thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
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contradiction ;
thesis ;
thesis ;
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thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
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contradiction ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b
D c= S
let Y ;
S `1 is Cauchy
q in X ;
V in X ;
y in N ;
x in T ;
m < n ;
m <= n ;
n > 1 ;
let r ;
t in I ;
n <= 4 ;
M is finite ;
let X ;
Y c= Z ;
A // M ;
let U ;
a in D ;
q in Y ;
let x ;
1 <= l ;
1 <= w ;
let G ;
y in N ;
f = {} ;
let x ;
x in Z ;
let x ;
F is one-to-one ;
e <> b ;
1 <= n ;
f is special ;
S misses C
t <= 1 ;
y divides m ;
P divides M ;
let Z ;
let x ;
y c= x ;
let X ;
let C ;
x _|_ p ;
o is monotone ;
let X ;
A = B ;
1 < i ;
let x ;
let u ;
k <> 0 ;
let p ;
0 < r ;
let n ;
let y ;
f is onto ;
x < 1 ;
G c= F ;
a >= 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 rng f ;
x in X ;
Y ` in Y ;
assume 0 < g ;
not c in Y ;
not v in L ;
2 in z `2 ;
assume f = g ;
N c= b ` ;
assume i < k ;
assume u = v ;
I = J ;
B ` = b ` ;
assume e in F ;
assume p > 0 ;
assume x in D ;
let i be element ;
assume F is onto ;
assume n <> 0 ;
let x be element ;
set k = z ;
assume o = x ;
assume b < a ;
assume x in A ;
a `2 <= b `2 ;
assume b in X ;
assume k <> 1 ;
f = Product l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is generated of squares ;
assume m > 0 ;
assume A c= B ;
X is lower
assume A <> {} ;
assume X <> {} ;
assume F <> {} ;
assume G is open ;
assume f is dilatation ;
assume y in W ;
y \not <= x ;
A ` in B ` ;
assume i = 1 ;
let x be element ;
x `2 = x `2 ;
let X be BCK-algebra ;
assume S is non empty ;
a in REAL ;
let p be set ;
let A be set ;
let G be _Graph , F be Subset-Family of G ;
let G be _Graph , F be Subset-Family of G ;
let a be Complex ;
let x be element ;
let x be element ;
let C be FormalContext , a , b be Element of C ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
let y be Real ;
X c= f . a ;
let y be element ;
let x be element ;
i be Nat ;
let x be element ;
n in NAT ;
let a be element ;
m in NAT ;
let u be element ;
i in NAT ;
let g be Function ;
Z c= NAT ;
l <= ma ;
let y be element ;
r2 in dom f ;
let x be element ;
k1 be Integer ;
let X be set ;
let a be element ;
let x be element ;
let x be element ;
let q be element ;
let x be element ;
assume f is being_homeomorphism ;
let z be element ;
a , b // K ;
let n be Nat ;
let k be Nat ;
B ` c= B ` ;
set s = |^ n ;
n >= 0 + 1 ;
k c= k + 1 ;
R1 c= R ;
k + 1 >= k ;
k c= k + 1 ;
let j be Nat ;
o , a // Y ;
R c= Cl G ;
Cl B = B ;
let j be Nat ;
1 <= j + 1 ;
arccot is_differentiable_on Z ;
exp_R is_differentiable_in x ;
j < i0 ;
let j be Nat ;
n <= n + 1 ;
k = i + m ;
assume C meets S ;
n <= n + 1 ;
let n be Nat ;
h1 = {} ;
0 + 1 = 1 ;
o <> b3 ;
f2 is one-to-one ;
support p = {} ;
assume x in Z ;
i <= i + 1 ;
r1 <= 1 ;
let n be Nat ;
a "/\" b <= a ;
let n be Nat ;
0 <= r0 ;
let e be Real , x be Point of V ;
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 dom f ;
cluster uparrow x -> means : cluster uparrow x -> be ;
X <> { x } ;
x in { x } ;
q , b // M ;
A . i c= Y ;
P [ k ] ;
2 to_power x in W ;
X [ 0 ] ;
P [ 0 ] ;
A = A |^ i ;
2 >= ks ;
G . y <> 0 ;
let X be RealNormSpace , x be Point of X ;
a in A ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , M be Subset of V ;
assume x in - M ;
k < s . a ;
not t in { p } ;
let Y be set , f be Function of Y , BOOLEAN ;
M , L are_equipotent ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded ;
rng f = Y ;
G8 c= L ;
assume x in Cl Q ;
m in dom P ;
i <= len Q ;
len F = 3 ;
Free p = {} ;
z in rng p ;
lim b = 0 ;
len W = 3 ;
k in dom p ;
k <= len p ;
i <= len p ;
1 in dom f ;
b `1 = a `1 + 1 ;
x `2 = a * y `2 ;
rng D c= A ;
assume x in K1 ;
1 <= i + ( i + 1 ) ;
1 <= i + ( i + 1 ) ;
px c= PI ;
1 <= ii & ii <= len f ;
1 <= ii & ii <= len f ;
LMP C in L ;
1 in dom f ;
let seq , seq1 , seq2 ;
set C = a * B ;
x in rng f ;
assume f is Lipschitzian ;
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 ;
FF 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 ] ;
if C c= f holds C c= f ;
xx is increasing & xx is increasing ;
let e2 be element ;
- b divides b ;
F c= \tau ( F ) ;
Gseq is non-decreasing implies seq is non-decreasing
Gseq is non-decreasing implies seq is non-decreasing
assume v in H . m ;
assume b in [#] B ;
let S be non void non empty ManySortedSign , f be Function of S , X ;
assume P [ n ] ;
assume union S is independent & A is finite ;
V is Subspace of V ;
assume P [ k ] ;
rng f c= NAT ( ) ;
assume ex_inf_of X , L ;
y in rng f ;
let s , I be set , f be FinSequence of I ;
b `1 c= b9 `1 & b `2 c= b9 `2 ;
assume not x in REAL + Q ;
A /\ B = { a } ;
assume len f > 0 ;
assume x in dom f ;
b , a // o , c ;
B in B-24 ;
cluster Product p -> non empty ;
z , x // x , p ;
assume x in rng N ;
cosec is_differentiable_in x & cosec is_differentiable_in x ;
assume y in rng S ;
let x , y be element ;
i2 < i1 & i2 < i2 implies i1 < i2
a * h in a * H ;
p , q in Y ;
Observe sqrt I ;
q1 in A1 & q2 in A2 ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A2 c= A1 ;
an < n ;
assume A c= dom f ;
Re ( f ) is_integrable_on M ;
let k , m be element ;
a , a \equiv b , b ;
j + 1 < k + 1 ;
m + 1 <= n1 ;
g is_differentiable_in x0 & g is_differentiable_in x0 ;
g is_continuous_in x0 & g is_continuous_in x0 ;
assume O is symmetric & 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 is_halting_on s , P ;
d , c // a , b ;
let t , u be set ;
let X be set with a non-empty ManySortedSet of X ;
assume k in dom s ;
let r be non negative Real ;
assume x in F | M ;
let Y be Subset of S ;
let X be non empty TopSpace , f be Function of X , X ;
[ a , b ] in R ;
x + w < y + w ;
{ a , b } >= c ;
let B be Subset of A , C be Subset of A ;
let S be non empty ManySortedSign ;
let x be variable , f be Function ;
let b be Element of X , f be Function of X , REAL ;
R [ x , y ] ;
x ` ` = x ;
b \ x = 0. X ;
<* d *> in D |^ 1 ;
P [ k + 1 ] ;
m in dom ( n - 1 ) ;
h2 . a = y ;
P [ n + 1 ] ;
Observe G * F is pre| ;
let R be non empty multMagma , I be Element of R ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p ;
assume f | X is lower ;
x in rng co & y in rng co ;
let x be Element of B ;
let t be Element of D ;
assume x in Q .vertices() ;
set q = s ^\ k ;
let t be VECTOR of X ;
let x be Element of A ;
assume y in rng p `2 ;
let M be mamamaid ;
let N be non empty Subset of M ;
let R be RelStr with finite or R is finite ;
let n , k be Nat ;
let P , Q be be be be be _ RelStr ;
P = Q /\ [#] S ;
F . r in { 0 } ;
let x be Element of X ;
let x be Element of X ;
let u be VECTOR of V ;
reconsider d = x as Int-Location ;
assume I does not destroy a ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < ( v - u ) ;
x <= c2 . x ;
x in F ` & y in F ` ;
Observe S --> T is ManySortedSet of I ;
assume t1 <= t2 & 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 <> A6 & A2 <> A6 ;
set i1 = i + 1 ;
assume a1 = b1 & a2 = b2 ;
dom g1 = A & dom g2 = A ;
i < len M + 1 ;
assume not -infty in rng G ;
N c= dom ( f1 + f2 ) ;
x in dom ( sec | [. 0 , PI / 2 .] ) ;
assume [ x , y ] in R ;
set d = ( x - y ) / ( x - y ) ;
1 <= len g1 & 1 <= len g2 ;
len s2 > 1 & len s2 > 1 ;
z in dom ( f1 + f2 ) ;
1 in dom D2 & 1 in dom D2 ;
( p `2 ) ^2 = 0 ;
j2 <= width G & j2 <= width G ;
len PI > 1 + 1 ;
set n1 = n + 1 ;
|. q-35 .| = 1 ;
let s be SortSymbol of S ;
gcd ( i , i ) = i ;
X1 c= dom f & X2 c= dom f ;
h . x in h . a ;
let G be cluster cluster -> functor of on ;
cluster m * n -> invertible ;
let k9 be Nat , n be Nat ;
i - 1 > m - 1 ;
R is transitive implies field R c= field R & field R c= field R
set F = <* u , w *> ;
( p `1 ) c= P3 `1 & p `2 c= P3 `2 ;
I is_halting_on t , Q ;
assume [ S , x ] is real ;
i <= len ( f2 | i ) ;
p is FinSequence of X ;
1 + 1 in dom g ;
Sum R2 = n * r ;
cluster f . x -> complex-valued ;
x in dom ( f1 - f2 ) ;
assume [ X , p ] in C ;
B9 c= ( X \/ Y ) ;
n2 <= ( 2 |^ ( n + 1 ) ) ;
A /\ cP ` c= A ` ;
cluster x -valued -> x -valued for Function ;
let Q be Subset-Family of S , f be Function of Q , S ;
assume n in dom g2 & m + 1 in dom g2 ;
let a be Element of R ;
t `2 in dom e2 & t `2 in dom e2 ;
N . 1 in rng N ;
- z in A \/ B ;
let S be SigmaField of X , f be Function of S , REAL ;
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 & s2 in r-5 ;
let z , z be complex number ;
n <= NN . m ;
LIN q , p , s ;
f . x = waybelow x /\ B ;
set L = |[ S \to T ]| ;
let x be non positive ExtReal ;
let m be Element of M ;
f in Union ( F1 , A ) ;
let K be add-associative right_zeroed right_complementable non empty doubleLoopStr , f be FinSequence of K ;
let i be Element of NAT , k be Nat ;
rng ( F * g ) c= Y
dom f c= dom x & dom f c= dom x ;
n1 < n1 + 1 & n2 < n2 + 1 ;
n1 < n1 + 1 & n2 < n2 + 1 ;
cluster ( T . X ) /\ T . X -> thesis ;
[ y2 , 2 ] = z ;
let m be Element of NAT , n be Element of NAT ;
let S be Subset of R ;
y in rng ( S . k ) ;
b = sup dom f & a = sup dom f ;
x in Seg ( len q ) ;
reconsider X = D ( ) as set ;
[ a , c ] in E1 ;
assume n in dom ( h2 ^ ) ;
w + 1 = ( a - 1 ) ;
j + 1 <= j + 1 + 1 ;
k2 + 1 <= k1 ;
let i be Element of NAT , k be Nat ;
Support u = Support p & Support u c= Support p ;
assume X is complete e e e ;
assume that f = g and p = q ;
n1 <= n1 + 1 & n2 <= n1 + 1 ;
let x be Element of REAL m , y be Element of REAL m ;
assume x in rng s2 & y in rng s2 ;
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 Real_Sequence , f be FinSequence ;
let k be Element of NAT , n be Nat ;
Integral ( M , P ) < +infty ;
let n be Element of NAT , x be Element of REAL ;
assume z in being atatlen } _ _ 0 ( A ) ;
let i be set ;
n - 1 = n-1 - 1 ;
len ( n - m ) = n ;
\mathop { \rm Set } ( Z , c ) c= F
assume x in X or x = X ;
x is midpoint of b , c ;
let A , B be non empty set , f be Function of A , B ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q & dom q = Seg n ;
let s be Element of E -tuples_on E ;
let B1 be Basis of x , B ;
Carrier ( L /\ L2 ) = {} ;
L1 /\ L2 /\ L2 = {} ;
assume downarrow x = downarrow y ;
assume b , c // b , c ;
LIN q , c , c ;
x in rng ( f . -129 ) ;
set n8 = n + j ;
let D7 be non empty set , f be FinSequence of D ;
let K be add-associative right_zeroed right_complementable non empty addLoopStr , f be Function of K , K ;
assume that f `1 = f and h `2 = h ;
R1 - R2 is total & R1 - R2 is total ;
k in NAT & 1 <= k ;
let a be Element of G ;
assume x0 in [. a , b .] ;
K1 ` is open & ( TOP-REAL 2 ) | K1 is open ;
assume a , b are_maximal in C ;
let a , b be Element of S ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = Integral ( M , f ) ;
cluster nson -> neson ;
not u in { ag } ;
the carrier of f c= B ( ) ;
reconsider z = x as Vector of V ;
cluster *> is \rangle for non empty RelStr ;
r (#) H is as as as as as as as as {} -defined Function of X , REAL ;
s . intloc 0 = 1 ;
assume that x in C and y in C ;
let U0 be strict non-empty MSAlgebra over S , o be OperSymbol of S ;
[ x , Bottom T ] is compact ;
i + 1 + k in dom p ;
F . i is stable Subset of M ;
r-35 in : ( ex y st y in : x = y ) ;
let x , y be Element of X ;
let A , I be such that A is such that I is such that A c= I ;
[ y , z ] in O ;
( } Macro i ) = 1 ;
rng Sgm A = A ;
q |- p -<* All ( y , q ) *> ;
for n holds X [ n ] ;
x in { a } & x in d ;
for n holds P [ n ] ;
set p = |[ x , y , z ]| ;
LIN o , a , b & LIN o , a , c ;
p . 2 = Z / Y ;
( D `2 ) ^2 = {} ;
n + 1 + 1 <= len g ;
a in [: NAT , NAT :] ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
let I be Ideal of L ;
set g = f1 + f2 , h = f1 + f2 ;
a <= max ( a , b ) ;
i-1 < len G + 1-1 ;
g . 1 = f . i1 ;
x `1 , y `2 ] in A2 ;
( f /* s ) . k < r ;
set v = VAL g ;
i - k + 1 <= S ;
cluster associative for non empty multMagma ;
x in support ( ( support t ) * ( support b ) ) ;
assume a in [: the carrier of G , the carrier of G :] ;
i `2 <= len ( y | i ) ;
assume p divides b1 + b2 & p divides b2 + b3 ;
p0 <= upper_bound M1 & ( for i st i in Seg n holds M * ( i , j ) <= 0 ) implies M1 is bounded
assume x in W-min ( X ) & y in W-min ( X ) ;
j in dom ( z | Seg n ) ;
let x be Element of D ( ) ;
IC s4 = l1 .= l1 .= l1 .= l1 ;
a = {} or a = { x } ;
set u9 = Vertices G , u9 = Vertices G , u9 = Vertices G ;
seq " is non-zero & seq " is non-zero implies seq " is non-zero
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
h-4 c= h-14 & hh2 c= h-14 ;
]. a , b .[ c= Z ;
X1 , X2 are_separated & X2 , X1 are_separated implies X1 union X2 , X2 union X1 union X2 are_separated
a in Cl ( union F \ G ) ;
set x1 = [ 0 , 0 ] ;
k + 1 - 1 = k - 1 ;
cluster -> real-valued for Relation of NAT , REAL ;
ex v st C = v + W ;
let IT be non empty thesis , F be Function of IT , IT ;
assume V is Abelian add-associative right_zeroed right_complementable non empty addLoopStr ;
XY \/ Y in \sigma ( L ) ;
reconsider x = x as Element of S ;
max ( a , b ) = a ;
sup B is upper & sup B = sup B ;
let L be non empty reflexive RelStr , x be Element of L ;
R is reflexive & X is transitive implies R is transitive
E , g |= ( g ) / ( g . H ) ;
dom G `2 /. y = a ;
( 1 - 4 ) / ( 1 - 4 ) >= - ( 1 / 4 ) ;
G . p0 in rng G & G . I in rng G ;
let x be Element of F , y be Element of F ;
D [ P , 0 ] & P [ 0 ] ;
z in dom id B & z in dom id B ;
y in the carrier of N & z in the carrier of N ;
g in the carrier of H & h in the carrier of G ;
rng ( f | X ) c= NAT & rng ( f | X ) c= dom f ;
j `2 + 1 in dom s1 ;
let A , B be strict Subgroup of G ;
let C be non empty Subset of REAL , f be PartFunc of REAL , REAL ;
f . z1 in dom h & f . z2 in dom h ;
P . k1 in rng P & P . k2 in rng P ;
M = ( A +* ( {} .--> a ) ) +* ( {} .--> a ) ;
let p be FinSequence of REAL , n be Nat ;
f . n1 in rng f & f . n2 in rng f ;
M . ( F . 0 ) in REAL ;
( h | [. a , b .] ) = b-a ;
assume the distance of V , Q is_V ;
