thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
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thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b
D c= S
let Y ;
S ` is convergent ;
q in X ;
V ;
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 , H ;
y in N ;
f = {} ;
let x ;
x in Z ;
let x ;
F is one-to-one ;
e <> b ;
1 <= n ;
f is special ;
S misses C
t <= 1 ;
y divides m ;
P divides M ;
let Z ;
let x ;
y c= x ;
let X ;
let C ;
x _|_ p ;
o is monotone ;
let X ;
A = B ;
1 < i ;
let x ;
let u ;
k <> 0 ;
let p ;
0 < r ;
let n ;
let y ;
f is onto ;
x < 1 ;
G c= F ;
a is_>=_than X ;
T is continuous ;
d <= a ;
p <= r ;
t < s ;
p <= t ;
t < s ;
let r ;
D <= E ;
assume e > 0 ;
assume 0 < g ;
p in X ;
x in X ;
Y ` in Y ;
assume 0 < g ;
not c in Y ;
not v in L ;
2 in z `2 ;
assume f = g ;
N c= b ` ;
assume i < k ;
assume u = v ;
I = J ;
B ` = b ` ;
assume e in F ;
assume p > 0 ;
assume x in D ;
let i be element ;
assume F is onto ;
assume n <> 0 ;
let x be element ;
set k = z ;
assume o = x ;
assume b < a ;
assume x in A ;
a `2 <= b `2 ;
assume b in X ;
assume k <> 1 ;
f = product l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is generated ;
assume m > 0 ;
assume A c= B ;
X is lower
assume A <> {} ;
assume X <> {} ;
assume F <> {} ;
assume G is open ;
assume f is dilatation ;
assume y in W ;
y \not <= x ;
A ` in B ` ;
assume i = 1 ;
let x be element ;
x `1 = x `1 ;
let X be BCK-algebra ;
assume S is non empty ;
a in REAL ;
let p be set ;
let A be set ;
let G be _Graph , S be Subset of G ;
let G be _Graph , S be Subset of G ;
let a be Complex ;
let x be element ;
let x be element ;
let C be FormalContext , a , b be Element of C ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
let y be Real ;
X c= f . a
let y be element ;
let x be element ;
let i be Nat ;
let x be element ;
n in NAT ;
let a be element ;
m in NAT ;
let u be element ;
i in NAT ;
let g be Function ;
Z c= NAT ;
l <= ma ;
let y be element ;
r2 in dom f ;
let x be element ;
k1 be Integer ;
let X be set ;
let a be element ;
let x be element ;
let x be element ;
let q be element ;
let x be element ;
assume f is being_homeomorphism ;
let z be element ;
a , b // K ;
let n be Nat ;
let k be Nat ;
B ` c= B ` ;
set s = b ;
n >= 0 + 1 ;
k c= k + 1 ;
R1 c= R ;
k + 1 >= k ;
k c= k + 1 ;
let j be Nat ;
o , a // Y ;
R c= Cl G ;
Cl B = B ;
let j be Nat ;
1 <= j + 1 ;
arccot is_differentiable_on Z ;
exp_R is_differentiable_in x ;
j < i0 ;
let j be Nat ;
n <= n + 1 ;
k = i + m ;
assume C meets S ;
n <= n + 1 ;
let n be Nat ;
h1 = {} ;
0 + 1 = 1 ;
o <> b3 ;
f2 is one-to-one ;
support p = {}
assume x in Z ;
i <= i + 1 ;
r1 <= 1 ;
let n be Nat ;
a "/\" b <= a ;
let n be Nat ;
0 <= r0 ;
let e be Real , x be Point of TOP-REAL 2 ;
not r in G . l
c1 = 0 ;
a + a = a ;
<* 0 *> in e ;
t in { t } ;
assume 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 Integer ;
let l be Nat ;
let i be Nat ;
let r ;
1 <= i2 ;
a "\/" c = c ;
let r be Real ;
let i be Nat ;
let m be Nat ;
x = p2 ;
let i be Nat ;
y < r + 1 ;
rng c c= E
Cl R is boundary ;
let i be Nat ;
R2 ;
cluster uparrow x -> closed ;
X <> { x } ;
x in { x } ;
q , b // M ;
A . i c= Y ;
P [ k ] ;
2 to_power x in W ;
X [ 0 ] ;
P [ 0 ] ;
A = A |^ i ;
2 >= seq . s ;
G . y <> 0 ;
let X be RealNormSpace , x be Point of X ;
a in X ;
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 set , X be set ;
M , L are_isomorphic ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded ;
rng f = Y ;
G8 c= L ;
assume x in Cl Q ;
m in dom P ;
i <= len Q ;
len F = 3 ;
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 <= i-32 ;
1 <= i-32 ;
pp c= PI ;
1 <= i & i <= len G ;
1 <= i & i <= len G ;
UMP C in L ;
1 in dom f ;
let seq , seq1 , seq2 ;
set C = a * B ;
x in rng f ;
assume f is_continuous_on X ;
I = dom A ;
u in dom p ;
assume a < x + 1 ;
s-7 is bounded ;
assume I c= P1 ;
n in dom I ;
let Q ;
B c= dom f ;
b + p \not _|_ a ;
x in dom g ;
F-14 is continuous ;
dom g = X ;
len q = m ;
assume A2 : x in A2 ;
cluster R \ S -> real-valued ;
sup D in S ;
x << sup D ;
b1 >= Z1 & b2 >= Z1 ;
assume w = 0. V ;
assume x in A . i ;
g in the \rm being carrier of X ;
y in dom t ;
i in dom g ;
assume P [ k ] ;
if C c= f , f
xx is increasing ;
let e2 be element ;
- b divides b ;
F c= \tau ( F ) ;
Gseq is non-decreasing ;
Gseq is non-decreasing ;
assume v in H . m ;
assume b in [#] B ;
let S be non void ManySortedSign , X be non-empty ManySortedSet of S ;
assume P [ n ] ;
assume union S is independent & finite S is finite ;
V is Subspace of V ;
assume P [ k ] ;
rng f c= NAT * ;
assume ex_inf_of X , L ;
y in rng f ;
let s , I be set , A be ManySortedSet of I ;
b ` c= b9 ` ;
assume not x in NAT ;
A /\ B = { a } ;
assume len f > 0 ;
assume x in dom f ;
b , a // o , c ;
B in B-24 ;
cluster product p -> non empty ;
z , x // x , p ;
assume x in rng N ;
cosec is_differentiable_in x & cosec is_differentiable_in x ;
assume y in rng S ;
let x , y be element ;
i2 < i1 or i2 < i2 ;
a * h in a * H ;
p , q in Y ;
redefine func sqrt I ;
q1 in A1 & q2 in A2 ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A2 c= A1 ;
an < n ;
assume A c= dom f ;
Re f is_integrable_on M ;
let k , m be element ;
a , a // b , b ;
j + 1 < k + 1 ;
m + 1 <= n1 ;
g is_differentiable_in x0 & g . x0 <> 0 ;
g is_continuous_in x0 & g is_continuous_in x0 ;
assume O is symmetric transitive ;
let x , y be element ;
let j0 be Nat ;
[ y , x ] in R ;
let x , y be element ;
assume y in conv A ;
x in Int V ;
let v be VECTOR of V ;
P3 halts_on s ;
d , c // a , b ;
let t , u be set ;
let X be set ;
assume k in dom s ;
let r be non negative Real ;
assume x in F | M ;
let Y be Subset of S ;
let X be non empty TopSpace , A be Subset of X ;
[ a , b ] in R ;
x + w < y + w ;
{ a , b } >= c ;
let B be Subset of A , C be Subset of B ;
let S be non empty ManySortedSign ;
let x be variable of f , g , h be element ;
let b be Element of X , x be Element of X ;
R [ x , y ] ;
x ` ` = x ;
b \ x = 0. X ;
<* d *> in D |^ 1 ;
P [ k + 1 ] ;
m in dom mn ;
h2 . a = y ;
P [ n + 1 ] ;
redefine func G * F -> l -| C ;
let R be non empty multMagma , a be Element of R ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p `2 ;
assume f | X is lower ;
x in rng co ;
let x be Element of B ;
let t be Element of D ;
assume x in Q .vertices() ;
set q = s ^\ k ;
let t be VECTOR of X ;
let x be Element of A ;
assume y in rng p `2 ;
let M be id mamaid id ;
let N be non empty holds the non empty Subset of M ;
let R be RelStr with finite is finite ;
let n , k be Nat ;
let P , Q be be be be be be be in ;
P = Q /\ [#] S ;
F . r in { 0 } ;
let x be Element of X ;
let x be Element of X ;
let u be VECTOR of V ;
reconsider d = x as FinSeq-Location ;
assume I does not destroy a ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < ( v - u ) / 2 ;
x <= c2 . x ;
x in F ` & y in F ` ;
redefine func S --> T -> * ;
assume t1 <= t2 & t2 <= t2 ;
let i , j be even Integer ;
assume F1 <> F2 & F2 <> F1 ;
c in Intersect ( union R ) ;
dom p1 = c & dom p2 = c ;
a = 0 or a = 1 ;
assume A1 : A <> A6 ;
set i1 = i + 1 ;
assume a1 = b1 & a2 = b2 ;
dom g1 = A & dom g2 = A ;
i < len M + 1 ;
assume not -infty in rng G ;
N c= dom f1 & N c= dom f2 ;
x in dom sec & x in dom sec ;
assume [ x , y ] in R ;
set d = ( x - y ) / ( x - y ) ;
1 <= len g1 & g1 /. 1 = g1 /. 1 ;
len s2 > 1 & len s2 > 1 ;
z in dom ( f1 (#) f2 ) ;
1 in dom ( D2 | Seg 1 ) ;
( p `2 / |. p .| - sn ) = 0 ;
j2 <= width G ;
len PI > 1 + 1 ;
set n1 = n + 1 ;
|. q-35 .| = 1 ;
let s be SortSymbol of S ;
if i = i holds i = 1 ;
X1 c= dom f & X2 c= dom f ;
h . x in h . a ;
let G be \it S1 with id of on ;
cluster m * n -> square ;
let kk be Nat , a be Int-Location ;
i - 1 > m - 1 ;
R is transitive in field R ;
set F = <* u , w *> ;
p-2 c= P3 & p`2 c= P3 ;
I is_closed_on t , Q ;
assume [ S , x ] is vertical ;
i <= len ( f2 | n ) ;
p is FinSequence of X ;
1 + 1 in dom g ;
Sum R2 = n * r ;
cluster f . x -> complex-valued ;
x in dom ( f1 + f2 ) ;
assume [ X , p ] in C ;
BF c= [ 3 , 2 ] ;
n2 <= ( 2 * n ) / ( 2 * n ) ;
A /\ cP c= A ` ;
cluster x -valued -> constant for Function ;
let Q be Subset-Family of S , P be Subset of S ;
assume n in dom g2 & m + 1 in dom g2 ;
let a be Element of R ;
t `2 in dom e2 & t `2 in dom e2 ;
N . 1 in rng N ;
- z in A \/ B ;
let S be SigmaField of X , X be non empty set ;
i . y in rng i ;
REAL c= dom f & f | X is bounded ;
f . x in rng f ;
mt <= ( r / 2 ) * 2 ;
s2 in r-5 & s2 in r-5 ;
let z , z be complex number ;
n <= ( N . m ) ;
LIN q , p , s ;
f . x = waybelow x /\ B ;
set L = [' S , T '] ;
let x be non positive ExtReal ;
let m be Element of M ;
f in union rng ( F1 . n ) ;
let K be add-associative right_zeroed right_complementable associative non empty doubleLoopStr , p be Polynomial of K ;
let i be Element of NAT ;
rng ( F * g ) c= Y ;
dom f c= dom x & dom g c= dom x ;
n1 < n1 + 1 ;
n1 < n1 + 1 ;
cluster 1. T -> 8 ;
[ y2 , 2 ] `2 = z ;
let m be Element of NAT ;
let S be Subset of R ;
y in rng ( S | A ) ;
b = upper_bound dom f & b = upper_bound dom f ;
x in Seg ( len q ) ;
reconsider X = D ( ) as set ;
[ a , c ] in E1 ;
assume n in dom h2 & n + 1 in dom h2 ;
w + 1 = ma1 ;
j + 1 <= j + 1 + 1 ;
k2 + 1 <= k1 ;
let i be Element of NAT ;
Support u = Support p & Support u c= Support p ;
assume X is complete complete holds X is complete
assume f = g & p = q ;
n1 <= n1 + 1 & n1 + 1 <= n2 ;
let x be Element of REAL , r be Real ;
assume x in rng ( s2 | X ) ;
x0 < x0 + 1 & x0 + 1 < r2 ;
len ( L5 * L ) = W ;
P c= Seg ( len A ) ;
dom q = Seg n & dom q = Seg n ;
j <= width M *' ;
let seq1 be real-valued FinSequence , seq2 be real-valued Real_Sequence ;
let k be Element of NAT ;
Integral ( M , P ) < +infty ;
let n be Element of NAT ;
assume z in z := . 0 ;
let i be set ;
n - 1 = n-1 ;
len ( n | m ) = n ;
or \mathop { Z } c= F ;
assume x in X or x = X ;
x is midpoint of b , c ;
let A , B be non empty set , f be Function ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q & i in dom q ;
let s be Element of E * ;
let B1 be Basis of x , B2 ;
L3 /\ L2 = {} ;
L1 /\ LSeg ( L2 , p2 ) = {} ;
assume downarrow x = downarrow y ;
assume b , c // b , c ;
LIN q , c , c ;
x in rng ( f . -129 ) ;
set n8 = n + j ;
let D7 be non empty set , f be FinSequence of D ;
let K be add-associative right_zeroed right_complementable non empty addLoopStr , p be Polynomial of K ;
assume f opp = f & h opp = h ;
R1 - R2 is total & R2 - R1 is total ;
k in NAT & 1 <= k ;
let a be Element of G ;
assume x0 in [. a , b .] ;
K1 ` is open & K1 ` is open ;
assume that a , b are_maximal in C ;
let a , b be Element of S ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = Integral ( M , f ) ;
cluster n\bf for \in neas ;
not u in { ag } ;
the carrier of f c= B ;
reconsider z = x as VECTOR of V ;
cluster the
r (#) H is as as as as as as as as as as as as <> of X ;
s . intloc 0 = 1 ;
assume that x in C and y in C ;
let U0 be strict universal algebra , A be non-empty MSAlgebra over S ;
[ x , Bottom T ] is compact ;
i + 1 + k in dom p ;
F . i is stable Subset of M ;
r-35 in : ( y in { x } ) & r-35 in { y } ;
let x , y be Element of X ;
let A , I be contradiction Element of X ;
[ y , z ] in [: O , O :] ;
( } = 1 implies ( Macro i ) . 0 = 1
rng Sgm A = A ;
q |- \! such that All ( y , p ) ;
for n holds X [ n ] ;
x in { a } & x in d ;
for n holds P [ n ] ;
set p = |[ x , y , z ]| ;
LIN o , a , b & LIN o , a , b ;
p . 2 = Z / Y ;
( D `2 ) ^2 = {} & ( D `2 ) ^2 = 1 ;
n + 1 + 1 <= len g ;
a in [: CQC-WFF ( 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 + f3 ;
a <= max ( a , b ) ;
i-1 < len G + 1-1 ;
g . 1 = f . i1 ;
x `2 , y `2 ] in A2 ;
( f /* s ) . k < r ;
set v = VAL g ;
i - k + 1 <= S ;
cluster associative for non empty multMagma ;
x in support ( support ( t ) ) ;
assume a in [: the carrier of G , the carrier of G :] ;
i `2 <= len ( y `2 ) ;
assume p divides b1 + b2 & p divides b2 ;
M <= upper_bound M1 & M <= upper_bound M2 ;
assume x in W-min ( X ) & y in L~ f ;
j in dom ( z | Seg n ) ;
let x be Element of D ( ) ;
IC s4 = l1 & IC s4 = l1 ;
a = {} or a = { x } ;
set uG = Vertices G , uG = Vertices G ;
seq " is non-zero & seq " is non-zero implies seq " is non-zero
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
h-4 c= h-14 & hF c= h-14 ;
]. a , b .[ c= Z ;
X1 , X2 are_separated implies X1 , X2 are_separated
a in Cl ( union ( F \ G ) ) ;
set x1 = [ 0 , 0 ] ;
k + 1 - 1 = k - 1 ;
cluster -> real-valued for Relation ;
ex v st C = v + W ;
let IT be non empty addLoopStr , a be Element of IT ;
assume V is Abelian add-associative right_zeroed right_complementable associative commutative distributive non empty doubleLoopStr ;
X-21 \/ Y in \sigma ( L ) ;
reconsider x = x as Element of S ;
max ( a , b ) = a ;
sup B is upper & sup B is upper ;
let L be non empty reflexive RelStr , X be Subset of L ;
R is reflexive transitive in X & R is transitive in X ;
E , g |= ( the_left_argument_of H ) ;
dom G `2 = a & dom G = b ;
( 1 / 4 ) * 4 >= - r ;
G . p0 in rng G & G . p1 in rng G ;
let x be Element of [: F , G :] , X be set ;
D [ P-6 , 0 ] ;
z in dom id B & z in dom id B ;
y in the carrier of N & y in the carrier of N ;
g in the carrier of H & h in the carrier of G ;
rng ( f | X ) c= NAT & rng ( f | X ) c= X ;
j `2 + 1 in dom s1 ;
