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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b
D c= S
let Y ;
S ` is bounded
q in X ;
V ;
y in N ;
x in T ;
m < n ;
m <= n ;
n > 1 ;
let r ;
t in I ;
n <= 4 ;
M is finite ;
let X ;
Y c= Z ;
A // M ;
let U , S ;
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 >= X ;
T is continuous ;
d <= a ;
p <= r ;
t < s ;
p <= t ;
t < s ;
let r ;
D <= E ;
assume e > 0 ;
assume 0 < g ;
p in X ;
x in X ;
Y ` in Y ;
assume 0 < g ;
not c in Y ;
not v in L ;
2 in z `1 ;
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 `1 <= b `1 ;
assume b in X ;
assume k <> 1 ;
f = Product l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is from of squares ;
assume m > 0 ;
assume A c= B ;
X is lower
assume A <> {} ;
assume X <> {} ;
assume F <> {} ;
assume G is open ;
assume f is dilatation ;
assume y in W ;
y \not <= x ;
A ` in B ` ;
assume i = 1 ;
let x be element ;
x `1 = x `1 ;
let X be BCK-algebra ;
assume S is non empty ;
a in REAL ;
let p be set ;
let A be set ;
let G be _Graph , a , b be Vertex of G ;
let G be _Graph , a , b be Vertex of G ;
let a be Complex ;
let x be element ;
let x be element ;
let C be FormalContext , a , b be Element of C ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
y be Real ;
X c= f . a
let y be element ;
let x be element ;
let i be Nat ;
let x be element ;
n in NAT ;
let a be element ;
m in NAT ;
let u be element ;
i in NAT ;
let g be Function ;
Z c= NAT ;
l <= ma ;
let y be element ;
r2 in dom f ;
let x be element ;
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 = Set ;
n >= 0 + 1 ;
k c= k + 1 ;
R1 c= R ;
k + 1 >= k ;
k c= k + 1 ;
let j be Nat ;
o , a // Y ;
R c= Cl G ;
Cl B = B ;
let j be Nat ;
1 <= j + 1 ;
arccot is_differentiable_on Z ;
exp_R is_differentiable_in x ;
j < i0 ;
let j be Nat ;
n <= n + 1 ;
k = i + m ;
assume C meets S ;
n <= n + 1 ;
let n be Nat ;
h1 = {} ;
0 + 1 = 1 ;
o <> b3 ;
f2 is one-to-one ;
support p = {} ;
assume x in Z ;
i <= i + 1 ;
r1 <= 1 ;
let n be Nat ;
a "/\" b <= a ;
let n be Nat ;
0 <= r0 ;
let e be Real , x be set ;
not r in G . l
c1 = 0 ;
a + a = a ;
<* 0 *> in e ;
t in { t } ;
assume F is not discrete ;
m1 divides m ;
B * A <> {} ;
a + b <> {} ;
p * p > p ;
let y be ExtReal ;
let a be Int-Location , i be Integer ;
let l be Nat ;
let i be Nat ;
let r ;
1 <= i2 ;
a "\/" c = c ;
let r be Real ;
let i be Nat ;
let m be Nat ;
x = p2 ;
let i be Nat ;
y < r + 1 ;
rng c c= E
Cl R is boundary ;
let i be Nat ;
R2 ;
cluster uparrow x -> ex x st x in X ;
X <> { x } ;
x in { x } ;
q , b // M ;
A . i c= Y ;
P [ k ] ;
2 to_power x in W ;
X [ 0 ] ;
P [ 0 ] ;
A = A |^ i ;
2 >= such that x >= sn ;
G . y <> 0 ;
let X be RealNormSpace , A be Subset of X ;
a in A ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , A be Subset of V ;
assume x in - M ;
k < s . a ;
not t in { p } ;
let Y be set , f be Function of Y , BOOLEAN ;
M , L are_isomorphic ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded ;
rng f = Y ;
G8 c= L ;
assume x in Cl Q ;
m in dom P ;
i <= len Q ;
len F = 3 ;
still_not-bound_in p = {} ;
z in rng p ;
lim b = 0 ;
len W = 3 ;
k in dom p ;
k <= len p ;
i <= len p ;
1 in dom f ;
b `1 = a `1 + 1 ;
x `1 = a * y `1 ;
rng D c= A ;
assume x in K1 ;
1 <= ii & ii <= len f ;
1 <= ii & ii <= len f ;
pcontradiction c= PI ;
1 <= ii & ii <= len G ;
1 <= ii & ii <= len G ;
LMP C in L ;
1 in dom f ;
let seq , seq1 , seq2 ;
set C = a * B ;
x in rng f ;
assume f is_continuous_on X ;
I = dom A ;
u in dom p ;
assume a < x + 1 ;
s-7 is bounded ;
assume I c= P1 ;
n in dom I ;
let Q ;
B c= dom f ;
b + p _|_ a ;
x in dom g ;
F-14 is continuous ;
dom g = X ;
len q = m ;
assume A2 is closed ;
cluster R \ S -> real-valued ;
sup D in S ;
x << sup D ;
b1 >= Z1 & b2 >= Z1 ;
assume w = 0. V ;
assume x in A . i ;
g in being being being being being being being being being being being empty set ;
y in dom t ;
i in dom g ;
assume P [ k ] ;
c= c= f ;
x9 is increasing & x9 is increasing ;
let e2 be element ;
- b divides b ;
F c= \tau ( F ) ;
Gseq is non-decreasing ;
Gseq is non-decreasing ;
assume v in H . m ;
assume b in [#] B ;
let S be non void ManySortedSign , A be non-empty MSAlgebra over S ;
assume P [ n ] ;
assume union S is independent & finite S is finite ;
V is Subspace of V ;
assume P [ k ] ;
rng f c= NAT ;
assume ex_inf_of X , L ;
y in rng f ;
let s , I be set , f be Function ;
b ` c= b9 & b ` c= b9 ;
assume not x in NAT + Q ;
A /\ B = { a } ;
assume len f > 0 ;
assume x in dom f ;
b , a // o , c ;
B in B-24 ;
cluster product p -> non empty ;
z , x // x , p ;
assume x in rng N ;
cosec is_differentiable_in x & cosec is_differentiable_in x ;
assume y in rng S ;
let x , y be element ;
i2 < i1 & i1 < i2 implies i2 < i1
a * h in a * H ;
p , q in Y ;
redefine func sqrt I -> left ideal ;
q1 in A1 & q2 in A2 ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A2 c= A1 ;
an < n ;
assume A c= dom f ;
Re f is_integrable_on M ;
let k , m be element ;
a , a // b , b ;
j + 1 < k + 1 ;
m + 1 <= n1 ;
g is_differentiable_in x0 & g is_differentiable_in x0 ;
g is_continuous_in x0 & g is_continuous_in x0 ;
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 , P ;
d , c // a , b ;
let t , u be set ;
let X be set ;
assume k in dom s ;
let r be non negative Real ;
assume x in F | M ;
let Y be Subset of S ;
let X be non empty TopSpace , A be Subset of X ;
[ a , b ] in R ;
x + w < y + w ;
{ a , b } >= 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 ;
let b be Element of X , a be Element of X ;
R [ x , y ] ;
x ` ` = x ;
b \ x = 0. X ;
<* d *> in D |^ 1 ;
P [ k + 1 ] ;
m in dom ( n - 1 ) ;
h2 . a = y ;
P [ n + 1 ] ;
redefine func G * F -> prenot | ;
let R be non empty multMagma , a be Element of R ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p ;
assume f | X is lower ;
x in rng co & y in rng co ;
let x be Element of B ;
let t be Element of D ;
assume x in Q .vertices() ;
set q = s ^\ k ;
let t be VECTOR of X ;
let x be Element of A ;
assume y in rng p `1 ;
let M be mamaid ;
let N be non empty Subset of M ;
let R be RelStr with finite holds R is finite ;
let n , k be Nat ;
let P , Q be be be be RelStr ;
P = Q /\ [#] S ;
F . r in { 0 } ;
let x be Element of X ;
let x be Element of X ;
let u be Vector of V ;
reconsider d = x as Int-Location ;
assume I is not \leq a ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < v8 ;
x <= c2 . x ;
x in F ` & y in F ` ;
redefine func S --> T -> such that S is such that T is be be x1 } ;
assume t1 <= t2 & t2 <= t2 ;
let i , j be even Integer ;
assume F1 <> F2 & F2 <> F1 ;
c in Intersect ( union R ) ;
dom p1 = c & dom p2 = c ;
a = 0 or a = 1 ;
assume A1 : x in A6 ;
set i1 = i + 1 ;
assume a1 = b1 & a2 = b2 ;
dom g1 = A & dom g2 = A ;
i < len M + 1 ;
assume not -infty in rng G ;
N c= dom ( f1 - f2 ) ;
x in dom ( sec | Z ) ;
assume [ x , y ] in R ;
set d = ( x - y ) / 2 ;
1 <= len ( g1 ^ g2 ) ;
len s2 > 1 & len s2 > 1 ;
z in dom ( f1 - f2 ) ;
1 in dom ( D2 | Seg 1 ) ;
( p `2 ) ^2 = 0 ;
j2 <= width G & j2 <= width G ;
len PI > 1 + 1 ;
set n1 = n + 1 ;
|. q .| = 1 ;
let s be SortSymbol of S ;
@ i = i ;
X1 c= dom f & X2 c= dom g ;
h . x in h . a ;
let G be \it let V be \mathbin { empty } ;
cluster m * n -> square ;
let k9 be Nat , n be Nat ;
i - 1 > m - 1 ;
R is transitive implies field R = field R
set F = <* u , w *> ;
p-2 c= P3 & p-2 c= P3 ;
I is_closed_on t , Q ;
assume [ S , x ] is vertical ;
i <= len ( f2 | 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= XX & BX c= BY ;
n2 <= ( 2 |^ ( n + 1 ) ) ;
A /\ [: P , Q :] c= A `
cluster x -valued for Function ;
let Q be Subset-Family of S , A be Subset of S ;
assume n in dom ( g2 ) ;
let a be Element of R ;
t `1 in dom ( e2 | dom ( e2 | dom p ) ) ;
N . 1 in rng N ;
- z in A \/ B ;
let S be SigmaField of X , T be non empty set ;
i . y in rng i ;
REAL c= dom f & dom f c= dom g ;
f . x in rng f ;
mt <= ( r / 2 ) ;
s2 in r-5 ( X , i ) ;
let z , z be complex number ;
n <= ( N . m ) ;
LIN q , p , s ;
f . x = waybelow x /\ B ;
set L = |[ S \to T ]| ;
let x be non positive ExtReal , r be positive ExtReal ;
let m be Element of M ;
f in union rng ( F1 | A ) ;
let K be add-associative right_zeroed right_complementable non empty doubleLoopStr , A be Subset of K ;
let i be Element of NAT , k be Element of NAT ;
rng ( F * g ) c= Y
dom f c= dom x & rng f c= dom y ;
n1 < n1 + 1 & n2 + 1 < n1 ;
n1 < n1 + 1 & n2 + 1 < n1 ;
cluster 1. T -> non empty ;
[ y2 , 2 ] `2 = z ;
let m be Element of NAT , n be Element of NAT ;
let S be Subset of R ;
y in rng ( S | A ) ;
b = sup dom f & a = sup dom f ;
x in Seg ( len q ) ;
reconsider X = D ( ) as set ;
[ a , c ] in E1 ;
assume n in dom ( h2 - h2 ) ;
w + 1 = ( a - 1 ) ;
j + 1 <= j + 1 + 1 ;
k2 + 1 <= k1 & k1 + 1 <= k2 ;
let i be Element of NAT ;
Support u = Support p & Support u = Support p ;
assume X is complete \frac of m , n ;
assume that f = g and p = q ;
n1 <= n1 + 1 & n2 + 1 <= n1 ;
let x be Element of REAL n ;
assume x in rng s2 & y in rng s2 ;
x0 < x0 + 1 & x0 + 1 < x0 + 1 ;
len L5 = W & len L5 = len L5 ;
P c= Seg ( len A ) ;
dom q = Seg n & dom q = Seg n ;
j <= width M *' ;
let r8 be real-valued Real_Sequence ;
let k be Element of NAT , n be Element of NAT ;
Integral ( M , P ) < +infty ;
let n be Element of NAT , x be set ;
assume z in being as as as as atlen of z ;
let i be set ;
n - 1 = n-1 ;
len ( n - m ) = n ;
\mathop { Z } c= F ;
assume x in X or x = X ;
x is midpoint of b , c ;
let A , B be non empty set , f be Function of A , B ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q & dom q = Seg i ;
let s be Element of E |^ \omega ;
let B1 be Basis of x , y ;
Carrier ( L2 ) /\ Carrier ( L2 ) = {} ;
L1 /\ L2 = {} implies L1 /\ L2 = {}
assume downarrow x = downarrow y ;
assume b , c // b , c ;
LIN q , c , c ;
x in rng ( f . -129 ) ;
set n8 = n + j ;
let D7 be non empty set , f be Function of D , REAL ;
let K be add-associative right_zeroed right_complementable non empty addLoopStr , M be Matrix of K ;
assume that f `1 = f and h `1 = h ;
R1 - R2 is total & R1 - R2 is total ;
k in NAT & 1 <= k ;
let a be Element of G ;
assume x0 in [. a , b .] ;
K1 ` is open & K1 ` is open ;
assume a , b are_maximal in C ;
let a , b be Element of S ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = Integral ( M , f ) ;
cluster -> nes<* A *> -> ns<*
not u in { ag } ;
the carrier of f c= B ;
reconsider z = x as Vector of V ;
let L be \rangle cluster the RelStr of L ;
r (#) H is being being being being non-zero Real_Sequence of X , Y ;
s . intloc 0 = 1 ;
assume that x in C and y in C ;
let U0 be strict universal non-empty MSAlgebra over S , A be Subset of U0 ;
[ x , Bottom T ] is compact ;
i + 1 + k in dom p ;
F . i is stable Subset of M ;
rr-35 in ( { y } ) ;
let x , y be Element of X ;
let A , I be such that A is such such such that x in A ;
[ y , z ] in [: O , O :] ;
card Macro i = 1 & card Macro i = 1 ;
rng Sgm A = A & rng Sgm A = A ;
q |- r -<* 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 & LIN o , a , b ;
p . 2 = Z |^ Y ;
( D . D ) `2 = {} ;
n + 1 + 1 <= len g ;
a in [: CQC-WFF ( Al ) , NAT :] ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
let I be Ideal of L ;
set g = f1 + f2 , h = f2 + f3 ;
a <= max ( a , b ) ;
i-1 < len G + 1-1 ;
g . 1 = f . i1 ;
x `1 , y `2 in A2 ;
( f /* s ) . k < r ;
set v = VAL g ;
i - k + 1 <= S ;
cluster associative for non empty multMagma ;
x in support ( ( support t ) \ support ( b ) ) ;
assume a in [: the carrier of G , the carrier of G :] ;
i `1 <= len ( y `1 ) ;
assume p divides b1 + b2 & p divides b1 + b2 ;
M1 <= sup M1 & M2 <= sup M1 implies M1 + M2 <= M2
assume x in W-min ( X ) & y in W-min ( X ) ;
j in dom ( z | i ) ;
let x be Element of D ( ) ;
IC s4 = l1 & IC s4 = l1 ;
a = {} or a = { x } ;
set uG = Vertices G , uG = Vertices G ;
seq " (#) ( seq " ) is non-zero ;
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
hcn c= hF & hF c= hy ;
]. a , b .[ c= Z ;
X1 , X2 are_separated implies X1 , X2 are_separated
a in Cl ( union F \ G ) ;
set x1 = [ 0 , 0 ] ;
k + 1 -' 1 = k ;
cluster -> real-valued for Relation of NAT , Q ;
ex v st C = v + W ;
let IT be non empty addLoopStr , a be Element of IT ;
assume V is Abelian add-associative right_zeroed right_complementable ;
XY \/ Y in \sigma ( L ) ;
reconsider x = x as Element of S ;
max ( a , b ) = a ;
sup B is upper & sup B = sup B ;
let L be non empty reflexive RelStr , S be non empty Subset of L ;
R is reflexive & X is transitive implies R is transitive
E , g |= the_right_argument_of H implies E , g |= H
dom G `2 = a & dom G = a ;
( 1 - 4 ) >= - r ;
G . p0 in rng G & G . p2 in rng G ;
let x be Element of FF , y be Element of F ;
D [ P , 0 , 0 ] ;
z in dom id ( B ) ;
y in the carrier of N & y in the carrier of N ;
g in the carrier of H & h in the carrier of H ;
rng ( f | [: the carrier of S , the carrier of T :] ) c= [: the carrier of S , the carrier of T :] ;
j `2 + 1 in dom s1 & j + 1 in dom s2 ;
let A , B be strict Subgroup of G ;
let C be non empty Subset of REAL , a be Real ;
f . z1 in dom h & f . z2 in dom h ;
P . k1 in rng P & P . k1 in rng P ;
M = A9 +* ( {} , A ) & M = A ;
let p be FinSequence of ( the carrier of K ) * ;
f . n1 in rng f & f . n1 in rng f ;
M . ( F . 0 ) in REAL ;
