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thesis ;
contradiction ;
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thesis ;
contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
contradiction ;
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contradiction ;
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contradiction ;
contradiction ;
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thesis ;
contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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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 A ;
V is open ;
y in N ;
x in T ;
m < n ;
m <= n ;
n > 1 ;
let r ;
t in I ;
n <= 4 ;
M is finite ;
let X ;
Y c= Z ;
A // M ;
let U ;
a in D ;
q in Y ;
let x ;
1 <= l ;
1 <= w ;
let G ;
y in N ;
f = {} ;
let x ;
x in Z ;
let x ;
F is one-to-one ;
e <> b ;
1 <= n ;
f is special ;
S misses C ;
t <= 1 ;
y divides m ;
P divides M ;
let Z ;
let x ;
y c= x ;
let X ;
let C ;
x _|_ p ;
o is monotone ;
let X ;
A = B ;
1 < i ;
let x ;
let u ;
k <> 0 ;
let p ;
0 < r ;
let n ;
let y ;
f is onto ;
x < 1 ;
G c= F ;
a is_>=_than X ;
T is continuous ;
d <= a ;
p <= r ;
t < s ;
p <= t ;
t < s ;
let r ;
D <= E ;
assume e > 0 ;
assume 0 < g ;
p in P ;
x in Z ;
Y in Y ;
assume 0 < g ;
not c in Y ;
not v in L ;
2 in z `1 ;
assume f = g ;
N c= b ` ;
assume i < k ;
assume u = v ;
I = J ;
B = b `1 ;
assume e in F ;
assume p > 0 ;
assume x in D ;
let i be element ;
assume F is onto ;
assume n <> 0 ;
let x be element ;
set k = z ;
assume o = x ;
assume b < a ;
assume x in A ;
a `1 <= b `1 ;
assume b in X ;
assume k <> 1 ;
f = product l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is from of SCM+FSA ;
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 <= 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 , e be Vertex of G ;
let G be _Graph , e be Vertex of G ;
let a be Complex ;
let x be element ;
let x be element ;
let C be FormalContext , a be Real ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
let y be Real ;
X c= f . a ;
let y be element ;
let x be element ;
let i be Nat ;
let x be element ;
n in NAT ;
let a be element ;
m in NAT ;
let u be element ;
i in NAT ;
let g be Function ;
Z c= NAT ;
l <= NAT ;
let y be element ;
r2 in dom f ;
let x be element ;
let k1 be Integer ;
let X be set ;
let a be element ;
let x be element ;
let x be element ;
let q be element ;
let x be element ;
assume f is being_homeomorphism ;
let z be element ;
a , b // K ;
let n be Nat ;
let k be Nat ;
B c= B ` ;
set s = \mathclose ;
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 ;
the arccot of arccot is_differentiable_on Z ;
the function exp is differentiable ;
j < i2 ;
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 <> b2 ;
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 <= r3 ;
let e be Real , x be Real ;
not r in G . l ;
c1 = 0 ;
a + a = a ;
<* 0 *> in e ;
t in { t } ;
assume F is not discrete ;
m1 divides m ;
B * A <> {} ;
a + b <> {} ;
p * p > p ;
let y be ExtReal ;
let a be Int-Location , I be Program of SCM+FSA ;
let l be Nat ;
let i be Nat ;
let r ;
1 <= i2 ;
a "\/" c = c ;
let r be Real ;
let i be Nat ;
let m be Nat ;
x = p2 ;
let i be Nat ;
y < r + 1 ;
rng c c= E ;
Cl R is discrete ;
let i be Nat ;
R2 is total ;
cluster uparrow x -> -> -> -> -> -> -> -> -> -> -> -> -> -> as ;
X <> { x } ;
x in { x } ;
q , b // M ;
A . i c= Y ;
P [ k ] ;
2 |^ x in W ;
X [ 0 ] ;
P [ 0 ] ;
A = A |^ i ;
s >= s ;
G . y <> 0 ;
let X be RealNormSpace , x be Point of X ;
a in A ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , W be Subspace of V ;
assume x in M ;
k < s . a ;
not t in { p } ;
let Y be functional , f be FinSequence of Y ;
M , L are_isomorphic ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded & L is lower-bounded ;
rng f = Y ;
G c= L ;
assume x in Cl Q ;
m in dom P ;
i <= len Q ;
len F = 3 ;
still_not-bound_in p = {} ;
z in rng p ;
lim b = 0 ;
len W = 3 ;
k in dom p ;
k <= len p ;
i <= len p ;
1 in dom f ;
b `1 = a `1 + 1 ;
x `1 = a * y `1 ;
rng D c= A ;
assume x in K1 ;
1 <= i1 ;
1 <= i1 ;
pU c= cos ;
1 <= i1 & i1 <= len G ;
1 <= i1 & i1 <= len G ;
w in L ;
1 in dom f ;
let seq ;
set C = a * B ;
x in rng f ;
assume f is Lipschitzian ;
I = dom A ;
u in dom p ;
assume a < x + 1 ;
seq 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 >= Z ;
assume w = 0. V ;
assume x in A . i ;
g in ComplexBoundedFunctions X ;
y in dom t ;
i in dom g ;
assume P [ k ] ;
-1 c= f ;
xx is increasing ;
let d2 be element ;
- b divides b ;
F c= \mathclose { F ( ) } ;
G1 is non-decreasing ;
G1 is non-decreasing ;
assume v in H . m ;
assume b in [#] B ;
let S be non void ManySortedSign , X be non-empty ManySortedSet of S ;
assume P [ n ] ;
assume union S is linearly independent & finite is finite ;
V is Subspace of V ;
assume P [ k ] ;
rng f c= NAT ;
assume inf X in L ;
y in rng f ;
let s , I be set , s be Element of S ;
b ` c= b9 ` ;
assume not x in NAT + Q ;
A /\ B = { a } ;
assume len f > 0 ;
assume x in dom f ;
b , a // o , c ;
B in B-24 ;
cluster product p -> non empty ;
z , x // x , p ;
assume x in rng N ;
cosec is_differentiable_in x ;
assume y in rng S ;
let x , y ;
i2 < i1 & i2 < i2 ;
a * h in H * H ;
p , q in Y ;
cluster sqrt I -> ideal ;
q1 in A1 & q1 in A2 ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A2 c= A1 & A1 c= A2 ;
\hbox { \boldmath n , m } < n ;
assume A c= dom f ;
Re ( f ) is_integrable_on M ;
let k , m ;
a , b \equiv b , b ;
j + 1 < k + 1 ;
m + 1 <= n1 ;
g is_differentiable_in x0 & g is_differentiable_in x0 ;
g is continuous ;
assume O is symmetric and transitive is transitive ;
let x , y ;
let j2 be Nat ;
[ y , x ] in R ;
let x , y ;
assume y in conv A ;
x in Int V ;
let v be VECTOR of V ;
P3 halts_on s ;
d , c // a , b ;
let t , u ;
let X be set ;
assume k in dom s ;
let r be non negative Real ;
assume x in F | M ;
let Y be Subset of S ;
let X be non empty TopSpace , A be Subset of X ;
[ a , b ] in R ;
x + w < y + w ;
{ a , b } >= c ;
let B be Subset of A , a be Element of B ;
let S be non empty ManySortedSign ;
let x be variable , f be Function ;
let b be Element of X , a be Element of X ;
R [ x , y ] ;
x ` ` = x ;
b \ x = 0. X ;
<* d *> in D |^ 1 ;
P [ k + 1 ] ;
m in dom ( ( n + 1 ) -tuples_on NAT ) ;
h2 . a = y ;
P [ n + 1 ] ;
cluster G * F -> object ;
let R be non empty multiplicative magma , a be Element of R ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p ;
assume f | X is lower ;
x in rng ( go | ( len pion1 ) ) ;
let x be Element of B ;
let t be Element of D ;
assume x in Q .vertices() ;
set q = s ^\ k ;
let t be VECTOR of X ;
let x be Element of A ;
assume y in rng p `1 ;
let M be void non empty id ;
let N be non empty multiplicative loop of M ;
let R be RelStr ;
let n , k ;
let P , Q ;
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 is not destroy a ;
let n , k ;
let x be Point of T ;
f c= f +* g ;
assume m < v8 ;
x <= c2 . x ;
x in F " ( A ` ) ;
cluster S --> T -> o -valued ;
assume t1 <= t2 & t2 <= t2 ;
let i , j ;
assume F1 <> F2 & F2 <> F2 ;
c in Intersect ( union R ) ;
dom ( p1 , p2 ) = c ;
a = 0 or a = 1 ;
assume A1 <> A2 & A2 <> C1 ;
set i1 = i + 1 ;
assume a1 = b1 & a2 = b2 ;
dom ( g1 | A ) = A ;
i < len M + 1 ;
assume not - \infty in rng G ;
N c= dom ( f1 | X ) ;
x in dom sec ;
assume [ x , y ] in R ;
set d = sqrt ( x , y ) ;
1 <= len ( g1 | A ) ;
len s2 > 1 ;
z in dom ( f1 | X ) ;
1 in dom ( D2 | A ) ;
( p `2 ) ^2 = 0 ;
j2 <= width G & j1 <= width G ;
len cos > 1 + 1 ;
set n1 = n + 1 ;
|. q9 .| = 1 ;
let s be SortSymbol of S ;
order ( i , i ) = i ;
X1 c= dom f & X2 c= dom f ;
h . x in h . a ;
let G be \times of lies ;
cluster m * n -> invertible ;
let k2 be Nat ;
i - 1 > m - 1 ;
R is transitive implies field R is transitive
set F = <* u , w *> ;
p-2 c= P1 & PP c= P2 ;
I is_closed_on t , Q ;
assume [ S , x ] is universal ;
i <= len ( f2 | i ) ;
p is FinSequence of X ;
1 + 1 in dom g ;
Sum R2 = n * r ;
cluster f . x -> complex-valued ;
x in dom ( f1 | X ) ;
assume [ X , p ] in C ;
B9 c= [: X , Y :] ;
n2 <= ( n2 + 1 ) / ( n2 + 1 ) ;
A /\ ( P /\ Q ) c= A ` ;
cluster x -valued -> x -valued ;
let Q be Subset-Family of S , a be Element of S ;
assume n in dom ( g2 | n ) ;
let a be Element of R ;
t `1 in dom ( e | A ) ;
N . 1 in rng N ;
- z in A \/ B ;
let S be SigmaField of X , X be set ;
i . y in rng i ;
NAT c= dom f ;
f . x in rng f ;
NAT <= sqrt ( r / 2 ) ;
s2 in r-5 ( r ) ;
let z , z be complex number ;
n <= N . m ;
LIN q , p , s ;
f . x = -' x /\ B ;
set L = [ S \to T , T ] ;
let x be non negative Real ;
let m be Element of M ;
f in union rng ( F1 | A ) ;
let K be add-associative right_zeroed right_complementable associative distributive non empty doubleLoopStr , a be Element of K ;
let i be Element of NAT ;
rng ( F * g ) c= Y ;
dom f c= dom x & rng f c= dom x ;
n1 < n1 + 1 ;
n1 < n1 + 1 ;
cluster [: T , X :] -> On ;
[ y2 , 2 ] = z ;
let m be Element of NAT ;
let S be Subset of R ;
y in rng ( Ss | A ) ;
b = sup ( dom f ) ;
x in Seg ( len q ) ;
reconsider X = [: D , D :] as set ;
[ a , c ] in E ;
assume n in dom h2 ;
w + 1 = [: a1 , b1 :] ;
j + 1 <= j + 1 ;
k2 + 1 <= k1 + 1 ;
let i be Element of NAT ;
Support u = Support p ;
assume X is complete \vert ;
assume f = g & p = q ;
n1 <= n1 + 1 ;
let x be Element of REAL n ;
assume x in rng ( s2 | A ) ;
x0 < x0 + 1 ;
len ( L + - K ) = W ;
P c= Seg ( len A ) & P c= Seg ( len A ) ;
dom q = Seg n & dom q = Seg n ;
j <= width M *' ;
let r8 be real-valued FinSequence ;
let k be Element of NAT ;
d < +infty ;
let n be Element of NAT ;
assume z in the_c\rbrace \ ( A \ A ) ;
let i be set ;
n - 1 = n - 1 ;
len ( n |-> -27 ) = n ;
diff ( Z , c ) c= F ;
assume x in X or x = X ;
x is midpoint of b , c ;
let A , B be non empty set , f be Function of A , B ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q ;
let s be Element of E |^ \omega ;
let B1 be Basis of x , y ;
3 /\ L2 = {} ;
L1 /\ L2 = {} ;
assume x = \mathopen { y } x } ;
assume b , c // b , c ;
LIN q , c , a ;
x in rng ( f | X2 ) ;
set n8 = n + j ;
let Dbe non empty set , x be Element of D ;
let K be add-associative right_zeroed right_complementable right_zeroed right_complementable distributive non empty doubleLoopStr , a be Element of K ;
assume f = f & h = h ;
R1 - R2 is total ;
k in NAT & 1 <= k ;
let a be Element of G ;
assume x0 in [. a , b .] ;
K1 ` is open Subset of TOP-REAL 2 ;
assume a , b ] in maximal L~ Cage ( C , n ) ;
let a , b be Element of S ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = \int f , M ;
cluster binary for Al ;
not u in { \hbox { \boldmath $ g } } ;
the carrier of f c= B ;
reconsider z = x as VECTOR of V ;
cluster the connectives of L -> strict for non empty RelStr ;
r (#) H is H be convergent ;
s . intloc 0 = 1 ;
assume that x in C and y in C ;
let U be strict non-empty MSAlgebra over S , x be Element of U1 ;
[ x , Bottom T ] is compact ;
i + 1 + k in dom p ;
F . i is stable Subset of M ;
r-35 in reconsider ry & x in HT ( y , r ) ;
let x , y be Element of X ;
let A , I be \mathclose of X ;
[ y , z ] in [: O , A :] ;
Shift ( i , k ) = 1 ;
rng Sgm A = A & card ( A ) = card A ;
q |- ( y |- All ( y , q ) ) => ( q => q ) ;
for n holds X [ n ] ;
x in { a } & x in d ;
for n holds P [ n ] ;
set p = [ x , y , z ] ;
LIN o , a , b ;
p . 2 = Z |^ Y ;
( ( D1 | 0 ) | ( D1 | D2 ) ) = {} ;
n + 1 + 1 <= len g ;
a in [: Al , Al :] & b in [: Al , NAT :] ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
let I be Ideal of L ;
set g = f1 + f2 , h = f2 + g2 ;
a <= max ( a , b ) ;
i-1 < len G + 1 ;
g . 1 = f . i1 ;
x `1 , y `2 in A2 ;
( f /* s ) . k < r ;
set v = VAL g ;
i - k + 1 <= S ;
cluster non empty multiplicative magma -> unital ;
x in support ( support ( t ) ) ;
assume a in [: G , H :] ;
i `1 <= ( y `1 ) `1 ;
assume p divides b1 + b2 ;
M1 <= ( sup M1 ) / ( sup M2 ) ;
assume x in ( W-min ( X ) ) .: ( X ) ;
j in dom ( z | ( Seg n ) ) ;
let x be Element of [: D , D :] ;
IC Comput ( P3 , s3 , k ) = IC Comput ( P3 , s3 , k ) ;
a = {} or a = { x } ;
set uG = Vertices G , CG = Vertices G ;
seq " is non-zero ;
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
hK c= ( h " ) . ( len h + 1 ) ;
]. a , b .[ c= Z ;
X1 , X2 , X3 , X2 , X3 , X3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5
a in Cl ( union F \ G ) ;
set x1 = [ 0 , 0 ] ;
k + 1 - 1 = k ;
cluster -> real-valued for Relation ;
ex v st C = v + W ;
let G1 be non zero Nat , a be Element of L ;
assume V is Abelian add-associative right_zeroed right_complementable distributive distributive commutative associative commutative associative commutative associative distributive ;
X \/ Y in sigma ( L ) ;
reconsider x = x as Element of S ;
max ( a , b ) = a ;
sup B is upper Subset of B ;
let L be non empty reflexive RelStr , X be Subset of L ;
R is reflexive & R is transitive implies R is transitive
E , H |= H ;
dom G ' = a ;
sqrt ( 1 - 4 ) >= - ( r / 4 ) ;
G . x0 in rng G ;
let x be Element of ( F . i ) -tuples_on the carrier of S ;
D [ ( P-6 , 0 ) , ( ( n + 1 ) .--> 0 ) ] ;
z in dom id ( B ) & z in id B ;
y in the carrier of N ( ) ;
g in the carrier of H ;
rng ( f | A ) c= NAT ;
j `1 + 1 in dom ( s1 + s2 ) ;
let A , B be strict Subgroup of G ;
let C be non empty Subset of REAL ;
f . z1 in dom h ;
P . ( k1 + 1 ) in rng P ;
M = ( A +* {} ) +* {} .= {} ;
let p be FinSequence of REAL ;
f . ( n1 + 1 ) in rng f ;
M . ( F . 0 ) in REAL ;
( for a , b holds ( for a holds a = b ) implies a = b
assume that the InternalRel of V is symmetric and Q is symmetric ;
let a be Element of ^ ( V , C ) ;
let s be Element of ( the carrier of P ) ;
let Pa be non empty transitive RelStr ;
let n be Nat ;