let a be Element of ^ ( V ) ;
let s be Element of P , x be Element of X ;
let PA be non empty RelStr , x be Element of Y ;
n be Nat ;
the carrier of g c= B & the carrier of g c= B ;
I = halt SCM ( R ) .= ( the InstructionsF of SCM ) . I ;
consider b being element such that b in B ;
set BM = BCS K , BM = BCS M ;
l <= ( GoB ( F . j ) ) . i ;
assume x in downarrow [ s , t ] ;
( x `2 ) / ( |. x .| ) in uparrow t ;
x in ( JumpParts JumpParts T ) . ( 1 , T ) ;
let h be Morphism of c , a ;
Y c= 1. ( K , the_rank_of ) & card Y = card X ;
A2 \/ A3 c= Carrier ( L ) \/ Carrier ( L ) ;
assume LIN o , a , b & LIN o , a , c ;
b , c // d1 , e2 & b , c // d2 , e2 ;
x1 , x2 , x3 , x4 , 8 , 7 , 8 , 8 , 8 , 6 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 9 , 8 , 8
dom <* y *> = Seg 1 & dom <* y *> = Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar s .| ;
[ x , x `2 ] in X ~ ;
for n being Nat holds 0 <= x . n
[' a , b '] = [. a , b .] ;
cluster -> -> sqrt closed 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 = [ y1 , y2 ] ;
let R , Q be ManySortedSet of A ;
set d = ( 1 / ( n + 1 ) ) ;
rng g2 c= dom W & rng g2 c= dom W ;
P . ( [#] Sigma \ B ) <> 0 ;
a in field R & a = b ;
let M be non empty Subset of V , v be Element of V ;
let I be Program of SCM+FSA , J be Program of SCM+FSA ;
assume x in rng ( ( S * R ) * S ) ;
let b be Element of the carrier of T ;
dist ( e , z ) - r > re ;
u1 + v1 in W2 & v1 in W1 + W2 ;
assume the carrier of L misses rng G ;
let L be lower-bounded antisymmetric transitive RelStr ;
assume [ x , y ] in a9 ;
dom ( A * e ) = NAT ;
let a , b be Vertex of G ;
let x be Element of Bool M , a be Element of Bool M ;
0 <= Arg a < 2 * PI ;
o , a9 // o , y & o , a // o , y ;
{ v } c= the carrier of l ;
let x be variable of A ;
assume x in dom ( uncurry f ) & y in dom ( uncurry f ) ;
rng F c= ( Product f ) |^ X ;
assume D2 . k in rng D & D . k in rng D ;
f " . p1 = 0 & f " . p2 = 0 ;
set x = the Element of X , y = the Element of Y ;
dom Ser ( G ) = NAT & rng Ser ( G ) = NAT ;
let n be Element of NAT , x be Element of REAL ;
assume LIN c , a , e1 ;
cluster -> natural for FinSequence of NAT ;
reconsider d = c , e = d as Element of L1 ;
( v2 |-- I ) . X <= 1 ;
assume x in the carrier of f & y in the carrier of f ;
conv @ S c= conv A & conv @ S c= conv @ S ;
reconsider B = b as Element of the carrier of T ;
J , v |= P \lbrack l , P \lbrack l , P \rbrack ;
redefine func J . i -> non empty TopSpace ;
ex_sup_of Y1 \/ Y2 , T & ex_sup_of Y1 \/ Y2 , T ;
W1 is_not field ( W1 + W2 ) implies R + ( W2 + W1 ) is_not field R
assume x in the carrier of R & y in the carrier of R ;
dom ( n --> 0 ) = Seg n & dom ( n --> 0 ) = Seg n ;
s4 misses s2 & s3 misses s2 ;
assume ( a 'imp' b ) . z = TRUE ;
assume that X is open and f = X --> d ;
assume [ a , y ] in Indices ( f | A ) ;
assume that function I c= J and function I c= K and I c= K ;
Im ( ( lim seq ) - ( lim seq ) ) = 0 ;
( ( ( - 1 ) (#) ( sin * cos ) ) `| Z ) . x <> 0 ;
sin * ( ( - 1 ) (#) ( ( id Z ) (#) ( ( - 1 ) (#) ( ( id Z ) (#) ( ( id Z ) (#) ( ( id Z ) (#) ( ( id Z ) (#) (
t3 . n = t3 . n & t2 . n = t3 . ( n + 1 ) ;
dom ( ( - F ) | A ) c= dom F ;
W1 . x = W2 . x & W2 . x = W2 . x ;
y in W { W ( ) } \/ W { W ( ) } ;
( k + 1 ) <= len ( v ^ w ) ;
x * a \equiv y * a . ( mod m ) ;
proj2 .: S c= proj2 .: P & proj2 .: S c= proj2 .: P ;
h . p4 = g2 . I & h . p4 = g2 . I ;
Gij `1 = U /. 1 & Gij `1 = G * ( 1 , k ) `1 ;
f . rr1 in rng f & rr2 in rng f ;
i + 1 + 1 + 1-1 <= len - 1 ;
rng F = rng ( F . 0 ) & rng ( F . 0 ) c= rng ( F . 0 ) ;
mode Subset of \HM { the } \HM { is well unital non empty multMagma ;
[ x , y ] in A ~ ;
x1 . o in L2 . o & x2 . o in L2 . o ;
the carrier of support ( m ) c= B ;
not [ y , x ] in id X & x in id X ;
1 + p .. f <= i + len f ;
seq ^\ k1 is lower & seq ^\ k1 is lower implies seq ^\ k1 is lower
len ( F . ( I . i ) ) = len I ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be complex number , f be FinSequence of REAL ;
Comput ( P , s , n ) = s ;
k <= k + 1 & k + 1 <= len p ;
reconsider c = {} T as Element of L ;
let Y be Element of \HM { the } \HM { InternalRel of T : not contradiction } ;
cluster directed-sups-preserving for Function of L , L ;
f . j1 in K . j1 & f . j2 in K . j1 ;
Observe J => y is total for I be Element of J ;
K c= 2 -tuples_on the carrier of T ;
F . b1 = F . b2 & F . b2 = F . b2 ;
x1 = x or x1 = y or x1 = z or x1 = z ;
pred a <> {} means : Def4 : ( a - 1 ) / 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 , c ;
reconsider m = x as Element of Funcs ( V , C ) ;
let f be non constant non trivial FinSequence of D ;
let FS2 be non empty \cal X , f be FinSequence of X ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in F-8 ;
reconsider pp = x , pp = y as Subset of m ;
let A , B , C be Element of R ;
Observe non empty for D-be \overline of V ;
rng c `1 misses rng ( e | rng e1 ) ;
z is Element of gr { x } & z is Element of gr { x } ;
not b in dom ( a .--> p1 ) ;
assume that k >= 2 and P [ k ] ;
Z c= dom ( ( - 1 / 2 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) / 2 ) ) ) ;
( L~ Q ) c= UBD A & ( L~ Q ) /\ L~ Q c= UBD A ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( ( 1 / 2 ) (#) ( f ^ ) ) ;
pred f = u means : Def4 : 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 = S2 and q = S1 and p = S2 ;
gcd ( n1 , n2 , n3 ) = 1 & gcd ( n1 , n2 , n3 ) = 1 ;
set o9 = a * ( ( - b ) * ( - c ) ) ;
seq . n < |. r1 .| & |. seq . n .| < r1 ;
assume that seq is increasing and r < 0 and seq is increasing ;
f . ( y1 , x1 ) <= a ;
ex c being Nat st P [ c ] & c < n ;
set g = { n / 1 : n in dom f } ;
k = a or k = b or k = c ;
aa , ax , by , z be set ;
assume that Y = { 1 } and s = <* 1 *> ;
I1 . x = f . x .= f . x .= 0 ;
W3 .last() = W3 . 1 & W .last() = W . ( 1 + 1 ) ;
cluster trivial -> trivial for subgraph of G , finite _Graph ;
reconsider u = u as Element of Bags X ;
A in B ^ ) implies A , B are_that A , B are_that B , A are_that A , B are_that B , A are_that A , B are_that B , A |^ ;
x in { [ 2 * n + 3 , k ] } ;
1 >= ( q `1 / |. q .| - cn ) / ( 1 + cn ) ;
f1 is_as as such that f2 is_as as such such that f1 = f2 & f2 is and f1 . 1 = f2 ;
( f `2 ) ^2 / ( |. q .| ) ^2 <= ( q `2 ) ^2 / ( |. q .| ) ^2 ;
h is_\! \smallfrown Cage ( C , n ) ;
( b `2 ) ^2 / ( p `2 ) ^2 <= ( p `2 ) ^2 / ( p `1 ) ^2 ;
let f , g be Function of X , Y ;
S * ( k , k ) <> 0. K ;
x in dom ( max ( - f , - f ) ) ;
p2 in NN . p1 & p2 in NN . p2 implies p2 in NN . p1
len ( the_left_argument_of H ) < len ( H ) ;
F [ A , FF . A ] implies F ( A , B ) = F ( A , B )
consider Z such that y in Z and Z in X ;
attr 1 in C means : Def4 : A c= C ;
assume r1 <> 0 or r2 <> 0 or r1 <> 0 ;
rng q1 c= rng ( C1 ^ C2 ) & rng ( C1 ^ C2 ) c= rng C1 ;
A1 , L , A2 , A3 , A3 , A1 , A2 be Element of L ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in R ( p , Sp ) & c in R ( p , Sp ) ;
then S is atomic & P-2 [ S ] ;
Cl Int [#] T = [#] T & Cl ( [#] T ) = [#] T ;
( f | A2 ) | A2 = f2 | A2 & ( f | A2 ) | A2 = f2 ;
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 ;
let X be Subset of S ~ ;
consider H1 such that H = 'not' H1 and H1 in X ;
1_ ( 1 ) c= ( ( t * p1 ) / ( t * p2 ) ) ;
0 * a = 0. R .= a * 0. R ;
A |^ ( 2 , 2 ) = A ^^ A ;
set vbeing /. n = ( v4 /. n ) * ( v4 /. n ) ;
r = 0. ( REAL-NS n ) & ||. ( f . n ) - ( f . n ) .|| < r ;
( f . p4 ) `1 >= 0 & ( f . p4 ) `2 >= 0 ;
len W = len ( W .last() ) .= len ( W .last() ) ;
f /* ( s * G ) is divergent_to+infty & f . ( s * G ) is divergent_to+infty ;
consider l being Nat such that m = F . l ;
t16 does not destroy b1 & W7 does not destroy b1 implies not ( t in dom ( t +* ( q +* ( q +* ( s +* ( s +* ( p , t ) ) ) ) ) )
reconsider Y1 = X1 , Y2 = X2 as SubSpace of X ;
consider w such that w in F and not x in w ;
let a , b , c , d be Real ;
reconsider i = i - 1 as non zero Element of NAT ;
c . x >= id L . x & c . x >= id L . x ;
\sigma ( T ) \/ omega ( T ) is Basis of T ;
for x being element st x in X holds x in Y
cluster [ x1 , x2 , x3 , x4 ] -> non pair ;
sup downarrow a /\ downarrow t is Ideal of T ;
let X be \hbox { NAT , F , G , H be Element of X ;
rng f = S1 \/ S1 & f in S1 -element ;
let p be Element of B , x be SortSymbol of S ;
max ( N1 , 2 ) >= N1 & max ( N1 , 2 ) >= N2 ;
0. X <= b |^ ( m * mm1 ) ;
assume that i in I and R1 . i = R ;
i = j1 & p1 = q1 & p2 = q2 implies q1 = q2
assume gR in the right of g & FR 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 , Tor = Y as non empty Subset of Tsuch that X = [: X , Y :] ;
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the Source of G ) -valued ;
n1 <= i2 + len g2 & i2 + len g2 <= i2 + len g2 ;
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume v in the carrier' of G2 & v in the carrier' of G2 ;
y = Re y + ( Im y ) * i ;
( - ( ( - 1 ) * p ) .| ) gcd p = 1 ;
x2 is_differentiable_on ]. a , b .[ & ( for x st x in ]. a , b .[ holds ( x - b ) (#) ( f `| ]. a , b .[ ) ) . x = f . x - ( f `| ]. a , b
rng M5 c= rng D2 & rng D c= rng D2 ;
for p be Real st p in Z holds p >= a
( ( cn ) * 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 ) \ominus C
h \equiv gg . ( mod P ) , g . ( mod P ) ;
reconsider i1 = i-1 - 1 , i2 = 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 i-7 = i , iq2 = j as Element of NAT ;
dom f c= [: C , D :] & dom f = [: C , D :] ;
x in ( ( the Sorts of B ) . n ) . ( ( the Sorts of B ) . n ) ;
len that len that len that len that len [ that x , y ] in Seg len ( f2 | i ) ;
( p 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 are_not zero and u , v , w , y ;
[ z , z ] in union rng ( F * G ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S , G = $1 .. S ;
LIN a1 , a3 , b1 & LIN a1 , a2 , b2 & LIN a2 , a3 , b3 ;
f " ( f .: x ) = { x } ;
dom ( w2 ) = dom r12 & dom ( r12 ) = dom r12 ;
assume that 1 <= i and i <= n and j <= n ;
( ( g2 ) . O ) `2 <= 1 ;
p in LSeg ( E . i , F . i ) ;
Ia * ( i , j ) = 0. K ;
|. f . ( s . m ) - g .| < g1 ;
q9 . x in rng ( ( q - p ) | ( Seg n ) ) ;
Carrier ( ( Carrier ( f ) ) misses Carrier ( f ) ` ) ;
consider c being element such that [ a , c ] in G ;
assume Nreal = o9 & for o be Element of O st o in o holds o <= i & o <= i ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= ( F |^ C ) " { C } ;
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [. f . j , f . ( j + 1 ) .] ;
pred 0 <= x & x <= 1 implies x ^2 <= x ^2 ;
p `2 - q `2 <> 0. TOP-REAL 2 & p `2 <> 0. TOP-REAL 2 ;
Observe aa] ( S , T ) is non empty ;
let x be Element of S ~ ;
cluster cluster cluster cluster cluster cluster cluster cluster cluster cluster cluster cluster \hbox { - } , F ) -> one-to-one ;
|. i .| <= - ( - 2 |^ n ) / ( n + 1 ) ;
the carrier of I[01] = dom P & the carrier of I[01] = dom P ;
} * ( n + 1 ) ! > 0 * n ;
S c= ( A1 /\ A2 ) /\ A3 \/ A3 /\ A3 /\ A3 /\ ( A2 \/ A3 ) ;
a3 , a4 // b3 , b2 & a3 , a4 // b3 , b2 ;
then dom A <> {} & dom A <> {} & rng A <> {} ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , Y , G implies x = y & x = y ;
set v2 = ( v4 /. ( i + 1 ) ) ;
x = r . n .= ( r . n ) * ( r . n ) ;
f . s in the carrier of S2 & f . s in the carrier of S2 ;
dom g = the carrier of I[01] & dom g = the carrier of I[01] ;
p in Lower_Arc ( P ) /\ Lower_Arc ( P ) ;
dom d2 = [: A2 , A2 :] & dom d2 = A2 ;
0 < ( p / ( ||. z .|| + 1 ) ) / ( ||. z .|| + 1 ) ;
e . ( m + 1 ) <= e . ( mm + 1 ) ;
B \ominus X \/ B \ominus Y c= B \ominus X /\ Y
- -infty < Integral ( M , Im ( g | B ) ) ;
cluster O \cup F -> \HM { \HM { F } } -> } for operation 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 & Proj ( i , n ) * g is_differentiable_on X ;
let x , y , z be Point of X , f be Function of X , Y ;
reconsider pp = p . x , pp = p . y as Subset of V ;
x in the carrier of Lin ( A ) & x in the carrier of Lin ( B ) ;
let I , J be parahalting Program of SCM+FSA , a be Int-Location ;
assume that - a is lower and b is lower and a in - b ;
Int Cl ( A /\ B ) c= Cl Int Cl ( A /\ B ) ;
assume for A being Subset of X holds Cl A = A ;
assume q in Ball ( |[ x , y ]| , r ) ;
( p2 `2 ) ^2 / ( p2 `2 ) ^2 <= ( p `2 ) ^2 / ( p `1 ) ^2 ;
Cl Q ` = [#] ( ( T | P ) | R ) ;
set S = the carrier of T , T = the carrier of T ;
set I8 = ' ( f |^ n ) , I8 = ' ( f |^ n ) , I8 = ' ( f |^ n ) , I8 = ' ( f |^ n ) , I8 = ' ( f |^ n ) , I8
len p - n = len ( an - n ) - ( n - 1 ) ;
A is Permutation of Swap ( A , x , y ) ;
reconsider n6 = ni - 1 as Element of NAT ;
1 <= j + 1 & j + 1 <= len ( s | Seg n ) ;
let qbeing { q , q9 , q , s , t , q , s , t , q , s , t , q , s , t , q , s , t , q , s , t , s , q , s , t , q
a1 in the carrier of S1 & a2 in the carrier of S2 ;
c1 /. n1 = c1 . n1 & c2 /. n2 = c1 . n2 & c1 /. n2 = c2 . n2 ;
let f be FinSequence of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
y = ( ( f * S8 ) . x ) * ( f . ( S8 . x ) ) ;
consider x being element such that x in Assume Assume that x in Assume Assume that A c= R ;
assume r in ( ( dist ( o ) ) .: P ) ;
set i2 = ( n + 1 ) - ( n + 1 ) ;
h2 . ( j + 1 ) in rng h2 & h2 . ( j + 1 ) in rng h2 ;
Line ( M29 , k ) = M . i & Line ( M29 , k ) = M . i ;
reconsider m = ( x - 2 ) / ( x - 1 ) as Element of ( len x - 1 ) -tuples_on REAL ;
let U1 , U2 be Subspace of U0 , a be Element of U1 ;
set P = Line ( a , d ) ;
len p1 < len p2 + 1 & len p1 + 1 <= len p2 ;
let T1 , T2 be being being being Scott + of L , x be Element of L ;
then x <= y & .. ( x ) c= .. ( y ) ;
set M = n -tuples_on the carrier of K , N = n -tuples_on the carrier of K ;
reconsider i = x1 , j = x2 , k = x3 as Nat ;
rng ( the_arity_of a ) c= dom ( H . ( s . a ) ) ;
z1 " = ( z1 " ) * ( z2 " ) .= ( z1 " ) * ( z2 " ) .= z2 " * ( z2 " ) * ( z2 " ) ;
x0 - r / 2 in L /\ dom f /\ dom ( f | X ) ;
then w is that rng w /\ L <> {} & rng w /\ L <> {} ;
set xx = xx ^ <* Z *> , Z = ( x , Z ) --> ( x , y ) ;
len w1 in Seg ( len w1 + len w2 ) & len w2 in Seg ( len w1 + len w2 ) ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of PFuncs ( V , { k } ) ;
