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 ;
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thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b
D c= S
let Y ;
S ` is convergent
q in X ;
V in X ;
y in N ;
x in T ;
m < n ;
m <= n ;
n > 1 ;
let r ;
t in I ;
n <= 4 ;
M is finite ;
let X ;
Y c= Z ;
A // M ;
let U ;
a in D ;
q in Y ;
let x ;
1 <= l ;
1 <= w ;
let G ;
y in N ;
f = {} ;
let x ;
x in Z ;
let x ;
F is one-to-one ;
e <> b ;
1 <= n ;
f is special ;
S misses C
t <= 1 ;
y divides m ;
P divides M ;
let Z ;
let x ;
y c= x ;
let X ;
let C ;
x _|_ p ;
o is monotone ;
let X ;
A = B ;
1 < i ;
let x ;
let u ;
k <> 0 ;
let p ;
0 < r ;
let n ;
let y ;
f is onto ;
x < 1 ;
G c= F ;
a is_>=_than X ;
T is continuous ;
d <= a ;
p <= r ;
t < s ;
p <= t ;
t < s ;
let r ;
D is_<=_than E ;
assume e > 0 ;
assume 0 < g ;
p in X ;
x in X ;
Y ` in Y ;
assume 0 < g ;
not c in Y ;
not v in L ;
2 in z `2 ;
assume f = g ;
N c= b ` ;
assume i < k ;
assume u = v ;
I = J ;
B ` ` = b ` ;
assume e in F ;
assume p > 0 ;
assume x in D ;
let i be element ;
assume F is onto ;
assume n <> 0 ;
let x be element ;
set k = z ;
assume o = x ;
assume b < a ;
assume x in A ;
a `2 <= b `2 ;
assume b in X ;
assume k <> 1 ;
f = product l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is generated of squares ;
assume m > 0 ;
assume A c= B ;
X is lower
assume A <> {} ;
assume X <> {} ;
assume F <> {} ;
assume G is open ;
assume f is dilatation ;
assume y in W ;
y \not <= x ;
A ` in B ` ;
assume i = 1 ;
let x be element ;
x `1 = x `1 ;
let X be BCK-algebra ;
assume S is non empty ;
a in REAL ;
let p be set ;
let A be set ;
let G be _Graph , W be Walk of G ;
let G be _Graph , W be Walk of G ;
let a be Complex ;
let x be element ;
let x be element ;
let C be FormalContext , a , b be Real ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
y be Real ;
X c= f . a
let y be element ;
let x be element ;
i be Nat ;
let x be element ;
n in NAT ;
let a be element ;
m in NAT ;
let u be element ;
i in NAT ;
let g be Function ;
Z c= NAT ;
l <= ma ;
let y be element ;
r2 in dom f ;
let x be element ;
k1 be Integer ;
let X be set ;
let a be element ;
let x be element ;
let x be element ;
let q be element ;
let x be element ;
assume f is being_homeomorphism ;
let z be element ;
a , b // K ;
let n be Nat ;
let k be Nat ;
B ` c= B ` ;
set s = - 1 ;
n >= 0 + 1 ;
k c= k + 1 ;
R1 c= R ;
k + 1 >= k ;
k c= k + 1 ;
let j be Nat ;
o , a // Y ;
R c= Cl G ;
Cl B = B ;
let j be Nat ;
1 <= j + 1 ;
arccot is_differentiable_on Z ;
exp_R is_differentiable_in x ;
j < i0 ;
let j be Nat ;
n <= n + 1 ;
k = i + m ;
assume C meets S ;
n <= n + 1 ;
let n be Nat ;
h1 = {} ;
0 + 1 = 1 ;
o <> b3 ;
f2 is one-to-one ;
support p = {}
assume x in Z ;
i <= i + 1 ;
r1 <= 1 ;
let n be Nat ;
a "/\" b <= a ;
let n be Nat ;
0 <= r0 ;
let e be Real , x be Point of TOP-REAL 2 ;
not r in G . l
c1 = 0 ;
a + a = a ;
<* 0 *> in e ;
t in { t } ;
assume not F is discrete ;
m1 divides m ;
B * A <> {} ;
a + b <> {} ;
p * p > p ;
let y be ExtReal ;
let a be Int-Location , I be Program of SCM+FSA ;
let l be Nat ;
let i be Nat ;
let r ;
1 <= i2 ;
a "\/" c = c ;
let r be Real ;
let i be Nat ;
let m be Nat ;
x = p2 ;
let i be Nat ;
y < r + 1 ;
rng c c= E
Cl R is boundary ;
let i be Nat ;
R2 in X ;
cluster waybelow x -> in non empty ;
X <> { x } ;
x in { x } ;
q , b // M ;
A . i c= Y ;
P [ k ] ;
2 to_power x in W ;
X [ 0 ] ;
P [ 0 ] ;
A = A |^ i ;
L~ L~ L~ L~ L~ L~ L~ L~ \subseteq ' ;
G . y <> 0 ;
let X be RealNormSpace , A be Subset of X ;
a in X ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , A be Subset of V ;
assume x in - - M ;
k < s . a ;
not t in { p } ;
let Y be set , f be Function of Y , BOOLEAN ;
M , L are_isomorphic ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded & L is upper-bounded ;
rng f = Y ;
( G . n ) c= L ;
assume x in Cl Q ;
m in dom P ;
i <= len Q ;
len F = 3 ;
Free p = {} ;
z in rng p ;
lim b = 0 ;
len W = 3 ;
k in dom p ;
k <= len p ;
i <= len p ;
1 in dom f ;
b `1 = a `1 + 1 ;
x `1 = a * y `2 ;
rng D c= A ;
assume x in K1 ;
1 <= i-32 ;
1 <= i-32 ;
p^ c= PI ;
1 <= i-15 ;
1 <= i-15 ;
LMP C in L ;
1 in dom f ;
let seq , n ;
set C = a * B ;
x in rng f ;
assume f is_continuous_on X ;
I = dom A ;
u in dom p ;
assume a < x + 1 ;
s-7 is bounded ;
assume I c= P1 ;
n in dom I ;
let Q ;
B c= dom f ;
b + p _|_ a ;
x in dom g ;
F-14 is continuous ;
dom g = X ;
len q = m ;
assume A2 is closed ;
cluster R \ S -> real-valued ;
sup D in S ;
x << sup D ;
b1 >= Z1 + 1 ;
assume w = 0. V ;
assume x in A . i ;
g in the carrier of Assume X ;
y in dom t ;
i in dom g ;
assume P [ k ] ;
@ 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 , A be non-empty MSAlgebra over S ;
assume P [ n ] ;
assume union S is finite independent ;
V is Subspace of V ;
assume P [ k ] ;
rng f c= NAT * ;
assume ex_inf_of X , L ;
y in rng f ;
let s , I be set , A be ManySortedSet of I ;
b ` ` c= b9 ` ;
assume not x in RAT ;
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 & cos is_differentiable_in sin . x ;
assume y in rng S ;
let x , y be element ;
i2 < i1 & i2 < i2 ;
a * h in a * H ;
p , q in Y ;
redefine func sqrt I -> Ideal of L ;
q1 in A1 & q2 in A2 ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A2 c= A1 ;
an < n & cn < cn ;
assume A c= dom f ;
Re ( f ) is_integrable_on M ;
let k , m be element ;
a , a \equiv b , b ;
j + 1 < k + 1 ;
m + 1 <= n1 ;
g is_differentiable_in x0 & g is_differentiable_in x0 ;
g is_continuous_in x0 & g is_differentiable_in x0 ;
assume O is symmetric transitive ;
let x , y be element ;
let j0 be Nat ;
[ y , x ] in R ;
let x , y be element ;
assume y in conv A ;
x in Int V ;
let v be VECTOR of V ;
P3 halts_on s , P +* I ;
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 B ;
let S be non empty ManySortedSign ;
let x be variable of f , A , B be Subset of f ;
let b be Element of X , x be Element of X ;
R [ x , y ] ;
x ` ` = x ` ;
b \ x = 0. X ;
<* d *> in D * ;
P [ k + 1 ] ;
m in dom ( mnx ) ;
h2 . a = y ;
P [ n + 1 ] ;
redefine func G * F -> ] ] ;
let R be non empty multMagma , A be Subset of R ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p ;
assume f | X is lower ;
x in rng co /\ rng co ;
let x be Element of B ;
let t be Element of D ;
assume x in Q .vertices() ;
set q = s ^\ k ;
let t be VECTOR of X ;
let x be Element of A ;
assume y in rng p `2 ;
let M be be be be be id mamaid id ;
let N be non empty for \mathop { \rm be Nat } ;
let R be RelStr with finite finite finite finite : number ;
let n , k be Nat ;
let P , Q be be be be be RelStr ;
P = Q /\ [#] S ;
F . r in { 0 } ;
let x be Element of X ;
let x be Element of X ;
let u be VECTOR of V ;
reconsider d = x as FinSeq-Location ;
assume I does not ^ <* a *> ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < v8 ;
x <= c2 . x ;
x in F ` & y in F ` ;
redefine func S --> T -> ) ;
assume t1 <= t2 & t2 <= t1 ;
let i , j be even Integer ;
assume that F1 <> F2 and F2 <> {} ;
c in Intersect ( union R ) ;
dom p1 = c & dom p2 = c ;
a = 0 or a = 1 ;
assume A1 <> [: A2 , A1 :] ;
set i1 = i + 1 ;
assume a1 = b1 & a2 = b2 ;
dom g1 = A & dom g2 = A ;
i < len M + 1 ;
assume not -infty in rng G ;
N c= dom ( f1 + f2 ) ;
x in dom ( sec | A ) ;
assume [ x , y ] in R ;
set d = x / y ;
1 <= len g1 & g1 /. 1 = g1 /. 1 ;
len s2 > 1 & len s2 > 1 ;
z in dom ( f1 | X ) ;
1 in dom ( D2 | ( len D2 ) ) ;
p `2 = 0 & p `2 = 0 ;
j2 <= width G & j2 <= width G ;
len PI > 1 + 1 ;
set n1 = n + 1 ;
|. q-35 .| = 1 ;
let s be SortSymbol of S ;
( gcd ( i , i ) ) = i ;
X1 c= dom f & X2 c= dom f ;
h . x in h . a ;
let G be non empty thesis , F be Subset of G ;
cluster m * n -> square ;
let kl be Nat , k be Nat ;
i - 1 > m - 1 ;
R is transitive implies field R is transitive
set F = <* u , w *> ;
p-2 c= P3 & p`2 c= P3 ;
I is_halting_on t , Q ;
assume [ S , x ] is thesis ;
i <= len f2 & i <= len f2 ;
p is FinSequence of X & q is FinSequence of X ;
1 + 1 in dom g ;
Sum R2 = n * r ;
cluster f . x -> complex-valued ;
x in dom ( f1 | X ) ;
assume [ X , p ] in C ;
BX c= ( X \/ Z ) ;
n2 <= ( 2 * n ) - 1 ;
A /\ cP c= A ` ;
cluster x -valued for Function ;
let Q be Subset-Family of S , P be Subset of Q ;
assume n in dom g2 & n + 1 in dom g2 ;
let a be Element of R ;
t `2 in dom e2 & t `2 in dom e2 ;
N . 1 in rng N ;
- z in A \/ B ;
let S be SigmaField of X , A be Subset of S ;
i . y in rng i ;
REAL c= dom f & dom f c= REAL ;
f . x in rng f ;
mt <= r / 2 ;
s2 in r-5 & s2 in r-5 ;
let z , z be complex number ;
n <= Nm . m ;
LIN q , p , s ;
f . x = waybelow x /\ B ;
set L = [' S , T '] ;
let x be non positive Real ;
let m be Element of M ;
f in union rng ( F1 . n ) ;
let K be add-associative right_zeroed right_complementable non empty doubleLoopStr , p be Polynomial of K ;
let i be Element of NAT , k be Nat ;
rng ( F * g ) c= Y
dom f c= dom x & dom g c= dom y ;
n1 < n1 + 1 & n1 + 1 < n2 ;
n1 < n1 + 1 & n1 + 1 < n2 ;
cluster 1. T , X , Y :] -> \overline X ;
[ y2 , 2 ] = z ;
let m be Element of NAT ;
let S be Subset of R ;
y in rng S29 & y in rng S29 ;
b = sup ( dom f ) ;
x in Seg ( len q ) ;
reconsider X = D ( ) as set ;
[ a , c ] in E1 ;
assume n in dom ( h2 . n ) ;
w + 1 = ma + 1 ;
j + 1 <= j + 1 + 1 ;
k2 + 1 <= k1 & k2 + 1 <= k2 ;
i be Element of NAT ;
Support u = Support p \/ { u } ;
assume X is complete for x being Element of X holds x is complete
assume f = g & p = q ;
n1 <= n1 + 1 & n2 <= n1 + 1 ;
let x be Element of REAL , r be Real ;
assume x in rng ( s2 | X ) ;
x0 < x0 + 1 & x0 < r2 ;
len Carrier ( L ) = W ;
P c= Seg ( len A ) ;
dom q = Seg n & dom q = Seg n ;
j <= width ( M @ ) ;
let seq1 be real-valued sequence of X , seq be real-valued sequence of X ;
let k be Element of NAT ;
Integral ( M , P ) < +infty ;
let n be Element of NAT ;
assume z in A := being being being being being being being being Element of X ;
i be set ;
n - 1 = n-1 - 1 ;
len n-27 = n & width n-27 = n ;
not ] ( Z , c ) c= F
assume x in X or x = X ;
x is element of b , c ;
let A , B be non empty set , f be Function of A , B ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q & dom q = Seg i ;
let s be Element of E |^ omega ;
let B1 be Basis of x , B2 be Basis of y ;
L3 /\ L2 = {} ;
L1 /\ LSeg ( p2 , p2 ) = {} ;
assume downarrow x = downarrow y ;
assume b , c // b , c ;
LIN q , c , c ;
x in rng f-1of ( f . x ) ;
set n*> = n + j ;
let D be non empty set , f be FinSequence of D ;
let K be right_zeroed non empty addLoopStr , V be Subset of K ;
assume f `1 = f & h `2 = h ;
R1 - R2 is .| -holds R1 + R2 is total ;
k in NAT & 1 <= k ;
let a be Element of G ;
assume x0 in [. a , b .] ;
K1 ` is open & ( TOP-REAL 2 ) | K1 is open ;
assume a , b are_maximal in C ;
let a , b be Element of S ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = Integral ( M , f ) ;
cluster n^ -> nefor ;
not u in { ag } ;
the carrier of f c= B \/ { x }
reconsider z = x as VECTOR of V ;
cluster the carrier of L -> -> -> be for set ;
r (#) H is " (#) ( H ) ;
s . intloc 0 = 1 & s . intloc 0 = 2 ;
assume that x in C and y in C ;
let U0 be strict universal algebra , 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 : ( y in : x in { y } ) ;
let x , y be Element of X ;
let A , I be for X be { of A } ;
[ y , z ] in [: O , O :] ;
( \subseteq dom Macro i ) & ( card Macro i ) in dom Macro i ;
rng Sgm ( A ) = A ;
q |- \! such that q |- r ;
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 , b , b ;
p . 2 = Z |^ Y ;
( MD1 ) `2 = {} & ( MD1 ) `2 = {} ;
n + 1 + 1 <= len g ;
a in [: NAT , { A } :] ;
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 ( t ) ) ;
assume a in [: G ( ) , G ( ) :] ;
i `1 <= len ( y-5 ) ;
assume p divides b1 + b2 ;
M . x0 <= sup M1 & M . x0 <= 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 = l2 ;
a = {} or a = { x } ;
set uG = Vertices G , c = Vertices G ;
seq " is non-zero & seq " is non-zero implies seq " is non-zero
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
hcn c= h-14 & hh2 c= h-14 ;
]. a , b .[ c= Z ;
X1 , X2 , x3 , x4 , x5 , x5 , x1 , x2 , x3 , x4 , x4 , x4 , x5 , x1 , x2 , x3 , x4 , x4 , x4 , x1
a in Cl ( union F \ G ) ;
set x1 = [ 0 , 0 ] ;
k + 1 - 1 = k - 1 ;
cluster -> real-valued for Relation ;
ex v st C = v + W ;
let IT be non empty addLoopStr , x be Element of IT ;
assume V is Abelian add-associative right_zeroed right_complementable ;
X-21 \/ Y in \sigma ( L ) ;
reconsider x = x as Element of S ;
max ( a , b ) = a ;
sup B is upper Subset of B & sup B is upper Subset of A ;
let L be non empty reflexive antisymmetric RelStr , X be Subset of L ;
R is reflexive & R is transitive ;
E , g |= ( the_left_argument_of H ) ;
dom G `2 /. y = a ;
1 / 4 >= - r / 4 ;
G . p0 in rng G & G . p0 in rng G ;
let x be Element of FF , y be Element of D ;
D [ P-6 , 0 ] ;
z in dom ( id B ) /\ dom ( id B ) ;
y in the carrier of N & y in the carrier of N ;
g in the carrier of H & g in the carrier of G ;
rng fbeing Subset of NAT & fset c= NAT ;
j `2 + 1 in dom ( s1 . f ) ;
let A , B be strict Subgroup of G ;
let C be non empty Subset of REAL ;
f . z1 in dom h & h . z2 in dom h ;
P . k1 in rng P & P . k2 in rng P ;