let A , B be strict Subgroup of G ;
let C be non empty Subset of REAL ;
f . z1 in dom h & f . z2 in dom h ;
P . k1 in rng P ;
M = AM +* {} & M = AM +* {} ;
let p be FinSequence of ( the carrier of K ) * ;
f . n1 in rng f & f . n2 in rng f ;
M . ( F . 0 ) in REAL ;
diameter [. a , b .[ = b-a ;
assume the distance of V , Q is_^ v ;
let a be Element of ^ ( V ) ;
let s be Element of PI ( ) ;
let Ps be non empty Poset RelStr ;
let n be Nat ;
the carrier of g c= B ;
I = halt SCM & I = halt SCM ;
consider b being element such that b in B ;
set BM = BCS K , BM = BCS K ;
l <= j & j <= j implies j <= j ;
assume x in downarrow [ s , t ] ;
( x `2 ) / ( x `2 ) in uparrow t ;
x in ( . T ) \/ ( { T } \/ { T } ) ;
let h be Morphism of c , a ;
Y c= 1. ( K , the_rank_of Y ) ;
A2 \/ A3 c= L1 \/ L2 & A2 \/ A3 c= L2 ;
assume LIN o , a , b & LIN o , a , b ;
b , c // d1 , e2 ;
x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 ;
dom <* y *> = Seg 1 & dom <* y *> = Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar s .| ;
[ x , x `2 ] in X ~ ;
for n being Nat holds 0 <= x . n
[' a , b '] = [. a , b .] ;
cluster -> \hbox closed for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 , q1 is_collinear & q1 , q2 , q2 is_collinear ;
dom M1 = Seg n & dom M2 = Seg n ;
x = [ x1 , x2 ] & y = [ y1 , y2 ] ;
let R , Q be ManySortedSet of A ;
set d = ( 1 / ( n + 1 ) ) ;
rng g2 c= dom W & rng g2 c= dom W ;
P . ( [#] Sigma \ B ) <> 0 ;
a in field R & a = b ;
let M be non empty Subset of V , V be Subset of V ;
let I be Program of SCM+FSA , a be Int-Location ;
assume x in rng ( S * ( R * S ) ) ;
let b be Element of the carrier of T ;
dist ( e , z ) - r-r > r-r ;
u1 + v1 in W2 & v1 in W2 ;
assume that the carrier of L misses rng G ;
let L be lower-bounded antisymmetric transitive antisymmetric RelStr ;
assume [ x , y ] in a9 ;
dom ( A * e ) = NAT ;
let a , b be Vertex of G ;
let x be Element of Bool M , i be Element of I ;
0 <= Arg a & PI < 2 * PI ;
o9 , a9 // o9 , b9 & o9 , c9 // a9 , b9 ;
{ v } c= the carrier of l & { v } c= the carrier of l ;
let x be bound of A ;
assume x in dom ( uncurry f ) & y in dom ( uncurry f ) ;
rng F c= ( product f ) |^ X
assume that D2 . k in rng D and D2 . k in rng D ;
f " . p1 = 0 & f " . p2 = 0 ;
set x = the Element of X , y = the Element of Y ;
dom Ser ( G ) = NAT & rng Ser ( G ) c= NAT ;
let n be Element of NAT ;
assume LIN c , a , e1 ;
cluster -> \/ for FinSequence of NAT ;
reconsider d = c as Element of L1 ;
( v2 |-- I ) . X <= 1 ;
assume x in the carrier of f & y in the carrier of f ;
conv @ S c= conv @ A & conv @ S c= conv @ A ;
reconsider B = b as Element of the carrier of T ;
J , v |= P ! ( P ! l ) ;
redefine func J . i -> non empty TopSpace ;
ex_sup_of Y1 \/ Y2 , T & ex_sup_of Y1 , T ;
W1 is_\HM { field W1 } & R is_not ] implies R is Relation
assume x in the carrier of R & y in the carrier of S ;
dom ( n --> 0 ) = Seg n & dom ( n --> 0 ) = Seg n ;
s4 misses s2 & s4 misses s2 ;
assume ( a 'imp' b ) . z = TRUE ;
assume that X is open and f = X --> d ;
assume [ a , y ] in an ;
assume that } c= J and function } c= J ;
Im ( ( lim seq ) - ( lim seq ) ) = 0 ;
( sin . x ) <> 0 & ( sin . x ) <> 0 ;
sin is_differentiable_on Z & cos is_differentiable_on Z implies sin * ( cos ^ ) is_differentiable_on Z
t3 . n = t3 . n & t2 . n = t2 . n ;
dom ( dom ( - F ) ) c= dom F ;
W1 . x = W2 . x & W2 . x = W2 . x ;
y in W .vertices() \/ W .vertices() \/ W .vertices() ;
( ( k + 1 ) + 1 ) <= len ( v | ( k + 1 ) ) ;
x * a \equiv y * a . ( mod m ) ;
proj2 .: S c= proj2 .: P & proj2 .: S c= proj2 .: P ;
h . p4 = g2 . I & h . I = g2 . I ;
G6 = ( U /. 1 ) `1 .= G * ( 1 , k ) `1 ;
f . rr1 in rng f & f . rr2 in rng f ;
i + 1 + 1 <= len f ;
rng F = rng ( F . ( len F ) ) .= rng F ;
mode Y. is well unital associative non empty multMagma ;
[ x , y ] in A ~ ;
x1 . o in L2 . o & x2 . o in L2 . o ;
the carrier of m c= B ;
not [ y , x ] in id X ;
1 + p .. f <= i + len f ;
seq ^\ k1 is lower & seq is lower ;
len ( F . ( len F ) ) = len I ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be Complex , a be complex number ;
Comput ( P , s , n ) = s ;
k <= k + 1 & k + 1 <= len p ;
reconsider c = {} T as Element of L ;
let Y be Element of of of of of \HM { the carrier of T } ;
cluster -> directed-sups-preserving for Function of L , L ;
f . j1 in K . j1 ;
redefine func J => y -> total Function equals J * ( J * J ) ;
K c= 2 -tuples_on the carrier of T ;
F . b1 = F . b2 & F . b2 = F . b2 ;
x1 = x or x1 = y or x1 = z ;
pred a <> {} means : Def1 : ( a / a ) * ( b / a ) = 1 ;
assume that a c= b and b in a ;
s1 . n in rng s1 & s1 . n in rng s1 ;
{ o , b2 } on C2 & { o , b2 } on C2 ;
LIN o , b , b9 & LIN o , b , c ;
reconsider m = x as Element of Funcs ( V , C ) ;
let f be non trivial FinSequence of D ;
let FF2 be non empty /\ the l of X ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in [: F-8 , F-8 :] ;
reconsider pp = x , pk2 = y as Subset of m ;
let A , B , C be Element of R ;
redefine func strict non empty b9 -\overline D -> strict for
rng c `2 misses rng ee & rng c `2 misses rng ee ;
z is Element of gr { x } & z is Element of gr { x } ;
not b in dom ( a .--> p1 ) ;
assume k >= 2 & P [ k ] ;
Z c= dom ( cot * cot ) & Z c= dom ( cot * cot ) ;
the component of Q c= UBD A & the component of Q c= UBD A ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( 1 / ( f ^ ) ) ;
pred f = u means : Def1 : a * f = a * u ;
for n holds P1 [ \mathop { n } ] ;
{ x . O : x in L } <> {} ;
let x be Element of V . s ;
let a , b be Nat ;
assume that S = S2 and p = p2 and q = q2 ;
gcd ( n1 , n2 , n1 ) = 1 & gcd ( n1 , n2 , n1 , n2 , n1 ) = 1 ;
set oI = a * ( 0. ( Z , n ) ) ;
seq . n < |. r1 .| & seq . m < r1 ;
assume that seq is increasing and r < 0 and seq is increasing ;
f . ( y1 , x1 ) <= a ;
ex c being Nat st P [ c ] ;
set g = { n to_power 1 } , n = n to_power 1 ;
k = a or k = b or k = c ;
( a , b ) `1 , ( b , g ) `2 ] in G ;
assume that Y = { 1 } and s = <* 1 *> ;
IS1 . x = f . x .= 0 .= 0 ;
W3 .last() = W3 . 1 & y = W3 . 2 ;
cluster trivial -> trivial finite for _Graph of G , finite _Graph ;
reconsider u = u as Element of Bags X ;
A in B ^ implies A , B are_that A , B are_that B , A are_
x in { [ 2 * n + 3 , k ] } ;
1 >= ( q `1 / |. q .| - cn ) / ( 1 - cn ) ;
f1 is_the _ _ _ _ { f } , T ;
( f `2 ) ^2 / ( |. q .| ) ^2 <= ( |. q .| ) ^2 / ( |. q .| ) ^2 ;
h is_the carrier of Cage ( C , n ) ;
( b `2 ) ^2 / ( |. p .| ) ^2 <= ( |. p .| ) ^2 / ( |. p .| ) ^2 ;
let f , g be ex X be \cdot of f , Y ;
S * ( k , k ) <> 0. K ;
x in dom max ( - f , - f ) ;
p2 in [: N , N :] & p2 in [: N , N :] ;
len ( the_left_argument_of H ) < len ( H ) & len ( H ) < len ( H ) ;
F [ A , F-14 . A ] ;
consider Z such that y in Z and Z in X ;
hence 1 in C & A c= C |^ A ;
assume r1 <> 0 or r2 <> 0 & r1 <> 0 ;
rng q1 c= rng C1 & rng q2 c= rng C2 ;
A1 , L , A3 , A3 , L be non empty set ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in C ( p , S9 ) & c in C ( p , S ) ;
then S is atomic ;
Cl Int [#] T = [#] T & Cl Int [#] T = [#] T ;
( f | A2 ) | A2 = f2 | A1 & ( f | A2 ) | A1 = f2 ;
0. M in the carrier of W & 0. M in the carrier of W ;
let v , v be Element of M ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ V & V c= Y \ Z ;
let X be Subset of S ~ ;
consider H1 such that H = 'not' H1 and H1 in X ;
1_ ( L ) c= ( t * p ) * ( p * ( p * ( 1 / p ) ) ) ;
0 * a = 0. R .= a * 0 .= 0 * a ;
A |^ ( 2 , 2 ) = A ^^ A ;
set vLX = ( v /. n ) `1 , vX = ( v /. n ) `1 , vX = ( v /. n ) `1 , vX = ( v /. n ) `1 , vX =
r = 0. ( REAL-NS n ) * ||. x - y .|| .= ||. x - y .|| ;
( f . p4 ) `1 >= 0 & ( f . p2 ) `2 >= 0 ;
len W = len ( W | ( len W ) ) .= len W ;
f /* ( s * G ) is divergent_to+infty ;
consider l being Nat such that m = F . l ;
t8 does not destroy b1 & not ( t is not empty & not ( t is not empty ) & not ( t is not empty ) & not ( t is not empty ) & not ( t is not empty & not ( t is not empty
reconsider Y1 = X1 , Y2 = X2 as SubSpace of X ;
consider w such that w in F and not x in w ;
let a , b , c , d be Real ;
reconsider i = i - 1 as non zero Element of NAT ;
c . x >= id ( L . x ) ;
\sigma ( T ) \/ omega ( T ) is Basis of T ;
for x being element st x in X holds x in Y
cluster [ x1 , x2 , x3 ] -> pair for set ;
downarrow a /\ downarrow t is Ideal of T ;
let X be \hbox { \mathbb N } , N be non empty set ;
rng f = S2 \rm such that rng f = S2 -element ( S , X )
let p be Element of B , x be Element of the carrier' of S ;
max ( N1 , 2 ) >= N1 & max ( N2 , 2 ) >= N2 ;
0. X <= b |^ ( m * mm1 ) ;
assume that i in I and R1 . i = R ;
i = j1 & p1 = q1 & p2 = q2 implies p1 = p2
assume gR in the right & gR in the carrier of g ;
let A1 , A2 be Point of S , B1 be Point of S ;
x in h " ( P /\ [#] T1 ) /\ [#] T2 ;
1 in Seg 2 & 1 in Seg 3 implies 1 in Seg 3
reconsider X-5 = X , Xholds = Y as non empty Subset of Tsuch that X is non empty and Y is non empty ;
x in ( the Arrows of B ) . i /\ the Arrows of C ;
cluster E-32 . n -> ( the carrier' of G ) -valued ;
n1 <= i2 + len g2 & n2 + len g2 <= len f ;
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume v in the carrier' of G2 & v in the carrier' of G2 ;
y = Re y + ( Im y ) * i ;
( - ( - 1 ) ) , p ] .| = 1 ;
x2 is_differentiable_on ]. a , b .[ & x2 in ]. a , b .] ;
rng M5 c= rng D2 & rng M5 c= rng ( D1 | Seg n ) ;
for p being Real st p in Z holds p >= a
( 1. ( X , n ) ) * f = proj1 * f .= f ;
( seq ^\ m ) . k <> 0 ;
s . ( G . ( k + 1 ) ) > x0 ;
( p -Path ( M , n ) ) . 2 = d ;
A \oplus ( B \ominus C ) = ( A \oplus B ) \ominus C
h \equiv gg . ( mod ( P , T ) ) ;
reconsider i1 = i-1 , i2 = 1 as Element of NAT ;
let v1 , v2 be VECTOR of V , v be VECTOR of V ;
for V being Subspace of V holds V is Subspace of [#] V
reconsider i = i , j = j as Element of NAT ;
dom f c= [: C , D :] & rng f c= [: C , D :] ;
x in ( the Sorts of B ) . n ;
len } in Seg ( len f2 + len g2 ) & len f1 = len f2 + 1 ;
pS1 c= the topology of T & pS2 c= the topology of T ;
]. r , s .] c= [. r , s .] ;
let B2 be Basis of T2 , B be Basis of T2 ;
G * ( B * A ) = id o1 & G * ( B * A ) = id o2 ;
assume that p , u , u , v is_collinear and u , v , w , y is_collinear ;
[ z , z ] in union rng ( F . z ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S , C = $1 .. S ;
LIN a1 , a3 , b1 & LIN a2 , a3 , b2 ;
f " ( f .: x ) = { x } ;
dom w2 = dom r12 & dom r12 = dom r12 ;
assume that 1 <= i and i <= n and j <= n ;
( ( ( g2 ) . O ) `2 ) / 2 <= 1 ;
p in LSeg ( E . i , F . i ) ;
IK * ( i , j ) = 0. K ;
|. f . ( s . m ) - g .| < g1 ;
q7 . x in rng ( q7 | Seg ( n + 1 ) ) ;
Carrier ( L\subseteq ( Carrier ( f ) ) ` ) misses ( Carrier ( f ) ) ` ;
consider c being element such that [ a , c ] in G ;
assume that N_ = o_ ( o , A ) and o_ ( o , A ) = {} ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= [: F |^ C-1 , F |^ C-1 :] ;
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [: f . j , f . j :] ;
pred 0 <= x & x / 2 <= x ;
p `2 - q `2 <> 0. TOP-REAL 2 & p `2 - q `2 = 0. TOP-REAL 2 ;
redefine func aa\circ ( S , T ) -> non empty set ;
let x be Element of S ~ ;
the Arrows of F is one-to-one & the Arrows of F is one-to-one ;
|. i .| <= - ( - 2 |^ n ) / ( n + 1 ) ;
the carrier of I[01] = dom P & the carrier of I[01] = dom P ;
} * ( n + 1 ) ! > 0 * n ;
S c= ( A1 /\ A2 ) /\ ( A1 /\ A2 ) ;
a3 , a4 // b3 , b3 or a3 , a4 // b3 , b3 ;
then dom A <> {} & dom A <> {} & A <> {} ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , Y & y Joins G2 , G2 ;
set v2 = ( v /. ( i + 1 ) ) `1 , v1 = ( v /. ( i + 1 ) ) `1 ;
x = r . n .= r4 . n .= r4 . n ;
f . s in the carrier of S2 & f . s in the carrier of S2 ;
dom g = the carrier of I[01] & rng g c= the carrier of I[01] ;
p in Lower_Arc ( P ) /\ Lower_Arc ( P ) ;
dom d2 = [: A2 , A2 :] & dom d2 = [: A2 , A2 :] ;
0 < ( p / ||. z .|| + 1 ) / ( ||. z .|| + 1 ) ;
e . ( m3 + 1 ) <= e . ( m3 + 1 ) ;
B \ominus X \/ B \ominus Y c= B \ominus X /\ Y
-infty < Integral ( M , Im ( g | B ) ) ;
cluster O := F -> \cdot \HM { d } -valued for operation of X ;
let U1 , U2 be non-empty MSAlgebra over S , B be MSAlgebra over S ;
Proj ( i , n ) * g is_differentiable_on X & g is_differentiable_on X ;
let x , y , z be Point of X , r be Real ;
reconsider pp = p . x , pp = p . y as Subset of V ;
x in the carrier of Lin ( A ) & y in the carrier of Lin ( B ) ;
let I , J be parahalting Program of SCM+FSA , a be Int-Location ;
assume - a is lower & b is lower implies a + b is lower ;
Int Cl A c= Cl Int Cl Int Cl A & Int Cl 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 / |. p2 .| <= ( p2 `2 / |. p2 .| - sn ) / ( 1 + sn ) ;
Cl ( Q ` ) = [#] ( ( T | P ) ` ) ;
set S = the carrier of T , T = the carrier of T ;
set I8 = for f be Element of ' ( n , m ) holds f is one-to-one ;
len p - n = len thesis & len p - n = len p - n ;
A is Permutation of Swap ( A , x , y ) ;
reconsider nx = nx - 1 as Element of NAT ;
1 <= j + 1 & j + 1 <= len ( s | [: Seg n , Seg n :] ) ;
let qm , qm be Let w of M , I be Element of M ;
( a in the carrier of S1 ) & ( a in the carrier of S2 ) ;
c1 /. n1 = c1 . n1 & c2 /. n1 = c2 . n1 ;
let f be FinSequence of TOP-REAL 2 , i , j be Nat ;
y = ( ( f * S ) . x ) . ( f . x ) ;
consider x being element such that x in an -`1 ;
assume r in ( ( dist ( o ) ) .: P ) ;
set i2 = ( n , i ) `1 , i1 = ( n , i ) `1 , i2 = ( n + 1 ) + 1 ;
