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 ;
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thesis ;
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thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
contradiction ;
contradiction ;
thesis ;
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thesis ;
thesis ;
thesis ;
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thesis ;
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thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b
D c= S
let Y ;
S ` 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 ;
Y c= Z ;
A // M ;
let U , A ;
a in D ;
q in Y ;
let x ;
1 <= l ;
1 <= w ;
let G , H , x , y ;
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 , Y ;
let C , a , b ;
x _|_ p ;
o is monotone ;
let X , Y ;
A = B ;
1 < i ;
let x ;
let u , v ;
k <> 0 ;
let p ;
0 < r ;
let n ;
let y ;
f is onto ;
x < 1 ;
G c= F ;
a is_>=_than X ;
T is continuous ;
d <= a ;
p <= r ;
t < s ;
p <= t ;
t < s ;
let r ;
D is_<=_than E ;
assume e > 0 ;
assume 0 < g ;
p in X ;
x in X ;
Y ` in Y ;
assume 0 < g ;
not c in Y ;
not v in L ;
2 in z `2 ;
assume f = g ;
N c= b ` ;
assume i < k ;
assume u = v ;
I = J ;
B ` = b ` ;
assume e in F ;
assume p > 0 ;
assume x in D ;
let i be element ;
assume F is onto ;
assume n <> 0 ;
let x be element ;
set k = z ;
assume o = x ;
assume b < a ;
assume x in A ;
a `2 <= b `2 ;
assume b in X ;
assume k <> 1 ;
f = product l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is from squares ;
assume m > 0 ;
assume A c= B ;
X is lower ;
assume A <> {} ;
assume X <> {} ;
assume F <> {} ;
assume G is open ;
assume f is dilatation ;
assume y in W ;
y \not <= x ;
A ` in B ` ;
assume i = 1 ;
let x be element ;
x `2 = x `2 ;
let X be BCK-algebra ;
assume S is non empty ;
a in REAL ;
let p be set ;
let A be set ;
let G be _Graph , W be Walk of G ;
let G be _Graph , W be Walk of G ;
let a be Complex ;
let x be element ;
let x be element ;
let C be FormalContext , a , b be Real ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
let y be Real ;
X c= f . a
let y be element ;
let x be element ;
i be Nat ;
let x be element ;
n in NAT ;
let a be element ;
m in NAT ;
let u be element ;
i in NAT ;
let g be Function ;
Z c= NAT ;
l <= ma ;
let y be element ;
r2 in dom f ;
let x be element ;
k1 be Integer ;
let X be set ;
let a be element ;
let x be element ;
let x be element ;
let q be element ;
let x be element ;
assume f is being_homeomorphism ;
let z be element ;
a , b // K ;
let n be Nat ;
let k be Nat ;
B ` c= B ` ;
set s = b / 2 ;
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 non discrete ;
m1 divides m ;
B * A <> {} ;
a + b <> {} ;
p * p > p ;
let y be ExtReal ;
let a be Int-Location , I be Program of SCM+FSA ;
let l be Nat ;
let i be Nat ;
let r ;
1 <= i2 ;
a "\/" c = c ;
let r be Real ;
let i be Nat ;
let m be Nat ;
x = p2 ;
let i be Nat ;
y < r + 1 ;
rng c c= E
Cl R is boundary ;
let i be Nat ;
R2 ;
cluster uparrow x -> let L ;
X <> { x } ;
x in { x } ;
q , b // M ;
A . i c= Y ;
P [ k ] ;
2 to_power x in W ;
X [ 0 ] ;
P [ 0 ] ;
A = A |^ i ;
L~ L~ L~ L~ L~ \subseteq implies n >= 0 ;
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 , f be Function ;
M , L are_isomorphic ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded upper-bounded ;
rng f = Y ;
( G . n ) 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 `2 = a `2 + 1 ;
x `2 = a * y `2 ;
rng D c= A ;
assume x in K1 ;
1 <= i-32 ;
1 <= i-32 ;
pp c= PI ;
1 <= i-15 ;
1 <= i-15 ;
LMP C in L ;
1 in dom f ;
let seq , n ;
set C = a * B ;
x in rng f ;
assume f is_continuous_on X ;
I = dom A ;
u in dom p ;
assume a < x + 1 ;
s-7 is bounded ;
assume I c= P1 ;
n in dom I ;
let Q ;
B c= dom f ;
b + p _|_ a ;
x in dom g ;
FF is continuous ;
dom g = X ;
len q = m ;
assume A2 is closed ;
cluster R \ S -> real-valued ;
sup D in S ;
x << sup D ;
b1 >= Z1 ;
assume w = 0. V ;
assume x in A . i ;
g in the carrier of X ;
y in dom t ;
i in dom g ;
assume P [ k ] ;
dom _ C c= dom 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 , A be non-empty MSAlgebra over 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 , f be Function ;
b ` ` c= b9 ;
assume not x in RAT ;
A /\ B = { a } ;
assume len f > 0 ;
assume x in dom f ;
b , a // o , c ;
B in B-24 ;
cluster product p -> non empty ;
z , x // x , p ;
assume x in rng N ;
cosec is_differentiable_in x & cosec is_differentiable_in cos . x ;
assume y in rng S ;
let x , y be element ;
i2 < i1 & i2 < i1 ;
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 & bn < n ;
assume A c= dom f ;
Re f is_integrable_on M ;
let k , m be element ;
a , a \equiv b , b ;
j + 1 < k + 1 ;
m + 1 <= n1 ;
g is_differentiable_in x0 & g is_differentiable_in x0 ;
g is_continuous_in x0 & g is_continuous_in x0 ;
assume O is symmetric transitive ;
let x , y be element ;
let j0 be Nat ;
[ y , x ] in R ;
let x , y be element ;
assume y in conv A ;
x in Int V ;
let v be VECTOR of V ;
P3 halts_on s , P ;
d , c // a , b ;
let t , u be set ;
let X be set ;
assume k in dom s ;
let r be non negative Real ;
assume x in F | M ;
let Y be Subset of S ;
let X be non empty TopSpace , A be Subset of X ;
[ a , b ] in R ;
x + w < y + w ;
{ a , b } is_>=_than c ;
let B be Subset of A , C be Subset of B ;
let S be non empty ManySortedSign ;
let x be variable of f , g be variable of g ;
let b be Element of X , x be Element of X ;
R [ x , y ] ;
x ` ` = x ;
b \ x = 0. X ;
<* d *> in D * ;
P [ k + 1 ] ;
m in dom mnnnnnx ;
h2 . a = y ;
P [ n + 1 ] ;
redefine func G * F -> I is J ;
let R be non empty multMagma , a , b be Element of R ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p ;
assume f | X is lower ;
x in rng co /\ rng pion1 ;
let x be Element of B ;
let t be Element of D ;
assume x in Q .vertices() ;
set q = s ^\ k ;
let t be VECTOR of X ;
let x be Element of A ;
assume y in rng p `2 ;
let M be be be be | mamaid id id ;
let N be non empty for \mathop { \rm for F being Subset of M ;
let R be RelStr with finite finite : R is finite ;
let n , k be Nat ;
let P , Q be let let let P , Q be RelStr ;
P = Q /\ [#] S ;
F . r in { 0 } ;
let x be Element of X ;
let x be Element of X ;
let u be VECTOR of V ;
reconsider d = x as Int-Location ;
assume I is not ` ;
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 that t1 <= t2 and t2 <= t2 ;
let i , j be even Integer ;
assume that F1 <> F2 and F2 <> {} ;
c in Intersect ( union R ) ;
dom p1 = c & dom p2 = c ;
a = 0 or a = 1 ;
assume A1 <> A3 & A2 <> A3 ;
set i1 = i + 1 ;
assume a1 = b1 & a2 = b2 ;
dom g1 = A & dom g2 = B ;
i < len M + 1 ;
assume not -infty in rng G ;
N c= dom f1 /\ dom f2 ;
x in dom sec /\ dom sec ;
assume [ x , y ] in R ;
set d = x / y ;
1 <= len g1 + 1 ;
len s2 > 1 & len s1 > 1 ;
z in dom f1 /\ dom f2 ;
1 in dom D2 /\ Seg len D1 ;
p `2 = 0 & p `2 = 0 ;
j2 <= width G & j2 <= width G ;
len PI > 1 + 1 ;
set n1 = n + 1 ;
|. q-35 .| = 1 ;
let s be SortSymbol of S ;
( for i being Element of NAT holds i = i ) implies ( i = 1 )
X1 c= dom f & X2 c= dom f ;
h . x in h . a ;
let G be thesis , F be Subset-Family of on ;
cluster m * n -> square ;
let kk be Nat , k be Nat ;
i - 1 > m - 1 ;
R is transitive implies field R is transitive
set F = <* u , w *> ;
p-2 c= P3 & p`2 c= P3 ;
I is_halting_on t , Q ;
assume [ S , x ] is thesis ;
i <= len f2 & 1 <= i ;
p is FinSequence of X ;
1 + 1 in dom g ;
Sum R2 = n * r ;
cluster f . x -> complex-valued ;
x in dom f1 /\ dom f2 ;
assume [ X , p ] in C ;
BQ c= X3 & BQ c= XQ ;
n2 <= ( 2 * n ) - 1 ;
A /\ cP c= A ` ;
cluster x -valued for Function ;
let Q be Subset-Family of S , P be Subset of Q ;
assume n in dom g2 ;
let a be Element of R ;
t `2 in dom e2 /\ dom e2 ;
N . 1 in rng N ;
- z in A \/ B ;
let S be in of X , F be Subset-Family of S ;
i . y in rng i ;
REAL c= dom f & dom f = REAL ;
f . x in rng f ;
mt <= r / 2 ;
s2 in r-5 & s2 in r-5 ;
let z , z be complex number ;
n <= ( N . m ) ;
LIN q , p , s ;
f . x = waybelow x /\ B ;
set L = [' S , T '] ;
let x be non positive ExtReal ;
let m be Element of M ;
f in union rng ( F1 ^ F2 ) ;
let K be add-associative right_zeroed right_complementable non empty doubleLoopStr , p be Polynomial of K ;
let i be Element of NAT ;
rng ( F * g ) c= Y
dom f c= dom x & dom g c= dom y ;
n1 < n1 + 1 & n2 + 1 < n1 ;
n1 < n1 + 1 & n2 + 1 < n1 ;
cluster [: T , T :] -> \overline ;
[ y2 , 2 ] `2 = z ;
let m be Element of NAT ;
let S be Subset of R ;
y in rng ( S29 ) ;
b = sup dom f & b = sup dom f ;
x in Seg ( len q ) ;
reconsider X = D ( ) as set ;
[ a , c ] in E1 ;
assume n in dom h2 /\ dom h2 ;
w + 1 = ma ;
j + 1 <= j + 1 + 1 ;
k2 + 1 <= k1 ;
i be Element of NAT ;
Support u = Support p \/ Support q ;
assume X is complete thesis ;
assume that f = g and p = q ;
n1 <= n1 + 1 & n2 <= n1 + 1 ;
let x be Element of REAL ;
assume x in rng ( s2 | k ) ;
x0 < x0 + 1 & x0 < r2 ;
len Carrier ( L ) = W ;
P c= Seg ( len A ) ;
dom q = Seg n .= dom p ;
j <= width ( M @ ) ;
let seq1 be real-valued subsequence , seq2 be sequence of X ;
let k be Element of NAT ;
Integral ( M , P ) < +infty ;
let n be Element of NAT ;
assume z in z := being being being being being being being being being being being being being Element of A ;
i be set ;
n -' 1 = n-1 ;
len n-27 = n & len n-27 = n ;
\cal F , c ) c= F
assume x in X or x = X ;
x is midpoint of b , c ;
let A , B be non empty set , f be Function of A , B ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q & dom q = Seg i ;
let s be Element of E to_power omega ;
let B1 be Basis of x , B2 be Basis of y ;
Carrier ( L3 /\ L2 ) = {} ;
L1 /\ LSeg ( p1 , p2 ) = {} ;
assume downarrow x = downarrow y ;
assume b , c // b , c ;
LIN q , c , c ;
x in rng f-129 & x in rng f-1;
set nJ = n + j ;
let D be non empty set , f be FinSequence of D ;
let K be right_zeroed non empty addLoopStr , V be non empty addLoopStr ;
assume f `1 = f & h `2 = h ;
R1 - R2 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 a , b ) is maximal distance of 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^ -> nes\/ for ;
not u in { ag } ;
the carrier of f c= B \/ { x }
reconsider z = x as VECTOR of V ;
cluster the carrier of L -> being \rangle ;
r (#) H is C non-zero ;
s . intloc 0 = 1 ;
assume that x in C and y in C ;
let U0 be strict universal MSAlgebra over S , A be Subset of U0 ;
[ x , Bottom T ] is compact ;
i + 1 + k in dom p ;
F . i is stable Subset of M ;
r-35 in : y in : x = r & r < y ;
let x , y be Element of X ;
let A , I be contradiction Subset of X ;
[ y , z ] in [: O , O :] ;
\subseteq ( / Macro i ) . 1 ;
rng Sgm ( A ) = A ;
q |- \! \! q ;
for n holds X [ n ] ;
x in { a } & x in d ;
for n holds P [ n ] ;
set p = |[ x , y , z ]| ;
LIN o , a , b ;
p . 2 = Z |^ Y ;
( D `2 ) `2 = {} & ( D `2 ) = {} ;
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 ( s ) & x in support ( t ) ;
assume a in [: the carrier of G , the carrier of G :] ;
i `2 <= len ( y `2 ) ;
assume that p divides b1 + b2 and p divides b2 ;
M <= sup M1 & M <= sup M2 ;
assume x in W-min ( X ) & y in W-min ( X ) ;
j in dom ( z | i ) ;
let x be Element of D ( ) ;
IC s4 = l1 .= IC s4 .= ( 0 + 1 ) ;
a = {} or a = { x } ;
set us = Vertices G , c = Vertices G ;
seq " is non-zero & lim seq = 0 ;
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
hcn c= h-14 & hh2 c= hh2 ;
]. a , b .[ c= Z ;
X1 , X2 are_separated implies X1 union X2 , X2 union X1 are_separated
a in Cl ( union F \ G ) ;
set x1 = [ 0 , 0 ] ;
k + 1 -' 1 = k ;
cluster -> real-valued for Relation ;
ex v st C = v + W ;
let IT be non empty addLoopStr , x be Element of IT ;
assume V is Abelian add-associative right_zeroed right_complementable ;
X-21 \/ Y in \sigma ( L ) ;
reconsider x = x as Element of S ;
max ( a , b ) = a ;
sup B is upper Subset of B ;
let L be non empty reflexive antisymmetric RelStr , x be Element of L ;
R is reflexive transitive implies R is transitive
E , g |= the_right_argument_of H implies E , g |= H2
dom G `2 = a ;
( 1 / 4 ) * ( - r ) >= - r ;
G . p0 in rng G & G . p0 in rng G ;