M = AM +* {} .= ( A \/ {} ) ;
let p be FinSequence of REAL , r be Real ;
f . n1 in rng f & f . n2 in rng f ;
M . ( F . 0 ) in REAL ;
Y - Y = b-a ;
assume the distance of V , Q is_{} ;
let a be Element of op ( V ) ;
let s be Element of PQ ( ) ;
let PA be non empty as non empty \rm RelStr ;
let n be Nat ;
the carrier of g c= B & the carrier of g c= A ;
I = halt SCM R & I = halt SCM R ;
consider b being element such that b in B ;
set BK = BCS ( K , n ) ;
l <= v . j ;
assume x in downarrow [ s , t ] ;
x `2 in uparrow t & x `2 in uparrow t ;
x in ( JumpParts T ) \/ { {} } ;
let h be Morphism of c , a ;
Y c= 1. ( K , the_rank_of Y ) ;
A2 \/ A3 c= ( Carrier ( L1 ) ) \/ ( Carrier ( L2 ) ) ;
assume LIN o , a , b & LIN o , b , b ;
b , c // d1 , e2 ;
x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , 7 , 8 , 8 , 8 be set ;
dom <* y *> = Seg 1 & dom <* y *> = Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar s .| ;
[ x , x `2 ] in [: X , X :] ;
for n be Nat holds 0 <= x . n
[' a , b '] = [. a , b .] ;
cluster -> strict for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 , q1 is_collinear & q2 , q2 , q2 is_collinear ;
dom M1 = Seg n & dom M2 = Seg n ;
x = [ x1 , x2 ] ;
let R , Q be ManySortedSet of A ;
set d = 1 / ( n + 1 ) ;
rng g2 c= dom W & rng g2 c= dom ( W + L ) ;
P . ( [#] Sigma \ B ) <> 0 ;
a in field R & a = b ;
let M be non empty Subset of V , V be Subset of M ;
let I be Program of SCM+FSA , a be Int-Location ;
assume x in rng ( the InternalRel of R ) ;
let b be Element of the carrier of the carrier of T ;
dist ( e , z ) - r-r > r-r ;
u1 + v1 in W2 & v1 in W1 + W2 ;
assume the carrier of L misses rng G ;
let L be lower-bounded antisymmetric transitive antisymmetric RelStr ;
assume [ x , y ] in a9 ;
dom ( A * e ) = NAT & dom ( A * e ) = NAT ;
let a , b be Vertex of G ;
let x be Element of Bool M , i be Element of I ;
0 <= Arg a & Arg a < 2 * PI ;
o , a9 // o , y & o , y // o , y ;
{ v } c= the carrier of l & { v } c= the carrier of l ;
let x be bound variable of A ;
assume x in dom ( uncurry f ) & y in dom ( uncurry f ) ;
rng F c= ( product f ) |^ X
assume D2 . k in rng D & D2 . k in rng D2 ;
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 & LIN c , e1 , e2 ;
cluster -> ordinal for FinSequence of NAT ;
reconsider d = c , e = d as Element of L1 ;
( v2 |-- I ) . X <= 1 ;
assume x in the carrier of f & y in the carrier of g ;
conv @ @ S c= conv @ A & conv @ S c= conv @ A ;
reconsider B = b as Element of the carrier of T ;
J , v |= P \lbrack Carrier l , P . l .] ;
redefine func J . i -> non empty TopSpace ;
ex_sup_of Y1 \/ Y2 , T & ex_sup_of Y2 , T ;
W1 is well field W1 & W2 is non empty implies W1 + W2 is non empty
assume x in the carrier of R & y in the carrier of R ;
dom n-16 = Seg n & dom n-16 = Seg n ;
( the carrier of X ) misses ss2 & ( the carrier of X ) misses ss2 ;
assume ( a 'imp' b ) . z = TRUE ;
assume that X is open and f = X --> d ;
assume [ a , y ] in an implies [ a , y ] in an
assume that I c= J and / I c= K and / I c= K ;
Im ( lim seq , 0 ) = 0 & Im ( lim seq , 0 ) = 0 ;
( ( sin - cos ) `| Z ) . x <> 0 ;
sin is_differentiable_on Z & cos is_differentiable_on Z implies cos is_differentiable_on Z & for x st x in Z holds cos . x <> 0
t3 . n = t3 . n .= s . n ;
dom ( ( - 1 ) (#) F ) c= dom F ;
W1 . x = W2 . 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 .: S c= proj2 .: P ;
h . p4 = g2 . I .= g . I ;
Gij `1 = U /. 1 .= G * ( 1 , k ) `1 ;
f . rr1 in rng f & f . rr2 in rng f ;
i + 1 + 1-1 <= len - 1 ;
rng F = rng ( F . ( len F ) ) ;
mode non empty multMagma is well unital associative non empty multMagma ;
[ x , y ] in [: A , { a } :] ;
x1 . o in L2 . o & x2 . o in L2 . o ;
the carrier of ( support m ) c= B ;
not [ y , x ] in id X ;
1 + p .. f <= i + len f ;
seq ^\ k1 is lower & seq ^\ k1 is lower implies seq ^\ k is lower
len ( F . ( i + 1 ) ) = len I ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be Complex , x be Element of REAL ;
Comput ( P , s , n ) = s ;
k <= k + 1 & k + 1 <= len p ;
reconsider c = {} T as Element of L ;
let Y be an empty Chain \in the carrier of T ;
cluster -> directed-sups-preserving for Function of L , L ;
f . j1 in K . j1 & f . j2 in K . j2 ;
redefine func J => y -> total for I -valued Function ;
K c= 2 |^ the carrier of T & K is finite ;
F . b1 = F . b2 .= G . b2 ;
x1 = x or x1 = y or x1 = z ;
attr a <> {} means a / a = 1 ;
assume that ca c= b and b in a ;
s1 . n in rng s1 & s1 . n in rng s1 ;
{ o , b2 } on C2 & { o , b2 } on C2 ;
LIN o , b , b9 & LIN o , b , b9 ;
reconsider m = x as Element of Funcs ( V , C ) ;
let f be non constant FinSequence of D ;
let FF2 be non empty element , F be non empty set ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in F-8 ;
reconsider pp2 = x , pp2 = y as Subset of m ;
let A , B , C be Element of R ;
redefine attr ex X being strict non empty as strict , normal ;
rng c `1 misses rng e`1 & rng e`1 misses rng e`2 ;
z is Element of gr { x } & z is Element of gr { x } ;
not b in dom ( a .--> p1 ) ;
assume that k >= 2 and P [ k ] ;
Z c= dom ( cot * ( ( id Z ) ^ ) ) ;
the component of Q c= UBD ( A ) & UBD ( Q ) c= UBD ( A ) ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( 1 / ( g - f ) ) ;
pred f = u means : R : a * f = a * u ;
for n holds P1 [ n ] implies P1 [ n + 1 ]
{ x . O : x in L } <> {} ;
let x be Element of V . s ;
let a , b be Nat ;
assume that S = S2 and p = p2 and q = p1 ;
gcd ( n1 , n2 , n3 ) = 1 & gcd ( n1 , n2 , n3 ) = 1 ;
set oZ = ( 2 * PI ) * ( i , j ) ;
seq . n < |. r1 .| & seq . n < r2 ;
assume that seq is increasing and r < 0 and seq is increasing ;
f . ( y1 , x1 ) <= a & f . ( y1 , x2 ) <= b ;
ex c being Nat st P [ c ] ;
set g = { n to_power 1 where n is Element of NAT : n <= 1 } ;
k = a or k = b or k = c ;
aa , a{ g , h } , f , g , h , g ;
assume Y = { 1 } & s = <* 1 *> ;
Is1 . x = f . x .= 0 .= 0 ;
W3 .last() = W3 . 1 .= v . 1 .= v . 1 ;
cluster trivial -> finite for Walk of G ;
reconsider u = u as Element of Bags X ;
A in B @ implies A , B are_that A , B are_
x in { [ 2 * n + 3 , k ] } ;
1 >= ( q `1 / |. q .| - cn ) / ( 1 - cn ) ;
f1 is_\HM { the Element of L~ f2 : f2 is_One } ;
( f `2 ) ^2 / ( |. q .| ) ^2 <= ( q `2 ) ^2 / ( |. q .| ) ^2 ;
h is_the carrier of Cage ( C , n ) ;
b `2 <= p `2 & p `2 <= ( p `2 + q `2 ) / 2 ;
let f , g be s1 ) Function of X , Y ;
S * ( k , k ) <> 0. K ;
x in dom max ( - ( f . x ) , r ) ;
p2 in Nv & p2 in Nv & p2 in Nv ;
len ( the_left_argument_of H ) < len ( H ) & len ( H ) < len ( H ) ;
F [ A , F-14 . A ] ;
consider Z such that y in Z and Z in X ;
attr 1 in C means : Def8 : A c= C |^ A ;
assume that r1 <> 0 or r2 <> 0 and r1 <> 0 ;
rng q1 c= rng C1 & rng q2 c= rng C2 ;
A1 , L , A3 , A3 , A2 , A3 , A3 , A2 be non empty set ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in 4 ( p , SQ ) & a in { p } ;
then S is \times l is atomic & P-2 [ S ] ;
Cl Int [#] T = [#] T & Int [#] T = [#] T ;
f12 | A2 = f2 | A2 & f12 | A2 = f2 | A2 ;
0. M in the carrier of W & 0. M in the carrier of W ;
let v , v be Element of M ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ V & V c= Y \ Z ;
let X be Subset of [: S , T :] ;
consider H1 such that H = 'not' H1 ;
1_ 1 c= ( t * ( \HM { the } \HM { e } ) ) ;
0 * a = 0. R .= a * 0 ;
A |^ ( 2 , 2 ) = A ^^ A ;
set v{ v } = ( v /. n ) `1 ;
r = 0. ( REAL-NS n ) & r = 0. ( REAL-NS n ) ;
( f . p4 ) `1 >= 0 & ( f . p4 ) `2 >= 0 ;
len W = len ( W | ( W .: ( W .: ( W ) ) ) ;
f /* ( s * G ) is divergent_to-infty & f /. ( s * G ) is divergent_to-infty ;
consider l being Nat such that m = F . l ;
t8 | W8 does not destroy b1 & t8 does not destroy b1 ;
reconsider Y1 = X1 , Y2 = X2 as SubSpace of X ;
consider w such that w in F and not x in w ;
let a , b , c , d be Real , p be Point of TOP-REAL 2 ;
reconsider i = i - 1 as non zero Element of NAT ;
c . x >= id ( L . x ) ;
\sigma ( T ) \/ omega ( T ) is Basis of T ;
for x being element st x in X holds x in Y
cluster [ x1 , x2 ] -> pair for set ;
downarrow a /\ downarrow t is Ideal of T & downarrow t is Ideal of T ;
let X be with_NAT non empty set , f be Function of X , NAT ;
rng f = ` implies dom ( S , X ) = I
let p be Element of B , the carrier' of S ;
max ( N1 , 2 ) >= N1 & max ( N2 , 2 ) >= N2 ;
0. X <= b |^ ( m * mm1 ) ;
assume that i in I and R1 . i = R ;
i = j1 & p1 = q1 & p2 = q2 implies p1 = p2
assume gR in the right of g & gR in the carrier of g ;
let A1 , A2 be Point of S , A be Subset of X ;
x in h " P /\ [#] T1 & x in h " P /\ [#] T2 ;
1 in Seg 2 & 1 in Seg 3 implies 1 in Seg 3
reconsider X-5 = X , X*> as non empty Subset of Tsuch that X = { m } ;
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the carrier' of G ) -defined ;
n1 <= i2 + len g2 & i2 <= len g2 implies ( ( f | ( i + 1 ) ) /. ( i + 1 ) ) `1 <= ( f | ( i + 1 ) ) `1
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume v in the carrier' of G2 & u in the carrier' of G2 ;
y = Re y + ( Im y ) * i ;
( / ( - 1 ) ) * ( - 1 ) gcd p = 1 ;
x2 is_differentiable_on ]. a , b .[ & ( for x st x in ]. a , b .[ holds x <> a ) implies x2 = a
rng M5 c= rng D2 & rng M5 c= rng D2 ;
for p being Real st p in Z holds p >= a
( the carrier of X ) = proj1 * ( f . x ) ;
( seq ^\ m ) . k <> 0 & ( seq ^\ m ) . k <> 0 ;
s . ( G . ( k + 1 ) ) > x0 ;
( p -Path M ) . 2 = d ;
A \oplus ( B \ominus C ) = ( A ++ 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 strict Subspace of V holds V is Subspace of [#] V
reconsider i-7 = i , im2 = j as Element of NAT ;
dom f c= [: C ( ) , D ( ) :] ;
x in ( the carrier of B ) /\ ( A /\ B ) ;
len } in Seg ( len f2 ) & len ( f1 ^ f2 ) = len f1 + len f2 ;
pp1 c= the topology of T & pp2 c= the topology of T ;
]. r , s .] c= [. r , s .] ;
let B2 be Basis of T2 , B be Basis of T2 ;
G * ( B * A ) = ( B * o1 ) * A ;
assume that p , u , u is_collinear and p , q , v is_collinear ;
[ z , z ] in union rng ( F . z ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S & $1 .. S = $1 .. S ;
LIN a1 , a3 , b1 & LIN a2 , a3 , b1 ;
f " ( f .: x ) = { x } ;
dom w2 = dom r12 & dom r12 = dom r12 ;
assume that 1 <= i and i <= n and j <= n ;
( ( ( g2 ) . O ) `2 ) ^2 <= 1 ;
p in LSeg ( E . i , F . i ) ;
Ix * ( i , j ) = 0. K ;
|. f . ( s . m ) - g .| < g1 ;
q9 . x in rng ( q | ( Seg n ) ) ;
Carrier ( Lxy ) misses Carrier ( Lxy ) ` \/ { x } ;
consider c being element such that [ a , c ] in G ;
assume Na9 = oin & o8 = oin & o8 = ov ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= ( F |^ Cj ) " { x } ;
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [. f . j , f . j .] ;
attr 0 <= x & x <= 1 & x ^2 <= x ;
p `2 - q `2 <> 0. TOP-REAL 2 & p `2 - q `2 <> 0. TOP-REAL 2 ;
redefine func \cal a] ( S , T ) -> non empty ;
let x be Element of [: S , T :] ;
( the carrier of F ) . ( a , b ) is one-to-one ;
|. i .| <= - ( - 2 to_power n ) & |. i .| <= - ( - 2 to_power n ) ;
the carrier of I[01] = dom P & the carrier of I[01] = dom ( P | P ) ;
} * ( n + 1 ) ! > 0 * } ;
S c= ( A1 /\ A2 ) /\ ( A2 /\ A3 ) ;
a3 , a4 // b3 , b2 & a3 , a4 // b3 , b3 ;
then dom A <> {} & dom A <> {} & rng A c= A ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , Y & y is Vertex of G2 implies x = y
set v2 = ( v /. ( i + 1 ) ) ;
x = r . n .= r4 . n .= r4 . n ;
f . s in the carrier of S2 & f . s in the carrier of S2 ;
dom g = the carrier of I[01] & rng g = the carrier of I[01] ;
p in Upper_Arc ( P ) /\ Lower_Arc ( P ) ;
dom d2 = [: A2 , A2 :] & dom d2 = [: A1 , A2 :] ;
0 < p / ( ||. z .|| + 1 ) ;
e . ( m3 + 1 ) <= e . m3 ;
B \ominus X \/ B \ominus Y c= B \ominus X /\ Y
-infty < Integral ( M , Im ( g | B ) ) ;
cluster O := F -> \HM { 0 } for operation of X ;
let U1 , U2 be non-empty MSAlgebra over S , A be non-empty MSAlgebra over S ;
Proj ( i , n ) * g is_differentiable_on X & Proj ( i , n ) * g is_differentiable_on X ;
let x , y , z be Point of X , p be Point of TOP-REAL n ;
reconsider px0 = p . x , px0 = q . x as Subset of V ;
x in the carrier of Lin ( A ) & y in the carrier of Lin ( A ) ;
let I , J be parahalting Program of SCM+FSA , a be Int-Location ;
assume - a is lower Subset of - X & - a is Element of - X ;
Int Cl A c= Cl Int Cl ( Cl A ) ;
assume for A being Subset of X holds Cl A = A ;
assume q in Ball ( |[ x , y ]| , r ) ;
p2 `2 <= p `2 & p `2 <= p2 `2 or p2 `2 <= p `2 & p `2 <= p2 `2 ;
Cl Q ` = [#] ( ( TOP-REAL 2 ) | P ) ;
set S = the carrier of T , T = the carrier of T ;
set I8 = for f be FinSequence of TOP-REAL n holds for n be Nat holds f . n = n ;
len p - n = len ( p - n ) - n ;
A is Permutation of Swap ( A , x , y ) ;
reconsider n6 = n} , n7 = n7 as Element of NAT ;
1 <= j + 1 & j + 1 <= len ( s . k ) ;
let q\subseteq be { q where q is Element of M : q in F } , T = { q where q is Element of M : q in F } ;
( a , b ) in the carrier of S1 & ( a , b ) in the carrier of S2 ;
c1 /. n1 = c1 . n1 & c2 /. n1 = c1 . n1 ;
let f be FinSequence of TOP-REAL 2 , p be Point of TOP-REAL 2 , r be Real ;
y = ( ( f * S9 ) . x ) . x ;
consider x being element such that x in be Element of be IC is Element of A ;
assume r in ( ( dist ( o ) ) .: P ) ;
set i2 = ( n , i ) `1 , h = ( n , i ) `2 ;
