thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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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 ;
V ;
y in N ;
x in T ;
m < n ;
m <= n ;
n > 1 ;
let r ;
t in I ;
n <= 4 ;
M is finite ;
let X ;
Y c= Z ;
A // M ;
let U ;
a in D ;
q in Y ;
let x ;
1 <= l ;
1 <= w ;
let G ;
y in N ;
f = {} ;
let x ;
x in Z ;
let x ;
F is one-to-one ;
e <> b ;
1 <= n ;
f is special ;
S misses C
t <= 1 ;
y divides m ;
P divides M ;
let Z ;
let x ;
y c= x ;
let X ;
let C ;
x _|_ p ;
o is monotone ;
let X ;
A = B ;
1 < i ;
let x ;
let u ;
k <> 0 ;
let p ;
0 < r ;
let n ;
let y ;
f is onto ;
x < 1 ;
G c= F ;
a is_>=_than X ;
T is continuous ;
d <= a ;
p <= r ;
t < s ;
p <= t ;
t < s ;
let r ;
D <= E ;
assume e > 0 ;
assume 0 < g ;
p in Y ;
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 ` <= b ` ;
assume b in X ;
assume k <> 1 ;
f = Product l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is generated ;
assume m > 0 ;
assume A c= B ;
X is lower
assume A <> {} ;
assume X <> {} ;
assume F <> {} ;
assume G is open ;
assume f is dilatation ;
assume y in W ;
y \not <= x ;
A ` in B ` ;
assume i = 1 ;
let x be element ;
x ` = x ` ;
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 , v be set ;
let G be _Graph , v be set ;
let a be Complex ;
let x be element ;
let x be element ;
let C be FormalContext , a , b be Element of C ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
let y be Real ;
X c= f . a
let y be element ;
let x be element ;
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 <= 1 ;
let x be element ;
let k1 be Integer ;
let X be set ;
let a be element ;
let x be element ;
let x be element ;
let q be element ;
let x be element ;
assume f is being_homeomorphism ;
let z be element ;
a , b // K ;
let n be Nat ;
let k be Nat ;
B ` c= B ` ;
set s = Following s ;
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 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 = {} & h2 = {} ;
0 + 1 = 1 ;
o <> b2 ;
f2 is one-to-one ;
support p = {}
assume x in Z ;
i <= i + 1 ;
r1 <= 1 & r2 <= 1 ;
let n be Nat ;
a "/\" b <= a ;
let n be Nat ;
0 <= r3 ;
let e be Real , x be Real ;
not r in G . l
c1 = 0 & c2 = 0 ;
a + a = a ;
<* 0 *> in e ;
t in { t } ;
assume F is not discrete ;
m1 divides m ;
B * A <> {} ;
a + b <> {} ;
p * p > p ;
let y be ExtReal ;
let a be Int-Location , I be Program of SCM+FSA ;
let l be Nat ;
let i be Nat ;
let r ;
1 <= i2 ;
a "\/" c = c ;
let r be Real ;
let i be Nat ;
let m be Nat ;
x = p2 ;
let i be Nat ;
y < r + 1 ;
rng c c= E
Cl R is boundary ;
let i be Nat ;
R2 is open ;
cluster uparrow x -> closed ;
X <> { x } ;
x in { x } ;
q , b // M ;
A . i c= Y ;
P [ k ] ;
2 |^ x in W ;
X [ 0 ] ;
P [ 0 ] ;
A = A |^ i ;
that pion1 >= s ;
G . y <> 0 ;
let X be RealNormSpace , x be Element of X ;
a ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , W be Subspace of V ;
assume x in - M ;
k < s . a ;
not t in { p } ;
let Y be set , x be set ;
M , L are_isomorphic ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded continuous ;
rng f = Y ;
GL c= L ;
assume x in Cl Q ;
m in dom P ;
i <= len Q ;
len F = 3 ;
still_not-bound_in p = {} ;
z in rng p ;
lim b = 0 ;
len W = 3 ;
k in dom p ;
k <= len p ;
i <= len p ;
1 in dom f ;
b `1 = a `1 + 1 ;
x ` = a * y ;
rng D c= A ;
assume x in K1 ;
1 <= i0 ;
1 <= i0 ;
p0 `1 c= PI ;
1 <= ii ;
1 <= ii ;
UMP C in L ;
1 in dom f ;
let seq ;
set C = a * B ;
x in rng f ;
assume f is Lipschitzian ;
I = dom A ;
u in dom p ;
assume a < x + 1 ;
seq is bounded ;
assume I c= P1 ;
n in dom I ;
let Q ;
B c= dom f ;
b + p _|_ a ;
x in dom g ;
F-14 is continuous ;
dom g = X ;
len q = m ;
assume A2 is closed ;
cluster R \ S -> real-valued ;
sup D in S ;
x << sup D ;
b1 >= d1 ;
assume w = 0. V ;
assume x in A . i ;
g in PreNorms X ;
y in dom t ;
i in dom g ;
assume P [ k ] ;
if C c= f holds C c= f
x4 is increasing ;
let e2 be element ;
- b divides b ;
F c= \tau ( F ) ;
Gseq is non-decreasing ;
Gseq is non-decreasing ;
assume v in H . m ;
assume b in [#] B ;
let S be non empty non void ManySortedSign , x be set ;
assume P [ n ] ;
assume union S is linearly-independent & card 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 of s , I ;
b ` c= ( b ` ) ` ;
assume not x in [: NAT , NAT :] ;
A /\ B = { a } ;
assume len f > 0 ;
assume x in dom f ;
b , a // o , c ;
B in B-24 ;
cluster Product p -> non empty ;
z , x // x , p ;
assume x in rng N ;
cosec is_differentiable_in x & cosec is_differentiable_in x ;
assume y in rng S ;
let x , y be element ;
i2 < i1 & i2 < i2 ;
a * h in a * H ;
p , q in Y ;
cluster sqrt I -> left ideal ;
q1 in A1 & q2 in A2 ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A2 c= A1 ;
|. cA1 . n .| < 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 . x0 in dom f ;
g is continuous & f . 0 = 0 ;
assume O is symmetric ;
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 , P3 ;
d , c // a , b ;
let t , u ;
let X be set ;
assume k in dom s ;
let r be non negative Real ;
assume x in F | M ;
let Y be Subset of S ;
let X be non empty TopSpace , A be Subset of X ;
[ a , b ] in R ;
x + w < y + w ;
{ a , b } >= c ;
let B be Subset of A , C be Subset of A ;
let S be non empty ManySortedSign ;
let x be variable , f be Function of x , A , g be Function ;
let b be Element of X , c be Element of X ;
R [ x , y ] ;
x ` = x ;
b \ x = 0. X ;
<* d *> in D |^ 1 ;
P [ k + 1 ] ;
m in dom ( ( n + 1 ) -tuples_on NAT ) ;
h2 . a = y ;
P [ n + 1 ] ;
cluster G * F -> pre\kern1pt ;
let R be non empty multMagma , x be Element of R ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p ;
assume f | X is lower ;
x in rng ( go ^' pion1 ) ;
let x be Element of B ;
let t be Element of D ;
assume x in Q .last() ;
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 void maid non empty id the carrier of M ;
let N be non empty \mathclose { \rm id } of M ;
let R be RelStr with finite non empty RelStr ;
let n , k be Nat ;
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 destroy a ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < ( v - u ) ;
x <= c2 . x ;
x in F ` ;
cluster S --> T -> product z2 ;
assume t1 <= t2 & t2 <= t2 ;
let i , j be even Nat ;
assume F1 <> F2 & F2 <> F1 ;
c in Intersect ( union R ) ;
dom p1 = c & dom p2 = c ;
a = 0 or a = 1 ;
assume A1 <> A2 & A2 <> A1 ;
set i1 = i + 1 ;
assume a1 = b1 & b1 = b2 ;
dom ( g1 | A ) = A ;
i < len M + 1 ;
assume not - +infty in rng G ;
N c= dom ( f1 + f2 ) ;
x in dom ( sec | Z ) ;
assume [ x , y ] in R ;
set d = ( x / y ) / 2 ;
1 <= len ( g1 ) & 1 <= len g1 ;
len s2 > 1 & len s2 > 0 ;
z in dom ( f1 + f2 ) ;
1 in dom ( D2 | ( len D2 -' 1 ) ) ;
( p `1 ) ^2 = 0 ;
j2 <= width G & j1 <= width G ;
len PI > 1 + 1 ;
set n1 = n + 1 ;
|. ( q `1 ) / ( 1 + ( q `2 ) ) ^2 .| = 1 ;
let s be SortSymbol of S ;
gcd ( i , i ) = i ;
X1 c= dom f & X2 c= dom f ;
h . x in h . a ;
let G be not empty RelStr ;
cluster m * n -> invertible ;
let k9 be Nat ;
i -' 1 > m ;
R is transitive implies R ~ is transitive
set F = <* u , w *> ;
p0 `1 c= P3 `1 & p `2 <= p `2 ;
I is_closed_on t , Q & I is_halting_on t , Q ;
assume [ S , x ] is vertical ;
i <= len ( f2 | i ) ;
p is FinSequence of X ;
1 + 1 in dom g ;
Sum R2 = n * r ;
cluster f . x -> complex-valued ;
x in dom ( f1 + f2 ) ;
assume [ X , p ] in C ;
BX c= X0 & BX c= X ;
n2 <= ( 2 |^ ( n + 1 ) ) ;
A /\ ( TOP-REAL 2 ) c= A `
cluster x -element for Function ;
let Q be Subset-Family of S , S be Subset of T ;
assume n in dom g2 & n <= len g2 ;
let a be Element of R ;
t `2 in dom e2 & t `2 = e ;
N . 1 in rng N ;
- z in A \/ B ;
let S be SetSequence of X , T be Subset of S ;
i . y in rng i ;
[: REAL , REAL :] c= dom f ;
f . x in rng f ;
not mt <= ( r / 2 ) * ( r / 2 ) ;
s2 in r-5 & s2 in r-5 ;
let z , w be complex number ;
n <= NN . m ;
LIN q , p , s ;
f . x = waybelow x /\ B ;
set L = [ S \to T ] ;
let x be non empty Real ;
let m be Element of M ;
f in union ( rng F1 ) ;
let K be add-associative right_zeroed right_complementable associative distributive non empty doubleLoopStr , p , q be Polynomial of K ;
let i be Element of NAT ;
rng ( F * g ) c= Y
dom f c= dom x & dom f c= dom x ;
n1 < n1 + 1 & n1 <= len f + 1 ;
n1 < n1 + 1 & n1 <= len f + 1 ;
cluster [: X , Y :] -> \overline the carrier of [: X , Y :] ;
[ y2 , 2 ] `2 = z ;
let m be Element of NAT ;
let S be Subset of R ;
y in rng ( S29 ) ;
b = sup ( dom f ) ;
x in Seg ( len q ) ;
reconsider X = [: D , D :] as set ;
[ a , c ] in E1 ;
assume n in dom h2 & n in dom h2 ;
w + 1 = ( a1 + b1 ) ;
j + 1 <= j + 1 ;
k2 + 1 <= k1 + 1 ;
let i be Element of NAT ;
Support u = Support p & Support u = Support p ;
assume X is complete complete complete ;
assume that f = g and p = q ;
n1 <= n1 + 1 & n1 <= len f + 1 ;
let x be Element of REAL ;
assume x in rng ( s2 | X ) ;
x0 < x0 + 1 & x0 < x0 + 1 ;
len ( ( L5 ) * ( A * B ) ) = W ;
P c= Seg ( len A ) & P [ len A ] ;
dom q = Seg n & dom q = Seg n ;
j <= width M *' ;
let rseq be real-valued Real_Sequence ;
let k be Element of NAT ;
Integral ( M , Im f ) < +infty ;
let n be Element of NAT ;
assume z in in in in in in in in \mathop { 0 } ( A ) ;
let i be set ;
n -' 1 = n-1 - 1 ;
len ( n + 1 ) = n ;
\mathbin { Z , c } c= F
assume x in X or x = X ;
x is midpoint of b , c ;
let A , B be non empty set , x be Element of A , y be Element of B ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q & i in dom q ;
let s be Element of E -tuples_on E ;
let B1 be Basis of x , B ;
Carrier ( L2 ) /\ L2 = {} ;
L1 /\ L2 = {} & L2 /\ L2 = {} ;
assume ]. x , y .[ = ]. y , z .[ ;
assume b , c // b , c ;
LIN q , c , c ;
x in rng ( ( f . x ) `1 ) ;
set nn8 = n + j ;
let D7 be non empty set , f be Function of D , REAL ;
let K be add-associative right_zeroed right_complementable associative distributive non empty doubleLoopStr , M be Matrix of K ;
assume that f ' = f and h = h ;
R1 - R2 is total & R2 is total ;
k in NAT & 1 <= k & k <= len f ;
let a be Element of G ;
assume x0 in [. a , b .] ;
K1 ` is open & K1 is open ;
assume a , b ] is_a_maximal in C ;
let a , b be Element of S ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = Integral ( M , Im f ) ;
cluster as \vert \vert -\vert for Ordinal ;
not u in { cg } ;
the carrier of f c= B ;
reconsider z = x as VECTOR of V ;
cluster the connectives Str of L -> strict ;
r (#) H is partial ;
s . intloc 0 = 1 & s . intloc 0 = 1 ;
assume that x in C and y in C ;
let U2 be strict non-empty MSAlgebra over S , x be Element of U2 ;
[ x , Bottom T ] is compact ;
i + 1 + k in dom p ;
F . i is stable Subset of M ;
r[. r , y .] in TRUE ( { y } , { z } ) ;
let x , y be Element of X ;
let A , I be R_ of X ;
[ y , z ] in O ;
( that that that that that that that card Macro i = 1 and card Macro i = card I ;
rng Sgm A = A & Sgm A = A ;
q |- All ( y , 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 ;
( DD ) `2 = {} ;
n + 1 + 1 <= len g ;
a in \times [: Al , Al :] ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
let I be Ideal of L ;
set g = f1 + f2 , h = f3 + 4 ;
a <= max ( a , b ) ;
i-1 < len G + 1-1 ;
g . 1 = f . i1 ;
x ` , y in A2 ;
( f /* s ) . k < r ;
set v = VAL g ;
i -' k + 1 <= S ;
cluster non empty multiplicative for multMagma ;
x in support ( support ( t ) ) ;
assume a in [: the carrier of G , the carrier of G :] ;
i `2 <= len ( y | i ) ;
assume p divides b1 + b2 & q divides b2 + b3 ;
M1 <= sup M1 & sup M2 <= sup M2 ;
assume x in W-min ( X ) ;
j in dom ( z | ( len z ) ) ;
let x be Element of D ( ) ;
IC Comput ( P3 , s3 , k ) = l1 + 1 ;
a = {} or a = { x } ;
set uu = Vertices G , uv = Vertices G , uv = Vertices G ;
seq " is non-zero & seq " is non-zero ;
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
hK c= h-14 & h c= rng h ;
]. a , b .[ c= Z ;
X1 , X2 are_separated & X2 is not 0 implies X1 is not 0
a in Cl ( F \ G ) ;
set x1 = [ 0 , 0 ] , x2 = [ 0 , 0 ] , x3 = [ 0 , 0 ] , x4 = [ 0 , 0 ] , x4 = [ 0 , 1
k + 1 -' 1 = k ;
cluster empty for Relation ;
ex v st C = v + W ;
let IT be non empty RelStr , x be Element of X ;
assume V is Abelian add-associative right_zeroed right_complementable associative ;
X9 \/ Y in \sigma ( L ) ;
reconsider x = x as Element of S ;
max ( a , b ) = a ;
sup B is upper & sup B is upper ;
let L be non empty reflexive RelStr , x be Element of L , y be Element of L ;
R is_reflexive transitive transitive transitive X & R is_transitive X implies R is_transitive X
E , g |= H implies H = H
dom G ' _ { y } = a ;
sqrt ( 1 - ( r / 4 ) ^2 ) >= - r ;
G . p0 in rng G ;
let x be Element of Fs , y be Element of F ;
D [ Initialize ( ( card I ) + 2 ) ] ;
z in dom ( id B ) & z in dom ( id B ) ;
y in the carrier of N & y in the carrier of N ;
g in the carrier of H & h . ( len g ) = g . ( len g ) ;
rng ( f | X ) c= [: NAT , NAT :] ;
j `2 + 1 in dom ( s1 . n ) ;
let A , B be strict Subgroup of G ;
let C be non empty Subset of REAL ;
f . z1 in dom h & f . z2 in dom h ;
P . k1 in rng P & P . k1 in rng P ;
M = ( ( A +* B ) +* ( A +* B ) ) +* ( B +* C ) ;
let p be FinSequence of REAL , a be Element of REAL ;
f . n1 in rng f & f . n1 in rng f ;
M . ( F . 0 ) in REAL ;
ind [. a , b .] = b ;
assume that the distance of V , Q and Q is open ;
let a be Element of ^ ( V ) ;
let s be Element of , t be Element of Q ;
let PL be non empty RelStr ;
let n be Nat ;
the carrier of g c= B & the carrier of g c= B ;
I = halt SCM+FSA .= ( the InstructionsF of SCM+FSA ) . IC SCM+FSA ;
consider b being element such that b in B ;
set BK = BCS K , BK = BCS K ;
l <= ( -> -> net ) . j ;
assume x in ]. [ s , t ] , [ t , t ] } ;
( x - t ) in ]. t , s .[ ;
x in JumpParts ( JumpParts T ) & x in { 0 } ;
let h be Morphism of c , a ;
Y c= [: { \bf R } , { \bf R } } , { \bf R } :] ;
A2 \/ A3 c= Carrier ( L1 ) \/ Carrier ( L2 ) ;
assume LIN o , a , b ;
b , c // d1 , d2 ;
x1 , x2 , x3 is_collinear & x1 , x2 , x3 is_collinear ;
dom <* y *> = Seg 1 & dom <* y *> = Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar .| , m = |. ar .| , n = |. ar .| , n = |. ar .| , m = |. l .| ;
[ x , x `2 ] in X ~ ;
for n being Nat holds 0 <= x . n
|[ a , b ]| = [. a , b .] ;