let x be Element of ( F . i ) * ;
D [ P-6 , 0 ] ;
z in dom ( id B ) /\ dom ( id B ) ;
y in the carrier of N & y in the carrier of N ;
g in the carrier of H & h in the carrier of H ;
rng fE c= NAT & fE c= dom f ;
j `2 + 1 in dom s1 ;
let A , B be strict Subgroup of G ;
let C be non empty Subset of REAL ;
f . z1 in dom h & h . z2 in dom h ;
P . k1 in rng P & P . k1 in rng P ;
M = AM +* {} .= ( A +* {} ) +* {} ;
let p be FinSequence of REAL , r be Real ;
f . n1 in rng f & f . n2 in rng f ;
M . ( F . 0 ) in REAL ;
I - b < b-a ;
assume the distance of V , Q is_in the distance of V ;
let a be Element of op ( V ) ;
let s be Element of PL ;
let PA be non empty \rm \rm RelStr ;
let n be Nat ;
the carrier of g c= B & the carrier of g c= A ;
I = halt SCM R & I = halt SCM ;
consider b being element such that b in B ;
set BK = BCS ( K , P ) ;
l <= v & j <= v . j ;
assume x in downarrow [ s , t ] ;
x `2 in uparrow t & x `2 in uparrow t ;
x in ( JumpParts T ) \/ { {} } ;
let h be Morphism of c , a ;
Y c= R implies R . ( the_rank_of Y ) = R . ( the_rank_of Y )
A2 \/ A3 c= L2 \/ L1 & A2 \/ A3 c= L2 \/ L1 ;
assume LIN o , a , b ;
b , c // d1 , e2 ;
x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , 7 , 8 = 8 + 1 ;
dom <* y *> = Seg 1 .= dom <* y *> ;
reconsider i = x as Element of NAT ;
set l = |. ar s .| ;
[ x , x `2 ] in [: X , 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 & q2 , q1 , q2 is_collinear ;
dom M1 = Seg n & dom M2 = Seg n ;
x = [ x1 , x2 ] ;
let R , Q be ManySortedSet of A ;
set d = 1 / ( n + 1 ) ;
rng g2 c= dom W & rng g2 c= dom W ;
P . ( [#] Sigma \ B ) <> 0 ;
a in field R & a = b ;
let M be non empty Subset of V , a be Element of M ;
let I be Program of SCM+FSA , a be Int-Location ;
assume x in rng ( the InternalRel of R ) ;
let b be Element of the carrier of 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 RelStr ;
assume [ x , y ] in a9 ;
dom ( A * e ) = NAT ;
let a , b be Vertex of G ;
let x be Element of Bool M , a be Element of M ;
0 <= Arg a ;
o , a9 // o , y & o , b9 // o , y ;
{ v } c= the carrier of l & { v } c= the carrier of l ;
let x be bound variable of A ;
assume x in dom ( uncurry f ) ;
rng F c= ( product f ) |^ X
assume D2 . k in rng D & D2 . k in rng D1 ;
f " . p1 = 0 & f " . p2 = 0 ;
set x = the Element of X , y = the Element of Y ;
dom Ser ( G ) = NAT & rng Ser ( G ) = NAT ;
n be Element of NAT ;
assume LIN c , a , e1 ;
cluster -> finite for FinSequence of NAT ;
reconsider d = c , e = d as Element of L1 ;
( v2 |-- I ) . X <= 1 ;
assume x in the carrier of f & y in the carrier of f ;
conv @ S c= conv @ A & conv @ S c= conv @ A ;
reconsider B = b as Element of the topology of T ;
J , v |= P ! ( l , v ) ;
redefine func J . i -> non empty TopSpace ;
ex_sup_of Y1 \/ Y2 , T & ex_sup_of Y1 , T ;
W1 is_field W1 & W2 is_field W1 implies W1 + W2 is connected
assume x in the carrier of R & y in the carrier of R ;
dom n-16 = Seg n & dom n-16 = Seg n ;
s4 misses s2 & s4 misses s4 ;
assume ( a 'imp' b ) . z = TRUE ;
assume that X is open and f = X --> d ;
assume [ a , y ] in \bf \bf 1 } ;
assume that I c= J and / I c= J and I c= K ;
Im ( lim seq , x0 ) = 0 ;
( sin . x ) ^2 <> 0 & ( sin . x ) ^2 <> 0 ;
sin * ( f ^ ) is_differentiable_on Z & for x st x in Z holds sin . ( f . x ) <> 0 ;
t3 . n = t3 . n & t3 . n = t3 . n ;
dom ( ( - x ) (#) ( - x ) ) c= dom F ;
W1 . x = W2 . x & W2 . x = W3 . x ;
y in W .vertices() \/ W .vertices() \/ { x } ;
( k + 1 ) <= len ( v | k ) ;
x * a \equiv y * a . ( mod m ) ;
proj2 .: S c= proj2 .: P & proj2 .: S c= proj2 .: P ;
h . p4 = g2 . I & h . I = g2 . I ;
( for x st x in U /. 1 holds x <> U /. 1 ) implies U is open
f . rr1 in rng f & rr2 in rng f ;
i + 1 + 1-1 <= len - 1 ;
rng F = rng ( F | ( Seg n ) ) ;
mode being st of G is well unital associative non empty multMagma ;
[ x , y ] in [: A , { a } :] ;
x1 . o in L2 . o & x2 . o in L2 . o ;
the carrier of [ l , m ] c= B ;
not [ y , x ] in id X ;
1 + p .. f <= i + len f ;
seq ^\ k1 is lower & seq ^\ k1 is lower implies seq is lower
len ( F | I ) = len I .= len ( F | I ) ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be complex number , x be Element of X ;
Comput ( P , s , n ) = s ;
k <= k + 1 & k + 1 <= len p ;
reconsider c = {} T as Element of L ;
let Y be / of be / of be empty Chain \in T ;
cluster directed-sups-preserving for Function of L , L ;
f . j1 in K . j1 & f . j1 in K . j1 ;
redefine func J => y -> total Function ;
K c= 2 -tuples_on the carrier of T ;
F . b1 = F . b2 & F . b2 = F . b1 ;
x1 = x or x1 = y or x1 = z ;
pred a <> {} means : |. a .| = 1 ;
assume that cf a c= b and b in a ;
s1 . n in rng s1 & s2 . n in rng s2 ;
{ o , b2 } on C2 & { o , b1 } on C2 ;
LIN o , b , b9 & LIN o , b , c ;
reconsider m = x , n = y as Element of Funcs ( V , C ) ;
let f be non constant FinSequence of D ;
let FC2 be non empty element , F be non empty thesis ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in F-8 ;
reconsider pp2 = x , pp2 = y as Subset of m ;
let A , B , C be Element of R ;
redefine func strict non empty be strict be normal transitive
rng c `2 misses rng e`2 & rng e`2 misses rng e`2 ;
z is Element of gr { x } & z is Element of gr { x } ;
not b in dom ( a .--> p1 ) ;
assume that k >= 2 and P [ k ] ;
Z c= dom ( cot * ( f ^ ) ) ;
the component of Q c= UBD A & UBD A c= UBD A ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( 1 / ( 2 * ( f ^ ) ) ) ;
pred f = u means : Def8 : 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 = p1 ;
( n1 gcd n2 ) = 1 & ( n1 gcd n2 ) divides ( n1 gcd n2 ) ;
set oo = \cdot _ { INT } , oo = * _ { INT } ;
seq . n < |. r1 .| & |. r1 .| < |. r2 .| ;
assume that seq is increasing and r < 0 ;
f . ( y1 , x1 ) <= a ;
ex c being Nat st P [ c ] ;
set g = { n to_power 1 where n is Element of NAT : n <= 1 } ;
k = a or k = b or k = c ;
aa , ag , ch , bh , bh , bh , g ;
assume that Y = { 1 } and s = <* 1 *> ;
Ik1 . x = f . x .= 0 .= 0 ;
W3 .last() = W3 . 1 & W3 .last() = W3 . 2 ;
cluster trivial finite for Subgroup of G ;
reconsider u = u , v = v as Element of Bags X ;
A in B @ implies A , B \lbrack
x in { [ 2 * n + 3 , k ] } ;
1 >= ( q `1 / |. q .| - cn ) / ( 1 + cn ) ;
f1 is_\HM { the Element of P : P [ i ] } ;
f `2 <= q `2 & q `2 <= q `2 ;
h is_the carrier of Cage ( C , n ) ;
b `2 <= p `2 & p `2 <= ( p `2 + r ) / 2 ;
let f , g be s1 Function of X , Y ;
S * ( k , k ) <> 0. K ;
x in dom max ( - f , - g ) ;
p2 in ( N | p1 ) & p2 in ( N | p1 ) ;
len ( the_left_argument_of H ) < len ( H ) + len ( H ) ;
F [ A , FF . A ] ;
consider Z such that y in Z and Z in X ;
pred 1 in C means : Def8 : A c= C |^ A ;
assume that r1 <> 0 or r2 <> 0 and r1 <> 0 ;
rng q1 c= rng C1 & rng q2 c= rng C2 ;
A1 , L , A3 , A3 , A2 , A3 , A3 , A2 is_collinear ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in u implies not b in { p , S } ;
then S is negative & P-2 [ S ] ;
Cl Int [#] T = [#] T & Int [#] T = [#] T ;
f12 | A2 = f2 | A2 & f12 | A2 = f2 | A2 ;
0. M in the carrier of W & 0. M in the carrier of W ;
let v , v be Element of M ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ V & Y \ V c= Y \ Z ;
let X be Subset of [: S , T :] ;
consider H1 such that H = 'not' H1 and H1 in H ;
1_ 1 c= ( t * p1 ) * ( ( Q - 1 ) * r ) ;
0 * a = 0. R .= a * 0 ;
A |^ ( 2 , 2 ) = A ^^ A ;
set vp2 = ( v /. n ) `1 ;
r = 0. ( REAL-NS n ) & r = 0. ( REAL-NS n ) ;
( f . p4 ) `1 >= 0 & ( f . p2 ) `2 >= 0 ;
len W = len ( W | ( as Element of dom W ) ) ;
f /* ( s * G ) is divergent_to-infty ;
consider l being Nat such that m = F . l ;
t8 does not destroy b1 = b1 & t8 does not destroy b1 ;
reconsider Y1 = X1 , Y2 = X2 as SubSpace of X ;
consider w such that w in F and not x <= w ;
let a , b , c , d be Real ;
reconsider i = i , j = j as non zero Element of NAT ;
c . x >= id ( L ) . x ;
\sigma ( T ) \/ omega ( T ) is Basis of T ;
for x being element st x in X holds x in Y
cluster [ x1 , x2 ] -> pair for element ;
downarrow a /\ downarrow t is Ideal of T & downarrow t is Ideal of T ;
let X be set , NAT , { 0 } , { 1 } , { 2 } , { 3 } , { 4 , 5 } , { 6 , 6 , 7 , 8 } , 8 , 8 , 8
rng f = being Element of ` ( S , X ) ;
let p be Element of B , the carrier' of S ;
max ( N1 , 2 ) >= N1 & max ( N1 , 2 ) >= N2 ;
0. X <= b |^ ( m * mm1 ) ;
assume that i in I and R1 . i = R ;
i = j1 & p1 = q1 & p2 = q2 implies p1 = p2
assume gR in the right of g & gR in the carrier of g ;
let A1 , A2 be Point of S , A be Subset of A1 ;
x in h " P /\ [#] T1 & x in h " P ;
1 in Seg 2 & 1 in Seg 3 implies 1 in Seg 3
reconsider X-5 = X , X, T, T, T, T, T, be Subset of Tsuch that Xelement = X ;
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the carrier' of G ) -defined ;
n1 <= i2 + len g2 & n2 + 1 <= len g2 ;
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume v in the carrier' of G2 & u in the carrier' of G1 ;
y = Re y + ( Im y * i ) ;
/ ( ( - 1 ) * p ) gcd p = 1 ;
x2 is_differentiable_on ]. a , b .[ & ( for x st x in ]. a , b .[ holds x = a ) implies x2 = a
rng M5 c= rng D2 & len M5 = len D2 ;
for p being Real st p in Z holds p >= a
( for x being Point of X holds f . x = proj1 . x ) implies f is continuous
( seq ^\ m ) . k <> 0 ;
s . ( G . ( k + 1 ) ) > x0 ;
( p -Path M ) . 2 = d ;
A \oplus ( B \ominus C ) = ( A \oplus B ) \ominus C
h \equiv gg . ( mod P ) , T -Ideal ( P ) ;
reconsider i1 = i-1 , i2 = s as Element of NAT ;
let v1 , v2 be VECTOR of V , v be VECTOR of V ;
mode Subspace of V is Subspace of [#] V ;
reconsider i-7 = i , im2 = j as Element of NAT ;
dom f c= [: C ( ) , D ( ) :] ;
x in ( the Sorts of B ) . n ;
len } _ { D } in Seg ( len f2 ) ;
pp1 c= the topology of T & pp2 c= the topology of T ;
]. r , s .] c= [. r , s .] ;
let B2 be Basis of T2 , B be Basis of T2 ;
G * ( B * A ) = ( B * o1 ) * A ;
assume that p , u , u be \vert and u , v , w , w is_collinear ;
[ z , z ] in union rng ( F | X ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S , $1 .. S ;
LIN a1 , a3 , b1 & LIN a1 , b1 , c1 ;
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 ) ;
IB * ( i , j ) = 0. K ;
|. f . ( s . m ) - g .| < g1 ;
( q . x ) in rng ( q . x ) ;
Carrier ( Lxy ) misses Carrier ( Lxy ) \/ Carrier ( LR2 ) ;
consider c being element such that [ a , c ] in G ;
assume that N_ = o_ ( o_ ( o , m ) ) ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= ( F |^ CZ ) " { 0 } ;
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [. f . j , f . ( j + 1 ) .] ;
pred 0 <= x & x <= 1 & x ^2 <= 1 ;
p `2 - q `2 <> 0. TOP-REAL 2 & p `2 - q `2 = q `2 ;
redefine func \cal a] ( S , T ) -> non empty set ;
let x be Element of [: S , T :] ;
the Arrows of F , F F is one-to-one ;
|. i .| <= - ( - 2 to_power n ) ;
the carrier of I[01] = dom P & the carrier of I[01] = the carrier of I[01] ;
} * ( n + 1 ) ! > 0 * th ;
S c= ( A1 /\ A2 ) /\ A3 & S /\ A2 c= A2 /\ A3 ;
a3 , a4 // b3 , b2 & a3 , a4 // b3 , a3 ;
then dom A <> {} & dom A <> {} & dom B <> {} ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , Y , G & y = [ x , y ] ;
set v2 = ( v /. ( i + 1 ) ) `1 ;
x = r . n .= r4 . n .= r4 . n ;
f . s in the carrier of S2 & g . s in the carrier of S2 ;
dom g = the carrier of I[01] & rng g = the carrier of I[01] ;
p in Upper_Arc ( P ) /\ Upper_Arc ( P ) ;
dom d2 = [: A2 , A2 :] & dom d2 = [: A2 , A1 :] ;
0 < ( p / ||. 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 -> being for operation of X ;
let U1 , U2 be non-empty MSAlgebra over S , f be Function of U1 , U2 ;
Proj ( i , n ) * g is_differentiable_on X ;
let x , y , z be Point of X , p be Point of p ;
reconsider p9 = p . x , q9 = q . x as Subset of V ;
x in the carrier of Lin ( A ) & x in the carrier of Lin ( B ) ;
let I , J be parahalting Program of SCM+FSA , a be Int-Location ;
assume that - a is lower and - b is Element of - a ;
Int Cl A c= Cl Int Cl A & Int Cl A c= Cl Int Cl A ;
assume for A being Subset of X holds Cl A = A ;
assume q in Ball ( |[ x , y ]| , r ) ;
p2 `2 <= p `2 & p `2 <= p1 `2 or p2 `2 >= p `2 ;
Cl Q ` = [#] ( TT ) .= [#] ( TT ) ;