h . [. a , b .[ = b-a ;
assume the distance of V , Q is Real ;
let a be Element of ^ ( V , C ) ;
let s be Element of [: P , Q :] ;
let PA be non empty RelStr , T be non empty RelStr ;
let n be Nat ;
the carrier of g c= B & the carrier of g c= B ;
I = halt SCM R & I = {} ;
consider b being element such that b in B ;
set BM = BCS ( K , n ) ;
l <= -> and ( every j holds l . j <= x ) ;
assume x in downarrow [ s , t ] ;
( x `2 ) ^2 in uparrow t ;
x in ( JumpParts T ) \ ( JumpParts T ) ;
let h be Morphism of c , a ;
Y c= 1. ( K , the_rank_of Y ) ;
A2 \/ A3 c= Carrier ( L1 ) \/ Carrier ( L2 ) ;
assume LIN o , a , b ;
b , c // d1 , e2 ;
x1 , x2 , x3 , x4 is_collinear & x2 , x3 , x4 is_collinear ;
dom <* y *> = Seg 1 & dom <* y *> = Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar s .| ;
[ x , x `1 ] in X ~ ;
for n being Nat holds 0 <= x . n
[' a , b '] = [. a , b .] ;
cluster -> g closed for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 , q1 is_collinear & p1 , p2 is_collinear ;
dom M1 = Seg n & dom M2 = Seg n ;
x = [ x1 , x2 , x3 , x4 ] ;
let R , Q be ManySortedSet of A ;
set d = ( 1 / ( n + 1 ) ) ;
rng ( g2 ) c= dom ( W (#) ( g1 ) ) ;
P . ( [#] Sigma \ B ) <> 0 ;
a in field R & a = b ;
let M be non empty Subset of V , A be Subset of V ;
let I be Program of SCM+FSA , a be Int-Location ;
assume x in rng ( ( R * S ) * R ) ;
let b be Element of the lattice of T ;
dist ( e , z ) > r-r ;
u1 + v1 in W2 & v1 + v2 in W1 ;
assume the carrier of L misses rng G ;
let L be lower-bounded antisymmetric transitive RelStr ;
assume [ x , y ] in a9 ;
dom ( A * e ) = NAT ;
let a , b be Vertex of G ;
let x be Element of Bool M , M be Subset of Bool M ;
0 <= Arg a & Arg a < 2 * PI ;
o9 , a19 // o9 , y & o9 , y9 // y , y ;
{ v } c= the carrier of l ;
let x be variable of A ;
assume x in dom ( uncurry f ) & y in dom ( uncurry f ) ;
rng F c= ( product f ) |^ X
assume D2 . k in rng D & D1 . k in rng D ;
f " . p1 = 0 & f . p2 = 0 ;
set x = the Element of X , y = the Element of X ;
dom Ser ( G ) = NAT & rng Ser ( G ) = NAT ;
let n be Element of NAT ;
assume LIN c , a , e1 ;
cluster finite for FinSequence of NAT ;
reconsider d = c , e = d as Element of L1 ;
( v2 |-- I ) . X <= 1 ;
assume x in the carrier of f & y in the carrier of f ;
conv @ A c= conv conv @ A & conv @ A c= conv @ A ;
reconsider B = b as Element of the topology of T ;
J , v |= P ! l & J , v |= l ;
redefine func J . i -> non empty TopSpace ;
ex_sup_of Y1 \/ Y2 , T & ex_sup_of Y1 , T ;
W1 is_\HM { 0 } & R is not empty implies R is connected
assume x in the carrier of R & y in the carrier of R ;
dom ( n | n ) = Seg n & dom ( n | n ) = Seg n ;
s4 misses s2 & s4 misses s2 ;
assume ( a 'imp' b ) . z = TRUE ;
assume that X is open and f = X --> d ;
assume [ a , y ] in an ;
assume that function I c= J and function I c= K and I c= J ;
Im ( seq . n ) = 0 & Im ( seq . n ) = 0 ;
( sin . x ) ^2 <> 0 & ( sin . x ) ^2 <> 0 ;
sin * ( sin - cos ) is_differentiable_on Z & Z c= dom ( sin - cos ) ;
t3 . n = t3 . n & i <= n ;
dom ( element - F ) c= dom F ;
W1 . x = W2 . x & W2 . x = W2 . x ;
y in W .vertices() \/ W .vertices() ;
( k + 1 ) <= len ( v | ( k + 1 ) ) ;
x * a divides y * a . mod m ;
proj2 .: S c= proj2 .: P & proj2 .: P c= P ;
h . p4 = g2 . I & h . I = g2 . I ;
G * ( 1 , k ) `1 = U /. 1 .= G * ( 1 , k ) `1 ;
f . rr1 in rng f & rr1 in rng f ;
i + 1 + 1-1 <= len - 1 ;
rng F = rng ( F | n ) & rng ( F | n ) = rng F ;
mode seq of A is well unital associative associative non empty multMagma ;
[ x , y ] in A ~ { a } ;
x1 . o in L2 . o & x2 . o in L2 . o ;
the carrier of m - m c= B ;
not [ y , x ] in id [: X , Y :] ;
1 + p .. f <= i + len f ;
seq ^\ k1 is lower & seq ^\ k1 is lower ;
len ( F | I ) = len I & len ( F | I ) = len I ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be complex number , a be complex number ;
Comput ( P , s , n ) = s ;
k <= k + 1 & k + 1 <= len p ;
reconsider c = {} T as Element of L ;
let Y be Element of of of of \HM { the } of T ;
cluster directed-sups-preserving for Function of L , L ;
f . j1 in K . j1 & f . j2 in K ;
redefine func J => y -> total for Function ;
K c= 2 |^ ( the carrier of T )
F . b1 = F . b2 & F . b2 = F . b2 ;
x1 = x or x1 = y or x1 = z ;
pred a <> {} means : Def2 : a = 1 ;
assume that card 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 , b9 ;
reconsider m = x as Element of Funcs ( V , C ) ;
let f be non constant non trivial FinSequence of D ;
let FS2 be non empty \cal let X , F be Function ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in F-8 ;
reconsider ps2 = x , ps2 = y as Subset of m ;
let A , B , C be Element of R ;
redefine func strict non empty for Subset of M ;
rng c `1 misses rng ( e | i ) ;
z is Element of gr { x } & z is Element of gr { x } ;
not b in dom ( a .--> p1 ) ;
assume that k >= 2 and P [ k ] ;
Z c= dom ( ( - 1 / 2 ) (#) ( ( id Z ) ^ ) ) ;
the component of Q c= UBD ( A ) & ( A /\ B ) c= UBD ( A ) ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( ( 1 / 2 ) (#) ( f ^ ) ) ;
pred f = u means : Def2 : a * f = a * u ;
for n holds P1 [ n ] implies P1 [ n + 1 ]
{ x . O : x in L } <> {} ;
let x be Element of V . s ;
let a , b be Nat ;
assume that S = S2 and p = p2 and S = S2 ;
gcd ( n1 , n2 , n1 , n2 ) = 1 & gcd ( n1 , n2 , n1 , n2 ) = 1 ;
set oi = a * ( - 1 , 1 ) ;
seq . n < |. r1 .| & |. seq . n .| < r1 ;
assume that seq is increasing and r < 0 ;
f . ( y1 , x1 ) <= a ;
ex c being Nat st P [ c ] ;
set g = { n |^ 1 : n in dom g } ;
k = a or k = b or k = c ;
a9 , b9 , c9 is_collinear & a9 , b9 , c9 is_collinear implies a9 , b9 , c9 is_collinear
assume that Y = { 1 } and s = <* 1 *> ;
IS1 . x = f . x .= f . x .= 0 ;
W3 .last() = W3 . 1 & W2 .first() = ( W2 . 1 ) `1 ;
cluster trivial -> trivial for M -connected Subset of G ;
reconsider u = u as Element of Bags X ;
A in B ^ implies A , B are_that A , B are_that A , B are_that A , B are_\frac B , A |^ ( n + 1 )
x in { [ 2 * n + 3 , k ] } ;
1 >= ( q `1 / |. q .| - cn ) ;
f1 is_TOP-REAL \notin the as \HM { means : Let : f is it & f is non empty ;
( f . q ) `2 <= ( q `2 ) ^2 ;
h is_the \! - } ( Cage ( C , n ) , i ) ;
( b `2 ) ^2 <= ( p `2 ) ^2 & ( p `2 ) ^2 <= ( p `2 ) ^2 ;
let f , g be |. f .| Function of X , Y ;
S * ( k , k ) <> 0. K ;
x in dom ( - f ) & x in dom ( - f ) ;
p2 in ( N . p1 ) & p2 in ( N . p1 ) ;
len ( the_left_argument_of H ) < len ( H ) ;
F [ A , F-14 . A ] ;
consider Z such that y in Z and Z in X ;
redefine pred 1 in C means : Def4 : A c= C ;
assume that r1 <> 0 or r2 <> 0 and r1 < r2 ;
rng q1 c= rng ( C1 ^ C2 ) & rng ( C1 ^ C2 ) c= rng ( C1 ^ C2 ) ;
A1 , A2 , A3 is_collinear & A1 , A2 , A3 is_collinear ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in Segment ( p , Sp , S ) ;
then S is negative & P-2 [ S ] ;
Cl Int [#] T = [#] ( T ) & Cl Int [#] ( T ) = [#] T ;
f12 | A2 = f2 & f12 | A2 = f2 | A2 ;
0. M in the carrier of W & 0. M in the carrier of W ;
let v , v be Element of M ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ V & V c= Y \ Z ;
let X be Subset of S ~ ;
consider H1 such that H = 'not' H1 and H1 in H ;
1_ 1 c= ( 1 - 1 ) * ( \rm div r ) ;
0 * a = 0. R .= a * 0. R ;
A |^ ( 2 , 2 ) = A ^^ ( A , 2 ) ;
set vbeing = ( vseq /. n ) `1 , v4 = ( vseq /. n ) `1 ;
r = 0. ( REAL-NS n , ||. f .|| ) ;
( f . p4 ) `1 >= 0 & ( f . p4 ) `2 >= 0 ;
len W = len ( W - ( i + 1 ) ) ;
f /* ( s * G ) is divergent_to+infty ;
consider l being Nat such that m = F . l ;
t8 does not destroy b1 & W7 does not destroy b1 implies not t8 on 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 ;
\sigma ( T ) \/ omega ( T ) is Basis of T ;
for x being element st x in X holds x in Y
cluster [ x1 , x2 , x3 , x4 ] -> pair ;
sup downarrow a /\ downarrow t is Ideal of T ;
let X be set , N be non empty set , f be Function of X , N ;
rng f = be \rm <* S , X *> ;
let p be Element of B , x be the SortSymbol of S ;
max ( N1 , 2 ) >= N1 & max ( N2 , 2 ) >= N2 ;
0. X <= b |^ ( m * mm1 ) ;
assume that i in I and R1 . i = R ;
i = j1 & p1 = q1 & p2 = q2 implies p1 = p2
assume gR in the right of g & gR in the carrier of g ;
let A1 , A2 be Point of S , A be Subset of S ;
x in h " P /\ [#] ( T1 | P ) ;
1 in Seg 2 & 1 in Seg 3 & 2 in Seg 3 ;
reconsider X-5 = X , Xe = Y as non empty Subset of Tsuch that X = T5 and X is non empty ;
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the \lbrack G , F ) -defined ;
n1 <= i2 + len ( g2 | i2 ) & n1 + len ( g1 | i2 ) = i2 ;
( 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 ;
len ( ( - 1 ) * p ) = 1 ;
x2 is_differentiable_on ]. a , b .[ & ]. a , b .[ c= dom f ;
rng ( M | D ) c= rng ( D2 | D ) ;
for p be Real st p in Z holds p >= a
( cn ) * ( f | K1 ) = proj1 * ( f | K1 ) ;
( seq ^\ m ) . k <> 0 ;
s . ( G . ( k + 1 ) ) > x0 ;
( p -Path M ) . 2 = d ;
A \oplus ( B \ominus C ) = ( A \oplus B ) \ominus C
h \equiv gg . ( mod P ) ;
reconsider i1 = i-1 - 1 , i2 = i - 1 as Element of NAT ;
let v1 , v2 be VECTOR of V , v be VECTOR of V ;
for V being Subspace of V holds V is Subspace of [#] V
reconsider ii = i , ii2 = j as Element of NAT ;
dom f c= [: C , D :] ;
x in ( the sequence of B ) . n & x in ( the rng of B ) . n ;
len \rbrace in Seg ( len ( f2 | i ) ) ;
pp1 c= the topology of T & pp1 c= the topology of T ;
]. r , s .[ c= [. r , s .] ;
let B2 be Basis of T2 , B be Basis of T2 ;
G * ( B * A ) = id o1 & G * ( B * A ) = id o2 ;
assume that p , u , v is_collinear and u , v , w is_collinear ;
[ z , z ] in union rng ( F | ( Seg n ) ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S , Element of S .. $1 ;
LIN a1 , a3 , b1 & LIN a1 , b1 , c1 ;
f " ( f .: x ) = { x } ;
dom ( w2 ) = dom r12 & dom ( r12 ) = dom r12 ;
assume that 1 <= i and i <= n and j <= n ;
( ( g2 ) . O ) `2 <= 1 ;
p in LSeg ( E . i , F . i ) ;
Ii * ( i , j ) = 0. K ;
|. f . ( s . m ) - g .| < g1 ;
q9 . x in rng ( q | i ) & q . x in rng ( q | i ) ;
Carrier ( LLet ) misses Carrier ( L7 ) \ Carrier ( L7 ) ;
consider c being element such that [ a , c ] in G ;
assume Npreal = o( o , a ) & for i be Element of NAT holds o . i = o . i ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= ( F /. C-1 ) ;
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [: [. f . j , f . j .] , Q :] ;
pred 0 <= x & x <= 1 implies x / 2 <= x / 2 ;
p `1 - q `1 <> 0. TOP-REAL 2 & p `2 - q `1 <> 0. TOP-REAL 2 ;
redefine func aaa] ( S , T ) -> non empty ;
let x be Element of S ~ ;
( \HM { the } \HM { object } \HM { of F ) is one-to-one ;
|. i .| <= - ( - 2 |^ n ) / ( n + 1 ) ;
the carrier of I[01] = dom P & the carrier of I[01] = dom P & the carrier of I[01] = dom P ;
} * ( n + 1 ) ! > 0 * PI ;
S c= ( A1 /\ A2 ) /\ ( A1 /\ A2 ) & S c= ( A1 /\ A2 ) ;
a3 , a4 // b3 , b2 & a3 , a4 // b3 , b2 ;
then dom A <> {} & dom A <> {} & rng A <> {} ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , Y & y Joins Y , x , G2 ;
set v2 = ( v /. ( i + 1 ) ) ;
x = r . n .= r4 . n .= r4 . n ;
f . s in the carrier of S2 & f . s in the carrier of S2 ;
dom g = the carrier of I[01] & rng g = the carrier of I[01] ;
p in Upper_Arc ( P ) /\ Lower_Arc ( P ) ;
dom d2 = [: A2 , A2 , A1 , A2 , B2 , i , j , k be Nat ;
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 -> Line for operation of X ;
let U1 , U2 be non-empty MSAlgebra over S , A be non-empty MSAlgebra over S ;
Proj ( i , n ) * g is_differentiable_on X & g is_differentiable_on X ;
x , y , z be Point of X , p be Point of X ;
reconsider p0 = p . x , p0 = p . x as Subset of V ;
x in the carrier of Lin ( A ) & y in the carrier of Lin ( A ) ;
let I , J be parahalting Program of SCM+FSA , a be Int-Location ;
assume that - a is lower and a in - ( - X ) ;
Int Cl Int Cl A c= Cl Int Cl Int Cl Int Cl Int A & Int Cl Int Cl Int Cl Int Cl Int Cl Int Cl A c= Cl Int Cl Int Cl Int Cl Int Cl Int Cl A ;
assume for A being Subset of X holds Cl A = A ;
assume q in Ball ( |[ x , y ]| , r ) ;
( p2 `2 ) ^2 <= ( p `2 ) ^2 & ( p2 `2 ) ^2 <= ( p `2 ) ^2 ;
Cl ( Q ` ) = [#] ( ( T | A ) ` ) ;
set S = the carrier of T , S = the carrier of T ;
set I8 = ' ( f |^ n , n ) ;
len p - n = len ( thesis - n ) .= len p - n ;
A is Permutation of Swap ( A , x , y ) ;
reconsider n6 = n- ( n - 1 ) as Element of NAT ;
1 <= j + 1 & j + 1 <= len ( s | j ) ;
let qbe { q } , q be Element of M ;
a1 in the carrier of S1 & a2 in the carrier of S1 & a3 in the carrier of S2 ;
c1 /. n1 = c1 . n1 & c2 /. n1 = c1 . n1 & c1 /. n1 = c1 . n1 ;
let f be FinSequence of TOP-REAL 2 , p be Point of TOP-REAL 2 , r be Real ;
y = ( ( f * ( S . n ) ) . x ) ;
consider x being element such that x in being being being element such that x in being being being element ;
assume r in ( ( dist ( o ) ) .: P ) ;
set i2 = ( n , i ) `1 , i1 = i + 1 ;
h2 . ( j + 1 ) in rng h2 & h2 . ( j + 1 ) in rng h2 ;
Line ( M29 , k ) = M . i ;
reconsider m = ( x - 1 ) / 2 as Element of ( len x - 1 ) / 2 ;
let U1 , U2 be non empty Subset of U0 , A be Subset of U1 ;
set P = Line ( a , d ) ;
len p1 < len p2 + 1 & len p1 + 1 < len p2 ;
let T1 , T2 be complete Scott Scott let L , f be closed Subset of L ;
then x <= y & x c= ( { y } ) ;
set M = n -{ m } ;
reconsider i = x1 , j = x2 , k = x3 as Nat ;
rng ( the_arity_of o ) c= dom H & dom ( the_arity_of o ) c= dom H ;
z1 " = ( z " ) * ( z " ) .= ( z " ) * ( z " ) ;
x0 - r / 2 in L /\ dom f & x0 - r / 2 in L /\ dom f ;