x . n = ( |. a . n .| ) * ( A . n ) ;
( p `1 ) ^2 / ( p `2 ) ^2 <= ( G * ( 1 , 1 ) ) ^2 / ( G * ( 1 , 1 ) ) ^2 ;
rng ( g ) c= L~ ( g ) /\ L~ ( g ) ;
reconsider k = i-1 * ( l + j ) as Nat ;
for n being Nat holds F . n is \HM { -infty } & F . n <> +infty ;
reconsider xx = xx , xx = xx , xx = xx , xx = xx , xx = 1 as Vector of M ;
dom ( f | X ) = X /\ dom f /\ X .= dom f /\ X ;
p , a // p , c & b , a // c , d ;
reconsider x1 = x , y1 = y , y2 = z as Element of REAL m ;
assume i in dom ( a * p ^ q ) ;
m . ( ag ) = p . ( cg ) ;
a / ( s . m - n ) / ( s . n - m ) <= 1 ;
S . ( n + k + 1 ) c= S . ( n + k ) ;
assume B1 \/ C1 = B2 \/ C2 & C2 \/ C2 = C1 \/ C2 ;
X . i = { x1 , x2 , x3 , x4 , x5 , 6 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 9 } ;
r2 in dom ( h1 + h2 ) & r1 in dom ( h1 + h2 ) ;
that - 0. R = a and b-0 = b ;
FF is_closed_on t3 , Q8 & I is_halting_on t3 , Q8 & I is_halting_on t3 , Q8 ;
set T = ^2 (# such that for x0 , x0 , x1 , x2 , x3 , x4 , x4 , x5 , x5 , x5 , W , W , W , N , W , S , S , W , S , N , S , W , W
Int Cl ( Int Cl R ) c= Int Cl R & Int Cl R c= Cl R ;
consider y being Element of L such that c . y = x ;
rng ( FF . x ) = { FF . x } & rng ( FF . x ) = { F . x } ;
G-23 ( { c } ) c= B \/ S \/ S ;
( f ^ ) is Relation of [: X , X :] , X & f is Function of [: X , X :] , X ;
set RE = the \mathclose of P , RE = the \mathclose of Q ;
assume that n + 1 >= 1 and n + 1 <= len M ;
let k2 be Element of NAT , k be Nat ;
reconsider p/. n = u as Element of ( TOP-REAL n ) | ( ( TOP-REAL n ) | P ) ;
g . x in dom f & x in dom g implies f . x = g . x
assume that 1 <= n and n + 1 <= len f1 ;
reconsider T = b * N as Element of G / ( N * N ) ;
len ( Pt ) <= len ( Pt ) - len ( Pt ) ;
x " in the carrier of A1 & x in the carrier of A2 ;
[ i , j ] in Indices ( ( A + B ) * ( i , j ) ) ;
for m be Nat holds Re ( F . m ) is simple ;
f . x = a . i .= a1 . k .= a1 . k ;
let f be PartFunc of REAL i , REAL n , x be Element of REAL m ;
rng f = the carrier of ( Carrier A ) * ( i , 1 ) ;
assume s1 = sqrt ( ( 2 * p ) ^2 + ( 2 * p ) ^2 ) ;
pred a > 1 & b > 0 & a / b > 1 ;
let A , B , C be Subset of [: I , B :] ;
reconsider X0 = X , Y0 = Y , Y0 = Z as RealNormSpace ;
let f be PartFunc of REAL , REAL , g be PartFunc of REAL , REAL ;
r * ( v1 |-- I ) . X < r * 1 ;
assume that V is Subspace of X and X is Subspace of V ;
let t-3 , tt2 be Relation of S , X ;
Q [ e-14 \/ { v-5 } , f ] implies f . ( v-5 ) = f . ( v-5 ) ;
g \circlearrowleft ( L~ z ) = z implies ( g /. 1 ) .. z = ( g /. 1 ) .. z
|. |[ x , v ]| - |[ x , y ]| .| = v12 ;
- f . w = - ( L * w ) ;
z - y <= x iff z <= x + y & z <= y - x ;
( 7 * p1 ) / ( 1 - e ) > 0 ;
assume X is BCK-algebra & 0 , 0 , 0 , 0 , 0 ;
F . 1 = v1 & F . 2 = v2 & F . 3 = v2 ;
( f | X ) . x2 = f . x2 & ( f | X ) . x2 = f . x2 ;
( ( ( ( ( 1 / 2 ) (#) ( ( #Z 2 ) * ( f1 + #Z 2 ) ) ) `| Z ) . x = f . x ;
i2 = ( f /. len f ) * ( len f ) .= f /. ( len f ) ;
X1 = X2 \/ ( X1 \ X2 ) & X2 = X2 \/ X1 \ X2 ;
[. a , b , 1_ G .] = 1_ G & a = 1_ G ;
let V , W be non empty VectSpStr over F_Complex , f be FinSequence of V ;
dom g2 = the carrier of I[01] & dom g2 = the carrier of I[01] & rng g2 c= the carrier of I[01] ;
dom f2 = the carrier of I[01] & dom f2 = the carrier of I[01] & rng f2 = the carrier of I[01] ;
( proj2 | X ) .: X = proj2 .: X & ( proj2 | X ) .: Y = proj2 .: Y ;
f . ( x , y ) = h1 . ( x `1 , y `2 ) ;
x0 - r < a1 . n & x0 < a1 . n ;
|. ( f /* s ) . k - G . ( n + 1 ) .| < r ;
len Line ( A , i ) = width A & width Line ( A , i ) = width A ;
Sbeing / op = ( S . g ) / op ( S . g ) ;
reconsider f = v + u as Function of X , the carrier of Y ;
intloc 0 in dom ( Initialized p +* ( 1 , k ) ) ;
i1 , i2 , i3 , i3 , a5 , a5 , 6 , 7 , 8 , 8 , 8 , 8 , 9 , 8 , 8 , 8 , 8 , 9 , 8 , 8 , 8 , 8 , 8 , 9 , 8 , 8 , 8 , 8 , 8 , 9 , 8 , 8
( ( arccos r ) * ( arccos ) ) + ( ( arccos r ) * ( 1 / 2 ) ) = ( PI / 2 ) + 0 ;
for x st x in Z holds f2 is_differentiable_in x & ( f2 * f1 ) . x > 0 ;
reconsider q2 = ( q - x ) / ( q - x ) as Element of REAL ;
( 0 qua Nat ) + 1 <= i + j1 + 1 ;
assume f in the carrier of [ X \to Omega Y , Omega Y ] ;
F . a = H / ( ( x , y ) / ( x , y ) ) ;
( {} T ) at ( C , u ) = TRUE & ( T . ( C , u ) ) at ( C , u ) = TRUE ;
dist ( ( a * seq ) . n , h ) < r / 2 ;
1 in the carrier of [. 0 , 1 .] & 1 in dom ( ( 1 - 1 ) (#) ( ( 1 - 1 ) (#) ( ( 1 - 1 ) (#) ( ( 1 - 1 ) (#) ( ( 1 - 1 ) (#) ( ( 1 - 1 ) (#) (
( p2 `1 - x1 ) - x1 > - g / 2 ;
|. r1 - `2 .| = |. a1 .| * |. or - W .| = |. a1 .| * |. or - W .| ;
reconsider S-14 = 8 , S-14 = 8 as Element of Seg 8 ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b
D0W .2 = DW .2 ( n ) + 1 ;
i1 = ( a + n ) + i2 & i2 = ( a + n ) + ( n + 1 ) ;
f . a [= f . ( f .: O1 "\/" f . ( a "\/" a ) ) ;
pred f = v & g = u & f + g = v + u ;
I . n = Integral ( M , F . n ) + Integral ( M , F . n ) ;
chi ( T1 , T1 ) . s = 1 & chi ( T2 , T2 ) . s = 1 ;
a = VERUM ( A ) or a = VERUM ( A ) or a = VERUM ( A ) ;
reconsider k2 = s . b3 , k2 = s . b3 as Element of NAT ;
( Comput ( P , s , 4 ) ) . GBP = 0 ;
L~ M1 meets L~ ( R * ( i , 1 ) ) & LSeg ( R * ( i , 1 ) , R * ( i , 1 ) ) meets L~ ( R * ( i , 1 ) ) ;
set h = ( the continuous Function of X , R ) , f = ( the Sorts of X ) * ( ( id X ) * ( f | X ) ) ;
set A = { L . ( ( k . n ) . x ) : not contradiction } ;
for H st H is atomic holds P [ H ] implies P [ H ]
set b\frac = S5 ^\ ( i + 1 ) , S5 = S5 ^\ ( i + 1 ) , S5 = ( S ^\ ( i + 1 ) ) ;
Hom ( a , b ) c= Hom ( a `1 , b `2 ) ;
( 1 - ( n + 1 ) ) / ( n + 1 ) < ( 1 - ( n + 1 ) ) / ( n + 1 ) ;
( l `1 ) = [ dom l , cod l ] & ( l `2 ) = dom l ;
y +* ( i , y /. i ) in dom g ;
let p be Element of QC-WFF ( Al ( ) ) , x be Element of D ( ) ;
X /\ X1 c= dom ( f1 - f2 ) /\ dom ( f2 - f1 ) ;
p2 in rng ( f /^ ( len p1 + 1 ) ) & p1 <> p2 implies p2 = f /^ ( len p1 + 1 )
1 <= indx ( D2 , D1 , j1 ) + 1 - 1 ;
assume x in ( ( ( TOP-REAL 2 ) | K0 ) \/ ( ( TOP-REAL 2 ) | K0 ) ) ;
- 1 <= ( ( f2 ) . O ) `2 & - 1 <= ( ( f2 ) . I ) `2 ;
let f , g be Function of I[01] , TOP-REAL 2 , a , b , c be Real ;
k1 - k2 = k1 - k2 - k2 .= k1 - k2 - k2 .= k1 - k2 - k2 ;
rng seq c= ]. x0 - r , x0 .[ & rng seq c= ]. x0 - r , x0 .[ ;
g2 in ]. x0 - r , x0 .[ & g2 in ]. x0 - r , x0 .[ ;
sgn ( p `1 , K ) = - ( 1_ K ) ;
consider u being Nat such that b = p |^ y * u ;
ex A being \mathopen or ex A being ' of W st a = Sum A & A c= A ;
Cl ( union ( H ) ) = union ( ( H ) . n ) .= ( ( H ) . n ) \/ ( ( H . n ) . n ) ;
len t = len t1 + len t2 & len t1 = len t2 + len t2 ;
v-29 = v + w |-- v + AA ;
v <> DataLoc ( ( t . GBP ) , 3 ) & v <> DataLoc ( ( t . GBP ) , 3 ) ;
g . s = sup ( d " { s } ) .= sup ( d " { s } ) ;
( \dot { y } ) . s = s . ( y , s ) ;
{ s : s < t } in REAL & t = {} implies s = t
s ` \ s = s ` \ ( s ` \ s ) .= ( s ` \ s ) \ ( s ` \ s ) ;
defpred P [ Nat ] means B + $1 in A & B + $1 in B ;
( 329 + 1 ) ! = 33331! * ( 329 + 1 ) ;
( 1. ( A , B ) ) * ( 1. ( A , B ) ) = 1. ( A , B ) ;
reconsider y = y as Element of ( len y ) -tuples_on the carrier of K ;
consider i2 being Integer such that y0 = p * i2 and i2 in dom f ;
reconsider p = Y | Seg k as FinSequence of ( len Y ) -tuples_on NAT ;
set f = ( S , U ) \mathop { I } , g = S . I ;
consider Z being set such that lim s in Z and Z in F ;
let f be Function of I[01] , TOP-REAL n , x be Point of TOP-REAL n , r be Real ;
( not 1 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. ( { 0. V } ) as Linear_Combination of A ;
set r = |. e .| + |. n .| + |. w .| + a * s ;
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 ) ;
||. ( x - g ) .|| < r2 & ||. ( x - g ) .|| < r2 ;
b9 , a9 // b9 , c9 & b9 , c9 // b9 , c9 & a , b // c , c9 ;
1 <= k2 -' k1 & k2 + 1 = k2 & k2 + 1 = k2 + 1 & k2 + 1 = k2 ;
( ( p `2 / |. p .| - cn ) / ( 1 + cn ) ) ^2 >= 0 ;
( ( q `2 / |. q .| - cn ) / ( 1 + cn ) ) ^2 < 0 ;
E-max C in cell ( RCage ( C , n ) , 1 , 1 ) /\ L~ Cage ( C , n ) ;
consider e being Element of NAT such that a = 2 * e + 1 ;
Re ( ( lim F ) | D ) = Re ( ( lim G ) . m ) ;
LIN b , a , c or LIN b , c , a & LIN b , c , c ;
p `2 , a `2 // a `2 , b `2 or p `2 , a `2 // b `2 , a `2 ;
g . n = a * Sum ( f | n ) .= f . n * ( f | n ) ;
consider f being Subset of X such that e = f and f is being being element ;
F | ( N2 , S ) = CircleMap * ( F | ( N2 , S ) ) ;
q in LSeg ( q , v ) \/ LSeg ( v , p ) ;
Ball ( m , r0 ) c= Ball ( m , s ) & Ball ( m , r0 ) c= Ball ( m , s ) ;
the carrier of (0). V = { 0. V } .= { 0. V } ;
rng ( ( ( - 1 ) (#) ( ( id Z ) ^ ) ) `| Z ) = [. - 1 , 1 .] ;
assume Re ( seq ) is summable & Im ( seq ) is summable & Im ( seq ) is summable ;
||. ( ( ( vseq . n ) - ( vseq . m ) ) - ( vseq . n ) ) .|| < e / 2 ;
set g = O --> 1 ;
reconsider t2 = t2 as ( 0 , 1 ) -element string of S2 , t2 = t2 as ( 0 , 1 ) -element string of S2 ;
reconsider xx = seq . ( n + 1 ) as sequence of REAL n , r be Real ;
assume that C meets L~ Cage ( C , n ) and C meets L~ Cage ( C , n ) and C meets L~ Cage ( C , n ) ;
- ( ( - 1 ) (#) ( ( - 1 ) (#) ( - 1 ) ) ) < F . n - ( - 1 ) * ( - 1 ) ;
set d1 = being being dist of dist ( x1 , z1 ) , d2 = dist ( x2 , z2 ) , d2 = dist ( y2 , z2 ) ;
2 |^ ( q -' 1 ) - 1 = 2 |^ ( q -' 1 ) - 1 ;
dom ( v | Seg len ( v | i ) ) = Seg len ( v | i ) ;
set x1 = - ( k2 + 2 ) , x2 = - ( k2 + 2 ) , x3 = - ( k2 + 2 ) , x4 = - ( k2 + 2 ) , x4 = - ( k2 + 2 ) , x4 = - ( k2 + 2 ) , x4 = - ( k2 + 2
assume for n being Element of X holds 0. X <= F . n & 0. X <= F . n ;
assume that 0 <= T-32 . i and T-32 . ( i + 1 ) <= 1 and T-32 . ( i + 1 ) <= 1 ;
for A being Subset of X holds c . ( c . A ) = c . A
the carrier of Carrier ( Carrier ( LT + L2 ) ) c= I2 & Carrier ( ( Carrier ( LT + L2 ) ) + ( Carrier ( LT + L2 ) ) c= I2 ;
'not' Ex ( x , p ) => All ( x , 'not' p ) is valid ;
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the normal w.r.t. of A ;
Z c= dom ( ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( f1 + f2 ) ) ) ;
|. 0. TOP-REAL 2 - ( q `2 / |. q .| - cn ) .| < r / 2 ;
ConsecutiveSet2 ( B , L ) c= ConsecutiveSet2 ( A , succ ( d , L ) ) ;
E = dom Carrier ( L ) & Carrier ( L ) c= E & Carrier ( L ) c= E ;
C / ( A + B ) = C / B * C / A ;
the carrier of W2 c= the carrier of V & the carrier of W1 c= the carrier of W2 implies W1 + W2 c= W2 + W1
I . IC s2 = P . IC s2 .= ( I +* ( IC s2 , n ) ) .= ( I . IC s2 ) ;
pred x > 0 means : Def4 : ( 1 - x ) / ( 1 - x ) = x / ( 1 - x ) ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) ;
consider p being Point of T such that C = [. p , q .] and p in C ;
b , c are_connected & - C , - ( - C , - C ) where C is Path of - C , - C is Path of - C , - C : C , - C are_connected } ;
assume f = id the carrier of O & g = id the carrier of O & h = id the carrier of O ;
consider v such that v <> 0. V and f . v = L * v ;
let l be Linear_Combination of {} ( ( the carrier of V ) \ { 0. V } ) ;
reconsider g = f " as Function of U2 , U1 , U2 ;
A1 in the points of ( G . k ) . ( X . k ) ;
|. - x .| = - ( - x ) .= - x .= - ( - x ) .= - x ;
set S = ) +* ( x , y , c ) ;
Fib ( n ) * ( 5 * Fib ( n ) ) / ( 5 * Fib ( n ) ) >= 4 * ( 5 * / / ( 5 * n ) ) ;
vM /. ( k + 1 ) = vM . ( k + 1 ) ;
0 mod i = - ( i * ( 0 qua Nat ) - ( i * ( 0 qua Nat ) ) ) ;
Indices M1 = [: Seg n , Seg n :] & len M2 = n & width M2 = n ;
Line ( S\mathopen ( i , j ) , j ) = S\mathopen ( j , i ) ;
h . ( x1 , y1 ) = [ y1 , x1 ] & h . ( y1 , y2 ) = [ y2 , x2 ] ;
|. f .| - ( Re ( |. f .| ) * ( ( card b ) * h ) ) is nonnegative ;
assume x = ( a1 ^ <* x1 *> ) ^ b1 & y = ( a1 ^ <* x1 *> ) ^ b1 ;
MW is_halting_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 - y ;
LIN c , q , b & LIN c , q , c & LIN c , q , b ;
f\rbrack . ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + z1 ) .= x1 + ( y1 + z1 ) ;
fE . a = ( f . a ) . a & v in InputVertices S & ( f . b ) . v 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 , E8 = Cage ( C , n ) ;
( p `1 ) ^2 >= ( ( E-max C ) `1 ) ^2 / ( ( E-max C ) `1 ) ^2 ;
consider p such that p = ( p . i ) `1 and s1 < p . i and p . i in rng s1 ;
|. ( f /* ( s * F ) ) . l - G . l .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len Line ( N , k + 1 + 1 ) = width N ;
f1 /* s1 is convergent & f2 /* s1 is convergent & lim ( f1 /* s1 ) = f1 . x0 & lim ( f2 /* s1 ) = f2 . x0 ;
f . x1 = x1 & f . y1 = y1 & f . y2 = y2 & f . y2 = x2 ;
len f <= len f + 1 & len f + 1 <> 0 & len f + 1 <> 0 ;
dom ( Proj ( i , n ) * s ) = REAL m & rng ( Proj ( i , n ) * s ) = REAL m ;
n = k * ( 2 * t ) + ( n mod ( 2 * k ) ) ;
dom B = 2 -tuples_on the carrier of V , C = { {} } --> 0. V ;
consider r such that r \not _|_ a and r _|_ x and r _|_ y ;
reconsider B1 = the carrier of Y1 , B2 = the carrier of Y2 , B1 = the carrier of X2 as Subset of X ;
1 in the carrier of [. 1 / 2 , 1 / 2 .] & 1 / 2 in dom ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( 1 / 2 ) ) ) ;
for L being complete LATTICE holds <* <* \mathclose { \rm c } , L *> , L *> *> is isomorphic
[ gi , gj ] in Ij \ ( Ij \ ( i \ { i } ) ) ;
set S2 = 1GateCircStr ( x , y , c ) , S2 = 1GateCircStr ( x , y , c ) ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and for r be Real st x0 < r ex g be Real st g < r & g < x0 & g < x0 & g < x0 & g in dom ( f1 + f2 ) ;
reconsider y = ( a " ) / ( F . ( a * F . ( a * b ) ) ) as Element of L ;
dom s = { 1 , 2 , 3 , 4 } & s . 1 = d1 & s . 2 = d2 ;
( min ( g , ( 1 - g ) ) ) . c <= h . c ;
set G2 = the subgraph of G , s3 = the \mathopen of G , v = the Vertex of G , e = the set of G , v = the Vertex of G , w = the set of G ;
reconsider g = f as PartFunc of REAL n , REAL-NS m , REAL-NS n ;
|. s1 . m - p .| / |. p .| < d / ( p - q ) / ( p - q ) ;
for x being element st x in ( \HM { u } ) holds x in ( \HM { the } \HM { carrier } \HM { of } L )
P = the carrier of ( TOP-REAL n ) | P & Q = the carrier of ( TOP-REAL n ) | P ;
assume that p01 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p01 ) and LSeg ( p1 , p01 ) /\ LSeg ( p01 , p2 ) = { p1 } ;
