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 ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b
D c= S
let Y ;
S ` is Cauchy
q in X ;
V 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 , S ;
a in D ;
q in Y ;
let x ;
1 <= l ;
1 <= w ;
let G , H ;
y in N ;
f = {} ;
let x ;
x in Z ;
let x ;
F is one-to-one ;
e <> b ;
1 <= n ;
f is special ;
S misses C
t <= 1 ;
y divides m ;
P divides M ;
let Z ;
let x ;
y c= x ;
let X ;
let C ;
x _|_ p ;
o is monotone ;
let X ;
A = B ;
1 < i ;
let x ;
let u ;
k <> 0 ;
let p ;
0 < r ;
let n ;
let y ;
f is onto ;
x < 1 ;
G c= F ;
a is_>=_than X ;
T is continuous ;
d <= a ;
p <= r ;
t < s ;
p <= t ;
t < s ;
let r ;
D is_<=_than E ;
assume e > 0 ;
assume 0 < g ;
p in X ;
x in X ;
Y ` in Y ;
assume 0 < g ;
not c in Y ;
not v in L ;
2 in z `2 ;
assume f = g ;
N c= b ` ;
assume i < k ;
assume u = v ;
I = J ;
B ` = b ` ;
assume e in F ;
assume p > 0 ;
assume x in D ;
let i be element ;
assume F is onto ;
assume n <> 0 ;
let x be element ;
set k = z ;
assume o = x ;
assume b < a ;
assume x in A ;
a `1 <= b `1 ;
assume b in X ;
assume k <> 1 ;
f = Product l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is from from from W1 , W2 ;
assume m > 0 ;
assume A c= B ;
X is lower
assume A <> {} ;
assume X <> {} ;
assume F <> {} ;
assume G is open ;
assume f is dilatation ;
assume y in W ;
y \not <= x ;
A ` in B ` ;
assume i = 1 ;
let x be element ;
x `1 = x `1 ;
let X be BCK-algebra ;
assume S is non empty ;
a in REAL ;
let p be set ;
let A be set ;
let G be _Graph , a , b be set ;
let G be _Graph , a , b be set ;
let a be Complex ;
let x be element ;
let x be element ;
let C be FormalContext , a , b , c , d be Element of C ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
let y be Real ;
X c= f . a
let y be element ;
let x be element ;
let i be Nat ;
let x be element ;
n in NAT ;
let a be element ;
m in NAT ;
let u be element ;
i in NAT ;
let g be Function ;
Z c= NAT ;
l <= ma ;
let y be element ;
r2 in dom f ;
let x be element ;
k1 be Integer ;
let X be set ;
let a be element ;
let x be element ;
let x be element ;
let q be element ;
let x be element ;
assume f is being_homeomorphism ;
let z be element ;
a , b // K ;
let n be Nat ;
let k be Nat ;
B ` c= B ` ` ;
set s = / ( 2 |^ n ) ;
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 * ( f1 - f2 ) 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 Element of REAL ;
not r in G . l
c1 = 0 ;
a + a = a ;
<* 0 *> in e ;
t in { t } ;
assume F is non discrete ;
m1 divides m ;
B * A <> {} ;
a + b <> {} ;
p * p > p ;
let y be ExtReal ;
let a be Int-Location , i be Integer ;
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 -> and x is directed ;
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 >= TOP-REAL s ;
G . y <> 0 ;
let X be RealNormSpace , x be Element of X ;
a in X ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , v be VECTOR 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 lower-bounded ;
rng f = Y ;
G8 c= L ;
assume x in Cl Q ;
m in dom P ;
i <= len Q ;
len F = 3 ;
still_not-bound_in p = {} ;
z in rng p ;
lim b = 0 ;
len W = 3 ;
k in dom p ;
k <= len p ;
i <= len p ;
1 in dom f ;
b `1 = a `1 + 1 ;
x `1 = a * y `2 ;
rng D c= A ;
assume x in K1 ;
1 <= i-32 ;
1 <= i-32 ;
px `1 c= PI / 2 ;
1 <= ii & 1 <= ii ;
1 <= ii & 1 <= ii ;
LMP C in L ;
1 in dom f ;
let seq , seq1 , seq2 , seq1 , seq2 , seq1 , seq2 , seq1 , seq2 , seq1 , seq2 , seq1 , seq2 , seq1 , seq2
set C = a * B ;
x in rng f ;
assume f is_continuous_on X ;
I = dom A ;
u in dom p ;
assume a < x + 1 ;
s-7 is bounded ;
assume I c= P1 ;
n in dom I ;
let Q ;
B c= dom f ;
b + p _|_ a ;
x in dom g ;
F-14 is continuous ;
dom g = X ;
len q = m ;
assume A2 is closed ;
cluster R \ S -> real-valued ;
sup D in S ;
x << sup D ;
b1 >= Z1 & b2 >= Z1 ;
assume w = 0. V ;
assume x in A . i ;
g in the carrier of X ;
y in dom t ;
i in dom g ;
assume P [ k ] ;
c= C c= f ;
x4 is increasing ;
let e2 be element ;
- b divides b ;
F c= \tau ( F ) ;
Gseq is non-decreasing ;
Gseq is non-decreasing ;
assume v in H . m ;
assume b in [#] B ;
let S be non void ManySortedSign , f be Function of S , X ;
assume P [ n ] ;
assume union S is finite & finite 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 Function ;
b ` c= b9 ` & b ` c= b ` ;
assume not x in I[01] + 1 ;
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 & sin . x <> 0 ;
assume y in rng S ;
let x , y be element ;
i2 < i1 or i2 < i1 ;
a * h in a * H ;
p , q in Y ;
redefine func sqrt I -> left ideal ;
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 transitive ;
let x , y be element ;
let j0 be Nat ;
[ y , x ] in R ;
let x , y be element ;
assume y in conv A ;
x in Int V ;
let v be Vector of V ;
P3 halts_on s , m ;
d , c // a , b ;
let t , u be set ;
let X be set ;
assume k in dom s ;
let r be non negative Real ;
assume x in F | M ;
let Y be Subset of S ;
let X be non empty TopSpace , A be Subset of X ;
[ a , b ] in R ;
x + w < y + w ;
{ a , b } is_>=_than c ;
let B be Subset of A , C be Subset of A ;
let S be non empty ManySortedSign ;
let x be variable of f , g , h be element ;
let b be Element of X , x be Element of X ;
R [ x , y ] ;
x ` ` = x ;
b \ x = 0. X ;
<* d *> in D |^ 1 ;
P [ k + 1 ] ;
m in dom ( mn ) ;
h2 . a = y ;
P [ n + 1 ] ;
redefine func G * F -> ] J ;
let R be non empty multMagma , a , b be Element of R ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p `2 ;
assume f | X is lower ;
x in rng co & y in rng co ;
let x be Element of B ;
let t be Element of D ;
assume x in Q .vertices() ;
set q = s ^\ k ;
let t be VECTOR of X ;
let x be Element of A ;
assume y in rng p `1 ;
let M be void mamaid ;
let N be non empty be non empty Subset of M ;
let R be RelStr with finite 1 ;
let n , k be Nat ;
let P , Q be be be be be be _ RelStr ;
P = Q /\ [#] S ;
F . r in { 0 } ;
let x be Element of X ;
let x be Element of X ;
let u be Vector of V ;
reconsider d = x as Int-Location ;
assume I does not destroy a ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < ( len v ) - 1 ;
x <= c2 . x ;
x in F ` & y in F ` ;
redefine func S --> T -> such such that S is such } -valued ;
assume that t1 <= t2 and t2 <= t2 ;
let i , j be even Integer ;
assume F1 <> F2 & F2 <> F1 ;
c in Intersect ( union R ) ;
dom p1 = c & dom p2 = c ;
a = 0 or a = 1 ;
assume A1 <> A2 & A2 is non empty ;
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 /\ dom f2 ;
x in dom ( sec | [. - PI / 2 , 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 & z in dom f2 ;
1 in dom ( D2 | 1 ) ;
( p `2 ) ^2 = 0 ;
j2 <= width G & 1 <= j2 ;
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 g ;
h . x in h . a ;
let G be u be points of on on ( k + 1 ) ;
cluster m * n -> invertible ;
let kk be Nat , x be set ;
i -' 1 > m ;
R is transitive & R is transitive implies R is transitive
set F = <* u , w *> ;
p-2 c= P3 & P3 c= P3 ;
I is_halting_on t , Q ;
assume [ S , x ] is real ;
i <= len ( f2 | ( i + 1 ) ) ;
p is FinSequence of X ;
1 + 1 in dom g ;
Sum R2 = n * r ;
cluster f . x -> complex-valued ;
x in dom f1 & x - h in dom f1 ;
assume [ X , p ] in C ;
BX c= ( X0 \/ { x } ) ;
n2 <= ( 2 |^ ( n + 1 ) ) ;
A /\ cP c= A ` ;
cluster -> x -valued for Function ;
let Q be Subset-Family of S , P be Subset of S ;
assume n in dom g2 & n < len g2 ;
let a be Element of R ;
t `1 in dom ( e2 | ( dom ( e2 | ( dom ( t2 | dom ( t2 | dom ( t2 | dom ( t2 | dom ( t2 | dom (
N . 1 in rng N ;
- z in A \/ B ;
let S be SigmaField of X , x be Element of X ;
i . y in rng i ;
REAL c= dom f & f | X is bounded ;
f . x in rng f ;
mt <= ( r / 2 ) ;
s2 in r-5 & s2 in r-5 ;
let z , z be complex number ;
n <= ( N . m ) ;
LIN q , p , s ;
f . x = waybelow x /\ B ;
set L = |[ S \to T ]| ;
let x be non positive Real ;
let m be Element of M ;
f in union rng F1 & g in union rng F2 ;
let K be add-associative right_zeroed right_complementable non empty doubleLoopStr , p be Polynomial of K ;
let i be Element of NAT ;
rng ( F * g ) c= Y
dom f c= dom x & x in dom y ;
n1 < n1 + 1 & n2 < n1 + 1 ;
n1 < n1 + 1 & n2 < n1 + 1 ;
cluster 1. T -> 8 -element ;
[ y2 , 2 ] `2 = z ;
let m be Element of NAT ;
let S be Subset of R ;
y in rng ( S29 | X ) ;
b = sup dom f & b = sup dom f ;
x in Seg ( len q ) ;
reconsider X = D ( ) as set ;
[ a , c ] in E1 ;
assume n in dom h2 & n + 1 in dom h2 ;
w + 1 = ( a - 1 ) / ( a - 1 ) ;
j + 1 <= j + 1 + 1 ;
k2 + 1 <= k1 & k2 + 1 <= len f ;
let i be Element of NAT ;
Support u = Support p & Support u c= Support p ;
assume X is complete \frac of m , n ;
assume that f = g and p = q ;
n1 <= n1 + 1 & n2 <= n1 + 1 ;
let x be Element of REAL , y be Element of REAL ;
assume x in rng ( s2 | X ) ;
x0 < x0 + 1 & x0 < x0 + 1 ;
len ( L5 ) = W ;
P c= Seg ( len A ) ;
dom q = Seg n & dom q = Seg n ;
j <= width M *' ;
let r8 be real-valued sequence of REAL ;
let k be Element of NAT ;
Integral ( M , P ) < +infty ;
let n be Element of NAT ;
assume z in z := is ] -> ] -) ;
let i be set ;
n -' 1 = n-1 ;
len ( ( n - m ) * ( n - m ) ) = n ;
\mathop { Z } c= F ;
assume x in X or x = X ;
x is midpoint of b , c ;
let A , B be non empty set , f be Function of A , B ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q & i in dom q ;
let s be Element of E |^ \omega ;
let B1 be Basis of x , y ;
L3 /\ L2 = {} ;
L1 /\ L2 = {} ;
assume downarrow x = downarrow y ;
assume b , c // b , c ;
LIN q , c , c ;
x in rng ( f . ( f . x ) ) ;
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 opp = f and h opp = h ;
R1 - R2 is total ;
k in NAT & 1 <= k ;
let a be Element of G ;
assume x0 in [. a , b .] ;
K1 ` is open & ( TOP-REAL 2 ) ` is open ;
assume a , b are_maximal in C ;
let a , b be Element of S ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = Integral ( M , f ) ;
cluster -> n] for <= nlen s\lbrack ;
not u in { ag } ;
the carrier of f c= B ;
reconsider z = x as Vector of V ;
cluster the bounded Str of L -> empty ;
r (#) H is C " ;
s . intloc 0 = 1 ;
assume that x in C and y in C ;
let U0 be strict universal universal MSAlgebra over S , A be Subset of U0 ;
[ x , Bottom T ] is compact ;
i + 1 + k in dom p ;
F . i is stable Subset of M ;
r-35 in : ex y st y in : x in W & y in W } ;
let x , y be Element of X ;
let A , I be such that I is such such that A is \rrangle commutative ;
[ y , z ] in [: O , O :] ;
card dom Macro i = 1 & card dom Macro i = 1 ;
rng Sgm A = A ;
q |- p \! \! q ;
for n holds X [ n ] ;
x in { a } & x in d ;
for n holds P [ n ] ;
set p = |[ x , y , z ]| ;
LIN o , a , b & LIN o , a , b ;
p . 2 = Z |^ Y ;
( D . ( j + 1 ) ) `2 = {} ;
n + 1 + 1 <= len g ;
a in [: dom ( Al ( ) ) , dom ( Al ( ) ) :] ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
let I be Ideal of L ;
set g = f1 + f2 , h = f2 + f3 ;
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 ( t ) ) ) ;
assume a in [: the carrier of G , the carrier of G :] ;
i `1 <= len ( y `1 ) ;
assume p divides b1 + b2 & p divides b2 + b2 ;
M1 <= sup M1 & sup M1 <= sup M2 ;
assume x in W-min ( X ) & y in W-min ( X ) ;
j in dom ( z | i ) ;
let x be Element of D ( ) ;
IC s4 = l1 & IC s4 = l1 ;
a = {} or a = { x } ;
set uG = Vertices G , uH = Vertices H ;
seq " is non-zero implies seq " is non-zero & lim seq = 0
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
hcn c= h-14 & hj c= dom h ;
]. a , b .[ c= Z ;
X1 , X2 are_separated implies X1 , X2 are_separated
a in Cl ( union F \ G ) ;
set x1 = [ 0 , 0 ] ;
k + 1 -' 1 = k ;
cluster -> real-valued for Relation of NAT , REAL ;
ex v st C = v + W ;
let IT be non empty zero Nat , a be Element of G ;
assume V is Abelian add-associative right_zeroed right_complementable ;
XY \/ Y in \sigma ( L ) ;
reconsider x = x as Element of S ;
max ( a , b ) = a ;
sup B is upper & sup B is upper ;
let L be non empty reflexive antisymmetric RelStr , X be Subset of L ;