h2 . ( j + 1 ) in rng h2 & h2 . ( j + 1 ) in rng h2 ;
Line ( M29 , k ) = M . i .= Line ( M29 , k ) ;
reconsider m = ( x - 2 ) / 2 as Element of ( the carrier of TOP-REAL 2 ) ;
let U1 , U2 be strict Subspace of U0 , U2 be strict Subspace of U0 ;
set P = Line ( a , d ) ;
len p1 < len p2 + 1 & p1 in LSeg ( p1 , p2 ) ;
let T1 , T2 be Scott Scott `1 of L , x be Element of L ;
then x <= y & ( x c= { y } ) & y c= { x } ;
set M = n -\hbox { m } , N = n -\hbox { m } , S = n -\hbox { m } , T = n -\hbox { n } , T = n -\hbox { m } , S = n -\hbox { m } , T
reconsider i = x1 , j = x2 , k = x3 as Nat ;
rng the_arity_of a9 c= dom H & rng the_arity_of a9 c= dom H ;
z1 " = ( z1 " * z2 ) " .= ( z1 " * z2 ) " .= ( z1 " * z2 ) " ;
x0 - r / 2 in L /\ dom f & x0 - r / 2 in dom f ;
then w is that rng w /\ L <> {} & rng w /\ L <> {} ;
set x-10 = x-9 ^ <* Z *> ^ <* Z *> ^ xZ ;
len w1 in Seg ( len w1 + 1 ) & w1 in Seg ( len w1 + 1 ) ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of |. <* V , { k } *> } ;
x . n = ( |. a . n .| ) * ( A * ) ;
( p `1 ) ^2 / ( G * ( len G , 1 ) `1 ) ^2 <= ( p `1 ) ^2 / ( G * ( len G , 1 ) `1 ) ^2 ;
rng ( g * f ) c= L~ ( g * f ) \/ L~ ( g * f ) ;
reconsider k = i-1 * ( l + j ) as Nat ;
for n being Nat holds F . n is \HM { +infty } is \HM { -infty }
reconsider xx = xx , xx = xx , xx = 1 as VECTOR of M ;
dom ( f | X ) = X /\ dom f .= X /\ dom f ;
p , a // p , c & b , a // c , c ;
reconsider x1 = x , y1 = y , y2 = z as Element of REAL m ;
assume i in dom ( a * p ^ q ) ;
m . ag = p . ( ag . ( a , b ) ) ;
a / ( s . m - s . n ) / 2 <= 1 ;
S . ( n + k + 1 ) c= S . ( n + k ) ;
assume that B1 \/ C1 = B2 \/ C2 and B2 \/ C2 = C1 \/ C2 ;
X . i = { x1 , x2 } . i .= x1 . i ;
r2 in dom ( h1 + h2 ) & r1 < r2 implies r1 < r2 & r2 < r2
- ( 0. R ) = a & b-0 = b ;
F8 is_closed_on t2 , Q8 & F8 is_halting_on t1 , Q1 ;
set T = -> in and { x0 } in { 0 , 1 } ;
Int Cl Int Cl R c= Int Cl R & Int Cl R c= Int Cl R ;
consider y being Element of L such that c . y = x ;
rng ( F<* x *> ) = { FF . x } & rng FF = { F . x } ;
G-23 " { c } c= B \/ S ;
f[#] is Relation of [: X , X :] , X & f is Function of X , X ;
set RQ = the Element of P , RQ = the Element of Q ;
assume that n + 1 >= 1 and n + 1 <= len M ;
let k2 be Element of NAT , k be Nat ;
reconsider pA = u , pA = v as Element of ( ( ( n + 1 ) -tuples_on ( ( n + 1 ) -tuples_on ( n + 1 ) ) ) -tuples_on ( ( n + 1 ) -tuples_on ( ( n + 1 ) -tuples_on ( ( n +
g . x in dom f & x in dom g implies x = f . x
assume that 1 <= n and n + 1 <= len f1 ;
reconsider T = b * N as Element of ^ ( G , N ) ;
len P\bf 1 <= len P-35 & P\bf 1 <= len P-35 ;
x " in the carrier of A1 & x " in the carrier of A2 ;
[ i , j ] in Indices ( A * ( i , j ) ) ;
for m be Nat holds Re ( F . m ) is simple function of S
f . x = a . i .= a1 . k .= a . k ;
let f be PartFunc of REAL i , REAL n , x be Element of REAL n ;
rng f = the carrier of ( A + ( - 1 ) * ( A + - 1 ) ) ;
assume s1 = sqrt ( |[ 2 , p ]| - r / 2 ]| - r / 2 ;
pred a > 1 & b > 0 & a / b > 1 ;
let A , B , C be Subset of IK , a be element ;
reconsider X0 = X , Y0 = Y as Point of X ;
let f be PartFunc of REAL , REAL , x be Element of REAL ;
r * ( v1 |-- I ) . X < r * 1 ;
assume that V is Subspace of X and X is Subspace of V ;
let t-3 , t-3 be Relation of P , P , t-3 be Relation ;
Q [ e-14 \/ { vN } , f . ( v-5 \/ { vN } ) ] ;
g \circlearrowleft ( W-min L~ z ) = z implies ( W-min L~ z ) .. z < ( E-max L~ z ) .. z
|. |[ x , v ]| - |[ x , y ]| .| = v\vert x - y .| ;
- f . w = - ( L * w ) .= - ( L * w ) ;
z - y <= x iff z <= x + y & y <= z ;
( 7 / ( 1 - e ) ) / ( 1 - e ) > 0 ;
assume X is BCK-algebra & 0 , 0 , 0 , 0 , 0 ;
F . 1 = v1 & F . 2 = v2 & F . 3 = v2 ;
( f | X ) . x2 = f . x2 & ( f | X ) . x2 = f . x2 ;
( ( tan * tan ) `| Z ) . x in dom ( sec * sec ) ;
i2 = ( f /. len f ) `1 & i1 = ( f /. len f ) `1 ;
X1 = X2 \/ ( X1 \ X2 ) & X2 = X1 \ ( X2 \ X1 ) ;
[. a , b , 1_ G .] = 1_ G & a * b = 1_ G ;
let V , W be non empty VectSpStr over F_Complex , f be FinSequence of V ;
dom g2 = the carrier of ( I[01] ) | P .= P .= P ;
dom f2 = the carrier of ( I[01] ) | P .= P .= the carrier of ( I[01] ) | P ;
( proj2 | X ) .: X = proj2 .: X .= proj2 .: X ;
f . ( x , y ) = h1 . ( x `1 , y `2 ) ;
x0 - r < a1 . n & x0 < x0 + r ;
|. ( f /* s ) . k - G3 .| < r ;
len Line ( A , i ) = width A & len Line ( B , i ) = width B ;
SFinSequence ^ ( S , g ) = ( S . g ) ^ ( S . g ) ;
reconsider f = v + u as Function of X , the carrier of Y ;
( intloc 0 ) in dom ( Initialized p ) & ( Initialized p ) . a = ( Initialized p ) . a ;
i1 does not destroy i & i2 does not destroy i & i1 does not destroy j implies not i1 does not destroy i2
arccos r + arccos r = ( PI / 2 + 0 ) / 2 ;
for x st x in Z holds f2 is_differentiable_in x & f2 . x > 0 ;
reconsider q2 = ( q - x ) / ( q - x ) as Element of REAL ;
( 0 qua Nat ) + 1 <= i + j1 & i + 1 <= len f ;
assume f in the carrier of [: X , Omega Y :] ;
F . a = H / ( ( x , y ) / ( x , y ) ) ;
( TRUE T ) at ( C , u ) = TRUE & ( T . ( u + 1 ) ) at ( C , u ) = TRUE ;
dist ( ( a * seq ) . n , h ) < r ;
1 in the carrier of [. 0 , 1 .] & 1 in dom ( f | [. 0 , 1 .] ) ;
( p2 `1 - x1 ) / ( 1 - x1 ) > - g / ( 1 - x1 ) ;
|. r1 - `2 .| = |. a1 .| * |. thesis .| ;
reconsider S-14 = 8 as Element of Seg 8 ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b
D0W .] = D0W .succ ( n + 1 ) ;
i1 = ma + n & i2 = K + n & i1 = K + n ;
f . a [= f . ( f . O1 "\/" f . a ) ;
pred f = v & g = u & f + g = v + u ;
I . n = Integral ( M , F . n ) ;
chi ( T1 , S ) . s = 1 & chi ( T2 , S ) . s = 2 ;
a = VERUM ( A ) or a = VERUM ( A ) ;
reconsider k2 = s . b3 , k2 = s . b3 as Element of NAT ;
( Comput ( P , s , 4 ) ) . GBP = 0 ;
L~ M1 meets L~ ( R4 ^ R4 ^ ( R4 ^ ( R4 ^ ( R4 ^ ( R4 ^ ( R4 ^ ( R4 ^ ( R4 ^ ( R4 ^ ( R4 ^ ( R4 ^ ( R4 ^ ( R4 ^
set h = the continuous Function of X , R , x be Point of X ;
set A = { L . ( ( k . n ) . x ) where k is Nat : not contradiction } ;
for H st H is atomic holds P7 [ H ] ;
set b9 = S5 ^\ ( i + 1 ) , S = S . ( i + 1 ) , T = S . ( i + 1 ) , T = S . ( i + 1 ) , S . ( i + 1 ) ;
Hom ( a , b ) c= Hom ( a opp , b ) ;
( 1 / ( n + 1 ) ) * ( 1 / ( n + 1 ) ) < ( 1 / ( n + 1 ) ) * ( 1 / ( n + 1 ) ) ;
( l `1 ) = [ dom l , cod l ] & ( l `2 ) = cod l ;
y +* ( i , y /. i ) in dom g ;
let p be Element of CQC-WFF ( Al ( ) ) , a be Element of CQC-WFF ( Al ( ) ) ;
X /\ X1 c= dom ( f1 - f2 ) /\ dom ( f1 - f2 ) ;
p2 in rng ( f /^ ( 1 + 1 ) ) & p2 in rng ( f /^ ( 1 + 1 ) ) ;
1 <= indx ( D2 , D1 , j1 ) & indx ( D2 , D1 , j1 ) + 1 <= len D1 ;
assume x in ( ( K /\ p3 ) \/ K /\ p4 ) /\ K /\ K ;
- 1 <= ( ( ( f2 ) . O ) `2 ) / ( 1 - sn ) ;
let f , g be Function of I[01] , ( TOP-REAL 2 ) | P , ( TOP-REAL 2 ) | P ;
k1 - k2 = k1 - k2 & k1 - k2 = k1 - k2 ;
rng seq c= ]. x0 , x0 + r .[ & rng seq c= ]. x0 , x0 + r .[ ;
g2 in ]. x0 , x0 + r .[ & g2 in ]. x0 , x0 + r .[ ;
sgn ( p `1 , K ) = - 1_ K & sgn ( p `2 , K ) = - 1_ K ;
consider u being Nat such that b = p |^ ( y * u ) ;
ex A being \/ of normal normal ' of T st a = Sum A ;
Cl ( union ( H ) ) = union ( ( union H ) \/ ( union G ) ;
len t = len t1 + len t2 & len t1 = len t2 + len t2 ;
v-29 = v + w |-- v + A8 ;
cv <> DataLoc ( t0 . GBP , 3 ) & cv <> DataLoc ( t0 . GBP , 3 ) ;
g . s = upper_bound ( d " { s } ) & g . s = upper_bound ( d " { s } ) ;
( \dot { y } ) . s = s . ( y , s ) ;
{ s : s < t } in [: NAT , NAT :] implies t = {} ;
s ` \ s = s ` \ 0. X .= 0. X ;
defpred P [ Nat ] means B + $1 in A & $1 + 1 in B ;
( 329 + 1 ) ! = 329 ! * ( 329 + 1 ) ;
1. ( A , succ A ) = 1. ( A , A ) & 1. ( A , A ) = 1. ( A , A ) ;
reconsider y = y as Element of ( len y ) -tuples_on COMPLEX ;
consider i2 being Integer such that y0 = p * i2 and i2 in dom f ;
reconsider p = Y | Seg k as FinSequence of ( the carrier of K ) * ;
set f = ( S , U ) \mathop { F . z } ;
consider Z being set such that lim s in Z and Z in F ;
let f be Function of I[01] , ( TOP-REAL n ) | P , ( TOP-REAL n ) | P ;
( SAT M ) . [ n + i , 'not' A ] <> 1 ;
ex r being Real st x = r & a <= r & r <= b ;
let R1 , R2 be Element of REAL n , R2 be Element of REAL n ;
reconsider l = 0. ( V ) as Linear_Combination of A ;
set r = |. e .| + |. n .| + |. w .| + a ;
consider y being Element of S such that z <= y and y in X ;
a be ' L L L L L L L L L 'or' ( a 'or' b ) = 'not' ( a 'or' b ) ;
||. ( x - g ) - ( g - f ) .|| < r2 / 2 ;
b9 , a9 // b9 , c9 & b9 , c9 // c9 , a9 & b9 , c9 // a9 , c9 ;
1 <= k2 -' k1 & k2 + 1 = k2 & k2 + 1 = k2 & k2 + 1 = k2 ;
( ( p `2 / |. p .| - sn ) / ( 1 - sn ) ) ^2 >= 0 ;
( ( q `2 / |. q .| - sn ) / ( 1 - sn ) ) ^2 < 0 ;
E-max C in cell ( RCage ( C , n ) , 1 , 1 ) & E-max L~ Cage ( C , n ) in L~ Cage ( C , n ) ;
consider e being Element of NAT such that a = 2 * e + 1 ;
Re ( ( lim F ) | D ) = Re ( ( lim G ) | D ) ;
LIN b , a , c or LIN b , c , a ;
p `1 , a // a `1 , b or p `2 , a // b `1 , a ;
g . n = a * Sum ( f | n ) .= f . n ;
consider f being Subset of X such that e = f and f is id and f is \lbrace p , q } ;
F | ( N2 , S ) = CircleMap * ( F | N2 , S ) .= ( F | N2 ) | N2 ;
q in LSeg ( q , v ) \/ LSeg ( v , p ) ;
Ball ( m , r0 ) c= Ball ( m , s ) & Ball ( m , s ) c= Ball ( m , s ) ;
the carrier of (0). V = { 0. V } & the carrier of V = { 0. V } ;
rng ( cos * ( sin * ( id Z ) ) ) = [. - 1 , 1 .] ;
assume that Re seq is summable and Im seq is summable and Im seq is summable and Im seq is summable ;
||. ( ( vseq . n ) - ( vseq . m ) ) - ( ( vseq . n ) - ( vseq . m ) ) .|| < e ;
set g = O --> 1 ;
reconsider t2 = t11 as string of S2 , t2 = <* 0 *> ^ <* 1 *> ^ ( t2 ^ <* 1 *> ^ t2 ) ;
reconsider x-29 = seq . n , xM = seq . m as sequence of REAL ;
assume that C meets ( L~ go \/ L~ pion1 ) and C meets ( L~ go \/ L~ pion1 ) ;
- ( ( 1 / 2 ) * x ) < F . n - ( ( 1 / 2 ) * x ) ;
set d1 = dist ( x1 , z1 ) , d2 = dist ( x2 , z1 ) , d2 = dist ( x2 , z1 ) , d2 = dist ( x2 , z1 ) , d2 = dist ( x2 , z1 ) ;
2 |^ ( x -' 1 ) = 2 |^ ( x -' 1 ) - 1 ;
dom ( v | ( len d6 ) ) = Seg ( len d6 ) .= dom ( v | ( dom d6 ) ) ;
set x1 = - k2 + |. k2 .| + 4 ;
assume for n being Element of X holds 0. <= F . n ;
assume that 0 <= T-32 . i and T-32 . ( i + 1 ) <= 1 ;
for A being Subset of X holds c . ( c . A ) = c . A
the carrier of ( Carrier ( L1 + L2 ) ) c= I2 & the carrier of ( Carrier ( L1 + L2 ) ) c= I2 ;
'not' Ex ( x , p ) => All ( x , 'not' p ) is valid ;
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the normal normal \Rightarrow of {} ;
Z c= dom ( ( - sin * f1 ) `| Z ) & Z c= dom ( ( - sin * f1 ) `| Z ) ;
|. 0. TOP-REAL 2 - q .| < r / 2 - q / 2 ;
\ \ { A , succ d } c= ConsecutiveSet2 ( A , st A c= L & A c= L holds A c= L ;
E = dom ( L (#) G ) & L (#) G is_measurable_on E & L (#) G is_measurable_on E ;
C / ( A + B ) = C / B * C / A ;
the carrier of W2 c= the carrier of V & the carrier of W1 c= the carrier of V ;
I . IC s2 = P . IC s2 .= Exec ( CurInstr ( P2 , s2 ) , n ) . IC s2 ;
pred x > 0 means : Def1 : ( 1 / x ) * ( 1 / x ) = x ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) \/ LSeg ( g , i ) ;
consider p being Point of T such that C = [: [. p , q .] , { p } :] ;
b , c are_connected & - C , - C - C + - C + ( - C , - C + - C ) + ( - C , - C + - C ) + ( - C , - C + - C ) + ( - C - D + D ) + D + - D
assume f = id the carrier of O1 & g = id the carrier of O2 ;
consider v such that v <> 0. V and f . v = L * v ;
let l be Linear_Combination of {} ( ( the carrier of V ) --> { 0. V } ) ;
reconsider g = f " as Function of U2 , U1 & g * f = id U2 ;
A1 in the Points of G_ ( k , X ) & A2 in the Points of k ;
|. - x .| = - ( - x ) .= - x .= - x ;
set S =
Fib ( n ) * ( 5 * Fib ( n ) - 1 ) >= 4 * ;
vM /. ( k + 1 ) = vM . ( k + 1 ) ;
0 mod i = - ( i * ( 0 qua Nat ) + 1 ) ;
Indices M1 = [: Seg n , Seg n :] & Indices M1 = [: Seg n , Seg n :] ;
Line ( S\mathopen { i , j } , j ) = S\mathopen { i , j } ;
h . ( x1 , y1 ) = [ y1 , x1 ] & h . ( y1 , y2 ) = [ y1 , y2 ] ;
|. f .| - Re ( |. f .| * ( card b * h ) ) is nonnegative ;
assume x = ( a1 ^ <* x1 *> ) ^ b1 & y = ( a1 ^ <* x1 *> ) ^ b1 ;
MI is_closed_on IExec ( I , P , s ) , P & MI is_halting_on s , P ;
DataLoc ( t3 . a , 4 ) = intpos ( 0 + 4 ) .= intpos ( 0 + 4 ) ;
x + y < - x + y & |. x .| = - x & |. x - y .| = - x ;
LIN c , q , b & LIN c , q , c & LIN c , q , c ;
f| ( 1 , t ) . ( 0 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + z1 ) .= x1 + ( y1 + z1 ) ;
fthe Sorts of A . a = flim . a & v in InputVertices S & [ f , g ] in ( the Sorts of A ) . a ;
( p `1 ) ^2 / ( ( E-max C ) `1 ) ^2 <= ( ( E-max C ) `1 ) ^2 / ( ( E-max C ) `1 ) ^2 ;
set R8 = Cage ( C , n ) \circlearrowleft E8 , R8 = Cage ( C , n ) ;
( p `1 ) ^2 >= ( E-max C ) `1 & ( E-max C ) `1 >= ( E-max C ) `1 ;
consider p such that p = pp and s1 < p & p <= i and i <= len f ;
|. ( f /* ( s * F ) ) . l - G3 .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len Line ( N , k + 1 + 1 ) = width N ;
f1 /* s1 is convergent & f2 /* s1 is convergent & lim ( f1 /* s1 ) = lim ( f1 /* s1 ) ;
f . x1 = x1 & f . y1 = y1 & f . y2 = y2 ;
len f <= len f + 1 & len f + 1 <> 0 & f . ( len f + 1 ) <> 0 ;
dom ( Proj ( i , n ) * s ) = REAL m & dom ( Proj ( i , n ) * s ) = REAL m ;
n = k * ( 2 * t ) + ( n mod 2 * k ) ;
dom B = 2 -tuples_on the carrier of V & rng B c= the carrier of V ;
consider r such that r \not _|_ a and r _|_ x and r _|_ y ;