( 0. X \ x ) |^ ( m * ( k + 1 ) ) = 0. X ;
let g be Element of Hom ( cod f , cod f ) ;
2 * a * b + ( 2 * c ) * d <= 2 * C1 * C2 ;
let f , g , h be Point of the carrier of X , f be Function of X , Y ;
set h = Hom ( a , g ) ;
then ( idseq n ) | Seg m = ( idseq n ) | Seg m & m <= n ;
H * ( g " * a ) in the right of H * ( g * a ) ;
x in dom ( ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( (
cell ( G , i1 , j2 -' 1 ) misses C & cell ( G , i1 , j2 -' 1 ) misses C ;
LE q2 , p4 , P , p1 , p2 & LE q2 , q1 , P , p1 , p2 & LE q2 , q2 , P , p1 , p2 & LE q2 , q2 , P , p1 , p2 ;
attr B is BDD of A means : Def4 : B c= BDD A ;
deffunc D ( set , set , set ) = union rng $2 & $2 in rng $2 & $2 = ( $1 + 1 ) * $2 ;
n + - n < len ( p + - n ) + ( - n ) ;
pred a <> 0. K means : Def4 : the_rank_of M = the_rank_of ( a * M ) ;
consider j such that j in dom \mathbb \cap dom ' ( I , J ) and I = len b1 + j ;
consider x1 such that z in x1 and x1 in P8 and x = [ x1 , x1 ] ;
for n ex r being Element of REAL st X [ n , r ] & r <= n ;
set CS1 = Comput ( P2 , s2 , i + 1 ) , CS2 = Comput ( P2 , s2 , i + 1 ) , CS2 = Comput ( P2 , s2 , i + 1 ) , CS2 = Comput ( P2 , s2 , i + 1 ) , CS2 = Comput ( P2 , s2
set cv = 3 / ( a , b , c ) / ( a , b , c ) ;
conv @ W c= union ( F .: ( E " W ) ) & conv ( F " W ) c= union ( F " ( E " W ) ) ;
1 in [. - 1 , 1 .] /\ dom ( ( ( - 1 ) (#) ( ( #Z 2 ) * ( f1 + #Z 2 ) ) ) ) ;
r3 <= s0 + ( r0 - ( v2 - v1 ) ) / ( 2 * ( v2 - v1 ) ) ;
dom ( f * f4 ) = dom f /\ dom ( f * f4 ) .= dom f /\ dom ( f * f4 ) ;
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 , gq = gq , gq = gq , gp = g1 as Point of TOP-REAL n ;
( T * h . ( s . x ) ) . x = T . ( h . ( s . x ) ) ;
I . ( L . ( J . x ) ) = ( I * L ) . ( J . x ) ;
y in dom <* *> & ( Frege *> ) . ( ( Frege ( A . o ) ) . ( x , y ) ) ;
for I being non degenerated commutative commutative distributive non empty doubleLoopStr holds I is commutative iff I is commutative
set s2 = s +* ( ( intloc 0 ) .--> 1 ) , P2 = P +* ( ( intloc 0 ) .--> 1 ) , P3 = P +* ( ( intloc 0 ) .--> 1 ) , s4 = P +* ( ( intloc 0 ) .--> 1 ) , P4 = P +* ( ( intloc 0 ) .--> 1 ) , P4 = Comput
P1 /. IC s1 = P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 ;
lim S1 in the carrier of [. a , b .] & lim S1 = b & lim S1 = a & lim S2 = b ;
v . ( l-13 . i ) = ( v *' ( lpp . i ) ) . i ;
consider n being element such that n in NAT and x = ( sn " ) . n ;
consider x being Element of c such that F1 . x <> F2 . x and F1 . x <> 0 ;
Funcs ( X , 0 , x1 , x2 , x3 , x4 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 9 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 9 , 8 , 8 , 7 , 8 , 8 ,
j + ( 2 * ( k + 1 ) ) + m1 > j + ( 2 * ( k + 1 ) ) ;
{ s , t } on A3 & { s , t } on A3 & { s , t } on B2 ;
n1 > len crossover ( p2 , p1 , n1 , n2 , n3 , n3 , n3 , n4 , n4 , n4 , W , W , W ) ;
( mg1 ) . ( HT ( mg2 , T ) ) = 0. L ;
then H1 , H2 are_that H , H1 are_isomorphic & card H1 , H1 is_|^ H2 ;
( ( N-min L~ f ) .. ( fp ) ) .. ( fp ) > 1 & ( E-max L~ f ) .. ( fp ) > 1 ;
]. s , 1 .[ = ]. s , 2 .] /\ [. 0 , 1 .] & ]. s , 1 .] c= [. 0 , 1 .] ;
x1 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) ) & x2 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) ) ;
let f1 , f2 be continuous PartFunc of REAL , the carrier of S , f be PartFunc of REAL , the carrier of S ;
DigA ( t-23 , z9 ) is Element of k -tuples_on ( the carrier of K ) ;
I which not 2222) = d\vert & I is not empty & not I is non empty ;
u9 ~ = { [ a , u9 ] , [ b , u9 ] } ;
( w | p ) | ( p | ( w | w ) ) = p ;
consider u2 such that u2 in W2 and x = v + u2 and u1 in W2 and u2 in W2 ;
for y st y in rng F ex n st y = a |^ n & F . n = a |^ n
dom ( ( g * ( {} V \dot \to C ) ) | K ) = K ;
ex x being element st x in ( ( ( ( ( U0 ) \/ A ) ) \/ B ) . s ) ;
ex x being element st x in ( ( \HM { the } \HM { carrier of S } ) \/ A ) . s ;
f . x in the carrier of [. - r , r / 2 .] & f . x in [. - r , r / 2 .] ;
( ( the carrier of X1 union X2 ) /\ ( ( the carrier of X1 ) \/ ( the carrier of X2 ) ) <> {} ;
L1 /\ LSeg ( p01 , p2 ) c= { p01 } /\ LSeg ( p01 , p2 ) ;
( b + ( be ) ) / 2 in { r : a < r & r < b } ;
ex_sup_of { x , y } , L & x "\/" y = sup { x , y } ;
for x being element st x in X ex u being element st P [ x , u ]
consider z being Point of GX such that z = y and P [ z ] and z in A and z <> y ;
( ( the sequence of ( ( the sequence of ( X ) ) | ( the carrier of X ) ) ) . ( ( the carrier of X ) | ( the carrier of X ) ) . ( ( the carrier of X ) /\ ( the carrier of X ) ) . ( ( the carrier of X ) | ( the carrier of
len ( w ^ w2 ) + 1 = len w + 2 + 1 .= len w + 2 + 1 ;
assume q in the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 & q <> 0. TOP-REAL 2 ;
f | E-4 ` = g | E-4 ` .= g | E-4 ` .= g | E-4 ` .= g | E-4 ` .= g | E-4 ;
reconsider i1 = x1 , i2 = x2 , z = x3 , n = x4 , m = x4 , n = 6 , n = 7 , m = 6 , n = 8 , m = 6 , n = 6 , m = 7 , n = 6 , m = 6 , n = 6 , m = 6 , n = 6 , m = 6 , n =
( a * A * B ) @ = ( a * ( A * B ) ) @ ;
assume ex n0 being Element of NAT st f |^ n0 is min & f . ( n0 + 1 ) = R ;
Seg len ( ( ( ( ( ( f2 ) * f1 ) ) | Seg ( len ( f2 ) ) ) ) ) = dom ( ( ( ( f1 ) * f2 ) | Seg ( len ( f2 ) ) ) ) ;
( Complement A1 ) . m c= ( Complement A1 ) . n & ( Complement A1 ) . n c= ( Complement A1 ) . n ;
f1 . p = p8 & g1 . ( p8 ) = d & g2 . ( p8 ) = d & f1 . ( ( - d ) * p ) = d ;
FinS ( F , Y ) = FinS ( F , dom ( F | Y ) ) ;
( x | y ) | z = z | ( y | x ) ;
( ( |. x .| ) |^ n ) / ( ( |. x .| ) |^ n ) <= ( ( |. x .| ) |^ n ) ;
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 W3 and W2 is Subspace of W1 and W1 /\ W2 is Subspace of W2 ;
||. ( t . x ) - ( t . x ) .|| = lim ( ||. ( x - y ) - ( x - y ) .|| ) ;
assume that i in dom D and f | A is lower and g | A is lower ;
( ( p `2 ) ^2 + ( p `2 ) ^2 ) <= ( ( - ( p `2 ) ) / ( 1 + ( p `2 ) ^2 ) ) ;
g | Sphere ( p , r ) = id ( Sphere ( p , r ) ) & g | Sphere ( p , r ) = id ( Sphere ( p , r ) ) ;
set N8 = ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ;
for T being non empty TopSpace holds T is countable countable implies the TopStruct of T is countable countable
width B |-> 0. K = Line ( B , i ) .= Line ( B , i ) .= B * ( i , i ) ;
pred a <> 0 means : Def4 : ( A \diffsym B ) c= ( A Y. ) \diffsym ( B Let a ) ;
then f is_is_\cal 2 2 2 , u & pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 ;
assume that a > 0 and a <> 1 and b > 0 and c > 0 and d > 0 and c > 0 and d > 0 ;
w1 , w2 in Lin { w1 , w2 } & w2 in Lin { w1 , w2 } implies w1 in { w1 , w2 }
p2 /. IC Comput ( p2 , s2 , k ) = p2 . IC Comput ( p2 , s2 , k ) .= ( IC Comput ( p2 , s2 , k ) ) .= ( card I + ( card I + 1 ) ) ;
ind ( T-10 | b ) = ind b .= ind B .= ind b .= ind b ;
[ a , A ] in the \mathclose of Line ( A ) & [ a , A ] in the carrier of Line ( A , 1 ) ;
m in ( the Arrows of C ) . ( o1 , o2 ) & ( the Arrows of C ) . ( o2 , o1 ) = ( the Arrows of C ) . ( o1 , o2 ) ;
( ( a , CompF ( PA , G ) ) ) . z = FALSE or ( ( a , CompF ( PA , G ) ) ) . z = FALSE ;
reconsider phi = phi /. 11 , phi = phi /. 2 , phi = phi /. 3 as Element of ( be Element of z1 ) ;
len s1 - ( len s2 - 1 ) > 0 + 1 - ( len s2 - 1 ) ;
delta ( D ) * ( f . ( upper_bound A ) - f . ( lower_bound A ) ) < r ;
[ f21 , f22 ] in the carrier' of A ~ & [ f22 , f22 ] in the carrier' of A ;
the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 = K1 & the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 = K1 ;
consider z being element such that z in dom g2 and p = g2 . z and q = g2 . z ;
[#] V1 = { 0. V1 } .= the carrier of ( ( the carrier of V1 ) \ { 0. V1 } ) \ { 0. V1 } .= { 0. V1 } ;
consider P2 being FinSequence such that rng P2 = M and P2 is one-to-one and len P2 = len M and for k be Nat st k in Seg len P2 holds P2 . k = F ( k ) ;
assume that x1 in dom ( f | X ) and ||. x1 - x0 .|| < s and s < ( f | X ) . x1 ;
h1 = f ^ ( <* p3 *> ^ <* p *> ) .= h ^ <* p *> .= h ^ <* p *> ;
c / ( |[ b , c ]| ) = c .= c / ( |[ a , c ]| ) .= c / ( |[ a , c ]| ) ;
reconsider t1 = p1 , t2 = p2 , t2 = p3 , t1 = p2 as Term of C , V ;
( 1 - ( 2 * PI ) ) / ( 1 - ( 2 * PI ) ) in the carrier of ( ( TOP-REAL 2 ) | P ) ;
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 ) ;
R . ( b - a ) = 2 * \cal - b .= 2 * PI - b .= 2 * PI - b ;
consider \rangle such that B = ( - 1 ) * C + ( - 1 ) * A and 0 <= 1 ;
dom g = dom ( ( the Sorts of A ) * ( a , I ) ) & dom ( ( the Sorts of A ) * ( b , I ) ) = dom ( ( the Sorts of A ) * ( b , I ) ) ;
[ P . ( l ) , P . ( l ) ] in => ( ( T . ( l ) ) , T . ( l ) ) ;
set s2 = Initialize s , P2 = P +* I ;
reconsider M = mid ( z , i2 , i1 ) , N = mid ( z , i2 , i1 ) as non empty Subset of TOP-REAL 2 ;
y in product ( ( Carrier J ) +* ( V , { 1 } ) ) ;
1 / ( |[ 0 , 1 ]| ) = 1 & 0 / ( |[ 0 , 1 ]| ) = 0 & 1 / ( |[ 0 , 1 ]| ) = 1 ;
assume x in the left of g or x in the left of g or x in the right of g & x = g ;
consider M being strict Subspace of A9 such that a = M and T is Subspace of M and M is o of M ;
for x st x in Z holds ( ( ( ( 1 / 2 ) (#) f ) `| Z ) . x <> 0 & ( ( ( 1 / 2 ) (#) f ) `| Z ) . x <> 0
len W1 + len W2 + m = 1 + len W3 + m + len W2 .= len ( W1 + W2 ) + len W2 ;
reconsider h1 = ( ( vseq . n ) - ( t - t ) ) as Lipschitzian LinearOperator of X , Y ;
( i mod len ( p + q ) ) + 1 in dom ( p + q ) ;
assume that s2 is from s1 , s1 and F in the |= of s2 and F in the v of s2 and F in the v of s2 ;
( ( gcd ( x , y ) ) * ( 1 , 3 ) ) = gcd ( x , y , 3 ) * ( 1 , 2 ) ;
for u being element st u in Bags n holds ( p `2 + m ) . u = p . u + m
for B being Subset of u9 st B in E holds A = B or A misses B or B misses E ;
ex a being Point of X st a in A & A /\ Cl { y } = { a } ;
set W2 = tree ( p ) \/ W1 , W1 = tree ( q ) , W2 = tree ( p ) , W1 = q ;
x in { X where X is Ideal of L : X is Ideal of L & 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 + b ) = + ( 1 / ( a + b ) ) * ( b + c ) ;
( ( X --> f ) . x ) = ( X --> dom f ) . x .= ( X --> dom f ) . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) ;
p => ( q => r ) => ( p => q ) in carrier ( A ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
- 1 + 1 <= ( ( C |^ ( n -' m ) ) - 1 ) + 1 ;
( reproj ( 1 , z0 ) ) . x in dom ( ( f1 (#) f2 ) . ( x - x0 ) ) ;
assume that b1 . r = { c1 } and b2 . r = { c2 } and c1 . r = c2 and c2 . r = c2 ;
ex P st a1 on P & a2 on P & a1 on P & a2 on P & a1 on P & a2 on P & a2 on P & a1 on P & a2 on P & a2 on P & a1 on P & a2 on P & a2 on P & a2 on P & a1 on P & a2 on P & a2 on P & a2 on P
reconsider gf = g `2 * f `2 , hg = h `2 * g `2 as strict Subgroup of X ;
consider v1 being Element of T such that Q = ( downarrow v1 ) ` and v1 in F and v2 in G ;
n in { i where i is Nat : i < n0 + 1 & i < n + 1 } ;
( F * ( i , j ) ) `2 >= ( F * ( m , k ) ) `2 ;
assume K1 = { p : p `1 >= cn & p <> 0. TOP-REAL 2 } & K1 <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 ;
ConsecutiveSet ( A , succ O1 ) = ( ConsecutiveSet ( A , O1 ) ) . ( succ O1 ) .= ( ConsecutiveSet ( A , O1 ) ) . ( succ O1 ) ;
set I1 = Macro ( a , intloc 0 ) , I2 = AddTo ( a , intloc 0 ) , I2 = AddTo ( a , intloc 0 ) , I2 = intloc 0 , I2 = intloc 0 , I2 = intloc 0 , I2 = intloc 0 , I2 = intloc 0 , I3 = intloc 0 , I5 = intloc 0 , I5 = intloc 0 , I5 = intloc 0 , I
for i be Nat st 1 < i & i < len z holds z /. i <> z /. 1 & z /. ( 1 + i ) <> z /. ( 1 + i )
X c= ( the carrier of L1 ) /\ ( the carrier of L2 ) & X c= ( the carrier of L1 ) /\ ( the carrier of L2 ) ;
consider xx be Element of GF ( p ) such that xx |^ 2 = a & xx |^ 2 = b ;
reconsider e3 = e4 , f5 = f5 , f6 = ( f . ( x + 1 ) ) * ( f . ( x + 1 ) ) as Element of D ;
ex O being set st O in S & C1 c= O & M . O = 0. ( M . O ) ;
consider n being Nat such that for m being Nat st n <= m holds S . m in U1 and S . m in U1 ;
f * g * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) * reproj ( i , x ) ) . x ;
defpred P [ Nat ] means A + succ $1 = succ A + ( succ $1 ) & A c= succ $1 implies A c= succ $1 ;
the left of - g = the left of - ( g - f ) .= ( - g ) * ( - f ) .= ( - g ) * ( - f ) ;
reconsider pp = x , pp = y , pp = z , pp = z , pp = w , pp = z , y2 = w , j1 = z , y2 = y , y2 = z as Point of TOP-REAL 2 ;
consider g2 such that g2 = y and x <= g2 and g2 <= x0 and x0 <= g2 and g2 <= x0 and g2 <= x0 ;
for n being Element of NAT ex r being Element of REAL st X [ n , r ] & X [ n , r ]
len ( x2 ^ y2 ) = len x2 + len y2 .= len x2 + len y2 + len y2 .= len x2 + len y2 + len y2 .= len x2 + len y2 + len y2 ;
for x being element st x in X holds x in the set of the set of f & x in the set of f & f . x = f . ( n + 1 ) ;
LSeg ( p01 , p2 ) /\ LSeg ( p1 , p01 ) = {} or LSeg ( p1 , p01 ) /\ LSeg ( p1 , p01 ) = {} ;
func for X be set means : Def4 : for f being Function of X , X holds it = <* h . ( id X ) , h . ( id X ) , h . ( id X ) *> ;
len ( ( { Gauge ( C , n ) * ( len Gauge ( C , n ) , 1 ) ) * ( len Gauge ( C , n ) , 1 ) ) <= len C ;
attr K is being being being being being being being being being Field of K , a , b , c being Element of K ;
consider o being OperSymbol of S such that t `1 . {} = [ o , the carrier of S ] and o in dom ( t `1 ) ;
for x st x in X ex y st x c= y & y in X & y in Y & x <> y
IC Comput ( P1 , s1 , k ) in dom ( Comput ( P2 , s2 , k ) +* I ) ;
pred q < s & r < s implies ]. r , s .] /\ ]. p , q .] c= ]. r , s .] & ]. s , q .] c= ]. p , q .] ;
consider c being Element of Class ( f , c ) such that Y = ( F . c ) `1 and c in { F . c , F . c , F . c } ;
func the ResultSort of S2 -> Function means : Def4 : the ResultSort of it = id the carrier' of S2 & the ResultSort of it = id the carrier' of S2 ;
set y9 = [ <* y , z *> , f2 ] , z9 = [ <* z , x *> , f3 ] , y9 = [ <* x , y *> , f3 ] ;
assume x in dom ( ( ( ( 1 / 2 ) (#) ( ( #Z 2 ) * ( f1 + #Z 2 ) ) ) `| Z ) ;
r-7 in Int cell ( GoB f , i , GoB f ) \ L~ f \/ L~ f \/ L~ f /\ L~ f \/ L~ f /\ L~ f \/ L~ f /\ L~ f ;