R is reflexive & R is transitive implies R is transitive
E , g |= ( the_left_argument_of H ) ;
dom G ' ( y ) = a ;
( 1 - 4 ) >= - r ;
G . p0 in rng G & G . p0 in rng G ;
let x be Element of FF , y be Element of FF ;
D [ ( E , 0 ) `1 , 0 ] ;
z in dom id ( B ) & z in dom id ( B ) ;
y in the carrier of N & y 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= X ;
j `2 + 1 in dom s1 & j `2 + 1 in dom s1 ;
let A , B be strict Subgroup of G ;
let C be non empty Subset of REAL ;
f . z1 in dom h & h . z2 in dom h ;
P . k1 in rng P & P . k2 in rng P ;
M = ( A +* {} ) +* ( A .--> {} ) ;
let p be FinSequence of REAL , i be Nat ;
f . n1 in rng f & f . n1 in rng f ;
M . ( F . 0 ) in REAL ;
ind [. a , b .[ = b-a ;
assume the distance of V , Q is v ;
let a be Element of ^ ( V ) ;
let s be Element of PP , a be Element of s ;
let PP be non empty Line ( D , i ) ;
n be Nat ;
the carrier of g c= B & g is one-to-one ;
I = halt SCM R & I = halt SCM R ;
consider b being element such that b in B ;
set BM = BCS ( K , n , k ) ;
l <= a1 . j ;
assume x in downarrow [ s , t ] ;
( x `2 ) ^2 in uparrow t ;
x in \in \in \in ( JumpParts T ) & x in { 0 } ;
let h be Morphism of c , a ;
Y c= 1. ( K , the_rank_of Y ) ;
A2 \/ ( 4 * A2 ) c= Carrier ( L ) \/ Carrier ( L ) ;
assume LIN o , a , b ;
b , c // d1 , e2 ;
x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 *> ;
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 -> -> -> -> -> -> -> -> -> non closed for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 , q1 is_collinear & q1 , q2 , q2 is_collinear ;
dom M1 = Seg n & dom M2 = Seg n ;
x = [ x1 , x2 , x3 ] ;
let R , Q be ManySortedSet of A ;
set d = ( 1 / ( n + 1 ) ) ;
rng g2 c= dom W & ( W is open implies W is open ) ;
P . ( [#] Sigma \ B ) <> 0 ;
a in field R & a = b ;
let M be non empty Subset of V , v be Element of V ;
let I be Program of SCM+FSA , a be Int-Location ;
assume x in rng ( such that R is open and x in R ;
let b be Element of the carrier of T ;
dist ( e , z ) - r > r-r ;
u1 + v1 in W2 & v1 + v2 in W1 ;
assume the carrier of L misses rng G ;
let L be lower-bounded antisymmetric transitive antisymmetric RelStr ;
assume [ x , y ] in a9 ;
dom ( A * e ) = NAT ;
let a , b be Vertex of G ;
let x be Element of Bool M ;
0 <= 2 * PI ;
o9 , a9 // o9 , y & o9 , b9 // o9 , y ;
{ v } c= the carrier of l ;
let x be bound of A ;
assume x in dom ( uncurry f ) & y in dom ( uncurry f ) ;
rng F c= ( product f ) |^ X ;
assume that D2 . k in rng D and 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 ;
assume LIN c , a , e1 ;
cluster -> increasing for FinSequence of NAT ;
reconsider d = c , e = d as Element of L1 ;
( v2 |-- I ) . X <= 1 ;
assume that x in the carrier of f and y in the carrier of g ;
conv @ A c= conv A & conv @ A c= conv A ;
reconsider B = b as Element of the carrier of T ;
J , v |= P \lbrack l , l2 .] ;
redefine func J . i -> non empty TopSpace ;
ex_sup_of Y1 \/ Y2 , T & ex_sup_of Y1 , T ;
W1 is_well field W1 & W2 is_well field W1 implies W1 + W2 is_field W1
assume x in the carrier of R & y in the carrier of R ;
dom ( n |-> ( i + 1 ) ) = Seg n ;
s4 misses s4 & s4 misses s5 ;
assume ( a 'imp' b ) . z = TRUE ;
assume that X is open and f = X --> d ;
assume [ a , y ] in 0. ( , f ) ;
assume that function I c= J and function I c= K ;
Im ( lim seq ) = 0 & Im ( lim seq ) = 0 ;
( ( sin * sin ) `| Z ) . x <> 0 ;
sin * sin is_differentiable_on Z & sin * cos is_differentiable_on Z ;
t3 . n = t3 . n .= 0 ;
dom ( element | ( dom F ) ) c= dom F ;
W1 . x = W2 . x .= W1 . x .= W2 . x ;
y in W .vertices() \/ W .vertices() ;
( k + 1 ) <= len ( v | k ) ;
x * a \equiv y * a . ( mod m ) ;
proj2 .: S c= proj2 .: P & proj2 .: P c= proj2 .: P ;
h . p4 = g2 . I .= ( h . p2 ) `2 ;
Gij `1 = ( U /. 1 ) `1 .= ( U /. 1 ) `1 ;
f . rp1 in rng f & f . rp1 in rng f ;
i + 1 + 1-1 <= len f - 1 ;
rng F = rng ( F | ( Seg n ) ) ;
mode seq is well unital associative commutative non empty multMagma ;
[ x , y ] in A ~ { a } ;
x1 . o in L2 . o & x2 . o in L2 . o ;
the carrier of m _ p c= B ;
not [ y , x ] in id X ;
1 + p .. f <= i + len f ;
seq ^\ k1 is lower & seq ^\ k1 is lower ;
len ( F | ( i -' 1 ) ) = len I - 1 ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be Complex , a be Element of COMPLEX ;
Comput ( P , s , n ) = s ;
k <= k + 1 & k + 1 <= len p ;
reconsider c = {} T as Element of L ;
let Y be Element of \vert the { of T .| ;
cluster -> directed-sups-preserving for Function of L , L ;
f . j1 in K . j1 & f . j2 in K . j1 ;
redefine func J => y -> total Function ;
K c= 2 |^ the carrier of T ;
F . b1 = F . b2 & F . b2 = F . b2 ;
x1 = x or x1 = y or x1 = z ;
pred a <> {} means : Def4 : a = 1 & a = 1 ;
assume that 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 or LIN o , b , b9 ;
reconsider m = x as Element of Funcs ( V , C ) ;
let f be non constant FinSequence of D ;
let FF2 be non empty non void TopSpace , f be Function of FF2 , FF2 ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in F-8 ;
reconsider pp2 = x as Subset of m -tuples_on REAL ;
let A , B , C be Element of R ;
redefine mode non empty strict real RelStr is strict for non empty sqrt 5 ;
rng c `1 misses rng ( e | ( rng c ) ) ;
z is Element of gr ( { x } ) ;
not b in dom ( a .--> p1 ) ;
assume that k >= 2 and P [ k ] ;
Z c= dom ( cot * cot ) /\ dom ( cot * cot ) ;
the component of Q c= UBD A & UBD ( Q ) c= UBD ( Q ) ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( ( 1 / 2 ) (#) ( f ^ ) ) ;
pred f = u , v , a , f ;
for n holds P1 [ n ] implies P1 [ n + 1 ]
{ x . O : x in L } <> {} ;
let x be Element of V . s ;
a , b be Nat ;
assume that S = S2 and p = p2 and S is open ;
gcd ( n1 , n2 , n1 , n2 ) = 1 & gcd ( n1 , n2 , n1 , n2 ) = 1 ;
set on = a * ( ( - 1 ) |^ 2 ) ;
seq . n < |. r1 .| & |. seq . n .| < r1 ;
assume that seq is increasing and r < 0 ;
f . ( y1 , x1 ) <= a & f . ( y1 , x1 ) <= b ;
ex c being Nat st P [ c ] & c <= n ;
set g = { n |^ 1 : n in NAT } ;
k = a or k = b or k = c ;
( a , b ) `1 , ( a , b ) `2 ] in R ;
assume that Y = { 1 } and s = <* 1 *> ;
IF1 . x = f . x .= f . x .= 0. F .= 0. F ;
W3 . ( W1 W1 W1 W1 W1 . 1 ) = W3 . 1 .= W2 . 1 ;
cluster trivial -> trivial for Walk of G , finite _Graph ;
reconsider u = u as Element of Bags X ;
A in B ^ implies A , B ^ , B ^ ;
x in { [ 2 * n + 3 , k ] } ;
1 >= ( q `1 / |. q .| - cn ) / ( 1 - cn ) ;
f1 is_as ] as ] such
( f . q ) `2 <= ( q `2 ) / ( |. q .| ) ;
h is_the carrier of Cage ( C , n ) ;
( b `2 ) ^2 / ( |. b .| ) ^2 <= ( |. b .| ) ^2 / ( |. b .| ) ^2 ;
let f , g be Function of X , Y ;
S * ( k , k ) <> 0. K ;
x in dom ( max ( - f , - f ) ) ;
p2 in ( ( N | p1 ) | ( p1 | p2 ) ) " { 0 } ;
len ( the_left_argument_of H ) < len ( H ) ;
F [ A , F-14 ( A , F ) . A ] ;
consider Z such that y in Z and Z in X ;
redefine attr 1 in C means : Def4 : A c= C |^ A ;
assume that r1 <> 0 or r2 <> 0 and r1 < 0 ;
rng q1 c= rng C1 & rng q1 c= rng ( f | X ) ;
A1 , L , A3 , A3 , A2 , A3 , A2 , A3 , A1 } ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in where p is Element of dom ( p | ( ( len p ) | ( len p ) ) ) : p in dom ( p | ( len p ) ) } ;
then S is atomic implies P-2 [ S ] ;
Cl Int ( [#] T ) = [#] T & Cl ( Int ( [#] T ) ) = [#] T ;
f12 | A2 = f2 | A2 .= ( f2 | A2 ) | A2 ;
0. M in the carrier of W & 0. M in the carrier of W ;
let v , v be Element of M ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ V & V \ V c= Y \ Z ;
let X be Subset of S ~ ;
consider H1 such that H = 'not' H1 and H1 in A ;
1_ 1 c= ( 1 * t ) * ( p1 - 1 ) ;
0 * a = 0. R .= a * 0. R .= a * 0. R ;
A |^ ( 2 , 2 ) = A ^^ A ;
set vY = ( vseq /. n ) `1 , vY = ( vseq /. n ) `1 ;
r = 0. ( REAL-NS n ) .= 0. ( REAL-NS n ) ;
( f . p4 ) `1 >= 0 & ( f . p4 ) `2 >= 0 ;
len W = len ( W .cut ( n , m ) ) ;
f /* ( s * G ) is divergent_to-infty ;
consider l being Nat such that m = F . l ;
t16 does not destroy b1 & not ( ex b1 , b2 st b1 in dom b1 & b2 in dom b2 & b1 <> b2 ) ;
reconsider Y1 = X1 , Y2 = X2 as SubSpace of X ;
consider w such that w in F and not x in w ;
let a , b , c , d be Real ;
reconsider i = i - 1 as non zero Element of NAT ;
c . x >= id ( the carrier of L ) . x ;
\sigma ( T ) \/ omega ( T ) is Basis of T ;
for x being element st x in X holds x in Y ;
cluster [ x1 , x2 , x3 , x4 ] -> pair ;
downarrow a /\ downarrow t is Ideal of T ;
let X be with_with NAT non empty set , f be Function of X , NAT ;
rng f = ` ex S being Element of ` ;
let p be Element of B , x be the bound sort of S ;
max ( N1 , 2 ) >= N1 & max ( N1 , 2 ) >= N2 ;
0. X <= b |^ ( m * mm1 ) ;
assume that i in I and R0 . i = R ;
i = j1 & p1 = q1 & p2 = q2 implies p1 = p2
assume gR in the right of g & FR in the carrier of g ;
let A1 , A2 be Point of S , A be Subset of S ;
x in h " P /\ [#] T1 & x in h " P ;
1 in Seg 2 & 1 in Seg 3 & 2 in Seg 3 ;
reconsider X-5 = X , X] as non empty Subset of T<* Y , Z *> ;
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the Target of G ) -valued ;
n1 <= i2 + len g2 & 1 <= i2 + len g2 ;
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume that v in the carrier' of G2 and v in the carrier' of G1 ;
y = Re y + ( Im y ) * i ;
( ( - 1 ) |^ ( p -' 1 ) ) gcd p = 1 ;
x2 is_differentiable_on ]. a , b .[ & ( for x st x in ]. a , b .[ holds x - b < x ) ;
rng ( M | ( len D ) ) c= rng ( D2 | ( len D ) ) ;
for p being Real st p in Z holds p >= a
( cn ) * ( f | K1 ) = proj1 * ( f | K1 ) ;
( 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 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 , ib = j as Element of NAT ;
dom f c= [: C , D :] & rng f c= D ;
x in ( the sequence of B ) . n ;
len \rbrace in Seg ( len ( f2 | ( i + 1 ) ) ) ;
pp1 c= the topology of T & pp1 c= the topology of T ;
]. r , s .[ c= [. r , s .] ;
let B2 be Basis of T2 , a be Element of T1 ;
G * ( B * A ) = ( id o1 ) * ( id o2 ) ;
assume that p , u , v , w , w is_collinear and u , v , w , w , y ;
[ z , z ] in union rng ( F | ( n + 1 ) ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S , $1 .. S = $1 .. S ;
LIN a1 , a3 , b1 & LIN a1 , b1 , c1 ;
f " ( f .: x ) = { x } ;
dom ( w2 ) = dom r12 & dom ( w2 ) = dom r12 ;
assume that 1 <= i and i <= n and j <= n ;
( ( g2 ) . O ) `2 <= 1 ;
p in LSeg ( E . i , F . i ) ;
Ii * ( i , j ) = 0. K ;
|. f . ( s . m ) -g .| < g1 ;
q7 . x in rng ( q7 | ( Seg n ) ) ;
Carrier ( LLet ) misses Carrier ( Lj ) ` ;
consider c being element such that [ a , c ] in G ;
assume that Na9 = o8 and o8 = o8 and o8 = o8 ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= ( F |^ C-1 ) " { C } ;
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [. f . j , f . j .] ;
pred 0 <= x & x <= 1 & x ^2 <= x ;
p `2 - q `2 <> 0. TOP-REAL 2 & p `1 - q `2 <> 0. TOP-REAL 2 ;
redefine func \subseteq aaa.: ( S , T ) -> non empty set ;
let x be Element of S ~ ;
( \HM { the } \HM { object } \HM { of F ) is one-to-one ;
|. i .| <= - ( 2 |^ n ) / ( 2 |^ n ) ;
the carrier of I[01] = dom P & P . 0 = P . 1 ;
} * ( n + 1 ) ! > 0 * n ;
S c= ( A1 /\ A2 ) /\ ( A1 /\ A2 ) \/ ( A1 /\ A2 ) /\ ( A1 /\ A2 ) ;
a3 , a4 // b3 , b2 or a3 , a4 // b3 , b2 ;
then dom A <> {} & dom A <> {} ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , Y , G & y in X & z in Y ;
set v2 = ( v /. ( i + 1 ) ) `1 , v1 = ( v /. ( i + 1 ) ) `1 ;
x = r . n .= r4 . n .= r4 . n ;
f . s in the carrier of S2 & f . s in the carrier of S2 ;
dom g = the carrier of I[01] & dom g = the carrier of I[01] ;
p in Lower_Arc ( P ) /\ Lower_Arc ( P ) ;
dom ( d2 * ( F | A ) ) = [: A , A :] ;
0 < ( p / ( ||. z .|| + 1 ) ) / ( ||. z .|| + 1 ) ;
e . ( mm + 1 ) <= e . ( m3 + 1 ) ;
B \ominus X \/ B \ominus Y c= B \ominus X /\ Y
-infty < Integral ( M , Im ( g | B ) ) ;
cluster O := F -> Line for Element of X ;
let U1 , U2 be non-empty MSAlgebra over S , a be Element of U1 ;
Proj ( i , n ) * g is_differentiable_on X ;
x , y , z be Point of X , p be Point of X ;