the carrier of g c= B ;
I = halt R .= ( the InstructionsF of R ) . ( l , k ) ;
consider b being element such that b in B ;
set BK = BCS ( K , n ) ;
l <= ( j . j ) `1 ;
assume x in \mathopen { [ s , t ] } ;
( x `2 ) / ( |. x .| ) in ]. t . t .| ;
x in dom ( T . 0 ) ;
let h be Morphism of c opp , a ;
Y c= [: Y , \bf T :] ;
A2 \/ ( A2 \/ A3 ) c= L1 \/ L2 ;
assume LIN o , a , b ;
b , c // d1 , d2 ;
x1 , x2 , x3 is_collinear & x1 , x2 , x3 is_collinear ;
dom <* y *> = Seg 1 & dom <* y *> = Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar s .| ;
[ x , x ] in X [: X , X :] ;
for n being Nat holds 0 <= x . n
[ a , b ] = [. a , b .] ;
cluster -> regular for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 , q1 , q2 is_collinear ;
dom ( M1 * M2 ) = Seg n & dom ( M2 * M2 ) = Seg n ;
x = [ x1 , x2 ] ;
let R , Q be ManySortedSet of A ;
set d = sqrt ( 1 + n ) ;
rng ( g2 | A ) c= dom W ;
P ( [#] ( Omega \ B ) ) \ B <> 0 ;
a in field R & a = b ;
let M be non empty Subset of V , V be Subset of V ;
let I be Program of SCM+FSA , s be State of SCM+FSA ;
assume x in rng ( R * S ) ;
let b be Element of the lattice of T ;
dist ( e , z ) > r-r ;
u1 + v1 in W2 & v1 + v2 in W2 ;
assume the support of L misses rng G ;
let L be lower-bounded antisymmetric antisymmetric antisymmetric antisymmetric RelStr ;
assume [ x , y ] in [: a , b :] ;
dom ( A * e ) = NAT ;
let a , b be Vertex of G ;
let x be Element of ( ( bool M ) . s ) ;
0 <= 2 * PI ;
o `1 , a9 // o `1 , b9 ;
{ v } c= the carrier of l ;
let x be variable of A ;
assume x in dom ( uncurry f ) ;
rng F c= ( product f ) |^ X ;
assume D2 . k in rng D ;
f " . p1 = 0 ;
set x = the Element of X ;
dom ( G . n ) = NAT ;
let n be Element of NAT ;
assume LIN c , a , a1 ;
cluster -> finite for FinSequence of NAT ;
reconsider d = c as Element of L1 ;
( v2 |-- I ) . X <= 1 ;
assume x in the carrier of f ;
conv @ @ @ c= conv @ A ;
reconsider B = b as Element of the carrier of T ;
J , v / ( l , k ) / ( l , k ) / ( l , k ) / ( l , k ) / ( l , k ) / ( l , k ) / (
cluster J . i -> non empty for TopSpace ;
ex_sup_of ( Y1 \/ Y2 , T ) , T ;
W1 is_well field ( W1 , W2 ) ;
assume x in the carrier of R ;
dom ( n |-> ( x | ( n + 1 ) ) ) = Seg n ;
s3 misses ( the Sorts of A ) . ( n + 1 ) ;
assume ( a 'imp' b ) . z = TRUE ;
assume that X is open and f = X --> d ;
assume [ a , y ] in Indices ( f ) ;
assume that that that that that that that I c= J and K c= K and I c= K ;
Im ( ( lim seq ) ^\ k ) = 0 ;
( ( the Sorts of sin ) . x ) . ( ( the Sorts of sin ) . x ) <> 0 ;
the function sin is differentiable & cos is differentiable implies sin | A is continuous
t6 . n = t6 . n ;
dom ( H | A ) c= dom F ;
W1 . x = W2 . x ;
y in W .vertices() \/ W .vertices() \/ W .vertices() ;
( ( k + 1 ) + 1 ) <= ( ( k + 1 ) + 1 ) + 1 ;
x * a \equiv y * a ;
proj2 .: P c= proj2 .: P ;
h . p3 = g2 . I ;
G * ( 1 , 1 ) `1 = ( G * ( 1 , 1 ) `1 ) `1 ;
f . ( r + 1 ) in rng f ;
i + 1 + 1 <= len One ;
rng F = rng ( F | A ) & rng ( F | A ) c= rng ( F | A ) ;
mode multiplicative magma is unital non empty doubleLoopStr ;
[ x , y ] in A [: { a } , { a } :] ;
x1 . o in L2 . ( o . o ) ;
the support of m c= B ;
not [ y , x ] in id ( X ) ;
1 + p .. f <= i + len f ;
seq ^\ k is lower implies seq is lower
len ( F | ( len F ) ) = len I ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be Complex , s1 be Real ;
Comput ( P , s , n ) = s ;
k <= k + 1 & 1 <= k ;
reconsider c = {} as Element of L ;
let Y be with_internal \cal d\cal of T ;
cluster -> directed-sups-preserving for Function of L , L ;
f . ( j1 , j2 ) in K . ( j1 , j2 ) ;
cluster J => y -> total for Function ;
K c= 2 -tuples_on the carrier of T ;
F . b1 = F . b2 .= F . b2 ;
x1 = x or x1 = y ;
attr a <> {} means : Def2 : sqrt a = 1 ;
assume that succ a c= b and b in a ;
s1 . n in rng ( s1 | n ) ;
{ o , b2 } on C2 & { o , b1 } on C2 ;
LIN o , b , b9 ;
reconsider m = x as Element of Funcs ( V , C ) ;
let f be non trivial FinSequence of D ;
let F be non empty TopSpace ;
assume h is being_homeomorphism & y = h . x ;
[ f . 1 , w ] in F ;
reconsider px = x as Subset of REAL m ;
A , B , C , D , E , F , J be Element of R ;
cluster strict non empty for SubSpace of G ;
rng c misses rng ( e `1 ) & rng ( e `2 ) misses rng ( e `2 ) ;
z is Element of gr ( { x } ) ;
not b in dom ( a .--> ( p . i ) ) ;
assume that k >= 2 and P [ k ] ;
Z c= dom ( ( cot * cot ) `| Z ) ;
the component of Q c= ( the Sorts of A ) \/ ( the Sorts of B ) ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( 1 / ( f . x ) ) ;
attr f = u * f means : Def2 : a * f = u * f ;
for n holds P1 [ n ] ;
{ x . O : x in L } <> {} ;
let x be Element of V . s ;
let a , b be Nat ;
assume that S = S2 and p = S2 and q = S2 ;
gcd ( n1 , n2 ) = 1 & gcd ( n2 , n2 ) = 1 ;
set o1 = ( n * ) * ( n + 1 ) ;
seq . n < |. r1 .| ;
assume that seq is increasing and r < 0 ;
f . ( y1 , y2 ) <= a ;
ex c being Nat st P [ c ] ;
set g = { n / ( 1 / n ) where n is Element of NAT : n in NAT } ;
k = a or k = b ;
a9 , b9 , c9 is_collinear & b9 , c9 , a9 is_collinear ;
assume that Y = { 1 } and s = <* 1 *> ;
I1 . x = f . x .= 0 ;
W4 ( ) = W . ( 1 + 1 ) ;
cluster -> trivial for Vertex of G ;
reconsider u = u as Element of Bags X ;
A in B ^ A implies A , B are_equipotent
x in { [ 2 * n + 3 , k ] } ;
1 >= sqrt ( ( q `1 / |. q .| - sn ) ) ;
f1 is in the carrier' of ( the carrier' of C ) , f2 ;
( f . q ) `2 <= ( q `2 ) ^2 ;
h is in the area of Cage ( C , n ) ;
( b / ( p `1 ) ) / ( p `2 ) <= ( p `2 ) / ( p `2 ) ;
let f , g be one-to-one Function of X , Y ;
S * ( k , k ) <> 0. ( K , k ) ;
x in dom ( f - g ) ;
p2 in ( ( 1 - p1 ) * ( 1 - p1 ) ) * ( ( 1 - p1 ) * ( 1 - p1 ) ) ;
len ( H ) < len H & H . ( len H ) < len H ;
F [ A , ( F . A ) ] ;
consider Z such that y in Z and Z in X ;
attr 1 in C means : Def2 : A c= C & A c= C ;
assume that r1 <> 0 or r2 <> 0 ;
rng ( q1 ^ q2 ) c= rng ( q1 ^ q2 ) ;
A1 , L , L , R , L ) , A , L , R , R , R , L ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in less_dom ( p , S ) ;
then S is negative implies P-2 [ S ] ;
Cl ( [#] T ) = [#] ( T | A ) ;
( f | A ) | ( ( f | A ) | A ) = ( f | A ) | A ;
0. ( M , m ) in the carrier of W ;
v , v be Element of M ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ V & Y \ V c= Y \ Z ;
let X be Subset of S , T be non empty Subset of S ;
consider H1 such that H = 'not' H1 and H1 in Free H ;
1_ ( L * t ) c= ( \frac { 1 } { t } ) * ( \frac { 1 } { t } ) ;
0 * a = 0. R .= a * 0 ;
A |^ 2 , 2 |^ 2 = A |^ 2 ;
set vX = ( v /. n ) * ( ( v /. n ) * ( v /. n ) ) ;
r = 0. ( \langle \cal E , \Vert * \Vert *> ) ;
( f . p3 ) `1 >= 0 ;
len W = len ( W | ( m + 1 ) ) ;
f /* ( s * G ) is divergent_to+infty ;
consider l being Nat such that m = F . l ;
t . intpos ( a + 1 ) not ( t . intpos ( b + 1 ) ) ;
reconsider Y1 = X1 as SubSpace of X ;
consider w such that w in F and not x in w ;
let a , b , c , d be Real ;
reconsider i = i 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 ;
sup a /\ \mathopen { t } , sup { t } } is Subset of T ;
let X be set , N be non empty set ;
rng f = Funcs ( S , X ) ;
let p be Element of B , x be being being SortSymbol of S ;
max ( N1 , 2 ) >= N1 & max ( N1 , 2 ) >= N2 ;
0. X <= ( b |^ m ) * ( ( m * ( n + 1 ) ) ) ;
assume i in I & R . i = R ;
i = j1 & ( p1 , p2 ) = q1 & ( p1 , p2 ) = q2 ;
assume gR in the carrier' of g & x in the carrier' of g ;
let A1 , A2 be Point of S ;
x in h " ( P ) /\ [#] ( T | P ) ;
1 in Seg 2 & 1 in Seg 3 & 1 in Seg 3 ;
reconsider A-5 = X as non empty Subset of T | ( X /\ Y ) ;
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the Target of G ) -valued ;
n1 <= i2 + ( len g2 + 1 ) & n2 + ( len g2 + 1 ) <= len g2 ;
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume v in the carrier' of G2 & v in the carrier' of G2 ;
y = Re ( y - y ) + ( Im ( y - z ) ) ;
( ( - ( ( - 1 ) * p ) ) gcd ( - 1 ) ) = 1 ;
x2 is_differentiable_on ]. a , b .[ & diff ( a , b ) is_differentiable_on ]. a , b .[ ;
rng ( M * ( i , j ) ) c= rng ( M * ( i , j ) ) ;
for p being Real st p in Z holds p >= a
\bf X \bf Y ( f ) = proj1 ( f ) ;
( seq ^\ m ) . k <> 0 ;
s . ( G . ( k + 1 ) ) > x0 ;
( p | ( M | ( N | ( M | ( N | ( N ) ) ) ) ) ) ) . 2 = d ;
A /\ ( B \/ C ) = ( A /\ B ) /\ C
h \equiv ( gg ) . ( ( g mod P ) . ( h . ( g . ( h . ( a ) ) ) ) ) ;
reconsider i1 = i - 1 as Element of NAT ;
let v1 , v2 be VECTOR of V ;
mode Linear_Combination of V is Subspace of [#] ( V ) ;
reconsider i1 = i - 1 as Element of NAT ;
dom f c= [: C , D :] & rng f c= [: D , D :] ;
x in ( the Sorts of B ) . n & x in ( the Sorts of B ) . n ;
len that len that len f2 in Seg ( len f2 ) and ( len f1 ) = len f2 + 1 ;
p1 c= the topology of T & the topology of T c= the topology of T ;
]. r , s .[ c= [. r , s .] ;
let B2 be Basis of T2 | ( the carrier of T2 ) ;
G * ( B * A ) = id ( o1 , o2 ) ;
assume that p , u , v , u is_collinear and u , v , w , y is_collinear ;
[ z , z ] in union rng ( F | ( { x } ) ) ;
( 'not' b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S , $1 = $1 .. S ;
LIN a1 , b1 , c1 & LIN a1 , b1 , c1 ;
f " ( f .: x ) = { x } ;
dom ( w | ( dom ( w | ( dom w | ( dom w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w |
assume that 1 <= i and i <= n and j <= n ;
( g2 . O ) `2 <= 1 ;
p in LSeg ( E . i , F . i ) ;
I1 * ( i , j ) = 0. ( K , n , n ) ;
|. f . ( s . m ) - g .| < g1 ;
q9 . x in rng ( ( q | i ) | ( Seg n ) ) ;
Lmin misses ( L | A ) ` .= ( L | A ) ` ;
consider c being element such that [ a , c ] in G ;
assume S99 = ( o /. ( meets o ) ) & ( o /. ( meets o ) ) = ( o /. ( meets o ) ) ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= ( F | C ) .: C ;
P . ( ( B2 \/ D2 ) . ( B2 , D2 ) ) <= 0 + 0 ;
f . j in [. f . j , f . j .] ;
attr 0 <= x & x <= 1 implies x / ( x / ( x / ( x / ( x / ( x / ( x / ( x / ( x / ( x / ( x / ( x / ( x / ( x /
p `1 <> 0. TOP-REAL 2 & q `2 <> 0. TOP-REAL 2 ;
cluster \vert \vert ( S , T ) -> non empty ;
let x be Element of S , T be Element of S ;
<^ F . a , F . b ^> is one-to-one ;
|. i .| <= - ( - 2 |^ n ) ;
the carrier of I[01] = dom ( ( TOP-REAL 2 ) | K1 ) ;
n * ( n + 1 ) ! > 0 * n ;
S c= ( A1 /\ A2 ) /\ ( A2 /\ A1 ) ;
a3 , a4 // b2 , b1 & a3 , a4 // b2 , b1 ;
then dom A <> {} & dom A <> {} & dom A <> {} ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , G , G ;
set v2 = v2 /. ( i + 1 ) , v2 = v2 /. ( i + 1 ) ;
x = r . n .= ( r . n ) . n ;
f . s in the carrier of S2 & f . s in the carrier of S2 ;
dom g = the carrier of I[01] & rng g c= the carrier of TOP-REAL 2 ;
p in Lower_Arc ( P ) /\ Lower_Arc ( P ) ;
dom ( d * A ) = [: A , A :] & dom ( d * A ) = [: A , A :] ;
0 < sqrt ( p `1 - z `1 ) + ( p `2 - z `2 ) ;
e . ( m + 1 ) <= e . ( m + 1 ) ;
B \/ X \/ B c= B \/ X
- +infty < \int ( g | B ) | A ;
cluster O \tt F -> o OperSymbol of X ;
let U1 , U2 be non-empty MSAlgebra over S , x be Element of S ;
Proj ( i , n ) * g is_differentiable_on X ;
let x , y , z be Point of X , f be Function of X , Y ;
reconsider p0 = p . x as Subset of V ;
x in the carrier of Lin ( A ) & x in Lin ( A ) ;
let I , J be parahalting Program of SCM+FSA , s be State of SCM+FSA ;
assume - a is lower implies - a is lower & b is lower
Int ( Cl A ) c= Cl ( Int ( Cl A ) ) ;
assume for A being Subset of X holds Cl A = A ;
assume q in Ball ( [ x , y ] , r ) ;
( p2 `2 ) / ( 1 + ( p2 `2 ) / ( 1 + ( p2 `2 ) ) ^2 ) <= ( p2 `2 ) / ( 1 + ( p2 `2 ) ) ^2 ;
Cl ( Q ` ) = [#] ( ( TOP-REAL 2 ) | P ) ;
set S = the carrier of T ;
set I8 = lines ( f , n ) , C8 = f | ( n + 1 ) ;
len p - n = len p - n + 1 .= len p - n + 1 ;
A is permutation of Funcs ( A , x , y ) ;
reconsider n6 = ni - 1 as Element of NAT ;
1 <= j + 1 & 1 <= ( s . ( j + 1 ) ) ;
let q9 , q9 be Element of M ( ) , q be Element of M ( ) ;
a1 in the carrier of S1 & a2 in the carrier of S2 ;
c1 /. ( n + 1 ) = c1 . ( n + 1 ) ;
let f be FinSequence of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
y = ( ( f * ( S . x ) ) ) . x ;
consider x being element such that x in " A ;
assume r in ( dist ( o , r ) ) .: P ;
set i2 = ( n + 1 ) -tuples_on the carrier' of S ;
h2 . ( j + 1 ) in rng h2 ;
Line ( Line ( M , k ) , i ) = M . i ;
reconsider m = sqrt ( x / 2 ) as Element of ( - 1 ) -tuples_on REAL ;
let U1 , U2 be non-empty Subspace of U0 ;
set P = Line ( a , d ) ;
len ( p1 + p2 ) < len ( p1 + p2 ) + len ( p2 + p3 ) ;
let T1 , T2 be complete TopStruct , T be Scott topological T ;
then x <= y & y c= connected ( x , y ) ;
set M = n ! ( m , n ) ;
reconsider i = x1 , j = x2 as Nat ;
rng ( the_arity_of o ) c= dom H & rng ( the_arity_of o ) c= dom H ;
z1 " = ( z1 " ) * ( z1 " ) ;
x0 - r / 2 in L /\ dom f ;
then w is strict for non empty SubSpace of S ;
set x-10 = ( x ^ <* Z *> ) ^ <* Z *> ;
len ( w1 ^ w2 ) in Seg ( len w1 + len w2 ) ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of PFuncs ( V , { k } ) ;
x . n = sqrt ( a . n ) ;
( p `1 ) / ( 1 + ( p `2 ) / ( 1 + ( p `2 ) ) ^2 ) <= ( p `2 ) / ( 1 + ( p `2 ) ) ^2 ;
rng ( g | A ) c= L~ ( g | A ) \/ ( L~ ( g | A ) ) ;
reconsider k = i-1 * ( l + 1 ) as Nat ;
for n being Nat holds F . n is \setminus implies F . n is \setminus implies F . n is \setminus
reconsider x9 = x9 as VECTOR of M ;
dom ( f | X ) = X /\ dom f ;
p , a // p , c & b , a // c , a ;
reconsider x1 = x as Element of REAL m m -tuples_on REAL n ;
assume i in dom ( a * p ^ q ) ;
m . ( \hbox { \boldmath $ g g g } ) = p . ( \hbox { \boldmath $ g $ } , S ) ;
a / ( s . m ) <= 1 / ( ( s . n ) ! ) ;
S . ( n + k + 1 ) c= S . ( n + k ) ;
assume ( B1 \/ B2 ) \/ ( B2 \/ C2 ) = ( B1 \/ B2 ) \/ ( B2 \/ C2 ) ;
X . i = { x1 , x2 } . i .= { x1 , x2 } . i ;
r2 in dom ( ( h1 + h2 ) | A ) & ( h1 + h2 ) | A is bounded ;
<* 0 *> = a & b--(#) ( ba ) = b ;