cluster empty -> non empty for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 in P & q2 in P & q1 , q2 in P ;
dom ( M1 * M2 ) = Seg n & dom ( M1 * M2 ) = Seg n ;
x = [ x1 , x2 ] & y = [ x2 , x3 ] ;
let R , Q be ManySortedSet of A ;
set d = ( 1 / ( n + 1 ) ) * ( 1 / ( n + 1 ) ) ;
rng g2 c= dom W & g2 . 0 = W . 1 ;
P ( ( [#] Omega ) \ B ) <> 0 ;
a in field R & a = b ;
let M be non empty Subset of V , v be Element of V ;
let I be Program of SCM+FSA , a be Int-Location ;
assume x in rng ( ( R * \ { 0 } ) * ( R * \ { 0 } ) ) ;
let b be Element of the lattice of T ;
dist ( e , z ) < r-r ;
u1 + v1 in W2 & v1 in W2 + W3 ;
assume the carrier of L misses ( rng G ) ;
let L be lower-bounded non empty RelStr ;
assume [ x , y ] in [: a9 , b9 :] ;
dom ( A * e ) = [: NAT , NAT :] ;
let a , b be Vertex of G ;
let x be Element of Bool M ;
0 <= Arg a * PI ;
o , a9 // o , y ;
{ v } c= the carrier of l & { v } c= the carrier of l ;
let x be variable of A ;
assume x in dom ( uncurry f ) ;
rng F c= ( product f ) |^ X
assume D2 . k in rng D ;
f " . p1 = 0 & f " . p2 = 0 ;
set x = the Element of X , y = the Element of Y ;
dom Ser ( G ) = [: NAT , NAT :] ;
let n be Element of NAT ;
assume LIN c , a , e1 ;
cluster finite for FinSequence of NAT ;
reconsider d = c as Element of L1 ;
( v2 |-- I ) . X <= 1 ;
assume x in the carrier of f & y in the carrier of f ;
conv ( A , conv ( A ) ) c= conv ( A , conv ( A ) ) ;
reconsider B = b as Element of the carrier of T ;
J , v / ( P ! l ) |= All ( x , y , G ) ;
cluster J . i -> non empty for TopSpace ;
ex_sup_of Y1 \/ Y2 , T & ex_sup_of Y1 , T ;
W1 is_field W1 & W2 is_field W2 implies W1 is_Lin field W2
assume x in the carrier of R & y in the carrier of S ;
dom ( nn ) = Seg n & dom ( n ) = Seg n ;
s4 misses s4 & not ( not ex s st s in dom ( f | X ) ) ;
assume ( a 'imp' b ) . z = TRUE ;
assume that X is open and f = X --> d ;
assume [ a , y ] in Indices ( f | ( Seg n ) ) ;
assume that that that that that that that that that that that that that that that that that that stop I c= J and card I c= K and card I = card J ;
Im ( seq , n ) = 0 & Im ( seq , n ) = 0 ;
( ( ( sin * sin ) `| Z ) . x ) <> 0 ;
sin * sin is_differentiable_on Z & cos * sin is_differentiable_on Z ;
t1 . n = t2 . n .= s . n ;
dom ( ( exp_R * F ) `| Z ) c= dom F ;
W1 . x = W2 . x .= W2 . x ;
y in W .last() \/ W { x } \/ W { y } ;
k9 <= len ( vk ) & k <= len ( vk ) ;
x * a \equiv y * a . ( mod m ) ;
proj2 .: S c= proj2 .: P & proj2 .: P c= proj2 .: P ;
h . p4 = g2 . I .= g2 . I .= g2 . I ;
G6 = U /. 1 .= ( U /. 1 ) `1 ;
f . ( r1 ) in rng f ;
i + 1 + 1 <= len f ;
rng F = rng ( F . 0 ) & F . 1 = F . 1 ;
mode multiplicative non empty multMagma is well unital non empty multiplicative loop structure ;
[ x , y ] in A ~ ;
x1 . o in L2 . o . ( the Sorts of A ) . o ;
the support of ( m + 1 ) c= B ;
not [ y , x ] in id X & [ y , x ] in id X ;
1 + p .. f <= i + len f ;
seq ^\ k1 is lower & seq . k1 is lower ;
len ( F . m ) = len I & len ( F . m ) = len I ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be Complex , s1 , s2 be Real ;
Comput ( P , s , n ) . x = s . x ;
k <= k + 1 & k <= len p ;
reconsider c = {} _ { T } as Element of L ;
let Y be with_empty connected Subset of T ;
cluster -> directed-sups-preserving for Function of L , L ;
f . j1 in K . j1 ;
cluster J => y -> total for Function ;
K c= 2 -tuples_on the carrier of T & K is open ;
F . b1 = F . b2 & F . b2 = F . b1 ;
x1 = x or x1 = y or x1 = z or x2 = z ;
pred a <> {} means a = 1 & a = 1 ;
assume that succ a c= b and b in a ;
s1 . n in rng s1 & s1 . n in rng s1 ;
{ o , b2 } on C2 & { o , b1 } on C2 ;
LIN o , b , b9 & LIN o , b , a9 ;
reconsider m = x as Element of Funcs ( V , C ) ;
let f be non constant FinSequence of D ;
let F2 be non empty *> ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in F-8 ;
reconsider p9 = x as Subset of m , p9 = x as Subset of m ;
let A , B , C be Element of R ;
cluster non empty for union empty \mathopen of X ;
rng c ' misses rng ( e | rng c ) ;
z is Element of gr ( { x } ) ;
not b in dom ( a .--> p1 ) ;
assume that k >= 2 and P [ k ] ;
Z c= dom ( ( ( - cot ) * ( arccot ) ) `| Z ) ;
the component of Q c= ( ( TOP-REAL 2 ) | A ) \/ ( ( TOP-REAL 2 ) | A ) ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( 1 / 2 ) & g2 in dom ( 1 / 2 ) ;
pred f = u means a * f = a * u ;
for n holds P1 [ n ] ;
{ x . O : x in L } <> {} ;
let x be Element of V . s ;
a , b be Nat ;
assume that S = S2 and p = S2 and p = S2 and q = S2 ;
gcd ( n1 , n2 ) = 1 & gcd ( n1 , n2 ) = 1 ;
set o9 = ( 2 * PI ) * ( 2 * PI ) ;
seq . n < |. r1 - r2 .| ;
assume that seq is increasing and r < 0 and seq is increasing ;
f . ( y1 , x1 ) <= a ;
ex c being Nat st P [ c ] ;
set g = { n / ( 1 / ( 2 |^ n ) ) } ;
k = a or k = b or k = c or k = d ;
a9 , b9 , c9 is_collinear & a9 , b9 , a9 is_collinear ;
assume that Y = { 1 } and s = <* 1 *> ;
I1 . x = f . x .= 0 .= 1 . x .= 1 ;
W4 .last() = W3 . 1 .= 3 ;
cluster trivial for subgraph of G ;
reconsider u = u as Element of Bags X ( ) ;
A in B ^ A implies A , B are_lim
x in { [ 2 * n + 3 , k ] } ;
1 >= ( q `1 / |. q .| - cn ) / ( 1 + cn ) ;
f1 is_' f2 & f2 is_Q implies f1 is_Q
( f . q ) `2 <= ( q `2 ) ^2 + ( q `2 ) ^2 ;
h is_the \! \! Cage ( C , n ) ;
( b - a ) / ( p / ( p / 2 ) ) <= ( p / 2 ) / ( p / 2 ) ;
let f , g be functions of X , Y ;
S * ( k , k ) <> 0. K ;
x in dom ( max ( - f , - g ) ) ;
p2 in NO . p1 & p2 in NO . p2 ;
len ( H ) < len ( H ) & len ( H ) < len H ;
F [ A , F . A ] ;
consider Z such that y in Z and Z in X ;
pred 1 in C means A c= C |^ A ;
assume that r1 <> 0 or r2 <> 0 and r1 <> 0 ;
rng ( q1 + q2 ) c= rng ( C1 + C2 ) ;
A1 , A2 , p3 is_collinear ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in ' ( p , SS ) ;
then S is negative & P-2 [ S ] ;
Cl ( Int [#] T ) = [#] T & Cl ( Int ( [#] T ) ) = [#] T ;
f12 | A2 = f2 | A2 & f1 | A2 = f2 | ( A1 \/ A2 ) ;
0. M in the carrier of W & 0. M in the carrier of W ;
let v , u be Element of M ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ V ;
let X be Subset of S ~ ;
consider H1 such that H = 'not' H1 and H1 is open ;
{ 1 } c= ( 1 / t ) * ( \HM { the } \HM { function } \HM { cos } )
0 * a = 0. R .= a * 0. R .= a * 0. R ;
A |^ 2 = A |^ ( A , 2 ) ;
set vY = ( vY ) /. n , vY = ( the carrier of Y ) . n ;
r = 0. ( \langle \cal E , \Vert \cdot \Vert \rangle ) .= ||. 0. ( REAL-NS n ) .|| ;
( f . p4 ) ^2 >= 0 & ( f . p4 ) ^2 >= 0 ;
len W = len ( W | ( len W ) ) ;
f /* ( s * G ) is divergent_to+infty ;
consider l being Nat such that m = F . l ;
t8 & W7 & W7 does not destroy b1 & not IC t1 <= len b1 & IC t1 <= len b1 & IC t1 <= len b1 ;
reconsider Y1 = X1 , Y2 = X2 as SubSpace of X ;
consider w such that w in F and not x in w ;
let a , b , c , d be Real ;
reconsider i = i as non zero Element of NAT ;
c . x >= id L . x .= x ;
\sigma ( T ) \/ \omega ( T ) is Basis of T ;
for x being element st x in X holds x in Y
cluster [ x1 , x2 ] -> pair for set ;
sup { a } /\ \mathopen { a , t } is Ideal of T ;
let X be with_with with NAT non empty set , f be set ;
rng f = TS \kern1pt dSym ( S , X ) ;
let p be Element of B , x be SortSymbol of S ;
max ( N1 , 2 ) >= N1 & max ( N1 , 2 ) >= N1 ;
0. X <= ( b |^ m ) * ( mm1 ) ;
assume that i in I and R1 . i = R . i ;
i = j1 & p1 = q1 & p2 = q2 & q1 = q2 ;
assume gR in the right & g in the carrier of g & f . x = g . ( x , y ) ;
let A1 , A2 be Subset of S , A2 be Subset of T ;
x in h " ( P /\ [#] T1 ) /\ [#] T2 ;
1 in Seg 2 & 1 in Seg 3 & 2 in Seg 3 ;
reconsider X9 = X as non empty Subset of TN , Y = X as non empty Subset of TN ;
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the carrier of G ) -defined ;
n1 <= i2 + len g2 & i2 <= len g2 + 1 & i1 <= len g2 ;
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume v in the carrier' of G2 & v in the carrier' of G2 ;
y = Re y + ( Im y ) * i ;
( exists ( - 1 ) ) gcd p ) = 1 ;
x2 is_differentiable ]. a , b .[ & ex r st f . r = ( f . r ) * ( f . r )
rng ( M * ( i , j ) ) c= rng ( D2 * ( i , j ) ) ;
for p being Real st p in Z holds p >= a
\bf X \bf Y \bf Y \rm \hbox { - } coordinate } ( f ) = proj1 ( f ) ;
( seq ^\ m ) . k <> 0 ;
s . ( G . ( k + 1 ) ) > x0 ;
( p \! \mathop { \rm \hbox { - } count } M ) . 2 = d ;
A // ( B // A // B // C // B // C // C ;
h \equiv gg . ( ( mod P ) , ( m mod P ) ) ;
reconsider i1 = i-1 as Element of NAT , i2 be Element of NAT ;
let v1 , v2 be VECTOR of V , v be VECTOR of V ;
for V being Subspace of V holds V is Subspace of V
reconsider i9 = i - 1 as Element of NAT ;
dom f c= [: C , D :] & [: D , C :] c= [: D , D :] ;
x in ( the Sorts of B ) . n ;
len _ in Seg len _ ( f1 ) & len ( f1 ^ f2 ) = len ( f1 ^ f2 ) ;
p9 c= the topology of T & p9 c= the topology of T implies p9 in the topology of T
]. r , s .[ c= [. r , s .] ;
let B2 be Basis of T2 , x be Element of T2 ;
G * ( B * A ) = ( id o1 ) * ( ( the Sorts of o2 ) * ( the Arity of o2 ) ) ;
assume that p , u , v is_collinear and u , v , w thesis ;
[ z , z ] in union rng ( F . z ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S , S = $1 .. S , T = $1 .. S ;
LIN a1 , a3 , b1 & LIN a1 , b1 , c1 ;
f " ( f .: ( f .: x ) ) = { x } ;
dom ( w2 ) = dom ( r (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( 1 / 2 ) ) ) ) ) ;
assume that 1 <= i and i <= n and j <= n ;
( ( g2 ) . O ) `2 <= 1 ;
p in LSeg ( E . i , F . i ) ;
Iu * ( i , j ) = 0. K & Iu * ( i , j ) = 0. K ;
|. f . s . m - g .| < g1 ;
q9 . x in rng ( q | ( len q ) ) ;
Carrier ( Lpion1 ) misses Carrier ( Lpion1 ) ;
consider c being element such that [ a , c ] in G ;
assume that for o being OperSymbol of S holds o in o iff o in o & o in o ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= ( F |^ ( C + 1 ) ) * ;
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [. f . j , f . j .] ;
pred 0 <= x & x ^2 <= x ^2 & x ^2 <= 1 ;
p `2 <> 0. TOP-REAL 2 & p `2 <> 0. TOP-REAL 2 ;
cluster [: { 0 } , T :] -> non empty ;
let x be Element of S ~ ;
( \mathop { \rm \hbox { - } F } ) . ( a , b ) is one-to-one ;
|. i .| <= - ( - 2 |^ n ) ;
the carrier of I[01] = dom P & the carrier of I[01] = dom P ;
n * ( n + 1 ) ! > 0 * n ;
S c= ( A1 /\ A2 ) /\ ( A1 \/ A2 ) ;
a3 , a4 // a3 , b3 & a3 , a4 // a3 , b3 ;
then dom A <> {} & A <> {} & A <> {} ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , Y & y in Y & z in X implies x in Y
set v2 = v2 /. ( i + 1 ) , v2 = v2 /. ( i + 1 ) ;
x = r . n .= ( r . n ) * ( r . n ) ;
f . s in the carrier of S2 & f . s in the carrier of S2 ;
dom g = the carrier of I[01] & rng g c= the carrier of I[01] ;
p in Lower_Arc ( P ) /\ Lower_Arc ( P ) ;
dom d2 = A2 & dom d2 = A2 & dom d2 = B2 & dom d2 = A2 & dom d2 = B2 & dom d2 = A2 ;
0 < ( p / |. z .| - 1 ) / ( |. z .| - 1 ) ;
e . ( m + 1 ) <= e . ( m + 1 ) ;
B \ominus X \/ B \ominus Y c= B \ominus X /\ Y
- - +infty < Integral ( M , Im ( g | B ) ) ;
cluster O := F -> \HM empty for OperSymbol of X ;
let U1 , U2 be non-empty MSAlgebra over S , U2 be non-empty MSAlgebra over S ;
Proj ( i , n ) * g is_differentiable_on X ;
let x , y , z be Point of X , p be Point of X ;
reconsider p9 = p . x as Subset of V . x ;
x in the carrier of Lin ( A ) & y in the carrier of Lin ( A ) ;
let I , J be parahalting Program of SCM+FSA , a , b be Int-Location ;
assume - a is lower & b is lower & a <= - b ;
Int Cl ( A ) c= Cl ( Cl ( A ) ) ;
assume for A being Subset of X holds Cl A = A ;
assume q in Ball ( |[ x , r ]| , r ) ;
( p2 `2 / p2 `1 ) ^2 <= ( p2 `1 ) ^2 ;
Cl Q ` = [#] ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) |
set S = the carrier of T , T = the carrier of S ;
set II = ' ( f , n ) , II = ' ( f , n ) , II = ' ( f , n ) , II = ' ( f , n ) , II = ' ( f , n ) , II
len ( p /^ n ) = len ( p | n ) .= len p ;
A is Permutation of Funcs ( A , x , y ) ;
reconsider nn6 = n6 as Element of NAT ;
1 <= j + 1 & j <= len ( s . j ) ;
let q9 , q9 be Element of M , q be Element of M ;
a9 in the carrier of S1 & b9 in the carrier of S2 ;
c1 /. n1 = c1 . n1 & c2 /. n1 = c2 . n1 ;
let f be FinSequence of TOP-REAL 2 , p , q be Point of TOP-REAL 2 ;
y = ( ( f * SS ) . x ) * ( ( f * SS ) . x ) ;
consider x being element such that x in \mathop { \rm [ A , B ] } ;
assume r in ( ( dist ( o ) ) ) .: P ;
set i2 = Gauge ( C , n ) , i1 = Gauge ( C , n ) ;
h2 . ( j + 1 ) in rng h2 & h2 . ( j + 1 ) in rng h2 ;
Line ( M , k ) . i = M . i ;
reconsider m = ( x / 2 ) / ( 2 |^ ( m + 1 ) ) as Element of ( Cl ( x / 2 ) ) ;
let U1 , U2 be non-empty non-empty MSAlgebra over S , x be Element of U1 , y be Element of U2 ;
set P = Line ( a , d ) ;
len p1 < len p2 + 1 & len p2 = len p1 + 1 ;
let T1 , T2 be Scott Scott Scott Point of L , a be Element of T ;
then x <= y & ( x in Support ( x ) ) & y in Support ( y ) ;
set M = n -{ m } , N = n -\hbox { m } ;
reconsider i = x1 , j = x2 as Nat ;
rng ( ( the_arity_of o ) . n ) c= dom ( H . n ) ;
z1 " = z1 " * ( z1 " ) .= z1 " " * ( z1 " ) " .= z1 " " * ( z1 " ) " ;
x0 - sqrt ( r ^2 + ( 2 * r ) ^2 ) in L /\ dom f ;
then w is strict string of S , L & w in ( S , L ) ;
set x-10 = ( x ^ <* Z *> ) ^ <* Z *> ;
len w1 in Seg len w1 & len w2 = len w2 & len w1 = len w2 ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of PFuncs ( V , { k } ) ;
x . n = ( |. a . n .| ) * ( |. a . n .| ) ;
( p `1 ) ^2 <= ( G * ( 1 , 1 ) ) ^2 ;
rng ( ( g | X ) ^ ( g | X ) ) c= L~ ( g | X ) ;
reconsider k = i-1 * ( l + 1 ) as Nat ;
for n being Nat holds F . n is Seg ( n + 1 )
reconsider x9 = x-10 as VECTOR of M , w be VECTOR of M ;
dom ( f | X ) = X /\ dom f /\ X .= X /\ dom f ;
p , a // p , c & b , a // c , a ;
reconsider x1 = x as Element of ( REAL m ) -tuples_on REAL m ;
assume i in dom ( a * p ) ;
m . ( |. g .| ) = p . ( |. g .| ) ;
a / ( s . m ) - ( a / ( s . n ) ) <= 1 ;
S . ( n + k ) c= S . ( n + k ) ;
assume B1 \/ B2 = B2 \/ C2 & B2 \/ C2 = B2 \/ C2 & B2 \/ C2 = B2 \/ C2 ;
X . i = { x1 , x2 } . i .= { x1 , x2 } . i ;
r2 in dom ( ( h1 + h2 ) (#) ( h2 + h2 ) ) ;
||. 0. R .|| = a & b--& b---` = b ;
F8 is closed & P8 is_halting_on t1 , Q & P8 is_halting_on t1 , Q ;
set T = cluster cluster non empty for non empty TopSpace ;
Int Cl ( Cl R ) c= Int Cl R & Cl R c= Cl R ;
consider y being Element of L such that c . y = x ;
rng ( ( F . x ) | ( F . y ) ) = { F . x } ;