reconsider pp = p . x , pp = p . x as Subset of V ;
x in the carrier of Lin ( A ) & x in the carrier of Lin ( A ) ;
let I , J be parahalting Program of SCM+FSA , a be Int-Location , k1 be Integer ;
assume that - a is lower and a is lower and b is lower ;
Int Cl A c= Cl Int Cl Int A & Int Cl A c= Cl Int Cl A ;
assume for A being Subset of X holds Cl A = A ;
assume q in Ball ( |[ x , y ]| , r ) ;
( p2 `2 ) ^2 / ( |. p2 .| ) ^2 <= ( p2 `2 ) ^2 / ( |. p2 .| ) ^2 ;
Cl Q ` = [#] ( TT | P ) .= P ;
set S = the carrier of T , S = the carrier of T ;
set I8 = ' ( f |^ n ) , I8 = ' ( f |^ n ) ;
len p - n = len ( p - n ) .= len p - n ;
A is Permutation of Swap ( A , x , y ) ;
reconsider n6 = n8 - 1 as Element of NAT ;
1 <= j + 1 & j + 1 <= len ( s | n1 ) ;
let q<* , q\mathopen { - } 1 } , q be State of M ;
( a - b ) in the carrier of S1 & ( - b ) in the carrier of S2 ;
c1 /. n1 = c1 . n1 & c2 /. n1 = c2 . n1 ;
let f be FinSequence of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
y = ( ( f * ( S . n ) ) | ( dom ( S . n ) ) ) | ( dom ( S . n ) ) ;
consider x being element such that x in an " A ;
assume r in ( ( dist ( o ) ) .: P ) ;
set i2 = ( n , h ) `1 , i1 = ( n , h ) `1 ;
h2 . ( j + 1 ) in rng h2 & h2 . ( j + 1 ) in rng h2 ;
Line ( ( M @ ) , k ) = M . i ;
reconsider m = ( x - 2 ) / ( x - 2 ) as Element of 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 being being Scott Scott Subset of L , x be Element of X ;
then x <= y & : x c= ( { y } ) \ { y } ;
set M = n -tuples_on the carrier of K ;
reconsider i = x1 , j = x2 , k = x3 as Nat ;
rng ( the_arity_of ( a ) ) c= dom H & ( the_arity_of ( a ) ) . o in dom ( H . o ) ;
z1 " = z9 " & ( z1 " ) * ( z1 " ) = z1 " * z1 " ;
x0 - r / 2 in L /\ dom f & x0 - r / 2 in L /\ dom f ;
then w is { phi } /\ rng w <> {} & not w in rng w ;
set x-10 = ( x9 ^ <* Z *> ) ^ <* Z *> ;
len w1 in Seg len w1 & len w2 = len w1 & 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 / ( 1 + ( p `1 / p `2 ) ^2 ) <= ( G * ( 1 , 1 ) `1 ) / ( 1 + ( p `1 / p `2 ) ^2 ) ;
rng ( g | ( len g ) ) c= L~ ( g | ( len g ) ) ;
reconsider k = i-1 * ( l + j ) as Nat ;
for n being Nat holds F . n is \HM { -infty } ;
reconsider x9 = x9 , y9 = y9 as Vector of M ;
dom ( f | X ) = X /\ dom f .= X ;
p , a // p , c & b , a // c , c ;
reconsider x1 = x , y1 = y , z1 = z as Element of REAL m ;
assume i in dom ( a * p ^ q ) ;
m . ( ( k + 1 ) + 1 ) = p . ( ( k + 1 ) + 1 ) ;
a / ( s . m - s . n ) / ( n + 1 ) <= 1 ;
S . ( n + k + 1 ) c= S . ( n + k ) ;
assume that B1 \/ C1 = B2 \/ C2 and B2 \/ C2 = A2 \/ B2 ;
X . i = { x1 , x2 , x3 , x4 } . i ;
r2 in dom ( h1 + h2 ) & r1 in dom ( h1 + h2 ) ;
- - 0. R = a & b-0 = b ;
FF is_closed_on t2 , Q2 & g is_halting_on t2 , Q2 & g is_halting_on t2 , Q2 ;
set T = -> inInInInIn\vert ( X , x0 ) , x0 ;
Int Cl ( Int R ) c= Int R & Int Cl ( Int R ) c= Cl R ;
consider y being Element of L such that c . y = x ;
rng ( FF | x ) = { F ( x ) } & rng ( F | x ) c= dom F ;
G-23 \ { c } c= B \/ S \/ S ;
f[#] A is Relation of [: X , X :] , X ;
set RP = the Element of P , Q = the Element of Q ;
assume that n + 1 >= 1 and n + 1 <= len M ;
let k2 be Element of NAT , i be Nat ;
reconsider p/* = u as Element of ( TOP-REAL n ) | ( ( TOP-REAL n ) | K1 ) ;
g . x in dom f & x in dom g implies ( g . x ) in dom f
assume that 1 <= n and n + 1 <= len f1 ;
reconsider T = b * N as Element of G / ( N , G ) ;
len ( ( P * ( i , j ) ) * ( P * ( i , j ) ) ) <= len ( P * ( i , j ) ) ;
x " in the carrier of A1 & x " in the carrier of A2 ;
[ i , j ] in Indices ( ( A @ ) * ( 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 , x be Element of REAL i ;
rng f = the carrier of element & { 0 } c= the carrier of element ;
assume s1 = sqrt ( ( 2 |^ 2 ) - ( p / 2 ) ) ;
pred a > 1 & b > 0 & a / b > 1 ;
let A , B , C be Subset of [: I , I :] ;
reconsider X0 = X , Y0 = Y as RealNormSpace , X be Subset of X ;
let f be PartFunc of REAL , REAL , x be Element of REAL ;
r * ( v1 |-- I ) . X < r * 1 ;
assume that V is Subspace of X and X is Subspace of V ;
let t-3 , t-3 , t-3 , t-3 be Relation ;
Q [ ( e-14 \/ { v } ) \/ { v } , f . ( k + 1 ) } ] ;
g \circlearrowleft ( W-min L~ z ) = z & ( W-min L~ z ) .. z < ( W-min L~ z ) .. z ;
|. |[ x , v ]| - |[ x , y ]| .| = v12 ;
- f . w = - ( L * w ) .= - ( L * w ) ;
z - y <= x iff z <= x + y & y <= z + y ;
( 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 ;
( ( tan | Z ) `| Z ) . x in dom ( sec | Z ) ;
i2 = ( f /. len f ) `1 .= ( f /. len f ) `1 .= ( f /. len f ) `1 ;
X1 = X2 \/ ( X1 \ X2 ) & X2 = X2 \/ ( X1 \ X2 ) ;
[. a , b , 1_ G .] = 1_ G & a * b = 1_ G ;
let V , W be non empty vector space over F_Complex , f be Function of V , W ;
dom g2 = the carrier of I[01] & dom ( ( g2 ) | K1 ) = the carrier of ( TOP-REAL 2 ) | K1 ;
dom ( f2 ) = the carrier of I[01] & rng ( f2 ) c= the carrier of I[01] ;
( proj2 | X ) .: X = proj2 .: X .= proj2 .: ( X /\ Y ) ;
f . ( x , y ) = h1 . ( x `1 , y `2 ) ;
x0 - r < a1 . n & x0 - r < x0 - r ;
|. ( f /* s ) . k - ( f /* s ) . k .| < r ;
len ( Line ( A , i ) ) = width A & width ( Line ( A , i ) ) = width A ;
SFinSequence / ( S * g ) = ( S . g ) / ( g * f ) ;
reconsider f = v + u as Function of X , the carrier of Y ;
intloc 0 in dom Initialized ( p +* ( intloc 0 ) ) & intloc 0 in dom Initialized ( p +* ( intloc 0 ) ) ;
i1 , i2 , i3 , i3 , Nat & I does not destroy b1 , b2 , b3 , b3 ;
arccos r + arccos r = ( cos ( 2 ) ) ^2 + 0 ;
for x st x in Z holds f2 is_differentiable_in x & f2 is_differentiable_in x & Z = dom ( f1 - f2 ) ;
reconsider q2 = ( q - x ) / ( |. q - x .| ) as Element of REAL ;
( 0 qua Nat ) + 1 <= i + j1 & i + 1 <= j + 1 ;
assume that f in the carrier of [: X , Omega Y :] and f in the carrier of [: X , Omega Y :] ;
F . a = H / ( ( x. ( x , y ) ) . a ) ;
true T at ( C , u ) = TRUE & not ( C , u ) in T ;
dist ( ( a * seq ) . n , h ) < r ;
1 in the carrier of [. 0 , 1 .] & 1 in dom ( f | [. 0 , 1 .] ) ;
( p2 `1 ) ^2 - x1 > - g / 2 - g / 2 ;
|. r1 - be Real .| = |. a1 .| * |. thesis - x .| ;
reconsider S-14 = 8 , S-14 = 8 as Element of Seg 8 ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b
D0W .order() = DW .1 + 1 ;
i1 = ( a + n ) & i2 = ( a + n ) * ( a + n ) ;
f . a [= f . ( f . O1 "\/" f . ( a "\/" a ) ) ;
pred f = v & g = u & f + g = v + u ;
I . n = Integral ( M , F . n ) ;
chi ( T1 , T1 ) . s = 1 & ( chi ( T1 , T1 ) ) . 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 \/ S ) & L~ ( R \/ S ) meets L~ ( R \/ S ) ;
set h = the continuous Function of X , R , x be Element of X ;
set A = { L . ( k . n ) : n in dom L } ;
for H st H is atomic holds P [ H ] ;
set b\HM = S5 \ ( i + 1 ) , S5 = ( i + 1 ) \ ( i + 1 ) ;
Hom ( a , b ) c= Hom ( a opp , b opp ) ;
( 1 - s ) / ( n + 1 ) < ( 1 - s ) / ( n + 1 ) ;
( l `1 ) `1 = [ dom l , cod l ] `1 .= ( l `1 ) `1 ;
y +* ( i , y /. i ) in dom g & y in dom g ;
let p be Element of [: Al ( ) , D ( ) :] ;
X /\ X1 c= dom ( f1 - f2 ) /\ dom ( f1 - f2 ) ;
p2 in rng ( f /^ ( len p1 -' 1 ) ) & p1 in rng ( f /^ ( len p1 -' 1 ) ) ;
1 <= indx ( D2 , D1 , j1 ) + 1 - 1 ;
assume that x in ( ( ( TOP-REAL 2 ) | K1 ) \/ ( ( TOP-REAL 2 ) | K1 ) ) /\ K1 ;
- 1 <= ( ( f2 ) . O ) `2 & - 1 <= ( ( f2 ) . I ) `2 ;
let f , g be Function of I[01] , TOP-REAL 2 , a , b , c , d be Real ;
k1 -' k2 = k1 - k2 & k2 -' k1 = k2 -' k1 + 1 ;
rng seq c= ]. x0 , x0 + r .[ & ( for n holds seq . n < x0 ) implies ( seq is convergent & lim seq = x0 )
g2 in ]. x0 - r , x0 + r .[ & x0 < g2 & g2 in ]. x0 - r , x0 .[ ;
sgn ( p `1 , K ) = - 1_ K .= - 1_ K ;
consider u being Nat such that b = p |^ y * u ;
ex A being as as as as as as as normal normal Ordinal st a = Sum A ;
Cl ( union ( H ) ) = union ( ( Cl ( H ) ) \ ( Cl ( H ) ) ) ;
len t = len t1 + len t2 .= len t1 + len t2 .= len t1 + len t2 ;
v-29 = v + w |-- ( A , v + A8 ) ;
cv <> DataLoc ( ( t . GBP ) , 3 ) & v . DataLoc ( ( t . GBP ) , 3 ) = s . DataLoc ( t . GBP , 3 ) ;
g . s = sup ( d " { s } ) .= s . s ;
( \dot y ) . s = s . ( \dot y . s ) ;
{ s : s < t } in INT implies t = {} & s = {} ;
s ` \ s = s ` \ ( 0. X \ s ) .= 0. X \ ( 0. X \ s ) .= 0. X ;
defpred P [ Nat ] means B + $1 in A & B + $1 in A ;
( 3be ! + 1 ) ! = 33222! * ( 3<* + 1 *> + 1 ) ;
( ( U succ A ) \ ( A \ { {} } ) ) = ( ( U U \ A ) \ ( A \ { {} } ) ) ;
reconsider y = y as Element of COMPLEX , i be Nat ;
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 REAL ;
set f = ( S , U ) \mathop { \rm \hbox { - } F } , F = S \! \mathop { - } S ;
consider Z being set such that lim s in Z and Z in F ;
let f be Function of I[01] , TOP-REAL n , a , b be Real ;
( SAT M ) . [ n + i , 'not' A ] <> 1 ;
ex r being Real st x = r & a <= r & r <= b ;
let R1 , R2 be Element of REAL n , x be Element of REAL n ;
reconsider l = 0. ( V ) , a = 0. ( V ) as Linear_Combination of A ;
set r = |. e .| + |. n + 1 .| + a * |. s .| + a ;
consider y being Element of S such that z <= y and y in X ;
a 'or' ( b 'or' c ) = 'not' ( ( a 'or' b ) 'or' c )
||. ( x9 - g ) * ( x - g ) .|| < r2 / 2 ;
b9 , a9 // b9 , c9 & b9 , c9 // b9 , c9 & b9 , c9 // b9 , c9 ;
1 <= k2 -' k1 & k1 + 1 = k2 or k1 + 1 = k2 & k2 = k2 + 1 ;
( ( p `2 / |. p .| - cn ) / ( 1 + cn ) ) ^2 >= 0 ;
( q `2 / |. q .| - cn ) / ( 1 + cn ) < 0 ;
E-max C in right_cell ( R , 1 ) & E-max L~ Cage ( C , 1 ) in rng R ;
consider e being Element of NAT such that a = 2 * e + 1 ;
Re ( ( lim F ) | D ) = Re ( ( lim G ) | D ) ;
LIN b , a , c or LIN b , c , a ;
p `1 , a // a `1 , b or p `1 , a // b `1 , a ;
g . n = a * Sum ( f | ( n + 1 ) ) .= f . n ;
consider f being Subset of X such that e = f and f is w ;
F | ( N2 ~ S ) = CircleMap * ( ( | N2 ) | ( N2 | N2 ) ) ;
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 } & the carrier of W = { 0. V } ;
rng ( ( - 1 / 2 ) (#) ( ( id Z ) ^ ) `| Z ) = [. - 1 , 1 .] ;
assume that Re seq is summable and Im seq is summable and Im seq is summable and Im seq is summable ;
||. ( vseq . n ) - ( vseq . n ) .|| < e / 2 ;
set g = O --> 1 ;
reconsider t2 = t11 as ( 0 , 0 ) string of S2 , D ;
reconsider x9 = seq . n , y9 = seq . n as sequence of REAL n ;
assume that C meets ( L~ go \/ L~ pion1 ) and L~ go meets L~ pion1 and L~ pion1 meets L~ pion1 ;
- ( ( - 1 ) / ( n + 1 ) ) < F . n - x ;
set d1 = \bf dist ( x1 , z1 ) , d2 = dist ( x2 , z1 ) , d2 = dist ( x1 , z2 ) ;
2 |^ ( 2 |^ 100 -' 1 ) = 2 |^ ( 100 -' 1 ) - 1 ;
dom ( ( v | ( Seg ( len ( v | i ) ) ) ) | ( Seg ( len ( v | i ) ) ) ) = Seg ( len ( v | i ) ) ;
set x1 = - k2 + |. k2 - k1 .| + 4 ;
assume for n being Element of X holds 0. X <= F . n ;
assume that 0 <= T-32 . i and T-32 . ( i + 1 ) <= 1 ;
for A being Subset of X holds c . ( c . A ) = c . A
the carrier of Carrier ( L\lbrace x } + L2 ) c= I2 & the carrier of L|^ ( x + 1 ) c= I2 ;
'not' Ex ( x , p ) => All ( x , 'not' p ) is valid ;
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the normal normal w.r.t. of {} ;
Z c= dom ( ( ( 1 / 2 ) (#) ( f1 - f2 ) ) `| Z ) ;
|. 0. TOP-REAL 2 - ( q `2 / |. q .| - cn ) .| < r / 2 ;
ConsecutiveSet2 ( B , succ ( d , ( A , d ) ) ) c= ConsecutiveSet2 ( A , succ ( d , ( A , d ) ) ) ;
E = dom L8 & L is measurable & L is measurable implies L (#) ( L (#) F ) is measurable
C |^ ( A + B ) = C |^ B * C |^ A ;
the carrier of W2 c= the carrier of V & the carrier of W1 c= the carrier of V ;
I . IC Comput ( P , s , m ) = P . IC Comput ( P , s , m ) .= ( card I + 2 ) ;
pred x > 0 means : Def4 : ( 1 - x ) |^ 2 = x |^ ( - 1 ) ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) \/ LSeg ( g , k ) ;
consider p being Point of T such that C = [. p , R .] and p in A ;
b , c are_connected & - C , - C - a + b .] ;
assume that f = id the carrier of O1 and g is Function of the carrier of O1 , the carrier of O1 ;
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 ) , ( ( U2 ) | ( the carrier of U1 ) ) ;
A1 in the carrier of ( G . k ) | ( X /\ k ) ;