h2 . ( j + 1 ) in rng h2 & h2 . ( j + 1 ) in rng h2 ;
Line ( M29 , k ) . i = M . i ;
reconsider m = ( x - 1 ) / 2 as Element of ExtREAL ;
let U1 , U2 be strict Subspace of U0 , a be Element of U1 ;
set P = Line ( a , d ) ;
len p1 < len p2 + 1 & len p2 + 1 <= len p1 + 1 ;
let T1 , T2 be Scott Scott topological B of L , x be Element of T ;
then x <= y & : x in : x in : y in : x in { y } ;
set M = n -finite ( m , n ) ;
reconsider i = x1 , j = x2 , k = x3 , l = x4 as Nat ;
rng ( ( the_arity_of o ) . n ) c= dom H & ( the_arity_of o ) . n in dom ( H . n ) ;
z1 " = z9 " & z2 " = z9 " & z1 = z2 " ;
x0 - r / 2 in L /\ dom f & f . x0 = r / 2 ;
then w is that rng w /\ L <> {} & ( S /\ L ) . w = {} ;
set x-10 = xx ^ <* Z *> , xT = ( xx ^ <* Z *> ) ^ <* Z *> ;
len w1 in Seg len w1 & len w1 in Seg len w1 & len w1 in Seg width w1 ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of thesis , k be Element of PFuncs ( V , { k } ) ;
x . n = ( |. a . n .| ) * ( x . n ) ;
p `1 <= Gik `1 & p `1 <= Gik `1 & p `1 <= G * ( 1 , 1 ) `1 ;
rng gg c= L~ ( g | ( L~ g ) ) & rng ( g | ( rng g ) ) c= rng ( g | ( rng g ) ) ;
reconsider k = i-1 * ( l + j ) as Nat ;
for n being Nat holds F . n is \HM { -infty } is \HM { -infty } ;
reconsider x-10 = x-7 , xq = xq as VECTOR of M ;
dom ( f | X ) = X /\ dom f .= X /\ dom f ;
p , a // p , c & b , a // c , c ;
reconsider x1 = x , y1 = y , y2 = z as Element of REAL m ;
assume i in dom ( a * p ^ q ) ;
m . ag = p . ag .= ( m + 1 ) . f ;
a / ( s . m - s . n ) / ( s . m - s . n ) <= 1 ;
S . ( n + k + 1 ) c= S . ( n + k ) ;
assume that B1 \/ C1 = B2 \/ C2 and C2 = C2 \/ C1 ;
X . i = { x1 , x2 } . i .= ( the carrier of X ) ;
r2 in dom ( h1 + h2 ) /\ dom ( h2 + h2 ) ;
\mathclose { 0. R } = a & b-0 = b ;
FF is_closed_on t2 , Q2 & I is_halting_on t2 , Q2 implies for k being Nat st k <= m holds I is_halting_on t1 , Q
set T = the = the topology of non ( X , x0 ) ;
Int Cl Int Cl R c= Int Cl R & Int Cl R c= Cl Int Cl R ;
consider y being Element of L such that c . y = x ;
rng ( F . x ) = { F . x } .= { F . x } ;
G-23 \ { c } c= B \/ S \/ S ;
f+ 1 is_X , ( X + 1 ) \ { x } ;
set RQ = the carrier of P , RQ = the carrier of Q ;
assume that n + 1 >= 1 and n + 1 <= len M ;
let k2 be Element of NAT , k be Nat ;
reconsider pq = u , pq = v as Element of ( then n + 1 ) -tuples_on the carrier of K ;
g . x in dom f & x in dom g implies f . x = g . x
assume that 1 <= n and n + 1 <= len f1 and f1 /. n = f1 /. ( n + 1 ) ;
reconsider T = b * N as Element of carr ( G ) ;
len PM <= len P-35 & len PM <= len P-35 implies PM . ( len M , i ) = M . ( i , j )
x " in the carrier of ( A1 \/ A2 ) & x " in the carrier of ( A1 \/ A2 ) ;
[ i , j ] in Indices ( A * ( i , j ) ) ;
for m being Nat holds Re ( F . m ) is simple function
f . x = a . i .= a1 . k ;
let f be PartFunc of REAL i , REAL , x be Element of REAL i ;
rng f = the carrier of ( ( Carrier A ) . i ) .= { i } ;
assume s1 = sqrt ( 2 * p ) - p / 2 ;
attr a > 1 & b > 0 & a / b > 1 ;
let A , B , C be Subset of IQ , a be Real ;
reconsider X0 = X , Y0 = Y , Y0 = Z as RealNormSpace ;
let f be PartFunc of REAL , REAL , x be Element of REAL ;
r * ( v1 |-- I ) . X < r * 1 ;
assume that V is Subspace of X and X is Subspace of V ;
let t-3 , t-4 be Relation of A , B ;
Q [ e-14 \/ { v-5 } , f . v-5 ] ;
g \circlearrowleft W-min L~ z = z implies ( W-min L~ z ) .. z < ( W-min L~ z ) .. z
|. |[ x , v ]| - |[ x , y ]| .| = v} ;
- f . w = - ( L * w ) .= - ( L * w ) ;
z - y <= x iff z <= x + y & y <= z - x
( 7 / p1 ) |^ ( 1 / e ) > 0 ;
assume X is BCK-algebra of 0 , 0 , 0 , 0 , 0 , 0 ;
F . 1 = v1 & F . 2 = v2 & F . 3 = v1 ;
( f | X ) . x2 = f . x2 .= ( f | X ) . x2 ;
( ( ( - tan ) `| Z ) . x ) in dom ( sec * ( ( id Z ) ^ ) ) ;
i2 = ( f /. len ( f | ( i + 1 ) ) ) ;
X1 = X2 \/ ( X1 \ X2 ) & X2 = ( X1 \ X2 ) \/ ( X2 \ X1 ) ;
[. a , b , 1_ G .] = 1_ G & 1_ G = 1_ G ;
let V , W be non empty VectSpStr over F_Complex , f be FinSequence of V ;
dom g2 = the carrier of I[01] & rng g2 = the carrier of I[01] & g2 . 0 = p1 ;
dom f2 = the carrier of I[01] & rng g2 = the carrier of I[01] & f2 . 0 = p1 ;
( proj2 | X ) .: X = proj2 .: ( proj2 .: X ) ;
f . ( x , y ) = h1 . ( x `1 , y `2 ) ;
x0 - r < a1 . n & a1 . n < x0 + r ;
|. ( f /* s ) . k - GM .| < r ;
len Line ( A , i ) = width A & width Line ( A , i ) = width A ;
S{ x } |^ n = ( S . g ) |^ n .= S . g ;
reconsider f = v + u as Function of X , the carrier of Y ;
intloc 0 in dom Initialized p & ( Initialized p ) . 0 in dom DataPart p ;
i1 := i2 := i3 does not contradiction & i2 does not destroy b1 implies ( i1 , i2 ) not f . ( i1 + 1 ) in dom ( i2 )
arccos r + arccos r = PI / 2 + 0 .= PI / 2 + 0 ;
for x st x in Z holds f2 is_differentiable_in x & ( f2 * f1 ) . x > 0
reconsider q2 = ( q - x ) / ( 1 - x ) as Element of REAL ;
( 0 qua Nat ) + 1 <= i + j1 & i + 1 <= len ( 0 qua Nat ) ;
assume f in the carrier of [' X , Omega Y '] ;
F . a = H / ( x , y ) . a ;
( ( the carrier of T ) at ( C , u ) ) . x = TRUE ;
dist ( ( a * seq ) . n , h ) < r ;
1 in the carrier of [. 0 , 1 .] & 1 in dom ( G | [. 0 , 1 .] ) ;
p2 `1 - x1 > - g & p2 `1 - x1 < p2 `1 - g & p2 `1 - x1 < p2 `1 - g ;
|. r1 - thesis .| = |. a1 .| * |. thesis .| ;
reconsider S-14 L = 8 as Element of Seg 8 & dom SL = Seg 8 ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b
D0W .succ ( n ) = D0W .2 + 1 ;
i1 = ma + n & i2 = K + n & h . i1 = 0. K ;
f . a [= f . ( f . O1 "\/" f . a ) ;
pred f = v & g = u , v + u ;
I . n = Integral ( M , F . n ) ;
chi ( T1 , S ) . s = 1 & chi ( T2 , S ) . s = 1 ;
a = VERUM ( A ) or a = VERUM ( A ) or a = VERUM ( A ) ;
reconsider k2 = s . b3 , k2 = s . b3 as Element of NAT ;
( Comput ( P , s , 4 ) ) . GBP = 0 ;
L~ M1 meets L~ R4 implies ( L~ R ) /\ L~ R4 = { R /. 1 }
set h = the continuous Function of X , R , x be Point of X ;
set A = { L . ( k . n ) where k is Nat : k <= n } ;
for H st H is atomic holds P [ H ] ;
set b`1 = S5 ^\ ( ia1 + 1 ) , ba1 = S5 ^\ ( ia1 + 1 ) ;
Hom ( a , b ) c= Hom ( a opp , b opp ) ;
1 / ( n + 1 ) < 1 / ( s " ) ;
l `1 = [ dom l , cod l ] & l `2 = cod l ;
y +* ( i , y /. i ) in dom g & y +* ( i , y ) in dom g ;
let p be Element of CQC-WFF ( Al ( ) ) , P [ p ] ;
X /\ X1 c= dom ( f1 - f2 ) /\ ( dom ( f1 - f2 ) ) ;
p2 in rng ( f /^ ( len p1 -' 1 ) ) & p2 in rng ( f /^ ( len p1 -' 1 ) ) ;
1 <= indx ( D2 , D1 , j1 ) & indx ( D2 , D1 , j1 ) <= len D2 ;
assume x in ( ( ( ( ( ( ( ( ( ( ( ( ( ( K ) \/ K0 ) ) ) \/ ( ( ( TOP-REAL 2 ) | D ) ) ) ) ) \/ ( ( TOP-REAL 2 ) | D ) ) ) ) ) ) ;
- 1 <= ( ( f2 ) . O ) `2 & ( ( f2 ) . I ) `2 <= 1 ;
let f , g be Function of I[01] , TOP-REAL 2 , a , b be Real ;
k1 -' k2 = k1 - k2 & k1 -' k2 = k2 - k2 implies k1 = k2
rng seq c= ]. x0 , +infty .[ & ( for n holds seq . n < x0 ) implies seq is convergent & lim seq = x0
g2 in ]. x0 - r , x0 + r .[ & g2 in ]. x0 - r , x0 + r .[ ;
sgn ( p `1 , K ) = - ( - 1_ K ) .= - ( - 1_ K ) ;
consider u being Nat such that b = p |^ y * u ;
ex A being subset of T st a = Sum A ;
Cl union ( H ) = union ( ( union H ) \/ ( union H ) ) ;
len t = len t1 + len t2 & len t = len t1 + len t2 & width t = width t2 + width t2 ;
v-29 = v + w |-- v + ( w |-- ( A , B ) ) ;
cv <> DataLoc ( t0 . GBP , 3 ) .= intpos ( 0 + 3 ) ;
g . s = sup ( d " { s } ) .= sup ( d " { s } ) ;
( \dot y ) . s = s . ( \dot y ) ;
{ s : s < t } in REAL implies t = {} or t = {}
s ` \ s = s ` \ ( 0. X ) .= ( 0. X ) \ ( 0. X ) ;
defpred P [ Nat ] means B + $1 in A implies B . $1 in A ;
( 339 + 1 ) ! = 3339 ! * ( 339 + 1 ) ;
U ( succ A ) = 1. T & ( U , A ) is Element of ( the carrier of T ) ;
reconsider y = y as Element of ( len y ) -tuples_on the carrier of K ;
consider i2 being Integer such that y0 = p * i2 and i2 in dom f ;
reconsider p = Y | Seg k as FinSequence of NAT , k be Nat ;
set f = ( S , U ) \mathop { z } , g = S S , z = U ;
consider Z being set such that lim s in Z and Z in F ;
let f be Function of I[01] , TOP-REAL n , x be Point of TOP-REAL n , r be Real ;
( ( ( M + i , 'not' A ) . [ n , 'not' A ] ) . 1 <> 1 ;
ex r being Real st x = r & a <= r & r <= b ;
let R1 , R2 be Element of ( n + 1 ) -tuples_on REAL , x be Element of REAL ;
reconsider l = 0. ( { v } ) , r = 0. ( A ) as Linear_Combination of A ;
set r = |. e .| + |. n .| + |. w .| + a ;
consider y being Element of S such that z <= y and y in X ;
a is being being being being being Element of 'not' ( b 'or' c ) = 'not' ( ( a 'or' b ) 'or' c )
||. ( x9 - gg ) .|| < r2 & ||. ( x9 - gg ) .|| < r ;
b9 , a9 // b9 , c9 & b9 , c9 // c9 , c9 implies b9 , c9 // c9 , a9
1 <= k2 -' k1 & k1 + 1 = k2 & k2 + 1 = k2 implies ( k2 + 1 ) <= k2
( ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) ^2 >= 0 ;
( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 < 0 ;
E-max C in right_cell ( Rv , 1 ) & E-max L~ Cage ( C , 1 ) in rng Rv ;
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 , a // a , b or p , a // b , a ;
g . n = a * Sum fs1 .= f . n ;
consider f being Subset of X such that e = f and f is strict ;
F | ( N2 , S ) = CircleMap * ( F | N2 ) .= ( F | N2 ) | ( N2 , S ) ;
q in LSeg ( q , v ) \/ LSeg ( v , p ) ;
Ball ( m , r0 ) c= Ball ( m , s ) & Ball ( x , r ) c= Ball ( x , s ) ;
the carrier of (0). V = { 0. V } & the carrier of (0). V = { 0. V } ;
rng ( ( cos | [. - 1 , 1 .] ) ) = [. - 1 , 1 .] ;
assume that Re seq is summable and Im seq is summable and Im seq is summable and Im seq is summable ;
||. ( vseq . n ) - ( vseq . n ) .|| < e / 2 ;
set g = O --> 1 ;
reconsider t2 = t11 , t2 = 0 as $ 0 as string of S2 , U ;
reconsider xd = seq . n , xd = seq . n as sequence of REAL ;
assume that C meets C and not E-max C meets L~ go and not E-max C in L~ pion1 and not E-max C in L~ pion1 ;
- ( Cl ( 1 - r ) ) < F . n - r . x ;
set d1 = thesis , d2 = dist ( x1 , z1 ) , d2 = dist ( x2 , z2 ) , d1 = dist ( y2 , z2 ) ;
2 |^ ( -' 00 -' 1 ) = 2 |^ ( -' 00 ) - 1 .= 2 |^ ( -' 00 ) ;
dom vG2 = Seg ( len d6 ) & dom v6 = Seg ( len d6 ) ;
set x1 = - k2 + |. k2 .| , x2 = - k2 + |. k2 .| , x3 = - k2 + 1 ;
assume for n being Element of X holds 0. <= F . n & F . n <= F . n ;
assume that 0 <= T-32 . i and T-32 . ( i + 1 ) <= 1 and T-32 . ( i + 1 ) <= 1 ;
for A being Subset of X holds c . ( c . A ) = c . A
the carrier of ( Carrier ( LT + L2 ) ) c= I2 & the carrier of ( Carrier ( LT + L2 ) ) c= I2 ;
'not' Ex ( x , p ) => All ( x , 'not' p ) is valid ;
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the normal normal \hbox { - } over {} ;
Z c= dom ( ( - 1 / ( sin * f1 ) ) `| Z ) ;
|. 0. TOP-REAL 2 - ( q `1 / |. q .| - cn ) .| < r / 2 - cn ;
A \ { A , B } c= A & A in not not L . ( A , succ ( d , B ) ) ;
E = dom Carrier ( f ) & ( for x be Element of E holds f . x is_measurable_on E ) implies ( f | E ) is_measurable_on E
C / ( A + B ) = C / B * C / A ;
the carrier of W2 c= the carrier of V & the carrier of W1 c= the carrier of V ;
I . IC ss2 = P . IC ss2 .= P . IC ss2 .= ( P +* I ) . IC s2 ;
attr x > 0 means : Def8 : 1 / x = x / ( - 1 ) ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) ;
consider p being Point of T such that C = [. p , R .] ;
b , c are_connected & - C , - C + - C + D + D + D + E + E + F + J + M + N + E + F + J + M + N + E + F + J + M + N + E + F + J + M + N +
assume f = id the carrier of OO & g = id the carrier of OO & h = id the carrier of OO ;
consider v such that v <> 0. V and f . v = L * v ;
let l be Linear_Combination of {} ( the carrier of V ) , r be Element of V ;
reconsider g = f " as Function of U2 , U1 , U2 ;
A1 in the carrier of G_ ( k , X ) & A2 in the carrier of G ;
|. - x .| = - ( - x ) .= - x .= - x .= - x ;
set S = is non empty ;
Fib ( n ) * ( 5 * Fib ( n ) -2 ) >= 4 * be Nat ;
vM /. ( k + 1 ) = vM . ( k + 1 ) ;
0 mod i = - ( i * ( 0 qua Nat ) ) .= - i * ( 0 qua Nat ) ;
Indices M1 = [: Seg n , Seg n :] & Indices M1 = [: Seg n , Seg n :] ;
Line ( St , j ) = St . j .= ( j - 1 ) + 1 ;
h . ( x1 , y1 ) = [ y1 , x1 ] & h . ( y1 , y2 ) = [ y2 , y1 ] ;
|. f .| - Re ( |. f .| * ( card b ) ) is nonnegative ;
assume x = ( a1 ^ <* x1 *> ) ^ b1 & y = ( a1 ^ <* x2 *> ) ^ b1 ;
MI is_closed_on IExec ( I , P , s ) , P & M is_halting_on s , P ;
DataLoc ( t3 . a , 4 ) = intpos ( 0 + 4 ) .= intpos ( 0 + 4 ) ;
x + y < - x + y & |. x .| = - x + y & |. y .| = - y ;
LIN c , q , b & LIN c , q , c & LIN c , q , b ;
f{} . ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + z1 ) .= y1 + ( y2 + z1 ) ;
f' . a = f{ a } & v in InputVertices S & [ x , v ] in InputVertices S ;
p `1 <= ( E-max C ) `1 & ( E-max C ) `1 <= ( E-max C ) `1 ;
set R8 = Cage ( C , n ) \circlearrowleft E8 , E7 = Cage ( C , n ) ;
p `1 >= ( E-max C ) `1 & ( E-max C ) `1 <= ( E-max C ) `1 ;
consider p such that p = p-20 and s1 < p /. i and p in L~ f ;
|. ( f /* ( s * F ) ) . l - GM .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len Line ( N , k + 1 + 1 ) = width N .= width N ;
f1 /* s1 is convergent & f2 /* s1 is convergent & ( f1 /* s1 ) . n = lim ( ( f1 /* s1 ) /* s1 ) ;
f . x1 = x1 & f . y1 = y1 & f . y2 = y2 ;
len f <= len f + 1 & len f + 1 <> 0 implies 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 A = the carrier of V ;