Gt " { c } c= B \/ S \/ S ;
fthere exists a relation of [: X , X :] , X st f = ( [: X , X :] --> ( f . x ) ) ;
set RQ = the Element of P , R = the Element of P , I = the Element of Q ;
assume that n + 1 >= 1 and n + 1 <= len M ;
let k2 be Element of NAT , n be Element of NAT ;
reconsider p9 = u as Element of ^ ( w , n ) , n = n as Element of ^ ( w , n ) ;
g . x in dom f & x in dom g implies x in dom g
assume that 1 <= n and n + 1 <= len f1 and f1 . n = f1 /. ( n + 1 ) ;
reconsider T = b * N as Element of G / ( N * N ) ;
len ( P\times ( P ) ) <= len ( ( P ^ ) ^ <* A *> ) + len ( ( P ^ <* A *> ) ^ <* A *> ) ;
x " in the carrier of A1 & y " in the carrier of A1 & x " in the carrier of A1 ;
[ i , j ] in Indices ( ( A @ ) @ ) ;
for m being Nat holds Re ( F . m ) is simple
f . x = a . i .= a1 . k .= a1 . k .= a1 . k ;
let f be PartFunc of REAL-NS i , REAL-NS n , x be Element of REAL n ;
rng f = the carrier of Lin ( A ) & f is one-to-one & f is one-to-one ;
assume s1 = sqrt ( 2 * ( p `1 / p `1 ) ^2 ) ;
pred a > 1 & b > 0 implies a |^ b > 1 ;
let A , B , C be Subset of [: I , J :] ;
reconsider X0 = X , Y = Y as RealNormSpace ;
let f be PartFunc of REAL , REAL , g be PartFunc of REAL , REAL ;
r * ( v1 |-- I ) . X < r * 1 ;
assume that V is Subspace of X and X is Subspace of V ;
tK , tK be Relation ;
Q [ e \/ { v2 } , f ] & [ e , f ] in [: f , g :] ;
g \circlearrowleft ( ( E-max L~ z ) .. z ) = z ;
|. |[ x , v ]| - |[ x , y ]| .| = v-2 ;
- f . w = - ( L * w ) .= - ( L * w ) ;
z -' y <= x iff z <= x + y & z <= z
sqrt ( 7 / ( 1 + e ) ^2 ) > 0 ;
assume X is BCI-algebra & X is BCK-algebra & 0 < 0 & 0 < 0 ;
F . 1 = v1 & F . 2 = v2 & F . 3 = v2 ;
( f | X ) . x2 = f . x2 ;
( ( ( tan * tan ) `| Z ) . x ) ^2 + ( tan * tan . x ) ^2 < 1 ;
i2 = ( f /. len f ) & i2 = ( f /. len f ) .. f ;
X1 = X2 \/ ( X1 \ X2 ) .= ( X1 union X2 ) \ ( X2 \ X1 ) ;
[. a , b , \bf 1_ G .] = 1_ G & [. a , b .] = 1_ G ;
let V , W be non empty VectSpStr over F_Complex , f be Function of V , W ;
dom g2 = the carrier of I[01] & 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] & g2 is one-to-one ;
( proj2 | X ) .: X = proj2 .: ( ( proj2 | X ) .: X ) ;
f . ( x , y ) = h1 . ( x , y ) ;
x0 - r1 < a1 . n - a1 . n ;
|. ( f /* s ) . k - G . ( k + 1 ) .| < r ;
len Line ( A , i ) = width A & width Line ( A , i ) = width A ;
Sthesis .: S = ( S . g ) .: S ;
reconsider f = v + u as Function of X , the carrier of Y ;
intloc 0 in dom Initialized Initialized Initialized ( s ) & IC Comput ( p , s , 0 ) in dom Initialized ( s ) ;
i1 <> i2 & i2 <> j2 & i1 <= i2 & i2 <= j2 & j2 <= i2 & i2 <> j2 & i1 <> i2 & i2 <> j2 & i1 <> i2 & i1 <> i2 & i2 <> j2 & i1 <> i2 & i2 <> j2 & i1 <> i2 & i2 <> j2 & i1 <> i2
( arccot r + 1 ) * ( 1 / ( 2 * r ) ) = ( PI / 2 ) * ( 1 / ( 2 * r ) ) ;
for x st x in Z holds f2 * ( exp_R * f ) is_differentiable_in x
reconsider q2 = ( q / x ) / ( q / x ) as Element of REAL ;
( 0 qua Nat ) + 1 <= i + j1 ;
assume f in the carrier of [ X , Omega Y ] ;
F . a = H / ( ( x , y ) / ( a , b ) ) ;
not ( ( ex u st u in T ) ) & ( u in T ) & ( u in T ) ;
dist ( ( a * seq ) . n , h ) < r ;
1 in the carrier of [. 0 , 1 .] & 1 <= len ( G . ( 1 , 1 ) ) ;
( p2 `1 - p1 `1 ) ^2 - p1 `1 / p1 `1 > ( p2 `1 - p1 `1 ) ^2 ;
|. r1 - r2 .| = |. a1 - a2 .| * |. a2 - x .| ;
reconsider S-14 = 8 as Element of Seg 8 ( ) ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b
DWW = DWW \! ( k + 1 ) ;
i1 = ( - a ) + n & i2 = ( - a ) + n ;
f . a [= f . ( f . O1 "\/" f . a ) ;
pred f = v & g = u , v = v + u , w = v + u ;
I . n = Integral ( M , F . n ) ;
[: [: T1 , T2 :] , [: T2 , T2 :] . s = 1 ;
a = VERUM ( A ) or a = VERUM ( A ) or a = VERUM ( A ) ;
reconsider k2 = s . b3 as Element of NAT , k1 , k2 being Element of NAT ;
( Comput ( P , s , 4 ) . GBP ) . GBP = 0 ;
L~ M1 meets L~ Rmin ( Rmin ( Rmin ( Rmin ( Rmin ( Rmin ( Rmin ( Rmin ( Rmin ( M1 , R ) , 1 ) , 1 ) , 1 ) , 1 ) ) ;
set h = the continuous Function of X , R , S be Function of X , R ;
set A = { L . ( k9 . n ) : not contradiction } ;
for H st H is negative holds P7 [ H ] ;
set breal = S5 . ( i + 1 ) , Swhere i is Element of NAT : i < ( len S ) + 1 } ;
Hom ( a , b ) c= Hom ( a opp , b opp ) ;
sqrt ( 1 - ( n + 1 ) ) < sqrt ( 1 - ( n + 1 ) ) ;
( l ) `1 = [ [ dom l , cod l ] , cod l ] `1 .= [ [ dom l , cod l ] , cod l ] ;
y +* ( i , y /. i ) in dom g & y . i = g . i ;
let p be Element of CQC-WFF ( Al ) , d be Element of CQC-WFF ( Al ) ;
X /\ X1 c= dom ( f1 - f2 ) /\ dom ( f2 - f3 ) ;
p2 in rng ( f /^ ( len f -' 1 ) ) & p2 in rng ( f /^ ( len f -' 1 ) ) ;
1 <= indx ( D2 , D1 , j1 ) + 1 ;
assume x in K1 /\ K1 & y in ( ( TOP-REAL 2 ) | K1 ) \/ ( ( ( TOP-REAL 2 ) | K1 ) ) /\ K1 ;
- 1 <= ( ( f2 . O ) . I ) `2 ;
let f , g be Function of I[01] , TOP-REAL 2 , R^1 , a , b , c be Real ;
k1 -' k2 = k1 - k2 + 1 & k1 -' k2 = k1 -' k2 + 1 ;
rng seq c= ]. x0 - r , x0 .[ & rng seq c= ]. x0 - r , x0 + r .[ ;
g2 in ]. x0 - r , x0 + r .[ & g2 in ]. x0 - r , x0 + r .[ ;
sgn ( p `1 , K ) = - ( - 1_ K ) .= - 1_ K .= - 1_ K ;
consider u being Nat such that b = ( p |^ y ) * u ;
redefine pred a is the ^ of A means : Def1 : a = Sum A ;
Cl ( ( Cl H ) \/ ( Cl H ) ) = union ( ( Cl H ) \/ ( Cl H ) ) ;
len t = len t1 + len t2 & len t1 = len t2 + len t2 ;
v-29 = v + w |-- ( A + B ) ;
cv <> DataLoc ( t1 . GBP , 3 ) & card I <> 0 ;
g . s = sup ( d " { s } ) ;
( \dot y ) . s = s . ( \dot y ) ;
{ s : s < t & t in Q + + } c= [: { 0 } , REAL+ :]
s ` \ s = s ` \ ( s ` \ ( s ` ) ) .= ( s ` \ ( s ` \ ( s ` ) ) ` ) ` ;
defpred P [ Nat ] means B + 1 in A & B + 1 in A ;
( 339 + 1 ) ! = 3139 * ( 339 + 1 ) ;
U . ( succ A ) = U . ( ( succ A ) . ( succ A ) ) ;
reconsider y = y as Element of ( len y ) -tuples_on COMPLEX ( D ) ;
consider i2 being Integer such that y2 = p * i2 and i2 in dom f and x = f . i2 ;
reconsider p = Y | ( Seg k ) as FinSequence of ( the carrier of G ) * ;
set f = ( S , U ) -TruthEval z , g = ( S , U ) -TruthEval z , F = ( S , U ) -TruthEval z , F = ( S , U ) -TruthEval z , F = ( S , U ) -TruthEval z , F = ( 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 , TOP-REAL n , a , b be Real ;
. [ n + i , 'not' A ] <> 1 ;
ex r being Real st x = r & a <= r & r <= b ;
R1 , R2 be Element of [: { 0 } , the carrier of n :] , the carrier of n :] ;
reconsider l = 1_ ( V ) as Linear_Combination of A , B ;
set r = |. e .| + |. w .| + |. w .| ;
consider y being Element of S such that z <= y and y in X ;
a 'or' b 'or' c = 'not' ( a 'or' b ) 'or' ( a 'or' c ) ;
||. ( x - g ) . m - g .|| < r2 & ||. ( x - g ) . m - g .|| < r2 ;
b9 , a9 // b9 , c9 & b9 , c9 // c9 , a9 ;
1 <= k2 -' k1 & k2 + 1 + 1 = k2 & k2 + 1 = k2 + 1 ;
sqrt ( ( p `1 / |. p .| - sn ) / ( 1 + sn ) ) ^2 >= 0 ;
sqrt ( ( q `1 / |. q .| - sn ) / ( 1 + sn ) ) ^2 < 0 ;
E-max C in cell ( RR , 1 , 1 ) & not C in cell ( RR , 1 , 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 `1 , a `2 // a `1 , b `2 or p `1 = b ;
g . n = a * Sum ( f | n ) .= f . n * f . n ;
consider f being Subset of X such that e = f and f is x0 and f is x0 ;
F | ( N2 \times S ) = CircleMap * ( ( CircleMap * F ) | ( N2 * J ) ) ;
q in LSeg ( q , v ) \/ LSeg ( v , p ) ;
Ball ( m , r1 ) c= Ball ( m , s ) & Ball ( m , s ) c= Ball ( m , s ) ;
the carrier of ( V ) = { 0. V } & the carrier of ( V ) = { 0. V } ;
rng ( ( ( ( - 1 ) (#) ( ( ( id Z ) ^ ) ) (#) ( ( id Z ) ^ ) ) ) `| Z ) = [. - 1 , 1 .] ;
assume that Re ( seq ) is summable and Im ( seq ) is summable and Im ( seq ) is summable and Im ( seq ) is summable ;
||. ( vseq . n ) - ( vseq . m ) .|| < e ;
set g = O --> 1 ;
reconsider t2 = t11 , t2 = the string of S2 , t2 = the string of S2 , E = the string of S2 ;
reconsider x9 = seq . n as sequence of ( TOP-REAL n ) * ;
assume that not E-max L~ go meets L~ go and not E-max L~ go meets L~ go and not go in L~ go and not go in L~ go and not go in L~ pion1 and not go in L~ co and not go in L~ co ;
- ( - 1 / ( n + 1 ) ) < F . n - ( F . x ) ;
set d1 = dist ( x1 , z1 ) , d2 = dist ( x2 , z1 ) , d2 = dist ( x2 , z1 ) ;
2 |^ ( 2 -' 1 ) = ( 2 |^ ( 2 -' 1 ) ) - 1 ;
dom ( ( - 1 ) (#) ( ( id Z ) ^ ) ) = Seg len ( ( - 1 ) (#) ( ( id Z ) ^ ) ) ;
set x1 = - ( k + 1 ) * |. x1 - x2 .| + |. x1 - x2 .| ;
assume for n being Element of X holds 0. <= F . n & 0. X <= F . n ;
assume that 0 <= T-32 . i and T-32 . ( i + 1 ) <= 1 and T-32 . ( i + 1 ) <= 1 ;
for A being Subset of X holds c . ( c . A ) = c . A
the carrier of ( Carrier ( L2 ) + L2 ) c= I1 & the carrier of ( Carrier ( L2 ) ) c= I2 ;
'not' All ( x , p ) => All ( x , p ) is valid ;
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the normal normal w.r.t. A ;
Z c= dom ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 1 / 2 ) ) ) ) ) ) * ( ( ( ( ( ( ( 1 / 2 ) ) ) * ( ( ( ( ( ( 1 / 2 ) ) ) ) * ( ( ( ( 1 /
|. 0. TOP-REAL 2 - ( q `1 / |. q .| - sn ) ) .| < r ;
ConsecutiveSet2 ( B , succ B ) c= ConsecutiveSet2 ( A , succ ( d , succ ( d , succ ( d , succ ( d , C ) ) ) ) ;
E = dom ( L . n ) & Carrier ( L . n ) c= 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 W2 & the carrier of W2 c= the carrier of W2 ;
I . IC Comput ( P , s , 0 ) = P . IC Comput ( P , s , 0 ) ;
pred x > 0 means : Def1 : 1 / ( x ^2 ) = x ^2 / ( x ^2 ) ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) \/ LSeg ( f , i ) ;
consider p being Point of T such that C = [. p , R .] and p in C ;
b , c are_connected & - a , b are_connected & - a , b are_connected & - a , b are_connected & - a , b are_connected ;
assume f = id the carrier of O & f is Function of the carrier of O , the carrier of O ;
consider v such that v <> 0. V and f . v = L * v ;
let l be Linear_Combination of {} ( ( the carrier of V ) \ { v } ) ;
reconsider g = f " as Function of U2 , U2 , E , F , G ;
A1 in the carrier of ( ( k + 1 ) , ( k + 1 ) , ( k + 1 ) ) ;
|. - x .| = - ( - x ) .= - x .= - x .= - x .= - x ;
set S = \mathop { \rm in in in in in in { x , y , c } ;
Fib ( n ) * ( 5 * Fib n ) >= 4 * log ( n , 5 ) ;
v3 /. ( k + 1 ) = v3 . ( k + 1 ) ;
0 mod i = - ( i * ( 0 qua Nat ) ) mod ( i + 1 ) ;
Indices M1 = [: Seg n , Seg n :] & width M1 = n & width M2 = n ;
Line ( SLine ( SLine ( M , j ) , j ) , i ) = SLine ( SLine ( M , j ) , i ) ;
h . ( x1 , y1 ) = [ y1 , y2 ] & [ y1 , y2 ] `1 = [ y2 , y1 ] `1 ;
|. f .| is_integrable_on Re ( |. f .| (#) ( ( |. b .| (#) h ) (#) ( |. a .| (#) ( |. b .| (#) h ) ) ) is nonnegative ;
assume x = ( a1 ^ <* b1 *> ) ^ <* b1 *> ^ ( a2 ^ <* b2 *> ) ;
MJ is_halting_on IExec ( I , P , s ) , P & I is_halting_on Initialized s , P ;
DataLoc ( t3 . a , 4 ) = intpos ( 0 + 4 ) .= intpos ( 0 + 4 ) ;
x + y < - x + y & |. x - y .| = - x + y ;
LIN c , q , b & LIN c , q , b ;
f\rbrack . ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + y2 ) .= x1 + ( y1 + y2 ) ;
f[ a , f ] . a = f[ a , f ] & v in InputVertices S & v in InputVertices S & [ v , f ] . a = v ;
( p `1 ) ^2 <= ( ( E-max C ) `1 ) ^2 + ( ( E-max C ) `1 ) ^2 ;
set R8 = Cage ( C , n ) \circlearrowleft E8 , E8 = Cage ( C , n ) ;
( p `1 ) ^2 >= ( ( E-max C ) `1 ) ^2 + ( ( E-max C ) `1 ) ^2 ;
consider p such that p = p9 and s1 < p and p < s2 and p in L~ f and f . p = f /. i ;
|. ( f /* ( s * F ) ) . l - G . l .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len Line ( N , k + 1 ) = width N & width Line ( N , k + 1 ) = width N ;
f1 /* s1 is convergent & f2 /* s1 is convergent & lim ( f2 * f1 ) = f2 . ( lim s1 ) ;
f . x1 = x1 & f . y1 = y1 & f . y2 = y2 & f . y2 = y2 ;
len f <= len f + 1 & len f + 1 <> 0 & len f + 1 <> 0 & len f + 1 <= len f ;
dom ( Proj ( i , n ) * s ) = REAL m & rng ( Proj ( i , n ) * s ) = REAL m ;
n = k * ( 2 * t + 1 ) + ( n mod ( 2 * t ) ) ;
dom B = 2 -tuples_on the carrier of V \ { {} } & rng B c= the carrier of V ;
consider r such that r _|_ a and r _|_ x and r _|_ y and r _|_ y ;
reconsider B1 = the carrier of X1 , B2 = the carrier of X2 as Subset of X ;
1 in the carrier of [. 1 / 2 , 1 .] & 1 / 2 <= 1 / 2 ;
for L being complete LATTICE holds C is complete implies L is isomorphic & L is isomorphic
[ go , go ] in [: I \ I , I \ { i } :] \ [: I , I :] ;
set S2 = 1GateCircStr ( x , y , c ) , S2 = 1GateCircStr ( x , y , c ) ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and for x st x in dom f1 holds f1 . x = - ( f1 . x ) ^2 ) and for x st x in dom f1 holds f1 . x = - ( f1 . x ) ^2 ;
reconsider y = ( a " ) / ( a " ) / ( a " ) as Element of L ;
dom s = { 1 , 2 , 3 } & s . 1 = d1 & s . 2 = d2 ;
( min ( g , max ( f , g ) ) ) . c <= h . c ;
set G2 = the subgraph of G , v = the Vertex of G , X = the carrier of G , Y = the carrier of G , X = the carrier of G , Y = the carrier of G , Y = the carrier of G , Z = the carrier of G , Z = the carrier of G , Y
reconsider g = f as PartFunc of REAL , REAL-NS n , REAL-NS n ;
|. s1 . m .| / |. p . m - ( lim s1 ) .| < d / ( p . m - ( lim s1 ) ) ;
for x being element st x in B holds x in ( E . u ) & x in ( E . v ) ;
P = the carrier of ( ( TOP-REAL n ) | K1 ) .= ( ( TOP-REAL n ) | K1 ) ;
assume p10 in LSeg ( p1 , p2 ) /\ LSeg ( p2 , p3 ) /\ LSeg ( p1 , p2 ) ;
( 0. X \ x ) |^ ( m + 1 ) = 0. X ;
let g be Element of Hom ( cod f , cod f ) ;
2 * a * b + ( 2 * c ) <= 2 * C1 + ( 2 * c ) ;
let f , g , h be PartFunc of X , the carrier of X , Y ;
set h = Hom ( a , g ) and k = g ;
then idseq ( n ) | Seg m = idseq ( m ) . ( n + m ) ;
H * ( g " * a ) in the carrier of H & H * ( g " * a ) in the carrier of H ;
x in dom ( ( ( - 1 ) (#) ( ( id Z ) ^ ) ) `| Z ) ;
Int cell ( G , i1 , j2 -' 1 ) misses C & Int cell ( G , i1 , j2 -' 1 ) misses C ;
LE q2 , p2 , P , p1 , p2 & LE q2 , p2 , P , p1 , p2 ;
attr B is an component means : Def1 : B c= BDD A & B c= BDD A ;
deffunc D ( set , set ) = union rng $2 & $2 = union rng $2 & $2 = union rng $2 ;
n + - n < len ( p + - n ) + ( - n ) ;
pred a <> 0. K means for M , N being Matrix of K holds the_rank_of ( a * M ) = the_rank_of ( a * M ) ;
consider j such that j in dom \mathbb J and I = len \cal J + j and I = len an + j ;
consider x1 such that z in x1 and x1 in ( P . n ) and x1 in ( P . m ) and x = [ x1 , x1 ] ;
for n holds X [ n , r ] ;
set C1 = Comput ( P2 , s2 , i + 1 ) , C2 = Comput ( P2 , s2 , i + 1 ) , C2 = Comput ( P2 , s2 , i + 1 ) , C2 = Comput ( P2 , s2 , i + 1 ) , C2 = Comput ( P2 , s2