then w is that rng w /\ L <> {} & rng w /\ L <> {} ;
set x9 = x9 ^ <* Z *> , y9 = <* Z *> ^ ( x ^ y ) ;
len w1 in Seg len w1 & len w2 = len w1 & len w1 = len w2 & len w2 = 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 * ( i , k ) ) ^2 <= ( G * ( i , k ) ) ^2 / ( G * ( i , k ) ) ^2 ;
rng ( g ) c= L~ ( g | ( len g -' 1 ) ) ;
reconsider k = i-1 * ( l + j ) as Nat ;
for n be Nat holds F . n is \HM { -infty } ;
reconsider x9 = x9 , y9 = y9 , z9 = z9 as Vector of M ;
dom ( f | X ) = X /\ dom f & dom ( f | X ) = X ;
p , a // p , c & b , a // c , c ;
reconsider x1 = x , y1 = y , z1 = z as Element of REAL m ;
assume i in dom ( a * p ^ q ) ;
m . ( ag ) = p . ( ag . ( ag ) ) ;
a / ( s . m - n ) / ( s . m - n ) <= 1 ;
S . ( n + k + 1 ) c= S . ( n + k ) ;
assume B1 \/ C1 = B2 \/ C2 & B2 \/ C2 = {} ;
X . i = { x1 , x2 , x3 , x4 , 8 } . i ;
r2 in dom ( h1 + h2 ) & r1 in dom ( h1 + h2 ) ;
- - 0. R = a & b-0 = b ;
F8 is_closed_on t8 , Q7 & F8 is_closed_on t8 , Q8 ;
set T = ^2 (# l , x0 , x1 , x2 , x3 , x4 #) ;
Int Cl Int Cl Int Cl R c= Int Cl R & Int Cl R c= Int Cl R ;
consider y being Element of L such that c . y = x ;
rng ( F/. x ) = { Ff . x } & rng ( Ff ) = { F . x } ;
G-23 \ { c } c= B \/ S & G c= S \/ S ;
f[#] A is Relation of [: X , Y :] , X & X c= dom f ;
set RE = the Point of P , RE = the Point of Q ;
assume that n + 1 >= 1 and n + 1 <= len M ;
let k2 be Element of NAT , k be Element of NAT ;
reconsider ppA = u , ppA = v , pA = w as Element of ( ( TOP-REAL n ) | ( i + 1 ) ) ;
g . x in dom f & x in dom g implies f . x in dom g
assume that 1 <= n and n + 1 <= len f1 ;
reconsider T = b * N as Element of ( G , N ) / ( N , i ) ;
len ( ( P ) . i ) <= len ( ( P ) . i ) ;
x " in the carrier of A1 & x " in the carrier of A1 ;
[ i , j ] in Indices ( ( A @ ) * ( i , j ) ) ;
for m be Nat holds Re ( F . m ) is simple ;
f . x = a . i .= a1 . k .= a . k ;
let f be PartFunc of REAL i , REAL n , x be Element of REAL n ;
rng f = the carrier of \bf 1 & rng f = the carrier of \bf 1 ;
assume s1 = sqrt ( ( |[ 2 , 0 ]| ) ^2 - ( p `1 ) ^2 ) ;
pred a > 1 & b > 0 & a > 0 implies a / b > 1 ;
let A , B , C be Subset of [: I , I :] ;
reconsider X0 = X , Y0 = Y , Y0 = Z as Subset of X ;
let f be PartFunc of REAL , REAL , x be Element of REAL ;
r * ( v1 |-- I ) . X < r * 1 ;
assume that V is Subspace of X and X is Subspace of V ;
let t-3 , t-3 be Relation of the carrier of S , the carrier of T ;
Q [ e \/ ( { v } \/ { v-5 } ) , f ] ;
g \circlearrowleft ( W-min L~ z ) = z & ( W-min L~ z ) .. z = z ;
|. |[ x , v ]| - |[ x , y ]| .| = vSet ;
- f . w = - ( L (#) w ) .= - ( L (#) w ) ;
z - y <= x iff z <= x + y & y <= z + y
( 7 / p1 ) |^ ( 1 / e ) > 0 ;
assume X is BCK-algebra & 0 , 0 , 0 , 0 , 0 ;
F . 1 = v1 & F . 2 = v2 & F . 3 = v2 ;
( f | X ) . x2 = f . x2 & ( f | X ) . x2 = f . x2 ;
( ( tan * tan ) `| Z ) . x in dom ( tan * tan ) ;
i2 = ( f /. len f ) `1 & i1 = ( f /. len f ) `1 ;
X1 = X2 \/ ( X1 \ X2 ) & X2 = X1 \ X2 implies X1 = X2
[. a , b , 1_ G .] = 1_ G & [. a , b , 1_ G .] = 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] & rng g2 = the carrier of I[01] & rng g2 = the carrier of I[01] ;
dom ( f2 ) = the carrier of I[01] & dom ( f2 ) = the carrier of I[01] ;
( proj2 | X ) .: X = proj2 .: ( X /\ Y ) ;
f . ( x , y ) = h1 . ( x `1 , y `2 ) ;
x0 - r < a1 . n & x0 < a1 . n ;
|. ( f /* s ) . k - G . k .| < r ;
len Line ( A , i ) = width A & width ( A @ ) = width A ;
Sbeing @ = ( S . g ) @ .= ( S . g ) @ ;
reconsider f = v + u as Function of X , the carrier of Y ;
intloc 0 in dom Initialized ( p +* I ) & IC p in dom I ;
i1 , i2 , j1 , j2 is_collinear & i does not destroy b1 implies ( i1 , i2 ) on ( i2 , j1 )
arccos . r + arccos . r = ( cos . ( PI / 2 ) ) ^2 + 0 ;
for x st x in Z holds f2 * ( f1 - #Z 2 ) is_differentiable_in x
reconsider q2 = ( q `1 ) / ( x - 1 ) , q2 = ( q `2 ) / ( x - 1 ) as Element of REAL ;
( 0 qua Nat ) + 1 <= i + j1 & i + 1 <= j ;
assume f in the carrier of [: X , Omega Y :] ;
F . a = H / ( ( x , y ) / ( x , y ) ) ;
( TRUE T ) at ( C , u ) = TRUE & ( T . ( C , u ) ) = TRUE ;
dist ( ( a * seq ) . n , h ) < r / 2 ;
1 in the carrier of [. 0 , 1 .] & 1 in the carrier of I[01] ;
( p2 `1 - x1 ) - x1 > - g & ( p2 `1 - x1 ) - g > - g ;
|. r1 - `2 .| = |. a1 .| * |. TOP-REAL n .| ;
reconsider S-14 = 8 as Element of Seg 8 , S be non empty set ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b
D0W -' 1 = D0W - 1 + 1 ;
i1 = a + n & i2 = K + n & i1 = K + n ;
f . a [= f . ( f . O1 "\/" f . a ) ;
pred f = v & g = u , h = v + u ;
I . n = Integral ( M , F . n ) ;
chi ( [: T1 , T2 :] , S ) . s = 1 ;
a = VERUM ( A ) or a = VERUM ( A ) ;
reconsider k2 = s . b3 , k2 = s . b3 as Element of NAT ;
( Comput ( P , s , 4 ) ) . GBP = 0 ;
L~ M1 meets L~ ( R4 * ( i , j ) ) ;
set h = the continuous Function of X , R , f be Function of X , R ;
set A = { L . ( k9 . n ) : n <= k } ;
for H st H is atomic holds P [ H ] ;
set b\HM = S5 \ ( i \ { i } ) , S5 = i \ { i } ;
Hom ( a , b ) c= Hom ( a `1 , b `2 ) ;
( 1 - s ) / ( n + 1 ) < ( 1 - s ) / ( n + 1 ) ;
( l `1 ) = [ dom l , cod l ] & ( l `2 ) = cod l ;
y +* ( i , y /. i ) in dom g & y in dom g ;
let p be Element of [: CQC-WFF ( Al ) , D ( ) :] ;
X /\ X1 c= dom ( f1 - f2 ) & X /\ X1 c= dom ( f1 - f2 ) ;
p2 in rng ( f /^ ( i -' 1 ) ) & p1 in rng ( f /^ i ) ;
1 <= indx ( D2 , D1 , j1 ) & 1 <= j1 ;
assume x in ( ( ( TOP-REAL 2 ) | K1 ) \/ ( ( TOP-REAL 2 ) | K1 ) ) ;
- 1 <= ( ( f2 ) . O ) `2 & - 1 <= ( ( f2 ) . O ) `2 ;
let f , g be Function of I[01] , ( TOP-REAL 2 ) | K1 , R^1 , a be Real ;
k1 - k2 = k1 - k2 & k1 - k2 = k1 - k2 - k2 ;
rng seq c= ]. x0 - r , x0 .[ & rng seq c= ]. x0 - r , x0 .[ ;
g2 in ]. x0 - r , x0 + r .[ & g2 in ]. x0 - r , x0 + r .[ ;
sgn ( p `1 , K ) = - 1_ K & sgn ( p `1 , K ) = - 1_ K ;
consider u being Nat such that b = p |^ y * u ;
ex A being as as as as as as normal Ordinal of W st a = Sum A ;
Cl ( union ( H ) ) = union ( ( Cl ( H ) ) /\ ( Cl ( H ) ) ;
len t = len t1 + len t2 & len t1 = len t1 + len t2 & len t2 = len t1 + len t2 ;
v-29 = v + w |-- ( A , v ) & vn = v + ( A + B ) ;
v ( ) <> DataLoc ( ( t . GBP ) , 3 ) & v ( ) <> DataLoc ( ( t . GBP ) , 3 ) ;
g . s = sup ( d " { s } ) & g . s = s ;
( \dot \dot ) . s = s . ( \dot y ) ;
{ s : s < t } in REAL implies t = {} & t = {} ;
s ` \ s = s ` \ ( 0. X \ s ) .= ( 0. X \ s ) \ ( 0. X \ s ) ;
defpred P [ Nat ] means B + $1 in A & B + $1 in A ;
( 329 + 1 ) ! = 3329 ! * ( 329 + 1 ) ;
( 1. ( A , B ) ) * ( 1. ( A , B ) ) = 1. ( A , B ) ;
reconsider y = y as Element of ( len y ) -tuples_on the carrier of K ;
consider i2 being Integer such that y0 = p * i2 and i2 in dom p and i2 in dom p ;
reconsider p = Y | Seg k as FinSequence of ( the carrier of K ) \ { i } ;
set f = ( S , U ) \mathop { {} } , g = S \! \mathop { {} } ;
consider Z be set such that lim s in Z and Z in F ;
let f be Function of I[01] , ( TOP-REAL n ) | K1 , R^1 , a be Real ;
( SAT M ) . [ n + i , 'not' A ] <> 1 ;
ex r being Real st x = r & a <= r & r <= b ;
let R1 , R2 be Element of ( n + 1 ) -tuples_on the carrier of K , a be Element of K ;
reconsider l = 0. ( K , n ) as Linear_Combination of A , n ;
set r = |. e .| + |. n .| + |. w .| + a ;
consider y being Element of S such that z <= y and y in X ;
a 'or' ( b 'or' c ) = 'not' ( ( a 'or' b ) 'or' c )
||. x9 - g .|| < r2 & ||. g - x .|| < r2 ;
b9 , a9 // b9 , c9 & b9 , c9 // b9 , c9 & c9 , c9 // b9 , c9 implies b9 , c9 // c9 , c9
1 <= k2 -' k1 & k1 + 1 = k2 & k2 + 1 = k1 & k1 + 1 = k2 implies ( k1 + 1 ) + 1 = k2
( ( p `2 / |. p .| - sn ) / ( 1 - sn ) ) ^2 >= 0 ;
( q `2 / |. q .| - sn ) < 0 & ( q `2 / |. q .| - sn ) < 0 ;
( E-max C ) `1 in LeftComp ( RCage ( C , n ) ) & ( E-max C ) `1 in L~ ( RCage ( C , n ) ) ;
consider e being Element of NAT such that a = 2 * e + 1 ;
Re ( lim F ) = Re Re ( lim G ) .= Re ( lim G ) ;
LIN b , a , c or LIN b , c , a ;
p `1 , a `2 // a `1 , b or p `1 , a `2 // b `1 , a ;
g . n = a * Sum ( f | 1 ) .= f . n * a ;
consider f being Subset of X such that e = f and f is Cage ;
F | ( N2 , S ) = CircleMap * ( ( F | N2 ) . ( n + 1 ) ) ;
q in LSeg ( q , v ) \/ LSeg ( v , p ) ;
Ball ( m , r0 ) c= Ball ( m , s ) & Ball ( m , s ) c= Ball ( m , r ) ;
the carrier of (0). V = { 0. V } & the carrier of (0). V = { 0. V } ;
rng ( ( - 1 ) (#) ( cos * sin ) ) = [. - 1 , 1 .] ;
assume that Re seq is summable and Im seq is summable and Im seq is summable ;
||. ( vseq . n ) - ( vseq . n ) .|| < e / 2 ;
set g = O --> 1 ;
reconsider t2 = t11 , t2 = 0 as 0 -started string of S2 , t2 = 0 as ( S , U ) If , D = { t } ;
reconsider x9 = seq . n , y9 = seq . n as sequence of REAL n ;
assume that that C meets ( L~ go \/ L~ pion1 ) and C meets ( L~ pion1 ) and C meets ( L~ pion1 ) ;
- ( ( 1 - r ) / 2 ) < F . n - r / 2 ;
set d1 = being element , d2 = dist ( x1 , z1 ) , d2 = dist ( x2 , z2 ) , d2 = dist ( x1 , z2 ) , d2 = dist ( x2 , z1 ) ;
2 |^ ( |. 00 .| -' 1 ) = 2 |^ ( |. 100 .| - 1 ) ;
dom ( v | ( len ( d | i ) ) ) = Seg len ( d | i ) ;
set x1 = - k2 + |. k2 .| + |. k1 .| + |. k2 .| + |. k .| + |. k .| ;
assume for n being Element of X holds 0. <= F . n & 0. <= F . n ;
assume that 0 <= T-32 . i and T-32 . ( i + 1 ) <= 1 ;
for A being Subset of X holds c . ( c . A ) = c . A
the carrier of Lconsider such that the carrier of Lconsider and L2 c= I2 and the carrier of Lconsider A c= I ;
'not' Ex ( x , p ) => All ( x , p ) is valid ;
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the normal w.r.t. of A ;
Z c= dom ( ( ( 1 / 2 ) (#) ( f1 + f2 ) ) `| Z ) ;
|. 0. TOP-REAL 2 - q `1 .| - |. q .| < r / 2 - r / 2 ;
\ { A , succ ( d , B ) } c= ConsecutiveSet2 ( A , succ ( d , B ) ) ;
E = dom L8 & L8 is_measurable_on E & L8 c= E & L8 c= E & L8 c= E & E c= dom L8 ;
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 implies W1 + W2 c= W2
I . IC s2 = P . IC s2 .= ( I . IC s2 ) .= ( I . IC s2 ) ;
pred x > 0 means : Def2 : ( 1 - x ) |^ ( 1 - x ) = x |^ ( 1 - x ) ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) ;
consider p being Point of T such that C = [. p , R .] and p in A ;
b , c are_connected & - C , - ( b , a ) + - ( b , c ) + - ( b , a ) + - ( b , c ) + - ( b , c ) + - ( b , c ) + - ( b , c ) + - ( b ,
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 ) , the carrier of V ;
reconsider g = f " as Function of U2 , U1 , U2 ;
A1 in the points of ( k + 1 ) , k + 1 , G ;
|. - x .| = - ( - x ) .= - x .= - x .= - x ;
set S = { is \in in in in { x , y , c } ;
Fib ( n ) * ( 5 * ( 5 / n ) ) >= 4 * ( 5 * n ) ;
vseq /. ( k + 1 ) = ( vseq . ( k + 1 ) ) `1 ;
0 mod i = - ( i * ( 0 qua Nat ) - i ) ;
Indices M1 = [: Seg n , Seg n :] & len M2 = n & width M1 = n & width M2 = n ;
Line ( S\mathopen { i , j } , j ) = S\mathopen ( j , i ) ;
h . ( x1 , y1 ) = [ y1 , x1 ] & h . ( y1 , y2 ) = [ y2 , x2 ] ;
|. f - Re ( |. f .| * ( card b - a ) ) .| is nonnegative ;
assume x = ( a1 ^ <* x1 *> ) ^ b1 & y = ( a1 ^ <* x2 *> ) ^ b1 ;
MW is_closed_on IExec ( I , P , s ) . a , P & MW is_halting_on s , P ;
DataLoc ( t3 . a , 4 ) = intpos ( 0 + 4 ) .= ( 0 + 4 ) ;
x + y < - x + y & |. x .| = - x + y & |. x .| = - x + y ;
LIN c , q , b & LIN c , q , c & LIN c , q , c ;
f'not' st . ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + z1 ) .= x1 + ( y1 + z1 ) ;
fproduct ( f . a ) = f\in . a & v in InputVertices S & v in InputVertices S ;
( p `1 ) ^2 <= ( ( ( E _ 2 ) ) `1 ) ^2 & ( ( ( E _ 2 ) ) `1 <= ( ( E _ 2 ) ) `1 ;
set R8 = Cage ( C , n ) \circlearrowleft E7 , E8 = Cage ( C , n ) ;
( p `1 ) ^2 >= ( ( ( ( E _ 2 ) ) `1 ) / ( 1 + ( p `2 ) ) ^2 ) ;
consider p such that p = p-20 and s1 < p and p < i and i < len p ;
|. ( 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 ) = width N ;
f1 /* s1 is convergent & f2 /* s1 is convergent & lim ( f1 /* s1 ) = x0 ;
f . x1 = x1 & f . y1 = y1 & f . y2 = y1 & f . y2 = y2 ;
len f <= len f + 1 & len f + 1 <> 0 & len f + 1 <> 0 implies f ^ <* p *> is not empty
dom ( Proj ( i , n ) * s ) = REAL m & rng ( Proj ( i , n ) * s ) = REAL m ;
n = k * ( 2 * t ) + ( n mod 2 ) ;
dom B = 2 -tuples_on the carrier of V , { {} } :] ;
consider r such that r _|_ a and r _|_ x and r _|_ y ;
reconsider B1 = the carrier of Y1 , B2 = the carrier of Y2 , B1 = the carrier of X1 as Subset of X ;
1 in the carrier of [. 1 / 2 , 1 .] & 1 / 2 in dom ( 1 / 2 ) ;
for L being complete LATTICE holds every \mathbb L , C , L st C , L are_isomorphic & C is isomorphic
[ gi , gj ] in Ii \ Ij ;
set S2 = 1GateCircStr ( x , y , c ) ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and for r st r < x0 ex g st r < g & g < x0 & g in dom ( f2 * f1 ) ;
reconsider y = ( a ` ) / ( F ` ) , z = ( a ` ) / ( F ` ) as Element of L ;
dom s = { 1 , 2 , 3 } & s . 1 = d1 & s . 2 = d2 ;
( min ( g , ( 1 - g ) ) ) . c <= h . c ;
set G2 = the as as as as as as as as as as as as as as of M -A1 , { v } , { v } } ;
reconsider g = f as PartFunc of REAL n , REAL-NS n , REAL-NS n ;
|. s1 . m - p .| / |. p .| < d / ( p `1 - p `2 ) ;
for x being element st x in \HM { u } holds x in \HM { the } \HM { carrier of S : x in consider t being element st t in \HM { the carrier of S } }
P = the carrier of ( TOP-REAL n ) | P & Q = the carrier of ( TOP-REAL n ) | P ;
assume that p11 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) and LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = { p1 } ;
