = { a "/\" a ` : a in X } ;
i - len f <= len f + len ( f /^ 1 ) - len ( f /^ 1 ) + len ( f /^ 1 ) - len ( f /^ 1 ) ;
for n ex x st x in N & x in N1 & h . n = x- ( x0 ) & h . n = x- ( x0 )
set s0 = ( \mathop { \it SCMPDS } , p , s ) . i , s1 = ( \mathop { \it true } ) . i , s2 = ( \mathop { \it true } ) . i , s2 = ( \mathop { \it true } ) . i , s2 = ( \mathop { \it true } ) . i , s2 = ( \mathop { \it true } ) . i ;
( for k holds p . k = 1 or p . 0 = - 1 ) or ( p . 0 = 1 & p . 1 = - 1 ) implies p is one-to-one
u + Sum L-18 in ( U \ { u } ) \/ { u + Sum L-18 } ;
consider x9 being set such that x in x9 and x9 in V and x9 in V and x = [ x9 , y9 ] ;
( p ^ ( q | k ) ) . m = ( q | k ) . ( mm ( m ) ) . ( mm ( m ) ) ;
g + h = gg + h1 & ||. g + h , X + g - h .|| = g + h - h ;
L1 is distributive & L2 is distributive implies L1 [: L1 , L2 :] is distributive & L2 is distributive & L1 [: L2 , L2 :] is distributive
pred x in rng f & y in rng ( f /^ x ) & f . x = f . ( x .. f ) ;
assume that 1 < p and p + ( 1 - p ) * q = 1 and 0 <= a and a <= b and b <= 0 ;
F* ( f , <* the carrier of C *> ) = rpoly ( 1 , the carrier of C ) *' t + 0. L .= 0. L ;
for X being set , A being Subset of X holds A ` = {} implies A = {} & A = {} & A = {}
( ( N-min X ) `1 ) ^2 <= ( ( ( N-min X ) `1 ) ^2 ) ^2 & ( ( ( N-min X ) `2 ) ^2 <= ( ( ( N-min X ) `2 ) `1 ) ^2 ;
for c being Element of the ] Element of the \cal A , a being Element of the \cal A holds c <> a implies c = a
s1 . GBP = ( Exec ( i2 , s2 ) ) . ( ( GBP + 1 ) + 1 ) .= ( Exec ( i2 , s1 ) ) . ( ( GBP + 1 ) + 1 ) .= 0 ;
for a , b being Real holds [ a , b ] in ( y >= 0 iff b >= 0 ) & ( b >= 0 implies a >= 0 ) implies b = 0
for x , y being Element of X holds x ` \ y ` = ( x \ y ) ` & y \ x = ( x \ y ) `
mode BCK-algebra of i , j , m , n , m , n , m , k being BCK-algebra of i , j , n , m , k being Nat ;
set x2 = |( ( Re ( y ) ) , ( Im ( y ) ) , ( Im ( y ) ) )| ;
[ y , x ] in dom u5 & ( u . ( y , x ) ) = g . ( y , x ) ;
]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .] c= A & upper_bound divset ( D , k ) = upper_bound A ;
0 <= \delta ( S2 . n ) & |. \delta ( S2 . n ) - 0 .| < ( e / 2 ) / 2 ;
( - ( q `1 / |. q .| - sn ) ) / ( 1 + sn ) <= ( - ( q `1 / |. q .| - sn ) ) / ( 1 + sn ) ;
set A = ( 2 / b ) / b ;
for x , y being set st x in R" & y in R" holds x , y \hbox { x , y } are_equipotent
deffunc FG2 ( Nat ) = b . ( $1 + 1 ) * ( M * G ) . $1 * ( M * ( $1 , 1 ) ) ;
for s being element holds s in contradiction ( f 'or' g ) iff s in non empty & s in non empty s \/ ^2
for S being non empty non void holds for S being non void non empty ManySortedSign st S is connected holds S is connected
max ( ( degree ( K ) ) , ( degree ( K ) ) ) >= 0 & ( degree ( K ) ) . ( ( degree ( K ) ) ) >= 0 ;
consider n1 being Nat such that for k holds seq . ( n1 + k ) < r + s and not ( seq is convergent & lim seq = x0 ) ;
Lin ( A /\ B ) is Subspace of Lin ( A /\ B ) & the carrier of Lin ( B ) = the carrier of Lin ( B ) ;
set n-15 = nnw '&' ( M . x qua Element of BOOLEAN ) , nw = M . ( M . x qua Element of BOOLEAN ) , nw = M . ( M . x ) ;
f " V in ( ex X st V in SL ) & f " V in D & f " V in D & f " V in D & f .: V c= D
rng ( ( a , b ) \mathbin { { + } \cdot } ( 1 , b ) ) c= { a , c , b } ;
consider y being many of G1 such that y ` = y and dom y ` = dom ( W ` ) and rng y ` = WG1 ;
dom ( 1 / f ) /\ ]. x0 - r , x0 + r .[ c= ]. x0 - r , x0 + r .[ ;
an in cell ( i , j , n , r ) & r is Element of [#] ( 1. ( K , n , j ) ) ;
v ^ ( n |-> 0 ) in Lin rng ( ( B | ( n -' 1 ) ) | ( n -' 1 ) ) ;
ex a , k1 , k2 st i = a /. k1 & j = a /. k1 & i = b /. k2 & j = b /. k2 ;
t . NAT = ( NAT .--> ( i1 -' 1 ) ) . ( ( i1 -' 1 ) + 1 ) .= ( i1 -' 1 ) .= ( i1 -' 1 ) + ( i1 -' 1 ) .= ( i1 -' 1 ) ;
assume that F is bbfamily and rng p = F and dom p = Seg ( n + 1 ) and rng p = Seg ( n + 1 ) and for n being Nat holds p . n = F ( n + 1 ) ;
not LIN b , b9 , a & not LIN b , a , c & LIN b , c , a
( L1 or L2 ) \& O c= ( L1 \HM { the } \HM { carrier of L1 } ) \HM { the } \HM { carrier of L1 } ) & ( L1 is finite implies L1 is finite ;
consider F be ManySortedFunction of E , REAL such that for d being Element of E holds F . d = F ( d ) and for d being Element of E holds F . d = G ( d ) ;
consider a , b such that a * ( \rrangle = b * ( -w ) and 0 < a and 0 < b and 0 < b ;
defpred P [ FinSequence of D , FinSequence of D ] means |. Sum ( $1 ) - ( 1 - ( $1 + 1 ) ) .| <= Sum ( |. $1 - 1 ) ;
u = cos ( x , y ) . v * x + cos ( cos ( x , y ) . v ) .= cos ( x , y ) . v * x .= v ;
dist ( ( seq . n ) + x , x + g ) <= dist ( ( seq . n ) , g ) + 0 ;
P [ p , |. p .| : p in [: the carrier of A , the carrier of A :] & id the carrier of A = id the carrier of A
consider X being Subset of CQC-WFF ( Al ( ) ) such that X c= Y and X is finite and X is finite ;
|. b .| * |. eval ( f , z ) .| >= |. b .| * |. eval ( f , z ) .| ;
1 < ( ( N-min L~ Cage ( C , n ) ) .. Cage ( C , n ) ) .. Cage ( C , n ) + 1 - 1 ;
l in { l1 where l1 is Real : g <= l1 & l1 <= h & g . l1 = ( h . l1 ) * ( l1 + 1 ) & l <= h . l1 & l <= h . l1 + 1 } ;
( Partial_Sums ( G ) . n ) vol <= ( Partial_Sums ( G ) . n ) * vol ( ( G . n ) ) ;
f . y = x .= x * ( 1_ L ) .= x * ( ( power L ) * ( y , 0 ) ) .= x * ( ( power L ) . ( y , 0 ) ) ;
NIC ( CurInstr ( <% i1 %> , i ) , \ { i1 } ) = { i1 , i2 } \/ { i1 , i2 } .= { i1 , i2 } ;
LSeg ( p10 , p2 ) /\ LSeg ( p1 , p11 ) = { p1 } /\ LSeg ( p1 , p2 ) \/ { p2 } ;
product ( ( the carrier of I-15 ) +* ( i , { 1 } ) ) in Z & product ( ( i , { 1 } ) +* ( i , { 1 } ) ) in Z ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) | ( the carrier of S1 ) .= Following ( s1 , n ) ;
( W-bound Q1 - 1 ) / 2 <= ( q `1 ) / 2 & ( q `2 ) / 2 <= ( q `2 ) / 2 ;
f /. i2 <> f /. ( len f -' 1 + len g -' 1 , f /. ( len f -' 1 + 1 ) ) ;
M , v / ( x. 3 , x. 4 ) / ( x. 0 , x. 4 ) |= H / ( x. 4 , x. 4 ) ;
len ( ( P ^ ) ^ <* A *> ) in dom ( ( P ^ ) ^ <* A *> ) .= dom ( ( P ^ ) ^ <* A *> ) .= dom ( P ^ <* A *> ) ;
A |^ ( m , n ) c= ( A |^ m , n ) |^ ( k , n ) & A |^ ( k , n ) |^ ( k + 1 ) = A |^ ( k , n ) ;
R |^ n \ { q : |. q - p .| < a & |. q - p .| < a } c= { q1 : q1 in R }
consider n1 being element such that n1 in dom p1 and y1 = p1 . n1 and p1 . n1 = p1 . n1 and p2 . n1 = p2 . n1 ;
consider X being set such that X in Q and for Z being set st Z in Q & Z in X holds X in Z ;
CurInstr ( P3 , Comput ( P3 , s2 , l ) ) <> halt SCMPDS .= halt SCMPDS .= ( halt SCMPDS ) . IC Comput ( P3 , s2 , l ) .= halt SCMPDS ;
for v being VECTOR of l1 holds ||. v .|| = upper_bound rng |. ( ||. v .|| ) . v - ( ||. v .|| ) . v .|
for phi st phi in X holds not phi in X & not phi in X & phi in X & not phi in X
rng ( Sgm dom ( f | ( dom f \ dom f ) ) ) c= dom ( f | ( dom f \ dom f ) ) .= dom ( f | ( dom f \ dom f ) ) ;