reconsider B1 = the carrier of Y1 , B2 = the carrier of Y2 as Subset of X ;
1 in the carrier of [. 1 / 2 , 1 .] & 1 / 2 <= 1 / 2 ;
for L being complete LATTICE holds L , lattice ( rng F , L ) are_isomorphic implies L is isomorphic
[ gi , gj ] in Ii \ ( I \ { i } ) & [ gj \ { i } , i \ { j } ] in I ;
set S2 = 1GateCircStr ( x , y , c ) , S2 = 1GateCircStr ( y , c , d ) ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and f1 . x0 = ( f2 * f1 ) . x0 ;
reconsider y = ( a ` ) / ( F . ( a , b ) ) , z = ( a ` ) / ( F . ( a , b ) ) as Element of L ;
dom s = { 1 , 2 , 3 } & s . 1 = d1 & s . 2 = d2 ;
( min ( g , ( 1 - g ) (#) f ) . c <= h . c ;
set G2 = the as \HM { finite } \HM { v } , { v } } , G2 = the set ;
reconsider g = f as PartFunc of REAL , REAL-NS n , REAL-NS n ;
|. s1 . m / p .| / |. p .| < d / p / p ;
for x being element st x in k holds x in ( ( 0 qua Function ) * t ) . x
P = the carrier of ( ( TOP-REAL n ) | P ) .= P ;
assume that p00 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ;
( 0. X \ x ) to_power ( m * ( k + 1 ) ) = 0. X ;
let g be Element of Hom ( cod f , cod f ) ;
2 * a * b + ( 2 * c * d ) <= 2 * C1 * C2 ;
let f , g , h be Point of the carrier of X , Y be non empty TopSpace ;
set h = Hom ( a , g (*) f ) ;
then idseq ( n ) | Seg m = idseq ( m ) & m <= n & n <= m ;
H * ( g " * a ) in the right F of H & H * ( g " * a ) in the right F of H ;
x in dom ( ( cos * sin ) `| Z ) & x - h / 2 < ( ( cos * sin ) `| Z ) . x ;
cell ( G , i1 , j2 -' 1 ) misses C & cell ( G , i2 , j2 -' 1 ) misses C ;
LE q2 , p4 , P & LE p1 , p2 , P & LE p2 , p3 , P & LE p2 , p3 , P ;
pred B is closed for A c= BDD A & B c= BDD A ;
deffunc D ( set , set ) = union rng $2 & $2 = union rng $2 ;
n + - n < len ( pwhere p is Element of NAT : p in dom ( p | n ) } ;
pred a <> 0. K means for M st the_rank_of M = the_rank_of ( a * M ) & the_rank_of M = the_rank_of M ;
consider j such that j in dom /\ /\ dom /\ /\ I and I = len } + j ;
consider x1 such that z in x1 and x1 in P8 and x2 in P8 and x1 <> x2 ;
for n ex r being Element of REAL st X [ n , r ] & r <= n ;
set CS1 = Comput ( P2 , s2 , i + 1 ) , CS2 = Comput ( P2 , s2 , i + 1 ) , CS2 = Comput ( P2 , s2 , i + 1 ) , CS2 = Comput ( P2 , s2 , i + 1 ) , CS2 = Comput ( P2 , s2
set cv = 3 / ( 3 / ( a , b , c ) ) , cv = 3 / ( a , b , c ) ;
conv @ W c= union ( F .: ( E " W ) ) & Int @ W c= union ( F .: ( E " W ) ) ;
1 in [. - 1 , 1 .] /\ dom ( ( #Z 2 ) * ( f1 + f2 ) ) ;
r3 <= s0 + ( r0 - ( |. v2 - v1 .| / 2 ) * ( 1 - ( v2 - v1 ) / 2 ) * ( v2 - v1 ) ) ;
dom ( f * f4 ) = dom f /\ dom f4 .= dom ( f * f4 ) /\ dom ( f * f4 ) ;
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 g9 = gp , gq = gq as Point of Euclid n1 ;
( T * h . s ) . x = T . ( h . s . x ) ;
I . ( L . ( J . x ) ) = ( I * L ) . ( J . x ) ;
y in dom <* *> -LSeg ( F , ( Frege A ) . o ) & y in dom ( F . ( ( Frege A ) . o ) ) ;
for I being non degenerated commutative Ring holds the carrier of I is commutative commutative non empty doubleLoopStr
set s2 = s +* Initialize ( ( intloc 0 ) .--> 1 ) , P2 = P +* I ;
P1 /. IC s1 = P1 . IC s1 .= P1 . IC s1 .= P1 . IC s2 ;
lim S1 in the carrier of [. a , b .] & S2 . ( lim S1 ) = a ;
v . ( ( l . i ) . i ) = ( v *' ( l . i ) ) . i ;
consider n being element such that n in NAT and x = ( sn " ) . n ;
consider x being Element of c such that F1 . x <> F2 . x and F1 . x <> F2 . x ;
Funcs ( X , 0 , x1 , x2 , x3 ) = { E } & card { E , F , G } = 1 ;
j + ( 2 * ( kk + 1 ) ) + m1 > j + ( 2 * ( kk + 1 ) ) ;
{ s , t } on A3 & { s , t } on B2 ;
n1 > len crossover ( p2 , p1 , n1 , n2 , n3 , n3 , n2 , n3 , n3 ) & n2 > len crossover ( p2 , p1 , n1 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n2 ) ;
mg1 . ( HT ( mg2 , T ) ) = 0. L & mg1 . ( HT ( mg2 , T ) ) = 0. L ;
then that H1 , H2 are_<* H , I *> and card H1 , H are_p / 2 ;
( ( N-min L~ f ) .. ( f /. 1 ) ) .. ( f /. len f ) > 1 ;
]. s , 1 .] = ]. s , 2 .] /\ [. 0 , 1 .] ;
x1 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) ) & x2 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) ) ;
let f1 , f2 be continuous PartFunc of REAL , the carrier of S , the carrier of S ;
DigA ( t-23 , z9 ) is Element of k -tuples_on ( k -tuples_on D ) ;
I \leq d2is & I = k2 & I is k2 & I is k2 ;
[: u9 , { u } :] = { [ a , u9 ] } & [: u9 , { u } :] = [: { a , v } , { v } :] ;
( w | p ) | ( p | ( w | w ) ) = p ;
consider u2 such that u2 in W2 and x = v + u2 and u1 in W2 and u2 in W3 ;
for y st y in rng F ex n st y = a |^ n & F . n = a |^ n
dom ( ( g * ( f . ( V \dot \to C ) ) | K ) = K ;
ex x being element st x in ( ( the Sorts of U0 ) \/ A ) . s ;
ex x being element st x in ( ( the Sorts of O1 ) \/ A ) . s ;
f . x in the carrier of [. - r , r .] & f . x in [. - r , r .] ;
( the carrier of X1 union X2 ) /\ ( the carrier of X2 ) <> {} & ( the carrier of X1 ) /\ ( the carrier of X2 ) <> {} ;
L1 /\ LSeg ( p1 , p2 ) c= { p1 } /\ LSeg ( p1 , p2 ) ;
( b + ( bC - a ) ) / 2 in { r : a < r & r < b } ;
ex_sup_of { x , y } , L & x "\/" y = sup { x , y } ;
for x being element st x in X ex u being element st P [ x , u ]
consider z being Point of G8 such that z = y and P [ z ] and z in G ;
( the sequence of ( ( the carrier of ( ( the carrier of X ) --> the carrier of Y ) ) --> the carrier of Y ) ) . ( x , y ) <= e ;
len ( w ^ w2 ) + 1 = len w + 2 + 1 .= len w + 1 ;
assume that q in the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 and q in the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 ;
f | E-4 ` = g | E-4 ` & f | E-4 = g | E-4 ;
reconsider i1 = x1 , i2 = x2 , j1 = x3 , j2 = x4 , i1 = i2 , i2 = j1 , j1 = j2 , j2 = j2 ;
( a * A * B ) @ = ( a * ( A * B ) ) @ ;
assume ex n0 being Element of NAT st f to_power n0 is min & f to_power n0 is min ;
Seg len ( ( the thesis ) --> ( f2 , f1 ) ) = dom ( ( the support of f2 ) --> ( f1 , f2 ) ) ;
( Complement A1 ) . m c= ( Complement A1 ) . n & ( Complement A2 ) . m c= ( Complement A2 ) . m ;
f1 . p = p9 & g1 . ( p9 , p ) = d & g1 . ( p9 , p ) = d ;
FinS ( F , Y ) = FinS ( F , dom ( F | Y ) ) .= FinS ( F , Y ) ;
( x | y ) | z = z | ( y | x ) ;
( |. x .| to_power n ) * ( n / ( 2 * n ) ) <= ( r2 |^ n ) * ( n / ( 2 * n ) ) ;
Sum ( F ) = Sum f & dom ( F ) = dom g & for x st x in dom F holds F . x = f . x ;
assume for x , y being set st x in Y & y in Y holds x /\ y in Y ;
assume that W1 is Subspace of W3 and W2 is Subspace of W3 and W3 is Subspace of W3 and W3 is Subspace of W2 ;
||. ( t-15 . x ) - ( x - y ) .|| = lim ||. ( x - y ) - ( x - y ) .|| ;
assume that i in dom D and f | A is lower and g | A is lower ;
( ( p `2 / |. p .| - sn ) / ( 1 - sn ) ) ^2 <= ( ( - 1 ) / ( 1 - sn ) ) / ( 1 - sn ) ;
g | Sphere ( p , r ) = id ( Sphere ( p , r ) ) & g | Sphere ( p , r ) = id ( Sphere ( p , r ) ) ;
set N8 = ( N-min L~ Cage ( C , n ) ) .. Cage ( C , n ) ;
for T being non empty TopSpace holds T is countable countable implies the TopStruct of T is countable
width B |-> 0. K = Line ( B , i ) .= B * ( i , i ) .= B * ( i , i ) ;
pred a <> 0 means : Def1 : ( A \ B ) Let a = ( A \ a ) \ ( B \ a ) ;
then f is_is_is_differentiable on 3 & pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 & pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 ;
assume that a > 0 and a <> 1 and b > 0 and b <> 1 and c > 0 and d > 0 ;
w1 , w2 in Lin { w1 , w2 } & w2 in Lin { w1 , w2 } ;
p2 /. IC s-7 = p2 . IC s-7 .= p2 . IC si2 .= ( p2 /. IC si2 ) + ( p2 /. IC si2 ) ;
ind ( T-10 | b ) = ind b .= ind B .= ind b .= ind b ;
[ a , A ] in the Points of G_ ( k , X ) & [ a , A ] in the \cdot of G ;
m in ( the Arrows of C ) . ( o1 , o2 ) & m in the Arrows of C ;
( ( a , CompF ( PA , G ) ) . z ) . z = FALSE ;
reconsider phi = phi /. 11 , phi = phi /. 2 as Element of ( len phi ) -tuples_on ( { 2 } * ) * ;
len s1 - ( len s2 - 1 ) * ( len s2 - 1 ) + 1 > 0 + 1 ;
delta ( D ) * ( f . ( upper_bound A ) - lower_bound A ) < r ;
[ f21 , f22 ] in [: the carrier' of A , the carrier' of B :] ;
the carrier of ( ( TOP-REAL 2 ) | K1 ) = K1 & the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 = K1 ;
consider z being element such that z in dom g2 and p = g2 . z and q = g2 . z ;
[#] V1 = { 0. V1 } .= the carrier of ( ( the carrier of V ) --> 0. V1 ) .= the carrier of ( ( the carrier of V ) --> 0. V1 ) ;
consider P2 be FinSequence such that rng P2 = M and P2 is one-to-one and P2 is one-to-one and P2 is one-to-one ;
assume that x1 in dom ( f | X ) and ||. x1 - x0 .|| < s and ||. x1 - x0 .|| < s ;
h1 = f ^ ( <* p3 *> ^ <* p *> ) .= h ^ <* p *> ^ <* p *> .= h ^ <* p *> ;
c / ( |[ b , c ]| ) = c .= |[ a , c ]| / ( |[ a , c ]| ) ;
reconsider t1 = p1 , t2 = p2 , t2 = p3 as Term of C , V , f be Function of C , V ;
( 1 / 2 ) * ( 1 / 2 ) in the carrier of [. 1 / 2 , 1 .] ;
ex W being Subset of X st p in W & W is open & h .: W c= V ;
( h . p1 ) `2 = C * ( p1 `2 + D ) `2 .= ( h . p2 ) `2 + D ;
R . b - R . b = 2 * - b .= 2 * - b .= b ;
consider 1 such that B = ( - 1 ) * C + ( - 1 ) * A and 0 <= 1 ;
dom g = dom ( ( the Sorts of A ) * ( the_arity_of o ) ) & dom g = dom ( ( the Sorts of A ) * ( the_arity_of o ) ) ;
[ P . ( l . ( k . m ) , P . ( l . m ) ] in => ( rng P ) ;
set s2 = Initialize s , P2 = P +* I ;
reconsider M = mid ( z , i2 , i1 ) , N = len z - 2 * ( i1 + 1 ) as Point of ( TOP-REAL 2 ) | ( L~ z ) ;
y in product ( ( the carrier of J ) +* ( V , { 1 } ) ) ;
1 / 2 * ( |[ 0 , 1 ]| ) = 1 & 0 / 2 * ( |[ 0 , 1 ]| ) = 0 ;
assume x in the left & x in the left of g or x in the right & y in the carrier of g ;
consider M being strict Subgroup of Abe strict Subgroup of V such that a = M and T is Subgroup of M ;
for x st x in Z holds ( ( ( #Z 2 ) * f ) `| Z ) . x <> 0 & ( ( #Z 2 ) * f ) `| Z ) . x = f . x
len W1 + len W2 + m = 1 + len W3 + m .= len W2 + len W3 + m + 1 ;
reconsider h1 = ( vseq . n - t-16 ) * ( vseq . n - t-16 ) as Lipschitzian Lipschitzian Lipschitzian LinearOperator of X , Y ;
( i mod len ( p + q ) ) + 1 in dom ( p + q ) ;
assume that s2 is_or F in the ex s1 st s1 in the \HM { F . s1 } & s2 in the { F . s1 } ;
( ( ( the Element of [ x , y ] ) * ( 1 , 3 ] ) * ( 1 , 2 ) ) = gcd ( x , y , 3 ) ;
for u being element st u in Bags n holds ( p + m ) . u = p . u
for B being Subset of u-5 st B in E holds A = B or A misses B or A misses B ;
ex a being Point of X st a in A & A /\ Cl { y } = { a } ;
set W2 = tree ( p ) \/ W1 , W3 = tree ( q ) , W2 = tree ( p ) , S = the Sorts of A ;
x in { X where X is Ideal of L : X is directed } or x in { X where X is Subset of L : X is directed } ;
the carrier of W1 /\ W2 c= the carrier of W1 & the carrier of W1 /\ W2 c= the carrier of W2 ;
( for a , b holds 1 * ( id a + b ) * id a = id a * ( 1 + b ) * ( id a )
( ( X --> f ) . x ) . x = ( X --> dom f ) . x .= ( X --> f ) . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) ;
p => ( q => r ) => ( p => q ) in TAUT ( A ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
- 1 + 1 <= ( ( - 2 |^ ( n -' m ) ) + 1 ) + 1 ;
( reproj ( 1 , z0 ) ) . x in dom ( f1 (#) f2 ) /\ dom ( f2 (#) f3 ) ;
assume that b1 . r = { c1 } and b2 . r = { c2 } and c2 . r = c2 . r ;
ex P st a1 on P & a2 on P & b on P & c on P & d on P ;
reconsider gf = g opp * f opp , hg = h opp * g opp as strict Element of X ;
consider v1 being Element of T such that Q = ( downarrow v1 ) ` and v1 in V and v2 in V ;
n in { i where i is Nat : i < n0 + 1 } & n in { i + 1 } ;
( F * ( i , j ) `2 ) >= ( F * ( m , k ) `2 ) `2 ;
assume K1 = { p : p `1 >= sn & p `2 >= 0 & p <> 0. TOP-REAL 2 } ;
ConsecutiveSet ( A , succ O1 ) = ( ConsecutiveSet ( A , O1 ) ) . ( A , O1 ) .= A . ( O1 , O2 ) ;
set IS1 = Macro ( a , intloc 0 ) , IS2 = SubFrom ( a , intloc 0 ) , IS2 = SubFrom ( a , intloc 0 ) , IS2 = SubFrom ( a , intloc 0 ) , IS2 = goto 2 , IS2 = goto 2 , IS2 = goto 2 , IS2 = goto 2 , IS2 = goto 2 , IS2 = goto 2 , IS2 = goto 2 ;
for i be Nat st 1 < i & i < len z holds z /. i <> z /. 1 ;
X c= ( the carrier of L1 ) /\ ( the carrier of L2 ) & X c= ( the carrier of L1 ) /\ the carrier of L2 ;
consider xx be Element of GF ( p ) such that xx |^ 2 = a & xx |^ 2 = b |^ 2 ;
reconsider e3 = e4 , f4 = f4 , f4 = f4 as Element of D * ;
ex O being set st O in S & C1 c= O & M . O = 0. ;
consider n being Nat such that for m being Nat st n <= m holds S . m in U1 and n <= m ;
f * g * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) * g ) . x ;
defpred P [ Nat ] means A + succ $1 = succ A & $1 in A + $1 & $1 in A + $1 ;
the left & - g = the left & the carrier of g = the carrier of g implies for x being Element of X holds x is left & g is left & f . x = ( - g ) . x
reconsider pp = x , pp = y , pp = z , \overline = x , pp = y , \overline = z , \overline = z , \overline = z as Point of Euclid 2 ;
consider ex ex g2 such that g2 = y & x <= g2 & g2 <= x0 & x0 < g2 & g2 in dom f ;
for n being Element of NAT ex r being Element of REAL st X [ n , r ] & r <= n
len ( x2 ^ y2 ) = len x2 + len y2 & len ( x2 ^ y2 ) = len x2 + len y2 ;
for x being element st x in X holds x in the set of ( the set of \HM { 0 } ) & ( x in X implies x = 1 ) ;
LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = {} & LSeg ( p1 , p2 ) /\ LSeg ( p2 , p2 ) = {} ;
func real ( X ) -> set means : Def1 : it = [: \mathop { h } , the carrier of X :] & it = [: the carrier of X , the carrier of X :] ;