( F . 8 ) is closed & ( F . 8 ) . 8 = ( F . 8 ) . 8 ;
set T = TopGGl ( X , x0 ) ;
Int ( Cl R ) c= Int ( R ) ;
consider y being Element of L such that c . y = x ;
rng ( ( F . x ) | ( rng F . x ) ) = { F . x } ;
G " { c } c= B \/ S \/ S \/ { c } ;
f is Relation of [: X , Y :] , X ;
set Rx = the Element of P ( ) , Rx = the Element of P ( ) ;
assume n + 1 >= 1 & n + 1 <= len M ;
let k2 be Element of NAT ;
reconsider pv = u as Element of GF ( n ) ;
g . x in dom f & x in dom g ;
assume that 1 <= n and n + 1 <= len f1 ;
reconsider T = b * N as Element of ( G G ) / N ;
len ( ( PX2 ) . ( i + 1 ) ) <= len ( PX2 ) ;
x " in the carrier of ( ( A * ) * ( x " ) ) ;
[ i , j ] in Indices ( ( A @ ) * ( i , j ) ) ;
for m being Nat holds Re ( F . m ) is simple ;
f . x = a . i .= a . k ;
let f be PartFunc of REAL n , REAL n ;
rng f = the carrier of Lin ( A ) & rng f c= the carrier of Lin ( A ) ;
assume s1 = sqrt ( 2 * ( p `2 / |. p .| - sn ) ) ;
attr a > 1 & b > 0 implies a / b > 1 ;
let A , B , C , D be Subset of TOP-REAL n ;
reconsider X1 = X , X2 = Y as real number ;
let f be PartFunc of REAL , REAL n ;
r * ( v1 , I ) . X < r * 1 ;
assume that V is Subspace of X and X is Subspace of V ;
let t-3 , t-3 be binary relation ;
Q [ ( e \/ { v } ) \/ { v } ] ;
g \circlearrowleft ( L~ z ) = z ;
|. [ x , v ] - [ x , y ] .| = vI ;
- f . w = - ( L * w ) ;
z - y <= x - y iff z <= x + y & z <= x + y ;
sqrt ( 7 * ( p1 + e ) ) > 0 ;
assume that X is BCK-algebra and 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0
F . 1 = v1 & F . 2 = v2 ;
( f | X ) . x2 = f . x2 ;
( ( ( tan * sec ) `| Z ) . x ) . x in dom ( sec * sec ) . x ;
i2 = ( f /. ( len f -' 1 ) ) /. ( len f -' 1 ) ;
X1 = X2 \/ ( X1 \ X2 ) ;
[. a , b .] = 1_ G & { a , b } = 1_ G ;
let V , W be non empty VectSpStr over K ;
dom g2 = the carrier of I[01] & rng g2 c= the carrier of TOP-REAL 2 ;
dom f2 = the carrier of I[01] & rng ( f2 | K1 ) c= the carrier of TOP-REAL 2 ;
( proj2 | X ) .: X = ( proj2 | X ) .: X ;
f . ( x , y ) = h1 . ( x , y ) ;
x0 - r < a1 . n - r ;
|. ( f /* s ) . k - ( f /* s ) . k .| < r ;
len Line ( A , i ) = width A & width ( Line ( B , i ) ) = width A ;
S.|| = ( S . g ) / ( g . h ) ;
reconsider f = v + u as Function of X , the carrier of Y ;
intloc 0 in dom Initialized ( p ) & IC ( p +* I ) in dom Initialized ( p +* I ) ;
i1 , i2 , j1 , j2 ) & ( i1 , i2 , j1 , j2 ) := i2 , j2 ;
set set that ( set r + 1 ) / 2 = ( cos r ) / 2 + 0 ;
for x st x in Z holds f2 is_differentiable_in x ;
reconsider q2 = sqrt ( q , x ) as Element of REAL n ;
( 0 qua Nat ) + 1 + 1 <= i + 1 ;
assume f in the carrier of [: X , [#] Y :] ;
F . a = H / ( { x } , y ) ;
true ( T , u ) = TRUE ;
dist ( ( a * seq ) . n , h ) < r ;
1 in the carrier of [. 0 , 1 .] & 1 <= r & r <= 1 ;
( p2 `2 ) - ( p1 `2 ) > - ( p1 `2 ) ;
|. r1 - q1 .| = |. a1 .| * |. q1 - q2 .| ;
reconsider S-14 = 8 as Element of Seg 8 ;
( A \/ B ) |^ b c= A |^ b \/ B |^ a ;
DkW = DkW ( ) + 1 .= DWW ( ) + 1 ;
i1 = [: NAT , n :] & i2 = [: NAT , n :] & j2 = [: NAT , n :] ;
f . a [= f .: ( f .: ( O , a ) ) ;
attr f = v & g = u + v ;
I . n = \int F . n , ( lim F ) | E ;
( ( \raise .4ex \hbox { $ \chi $ } } , T ) . s ) . s = 1 ;
a = VERUM ( A ) or a = VERUM ( A ) or a = VERUM ( A ) ;
reconsider k2 = s . b2 , k2 = s . b2 as Element of NAT ;
( Comput ( P , s , 4 ) ) . intpos ( 0 + 4 ) = 0 ;
L~ ( M1 * M2 ) meets L~ ( M2 * M2 ) ;
set h = the continuous Function of X , R ;
set A = { L . ( ( k + 1 ) \ A . ( k + 1 ) where k is Element of NAT : not contradiction } ;
for H st H is negative holds P [ H ] ;
set bmin = Smin ( i , -' ( i + 1 ) ) ;
Hom ( a , b ) c= Hom ( a , b ) ;
sqrt ( 1 - n + 1 ) < sqrt ( 1 - s ) ;
( l , 1 ) `2 = [ [ [ dom l , cod l ] , [ a , cod l ] ] , [ a , cod l ] ] ;
y +* ( i , y /. i ) in dom g ;
let p be Element of QC-WFF ( Al ) , x be Element of QC-WFF ( Al ) ;
X /\ X1 c= dom ( f1 - f2 ) /\ dom ( f2 - f1 ) ;
p2 in rng ( f /^ ( 1 + 1 ) ) & p2 in rng ( f /^ ( 1 + 1 ) ) ;
1 <= indx ( D2 , D1 , j1 ) & indx ( D2 , D1 , j1 ) <= 2 ;
assume x in ( K /\ L ) \/ ( K /\ L ) /\ K ;
- 1 <= ( f2 . O ) . O & ( f2 . O ) . I = ( f2 . O ) . I ;
let f , g be Function of I[01] , TOP-REAL 2 ;
k1 - ( k + 1 ) = ( k - 1 ) - ( k + 1 ) ;
rng ( seq + c ) c= ]. x0 - r , x0 .[ ;
g2 in ]. x0 - r , x0 + r .[ ;
sgn ( p `1 , K ) = - ( - ( - 1 ) * ( - 1 ) * ( - 1 ) ) ;
consider u being Nat such that b = ( p |^ y ) * u ;
There exists a normal sequence of REAL st a = Sum A & Sum A = Sum A ;
Cl ( ( Cl ( H ) ) | ( union ( H ) ) ) = union ( ( Cl ( H ) | ( union ( H ) ) ) ) ;
len t = len ( t1 + t2 ) + len ( t2 + t1 ) ;
v = v + w & v + w = v + ( A + B ) ;
v <> DataLoc ( t . GBP , 3 ) & v . DataLoc ( t . GBP , 3 ) = 0 ;
g . s = sup ( d " { s } ) ;
( \dot { y , s ) . s = s . ( y , s ) ;
{ s : s < t } in NAT & t = {} + 1 } ;
s ` \ s = s ` \ ( 0. X ) .= s ` \ ( 0. X ) ;
defpred P [ Nat ] means B + $1 in A ;
( 319 + 1 ) ! = ( 319 + 1 ) ! * ( 319 + 1 ) ;
U _ { A , succ A } = U ( A , succ A ) ;
reconsider y = y as Element of ( len y ) -tuples_on the carrier of K ;
consider i2 being Integer such that y = p * i2 + ( 1 - i2 ) * i2 ;
reconsider p = Y | Seg ( k + 1 ) as FinSequence of ( the carrier of K ) ;
set f = ( S , U ) \! \mathop { \rm \hbox { - } string } ;
consider Z be set such that lim s in Z and Z in F ;
let f be Function of ( TOP-REAL n ) | K1 , R^1 ;
( SAT M ) . [ n + i , 'not' A ] <> 1 ;
ex r being Real st x = r & a <= r & r <= b ;
R1 , R2 be Element of REAL n , a be Element of REAL n ;
reconsider l = 0. ( V , m ) as Linear_Combination of A ;
set r = |. e .| + |. w .| ;
consider y being Element of S such that z <= y and y in X ;
a in ( a "\/" b ) 'or' ( a 'or' c ) = 'not' ( ( a 'or' b ) 'or' ( a 'or' c ) ) ;
||. ( x1 - x2 ) - ( g - x2 ) .|| < r2 ;
b9 , c9 // b9 , c9 & a9 , c9 // c9 , c9 ;
1 <= ( ( k - 1 ) - ( k + 1 ) ) / ( ( k + 1 ) - ( k + 1 ) ) / ( ( k + 1 ) - ( k + 1 ) ) ) = ( k + 1 ) / ( ( k + 1 ) - ( k + 1 ) ) ;
sqrt ( ( p `1 / |. p .| - sn ) / ( 1 + sn ) ) ^2 ) >= 0 ;
sqrt ( ( q `1 / |. q .| - sn ) / ( 1 + sn ) ) < 0 ;
W-min ( C , 1 ) in Support ( ( R /. 1 ) * ( R /. 1 ) ) ;
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 , a , c ;
p `1 , a // a `1 , b `2 or p `2 = b ;
g . n = a * Sum ( f | n ) .= f . n * ( a | n ) ;
consider f being Subset of X such that e = f and f is x1 ;
F | ( [: S2 , S :] ) = ( F * ( S1 , S2 ) ) | ( S2 , S2 ) ;
q in LSeg ( q , v ) \/ LSeg ( p , v ) ;
Ball ( m , r ) c= Ball ( m , s ) ;
the carrier of ( V | A ) = { 0. V } ;
rng ( ( id ( the carrier of TOP-REAL 2 ) ) | K1 ) = [. - 1 , 1 .] ;
assume that Re ( seq ) is summable and Im ( seq ) is summable ;
||. ( v . n - t . n ) .|| < e ;
set g = O --> 1 ;
reconsider t2 = ( t | 11 ) as 0 -started string of S2 ;
reconsider xA = seq as sequence of ( TOP-REAL n ) | A ;
assume that W-min L~ Cage ( C , n ) meets L~ Cage ( C , n ) and not W-min L~ Cage ( C , n ) meets L~ Cage ( C , n ) ;
- ( ( 1 / ( n + 1 ) ) (#) ( 1 / ( n + 1 ) ) ) < F . n ;
set d1 = \mathopen ( x1 , y1 , y2 ) , d2 = dist ( x1 , y1 , y2 ) ;
2 to_power ( 2 to_power 1 ) = ( 2 to_power 1 ) * ( 2 to_power 1 ) ;
dom ( v | ( len ( d | i ) ) ) = Seg ( len ( d | i ) ) .= Seg ( len ( d | i ) ) ;
set x1 = - ( ( k + 1 ) / ( k + 1 ) ) , x2 = ( k + 1 ) / ( k + 1 ) ;
assume for n being Element of NAT holds 0. ( X , X ) <= F . n ;
assume that 0 <= ( T . i ) . ( i + 1 ) and T . ( i + 1 ) <= 1 ;
for A being Subset of X holds c . A = c . A
the carrier of ( ( Carrier ( L ) + R ) ) /\ ( Carrier ( L ) + R ) c= I ;
'not' All ( x , p ) => ( 'not' All ( x , p ) ) is valid ;
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the carrier of V ;
Z c= dom ( ( ( exp_R * f1 ) `| Z ) ;
|. 0. TOP-REAL 2 - ( q `2 / |. q .| - sn ) .| < r ;
ConsecutiveSet ( A , succ B ) c= ConsecutiveSet ( A , succ C ) ;
E = dom ( L | A ) & ( for n holds L . n = 0 ) implies ( for n holds L . n = 0 ) & for n st n in dom L holds L . n = 0 ) & for n st n in dom L holds L . n = 0 ) implies for
C |^ ( A + B ) = C |^ ( A + B ) ;
the carrier of W2 c= the carrier of V & the carrier of V c= the carrier of V ;
I . IC Comput ( P , s , k ) = P . IC Comput ( P , s , k ) .= P . IC Comput ( P , s , k ) ;
attr x > 0 means : Def2 : sqrt ( 1 - x ) = x / ( 1 - x ) ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) ;
consider p being Point of T such that C = [. p , q .] ;
b , c , d & a , b , c is_collinear & b , c , d is_collinear implies a , b , c is_collinear & b , c , d is_collinear & a , c , d , b is_collinear & a , c , d is_collinear & b , c , d , b is_collinear & a , c
assume f = id ( the carrier of C ) , ( id ( the carrier' of C ) ) . ( f . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . x ) ) ) ) ) ) ) ) ) ;
consider v such that v <> 0. V and f . v = L * v ;
let l be Linear_Combination of {} ( ( the carrier of V ) --> 0. ( V ) ) ;
reconsider g = f " as Function of ( the carrier of U1 ) , the carrier of U2 ;
A1 in the points of ( G . k ) & A2 : not ex X being Subset of ( G . k ) st X = { X } ;
|. - x .| = - ( - x ) .= - ( - x ) .= - ( - x ) ;
set S = 1GateCircStr ( x , y , c ) ;
Fib ( n ) * ( 5 * n ) >= 4 * n ;
vk /. ( k + 1 ) = vk . ( k + 1 ) ;
0 mod i = - ( i * ( 0 qua Nat ) ) ;
Indices ( M1 @ ) = [: Seg n , Seg n :] & width ( M2 @ ) = width ( M2 @ ) ;
Line ( Sj , j ) = Sj . ( j , i ) ;
h . ( x1 , y1 ) = [ y1 , y2 ] ;
|. f .| (#) ( |. b .| (#) ( ( ( ( ( ( f ) ) (#) h ) ) (#) h ) ) | A ) is nonnegative ;
assume x = ( a1 ^ <* b1 *> ) ^ <* b2 *> ;
M is closed implies ( for s being State of SCM+FSA , P being Subset of SCM+FSA holds P [ s , P ] ) & P [ s , P ]
DataLoc ( t3 . a , 4 ) = intpos ( 0 + 4 ) ;
x + y < - x + y & |. x - y .| = - x ;
LIN c , q , b & LIN c , b , q ;
f . ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + y2 ) ;
f . a = ( f . a ) & v in InputVertices S ;
( p `1 ) ^2 <= ( W-min ( C ) ) ^2 ;
set R8 = Cage ( C , n ) \circlearrowleft E8 , E8 = Cage ( C , n ) \circlearrowleft E8 ;
( p `1 ) ^2 >= ( ( W-min C ) ^2 + ( E-max C ) ^2 ;
consider p such that p = ( p | i ) and s1 < p /. i ;
|. ( f /* ( s * F ) ) . l - ( f /* s ) . l .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len Line ( N , k + 1 ) = width N + 1 ;
f1 /* ( f1 /* s ) is convergent & f2 /* ( f1 /* s ) is convergent ;
f . ( x1 , y1 ) = x1 & f . ( y1 , y2 ) = x2 ;
len f <= len f + 1 & len f + 1 <> 0 ;
dom ( Proj ( i , n ) * s ) = [: the carrier of TOP-REAL m , the carrier of TOP-REAL m :] ;
n = k * ( 2 * t + n ) + ( n mod ( 2 * t ) ) ;
dom B = 2 -tuples_on the carrier of V \ { {} } & { {} } c= the carrier of V ;
consider r such that r \not _|_ a and r _|_ x ;
reconsider B1 = the carrier of ( Y | ( the carrier of X ) ) as Subset of X ;
1 in the carrier of [. 1 , 1 .] & ( 1 - 1 ) * ( 1 - 1 ) <= 1 ;
for L being complete LATTICE holds L , L are_isomorphic implies L , L are_isomorphic & L , L are_isomorphic
[ gi , gj ] in [: I , I :] \ ( I \ J ) ;
set S2 = 1GateCircStr ( x , y , c ) ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and f1 . x0 = lim ( f1 , x0 ) ;
reconsider y = ( a " ) / ( b " ) as Element of L ;
dom s = { 1 , 2 , 3 } & s . 1 = d & s . 2 = d ;
( min ( g , max ( f , g ) ) ) . c <= h . c ;
set G1 = the Vertex of G , G2 = the Vertex of G , G2 = the Vertex of G , G2 = the Vertex of G ;
reconsider g = f as PartFunc of REAL n , REAL n ;
|. s1 . m - p . ( n + 1 ) .| < d / ( n + 1 ) ;
for x being element st x in one-to-one ( u , t ) holds x in Q ( u , t ) ;
P = the carrier of ( ( TOP-REAL n ) | K1 ) | K1 .= K1 ;
assume that p1 in LSeg ( p1 , p2 ) /\ LSeg ( p2 , p3 ) and p2 in LSeg ( p3 , p4 ) /\ LSeg ( p3 , p4 ) ;
( 0. X \ x ) |^ ( m + 1 ) = 0. X ;
let g be Element of Hom ( cod f , cod g ) ;
2 * a + b * c + ( 2 * a * b ) <= 2 * ( a * b ) ;
let f , g , h be Point of the \overline of X , Y ;
set h = Hom ( a , g ) ;
then idseq ( n , m ) | Seg n = idseq ( m , n ) ;
H * ( g " * a ) in the carrier of H ;
x in dom ( ( id ( the carrier of C ) ) | K1 ) ;
cell ( G , i1 , j1 -' 1 ) misses C & not ex n st G * ( i1 -' 1 , j1 ) misses C & not contradiction ;
LE q2 , q1 , P , p1 , p2 & LE q1 , q2 , P , p1 , p2 , p1 , p2 ;
attr B is component means : Def2 : B c= BDD A & B c= BDD A ;
deffunc D ( set , set , set ) = union ( rng $2 ) ;
n + - n < len ( p - q ) + ( - n ) ;
attr a <> 0. ( K , len M ) means : Def2 : rk ( a , M ) = rk ( a , M ) ;
consider j such that j in dom |^ m and I = len b1 + j ;
consider x1 such that z in x1 and x1 in ( P * ( x1 , x2 ) ) and ( P * ( x1 , x2 ) ) . ( x1 , x2 ) = P * ( x1 , x2 ) ;
for n ex r being Element of REAL st X [ n , r ]
set C1 = Comput ( P2 , s2 , i + 1 ) , C1 = Comput ( P2 , s2 , i + 1 ) ;
set v = 3 / ( a , b , c ) , w = 4 / ( a , b , c ) , f = 5 / ( a , b , c ) , g = 5 / ( a , b , c ) ;
conv ( @ W ) c= union ( F .: ( E .: W ) ) ;
1 in [. - 1 , 1 .] /\ dom ( ( - 1 ) (#) ( ( id ( ( - 1 ) (#) ( id ( ( - 1 ) ) ) ) ) ) ) ;
s3 <= s2 + ( s2 - s1 ) * ( s2 - s1 ) ;
dom ( f * ( f1 | A ) ) = dom f /\ A .= A /\ B ;
dom ( f * G ) = dom ( l (#) F ) /\ Seg k .= Seg k /\ Seg k ;
rng ( s ^\ k ) c= dom f1 \ { x0 } & rng ( s ^\ k ) c= dom f2 \ { x0 } ;
reconsider g2 = ( gp ) . ( n + 1 ) as Point of ( TOP-REAL n ) | P ;
( T * h . s ) . x = T . ( h . s . x ) ;
I . ( J . ( J . x ) ) = ( I * L ) . ( J . x ) ;
y in dom ( ( the Sorts of A ) * ( ( the Arity of S ) . ( o , y ) ) ) ;
for I being non degenerated commutative doubleLoopStr holds I is commutative
set s2 = s +* ( intloc 0 , 1 ) ;
P1 /. IC Comput ( P1 , s1 , k ) = P1 . IC Comput ( P2 , s2 , k ) .= P1 . IC Comput ( P2 , s2 , k ) ;
lim ( S1 , a ) in the carrier of \lbrack a , b .] & lim ( S1 , a ) in the carrier of \lbrack a , b .] ;
v . ( l . i ) = ( v *' ) . ( l . i ) ;