set \cal v = 3 / ( 2 , <* a , b *> ) , w = 3 / ( 2 , <* b , c *> ) , n = 2 / ( 2 , <* a , c *> ) , m = 3 / ( 2 , <* a , b *> ) , n = 3 / ( 2 ,
conv ( @ W ) c= union ( ( F .: ( E " W ) ) ) ;
1 in [. - 1 , 1 .] /\ dom ( ( arccot ) (#) ( arccot ) ) ;
r3 <= s1 + ( r2 - s2 ) * ( 1 / 2 ) ;
dom ( f (#) ( ( f (#) g ) `| Z ) ) = dom f /\ dom ( ( f (#) g ) `| Z ) ;
dom ( f * G ) = dom ( l (#) F ) /\ Seg k .= Seg k ;
rng ( s ^\ k ) c= dom f1 \ { x0 } & rng ( s ^\ k ) c= dom f1 \ { x0 } ;
reconsider g4 = go . p as Point of ( TOP-REAL n ) | K1 , R^1 ;
( T * h . s ) . x = T . ( h . s ) ;
I . ( J . x ) = ( I * L ) . x ;
y in dom ( ( Frege the multF of A ) . o ) & y in ( ( Frege the e of A ) . o ) ;
for I being non degenerated doubleLoopStr holds I is commutative iff I is commutative
set s2 = s +* Initialize ( ( intloc 0 ) .--> 1 ) , P2 = P +* Initialize ( ( intloc 0 ) .--> 1 ) , P2 = P +* stop I , s2 = Comput ( P2 , s2 , 1 ) , P2 = P +* stop I , s2 = Comput ( P2 , s2 , 1 ) , P4 = P3 ;
P1 /. IC Comput ( P1 , s1 , k ) = P1 . IC Comput ( P1 , s1 , k ) .= P1 . IC Comput ( P1 , s1 , k ) ;
lim S1 in the carrier of [. a , b .] & lim S1 in [. a , b .] ;
v . i = ( v *' lU ) . i .= ( v *' l ) . i ;
consider n being element such that n in NAT and x = seq . n and x = seq . n ;
consider x being Element of c such that F1 . x <> F2 . x and F1 . x = F2 ( x ) ;
card cluster cluster cluster cluster cluster cluster cluster empty , 0 , x1 , x2 , x3 , x3 , x4 , x1 , x2 , x3 , x4 , x1 , x2 , x3 , x4 , x1 , x2 , x3 , x4 , x2 , x3 , x4 , x1 , x2 , x3 , x4 , x1 , x2 , x3 , x4 ,
j + ( 2 * ( k + 1 ) ) + m1 > j + ( 2 * ( k + 1 ) ) ;
{ s , t } on A2 & { s , t } on A3 & { s , t } on A3 ;
n1 > len crossover ( p2 , p1 , n1 , n2 ) & n1 <= len crossover ( p2 , p1 , n1 ) ;
( ( ( ( ( g2 ) ) . HT ( g2 , T ) ) ) *' ( ( g2 ) . HT ( g2 , T ) ) ) = 0. L ;
then H1 , H2 are_group & ( H1 , H2 ) is are group & ( H1 , H2 ) , ( H1 , H2 ) U U U U U U U U ) ;
( E-max L~ f ) .. f > 1 & ( E-max L~ f ) .. f > 1 ;
]. s , 1 .[ = ]. s , 2 .[ /\ [. 0 , 1 .] ;
x1 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) ) & x2 in ( ( TOP-REAL 2 ) | ( L~ g ) ) ;
let f1 , f2 be continuous PartFunc of REAL , REAL , x0 be Point of S , x1 be Point of S ;
DigA ( tmax ( k , ( k + 1 ) , z ) , ( k + 1 ) ) is Element of k -tuples_on k -tuples_on D ;
I : I = d_ 2 & I = k2 & I = k2 & I = k2 & I = k2 & I = k2 & I = k2 & I = k2 & I = k2 ;
uI ~ = { [ a , uI ] , [ a , uI ] } ;
( w | p ) | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w , p ) ) ) ) ) ) ) ) ) = p ;
consider u2 such that u2 in W2 and x = v + u and u in W2 and v in W2 and u = v + u2 ;
for y st y in rng F ex n st y = a |^ n & n >= 1
dom ( ( g * ( ( id V ) \dot \to C ) ) | K ) = K ;
ex x being element st x in ( ( the Sorts of U2 ) \/ ( the Sorts of U2 ) ) . s ;
ex x being element st x in ( ( the Sorts of O ) \/ A ) . s & x in ( the Sorts of O ) . s ;
f . x in the carrier of [. - r , 0 .] & f . x in [. - r , 1 .] ;
( the carrier of X1 union X2 ) /\ ( the carrier of X2 ) <> {} & ( the carrier of X1 union X2 ) /\ ( the carrier of X2 ) <> {} ;
L1 /\ LSeg ( p01 , p2 ) c= { p2 } /\ { p2 } ;
sqrt ( b + ( bC ) / 2 ) in { r : a < r & r < b + ( - C ) / 2 } ;
ex_sup_of { x , y } , L & x "\/" y = sup { x , y } ;
for x being element st x in X ex u being element st P [ x , u ]
consider z being Point of G8 such that z = y and P [ z ] and P [ z ] ;
( the Lipschitzian thesis of ( ( the sequence of \overline the carrier of X ) ) . ( thesis , the carrier of X ) ) <= e ;
len ( w ^ w2 ) + 1 = len w + ( len w + 1 ) ;
assume q in the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 & q in the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 ;
f | E-4 = g | E-4 .= ( g | EK ) | EK .= g | EK .= f | EK ;
reconsider i1 = x1 , i2 = x2 , j2 = x3 , i = x4 , i1 = x2 , i2 = x3 , j1 = x4 , i2 = x3 , j2 = x4 , i1 = [ x1 , x2 ] , i2 = [ x2 , x3 ] , i2 = [ x1 , x3 ] , j2 = [ x2 , x3 ] , i1 = [ x2 , x3
( a * A ) @ = ( a * A ) @ ;
assume ex n1 being Element of NAT st f .: n1 is succ & f . n1 is succ & f . n1 is succ ;
Seg len ( ( ( f | ( len f -' 1 ) ) ^ ( f | ( len f -' 1 ) ) ) ) = dom ( ( f | ( len f -' 1 ) ) ^ ( f | ( len f -' 1 ) ) ) ;
( ( Complement A1 ) . m ) c= ( ( Complement A1 ) . n ) * ( ( Complement A1 ) . m ) ;
f1 . p = p0 & g1 . ( p . ( p . ( p . ( p . m ) ) ) ) = d & g1 . ( p . ( p . m ) ) = d ;
FinS ( F , Y ) = FinS ( F , dom ( F | Y ) ) ;
( x | y ) | z = z | ( y | x ) ;
sqrt ( |. x .| ) ^2 <= ( r2 ^2 + ( r1 ^2 ) ^2 ) ;
Sum ( F ) = Sum f & dom ( F | X ) = dom g & dom ( F | X ) = dom g ;
assume for x , y being set st x in Y holds x /\ y in Y ;
assume that W1 is Subspace of W2 and W2 is Subspace of W2 and W1 is Subspace of W2 and W2 is Subspace of W1 and W2 is Subspace of W2 ;
||. ( ( vseq . x ) - ( vseq . x ) ) .|| = lim ||. ( vseq . x ) - ( vseq . x ) ) ;
assume that i in dom D and f | A is lower and g | A is lower and g | A is lower and g | A is lower and g | A is lower ;
sqrt ( ( p `1 ) ^2 + ( p `2 ) ^2 ) <= sqrt ( ( p `1 ) ^2 + ( p `2 ) ^2 ) ;
g | Sphere ( p , r ) = id Ball ( p , r ) & g | Ball ( p , r ) = id Ball ( p , r ) ;
set Nmin = ( E-max L~ Cage ( C , n ) ) .. ( Cage ( C , n ) ) ;
for T being non empty TopSpace holds T is countable implies T is countable & the TopStruct of T is countable
width B |-> 0. K = Line ( B , i ) .= Line ( B , i ) .= B * ( i , j ) .= B * ( i , j ) ;
pred a <> 0 means A c= ( A ++ B ) ++ a & ( A ++ B ) there a st a in A \subseteq ( A ++ B ) exists a st a in A \subseteq ( A ++ B ) exists a ;
then f is_partial differentiable u , 3 & pdiff1 ( f , 3 ) is_partial_differentiable_in u , 3 & pdiff1 ( f , 3 ) is_partial_differentiable_in u , 3 ;
assume that a > 0 and a <> 1 and b > 0 and a <> 1 and c <> 0 and a <> 1 and b <> 0 and c <> 0 and a <> 0 and a <> 0 and a <> 0 and b <> 0 and a <> 0 ;
w1 , w2 in Lin { w1 , w2 } & w2 in Lin { w1 , w2 } ;
p2 /. IC Comput ( p2 , s2 , k ) = p2 . IC Comput ( p2 , s2 , k ) .= p2 . IC Comput ( p2 , s2 , k ) ;
ind ( ( T | b ) | b ) = ind b .= ind B - ind ( T | b ) .= ind B - ind ( T | b ) .= ind B - ind B ;
[ a , A ] in the InternalRel of Line ( Line ( D , A ) , [ a , A ] ) ;
m in ( the Arrows of C ) . ( o1 , o2 ) & m in ( the Arrows of C ) . ( o1 , o2 ) ;
( ( CompF ( PA , G ) ) . z ) = FALSE ;
reconsider phi = phi /. 11 , phi = phi /. 2 , phi = ( I , {} ) \HM { {} } ;
len s1 - ( len s2 - 1 ) * ( len s2 - 1 ) > 0 + 1 ;
\delta ( D * f . ( sup A ) - f . ( sup A ) ) < r ;
[ f21 , f22 ] in the carrier' of A & [ f21 , f22 ] in the carrier' of A & [ f21 , f22 ] in the carrier' of A ;
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 p = g2 . z ;
[#] V1 = { 0. V1 } .= { 0. V1 } .= { 0. V1 } .= { 0. V1 } .= { 0. V1 } ;
consider P2 be FinSequence such that rng P2 = M and P2 is one-to-one and rng P2 = M and P2 is one-to-one and rng P2 = M and P2 is one-to-one and rng P2 = M ;
assume that x1 in dom ( f | X ) and ||. x1 - x0 .|| < s and ||. x1 - x0 .|| < s ;
h1 = f ^ ( <* p3 *> ^ <* p3 *> ) .= h ^ ( <* p3 *> ^ <* p3 *> ) .= h ^ ( <* p3 *> ^ <* p3 *> ) .= h ;
c / ( |[ b , c ]| ) = c / ( |[ a , c ]| ) .= c / ( |[ a , c ]| ) .= c / ( |[ a , c ]| ) .= c / ( |[ a , c ]| ) ;
reconsider t1 = p1 , t2 = p2 , t2 = p2 , t2 = p3 as Term of C , V , f , g be Morphism of C , V ;
sqrt ( 1 - ( 2 * ( 1 + ( 1 + ( 1 - ( 2 * ( 1 / 2 ) ) ) / 2 ) ) ^2 ) ) ) in the carrier of ( ( ( ( 1 - ( 2 * ( 1 / 2 ) ) / 2 ) ) ;
ex W being Subset of X st p in W & W is open & h .: W c= V ;
( h . p1 ) `2 = C * ( p1 `1 + D ) / C * D + D * E ;
R . b ` = 2 * PI .= 2 * PI .= 2 * PI .= 2 * PI .= 2 * PI ;
consider I1 such that B = - I1 * C + ( - I1 ) * C and 0 <= I1 and I1 <= 1 ;
dom g = dom ( ( ( the Sorts of A ) * ( the Sorts of A ) ) * ( the Arity of S ) ) ;
[ P . l1 , P . l2 ] in [: T , T :] & [ P . l1 , P . l2 ] in [: T , T :] ;
set s2 = Initialize s , P2 = P +* stop I ;
reconsider M = mid ( z , i2 , i1 ) , i2 = mid ( z , i2 , i1 ) , j2 = ( z ) .. z , i2 = ( z ) .. z ;
y in product ( ( Carrier J ) +* ( V , { 1 } ) ) ;
1 / ( |[ 0 , 1 ]| ) = 1 & 1 / ( |[ 0 , 1 ]| ) = 0 ;
assume x in the left left of g or x in the left or x in the left of g or x in the right of g ;
consider M being strict non-empty non-empty non-empty MSAlgebra over A9 such that a = M and T is strict and T is strict and for i being Element of I holds T . i = the carrier of M ;
for x st x in Z holds ( ( ( ( ( ( ( ( exp_R + f ) ) * ( ( exp_R + f ) ^ ) ) `| Z ) ) `| Z ) . x ) <> 0
len W1 + len W2 = 1 + len W2 + m & len W2 + len W1 = len W2 + len W1 + m ;
reconsider h1 = ( vseq . n - t1 ) * ( vseq . n ) as Lipschitzian Lipschitzian LinearOperator of X , Y ;
( i mod len ( p + q ) + 1 ) in dom ( p + q ) ;
assume that s2 is negative and F in the carrier of ( s2 ) and F in the carrier of ( s2 ) and F is the carrier of ( s2 ) and F is the carrier of ( s2 ) and F is the carrier of ( s2 ) ;
( ( \mathop { \rm gcd } ( x , y , z ) ) ) * ( ( \mathop { \rm gcd } ( x , y , z ) ) * ( ( \mathop { \rm gcd } ( x , y , z ) ) * ( ( \mathop { \rm gcd } ( x , y , z ) ) ) ) ) = gcd ( x , y , z
for u being element st u in Bags n holds ( p `2 + m ) . u = p . u
for B being Subset of u-5 st B in E holds A = B or A misses B or A misses B ;
ex a being Point of X st a in A & A /\ Cl { y } = { a } ;
set W2 = InnerVertices finite ( p ) \/ ( <* x *> \/ <* y *> ) ;
x in { X where X is Ideal of L : X is ideal of L & X is directed } ;
the carrier of W1 /\ W2 c= the carrier of W1 & the carrier of W1 /\ W2 c= the carrier of W2 & the carrier of W1 /\ W2 c= the carrier of W1 /\ W2 ;
( not ( 1 / a + b ) * id a ) * id a = ( 1 / a ) * ( a * ( b * a ) ) ;
( ( X --> f ) . x ) . x = ( X --> dom f ) . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) /\ LSeg ( g , m ) ;
p => ( q => r ) => ( p => ( q => r ) ) in TAUT ( A ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
- 1 + 1 <= ( ( m + 1 ) / ( m + 1 ) ) + 1 ;
( reproj ( 1 , z0 ) ) . x in dom ( f1 (#) ( f2 * f1 ) ) ;
assume that b1 . r = { c1 . r } and b2 . r = { c2 . r } and c2 . r = c2 . r and c2 . r = c2 . r ;
ex P st a1 on P & a2 on P & a1 on P & a2 on P & a1 on P & a2 on P & a1 on P & a2 on P & a1 on P & a2 on P & a1 on P & a2 on P & a1 on P & a2 on P & a1 on P & a2 on P & a1 on P & a2 on P
reconsider gf = g `2 * f as strict Subgroup of X , Y ;
consider v1 being Element of T such that Q = ( downarrow v1 ) ` and v1 in V and v1 in V ;
n in { i where i is Nat : i < n1 + 1 & n1 < len f + 1 } ;
( F * ( i , j ) ) `2 >= ( F * ( m , k ) ) `2 ;
assume K1 = { p : ( p `1 / |. p .| - sn ) / ( 1 + sn ) >= sn & p <> 0. TOP-REAL 2 } ;
ConsecutiveSet ( A , succ O1 ) = ( ( ConsecutiveSet ( A , O1 ) ) * ( ( ConsecutiveSet ( A , O1 ) ) * ( ( ConsecutiveSet ( A , O1 ) ) * ( ( ConsecutiveSet ( A , O1 ) ) * ( ( ConsecutiveSet ( A , O1 ) ) * ( ( ConsecutiveSet ( A , O1 ) ) * ( ( ConsecutiveSet ( A , O1 ) ) * ( ( ConsecutiveSet ( A , O1 ) ) ) )
set I1 = Macro ( a , intloc 0 ) , i2 = Macro ( a , intloc 0 ) , i1 = [ a , intloc 0 ] , i2 = [ a , intloc 0 ] , i1 = [ a , intloc 0 ] , i2 = [ a , intloc 0 ] , i2 = [ a , intloc 0 ] , i1 = [ a , intloc 0 ] , i2 = [ a , intloc 0 ] , i2 = [ a
for i being Nat st 1 < i & i < len z holds z /. i <> z /. ( i + 1 )
X c= ( the carrier of L1 ) & the carrier of L2 c= ( the carrier of L2 ) & the carrier of L2 c= the carrier of L2 ;
consider x9 being Element of GF ( p ) such that x9 |^ 2 = a and x9 |^ 2 = a |^ ( 2 * 1 ) ;
reconsider ee = e1 , fe = f . ( x , y ) , fe = f . ( x , y ) , fe = f . ( y , z ) , fe = f . ( y , z ) , fe = f . ( y , z ) , fe = f . ( z , y ) , fe = f . ( y , z ) , e = f . ( y
ex O being set st O in S & C1 c= O & M . O = 0. ( Cl ( f . O ) ) ;
consider n be Nat such that for m being Nat st n <= m holds S . m in U1 and S . m in U1 ;
f * g * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) ) . x ;
defpred P [ Nat ] means A + succ $1 = succ ( A + $1 ) & A = ( A + $1 ) --> ( A + $1 ) ;
the left left \/ of ( - g ) = the left \/ of ( - g ) .= the left or the right of ( - g ) = the left or the right of ( - g ) = the left or the right of ( - g ) = the left of ( - g ) ;
reconsider p0 = x , p2 = y , p3 = z as Point of ( TOP-REAL 2 ) | LSeg ( p1 , p2 ) ;
consider g2 such that g2 = y and x = g2 and x <= g2 and g2 <= x0 and x0 <= g2 and g2 <= x0 and g2 <= x0 and x0 <= g2 and g2 <= x0 and g2 <= x0 and x0 <= x0 and g2 <= x0 ;
for n being Element of NAT holds X [ n , r ] implies ex r being Element of REAL st X [ n , r ]
len ( x2 ^ y2 ) = len x2 + len y2 & len ( x2 ^ y2 ) = len ( x2 ^ y2 ) + len ( y2 ^ y1 ) ;
for x being element st x in X holds x in the set of \HM { the } \HM { set } , \HM { is } \HM { Real } : x in X }
LSeg ( p10 , p2 ) /\ LSeg ( p1 , p2 ) = {} & LSeg ( p10 , p2 ) /\ LSeg ( p2 , p2 ) = {} ;
func mid ( X , X ) -> set means : Def1 : it = ( \mathop { \rm R , X ) --> 0. X ) & it = ( ( id X ) --> 0. X ) +* ( id X ) ;
len ( ( Gauge ( C , n ) * ( len Gauge ( C , n ) , 1 ) ) ) <= len ( ( the Sorts of Gauge ( C , n ) ) * ( len Gauge ( C , n ) , 1 ) ) ;
attr K is Field means : Def2 : a <> 0. K & v . ( a |^ i ) = i * v . ( a |^ i ) ;
consider o being OperSymbol of S such that t . {} = [ o , the carrier of S ] and t . {} = [ o , the carrier of S ] ;
for x st x in X ex y st x c= y & y in X & y in f . x
IC Comput ( P3 , s3 , k ) in dom Comput ( P3 , s3 , k ) ;
pred q < s & r < s implies ]. r , s .] c= ]. p , q .[
consider c being Element of Class ( f , 3 ) such that Y = ( F . c ) `1 and c in X ;
the Arity of S2 = id ( the carrier of S2 ) & the Arity of S2 = id ( the carrier of S2 ) ;
set y9 = [ <* y , z *> , f2 ] , z9 = [ <* z , x *> , f3 ] , \mathopen ( <* y , z *> , f2 ) ] ;
assume x in dom ( ( ( ( ( ( ( ( ( ( ( arccot ) ) * ( arccot ) ) * ( ( ( ( ( ( ( ( ( ( ( arccot ) ) * ( arccot ) ) ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( A A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
r-7 in Int cell ( GoB f , i , width GoB f ) \ ( ( GoB f ) * ( i + 1 , width GoB f ) ) ;
( q `2 / |. q .| - sn ) / ( 1 + sn ) >= ( ( ( ( ( q `2 ) / |. q .| - sn ) / ( 1 + sn ) ) / ( 1 + sn ) ) ^2 ;
set Y = { a "/\" a ` : a in X } ;