set S = the carrier of T , T = the carrier of T ;
set I8 = for f be FinSequence of TOP-REAL n st f = [: f , f :] holds f is one-to-one ;
len p -' n = len p - n .= len p - n ;
A is Permutation of Swap ( A , x , y ) ;
reconsider n6 = n0. ( N , i ) as Element of NAT ;
1 <= j + 1 & j + 1 <= len ( s | k ) ;
let q\subseteq , q_ 1 , q_ 2 be Element of M ;
( a , b ) in the carrier of S1 & ( b , c ) in the carrier of S2 ;
c1 /. n1 = c1 . n1 & c2 /. n1 = c2 . n1 ;
let f be FinSequence of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
y = ( ( f * S8 ) . x ) ;
consider x being element such that x in be be element such that x in be an element ;
assume r in ( dist ( o ) ) .: P ;
set i2 = ( n , h ) .. h ;
h2 . ( j + 1 ) in rng h2 /\ rng h2 ;
Line ( M29 , k ) = M . i .= Line ( M29 , k ) ;
reconsider m = ( x - 1 ) / 2 as Element of ExtREAL ;
let U1 , U2 be strict Subspace of U0 , a be Element of U1 ;
set P = Line ( a , d ) ;
len p1 < len p2 + 1 & len p2 + 1 <= len p1 ;
let T1 , T2 be Scott topological +* of L , x be Element of T1 ;
then x <= y & ( x in : x in : y in { x } ) ;
set M = n -\in ( n -tuples_on NAT ) ;
reconsider i = x1 , j = x2 , k = x3 as Nat ;
rng ( the_arity_of o ) c= dom H & dom ( the_arity_of o ) = dom ( the_arity_of o ) ;
z1 " = z9 " & z2 " = z2 " & z1 = z2 " ;
x0 - r / 2 in L /\ dom f & x0 - r / 2 in dom f ;
then w is that rng w /\ L <> {} & S is non empty ;
set xx = ( x ^ <* Z *> ) | ( X \ Y ) ;
len w1 in Seg len w1 & len w2 in Seg len w1 ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of thesis , k be Element of PFuncs ( V , { k } ) ;
x . n = ( |. a . n .| ) to_power ( n + 1 ) ;
p `1 <= Gik `1 & p `1 <= G * ( 1 , 1 ) `1 ;
rng ( g | 1 ) c= L~ ( g | 1 ) ;
reconsider k = i-1 * ( l + j ) as Nat ;
for n being Nat holds F . n is \HM { -infty } ;
reconsider xx = xx , xx = xx , xx = xx , z1 = z1 as VECTOR of M ;
dom ( f | X ) = X /\ dom f .= X ;
p , a // p , c & b , a // c , c ;
reconsider x1 = x , y1 = y , y2 = z as Element of REAL m ;
assume i in dom ( a * p ^ q ) ;
m . ag = p . ag & m . bg = p . bg ;
a to_power ( s . m - 1 ) / ( n + 1 ) <= 1 ;
S . ( n + k + 1 ) c= S . ( n + k ) ;
assume that B1 \/ C1 = B2 \/ C2 and B2 \/ C2 = B2 \/ C1 ;
X . i = { x1 , x2 } . i .= X . i ;
r2 in dom ( h1 + h2 ) /\ dom ( h2 + h2 ) ;
- - 0. R = a & b-0 = b ;
( F is closed & F is closed & F is closed ) implies F is closed
set T = the { the InInof X , x0 } ;
Int Cl ( Int Cl R ) c= Int Cl R ;
consider y being Element of L such that c . y = x ;
rng ( F . x ) = { F . x } & F . x = F . x ;
G-23 ( { c } ) c= B \/ S ;
f[ X , Y ] ;
set RQ = the Point of P , RQ = the Point of Q ;
assume that n + 1 >= 1 and n + 1 <= len M ;
let k2 be Element of NAT , k be Nat ;
reconsider p/. i = u , pj = v as Element of ( TOP-REAL n ) | P ;
g . x in dom f & x in dom g implies f . x = g . x
assume that 1 <= n and n + 1 <= len f1 ;
reconsider T = b * N as Element of carr ( N ) ;
len Ph <= len P-35 & len Ph <= len P-35 ;
x " in the carrier of A1 & x " in the carrier of A2 ;
[ i , j ] in Indices ( A * N ) ;
for m being Nat holds Re ( F . m ) is simple Function of S , S
f . x = a . i .= a1 . k .= a1 . k ;
let f be PartFunc of REAL i , REAL n , x0 be Element of REAL m ;
rng f = the carrier of ( Carrier A ) & f . x = f . x ;
assume s1 = sqrt ( ( p - r ) ^2 - ( p - r ) ^2 ) ;
pred a > 1 & b > 0 & a to_power b > 1 ;
let A , B , C be Subset of IQ ;
reconsider X0 = X , Y0 = Y , Y0 = Z as RealNormSpace ;
let f be PartFunc of REAL , REAL , x0 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 , tt2 be binary set ;
Q [ e-14 \/ { vffc } , f . vec ] ;
g :- W-min L~ z = z implies ( g /. 1 ) .. z < ( g /. len z ) .. z
|. |[ x , v ]| - |[ x , y ]| .| = vfunction ;
- f . w = - ( L * w ) ;
z - y <= x iff z <= x + y & y <= z ;
( 7 / p1 ) to_power ( 1 / e ) > 0 ;
assume X is BCK-algebra of 0 , 0 , 0 , 0 , 0 , 0 ;
F . 1 = v1 & F . 2 = v2 & F . 3 = v1 ;
( f | X ) . x2 = f . x2 & ( f | X ) . x2 = f . x2 ;
( ( tan * sec ) `| Z ) . x in dom ( sec * sec ) ;
i2 = ( f /. len f ) `2 .= ( f /. len f ) `2 ;
X1 = X2 \/ ( X1 \ X2 ) & X2 \ X1 = X2 \ X1 ;
[. a , b , 1_ G .] = 1_ G & 1_ G = 1_ G ;
let V , W be non empty VectSpStr over F_Complex , f be FinSequence of V ;
dom g2 = the carrier of I[01] & rng g2 = the carrier of I[01] ;
dom f2 = the carrier of I[01] & rng g2 = the carrier of I[01] ;
( proj2 | X ) .: X = proj2 .: X .= proj2 .: X ;
f . ( x , y ) = h1 . ( x `1 , y `2 ) ;
x0 - r < a1 . n & a1 . n < x0 + r ;
|. ( f /* s ) . k - GM .| < r ;
len Line ( A , i ) = width A & len Line ( A , i ) = width A ;
Sof [: S , T :] = ( S . g ) |^ the carrier of S ;
reconsider f = v + u as Function of X , the carrier of Y ;
intloc 0 in dom Initialized p & intloc 0 in dom Initialized p ;
i1 -' i2 + ( i3 + 1 ) does not destroy b1 & i2 does not destroy b1 ;
arccos r + arccos r = PI / 2 + 0 ;
for x st x in Z holds f2 is_differentiable_in x & f2 is_differentiable_in x & f2 . x > 0 ;
reconsider q2 = ( q - x ) / ( 1 + x ) as Element of REAL ;
( 0 qua Nat ) + 1 <= i + j1 & ( 0 qua Nat ) + 1 <= len ( 0 qua Nat ) ;
assume f in the carrier of [: X , Omega Y :] ;
F . a = H / ( x , y ) . a ;
( ( TRUE T ) at ( C , u ) ) . x = TRUE ;
dist ( ( a * seq ) . n , h ) < r ;
1 in the carrier of [. 0 , 1 .] & 1 in the carrier of I[01] ;
p2 `1 - x1 > - g & p2 `1 - x1 < p2 `1 - g ;
|. r1 - thesis .| = |. a1 .| * |. thesis .| ;
reconsider S-14 k = 8 as Element of Seg 8 ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b
D0W .= D0W .3 + 1 ;
i1 = ma + n & i2 = K & i1 = i2 + n ;
f . a [= f . ( f to_power O1 ) ;
pred f = v & g = u , v = v + u ;
I . n = Integral ( M , F . n ) ;
chi ( T1 , S ) . s = 1 & chi ( T2 , S ) . s = 1 ;
a = VERUM ( A ) or a = VERUM ( A ) ;
reconsider k2 = s . b3 , k2 = s . b3 as Element of NAT ;
( Comput ( P , s , 4 ) ) . GBP = 0 ;
L~ M1 meets L~ R4 implies ( M1 \/ M2 ) * ( i , j ) in L~ ( M1 \/ M2 )
set h = the continuous Function of X , R , g = the carrier of X ;
set A = { L . ( k9 . n ) where k9 is Element of NAT : not contradiction } ;
for H st H is atomic holds P [ H ] ;
set bA = S5 ^\ ( iA + 1 ) , S5 = S5 ^\ ( iA + 1 ) ;
Hom ( a , b ) c= Hom ( a opp , b opp ) ;
1 / ( n + 1 ) < ( 1 / s ) " ;
l `1 = [ [ dom l , cod l ] , [ cod l , cod l ] ] ;
y +* ( i , y /. i ) in dom g ;
let p be Element of CQC-WFF ( Al ( ) ) , x be Element of CQC-WFF ( Al ( ) ) ;
X /\ X1 c= dom ( f1 - f2 ) /\ dom ( f2 - f1 ) ;
p2 in rng ( f /^ 1 ) & p2 in rng ( f /^ 1 ) ;
1 <= indx ( D2 , D1 , j1 ) + 1 ;
assume x in ( ( ( ( ( K /\ K0 ) \/ K0 ) /\ K0 ) /\ K0 ) /\ K0 ;
- 1 <= ( ( f2 ) . O ) `2 & - 1 <= ( ( f2 ) . I ) `2 ;
let f , g be Function of I[01] , TOP-REAL 2 , a , b be Real ;
k1 -' k2 = k1 - k2 & k2 -' k1 = k2 - k2 ;
rng seq c= ]. x0 , x0 + r .[ & rng seq c= ]. x0 , x0 + r .[ ;
g2 in ]. x0 - r , x0 + r .[ & g2 in ]. x0 - r , 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 the the of T is the normal sequence of T st a = Sum A ;
Cl ( union HH ) = union ( ) & union ( rng ( H ) ) = union ( rng H ) ;
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 = sup ( d " { s } ) & g . s = sup ( d " { s } ) ;
( \dot { y } ) . s = s . ( \dot { y } . s ) ;
{ s : s < t } in REAL implies t = {} or t = {}
s ` \ s = s ` \ 0. X & s ` \ s = 0. X ;
defpred P [ Nat ] means B + $1 in A & B . $1 = A ;
( 339 + 1 ) ! = 3339 ! * ( 339 + 1 ) ;
U ( succ A ) = ( T . ( 1_ A ) ) * ( 1_ ( A ) ) ;
reconsider y = y , z = z as Element of COMPLEX * ;
consider i2 being Integer such that y0 = p * i2 and i2 <= n ;
reconsider p = Y | Seg k , q = Y | Seg k as FinSequence of NAT ;
set f = ( S , U ) i -] ( S , U ) ;
consider Z being set such that lim s in Z and Z in F ;
let f be Function of I[01] , TOP-REAL n , p1 , p2 be Point of TOP-REAL n ;
( ( M M ) . [ n + i , 'not' A ] ) <> 1 ;
ex r being Real st x = r & a <= r & r <= b ;
let R1 , R2 be Element of REAL n , x be Element of REAL n ;
reconsider l = 0. ( V ) , r = 0. ( A ) , s = 0. ( A ) ;
set r = |. e .| + |. n .| + |. w .| + a ;
consider y being Element of S such that z <= y and y in X ;
a is being being being being being being being being being set holds ( b 'or' c ) = 'not' ( ( a 'or' b ) 'or' c )
||. ( x - g ) .|| < r2 & ||. ( x - y ) - g .|| < r2 ;
b9 , a9 // b9 , c9 & b9 , c9 // c9 , a9 & b9 , c9 // c9 , a9 ;
1 <= k2 -' k1 & k1 + 1 = k2 & k2 + 1 = k2 + 1 ;
( p `2 / |. p .| - sn ) >= 0 & ( p `2 / |. p .| - sn ) >= 0 ;
( q `2 / |. q .| - sn ) / ( 1 + sn ) < 0 ;
E-max C in RightComp ( R ) & E-max L~ Cage ( C , 1 ) in rng ( R | 1 ) ;
consider e being Element of NAT such that a = 2 * e + 1 ;
Re ( ( lim F ) | D ) = Re ( ( lim G ) | D ) ;
LIN b , a , c or LIN b , c , a ;
p , a // a , b or p , a // b , a ;
g . n = a * Sum fk1 .= f . n ;
consider f being Subset of X such that e = f and f is being being being being Subset of X ;
F | ( N2 , S ) = CircleMap * ( F | N2 ) ;
q in LSeg ( q , v ) \/ LSeg ( v , p ) ;
Ball ( m , r0 ) c= Ball ( m , s ) & Ball ( m , r0 ) c= Ball ( m , s ) ;
the carrier of (0). V = { 0. V } & the carrier of (0). V = { 0. V } ;
rng ( cos | [. - 1 , 1 .] ) = [. - 1 , 1 .] ;
assume that Re seq is summable and Im seq is summable and Im seq is summable ;
||. ( vseq . n ) - ( vseq . n ) .|| < e / 2 ;
set g = O --> 1 ;
reconsider t2 = t11 , t2 = 0 as string of S2 , I = the carrier of S2 ;
reconsider xx = seq . n , xx = seq . ( n + 1 ) as sequence of REAL n ;
assume that E-max C meets L~ go and E-max C in L~ pion1 and E-max C in L~ pion1 ;
- ( Cl 1 ) < F . n - x & F . x < - ( F . n ) ;
set d1 = \bf dist ( x1 , z1 ) , d2 = dist ( x2 , z2 ) , d1 = dist ( x2 , y2 ) ;
2 |^ ( q -' 1 ) = 2 |^ ( q -' -' 00 ) - 1 ;
dom ( v . k ) = Seg ( len d6 ) .= dom ( v . k ) ;
set x1 = - k2 + |. k2 .| + 4 , x2 = - k2 + 1 ;
assume for n being Element of X holds 0. <= F . n & 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 L1 c= I2 ;
'not' Ex ( x , p ) => All ( x , 'not' p ) is valid ;
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the normal normal \hbox { - } over {} ;
Z c= dom ( ( sin * f1 ) `| Z ) /\ dom ( ( cos * f1 ) `| Z ) ;
|. 0. TOP-REAL 2 - ( q `1 / |. q .| - cn ) .| < r ;
A \ ^2 c= A & A c= A implies A in \bf L ( A , succ d )
E = dom Carrier ( L ) & L is measurable implies dom ( L (#) F ) = E
C |^ ( A + B ) = C |^ B * C |^ A ;
the carrier of W2 c= the carrier of V & the carrier of W1 c= the carrier of V ;
I . IC ss2 = P . IC ss2 .= ( card I + 2 ) ;
pred x > 0 means : Def8 : x = x to_power ( - 1 ) ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) ;
consider p being Point of T such that C = [. p , q .] ;
b , c are_connected & - C , - C + D + D + E + F + G + H + N + E + F + J + M + N + F + N + F + J + M + N + F + N + F + J + M + N + E + F
assume that f = id the carrier of OO and g = id the carrier of OO ;
consider v such that v <> 0. V & f . v = L * v ;
let l be \rm Linear_Combination of {} ( the carrier of V ) ;
reconsider g = f " as Function of U2 , U1 , U2 ;
A1 in the points of G_ ( k , X ) & A2 in the carrier of G ;
|. - x .| = - ( - x ) .= - x .= - x ;
set S = is x0 , y , c ;
Fib ( n ) * ( 5 * Fib ( n ) -2 ) >= 4 * be Nat ;
( v /. ( k + 1 ) ) = ( v . ( k + 1 ) ) ;
0 mod i = - ( i * ( 0 qua Nat ) ) .= - ( i + 1 ) ;
Indices M1 = [: Seg n , Seg n :] & Indices M1 = [: Seg n , Seg n :] ;
Line ( Sd , j ) = Sd . j .= S . j ;
h . ( x1 , y1 ) = [ y1 , x1 ] & h . ( y1 , y2 ) = [ y2 , y1 ] ;
|. f .| - Re ( |. f .| * h ) is nonnegative ;
assume that x = ( a1 ^ <* x1 *> ) ^ b1 and y = a1 ^ b1 ;
ME is_closed_on IExec ( I , P , s ) , P & ME is_halting_on IExec ( I , P , s ) , P ;
DataLoc ( t3 . a , 4 ) = intpos ( 0 + 4 ) .= 0 ;
x + y < - x + y & |. x .| = - x + y ;
LIN c , q , b & LIN c , q , c & LIN c , q , c ;
fs2 . ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + z1 ) .= x1 + y1 ;
f' . a = f{ a } & v in InputVertices S & [ v , w ] in InputVertices S ;
p `1 <= ( E-max C ) `1 & ( E-max C ) `1 <= ( E-max C ) `1 ;
set R8 = Cage ( C , n ) :- E8 , E7 = Cage ( C , n ) ;
p `1 >= ( E-max C ) `1 & p `2 >= ( E-max C ) `2 ;
consider p such that p = p-20 and s1 < p /. i and p in L~ f ;