ex c being FinSequence of D ( ) st len c = k & ( P [ c ] implies a = c ) & ( P [ c ] implies a = c ) ;
( the_arity_of ( a , b , c ) ) = <* ( F . ( b , c ) ) *> .= <* ( F . ( b , c ) ) *> ;
consider f1 be Function of the carrier of X , REAL such that f1 = |. f .| and f1 is continuous and f1 is continuous ;
a1 = b1 & a2 = b2 or a1 = b2 & b1 = b2 & b2 = b3 & b1 = b2 & b2 = b3 & b3 = b3 & b2 = b3 ;
D2 . indx ( D2 , D1 , n1 ) = D1 . indx ( D2 , D1 , n1 ) .= D1 . indx ( D2 , D1 , n1 ) .= D1 . indx ( D2 , D1 , n1 ) ;
f . ( ||. r .|| ) = ||. |[ r , s ]| . 1 .= <* r , s ]| . 1 .= <* r *> . 1 .= <* r *> . 1 .= r ;
consider n being Nat such that for m being Nat st n <= m holds Cseq . m = Cseq . m and Cseq . n = Cseq . m ;
consider d being Real such that for a , b being Real st a in X & b in Y holds a <= d & b <= d ;
||. L /. h - ( K * |. h .| ) + ( K * |. h .| ) + ( K * |. h .| ) <= x0 + ( K * |. h .| ) ;
attr F is commutative associative means : Def: for b being Element of X holds F \hbox { b } = f . b ;
p = - 1 * ( p0 + 0. TOP-REAL 2 ) + 0. TOP-REAL 2 .= 1 * ( ( - 1 ) * ( p2 + p1 ) ) .= 1 * ( ( - 1 ) * ( p2 + p1 ) ) .= 1 * ( ( - 1 ) * ( p2 + p1 ) ) .= 1 * ( ( - 1 ) * ( p2 + p1 ) ) .= ( - 1 ) * ( p2 + p1 ) ;
consider z1 such that b , x3 , x3 , x1 , x2 , x3 , x3 , x4 is_collinear and o , x1 , x2 , x3 is_collinear and o <> x2 ;
consider i such that Arg ( ( Rotate ( s ) ) . q ) = s + Arg q and 0 < ( 2 * PI * i ) ;
consider g such that g is one-to-one and dom g = card ( f . x ) and rng g c= f . x and g is one-to-one ;
assume that A = P2 \/ Q2 and P <> {} and Q = {} and Q = {} and Q = {} and Q = {} and Q = {} and Q = {} and Q = {} and Q is closed and Q is closed and Q is closed and Q is closed and Q is closed and Q is closed and Q is closed and Q is closed ;
attr F is associative means : Def: F .: ( F .: ( f , g ) , h ) = F .: ( f , g ) ;
ex x being Element of NAT st m = x `1 & x in z & x in { i } or m in { i } & m in { i } ;
consider k2 being Nat such that k2 in dom P-2 and l = P-2 . ( k + 1 ) and ( k + 1 ) <= ( k + 1 ) ;
seq = r * seq implies for n holds seq . n = r * seq . n & seq is convergent & lim seq = r * seq . n
F1 . [ ( id a ) * [ a , [ a , a ] ] = [ f * ( id a ) , f * ( id a ) ] ;
{ p } "\/" D2 = { p "\/" y where y is Element of L : y in D2 & p in D1 & y in D2 } ;
consider z being element such that z in dom ( ( dom _ \kappa F ( \kappa ) ) * ( i , 1 ) ) and ( ( dom _ \kappa F ( \kappa ) ) . z = y ;
for x , y being element st x in dom f & y in dom f & f . x = f . y holds x = y
Int cell ( G , i , j ) = { |[ r , s ]| : r <= G * ( 0 + 1 , 1 ) `1 & G * ( 0 , 1 ) `2 <= s } ;
consider e being element such that e in dom ( T | ( E . 1 ) ) and ( T | ( E . 1 ) ) . e = v ;
( F ` * b1 ) . x = ( ( Mx2Tran ( J , T ) ) * ( BY. , T ) ) . \mathclose ( thesis ) .= ( ( Mx2Tran ( J , T ) ) * ( BY. , j ) ) . j ;
- 1 / ( - 1 ) = ( mD ) . n .= ( mD ) . n .= ( ( - 1 ) (#) D ) . n .= ( ( - 1 ) (#) D ) . n .= ( ( - 1 ) (#) D ) . n .= ( - 1 ) (#) D ;
attr for x being set st x in dom f /\ dom g holds g . x <= f . x & g . x <= f . x ;
len ( f1 . j ) = len ( f2 /. j ) .= len ( f2 /. j ) .= len ( f2 /. j ) .= len ( ( f2 /. j ) ) .= len ( ( f2 /. j ) ) ;
All ( All ( a , A , G ) , B , G ) divides Ex ( All ( a , A , G ) , A , G ) ;
LSeg ( E . k , F . ( k + 1 ) ) c= Cl RightComp Cage ( C , k + 1 ) \/ RightComp Cage ( C , k + 1 ) ;
x \ a |^ m = x \ ( a |^ k * a ) .= ( x \ a ) |^ ( k + 1 ) .= ( x \ a ) |^ ( k + 1 ) ;
k -Y. -inininC = ( commute ( I1 . k ) ) . ( k + 1 ) .= ( commute ( I1 . k ) ) . ( k + 1 ) .= ( ( commute ( I1 . k ) ) . i ) .= ( ( commute ( I1 . k ) ) . i ) .= ( ( commute ( I1 . k ) ) . i ) . i ;
for s being State of A2 holds Following ( s , n ) . ( 0 + 1 ) + ( s . n ) * ( 2 * n + 1 ) is stable ;
for x st x in Z holds f1 . x = a / x ^2 & ( f1 - f2 ) . x = a / x ^2 & ( f1 - f2 ) . x = a / x * ( a - b )
support ( ( support ( n ) ) \/ support ( support ( m ) ) ) c= support ( ( support ( n ) ) \/ support ( ( support ( m ) ) ) ) ;
reconsider t = u as Function of ( the carrier of A ) , ( the carrier of B ) | the carrier of the carrier of C , the carrier of B ;
- ( a * sqrt ( 1 + b ^2 ) ) <= - ( b * sqrt ( 1 + a ^2 ) ) ;
phi /. ( succ b1 ) = g . a & phi /. ( a . a ) = g . ( g . a ) & phi . ( a . a ) = f . ( g . a ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( F ^ <* p *> ) and i = len ( F ^ <* p *> ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , M } = { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 } ;
the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U2 /\ ( U2 "\/" U2 ) .= the Sorts of U2 /\ ( U2 \/ W3 ) .= the Sorts of U2 /\ ( U2 \/ W3 ) ;
( - ( 2 * a ) * ( b - sqrt ( 2 * a ) ) + b ^2 ) ) / ( - ( 2 * a ) ) > 0 ;
consider W00 such that for z being element holds z in W00 iff z in N ~ & not ( ex x being element st x in N & x in N & not ( x in N & x in N & y = [ x , y ] ) ;
assume that ( the Arity of S ) . o = <* a *> and ( the Arity of S ) . o = r and ( the Arity of S ) . o = <* a *> ;
Z = dom ( ( exp_R * ( arccot ) ) `| Z ) /\ dom ( ( arccot * ( arccot ) ) ^2 ) .= dom ( ( exp_R * ( arccot ) ) ^2 ) ;
lim ( f , SS1 ) is convergent & lim ( ( f , SS1 ) . n ) = integral ( f , S ) - lim ( ( f , S ) . n ) ;
( X [ a9 , f ( ) => ( g ( ) => ( x , y ) ) ) => ( ( x1 => x2 ) => ( x2 => x1 ) ) in TAUT ( \rm \hbox { - } WFF } ;
len ( M2 * ( M1 * M2 ) ) = n & width ( M2 * ( M1 * M2 ) ) = n & len ( M2 * ( M1 * M2 ) ) = n ;
attr X1 union X2 is open SubSpace of X & X1 , X2 , x3 , x4 , x4 , x5 , x5 , x5 , 7 , 8 , 8 , a9 , 8 , a9 , 8 , a9 , b9 , c9 , a9 , b9 , c9 be Subset of X ;
for L being lower-bounded antisymmetric RelStr , X being non empty Subset of L holds X "\/" { Top L } = { Top L }
reconsider f-1= ( F . ( ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . b . b ) b . b
consider w being FinSequence of I such that the InitS of M , <* s *> ^ w ^ w ^ w ^ w ^ y ^ y ^ w ^ y ^ w ^ y ^ w ^ y ;
g . ( a |^ 0 ) = g . ( 1_ G ) .= g . ( 1_ G ) .= ( g . ( a * b ) ) * ( g . ( a * b ) ) .= g . ( a * b ) .= ( g . ( a * b ) ) * ( g . ( a * b ) ) ;
assume for i being Nat st i in dom f ex z being Element of L st f . i = rpoly ( 1 , z ) & z in dom z & f . i = rpoly ( 1 , z ) ;
ex L being Subset of X st L = L & for K being Subset of X st K in C & K /\ K <> {} & K /\ L <> {} ;
( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C1 & ( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C1 ;
reconsider o9 = o `1 , p = o `2 , q = o `2 , r = o `2 as Element of TS ( ( the Sorts of A ) * ( the_arity_of o ) ) ;
1 * x1 + ( 0 * x2 + 0 * x3 ) + ( 0 * x2 + 0 * x3 ) = x1 + ( 0 * x2 + 0 * x3 ) .= x1 + ( 0 * x2 + 0 * x3 ) .= x1 + ( 0 * x2 + 0 * x3 ) .= x1 + ( 0 * x2 + 0 * x3 ) .= x1 + ( 0 * x2 + 0 * x3 ) ;