len ( the _ of where C , CC is Element of X : 1 <= len CC & C = ( the _ of C ) * ( len CC , len CC ) ) `1 ;
pred K is has a , n & a <> 0. K implies v . ( a |^ i ) = i * v . a ;
consider o being OperSymbol of S such that t `2 . {} = [ o , the carrier of S ] and o in rng p ;
for x st x in X ex y st x c= y & y in X & y is - 1 / ( f . x )
IC Comput ( P-6 , sd , k ) in dom ( PJ +* I ) & IC Comput ( PJ , sd , k ) in dom I ;
pred q < s & r < s & ]. r , s .] \not c= ]. p , q .] & s <= q ;
consider c being Element of Class ( f , c ) such that Y = ( F . c ) `1 and c in X ;
func the ResultSort of S2 -> Function of the carrier' of S2 , the carrier' of S2 means : Def1 : for x being set st x in the carrier' of S2 holds it . x = F ( x ) ;
set y9 = [ <* y , z *> , f2 ] , z9 = [ <* z , x *> , f3 ] ;
assume x in dom ( ( ( - ( #Z 2 ) * ( f1 + f2 ) ) `| Z ) & x in dom ( ( #Z 2 ) * ( f1 + f2 ) ) ;
r-7 in Int cell ( GoB f , i , GoB f ) \ { ( GoB f ) * ( i + 1 , 1 ) } & r-7 in L~ f ;
q `2 >= ( ( Cage ( C , n ) * ( i + 1 ) ) `2 ) / 2 ;
set Y = { a "/\" a ` : a ` in X } ;
i - len f <= len f + len f1 - len f & i - len f <= len f + len f - len f ;
for n ex x st x in N & x in N1 & h . n = - ( x0 - r ) * ( x - x0 )
set s0 = ( ( a , I , p , s ) +* ( i , I ) ) . i ;
p . k . 0 = 1 or p . k . 0 = - 1 & p . 0 = 1 & p . 0 = - 1 ;
u + Sum ( L-18 ) in ( U \ { u } ) \/ { u + Sum ( L-18 ) } ;
consider xx being set such that x in xx and xx in V1 and xx in V1 and x = [ xx , xx ] ;
( p ^ ( q | k ) ) . m = ( q | k ) . ( - len p ) .= p . m ;
g + h = gg + h1 & Comput ( g + h , X , X ) = g + h ;
L1 is distributive & L2 is distributive implies [: L1 , L2 :] is distributive & [: L1 , L2 :] is distributive & [: L1 , L2 :] is distributive
pred x in rng f & y in rng ( f | x ) & f . x = f . y implies x = y ;
assume that 1 < p and ( 1 - p ) * ( 1 - q ) = 1 and 0 <= a and a <= b ;
F* ( f , sA1 ) = rpoly ( 1 , M *' t ) + 0. F_Complex .= 0. F_Complex ;
for X being set , A being Subset of X holds A ` = {} implies A = X & A is closed
( ( N-min X ) `1 <= ( ( ( E-max X ) `1 ) / 2 ) `1 ;
for c being Element of the \rbrack of A , a being Element of the free of A holds c <> a implies c = a
s1 . GBP = ( Exec ( i2 , s2 ) ) . GBP .= Exec ( i2 , s2 ) . GBP .= s2 . GBP .= Exec ( i2 , s2 ) . GBP .= s . GBP ;
for a , b being Real holds |[ a , b ]| in ( y >= 0 ) implies b >= 0 & a >= 0 & b >= 0
for x , y being Element of X holds x ` \ y = ( x \ y ) ` & y \ x = 0. X ;
mode BCK-algebra of i , j , m , n , m , n , m , n , m , m , n ;
set x2 = |( Re ( y - x ) , Im ( y - x ) )| ;
[ y , x ] in dom u5 & u5 . ( y , x ) = g . y ;
]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .] c= A & upper_bound divset ( D , k ) = lower_bound A ;
0 <= delta ( S2 . n ) & |. delta ( S2 . n ) .| < ( e / 2 ) * ( 2 * n ) ;
( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) <= ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ;
set A = ( 2 / b-a ) ;
for x , y being set st x in RO & y in RO holds x , y are_> ;
deffunc FF2 ( Nat ) = b . ( $1 * ( M * G ) . $1 ) * ( M * G ) . $1 ;
for s being element holds s in ( ( f 'or' g ) . s iff s in ( f \/ g ) . s
for S being non empty non void non empty non void non empty ManySortedSign st S is connected holds S is connected
max ( degree ( ( z `1 ) / |. z .| - cn ) , degree ( ( z `2 / |. z .| - cn ) / |. z .| - cn ) ) >= 0 ;
consider n1 being Nat such that for k holds seq . ( n1 + k ) < r + s and seq . ( n1 + k ) < r + s ;
Lin ( A /\ B ) is Subspace of Lin ( A ) & Lin ( B ) is Subspace of Lin ( B ) ;
set n-15 = n-13 '&' ( M . ( x , y ) qua Element of BOOLEAN ) , n-15 = M . ( x , y ) ;
f " V in the topology of X & f " V in D & f " V in D implies f " V in D & f " V in D
rng ( ( a ^\ c ) \mathbin { + } ( 1 , b ) ) c= { a , c , b } ;
consider y being | of G1 such that y ` = y and dom y ` = WW: G . y ;
dom ( 1 / f ) /\ ]. x0 - r , x0 .[ c= ]. x0 - r , x0 .[ /\ dom f ;
as Matrix of i , j , n , r , - r , n , - r , n , - r ;
v ^ ( ( n |-> 0 ) ^ ( ( n |-> 0 ) ^ ( ( n |-> 0 ) ^ ( n |-> 0 ) ) ) in Lin rng ( ( n |-> 0 ) ^ ( n |-> 0 ) ) ;
ex a , k1 , k2 st i = a /. k1 & i = b /. k2 & k2 = c /. k2 & k2 = c . k2 ;
t . NAT = ( NAT .--> succ i1 ) . NAT .= succ i1 .= succ ( 5 + 1 ) .= ( NAT + 1 ) * ( NAT + 1 ) ;
assume that F is bbfamily and rng p = F and dom p = Seg ( n + 1 ) and for i be Nat st i in Seg ( n + 1 ) holds p . i = F . i ;
not LIN b , b9 , a & not LIN a , a9 , c & LIN a , a9 , c & LIN a , a9 , c ;
( L1 \HM { L2 } \HM { O } ) \& O c= ( L1 \& O ) \& ( L2 \& O ) ;
consider F be ManySortedSet of E such that for d be Element of E holds F . d = F ( d ) ;
consider a , b such that a * ( u - w ) = b * ( -w ) and 0 < a & 0 < b ;
defpred P [ FinSequence of D ] means |. Sum $1 .| <= Sum |. $1 .| & $1 <= len |. $1 .| ;
u = cos . ( x , y ) * x + ( cos . ( x , y ) * y ) * y .= v ;
dist ( ( seq . n ) + x , g + x ) <= dist ( ( seq . n ) , g ) + 0 ;
P [ p , |. p .| ^ <* {} *> , {} ] & p = id ( the Sorts of A ) . p ;
consider X being Subset of CQC-WFF ( Al ) such that X c= Y and X is finite and not X is non ininthesis ;
|. b .| * |. eval ( f , z ) .| >= |. b .| * |. eval ( f , z ) .| ;
1 < ( ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ) .. Cage ( C , n ) ;
l in { l1 where l1 is Real : g <= l1 & l1 <= h & h <= g } & l <= f . l ;
vol ( ( G . n ) vol ) <= vol ( ( ( G . n ) vol ) , ( G . n ) / 2 ) ;
f . y = x .= x * 1. ( L , y ) .= x * ( power ( L , y ) ) .= x * ( 1. ( L , y ) ) ;
NIC ( <% i1 , i2 %> , ( 0 + 1 ) ) = { i1 , succ i2 } & ( 0 + 1 ) in { i1 , succ i2 } ;
LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = { p1 } /\ LSeg ( p1 , p2 ) .= { p1 } /\ LSeg ( p1 , p2 ) ;
Product ( ( the carrier of I-15 ) +* ( i , { 1 } ) ) in Z ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) .= Following ( s1 , n ) .= Following ( s2 , n ) ;
W-bound ( Qs2 ) <= ( q1 `1 ) / ( 2 * ( ( q1 `1 ) / ( 2 * ( ( q2 `1 ) / ( 2 * ( q2 `1 ) / ( 2 * ( q2 `1 ) / ( 2 * ( q2 `1 ) / ( 2 * ( q2 `1 ) / ( 2 * ( q2 `1 ) / ( 2 * ( q2 `1 ) / ( 2 * ( q2 `1 ) / ( 2 * ( q2 `1 ) / ( 2 *
f /. i2 <> f /. ( ( i1 + len g -' 1 ) + ( len g -' 1 ) ) & f /. i2 = f /. ( i1 + 1 ) ;
M , f / ( x. 3 , m ) / ( x. 4 , m ) / ( x. 4 , m ) |= H ;
len ( ( P ^ Q ) ^ <* P . ( len P + 1 ) *> ) in dom ( ( P ^ Q ) ^ <* P . ( len P + 1 ) *> ) ;
A |^ ( mn ) c= A |^ ( m , n ) & A |^ ( ( k + 1 ) + 1 ) c= A |^ ( k , n ) ;
( REAL n \ { q : |. q .| < a } c= { q1 : |. q1 .| >= a }
consider n1 being element such that n1 in dom p1 and y1 = p1 . n1 and p1 . n1 = p2 . n1 ;
consider X being set such that X in Q and for Z being set st Z in Q & Z <> X holds X \not c= Z ;
CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA & CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA ;
for v be VECTOR of l1 holds ||. v .|| = upper_bound rng |. ( id the carrier of V ) . v .| & ||. v .|| = upper_bound rng |. ( id the carrier of V ) . v .|
for phi st phi in X holds phi in X & not phi in X & not phi in X & not phi in X & phi in X
rng ( Sgm dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f |
ex c being FinSequence of D ( ) st len c = k & P [ c ] & a = c ^ <* d *> ;
the_arity_of ( a , b , c ) = <* o , o , c *> & o <> c & o <> c & o <> c ;
consider f1 be Function of the carrier of X , REAL such that f1 = |. f .| and f1 is continuous ;
a1 = b1 & a2 = b2 or a1 = b2 & a2 = b2 & a3 = b3 or a1 = b3 & a2 = b2 & a3 = b3 or a3 = b3 & a4 = 6 ;
D2 . indx ( D2 , D1 , n1 + 1 ) = D1 . ( n1 + 1 ) & D2 . ( n1 + 1 ) = D2 . ( n1 + 1 ) ;
f . ( ||. r .|| ) = ||. ||. r .|| /. 1 .= ||. r .|| . 1 .= ||. r .|| /. 1 .= x ;
consider n being Nat such that for m being Nat st n <= m holds C-25 . m = C-25 . m and C [ n , m ] ;
consider d being Real such that for a , b being Real st a in X & b in Y holds a <= d & d <= b ;
||. L /. h - ( K * |. h .| ) + ( K * |. h .| ) * ( h - f ) <= p0 + K * ( h - f ) ;
attr F is commutative means : Def1 : for b being Element of X holds F -\hbox { b } = f . b & f . b = f . b ;
p = - ( - ( 2 * p0 + 0. TOP-REAL 2 ) * p1 ) .= 1 * p1 + 0. TOP-REAL 2 .= ( - 1 ) * p1 + 0. TOP-REAL 2 .= ( - 1 ) * p1 + 0. TOP-REAL 2 ;
consider z1 such that b `1 , x3 , x3 , x4 , x5 , x5 , x5 , x5 , v2 and o , x1 , x2 is_collinear and o <> x1 ;
consider i such that Arg ( ( Rotate ( s , r ) ) . q ) = s + Arg q + ( 2 * PI * i ) ;
consider g such that g is one-to-one and dom g = card f and rng g = f . x and g is one-to-one and g is one-to-one ;
assume that A = P2 \/ Q2 and P2 <> {} and Q2 <> {} and Q2 misses Q2 and ( for x st x in dom P2 holds P2 . x = P2 . x ) and ( for x st x in dom P2 holds P2 . x = P2 . x ) ;
attr F is associative means : Def1 : F .: ( F .: ( f , g ) , h ) = F .: ( f , F .: ( g , h ) ) ;
ex x being Element of NAT st m = x `1 & x in z `1 & m < i or m in { i } & i in { i } ;
consider k2 being Nat such that k2 in dom P-2 and l in P-2 and Pk1 . k2 = P-2 . k2 ;
seq = r (#) seq implies for n holds seq . n = r * seq . n & seq is convergent & lim seq = r * seq . n
F1 . [ ( id a ) , [ a , a ] ] = [ f * ( id a ) , f * ( id a ) ] .= [ f * ( id a ) , f * ( id a ) ] ;
{ p } "\/" D2 = { p "\/" y where y is Element of L : y in D2 } & { p , q } "\/" { p , q } = { p "\/" q where p is Element of L : p in D1 } ;
consider z be element such that z in dom ( ( dom F ) . ( ( the Sorts of A ) . ( i + 1 ) ) ) and ( ( F . i ) . ( ( the Sorts of A ) . ( i + 1 ) ) = y ;
for x , y being element st x in dom f & f . x = f . y holds x = y
cell ( G , i , j ) = { |[ r , s ]| : r <= G * ( 0 + 1 , 1 ) `1 } ;
consider e being element such that e in dom ( T | E1 ) and ( T | E1 ) . e = v ;
( F `1 * b1 ) . x = ( Mx2Tran ( ( J , BZ , BZ ) . ( i + 1 ) ) * ( ( BZ ) /. j ) ;
- 1 / ( - 1 / 2 ) = mm (#) D | n .= mm (#) ( - 1 / 2 ) .= Det M .= Det M ;
pred for x being set st x in dom f /\ dom g holds g . x <= f . x & - ( g . x ) <= f . x ;
len ( f1 . j ) = len f2 /. j .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) ;
All ( 'not' All ( a , A , G ) , B , G ) '<' Ex ( 'not' All ( a , B , G ) , A , G ) ;
LSeg ( E . k0 , F . k0 ) c= Cl RightComp Cage ( C , k + 1 ) & LSeg ( E , k + 1 ) c= L~ Cage ( C , k + 1 ) ;
x \ a |^ m = x \ ( a |^ k * a ) .= ( x \ ( a |^ k ) ) \ a ;
k -ininininininininininthe carrier of S = ( commute ( I ) ) . k .= ( ( commute ( I ) ) . k ) . i .= ( ( the Sorts of A ) * ( the_arity_of ( I ) ) . k ) . i ;
for s being State of Aan holds Following ( s , n ) . ( 0 + 2 * n + 1 ) is stable ;
for x st x in Z holds f1 . x = a / 2 & ( f1 - f2 ) . x <> 0 & ( f1 - f2 ) . x <> 0 & ( f1 - f2 ) . x <> 0 ;
support ( support ( n ) \/ support ( m ) ) c= support ( max ( n , m ) ) \/ support ( m ) \/ 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 b1 ) = g . a & phi /. ( g . a ) = f . ( g . a ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( ( F ^ <* p *> ) ^ <* p *> ) and i = len ( F ^ <* p *> ) + 1 ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 } = { x1 } \/ { x2 } \/ { x3 , x4 , x5 , x5 , x5 , x5 } \/ { x5 , x5 , x5 , x5 , x5 } ;
the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U1 & the Sorts of U2 /\ ( U2 "\/" U2 ) c= the Sorts of U2 ;
( - ( 2 * a * ( b - 2 * a ) ) / b + - ( 2 * a * b ) / b ) > 0 ;
consider W00 such that for z being element holds z in W00 iff z in N & P [ z ] & P [ z ] ;
assume that ( the Arity of S ) . o = <* a *> and ( the ResultSort of S ) . o = r and ( the ResultSort of S ) . o = <* r *> and r = <* r *> and r = <* r *> and s = <* r *> and r = <* r *> ;
Z = dom ( ( exp_R * ( arccot - f1 ) ) / ( ( #Z 2 ) * ( f1 + #Z 2 ) ) ) ;
sum ( f , SS1 ) is convergent & lim ( sum ( f , SS1 ) ) = integral ( f , SS2 ) ;
( for a9 holds ( ( a9 . f ) => ( g9 . ( x , y ) ) ) => ( x9 => ( a9 . y ) ) in -> } implies for x holds x in [: the carrier of L , the carrier of L :]
len ( M2 * M3 ) = n & width ( M2 * M3 ) = n & width ( M2 * M3 ) = n & width ( M2 * M3 ) = n ;
attr X1 \/ X2 is open SubSpace of X means : Def1 : X1 , X2 are_separated & X2 , X1 are_separated & X1 , X2 are_separated & X1 , X2 are_separated & X2 , X2 are_separated ;
for L being upper-bounded antisymmetric RelStr , X being non empty Subset of L holds X "\/" { Top L } = { Top L } & X "\/" { Top L } = { Top L }
reconsider f-129 = F1 . ( b `2 , b `2 ) , f29 = F2 . ( b `2 , b `2 ) as Function of [: M , M :] , M ;
consider w being FinSequence of I such that the InitS of M is_\HM { <* s *> ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ y ;
g . ( a |^ 0 ) = g . ( 1_ G ) .= 1_ H .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) ;
assume for i be Nat st i in dom f ex z be Element of L st f . i = rpoly ( 1 , z ) & f . i = z ;
ex L being Subset of X st Carrier ( L ) = C & for K being Subset of X st K in C holds L /\ K <> {} ;
( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C1 & ( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C2 ;
reconsider oY = o `1 , oY = p `2 , oY = o `2 , oY = o `2 , oY = o `2 , oY = o `2 , oY = o `2 ;
1 * x1 + ( 0 * x2 ) + ( 0 * x3 ) = x1 + <* \underbrace ( 0 , \dots , 0 ) , 0 , 0 *> .= x1 + 0 * x2 + 0 * x3 ;
EK " . 1 = ( EK qua Function ) " . 1 .= ( 1 - 2 ) / 2 .= ( 1 - 2 ) / 2 .= ( 1 - 2 ) / 2 ;
reconsider u1 = the carrier of U1 /\ ( U1 "\/" U2 ) as non empty Subset of U0 ;
( ( x "/\" z ) "\/" ( x "/\" y ) ) "\/" ( z "/\" y ) <= ( x "/\" ( z "\/" y ) ) "\/" ( z "/\" ( x "\/" y ) ) ;