( q `2 ) ^2 >= ( ( Cage ( C , n ) /. ( i + 1 ) ) `2 ;
set Y = { a "/\" a ` : a in X } ;
i - len f <= len f + len f1 - len f + len f - len f + len f - len f + len f - len f + len f - len f + len f - len f + len f - len f + len f - len f + len f - len f + len f - len f + len f - len f + len f - len f + len f -
for n ex x st x in N & x in N1 & h . n = x- ( x0 - r / 2 )
set s0 = ( ( \mathop { \it true } ( a , I , p , s ) ) +* ( i , n ) ) . a ;
( p . k ) . 0 = 1 or ( p . k ) . 0 = - 1 or p . k = - 1 & p . k = - 1 ;
u + Sum ( L-18 in U \ { u } ) \/ { u + Sum ( L-18 ) } ;
consider xx being set such that x in xx and xx in V1 and x = [ xx , xx ] ;
( p ^ ( q | k ) ) . m = ( q | k ) . ( ( len p ) - k ) ;
g + h = gg + h1 & A1 + h = g + h + X & A2 + h = g + h + X ;
L1 is distributive & L2 is distributive implies L1 ~ is distributive & L2 ~ is distributive & L1 ~ is distributive & L2 is distributive & L1 is distributive & L2 is distributive & L1 is distributive & L2 is distributive implies L1 * L2 is distributive
pred x in rng f & y in rng ( f | x ) implies f . x = f . y & f . y = f . x ;
assume that 1 < p and ( 1 - p ) * q + ( 1 - p ) * q = 1 and 0 <= a and a <= 1 and p <= 1 ;
F* ( f , <* <* *> *> ) = rpoly ( 1 , the carrier of F_Complex ) *' + ( 1. F_Complex ) *' .= 1. F_Complex ;
for X being set , A being Subset of X holds A ` = {} implies A = {} & A = {} & A = {} ;
( ( E-max X ) `1 ) ^2 / ( ( E-max X ) `1 ) ^2 <= ( ( E-max X ) `1 ) ^2 / ( ( E-max X ) `1 ) ^2 / ( ( E-max X ) `1 ) ^2 ;
for c being Element of the Sorts of A , a being Element of the Sorts of A holds c <> a implies c <> a
s1 . GBP = ( Exec ( i2 , s2 ) ) . GBP .= Exec ( i2 , s2 ) . GBP .= Exec ( i2 , s2 ) . GBP .= 0 ;
for a , b being Real holds [ a , b ] in ( y >= 0 ) implies b >= 0 & a >= 0 & b >= 0
for x , y being Element of X holds x ` \ y = ( x \ y ) ` & y \ x = ( x \ y ) ` implies x = y
mode BCK-algebra of i , j , m , n , m , n , m , m , n , m , n , m , m , n , m , m , n , m , n , m , n , m , m , n , m , n , m , n ;
set x2 = |( ( Re y ) , ( Im x ) * ( Im y ) )| ;
[ y , x ] in dom u9 & u9 . ( y , x ) = g . y & u9 . ( y , x ) = g . y ;
]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .] c= A & A c= B & B c= A ;
0 <= delta ( S2 . n ) & |. delta ( S2 . n ) .| < ( e / 2 ) / 2 ;
( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) <= ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ;
set A = ( 2 / ( 2 * b-a ) ) ;
for x , y being set st x in R" holds x , y are_\hbox { - } implies x , y are_\hbox { - }
deffunc FF ( Nat ) = b . ( $1 + 1 ) * ( M * G . $1 ) ;
for s being element holds s in -> Element of -> Element of -> Element of S holds s in -> Element of -> Element of S \/ x2 ;
for S being non empty non void non void non empty RelStr for T being connected non empty TopSpace st S is connected holds S is connected iff T is connected
max ( ( degree ( z ) ) / ( 1 - ( degree ( z ) ) / ( 1 - ( degree ( z ) ) / ( 1 - ( degree ( z ) ) / ( 1 - ( degree ( z ) ) / ( 1 - ( degree ( z ) ) / ( 1 - ( degree ( z ) ) / ( 1 - ( degree ( z ) ) / ( 1 - ( degree ( z
consider n1 being Nat such that for k holds seq . ( n1 + k ) < r + s and for n holds seq . ( n + k ) < r + s ;
Lin ( A /\ B ) is Subspace of Lin ( A ) & Lin ( B ) is Subspace of Lin ( B ) ;
set n-15 = n-13 '&' ( M . x qua Element of BOOLEAN ) , n-15 = M . ( M . x ) , n-15 = M . ( M . x ) , n-15 = M . ( M . x ) , n-15 = M . ( M . x ) , n-15 = M . ( M . x ) , n-15 = M . ( n + 1 ) ;
f " V in such that V ( X ) in D ( ) & f " V in D ( Y , p ) & p in D ( ) & p in D ( ) ;
rng ( ( ( a , c ) sequence of c ) +* ( 1 , b ) ) c= { a , c , b , c } ;
consider y being connected Walk of G1 such that y `1 = y and dom y `1 = WW: 2 and dom y `1 = WWW: 2 ;
dom ( ( 1 / f ) (#) ( ]. x0 - r , x0 .[ ) ) c= ]. x0 - r , x0 .[ /\ dom ( f | ]. x0 , x0 .[ ) ;
as Matrix of i , j , n , r , s be Element of v1 ( i , j , n , - r ) ;
v ^ ( ( n |-> 0 ) ^ ( n |-> 0 ) ) in Lin ( rng ( ( B | c1 ) +* ( n |-> 0 ) ) ) ;
ex a , k1 , k2 st i = a := k1 & k2 = b := k2 & k2 = b := k2 & k2 = b := k2 ;
t . ( NAT ) = ( NAT .--> succ i1 ) . ( NAT + 1 ) .= succ i1 .= succ i1 .= succ i1 .= succ i1 .= succ i1 .= succ i1 ;
assume that F is bbfamily and rng p = F and dom p = Seg ( n + 1 ) and for k be Nat st k in Seg ( n + 1 ) holds p . k = F ( k ) ;
not LIN b , b9 , a & not LIN a , b , c & LIN a , b , c & LIN a , b , c & LIN b , c , a & LIN a , c , b ;
( L1 v v v or L2 . O c= ( L1 => L2 ) . O ) & ( L1 . O ) => ( L2 . O ) in L ;
consider F being ManySortedSet of E such that for d being Element of E holds F . d = F ( d ) and for d be Element of E holds F . d = G ( d ) ;
consider a , b such that a * ( contradiction ) = b * ( -w ) and 0 < a and a < b ;
defpred P [ FinSequence of D ] means |. Sum ( $1 ) .| <= Sum ( $1 ) & ( Sum ( $1 ) ) = Sum ( $1 ) ;
u = cos / ( x , y ) . v * x + ( cos / ( x , y ) . v ) .= v ;
dist ( seq . n ) . ( n + x , g + x ) <= dist ( ( seq . n ) . ( n + x , g ) ) + 0 ;
P [ p , |. p .| ^ |. p .| ^ |. p .| , ( id the Sorts of A ) . p ] ;
consider X being Subset of Al such that X c= Y and X is non empty and X is non empty and X is non empty and X is non empty ;
|. b .| * |. eval ( f , z ) .| >= |. b .| * |. eval ( f , z ) .| ;
1 < ( ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ) .. Cage ( C , n ) ;
l in { l1 where l1 is Real : g <= l1 & l1 <= h . l1 & l1 <= h . l1 } ;
vol ( ( G . n ) . ( vol ( G . n ) ) ) <= ( ( G . n ) . ( vol ( G . n ) ) ) * vol ( G . n ) ;
f . y = x .= x * 1. L .= x * ( 1. L ) .= x * ( ( power L ) . ( y , 0 ) ) ;
NIC ( ( a , i1 ) --> ( i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ i1 ) ) ) ) ) ) ) ) = { i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1 , succ ( i1
LSeg ( p01 , p2 ) /\ LSeg ( p1 , p01 ) = { p1 } /\ LSeg ( p1 , p01 ) .= { p1 } /\ LSeg ( p1 , p01 ) ;
Product ( ( ( the support of I-15 ) +* ( i , { 1 } ) ) +* ( i , { 1 } ) ) in Z ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) +* Following ( s2 , n ) .= Following ( s1 , n ) +* Following ( s2 , n ) ;
( W-bound Q2 ) / ( 1 - ( p `1 / |. p .| - cn ) / ( 1 - cn ) ) <= ( ( q `1 / |. q .| - cn ) / ( 1 - cn ) ) / ( 1 - cn ) ;
f /. i2 <> f /. ( ( i1 + len g -' 1 ) + ( len g -' 1 ) ) ;
M , v / ( x. 3 , a ) / ( x. 4 , a ) / ( x. 0 , a ) / ( x. 4 , a ) / ( x. 0 , a ) / ( x. 4 , a ) / ( x. 0 , a ) / ( x. 0 , a ) / ( x. 4 , a ) / ( x. 0 , a ) / ( x. 0 , a ) / ( x. 4 , a ) / ( x. 0 , a ) ) ;
len ( P ^ Q ) in dom ( P ^ Q ) & len ( P ^ Q ) = len ( P ^ Q ) + len ( P ^ Q ) ;
A |^ ( n , m ) c= A |^ ( m , n ) & A |^ ( k , m ) c= A |^ ( k , l ) ;
R |^ n \ { q : |. q .| < a } c= { q1 : |. q1 .| >= a }
consider n1 being element such that n1 in dom p1 and y1 = p1 . ( n1 + 1 ) and 1 <= n1 and n1 <= len p1 ;
consider X being set such that X in Q and for Z being set st Z in Q & Z <> X holds X \not c= Z ;
CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA .= halt SCM+FSA .= halt SCM+FSA .= halt SCM+FSA .= halt SCM+FSA ;
for v be VECTOR of l1 holds ||. v .|| = upper_bound rng |. ( the Sorts of l ) . v .| & ||. v .|| = upper_bound rng |. ( the Sorts of l ) . v .|
for phi holds phi in X implies phi in X & phi in X & phi in X & phi in X & phi in X implies phi in X
rng ( ( ( Sgm dom ( f | dom ( f | dom ( f | dom ( f | dom ( i + 1 ) ) ) ) ) | dom ( ( f | dom ( f | dom ( i + 1 ) ) ) ) ) ) c= dom ( ( f | dom ( f | dom ( i + 1 ) ) ) | dom ( ( f | dom ( i + 1 ) ) ) | dom ( ( f | dom ( i + 1
ex c being FinSequence of D ( ) st len c = k & P [ c ] & a = c & b = c ;
the_arity_of ( ( a , b , c ) --> ( b , c ) ) = <* \in ( b , c ) --> ( a , b ) , ( c , d ) --> ( b , d ) *> ;
consider f1 be Function of the carrier of X , REAL such that f1 = |. f .| and f1 is continuous and f1 . 0 = ( f1 - f2 ) . 1 ;
a1 = b1 & a2 = b2 or a1 = b1 & a2 = b2 & a3 = b3 & a4 = b2 & a4 = b3 & a5 = 6 & a5 = 6 & a5 = 6 & a5 = 6 & a5 = 7 & 8 = 6 & 8 = 6 & 8 = 6 & 8 = 6 & 8 = 6 & 8 = 6 & 8 = 6 & 8 = 6 & 8 = 6 & 7 = 6 & 8 = 6 & 8 = 6
D2 . indx ( D2 , D1 , n1 + 1 ) = D1 . ( n1 + 1 ) & D2 . indx ( D2 , D1 , n1 ) = D2 . ( n1 + 1 ) ;
f . ( ]. r , r .[ ) = ( |. r .| ) /. 1 .= <* r .| /. 1 .= <* r *> . 1 .= x ;
consider n being Nat such that for m being Nat st n <= m holds C-25 . m = C-25 . m and C . m = C-25 . ( m + 1 ) ;
consider d be Real such that for a , b be Real st a in X & b in Y holds a <= d & b <= d holds a <= b ;
||. L /. h .|| - ( K * |. h .| ) + ( K * |. h .| ) - ( K * |. h .| ) <= p0 + ( K * |. h .| ) ;
attr F is commutative associative means : Def4 : for b being Element of X holds F \hbox { b } = f . b & F . ( b , f . b ) = f . b ;
p = - ( - ( p0 + 0. TOP-REAL 2 ) ) * ( ( p0 - 0. TOP-REAL 2 ) * ( ( p0 - 0. TOP-REAL 2 ) * ( ( p0 - 0. TOP-REAL 2 ) * ( p - 0. TOP-REAL 2 ) ) ) ) .= 1 * ( ( p - 0. TOP-REAL 2 ) * ( p - 0. TOP-REAL 2 ) ) .= ( ( p - 0. TOP-REAL 2 ) * ( p - 0. TOP-REAL 2 ) ) * ( p -
consider z1 such that b , x1 , z1 is_collinear and o , x1 , z1 is_collinear and o , x1 , z1 is_collinear and o , x1 , z1 is_collinear and o <> z1 and o <> z1 and o <> z2 ;
consider i such that Arg ( ( Rotate ( s ) ) . q ) = s + Arg ( q ) + ( 2 * PI * i ) ;
consider g such that g is one-to-one and dom g = f . x and rng g c= f . x and g . x = f . x ;
assume that A = P2 \/ Q2 and P2 <> {} and P2 <> {} and P2 <> {} and P2 <> {} and P2 <> {} and P2 <> {} and P2 <> {} and P2 <> {} and P2 <> {} and P2 <> {} and P2 <> {} and P2 <> {} and P2 <> {} and P2 <> {} and P2 <> {} and P2 <> {} and P2 <> {} and P2 <> {} and P2 <> {} and P2 <> {} and P2 <> {} and P2
attr F is associative means : Def4 : F .: ( F .: ( f , g ) , h ) = F .: ( f , g ) ;
ex x being Element of NAT st m = x `1 & x in z `1 & m < i or m in { i } & i in { i } ;
consider k2 being Nat such that k2 in dom P-2 and l = P-2 . k2 and ( l + 1 ) <= k2 + 1 ;
seq = r * seq implies for n holds seq . n = r * seq . n & seq . n = r * seq . ( n + 1 )
F1 . [ ( ( id a ) * [ a , b ] ) , ( id b ) * [ b , a ] ] = f * ( id a ) ;
{ p } "\/" D2 = { p "\/" y where y is Element of L : y in D2 } & { p "\/" q where q is Element of L : q in D2 } = { p "\/" q : q in D2 } ;
consider z being element such that z in dom ( ( dom F ) . 0 ) and ( ( F . 0 ) . 1 ) . z = y ;
for x , y being element st x in dom f & y in dom f & f . x = f . y holds x = y
cell ( G , i , 1 ) = { |[ r , s ]| : r <= G * ( 0 + 1 , 1 ) `1 } ;
consider e being element such that e in dom ( T | E1 ) and ( T | E1 ) . e = v and ( T | E1 ) . e = v ;
( F `1 ) * b1 . x = ( Mx2Tran ( J , b1 , b2 ) ) . ( \mathbb j + len ( b1 . j ) ) .= ( ( Mx2Tran J ) * b2 ) . j ;
- 1 / ( - 1 / 2 ) = ( - m ) (#) D .= ( - m ) (#) D .= ( - m ) (#) D .= ( - m ) (#) D .= ( - m ) (#) D .= ( - m ) (#) D ;
attr x be set means : Def4 : 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 ) . ( i + 1 ) ) .= len ( ( f2 . j ) . ( i + 1 ) ) ;
All ( 'not' All ( a , A , G ) , B , G ) '<' Ex ( 'not' All ( 'not' All ( a , B , G ) , A , G ) , B , G ) ;
LSeg ( E . ( k + 1 ) , F . ( k + 1 ) ) c= Cl RightComp Cage ( C , n + 1 ) /\ L~ Cage ( C , n + 1 ) ;
x \ a |^ m = x \ ( a |^ k * a ) .= ( ( x \ a ) |^ k ) \ a ;
k >= ( ( commute th ) . k ) . ( ( commute >= 0 ) . k ) .= ( commute ( I ) . k ) . ( ( commute I ) . k ) .= ( commute I ) . ( ( commute I ) . k ) .= ( commute I ) . ( ( commute I ) . k ) .= ( commute I ) . ( ( commute I ) . k ) ;
for s being State of Athere being State of A1 holds Following ( s , n ) . ( 0 + ( 2 * n + 1 ) ) is stable ;
for x st x in Z holds f1 . x = a / ( 2 * x ) & ( f1 - f2 ) . x <> 0 & ( f1 - f2 ) . x <> 0 ) implies f1 - f2 <> 0
support ( ( support ( m ) ) \/ support ( ( m ) ) c= support ( ( m ) ) /\ support ( ( m ) ) ) ;
reconsider t = u as Function of ( the carrier of A ) , ( the carrier' of B ) * the Arity of C , s be Element of the carrier' of C ;
- ( a * sqrt ( 1 + ( a ^2 + b ^2 ) ) / ( 1 + ( a ^2 + b ^2 ) ) ) <= - ( b * sqrt ( 1 + ( a ^2 + b ^2 ) ) / ( 1 + ( a ^2 + b ^2 ) ) ) ;
phi /. ( succ b1 ) = g . a & phi /. ( succ b1 ) = f . ( g . a ) & phi /. ( succ b1 ) = f . ( g . a ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( ( F ^ <* p *> ) . i ) and i <> j ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , 6 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 ,
the Sorts of ( U1 /\ ( U1 "\/" U2 ) ) c= the Sorts of ( U1 "\/" U2 ) ;
( - ( 2 * a ) * ( b - a ) ) / ( 2 * a ) + ( - 2 * a ) / ( 2 * a ) > 0 ;
consider W00 such that for z being element holds z in W00 iff z in N ~ & P [ z ] and P [ z ] ;
assume that ( the Arity of S ) . o = <* a *> and ( the Arity of S ) . o = r and ( the Arity of S ) . o = <* a *> and ( the Arity of S ) . o = r ;
Z = dom ( ( ( exp_R * ( ( ( #Z n ) * ( f1 + #Z n ) ) ) / ( f1 + #Z n ) ) ) ;
sum ( f , SS1 ) is convergent & lim ( upper_volume ( f , SS1 ) ) = integral ( f , S ) - integral ( f , S ) ;
( for a9 holds ( a . ( f . ( a ) ) => ( g . ( x9 ) ) ) in
len ( M2 * M3 ) = n & width ( M2 * M3 ) = n & width ( M2 * M3 ) = n & width ( M2 * M3 ) = n & width ( M2 * M3 ) = n ;
attr X1 union X2 is open means : Def4 : X1 , X2 are_separated & X2 , X1 are_separated & X1 , X2 are_separated & X2 , X1 are_separated implies X1 , X2 are_separated & X2 , X1 union X2 are_separated ;
for L being lower-bounded antisymmetric RelStr , X being non empty RelStr , X being non empty Subset of L holds X "\/" { Top L } = { Top L }
reconsider f-1-1be ( b ) `2 = ( F . ( b ) ) `2 , f-1b = ( F . ( b ) ) `2 , f-1b = ( F . ( b ) ) `2 , f-1b = ( F . ( b ) ) `2 ;
consider w being FinSequence of I such that the InitS of M is_ststst<* s *> ^ w ^ w ^ q and q ^ w ^ w ^ w ^ w ^ q in dom q ;
g . ( a |^ 0 ) = g . ( 1_ G ) .= 1_ G .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) ;
assume for i being Nat st i in dom f ex z being Element of L st f . i = rpoly ( 1 , z ) & z in dom f & f . i = rpoly ( 1 , z ) ;
ex L being Subset of X st Carrier L = L & for K being Subset of X st K in C holds L /\ K <> {} ;
( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C1 & ( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C1 ;