Eg " . 1 = ( ( E qua Function ) qua Function ) . 1 .= ( ( E qua Function ) " ) . 1 .= ( ( E qua Function ) " ) . 1 .= ( ( E qua Function ) " ) . 1 .= ( E qua Function ) . 1 .= ( E " ) . 1 .= ( E " ) . 1 .= ( E " ) . 1 .= ( E " ) . 1 .= ( E " ) . 1 ;
reconsider u1 = the carrier of U1 /\ ( U1 "\/" U2 ) , v1 = the carrier of U2 /\ ( U1 "\/" U2 ) as non empty Subset of U2 ;
( ( x "/\" z ) "\/" ( x "/\" y ) ) "\/" ( z "/\" y ) <= ( x "/\" ( z "/\" y ) ) "\/" ( z "/\" ( x "/\" y ) ) ;
|. f . ( s1 . ( l1 + 1 ) ) - ( s1 . ( l1 + 1 ) ) - ( s1 . ( l1 + 1 ) ) .| < ( 1 - M ) * ( 1 - M ) ;
LSeg ( ( Lower_Seq ( C , n ) ) * ( i , j ) , ( Gauge ( C , n ) ) * ( i + 1 , j ) ) is vertical ;
( f | Z ) /. x - ( f | Z ) /. x0 = L /. ( x- ( x - x0 ) ) + R /. ( x- ( x - x0 ) ) ;
g . c * ( - g . c ) + f . c <= h . c * ( ( - g ) . c ) + f . c ;
( f + g ) | divset ( D , i ) = f | divset ( D , i ) + g | divset ( D , i ) .= f | divset ( D , i ) + g | divset ( D , i ) ;
assume that ColVec2Mx ( f ) in the carrier of Lin ( A ) and ColVec2Mx ( f ) in the carrier of A and f . 1 = width A and f . 1 = Line ( A , 1 ) and f . len f = Line ( A , 1 ) ;
len ( - ( M1 + M2 ) ) = len M1 & width ( - ( M1 + M2 ) ) = width M1 & width ( - ( M1 + M2 ) ) = width M1 ;
for n , i being Nat st i + 1 < n & i < n holds [ i , i + 1 ] in the InternalRel of ( TOP-REAL n ) | ( the carrier of ( TOP-REAL n ) | ( the carrier of ( TOP-REAL n ) | ( the carrier of TOP-REAL n ) ) )
pdiff1 ( f1 , 2 ) is_partial_differentiable_in u0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in u0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in u0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in u0 , 1 & pdiff1 ( f1 , 2 ) is_partial_differentiable_in 1 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in 1 , 2 ;
pred a <> 0 & b <> 0 & Arg a = Arg b & Arg b = Arg a & Arg a = Arg b & Arg b = Arg a & Arg b = Arg b ;
for c being set st not c in [. a , b .] holds not c in Intersection ( the partial of f , a , b ) & not c in Intersection ( the partial of f , a , b )
assume that V1 is linearly-independent and V1 is linearly-independent and V1 = { v + u : v in V1 & u in V1 & v in V1 & u in V1 & v in V1 & u in V1 & v in V1 & u in V1 & v in V1 & u in V1 & v in V1 & u in V1 & v in V1 & v in V1 & u in V1 & v in V1 & v in V1 ;
z * x1 + ( 1 - z ) * x2 in M & z * ( x1 - z ) + ( 1 - z ) * x2 in N ;
rng ( ( ( P qua Function ) " ) * S ) = Seg ( card ( P * S ) ) .= Seg ( card ( P * S ) ) .= Seg ( card ( S * S ) ) .= Seg ( card ( S * S ) ) ;
consider s2 being complex number such that s2 is convergent and b = lim s2 and ( for n holds s2 . n = lim ( s2 . n ) ) & ( for n holds s2 . n = ( lim s2 ) * ( lim s2 ) ) ;
h2 " . n = h2 . n " & 0 < ( - 1 ) / ( n + 1 ) & 0 < ( - 1 ) / ( n + 1 ) & 0 < ( - 1 ) / ( n + 1 ) ;
( Partial_Sums ||. seq .|| ) . m = ||. ( seq . m ) - ( seq . n ) .|| .= ||. ( seq . m ) - ( seq . n ) .|| .= 0 ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b ;
- v = ( - 1 ) * v & - w = ( - 1 ) * v & - w = ( - 1 ) * v + ( - 1 ) * w ;
sup ( ( k .: D ) .: D ) = sup ( ( k .: D ) .: D ) .= sup ( ( k .: D ) .: D ) .= ( k .: D ) .: D .= ( k .: D ) .: D .= ( k .: D ) .: D ;
A |^ ( k , l ) ^ ^^ ( A |^ ( n , .. A ) ) = ( ( A |^ n ) * ( A |^ ( k , .. A ) ) ) ^ ( A |^ ( k , .. A ) ) ;
for R being add-associative right_zeroed right_complementable non empty addLoopStr , I , J being Subset of R holds I + ( J + K ) = ( I + J ) + K
( f . p ) `1 = ( ( p `1 ) ) ^2 + ( ( p `2 ) ) ^2 .= ( ( p `1 ) ) ^2 + ( ( p `2 ) ) ^2 ;
for a , b being non zero Nat st a , b are_congruent_mod p holds ( a * b ) mod p = ( ( a * b ) div p ) + ( ( a * b ) div p )
consider A5 being countable set such that r is countable & A is Element of CQC-WFF ( Al ( ) ) & A5 is ( len A5 ) -element & A5 is ( len A5 ) -element ;
for X being non empty addLoopStr , M being Subset of X , x , y being Point of X st x in M & y in M holds x + y in M + M
{ [ x1 , x2 , x3 , x4 , x5 ] , [ x1 , x2 , x3 , x4 ] } c= { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 } ;
h . ( f . O ) = |[ A * ( f . O ) `1 + B , C * ( f . O ) `2 + D ]| ;
( Gauge ( C , n ) * ( k , i ) ) in L~ Lower_Seq ( C , n ) /\ L~ Lower_Seq ( C , n ) ;
cluster m , n ) -> prime implies ( the carrier of K ) \ ( m , n ) is natural & ( for p being Nat st p in m holds p divides n ) & ( for p being Nat st p in m holds p divides n ) implies ( m divides n ) implies ( m divides n )
( f * F ) . x1 = f . ( F . x1 ) & ( f * F ) . x2 = f . ( F . x2 ) & ( f * F ) . x2 = f . ( F . x2 ) ;
for L being LATTICE , a , b , c being Element of L st a \ b <= c & b \ c <= c holds a \ b <= c
consider b being element such that b in dom ( H / ( ( x , y ) \leftarrow ( x , y ) ) ) and z = H / ( ( x , y ) \leftarrow ( x , y ) ) . b ;
assume that x in dom ( F * g ) and y in dom ( F * g ) and ( F * g ) . x = ( F * g ) . y ;
assume ex e being element st e Joins W . 1 , W . 1 , W . 5 , G & e in W . 3 & W . 4 = W . 3 ;
( ] holds ( exists h h h ) . ( 2 * n ) . x = ( ( h h ) . n ) * ( 2 * n ) . x + ( h . n ) * ( 2 * n ) . x ;
j + 1 = j + ( len h11 + 2 ) .= i + 1 + 1 - 1 .= i + 1 - 1 .= i + 1 - 1 .= i + 1 - 1 ;
( *' ( S *' ) ) . f = *' ( S *' ) . ( ( S *' ) . f ) .= S . ( ( S *' ) . f ) .= S . ( ( S *' ) . f ) .= S . ( ( S *' ) . f ) .= S *' ( ( S *' ) . f ) ;
consider H such that H is one-to-one & rng H = the carrier of L2 and Sum ( L2 * H ) = Sum ( L2 * H ) and Sum ( L1 * H ) = Sum ( L2 * H ) ;
attr R is max means : Def: for p , q st p in R & q in R & p <> q holds ex P st P = R & q = R /. 1 & P is closed ;
dom ( product ( X --> f ) ) = meet ( dom ( X --> f ) ) .= meet ( dom ( X --> f ) ) .= meet ( X --> f ) .= meet ( X --> f ) .= meet ( X --> f ) .= dom f .= dom f .= dom f ;
upper_bound ( proj2 .: ( Lower_Arc ( C ) /\ Lower_Arc ( C ) /\ Vertical_Line w ) ) <= upper_bound ( proj2 .: ( C /\ Vertical_Line w ) ) & upper_bound ( proj2 .: ( C /\ Vertical_Line w ) ) <= upper_bound ( proj2 .: ( C /\ \mathbin { w } ) ) ;
for r be Real st 0 < r ex n be Nat st for m being Nat st n <= m holds |. S . m - p .| < r
i * fN - fN = i * ( f - y ) - ( i * ( f - y ) ) .= i * ( f - y ) - ( i * ( f - y ) ) .= i * ( f - y ) ;
consider f being Function such that dom f = 2 -tuples_on X ( ) & for Y being set st Y in 2 -tuples_on X ( ) holds f . Y = F ( Y ) ;
consider g1 , g2 being element such that g1 in [#] Y & g2 in union C and g = [ g1 , g2 ] and g1 in C and g2 in C and g2 in C and g2 in C and g2 in C ;
func d \! \mathop n -> Nat means : Def7 : d |^ n divides n & ( d |^ n ) * ( d |^ n ) divides n * ( d |^ n ) ;
fy1 . [ 0 , t ] = f . [ 0 , t ] .= f . [ 0 , t ] .= ( - P ) . ( 2 * ( x , t ) ) .= a * ( - P ) .= a * ( - P ) .= a * ( - P ) .= a * ( - P ) ;