consider r such that r \not _|_ a and r \not _|_ x and r \not _|_ 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 * ( 1 / 2 ) <= 1 ;
for L being complete LATTICE for A being non empty Subset of rng \mathbb L holds L , A are_isomorphic implies L is complete
[ gi , gj ] in Ii \ Ij implies gi \ gj c= Ii \ Ij
set S2 = 1GateCircStr ( x , y , c ) ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and for r st r < x0 ex g st r < g & g < x0 & g in dom ( f2 * f1 ) ;
reconsider y = ( a ` ) / ( F . ( x , y ) ) as Element of L ;
dom s = { 1 , 2 , 3 } & s . 1 = d1 & s . 2 = d2 ;
( min ( g , ( 1 - 1 ) ) (#) f ) . c <= h . c ;
set G3 = the \frac of G , v , w be Vertex of G , v be Vertex of G ;
reconsider g = f as PartFunc of REAL , REAL-NS n , x be Element of REAL n ;
|. s1 . m / p .| / |. p .| < d / p / ( |. p .| + 1 ) ;
for x being element st x in ( ( 1 - u ) * t ) holds x in ( ( 1 - u ) * t )
P = the carrier of ( ( TOP-REAL n ) | D ) | P .= ( ( TOP-REAL n ) | D ) | P ;
assume that p10 in LSeg ( p1 , p2 ) /\ LSeg ( p2 , p2 ) and p2 in LSeg ( p1 , p2 ) /\ LSeg ( p2 , p2 ) ;
( 0. X \ x ) |^ ( m * ( k + 1 ) ) = 0. X ;
let g be Element of Hom ( cod f , \square ) ;
2 * a * b + ( 2 * c * d ) <= 2 * C1 * C2 ;
let f , g , h be Point of the carrier of the carrier of X , Y , h be Function of X , Y ;
set h = Hom ( a , g (*) f ) ;
then idseq ( n ) | Seg m = idseq ( m ) & m <= n & n <= m ;
H * ( g " * a ) in the right * ( the right * a ) ;
x in dom ( ( - cos * sin ) `| Z ) & ( - cos * sin ) `| Z ) . x = - sin . x / ( cos . x ) ^2
cell ( G , i1 , j2 -' 1 ) misses C & cell ( G , i1 , j2 -' 1 ) misses C ;
LE q2 , p4 , P , p1 , p2 & LE q2 , p , P , p1 , p2 implies LE q2 , p , P , p1 , p2
attr B is an component of A means : Def8 : B c= BDD A ;
deffunc D ( set , set ) = union rng $2 & union rng $2 = union rng $1 & union rng $2 = union rng $1 ;
n + - n < len ( pthesis + - n ) + - n & n + - n < len ( p ^ <* n *> ) ;
attr a <> 0. K means for M being Matrix of K holds the_rank_of M = the_rank_of ( a * M ) ;
consider j such that j in dom /\ /\ dom /\ I and I = len k + j ;
consider x1 such that z in x1 and x1 in ( P . x1 ) and ( P . x1 ) `1 = x ;
for n ex r being Element of REAL st X [ n , r ]
set CP1 = Comput ( P2 , s2 , i + 1 ) , CP2 = P2 ;
set cv = 3 / 4 , c/ ( 2 * PI ) , cv = 3 / 4 , cv = 5 / 4 ;
conv @ W c= union ( F .: ( E " W ) ) ;
1 in [. - 1 , 1 .] /\ dom ( arccot * ( arccot ) ) implies ( arccot * ( arccot ) ) . 1 = - 1
r3 <= s0 + ( r0 - ( |. v2 - v1 .| + 1 ) ) / ( 2 * ( ( v2 - v1 ) + 1 ) ) ;
dom ( f * f4 ) = dom f /\ dom f4 .= dom ( f * f4 ) /\ dom ( g * 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 , gr = gr as Point of TOP-REAL n ;
( T * h . s . x ) = T . ( h . s . x ) ;
I . ( L . ( J . x ) ) = ( I * L ) . ( J . x ) ;
y in dom ( the mapping of F ) implies commute ( Frege ( Frege ( A . o ) ) ) = dom ( Frege ( A . o ) ) ;
for I being non degenerated integral , I being commutative Ideal of R holds the carrier of I is commutative doubleLoopStr
set s2 = s +* Initialize ( ( intloc 0 ) .--> 1 ) , P2 = P +* I ;
P1 /. IC s1 = P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 ;
lim S1 in the carrier of [. a , b .] & lim S1 in the carrier of [. a , b .] ;
v . ( l-13 . i ) = ( v *' l+ ( v *' l-13 ) ) . i ;
consider n being element such that n in NAT and x = ( sn " ) . n ;
consider x being Element of c such that F1 . x <> F2 . x and F2 . x <> 0 ;
card ( card ( X , 0 , x1 , x2 , x3 ) ) = { E } & card ( X , 0 , x1 , x2 , x3 } ) = 1 ;
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 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n3
mg . HT ( mg , T ) = 0. L & mg . HT ( mg , T ) = 0. L ;
then H1 , H2 are_) & card H1 , ( card H2 ) * card ( H1 , { x } ) * card ( H2 , { x } ) * card ( H1 , { x } ) * card ( H2 , { x } ) = H ;
( N-min L~ ff ) .. ff > 1 & ( N-min L~ ff ) .. ff > 1 & ( N-min L~ ff ) .. ff > 1 ;
]. s , 1 .] = ]. s , 2 .] /\ [. 0 , 1 .] .= [. 0 , 1 .] ;
x1 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) ) & x2 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) ) ;
let f1 , f2 be continuous PartFunc of REAL , the carrier of S , the carrier of T ;
DigA ( t-23 , z9 ) is Element of k -tuples_on ( the carrier of K ) ;
I V V V V V V V V V V V V V V \mathop { d } = V ( k2 ) & V ( k2 ) = V ( k2 ) ;
[: { u } , { u9 } :] = { [ a , u9 ] } \/ { [ a , u9 ] } ;
( w | p ) | ( p | ( w | w ) ) = p ;
consider u2 such that u2 in W2 and x = v + u2 and u2 in W2 and u2 in W1 ;
for y st y in rng F ex n st y = a |^ n & P [ n , y ]
dom ( ( g * ( f . x ) ) | K ) = K & dom ( ( g * ( f . x ) ) | K ) = K ;
ex x being element st x in ( ( the Sorts of U0 ) \/ A ) . s ;
ex x being element st x in ( ( and OO ) \/ A ) . s ;
f . x in the carrier of [. - r , r .] & f . x in [. - r , r .] ;
( the carrier of X1 union X2 ) /\ ( ( the carrier of X1 ) \/ ( the carrier of X2 ) ) <> {} ;
L1 /\ LSeg ( p10 , p2 ) c= { p10 } /\ LSeg ( p1 , p2 ) ;
( b + b\cap bs ) in { r : a < r & r < b } ;
ex_sup_of { x , y } , L & x "\/" y = sup { x , y } ;
for x being element st x in X ex u being element st P [ x , u ]
consider z being Point of GX such that z = y and P [ z ] and z in A ;
( the carrier of ( ( the carrier of ( ( the carrier of X ) ) ) ) ) . ( - x ) <= e ;
len ( w ^ w2 ) + 1 = len w + 2 + 1 .= len w + 1 + 1 ;
assume q in the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 & q <> 0. TOP-REAL 2 ;
f | E-4 ` = g | E-4 ` & g | E-4 ` = g | Ed ` & g | Ed = g | Ed ;
reconsider i1 = x1 , i2 = x2 , j2 = x3 , j1 = x4 , j2 = x4 as Element of NAT ;
( a * A * B ) @ = ( a * ( A * B ) ) @
assume ex n0 being Element of NAT st f to_power n0 is + 1 & f . n0 is + 1 ;
Seg len ( ( the carrier of G ) --> { 1 } ) = dom ( ( the carrier of G ) --> { 1 } ) ;
( Complement ( ( Complement ( A . m ) ) ) . n ) . x c= ( ( Complement ( A . m ) ) . n ) . x ;
f1 . p = p9 & g1 . p = d & g1 . p = d & g2 . p = b ;
FinS ( F , Y ) = FinS ( F , dom ( F | Y ) ) ;
( x | y ) | z = z | ( y | x ) ;
( |. x .| |^ n ) / ( n + 1 ) <= ( r2 |^ n ) / ( n + 1 ) ;
Sum ( F ) = Sum f & dom ( F ) = dom g & for x be Element of X holds ( F . x ) . x = F . x ;
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 W3 is Subspace of W3 and W3 is Subspace of W3 ;
||. t-15 . x .|| = lim ||. ( x - y ) .|| .= ||. ( x - y ) .|| .= ||. x - y .|| ;
assume that i in dom D and f | A is lower bounded and g | A is lower bounded and g | A is lower ;
( ( p `2 ) ^2 - 1 ) * ( - ( - ( - ( p `2 / |. p .| - sn ) ) ) ) <= ( - ( - ( p `2 / |. p .| - sn ) ) ) ;
g | Sphere ( p , r ) = id Sphere ( p , r ) & g | Sphere ( p , r ) = id Sphere ( p , r ) ;
set N8 = N-min L~ Cage ( C , n ) , N8 = N-min L~ Cage ( C , n ) , N8 = \cal L ( Cage ( C , n ) , 1 ) ;
for T being non empty TopSpace holds T is countable countable implies the TopStruct of T is countable countable
width B |-> 0. K = Line ( B , i ) .= B * ( i , i ) .= B * ( i , j ) ;
attr a <> 0 means a <> 0 implies ( A \+\ B ) \ a = ( A \ a ) \+\ ( B \ a ) ;
then f is_\mathbin { \frac 2 } 3 , pdiff1 ( f , 1 ) & pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 ;
assume that a > 0 and a <> 1 and b > 0 and b <> 1 and c > 0 ;
w1 , w2 in Lin { w1 , w2 } & w2 in Lin { w2 , w1 } ;
p2 /. IC s = p2 . IC s .= ( IC Comput ( p2 , s2 , k ) ) .= ( card I + card J ) ;
ind ( T-10 | b ) = ind b .= ind B - ind ( T-10 | b ) .= ind B - ind ( T-10 | b ) ;
[ a , A ] in the carrier of [ the carrier of * ( A , B ) , the carrier of G ( ) ] ;
m in ( the Arrows of C ) . ( o1 , o2 ) & ( the Arrows of C ) . ( o1 , o2 ) = ( the Arrows of C ) . ( o1 , o2 ) ;
( ( a , CompF ( PA , G ) ) . z ) . x = FALSE & ( ( a , CompF ( PA , G ) ) . z ) . x = FALSE ;
reconsider phi = phi /. 11 , phi = phi /. 22 , phi = phi /. 11 as Element of ( S , U ) * ;
len s1 - 1 * ( len s2 - 1 ) + 1 > 0 + 1 * ( len s2 - 1 ) ;
delta ( D ) * ( f . ( upper_bound A ) - f . ( lower_bound A ) ) < r ;
[ f21 , f22 ] in the carrier' of [: A , B :] & f22 = [: A , B :] ;
the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 = K1 & the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 = K1 ;
consider z being element such that z in dom g2 and p = g2 . z and q . z = x ;
[#] V1 = { 0. V1 } .= the carrier of (0). V1 .= the carrier of (0). V1 .= the carrier of V1 ;
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 s in dom ( f | X ) and ||. x1 - x0 .|| < s ;
h1 = f ^ ( <* p3 *> ^ <* p *> ) .= h ^ <* p *> .= h ^ <* p *> ;
c /. |[ b , c ]| = c .= |[ |[ a , c ]| , |[ b , c ]| ]| ;
reconsider t1 = p1 , t2 = p2 , t1 = p3 , t2 = p2 , t2 = p3 , t1 = p2 , t2 = p3 , t2 = p1 , t1 = p2 , t2 = p3 , t2 = p1 , t2 = p2 , d = p3 , t2 = p3 , d = p1 , t2 = p3 , .: = { p2 , d
1 / 2 in the carrier of [. 1 / 2 , 1 .] & 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 .= C * ( p1 `2 ) + D ;
R . b - a + b = 2 * - b .= 2 * - b .= b ;
consider \vert 1 - \vert 1 .| such that B = - 1 * ] + ( 1 - \vert 1 .| ) * A ;
dom g = dom ( ( the Sorts of A ) * ( a , I ) ) & dom ( ( the Sorts of A ) * ( a , I ) ) = dom ( ( the Sorts of A ) * ( a , I ) ) ;
[ P . ( l ) , P . ( l ) ] in ( the carrier of TT ) \/ { [ l . ( l + 1 ) , P . ( l + 1 ) ] } ;
set s2 = Initialize s , P2 = P +* I ;
reconsider M = mid ( z , i2 , i1 ) , N = L~ z , M = LSeg ( i2 , i1 ) , N = L~ z ;
y in product ( ( Carrier J ) +* ( V , { 1 } ) ) ;
1 / ( |[ 0 , 1 ]| ) = 1 & 0 / ( |[ 0 , 1 ]| ) = 0 ;
assume x in the left of g or x in the right & y in the right & x in the carrier of g ;
consider M being strict Subspace of Aex T being strict Subspace of M st a = M & T is strict Subspace of M ;
for x st x in Z holds ( ( #Z n ) + f ) . x <> 0 & ( #Z n ) + f . x <> 0
len W1 + len W2 + m = 1 + len W3 + m .= len W3 + m + 1 .= len W3 + m + 1 ;
reconsider h1 = ( vseq . n ) - ( t-16 . n ) as Lipschitzian LinearOperator of X , Y ;
( - ( i mod len ( p + q ) ) + 1 ) in dom ( p + q ) ;
assume that s2 is or s1 is_the { of s1 : F in the topology of s2 } and s2 is finite and F in the topology of s2 ;
( ( ( ( q - y ) / 2 ) * ( x - y ) ) * ( x - y ) ) = ( ( x - y ) / 2 ) * ( x - y ) ;
for u being element st u in Bags n holds ( p `2 + m ) . u = p . u
for B being Subset of u-5 st B in E holds A = B or A misses B or A misses B
ex a being Point of X st a in A & A /\ Cl { y } = { a } ;
set W2 = tree ( p ) \/ ( W1 \/ W2 ) ;
x in { X where X is Ideal of L |^ \rm op ( L ) : X in D } ;
the carrier of W1 /\ W2 c= the carrier of W1 & the carrier of W1 /\ W2 c= the carrier of W2 & the carrier of W1 /\ W2 c= the carrier of W2 ;
( for a , b being Element of L holds a * ( a + b ) = ( a + b ) * ( a + b )
( ( X --> f ) . x ) . x = ( X --> dom f ) . x .= ( ( X --> dom f ) . x ) . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) , y = 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 <= ( ( i - 2 ) |^ ( n -' m ) + 1 ) - 1 + 1 ;
( reproj ( 1 , z0 ) ) . x in dom ( f1 (#) f2 ) /\ dom ( f2 (#) f3 ) ;
assume that b1 . r = { c1 } and b2 . r = { c2 } and b1 . r = { c2 } ;
ex P st a1 on P & a2 on P & b on P & c on P & a , b , P & c , d , P is_collinear ;
reconsider gf = g `1 * f `2 , hg = h `2 * g `2 as strict Element of X ;
consider v1 being Element of T such that Q = ( downarrow v1 ) ` and v1 in V and v1 in V ;
n in { i where i is Nat : i < n0 + 1 & i < n0 + 1 } ;
( F * ( i , j ) ) `2 >= ( F * ( m , k ) ) `2 ;
assume K1 = { p : p `1 >= cn * |. p .| & p `2 <= 0 & p <> 0. TOP-REAL 2 } ;
ConsecutiveSet ( A , succ O1 ) = ( ConsecutiveSet ( A , O1 ) ) .: ( A , O1 ) ;
set Is1 = in dom Macro ( a , intloc 0 ) , Is2 = SubFrom ( a , intloc 0 ) , Is2 = SubFrom ( a , intloc 0 ) , Is2 = SubFrom ( a , intloc 0 ) ;
for i be Nat st 1 < i & i < len z holds z /. i <> z /. 1 ;
X c= ( the carrier of L1 ) \/ ( the carrier of L2 ) & ( the carrier of L1 ) \/ ( the carrier of L2 ) c= the carrier of L2 ;
consider xx being Element of GF ( p ) such that xx |^ 2 = a and xx |^ 3 = b ;
reconsider eM = e4 , fN = f-5 , fN = f-5 , fN = fN as Element of D ;
ex O being set st O in S & C1 c= O & M . O = 0. ;
consider n being Nat such that for m being Nat st n <= m holds S . m in U1 and S . n in U2 ;
f * g * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) . x ) ;
defpred P [ Nat ] means A + succ $1 = succ A & ( A + ) = ( succ A ) + ( succ $1 ) ;
the left of - g = the left of g & the carrier of - g = the carrier of g & the carrier of - g = the carrier of g ;
reconsider pM = x , pM = y , pM = z , pM = w , pM = y as Point of TOP-REAL 2 ;
consider ' being Real such that p4 = y and x <= ' and for x0 being Element of REAL st x0 in dom f & x0 <= x0 holds f . x0 <= f . 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 & width ( x2 ^ y2 ) = width ( x2 ^ y2 ) ;
for x being element st x in X holds x in the set of positive iff ( for n being Nat holds x . n = ( n - 1 ) * x )
LSeg ( p11 , p2 ) /\ LSeg ( p1 , p2 ) = {} & LSeg ( p1 , p2 ) /\ LSeg ( p2 , p2 ) = {} ;
func union ( X ) -> set equals ( the carrier of X ) \/ ( the carrier of X ) & ( the carrier of X ) \/ ( id X ) = ( the carrier of X ) ;
len ( { ( C /. ( len C /. 1 ) ) , ( C /. ( len C -' 1 ) ) } ) <= len ( C /. ( len C -' 1 ) ) ;
attr K is with_a , a , b be Element of K , i be Nat holds v . ( a |^ i ) = i * v . a ;