i -' len f + ( len f -' 1 ) <= len f + ( len f -' 1 ) ;
for n holds ex x st x in N1 & x in N1 & h . n = - ( x - h )
set s0 = ( \mathop { \it SCMPDS } , p , s ) . i , p = ( \mathop { \it true } , p ) . i , s = ( \mathop { \it true } , p ) . i , q = ( \mathop { \it true } , s ) . i ;
( p . k ) . 0 = 1 or ( p . k ) . 0 = - 1 or ( p . k ) . 1 = 1 ;
u + Sum ( L-18 ) in ( U \ { u } ) \/ { u + Sum ( L ) } ;
consider x9 being set such that x in x9 and x9 in V1 and x9 in V1 and x = [ x9 , y9 ] ;
( p ^ q ) . m = ( q | k ) . ( mm + 1 ) .= ( q | ( Seg ( len p ) ) ) . ( mm + 1 ) ;
g + h = gg + h + ( h + c ) & for n holds l . n = g . n + h . n
L1 is distributive LATTICE & L2 is distributive implies L1 & L2 * L2 is distributive & L2 * L2 is distributive & L2 * L2 is distributive
pred x in rng f & y in rng ( f | x ) & f . x = f . y implies x in rng ( f | y ) ;
assume that 1 < p and p + sqrt ( 1 + ( p `1 / p `2 ) ^2 ) = 1 and 0 <= p `1 and p `1 <= 1 and p `2 <= 1 and p `2 <= 1 and p `2 <= 1 ;
FM * ( f , <* the carrier of C *> *' t ) = rpoly ( 1 , ( the carrier of C ) *' t ) .= <* 0. C *> ;
for X being set , A being Subset of X holds A ` = {} implies A = {} implies A = {} & A = {}
( ( ( E-max L~ f ) .. f ) .. f ) .. f <= ( ( E-max L~ f ) .. f ) .. f ;
for c being Element of the Sorts of A , a being Element of the bound of A holds c <> a
s1 . intloc 0 = ( Exec ( i2 , s2 ) ) . intloc 0 .= s2 . intloc 0 .= s2 . intloc 0 .= s2 . intloc 0 .= s2 . intloc 0 .= 1 ;
for a , b , c , d being Real holds [ a , b ] in ( y iff a >= 0 & b >= 0 & c >= 0 & d >= 0 & a >= 0 & b >= 0 & d >= 0 & a >= 0 & b >= 0
for x , y being Element of X holds x ` \ y = ( x \ y ) ` & y \ x = ( x \ y ) `
mode BCK-algebra of i , j , m , n , m , n , m ) is BCK-algebra of i , j , n , m , m , n , m , n , m , m , n ;
set x2 = |. ( Re y ) . x .| , y2 = |. ( Im y ) . x .| ;
[ y , x ] in dom u5 & u5 . ( y , x ) = g . ( y , x ) ;
]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .] c= A & [. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .] c= A ;
0 <= ( \delta ( S2 ) ) . n & |. delta ( S2 ) . n - 0 .| < e / 2 ;
( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) <= ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ;
set A = ( 2 / 2 ) * ( 2 * PI ) ;
for x , y being set st x in R" holds x , y in R" & y , x ] in R
deffunc F ( Nat ) = b . ( $1 + 1 ) * ( M * ( $1 , m ) ) ;
for s being element holds s in ^2 ( f \/ g ) iff s in |= ( f \/ g ) \/ |= ( f \/ g )
for S being non empty non void finite void finite void void transitive for T being non empty TopSpace st S is connected holds S is connected
max ( ( |. z .| ) , ( |. z .| ) ) / ( |. z .| ) ) >= 0 ;
consider n1 being Nat such that for k holds seq . ( n1 + k ) < r + s ;
Lin ( A /\ B ) is Subspace of Lin ( A ) & Lin ( B ) is Subspace of Lin ( B ) & Lin ( A ) = Lin ( B ) ;
set n-15 = nnM , M = ( M . x ) `1 , ni2 = ( M . x ) `2 , ni2 = ( M . x ) `2 , ni2 = ( M . x ) `1 , ni2 = ( M . x ) `1 , ni2 = ( M . y ) `1 , ni2 = ( M . x ) `1 , ni2 = ( M . y
f " V in Hom ( X , p ) & f " ( f " V ) in D & f " ( f .: ( X , p ) ) in D ;
rng ( ( a , c ) *> \mathbin { + } \cdot ( c , b ) ) c= { a , c } ;
consider y being subgraph of G1 such that y ` = y and dom y = WWWWWWWWWbound and y in WWWWWWWWWWW\mathop ( G1 , G2 ) ;
dom ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) * ( ( 1 / 2 ) * ( 1 / 2 ) ) ) ) ) ) c= ]. - 1 , 1 .[ ;
in LSeg ( i , j , n ) & ( i , j , n ) is Subset of Gauge ( C , n ) & ( i , j ) in C implies ( i , j ) in C
v ^ ( n\rangle ^ <* 0 *> ^ ( B ^ ( A ^ <* 1 *> ) ) in Lin ( ( rng ( B ^ ( A ^ <* 1 *> ) ) ) ) ;
ex a , k1 , k2 st i = a & i = b := k2 & j = c := k2 ;
t . NAT = ( ( ( NAT .--> NAT ) .--> NAT ) . NAT ) . NAT .= ( ( ( the Sorts of A ) * ( the Sorts of A ) ) . NAT ) . NAT .= ( ( the Sorts of A ) . NAT ) . NAT .= ( the Sorts of A ) . NAT .= ( the Sorts of A ) . NAT ;
assume that F is bbfamily and rng p = F and rng p = Seg ( n + 1 ) and rng p = Seg ( n + 1 ) and rng p = Seg ( n + 1 ) ;
not LIN b , b9 , a & not LIN b , a , c & LIN a , b , c
( L1 or L2 ) . O c= ( L1 . O ) and ( L2 . O ) . O = ( L1 . O ) . O ;
consider F be ManySortedFunction of E , F such that for d being Element of E holds F . d = F ( d ) and F . d = F ( d ) ;
consider a , b such that a * ( u1 - w ) = b * ( -B ) and 0 < a and a < b and b < 1 ;
defpred P [ FinSequence of D ] means |. Sum ( $1 ) .| <= Sum ( |. $1 .| ) & |. $1 .| <= Sum ( |. $1 .| ) ;
u = PI / ( x , y ) . v * x + ( PI / ( y , v ) ) . v .= v ;
dist ( seq . n + x , x + g ) <= dist ( seq . n , g ) + ( g . n ) ;
P [ p , |. p .| : p in [: the carrier of A , the carrier of A :] & [ p , id the carrier of A ] in 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 ininand X is inin;
|. b .| * |. eval ( f , z ) .| >= |. b .| * |. eval ( f , z ) .| ;
1 < ( ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ) .. Gauge ( C , n ) ;
l in { l1 where l1 is Real : g <= l1 & l1 <= h . l1 & l <= h . l1 & l1 <= h . l1 } ;
( Partial_Sums ( G . n ) ) . ( n + 1 ) <= ( Partial_Sums ( G . n ) ) . ( n + 1 ) ;
f . y = x .= x .= x .= 1_ L .= 1_ L .= 1_ L .= 1_ L .= 1_ L .= 1_ L .= 1_ L ;
NIC ( halt SCM+FSA , |. i1 .| ) = { i1 , i2 } & { i1 , i2 } in { i1 , i2 } ;
LSeg ( p01 , p2 ) /\ LSeg ( p1 , p2 ) = { p1 } /\ LSeg ( p2 , p2 ) /\ LSeg ( p1 , p2 ) ;
Product ( ( ( the support of I1 ) +* ( i , { 1 } ) ) +* ( i , { 2 } ) ) in Z ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) +* ( the Sorts of S2 ) ;
( for q being Element of W holds q `1 <= ( q `1 ) / 2 ) * ( ( q `1 ) / 2 ) & ( q `1 ) / 2 <= ( q `1 ) / 2 ;
f /. i2 <> f /. ( ( i1 + len g -' 1 ) + 1 ) & f /. ( ( i1 + len g -' 1 ) + 1 ) = f /. ( ( i1 + len g -' 1 ) + 1 ) ;
M , v / ( x. 3 , x. 4 ) / ( x. 0 , x. 4 ) / ( x. 0 , x. 4 ) / ( x. 0 , x. 4 ) / ( x. 4 , x. 0 ) / ( x. 0 , x. 4 ) / ( x. 4 , x. 0 ) |= H ;
len ( ( P ^ Q ) ^ ( P ^ Q ) ) in dom ( ( P ^ Q ) ^ ( P ^ Q ) ) ;
A |^ ( m , n ) c= ( A |^ m ) |^ ( k , l ) & A |^ ( m , l ) c= ( A |^ k ) |^ ( k , l ) ;
( ( TOP-REAL n ) \ { q : |. q .| ) ^2 >= a } c= { q : |. q .| >= a }
consider n1 being element such that n1 in dom p1 and y1 = p1 . n1 and p1 . n1 = p1 . n1 and p2 . n1 = p2 . n1 and p1 . n1 = p2 . n1 ;
consider X be set such that X in Q and for Z being set st Z in Q holds Z in X and Z c= Z ;
CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA & CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA & CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA ;
for v being VECTOR of l1 holds ||. v .|| = upper_bound ( |. v .| ) & ||. v .|| = upper_bound ( |. v .| )
for phi holds phi in X implies phi . phi in X & phi . phi in X & phi . phi in Y
rng ( Sgm dom ( ( Sgm dom ( f | dom ( g | dom ( f | dom ( g | dom ( g | dom ( g | dom ( g | dom ( g | dom ( g | dom ( g | dom ( g | dom ( g | dom ( g | dom ( g | dom ( g | dom ( g | dom ( g | dom ( g | dom ( g | dom ( g | dom ( g | dom ( g |
ex c being FinSequence of D ( ) st len c = k & for k st 1 <= k & k <= len c holds P [ k , c . k ] ;
the_result_sort_of ( a , b , c ) = <* <* \in ( the carrier of C ) , {} , {} *> , <* {} , <* {} *> *> *> , <* {} , <* {} *> *> *> , <* <* {} , {} *> *> *> ;
consider f1 being Function of the carrier of X , REAL such that f1 = |. f .| and f1 is continuous and rng f1 c= X and f1 is continuous and rng f1 c= X ;
a1 = b1 & a2 = b2 & a3 = b3 or a1 = b3 & a2 = b3 & a3 = b3 & a4 = b3 & b3 = b3 & b3 = b3 & b3 = b3 & b3 = b3 & b3 = b3 & b3 = b3 & b3 = b3 & b3 = 6 & b3 = 6 & 8 = 8 & 8 = 6 & 8 = 8 & 8 = 8 & 8 = 8 & 8 = 8 & 8 = 7 & 8 = 8
D2 . indx ( D2 , D1 , n1 ) = D1 . ( n1 + 1 ) .= D2 . ( n1 + 1 ) .= D2 . ( n1 + 1 ) ;
f . ( ||. r .|| ) = ||. ||. r .|| ) /. 1 .= ||. r .|| . 1 .= ||. r .|| . 1 .= ||. r .|| . 1 .= ||. r .|| . 1 .= ||. r .|| . 1 .= r .= r . 1 ;
consider n being Nat such that for m being Nat st n <= m holds C-25 . m = C-25 . m and C-25 . n = C-25 . m ;
consider d be Real such that for a , b being Real st a in X & b in Y holds a <= b & b <= d ;
||. L /. h .|| - ( K * |. h .| ) + ( K * |. h .| ) <= x0 + ( K * |. h .| ) ;
attr F is commutative associative means : Def1 : for b being Element of X holds F \hbox { b } -\hbox { b } = f . b ;
p = ( 1 - ( p `2 / p `1 ) ) * ( p `1 / p `1 ) .= 1 * ( p `1 / p `1 ) .= ( p `1 / p `1 ) * ( p `2 / p `1 ) .= ( p `1 / p `1 ) * ( p `2 / p `1 ) .= ( p `1 / p `1 ) * ( p `2 / p `1 ) ;
consider z1 such that b , x1 , x3 is_collinear and ( for x st x in X & x1 <> x2 & x <> x3 & x <> x3 & x <> x3 & x <> x3 & x <> x3 & x <> x3 & x <> x3 & x <> x3 & x <> x3 & x <> x3 & x <> x3 & x <> x3 & x <> x3 & x <> x3 & x <> x3 & x <> x3 & x <> x3 & x <>
consider i such that Arg ( ( Rotate ( s , 2 ) ) . q ) = s + Arg ( ( Rotate ( s , 2 ) ) . q ) ;
consider g such that g is one-to-one and dom g = card ( f . x ) and rng g c= dom ( f . x ) and g . x = f . x and g . x = f . x ;
assume that A = P2 \/ Q2 and and ( for i st i in dom P2 holds P2 . i = ( i + 1 ) and ( i + 1 ) in dom P2 and ( i + 1 ) in dom P2 and ( i + 1 ) in dom P2 ;
attr F is associative means : Def1 : F .: ( F .: ( f , g ) ) = F .: ( f , g ) ;
ex x being Element of NAT st m = x & x `1 = z & x `2 < i & i < m or m = i & i = m or m = i & i = m or m = i & i = m ;
consider k2 being Nat such that k2 in dom Pk and l in dom Pk and P [ k2 , l . k2 ] ;
seq = r * seq implies for n holds seq . n = r * seq . n & seq . n = r * seq . n
F1 . [ id a , id a ] = [ f * ( id a ) , f * ( id a ) ] ;
{ p } "\/" D2 = { p "\/" y where y is Element of L : y in D2 & p in D2 } ;
consider z being element such that z in dom ( ( dom _ F ) . 0 ) and ( ( the Sorts of F ) . 0 ) . 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 , 1 ) = { |[ r , s ]| : r <= G * ( i + 1 , 1 ) `1 & G * ( i + 1 , 1 ) `2 <= s & s <= G * ( i + 1 , 1 ) `2 } ;
consider e being element such that e in dom ( T | E1 ) and ( T | E1 ) . e = v and ( T | E1 ) . e = v ;
( F ` * b1 ) . x = ( Mx2Tran ( J , b1 , b2 ) ) . ( ||. b1 .|| , ||. b2 .|| ) . ( ||. b1 .|| , ||. b2 .|| ) . ( ||. b1 .|| , ||. b2 .|| ) . ( ||. b1 .|| , ( J , b1 , b2 ) ) . ( ||. b2 .|| , ( J , b2 ) ) . ( ||. b1 .|| ) ) ;
- 1 / ( - 1 ) = ( ( ( - 1 ) (#) D ) (#) D ) (#) ( ( - 1 ) (#) D ) (#) ( ( - 1 ) (#) D ) ) .= ( ( - 1 ) (#) D ) (#) ( ( - 1 ) (#) D ) .= ( ( - 1 ) (#) D ) (#) ( ( - 1 ) (#) D ) .= ( ( - 1 ) (#) D ) (#) ( ( - 1 ) (#) D )
pred for x being set st x in dom f /\ dom g holds g . x <= f . x & g . x <= f . x ;
len ( f1 . j ) = len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) ;
All ( All ( All ( a , A , G ) , B , G ) , G ) '<' All ( All ( a , B , G ) , A , G ) ;
LSeg ( E . k2 , F . k2 ) c= Cl RightComp Cage ( C , n + 1 ) & Cl RightComp Cage ( C , n + 1 ) c= Cl RightComp Cage ( C , n + 1 ) ;
x \ a |^ m = x \ ( ( a |^ k ) * a ) .= ( ( a |^ k ) * a ) * a .= ( ( a |^ k ) * a ) * a ;
k -{ I } -inininthe carrier of S = ( ( commute I ) --> ( 1 , the carrier of S ) ) . k .= ( ( ( commute I ) --> ( 1 , the carrier of S ) ) . k ) . k .= ( ( ( ( the Sorts of A ) . ( 1 , the carrier of S ) ) . k ) . k ;
for s being State of A2 holds Following ( s , n ) . ( n + 1 ) is stable ;
for x st x in Z holds f1 . x = a ^2 & ( f1 - f2 ) . x <> 0 & ( f1 - f2 ) . x <> 0 & ( f1 - f2 ) . x <> 0 ) implies f1 - f2 . x <> 0
support ( \mathop { \rm support } ( n ) ) \/ support ( \mathop { \rm support } ( m ) ) c= support ( ( support ( m ) ) ) \/ support ( m ) ) ;
reconsider t = u as Function of ( the carrier of A ) , the carrier of B ;
- ( a * sqrt ( 1 + ( b ^2 + a ^2 ) ) ) <= - ( - ( b ^2 + b ^2 ) ) ;
phi /. ( a . a ) = g . a & phi . ( a . a ) = f . ( g . a ) & phi . ( a . a ) = f . ( g . a ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( F ^ <* p *> ) and i = len ( F ^ <* p *> ) and j = len ( F ^ <* p *> ) and i = j ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x6 , x1 , x2 } = { x1 , x2 , x3 , x4 , x5 , x5 , x2 } \/ { x1 , x2 , x3 , x4 , x4 , x5 , x5 , x5 , x5 , x5 } ;
the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U2 & the Sorts of U2 /\ ( U2 "\/" U2 ) c= the Sorts of U1 /\ ( U2 "\/" U2 ) ;
( - ( 2 * a ) * ( b - a ) ) / ( 2 * a ) + ( - ( 2 * a ) * ( b - a ) ) > 0 ;
consider seq1 such that for z being element holds z in seq1 iff z in seq1 ~ iff z in N ~ & P [ z ] ;
assume that ( the Arity of S ) . o = <* a *> and ( the Arity of S ) . o = r and ( the Arity of S ) . o = r and ( the Arity of S ) . o = r ;
Z = dom ( ( exp_R * ( arccot * ( arccot * ( arccot * ( arccot * ( arccot * ( arccot * ( arccot * ( arccot * ( arccot * ( arccot * ( arccot * ( arccot * ( 1 / 2 ) ) ) ) ) ) ) ) ) ) ) ;
integral ( f , S1 , S2 ) is convergent & lim ( f , S1 ) = lim ( f , S1 ) & lim ( f , S1 ) = lim ( f , S1 ) ;
( for a9 , b9 st a9 . ( f . ( f . ( x , y ) ) => ( f . ( x , y ) ) ) in as Element of \mathclose { x } & ( f . ( x , y ) ) => ( f . ( x , y ) ) in \mathclose { x }
len ( M2 * M1 ) = n & width ( M2 * M2 ) = n & width ( M2 * M1 ) = n & width ( M2 * M2 ) = n ;
attr X1 \/ X2 is open means : Def1 : X1 is open & X2 is open & X1 is open & X2 is open & X1 is open & X2 is open & X1 is open & X2 is open & X1 is open & X2 is open & X1 is open & X2 is open & X1 is open & X2 is open ;
for L being lower-bounded antisymmetric RelStr for X being Subset of L holds X "\/" { Bottom L } = { Bottom L } & X "\/" { Bottom L } = { Bottom L }
reconsider f-1J = F2 . ( ( F . b ) . ( ( F . c ) . ( F . d ) ) ) as Function of ( ( the Sorts of A ) . ( ( F . c ) . ( F . d ) ) , M ;
consider w being FinSequence of I such that the carrier of M = the carrier of M and the q empty empty ( <* s *> ^ w ) ^ w ^ ( q , <* s *> ) ) ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^
g . ( a |^ 0 ) = g . ( 1_ G ) .= 1_ G .= g . ( 1_ G ) .= g . ( 1_ G ) .= g . ( 1_ G ) .= g . ( 1_ G ) .= g . ( 1_ G ) .= g . ( 1_ G ) ;
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 <> {} & K /\ L <> {} & K /\ L <> {} ;
( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C1 & the carrier' of C2 = the carrier' of C2 & the carrier' of C2 = the carrier' of C2 & the carrier' of C2 = the carrier' of C2 ;
reconsider o9 = o `2 , p = ( the Sorts of A ) . ( ( the Sorts of A ) . o ) as Element of TS ( ( the Sorts of A ) . v , ( the Sorts of A ) . v ) ;