t = h . D ( ) or t = h . B ( ) or t = h . C ( ) or t = h . E ( ) or t = h . F ( ) or t = h . J ( ) ;
consider m1 being Nat such that for n st n >= m1 holds dist ( ( seq . n ) , ( seq . n ) ) < 1 / ( ( seq . n ) + ( seq . n ) ) ;
( ( q `1 ) / |. q .| ) ^2 / ( 1 + ( q `2 ) ) ^2 <= ( ( q `1 ) ) ^2 / ( 1 + ( q `2 ) ) ^2 ;
h0 . ( i + 1 + 1 ) = h21 . ( i + 1 + 1 + 1 -' len h11 + 2 -' 1 ) .= h21 . ( i + 1 + 1 -' len h11 ) ;
consider o being Element of the carrier' of S , x2 being Element of { [ o , x2 ] } such that a = [ o , x2 ] and [ o , x2 ] in o and [ o , x2 ] in o ;
for L being RelStr , a , b being Element of L holds a <= { b } iff a <= b & b <= a & a <= b
||. h1 . n .|| = ||. h1 . n - ( ||. g . n - g .|| ) .= ||. ( h1 . n - g . n ) - ( ||. g . n - g .|| ) .= ||. g . n - g . n .|| .= ||. g . n - g . n .|| .= ||. g . n - g . n .|| ;
( - ( exp_R * exp_R ) ) . x = f . x - ( exp_R . x ) .= ( - 1 ) * exp_R . x .= ( - 1 ) * exp_R . x .= ( - 1 ) * exp_R . x .= ( - 1 ) * exp_R . x ;
pred r = F .: ( p , q ) means : Def21 : len r = min ( len p , len q ) ;
( ( r8 / 2 ) ^2 + ( r8 / 2 ) ^2 ) + ( ( r8 / 2 ) ^2 ) <= ( ( r / 2 ) ^2 + ( r / 2 ) ^2 ) + ( ( r / 2 ) ^2 ) ;
for i being Nat , M being Matrix of n , K st i in Seg n & i in Seg n holds Det ( M , i ) = Sum ( \cal L ( M , i ) )
then a " <> 0. R implies a " * ( a * v ) = 1 * v & a " * ( a * v ) = 1 * v ;
p . ( j -' 1 ) * ( q *' r ) . ( i + 1 -' j ) = Sum ( p . ( j -' 1 ) * ( q *' r ) ) .= p . ( j -' 1 ) * ( q . ( j -' 1 ) * ( q . ( j -' 1 ) ) ;
deffunc F ( Nat ) = L . 1 + L . ( ( R /* h ) ^\ n ) * ( ( R /* h ) ^\ n ) " ) . $1 - L . ( ( R /* h ) ^\ n ) . $1 ;
assume that the carrier of H2 = f .: ( the carrier of H1 ) and the carrier of H2 = f .: ( the carrier of H2 ) and the carrier of H2 = f .: ( the carrier of H2 ) and the carrier of H2 = f .: ( the carrier of H2 ) ;
Args ( o , Free ( S , X ) ) = ( ( the Sorts of Free ( S , X ) ) * ( the Arity of S ) ) . o .= ( ( the Sorts of Free ( S , X ) ) * ( the Arity of S ) ) . o ;
H1 = n + 1 / ( |. 2 to_power ( n + 1 ) .| + h to_power ( n + 1 ) ) .= n + 1 / ( n + 1 ) ;
( O = 0 & 3 = 0 & 1 <= O & O = 1 & O = 0 or O = 1 & O = 0 & O = 1 & O = 0 & O = 1 & O = 0 & O = 1 & O = 0 & O = 1 & O = 1 & O = 0 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O =
F1 .: ( dom F1 /\ dom F2 ) = F1 .: ( dom F1 /\ dom F2 ) .= F1 .: ( dom F1 /\ dom F2 ) .= { f . ( n + 1 ) } .= { f . ( n + 1 ) } .= { f . ( n + 1 ) } ;
attr b <> 0 & d <> 0 & b <> d & ( b <> d & ( a - b ) = ( ( e - b ) / ( d - c ) ) / ( d - c ) ;
dom ( ( f +* g ) | D ) = dom ( ( f +* g ) | D ) /\ D .= dom ( ( f +* g ) | D ) /\ D .= dom ( ( f +* g ) | D ) /\ D .= dom ( ( f +* g ) | D ) /\ D .= dom ( ( f +* g ) | D ) ;
for i being set st i in dom g ex u , v being Element of L st g /. i = u * a & u in B & v in C & v in C & v in C & w in C
g ` * P * g " = g ` * ( g * P ) .= g ` * ( g * P ) .= g ` * ( g * P ) .= g * ( g * P ) .= g * ( g * P ) ;
consider i , s1 such that f . i = s1 and ( ex s st s = s1 & ( for i st i in dom s1 holds s1 . i = s1 . i ) & ( for i st i in dom s1 holds s1 . i = s1 . i ) & s1 is convergent ;
h5 | ]. a , b .[ = ( g | Z ) | ]. a , b .[ .= ( g | Z ) | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ ;
[ s1 , t1 ] , [ s2 , t2 ] are_connected & [ s1 , t2 ] , [ s2 , t2 ] are_connected & [ s1 , t2 ] , [ s2 , t2 ] are_connected ;
then H is negative & H is negative & H is negative & H is non empty & H is conjunctive & H is conjunctive -gex F being FinSequence st F is conjunctive -\mathopen of H ;
attr f1 is total means : Def21 : ( 1 - f1 ) (#) ( f2 * f1 ) is total & ( - f1 ) (#) ( f2 * f1 ) = ( ( - f1 ) (#) ( f2 * f1 ) ) (#) ( ( - f1 ) (#) ( f2 * f1 ) ) ;
z1 in W2 ` & ( for z2 st z2 in W2 holds ( z1 = z2 ) & ( z1 = z2 ) & ( z2 = z2 ) & ( z2 = z2 implies z1 = z2 ) implies z1 = z2 )
p = 1 * p .= a " * a * p .= a " * ( b " * p ) .= a " * ( b " * p ) .= a " * ( b " * p ) .= a " * ( b " * p ) .= a " * ( b " * p ) ;
for rseq be Real_Sequence for K be Real st for n be Nat st n <= K holds rseq . n <= K * ( K . n ) & upper_bound rng ( K . n ) <= K * ( K . n )
\hbox { W-min C , E-max C , W-min C , W-min C , W-min C , W-min C , W-min C , W-min C } meets L~ go \/ L~ pion1 or \hbox { W-min C , E-max C , W-min C , W-min C , W-min C , W-min C , W-min C , W-min C } meets L~ pion1 or not C in L~ pion1 or C in L~ pion1 & C in L~ pion1 & C in L~ pion1 & C in L~ pion1 & C in L~ pion1 & C in L~ pion1 & C in L~ pion1 & C in L~ pion1 & C in L~ pion1 & C in L~ pion1 & C in L~ pion1 & C in
||. ( f . g . ( k + 1 ) - g . ( k + 1 ) ) - g . ( g . ( k + 1 ) ) .|| <= ||. g . ( g . k ) - g . ( k + 1 ) .|| * K ;
assume h = ( ( B .--> ( B .--> C ) +* ( D .--> E ) ) +* ( E .--> F ) ) +* ( F .--> J ) +* ( M .--> N ) +* ( M .--> N ) +* ( M .--> N ) +* ( M .--> N ) ) +* ( M .--> N ) ;
|. ( ( lower ( H . n ) , T . m ) - ( ( lower ( H . n ) , T . m ) ) ) . k - ( ( lower ( H . n , T . m ) ) - ( ( lower ( H . n ) , T . m ) ) ) . k .| <= e * ( e * ( b-a . n ) ) - ( e * ( b-a . m ) ) ;
( ( the Subset of C ) . ( i , 1 ) ) . e = [ ( the Subset of C ) . ( i , 1 ) , ( the carrier of C ) . ( i , 1 ) ] .= [ ( the Subset of C ) . ( i , 1 ) , ( the carrier of C ) . ( i , 1 ) ] ;
{ x1 , x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 } = { x1 , x2 , x3 , x5 } ;
assume that A = [. 0 , 2 * PI .] and integral ( A , ( ( exp_R . n ) * cos ) , A ) = 0 and ( integral ( A , ( ( exp_R . n ) * cos ) ) `| A ) = 0 ;
p `1 is Permutation of dom ( f1 /. i ) & p `1 = ( Sgm Y ) . ( p `1 ) & p `2 = ( Sgm Y ) . ( p `2 ) * ( Sgm Y ) . ( p `2 ) " ;
for x , y st x in A & y in A holds |. ( 1 / ( f . x ) - ( 1 / ( f . y ) ) ) .| <= 1 * |. f . x - ( 1 / ( f . y ) ) - ( 1 / ( f . y ) ) .|
( ( p2 `2 ) * ( 1 - sn ) ) = |. q2 .| * ( ( ( p2 `2 ) ) * ( 1 - sn ) ) .= ( ( p2 `2 ) ) * ( 1 - sn ) ;
for f being PartFunc of the carrier of C , REAL st dom f is compact & rng f c= dom f & f is continuous & rng f c= dom f holds f is continuous
assume for x being Element of Y st x in EqClass ( z , CompF ( B , G ) ) holds ( Ex ( a , A , G ) ) . x = TRUE ;
consider FM such that dom FM = n1 and for k be Nat st k in n1 holds ( for k be Nat st k in n1 holds ( F . k ) . k = ( F . k ) . ( M . k ) ) ;
ex u , u1 st u <> u1 & u , u1 , u1 , v1 , u1 , v1 , u1 , v1 , u1 , v1 , u1 , v1 , v2 , v1 , u1 , v1 , v2 , u1 , v1 , v2 , v1 , u1 , v1 , v2 , u1 , v1 , v2 , v1 , u1 , v2 , v1 , v2 , u1 , v1 , v2 , u1 , v1 , v2 , u1 , v1 , v2 , u1 , v1 , v2 , u1 , v1 , v2 , v1 , v2 , v1 , v2 , v1 , v2 , u1 , v2 , u1 , v1 , v2 , v1
for G being Group , A , B being non empty Subgroup of G , N being normal Subgroup of G holds ( N N ) * ( N ~ A ) = N ~ A * N ` B