|. - x .| = - ( - x ) .= - x .= - x .= - x ;
set S = in in in in in in { x , y , c } ;
Fib ( n ) * ( 5 * Fib ( n ) - 1 ) >= 4 * ( 5 * n ) - 1 ;
vseq /. ( k + 1 ) = ( vseq /. ( k + 1 ) ) `1 ;
0 mod i = - ( i * ( 0 qua Nat ) ) / ( i - 1 ) ;
Indices M1 = [: Seg n , Seg n :] & len M1 = n & width M2 = n ;
Line ( S\mathopen { - } j } , j ) = S\mathopen { - j } ;
h . ( x1 , y1 ) = [ y1 , x1 ] & h . ( y1 , y2 ) = [ y1 , x1 ] ;
|. f - Re ( |. f .| * ( card b ) ) .| is nonnegative ;
assume that x = ( a1 ^ <* x1 *> ) ^ b1 and y = ( a1 ^ <* x1 *> ) ^ b1 ;
ME 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 ;
LIN c , q , b & LIN c , q , c ;
f\rbrace . ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + z1 ) .= x1 + ( y1 + z1 ) ;
flet a . a = ( f . a ) . a & v in InputVertices S & 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 , E7 = Cage ( C , n ) ;
( p `1 ) ^2 >= ( E-max C ) `1 & ( E-max C ) `1 <= ( E-max C ) `1 ;
consider p such that p = p-20 and s1 < p and p < i and i < len p ;
|. ( f /* ( s * F ) ) . l - GM /. 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 ) = lim ( f1 /* s1 ) ;
f . x1 = x1 & f . y1 = y1 & f . y2 = y1 & f . y2 = y2 ;
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 & rng B c= the carrier of V ;
consider r such that r _|_ a and r _|_ x and r _|_ y ;
reconsider B1 = the carrier of Y1 , B2 = the carrier of Y2 as Subset of X ;
1 in the carrier of [. 1 / 2 , 1 .] & 1 / 2 in dom ( ( 1 / 2 ) (#) ( ( 1 / 2 ) * ( 1 / 2 ) ) ) ;
for L being complete LATTICE holds <* <* \mathbb L *> , <* a *> *> , L #) is isomorphic
[ gi , gj ] in Ii \ Ij " { i } & [ gj , gj ] in I ;
set S2 = 1GateCircStr ( x , y , c ) , S2 = 1GateCircStr ( y , c , d ) ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and for r st r in dom f1 & r < x0 holds f1 . r < f2 . x0 ;
reconsider y = ( a " ) / ( F . ( F . ( n + 1 ) ) ) as Element of L ;
dom s = { 1 , 2 , 3 } & s . 1 = d1 & s . 2 = d2 ;
( min ( g , ( 1 - g ) ) * f ) . c <= h . c ;
set G2 = the as finite _Graph of G , { v } , { w } } , G = the \langle of G , { v } , { w } } ;
reconsider g = f as PartFunc of REAL , REAL-NS n , REAL-NS n ;
|. s1 . m / p .| < d / p / p & |. s1 . m - 1 .| < d / p ;
for x being element st x in ( q u ) holds x in ( q * t ) iff x in ( q * t )
P = the carrier of ( TOP-REAL n ) | P & Q = ( TOP-REAL n ) | P ;
assume that p11 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) and LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) <> {} ;
( 0. X \ x ) |^ ( m * k + 1 ) = 0. X ;
let g be Element of Hom ( cod f , dom g ) ;
2 * a * b + ( 2 * c * d ) <= 2 * C1 * C2 ;
let f , g , h be Point of the carrier of X , g be Point of Y ;
set h = Hom ( a , g (*) f ) ;
then idseq ( n ) | Seg m = idseq ( m ) | Seg n ;
H * ( g " * a ) in the carrier of H & g * ( g " * a ) in the carrier of H ;
x in dom ( ( - 1 / 2 ) (#) ( sin * cos ) ) `| Z ) ;
cell ( G , i1 , j2 -' 1 ) misses C & cell ( G , i1 , j2 -' 1 ) misses C ;
LE q2 , p4 , P , p1 , p2 & LE q2 , p2 , P , p1 , p2 ;
attr B is BDD of A means : Def1 : B c= BDD A ;
deffunc D ( set , set ) = union rng $2 & $2 in rng $2 & $2 in rng $2 ;
n + - n < len ( ( p | ( n + 1 ) ) | ( n + 1 ) ) ;
attr a <> 0. K means : Def4 : the_rank_of M = the_rank_of a & the_rank_of M = 0. K ;
consider j such that j in dom /\ /\ /\ dom ( k | i ) and I = len } + j ;
consider x1 such that z in x1 and x1 in P8 and x1 in P8 and x = [ x1 , x1 ] ;
for n ex r being Element of REAL st X [ n , r ] & X [ n , r ]
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 / 3 * ( a - b ) / 3 ;
conv @ W c= union ( F .: ( E " ( W " ( W ) ) ) ) ;
1 in [. - 1 , 1 .] /\ dom ( ( - 1 ) (#) ( ( #Z 2 ) * ( f1 ) ) `| Z ) ;
r3 <= s0 + ( r0 - ( v2 - ( v2 - ( v2 - ( v2 - ( v2 - ( v2 - ( v2 - ( v2 - ( v2 - ( v2 - ( v2 - ( v2 - ( v2 - ( v2 - ( v2 - ( v2 - ( v2 - ( v2 - ( v2 - (
dom ( f * f4 ) = dom f /\ dom ( f4 * f4 ) .= dom f /\ dom 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 = g as Point of ( TOP-REAL n1 ) | K1 ;
( T * h . s ) . x = T . ( h . s . x ) ;
I . ( L . ( J . x ) ) = ( I * L ) . ( J . x ) ;
y in dom <* *> *> & ( Frege ( Frege ( A . o ) ) . y = ( commute ( A . o ) ) . y ;
for I being non degenerated commutative commutative commutative commutative commutative associative commutative distributive non empty doubleLoopStr holds I is commutative iff I is commutative
set s2 = s +* Initialize ( ( intloc 0 ) .--> 1 ) , P1 = P +* Initialize ( ( intloc 0 ) .--> 1 ) ;
P1 /. IC Comput ( P1 , s1 , k ) = P1 . IC Comput ( P1 , s1 , k ) .= P1 . IC Comput ( P1 , s1 , k ) ;
lim S1 in the carrier of [. a , b .] & lim S1 in the carrier of [. a , b .] ;
v . ( l-13 . i ) = ( v *' ( lpp . i ) ) . i ;
consider n being element such that n in NAT and x = ( cn " ) . n ;
consider x being Element of c such that F1 . x <> F2 ( ) and F2 . x <> 0 ;
Funcs ( X , 0 , x1 , x2 , x3 , x4 ) = { EF } & EF ( 0 ) = EF ( 0 ) ;
j + ( 2 * ( k + 1 ) ) + m1 > j + ( 2 * ( k + 1 ) ) ;
{ s , t } on A3 & { s , t } on B2 & { s , t } on B2 ;
n1 > len crossover ( p2 , p1 , n1 ) & n2 < len crossover ( p2 , p1 , n1 ) ;
mg1 . HT ( ( mg2 ) , T ) = 0. L .= ( - 1 ) * ( ( - 1 ) * ( b * ( b * ( b * ( b * a ) ) ) ) ) ;
then H1 , H2 are_or ( H , G ) / ( x , y ) , ( H , G ) / ( x , y ) " ;
( ( N-min L~ f ) .. ( f | 1 ) ) .. ( ( f | 1 ) .. ( f | 1 ) ) > 1 ;
]. s , 1 .[ = ]. s , 2 .[ /\ [. 0 , 1 .] ;
x1 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) | ( L~ g ) ) & x2 in ( ( TOP-REAL 2 ) | ( L~ g ) ) | ( L~ g ) ;
let f1 , f2 be continuous PartFunc of REAL , the carrier of S , the carrier of S ;
DigA ( tmax ( n , ( k + 2 ) ) , i ) is Element of k -tuples_on NAT ;
I gcd 223 = d2element & I gcd 223 = k2 & I <> k2 & I <> k2 ;
u9 ~ { u9 } = { [ a , u9 ] , [ a , u9 ] } & u9 in { [ a , u9 ] } ;
( w | p ) | ( p | ( w | ( w | ( w | w ) ) ) ) = p ;
consider u2 such that u2 in W2 and x = v + u2 and u1 in W2 and u2 in W1 and u2 in W2 ;
for y st y in rng F ex n st y = a |^ n & a |^ n in X ;
dom ( ( g * ( {} \dot \to C ) ) | K ) = K ;
ex x being element st x in ( ( ( ( the Sorts of U0 ) \/ A ) * ( the_arity_of o ) ) . s ) ;
ex x being element st x in ( ( ( ( O \/ A ) \/ A ) \/ B ) . s ) & x in ( ( O \/ B ) . s ) ;
f . x in the carrier of [. - r , r .] & f . x in [. - r , r .] ;
( the carrier of X1 union X2 ) /\ ( ( the carrier of X2 ) \/ ( the carrier of X3 ) ) <> {} ;
L1 /\ LSeg ( p11 , p2 ) c= { p11 } /\ LSeg ( p11 , p2 ) \/ { p11 } ;
( b + ( b-2 ) ) / 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 in B ;
( the sequence of ( being sequence of iff the carrier of X ) ) . ( id the carrier of X ) <= e ;
len ( w ^ w2 ) + 1 = len w + 2 + 1 .= len w + 2 ;
assume that q in the carrier of ( TOP-REAL 2 ) | K1 and q in the carrier of ( TOP-REAL 2 ) | K1 ;
f | E-4 ` = g | E-4 ` .= g | E-4 ` .= g | E-4 ` .= g | E-4 ` ;
reconsider i1 = x1 , i2 = x2 , j1 = y2 , j2 = z2 , j1 = y2 , j2 = z2 , j2 = z2 , j1 = y2 , j2 = z2 ;
( a * A * B ) @ = ( a * ( A * B ) ) @ ;
assume ex n0 being Element of NAT st f to_power ( n0 + 1 ) is Seg n ;
Seg len ( ( ( ( f2 ) | ( i -' 1 ) ) ^ ( ( f1 ) | ( i -' 1 ) ) ^ ( f2 | ( i -' 1 ) ) ) ) = dom ( ( ( f1 ) | ( i -' 1 ) ) ^ ( f2 | ( i -' 1 ) ) ) ;
( Complement ( A * B ) ) . m c= ( Complement ( A * B ) ) . n ;
f1 . p = p9 & g1 . ( p9 , q9 ) = d & g1 . ( p9 , q9 ) = d ;
FinS ( F , Y ) = FinS ( F , dom ( F | Y ) ) ;
( x | y ) | z = z | ( y | x ) ;
( ( |. x .| to_power n ) / ( n + 1 ) ) ^2 <= ( ( r2 ) to_power n ) / ( n + 1 ) ;
Sum ( F ) = Sum f & dom ( F | ( len F ) ) = dom g & rng ( F | ( len F ) ) = dom g ;
assume for x , y being set st x in Y & y in Y holds x /\ y in Y ;
assume that W1 is Subspace of W3 and W2 is Subspace of W3 and W1 is Subspace of W2 ;
||. ( ( t . x ) - ( t . x ) ) - ( ( t . x ) - ( t . x ) ) .|| = lim ||. ( ( t . x ) - ( t . x ) ) - ( t . x ) .|| ;
assume that i in dom D and f | A is lower and g | A is lower ;
( ( p `2 ) ^2 - ( p `2 ) ^2 ) <= ( ( - 1 ) * ( 1 + ( p `2 / p `1 ) ^2 ) ) ;
g | Sphere ( p , r ) = id ( Sphere ( p , r ) ) .= id ( Sphere ( p , r ) ) ;
set N8 = ( N-min L~ Cage ( C , n ) ) .. Cage ( C , n ) ;
for T being non empty TopSpace holds T is countable implies the TopStruct of T is countable countable
width B |-> 0. K = Line ( B , i ) .= B * ( i , i ) .= B * ( i , i ) ;
pred a <> 0 means : Def4 : ( A \ B ) Y. = ( A Y. ) \ ( B Y. ) ;
then f is_is_is_is_is_differentiable on u , 3 & pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 ;
assume that a > 0 and a <> 1 and a > 0 and b <> 1 and c > 0 ;
w1 , w2 in Lin { w1 , w2 } & w2 in Lin { w1 , w2 } ;
p2 /. IC s = p2 . IC s .= p2 . IC s .= p2 . IC s .= p2 . IC s .= p2 . IC s ;
ind ( T-10 | b ) = ind b .= ind B .= ind B .= ind b ;
[ a , A ] in the carrier of Line ( AS , b ) & [ a , A ] in the carrier of Line ( AS , b ) ;
m in ( the Arrows of C ) . ( o1 , o2 ) & ( the Arrows of C ) . ( o2 , o1 ) = ( the Arrows of C ) . ( o2 , o2 ) ;
( ( a , CompF ( PA , G ) ) . z ) . TRUE = FALSE .= FALSE ;
reconsider phi = phi , phi = phi , phi = phi , phi = phi , phi = phi , N = phi , H = S S , N = S S , N = S , F = S S , F = S S , F = S S , N = S , F = S S , F = S S , N = S , F = S
len s1 - ( len s2 - 1 ) * ( len s2 - 1 ) + 1 > 0 + 1 ;
delta ( D * ( f . ( upper_bound A ) - f . ( lower_bound A ) ) ) < r ;
[ f21 , f22 ] in the carrier' of A ~ & [ f21 , f22 ] in the carrier' of A ~ ;
the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 = K1 & ( ( TOP-REAL 2 ) | K1 ) | K1 = K1 ;
consider z being element such that z in dom g2 and p = g2 . z and x = g2 . z ;
[#] V1 = { 0. V } .= the carrier of ( W1 + W2 ) /\ the carrier of ( W1 + W2 ) .= the carrier of ( W1 + W2 ) ;
consider P2 being FinSequence such that rng P2 = M and P2 is one-to-one and P2 is one-to-one and P2 is one-to-one ;
assume that x1 in dom ( f | X ) and ||. x1 - x0 .|| < s and ||. x1 - x0 .|| < s ;
h1 = f ^ ( <* p3 *> ^ <* p *> ) .= h ^ <* p *> .= h ;
c / ( |[ b , c ]| ) = c .= c / ( |[ a , c ]| ) .= c / ( |[ a , c ]| ) .= c ;
reconsider t1 = p1 , t2 = p2 , t2 = p3 as Term of C , V ;
( 1 - 2 ) * ( 1 - 2 ) in the carrier of [. 1 , 1 .] & ( 1 - 2 ) * ( 1 - 2 ) in the carrier of I[01] ;
ex W being Subset of X st p in W & W is open & h .: W c= V ;
( h . p1 ) `2 = C * ( p1 `2 ) + D .= C * ( p1 `2 ) + D ;
R . ( b - b ) = 2 * PI * b .= 2 * PI * b .= b ;
consider 1 such that B = 1- 1 * ] + ( 1 - 1 ) * A and 0 <= 1 ;
dom g = dom ( ( ( the Sorts of A ) * ( a , I ) ) * ( ( the Sorts of A ) * ( a , I ) ) ) ;
[ P . ( l7 , P . ( l7 , m ) ) , P . ( l7 , m ) ] in => ( ( P . ( l7 , m ) ) , T ) ;
set s2 = Initialize s , P1 = P +* stop I ;
reconsider M = mid ( z , i2 , i1 ) as non empty Subset of TOP-REAL 2 ;
y in product ( ( Carrier J ) +* ( V , { 1 } ) +* ( { 1 } , { 1 } ) ) ;
1 / ( |[ 0 , 1 ]| ) = 1 & 0 / ( |[ 0 , 1 ]| ) = 0 ;
assume that x in the left of g and x in the right & y in the right of g and x in the carrier of g ;
consider M being strict Subgroup of A9 such that a = M and T is strict Subgroup of M ;
for x st x in Z holds ( ( ( ( 1 / 2 ) (#) ( f + #Z 2 ) ) * ( f + #Z 2 ) ) `| Z ) . x <> 0
len W1 + len W2 + m = 1 + len W3 + m .= len W3 + len W2 + m .= len W3 + m + 1 ;
reconsider h1 = ( vseq . n - t-16 ) as Lipschitzian LinearOperator of X , Y ;
( - ( len p + q ) ) + 1 in dom ( p + q ) ;
assume that s2 is \langle s1 *> and F in the { of s2 : s1 in the { of s2 : s2 in the carrier of s2 } ;
( ( for x , y being Element of NAT holds x , y ) * ( x , z ) = gcd ( x , y , z )
for u being element st u in Bags n holds ( p *' ) . u = p . u & ( p *' ) . u = p . u
for B be Subset of u-5 st B in E holds A = B or A = B or B misses B ;
ex a being Point of X st a in A & A /\ Cl { y } = { a } ;
set W2 = tree ( p ) \/ W1 and W1 = p \/ W2 ;
x in { X where X is Ideal of L : X is Ideal of L } ;
the carrier of W1 /\ W2 c= the carrier of W1 & the carrier of W1 /\ W2 c= the carrier of W1 & W1 /\ W2 c= the carrier of W2 ;