consider n being element such that n in NAT and x = seq . n ;
consider x being Element of c such that F1 . x <> F ( x ) and F1 . x <> F ( x ) ;
Choose ( X , 0 , x1 , x2 , x3 , x4 ) = { E , F } ;
j + ( 2 * ( k + 1 ) ) + m1 > j + ( 2 * ( k + 1 ) ) ;
{ s , t } on ( B1 \/ B2 ) & { s , t } on ( B1 \/ B2 ) ;
n1 > len ( p2 , p1 , p2 , n1 , n2 , n2 , n2 , n2 , n3 , n2 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( g2 g2 ) ) ) ) ) ) ) ) | ( ( ( ( g2 ) ) | ( ( g2 | ( g2 | ( g2 | ( g2 | ( g2 | ( g2 | ( g2 | ( g2 | ( g2 | ( g2 | ( g2 | ( g2
then H1 , H2 are_isomorphic & card H1 , H2 are_isomorphic & card H1 , H2 are_isomorphic ;
( W-min L~ f ) .. ( ( W-min L~ f ) .. ( f ) ) > 1 ;
]. s , 1 / 2 .[ = ]. s , 1 .] /\ ]. 0 , 1 .] ;
x1 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) | ( L~ g ) ) ;
let f1 , f2 be continuous PartFunc of REAL , REAL n ;
( DigA ( ti1 , z ) ) . ( ( k + 1 ) -tuples_on ( the carrier of S ) ) is Element of k -tuples_on ( the carrier of S ) ;
I = ( d , k1 ) := k2 & I = ( d , k1 ) := k2 & I = k2 & I = k2 ;
uu = { [ a , u ] , [ b , u ] } ;
( w | p ) | ( w | p ) = p ;
consider u2 such that v2 in W2 and x = v + ( u + v2 ) ;
for y st y in rng F ex n st y = a |^ n & P [ a , n ]
dom ( ( g * ( id V ) ) | K ) = K ;
ex x being element st x in ( ( the Sorts of U1 ) \/ A ) . s & A . s = ( the Sorts of U1 ) . s ;
ex x being element st x in ( ( ( the Sorts of O ) \/ A ) . s ) . s & ( the Sorts of O ) . s = ( the Sorts of A ) . s ;
f . x in the carrier of [. r , s .] & f . x in [. p , q .] ;
( the carrier of X1 union X2 ) /\ ( the carrier of X2 ) <> {} ;
L1 /\ LSeg ( p1 , p2 ) c= { p1 } /\ L1 /\ L2 ;
sqrt ( b + ( b-2 ) / 2 ) in { r where r is Real : a < r } ;
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 ( G | A ) such that z = y and P [ z , x ] ;
( the addF of ( \rm seq ) ) . ( e , e ) <= e ;
len ( w ^ ( w ^ ( w ^ ( w ^ ( w ^ ( w ^ ( w ^ ( w ^ ( w ^ ( w ^ ( w ^ ( w ^ ( w ^ ( w ^ ( w ^ ( w ^ ( w ^ ( w ^ ( w ^ ( w ^ ( w ^ ( w ^ (
assume q in the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 & q in K1 ;
f | ( ( TOP-REAL 2 ) | K1 ) = g | K1 .= ( ( TOP-REAL 2 ) | K1 ) | K1 .= ( TOP-REAL 2 ) | K1 ;
reconsider i1 = x1 , i2 = x2 , j2 = x3 , j1 = x4 ( x1 , x2 , x3 ) as Element of NAT ;
( a * A ) / ( a * B ) = ( a * A ) / ( a * B ) ;
assume ex n being Element of NAT st f |^ n is \lbrace 0 } ;
Seg ( len ( ( f2 | i ) | ( len ( f2 | i ) ) ) ) = dom ( ( f2 | i ) | ( dom ( f2 | i ) ) ) ;
( ( Complement ( A ) ) . m ) . n c= ( ( Complement ( A ) ) . n ) . m ;
f1 . p = ( ( f1 | A ) . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p . (
FinS ( F , Y ) = FinS ( F , Y ) .= FinS ( F , Y ) ;
( x | y ) | z = ( y | x ) | ( y | x ) ;
sqrt ( |. x .| ^2 + ( |. x .| ) ^2 ) <= sqrt ( ( |. x .| ) ^2 + ( |. x .| ) ^2 ) ;
Sum ( F ) = Sum f & dom ( F ) = dom g & dom ( F ) = dom g ;
assume for x , y , z being set st x in Y holds x /\ y in Y ;
assume that W1 is Subspace of W2 and W2 is Subspace of W1 and W1 is Subspace of W2 ;
||. ( t . x ) .|| = lim ( ||. x .|| ) .= lim ( ||. x .|| ) ;
assume that i in dom D and f | A is lower and g | A is lower ;
sqrt ( ( p `1 ) ^2 - ( p `2 ) ^2 ) <= sqrt ( ( p `2 ) ^2 - ( p `2 ) ^2 ) ;
g | Sphere ( p , r ) = id ( the carrier of TOP-REAL n ) ;
set N8 = W-min ( C , n ) , S8 = W-min ( C , n ) ;
for T being non empty TopSpace holds T is countable implies the TopStruct of T is countable
width B |-> 0. ( K , n , n ) .= width ( B @ ) .= width ( B @ ) ;
attr a <> 0 means : Def2 : ( A /\ B ) +^ a = ( A /\ a ) +^ ( B /\ a ) ;
then f is partial u0 u0 u0 u0 & pdiff1 ( f , 1 ) is_partial_differentiable_in u0 , 1 ;
assume that a > 0 and a <> 1 and b <> 0 and c <> 0 ;
w1 , w2 in Lin ( { w1 , w2 } , { w2 , w1 } ) ;
p2 /. IC Comput ( p2 , s2 , k ) = p2 . IC Comput ( p2 , s2 , k ) .= p2 . IC Comput ( p2 , s2 , k ) ;
ind ( T | b ) = ind b .= ind ( T | b ) .= ind ( T | b ) ;
[ a , A ] in the InternalRel of Line ( A , 1 ) & [ a , A ] in Indices ( Line ( A , 1 ) ) ;
m in ( the Arrows of C ) . ( o1 , o2 ) & ( the Arrows of C ) . ( o2 , o2 ) = ( the Arrows of C ) . ( o2 , o2 ) ;
( ( 'not' a , CompF ( PA , G ) ) ) . z = TRUE ;
reconsider phi = phi , phi = phi , phi = phi , phi = phi , phi = phi , phi = phi , phi = phi , phi = phi , phi = phi , phi = phi , phi = phi , phi = phi , phi = phi , phi = S , phi = S , phi = S , phi = S , phi = S , phi =
len ( s1 - ( len s2 - 1 ) ) * ( len s1 - 1 ) + 1 > 0 + 1 ;
\delta ( f , D ) * ( ( f . ( upper_bound A ) ) - ( f . ( lower_bound A ) ) ) < r ;
[ f , f ] in the InternalRel of A & [ f , f ] in the InternalRel of A & [ f , f ] in the InternalRel of B ;
the carrier of ( ( TOP-REAL 2 ) | K1 ) = K1 & ( TOP-REAL 2 ) | K1 = K1 ;
consider z being element such that z in dom g2 and p = g2 . z ;
[#] ( ( V | ( V | ( V | ( V | ( V | ( V | ( V | ( V | ( V | ( V | ( V | ( V | ( V | ( V | ( V | ( V | ( V | ( V | ( V | ( V | ( V | ( V | ( V | ( V | ( V | ( V |
consider P2 being FinSequence such that rng P2 = M and P2 is one-to-one and P2 is one-to-one ;
assume that x1 in dom ( f | X ) and ||. x1 - x0 .|| < s ;
h1 = f ^ ( <* p3 *> ^ <* p3 *> ) .= h ^ <* p3 *> ^ <* p3 *> ^ <* p3 *> ^ <* p3 *> ^ <* p3 *> ^ <* p3 *> ^ <* p3 *> ^ <* p3 *> ^ <* p3 *> ^ <* p3 *> ^ <* p3 *> ^ <* p3 *> ^ <* p3 *> ^ <* p3 *> ^ <* p3 *> ^ <* p3 *> ^
c /. [ b , c ] = c /. [ a , c ] .= c /. ( a , b ) ;
reconsider t1 = p1 , t2 = p2 , t2 = p3 as Term of C , V ;
sqrt ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( q ) ) ) ) ) ) ) ) ) ) ) ) ) in REAL ;
ex W being Subset of X st p in W & h .: W c= V ;
( h . p1 ) `2 = C * ( ( p1 `2 ) + D ) `2 + D * ( p1 `2 ) ) + D ;
R . b = 2 * PI .= 2 * PI .= 2 * PI .= 2 * PI ;
consider 1- 1- 1 = ( 1 - lambda ) * C + ( - 1 ) * A and 0 <= 1 ;
dom g = dom ( ( the Sorts of A ) * ( the Arity of S ) ) ;
[ P . ( l , P . ( l , T . ( l , T . ( l , T . ( l , T . ( l , T . ( l , T . ( l , T . ( l , T . ( l , T . ( l , T . ( l , T . ( l , T ) ) ) ) ) ) ) ) ) ] in
set s2 = Initialize ( s ) , P2 = P +* I ;
reconsider M = mid ( z , i2 , i1 ) as Element of REAL 2 ;
y in product ( ( the support of J ) +* { 1 , 0 } ) ;
1 / ( 0 , 1 ) = 1 & 0 < 1 / ( 0 , 1 ) ;
assume x in the left left of g or x in the carrier' of g ;
consider M being strict Subgroup of ( A , B ) such that a = M and M is Subgroup of A ;
for x st x in Z holds ( ( ( ( for x st x in Z holds exp_R . x ) + exp_R . x ) ) & ( for x st x in Z holds exp_R . x = 0 ) implies exp_R . x <> 0
len ( W1 + ( len W1 + m ) ) = 1 + ( len W1 + m ) .= ( len W1 + m ) + ( len W1 + m ) ;
reconsider h1 = ( v . n - t . n ) as Lipschitzian Lipschitzian Lipschitzian Function of X , Y ;
( i mod ( len p + 1 ) ) + 1 in dom ( p + q ) ;
assume that s2 is negative and F in the InternalRel of ( the InternalRel of S ) and F in the InternalRel of ( the InternalRel of S ) and F in the InternalRel of ( the InternalRel of S ) \/ the InternalRel of S ;
( ( ( ( x , y , z ) ) / ( x , y , z ) ) / ( x , y , z ) ) / ( x , z ) = gcd ( x , y , z ) ;
for u being element st u in Bags n holds ( p *' m ) . u = p . u
for B being Subset of u st B in E holds A = B or A misses B ;
ex a being Point of X st a in A & A /\ Cl { y } = { a } ;
set W2 = [: p , q :] \/ [: q , p :] ;
x in { X where X is Subset of L : X c= { X } } ;
the carrier of W1 /\ ( the carrier of W2 ) c= the carrier of W1 /\ ( the carrier of W2 ) ;
( id ( a + b ) ) * id ( a + b ) = id ( a + b ) * id ( a + b ) ;
( dom ( X --> f ) ) . x = ( X --> f ) . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) ;
p => ( q => r ) => ( p => ( q => r ) ) in TAUT ( A ) ;
set cos = LSeg ( G * ( i1 , j1 ) , G * ( i1 , j1 ) ) ;
set cos = LSeg ( G * ( i1 , j1 ) , G * ( i1 , j1 ) ) ;
- 1 + 1 <= sqrt ( ( m - n ) / ( m - n ) ) + 1 ;
( reproj ( 1 , z ) ) . x in dom ( f1 * f2 ) ;
assume that b1 . r = { c1 } and b2 . r = { c2 . r } ;
ex P st a1 on P & a2 on P & a1 , b1 , b2 is_collinear & b1 , c1 , b1 is_collinear & b1 , c1 , b1 is_collinear & b2 <> b1 , b1 , b2 is_collinear & b1 , c1 , b1 is_collinear & b1 , c1 , b1 is_collinear & b1 , c1 , b2 is_collinear & b1 , c1 , b1 is_collinear & b1 , c1 , b1
reconsider gf = g * f , h1 = h * f as strict Subgroup of X ;
consider v1 being Element of T such that Q = ( \mathopen { \downarrow } v1 ) ` and v1 in ( the InternalRel of T ) ` ;
n in { i where i is Nat : i < n + 1 } ;
( F /. ( i , j ) ) `2 >= ( ( F /. ( m , k ) ) `2 ) `2 ;
assume K = { p : ( p `1 / |. p .| - sn ) / ( 1 + sn ) >= sn } ;
ConsecutiveSet ( A , succ O1 ) = ( ConsecutiveSet ( A , succ O1 ) ) ^ ( ConsecutiveSet ( A , succ O1 ) ) ;
set I1 = Macro ( a , intloc 0 ) , I2 = intloc 0 , I2 = intloc 0 , I1 = intloc 0 , I1 = intloc 0 , I2 = intloc 0 , I1 = intloc 0 , I1 = intloc 0 , I1 = intloc 0 , I1 = intloc 0 , I1 = intloc 0 , I1 = [ a , I1 ] , I2 , I2 ] ;
for i being Nat st 1 < i & i < len z holds z /. i <> z /. 1
X c= ( the carrier of L1 ) /\ ( the carrier of L2 ) & X c= ( the InternalRel of L1 ) /\ ( the InternalRel of L2 ) ;
consider x9 being Element of GF ( p ) such that x9 |^ ( p ) = a and x9 in C ;
reconsider e1 = e1 , e2 = f , e2 = g as Element of D ( ) ;
ex O being set st O in S & ( for C being set st C c= O holds M . C = 0. ( ( the carrier of V ) | ( the carrier of V ) ) ) & M . C = 0. ( V ) ;
consider n be Nat such that for m being Nat st n <= m holds S . m in U ;
f * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) * x ) . x ;
defpred P [ Nat ] means A + ( $1 + 1 ) = succ ( $1 + 1 ) ;
the left left of - ( - g ) = the carrier' of ( - g ) .= the carrier' of ( - g ) ;
reconsider pX2 = x , pX2 = y as Point of TOP-REAL 2 ;
consider g2 such that g2 = y and x <= g2 and g2 <= x and x <= g2 and g2 <= x and g2 <= x and g2 <= x ;
for n being Element of NAT holds X [ n , r . n ] ;
len ( x2 ^ y2 ) = len ( x2 ^ y2 ) + len ( y2 ^ z2 ) .= len ( x2 ^ y2 ) + len ( y2 ^ z2 ) ;
for x being element st x in X holds x in the set of ( the set of n ) --> ( 0 , 0 )
LSeg ( p1 , p2 ) /\ LSeg ( p2 , p1 ) = {} ;
func ( -1 ) -> set equals ( id ( X ) ) | ( ( id ( X ) ) | ( X ) ) | ( X ) ;
len ( ( the _ of ( C , n ) /. 1 ) ) <= len ( ( the _ of C , n ) /. 1 ) ;
attr K is L means : Def2 : a <> 0. K & v . ( a , b ) = i * v ;
consider o being OperSymbol of S such that t . {} = [ o , the carrier of S ] and o . {} = [ o , the carrier of S ] ;
for x st x in X ex y st y c= X & y in X & f . x = b ;
IC Comput ( P-6 , Comput ( P-6 , m1 , k ) , k ) in dom ( PmmE ) ;
attr q < s means : Def2 : ]. r , s .] c= ]. p , q .] ;
consider c being Element of Class ( f , c ) such that Y = ( F . c ) . ( F . c ) ;
func the ResultSort of S2 -> ResultSort of S1 , the carrier' of S2 means : Def2 : the ResultSort of S1 = id S2 & the ResultSort of S2 = id S2 ;
set y9 = [ <* y , z *> , f2 ] , f3 ] ;
assume x in dom ( ( ( ( ( id Z ) (#) ( ( id Z ) * ( ( id Z ) * ( ( id Z ) ) ) ) `| Z ) ) `| Z ) ;
r-7 in Int cell ( GoB f , i , width GoB f ) \ { G * ( i , 1 ) where i , 1 ) `1 is Nat : G * ( i + 1 , 1 ) `1 <= r } ;
( q `1 ) ^2 >= ( ( W-min ( C , n ) ) ) ^2 ;
set Y = { a "/\" a : a in X } ;
i - len f <= len f + ( len f -' 1 ) - len f + ( len f -' 1 ) ;
for n ex x st x in N & x in N & h . n = U ( n )
set s = ( \it true ( a , I , J ) ) . i , p = ( s , p ) . i , s = ( s , p ) . i , p ) . i ;
( p . k ) . 0 = 1 or ( p . 0 ) . 0 = 1 or ( p . 0 ) . 0 = 0 ;
u + Sum ( L \ { u } ) in ( U \ { u } ) \/ { u + Sum ( L \ { u } ) ;
consider x9 be set such that x in x9 and x9 in ( the carrier of S ) and x9 in ( the carrier of S ) /\ ( the carrier of S ) ;
( p ^ q ) . m = ( q | ( Seg ( len p ) ) ) . ( ( Seg ( len p ) ) ) . ( Seg ( ( len p ) + 1 ) ) ) ;
g + h = ( gg + h ) + ( g + X ) & f + h = g + h ;
L1 is distributive & L2 is distributive implies for x being Element of L1 holds L1 . x = L2 . x & L2 . x = L2 . x
attr x in rng f means : Def2 : y in rng f & x in rng f implies f . y = f . x ;
assume that 1 < p and sqrt ( 1 + ( p `2 / q ) ^2 ) = 1 and 0 <= ( p `2 / q ) ^2 ;
F* ( f , t ) = rpoly ( 1 , t ) *' + ( t *' ) *' .= ( 1 *' ) *' + ( t *' ) *' ;
for X being set , A being Subset of X holds A ` = {} implies A ` = {}
( W-min ( X ) ) `1 <= ( ( W-min ( X ) ) ) `1 ;
for c being Element of the bound of A holds c <> a & c <> a ;
s1 . ( ( the Sorts of A ) . ( the Sorts of A ) . ( the Sorts of B ) . ( the Arity of S ) . ( the Arity of S ) ) = 0 ;
for a , b , c being Real holds [ a , b ] in ( y iff a >= 0 ) & b >= 0 implies a >= 0
for x , y being Element of X holds x \ ( x \ y ) = ( x \ y ) \ ( x \ y )
mode BCK-algebra of i , j , m , n , m , n , k be Nat , i , j be Element of NAT , m be Element of NAT , n be Element of NAT , n be Element of NAT , m be Element of NAT ;
set x2 = ( Re ( y , Im ( y , z ) ) ) | ( Im ( y , z ) ) ;
[ y , x ] in dom ( u . ( y , x ) ) & u . ( y , x ) = g . ( y , x ) ;
]. lower_bound divset ( D , k ) - lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .] c= A ;
0 <= \delta ( S2 . n , S2 . n ) & |. S2 . n - S2 . n .| < r / 2 ;
( - ( q `1 / |. q .| - sn ) ) / ( 1 + sn ) <= ( - ( q `1 / |. q .| - sn ) ) / ( 1 + sn ) ;
set A = sqrt ( 2 * PI ) ;
for x , y being set st x in R1 & y in R2 holds x , y are_connected holds x , y are_connected
deffunc F ( Nat ) = b . ( $1 + 1 ) * ( M . ( $1 + 1 ) ) ;
for s being element holds s in |= ( f 'or' g ) iff s in |= ( f 'or' g ) \/ |= ( f 'or' g )
for S being non void non empty non void ManySortedSign holds S is connected implies S is connected
max ( ( z `1 ) / ( |. z .| ) , ( |. z .| ) / ( |. z .| ) ) >= 0 ;
consider n1 be Nat such that for k holds seq . ( n + k ) < r + s ;
Lin ( A /\ B ) is Subspace of Lin ( A /\ B ) & Lin ( B ) is Subspace of Lin ( A ) ;
set n-15 = ( M . ( x , n ) ) '&' ( M . ( x , n ) ) , ( M . ( x , n ) ) = ( M . ( x , n ) ) '&' ( M . ( x , n ) ) ) ;