1 * ( x1 + 0 ) + ( 0 * x2 ) = x1 + ( 0 * x2 ) .= x1 + ( 0 * x2 ) .= x1 + ( 0 * x2 ) .= x1 + ( 0 * x2 ) .= x1 + ( 0 * x2 ) .= x1 + ( 0 * x2 ) .= x1 + ( 0 * x2 ) ;
Eq " . 1 = ( Eq " ) . 1 .= ( ( E . 1 ) " ) . 1 .= ( E . 1 ) " .= ( E . 1 ) " .= ( E . 1 ) " .= ( E . 1 ) " .= ( E . 1 ) " .= ( E . 1 ) " .= ( E . 1 ) " ;
reconsider u1 = the carrier of ( U1 /\ U2 ) , u2 = the carrier of ( U1 "\/" U2 ) , v2 = the carrier of ( U1 "\/" U2 ) , u1 = the carrier of ( U1 "\/" U2 ) ;
( ( x "/\" z ) "\/" ( x "/\" y ) ) "\/" ( ( x "/\" z ) "\/" ( z "/\" y ) ) <= ( x "/\" ( z "\/" y ) ) "\/" ( z "\/" ( x "\/" y ) ) ;
|. f . ( s1 . ( l1 + 1 ) ) - f . ( l1 + 1 ) .| < r / ( 1 - r ) ;
LSeg ( ( Gauge ( C , n ) * ( i , j ) , ( Gauge ( C , n ) * ( i + 1 ) ) ) is vertical & LSeg ( ( Gauge ( C , n ) * ( i + 1 , j ) , ( Gauge ( C , n ) * ( i + 1 ) ) is vertical ;
( f | Z ) /. x - ( f | Z ) /. x0 = L /. ( x- ( 2 * x ) + R /. ( x- ( 2 * x ) + R ) /. x0 ) ;
g . c * ( - g . c ) * f + f . c <= h . c * ( - f . c ) + f . c ;
( f + g ) | divset ( D , i ) = f | divset ( D , i ) + g | divset ( D , i ) ;
assume that which that is being_line and f is being_line and len ( f . 0 ) = width A and width ( f . 1 ) = width A and width ( f . 1 ) = width A and width ( f . 1 ) = width A and width ( f . 1 ) = width A and width ( f . 1 ) = width A and width ( f . 1 ) = width A ;
len ( - ( - ( - ( - ( - ( - ( - ( - - ( - - ( - ( - - ( - - ( - - ( - - ( - - ( - - ( / - / / / / 2 ) ) / 2 ) ) / 2 ) ) - 1 ) ) / ( 2 * ( - ( - ( - ( - 1 / 2 ) / 2 ) ) ^2 ) ) ) = len ( - ( - ( - ( - ( 1 / 2 ) / 2 ) / 2
for n , i being Nat st i + 1 < n holds [ i , i ] in the InternalRel of ( ( TOP-REAL n ) | ( the carrier of TOP-REAL n ) )
pdiff1 ( f1 , 2 ) is_partial_differentiable_in u0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in u0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in u0 , 2 & pdiff1 ( f1 , 2 ) is_partial_differentiable_in u0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in u0 , 2 & pdiff1 ( f2 , 2 ) = 3 implies pdiff1 ( f1 , 2 ) * pdiff1 ( f2 , 2 ) * pdiff1 ( f2 , 2 )
pred a <> 0 & b <> 0 & Arg a = Arg b & Arg a = Arg a & Arg ( a ) = Arg a & Arg ( a ) = Arg a & Arg ( a ) = Arg a & Arg ( a ) = Arg a & Arg ( a ) = Arg a ;
for c being set st not c in [. a , b .[ holds not c in Intersection ( ( the open m ) \ ( the carrier of a ) )
assume that V1 is linearly closed and V2 is closed and V1 = { v + u : v in V1 & u in V1 & v in V1 & u in V1 & v in V1 & u in V1 and v in V1 and u in V1 and v in V1 and u in V1 and v in V1 and u = v + v and u in V1 and v in V1 and u = v + v ;
z * x1 + ( 1 - z ) * x2 in M & z * x1 + ( 1 - z ) * x2 in N ;
rng ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) = Seg ( card ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
consider s2 being Real_Sequence such that s2 is convergent and b = lim s2 and for n holds s2 . n <= lim s2 . n and s2 . n <= x0 ;
h2 " . n = h2 . n / ( 2 |^ ( n + 1 ) ) & 0 < h2 . n / ( 2 |^ ( n + 1 ) ) & 0 < h2 . n / ( 2 |^ ( n + 1 ) ) ;
Partial_Sums ( ||. seq .|| ) . m = ||. ( ||. seq .|| ) . m .|| .= ||. ( ||. seq .|| ) . m - ( ||. seq .|| ) . m .|| .= ||. ( ||. seq .|| ) . m - ( ||. seq .|| ) . m .|| .= ||. ( ||. seq .|| ) . m - ( ||. seq .|| ) . m ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b ;
- v = ( - 1 ) * v & - ( - 1 ) * v = ( - 1 ) * v + ( - 1 ) * v .= ( - 1 ) * v .= ( - 1 ) * v .= ( - 1 ) * v .= ( - 1 ) * v .= ( - 1 ) * v ;
sup ( ( k .: D ) .: D ) = sup ( ( k .: D ) .: D ) .= sup ( ( k .: D ) .: D ) .= sup ( ( k .: D ) .: D ) .= sup ( ( k .: D ) .: D ) .= sup ( ( k .: D ) .: D ) .= sup ( k .: D ) .= k ;
( A |^ k , l ) |= ( A |^ ( n , .. A ) ) ^ ( A |^ ( n , .. A ) ) ;
for R being add-associative right_zeroed right_complementable Abelian associative distributive non empty doubleLoopStr , I , J being Subset of R holds I + J = ( I + J ) + K
( f . p ) `1 = ( p `1 ) ^2 + ( p `2 ) ^2 ;
for a , b being non zero Nat st a , b are_relative_prime & a , b are_relative_prime holds support ( a * b ) = support ( a ) + support ( b )
consider A5 being countable such that r is countable and r is countable and ( for i being Nat holds A5 . i = ( A . i ) `1 ) `1 and ( A . i ) `1 = ( A . i ) `1 ;
for X being non empty addLoopStr , M , N being Subset of X , x , y being Point of X st x in M holds x + y in M + N
{ [ x1 , x2 ] , [ y1 , y2 ] } c= { x1 , x2 } & [ x1 , y1 ] in [: { x1 } , { x2 } :] ;
h . ( f . O ) = |[ A * ( f . O ) + B * ( f . I ) , C * ( f . I ) + D * ( f . I ) + D * ( f . I ) + D * ( f . I ) + D * ( f . I ) + D * ( f . I ) + D * ( f . I ) + D * ( f . I ) + D * ( f . I ) + D *
( Gauge ( C , n ) * ( i , i ) ) in L~ Upper_Seq ( C , n ) /\ L~ Cage ( C , n ) /\ L~ Cage ( C , n ) ;
cluster m , n ) gcd n -> prime implies for Nat st m divides n & n divides m & m divides p & p divides q & p divides q & q divides ( m + 1 ) & p divides q
( f * F ) . x1 = f . ( F . x1 ) & ( f * F ) . x2 = f . ( F . x2 ) & ( f * F ) . x2 = f . ( F . x2 ) ;
for L being Lattice , a , b , c being Element of L st a \ b <= c & a \ b <= c holds a \ b <= c
consider b being element such that b in dom ( H / ( x. 0 ) ) and z = H / ( x. 0 ) and z = H / ( x. 0 ) . b ;
assume that x in dom ( F (#) g ) and y in dom ( F (#) ( G (#) g ) ) and ( F (#) ( G (#) g ) ) . x = ( F (#) ( G (#) g ) ) . y ;
assume ex e being element st e Joins W . 1 , W . 2 , G & e in G & e in W . 3 & e in G . 4 ;
( exists h st h = ( exists f st f = ( h ) * n ) . ( 2 * n ) ) & ( h * f ) . ( 2 * n ) = ( h * f ) . ( 2 * n ) ;
j + 1 = ( i + 1 ) -' 2 + 1 .= i + ( len h2 -' 1 ) + 2 .= i + ( len h2 -' 1 ) ;
( *' S ) . f = *' S . ( S . f ) .= S . ( S . f ) .= S . ( S . f ) .= S . ( S . f ) .= S . ( S . f ) .= S . f ;
consider H such that H is one-to-one and rng H = the carrier of L2 and Sum ( L2 ) = Sum ( L2 ) and Sum ( L2 ) = Sum ( L2 ) and Sum ( L2 ) = Sum ( L2 ) and Sum ( L2 ) = Sum ( L2 ) ;
attr R is <= such that R is : verify : for p , q st p in R & q <> q ex P st P [ p , q ] & P [ p , q ] ;
dom ( product ( X --> f ) ) = meet ( ( X --> f ) --> f ) .= meet ( ( X --> f ) ) .= meet ( ( X --> f ) ) .= meet ( ( X --> f ) ) .= meet ( ( X --> f ) ) .= meet ( X --> f ) .= meet ( X --> f ) .= meet ( X --> f ) ;
sup ( ( proj2 .: ( Upper_Arc C ) ) /\ ( ( proj2 .: ( Upper_Arc C ) ) ) ) <= sup ( ( proj2 .: ( ( proj2 .: ( C ) ) ) /\ ( ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) ) | ( ( TOP-REAL 2 ) ) ) ) ) ;
for r be Real st 0 < r ex n be Nat st for m be Nat st n <= m holds |. S . m - 0 .| < r
i * ( fN - fN ) = i * ( f . ( y - z ) ) .= i * ( f . ( y - z ) ) .= i * ( f . ( y - z ) ) .= i * ( f . ( y - z ) ) .= i * ( f . ( y - z ) ) ;
consider f being Function such that dom f = 2 -tuples_on X ( ) and 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 C and g2 in C and g1 = [ g1 , g2 ] and g2 in C and g2 in C and g1 in C and g2 in C and g2 in C and g2 in C and g2 in C and g2 in C and g2 in C and g2 in C and g2 in C and g2 in C and g2 in C and g in C and g2 in C ;
func d \! \mathop { n } -> Nat means : Def1 : for m holds it |^ m divides n & it |^ m divides n & it |^ m divides n & it |^ m divides n ;
f[ 0 , t ] = f . [ 0 , t ] .= ( - P ) . [ 0 , t ] .= ( - P ) . [ 0 , t ] .= a . [ 0 , t ] .= a . [ 0 , t ] .= a . [ 0 , t ] .= a . [ 0 , t ] .= a . [ 0 , t ] .= a . [ 0 , t ] .= a . [ 0 , t ] .= a . [ 0 , t ] .= a . [ 0 , t ] .= a
t = h . D or t = h . C or t = h . D or t = h . E or t = h . F or t = h . M or t = h . M or t = h . N or t = h . N or t = h . N ;
consider m1 be Nat such that for n st n >= m1 holds dist ( ( seq . n ) , ( seq . n ) ) < 1 / ( ( seq . n ) + ( seq . m ) ) ;
sqrt ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 <= ( ( q `1 / |. q .| - cn ) ) ^2 / ( 1 + cn ) ^2 ;
h0 . ( i + 1 ) = ( h . ( i + 1 ) + ( h . ( i + 1 ) ) ) . ( i + 1 ) ;
consider o being Element of the carrier' of S , x2 being Element of { the carrier of S } such that a = [ o , x2 ] and a = [ o , x2 ] ;
for L being RelStr , a , b being Element of L holds a <= b iff a <= b & b <= a & a <= b & b <= a & a <= b implies a <= b
||. h1 .|| . n = ||. h1 . n .|| .= ||. h1 . n .|| .= ||. h1 . n .|| .= ||. h1 . n .|| .= ||. h1 . n .|| .= ||. h1 . n .= ||. h1 . n .|| .= ||. h1 . n .|| .= ||. h1 . n .|| .= ||. h1 . n ;
( ( ( - ( ( exp_R - exp_R ) ) * ( exp_R - exp_R ) ) ) `| Z ) . x = f . x - ( exp_R . x ) ^2 .= ( - ( exp_R . x ) ^2 ) / ( exp_R . x ) ^2 .= ( - 1 ) / ( exp_R . x ) ^2 ;
pred r = F .: ( p , q ) means : Def1 : len r = len ( F . ( len p ) ) ;
sqrt ( ( r / 2 ) ^2 + ( r / 2 ) ^2 ) <= sqrt ( ( r / 2 ) ^2 + ( r / 2 ) ^2 ) + sqrt ( ( r / 2 ) ^2 + ( r / 2 ) ^2 ) ;
for i being Nat , M being Matrix of K , K for i , j being Nat st i in Seg n holds Det ( M @ ) = Sum ( ( M @ ) @ ) * ( i , j )
then a <> 0. R & a " * ( a * v ) = 1 / ( a * v ) & a " * ( a * v ) = a " * ( a * v ) ;
p . ( j -' 1 ) * ( q *' r ) . ( i + 1 ) = Sum ( p . ( j -' 1 ) ) * ( q . ( j + 1 ) ) ;
deffunc F ( Nat ) = L . 1 + L . ( ( R /* h ) " ) * ( ( R /* h ) " ) . ( ( h ^\ n ) . ( ( h ^\ n ) . ( ( h + c ) " ) ) ;
assume that the carrier of H2 = f .: ( the carrier of H1 ) and the carrier of H2 = f .: ( the carrier of H2 ) and the carrier of H2 = f .: ( the carrier of H2 ) 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 ;
Args ( o , Free ( S , X ) ) = ( ( ( the Sorts of Free ( S , X ) ) * ( the Arity of S ) ) . o ) ;
H1 = n + 1 & n + 1 <= ( 2 to_power ( n + 1 ) ) + ( 2 to_power ( n + 1 ) ) .= n + 1 ;
( O . O ) `1 = 0 & ( O . I ) `1 = 1 & ( O . I ) `1 = 0 & ( O . I ) `1 = 1 & ( O . I ) `1 = 1 & ( O . I ) `1 = 1 & ( O . I ) `1 = 1 ;
F1 .: ( dom F1 /\ dom F2 ) = F1 .: ( dom F1 /\ dom F2 ) .= { f /. ( n + 1 ) } .= { f /. ( n + 1 ) } .= { f /. ( n + 1 ) } .= { f /. ( n + 1 ) } .= { f /. ( n + 1 ) } ;
pred b <> 0 & d <> 0 & b <> d & a = ( id a ) / ( b - d ) & ( a - b ) / ( b - d ) = ( <* b *> ) / ( b - d ) ;
dom ( ( f +* g ) | D ) = dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) .= dom ( f +* g ) /\ D .= dom ( f +* g ) .= dom ( f +* g ) /\ D
for i being set st i in dom g ex u , v being Element of B st g /. i = u * v & ex a being Element of B st g /. i = a * v & a * v = a * v
g `2 * P `1 = g `2 * ( g `2 ) * ( g `2 ) .= g `2 * ( g `2 ) * ( g `2 ) * ( g `2 ) * ( g `2 ) * ( g `2 ) * ( g `2 ) * ( g `2 ) * ( g `2 ) * ( g `2 ) * ( g `2 ) * ( g `2 ) * ( g `2 ) * ( g `2 ) * ( g `2 ) ) ;
consider i , s1 such that f . i = s1 and not ( f . i = s1 & not ( f . i = s1 ) & ( not f . i = s1 ) & ( not f . i = s1 ) & ( not f . i = s1 ) & not f . i = s1 ) ;
h5 | ]. a , b .[ = ( g | ]. a , b .[ ) | ]. a , b .[ .= ( g | ]. a , b .[ ) | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ ;
[ s1 , t1 ] in R & [ s2 , t2 ] in R & [ s1 , t2 ] in R & [ s2 , t2 ] in R & [ s2 , t2 ] in R & [ s1 , t2 ] in R & [ s1 , t2 ] in R & [ s2 , t2 ] in R ;
then H is negative & H is negative & H is non negative & H is non negative & H is non negative ;
attr f1 is total means : Def1 : for c , d being total Function st f1 is total & f2 is total holds ( f1 + f2 ) . c = ( f1 + f2 ) . c + ( f2 + f3 ) . d ;
z1 in W2 ` or z1 = W2 ` or z1 = W2 ` or z1 in W2 ` & z2 in W2 ` & z1 in W2 ` & z2 in W2 ` & z1 in W2 ;
p = 1 * p .= a " * a " * p .= a " * ( a " * p ) .= a " * ( a " * p " * p ) .= a " * ( a " * p " * p ) .= a " * ( a " * p " * p ) .= a " * ( a " * p " * p " * p " * p " ;
for seq1 be Real_Sequence , K be Real st for n be Nat st n <= m holds seq1 . n <= K holds upper_bound rng ( seq . n ) <= K
not ( E-max C in L~ go or ( E-max C ) in L~ go or ( E-max C ) in L~ go or ( E-max C ) in L~ go or ( E-max C ) in L~ go or ( E-max C ) in L~ go or ( E-max C ) meets L~ go or ( E-max C ) meets L~ go or ( E-max C ) meets L~ go or ( E-max C ) meets L~ go or ( E-max C ) meets L~ go or ( E-max C ) meets L~ go or ( E-max C in L~ go or ( E-max C in L~ go or ( E-max C in L~ go or not E-max C in L~ go or ( E-max
||. f . ( g . ( k + 1 ) ) - g . ( k + 1 ) .|| <= ||. g . ( k + 1 ) - g . ( k + 1 ) .|| * K ;
assume h = ( ( B .--> ( C .--> D ) ) +* ( C .--> E ) ) +* ( D .--> E ) +* ( E .--> F ) +* ( F .--> J ) +* ( M .--> N ) +* ( M .--> N ) +* ( N .--> N ) +* ( M .--> N ) +* ( M .--> N ) ) +* ( M .--> N ) ;
|. ( ( Carrier ( H ) . n ) (#) ( ( the Sorts of A ) . m ) - ( ( the Sorts of A ) . n ) ) .| <= e * ( ( the Sorts of A ) . m ) ;
( ( ( the Sorts of A ) . * ( the Arity of S ) ) . v ) = [ [ ( the Arity of S ) . v , ( the Sorts of A ) . v ] ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x1 , x2 , x3 , x4 , x1 , x2 } = { x1 , x2 , x3 , x4 , x1 , x2 } .= { x1 , x2 , x3 , x4 , x2 } .= { x1 , x2 , x3 , x4 , x2 } ;
assume that A = [. 0 , 2 * PI .] and <* A , B *> = <* 0 , 1 *> and <* A *> = <* 0 , 1 *> and <* A *> = <* 0 , 1 *> and <* A *> = <* 1 *> and <* A *> = <* 1 *> ;
p `2 is Permutation of dom f1 & p `2 = ( Sgm Y ) . i & p `2 = ( Sgm Y ) . i implies p `2 = ( Sgm Y ) . i
for x , y st x in A holds |. ( 1 / 2 ) * ( ( 1 / 2 ) * ( ( 1 / 2 ) * ( 1 / 2 ) ) ) .| <= 1 * |. ( 1 / 2 ) * ( 1 / 2 ) .|
( p2 `2 ) ^2 = |. q2 .| * ( ( q2 `2 / |. q2 .| - sn ) / ( 1 + sn ) ) ^2 .= ( q2 `1 / |. q2 .| - sn ) / ( 1 + sn ) ;
for f being PartFunc of the carrier of Cmin st dom f is compact & rng f c= dom f holds f . ( len f ) = f . ( len f ) & f . ( len f ) = f . ( len f )
assume for x being Element of Y st x in EqClass ( z , CompF ( B , G ) ) holds ( All ( a , CompF ( B , G ) ) ) . x = TRUE ;
consider F3 such that dom ( F . n ) = n1 and for k be Nat st k in n1 holds Q [ k , F . k ] ;
ex u , u1 st u <> u1 & u , u1 // v , v1 & u , u1 // v , v1 & u1 , v1 // v , u1 & u , v1 // v , u1 & u , u1 // v , v1 & u , u1 // v , u1 & v , v1 // v , u1 ;
for G being Group , A , B being strict Subgroup of G , N being normal Subgroup of G holds ( N , A ) ` * ( N , B ) = N ` * ( N , B )
for s be Real st s in dom F holds F . s = \int F . ( R . s + F . ( f . s ) ) ^2
width ( ( exists f1 , b1 , b2 st ( for i st i in dom f1 holds f1 . i = ( ( f2 . i ) * ( ( f1 . i ) * ( f1 . i ) ) ) ) * ( ( f2 . i ) * ( ( f2 . i ) * ( ( f2 . i ) * ( ( f2 . i ) * ( ( f2 . i ) * ( ( f2 . i ) * ( ( f2 . i ) * ( ( f2 . i ) ) ) ) ) ) ) ) ) ) ) = len ( ( f2 . i ) ) .= len ( ( f2 .