reconsider o9 = o `1 , t = o `2 , p = o `1 , q = o `2 , r = o `2 , s = o `2 , s = o `2 , t = o `2 , s = s , w = t , w = s , s = t , w = t , s = s , w = t , w = s , w = t , s = s , w = t , s = s , w = t , s = s , w = s , w = s ,
1 * x1 + ( 0 * x2 + ( 0 * x3 ) ) + ( 0 * x2 + ( 0 * x3 ) ) = x1 + ( ( 0 * x2 ) + ( 0 * x3 ) ) .= x1 + ( ( 0 * x3 ) + ( 0 * x4 ) ) .= x1 + ( ( 0 * x2 ) + ( 0 * x3 ) ) .= x1 + x2 ;
EP " . 1 = ( EP " ) . 1 .= ( ( ( P " ) * ( ( P " ) * ( P " ) ) ) * ( 1 - 1 ) ) .= ( ( 1 - 1 ) * ( ( P " ) * ( P " ) ) ) * ( 1 - 1 ) ;
reconsider u1 = the carrier of U1 /\ ( U1 "\/" U2 ) , v1 = the carrier of U1 , v2 = the carrier of U2 , f = ( U1 "\/" U2 ) "\/" ( U1 "\/" U2 ) as non empty Subset of U0 ;
( ( x "/\" z ) "\/" ( x "/\" y ) ) "\/" ( z "/\" y ) <= ( x "/\" ( z "\/" y ) ) "\/" ( z "/\" ( y "\/" x ) ) ;
|. f . ( s1 . ( l1 + 1 ) ) - f . ( s1 . l1 ) .| < ( 1 / ( M . ( l1 + 1 ) ) - f . ( s1 . l1 ) ) ;
LSeg ( ( Upper_Seq ( C , n ) ) * ( i , j ) , ( Upper_Seq ( C , n ) ) * ( i + 1 , j ) ) is vertical ;
( f | Z ) /. x - ( f | Z ) /. x0 = L /. ( x- - x0 ) + R /. ( x- - x0 ) ;
g . c * ( - g . c ) + f . c * f . c <= h . c * ( - f . 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 set of \HM { the } \HM { set , b , c , d } and len f = width A and width f = width B and width A = width B and width A = width B ;
len ( - M1 ) = len M1 & width ( - M2 ) = width M1 & width ( - M2 ) = width M2 & width ( - M2 ) = width M1 ;
for n , i being Nat st i + 1 < n holds [ i , i + 1 ] in the InternalRel of ( \cal E ( n ) ) \ the InternalRel of ( TOP-REAL n ) \ the InternalRel of ( TOP-REAL n ) )
pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 ;
pred a <> 0 & b <> 0 & Arg a = Arg b & Arg ( - a ) = Arg ( - b ) & Arg ( - a ) = Arg ( - b ) & Arg ( - a ) = Arg ( - b ) ;
for c being set st not c in [. a , b .] holds not c in Intersection ( the } of
assume that V1 is linearly closed and V2 is closed and V = { v + u : v in V1 & u in V1 & u in V1 } and v in V1 and u in V1 and v in V1 and u in V1 and v in V1 ;
z * x1 + ( 1 - z ) * x2 in M & z * ( x1 + x2 ) in N & ( z * x1 + ( 1 - z ) * x2 ) in N ;
rng ( ( ( P qua Function ) " ) * S6 ) = Seg ( card d6 ) .= Seg ( card d6 ) .= Seg ( card d6 ) .= Seg ( card d6 ) .= Seg ( card d6 ) ;
consider s2 being rational Real_Sequence such that s2 is convergent and b = lim s2 and for n holds s2 . n <= b . n and s2 . n <= b . n ;
h2 " . n = h2 . n " & 0 < ( - 1 / ( n + 1 ) ) / ( n + 1 ) & 0 < ( - 1 / ( n + 1 ) ) / ( n + 1 ) ;
( Partial_Sums ( ||. ( seq1 ) .|| ) . m ) = ||. ( seq1 . m ) - ( seq2 . m ) .|| .= ||. ( seq1 . m ) - ( seq2 . m ) .|| .= ||. ( seq1 . m ) - ( seq2 . m ) .|| .= ||. ( seq1 . m ) - ( seq2 . m ) .|| .= 0 ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b .= Comput ( P2 , s2 , 1 ) . b ;
- v = ( - 1_ G ) * v & - w = ( - 1_ G ) * v & - w = ( - 1_ G ) * v & - w = - ( - 1_ G ) * v ;
sup ( ( k .: D ) .: D ) = sup ( ( k .: D ) .: D ) .= sup ( ( k .: D ) .: D ) .= sup ( ( k .: D ) .: D ) .= sup ( ( k .: D ) .: D ) ;
A |^ ( k , l ) ^^ ( A |^ ( n , l ) ) = ( A |^ ( n , l ) ) ^^ ( A |^ ( k , l ) ) ;
for R being add-associative right_zeroed right_complementable non empty addLoopStr , I , J , K 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 + ( p `2 ) ^2 ;
for a , b being non zero Nat st a , b are_congruent_mod n holds support ( a * b ) = support ( a ) + support ( b ) & support ( a * b ) = support ( a ) + support ( b )
consider A5 being countable Nat such that r is countable and A5 is ( len A5 ) -element & ( for n being Nat holds A5 . n = ( n + 1 ) ! ) & ( n + 1 ) in dom ( ( n + 1 ) ! ) ;
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 + N
{ [ x1 , x2 ] , [ y1 , y2 ] } c= { x1 , x2 , x3 , x4 , x5 , x5 , D2 , 7 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 ,
h . ( f . O ) = |[ A * ( f . O ) `1 + B , C * ( f . O ) `2 + D * ( f . I ) `2 ]| ;
( Gauge ( C , n ) * ( k , i ) ) `2 in L~ Upper_Seq ( C , n ) /\ L~ Upper_Seq ( C , n ) ;
cluster m , n are_relative_prime -> prime for Nat means : Def4 : for p being prime Nat holds p divides m & p divides n & p divides n implies p 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 ) / ( x , y ) ) ) and z = H / ( ( x , y ) / ( x , y ) ) . b ;
assume that x in dom ( F * g ) and y in dom ( F * g ) and ( F * g ) . x = ( F * g ) . y ;
assume ex e being element st e Joins W . 1 , W . 5 , W . ( 5 + 1 ) , G or e Joins W . 3 , W . ( 5 + 1 ) , G ;
( \cal x0 ) / ( h . n ) . x = ( r (#) ( f . n ) ) / ( 2 * n ) . x + ( r (#) ( f . n ) ) / ( 2 * n ) . x ;
j + 1 = ( len h11 + 2 ) - len h11 + 2 .= i + ( 1 + 2 ) - len h11 + 2 - 2 .= i + 2 - 2 + 2 - 1 ;
^ ( S /* ( S /* f ) ) . f = S *' ( S ^ ) .= S . ( ( S ^ ) . f ) .= S . ( f . f ) .= S . ( f . f ) ;
consider H such that H is one-to-one and rng H = the carrier of L2 and Sum ( L2 * H ) = Sum ( L2 ) and Sum ( L2 * H ) = Sum ( L2 ) ;
attr R is <= .| means : Def4 : p in R & p <> q & q <> p & p <> q & q <> p & p <> q & q <> p & p <> q & q <> p & p <> q ;
dom Product ( ( X --> f ) ) = meet ( dom ( X --> f ) ) .= meet ( ( X --> f ) . ( dom f ) ) .= meet ( X --> f ) .= dom f .= dom f .= dom f ;
sup ( ( proj2 .: Lower_Arc C ) /\ Lower_Arc C ) <= sup ( ( proj2 .: ( Lower_Arc C ) ) /\ Vertical_Line ( w ) ) & sup ( ( proj2 .: ( Lower_Arc C ) /\ Vertical_Line w ) ) <= sup ( ( proj2 .: ( Lower_Arc C ) ) /\ Vertical_Line w ) ;
for r be Real st 0 < r ex n be Nat st for m be Nat st n <= m holds |. S . m - p .| < r / 2 ;
i * ( f - f-28 ) = i * ( f - ( i * y ) ) .= i * ( f - ( i * y ) ) .= i * ( f - ( i * 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 and g2 in union C and g = [ g1 , g2 ] and g1 = [ g1 , g2 ] ;
func d |-count n -> Nat means : Def4 : d |^ ( n + 1 ) divides n & ( d |^ ( n + 1 ) ) divides ( d |^ ( n + 1 ) ) & ( d |^ ( n + 1 ) ) divides ( ( d |^ n ) * ( ( d |^ n ) * ( ( d |^ n ) * ( ( d |^ n ) * ( ( d |^ n ) * ( d |^ n ) ) ) ) ;
f\rbrack . [ 0 , t ] = f . [ 0 , t ] .= f . [ 0 , t ] .= ( - P ) . ( 2 * x ) .= a ;
t = h . D or t = h . B or t = h . C or t = h . D or t = h . E or t = h . F or t = h . J or t = h . M ;
consider m1 being Nat such that for n st n >= m1 holds dist ( ( seq . n ) - ( seq . m ) ) < 1 / ( ( seq . n ) - ( seq . m ) ) ;
( ( q `1 ) / |. q .| ) ^2 / ( 1 + cn ) ^2 <= ( ( q `2 ) / ( 1 + cn ) ) ^2 / ( 1 + cn ) ^2 / ( 1 + cn ) ^2 ;
h0 . ( i + 1 + 1 ) = h0 . ( i + 1 + 1 -' len h11 + 2 -' 1 ) .= h0 . ( i + 1 + 2 - 1 ) ;
consider o being Element of the carrier' of S , x2 being Element of { the carrier' of S } such that a = [ o , x2 ] and o <> x2 ;
for L being RelStr , a , b being Element of L holds a <= { b } iff a <= b & b <= a & a <= b & b <= a & a <= b implies a <= b
||. h1 .|| . n = ||. h1 . n .|| .= ||. h . n .|| .= ||. h . n .|| .= ||. h . n .|| .= ||. h . n .|| .= ||. h . n .|| .= ||. h . n .|| .= ||. h . n .|| ;
( ( - ( ( #Z n ) * ( exp_R + f ) ) `| Z ) . x = f . x - ( exp_R . x ) / ( exp_R . x ) ^2 ) .= ( - ( exp_R . x ) / ( exp_R . x ) ^2 ) / ( exp_R . x ) ^2 ;
pred r = F .: ( p , q ) means : Def4 : len r = min ( len p , len q ) ;
( r / 2 ) ^2 + ( r / 2 ) ^2 / 2 <= ( r / 2 ) ^2 + ( r / 2 ) ^2 / 2 ;
for i being Nat , M being Matrix of n , K st i in Seg n holds Det ( M , i ) = Sum ( ( L * M ) * ( i , j ) )
then a <> 0. R & a " * ( a * v ) = 1 / ( a * v ) & a " * ( a * v ) = 1 / ( a * v ) ;
p . ( j - 1 ) * ( q *' ) . ( i + 1 - j ) = Sum ( p . ( j - 1 ) * ( q . ( i - 1 ) ) ) ;
deffunc F ( Nat ) = L . 1 + ( ( R /* h ^\ n ) * ( h ^\ n ) ) . ( h ^\ n ) " .= ( ( R /* h ) ^\ n ) . ( h ^\ n ) " ;
assume that the carrier of H2 = f .: ( the carrier of H1 ) and the carrier of H2 = f .: ( the carrier of H2 ) and the carrier of H1 = the carrier of H2 and the carrier of H2 = the carrier of H2 and the carrier of H1 = the carrier of H2 and the carrier of H1 = the carrier of H2 ;
Args ( o , Free ( S , X ) ) = ( ( the Sorts of Free ( S , X ) ) * the Arity of S ) . o .= ( ( the Sorts of Free ( S , X ) ) * the Arity of S ) . o ;
H1 = n + 1 - ( 2 to_power ( n + 1 ) ) .= n + 1 - ( 2 to_power ( n + 1 ) ) .= n + 1 - ( 2 to_power ( n + 1 ) ) ;
( O = 1 & 3 ) = 0 & ( O = 1 or O = 1 or O = 1 or O = 1 & O = 1 or O = 1 & O = 1 & O = 1 or O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 ) & O = O & O = O & O = O & O = O & O = O & O = I & O = I & O = I = I & O = I & O = I = I & O = I
F1 .: ( dom F1 /\ dom F2 ) = F1 .: ( dom F1 /\ dom F2 ) .= { f /. ( n + 1 ) } .= { f /. ( n + 2 ) } .= { f /. ( n + 1 ) } .= { f /. ( n + 2 ) } ;
pred b <> 0 & d <> 0 & b <> 0 & ( a = e & b = e ) implies ( a / b ) / ( b / ( d - c ) ) = ( ( - e ) / ( d - c ) ) / ( d - c ) ;
dom ( ( f +* g ) | D ) = dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) ;
for i be set st i in dom g ex u , v being Element of L st g /. i = u * a & v in B & u in A & v in B & v in A & u in B ;
g `2 * P `2 * g `2 = g `2 * ( g `2 * P `2 ) * ( g `2 * P `2 ) " .= g `2 * ( g `2 * P `2 ) * ( g `2 * P `2 ) " .= g `2 * ( g `2 * P `2 ) * ( g `2 * P `2 ) ;
consider i , s1 such that f . i = s1 and not ( ex s st s in dom ( s1 . i ) & ( not s1 in dom ( s2 . i ) & s <> s1 ) & not s1 in { s2 . i } ;
h5 | ]. a , b .[ = ( g | ]. a , b .[ ) | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ ;
[ s1 , t1 ] , [ s2 , t2 ] Point ( X | Y ) & [ s2 , t2 ] , [ t2 , t2 ] Point ( X | Y ) . [ t2 , t2 ] ] <= [ s2 , t2 ] ;
then H is atomic & H is non negative & H is non ] & H is non empty implies H is non -\sqrt @ for f , g ;
attr f1 is total means : Def4 : ( 1 - ( f1 + f2 ) ) (#) ( f1 - f2 ) = f1 . c * ( f2 - f3 ) (#) ( f1 - f3 ) ;
z1 in W2 " ( W2 " ( { z } ) ) or z1 = W2 " ( W2 " { z } ) & z2 in W2 " { z } & z in W2 " { z } ;
p = 1 * p .= a " * p .= a " * a * p .= a " * ( b * q ) .= a " * ( b * q ) .= a " * ( b * q ) * q .= ( a * ( b * q ) ) * q ;
for r9 be Real_Sequence , K be Real st for n be Nat holds seq1 . n <= K holds upper_bound rng ( seq1 ^\ K ) <= K * ( seq1 ^\ K )
C meets L~ go \/ L~ pion1 or C meets L~ pion1 or C meets L~ co or C meets L~ co or C meets L~ co or C meets L~ co or C meets L~ co or C meets L~ co or C meets L~ co or C meets L~ co or C meets L~ co or C meets L~ co or C meets L~ co or C meets L~ co or C meets L~ co or C meets L~ co or C meets L~ co or C meets L~ co or C meets L~ co or C meets L~ Cage or C meets L~ co or C meets L~ co or C meets L~ co or C meets L~ co or C meets L~
||. f . ( g . ( k + 1 ) ) - g . ( k + 1 ) .|| <= ||. g . 1 - g . ( k + 1 ) .|| * ( K * K / ( 1 + 1 ) ) ;
assume h = ( ( B .--> B ) +* ( C .--> D ) +* ( E .--> F ) +* ( F .--> J ) +* ( F .--> J ) +* ( M .--> N ) +* ( F .--> N ) +* ( F .--> N ) +* ( M .--> N ) +* ( F .--> N ) +* ( M .--> N ) +* ( F .--> N ) +* ( M .--> N ) ) +* ( M .--> N ) ;
|. ( ( upper_volume ( H . n ) ) `| A ) . k - ( ( upper_volume ( H , n ) ) . k ) .| <= e * ( ( delta ( H , n ) ) . k - ( delta ( H , n ) ) . k ) ;
( (
{ x1 , x1 , x1 , x2 , x3 , x4 , x5 , x5 , 6 , 7 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 7
assume that A = [. 0 , 2 * PI .] and integral ( A , ( ( ( #Z n ) * ( f1 + #Z n ) ) * ( ( f1 + #Z n ) * ( ( f1 + #Z n ) * ( ( f1 + #Z n ) * ( f1 + #Z n ) ) ) ) = 0 ;
p `2 is Permutation of dom f1 & p `2 * p `2 = ( Sgm Y ) . i " * p " * p " .= ( Sgm Y ) . i " * p " * p " * p " ;
for x , y st x in A holds |. ( 1 / ( f . x - 1 ) ) * ( f . y - 1 / ( f . y ) ) .| <= 1 * |. f . x - 1 / ( f . y ) .|
( p2 `2 ) ^2 = |. q2 .| * ( ( q2 `2 ) ^2 + ( q2 `2 ) ^2 ) .= ( q2 `2 ) * ( ( q2 `2 ) ^2 + ( q2 `2 ) ^2 ) .= ( q2 `2 ) * ( ( q2 `2 ) ^2 + ( q2 `2 ) ^2 ) .= ( q2 `2 ) * ( ( q2 `2 ) ^2 + ( q2 `2 ) ^2 ) ;
for f be PartFunc of the carrier of CNS , REAL st dom f is compact & f | X is continuous holds rng f = dom f & f | X is continuous & f | X is continuous & f | X is continuous & f | X is continuous & f | X is continuous & f | X is continuous & f | X is continuous & f | X is continuous
assume for x being Element of Y st x in EqClass ( z , CompF ( B , G ) ) holds ( Ex ( a , A ) ) . x = TRUE ;
consider FF such that dom FF = n1 and for k be Nat st k in n1 holds Q [ k , FF . k ] and for k be Nat st k in n1 holds Q [ k , FF . k ] ;
ex u , u1 st u <> u1 & u , u1 / ( a , v ) / ( a , v ) / ( a , b ) / ( a , b ) / ( a , b ) / ( a , b ) / ( a , b ) / ( a , b ) / ( a , b ) / ( a , b ) / ( a , b ) / ( a , b ) / ( a , b ) / ( a , b ) / ( a , b ) / ( a , b ) / ( a , b ) / ( a , b ) / ( a , b )
for G being Group , A , B being non empty Subset of G holds ( N ` A ) * ( N ` A ) = N ` A * ( N ` B )
for s be Real st s in dom F holds F . s = integral ( R / ( f + g ) , ( f - g ) / ( f - g ) ) / ( f - g ) . x
width AutMt ( f1 , b1 , b2 , b3 ) = len b2 .= len ( b2 * ( b1 , b2 ) ) .= len ( b1 * ( b2 , b3 ) ) .= len ( b1 * ( b2 , b3 ) ) .= len ( b1 * ( b2 , b3 ) ) .= len ( b1 * ( b2 , b3 ) ) ;
f | ]. - PI / 2 , PI / 2 .[ = f & f | ]. - PI / 2 , PI / 2 .[ = f | ]. - PI / 2 , PI / 2 .[ & f | ]. - PI / 2 , PI / 2 .[ = f | ]. - PI / 2 , PI / 2 .[ ;
assume that X is closed and a in X and a in X and y in a and x in { [ n , x ] } \/ y and x in X ;
Z = dom ( ( ( ( 1 / 2 ) (#) ( ( #Z 2 ) * ( f1 + #Z 2 ) ) ) `| Z ) /\ dom ( ( ( 1 / 2 ) (#) ( f1 + #Z 2 ) ) (#) ( f1 + #Z 2 ) ) ;
func Let ( V ) -> Subset of V means : Def4 : 1 <= k & k <= len l & for k st 1 <= k & k <= len l holds it . k in V . k ;
for L being non empty TopSpace , N being net of L , M being net of N st M is net of N & M is net of N holds M is net of N
for s being Element of NAT holds ( ( ( ( id the carrier of X ) + ( id the carrier of X ) ) + ( id the carrier of X ) ) . s = ( ( ( id the carrier of X ) + ( id the carrier of X ) ) + ( id the carrier of X ) ) . s
then z /. 1 = ( N-min L~ z ) .. z & ( E-max L~ z ) .. z < ( E-max L~ z ) .. z ;
len ( p ^ <* ( 0 qua Real ) * ( 0 qua Real ) *> ) = len p + len <* ( 0 qua Real ) * ( 0 qua Real ) *> .= len p + 1 .= len p + 1 ;
assume that Z c= dom ( ( - ( ln * f ) ) `| Z ) and for x st x in Z holds f . x = x / ( a + x ) ^2 and f . x > 0 ;