consider o being OperSymbol of S such that t `2 . {} = [ o , the carrier of S ] and t `2 = [ o , the carrier of S ] ;
for x st x in X ex y st x c= y & y in X & y is - f . x
IC Comput ( P-6 , sd , k ) in dom ( Pd +* I ) & IC Comput ( Pd , sd , k ) in dom I ;
attr q < s means : : : r < s & s < q & ]. r , s .] c= ]. p , q .] ;
consider c being Element of Class ( f , c ) such that Y = ( F . c ) `1 and [ x , c ] in R ;
func the ResultSort of S2 -> Function of the carrier' of S2 , the carrier' of S2 means : Def: for x being Element of the carrier' of S2 holds it . x = id the carrier' of S2 ;
set y9 = [ <* y , z *> , f2 ] ;
assume x in dom ( ( ( #Z 2 ) * ( arccot ) ) `| Z ) & ( ( #Z 2 ) * ( arccot ) ) `| Z ) . x = ( ( #Z 2 ) * ( arccot ) ) . x ;
r-7 in Int cell ( GoB f , i , GoB f ) \ { ( GoB f ) * ( i , j ) } & ri2 in cell ( GoB f , i + 1 , j ) ;
q `2 >= ( ( Cage ( C , n ) /. ( i + 1 ) ) `2 ) / ( ( Cage ( C , n ) /. ( i + 1 ) ) `2 ) ;
set Y = { a "/\" a ` : a in X } ;
i - len f <= len f + len f1 - len f + len f - len f + 1 - len f + 1 - len f + 1 - len f + 1 - len f + 1 - len f + 1 <= len f - len f + 1 - len f + 1 - len f + 1 - len f + 1 - len f + 1 - len f + 1 ;
for n ex x st x in N & x in N1 & h . n = x- ( x0 - x )
set | 0 = ( \mathop { a , I , p , s ) . i , C = p +* I ;
p ( k ) . 0 = 1 or p ( k ) . 0 = - 1 & p ( k ) . 0 = 1 & p ( k ) . 1 = - 1 ;
u + Sum L-18 in ( U \ { u } ) \/ { u + Sum L-18 } ;
consider xx being set such that x in xx and xx in Vd and x = [ xx , xx ] ;
( p ^ ( q | k ) ) . m = ( q | k ) . ( - len p ) .= p . ( - len p ) ;
g + h = gg + hg1 & Nat ( g + h , X ) = g + h + h ;
L1 is distributive & L2 is distributive implies [: L1 , L2 :] is distributive & [: L1 , L2 :] is distributive & [: L1 , L2 :] is distributive
attr x in rng f means : Def8 : y in rng ( f \leftarrow x ) implies f / x , y / x |= f . y ;
assume that 1 < p and 1 / p + 1 / q = 1 and 0 <= a and a <= b and 0 <= b ;
F* ( f , M ) = rpoly ( 1 , M ) *' t + 1. F_Complex .= 1. F_Complex + 1. F_Complex .= 1. F_Complex ;
for X being set , A being Subset of X holds A ` = {} implies A = X & A = {} or A = {}
( ( N-min X ) `1 ) ^2 + ( ( ( N-min X ) `1 ) ^2 ) <= ( ( ( ( N-min X ) `1 ) `1 ) ^2 + ( ( ( N-min X ) `2 ) ^2 ) ) ^2 ;
for c being Element of the Sorts of A , a being Element of the free of A holds c <> a implies c <> a
s1 . GBP = ( Exec ( i2 , s2 ) ) . GBP .= s2 . GBP .= Exec ( i2 , s2 ) . GBP .= 0 ;
for a , b being Real holds |[ a , b ]| in ( y >= 0 ) -plane implies b >= 0 & a <= b
for x , y being Element of X holds x ` \ y = ( x \ y ) ` & y = ( x \ y ) `
mode BCK-algebra of i , j , m , n , m , n be Nat , i , j be Element of NAT , j be Nat ;
set x2 = |( Re ( y - y ) , Im ( x - y ) )| ;
[ y , x ] in dom u5 & u5 . ( y , x ) = g . y ;
]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .[ c= A & upper_bound divset ( D , k ) = upper_bound divset ( D , k ) ;
0 <= delta ( S2 . n ) & |. delta ( S2 . n ) .| < e / 2 implies 0 < e / 2
( - ( q `1 / |. q .| - cn ) ) ^2 <= ( - ( q `1 / |. q .| - cn ) ) ^2 + ( - ( q `1 / |. q .| - cn ) ) ^2 ;
set A = 2 / b-a ;
for x , y being set st x in R" holds x , y are_\hbox { - } f . x , f . y -\hbox { - } f . y }
deffunc FF2 ( Nat ) = b . $1 * ( M * G ) . $1 & ( M * G ) . $1 = ( M * G ) . $1 ;
for s being element holds s in -> element iff s in -> Element of ( -> Element of S ) \/ ( \rm \rm \rm \rm \rm \rm x0 } )
for S being non empty non void non void holds S is connected iff S is connected & S is connected
max ( degree ( z `1 ) , degree ( z `2 ) ) >= 0 & degree ( z `2 ) <= degree ( z `1 ) + degree ( z `2 ) ;
consider n1 being Nat such that for k holds seq . ( n1 + k ) < r + s and for n holds seq . ( n + k ) < r + s ;
Lin ( A /\ B ) is Subspace of Lin ( A ) & Lin ( B ) is Subspace of Lin ( B ) ;
set n-15 = n-13 '&' ( M . x qua Element of BOOLEAN ) , nL = M . ( x qua Element of BOOLEAN ) , nL = M . ( x , n ) ;
f " V in the topology of X & f " V in D & f " V in D & f " V in the topology of X & f .: V in D ;
rng ( ( a the function T ) +* ( 1 , b ) ) c= { a , c , b } ;
consider y being connected Walk of G1 such that y `1 = y and dom y `1 = WWthesis & y `2 = WW
dom ( 1 / f ) /\ ]. -infty , x0 .[ c= ]. -infty , x0 .[ & ( 1 / f ) | ]. x0 , x0 + r .[ is continuous ;
f2 is v of T ( i , j , n , r ) is Element of T ( i , j , n , - r ) ;
v ^ ( n-3 |-> 0 ) in Lin ( ( B | c1 ) \/ ( B | c2 ) ) & v ^ ( n-3 |-> 0 ) in Lin ( B ) ;
ex a , k1 , k2 st i = a /. k1 & j = b /. k2 & k1 = a /. k2 & k2 = b . k2 ;
t . NAT = ( NAT .--> succ i1 ) . NAT .= succ ( 5 .--> succ i1 ) .= succ ( 5 .--> succ i1 ) .= succ 5 ;
assume that F is bbfamily and rng p = F and dom p = Seg ( n + 1 ) and for i being Nat st i in Seg ( n + 1 ) holds p . i = F . i ;
( not b , b9 , a is_collinear ) & not LIN a , a9 , c & not LIN a , a9 , b & not LIN b , b9 , c
( L1 , O ) := O c= ( L1 , O ) \& ( L2 , O ) , ( L2 , O ) \& ( 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 * ( /. v ) = b * ( - w ) and 0 < a and a < b ;
defpred P [ FinSequence of D ] means |. Sum $1 .| <= Sum |. $1 .| & Sum ( $1 ) <= Sum ( |. $1 .| ) ;
u = cos / ( x , y ) . v * x + ( cos / ( x , y ) . v * y ) .= v ;
dist ( ( seq . n ) + x , g + x ) <= dist ( ( seq . n ) , g ) + 0 ;
P [ p , |. p .| (#) |. p .| , {} ] & P [ p , id ( the Sorts of A ) ] implies P [ p ]
consider X being Subset of CQC-WFF ( Al ( ) ) such that X c= Y and X is finite and X is inininand X is ininand X is ininand X is non empty ;
|. b .| * |. eval ( f , z ) .| >= |. b .| * |. eval ( f , z ) .| ;
1 < ( ( ( N-min L~ Cage ( C , n ) ) .. Cage ( C , n ) ) .. Cage ( C , n ) ) .. Cage ( C , n ) ;
l in { l1 where l1 is Real : g <= l1 & l1 <= h & h <= g } implies ex l st l = { l1 }
vol ( ( G . n ) vol ) <= ( Partial_Sums ( ( G . n ) vol ) ) vol ( ( G . n ) vol ) ) ;
f . y = x .= x * 1. L .= x * ( power L ) . ( y , 0 ) .= x * ( power L ) . ( y , 0 ) ;
NIC ( <% i1 , i2 , j2 %> , k ) = { i1 , succ ( i1 , k ) } .= { succ ( i1 , k ) } .= { succ ( i1 , k ) } ;
LSeg ( p10 , p2 ) /\ LSeg ( p1 , p2 ) = { p1 } & LSeg ( p1 , p2 ) /\ LSeg ( p2 , p2 ) = { p2 } ;
Product ( ( the carrier of I-15 ) +* ( i , { 1 } ) ) in ( ( the carrier of I-15 ) \/ ( i , { 1 } ) ) ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) .= Following ( s1 , n ) .= Following ( s2 , n ) ;
W-bound Qs2 <= q1 `1 & ( for q being Point of TOP-REAL 2 st q in Qs2 holds q `1 <= ( q `1 ) / 2 ) & ( q `1 <= ( q `1 ) / 2 )
f /. i2 <> f /. ( ( len f + len g -' 1 ) -' 1 ) & f /. i2 = f /. ( ( len f + 1 ) -' 1 ) ;
M , f / ( x. 3 , a ) / ( x. 4 , a ) / ( x. 4 , a ) / ( x. 4 , a ) / ( x. 4 , a ) |= H ;
len ( ( P ^ ) ) in dom ( ( P ^ ) ) & len ( ( P ^ ) ) = len ( P ^ ) + len ( P ^ ) ;
A |^ ( n , n ) c= A |^ ( m , n ) & A |^ ( k , l ) c= A |^ ( k , l ) ;
( TOP-REAL n ) \ { q : |. q .| < a } c= { q1 : |. q1 .| >= a }
consider n1 being element such that n1 in dom p1 and y1 = p1 . n1 and p1 . n1 = p2 . n1 and p1 . n1 = p2 . n1 ;
consider X being set such that X in Q and for Z being set st Z in Q & Z <> X holds X \not c= Z ;
CurInstr ( P3 , Comput ( P3 , s2 , l ) ) <> halt SCM+FSA & CurInstr ( P3 , Comput ( P3 , s2 , l ) ) <> halt SCM+FSA ;
for v be VECTOR of l1 , w be Element of REAL holds ||. v - w .|| = upper_bound rng |. ( ( the carrier of V ) --> { w } ) .|
for phi holds phi in X implies ( phi in X & not phi in X & phi in X ) & ( phi in X implies phi in X )
rng ( Sgm dom ( f-6 | dom ( f-9 | dom ( f-9 | dom ( f-9 | dom ( f-9 | dom f-9 | dom f-9 ) ) ) ) ) c= dom f-6 ;
ex c being FinSequence of D ( ) st len c = k & ( P [ c ] ) & ( P [ c ] implies P [ c ] ) ;
( the_arity_of ( a , b , c ) ) = <* \mathop ( b , c ) , \mathop ( a , b ) , \mathop ( a , c ) *> ;
consider f1 being Function of the carrier of X , REAL such that f1 = |. f .| and f1 is continuous and for x being Point of X holds f1 . x = x ;
a1 = b1 & a2 = b2 or a1 = b1 & a2 = b2 & a3 = b3 or a1 = b1 & a3 = b3 & a4 = b2 ;
D2 . ( indx ( D2 , D1 , n1 + 1 ) + 1 ) = D1 . ( n1 + 1 ) .= D1 . ( n1 + 1 ) ;
f . ( ||. r .|| ) = ||. |[ r , r ]| .|| /. 1 .= <* r *> . 1 .= <* r *> . 1 .= x ;
consider n being Nat such that for m being Nat st n <= m holds C-25 . n = C-25 . m and C-25 . m = C-25 . m ;
consider d being Real such that for a , b being Real st a in X & b in Y holds a <= d & d <= b ;
||. L /. h .|| - ( K * |. h .| ) + ( K * |. h .| ) - ( K * |. h .| ) <= p0 + ( K * |. h .| ) ;
attr F is commutative associative means : Def8 : for b being Element of X holds F -Sum { b } = f . b ;
p = - ( - p0 + 0. TOP-REAL 2 ) .= 1 * ( p0 `1 ) + 0. TOP-REAL 2 .= ( - p `1 ) * ( p0 `2 ) .= ( - p `1 ) * ( p0 `2 ) .= ( - p `1 ) * ( p0 `2 ) ;
consider z1 such that b `1 , x3 , x3 is_collinear and o , x1 , z1 is_collinear and o , x1 , z1 is_collinear and o <> z1 and o <> z1 ;
consider i such that Arg ( Rotate ( s ) ) . q = s + Arg q + ( 2 * PI * i ) and Arg ( Rotate ( s ) ) . i = PI ;
consider g such that g is one-to-one and dom g = card ( f . x ) and rng g = f . x and for x st x in dom f holds g . x = F ( x ) ;
assume that A = P2 \/ Q2 and P2 <> {} and Q2 <> {} and P2 <> {} and Q2 <> {} and Q2 <> {} and P2 <> {} and Q2 <> {} and P2 <> {} and P2 <> {} ;
attr F is associative means : Def8 : F .: ( F .: ( f , g ) , h ) = F .: ( f , F .: ( g , h ) ) ;
ex x being Element of NAT st m = x `2 & x `1 in z `1 & x `2 < i or m in { i } ;
consider k2 being Nat such that k2 in dom P-2 and l in P-2 . k2 and for k being Nat st k in dom P-2 holds P-2 . k = P-2 . ( k2 + 1 ) ;
seq = r (#) seq implies for n holds seq . n = r * seq . n & seq . n = r * seq . n & seq . n = r * seq . n
F1 . [ ( id a ) , [ a , a ] ] = [ f * ( ( id a ) , [ a , b ] ) , f * ( id a ) ] ;
{ p } "\/" D2 = { p "\/" y where y is Element of L : y in D2 & p in D2 } \/ { p } ;
consider z being element such that z in dom ( ( the Sorts of F ) * ( the Arity of S ) ) and ( ( the Sorts of F ) * ( the Arity of S ) ) . z = y ;
for x , y being element st x in dom f & y in dom f & f . x = f . y holds x = y
cell ( G , i , 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 `1 * b1 ) . x = ( Mx2Tran ( J , ' Z , ' ) ) . ( thesis /. j ) .= ( Mx2Tran J ) . ( T /. j ) ;
- 1 / ( - ( - 1 ) ) = mm (#) D | n .= mm (#) D .= mm (#) ( - ( 1 / ( - 1 ) ) ) .= Det M ;
attr x be set means : Def8 : for x be set st 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 ) .= len ( f2 . j ) ;
All ( All ( 'not' a , A , G ) , B , G ) '<' Ex ( 'not' All ( a , B , G ) , A , G ) ;
LSeg ( E . k0 , F . k0 ) c= Cl RightComp Cage ( C , k0 + 1 ) & LSeg ( E . k0 , F . k0 ) c= RightComp Cage ( C , k0 + 1 ) ;
x \ a |^ m = x \ ( a |^ k * a ) .= ( x \ a ) |^ k * a .= ( x \ a ) |^ k ;
k -th ininininininin1 = ( commute ( I-5 . k ) ) . k .= ( commute ( I-5 . k ) ) . ( ( commute ( I-5 . k ) ) . i ) .= ( ( commute ( IU . k ) ) . i ) . i ;
for s being State of Aex n being Nat holds Following ( s , n ) . ( 0 + ( n + 2 ) * n + 1 ) is stable
for x st x in Z holds f1 . x = a / 2 & ( f1 - f2 ) . x <> 0 & ( f1 - f2 ) . x <> 0
support ( thesis ) \/ support ( ( support ( m ) ) ) c= support ( max ( n , ( support ( m ) ) ) ) \/ support ( ( support ( m ) ) ) ;
reconsider t = u as Function of ( the carrier of A ) , ( the carrier of B ) , the carrier' of C ;
- ( a * sqrt ( 1 + b ^2 ) ) <= - ( b * sqrt ( 1 + a ^2 ) ) ;
phi ( succ b1 ) . a = g . a & phi ( b ) . ( g . a ) = f . ( g . a ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( ( F ^ <* p *> ) . i ) and i <> j ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , 7 , 8 } = { x1 } \/ { x2 , x3 , x4 , x5 , 8 }
the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U1 & the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U1 & the Sorts of U1 /\ ( U1 \/ U2 ) c= the Sorts of U2 ;
( - ( 2 * a * ( b - a ) ) ) ^2 + b ^2 - Let ( a , b , c ) ^2 > 0 ;
consider W00 such that for z being element holds z in W00 iff z in [: N , N :] & P [ z ] and P [ z ] ;
assume ( the Arity of S ) . o = <* a *> & ( the ResultSort of S ) . o = r & ( the ResultSort of S ) . o = <* r *> & ( the ResultSort of S ) . o = r ;
Z = dom ( ( exp_R * ( arccot ) ) `| Z ) /\ dom ( ( arccot * ( arccot ) ) `| Z ) .= dom ( ( exp_R * ( arccot ) ) `| Z ) ;
sum ( f , SS1 ) is convergent & lim ( \HM { x } ) = integral ( f , SS1 ) & lim ( f , SS2 ) = integral ( f , SS2 ) ;
( X ( ) . ( a9 => g ) ) => ( x9 => ( x9 => x9 ) ) in ( ( the carrier of L ) \ { x } ) \/ ( ( the carrier of L ) \ { x } ) ;
len ( M2 * M3 ) = n & width ( M2 * M3 ) = n & width ( M2 * M3 ) = n & width ( M2 * M3 ) = n & width ( M2 * M3 ) = n ;
attr X1 union X2 means : such that X1 is open SubSpace of X & X2 is open SubSpace of X & X1 is open SubSpace of X & X2 is open SubSpace of X & X2 is open SubSpace of X ;
for L being upper-bounded antisymmetric RelStr for X being non empty Subset of L for X being non empty Subset of L holds X "\/" { Top L } = { Top L }
reconsider f-129 = ( F2 . b ) `2 , f-129 = ( F2 . b ) `2 , f-129 = ( F2 . b ) `2 , f-129 = ( F2 . b ) `2 as Function of M , M ;
consider w being FinSequence of I such that the InitS of M , the InitS of M -{ s } ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w