for s be Real st s in dom F holds F . s = integral ( R , ( R + g ) * e ) , ( ( R + g ) * e ) . x ) dx
width AutMt ( f1 , b1 , b2 , b3 , b3 , b2 , b3 , b3 , b2 ) = len b1 .= len b2 .= len b1 .= len b2 .= len b1 .= len b2 .= len b2 .= len b2 .= len b2 .= len b2 .= len b1 .= len b2 .= len b2 .= len b2 ;
f | ]. - PI / 2 , PI / 2 .[ = f & dom ( f | [. - PI / 2 , PI / 2 .] ) = dom f & rng ( f | [. - PI / 2 , PI / 2 .] ) = ]. - PI / 2 , PI / 2 .] ;
assume that X is closed and a in X and a in X and y in a ^ f and x in X and y in X and x in X and y in X ;
Z = dom ( ( ( id Z ) * ( arctan ) ) `| Z ) /\ dom ( ( id Z ) * ( arctan ) ) .= dom ( ( id Z ) * ( arctan ) ) /\ dom ( ( id Z ) * ( exp_R ) ) ;
func ^2 ( l . 1 ) -> Subset of V means : Def: for k st 1 <= k & k <= len l holds it . k = V ( k ) & ( for k st 1 <= k & k <= len l holds it . k = V ( k ) ) ;
for L being non empty TopStruct , N being net of L , M being net of L st M is net of N for x being Element of L st x in M & x in N holds M . x in N . x
for s being Element of NAT holds ( ( id NAT ) + ( id the carrier of X ) + ( id the carrier of X ) ) . s = ( ( id NAT ) + ( id the carrier of X ) ) . s
then z /. 1 = ( N-min L~ z ) .. z & ( N-min L~ z ) .. z < ( ( N-min L~ z ) .. z ) .. z & ( N-min L~ z ) .. z < ( ( N-min L~ z ) .. z ) .. z ;
len ( p ^ <* ( 0 qua Real ) * ( 0 qua Real ) *> ) = len p + len <* ( 0 qua Real ) * ( 0 qua Real ) *> .= len p + 1 .= len p + 1 ;
assume that Z c= dom ( ( - 1 / 2 ) (#) f ) and for x st x in Z holds f . x = x and f . x > 0 and f . x > 0 ;
for R being add-associative right_zeroed right_complementable distributive non empty doubleLoopStr , I being Subset of R , J being Subset of R , I being Subset of R holds ( I + J ) *' ( I /\ J ) c= I /\ J
consider f being Function of [: B1 , B2 :] , B2 such that for x being Element of B1 holds f . x = F ( x ) and f is one-to-one and for x being Element of B1 holds f . x = F ( x ) ;
dom ( x2 + y2 ) = Seg len x .= Seg len x .= Seg len ( ( x2 + y2 ) + ( y2 + z2 ) ) .= Seg len ( ( x2 + y2 ) + ( y2 + z2 ) ) .= Seg len ( ( x2 + y2 ) + ( y2 + z2 ) ) .= dom ( ( x2 + y2 ) + ( y2 + z2 ) ) ;
for S being such that for C being cluster B , B being object of C holds ( S . ( id B ) ) . ( id C ) = id ( ( Obj S ) . ( id B ) ) . ( id B )
ex a st a = a2 & a in dom f /\ dom ( f | A ) & ( for x st x in A /\ A /\ B holds f . x = f2 . x ) implies A = {} & B = {} & A = {} & B = {} & A = {} implies B = {}
a in Free ( H2 / ( x. 4 , x. k ) ) \/ ( H2 / ( x. 4 , x. k ) ) \/ ( H2 / ( x. k , x. k ) ) ;
for C1 , C2 , f , g being stable Function , f being Function of C1 , C2 st f is stable & g is stable & f is stable holds f is stable & g is stable & g is stable
( W-min L~ Cage ( C , n ) ) `1 = ( W-min L~ Cage ( C , n ) ) `1 .= ( W-min L~ Cage ( C , n ) ) `1 .= ( W-min L~ Cage ( C , n ) ) `1 .= ( W-min L~ Cage ( C , n ) ) `1 .= ( W-min L~ Cage ( C , n ) ) `1 ;
Suppose u = <* x0 , y0 *> and f is partial & f is partial differentiable of // *> & u = <* x0 , y0 *> & SVF1 ( 3 , f , u ) = SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) ; Then SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) = SVF1 ( 3 , pdiff1 ( f , 1 ) , u )
then ( t . {} ) `1 in Vars & ex x being Element of Vars st x = ( t . {} ) `1 & ( t . {} ) `2 = ( t . {} ) `1 & ( t . {} ) `2 = ( t . {} ) `2 ;
Valid ( p '&' p , J ) . v = Valid ( p , J ) . v '&' Valid ( p , J ) . v .= Valid ( p , J ) . v '&' Valid ( p , J ) . v .= Valid ( p , J ) . v .= Valid ( p , J ) . v ;
assume for x , y being Element of S st x <= y for a , b being Element of T st a = f . x & b = f . y holds a >= b ;
func Class ( R , a ) -> Subset-Family of R means : Def6 : for A being Subset of R holds A in it iff ex a being Element of R st a in A & A c= a & a in A ;
defpred P [ Nat ] means ( ( ( ( ( ( G ) ) | ( ( ( G ) | ( $1 -' 1 ) ) ) ) . ( 1 + 1 ) ) `1 ) `1 c= G . ( ( ( G ) | ( $1 -' 1 ) ) . ( 1 + 1 ) ) `1 ;
assume that dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W2 ) = 0 and dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W2 ) = 0 and dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W1 ) = 0 ;
ma_\hbox ( m . t ) = ( m . t ) . {} .= ( m . t ) . {} .= ( m . t ) `1 .= ( m . t ) `1 .= ( m . t ) `1 .= ( m . t ) `1 .= ( m . t ) `1 .= m . t ;
d11 = x9 ^ <* d22 *> .= f . ( y9 , d22 ) .= f . ( y9 , d22 ) .= f . ( y9 , d22 ) .= f . ( y9 , d22 ) .= f . ( y9 , d22 ) .= ( d ^ <* d22 *> ) . ( y9 , d22 ) .= d ^ d22 ;
consider g such that x = g and dom g = dom ( f | X ) and for x being element st x in dom ( f | X ) holds g . x = ( f | X ) . x ;
x + 0. F_Complex = x + len x |-> 0. F_Complex .= x + len x |-> 0. F_Complex .= ( x , x ) ^ ( x |-> 0. F_Complex ) .= ( x *' ) ^ ( x |-> 0. F_Complex ) .= x ` ^ ( x \ 0. F_Complex ) .= x ` ^ ( x \ 0. F_Complex ) .= x ` ;
( k -' ( k -' 1 ) + 1 ) in dom ( ( f | ( k -' 1 ) ) | ( k -' 1 ) ) /\ ( ( f | ( k -' 1 ) ) | ( k -' 1 ) ) ;
assume that P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P = { p1 } and P1 = { p2 } and P2 = { p1 } and P = { p2 } and Q = { p1 } \/ Q and Q = { p2 } \/ Q and Q = { p1 } \/ Q and Q = { p2 } \/ Q and Q = { p2 } \/ Q and Q = { p2 } \/ Q and Q = { p2 } \/ Q and Q = { p2 } \/ Q and Q = { p1 } \/ { p2 } and Q = { p2 } \/ { p2 } \/ { p2 } and Q = { p2 } \/ { p2 } \/ { p2 } and Q = { p2 } \/ { p2 } \/ { p2 } \/ { p2 } and Q
reconsider a1 = a , b1 = b , c1 = c , c1 = d , c2 = p , c1 = p , c2 = q , c1 = p , c2 = r , c2 = s , c1 = p , c2 = q , c1 = s , c2 = p , c2 = s , c1 = p , c2 = q as Element of A ;
reconsider set thesis t1f = G1 . ( t , b ) * F1 . f , FFf = G1 . ( t , a ) * F1 . f as Morphism of ( G1 * F1 ) . ( t , a ) , ( G1 * F2 ) . f ;
LSeg ( f , i + i1 -' 1 ) = LSeg ( f /. ( i + 1 -' 1 ) , f /. ( i + 1 -' 1 + 1 -' 1 ) ) \/ LSeg ( f /. ( i + 1 -' 1 ) , f /. ( i + 1 -' 1 + 1 -' 1 ) ) ;
Integral ( P . m , P . m ) | dom ( P . n ) <= Integral ( P . n , P . m ) & Integral ( P . n , P . m ) <= Integral ( M , P . m ) ;
assume that dom f1 = dom f2 and for x , y being element st [ x , y ] in dom f1 & [ x , y ] in Indices f2 holds f1 . [ x , y ] = f2 . [ x , y ] ;
consider v such that v = y and dist ( u , v ) < min ( ( - ( G * ( i , 1 ) ) `1 ) , ( G * ( i , 1 ) `2 ) - ( G * ( i + 1 , 1 ) `2 ) ;
for G being Group , H being Subgroup of G , a being Element of G , b being Element of H st a = b & b = a |^ b holds a |^ b = b |^ ( a * b )
consider B being Function of Seg ( S + L ) , the carrier of V1 such that for x being element st x in Seg ( S + L ) holds P [ x , B . x ] ;
reconsider K1 = { p9 where p9 is Point of TOP-REAL 2 : P [ p9 ] & p9 `1 <= p9 `1 & p9 `2 <= 0 & p9 `2 <= 0 & p9 `2 <= 0 & p9 `2 <= 0 & p9 <> 0. TOP-REAL 2 } as Subset of TOP-REAL 2 ;