( 0. X \ x ) |^ ( m * ( k + 1 ) ) = 0. X ;
let g be Element of Hom ( cod f , cod f ) ;
2 * a * b + ( 2 * c ) * d <= 2 * C1 * C2 ;
let f , g , h be Point of the complex normed space of X , Y , f be Function of X , Y , g be Function of X , Z ;
set h = Hom ( a , g ) ;
then Seg ( n ) | Seg m = idseq ( m ) & m <= n ;
H * ( g " * a ) in the right of H & H * ( g " * a ) in the right of H ;
x in dom ( ( id Z ) (#) ( sin - cos ) ) & Z c= dom ( ( id Z ) (#) ( sin - cos ) ) ;
cell ( G , i1 , j2 -' 1 ) misses C & cell ( G , i1 , j1 -' 1 ) misses C ;
LE q2 , p4 , P & LE p1 , p2 , P & LE p2 , p3 , P & LE p1 , p2 , P & LE p2 , p3 , P ;
attr B is BDD of A means : Def4 : B c= BDD A ;
deffunc D ( set , set ) = union rng $2 & $2 = union rng $2 ;
n + - n < len ( p + - n ) + ( - n ) ;
pred a <> 0. K means : Def2 : for M holds the_rank_of M = rk ( a * M , i ) ;
consider j such that j in dom \mathbb B and I = len } + j and I = len ( k + 1 ) ;
consider x1 such that z in x1 and x1 in P and x2 in P and x = [ x1 , x2 ] ;
for n ex r being Element of REAL st X [ n , r ] ;
set CS1 = Comput ( P2 , s2 , i + 1 ) , CS2 = Comput ( P2 , s2 , i + 1 ) ;
set cv = 3 / ( 2 * PI ) , cv = 4 / ( 2 * PI ) , cv = 5 / ( 2 * PI ) , cv = 4 / ( 2 * PI ) , cv = 5 / ( 2 * PI ) , cv = 5 / ( 2 * PI
conv @ W c= union ( ( F .: ( E " W ) ) /\ conv ( E " W ) ) ;
1 in [. - 1 , 1 .] /\ dom ( ( arccot ) * ( f1 - f2 ) ) ;
r3 <= s0 + ( r0 - ( 1 - r ) * ( 1 - r ) * ( 1 - r ) * ( 1 - r ) * ( 1 - r ) * ( 1 - r ) * ( 1 - r ) * ( 1 - r ) * ( 1 - r ) * ( 1 - r
dom ( f (#) ( f4 ) ) = dom f /\ dom ( f | dom ( f1 | dom ( f2 | dom ( f2 | dom ( f2 | dom ( f2 | dom ( f2 | dom ( f2 | dom ( f2 | dom ( f2 | dom ( f2 | dom ( f2 | dom ( f2 | dom
dom ( f (#) G ) = dom ( l (#) F ) /\ Seg k .= Seg k ;
rng ( s ^\ k ) c= dom f1 \ { x0 } & rng ( s ^\ k ) c= dom f2 \ { x0 } ;
reconsider gg = gp , gq = gq as Point of ( TOP-REAL n1 ) | ( ( TOP-REAL n1 ) | K1 ) ;
( T * h . s ) . x = T . ( h . s . x ) ;
I . ( L . J . x ) = ( I * L ) . x ;
y in dom be let *> <* ( Frege ( A . o ) ) . ( ( commute ( A . o ) ) . ( x , y ) ) *> ;
for I being non degenerated commutative commutative commutative commutative associative distributive non empty doubleLoopStr holds I is commutative
set s2 = s +* Initialize ( ( intloc 0 ) .--> 1 ) , P2 = P +* Initialize ( ( intloc 0 ) .--> 1 ) ;
P1 /. IC s1 = P1 . IC s1 & P1 . IC s1 = P1 . IC s1 & P1 . IC s1 = P1 . IC s1 ;
lim S1 in the carrier of [. a , b .] & lim S1 in the carrier of [. a , b .] ;
v . ( lpp . i ) = ( v *' lpp ) . i .= ( v *' lpp ) . i ;
consider n being element such that n in NAT and x = ( sn - 1 ) . n ;
consider x being Element of c such that F1 . x <> F2 . x and x in A and F2 . x <> 0 ;
Funcs ( X , 0 , x1 , x2 , x3 , x4 , 6 , 7 , 8 ) = { E8 } ;
j + ( 2 * ( k - m ) ) + m1 > j + ( 2 * ( k - m ) ) ;
{ s , t } on A3 & { s , t } on B2 & { s , t } on B2 ;
n1 > len crossover ( p2 , p1 , n1 , n2 , n3 , n3 , n1 , n2 , n3 , n3 , n2 , n3 , n3 , n1 , n2 , n3 , n3 , n2 , n3 , n3 , n1 , n2 , n3 , n3 , n3 , n2 , n3 , n3 , n4 , n4 , n1 , n2
( mg1 ) . HT ( ( ( ( ( ( ( g2 ) ) . HT ( g2 , T ) ) , T ) ) , T ) = 0. L ;
then H1 , H2 are_that H , H1 are_that H , H1 are_that H , H1 / ( x , y ) / ( x , y ) ;
( ( us L~ f ) .. ( f ) ) .. ( f ) > 1 & ( ( f ) .. ( 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 , the carrier of S , the carrier of S ;
DigA ( t-23 , z9 ) is Element of k -tuples_on INT & DigA ( tmax , z ) = k - 1 ;
I \mathop { d , 223 } = d/ 2 & I \mathop { d , l } = k2 & I is non empty ;
u9 ~ = { [ a , u9 ] } & u9 in { [ a , u9 ] } ;
( w | p ) | ( p | ( w | w ) ) = p ;
consider u2 such that u2 in W2 and x = v + u2 and u2 in W2 and u = v + u2 ;
for y st y in rng F ex n st y = a |^ n & for x st x in F holds x in A
dom ( ( g * ( M , j ) ) | K ) = K & dom ( ( g * ( M , j ) ) | K ) = K ;
ex x being element st x in ( ( ( U0 ) \/ A ) . s ) & x in ( ( U0 ) . s ) ;
ex x being element st x in ( ( that ( for O being element st O in O holds O in A ) ) . s ) ;
f . x in the carrier of [. - r , r .] & f . x in the carrier of [. - r , r .] ;
( 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= { p01 } & LSeg ( p01 , p2 ) c= { p01 } ;
( b + ( be - b ) ) / 2 in { r : a < r & r < b } ;
ex_sup_of { x , y } , L & x "\/" y = sup { x , y } ;
for x being element st x in X ex u being element st P [ x , u ]
consider z being Point of GX such that z = y and P [ z ] and z in A and z in B ;
( the sequence of number ) . ( x - y ) <= e / 2 & e / 2 <= e / 2 ;
len ( w ^ w2 ) + 1 = len w + 2 + 1 & len ( w ^ w2 ) = len w + 2 ;
assume q in the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 & q in the carrier of ( ( TOP-REAL 2 ) | K1 ) ;
f | Eq ` = g | Eq ` & g | Eq ` = g | Eq ` & g | Eq = g | Eq ` ;
reconsider i1 = x1 , i2 = x2 , j1 = y2 , j2 = x3 , j1 = x4 as Element of NAT ;
( a * A * B ) @ = ( a * ( A * B ) ) @ ;
assume ex n0 being Element of NAT st f |^ ( n0 - 1 ) is Seg n & f .: ( f .: n0 ) is Seg n ;
Seg len ( ( ( f1 ^ f2 ) | i ) ) = dom ( ( ( f1 ^ f2 ) | i ) ) ;
( Complement A1 ) . m c= ( Complement A1 ) . n & ( Complement A1 ) . m c= ( Complement A1 ) . n ;
f1 . p = p8 & g1 . ( p8 ) = d & g1 . ( p8 ) = d & g2 . ( p8 ) = d ;
FinS ( F , Y ) = FinS ( F , dom ( F | Y ) ) ;
( x | y ) | z = z | ( y | x ) ;
( |. x .| to_power n ) / ( 1 - r ) <= ( r2 / ( 2 |^ n ) ) ;
Sum ( F ) = Sum f & dom ( F ) = dom g & rng ( F ) = dom g ;
assume for x , y being set st x in Y & y in Y holds x /\ y in Y ;
assume that W1 is Subspace of W3 and W2 is Subspace of W1 and W1 is Subspace of W2 and W2 is Subspace of W1 ;
||. ( ( t . x ) .|| ) = lim ||. ( ( x . x ) .|| ) .= ||. ( x . x ) .|| .= ||. ( x . x ) .|| ;
assume that i in dom D and f | A is lower and g | A is lower ;
( p `2 / |. p .| - cn ) <= ( - ( 1 - cn ) ) / ( 1 - cn ) ;
g | Sphere ( p , r ) = id ( Sphere ( p , r ) ) & g | Sphere ( p , r ) = id ( Sphere ( p , r ) ) ;
set N8 = ( ( L~ Cage ( C , n ) ) | ( L~ Cage ( C , n ) ) ) ;
for T being non empty TopSpace holds T is countable implies the TopStruct of T is countable
width B |-> 0. K = Line ( B , i ) .= B * ( i , i ) .= B * ( i , i ) ;
pred a <> 0 implies ( A Let B ) Y. = ( A Y. ) /\ ( B Y. ) ;
then f is_PartFunc of REAL , REAL & pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 ;
assume that a > 0 and a <> 1 and b > 0 and b <> 0 and c <> 0 ;
w1 , w2 , w1 is_collinear & w2 in { w1 , w2 } implies w1 + w2 in { w1 , w2 }
p2 /. IC s = p2 . IC s .= ( ( IC s ) + 1 ) .= ( ( IC s ) + 1 ) ;
ind ( T-10 | b ) = ind b .= ind B - ind b .= ind B - ind b ;
[ a , A ] in the Indices of Line ( A9 , b ) & [ a , A ] in Indices Line ( A9 , b ) ;
m in ( the Arrows of C ) . ( o1 , o2 ) & ( the Arrows of C ) . ( o2 , o2 ) = ( the Arrows of C ) . ( o2 , o2 ) ;
( a 'imp' ( CompF ( PA , G ) , B ) ) . z = FALSE ;
reconsider phi = phi , phi = phi , phi = phi , phi = phi , phi = phi , N = { phi } , N = { phi } , N = { phi } , M = { phi } , N = { phi } , N = { phi } , N = { phi } , M = { phi } , N = { phi } ,
len s1 - ( len s2 - 1 ) + 1 > 0 + 1 - 1 ;
delta ( D ) * ( f . ( upper_bound A ) - lower_bound A ) < r ;
[ f21 , f22 ] in the carrier' of A ~ & [ f22 , f22 ] in the carrier' of A ;
the carrier of ( ( TOP-REAL 2 ) | K1 ) | 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 q = g2 . z ;
[#] V1 = { 0. V1 } .= the carrier of (0). V1 .= the carrier of (0). V1 .= [#] V1 ;
consider P2 be FinSequence such that rng P2 = M and P2 is one-to-one and P2 is one-to-one ;
assume that x1 in dom ( f | X ) and ||. x1 - x0 .|| < s and ||. x1 - x0 .|| < s ;
h1 = f ^ ( <* p3 *> ^ <* p *> ) .= h ^ <* p *> .= h ;
c / ( |[ b , c ]| ) = c .= c / ( |[ a , c ]| ) .= c / ( |[ a , c ]| ) ;
reconsider t1 = p1 , t2 = p2 , t2 = p1 , t1 = p2 , t2 = p3 as Term of C , V ;
( 1 - 1 ) * ( 1 - 1 ) in the carrier of [. 1 , 1 .] & ( 1 - 1 ) * ( 1 - 1 ) in the carrier of I[01] ;
ex W being Subset of X st p in W & W is open & h .: W c= V ;
( h . p1 ) `2 = C * ( p1 `2 ) + D * ( p1 `2 ) ;
R . b - a * b = 2 * - b * b .= 2 * b - b * a .= b ;
consider ] such that B = 1- 1 * C + ( 1 - 1 ) * A and 0 <= 1 ;
dom g = dom ( ( ( the Sorts of A ) * ( the_arity_of o ) ) * ( the Arity of S ) ) ;
[ P . ( l6 ) , P . ( l6 ) ] in => ( ( T . ( l6 ) ) , T . ( l6 ) ) ;
set s2 = Initialize s , P2 = P +* I ;
reconsider M = mid ( z , i2 , i1 ) , N = L~ z as non empty Subset of TOP-REAL 2 ;
y in product ( ( Carrier J ) +* ( V , { 1 } ) ) ;
1 / ( |[ 0 , 1 ]| ) = 1 & 0 / ( |[ 0 , 1 ]| ) = 0 ;
assume x in the left & x in the left & y in the right of g implies x + y in the right of g & x + y in the right & x + y in the right of g ;
consider M being strict Subgroup of A9 such that a = M and T is strict Subgroup of M and M is strict Subgroup of M ;
for x st x in Z holds ( ( ( 1 / 2 ) (#) f ) `| Z ) . x <> 0 & f . x > 0 ;
len W1 + len W2 + m = 1 + len W2 + len W1 & len ( W1 + W2 ) = len W1 + len W2 + len W2 ;
reconsider h1 = ( vseq . n - tf1 ) . t as Lipschitzian LinearOperator of X , Y ;
( - ( len p + q ) ) + 1 in dom ( p + q ) ;
assume that s2 is negative and F is for s1 , s2 st s1 in the { s2 } & s2 in the { s1 } & s1 <> s2 & s2 in the { s2 } holds s2 in the { s1 } ;
( ( the thesis of M ) * ( x , y ) ) `1 = gcd ( x , y , z ) & ( ( the Element of M ) * ( x , y ) ) `2 = gcd ( x , y , z ) ;
for u being element st u in Bags n holds ( p `1 + m ) . u = p . u + m . u
for B being Subset of u-5 st B in E holds A = B or A misses B or A misses B ;
ex a being Point of X st a in A & A /\ Cl { y } = { a } ;
set W2 = tree ( p ) \/ W1 , W1 = tree ( q ) , W2 = tree ( p ) ;
x in { X where X is Ideal of L : X is Ideal of L } & x in { X where X is Ideal of L : X is Ideal of L } ;
the carrier of W1 /\ W2 c= the carrier of W1 & the carrier of W1 /\ W2 c= the carrier of W2 implies W1 + W2 c= W2
( for a , b holds a + b * id a = 1. ( K , n ) ) & ( a + b ) * id a = id ( the carrier of K )
( ( X --> f ) . x ) . x = ( X --> dom f ) . x .= ( X --> f ) . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) ;
p => ( q => r ) => ( p => q ) in TAUT ( A ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
- 1 + 1 <= ( ( - 1 ) |^ ( n -' m ) ) + 1 ;
( reproj ( 1 , z0 ) ) . x in dom ( f1 (#) f2 ) & ( f1 (#) f2 ) . x in dom ( f1 (#) f2 ) ;
assume that b1 . r = { c1 } and b2 . r = { c2 } and b1 . r = { c2 } ;
ex P st a1 on P & a2 on P & b on P & c on P & a on P & b on P & c on P & a on P & b on P & c on P & a on P & b on P & c on P & b on P & a on P & b on P & c on P & b on P
reconsider gf = g `1 * f `2 , hf = h `1 * g `2 as strict non empty Element of X ;
consider v1 being Element of T such that Q = ( downarrow v1 ) ` and v1 in F and v2 in { v } ;
n in { i where i is Nat : i < n + 1 & n < ( n + 1 ) + 1 } ;
( F * ( i , j ) ) `2 >= ( ( F * ( m , k ) ) `2 ) ;
assume K1 = { p : p `1 >= sn & p `2 >= 0 & p <> 0. TOP-REAL 2 } ;
ConsecutiveSet ( A , succ O1 ) = ( ConsecutiveSet ( A , O1 ) ) . ( succ O1 ) .= ( ConsecutiveSet ( A , O1 ) ) . ( succ O1 ) ;
set IS1 = Macro ( a , intloc 0 ) , IS2 = AddTo ( a , intloc 0 ) , IS2 = goto ( 0 + 1 ) , IS2 = goto ( 0 + 1 ) , IS2 = goto ( 0 + 1 ) , IS2 = goto ( 0 + 1 ) , IS2 = goto ( 0 + 1 ) , IS2 = goto ( 0 + 1 ) ;
for i be Nat st 1 < i & i < len z holds z /. i <> z /. 1 & z /. i <> z /. ( i + 1 )
X c= ( the carrier of L1 ) /\ ( the carrier of L2 ) & ( the carrier of L1 ) /\ ( the carrier of L2 ) c= the carrier of L1 ;
consider x9 being Element of GF ( p ) such that x9 |^ 2 = a and x9 |^ 3 = a ;
reconsider ee = ee , f = f , fe = f , f = f , e = f , f = g , e = f , f = g ;
ex O being set st O in S & C1 c= O & M . O = 0. ( M , f . O ) ;
consider n be Nat such that for m be Nat st n <= m holds S . m in U1 and S . m in U2 ;
f (#) g * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) . x ) ;
defpred P [ Nat ] means A + succ $1 = succ A & A + succ $1 = succ $1 & A = succ $1 & A = succ $1 ;
the left & - g = the left & - g = the left of g implies - ( - ( - g ) ) = - ( - ( - g ) )
reconsider p\mathopen { x } , p\mathopen { x } , p\mathopen { x } , p , q } = y as Point of TOP-REAL 2 ;
consider g3 such that g3 = y and x <= y and x <= y and y <= x0 and x0 <= g2 and g2 <= x0 ;
for n being Element of NAT ex r being Element of REAL st X [ n , r ] & X [ n , r ]
len ( x2 ^ y2 ) = len x2 + len y2 & len ( x2 ^ y2 ) = len x2 + len y2 & len ( x2 ^ y2 ) = len x2 + len y2 ;
for x being element st x in X holds x in the set of ( the set of f ) & x in the set of ( the set of f ) . ( n + 1 ) holds x in X
LSeg ( p11 , p2 ) /\ LSeg ( p1 , p2 ) = {} & LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = {} ;
func \HM { \rm such that X is finite & Y is non empty & F is st X is finite & Y is non empty & F is non empty ;
len ( CR ( Cf ) /. 1 , Cf /. ( len Cf ) ) <= len ( Cf ) ;
pred K is Set means : Def2 : a <> 0. K & a * v = v . a * v . a ;
consider o being OperSymbol of S such that t `1 . {} = [ o , the carrier of S ] and o in rng p ;
for x st x in X ex y st x c= y & y in X & y in X & y is NAT & f . x = f . y
IC Comput ( P1 , s , k ) in dom ( ( n + 1 ) .--> ( n + 1 ) ) ;
pred q < s & r < s implies ]. p , s .[ /\ ]. p , q .[ c= ]. p , q .[ ;
consider c being Element of Class f such that Y = ( F . c ) `1 and c in Class ( F . c , 3 ) ;