f " ( V ) in Funcs ( X , X ) & f " ( V , p ) in D ;
rng ( ( a , c ) := ( b , c ) ) c= { a , b , c } ;
consider y being Vertex of G1 such that y = y and dom y = dom the InternalRel of G1 and y in dom the InternalRel of G2 and y in dom the InternalRel of G2 ;
dom ( 1 / ( f | ]. 0 , PI / 2 .[ ) ) /\ ]. 0 , PI .[ c= ]. 0 , PI .[ ;
AffineMap ( i , j , n , r ) is Element of Funcs ( i , j , n , n ) ;
v ^ ( n |-> 0 ) in Lin ( ( ( B | ( n + 1 ) ) |-> 0 ) ) ;
ex a , k1 , k2 st i = a & ( ex k1 , k2 st i = a & k2 = b & k2 = c & ( ex k2 , k2 st i = b & k2 = c & k2 = c & k2 = d ) & ( ex k2 , k2 st i = c & k2 = d & k2 = d & k2 = c ) ) ;
t . ( NAT , NAT ) = ( NAT .--> ( the Sorts of A1 ) . ( NAT , NAT ) ) . ( NAT + 1 ) .= ( the Sorts of A2 ) . ( NAT + 1 , NAT ) .= ( the Sorts of A1 ) . ( NAT + 1 ) .= ( the Sorts of A2 ) . ( NAT + 1 ) ;
assume that F is bbbe Subset-Family of X and rng p = dom p and dom p = Seg ( n + 1 ) and dom p = Seg ( n + 1 ) and dom p = Seg ( n + 1 ) ;
not LIN b `1 `1 , a & not LIN b , a , c ;
( L1 \/ L2 ) \ O c= ( L1 \/ L2 ) \ O & ( L1 \/ L2 ) \ O c= ( L1 \/ L2 ) \ O ;
consider F be ManySortedSet of E such that for d being Element of D holds F . d = F ( d ) ;
consider a , b such that a * ( w , v ) = b * ( w , w ) and 0 < b ;
defpred P [ FinSequence of D ] means |. Sum ( $1 ) .| <= Sum ( $1 ) ;
u = cos / ( x , y ) + cos / ( x , y ) .= cos / ( x , y ) + cos / ( x , y ) .= v ;
dist ( seq . n + x , x + g ) <= dist ( seq . n , x + g ) + 0 ;
P [ p , |. p .| ] , [ {} , id ( the carrier of A ) ] ;
consider X be Subset of QC-WFF ( Al ) such that X c= Y and X is non empty and X is finite and X is non empty ;
|. b .| * |. eval ( f , z ) .| >= |. b .| * |. ( f . z ) .| ;
1 < ( W-min L~ Cage ( C , n ) ) .. ( Cage ( C , n ) ) .. ( Cage ( C , n ) ) ;
l in { ( ( l1 - 1 ) * ( l - 1 ) ) * ( ( l - 1 ) * ( l - 1 ) ) + ( l - 1 ) * ( l - 1 ) * ( l - 1 ) ) ;
( Partial_Sums ( G ) ) . n <= ( Partial_Sums ( G ) ) . n ;
f . y = x * ( x * y ) .= x * ( y * ( x * y ) ) .= x * ( y * ( x * y ) ) ;
NIC ( i1 , i2 ) = { i1 , i2 , j1 , j2 } & Exec ( i2 , j2 ) = { i1 , i2 , j2 } ;
LSeg ( p1 , p2 ) /\ LSeg ( p2 , p1 ) = { p1 } /\ LSeg ( p2 , p2 ) ;
product ( ( the Sorts of ( I , <* i *> ) ) +* ( { i } , <* 1 *> ) ) . ( i + 1 ) in INT ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s , n ) | ( the carrier of S2 ) .= Following ( s , n ) ;
W-min ( P ) <= ( ( q `2 ) / 2 ) * ( ( q `2 ) / 2 ) ;
f /. ( i + 1 ) <> f /. ( i1 + len g -' 1 ) ;
M , ( ( f | ( { x } ) | ( { x } ) ) | ( { x } ) ) / ( H . ( { x } ) | ( { x } ) ) ) / ( H . ( { x } ) ) ) / ( H . ( { x } ) ) ) / ( H . ( { x } ) ) ) ;
len ( ( P1 ^ P2 ) ^ ( P2 ^ s2 ) ) in dom ( P1 ^ P2 ) ;
( A |^ ( m , n ) ) c= ( A |^ ( m , n ) ) * ( A |^ ( k , n ) ) & ( A |^ ( k , n ) ) c= ( A |^ ( k , n ) ) * ( A |^ ( k , n ) ) ;
( R |^ n ) \ { q : |. q .| < a } c= { q where q is Point of TOP-REAL n : |. q .| < a } ;
consider n1 be element such that n1 in dom ( p1 | n1 ) and ( p1 | n1 ) . n1 = ( p1 | n1 ) . n1 ;
consider X be set such that X in Q and for Z being set st Z in Q holds X c= Z ;
CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA ;
for v being VECTOR of l holds ||. v .|| = upper_bound ( |. v .| ) & ||. v .|| = upper_bound ( |. v .| )
for phi holds phi in X & phi in X & phi in X & phi in X implies phi is X
rng ( ( Sgm ( dom ( f | ( dom f ) ) ) | ( dom ( f | ( dom ( f | ( dom ( g | ( dom ( g | ( ( g | ( ( g | ( ( ( ( ( ( ( ( ( ( ( ( ( ( f | ( A ) ) ) ) ) ) ) ) ) ) ) ) ) ) | ( dom ( ( g | ( A | ( A | ( A ) ) ) )
ex c being FinSequence of D st len c = k & P [ c ] & a = c . ( len c ) & a = b . ( len c ) ;
Args ( a , b , c ) = <* Hom ( a , b , c ) , <* {} , {} *> , <* {} , {} *> *> ;
consider f1 be Function of the carrier of X , REAL such that f1 = |. f .| and f1 is continuous ;
a1 = b1 & a2 = b2 & a1 = b1 & b1 = b2 & b2 = b2 & b1 = b2 & b2 = b1 & b1 = b2 & b2 = b1 & b2 = b1 & b1 = b2 & b2 = b2 & b1 = b2 & b2 = b1 & b2 = b1 & b2 = b1 & b1 = b2 & b2 = b1 & b1 = b2 & b1 = b2 & b2 = b1 & b1 = b2 & b2 = b1 & b1 = b2
D2 . indx ( D2 , D1 , n1 ) = ( D1 | indx ( D2 , D1 , n1 ) ) . ( n1 + 1 ) .= D1 . ( n1 + 1 ) ;
f . ( |. r .| ) = ||. r . 1 - f . 1 .|| .= <* r . 1 *> .= <* r . 1 *> ;
consider n be Nat such that for m being Nat st n <= m holds C . m = C-25 . m ;
consider d being Real such that for a , b being Real st a in X & b in Y holds a <= b ;
||. L /. h - ( L /. h ) .|| <= ||. ( L /. h - L /. h ) .|| + ||. ( L /. h - L /. h ) .|| ;
attr F is commutative means : Def2 : for b being Element of X holds F . b = f . b ;
p = 1 * ( p1 + 0. TOP-REAL 2 ) + 0. TOP-REAL 2 .= 1 * ( p1 + p2 ) + 0. TOP-REAL 2 .= 1 * ( p1 + p2 ) + 0. TOP-REAL 2 ;
consider z1 such that b `1 , z1 , z2 , z2 , z1 , z2 is_collinear and ( o , z1 , z2 ) , ( o , z1 , z2 ) , ( o , z2 ) , ( o , z1 ) , ( o , z2 ) , ( o , z2 ) is_collinear ;
consider i such that Arg ( ( Rotate ( s , q ) ) . ( q - p ) ) = s + ( 2 * PI * ( i - p ) ) ;
consider g such that g is one-to-one and dom g = card ( f . x ) and rng g c= dom ( f . x ) ;
assume that A = ( P2 \/ Q ) and ( ( P1 \/ Q ) /\ ( P2 \/ Q ) = {} and ( P1 \/ Q ) /\ ( P1 \/ Q ) = {} ;
attr F is associative means : Def2 : F .: ( f , g ) = F .: ( f , g ) ;
ex x being Element of NAT st m = x `1 & x in { i } & m in { i } ;
consider k2 be Nat such that k2 in dom ( ( P . ( k + 1 ) ) | ( Seg ( k + 1 ) ) ) and l = ( P . ( k + 1 ) ) | ( Seg ( k + 1 ) ) ;
seq = r * ( seq . n ) implies for n holds seq . n = r * ( seq . n )
F1 . [ id a , [ a , b ] ] = [ f . [ a , b ] , [ a , b ] ] ;
{ p } "\/" D2 = { p "\/" q where q is Element of L : q in D & p in D } ;
consider z being element such that z in dom ( ( the _ of F ) . ( ( the _ of F ) . ( x , z ) ) ) and ( ( the _ of F ) . ( x , z ) ) . z = y ;
for x , y , 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 & G * ( 1 , 1 ) `2 <= s & s <= G * ( 1 , 1 ) `2 } ;
consider e being element such that e in dom ( T | ( E , m ) ) and ( T | ( E , m ) ) . e = v ;
( F *' b1 ) . x = ( ( Mx2Tran ( J , b1 , b2 ) ) . x , b2 ) . ( <* b1 , b2 *> , <* b2 *> ) . ( <* b2 , b2 *> ) ) ;
- ( 1 / ( n + 1 ) ) = ( ( - 1 ) (#) ( D * ( n + 1 ) ) ) * ( ( - 1 ) * ( D * ( n + 1 ) ) ) .= ( - 1 ) * ( ( - 1 ) * ( D * ( n + 1 ) ) ) .= ( - 1 ) * ( ( - 1 ) * ( D * ( n + 1 ) ) ) .= Det ( M *
attr x in dom f /\ dom g /\ dom g means : Def2 : for x st x in dom f /\ dom g holds it . x <= f . x ;
len ( f1 . j ) = len ( f1 . j ) .= len ( f1 . j ) .= len ( f1 . j ) .= len ( f1 . j ) .= len ( f1 . j ) ;
All ( 'not' All ( a , A , G ) , B , G ) '&' All ( a , B , G ) '&' All ( a , B , G ) '&' ( 'not' a , B , G ) '&' ( 'not' a , B ) '&' ( 'not' a , B ) '&' ( 'not' a , B ) '&' ( 'not' a , B ) )
LSeg ( E . ( k + 1 ) , F . ( k + 1 ) ) c= Cl ( ( Cage ( C , k + 1 ) ) . ( k + 1 ) ) ;
x \ ( a |^ m ) = x \ ( ( x |^ k ) * a ) .= ( x \ ( a |^ k ) ) \ a ;
k in ( ( the Sorts of U1 ) * ( the Arity of S ) ) . k .= ( ( the Sorts of U1 ) * ( the Arity of S ) ) . k .= ( ( the Arity of S ) * ( the Arity of S ) ) . k .= ( ( the Arity of S ) * ( the Arity of S ) ) . k ) .= ( ( the Arity of S ) * ( the Arity of S ) ) . k ;
for s being State of A holds Following ( s , 0 ) . ( n + 1 ) + ( n + 1 ) is stable ;
for x st x in Z holds ( f1 - f2 ) . x = a / ( x - a ) & ( f1 - f2 ) . x <> 0 ;
support ( ( support ( n ) ) \/ support ( support ( n ) ) ) c= support ( ( support ( n ) ) \/ support ( n ) ) ;
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 ( a , b ) ) = g . ( a , b ) & phi . ( a , b ) = f . ( g . ( a , b ) ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( F ^ <* p *> ) and i in dom ( F ^ <* p *> ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 ,
the Sorts of ( U1 /\ U2 ) /\ ( the Sorts of U2 ) c= the Sorts of U1 /\ ( the Sorts of U2 ) ;
( - ( 2 * a * b + sqrt ( 2 * a * b + c ) ) / ( 2 * a * b + c ) ) / ( 2 * a * b + c ) ) > 0 ;
consider W such that for z being element holds z in ( W | N ) iff z in ( W | N ) . ( z , x ) ;
assume ( the Arity of S ) . ( o , the Arity of S ) = <* a , b *> & ( the Arity of S ) . ( o , the Arity of S ) = <* b , c *> ;
Z = dom ( ( exp_R * ( f1 + #Z 2 ) ) ) /\ dom ( ( exp_R * ( f1 + #Z 2 ) ) ) ;
lim ( f , S ) is convergent & lim ( f , S ) = lim ( f , S ) ;
( X . ( f . x ) ) => ( ( f . x ) => ( ( f . x ) => ( f . x ) ) ) in X ;
len ( M2 * M2 ) = n & width ( M2 * M2 ) = n ;
attr X1 union X2 means : Def2 : X1 , X2 , X3 , X3 , X2 , X3 , X3 , x5 , x5 , x5 , v2 , x5 , v2 , w1 , w1 , w1 , w2 , w1 , w2 , w1 , w1 , w2 , w1 , w1 , w1 , w2 , w1 , w2 , w1 , w1 , w2 , w1 , w2 , w1 , w1 , w2 , w1 , w1 , w2 , w1 , w1 , w2 , w1 , w1 , w1 , w2
for L being lower-bounded antisymmetric antisymmetric antisymmetric antisymmetric RelStr , X being Subset of L holds X "\/" { "/\" ( X , L ) } = { "/\" ( X , L ) }
reconsider f29 = ( F . ( b , c ) ) . ( ( F . ( b , c ) ) . ( ( F . ( b , c ) ) . ( ( F . ( b , c ) ) ) . ( ( F . ( b , c ) ) . ( ( F . ( b , c ) ) . ( ( F . ( b , c ) ) . ( ( F . ( b , c ) ) . ( ( F . c ) )
consider w being FinSequence of I such that the InitS of M , the InitS of S , w ^ <* s *> ;
g . ( a |^ 0 ) = g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) ;
assume for i being Nat st i in dom f ex z being Element of L st f . i = rpoly ( 1 , z ) . ( i + 1 ) ;
ex L being Subset of X st L = L & for K being Subset of X st K in L holds K /\ L <> {} ;
( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C2 & ( the carrier' of C2 ) /\ ( the carrier' of C2 ) c= the carrier' of C2 ;
reconsider o9 = o `1 , y9 = ( the Sorts of A ) . ( ( the Sorts of A ) . ( o , the carrier of S ) ) . ( o , the carrier of S ) as Element of TS ( S ) ;
1 * ( x1 + x2 ) + ( 0 * x1 + 0 * x2 ) = x1 + 0 * x2 + 0 * x2 .= x1 + 0 ;
( E " ) . 1 = ( E qua Function ) . 1 .= ( E qua Function ) . 1 .= ( E qua Function ) . 1 .= ( E qua Function ) . 1 .= ( E qua Function ) . 1 .= ( E qua Function ) . 1 .= ( E qua Function ) . 1 .= ( E qua Function ) . 1 .= ( E qua Function ) . 1 .= ( E qua Function ) . 1 .= ( E qua Function ) . 1 .= ( E qua Function ) . 1 .= ( E
reconsider u1 = the carrier of ( U1 /\ ( U1 "\/" U2 ) ) as non empty Subset of ( U1 /\ ( U1 "\/" U2 ) ) ;
( x "/\" z ) "\/" ( x "/\" ( y "/\" z ) ) <= ( x "/\" ( y "/\" z ) ) "\/" ( y "/\" ( x "/\" z ) ) ;
|. f . ( ( s1 . ( l + 1 ) ) - f . ( l + 1 ) ) .| < r / ( 1 / ( l + 1 ) ) ;
LSeg ( ( W-min ( C , n ) /. ( i + 1 ) , ( W-min ( C , n ) /. ( i + 1 ) ) , ( W-min ( C , n ) /. ( i + 1 ) ) ) is vertical ;
( f | Z ) /. x - ( f | Z ) /. x = L /. ( x - x ) + R /. ( x - x ) ;
g . c * ( f . c ) + ( f . c ) * f . c <= h . c * ( f . c ) + ( f . c ) * f . c ;
( f + g | divset ( D , i ) ) | divset ( D , i ) = f | divset ( D , i ) ;
assume that width ( f ) in the carrier of A and width ( f @ ) = width A and width ( f @ ) = width A ;
len ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - - ( - ( - - ( - - ( - - ( - - ( - - ( - - ( - ( - - ( - - ( - ( - ( - ( - ( - - ( - ( - - ( - - ( - - ( - - ( - - ( - - - ( - - ( - - ( - ( - - ( -
for n , i being Nat , i being Nat st i + 1 < n & i in the InternalRel of G holds [ i , i ] in the InternalRel of G & [ i , j ] in the InternalRel of G
pdiff1 ( f1 , 2 ) is_partial_differentiable_in u0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in u0 , 1 & pdiff1 ( f1 , 2 ) is_partial_differentiable_in u0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in u0 , 1 & pdiff1 ( f1 , 2 ) is_partial_differentiable_in u0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in u0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in 1 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in 1 , 1 ;
attr a <> 0 & b <> 0 & a = - b & b = - a & - a = - b & - b = - a ;
for c being set st c in [. a , b .] holds not c in Intersection ( the InternalRel of a , b )
assume that V1 is linearly closed and V1 is closed and V1 in V1 and V2 in V1 and V1 in V2 and V1 in V2 and V2 in V1 and V1 in V2 and V1 in V2 ;
z * ( x1 + ( 1 - z ) * ( x2 + - z ) ) in M & ( z * ( x1 + - z ) * ( x2 + z ) ) in N ;
rng ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( R ) ) ) ) ) ) ) ) ) ) ) " ) ) ) ) ) ) ) ) ) ) = Seg ( ( ( ( ( ( R * ( ( R * ( ( R | ( n + 1 ) ) ) ) ) ) ) ) ) .= ( ( ( R * ( ( R | ( n + 1 ) ) ) ) ) ) ) ;
consider s2 being convergent convergent Real_Sequence such that b is convergent and lim s2 = lim s2 and for n holds ( for n holds s2 . n = lim s2 ) ;
h2 " . n = h2 . n & 0 < h2 . ( n + 1 ) ;
( Partial_Sums ( |. ( ( r .| ) ) ) ) . m ) = ( |. ( r .| ) ) . m ) . m .= ( ( r .| ) . m ) . m .= ( r (#) ( ( r (#) ( |. r .| ) ) ) ) . m .= ( ( r (#) ( |. r .| ) ) ) . m ;
( Comput ( P1 , s1 , 1 ) ) . b = ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b ;
- v = ( - 1 ) * v & - ( - 1 ) * v = ( - 1 ) * w + ( - 1 ) * w ;
sup ( ( ( k | D ) .: ( k | ( k | ( k | ( k | ( k | ( k | ( k | ( k | ( k | ( k | ( k | ( k | ( k | ( k | ( k | ( k | ( k | ( k | ( k | ( k | ( k | ( k | ( k | ( k | ( k | ( k | ( k | ( k | ( k | ( k | ( k | ( k | ( k
( A |^ k , l ) .. ( A |^ ( k , l ) ) = ( ( A |^ k , l ) .. ( A |^ k , l ) ) .. ( A |^ k , l ) ;
for R being add-associative right_zeroed right_complementable right_zeroed right_zeroed commutative associative distributive non empty doubleLoopStr , I being Subset of R , J being Subset of R , K being Subset of R st I + J = ( I + J ) \ K holds K + L = ( I + J ) \ K
( f . p ) `1 = sqrt ( ( p `1 ) ^2 + ( p `2 ) ^2 ) ;
for a , b being non zero Nat , a , b being Element of NAT holds ( a * b ) * ( a * b ) = ( a * b ) * ( a * c )
consider A5 being set such that r is countable & ( ex A being Subset of Al st ( A = { x } ) & ( ex S being set st S is m ) & ( ex m being Nat st S is m ) & ( ex S being set st S = m ) & ( S is m ) & ( S is m ) & ( S is m ) ) & ( S is m ) ) ;