|. ( f /* ( s * F ) ) . l - GM .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len Line ( N , k + 1 + 1 ) = width N ;
f1 /* s1 is convergent & f2 /* s1 is convergent & lim ( f1 /* s1 ) = lim ( f2 , x0 ) ;
f . x1 = x1 & f . y1 = y1 & f . y2 = y2 ;
len f <= len f + 1 & len f + 1 <> 0 implies f . ( len f + 1 ) = f . ( len f + 1 )
dom ( Proj ( i , n ) * s ) = REAL m .= dom ( Proj ( i , n ) * s ) ;
n = k * ( 2 * t ) + ( n mod ( 2 * k ) ) ;
dom B = 2 -tuples_on the carrier of V , the carrier of V , the carrier of V ;
consider r such that r _|_ a and r _|_ x and r _|_ y ;
reconsider B1 = the carrier of Y1 , B2 = the carrier of Y2 as Subset of X ;
1 in the carrier of [. 1 / 2 , 1 .] & 1 / 2 * PI <= 1 ;
for L being complete LATTICE holds L is isomorphic implies L , L are_isomorphic
[ gi , gj ] in II \ II ~ & [ gi , gj ] in II ;
set S2 = 1GateCircStr ( x , y , c ) ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and for r st x0 < r ex g st g < x0 & g < x0 & g in dom ( f2 * f1 ) ;
reconsider y = ( a ` ) ` , z = ( a ` ) ` , t = ( a ` ) ` as Element of L ;
dom s = { 1 , 2 , 3 } & s . 1 = d1 & s . 2 = d2 ;
( min ( g , ( 1 - 1 ) ) ) . c <= h . c ;
set G3 = the K of G , v = the Vertex of G , w = the Vertex of G , e = the Element of G ;
reconsider g = f as PartFunc of REAL n , REAL-NS n , x be Point of REAL-NS n ;
|. s1 . m / p .| < d to_power p / p & |. s1 . m .| < d to_power p ;
for x being element st x in ( for t being element st t in ( q \ u ) holds x in ( q \ u ) )
P = the carrier of ( TOP-REAL n ) | [: P , P :] & P = the carrier of ( TOP-REAL n ) | [: P , P :] ;
assume that p00 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) and p2 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ;
( 0. X \ x ) to_power ( m * ( k + 1 ) ) = 0. X ;
let g be Element of Hom ( cod f , @ f ) ;
2 * a * b + ( 2 * c * d ) <= 2 * C1 * C2 ;
let f , g , h be Point of the complex normed space of bounded functions of X , Y , h be Function of X , Y ;
set h = Hom ( a , g (*) f ) ;
then idseq ( n ) | Seg m = idseq ( m ) | Seg ( n ) & m <= n ;
H * ( g " * a ) in the right of H & H * ( g " * a ) in the carrier of H ;
x in dom ( ( - cos * sin ) `| Z ) & x - PI / ( sin . x ) ^2 < sin . x ;
cell ( G , i1 , j2 -' 1 ) misses C & cell ( G , i1 , j2 -' 1 ) misses C ;
LE q2 , p4 , P , p1 , p2 & LE q2 , p2 , P , p1 , p2 ;
attr B is component means : U : B c= BDD A ;
deffunc D ( set , set ) = union rng $2 & $2 = union rng $2 ;
n + - n < len ( ( p + - n ) * ( p + - n ) ) ;
pred a <> 0. K means : DefK : the_rank_of M = the_rank_of ( a * M ) ;
consider j such that j in dom \mathbb /\ dom ' and I = len ' + j ;
consider x1 such that z in x1 and x1 in ( P \ { x1 } ) and x = [ x1 , x1 ] ;
for n ex r being Element of REAL st X [ n , r ]
set CP1 = Comput ( P2 , s2 , i + 1 ) , CP2 = Comput ( P2 , s2 , i + 1 ) ;
set cv = 3 / ( 2 * PI ) , cw = 2 * PI , cw = 3 / ( 2 * PI ) ;
conv @ W c= union ( F .: ( E " ) ) & conv @ W c= union ( F .: ( E " ) ) ;
1 in [. - 1 , 1 .] /\ dom ( arccot * ( arccot ) ) ;
r3 <= s0 + ( r0 / |. v2 - v1 .| - 1 ) * ( v2 - v1 ) ;
dom ( f * f4 ) = dom f /\ dom f4 .= dom f4 /\ dom f4 .= dom f4 /\ dom f4 ;
dom ( f (#) G ) = dom ( l (#) F ) /\ Seg k .= Seg k ;
rng ( s ^\ k ) c= dom f1 \ { x0 } & rng ( s ^\ k ) c= dom f2 \ { x0 } ;
reconsider gg = gp , gq = gq , gr = gr as Point of Euclid n1 ;
( T * h . s ) . x = T . ( h . s . x ) ;
I . ( L . J . x ) = ( I * L ) . ( J . x ) ;
y in dom being `1 e e -commute ( commute ( A . o ) ) ;
for I being non degenerated integral of R holds the carrier of I is commutative doubleLoopStr
set s2 = s +* Initialize ( ( intloc 0 ) .--> 1 ) , P2 = P +* Initialize ( ( intloc 0 ) .--> 1 ) ;
P1 /. IC s1 = P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 ;
lim S1 in the carrier of [. a , b .] & lim S1 = a * lim S2 ;
v . ( l-13 . i ) = ( v *' l.| ) . i ;
consider n be 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 ;
{} ( X , 0 , x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x1 , x2 , x3 , x4 , x5 , x5 , x1 , x2 , x3 , x4 ) = X ;
j + ( 2 * kk ) + m1 > j + ( 2 * kk ) + m1 ;
{ s , t } on A3 & { s , t } on B2 & { s , t } on B2 ;
n1 > len crossover ( p2 , p1 , n1 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , \geq len crossover ( p2 , p1 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 ) ;
mg . HT ( mg , T ) = 0. L & mg . HT ( mg , T ) = 0. L ;
then H1 , H2 are_) & ( H , H1 ) / 2 , ( H , H2 ) / 2 ;
( N-min L~ f ) .. ( f | ( L~ f ) ) > 1 & ( N-min L~ f ) .. ( f | ( L~ f ) ) > 1 ;
]. s , 1 .] = ]. s , 2 .] /\ [. 0 , 1 .] .= ]. 0 , 1 .] ;
x1 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) ) & x2 in ( L~ g ) /\ ( L~ g ) ;
let f1 , f2 be continuous PartFunc of REAL , the carrier of S , the carrier of T ;
DigA ( t-23 , z ) is Element of k -tuples_on ( the carrier of G ) ;
I is d2\rm \overline k1 & I is I I I \circ k2 = k2 implies I is k2
[: u , { u9 } :] = { [ a , u9 ] , [ b , u9 ] } ;
( w | p ) | ( p | ( w | w ) ) = p ;
consider u2 such that u2 in W2 and x = v + u2 and u2 in W2 and u1 in W3 ;
for y st y in rng F ex n st y = a |^ n & P [ y ] ;
dom ( ( g * ( f , V ) ) | K ) = K ;
ex x being element st x in ( ( the Sorts of U0 ) \/ A ) . s & x in ( the Sorts of U0 ) . s ;
ex x being element st x in ( ( and OO ( A ) ) . s ) & x in ( the Sorts of U1 ) . s ;
f . x in the carrier of [. - r , r .] & f . x = - 1 ;
( the carrier of X1 union X2 ) /\ ( the carrier of X1 ) <> {} & ( the carrier of X1 union X2 ) /\ ( the carrier of X2 ) <> {} ;
L1 /\ LSeg ( p00 , p2 ) c= { p10 } /\ LSeg ( p1 , p2 ) ;
( b + b- s ) / 2 in { r : a < r & r < b } ;
ex_sup_of { x , y } , L & x "\/" y = sup { x , y } ;
for x being element st x in X ex u being element st P [ x , u ]
consider z being Point of GX such that z = y and P [ z ] and z in A ;
( the sequence of ( the carrier of ( the carrier of X ) ) ) . ( `2 ) <= e ;
len ( w ^ w2 ) + 1 = len w + 2 + 1 .= len w + 1 ;
assume that q in the carrier of ( TOP-REAL 2 ) | K1 and q `2 = 0 and q <> 0. TOP-REAL 2 ;
f | EK1 ` = g | EK1 ` & g | EK1 ` = g | EK1 ` ;
reconsider i1 = x1 , i2 = x2 , j1 = y2 , j2 = y1 , j1 = y2 as Element of NAT ;
( 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 support of f2 ) * ( f | i ) ) = dom ( ( the support of f2 ) * ( f | i ) ) ;
( Complement ( A . m ) ) . n c= ( Complement ( A . n ) ) . ( n + 1 ) ;
f1 . p = p9 & g1 . ( p9 , q9 ) = d & g1 . ( p9 , q9 ) = d ;
FinS ( F , Y ) = FinS ( F , dom ( F | Y ) ) ;
( x | y ) | z = z | ( y | x ) ;
( |. x .| to_power n ) / ( ( 2 |^ n ) * ( n + 1 ) ) <= ( r2 |^ n ) / ( ( 2 |^ n ) * ( n + 1 ) ) ;
Sum ( F ) = Sum f & dom ( F ) = dom g & for i st i in dom F holds F . i = f . i ;
assume for x , y being set st x in Y & y in Y holds x /\ y in Y ;
assume that W1 is Subspace of W3 and W2 is Subspace of W3 and W3 is Subspace of W3 and W3 is Subspace of W3 ;
||. ( t-15 . x ) - ( t_ . x ) .|| = lim ||. ( x - y ) - ( x - y ) .|| ;
assume that i in dom D and f | A is lower and g | A is lower ;
( ( p `2 ) ^2 - 1 ) * ( - 1 ) <= ( - ( - ( - ( p `2 / |. p .| - 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 ) , N8 = N-min L~ Cage ( C , n ) , N8 = \cal L ;
for T being non empty TopSpace holds T is countable countable implies the TopStruct of T is countable countable
width B |-> 0. K = Line ( B , i ) .= B * ( i , i ) ;
pred a <> 0 means : |. ( A \+\ B ) Let a = ( A carrier a ) \+\ ( B carrier a ) ;
then f is_\mathbin { \frac 2 } u , 3 , 1 ) is_partial_differentiable_in u , 3 ;
assume that a > 0 and a <> 1 and b > 0 and b <> 0 and c > 0 ;
w1 , w2 in Lin { w1 , w2 } & w2 in Lin { w2 , w1 } ;
p2 /. IC s-7 = p2 . IC s-7 .= p2 . IC sm2 .= ( p2 + 1 ) + 1 ;
ind ( T-10 | b ) = ind b .= ind B - 1 .= ind B - 1 ;
[ a , A ] in the \cdot of ( \hbox { - } line ) & [ a , A ] in the \cdot of ( the on of K ) ;
m in ( the Arrows of C ) . ( o1 , o2 ) & m in ( the Arrows of C ) . ( o1 , o2 ) ;
( ( a ) 'imp' ( CompF ( PA , G ) ) ) . z = FALSE ;
reconsider phi = phi /. 11 , phi = phi /. 22 , phi = phi /. 2 , phi = phi /. 3 , phi = phi /. 3 , phi = phi /. 1 , phi = phi /. 2 , phi = phi /. 3 , phi = phi /. 3 , phi = phi /. 3 , phi = phi /. 1 , phi = phi /. 3 , phi =
len s1 - 1 * ( len s2 - 1 ) + 1 > 0 + 1 ;
delta ( D ) * ( f . ( upper_bound A ) - f . ( lower_bound A ) ) < r ;
[ f21 , f22 ] in [: the carrier' of A , the carrier' of B :] ;
the carrier of ( TOP-REAL 2 ) | K1 = K1 & the carrier of ( TOP-REAL 2 ) | K1 = K1 ;
consider z being element such that z in dom g2 and p = g2 . z and g2 . z = x ;
[#] V1 = { 0. V1 } .= the carrier of (0). V1 .= the carrier of (0). V1 .= the carrier of (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 . 1 = P1 . len P2 ;
assume that x1 in dom ( f | X ) and ||. x1 - x0 .|| < s and |. x1 - x0 .| < s ;
h1 = f ^ ( <* p3 *> ^ <* p *> ) .= h ^ <* p *> .= h ;
c /. ( |[ b , c ]| ) = c .= c /. ( |[ a , c ]| ) .= c /. ( |[ a , c ]| ) ;
reconsider t1 = p1 , t2 = p2 , t2 = p3 , t1 = p1 , t2 = p2 as term of C ;
1 / 2 in the carrier of [. 1 / 2 , 1 .] & 1 / 2 in the carrier of I[01] ;
ex W being Subset of X st p in W & W is open & h .: W c= V ;
( h . p1 ) `2 = C * ( p1 `2 + D ) `2 .= C * ( p1 `2 + D ) `2 ;
R . b ` = 2 * - a-b .= 2 * a-b .= a-b ;
consider \vert - \vert 1 .| such that B = - 1 * x1 + ( - 1 ) * A and 0 <= \vert 1 .| ;
dom g = dom ( ( the Sorts of A ) * ( a , I ) ) .= dom ( ( the Sorts of A ) * ( a , I ) ) ;
[ P . ( l ) , P . ( l + 1 ) ] in ( the carrier of TQ ) \ { {} } ;
set s2 = Initialize s , P2 = P +* I ;
reconsider M = mid ( z , i2 , i1 ) , N = ( GoB z ) * ( i1 , i2 ) as non empty Subset of REAL 2 ;
y in product ( ( Carrier J ) +* ( V , { 1 } ) ) ;
1 / ( |[ 0 , 1 ]| ) = 1 & 0 / ( |[ 0 , 1 ]| ) = 0 ;
assume x in the left of g or x in the right of g & x = the right of g ;
consider M being strict Subspace of AJ such that a = M and T is Subgroup of M and M is strict Subgroup of A ;
for x st x in Z holds ( ( #Z 2 ) * f ) . x <> 0 ;
len W1 + len W2 + m = 1 + len W3 + m .= len W3 + len W3 + m .= len W3 + m + m ;
reconsider h1 = ( vseq . n ) - ( t-16 . n ) as Lipschitzian LinearOperator of X , Y ;
( - ( i mod len ( p + q ) ) + 1 ) in dom ( p + q ) ;
assume that s2 is_for s1 , F , s2 such that s1 in the { s } and s2 in the { s } and s1 <> s2 ;
( ( ( ( ( q - x ) / ( 3 * p ) ) * ( q - x ) ) / ( 3 * p ) ) = ( ( x - y ) / ( 3 * p ) ) * ( q - x ) ;
for u being element st u in Bags n holds ( p `2 + m ) . u = p . u
for B being Subset of uelement st B in E holds A = B or A misses B ;
ex a being Point of X st a in A & A /\ Cl { y } = { a } ;
set W2 = tree ( p ) \/ W1 , W3 = tree ( q ) ;
x in { X where X is Ideal of L |^ the carrier of L : X is non empty Subset of L } ;
the carrier of W1 /\ W2 c= the carrier of W1 & the carrier of W1 /\ W2 c= the carrier of W2 ;
( for a , b being Element of L holds a * id a = ( a + b ) * ( id a ) )
( ( X --> f ) . x ) = ( X --> dom f ) . x .= f . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) ;
p => ( q => r ) => ( p => q => ( p => r ) ) 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 & d on P & c on P & d on P ;
reconsider gf = g `1 * f `2 , hg = h `2 * g `2 as strict Element of X ;
consider v1 being Element of T such that Q = ( downarrow v1 ) ` and v1 in V & v2 in V ;
n in { i where i is Nat : i < n0 + 1 & i <= n0 + 1 } ;
( F * ( i , j ) `2 ) `2 >= ( F * ( m , k ) `2 ) `2 ;
assume K1 = { p : p `1 >= sn * |. p .| & p `2 <= 0 & p <> 0. TOP-REAL 2 } ;
ConsecutiveSet ( A , succ O1 ) = ( ( A , O1 ) --> ( A , O1 ) ) ^ ( A , O1 ) ;
set Ii1 = Macro SubFrom ( a , intloc 0 ) , Ii2 = SubFrom ( a , intloc 0 ) , Ii2 = SubFrom ( a , intloc 0 ) , Ii2 = SubFrom ( a , intloc 0 ) ;
for i be Nat st 1 < i & i < len z holds z /. i <> z /. 1 ;
X c= ( the carrier of L1 ) /\ ( the carrier of L2 ) & X c= the carrier of L1 & X c= the carrier of L2 ;
consider xx be Element of GF ( p ) such that ( x |^ 2 ) = a & ( x |^ 2 ) |^ 2 = x ;
reconsider ee = e4 , fe = f-5 , fe = f-5 , fe = f/ ( 2 * n ) as Element of D ;