( ( ( N - S ) - ( S - S ) / ( m + 1 ) ) / ( m + 1 ) ) / ( m + 1 ) <= ( ( N - S ) / ( m + 1 ) ) / ( m + 1 ) ;
for x being Element of X , n be Nat st x in E holds |. ( Re F ) . n - ( Im F ) . x .| <= P . x & |. ( Im F ) . n - ( Im F ) . x .| <= P . x
len @ as = len ( @ @ @ ) + len <* [ 2 , 0 ] *> .= len ( @ @ ) + len <* 0 *> .= len @ @ <* 2 *> .= len @ @ ) + len @ <* 1 *> .= len @ @ <* 2 *> ;
v / ( x. 3 , m1 ) / ( x. 0 , m1 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) = ( x. 3 ) / ( x. 4 , m2 ) ;
consider r being Element of M such that M , v / ( x. 3 , m ) / ( x. 4 , m ) / ( x. 3 , m ) / ( x. 4 , m ) / ( x. 3 , m ) / ( x. 4 , m ) / ( x. 3 , m ) ) |= r and m = r ;
func w1 \ w2 -> Element of ( Union G ) * means : Def14 : for w1 , w2 being Element of ( ( the carrier of G ) \ { w1 } ) holds it . ( ( the carrier of G ) \ { w2 } ) = ( ( the carrier of G ) \ { w1 } ) . ( w1 , w2 ) ;
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= ( Exec ( n2 , s1 ) ) . b2 .= ( Exec ( n2 , s1 ) ) . b2 .= ( Exec ( n2 , s1 ) ) . b2 .= ( Exec ( n2 , s1 ) ) . b2 .= ( Exec ( n2 , s1 ) ) . b2 .= ( Exec ( n2 , s1 ) ) . b2 .= ( Exec ( n2 , s1 ) ) . b2 .= ( s2 . b2 ) ;
for n , k being Nat holds 0 <= ( Partial_Sums ( |. seq .| ) ) . ( n + k ) - ( Partial_Sums ( |. seq .| ) ) . ( n + k ) + ( Partial_Sums ( |. seq .| ) ) . ( n + k ) - ( Partial_Sums ( |. seq .| ) ) . n
set F = S \! \mathop { {} } , G = S \! \mathop { {} } , F = S \! \mathop { {} } , N = S \! \mathop { {} } , A = S \! \mathop { {} } , B = S \! \mathop { {} } , C = S \! \mathop { {} } , D = S \! \mathop { {} } ;
( Partial_Sums ( seq ) ) . ( n + 1 ) + Partial_Sums ( seq ) . ( n + 1 ) + Partial_Sums ( seq ) . ( n + 1 ) >= ( Partial_Sums ( seq ) ) . ( n + 1 ) + Partial_Sums ( seq ) . ( n + 1 ) ;
consider L , R such that for x st x in N holds ( f | Z ) . x - ( f | Z ) . x0 = L . ( x- Z ) + R . ( x- Z ) . ( x- Z ) ;
func the closed \HM { a , b , c , d , e , f , g , h , i , f , i , g , h , i , h , i ) -> Subset of TOP-REAL 2 ;
a * b ^2 + ( a * c ) + ( b * c ) + ( b * c ) + ( b * c ) + ( c * c ) + ( c * c ) >= 6 * 6 * a * b * c ;
v / ( x1 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) = v / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) ;
( \mathop { Q ^ <* x *> , M ) +* ( <* x *> , M ) = ( \mathop { Q , M *> ) +* ( ( z ^ <* x *> --> TRUE ) ) +* ( ( z ^ <* TRUE *> --> TRUE ) --> TRUE ) .= ( ( z ^ <* x *> --> TRUE ) +* ( x --> TRUE ) ) +* ( ( z ^ <* TRUE *> --> TRUE ) --> TRUE ) ;
Partial_Sums ( F ) = ( r |^ n1 ) * Partial_Sums ( C ) .= ( C |^ n1 ) * Partial_Sums ( C |^ n1 ) .= C . ( ( n + 1 ) + 1 ) .= C . ( ( n + 1 ) + 1 ) .= C . ( ( n + 1 ) + 1 ) .= C . ( ( n + 1 ) + 1 ) .= C . ( ( n + 1 ) + 1 ) .= C . ( ( n + 1 ) + 1 ) .= C . ( ( n + 1 ) + 1 ) .= C . ( ( n + 1 ) + 1 .= C . ( ( n + 1 ) + 1 ) .= C . ( ( n + 1 ) + 1 ) .= C
( ( GoB f ) * ( len GoB f , 2 ) ) `1 = ( ( GoB f ) * ( len GoB f , 1 ) ) `1 .= ( ( GoB f ) * ( len GoB f , 1 ) ) `1 .= ( GoB f ) * ( len GoB f , 1 ) `1 .= ( GoB f ) * ( len GoB f , 1 ) `1 ;
defpred X [ Element of NAT ] means ( Partial_Sums ( s ) ) . $1 = ( Partial_Sums ( a ) ) . ( $1 + 1 ) * ( Partial_Sums ( s ) ) . ( $1 + 1 ) + ( Partial_Sums ( s ) ) . ( $1 + 1 ) * ( $1 + 1 ) ;
( the Arity of S ) . g = ( the Arity of S ) . g .= ( [ ( the Arity of S ) . g , ( the Arity of S ) . g ] ) `1 .= ( ( the Arity of S ) . g ) `1 .= ( the Arity of S ) . g .= ( the Arity of S ) . g .= ( the Arity of S ) . g ;
( X ~ ) ^ ^ Z is_differentiable_on X ~ & ( X ~ ) ^ Z c= ( X ~ ) ^ ( Z ~ ) implies ( X ~ ) ^ Z = ( X ~ ) ^ ( Z ~ )
for a , b being Element of S , s being Element of NAT st s = n & a = F . n & b = F . n & b = F . n & s = N . ( n + 1 ) \ G . ( n + 1 )
E , f |= All ( All ( x , All ( x , p ) ) , ( All ( x , p ) ) '&' ( All ( x , p ) '&' ( All ( x , p ) ) ) ) => ( All ( x , p ) '&' ( All ( x , p ) '&' ( All ( x , p ) '&' ( All ( x , p ) ) '&' ( All ( x , p ) ) ) ) ;
ex R2 being 1-sorted st R2 = ( p | ( n + 1 ) ) . i & ( the carrier of p ) . i = the carrier of ( p | ( n + 1 ) ) & ( the carrier of p ) . i = the carrier of ( p | ( n + 1 ) ) . i ;
[. a , b + sqrt ( 1 - ( k + 1 ) ) / ( 2 * ( k + 1 ) ) , ( k + 1 ) / ( 2 * ( k + 1 ) ) .] is Element of the partial function of REAL , REAL ;
Comput ( P , s , 2 + 1 ) = Exec ( P . 2 , Comput ( P , s , 2 ) ) .= Exec ( ( a , k1 ) := ( card I + 2 ) ) .= Exec ( ( a , k1 ) := ( card I + 2 ) , Comput ( P , s , 2 ) ) ;
card ( h1 | k ) = ( power ( K , n ) ) . k .= ( ( - 1_ K ) * ( - 1_ K ) ) * Sum u .= ( ( - 1_ K ) * ( - 1_ K ) ) * u .= ( ( - 1_ K ) * ( - 1_ K ) ) * u .= ( ( - 1_ K ) * u ) * u .= ( ( - 1_ K ) * u ) ;
( f (#) g ) /. c = f /. c * ( g /. c ) " .= ( f /. c ) * ( g /. c ) .= ( f /. c ) * ( g /. c ) .= ( f (#) ( g /. c ) ) * ( g /. c ) .= ( f (#) ( g /. c ) ) * ( g /. c ) .= ( f (#) ( g /. c ) ) * ( g /. c ) ;
len ( C - ( len ( C , n ) ) - len ( Gauge ( C , n ) ) = len ( C , n ) - len ( Gauge ( C , n ) ) .= len ( C , n ) - len ( Gauge ( C , n ) ) .= len ( C , n ) - len ( Gauge ( C , n ) ) ;
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) .= dom ( r (#) f ) /\ X ) .= dom ( r (#) f ) /\ X ) .= dom ( r (#) f ) /\ X ) .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom (
defpred P [ Nat ] means for n holds 2 * Fib ( n + $1 ) = Fib ( n ) * Fib ( n + $1 ) + ( 5 * Fib ( n + $1 ) ) + ( 5 * Fib ( n + $1 ) ) * Fib ( n + $1 ) ;
consider f being Function of [: { n + 1 } , INT :] , INT such that f = f and f is onto and f is onto and for n being Nat st n < 1 holds f " { n } = { n } and f " { n } = { n } ;
consider c9 being Function of S , BOOLEAN such that c9 = chi ( S , BOOLEAN , S ) and dom ( ( A \/ B ) * ( A \/ B ) ) = dom ( Prob * ( A \/ B ) ) and ( ( A \/ B ) * ( A \/ B ) ) . ( A \/ B ) = Prob * ( A \/ B ) ;
consider y being Element of Y ( ) such that a = "\/" ( { F ( x , y ) where x is Element of X ( ) : P [ x , y ] } , L ( ) & P [ y , x ] ;
assume that A c= dom f and f = ( ( - 1 ) (#) ( ( id Z ) ^ ) ) * ( ( id Z ) ^ ) + ( ( id Z ) ^ ) * ( ( id Z ) ^ ) + ( id Z ) ^ ) = f . ( x + 1 ) * ( ( id Z ) ^ ) . x ;
( f /. i ) `2 = ( ( GoB f ) * ( 1 , j2 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 ) ) `2 ;
dom Shift ( Seq q2 , len q1 ) = { j + len ( Seq q2 ) + len ( Seq q2 ) + len ( Seq q2 ) + len ( Seq q2 ) .= { j + len ( Seq q2 ) + len ( Seq q2 ) ;
consider G1 , G2 , G2 , C being Element of V such that G1 <= G2 & G2 <= G2 & G1 <= G2 & G2 = { G1 where G1 is Element of V : G1 in A & G2 in B & f . x = G1 * ( G1 , G2 ) & f . x = G2 * ( G1 , G2 ) & f . x = G2 * ( G1 , G2 ) ;
func - f -> PartFunc of C , V means : Def5 : dom it = dom f & for c st c in dom it holds it /. c = - f /. c & for c st c in dom it holds it /. c = - f /. c ;