( for a , b being Element of L holds ( a + b ) * id a = ( a + b ) * id a
( ( dom X --> f ) | X ) . x = ( X --> dom f ) . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) ;
p => ( q => r ) => ( p => q ) in TAUT ( A ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
- 1 + 1 <= ( - 2 |^ ( n -' m ) ) + 1 ;
( reproj ( 1 , z0 ) ) . x in dom ( f1 (#) f2 ) /\ dom ( f2 (#) f3 ) ;
assume that b1 . r = { c1 } and b2 . r = { c2 } and b2 . r = c1 . c2 ;
ex P st a1 on P & a2 on P & b on P & c on P & d on P & a on P & b on P ;
reconsider gf = g opp * f opp , hf = h opp * g opp as strict non empty Element of X ;
consider v1 being Element of T such that Q = ( downarrow v1 ) ` and v1 in ( downarrow v2 ) ` and v1 in ( downarrow v1 ) ` ;
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 `2 >= 0 & p <> 0. TOP-REAL 2 } ;
ConsecutiveSet ( A , succ O1 ) = ( ConsecutiveSet ( A , O1 ) ) . ( ( succ O1 ) . ( ( succ O1 ) . ( ( succ O1 ) . ( A , O1 ) ) . ( ( succ O1 ) . ( A , O1 ) ) ) ;
set I1 = Macro ( a , intloc 0 ) , for s being State of SCM+FSA , a , intloc 0 st s = [ a , intloc 0 , intloc 0 ] holds I = P ;
for i be Nat st 1 < i & i < len z holds z /. i <> z /. 1 & z /. i <> z /. 1
X c= ( the carrier of L1 ) /\ ( the carrier of L2 ) & ( the carrier of L1 ) /\ ( the carrier of L2 ) c= the carrier of L2 ;
consider x9 being Element of GF ( p ) such that x9 |^ 2 = a and x9 |^ 2 = b ;
reconsider ee = ee , fe = f , fe = g , fe = h as Element of D ;
ex O being set st O in S & C1 c= O & M . O = 0. ( X , Y ) ;
consider n being Nat such that for m being Nat st n <= m holds S . m in U1 & S . m in U2 ;
f * g * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) * g ) . x ;
defpred P [ Nat ] means A + succ $1 = succ A & A + succ $1 = succ $1 & A + succ $1 = succ $1 ;
the left of - g = the left of g & the carrier of - g = the carrier of g ;
reconsider pp = x , pp = y , pp = z , pp = w , pp = z , pp = w , pp = y , pp = z , pp = w , pp = z ;
consider g3 such that g3 = y and x <= y 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 ^ y2 ) = len x2 + len y2 ;
for x being element st x in X holds x in the set of the set of positive iff x is set & f . x = ( n + 1 ) / ( n + 1 )
LSeg ( p11 , p2 ) /\ LSeg ( p1 , p11 ) = {} & LSeg ( p1 , p11 ) /\ LSeg ( p11 , p2 ) = {} ;
func where X is set means : -> set means : -> Element of it = for f being Function of X , X holds it = ( id X ) " ( id X ) ;
len | ( ( CR ( C , n ) ) /. 1 ) <= len ( C | ( len C -' 1 ) ) ;
attr K is a1 means : Assume a <> 0. K & a * ( a |^ i ) = i * v . a ;
consider o being OperSymbol of S such that t . {} = [ o , the carrier of S ] and t . {} = [ o , the carrier of S ] ;
for x st x in X ex y st x c= y & y in X & y in X & y in Y ) holds ex f st f is x & f . x = f . y
IC Comput ( P-6 , sd , k ) in dom ( ( k + 1 ) .--> ( k + 1 ) ) ;
pred q < s & r < s implies ]. r , s .[ \not c= ]. p , q .[ & s < q ;
consider c being Element of Class ( f , c ) such that Y = ( F . c ) `1 and c in X ;
func the ResultSort of S2 = id the carrier' of S2 , the carrier' of S2 = id the carrier' of S2 , the carrier' of S2 ;
set y-13 = [ <* y , z *> , f2 ] , y] ;
assume that x in dom ( ( ( 1 / 2 ) (#) ( ( #Z 2 ) * ( f1 + #Z 2 ) ) ) `| Z ) ;
r-7 in Int cell ( GoB f , i , GoB f ) \ L~ f & r-7 in L~ f \ L~ f ;
( q `2 ) ^2 >= ( ( Cage ( C , n ) ) * ( i + 1 ) ) `1 ;
set Y = { a "/\" a ` : a ` in X } ;
i -' len f <= len f + len ( f | ( len f -' 1 ) ) - len ( f | ( len f -' 1 ) ) ;
for n ex x st x in N & x in N1 & h . n = x- ( x0 , x0 )
set s0 = ( ( a , I , p , s ) +* ( a , I , p ) ) . i , s0 = s . i , s0 = s . i ;
p . ( k + 0 ) = 1 or p . ( k + 0 ) = - 1 or p . ( k + 0 ) = 1 ;
u + Sum L-18 in ( U \ { u } ) \/ { u + Sum L-18 } ;
consider x9 be set such that x in x9 and x9 in V1 and x9 in V1 and f . x9 = f . x9 ;
( p ^ ( q | k ) ) . m = ( q | k ) . ( len p + m ) .= ( len p ) + m ;
g + h = gg + hh & A1 + h = g + h ;
L1 is distributive & L2 is distributive implies L1 ~ is distributive & L2 ~ is distributive & for x being Element of L1 holds L1 ~ x is distributive iff L2 ~ x is distributive
pred x in rng f & y in rng ( f | x ) implies f | x = f | y ;
assume that 1 < p and ( 1 - p ) * q + ( 1 - p ) * q = 1 and 0 <= a and a <= b ;
F* ( f , <* <* *> *> ) = rpoly ( 1 , the carrier of F_Complex ) *' .= <* 0. F_Complex *> ;
for X being set , A being Subset of X , B being Subset of X holds A ` = {} implies A = {} & B = {}
( ( N-min X ) `1 <= ( ( E-max X ) `1 ) / 2 & ( ( E-max X ) `1 ) / 2 <= ( ( E-max X ) `1 ) / 2 ;
for c being Element of the bound is Element of the bound is Element of the bound is Element of the bound is Element of A holds c <> a
s1 . GBP = ( Exec ( i2 , s2 ) ) . GBP .= Exec ( i2 , s2 ) . GBP .= s2 . GBP .= s . GBP .= s . GBP ;
for a , b being Real holds |[ a , b ]| in ( y >= 0 ) implies b >= 0
for x , y being Element of X holds x ` \ y = ( x \ y ) ` & y \ x = ( x \ y ) `
mode BCK-algebra of i , j , m , n , m , n , m , m , n , m , n , m , m , n , m , n , m , m , n ;
set x2 = |( ( Re y ) , ( Im y ) )| ;
[ y , x ] in dom ( ( u | y ) | x ) & ( u | y ) . ( y , x ) = g . y ;
]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .[ c= A & A c= divset ( D , k ) ;
0 <= delta ( S2 . n ) & |. delta ( S2 . n ) .| < ( e / 2 ) ;
( - ( q `1 / |. q .| - cn ) ) ^2 <= ( - ( q `1 / |. q .| - cn ) ) ^2 ;
set A = ( 2 / 2 ) * b-a ;
for x , y being set st x in R" holds x , y are_let x , y ;
deffunc FF2 ( Nat ) = b . $1 * ( M * G . $1 ) * ( M * G . $1 ) ;
for s being element holds s in -> Element of \mathclose ( { 0 } \/ R ) iff s in _ ( dom f \/ R )
for S being non empty non void non void non empty void void void thesis st S is connected holds S is connected
max ( degree ( ( z `1 ) / |. z .| - cn ) / ( 1 + cn ) ) >= 0 ;
consider n1 being Nat such that for k holds seq . ( n1 + k ) < r + s . ( n1 + k ) ;
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 ) ) . ( ( M . x qua Element of BOOLEAN ) ) ;
f " V in ' ( X ) & f " V in D & f " V in D & f " V in D & f " V in D & f .: V c= D ;
rng ( ( a ^\ c ) +* ( 1 , b ) +* ( 1 , c ) ) c= { a , c , b } ;
consider y being as as as Point of G1 such that y ` = y and dom y ` = WWX and y is WWX ;
dom ( ( 1 / ( f . x ) ) (#) ( f | ]. x0 , x0 + r .[ ) ) c= ]. x0 , x0 + r .[ ;
as Element of v1 , j , r , n , - r , - r , n , - r ;
v ^ ( ( n |-> 0 ) ^ ( ( n |-> 0 ) ^ ( ( n |-> 0 ) ^ ( ( n |-> 0 ) ^ ( ( n |-> 0 ) ^ ( ( n --> 0 ) ^ ( ( n --> 0 ) ^ ( ( n --> 0 ) ^ ( ( n --> 0 ) ^ ( ( n --> 0 ) ^ ( ( n --> 0 ) ^ ( ( n --> 0
ex a , k1 , k2 st i = a := k1 & i = a := k2 & i = a := k2 & i <> k2
t . NAT = ( NAT .--> succ i1 ) . NAT .= ( NAT .--> succ i1 ) . NAT .= ( NAT .--> succ i1 ) . NAT .= ( NAT .--> NAT ) . NAT .= NAT ;
assume that F is bbfamily of X and rng p = F and dom p = Seg ( n + 1 ) and for k be Nat st k in Seg ( n + 1 ) holds p . k in Seg ( n + 1 ) ;
not LIN b , b9 , a & LIN a , a9 , c & LIN a , a9 , c & LIN a , a9 , c ;
( L1 gcd L2 ) \& O c= ( L1 => L2 ) \& O & ( L2 => L2 ) \& O c= ( L1 \& L2 ) \& O ;
consider F being ManySortedSet of E such that for d being Element of E holds F . d = F ( d ) and for d being Element of E holds F . d = G ( d ) ;
consider a , b such that a * ( x - v ) = b * ( -w ) and 0 < a and a < b ;
defpred P [ FinSequence of D ] means |. Sum ( $1 ) .| <= Sum ( |. $1 .| ) ;
u = cos . ( x , y ) * x + ( cos . ( x , y ) * y ) .= v ;
dist ( seq . n , x + g , x + g ) <= dist ( seq . n , g ) + 0 ;
P [ p , |. p .| ^ |. p .| , {} ] implies P [ p , {} ]
consider X being Subset of Al ( ) such that X c= Y and X is finite and X is finite and P [ X ] ;
|. 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 & g <= l1 & l1 <= h } ;
vol ( ( G . n ) vol ) <= card ( ( ( G . n ) vol ) | ( G . n ) ) ;
f . y = x .= x * 1_ L .= x * ( power L ) . ( y , 0 ) .= x * ( power L ) . ( y , 0 ) ;
NIC ( ( a , k1 ) --> ( k2 , k2 ) ) = { i1 , succ ( ( a , k1 ) --> ( k2 , k2 ) ) }
LSeg ( p11 , p2 ) /\ LSeg ( p1 , p2 ) = { p1 } & LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = { p1 } ;
Product ( ( ( Carrier ( I ) ) +* ( i , { 1 } ) ) +* ( i , { 1 } ) ) in Z ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) .= Following ( s1 , n ) ;
W-bound ( Qb ) <= ( q1 `1 ) / 2 & ( q1 `1 ) / 2 <= ( q1 `1 ) / 2 ;
f /. i2 <> f /. ( len f -' 1 + len g -' 1 , f /. ( len f -' 1 ) ) ;
M , v |= ( f / ( x. 3 , x. 4 ) ) / ( x. 4 , x. 0 ) ;
len ( ( P ^ Q ) ^ ) in dom ( ( P ^ ) ^ ) & len ( ( P ^ Q ) ^ ) in dom ( ( P ^ Q ) ^ ) ;
A |^ ( m , n ) c= A |^ ( m , n ) & A |^ ( k , m ) c= A |^ ( k , n ) ;
R |^ n \ { q : |. q .| < a } c= { q1 : |. q1 .| >= a }
consider n1 being element such that n1 in dom p1 and y1 = p1 . n1 and p1 . n1 = p2 . n1 and p2 . n1 = p2 . n1 ;
consider X being set such that X in Q and for Z being set st Z in Q & Z <> X holds not X in Z ;
CurInstr ( P3 , Comput ( P3 , s2 , l ) ) <> halt SCM+FSA & CurInstr ( P3 , Comput ( P3 , s2 , l ) ) <> halt SCM+FSA ;
for v be VECTOR of l1 holds ||. v - v .|| = upper_bound rng |. ( id the carrier of l1 ) . v .|
for phi st phi in X holds phi in X & not phi in X & not phi in X & not phi in X & not phi in X ;
rng ( ( Sgm dom ( ( f | ( dom ( f | ( dom ( f | ( dom ( f | ( dom ( f | ( dom ( f | ( dom ( f | ( dom ( f | | | ( dom ( f | | dom ( | | | dom ( ( | | dom ( | | dom ( | | | dom ( | | | dom ( | | | dom ( | | dom ( | | | dom ( | | |
ex c being FinSequence of D ( ) st len c = k & a = c & a = c & b = c ;
the_arity_of ( ( a , b , c ) , g ) = <* \in Hom ( b , c ) , Hom ( c , d ) *> ;
consider f1 be Function of the carrier of X , REAL such that f1 = |. f .| and f1 is continuous ;
a1 = b1 & a2 = b2 or a1 = b1 & a2 = b2 & a3 = b2 & a4 = b1 & a5 = b2 & a5 = b3 ;
D2 . ( indx ( D2 , D1 , n1 ) + 1 ) = D1 . ( n1 + 1 ) & D2 . ( n1 + 1 ) = D2 . ( n1 + 1 ) ;
f . ( ( ||. r .|| ) * ( ||. r .|| ) ) = ||. r .|| . 1 .= ||. r .|| . 1 .= ||. r .|| . 1 .= ||. r .|| . 1 ;
consider n being Nat such that for m being Nat st n <= m holds C-25 . m = C-25 . m and C-25 . n = 0 ;
consider d being Real such that for a , b being Real st a in X & b in Y holds a <= b & b <= d ;
||. L /. h - K * ( K * |. h .| ) + K * ( K * |. h .| ) - K * ( K * |. h .| ) ) <= p0 + K ;
attr F is commutative associative means : Def4 : for b being Element of X holds F \hbox { b } = f . b ;
p = - 1 * p0 + 0. TOP-REAL 2 .= 1 * p0 + 0. TOP-REAL 2 .= 1 * p0 + 0. TOP-REAL 2 .= ( 1 - ( 1 - ( ( 1 - ( a `2 / b ) ) / ( 1 - sn ) ) ) * ( 1 - sn ) ) .= ( 1 - sn ) * ( 1 - sn ) ;
consider z1 such that b , x3 , x1 , x1 , x2 , x1 , x1 , x2 , x1 , x1 , x1 , x2 , x1 , x1 , x1 , x2 , x1 , x1 , x2 , x1 , x1 , x2 , x1 , x2 , x1 , x1 , x2 , x1 , x2 , x1 , x1 , x2 , x1 , x2 , x1 , x2 , x1 , x2 , x1 , x2 , x1 , x2 , x1 , x2 , x1
consider i such that Arg ( ( Rotate ( s ) ) . q ) = s + Arg ( q ) + ( 2 * PI * PI * i ) ;
consider g such that g is one-to-one and dom g = card f and rng g = x and rng g = { x } and g is one-to-one ;
assume that A = P2 \/ Q2 and P2 <> {} and ( for x st x in P2 holds x <> Q & x <> Q & Q [ x ] ) and Q [ Q ] ;
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 & x < i or m < i & i < z `1 & i < m ;
consider k2 being Nat such that k2 in dom ( ( P . k2 ) `1 ) and l in dom ( ( P . k2 ) `1 ) and ( P . k2 = ( P . k2 ) `2 ;
seq = r * seq implies for n holds seq . n = r * seq . n & seq is convergent & lim seq = r * seq . n
F1 . [ [ id a , a ] , F2 . [ a , a ] ] = [ f * ( id a ) , F2 . [ a , b ] ] ;
{ p } "\/" D2 = { p "\/" y where y is Element of L : y in D2 & p in D2 } ;
consider z being element such that z in dom ( ( dom F ) | ( dom F ) ) and ( ( dom F ) | ( dom F ) ) = 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 , j ) = { |[ 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 @ b1 ) . x = ( Mx2Tran ( J , T , ( BT , ( BT , ( T , m ) ) ) ) ) . ( Y. /.