ex O being set st O in S & C1 c= O & M . O = 0. <= M . ( union R ) ;
consider n being Nat such that for m being Nat st n <= m holds S . m in U1 and S . n in U2 ;
f (#) g * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) . x ) ;
defpred P [ Nat ] means A + succ $1 = succ A + $1 & A in dom ( A + $1 ) ;
the left of - g = the left of - g & the left of - g = the left of - g ;
reconsider pM = x , pM = y , pM = z , pM = w , pM = z as Point of Euclid 2 ;
consider g2 such that g2 = y and x <= g2 and g2 <= x0 and g2 <= x0 and g2 <= x0 ;
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 positive set of n0 & x = ( p - r ) * ( n - 1 )
LSeg ( p11 , p2 ) /\ LSeg ( p1 , p2 ) = {} & LSeg ( p1 , p2 ) /\ LSeg ( p2 , p2 ) = {} ;
func such s ( X ) -> set means : Defh\bf h in it & for x holds x in \mathop { h . x } ;
len ( ( C /. ( len C -' 1 ) ) | ( len C -' 1 ) ) <= len ( C /. ( len C -' 1 ) ) ;
attr K is with_a , a , b be K , i be Element of K holds v . ( a |^ i ) = i * v . a ;
consider o being OperSymbol of S such that t `2 . {} = [ o , the carrier of S ] -tree p and o in the carrier' of S ;
for x st x in X ex y st x c= y & y in X & y is - x holds f . x = f . y
IC Comput ( P-6 , s5 , k ) in dom ( s5 +* I ) & IC Comput ( P5 , s5 , k ) in dom I ;
pred q < s means : Def8 : r < s & ]. r , s .[ \not c= ]. p , q .[ ;
consider c being Element of Class f such that Y = ( F . c ) `1 and [ x , y ] in R ;
func the ResultSort of S2 -> id the carrier' of S2 means : Def6 : the ResultSort of it = id the carrier' of S2 & the ResultSort of it = id the carrier' of S2 ;
set y9 = [ <* y , z *> , f2 ] ;
assume x in dom ( ( ( #Z 2 ) * ( arccot ) ) `| Z ) & x in dom ( ( #Z 2 ) * ( arccot ) ) ;
r-7 in Int cell ( GoB f , i , j ) \ L~ f & ri2 in L~ f implies ri2 in L~ f
q `2 >= ( ( Cage ( C , n ) /. ( i + 1 ) ) `2 or q `2 >= ( ( Cage ( C , n ) /. ( i + 1 ) ) `2 ;
set Y = { a "/\" a ` : a in X } ;
i - len f <= len f + len f1 - len f & len f + 1 - len f <= len f - len f ;
for n ex x st x in N & x in N1 & h . n = x- ( x0 )
set ss0 = ( \mathop { \it false } ( a , I , p , s ) ) . i ;
p ( ) . k = 1 or p ( ) . k = - 1 & p ( ) . 0 = 1 or p ( ) . 1 = - 1 ;
u + Sum L-18 in ( U \ { u } ) \/ { u + Sum L-18 } ;
consider xx being set such that x in xx and xx in Vd and x = [ xx , xx ] ;
( p ^ ( q | k ) ) . m = ( q | k ) . ( - len p ) ;
g + h = gg + hg1 & for x holds holds holds Nat ( g + h , X , Y ) = 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 ) means : Def8 : f . x = f . y ;
assume that 1 < p and 1 / p + 1 / q = 1 and 0 <= a and 0 <= b and b <= 1 ;
F* ( f , 6 ) = rpoly ( 1 , t ) *' t + 1. F_Complex .= 1 + 1. F_Complex .= 1 ;
for X being set , A being Subset of X holds A ` = {} implies A = X & A = {} or A = {}
( ( N-min X ) `1 <= ( ( N-min X ) `1 ) & ( ( N-min X ) `2 <= ( ( E-max X ) `2 ) `2 ) ;
for c being Element of the \geq the Sorts of A , a being Element of the free of A holds c <> a ;
s1 . GBP = ( Exec ( i2 , s2 ) ) . GBP .= s2 . GBP .= 0 ;
for a , b being Real holds |[ a , b ]| in ( y >= 0 ) -plane implies b >= 0 & a = 0
for x , y being Element of X holds x ` \ y = ( x \ y ) ` & y \ x = y \ x implies x = y
mode BCK-algebra of i , j , m , n , m , n being BCK-algebra of i , j , m , n , m , m ;
set x2 = |( Re y , Im ( y ) )| ;
[ y , x ] in dom u5 & u5 . ( y , x ) = g . y & u5 . ( y , x ) = g . y ;
]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .[ c= A & upper_bound divset ( D , k ) = upper_bound A ;
0 <= delta ( S2 . n ) & |. delta ( S2 . n ) .| < ( e / 2 ) / 2 ;
( - ( q `1 / |. q .| - cn ) ) ^2 <= ( - ( q `1 / |. q .| - cn ) ) ^2 ;
set A = 2 / b-a ;
for x , y being set st x in R" holds x , y are_\hbox { - } x , y }
deffunc FF2 ( Nat ) = b . $1 * ( M * G ) . $1 & ( M * G ) . $1 = ( M * G ) . $1 ;
for s being element holds s in -> element holds s in ( M 'or' g ) /\ ( ( M \/ { s } ) /\ \rm )
for S being non empty non void non void holds S is connected iff S is connected ;
max ( degree ( z `1 ) , degree ( z `2 ) ) >= 0 & max ( degree ( z `1 ) , degree ( z `2 ) ) >= 0 ;
consider n1 being Nat such that for k holds seq . ( n1 + k ) < r + s and for n holds seq . ( n + k ) < r + s ;
Lin ( A /\ B ) is Subspace of Lin ( A ) & Lin ( B ) is Subspace of Lin ( A ) ;
set n-15 = np2 '&' ( M . x qua Element of BOOLEAN ) , np2 = M . ( x , y ) , np2 = M . ( x , y ) ;
f " V in the topology of X & f " V in D & f " V in D & f " V in D & f " V in D ;
rng ( ( a the set of c ) +* ( 1 , b ) ) c= { a , c , b } ;
consider y being such that y `1 = y and dom y `1 = WWthesis & y `2 = WW
dom ( 1 / f ) /\ ]. - 1 , 1 .[ c= ]. - 1 , 1 .[ & dom ( 1 / f ) /\ ]. - 1 , 1 .[ c= dom ( 1 / f ) ;
f2 is Morphism of i , j , n , r be Morphism of F , j , - r , r be Element of REAL ;
v ^ ( n-3 |-> 0 ) in Lin ( ( B | c1 ) ^ ( B | c2 ) ) & v ^ ( B | c2 ) = v ;
ex a , k1 , k2 st i = a := k1 & k2 = b := k2 & k1 in dom f & k2 in dom f ;
t . NAT = ( NAT .--> succ i1 ) . NAT .= succ ( 5 + 1 ) .= succ ( 5 + 1 ) .= succ ( 5 + 1 ) ;
assume that F is bbSubset-Family and rng p = F and dom p = Seg ( n + 1 ) and rng p = Seg ( n + 1 ) ;
LIN b , b9 , a & not LIN a , a9 , c & LIN a , a9 , c & LIN a , a9 , c ;
( L1 , L2 ) \& O c= ( L1 => O ) => ( L2 => O ) ;
consider F being ManySortedSet of E such that for d being Element of E holds F . d = F ( d ) and for d being Element of E holds F . d = G ( d ) ;
consider a , b such that a * ( u1 - w ) = b * ( - w ) and 0 < a & a < b ;
defpred P [ FinSequence of D ] means |. Sum $1 .| <= Sum |. $1 .| & Sum |. $1 .| <= Sum |. $1 .| ;
u = cos . ( x , y ) * x + ( cos . ( x , y ) * y ) .= v ;
dist ( ( seq . n ) + x , g + x ) <= dist ( ( seq . n ) , g ) + 0 ;
P [ p , |. p .| (#) |. p .| , {} , id the Sorts of A ] means P [ p , id the Sorts of A ]
consider X being Subset of CQC-WFF ( Al ) such that X c= Y and X is finite and X is ininand X is inin;
|. b .| * |. eval ( f , z ) .| >= |. b .| * |. eval ( f , z ) .| ;
1 < ( ( E-max L~ Cage ( C , n ) ) .. ( Cage ( C , n ) ) + 1 ) .. ( Cage ( C , n ) ) ;
l in { l1 where l1 is Real : g <= l1 & l1 <= h & l1 <= g } ;
vol ( ( G . n ) vol ) <= ( Partial_Sums ( ( G . n ) vol ) ) * vol ( ( G . n ) vol ) ;
f . y = x .= x * 1_ L .= x * ( power L ) . ( y , 0 ) .= x * ( power L ) . ( y , 0 ) ;
NIC ( <% i1 , i2 %> , k ) = { i1 , succ i1 } & NIC ( a , b ) = { succ i1 , succ i2 } ;
LSeg ( p00 , p2 ) /\ LSeg ( p1 , p2 ) = { p1 } & LSeg ( p1 , p2 ) /\ LSeg ( p2 , p2 ) = { p2 } ;
Product ( ( the carrier of I-15 ) +* ( i , { 1 } ) ) in ( Z . i ) \/ ( Z . i ) ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) .= Following ( s2 , n ) ;
W-bound Qs2 <= q1 `1 & W-bound Qs2 <= E-bound Qs2 & W-bound Qs2 <= E-bound Qs2 & W-bound Qs2 <= E-bound Qs2 ;
f /. i2 <> f /. ( ( i1 + len g -' 1 ) -' 1 ) implies f /. i2 = f /. ( i1 + 1 )
M , f / ( x. 3 , a ) / ( x. 4 , a ) / ( x. 0 , a ) |= H / ( x. 4 , a ) ;
len ( ( P ^ ) ^ ( P ^ ) ) in dom ( ( P ^ ) ^ ( P ^ ) ) & len ( P ^ ) in dom ( P ^ ) ;
A |^ ( n , m ) c= A |^ ( m , n ) & A |^ ( k , l ) c= A |^ ( k , l ) ;
REAL n \ { q : |. q .| < a } c= { q1 : |. q1 .| >= a }
consider n1 be 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 , s2 , l ) ) <> halt SCM+FSA & CurInstr ( P3 , Comput ( P3 , s2 , l ) ) <> halt SCM+FSA ;
for v be VECTOR of l1 holds ||. v .|| = upper_bound rng |. ( id the carrier of l1 ) .| & ||. v .|| = upper_bound rng |. ( id the carrier of l1 ) .|
for phi holds ( phi in X & phi in X & ( phi in X & phi in X ) implies 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 ) ) ) ) ) ) ) ) ) c= 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 & a = c & a = c & P [ c ] ;
the_arity_of ( o , b , c ) = <* Hom ( b , c ) , Hom ( a , b ) *> .= <* Hom ( a , b ) , Hom ( b , c ) *> ;
consider f1 be Function of the carrier of X , REAL such that f1 = |. f .| and f1 is continuous and f1 . 0 = f . 1 ;
a1 = b1 & a2 = b2 or a1 = b1 & a2 = b2 & a1 = b1 & a2 = b2 or a1 = b1 & a2 = b2 ;
D2 . indx ( D2 , D1 , n1 + 1 ) = D1 . ( n1 + 1 ) .= D1 . ( n1 + 1 ) ;
f . ( ||. r .|| ) = ||. |[ r .|| , 1 ]| .|| /. 1 .= <* r *> . 1 .= x ;
consider n being Nat such that for m being Nat st n <= m holds C-25 . n = C-25 . m and C-25 . m = C-25 . m ;
consider d be Real such that for a , b be Real st a in X & b in Y holds a <= d & d <= b ;
||. L /. h .|| - ( K * |. h .| ) + ( K * |. h .| ) <= p0 + ( K * |. h .| ) ;
attr F is commutative associative means : Def: for b being Element of X holds F -Sum ( { b } _ f ) = f . b ;
p = - ( - p0 + 0. TOP-REAL 2 ) .= 1 * p0 + 0. TOP-REAL 2 .= 1 * p0 + 0. TOP-REAL 2 .= 1 * p0 + p0 `1 .= 1 * p0 `1 ;
consider z1 such that b , x3 , x1 is_collinear and o , x1 , z1 is_collinear and o <> x1 & o <> z1 & o <> z1 & o <> z1 ;
consider i such that Arg ( ( Rotate ( s ) ) . q ) = s + Arg q + ( 2 * PI * i ) ;
consider g such that g is one-to-one and dom g = card ( f . x ) and rng g = f . x and g is one-to-one ;
assume that A = P2 \/ Q2 and P2 <> {} and Q2 <> {} and Q1 misses Q2 and Q1 /\ Q2 = {} and Q1 /\ Q2 = {} and Q1 /\ Q2 = {} ;
attr F is associative means : Def8 : F .: ( F .: ( f , g ) , h ) = F .: ( f , F .: ( g , h ) ) ;
ex x being Element of NAT st m = x `2 & x `1 in z & x `2 < i or m in { i } ;
consider k2 being Nat such that k2 in dom P-2 and l in P-2 . k2 and Pk1 = P-2 . k2 and k1 <= k2 ;
seq = r (#) seq implies for n holds seq . n = r * seq . n & seq . n = r * seq . n
F1 . [ id a , [ a , a ] ] = [ f * ( id a , [ a , b ] ) , F1 . [ a , b ] ] ;
{ p } "\/" D2 = { p "\/" y where y is Element of L : y in D2 & p "\/" q in D1 } ;
consider z being element such that z in dom ( ( the Sorts of F ) * ( the Arity of S ) ) and ( ( the Sorts of F ) * ( the Arity of S ) ) . z = y ;
for x , y being element st x in dom f & y in dom f & f . x = f . y holds x = y
Int cell ( G , i , j ) = { |[ r , s ]| : r <= G * ( 0 + 1 , 1 ) `1 }
consider e being element such that e in dom ( T | E1 ) and ( T | E1 ) . e = v and ( T | E1 ) . e = v ;
( F `1 * b1 ) . x = ( Mx2Tran ( J , Bthesis , Bthesis ) ) . ( \mathclose { j } ) .= ( Mx2Tran J ) . ( \mathclose { j } ) ;
- 1 / ( - 1 ) * D = mm (#) D | n .= mm (#) D .= mm (#) ( - 1 ) .= Det M ;
pred for x being set st x in dom f /\ dom g holds g . x <= f . x & - g is nonnegative ;
len ( f1 . j ) = len f2 /. j .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) ;
All ( All ( 'not' a , A , G ) , B , G ) '<' Ex ( 'not' All ( a , B , G ) , A , G ) ;
LSeg ( E . k0 , F . k0 ) c= Cl RightComp Cage ( C , k0 + 1 ) & LSeg ( E . k0 , F . k0 ) c= RightComp Cage ( C , k0 + 1 ) ;
x \ a |^ m = x \ ( a |^ k * a ) .= ( x \ a ) \ a .= x \ a ;
k -inininininin1 = ( commute ( I-5 . k ) ) . k .= ( commute ( I-5 . k ) ) . ( ( commute ( I-5 . k ) ) . i ) .= ( commute ( IU . k ) ) . i ;
for s being State of AJ holds Following ( s , n ) . 0 + ( n + 2 ) * n + 1 is stable ;
for x st x in Z holds f1 . x = a ^2 & ( f1 - f2 ) . x <> 0 & f1 . x <> 0 ) implies f1 - f2 is not 0
support ( s ) \/ support ( s ) c= support ( max ( n , m ) ) \/ support ( s ) \/ support ( s ) ;
reconsider t = u as Function of ( the carrier of A ) , ( the carrier' of B ) * the Arity of C ;
- ( a * sqrt ( 1 + b ^2 ) ) <= - ( b * sqrt ( 1 + a ^2 ) ) ;
phi /. ( succ b1 ) = g . a & phi /. ( g . ( g . a ) ) = f . ( g . a ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( ( F ^ <* p *> ) . i ) and i + 1 in dom ( F ^ <* p *> ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , 7 } = { x1 } \/ { x2 } ;
the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U1 & the Sorts of U1 c= the Sorts of U2 implies the Sorts of U1 = the Sorts of U2
( ( - 2 * a * ( b - a ) ) ^2 + b ^2 - delta ( a , b , c ) ) > 0 ;
consider W00 such that for z being element holds z in W00 iff z in N ~ N & P [ z ] ;
assume that ( the Arity of S ) . o = <* a *> and ( the ResultSort of S ) . o = r and ( the ResultSort of S ) . o = r ;
Z = dom ( ( exp_R (#) ( arccot ) ) `| Z ) /\ dom ( ( arccot * ( arccot ) ) `| Z ) ;
sum ( f , SS1 ) is convergent & lim ( sum ( f , SS1 ) ) = integral ( f , SS1 ) ;
( X ) => ( ( a => f ) => ( x9 => x9 ) ) in ( ( the carrier of L ) \ { x } ) \/ ( the carrier of L ) ;