for X being non empty addLoopStr , M being Subset of X , x being Point of X st x in M holds x + y in M + M
{ [ x1 , x2 ] , [ y1 , y2 ] } c= { [ x1 , x2 ] , [ y1 , y2 ] } & { [ y1 , y2 ] } c= { [ y1 , y2 ] , [ y1 , y2 ] } ;
h . O = |[ A * ( f . O ) + B * ( f . O ) + C * ( f . O ) + D * ( f . O ) + D * ( f . O ) + D * ( f . O ) + D * ( f . O ) + D * ( f . O ) + D * ( f . O ) + D * ( f . O ) + D * ( f . O ) + D * ( f . O
( Gauge ( C , n ) * ( i , i ) ) /. ( k + 1 ) in L~ Cage ( C , n ) /\ L~ Cage ( C , n ) ;
cluster m , n -> prime ;
( f * F ) . ( x1 , x2 ) = f . ( F . ( x1 , x2 ) ) & ( f * F ) . ( x2 , y2 ) = f . ( x2 , y2 ) ;
for L being complete LATTICE , a , b being Element of L holds a \ b <= c implies a \ b <= c \ ( a \ b )
consider b being element such that b in dom ( H / ( x , y ) ) and z = H / ( x , y ) ;
assume that x in dom ( F * g ) and y in dom ( F * g ) and ( F * g ) . x = ( F * g ) . y ;
assume that not e Joins W . 1 , W . 1 , G and W . 2 in G . ( 3 + 1 ) and W . 1 in G . ( 3 + 1 ) ;
( ( h (#) f ) . n ) . x = ( h (#) f ) . ( x + h . x ) ;
j + 1 = ( i + 1 ) + 1 .= i + ( len ( h1 ) - 1 ) .= i + ( len h1 - 1 ) .= i + ( len h1 - 1 ) ;
S *' ( f *' ) = S *' ( f *' ) .= S *' ( f *' ) .= ( S *' ) *' ( f *' ) .= ( S *' ) *' ( f *' ) .= ( S *' ) *' ( f *' ) ;
consider H such that H is one-to-one and rng H = Carrier ( L ) and Sum ( L ) = Sum ( L ) ;
attr R is indices means : Def2 : for p , q st p in R & q in R & p <> q holds ex P st P = p & P c= R & P c= R & Q c= R & P c= R ;
dom ( product ( X --> f ) ) = meet ( X --> f ) .= meet ( X --> f ) .= meet ( X --> f ) .= meet ( X --> f ) .= ( meet ( X --> f ) ) . f .= ( meet ( X --> f ) ) . f ;
sup ( ( proj2 .: Lower_Arc C ) /\ Lower_Arc ( C ) ) <= upper_bound ( ( proj2 .: C ) /\ Lower_Arc ( C ) ) ;
for r be Real st 0 < r ex n be Nat st for m be Nat st n <= m holds |. S . m - 0 .| < r ;
i * ( ( f - g ) - ( f - g ) ) = i * ( ( f - g ) * ( ( f - g ) - ( f - g ) ) ) .= i * ( ( f - g ) * ( ( f - g ) * ( ( f - g ) - ( f - g ) ) ) .= i * ( ( f - g ) * ( f - g ) ) * ( ( f - g ) ) ) ;
consider f being Function such that dom f = 2 -tuples_on X and for Y being set st Y in 2 -tuples_on Y holds f . Y = F ( Y ) ;
consider g1 , g2 being element such that g1 in [#] ( Y | B ) and g2 in ( the carrier of Y ) and g1 = [ g2 , g2 ] ;
func d \! \mathop { n } -> Nat means : Def2 : for p being Element of D holds it . ( n + 1 ) = n & it . ( n + 1 ) divides n ;
f . [ 0 , t ] = f . ( [ 0 , t ] , [ 0 , t ] ) .= ( - ( - ( 1 , t ) ) ) * ( ( - t ) . ( 0 , t ) ) .= - ( - ( t ) . ( 0 , t ) ) .= - ( ( - t ) . ( 0 , t ) ) .= - ( - t ) . ( 0 , t ) ) .= - ( - t ) . ( 0 , t ) ;
t = h . D or t = h . E or t = h . F or t = h . J or t = h . E ;
consider m1 be Nat such that for n be Nat st n >= m1 holds dist ( ( seq . n ) . n , ( seq . n ) . x ) < 1 ;
sqrt ( ( q `1 / |. q .| - sn ) / ( 1 + sn ) ) ^2 ) <= sqrt ( ( q `1 / |. q .| - sn ) ) ^2 ) ;
h1 . ( i + 1 + 1 ) = h1 . ( i + 1 + 1 ) .= h1 . ( i + 1 + 1 ) ;
consider o being Element of the carrier' of S , x2 being Element of { [ o , x2 ] } such that a = [ o , x2 ] , [ o , y2 ] } ;
for L being RelStr , a , b being Element of L holds a <= b iff a <= b & b <= a & a <= b & b <= a & a <= b
||. ( h1 " ) . n - ( h1 " ) . n .|| = ||. ( h1 " ) . n - ( ( h1 " ) . n ) .= ||. ( h " ) . n - ( ( h " ) . n ) .|| .= ||. ( h " ) . n - ( ( h " ) . n ) .|| ;
( ( - ( ( ( the Sorts of T ) * ( f | A ) ) ) `| ( ( the Sorts of T ) * ( f | A ) ) `| ( ( the Sorts of T ) * ( f | A ) ) `| ( ( the Sorts of T ) * ( f | A ) ) ) `| ( ( ( the Sorts of T ) * ( f | A ) ) `| ( ( the Sorts of T ) ) ) `| ( ( ( ( the Sorts of T ) ) . ( f | A ) ) = - ( ( the
attr r = F .: ( p , q ) means : Def2 : len r = len ( p , q ) ;
sqrt ( ( r / 2 ) ^2 + ( r / 2 ) ^2 ) <= sqrt ( ( r / 2 ) ^2 + ( r / 2 ) ^2 ) + ( r / 2 ) ^2 ) ;
for i being Nat , M being Matrix of n , K , a being Element of K st i in Seg n & a in Seg n holds Det ( M , i ) = Sum ( Line ( M , i ) , a )
then a " * ( a " * v ) = 1 & a " * ( a " * v ) = 1 ;
p . ( j -' 1 ) * ( q /. ( i -' 1 ) ) = Sum ( p . ( j -' 1 ) ) * ( q /. ( j -' 1 ) ) ;
deffunc F ( Nat ) = L . 1 + ( ( R /* h ) /* ( h + c ) ) * ( ( R /* h ) /* ( h + c ) ) ;
assume that the carrier of ( H1 | ( the carrier of H2 ) ) = f .: ( the carrier of H1 ) and the InternalRel of ( H1 | ( the carrier of H2 ) ) = f .: ( the carrier of H2 ) ;
Args ( o , ( the Sorts of Free ( S , X ) ) * ( the Arity of S ) ) . ( ( the Arity of S ) . ( ( the Arity of S ) . ( o , ( the Arity of S ) . ( o , ( the Arity of S ) ) . ( o , ( the Arity of S ) . ( o , ( the Arity of S ) . ( o , ( the Arity of S ) ) ) ) ) ) = ( ( the Arity of S ) . ( o , ( the Arity of S ) ) ) . ( o , ( the Arity of S ) ) ) ;
H1 = n + 1 + ( |. 2 to_power ( n + 1 ) .| ) .= n + 1 + ( n + 1 ) .= n + 1 + ( n + 1 ) ;
( O O ) . ( O , O ) = 0 & ( O O ) . ( O , O ) = 1 & ( O O ) . ( O , O ) = 0 ;
F1 .: ( dom ( F1 /\ F2 ) ) = ( F1 | ( dom ( F1 | ( n + 1 ) ) ) .= ( F1 | ( n + 1 ) ) ) .: ( ( n + 1 ) -tuples_on ( F1 | ( n + 1 ) ) ) .= ( F1 | ( n + 1 ) ) ) .: ( ( n + 1 ) -tuples_on ( F1 | ( n + 1 ) ) .= ( F1 | ( n + 1 ) ) .: ( n + 1 ) ;
attr b <> 0 & d <> 0 & b <> 0 & a = b & b = sqrt ( b , a ) & c = sqrt ( b , a ) & d = sqrt ( b , a ) ;
dom ( ( f +* g ) | D ) = dom ( f +* g ) /\ D .= ( f +* g ) | D .= ( f +* g ) | D .= ( f +* g ) | D .= ( f +* g ) | D .= ( f +* g ) | D .= ( f +* g ) | D .= ( f +* g ) | D .= ( f +* g ) | D .= ( f +* g ) | D .= ( f +* g ) | D .= ( f +* g ) | D .= ( f +* g ) | D .= ( f +* g ) | D
for i being set st i in dom g ex a being Element of L st a /. i = u * a & ex u being Element of L st a /. i = u * a & u in A * a ;
g * P = g * P * ( g * P ) .= g * ( P * P ) .= g * ( P * P ) .= ( g * P ) * ( P * P ) ;
consider i , s1 such that f . i = s1 and not ( ex s being Element of S st s . i = s1 & ( not ex i being Element of S st ( ex s being Element of S st s . i = s1 & ( not i in dom s1 ) & not thesis ) ;
( h | ]. a , b .[ ) | ]. a , b .[ = ( g | ]. a , b .[ ) | ]. a , b .[ .= ( g | ]. a , b .[ ) | ]. a , b .[ .= ( g | ]. a , b .[ ) | ]. a , b .[ ) | ]. a , b .[ .= ( g | ]. a , b .[ ) | ]. a , b .[ ) | ]. a , b .[ ;
[ s1 , t1 ] , [ s2 , t2 ] ] , [ s2 , s2 ] ] in R & [ s2 , t2 ] in R ;
then H is negative means : then H is negative & H is negative & H is negative & H is negative & H is negative & H is negative & H is negative ;
attr f1 is total means : Def2 : for c being Element of C holds ( f1 /* c ) . c = ( f1 /* ( f1 /* c ) ) . c ;
( z1 in ( z1 + z2 ) \ ( z1 + z2 ) or ( z1 + z2 ) \ ( z1 + z2 ) = ( z1 + z2 ) \ ( z1 + z2 ) \ ( z1 + z2 ) ;
p = 1 * p .= a * ( b * q ) .= a * ( b * q ) .= a * ( b * q ) .= a * ( b * q ) .= a * ( b * q ) ;
for rbeing sequence of REAL , K being Real st for n being Nat holds ( for m being Nat holds ( for n being Nat holds ( for n being Nat holds n <= m holds ( for n being Nat holds n <= m ) implies ( for n be Nat holds ( for n be Nat holds n <= m ) implies ( for n be Nat holds ( for n be Nat holds n <= m ) implies ( for n be Nat holds ( n <= m ) implies n <= n ) implies ( n <= m ) implies ( n <= m ) implies ( n <= n ) implies ( n <= m ) implies ( n
<* W-min ( C ) , W-min ( C ) *> meets L~ Cage ( C , n ) or not LIN W-min ( C , n ) , W-min ( C , n ) , W-min ( C , n ) , W-min ( C , n ) ;
||. f . ( g . ( k + 1 ) ) .|| <= ||. g . ( k + 1 ) - g . ( k + 1 ) .|| ;
assume h = ( B .--> C ) +* ( D .--> E ) +* ( C .--> F ) +* ( D .--> J ) +* ( E .--> F ) +* ( F .--> J ) +* ( M .--> J ) +* ( F .--> J ) +* ( M .--> F ) +* ( M .--> J ) +* ( F .--> J ) +* ( M .--> J ) ) +* ( M .--> N ) +* ( F .--> N ) +* ( M .--> N ) ) ;
|. ( ( H . n ) || ( A . m ) ) - ( ( H . n ) || ( A . m ) ) . k - ( H . k ) ) .| <= e * ( e * r2 ) ;
( ( the Sorts of Free ( S , X ) ) . v ) . e = [ [ ( the Sorts of Free ( S , X ) ) . e , ( the Sorts of Free ( S , X ) ) . e ] ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 } } = { 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 ,
for A , B st A = [. 0 , PI .] holds integral ( ( sin (#) cos ) | A ) = 0 & integral ( sin , A ) | A = 0
p is permutation of dom ( f1 | i ) & p * ( ( Sgm ( Seg n ) ) " ) = ( ( Sgm ( Seg n ) ) * Sgm ( Seg n ) ) * Sgm ( Seg n ) ;
for x , y , y st x in A holds |. ( 1 / 2 ) * ( f . x - f . y ) .| <= 1 * |. f . x - f . y .|
( p2 `2 ) = |. q2 .| * sqrt ( ( p2 `2 ) ^2 + ( p2 `2 ) ^2 ) ;
for f being PartFunc of the carrier of C , the carrier of C st f is continuous & rng f c= the carrier of C holds f is continuous iff ( for x being Point of C st x in dom f holds f . x = ( f . x ) `1 ) `1
assume for x being Element of Y st x in EqClass ( z , CompF ( B , CompF ( C , CompF ( D , CompF ( D , CompF ( D , CompF ( D , CompF ( D , CompF ( D , CompF ( D , CompF ( D , CompF ( D , CompF ( D , CompF ( D , CompF ( D , CompF ( D , len ( D , n ) ) ) ) ) ) ) ) ) holds ( ( ( a , B ) ) ) ) ) . x = TRUE ;
consider F3 such that dom ( F . ( n + 1 ) ) = Seg ( n + 1 ) and for k being Nat st k in dom ( F . ( n + 1 ) ) holds Q [ k , F . ( n + 1 ) ] ;
ex u , u1 st u <> u1 & u , v // u1 , v1 & u1 , v1 // u1 , v1 & v1 , u1 // v1 , v2 & u1 , v1 // v1 , v2 & v1 , v2 // v1 , v2 & u1 , v1 // v1 , v2 & v1 , v2 // u1 , v1 & u1 , v1 // u1 , v1 & u1 , v1 // u1 , v1 ;
for G being Group , A being non empty Subset of G , N being normal Subgroup of G holds ( N * A ) * ( N * B ) = N * ( A * B )
for s be Real st s in dom F holds F . s = integral ( R , f ) & ( F . s ) . s = integral ( R , f ) + ( ( f + g ) . s ) )
width ( ( f1 * ( b1 , b2 , b1 , b2 ) ) ) = len ( ( f1 * ( b1 , b2 , b1 , b2 ) ) ) .= width ( ( f1 * ( b1 , b2 , b2 ) ) ) .= width ( ( f1 * ( b1 , b2 , b2 ) ) ) .= width ( ( f1 * ( b1 , b2 ) ) ) .= width ( ( f1 * ( b1 , b2 ) ) ;
f | ]. - PI / 2 , PI .[ = f | ]. - PI / 2 , PI .[ & f | ]. - PI / 2 , PI .[ = f | ]. - PI / 2 , PI .[ ;
assume that X is closed w.r.t. A2 and a in X and a in X and y in X & x in X & y in X and x in X and y in X and x in X ;
Z = dom ( ( ( ( exp_R * exp_R ) ) (#) ( exp_R * exp_R ) ) `| Z ) /\ dom ( ( exp_R * exp_R ) `| Z ) ;
func [: l ( ) , l ( ) -> Subset of V ( ) equals { [ l ( ) , l ( ) ] : 1 <= l ( ) & l ( ) = [ l ( ) , l ( ) ] } ;
for L being non empty TopSpace , N being net of L , M being net of L , N being net of L st N is net of L holds N is convergent & for c being Element of L st c in N holds c is convergent & N is convergent & c in N holds c is convergent
for s being Element of NAT holds ( ( id ( the carrier of V ) ) + ( id ( the carrier of V ) ) + ( id ( the carrier of V ) ) ) . s = ( id ( the carrier of V ) ) . s
then z /. 1 = W-min ( L~ z ) & ( W-min L~ z ) .. z < ( W-min L~ z ) .. z ;
len ( p ^ <* 0 qua Real *> ) = len p + len <* 0 qua Real *> .= len p + 1 .= len p + 1 ;
assume that Z c= dom ( ( - ( ( ( ( exp_R * f ) ) `| Z ) ) `| Z ) and for x st x in Z holds f . x = exp_R . x - 1 / ( x + a ) ) and for x st x in Z holds f . x = a ;
for R being add-associative right_zeroed right_complementable associative associative distributive non empty doubleLoopStr , I being Subset of R , J being Subset of R , I being Subset of R , J being Subset of R , I being Subset of R st I c= I holds ( I + J ) *' ( I + J ) c= I /\ J
consider f being Function of [: B1 , B2 :] , the carrier of B2 such that for x being Element of the carrier of B1 holds f . x = F ( x , x ) ;
dom ( x2 + y2 ) = Seg ( len x + ( len y ) ) .= Seg ( len x + len y ) .= Seg ( len x + len y ) .= dom x /\ Seg ( len y ) .= dom x /\ Seg ( len y ) .= dom x /\ Seg ( len y ) .= dom x /\ Seg ( len y ) ;
for S being Functor of C , B , C being category , a being object of C holds ( id ( C ) ) . ( id ( C ) ) = id ( C ) . ( id ( C ) )
ex a st a = a2 & a in f /\ ( { x } ) & f . a = f . ( a , a ) & f . ( a , b ) = f . ( a , b ) ;
a in ( ( H / ( ( ( ( ( ( H / ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 4 4 4 4 ) 4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) | ( ( ( ( H / ( ( ( ( ( ( ( ( ( ( 4 4 ) ) ) ) | ( ( H / ( 4 * ( 4 * ( 4 * ( 4 * ( 4 * ( 4 * ( 4 * ( 4 * ( 4 * ( 4 * ( 4 * ( 4 * ( 4 * ( 4 * ( 4 * ( 4 * ( 4 * ( 4
for C1 , C2 being stable non empty set , f being Function of C1 , C2 holds ( ( f . ( g . ( C , f ) ) ) is stable iff ( f . ( C , g ) ) . ( f . ( C , g ) ) = ( f . ( C , g ) ) . ( f . ( C , g ) ) )
( W-min ( P ) \/ L~ ( go /. 1 ) ) `1 = ( W-min ( P ) ) \/ ( W-min ( P ) ) \/ ( W-min ( P ) ) ;
assume that u = <* x0 , y0 , y0 , z0 *> and f is partial & u = y0 & f is partial & f is PartFunc of x0 , y0 ;
then ( t . {} ) `1 in Vars & ex x being Element of Vars st x = ( t . {} ) `1 & ( t . {} ) `2 = ( t . {} ) `1 ;
Valid ( p '&' q , J ) . v = Valid ( p , J ) . v .= Valid ( q , J ) . v .= Valid ( p , J ) . v .= Valid ( q , J ) . v ;
assume for x , y being Element of S st x <= y for a being Element of T st a = f . x & b = f . y holds a >= f . y ;
func Class ( R , A ) -> Subset-Family of R means : Def2 : for a being Element of R holds it . a = Class ( R , a ) ;
defpred P [ Nat ] means ( ( the Target of G ) . $1 ) `1 c= ( the Target of G ) . $1 ;
assume that dim ( U1 , U2 ) = 0 and dim ( U1 , U2 ) = 0 and dim ( U1 , U2 ) = 0 and dim ( U1 , U2 ) = 0 ;
( the non empty set of m ) . ( m . {} , t . {} ) = ( m . {} , the m of C ) . {} .= m . {} .= m . {} ;
d = ( ( y ^ <* d *> ) ^ <* d *> ) . ( ( y ^ <* d *> ) . ( ( y ^ <* d *> ) /. ( ( y ^ <* d *> ) /. ( y ^ <* d *> ) ) ) .= ( y ^ <* d *> ) . ( ( y ^ <* d *> ) /. ( y ^ <* d *> ) ) .= ( y ^ <* d *> ) /. ( y ^ <* d *> ) .= ( y ^ <* d *> ) /. ( y ^ <* d *> ) .= ( y ^ <* d *> ) /. ( y ^ <* d *> ) .= ( y ^ <* d *> ) /. ( y ^ <* d *> ) /. ( y ^ <* d *> ) .= ( y ^ <* d *> ) /.