f | ]. - PI / 2 , PI / 2 .[ = f & f | ]. - PI / 2 , PI .[ = f | ]. - PI / 2 , PI .[ & f | ]. - PI / 2 , PI .[ = f | ]. - PI / 2 , PI .[ ;
assume that X is closed and a in X and a in a and y in a and not ex x st x in a & y in X & not x in { [ n , x ] } and y in a ;
Z = dom ( ( ( ( ( ( ( ( ( ( ( exp_R - arccot ) ) * ( ( ( exp_R - arccot ) ) * ( ( ( exp_R - arccot ) * ( ( ( exp_R - arccot ) * ( ( ( exp_R - arccot ) / ( 1 + ( ( ( exp_R - arccot ) ) * ( ( ( exp_R - arccot ) / ( 1 + ( ( exp_R - arccot ) ) ) ) ) ) ) ) ) ) ) ;
func VERUM ( V ) -> Subset of V means : Def1 : for k st 1 <= k & k <= len it holds it . k = ( l . k ) . k ;
for L being non empty TopSpace , N being net of L , M being net of L , L st M is convergent holds for c being Element of L st c in M holds c in N holds M is convergent iff for c being Element of L st c in M holds c <= ( N . c ) * ( N . c )
for s being Element of NAT holds ( ( ( id NAT ) (#) ( ( id NAT ) (#) ( ( id NAT ) (#) ( ( id NAT ) (#) ( ( id NAT ) (#) ( ( id NAT ) \ ( id NAT ) ) ) ) ) ) . s = ( ( ( ( id NAT ) (#) ( ( id NAT ) * ( ( id NAT ) * ( ( id NAT ) * ( ( id NAT ) * ( ( id NAT ) * ( ( id NAT ) ) ) ) ) ) ) . s ) ) . s )
then z /. 1 = ( E-max L~ z ) .. z & ( E-max L~ z ) .. z < ( E-max L~ z ) .. z ;
len ( p ^ <* ( 0 qua Real ) *> ) = len p + len <* ( 0 qua Real ) *> .= len p + 1 .= len p + 1 ;
assume that Z c= dom ( ( ( - ( ( ( - ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( A A A ) ) ) ) ) ) ) ) ) ) ) ) ) ) `| Z ) and for x st x in Z holds f . x = exp_R . x ) and f . x = - 1 ;
for R being add-associative right_zeroed right_complementable distributive associative associative associative distributive non empty doubleLoopStr , I , J being Subset of R holds ( I + J ) *' ( I + J ) c= I /\ J
consider f being Function of B1 , B2 such that for x being Element of B1 holds f . x = F ( x ) and f . x = F ( x ) and f . x = F ( x ) ;
dom ( x2 + y2 ) = Seg len x .= Seg len x .= dom ( x ^ <* y *> ) .= Seg len ( x ^ <* y *> ) .= Seg len ( x ^ <* y *> ) .= dom ( x ^ <* y *> ) .= Seg len ( x ^ <* y *> ) .= Seg len ( x ^ <* y *> ) .= dom ( x ^ <* y *> ) ;
for S being Functor of C , B , c being Object of C holds ( for f being Morphism of C holds ( id B ) . f = id ( ( the carrier of B ) . f ) iff ( id C ) . f = id ( ( the carrier of C ) . f )
ex a st a = a2 & a in f6 & a in f6 & not LIN f . a , f . b & for x st x in Z holds f . x = \/ ( { f . x } , f . x ) & f . x = \/ { f . x } ;
a in Free ( H2 , ( x. 4 ) ) '&' ( ( x. 4 ) '&' ( ( x. 4 ) '&' ( x. 4 ) ) ) & a in Free ( H2 , ( x. 4 ) '&' ( ( x. 4 ) '&' ( ( x. 4 ) '&' ( ( x. 4 ) '&' ( ( x. 4 ) '&' ( ( x. 4 ) '&' ( ( x. 4 ) '&' ( ( x. 4 ) '&' ( x. 4 ) ) ) ) ) ;
for C1 , C2 being non empty set , f being Function of C1 , C2 st f = ( the carrier of C2 ) holds f = g & f = g & f = g
( W-min L~ go ) .. go + ( E-max L~ go ) .. go = ( W-bound L~ go + W-bound L~ pion1 ) .. go + ( W-bound L~ go ) .. go + ( W-bound L~ pion1 ) .. go ;
assume that u = <* x0 , y0 , z0 *> and f is PartFunc of 3 , u and f is PartFunc of 3 , u and u = u and f . 3 = u and f . 1 = u and f . 2 = v and f . 3 = u and f . 1 = v and f . 1 = v and f . 1 = u and f . 2 = v ;
then ( t . {} in Vars & ex x being Element of Vars st x = ( t . {} ) . {} & ( t . {} ) term ( C ) ) . {} & ( t . {} = ( t . {} ) term ( C ) . {} ;
Valid ( p '&' p , J ) . v = Valid ( p , J ) . v .= Valid ( p , J ) . v .= Valid ( p , J ) . v .= Valid ( p , J ) . v .= Valid ( p , J ) . v .= 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 >= a ;
func Class R -> Subset-Family of R means : Def1 : for A being Subset of R holds it = Class ( R , A ) iff ex a being Element of R st a in A & a in A & a in A & a in A & a = R . a ;
defpred P [ Nat ] means ( ( ( ( ( the Source of G ) . ( n + 1 ) ) . ( n + 1 ) ) . ( n + 1 ) ) c= G . ( ( the Element of G ) . ( n + 1 ) ) . ( n + 1 ) ) ;
assume that dim ( W1 ) = 0 and dim ( W2 ) = 0 and dim ( W1 ) = 0 and dim ( W2 ) = 0 and dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W1 ) = 1 and dim ( W2 ) = 0 and dim ( W1 ) = 1 and dim ( W1 ) = 1 and dim ( W1 ) = 0 and dim ( W2 ) = 1 and dim ( W1 ) = 1 and dim ( W2 ) = 0 and dim ( W1 ) = 1 and dim ( W1 ) = 1 and dim ( W2 ) = 1 and dim ( W2 ) = 1 and dim ( W1 ) = 0 and dim ( W1 ) = 1 and dim ( W2 ) = 1 and dim ( W2 ) = 1 and dim ( W1 ) = 0
not ( ex m st m in ( m . t ) . {} = ( m . t ) . {} & ( m . t ) . {} = ( m . t ) . {} .= ( m . t ) . {} .= ( m , the carrier of C ) . {} .= m . {} .= m . {} ;
d11 = ( x9 ^ <* d *> ) . ( ( y9 ^ <* d *> ) . ( x , y ) ) .= f . ( ( y9 ^ <* d *> ) . ( x , y ) ) .= f . ( ( y9 ^ <* d *> ) . ( x , y ) ) .= f . ( ( x9 ^ <* d *> ) . ( x , y ) ) .= f . ( ( x9 ^ <* d *> ) . ( x , y ) ) .= f . ( ( x9 ^ <* d *> ) .= f . ( x9 ^ <* d *> ) .= f . ( y9 ^ <* d *> ) . ( x , y ) .= f . ( x9 ^ <* d *> ) .= f . ( x9 ^ <* d *> ) .= f . (
consider g such that x = g and dom g = dom ( f . 0 ) and for x being element st x in dom ( f . 0 ) holds g . x = ( f . 0 ) . x ;
x + 0. F_Complex = x + ( len x |-> 0. F_Complex ) .= ( x + 0. F_Complex ) .= ( x + 0. F_Complex ) .= x + 0. F_Complex .= x + 0. F_Complex .= x + 0. F_Complex .= x + 0. F_Complex .= x + 0. F_Complex .= x + 0. F_Complex .= x + 0. F_Complex .= x + 0. F_Complex ;
( ( f /. ( len f -' 1 ) ) + 1 ) in dom ( f /. ( len f -' 1 ) ) & ( f /. ( len f -' 1 ) ) . ( len f -' 1 ) = ( f /. ( len f -' 1 ) ) . ( len f -' 1 ) ;
assume that P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p2 , p3 and P1 is_an_arc_of p2 , p4 and P1 is_an_arc_of p2 , p4 and p2 in LSeg ( p1 , p2 ) and p2 in LSeg ( p2 , p3 ) and p1 in LSeg ( p2 , p1 ) and p2 in LSeg ( p1 , p2 ) and p2 in LSeg ( p2 , p3 ) and p2 in LSeg ( p1 , p2 ) and p2 in LSeg ( p2 , p3 ) and p2 in LSeg ( p2 , p4 ) and p2 in LSeg ( p1 , p2 ) and p2 in LSeg ( p2 , p4 ) and p2 in LSeg ( p1 , p2 ) and p2 in LSeg ( p2 , p4 ) and p2 in LSeg ( p2 , p4 ) and p2 in LSeg ( p1 , p2 ) and p2 in LSeg ( p2 ) and
reconsider a1 = a , b1 = b , c1 = c , c2 = d , c2 = a , c2 = d , c2 = c , c2 = d , c2 = d , c2 = c , c2 = d , c2 = d , c2 = d , c2 = c , c2 = d , c2 = d , c2 = d , c2 = c , c2 = d , c2 = d , \overline { a , b } , c2 = d , 8 = c , 8 = d , 8 = d , 7 = d , 8 = d , 8 = d , 8 = d , 8 = c , 8 = d , 8 = d , 8 = c , 8 = d , 8 = d , 8 = d , 8 = c , 8 = c , 8 = d
reconsider GtFFFFFFf = G1 . t as Morphism of ( G1 * F1 ) , ( G2 * F2 ) . a , ( G2 * F2 ) . a ;
LSeg ( f , i + i1 -' 1 ) = LSeg ( f /. ( i + 1 ) , f /. ( i + 1 ) ) \/ LSeg ( f /. ( i + 1 ) , f /. ( i + 1 ) ) ;
Integral ( P . m , P . n ) | dom ( P . m ) <= Integral ( M . n , P . m ) & Integral ( M . n , P . m ) <= Integral ( M . n , P . m ) ;
assume that dom f1 = dom f2 and for x , y being element st [ x , y ] in dom f2 holds f1 . [ x , y ] = f2 . [ x , y ] and f2 . [ x , y ] = f2 . [ y , x ] ;
consider v such that v = y and dist ( u , v ) < min ( ( - ( G * ( i , 1 ) `1 ) ) , ( G * ( i + 1 , 1 ) `2 ) ;
for G being Group , H being Subgroup of G , a being Element of G , b being Element of H st a = b holds a |^ ( a , b ) = ( b |^ a ) |^ ( b , a )
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 = { p0 where p2 is Point of TOP-REAL 2 : P [ p2 ] & p2 `1 <= 0 & p2 `1 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 } as Subset of ( TOP-REAL 2 ) | LSeg ( p1 , p2 ) ;
sqrt ( ( ( W-bound C ) / 2 ) ^2 + ( W-bound C ) / 2 ) ^2 ) <= sqrt ( ( W-bound C ) ^2 + ( W-bound C ) ^2 ) + ( W-bound C ) ^2 ;
for x be Element of X , n be Nat st x in E holds |. ( Re F ) . n .| <= P . x & |. ( Im F ) . n .| <= P . x
len ( @ ( @ ( @ p ) ) ) = len ( @ ( @ p ) ) + len ( <* ( @ p ) *> ) .= len ( @ ( @ p ) ) + len ( <* p *> ) ) .= len ( @ ( @ p ) ) + len ( <* p *> ) .= len ( @ p ) + len ( <* p *> ) .= len ( <* p *> ) ;
v / ( x. 3 , m1 ) / ( x. 4 , x. 0 ) / ( x. 4 , x. 0 ) / ( x. 4 , x. 0 ) / ( x. 4 , x. 0 ) / ( x. 4 , x. 0 ) / ( x. 4 , x. 0 ) = ( x. 4 ) / ( x. 0 , x. 0 ) ;
consider r being Element of M such that M , v / ( x. 3 , m ) / ( x. 4 , m ) / ( x. 0 , m ) / ( x. 4 , m ) / ( x. 0 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 0 , m ) / ( x. 4 , m ) ) / ( x. 4 , m ) = r / ( x. 4 , m ) / ( x. 0 , m ) / ( x. 4 , m ) ;
func w1 \ w2 -> Element of Union ( G , the carrier of G ) means : Def1 : for w being Element of ( G , v being Element of ( G , w ) * holds it . ( w , v ) = ( ( the Element of G ) \ ( the carrier of G ) ) . ( w , v ) ;
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= Exec ( i2 , s2 ) . b1 .= Exec ( i2 , s2 ) . b1 .= Exec ( i2 , s2 ) . b1 .= s . b1 .= s . b1 .= s . b1 .= s . b1 .= s . b1 .= s . b1 .= s . b1 .= s . b1 ;
for n , k being Nat holds 0 <= ( Partial_Sums |. seq .| ) . ( n + k ) - ( Partial_Sums ( |. seq .| ) . n )
set F = S \! \mathop { {} } ;
Partial_Sums ( seq ) . K + Partial_Sums ( seq ) . K + Partial_Sums ( seq ) . K + Partial_Sums ( seq ) . K + Partial_Sums ( seq ) . K >= ( Partial_Sums ( seq ) . K + Partial_Sums ( seq ) . K + Partial_Sums ( seq ) . K ;
consider L , R such that for x st x in N holds ( f | Z ) . x - ( f | Z ) . ( - 1 ) = L . ( - 1 ) + R . ( - 1 ) ;
the closed \HM { p1 } = ( the carrier of rectangle ( a , b , c ) ) \/ ( the carrier of Closed-Interval-TSpace ( a , b , c ) ) .= ( the carrier of Closed-Interval-TSpace ( a , b , c ) ) \/ ( the carrier of Closed-Interval-TSpace ( a , b , c ) ) ;
a * b ^2 + ( a * c ) ^2 + ( b * c ) ^2 + ( a * c ) ^2 >= 6 * a * c + ( b * c ) ^2 + ( a * c ) ^2 + ( a * c ) ^2 + ( a * c ) ^2 + ( a * c ) ^2 + ( a * b ) ^2 >= 6 * a * c + ( a * c ) ^2 + ( a * b ) ^2 ;
v / ( x1 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) = v / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) ;
Rotate ( Q ^ <* x *> , M ) = ( ( Q +* <* x *> ) +* ( ( Q +* <* x *> ) --> TRUE ) ) +* ( ( Q +* <* x *> --> TRUE ) ) +* ( ( Q +* <* x *> --> TRUE ) ) +* ( ( Q +* <* x *> --> TRUE ) ) ) ;
Sum ( ( F |^ n1 ) * Sum ( C |^ n1 ) ) = ( r |^ n1 ) * Sum ( C |^ n1 ) .= ( ( C |^ n1 ) * ( C |^ n1 ) ) * ( C |^ n1 ) .= ( ( C |^ n1 ) * ( C |^ n1 ) ) * ( C |^ n1 ) .= ( C |^ n1 ) * ( C |^ n1 ) .= ( C |^ n1 ) * ( C |^ n1 ) .= ( C |^ n1 ) * ( C |^ n1 ) .= ( C |^ n1 ) * ( C |^ n1 ) * ( C |^ n1 ) .= ( C |^ n1 ) * ( C |^ n1 ) * ( C |^ n1 ) .= ( C |^ n1 ) * ( C
( ( GoB f ) * ( len GoB f , 1 ) ) `1 = ( ( GoB f ) * ( 1 , 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 * ( s . $1 ) ) * ( a * ( s . $1 ) ) * ( a * ( s . $1 ) ) * ( a * ( s . $1 ) ) * ( a * ( s . $1 ) ) * ( a * ( s . $1 ) ) ) ;
( the_arity_of g ) . x = ( ( the Arity of S ) . ( g . x ) ) . ( ( the Arity of S ) . ( g . x ) ) .= ( ( the Arity of S ) . ( g . x ) ) . ( ( the Arity of S ) . ( g . x ) ) .= ( ( the Arity of S ) . ( g . x ) ) . ( ( the Arity of S ) . x ) .= ( ( the Arity of S ) . ( ( g . x ) ) . ( ( ( the Arity of S ) . x ) ;
( X \times Y ) c= X |^ Z & card ( ( X \times Y ) \ Y ) = card X |^ Z & card ( ( X \times Y ) \ Y ) = card X |^ Z ;
for a , b being Element of S , s being Element of NAT st s = F . n & a = F . n & b = F . ( n + 1 ) holds b = N . ( n + 1 )
E , f / ( x. 2 , x. 3 ) |= All ( x. 3 , x. 0 ) '&' ( x. 3 , x. 4 ) '&' ( x. 0 , x. 0 ) '&' ( x. 4 , x. 0 ) ) '&' ( x. 0 , x. 0 ) '&' ( x. 4 , x. 0 ) ) ;
ex R2 being 1-sorted st R2 = ( p | ( n2 -' 1 ) ) . i & ( the carrier of R1 ) . i = ( the carrier of R2 ) . i & ( the carrier of R2 ) . i = ( the carrier of R2 ) . i ;
[. a , b + sqrt ( 1 / ( k + 1 ) ) , ( ( the partial } \HM { z } ) + ( ( the partial } \HM { w ) ) + ( ( the partial } \HM { w } ) + ( ( the partial } \HM { w } ) + ( ( the partial } \HM { { z } ) + ( { w } ) ) ) ) . k } is Element of the partial } ;
Comput ( P , s , 2 + 1 ) . IC SCM+FSA = Exec ( P , Comput ( P , s , 2 ) ) . IC SCM+FSA .= Exec ( i , Comput ( P , s , 2 ) ) . IC SCM+FSA .= succ IC Comput ( P , s , 2 ) . IC SCM+FSA .= succ IC Comput ( P , s , 2 ) . IC SCM+FSA .= IC Comput ( P , s , 2 ) . IC SCM+FSA ;
card ( h1 ) . k = ( power F_Complex ) . ( - 1_ F_Complex ) .= ( ( - 1_ F_Complex ) * ( - 1_ F_Complex ) ) . k .= ( ( - 1_ F_Complex ) * ( - 1_ F_Complex ) ) . k .= ( ( - 1_ F_Complex ) * ( - 1_ F_Complex ) ) . k .= ( ( - 1 ) * ( - 1_ F_Complex ) ) . k .= ( ( - 1 ) * ( - 1 ) ) * ( - 1 ) .= ( ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1
( f * g ) /. c = f /. c * ( g /. c ) " .= ( f /. c ) * ( g /. c ) " .= ( f /. c ) * ( g /. c ) " .= ( f /. c ) * ( g /. c ) " .= ( f /. c ) * ( g /. c ) " .= ( f /. c ) * ( g /. c ) " ;
len ( ( C | ( len C -' 1 ) ) ^ ( ( C | ( len C -' 1 ) ) ) ) = len ( ( C | ( len C -' 1 ) ) ^ ( ( C | ( len C -' 1 ) ) ) ) .= len ( ( C | ( len C -' 1 ) ) ) + ( C | ( len C -' 1 ) ) ;
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) ;
defpred P [ Nat ] means for n holds 2 * Fib ( n + 1 ) = Fib ( n + 1 ) * Fib ( n + 1 ) + ( 2 * Fib ( n + 1 ) ) * Fib ( n + 1 ) * Fib ( n + 1 ) * Fib ( n + 1 ) + ( 2 * Fib ( n + 1 ) ) * Fib ( n + 1 ) * Fib ( n + 1 ) * Fib ( n + 1 ) * Fib ( n + 1 ) * Fib ( n + 1 ) * Fib ( n + 1 ) * Fib ( n + 1 ) * Fib ( n + 1 ) * Fib ( n + 1 ) * Fib ( n + 1 ) * Fib ( n + 1 ) * Fib ( n + 1 ) * Fib ( n + 1 ) * Fib
consider f being Function of [: INT , INT :] , INT such that f = f and f is onto and f is onto and f . 0 = n and f . 1 = n and f . 1 = n and f . 1 = n and f . 1 = n and f . 1 = n and f . 1 = n ;
consider c9 being Function of S , BOOLEAN such that c9 = ( chi ( S , BOOLEAN ) ) . ( A \/ B ) and E = ( chi ( S , BOOLEAN ) ) . ( A \/ B ) and E = ( ( A \/ B ) . ( A \/ B ) ) . ( A \/ B ) ;
consider y being Element of Y ( ) such that a = "\/" ( { F ( x , y ) where x is Element of Y ( ) , y is Element of Y ( ) : P [ x , y ] } , L ) and P [ y , x ] ;
assume that A c= Z and f = ( ( ( id Z ) (#) ( ( exp_R - arccot ) * ( ( exp_R - arccot ) * ( ( exp_R - arccot ) * ( ( exp_R - arccot ) * ( ( exp_R - arccot ) * ( ( exp_R - arccot ) * ( ( exp_R - arccot ) * ( ( exp_R - arccot ) * ( ( exp_R - arccot ) * ( ( exp_R - arccot ) * ( ( exp_R - arccot ) * ( ( exp_R - arccot ) ) ) ) ) ) ) ) ) ) ) and Z ) = dom ( ( exp_R - arccot ) ) and Z ) and Z ) = dom ( ( exp_R - arccot ) * ( ( ( exp_R - arccot ) * ( ( exp_R - arccot ) * ( ( ( exp_R - arccot ) ) ) ) ;
( ( GoB f ) * ( i , j2 ) ) `2 = ( ( GoB f ) * ( i + 1 , j2 ) ) `2 .= ( ( GoB f ) * ( i + 1 , j2 ) ) `2 .= ( ( GoB f ) * ( i + 1 , j2 ) ) `2 .= ( ( GoB f ) * ( i + 1 , j2 ) ) `2 .= ( ( GoB f ) * ( i + 1 , j2 ) ) `2 ;
dom Shift ( q , len q ) = { j + len Seq q where q is Nat : q in dom Seq ( q , len q ) & q in dom Seq ( q , len q ) & len Seq ( q , len q ) = len q + 1 & len ( q , len q ) = len q ;
consider G1 , G2 being Element of V such that G1 <= G2 and G2 <= G2 and G2 <= G2 and G2 <= G2 and G2 <= G2 and G2 <= G2 and G2 <= G2 and G2 <= G2 and G2 <= G2 and G2 <= G2 and G2 <= G2 and G2 <= G2 and G2 <= G2 and G2 <= G2 and G2 <= G2 and G2 <= G2 and G2 <= G2 ;
func - f -> PartFunc of C , V means : Def1 : dom it = dom f & for c st c in dom it holds it . c = - f . c & it . c = - f . c ;
consider phi such that phi is increasing and phi is increasing and for a st phi . a = a & a <> {} & a in L holds L . a = union ( L . a ) and L . a = L . ( union ( L . 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 ) and f /. ( i1 + 1 ) = ( GoB f ) * ( i1 , j1 ) ;
consider i , n such that n <> 0 and sqrt ( p . n ) = sqrt ( i / n ) and for n1 being Nat st n1 <> 0 & i < n holds |. p . n1 .| = ( i / n ) * ( i / n ) ;