|. f . ( s1 . ( l1 + 1 ) - s1 . ( s1 . l1 ) ) - f . ( s1 . l1 ) .| < ( 1 / |. M .| + 1 / 2 ) ;
LSeg ( ( Cage ( C , n ) /. ( i + 1 ) , ( Cage ( C , n ) /. ( i + 1 ) ) ) is vertical ;
( f | Z ) /. x - ( f | Z ) /. x0 = L /. ( x- ( f /. ( x - x0 ) ) + R /. ( x - x0 ) ) ;
g . c * ( - g . c * f . c + f . c ) <= h . c * ( ( - g . c ) * f . c ) + f . c ;
( f + g ) | divset ( D , i ) = f | divset ( D , i ) + g | divset ( D , i ) ;
assume that ColVec2Mx f in the set of ColVec2Mx A and len f = width A and width ( ColVec2Mx b ) = width A and width ( ColVec2Mx f ) = width A and width ( ColVec2Mx f ) = width A and width ( ColVec2Mx f ) = width A ;
len ( - M3 ) = len M1 & width ( - M3 ) = width M1 & width ( - M3 ) = width M1 & width ( - M3 ) = width M1 ;
for n , i being Nat st i + 1 < n holds [ i , i + 1 ] in the InternalRel of ( ( the InternalRel of ( n + 1 ) ) --> ( i , i + 1 ) )
pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in u0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in u0 , 2 & pdiff1 ( f2 , 2 ) . 2 = pdiff1 ( f1 , 2 ) . 2 ;
pred a <> 0 & b <> 0 & Arg a = Arg b & Arg ( - a ) = Arg ( - b ) & Arg ( - a ) = Arg ( - a ) & Arg ( - a ) = Arg ( - a ) ;
for c being set st not c in [. a , b .] holds not c in Intersection ( the open set of a , b ) & not c in Intersection ( the topology of a , b )
assume that V1 is closed and V2 is closed and V2 = { v + u : v in V1 & u in V1 & u in V1 } and V1 is closed and V1 is closed and V2 is closed and V1 is closed and V2 is closed and V1 is closed and V2 is closed ;
z * x1 + ( 1 / 2 * x2 ) * x2 in M & z * y1 + ( 1 / 2 * x2 ) * y2 in N implies z * y1 + ( 1 / 2 * x2 ) * y2 in N
rng ( ( PS1 qua Function ) " * SS2 ) = Seg card ( ( ( card dS2 ) " * SS2 ) * ( SS2 " * ( id dom ( PS2 ) " * ( id dom ( PS2 ) " * ( id dom ( PS2 ) " * ( id dom ( PS2 ) " * ( id dom ( PS2 ) " * ( id dom ( PS2 ) " * ( id dom ( PS2 ) " * ( id dom ( PS2 ) ) ) ) ) ) ) )
consider s2 being rational Real_Sequence such that s2 is convergent and b = lim s2 and for n holds s2 . n <= b * ( n + 1 ) and s2 is convergent & lim s2 = lim s2 ;
h2 " . n = h2 . n " & 0 < - ( 1 / 2 ) / 2 & 0 < - ( 1 / 2 ) / 2 & - ( 1 / 2 ) / 2 < - ( 1 / 2 ) / 2 ;
( Partial_Sums ( ||. seq1 .|| ) . m ) = ||. seq1 .|| . m .= ||. seq1 .|| . m .= 0 * ||. seq .|| . m .= 0 * ||. seq .|| . m .= 0 * ||. seq .|| . m .= 0 * ||. seq .|| . m ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= Comput ( P2 , s2 , 1 ) . b .= Comput ( P2 , s2 , 1 ) . b .= Comput ( P2 , s2 , 1 ) . b ;
- v = ( - 1_ G ) * v & - w = ( - 1_ G ) * v & - w = ( - 1_ G ) * v & - w = 0. G ;
upper_bound ( ( k .: D ) .: ( k .: D ) = upper_bound ( ( k .: D ) .: ( k .: D ) ) .= k . ( sup D ) .= sup D ;
A |^ ( k , l ) ^^ ( A |^ ( n , .. A ) ) = ( A |^ ( k , .. A ) ) ^^ ( A |^ ( k , .. A ) ) ;
for R being add-associative right_zeroed right_complementable non empty addLoopStr , I , J being Subset of R holds I + ( J + K ) = ( I + J ) + K
( f . p ) `1 = ( p `1 / sqrt ( 1 + ( p `2 / p `1 ) ^2 ) ) / sqrt ( 1 + ( p `2 / p `1 ) ^2 ) ;
for a , b being non zero Nat st a , b are_relative_prime holds support ( a * b ) = support ( a * b ) + support ( b * a )
consider A5 being countable Nat such that r is Element of CQC-WFF ( Al ) & A5 is ( not empty & not contradiction ) & not ( ex n being Nat st n < len A5 & not n in dom ( A . n ) ) ;
for X being non empty addLoopStr , M being Subset of X , x , y being Point of X st y in M holds x + y in M + M
{ [ x1 , x2 ] , [ y1 , y2 ] , [ y2 , x2 ] } c= { [ x1 , y1 ] , [ y1 , y2 ] } \/ { [ y1 , y2 ] , [ y1 , y2 ] } ;
h . ( f . O ) = |[ A * ( f . O ) `1 + B , C * ( f . O ) `2 + D ]| ;
( Gauge ( C , n ) * ( k , i ) ) * ( k , i ) in L~ Upper_Seq ( C , n ) /\ L~ Upper_Seq ( C , n ) ;
cluster m , n are_relative_prime means : Def1 : for p being prime Nat holds p divides m & not p divides n & p divides n & not p divides n ;
( f * F ) . x1 = f . ( F . x1 ) & ( f * F ) . x2 = f . ( F . x2 ) ;
for L being LATTICE , a , b , c being Element of L st a \ b <= c & b \ a <= c holds a \+\ b <= c
consider b being element such that b in dom ( H / ( x. 0 , y ) ) and z = H / ( x. 0 , y ) . b ;
assume that x in dom ( F * g ) and y in dom ( F * g ) and ( F * g ) . x = ( F * g ) . y ;
assume ex e being element st e Joins W . 1 , W . 5 , G & e Joins W . 3 , G & e . 7 = G . 3 ;
( ( h (#) f ) . ( 2 * n ) ) . x = ( ( h (#) f ) . ( 2 * n ) ) . ( 2 * n + ( n * n ) * h . ( 2 * n + 1 ) ) ;
j + 1 = ( j - len h11 + 2 ) + 1 .= i + 1 - len h11 + 2 - 1 .= i + 2 - 1 + 2 - 1 ;
*' ( S *' ) . f = S *' . ( ( ^ ^ ) . f ) .= S . ( ( ^ ^ ) . f ) .= S . ( f . f ) .= S . ( f . f ) ;
consider H such that H is one-to-one and rng H = the carrier of L2 and Sum ( L2 * H ) = Sum ( L2 * H ) and Sum ( L2 * H ) = Sum ( L2 * H ) ;
pred R is >= a & p in R & p <> q & p <> q & P [ p , q ] & P [ q , p ] & P [ p , q ] ;
dom product ( X --> f ) = meet ( dom ( X --> f ) ) .= meet ( X --> dom f ) .= meet ( X --> dom f ) .= meet ( X --> dom f ) .= dom f /\ dom f .= dom f /\ dom f .= dom f /\ dom f .= dom f /\ dom f .= dom f /\ dom f .= dom f /\ dom f ;
upper_bound ( proj2 .: ( Lower_Arc C /\ Lower_Arc C /\ Lower_Arc C ) ) <= upper_bound ( proj2 .: ( C /\ Vertical_Line w ) /\ proj2 .: ( C /\ Vertical_Line w ) ) ;
for r be Real st 0 < r ex n be Nat st for m be Nat st n <= m holds |. S . m - pp .| < r
i * ( f - fN ) = i * ( f - ( i * y9 ) ) .= i * ( f - ( i * y9 ) ) .= i * ( f - ( i * y9 ) ) ;
consider f being Function such that dom f = 2 -tuples_on X & for Y being set st Y in 2 -tuples_on X holds f . Y = F ( Y ) ;
consider g1 , g2 being element such that g1 in [#] Y and g2 in union C and g = [ g1 , g2 ] and g1 in C and g2 in C and g2 in D ;
func d |-count n -> Nat means : Def1 : d divides n & it |^ ( n + 1 ) divides n & d |^ ( n + 1 ) divides n & it |^ ( n + 1 ) divides n ;
f\rbrack . [ 0 , t ] = f . [ 0 , t ] .= ( - P ) . ( 2 * x ) .= a * ( - P . ( x , t ) ) .= a * ( - P . ( x , t ) ) .= a * ( - P . ( x , t ) ) .= a * ( - P . ( x , t ) ;
t = h . D or t = h . B or t = h . C or t = h . E or t = h . F or t = h . J ;
consider m1 be Nat such that for n st n >= m1 holds dist ( ( seq . n ) - ( seq . n ) ) < 1 / ( 2 * ( n + 1 ) ) ;
( ( q `1 ) / |. q .| ) ^2 <= ( ( q `2 ) / |. q .| ) ^2 + ( ( q `2 / |. q .| ) / |. q .| ) ^2 ;
h0 . ( i + 1 + 1 ) = h21 . ( i + 1 + 1 -' len h11 + 2 -' 1 ) .= h21 . ( i + 1 -' len h11 + 2 -' 1 ) ;
consider o being Element of the carrier' of S , x2 being Element of { [ o , x2 ] } such that a = [ o , x2 ] and o in { [ o , x2 ] } ;
for L being RelStr , a , b being Element of L holds a <= { b } iff a <= b & b <= a & a <= b & b <= a
||. h1 .|| . n = ||. h1 . n .|| .= ||. h . n .|| .= ||. h . n .|| .= ||. h . n .|| .= ||. h . n .|| .= ||. h . n .|| ;
( ( - ( #Z 2 ) * ( exp_R * f ) ) . x = f . x - ( exp_R * f ) . x .= ( - 1 ) * ( exp_R * f ) . x .= ( - 1 ) * ( exp_R * f ) . x ;
pred r = F .: ( p , q ) means : Def1 : len r = min ( len p , len q ) & for i st i in dom r holds r . i = F . i ;
( rbeing / 2 ) ^2 + ( re / 2 ) ^2 <= ( r / 2 ) ^2 + ( r / 2 ) ^2 ;
for i being Nat , M being Matrix of n , K st i in Seg n holds Det M = Sum ( \cal M ) & Det M = Sum ( Det ( M @ ) )
then a <> 0. R & a " * ( a * v ) = 1 * v & a " * v = 1 * v & a * v = 1 * v ;
p . ( j - 1 ) * ( q *' r ) . ( i + 1 - j ) = Sum ( p . ( j -' 1 ) * r3 ) ;
deffunc F ( Nat ) = L . 1 + ( ( R /* ( h ^\ n ) ) * ( h ^\ n ) " ) . $1 ;
assume that the carrier of H = f .: ( the carrier of H1 ) and the carrier of H = f .: ( the carrier of H2 ) and the carrier of H = the carrier of H2 and the carrier of H = the carrier of H2 and the carrier of H = the carrier of H2 ;
Args ( o , Free ( S , X ) ) = ( ( the Sorts of Free ( S , X ) ) * the Arity of S ) . o .= ( the Sorts of Free ( S , X ) ) . o ;
H1 = n + 1 -\hbox { 2 |^ ( n + 1 ) } .= n + 1 -\hbox { 2 |^ ( n + 1 ) } .= n + 1 -\hbox { 2 |^ ( n + 1 ) } ;
( O = 1 & O = 0 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 ;
F1 .: ( dom F1 /\ dom F2 ) = F1 .: ( dom F1 /\ dom F2 ) .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } ;
pred b <> 0 & d <> 0 & b <> d & ( a - b ) / ( d - c ) = ( - ( b - b ) / ( d - c ) ) / ( d - c ) ;
dom ( ( f +* g ) | D ) = dom ( f +* g ) /\ D .= dom f \/ dom g /\ D .= dom f /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) ;
for i be set st i in dom g ex u , v being Element of L st g /. i = u * a & v in B & u in C & v in C
g `2 * P `2 * g `2 = g `2 * ( g `2 * P `2 ) * g `2 .= g `2 * ( g `2 * P `2 ) * ( g `2 * P `2 ) ;
consider i , s1 such that f . i = s1 and if i <> 1 & i <> 1 & f . ( i + 1 ) <> s1 & f . ( i + 1 ) <> s1 & i <> 1 & i <> 1 & j <> 1 & j <> 1 & j <> 1 & j <> 1 & j <> 1 & j <> 1 & j <> 1 & j <> 1 & j <> 1 ;
h5 | ]. a , b .[ = ( g | Z ) | ]. a , b .] .= g | ]. a , b .] .= g | ]. a , b .] .= g | ]. a , b .] ;
[ s1 , t1 ] , [ s2 , t2 ] Point ( ( TOP-REAL 2 ) | P ) & [ s2 , t2 ] , [ t2 , t2 ] Point ( ( TOP-REAL 2 ) | P ) . ( s1 , t2 ) , t2 ] ;
then H is negative & H is non negative & H is non empty & H is non empty & H is non empty implies H is not -g+* for p being Element of H holds not p is not negative -g+* ( H , X ) is not negative
attr f1 is total means : Def1 : ( f1 (#) f2 ) is total & ( f1 (#) f2 ) . c = f1 . c * f2 . c & ( f1 (#) f2 ) . c = f1 . c * ( f2 . c ) " ;
z1 in W2 " ( W2 " ( W1 ) ) or z1 = z2 " ( W2 " ( W2 ) ) & not z1 in W2 " ( W2 " ( W1 ) ) & not z1 in W2 " ( W2 " ( W2 ) ) ;
p = 1 * p .= a " * a * p .= a " * ( b * q ) .= a " * ( b * q ) .= a " * ( b * q ) .= a " * ( b * q ) ;
for seq1 be Real_Sequence , K be Real st for n be Nat holds seq1 . n <= K holds upper_bound rng seq1 <= upper_bound rng seq2
x0 meets ( L~ go \/ L~ pion1 ) or x0 meets ( L~ go \/ L~ pion1 ) or x0 in ( L~ go \/ L~ pion1 ) \/ ( L~ co \/ L~ co ) or x0 in ( L~ go \/ L~ pion1 ) \/ ( L~ co \/ L~ co ) or x0 in L~ co or x0 in L~ co ;
||. f . ( g . ( k + 1 ) - g . ( k + 1 ) ) - f . ( K * ( K to_power k ) ) .|| <= ||. g . ( K * ( K to_power k ) - g . ( K to_power k ) ) .|| ;
assume h = ( ( ( B .--> B ' ) +* ( C .--> D ) +* ( E .--> F ) ) +* ( F .--> J ) +* ( J .--> M ) +* ( M .--> N ) +* ( M .--> N ) ) +* ( M .--> N ) +* ( M .--> N ) +* ( M .--> N ) +* ( M .--> N ) ) ;
|. ( ( delta ( H . n , T ) || A ) . k - ( ( delta ( H , T ) || A ) . k ) * ( ( delta ( H , T ) || A ) . k ) .| <= e * ( ( b-a ( H , T ) . k ) * ( ( delta ( H , T ) . k ) * ( ( delta ( H , T ) . k ) * ( 1 / 2 ) ) * ( 1 / 2 ) ) ) ;
( ( the Sorts of A ) . ( i , j ) ) . e = [ ( the Sorts of A ) . ( ( the Sorts of B ) . ( i , j ) ) , ( the Sorts of B ) . ( ( the Sorts of B ) . ( i , j ) ) ] ;
{ x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x2 , x1 } } = { x1 , x1 } \/ { x1 , x2 } .= { x1 } \/ { x2 , x1 } ;
assume that A = [. 0 , 2 * PI .] and integral ( ( #Z n ) * ( sin + cos ) , A ) = 0 and integral ( ( #Z n ) * ( sin + cos ) , A ) = 0 ;
p `2 is Permutation of dom f1 & p `2 " = ( Sgm Y ) * p " & p `2 * Sgm X = ( Sgm Y ) * Sgm X & p `2 * Sgm X = Sgm X ;
for x , y st x in A & y in A holds |. ( 1 / ( f . x - f . y ) ) * ( f . y - f . x ) .| <= 1 * |. ( f . x - f . y ) .|
( p2 `2 ) = |. q2 .| * ( ( ( q2 `2 / |. q2 .| - sn ) / ( 1 - sn ) ) / ( 1 - sn ) ) .= ( ( ( ( q2 `2 / |. q2 .| - sn ) / ( 1 - sn ) ) / ( 1 - sn ) ) / ( 1 - sn ) ) / ( 1 - sn ) ;
for f be PartFunc of the carrier of CNS , REAL st dom f is compact & f is continuous holds rng f is compact & f /. ( len f ) = f /. ( len f )
assume for x being Element of Y st x in EqClass ( z , CompF ( B , G ) ) holds ( Ex ( a , A , G ) ) . x = TRUE ;
consider F3 such that dom F3 = n1 and for k be Nat st k in n1 holds Q [ k , F3 . k ] and for k be Nat st k in n1 holds Q [ k , F3 . k ] ;
ex u , u1 st u <> u1 & u , u1 // v , v1 & u , u1 // v , v1 & u , v1 // u1 , v2 & u1 , v1 // v , v2 & u1 , v1 // v , v2 & u1 , v1 // v , v2 & u , u1 // v , v1 & u1 , v1 // v , v2 ;
for G being Group , A , B being non empty Subgroup of G , N being normal Subgroup of G holds ( N -is normal Subgroup of A ) * ( N -B ) = N -B
for s be Real st s in dom F holds F . s = integral ( ( R to_power 0 ) (#) ( f to_power ( 0 + 1 ) ) (#) ( f to_power ( 0 + 1 ) ) ) ;
width AutMt ( f1 , b1 , b2 ) = len b2 .= len ( ( f2 * ( b1 , b2 ) ) * ( b2 * ( b2 , b2 ) ) ) .= width ( ( f2 * ( b1 , b2 ) ) * ( b2 * ( b2 , b2 ) ) ) ;
f | ]. - PI / 2 , PI / 2 .[ = f & dom f = ]. - 1 , 1 .[ & rng f c= ]. - 1 , 1 .[ & f | ]. - 1 , 1 .] is continuous ;
assume that X is closed w.r.t. Z and a in X and a in X and y in a ^ f and x in { { [ n , x ] } \/ { y } } \/ { x } ;
Z = dom ( ( ( - 1 / 2 ) (#) ( ( #Z 2 ) * ( f1 + #Z 2 ) ) ) `| Z ) /\ dom ( ( #Z 2 ) * ( f1 + #Z 2 ) ) ;
func TAUT ( V ) -> Subset of V means : Def1 : for k st 1 <= k & k <= len l holds it . k in V & l . k in V ;
for L being non empty TopSpace , N being net of L , M being net of L , c being Point of N st c is \frac of M holds c is \mathbin { -mod of M , M
for s being Element of NAT holds ( ( id C\mathop ( v , C\mathop ( v , C\mathop ( v , C\mathop ( v , C\mathop ( C\mathop ( v , C\mathop ( v , C\mathop ( v , C\mathop ( v , C\mathop ( C\mathop ( v , C\mathop ( Cmin ( v , Cmin ( v , Cmin ( v , Cmin ( v , Cmin ( v , Cmin ) ) ) ) ) ) ) ) ) ) ) ) ) . s = ( ( s , s ) ) . s ) ) . s ) . ( s , ( ( s ,
then z /. 1 = ( N-min L~ z ) .. z & ( ( N-min L~ z ) .. z < ( ( E-max L~ z ) .. z ) .. z ;