consider g such that x = g and dom g = dom f and for x being element st x in dom f holds f . x = ( f . x ) `1 ;
x + 0. ( ( the carrier of V ) ) = x + 0. ( V , the carrier of V ) .= x + 0. ( V , the carrier of V ) .= x + 0. ( V , the carrier of V ) .= x + 0. ( V , the carrier of V ) .= x + 0. ( V , the carrier of V ) ;
( ( f | ( k -' 1 ) ) + ( f | ( k -' 1 ) ) ) in dom ( ( f | ( k -' 1 ) ) | ( k -' 1 ) ) ) ;
assume that P1 is_an_arc_of p1 , p2 and ( for p being Point of TOP-REAL 2 st p in P1 & p in P1 & q in P1 & p <> 0. TOP-REAL 2 holds P /\ LSeg ( p1 , p2 ) = { p } and P /\ LSeg ( p , q ) = { p } and P /\ LSeg ( p , q ) = { p } and P /\ LSeg ( p , q ) = { p } and P /\ LSeg ( q , p ) = { p } and P /\ LSeg ( p , q ) = { p } and P /\ LSeg ( p , q ) /\ LSeg ( p , q ) = { p } and P /\ LSeg ( p , q ) = { p } and P /\ LSeg ( p , q ) = { p } and
reconsider a1 = a , b1 = b , c1 = c , c2 = d , c2 = c , c2 = d , c2 = d , c2 = d , c2 = c , c2 = d , c2 = d , c2 = d , c2 = c , c2 = d , c2 = d , c2 = d , c2 = d , c2 = d , c2 = c , c2 = d , c2 = d , c2 = d , 6 = c , 8 = d , 8 = d , 6 = d , 6 = d , 6 = d , 6 = d , 6 = d , 6 = d , 6 = d , 6 = d , 6 , 6 = d , 6 , 7 = d , 6 , 6 , 7 = 6 , 6 = 6 , 6 ,
reconsider ssbPPf = ( t . b ) * ( ( t . a ) * ( t . b ) ) as Morphism of ( ( ( t . a ) * ( t . b ) ) ) * ( ( t . a ) * ( t . b ) ) ;
LSeg ( f , i + i1 -' 1 ) = LSeg ( f /. ( i + i1 -' 1 ) , f /. ( i + 1 -' 1 ) ) ;
\int ( P . m ) | dom ( P . n ) , ( P . m ) | dom ( P . n ) | dom ( P . m ) ) <= \int ( P . n , ( P . m ) | dom ( P . n ) ) ;
assume that dom f1 = dom f2 and for x , y being element st [ x , y ] in f1 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 ) ;
for G being Group , H being Element of G holds a = b iff a = b & a = c & b = c & a = c & b = d & a = d ;
consider B being Function of ( Seg ( S + L ) ) , the carrier of V such that for x being element st x in Seg ( S + L ) holds P [ x , B . x , L . x ] ;
reconsider K1 = { p where p is Point of TOP-REAL 2 : P [ p ] } as Subset of TOP-REAL 2 ;
sqrt ( ( W-min C ) - ( W-min C ) ) / ( ( W-min C ) - ( W-min C ) ) <= sqrt ( ( W-min C ) - ( W-min C ) ) / ( ( W-min C ) - ( W-min C ) ) / ( ( W-min C ) - ( W-min C ) ) ;
for x being Element of X , n being Nat st x in E holds |. ( Re F ) . n .| <= P . x & |. ( Im F ) . n .| <= P . x & |. ( Im F ) . x .| <= P . x ;
len ( @ @ q ) = len ( @ ( <* 2 *> ^ q ) ) + len ( @ q ) .= len ( @ ( q ^ <* 0 *> ) ) + len ( @ q ) .= len ( @ ( q ^ <* 0 *> ) ) + len ( @ q ) ;
v / ( ( ( x , m1 ) / ( x , m1 ) ) / ( x , m1 ) ) / ( ( x , m1 ) / ( x , m1 ) ) ) / ( x , m1 ) = ( m , m1 ) / ( x , m1 ) ;
consider r being Element of M such that M , ( ( { x } \leftarrow ( m ) ) / ( ( { x } \leftarrow ( m ) ) / ( ( x , m ) / ( x , m ) ) / ( x , m ) ) / ( x , m ) / ( x , m ) / ( x , m ) ) / ( x , m ) ) / ( x , m ) / ( x , m ) ) / ( x , m ) ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) ) / ( x
func ( G \ ( { w } ) ) \ ( { w } ) -> Element of ( G \ { w } ) \ { w } ) ;
s2 . ( b , s ) = ( Exec ( n , s1 ) . b , s ) . ( b , s ) .= Exec ( n , s ) . b .= s . b ;
for n , k holds 0 <= ( Partial_Sums ( |. seq .| ) ) . ( n + k ) - Partial_Sums ( |. seq .| ) . ( n + k )
set F = S \! \mathop { 0 } ;
( Partial_Sums ( s ) ) . ( n + 1 ) + Partial_Sums ( s ) . ( n + 1 ) >= ( Partial_Sums ( s ) ) . ( n + 1 ) + Partial_Sums ( s ) . ( n + 1 ) ;
consider L , R such that for x st x in N holds ( f | Z ) . x - ( f | Z ) . x = L . ( x - b ) + R . ( x - b ) ;
func the closed Subset of Closed-Interval-TSpace ( a , b , c ) -> Subset of TOP-REAL 2 equals ( the InternalRel of Closed-Interval-TSpace ( a , b , c , d ) ) \ { 0 } ;
a * b + ( b * c + ( a * b + c ) ) + ( b * a + c * a + c * b + d * b + d * a + c * b + d * a + d * b + c * a + d * b + d * a + d * b + c * a + d * b + d * b + d * a + c * b + d * b + d * a + d * b + c * b + d * b + d * b + d * a + d * b + d * b + d * b + d * b + c * b + d * b + d * a + d * b + d *
v / ( x1 , m1 ) / ( x2 , m1 ) = v / ( x1 , m1 ) / ( x2 , m1 ) ;
( ( Q ^ <* x *> ) ^ <* ( x ^ <* y *> ) *> ) . ( ( Q ^ <* x *> ) . ( ( Q ^ <* x *> ) . ( ( Q ^ <* x *> ) . ( ( Q ^ <* x *> ) . ( ( Q ^ <* y *> ) . ( ( Q ^ <* x *> ) . ( ( Q ^ <* x *> ) . ( ( Q ^ <* y *> ) ) . ( ( Q ^ <* y *> ) ) ) ) ) ) ) ) = ( ( Q ^ <* x *> ) ) . ( ( Q ^ <* x *> ) ) ) . ( ( Q ^ <* x *> ) ) ) . ( ( Q ^ <* x *> )
Partial_Sums ( F ) = ( r / ( n + 1 ) ) * Partial_Sums ( F ) . ( n + 1 ) .= ( ( r / ( n + 1 ) ) ) * Partial_Sums ( F ) ) . ( n + 1 ) .= ( ( r / ( n + 1 ) ) * Partial_Sums ( F ) ) . ( n + 1 ) .= ( r / ( n + 1 ) ) * ( ( n + 1 ) ) * ( n + 1 ) ;
( ( GoB f ) * ( len GoB f , 1 ) ) `1 = ( ( GoB f ) * ( len GoB f , 1 ) ) `1 .= ( ( GoB f ) * ( 1 , 1 ) ) `1 ;
defpred X [ Element of NAT ] means ( Partial_Sums s ) . $1 = ( Partial_Sums ( s ) ) . $1 + ( Partial_Sums ( s ) ) . $1 ;
the_arity_of g = ( the Arity of S ) . ( ( the Arity of S ) . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g
( X [: Y , Z :] ) \/ ( X \/ Y ) c= X \/ ( Y \/ Z ) & ( X c= Y implies X c= Y & Z c= Y & Z c= X & Z c= Y & Z c= Y & Z c= X & Z c= Y & Z c= Y & Z c= Y & Z c= X & Z c= Y & Z c= Y & Z c= X & Z c= Y & Z c= Y & Z c= X & Z c= Y & Z c= Y & Z c= Y & Z c= Y & Z c= Y & Z c= Y & Z c= Y & Z c= Y & Z c= Y & Z c= Y & Z c= Y & Z c= Y & Z c= Y & Z implies Z c= Y & Z c=
for a , b being Element of S , s being Element of S st s = n & a = F . ( n + 1 ) & b = F ( n ) holds s = G ( n + 1 ) \ G ( n + 1 ) \ G ( n + 1 ) \ G ( n + 1 ) \ G ( n + 1 ) ;
E , f / ( ( ( not ( ( x , y , c ) ) / ( x , y ) ) / ( x , y ) ) ) / ( ( x , y , c ) / ( x , y ) ) ) / ( x , y ) / ( x , y ) ) / ( x , y ) = ( x , y , c ) / ( x , y ) ) / ( x , y ) ;
ex R2 being 1-sorted st R2 = ( p | ( n + 1 ) ) & ( ( the _ of n + 1 ) * ( p | ( n + 1 ) ) ) = ( the _ of n + 1 ) * ( p | ( n + 1 ) ) ;
[. a , b + sqrt ( 1 + ( 1 + ( b + c ) ) ) .] is Element of the partial sets of REAL , REAL ;
Comput ( P , s , 2 + 1 ) = Exec ( P , Comput ( P , s , 2 ) ) .= Exec ( P , Comput ( P , s , 2 ) ) .= Exec ( P , Comput ( P , s , 2 ) ) .= Exec ( P , Comput ( P , s , 2 ) , 2 ) .= Exec ( P , s , 2 ) ) ;
card ( h1 . k ) = ( - ( ( - ( z . k ) ) * ( ( - ( z . k ) ) * ( ( z . k ) ) * ( ( z . k ) ) * ( ( z . k ) ) * ( ( z . k ) ) * ( ( z . k ) ) ) ) .= ( - ( z . k ) ) * ( ( z . k ) ) * ( ( z . k ) ) * ( ( z . k ) ) .= ( - ( ( z . k ) ) * ( ( z . k ) ) * ( ( z . k ) ) ) * ( ( z . k ) ) * ( ( z . k ) ) * ( ( z . k
sqrt ( f /. c ) = f /. c * ( g /. c ) .= ( f /. c ) * ( g /. c ) .= ( f /. c ) * ( g /. c ) .= ( f /. c ) * ( g /. c ) .= ( f /. c ) * ( g /. c ) ;
len ( ( C - len Gauge ( C , n ) ) - len ( Gauge ( C , n ) ) = len ( ( C - |^ ( n + 1 ) ) ) - len ( Gauge ( C , n ) ) ;
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom ( ( r (#) f ) | X ) .= X /\ dom ( f | X ) .= X /\ X .= X /\ dom f .= X /\ dom f .= X /\ dom f ;
defpred P [ Nat ] means for n being Nat holds 2 * n + 2 * n + 1 = Fib ( n + 1 ) + Fib ( n + 1 ) * Fib ( $1 + 1 ) * Fib ( $1 + 1 ) + Fib ( n + 1 ) * Fib ( $1 + 1 ) + Fib ( n + 1 ) * Fib ( n + 1 ) * Fib ( n + 1 ) ;
consider f being Function of [: ( n + 1 ) , ( n + 1 ) -tuples_on NAT , ( n + 1 ) -tuples_on NAT such that f = f and f is one-to-one and rng f c= { n + 1 } ;
consider c1 be Function of S , BOOLEAN such that c1 = ( A \/ B ) | ( A \/ B ) and ( for A being Element of S holds ( A \/ B ) . ( A \/ B ) = A ( B \/ C ) and ( A \/ C ) . ( A \/ C ) = A ( B \/ C ) ;
consider y being Element of [: Y , the carrier of X :] such that a = "\/" ( { x } , the InternalRel of Y ) and y in { x } and P [ y , x ] ;
assume that A c= Z and f = ( ( - 1 ) (#) ( ( id Z ) (#) ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R + ( exp_R +
( ( f /. i ) `1 ) `1 = ( ( GoB f ) * ( 1 , 1 ) ) `1 .= ( ( GoB f ) * ( 1 , 1 ) ) `1 .= ( GoB f ) * ( 1 , 1 ) `1 ;
dom ( Shift ( q , len ( q , len ( q , len ( p , q ) ) ) ) ) = { j + 1 where j is Nat : j in dom ( q , len ( q , len ( p , q ) ) ) } ;
consider G1 , G2 being Element of V such that G1 <= G2 and G1 <= G2 and for f being Morphism of G1 , G2 st f in G1 & g in G2 & f in G2 holds f = g & f = g & g = f & f = g ;
func - f -> PartFunc of C means : Def2 : for c being Element of C st c in dom f holds it . c = - f . c & for c being Element of C st c in dom f holds it . c = - f . c ;
consider phi such that phi . a = a and for a st a in a holds phi . a = a and for a st a in union L holds phi . a = H ( a ) ;
consider i1 , j1 such that [ i1 , j1 ] in Indices GoB f and [ i1 , j1 ] in Indices GoB f and [ i1 , j1 ] in Indices GoB f and f /. ( i1 + 1 ) = ( GoB f ) * ( i1 , j1 ) ;
consider i , n such that n <> 0 and n <> 0 & n <> 0 & n <> 0 & n <> 0 & n <> 0 & n <> 0 & n <> 0 & n <> 0 & n <> 0 & n <> 0 implies n = 0 & n = 0 & n = 0 & n = 0 & n = 0 & n = 0 & n = 0 & n = 0 ;
assume that 0 in Z and Z c= dom ( ( ( 1 - ( exp_R + f1 ) ) (#) ( ( exp_R + f2 ) (#) ( exp_R + f2 ) ) ) ) and for x st x in Z holds ( ( 1 - ( exp_R + f2 ) ) (#) ( ( exp_R + f2 ) ) ) `| Z ) . x = - 1 ;
cell ( G1 , i1 -' 1 , j1 -' 1 ) \ ( ( the InternalRel of G1 ) \ ( the InternalRel of G2 ) \ ( the InternalRel of G1 ) \ ( the InternalRel of G2 ) \ ( the InternalRel of G1 ) \ ( the InternalRel of G2 ) \ ( the InternalRel of G1 ) \ ( the InternalRel of G2 ) ) c= ( the InternalRel of G1 ) \ ( the InternalRel of G2 ) \ ( the InternalRel of G2 ) \ ( the InternalRel of G2 ) ;
ex Q being open Subset of X st s = Q & ex F being Subset-Family of X st ( for x being Point of X st x in Q holds F . x = union ( F ) & ( for x being Point of X st x in Q holds F . x = union ( F ) . x ) ) & ( for x being Point of X st x in Q holds F . x = union ( F . x ) ) & ( for x being Point of X st x in Q holds F . x = union ( F . x ) = union ( F . x ) ) & ( for x being Point of X st x in Q holds F . x = union ( F . x ) & ( for x being Point of X st x in union ( F . x ) & ( for x being Point of X st x in Q
gcd ( A , the carrier of A ) , ( the InternalRel of A ) /\ ( the InternalRel of B ) = 1 ;
R8 = ( the _ of ( ( the Sorts of ( s2 + 1 ) ) . ( ( the Sorts of ( s2 + 1 ) ) . ( m + 1 ) ) ) ) . ( ( the Sorts of ( s2 + 1 ) ) . ( m + 1 ) ) .= ( ( the Sorts of ( s2 + 1 ) ) . ( m + 1 ) ) ) . ( m + 1 ) .= ( the Sorts of ( s2 + 1 ) ) . ( m + 1 ) .= ( ( the Sorts of ( s2 + 1 ) ) . ( m + 1 ) .= ( ( the Sorts of B ) . ( m + 1 ) .= ( ( the Sorts of B ) . ( m + 1 ) ) . ( m + 1 ) .= ( ( the Sorts of B ) . ( m + 1 ) ) . (
CurInstr ( P-6 , Comput ( PE , s , m1 + 1 ) ) = CurInstr ( PE , Comput ( PE , s , m1 + 1 ) ) .= CurInstr ( PE , Comput ( PE , s , m1 + 1 ) ) .= CurInstr ( PE , Comput ( PE , s , m1 + 1 ) ) .= CurInstr ( PE , Comput ( PE , s , m1 + 1 ) ) .= CurInstr ( PE , Comput ( PE , Comput ( PE , s , m1 ) ) .= CurInstr ( PE , Comput ( PE , s , m1 + 1 ) .= CurInstr ( PE , Comput ( PE , s , m1 ) ) .= CurInstr ( PE , Comput ( PE , s , m1 ) .= CurInstr ( PE , Comput ( PE , s ,
P1 /\ ( ( ( P1 \/ P2 ) \/ P1 /\ P2 ) = ( ( P1 \/ P2 ) \/ ( P1 \/ P2 ) /\ ( P1 \/ P2 ) ) \/ ( ( P1 \/ P2 ) /\ ( P1 \/ P2 ) ) .= ( P1 \/ P2 ) /\ ( P1 \/ P2 ) \/ ( P1 /\ P2 ) ;
func still ( f ) -> Subset of the carrier of Al means : Def2 : for p being Element of the carrier' of Al holds it . p = f . ( p . ( p . ( p . ( p . ( p . ( p . ( p . ( p ) ) ) ) ) ) ) ) ;
for a , b being Element of COMPLEX , f being Polynomial of F_Complex st |. a .| > 1 & |. f .| >= 1 & f . ( a , b ) = 1 holds f . ( a , b ) = 1
defpred P [ Nat ] means ( 1 <= $1 & $1 <= len g implies ( for i st i in dom g holds g . i = ( G * ( i , j ) ) ) . ( $1 , i ) & ( G * ( i , j ) ) . ( $1 , i ) = ( G * ( i , j ) ) . ( $1 , i ) ) ;
cluster C1 , C2 , C2 , C2 , C2 , C2 , C2 , C2 , C2 , C2 , C2 , C2 , C2 , C2 , cin , cin , cin , cin *> -> stable implies C2 is stable & C2 is stable & C2 is stable & C2 is stable & C2 is stable & C2 is stable & C2 is stable & C2 is stable & C2 is stable & C2 is stable & C2 is stable & C2 is stable & C2 is stable & C2 is stable & C2 is stable & C2 is stable & C2 is stable & C2 is stable & C2 is stable & C2 is stable & C2 is stable & C2 is stable & C2 is stable & C2 is stable & C2 is stable & C2 is stable & C2 is stable & C2 is stable & C2 is stable & C2 is stable & C2 is stable & C2 is stable &