func the ResultSort of S2 -> Function of the carrier' of S1 , the carrier' of S2 means : Def4 : for x being set st x in the carrier' of S2 holds it . x = ( the ResultSort of S1 ) . x ;
set yxy = [ <* y , z *> , f2 ] ;
assume x in dom ( ( ( 1 / 2 ) (#) ( ( #Z 2 ) * ( f1 + #Z 2 ) ) `| Z ) ;
r-7 in Int cell ( GoB f , i , GoB f ) \ L~ f & rri2 in Int cell ( GoB f , i , GoB f ) ;
( q `2 ) ^2 >= ( ( Cage ( C , n ) * ( i + 1 ) ) `1 ;
set Y = { a "/\" a ` : a ` in X } ;
i - len f <= len f + len f1 - len f - len f + len f - len f + len f - len f + len f - len f + len f - len f + len f - len f + len f - len f + len f - len f + len f - len f + len f - len f - len f + len f - len f + len f -
for n ex x st x in N & x in N1 & h . n = - x0 & h . n = x0
set s0 = ( ( a , I , p , s ) := ( a , I , p ) ) . i ;
p . k . 0 = 1 or p . 0 = - 1 or p . 0 = 1 & p . 0 = - 1 ;
u + Sum L-18 in ( U \ { u } ) \/ { u + Sum L-18 } ;
consider x9 being set such that x in x9 and x9 in V1 and x9 in V1 and f . x9 = f . x9 ;
( p ^ ( q | k ) ) . m = ( q | k ) . ( : len p = len p + len q ) ;
g + h = gg + h1 & \rm A1 + A2 = g + h & h + c = g + h + c ;
L1 is distributive & L2 is distributive implies L1 , L2 is distributive & for x being Element of L1 holds L1 , L2 be distributive non empty Poset holds L1 , L2 let L1 , L2 be distributive LATTICE st L1 , L2 let x , y be Element of L1 holds x "/\" y = x "/\" y
pred x in rng f & y in rng ( f /^ x ) implies f / x = f / y & f / y = f / ( y - x ) ;
assume that 1 < p and ( 1 - p ) * q + ( 1 - p ) * q = 1 and 0 <= a and a <= b ;
F* ( f , <* A1 *> ) = rpoly ( 1 , <* A1 *> ) *' t + rpoly ( 1 , <* A1 *> ) ;
for X being set , A being Subset of X holds A ` = {} implies A = {} & A = {} & A = {} & A = {} ;
( ( ( ( ( N ) ) | X ) ) `1 <= ( ( ( ( ( ( N ) ) | X ) ) `1 ) ) `1 ;
for c being Element of the \rbrack of A , a being Element of the bound is Element of the bound A holds c <> a
s1 . GBP = ( Exec ( i2 , s2 ) ) . GBP .= s2 . GBP .= s2 . GBP .= s2 . GBP .= s . GBP ;
for a , b being Real holds |[ a , b ]| in ( y >= 0 implies b >= 0 ) & a >= 0 implies 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 , k being Nat ;
set x2 = |( Re ( y - x ) , Im ( y - x ) )| ;
[ y , x ] in dom ( u . ( y , x ) ) & ( u . ( y , x ) ) . ( y , x ) = g . y ;
]. 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 ) .| < ( e / 2 ) * ( e / 2 ) ;
( - ( q `1 / |. q .| - cn ) ) / ( 1 - cn ) <= ( - ( q `1 / |. q .| - cn ) ) / ( 1 - cn ) ;
set A = ( 2 / 2 ) * ( b-a ) ;
for x , y being set st x in R" holds x , y are_\hbox { - } ;
deffunc FF2 ( Nat ) = b . ( $1 - 1 ) * ( M * ( $1 - 1 ) ) ;
for s being element holds s in -> Element of \mathclose { f } iff s in -> Element of \mathclose { f } \/ _ f
for S being non empty non void void void holds S is connected iff S is connected
max ( ( degree ( z ) ) , ( degree ( z ) ) / 2 ) >= 0 & ( |. z .| ) ^2 >= 0 ;
consider n1 be Nat such that for k holds seq . ( n1 + k ) < r + s and for n holds seq . ( n + k ) < r + s ;
Lin ( A /\ B ) is Subspace of Lin ( A ) & Lin ( B ) is Subspace of Lin ( A ) & Lin ( B ) is Subspace of Lin ( A ) ;
set n-15 = n-13 '&' ( M . ( x qua Element of BOOLEAN ) qua Element of BOOLEAN ) , n-15 = M . ( x qua Element of BOOLEAN ) ;
f " V in Hom ( X , L ) & f " V in D & f " V in D & f " V in D & f .: V c= D ;
rng ( ( a ^\ c ) +* ( 1 , b ) ) c= { a , c , b } ;
consider y being as as as as as as WWof G1 such that y ` = y and dom y ` = WWWWWWLet ;
dom ( ( 1 / 2 ) (#) f ) /\ ]. x0 - r , x0 .[ c= ]. x0 - r , x0 .[ & ]. x0 - r , x0 .[ c= dom f ;
as Subset of as Subset of as Subset of as Subset of as Subset of as ( i , j , n , - r ) ;
v ^ ( ( n |-> 0 ) ^ ( n |-> 0 ) ) in Lin ( ( ( B | ( n -' 1 ) ) ^ ( n |-> 0 ) ) ) ;
ex a , k1 , k2 st i = a := k1 & i = a := k2 & j = a := k1 & i <> j & j <> i implies ex a , b st i = a := b & j <> i & j <> i & i <> j
t . NAT = ( NAT .--> succ i1 ) . NAT .= ( NAT .--> succ i1 ) . NAT .= ( 5 .--> succ i1 ) . NAT .= ( 5 .--> succ i1 ) . NAT .= ( 5 .--> succ i1 ) . NAT ;
assume that F is bbfamily of X and rng p = Seg ( n + 1 ) and dom p = Seg ( n + 1 ) ;
LIN b , b9 , a & not LIN a , a9 , c & LIN a , a9 , c & LIN a , a9 , c & LIN b , b9 , c9 & LIN a , a9 , c & LIN b , b9 , c & LIN a , a9 , c & LIN b , b9 , c9 & LIN a , b9 , c & LIN b , c9 , c ;
( L1 over L2 ) \& O c= ( L1 Let O ) Let ( L2 Let O ) Let ( L1 Let O , O ) ;
consider F be ManySortedSet of E such that for d be Element of E holds F . d = F ( d ) and for d be Element of E holds F . d = F ( d ) ;
consider a , b such that a * ( contradiction w ) = b * ( -w ) and 0 < a & a < b ;
defpred P [ FinSequence of D ] means |. Sum $1 .| <= Sum |. $1 .| & for k st k in dom $1 holds |. Sum ( $1 ) .| <= Sum |. $1 .| ;
u = cos ^ ( x , y ) . v * x + ( cos ^ ( x , y ) . v * y ) .= v ;
dist ( ( seq . n ) + x , x + g ) <= dist ( ( seq . n ) , x + g ) + 0 ;
P [ p , |. p .| ^ |. p .| , {} ] & [ p , {} ] 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 finite and X is finite ;
|. b .| * |. eval ( f , z ) .| >= |. b .| * |. eval ( f , z ) .| ;
1 < ( ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ) .. Cage ( C , n ) ;
l in { l1 where l1 is Real : g <= l1 & l1 <= h & for k st k <= l & k <= h holds |. ( f | l ) . k - f /. k .| < r } ;
Int ( ( G . n ) vol ) <= ( Cl ( ( G . n ) vol ) ) * vol ( ( G . n ) vol ) ;
f . y = x .= x * 1_ L .= x * ( 1_ L ) .= x * ( power L ) .= x * ( power L ) ;
NIC ( ( a , i1 ) --> ( i , k ) , k ) = { i1 , succ ( k , i ) } .= { i1 , succ ( k , i ) } ;
LSeg ( p01 , p2 ) /\ LSeg ( p1 , p2 ) = { p1 , p2 } & LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = { p1 } ;
Product ( ( the support of I-15 ) +* ( i , { 1 } ) ) in Z1 & ( the support of ( the Sorts of I-15 ) +* ( i , { 1 } ) ) in Z1 ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) | ( the carrier of S1 ) ;
W-bound ( Qs2 ) <= ( q1 `1 ) / 2 & ( q1 `1 <= ( q1 `1 ) / 2 & ( q1 `2 <= ( q1 `2 ) / 2 & ( q1 `2 <= ( q1 `2 ) / 2 ) ) / 2 ;
f /. i2 <> f /. ( ( i1 + len g -' 1 ) -' len f + 1 -' 1 , f /. ( i1 + 1 -' len f + 1 ) ) ;
M , v / ( x. 3 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) ;
len ( ( P ^ Q ) ^ ( P ^ Q ) ) in dom ( ( P ^ Q ) ^ ( P ^ Q ) ) ;
A |^ ( m , n ) c= A |^ ( m , n ) & A |^ ( k , n ) c= A |^ ( k , l ) ;
R |^ n \ { q : |. q .| < a } c= { q1 : |. q1 .| >= a }
consider n1 being element such that n1 in dom p1 and y1 = p1 . n1 and p1 . n1 = p2 . n1 and for n being Nat holds p1 . n = p2 . ( n + 1 ) ;
consider X be set such that X in Q and for Z be set st Z in Q & Z <> X holds X c= Z ;
CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA & CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA ;
for v be VECTOR of l1 holds ||. v .|| = upper_bound rng |. ( id the carrier of V ) . v .| & ||. v .|| = upper_bound rng |. ( id the carrier of V ) . v .|
for phi holds phi in X implies phi in X & not phi in X & not phi in X & not phi in X & not phi in X
rng ( ( Sgm dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f
ex c being FinSequence of D ( ) st len c = k & P [ c ] & a = c & a = c ;
the_arity_of o = <* o , b , c *> & o = <* o , c , o *> & o <> c & o <> b & o <> c & o <> c & o <> b & o <> c & o <> a & o <> b & o <> c & o <> b & o <> c & o <> a & o <> b & a <> b implies o , c // a , b
consider f1 be Function of the carrier of X , REAL such that f1 = |. f .| and f1 is continuous and f1 is continuous ;
a1 = b1 & a2 = b2 or a1 = b1 & a2 = b2 & a3 = b3 & a4 = b3 & a5 = b2 & a5 = b3 & a5 = b3 & 6 = 6 & 7 = 7 & 8 = 8 & 8 = 8 & 8 = 8 & 7 = 8 & 8 = 8 & 8 = 8 & 8 = 8 & 8 = 8 & 8 = 7 & 8 = 8 & 8 = 8 ;
D2 . ( indx ( D2 , D1 , n1 ) + 1 ) = D1 . ( n1 + 1 ) & D2 . ( indx ( D2 , D1 , n1 ) + 1 ) = D2 . ( n1 + 1 ) ;
f . ( ||. |[ r , s ]| ) = ||. ||. |[ r , s ]| .|| , ||. r , s .|| .|| .= ||. r .|| .= ||. r .|| ;
consider n be Nat such that for m be Nat st n <= m holds C-25 . m = C-25 . m and C-25 . m = C-25 . m ;
consider d be Real such that for a , b be Real st a in X & b in Y holds a <= d & a <= b ;
||. L /. h - ( K * |. h .| ) + ( K * |. h .| ) + ( K * |. h .| ) <= p0 + ( K * |. h .| ) ;
attr F is commutative associative means : Def2 : for b being Element of X holds F - Sum ( F - f ) = f . b - f . b ;
p = - ( - 1 ) * p0 + 0. TOP-REAL 2 .= 1 * ( p0 `1 ) + 0. TOP-REAL 2 .= ( 1 - 1 ) * p1 + 0. TOP-REAL 2 .= ( 1 - 1 ) * p1 + 0. TOP-REAL 2 ;
consider z1 such that b , x1 , x3 is_collinear and o , x1 , z1 is_collinear and o , x1 , z1 is_collinear and o , x1 , z1 is_collinear and o , x1 , z1 is_collinear ;
consider i such that Arg ( ( Rotate ( s , r ) ) . q ) = s + Arg q + ( 2 * PI * i ) ;
consider g such that g is one-to-one and dom g = card f and rng g = f . x and rng g c= f . x and for x st x in dom g holds g . x = f . x ;
assume that A = P2 \/ Q2 and P2 <> {} and Q2 <> {} and Q1 is closed and for x being set st x in P1 & x in P2 holds x in P2 ;
attr F is associative means : Def2 : F .: ( F .: ( f , g ) , h ) = F .: ( f , g ) ;
ex x being Element of NAT st m = x `1 & x in z & x < z or m < i & i < z & i < z ;
consider k2 being Nat such that k2 in dom ( P . ( k2 + 1 ) ) and l in Pk and i = ( k + 1 ) + k2 ;
seq = r (#) seq implies for n holds seq . n = r * seq . n & seq is bounded & lim seq = r * seq . n & lim seq = r * seq . n
F1 . [ id a , [ a , a ] ] = [ f * ( id a , a ) , f * ( id a , b ) ] ;
{ 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 ) | ( dom F ) ) and ( ( dom F ) | ( dom F ) ) . z = y ;
for x , y being element st x in dom f & y in dom f & f . x = f . y holds x = y
Int cell ( G , i , j ) = { |[ r , s ]| : r <= G * ( 0 + 1 , 1 ) `1 } ;
consider e being element such that e in dom ( T | E1 ) and ( T | E1 ) . e = v and ( T | E1 ) . e = v ;
( F `1 * b1 ) . x = ( Mx2Tran ( J , b1 , b2 ) ) . ( T /. j ) .= ( Mx2Tran ( J , b1 , b2 ) ) . j ;
- 1 / ( - 1 / ( n + 1 ) ) = ( m (#) D ) | n .= ( - m ) (#) D .= ( - m ) (#) D .= ( - m ) (#) D .= ( - m ) (#) D ;
pred for x be 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 ) ;
All ( 'not' All ( 'not' a , A , G ) , B , G ) '<' Ex ( 'not' Ex ( 'not' a , B , G ) , A , G )
LSeg ( E . k , F . ( k + 1 ) ) c= Cl RightComp ( Cage ( C , n + 1 ) , i ) ;
x \ a |^ m = x \ ( a |^ k * a ) .= ( x \ ( a |^ k ) ) \ a .= ( x \ ( a |^ k ) ) \ a ;
k -inininininininthe Element of S , I , J be Element of S , i be Element of I , k be Element of I , i be Element of I ;
for s being State of A9 holds Following ( s , n ) . 0 + ( n + 2 ) * ( n + 2 ) 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 ( ( support ( m ) ) \/ support ( m ) c= support ( ( m ) \ ( support ( n ) ) ) & support ( ( m ) \ ( support ( n ) ) ) c= support ( m ) ;
reconsider t = u as Function of ( the carrier of A ) , ( the carrier of B ) * , the carrier' of C ;
- ( a * sqrt ( 1 + b ^2 ) ) <= - ( b * sqrt ( 1 + a ^2 ) ) ;
phi /. ( succ b1 ) = g . a & phi /. ( succ b1 ) = f . a & phi . ( succ b1 ) = f . ( g . a ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( ( F ^ <* p *> ) ^ <* p *> ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 } = { x1 } \/ { x2 , x3 , x4 , x5 } .= { x1 } \/ { x2 , x3 , x4 , x4 } ;
the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U1 & the Sorts of U2 c= the Sorts of U2 implies ( the Sorts of U1 ) \/ ( the Sorts of U2 ) c= the Sorts of U1
( - ( 2 * a - ( 2 * a - b ) ) / 2 ) + b / 2 - a / 2 > 0 ;
consider W00 such that for z being element holds z in W00 iff z in N & P [ z , W00 . z ] ;
assume that ( the Arity of S ) . o = <* a *> and ( the ResultSort of S ) . o = r and ( the ResultSort of S ) . o = <* a *> and ( the ResultSort of S ) . o = r ;
Z = dom ( ( exp_R * ( ( #Z n ) * ( f1 + #Z n ) ) ) `| Z ) & Z = dom ( ( #Z n ) * ( f1 + #Z n ) ) ;
sum ( f , SS1 ) is convergent & lim ( f , SS1 ) = integral ( f , SS1 ) & lim ( f , SS1 ) = integral ( f , S ) ;
\cal X [ a9 , g . ( f , g ) => ( x9 => ( f , x9 ) ) ] in ( J , the carrier of l ) ;
len ( M2 * ( M1 - M2 ) ) = n & width ( M2 * ( M1 - M2 ) ) = n & width ( M2 * ( M1 - M2 ) ) = n & width ( M2 * ( M1 - M2 ) ) = n ;
attr X1 union X2 is open means : Def4 : X1 , X2 are_separated & X1 , X2 are_separated & X2 , X1 are_separated & X1 , X2 are_separated & X2 , X1 are_separated & X1 , X2 are_separated implies X1 , X2 are_separated ;
for L being upper-bounded antisymmetric RelStr , X being non empty RelStr , X being non empty Subset of L holds X "\/" { Top L } = { Top L }
reconsider f-129 = F2 . ( ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . b ) ) ) ) ) ) ) ) )
consider w being FinSequence of I such that the InitS of M , w ^ <* s *> ^ w ^ w ^ q = q ^ w ^ ( <* s *> ^ w ) ;
g . ( a |^ 0 ) = g . ( 1_ G ) .= 1_ H .= 1_ H .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) ;
assume for i be Nat st i in dom f ex z being Element of L st f . i = rpoly ( 1 , z ) & for i be Element of NAT st i in dom z holds z . i = rpoly ( 1 , z ) ;
ex L being Subset of X st Carrier ( L ) = L & for K being Subset of X st K in C holds L /\ K <> {} & L /\ K <> {} ;
( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C1 & ( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C2 implies ( the carrier' of C1 ) /\ ( the carrier' of C2 ) = the carrier' of C2
reconsider oY = o `1 , o = o `1 , tY = o `2 , o = o `1 , o = o `2 , o = o `1 , o = o `2 , Y = o `2 , X = ( the Sorts of A ) . o , Y = ( the Sorts of A ) . o ;