len ( p ^ <* ( 0 qua Real ) *> ^ <* ( 0 qua Real ) + 1 *> ) = len p + 1 .= len p + 1 .= len p + 1 .= len p + 1 ;
assume that Z c= dom ( - ( ln * f ) ) and for x st x in Z holds f . x = x and f . x > 0 and Z c= dom f and f . x > 0 ;
for R being add-associative right_zeroed right_complementable associative commutative associative commutative distributive non empty doubleLoopStr , I being Ideal of R , J being Subset of R holds ( I + J ) *' ( I /\ J ) c= I /\ J
consider f being Function of [: B1 , B2 :] , B2 such that for x being Element of B1 holds f . x = F ( x ) and f . x = F ( x ) ;
dom ( x2 + y2 ) = Seg len x .= Seg len x .= len ( x2 ^ ( y ^ z ) ) .= dom ( x ^ ( y ^ z ) ) .= dom ( x ^ ( y ^ z ) ) .= dom ( x ^ ( y ^ ( y ^ z ) ) ) .= dom ( x ^ ( y ^ ( y ^ z ) ) ) ;
for S being card Functor of C , B for c being Object of C holds card S . ( id c ) = id ( ( Obj S ) . ( id c ) ) & id ( ( Obj S ) . ( id c ) ) = id ( ( Obj S ) . ( id c ) )
ex a st a = a2 & a in f6 /\ f5 & for x st x in \mathop { f . x } holds { f . x , f . x } = }
a in Free ( ( H / ( x. 4 , x. k ) ) '&' ( ( H / ( x. k , x. k ) ) '&' ( ( x. k , x. k ) / ( x. k , x. k ) ) ) ) ;
for C1 , C2 being v1 , f , g being stable Function of C1 , C2 st ( for x being set st x in C1 holds f . x = g . x ) holds f = g
( W-min L~ go \/ L~ pion1 ) `1 = W-bound L~ go \/ E-bound L~ pion1 & ( W-min L~ go \/ L~ co ) `1 = W-bound L~ go & ( W-min L~ go \/ L~ co ) `1 = W-bound L~ Cage ( C , n ) or W-bound L~ go = W-bound L~ Cage ( C , n ) & W-bound L~ Cage ( C , n ) = W-bound L~ Cage ( C , n ) ;
assume that u = <* x0 , y0 , z0 *> and f is_assume u , 3 and SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) . z = SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) . z ;
then ( t . {} ) `1 in Vars & ex x being Element of Vars st x = ( t . {} ) `1 & t . {} = x & t . {} = x & t . {} = y & t . {} = y ;
Valid ( p '&' p , J ) . v = Valid ( p , J ) . v '&' Valid ( q , J ) . v .= Valid ( p , J ) . v '&' Valid ( q , J ) . v .= Valid ( p , J ) . v ;
assume for x , y being Element of S st x <= y for a , b being Element of T st a = f . x & b = f . y holds a >= b ;
func Class R -> Subset-Family of R means : Def1 : for A being Subset of R holds A in it iff ex a being Element of R st a in Class ( R , a ) & it c= Class ( R , a ) ;
defpred P [ Nat ] means ( ( ( \HM { the } \HM { vertices } \HM { v } ) | ( ( \HM { the } \HM { vertices } \HM { v } ) | ( ( the carrier' of G ) | ( { v } ) ) ) ) ) . ( $1 + 1 ) = G . ( $1 + 1 ) ;
assume that dim ( U1 ) = 0 and dim ( U2 ) = 0 and dim ( U2 ) = 0 and dim ( U2 ) = 0 and dim ( U2 ) = 0 and dim ( U2 ) = 0 and dim ( U2 ) = 0 and dim ( U2 ) = 0 and dim ( U2 ) = 0 ;
mas ( m . t ) = ( m . t ) `1 .= ( [ m . t , the carrier of C ] `1 ) `1 .= [ m . t , the carrier of C ] `1 .= m . t ;
d11 = xx ^ d22 .= f . ( ( y , d22 ) . ( y , 2 ) ) .= f . ( ( y , 2 ) . ( y , 2 ) ) .= d22 ^ d22 .= d22 ^ d22 .= d22 ^ d22 ;
consider g such that x = g and dom g = dom f and for x being element st x in dom f holds g . x in f . x and f . x in f . x ;
x + 0. F_Complex = x + len x |-> 0. F_Complex .= ( x + len x ) |-> 0. F_Complex .= ( x + ( len x ) |-> 0. F_Complex ) ^ ( x |-> 0. F_Complex ) .= x `1 + ( x * x ) .= x `1 + x `1 ;
( ( k -' ( k -' 1 ) ) + 1 ) in dom ( f | ( ( k -' 1 ) + 1 ) ) /\ dom ( f | ( k -' 1 ) ) ;
assume that P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 = { p1 , p2 } \/ { p2 , p3 } and P = { p1 , p2 , p3 } and P = { p1 , p2 , p3 } and P = { p2 , p3 , p3 } and P = { p1 , p2 , p3 } and P = P \/ { p2 , p3 } and P = P \/ { p1 , p2 , p3 } and P = { p2 , p3 , p4 } and P = P \/ { p1 , p2 , p3 } and P = P \/ P and P = { p2 , p3 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 ,
reconsider a1 = a , b1 = b , c1 = c , b1 = p `1 , c1 = p `2 , c2 = p `2 , c2 = p `2 , c1 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c1 = p `2 , c2 = p `2 , c2 = p `2 , c1 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c1 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2
reconsider being } = G1 . ( t . b * F1 . f ) as Morphism of ( G1 * F1 ) . a , ( G1 * F2 ) . b * F2 . b , ( G1 * F2 ) . b ;
LSeg ( f , i + i1 -' 1 ) = LSeg ( f /. ( i + i1 -' 1 ) , f /. ( i + i1 -' 1 + 1 ) ) .= LSeg ( f , i + i1 -' 1 + 1 ) ;
Integral ( M , P . m ) | dom ( P . n -d ) <= Integral ( M , P . n ) * Integral ( M , P . m ) ;
assume that dom f1 = dom f2 and for x , y being element st [ x , y ] in dom f1 & [ x , y ] in dom f2 holds f1 . ( x , y ) = f2 . ( x , y ) ;
consider v such that v = y and dist ( u , v ) < min ( ( G * ( i , 1 ) `1 - G * ( i + 1 , 1 ) `1 ) , ( G * ( i + 1 , 1 ) `1 - G * ( i + 1 , 1 ) `1 ) / 2 ) ;
for G being Group , H being Subgroup of G , a being Integer , b being Integer st a = b holds for i being Integer st i in dom b holds a |^ i = b |^ i & b |^ i = b |^ i
consider B being Function of Seg ( S + L ) , the carrier of V1 such that for x being element st x in Seg ( S + L ) holds P [ x , B . x ] ;
reconsider K1 = { p9 where p9 is Point of TOP-REAL 2 : P [ p9 ] & p9 `2 <= p9 `2 & p9 `2 <= p9 `2 & p9 `2 <= p9 `2 & p9 `2 <= - p9 `2 } as Subset of ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( K \ { p } ) ) ;
( ( ( ( N - S ) / 2 ) * ( 2 |^ m ) - ( ( ( N - S ) / 2 ) * ( 2 |^ m ) - ( ( N - S ) / 2 ) * ( 2 |^ m ) ) / 2 ) ) / 2 <= ( ( ( N - S ) / 2 ) / 2 ) / 2 ;
for x be Element of X , n be Nat st x in E holds |. Re ( F . n ) .| <= P . x & |. Im ( F . n ) .| <= P . x & |. Im ( F . n ) .| <= P . x
len @ ( @ ( @ @ p ^ @ q ) ) = len ( @ ( @ p ^ @ q ) ) + len <* [ 2 , 0 ] *> .= len ( @ p ^ @ q ) + len @ q .= len @ ( @ p ^ @ q ) + len @ q ;
v / ( x. 3 , m1 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / (
consider r be Element of M such that M , v2 / ( x. 3 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 3 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) ) / ( x. 4 , m3 ) / ( x. 3 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3
func w1 \ w2 -> Element of Union ( G , R8 ) means : Def1 : for w1 , w2 being Element of Union ( G , R8 ) holds it . ( w1 , w2 ) = ( ( ( ( the NN8 of G ) * ( the Sorts of R8 ) * ( the Sorts of R8 ) ) . ( w1 , w2 ) ) . ( w1 , w2 ) ;
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= s1 . b2 .= s2 . b2 .= Exec ( n2 , s2 ) . b2 .= Exec ( n2 , s2 ) . b2 .= s2 . b2 .= s . b2 .= s . b2 .= s . b2 .= s . b2 ;
for n , k being Nat holds 0 <= ( Partial_Sums ( |. seq .| ) . ( n + k ) - Partial_Sums ( |. seq .| ) . n + Partial_Sums ( |. seq .| ) . ( n + k ) - Partial_Sums ( |. seq .| ) . n
set F = S -\mathop { 0 } ;
( Partial_Sums ( seq ) ) . ( n + 1 ) + Partial_Sums ( seq ) . ( n + 1 ) >= ( Partial_Sums ( seq ) . ( n + 1 ) + Partial_Sums ( seq ) . ( n + 1 ) ) + 0 ;
consider L , R such that for x st x in N holds ( f | Z ) . x - ( f | Z ) . x0 = L . ( x- ( f , Z ) . ( x - x0 ) ) + R . ( x - x0 ) ;
func the closed \HM { a , b , c , d } -> Subset of \HM { a , b , c , d } : a = b & b = c & c = d & d = d } & a = b & c = d & d = d & d = b & a = c & b = d & c = d & d = d & a = b & c = d & d = d } ;
a * b ^2 + ( a * c ^2 + b * a * c + ( b * c ) + ( c * a ) + ( c * a ) + ( c * a ) + ( c * a ) + ( c * a ) >= 6 * a * b * c ;
v / ( x1 , m1 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m1 ) ) = v / ( x2 , m1 ) ;
Rotate ( Q ^ <* x *> , M ) = ( Rotate ( Q , ( M ^ <* 0 *> ) , ( M ^ <* x *> ) ) . ( ( M ^ <* 0 *> ) --> FALSE ) .= ( M ^ <* 0 *> , ( M ^ <* 1 *> ) --> FALSE ) . ( ( M ^ <* 1 *> ) --> TRUE ) .= ( M ^ <* 0 *> , 0 ) --> TRUE ;
Sum ( FM . n1 ) = r |^ n1 * Sum CM .= C . ( n1 + 1 ) * ( C . n1 ) .= C . ( n1 + 1 ) * ( C . n1 ) .= C . ( n1 + 1 ) * ( C . n1 ) .= C . ( n1 + 1 ) * ( C . n1 ) ;
( ( GoB f ) * ( len GoB f , 2 ) ) `1 = ( ( GoB f ) * ( len GoB f , 1 ) ) `1 .= ( ( GoB f ) * ( len GoB f , 1 ) ) `1 .= ( ( GoB f ) * ( len GoB f , 1 ) ) `1 ;
defpred X [ Element of NAT ] means ( Partial_Sums ( s ) . $1 ) * ( $1 + 1 ) = ( a * ( $1 + 1 ) * ( $1 + 1 ) * ( $1 + 1 ) * ( $1 + 1 ) * ( $1 + 1 ) * ( $1 + 1 ) * ( $1 + 1 ) * ( $1 + 1 ) * ( $1 + 1 ) * ( $1 + 1 ) * ( $1 + 1 ) * ( $1 + 1 ) ;
the_arity_of g = ( the Arity of S ) . g .= [ ( the Arity of S ) . g , ( the Arity of S ) . g ] .= [ g , ( the ResultSort of S ) . g ] .= [ g , ( the ResultSort of S ) . g ] .= [ g , ( the ResultSort of S ) . g ] .= [ g , ( the ResultSort of S ) . g ] ;
( X ~ ) c= X ~ & card ( X ~ ) = card X & card ( X ~ ) = card X & card ( X ~ ) = card X ;
for a , b being Element of S , s being Element of NAT st s = n & a = F . n & b = F . ( n + 1 ) holds b = N . ( n + 1 ) \ G . s
E , f |= All ( x. 2 , ( x. 2 ) '&' ( x. 2 , ( x. 2 ) '&' ( x. 2 ) '&' ( x. 2 , ( x. 2 ) '&' ( x. 1 , x. 2 ) '&' ( x. 2 ) '&' ( x. 2 , x. 2 ) ) ) ) ;
ex R2 being 1-sorted st R2 = ( p | ( n -' ( n -' 1 ) ) ) . i & ( ( the carrier of R1 ) | ( n -' 1 ) ) . i = the carrier of R2 & ( ( the carrier of R2 ) | ( n -' 1 ) ) . i = the carrier of R2 ;
[. a , b + 1 / ( k + 1 ) .[ is Element of the _ of the carrier of X & ( the partial F of partial f ) . k is Element of the carrier of X & ( the partial F of partial f ) . k is Element of the carrier of X & ( the Sorts of L~ f ) . k is Element of the carrier of X ;
Comput ( P , s , 2 + 1 ) . a = Exec ( P . 2 , Comput ( P , s , 2 ) . a ) .= Exec ( a3 , s ) . a .= s . a ;
card ( h1 ) . k = power F_Complex * ( - 1_ F_Complex ) * Sum u .= ( - 1_ F_Complex ) * Sum u .= ( - ( - 1_ F_Complex ) * u ) * Sum u .= ( ( - ( - 1_ F_Complex ) * u ) * u ) * ( - ( - 1_ F_Complex ) * v ) ;
( f / g ) /. c = f /. c * ( g /. c ) " .= f /. c * ( ( 1 / 2 ) * g /. c ) .= ( f (#) g ) /. c * ( ( 1 / 2 ) * g ) /. c .= ( f (#) g ) /. c * ( g (#) g ) /. c ;
len Cs - len ( -> FinSequence of ( the carrier of ( ( ( the carrier of ( ( the carrier of ( ( the carrier of ( ( the carrier of ( ( the carrier of ( ( the carrier of ( ( the carrier of ( the carrier of ( the carrier of ( the carrier of C ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) = len ( ( the _ of ( ( the carrier of ( the carrier of ( the carrier of ( ( the carrier of ( ( the carrier of ( ( the carrier of ( ( the carrier of ( ( the carrier of ( ( the carrier of ( ( the carrier of ( ( the carrier of ( ( the carrier of ( ( the carrier of ( ( the carrier of ( ( the carrier of ( ( the carrier of ( ( the carrier
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom f /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= X /\ X .= X /\ X .= dom f /\ X ;
defpred P [ Nat ] means 2 * Fib ( n + $1 ) = Fib ( n ) * Fib ( n + $1 ) + ( 5 * Fib ( n + $1 ) * Fib ( n + $1 ) * Fib ( n + $1 ) * Fib ( n + $1 ) * Fib ( n + $1 ) * Fib ( n + $1 ) * Fib ( n + $1 ) ;
consider f being Function of INT , INT such that f = f ` and f is onto and n < n and f is onto and n < len f and f " { f . n } = { n } and f " { f . n } = { n } ;
consider c9 being Function of S , BOOLEAN such that c9 = chi ( A \/ B , S ) and E7 . ( A \/ B ) = Prob ( c , S ) and E7 . ( A \/ B ) = Prob ( c , S ) ;
consider y being Element of Y ( ) such that a = "\/" ( { F ( x , y ) where x is Element of X ( ) : P [ x ] } , T ( ) ) and Q [ y ] and P [ y ] and P [ y ] ;
assume that A c= Z and f = ( ( - 1 ) (#) ( sin * ( ( #Z 2 ) * ( sin * ( f1 + f2 ) ) / ( sin * ( f1 + f2 ) / ( sin * ( f1 + f2 ) ) ^2 ) ) ) ) and Z c= dom ( ( - 1 ) (#) ( sin * ( f1 + f2 ) / ( sin * ( f1 + f2 ) / ( sin * ( f1 + f2 ) / ( f1 * f2 ) ) ^2 ) ) ;
( f /. i ) `2 = ( ( GoB f ) * ( 1 , j2 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 + 1 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 + 1 ) ) `2 ;
dom Shift ( Seq q2 , len Seq q1 ) = { j + len Seq q1 where j is Nat : j in dom Seq q1 & len Seq q1 = len Seq q1 } & len Seq q1 = len Seq q1 & len Seq q1 = len Seq q2 & len Seq q1 = len Seq q1 & len Seq q1 = len Seq q1 ;
consider G1 , G2 , G3 being Element of V such that G1 <= G2 & G2 <= G2 and f = G1 * ( G1 , G2 ) and g = G2 * ( G1 , G2 ) and f = G1 * ( G1 , G2 ) and g = G2 * ( G1 , G2 ) ;
func - f -> PartFunc of C , V means : Def1 : dom it = dom f & for c st c in dom it holds it /. c = - f /. c & it /. c = - f /. c ;
consider phi such that phi is increasing and for a st phi . a = a & {} <> a for v st v in union ( ( union L ) | ( union L ) ) holds union L |= H & for a st a in union L holds L . a c= v ;
consider i1 , j1 such that [ i1 , j1 ] in Indices GoB f and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) ;
consider i , n such that n <> 0 and sqrt ( p . n1 ) = ( i - n ) * ( i - 1 ) and for n1 being Nat st n1 <> 0 & n1 <= n holds n <= i & n <= n ;
assume that not 0 in Z and Z c= dom ( ( - 1 ) (#) ( ( #Z 2 ) * ( f1 + f2 ) ) ) and for x st x in Z holds 1 / ( ( #Z 2 ) * ( f1 + f2 ) ) . x > - 1 / ( ( #Z 2 ) * ( f1 + f2 ) ) . x ;
cell ( G1 , i1 -' 1 , 2 -' 1 ) \ ( ( Y -' 1 ) * ( Y -' 1 ) + 2 -' 1 ) c= ( ( L~ f1 ) \ ( L~ f1 ) \/ L~ f2 ) \/ ( L~ f2 ) ;
ex Q1 being open Subset of X st s = Q1 & ex Q1 being Subset-Family of Y st [: Q1 , Q1 :] c= F & [: Q1 , Q1 :] c= union ( F .: [: Y , X :] ) & [: Q1 , Q :] c= union ( F .: [: Y , X :] ) & [: Q1 , Q :] c= union ( F .: [: Y , X :] ) ;
gcd ( A1 , ( the carrier of A ) --> ( r1 , r2 ) , gcd ( A1 , r2 ) ) = 1 & gcd ( A1 , ( the carrier of A ) --> ( r2 , ( the carrier of A ) --> ( r1 , r2 ) ) , 1 ) = 1 ;