( ||. f .|| | X ) . c = ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| ;
|. q .| ^2 = ( ( q `1 / |. q .| - sn ) / ( 1 + sn ) ) ^2 + ( ( q `2 / |. q .| - sn ) ) ^2 + ( ( q `2 / |. q .| - sn ) ) ^2 ) & 0 < ( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 ;
for F being Subset-Family of T st F is open & {} in F & for A being Subset of T st A in F & A in F holds A misses ( A \/ B ) & ( for i being Nat st i in I holds A misses ( B \/ C ) or A c= ( B \/ C ) . i ) & ( for i being Element of I st i in I holds A misses ( B \/ C ) . i ) holds A misses ( B \/ C ) . i )
assume that len F >= 1 and F . ( len F ) = k + 1 and for k being Element of NAT st k in dom F holds F . ( k + 1 ) = g . ( k + 1 ) ;
i |^ ( ( order ( n , m ) - i ) ) * ( ( order ( n , m ) - ( i |^ n ) ) ) = ( i |^ ( n + 1 ) ) * ( i + 1 ) .= ( i |^ ( n + 1 ) ) * ( i + 1 ) ;
consider q being oriented Chain of G such that r = q and q <> {} and q in rng ( p ^ q ) and for q being Element of the carrier' of G st q in rng ( p ^ q ) holds q . ( len q + 1 ) = ( p ^ q ) . ( len q + 1 ) ;
defpred P [ Element of NAT ] means ( ( for x , y being Element of NAT st x <= y holds ( ( ( f . $1 ) . ( x , y ) ) . ( y , z ) ) . ( ( f . $1 ) . ( x , z ) ) . ( y , z ) ) = ( ( ( f . $1 ) . ( y , z ) ) . ( ( f . $1 ) . ( y , z ) ) ) ;
for A , B being Matrix of n , K , a , b , c being Element of K st len ( A * B ) = n & width ( A * B ) = n & width ( A * B ) = n & width ( A * B ) = n & width ( B * B ) = n & width ( A * B ) = n & width ( A * B ) = n ;
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 being Element of R st a . i = s . i & a in I & b . i = s . i ;
func |. x , y .| -> Element of COMPLEX equals |. ( x | ( |. x .| ) ) * ( |. y .| ) .| ;
consider g be FinSequence of ( the carrier of F ) | A , the carrier of F such that ( for n being Nat holds g . n = A & ( for n being Nat holds g . n = ( the Sorts of F ) . ( n + 1 ) ) & ( for n being Nat holds g . n = ( the Sorts of F ) . ( n + 1 ) ) ;
then n1 >= len ( p1 , p2 , n1 , n2 , n2 , n2 , n3 ) & n2 <= len ( p1 , p2 , n2 , n3 , n2 , n3 ) ;
( q `1 ) * a <= ( q `1 ) * a & ( - q `1 ) * b <= ( - q `1 ) * b & - q `1 <= - q `1 or - q `1 <= - q `1 or - q `1 <= - q `1 & - q `2 <= - q `1 or - q `2 <= - q `1 & - q `2 <= - q `2 & - q `2 <= - q `1 & - q `2 <= - q `1 & - q `2 <= - q `1 & - q `2 <= - q `1 & - q `2 <= - q `1 & - q `2 <= - q `1 <= - q `1 & - q `2 <= - q `1 & - q `2 <= - q `1 & - q `2 <= - q `1 or - q `2 <= - q `1 & - q `2 <= - q `1 & - q `2 <= - q `1 or - q `2 <= - q `1
( F . ( len ( p | ( len p ) ) ) ) . ( len p + 1 ) = ( F . ( len p ) ) ) . ( len p + 1 ) .= ( F . ( len p ) ) . ( len p + 1 ) .= ( F . ( len p ) ) ) . ( len p + 1 ) .= ( F . ( len p ) ) . ( len p + 1 ) .= ( F . ( len p ) ) /. ( len p + 1 ) .= ( F . ( len p + 1 ) .= ( F . ( len p + 1 ) .= ( F . ( len p + 1 ) .= ( F . ( len p + 1 ) ) . ( len p + 1 ) .= ( F . ( len p + 1 ) ) /. ( len p + 1 ) .= ( F . ( len p + 1 ) .= ( F
consider k1 being Nat such that ( a + k1 ) = 1 and ( a := k1 ) := ( a , k1 ) = ( a := k1 ) := ( a := k1 ) ) ^ ( a := k1 ) ;
consider B8 being Subset of ( ( A \/ B ) , C ) , C8 being Subset of ( A \/ B ) , C8 , C8 being Subset of ( A \/ B ) , C8 , C8 being Subset of ( A \/ B ) such that ( A \/ B ) , C ) , ( B \/ C ) , ( C \/ D ) , D ) are_equipotent and ( A , B ) , C , D ) is finite and ( A , D , E , E , f being Function of A , B , f being Function of A , B , f , f being Function of A , f , f , f , g being Function of A , B , g being Function of A , B , g being Function of A , B , g being Function of A , C , g being Function of A , C , g being Function of A , D , g being
v2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . F2 . F2 ) F2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) = ( ( ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . ( F2 . (
dom IExec ( I , P , s ) = the carrier of SCMPDS ( ) \/ ( the carrier of SCMPDS ) .= the carrier of SCMPDS ( ) \/ ( the carrier of SCMPDS ) .= the carrier of SCMPDS ( ) \/ ( the carrier of TOP-REAL 2 ) .= the carrier of TOP-REAL 2 ;
ex d1 be Real st d1 > 0 & for h be Real st h in { 0 } & |. h .| < 1 holds |. ( h " ) . ( h . ( n + 1 ) ) - ( h " ) . ( n + 1 ) .| < e ;
LSeg ( G * ( len G , 1 ) + |[ 1 , 0 ]| , |[ 1 , 0 ]| ) c= Int cell ( G * ( len G , 1 ) , |[ 1 , 0 ]| , |[ 1 , 0 ]| ) \/ |[ 1 , 0 ]| ;
LSeg ( mid ( h , i1 , i2 ) , i ) = LSeg ( h /. ( i1 + 1 ) , h /. ( i1 + 1 ) ) .= LSeg ( h /. ( i1 + 1 ) , h /. ( i1 + 1 ) ) .= ( h /. ( i1 + 1 ) ) ;
A = { q where q is Point of TOP-REAL 2 : LE ( q , q , P , p1 , p2 ) , P , p1 , p2 & LE q , q , P , p1 , p2 & LE q , q , P , p2 , p1 , p2 & LE q , p , P , p2 , p1 , p2 & LE q , p , P , p2 , p1 , p2 , P , p1 , p2 , p2 & LE q , p , P , p2 , p1 , p2 , p2 , P , p1 , p2 , p2 , P , p1 , p2 , p2 , p1 , p2 , P , p1 , p2 , P , p1 , p2 , p1 , p2 , P , p1 , p2 , p1 , p2 , p1 , p2 , p1 , p2 , P , p1 , p2 , P , p1 , p2 , p2 , p1 , p2 , p1 , p2 , p1 , p2 , p1 , p2 , p1 , p2 , p1 , p1 , p2
( - x ) | ( - y ) = ( - ( x | ( - y ) ) ) | ( - ( y | ( - y ) ) ) .= ( - ( x | ( - y ) ) ) | ( - ( y | ( - y ) ) ) ) | ( - ( y | ( - y ) ) ) .= ( - ( x | ( - y ) ) ) | ( - ( y | ( - y ) ) ) ) .= ( - ( ( - ( y | ( - y | ( - y | ( - y ) ) ) ) | ( - ( - y ) ) ) | ( - ( - y | ( - y | ( - y ) ) ) | ( - y | ( - y ) ) ) | ( - y | ( - y | ( - y | ( - y ) ) .= ( - ( - y | ( - y | ( - y | ( - y | ( - y |
0 * sqrt ( 1 + ( p `2 / |. p .| - sn ) ) = sqrt ( 1 + ( p .| - sn ) ) * sqrt ( 1 + sn ) ;
sqrt ( ( U - ( 1 / 2 ) * ( 1 / 2 ) ) ) = ( ( 1 / 2 ) * ( 1 / 2 ) ) ) * ( ( 1 / 2 ) * ( 1 / 2 ) ) .= ( ( 1 / 2 ) * ( 1 / 2 ) ) * ( 1 / 2 ) .= ( ( 1 / 2 ) * ( 1 / 2 ) ) * ( 1 / 2 ) ) .= ( ( 1 / 2 ) * ( 1 / 2 ) ) * ( 1 / 2 ) ) * ( 1 / 2 ) * ( 1 / 2 ) .= ( ( 1 / 2 ) * ( 1 / 2 ) ) * ( 1 / 2 ) ) * ( 1 / 2 ) ) * ( 1 / 2 ) * ( ( 1 / 2 ) * ( 1 / 2 ) .= ( ( 1 / 2 ) * ( ( 1 / 2 ) .= ( ( 1 / 2 ) * ( ( 1 / 2 ) )
func Shift ( f , h ) -> PartFunc of REAL , REAL means : Def2 : dom it = dom f & for x being Element of REAL n holds it . x = ( f + h ) . x & ( f + h ) . x = ( f + h ) . x + ( f + h ) . x ;
assume that 1 <= k and k + 1 <= len f and [ i , j ] in Indices G and [ i , j ] in Indices G and [ i , j ] in Indices G and [ i , j ] in Indices G and [ i , j ] in Indices G and [ i , j ] in Indices G and [ i , j ] in Indices G and [ i , j ] in Indices G and [ i , j ] in Indices G and [ i , j ] in Indices G and [ i , j ] in Indices G and [ i , j ] in Indices G and [ i , j ] in Indices G and [ i , j ] in Indices G * G * ( i , j ] in Indices G * ( i , j ] in Indices G * * ( i , j ] in Indices G * ( i , j ] in Indices G * * ( i , j ] in Indices G * ( i , j ] in Indices G *
assume that not y in Free H and not x in Free H and y in Free H and not x in Free H and y in Free H and x in Free H and y in Free H ;
defpred P1 [ Element of NAT ] means ( $1 = p implies ( $1 = 1 implies ( $1 = 2 ) |^ ( $1 ) ) & ( $1 = 1 ) |^ ( $1 ) ) & ( $1 is prime implies ( $1 is prime implies ( $1 is prime ) & ( $1 is prime ) & ( $1 is prime implies $1 is prime ) & ( $1 is prime ) & ( $1 is prime ) & ( $1 is prime ) & ( $1 is prime ) & ( $1 is prime ) & ( $1 is prime ) & ( $1 is prime implies ( $1 is prime implies $1 is prime ) & ( $1 is prime ) & ( $1 is prime ) & ( $1 is prime ) & ( $1 is prime implies $1 is prime ) & ( $1 is prime ) & ( $1 is prime implies ( $1 is prime ) & ( $1 is prime ) & ( $1 is prime ) & ( $1 is prime ) & ( $1 is prime ) & ( $1 is prime & ( $1 is prime ) & ( $1 is prime implies $1 is prime ) & ( $1
func \sigma ( C ) -> non empty Subset-Family of X means : Def2 : for A being Subset of X holds it . A c= A & for C being Subset of X st C in it holds C . C = C . ( A \/ C ) ;
[#] ( ( ( ( dist ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( .| .| .| .| ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) | Q ) ) ) = ( ( ( ( ( ( ( ( R | Q ) ) | Q ) ) | Q ) ) | Q ) | Q ) | Q ) ;
rng ( F | [: S , T :] ) = {} or rng ( F | [: S , T :] ) = { 1 , 2 } or rng ( F | [: S , T :] ) = { 1 , 2 } ;
( 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 Subset of TOP-REAL 2 such that P1 /\ P2 = { p1 , p2 } and P1 /\ P2 = { p1 , p2 } and P2 /\ LSeg ( p1 , p2 ) = { p1 , p2 } and P1 /\ P2 = { p2 } ;
f . ( p2 , p2 ) = |[ ( ( p2 `1 ) / ( 1 + ( p2 `2 ) ) ^2 ) , ( p2 `2 ) ^2 ]| ;
( AffineMap ( a , X ) " ) . x = ( AffineMap ( a , X ) " ) . x .= ( ( proj ( a , X ) " ) . x ) * u .= 0. TOP-REAL n .= 0 ;
for T being non empty TopSpace , A , B being closed Subset of T , p being Point of T , r being Real st A <> {} & B <> {} & A misses B holds A misses B & B misses C & C misses D & A misses D & B misses D & C misses D & C misses D & D misses E & A misses E & B misses D & C misses D & A misses D & B misses D & C misses D & B misses D & A misses D & B misses D & B misses D & C misses D & C misses D & C misses D & C misses D & C misses D & C misses D & C misses D & A misses D & C misses D & A misses D & B misses D & C misses D & A misses D & B in D & C misses D & C misses D & C misses D & C misses D & A misses D & A in D & A in D & A in D & C misses D & C misses D & D misses D & C misses D & B misses D & C misses D & C misses D & C misses D & D in D
for i being strict Subgroup of G , i being strict normal Subgroup of G st i + 1 in dom F & for x being Element of G st x in dom F ex y being Element of G st P [ x , y ] & P [ x , y ] holds P [ x , y ]
for x st x in Z holds ( ( ( exp_R * exp_R ) `| Z ) . x ) / ( exp_R . x ) = ( exp_R . x ) / ( exp_R . x ) ^2 + ( exp_R . x ) ^2 ) / ( exp_R . x ) ^2
synonym f /* a -> convergent means : Def2 : for x st x in dom f holds f . x = lim ( f /* a , lim ( f /* a ) ) & for n holds f . ( x - x0 ) = lim ( f /* a , lim ( a , x0 ) ) ;
then X1 , X2 , X1 , X2 , X2 , X1 , X2 , X2 , X1 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X1 , X2 , X2 , X1 , X2 , X2 , X1 , X2 , X2 , X1 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X1 , X2 , X2 , X1 , X2 , X2 , X1 , X2 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X1 , X2 , X2 , X1 , X2 , X1 , X2 , X2 , X1 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X1 , X2 ,
ex N be Neighbourhood of x0 st N c= dom ( SVF1 ( 1 , f , u ) ) & R . ( h . ( h . ( f . ( f . ( f . ( f . ( f . ( f . ( f . ( f . ( f . ( f . ( f . 0 ) ) ) ) ) ) ) ) ) ) ) & ( ex L st L = ( SVF1 ( 1 , f , u ) ) ) ) . ( h . ( f . ( f . 0 ) ) ) ) ;
sqrt ( ( p2 `2 / |. p2 .| - sn ) / ( 1 + sn ) ) ^2 + ( ( p2 `2 / |. p2 .| - sn ) ) ^2 + ( p2 `2 / |. p2 .| - sn ) ) ^2 >= ( ( p2 `2 / |. p2 .| - sn ) ) ^2 ;
( ( ( 1 - ( f1 * ( f1 - f2 ) ) ) / ( ( f1 - f2 ) / ( f1 - f2 ) ) ) ) / ( ( f1 - f2 ) / ( f1 - f2 ) ) ) = ( ( ( f1 * ( f1 - f2 ) ) / ( f1 - f2 ) ) / ( f1 - f2 ) ) / ( f1 - f2 ) ) & ( ( f1 * ( f1 - f2 ) ) / ( f1 - f2 ) ) ) / ( f1 - f2 ) ) / ( f1 - f2 ) ) / ( f1 - f2 ) ) / ( f1 - f2 ) ) / ( f1 - f2 ) = ( ( ( f2 * ( f1 - f2 ) ) / ( f1 - f2 ) ) / ( f1 - f2 ) ) / ( f1 - f2 ) ) / ( f1 - f2 ) ) / ( f1 - f2 ) ) / ( f1 - f2 ) ) / ( f1 - f2 ) ) = ( ( ( f2 * ( f1 - f2 ) ) / ( f1 - f2 ) / ( f1 - f2 ) ) / ( f1 - f2 ) ) / ( f1 - f2 ) ) /
assume that for x holds f . x = ( ( - 1 ) (#) ( sin + cos ) ) . x and for x st x in Z holds ( for x st x in Z holds f . x = - 1 ) and for x st x in Z holds f . x = - 1 ;
consider [: X1 , Y1 :] such that t = [: X1 , Y1 :] and [: X1 , Y1 :] in [: X2 , Y2 :] and ex Y1 being Subset of [: X1 , Y1 :] st Y1 = [: Y1 , Y2 :] & [: Y1 , Y2 :] in [: Y1 , Y2 :] & [: Y1 , Y2 :] in [: Y1 , Y2 :] ;
card ( S . n ) = card { d ( n + 1 ) + b ( n + 1 ) where d is Element of GF ( p ) : d in { a } & b in { d } & c in { d } } ;
sqrt ( ( ( - ( ( - ( - ( - ( i - n ) / 2 ) ) / ( 2 * ( i - n ) ) / ( 2 * ( i - n ) ) / ( 2 * ( i - n ) ) ) / ( 2 * ( i - n ) ) ) ) ) ) ) = sqrt ( ( - ( i - n ) / ( 2 * ( i - n ) ) ) / ( 2 * ( i - n ) ) ) ) ;