1 * x1 + ( 0 * x2 ) + ( 0 * x3 ) = x1 + ( 0 * x2 ) .= x1 + ( 0 * x2 ) .= x1 + ( 0 * x3 ) .= x1 + ( 0 * x2 ) .= x1 + x2 ;
Ea " . 1 = ( Ea qua Function ) " . 1 .= ( Ea " ) . 1 .= ( ( Ea ) " ) . 1 .= ( Ea ) . 1 .= ( Ea ) . 1 ;
reconsider u1 = the carrier of U1 /\ ( U1 "\/" U2 ) , v1 = the carrier of U1 /\ ( U1 "\/" U2 ) as non empty Subset of U0 ;
( ( x "/\" z ) "\/" ( x "/\" y ) ) "\/" ( z "/\" y ) <= ( x "/\" ( z "\/" y ) ) "\/" ( z "/\" y ) ;
|. f . ( s1 . ( l1 + 1 ) ) - f . ( s1 . ( l1 + 1 ) ) .| < ( 1 / |. M . ( l + 1 ) .| ) ;
LSeg ( ( Upper_Seq ( C , n ) ) * ( i , j ) , ( \lbrace p } ) * ( i , j ) ) is vertical ;
( f | Z ) /. x - ( f | Z ) /. x0 = L /. ( x- x ) + R /. ( x- x ) ;
g . c * ( - g . c ) + f . c <= h . c * ( - f . c ) + f . c ;
( f + g ) | divset ( D , i ) = f | divset ( D , i ) + g | divset ( D , i ) .= f | divset ( D , i ) + g | divset ( D , i ) ;
assume that ColVec2Mx f in the set of \HM { the } \HM { set , b is } and len ColVec2Mx f = width A and width ColVec2Mx f = width A and width ColVec2Mx f = width A and width ColVec2Mx f = width A ;
len ( - M1 ) = len M1 & width ( - M1 ) = width M1 & width ( - M1 ) = width M2 & width ( - M1 ) = width M2 & width ( - M2 ) = width M2 ;
for n , i being Nat st i + 1 < n & i < n holds [ i , i + 1 ] in the InternalRel of [: the InternalRel of thesis , the InternalRel of R :]
pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 ;
pred a <> 0 & b <> 0 & Arg a = Arg b & Arg b = Arg a & Arg a = Arg b & Arg b = Arg a & Arg a = Arg b & Arg b = Arg a & Arg a = Arg b & Arg b = Arg a & Arg b = Arg a & Arg a <> 0 & Arg b <> 0 & Arg b <> 0 & Arg a <> 0 & Arg b <> 0 & Arg a <> 0 ;
for c being set st not c in [. a , b .] holds not c in Intersection ( the open of a , b ) & not c in Intersection ( the open of a , b )
assume that V1 is linearly-independent and V2 is closed and V2 = { v + u : v in V1 & u in V1 & v in V1 } and V1 = { v + u : u in V1 & v in V1 } ;
z * x1 + ( 1 - z ) * x2 in M & z * y1 + ( 1 - z ) * y2 in N implies z * y1 + ( 1 - z ) * y2 in N
rng ( ( ( ( ( P qua Function ) ) * ( P * ( S , R ) ) " ) ) ) = Seg card ( ( ( P * ( S * ( S , R ) ) ) " ) ) ;
consider s2 being Integer such that s2 is convergent and b = lim s2 and for n holds s2 . n <= b and s2 . n <= b ;
h2 " . n = h2 . n " & 0 < - ( 1 / 2 ) & 0 < ( - ( 1 / 2 ) ) / 2 & ( - ( 1 / 2 ) ) < ( - ( 1 / 2 ) ) / 2 ;
( Partial_Sums ||. seq .|| ) . m = ||. ( seq . m ) - ( seq . m ) .|| .= ||. ( seq . m ) - ( seq . m ) .|| .= 0 ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b ;
- v = - 1_ G & - w = - 1_ G & - v = - 1_ G implies ( - v ) * w = - ( 1_ G ) * w & ( - v ) * w = - ( 1_ G ) * v
sup ( ( k .: D ) .: D ) = sup ( ( k .: D ) .: D ) .= k . ( sup D ) .= sup ( ( k .: D ) .: D ) .= sup ( k .: D ) ;
A |^ ( k , l ) ^^ ( A |^ ( n , .. A ) ) = ( A |^ ( n , .. A ) ) ^^ ( A |^ ( k , .. A ) ) ;
for R being add-associative right_zeroed right_complementable non empty addLoopStr , I , J , K being Subset of R holds I + ( J + K ) = ( I + J ) + K
( f . p ) `1 = ( p `1 ) ^2 + ( p `2 ) ^2 .= ( 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 cluster [: a , b :] = [: \hbox { a } , the carrier of p :] & a , b are_relative_prime & b , a are_relative_prime holds [: n , b :] = [: n , n :]
consider A9 being countable Subset of Al such that r is countable and A9 is non empty and A9 is finite and for i being Nat holds A9 is finite & i <= len ( A | i ) ;
for X being non empty addLoopStr , M , N being Subset of X , x , y being Point of X st x in M & y in M holds x + y in x + y
{ [ x1 , x2 ] , [ y1 , y2 ] } c= { [ x1 , y1 ] , [ x2 , y2 ] }
h . ( f . O ) = |[ A * ( f . O ) `1 + B , C * ( f . O ) `2 + D ]| ;
( Gauge ( C , n ) * ( k , i ) ) `2 in L~ ( Upper_Seq ( C , n ) ) /\ L~ ( Upper_Seq ( C , n ) ) ;
cluster m , n are_relative_prime implies for Nat holds ( for Nat st m divides n & not ( m divides n & not m divides n & not m divides n ) & not ( ex p being prime Nat st p divides n & not m divides n ) & not p divides n ) & not ( m divides n & not p divides n ) ;
( 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 & b \ a <= c holds a "/\" b <= c
consider b being element such that b in dom ( H / ( x. 0 , y ) ) and z = H / ( x. 0 , y ) . b ;
assume that x in dom ( F (#) g ) and y in dom ( F (#) g ) and ( F (#) g ) . x = ( F (#) g ) . y ;
assume ex e being element st e Joins W . 1 , W . 5 , G & e Joins W . 3 , G & W . 7 in G ;
( \cal o ) . ( 2 * n ) . x = ( o holds h . ( 2 * n ) ) . x + ( h . ( 2 * n ) ) . x ;
j + 1 = ( i - len h11 + 2 ) + 2 .= i + ( 1 - len h11 + 2 - len h11 + 2 - len h11 + 2 - len h11 + 2 - len h11 + 2 - len h11 + 2 - len h11 + 2 - len h11 + 2 - len h11 + 2 - len h11 + 2 - len h11 + 2 - 1 ;
*' ( 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 ) ;
attr R is max means : Def4 : for p , q st p in R & q in R & p <> q holds ex P st P is special & p in P & q in P & p <> q ;
dom product ( X --> f ) = meet ( dom ( X --> f ) ) .= meet ( dom ( X --> f ) ) .= meet ( X --> f ) .= meet ( X --> f ) .= meet ( X --> f ) .= dom f ;
sup ( proj2 .: ( Upper_Arc C ) /\ \mathop { \rm UpperArc ( C ) /\ \mathop { \rm UpperArc } ( C ) ) <= sup ( proj2 .: ( C /\ \mathop { w } ) ) ;
for r be Real st 0 < r ex n be Nat st for m be Nat st n <= m holds |. S . m - p .| < r
i * ( f - fN ) = i * ( f - ( i - y ) ) .= i * ( f - ( i - y ) ) .= i * ( f - ( i - y ) ) .= i * ( f - ( i - y ) ) ;
consider f being Function such that dom f = 2 -tuples_on X and for Y be set st Y in 2 -tuples_on X holds f . Y = F ( Y ) ;
consider g1 , g2 being element such that g1 in [#] Y and g2 in union C and g = [ g1 , g2 ] and g1 in union C and g2 in A and g2 in B ;
func d div n -> Nat means : Def4 : d divides n & it divides n & ( d |^ n ) divides ( d |^ n ) & ( d |^ n divides ( d |^ n ) ) ;
f\in f . [ 0 , t ] = f . [ 0 , t ] .= ( - P ) . [ 2 * x , t ] .= ( - P ) . [ 2 * x , t ] .= a ;
t = h . D or t = h . B or t = h . C or t = h . E or t = h . F or t = h . J ;
consider m1 be Nat such that for n st n >= m1 holds dist ( ( seq . n ) , ( seq . n ) ) < 1 / ( n + 1 ) ;
( q `1 / |. q .| - cn ) / ( 1 - cn ) <= ( ( q `1 / |. q .| - cn ) / ( 1 - cn ) ) / ( 1 - cn ) ;
h0 . ( i + 1 + 1 ) = h21 . ( i + 1 + 1 -' len h11 + 2 -' 1 ) .= h21 . ( i + 1 -' len h11 + 2 -' 1 ) ;
consider o being Element of the carrier' of S , x2 being Element of { the carrier' of S } such that a = [ o , x2 ] and o <> [ o , x2 ] ;
for L being RelStr , a , b being Element of L holds a <= { b } iff a <= b & a <= b & b <= a & a <= b & b <= a ) implies a <= b
||. h1 .|| . n = ||. h1 . n .|| .= ||. h . n .|| .= ||. h . n .|| .= ||. h . n .|| .= ||. h . n .|| .= ||. h . n .|| ;
( ( - ( exp_R * f ) ) `| Z ) . x = f . x - ( exp_R . x ) / ( exp_R . x ) .= ( - 1 ) / ( exp_R . x ) .= ( - 1 ) / ( exp_R . x ) .= ( - 1 ) / ( exp_R . x ) ;
pred r = F .: ( p , q ) means : Def4 : len r = min ( len p , len q ) ;
( r\mathbin { r / 2 } ) ^2 + ( r8 / 2 ) ^2 <= ( r / 2 ) ^2 + ( r / 2 ) ^2 ;
for i being Nat , M , N being Matrix of n , K st i in Seg n & i in Seg n holds Det ( M @ ) = Sum ( ( 1 , i ) * ( - 1 , j ) )
then a <> 0. R & a " * ( a * v ) = 1 * v & a " * ( a * v ) = 1 * v & a " * ( a * v ) = 1 * v ;
p . ( j -' 1 ) * ( q *' r ) . ( i + 1 -' j ) = Sum ( p . ( j -' 1 ) * ( q . ( j -' 1 ) ) ) ;
deffunc F ( Nat ) = L . 1 + ( ( R /* ( h ^\ n ) ) * ( R /* ( h ^\ n ) ) " ) ;
assume that the carrier of H2 = f .: the carrier of H1 and the carrier of H1 = f .: the carrier of H2 and the carrier of H2 = the carrier of H1 and the carrier of H1 = the carrier of H2 and the carrier of H2 = the carrier of H2 and the carrier of H1 = the carrier of H2 and the carrier of H2 = the carrier of H2 ;
Args ( o , Free ( S , X ) ) = ( ( the Sorts of Free ( S , X ) ) * ( the Arity of S ) ) . o .= ( ( the Sorts of Free ( S , X ) ) * ( the Arity of S ) ) . o ;
H1 = n + 1 - ( |. 2 to_power ( n + 1 ) .| ) .= n + 1 - ( |. 2 to_power ( n + 1 ) .| ) .= n + 1 - ( |. 2 to_power ( n + 1 ) .| ) ;
( O = 0 & 3 ) = 0 & 3 = 1 & 4 = 1 & 5 = 2 & 6 = 3 & 7 = 4 & 6 = 5 & 8 = 6 & 7 = 8 & 8 = 8 & 8 = 7 & 8 = 8 & 8 = 8 & 7 = 8 & 8 = 8 & 8 = 8 & 8 = 8 & 8 = 8 & 8 = 8 & 8 = 9 & 8 = 8 & 8 = 8 & 9 = 8 & 8 = 8 & 8 = 8 & 8 = 8 & 8 = 8 & 8 = 8 & 7 = 8 & 8
F1 .: ( dom F1 /\ dom F2 ) = F1 .: ( dom F1 /\ dom F2 ) .= { f /. ( n + 1 ) } .= { f /. ( n + 1 ) } .= { f /. ( n + 1 ) } ;
pred b <> 0 & d <> 0 & b <> d & b <> d & a = ( - e ) / ( 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 ) ;
for i be set st i in dom g ex u , v be Element of L st g /. i = u * v & u in B & v in B & v in B & u in B & v + u in B
g `1 * P `1 * g `1 = g `1 * ( g `1 * P `2 ) .= g `1 * ( g `2 * P `1 ) .= g `1 * ( g `2 * P `1 ) .= g `1 * ( g `2 * P `1 ) ;
consider i , s1 such that f . i = s1 and not i in s1 and not i in { s1 } and not i in { s1 } and not i in { s2 } and not i in { s2 } and not j in { s2 } and not i in { s2 } and not j in { s2 , s1 } ;
h5 | ]. a , b .[ = ( g | ]. a , b .[ ) | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ ;
[ s1 , t1 ] , [ s2 , t2 ] & [ s2 , t2 ] , [ s2 , t2 ] <= [ s2 , t2 ] & [ s2 , t2 ] in the InternalRel of TOP-REAL 2 ;
then H is negative & H is not negative & H is not negative & H is not negative implies H is not negative -g\mathopen H & H is not negative ;
attr f1 is total means : Def4 : ( 1 - r ) (#) ( f1 (#) f2 ) is total & for c holds ( f1 (#) f2 ) . c = f1 . c * ( f2 . c ) " ;
z1 in W2 ` & z2 = W2 ` & ( for z st z in W2 holds z in W1 & z in W2 holds not z in W2 ) implies ( for z st z in W1 & z in W2 holds z in W1 & z in W2 )
p = 1 * p .= a " * a * p .= a " * a * p .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= a * ( b " * q ) .= a * q ;
for r be Real , K be Real st for n be Nat holds rseq . n <= K holds upper_bound rng ( seq1 . n ) <= K * ( upper_bound rng ( seq1 ^\ K ) )
( for C being Subset of TOP-REAL 2 holds E-max C meets L~ ( go ) \/ L~ ( co ) or C meets ( L~ ( go ) \/ L~ ( co ) ) or C meets ( L~ ( go ) \/ L~ ( co ) ) ;
||. f . ( g . ( k + 1 ) ) - g . ( g . k ) .|| <= ||. g . 1 - g . 0 .|| * ( K * ( K / ( k + 1 ) ) ) ;
assume h = ( ( B .--> B ' ) +* ( C .--> D ) +* ( E .--> F ) +* ( F .--> J ) +* ( J .--> M ) +* ( F .--> N ) +* ( M .--> N ) +* ( F .--> N ) +* ( M .--> N ) +* ( F .--> N ) ) ;
|. ( ( lower . n ) || ( A . n ) ) . k - ( ( lower . n ) || ( A . n ) ) . k .| <= e * ( ( b-a . n ) || ( A . n ) ) ;
( ( being being such such that being being such that ( for i being Nat holds i in I holds f . i = [ the |^ of I , the carrier of S ] ) holds ( for i being Element of I holds f . i = ( the
{ x1 , x1 , x2 , x3 , x4 , x5 , x5 , 6 , 7 , 8 } = { x1 , x2 , x3 , x4 , 7 , 8 } .= { x1 , x2 , x3 , x4 , 6 , 7 } .= { x1 , x2 , x3 , x4 } ;
assume that A = [. 0 , 2 * PI .] and integral ( ( ( #Z n ) * sin ) , A ) = 0 and integral ( ( #Z n ) * sin ) = 0 ;
p `1 is Permutation of dom f1 & p `1 = ( Sgm Y ) . i & p `2 = ( Sgm Y ) . i implies p `1 * Sgm X = p `1 * Sgm Y
for x , y st x in A & y in A holds |. ( 1 / ( f . x - 1 ) ) * ( f . y - 1 ) .| <= 1 * |. f . x - f . y .|
( p2 `2 ) ^2 = |. q2 .| * ( ( q `2 / |. q .| - sn ) / ( 1 - sn ) ) .= |. q .| * ( ( q `2 / |. q .| - sn ) / ( 1 - sn ) ) ;
for f be PartFunc of the carrier of CNS , REAL st dom f is compact & f is compact & f is continuous holds rng f = dom f & f is compact & f is bounded & f is bounded & f is bounded & f is bounded & f is bounded & f is bounded & f is bounded holds f is compact
assume for x being Element of Y st x in EqClass ( z , CompF ( B , G ) ) holds ( Ex ( a , A , G ) ) . x = TRUE ;
consider FM such that dom FM = n1 and for k be Nat st k in n1 holds Q [ k , FM . k ] and for k be Nat st k in n1 holds Q [ k , FM . k ] ;
ex u , u1 st u <> u1 & u , u1 // v , v1 & u , u1 // v , v1 & u , u1 // v , v1 & v , v1 // v , u1 & u , u1 // v , v1 & v , v1 // v , u1 & v , v1 // v , u1 implies u , u1 // v , v1
for G being Group , A , B being non empty Subset of G , N being normal Subgroup of G holds ( N ` A ) * ( N ` ) = N ` A * B
for s be Real st s in dom F holds F . s = integral ( R / > ( R / > ) (#) integral ( f , ( f + g ) / ( f - g ) ) , ( f - g ) / ( f - g ) ) ;
width AutMt ( f1 , b1 , b2 ) = len ( ( f1 - f2 ) * ( f1 - f2 ) ) .= len ( ( f1 - f2 ) * ( f1 - f2 ) ) .= len ( ( f1 - f2 ) * ( f1 - f2 ) ) .= len ( ( f1 - f2 ) * ( f1 - f2 ) ) ;
f | ]. - PI / 2 , PI / 2 .[ = f & dom f = ]. - PI / 2 , PI / 2 .[ & for x st x in ]. - PI / 2 , PI .[ holds f . x = f . x - 1 / 2 & for x st x in ]. - PI , PI / 2 .[ holds f . x = - 1 & f . x = 1
assume that X is closed and a in X and a c= X and a in X and y in X and x in a and y in X and x in X ;
Z = dom ( ( ( 1 / 2 ) (#) ( ( #Z 2 ) * ( f1 + #Z 2 ) ) ) `| Z ) & Z = dom ( ( 1 / 2 ) (#) ( ( #Z 2 ) * ( f1 + #Z 2 ) ) ) ;
func Let ( V , l ) -> Subset of V means : Def4 : for k st 1 <= k & k <= len l holds it . k in V & l . k in V ;
for L being non empty TopSpace , N being net of L , M being net of N , N being net of L st c is inf of N & M is open & N c= M holds c in N
for s being Element of NAT holds ( ( id the carrier of C\mathop { v } ) + ( id the carrier of C\mathop { v } ) ) . s = ( ( id the carrier of C\mathop { v } ) + ( id the carrier of CV ) ) . s
then z /. 1 = ( ( us L~ z ) .. z ) .. z & ( ( ( ( L~ z ) .. z ) .. z ) .. z < ( ( ( L~ z ) .. z ) .. z ) .. z ) .. z ;