assume that not 0 in Z and Z c= dom ( ( ( arccot ) (#) ( arccot ) ) `| Z ) and for x st x in Z holds ( ( arccot ) (#) ( arccot ) ) `| Z ) . x = - 1 / ( ( arccot ) . x ) ^2 and for x st x in Z holds ( ( arccot ) (#) ( arccot ) ) `| Z ) . x = - 1 / ( 1 + x ^2 ) ;
cell ( G1 , i1 -' 1 , j2 -' 1 ) \ ( ( i1 + 1 ) -' 1 ) c= ( ( ( i1 + 1 ) -' 1 ) \ ( ( ( i1 + 1 ) -' 1 ) + ( ( ( i1 + 1 ) -' 1 ) -' 1 ) + ( ( ( i1 + 1 ) -' 1 ) -' 1 ) + ( ( i1 + 1 ) -' 1 ) -' 1 ) \ ( ( i1 + 1 ) -' 1 ) ) ;
ex Q1 being open Subset of X st s = Q1 & ex F being Subset-Family of Y st ( for x being set st x in F holds F . x = union ( F . x ) ) & ( for x being set st x in F holds F . x = union ( F . x ) ) & ( for x being set st x in F holds F . x = union ( F . x ) ) & ( F . x = union ( F . x ) ) ;
gcd ( A1 , A2 ) = ( 1. ( A , A1 ) ) * ( A1 , A2 ) .= ( 1_ ( A , A1 ) ) * ( A1 , A2 ) .= ( 1_ ( A , A1 ) ) * ( A1 , A2 ) .= ( 1_ ( A , A1 ) ) * ( A1 , A2 ) ;
R8 = ( ( ( the Sorts of s2 ) . ( m2 + 1 ) ) . ( m2 + 1 ) ) . ( m2 + 1 ) .= [ ( the Sorts of s2 ) . ( m2 + 1 ) , ( the Sorts of s2 ) . ( m2 + 1 ) ] .= [ [ 3 , 0 ] .= [ 3 , 0 ] .= [ 3 , 0 ] ;
CurInstr ( P3 , Comput ( P3 , s3 , m1 + 1 ) ) = CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= halt SCMPDS .= halt SCMPDS .= halt SCMPDS .= halt SCMPDS .= halt SCMPDS ;
P1 /\ P2 = ( { p1 } \/ LSeg ( p1 , p2 ) \/ LSeg ( p2 , p2 ) ) \/ ( LSeg ( p1 , p2 ) \/ LSeg ( p2 , p2 ) ) \/ ( LSeg ( p1 , p2 ) \/ LSeg ( p2 , p2 ) \/ LSeg ( p2 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p2 , p2 ) \/ LSeg ( p2 , p2 ) \/ LSeg ( p2 , p2 ) \/ LSeg ( p2 , p2 ) ) ;
func not f in the still of Al means : Def1 : ex i st ( for p st p in dom f ex a st a in dom f & f . p = ( f . a ) . ( p . a ) ) & ( for i st i in dom f ex a st a in dom f & f . i = ( f . a ) . ( p . i ) ) ;
for a , b being Element of COMPLEX st |. a .| > |. b .| & |. a .| >= 1 holds a * ( - b ) is ] & ( - b ) * ( - a ) is ]
defpred P [ Nat ] means ( 1 <= $1 & $1 <= len g implies for i st [ i , j ] in Indices G & [ i , j ] in Indices G & G * ( i , j ) = G * ( i , j ) holds G * ( i , j ) = G * ( i , j ) ;
assume that C1 , C2 , C2 *> : f . ( len f + 1 ) = ( s1 . ( len f ) ) and for i being Nat holds s1 . ( len f + 1 ) = ( s1 . ( len f ) ) and s2 . ( len f + 1 ) = s2 . ( len f ) ;
( ||. f .|| ) | X = ||. f .|| .= ||. f .|| .= ||. f .|| .= ||. f .|| .= ||. f .|| .= ||. f .|| .= ||. f .|| .= ||. f .|| .= ||. f .|| .= ||. f .|| .= ||. f .|| .= ||. f .|| .= ||. f .|| .= ||. f .|| .= ||. f .|| .= ||. f .|| ;
|. q .| = ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) / ( 1 + cn ) & 0 < ( q `1 / |. q .| - cn ) / ( 1 + cn ) ;
for F being Subset-Family of T7 st F is open & not {} in F & for A , B being Subset of T7 st A in F & B in F & A in F & A misses B holds card A = card B & card A = card B & card A = card B & card A = card B & card A = card B & card A = card B & card A = card B & card B = card A & card B = card B & card A = card B & card B = card B = card B & card A = card B & card B = card B & card A = card A c= card B & card B = card A & card B = card A & card B = card B & card B = card A c= card A & card B = card B & card B
assume that len F >= 1 and len F = k + 1 and len F = len G and for k st k in dom F holds F . k = g . k and for k st k in dom F holds F . k = g . k and for k st k in dom F holds F . k = f . k ;
i |^ ( ( mod n ) |^ ( i + 1 ) ) = ( i |^ s ) * ( i |^ ( i + 1 ) ) .= ( i |^ ( i + 1 ) ) * ( i |^ ( i + 1 ) ) .= ( i |^ ( i + 1 ) ) * ( i |^ ( i + 1 ) ) .= ( i |^ ( i + 1 ) ) * ( i |^ ( i + 1 ) ) .= ( i |^ ( i + 1 ) * ( i + 1 ) * ( i |^ ( i + 1 ) * ( i + 1 ) * ( i + 1 ) * ( i |^ ( i + 1 ) * ( i + 1 ) * ( i + 1 ) * ( i + 1 ) .= ( i + 1 ) * ( i |^ ( i + 1 )
consider q being oriented oriented Chain of G such that r = q and q <> {} and q <> {} and ( F . ( len q ) ) = ( ( F . len q ) ) `1 and ( F . len q ) `1 = ( F . len q ) `1 and ( F . len q ) `1 = ( F . len q ) `1 and ( F . len q ) `1 = ( F . len p ) `1 ;
defpred P [ Element of NAT ] means ( ( ( g , Z ) Z ) . ( len g + 1 ) ) . ( len ( ( g , Z ) . ( len g + 1 ) ) ) = ( ( ( ( f , Z ) . ( len ( g , Z ) ) . ( len ( f , Z ) ) ) ) . ( len ( f , Z ) ) ;
for A , B being Matrix of n , REAL holds len ( A * B ) = len A & width ( A * B ) = width A & width ( A * B ) = width A & width ( A * B ) = width B & width ( A * B ) = width A & width ( A * B ) = width B & width ( A * B ) = width B
consider s being FinSequence of the carrier of R such that Sum s = u and for i being Element of NAT st 1 <= i & i <= len s ex a , b being Element of R st s . i = a * b & a in I & b in J & a <= b & b <= I & a <= b & b <= I ;
func |. ( x , y ) .| -> Element of COMPLEX equals |. ( Re x ) * ( ( Im y ) * ( Im y ) ) + ( Im y ) * ( Im y ) ) ;
consider g2 being FinSequence of Fu such that g2 is continuous and rng g2 c= A and g2 . 0 = A and g2 . 1 = ( g . 1 ) and g2 . ( len g2 ) = ( g . 1 ) `1 and g2 . ( len g2 ) = ( g . 1 ) `1 and g2 . ( len g2 ) = ( g . len g2 ) `1 ;
then n1 >= len p1 & n2 >= len p1 & n1 <= len p2 & n1 <= len p1 & n1 <= len p2 & n1 <= len p1 & n1 <= len p2 & n1 <= len p1 & n1 <= len p2 & n1 <= len p2 & n1 <= len p1 & n1 <= len p2 & n1 <= len p2 & n1 <= len p2 & n1 <= len p2 & n1 <= len p2 & n1 <= len p2 & n1 <= len p2 & n1 <= len p2 & n1 <= len p2 & n1 <= len p2 & n1 <= len p2 & n1 <= len p2 & n1 <= len p2 & n1 <= len p2 & n1 <= len p2 & n1 <= len p2 & n1 <= len p2 & n1 <= len p2 & n1 <= len p2 & n1 <= len p2 & n1 <= len p2 & n1 <= len p2 & n1 <= len p2 & n1 <= len p2 & n1 <= len p2 & n1 <= len p2
( q `1 ) * a <= ( q `1 ) * ( q `2 ) & ( q `1 ) * ( q `2 ) <= ( q `1 ) * ( q `2 ) & ( q `1 ) * ( q `2 ) <= ( q `1 ) * ( q `2 ) ;
( F6 . ( len p + 1 ) ) = ( F6 . ( len p + 1 ) ) .= ( ( F6 ) . ( len p + 1 ) ) .= ( ( ( the Sorts of A ) . ( len p + 1 ) ) . ( len p + 1 ) ) .= ( ( the Sorts of A ) . ( len p + 1 ) ) . ( len p + 1 ) .= ( ( the Sorts of A ) . ( len p + 1 ) ) . ( len p + 1 ) .= ( ( the Sorts of A ) . ( len p + 1 ) . ( len p + 1 ) . ( len p + 1 ) . ( len p + 1 ) ) . ( len p + 1 ) .= ( ( the Sorts of A ) . ( len p + 1 ) .= ( ( ( p . ( len p + 1 ) .= ( ( the
consider k1 being Nat such that k1 + k = 1 and a := k = ( <* a *> , <* a *> ) := ( <* a *> , <* a *> ) ) ^ ( <* a *> --> ( <* a *> ) ) ;
consider B8 being Subset of [: B1 , B2 :] , B1 being Subset of [: B2 , B2 :] such that B1 is finite and B2 is finite and B1 is finite and card B1 = card B2 and card B1 = card B2 and card B2 = card B1 and card B2 = card B2 and card B2 = card B2 and card B1 = card B2 and card B2 = card B2 and card B2 = card B2 and card B1 = card B2 and card B1 = card B2 and card B1 = card B2 = card B1 and card B2 = card B2 and card B2 = card B2 = card B2 and card B2 = card B2 = card B2 and card B1 = card B2 = card B2 and card B2 = card B2 and card B2 = card B2 and card B2 = card B2 and card B1 = card B2 and card B2 = card B1 and card B2 = card B2 = card B2 and card B2 = card B2 = card
v2 . b2 = ( curry ( F2 , g ) * ( ( curry ( F2 , g ) ) * ( ( curry ( F2 , g ) ) * ( ( ( curry ( F2 , g ) ) * ( ( curry ( F2 , g ) ) * ( ( curry ( F2 , g ) ) * ( ( curry ( F2 , g ) ) * ( ( id B ) * ( ( F2 , f ) ) * ( id B ) ) ) ) ) ) .= ( ( ( ( F2 , f ) ) * ( ( F2 , g ) ) ) . ( ( id B ) ) ) . b2 ) .= ( ( ( F2 , g ) ) . ( ( F2 , g ) ) . b2 ) . b2 ) . b2 ) . b2 .= ( ( ( ( ( ( ( ( F2 , f ) ) . b2 ) . b2 ) . b2 ) .= ( ( F2 , g
dom IExec ( I , P , Initialize s ) = the carrier of SCMPDS , P = P +* Initialize s , P2 = P +* stop I , s2 = P +* stop I , P2 = P +* stop I , P3 = Comput ( P3 , s3 , 1 ) , P4 = Comput ( P3 , s3 , 1 ) , P4 = P3 +* stop I , P4 = P3 +* stop I , s4 = P3 +* stop I , P4 = P3 +* stop I , P4 = P3 , P4 = P3 , s4 = P3 , P4 = P3 , P4 = P3 , P4 = P3 , P4 = P3 , P4 = P3 , P4 = P3 , P4 = P3 +* stop I , P4 = P3 +* stop I , P4 = P3 , P4 = P3 , P4 = P3 , P4 = P3 , P4 = P3 = P3 , P4 = P3 , P4 = P3 , P4 = P3 , P4 = P3 , P4
ex d-32 be Real st di0 > 0 & for h be Real st h > 0 & |. h .| < 1 holds |. h .| " * ||. ( R + R1 ) /. h .|| < e
LSeg ( G * ( len G , 1 ) , G * ( len G , 1 ) ) c= Int cell ( G , len G , width G ) \/ { G * ( len G , 1 ) } \/ { G * ( len G , 1 ) } ;
LSeg ( mid ( h , i1 , i2 ) , i1 -' 1 ) = LSeg ( h /. ( i1 -' 1 ) , h /. ( i1 -' 1 ) ) .= LSeg ( h /. ( i1 -' 1 ) , h /. ( i1 -' 1 ) ) .= LSeg ( h /. ( i1 -' 1 ) , h /. ( i1 -' 1 ) ) .= LSeg ( h /. ( i1 -' 1 ) , h /. ( i1 -' 1 ) ) ;
A = { q where q is Point of TOP-REAL 2 : LE q , q , P , p1 , p2 & LE q , q , P , p1 , p2 & LE q , q , P , p1 , p2 & LE q , q , P , p2 } ;
( ( - x ) .|. y ) = ( - 1 ) * ( ( - x ) .|. y ) .= ( - 1 ) * ( ( - x ) .|. y ) .= ( - 1 ) * ( ( - x ) .|. y ) .= ( - 1 ) * ( - 1 ) .= ( - 1 ) * ( - 1 ) .= ( - 1 ) * ( - 1 ) .= ( - 1 ) * ( - 1 ) * ( - 1 ) .= ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( 1 / ( - 1 ) .= ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1
0 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) = sqrt ( ( p `1 / p `2 ) ^2 + ( p `2 / p `1 ) ^2 ) * sqrt ( 1 + ( p `2 / p `2 ) ^2 ) ;
( ( ( - 1 ) (#) ( ( ( ( ( ( ( ( ( ( ( q `1 ) ) ) ) * ( ( q `1 ) ) / ( 1 + ( q `1 / q `1 ) ^2 ) ) ) ) ) ) ) / ( 1 + ( q `2 / q `1 ) ^2 ) ) ) = ( ( ( q `1 ) * ( q `2 ) ) / ( 1 + ( q `2 / q `1 ) ^2 ) ) / ( 1 + ( q `2 ) ^2 ) .= ( ( q `1 ) ^2 ) / ( 1 + ( q `1 ) ^2 ) / ( 1 + ( q `1 ) ^2 ) .= ( ( q `1 ) ^2 ) / ( 1 + ( q `1 ) ^2 ) / ( 1 + ( q `1 ) ^2 ) ^2 .= ( ( q `1 ) ^2 ) / ( 1 + ( q `1 ) ^2 ) ) / ( 1 + ( q `1 ) ^2 ) ) / ( 1 + ( q `1 )
func Shift ( f , h ) -> PartFunc of REAL , REAL means : Def1 : for x st x in dom it holds it . x = ( - h ) . x & it . x = ( - h ) . x & for x st x in dom it holds it . x = ( - h ) . x + ( - h ) . x ;
assume that 1 <= k and k + 1 <= len f and [ i , j ] in Indices GoB f and [ i + 1 , j ] in Indices GoB f and [ i + 1 , j ] in Indices GoB f and [ i + 1 , j ] in Indices GoB f and f /. k = G * ( i + 1 , j ) and f /. ( k + 1 ) = G * ( i + 1 , j ) ;
assume that not y in Free H and not x in Free ( H ) and not x in Free ( H ) and not y in Free ( H ) and not x in Free ( H ) and not y in Free ( H ) and not x in Free ( H ) and not y in Free ( H ) and not y in Free ( H ) ;
defpred P11 [ Nat ] means ( p |^ $1 ) * ( p |^ $1 ) = ( p |^ $1 ) * ( p |^ $1 ) & ( p |^ $1 ) * ( p |^ $1 ) = ( p |^ $1 ) * ( p |^ $1 ) ;
func \sigma ( C ) -> non empty Subset-Family of X means : Def1 : for A , B being Subset of X holds it . A c= it . ( A \/ B ) & for A , B being Subset of X holds it . ( A , B ) <= it . ( A , B ) ;
[#] ( ( ( ( dist ( o ) ) .: Q ) ) ) = ( ( ( dist ( o ) ) .: Q ) .: Q ) & ( ( dist ( o ) ) .: Q ) .: Q ) /\ ( ( dist ( o ) ) .: Q ) = ( ( dist ( o ) ) .: Q ) /\ ( ( dist ( o ) ) .: Q ) ;
rng ( F | ( [: S , T :] ) ) = { 1 } or rng ( F | ( [: S , T :] ) ) = { 1 } or rng ( F | [: S , T :] ) = { 1 } or rng ( F | [: S , T :] ) = { 1 } or rng ( F | [: S , T :] ) = { 1 } ;
( f " ( ( rng f ) . i ) ) . ( ( ( f . i ) . ( f . i ) ) ) = f . i " . ( ( f . i ) . ( f . i ) ) .= ( f . i ) " . ( ( f . i ) . ( f . i ) ) .= ( f . i ) " . ( ( f . i ) . ( f . i ) ) .= ( f . i ) " . ( f . i ) " . ( f . i ) " . ( f . i ) " . ( f . i ) " . ( f . i ) " . ( f . i ) " . ( f . i ) " . ( f . i ) " . ( f . i ) " . ( f . i ) " . ( f . i ) .= ( f . i ) " . ( f . i ) " . ( f . i ) .= ( f . i ) " . ( f . i ) " . ( f . i ) .= ( f
consider P1 , P2 being Subset of TOP-REAL 2 such that P1 /\ P2 = { p1 } and P1 = { p2 } and P1 = { p2 } and P1 = { p2 } and P1 = { p2 } and P2 = { p1 } and P1 = { p2 } and P1 = { p2 } and p2 = { p2 } and p2 = { p2 } and p1 in { p2 } and p2 in { p2 } and p2 in { p2 } and p1 in { p2 } and p2 in LSeg ( p1 , p2 } and p2 in LSeg ( p1 , p2 , p3 } and p2 in LSeg ( p1 , p2 } and p2 in LSeg ( p1 , p2 } and p2 in LSeg ( p2 in LSeg ( p1 , p2 , p3 } and p2 in LSeg ( p1 , p2 , p4 ) and p2 in LSeg ( p1 , p2 , p3 , p3 } and p2 in LSeg ( p1 , p2 , p3 } and p2 in LSeg ( p1 , p2 , p4 ) and p2 in LSeg ( p2 , p3 , p4 ) and p2 in LSeg ( p2 , p3 ,
f . p2 = |[ ( ( p2 `1 ) / |. p2 .| - cn ) / ( 1 + cn ) , ( p2 `1 / |. p2 .| - cn ) / ( 1 + cn ) ]| ;
( AffineMap ( a , X ) ) " . x = ( \HM { the } \HM { carrier } \HM { of X : a in ( ( TOP-REAL n ) \ { 0 } ) \ { 0 } ) . x .= ( ( ( TOP-REAL n ) \ { 0 } ) \ { 0 } ) . x .= ( ( TOP-REAL n ) \ { 0 } ) . x .= 0. X ;
for T being non empty normal TopSpace , A , B being Subset of T , r being Real st A <> {} & A misses B & B misses A holds r in ( Cl ( A ) ) \/ ( ( Cl ( B ) ) \/ ( Cl ( B ) ) ) & for p being Point of T st p in ( Cl ( A ) ) \/ ( Cl ( B ) ) holds p in ( Cl ( B ) ) \/ ( Cl ( B ) )
for i , j st i + 1 in dom F for G1 , G2 being strict Subgroup of G st G1 = F . ( i + 1 ) & G2 = F . ( i + 1 ) & G2 = F . ( i + 1 ) holds G1 = G2 & G2 = F . ( i + 1 )
for x st x in Z holds ( ( ( arccot ) (#) ( arccot * ( arccot * ( arccot * ( arccot * ( arccot * ( arccot * x ) ) ) ) ) `| Z ) . x = ( ( ( arccot * ( arccot * x ) ) / ( 1 + x ^2 ) ) * ( ( arccot * ( arccot * x ) ^2 ) )
synonym f /* a -> convergent means : Def2 : for for a , b st a in dom f & b in dom f & a < b ex f st f . a = ( f /* a ) . b & for a st a in dom f holds f . a = ( f /* a ) . ( f /* a ) ;
then X1 misses ( X1 union X2 ) & ( X1 union X2 ) misses ( X1 union X2 ) & ( X1 union X2 ) misses ( X1 union X2 ) & ( X1 union X2 ) misses ( X1 union X2 ) & ( X1 union X2 ) misses ( X1 union X2 ) & ( X1 union X2 ) misses ( X1 union X2 ) ;
ex N be Neighbourhood of x0 st N c= dom SVF1 ( 1 , f , u ) & ex L be Neighbourhood of x0 st for x st x in N holds ( SVF1 ( 1 , f , u ) ) . x - ( SVF1 ( 1 , f , u ) ) . x = ( SVF1 ( 1 , f , u ) ) . x - ( SVF1 ( 1 , f , u ) ) . x
sqrt ( ( p2 `1 / p2 `1 ) ^2 + ( p2 `2 / p2 `1 ) ^2 ) >= sqrt ( ( p2 `1 / p2 `1 ) ^2 + ( p2 `2 / p2 `2 ) ^2 ) ;
( ( 1 / ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( q `1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) / ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
assume that for x holds f . x = ( ( - 1 ) (#) ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin
consider X1 being Subset of Y , Y1 being Subset of X such that t = X1 and Y1 in Y1 and Y1 in Y2 and ex Y1 being Subset of Y st Y1 = Y1 & Y1 = Y1 & Y1 in Y2 & Y1 in Y2 & Y1 in Y2 and Y1 = Y1 and Y1 in Y2 and Y1 in Y2 and Y1 = Y2 and Y1 = Y2 and Y1 = Y2 and Y1 = Y2 and Y1 = Y2 and Y1 = Y2 and Y1 = Y2 and Y1 = Y2 and Y2 = Y2 and Y1 = Y2 and Y1 = Y2 and Y1 = Y2 and Y1 = Y2 and Y2 = Y2 and Y2 = Y2 and Y2 = Y1 and Y1 = Y2 and Y1 = Y2 and Y2 = Y2 and Y2 = Y2 and Y2 = Y2 and Y2 = Y2 and Y2 = Y2 and Y2 = Y2 and Y2 = Y2 and Y2 = Y2 and Y2 = Y2 and Y2 = Y2 and Y2 and Y2 = Y2 and Y2 and Y2 = Y2 and Y2 and Y2 = Y2 and Y2 = Y2 and Y2 and Y2 = Y2 and Y2 = Y2 and Y2 = Y2 and Y2 = Y2 and Y2 = Y2 and Y2 = Y2 and Y2 = Y2 and Y2 and Y2 =
card ( S . n ) = card ( { [ d , c ] } + ( a * d ] } ) .= card ( { [ d , c ] } , [ d , b ] } , [ d , c ] } , [ d , c ] } , [ d , c ] } , [ d , c ] } ;
sqrt ( ( ( W-bound D ) / ( 2 |^ ( m + 1 ) ) - ( W-bound D ) / ( 2 |^ ( m + 1 ) ) ) ) = ( ( W-bound D ) / ( 2 |^ ( m + 1 ) - ( W-bound D ) / ( 2 |^ ( m + 1 ) ) ) / ( 2 |^ ( m + 1 ) - ( W-bound D ) / ( 2 |^ ( m + 1 ) ) ) .= ( ( W-bound D ) - ( W-bound D ) - ( W-bound D ) ) / ( 2 |^ ( m + 1 ) ) / ( 2 |^ ( m + 1 ) - ( W-bound D ) - ( 2 |^ ( m + 1 ) ) / ( 2 |^ ( m + 1 ) ) / ( 2 |^ ( m + 1 ) ) / ( 2 |^ ( m + 1 ) ) / ( 2 |^ ( m + 1 ) ) / ( 2 |^ ( m + 1 ) ) .= ( ( ( 2 |^ ( m + 1 ) - ( 2 |^ ( m + 1 ) ) / ( 2 |^ ( m + 1 ) ) .= ( ( 2 |^ ( m + 1 ) )