R8 = ( ( ( the _ of s2 ) . ( m2 + 1 ) ) . ( m2 + 1 ) ) . ( m2 + 1 ) .= ( ( the _ of s2 ) . ( m2 + 1 ) ) . ( m2 + 1 ) .= [ 3 , ( the _ of s2 ) . ( m2 + 1 ) ] .= [ 3 , ( the InternalRel of s2 ) . ( m2 + 1 ) ] ;
CurInstr ( P-6 , Comput ( P-6 , s2 , m3 + 1 ) ) = CurInstr ( P3 , Comput ( P3 , s3 , m3 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m3 ) ) .= halt SCMPDS .= CurInstr ( P3 , Comput ( P3 , s3 , m3 ) , m3 ) .= halt SCMPDS ;
P1 /\ P2 = ( { p1 } \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p2 , p1 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p2 , p1 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p2 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p2 , p1 ) \/ LSeg ( p2 , p2 ) ;
func not bound in the Sorts of Al means : Def1 : a in it iff ex i st i in dom f & a = f . i & ( for i st i in dom f ex p st p in dom f & f . i = f . p ) & ( for i st i in dom f holds f . i = p ) ;
for a , b being Element of F_Complex , f being FinSequence of F_Complex st |. a .| > |. b .| & f is \cup g implies a * ( - b * ( - b * f ) ) is \cup S & f is \cup S & f is \cup S & f is \cup S is \/ S & f is ]
defpred P [ Nat ] means 1 <= $1 & $1 <= len g implies G * ( i , j ) `1 = g * ( $1 , 1 ) `1 & G * ( i , j ) `2 < g * ( i + 1 , j ) `2 ;
assume that C1 , C2 are_\HM { f , g } and g = f * g and for s1 being State of C1 , s2 being State of C2 , s1 being State of C2 st s1 = g * f holds s1 is stable and s2 is stable and for n being Nat st n in dom f holds f . n is stable ;
( ||. f .|| | X ) . c = ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| ;
|. q .| ^2 = ( q `1 ) ^2 + ( q `2 ) ^2 & 0 + ( q `2 ) ^2 < ( q `1 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 ;
for F being Subset-Family of T7 st F is open & {} in F & for A , B being Subset of T7 st A in F & B <> {} & A <> {} & B in F & A misses B holds card F c= card G
assume that len F >= 1 and len F = k + 1 and len F = len G and for k st k in dom F holds H . k = g . ( F . k , G . k ) and for k st k in dom F holds H . k = g . ( F . k , G . k ) ;
i |^ ( ( mod n ) - i |^ s ) = i |^ ( s + k - i ) * i .= i |^ ( s * i - i |^ k ) * i .= i |^ ( s * i - i |^ k ) * i .= i |^ ( s * i - i |^ k ) * i - i |^ k * i ;
consider q being oriented Chain of G such that r = q and q <> {} and F8 . ( q . 1 ) = v1 and rng q c= rng ( p ^ q ) and rng q c= rng ( p ^ q ) and rng q c= rng ( p ^ q ) and rng q c= rng ( p ^ q ) ;
defpred P [ Element of NAT ] means $1 <= len ( ) implies ( ( ( g , Z ) ^ <* I *> ) . ( len ( g , Z ) + 1 ) ) . ( len ( g , Z ) + 1 ) = ( ( ( f , Z ) ^ <* I *> ) . ( len ( g , Z ) + 1 ) ) . ( len ( f , Z ) + 1 ) ;
for A , B being Matrix of n , REAL holds len ( A * B ) = len A & width ( A * B ) = width B & width ( A * B ) = width A & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B
consider s being FinSequence of the carrier of R such that Sum s = u and for i being Element of NAT st 1 <= i & i <= len s ex a , b being Element of R st s . i = a * b & a in I & b in J & b in J & a * b in J ;
func |( x , y )| -> Element of COMPLEX means : Def1 : for i be Nat holds it . i = |( Re ( x , y ) , ( Re ( x , y ) ) , ( Im ( x , y ) ) )| + ( i * ( Re ( x , y ) ) , ( Im ( x , y ) ) * ( Re ( x , y ) ) , ( Im ( x , y ) ) * ( x , y ) ) )| ;
consider g2 being FinSequence of FH such that g2 is continuous and rng g2 c= A & g2 . 1 = x1 & g2 . len g2 = x2 & for k st k in dom g2 holds g2 . k = F ( k ) and g2 . k = F ( k ) and g2 . ( len g2 ) = G ( k ) ;
then n1 >= len p1 & crossover ( p1 , p2 , n1 , n2 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , F , J ) = crossover ( p1 , p2 , n1 , n2 , n3 , n3 , n3 , F , J ) ;
( q `1 * a <= q `1 & - q `1 <= q `2 & - q `2 <= q `1 or q `1 * a & q `2 * a <= q `2 & q `2 * a <= - q `2 & q `2 * a <= - q `2 ) & - q `2 * a <= q `2 * a ;
Fs . ( p9 . ( len p9 ) ) = Fs . ( p . ( len p9 ) ) .= ( ( v . ( len p9 ) ) * ( v . ( len p9 ) ) ) * ( v . ( len p9 ) ) .= ( v . ( len p9 ) ) * ( v . ( len p9 ) ) * ( v . ( len p9 ) ) .= ( v . ( len p9 ) * ( v . ( len p9 ) ) * ( v . ( len p9 ) ) .= v . ( len p9 ) * ( v . ( len p9 ) * ( v . ( len p9 ) * ( v . ( len p9 ) * ( v . ( len p9 ) ) * ( v . ( len p9 ) .= v . ( len p9 ) * ( v . ( len p9 ) .= ( v . ( len p9 ) .= ( v . ( len p9
consider k1 being Nat such that k1 + k = 1 and a := k = ( <* a := intloc 0 *> ^ ( ( intloc 0 ) --> 1 ) ^ ( ( intloc 0 ) .--> 1 ) ) ^ ( ( intloc 0 ) .--> 1 ) ) ^ ( ( intloc 0 ) .--> 1 ) ^ ( ( intloc 0 ) .--> 1 ) ;
consider B8 being Subset of B1 , y8 being Function of B1 , y2 such that B8 is finite and B8 = the carrier of ( B1 \/ B2 ) and for x being set st x in B8 holds B8 ( x , y ) = \frac { 0 } { 1 } and B . x = \frac { 0 } { 1 } and B . y = 1 / 2 ;
v2 . b2 = ( ( curry F2 ) * ( ( ( F . b2 ) * ( ( F . b2 ) * ( F . b2 ) ) ) ) . ( ( ( ( F . b2 ) * ( F . b2 ) ) * ( F . b2 ) ) . ( ( F . b2 ) * ( F . b2 ) ) . ( ( F . b2 ) * ( F . b2 ) . ( ( F . b2 ) * ( F . b2 ) ) . ( ( F . b2 ) . ( b1 , b2 ) ) .= ( ( F . b2 ) . ( b1 , b2 ) ) * ( F . b2 ) .= ( ( b2 , b1 ) * ( F . b2 ) * ( F . b2 ) * ( F . b2 ) ) * ( F . b2 ) .= ( ( b2 , b2 ) * ( F . b2 ) * ( F . b2 ) *
dom IExec ( I-35 , P , Initialize s ) = the carrier of SCMPDS .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ;
ex dbeing Real st d-32 > 0 & for h be Real st h <> 0 & |. h .| < e holds |. h .| " * ||. ( R2 * ( R1 + R2 ) ) /. h .|| < e / ( 2 * ||. R1 /. h .|| ) * ||. ( R2 * ( R1 + R2 ) ) /. h .|| ;
LSeg ( G * ( len G , 1 ) + |[ - 1 , 1 ]| , G * ( len G , 1 ) + |[ 1 , 0 ]| ) c= Int cell ( G , len G , 0 ) \/ { |[ 1 , 0 ]| } \/ { |[ 1 , 0 ]| } ;
LSeg ( mid ( h , i1 , i2 ) , i ) = LSeg ( h /. ( i + 1 -' 1 ) , h /. ( i + 1 -' 1 + 1 ) ) .= LSeg ( h , i + 1 -' 1 ) .= LSeg ( h , i + 1 -' 1 ) .= LSeg ( h , i + 1 -' 1 ) ;
A = { q where q is Point of TOP-REAL 2 : LE p1 , p2 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p1 , p2 , P & LE p2 , p1 , P & LE p2 , p1 , P } , P & LE p1 , p2 , P & LE p2 , p2 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p1 , p1 , P & LE p1 , p2 , P & LE p1 , p2 , P & LE p1 , p2 , P &
( ( - x ) .|. y ) = - ( ( 1 - x ) .|. y ) * ( x .|. y ) .= - ( ( - 1 ) * ( x .|. y ) ) * ( x .|. y ) .= - ( ( - 1 ) * ( x .|. y ) ) * ( x .|. y ) .= - ( ( - 1 ) * ( x .|. y ) * ( x .|. y ) .= - ( ( - 1 ) * ( x .|. y ) * ( x .|. y ) .= - ( - 1 ) * ( x .|. y ) * ( x .|. y ) * ( x .|. y ) * ( x .|. y ) * ( x .|. y ) * ( x .|. y ) * ( x .|. y ) * ( x .|. y ) * ( x .|. y ) * ( x .|. y ) .= - ( ( x .|. y ) * ( x .|. y ) * ( x .|. ( - ( - 1 ) * ( x .|.
0 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) = ( p `2 / sqrt ( 1 + ( p `2 / p `1 ) ^2 ) ) * sqrt ( 1 + ( p `2 / p `1 ) ^2 ) .= ( p `2 / sqrt ( 1 + ( p `2 / p `1 ) ^2 ) ) * sqrt ( 1 + ( p `2 / p `1 ) ^2 ) ;
( ( U . n ) * ( W . n ) * ( W . n ) = ( ( ( U . n ) * ( W . n ) ) * ( W . n ) ) * ( W . n ) .= ( ( U . n ) * ( W . n ) ) * ( W . n ) * ( W . n ) * ( W . n ) * ( W . n ) * ( W . n ) .= ( ( U . n ) * ( W . n ) * ( W . n ) * ( W . n ) * ( W . n ) * ( W . n ) * ( W . n ) * ( W . n ) * ( W . n ) * ( W . n ) * ( W . n ) * ( W . n ) * ( W . n ) * ( W . n ) * ( W . n ) * ( W . n ) * ( W . n ) * ( W .
func Shift ( f , h ) -> PartFunc of REAL , REAL means : Def1 : dom it = dom f & for x st x in dom it holds it . x = - h . x & for x st x in dom it holds it . x = - f . x + f . x * h . x ;
assume that 1 <= k and k + 1 <= len f and [ i , j ] in Indices G and f /. k = G * ( i + 1 , j ) and f /. k = G * ( i + 1 , j ) and f /. ( k + 1 ) = G * ( i , j ) and f /. ( k + 1 ) = G * ( i , j ) ;
assume that not y in Free H and not x in Free H and not x in Free ( H ) and not x in Free ( H ) and not x in Free ( H ) and not x in Free ( H ) and not x in Free ( H ) and not x in Free ( H ) and not x in Free ( H ) ;
defpred P11 [ Element of NAT , \HM { Element of NAT ] means P [ $1 ] & ( $1 <= 2 implies $2 = ( p |^ 2 ) * ( $1 |^ 2 ) ) & ( $1 |^ 2 ) * ( $1 |^ 3 ) < ( p |^ 2 ) * ( $1 |^ 3 ) ;
func \sigma ( C ) -> non empty Subset-Family of X means : Def1 : for A being Subset of X holds A in it iff for W being Subset of X , Z being Subset of X st W c= Z & Z c= W holds C . W <= C . Z + C . Z ;
[#] ( ( dist ( ( dist ( P ) ) .: Q ) ) = ( dist ( ( dist ( P ) ) .: Q ) ) .: Q & lower_bound ( [#] ( ( dist ( P ) .: Q ) .: Q ) ) = lower_bound ( ( dist ( P ) .: Q ) .: Q ) ;
rng ( F | ( [: S , S :] ) ) = {} or rng ( F | [: S , S :] ) = { 1 } or rng ( F | [: S , S :] ) = { 2 } or rng ( F | [: S , S :] ) = { 1 } or rng ( F | [: S , S :] ) = { 2 } or rng ( F | [: S , S :] ) = { 2 } ;
( f " rng ( f * ( f * g ) ) ) . i = f . i " * ( ( f * ( f * g ) ) . i ) .= f . i * ( ( f * ( f * g ) ) . i ) .= ( f * ( f * g ) ) . i .= ( f * ( f * g ) ) . i .= ( f * ( f * g ) ) . i ;
consider P1 , P2 being non empty Subset of TOP-REAL 2 such that P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p2 , p2 and P1 /\ P2 = { p1 , p2 } and P1 /\ P2 = { p1 , p2 } and P1 /\ P2 = { p2 , p1 } and P1 /\ P2 = { p2 , p1 } and P = { p2 , p1 } and P = { p2 , p1 } and P = { p2 , p1 } and P = { p2 , p1 } and P = { p2 , p1 , p2 , p3 , p3 , p2 , p3 } and P = P \/ P and P = P \/ P and P = P \/ P and P = P \/ P and P = P \/ P and P = P \/ P and P = P \/ P and P = P \/ P and Q = P \/ P and Q = P \/ P and Q = P \/ P and Q = P \/ P and Q = P \/ P and Q = P \/ P and P = P \/ P and P = P \/ P and P = P \/
f . p2 = |[ ( p2 `1 / |. p2 .| - cn ) / ( 1 + cn ) , ( p2 `2 / |. p2 .| - cn ) / ( 1 + cn ) ]| .= |[ ( p2 `1 / |. p2 .| - cn ) / ( 1 + cn ) , ( p2 `2 / |. p2 .| - cn ) / ( 1 + cn ) ]| ;
( ( ( a , X ) " ) * ( ( ( AffineMap ( a , X ) " ) * ( x - a ) ) " ) . x = ( ( a , X ) " ) * ( x - a ) ) * u .= ( ( a , X ) " ) * u + a * v .= ( ( a , X ) " ) * u + a * v .= ( ( a , X ) " ) * u + a * v .= ( a , X ) * u + a * v + a * v + a * v + a * v + a * w + a * w + a * w + a * w + a * w + a * w + a * w + a * w + a * w + a * w + a * w + a * w + a * v .= ( a * w ) * v .= ( ( a , X ) * v .= ( ( a , X ) * w + a * w + a * w .= ( a * w + a * w + a * w + a * w + a * w + a * v + a *
for T being non empty normal TopSpace , A , B being closed Subset of T st A <> {} & A misses B for p being Point of T , r being Real st p in ( ( in G ) \ A ) & r in ( ( \mathbb G ) \ A ) & p in ( ( \mathbb G ) \ A ) holds p in ( ( ( \mathbb G ) \ A ) \ B )
for i , j st i + 1 in dom F for G1 , G2 being strict normal Subgroup of G st G1 = F . i & G2 = F . ( i + 1 ) & G2 = F . ( i + 1 ) holds G1 is strict Subgroup of G1 & G2 is strict Subgroup of G2
for x st x in Z holds ( ( ( #Z 2 ) * ( arctan - arccot ) `| Z ) . x = ( ( #Z 2 ) * ( arctan - arccot ) `| Z ) . x / ( 1 + x ^2 )
synonym f is_or f /* a = lim ( f , x0 ) & f . x0 = lim ( f , x0 ) & for a st a in dom f & a < x0 ex b st b < b & f . a = lim ( f , x0 ) & f . b = lim ( f , x0 ) ;
then X1 , X2 are_separated & ( X1 misses X2 & Y1 misses Y2 or Y1 misses Y2 & Y2 misses X2 & ( X1 misses Y2 or Y1 misses Y2 & Y1 misses Y2 & Y2 misses X2 & Y1 misses Y2 & Y1 misses Y2 & Y1 misses Y2 & Y1 misses Y2 & Y1 misses Y2 & Y1 misses Y2 & Y1 misses Y2 & Y1 misses Y2 & Y1 misses Y2 & Y1 misses Y2 & Y1 misses Y2 & Y1 misses Y2 & Y1 misses Y2 & Y1 misses Y2 & Y1 misses Y2 & Y1 misses Y2 & Y1 misses Y2 & Y1 misses Y2 & Y1 misses Y2 & Y1 misses Y2 & Y1 misses Y2 & Y1 misses Y2 & Y1 misses Y2 & Y1 misses Y2 & Y1 misses Y2 & Y2 misses Y2 & Y1 misses Y2 & Y2 misses Y2 & Y1 misses Y2 & Y1 misses Y2 & Y1 misses Y2 & Y2 misses Y2 & Y2 misses Y2 & Y2 misses Y2 & Y2 misses Y2 & Y2 misses Y2 & Y2 misses Y2 & Y2 misses Y2 & Y2 misses Y2 & Y2 misses Y2 & Y2 misses Y2 & Y2 misses Y2 & Y2 misses Y2 & Y2 misses Y2 & Y2 misses Y2 & Y2 misses Y2 & Y2 misses Y2 & Y2 misses Y2 & Y2
ex N being Neighbourhood of x0 st N c= dom SVF1 ( 1 , f , u ) & ex L , R st for x st x in N holds SVF1 ( 1 , f , u ) . x - SVF1 ( 1 , f , u ) . x0 = L . ( x - u ) + R . ( x - u ) ;
( p2 `1 ) * sqrt ( 1 + ( p3 `2 / p3 `1 ) ^2 ) >= ( ( p2 `1 ) * sqrt ( 1 + ( p3 `2 / p3 `1 ) ^2 ) ) * sqrt ( 1 + ( p3 `2 / p3 `1 ) ^2 ) ;
( ( 1 / ( t1 * ||. f1 .|| ) ) to_power ( n + 1 ) ) . x = ( ( 1 / ( t1 * ||. g1 .|| ) ) to_power ( n + 1 ) ) . x & ( ( 1 / ( t2 * ||. g1 .|| ) ) to_power ( n + 1 ) ) . x = ( ( 1 / ( t2 * ||. g1 .|| ) ) to_power ( n + 1 ) ;
assume that for x holds f . x = ( ( - sin * ( x + h ) ) (#) ( sin * ( x + h ) ) - ( sin * ( x + h ) ) (#) ( sin * ( x + h ) ) and x - h / 2 = ( - sin * ( x + h ) ) (#) ( sin * ( x + h ) ) ;
consider Xj1 being Subset of Y , Y1 being Subset of X such that t = [: Y1 , Y1 :] and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open ;
card S . n = card { [: d , Y :] + ( a * d ) * b where d is Element of GF ( p ) , Y is Element of GF ( p ) : [ d , b ] in R } .= { 1 } \/ { d , b } .= { d , b } \/ { d , c } .= { 1 , d } \/ { 1 , c } ;
( W-bound D - W-bound D ) * ( ( W-bound D - W-bound D ) / 2 * ( ( W-bound D - W-bound D ) / 2 * ( ( W-bound D - W-bound D ) / 2 * ( ( W-bound D - W-bound D ) / 2 * ( ( W-bound D - W-bound D ) / 2 * ( ( W-bound D - W-bound D ) / 2 * ( ( W-bound D - W-bound D ) / 2 * ( ( W-bound D - W-bound D ) / 2 * ( ( W-bound D - W-bound D ) / 2 * ( ( W-bound D ) / 2 * ( ( W-bound D ) / 2 * ( ( W-bound D ) / 2 * ( ( W-bound D - W-bound D ) / 2 * ( ( W-bound D ) / 2 * ( ( W-bound D ) / 2 * ( i - W-bound D ) / 2 ) / 2 ) / 2 ) * ( i - j ) / 2 ) * ( i - j ) / 2 ) ) = ( W-bound D ) / 2 ) / 2 ) * ( i - j ) / 2 ) ) * ( i - j ) ) * ( ( i - j ) / 2 ) * ( i