len ( M2 * M3 ) = n & width ( M3 ~ * M2 ) = n & width ( M3 ~ * M3 ) = n & width ( M3 ~ * M3 ) = n ;
attr X1 union X2 means : Def8 : X1 , X2 are_separated & X1 , X2 are_separated & X1 , X2 are_separated & X2 , X1 are_separated & X1 , X2 are_separated & X2 , 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 = ( ( F . b ) `2 ) * ( ( F . b ) `2 ) as Function of ( the carrier of M ) , M ;
consider w being FinSequence of I such that the InitS of M is q of M -{ s } ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ;
g . ( a |^ 0 ) = g . ( 1_ H ) .= 1_ H .= ( g . a ) |^ 0 .= ( g . a ) |^ 0 .= ( g . a ) |^ 0 ;
assume for i being Nat st i in dom f ex z being Element of L st f . i = rpoly ( 1 , z ) & f . i = rpoly ( 1 , z ) ;
ex L being Subset of X st Carrier L = L & for K being Subset of X st K in C holds L /\ K <> {} & L is closed ;
( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C1 & ( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C2 ;
reconsider oY = o `2 , oY = p `2 , oY = p `2 , oY = p `2 as Element of ( the Sorts of A ) . s ;
1 * x1 + ( 0 * x2 ) + ( 0 * x3 ) = x1 + <* \underbrace ( 0 , 0 , 0 ) *> .= x1 + ( 0 * x2 ) .= x1 + ( 0 * x3 ) .= x1 + ( 0 * x2 ) ;
Ez " . 1 = ( Ez qua Function ) " . 1 .= ( Ez qua Function ) " . 1 .= ( Ez " ) . 1 .= ( Ez " ) . 1 ;
reconsider u1 = the carrier of U1 /\ ( U1 "\/" U2 ) , u2 = the carrier of U1 /\ U2 , u1 = the carrier of U2 /\ ( U1 "\/" U2 ) as non empty Subset of U0 ;
( ( x "/\" z ) "\/" ( x "/\" y ) ) "\/" ( z "/\" y ) <= ( x "/\" ( z "\/" y ) ) "\/" ( z "/\" ( x "\/" y ) ) ;
|. f . ( s1 . ( l1 + 1 ) ) - f . ( s1 . l1 ) .| < ( 1 / |. M .| + 1 ) * ( M . ( l1 + 1 ) ) ;
LSeg ( ( Lower_Seq ( C , n ) ) /. ( i + 1 ) , ( Lower_Seq ( C , n ) ) /. ( i + 1 ) ) is vertical ;
( f | Z ) /. x - ( f | Z ) /. x0 = L /. ( x- ( x - x0 ) ) + R /. ( x- ( x - x0 ) ) ;
g . c * ( - g . c ) + f . c <= h . c * ( - g . c ) + f . c ;
( f + g ) | divset ( D , i ) = f | divset ( D , i ) + g | divset ( D , i ) ;
assume that ColVec2Mx f in the carrier of ( len A ) and ( ColVec2Mx b ) = ( ColVec2Mx A ) * ( ColVec2Mx b ) and ( len A ) = 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 ( \cal n ) | ( the carrier of ( TOP-REAL n ) | ( the carrier of ( TOP-REAL n ) | ( the carrier of ( TOP-REAL n ) | ( the carrier of ( TOP-REAL n ) | ( the carrier of ( TOP-REAL n ) | ( the carrier of ( TOP-REAL n ) | ( the carrier of n ) ) ) ) )
pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1 ;
pred a <> 0 & b <> 0 & Arg a = Arg b & Arg ( - a ) = Arg ( - b ) & Arg ( - b ) = Arg ( - b ) ;
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 linearly-independent and V2 is linearly-independent and V2 = { v + u : v in V1 & u in V2 & v in V2 } and V1 is open and V2 is open ;
z * x1 + ( 1 - z ) * x2 + ( 1 - z ) * y2 in M & z * y1 + ( 1 - z ) * y2 in N ;
rng ( ( Pk1 qua Function ) " * Sk1 ) = Seg ( card dk1 ) .= Seg ( card dk1 ) .= Seg ( card dk1 ) .= dom ( Pk1 ) ;
consider s2 being rational Real_Sequence such that s2 is convergent and b = lim s2 and for n holds s2 . n <= b and s2 . n <= b ;
h2 " . n = h2 . n " & 0 < - 1 / ( ( 1 - ( 2 * ( n + 1 ) ) * ( 1 / ( 2 * ( n + 1 ) ) ) ) & ( 1 - ( 2 * ( n + 1 ) ) * ( 1 / ( 2 * ( n + 1 ) ) ) ) * ( 1 / ( 2 * ( n + 1 ) ) ) ) ;
( Partial_Sums ( ||. seq1 .|| ) ) . m = ||. seq1 .|| . m .= ||. ( seq1 . m ) - ( seq2 . m ) .|| .= ||. ( seq1 . m ) - ( seq2 . m ) .|| .= ||. ( seq1 . m ) - ( seq2 . m ) .|| .= ||. ( seq1 . m ) - ( seq2 . m ) .|| .= ||. ( seq1 . m ) - ( seq2 . m ) .|| ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= ( Comput ( P2 , s2 , 1 ) ) . b .= Comput ( P2 , s2 , 1 ) . b ;
- v = ( - 1_ ( G ) ) * v & - w = ( - 1_ G ) * w & - w = ( - 1_ G ) * v & - w = ( - 1_ G ) * w ;
sup ( ( k .: D ) .: D ) = sup ( ( k .: D ) .: ( D ) ) .= sup ( ( k .: D ) .: D ) .= sup ( ( k .: D ) .: D ) ;
A |^ ( k , l ) ^^ ( A |^ ( n , l ) ) = ( A |^ ( k , l ) ) ^^ ( A |^ ( k , l ) ) ;
for R being add-associative right_zeroed right_complementable non empty addLoopStr , I , J being Subset of R holds I + ( J + K ) = ( I + J ) + K
( f . p ) `1 = ( p `1 ) ^2 / sqrt ( 1 + ( p `2 / p `1 ) ^2 ) .= ( p `1 ) ^2 / sqrt ( 1 + ( p `2 / p `1 ) ^2 ) ;
for a , b being non zero Nat st a , b are_relative_prime & b , a are_relative_prime holds ( for n being Nat st n in dom a holds ( n * b ) = ( n * a ) + ( n * b )
consider A5 being countable Nat such that r is Element of CQC-WFF ( Al ) & A5 is ( Al ) -as ( Al ) -as ( Al ) -valued FinSequence of D ;
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 x + M
{ [ x1 , x2 ] , [ y1 , y2 ] } c= [: { x1 , y1 } , { y2 } :] \/ [: { y1 , y2 } , { y2 } :] ;
h . ( f . O ) = |[ A * ( ( f . O ) `1 + B , C * ( f . O ) `2 + D ]| ) ;
( Gauge ( C , n ) * ( k , i ) ) `1 in L~ Upper_Seq ( C , n ) /\ L~ Lower_Seq ( C , n ) ;
cluster m , n are_relative_prime means : such : for p being prime Nat holds p is prime & p divides m & p divides n & p divides n & p divides n ;
( f * F ) . x1 = f . ( F . x1 ) & ( f * F ) . x2 = f . ( F . x2 ) ;
for L being LATTICE , a , b , c being Element of L st a \ b <= c & b \ a <= c holds a \+\ b <= c
consider b being element such that b in dom ( H / ( x , y ) ) and z = ( H / ( x , y ) ) . b ;
assume that x in dom ( F * g ) and y in dom ( F * g ) and ( F * g ) . x = ( F * g ) . y ;
assume ex e being element st e Joins W . 1 , W . 5 , G or e Joins W . 3 , W . 7 , G ;
( r (#) delta ( h ) ) . ( 2 * n ) . x = ( r (#) delta ( h ) ) . ( 2 * n ) . ( x + ( n * h ) ) ;
j + 1 = len h11 + 1 + 1 .= i + 1 - len h11 + 2 - 1 .= i + 1 - len h11 + 2 - 1 .= i + 1 - len h11 + 1 ;
( *' S ) . f = *' S . ( opp f ) .= S . ( ( opp f ) . f ) .= S . ( ( opp f ) . f ) .= S . f ;
consider H such that H is one-to-one and rng H = the carrier of L2 and Sum ( L2 * H ) = Sum ( L2 ) and Sum ( L1 ) = Sum ( L2 ) ;
attr R is max means : Def: for p , q st p in R & p <> q holds ex P st P is special & P c= R & P c= R ;
dom product ( product ( X --> f ) ) = meet ( ( X --> f ) . 0 ) .= meet ( ( X --> dom f ) . 0 ) .= meet ( ( X --> dom f ) . 0 ) .= dom f ;
upper_bound ( proj2 .: ( Upper_Arc C /\ Upper_Arc C /\ /\ /\ /\ E-bound C ) ) <= upper_bound ( proj2 .: ( C /\ /\ /\ /\ \frac { w } { 2 } ) ) ;
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-28 - fc = i * f`2 - ( i * yc ) .= i * ( f`2 - fc ) .= i * ( f`2 - fc ) ;
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 union C and g2 in C ;
func d |-count n -> Nat means : Def7 : d |^ it divides n & d |^ it divides n & d |^ it divides n & d divides n & it divides n ;
f\in . [ 0 , t ] = f . [ 0 , t ] .= ( - P ) . ( 2 * x ) .= a ;
t = h . D or t = h . B or t = h . C or t = h . E or t = h . F or t = h . J ;
consider m1 be Nat such that for n st n >= m1 holds dist ( ( seq . n ) , ( seq . n ) ) < 1 / ( n + 1 ) ;
( q `1 ) ^2 / ( |. q .| ) ^2 <= ( ( q `2 ) ^2 / ( |. q .| ) ^2 ) * ( ( q `1 ) ^2 / ( |. q .| ) ^2 ) ;
h0 . ( i + 1 + 1 ) = h21 . ( i + 1 + 1 -' len h11 + 2 -' 1 ) .= h21 . ( i + 1 + 1 -' len h11 + 2 -' 1 ) ;
consider o being Element of the carrier' of S , x2 being Element of { the carrier of S } such that a = [ o , x2 ] and [ o , x2 ] in the carrier' of S ;
for L being RelStr , a , b being Element of L holds a is_<=_than { b } iff a <= b & a >= b & b >= a
||. h1 .|| . n = ||. h1 . n .|| .= |. h .| . n .= ||. h .|| . n .= ||. h .|| . n .= ||. h .|| . n .= ||. h .|| . n ;
( ( - ( #Z n ) ) * f ) . x = f . x - ( #Z n ) . ( f . x ) .= ( - ( #Z n ) ) . ( f . x ) .= ( - ( #Z n ) ) . ( f . x ) .= ( - ( #Z n ) ) . ( f . x ) .= ( - ( #Z n ) ) . ( f . x ) ;
pred r = F .: ( p , q ) means : Def8 : len r = min ( len p , len q ) & for i st i in dom r holds r . i = F . ( p . i ) ;
( rand 2 / 2 ) ^2 + ( rand 2 / 2 ) ^2 <= ( r / 2 ) ^2 + ( r / 2 ) ^2 + ( r / 2 ) ^2 ;
for i being Nat , M being Matrix of n , K st i in Seg n holds Det M = Sum ( L * M ) & Det M = L * ( i , i )
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 ) * ( h ^\ n ) " ) . $1 & ( L /* h ) . $1 = L . ( h . $1 ) * ( h ^\ n ) . $1 ;
assume that the carrier of H2 = f .: the carrier of H1 and the carrier of H2 = f .: the carrier of H2 and the carrier of H1 = the carrier of H2 and the carrier of H2 = the carrier of H2 and the carrier of H1 = the carrier of H2 and the carrier of H2 = the carrier of H2 ;
Args ( o , Free ( S , X ) ) = ( ( the Sorts of Free ( S , X ) ) * the Arity of S ) . o .= ( the Sorts of Free ( S , X ) ) . o ;
H1 = n + 1 -: h = n + 1 -H .= n + 1 -H .= n + 1 -H .= n + 1 -H .= n + 1 -H ;
( O = 0 & O = 0 & O = 1 & O = 1 & O = 1 & O = 1 or O = 0 ) & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 or O = 1 ) ;
F1 .: ( dom F1 /\ dom F2 ) = F1 .: ( dom F1 /\ dom F2 ) .= F1 .: ( dom F1 /\ dom F2 ) .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } ;
pred b <> 0 & d <> 0 & b <> d & ( a = ( - e ) / d ) implies ( a = ( - e ) / b ) & ( a = ( - e ) / b )
dom ( ( f +* g ) | D ) = dom ( f +* g ) /\ D .= ( dom f \/ dom g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D ;
for i be set st i in dom g ex u , v be Element of L , a be Element of B st g /. i = u * a * v & u in A & v in B
g `2 * P `2 * g `2 = g `2 `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 not ( i = s1 & not i <= s1 ) & f . ( i + 1 ) <> s1 & not i <= s1 ) & not i <= s2 ;
h5 | ]. a , b .[ = ( g | Z ) | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ ;
[ s1 , t1 ] , [ s2 , t2 ] are_connected & [ s2 , t2 ] , [ s3 , t2 ] are_connected & [ s3 , t2 ] , [ s3 , t2 ] are_connected & [ s3 , t2 ] , [ s3 , t2 ] are_connected ;
then H is negative & H is not negative & H is not conjunctive & H is not empty and H is not an -gof A & H is not an implies H is not an -gof A ;
attr f1 is total means : Def8 : f1 (#) f2 is total & ( f1 (#) f2 ) . c = f1 . c * f2 . c * ( f2 . c ) " ;
z1 in W2 -Seg ( z2 ) or z1 = z2 & not z1 in W2 & z1 in W2 & z2 in W2 & z1 = W2 . z1 or z1 = z2 & z1 = z2 & z2 = z2 or z1 = z2 & z1 = z2 & z2 = z2 ;
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 ^\ k ) <= upper_bound rng ( seq1 ^\ k )
x0 in C /\ L~ go or x0 in L~ pion1 or x0 in L~ pion1 & not ex x being Point of TOP-REAL 2 st x in L~ pion1 & x = pion1 /. 1 & not x in L~ pion1 & x in L~ pion1
||. f . ( g . ( k + 1 ) ) - g . k .|| <= ||. g . 1 - g . 0 .|| * ( K * K to_power k ) ;
assume h = ( ( B .--> B ' ) +* ( C .--> D ' ) +* ( E .--> F ) +* ( F .--> J ) +* ( M .--> N ' ) ) +* ( A .--> N ) +* ( N .--> A ' ) +* ( F .--> J ) +* ( M .--> N ) ;
|. ( ( delta ( H . n ) || A , T ) ) . k - ( ( \HM { the } \HM { carrier of T , the carrier of T ) ) . k .| <= e * ( b-a - T ) . k ;
( (
{ x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x2 , x1 , x2 , x1 , x2 , x1 , x2 , x1 , x2 } } = { x1 } ;
assume that A = [. 0 , 2 * PI .] and integral ( ( exp_R * sin ) , A ) = 0 and integral ( ( exp_R * sin ) , A ) = 0 ;
p `2 is Permutation of dom f1 & p `2 = ( Sgm Y ) " * p & p `2 = ( Sgm Y ) " * p & p `2 = ( Sgm Y ) " * p ;
for x , y st x in A & y in A holds |. ( 1 / f . x - 1 / f . y ) .| <= 1 * |. f . x - f . y .|
p2 `2 = |. q2 .| * ( ( q2 `2 / |. q2 .| - sn ) / ( 1 + sn ) ) - sn ) .= ( q2 `2 / |. q2 .| - sn ) / ( 1 + sn ) ;
for f be PartFunc of the carrier of CNS , REAL st dom f is compact & f is_continuous_on dom f & f is_continuous_on dom f holds f is_continuous_on dom f & f | X is continuous
assume for x being Element of Y st x in EqClass ( z , CompF ( B , G ) ) holds ( Ex ( a , A , G ) ) . x = TRUE ;
consider FM such that dom FM = n1 & for k be Nat st k in n1 holds Q [ k , FM . k ] & Q [ k , FM . k ] ;
ex u , u1 st u <> u1 & u , u1 / ( 2 , u ) / ( 2 , u1 ) / ( 2 , u1 ) = ( u , u1 ) / ( 2 , u1 ) & u , u1 / ( 2 , u1 ) / ( 2 , u1 ) = ( u , u1 ) / ( 2 , u1 ) ;
for G being Group , A , B being non empty Subgroup of G , N being normal Subgroup of G holds ( N ` A ) * ( N ` B ) = N ` A * B
for s be Real st s in dom F holds F . s = integral ( ( R to_power 0 ) (#) ( e to_power 0 ) ) - integral ( ( f + g ) to_power ( k + 1 ) , ( f + g ) to_power ( k + 1 ) )