consider phi such that phi is increasing and for a st phi . a = a & {} <> a & for v st v in a holds L , L |= H holds L , L |= H and L , L |= H ;
consider i1 , j1 such that [ i1 , j1 ] in Indices GoB ( f ) and f /. ( k + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( k + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( k + 1 ) = ( GoB f ) * ( i1 , j1 ) ;
consider i , n such that n <> 0 and sqrt p = ( i |^ n ) * ( i |^ n ) and for i being Integer st i <> 0 & i < n holds ( i |^ n ) * ( i |^ n ) = ( i |^ n ) * ( i |^ n ) ;
assume that not 0 in Z and Z c= dom ( ( ( arccot ) * ( ( arccot ) * f ) ) `| Z ) and for x st x in Z holds ( ( ( id Z ) * ( f + g ) ) `| Z ) . x > - 1 ;
cell ( G1 , i1 -' 1 , 1 -' 1 ) \ LSeg ( ( m -' 1 ) / 2 , ( m -' 1 ) / 2 ) c= ( L~ f1 ) \ { ( m -' 1 ) / 2 } \/ ( L~ f1 ) \/ ( L~ f1 ) ;
ex Q1 being open Subset of X st s = Q1 & ex Q1 being Subset of X st Q c= F & ( for x being set st x in Q holds ( x in Q ) & ( x in Q & Q is finite ) & ( x in Q implies x in Q ) implies ( x in Q ) & ( x in Q ) implies x in Q )
( the gcd of A ) . ( ( the _ of A ) . ( r1 , r2 ) , ( the _ of A ) . ( r2 , s2 ) ) = 1 / ( ( the _ of A ) . ( r2 , s2 ) ) .= ( the thesis of R ) . ( r2 , s2 ) ;
R8 = ( ( the _ of ( s2 ) ) . ( 1 + 1 ) ) . ( m1 + 1 ) .= ( ( the _ of ( s2 ) ) . ( m1 + 1 ) ) . ( m1 + 1 ) .= ( ( the _ of ( s2 ) ) . ( m1 + 1 ) ) . ( m1 + 1 ) .= ( ( the _ of ( s2 ) ) . ( m1 + 1 ) ) . m1 .= ( the _ of ( s2 ) ) . m1 ;
CurInstr ( P-6 , Comput ( PE , s1 , m1 + 1 ) ) = CurInstr ( PE , Comput ( PE , s1 , m1 + 1 ) ) .= CurInstr ( PE , Comput ( PE , s1 , m1 ) ) .= CurInstr ( PE , Comput ( PE , s1 , m1 ) ) .= CurInstr ( PE , Comput ( PE , s1 , m1 ) ) .= halt E ;
P1 /\ P2 = ( { p1 } \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p2 , p2 ) \/ LSeg
func -> Subset of the Sorts of A means : Def5 : a in it iff ex p st p in dom f & a = f . p & p in it & a = f . p ;
for a , b being Element of F_Complex st |. a .| > |. b .| & |. b .| >= 1 & |. a .| >= 1 holds ( a * f ) . ( len f ) = and and ( a * f ) is >= >= 0 implies ( a * f ) is >= )
defpred P [ Nat ] means 1 <= $1 & $1 <= len g & for i , j st [ i , j ] in Indices G & G * ( i , j ) = g /. ( i + 1 ) & G * ( i , j ) = g /. ( j + 1 ) ;
assume that C1 , C2 , f , g , h being MSAlgebra over f , g being \rm p being State of C1 , s being State of C2 , t being State of C2 , f being State of C2 , g being State of C2 , s being State of C2 , t being State of C2 , f being State of C2 , g being State of s , s being State of t , p being State of s , g being State of s , s being State of t , t being State of t , s being State of s , g being State of s being State of s being State of t , p being State of s being State of t , g being State of s , p being State of s , q being State of s , q being State of s , r being State of s , q being State of t
( ||. f .|| | X ) . c = ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| * ||. f /. c .|| .= ||. f /. c .|| * ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| * ||. f /. c * ||. f /. c .|| .= ||. f /. c .|| .= ||. f /.
|. q .| ^2 = ( ( q `1 ) ^2 + ( q `2 ) ^2 ) + ( ( q `2 ) ^2 ) ^2 + ( q `2 ) ^2 ^2 + ( q `2 ) ^2 ^2 + ( q `2 ) ^2 ^2 ^2 < ( ( q `2 ) ) ^2 + ( q `2 ) ^2 ;
for F being Subset-Family of [: T , T :] st F is open & not {} in F & for A , B being Subset of T st A in F & B in F & A c= B holds A misses B & B c= A & A misses B implies A misses B
assume that len F >= 1 and len F = k + 1 and len F = len G and for k st k in dom F holds F . k = G * ( k , 1 ) and for k st k in dom F holds F . k = G * ( k , 1 ) ;
i |^ ( ( order n ) - i ) |^ s = i |^ ( s + k ) - i |^ ( s + k ) .= i |^ ( s + k ) - i |^ ( s + k ) .= i |^ ( s + k ) - i |^ ( s + k ) .= i |^ ( s + k ) - i |^ ( s + k ) ;
consider q being oriented oriented Chain of G such that r = q and q <> {} and q in rng ( p ^ q ) and ( for q st q in rng p holds q in rng p & q in rng p holds q in rng p & q in rng p implies q is oriented & p is oriented & q is oriented & p is oriented ;
defpred P [ Element of NAT ] means $1 <= len I implies ( \mathop { g , Z , I ) . $1 = ( \mathop { \rm \cal f } ( g , Z , I ) ) . ( len I + 1 ) & ( \mathop { \rm \cal f } ( g , Z , I ) ) . ( len I + 1 ) = ( \mathop { \rm \cal f } ( g , Z , I ) ) . ( len I + 1 ) ;
for A , B being square Matrix of n , K holds len ( A * B ) = len A & width ( A * B ) = width A & width ( B * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B
consider s being FinSequence of the carrier of R such that Sum s = u & for i being Element of NAT st 1 <= i & i <= len s ex a , b being Element of R st s . i = a * b & a in I & b in J & s . i = b * a ;
func |( ( x , y ) , ( x , y ) )| -> Element of COMPLEX equals |( ( ( Re x ) , ( Re y ) ) , ( Im y ) , ( Im y ) + ( Im y ) , ( Im y ) ) + ( Im y ) , ( Im y ) + ( Im y ) + ( Im y ) ) + ( Im y ) ;
consider g2 being FinSequence of REAL such that g2 is continuous and rng g2 c= A & rng g2 c= A & g2 is one-to-one & rng g2 c= A & rng g2 c= A & g2 is one-to-one and rng g2 c= A and g2 is one-to-one and rng g2 c= A and g2 is one-to-one and rng g2 c= A and g2 is one-to-one and rng g2 c= A and g2 is one-to-one and rng g2 c= A and g2 is one-to-one ;
then n1 >= len p1 & crossover ( p1 , p2 , n1 , n2 , n3 , n3 , n3 , n3 , n3 , n4 , n4 , n4 , n4 , n1 , n3 , n3 , n4 , n4 , n4 , n3 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , p2 , n3 , n4 , crossover , p1 , p2 , n1 , n2 , n3 , n4 , n4 , n4 , n4 , p2 , n3 , n1 , n3 , n4 , n4 , to , n4 , to , n3 , to , n3 , n4 , n4 , n4 , n3 , n4 , G1 , to , n3 , to , to , to , to , to , to , to , to , to , n3 , to , n3 , n4 , n4 , n4 , to , F , to , n3 , n3 , F , F , F , F , F , F , F , F , F
( q `1 ) * a <= ( q `1 ) * ( q `2 ) & ( - q `1 ) * ( q `2 ) <= ( q `1 ) * ( q `2 ) & ( - q `1 ) * ( q `2 ) >= 0 ;
Ft . ( ( p . len p ) + 1 ) = ( ( p . len p ) ) . ( len p ) .= ( ( p . len p ) + 1 ) .= ( ( p . len p ) + 1 ) * ( ( p . len p ) + 1 ) .= ( v . 1 ) * ( v . 1 ) .= v . 1 ;
consider k1 being Nat such that k1 + k = 1 and a = ( <* a *> --> ( k + 1 ) ) ^ <* a *> ^ <* b *> ^ <* a *> ^ <* b *> ^ <* b *> ;
consider B8 being Subset of [: B1 , B2 :] , y1 being Function of B1 , B2 such that B1 is finite and y1 is finite and for x being set st x in B1 holds B1 . x = \frac { B1 . x } and B1 is finite and B1 is finite and B1 is finite ;
v2 . b2 = ( ( curry ( F2 , g ) * ( ( curry ( F2 , g ) ) * ( ( curry ( F2 , g ) ) * ( id ( dom F2 ) ) ) ) ) . b2 .= ( ( curry ( F2 , g ) ) * ( ( curry ( F2 , g ) ) * ( id ( dom F2 ) ) ) ) . b2 .= ( ( curry ( F2 , g ) ) * ( id ( dom F2 ( F2 , g ) ) ) ) . b2 .= ( ( ( F2 , g ) ) . ( ( ( curry ( F2 , g ) ) * ( id ( dom F2 ) ) ) . ( ( ( F2 , g ) ) ) . ( ( ( curry ( F2 , g ) ) . ( ( curry ( F2 , g ) ) . ( ( ( F2 , g ) ) . ( ( curry ( F2 , g ) ) .