g . ( a |^ 0 ) = g . ( 1_ G ) .= 1_ H .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) ;
assume for i being Nat st i in dom f ex z being Element of L st f . i = rpoly ( 1 , z ) & f . i = rpoly ( 1 , z ) ;
ex L being Subset of X st Carrier L = L & for K being Subset of X st K in C holds L /\ K <> {} ;
( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C1 & ( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C2 & ( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C2 ;
reconsider o-21 = o `2 as Element of TS ( ( the Sorts of A ) . v ) , ( the Sorts of A ) . v ;
1 * x1 + ( 0 * x2 ) + ( 0 * x3 ) = x1 + <* \underbrace ( 0 , 0 ) *> .= x1 + ( 0 * x2 ) .= x1 + ( 0 * x3 ) .= x1 + ( 0 * x2 ) ;
Ex " . 1 = ( Ex qua Function ) " . 1 .= ( ( E qua Function ) " ) . 1 .= ( ( E qua Function ) " ) . 1 .= ( ( E qua Function ) " ) . 1 .= ( ( E qua Function ) " ) . 1 ;
reconsider u1 = the carrier of U1 /\ ( U1 "\/" U2 ) , v1 = the carrier of U1 /\ ( U1 "\/" U2 ) as non empty Subset of U0 ;
( ( x "/\" z ) "\/" ( x "/\" y ) ) "\/" ( z "/\" y ) <= ( x "/\" ( z "\/" y ) ) "\/" ( z "/\" ( x "\/" y ) ) ;
|. f . ( s1 . ( l1 + 1 ) ) - f . ( s1 . l1 ) .| < 1 / ( |. M .| + 1 ) ;
LSeg ( ( Lower_Seq ( C , n ) ) * ( i , ( n + 1 ) ) , ( ( 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 * ( - ( g . c ) * f . c ) <= h . c * ( - ( g . c ) * f . c ) ;
( f + g ) | divset ( D , i ) = f | divset ( D , i ) + g | divset ( D , i ) .= f | divset ( D , i ) ;
assume that ColVec2Mx f in the set of the carrier of A and ColVec2Mx b in the carrier of ( ( len A ) ) \ ( ( len A ) \ { i } ) and len f = width A and width A = width A ;
len ( - M3 ) = len M1 & width ( - M3 ) = width M1 & width ( - M3 ) = width M1 & width ( - M3 ) = width M1 & width ( - M3 ) = width M1 ;
for n , i being Nat st i + 1 < n holds [ i , i + 1 ] in the InternalRel of ( ( the carrier of TOP-REAL n ) \ { i } )
pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1 implies pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1
attr a <> 0 & b <> 0 & Arg a = Arg b & Arg b = Arg b implies Arg ( - a ) = Arg ( - b ) & Arg ( - b ) = Arg ( - b )
for c being set st not c in [. a , b .] holds not c in Intersection ( the topology of X , a ) & not c in Intersection ( the topology of X , b )
assume that V1 is linearly-independent and V2 is linearly-independent and V is linearly-independent and V1 in V1 and u in V1 and v in V1 and V1 in V2 and V2 c= V1 ;
z * x1 + ( 1 - z ) * x2 in M & z * y1 + ( 1 - z ) * y2 in N implies z * y1 + ( 1 - z ) * y2 in M
rng ( ( PS1 qua Function ) " * SS2 ) = Seg card dS2 & dom ( PS2 qua Function ) = Seg card dS2 & rng ( PS2 ) = Seg card dS2 ;
consider s2 being rational Real_Sequence such that s2 is convergent and b = lim s2 and for n holds s2 . n <= b and for n holds s2 . n <= b . n ;
h2 " . n = h2 . n " & 0 < - 1 / ( ( 1 - ( ( 1 - ( ( 1 - ( 2 / ( n + 1 ) ) ) |^ n ) ) ) ) & ( - ( ( 1 - ( ( 1 / ( n + 1 ) ) |^ n ) ) |^ n ) ) |^ n = ( ( 1 - ( 1 / ( n + 1 ) ) |^ n ) ) |^ n )
( Partial_Sums ( ||. seq1 .|| ) ) . m = ||. seq1 .|| . m .= ||. ( seq1 . m ) - ( seq2 . m ) .|| .= ||. ( seq1 . m ) - ( seq2 . m ) .|| .= 0 ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b ;
- v = ( - 1_ ( G ) ) * v & - w = ( - 1_ ( G ) ) * w & ( - w ) * v = ( - 1_ ( G ) ) * w & ( - w ) * w = ( - 1_ ( G ) ) * w ;
sup ( ( k .: D ) .: D ) = sup ( ( k .: D ) .: ( D ) ) .= k . ( sup D ) .= sup ( D . ( D . ( D . ( D . ( D . ( D . ( D . ( D . ( D . ( D . ( D . ( D . ( D . ( D . ( D . ( D . ( D ) ) ) ) ) ) ) ) ) ) ) ) ) ;
A |^ ( k , l ) ^^ ( A |^ ( n , l ) ) = ( A |^ ( k , l ) ) ^^ ( A |^ ( k , l ) ) ;
for R being add-associative right_zeroed right_complementable non empty addLoopStr , I , J being Subset of R holds I + ( J + K ) = ( I + J ) + K
( f . p ) `1 = ( p `1 ) ^2 / sqrt ( 1 + ( p `2 / p `1 ) ^2 ) .= ( p `1 ) ^2 / sqrt ( 1 + ( p `2 / p `1 ) ^2 ) ;
for a , b being non zero Nat for a , b being Element of NAT st a , b are_relative_prime holds ( for n being Nat holds ( n * b ) . n = ( ( n * a ) + ( n * b ) ) . n ) holds ( n * a ) . n = ( ( n * a ) + ( n * b ) ) . n
consider A5 being countable set such that r is Element of CQC-WFF ( Al ) & A5 is ( Al ) -V & ( not ( ex A being Subset of Al st A is open & A is open ) ) & ( A is open ) implies A is ( ( A is open ) ) & A is ( ( A is open ) ) implies A is ( ( A ) ) ` ) ;
for X being non empty addLoopStr for M being Subset of X , x , y being Point of X st y in M holds x + y in x + M
{ [ x1 , x2 ] , [ y1 , y2 ] } c= [: { x1 , y1 } , { y2 } :] \/ [: { x2 , y2 } , { y2 } :] ;
h . ( f . O ) = |[ A * ( ( f . O ) `1 ) + B , C * ( ( f . O ) `2 ) + D ]| ;
( Gauge ( C , n ) * ( k , i ) ) in L~ Lower_Seq ( C , n ) /\ L~ Lower_Seq ( C , n ) implies ( Lower_Seq ( C , n ) * ( k , i ) ) `1 = ( Gauge ( C , n ) * ( k , i ) ) `1
cluster m , n are_relative_prime means : such : for p being prime Nat holds it is prime & for n being Nat holds it . ( p |^ n ) = m & p divides n ;
( f * F ) . x1 = f . ( F . x1 ) & ( f * F ) . x2 = f . ( F . x2 ) ;
for L being LATTICE , a , b , c being Element of L st a \ b <= c & b \ a <= c holds a \+\ b <= c
consider b being element such that b in dom ( H / ( x , y ) ) and z = ( H / ( x , y ) ) . b and ( H / ( x , y ) ) . b = x ;
assume that x in dom ( F * g ) and y in dom ( F * g ) and ( F * g ) . x = ( F * g ) . y ;
assume ex e being element st e Joins W . 1 , W . 5 , G or e Joins W . 3 , W . 7 , G ;
( ( r (#) f ) | A ) . ( 2 * n ) . x = ( r (#) ( f | A ) ) . ( 2 * n ) . x + ( r (#) ( f | A ) ) . ( x + h ) ;
j + 1 = ( len h11 + 2 ) - 1 + 1 .= i + 1 - 1 + 2 .= i + 1 - 1 + 2 .= i + 1 - 1 + 1 .= i + 1 - 1 + 1 ;
( *' ( S *' ) ) . f = *' ( S *' ) . ( ( opp f ) . f ) .= S . ( ( opp f ) . f ) .= S . ( ( *' f ) . f ) ;
consider H such that H is one-to-one and rng H = the carrier of L2 and Sum ( L2 * H ) = Sum ( L2 ) and for k st k in dom ( L1 * H ) holds Sum ( L1 ) = Sum ( L2 ) ;
attr R is b2 means : Def8 : for p , q st p in R & q <> q holds ex P st P is special & P c= R & P c= R & P c= R ;
dom product ( product ( X --> f ) ) = meet ( ( X --> f ) . ( ( X --> f ) . ( ( X --> f ) . ( ( X --> f ) . ( ( X --> f ) . ( ( X --> f ) . ( ( X --> f ) . ( ( X --> f ) . ( ( X --> f ) . ( ( X --> f ) . ( ( X --> f ) . ( X --> f ) ) ) ) ) ) ) ) ) ) )
upper_bound ( proj2 .: ( Upper_Arc ( C ) /\ Upper_Arc ( w ) ) ) <= upper_bound ( proj2 .: ( C /\ Vertical_Line ( w ) ) ) & upper_bound ( proj2 .: ( C /\ Vertical_Line ( w ) ) ) <= upper_bound ( proj2 .: ( C /\ [ w , w ] ) ) ;
for r be Real st 0 < r ex n be Nat st for m be Nat st n <= m holds |. S . m - pp .| < r
i * fN - fN = i * fN - ( i * yN ) .= i * ( fN - ( i * yN ) ) .= i * ( fN - ( i * yN ) ) ;
consider f being Function such that dom f = 2 -tuples_on X & for Y being set st Y in 2 -tuples_on X holds f . Y = F ( Y ) ;
consider g1 , g2 being element such that g1 in [#] Y and g2 in union C and g = [ g1 , g2 ] and g1 in ( the carrier of X ) and g2 in ( the carrier of Y ) ;
func d |-count n -> Nat means : - : d |^ n divides n & it |^ ( n + 1 ) divides n & d |^ ( n + 1 ) divides n & it divides n ;
f\in . [ 0 , t ] = f . [ 0 , t ] .= ( - P ) . ( 2 * x ) .= ( - P ) . ( x `1 ) .= a ;
t = h . D or t = h . B or t = h . C or t = h . E or t = h . F or t = h . J ;
consider m1 be Nat such that for n st n >= m1 holds dist ( ( seq . n ) , ( seq . n ) ) < 1 / ( n + 1 ) ;
( ( q `1 ) / |. q .| ) ^2 <= ( ( q `1 ) / |. q .| ) ^2 & ( ( q `2 ) / |. q .| ) ^2 <= ( ( q `1 ) / |. q .| ) ^2 ;
h0 . ( i + 1 + 1 ) = h21 . ( i + 1 + 1 - len h11 + 2 - 1 ) .= h11 . ( 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 , x2 ] in the carrier' of S ;
for L being RelStr , a , b being Element of L holds a <= { b } iff a <= b & a >= b & a >= b
||. h1 .|| . n = ||. h1 . n .|| .= |. h . n .| .= |. h . n .| .= |. h . n .| .= |. h . n .| .= ||. h . n .|| .= ||. h . n .|| .= ||. h . n .|| ;
( ( - ( #Z n ) ) * ( #Z n ) ) . x = f . x - ( #Z n ) . x .= ( - ( #Z n ) ) . x .= ( - ( #Z n ) ) * ( #Z n ) ) . x ;
attr r = F .: ( p , q ) means : : : len r = min ( len p , len q ) & for i being Nat st i in dom r holds r . i = F . ( p . i ) ;
( rM / 2 ) ^2 + ( rM / 2 ) ^2 + ( rM / 2 ) ^2 <= ( r ^2 ) ^2 + ( r ^2 ) ^2 + ( r ^2 ) ^2 ;
for i being Nat , M being Matrix of n , K st i in Seg n holds Det M = Sum ( ( the carrier of K ) | ( i , j ) )
then a <> 0. R & a " * ( a * v ) = 1 / a * v & a " * ( a * v ) = 1 / a * v & a * v = 1 / a * v ;
p . ( j - 1 ) * ( q *' r ) . ( i + 1 - j ) = Sum ( p . ( j - 1 ) * r3 ) .= Sum ( p ) - Sum ( q ) ;
deffunc F ( Nat ) = L . 1 + ( ( R /* ( h ^\ n ) ) * ( h ^\ n ) " ) . $1 & ( ( R /* ( h ^\ n ) ) " ) . $1 = L . ( h . ( h ^\ n ) " ) ;
assume that the carrier of H1 = f .: the carrier of H1 and the carrier of H2 = f .: the carrier of H2 and the carrier of H1 = the carrier of H2 and the carrier of H1 = the carrier of H2 and the carrier of H1 = the carrier of H2 and the carrier of H1 = the carrier of H2 ;
Args ( o , Free ( S , X ) ) = ( ( the Sorts of Free ( S , X ) ) * the Arity of S ) . o .= ( the Sorts of Free ( S , X ) ) . o ;
H1 = n + 1 -H .= n + 1 -H .= n + 1 -H .= n + 1 -H .= n -H .= n -H .= n -H .= n -H .= n -H ;
( O = 0 & O = 0 & O = 1 & O = 1 & O = 1 or O = 1 ) & O = { O , O } & O = { O , O } ;
F1 .: ( dom F1 /\ dom F2 ) = F1 .: ( 1 / 2 ) .= F1 .: ( 1 / 2 ) .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } ;
attr b <> 0 & d <> 0 & b <> d & ( a - b ) / ( d - b ) = ( - ( e - b ) ) / ( d - b ) ;
dom ( ( f +* g ) | D ) = dom ( f +* g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D ;
for i be set st i in dom g ex u , v be Element of L st g /. i = u * a & u in A & v in B & u in C
g `2 * P `2 * g `2 " = g `2 * ( g `2 * P `2 ) * g `2 .= g `2 * ( g `2 * P `2 ) * ( g `2 * P `2 ) .= g `2 * ( g `2 * P `2 ) ;
consider i , s1 such that f . i = s1 and not ( i in dom s1 & f . ( i + 1 ) <> s1 . ( i + 1 ) ) and not ( i in dom s1 & s1 . ( i + 1 ) <> s1 . ( i + 1 ) ) ;
h5 | ]. a , b .] = ( g | Z ) | ]. a , b .] .= g | ]. a , b .] .= g | ]. a , b .] .= g | ]. a , b .] ;
[ s1 , t1 ] , [ s2 , t2 ] are_connected & [ s2 , t2 ] , [ t2 , t2 ] ] , [ s2 , t2 ] are_connected & [ s2 , t2 ] in Indices G & [ s2 , t2 ] in Indices G & [ s2 , t2 ] in Indices G & [ s2 , t2 ] in Indices G & [ s2 , t2 ] in Indices G & [ s2 , t2 ] in Indices G & [ s2 , t2 ] in Indices G & [ s2 , t2 ] in Indices G & [ s2 , t2 ] in Indices G & [ s2 , t2 ] in Indices G & [ s2 ,
then H is negative means : then H is not negative & H is not negative & H is not negative -g\mathopen \neq H & H is not negative -gOne ;
attr f1 is total means : Def8 : 1 / ( f1 + f2 ) is total & ( f1 + f2 ) . c = f1 . c * f2 . c + f2 . c * ( f1 . c ) " ;
z1 in W2 -Seg ( z2 ) or z1 = z2 & not z1 in W2 & ( z1 in W2 & z2 in W1 implies z1 in W2 & z2 in W1 & z1 in W2 & z2 in W1 & z1 in W2 ) implies z1 in W2
p = 1 * p .= a " * a * p .= a " * ( b * q ) .= a " * ( b * q ) .= a " * ( b * q ) .= a " * ( b * q ) ;
for seq1 be sequence of REAL for K be Real st for n be Nat holds seq1 . n <= K holds upper_bound ( rng seq1 ) <= upper_bound ( rng seq2 ) & upper_bound ( rng seq1 ) <= upper_bound ( rng seq1 )
x0 in x0 or x0 in L~ go \/ L~ pion1 or p in L~ pion1 or p in L~ pion1 & p in L~ pion1 or p in L~ pion1 & p in L~ pion1 or p in L~ pion1 & p in L~ pion1 or p in L~ pion1 & p in L~ pion1 or p in L~ pion1 & p in L~ pion1 & p in L~ pion1 or p in L~ pion1 & p in L~ pion1 ;
||. f . ( g . ( k + 1 ) ) - g . ( g . k ) .|| <= ||. g . 1 - g . 0 .|| * ( K to_power k ) ;
assume h = ( ( B .--> B ' +* ( C .--> D ) +* ( E .--> F ) +* ( F .--> J ) ) +* ( M .--> N ) +* ( M .--> N ) +* ( M .--> N ) +* ( M .--> N ) +* ( M .--> N ) ) +* ( M .--> N ) +* ( M .--> A ) ;
|. ( ( ( ( ( ( H . n ) || A ) || A ) ) . k - ( ( ( ( H . n ) || A ) || A ) . k ) ) .| <= e * ( b-a ) ;
( ( the Sorts of A ) . ( i ) ) . e = [ the ; of v , the carrier of ( the carrier of ( X . i ) ) , the carrier of ( ( the Sorts of A ) . i ) , the carrier of ( ( the Sorts of A ) . i ) ] ;
{ x1 , x1 , x1 , x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , 7 , 8 } = { x1 , x2 , x3 } \/ { x4 , x5 , 7 } .= { x1 } \/ { x2 , x3 } .= { x1 , x2 } \/ { x2 , x3 } ;
assume that A = [. 0 , 2 * PI .] and integral ( cos , A ) = 0 and integral ( cos , A ) = 0 and integral ( cos , A ) = 0 and integral ( cos , A ) = 0 ;
p `2 is Permutation of dom ( f1 /. i ) & p `2 " = ( ( Sgm Y ) /. i ) " * ( p /. i ) " * ( Sgm X ) /. i ;
for x , y st x in A & y in A holds |. 1 / ( f . x ) - 1 / ( f . y ) .| <= 1 * |. f . x - f . y .|
p2 `2 = |. q2 .| * ( ( q2 `2 / |. q2 .| - sn ) / ( 1 - sn ) ) .= ( q2 `2 / |. q2 .| - sn ) / ( 1 - sn ) .= ( q2 `2 / |. q2 .| - sn ) / ( 1 - sn ) ;
for f be PartFunc of the carrier of CNS , REAL st dom f is compact & f is_continuous_on dom f & f is_continuous_on dom f & f is_continuous_on dom f holds rng ( f | X ) is compact & rng ( f | X ) c= dom f
assume for x being Element of Y st x in EqClass ( z , CompF ( B , G ) ) holds ( Ex ( a , A ) ) . x = TRUE ;
consider FF such that dom FF = n1 and for k be Nat st k in n1 holds Q [ k , FF . k ] and for k be Nat st k in n1 holds Q [ k , FF . k ] holds Q [ k , FF . k ] ;