len ( p ^ <* ( 0 qua Real ) * ( 0 qua Real ) *> ) = len p + len <* ( 0 qua Real ) * ( 0 qua Real ) *> .= len p + 1 .= len p + 1 ;
assume that Z c= dom ( - ( ln * f ) ) and for x st x in Z holds f . x = x and for x st x in Z holds f . x = x and f . x > 0 ;
for R being add-associative right_zeroed right_complementable Abelian distributive non empty doubleLoopStr , I , J being Subset of R , I being Subset of R holds ( I + J ) *' ( I /\ J ) c= I /\ J implies I /\ J c= I /\ J
consider f being Function of [: B1 , B2 :] , ( for x being Element of B1 , y being Element of B2 holds f . x = F ( x , y ) ) and f is Function of B1 , B2 ;
dom ( x2 + y2 ) = Seg len x .= Seg len ( x ^ ( y - z ) ) .= dom ( x ^ ( y - z ) ) .= dom ( x ^ ( y - z ) ) .= dom ( x ^ ( y - z ) ) .= dom ( x ^ ( y - z ) ) .= dom ( x ^ ( y - z ) ) ;
for S being not not not F is not a functor of C , B , c being Object of C holds card S . ( id c ) = id ( ( the Arrows of C ) . ( id c ) )
ex a st a = a2 & a in f /\ ( f " ) & a in f " ( f " ) & b in f " ( f " ) & a in f " ( f " ) & b in f " ( f " ) & a in f " ( f " { b } ) & b in f " { a } implies a = b
a in Free ( H2 / ( x. 4 , x. k ) ) '&' ( H2 / ( x. 4 , x. k ) ) & ( H1 / ( x. 4 , x. k ) ) / ( x. 4 , x. k ) = ( H1 / ( x. 4 , x. k ) ) / ( x. 4 , x. k ) ;
for C1 , C2 being set , f being stable Function of C1 , C2 st f = @ g & f is stable holds f = g & f = g implies f = g
( W-min L~ go ) `1 = W-bound L~ ( go \/ L~ pion1 ) & ( W-min L~ go ) `1 = E-bound L~ ( go \/ L~ pion1 ) & ( W-min L~ go ) `1 = E-bound L~ ( go \/ L~ pion1 ) & ( W-min L~ go ) `1 = E-bound L~ ( go \/ L~ pion1 ) ;
consider u such that u = <* x0 , y0 , z0 *> and f is partial & u in dom f & for z be element st z in dom f holds SVF1 ( 3 , f , u ) . z - SVF1 ( 3 , f , u ) . z = L . z - R . z ;
then ( ex x being Element of Vars st x = ( t . {} ) `1 & ex x being Element of Vars st x = ( t . {} ) `1 & t = ( t . {} ) `1 & t . {} = ( t . {} ) `2 & t . {} = ( t . {} ) `1 ;
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 ;
assume for x , y being Element of S st x <= y & for a , b being Element of T st a = f . x & b = f . y holds a >= b ;
func Class R -> Subset-Family of R means : Def4 : for A being Subset of R holds A in it iff ex a being Element of R st a in it & A c= a & it c= a ;
defpred P [ Nat ] means ( ( ( ( ) `2 ) | ( ( ) ) \ { x } ) ) . $1 c= G ( ( ( ( ( ) | ( ( ( ) ) \ { x } ) ) ) ) . $1 ) & ( ( ( ( ( G ) | ( { x } ) ) ) ) . $1 = G ( ( ( G | ( { x } ) ) ) . $1 ) ;
assume that dim W1 = 0 and dim W2 = 0 and dim W1 = 0 and dim W2 = 0 and dim W1 = 0 and dim W2 = 0 and dim W1 = 0 and dim W2 = 0 and dim W1 = 0 and dim W2 = 0 and 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 = 0 and dim W1 = 0 ;
mamas ( m , t ) . {} = ( m . t ) `1 .= ( m . {} ) `1 .= ( m . t ) `1 .= ( m . t ) `1 .= m . t ;
d11 = x9 ^ d22 .= f . ( y11 , d22 ) .= f . ( y11 , d22 ) .= f . ( y22 , d22 ) .= ( f | ( y22 ) ) . ( y22 , d22 ) .= ( f | ( y22 ) ) . ( d22 , d22 ) .= ( f | ( y11 ) ) . ( d11 , d22 ) .= d22 ;
consider g such that x = g and dom g = dom ( f | A ) and for x being element st x in dom ( f | A ) holds g . x in f . x and g . x in f . x ;
x + 0. F_Complex = x + len x .= x + len x .= ( x + len x ) |-> 0. F_Complex .= ( x + len x ) |-> 0. F_Complex .= x + ( 0. F_Complex ) .= x + 0. F_Complex ;
( ( k -' ( k -' 1 ) ) + 1 ) in dom ( f | ( ( k -' 1 ) + 1 ) ) /\ dom ( f | ( ( k -' 1 ) + 1 ) ) ;
assume that P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 = P1 \/ P2 and P1 = { p1 , p2 } and P1 = { p1 , p2 } and P2 = { p1 , p2 } and P1 = { p2 , p1 } and P2 = { p1 , p2 } and P1 = { p2 , p1 } and P2 = { p1 , p2 } and P1 = { p2 , p1 } and P2 = { p1 , p2 } and P1 = { p2 , p1 , p2 } and P2 = { p1 , p2 , p3 , p3 } and P2 = { p1 , p2 , p3 , p3 , p1 , p2 , p3 , p2 , p3 , p2 , p3 , p4 , p1 , p2 , p3 , p4 , p1 , p3 , p4 , p4 , p4 ,
reconsider a1 = a , b1 = b , c1 = c , c1 = d , c1 = c , c2 = d , c1 = d , c2 = c , c1 = d , c2 = c , c1 = d , c2 = d , c1 = c , c2 = d , c1 = d , c1 = d , c2 = c , c1 = d , c1 = d , c2 = c , c1 = d , c2 = c , c1 = d , c1 = d , c2 = c , c1 = d , c1 = d , c1 = d , c2 = d , c2 = d , c1 = d , c1 = d , c1 = d , c2 = d , c2 = d , c1 = d , c1 = d , c1 = c = d , c2 = c ,
reconsider Gtbf = G1 . ( t , [. b , a .] ) * F1 . f as Morphism of ( G1 * F1 ) . a , ( G1 * F2 ) . b ;
LSeg ( f , i + i1 -' 1 ) = LSeg ( f /. ( i + i1 -' 1 ) , f /. ( i + i1 -' 1 + 1 -' 1 ) ) ;
Integral ( M , P . m ) | dom ( P . n -| ( dom P ) ) <= Integral ( M , P . n ) & Integral ( M , P . m ) <= Integral ( M , P . n ) ;
assume that dom f1 = dom f2 and for x , y being element st [ x , y ] in dom f1 & [ x , y ] in dom f2 holds f1 . ( x , y ) = f2 . ( x , y ) and f2 . ( x , y ) = f2 . ( x , y ) ;
consider v such that v = y and dist ( u , v ) < min ( ( ( G * ( i , 1 ) ) `1 , ( G * ( i + 1 , 1 ) ) `2 ) ;
for G being Group , H being Subgroup of G , a being Element of H st a = b holds for i being Integer st i = b holds a |^ i = b |^ i & a |^ i = b |^ i
consider B being Function of Seg ( S + L ) , the carrier of V1 such that for x being element st x in Seg ( S + L ) holds P [ x , B . x ] and P [ B , B . x ] ;
reconsider K1 = { p0 where 7 is Point of TOP-REAL 2 : P [ 7 ] } , K1 = { p where p is Point of TOP-REAL 2 : P [ 7 ] } as Subset of ( TOP-REAL 2 ) | K1 ;
( ( S-bound C - S-bound C ) / 2 ) * ( ( S-bound C - S-bound C ) / 2 ) <= ( ( S-bound C - S-bound C ) / 2 ) * ( ( S-bound C - S-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 & |. Im ( F . n ) .| <= P . x
len @ ( @ ( @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
v / ( x. 3 , m1 ) / ( x. 4 , m1 ) / ( x. 4 , m1 ) / ( x. 4 , m1 ) / ( x. 4 , m1 ) / ( x. 4 , m1 ) / ( x. 4 , m1 ) / ( x. 4 , m1 ) / ( x. 4 , m1 ) / ( x. 4 , m1 ) / ( x. 4 , m1 ) = m3 ;
consider r being Element of M such that M , v |= ( v2 , ( x. 3 ) ) / ( x. 4 , m ) / ( x. 4 , n ) / ( x. 4 , n ) / ( x. 4 , n ) / ( x. 4 , n ) / ( x. 4 , n ) / ( x. 4 , n ) / ( x. 4 , n ) ;
func w1 \ w2 -> Element of Union ( G , R8 ) equals ( ( ( ( H\in G ) \ R8 ) * ( i , w ) ) ) . ( w1 , w2 ) & ( ( ( H\in G ) \ ( i , w ) ) * ( w1 , w2 ) ) . ( w1 , w2 ) = ( ( ( H8 \ ( G , R8 ) ) * ( i , w2 ) ) ) . ( w1 , w2 ) ;
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= ( ( s2 ) . b2 ) . b2 .= ( s2 . b2 ) . b2 .= ( s2 . b2 ) . b1 .= ( s2 . b2 ) . b1 .= ( s2 . b2 ) . b1 .= ( s2 . b2 ) . b1 .= ( s2 . b1 ) . b1 .= ( s2 . b1 ) . b2 ;
for n , k be Nat holds 0 <= ( Partial_Sums ( |. seq .| ) . ( n + k ) - Partial_Sums ( |. seq .| ) . n + Partial_Sums ( |. seq .| ) . ( n + k ) - Partial_Sums ( |. seq .| ) . n ;
set F = S \! \mathop { 0 } ;
( Partial_Sums ( seq ) . n + Partial_Sums ( seq ) . K + Partial_Sums ( seq ) . ( n + 1 ) ) + Partial_Sums ( seq ) . ( n + 1 ) >= ( Partial_Sums ( seq ) . n + Partial_Sums ( seq ) . ( n + 1 ) ) + Partial_Sums ( seq ) . ( n + 1 ) ;
consider L , R such that for x st x in N holds ( f | Z ) . x - ( f | Z ) . x0 = L . ( x- ( x0 - R ) ) + R . ( x- ( x0 - R ) ) ;
func the closed \HM { a , b , c , d , e , f , g , h be Subset of TOP-REAL 2 , a , b , c , d be Real st a in \HM { c , d , e , f , g , h being Point of TOP-REAL 2 holds f . ( c , d , g ) = f . ( c , d , g )
a * b / 2 + ( a * c ) / 2 + ( b * c ) / 2 + ( b * c ) / 2 + ( a * c ) / 2 + ( b * c ) / 2 + ( b * c ) / 2 >= 6 * a * b * c ;
v / ( x1 , m1 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x3 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) = v / ( x2 , m1 ) / ( x2 , m1 ) ;
o ( Q ^ <* x *> , M ) = ( Rotate ( Q , M1 ) +* ( i , TRUE ) +* ( card { x } --> TRUE , M ) ) +* ( ( card { x } --> TRUE , M ) +* ( ( card { x } --> TRUE , M ) ) ) ;
Sum ( FM . n1 ) = r |^ n1 * Sum ( CM . n1 ) .= C . n1 * ( CM . n1 ) .= CM . n1 * ( CM . n1 ) .= CM . n1 * ( CM . n1 ) .= CM . n1 * ( CM . n1 ) .= CM . n1 * ( CM . n1 ) ;
( ( GoB f ) * ( len GoB f , 1 ) ) `1 = ( ( GoB f ) * ( len GoB f , 1 ) ) `1 .= ( ( GoB f ) * ( len GoB f , 1 ) ) `1 .= ( ( GoB f ) * ( len GoB f , 1 ) ) `1 .= ( ( GoB f ) * ( len GoB f , 1 ) ) `1 ;
defpred X [ Element of NAT ] means ( Partial_Sums s ) . $1 = ( a * ( $1 + 1 ) ) / ( $1 + 1 ) & ( Partial_Sums s ) . $1 = ( a * ( $1 + 1 ) ) / ( $1 + 1 ) ;
( the_arity_of g ) . g = ( the Arity of S ) . ( g . g ) .= ( ( the Arity of S ) . g ) . g .= ( ( the Arity of S ) . g ) . g .= ( ( the Arity of S ) . g ) . g .= ( ( the Arity of S ) . g ) . g ;
( X ~ ) c= X * Z & card Y c= card Z implies card ( ( X ~ ) \/ ( Y ~ ) ) = card X * Z
for a , b being Element of S , s being Element of NAT st s = n & a = F . n & b = F . n & a = N . ( n + 1 ) holds b = N . ( n + 1 ) \ G . ( n + 1 )
E , f |= All ( All ( x , All ( x , p ) ) , ( ( x. ( x , p ) ) '&' ( x. ( x , p ) ) ) '&' ( ( x. ( x , p ) ) '&' ( x. ( x , p ) ) ) ;
ex R2 being 1-sorted st R2 = ( p | ( n + 1 ) ) . i & ( the carrier of p ) = the carrier of R1 & ( the carrier of p ) = the carrier of R2 & ( the carrier of p ) = the carrier of R2 & ( the carrier of p ) = the carrier of R2 ) & ( the carrier of p ) = the carrier of R2 ) ;
[. a , b + ( 1 / k ) .[ is Element of the \circ of f & ( the partial of f ) . k is Element of the carrier of f & ( the partial of f ) . k is Element of the carrier of f & ( the partial of f ) . k is Element of the carrier of f & ( the distance of f ) . k is Element of the carrier of f ) ;
Comput ( P , s , 2 + 1 ) = Exec ( P . 2 , Comput ( P , s , 2 ) ) .= Exec ( a3 , s ) . 2 .= s . 2 .= s . 3 ;
card ( h1 ) . k = power ( F_Complex , n ) . ( ( - 1_ F_Complex ) * ( - 1_ F_Complex ) ) .= ( ( - 1_ F_Complex ) * ( - 1_ F_Complex ) ) * Sum u .= ( ( - 1_ F_Complex ) * ( - 1_ F_Complex ) ) * u .= ( ( - 1_ F_Complex ) * u ) * v .= ( ( - 1_ F_Complex ) * u ) ;
( 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 ) .= ( f /. c ) * ( g /. c ) ;
len Cf - len ( ( f | ( len ( f | ( len ( f | ( len f -' 1 ) ) ) ) ) ) = len ( ( f | ( len ( f | ( len f -' 1 ) ) ) ) ) .= len ( ( f | ( len ( f | ( len f -' 1 ) ) ) ) ) ;
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom ( f | X ) /\ X .= dom ( f | X ) /\ X .= dom ( f | X ) .= dom ( f | X ) .= dom ( f | X ) .= dom ( f | X ) .= dom ( f | X ) .= dom ( f | X ) .= dom ( f | X ) .= dom ( f | X ) .= dom ( f | X ) ;
defpred P [ Nat ] means for n holds 2 * Fib ( n + $1 ) = Fib ( n ) * Fib ( n ) + ( 5 * Fib ( n + $1 ) ) * Fib ( n + $1 ) + ( 5 * Fib ( n + $1 ) ) * Fib ( n + $1 ) ;
consider f being Function of INT , INT such that f = f and f is onto and f is onto and for n st n < k holds f " { f . n } = { n + 1 } and f " { f . n } = { n } ;
consider c9 being Function of S , BOOLEAN such that c9 = chi ( S , B ) and E7 = Prob and E7 = Prob and E7 = Prob and E7 = Prob and E7 = Prob and E7 = Prob and E7 = Prob and E7 = Prob and E8 = Prob and E7 = Prob and E7 = Prob and E7 = the Sorts of A ;
consider y being Element of Y ( ) such that a = "\/" ( { F ( x , y ) where x is Element of X ( ) : P [ x ] } , R ( y , z ) ) and Q [ y , z ] ;
assume that A c= Z and Z c= dom f and f = ( ( ( 1 / 2 ) (#) ( f1 - #Z 2 ) ) `| Z ) and for x st x in Z holds f . x = 1 / ( ( x + 0 ) * ( f1 - #Z 2 ) ) and f | Z is continuous ;
( f /. i ) `2 = ( ( GoB f ) * ( 1 , j2 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 + 1 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 + 1 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 + 1 ) ) `2 ;
dom Shift ( q2 , len Seq q1 ) = { j + len Seq q1 where j is Nat : j in dom Seq q1 & len Seq q1 = len Seq q2 } & len Seq q1 = len Seq q2 & len Seq q2 = len Seq q1 & len Seq q2 = len Seq q2 & len Seq q2 = len Seq q2 & len Seq q2 = len Seq q2 & len Seq q2 = len Seq q2 & len Seq q2 = len Seq q2 & len Seq q2 = len Seq q2 & len Seq q2 = len Seq q2 & len Seq q2 = len Seq q2 = len Seq q2 & len Seq q2 = len Seq q2 & len Seq q2 = len Seq q2 & len Seq q2 = len Seq q2 & len Seq q2 = len Seq q2 & len Seq q2 = len Seq q2 = len Seq q2 = len
consider G1 , G2 , G3 being Element of V such that G1 <= G2 and G2 <= G1 and f is Morphism of G1 , G2 and g is Morphism of G2 , G2 and for f being Morphism of G1 , G2 st f in G1 & g = f holds f * g = g * f ;
func - f -> PartFunc of C , V means : Def4 : dom it = dom f & for c st c in dom it holds it /. c = - f /. c & for c be element st c in dom it holds it /. c = - f /. c ;
consider phi such that phi is increasing and for a st phi . a = a & {} <> a & for v st v in a & v <> a holds union L , v |= H iff for v1 st v1 <> v & v1 , v |= H holds L , v |= H ;
consider i1 , j1 such that [ i1 , j1 ] in Indices GoB f and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) ;
consider i , n such that n <> 0 and sqrt p = ( i - 1 ) * ( i - 1 ) and for n1 , n2 being Integer st n1 <> 0 & n2 <> 0 & n1 < n & n2 <> 0 & n2 < n & n < n holds |. p . ( n1 + 1 ) - ( n2 - 1 ) * ( i - 1 ) .| < p . ( n1 + n2 ) ;
assume that not 0 in Z and Z c= dom ( ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) * ( f1 + f2 ) ) ) `| Z ) and for x st x in Z holds ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( f1 + f2 ) ) `| Z ) . x = - 1 / 2 and for x st x in Z holds ( ( 1 / 2 ) (#) ( f1 + f2 ) ) `| Z ) . x = 1 / 2 and ( ( 1 / 2 ) . x = 1 / 2 ) and ( ( 1 / 2 ) * ( f1 + f2 ) . x = 1 / 2 ) . x = 1 / 2 and ( ( 1 / 2 ) . x = 1 / 2 ) .