for R being add-associative right_zeroed right_complementable associative non empty doubleLoopStr , I being Ideal of R , J being Subset of R , I being Ideal of R , I being Ideal of R holds ( I + J ) *' ( I /\ J ) c= I /\ J
consider f being Function of [: B1 , B2 :] , B2 such that for x being Element of B1 , y being Element of B2 holds f . ( x , y ) = F ( x , y ) ;
dom ( x2 + y2 ) = Seg len x .= Seg len ( x2 + y2 ) .= Seg len ( x2 + y2 ) .= Seg len ( x2 + y2 ) .= Seg len ( x2 + y2 ) .= Seg len ( x2 + y2 ) .= Seg len ( x2 + y2 ) .= Seg len ( x2 + y2 ) .= dom ( x2 + y2 ) ;
for S being -1 Functor of C , B , c being Object of C holds card S . ( id c ) = id ( ( Obj S ) . ( id c ) ) & S . ( id c ) = id ( ( Obj S ) . ( id c ) )
ex a st a = a2 & a in dom ( f | ( rng f ) ) & ( for x st x in dom ( f | ( rng f ) ) holds f . ( x , a ) = \rbrace implies f . ( x , a ) = lim ( f , a ) ;
a in Free ( ( H / ( x. 4 , x. k ) ) '&' ( ( H / ( x. 3 , x. k ) ) / ( x. k , x. k ) ) ) ;
for C1 , C2 being 1 , f being Function of C1 , C2 st ( ex g being Function of C1 , C2 st f = g & g is stable ) holds f = g & f = g
( W-min L~ go ) `1 = ( W-min L~ go ) `1 .= ( W-bound L~ go ) * ( 1 , 1 ) `1 .= ( W-bound L~ Cage ( C , n ) ) * ( 1 , 1 ) `1 .= ( W-bound L~ Cage ( C , n ) ) `1 .= ( W-bound L~ Cage ( C , n ) ) `1 ;
assume that u = <* x0 , y0 , z0 *> and f is_partial_differentiable_in z0 , 2 and SVF1 ( 3 , f , u ) is_partial_differentiable_in z0 , 3 and SVF1 ( 3 , f , u ) . 3 = SVF1 ( 3 , f , u ) . 3 ;
then ( t . {} ) `1 in Vars & ex x being Element of Vars st x = ( t . {} ) `1 & t . {} = x & t . {} = y & t . {} = s ;
Valid ( p '&' p , J ) . v = Valid ( p , J ) . v '&' Valid ( p , J ) . v .= Valid ( p , J ) . v '&' Valid ( q , J ) . v .= Valid ( p , J ) . v ;
assume for x , y being Element of S st x <= y for a , b being Element of T st a = f . x & b = f . y holds a >= b & b >= a ;
func Class R -> Subset-Family of R means : Def4 : for A being Subset of R holds A in it iff ex a being Element of R st a in it & A c= a & it . a = Class ( R , a ) ;
defpred P [ Nat ] means ( ( \HM { the } \HM { \Rightarrow ( G ) ) . $1 ) `1 c= G . ( ( the Source of G ) . $1 ) `1 , G . ( ( the Source of G ) . $1 ) `2 ) ;
assume that 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 and dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W1 ) = 0 ;
mama-\hbox { m ( t ) -> Element of C ) , m ( ) -> Element of C ( ) , A ( ) -> Element of C ( ) , t ( ) -> Element of C ( ) , f ( set ) ] means for m being Element of C ( ) holds it . m = [ m , the carrier of C ] ;
d11 = xx ^ d22 .= f . ( ( y9 , d22 ) . ( x , y ) ) .= f . ( ( y9 , d22 ) . ( x , y ) ) .= ( f ^ d22 ) . ( x , y ) .= ( f ^ d22 ) . ( x , y ) .= ( f ^ d22 ) . ( x , y ) .= ( f ^ <* x *> ) . ( x , y ) .= ( f ^ <* x , y *> ) . ( x , y , y *> .= ( f ^ <* x , y *> .= ( f ^ <* x , y *> .= ( f ^ <* x , y *> ) . ( x , y *> ) . ( x , y *> .= ( f ^ <* x
consider g such that x = g and dom g = dom ( f . 0 ) and for x being element st x in dom ( f . 0 ) holds g . x in f . x ;
x + 0. F_Complex = x + len x .= ( x + len x ) |-> 0. F_Complex .= ( x + len x ) |-> 0. F_Complex .= ( x + len x ) |-> 0. F_Complex .= ( x + len x ) |-> 0. F_Complex .= x + ( x + 0. F_Complex ) .= x + ( x + 0. F_Complex ) .= x + 0. F_Complex .= x + 0. F_Complex .= x + 0. F_Complex .= x + 0. F_Complex ;
( ( k -' ( k + 1 ) ) + 1 ) in dom ( f /. ( ( k -' 1 ) + 1 ) ) /\ dom ( f /^ ( k + 1 ) ) ;
assume that P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 = { p1 , p2 } and P2 = { p1 , p2 } and P1 = { p1 , p2 } and P2 = { p1 , p2 } and P1 = { p2 , p1 } and P2 = { p1 , p2 } and P1 = P2 and P2 = { p1 , p2 } and P2 = { p1 , p2 } and P1 = P2 and P2 = { p2 , p1 , p2 } and P1 = P2 and P2 = { p1 , p2 , p2 , p3 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 ;
reconsider a1 = a , b1 = b , c1 = c , c1 = d , c2 = c , c1 = d , c2 = p , c1 = p , c2 = q , c2 = p , c1 = d , c2 = p , c2 = q , c1 = p , c2 = d , c2 = p , c1 = q , c2 = p , c2 = q , c2 = p , c2 = q , c2 = p , c1 = q , c2 = p , c1 = q , c2 = p , c1 = q , c2 = p , c1 = p , c2 = q , c1 = p , c2 = p , c1 = q , c2 = p , c1 = p , c1 = q , c2 = p , c1 = p , c1 = q , c2 =
reconsider thesis thesis tbbb1f = G1 . ( t , b ) * F1 . ( f . ( a * b ) ) as Morphism of ( G1 * F1 ) . ( b * a ) , ( G1 * F2 ) . ( b * a ) ;
LSeg ( f , i + i1 -' 1 ) = LSeg ( f /. ( i + i1 -' 1 ) , f /. ( i + i1 -' 1 + 1 ) ) .= LSeg ( f /. ( i + i1 -' 1 ) , f /. ( i + i1 -' 1 + 1 ) ) ;
Integral ( P `1 , m ) | dom ( P . n ) <= Integral ( M . n , P . m ) & Integral ( M . n ) <= Integral ( M . n , P . m ) ;
assume that dom f1 = dom f2 and for x , y being element st [ x , y ] in dom f1 & [ x , y ] in dom f2 holds f1 . ( x , y ) = f2 . ( x , y ) ;
consider v such that v = y and dist ( u , v ) < min ( ( G * ( i , 1 ) ) `1 , ( G * ( i + 1 , 1 ) `1 ) / 2 ) ;
for G being Group , H being Subgroup of G , a being Element of G st a = b holds for i being Integer holds a |^ i = b |^ ( i + 1 ) & a |^ ( i + 1 ) = b |^ ( i + 1 )
consider B being Function of Seg ( S + L ) , the carrier of V1 such that for x being element st x in Seg ( S + L ) holds P [ x , B . x ] ;
reconsider K1 = { ( p ) _ { \bf 7 } } , ( p ) _ { \bf 2 } } = { p where p is Point of TOP-REAL 2 : P [ p ] & p <> 0. TOP-REAL 2 } , K1 = { p : p <> 0. TOP-REAL 2 } , K0 = { p : p <> 0. TOP-REAL 2 } , K0 = { p : p <> 0. TOP-REAL 2 } , K1 = { p : p <> 0. TOP-REAL 2 } , K0 = { p : p <> 0. TOP-REAL 2 } , K0 = { p : p <> 0. TOP-REAL 2 } , K0 = { p : p <> 0. TOP-REAL 2 } , K0 = 0. TOP-REAL 2 } , K0 = { p : p <>
( ( ( ( ( N - S ) / 2 ) - ( S - S ) / 2 ) - ( S - S ) / 2 ) / 2 ) <= ( ( ( N - S ) / 2 ) - ( ( N - S ) / 2 ) / 2 ) / 2 ;
for x be Element of X , n be Nat st x in E holds |. ( Re F ) . n .| <= P . x & |. ( Im F ) . n .| <= P . x & |. ( Im F ) . n .| <= P . x
len ( @ @ ) = len ( ( @ @ p ) ^ <* 0 *> ) + len <* [ 2 , 0 ] *> .= len ( ( @ p ) ^ <* 1 *> ) + len <* 1 *> .= len ( ( @ p ) ^ <* 1 *> ) + len <* 1 *> .= len ( ( @ p ) ^ <* 1 *> ) + len <* 1 *> ) ;
v / ( x. 3 , m1 ) / ( x. 0 , m1 ) / ( x. 4 , m2 ) / ( x. 0 , m1 ) / ( x. 4 , m1 ) / ( x. 0 , m1 ) / ( x. 4 , m2 ) / ( x. 0 , m1 ) / ( x. 4 , m1 ) / ( x. 0 , m1 ) / ( x. 4 , m1 ) / ( x. 0 , m1 ) / ( x. 4 , m1 ) ) / ( x. 0 , m1 ) / ( x. 4 , m1 ) / ( x. 0 , m1 ) / ( x. 4 , m1 ) / ( x. 0 , m1 ) / ( x. 4 , m1 ) / ( x. 4 , m1 ) / (
consider r being Element of M such that M , v / ( x. 3 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 3 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 3 , m ) / ( x. 4 , m ) ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m
func w1 \ w2 -> Element of Union ( G , R8 ) means : Def4 : for w1 , w2 being Element of Union ( G , R8 ) holds it . ( ( H8 ) . w1 , ( ( H8 ) * ( G , R8 ) ) . w2 ) = ( ( H8 ) * ( G , R8 ) ) . w1 ;
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= ( Exec ( n2 , s1 ) ) . b2 .= ( ( Exec ( n2 , s2 ) ) +* ( i , - 1 ) ) . b2 .= ( ( Exec ( n2 , s2 ) ) +* ( i , - 1 ) ) . b2 .= ( Exec ( n2 , s2 ) ) . b2 .= ( Exec ( n2 , s2 ) ) . b2 .= ( Exec ( n2 , s2 ) ) . b2 .= ( Exec ( n2 , s2 ) ) . b2 ;
for n , k be Nat holds 0 <= ( Partial_Sums ( |. seq .| ) ) . ( n + k ) - ( Partial_Sums ( |. seq .| ) ) . ( n + k ) + ( Partial_Sums ( |. seq .| ) ) . ( n + k ) - ( Partial_Sums ( |. seq .| ) ) . ( n + k ) ;
set F = S -\mathop { N } , G = S -\mathop { N } ;
( Partial_Sums ( seq ) ) . ( K + 1 ) + Partial_Sums ( seq ) . ( K + 1 ) >= ( Partial_Sums ( seq ) ) . ( K + 1 ) + Partial_Sums ( seq ) . ( K + 1 ) ;
consider L , R such that for x st x in N holds ( f | Z ) . x - ( f | Z ) . x0 = L . ( x- ( x0 - R ) / ( x - x0 ) ) + R . ( x- x0 ) ;
func \HM { closed \HM { a } , b , c , d : a in the \HM { the } \HM { carrier of \HM { of ( ( ( a , b ) `1 , c ) , d } ) & b in the carrier of ( ( ( ( a , b ) `1 , c ) ) / 2 } ) & c in the carrier of ( ( ( a , b ) `2 ) ) ;
a * b ^2 + ( a * c ) ^2 + ( b * c ) ^2 + ( c * a ) ^2 + ( c * b ) ^2 + ( c * a ) ^2 >= 6 * a * b * c + ( c * b ) ^2 + ( c * a ) ^2 ;
v / ( x1 , m1 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) ) = v / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) /
.: ( Q ^ <* x *> , M ^ <* x *> ) = ( .: ( Q , M ) +* ( 1 , FALSE ) ) +* ( ( M , M ) +* ( 1 , FALSE ) ) +* ( ( M , M ) +* ( 1 , TRUE ) ) +* ( ( M , M ) +* ( 1 , TRUE ) ) +* ( ( M , M ) +* ( 1 , TRUE ) ) ;
Sum ( FF ) = r |^ ( n1 + 1 ) * Sum ( Cz ) .= C . ( n1 + 1 ) * ( Cz ) .= C . ( n1 + 1 ) * ( Cz ) .= C . ( n1 + 1 ) * ( Cz ) .= C . ( n1 + 1 ) * ( ( n + 1 ) + 1 ) .= C . ( n1 + 1 ) ;
( ( GoB f ) * ( len GoB f , 2 ) ) `1 = ( ( GoB f ) * ( len GoB f , 1 ) ) `1 .= ( ( GoB f ) * ( 1 , 1 ) ) `1 .= ( ( GoB f ) * ( 1 , 1 ) ) `1 .= ( ( GoB f ) * ( 1 , 1 ) ) `1 .= ( ( GoB f ) * ( 1 , 1 ) ) `1 ;
defpred X [ Element of NAT ] means ( Partial_Sums s ) . $1 = ( a * ( $1 + 1 ) ) / ( ( $1 + 1 ) + 1 ) * ( ( $1 + 1 ) + 1 ) / ( $1 + 1 ) * ( ( $1 + 1 ) + 1 ) / ( $1 + 1 ) * ( $1 + 1 ) ;
( the_arity_of g ) . o = ( the Arity of S ) . ( g . o ) .= ( ( the Arity of S ) . ( g . o ) ) . ( g . o ) .= ( ( the Arity of S ) . o ) . ( g . o ) .= ( ( the Arity of S ) . o ) . ( g . o ) .= ( ( the Arity of S ) * g ) . ( g . o ) ;
( X ~ ) \ Z tolerates X \ Y & card ( ( X ~ ) \ Z ) = card ( X ~ ) & card ( ( X ~ ) \ Z ) = card ( X ~ ) ;
for a , b being Element of S , s being Element of NAT st s = n . n & a = F . n & b = F . ( n + 1 ) holds b = N . ( s . a ) \ G . s ;
E , f |= All ( x. 2 , ( x. 2 ) ) '&' ( ( x. 2 ) '&' ( x. 1 ) ) '&' ( x. 2 ) '&' ( x. 1 ) '&' ( x. 1 ) '&' ( x. 1 ) '&' ( x. 2 ) '&' ( x. 1 ) '&' ( x. 1 ) ) ;
ex R2 be 1-sorted st R2 = ( p | ( n + 1 ) ) . i & ( the carrier of p ) . i = the carrier of R2 & ( the carrier of p ) . i = the carrier of R2 & ( the carrier of p ) . i = the carrier of R2 ;
[. a , b + 1 / ( k + 1 ) .[ is Element of the , ( the partial of \HM { f } ) . ( k + 1 ) & ( the partial of \HM { f } ) . ( k + 1 ) is Element of the \overline of ( the partial of a ) . ( k + 1 ) & ( the partial of a ) . k is Element of REAL ;
Comput ( P , s , 2 + 1 ) = Exec ( P . 2 , Comput ( P , s , 2 ) ) .= Exec ( a3 , Comput ( P , s , 2 ) ) .= Exec ( a3 , s ) +* ( ( a , a ) .--> 1 ) .= Exec ( a3 , s ) +* ( ( a , a ) .--> 1 ) ;
card ( h1 ) . k = power ( F_Complex ) . ( ( - 1_ F_Complex ) * ( - 1_ F_Complex ) ) .= ( ( - 1_ F_Complex ) * ( - 1_ F_Complex ) ) . k .= ( ( - 1_ F_Complex ) * ( - 1_ F_Complex ) ) . k .= ( ( - ( - 1_ F_Complex ) ) * ( - 1_ F_Complex ) ) . k ;
( f - g ) /. c = f /. c * ( g /. c ) " .= f /. c * ( ( 1 - g ) * ( 1 - g ) ) .= ( f * ( 1 - g ) ) /. c .= ( f * ( 1 - g ) ) /. c ;
len Cv - len ( ( the carrier of ( C ) ) * ( len ( the R ) ) ) = len ( ( the carrier of ( C ) ) * ( len ( the R ) ) ) .= len ( ( the R ) * ( len ( the R ) ) ) .= len ( ( the R ) * ( len ( the R ) ) ) ;
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom ( f | X ) /\ X .= dom ( f | X ) /\ X .= dom ( f | X ) .= dom ( f | X ) /\ X .= dom ( f | X ) .= dom ( f | X ) .= dom ( f | X ) .= dom ( f | X ) .= dom ( f | X ) .= dom ( f | X ) .= dom ( f | X ) /\ dom ( f | X ) .= dom ( f | X ) /\ dom ( f | X ) .= dom ( f | X ) .= dom ( f | X ) .= dom ( f | X ) .= dom ( f | X ) /\ dom ( f | X ) .= dom ( f | X ) /\ dom
defpred P [ Nat ] means for n holds 2 * Fib ( n + $1 ) = Fib ( n ) * Fib ( n ) + ( 5 * Fib ( n ) ) * Fib ( n ) + ( 5 * Fib ( n ) ) * Fib ( n ) + ( 5 * Fib ( n ) ) * Fib ( n ) ;
consider f being Function of REAL n , Z such that f = f ' and f is onto and n < m and Z = f " { f . n } and for n st n in Seg m holds f . ( n + 1 ) = { f . n } and f . ( n + 1 ) = { f . n } ;
consider c9 being Function of S , BOOLEAN such that c9 = chi ( S , B ) . ( A \/ B ) and E . ( A \/ B ) = Prob . ( ( A \/ B ) . ( A \/ B ) ) and E . ( A \/ B ) = Prob . ( ( A \/ B ) . ( A \/ B ) ) ;
consider y being Element of Y ( ) such that a = "\/" ( { F ( x , y ) where x is Element of X ( ) : P [ x ] } , y ) and Q [ y ] and P [ y ] ;
assume that A c= Z and f = ( ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) ) ) ) ) ) ) ) ) ) and Z = dom ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 )
( 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 ( Seq q1 , len Seq q2 ) , j ) = { j + len Seq q1 where j is Nat : j in dom Seq q1 & j in dom Seq q1 } \/ dom Seq q2 \/ dom Seq q2 \/ dom Seq q2 \/ dom Seq q2 & j in dom Seq q1 \/ dom Seq q2 & len Seq q1 = len Seq q2 + len Seq q2 ;
consider G1 , G2 , G3 being Element of V such that G1 <= G2 and G2 <= G2 and f = G1 * ( G1 , G2 ) and g = G2 * ( G2 , G2 ) and f = G1 * ( G2 , G2 ) and g = G2 * ( G1 , G2 ) ;
func - f -> PartFunc of C , V means : Def4 : dom it = dom f & for c be Element of C st c in dom it holds it /. c = - f /. c & for c be Element of 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 <> {} holds ( v in rng L iff not v in { v } ) & not v in { v } implies L . ( v , a ) |= L ;
consider i1 , j1 such that [ i1 , j1 ] in Indices GoB f and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) ;
consider i , n such that n <> 0 and sqrt p = ( i - n ) * ( i - 1 ) and for n1 being Nat st n1 <> 0 & |. p .| < n holds |. p .| <= ( |. p .| ) * ( i - 1 ) ;
assume that not 0 in Z and Z c= dom ( ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( 1 / 2 ) ) ) ) ) ) ) ) and for x st x in Z ) holds ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( 1 / 2 ) ) )
cell ( G1 , i1 -' 1 , ( 2 |^ ( m -' 1 ) ) * ( Y1 -' 1 ) ) \ L~ ( ( Y1 + Y2 ) * ( 1 , ( Y1 + 1 ) ) * ( 1 , ( Y1 + 1 ) ) * ( Y1 + 1 ) ) c= BDD L~ ( Y1 + Y2 ) ;
ex Q1 being open Subset of X st s = Q1 & ex FF being Subset-Family of Y st Q1 c= F & ( for x being Element of Y st x in Q1 holds ( x in Q1 implies x in Q1 ) & ( x in Q1 implies x in Q1 ) & ( for x being Element of Y st x in Q1 holds x in Q1 implies x in Q1 ) ;
gcd ( ( 1. ( A , B ) ) * ( 1. ( A , B ) ) , ( 1. ( A , B ) ) * ( 1. ( A , B ) ) ) = 1. ( A , B ) * ( 1. ( A , B ) ) .= 1. ( A , B ) * ( 1. ( A , B ) ) ;