- 1 = mm1 (#) D .= mm1 (#) D .= mm1 (#) ( - 1 ) .= ( - 1 ) (#) ( - 1 ) .= Det M ;
attr x be set means : Def4 : x in dom f /\ dom g holds g . x <= f . x ;
len ( f1 . j ) = len ( f2 /. j ) .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) ;
All ( 'not' All ( a , A , G ) , B , G ) '<' Ex ( 'not' Ex ( a , B , G ) , A , G )
LSeg ( E . i0 , F . i0 ) c= Cl RightComp Cage ( C , ( i0 + 1 ) ) ` ) & LSeg ( E , k ) c= RightComp Cage ( C , ( i0 + 1 ) ) ` ;
x \ a |^ m = x \ ( a |^ k * a ) .= ( x \ a ) |^ k .= ( x \ a ) |^ k ;
k = ( ( ( I -in1 ) -inin1 ) | ( I * ( I * ( I * ( I * ( I * ( J * J ) ) ) ) ) ) ) ) . k .= ( ( commute ( I * ( J * J ) ) ) ) . ( ( I * ( J * J ) ) . k ) ;
for s being State of Aj holds Following ( s , n . 0 + ( n + 2 ) * n + 1 ) is stable ;
for x st x in Z holds f1 . x = a ^2 & ( f1 - f2 ) . x <> 0 & ( f1 - f2 ) . x <> 0 ) implies f1 - f2 is continuous
support ( ( support ( m ) ) \/ 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 carrier of F , ( the carrier' of C ) * the carrier of F ;
- ( a * sqrt ( 1 + b ^2 ) ) <= - ( b * sqrt ( 1 + a ^2 ) ) ;
phi /. ( succ b1 ) = g . a & phi /. ( a . a ) = f . ( g . a ) & phi /. ( a . a ) = f . ( g . a ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( ( F ^ <* p *> ) ^ ( F ^ <* p *> ) ) ;
{ x1 , x2 , x3 , x4 } = { x1 } \/ { x2 , x3 , x4 } .= { x1 } \/ { x2 , x3 , x4 } ;
the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U1 & the Sorts of U2 c= the Sorts of U2 implies ( U1 "\/" U2 ) /\ ( U1 "\/" U2 ) c= the Sorts of U1
( - ( 2 * a * ( b - a ) ) ^2 + b ^2 ) - ( - 2 * a * b ) ^2 > 0 ;
consider W00 such that for z being element holds z in W00 iff z in N ~ & P [ z ] ;
assume that ( the ResultSort of S ) . o = <* a *> and ( the ResultSort of S ) . o = r and ( the ResultSort of S ) . o = a ;
Z = dom ( ( ( ( ( - 1 / ( n + 1 ) ) (#) ( f1 + #Z 2 ) ) * ( f1 + #Z 2 ) ) ) `| Z ) ;
sum ( f , SS1 ) is convergent & lim ( upper_volume ( f , SS1 ) ) = integral ( f , SS1 ) ;
( X [ a9 , f . ( f . ( x , y ) ) => ( g . ( x , y ) ) ] ) in cluster ( as sequence of l ) ;
len ( M2 * ( M3 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2 * ( M2
attr X1 union X2 is open SubSpace of X means : Def1 : X1 , X2 are_separated & X2 , X1 union X2 are_separated & X1 , X2 are_separated ;
for L being lower-bounded antisymmetric antisymmetric RelStr , X being non empty Subset of L holds X "\/" { Top L } = { Top L }
reconsider f-1-1S = ( F . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . b ) ) ) ) )
consider w being FinSequence of I such that the InitS of M , the InitS \HM { s } ^ w ^ w ^ q ^ q ^ w ^ q ^ w ^ q ^ w ^ q = q ^ w ^ w ^ q ;
g . ( a |^ 0 ) = g . ( 1_ G ) .= 1_ H .= ( g . ( a |^ 0 ) ) * ( g . ( a |^ 0 ) ) .= ( g . ( a |^ 0 ) ) * ( g . ( a |^ 0 ) ) .= ( g . ( a |^ 0 ) ) * ( g . ( a |^ 0 ) ) ;
assume for i be Nat st i in dom f ex z being Element of L st f . i = rpoly ( 1 , z ) . i & z in D ;
ex L being Subset of X st Carrier ( L ) = L & for K being Subset of X st K in C holds K /\ K <> {} ;
( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C1 & ( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C2 ;
reconsider oY = o `2 , op = p `2 , oq = ( the Sorts of A ) * ( the_arity_of o ) as Element of TS ( ( the Sorts of A ) * ( the_arity_of o ) ) ;
1 * x1 + ( 0 * x2 + x3 * x3 ) + ( 0 * x2 + x3 * x4 ) = x1 + ( 0 * x2 + x3 * x3 ) .= x1 + ( 0 * x3 ) .= x1 ;
Ea " . 1 = ( Ea qua Function ) " . 1 .= ( ( ( 1 - a ) " ) * ( ( 1 - a ) " ) ) . 1 .= ( ( 1 - a ) " ) * ( ( 1 - a ) " ) .= ( ( 1 - a ) " ) * ( ( 1 - a ) " ) .= ( 1 - a ) " ;
reconsider u1 = the carrier of ( U1 /\ ( U1 "\/" U2 ) ) , ( U1 "\/" U2 ) = the carrier of ( U0 /\ ( U1 "\/" U2 ) ) ;
( ( x "/\" z ) "\/" ( x "/\" y ) ) "\/" ( z "/\" y ) <= ( x "/\" ( z "\/" y ) ) "\/" ( z "/\" y ) ;
|. f . ( s1 . ( l1 + 1 ) ) - f . ( s1 . l1 ) .| < ( 1 / M ) * ( M . l1 ) + 1 / M . l1 ;
LSeg ( ( Lower_Seq ( C , n ) ) * ( i , j ) , ( Lower_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 ) ;
( ( ColVec2Mx f ) in the carrier of A & ColVec2Mx b in the carrier of A & ColVec2Mx b = ( ColVec2Mx b ) * ( ColVec2Mx b ) ) & ( len f = len b implies f . b = 0. K ) ;
len ( - ( M1 * M2 ) ) = len M1 & width ( - ( M1 * M2 ) ) = width M1 & width ( - ( M1 * M2 ) ) = width M1 ;
for n , i being Nat st i + 1 < n & i in the InternalRel of TOP-REAL n holds [ i , i + 1 ] in the InternalRel of TOP-REAL n
pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 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 b = Arg a & Arg a = Arg b ;
for c being set st not c in [. a , b .] holds not c in Intersection ( the open of a , b ) & c in Intersection ( the open set of a , b )
assume that V1 is linearly-independent and V2 is closed and V = { v + u : v in V1 & u in V1 & v in V1 } ;
z * x1 + ( 1 - ( 1 - ( a * x2 ) ) ) in M & z * y1 + ( 1 - ( a * x2 ) ) ) in N ;
rng ( ( ( ( P qua Function ) " ) * ( S * ( S * ( S * T ) ) ) ) * ( S * T ) ) ) = Seg card ( ( ( P * T ) " ) * ( S * T ) ) ;
consider s2 being integer number such that s2 is convergent and b = lim s2 and for n holds s2 . n <= b . n and s2 . n <= x0 ;
h2 " . n = h2 . n " & 0 < - ( ( 1 / 2 ) |^ n ) & 0 < ( ( 1 / 2 ) |^ n ) / ( n + 1 ) ;
Partial_Sums ( ||. ( seq ^\ k ) . m - ( seq ^\ k ) . m .|| ) = ||. ( ( seq ^\ k ) . m - ( seq ^\ k ) . m ) - ( seq ^\ k ) . 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 = - v & - w = - v & - v = - w ;
sup ( ( k .: D ) .: D ) = sup ( ( k .: D ) .: D ) .= sup ( ( k .: D ) .: D ) .= k . ( sup D ) .= ( sup D ) . ( sup D ) ;
A |^ ( k , l ) ^^ ( A |^ ( n , .. A ) ) = ( A |^ ( n , .. A ) ) ^^ ( A |^ ( k , .. A ) ) ;
for R being add-associative right_zeroed right_complementable non empty addLoopStr , I , J , K being Subset of R , I , J being Subset of R holds I + J = ( I + J ) + K
( f . p ) `1 = ( p `1 ) ^2 + ( p `2 ) ^2 .= ( p `1 ) ^2 + ( p `2 ) ^2 ;
for a , b being non zero Nat st a , b are_relative_prime holds ( a * b ) gcd ( a * b ) = [ [ a , b ] , [ b , a ] ] & ( a * b ) gcd ( a * b ) = [ a , b ]
consider A9 being countable Nat such that r is countable and ( for A being Subset of Al holds A is closed & A is closed ) & ( A is Al implies A is Al ) ;
for X being non empty addLoopStr , M being Subset of X , x , y being Point of X st y in M & x + y in M holds x + y in M + M
{ [ x1 , x2 ] , [ y1 , y2 ] , [ y1 , y2 ] } c= { x1 , y1 } \/ { x2 , y2 } ;
h . ( f . O ) = |[ A * ( f . O ) `1 + B , C * ( f . O ) `2 + D ]| ;
( Gauge ( C , n ) * ( k , i ) ) * ( k , i ) in L~ Upper_Seq ( C , n ) /\ L~ Upper_Seq ( C , n ) ;
cluster m , n are_relative_prime for Nat , p be prime Nat , n be Nat , m be Nat , p be prime Nat , m be Nat , n be Nat , p be non zero Nat ;
( 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 & a \ c <= c holds a \ b <= c
consider b being element such that b in dom ( H / ( x. 0 , y ) ) and z = ( H / ( x. 0 , 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 , G or e Joins W . 3 , W . 5 , G ;
( \cal x0 ) . ( 2 * n ) . x = ( L (#) ( h . n ) ) . ( 2 * n + 1 ) .= ( L (#) ( h . n ) ) . x ;
j + 1 = ( len h11 + 2 ) - 1 .= i + 1 - 2 .= i + 1 - 2 - 1 .= 2 - 2 - 1 ;
*' ( S *' ) . f = S *' *' ( S *' ) . f .= S *' ( S *' ) . f .= S *' ( S *' ) . f .= S *' ( S *' ) .= S *' ( S *' ) ;
consider H such that H is one-to-one and rng H = the carrier of L2 and Sum ( L2 (#) H ) = Sum ( L2 (#) H ) and Sum ( L2 (#) H ) = Sum ( L2 (#) H ) ;
attr R is b2 means : Def1 : for p , q st p in R & q <> q holds ex P st P is special arc p , q & p in P & q in R ;
dom product ( X --> f ) = meet ( dom ( X --> f ) ) .= meet ( ( X --> dom f ) . 0 ) .= meet ( X --> f ) .= meet ( X --> f ) .= meet ( X --> f ) .= dom f ;
upper_bound ( proj2 .: ( Lower_Arc ( C ) /\ Lower_Arc ( C ) /\ Lower_Arc ( C ) ) ) <= upper_bound ( proj2 .: ( Lower_Arc ( C ) /\ Lower_Arc ( C ) /\ VerticalLine ( C ) ) ) ;
for r be Real st 0 < r ex n be Nat st for m be Nat st n <= m holds |. S . m - p .| < r
i * ( f - f ) = i * ( f - f ) .= i * ( f - f ) .= i * ( f - f ) .= i * ( f - f ) ;
consider f being Function such that dom f = 2 -tuples_on X and for Y be set st Y in 2 -tuples_on X holds f . Y = F ( Y ) ;
consider g1 , g2 being element such that g1 in [#] Y and g2 in union C and g = [ g1 , g2 ] and g1 in union C and g2 in A and g2 in B ;
func d gcd n -> Nat means : Def1 : d divides n & ( d |^ ( n + 1 ) ) divides ( d |^ n ) & ( d |^ ( n + 1 ) ) divides ( d |^ n ) ;
f\in f . [ 0 , t ] = f . [ 0 , t ] .= ( - P ) . ( 2 * x ) .= a ;
t = h . D or t = h . B or t = h . C or t = h . E or t = h . F ;
consider m1 be Nat such that for n st n >= m1 holds dist ( ( seq . n ) , ( seq . n ) ) < 1 / ( n + 1 ) ;
( q `1 / |. q .| - cn ) ^2 <= ( q `1 / |. q .| - cn ) / ( 1 + cn ) ;
h0 . ( i + 1 + 1 ) = h21 . ( i + 1 + 1 -' len h11 + 2 -' 1 ) .= h21 . ( i + 1 + 1 -' len h11 + 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 in { [ o , x2 ] } ;
for L being RelStr , a , b being Element of L holds a <= { b } iff a <= b & b <= a & a <= b & b <= a
||. h1 . n - h1 . n .|| = ||. h1 . n - h1 . n .|| .= ||. h . n - h1 . n .|| .= ||. h . n - h . n .|| .= ||. h . n - h . n .|| ;
( ( - ( exp_R * f ) ) `| Z ) . x = f . x - ( exp_R * f ) . x .= ( - ( exp_R * f ) ) . x .= ( - ( exp_R * f ) ) . x ;
pred r = F .: ( p , q ) means : Def4 : len r = min ( len p , len q ) ;
( rmin ( 2 * r ) ^2 + ( rbeing / 2 ) ^2 ) <= ( r / 2 ) ^2 + ( r / 2 ) ^2 ;
for i being Nat , M being Matrix of n , K st i in Seg n holds Det ( M @ ) = Sum ( ( Det M ) @ )
then a <> 0. R & a " * ( a * v ) = 1 / a * v & a " * ( a * v ) = 1 / a * v & a * v = 0. R ;
p . ( j -' 1 ) * ( q *' r ) . ( i + 1 -' j ) = Sum ( p . ( j -' 1 ) * ( q . ( j -' 1 ) ) ) ;
deffunc F ( Nat ) = L . 1 + ( ( R /* h ) ^\ n ) * ( h ^\ n ) " .= ( R /* h ) . ( n + 1 ) * ( h ^\ n ) " ;
assume that the carrier of H2 = f .: ( the carrier of H1 ) and the carrier of H2 = f .: ( the carrier of H2 ) and the carrier of H2 = the carrier of H2 and the carrier of H2 = the carrier of H2 ;
Args ( o , Free ( S , X ) ) = ( ( the Sorts of Free ( S , X ) ) * ( the Arity of S ) ) * ( the Arity of S ) ;
H1 = n + 1 - ( |. 2 to_power ( n + 1 + h ) .| ) .= n + 1 - ( 2 to_power ( n + 1 ) ) .= n + 1 - ( 2 to_power ( n + 1 ) ) ;
( O = 0 & 3 ) = 0 & 3 = 1 & ( O = 0 or O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 ;
F1 .: ( dom F1 /\ dom F2 ) = F1 .: ( dom F1 /\ dom F2 ) .= F1 .: ( dom F1 /\ dom F2 ) .= { f /. ( n + 2 ) } .= f /. ( n + 2 ) .= f /. ( n + 2 ) ;
pred b <> 0 & d <> 0 & b <> d & b <> d & ( a - b ) / ( d - c ) = ( - ( e - d ) ) / ( d - c ) ;
dom ( ( f +* g ) | D ) = dom ( ( f +* g ) | D ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) ;
for i be set st i in dom g ex u , v be Element of L st g /. i = u * a & v in B & u in C & v in C & v in D & u in D ;
g `2 * P `2 * g " = g `2 * ( g * P " ) * g " .= g `2 * ( g * P " ) .= g " * ( g * P " ) .= g " * ( g * P " ) ;
consider i , s1 such that f . i = s1 and not ( ex s st s in dom s1 & not ( ex s st s in dom s1 & s in dom s2 ) & not ( ex i st i < 1 & s in dom s2 ) ;
h5 | ]. a , b .[ = ( g | ]. a , b .[ ) | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= h5 | ]. a , b .[ .= h5 | ]. a , b .[ ;
[ s1 , t1 ] , [ s2 , t2 ] , [ t2 , t2 ] connected & [ s2 , t2 ] , [ t2 , t2 ] <= [ s2 , t2 ] ;
then H is negative & H is non negative & H is non negative & H is non negative implies H is not negative -g\mathopen H ;
attr f1 is total means : Def1 : for c be Element of C holds ( f1 (#) f2 ) . c is total & ( f1 (#) f2 ) . c = f1 . c * f2 . c " ;
z1 in W2 ` & z1 = z2 ` & ( z1 = z2 implies ( z1 = z2 & z2 = z2 ) & ( z1 = z2 ) & ( z1 = z2 implies z1 = z2 ) ;
p = 1 * p .= a " * a * p .= a " * a * p .= a " * ( b * q ) .= a " * ( b * q ) .= a " * ( b * q ) .= a " * ( b * q ) ;
for r9 be Real_Sequence for K be Real st for n be Nat holds ||. ( seq . n ) - ( seq . n ) .|| <= K holds upper_bound rng ( ( seq ^\ n ) - ( seq . n ) ) <= K
( for x being Element of TOP-REAL 2 st x in L~ go holds x in L~ go \/ L~ pion1 or x in L~ pion1 ) or ( x in L~ go \/ L~ pion1 ) & ( x in L~ pion1 or x in L~ pion1 or x in L~ pion1 ) ;
||. f . ( g . ( k + 1 ) ) - g . ( g . ( k + 1 ) ) .|| <= ||. g . 1 - g . 0 .|| * ( K * ( K * ( K * ( k + 1 ) ) ) / ( 1 + K * ( K * ( k + 1 ) ) ) ) ;
assume h = ( ( B .--> B ' ) +* ( D .--> E ' ( F .--> J ) ) +* ( E .--> F ' ( J .--> M ' ) ) +* ( F .--> J ' ( M .--> A ' ) ) +* h +* ( N .--> A ' ) ) ;