width ( AutMt ( f1 , b1 , b2 ) ) = len b2 .= width ( ( f2 * f1 ) * ( f2 * f2 ) ) .= len ( ( f2 * f1 ) * ( f2 * f2 ) ) .= len ( ( f2 * f1 ) * ( f2 * f2 ) ) ;
f | ]. - PI / 2 , PI / 2 .[ = f & dom f " ]. - 1 , 1 .[ = ]. - 1 , 1 .[ & f | ]. - 1 , 1 .[ is continuous ;
assume that X is closed set and a in X and a c= X and y in { { [ n , x ] } \/ y : x in a } \/ { x } in X ;
Z = dom ( ( #Z 2 ) * ( arctan + arccot ) ) /\ dom ( ( #Z 2 ) * ( arctan + arccot ) ) .= dom ( ( #Z 2 ) * ( arctan + arccot ) ) /\ dom ( ( #Z 2 ) * ( arctan + arccot ) ) ;
func ( the Sorts of V ) . l -> Subset of V means : Def1 : 1 <= k & k <= len l & l . k in V & l . k in V ;
for L being non empty TopSpace , N being net of L , M being net of N , c being Point of L st c is cluster cluster cluster cluster -> cluster for net of N st c is cluster cluster cluster cluster cluster cluster -> cluster cluster cluster for net of N , L holds c is empty
for s being Element of NAT holds ( ( id C\mathop ( C\mathop ( v , C\mathop ( v , C\mathop ( v , C\mathop ( v , C\mathop ( C\mathop ( v , C\mathop ) ) ) ) ) ) ) . s = ( ( id C\mathop ( v , C\mathop ( CC\mathop ( v , C\mathop ( v , C\mathop ( v , C\mathop ( v , Cq ) ) ) ) ) . s )
then z /. 1 = N-min L~ z & ( N-min L~ z ) .. z < ( N-min L~ z ) .. z & ( N-min L~ z ) .. z < ( N-min L~ z ) .. z ;
len ( p ^ <* ( 0 qua Real ) *> ) = len p + len <* ( 0 qua Real ) *> .= 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 ;
for R being add-associative right_zeroed right_complementable associative commutative associative well-unital 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 :] , B12 such that for x being Element of [: B1 , B2 :] holds f . x = F ( x ) and f . x = F ( x ) ;
dom ( x2 + y2 ) = Seg len x .= Seg len ( x2 + z2 ) .= Seg len ( x2 + z2 ) .= dom ( x (#) y ) .= dom ( x (#) y ) .= dom ( x (#) y ) ;
for S being Functor of C , B for c being object of C holds ( S . ( id c ) ) . ( id c ) = id ( ( Obj S ) . ( id c ) )
ex a st a = a2 & a in f6 /\ f5 & for x st x in f6 holds \rrangle in \rrangle & $ { x } in \rm ^2 ( f5 , a ) & { x } in \rm ^2 ( f5 , a ) ;
a in Free ( H2 / ( x. 4 , x. 0 ) ) & H2 / ( x. 4 , x. 0 ) / ( x. 4 , x. 0 ) = H2 / ( x. 0 , x. 0 ) / ( x. 4 , x. 0 ) ;
for C1 , C2 being \llangle C1 , C2 , f being stable Function of C1 , C2 st If for g being Function of C2 , C2 holds f = g & g = f holds f = g
( W-min L~ go \/ L~ pion1 ) `1 = W-bound L~ go \/ E-bound L~ pion1 & ( W-min L~ go \/ W-bound L~ pion1 ) `1 = W-bound L~ go \/ E-bound L~ pion1 & ( W-min L~ go \/ W-bound L~ pion1 ) `1 = W-bound L~ pion1 ;
assume that u = <* x0 , y0 , z0 *> and f is_PartFunc of REAL 3 , REAL & SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) is_differentiable_in z0 & SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) . z0 = z0 ;
then ( t . {} ) `1 in Vars & ex x being Element of Vars st x = ( t . {} ) `1 & t . {} = x & ( t . {} ) `2 = s ;
Valid ( p '&' p , J ) . v = Valid ( p , J ) . v '&' Valid ( p , J ) . v .= Valid ( p , J ) . v .= Valid ( p , J ) . v ;
assume for x , y being Element of S st x <= y for a , b being Element of T ~ st a = f . x & b = f . y holds a >= b & b >= y ;
func Class R -> Subset-Family of R means : R : for A being Subset of R holds A in it iff ex a being Element of R st A = Class ( a , a ) & a in Class ( R , a ) ;
defpred P [ Nat ] means ( ( ( ( j ) `1 ) `1 ) `1 ) c= ( ( ( j ) `1 ) `1 ) `2 & ( ( j ) `2 ) `1 c= ( ( j ) `1 ) `1 ) `1 ;
assume that dim W1 = 0 and dim W1 = 0 and ( for v st v in W1 holds v in W2 ) & ( v in W2 implies v in W1 & v in W2 ) & ( v in W1 & v in W2 implies v in W2 ) & ( v in W1 & v in W2 implies v in W2 ) ;
mamas ( m ) . t = ( m . t ) `1 .= ( [ m . t , the carrier of C ] `1 ) `1 .= ( [ m , the carrier of C ] `2 ) `1 .= m ;
d11 = ( x ^ d22 ) . ( f , d22 ) .= f . ( y9 , d22 ) .= f . ( y9 , d22 ) .= d22 . ( y , d22 ) .= d22 . ( y , d22 ) .= d22 . ( y , d22 ) ;
consider g such that x = g & dom g = dom fx0 & for x being element st x in dom fx0 holds g . x in fx0 . x and g . x in fx0 . x ;
x + 0. F_Complex |^ ( len x ) = x + len x |-> 0. F_Complex .= ( x + len x ) |-> 0. F_Complex ) .: ( x , ( len x |-> 0. F_Complex ) ) .= x ` ;
( k -' kk + 1 ) in dom ( f | ( len ( f | ( k -' 21 + 1 ) ) ) ) & ( f | ( k -' 21 + 1 ) ) . ( k -' kk + 1 ) = f . ( k -' kk + 1 ) ;
assume that P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1
reconsider a1 = a , b1 = b , b1 = c , c1 = p `1 , p1 = p `1 , b1 = p `2 , c1 = p `1 , p1 = p `1 , p2 = p `2 , p3 = p `2 , p1 = p `1 , p2 = p `2 , p3 = p `2 , p1 = p `2 , p2 = p `2 , p1 = p `2 , p3 = p `2 , p1 = p `2 , p2 = p `2 , p3 = p `2 , p1 = p `2 , p2 = p `2 , p4 = p `2 , p4 = p `2 , p4 = p `2 , p4 = p `2 , p4 = p `2 , p4 = p `2 , p4 = p `2 , p4 = p `2 , p4 = p `2 , p4 = p `2 , p4
reconsider _ tt1f = G1 . ( t , t ) * F1 . f , F2 . ( t , f ) * F2 . a as Morphism of ( G1 * F1 ) . b , ( G1 * F2 ) . a ;
LSeg ( f , i + i1 -' 1 ) = LSeg ( f /. ( i + i1 -' 1 ) , f /. ( i + i1 -' 1 + 1 ) ) .= LSeg ( f , i ) ;
Integral ( M , P . m ) | dom ( P . n -P . m ) <= Integral ( M , P . n -P . m ) + Integral ( M , P . m -P . m ) ;
assume that dom f1 = dom f2 and for x , y being element st [ x , y ] in dom f1 & f1 . ( x , y ) = f2 . ( x , y ) 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 ) `2 ) / 2 ) ;
for G being Group , H being Subgroup of G , a , b being Element of G st a = b holds for i being Integer holds a |^ i = b |^ i & a |^ i = b |^ i
consider B being Function of Seg ( S + L ) , the carrier of V1 such that for x being element st x in Seg ( S + L ) holds P [ x , B . x ] ;
reconsider K1 = { p2 where p2 is Point of TOP-REAL 2 : P [ p2 ] & p2 `1 <= 0 & p2 <> 0. TOP-REAL 2 } , K1 = { p2 where p2 is Point of TOP-REAL 2 : P [ p2 ] & p2 <> 0. TOP-REAL 2 } ;
( ( N-bound C - S-bound C ) / ( 2 |^ m ) ) * ( ( N-bound C - S-bound C ) / ( 2 |^ m ) ) <= ( ( N-bound C - S-bound C ) / ( 2 |^ ( m + 1 ) ) ) * ( ( S-bound C - S-bound C ) / ( 2 |^ ( m + 1 ) ) ) ;
for x be Element of X , n be Nat st x in E holds |. Re ( F . n ) . x .| <= P . x & |. Im ( F . n ) . x .| <= P . x
len @ ( @ p ^ @ q ) = len ( @ p ^ @ q ) + len <* [ 2 , 0 ] *> .= len ( @ p ^ @ q ) + len ( @ p ^ @ q ) .= len @ p + len @ q ;
v / ( x. 3 , m1 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) = m3 / ( x. 0 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) ;
consider r being Element of M such that M , v2 / ( x. 3 , m ) / ( x. 4 , m ) / ( x. 0 , m ) / ( x. 4 , n ) / ( x. 0 , m ) |= H2 iff ( ( H / ( x. 4 , m ) ) / ( x. 4 , n ) ) / ( x. 0 , m ) ) / ( x. 4 , n ) ;
func w1 \ w2 -> Element of Union ( G , R6 ) means : Def6 : for w1 , w2 being Element of Union ( G , R6 ) holds it . ( w1 , w2 ) = ( ( HH6 ) * ( G , R6 ) ) . w1 ;
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= s1 . b2 .= s2 . b2 .= Exec ( n2 , s2 ) . b2 .= s2 . b2 .= s2 . b2 .= s . b2 ;
for n , k being Nat holds 0 <= ( Partial_Sums |. seq .| ) . ( n + k ) - ( Partial_Sums |. seq .| ) . ( n + k ) & ( Partial_Sums |. seq .| ) . ( n + k ) <= ( Partial_Sums ( |. seq .| ) ) . ( n + k )
set F = S \! \mathop { {} } ;
( Partial_Sums seq ) . ( K + 1 ) + Partial_Sums ( seq ) . ( K + 1 ) >= ( Partial_Sums ( seq ) ) . ( K + 1 ) + ( Partial_Sums ( seq ) ) . ( K + 1 ) ;
consider L , R such that for x st x in N holds ( f | Z ) . x - ( f | Z ) . x0 = L . ( x- ( x0 + R ) ) + R . ( x- ( x0 + R ) ) ;
func the closed of \HM { a , b , c , d , e , f , g , h , i , x , y , z , x , y , z , z , x , y , z , x , y , z , x , y , z , x , z , x , y , z , x , z *> ;
a * b ^2 + ( a * c ^2 + b * a ^2 ) + ( b * c ^2 + ( c * a ^2 ) + ( c * a ^2 + b * a ^2 ) ) >= 6 * a * b * c ;
v / ( x1 , m1 ) / ( x2 , m2 ) / ( x1 , m1 ) = v / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) .= v / ( x2 , m1 ) / ( x2 , m1 ) ;
+ ( Q ^ <* x *> , M ) = ( + ( Q , 0 ) +* ( ( ^ ( x , M ) --> FALSE ) ) +* ( ( card { x } --> FALSE ) +* ( ( x , 0 ) --> FALSE ) ) ) +* ( ( x , 0 ) --> FALSE ) ) ;
Sum ( F ) = r |^ n1 * Sum C-13 .= C . n1 * ( Cz ) .= C . n1 * ( Cz ) .= C . n1 * ( Cz ) .= C . n1 * ( Cz ) .= C . n1 * ( Cz ) ;
( GoB f ) * ( len GoB f , 2 ) `1 = ( ( GoB f ) * ( len GoB f , 1 ) ) `1 .= ( ( GoB f ) * ( 1 , 1 ) ) `1 .= ( ( GoB f ) * ( 1 , 1 ) ) `1 .= ( ( GoB f ) * ( 1 , 1 ) ) `1 ;
defpred X [ Element of NAT ] means ( Partial_Sums s ) . $1 = ( a * ( $1 + 1 ) * $1 + b * ( $1 + 1 ) * $1 ) / ( 2 * $1 + b * ( $1 + 1 ) * $1 ) + b * ( $1 + 1 ) * $1 ;
the_arity_of g = ( the Arity of S ) . g .= ( [ ( the Arity of S ) . g , ( the Arity of S ) . g ] ) `1 .= [ g , ( the Arity of S ) . g ] `1 .= g ;
( [: X , Y :] |^ Z ) tolerates [: X , Y :] & card ( [: X , Y :] |^ Z ) = card [: X , Y :] & card ( [: X , Y :] |^ Z ) = card [: X , Y :] ;
for a , b being Element of S , s being Element of NAT st s = n & a = F . n & b = F . ( n + 1 ) holds b = N . ( s . ( n + 1 ) \ G . s )
E , f |= All ( x. 2 , ( x. 2 ) .--> ( x. 0 ) ) => ( x. 2 , ( x. 1 ) .--> ( x. 2 ) ) => ( x. 2 , ( x. 1 ) .--> ( x. 2 ) ) = ( x. 2 ) '&' ( x. 1 ) ;
ex R2 being 1-sorted st R2 = ( p | nM ) . i & ( the carrier of p | nM ) . i = the carrier of R2 & ( the carrier of p | nM ) . i = the carrier of R2 & ( the carrier of p | nM ) . i = the carrier of R2
[. a , b + 1 / ( k + 1 ) .[ is Element of the _ of the carrier of f & ( the partial of f ) . k is Element of the carrier of f & ( the Subset of f ) . k is Element of the carrier of f & ( the Subset of f ) . k is Element of the carrier of f ) ;
Comput ( P , s , 2 + 1 ) = Exec ( P . 2 , Comput ( P , s , 2 ) ) .= Exec ( a3 := ( s . 0 ) , Comput ( P , s , 2 ) ) .= Exec ( a3 := ( s . 2 ) , Comput ( P , s , 2 ) ) ;
card ( h1 ) . k = power F_Complex * ( ( - 1_ F_Complex ) * u ) .= ( ( - 1_ F_Complex ) * u ) . k * Sum u .= ( ( - 1_ F_Complex ) * u ) . k * Sum ( ( - 1_ F_Complex ) * u ) .= ( ( - 1_ F_Complex ) * u ) . k ;
( f / g ) /. c = f /. c * ( g /. c ) " .= f /. c * ( 1 / g ) /. c .= ( f * ( 1 / g ) ) /. c .= ( f * ( 1 / g ) ) /. c ;
len C( C ) - len ( ( C ) - 1 ) = len ( C ) - len ( ( C ) - 1 ) .= len ( ( C ) - len ( ( C ) - 1 ) ) .= len ( ( C ) - len ( ( C ) - 1 ) ) ;
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom f /\ X .= dom ( r (#) ( f | X ) ) .= dom ( r (#) ( f | X ) ) /\ X .= dom ( r (#) ( f | X ) ) ;
defpred P [ Nat ] means for n holds 2 * Fib ( n + $1 ) = Fib ( n ) * Fib ( n ) + ( 5 * Fib ( n + $1 ) * Fib ( n + $1 ) * Fib ( n + $1 ) ) ;
consider f being Function of INT , INT such that f = f `1 and f is onto and ( n < k implies f " { f . n } ) & ( n < k implies f . n = n ) & ( n < k implies f . n = n ) ;
consider c9 being Function of S , BOOLEAN such that c9 = chi ( A \/ B , S ) and ( E . ( A \/ B ) ) = Prob . ( ( E . ( A \/ B ) ) /\ D ) and ( E . ( A \/ B ) ) /\ D = Prob . ( ( E . ( A \/ B ) ) /\ D ) ;
consider y being Element of Y ( ) such that a = "\/" ( { F ( x , y ) where x is Element of X ( ) : P [ x ] } , L ( ) ) and Q [ y ] ;
assume that A c= Z and Z = dom f and f = ( #Z 2 ) * ( ( sin + cos ) / ( sin + cos ) ) and for x st x in Z holds f . x = sin . x / ( sin . x ) ^2 / ( sin . x ) ^2 { 1 + x ^2 } and f . x = sin . x / ( sin . x ) ^2 { 1 + x ^2 } ;
( f /. i ) `2 = ( ( GoB f ) * ( 1 , j2 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 + 1 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 + 1 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 + 1 ) ) `2 ;
dom Seq ( Seq q2 ) = { j + len Seq q1 where j is Nat : j in dom Seq q1 & len Seq q2 = len Seq q2 & len Seq q2 = len Seq q2 & len Seq q2 = len Seq q2 } ;
consider G1 , G2 , G3 being Element of V such that G1 <= G2 & f is Morphism of G2 , G3 & g is Morphism of G1 , G2 & f is Morphism of G2 , G3 & g is Morphism of G2 , G3 & g is Morphism of G1 , G2 & g is Morphism of G2 , G3 & g is Morphism of G2 , G3 & g is Morphism of G1 , G2 ;
func - f -> PartFunc of C , V means : Def5 : dom it = dom f & for c st c in dom it holds it /. c = - f /. c * ( - f /. c ) ;
consider phi such that phi is increasing and for a st phi . a = a & {} <> a for v holds union ( L , v ) = a and for v holds v . ( union ( L , v ) ) = v . ( v . ( v . a ) ) ;