R8 = ( ( ( Following ( s2 , 1 ) ) . ( m1 + 1 ) ) . ( m2 + 1 ) .= ( ( Following ( s2 , m1 ) ) . ( m2 + 1 ) ) . ( m2 + 1 ) .= ( ( Following ( s2 , m1 ) ) . ( m2 + 1 ) ) . ( m2 + 1 ) .= [ 3 , 4 ] ;
CurInstr ( P3 , Comput ( P3 , Comput ( P3 , s3 , m1 + 1 ) ) ) = CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= halt SCMPDS .= halt SCMPDS .= halt SCMPDS .= ( CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) ) .= halt SCMPDS .= halt SCMPDS .= ( halt SCMPDS ) .= halt SCMPDS .= ( halt SCMPDS ) ;
P1 /\ P2 = ( { p1 } \/ LSeg ( p1 , p2 ) ) \/ ( LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) ) \/ ( LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) ) ;
func the still of f -> Subset of the Sorts of A means : Def4 : a in it iff ex p , q st p in dom f & q in it & p = f . ( i + 1 ) & q in it & p <> q ;
for a , b being Element of F_Complex st |. a .| > |. b .| & |. b .| > 1 holds f . ( ( f . a ) * ( f . b ) ) = ( f . b ) * ( ( f . a ) * ( f . b ) )
defpred P [ Nat ] means 1 <= $1 & $1 <= len g implies for i , j st [ i , j ] in Indices G & G * ( i , j ) = g * ( i , j ) & G * ( i , j ) = g * ( $1 , j ) ;
assume that C1 , C2 , f being <* of C1 , g , h being State of C1 , f being Function of C1 , C2 such that C1 = C2 and C2 = f and h = g and f = g and g = h and f = g ;
( ||. 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 .|| * ||. c .|| * ||. c .|| * ||. c .|| .= ||. f /. c .|| * ||. c .|| * ||. c .|| .= ||. c .|| * ||. c .|| .= ||. c .|| * ||. c .|| * ||. c .|| .= ||. c .|| *
|. q .| ^2 = ( q `1 ) ^2 + ( q `2 ) ^2 * ( 1 + ( q `2 / q `1 ) ^2 ) & 0 + ( q `2 / q `1 ) ^2 < ( q `1 ) ^2 + ( q `2 / q `1 ) ^2 + ( q `2 / q `1 ) ^2 ;
for F being Subset-Family of T7 st F is open & {} in F & for A , B being Subset of T7 st A in F & B <> {} & A <> {} & B <> {} holds card F = card G & card G = card A & card G = card B c= card G & card G = card A & card G = card A & card G = card B & card G = card A & card G = card B c= card G & card G c= card G & card G c= card G & card G = card G & card G c= card G & card G c= card G & card G c= card G & card G c= card G & card G c= card G & card G c= card G & card G c= card G & card G c= card G & card F c= card G &
assume that len F >= 1 and len F = k + 1 and len G = k and for k st k in dom F holds F . k = G . ( k , 1 ) and for i st i in dom F & i <> k holds F . ( i , k ) = g . ( k , i ) ;
i |^ ( ( ( ( p |^ n ) |^ ( ( p |^ k ) - i ) ) * ( ( p |^ k ) - i ) ) * ( ( p |^ k - i ) * ( ( p |^ k - i ) * ( ( p |^ k - i ) * ( ( p |^ k - i ) * ( ( p |^ k - i ) * ( ( p |^ k - i ) * ( p |^ k ) ) ) ) ) ) .= i |^ ( ( p |^ k - i ) * ( ( p |^ k - i ) ) * ( ( p |^ k - i ) ) * ( ( p |^ k - i ) * ( ( p |^ k ) - ( p |^ k ) * ( p |^ k ) ) ) ;
consider q being oriented oriented Chain of G such that r = q and q <> {} and ( F . ( q . 1 ) = v1 & ( F . ( q . 1 ) ) `1 = v2 & ( F . ( q . 1 ) ) `1 = v2 & ( F . ( q . 1 ) ) `1 = v2 ;
defpred P [ Element of NAT ] means $1 <= len ( ( f , Z ) +* ( i , I ) ) . ( $1 + 1 ) = ( ( ( f , Z ) +* ( i , I ) ) +* ( $1 , I ) ) . ( $1 + 1 ) ;
for A , B being Matrix of n , REAL holds len ( A * B ) = len A & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B
consider s being FinSequence of the carrier of R such that Sum s = u and for i being Element of NAT st 1 <= i & i <= len s holds s . i = a * b & s . ( len s ) = a * b ;
func |( x , y )| -> Element of COMPLEX means : Def4 : |( ( x , y ) , ( Re ( x ) ) , ( Im ( x ) ) )| = ( Re ( x ) ) * ( ( Im ( x ) ) * ( Im ( y ) ) ) + ( ( Im ( x ) ) * ( Im ( y ) ) ) ;
consider g2 being FinSequence of FF such that g2 is continuous and rng g2 c= A and g2 . 1 = x1 and g2 . ( len g2 ) = x2 and g2 . ( len g2 ) = x2 and g2 . ( len g2 ) = x2 and g2 . ( len g2 ) = x2 and g2 . ( len g2 ) = x2 ;
then n1 >= len p1 & n2 >= len p1 & n3 ( p1 , p2 , n1 , n2 , n3 , n3 , n3 , n4 , n4 , n4 , n4 , n4 , n4 , W , W , W , W , N , W , W , S , N , S , N , S , ( N , S , N ) ) = crossover ( p1 , p2 , n1 , n2 , n3 , n3 , n3 , n3 , W , W , W , W , W , W , W , W , W , W , W , W , W , ( N , S , W , W , W , W , W , W , W , ( ( S , S , W , W , W , W , W , W , ( N , S , W ) ) ) ;
( q `1 ) * a <= ( q `1 ) * a & ( q `2 ) * a <= ( q `1 ) * a or q `1 >= ( q `1 ) * a & q `1 >= ( q `2 ) * a & q `1 >= ( q `1 ) * a or q `1 >= ( q `1 ) * a & q `1 >= ( q `1 ) * a & q `2 >= 0 & q `2 >= 0 & q `2 >= 0 & q `2 >= 0 & q `2 >= 0 & q `2 >= 0 & q `2 >= 0 & q `2 >= 0 & q `2 >= 0 & q `2 >= 0 & q `1 >= 0 & q `1 >= 0 & q `1 >= 0 & q `1 >= 0 & q `1 >= 0 & q `1 >= 0 & q `1 >= 0 & q `1 >= 0 & q `1 >= 0 & q `1 >= 0 & q `1 >= 0 & q `1
( F . ( p . ( len p ) ) ) . ( len p ) = ( F . ( p . ( len p ) ) ) . ( len p ) .= ( ( F . ( len p ) ) ) . ( len p ) .= ( ( F . ( len p ) ) ) . ( len p ) .= ( ( F . ( len p ) ) ) . ( len p ) .= ( ( F . ( len p ) ) ) . ( len p ) ;
consider k1 being Nat such that k1 + k = 1 and a := k = ( <* a *> ^ ( ( a := k ) := ( a := k ) ) ) ^ ( ( a := k ) := ( a := k ) ) ;
consider B9 being Subset of B1 , y9 being Function of B1 , B2 such that B9 is finite and D2 = rng ( f | B1 ) and for x being Element of B1 holds ( f | B1 ) . ( x , y ) = \bf T ( 0 , 1 , 0 , 0 , 1 , 0 , 1 , 0 , 0 , 1 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 0 , 1 , 0 , 1 , 1 , 0 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 ,
v2 . b2 = ( curry ( F2 , g ) * ( ( curry F ) . b2 ) ) . b2 .= ( ( curry F ) . b2 ) . b2 .= ( ( curry F ) . b2 ) . b2 .= ( ( curry F ) . b2 ) . b2 .= ( ( curry F ) . b2 ) . b2 .= ( ( curry F ) . b2 ) . b2 .= ( ( curry F ) . b2 ) . b2 .= ( ( curry F ) . b2 ) . b2 .= ( ( ( curry F ) . b2 ) . b2 .= ( ( curry F ) . b2 ) . b2 .= ( ( ( ( curry F ) . b2 ) . b2 .= ( ( curry F ) . b2 ) . b2 .= ( ( curry F ) . b2 ) . b2 .= ( ( ( F ) . b2 ) . b2 ) . b2 .= ( ( ( ( F ) . b2 ) .
dom IExec ( I-35 , P , Initialize s ) = the carrier of SCMPDS .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ;
ex d-32 be Real st d-32 > 0 & for h be Real st h <> 0 & |. h .| < e holds |. h .| " * ||. ( h + R1 ) . h .|| < e / ( 1 + 1 ) * ||. ( h + R1 ) . h .|| ) & ||. ( h + R1 ) . h .|| < e / ( 1 + 1 ) ;
LSeg ( G * ( len G , 1 ) + |[ 1 , - 1 ]| , G * ( len G , 1 ) + |[ 1 , 0 ]| ) c= Int cell ( G , len G , 1 ) \/ { G * ( len G , 1 ) + |[ 1 , 0 ]| } ;
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 ) ) ;
A = { q where q is Point of TOP-REAL 2 : LE q , p , P , p1 , p2 & LE q , p , P , p1 , p2 & LE q , p , P , p1 , p2 & LE p , q , P , p1 , p2 & LE q , p , P , p1 , p2 & LE p , q , P , p1 , p2 & LE p , q , P , p1 , p2 & LE p , q , P , p1 , p2 & LE p , q , P , p1 , p2 , P , p2 & LE p , q , P , p1 , p2 , P , p1 , p2 & LE p , q , P , p1 , p2 & LE p , q , P , p1 , p2 & LE p , q , P , P , p1 , p2 , p2 & LE p , p1 , P , p1 , p2 , p1 , p2 & LE p , p1 , P , p1 , p2 & LE p
( ( - x ) .|. y ) = - ( ( 1 - x ) .|. y ) * ( ( - x ) .|. y ) .= ( ( - 1 ) .|. y ) * ( ( - x ) .|. y ) .= ( ( - 1 ) .|. y ) * ( ( - x ) .|. y ) .= ( ( - 1 ) .|. y ) * ( ( - x ) .|. y ) .= ( ( - 1 ) .|. y ) * ( - y ) ;
0 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) = ( p `1 / p `2 ) ^2 + ( p `2 / p `2 ) ^2 * ( 1 + ( p `2 / p `1 ) ^2 ) .= ( p `1 / p `2 ) ^2 * ( 1 + ( p `2 / p `1 ) ^2 ) ;
( ( U * ( W - p ) ) * ( W - p ) ) = ( ( U * ( W - p ) ) * ( W - p ) ) * ( W - p ) .= ( ( U * ( W - p ) ) * ( W - p ) ) * ( W - p ) .= ( ( U * ( W - p ) ) * ( W - p ) ) * ( W - p ) .= ( ( U * ( W - p ) ) * ( W - p ) ) * ( W - p ) * ( W - p ) ) * ( W - p ) * ( W - p ) * ( W - p ) * ( W - p ) ) * ( W - p ) * ( W - p ) * ( W - p ) * ( W - p ) .= ( ( ( p ) * ( W - p ) * ( W - p ) * ( W - p ) * ( W -
func Shift ( f , h ) -> PartFunc of REAL m , REAL means : Def4 : dom it = dom f & for x be Element of REAL m st x in dom it holds it . x = - 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 , j ) and f /. k = 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 & y in Free H or x in Free H ) and not ( x in Free H & y in Free H ) and not ( x in Free H & y in Free H ) ;
defpred P11 [ Element of NAT , Element of NAT ] means ( p |-count ( $1 + 1 ) ) * ( ( p |-count ( $1 + 1 ) ) * ( ( p |-count ( $1 + 1 ) ) * ( ( p |-count ( $1 + 1 ) ) ) ) & ( p |-count ( $1 + 1 ) ) * ( ( p |-count ( $1 + 1 ) ) * ( ( p |-count ( $1 + 1 ) ) ) ) ;
func \sigma ( C ) -> non empty Subset-Family of X means : Def4 : for A , B being Subset of X st A c= it & B c= it holds C . ( A \/ B ) <= C . ( A \/ B ) & C . ( A \/ B ) <= C . ( A \/ B ) ;
[#] ( ( dist ( ( dist ( > ) ) ) .: Q ) = ( ( dist ( ( dist ( P ) ) ) .: Q ) ) & lower_bound ( ( dist ( P ) ) .: Q ) = lower_bound ( ( dist ( P ) ) .: Q ) & lower_bound ( ( dist ( P ) ) .: Q ) = lower_bound ( ( dist ( Q ) ) .: Q ) ;
rng ( F | ( ( S | ( S ) ) ) ) = {} or rng ( F | ( ( S | ( S ) ) ) ) = { 1 } or rng ( F | ( S /\ ( S /\ ( S /\ ( T ) ) ) ) ) = { 1 , 2 } or rng ( F | ( S /\ ( T /\ T ) ) ) = { 1 , 2 , 3 } ;
( f " ( rng ( f * ( ( f * ( f * g ) ) ) ) ) . i = f . i .= ( ( f * ( f * g ) ) * ( f * ( f * g ) ) ) . i .= ( f * ( f * g ) ) . i .= ( f * ( g * g ) ) . i .= ( f * g ) . i ;
consider P1 , P2 being non empty Subset of TOP-REAL 2 such that P1 /\ P2 = { p1 , p2 } and P = { p1 , p2 } and P1 <> p2 and P2 = { p1 , p2 } and P = { p1 , p2 } and P = { p2 , p1 } and P = { p1 , p2 } and Q = { p2 , p1 } and P = Q and Q = { p1 , p2 } and P = Q and Q = { p1 , p2 } and Q = { p2 , p1 , p2 , p1 , p2 , p3 } and P = { p2 , p1 , p3 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 } and P = { p1 , p4 } and P = { p1 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4
f . p2 = |[ ( p2 `1 ) ^2 + ( p2 `2 ) ^2 * ( p2 `2 ) ^2 , ( p2 `2 ) ^2 * ( p2 `2 ) ^2 + ( p2 `2 ) ^2 * ( p2 `2 ) ^2 + ( p2 `2 ) ^2 * ( p2 `2 ) ^2 , ( p2 `2 ) ^2 * ( p2 `2 ) ^2 + ( p2 `2 ) ^2 * ( p2 `2 ) ^2 ;
( ( \lbrace a , X } ) * ( ( \lbrace a , X } ) " ) ) . x = ( ( \lbrace a , X \rbrace qua Function ) . x ) * ( ( ( a , X ) * ( ( 1 , X ) " ) ) . x .= ( ( ( 1 , X ) --> u ) * ( 1 , X ) ) . x .= ( ( ( 1 , X ) --> u ) * ( 1 , X ) ) . x .= ( ( ( 1 , X ) --> u ) . x ) * ( ( 1 , u ) . x ) * ( ( 1 , u ) ) . x ) * ( ( 1 , u ) . x ) * ( ( 1 , u ) * ( 1 , u ) . x ) * ( 1 , u ) . x ) * ( ( ( ( ( 1 , u ) . x ) " * ( ( ( ( 1 , u ) * ( ( 1 , u ) . x ) * ( ( 1 , u ) . x ) * ( ( 1 , u ) . x ) .= ( ( ( 1 , u ) . x ) * ( (
for T being non empty normal TopSpace , A , B being closed Subset of T st A <> {} & A misses B & B misses A & A misses B holds A /\ B c= B & A /\ ( B /\ C ) c= A & A /\ ( ( in \in \mathbb R ) \ B ) implies A /\ B c= B & A /\ ( ( in \mathbb R ) \ B )
for i , j st i in dom F for G1 , G2 being strict normal Subgroup of G st i = F . i & j = G . j & i in dom G1 & j in dom G2 holds G1 . i = G1 . ( i + 1 ) & G2 . ( i + 1 ) = G1 . ( i + 1 ) & G1 . ( i + 1 ) = G2 . ( i + 1 )
for x st x in Z holds ( ( ( ( 1 / 2 ) (#) ( ( #Z 2 ) * ( f1 + #Z 2 ) ) ) `| Z ) . x = ( ( 1 / 2 ) (#) ( ( #Z 2 ) * ( f1 + #Z 2 ) ) ) `| Z ) . x / ( ( 1 + 2 ) (#) ( ( #Z 2 ) * ( f1 + #Z 2 ) ) ) . x
attr f is right & x0 in dom ( f /* a ) & ( for x st x in dom f holds f . x - f . x0 = ( f /* a ) . x ) implies f is convergent & lim ( f , x0 ) = ( f /* a ) . x0 - ( f /* a ) . x0 ;
then X1 , X2 are_separated or X1 misses X2 or X2 misses X1 or X1 misses X2 & X2 misses X2 or X1 misses X2 & X2 misses X1 or X1 misses X2 & X2 misses X1 or X1 misses X2 & X2 misses X2 or X1 misses X2 & X2 misses X2 or X1 misses X2 & X2 misses X2 or X1 misses X2 & X2 misses X1 or X2 misses X1 & X2 misses X2 & X1 misses X2 & X2 misses X2 & X1 misses X2 & X2 misses X1 & X2 misses X1 & X1 misses X2 or X2 misses X2 & X1 misses X2 or X2 misses X2 or X1 union X2 & X2 misses X2 or X2 misses X2 & X2 misses X2 or X2 misses X2 & X2 misses X2 & X2 misses X2 & X1 union X2 & X2 misses X2 & X1 union X2 & X1 misses X2 & X2 misses X2 & X2 misses X1 & X1 union X2 & X2 misses X2 or X1 union X2 & X2 misses X2 & X1 union X2 & X2 misses X2 or X1 union X2 & X2 misses X2 or X2 misses X2 union X2 & X2 misses X2 or X1 union X2 misses X2 union X2 union X2 & X2 misses X1 union X2 & X2 meets X2 & X1 union X2 misses X2
ex N be Neighbourhood of x0 st N c= dom SVF1 ( 1 , f , u ) & ex L be ( for x be Element of REAL m st x in N holds ( SVF1 ( 1 , f , u ) ) . x - SVF1 ( 1 , f , u ) . x0 = L . ( x - x0 ) + R . ( x - x0 ) ;
( ( p2 `1 ) * sqrt ( 1 + ( p3 `2 ) ^2 ) ) / ( ( p2 `1 ) * sqrt ( 1 + ( p3 `2 ) ^2 ) ) >= ( ( p2 `1 ) * sqrt ( 1 + ( p3 `2 ) ^2 ) / ( ( p2 `1 ) ^2 ) ;
( ( 1 - t1 ) (#) ( ( 1 - t1 ) (#) ( ( 1 - t1 ) (#) ( ( 1 - t1 ) (#) ( ( 1 - t1 ) (#) ( ( 1 - t1 ) (#) ( ( 1 - t1 ) (#) ( ( 1 - t1 ) (#) ( ( 1 - t1 ) (#) ( ( 1 - t1 ) (#) ( ( 1 - t1 ) (#) ( ( 1 - t1 ) (#) ( ( 1 - t1 ) (#) ( ( 1 - t1 ) (#) ( ( 1 - t1 ) * ( ( 1 - t1 ) * ( ( 1 - t1 ) * ( ( 1 - t1 ) * ( ( 1 - t1 ) * ( ( 1 - t1 ) * ( 1 - t1 ) ) * ( 1 - t1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) `| ( ( 1 - t1 ) ) ) `| ( ( 1 - t1 ) ) . x = ( ( 1 - t1 ) (#) ( ( 1 - t1 ) ) ) . x ) ) / ( ( 1 - t1 ) ) ) / ( ( 1 - t1 ) (#) ( ( 1 - t1 ) ) / (
assume that for x holds f . x = ( ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) ) ) ) ) ) ) ) `| Z ) ) and for x ) ) ) . x ) ) and for x st x in Z ) holds ( ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) ) `| Z ) . x ) - ( ( 1 / 2 ) ) . x ) - ( ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) ) ) `| Z ) . x ) ) and for x holds
consider X1 being Subset of Y , Y1 being open Subset of X such that t = X1 and Y1 in A and Y1 is open and for Y1 being Subset of X st Y1 in A & Y1 is open holds Y1 /\ Y1 = Y1 & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 c= Y1 & Y1 is open & Y1 is open ;
card ( S . n ) = card { [: d , ( a |^ 3 ) + b , d :] where d is Element of GF ( p ) : [ d , ( a |^ 3 ) + b ] in R } .= 1 + ( R * ( a |^ 3 ) ) .= ( R * ( a |^ 3 ) ) * ( R * ( a |^ 3 ) ) ;
( ( W-bound D - W-bound D ) / ( ( i1 - 1 ) / ( m - 1 ) ) / ( ( m - 1 ) / ( m - 1 ) ) * ( ( m - 1 ) / ( m - 1 ) ) = ( ( W-bound D - ( m - 1 ) ) / ( m - 1 ) ) / ( ( m - 1 ) / ( m - 1 ) ) * ( ( m - 1 ) / ( m - 1 ) ) ;