ex u , u1 st u <> u1 & u , u1 / ( 2 , v ) / ( 2 , v1 ) / ( 2 , v1 ) / ( 2 , v1 ) / ( 2 , v1 ) = ( u + u1 ) / ( 2 , v1 ) & u , u1 / ( 2 , v1 ) / ( 2 , v1 ) / ( 2 , v2 ) = ( u + u1 ) / ( 2 , v1 )
for G being Group , A , B being non empty Subset of G , N being normal Subgroup of G holds ( N ` A ) * ( N ` B ) = N ` A * ( N ` B )
for s be Real st s in dom F holds F . s = integral ( R / ( f - g ) ) - Integral ( M , ( f + g ) / ( f - g ) / ( f - g ) ) . x
width AutMt ( f1 , b1 , b2 ) = len b2 .= width b2 .= width ( ( f2 , b1 , b2 ) * ( i , j ) ) .= width ( ( f2 , b1 , b2 ) * ( i , j ) ) ;
f | ]. - PI / 2 , PI / 2 .[ = f & dom f " ]. - PI / 2 , PI / 2 .[ = ]. - 1 , 1 .[ & for x st x in ]. - 1 , 1 .[ holds f . x = 1 / 2 * x + 1 / 2 * x + 1 / 2 * x + 1 / 2 * x + 1 / 2 * x + 1 / 2 * x + 1 / 2 * x + 1 / 2 * x + 1 / 2 * x + 1 / 2 * x + 1 / 2 * x + 1 / 2 * x + 1 /
assume that X is closed w.r.t. ex a st a in X & a c= X and y in a |^ ( n + 1 ) and { x } \/ { y } in a ;
Z = dom ( ( ( #Z 2 ) * ( arctan ) ) `| Z ) /\ dom ( ( #Z 2 ) * ( arctan ) + ( #Z 2 ) * ( arctan ) ) .= dom ( ( #Z 2 ) * ( arctan ) + ( #Z 2 ) * ( arctan ) ) ;
func TAUT ( V ) -> Subset of V means : - l . 1 = { l . k : 1 <= k & k <= len l & l . k in V } ;
for L being non empty TopSpace , N being net of L , M being net of N , c being Point of L st c is Point of M & M is in the carrier of N holds c is Point of M
for s being Element of NAT holds ( ( ( ( id C\mathop ) . v ) + ( ( id C\mathop { C\mathop . v } ) + ( C\mathop { C\mathop { v } ) ) ) . s ) . s = ( ( ( ( id C\mathop { v } ) + ( C\mathop { C\mathop { v } ) ) + ( C\mathop { v } ) ) . s ) . s
then z /. 1 = N-min L~ z & ( N-min L~ z ) .. z < ( N-min L~ z ) .. z & ( N-min L~ z ) .. z < ( ( N-min L~ z ) .. z ) .. z ;
len ( p ^ <* ( 0 qua Real ) *> ) = len p + len <* ( 0 qua Real ) *> .= len p + 1 .= len p + 1 .= len <* ( 0 qua Real ) *> .= len p + 1 ;
assume that Z c= dom ( - ( ln * f ) ) and for x st x in Z holds f . x = x & f . x > 0 and for x st x in Z holds f . x > - 1 & f . x < 1 ;
for R being add-associative right_zeroed right_complementable right complementable non empty doubleLoopStr , I being Subset of R , J being Subset of R , I being Ideal of R , J being Subset of R holds ( I + J ) *' ( I /\ J ) c= I /\ J
consider f being Function of [: B1 , B2 :] , B12 such that for x being Element of B1 , y being Element of B2 holds f . x = F ( x ) and f . y = F ( y ) ;
dom ( x2 + y2 ) = Seg len x .= Seg len ( x2 + z2 ) .= Seg len ( x2 + z2 ) .= Seg len ( x + y ) .= dom ( x + y ) .= dom ( x + y ) ;
for S being Functor of C , B for c being Object of C holds card S . ( id c ) = id ( ( Obj S ) . ( id c ) ) & ( Obj S ) . ( id c ) = id ( ( Obj S ) . ( id c ) )
ex a st a = a2 & a in dom f9 /\ f5 & for x st x in dom f9 holds \rrangle in \rrangle & \rrangle in \mathop { \rm seq } ( a , b ) & card { a } = card { a }
a in Free ( ( H2 / ( x. 4 , x. k ) ) '&' ( H2 / ( x. 4 , x. k ) ) ) & ( ( H1 / ( x. 4 , x. k ) ) '&' ( H2 / ( x. 4 , x. k ) ) ) '&' ( H2 / ( x. 4 , x. k ) ) ) in Free ( x. 4 , x. k ) ;
for C1 , C2 being is ] & f is stable Function of C1 , C2 for g being Function of C2 , C2 st ( for x being set st x in C1 holds f . x = g . x ) holds f = g
( W-min ( L~ go \/ L~ pion1 ) ) `1 = W-bound ( L~ go \/ L~ pion1 ) & ( W-min ( L~ go \/ L~ pion1 ) ) `1 = W-bound ( L~ go \/ L~ pion1 ) & ( W-min ( L~ go \/ L~ pion1 ) ) `1 = W-bound ( L~ go \/ L~ pion1 ) & ( W-min ( L~ go \/ L~ pion1 ) ) `1 = W-bound ( L~ go \/ L~ pion1 ) ;
assume that u = <* x0 , y0 , z0 *> and f is_Assume u is_\/ pdiff1 ( f , 1 ) and SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) is_differentiable_in z0 and SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) is_differentiable_in z0 ;
then ( t . {} ) `1 in Vars & ex x being Element of Vars st x = ( t . {} ) `1 & t . {} = x & ( t . {} ) `2 = s & ( t . {} ) `2 = s ;
Valid ( p '&' p , J ) . v = Valid ( p , J ) . v '&' Valid ( p , J ) . v .= Valid ( p , J ) . v '&' Valid ( q , J ) . v .= Valid ( p , J ) . v ;
assume for x , y being Element of S st x <= y for a , b being Element of T st a = f . x & b = f . y holds a >= b ;
func Class R -> Subset-Family of R means : R : for A being Subset of R holds A in it iff ex a being Element of R st A = Class ( a , a ) & for A being Subset of R st A in Class ( R , a ) holds it . A = a ;
defpred P [ Nat ] means ( ( ( \HM { the } \HM { vertices } ) \/ ( \HM { the } \HM { vertices } \HM { of G ) ) \ { v } ) c= G } implies ( ( the non empty Subset of G ) \ { v } ) is non empty
assume that dim ( W1 ) = 0 and dim ( U1 ) = 0 and for v being Element of V st v in W1 holds ( v in W1 & ( v in W2 implies v in W1 ) & ( v in W2 ) ) & ( v in W1 implies v in W2 ) & ( v in W1 implies v in W2 ) & ( v in W2 implies v in W1 ) ) ;
mamas ( m ) . t = ( m . t ) `1 .= [ [ m . t , the carrier of C ] `1 , the carrier of C ] `1 .= [ m . t , the carrier of C ] `2 .= m ;
d11 = ( x9 ^ d22 ) . ( y9 , d22 ) .= f . ( y9 , d22 ) .= f . ( y9 , d22 ) .= ( f | ( i + 1 ) ) . d22 .= ( f | ( i + 1 ) ) . d22 .= ( f | ( i + 1 ) ) . d22 .= ( f | ( i + 1 ) ) . d22 ;
consider g such that x = g and dom g = dom fx0 and for x being element st x in dom fx0 holds g . x in fx0 and g . x in fx0 ;
x + 0. F_Complex |^ ( len x ) = x + len x |-> 0. F_Complex .= ( x + len x ) |-> 0. F_Complex .= ( x + len x |-> 0. F_Complex ) .= x ` .= x ` ;
( k -' kk + 1 ) in dom ( f | ( ( k -' 1 ) -' 1 ) ) & ( f | ( ( k -' 1 ) -' 1 ) ) . ( k -' 1 ) = ( f | ( k -' 1 ) ) . ( k -' 1 ) ;
assume that P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P = { p1 , p2 } and P = { p2 , p1 } and P /\ P2 = { p1 , p2 } and P /\ P2 = { p2 , p1 } and P /\ P2 = { p2 , p1 } and P /\ P2 = { p2 , p1 } and P /\ P2 = { p2 , p1 } and P /\ P2 = { p2 , p1 } ;
reconsider a1 = a , b1 = b , b1 = b , c1 = p `1 , c1 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c1 = p `1 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c1 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , _ { p `1 , c1 } } ;
reconsider being ttb1f = G1 . ( t . b ) * F1 . f as Morphism of ( G1 * F1 ) . a , ( G1 * F2 ) . b * F2 . b ;
LSeg ( f , i + i1 -' 1 ) = LSeg ( f /. ( i + i1 -' 1 ) , f /. ( i + i1 -' 1 ) ) .= LSeg ( f , i + i1 -' 1 ) ;
Integral ( M , P . m ) | dom ( P . n -P . m ) <= Integral ( M , P . n -P . m ) | dom ( P . n -P . m ) ;
assume that dom f1 = dom f2 and for x , y being element st [ x , y ] in dom f1 & [ y , x ] in dom f2 holds f1 . ( x , y ) = f2 . ( x , y ) ;
consider v such that v = y and dist ( u , v ) < min ( ( G * ( i , 1 ) `1 ) - ( G * ( i + 1 , 1 ) `1 ) , ( G * ( i + 1 , 1 ) `2 - ( G * ( i + 1 , 1 ) `2 ) ) / 2 ) ;
for G being Group , H being Subgroup of G , a being Element of H st a = b holds for i being Integer , b being Integer st a |^ i = b holds for i being Integer holds 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 = { p9 where p9 is Point of TOP-REAL 2 : P [ p9 ] } , K1 = { p where p is Point of TOP-REAL 2 : P [ p ] } , K1 = { p where p is Point of TOP-REAL 2 : P [ p ] } , K1 = { p : p `2 <= 0 } , K1 = { p : p <> 0. TOP-REAL 2 } ;
( ( N-bound C ) - ( S-bound C ) ) / ( 2 |^ m ) <= ( ( N-bound C ) - ( S-bound C ) ) / ( 2 |^ m ) & ( ( S-bound C ) - ( S-bound C ) ) / ( 2 |^ m + 1 ) <= ( ( N-bound C ) - ( S-bound C ) ) / ( 2 |^ m + 1 ) ;
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 ) + len <* [ 2 , 0 ] *> .= len @ ( @ p ) + len @ ( @ q ) .= len @ ( @ p ) + 1 ;
v / ( x. 3 , m1 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 ) = m3 ;
consider r being Element of M such that M , v2 / ( x. 3 , m ) / ( x. 4 , m ) / ( x. 4 , n ) / ( x. 4 , n ) / ( x. 4 , n ) / ( x. 4 , n ) / ( x. 4 , n ) / ( x. 4 , n ) / ( x. 4 , n ) / ( x. 4 , n ) / ( x. 4 , n ) ;
func w1 \ w2 -> Element of Union ( G , R6 ) means : such that for w1 being Element of Union ( G , R6 ) holds it . ( w1 , w2 ) = ( ( ( ( the - Kin G ) | ( i , w ) ) | ( i , w ) ) ) . ( w1 , w2 ) ;
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= s1 . b2 .= s2 . b2 .= Exec ( n2 , s2 ) . b2 .= s2 . b2 .= Exec ( n2 , s2 ) . b2 .= s2 . b2 .= s . b2 ;
for n , k be Nat holds 0 <= ( Partial_Sums |. seq .| ) . ( n + k ) - Partial_Sums ( |. seq .| ) . ( n + k ) & Partial_Sums ( |. seq .| ) . n <= Partial_Sums ( |. seq .| ) . ( n + k )
set F = S \! \mathop { {} } ;
( Partial_Sums ( seq ) ) . ( K + 1 ) + 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 closed -> closed Subset of TOP-REAL 2 equals ( the carrier of rectangle ( a , b , c , d ) ) \/ ( the carrier of rectangle ( a , b , c , d ) ) \/ ( the carrier of rectangle ( a , b , c , d ) ) ;
a * b ^2 + ( a * c ^2 + b * a ^2 ) + ( b * c ^2 + c * a ^2 ) + ( c * a ^2 + b * c ^2 ) >= 6 * a * b * c * c * a * b * c ;
v / ( x1 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m1 ) ;
+ ( Q ^ <* x *> , M ) = ( + ( Q , M ) +* ( ( L , M ) --> ( ( L , M ) --> FALSE ) ) +* ( ( card { x } --> FALSE ) --> FALSE ) ) .= ( ( ( L , M ) --> FALSE ) +* ( ( L , M ) --> FALSE ) ) +* ( ( L , M ) --> FALSE ) ) ;
Sum ( F ) = r |^ n1 * Sum ( C-13 ) .= 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 ) `2 = ( GoB f ) * ( 1 , 1 ) `2 ;
defpred X [ Element of NAT ] means ( Partial_Sums s ) . $1 = ( a * ( $1 + 1 ) ) * ( $1 + 1 ) + b * ( $1 + 1 ) * ( $1 + 1 ) * ( $1 + 1 ) ) + b * ( $1 + 1 ) * ( $1 + 1 ) ;
( the_arity_of g ) . x = ( the Arity of S ) . g .= ( [ ( the Arity of S ) . g , ( the Arity of S ) . g ] ) `1 .= ( the Arity of S ) . g .= ( the Arity of S ) . g .= ( the Arity of S ) . g ;
( [: X , Y :] |^ Z ) tolerates [: X , Y :] & card ( ( X , Y ) |^ Z ) = card [: X , Y :] & card ( ( X , Y ) |^ 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 + 1 ) holds b = N . ( s . n ) \ G . s ;
E , f |= All ( x. 2 , ( x. 2 ) / ( x. 0 , ( x. 2 ) ) ) '&' ( x. 2 , ( x. 1 ) / ( x. 2 , ( x. 1 ) ) ) '&' ( x. 2 , ( x. 1 ) / ( x. 0 , ( x. 1 ) ) ) ) ;
ex R2 being 1-sorted st R2 = ( p | n-11 ) . i & ( the carrier of p | n-11 ) . i = the carrier of R2 & ( the carrier of p | n-11 ) . i = the carrier of R2 . i & ( the carrier of p | n-11 ) . i = the carrier of R2 . i ;
[. a , b + 1 / ( k + 1 ) .[ is Element of the carrier of the carrier of X & ( the partial of f ) . ( k + 1 ) is Element of the carrier of X & ( the partial of f ) . ( k + 1 ) is Element of the carrier of X & ( the carrier of f ) . ( k + 1 ) is Element of the carrier of X ;
Comput ( P , s , 2 + 1 ) = Exec ( P . 2 , Comput ( P , s , 2 ) ) .= Exec ( a3 := ( s . a ) , Comput ( P , s , 2 ) ) .= Exec ( a3 := ( s . a ) , Comput ( P , s , 2 ) ) ;
card ( h1 ) . k = power ( F_Complex ) . ( - 1_ F_Complex , k ) * Sum u .= ( ( - 1. F_Complex ) * ( - 1_ F_Complex ) ) * u .= ( ( - 1. F_Complex ) * ( - 1_ F_Complex ) ) * u .= ( ( - ( 1. F_Complex ) ) * u ) . k ;
( f / g ) /. c = f /. c * ( g /. c ) " .= f /. c * ( 1 / g ) /. c .= ( f (#) ( 1 / g ) ) /. c .= ( f (#) ( 1 / g ) ) /. c ;
len ( C - the carrier of ( ( len C ) - 1 ) ) = len ( C - the carrier of ( ( len C ) - 1 ) ) .= len ( C - the carrier of ( ( len C ) - 1 ) ) .= len ( C - the carrier of ( ( len C ) - 1 ) ) .= len ( C - the carrier of ( ( len C ) - 1 ) ) ;
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom f /\ X .= dom ( f | X ) /\ X .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) ;
defpred P [ Nat ] means for n holds 2 * Fib ( n + $1 ) = Fib ( n ) * Fib ( n + $1 ) + ( 5 * Fib ( n + $1 ) ) * Fib ( n + $1 ) + ( 5 * Fib ( n + $1 ) ) * Fib ( n + $1 ) ;
consider f being Function of INT , INT such that f = f `1 and f is onto and for n being Nat st n < k + 1 holds f " { f . n } = { n } and f " { f . n } = { n } ;
consider vs being Function of S , BOOLEAN such that vs = chi ( A \/ B , S ) and ( for A being Element of S holds E . ( A \/ B ) = Prob . A ) and ( E . ( A \/ B ) ) = Prob . ( A \/ B ) and ( E . ( A \/ B ) ) = Prob . ( A \/ B ) ;
consider y being Element of Y ( ) such that a = "\/" ( { F ( x , y ) where x is Element of X ( ) : P [ x ] } , L ( ) ) and Q [ y ] ;
assume that A c= Z and Z = dom f and f = ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( f1 + #Z 2 ) ) ) ) / ( ( #Z 2 ) * ( f1 + #Z 2 ) ) and for x st x in Z holds ( ( #Z 2 ) * ( f1 + #Z 2 ) ) / ( ( #Z 2 ) * ( f1 + #Z 2 ) ) ^2 ) = f . x ;
( f /. i ) `2 = ( ( GoB f ) * ( 1 , j2 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 ) ) `2 ;
dom Shift ( Seq q2 , len Seq q1 ) = { j + len Seq q1 where j is Nat : j in dom Seq q1 & len Seq q2 = len Seq q2 & len Seq q2 = len Seq q2 } .= dom Seq q2 \/ dom Seq q2 \/ dom Seq q2 .= dom Seq q2 \/ dom Seq q2 \/ dom Seq q2 \/ dom Seq q2 ;
consider G1 , G2 , G3 being Element of V such that G1 <= G2 and G2 <= G2 and f is Morphism of G1 , G2 and g is Morphism of G2 , G3 and for x being Element of G1 , y being Element of G2 st x in G1 & y in G2 holds f . x = G1 * ( x , y ) ;
func - f -> PartFunc of C , V means : - it = dom f & for c st c in dom it holds it /. c = - f /. c & for c st c in dom it holds it /. c = - f /. c + f /. c ;
consider phi such that phi is increasing and for a st phi . a = a & {} <> a for v holds union ( L , v ) = a and for v holds union ( L , v ) = v and union ( L , v ) = v and L . a , v |= H ;