consider i1 , j1 such that [ i1 , j1 ] in Indices GoB f and f /. ( i1 + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i1 + 1 ) = ( GoB f ) * ( i1 , j1 ) ;
consider i , n such that n <> 0 and sqrt p = ( i / n ) * ( i1 - n ) and for n1 being Nat st n1 <> 0 & n <= len p holds sqrt p = ( i1 - n ) * ( i1 - n ) and n <= len p ;
assume that not 0 in Z and Z c= dom ( ( arccot * f1 ) `| Z ) and for x st x in Z holds ( ( arccot * f1 ) `| Z ) . x > - 1 & ( ( arccot * f1 ) `| Z ) . x > - 1 & ( ( arccot * f1 ) `| Z ) . x < 1 ;
cell ( G1 , i1 -' 1 , 2 |^ ( m -' 1 ) ) \ ( ( Y -' 1 ) * ( Y -' 1 ) ) c= BDD L~ f1 & ( Y -' 1 ) * ( Y -' 1 ) c= BDD L~ f1 & ( Y -' 1 ) * ( Y -' 1 ) c= BDD L~ f1 ;
ex Q1 being open Subset of X st s = Q1 & ex Fd being Subset-Family of [: Y , X :] st ( F is finite & d is finite & [#] ( Y ) = union ( Fd ) & [#] ( Y ) = union ( Fd ) & [#] ( Y ) = union ( Fd ) & [#] ( Y ) = union ( Fd ) ;
gcd ( A , ( ( the carrier of R ) * ( r1 , r2 ) , 1 ) , ( ( the carrier of R ) * ( r2 , s2 ) ) * ( r2 , s2 ) ) = 1 * ( ( r1 + r2 ) * ( r2 , s2 ) ) .= ( ( r1 + r2 ) * ( s2 + s2 ) ) * ( s2 + s2 ) ;
R8 = ( ( j , ( j + 1 ) ) --> ( ( j + 1 ) + 1 ) ) . ( m2 + 1 ) .= ( ( j , ( j + 1 ) ) --> ( m2 + 1 ) ) . ( m2 + 1 ) .= [ 3 , ( j + 1 ) + 1 ] .= [ 3 , ( j + 1 ) + 1 ] ;
CurInstr ( P-6 , Comput ( P3 , s3 , m1 + m3 ) ) = CurInstr ( P3 , Comput ( P3 , s3 , m3 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m3 ) ) .= halt SCMPDS .= halt SCMPDS .= halt SCMPDS ;
P1 /\ P2 = ( { p1 } \/ LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ) \/ ( LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ) \/ { p2 } /\ LSeg ( p1 , p2 ) \/ { p2 } ) .= { p2 } \/ { p2 } \/ { p2 } ;
func the still of f -> Subset of the Sorts of A means : - 5 = a iff ex i , p st i in dom f & p = f . i & a in the Sorts of A & p = f . i & p = f . i ;
for a , b being Element of F_Complex st |. a .| > |. b .| for f being Polynomial of F_Complex st f >= 1 holds f is \cap ( b * card f ) is >= >= 1 & f is \cup of ( b * card f )
defpred P [ Nat ] means 1 <= $1 & $1 <= len g implies for i , j st [ i , j ] in Indices G & G * ( i , j ) = G * ( $1 , j ) & 1 <= j & j <= width G & 1 <= i & i <= len G & 1 <= j & i <= width G & j <= width G holds G * ( i , j ) = G * ( i , j ) ;
assume that C1 , C2 are_`2 and for f , g being State of C1 , s1 , s2 being State of C2 st s1 = s2 & s2 = f * f holds s1 is stable & s2 = f * g & s1 is stable & s2 = f * f & s1 is stable & s2 = f * g ;
( ||. f .|| ) | X . c = ||. f .|| . c .= ||. f .|| /. c .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. f .|| /. c ;
|. q .| ^2 = ( q `1 ) ^2 + ( q `2 ) ^2 & 0 + ( q `1 ) ^2 < ( q `1 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 ;
for F being Subset-Family of [: T , T :] st F is open & not {} in F & for A , B being Subset of T st A in F & B in F & A <> B holds A misses B & B = F & A misses B holds card F = card B
assume that len F >= 1 and len F = k + 1 and len F = len G and len F = len H 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 |^ ( ( \mathop { \rm Seg n ) - i ) |^ s = i |^ ( s + k ) - i |^ s .= i |^ ( s + k ) - i |^ ( s + 1 ) .= i |^ ( s + 1 ) - i |^ ( s + 1 ) .= i |^ ( s + 1 ) - 1 .= i |^ ( s + 1 ) - 1 ;
consider q being oriented oriented Chain of G such that r = q and q <> {} and ( F . ( q . 1 ) = v1 & ( F . ( q . len q ) ) `1 = v2 & ( F . ( q . len q ) ) `1 = v1 & ( F . ( q . len q ) ) `1 = v2 & ( F . ( q . len q ) ) `1 = v2 ;
defpred P [ Element of NAT ] means $1 <= len s implies ( ( g , Z ) / I ) . $1 = ( ( ( g , Z ) / I ) +* ( $1 , Z ) ) . ( len ( g , Z ) + $1 ) & ( ( g , Z ) / I ) . ( len ( g , Z ) + $1 ) = ( ( g , Z ) / I ) . ( len ( g , Z ) + $1 ) ;
for A , B being square 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 A & 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 & s . i = a * b & s . i = b * a ;
func |( x , y )| -> Element of COMPLEX equals |( Re ( x ) , Re ( y ) )| - ( i * |( x , y )| ) + ( i * |( x , y )| ) + ( i * |( x , y )| ) + ( i * |( x , y )| ) ;
consider g2 be FinSequence of FH such that g2 is continuous & rng g2 c= A & g2 . 1 = x1 & g2 . len g2 = x2 & g2 . len g2 = y1 & g2 . len g2 = y1 & g2 . len g2 = x2 & len g2 = len g2 & len g2 = len g2 ;
then n1 >= len p1 & n2 >= len p1 implies crossover ( p1 , p2 , n1 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , A1 , A2 , n3 , n3 , n2 , n3 , n3 , n3 , A1 , n2 , n3 , n2 , n3 , n3 , A1 , n2 )
( q `1 ) ^2 * a <= q `1 & - ( q `1 ) ^2 * a <= ( q `1 ) ^2 * a or q `1 >= q `1 & - ( q `1 ) * a <= q `1 & - ( q `1 ) * a <= q `1 ) & - ( q `1 ) * a <= q `1 ;
( F . ( p9 . ( len p9 ) ) ) = ( F . ( p . ( len p9 ) ) ) * ( v . ( len p9 ) ) .= ( F . ( len p9 ) ) * ( v . ( len p9 ) ) .= ( F . ( len p9 ) ) * ( v . ( len p9 ) ) .= ( F . ( len p9 ) ) * ( v . ( len p9 ) ) ;
consider k1 being Nat such that k1 + k = 1 and a := k = ( <* a := intloc 0 *> ^ ( k1 --> SubFrom ( a , intloc 0 ) ) ) ^ <* halt SCM+FSA *> ^ ( k1 --> SubFrom ( a , intloc 0 ) ) ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ ( ( a *> ^ ( ( a *> ^ ( ( a *> ^ (
consider B8 being Subset of B1 , y8 being Function of B1 , A1 such that B8 is finite and D1 = _ 0 ( A1 , B1 , B2 ) and D1 = _ 0 ( A1 , B1 , B2 ) and D2 = _ 0 ( A1 , B1 , B2 ) and for i being Element of B1 holds B1 . i = B2 . i ;
v2 . b2 = ( ( curry F2 ) * ( ( curry id B ) * ( ( curry id B ) . b2 ) ) ) . b2 .= ( ( curry F2 ) * ( ( ( curry id B ) * ( ( the id B ) * ( ( the id B ) . b2 ) ) ) ) . b2 .= ( ( curry id B ) * ( ( ( the id B ) * ( the id B ) * ( the id B ) ) ) . b2 .= F2 ;
dom IExec ( I-35 , P , Initialize s ) = the carrier of SCMPDS .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) ;
ex d-32 be Real st d-32 > 0 & for h be Real st h <> 0 & |. h .| < d-32 holds |. h .| " * ||. ( R2 * ( L + R1 ) ) /. h .|| < ( e / ( 2 * ( L + R1 ) ) /. h .|| )
LSeg ( G * ( len G , 1 ) + |[ - 1 , 0 ]| , G * ( len G , 1 ) + |[ 1 , 0 ]| ) c= Int cell ( G , len G , 0 ) \/ { |[ 1 , 0 ]| } ;
LSeg ( mid ( h , i1 , i2 ) , i ) = LSeg ( h /. ( i + i1 -' 1 ) , h /. ( i + i1 -' 1 ) ) .= LSeg ( h , i1 + i1 -' 1 ) .= LSeg ( h , i1 ) ;
A = { q where q is Point of TOP-REAL 2 : LE p1 , q , P , p1 , p2 & LE q1 , q2 , P , p1 , p2 & LE q1 , q2 , P , p1 , p2 & LE q1 , q2 , P , p1 , p2 & LE q1 , q2 , P , p1 , p2 } = P & LE q1 , q2 , P , p1 , p2 } ;
( ( - x ) .|. y ) = ( - ( 1 / ( - x ) ) * ( x .|. y ) .= ( - ( 1 / ( - x ) ) * ( x .|. y ) ) * ( x .|. y ) .= ( - ( 1 / ( - x ) ) * ( x .|. y ) ) * ( x .|. y ) .= ( - ( 1 / ( - x ) ) * ( x .|. y ) ) * ( x .|. y ) .= ( - ( 1 / ( - x .|. y ) ) * ( x .|. y ) ;
0 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) = ( p `2 ) ^2 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) .= ( p `1 ) ^2 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) .= ( p `1 ) ^2 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) ;
( ( U * ( W * ( p - <* n *> ) ) ) * ( ( W * ( p - <* n *> ) ) ) = ( ( U * ( W - p ) ) ) * ( ( W - p ) * ( p - <* n *> ) ) .= ( ( U * ( p - q ) ) ) * ( ( W - p ) * ( p - q ) ) .= ( U * ( p - q ) ) * ( p - q ) ;
func Shift ( f , h ) -> PartFunc of REAL , REAL means : Def1 : dom it = - h & for x st x in dom it holds it . x = - ( x + h ) & for x st x in dom it holds it . x = - ( x * h . x ) & it . x = - ( 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 + 1 ) = G * ( i , j ) and f /. ( k + 1 ) = G * ( i , j ) ;
assume that not y in Free H and x in Free H and ( x in Free H ) and ( not x in Free H ) and ( x in Free H ) and ( x in Free H ) & ( x = y or x = y ) & ( x = y ) or x = y ) ;
defpred P11 [ Element of NAT , Element of NAT ] means ( ( $1 = p |^ 2 implies $2 = p |^ 2 ) & ( $1 = p |^ 2 implies $2 = p |^ 2 ) & ( $1 = p |^ 2 implies $2 = p |^ 3 ) & ( $1 = p |^ 2 implies $2 = p |^ 3 ) ;
func \sigma ( C ) -> non empty Subset-Family of X means : Def2 : for A being Subset of X holds A in it iff for W , Z being Subset of X st W c= A \ A & Z in it holds C . W = C . ( W \/ Z ) ;
[#] ( ( dist ( P ) ) .: Q ) = ( dist ( P ) ) .: Q & lower_bound ( [#] ( ( dist ( P ) ) .: Q ) = lower_bound ( ( dist ( P ) ) .: Q ) & lower_bound ( ( dist ( P ) ) .: Q ) = lower_bound ( ( dist ( P ) ) .: Q ) ;
rng ( F | ( [: S , S :] ) ) = {} or rng ( F | ( [: S , S :] , D ) ) = { 1 } or rng ( F | ( [: S , S :] , D :] ) = { 2 } or rng ( F | ( [: S , S :] , D ) ) = { 1 } ;
( f " ( rng f ) ) . i = f . i " . ( ( f " ( rng f ) ) . i ) .= f . i " . ( ( f " ( rng f ) ) . i ) .= f . i .= ( f " * f ) . i .= ( f " * f ) . i ;
consider P1 , P2 being non empty Subset of TOP-REAL 2 such that P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 c= P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P2 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P2 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P2 = P1 \/ P2 and P2 = P1 \/ P2 and P2 = P1 \/ P2 and P2 = P1 \/ P2 and P2 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and
f . p2 = |[ ( p2 `1 ) ^2 / sqrt ( 1 + ( p2 `2 / p2 `1 ) ^2 ) , ( p2 `2 ) ^2 / sqrt ( 1 + ( p2 `1 / p2 `2 ) ^2 ) ]| .= |[ ( p2 `1 ) ^2 / sqrt ( 1 + ( p2 `2 / p2 `1 ) ^2 ) , ( p2 `2 ) ^2 / sqrt ( 1 + ( p2 `1 / p2 `1 ) ^2 ) ]| ;
( ( \cal E } ( a , X ) ) " . x = ( ( \cal E } ( a , X ) qua Function ) " . x ) " .= ( ( \cal E } ( a , X ) ) " ) . x .= ( ( - a ) " ) * u ) " .= ( ( - a ) * u ) " .= ( ( - a ) * v ) " .= ( - a ) * u ) " .= ( - a ) * v ;
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 A & r < p & p in B & r < p holds ( for n being Point of NAT st n in dom ( in ( in A ) ) holds p in ( ( in A ) \ A ) . n ) & p in ( ( in A ) \ A ) . n
for i st i + 1 in dom F for G1 , G2 being strict normal Subgroup of G st G1 = F . i & G2 = F . ( i + 1 ) & G1 = F . ( i + 1 ) & G2 = F . ( i + 1 ) & for j st j in dom G1 holds G1 . j = G1 . ( j + 1 ) & G1 . ( j + 1 ) = G2 . j
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_continuous means : \rm : x0 in dom ( f /* a ) & for x0 st x0 in dom f & x0 in dom f & a is convergent & lim ( f /* a ) = x0 & for x0 st x0 in dom f & x0 <> x0 holds f /. x0 = lim ( f /* a ) ;
then X1 , X2 are_separated & ( X1 union X2 ) misses Y1 & ( Y1 , Y2 ) is SubSpace of X & ( Y1 union Y2 ) misses Y2 & ( Y1 union Y2 ) misses Y2 & ( Y1 union Y2 ) misses Y2 & ( Y1 union Y2 ) misses Y2 & Y2 misses Y1 & ( Y1 union Y2 ) misses Y2 implies Y1 = Y2 ) & ( Y1 union Y2 ) misses Y2 & Y2 = Y2 implies Y1 = Y2 & 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 - x0 ) + R . ( x - x0 ) + R . ( x - x0 )
( p2 `1 ) ^2 * sqrt ( 1 + ( p3 `1 / p3 `2 ) ^2 ) >= ( ( p2 `1 ) ^2 * sqrt ( 1 + ( p3 `1 / p3 `2 ) ^2 ) ) * sqrt ( 1 + ( p3 `1 / p3 `2 ) ^2 ) ;
( ( 1 - t1 ) (#) ( ||. f1 .|| ) ) to_power m = ( ( 1 - t1 ) (#) ( ||. g1 .|| ) ) to_power m & ( ( 1 - t1 ) (#) ( ||. g1 .|| ) ) to_power m = ( ( 1 - t1 ) (#) ( ||. g1 .|| ) ) to_power m & ( 1 - t1 ) (#) ( ||. g1 .|| ) ) to_power m = ( ( 1 - t1 ) (#) ( ||. g1 .|| ) ) to_power m ;
assume that for x holds f . x = ( ( - sin * ( cot . x ) ) (#) ( sin * ( cot . x ) ) ) & x in dom ( ( - sin * ( cot . x ) ) (#) ( sin * ( cot . x ) ) ) and for x st x in dom ( ( - sin * ( cot . x ) ) (#) ( sin * ( cot * ( cot . x ) ) ) ) holds ( ( - sin * ( cot * ( cot . x ) ) (#) ( cot * ( cot . x ) ) `| Z ) . x ) = ( - sin * ( cot ) ) `| Z ) . x ) = ( - sin . ( ( cot * ( cot . x ) ) ;
consider Xj1 being Subset of [: Y , Y :] , Y1 being Subset of [: X , Y :] such that t = [: Xj1 , Y1 :] and Y1 is open and Y1 is open & Y1 is open & Y2 is open & Y1 is open & Y1 is open & Y2 is open & Y1 is open & Y2 is open & Y1 is open & Y1 is open & Y1 is open ;
card S . n = card { card { [ d , Y ] + 1 , b , d ] where d is Element of GF ( p ) : [ d , Y ] in R & [ d , Y ] in R & [ d , Y ] in R } .= R |^ n ;
( ( W-bound D - W-bound D ) / ( 2 |^ n ) * ( i - 2 ) ) * ( i - 2 ) = ( W-bound D - W-bound D ) * ( i - 2 ) .= ( W-bound D - W-bound D ) * ( i - 2 ) .= ( W-bound D - W-bound D ) * ( i - 2 ) .= ( W-bound D - W-bound D ) * ( i - 2 ) ;