|. ( ( upper_volume ( H . n , T ) || A ) . k - ( upper_volume ( H . n , T ) | A ) . k ) .| <= e * ( b-a * ( b-a . n ) ) ;
( (
{ x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 } } = { x1 , x1 } \/ { x1 , x1 , x1 } .= { x1 , x1 } ;
consider A such that A = [. 0 , 2 * PI .] and integral ( ( exp_R (#) sin ) , A ) = 0 and ( ( exp_R (#) cos ) | A ) . x = 0 ;
p `2 is Permutation of dom f1 & p `2 " = ( Sgm Y ) " * p " .= ( Sgm Y ) " * p .= ( Sgm Y ) " * p ;
for x , y st x in A & y in A holds |. ( 1 / ( x - y ) ) - ( 1 / ( x - y ) ) .| <= 1 * |. ( f . x - f . y ) .|
( ( p2 `2 ) ^2 ) = |. q2 .| * ( ( ( q2 `2 ) ^2 - ( q2 `2 ) ) / ( 1 + ( q2 `2 / |. q2 .| - cn ) ) ^2 ) ;
for f be PartFunc of the carrier of CNS , REAL , g be PartFunc of the carrier of CNS , REAL , x be Element of REAL st dom f = the carrier of x & f | X is continuous holds f | X is continuous
assume for x being Element of Y st x in EqClass ( z , CompF ( B , G ) ) holds ( Ex ( a , CompF ( B , G ) ) ) . 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 // v , v1 & u , u1 // v , v1 & u1 , v1 // v , v2 & u2 , u1 // v , v2 & u2 , v2 // v , v1 & u2 , v2 // v , v2 ;
for G being Group , A , B being non empty Subgroup of G , N being normal Subgroup of G holds ( N : N is normal & N is normal Subgroup of A ) implies N is normal Subgroup of A
for s be Real st s in dom F holds F . s = integral ( R , ( f + g ) (#) ( e - e ) ) ;
width AutMt ( f1 , b1 , b2 ) = len ( ( b * b1 ) * ( b * b2 ) ) .= len b1 .= len b1 .= len b1 .= len b1 .= len b2 .= len b2 ;
f | ]. - PI / 2 , PI / 2 .[ = f & ( for x st x in ]. - PI / 2 , PI / 2 .[ holds f . x = - 1 ) implies f | ]. - PI / 2 , PI / 2 .[ = ( - 1 ) (#) ( ( id ]. - PI / 2 , PI / 2 .[ ) )
assume that X is closed and a in X and a in X and a in X and y in a and x in a and y in a and x in X ;
Z = dom ( ( ( 1 / 2 ) (#) ( ( arctan ) * ( arctan ) ) ) `| Z ) /\ dom ( ( ( 1 / 2 ) (#) ( ( arctan ) * ( arctan ) ) `| Z ) ) ;
func [: l , l :] -> Subset of V means : Def4 : 1 <= k & k <= len l & l . k in V & l . k in V ;
for L being non empty TopSpace , N being net of L , M being net of L st M is net of N for c being Element of N st c is inf of M holds M . c is net of N
for s being Element of NAT holds ( ( ex v being Element of Csequence ( X ) ) . s + ( id ( the carrier of X ) ) . s ) . s = ( ( id ( the carrier of X ) ) . s ) . s ) . v
then z /. 1 = ( N-min L~ z ) .. z & ( N-min 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 ) ) and for x st x in Z holds f . x = x and f . x > 0 ;
for R being add-associative right_zeroed right_complementable commutative associative commutative associative commutative distributive non empty doubleLoopStr , I , J being Subset of R , I being Subset of R , J being Subset of R , I being Subset of R holds ( I + J ) *' I c= I /\ J
consider f being Function of [: B1 , B2 :] , B2 such that for x being Element of [: B1 , B2 :] holds f . x = F ( x ) and f . x = F ( x ) ;
dom ( x2 + y2 ) = Seg len x .= len ( x2 ^ y2 ) .= len ( x2 ^ y2 ) .= len ( x2 ^ y2 ) .= len ( x ^ ( y ^ z2 ) ) .= len ( x ^ ( y ^ z2 ) ) .= len ( x ^ ( y ^ z2 ) ) .= len ( x ^ ( y ^ z2 ) ) .= len ( x ^ ( y ^ z2 ) ) ) ;
for S being card being finite Functor of C , B for c being Object of C holds card S = id ( ( the carrier of B ) /\ ( the carrier of C ) )
ex a st a = a2 & a in f /\ ( f " { 0 } ) & ( for x st x in f " { 0 } holds f . x = {} ) ;
a in Free ( ( H2 / ( x. 4 , x. 0 ) ) '&' ( H2 / ( x. 0 , x. 4 ) ) '&' ( H2 / ( x. 0 , x. 4 ) ) ) ;
for C1 , C2 being \in the carrier of C1 , f , g being stable Function of C1 , C2 st C2 = g holds f = g iff for g being Function of C2 , C2 st g in C2 holds f = g
( W-min L~ go \/ L~ pion1 ) `1 = W-bound L~ go \/ W-bound L~ pion1 .= W-bound L~ pion1 \/ W-bound L~ pion1 .= W-bound L~ go \/ W-bound L~ pion1 .= W-bound L~ pion1 \/ W-bound L~ pion1 .= W-bound L~ pion1 \/ W-bound L~ pion1 .= W-bound L~ pion1 \/ W-bound L~ co .= W-bound L~ pion1 ;
consider u such that u = <* x0 , y0 , z0 *> and f partial u , z & SVF1 ( 3 , f , u ) /. 3 = SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) ;
then ( t . {} ) `1 in Vars & ex x being Element of Vars st x in Vars & t . {} = x & t . {} = y & t . {} = x & t . {} = y & t . {} = z ;
Valid ( p '&' p , J ) . v = Valid ( p , J ) . v '&' Valid ( p , J ) . v .= Valid ( p , J ) . v '&' Valid ( p , J ) . v .= Valid ( p , J ) . v ;
assume for x , y being Element of S st x <= y for a , b being Element of T st a = f . x & b = f . y holds a >= b ;
func Class R -> Subset-Family of R means : an : for A being Subset of R holds A in it iff ex a being Element of R st a in it & a in A & R . a = Class ( R , a ) ;
defpred P [ Nat ] means ( ( ( ( ( ) `2 ) | ( ( ) `1 ) | ( dom ( ( G ) ) | ( dom ( G ) ) ) ) ) . $1 ) `1 c= G . ( ( ( ( G ) | ( dom ( G ) ) | ( dom ( G ) ) ) . $1 ) `1 ) ;
assume that dim W1 = 0 and dim W2 = 0 and dim W1 = 0 and dim W2 = 0 and dim W1 = 0 and dim W2 = 0 and dim W1 = 0 and dim V = 0 and dim V = 0 and dim V = 0 and dim V = 0 and dim V = 0 and dim V = 0 ;
mam in mam ( t ) . {} = ( m . {} ) `1 .= ( [ m , the carrier of C ] `1 ) `1 .= ( m . {} ) `1 .= m . {} .= m . {} .= m . {} .= m . {} .= m . {} .= m . {} ;
d11 = x9 ^ d22 .= f ^ ( ( y , d22 ) | ( y , d22 ) ) .= f ^ ( ( y , d22 ) | ( y , d22 ) ) .= ( y ^ ( y , d22 ) ) .= ( y ^ ( y , d22 ) ) .= ( y ^ ( y ^ ( y , d22 ) ) ) .= ( d ^ ( y , d22 ) ) .= d22 ;
consider g such that x = g and dom g = dom ( f | X ) and for x being element st x in dom ( f | X ) holds g . x in f . x and g . x in f . x ;
x + 0. F_Complex |^ ( len x , len x ) = x + x .= ( x + 0. F_Complex ) * ( x , len x ) .= ( x + 0. F_Complex ) * x .= x `1 ;
( ( k -' ( k -' 1 ) ) + 1 ) in dom ( f | ( ( k -' 1 ) + 1 ) ) | ( ( k -' 1 ) + 1 ) ) ;
assume that P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 = P \/ Q and P1 = P \/ Q and P1 = P \/ Q and P1 = P \/ Q and P2 = P \/ Q and P1 \/ Q = P \/ Q and P1 \/ Q = P \/ Q and P1 \/ Q = P \/ Q and P1 \/ Q = P \/ Q and P1 \/ Q = P \/ Q ;
reconsider a1 = a , b1 = b , b1 = c , c1 = d , c2 = p , c1 = p , c2 = d , c2 = p , c1 = q , c2 = d , c1 = p , c2 = q , c2 = d , c1 = p , c2 = q , c2 = p , c1 = q , c2 = d , c1 = p , c2 = q , c2 = d , c2 = p , c1 = q , c2 = d ;
reconsider thesis , } Fb1f = G1 . ( t , b ) * F1 . ( f . a ) as Morphism of ( G1 * F1 ) . ( ( F2 * F2 ) . b ) , ( G1 * F2 ) . ( f . b ) ;
LSeg ( f , i + i1 -' 1 ) = LSeg ( f /. ( i + i1 -' 1 ) , f /. ( i + 1 -' 1 ) ) .= LSeg ( f /. ( i + i1 -' 1 -' 1 ) , f /. ( i + 1 -' 1 ) ) ;
Integral ( M , ( P . m ) | dom ( P . n ) ) <= Integral ( M , ( P . m ) | dom ( P . m ) ) ;
assume that dom f1 = dom f2 and for x , y being element st x in dom f1 & y in dom f2 & f1 . x = f2 . y holds f1 . x = f2 . y ;
consider v such that v = y and dist ( u , v ) < min ( ( G * ( i , 1 ) `1 ) , ( G * ( i + 1 , 1 ) `1 ) ) ;
for G being Group , H being Subgroup of G , a being Element of H st a = b holds for i being Integer holds a |^ i = b |^ i & a |^ i = b |^ i
consider B being Function of Seg ( S + L ) , the carrier of V1 such that for x being element st x in Seg ( S + L ) holds P [ x , B . x ] ;
reconsider K1 = { ( p ) where p is Point of TOP-REAL 2 : P [ p ] & p `1 <= 0 & p `2 <= 0 & p <> 0. TOP-REAL 2 } as Subset of TOP-REAL 2 ;
( ( ( ( N - S ) / ( 2 |^ m ) - ( 2 |^ n ) ) / ( 2 |^ m ) ) / ( 2 |^ m ) ) ^2 <= ( ( ( N - S ) / ( 2 |^ m ) - ( 2 |^ m ) ) / ( 2 |^ m ) ) / ( 2 |^ m ) ;
for x be Element of X , n be Nat st x in E holds |. Re ( F . n ) .| . x <= P . x & |. Im ( F . n ) .| . x <= P . x
len @ p = len ( @ ( @ p ^ q ) ) + len <* [ 2 , 0 ] *> .= len ( @ ( @ p ^ @ q ) ) + len ( @ q ^ @ p ) .= len ( @ p ^ @ q ) + len ( @ q ^ @ q ) .= len ( @ p ^ @ q ) ;
v / ( x. 3 , m1 ) / ( x. 0 , m1 ) / ( x. 4 , m1 ) / ( x. 4 , m2 ) / ( x. 4 , m1 ) / ( x. 0 , m1 ) = m1 / ( x. 4 , m2 ) ;
consider r being Element of M such that M , v |= ( ( x. 3 , x. 4 ) / ( x. 0 , m ) ) / ( x. 4 , m ) |= H2 and ( ( x. 4 , x. 4 ) / ( x. 4 , m ) ) / ( x. 4 , m ) = r ;
func w1 \ w2 -> Element of Union ( G , R^2 ) means : such : for w1 , w2 being Element of Union ( G , R^2 ) holds it . ( w1 , w2 ) = ( ( the HKsp2 of G ) * ( w1 , w2 ) ) . ( w1 , w2 ) ;
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= ( Exec ( n2 , s2 ) ) . b2 .= ( Exec ( n2 , s2 ) ) . b2 .= ( Exec ( n2 , s2 ) ) . b2 .= ( Exec ( n2 , s2 ) ) . b2 .= ( Exec ( n2 , s2 ) ) . b2 .= ( s ) . b2 .= ( s ) . b2 ;
for n , k being Nat holds 0 <= ( Partial_Sums ( |. seq .| ) . ( n + k ) - Partial_Sums ( |. seq .| ) . n ) implies ( Partial_Sums ( |. seq .| ) . n ) - Partial_Sums ( |. seq .| ) . k = ( Partial_Sums ( |. seq .| ) . n )
set F = S \! \mathop { \vert } , F = S \! \mathop { \vert } } ;
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 ) ;
func the closed \HM { a , b , c , d } = ( the \HM { a , b , c } ) ` .= ( the carrier of \HM { a , b , c } ) ` .= ( the carrier of TOP-REAL 2 ) ` ;
a * b ^2 + ( a * c ^2 + ( b * a ^2 + c ) ) + ( b * a ^2 + ( c * a ^2 + c * b ^2 ) ) >= 6 * a * a * b * c ;
v / ( ( x1 , m1 ) / ( x2 , m2 ) ) / ( ( x1 , m1 ) / ( x2 , m2 ) ) = v / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) ;
consider consider consider consider consider consider consider consider consider such that len ( Q ^ <* x *> ) , M such that len ( Q ^ <* x *> ) = len ( Q +* ( M ^ <* y *> --> FALSE ) ) and len ( Q ^ <* y *> --> TRUE ) = len ( Q ^ <* y *> --> FALSE ) ;
Sum ( F |^ n1 ) = r |^ n1 * Sum ( C |^ n1 ) .= C . n1 * ( C |^ n1 ) .= C . n1 * ( C |^ n1 ) .= C . n1 * ( C |^ n1 ) .= C . n1 * ( C |^ n1 ) .= C . n1 * ( C |^ n1 ) .= C . n1 ;
( GoB f ) * ( len GoB f , 2 ) `1 = ( GoB f ) * ( len GoB f , 1 ) `1 .= ( GoB f ) * ( len GoB f , 1 ) `1 .= ( GoB f ) * ( len GoB f , 1 ) `1 .= ( GoB f ) * ( len GoB f , 1 ) `1 .= ( GoB f ) * ( len GoB f , 1 ) `1 ;
defpred X [ Element of NAT ] means ( Partial_Sums ( s ) . $1 = ( a * ( $1 + 1 ) ) * ( 2 |^ $1 ) + b * ( 2 |^ $1 ) * ( 2 |^ $1 ) ) + b * ( 2 |^ $1 ) ;
the_arity_of g = ( the Arity of S ) . g .= ( ( the Arity of S ) * g ) . g .= ( ( the Arity of S ) * g ) . g .= ( the Arity of S ) . g .= ( the Arity of S ) . g ;
( X ~ Y ) |^ Z tolerates X ^2 & card ( ( X ~ Y ) |^ Z ) = card ( X ~ Z ) & card ( ( X ~ Z ) |^ Z ) = card ( X ~ Z ) ;
for a , b being Element of S , s being Element of NAT st s = n & a = F . n & b = F . n & b = G . n holds b = N . ( n + 1 ) \ G . n
E , f |= All ( x. 2 , ( x. 2 ) .--> ( x. 2 ) ) '&' ( ( x. 2 ) .--> ( x. 2 ) ) '&' ( x. 0 ) '&' ( x. 2 ) '&' ( x. 1 ) ) '&' ( ( x. 2 ) '&' ( x. 1 ) ) '&' ( x. 2 ) ) ;
ex R2 being 1-sorted st R2 = ( p | ( n + ( n + 1 ) ) ) & ( for i be Element of NAT st i in Seg n holds p . i = the carrier of R2 ) & ( for i be Element of NAT st i in Seg n holds p . i = 0. ) ;
[. a , b + ( 1 / ( k + 1 ) ) .[ is Element of the , REAL & ( ( the partial of f ) * ( ( the partial of f ) | k ) ) | ( ( the carrier of ( k + 1 ) ) \ ( ( the partial of f ) | k ) ) | ( ( the carrier of ( k + 1 ) ) \ ( ( k + 1 ) \ ( k + 1 ) ) ) ) ;
Comput ( P , s , 2 + 1 ) = Exec ( P . 2 , Comput ( P , s , 2 ) ) .= Exec ( a3 , Comput ( P , s , 2 ) ) .= Exec ( a3 , s ) ;
card ( h1 . k ) = power ( F_Complex ) . ( ( - 1_ F_Complex ) * ( - 1_ F_Complex ) ) .= ( - 1_ F_Complex ) . ( ( - 1_ F_Complex ) * ( - 1_ F_Complex ) ) .= ( - 1_ F_Complex ) . ( ( - 1_ F_Complex ) * ( - 1_ F_Complex ) ) .= ( - 1_ F_Complex ) . ( ( - 1_ F_Complex ) * ( - 1_ F_Complex ) ) .= ( - ( - 1_ F_Complex ) ) ) . k ;
( f (#) g ) /. c = f /. c * ( g /. c ) " .= ( f (#) ( g /. c ) ) " .= ( f (#) ( g * ( g /. c ) ) ) " .= ( f (#) ( g * ( g * ( g * c ) ) ) ) " .= ( f (#) ( g * ( g * c ) ) ) ) ;
len Cv - len ( ( C | ( len C -' 1 ) ) | ( len C -' 1 ) ) = len C - len ( ( C | ( len C -' 1 ) ) | ( len C -' 1 ) ) .= len C - len ( ( C | ( len C -' 1 ) ) | ( len C -' 1 ) ) .= len C - len ( C | ( len C -' 1 ) ) ;
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom ( f | X ) /\ X .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) .= X ;
defpred P [ Nat ] means for n holds 2 * Fib ( n + $1 ) = Fib ( n ) * Fib ( n + $1 ) + ( 5 * Fib ( n + $1 ) ) ;
consider f being Function of INT , INT such that f = f ' and f is onto and n < k and f is onto and n < k and f | n is increasing ;
consider vs be Function of S , BOOLEAN such that vs = chi ( A \/ B , S ) and E7 : ( for A , B being Element of S holds E7 . A = Prob ( B \/ D , S ) ) & E7 : for C being Element of S holds C in rng ( ( F * ( C \/ D ) ) | B ) ;