consider i1 , j1 such that [ i1 , j1 ] in Indices GoB f and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) ;
consider i , n such that n <> 0 and sqrt p = ( i - n ) / ( i + 1 ) and for i1 being Nat , i2 being Integer st ( i1 <> 0 & i2 <> 0 or [ i1 , i2 ] in Indices G ) holds n <= i2 & i2 <= n ;
assume that not 0 in Z and Z c= dom ( ( arccot * ( 1 / 2 ) ) ) and for x st x in Z holds ( ( 1 / 2 ) (#) ( ( arccot * ( 1 / 2 ) ) ) `| Z ) . x > - 1 & ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) * ( 1 / 2 ) ) ) `| Z ) . x ;
cell ( G1 , i1 -' 1 , ( 2 |^ m -' 1 ) ) \ ( ( Y -' 1 ) * ( ( Y -' 1 ) + ( Y -' 1 ) ) ) c= BDD L~ f & ( ( Y -' 1 ) + ( Y -' 1 ) ) \ ( ( Y -' 1 ) + ( Y -' 1 ) ) \ ( ( Y -' 1 ) + ( Y -' 1 ) ) ) c= BDD L~ f
ex Q1 being open Subset of X st s = Q1 & ex Q being Subset-Family of [: Y , X :] st Q c= F & Q is finite & [#] ( Y | Q ) = union Q & [#] ( Y | Q ) = union Q & [#] ( Y | Q ) = union Q & [#] ( Y | Q ) = union Q ;
gcd ( A-27 , ( the carrier of A ) , s1 , s2 , s2 , Amp ) = 1. R & gcd ( A-27 , ( the carrier of A ) , s2 , s2 , Amp ) = 1. R & gcd ( A-27 , ( the carrier of A ) , s2 , Amp ) = 1. R ;
R8 = ( ( j , ( j + 1 ) ) ) . ( m2 + 1 ) .= ( ( j + 1 ) + 1 ) . ( m2 + 1 ) .= ( ( j + 1 ) + 1 ) . ( m2 + 1 ) .= [ 3 , ( j + 1 ) + 1 ] .= [ 3 , ( j + 1 ) + 1 ] ;
CurInstr ( P-6 , Comput ( P-6 , s2 , m1 + m3 ) ) = CurInstr ( P3 , Comput ( P3 , s3 , m3 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m3 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m3 ) ) .= halt SCMPDS .= halt SCMPDS .= halt SCMPDS ;
P1 /\ P2 = ( { p1 } \/ LSeg ( p1 , p2 ) /\ LSeg ( p11 , p2 ) ) \/ ( LSeg ( p2 , p2 ) /\ LSeg ( p2 , p2 ) ) \/ ( LSeg ( p2 , p2 ) /\ LSeg ( p2 , p2 ) ) \/ { p2 } ) .= { p2 } \/ { p2 } \/ { p2 } ;
func the still of f -> Subset of the Sorts of Al means : - ex a , b st a in it & b in it & a in the carrier of f & p in the carrier of f & a in the carrier of f & b in the carrier of f & p in the carrier of f & a in the carrier of f & b in the carrier of f ;
for a , b being Element of F_Complex for f being Polynomial of F_Complex st |. a .| > |. b .| for a being Polynomial of F_Complex st a >= 1 & f is \cap L~ f holds a * ( - b ) is \/ { f . ( - b ) } is \/ { f . ( - b ) }
defpred P [ Nat ] means 1 <= $1 & $1 <= len g implies for i , j st [ i , j ] in Indices G & G * ( i , j ) = g . ( $1 + 1 ) & 1 <= j & j <= width G & j <= width G & G * ( i , j ) = G * ( i , j ) holds j < i ;
assume that C1 , C2 are_`2 and for f being State of C1 , g being State of C2 , s1 , s2 being State of C2 st s1 = s2 & s2 = s2 holds s1 is stable iff for f being Function of C1 , C2 st f = s2 holds s2 is stable iff f is stable & f is stable & f is stable & f is stable & f is stable & f is stable & f is stable & f is stable ;
( ||. f .|| | X ) . c = ||. f .|| . c .= ||. f /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| ;
|. q .| ^2 = ( q `1 ) ^2 + ( q `2 ) ^2 & 0 + ( q `1 ) ^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 [: T , T :] st F is open & not {} in F & for A , B being Subset of [: T , T :] st A in F & B in F & A <> B & B <> {} holds card F = card ( A \/ B ) & card F = card ( A \/ B )
assume that len F >= 1 and len F = k + 1 and len F = len G and for k st k in dom F holds H . k = g . k and for k st k in dom F holds H . k = g . k * G . k and for k st k in dom F holds H . k = g . k * F . k ;
i |^ ( ( mod n ) - i |^ s ) = i |^ ( s + k ) - i |^ s .= i |^ ( s + k ) - i |^ s .= i |^ ( s + k ) - i |^ s .= i |^ ( s + k ) - i |^ s * 1 .= i |^ ( s + k ) - i |^ s * 1 ;
consider q being oriented oriented Chain of G such that r = q and q <> {} and ( F . ( q . 1 ) = v1 ) and ( F . ( q . len q ) = v2 ) & ( F . ( q . 1 ) = v2 ) & ( F . ( len q ) = v2 ) & ( F . ( len q ) = v2 ) ;
defpred P [ Element of NAT ] means $1 <= len ( I ) implies ( ( g ) ^ ( I , Z ) ) . $1 = ( ( ( g , Z ) ^ ( I , Z ) ) . ( len ( G , Z ) ) ) . ( len ( G , Z ) + $1 ) ;
for A , B being square Matrix of n , REAL for x being Element of REAL n holds len ( A * B ) = len A & width ( A * B ) = width B & width ( A * B ) = width A & width ( A * B ) = width B & width ( A * B ) = width A
consider s being FinSequence of the carrier of R such that Sum s = u and for i being Element of NAT st 1 <= i & i <= len s ex a , b being Element of R st s . i = a & s . i = b & a in I & b in J & s . i in J ;
func |( x , y )| -> Element of COMPLEX equals |( ( Re x ) , ( Re y ) )| - ( ( Re y ) * |( x , y )| ) + ( ( Im y ) * |( x , y )| ) + ( ( Im y ) * |( x , y )| ) ;
consider g9 being FinSequence of F such that for x being Element of NAT st x0 in A & ( for y be Element of NAT st y in A holds g . y = A . ( x , y ) ) & ( for y be Element of NAT st y in A holds g . y = F . ( y , x ) ) & ( for y be Element of NAT st y in A holds g . y = F ( y ) ) holds < g9 ;
then n1 >= len p1 & n2 >= len p1 implies crossover ( p1 , p2 , n1 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n2 , n3 , n2 , n3 , n3 , n2 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n2 , n3 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n2 , n3 , n3 ,
( q `1 * a ) ^2 <= ( q `1 ) ^2 & - ( q `1 ) * a <= ( q `1 ) ^2 & ( q `1 ) ^2 + ( q `2 ) ^2 * a <= ( q `1 ) ^2 & ( q `1 ) ^2 + ( q `2 ) ^2 <= ( q `1 ) ^2 + ( q `2 ) ^2 ;
( F . ( len pp ) ) . ( len pp ) = ( F . ( p . ( len pp ) ) ) . ( len pp + 1 ) .= ( F . ( len pp + 1 ) ) . ( len pp + 1 ) .= ( F . ( len pp + 1 ) ) . ( len pp + 1 ) .= ( F . ( len pp + 1 ) ) . ( len pp + 1 ) .= ( F . ( len pp + 1 ) ) . ( len pp + 1 ) ;
consider k1 being Nat such that k1 + k = 1 and a := k = ( <* a := intloc 0 *> ^ ( ( intloc 0 ) --> 1 ) ) ^ ( ( intloc 0 ) --> 1 ) ^ <* halt SCM+FSA *> ^ ( ( intloc 0 ) --> 1 ) ^ <* halt SCM+FSA *> ) ;
consider B9 being Subset of B1 , y9 being Function of B1 , B2 such that [: B1 , B2 :] is finite and D1 = \frac ( 0 , B ) . ( y , y1 ) and for x being set st x in B1 holds [ x , y ] in the carrier of B1 and [ y , x ] in the carrier of B2 ;
v2 . b2 = ( ( curry F2 ) * ( ( curry F2 ) * ( ( the > id B ) . b2 ) ) ) . b2 .= ( ( ( curry F2 ) * ( ( ( ( ( ( ( ( the carrier of B ) ) * ( ( ( the carrier of B ) * ( ( the carrier of A ) * ( the carrier of B ) ) ) ) ) ) ) ) ) . b2 .= F2 . b2 ;
dom IExec ( I-35 , P , Initialize s ) = the carrier of SCMPDS .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) ;
ex d-32 be Real st d-32 > 0 & for h be Real st h <> 0 & |. h .| < d-32 & |. h .| < ( L + R1 ) * ( ( L + R1 ) * ( h + R1 ) ) & |. h .| " * ( ( L + R1 ) * ( h + R1 ) ) < e / ( ( L + R1 ) * ( h + R1 ) )
LSeg ( G * ( len G , 1 ) + |[ - 1 , 0 ]| , G * ( len G , 1 ) + |[ 1 , 0 ]| ) c= Int cell ( G , len G , 0 ) \/ { G * ( len G , 1 ) + |[ 1 , 0 ]| } ;
LSeg ( mid ( h , i1 , i2 ) , i ) = LSeg ( h /. ( i + i1 -' 1 ) , h /. ( i + i1 -' 1 ) ) .= LSeg ( h , i + i1 -' 1 ) .= LSeg ( h , i + i1 -' 1 ) .= LSeg ( h , i + i1 -' 1 ) .= LSeg ( h , i + i1 -' 1 ) ;
A = { q where q is Point of TOP-REAL 2 : LE p1 , q , P , p1 , p2 & LE p2 , q , P , p1 , p2 & LE q , p1 , P , p1 , p2 & LE q , p1 , P , p1 , p2 & LE q , p1 , P , p1 , p2 } ;
( ( - x ) .|. y ) = ( - ( 1 / ( ( - x ) * y ) ) ) * ( x .|. y ) .= ( - ( 1 / ( ( - x ) * y ) ) ) * ( x .|. y ) .= ( ( - ( 1 / ( ( - x ) * y ) ) * y ) ) * ( x .|. y ) .= ( ( - ( - 1 / ( - x ) ) ) * y ) ;
0 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) = ( p `2 ) ^2 / sqrt ( 1 + ( p `1 / p `2 ) ^2 ) .= ( p `2 ) ^2 / sqrt ( 1 + ( p `1 / p `2 ) ^2 ) ) * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) ;
( ( U - W ) * ( *> - p ) ) * ( *> - p ) = ( ( U - W ) * ( W - p ) ) * ( ( W - p ) * ( W - p ) ) ) .= ( ( U - W ) * ( W - p ) ) * ( ( W - p ) * ( W - p ) ) .= ( W - p ) * ( W - p ) ) * ( W - p ) ;
func Shift ( f , h ) -> PartFunc of REAL , REAL means : - : dom it = - h & for x st x in - h holds it . x = ( - h ) . x & for x st x in dom it holds it . x = ( - h ) . x + ( - h . x ) * ( - h . x ) ;
assume that 1 <= k and k + 1 <= len f and [ i , j ] in Indices G and [ i + 1 , j ] in Indices G and f /. k = G * ( i + 1 , j ) and f /. k = G * ( i , j ) and f /. k = G * ( i , j ) ;
assume that not y in Free H and x in Free H and ( not x in Free H ) and ( not x in Free H & y in Free H ) and ( not x in Free H & y in Free H ) and ( x in Free H ) or x in Free H ) and ( x in Free H ) and ( x in Free H ) or x in Free H ) ;
defpred P11 [ Element of NAT , Element of NAT ] means ( P [ $1 ] implies ( not P [ $1 ] ) & ( not P [ $1 ] ) & ( not P [ $1 ] implies $2 = p |^ $1 ) & ( not P [ $1 ] ) & ( not P [ $1 ] implies $2 = p |^ $1 ) & ( not P [ $1 ] ) & ( not P [ $1 ] implies $2 = p |^ $1 ) ) ;
func \sigma ( C ) -> non empty Subset-Family of X means : - C in it iff for A being Subset of X holds A in it iff for W being Subset of X , Z being Subset of X st W c= A \ A & Z in it holds C . W <= C . Z + C . Z ;
[#] ( ( dist ( P ) ) .: Q ) = ( dist ( P ) ) .: Q & lower_bound ( ( dist ( P ) ) .: Q ) = ( lower_bound ( ( dist ( P ) ) .: Q ) ) & lower_bound ( ( dist ( P ) ) .: Q ) = lower_bound ( ( dist ( P ) ) .: Q ) ;
rng ( F | ( [: S , S :] ) ) = {} or rng ( F | ( [: S , T :] ) ) = { 1 } or rng ( F | ( [: S , T :] ) ) = { 2 } or rng ( F | ( [: S , T :] ) ) = { 1 } ;
( f " ( rng f ) ) . i = f . i " . ( ( f . i ) " . ( f . i ) ) .= ( f . i ) " . ( f . i ) .= ( f . i ) " . ( f . i ) .= ( f . i ) " . ( f . i ) .= ( f . i ) " . ( f . i ) .= ( f . i ) " . ( f . i ) ;
consider P1 , P2 being non empty Subset of TOP-REAL 2 such that P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and for C being Subset of TOP-REAL 2 st C = { p1 , p2 } & C = { p2 } holds P1 \/ P2 = { p1 , p2 } and P2 is closed and P2 is closed and C = { p1 , p2 } ;
f . p2 = |[ ( p2 `1 ) ^2 / ( 1 + ( p2 `2 / p2 `1 ) ^2 ) , ( p2 `2 ) ^2 / ( 1 + ( p2 `2 / p2 `1 ) ^2 ) ]| .= |[ ( p2 `1 ) ^2 / ( 1 + ( p2 `2 / p2 `1 ) ^2 ) , ( p2 `2 ) ^2 / ( 1 + ( p2 `2 / p2 `1 ) ^2 ) ]| ;
( ( -\mathbin { a , X ) ) " . x = ( ( --st ( a , X ) qua Function ) " ) . x .= ( ( - a ) qua Function ) . x .= ( - a ) . x + ( - a ) . x .= ( - a ) . x + ( - a ) . x .= ( - a ) . x .= ( - a ) . x + ( - a ) . x .= ( - a ) . x .= ( - a ) . x .= ( - a ) . x .= ( - a ) . x .= ( - a ) . x .= ( - a ) . x .= ( - a ) . x .= ( - a ) . x .= ( - a ) . x .= ( - a ) . x .= ( - a ) . x .= ( - a ) . x .= ( - a ) . x .= ( ( - a ) . x .= ( - a ) . x .= ( - a ) . x .= ( - a ) . x .= ( - a ) . x .= ( - a ) . x .= ( - a ) . x .=
for T being non empty normal TopSpace , A , B being closed Subset of T , r being Real st A <> {} & A misses B for p being Point of T , r being Point of T , s being Point of ( in > G ) , r being Point of ( ( in > G ) ) . p , s being Point of ( ( in > G ) ) . p ) holds ( ( in > G ) . p )
for i , j st i + 1 in dom F for G1 , G2 being strict normal Subgroup of G st G1 = F . i & G2 = F . j & G2 = F . ( i + 1 ) & for k being Nat st k in dom G1 holds G1 . k is strict Subgroup of G1 . k & G2 is strict Subgroup of G2 . k holds G1 is strict Subgroup of G1
for x st x in Z holds ( ( ( #Z 2 ) * ( arctan - arccot ) ) `| Z ) . x = ( ( 2 * x ) / ( 1 + x ^2 ) ) * ( arctan - arccot ) . x / ( 1 + x ^2 )
synonym f is_right continuous means : \rm : x0 in dom ( f /* a ) & for a st a in dom f & for n st n in dom f & n <= m holds f /. n - f /. x0 in dom ( f | ]. x0 , x0 + r .[ ) & for n st n <= m holds f /. n - f /. x0 in { 0 } ;
then ( X1 , X2 ) is closed & ( Y1 misses Y2 or ex Y2 being non empty SubSpace of X st Y1 is open & Y2 is open & ( Y2 is closed or Y1 is closed ) & ( Y2 is closed ) & ( Y1 is closed implies Y2 is closed ) & ( Y1 is closed implies Y2 is closed ) & ( Y2 is closed implies Y2 is closed ) & ( Y1 is closed implies Y2 is closed ) ) ;
ex N being 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- ( 1 , f , u ) . x - SVF1 ( 1 , f , u ) . x0 ) + R . ( x - x0 )
( p2 `1 ) ^2 * sqrt ( 1 + ( p3 `1 / p3 `2 ) ^2 ) >= ( ( p3 `1 ) ^2 * ( p3 `1 ) ^2 + ( p3 `2 ) ^2 * ( p3 `2 ) ^2 ) ) * sqrt ( 1 + ( p3 `1 ) ^2 ) ;
( ( 1 / t1 ) (#) ( ||. f1 .|| ) ) . x = ( ( 1 / t1 ) * ( ||. g1 .|| ) ) . x & ( ( 1 / t2 ) (#) ( ||. f1 .|| ) ) . x = ( ( 1 / t ) * ( ||. g1 .|| ) ) . x & ( ( 1 / t ) (#) ( ||. g1 .|| ) ) . x = ( ( 1 / t ) * ( ||. g1 .|| ) ) . x ;
assume for x holds f . x = ( ( - sin * ( x + h ) ) (#) ( cos * ( x + h ) ) ) . ( x + h ) & x + h in dom ( ( - sin * ( x + h ) ) (#) ( cos * ( x + h ) ) ) & ( ( - sin * ( x + h ) ) (#) ( cos * ( x + h ) ) ) ) . x = ( 1 / ( sin * ( x + h ) ) * ( x + h ) ) / ( x + h ) ) / ( x + h ) ) ^2 / ( x + h ) ) ^2 ;
consider X-23 being Subset of Y , Y1 being Subset of X such that t = [: X-22 , Y1 :] and Y1 is open and ex Y1 being Subset of Y 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 & Y1 is open & Y1 is open & Y1 is open & Y1 is open ;
card S . n = card { [: d , Y :] + ( a * d ) + b where d is Element of GF ( p ) , Y is Element of GF ( p ) : [ d , Y ] in R } .= { d , Y } .= { d , Y } \/ { b } .= { d , Y } \/ { b } .= { d , b } ;
( W-bound D - W-bound D ) * ( i1 - W-bound D ) / 2 * ( i - 2 ) * ( i - 2 ) = ( W-bound D - W-bound D ) * ( i - 2 ) * ( i - 2 ) .= ( W-bound D - W-bound D ) * ( i - 2 ) * ( i - 2 ) .= ( W-bound D - 2 ) * ( i - 2 ) * ( i - 2 ) ;