cell ( G1 , i1 -' 1 , j1 -' 1 ) \ ( ( Y -' 1 ) * ( i1 -' 1 ) + ( Y -' 1 ) * ( i1 -' 1 ) ) c= BDD L~ f & ( ( Y -' 1 ) * ( i1 -' 1 ) ) \ ( ( Y -' 1 ) * ( i1 -' 1 ) ) c= BDD L~ f ;
ex Q1 being open Subset of X st s = Q1 & ex Q1 being Subset of Y st s c= Q1 & Q1 is open & ( for F being Subset-Family of Y st F c= F & F is open & F is open & A c= F & Q1 c= F & Q1 c= F & Q1 c= F & Q1 c= F & Q1 c= F & Q1 c= F & Q1 c= F & Q1 c= F & Q1 c= F & Q1 c= F & Q1 c= F & Q1 c= F & Q1 is open & Q1 c= F & Q1 c= F & Q1 c= F & Q1 c= F & Q1 c= F & Q1 c= F & Q1 c= F & Q1 c= F & Q1 c= F & Q1 c= F & Q1 c= F & Q1 c= F & Q1 c= F & Q1 c= F & Q1 c= F & Q1 c= F & Q1 c= F &
gcd ( A1 , A2 , s1 , s2 , Amp ) = 1 & gcd ( A1 , A2 , s1 , s2 , Amp ) = 1 & gcd ( A1 , A2 , s1 , s2 , Amp ) = 1 & gcd ( A2 , s2 , s1 , Amp ) = 1 & gcd ( A1 , A2 , s1 , s2 , Amp ) = 1 & gcd ( A2 , A1 , s1 , r2 , s2 ) = 1 ;
R8 = ( ( the \ s2 ) . ( m1 + 1 ) ) . ( m2 + 1 ) .= ( ( the the Sorts of s2 ) . ( m2 + 1 ) ) . ( m2 + 1 ) .= ( ( the Sorts of s2 ) . ( m2 + 1 ) ) . ( m2 + 1 ) .= ( 3 + 1 ) * ( ( the Sorts of s2 ) . ( m2 + 1 ) ) .= ( 3 * ( ( 1 + 1 ) ) * ( 1 + 1 ) ) .= ( 3 * ( 1 + 1 ) ) * ( 1 + 1 ) * ( 1 + 1 ) * ( 1 + 1 ) * ( 1 + 1 ) * ( 1 + 1 ) * ( 1 + 1 ) * ( 1 + 1 ) * ( 1 + 1 ) * ( 1 + 1 ) * ( 1
CurInstr ( P-6 , Comput ( P3 , s3 , m1 + 1 ) ) = CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= halt SCMPDS .= ( halt SCMPDS ) ;
P1 /\ P2 = ( { p1 } \/ LSeg ( p1 , p2 ) ) \/ ( LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ) .= { p1 , p2 } \/ ( LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ) .= { p1 , p2 } \/ { p2 , p1 } ;
func -> Subset of the Sorts of A means : Def4 : a in it iff ex i st i in dom f & ex p st p in it & a = f . i & p in it & a c= p & p c= it ;
for a , b being Element of F_Complex , f being FinSequence of COMPLEX st |. a .| > |. b .| & f is non zero & a = 1 holds f * ( - ( b * f ) ) is non ] implies f * ( - ( b * f ) ) is non ]
defpred P [ Nat ] means 1 <= $1 & $1 <= len g implies for i , j st [ i , j ] in Indices G & G * ( i , j ) = g . ( i , j ) & G * ( i , j ) = g . ( $1 , i ) & G * ( i , j ) = g . ( i , j ) ;
assume that C1 , C2 , f is_collinear and for s1 , s2 being State of C1 , s2 being State of C2 , s1 being State of C1 , s2 being State of C2 st s1 = f & s2 = f holds s1 is stable and s2 is stable and s1 = s2 and s1 = s2 and s2 = s2 ;
( ||. f .|| | X ) . c = ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| ;
|. q .| ^2 = ( ( q `1 ) ^2 + ( q `2 ) ^2 ) & 0 + ( ( q `2 ) ^2 + ( q `2 ) ^2 ) < ( ( q `1 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 ;
for F being Subset-Family of T7 st F is open & {} in F & not {} in F & for A , B being Subset of T7 st A in F & B in F & A <> B & A c= B holds card F c= i & card ( F \/ B ) c= i
assume that len F >= 1 and len F = k + 1 and len F = k and for i st i in dom F & i in dom F holds H . i = g . i and for k st k in dom F & k <> i & k <> i & k <> i holds H . k = g . ( k + 1 ) ;
i |^ ( Let ( Let n + 1 ) - i ) = i |^ ( s + k - i ) .= i |^ ( s + ( k - i ) ) - i .= i |^ ( s + ( k - i ) ) - i .= i |^ ( s + ( k - i ) ) ;
consider q being oriented oriented oriented Chain of G such that r = q and q <> {} and ( for q being oriented Chain of G st q <> {} & q is oriented & q is oriented holds ( for p being Point of G st p in rng p & p <> q holds p . q = v ) & ( for q being oriented of G st q in rng p holds q . q = v ) & ( for q being oriented of G st q in rng p holds q . q = v ) & ( for p being oriented of G holds q . p = v ) implies p . q = v ) & ( for q being oriented oriented L holds q . q = v ) implies p . q = v implies p . q = v ) & ( for q being oriented oriented of G st q is oriented L implies
defpred P [ Element of NAT ] means $1 <= len ( I ) implies ( g . $1 ) . $1 = ( ( I , Z ) ^ <* ( g . $1 ) . ( len g ) *> ) . $1 + ( g . $1 ) . ( len g ) ;
for A , B being Matrix of n , REAL holds len ( A * B ) = len A & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B
consider s being FinSequence of the carrier of R such that Sum s = u and for i being Element of NAT st 1 <= i & i <= len s ex a , b being Element of R st a = u * b & a in I & b in J & a * s in J & b * s in J ;
func |( x , y )| -> Element of COMPLEX equals |( ( Re x ) , ( Im y ) )| , ( Im y ) | REAL , ( Im y ) | REAL , ( Im y ) | [. x , y .] , ( Im y ) | [. x , y .] ;
consider g2 being FinSequence of ( F . i ) * such that g2 is continuous and rng g2 c= A and for k st k in dom g2 & k <= len g2 holds g2 . k = x0 and g2 . k = x0 and g2 . k = x0 and g2 is one-to-one and for k st k in dom g2 holds g2 . k = x0 ;
then n1 >= len p1 & n2 >= len p1 implies crossover ( p1 , p2 , n1 , n2 , n3 , n3 , n1 , n2 , n3 , n3 , n3 , n1 , n2 , n3 , n3 , n3 , n1 , n2 , n3 , n3 , n3 , n1 , n2 , n3 , n3 , n3 , n3 , n1 , n2 , n3 , n3 , n2 , n3 , n3 , n1 , n2 , n3 , n3 , n3 , n1 , n2 , n3 , n2 , n3 , n3 , n3 , n2 , n3 , n2 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , 9 , n3 , n3 , n3 , 9 , n3 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n4 , n2 , n3 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n2 , n3 ,
( q `1 ) * a <= ( q `1 ) * a & - ( q `1 ) * a <= ( q `1 ) * a & - ( q `1 ) * a <= ( q `1 ) * a & - ( q `1 ) * a <= ( q `1 ) * a & - ( q `1 ) * a <= ( q `1 ) * a ;
Fv . ( pv . ( len pv ) ) = ( Fv . ( len ( pv . ( len ( pv . ( len ( pv . ( len ( pv . i ) ) ) ) ) ) ) ) .= ( ( ( F . ( len ( ( F . i ) ) ) ) ) . ( ( len ( ( F . i ) ) ) .= ( ( v . i ) ) . ( ( v . i ) ) .= ( ( v . i ) ) . ( ( ( v . i ) ) * ( ( v . i ) * ( ( v . i ) * ( ( v . i ) * ( ( v . i ) ) * ( ( v . i ) * ( ( v . i ) * ( ( v . i ) ) * ( ( ( ( v . i ) ) ) ) .= ( ( ( ( v
consider k1 being Nat such that k1 + k = 1 and a = ( <* a *> ^ ( k --> a ) ) ^ ( ( k + 1 ) --> a ) and ( k + 1 ) <> 0 implies ( ( k + 1 ) --> a ) ^ ( ( k + 1 ) --> a ) = ( k + 1 ) --> a ;
consider B8 being Subset of B1 , y8 being Subset of B1 such that B8 is finite and y1 = the carrier of ( B1 | A1 ) and D1 = the carrier of ( B1 | A1 ) and D2 = the carrier of ( B1 | A1 ) and for x being set st x in B1 holds ( x in B1 & x in B1 implies x = y ) ;
v2 . b2 = ( ( curry F2 ) * ( ( curry F2 ) . b2 ) ) * ( ( ( curry F2 ) . b2 ) ) .= ( ( ( ( curry F2 ) . b1 ) * ( ( ( ( curry F2 ) . b2 ) ) ) ) * ( ( ( ( ( curry F2 ) . b1 ) ) * ( ( ( ( ( curry F2 ) . b1 ) ) ) ) .= ( ( ( ( ( ( curry F2 ) . b1 ) ) ) ) ) ) * ( ( ( ( ( ( id F2 ) . b1 ) ) ) ) ) .= ( ( ( ( ( ( ( the .|| of B ) . b1 ) ) ) * ( ( ( ( ( ( ( ( the Arrows of B ) . b1 ) ) ) ) ) * ( ( ( B ) . b1 ) ) ) .= ( ( ( B ) . b1 ) ) * ( ( ( (
dom IExec ( I-35 , P , Initialize s ) = the carrier of SCMPDS .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ;
ex d-32 be Real st d\llangle h , d] > 0 & for h be Real st h > 0 & |. h .| < d[ h , R] holds |. ( ( R * L ) . h ) . ( ( R * L ) . h ) - ( R * L ) . h .| < e
LSeg ( G * ( len G , 1 ) + |[ - 1 , 1 ]| , G * ( len G , 1 ) + |[ - 1 , 1 ]| ) c= Int cell ( G , len G , width G ) \/ { |[ - 1 , 0 ]| } ;
LSeg ( mid ( h , i1 , i2 ) , i , j ) = LSeg ( h /. ( i + i1 -' 1 ) , h /. ( i + i1 -' 1 ) ) .= LSeg ( h , i -' i1 -' 1 ) .= LSeg ( h , i -' 1 ) .= LSeg ( h , i -' 1 ) ;
A = { q where q is Point of TOP-REAL 2 : LE p1 , p2 , P , p1 , p2 & LE p1 , p2 , P , p1 , p2 & LE p2 , p1 , P , p1 , p2 & LE p1 , p2 , P , p1 , p2 & LE p2 , p1 , P , p1 , p2 & LE p1 , p2 , P , p1 , p2 & LE p2 , p1 , P , p1 , p2 & LE p2 , p1 , P , p1 , p2 & LE p1 , p2 , P , p1 , p2 & LE p1 , p2 , P , p1 , p2 & LE p1 , p1 , P , p1 , p2 & LE p1 , p2 , P , p1 , p2 & LE p1 , p2 , P , p1 , p2 & LE p1 , p2 , P , p1 , p2 , P , p1 , p2 , p1 , p2 , p1 , p2 , P , p1 , p2 , p2 , P , p1 , p2 , p2 , p1
( ( - x ) .|. y ) = - ( ( 1 - x ) .|. y ) .= ( - ( 1 - x ) .|. y ) * ( ( - x ) .|. y ) .= ( - ( 1 - x ) .|. y ) * ( ( - x ) .|. y ) .= ( - ( 1 - x ) .|. y ) * ( - ( x ) .|. y ) .= ( - ( 1 - x ) .|. y ) * ( - x ) ;
0 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) = ( ( p `1 / p `2 ) ^2 + ( p `2 / p `2 ) ^2 ) .= ( p `1 ) ^2 * sqrt ( 1 + ( p `2 / p `1 ) ^2 ) .= ( p `1 ) ^2 * sqrt ( 1 + ( p `2 / p `1 ) ^2 ) ;
( ( U - 7 ) * ( W - p ) ) * ( W - p ) = ( ( U - n ) * ( W - p ) ) * ( W - p ) .= ( ( U - n ) * ( W - p ) ) * ( W - p ) .= ( ( U - n ) * ( W - p ) ) * ( W - p ) .= ( ( U - n ) * ( W - p ) ) * ( W - p ) .= ( ( U - p ) * ( W - p ) ;
func Shift ( f , h ) -> PartFunc of REAL , REAL means : Def4 : dom it = dom f & for x be Element of REAL holds it . x = - h . x & for x be Element of REAL 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 G and [ i + 1 , j ] in Indices G and f /. k = G * ( i , j ) and f /. k = G * ( i + 1 , j ) and f /. k = G * ( i + 1 , j ) and f /. k = 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 y in Free H and not x in Free H and not y in Free H and not y in Free H and not x in Free H and y in Free H and not y in Free H and not x in Free H and y in Free H and y in Free H and y in Free H and not y in Free H and not y in Free H and not y in Free H and not y in Free H and not y in Free H and not y in Free H and not y in Free H and not y 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 x in Free H and not x in Free H and not x in Free H and not x in Free H and not x in Free H and not x in Free H and not x in Free H and not
defpred P11 [ Element of NAT , Element of NAT , Element of NAT ] means ( for k being Element of NAT st k < $1 & k <= $1 holds $2 |^ k = ( p |^ $1 ) |^ ( k -' 1 ) ) & ( ( p |^ $1 ) |^ ( k -' 1 ) ) = ( p |^ k ) |^ ( k -' 1 ) ;
func \sigma ( C ) -> non empty Subset-Family of X means : Def4 : for A being non empty Subset of X holds A in it iff for W being non empty Subset of X st W c= A & W is open & for A being Subset of X holds A c= W implies C c= W ;
[#] ( ( dist ( P ) ) .: Q ) = ( ( dist ( P ) ) .: Q ) & lower_bound ( ( dist ( P ) ) .: Q ) = lower_bound ( ( dist ( P ) ) .: Q ) & lower_bound ( ( dist ( P ) ) .: Q ) = lower_bound ( ( dist ( P ) ) .: Q ) ;
rng ( F | [: S , T :] ) = {} or rng ( F | [: S , T :] ) = { 1 } or rng ( F | [: S , T :] ) = { 1 } or rng ( F | [: S , T :] ) = { 2 } ;
( f " ( rng f ) ) . i = f . i " . i .= ( f " ) . i .= ( f " ) . i .= ( f " ) . i .= ( f " ) . i .= ( f " ) . i .= ( f " ) . i .= ( f " ) . i .= ( f " ) . i .= ( f " ) . i .= ( f " ) . i ;
consider P1 , P2 being non empty Subset of TOP-REAL 2 such that P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p2 , p1 and C = { p1 , p2 } and C = { p1 , p2 } and P = { p1 , p2 } and C = { p2 , p1 } and P = { p2 , p1 , p2 } and C = { p1 , p2 , p3 } and C = { p1 , p2 , p3 } and C = { p1 , p2 , p3 } and C = { p2 , p3 , p4 } and C = { p1 , p2 , p3 } and C = { p1 , p2 , p3 , p4 } and C = { p2 , p3 , p4 , p4 , p4 , p4 , p4 , p4 , p4 } and C = { p1 , p2 , p4 } and C = { p1 , p2 , p4 , p4 , p4 , p4 , p4 , p4 and C = { p1 , p1 , p2 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 ,
f . p2 = |[ ( p2 `1 ) ^2 + ( p2 `2 ) ^2 * ( p2 `1 ) ^2 + ( p2 `2 ) ^2 * ( p2 `1 ) ^2 + ( p2 `2 ) ^2 * ( p2 `1 ) ^2 + ( p2 `2 ) ^2 * ( p2 `1 ) ^2 ;
( ( \mathbin { \rm to } ( a , X ) ) " ) . x = ( ( \mathop { \rm \mathbin { - } a } ) . x ) " .= ( ( \mathop { \rm \mathbin { - } a } ) . x ) " .= ( ( ( \mathop { \rm AffineMap ( a , X ) ) . x ) ) * ( ( \mathop { \rm AffineMap ( a , X ) ) . x ) .= ( ( \mathop { \rm AffineMap ( a , X ) ) . x ) " .= ( ( ( 0 , X ) ) . x ) " .= ( ( 0 , X ) . x ) " .= ( ( 0 , X ) ) . x ) " .= ( ( 0 , X ) . x ) " .= ( ( 0 , X ) . x ) " .= ( ( 0 , X ) . x ) * ( ( ( 1 , X ) * ( ( ( the carrier of X ) . x ) * ( ( the carrier of X ) . x ) * ( ( ( 0 , X ) . x ) * ( ( ( ( 1 , X ) . x ) " .= ( ( 0 , X ) .
for T being non empty normal TopSpace , A , B being closed Subset of T , A , B being closed Subset of T st A <> {} & A misses B & B misses A & A c= B & B c= A & A c= B & B c= B & A c= B & B c= B holds ( NAT + 1 ) * ( A + B ) = ( NAT + 1 ) * ( B + 1 )
for i st i in dom F for G1 , G2 being strict normal Subgroup of G st G1 = F . i & G2 = F . i & G1 is strict normal Subgroup of G & G2 is normal holds G1 * ( i , [#] G ) = G1 * ( i , [#] G ) & G1 * ( i , [#] G ) = G1 * ( i , [#] G )
for x st x in Z holds ( ( ( 1 / 2 ) (#) ( ( #Z 2 ) * ( f1 - #Z 2 ) ) `| Z ) . x = ( ( 1 / 2 ) * ( f1 - #Z 2 ) ) `| Z ) . x
synonym f is right continuous means : Def2 : x0 in dom ( f /* a ) & for x st x in dom f & x in dom f & x0 in dom f & for a st a in dom f & a in dom f & f . a = ( f /* a ) . x holds f . ( a - x0 ) < r ;
then X1 , X2 are_separated & X1 misses X2 & X2 misses X1 & X1 , X2 are_separated & X2 , X1 are_separated & X1 , X2 are_separated & X2 , X1 are_separated & X1 , X2 are_separated & X2 , X2 are_separated & X1 , X2 are_separated & X2 , X2 are_separated & X1 , X2 are_separated & X1 , X2 are_separated & X2 , X2 are_separated & X1 , X2 are_separated & X1 , X2 are_separated implies X1 , X2 are_separated & X2 , X1 are_separated & X2 , X1 are_separated & X1 , X2 are_separated & X2 , X2 are_separated & X1 , X2 are_separated & X1 , X2 are_separated & X1 , X2 are_separated & X1 , X2 are_separated & X2 , X2 are_separated & X1 , X2 are_separated & X1 , X2 , X2 , & X1 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X1 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 , X2 ,
ex N be Neighbourhood of x0 st N c= dom SVF1 ( 1 , f , u ) & ex L , R st for x st x in N holds ( SVF1 ( 1 , f , u ) ) . x - ( SVF1 ( 1 , f , u ) ) . x0 = L . ( x - x0 ) + R . ( x - x0 )
( ( p2 `1 ) ^2 ) * sqrt ( 1 + ( p2 `1 ) ^2 ) + ( ( p2 `2 ) ^2 * ( ( p2 `1 ) ^2 + ( p2 `2 ) ^2 ) ) >= ( ( p2 `1 ) ^2 + ( p2 `2 ) ^2 ) ;
( ( 1 - t ) * ||. f1 .|| ) / ( n + 1 ) = ( ( 1 - t ) * ( ( 1 - t ) / ( n + 1 ) ) ) / ( n + 1 ) & ( ( 1 - t ) * ( ( 1 - t ) / ( n + 1 ) ) ) / ( n + 1 ) = ( ( 1 - t ) / ( n + 1 ) ) / ( n + 1 ) ;
assume that for x holds f . x = ( ( - 1 / 2 ) (#) ( sin - cos ) ) `| Z and for x st x in Z holds ( ( - 1 / 2 ) (#) ( sin - cos ) ) `| Z = ( ( - 1 / 2 ) (#) ( sin - cos ) ) `| Z and for x st x in Z holds ( ( ( - 1 / 2 ) (#) ( sin - cos ) ) `| Z ) . x = 1 / 2 and f . x = 1 / 2 and f . x = 1 / 2 and f . x = 1 / 2 and f . x = 1 / 2 and f . x = 1 / 2 and f . x = 1 and f . x = 1 / 2 and f . x = 1 and f . x = 1 and f . x = 1 and f . x = 1 and f . x = 1 and f . x = 1 and f . x = 1 and f . x = 1 and f . x = 1 and f . x = 1 and f . x = 1 and f . x = 1 and f . x = 1 and f .
consider X1 being Subset of Y , Y1 being Subset of X such that t = X1 and Y1 in A and Y1 is open and ex Y1 being Subset of X st Y1 = Y1 & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 c= Y1 & Y1 c= Y1 ;
card S . n = card { [: d , Y :] + b * a , d * b , c * a , d * b , c * b , d * a , b * c , d * b , c * a , b * c , d * b , c * a , d * b , c * b , d * a , b * c , d * b , c * a , b * c , d * b , d * b , c * a , b * b , d * b , c * a , d * b , c * b , d * b , d * b , d * b , d * b , c * b , c * b , d * b , c * a , d * b , c * a , d * b , c * a , c * a , c * a , c * a , c * a , c * a , c * c , d * a , b * c , c * a , b * c , d * a , c * a , c * a , c * a , c * a , c * a ,
( ( W-bound D - W-bound D ) / 2 ) * ( ( W-bound D - W-bound D ) / 2 ) = ( ( W-bound D - W-bound D ) / 2 ) * ( ( W-bound D - W-bound D ) / 2 ) .= ( ( W-bound D - W-bound D ) / 2 ) * ( ( W-bound D - W-bound D ) / 2 ) .= ( ( W-bound D - W-bound D ) / 2 ) * ( ( W-bound D - W-bound D ) / 2 ) ;