dom IExec ( I , P , Initialize s ) = the carrier of SCMPDS .= dom IExec ( I , P , Initialize s ) .= dom IExec ( I , P , Initialize s ) .= dom IExec ( I , P , Initialize s ) .= dom IExec ( I , P , Initialize s ) .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) ;
ex d-32 be Real st d-32 > 0 & for h be Real st h > 0 & |. h .| < d & |. h .| < e & |. h .| < d holds |. h .| " * ||. R /. h - R /. h .|| < e
LSeg ( G * ( len G , 1 ) + |[ 1 , 1 ]| , G * ( len G , 1 ) + |[ 1 , 0 ]| ) c= Int cell ( G , len G , 1 ) \/ { G * ( len G , 1 ) } \/ { G * ( len G , 1 ) } ;
LSeg ( mid ( h , i1 , i2 ) , i ) = LSeg ( h /. ( i + 1 ) , h /. ( i + 1 ) ) .= LSeg ( h /. ( i + 1 ) , h /. ( i + 1 ) ) .= LSeg ( h /. ( i + 1 ) , h /. ( i + 1 ) ) .= LSeg ( h /. ( i + 1 ) , h /. ( i + 1 ) ) .= LSeg ( h /. ( i + 1 ) , h /. ( i + 1 ) , h /. ( i + 1 ) ;
A = { q where q is Point of TOP-REAL 2 : LE q , p , P & LE q , p , P & LE p , p , P & LE q , p , P & LE p , q , P & LE p , q , P & LE p , q , P & LE q , p , P & LE p , q , P & LE q , p , P & LE p , q , P & LE q , p , P & LE p , q , P & LE p , q , P , p1 , P , p1 , p2 , P , p1 , p2 , P , p2 , P , p1 , p2 , p2 , P , p2 , p2 , p1 , p2 , p2 , p2 , p1 , p2 , p1 , P , p2 , p1 , P , p2 , p1 , p2 , p1 , p2 , p1 , p2 , p1 , p2 , p2 , p2 , p2 , p2 , p1 , p2 , P , p1
( ( - x ) .|. y ) = ( - ( 1 - 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 ) ) * ( x .|. y ) ) * ( x .|. y ) .= ( ( - ( x .|. y ) ) * ( x .|. y ) .= ( ( - ( x .|. y ) ) * ( x .|. y ) .= ( ( - ( x .|. y ) ) * ( x .|. y ) * ( x .|. y ) .= ( ( -
0 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) = ( ( p `1 / |. p .| - cn ) / ( 1 + cn ) ) ^2 .= ( ( p `1 / |. p .| - cn ) / ( 1 + cn ) ) ^2 ;
( ( U - 1 ) * ( ( W - 1 ) * ( W - 1 ) ) ) * ( ( W - 1 ) * ( W - 1 ) ) = ( ( U - 1 ) * ( W - 1 ) ) * ( W - 1 ) .= ( ( U - 1 ) * ( W - 1 ) ) * ( W - 1 ) .= ( ( U - 1 ) * ( W - 1 ) ) * ( W - 1 ) .= ( ( U - 1 ) * ( W - 1 ) ) * ( W - 1 ) * ( W - 1 ) .= ( U - 1 ) * ( W - 1 ) * ( W - 1 ) .= ( U - 1 ) * ( W - 1 ) * ( W - 1 ) * ( W - 1 ) .= ( U - 1 ) * ( W - 1 ) * ( W - 1 ) .= ( ( ( W - 1 ) * ( W - 1 ) .= (
func Shift ( f , h ) -> PartFunc of REAL , REAL means : Def: dom it = dom ( f + h ) & for x st x in dom it holds it . x = ( f + h ) . x + h . x * f . x ;
assume that 1 <= k and k + 1 <= len f and [ i , j ] in Indices G and [ i + 1 , j ] in Indices G and f /. k = G * ( i + 1 , j ) and f /. ( k + 1 ) = G * ( i + 1 , j ) and f /. k = G * ( i , j ) ;
assume that not y in Free H and x in Free H and not x in Free ( H ) and not x in Free ( H ) and not x in Free ( H ) and not x in Free ( H ) and not x in Free ( H ) and not x in Free ( H ) and not x in Free ( H ) ;
defpred P11 [ Element of NAT , Element of NAT , Element of NAT , set ] means ( for p being prime Element of NAT st p in $1 & p in $1 holds $2 |^ p = ( a |^ p ) |^ ( $1 -' 1 ) ) * ( a |^ p ) ;
func \sigma ( C ) -> non empty Subset-Family of X means : Def8 : for A being Subset of X holds A in it iff for C being Subset of X st C in it & A c= C holds it . C = C \ A & for A being Subset of X st A in C holds A c= C holds it . A = C ;
[#] ( ( dist ( P ) ) .: Q ) = ( ( dist ( P ) ) .: Q ) .: Q & ( lower_bound ( ( dist ( P ) ) ) .: Q ) .: Q = ( lower_bound ( ( dist ( P ) ) ) .: Q ) .: Q ) ;
rng ( F | ( [: S , T :] , 2 ) = {} or rng ( F | ( S , 2 ) ) = { 1 } or rng ( F | ( S , 2 ) ) = { 1 } or rng ( F | ( S , 2 ) ) = { 1 } or rng ( F | ( S , 2 ) ) = { 1 } ;
( f " ( rng ( f | ( rng f ) ) ) . i = f . i .= ( f . i ) " ( f . i ) .= ( f . i ) " ( f . i ) .= ( f . i ) " ( f . i ) .= ( f . i ) " ( f . i ) .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i ) ;
consider P1 , P2 being non empty Subset of TOP-REAL 2 such that P1 /\ { p1 } = { p1 } and P1 = { p1 } and P2 = { p2 } and P = { p1 } and P1 is closed and P2 is closed and P = { p2 } and P1 is closed and P2 is closed and p1 in P and p2 in P and p1 in P and p2 in P and p1 <> p2 and p2 in P and p1 <> p2 ;
f . p2 = |[ ( ( ( p2 `1 ) / |. p2 .| ) ^2 + ( ( p2 `1 ) / |. p2 .| ) ^2 ) , ( ( p2 `2 ) ) ^2 * ( ( p2 `2 ) ) ^2 + ( ( p2 `2 ) ) ^2 ) ]| ;
( AffineMap ( a , X ) ) " . x = ( ( AffineMap ( a , X ) ) qua Function ) . x .= ( ( AffineMap ( a , X ) ) qua Function ) . x .= ( ( AffineMap ( a , X ) ) " ) . x .= ( - a ) * x .= ( - a ) * x .= ( - a ) * x .= ( - a ) * x .= ( - a ) * x .= ( - a ) * x .= ( - a ) * x .= ( ( - a ) * x .= ( ( - a ) * x .= ( ( - a ) * x + ( ( - a ) * x + ( ( - a ) * x + ( - a ) * x + ( ( - a ) * x .= ( ( - a ) * x ) . x .= ( ( - a ) * x ) . x + ( ( ( ( - a ) * x ) . x + ( - a ) . x .= ( ( - a ) * x ) . x + ( - a ) . x .= ( ( ( - a ) * x ) . x + (
for T being non empty normal TopSpace , A , B being closed Subset of T , A being Subset of T st A <> {} & A in B & B in A & A c= B holds A is Subset of ( ( \in G ) | A ) | B ) & ( for p being Point of T st p in A holds p in ( ( \in G ) | B ) . p ) implies p in ( ( ( \in G ) | B ) . p )
for i , j being strict Subgroup of G st i + 1 in dom F for G1 being strict Subgroup of G , G2 being strict Subgroup of G st G1 = F . i & G2 = F . ( i + 1 ) & G1 is strict Subgroup of G2 holds G1 is strict Subgroup of G2
for x st x in Z holds ( ( ( ( id Z ) * arccot ) `| Z ) . x = ( ( id Z ) * ( arccot ) ) `| Z ) . x - ( ( id Z ) * ( f `| Z ) . x ) / ( 1 + x ^2 )
synonym f /* ( f /* a ) = lim ( f /* a ) & ( for x0 st x0 in dom f holds f . ( n + 1 ) = ( f /. x0 ) - ( f /. x0 ) ) & ( for n st n in dom f holds f /. n = ( f /. x0 ) - ( f /. x0 ) ) ;
then X1 , X2 , X1 , X2 , X2 , Y2 , X2 , X1 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X1 ,
ex N being Neighbourhood of x0 st N c= dom SVF1 ( 1 , f , u ) & ex L be R st for x st x in N holds ( SVF1 ( 1 , f , u ) ) . x - L . ( x - x0 ) = L . ( x - x0 ) + R . ( x - x0 ) ;
sqrt ( ( ( ( ( ( ( ( Q ) ) ) * sqrt ( 1 + ( Q `1 ) ) ^2 ) ) ^2 ) + ( ( ( ( Q ) ) ^2 ) ^2 ) ) ^2 ) >= ( ( ( ( Q ) ) ^2 ) ^2 + ( ( ( Q ) ) ^2 ) ^2 ) ;
( ( 1 - t ) * ||. ( t - g ) . n .|| ) / ( ( 1 - t ) * ||. ( t - g ) . n .|| ) = ( ( 1 - t ) * ||. ( t - g ) . n .|| ) / ( ( 1 - t ) . n ) & ( ( 1 - t ) * ||. g - g .|| ) * ||. g - g .|| < r ;
assume that for x holds f . x = ( ( 1 - sin ( x ) ) (#) ( sin ( x ) ) - sin ( x ) ) and for x holds x in dom ( ( 1 - sin ( x ) ) (#) ( sin ( x ) ) - sin ( x ) ) and f . x = ( ( 1 - sin ( x ) ) (#) ( sin ( x ) ) ) - ( sin ( x ) ) ) ;
consider [: X , Y :] , Y1 being open Subset of [: X , Y :] such that t = [: [: X , Y :] , Y1 :] and [: X , Y :] = [: Y1 , Y :] and [: Y1 , Y :] is open and [: Y1 , Y :] is open and [: Y1 , Y :] is open and [: Y1 , Y :] is open ;
card ( S . n ) = card { [ d , ( d * a ) + 1 , ( d * b ) + 1 ] where d is Element of GF ( p ) : [ d , ( d * b ) + 1 ] in Indices ( p ) & d in { ( d * a ) + 1 } } ;
( ( W-bound D - W-bound D ) / ( 2 |^ ( n + 1 ) ) - ( W-bound D - W-bound D ) ) / ( 2 |^ ( m + 1 ) ) = ( W-bound D - W-bound D ) / ( 2 |^ ( m + 1 ) ) .= ( W-bound D - W-bound D ) / ( 2 |^ ( m + 1 ) ) .= ( W-bound D - W-bound D ) / ( 2 |^ ( m + 1 ) ) ;