consider y being Element of Y ( ) such that a = "\/" ( { F ( x , y ) where y is Element of X ( ) : P [ x , y ] } , T ( ) ) and Q [ y , x ] ;
assume that A c= Z and f = ( ( - 1 / 2 ) (#) ( ( exp_R * f1 ) / ( exp_R * f1 ) ) ) `| Z ) = f ;
( 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 .= ( GoB f ) * ( 1 , j2 ) `2 ;
dom Shift ( Seq q2 , len Seq q1 ) = { j + len Seq q1 where j is Nat : 1 <= j & j <= len Seq q1 } & len Seq q1 = len q1 + len Seq q2 } ;
consider G1 , G2 , G2 being Element of V such that G1 <= G2 and f is Morphism of G2 , G2 and g is Morphism of G1 , G2 and g is Morphism of G2 , G2 and f is Morphism of G1 , G2 and g is Morphism of G2 , G2 ;
func - f -> PartFunc of C , V means : Def1 : dom it = dom f & for 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 holds for v st v in a holds union ( L | ( union rng L ) ) in H iff for v st v in ( union rng L ) | ( union { a } ) holds L . v |= H ;
consider i1 , j1 such that [ i1 , j1 ] in Indices GoB f and f /. ( i1 + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i1 + 1 ) = ( GoB f ) * ( i1 , j1 ) ;
consider i , n such that n <> 0 and sqrt p = ( i - n ) / ( n + 1 ) and for n1 being Nat st n1 <> 0 & n1 < n holds |. p . n1 - p . n1 .| < p . n1 ;
assume that not 0 in Z and Z c= dom ( ( ( 1 / 2 ) (#) ( ( 1 / 2 ) * ( f1 - #Z 2 ) ) ) `| Z ) and for x st x in Z holds ( ( 1 / 2 ) (#) ( f1 - #Z 2 ) ) `| Z ) . x > - 1 & f . x < 1 ;
cell ( G1 , i1 -' 1 , j1 -' 2 ) \ ( ( m -' 1 ) / 2 |^ ( ( m -' 1 ) -' 1 ) * ( ( m -' 1 ) / 2 |^ ( ( m -' 1 ) + 2 ) ) * ( ( m -' 1 ) / 2 |^ ( ( m -' 1 ) + 2 ) ) ) c= BDD C ;
ex Q1 being open Subset of X st s = Q1 & ex Q1 being Subset of Y st Q c= F & ( for F being Subset-Family of Y st F is open & F is open holds Q [ F , Q1 ] ) & ( for P being Subset of X st P in F holds P is open implies Q is open ) implies Q is open )
gcd ( ( ( the carrier of A ) * ( gcd ( r1 , r2 , s1 , s2 ) ) , ( the carrier of A ) * ( gcd ( r1 , r2 , s2 , s2 ) ) , ( the carrier of A ) * ( gcd ( r1 , r2 , s2 , s2 ) ) ) , 1 ) = 1 ;
R8 = ( ( the _ of ( the is the / of n ) ) . ( m1 + 1 ) ) . m2 .= ( ( the _ of ( the / 2 ) ) . ( m1 + 1 ) ) . m2 .= [ 3 , ( the _ of ( the carrier of n ) ) . m1 + 1 ) ] .= [ 3 , ( the _ 3 ) . m1 + 1 ] ;
CurInstr ( P3 , Comput ( P3 , s , m1 + 1 ) ) = CurInstr ( P3 , Comput ( P3 , s , m1 + 1 ) ) .= CurInstr ( P3 , Comput ( P3 , s , m1 ) ) .= 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 ) \/ { p2 } /\ LSeg ( p1 , p2 ) \/ { p2 } /\ LSeg ( p1 , p2 ) \/ { p2 } /\ LSeg ( p1 , p2 ) \/ { p2 } /\ LSeg ( p1 , p2 ) \/ { p2 } ;
func -> Subset of the bound of Al means : Assume ex a , b st a in dom f & b in dom f & a in dom f & f . a = f . b & a in dom f & b in dom f & f . b = f . a ;
for a , b being Element of F_Complex st |. a .| > |. b .| & f is non zero holds a * ( - b ) is non zero implies a * ( - b ) is non ]
defpred P [ Nat ] means 1 <= $1 & $1 <= len g implies for i , j st [ i , j ] in Indices G & [ i , j ] in Indices G & [ i , j ] in Indices G holds G * ( i , j ) = G * ( i , j ) ;
assume that C1 , C2 are_as seq of C1 and g is reflexive and for s being State of C1 , t being State of C2 , f being Function of C2 , C2 st f = g * s holds ( for s being State of C1 , t being State of C2 holds s is stable iff s is stable ) ;
( ||. f .|| | X ) . c = ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| ;
|. q .| ^2 = ( q `1 ) ^2 + ( q `2 ) ^2 & 0 + ( q `2 ) ^2 < ( q `1 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 ;
for F being Subset-Family of T7 st F is open & {} in F & not {} in F & for A , B being Subset of T7 st A in F & B in F & A <> B holds card A = k & card B c= k implies card F c= k
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 , i ) and for k st k in dom F holds F . k = g . ( k , i ) ;
i |^ ( Let ( Let ( Let n ) - ( p |^ s ) ) |^ i ) = i |^ ( s + k ) - ( p |^ k ) * ( i |^ k ) .= i |^ ( s * k - ( p |^ k ) ) * ( i |^ k ) .= i |^ ( s * k - ( p |^ k ) ) * ( i |^ k ) ;
consider q being oriented Chain of G such that r = q and q <> {} and F . ( q . 1 ) = v1 and rng q c= rng ( p | ( len p ) ) and for i being Nat st i in dom p holds p . i = v2 and q . ( len q ) = v2 and rng q c= rng ( p | ( len p ) ) ;
defpred P [ Element of NAT ] means $1 <= len ( g , Z ) . ( len g + $1 ) = ( ( \mathop { \rm Y , Z ) ^ ( g , I ) ) . ( len g + $1 ) ;
for A , B being square Matrix of n , m , D , B being Matrix of n , m , D , C being Matrix of n , m , D st len A = n & width B = n & width C = m holds A * C = B * C
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 ex a , b being Element of R st s . i = a * b & a in I & b in J & s . i = b * a ;
func |( x , y )| -> Element of COMPLEX equals |( ( Re x ) , ( Re y ) )| - ( ( Im y ) ^2 + ( Im y ) ^2 ) , ( Im y ) ^2 + ( ( Im y ) ^2 + ( ( Im y ) ^2 + ( Im y ) ^2 ) ) ;
consider consider consider consider consider consider consider \mathop be FinSequence of F such that g is continuous and rng g c= A and for x be Element of X st x in A & g . x = x1 and g . x = y1 and g . x = y1 and g . x = y2 and g . x = y1 ;
then n1 >= len p1 & crossover ( p1 , p2 , n1 , n2 , n3 ) = crossover ( p1 , p2 , n1 , n2 , n3 ) & crossover ( p1 , p2 , n1 , n2 , n3 ) = crossover ( p1 , p2 , n1 , n2 , n3 ) ;
( q `1 ) * a <= ( q `1 ) * a & - ( q `1 ) * a <= ( q `1 ) * a or q `1 = - ( q `1 ) * a & q `1 = - ( q `2 ) * a or q `1 = - ( q `1 ) * a & q `2 = - ( q `2 ) * a ;
Fv . ( ( len ( p . ( len p ) ) ) `1 ) = Fv . ( p . ( len p ) ) `1 .= ( ( p . ( len p ) ) `1 ) `1 .= ( ( p . ( len p ) ) `1 ) `1 .= ( ( p . ( len p ) ) `1 ) `1 .= ( ( p . ( len p ) ) `1 ) `1 .= ( v . ( len p ) ) `1 .= ( v . ( len p ) `1 ;
consider k1 being Nat such that k1 + k = 1 and a := k = ( <* a := intloc 0 *> ^ ( ( intloc 0 ) .--> 1 ) ) ^ ( ( intloc 0 ) .--> 1 ) ) and ( a := intloc 0 ) ^ ( ( intloc 0 ) .--> 1 ) = ( ( intloc 0 ) .--> 1 ) ^ ( ( intloc 0 ) .--> 1 ) ;
consider B8 being Subset of B1 , y8 being Function of B1 , A1 such that B8 is finite and D = the carrier of ( B1 | A1 ) \/ ( ( B1 | A1 ) \/ ( B1 | A2 ) ) and D is finite and f is x and f . 0 = ( the carrier of ( B1 | A1 ) ) \/ ( ( B1 | A1 ) \/ ( B1 | A2 ) ) ;
v2 . ( b2 , g ) = ( curry ( F2 , g ) * ( ( curry thesis ) . b2 ) ) * ( ( curry thesis ) . b2 ) .= ( ( curry thesis ) * ( ( curry id id ( B ) ) . b2 ) ) * ( ( curry id id ( B ) ) . b2 ) .= ( ( curry id ( B ) ) . b2 ) * ( ( curry id ( B ) ) . b2 ) .= ( F2 id ( B ) ) . b2 ;
dom IExec ( I , P , Initialize s ) = the carrier of SCMPDS .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) .= dom ( 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 .| " * ||. ( R + R1 ) /. h .|| < e / ( |. h .| + R ) ;
LSeg ( G * ( len G , 1 ) + |[ 1 , - 1 ]| , G * ( len G , 1 ) + |[ 1 , 0 ]| ) c= Int cell ( G , len G + |[ 1 , 0 ]| , G * ( len G , 1 ) + |[ 1 , 0 ]| ) \/ Int cell ( G , len G + 1 , 1 ) ;
LSeg ( mid ( h , i1 , i2 ) , i ) = LSeg ( h /. ( i1 -' 1 ) , h /. ( i1 -' 1 ) ) .= LSeg ( h /. ( i1 -' 1 ) , h /. ( i1 -' 1 ) ) .= LSeg ( h /. ( i1 -' 1 ) , h /. ( i1 -' 1 ) ) .= LSeg ( h /. ( i1 -' 1 ) , h /. ( i1 -' 1 ) ) ;
A = { q where q is Point of TOP-REAL 2 : LE p1 , p2 , P & LE p1 , p2 , P & LE p2 , p1 , P & LE p1 , p2 , P & LE p2 , p1 , P & LE p1 , p2 , P & LE p2 , p1 , P & LE p1 , p2 , P & LE p2 , p1 , P & LE p2 , p1 , P } ;
( ( - x ) .|. y ) = - ( ( 1 - x ) .|. y ) * ( x .|. y ) .= ( - ( 1 - x ) .|. y ) * ( x .|. y ) .= ( - ( 1 - x ) .|. y ) * ( x .|. y ) .= ( - ( 1 - x ) .|. y ) * ( x .|. y ) .= ( - ( 1 - x ) .|. y ) ;
0 * sqrt ( 1 + ( p `1 / |. p .| - cn ) / ( 1 + cn ) ) ^2 = ( p `1 / |. p .| - cn ) / ( 1 + cn ) ;
( ( U | W ) * ( W * ( 1 / 2 ) ) ) * ( ( W * ( 1 / 2 ) ) * ( 1 / 2 ) ) = ( ( U | W ) * ( 1 / 2 ) ) * ( ( W * ( 1 / 2 ) ) * ( 1 / 2 ) ) .= ( ( U | W ) * ( 1 / 2 ) ) * ( 1 / 2 ) .= ( U * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * (
func Shift ( f , h ) -> PartFunc of REAL , REAL means : Assume for x be Element of REAL holds it . x = - h . x & for x be Element of REAL st x in dom it holds it . x = - h . x ;
assume that 1 <= k and k + 1 <= len f and [ i + 1 , j ] in Indices G and f /. k = G * ( i + 1 , j ) and f /. k = G * ( i + 1 , j ) and f /. k = G * ( i + 1 , j ) ;
assume that not y in Free H and not x in Free H and not ( ( H / ( x. 0 ) ) = ( ( Free H ) \ { y } ) \/ { y } ) and not ( not x in Free H & y in Free H & not x in Free H ) ;
defpred P11 [ Element of NAT , Element of NAT , Element of NAT , Element of NAT ] means ( $1 |^ p ) |^ $1 = ( $1 |^ p ) |^ $1 & ( $1 |^ p ) |^ $1 = ( $1 |^ p ) |^ ( p |^ $1 ) & ( $1 |^ p ) |^ ( $1 |^ $1 ) = ( $1 |^ p ) |^ ( p |^ $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 it . ( A \ B ) <= C . ( A \ B ) ;
[#] ( ( dist ( ( ( Euclid n ) | P ) ) .: Q ) ) = ( dist ( ( dist ( ( Euclid n ) | P ) , ( dist ( ( Euclid n ) | Q ) ) .: Q ) ) ) .: Q & lower_bound ( ( dist ( ( Euclid n ) | P ) ) .: Q ) = lower_bound ( ( dist ( ( Euclid n ) | P ) ) .: Q ) ;
rng ( F | ( [: S , S :] | ( { 0 } , { 1 } :] ) ) = {} or rng ( F | ( [: S , S :] | ( { 1 } , { 1 } :] ) ) = { 1 } or rng ( F | ( [: S , S :] | ( { 1 } , { 1 } ) ) ) = { 2 } ;
( f " ( rng ( f | ( rng f ) ) ) ) . i = f . i " .= ( f | ( rng f ) ) . i .= ( f | ( rng f ) ) . i .= ( f | ( rng f ) ) . i .= ( f | ( rng f ) ) . i .= ( f | ( rng f ) ) . i .= ( f | ( rng f ) ) . i .= ( f | ( rng f ) ) . i ;
consider P1 , P2 being non empty Subset of TOP-REAL 2 such that P1 /\ P2 = { p1 , p2 } and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 is_an_arc_of p2 , p1 and P2 is_an_arc_of p1 , p2 ;
f . p2 = |[ ( p2 `1 ) ^2 + ( p2 `2 ) ^2 * ( 1 + ( p2 `1 / p2 `2 ) ^2 ) , ( p2 `2 ) ^2 * ( 1 + ( p2 `1 / p2 `2 ) ^2 ) ]| ;
( ( ( AffineMap ( a , X ) ) " ) ) . x = ( ( the \HM { \bf qua } \HM { Function } ) " ) . x .= ( ( the carrier of ( X | X ) ) " ) . x .= ( ( the carrier of ( X | X ) ) . x ) .= ( ( the carrier of ( X | X ) ) . x ) .= ( ( the carrier of ( X | X ) ) . x ) .= 0. X ;
for T being non empty normal TopSpace , A , B being closed Subset of T , p being Point of T st A <> {} & B misses B holds A is non empty iff for p being Point of T , r being Real st p in B & r in B holds p in A & r in B & p in B
for i st i in dom F for G1 , G2 being strict normal Subgroup of G st G1 = F . i & G2 = F . i & G1 = F . ( i + 1 ) holds G1 is strict Subgroup of G & G2 is strict Subgroup of G
for x st x in Z holds ( ( ( ( 1 / 2 ) (#) ( ( arctan ) * ( f1 - #Z 2 ) ) ) `| Z ) . x = ( ( ( 1 / 2 ) (#) ( ( arctan ) * ( f1 - #Z 2 ) ) ) `| Z ) . x / ( 1 + x ^2 )
synonym f is right continuous means : Assume for x0 st x0 in dom f & x0 in dom f & f . x0 in dom f & for a st a in dom f & a in dom f holds f . a - f . x0 in dom f & f . a in dom f ;
then X1 , X2 are_separated or ( X1 union X2 ) misses X2 & ( X1 , X2 are_separated & X2 , X1 union X2 are_separated ) & ( X1 , X2 are_separated & X2 , X1 union X2 are_separated ) & ( X1 , X2 are_separated & X2 , X2 are_separated implies X1 , X2 are_separated ) ;
ex N be Neighbourhood of x0 st N c= dom SVF1 ( 1 , f , u ) & ex L , R st for x st x in N holds SVF1 ( 1 , f , u ) . x - SVF1 ( 1 , f , u ) . x0 = L . ( x - x0 ) + R . ( x - x0 )
( ( p2 `1 ) ^2 * sqrt ( 1 + ( p3 `1 / p3 `2 ) ^2 ) ) / sqrt ( 1 + ( p3 `1 / p3 `2 ) ^2 ) >= ( ( p2 `1 ) ^2 * sqrt ( 1 + ( p3 `2 / p3 `1 ) ^2 ) ) / sqrt ( 1 + ( p3 `1 / p3 `1 ) ^2 ) ;
( ( 1 / t1 ) (#) ( ||. f1 .|| " ) ) . x = ( ( 1 / t ) * ( ||. f1 .|| " ) ) . x & ( ( 1 / t ) * ( ||. f1 .|| " ) ) . x = ( ( 1 / t ) * ( ||. f1 .|| " ) ) . x ;
assume that for x holds f . x = ( ( - 1 / 2 ) (#) ( sin - cos ) ) `| Z and for x st x in Z holds ( ( - 1 / 2 ) (#) ( sin - cos ) ) `| Z ) . x = - 1 / ( sin . x ) ^2 and f . x = 1 / ( sin . x ) ^2 and f . x = 1 / ( sin . x ) ^2 ) ;
consider X-23 being Subset of Y , Y1 being Subset of X such that t = [: X-23 , Y1 :] and Y1 is open and ex Y1 being Subset of X st Y1 = [: Y1 , Y1 :] & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open ;
card ( S . n ) = card { [: d , b , d :] + b where d is Element of GF ( p ) : [ d , b ] in the carrier of GF ( p ) & [ d , b ] in the carrier of GF ( p ) } .= p ;
( W-bound D - W-bound D ) * ( 2 |^ ( n - 1 ) ) = ( W-bound D - W-bound D ) * ( 2 |^ ( n - 1 ) ) .= ( W-bound D - W-bound D ) * ( 2 |^ ( n - 1 ) ) .= ( W-bound D - W-bound D ) * ( 2 |^ ( n - 1 ) ) .= ( W-bound D - W-bound D ) * ( 2 |^ n ) ;